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\section{ Introduction}
Tunneling between topologically different configurations of the gauge
field, described semiclassically by instantons \cite{BPST}, dominate
the physics of light quarks. In early works (summarized e.g. in
\cite{CDG}) instantons were treated as a dilute gas, while later it was
recognised that the instanton
ensemble resemble rather a strongly interacting ``instanton liquid"
\cite{Shuryak_82,DP}.
During the last years calculations of
the correlation functions \cite{SV} and Bethe-Salpeter wave functions \cite{SS}
for various mesonic and baryonic channels were made
along these lines. The results agree surprisingly well both with
phenomenology
\cite{Shuryak_cor}
and lattice simulations \cite{Negele}. Parameters of the ''instanton liquid"
were also reproduced
(by the ''cooling" method) directly from the (quenched) lattice configurations
\cite{Negele_DALLAS}.
In addition to that, it was found that correlation functions
as well as hadronic wave functions in most channels remain
practically unchanged after ''cooling". In particular, main mesons and e.g.
the nucleon remains bound, with about the same mass and wave function.
This confirms that the
agreement of previous lattice calculations with the instanton model
was not accidental, and instantons indeed are
the most important non-perturbative phenomena in QCD.
Investigations of the finite temperature case were started in \cite{HS},
where classical {\it caloron}
solution was found. Although the solution depends on T,
the action is T-independent. Furthermore, it was argued
by one of us \cite{Shuryak_conf}
that at high T the specific charge renormalization and the
Debye-type screening of the electric field in quark-gluon
plasma should suppress instantons with size $\rho > 1/T$. Pisarski and Yaffe
\cite{PY} have evaluated the T-dependence in the one-loop approximation.
The physical nature of their result and its applicability region will be
discussed below.
Last years the studies of the finite temperature
had focused especially on the region around the chiral
phase transition $T\approx T_c$. The first attempt to understand this
phase transition as a rearrangement of the instanton liquid, going from
a random phase at low temperatures to a strongly correlated ``molecular"
phase at high temperatures was made in \cite{IS}. Recently
this idea was recently made more quantitative in \cite{IS2,ssv},
and although the detailed comparison to lattice thermodynamics and correlation
functions is yet to be made, the first results show overall agreement,
indicating that the mechanism of chiral restoration is basically
understood.
The particular topic of this paper, the {\it temperature
dependence of the instanton
density} n(T), is certainly an important ingredient of all this development.
However, it has attracted surprisingly little attention in literature.
The only attempts to determine this quantity by ``cooling" of the lattice
configurations (same method as was used at T=0) were made in
refs.\cite{lattice}. The only statement one can probably
make using these data is that the density has
{\it no} significant T-dependence, till $T\sim T_c$.
Unfortunately,
accuracy of this statement remains at the level of 50\%, at best.
\section{ Instantons at low temperatures}
As far as we know, the modification of the instanton density in this limit
was never considered before.
However, the general physical picture at low $T<<T_c$ is well
known: the heat bath is just a dilute gas of the lightest hadrons,
the pions. The problem is especially clear in the chiral limit, in which
quark and pion masses are neglected, and this case is assumed in what follows.
Due to their Goldstone nature, the large wavelength pions are nothing else
but small
collective distortions of the quark condensate. Therefore, one can always
translate the average values over the {\it
pion state}
as the {\it vacuum expectation value} (VEV) of a different but related
quantity.
Let us now consider a tunneling event, described semiclassically
by the instanton solution.
As discovered by 't Hooft \cite{Hooft}, it can only take place if certain
rearrangements in the fermionic sector are made, which can be described by
some effective Lagrangian with $2 N_f$ fermionic legs. For
simplicity, in this work we assume that the number
of light flavors $N_f=2$, disregarding strange and heavier quarks.
The situation can be further simplified by consideration of
{\it small-size} instantons $\rho \Lambda_{QCD}<<1$, for which
this Lagrangian
can be considered as a {\it local} operator.
What follows from 't Hooft Lagrangian, after averaging
over the instanton orientations is made, is \cite{SVZ_79}:
\begin{eqnarray}
\Delta {\cal L}= \int d\rho d_0(\rho)({4 \over 3}\pi^2 \rho^3)^2\{\bar q
\Gamma_+ q
\bar q \Gamma_- q+{3\over 32}\bar q \Gamma_+^a q \bar q \Gamma_-^a q-
{9\over 128}\bar q
\Gamma_+^{a\mu \nu} q \bar q \Gamma_-^{a\mu \nu} q\}
\end{eqnarray}
where the definition of the operators involved is as follows
\begin{eqnarray}
\Gamma_{\pm}=({1-\gamma_5 \over 2})\otimes ({1\pm\tau_3 \over 2})
\end{eqnarray}
\begin{eqnarray}
\Gamma_{\pm}^a=({1-\gamma_5 \over 2})\otimes ({1\pm\tau_3 \over 2})\otimes t^a
\end{eqnarray}
\begin{eqnarray}
\Gamma_{\pm}^{a\mu \nu}=({1-\gamma_5 \over 2})\sigma_{\mu \nu}\otimes
({1\pm\tau_3 \over 2})\otimes t^a
\end{eqnarray}
At T=0 the instanton density is therefore proportional to the VEV of
$\Delta {\cal L}$, and the only thing which changes at low T is clearly the
modification of the quantities above.
How to do this
technically was actually clarified by PCAC-related paper in 60's.
The necessary formulae can be found
e.g. in the recent paper by Eletsky \cite{Eletsky}, where
different set of four-fermion
operators (appearing in QCD sum rules for vector and axial currents)
was considered.
The general expression is
$$
\langle (\bar q A q)(\bar q B q)\rangle _T=
\langle (\bar q A q)(\bar q B q)\rangle _0
-{ T^2 \over 96 F^2_\pi}\langle (\bar q\{\Gamma^a_5 \{\Gamma_5^a A\}\} q)
(\bar q B q)\rangle _0
$$
\begin{eqnarray}
-{ T^2 \over 96 F^2_\pi}\langle (\bar q A q)(\bar q\{\Gamma^a_5
\{\Gamma^a_5 B\}\} q)\rangle _0
-{ T^2 \over 48 F^2_\pi}\langle (\bar q \{\Gamma^a_5 A\} q)(\bar q
\{\Gamma^a_5 B\} q)\rangle _0
\end{eqnarray}
where A,B are arbitrary flavor-spin-color matrices and $\Gamma_5^a=\tau^a
\gamma_5$.
Here and below flavor matrices are shown as $\tau_a$, and color one
as $t^a$.
For generality, there are six different operators
of different spin-flavor-color structure
involved (see
the Table). Their T-dependence in $O(T^2)$ order can be found from the
expression above, and it is also listed in the Table.
Generally, the operators mix, and it is convenient to group those combinations
which do not. Returning to the effective Lagrangian, one can see
that there are only two combinations which
are actually relevant
\begin{eqnarray}
K_1={\cal O}^A_1+{3\over 32}{\cal O}^B_1-{9\over 128}{\cal O}^C_1 \\
K_2={\cal O}^A_2+{3\over 32}{\cal O}^B_2-{9\over 128}{\cal O}^C_2
\end{eqnarray}
The final result for the instanton density
at low temperature T therefore contains two constants, the
{\it vacuum} averages of these operators
\begin{eqnarray}
dn(T) = {d\rho \over \rho^5} d(\rho) ({4\over 3}\pi^2 \rho^3 )^2
\left[\langle K_1\rangle _0{1\over 4}(1-{T^2 \over 6 F^2_\pi})
-\langle K_2\rangle _0{1\over 12}(1+{T^2 \over 6 F^2_\pi})\right]
\end{eqnarray}
Although two VEV's which appear here are unknown (and are subject to
further investigations), it is clear that
the total T-dependence should be {\it rather weak}: it is bound to be
$ 1 + a {T^2 \over F^2_\pi}$ with a in the
strip $a=(- 1/6,1/6)$.
The so called {\it vacuum dominance} (VD)
hypothesis \cite{SVZ_79} was used
in various applications (such as QCD sum rules and weak decays) for
evaluation of VEV's of various operators. It leads to VEV's and the
$O(T^2)$ corrections also indicated in the Table 1. Remarkably enough
\footnote{
(However, the reader should be warned that in general this approximation is
not supposed to hold or
be very accurate, particularly for the operators considered, which are related
with instantons.
},
in this case the T-dependence {\it exactly cancel}.
Returning to discussion of our general result (8), we comment that
this result by itself rules out some possible picture of low-T vacuum
structure. In particular, the so called ``random instanton liquid model"
(RILM) was shown to be a reasonable approximation for the T=0 case \cite{SV}.
One may wander if the same model can describe the T-dependence at low T.
If it is the case, the quark condensate
should scale with the instanton density as $n_{inst}^{1/2}$, see
\cite{Shuryak_82}. Being combined with the well known chiral
theory result
\begin{eqnarray}
\langle \bar q q\rangle _T=\langle \bar q q\rangle _0 (1 - {T^2
\over 8 F^2_\pi})
\end{eqnarray}
these two formulae lead to\footnote{
Note that the same result can be obtained by a naive assumption, which was used
in some works on QCD sum rules
in the past: namely, that average of {\it all} four-fermion operators
have T-dependence as the {\it square} of the condensates.}
$ n_{inst}(T)/n_{inst}(0)=(1 - {T^2 \over 4 F^2_\pi})$.
However, as this estimate happen to be {\it outside}
of the
strip indicated above, this possibility is definitely {\it
ruled out}. It means that, even if RILM is a
perfect model at T=0, it cannot be so for even small T. This conclusion
agrees very well with other studies of the instanton ensemble, such as
\cite{IS2}, which emphasize the role of correlations built up with growing T
in the ensemble of instantons.
\section{ Instantons at high temperatures}
QCD vacuum at high temperatures undergoes a phase transition into a new
phase, called the {\it quark-gluon plasma}\cite{Shuryak_QGP}. Although
{\it virtual} gluons {\it antiscreen}
the external charge (the asymptotic freedom), the {\it real} gluons of the
perturbative heat bath {\it screen} it, leading to the well known
expression for the Debye screening mass \cite{Shuryak_QGP}:
\begin{eqnarray}
M_D^2= (N_c/3+N_f/6) g^2 T^2
\end{eqnarray}
where $N_c,N_f$ are the numbers of colors and flavors, respectively.
''Normal" O(1) electric fields are therefore screened at distances $1/gT$,
while stronger ''non-perturbative" fields of the instantons $O(1/g)$
should be screened already at scale 1/T.
Quantitative behaviour of the
instanton density at high temperatures was determined in ref. \cite{PY}.
\begin{eqnarray}
dn(\rho,T)= dn(\rho,T=0)\exp \left\{ -{1\over 3}\lambda^2(2N_c+N_f)-12
A(\lambda)\left[1+{1\over 6}(N_c-N_f) \right] \right\}
\end{eqnarray}
where $\lambda=\pi\rho T$, and $A(\lambda)=-{1\over 36}\lambda^2
+o(\lambda^2)$.
Therefore, at high temperatures the contribution of small size
instantons such as $T>> 1/\rho$ is
exponentially suppressed. As a result, the
instanton-induced contribution to physical
quantities like energy density (or pressure, etc) become of the order
of\footnote{Here we consider only the pure glue theory. In the theory with
massless fermions individual instantons are impossible, and only
``instanton-antiinstanton" molecules can appear at high temperatures.}
\begin{eqnarray}
\epsilon(T) \sim \int^{1/T}_0 {d\rho \over \rho^5} (\rho \Lambda)^{(11 N_c/3)}
\sim T^4 (\Lambda/T)^{(11 N_c/3)}
\end{eqnarray}
which is small compared to that of ideal gas $\epsilon(T)_{ideal}\sim T^4$.
Although the Pisarski-Yaffe formula
contains only
the dimensionless parameter $\lambda$, its applicability is limited by {\it
two separate} conditions:
\begin{eqnarray}
\rho << 1/\Lambda, \,\,\,\,\,\,\, T >> \Lambda
\end{eqnarray}
The former condition ensure semiclassical treatment of the tunneling,
while the latter
is needed to justify perturbative treatment of the heat bath.
In this section we would like to discuss applicability conditions of these
well-known results in greater details.
Our first point is that the one-loop effective action
discussed by Pisarski and Yaffe actually
consists of {\it two parts} with very different
physical origin and interpretation. To show that in the simplest case,
consider the determinant corresponding to scalar
isospin 1/2 field\footnote{The determinants of the actual quadratic
fluctuations of the {\it quark} and {\it gluon} fields (modulo the factor
corresponding to zero fermion
modes) can be expressed via the
determinants of {\it scalar} fields with isospins 1/2 and 1 \cite{PY}.}
and rewrite them as follows:
\begin{eqnarray}
\delta =Tr_T[\log({-D^2(A(\rho,T))\over -\partial^2})]-
Tr[\log({-D^2(A(\rho))\over -\partial^2})]=\delta_1+\delta_2
\end{eqnarray}
\begin{eqnarray}
\delta_1 =Tr_T[\log({-D^2(A(\rho))\over -\partial^2})]-
Tr[\log({-D^2(A(\rho))\over -\partial^2})]
\end{eqnarray}
\begin{eqnarray}
\delta_2 =Tr_T[\log({-D^2(A(\rho,T))\over -\partial^2})]-
Tr_T[\log({-D^2(A(\rho))\over -\partial^2})]
\end{eqnarray}
Here $Tr_T$ is a trace over all matrix structures, plus integration over
$\cal M$ -- the strip in $R^4$ with span in the $\tau$ direction of $1/T$.
$A(\rho,T)$ is the caloron field and $A(\rho)$ is the instanton field.
Two contributions introduced in this way,
$\delta_1,\delta_2$, are the origin of two terms in the resulting formula
(11).
As it was shown in ref.\cite{Shuryak_A4}, the first term can be expressed
via the {\it forward scattering amplitude} of heath bath constituents,
on the instanton field.
Therefore its physical origin is clear: $\rho^2$ comes from the
scattering amplitude, while the temperature factor $T^2$ enters via the
standard thermal integral over the particle momenta:
\begin{eqnarray}
\delta_1=\int{ d^3p\over(2\pi)^3}{1\over 2p(\exp(p/T)-1)} TrT(p,p)
\end{eqnarray}
Let us show how it works using the example of a ``scalar quark",
which is simpler than realistic spinor and vector particles considered in
\cite{Shuryak_A4}.
One can evaluate $T(p,p)$, the forward scattering amplitude of a scalar
quark on the instanton
field, using standard Leman-Simansik-Zimmermann reduction formula:
\begin{eqnarray}
Tr T(p,p)=\int d^4x d^4y \ e^{ip.(x-y)} Tr(\partial ^2_x \Delta_{1\over2}(x,y)
\partial^2_y)
\end{eqnarray}
where $\Delta_{1\over2}$ is the (isospin 1/2) scalar quark propagator
\cite{BCCL}:
\begin{eqnarray}
\Delta_{1\over2}(x,y)={x^2 y^2+\rho^2 x.\tau y.\tau^\dagger \over
4\pi^2 (x-y)^2 x^2 y^2 (1+\rho^2/x^2)^{1\over2}(1+\rho^2/y^2)^{1\over2}}
\end{eqnarray}
By rescaling (18) as $\xi=px, \eta=py$, subtracting the trace of the free
propagator and going to the physical pole $p^2=0$, one gets:
\begin{eqnarray}
Tr T(p,p)=\int d^4\xi d^4\eta e^{in.(x-y)} {\rho^2\over
2\pi^2(\xi-\eta)^2}\left({\xi .\eta \over
\xi^2 \eta^2}-{1\over 2\xi^2}-{1\over 2\eta^2}\right)=-4\pi^2\rho^2
\end{eqnarray}
As it is just constant, there is no problem with its analytic continuation
to small Minkowski momenta of scattered quarks, and
plugging (20) into (17) we have:
\begin{eqnarray}
\delta_1={1 \over 3}\eta\lambda^2,\ \eta= \left\{ \matrix{ 1 & for& periodic
& fields \cr -1/2 & for& antiperiodic& fields \cr}\right\}
\end{eqnarray}
Note also, that this scattering amplitude
has the same origin (and the same dependence
on $N_c,N_f$) as the Debye mass.
Although formally any result obtained
by the perturbative expansion demand smallness
of the effective charge $g(T)<<1$, it is not clear
in practice what this condition actually imply.
However, we conjecture that accuracy of calculation sketched above
is controlled by the same effects as the
accuracy of {\it perturbative calculation of the
Debye mass} by itself.
If so, one can use available lattice studies
of the screening phenomena (e.g.\cite{screening}) and check at which T
their results start to agree with the perturbative formula (10).
We then conclude, based on available lattice data, that
Debye mass and instanton suppression formula (11) should be valid
above
$T> T_{pert}=3 T_c \approx 500$ MeV).
How strong can this suppression be, at that point? Using a canonical
``instanton liquid" size of the instanton $\rho\approx1/3$ fm, one gets
suppression on the level $10^{-3}$, from the $\delta_1$ term alone. It suggests
a very dramatic behaviour in the interval from 1 to 3 $T_c$.
Let us further speculate about
the magnitude of $\delta_1$ contribution for lower
temperatures. In the interval between $T_c$ and $T_{pert}$
it is expected on general ground (and observe on the lattice) that
the Debye mass $M_D\rightarrow 0$ at the critical temperature: screening is
gone together with the plasma. However, another suppression mechanism should
substitute it {\it below} $T_c$,
namely the one due to scattering of {\it hadrons} on the instanton.
This is what we have done above for the low-T case,in which only the
soft pions should be included.
At this time,
we do not know how to estimate this effect including other hadrons.
Now we turn to discussion of the second term $\delta_2$ in (14),
$A(\lambda)$ in \cite{PY}, which was
actually first obtained by Brown and Creamer
in \cite{BC}. At small $\lambda$ it leads to the following correction
\begin{eqnarray}
\delta_2=-(1/36)\lambda^2+o(\lambda^2)
\end{eqnarray}
and thus it has the same sign as $\delta_1$ and parametric magnitude, just
numerically smaller coefficient.
(In the isospin 1 case $\delta_1=(4/3)\lambda^2$ and
$\delta_2=-(4/9)\lambda^2+o(\lambda^2).$)
The splitting of the variation of
the effective action into two physically different contributions is
a generic phenomenon.
This second term has different physical
origin, because it is connected to
a {\it quantum correction} to the colored current, times the
{\it T-dependent variation} of
the instanton field, the difference between the caloron and the instanton.
Thus,
the finite T
effects
not only lead to appearance of a usual (perturbative) heat bath, but they
also modify strong ($O(1/g)$) {\it classical gauge field} of the instanton.
In Matsubara formalism this is described by the
non-linear ``interference" of the
instanton field with its ``mirror images", in the (imaginary) time direction.
Let us
conjecture, that for $T<T_c$ this suppression mechanism is actually
irrelevant, for the following reason. It is well known that in this T domain
all gluonic correlators decay strongly with distance, because all physical
``glueballs" states are very heavy.
It should make
any interaction with the ``mirror images" (at distance $\beta=1/T$)
virtually impossible. Estimating this effect as
$\exp(-M_{glueball}/T)$, where $M_{glueball}\sim
1.6 GeV$ is the mass of the
lightest glueball, one gets even at $T=T_c\sim 140 MeV$ a suppression factor
of the order of $10^{-5}$.
\section{ Summary and discussion}
We have studied the change of the instanton density at {\it low} and {\it
high}
temperatures.
In the former case, $T<<T_c$, we have
considered the heat bath as being
made of dilute soft pions.
Applying PCAC methods (in the chiral limit) we have derived strict
result (8) for the instanton density n(T) at low T.
It implies very weak T dependence, which agrees with
available lattice measurements inside their (so far rather poor) accuracy.
It also
contradicts to some naive models, for example it shows that the
''random instanton liquid model", presumably a good description of the
QCD vacuum, can not be true even at low T.
Fortunately, it perfectly
agrees with the current ideas about the finite-T QCD
\cite{IS2}, pointing out that quark-induced instanton-
antiinstanton correlations are building up with T, till only
instanton - antiinstanton ''molecules" remain for $T>T_c$.
Our discussion of the
{\it high} temperatures can be summarized as follows.
For very high $T>T_{pert}\sim 3 T_c$ the perturbative result of Pisarski and
Yaffe \cite{PY} holds.
Furthermore,
we have pointed out that it consists of two parts, $\delta_1,\delta_2$,
with different
underlying physics. The
first one is directly connected to
occupation densities of quarks and gluons from
the
plasma. It is the same effect as lead
e.g.
to the Debye screening mass, so one knows from lattice data at which T
this part of the instanton suppression can be trusted. We
also claimed that it weakens toward $T_c$, but at the same
time scattering of hadrons on instantons should appear at $T<T_c$, and
we do not know how to take it into account (except for soft pions).
The second term $\delta_2$
originates from the {\it T-dependent
variation of the classical field}, coupled
to a quantum correction to the colored current. We
expect this effect to become exponentially small
for $T<T_c$, but we do not know its T-dependence in the strip 1-3 $T_c$, where
it can be very strong.
Finally, let us repeat once more, that
understanding of the temperature dependence of the
instanton density is of crucial importance for understanding
of non-perturbative
phenomena at and around $T_c$. Surprisingly little efforts has been made
to clarify this question. In particularly, we call upon
lattice community to make quantitative measurements of n(T), which can
be done by well known methods.
{\bf Acknowledgements}
One of us (E.S.) acknowledge helpful discussions with A.DiGiacomo,
M.~C. Chu and V.Eletsky.
This work is supported in part by the US Department
of Energy under Grant No. DE-FG02-88ER40388 and
No. DE-FG02-93ER40768.
\section{ Table}
\moveleft 1. cm
\vbox{\offinterlineskip
\halign{\strut \vrule \ \hfil # \hfil \ & \vrule \ \hfil # \hfil \ & \vrule
\ \hfil # \hfil \ & \vrule \ \hfil # \hfil \ \vrule \cr
\noalign{\hrule}
Operator & Coeff. & T-renormalization & Vacuum Dominance\cr
& in ${\cal L}$ & &Hypothesis \cr
\noalign{\hrule}
${\cal O}^A_1=$
& ${1\over 4}$ & $\langle {\cal O}^A_1\rangle _0-$
& ${\langle \bar u u\rangle ^2_0\over 144}(132-360{T^2
\over 12 F^2_\pi})$ \cr
$\bar q {1-\gamma_5 \over 2} q \bar q {1-\gamma_5 \over 2} q$ & &
$(3\langle {\cal O}^A_1\rangle _0+\langle {\cal
O}^A_2\rangle _0){T^2\over 12 F^2_\pi}$ & \cr
\noalign{\hrule}
${\cal O}^A_2=$ & $-{1\over12}$ & $\langle {\cal O}^A_2\rangle _0-$ &
${\langle \bar u u\rangle ^2_0\over 144}(-36-360{T^2
\over 12 F^2_\pi})$ \cr
$\bar q {1-\gamma_5 \over 2}\tau^a q \bar q {1-\gamma_5 \over
2}\tau^a q$ & & $(3\langle {\cal O}^A_1\rangle _0+\langle {\cal
O}^A_2\rangle _0){T^2\over 12 F^2_\pi}$ & \cr
\noalign{\hrule}
${\cal O}^B_2=$ & ${3\over128}$ & $\langle {\cal O}^B_1\rangle _0-$
& ${\langle \bar u u\rangle ^2_0\over 144}(-64+384{T^2
\over 12 F^2_\pi})$\cr
$\bar q {1-\gamma_5 \over 2}t^i q \bar q {1-\gamma_5 \over
2}t^i q$ & & $(3\langle {\cal O}^B_1\rangle _0+\langle {\cal
O}^B_2\rangle _0){T^2\over 12 F^2_\pi}$ & \cr
\noalign{\hrule}
${\cal O}^B_2=$ & $-{1\over 128}$ & $ \langle {\cal O}^B_2\rangle _0-$
& ${\langle \bar u u\rangle ^2_0\over 144}(192+384{T^2
\over 12 F^2_\pi})$ \cr
$\bar q {1-\gamma_5 \over 2}t^i \tau^a q \bar q {1-\gamma_5
\over 2}t^i \tau^a q$ & & $(3\langle {\cal O}^B_1\rangle _0+\langle {\cal
O}^B_2\rangle _0){T^2\over 12 F^2_\pi}$ & \cr
\noalign{\hrule}
${\cal O}^C_1=$
& $-{9\over 512}$ & $ \langle {\cal O}^C_1\rangle _0-$ &
${\langle \bar u u\rangle ^2_0\over 144}(768-4608{T^2
\over 12 F^2_\pi})$ \cr
$\bar q {1-\gamma_5 \over 2}\sigma_{\mu \nu}t^i
q \bar q {1-\gamma_5 \over 2}\sigma_{\mu \nu}t^i q$ &
&$(3\langle {\cal O}^C_1\rangle _0+\langle {\cal
O}^C_2\rangle _0){T^2\over 12 F^2_\pi}$ & \cr
\noalign{\hrule}
${\cal O}^C_2=$& ${9\over 1536}$ & $\langle {\cal O}^C_2\rangle _0-$
& ${\langle \bar u u\rangle ^2_0\over 144}(2304-4608{T^2
\over 12 F^2_\pi})$ \cr
$\bar q {1-\gamma_5 \over 2}\sigma_{\mu \nu}t^i \tau^a
q \bar q {1-\gamma_5 \over 2}\sigma_{\mu \nu}t^i \tau^a q$ & &
$(3\langle {\cal O}^C_1\rangle _0+\langle {\cal
O}^C_2\rangle _0){T^2\over 12 F^2_\pi}$ & \cr
\noalign{\hrule}
}}
\newpage
|
2106.12834
|
\section{Languages of South Africa}
\label{sec:african}
The majority of languages in Africa are considered under-resourced~\cite{orife+etal_ICLR20}. %
This includes the eleven
official languages of South Africa.
As show in Figure~\ref{fig:hierarchy}, nine of these languages %
belong to the larger Southern Bantu family: isiZulu (Zul), isiXhosa (Xho), Sepedi (Nso), Setswana (Tsn), Sesotho (Sot), Xitsonga (Tso), siSwati (Ssw), Tshivenda (Ven) %
and isiNdebele (Nbl).
Many of these languages are also spoken %
in countries neighbouring South Africa.
From the Southern Bantu family there exist two principal, %
Nguni-Tsonga and Sotho-Makua-Venda. %
In the Sotho-Makua-Venda subfamily, Ven is somewhat of a standalone.
The other two languages, Afrikaans (Afr) and English (Eng), are Germanic languages from the %
Indo-European %
family.
All of these languages are considered under-resourced, except for Eng~\cite{barnard+etal_sltu14}.
To give an intuitive idea of how related these languages are, most of the Bantu languages that are grouped together at the lowest level of the hierarchy %
would (to an %
extent) be intelligible to a native speaker of another languages in the same group.
\begin{figure}[t]
\centering
\includegraphics[width=0.99\linewidth]{figures/language_hierarchy.pdf}
\captionsep
\caption[Caption]{A family tree for the official South African languages.\footnotemark
}
\label{fig:hierarchy}
\vspace{-5mm}
\end{figure}
\footnotetext{\scriptsize{\url{https://southafrica-info.com/arts-culture/11-languages-south-africa/}}}
\section{Conclusion}
\label{sec:conclusion}
We investigated the effect of training language choice when applying a multilingual acoustic word embedding model to a zero-resource language.
Using word discrimination and query-by-example search tasks on languages spoken in Southern Africa, we
showed that training a multilingual model on languages related to the target is beneficial.
We observed gains in absolute scores, but also in data efficiency: you can achieve similar performance with much less data when training on multiple languages from the same family as the target.
We showed that even including just one related language
already gives a large
gain.
From a practical perspective, these results indicate that one should prioritise collecting data from related languages (even in modest quantities) rather than collecting more extensive datasets from diverse unrelated families, when building multilingual acoustic word embedding models for a zero-resource language.
\vspace{2pt}
{\eightpt
\noindent \textbf{Acknowledgements.}
This work is supported by the South African NRF (120409), a Google Faculty Award, and support from the Stellenbosch University School of Data Science and Computational Thinking.
We thank Ewald van der Westhuizen for the NCHLT forced alignments.}
\section{Experimental setup}
\label{sec:experiments}
\textbf{Data.} We perform all our experiments on the NCHLT speech corpus~\cite{barnard+etal_sltu14}, which provides wide-band speech from each of the eleven official South African languages (\S\ref{sec:african}).
We use a version of the corpus
where all repeated utterances were removed, leaving
roughly 56 hours of speech from around 200 speakers in each language.
We use the default training, validation and test sets.
We treat six of the languages as well-resourced:
Xho$^*$, Ssw$^*$, Nbl$^*$, Nso$^\dagger$, Tsn$^\dagger$ and Eng$^\ddagger$.
We use labelled data from the well-resourced languages to train a single supervised multilingual model and then apply the model to the target zero-resource languages.
More specifically,
we extract 100k true positive word pairs using forced alignments for each training language.
We select Zul$^*$, Sot$^\dagger$, Afr$^\ddagger$, and Tso as our zero-resource languages and use another language, Ven, for validation of each model.\footnote{We use superscripts to indicate the different language families: $^*$Nguni, $^\dagger$Sotho-Tswana, $^\ddagger$Germanic.}
Training and evaluation languages are
carefully selected such that for each evaluation language at least one language from the same
family is part of the training languages, except for Tso which is in a group of its own.
\label{subsec:awe_models}
\textbf{Models.}
All speech audio is parametrised as
13-dimensional
static Mel-frequency cepstral coefficients.
The encoder unit of the \tablesystem{ContrastiveRNN} models (\S\ref{sec:model}) consists of three unidirectional RNNs with 400-dimensional hidden vectors, with an embedding size of $M = 130$
dimensions.
Models are optimised using Adam optimisation~\cite{kingma+ba_iclr15} with a learning rate of $0.001$.
The temperature parameter $\tau$ in~\eqref{eqn:contrastive_loss} is set to 0.1.
\textbf{Evaluation.}
We consider two tasks for evaluating the performance of the AWE models. First, we
use the \textit{same-different}
word discrimination task~\cite{carlin+etal_icassp11} to measure the intrinsic quality of the
AWEs.
To evaluate a particular AWE model, a set of isolated test word segments is embedded.
We use roughly 7k isolated word instances per language from the test data.
For every word pair in this set, the cosine distance between their embeddings is calculated.
Two words can then be classified as being of the same or different type based on some distance threshold, and a precision-recall curve is obtained by varying the threshold.
The area under this curve is used as final evaluation metric, referred to as the average precision (AP).
We are particularly interested in obtaining embeddings that are speaker invariant.
As in~\cite{hermann+etal_csl21},
we therefore calculate AP by only taking the recall over instances of the same word spoken by different speakers.
The second task is QbE (\S\ref{sec:qbe}).
For each evaluation language we use approximately two hours of test
utterances as the search
collection.
Sub-segments for the utterances in the speech collection are obtained by embedding
windows stretching from 20 to 60 frames with a 3-frame overlap.
For each evaluation language we randomly
draw instances of 15 spoken query word types from a disjoint speech set (the development set---which we never use for any validation experiments) where we only consider query words with
at least 5 characters
for Afr and Zul and 3 for Sot.
There are
between 6 and 51 occurrences of each query word.
For each QbE test, we ensure that the relevant
multilingual AWE model
has not seen any of the search or query data
during training or validation.
We report precision at ten
($P@10$),
which is the fraction of the ten top-scoring retrieved utterances from the search collection that
contains the given query.
\section{Introduction}
\label{sec:introduction}
Developing robust speech systems for zero-resource languages---where no transcribed speech resources are available for model training---remains a challenge.
Although full speech recognition is not possible in most zero-resource settings, researchers have proposed methods for applications such as speech search~\cite{levin+etal_icassp15,huang+etal_arxiv18,yuan+etal_interspeech18},
word discovery~\cite{park+glass_taslp08,jansen+vandurme_asru11,ondel+etal_interspeech19,rasanen+blandon_arxiv20}, and segmentation and clustering~\cite{kamper+etal_asru17,seshadri+rasanen_spl19,kreuk+etal_interspeech20}, making use of only
unlabelled speech audio.
Many of these applications require speech segments of different lengths to be compared.
This is conventionally done using alignment (e.g.\ with dynamic time warping).
But this can be slow and inaccurate.
Acoustic word embedding (AWE) models map a
variable duration speech segment
to a fixed dimensional vector~\cite{levin+etal_asru13}.
The goal is to map instances of the same word type to similar vectors.
Segments can then be efficiently compared by calculating the distance
in the embedding space.
Given the advantages AWEs have over
alignment methods, several AWE models have
been proposed~\cite{bengio+heigold_interspeech14,he+etal_iclr17,audhkhasi+etal_stsp17,wang+etal_icassp18,chen+etal_slt18,holzenberger+etal_interspeech18,chung+glass_interspeech18,haque+etal_icassp19,shi+etal_slt21,palaskar+etal_icassp19,settle+etal_icassp19,jung+etal_asru19}. Many of these are for the supervised setting, using labelled data to train a discriminative model.
For the zero-resource setting, a number of unsupervised AWE approaches have also been explored, many relying on autoencoder-based neural models trained on unlabelled data in the target language~\cite{chung+etal_interspeech16,kamper+etal_icassp16,holzenberger+etal_interspeech18,kamper_icassp19}.
However, there still exists a large performance gap between these unsupervised models and
their supervised counterparts~\cite{levin+etal_asru13,kamper_icassp19}.
A recent alternative for obtaining AWEs on a zero-resource language is to use multilingual transfer learning~\cite{ma+etal_arxiv20,kamper+etal_icassp20,kamper+etal_taslp21,hu+etal_slt21, hu+etal_interspeech20}.
The goal is
to have the benefits of supervised learning by training a model on labelled data from multiple well-resourced languages, but to then
apply
the model to an unseen target zero-resource language without fine-tuning it---a form of \textit{transductive} transfer learning~\cite{ruder_phd19}.
This multilingual transfer approach has been shown to outperform unsupervised monolingual AWE models~\cite{kamper+etal_taslp21,jacobs+etal_slt21}.
Although
there is clear benefit in applying multilingual AWE models
to an unseen zero-resource language, it is still unclear how the particular choice of training languages affects subsequent performance.
Preliminary experiments~\cite{kamper+etal_taslp21} show improved scores
when training a monolingual model on one language and applying it to another from the same family.
But this has not been investigated systematically and there are still several unanswered questions:
Does the benefit of training on related languages diminish as we train on more languages (which might or might not come from the same family as the target zero-resource language)?
When training exclusively on related languages, does performance suffer when adding an unrelated language?
Should we prioritise data set size or language diversity when collecting data for multilingual AWE transfer?
We try to answer these questions using
a corpus of under-resourced languages spoken in Southern Africa.
These languages can be grouped into different families
based on their linguistic links.
We specifically
want to see whether it is beneficial to use closely related languages when training
a multilingual model catered to a specific zero-resource language, similar to~\cite{yi+etal_taslp19}.
We divide the corpus into training and test languages, with (some of) the test languages coming from families that also occur in
training.
We conduct several experiments where we add data from different language families, and also
control for the amount of data per language.
AWEs are evaluated
in an isolated word discrimination task and in query-by-example (QbE) speech search on full utterances.
To our knowledge, only one other
study~\cite{hu+etal_slt21} has done
AWE-based QbE
using multilingual transfer.
Our main findings are as follows.
(i)~Training a multilingual model using languages that are closely related to the target
improves
performance.
This is true, not just
because of the increase in data, but because of
language diversity.
(ii)~When
we systematically add training languages, the largest improvements are
gained from adding a single related language. Adding more related languages
gives small gains.
Adding unrelated languages generally gives small or no
gains, but also doesn't hurt.
(iii)~For a
target language, the
performance of a multilingual model trained on unrelated languages can be matched by a model trained with much less data from multiple related~languages.
\section{Acoustic word embedding model}
\label{sec:model}
We use the \tablesystem{ContrastiveRNN} AWE model of~\cite{jacobs+etal_slt21}.\footnote{We extend the code available at \scriptsize{\url{https://github.com/christiaanjacobs/globalphone_awe_pytorch}}.}
It performed the best of the model variants considered for multilingual transfer in~\cite{jacobs+etal_slt21}. %
The model consists of an encoder recurrent neural network (RNN) %
that produces fixed-dimensional representations from variable-length speech segments.
It is trained %
to minimise %
the distance between embeddings from %
speech segments %
of the same word type while %
maximising the distance between embeddings from multiple words of a different type.
Let's use $X = \mathbf{x}_1, \mathbf{x}_2, \ldots, \mathbf{x}_T$ to denote a sequence of speech features.
Formally, given %
speech segments
$X_a$ and $X_p$ containing instances of the same word type and multiple negative examples $X_{n_{1}}, \ldots, X_{n_{K}}$, the \tablesystem{ContrastiveRNN} produces embeddings $\mathbf{z}_a, \mathbf{z}_p, \mathbf{z}_{n_{1}}, \ldots, \mathbf{z}_{n_{K}}$
(subscripts indicate anchor, positive and negative, respectively).
Each embedding is a fixed dimensional vector $\mathbf{z} \in \mathbb{R}^M$.
The model is illustrated in Figure~\ref{fig:contrastiveRNN}.
Let $\text{sim}(\mathbf{u}, \mathbf{v}) = \mathbf{u}^{\top}\mathbf{v}/\norm{\mathbf{u}}\norm{\mathbf{v}}$ denote the cosine similarity between vectors $\mathbf{u}$ and $\mathbf{v}$.
The loss given a positive pair $(X_a, X_p)$ and the set of negative examples is then defined as~\cite{chen+etal_icml20}:
\vspace*{-2.5pt}
\begin{equation}
J = -\text{log}\frac{\text{exp}\big\{\text{sim}(\mathbf{z}_a, \mathbf{z}_p)/\tau\big\}}{\sum_{j \in \{p, n_1, \hdots, n_K\}}^{}\text{exp}\big\{\text{sim}(\mathbf{z}_a, \mathbf{z}_j)/\tau\big\}}\,\text{,}
\label{eqn:contrastive_loss}
\end{equation}
where $\tau$ is a temperature parameter, tuned on development data.
\begin{figure}[t]
\centering
\includegraphics[width=0.99\linewidth]{figures/contrastive_rnn.pdf}
\captionsep
\caption{The \tablesystem{ContrastiveRNN}.
The model is trained to minimise the distance between the anchor and positive item $d(\mathbf{z}_a, \mathbf{z}_p)$ while maximising the distance between the anchor and multiple negatives $(\mathbf{z}_a, \mathbf{z}_{n_{k}})$.}
\label{fig:contrastiveRNN}
\vspace{-5mm}
\end{figure}
In the zero-resource setting we don't have labelled data in the target {language} %
to construct the positive and negative %
word pairs required for training.
We therefore follow the approach of~\cite{kamper+etal_taslp21}, and train a multilingual model on ground truth word pairs (extracted from forced alignments) from a number of %
languages for which we have labelled data.
Subsequently, at test time, we apply the encoder RNN from the multilingual model to extract AWEs for speech from
the target zero-resource language.
\section{Query-by-example speech search}
\label{sec:qbe}
For evaluating the different AWE {models}
we use
an isolated word discrimination task~\cite{carlin+etal_icassp11}.
Recent findings~\cite{algayres+etal_interspeech20} suggest, however, that this evaluation is not always indicative of downstream system performance.
We therefore also perform query-by-example (QbE) speech search, which in contrast to
{the word discrimination task}, does not assume a test set of isolated words, but instead operates on full unsegmented utterances.
Concretely, QbE speech search is the task of identifying the utterances in a speech collection that contain instances of a given spoken query.
A number of approaches have been put forward for AWE-based QbE~\cite{levin+etal_icassp15, chen+etal_icassp15, settle+etal_interspeech17}.
Here we use the simplified approach from~\cite{kamper+etal_icassp19}.
Using an AWE model, we first embed the query segment.
If we knew the word boundaries in the search collection, we could embed each of the words in an utterance and simply look up the closest embeddings to the query.
Instead, because we do not have word boundaries, each utterance is split into overlapping segments from some minimum to some maximum duration.
Each segment from each utterance is then embedded separately using the AWE model.
Finally, to do the QbE task, the query embedding is compared to each of the utterance sub-segment embeddings (using cosine distance), and the minimum distance over the utterance is then taken as the score for whether the utterance contains the given query.
\section{Experimental results}
\label{sec:results}
\begin{figure}[!t]
\centering
\includegraphics[width=0.99\linewidth]{figures/cross_lingual_heatmap.pdf}
\captionsep
\caption{AP (\%)
when training a monolingual supervised \tablesystem{ContrastiveRNN} on each language (rows) and then evaluating it on each of the other languages (columns). Heatmap colours are normalised for each evaluation language (i.e. per column).}
\label{fig:heatmap}
\vspace{-4mm}
\end{figure}
\subsection{Cross-lingual evaluation}
Before looking at multilingual modelling, we first consider a cross-lingual evaluation where we treat each language as a training language, train a supervised monolingual AWE model, and then apply it to every other language.
This allows us to see the effect of training on related languages in a pairwise fashion.
The results are shown in Figure~\ref{fig:heatmap}.
For each evaluation language excluding Tso and Ven, which are in family groups of their own, %
the best results are achieved from models trained on a language from the same family.
E.g.\ on Zul, Xho is the best training language giving an AP of 58.5\%.
Eng is the only exception where the model trained on Sot performs better than using Afr, the other Germanic language.
Although Ven is in its own group at the lowest layer of the family tree in Figure~\ref{fig:hierarchy}, some of the best results when evaluating on Ven are obtained using models trained on Sotho-Tswana languages (Sot, Tsn, Nso), which are in the same family at a higher level.
We also see that for all nine Bantu evaluation languages, worst performance is obtained from the two Germanic models (Afr, Eng).
\begin{table}[!t]
\mytable
\caption{AP (\%) on test data for multilingual models trained on different combinations of well-resourced languages. Models are applied to two zero-resource languages from different language families, Nguni and Sotho-Tswana. For each training language 100k word pairs were extracted.}
\captionsep
\vspace{-1mm}
\eightpt
\begin{tabularx}{1\linewidth}{Lcc}
\toprule
Multilingual model & Zul$^*$ & Sot$^\dagger$ \\
\midrule
\underline{\textit{Nguni:}} & &\\[2pt]
Xho$^*$ + Ssw$^*$ + Nbl$^*$ &\textbf{68.6} &--- \\
Xho$^*$ + Ssw$^*$ + Eng$^\ddagger$ &60.9 &--- \\
Xho$^*$ + Nso$^\dagger$ + Eng$^\ddagger$ &55.7 &--- \\
Tsn$^*$ + Nso$^\dagger$ + Eng$^\ddagger$ &37.5 &--- \\
Xho$^*$ + Ssw$^*$ + Nbl$^*$ {(\scriptsize{$10$\% subset})} &58.6 &--- \\[2pt]
\underline{\textit{Sotho-Tswana:}} \\[2pt]
Nso$^\dagger$ + Tsn$^\dagger $&--- &\textbf{76.7} \\
Nso$^\dagger$ + Eng$^\ddagger$ &--- &64.8 \\
Xho$^*$ + Ssw$^*$ &--- &51.9 \\
Xho$^*$ + Eng$^\ddagger$ &--- &52.5 \\
Nso$^\dagger$ + Tsn$^\dagger$ {(\scriptsize{$10$\% subset})} &--- &58.4 \\
\bottomrule
\end{tabularx}
\label{tbl:multilingual}
\vspace{-4mm}
\end{table}
\begin{figure*}[t]
\centering
\includegraphics[scale=0.31]{figures/multilingual_increment.pdf}
\captionsep
\caption{Same-different results %
from two sequences of multilingual models, trained by adding one language at a time. For each training language, 100k positive word pairs are used, which is indicated on the $x$-axis.} %
\label{fig:multi_increment}
\vspace{-3.5mm}
\end{figure*}
\begin{figure}[t]
\centering
\includegraphics[scale=0.31]{figures/qbe_zul.pdf}
\captionsep
\caption{QbE results %
on %
Zul %
using the same sequences of multilingual models as in Figure~\ref{fig:multi_increment}.}
\label{fig:qbe_increment}
\vspace{-4mm}
\end{figure}
\subsection{Multilingual evaluation}
The cross-lingual experiment above was in large part our inspiration for the subsequent analysis of multilingual models.
Concretely, we hypothesise that even better performance can be achieved by training on multiple languages from the same family as the target zero-resource language, and that this would be superior to multilingual models trained on unrelated languages.
Focusing on two evaluation languages, Zul and Sot from distinct language families, we
investigate this hypothesis
by training multilingual models with different language combinations, as shown in Table~\ref{tbl:multilingual}.
Firstly, we see the best result on a %
language when all the training languages comes from the same family as the target.
Secondly, we see how the performance %
gradually decrease as the number of training languages related to the target drop.
Furthermore, notice the performance boost from %
including even just one training language related to the target compared to not including any.
E.g.\ on Sot we see a increase of more than 12\% absolute when adding just one related language (from 52.5\% and 51.9\% to 64.8\%). %
To further demonstrate the benefit of using training languages from the same family, %
we train a multilingual model for each evaluation %
language on all its related languages using a 10\% subset of the original data.
For both Zul and Sot, %
the subset models outperform the models where no related languages are used. %
E.g.\ on Zul, the Xho+Ssw+Nbl subset model outperforms the full Tsn+Nso+Eng model (no related languages) by more than 20\% in AP.
Moreover, this subset model (58.6\% in AP) even outperforms the Xho+Nso+Eng model (55.7\%) where all the training data from one related language are included and almost matches the AP when using two related languages (Xho+Ssw+Eng, 60.9\%).
These comparisons do more than just show the benefit of training on related languages: they also show that it is beneficial to train on a diverse set of related languages. %
\subsection{Adding more languages} %
In the above experiments, we controlled for the amount of data per language and saw that training on languages from the same family improves multilingual transfer.
But this raises a question:
will adding additional unrelated languages harm performance? %
To answer this, we systematically train two sequences of multilingual models on all six well-resourced languages, %
evaluating each target language as a new training language is added.
Same-different results for all five evaluation languages are shown in Figure~\ref{fig:multi_increment} and QbE results for Zul are shown in Figure~\ref{fig:qbe_increment}. As for Zul, the trends in the same-different and QbE results track each other closely for the other evaluation languages (so these are not shown here).
In the fist sequence of multilingual models (green), we start by adding the three Nguni languages (Xho, Ssw, Nbl), followed by the two Sotho-Tswana languages (Nso, Tsn), and lastly the Germanic training language (Eng).
The second sequence (orange) does not follow a systematic procedure.
On Zul, the green sequence, which starts with a related language (Xho), initially achieves a higher score compared to the orange sequence in both Figures~\ref{fig:multi_increment} and~\ref{fig:qbe_increment}.
Then, the score gradually increase by adding more related languages (Ssw, Nbl). %
Thereafter, adding additional unrelated languages (Nso, Tsn, Eng) show no performance increase.
In fact, AP decreases slightly after adding the two Sotho-Tswana languages (Nso, Tsn), but not significantly.
The orange sequence starts low on Zul until the first related language (Ssw) is added, causing a sudden increase.
Adding the Germanic language (Eng) has little effect.
Adding the last two related languages (Nbl, Xho) again causes the score to increase.
A similar trend follows for Sot and Afr, where adding related languages causes a noticeable performance increase, especially when adding the first related language; after this, performance seem to plateau %
when adding more unrelated languages.
(Afr is the one exception, with a drop when adding the last language in the orange sequence).
On Tso, which does not have any languages from the same family in the training set, AP gradually increases in both sequences without any sudden jumps.
Although Ven isn't in the same family as Nso and Tsn at the lowest level of the tree in Figure~\ref{fig:hierarchy}, it belongs to the same family {(Sotho-Tswana)} at a higher level. %
This explains why it
closely tracks the Sot results.
Summarising these results, we see that adding unrelated languages generally does not decrease scores, but also does not provide a big benefit (except if it is one of the earlier languages in the training sequence, where data is still limited). In contrast, it seems that training on languages from the same family is again beneficial; this is especially the case for the first related language, irrespective of where it is added in the sequence.
|
1910.00924
|
\subsection*{Note to the referee and other readers} This appendix contains material that is not
intended for publication because it has too many details, is too reptitive
and also duplicative of material in the main paper. It is planned to be
included with the pdf electronicly distributed by the author
so that readers can see the results of the computer
programs in fuller detain than can be justified in a publication, even an electronic one.
\section{Proofs of various items}
Here is the special case of
\cite[Thm. 4.1]{MR2363058} that we need.
\begin{prop}\label{propAutosOfZphatk} Let $1\le m$ and $p$ a prime. then
$L=\text{$\mathbb Z$}_{p^m}$ has $p-1$ automorphisms.
\end{prop}
\begin{proof}
Every element $x\in L$ has the form
\[
x=
a_0+a_1p+\cdots a_{m-1}p^{m-1}
\]
where $0\le a_k<p$ for $0\le k <m$.
Every automorphism $T:L\to L$ is determined by $T(1)$. Using the facts that $1$ is
a generator of $L$ and $T$ is an automorphism, we see that
\begin{align}
T(x) &= T\prn{a_0+a_1p+\cdots+a_{m-1}p^{m-1}}
\\&
= T(a_0)+T(a_1)p+\cdots+T(a_{m-1})p^{m-1}.
\end{align}
\end{proof}
\begin{prop}There is no real extreme measure on $\{0,1,2,4\}\subset\Zp 7{}.$
\end{prop}
\begin{proof}
\textit{Look at the 4 possible measures.}
Case I.
$\mu=\delta_0+\delta_1+\delta_2-\delta_4.$
$M=\begin{pmatrix}
1&1&1&0&-1&0&0\\
0&1&1&1&0&-1&0\\
\dots
\end{pmatrix}
$
Row 2 inner row 1 is $=2\ne0$.
\smallskip
Case II.
$\mu=\delta_0+\delta_1-\delta_2+\delta_4.$
$M=\begin{pmatrix}
1&1&-1&0&1&0&0\\
0&1&1&-1&0&1&0\\
0&0&1&1&-1&0&1\\
\dots
\end{pmatrix}
$
Row 3 inner row 1 is $-2=\ne 0$.
\smallskip
Case III.
$\mu=\delta_0-\delta_1+\delta_2+\delta_4.$
$M=\begin{pmatrix}
1&-1&1&0&1&0&0\\
0&1&-1&+1&0&1&0\\
0&0&1&-1&1&0&1\\
\dots
\end{pmatrix}
$
Row 2 inner row 1 is $=-2\ne 0$.
\smallskip
Case IV.
$\mu=-\delta_0+\delta_1+\delta_2+\delta_4.$
$M=\begin{pmatrix}
-1&1&1&0&1&0&0\\
0&-1&1&1&0&1&0\\
0&0&-1&1&1&0&1\\
1&0&0&-1&1&1&0\\
\dots
\end{pmatrix}
$
Row 3 inner row 1 is $=-1\ne 0$.
\qedhere
\end{proof}
\section{Proofs of extremalities for cyclic groups}
In this section we give proofs of some of the extremalities claimed in and of related results.
We also show that the extreme sets here are neither sums of other extreme sets nor of a subgroup and a set.
To save the reader from flipping between \cite{ABeastiary} and this document, we sometimes include details
from \cite{ABeastiary}.
\subsection{Sets with three elements}
\begin{prop}\label{prop3inZ3}
$\Zp{3}{}$ is extreme. \newline \end{prop}
\begin{proof}
Let $\mu = \delta_{0}+e^{4\pi i/3}\delta_{1}+\delta_{2}
$. Then
\begin{align*}
\mu * \tilde \mu & = \big(1+1+1\big)\delta_{0}
\\&\qquad
+\big(1+e^{4\pi i/3}+e^{2\pi i/3}\big)\delta_{1}
\\&\qquad
+\big(e^{2\pi i/3}+e^{4\pi i/3}+1\big)\delta_{2}\\&=3\delta_{0}\qedhere
\end{align*}
\end{proof}
In \Zp{4}{} there is only one, up to translation, and it is extreme:
\begin{prop}\cite{MR627683}\label{prop3in4}
$\{0,1,2\}\subset \Zp4{}$ is extreme.
\end{prop}
\begin{proof}
Let $\nu = \delta(0) + e^{3\pi i/4}\delta(1) +i\delta(2)$.
Then
\begin{align*}
\widehat\nu(0) &= 1- \frac{\sqrt2}2 + \frac{\sqrt2}2i +i, \text{ so }
\\
|\widehat\nu(0)| &= \big|(1-\frac{\sqrt2}2)^2 +(1+\frac{\sqrt2}2)^2\big|^{1/2} =\sqrt3.
\\
\widehat\nu(1) &= 1 -\frac{\sqrt2}2 -\frac{\sqrt2}2i -i, \text{ so }
|\widehat\nu(1)|= \sqrt3\\
\widehat\nu(2) & = 1 +\frac{\sqrt2}2 -\frac{\sqrt2}2i +i \text{, so } |\widehat\nu(2)|= \sqrt3.
\\
\widehat\nu(3) & = 1 +\frac{\sqrt2}2 +\frac{\sqrt2}2i -i, \text{ so }
|\widehat\nu(3)| = \sqrt3.\qedhere
\end{align*}
\end{proof}
Another extreme measure is $\mu=\delta(0) - e^{3\pi i/4}\delta(1) + i\delta(2)
=\delta(0) + e^{7\pi i/4}\delta(1) + i\delta(2)$.
All extremal measures have one of the forms, $\nu$ or $\mu$:
Suppose $\mu=\delta_0+a\delta_1+b\delta_2$ is extreme on $\{0,1,2\}\subset \Zp4{}$. Then
$\mu*\tilde\mu=3\delta_0+(a+\bar ab)\delta_1 + (b+\bar b)\delta_2 +
(\bar a+a\bar b)\delta_3=3\delta_0$.
Hence, $b+\bar b=0$ so $b=\pm i$. Assume $b=i$. Then
$\bar a-ia=0$ so\footnote{\
Let $a= x+iy$. Then $\bar{a}-ia= x-iy-ix+y=0$ means $x+y=0$. Since
$|a|=1$, $x= \pm\sqrt2/2$.}
either $a=\pm e^{\pm\pi i/4}$ or $a =\pm e^{\pm 3\pi i/4}$.
Since $y=-x$, we have $a=\pm \exp(3\pi /4).$
\medskip
Every three-element extreme set can be obtained from \Zp{3}{} and a 3-three element subset of \Zp{4}{}
by the operations of group automorphism, passing to a subgroup, and translation
\cite[3.1(ii)]{MR627683}. Here is a proof of part of \cite[3.1(ii)]{MR627683}.
The complications of the proof here illustrate
why the proofs
of \cite[3.1-3.3]{MR627683} occupied 200 pages of manuscript.
\begin{prop}[\cite{MR627683}]\label{prop3eltsets}
If $3\le k$ and $E=\{0,a,b\}$ is extreme in $\Zp{k}{}$ then either $E$ is a subgroup or it is a three-element
subset of a four-element subgroup.
\end{prop}
\begin{proof} Suppose $E=\{0, a,b\}\subset \Zp{k}{}$ is extreme. Let $\mu = \delta_0 +\delta_a+\delta_b.$
We may assume $0<a<b<k$. Then
\begin{equation}\label{eq1prop3eltsets}
\mu*\tilde\mu= 3\delta_0+\delta_a+\delta_b +\delta_{-a}+\delta_{-b}+\delta_{a-b}+
\delta_{b-a}.
\end{equation}
If $a\ne -b$ and $a\ne -a$, then we have four pointmasses, $\delta_{\pm a},$ $ \delta_{\pm b}$ which cannot all
be matched by $\delta_{a-b},$ $\delta_{b-a}$ and $E$ is not extreme by \cite[Cor. 2.3]{ABeastiary}.
\smallbreak
Case I: Suppose $a=-a$, that is, $a=k-a.$ Then $k=2a$ and $b\ne -b$ so $a-b\ne a+b$.
Thus,\[
\mu*\tilde\mu= 3\delta_0+2\delta_a+\delta_b +\delta_{-b}+\delta_{a-b}+
\delta_{b+a}.
\]
If $a-b=b$ then $a=2b$ and $k=4b$ so we have three elements of a 4 element subgroup, $\{0, b, 2b\}\subset\Zp{4b}{}.$
Therefore, we may assume $a-b\ne b$.
Clearly $a-b$ is distinct from $0$, $a$, $b$, and $a+b$. Hence $\mu$ is not extreme in this case.
\smallbreak
Case II: $b=-b$, so $a\ne-a$. Then
\[
\mu*\tilde\mu= 3\delta_0+2\delta_b +\delta_a+\delta_{-a}+\delta_{a+b}+
\delta_{b-a}
\]
or
\[
\mu*\tilde\mu= 3\delta_0+2\delta_b +\delta_a+\delta_{-a}+\delta_{a+b}+
\delta_{-b-a}.
\]
If $a=-b-a$, then $b=2a$ and $k=4a$ and we are again in the situation of a 3-element subset of a 4-element subgroup.
Therefore we may assume $a\ne -b-a$, in which case $\delta_{-b-a}$ appears only once in \eqref{eq1prop3eltsets} and $E$ is not extreme.
\smallbreak
Case III:
$a=-b$ so
\[
\mu*\tilde \mu = 3\delta_0+2\delta_a+2\delta_b + \delta_{a-b}+\delta_{b-a}.
\]
We have two subcases. First, suppose $a= b-a$. Then since $a=-b$, $a= b--b=2b=-2a$ so $3a=0$. Hence $E=\{0,a,2a\}$ and $k=3a$, that is, $E$ is a subgroup.
Second subcase: $a-b=b-a$. We still assume $a=-b$.
Since $a\ne b$ we must have $a-b=\ell$ has order 2, and so $k=2\ell.$ Now $a-b = \ell = -2b = 2a$ $\mod\,2\ell$. Hence, $2a=\ell,$
and $a$ has order 4. Thus, $E=\{0,a, 2a\}\subset\Zp{4a}{}$ and $E$ is a three element subset of a 4-element subgroup. That takes care
of the third and final case.
\end{proof}
\subsection{Sets with four elements}
\begin{prop}\label{prop4in4}
$\Zp{4}{}$ is extreme. \newline \end{prop}
\begin{proof}
Let $\mu = \delta_{0}+e^{5\pi i/3}\delta_{1}+\delta_{2}+e^{2\pi i/3}\delta_{3}
$. Then
\begin{align*}
\mu * \tilde \mu & = \big(1+1+1+1\big)\delta_{0}
\\&\qquad
+\big(e^{4\pi i/3}+e^{5\pi i/3}+e^{\pi i/3}+e^{2\pi i/3}\big)\delta_{1}
\\&\qquad
+\big(1-1+1-1\big)\delta_{2}
\\&\qquad
+\big(e^{\pi i/3}+e^{5\pi i/3}+e^{4\pi i/3}+e^{2\pi i/3}\big)\delta_{3}\\&=4\delta_{0 }.\qedhere
\end{align*}
\end{proof}
\begin{prop}
\cite{MR627683}
\label{prop4in5}
$\{0,1,2,3\}$ is an extreme subset of $\Zp5{}$.
\end{prop}
\begin{proof}
Here is an extremal measure:
\begin{comment}\,\footnote{\ There are others. For example: $\mu = \delta_0+ e^{22\pi i/15}\delta_1 +e^{48\pi i/15}\delta_2 +e^{6\pi i/15}\delta_3.$
Then calculation shows that
\begin{align*}
\mu*\tilde\mu&= 4\delta_0
+e^{2\pi i/15}\big(-e^{\pi i/3}-e^{-\pi/3} +1\big)\delta_1
\\ &+e^{4\pi i/15}\big(1+e^{-20\pi i/15}+e^{-10\pi i/15}\big)\delta_2
\\
&+e^{\pi i/15}\big(e^{5\pi i/15}-e^{-5\pi i/15}+ e^{15\pi i/15}\big)\delta_3
\\
&+e^{3\pi i/15}\big(-e^{-10\pi i/15}-1+e^{-5\pi i/15}\big)\delta_4.
\end{align*}
The coefficient of each of $\delta_1$ to $\delta_4$ has a factor which is a sum of the third roots of three (or of their complex conjugates), which sum to zero.
\end{comment}
$\mu = \delta_0+e^{2\pi i/3}(\delta_1+\delta_2) +\delta_3$.
Then
$
\tilde\mu = \delta_0 +e^{-2\pi i/3}(\delta_3+\delta_4)+\delta_2$, so
\[\mu*\tilde\mu = 4\delta_0+(1+e^{2\pi i/3} +e^{-2\pi/3})(\delta_1+\delta_2+\delta_3+\delta_4)\\
= 4\delta_0. \qedhere
\]
\end{proof}
\begin{comment}
\begin{prop}\label{prop4in7}
$\{0,1,2,4\}$ is extreme in $\Zp7{}$.
\end{prop}
\end{comment}
\begin{prop}
\cite{MR627683}
\label{prop4in7}
$\{0, 1, 2, 4\}$ is extreme in \Zp{7}{}.
\end{prop}
\begin{proof}
Let $\mu = \delta_{0} - \delta_{1} - \delta_{2} - \delta_{4}
$. Then
\begin{align*}
\mu * \tilde \mu & = 4\delta_{0} + \big( -1 + 1\big)\delta_{1}
+ \big( -1 + 1\big)\delta_{2} + \big( -1 + 1\big)\delta_{3}
\\&\qquad
%
+ \big(1-1\big)\delta_{4} + \big( -1 + 1\big)\delta_{5}
+ \big( -1 + 1\big)\delta_{6}
=4\delta_0.\qedhere
\end{align*}
\end{proof}
\begin{proof}[Alternative proof of \propref{prop4in7}]
Let $z =e^{2\pi i/7}$.
We compute the transform of $\mu$, using arithmetic mod 7 in
the exponents of $z$.
\begin{align*}
\widehat\mu(0) &= 1 -1-1-1=-2.\\
|\widehat\mu(1)|^2 &= -z^4 + z^3 + z^{-3} - z^{-4} + 4
\\ \notag
&= -z^4+z^3+z^4-z^3+4=4,
\\
|\widehat\mu(2)|^2 &=
-z^8 + z^6 + z^{-6} - z^{-8} + 4=4
\\
|\widehat\mu(3)|^2 &=
-z^5-z^2-z^5-z^2+4=4,
\\
|\widehat\mu(4)|^2 &=
-z^2+z^5+z^2-z^5+4=4
\text{ and}
\\
|\widehat\mu(6)|^2 &=
-z^3+z^4+z^3-z^4+4=4.
\qedhere
\end{align*}
\end{proof}
\bigbreak
\subsection{Sets with five elements}
\begin{prop}\label{prop5in6}
\cite{MR627683}
$\{0,1,2,3,4\}$ is extreme in \Zp{6}{}.
\end{prop}
\begin{proof} Let $\mu = \delta_{0} + e^{ 3\pi i / 2}\delta_{1} + -\delta_{2} + e^{ 3\pi i / 2}\delta_{3} + \delta_{4}$.
Then
\begin{align*}
\mu * \tilde\mu&= 5\delta_{0} + \big( -i - i+ i + i \big)\delta_{1}
+ \big(1-1+1-1\big)\delta_{2}
\\& \qquad + \big( i - i- i+ i \big)\delta_{3}
+ \big( -1 +1-1+ 1\big)\delta_{4}
\\& \qquad
+ \big( i + i - i+ e^{ 3 \pi i}\big)\delta_{5}
\\&=5\delta_0.\qedhere
\end{align*}
\end{proof}
\begin{prop}
\cite{MR627683}
\label{prop5in12} The subsets $E=\{0,2,4,6,8\}$, $F=\{0,2,3,4,7\}\subset \Zp{12}{}$
are not equivalent but both are extreme.
\end{prop}
\begin{remark} This does not contradict \cite{MR627683}, since one set is in a subgroup.
\end{remark}
\begin{proof}
Non-equivalence:
the group automorphisms of \Zp{12}{} are multiplication by 5, 7 and 11 (all mod 12). Each of them takes odd elements of \Zp{12}{} to odd elements and even elements to even elements. Translations either take evens to evens and odds to odds or evens to odds and odds to evens. Thus, no combination of group automorphisms and translations can take
$E$, whose image will contain either only evens or only odds, onto $F$, which contains both evens and odds.
$\{0,2,4,6,8\}$ is extreme because it is a five-element subset of the the coset $\{0,2,4,6,8,10\}$ in \Zp{12}{}.
For the second set, let $\mu =
\delta_{0} + e^{ 3\pi i / 2}\delta_{2} + e^{ 5\pi i / 4}\delta_{3} + \delta_{4} + e^{\pi i / 4}\delta_{7}
.$
Then
\begin{align*} \mu * \tilde\mu&= 5\delta_{0} + \big( e^{ 7 \pi i / 4}+ e^{ 3 \pi i / 4}\big)\delta_{1}
+ \big( -i + i \big)\delta_{2}
\\& \qquad
+ \big( e^{ 5 \pi i / 4}+ e^{\pi i / 4}\big)\delta_{3}
+ \big(1-1\big)\delta_{4}
\\& \qquad + \big( e^{ 7 \pi i / 4}+ e^{ 3 \pi i / 4}\big)\delta_{5}
+ \big( e^{ 5 \pi i / 4}+ e^{\pi i / 4}\big)\delta_{7}
\\& \qquad + \big(1-1\big)\delta_{8}
+ \big( e^{ 3 \pi i / 4}+ e^{ 7 \pi i / 4}\big)\delta_{9}
\\& \qquad + \big( i - i\big)\delta_{10}
+ \big( e^{\pi i / 4}+ e^{ 5 \pi i}\big)\delta_{11}
\\&= 5\delta_0.
\end{align*}
\end{proof}
\subsection{Sets with 6 elements}
There are no extreme three element subsets of \Zp{10}{} nor of \Zp{14}{}, so the sets in the next two results
cannot be the sum of an extreme set with a two-element coset (all cosets having two elements in those groups).
The 6 element set $\{(0, 0), (0, 1), (0, 2), (0, 3), (0, 4), (0, 7) \}$ is extreme in $\Zp{2}{} \times \Zp{10}{}$. \newline
\begin{proof}
Let $\mu = \delta_{(0,0)}+e^{53\pi i/30}\delta_{(0,1)}+e^{8\pi i/15}\delta_{(0,2)}+e^{29\pi i/30}\delta_{(0,3)}+e^{2\pi i/5}\delta_{(0,4)}+e^{\pi i/30}\delta_{(0,7)}
$. Then
\begin{align*}
\mu * \tilde \mu & = \big(1+1+1+1+1+1\big)\delta_{(0,0)}
\\&\qquad
+\big(e^{53\pi i/30}+e^{23\pi i/30}+e^{13\pi i/30}+e^{43\pi i/30}\big)\delta_{(0,1)}
\\&\qquad
+\big(e^{8\pi i/15}+e^{6\pi i/5}+e^{28\pi i/15}\big)\delta_{(0,2)}
\\&\qquad
+\big(e^{59\pi i/30}+e^{29\pi i/30}+e^{19\pi i/30}+e^{49\pi i/30}\big)\delta_{(0,3)}
\\&\qquad
+\big(e^{26\pi i/15}+e^{2\pi i/5}+e^{16\pi i/15}\big)\delta_{(0,4)}
\\&\qquad
+\big(i-i\big)\delta_{(0,5)}
\\&\qquad
+\big(e^{8\pi i/5}+e^{14\pi i/15}+e^{4\pi i/15}\big)\delta_{(0,6)}
\\&\qquad
+\big(e^{31\pi i/30}+e^{41\pi i/30}+e^{11\pi i/30}+e^{\pi i/30}\big)\delta_{(0,7)}
\\&\qquad
+\big(e^{22\pi i/15}+e^{4\pi i/5}+e^{2\pi i/15}\big)\delta_{(0,8)}
\\&\qquad
+\big(e^{7\pi i/30}+e^{37\pi i/30}+e^{47\pi i/30}+e^{17\pi i/30}\big)\delta_{(0,9)}\\&=6\delta_{(0 , 0)}\qedhere
\end{align*}
\end{proof}
\begin{prop}\label{prop6in10} The 6-element set
$\{0,1,2,3,4,7\} $ is extreme in \Zp{10}{}.
\end{prop}
\begin{proof}
Let $\mu =
\delta_{0} + e^{ 5\pi i / 6}\delta_{1} + e^{ 2\pi i / 3}\delta_{2} + e^{ 5\pi i / 6}\delta_{3} + e^{ 2\pi i / 1}\delta_{4} + e^{\pi i / 6}\delta_{7}
.$
Then
\begin{align*} \mu * \tilde\mu&= 6\delta_{0} + \big( e^{ 5 \pi i / 6}+ e^{ 11 \pi i / 6}+ e^{\pi i / 6}+ e^{ 7 \pi i / 6}\big)\delta_{1}
+ \big( e^{ 2 \pi i / 3}+ 1+ e^{ 4 \pi i / 3}\big)\delta_{2}
\\& \qquad
+ \big( e^{ 11 \pi i / 6}+ e^{ 5 \pi i / 6}+ e^{ 7 \pi i / 6}+ e^{\pi i / 6}\big)\delta_{3}
+ \big( e^{ 2 \pi i / 3}+ 1+ e^{ 4 \pi i / 3}\big)\delta_{4}
\\& \qquad
+ \big( i - i\big)\delta_{5}
+ \big( 1+ e^{ 2 \pi i / 3}+ e^{ 4 \pi i / 3}\big)\delta_{6}
\\& \qquad
+ \big( e^{ 7 \pi i / 6}+ e^{ 5 \pi i / 6}+ e^{ 11 \pi i / 6}+ e^{\pi i / 6}\big)\delta_{7}
+ \big( e^{ 4 \pi i / 3}+ 1+ e^{ 2 \pi i / 3}\big)\delta_{8}
\\& \qquad + \big( e^{ 7 \pi i / 6}+ e^{\pi i / 6}+ e^{ 11 \pi i / 6}+ e^{ 5 \pi i}\big)\delta_{9}
\\&=6\delta_0.
\qedhere
\end{align*}
\end{proof}
\begin{prop}\label{prop6in14}
The set $\{ 0,1,2,3,4,7 \}$ is extreme in \Zp{14}{}.
\end{prop}
\begin{proof}
Let
$ \mu =\delta_{0} + e^{ 10\pi i / 12}\delta_{1} + e^{ 8\pi i / 12}\delta_{2} + e^{ 10\pi i / 12}\delta_{3} + \delta_{4} + e^{ 2\pi i / 12}\delta_{7}
.$
Then
\begin{align*}
\mu * \tilde \mu &= 6\delta_{0} + \big( i + i - i- i\big)\delta_{1}
+ \big( e^{\pi i / 1}+ 1+ e^{\pi i / 1}+ 1\big)\delta_{2}
\\& \qquad + \big( -i + i + i - i\big)\delta_{3}
+ \big( 1+ e^{\pi i / 1}+ 1+ e^{\pi i / 1}\big)\delta_{4}
\\& \qquad + \big( -i + i + i - i\big)\delta_{5}
+ \big( e^{\pi i / 1}+ 1+ e^{\pi i / 1}+ 1\big)\delta_{6}
\\& \qquad + \big( i - i+ i + i - i+ i - i- i\big)\delta_{7}
\\& \qquad + \big( e^{\pi i / 1}+ 1+ e^{\pi i / 1}+ 1\big)\delta_{8}
+ \big( -i + i + i - i\big)\delta_{9}
\\&\qquad + \big( e^{\pi i / 1}+ 1+ e^{\pi i / 1}+ 1\big)\delta_{10}
+ \big( -i - i+ i + i \big)\delta_{11}
\\& \qquad
+ \big( e^{\pi i / 1}+ 1+ e^{\pi i / 1}+ 1\big)\delta_{12}
+ \big( -i - i+ i + e^{\pi i / 1}\big)\delta_{13}
\\& = 6\delta_0.\qedhere
\end{align*}
\end{proof}
\subsection{Sets with 7 elements}\label{subsec7elts}
Seven-element exceptional extreme sets are somewhat more plentiful; we have
found one in \Zp{12}{}, two non-equivalent ones in \Zp{16}{} (and, as expected, the 7 element subset of the 8-element subgroup)
and one in \Zp{19}{}.
\begin{prop}\label{prop7in8} The 7-element set
$\{0,1,2,3,4,5,6\}$ is extreme in \Zp8{}
\end{prop}
\begin{proof}
Let $\mu = \delta_{0}+e^{4\pi i/3}\delta_{1}+e^{4\pi i/3}\delta_{2}+e^{2\pi i/3}\delta_{3}+e^{4\pi i/3}\delta_{4}+e^{4\pi i/3}\delta_{5}+\delta_{6}
$. Then
\begin{align*}
\mu * \tilde \mu & = \big(1+1+1+1+1+1+1\big)\delta_{0}
\\&\qquad
+\big(e^{4\pi i/3}+1+e^{4\pi i/3}+e^{2\pi i/3}+1+e^{2\pi i/3}\big)\delta_{1}
\\&\qquad
+\big(1+e^{4\pi i/3}+e^{4\pi i/3}+1+e^{2\pi i/3}+e^{2\pi i/3}\big)\delta_{2}
\\&\qquad
+\big(e^{2\pi i/3}+e^{4\pi i/3}+e^{2\pi i/3}+1+1+e^{4\pi i/3}\big)\delta_{3}
\\&\qquad
+\big(e^{2\pi i/3}+1+e^{4\pi i/3}+e^{4\pi i/3}+1+e^{2\pi i/3}\big)\delta_{4}
\\&\qquad
+\big(e^{4\pi i/3}+1+1+e^{2\pi i/3}+e^{4\pi i/3}+e^{2\pi i/3}\big)\delta_{5}
\\&\qquad
+\big(e^{2\pi i/3}+e^{2\pi i/3}+1+e^{4\pi i/3}+e^{4\pi i/3}+1\big)\delta_{6}
\\&\qquad
+\big(e^{2\pi i/3}+1+e^{2\pi i/3}+e^{4\pi i/3}+1+e^{4\pi i/3}\big)\delta_{7}\\&=7\delta_{(0 , 0}\qedhere
\end{align*}
\end{proof}
\begin{comment}
\begin{proof}
Let
$\nu=\delta_0+
e^{11\pi i/6}\delta_1+
e^{14\pi i/6} \delta_2
+e^{13\pi i/6}\delta_3
+e^{8\pi i/6}\delta_4
+e^{11\pi i/6}\delta_5
+e^{6\pi i/6}\delta_6.
$
Then
\begin{align*}
\nu*\widetilde\nu &= 7\delta_0 +
\big( e^{ 11 \pi i / 6}+ i + e^{ 11 \pi i / 6}+ e^{ 7 \pi i / 6}+ i + e^{ 7 \pi i / 6}\big)\delta_{1}
\\& \qquad + \big( e^{pi i}+ e^{\pi i / 3}+ e^{\pi i / 3}+ e^{pi i}+ e^{ 5 \pi i / 3}+ e^{ 5 \pi i / 3}\big)\delta_{2}
\\& \qquad + \big( e^{\pi i / 6}+ e^{ 5 \pi i / 6}+ e^{\pi i / 6}- i- i+ e^{ 5 \pi i / 6}\big)\delta_{3}
\\& \qquad + \big( e^{ 2 \pi i / 3}+ 1+ e^{ 4 \pi i / 3}+ e^{ 4 \pi i / 3}+ 1+ e^{ 2 \pi i / 3}\big)\delta_{4}
\\& \qquad + \big( e^{ 11 \pi i / 6}+ i + i + e^{ 7 \pi i / 6}+ e^{ 11 \pi i / 6}+ e^{ 7 \pi i / 6}\big)\delta_{5}
\\& \qquad + \big( e^{ 5 \pi i / 3}+ e^{ 5 \pi i / 3}+ e^{pi i}+ e^{\pi i / 3}+ e^{\pi i / 3}- 1 \big)\delta_{6}
\\& \qquad + \big( e^{\pi i / 6}- i+ e^{\pi i / 6}+ e^{ 5 \pi i / 6}- i+ e^{ 5 \pi i}\big)\delta_{7}
\\&=7\delta_0. \qedhere
\end{align*}
\end{comment}
\begin{comment}\footnote{\ An extreme measure on ${\0,\dots,5\}\subset\Zp8{}$
is $\mu = \delta_0 +e^{4\pi i/3}\delta_1+e^{4\pi i/3}\delta_2
+e^{2\pi i/3}\delta_3 + e^{4\pi i/3}\delta_4+e^{4\pi i/3}\delta_5+\delta_6$.
Indeed,
\begin{align*}
\mu * \tilde \mu = 7\delta_0 &+\prnb{e^{4\pi i/3} + e^{-4\pi/3}+1+e^{2\pi i/3}+e^{-2\pi i/3} +1}\delta_1 \\
&+\prnb{ 1+e^{-4\pi i/3}+e^{2\pi i/3} +1 + e^{-2\pi i/3}+e^{4\pi i/3} }\delta_2 \\
&+\prnb{e^{2\pi i/3} +e^{-4\pi i/3} + e^{-2\pi i/3} +1+e^{4\pi i/3} +1}\delta_3 \\
&+\prnb{ e^{4\pi i/3} + e^{4\pi i/3} +1 + e^{-4\pi i/3} + e^{-4\pi i/3} +1 }\delta_4 \\
&\cdots\\
&=7\delta_0.
\end{align*}
}
\end{proof}
\end{comment}
\begin{rem}
Extreme measures are not unique, even when they are required to have mass 1 at the identity.
Another extreme measure on $\{0,\dots,6\}\subset \Zp{8}{}$
is \[
\nu=\delta_0+
e^{11\pi i/6}\delta_1+
e^{14\pi i/6} \delta_2
+e^{13\pi i/6}\delta_3
+e^{8\pi i/6}\delta_4
+e^{11\pi i/6}\delta_5
+e^{6\pi i/6}\delta_6.
\]
We omit the proof that $\nu*\widetilde{\nu} =7\delta_0.$
\begin{comment}
Indeed,
\begin{align*}
\nu*\widetilde\nu &= 7\delta_0 +
\big( e^{ 11 \pi i / 6}+ i + e^{ 11 \pi i / 6}+ e^{ 7 \pi i / 6}+ i + e^{ 7 \pi i / 6}\big)\delta_{1}
\\& \qquad + \big( e^{pi i}+ e^{\pi i / 3}+ e^{\pi i / 3}+ e^{pi i}+ e^{ 5 \pi i / 3}+ e^{ 5 \pi i / 3}\big)\delta_{2}
\\& \qquad + \big( e^{\pi i / 6}+ e^{ 5 \pi i / 6}+ e^{\pi i / 6}- i- i+ e^{ 5 \pi i / 6}\big)\delta_{3}
\\& \qquad + \big( e^{ 2 \pi i / 3}+ 1+ e^{ 4 \pi i / 3}+ e^{ 4 \pi i / 3}+ 1+ e^{ 2 \pi i / 3}\big)\delta_{4}
\\& \qquad + \big( e^{ 11 \pi i / 6}+ i + i + e^{ 7 \pi i / 6}+ e^{ 11 \pi i / 6}+ e^{ 7 \pi i / 6}\big)\delta_{5}
\\& \qquad + \big( e^{ 5 \pi i / 3}+ e^{ 5 \pi i / 3}+ e^{pi i}+ e^{\pi i / 3}+ e^{\pi i / 3}- 1 \big)\delta_{6}
\\& \qquad + \big( e^{\pi i / 6}- i+ e^{\pi i / 6}+ e^{ 5 \pi i / 6}- i+ e^{ 5 \pi i}\big)\delta_{7}
\\&=7\delta_0. \qedhere
\end{align*}
\end{comment}
\end{rem}
\begin{prop}\label{prop7in12} The 7-element set
$\{0,1,2,5,6,8,9\}$ is extreme in \Zp{12}{}.
\end{prop}
\begin{proof} Let $\mu =\delta_{0} + e^{ 7\pi i / 12}\delta_{1} + \delta_{3} + e^{ 5\pi i / 6}\delta_{4} + i \delta_{6} + e^{ 7\pi i / 12}\delta_{7} + i \delta_{9} + e^{ 5\pi i / 6}\delta_{10}
$.
Then
\begin{align*} \mu*\tilde \mu &=7\delta_{0} + \big( e^{ 7 \pi i / 12}+ e^{ 5 \pi i / 6}+ e^{\pi i / 12}+ e^{\pi i / 3}\big)\delta_{1} \\& \qquad + \big( e^{ 7 \pi i / 6}+ e^{ 17 \pi i / 12}+ e^{ 5 \pi i / 3}+ e^{ 23 \pi i / 12}\big)\delta_{2} \\& \qquad + \big( -i + e^{ 7 \pi i / 4}+ 1+ e^{\pi i / 4}+ i + e^{ 7 \pi i / 4}+ 1+ e^{\pi i / 4}\big)\delta_{3} \\& \qquad + \big( e^{\pi i / 12}+ e^{ 5 \pi i / 6}+ e^{ 7 \pi i / 12}+ e^{\pi i / 3}\big)\delta_{4} \\& \qquad + \big( e^{ 17 \pi i / 12}+ e^{ 7 \pi i / 6}+ e^{ 23 \pi i / 12}+ e^{ 5 \pi i / 3}\big)\delta_{5} \\& \qquad + \big( -i + 1- i+ 1+ i + 1+ i + 1\big)\delta_{6} \\& \qquad + \big( e^{\pi i / 12}+ e^{\pi i / 3}+ e^{ 7 \pi i / 12}+ e^{ 5 \pi i / 6}\big)\delta_{7} \\& \qquad + \big( e^{ 7 \pi i / 6}+ e^{ 17 \pi i / 12}+ e^{ 5 \pi i / 3}+ e^{ 23 \pi i / 12}\big)\delta_{8} \\& \qquad + \big( 1+ e^{ 7 \pi i / 4}- i+ e^{\pi i / 4}+ 1+ e^{ 7 \pi i / 4}+ i + e^{\pi i / 4}\big)\delta_{9} \\& \qquad + \big( e^{ 7 \pi i / 12}+ e^{\pi i / 3}+ e^{\pi i / 12}+ e^{ 5 \pi i / 6}\big)\delta_{10} \\& \qquad + \big( e^{ 17 \pi i / 12}+ e^{ 7 \pi i / 6}+ e^{ 23 \pi i / 12}+ e^{ 5 \pi i}\big)\delta_{11}
\\&=7\delta_0.
\end{align*}
Each sum in the parenthese above is zero.
\begin{comment}\footnote{ Another extreme measure is: $ \mu =\delta_{0} + e^{ 17\pi i / 12}\delta_{1} + i \delta_{2} + e^{ 13\pi i / 12}\delta_{5} + e^{ 3\pi i / 2}\delta_{6} + -\delta_{8} + e^{ 13\pi i / 12}\delta_{9}
.$
Then
\begin{align*}
\mu*\tilde \mu =7\delta_{0}& + \big( e^{ 17 \pi i / 12}+ e^{ 13 \pi i / 12}+ e^{ 5 \pi i / 12}+ e^{\pi i / 12}\big)\delta_{1} + \big( i - i\big)\delta_{2}
\\& + \big( e^{ 11 \pi i / 12}+ e^{ 7 \pi i / 12}+ e^{ 23 \pi i / 12}+ e^{ 19 \pi i / 12}\big)\delta_{3} + \big( e^{pi i}+ e^{\pi i / 3}+ e^{ 5 \pi i / 3}+ e^{pi i}+ 1\big)\delta_{4}
\\& + \big( e^{ 5 \pi i / 12}+ e^{ 17 \pi i / 12}+ e^{ 13 \pi i / 12}+ e^{\pi i / 12}\big)\delta_{5} + \big( i - i- i+ i \big)\delta_{6}
\\& + \big( e^{ 11 \pi i / 12}+ e^{ 23 \pi i / 12}+ e^{ 19 \pi i / 12}+ e^{ 7 \pi i / 12}\big)\delta_{7} + \big( e^{\pi i / 3}+ e^{pi i}+ 1+ e^{pi i}+ e^{ 5 \pi i / 3}\big)\delta_{8}
\\& + \big( e^{ 17 \pi i / 12}+ e^{\pi i / 12}+ e^{ 5 \pi i / 12}+ e^{ 13 \pi i / 12}\big)\delta_{9} + \big( -i + i \big)\delta_{10}
\\& + \big( e^{ 7 \pi i / 12}+ e^{ 11 \pi i / 12}+ e^{ 19 \pi i / 12}+ e^{ 23 \pi i}\big)\delta_{11}\\&
= 7\delta_0.
\end{align*}
}
\end{comment}
\qedhere
\end{proof}
\begin{prop}\label{prop7in16} Each of 7-element sets
\begin{enumerate}
\item
$\{0,2,4,6,8,10,12\}$ and
\item $\{0,1,2,4,5,7,11\}$
\end{enumerate}
is extreme in \Zp{16}{}.
They are not equivalent.
\end{prop}
\begin{proof}
The sets are not equivalent because the automorphisms
(multiplication by odd integers) preserve the parity of
elements and translation switches or leaves fixed the
parity. Since the first set has both odd and even elements
so will every set equivalent to it. (We will use this argument
again in \propref{prop8in16}.)
As for extremality, first note that $\{0,2,4,6,8,10,12\}$ is extreme because it is a 7 element subset of an 8 element coset
and so extreme by \propref{prop7in8}. For the other set, set
\[\mu =
\delta_{0} -\delta_{1} + e^{ 4\pi i / 3}\delta_{2} + \delta_{4} + e^{ 5\pi i / 3}\delta_{5} + e^{ 5\pi i / 3}\delta_{7} + -\delta_{11}.
\]
Then
\begin{align*} \mu*\tilde \mu = 7\delta_{0} & + \big( -1 + e^{\pi i / 3}+ e^{ 5 \pi i / 3}\big)\delta_{1} + \big( e^{ 4 \pi i / 3}+ e^{ 2 \pi i / 3}+ 1\big)\delta_{2} \\& + \big( -1 + e^{\pi i / 3}+ e^{ 5 \pi i / 3}\big)\delta_{3} + \big( 1+ e^{ 2 \pi i / 3}+ e^{ 4 \pi i / 3}\big)\delta_{4} \\& + \big( -1 + e^{ 5 \pi i / 3}+ e^{\pi i / 3}\big)\delta_{5} + \big( 1+ e^{ 2 \pi i / 3}+ e^{ 4 \pi i / 3}\big)\delta_{6} \\& + \big( e^{\pi i / 3}+ e^{ 5 \pi i / 3}- 1 \big)\delta_{7} + \big( e^{\pi i / 3}- 1 + e^{ 5 \pi i / 3}\big)\delta_{9} \\& + \big( e^{ 4 \pi i / 3}+ e^{ 2 \pi i / 3}+ 1\big)\delta_{10} + \big( e^{\pi i / 3}+ e^{ 5 \pi i / 3}- 1 \big)\delta_{11} \\& + \big( 1+ e^{ 4 \pi i / 3}+ e^{ 2 \pi i / 3}\big)\delta_{12} + \big( -1 + e^{ 5 \pi i / 3}+ e^{\pi i / 3}\big)\delta_{13} \\& + \big( e^{ 2 \pi i / 3}+ e^{ 4 \pi i / 3}+ 1\big)\delta_{14} + \big( -1 + e^{ 5 \pi i / 3}- 1 \big)\delta_{15} \\&=7\delta_0.
\qedhere
\end{align*}
\end{proof}
\begin{prop}\label{prop7in19} The 7-element set
$\{0,1,2,5,12,13,15\}$ is extreme in \Zp{19}{}.
\end{prop}
\begin{proof}
Let $\mu =\delta_{0} + e^{ 4\pi i / 3}\delta_{1} + \delta_{2} + \delta_{5} + e^{ 4\pi i / 3}\delta_{12} + e^{ 4\pi i / 3}\delta_{13} + e^{\pi i / 3}\delta_{15}.$
Then
\begin{align*} \mu*\tilde \mu &=
7\delta_{0} + \big( e^{ 4 \pi i / 3}+ e^{ 2 \pi i / 3}+ 1\big)\delta_{1}
+ \big( 1+ e^{\pi i }\big)\delta_{2}
\\& \qquad + \big( 1+ e^{\pi i }\big)\delta_{3}
+ \big( e^{ 5 \pi i / 3}+ e^{ 2 \pi i / 3}\big)\delta_{4} \\& \qquad + \big( e^{\pi i }+ 1\big)\delta_{5}
+ \big( e^{ 2 \pi i / 3}+ e^{ 5 \pi i / 3}\big)\delta_{6} \\& \qquad + \big( e^{ 2 \pi i / 3}+ 1+ e^{ 4 \pi i / 3}\big)\delta_{7}
+ \big( 1+ e^{ 2 \pi i / 3}+ e^{ 4 \pi i / 3}\big)\delta_{8} \\& \qquad + \big( e^{ 2 \pi i / 3}+ e^{ 5 \pi i / 3}\big)\delta_{9}
+ \big( e^{ 4 \pi i / 3}+ e^{\pi i / 3}\big)\delta_{10} \\& \qquad + \big( e^{ 2 \pi i / 3}+ 1+ e^{ 4 \pi i / 3}\big)\delta_{11}
+ \big( e^{ 2 \pi i / 3}+ e^{ 4 \pi i / 3}+ 1\big)\delta_{12} \\& \qquad + \big( e^{ 4 \pi i / 3}+ e^{\pi i / 3}\big)\delta_{13}
+ \big( 1+ e^{\pi i }\big)\delta_{14} \\& \qquad + \big( e^{ 4 \pi i / 3}+ e^{\pi i / 3}\big)\delta_{15}
+ \big( 1+ e^{\pi i }\big)\delta_{16} \\& \qquad + \big( 1+ e^{\pi i }\big)\delta_{17}
+ \big( e^{ 2 \pi i / 3}+ e^{ 4 \pi i / 3}+ 1\big)\delta_{18}
\\&= 7\delta_0. \qedhere
\end{align*}
\end{proof}
We have found no other 7-element extreme sets in cyclic groups of order below 20.
\subsection{Sets with 8 elements}
\begin{prop}\label{prop8inZ9}
The 8 element set $\{0, 1, 2, 3, 4, 5, 6, 7 \}$ is extreme in $\Zp{9}{}$. \newline \end{prop}
\begin{proof}[First proof]
Let $\mu = \delta_{0}+e^{10\pi i/7}\delta_{1}+e^{4\pi i/7}\delta_{2}+e^{2\pi i/7}\delta_{3}+e^{2\pi i/7}\delta_{4}+e^{4\pi i/7}\delta_{5}+e^{10\pi i/7}\delta_{6}+\delta_{7}
$. Then
\begin{align*}
\mu * \tilde \mu & = \big(1+1+1+1+1+1+1+1\big)\delta_{0}
\\&\qquad
+\big(e^{10\pi i/7}+e^{8\pi i/7}+e^{12\pi i/7}+1+e^{2\pi i/7}+e^{6\pi i/7}+e^{4\pi i/7}\big)\delta_{1}
\\&\qquad
+\big(1+e^{4\pi i/7}+e^{6\pi i/7}+e^{12\pi i/7}+e^{2\pi i/7}+e^{8\pi i/7}+e^{10\pi i/7}\big)\delta_{2}
\\&\qquad
+\big(e^{4\pi i/7}+e^{10\pi i/7}+e^{2\pi i/7}+e^{6\pi i/7}+1+e^{8\pi i/7}+e^{12\pi i/7}\big)\delta_{3}
\\&\qquad
+\big(e^{10\pi i/7}+1+e^{4\pi i/7}+e^{2\pi i/7}+e^{8\pi i/7}+e^{6\pi i/7}+e^{12\pi i/7}\big)\delta_{4}
\\&\qquad
+\big(e^{12\pi i/7}+e^{6\pi i/7}+e^{8\pi i/7}+e^{2\pi i/7}+e^{4\pi i/7}+1+e^{10\pi i/7}\big)\delta_{5}
\\&\qquad
+\big(e^{12\pi i/7}+e^{8\pi i/7}+1+e^{6\pi i/7}+e^{2\pi i/7}+e^{10\pi i/7}+e^{4\pi i/7}\big)\delta_{6}
\\&\qquad
+\big(e^{10\pi i/7}+e^{8\pi i/7}+e^{2\pi i/7}+e^{12\pi i/7}+e^{6\pi i/7}+e^{4\pi i/7}+1\big)\delta_{7}
\\&\qquad
+\big(e^{4\pi i/7}+e^{6\pi i/7}+e^{2\pi i/7}+1+e^{12\pi i/7}+e^{8\pi i/7}+e^{10\pi i/7}\big)\delta_{8}\\&=8\delta_{0 }\qedhere
\end{align*}
\end{proof}
\begin{proof}[Second proof] Let \begin{align*}\mu =
\delta_{0} + & e^{ 20\pi i / 21}\delta_{1} + e^{ 64\pi i / 21}\delta_{2} + e^{ 20\pi i / 7}\delta_{3} + e^{ 32\pi i / 21}\delta_{4} \\ &\qquad + e^{ 64\pi i / 21}\delta_{5} + e^{ 16\pi i / 7}\delta_{6} + e^{ 8\pi i / 3}\delta_{7}
.\end{align*}
Then
\begin{align*} \mu * \tilde\mu&= 8\delta_{0} + \big( e^{ 20 \pi i / 21}+ e^{ 2 \pi i / 21}+ e^{ 38 \pi i / 21}+ e^{ 2 \pi i / 3}+ e^{ 32 \pi i / 21}
\\& \qquad \qquad+ e^{ 26 \pi i / 21}+ e^{ 8 \pi i / 21}\big)\delta_{1}
\\& \qquad
+ \big( e^{ 4 \pi i / 3}+ e^{ 22 \pi i / 21}+ e^{ 40 \pi i / 21}+ e^{ 10 \pi i / 21}
\\& \qquad \qquad+ e^{ 4 \pi i / 21}+ e^{ 16 \pi i / 21}+ e^{ 34 \pi i / 21}\big)\delta_{2}
\\& \qquad
+ \big( e^{ 12 \pi i / 7}+ e^{ 2 \pi i / 7}+ e^{ 6 \pi i / 7}+ e^{ 4 \pi i / 7}
\\& \qquad \qquad+ 1+ e^{ 10 \pi i / 7}+ e^{ 8 \pi i / 7}\big)\delta_{3}
\\& \qquad
+ \big( e^{ 20 \pi i / 21}+ e^{ 2 \pi i / 3}+ e^{ 8 \pi i / 21}+ e^{ 32 \pi i / 21}
\\& \qquad \qquad+ e^{ 2 \pi i / 21}+ e^{ 26 \pi i / 21}+ e^{ 38 \pi i / 21}\big)\delta_{4}
\\& \qquad
+ \big( e^{ 10 \pi i / 21}+ e^{ 40 \pi i / 21}+ e^{ 16 \pi i / 21}+ e^{ 4 \pi i / 21}
\\& \qquad \qquad+ e^{ 22 \pi i / 21}+ e^{ 4 \pi i / 3}+ e^{ 34 \pi i / 21}\big)\delta_{5}
\\& \qquad
+ \big( e^{ 8 \pi i / 7}+ e^{ 10 \pi i / 7}+ 1+ e^{ 4 \pi i / 7}+ e^{ 6 \pi i / 7}+ e^{ 2 \pi i / 7}+ e^{ 12 \pi i / 7}\big)\delta_{6}
\\& \qquad
+ \big( e^{ 20 \pi i / 21}+ e^{ 2 \pi i / 21}+ e^{ 32 \pi i / 21}+ e^{ 38 \pi i / 21}
\\& \qquad \qquad+ e^{ 26 \pi i / 21}+ e^{ 8 \pi i / 21}+ e^{ 2 \pi i / 3}\big)\delta_{7}
\\& \qquad + \big( e^{ 22 \pi i / 21}+ e^{ 40 \pi i / 21}+ e^{ 4 \pi i / 21}
\\& \qquad \qquad+ e^{ 4 \pi i / 3}+ e^{ 10 \pi i / 21}+ e^{ 16 \pi i / 21}+ e^{ 34 \pi i}\big)\delta_{8}
\\&= 9\delta_0.
\qedhere
\end{align*}
\end{proof}
\begin{prop}\label{prop8in10} The 8-element set
$\{0,1,2,3,5,6,7,8\}$ is extreme in \Zp{10}{}.
\end{prop}
\begin{proof}[Computational proof of \propref{prop8in10}]
Let $\mu=\delta_{0} + e^{ 7\pi i / 6}\delta_{1} + e^{ 2\pi i / 3}\delta_{2} + i \delta_{3} + i \delta_{5} + e^{ 2\pi i / 3}\delta_{6} + e^{ 7\pi i / 6}\delta_{7} + \delta_{8} $.
Then
\begin{align*}
\mu*\tilde\mu &= 8\delta_{0} + \big( e^{ 7 \pi i / 6}- i+ e^{ 11 \pi i / 6}+ e^{\pi i / 6}+ i + e^{ 5 \pi i / 6}\big)\delta_{1}
\\& \qquad + \big( 1+ e^{ 2 \pi i / 3}+ e^{ 4 \pi i / 3}+ 1+ e^{ 2 \pi i / 3}+ e^{ 4 \pi i / 3}\big)\delta_{2}
\\& \qquad + \big( e^{ 5 \pi i / 6}+ e^{ 7 \pi i / 6}+ i + e^{ 11 \pi i / 6}+ e^{\pi i / 6}- i\big)\delta_{3}
\\& \qquad + \big( e^{ 4 \pi i / 3}+ 1+ e^{ 2 \pi i / 3}+ e^{ 4 \pi i / 3}+ 1+ e^{ 2 \pi i / 3}\big)\delta_{4}
\\& \qquad + \big( -i + i - i+ i + i - i+ i - i\big)\delta_{5}
\\& \qquad + \big( e^{ 2 \pi i / 3}+ 1+ e^{ 4 \pi i / 3}+ e^{ 2 \pi i / 3}+ 1+ e^{ 4 \pi i / 3}\big)\delta_{6}
\\& \qquad + \big( -i + e^{\pi i / 6}+ e^{ 11 \pi i / 6}+ i + e^{ 7 \pi i / 6}+ e^{ 5 \pi i / 6}\big)\delta_{7}
\\& \qquad + \big( e^{ 4 \pi i / 3}+ e^{ 2 \pi i / 3}+ 1+ e^{ 4 \pi i / 3}+ e^{ 2 \pi i / 3}+ 1\big)\delta_{8}
\\& \qquad + \big( e^{ 5 \pi i / 6}+ i + e^{\pi i / 6}+ e^{ 11 \pi i / 6}- i+ e^{ 7 \pi i }\big)\delta_{9}
\\&=9\delta_0.\qedhere
\end{align*}
\end{proof}
\begin{proof}[Direct proof of \propref{prop8in10}] The set is the sum of a 4-element set in the 5-element
subgroup $\{0,2,4,6,8\}$ of \Zp{10}{} and the subgroup $\{0,5\}$:
$\{0,1,2,3,5,6,7,8\} = \{0,2,6,8\}+\{0,5\}=\{0,2,6,8,5,7,11,13\} = \{0,2,6,8,5,7,1,3\}$ $\mod 10$.
\end{proof}
Turning to \Zp{12}{}, we observe that an 8-element set there could be a sum of a 4-element set and a 2-element set.
However, we cannot get an extreme set that way, though there is an 8-element extreme set in \Zp{12}{}, as the
following Lemma and Proposition show.
\begin{lem}\label{lemsumsinZtwlv}
\Zp{12}{} has no 8-element set that is a sum of a four-element extreme set and a coset.
\end{lem}
\begin{proof}
Suppose $E= A+B$ has 8 elements, where $A$ has 4 elements and $B$ has two. We may assume $0\in A$ and $B=\{0,6\}$.
Clearly $A$ cannot contain 6. \refitem{remitOneRepGen} and calculation (both by hand and machine) show that if a 4-element set in \Zp{12}{}
lacks 6, it is not
extreme.\footnote{\,
The only 4-element extreme sets of \Zp{12}{} (up to equivalence)
are $\{0,1,6,7\},$ $\{0, 2,6,8\}$ and $\{0,3,6,9\}$, all containing 6. Several other four-element sets pass the test of
\refitem{remitOneRepGen} but are not extreme.
\end{proof}
\begin{prop}\label{prop8in12} The 8-element set
$\{0,1,3,4,6,7,9,10\}$
is extreme in \Zp{12}{} and is not the sum of two extreme sets in spite of being the sum $ \{0,1,3,4\} +\{0,6\}$
and also the sum $\{0,3,6,9\}+\{0,1\}$.
\end{prop}
\begin{proof}Let $\mu = \delta_{0} + e^{ 7\pi i / 6}\delta_{1} + \delta_{3} + e^{ 5\pi i / 3}\delta_{4} + e^{\pi i }\delta_{6} + e^{ 7\pi i / 6}\delta_{7} + e^{\pi i }\delta_{9} + e^{ 5\pi i / 3}\delta_{10}$.
Indeed, each of the sums in parentheses below is a net zero:
\begin{align*} \mu*\tilde \mu = 8\delta_{0} & + \big( e^{ 7 \pi i / 6}+ e^{ 5 \pi i / 3}+ e^{\pi i / 6}+ e^{ 2 \pi i / 3}\big)\delta_{1}
\\&
+ \big( e^{\pi i / 3}+ e^{ 5 \pi i / 6}
+ e^{ 4 \pi i / 3}+ e^{ 11 \pi i / 6}\big)\delta_{2}
\\
& + \big( e^{\pi i }- i+ 1+ i + e^{\pi i }- i+ 1+ i \big)\delta_{3}
\\
& + \big( e^{\pi i / 6}+ e^{ 5 \pi i / 3}+ e^{ 7 \pi i / 6}+ e^{ 2 \pi i / 3}\big)\delta_{4}
\\&
+ \big( e^{ 5 \pi i / 6}+ e^{\pi i / 3}+ e^{ 11 \pi i / 6}+ e^{ 4 \pi i / 3}\big)\delta_{5}
\\
& + \big( e^{\pi i }+ 1+ e^{\pi i }+ 1+ e^{\pi i }+ 1+ e^{\pi i }+ 1\big)\delta_{6}
\\&
+ \big( e^{\pi i / 6}+ e^{ 2 \pi i / 3}+ e^{ 7 \pi i / 6}+ e^{ 5 \pi i / 3}\big)\delta_{7}
\\
& + \big( e^{\pi i / 3}+ e^{ 5 \pi i / 6}+ e^{ 4 \pi i / 3}+ e^{ 11 \pi i / 6}\big)\delta_{8}
\\
& + \big( 1- i+ e^{\pi i }+ i + 1- i+ e^{\pi i }+ i \big)\delta_{9}
\\&
+ \big( e^{ 7 \pi i / 6}+ e^{ 2 \pi i / 3}+ e^{\pi i / 6}+ e^{ 5 \pi i / 3}\big)\delta_{10}
\\&
+ \big( e^{ 5 \pi i / 6}+ e^{\pi i / 3}+ e^{ 11 \pi i / 6}+ e^{ 4 \pi i }\big)\delta_{11}
\\& = 8\delta_0.
\end{align*}
Now apply \lemref{lemsumsinZtwlv} and the fact that $\{0,1\}$ is not extreme in \Zp{12}{}\footnote{The difference
$\{0,1\}-\{0,1\}$ produces the terms $0-0$, 1-1, 0-1, 1-0 and so $\{0,1\} \subset \Zp{12}{} $ fails the test of \refitem{remitOneRepGen}.}
\end{proof}
\begin{prop}\label{prop8in16} Each of the 8-element sets
\begin{enumerate}
\item
$\{0, 1, 4, 5, 8, 9, 12, 13\}$
\item $\{0,2,4,6,8,10,12,14\}$
\end{enumerate}
is extreme in \Zp{16}{}. They are not equivalent.
\end{prop}
\begin{proof}
The sets are not equivalent because the autormorphisms
(multiplication by odd integers) preserve the parity of
elements and translation switches or leaves fixed the
parity. Since the first set has both odd and even elements,
so will every set equivalent to it.
The second set is extreme because it is a coset.
For the extremality of the first set, let $\mu = \delta_{0} + e^{ 22 \pi i / 12}\delta_{1} + e^{ 18 \pi i / 12}\delta_{4} + e^{ 22 \pi i / 12}\delta_{5} +\delta_{8} + e^{ 10 \pi i / 12}\delta_{9} + e^{ 18 \pi i / 12}\delta_{12} + e^{ 10 \pi i / 12}\delta_{13}
$. Then
\begin{align*}
\mu * \tilde \mu & = 8\delta_{0} + \big( e^{ 11 \pi i / 6}+ e^{\pi i / 3}+ e^{ 5 \pi i / 6}+ e^{ 4 \pi i / 3}\big)\delta_{1}
\\&\qquad
+ \big( e^{ 7 \pi i / 6}+ e^{ 5 \pi i / 3}+ e^{\pi i / 6}+ e^{ 2 \pi i / 3}\big)\delta_{3} \\&\qquad + \big( e^{\pi i / 2}+ e^{\pi i}+ e^{ 3 \pi i / 2}+ 1+ e^{\pi i / 2}+ e^{\pi i}+ e^{ 3 \pi i / 2}+ 1\big)\delta_{4} \\&\qquad + \big( e^{\pi i / 3}+ e^{ 11 \pi i / 6}+ e^{ 4 \pi i / 3}+ e^{ 5 \pi i / 6}\big)\delta_{5}
\\&\qquad
+ \big( e^{ 7 \pi i / 6}+ e^{ 2 \pi i / 3}+ e^{\pi i / 6}+ e^{ 5 \pi i / 3}\big)\delta_{7} \\&\qquad + \big( 1+ e^{\pi i}+ 1+ e^{\pi i}+ 1+ e^{\pi i}+ 1+ e^{\pi i}\big)\delta_{8} \\&\qquad + \big( e^{ 11 \pi i / 6}+ e^{\pi i / 3}+ e^{ 5 \pi i / 6}+ e^{ 4 \pi i / 3}\big)\delta_{9} \\&\qquad + \big( e^{\pi i / 6}+ e^{ 2 \pi i / 3}+ e^{ 7 \pi i / 6}+ e^{ 5 \pi i / 3}\big)\delta_{11} \\&\qquad + \big( e^{\pi i / 2}+ 1+ e^{ 3 \pi i / 2}+ e^{\pi i}+ e^{\pi i / 2}+ 1+ e^{ 3 \pi i / 2}+ e^{\pi i}\big)\delta_{12} \\&\qquad + \big( e^{\pi i / 3}+ e^{ 11 \pi i / 6}+ e^{ 4 \pi i / 3}+ e^{ 5 \pi i / 6}\big)\delta_{13} \\&\qquad + \big( e^{\pi i / 6}+ e^{ 5 \pi i / 3}+ e^{ 7 \pi i / 6}+ e^{ 2 \pi i}\big)\delta_{15}
\\&=8\delta_0.\qedhere
\end{align*}
\end{proof}
\break
\subsection{Sets with 9 elements}
\begin{prop}\label{prop9in10}
$E=\{0,1,2,3,4,5,6,7,8\}$ is extreme in \Zp{10}{}.
\end{prop}
\begin{proof} Let $\mu =
\delta_{0} + e^{ 3\pi i / 2}\delta_{1} + \delta_{2} + i \delta_{3} + -\delta_{4} + i \delta_{5} + \delta_{6} + e^{ 3\pi i / 2}\delta_{7} + \delta_{8}
.$
Then
\begin{align*} \mu * \tilde\mu&= 9\delta_{0} + \big( -i + i + i + i - i- i- i+ i \big)\delta_{1} \\& \qquad + \big( 1+1-1- 1 +1-1- 1 + 1\big)\delta_{2} \\& \qquad + \big( i - i+ i - i+ i - i+ i - i\big)\delta_{3} \\& \qquad + \big( 1+ 1+1-1- 1 +1-1- 1 \big)\delta_{4} \\& \qquad + \big( -i - i+ i + i + i + i - i- i\big)\delta_{5} \\& \qquad + \big( -1 - 1 +1-1- 1 + 1+ 1+ 1\big)\delta_{6} \\& \qquad + \big( -i + i - i+ i - i+ i - i+ i \big)\delta_{7} \\& \qquad + \big(1-1- 1 +1-1- 1 + 1+ 1\big)\delta_{8} \\& \qquad + \big( i - i- i- i+ i + i + i + e^{ 3 \pi i}\big)\delta_{9}
\\&= 9\delta_0.
\end{align*}
\end{proof}
\begin{prop}\label{prop9in12}
$\{0,1,2,3,4,5,6,7,8\}$ and $\{0,1,2,4,5,6,8,9,10\}$ are extreme in \Zp{12}{}. They are non-equivalent.
\end{prop}
\begin{proof}
There are two ways to prove non-equivalence. First, by a tedious calculation (which we delegated to a computer). The second is to show that one of the sets is the sum
of a coset with a three-element set and the other is not, which we do in the next paragraph.
The only 3-element subgroup in \Zp{12}{} is $\{0,4,8\}$. If $\{0,4,8\}+\{a\}
\subset\{0,1,2,3,4,5,6,7,8\}$, then $a\ne\, 1,2,3$,
since $8+1=9,8+2=10, 8+3=11$ are
not in $E.$ Similarly,
$a\ne 5,6,7,9,10,11$
since none of those elements is in $E$. Thus, $a\in \{0,4,8\}$ and the first set is seen not be a sum.
Of course, $\{0,4,8\}+\{0,1,2\}=\{0,1,2,4,5,6,8,9,10\}$ and both final conclusions follow.
(1). For the extremality of the first set, let $\mu =
\delta_{0} + e^{ 23\pi i / 12}\delta_{1} + e^{ 3\pi i / 2}\delta_{2} + e^{ 17\pi i / 12}\delta_{3} + \delta_{4} + e^{\pi i / 4}\delta_{5} + e^{ 7\pi i / 6}\delta_{6} + e^{ 5\pi i / 12}\delta_{7} + e^{ 4\pi i / 3}\delta_{8}
.$
Then
\begin{align*}\mu*\tilde\mu &= 9\delta_0+ \big( e^{ 23 \pi i / 12}+ e^{ 19 \pi i / 12}+ e^{ 23 \pi i / 12}+ e^{ 7 \pi i / 12}+ e^{\pi i / 4}
\\&\quad\qquad+ e^{ 11 \pi i / 12}+ e^{ 5 \pi i / 4}+ e^{ 11 \pi i / 12}\big)\delta_{1}
\\&\qquad + \big( -i - i+ i + e^{ 5 \pi i / 6}+ e^{ 7 \pi i / 6}+ e^{\pi i / 6}+ e^{\pi i / 6}\big)\delta_{2}
\\&\qquad + \big( e^{ 17 \pi i / 12}+ e^{\pi i / 12}+ e^{ 3 \pi i / 4}+ e^{ 7 \pi i / 4}+ e^{ 5 \pi i / 12}+ e^{ 13 \pi i / 12}\big)\delta_{3}
\\&\qquad + \big( e^{ 2 \pi i / 3}+ 1+ e^{\pi i / 3}+ e^{ 5 \pi i / 3}+ e^{ 1 \pi i}+ e^{ 4 \pi i / 3}\big)\delta_{4}
\\&\qquad + \big( e^{ 19 \pi i / 12}+ e^{ 7 \pi i / 12}+ e^{\pi i / 4}+ e^{ 5 \pi i / 4}+ e^{ 11 \pi i / 12}+ e^{ 23 \pi i / 12}\big)\delta_{5}
\\&\qquad + \big( e^{ 5 \pi i / 6}- i+ e^{\pi i / 6}+ e^{ 7 \pi i / 6}+ i + e^{ 11 \pi i / 6}\big)\delta_{6}
\\&\qquad + \big( e^{ 7 \pi i / 4}+ e^{ 3 \pi i / 4}+ e^{ 13 \pi i / 12}+ e^{\pi i / 12}+ e^{ 5 \pi i / 12}+ e^{ 17 \pi i / 12}\big)\delta_{7}
\\&\qquad + \big( 1+ e^{ 5 \pi i / 3}+ e^{\pi i / 3}+ e^{ 1 \pi i}+ e^{ 2 \pi i / 3}+ e^{ 4 \pi i / 3}\big)\delta_{8}
\\&\qquad + \big( e^{ 7 \pi i / 12}+ e^{ 23 \pi i / 12}+ e^{ 5 \pi i / 4}+ e^{\pi i / 4}+ e^{ 19 \pi i / 12}+ e^{ 11 \pi i / 12}\big)\delta_{9}
\\&\qquad + \big( i + i - i+ e^{ 7 \pi i / 6}+ e^{ 5 \pi i / 6}+ e^{ 11 \pi i / 6}+ e^{ 11 \pi i / 6}\big)\delta_{10}
\\&\qquad + \big( e^{\pi i / 12}+ e^{ 5 \pi i / 12}+ e^{\pi i / 12}+ e^{ 17 \pi i / 12}+ e^{ 7 \pi i / 4}
\\&\quad\qquad+ e^{ 13 \pi i / 12}+ e^{ 3 \pi i / 4}+ e^{ 13 \pi i}\big)\delta_{11}
\\&=9\delta_0.
\end{align*}
(2). Let $
\mu = \delta_{0} + e^{ 10\pi i / 6}\delta_{1} + e^{ 10\pi i / 6}\delta_{2} + e^{ 8\pi i / 6}\delta_{4} + e^{ 102\pi i / 6}\delta_{5} + e^{ 2\pi i / 6}\delta_{6} + e^{ 4\pi i / 6}\delta_{8} + e^{ 10\pi i / 6}\delta_{9} + e^{ 6\pi i / 6}\delta_{10}
.$
Then
\begin{align*}\mu * \tilde\mu&= 9\delta_{0} + \big( e^{ 5 \pi i / 3}+ 1+ e^{ 5 \pi i / 3}+ e^{ 4 \pi i / 3}+ e^{\pi i}+ e^{ 4 \pi i / 3}\big)\delta_{1} \\&\qquad + \big( e^{\pi i}+ e^{ 5 \pi i / 3}+ e^{ 5 \pi i / 3}+ e^{\pi i}+ e^{\pi i / 3}+ e^{\pi i / 3}\big)\delta_{2}
\\&\qquad + \big( e^{\pi i / 3}+ e^{ 2 \pi i / 3}+ e^{ 5 \pi i / 3}+ e^{ 4 \pi i / 3}+ e^{ 5 \pi i / 3}+ e^{ 4 \pi i / 3}\big)\delta_{3} \\&\qquad + \big( e^{ 4 \pi i / 3}+ 1+ e^{ 2 \pi i / 3}+ e^{ 4 \pi i / 3}+ e^{ 4 \pi i / 3}+ e^{ 2 \pi i / 3}+ e^{ 4 \pi i / 3}
\\& \quad\qquad + e^{ 2 \pi i / 3}+ e^{ 2 \pi i / 3}\big)\delta_{4}
\\&\qquad + \big( e^{\pi i}+ 1+ e^{\pi i}+ e^{ 2 \pi i / 3}+ e^{\pi i / 3}+ 1\big)\delta_{5} \\&\qquad + \big( e^{ 5 \pi i / 3}+ e^{\pi i}+ e^{\pi i / 3}+ e^{\pi i / 3}+ e^{\pi i}+ e^{ 5 \pi i / 3}\big)\delta_{6}
\\&\qquad + \big( e^{\pi i}+ e^{ 4 \pi i / 3}+ e^{ 5 \pi i / 3}+ 1+ e^{\pi i}+ 1\big)\delta_{7} \\&\qquad + \big( e^{ 2 \pi i / 3}+ e^{ 2 \pi i / 3}+ e^{ 4 \pi i / 3}+ e^{ 2 \pi i / 3}+ e^{ 4 \pi i / 3}\\& \quad\qquad + e^{ 4 \pi i / 3}+ e^{ 2 \pi i / 3}+ 1+ e^{ 4 \pi i / 3}\big)\delta_{8}
\end{align*}
\begin{align*}
\phantom{\mu * \tilde\mu}&\qquad + \big( e^{\pi i / 3}+ e^{ 2 \pi i / 3}+ e^{\pi i / 3}+ e^{ 2 \pi i / 3}+ e^{ 5 \pi i / 3}+ e^{ 4 \pi i / 3}\big)\delta_{9} \\&\qquad + \big( e^{\pi i / 3}+ e^{\pi i / 3}+ e^{\pi i}+ e^{ 5 \pi i / 3}+ e^{ 5 \pi i / 3}+ e^{\pi i}\big)\delta_{10} \\&\qquad + \big( e^{\pi i / 3}+ 1+ e^{\pi i / 3}+ e^{ 2 \pi i / 3}+ e^{\pi i}+ e^{ 2 \pi i}\big)\delta_{11}
\\&=9\delta_0.
\end{align*}
\end{proof}
\begin{prop}\label{prop9in13} The 9-element set
$\{0, 1, 2, 3, 4, 5, 7, 9, 10\}$ is extreme in \Zp{13}{}.
\end{prop}
\begin{proof}
Let $\mu = \delta_{0} +\delta_{1} - \delta_{2} - \delta_{3} +\delta_{4} - \delta_{5} - \delta_{7} - \delta_{9} - \delta_{10}
$. Then
\begin{align*}
\mu * \tilde \mu & = 9\delta_{0} + \big(1-1+1-1- 1 + 1\big)\delta_{1}
+ \big( -1 - 1 - 1 + 1+ 1+ 1\big)\delta_{2} \\&\qquad + \big( -1 - 1 + 1+1-1+ 1\big)\delta_{3}
+ \big( -1 - 1 +1-1+ 1+ 1\big)\delta_{4} \\&\qquad + \big( -1 +1-1+1-1+ 1\big)\delta_{5}
+ \big( -1 + 1+1-1+1-1\big)\delta_{6} \\&\qquad + \big( -1 +1-1- 1 + 1+ 1\big)\delta_{7}
+ \big( -1 +1-1+1-1+ 1\big)\delta_{8} \\&\qquad + \big(1-1+ 1+1-1- 1 \big)\delta_{9}
+ \big( -1 + 1+1-1+1-1\big)\delta_{10} \\&\qquad + \big( -1 - 1 - 1 + 1+ 1+ 1\big)\delta_{11} \
+ \big(1-1+1-1- 1 + 1\big)\delta_{12}
\\&=9\delta_0.\qedhere
\end{align*}
\end{proof}
\begin{comment}
\begin{prop}
$\{0,1,3,4,6,7,9,10\}$
is extreme in \Zp{12}{}.
\end{prop}
\begin{proof}
Let $\mu =
\delta_{0} + e^{ 7\pi i / 6}\delta_{1} + \delta_{3} + e^{ 5\pi i / 3}\delta_{4} + e^{\pi i }\delta_{6} + e^{ 7\pi i / 6}\delta_{7} + e^{\pi i }\delta_{9} + e^{ 5\pi i / 3}\delta_{10}.
$
Then
\begin{align*} \mu*\tilde\mu =8\delta_{0}& + \big( e^{ 7 \pi i / 6}+ e^{ 5 \pi i / 3}+ e^{\pi i / 6}+ e^{ 2 \pi i / 3}\big)\delta_{1}
+ \big( e^{\pi i / 3}+ e^{ 5 \pi i / 6}+ e^{ 4 \pi i / 3}+ e^{ 11 \pi i / 6}\big)\delta_{2}
\\& + \big( e^{\pi i }- i+ 1+ i + e^{\pi i }- i+ 1+ i \big)\delta_{3}
\\& + \big( e^{\pi i / 6}+ e^{ 5 \pi i / 3}+ e^{ 7 \pi i / 6}+ e^{ 2 \pi i / 3}\big)\delta_{4}
+ \big( e^{ 5 \pi i / 6}+ e^{\pi i / 3}+ e^{ 11 \pi i / 6}+ e^{ 4 \pi i / 3}\big)\delta_{5}
\\& + \big( e^{\pi i }+ 1+ e^{\pi i }+ 1+ e^{\pi i }+ 1+ e^{\pi i }+ 1\big)\delta_{6}
\\& + \big( e^{\pi i / 6}+ e^{ 2 \pi i / 3}+ e^{ 7 \pi i / 6}+ e^{ 5 \pi i / 3}\big)\delta_{7}
+ \big( e^{\pi i / 3}+ e^{ 5 \pi i / 6}+ e^{ 4 \pi i / 3}+ e^{ 11 \pi i / 6}\big)\delta_{8}
\\& + \big( 1- i+ e^{\pi i }+ i + 1- i+ e^{\pi i }+ i \big)\delta_{9}
+ \big( e^{ 7 \pi i / 6}+ e^{ 2 \pi i / 3}+ e^{\pi i / 6}+ e^{ 5 \pi i / 3}\big)\delta_{10}
\\& + \big( e^{ 5 \pi i / 6}+ e^{\pi i / 3}+ e^{ 11 \pi i / 6}+ e^{ 4 \pi i }\big)\delta_{11} = 8\delta_0.
\qedhere
\end{align*}
\end{proof}
\end{comment}
\begin{comment
\begin{prop}
$\{0,1,2,4,7,8,9,11\}$ is extreme in \Zp{14}{}.
\end{prop}
\begin{proof}
Let $\mu =
\delta_{0} + i \delta_{1} + -\delta_{2} + -\delta_{4} + e^{ 3\pi i / 2}\delta_{7} + -\delta_{8} + i \delta_{9} + i \delta_{11}$. Then
\begin{align*} \mu * \tilde\mu=
8\delta_{0} & + \big( i + i - i- i\big)\delta_{1}
+ \big( -1 +1-1+ 1\big)\delta_{2}
\\& + \big( -i + i + i - i\big)\delta_{3}
+ \big(1-1+1-1\big)\delta_{4} \\& + \big( -i + i + i - i\big)\delta_{5}
+ \big( -1 +1-1+ 1\big)\delta_{6} \\& + \big( i - i+ i + i - i+ i - i- i\big)\delta_{7} \\& + \big( -1 +1-1+ 1\big)\delta_{8}
+ \big( -i + i + i - i\big)\delta_{9} \\& + \big( -1 +1-1+ 1\big)\delta_{10}
+ \big( -i - i+ i + i \big)\delta_{11} \\& + \big( -1 +1-1+ 1\big)\delta_{12}
+ \big( -i - i+ i - 1 \big)\delta_{13} \\& = 8\delta_0.
\qedhere
\end{align*}
\end{proof}
\end{comment}
\subsection{Sets with 10 elements}
The only 10-element extreme sets we have found are sums of 5-element cosets with 2-element cosets and 5-element seubsets of
6-element cosets with 2-element cosets.
\subsection{Sets with 11 elements}
\begin{prop}\label{prop11inZ12}
The 11-element set $\{0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 \}$ is extreme in $\Zp{12}{}$.
\newline \end{prop}
\begin{proof}
Let $\mu = \delta_{0}+e^{7\pi i/4}\delta_{1}-i\delta_{2}+e^{5\pi i/4}\delta_{3}+\delta_{4}+e^{7\pi i/4}\delta_{5}+i\delta_{6}+e^{\pi i/4}\delta_{7}-\delta_{8}+e^{7\pi i/4}\delta_{9}+i\delta_{10}
$. Then
\begin{align*}
\mu * \tilde \mu & = \big(1+1+1+1+1+1+1+1+1+1+1\big)\delta_{0}
\\&\qquad
+\big(e^{7\pi i/4}+e^{7\pi i/4}+e^{7\pi i/4}+e^{3\pi i/4}+e^{7\pi i/4}+e^{3\pi i/4}
\\&\qquad\qquad +e^{7\pi i/4}+e^{3\pi i/4}+e^{3\pi i/4}+e^{3\pi i/4}-i-i-i+i+
\\&\qquad\qquad
i+i+i+i-i-i+e^{\pi i/4}
+e^{5\pi i/4}+e^{5\pi i/4}+e^{\pi i/4}+e^{\pi i/4}+e^{5\pi i/4}
\\&\qquad\qquad
+e^{\pi i/4}+e^{5\pi i/4}+e^{5\pi i/4}
\\&\qquad\qquad
+e^{\pi i/4}-1+1-1+1+1-1-1-1+1+1
\\&\qquad\qquad
+e^{7\pi i/4}+e^{3\pi i/4}+e^{7\pi i/4}+e^{3\pi i/4}+e^{7\pi i/4}+e^{3\pi i/4}+e^{3\pi i/4}
\\&\qquad\qquad+e^{7\pi i/4}+e^{7\pi i/4}+e^{3\pi i/4}-i-i+i-i-i+i+i-i+i+i\big)\delta_{6}
\\&\qquad
+\big(e^{\pi i/4}+e^{5\pi i/4}+e^{5\pi i/4}+e^{\pi i/4}+e^{\pi i/4}+e^{5\pi i/4}
\\&\qquad\qquad
+e^{\pi i/4}+e^{5\pi i/4}+e^{\pi i/4}+e^{5\pi i/4}\big)\delta_{7}
\\&\qquad
+\big(1+1-1-1-1+1+1-1+1-1\big)\delta_{8}
\\&\qquad
+\big(e^{3\pi i/4}+e^{7\pi i/4}+e^{7\pi i/4}+e^{3\pi i/4}+e^{7\pi i/4}+e^{3\pi i/4}+e^{3\pi i/4}
\\&\qquad\qquad+e^{7\pi i/4}+e^{7\pi i/4}+e^{3\pi i/4}\big)\delta_{9}
\\&\qquad
+\big(i+i-i-i-i-i-i+i+i+i\big)\delta_{10}
\\&\qquad
+\big(e^{\pi i/4}+e^{\pi i/4}+e^{\pi i/4}+e^{5\pi i/4}+e^{\pi i/4}+e^{5\pi i/4}+e^{\pi i/4}
\\&\qquad\qquad+e^{5\pi i/4}+e^{5\pi i/4}+e^{5\pi i/4}\big)\delta_{11}\\&=11\delta_{0 }\qedhere
\end{align*}
\end{proof}
\subsection{Sets with 12 elements}
The only 12-element extreme sets we have found are sums of 6-element cosets with 2-element cosets.
\subsection{Sets with 13 elements}
$ \delta_{0}+\delta_{12}+e^{4\pi i/3}(\delta_{1}+ \delta_{3} + \delta_{4} +\delta_{8}+ \delta_{9}+\delta_{11})
+e^{2\pi i/3}(\delta_{2}+ \delta_{5}+ \delta_{6}+ \delta_{7}+ \delta_{10})$
Let $\mu = \delta_{(0,0)}+e^{4\pi i/3}\delta_{(0,1)}+e^{2\pi i/3}\delta_{(0,2)}+e^{4\pi i/3}\delta_{(0,3)}+e^{4\pi i/3}\delta_{(0,4)}+e^{2\pi i/3}\delta_{(0,5)}+e^{2\pi i/3}\delta_{(0,6)}+e^{2\pi i/3}\delta_{(0,7)}+e^{4\pi i/3}\delta_{(0,8)}+e^{4\pi i/3}\delta_{(0,9)}+e^{2\pi i/3}\delta_{(0,10)}+e^{4\pi i/3}\delta_{(0,11)}+\delta_{(0,12)}
$. Then \begin{align*}
\mu * \tilde \mu & = \big(1+1+1+1+1+1+1+1+1+1+1+1+1\big)\delta_{(0,0)}
\\&\qquad
+\big(e^{4\pi i/3}+e^{4\pi i/3}+e^{2\pi i/3}+1+e^{4\pi i/3}+1+1+e^{2\pi i/3}+1+e^{4\pi i/3}+e^{2\pi i/3}+e^{2\pi i/3}\big)\delta_{(0,1)}
\\&\qquad
+\big(1+e^{2\pi i/3}+1+e^{2\pi i/3}+e^{4\pi i/3}+e^{4\pi i/3}+1+e^{2\pi i/3}+e^{2\pi i/3}+e^{4\pi i/3}+1+e^{4\pi i/3}\big)\delta_{(0,2)}
\\&\qquad
+\big(e^{2\pi i/3}+e^{4\pi i/3}+e^{4\pi i/3}+1+1+e^{4\pi i/3}+e^{4\pi i/3}+e^{2\pi i/3}+e^{2\pi i/3}+1+1+e^{2\pi i/3}\big)\delta_{(0,3)}
\\&\qquad
+\big(e^{4\pi i/3}+1+e^{2\pi i/3}+e^{4\pi i/3}+e^{4\pi i/3}+1+e^{4\pi i/3}+1+e^{2\pi i/3}+1+e^{2\pi i/3}+e^{2\pi i/3}\big)\delta_{(0,4)}
\\&\qquad
+\big(e^{2\pi i/3}+e^{2\pi i/3}+e^{4\pi i/3}+e^{4\pi i/3}+e^{2\pi i/3}+e^{4\pi i/3}+1+1+1+1+e^{2\pi i/3}+e^{4\pi i/3}\big)\delta_{(0,5)}
\\&\qquad
+\big(e^{2\pi i/3}+1+1+1+e^{4\pi i/3}+e^{2\pi i/3}+e^{4\pi i/3}+e^{2\pi i/3}+1+e^{4\pi i/3}+e^{2\pi i/3}+e^{4\pi i/3}\big)\delta_{(0,6)}
\\&\qquad
+\big(e^{4\pi i/3}+1+e^{4\pi i/3}+e^{2\pi i/3}+1+e^{2\pi i/3}+e^{2\pi i/3}+1+e^{2\pi i/3}+e^{4\pi i/3}+1+e^{4\pi i/3}\big)\delta_{(0,7)}
\\&\qquad
+\big(e^{4\pi i/3}+e^{2\pi i/3}+e^{4\pi i/3}+1+e^{2\pi i/3}+e^{4\pi i/3}+e^{2\pi i/3}+e^{4\pi i/3}+1+1+1+e^{2\pi i/3}\big)\delta_{(0,8)}
\\&\qquad
+\big(e^{4\pi i/3}+e^{2\pi i/3}+1+1+1+1+e^{4\pi i/3}+e^{2\pi i/3}+e^{4\pi i/3}+e^{4\pi i/3}+e^{2\pi i/3}+e^{2\pi i/3}\big)\delta_{(0,9)}
\\&\qquad
+\big(e^{2\pi i/3}+e^{2\pi i/3}+1+e^{2\pi i/3}+1+e^{4\pi i/3}+1+e^{4\pi i/3}+e^{4\pi i/3}+e^{2\pi i/3}+1+e^{4\pi i/3}\big)\delta_{(0,10)}
\\&\qquad
+\big(e^{2\pi i/3}+1+1+e^{2\pi i/3}+e^{2\pi i/3}+e^{4\pi i/3}+e^{4\pi i/3}+1+1+e^{4\pi i/3}+e^{4\pi i/3}+e^{2\pi i/3}\big)\delta_{(0,11)}
\\&\qquad
+\big(e^{4\pi i/3}+1+e^{4\pi i/3}+e^{2\pi i/3}+e^{2\pi i/3}+1+e^{4\pi i/3}+e^{4\pi i/3}+e^{2\pi i/3}+1+e^{2\pi i/3}+1\big)\delta_{(0,12)}
\\&\qquad
+\big(e^{2\pi i/3}+e^{2\pi i/3}+e^{4\pi i/3}+1+e^{2\pi i/3}+1+1+e^{4\pi i/3}+1+e^{2\pi i/3}+e^{4\pi i/3}+e^{4\pi i/3}\big)\delta_{(0,13)}\\&=13\delta_{(0 , 0)}\qedhere
\end{align*}
\begin{prop}\label{prop13in14}
$\{0-12\}$ is extreme in \Zp{14}{}.
\end{prop}
xxx
\begin{proof} Let
$\mu =
\delta_{0} + e^{ 7\pi i / 4}\delta_{1} + \delta_{2} + e^{ 3\pi i / 4}\delta_{3} + e^{ 3\pi i / 2}\delta_{4} + e^{ 5\pi i / 4}\delta_{5}
+ i\delta_{6} + e^{ \pi i / 4}\delta_{7} + e^{ 3\pi i / 4}\delta_{8} + e^{ 7\pi i / 4}\delta_{9} + \delta_{10} + e^{ 3\pi i / 4}\delta_{11} + \delta_{12}.
Then
\begin{align*}\mu * \tilde\mu&= 13\delta_{0} + \big( e^{ 7 \pi i / 4}+ e^{ 9 \pi i / 4}+ e^{ 3 \pi i / 4}+ e^{ 11 \pi i / 4}+ e^{ 7 \pi i / 4}+ e^{ 7 \pi i / 4}
\\& \quad\qquad + e^{ 5 \pi i / 4}+ e^{ 13 \pi i / 4}+ e^{ 9 \pi i / 4}+ e^{ 9 \pi i / 4}+ e^{ 3 \pi i / 4}+ e^{ 13 \pi i / 4}\big)\delta_{1}
\\& \qquad + \big( 1+ 1+ e^{ \pi i}- i+ e^{ 5 \pi i / 2}- i+ e^{ \pi i}
\\& \quad\qquad + e^{ 5 \pi i / 2}+ e^{ 7 \pi i / 2}+ e^{ 5 \pi i / 2}+ e^{ \pi i}+ 1\big)\delta_{2}
\\& \qquad + \big( e^{ 5 \pi i / 4}+ e^{ 7 \pi i / 4}+ e^{ 3 \pi i / 4}+ e^{ 7 \pi i / 4}+ e^{ 5 \pi i / 4}+ e^{ 9 \pi i / 4}+ e^{ 3 \pi i / 4}
\\& \quad\qquad+ e^{ 9 \pi i / 4}+ e^{ 11 \pi i / 4}+ e^{ 15 \pi i / 4}+ e^{ 5 \pi i / 4}+ e^{ 9 \pi i / 4}\big)\delta_{3}
\\& \qquad + \big( 1+ e^{ 3 \pi i}+ 1- i- i+ e^{ \pi i}- i+ 1
\\& \quad\qquad
+ e^{ 5 \pi i / 2}+ e^{ 3 \pi i}+ e^{ 5 \pi i / 2}+ e^{ 5 \pi i / 2}\big)\delta_{4}
\\& \qquad + \big( e^{\pi i / 4}+ e^{ 7 \pi i / 4}+ e^{ 13 \pi i / 4}+ e^{ 3 \pi i / 4}+ e^{ 5 \pi i / 4}+ e^{ 5 \pi i / 4}+ e^{\pi i / 4}
\\& \quad\qquad
+ e^{ 11 \pi i / 4}+ e^{ 7 \pi i / 4}+ e^{ 15 \pi i / 4}\big)\delta_{5}
\\& \qquad + \big( i + 1+ 1+ 1- i+ e^{ \pi i}+ i
\\& \quad\qquad
- i+ e^{ 3 \pi i}+ e^{ 5 \pi i / 2}- i+ e^{ 3 \pi i}\big)\delta_{6}
\\& \qquad + \big( e^{ 7 \pi i / 4}+ e^{ 9 \pi i / 4}+ e^{ 9 \pi i / 4}+ e^{ 3 \pi i / 4}+ e^{ 11 \pi i / 4}+ e^{ 5 \pi i / 4}
\\& \quad\qquad
+ e^{\pi i / 4}+ e^{ 7 \pi i / 4}+ e^{ 7 \pi i / 4}+ e^{ 13 \pi i / 4}+ e^{ 5 \pi i / 4}+ e^{ 11 \pi i / 4}\big)\delta_{7}
\\& \qquad + \big( e^{ \pi i}+ e^{ 7 \pi i / 2}+ e^{ 5 \pi i / 2}+ e^{ \pi i}- i
\\& \quad\qquad
+ e^{ 5 \pi i / 2}+ e^{ \pi i}- i+ 1+ 1+ 1+ e^{ 5 \pi i / 2}\big)\delta_{8}
\\& \qquad + \big( e^{ 3 \pi i / 4}+ e^{ 11 \pi i / 4}+ e^{ 15 \pi i / 4}+ e^{ 5 \pi i / 4}+ e^{ 7 \pi i / 4}+ e^{ 5 \pi i / 4}
\\& \quad\qquad
+ e^{ 9 \pi i / 4}+ e^{\pi i / 4}+ e^{ 7 \pi i / 4}+ e^{ 9 \pi i / 4}+ e^{ 3 \pi i / 4}+ e^{ 13 \pi i / 4}\big)\delta_{9}
\\& \qquad + \big( i + e^{ 5 \pi i / 2}+ e^{ 3 \pi i}+ e^{ 5 \pi i / 2}+ 1- i+ e^{ \pi i}
\\& \quad\qquad
- i- i+ 1+ e^{ \pi i}+ 1\big)\delta_{10}
\\& \qquad + \big( e^{ 5 \pi i / 4}+ e^{ 9 \pi i / 4}+ e^{ 11 \pi i / 4}+ e^{ 7 \pi i / 4}+ e^{ 13 \pi i / 4}+ e^{ 7 \pi i / 4}
\\& \quad\qquad
+ e^{ 5 \pi i / 4}+ e^{\pi i / 4}+ e^{ 11 \pi i / 4}+ e^{ 7 \pi i / 4}+ e^{ 3 \pi i / 4}+ e^{ 9 \pi i / 4}\big)\delta_{11}
\\& \qquad + \big( 1+ e^{ 3 \pi i}+ e^{ 5 \pi i / 2}- i+ e^{ 5 \pi i / 2}
\\& \quad\qquad
+ e^{ 3 \pi i}- i+ i - i+ e^{ 3 \pi i}+ 1+ 1\big)\delta_{12}
\\& \qquad + \big( e^{\pi i / 4}+ e^{ 7 \pi i / 4}+ e^{ 13 \pi i / 4}+ e^{ 5 \pi i / 4}+ e^{ 9 \pi i / 4}
+ e^{ 9 \pi i / 4}+ e^{ 11 \pi i / 4}\\& \quad\qquad+ e^{ 3 \pi i / 4}+ e^{ 7 \pi i / 4}+ e^{ 7 \pi i / 4}+ e^{ 13 \pi i / 4}+ e^{ 3 \pi i}\big)\delta_{13} \
\\&= 13\delta_0.
\qedhere
\end{align*}
\end{proof}
\subsection{Sets with 14 to 16 elements}
Except for the special case in the next subsection, the computatational time needed to search for extreme
sets with more than 13 elements is prohibitive at this writing.
\begin{prop}\label{proppp16inZ16}
The 16 element set $\{0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15 \}$ is extreme in $\Zp{16}{}$. \newline \end{prop}
\begin{proof}
Let $\mu = \delta_{0}+\delta_{1}+i\delta_{2}+\delta_{3}+\delta_{4}+i\delta_{5}+\delta_{6}-\delta_{7}+\delta_{8}-\delta_{9}-i\delta_{10}+\delta_{11}+\delta_{12}-i\delta_{13}-\delta_{14}+\delta_{15}
$.
Then
\begin{align*}
\mu * \tilde \mu & = \big(1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1\big)\delta_{0}
\\&\qquad
+\big(1+1+i-i+1+i-i-1-1-1+i+i+1-i-i-1\big)\delta_{1}
\\&\qquad
+\big(-1+1+i+1-i+i+1+i+1+1-i-1+i-i-1+i\big)\delta_{2}
\\&\qquad
+\big(i-1+i+1+1+1+1-1-i-1+i+1-1+1-1+1\big)\delta_{3}
\\&\qquad
+\big(1+i-i+1+1+i-i-1+1+i-i-1+1+i-i+1\big)\delta_{4}
\\&\qquad
+\big(1+1-1-1+1+i+1+i+1-1-1+1-1-i+1+i\big)\delta_{5}
\\&\qquad
+\big(i+1+i+i-1+i+1-1-i-1-i-i+1+i-1-1\big)\delta_{6}
\\&\qquad
+\big(-1+i+i+1+i-i+1-1+1+i-i+1-i-i+1+1\big)\delta_{7}
\\&\qquad
+\big(1-1-1+1+1-1-1-1+1-1-1+1+1-1-1-1\big)\delta_{8}
\\&\qquad
+\big(-1+1-i+i+1+i+i+1+1-1-i-i+1-i+i+1\big)\delta_{9}
\\&\qquad
+\big(1-1+i-1+i+i+1-i-1-1-i+1-i-i-1-i\big)\delta_{10}
\\&\qquad
+\big(-i+1-i+1-1-1+1-1+i+1-i+1+1-1-1+1\big)\delta_{11}
\\&\qquad
+\big(1-i+i-1+1-i+i-1+1-i+i+1+1-i+i+1\big)\delta_{12}
\\&\qquad
+\big(1+1+1+1-1+i-1-i+1-1+1-1+1-i-1-i\big)\delta_{13}
\\&\qquad
+\big(-i+1+i-i+1-i+1+1+i-1-i+i-1-i-1+1\big)\delta_{14}
\\&\qquad
+\big(1-i+i+1-i+i-1-1-1-i-i+1+i+i-1+1\big)\delta_{15}\\&=16\delta_{0 }\qedhere
\end{align*}
\end{proof}
\begin{prop}\label{prop16inZ17}
The 16 element set $\{0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15 \}$ is extreme in $ \Zp{17}{}$. \newline \end{prop}
\begin{proof}
Let $\mu = \delta_{0}+e^{6\pi i/5}\delta_{1}+e^{6\pi i/5}\delta_{2}+e^{8\pi i/5}\delta_{3}+e^{2\pi i/5}\delta_{4}+e^{8\pi i/5}\delta_{5}+\delta_{6}+e^{2\pi i/5}\delta_{7}+e^{2\pi i/5}\delta_{8}+\delta_{9}+e^{8\pi i/5}\delta_{10}+e^{2\pi i/5}\delta_{11}+e^{8\pi i/5}\delta_{12}+e^{6\pi i/5}\delta_{13}+e^{6\pi i/5}\delta_{14}+\delta_{15}
$. Then
\begin{align*}
\mu * \tilde \mu & = 16\delta_{0}
+\big(3e^{6\pi i/5}+3+3e^{2\pi i/5}+3e^{4\pi i/5}+3e^{8\pi i/5}\big)
\big(\delta_{1}
+\delta_{2}+\cdots+\delta_{16}\big)\\&=16\delta_{0 }\qedhere
\end{align*}
\end{proof}
\begin{prop}\label{prop16inZ16}
The 16 element set $\{0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15 \}$ is extreme in $ \times \times \times \times \times \times \times \times \times \times \times \times \times \Zp{16}{}$. \newline \end{prop}
\begin{proof}
Let $\mu = \delta_{0}+\delta_{1}+\delta_{2}-i\delta_{3}-\delta_{4}+i\delta_{5}+\delta_{6}-\delta_{7}+\delta_{8}-\delta_{9}+\delta_{10}-i\delta_{11}-\delta_{12}+i\delta_{13}+\delta_{14}+\delta_{15}
$. Then
\begin{align*}
\mu * \tilde \mu & = \big(1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1\big)\delta_{0}
\\&\qquad
+\big(1+1+1-i-i-i-i-1-1-1-1-i-i-i-i+1\big)\delta_{1}
\\&\qquad
+\big(1+1+1-i-1-1-1+i+1+1+1+i-1-1-1-i\big)\delta_{2}
\\&\qquad
+\big(-i+1+1-i-1+i+i+1-i-1-1-i+1+i+i-1\big)\delta_{3}
\\&\qquad
+\big(-1-i+1-i-1+i+1-i-1+i+1+i-1-i+1+i\big)\delta_{4}
\\&\qquad
+\big(i-1-i-i-1+i+1-1+i+1-i-i+1+i-1+1\big)\delta_{5}
\\&\qquad
+\big(1+i-1-1-1+i+1-1+1-i-1-1-1-i+1-1\big)\delta_{6}
\\&\qquad
+\big(-1+1+i+i+i+i+1-1+1-1+i+i+i+i-1+1\big)\delta_{7}
\\&\qquad
+\big(1-1+1+1+1+1+1-1+1-1+1+1+1+1+1-1\big)\delta_{8}
\\&\qquad
+\big(-1+1-1-i-i-i-i-1+1-1+1-i-i-i-i+1\big)\delta_{9}
\\&\qquad
+\big(1-1+1+i-1-1-1+i+1-1+1-i-1-1-1-i\big)\delta_{10}
\\&\qquad
+\big(-i+1-1-i+1+i+i+1-i-1+1-i-1+i+i-1\big)\delta_{11}
\\&\qquad
+\big(-1-i+1+i-1-i+1-i-1+i+1-i-1+i+1+i\big)\delta_{12}
\\&\qquad
+\big(i-1-i-i+1+i-1-1+i+1-i-i-1+i+1+1\big)\delta_{13}
\\&\qquad
+\big(1+i-1-1-1-i+1+1+1-i-1-1-1+i+1+1\big)\delta_{14}
\\&\qquad
+\big(1+1+i+i+i+i-1-1-1-1+i+i+i+i+1+1\big)\delta_{15}\\&=16\delta_{0 }\qedhere
\end{align*}
\end{proof}
\subsection{Sets with 17 elements}
\begin{prop}\label{prop17in18}
$\{0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16\}$ is extreme in \Zp{18}{}.
\end{prop}
\begin{proof}
Let $\mu = \delta_{(0,0)}-i\delta_{(1,0)}-\delta_{(2,0)}-i\delta_{(3,0)}+\delta_{(4,0)}-i\delta_{(5,0)}-\delta_{(6,0)}+i\delta_{(7,0)}-\delta_{(8,0)}+i\delta_{(9,0)}-\delta_{(10,0)}-i\delta_{(11,0)}+\delta_{(12,0)}-i\delta_{(13,0)}-\delta_{(14,0)}-i\delta_{(15,0)}+\delta_{(16,0)}
$. Then
\begin{align*}
\mu * \tilde \mu & = \big(1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1\big)\delta_{(0,0)}
\\&\qquad
+\big(-i-i+i+i-i-i-i+i-i+i+i+i-i-i+i+i
\\&\qquad\qquad+1-1+1-1+1-1-1+1+1+1-1-1+1-1+1-1
\\&\qquad\qquad+i-i-i+i+i-i+i-i-i+i+i-i+i-i-i+i
\\&\qquad\qquad-1+1-1+1+1+1-1-1-1+1-1-1-1+1+1+1
\\&\qquad\qquad+i+i-i-i-i-i-i-i+i-i+i-i+i+i+i+i
\\&\qquad\qquad+1+1+1+1+1-1-1+1-1-1+1-1-1+1-1-1
\\&\qquad\qquad+i-i-i+i+i-i+i-i-i-i-i+i+i+i+i-i
\\&\qquad\qquad-1+1-1+1-1+1-1-1-1+1+1+1+1+1-1-1
\\&\qquad\qquad-i+i-i-i+i+i-i+i+i-i+i+i-i-i+i-i
\\&\qquad\qquad-1-1+1+1+1+1+1-1-1-1+1-1+1-1+1-1
\\&\qquad\qquad-i+i+i+i+i-i-i-i-i+i-i+i+i-i-i+i
\\&\qquad\qquad-1-1+1-1-1+1-1-1+1-1-1+1+1+1+1+1\big)\delta_{(12,0)}
\\&\qquad
+\big(i+i+i+i-i+i-i+i-i-i-i-i-i-i+i+i\big)\delta_{(13,0)}
\\&\qquad
+\big(1+1+1-1-1-1+1-1-1-1+1+1+1-1+1-1\big)\delta_{(14,0)}
\\&\qquad
+\big(i-i-i+i-i+i+i-i-i+i-i+i+i-i-i+i\big)\delta_{(15,0)}
\\&\qquad
+\big(-1+1-1+1-1-1+1+1+1-1-1+1-1+1-1+1\big)\delta_{(16,0)}
\\&\qquad
+\big(i+i-i-i+i+i+i-i+i-i-i-i+i+i-i-i\big)\delta_{(17,0)}\\&=17\delta_{(0 , 0)}\qedhere
\end{align*}
\end{proof}
\begin{rem} \label{remRealExtremtransform}
We note that if $\mu$ above is multiplied by $\delta_{-8}$, the resulting measure is
self-adjoint (that is, $(\delta_{-8}*\mu)\widetilde{\ \ } =\delta_{-8}*\mu$). Hence, the set
$\{-8,\dots,8\}$ is the support of an extreme measure with real transform. See
\propref{propExtrLessSingleton} for a related result.
\end{rem}
\newpage
\section{Proofs of extremality for non-cyclic groups}\label{secnoncyclicgps}
\subsection{\Zp23}
\begin{prop}
\cite{MR627683}
\label{prop5inZp2.Zp2.Zp2} The 5-element set
$\{(0, 0, 0), (0, 0, 1), (0, 1, 0), (0, 1, 1), (1, 0, 0) \}$
is extreme in \Zp{2}{3}.
\end{prop}
\begin{proof}
Let $\mu = \delta_{(0,0,0)}-\delta_{(0,0,1)}-\delta_{(0,1,0)}-\delta_{(0,1,1)}+i\delta_{(1,0,0)}
$. Then
\begin{align*}
\mu * \tilde \mu & = \big(1+1+1+1+1\big)\delta_{(0,0,0)}
\\&\qquad
+\big(-1-1+1+1\big)\delta_{(0,0,1)}
\\&\qquad
+\big(-1+1-1+1\big)\delta_{(0,1,0)}
\\&\qquad
+\big(-1+1+1-1\big)\delta_{(0,1,1)}
\\&\qquad
+\big(-i+i\big)\delta_{(1,0,0)}
\\&\qquad
+\big(i-i\big)\delta_{(1,0,1)}
\\&\qquad
+\big(i-i\big)\delta_{(1,1,0)}
\\&\qquad
+\big(i-i\big)\delta_{(1,1,1)}\\&=5\delta_{(0 , 0, 0)}\qedhere
\end{align*}
\end{proof}
\begin{prop}\label{prop6inZ2.X.Z2.X.Z2}
The 6 element set $\{(0, 0, 0), (0, 0, 1), (0, 1, 0), (0, 1, 1), (1, 0, 0), (1, 0, 1) \}$ is extreme in $\Zp{2}{3}$. \newline \end{prop}
\begin{proof}
Let $\mu = \delta_{(0,0,0)}-i\delta_{(0,0,1)}+e^{7\pi i/4}\delta_{(0,1,0)}+e^{3\pi i/4}\delta_{(0,1,1)}+e^{\pi i/4}\delta_{(1,0,0)}+e^{\pi i/4}\delta_{(1,0,1)}
$. Then
\begin{align*}
\mu * \tilde \mu & = \big(1+1+1+1+1+1\big)\delta_{(0,0,0)}
\\&\qquad
+\big(i-i-1-1+1+1\big)\delta_{(0,0,1)}
\\&\qquad
+\big(e^{\pi i/4}+e^{3\pi i/4}+e^{7\pi i/4}+e^{5\pi i/4}\big)\delta_{(0,1,0)}
\\&\qquad
+\big(e^{5\pi i/4}+e^{7\pi i/4}+e^{\pi i/4}+e^{3\pi i/4}\big)\delta_{(0,1,1)}
\\&\qquad
+\big(e^{7\pi i/4}+e^{5\pi i/4}+e^{\pi i/4}+e^{3\pi i/4}\big)\delta_{(1,0,0)}
\\&\qquad
+\big(e^{7\pi i/4}+e^{5\pi i/4}+e^{3\pi i/4}+e^{\pi i/4}\big)\delta_{(1,0,1)}
\\&\qquad
+\big(-i+i+i-i\big)\delta_{(1,1,0)}
\\&\qquad
+\big(-i+i-i+i\big)\delta_{(1,1,1)}\\&=6\delta_{(0 , 0, 0)}\qedhere
\end{align*}
\end{proof}
For \Zp23, a seven-element set has PSC at most 2.777127870 unlike seven-element subsets of \Zp8.
\subsection{\Zp32}
\begin{prop}\label{prop7inZp3.Zp3}
Each of the two 7-element sets below is extreme in \Zp{ 3}{2}.
\begin{enumerate}
\item $\{(0, 0), (0, 1), (0, 2), (1, 0), (1, 1), (2, 0), (2, 2) \}$
\item $\{(0, 0), (0, 1), (0, 2), (1, 0), (1, 1), (1, 2), (2, 0) \}$
\end{enumerate}
\end{prop}
\begin{proof}
(1). Let $\mu=\delta_{(0,0)}+e^{5\pi i/3}\delta_{(0,1)}+e^{2\pi i/3}\delta_{(0,2)}+e^{5\pi i/3}\delta_{(1,0)}+\delta_{(1,1)}+e^{2\pi i/3}\delta_{(2,0)}+e^{\pi i/3}\delta_{(2,2)}
$. Then
\begin{align*}
\mu * \tilde \mu & = \big(1+1+1+1+1+1+1\big)\delta_{(0,0)}
\\&
\quad+\big(e^{4\pi i/3}+e^{5\pi i/3}+e^{\pi i/3}+e^{\pi i/3}\big)\delta_{(0,1)}
\\&
\quad+\big(e^{\pi i/3}+e^{2\pi i/3}+e^{5\pi i/3}+e^{5\pi i/3}\big)\delta_{(0,2)}
\\&
\quad+\big(e^{4\pi i/3}+e^{\pi i/3}+e^{5\pi i/3}+e^{\pi i/3}-\big)\delta_{(1,0)}
\\&
\quad+\big(e^{5\pi i/3}+e^{\pi i/3}\big)\delta_{(1,1)}
\\&
\quad+\big(e^{4\pi i/3}+e^{4\pi i/3}+e^{2\pi i/3}+e^{2\pi i/3}\big)\delta_{(1,2)}
\\&
\quad+\big(e^{\pi i/3}+e^{5\pi i/3}+e^{2\pi i/3}+e^{5\pi i/3}\big)\delta_{(2,0)}
\\&
\quad+\big(e^{2\pi i/3}+e^{4\pi i/3}+e^{4\pi i/3}+e^{2\pi i/3}\big)\delta_{(2,1)}
\\&
\quad+\big(e^{5\pi i/3}+e^{\pi i/3}\big)\delta_{(2,2)}\\&=7\delta_{(0 , 0)}.
\end{align*}
(2). Let
$\mu=\delta_{(0,0)}+e^{4\pi i/3}\delta_{(0,1)}+e^{4\pi i/3}\delta_{(0,2)}+e^{5\pi i/3}\delta_{(1,0)}+e^{\pi i/3}\delta_{(1,1)}+e^{\pi i/3}\delta_{(1,2)}+\delta_{(2,0)}
$. Then
\begin{align*}
\mu * \tilde \mu & = \big(1+1+1+1+1+1+1\big)\delta_{(0,0)}
\\&
\quad+\big(e^{2\pi i/3}+e^{4\pi i/3}+1+e^{4\pi i/3}+e^{2\pi i/3}+1\big)\delta_{(0,1)}
\\&
\quad+\big(e^{2\pi i/3}+1+e^{4\pi i/3}+e^{4\pi i/3}+1+e^{2\pi i/3}\big)\delta_{(0,2)}
\\&
\quad+\big(1+e^{5\pi i/3}-1-1+e^{\pi i/3}\big)\delta_{(1,0)}
\\&
\quad+\big(e^{4\pi i/3}+e^{\pi i/3}+e^{\pi i/3}-1+e^{5\pi i/3}\big)\delta_{(1,1)}
\\&
\quad+\big(e^{4\pi i/3}+e^{\pi i/3}-1+e^{\pi i/3}+e^{5\pi i/3}\big)\delta_{(1,2)}
\\&
\quad+\big(e^{\pi i/3}-1-1+e^{5\pi i/3}+1\big)\delta_{(2,0)}
\\&
\quad+\big(e^{5\pi i/3}+e^{5\pi i/3}-1+e^{\pi i/3}+e^{2\pi i/3}\big)\delta_{(2,1)}
\\&
\quad+\big(e^{5\pi i/3}-1+e^{5\pi i/3}+e^{\pi i/3}+e^{2\pi i/3}\big)\delta_{(2,2)}\\&=7\delta_{(0 , 0)}.
\end{align*}
\qedhere
\end{proof}
\begin{prop}\label{prop7inZ3.X.Z3}
The 7 element set $\{(0, 0), (0, 1), (0, 2), (1, 0), (1, 1), (1, 2), (2, 0) \}$ is extreme in $\Zp{3}{2}$. \newline \end{prop}
\begin{proof}
Let $\mu = \delta_{(0,0)}+e^{4\pi i/3}\delta_{(0,1)}+e^{4\pi i/3}\delta_{(0,2)}+e^{\pi i/3}\delta_{(1,0)}-\delta_{(1,1)}-\delta_{(1,2)}+e^{4\pi i/3}\delta_{(2,0)}
$. Then
\begin{align*}
\mu * \tilde \mu & = \big(1+1+1+1+1+1+1\big)\delta_{(0,0)}
\\&\qquad
+\big(e^{2\pi i/3}+e^{4\pi i/3}+1+e^{4\pi i/3}+e^{2\pi i/3}+1\big)\delta_{(0,1)}
\\&\qquad
+\big(e^{2\pi i/3}+1+e^{4\pi i/3}+e^{4\pi i/3}+1+e^{2\pi i/3}\big)\delta_{(0,2)}
\\&\qquad
+\big(e^{2\pi i/3}+e^{\pi i/3}+e^{5\pi i/3}+e^{5\pi i/3}-1\big)\delta_{(1,0)}
\\&\qquad
+\big(1-1-1+e^{5\pi i/3}+e^{\pi i/3}\big)\delta_{(1,1)}
\\&\qquad
+\big(1-1+e^{5\pi i/3}-1+e^{\pi i/3}\big)\delta_{(1,2)}
\\&\qquad
+\big(e^{5\pi i/3}+e^{\pi i/3}+e^{\pi i/3}-1+e^{4\pi i/3}\big)\delta_{(2,0)}
\\&\qquad
+\big(-1-1+e^{\pi i/3}+e^{5\pi i/3}+1\big)\delta_{(2,1)}
\\&\qquad
+\big(-1+e^{\pi i/3}-1+e^{5\pi i/3}+1\big)\delta_{(2,2)}\\&=7\delta_{(0 , 0)}\qedhere
\end{align*}
\end{proof}
\subsection{\Zp{4}{2}.}
The following is a variant of
\cite[3.3 (i) ]{MR627683}.
\begin{prop}\label{prop5inZ4.X.Z4}
Each of the following 5-element sets is extreme in $ \Zp{4}{2}$.
\begin{enumerate}
\item $\{(0, 0), (0, 1), (0, 2), (0, 3), (2, 0) \}$
\item $\{(0, 0), (0, 1), (0, 2), (2, 0), (2, 2) \}$
\end{enumerate}
\end{prop}
\begin{proof} (1).
Let $\mu = \delta_{(0,0)}-i\delta_{(0,1)}+\delta_{(0,2)}+i\delta_{(0,3)}+i\delta_{(2,0)}
$. Then
\begin{align*}
\mu * \tilde \mu & = \big(1+1+1+1+1\big)\delta_{(0,0)}
+\big(-i-i+i+i\big)\delta_{(0,1)}
\\&\qquad
+\big(1-1+1-1\big)\delta_{(0,2)}
\\&\qquad
+\big(i-i-i+i\big)\delta_{(0,3)}
+\big(-i+i\big)\delta_{(2,0)}
\\&\qquad
+\big(-1+1\big)\delta_{(2,1)}
+\big(-i+i\big)\delta_{(2,2)}
+\big(1-1\big)\delta_{(2,3)}\\&=5\delta_{(0 , 0)}.
\end{align*}
\
(2).
Let $\mu = \delta_{(0,0)}-\delta_{(0,1)}-\delta_{(0,2)}+i\delta_{(2,0)}+i\delta_{(2,2)}
$. Then
\begin{align*}
\mu * \tilde \mu & = \big(1+1+1+1+1\big)\delta_{(0,0)}
+\big(-1+1\big)\delta_{(0,1)}
\\&\qquad
+\big(-1-1+1+1\big)\delta_{(0,2)}
+\big(-1+1\big)\delta_{(0,3)}
\\&\qquad
+\big(-i+i+i-i\big)\delta_{(2,0)}
+\big(i-i\big)\delta_{(2,1)}
\\&\qquad
+\big(-i+i-i+i\big)\delta_{(2,2)}
+\big(i-i\big)\delta_{(2,3)}\\&=5\delta_{(0 , 0)}\qedhere
\end{align*}
\end{proof}
\begin{prop}\label{prop6inZ4.X.Z4}
Each of the 6-element sets
\begin{enumerate}
\item $\{(0, 0), (0, 1), (0, 2), (0, 3), (1, 0), (1, 2) \}$
\item $\{(0, 0), (0, 1), (0, 2), (1, 0), (1, 1), (1, 2) \}$
\end{enumerate}
is extreme in $\Zp2{} \times \Zp{4}{}$.
\end{prop}
\begin{rem}
The 2018/10/30 version of my program (findNonCyclic) did not find those two sets equivalent. A better program might, however.
\end{rem}
\begin{proof} (1).
Let $\mu = \delta_{(0,0)}+e^{7\pi i/4}\delta_{(0,1)}+i\delta_{(0,2)}+e^{3\pi i/4}\delta_{(0,3)}+e^{3\pi i/4}\delta_{(1,0)}+e^{3\pi i/4}\delta_{(1,2)}
$. Then
\begin{align*}
\mu * \tilde \mu & = \big(1+1+1+1+1+1\big)\delta_{(0,0)}
\\&\qquad
+\big(e^{5\pi i/4}+e^{7\pi i/4}+e^{3\pi i/4}+e^{\pi i/4}\big)\delta_{(0,1)}
\\&\qquad
+\big(-i-1+i-1+1+1\big)\delta_{(0,2)}
\\&\qquad
+\big(e^{\pi i/4}+e^{5\pi i/4}+e^{7\pi i/4}+e^{3\pi i/4}\big)\delta_{(0,3)}
\\&\qquad
+\big(e^{5\pi i/4}+e^{7\pi i/4}+e^{3\pi i/4}+e^{\pi i/4}\big)\delta_{(1,0)}
\\&\qquad
+\big(-1+1+1-1\big)\delta_{(1,1)}
\\&\qquad
+\big(e^{5\pi i/4}+e^{7\pi i/4}+e^{\pi i/4}+e^{3\pi i/4}\big)\delta_{(1,2)}
\\&\qquad
+\big(-1+1-1+1\big)\delta_{(1,3)}\\&=6\delta_{(0 , 0)}\qedhere
\end{align*}
(2).
Let $\mu = \delta_{(0,0)}+e^{23\pi i/12}\delta_{(0,1)}-i\delta_{(0,2)}+e^{\pi i/3}\delta_{(1,0)}+e^{11\pi i/12}\delta_{(1,1)}+e^{11\pi i/6}\delta_{(1,2)}
$. Then
\begin{align*}
\mu * \tilde \mu & = \big(1+1+1+1+1+1\big)\delta_{(0,0)}
\\&\qquad
+\big(e^{23\pi i/12}+e^{19\pi i/12}+e^{7\pi i/12}+e^{11\pi i/12}\big)\delta_{(0,1)}
\\&\qquad
+\big(i-i+i-i\big)\delta_{(0,2)}
\\&\qquad
+\big(e^{\pi i/12}+e^{5\pi i/12}+e^{17\pi i/12}+e^{13\pi i/12}\big)\delta_{(0,3)}
\\&\qquad
+\big(e^{5\pi i/3}-1+e^{5\pi i/3}+e^{\pi i/3}-1+e^{\pi i/3}\big)\delta_{(1,0)}
\\&\qquad
+\big(e^{19\pi i/12}+e^{7\pi i/12}+e^{11\pi i/12}+e^{23\pi i/12}\big)\delta_{(1,1)}
\\&\qquad
+\big(e^{\pi i/6}+e^{7\pi i/6}+e^{5\pi i/6}+e^{11\pi i/6}\big)\delta_{(1,2)}
\\&\qquad
+\big(e^{13\pi i/12}+e^{\pi i/12}+e^{5\pi i/12}+e^{17\pi i/12}\big)\delta_{(1,3)}\\&=6\delta_{(0 , 0)}\qedhere
\end{align*}
\end{proof}
\subsection{\Zp43}
These groups have 25 or more elements, too many to find equivalence classes for sets with 6 or more elements in a
reasonable time.
\subsection{$\Zp{2}{}\times\Zp4{}$}
\begin{prop}\label{prop4inZ2.X.Z4}
The 4 element set $\{(0, 0), (0, 1), (1, 3), (1, 0) \}$ is extreme in $\Zp{2}{} \times \Zp{4}{}$. \newline \end{prop}
\begin{proof}
Let $\mu = \delta_{(0,0)}+i\delta_{(0,1)}-\delta_{(1,3)}+i\delta_{(1,0)}
$. Then
\begin{align*}
\mu * \tilde \mu & = \big(1+1+1+1\big)\delta_{(0,0)}
\\&\qquad
+\big(i-i\big)\delta_{(0,1)}
\\&\qquad
+\big(-i+i\big)\delta_{(0,3)}
\\&\qquad
+\big(-i+i\big)\delta_{(1,0)}
\\&\qquad
+\big(-1+1\big)\delta_{(1,1)}
\\&\qquad
+\big(-i+i\big)\delta_{(1,2)}
\\&\qquad
+\big(-1+1\big)\delta_{(1,3)}\\&=4\delta_{(0 , 0)}\qedhere
\end{align*}
\end{proof}
\begin{prop}\label{prop5inZ2.X.Z4}
Each of the following 5-element sets is extreme in \Zp2{}\Times\Zp4{}.
\begin{enumerate}
\item $\{(0, 0), (0, 1), (0, 2), (1, 0), (1, 2) \}$
\item $\{(0, 0), (0, 1), (0, 2), (0, 3), (1, 0) \}$
\end{enumerate}
\end{prop}
\begin{proof}
(1).
\begin{comment}
Let $\mu = \delta_{(0,0)}+\delta_{(0,1)}-\delta_{(0,2)}+i\delta_{(1,1)}+i\delta_{(1,3)}
$. Then
\begin{align*}
\mu * \tilde \mu & = \big(1+1+1+1+1\big)\delta_{(0,0)}
+\big(1-1\big)\delta_{(0,1)}
+\big(-1-1+1+1\big)\delta_{(0,2)}
\\&\qquad
+\big(1-1\big)\delta_{(0,3)}
+\big(-i+i\big)\delta_{(1,0)}
+\big(-i+i+i-i\big)\delta_{(1,1)}
\\&\qquad
+\big(-i+i\big)\delta_{(1,2)}
+\big(-i+i-i+i\big)\delta_{(1,3)}\\&=5\delta_{(0 , 0)}.
\end{align*}
(2).
\end{comment}
Let $\mu = \delta_{(0,0)}-\delta_{(0,1)}-\delta_{(0,2)}+i\delta_{(1,0)}+i\delta_{(1,2)}
$. Then
\begin{align*}
\mu * \tilde \mu & = \big(1+1+1+1+1\big)\delta_{(0,0)}
+\big(-1+1\big)\delta_{(0,1)}
+\big(-1-1+1+1\big)\delta_{(0,2)}
\\&\qquad
+\big(-1+1\big)\delta_{(0,3)}
+\big(-i+i+i-i\big)\delta_{(1,0)}
+\big(i-i\big)\delta_{(1,1)}
\\&\qquad
+\big(-i+i-i+i\big)\delta_{(1,2)}
+\big(i-i\big)\delta_{(1,3)}\\&=5\delta_{(0 , 0)}.
\end{align*}
(2).
Let $\mu = \delta_{(0,0)}-i\delta_{(0,1)}+\delta_{(0,2)}+i\delta_{(0,3)}+i\delta_{(1,0)}
$. Then
\begin{align*}
\mu * \tilde \mu & = \big(1+1+1+1+1\big)\delta_{(0,0)}
+\big(-i-i+i+i\big)\delta_{(0,1)}
\\&\qquad
+\big(1-1+1-1\big)\delta_{(0,2)}
+\big(i-i-i+i\big)\delta_{(0,3)}
+\big(-i+i\big)\delta_{(1,0)}
\\&\qquad
+\big(-1+1\big)\delta_{(1,1)}
+\big(-i+i\big)\delta_{(1,2)}
+\big(1-1\big)\delta_{(1,3)}\\&=5\delta_{(0 , 0)}\qedhere
\end{align*}
\end{proof}
\begin{prop}\label{prop6inZ2.X.Z4}
Each of the following 6-element sets
is extreme in $\Zp{2}{} \times \Zp{4}{}$.
\begin{enumerate}
\item$\{(0, 0), (0, 1), (0, 2), (0, 3), (1, 0), (1, 2) \}$
\item $\{(0, 0), (0, 1), (0, 2), (1, 0), (1, 1), (1, 2) \}$
\end{enumerate}
\end{prop}
\begin{proof}
(1.)
Let $\mu = \delta_{(0,0)}+e^{7\pi i/4}\delta_{(0,1)}+i\delta_{(0,2)}+e^{3\pi i/4}\delta_{(0,3)}+e^{3\pi i/4}\delta_{(1,0)}+e^{3\pi i/4}\delta_{(1,2)}
$. Then
\begin{align*}
\mu * \tilde \mu & = \big(1+1+1+1+1+1\big)\delta_{(0,0)}
\\&\qquad
+\big(e^{5\pi i/4}+e^{7\pi i/4}+e^{3\pi i/4}+e^{\pi i/4}\big)\delta_{(0,1)}
\\&\qquad
+\big(-i-1+i-1+1+1\big)\delta_{(0,2)}
\\&\qquad
+\big(e^{\pi i/4}+e^{5\pi i/4}+e^{7\pi i/4}+e^{3\pi i/4}\big)\delta_{(0,3)}
\\&\qquad
+\big(e^{5\pi i/4}+e^{7\pi i/4}+e^{3\pi i/4}+e^{\pi i/4}\big)\delta_{(1,0)}
+\big(-1+1+1-1\big)\delta_{(1,1)}
\\&\qquad
+\big(e^{5\pi i/4}+e^{7\pi i/4}+e^{\pi i/4}+e^{3\pi i/4}\big)\delta_{(1,2)}
+\big(-1+1-1+1\big)\delta_{(1,3)}\\&=6\delta_{(0 , 0)}.
\end{align*}
(2).
Let $\mu = \delta_{(0,0)}+e^{23\pi i/12}\delta_{(0,1)}-i\delta_{(0,2)}+e^{\pi i/3}\delta_{(1,0)}+e^{11\pi i/12}\delta_{(1,1)}+e^{11\pi i/6}\delta_{(1,2)}
$. Then
\begin{align*}
\mu * \tilde \mu & = \big(1+1+1+1+1+1\big)\delta_{(0,0)}
\\&\qquad
+\big(e^{23\pi i/12}+e^{19\pi i/12}+e^{7\pi i/12}+e^{11\pi i/12}\big)\delta_{(0,1)}
\\&\qquad
+\big(i-i+i-i\big)\delta_{(0,2)}
\\&\qquad
+\big(e^{\pi i/12}+e^{5\pi i/12}+e^{17\pi i/12}+e^{13\pi i/12}\big)\delta_{(0,3)}
\\&\qquad
+\big(e^{5\pi i/3}-1+e^{5\pi i/3}+e^{\pi i/3}-1+e^{\pi i/3}\big)\delta_{(1,0)}
\\&\qquad
+\big(e^{19\pi i/12}+e^{7\pi i/12}+e^{11\pi i/12}+e^{23\pi i/12}\big)\delta_{(1,1)}
\\&\qquad
+\big(e^{\pi i/6}+e^{7\pi i/6}+e^{5\pi i/6}+e^{11\pi i/6}\big)\delta_{(1,2)}
\\&\qquad
+\big(e^{13\pi i/12}+e^{\pi i/12}+e^{5\pi i/12}+e^{17\pi i/12}\big)\delta_{(1,3)}\\&=6\delta_{(0 , 0)}\qedhere
\end{align*}
\end{proof}
\begin{comment}
\subsection{$\Zp{2}{}\times\Zp{4}{2}$}
\marginpar{\tiny How is this Prop related to \propref{prop6inZ4.X.Z4} and
\propref{prop6inZp2.Zp4.Zp4}?}
\begin{prop}\label{prop6inZp2.Zp4.Zp4}
Each of the following 6-element sets is extreme in $\Zp{2}{} \times \Zp{4}{2}$.
\begin{enumerate}
\item $\{(0, 0, 0), (0, 0, 1), (0, 0, 2), (0, 0, 3), (1, 0, 0), (1, 0, 2) \}$
\item $\{(0, 0, 0), (0, 0, 1), (0, 0, 2), (0, 0, 3), (1, 1, 0), (1, 3, 2) \}$
\item $\{(0, 0, 0), (0, 0, 1), (0, 0, 2), (0, 0, 3), (1, 2, 0), (1, 2, 2) \}$
\item $\{(0, 0, 0), (0, 0, 1), (0, 0, 2), (0, 0, 3), (1, 1, 0), (1, 3, 0) \}$
\end{enumerate}
\end{prop}
\begin{proof}
(1).
Let $\mu = \delta_{(0,0,0)}+e^{7\pi i/4}\delta_{(0,0,1)}+i\delta_{(0,0,2)}+e^{3\pi i/4}\delta_{(0,0,3)}+e^{3\pi i/4}\delta_{(1,0,0)}+e^{3\pi i/4}\delta_{(1,0,2)}
$. Then
\begin{align*}
\mu * \tilde \mu & = \big(1+1+1+1+1+1\big)\delta_{(0,0,0)}\\&\qquad
+\big(e^{5\pi i/4}+e^{7\pi i/4}+e^{3\pi i/4}+e^{\pi i/4}\big)\delta_{(0,0,1)}
\\&\qquad
+\big(-i-1+i-1+1+1\big)\delta_{(0,0,2)}
\\&\qquad
+\big(e^{\pi i/4}+e^{5\pi i/4}+e^{7\pi i/4}+e^{3\pi i/4}\big)\delta_{(0,0,3)}
\\&\qquad
+\big(e^{5\pi i/4}+e^{7\pi i/4}+e^{3\pi i/4}+e^{\pi i/4}\big)\delta_{(1,0,0)}
\\&\qquad
+\big(-1+1+1-1\big)\delta_{(1,0,1)}
\\&\qquad
+\big(e^{5\pi i/4}+e^{7\pi i/4}+e^{\pi i/4}+e^{3\pi i/4}\big)\delta_{(1,0,2)}
\\&\qquad
+\big(-1+1-1+1\big)\delta_{(1,0,3)}\\&=6\delta_{(0 , 0, 0)}.
\end{align*}
(2).
Let $\mu = \delta_{(0,0,0)}+\delta_{(0,0,1)}+\delta_{(0,0,2)}-\delta_{(0,0,3)}+e^{\pi i/4}\delta_{(1,1,0)}+e^{3\pi i/4}\delta_{(1,3,2)}
$. Then
\begin{align*}
\mu * \tilde \mu & = \big(1+1+1+1+1+1\big)\delta_{(0,0,0)}
+\big(-1+1+1-1\big)\delta_{(0,0,1)}
\\&\qquad
+\big(1-1+1-1\big)\delta_{(0,0,2)}
+\big(1+1-1-1\big)\delta_{(0,0,3)}
\\&\qquad
+\big(-i+i\big)\delta_{(0,2,2)}
+\big(e^{5\pi i/4}+e^{\pi i/4}\big)\delta_{(1,1,0)}
\\&\qquad
+\big(e^{\pi i/4}+e^{5\pi i/4}\big)\delta_{(1,1,1)}
+\big(e^{5\pi i/4}+e^{\pi i/4}\big)\delta_{(1,1,2)}
\\&\qquad
+\big(e^{5\pi i/4}+e^{\pi i/4}\big)\delta_{(1,1,3)}
+\big(e^{7\pi i/4}+e^{3\pi i/4}\big)\delta_{(1,3,0)}
\\&\qquad
+\big(e^{7\pi i/4}+e^{3\pi i/4}\big)\delta_{(1,3,1)}
+\big(e^{7\pi i/4}+e^{3\pi i/4}\big)\delta_{(1,3,2)}
\\&\qquad
+\big(e^{3\pi i/4}+e^{7\pi i/4}\big)\delta_{(1,3,3)}\\&=6\delta_{(0 , 0, 0)}.
\end{align*}
(3).
Let $\mu = \delta_{(0,0,0)}+e^{7\pi i/4}\delta_{(0,0,1)}+i\delta_{(0,0,2)}+e^{3\pi i/4}\delta_{(0,0,3)}+e^{3\pi i/4}\delta_{(1,2,0)}+e^{3\pi i/4}\delta_{(1,2,2)}
$. Then
\begin{align*}
\mu * \tilde \mu & = \big(1+1+1+1+1+1\big)\delta_{(0,0,0)}
\\&\qquad
+\big(e^{5\pi i/4}+e^{7\pi i/4}+e^{3\pi i/4}+e^{\pi i/4}\big)\delta_{(0,0,1)}
\\&\qquad
+\big(-i-1+i-1+1+1\big)\delta_{(0,0,2)}
\\&\qquad
+\big(e^{\pi i/4}+e^{5\pi i/4}+e^{7\pi i/4}+e^{3\pi i/4}\big)\delta_{(0,0,3)}
\\&\qquad
+\big(e^{5\pi i/4}+e^{7\pi i/4}+e^{3\pi i/4}+e^{\pi i/4}\big)\delta_{(1,2,0)}
\\&\qquad
+\big(-1+1+1-1\big)\delta_{(1,2,1)}
\\&\qquad
+\big(e^{5\pi i/4}+e^{7\pi i/4}+e^{\pi i/4}+e^{3\pi i/4}\big)\delta_{(1,2,2)}
\\&\qquad
+\big(-1+1-1+1\big)\delta_{(1,2,3)}\\&=6\delta_{(0 , 0, 0)}.
\end{align*}
(4).
Let $\mu = \delta_{(0,0,0)}-\delta_{(0,0,1)}-\delta_{(0,0,2)}-\delta_{(0,0,3)}+e^{3\pi i/4}\delta_{(1,1,0)}+e^{3\pi i/4}\delta_{(1,3,0)}
$. Then
\begin{align*}
\mu * \tilde \mu & = \big(1+1+1+1+1+1\big)\delta_{(0,0,0)}
+\big(-1-1+1+1\big)\delta_{(0,0,1)}
\\&\qquad
+\big(-1+1-1+1\big)\delta_{(0,0,2)}
+\big(-1+1+1-1\big)\delta_{(0,0,3)}
\\&\qquad
+\big(1+1\big)\delta_{(0,2,0)}
+\big(e^{5\pi i/4}+e^{3\pi i/4}\big)\delta_{(1,1,0)}
\\&\qquad
+\big(e^{\pi i/4}+e^{7\pi i/4}\big)\delta_{(1,1,1)}
+\big(e^{\pi i/4}+e^{7\pi i/4}\big)\delta_{(1,1,2)}
\\&\qquad
+\big(e^{\pi i/4}+e^{7\pi i/4}\big)\delta_{(1,1,3)}
+\big(e^{5\pi i/4}+e^{3\pi i/4}\big)\delta_{(1,3,0)}
\\&\qquad
+\big(e^{\pi i/4}+e^{7\pi i/4}\big)\delta_{(1,3,1)}
+\big(e^{\pi i/4}+e^{7\pi i/4}\big)\delta_{(1,3,2)}
\\&\qquad
+\big(e^{\pi i/4}+e^{7\pi i/4}\big)\delta_{(1,3,3)}\\&=6\delta_{(0 , 0, 0)}.
\qedhere
\end{align*}
\end{proof}
\end{comment}
\subsection{\Zp{2}{2}\Times\Zp{3}{}}
\begin{prop}\label{prop8inZ2.X.Z2.X.Z3}
The 8 element set
\[
\{(0, 0, 0), (0, 0, 1), (0, 1, 0), (0, 1, 1), (1, 0, 0), (1, 0, 1), (1, 1, 0), (1, 1, 1) \}
\]
is extreme in $\Zp{2}{2}\Zp{3}{}$. \newline \end{prop}
\begin{proof}
Let $\mu = \delta_{(0,0,0)}+e^{7\pi i/15}\delta_{(0,0,1)}-i\delta_{(0,1,0)}+e^{29\pi i/30}\delta_{(0,1,1)}+i\delta_{(1,0,0)}+e^{29\pi i/30}\delta_{(1,0,1)}-\delta_{(1,1,0)}+e^{7\pi i/15}\delta_{(1,1,1)}
$. Then
\begin{align*}
\mu * \tilde \mu & = \big(1+1+1+1+1+1+1+1\big)\delta_{(0,0,0)}
\\&\qquad
+\big(e^{7\pi i/15}+e^{22\pi i/15}+e^{7\pi i/15}+e^{22\pi i/15}\big)\delta_{(0,0,1)}
\\&\qquad
+\big(e^{23\pi i/15}+e^{8\pi i/15}+e^{23\pi i/15}+e^{8\pi i/15}\big)\delta_{(0,0,2)}
\\&\qquad
+\big(i-i-i+i-i+i+i-i\big)\delta_{(0,1,0)}
\\&\qquad
+\big(e^{29\pi i/30}+e^{29\pi i/30}+e^{59\pi i/30}+e^{59\pi i/30}\big)\delta_{(0,1,1)}
\\&\qquad
+\big(e^{31\pi i/30}+e^{31\pi i/30}+e^{\pi i/30}+e^{\pi i/30}\big)\delta_{(0,1,2)}
\\&\qquad
+\big(-i-i+i+i+i+i-i-i\big)\delta_{(1,0,0)}
\\&\qquad
+\big(e^{59\pi i/30}+e^{59\pi i/30}+e^{29\pi i/30}+e^{29\pi i/30}\big)\delta_{(1,0,1)}
\\&\qquad
+\big(e^{31\pi i/30}+e^{31\pi i/30}+e^{\pi i/30}+e^{\pi i/30}\big)\delta_{(1,0,2)}
\\&\qquad
+\big(-1+1-1+1-1+1-1+1\big)\delta_{(1,1,0)}
\\&\qquad
+\big(e^{22\pi i/15}+e^{7\pi i/15}+e^{22\pi i/15}+e^{7\pi i/15}\big)\delta_{(1,1,1)}
\\&\qquad
+\big(e^{23\pi i/15}+e^{8\pi i/15}+e^{23\pi i/15}+e^{8\pi i/15}\big)\delta_{(1,1,2)}\\&=8\delta_{(0 , 0, 0)}\qedhere
\end{align*}
\end{proof}
\begin{comment}
\subsection{Perhaps to do}
\begin{verbatim}
As results are obtained for these groups,
these sections will have material added to them and
be moved into the regular text.
\subsection{$\Zp{2}{}\times\Zp{5}{2}$}
\subsection{$\Zp22 \times\Zp3{}{}$}
\subsection{$\Zp22 \times\Zp4{}{}$}
\subsection{$\Zp22 \times\Zp5{}{}$}
\subsection{$\Zp22 \times\Zp7{}{}$}
\subsection{$\Zp{2}{}\times\Zp{3}{2}$}
\subsection{$\Zp{3}{2}\times\Zp{4}{}$}
\subsection{$\Zp{4}{2}\times\Zp{3}{}$}
\end{verbatim}
\end{comment}
\bibliographystyle{amsplain}
\section{Introduction} \label{secintro}
Let $G$ be a locally compact abelian group with dual group $\widehat G$. Haar measure on $G$ will be
denoted by $m_G$ and is counting measure if $G$ is discrete.
\mpar{Removed all marginal comments earlier than 2019-08-06 \\
as of 2019-08-07}
The \emph{Sidon (or Helson) constant}, $S(E)$, of a set $E\subset G$ is
the infimum of constants,
$C$, such that $\|\mu\|\le C\|\widehat\mu\|_\infty$ for all
(non-zero) measures $\mu$ concentrated
on $E$. Here $\widehat\mu$
is the Fourier(-Stieltjes) transform of $\mu$.
The Sidon constant, $S(E)$, is always at least $\sqrt{\#E}$.
\begin{definition} A non-zero measure $\mu$ on the discrete abelian group $G$ is \emph{extreme}
(or \emph{extremal})
if $\|\widehat\mu\|_\infty = \|\mu\|/\sqrt{\#\supp\mu}$,
where $\supp\mu$ is the support of $\mu$.
A finite set $E$ is \emph{extreme} (or \emph{extremal})
if it is the support of an extreme measure.
If $\mu$ is an extreme measure with support $E$, we shall say $\mu$ is \emph{extreme for} $E$.
\end{definition}
The definition of ``extreme'' includes the finiteness of the set.
Finite abelian groups are extreme \cite{MR0458059}. Other extreme sets were given in \cite{MR627683}. The
contributions here are:
\begin{itemize}
\item
more examples of extreme
sets but with most of the (mostly both tedious and obvious) verifications left to the
appendix
\item
a proof that for each integer $N$
there are only a finite number of groups of prime
order that contain $N$-element extreme sets (\thmref{thmLimExtremeSetPrimeProds});
\item
some conjectures and questions.
\end{itemize}
Throughout this paper, $G$ will be a discrete abelian group (almost always finite) with counting measure
and $\Gamma$ the dual of $G$ with Haar measure $m_\Gamma(\Gamma) = 1$. So here the
Plancherel theorem says $\sum_{g\in G}|\mu(\{g\})|^2
= \int_\Gamma|\widehat\mu(\gamma)|^2dm_{\Gamma}.$
When $G$ is finite, the
Plancherel theorem becomes
\[
\sum_{g\in G}|\mu(\{g\})|^2 = (\#G)^{-1}\sum_{\gamma\in\Gamma}|\widehat\mu(\gamma)|^2.
\]
The set of (regular, Borel) measures on a set $E\subset G$ will be
denoted by $M(E)$.
\bigbreak
\subsection*{Organization of this paper} General properties of extreme sets are covered in
\secref{secTheory}. That includes conditions necessary for extremality (several) and sufficient conditions. The section
concludes with \thmref{thmLimExtremeSetPrimeProds}, which states that
for each $N$ there is a bound on primes $p_1, p_K$ such that an $N$-element set is
contained in the product of $K$ cyclic groups of respective order $p_1, \dots, p_K$.
Our ``Beastiary'' is in \secref{secBeastiary}: tables of extreme sets with sample extreme measures, along with commentary
on those sets.
Section \ref{secConjectures} contains conjectures, questions, and proofs of some of the results summarized in the
tables of the
previous section.
The final section discusses the computer programs used in the project.
\section{Theory}\label{secTheory}
\subsection{Properties of extreme measures}
\begin{lemma}\label{lemExtrHatConstant}
Let $\mu$ be an extreme measure on the discrete abelian group $G$. Then $|\widehat\mu|$ is constant.
\end{lemma}
\begin{proof} See
\cite[Lemma 1.1]{MR627683} for a different proof.
Let $E$ be the support of $\mu$. We may assume $\|\mu\|=N=\#E$. Then $\|\widehat\mu\|_\infty = \sqrt N$.
If there is $\gamma\in \Gamma$ such that $|\widehat\mu(\gamma)|<\sqrt N$, then
there is an open neighbourhood $U$ of $\gamma$ such that $|\widehat\mu|<\sqrt N$ on $U$. Thus,\
\[
\int_\Gamma|\widehat\mu(\gamma)|^2=\Big(\int_U+\int_{\Gamma\backslash U}\Big)|\widehat\mu|^2d\gamma<
Nm_\Gamma(U)+Nm_\Gamma(\Gamma\backslash U)<N.
\]
The Plancherel theorem now tells us that $\sum_{g\in E}|\mu(\{g\})|^2<N.$ We
use Jensen's inequality for $\varphi (x) = x^2$, weights $a_g$ and values $y_g$:
\[
\Big(\frac{\sum a_g y_g}{\sum a_g}\Big)^2\le \frac{\sum a_g y_g^2}{\sum a_g}.
\]
Set $a_g=1/N$ and $y_g = |\mu(\{g\})|$ for $g\in E$ . Then $\sum a_g=1$ and
\begin{align*}
N^{-2}\|\mu\|^2
&=\Big(\frac{N^{-1}\sum_g |\mu(\{g\})|}{NN^{-1}}\Big)^2
\ \overset{\text{\tiny Jensen}}{\le\phantom{|}}\ \frac{N^{-1}|\sum_g \mu(\{g\})|^2}{1}
< N^{-1}N
=
1,
\end{align*}
so $\|\mu\|^2<N^2$, contradicting the assumption that $\|\mu\|=N$.
\end{proof}
The direction (1) $\Rightarrow$ (2) of following result
is \cite[Lemma 1.1]{MR627683}; the other
direction must be known but we are unable to give a reference.
\begin{theorem}\label{thmExtrMsConstant} Let $\mu$ be a discrete measure on an abelian group. Then the following are equivalent.
\begin{enumerate}
\item
$\mu$ is an extreme measure.
\item
$|\mu(x)|$ is constant on $\Supp\mu$ and
$|\widehat\mu|$ is constant on the dual group.
\end{enumerate}
\end{theorem}
\begin{proof}[Proof of \thmref{thmExtrMsConstant}]
(1) $\Rightarrow$ (2). See
\cite[Lemma 1.1]{MR627683} for a different proof.
Let $\mu$ have support $E$. We may assume that
$\|\mu\|=N=\#E$ and $|\widehat\mu|=\sqrt{N}$ on $\Gamma$. For $g\in E$ let $\varepsilon_g=1-|\mu(\{g\})|$.
Then $N=\sum_{g\in E}|\mu(\{g\})| = \sum_{g\in E}(1-\varepsilon_g)$,
so $\sum\varepsilon_g=0$.
Let $f=\frac{d\mu}{dm_G}$, so
$\int hd\mu = \int hf\,dm_G$ for all $h:G\to\mathbb C$.
Then
\begin{align*}
\int |f|^2dm_G &= \sum_{g\in E}|f(g)|^2
=\sum_{g\in E}(1-\varepsilon_g)^2
\\
&=\sum_{g\in E}(1-2\varepsilon_g+\varepsilon_g^2)
={N} +N\sum\varepsilon_g^2.
\end{align*}
Hence,
\begin{equation}\label{eqL2ofmu}
\sum_g|\varepsilon_g|>0 \text{ implies }
\|f\|_2>\sqrt{{N}}.
\end{equation}
On the other hand, $\widehat f= \widehat\mu$ on $\Gamma$ and
so (using Plancherel and extremality)
\begin{align}
{N}+N\sum\varepsilon_g^2 =\int |f|^2dm_G &=
\int|\widehat f|^2 dm_{\Gamma}=
\int_\Gamma|\widehat\mu|^2 dm_\Gamma
= N,
\end{align}
so the $\varepsilon_g$ are all zero, by \eqref{eqL2ofmu}.
That proves that $|\mu(\{g\})|\equiv 1$
on $E$.
(2) $\Rightarrow$ (1).
Suppose $|\mu|$ is constant on its support, $E$,
and that $|\widehat\mu|$ is constant on $\Gamma$.
We may
assume
that $|\mu|=1$ everywhere on $E$.
As before, let $f=\frac{d\mu}{dm_G}$
so $f(g)=\mu(\{g\})$ for all $g \in G$ and $\widehat\mu=\widehat f$. Then
\begin{align}
\|\mu\|=N, \int|f|^2dm_G= N = \int \|\widehat\mu\|_\infty^2 dm_\Gamma =
\|\widehat\mu\|_\infty^2.
\end{align}
Hence $N = \|\widehat\mu\|_\infty^2$ and $\|\widehat\mu\|_\infty =\sqrt N$, so $\mu$ is extremal.
\end{proof}
The Pontriagin duality theorem and \thmref{thmExtrMsConstant} immediately yield:
\begin{cor}\label{corHatIsExtreme}
If $\mu$ is extreme for the finite abelian group $G$, then
\widehat\mu\, m_{\widehat G}$ is
extreme for $\widehat G$.
\end{cor}
\medbreak
The following corollary helps eliminate candidate sets for extremality and has been a useful tool in
the search for small extreme subsets of larger groups.
\begin{cor}\label{corExtrCnxln}
\mpar{\ \\ Statement and proof revised. \\2019-08-07}
Let $E$ be finite with $\#E>1$. Let $\nu= \sum_{x\in E}\delta_g$. If $E$ is the support of a non-zero measure $\mu$
such that $|\widehat\mu|$ is constant on $\Gamma$ (in particular, if $E$ is extreme),
then every point mass in $\nu*\tilde \nu$ has coefficient either at least 2 or 0.
\end{cor}
\begin{proof}
Because $|\widehat\mu|$ is constant, $\widehat\mu \,\overline{\widehat\mu}$ is constant and
so
\[
\mu*\widetilde\mu=|\widehat\mu(0)|^2\delta_0=\sum_g|\mu(\{g\})|^2.
\]
Hence,
for every $g\in (E-E)\backslash\{0\}$, $\mu*\tilde \mu(\{g\})=0$, so there must be cancellation of $\mu*\widetilde\mu$'s masses at $g$, which requires
that $\nu*\tilde\nu(\{g\})>1$.
\end{proof}
\begin{lem}\label{lemNoExtrInZn}
Let $M\ge 1$. Suppose $E$ is a compact subset of the Hilbert space $\mathcal H$
and has two or more
elements.
Then there is an element of $E-E$ that has only one
representation as a difference of two elements of $E$.
\end{lem}
\begin{proof}
Note that the hypotheses and conclusion of the lemma hold for $E$ if and only if they hold for any translate of $E$ if and only if they hold for any rescaling of $E$, that is, replacing $E$
with $sE=\{sx:x\in E\}$, for any $s>0$.
Because $E \times E$ is compact, the set $E$ has a finite diameter $D$:
$$
D=\sup\setof{\Vert x-y\Vert}{x, \, y \in E}<\infty.
$$
Here $\Vert\cdot\Vert_{\mathcal H}$ is the norm for $\mathcal H$.
By compactness, there are $u \in E$ and $v \in E$ such that $\Vert u-v \Vert=D$.
Because $E$ has at least 2 members, $D \ne 0$ and $u \ne v$.
By replacing $E$ with $E-u$ if necessary, we may assume $u=\mathbf 0$, the identity
of $\mathcal H$.
Thus, we may also assume $\mathbf 0\in E$.
By replacing (the possibly translated) $E$ with $\frac 1 D E$,
we may assume $D=1$. Let $\mathcal B$ be an
orthonormal basis of $\mathcal H$ whose first element is $v$. We note that
the coordinates of $v$ with respect to $\mathcal B$ are $(1,0,\dots,0)$.
Since $\mathbf 0\in E$, $E\subset E-E$.
For each element $g\in\mathcal H$ and $1\le m\le M$, let $t(g)$
be the first coefficient with respect to $\mathcal B$, that
is, $t(g)= \langle g,v\rangle.$
Since $E$ has diameter 1, $1\ge \|g\|\ge t(g)$
for all $g\in E$. Furthermore, $t(g)\ge 0$ for all $g\in E$
since otherwise such a $g$ would have distance greater than 1 from $v$.
Now suppose $v=x-y$ where $x,y\in E.$
Then if $t(y) = 0$, we have $y={\mathbf 0}$, since otherwise $\|y-v\|>1$.
Also, if $t(x) = 1$, then $x=v$, since otherwise $\|g-\mathbf 0\|>1$.
We can now show that $v$ is an element of $E-E$ with only one
representation, $v= v-\mathbf 0$, as a difference of two elements of $E$.
\begin{enumerate}
\item If $x\in E$ and $y\in E\backslash\{{\mathbf{ 0}}\}$,
then $t(y)>0$, so
$t(x-y) = t(x) -t(y)\le 1-t(y)<1\ne t(v)$.
Hence, $x-y\ne v$.
\item If $x\in E\backslash \{v\}$ and $y\in E$,
then $t(x-y)= t(x) -t(y) <1-t(y)<1$ so $x-y \ne v$.
\end{enumerate}
It now follows that the only representation of $v$ in $E-E$ is as
$v-\mathbf 0$.
\end{proof}
The following is immediate from \corref{corExtrCnxln} and \lemref{lemNoExtrInZn}.
\begin{cor} \label{corExtremeRm}
Let $1\le M<\infty$.
\begin{enumerate}
\item \label{itcorExtremeRm1}
Let $\mu$ be a finitely supported measure on $R^M$ such that
$|\widehat\mu|$ is constant. Then $\mu$ is a point mass.
\item
The only extreme subsets of $\R^M$ $($hence, of $\text{$\mathbb Z$}^M)$ are singletons.
\end{enumerate}
\end{cor}
A measure $\mu$ such that $|\widehat\mu|$ is constant will be called a
\emph{transform with constant absolute value} (\emph{TCAV)} measure. Clearly TCAV is
equivalent to $\mu*\widetilde\mu=a\delta_0$ for
some $a$. If a set is the support of a TCAV measure it will be called a TCAV set.
We do not know if being TCAV implies that a set is also extreme, as our only
examples distinguising ``extreme'' from ``
TCAV'' applies to the measures in the next remark.
\begin{rem}\label{remTCAVNotExtr}
Computation shows that the measure
\mpar{This remark is new; its unenlightening proof in the source, commented out.
\\2019-08-18}
$\mu=\delta_0+(1+\sqrt3 i)\delta_1+(1-\sqrt3 i)\delta_2$
is TCAV on \Zp3{}
and $\nu=\delta_0+\varepsilon i\delta_1$ is TCAV on \Zp2{}, for all
$0\le \varepsilon<\infty$. They are not extreme. The transform of $\mu$ takes on the values
$\pm 3$ and that of $\nu$ the values $1\pm i\varepsilon$.
\begin{comment}
$\mu$: Indeed,
\begin{align*}
\mu*\widehat\mu=\widehat\mu^2&=9\delta_0+
(1+i\sqrt3+1+i\sqrt3+1-2i\sqrt3 -3)\delta_1
\\&\qquad
+(1-i\sqrt3+1+2i\sqrt3-3 +1-i\sqrt3)\delta_2\\
&=9\delta_0.
\end{align*}
$\nu$: $\widehat\nu(0)=1+i\varepsilon$ and $\widehat\nu(1) = 1-i\varepsilon$.
The measures $\mu$ and $\nu$ are not extreme since the absolute values of their
masses vary.
\end{comment}
The example $\nu$ shows that the limit argument of \lemref{lemLimExtremeSet} can fail for TCAV measures.
\end{rem}
A set that is the sum (product) of two extreme sets is extreme if its cardinality is the product of the
cardinalities of the summands (Proof: take the convolution of an extreme measure on each set).
However, sets of the form $\text{$\mathbb Z$}\times F$ where $F$ is a finite group contain extreme sets of cardinality $(\#F)^2$;
such sets cannot be contained in a coset of a finite group; see \propref{propUnionCosets}.
\subsection{Subgroups and quotients}
The following is a simplified version of \cite[Theorem 2.1 (i)]{MR627683}\label{thmGR2.1}.
\begin{prop}\label{propUnionCosets}
Let $H\subset G$ be a subgroup of cardinality $N$. If $g_n+H$, $1\le n\le N$,
are distinct cosets of $H$, then $\bigcup _1^Ng_n+H$ is extreme.
\end{prop}
\begin{proof}
Let $m_H$ be Haar measure on $H$ with $\|m_H\|=N$.
Let $\lambda_1,\dots,\lambda_N$ be
elements of $\Gamma$ whose restrictions to $H$ are the $N$ characters of $H$.
Then $\widehat{m_H}$ is $N$ times the characteristic function
of $H^\perp$ and $\widehat\nu_n =\widehat{\lambda_nm_H}$
is the $N$
times characteristic function of
$\lambda_n+H^\perp$, $1\le n\le N$. Hence
\begin{equation}\label{eqZeroProducts}
\widehat \nu_n\widehat\nu_\ell =0 \ (1\le n\ne\ell\le N.
\end{equation}
Let $\tau_n=
\delta_{g_n}*
(\lambda_n m_H)$,
$1\le n\le N$ and $\mu = \sum_1^N\tau_n$.
Then,
$|\widehat\mu| \equiv N$ everywhere, the support of $\mu$ is
the union of $N$ disjoint cosets of
size $N$, so $\|\mu\|= N^2$ and $\mu$
is extreme.
\end{proof}
The following was suggested by L.~T.~Ramsey and is included with permission.
\begin{prop
\label{propRamseyProjection}
Let $H$ be a subgroup of $G$ and $E\subset G$ extreme. If $\#(E/H) =\#E$, then $E/H$ is extreme in $G/H$.
\end{prop}
\begin{proof}
Let $\mu$ be an extreme measure on $E$, with $\|\mu\|=\#E$ and $\|\widehat\mu\|_\infty=\sqrt{\#E}$.
Let $\nu$ be the measure on $E/H$ given by $\nu(g+H)=\mu(\{g\})$ for $g\in E$.
If $\gamma\in H^\perp=(G/H)\widehat{\ }\subset \widehat G$, then
\begin{align*}
\widehat\nu(\gamma) =
&\sum_{g+H\in E/H} \langle-\gamma,g+H\rangle\nu(g+H)
\\
= &\ \quad\sum_{g\in E}\quad \ \langle-\gamma,g\rangle\mu(\{g\})
\\
= &\quad\ \widehat\mu(\gamma).
\end{align*}
Thus, $\|\nu\|=\#(E/H)=\#E$ and $\|\widehat\nu\|_\infty=\sqrt{\#(E/H)}.$ Therefore $\nu$ and $E/H$ are extreme.
\end{proof}
\begin{rem} \label{remRamseyNoLifting}
The converse is false. Indeed, let $E=\{0,1,2\}\subset G=\Zp{8}{}$ and $H=\{0,4\}.$
Then $\{0,1,2\} $ fails the test of \corref{corExtrCnxln}: 2 and 6 have unique representations in the difference set,
so $\{0,1,2\}$ is not extreme in \Zp8{}.\footnote{Alternatively, apply
\cite[3.1]{MR627683}'s enumeration of three-element extreme sets, or just use the computer, which quickly
confirms that $\{0,1,2\}$ is not extreme in \Zp8{}.}
Now
let $\tau:G \to G/H$ be the natural mapping. Then $\tau(E)=\{0,1,2\}$ a
three element set in $\Zp4{}$ and therefore extreme.
\end{rem}
\subsection{Automorphisms and equivalent sets}
A group of prime order $p$ has $p-1$ automorphisms, given by multiplication by integers $1\le j<p$.
Every element (except the identity) is moved by every non-trivial automorphism.
A group of prime power order $p^k$ also has $p-1$ automorphisms
\cite[Thm. 4.1]{MR2363058}.\footnote{A proof of this fact is
sketched in \cite{ABeastiaryAppendix}.}
See \cite{MR2666671, MR2363058, MR2103185, MR1512510}
for more on automorphisms of
finite abelian groups.
Two subsets, $E,F$ of a group $G$ are \emph{equivalent}
if one of them can be obtained from the other
by a sequence of group automorphisms and translations.
It is immediate that
all the sets in an equivalence class have
the same Sidon constant. It is not true that
having the same Sidon constant and cardinality implies two subsets
are equivalent; examples are provided by
\propref{prop5in12} and the subsets of \Zp2{}\Times\Zp4{}
given in \cite[3.3 (i)]{MR627683}.
Whether two \emph{non}-extreme sets in different
equivalence classes can have the same Sidon
constant is unknown. Our computer program has not
found any for sets of cardinality up to 7 in
groups of order less than 30.
\bigbreak
In groups of prime order, all two-element sets are equivalent.
On the other hand, in the group $\Zp 7{}$ there are two
equivalence classes of three element sets, one
generated by $\{0,1,2\}$ and the other
by $\{0,1,3\}$. Since equivalent sets have
equal Sidon constants, and those two sets have different constants, that gives a different proof that
those two sets are not equivalent.
It is often useful to list the elements of the equivalence classes
for each finite group and our programs that search for extreme sets do more-or-less that as a
preliminary step
\subsection{The pseudo-Sidon constant (PSC)}
It is easier to write programs to calculate the infimum
\mpar{Revised defn of PSC to reflect what computer program does.
\\
Don't know if $PSC(E)=1/S(E)$.
\\2019-08-23} \begin{equation}\label{eqPSconstDef}
PSC(E) = \inf\{ \|\widehat\mu\|_\infty: \text{supp } \mu = E, |\mu(\{x\})|=1
\ \forall x\in E\}
\end{equation}
than to compute the Sidon constant directly and select sets for which it is extreme:.
We call $PSC(E)$ the \emph{pseudo-Sidon constant (PS constant, PSC)} of $E$ and note the trivial:
\begin{prop}
$E$ is extremal if and only if $PSC(E)=S(E)$, which occurs if and only if $PSC(E)=\sqrt{\#E}$.
\end{prop}
When we want to be clear about the group which contains $E$, we will write $PSC(E,G)$.
\subsubsection{The PSC and sets with two elements}\label{subsec2elts}
That the PSC can give an interesting (or odd) result is evidenced by
Part 2 of the following (which does not extend in any obvious way, though the computer
does suggest the existence of other sets with PSCs of $\sqrt3$ and of others with integer PSCs).
Part (3) is \cite[3.4(ii)]{MR627683} and Part (1), if new, is also obvious.
\begin{prop} \label{propTwoEltsinZedthree}
\begin{enumerate}\item
$PSC(\{0,1\},\Zp n{})$ increases monotonically to 2 as $n\to\infty$.
\item $PSC( \{0, 1 \},\Zp3{})=\sqrt3$.
\item \label{it2EltCyclic} The only two-element extreme sets are cosets.
\end{enumerate}
\end{prop}
\begin{proof} (1).
Let $\mu = \delta_0 +e^{2\pi i\theta}\delta_1$ on \Zp{n}{}.
Then
\begin{equation}\label{eqhatnuTwoElts}
\widehat\mu(k)= 1+e^{2\pi i(\theta-\frac{k}{n})}, \text{ for $k$ in the dual of \Zp{n}{} (i.e., $0\le k<n$).}
\end{equation}
It is clear from \eqref{eqhatnuTwoElts} that the minimum (over $\theta$) of
$\sup_k|\widehat\mu(k)|$ will occur only when $\theta$ is an odd
multiple of $\frac{1}{2n}$ and that the minimum will be
$|1+e^{2\pi i/2n}|$. Clearly, $|1+e^{2\pi i/2n}|\to2$ monotonically.
(2). Let $\nu = \delta_0+e^{2\pi i/3}\delta_1$. It will be clear from the next paragraph that
$\|\widehat\nu||_\infty = \sqrt3,$
so the PSC is at most $\sqrt3$.
Note that if, more generally, $\mu = \delta_0+ e^{i\theta}\delta_1$, then $|\widehat\mu(0)|^2 =
{2 +2\cos\theta}$. Hence $|\widehat\mu(0)|\le \sqrt3 $ if and only if
\begin{equation}\label{eqTwoEltSet1}
\frac\pi3\le\theta\le \frac{5\pi}3 \mod 2\pi.
\end{equation}
Now, $\widehat\mu(1) = 1 + e^{(\theta - 2\pi/3)i}$, so $|\widehat\mu(1)|\le \sqrt3$ if and only if
\begin{equation}\label{eqTwoEltSet2}
-\frac{\pi}3\le \theta\le {\pi} \mod 2\pi.
\end{equation}
Finally, $
\widehat\mu(2) = 1 + e^{(\theta-4\pi /3)i},
$
so $|\widehat\mu(2)|\le\sqrt3$ if and only if
\begin{equation}\label{eqTwoEltSet3}
- \pi\le\theta\le\frac{\pi}3 \mod 2\pi.
\end{equation}
Putting \eqref{eqTwoEltSet1}-\eqref{eqTwoEltSet3} together, we see that
$\theta=\pm \frac\pi3$ and $||\widehat\mu||_\infty = \sqrt3.$
(3). \cite[3.4 (ii)]{MR627683} It is enough to show that $\{0,k\}\subset \Zp{m}{}$ is extreme if and only if $k$ divides $m$ and $m/k=2$.
But if $\nu = \delta_0+\alpha\delta_k$ has $\|\widehat\nu\|_\infty=\sqrt2$ and, in particular,
$|\widehat\nu(0)|=|1+\alpha|=\sqrt2$, then $\alpha=\pm i$. We may assume $\alpha=i$. Hence, $\widehat\nu(\gamma) = 1
+ i\langle \gamma,k\rangle$ for $\gamma$ in the dual of \Zp{m}{}. Therefore $\langle\gamma, k\rangle=\pm1$ for all $\gamma$.
Hence, $k$ has order 2, that is, $m/k=2$.
Here is a second proof of (3):
let $\mu=\delta_0+\delta_g$. Then $\mu*\tilde \mu = 2\delta_0+\delta_g +\delta_{-g}$. If $g\ne -g$, then
$E$ fails the test of \corref{corExtrCnxln}. Since subgroups are extreme, the conclusion follows.
\end{proof}
\begin{rem}
We note that the transform of $\nu $ in (2) does not
have a constant absolute value, as is to
be expected from \thmref{thmExtrMsConstant}.
\end{rem}
\subsection{Limits of extreme sets of a given cardinality}
\label{subsecInfiniteNumOfSets}
This section is about limits of extreme sets. The term ``limit'' needs clarification:
Let a sequence of sets $E_j\subset\text{$\mathbb T$}$ and $E\subset\text{$\mathbb T$}$ be given. We say $E_j\to E$
if
\[
\max\prnb{\sup_{x\in E_j}\,\inf_{y\in E}|x-y|,\ \sup_{y\in E}\,\inf_{x\in E_j}|x-y|}.
\to0
\]
This is, of course, the Hausdorff distance.\footnote{Alternatively, let $\mu_j$ be counting measure on $E_j$ for each $j$.
If $\mu_j\to\mu$ weak* in $M(\text{$\mathbb T$})$ and $E=\Supp \mu
, then
$E$ is the of the $E_j$.}
A finite set in the group \Zp m{} can be thought as a subset of $\text{$\mathbb Z$} \mod m$ or it can be
identified with the subset of $\{e^{2\pi i k/m}:0\le k<m\}\subset \text{$\mathbb T$}$, where $\text{$\mathbb T$} \,(= \R \mod 2\pi)$
is the circle group. When we speak of ``limits'',
we are thinking of the latter representation. However to avoid the clutter (and eyestrain) of
many exponentials, we shall often write ``$k$'' where we mean ``$e^{2\pi i k/m}$'' and
$m$ is implicit.
\begin{lem}\label{lemLimExtremeSet}
Suppose $N>5$ and that there exist primes $2\le p_1<p_2<\dots$
such that
each
\Zp{p_k}{} contains a set $E_k$
of cardinality $N$ that is the support of an extreme measure.
Then there exists an extreme set $F\subset \mathbb T$ of cardinality $N$,
a subsequence $p_{k_\ell}$ and sets
$F_\ell\subset\Zp{p_{k_\ell}}{}
$ of cardinality $N$ such that $F$ is the limit of the $F_\ell$, $\#F=N$ and
$F$ is the support of an extreme measure.
\end{lem}
\begin{proof}
We recall that if $p_\ell$ is prime, then \Zp{p_\ell}{} has $p_\ell$ automorphisms.
By renumbering as we go and translating if needed, we may assume $k_\ell = \ell$ for all $\ell$, as well as
\begin{equation} \label{eqnSizepell}
0\in E_\ell \text{ and }
p_1> 300\, N^2.
\end{equation}
\mpar{deleted the repititious ``We may assume $0\in E$.''. \\
Added a comma after ``$N$''\\
2019-08-08}
We now count automorphisms. Fix an $\ell$.
Write the elements of $E_\ell$ as $\{e^{2\pi g_{\ell,n}/p_\ell}:0\le n< N\}$.
For $1\le m< n\le N$,
let $H_{m,n}$ be the set of automorphisms $T$ such that
\begin{equation}\label{eqnSizepe1}
|Tg_{\ell,m}-Tg_{\ell,n}|\le \frac{2\pi}{10N^2}.
\end{equation}
The reader will note that here we are using both the representation of \Zp{p_\ell}{} as
$\text{$\mathbb Z$} \mod p_\ell$ (using the $g_\ell$ and $T$) and as a subgroup
of $\text{$\mathbb T$}$ (to calculate the distance
between points as in \eqref{eqnSizepe1}).
Since there
\mpar{Inserted ``there''\\2019-08-08}
are at most $\frac{1+2p_\ell}{10 N^2}$ elements of $G$ that close
\mpar{Deleted the ``3+'' and added ``+1'', both here and \textit{ff}.\\2019-08-12}
(half on one ``side'', half on the other ``side'' and one in the middle),
there are at most that many automorphisms that carry the two points that close to each other.
Hence $\bigcup_{1\le m<n\le N}H_{m,n}$ has cardinality at most
$\binom{N}{2}\frac{3 p_\ell}{10N^2}$, so there are at least\footnote{\, We use
\eqref{eqnSizepell} and replaced ``$1+2p_\ell$'' with ``$2p_\ell$''. }
\begin{equation}\label{eqnRemainAutosI}
p_\ell-\frac{3p_\ell}{10}\ge \frac{7p_\ell}{10}\ge 210\,N^2
\end{equation}
automorphisms which keep the elements of $E_\ell$ separated by at least $\frac{2\pi}{10N^2}$.
Therefore, for all sufficiently large $\ell$, there exists an automorphism $S_\ell$ of
\Zp{p_\ell}{}
such that the elements of $F_\ell = S_\ell E_\ell$ are all
at least $\frac{2\pi}{10N^2}$ apart from each other.
Now for the limits. For each $\ell$ put an extreme measure $\mu_\ell$ on $F_\ell$
(now $F_\ell\subset \text{$\mathbb T$}$) with $\|\mu_\ell\|=N$
and $\|\widehat{\mu_\ell}\|_\infty =\sqrt N$.
We can find a subsequence $\mu_{\ell_k}$ which converges weak-*
in $M(\mathbb T)$ to a measure $\mu$. Since $|\mu _\ell| \equiv 1$
on $F_\ell$ and
because the elements in the supports of the $\mu_{\ell_k}$ stay apart, the support of $\mu$ has $N$
elements, $\|\mu\|=N$ and $\mu$ is extreme. The support $F$ of $\mu$ is our limit set and is also thus extreme.
\end{proof}
\begin{rem} There is no guarantee that the limit set is contained in a group of prime order, and, in fact, the next result arranges for the limit set \emph{not}
to be contained in a group of finite order.
\remref{remTCAVNotExtr} shows that the limit argument above can fail
for TCAV measures since their masses are not necesssarily bounded away from 0.
\end{rem}
\begin{lem}\label{lemLimExtremeSetII} Let $N>1$. Then there does not exist an
infinite number of primes $p_k$ such that \Zp {p_k}{} contains
an extreme set
of cardinality $N$.
\end{lem}
\begin{proof} This is a continuation of the proof of \lemref{lemLimExtremeSet} and we retain the notation of that proof, the assumptions \eqref{eqnSizepell}-\eqref{eqnSizepe1} and
the properties of the limit set given by \lemref{lemLimExtremeSet}.
In particular we see from \eqref{eqnRemainAutosI} that there are $\frac{7}{10}p_\ell$ autormorphisms that
keep the points of $E_\ell$ separated for each $\ell$.
Because of \cite[3.1-3.3 and 3.4 (ii)]{MR627683}, we may assume $N>5$.
Since in a group of prime order, all doubletons are equivalent, we
may assume that for each $\ell$, $g_{\ell,1} = 0$ and $g_{\ell,2} = 1$.
Enumerate \emph{all} the primes in increasing order as $2=q_1<q_2<\cdots$ and enumerate
the elements of the sets $E_\ell=\{0, g_{\ell,2}, \dots, g_{\ell, N}\}$.
For $K=1,\dots,$ let $H_K$ be the cyclic subgroup of $\mathbb T$ of cardinality $M_K =\prod_{k=1}^K q_k^K$.
By passing to a subsequence of the $p_\ell$, we may assume
that $p_\ell > M_\ell$ for all $\ell$.
For each $\delta>0$ and $2\le n \le N$,
the set of automorphisms $S$ of \Zp{p_\ell}{}
such that $S(g_{\ell, n})$ is within $\delta$ of an element of $H_K\backslash \{0\}$
has size at most $2M_K\delta p_{\ell}$.
Let $\delta_K = \frac{1}{2M_K (N-1) 10^K}$. Then there are
$\frac{p_\ell}{10^K}$
automorphisms that carry a non-zero element of $E$ to within
$\delta$ of an element of \Zp{M_K}{}.
Thus, there are at most
\begin{equation}\label{eqCountAutos}
\sum_1^\ell \frac{p_\ell}{10^K}<\frac{p_\ell}{9}
\end{equation}
automorphisms that carry a non-zero element of $E_\ell$ to within $\delta_K$ of \Zp{M_K}{}
for $1\le K<\ell$.
Using \eqref{eqnRemainAutosI}, we see there are at least
\begin{equation}\label{eqCountAutos2}
\frac{6p_\ell}{9}
\end{equation}
automormorphisms that both separate the
elements of $E_\ell$ from each other by at least $\frac{2\pi}{10N^2}$ and also
keep the non-zero elements
at least $\delta_K$ away from elements of \Zp{M_K}{} for $1\le K<\ell.$
For each $1\le\ell<\infty$ chose an automorphism $S_\ell$ of \Zp{p_\ell}{} such
that the elements of $F_\ell=S_\ell E_\ell$ are separated by $\frac{2\pi}{10N^2}$ and for each $1\le K<\ell$ the non-zero elements of $F_\ell$ are at least $\delta_K$
away from the elements of \Zp{M_K}{}.
Then the limit set $F\subset\text{$\mathbb T$}$ of any convergent subsequence of the
$F_\ell= S_\ell E_\ell$ contains no element of any $H_K\backslash \{0\}$
for each $K=1, 2,\dots$.
Let $F$ be such a limit.
\lemref{lemLimExtremeSet} shows $F$ is extreme.
Let $H $ be the group generated by $F$. Then $H=\text{$\mathbb Z$}^{r}\times L$ where $L$ is finite and $1<r<\infty$.
Since $0\in F_\ell$ for all $\ell$ we must have
$0\in F$ but
\begin{equation}\label{eqnSizePF}
F\cap \prnb{\{0\}\times L} \text{ contains only the identity of } H
\end{equation}
because $F\backslash\{0\}$ contains no elements of finite order.
Let $\mu$ be an extreme measure on $F$ with $\|\mu\|=\#E=\#F=N$.
We may assume $\mu$ has
mass 1 at the identity.
Let $P: H\rightarrow\text{$\mathbb Z$}^r$
the the natural projection of abelian groups and $ P'$ the corresponding
projection of measures
$M(H)\rightarrow M(\text{$\mathbb Z$}^{r}).$ Then $(P'\mu)\widehat{\ }= \widehat\mu_{|L^\perp}$. That is,
$(P'\mu)\widehat{\ }$ has constant absolute value of $A>\sqrt 5$ on $\text{$\mathbb Z$}$.\footnote{$|\widehat\mu|$ is constant on $\text{$\mathbb Z$}$ since it is the weak-* limit of extreme measures
on $\Zp{p_\ell}{}\subset\text{$\mathbb T$}$. Of course $\text{$\mathbb Z$}$ is dense in $\widehat H$.}
Because of \eqref{eqnSizePF}, $(P'\mu)(\{0\}) = \mu(\{0\}=1$
(here ``0'' is -- abusively -- the identity of
the group in question). Since the transform of $P'\mu$ has absolute value $A>1\ge (P'\mu)(\{0\})$, $P'\mu$
cannot be supported only on the identity of $\text{$\mathbb Z$}^r$. But the transform of $P'\mu$ has
constant absolute value. This contradicts
\corref{corExtremeRm} \itref{itcorExtremeRm1}
and completes the proof.
\end{proof}
\begin{thm}\label{thmLimExtremeSetPrimeProds} Let $N>1$. Then there does not exist an
infinite number of groups
$H_k= \Zp{p_{k,1}}{}\times\cdots\times \Zp{p_{k, m_k}}{} $ (all $p_{k,m}$ being prime and $m_k\ge 1$) such
that each $H_k$ contains
an extreme set
of cardinality $N$.
\end{thm}
\begin{proof}[Proof of \thmref{thmLimExtremeSetPrimeProds}]
We may assume $N>5$ since $1\le N\le 5$ are taken care of by
\cite[3.1-3.3 and 3.4(ii)]{MR627683}.
Let $E\subset H_k$ have $N$ elements.
It is clear that at most $L=\binom{N}{2}$ coordinates are enough to distinguish
the elements of $E.$ We may assume those
coordinates are the first $L$ ones.
Projecting $\Zp{p_{k,1}}{}\times\cdots\times\Zp{p_{k, m_k}}{}$
onto $\Zp{ p_{k,1}}{} \times\cdots\times \Zp{p_{L,m_L}}{}$
will map any extreme measure on
$E$ to an extreme measure on the image of $E$, since the two norms
$\|\mu\|$ and $\|\widehat \mu\|_\infty$ are preserved.
Thus, we may assume that the $H_k$ have at most $L$ factors.
The obvious coordinate-wise
form of the proof of \lemref{lemLimExtremeSetII} gives the required conclusion: a subsequence
of $ (p_{k,1},\dots, p_{k,L})$ can be chosen so that in each coordinate the
projections of the \Zp{p_{k, \ell}}{} accumulate only at the identity and at elements of infinite order and
stay separated by a fixed amount.
Thus, in the product, the accumulation points are the identity and elements of infinite order. The proof now concludes as
from \eqref{eqnSizePF}, \textit{mutatis mutandi}.
\end{proof}
\begin{rem}
The previous proof fails for groups of prime power order because
there are insufficient automorphisms.
\end{rem}
\section{A beastiary with remarks}\label{secBeastiary}
Here are i) a table of extreme sets previously known from \cite{MR0458059,MR627683}, ii) two tables of new extreme sets (``regular'' and ``irregular'') in cyclic groups,
and iii) a table of new extreme sets in non-cyclic groups. The tables are accompanied by some remarks.
\begin{table}[b]\begin{center}
\begin{tabular} {| c| c| c|}
\hline
Group & Set (size) & Extreme measure \\ \hline
$G$& $G$ ($\#G$) & See \cite{MR0458059}\\ \hline
\Zp{4}{} & 0\dots 2 .\phantom{0} (3) \phantom{*} &$ \delta_0 + e^{3\pi i/4}\delta_1 +i\delta_2 $\\ \hline
\Zp{5}{} & 0\dots 3.\phantom{0} (4) \phantom{*} & $ \delta_0+\delta_3+e^{2\pi i/3}(\delta_1+\delta_2) $ \\ \hline
\Zp{6}{}& 0\dots 4. (5)& $\delta_{0} + \delta_{4} -\delta_{2} + e^{ 3\pi i / 2}(\delta_{1} + \delta_{3}) $
\\ \hline
\Zp{7}{} & 0\dots 2, 4. (4) & $ \delta_{0} - \delta_{1} - \delta_{2} - \delta_{4} $ \\ \hline
\Zp{8}{}& 0\dots 6. (7)& $ \delta_{0}+\delta_{6}+e^{2\pi i/3}(\delta_{1}+ \delta_{2}+ \delta_{4}+ \delta_{5} )
$\\&&$
+e^{4\pi i/3}\delta_{3}$
\\ \hline
\Zp{2}{3}& (0, 0, 0), (0, 0, 1), (0, 1, 0), &$ \delta_{(0,0,0)}-\delta_{(0,0,1)}-\delta_{(0,1,0)}$
\\
& (0, 1, 1), (1, 0, 0. (5) & $-\delta_{(0,1,1)}+i\delta_{(1,0,0)}$
\\ \hline
\Zp2{}\Times\Zp4{}& (0,0), (0,1), (1,3), (1,0). (4) & $\delta_{(0,0)}+i(\delta_{(0,1)}+\delta_{(1,0)})-\delta_{(1,3)}
$\\ \hline
\Zp{12}{}& 0, 1, 2, & $ \delta_{0} + \delta_{4} + e^{ 3\pi i / 2}\delta_{2} + e^{ 5\pi i / 4}\delta_{3}
$\\ & 5, 10. (5) & $
+ e^{\pi i / 4}\delta_{7} $
\\ \hline
\Zp2{}\Times\Zp4{}&(0,0), (0,2), (1,0), & $ \delta_{(0,0)}-\delta_{(0,1)}-\delta_{(0,2)}$
\\&(1,2), (0,1). (5)&$+i(\delta_{(1,0)}+\delta_{(1,2)}) $
\\ \hline
\Zp23&(0,0,0), (1,0,0), (0,1, 0),&$ \delta_{(0,0,0)}-\delta_{(0,0,1)}-\delta_{(0,1,0)}$
\\& (0,0,1), (1,1,0). (5)& $-\delta_{(0,1,1)}+i\delta_{(1,0,0)} $
\\ \hline
\end{tabular}
\vskip.125in
\caption{Extreme sets from \cite{MR0458059,MR627683}}
\label{tableFromCite}
\end{center}
\end{table}
\begin{comment}
\begin{prop}\label{prop11inZ12}
The 11 element set $\{0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 \}$ is extreme in $ \times \times \times \times \times \times \times \times \times \Zp{12}{}$. \newline \end{prop}
\begin{proof}
Let $\mu = \delta_{0}+e^{4\pi i/5}\delta_{1}+e^{2\pi i/5}\delta_{2}+e^{8\pi i/5}\delta_{3}+e^{8\pi i/5}\delta_{4}+\delta_{5}+e^{8\pi i/5}\delta_{6}+e^{8\pi i/5}\delta_{7}+e^{2\pi i/5}\delta_{8}+e^{4\pi i/5}\delta_{9}+\delta_{10}
$. Then
\begin{align*}
\mu * \tilde \mu & = \big(1+1+1+1+1+1+1+1+1+1+1\big)\delta_{0}
\\&\qquad
+\big(e^{4\pi i/5}+e^{8\pi i/5}+e^{6\pi i/5}+1+e^{2\pi i/5}+e^{8\pi i/5}+1+e^{4\pi i/5}+e^{2\pi i/5}+e^{6\pi i/5}\big)\delta_{1}
\\&\qquad
+\big(1+e^{2\pi i/5}+e^{4\pi i/5}+e^{6\pi i/5}+e^{2\pi i/5}+1+e^{8\pi i/5}+e^{4\pi i/5}+e^{6\pi i/5}+e^{8\pi i/5}\big)\delta_{2}
\\&\qquad
+\big(e^{6\pi i/5}+e^{4\pi i/5}+e^{8\pi i/5}+e^{4\pi i/5}+e^{8\pi i/5}+1+1+e^{2\pi i/5}+e^{6\pi i/5}+e^{2\pi i/5}\big)\delta_{3}
\\&\qquad
+\big(e^{8\pi i/5}+1+e^{2\pi i/5}+e^{8\pi i/5}+e^{6\pi i/5}+e^{6\pi i/5}+1+e^{4\pi i/5}+e^{4\pi i/5}+e^{2\pi i/5}\big)\delta_{4}
\\&\qquad
+\big(e^{2\pi i/5}+e^{2\pi i/5}+e^{8\pi i/5}+e^{8\pi i/5}+1+e^{4\pi i/5}+e^{6\pi i/5}+e^{4\pi i/5}+e^{6\pi i/5}+1\big)\delta_{5}
\\&\qquad
+\big(e^{2\pi i/5}+e^{6\pi i/5}+1+e^{4\pi i/5}+e^{8\pi i/5}+e^{8\pi i/5}+e^{4\pi i/5}+1+e^{6\pi i/5}+e^{2\pi i/5}\big)\delta_{6}
\\&\qquad
+\big(1+e^{6\pi i/5}+e^{4\pi i/5}+e^{6\pi i/5}+e^{4\pi i/5}+1+e^{8\pi i/5}+e^{8\pi i/5}+e^{2\pi i/5}+e^{2\pi i/5}\big)\delta_{7}
\\&\qquad
+\big(e^{2\pi i/5}+e^{4\pi i/5}+e^{4\pi i/5}+1+e^{6\pi i/5}+e^{6\pi i/5}+e^{8\pi i/5}+e^{2\pi i/5}+1+e^{8\pi i/5}\big)\delta_{8}
\\&\qquad
+\big(e^{2\pi i/5}+e^{6\pi i/5}+e^{2\pi i/5}+1+1+e^{8\pi i/5}+e^{4\pi i/5}+e^{8\pi i/5}+e^{4\pi i/5}+e^{6\pi i/5}\big)\delta_{9}
\\&\qquad
+\big(e^{8\pi i/5}+e^{6\pi i/5}+e^{4\pi i/5}+e^{8\pi i/5}+1+e^{2\pi i/5}+e^{6\pi i/5}+e^{4\pi i/5}+e^{2\pi i/5}+1\big)\delta_{10}
\\&\qquad
+\big(e^{6\pi i/5}+e^{2\pi i/5}+e^{4\pi i/5}+1+e^{8\pi i/5}+e^{2\pi i/5}+1+e^{6\pi i/5}+e^{8\pi i/5}+e^{4\pi i/5}\big)\delta_{11}\\&=11\delta_{0 }\qedhere
\end{align*}
\end{proof}
\end{comment}
\begin{comment}
\begin{prop}\label{prop8inZ9}
The 8 element set $\{0, 1, 2, 3, 4, 5, 6, 7 \}$ is extreme in $ \times \times \times \times \times \times \Zp{9}{}$. \newline \end{prop}
\begin{proof}
Let $\mu = \delta_{0}+\delta_{7}
+e^{10\pi i/7}(\delta_{1}+\delta_{6})
+e^{4\pi i/7}(\delta_{2}+\delta_{5})
+e^{2\pi i/7}(\delta_{3}+\delta_{4})
$. Then
\begin{align*}
\mu * \tilde \mu & = \big(1+1+1+1+1+1+1+1\big)\delta_{0}
\\&\qquad
+\big(e^{10\pi i/7}+e^{8\pi i/7}+e^{12\pi i/7}+1+e^{2\pi i/7}+e^{6\pi i/7}+e^{4\pi i/7}\big)\delta_{1}
\\&\qquad
+\big(1+e^{4\pi i/7}+e^{6\pi i/7}+e^{12\pi i/7}+e^{2\pi i/7}+e^{8\pi i/7}+e^{10\pi i/7}\big)\delta_{2}
\\&\qquad
+\big(e^{4\pi i/7}+e^{10\pi i/7}+e^{2\pi i/7}+e^{6\pi i/7}+1+e^{8\pi i/7}+e^{12\pi i/7}\big)\delta_{3}
\\&\qquad
+\big(e^{10\pi i/7}+1+e^{4\pi i/7}+e^{2\pi i/7}+e^{8\pi i/7}+e^{6\pi i/7}+e^{12\pi i/7}\big)\delta_{4}
\\&\qquad
+\big(e^{12\pi i/7}+e^{6\pi i/7}+e^{8\pi i/7}+e^{2\pi i/7}+e^{4\pi i/7}+1+e^{10\pi i/7}\big)\delta_{5}
\\&\qquad
+\big(e^{12\pi i/7}+e^{8\pi i/7}+1+e^{6\pi i/7}+e^{2\pi i/7}+e^{10\pi i/7}+e^{4\pi i/7}\big)\delta_{6}
\\&\qquad
+\big(e^{10\pi i/7}+e^{8\pi i/7}+e^{2\pi i/7}+e^{12\pi i/7}+e^{6\pi i/7}+e^{4\pi i/7}+1\big)\delta_{7}
\\&\qquad
+\big(e^{4\pi i/7}+e^{6\pi i/7}+e^{2\pi i/7}+1+e^{12\pi i/7}+e^{8\pi i/7}+e^{10\pi i/7}\big)\delta_{8}\\&=8\delta_{0 }\qedhere
\end{align*}
\end{proof}
\end{comment}
\begin{table}[]\begin{center}
\begin{tabular} {| c| c| c|}
\hline
Group & Set (size) & Extreme measure \\ \hline
\Zp{k}{}& $0\dots(k-2).\ (k-1)$, & See table above and \cite{MR0458059,MR627683} \\
&for $4\le k <6$ \& $k=8 $ &\\ \hline
\Zp{9}{} & 0\dots 7.\phantom{0} (8) \phantom{*} &$ \delta_{0}+\delta_{7}
+e^{10\pi i/7}(\delta_{1}+\delta_{6})
$\\&&$
+e^{4\pi i/7}(\delta_{2}+\delta_{5})
+e^{2\pi i/7}(\delta_{3}+\delta_{4})$
\\ \hline
\Zp{10}{} & 0\dots 8.\phantom{0} (9) \phantom{*} & $\delta_{0} + \delta_{2}+ \delta_{6} + \delta_{8} -i(\delta_{1} + \delta_{7}) $\\&&$ + i (\delta_{3} +\delta_{5}) -\delta_{4} $ \\ \hline
\Zp{12}{} & 0\dots 10. (11) \phantom{*} & $\delta_{0} +\delta_{4}
-\delta_{8}
+e^{7\pi i/4}(\delta_{1}+\delta_{5}+\delta_{9})
$\\&&$
-i\delta_{2}
+e^{5\pi i/4}\delta_{3}
+i(\delta_{6}+\delta_{10})+e^{\pi i/4}\delta_{7}
$ \\ \hline
\begin{comment
Another measure:
$\delta_{0} + \delta_{2} + \delta_{10}+ \delta_{12}
+ e^{ 7\pi i / 4}(\delta_{1} + \delta_{9} )
$\\&&$
+ e^{ 3\pi i / 4}(\delta_{3} + \delta_{8} + \delta_{11} )
$\\&&$
+ e^{ 3\pi i / 2}\delta_{4} + e^{ 5\pi i / 4}\delta_{5}
+ i\delta_{6} + e^{ \pi i / 4}\delta_{7}$
\end{comment}
\Zp{14}{} & 0\dots 12. (13) \phantom{*} & $ \delta_{0}+\delta_{12}
$\\&&$
+e^{4\pi i/3}(\delta_{1}+ \delta_{3} + \delta_{4} +\delta_{8}+ \delta_{9}+\delta_{11})
$\\&&$
+e^{2\pi i/3}(\delta_{2}+ \delta_{5}+ \delta_{6}+ \delta_{7}+ \delta_{10})$
\\ \hline
\Zp{17}{}&0\dots 15. (16) \phantom{*}& $\delta_{0} +\delta_{9} +\delta_{15}+e^{6\pi i/5}(\delta_{1}+\delta_{2}+\delta_{13}+\delta_{14})
$\\&&$+e^{8\pi i/5}(\delta_{3}+\delta_{5} +\delta_{10}+ \delta_{12} )
$\\&&$
+e^{2\pi i/5}(\delta_{4}
+\delta_{6}+\delta_{7}
+\delta_{8}+\delta_{11})
$
\\ \hline
\Zp{18}{} & 0\dots 16. (17) \phantom{*} & $\delta_{0}+\delta_{4}+\delta_{12}+\delta_{16}
$\\&&$+i\delta_{7}+i\delta_{9}
-\delta_{2}
-\delta_{6}-\delta_{8}
-\delta_{10}-\delta_{14}
$\\&&$
-i(\delta_{1}+\delta_{3}+\delta_{5} +\delta_{11}+\delta_{13} +\delta_{15})
$\\ \hline
\Zp{20}{}&0\dots 18. \phantom{*}(19)& $\delta_{0} +\delta_{5} +\delta_{6} +\delta_{8}+\delta_{9}+\delta_{10} + \delta_{12}
+\delta_{13}$\\&&$
+e^{4\pi i/3}(\delta_{1}+\delta_{3} +\delta_{4} +\delta_{14}+\delta_{15} +\delta_{17})
$\\&&$
+e^{2\pi i/3}(\delta_{2}+\delta_7+\delta_{11}+\delta_{16}+\delta_{18}) $
\\ \hline
\end{tabular}
\vskip.125in
\caption{``Regular'' extreme sets in cyclic groups}
\label{tableRegular}
\end{center}
\end{table}
\begin{comment}\begin{prop}\label{prop7inZ8}
The 7 element set $\{0, 1, 2, 3, 4, 5, 6 \}$ is extreme in $ \times \times \times \times \times \Zp{8}{}$. \newline \end{prop}
\begin{proof}
Let $\mu = \delta_{0}+\delta_{6}+e^{2\pi i/3}(\delta_{1}+ \delta_{2}+ \delta_{4})+ \delta_{5} )
+e^{4\pi i/3}\delta_{3}
$. Then
\begin{align*}
\mu * \tilde \mu & = \big(1+1+1+1+1+1+1\big)\delta_{0}
\\&\qquad
+\big(e^{2\pi i/3}+1+e^{2\pi i/3}+e^{4\pi i/3}+1+e^{4\pi i/3}\big)\delta_{1}
\\&\qquad
+\big(1+e^{2\pi i/3}+e^{2\pi i/3}+1+e^{4\pi i/3}+e^{4\pi i/3}\big)\delta_{2}
\\&\qquad
+\big(e^{4\pi i/3}+e^{2\pi i/3}+e^{4\pi i/3}+1+1+e^{2\pi i/3}\big)\delta_{3}
\\&\qquad
+\big(e^{4\pi i/3}+1+e^{2\pi i/3}+e^{2\pi i/3}+1+e^{4\pi i/3}\big)\delta_{4}
\\&\qquad
+\big(e^{2\pi i/3}+1+1+e^{4\pi i/3}+e^{2\pi i/3}+e^{4\pi i/3}\big)\delta_{5}
\\&\qquad
+\big(e^{4\pi i/3}+e^{4\pi i/3}+1+e^{2\pi i/3}+e^{2\pi i/3}+1\big)\delta_{6}
\\&\qquad
+\big(e^{4\pi i/3}+1+e^{4\pi i/3}+e^{2\pi i/3}+1+e^{2\pi i/3}\big)\delta_{7}\\&=7\delta_{0 }\qedhere
\end{align*}
\end{proof}
\begin{prop}\label{prop19inZ2.X.Z20}
The 19 element set $\{(0, 0), (0, 1), (0, 2), (0, 3), (0, 4), (0, 5), (0, 6), (0, 7), (0, 8), (0, 9), (0, 10), (0, 11), (0, 12), (0, 13), (0, 14), (0, 15), (0, 16), (0, 17), (0, 18) \}$ is extreme in $\Zp{2}{} \times \times \times \times \times \times \times \times \times \times \times \times \times \times \times \times \times \Zp{20}{}$. \newline \end{prop}
\begin{proof}
Let $\mu = \delta_{(0,0)}+e^{4\pi i/3}\delta_{(0,1)}+e^{2\pi i/3}\delta_{(0,2)}+e^{4\pi i/3}\delta_{(0,3)}+e^{4\pi i/3}\delta_{(0,4)}+\delta_{(0,5)}+\delta_{(0,6)}+e^{2\pi i/3}\delta_{(0,7)}+\delta_{(0,8)}+\delta_{(0,9)}+\delta_{(0,10)}+e^{2\pi i/3}\delta_{(0,11)}+\delta_{(0,12)}+\delta_{(0,13)}+e^{4\pi i/3}\delta_{(0,14)}+e^{4\pi i/3}\delta_{(0,15)}+e^{2\pi i/3}\delta_{(0,16)}+e^{4\pi i/3}\delta_{(0,17)}+\delta_{(0,18)}
$. Then
\begin{align*}
\mu * \tilde \mu & = \big(1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1\big)\delta_{(0,0)}
\\&\qquad
+\big(e^{4\pi i/3}+e^{4\pi i/3}+e^{2\pi i/3}+1+e^{2\pi i/3}+1+e^{2\pi i/3}+e^{4\pi i/3}+1+1+e^{2\pi i/3}+e^{4\pi i/3}+1+e^{4\pi i/3}+1+e^{4\pi i/3}+e^{2\pi i/3}+e^{2\pi i/3}\big)\delta_{(0,1)}
\\&\qquad
+\big(1+e^{2\pi i/3}+1+e^{2\pi i/3}+e^{2\pi i/3}+e^{2\pi i/3}+e^{2\pi i/3}+1+e^{4\pi i/3}+1+e^{2\pi i/3}+1+e^{4\pi i/3}+e^{4\pi i/3}+e^{4\pi i/3}+e^{4\pi i/3}+1+e^{4\pi i/3}\big)\delta_{(0,2)}
\\&\qquad
+\big(e^{2\pi i/3}+e^{4\pi i/3}+e^{4\pi i/3}+1+e^{4\pi i/3}+e^{2\pi i/3}+e^{4\pi i/3}+1+1+e^{4\pi i/3}+e^{2\pi i/3}+1+1+e^{2\pi i/3}+e^{4\pi i/3}+e^{2\pi i/3}+1+e^{2\pi i/3}\big)\delta_{(0,3)}
\\&\qquad
+\big(e^{4\pi i/3}+1+e^{2\pi i/3}+e^{4\pi i/3}+e^{2\pi i/3}+e^{4\pi i/3}+e^{4\pi i/3}+e^{2\pi i/3}+1+1+1+1+1+e^{4\pi i/3}+e^{2\pi i/3}+e^{2\pi i/3}+e^{4\pi i/3}+e^{2\pi i/3}\big)\delta_{(0,4)}
\\&\qquad
+\big(e^{2\pi i/3}+e^{2\pi i/3}+e^{4\pi i/3}+e^{4\pi i/3}+1+e^{2\pi i/3}+1+e^{2\pi i/3}+e^{2\pi i/3}+1+e^{2\pi i/3}+e^{4\pi i/3}+1+e^{4\pi i/3}+e^{4\pi i/3}+1+e^{4\pi i/3}+1\big)\delta_{(0,5)}
\\&\qquad
+\big(e^{2\pi i/3}+1+1+1+e^{4\pi i/3}+1+e^{4\pi i/3}+e^{4\pi i/3}+e^{2\pi i/3}+e^{2\pi i/3}+e^{2\pi i/3}+1+e^{4\pi i/3}+e^{4\pi i/3}+e^{4\pi i/3}+e^{2\pi i/3}+e^{2\pi i/3}+1\big)\delta_{(0,6)}
\\&\qquad
+\big(1+1+e^{4\pi i/3}+e^{2\pi i/3}+1+1+e^{2\pi i/3}+e^{2\pi i/3}+e^{4\pi i/3}+e^{2\pi i/3}+e^{4\pi i/3}+1+1+e^{2\pi i/3}+e^{4\pi i/3}+e^{2\pi i/3}+e^{4\pi i/3}+e^{4\pi i/3}\big)\delta_{(0,7)}
\\&\qquad
+\big(1+e^{4\pi i/3}+e^{4\pi i/3}+1+e^{2\pi i/3}+e^{2\pi i/3}+1+1+e^{2\pi i/3}+e^{4\pi i/3}+e^{4\pi i/3}+e^{2\pi i/3}+1+e^{4\pi i/3}+e^{2\pi i/3}+e^{2\pi i/3}+e^{4\pi i/3}+1\big)\delta_{(0,8)}
\\&\qquad
+\big(e^{4\pi i/3}+e^{4\pi i/3}+e^{2\pi i/3}+1+1+e^{4\pi i/3}+e^{2\pi i/3}+e^{2\pi i/3}+1+e^{2\pi i/3}+1+e^{2\pi i/3}+e^{2\pi i/3}+e^{4\pi i/3}+e^{4\pi i/3}+1+e^{4\pi i/3}+1\big)\delta_{(0,9)}
\\&\qquad
+\big(1+e^{2\pi i/3}+e^{2\pi i/3}+e^{4\pi i/3}+1+e^{2\pi i/3}+e^{4\pi i/3}+e^{4\pi i/3}+1+1+e^{4\pi i/3}+e^{4\pi i/3}+e^{2\pi i/3}+1+e^{4\pi i/3}+e^{2\pi i/3}+e^{2\pi i/3}+1\big)\delta_{(0,10)}
\\&\qquad
+\big(1+e^{4\pi i/3}+1+e^{4\pi i/3}+e^{4\pi i/3}+e^{2\pi i/3}+e^{2\pi i/3}+1+e^{2\pi i/3}+1+e^{2\pi i/3}+e^{2\pi i/3}+e^{4\pi i/3}+1+1+e^{2\pi i/3}+e^{4\pi i/3}+e^{4\pi i/3}\big)\delta_{(0,11)}
\\&\qquad
+\big(1+e^{4\pi i/3}+e^{2\pi i/3}+e^{2\pi i/3}+e^{4\pi i/3}+1+e^{2\pi i/3}+e^{4\pi i/3}+e^{4\pi i/3}+e^{2\pi i/3}+1+1+e^{2\pi i/3}+e^{2\pi i/3}+1+e^{4\pi i/3}+e^{4\pi i/3}+1\big)\delta_{(0,12)}
\\&\qquad
+\big(e^{4\pi i/3}+e^{4\pi i/3}+e^{2\pi i/3}+e^{4\pi i/3}+e^{2\pi i/3}+1+1+e^{4\pi i/3}+e^{2\pi i/3}+e^{4\pi i/3}+e^{2\pi i/3}+e^{2\pi i/3}+1+1+e^{2\pi i/3}+e^{4\pi i/3}+1+1\big)\delta_{(0,13)}
\\&\qquad
+\big(1+e^{2\pi i/3}+e^{2\pi i/3}+e^{4\pi i/3}+e^{4\pi i/3}+e^{4\pi i/3}+1+e^{2\pi i/3}+e^{2\pi i/3}+e^{2\pi i/3}+e^{4\pi i/3}+e^{4\pi i/3}+1+e^{4\pi i/3}+1+1+1+e^{2\pi i/3}\big)\delta_{(0,14)}
\\&\qquad
+\big(1+e^{4\pi i/3}+1+e^{4\pi i/3}+e^{4\pi i/3}+1+e^{4\pi i/3}+e^{2\pi i/3}+1+e^{2\pi i/3}+e^{2\pi i/3}+1+e^{2\pi i/3}+1+e^{4\pi i/3}+e^{4\pi i/3}+e^{2\pi i/3}+e^{2\pi i/3}\big)\delta_{(0,15)}
\\&\qquad
+\big(e^{2\pi i/3}+e^{4\pi i/3}+e^{2\pi i/3}+e^{2\pi i/3}+e^{4\pi i/3}+1+1+1+1+1+e^{2\pi i/3}+e^{4\pi i/3}+e^{4\pi i/3}+e^{2\pi i/3}+e^{4\pi i/3}+e^{2\pi i/3}+1+e^{4\pi i/3}\big)\delta_{(0,16)}
\\&\qquad
+\big(e^{2\pi i/3}+1+e^{2\pi i/3}+e^{4\pi i/3}+e^{2\pi i/3}+1+1+e^{2\pi i/3}+e^{4\pi i/3}+1+1+e^{4\pi i/3}+e^{2\pi i/3}+e^{4\pi i/3}+1+e^{4\pi i/3}+e^{4\pi i/3}+e^{2\pi i/3}\big)\delta_{(0,17)}
\\&\qquad
+\big(e^{4\pi i/3}+1+e^{4\pi i/3}+e^{4\pi i/3}+e^{4\pi i/3}+e^{4\pi i/3}+1+e^{2\pi i/3}+1+e^{4\pi i/3}+1+e^{2\pi i/3}+e^{2\pi i/3}+e^{2\pi i/3}+e^{2\pi i/3}+1+e^{2\pi i/3}+1\big)\delta_{(0,18)}
\\&\qquad
+\big(e^{2\pi i/3}+e^{2\pi i/3}+e^{4\pi i/3}+1+e^{4\pi i/3}+1+e^{4\pi i/3}+e^{2\pi i/3}+1+1+e^{4\pi i/3}+e^{2\pi i/3}+1+e^{2\pi i/3}+1+e^{2\pi i/3}+e^{4\pi i/3}+e^{4\pi i/3}\big)\delta_{(0,19)}\\&=19\delta_{(0 , 0)}\qedhere
\end{align*}
\end{proof}
}
$\mu = \delta_{(0,0)}+e^{6\pi i/5}\delta_{(0,1)}+e^{6\pi i/5}\delta_{(0,2)}+e^{8\pi i/5}\delta_{(0,3)}+e^{2\pi i/5}\delta_{(0,4)}+e^{8\pi i/5}\delta_{(0,5)}+\delta_{(0,6)}+e^{2\pi i/5}\delta_{(0,7)}+e^{2\pi i/5}\delta_{(0,8)}+\delta_{(0,9)}+e^{8\pi i/5}\delta_{(0,10)}+e^{2\pi i/5}\delta_{(0,11)}+e^{8\pi i/5}\delta_{(0,12)}+e^{6\pi i/5}\delta_{(0,13)}+e^{6\pi i/5}\delta_{(0,14)}+\delta_{(0,15)}
$.
\end{comment}
\begin{table}[]
\begin{center}
\begin{tabular} {| c| c| c|}
\hline
Group & Set (size) & Extreme measure
\\ \hline
\begin{comment}
\Zp{7}{} &0\dots 2,4 (4) * \dag&$\delta_{0} - \delta_{1} - \delta_{2} - \delta_{4} $
\\ \hline
\Zp{12}{}&0, 2, 3, 4, 7 * (5) & $ \delta_{0} + e^{ 3\pi i / 2}\delta_{2} + e^{ 5\pi i / 4}\delta_{3} + \delta_{4} + e^{\pi i / 4}\delta_{7}$
\\ \hline
\end{comment}
\Zp{10}{} & 0\dots 4,7 (6) * \dag & $\delta_{0} + e^{ 5\pi i / 6}\delta_{1} + e^{ 2\pi i / 3}\delta_{2} + e^{ 5\pi i / 6}\delta_{3} $\\&&$ + e^{ 2\pi i / 1}\delta_{4} + e^{\pi i / 6}\delta_{7}$
\\ \hline
\Zp{14}{}&0,1,2,3,4,7 (6) \dag & $\delta_{0} + e^{ 10\pi i / 12}\delta_{1} + e^{ 8\pi i / 12}\delta_{2} $\\&&$ + e^{ 2\pi i / 12}\delta_{7}$
\\ \hline
\Zp{12}{} &0\dots 2,5,6,8,9 (7) \dag & $\delta_{0} + e^{ 7\pi i / 12}\delta_{1} + \delta_{3} + e^{ 5\pi i / 6}\delta_{4} $\\&&$ + i \delta_{6} + e^{ 7\pi i / 12}\delta_{7} + i \delta_{9} + e^{ 5\pi i / 6}\delta_{10}$
\\ \hline
\Zp{16}{}&0\dots 2,4,5,7,11 (7) & $ \delta_{0} -\delta_{1} + e^{ 4\pi i / 3}\delta_{2} + \delta_{4} + e^{ 5\pi i / 3}\delta_{5} $\\&&$ + e^{ 5\pi i / 3}\delta_{7} + -\delta_{11} $\\ \hline
\Zp{19}{}&{0\dots 2,5,12,13,15} (7) & $\delta_{0} + e^{ 4\pi i / 3}\delta_{1} + \delta_{2} + \delta_{5} + e^{ 4\pi i / 3}\delta_{12} $\\&&$ + e^{ 4\pi i / 3}\delta_{13} + e^{\pi i / 3}\delta_{15}$
\\ \hline
\Zp{12}{} & 0, 1, 3, 4, 6, 7, 9, 10 (8) \dag & $\delta_{0} + e^{ 7\pi i / 6}\delta_{1} + \delta_{3} + e^{ 5\pi i / 3}\delta_{4} + e^{\pi i }\delta_{6} $\\&&$ + e^{ 7\pi i / 6}\delta_{7} + e^{\pi i }\delta_{9} + e^{ 5\pi i / 3}\delta_{10}$
\\ \hline
\Zp{16}{}&0, 1, 4, 5, 8, 9, 12, 13 (8) \dag & $\delta_{0} + e^{ 22 \pi i / 12}\delta_{1} + e^{ 18 \pi i / 12}\delta_{4} $\\&&$+ e^{ 22 \pi i / 12}\delta_{5} +\delta_{8} + e^{ 10 \pi i / 12}\delta_{9} $\\&&$ + e^{ 18 \pi i / 12}\delta_{12} + e^{ 10 \pi i / 12}\delta_{13}
$
\\ \hline
\Zp{12}{} & 0\dots 8 (9) \dag & $\delta_{0} + e^{ 23\pi i / 12}\delta_{1} + e^{ 3\pi i / 2}\delta_{2}
$\\&&$
+ e^{ 17\pi i / 12}\delta_{3} + \delta_{4} + e^{\pi i / 4}\delta_{5} $\\&&$ + e^{ 7\pi i / 6}\delta_{6} + e^{ 5\pi i / 12}\delta_{7} + e^{ 4\pi i / 3}\delta_{8}$
\\ \hline
\Zp{12}{} &0,1,2,4,5,6,8,9,10 (9) \dag &$
\delta_{0} + e^{ 10\pi i / 6}\delta_{1} + e^{ 10\pi i / 6}\delta_{2} + e^{ 8\pi i / 6}\delta_{4}
$\\&&$ + e^{ 102\pi i / 6}\delta_{5} + e^{ 2\pi i / 6}\delta_{6} $\\&&$ + e^{ 4\pi i / 6}\delta_{8} + e^{ 10\pi i / 6}\delta_{9} + e^{ 6\pi i / 6}\delta_{10}$\\ \hline
\Zp{13}{} &0\dots 5, 7, 9, 10 (9) & $\delta_{0} +\delta_{1} - \delta_{2} - \delta_{3} +\delta_{4} - \delta_{5} - \delta_{7}$\\&&$ - \delta_{9} - \delta_{10} $
\\ \hline
\end{tabular}
\vskip.125in
\caption{``Irregular'' extreme sets in cyclic groups by set size}
\label{tableexceptional}
\end{center}
\end{table}
\
\begin{table}[t]
\begin{center}
\begin{tabular} {| c| c| c|}
\hline
Group & Set size\ & Extreme measure
\\ \hline
\begin{comment}
\Zp{2}{3} &
5 * \dag &$\delta_{(0,0,0)}-\delta_{(0,0,1)}-\delta_{(0,1,0)}-\delta_{(0,1,1)}+i\delta_{(1,0,0)}$
\\ \hline
\end{comment}
\Zp{2}{3} & 6&$\delta_{(0,0,0)}-i\delta_{(0,0,1)}+e^{7\pi i/4}\delta_{(0,1,0)}
$\\&&$
+e^{3\pi i/4}\delta_{(0,1,1)}+e^{\pi i/4}\delta_{(1,0,0)}+e^{\pi i/4}\delta_{(1,0,1)}$ \\ \hline
\Zp42& 6 \dag & $\delta_{(0,0)}+e^{7\pi i/4}\delta_{(0,1)}+i\delta_{(0,2)}+e^{3\pi i/4}\delta_{(0,3)}
$\\&&$
+e^{3\pi i/4}\delta_{(1,0)}+e^{3\pi i/4}\delta_{(1,2)} $ \\ \hline
\Zp42& 6 & $ \delta_{(0,0)}+e^{23\pi i/12}\delta_{(0,1)}-i\delta_{(0,2)}+e^{\pi i/3}\delta_{(1,0)}
$\\&&$
+e^{11\pi i/12}\delta_{(1,1)}+e^{11\pi i/6}\delta_{(1,2)} $ \\ \hline
\Zp2{}\Times\Zp4{}& 6 & $\delta_{(0,0)}+e^{7\pi i/4}\delta_{(0,1)}+i\delta_{(0,2)}+e^{3\pi i/4}\delta_{(0,3)}
+e^{3\pi i/4}\delta_{(1,0)}
$\\&&$
+e^{3\pi i/4}\delta_{(1,2)}$ \\ \hline
\Zp2{}\Times\Zp4{}& 6 & $ \delta_{(0,0)}+e^{23\pi i/12}\delta_{(0,1)}-i\delta_{(0,2)}
+e^{\pi i/3}\delta_{(1,0)}
$\\&&$
+e^{11\pi i/12}\delta_{(1,1)}+e^{11\pi i/6}\delta_{(1,2)} $\\ \hline
\Zp3{2} & 7 & $ \delta_{(0,0)}+e^{5\pi i/3}\delta_{(0,1)}+e^{2\pi i/3}\delta_{(0,2)}+e^{5\pi i/3}\delta_{(1,0)}
$\\&&$
+\delta_{(1,1)}+e^{2\pi i/3}\delta_{(2,0)}+e^{\pi i/3}\delta_{(2,2)}$ \\ \hline
\Zp2{2}\Times\Zp3{}& 8 &
$\delta_{(0,0,0)}+e^{7\pi i/15}\delta_{(0,0,1)}-i\delta_{(0,1,0)}
+e^{29\pi i/30}\delta_{(0,1,1)}
$\\&&$
+i\delta_{(1,0,0)}
+e^{29\pi i/30}\delta_{(1,0,1)}-\delta_{(1,1,0)}+e^{7\pi i/15}\delta_{(1,1,1)}$
\\ \hline
\Zp4{}\Times\Zp{3}{}& 8 &$\delta_{(0,0)}-i\delta_{(1,0)}+\delta_{(2,0)}-i\delta_{(3,0)}+e^{11\pi i/6}\delta_{(0,1)}
$\\&&$
+e^{11\pi i/6}\delta_{(1,1)}+e^{5\pi i/6}\delta_{(2,1)}+e^{5\pi i/6}\delta_{(3,1)}$
\\ \hline
\end{tabular}
\vskip.125in
\caption{Extreme sets in some non-cyclic groups}
\label{tablenoncyclic}
\end{center}
\end{table}
\begin{rems}\label{remsReTables}
\begin{enumerate}
A dagger (\dag) indicates that the set is discussed in Section \secref{secProofs}.
\item
In Table \ref{tableFromCite} is, among other things, is an example of an extreme
four-element set in \Zp7{}. Seven being prime, that four-element
set is neither a coset nor a subset of a five-element coset. All the 4-element
extreme subsets of \Zp7{} are equivalent.
We have not investigated the situation in other groups of prime order, but one could make \conjref{conjexceptionalZprime}.
\item \label{itSixInSeven} Computer calculations show that the six-element set in \Zp7{} is
not extremal and hint that the ten-element set in \Zp{11}{} and the 14-element set in \Zp{15}{} are
also
\emph{not} extremal. See \secref{secComputer} for more on the computer programs and their limitations.
\item We have found some cyclic
groups contain two non-equivalent extreme sets of the same cardinality. We provide proofs
for that assertion below.
\item Computer calculation with the 12-element set in \Zp{13}{} suggest that it is
extreme.
However, the masses suggested by that
computer calculation do not give a clear indication of what might be an extreme measure,
those masses involving exponents whose denominator is $2^{22}$. It is possible that replacing
one or more of the factors of 2 with 11 would produce a better result, but using
such a large factor would slow the program down impossibly. Hence, this set does not appear in
Table \ref{tableexceptional}.
\item\label{remRealExtremtransform}
Consideration of the 17-element set in \Zp{18}{} was occasioned
by the thought that the convolution of
$\mu*\tilde\mu$, when $\mu$ is extreme on the 17-element set, would produce 16 terms at each of
the elements $1,\dots,17$ and that therefore we had the possibility of cancellation if the masses involved were $\pm 1$, $\pm i$ only, which would make for the most rapid machine computation, and that occurred. Something similar might hold for the 65-element set in \Zp{66}{} but the calculation here would likely take $2^{49}$ times as long.
We note that if $\mu$ above is multiplied by $\delta_{-8}$, the resulting measure is
self-adjoint (that is, $(\delta_{-8}*\mu)\widetilde{\ \ } =\delta_{-8}*\mu$). Hence, the set
$\{-8,\dots,8\}$ is the support of an extreme measure with real transform.
A similar thought worked for 5th (resp. 3rd) roots of unity in the case of
16 elements in \Zp{17}{} (resp. 19 elements in \Zp{20}{}).
In general, if $k$ has a small factor, $j$, then there is the possibility that $j$th roots will appear
as masses of an extreme measure on the set of $k+1$ elements
in \Zp{k+2}{} and be quick for the computer to find if $j$ is sufficiently small.
That won't work for 12 elements
in \Zp{13} nor for 18 elements in \Zp{19}{}.
\item If the extreme measure given for $\{0-16\}\subset\Zp{18}{}$ is multiplied by $\delta_{-8}$, the resulting measure is
self-adjoint (that is, $(\delta_{-8}*\mu)\widetilde{\ \ } =\delta_{-8}*\mu$). Hence, the set
$\{-8,\dots,8\}$ is the support of an extreme measure with real transform.
\enumisave
\end{enumerate}
\medbreak
That suggests \conjref{conjnlessone} below.
\medbreak
\begin{enumerate}\setcounter{enumi}{\value{keepenumi}}
\item We have found no other six-element extreme sets (other than cosets and homomorphic images of $\{0,1,2,3,4,7\}$) in groups whose order is a multiple of $10$) in any cyclic group of order at most 29. (That search took nearly 8 days of continuous computation.)
\enumisave
\end{enumerate}
\medbreak
That suggests \conjref{conj6elts}.
\medbreak
\begin{enumerate}\setcounter{enumi}{\value{keepenumi}}
\item Sets of the form $\{0,g\}+\{0, n\}$ in \Zp{2n}{} and $1\le g<n$ are extremal according to
\cite[Theorem 2.1]{MR627683} and are also omitted from the Table \ref{tableexceptional},
as are images of 4-element sets in cosets of size 5 and 7.
We also omitted sets contained in subgroups and sets of cardinality a perfect square that
are obtained by application of \thmref{thmGR2.1} as well as
those extreme sets that are the sum of an extreme set with a coset. For example,
$\{0,1,2,3,5,6,7,8\}\subset \Zp{10}{}$ is the sum $\{0,5\}+\{0,2,6,8\}$.
Similarly,
$\{0,3,6,12\}+\{0,5,10\}\subset \Zp{15}{}$.
Those sort of examples will complicate
the process of finding all extreme sets of composite cardinality.
\item Further complicating matters is that the program gives
\[ \delta_0+e^{2\pi i/3}(\delta_1+\delta_2)+\delta_3\] a
s an extreme measure on $\{0,1,2,3\}\subset\Zp5{} $ if one starts the search
with third roots of unity, but if one starts with 15th roots of unity, $\delta_{0}+e^{2\pi i/15}\delta_{1}+e^{14\pi i/15}\delta_{2}+e^{2\pi i/5}\delta_{3}
$ is produced.
Similarly, starting the search for 11 elements in \Zp{12}{} with mesh 5 gives an extreme measure with
10th roots of unity (in contrast to the one given in
Table \ref{tableRegular}, which had 8th roots):
$ \delta_{0}+\delta_{5}+\delta_{10}
+e^{4\pi i/5}(\delta_{1}+\delta_{9})
+e^{2\pi i/5}(\delta_{2} +\delta_{8})
+e^{8\pi i/5}(\delta_{3}+\delta_{4}
)+e^{8\pi i/5}(\delta_{6}+\delta_{7})$.
Also, different variants of the search program can give different extreme measures: a late
variant of the search program gave the
extreme measure in Table \ref{tableRegular} for the 13-element set in \Zp{14}{}, $\delta_{0} + \delta_{2} + \delta_{10}+ \delta_{12}
+ e^{ 7\pi i / 4}(\delta_{1} + \delta_{9} )
+ e^{ 3\pi i / 4}(\delta_{3} + \delta_{8} + \delta_{11} )
+ e^{ 3\pi i / 2}\delta_{4} + e^{ 5\pi i / 4}\delta_{5}
+ i\delta_{6} + e^{ \pi i / 4}\delta_{7}$
was given by an earlier variant.
\item The 8-element set in \Zp{12}{} is \emph{not} the sum of a coset and an extreme set; see \lemref{lemsumsinZtwlv}
and \propref{prop8in12}.
\item As nearly as we and our (probably poor) programs can tell, the
two 6-element sets in $\Zp2{} \times \Zp{4}{}$ given in Table \ref{tablenoncyclic} are not equivalent.
\end{enumerate}
\end{rems}
\conjref{conjnlessone} does not ``explain'' the extremality of $\{0,1,2,4\}$ in \Zp{7}{}, the extremality of $\{0,1,2,3,4,7\}$ in \Zp{10}{}, nor the extremality of $\{0,1,2,5,6,8,9\}$ in \Zp{12}{}. The computer has looked at subsets of \Zp{11}{} up to size 8 and not found any extreme ones.
\section{Conjectures, questions, and a few proofs of extremality}\label{secConjectures}
\subsection{Conjectures and questions}\label{subsecconjs}
All of these conjectures and questions are suggested by the examples so far or by
numerical evidence -- or by the desire to know how long a computer run will take.
\begin{conj}\label{conjnlessone}
An $n-1$ element subset of \Zp{n}{} is extreme iff $n$ is not congruent to 3$\mod$ 4.
\end{conj}
\begin{conj} \label{conj6elts}
The only six-element extreme sets in cyclic groups are \Zp{6}{}, $\{0,1,2,3,4,7\}$ $\subset\Zp{10}{}$,
and their images under
group homomorphism and translation.
\end{conj}
\begin{conj} \label{conjexceptionalZprime}
If $p$ is prime and $1<n<p$, all the extreme $n$-element subsets of \Zp p{} are equivalent.
\end{conj}
\begin{conj}\label{conjDiffPS}
If two subsets of \Zp{k}{} have the same cardinality, the same Sidon constant and are in different equivalence classes,
then they are extreme.
\end{conj}
\begin{conj}\label{conjprimecards}
Extreme sets with prime cardinality appear ``more'' frequently than sets whose cardinality is not prime
and not a perfect square.
\end{conj}
\begin{ques}
What is the rate of growth of the number of distinct extreme sets of cardinality $N$ in terms of $N$.
``Distinct'' means ``not carried to one another by group injections, automorphisms or translations. ''
\end{ques}
\begin{ques}
Is there an explicit formula for the number of equivalance classes of size $k$ in a group of order $n$?
Or an order of growth in terms of $k$ and $n$?
\end{ques}
\begin{ques}\label{quesRealTransf}
What extreme sets support extreme measures with real transforms?
\end{ques}
\subsection{Extremal sets and their measures for cyclic groups} \label{secProofs}
In this subsection we give a few sample proofs of some of the extremalities claimed earlier and of related results.
In almost all cases of extremality, we
leave it to the reader to show that the measure in the relevant table is indeed extreme.
We also show that the extreme sets here are neither sums of other extreme sets nor of a subgroup and a set as in
\thmref{thmGR2.1}.
We begin with proving a few extremality results from \cite{MR627683} to illustrate what was omitted from that paper.
\begin{prop}\label{prop3inZ3}
$\Zp{3}{}$ and $\mu = \delta_{0}+e^{4\pi i/3}\delta_{1}+\delta_{2}
$ are extreme. \newline \end{prop}
{
\begin{proof}
$\mu * \tilde \mu
= \big(1+1+1\big)\delta_{0}
+\big(1+e^{4\pi i/3}+e^{2\pi i/3}\big)\delta_{1}
+\big(e^{2\pi i/3}+e^{4\pi i/3}+1\big)\delta_{2}
=3\delta_{0}
$
\end{proof}
}
\begin{prop}\cite{MR627683}\label{prop3in4}
$\{0,1,2\}\subset \Zp4{}$ and $\nu = \delta(0) + e^{3\pi i/4}\delta(1) +i\delta(2)$ are extreme.
\end{prop}
\omitproof{
\begin{proof}
Then
\begin{align*}
\widehat\nu(0) &= 1- \frac{\sqrt2}2 + \frac{\sqrt2}2i +i, \text{ so }
\\
|\widehat\nu(0)| &= \big|(1-\frac{\sqrt2}2)^2 +(1+\frac{\sqrt2}2)^2\big|^{1/2} =\sqrt3.
\\
\widehat\nu(1) &= 1 -\frac{\sqrt2}2 -\frac{\sqrt2}2i -i, \text{ so }
|\widehat\nu(1)|= \sqrt3\\
\widehat\nu(2) & = 1 +\frac{\sqrt2}2 -\frac{\sqrt2}2i +i \text{, so } |\widehat\nu(2)|= \sqrt3.
\\
\widehat\nu(3) & = 1 +\frac{\sqrt2}2 +\frac{\sqrt2}2i -i, \text{ so }
|\widehat\nu(3)| = \sqrt3.\qedhere
\end{align*}
\end{proof}
}
Another extreme measure is $\mu=\delta(0) - e^{3\pi i/4}\delta(1) + i\delta(2)
=\delta(0) + e^{7\pi i/4}\delta(1) + i\delta(2)$.
All extremal measures have one of the forms, $\nu$ or $\mu$:
Suppose $\mu=\delta_0+a\delta_1+b\delta_2$ is extreme on $\{0,1,2\}\subset \Zp4{}$. Then
$\mu*\tilde\mu=3\delta_0+(a+\bar ab)\delta_1 + (b+\bar b)\delta_2 +
(\bar a+a\bar b)\delta_3=3\delta_0$.
Hence, $b+\bar b=0$ so $b=\pm i$. Assume $b=i$. Then
$\bar a-ia=0$ so\footnote{\
Let $a= x+iy$. Then $\bar{a}-ia= x-iy-ix+y=0$ means $x+y=0$. Since
$|a|=1$, $x= \pm\sqrt2/2$.}
either $a=\pm e^{\pm\pi i/4}$ or $a =\pm e^{\pm 3\pi i/4}$.
Since $y=-x$, we have $a=\pm \exp(3\pi /4).$
\medskip
Every three-element extreme set can be obtained from \Zp{3}{} and a 3-three element subset of \Zp{4}{}
by the operations of group automorphism, passing to a subgroup, and translation
\cite[3.1(ii)]{MR627683}. See \cite{ABeastiaryAppendix} for a proof of part of \cite[3.1(ii)]{MR627683}. The complications of the proof there illustrate
why the proofs
of \cite[3.1-3.3]{MR627683} occupied 200 pages of manuscript.
\subsection{Non-equivalent extreme sets}
\begin{prop}
\cite{MR627683}
\label{prop5in12} The subsets $E=\{0,2,4,6,8\}$, $F=\{0,2,3,4,7\}$ of \Zp{12}{} are extreme but
not equivalent.
\end{prop}
\begin{proof}
Non-equivalence:
the group automorphisms of \Zp{12}{} are multiplication by 5, 7 and 11 (all mod 12). Each of them takes odd elements of \Zp{12}{} to odd elements and even elements to even elements. Translations either take evens to evens and odds to odds or evens to odds and odds to evens. Thus, no combination of group automorphisms and translations can take
$E$, whose image will contain either only evens or only odds, onto $F$, which contains both evens and odds.
$\{0,2,4,6,8\}$ is extreme because it is a five-element subset of the the coset $\{0,2,4,6,8,10\}$ in \Zp{12}{}.
The proof of the extremality of the second set is in \cite{ABeastiaryAppendix}.
\end{proof}
\begin{prop}\label{prop7in16} Each of 7-element sets
\begin{enumerate}
\item
$\{0,2,4,6,8,10,12\}$ and
\item $\{0,1,2,4,5,7,11\}$
\end{enumerate}
is extreme in \Zp{16}{}.
They are not equivalent.
\end{prop}
\begin{proof}
The sets are not equivalent because the automorphisms
(multiplication by odd integers) preserve the parity of
elements and translation switches or leaves fixed the
parity. Since the first set has both odd and even elements
so will every set equivalent to it. (We will use this argument
again in \propref{prop8in16}.)
As for extremality, first note that $\{0,2,4,6,8,10,12\}$ is extreme because it is a 7 element subset of an 8 element coset
and so extreme. For the other set, use the measure in Table \ref{tableexceptional}. Further details are
in \cite{ABeastiaryAppendix}.
\end{proof}
\begin{prop}\label{prop8in12} The 8-element set
$\{0,1,3,4,6,7,9,10\}$
is extreme in \Zp{12}{} and is not the sum of two extreme sets in spite of being the sum $ \{0,1,3,4\} +\{0,6\}$
and also the sum $\{0,3,6,9\}+\{0,1\}$.
\end{prop}
\begin{proof}
The reader can verify for herself that the measure given in Table \ref{tableexceptional} for this set is extreme.
In the proof of non-equivalency we apply \lemref{lemsumsinZtwlv} below and the fact that $\{0,1\}$ is not extreme in \Zp{12}{}\footnote{The difference
$\{0,1\}-\{0,1\}$ produces the terms $0-0$, 1-1, 0-1, 1-0 and so $\{0,1\} \subset \Zp{12}{} $ fails the test of \corref{corExtrCnxln}.}
\end{proof}
\begin{lem}\label{lemsumsinZtwlv}
\Zp{12}{} has no 8-element set that is a sum of a four-element extreme set and a coset.
\end{lem}
\begin{proof}
Suppose $E= A+B$ has 8 elements, where $A$ has 4 elements and $B$ has two. We may assume $0\in A$ and $B=\{0,6\}$.
Clearly $A$ cannot contain 6. \corref{corExtrCnxln} and calculation (both by hand and machine) show that if a 4-element set in \Zp{12}{}
lacks 6, it is not
extreme.\footnote{\,
The only 4-element extreme sets of \Zp{12}{} (up to equivalence)
are $\{0,1,6,7\},$ $\{0, 2,6,8\}$ and $\{0,3,6,9\}$, all containing 6. Several other four-element sets pass the test of
\corref{corExtrCnxln} but are not extreme.
\end{proof}
\begin{prop}\label{prop8in16} The 8-element set
$\{0, 1, 4, 5, 8, 9, 12, 13\}$
is extreme in \Zp{16}{} even though it is not equivalent to a coset of $\{0,2,4,6, 8,10,12,14\}$.
\end{prop}
\begin{proof}
The set is not equivalent to a coset because the autormorphisms
(multiplication by odd integers) preserve the parity of
elements and translation switches or leaves fixed the
parity. Since this set has both odd and even elements,
so will every set equivalent to it.
We leave to the reader that the measure given in Table \ref{tableexceptional} is extreme.
\end{proof}
\begin{prop}\label{prop9in12} The 9-element sets
$\{0,1,2,3,4,5,6,7,8\}$ and \\$\{0,1,2,4,5,6,8,9,10\}$ are extreme in \Zp{12}{}. They are non-equivalent.
\end{prop}
\begin{proof}
There are two ways to prove non-equivalence. First, by a tedious calculation (which we delegated to a computer). The second is to show that one of the sets is the sum
of a coset with a three-element set and the other is not, which we do in the next paragraph.
The only 3-element subgroup in \Zp{12}{} is $\{0,4,8\}$. If $\{0,4,8\}+\{a\}
\subset\{0,1,2,3,4,5,6,7,8\}$, then $a\ne\, 1,2,3$,
since $8+1=9,8+2=10, 8+3=11$ are
not in $E.$ Similarly,
$a\ne 5,6,7,9,10,11$
since none of those elements is in $E$. Thus, $a\in \{0,4,8\}$ and the first set is seen not be a sum.
Of course, $\{0,4,8\}+\{0,1,2\}=\{0,1,2,4,5,6,8,9,10\}$ and both final conclusions follow.
Proofs that the measures given in Table \ref{tableexceptional} are extreme are given in
\cite{ABeastiaryAppendix}.
\end{proof}
\begin{comment}
\begin{prop}\label{prop6inZ2.X.Z4}
Each of the following 6-element sets
is extreme in $\Zp{2}{} \times \Zp{4}{}$.
\begin{enumerate}
\item$\{(0, 0), (0, 1), (0, 2), (0, 3), (1, 0), (1, 2) \}$ with extreme measure
$\mu = \delta_{(0,0)}+e^{7\pi i/4}\delta_{(0,1)}+i\delta_{(0,2)}+e^{3\pi i/4}\delta_{(0,3)}+e^{3\pi i/4}\delta_{(1,0)}+e^{3\pi i/4}\delta_{(1,2)}
$
\item $\{(0, 0), (0, 1), (0, 2), (1, 0), (1, 1), (1, 2) \}$ with extreme measure
$\mu = \delta_{(0,0)}+e^{23\pi i/12}\delta_{(0,1)}-i\delta_{(0,2)}+e^{\pi i/3}\delta_{(1,0)}+e^{11\pi i/12}\delta_{(1,1)}+e^{11\pi i/6}\delta_{(1,2)}
$.
\end{enumerate}
\end{prop}
\omitpf{
\begin{proof}
(1.)
Let $\mu = \delta_{(0,0)}+e^{7\pi i/4}\delta_{(0,1)}+i\delta_{(0,2)}+e^{3\pi i/4}\delta_{(0,3)}+e^{3\pi i/4}\delta_{(1,0)}+e^{3\pi i/4}\delta_{(1,2)}
$. Then
\begin{align*}
\mu * \tilde \mu & = \big(1+1+1+1+1+1\big)\delta_{(0,0)}
\\&\qquad
+\big(e^{5\pi i/4}+e^{7\pi i/4}+e^{3\pi i/4}+e^{\pi i/4}\big)\delta_{(0,1)}
\\&\qquad
+\big(-i-1+i-1+1+1\big)\delta_{(0,2)}
\\&\qquad
+\big(e^{\pi i/4}+e^{5\pi i/4}+e^{7\pi i/4}+e^{3\pi i/4}\big)\delta_{(0,3)}
\\&\qquad
+\big(e^{5\pi i/4}+e^{7\pi i/4}+e^{3\pi i/4}+e^{\pi i/4}\big)\delta_{(1,0)}
+\big(-1+1+1-1\big)\delta_{(1,1)}
\\&\qquad
+\big(e^{5\pi i/4}+e^{7\pi i/4}+e^{\pi i/4}+e^{3\pi i/4}\big)\delta_{(1,2)}
+\big(-1+1-1+1\big)\delta_{(1,3)}\\&=6\delta_{(0 , 0)}.
\end{align*}
(2).
Let $\mu = \delta_{(0,0)}+e^{23\pi i/12}\delta_{(0,1)}-i\delta_{(0,2)}+e^{\pi i/3}\delta_{(1,0)}+e^{11\pi i/12}\delta_{(1,1)}+e^{11\pi i/6}\delta_{(1,2)}
$. Then
\begin{align*}
\mu * \tilde \mu & = \big(1+1+1+1+1+1\big)\delta_{(0,0)}
\\&\qquad
+\big(e^{23\pi i/12}+e^{19\pi i/12}+e^{7\pi i/12}+e^{11\pi i/12}\big)\delta_{(0,1)}
\\&\qquad
+\big(i-i+i-i\big)\delta_{(0,2)}
\\&\qquad
+\big(e^{\pi i/12}+e^{5\pi i/12}+e^{17\pi i/12}+e^{13\pi i/12}\big)\delta_{(0,3)}
\\&\qquad
+\big(e^{5\pi i/3}-1+e^{5\pi i/3}+e^{\pi i/3}-1+e^{\pi i/3}\big)\delta_{(1,0)}
\\&\qquad
+\big(e^{19\pi i/12}+e^{7\pi i/12}+e^{11\pi i/12}+e^{23\pi i/12}\big)\delta_{(1,1)}
\\&\qquad
+\big(e^{\pi i/6}+e^{7\pi i/6}+e^{5\pi i/6}+e^{11\pi i/6}\big)\delta_{(1,2)}
\\&\qquad
+\big(e^{13\pi i/12}+e^{\pi i/12}+e^{5\pi i/12}+e^{17\pi i/12}\big)\delta_{(1,3)}\\&=6\delta_{(0 , 0)}\qedhere
\end{align*}
\end{proof}
}
\mpar{How is this Prop related to \propref{prop6inZ4.X.Z4} and
\propref{prop6inZp2.Zp4.Zp4}?}
\begin{prop}\label{prop6inZp2.Zp4.Zp4}
Each of the following 6-element sets is extreme in $\Zp{2}{} \times \Zp{4}{2}$.
\begin{enumerate}
\item $\{(0, 0, 0), (0, 0, 1), (0, 0, 2), (0, 0, 3), (1, 0, 0), (1, 0, 2) \}$
\item $\{(0, 0, 0), (0, 0, 1), (0, 0, 2), (0, 0, 3), (1, 1, 0), (1, 3, 2) \}$
\item $\{(0, 0, 0), (0, 0, 1), (0, 0, 2), (0, 0, 3), (1, 2, 0), (1, 2, 2) \}$
\item $\{(0, 0, 0), (0, 0, 1), (0, 0, 2), (0, 0, 3), (1, 1, 0), (1, 3, 0) \}$
\end{enumerate}
\end{prop}
\end{comment}
\section{The computer programs}\label{secComputer}
\subsection{General descriptions}
We have two sets of programs: those searching for extreme sets and those checking
that a putative extreme set is ex\-treme. In each set there are
separate programs for cyclic
and non-cyc\-lic groups. The search programs for cyclic groups are of two types: those which examine the Fourier-Stieltjes transforms
of measures, and those which look at the convolution $\mu*\widetilde\mu$ of candidate measures to see
if the coefficients of that product are (close to) zero except at the identity.
The search programs divide into two classes of two streams each:
\begin{itemize}
\item findBest -- searchs for measures with minimal FST
\item findX -- searches for extreme measures (and keeps fewer candidates than findBest)
\end{itemize}
In each case
the program saves a list of promising candidates and discards less promising ones, assuming
there are not too many ``promising'' ones. Each saved
measure is used as a starting point for an increased mesh, usually doubled, at the next pass.\
The programs are not perfect, are in my poor C, and have been run
only in (Debian 8 \& 9) Linux BASH terminal windows --
there is no proper user interface, much less a GUI front end.
One edits the source code slightly and recompiles for each group and set size(s). Whether my programs
can be run
in other operating systems without changes I do not know, though
for other versions of Linux the answer is almost surely yes.
The programs that look for extreme sets can be assigned to use up to 16 GB of resident RAM for
storing promising measures for each group/setsize pair. Those programs are easily modified to
use less RAM, in which case they
will also run faster and give less reliable results (see next section).
\subsection{Confidence in the results} This depends on the answer a search program gives.
Each time a program
said, ``this set is extreme and here is an extreme measure,'' the program was correct. If the program
said, ``there's no extreme measure on this set,'' one can be confident \emph{only if} a) the
program has looked at all possible measures directly, or b) if the program shows that
the measures not considered cannot be extreme because
they are too close to non-extreme ones (a little differential calculus is useful in
setting up the relevant inequalities). For example, in the case of six elements in \Zp7{}, the
search programs showed that the square of the PSC is at least 6.74670307754671
with an error of at most 0.000467467. Hence, the set is not extreme. Alternative calculations, one written in $C$ by the author
and the other
in Mathematica by L. T. Ramsey, show that each candidate measure $\mu$ is such that $\mu*\tilde\mu-6\delta_0$
has at least one coefficient that's greater than 0.3.
The programs (whether looking at the FST or looking at $\mu*\widetilde\mu-(setsize)\delta_0$) discard
candidates if there are too many to keep for further examination. This means a ``no extreme set here''
result cannot be relied upon.
\subsubsection{ 10 elements in \Zp{11}{}} The case of 10 elements in \Zp{11}{} is instructive. The
search for an extremal measure
reported discarding many billions of candidates. Not discarding candidates would have required
least 43 GB of RAM to get beyond an initial mesh of 8 (that took 1 minute 23 seconds)
and 19 terabytes of RAM with an initial mesh of 16 (which took 11.5 hours). Writing candidates to disk and reading them back are even more time-consuming activities in and of themselves.
Increasing the starting fineness of the initial search
(a possible way to reduce the potential number of candidates stored) will increase the time
needed for the first pass: starting with mesh = 16 takes about 11.5 hours and starting at 32
will take more than 9 months, with no assurance that the number of candidates needed to look at will be fewer
than for mesh = 16.
Looking at the mesh 8 case again: there were about 80,000,000 candidates discarded. To search each of them
to final mesh of 64 I estimate would take a bit more than two centuries.
\subsubsection{More generally, $n-1$ elements in \Zp{n}{}} Each point mass in the expansion of $\nu=\mu*\tilde\mu$
(other than at the identity) has $n-1$ terms. For them to sum to zero they must involve either the
$(n-2)^{th}$ roots of unity or $k^{th}$ roots of unity where $k$ divides $n-1$, possibly rotated.
Thus, the most promising search involves starting with one of those $k$s. Of course, for 12 elements in \Zp{13}{},
this means starting with an initial mesh of 11, which would take several years to complete. We started the
search for an extremal measure on the 23-element subset of \Zp{24}{}. It took 23.5 seconds to complete
that initial search. Assuming that 23 of those seconds were in setting up, that means that some 6000
millenia would be needed (for our computer program on our hardware) to do mesh 4 with no discarding of
candidates. Alternatively, to process, say, 10 million candidates saved by the mesh 2 pass for mesh 4
would take mere additional 38 years.
Of course, in $n$ is not too large and $n-2$ has a small
factor, finding an extremal measure may be possible, as the cases of $n$=12, 14, 17, 18, and 20
show.
\smallbreak
\subsection{Can we go further?}
Not much. As the above indicates, a ``not extreme'' report for a set with 10 or
more elements cannot be believed and furthermore, the time and memory demands increase so rapidly that, absent
more than several orders of magnitude in computational power,
there will be no computationally trustworthy ``not extreme'' results
for these larger sets in the forseeable future.
``Is extreme'' results \emph{are} easier to come by as indicated, but of course each additional element in the
set increases search time by at a factor of 2 for initial mesh = 2,
and initial meshes greater than 3 are not feasible
for larger sets.
\subsubsection{Summary of particular cases}
If an $n-1$ element subset of \Zp{n}{} ($n\le 15$) does not appear in Table \ref{tableRegular}, then the comuter
program has given an unreliable ``not extreme'' conclusion.
\begin{comment}
------------------------
To do 2019-06-26
3 in Z4 (done) Not extreme
4 in Z5 (done) Extreme 3rd roots of 1
5 in Z6 (done) Extreme 4th roots of 1
6 in Z7 (done) not extreme
7 in Z8 (done) Extreme 12th roots of 1
8 in Z9 (done) extreme 42nd roots of 1
9 in Z10 (done) extreme 6th roots of 1
10 in Z11 3rd roots of 1? has been run but lots of rollovers
11 in Z12 (done) Extr 8th roots of 1
12 in Z13 11th roots of 1
13 in Z14 (done) Extr 3rd roots of 1
14 in Z15 13th roots of 1 running
15 in Z16 ready to go with 2nd roots of 1
16 in Z17 (done) Extr 10th roots of 1
17 in Z18 (done) Extr 4th roots of 1
18 in Z19 17th roots of 1
19 in Z20 (done) Extreme 3rd roots of 1
20 in Z21 19th roots of 1
21 in Z22 4th \& 5th roots of 1 running
23 in Z24 2nd \& 11th roots of 1.
\end{comment}
\subsection{More on searches}
\subsubsection*{Bounding the PSC} This gives an upper bound of the PSC.
If this type of search gives PSC equal to the square root
of the setsize, we have found an extremal set. If the search fails to show the
PSC is the root of the set size, we cannot conclude that the
set is not extremal unless nothing was thrown away and the search is fine enough.
\subsubsection*{Bounding the mass of $\nu*\widetilde\nu-(setsize)\delta_0$}
Assume $\mu$ is extremal, so $\mu*\widetilde\mu = (setsize)\delta_0$.
Let $\nu$ be a test measure\footnote{\ Both measures here have mass $setsize$ with weights of equal
absolute value at each support point and unit point mass at the identity.}. Then
\begin{equation}\label{eqClose1}
\|(setsize)\delta_0-\nu*\widetilde\nu \| =\|\mu*\tilde\mu -\nu*\tilde\nu\|\le 2 (setsize) \|\mu-\nu\|.
\end{equation}
The above is in turn is bounded on the right by
\begin{equation}\label{eqClose2}
\varepsilon =
2 \,(setsize)\, (setsize -1) \frac{2\pi }{mesh},
\end{equation}
when $\nu$ is the measure closest to $\mu$ in our search
and $mesh$ is the number of distinct masses considered at each point of $E$ (giving rise to time estimates
of the order of $(mesh)^{setsize-1}$).
If all the measures looked at with a particular $mesh$ satisfy
\begin{equation}\|\nu*\widetilde\nu\| > setsize + \varepsilon,
\end{equation}
then no finer search (i.e., with larger $mesh$'s) will produce an extremal measure . Hence,
there can be no extremal measure on that set.
This means that we make our lists of measures that satisfy
\[
\|\nu *\widetilde\nu\| \in [setsize-PRECISION, setsize +\varepsilon+PRECISION].
\]
Now, to speed up the program, instead of calculating $|\nu*\widetilde\nu(\{g\})|$,
which would involve two squares and a square root for each of $grouporder-1$ points,
we calculate
\[
|\Re \prn{\nu*\widetilde\nu(\{g\}}| + |\Im\prn{\nu*\widetilde\nu(\{g\}}|
\]
at $grouporder-1$ points.
That introduces a factor of
$\sqrt{2}$
and means that we make our lists using measures that satisfy
\[
\|\nu *\widetilde\nu\| \in [setsize-PRECISION, setsize +\varepsilon\sqrt2+PRECISION].
\]
I usually set PRECISION at $10^{-7}$. Changing it by a factor of 10 does not affect results much.
\subsection{\ \ Time complexity} \label{secTimeComplexity} As discussed above, I do not have much hope of going beyond sets with 10 or 11 elements in groups of size below 25 (the 17-element set
in \Zp{18}{} was a lucky guess): my (uncompleted) searches for 9 element sets
have taken several weeks using a reasonably fast CPU (Ryzen 7). Doubling or tripling the speed would add
perhaps one or two elements to the set size that my program can do in a reasonable time\footnote{\ Defined as the interval between power cuts in Whitehorse of more than 15 minutes, 15 minutes being the
estimated capacity of my UPS.}, as computation time is
at best exponential in set size (that is, $O(L\cdot mesh^{setsize)} $, where $L$ is the number of equivalence classes, which could be several hundred) plus another bit that's \proofBitFormat{binom(groupsize, set size)}. the binomial part comes from the search for equivalence classes.
If the program takes more than 2 days to find equivalence classes for a particular choice of group and setsize, I
kill it.
The situation is worse for non-cyclic groups because adding a
factor doubles (or worse) the size of the group and
vastly increases the number of ``equivalence'' classes the program
generates as well as the time to generate them.
Here are some examples, given to assuage my conscience about killing searches.
\begin{itemize}
\item The search for 7-element sets in $\Zp{2}{2}\times\Zp5{}$ (20 group elements)
produced 58 ``equivalence'' classes and took about 5 days to complete with a max mesh of 720.
\item Big sets take time. The search for 9-element sets in \Zp42 got into the
second class of 40 on its 7th day and was then killed. Estimated time to complete is thus 200-280 days.
\item Big groups lead to big problems. The search for 6-element sets in \Zp72 (49 group elements) spent several
days trying to generate the list of equivalence classes
before I killed it. I have no idea how many equivalence classes it
would have found nor how long to compute all their PSCs.
\end{itemize}
\bibliographystyle{amsplain}
|
2104.06947
|
\section{Introduction}
Billiards are a ubiquitous source of models in physics, in particular in statistical mechanics. The study of the ergodic properties of billiards is of paramount importance for such applications and also a source of innovative ideas in ergodic theory. In particular, starting with at least \cite{Kry}, it has become clear that a quantitive estimate of the speed of convergence to equilibrium is pivotal for this research program. The first strong result of this type dates back to Bunimovich, Sinai and Chernov \cite{bsc} in 1990, but it relies on a Markov-partition-like technology that is not very well suited to producing optimal results. The next breakthrough is due to Lai-Sang Young \cite{You98, You99} who put forward two techniques, towers and coupling, well suited to study the decay of correlations of a large class of systems, billiards included. The idea of coupling was subsequently refined by Dolgopyat \cite{Do04a,Do04b, Do05} who introduced the notion of standard pairs, which have proved a formidable tool to study the statistical properties of dynamical systems in general and billiards in particular \cite{Ch06, Ch08, CD, chernov zhang}. See \cite[Chapter~7]{chernov book} for a detailed exposition of these ideas and related references.
In the meantime another powerful idea has appeared, following the seminal work of Ruelle \cite{RS75, Ruelle76} and Lasota-Yorke \cite{LY73}, to study the spectral properties of the associated transfer operator acting on
spaces of functions adapted to the dynamics. After some preliminary attempts \cite{Fr86, Rugh, Kitaev}, the functional approach for hyperbolic systems was launched by the seminal paper \cite{BKL}, which was quickly followed and
refined by a series of authors, including \cite{Ba1, gouezel liverani, BaT, gouezel liverani2}.
Such an approach, when applicable, has provided the strongest results so far, see \cite{Babook} for a recent review. In particular, building on a preliminary result by Demers and Liverani \cite{DL08}, it has been applied to billiards by Demers and collaborators \cite{demzhang11, demzhang13, demzhang14, dem bill, max, thermo}.
This has led to manifold results, notably the proof of exponential decay of correlations for certain billiard flows \cite{BDL}.
Yet, lately there has been a growing interest in non-stationary systems, when the dynamical system changes with time. Since most systems of interest are not isolated, not even in first approximation, the possibility of a change to
the system due to external factors clearly has physical relevance. Another important scenario in which non-stationarity appears is in dynamical systems in random media, e.g. \cite{AL}. The functional approach as such seems not to be well suited to treat these situations since it is based on the study of an operator via spectral theory.
In the non-stationary case a single operator is substituted by a product of different operators and spectral theory
does not apply.
There exist several approaches that can be used to overcome this problem, notably:
\begin{enumerate}
\item consider random systems; in this case, especially in the annealed case, it is possible to recover an averaged transfer operator to which the theory applies. More recently, the idea has emerged to study quenched systems via infinite dimensional Oseledets theory, see e.g. \cite{DFGV1, DFGV2} and references therein;
\item consider only very slowing changing systems that can be treated using the perturbation theory in \cite{KL99, gouezel liverani}. For example, see \cite{DS}, and references therein, for some recent work in this direction;
\item use the technology of standard pairs, which has the advantage of being very flexible and applicable to the non-stationary case \cite{young zhang}. Note that the standard pair technology and the previous perturbation ideas can be profitably combined together, see \cite{DS16, DS18, DLPV};
\item use the cone and Hilbert metric technology introduced in \cite{liv95, liv95b, LM}, which has also been extended
to the random setting \cite{AL, atnip}.
\end{enumerate}
The first two approaches, although effective, impose severe limitations on the class of nonstationary systems that can be studied. The second two approaches are more general and seem more or less equivalent. However, coupling arguments are often cumbersome to write in detail and usually provide weaker quantitative estimates compared to the cone method.
Therefore, in the present article we develop the the cone method and demonstrate that it can be successfully applied to billiards. Indeed, we introduce a relatively simple cone that is contracted by a large class of billiards. This implies that one can easily prove a loss of memory result for sequences of billiard maps. To show that the previous results have concrete applications we devote one third of this paper to developing applications to several physically relevant classes of models.
We emphasize that the present paper does not exhaust the possible applications of the present ideas. To have a more complete theory one should consider, to mention just a few, billiards with corner points, billiards with electric or magnetic fields, billiards with more general reflection laws, measures different from the SRB measure (that is transfer operators with generalized potentials as in \cite{max, thermo}), etc. We believe that most of these cases can be treated by small modifications of the present theory; however, the precise implementation does require a
non-negligible amount of work and hence exceeds the scope of this presentation, which aims only at introducing the basic ideas and producing a viable cone for dispersing billiards.
The plan of the paper is as follows.
In Section \ref{setting} we introduce the class of billiards from which we will draw our sequential dynamics and summarize our main analytical results regarding cone contraction.
In Section~\ref{sec:hyp} we present the uniform properties of hyperbolicity and singularity sets enjoyed
by our class of maps, listed as {\bf (H1)}-{\bf (H5)}; we also prove a growth lemma for our sequences of maps and introduce one of our main characters, the transfer operator.
In Section \ref{sec:conedef} we introduce our protagonist, the cone (see Section \ref{sec:cone_dist}).
Section \ref{sec:cone} is devoted to showing that the cone so defined is invariant under the action of the transfer operators of the billiards in question.
In Section \ref{sec:L contract} we show that in fact the cone is eventually strictly invariant (the image has finite diameter in the associated Hilbert metric) thanks to some mixing properties of the dynamics on a finite scale.
The strict cone contraction implies exponential mixing for a very large class of observables and densities as is explained in Section \ref{sec:exp-mix}.
Finally, Section \ref{sec:appl} contains the announced applications, first to sequential systems with holes (open systems), then to chaotic scattering and finally to the Random Lorentz gas.
\section{Setting and Summary of Main Results }
\label{setting}
Since we are interested in studying sequential billiards, below we define a set of billiard tables that will have
uniform hyperbolicity constants, following \cite{demzhang13}. Other classes of billiards are also studied in
\cite{demzhang13}, such as infinite horizon billiards, billiards under small external forces and some types of
nonelastic reflections. While such classes of billiards are amenable to the present technique, we do not
treat the most general case here since the greater number of technicalities would obscure the main ideas we are
trying to present.
\subsection{Families of billiard tables with uniform properties}
\label{sec:bill family}
We first choose $K \in \mathbb{N}$
and numbers $\ell_i >0$, $i = 1, \ldots K$. Let
$M = \cup_{i=1}^K I_i \times [- \frac{\pi}{2}, \frac{\pi}{2}]$, where $I_i$ is an interval of length $\ell_i$ for each $i$.
$M$ will be the phase space common to our collection of billiard maps.
Given $K$ and $\{ \ell_i \}_{i=1}^K$,
we use the notation $Q = Q(\{ B_i \}_{i=1}^K)$ to denote the
billiard table $\mathbb{T}^2 \setminus (\cup_{i=1}^K B_i)$, where each $B_i$ is a convex set whose boundary has
arclength $\ell_i$. We assume that the scatterers $B_i$ are pairwise disjoint and that each
$\partial B_i$ is a $C^3$ curve with strictly positive curvature.
The billiard flow is defined by the motion of a point particle traveling at unit speed in
$Q := \mathbb{T}^2 \setminus (\cup_i B_i)$ and reflecting elastically at collisions.
The associated billiard map $T$ is the discrete-time collision map which maps a point on
$\partial Q$ to its next collision. Parameterizing $\partial Q$ according to an arclength parameter
$r$ (oriented clockwise on each obstacle $B_i$) and denoting by $\varphi$ the angle made
by the post-collision velocity vector and the outward pointing normal to the boundary yields
the cannonical coordinates for the phase space $M$ of the billiard map. In these coordinates,
$M = \cup_i I_i \times [-\pi/2, \pi/2]$, as defined previously.
For $x = (r,\varphi) \in M$, let $\tau(x)$ denote the time until the next collision for $x$ under the
flow. We assume that $\tau$ is bounded on $M$, i.e. the billiard has finite horizon.
Thus since the scatterers are disjoint, there exist constants $\tau_{\min}(Q), \tau_{\max}(Q) > 0$ depending on
the configuration $Q$ such that $\tau_{\min}(Q) \le \tau(x) \le \tau_{\max}(Q) < \infty$
for all $x \in M$.
Moreover, by assumption there exists $\mathcal{K}_{\min}(Q), \mathcal{K}_{\max}(Q) >0$ such that if $\mathcal{K}(r)$ denotes the curvature
of the boundary at coordinate $r$, then $\mathcal{K}_{\min}(Q) \le \mathcal{K}(r) \le \mathcal{K}_{\max}(Q)$. Finally, let
$E_{\max}(Q)$ denote the maximum value of the $C^3$ norm of the curves comprising $\partial Q$.
Now fix $\tau_*, \mathcal{K}_*, E_* \in \mathbb{R}^+$, and let $\mathcal{Q}(\tau_*, \mathcal{K}_*, E_*)$ denote the collection
of all billiard tables $Q( \{ B_i \}_{i=1}^K )$ such that
\[
\tau_* \le \tau_{\min}(Q) \le \tau_{\max}(Q) \le \tau_*^{-1}, \; \;
\mathcal{K}_* \le \mathcal{K}_{\min}(Q) \le \mathcal{K}_{\max}(Q) \le \mathcal{K}_*^{-1}, \mbox{ and } E_{\max}(Q) \le E_*.
\]
Let $\mathcal{F}(\tau_*, \mathcal{K}_*, E_*)$ denote the set of billiard maps corresponding to the tables
in $\mathcal{Q}(\tau_*, \mathcal{K}_*, E_*)$. Note that all $T \in \mathcal{F}(\tau_*, \mathcal{K}_*, E_*)$ have the same phase space $M$
since we have fixed $K$ and the arclengths $\{ \ell_i \}_{i=1}^K$.
It is a standard fact that all $T \in \mathcal{F}(\tau_*, \mathcal{K}_*, E_*)$ preserve the same smooth invariant probability
measure, $d\mu_{\mbox{\tiny SRB}} = c \cos \varphi \, dr \, d\varphi$, where
$c = \frac{1}{2|\partial Q|} = \frac{1}{2 \sum_{i=1}^K \ell_i}$ is the normalizing constant \cite{chernov book}.
It is proved\footnote{The abstract set-up in \cite{demzhang13} also allows billiard tables with infinite
horizon and those subjected to external forces, but we are not concerned with the most general case here.} in \cite[Theorem~2.7]{demzhang13} that all $T \in \mathcal{F}(\tau_*, \mathcal{K}_*, E_*)$ satisfy properties
(H1)-(H5) of that paper with uniform constants depending only on $\tau_*, \mathcal{K}_*$ and $E_*$. We recall
the relevant properties in Section~\ref{sec:hyp} that we shall use throughout the paper.
Next, we define a notion of distance in $\mathcal{Q}(\tau_*, \mathcal{K}_*, E_*)$ as follows.
Each table $Q$ comprises $K$ obstacles $B_i$. Each $\partial B_i$ can be parametrized according to arclength
by a function $u_i : I_i \to \mathbb{R}^2$ (unfolding $\mathbb{T}^2$). Since two arclength parametrizations of $\partial B_i$ can differ only in their
starting point, the collection $u_{i,\theta}$, $\theta \in [0, 2\pi)$, denotes the set of parametrizations associated with
$\partial B_i$. Let $\Pi_K$ denote the set of permutations on $\{ 1, \ldots K \}$. Then denoting
by $\tilde{u}_{i,\theta}$ the parametrizations for obstacles in $\widetilde Q$, define
\begin{equation}
\label{eq:d def}
\mathbbm d(Q, \widetilde{Q}) = \inf_{\pi \in \Pi_K} \inf_{\theta \in [0, 2\pi)} \sum_{i=1}^K |u_{i,0} - \tilde{u}_{\pi(i), \theta}|_{C^2(I_i, \mathbb{R}^2)} \, .
\end{equation}
Fix $Q_0 \in \mathcal{Q}(\tau_*, \mathcal{K}_*, E_*)$ and choose $\kappa \le \frac{1}{2} \min \{ \tau_*, \mathcal{K}_* \}$.
Let $\mathcal{Q}(Q_0, E_*, \kappa)$ denote the set of
billiard tables $Q$ with $\mathbbm d(Q, Q_0) < \kappa$ and $E_{\max}(Q) \le E_*$, $\tau_{\max} \le 2\tau_*$.\footnote{Indeed, the distance $\mathbbm d$ allows configurations to move from finite to infinite horizon (see \cite[Section~6.2]{demzhang13}), but we will not need that here as we will restrict ourselves to finite horizon configurations.}
Let $\mathcal{F}(Q_0, E_*; \kappa)$
denote the corresponding set of billiard maps. The following result
is \cite[Theorem~2.8]{demzhang13}.
\begin{prop}
\label{prop:close maps}
Let $Q_0 \in \mathcal{Q}(\tau_*, \mathcal{K}_*, E_*)$. There exists $C>0$ such that for all $\kappa \le \frac 12 \min \{ \tau_*, \mathcal{K}_* \}$,
$\mathcal{F}(Q_0, E_*; \kappa) \subset \mathcal{F}(\tau_*/2, \mathcal{K}_*/2, E_*)$, and
$d_{\mathcal{F}}(T_1, T_2) \le C \kappa^{1/3}$ for all $T_1, T_2 \in \mathcal{F}(Q_0, E_*; \kappa)$, where
$d_{\mathcal{F}}(\cdot, \cdot)$ is defined in Section~\ref{sec:map distance}.
\end{prop}
We use an (uncountable) index set $\mathcal{I}(\tau_*, \mathcal{K}_*, E_*)$, identifying $\iota \in \mathcal{I}(\tau_*, \mathcal{K}_*, E_*)$ with a map $T_\iota \in \mathcal{F}(\tau_*, \mathcal{K}_*, E_*)$.
Choosing a sequence $(\iota_j)_{j \in \mathbb{N}} \subset \mathcal{I}(\tau_*, \mathcal{K}_*, E_*)$,
we will be interested the dynamics of
\begin{equation}
\label{eq:comp}
T_n := T_{\iota_n} \circ \cdots \circ T_{\iota_2} \circ T_{\iota_1} \, , \quad n \in \mathbb{N} \, .
\end{equation}
If we choose $\iota_j = \iota$ for each $j$, then $T_n = T_\iota^n$, the iterates of a single map.
\subsection{Main analytical results: Cone contraction and loss of memory}
\label{sec:main}
As announced in the introduction, the main analytical tool developed in this paper is the construction of a convex
cone of functions $\mathcal{C}_{c,A,L}(\delta)$, depending on parameters $\delta >0$, $c, A, L > 1$, as defined in
Section~\ref{sec:cone_dist}, that is contracted under the sequential action of the
transfer operators $\mathcal{L} f = f \circ T^{-1} $, defined in Section~\ref{sec:transfer} for $T \in \mathcal{F}(\tau_*, \mathcal{K}_*, E_*)$.
For a sequence of maps $T_n$ as in \eqref{eq:comp}, define $\mathcal{L}_n f = f \circ T_n^{-1}$.
In order to state our main result on cone contraction,
we define open neighborhoods in $\mathcal{F}(\tau_*, \mathcal{K}_*, E_*)$ using the distance $d_{\mathcal{F}}$
between maps defined in
Section~\ref{sec:map distance}. Let $T \in \mathcal{F}(\tau_*, \mathcal{K}_*, E_*)$, and for
$0 < \kappa < \frac 12 \min \{ \tau_*, \mathcal{K}_*, E_* \}$, define\footnote{ By
Proposition~\ref{prop:close maps}, $\mathcal{F}(T, \kappa)$ contains the billiard maps corresponding to all billiard
tables whose obstacles are $C^2$ close in the metric $\mathbbm d$ to the billiard table $Q$ corresponding to $T$.}
\begin{equation}
\label{eq:close d 1}
\mathcal{F}(T, \kappa) = \{ \widetilde{T} \in \mathcal{F}(\tau_*, \mathcal{K}_*, E_*) : d_{\mathcal{F}}(\widetilde{T}, T) < \kappa \} \, .
\end{equation}
We will denote the index set corresponding to $\mathcal{F}(T, \kappa)$ by $\mathcal{I}(T, \kappa) \subset \mathcal{I}(\tau_*, \mathcal{K}_*, E_*)$.
Thus $\iota \in \mathcal{I}(T, \kappa)$ if and only if $T_\iota \in \mathcal{F}(T, \kappa)$.
\begin{theorem}
\label{thm:main}
Suppose $c, A$ and $L$ satisfy the conditions of Section~\ref{sec:conditions}, and that $\delta>0$ satisfies \eqref{eq:delta_0 ineq} and \eqref{eq:A-cond-s4}.
Let $N_{\mathcal{F}} := N(\delta)^- + k_*n_*$ be from Theorem~\ref{thm:cone contract} and let $\kappa>0$ be from
Lemma~\ref{lem:proper cross}(b).
Then there exists $\chi < 1$ and such that if $n \ge N_{\mathcal{F}}$, $T \in \mathcal{F}(\tau_*, \mathcal{K}_*, E_*)$
and $\{ \iota_j \}_{j=1}^n \subset \mathcal{I}(T, \kappa)$,
then $\mathcal{L}_n \mathcal{C}_{ c, A, L}(\delta) \subset \mathcal{C}_{ \chi c, \chi A, \chi L}(\delta) $.
In addition, for any $\chi \in \left( \max \{ \frac 12, \frac 1L, \frac{1}{\sqrt{A-1}} \}, 1 \right)$, the cone
$\mathcal{C}_{\chi c, \chi A, \chi L}(\delta)$ has diameter at most $\log \left( \frac{(1+\chi)^2}{1-\chi)^2} \chi L \right) < \infty$ in
$\mathcal{C}_{c, A, L}(\delta)$, provided $\delta>0$ is chosen sufficiently small to satisfy \eqref{eq:delta cond}.
\end{theorem}
The first statement of this theorem is proved in two steps: first, Proposition~\ref{prop:almost} shows that the parameters
$c$ and $A$ contract due to the uniform hyperbolicity properties {\bf (H1)}-{\bf (H5)} of the maps in
$\mathcal{F}(\tau_*, \mathcal{K}_*, E_*)$, subject to the constraints listed in
Section~\ref{sec:conditions} (all that is needed is $\{ \iota_j \}_{j=1}^n \subset \mathcal{I}(\tau_*, \mathcal{K}_*, E_*)$, and not
the stronger assumption $\{ \iota_j \}_{j=1}^n \subset \mathcal{I}(T, \kappa)$); second, Theorem~\ref{thm:cone contract} proves the contraction of $L$ using
the uniform mixing property of maps $\widetilde{T} \in \mathcal{F}(T, \kappa)$ as expressed by Lemma~\ref{lem:proper cross}.The second statement of Theorem~\ref{thm:main} is proved by
Proposition~\ref{prop:diameter}.
From this theorem follow our results on exponential loss of memory for sequential systems of billiard maps.
In the case that $T_{\iota_j} = T$ for each $j$, these results read as exponential decay of correlations and convergence
to equilibrium. Since our maps $T \in \mathcal{F}(\tau_*, \mathcal{K}_*, E_*)$ all preserve the measure $\mu_{\mbox{\tiny SRB}}$,
we also obtain a type of convergence to equilibrium in the sequential case (see, for example, \eqref{eq:avg conv}).
In order to state our result for the sequential system, we define the notion of an {\em admissible sequence} of maps from
$\mathcal{F}(\tau_*, \mathcal{K}_*, E_*)$.
As before, let $N_\mathcal{F}$ from Theorem~\ref{thm:cone contract} and let $\kappa>0$ be from
Lemma~\ref{lem:proper cross}(b).
\begin{defin}
We call a sequence $( \iota_j )_j$, $\iota_j \in \mathcal{I}(\tau_*, \mathcal{K}_*, E_*)$, {\em admissible} if
there exists a sequence $(N_k)_{k \ge 0}$ with $N_k \ge N_\mathcal{F}$, such that
for all $k \ge 0$ and $j \in [kN_k, (k+1)N_k-1]$,
$T_{\iota_j} \in \mathcal{F}(T_{\iota_{kN_k}}, \kappa)$.
\end{defin}
Thus an admissible sequence is a sequence which remains
in a $\kappa$ neighborhood of a fixed map for $N_k \ge N_\mathcal{F}$ iterates at a time, but which may undergo a large
change between such blocks.
We first state our results regarding loss of memory, both with respect to $\mu_{\mbox{\tiny SRB}}$ and leafwise: the difference of integrals along individual stable curves converge
to 0 exponentially fast along any admissible sequence. Let $\mathcal{W}^s(\delta)$ denote the set of cone
stable curves $\mathcal{W}^s$ defined in Section~\ref{sec:uni hyp}, having length between $\delta$ and $2\delta$.
We denote by $\mu_{\mbox{\tiny SRB}}(f) = \int_M f \, d\mu_{\mbox{\tiny SRB}}$ and by $|W|$ the (Euclidean) length of a stable curve $W$.
\begin{theorem}
\label{thm:memory state}
Let $\mathcal{C}_{c,A,L}(\delta)$ satisfy the assumptions of Theorem~\ref{thm:main} and suppose that, in addition,
$\delta>0$ satisfies the assumption of Lemma~\ref{lem:order}.
There exists $C>0$ and $\vartheta<1$ such that for all admissible sequences $(\iota_j)_j$, all $n \ge 0$, and all
$f, g \in C^1(M)$ with $\mu_{\mbox{\tiny SRB}}(f) = \mu_{\mbox{\tiny SRB}}(g)$:
\begin{itemize}
\item[a)] For all all $W \in \mathcal{W}^s(\delta)$ and all $\psi \in C^1(W)$, we have
\[
\left| \fint_{W} \mathcal{L}_n f \, \psi \, dm_{W} - \fint_{W} \mathcal{L}_n g \, \psi \, dm_{W} \right|
\le C \vartheta^n \, |\psi|_{C^1} \min \{ {|\:\!\!|\:\!\!|} f {|\:\!\!|\:\!\!|}_+, {|\:\!\!|\:\!\!|} g {|\:\!\!|\:\!\!|}_+ \} \, ;
\]
\item[b)] For all $\psi \in C^1(M)$,
\[
\left| \int_M \mathcal{L}_n f \, \psi \, d\mu_{\mbox{\tiny SRB}} - \int_M \mathcal{L}_n g \, \psi \, d\mu_{\mbox{\tiny SRB}} \right|
\le C \vartheta^n |\psi|_{C^1(M)} \min \{ {|\:\!\!|\:\!\!|} f {|\:\!\!|\:\!\!|}_+ , {|\:\!\!|\:\!\!|} g {|\:\!\!|\:\!\!|}_+ \} \, .
\]
\end{itemize}
\end{theorem}
Since all our maps preserve the same invariant measure $\mu_{\mbox{\tiny SRB}}$, we obtain additionally an
equidistribution result for stable curves as well as convergence to equilibrium
along admissible sequences.
\begin{theorem}
\label{thm:equi state}
Under the hypotheses of Theorem~\ref{thm:memory state},
there exists $C>0$ such that for all admissible sequences $(\iota_j)_j \subset \mathcal{I}(\tau_*, \mathcal{K}_*, E_*)$,
all $f, g \in C^1(M)$ with $\mu_{\mbox{\tiny SRB}}(f) = \mu_{\mbox{\tiny SRB}}(g)$, and all $n \ge 0$,
\begin{itemize}
\item[a)] For all $W_1, W_2 \in \mathcal{W}^s(\delta)$ and all $\psi_i \in C^1(W_i)$ with $\fint_{W_1} \psi_1 = \fint_{W_2} \psi_2$, we have
\[
\left| \fint_{W_1} \mathcal{L}_n f \, \psi_1 \, dm_{W_1} - \fint_{W_2} \mathcal{L}_n g \, \psi_2 \, dm_{W_2} \right|
\le C \vartheta^n \, (|\psi_1|_{C^1} + |\psi_2|_{C^1} ) \mu_{\mbox{\tiny SRB}}(f) \, ;
\]
in particular, for all $W \in \mathcal{W}^s(\delta)$ and $\psi\in C^1(W)$,
\[
\left| \fint_W \mathcal{L}_n f \, \psi \, dm_W - \mu_{\mbox{\tiny SRB}}(f) \fint_W \psi \, dm_W \right|
\le C \vartheta^n \, |\psi|_{C^1} \mu_{\mbox{\tiny SRB}}(f) \, ;
\]
\item[b)] for all $\psi \in C^1(M)$,
\[
\left| \int_M f \, \psi \circ T_n \, d\mu_{\mbox{\tiny SRB}} - \int_M f \, d\mu_{\mbox{\tiny SRB}} \int_M \psi \, d\mu_{\mbox{\tiny SRB}} \right|
\le C \vartheta^n |\psi|_{C^1(M)} \mu_{\mbox{\tiny SRB}}(f) \, .
\]
\end{itemize}
\end{theorem}
Theorems~\ref{thm:memory state} and \ref{thm:equi state} are proved in Section~\ref{sec:exp-mix},
specifically in Theorems~\ref{thm:memory} and \ref{thm:equi} and Corollary~\ref{cor:extend}.
\begin{remark}
\label{rmk:stenlund}
Theorem~\ref{thm:memory state} has some overlap with \cite{young zhang}, which also considers
sequential billiards in which scatterers shift slightly between collisions. Note, however, that our
definition of admissible sequence allows abrupt and large changes in the configuration of scatterers
within the family $\mathcal{F}(\tau_*, \mathcal{K}_*, E_*)$ every $N_{\mathcal{F}}$ iterates, compared to the slowly changing
requirement throughout \cite{young zhang}. This may seem like merely a technical difference due to the
cone technique, yet it is precisely this ability to introduce occasional large changes in the dynamics
that allows us to apply our results to the chaotic scattering problem and random Lorentz gas described
in Section~\ref{sec:appl}.
\end{remark}
With these convergence results in hand, we are able to provide three applications to concrete
problems of physical interest: sequential open systems in Section~\ref{sequential},
a chaotic scattering problem without assuming a no-eclipse condition in Section~\ref{sec:scattering},
and a variant of the random Lorentz gas in Section~\ref{sec:lorentz}.
\section{Uniform Hyperbolicity, Singularities and Transfer Operators}
\label{sec:hyp}
\subsection{Uniform properties for $T \in \mathcal{F}(\tau_*, \mathcal{K}_*, E_*)$}
\label{sec:uni hyp}
Fixing $K$ and $\{ \ell_i \}_{i=1}^K$, we recall some fundamental properties of billiard maps $T \in \mathcal{F}(\tau_*, \mathcal{K}_*, E_*)$ that
depend only on the quantities $\tau_*$, $\mathcal{K}_*$ and $E_*$. In order to better align with the abstract framework in
\cite{demzhang13}, we also label our properties {\bf (H1)}-{\bf (H5)}, although our set-up here is simpler than in
\cite{demzhang13}. We recall the corresponding index set $\mathcal{I}(\tau_*, \mathcal{K}_*, E_*)$ from Section~\ref{sec:bill family}
and the notation $T_n$ from \eqref{eq:comp}.
\medskip
\noindent
{\bf (H1) Hyperbolicity and Singularities.}
The (constant) family of cones
\[
C^s(x) = \{ (dr, d\varphi) \in \mathbb{R}^2 : -\mathcal{K}_*^{-1} - \tau_*^{-1} \le d\varphi/dr \le - \mathcal{K}_* \},
\quad \mbox{for $x \in M$,}
\]
is strictly invariant, $DT^{-1}C^s(x) \subset C^s(T^{-1}x)$, for all $T \in \mathcal{F}(\tau_*, \mathcal{K}_*, E_*)$.
Moreover,
$T^{-1}$ enjoys uniform expansion of vectors in the stable cone: Set $\Lambda = 1 + 2\mathcal{K}_* \tau_* > 1$; then
there exists $C_1 \in (0,1]$ and such that,
\begin{equation}
\label{eq:hyp}
\| DT_n^{-1}(x) v \| \ge C_1 \Lambda^n \| v \|, \qquad \mbox{for all $v \in C^s(x)$}.
\end{equation}
$T$ has a family of unstable cones $C^u$ defined similarly, but with
$\mathcal{K}_* \le d\varphi/dr \le \mathcal{K}_*^{-1} + \tau_*^{-1}$.
Due to the unbounded expansion of $DT$ near tangential collisions, we define the standard
homogeneity strips, following \cite{bsc}. For some $k_0 \in \mathbb{N}$, to be chosen later in \eqref{eq:one step},
we define
\begin{equation}
\label{eq:Hk}
\mathbb{H}_{\pm k} = \{ (r, \varphi) \in M : (k+1)^{-2} \le |\pm \tfrac{\pi}{2} - \varphi| \le k^{-2} \} ,
\mbox{ for all } k \ge k_0.
\end{equation}
Set $\mathcal{S}_0 = \{ (r, \varphi) \in M : \varphi = \pm \frac{\pi}{2} \}$. The singularity set for $T_n$
is denoted by $\mathcal{S}_n^{T_n} = \cup_{i=0}^{n} T_i^{-1} \mathcal{S}_0$, for $n \in \mathbb{Z}$.
On $M \setminus \mathcal{S}_n^{T_n}$, $T_n$ is a $C^2$ diffeomorphism onto its image.
There exists a constant, which we still call $C_1>0$, such that
\[
\frac{C_1}{\cos \varphi(Tx)} \le \frac{\| DT(x) v \|}{\| v \|} \le \frac{1}{C_1 \cos \varphi(Tx)} \, , \quad \mbox{for $x \notin \mathcal{S}_0$.}
\]
In order to achieve bounded distortion, we will consider the boundaries of the homogeneity strips
as an extended singularity set for $T$. To this end, define $\mathcal{S}_0^{\mathbb{H}} = \mathcal{S}_0 \cup (\cup_{k \ge k_0}
(\partial \mathbb{H}_k \cup \partial \mathbb{H}_{-k}))$, and
$\mathcal{S}_n^{\mathbb{H}} =\cup_{i=0}^{n} T_i^{-1} \mathcal{S}_0^{\mathbb{H}}$,
for $n \in \mathbb{Z}$.
\medskip
\noindent
{\bf (H2) Families of Stable and Unstable Curves.}
We call a curve $W \subset M$ a stable curve if for each $x \in W$, the tangent vector to $W$ at $x$ belongs to
$C^s$. A stable curve is called homogeneous if it lies in one homogeneity strip or outside their union.
Denote by $\mathcal{W}^s$ the set of homogeneous stable curves with length
at most $\delta_0$ (defined by \eqref{eq:one step}) and with curvature at most $\bar B$.
By \cite[Proposition~4.29]{chernov book}, we may choose $\bar B$ sufficiently large that
$T^{-1} \mathcal{W}^s \subset \mathcal{W}^s$, up to subdividing the curves of length larger than $\delta_0$,
for all $T \in \mathcal{F}(\tau_*, \mathcal{K}_*, E_*)$.
Similarly, we define an analogous set of homogeneous unstable curves by $\mathcal{W}^u$.
\medskip
\noindent
{\bf (H3) One-Step Expansion.} Defining the adapted norm $\| v \|_*$, $v = (dr, d\varphi)$ as in
\cite[Sect.~5.10]{chernov book}, we have
$\| DT^{-1}(x) v \|_* \le \Lambda \| v \|_*$ for all $v \in C^s(x)$, wherever $DT^{-1}$ is defined.
For $W \in \mathcal{W}^s$, let $V_i$ denote the maximal homogeneous components of $T^{-1}W$.
Then by \cite[Lemma~5.56]{chernov book}, there exists $\theta_0 \in (\Lambda, 1)$ and
a choice of $k_0$ for the homogeneity strips
and $\delta_0>0$ such that,
\begin{equation}
\label{eq:one step}
\sup_{T \in \mathcal{F}(\tau_*, \mathcal{K}_*, E_*)} \sup_{\substack{W \in \mathcal{W}^s \\ |W| \le \delta_0}}
\sum_i |J_{V_i}T|_* \le \theta_0 \, ,
\end{equation}
where $| J_{V_i}T |_*$ denotes the supremum of the Jacobian of $T$ along $V_i$ in the adapted metric.
Since the stable/unstable cones are global and bounded away from one another, the adapted metric can be
extended so that it is uniformly uniformly equivalent to the Euclidean metric: There exists
$C_0 \ge 1$ such that $C_0^{-1} \| v \| \le \| v \|_* \le C_0 \| v \|$ for all $v \in \mathbb{R}^2$.
\medskip
\noindent
{\bf (H4) Distortion Bounds.}
Suppose $W \in \mathcal{W}^s$ and $\{ \iota_j \}_{j=1}^n \subset \mathcal{I}(\tau_*, \mathcal{K}_*, E_*)$ are such that $T_jW \in \mathcal{W}^s$ for $j=0, \ldots n$.
There exists $C_d>0$,
independent of $W$ and $\{ \iota_j \}_{j=1}^n$,
such that for all $x, y \in W$,
\begin{equation}
\label{eq:distortion}
| \log J_WT_n(x) - \log J_WT_n(y)| \le C_d d(x,y)^{1/3},
\end{equation}
where $J_WT_n$ is the (stable) Jacobian of $T_n$ along $W$ and $d(\cdot, \cdot)$ denotes
arclength on $W$ with respect to the metric $dr^2+d\varphi^2$.
Similar bounds hold for stable Jacobians lying on the same unstable curve.
Suppose $V_1$, $V_2 \in \mathcal{W}^s$ are such that $T_jV_1, T_jV_2 \in \mathcal{W}^s$ for
$0 \le j \le n$, in particular they are not cut by any singularity, and there exists a foliation of unstable curves $\{ \ell_x \}_{x \in V_1} \subset \mathcal{W}^u$ creating a one-to-one correspondence between $V_1$ and $V_2$ and such that $\{ T_n( \ell_x) \}_{x \in V_1} \subset \mathcal{W}^u$ creates a one-to-one correspondence between $T_nV_1$ and $T_nV_2$.
For $x \in V_1$, define $\bar{x} = \ell_x \cap V_2$. Then there exists $C_d>0$, independent
of $\{ \iota_j \}_{j=1}^n$, $V_1$, $V_2$, and $x$, such that,
\begin{equation}
\label{eq:u dist}
|\log J_{V_1}T_n(x) - \log J_{V_2}T_n(\bar{x})| \le C_d(d(x, \bar{x})^{1/3} + \phi(x, \bar{x})),
\end{equation}
where $\phi(x, \bar{x})$ denotes the angle between the tangent vectors to $V_1$ and $V_2$
at $x$ and $\bar{x}$, respectively. For simplicity, we use the same symbol $C_d$ to represent the
distortion constants in \eqref{eq:distortion} and \eqref{eq:u dist}.
The proofs for these distortion bounds in this form for a single map can be found in \cite[Appendix A]{demzhang11}
(see also \cite[Section 5.8]{chernov book}). The analogous bounds for sequences of maps in
$\mathcal{F}(\tau_*, \mathcal{K}_*, E_*)$
are proved in \cite[Lemma~3.3]{demzhang13}. The constant $C_d$ depends only on the choice of
$k_0$ from {\bf (H3)} and the hyperbolicity constants $C_1$ and $\Lambda$ from {\bf (H1)}.
\medskip
\noindent
{\bf (H5) Invariant measure.}
All $T \in \mathcal{F}(\tau_*, \mathcal{K}_*, E_*)$ preserve the same invariant measure,
$d\mu_{\mbox{\tiny SRB}} = c \cos \varphi \, dr \, d\varphi$, where
$c = \frac{1}{2|\partial Q|} = \frac{1}{2 \sum_{i=1}^K \ell_i}$ is the normalizing constant \cite{chernov book}.
\begin{remark}
Property {\bf (H5)} is enjoyed by the class of maps we have chosen, but it is not necessary for this technique to work.
Indeed, \cite{demzhang13} replaces this condition by: There exists $\eta>0$ so that $1+\eta$ is sufficiently small compared to the hyperbolicity constant $\Lambda$ from {\bf (H1)}, such that $(J_{\mu_{\mbox{\tiny SRB}}}T)^{-1} \le 1+\eta$, where
$J_{\mu_{\mbox{\tiny SRB}}}T$ is the Jacobian of $\mu_{\mbox{\tiny SRB}}$ with respect to $T$.
Thus $T$ does not have to preserve $\mu_{\mbox{\tiny SRB}}$, but in this sense must be close to a map that does. This permits the
application of the current technique to billiards under small external forces and nonelastic reflections, as described
in \cite[Section 2.4]{demzhang13}. See also \cite{Ch08, zhang}. Note however, that while
Theorem~\ref{thm:memory state} will continue to hold in this generalized context,
Theorem~\ref{thm:equi state} will not hold
once there is no common invariant measure.
\end{remark}
\subsection{Distance between maps}
\label{sec:map distance}
Following \cite{demzhang13}, we define a distance in $\mathcal{F} = \mathcal{F}(\tau_*, \mathcal{K}_*, E_*)$ as follows.
For $T \in \mathcal{F}$ and $\varepsilon >0$, let $N_\varepsilon(\mathcal{S}^T_{-1})$ denote the $\varepsilon$-neighborhood in $M$ of the
singularity set $\mathcal{S}_{-1}^T$ for $T^{-1}$. For $T, \widetilde{T} \in \mathcal{F}(\tau_*, \mathcal{K}_*, E_*)$,
we write $d_{\mathcal{F}}(T, \widetilde{T}) \le \varepsilon$ if the maps are close away from their singularity sets in the following sense:
For $x \notin N_\varepsilon(\mathcal{S}_{-1}^T \cup \mathcal{S}_{-1}^{\widetilde{T}})$,
\begin{itemize}
\item[(C1)] $d(T^{-1}x, \widetilde{T}^{-1}x) \le \varepsilon$;
\item[(C2)] $\max \left\{ \left| \frac{J_WT(x)}{J_W\widetilde{T}(x)} -1 \right|, \left| \frac{J_W\widetilde{T}(x)}{J_WT(x)} -1 \right| \right\} \le \varepsilon$, for all $W \in \mathcal{W}^s$ and $x \in W$;
\item[(C3)] $\| DT^{-1}(x) v - D\widetilde{T}^{-1}(x) v \| \le \sqrt{\varepsilon}$, for any unit vector $v \in C^s(x)$.
\end{itemize}
Remark that this distance does not require that $\mathcal{S}_{-1}^T$ and $\mathcal{S}_{-1}^{\widetilde{T}}$ be close as sets. This is to account
for the fact that if we slide or deform scatterers from configuration $Q$ to configuration $\widetilde Q$,
$\mathcal{S}_{-1}^{\widetilde{T}}$ may acquire new connected components if the movement occurs near a double or triple tangency.
\subsection{Growth lemma}
Although all maps in $\mathcal{F}(\tau_*, \mathcal{K}_*, E_*)$ enjoy the uniform properties {\bf (H1)}-{\bf (H5)},
in Section~\ref{sec:scale}, we will find it convenient to increase the
contraction provided in \eqref{eq:one step} by
replacing $T$ with a higher iterate $T_n$ and choosing $\delta_0$
sufficiently small so that \eqref{eq:one step} holds for $T _*= T_n$ with constant $\theta_0^n$.
This is possible since if $W$ is a stable curve, then there exists $C>0$, depending only on the
family $\mathcal{F}(\tau_*, \mathcal{K}_*, E_*)$, such that
$|T^{-1}W| \le C|W|^{1/2}$ \cite[Exercise~4.50]{chernov book}.
Thus we may choose $\delta_0$ so small
that no connected component of $T_k^{-1}(W)$ is longer than $\delta_0$ for $k=0, \ldots, n$.
Since no artificial subdivisions are necessary, we apply \eqref{eq:one step} inductively in $k$
to obtain the desired contraction.
Choose
$\bar n$ and fix $\delta_0 \in (0,1)$ such that $\theta_1 := \theta_0^{\bar n}$ satisfies
\begin{equation}
\label{eq:theta_1}
3C_0 \frac{\theta_1}{1 - \theta_1} \leq \frac 14 \qquad \mbox{and} \qquad
\sup_{\substack{W \in \mathcal{W}^s \\ |W| \le \delta_0}} \sum_{V_i} |J_{V_i}T_{\bar n}|_* \leq \theta_1,
\end{equation}
where $V_i$ are the homogeneous components of $T_{\bar n}^{-1}W$ and $C_0 \ge 1$ is from {\bf (H3)}.
Note that if we shrink $\delta_0$ further, then \eqref{eq:theta_1} will continue to hold for the same
value of $\bar n$.
We shall work with the map $T_*= T_{\bar n}$ throughout the following. To simplify notation we will call $T_*$ again $T$ as no confusion can arise.
The following growth lemma is contained in
\cite[Lemma~5.5]{demzhang13}, but we include the proof of item (b)
here for convenience and to draw out the
explicit dependence on the constants.
For $W \in \mathcal{W}^s$, for $T_n$ as in \eqref{eq:comp} we denote by $\mathcal{G}_n(W)$ the homogeneous components of $T_n^{-1}W$,
where we have subdivided the elements of $T_n^{-1}W$ longer than $\delta_0$ into elements with length between
$\delta_0$ and $\delta_0/2$ so that $\mathcal{G}_n(W)\subset \mathcal{W}^s$. We call $\mathcal{G}_n(W)$ the $n$th generation
of $W$.
Let $\mathcal{I}_n(W)$ denote the set of curves $W_i \in \mathcal{G}_n(W)$ such that $T_j(W_i)$ is not contained
in an element of $\mathcal{G}_{n-j}(W)$ having length at least $\delta_0/3$ for any $j = 0, \ldots n$
\begin{lemma}
\label{lem:full growth}
There exists $\bar{C}_0 > 0$ such that for all $W \in \mathcal{W}^s$ and $n \ge 0$ and $\{ \iota_j \}_{j=1}^n$,
\begin{itemize}
\item[a)] $\displaystyle
\sum_{W_i \in \mathcal{I}_n(W)} |J_{W_i}T_n|_{C^0(W_i)} \le C_0 \theta_1^n$;
\item[b)] $\displaystyle
\sum_{W_i \in \mathcal{G}_n(W)} |J_{W_i}T_n|_{C^0(W_i)} \leq \bar{C}_0 \delta_0^{-1} |W| + C_0 \theta_1^n$.
\end{itemize}
\end{lemma}
\begin{proof}
Item (a) follows by induction on $n$ from \eqref{eq:theta_1} and the constant
$C_0$ from {\bf (H3)} comes from translating from
the adapted metric to the Euclidean metric at the last step. We focus on proving item (b).
For $W \in \mathcal{W}^s$, let $L_k(W) \subset \mathcal{G}_k(W)$ denote those elements of $\mathcal{G}_k(W)$ having
length at least $\delta_0/3$. For $k \le n$ and $W_i \in \mathcal{G}_n(W)$,
we say that $V_j \in L_k(W)$ is the most recent long ancestor of $W_i$ if $k \le n$ is the largest time
that $T_{n-k}W_i$ is contained in an element of $L_k(W)$. Then by definition, $W_i \in \mathcal{I}_{n-k}(V_j)$.
Note that if $W_i \in L_n(W)$, then $k=n$ and $W_i = V_j$. Now we estimate,
\[
\begin{split}
\sum_{W_i \in \mathcal{G}_n(W)} |J_{W_i}T_n|_{C^0(W_i)}
& \le \sum_{k=1}^n \sum_{V_j \in L_k(W)} \sum_{W_i \in \mathcal{I}_{n-k}(V_j)} |J_{W_i}T_{n-k}|_{C^0(W_i)}
|J_{V_j}T_k|_{C^0(V_j)} \\
& \qquad + \sum_{W_i \in \mathcal{I}_n(W)} |J_{W_i}T_n|_{C^0(W_i)} \\
& \le \sum_{k=1}^n \sum_{V_j \in L_k(W)} C_0 \theta_1^{n-k} e^{C_d \delta_0^{1/3}} \frac{ |T_kV_j|}{|V_j|}
+ C_0\theta_1^n \, ,
\end{split}
\]
where we have used item (a) of the lemma to sum over $W_i \in \mathcal{I}_{n-k}(W)$ and \eqref{eq:distortion}
to replace $|J_{V_j}T_k|_{C^0(V_j)}$ with $\frac{|T_kV_j|}{|V_j|}$. Now since
$\cup_{V_j \in L_k(W)} T_kV_j \subset W$, and $|V_j| \ge \delta_0/3$, we have
\[
\sum_{W_i \in \mathcal{G}_n(W)} |J_{W_i}T_n|_{C^0(W_i)}
\le \sum_{k=1}^n C_0 \theta_1^{n-k} 3 \delta_0^{-1} |W| e^{C_d \delta_0^{1/3}} + C_0 \theta_1^n \, ,
\]
which proves the lemma with $\bar C_0 := \frac{3C_0}{1-\theta_1} e^{C_d \delta_0^{1/3}}$.
\end{proof}
\begin{remark}
\label{rem:improve}
It is not necessary to work with $T = T_{\bar n}$ in Lemma~\ref{lem:full growth}.
It follows equally well from
\eqref{eq:one step} with $\theta_1$ replaced by $\theta_0$. Moreover, if $|W| \ge \delta_0/3$, then
all pieces $W_i \in \mathcal{G}_n(W)$ have a long ancestor and can be included in the sum over $k$; in this
case, the second term on the right side of item (b) is not needed, and the value of $\bar C_0$ remains unchanged.
\end{remark}
\subsection{Transfer operator}
\label{sec:transfer}
We define the transfer operator $\mathcal{L}$ associated with $T$ acting on scales of
spaces of distributions as in \cite{demzhang11}. For $\{ \iota_j \}_{j=1}^n \subset \mathcal{I}(\tau_*, \mathcal{K}_*, E_*)$,
we denote by $T_n^{-1}\mathcal{W}^s$ the set of
curves $W \in \mathcal{W}^s$ such that $T_jW \in \mathcal{W}^s$ for all $j = 0, \ldots n$.
For $\alpha \le 1/3$, let $C^\alpha(T_n^{-1}\mathcal{W}^s)$ denote
the set of complex valued functions on $M$ that are H\"older continuous on elements of
$T_n^{-1}\mathcal{W}^s$. Then for $\psi \in C^\alpha(\mathcal{W}^s)$, we have
$\psi \circ T_n \in C^\alpha(T_n^{-1}\mathcal{W}^s)$ (see Lemma~\ref{lem:test contract}(a)).
Define
\[
\mathcal{L}_n \mu(\psi) = \mu(\psi \circ T_n) , \mbox{ for $\mu \in (C^\alpha(T_n^{-1}\mathcal{W}^s ))^*$ .}
\]
This defines $\mathcal{L}_{T_{\iota_n}} : (C^\alpha(T_n^{-1}\mathcal{W}^s))^* \to (C^\alpha(T_{n-1}^{-1}\mathcal{W}^s))^*$ for any
$n \ge 1$. See \cite{demzhang11} for details.
Recall that by {\bf (H5)}, all our maps $T$ preserve the smooth invariant measure $d\mu_{\mbox{\tiny SRB}} = c \cos \varphi dr d\varphi$,
where $c$ is the normalizing constant.
When $d\mu = f d\mu_{\mbox{\tiny SRB}}$ is a measure absolutely continuous with respect to $\mu_{\mbox{\tiny SRB}}$,
we identify $\mu$ with its density $f$. With this identification, the transfer operator acting on densities has
the following familiar expression,
\[
\mathcal{L}_T f = f \circ T^{-1} ,
\]
and so $\mathcal{L}_n f = \mathcal{L}_{T_{\iota_n}} \cdots \mathcal{L}_{T_{\iota_1}} f$, pointwise.
We choose this identification of functions in order to simplify our later work: using the
reference measure $\mu_{\mbox{\tiny SRB}}$,
the Jacobian of the transformation is 1, making $\mathcal{L}$ simpler to work with.
\section{Cones and Projective Metrics}\label{sec:conedef}
Given a closed,\footnote{\label{foo:close} Closed here means that for all $f,g \in\mathcal{C}$ and sequence $\{\alpha_n\}\subset \mathbb{R}$ such that $\lim_{n\to\infty}\alpha_n=\alpha$ and $g+\alpha_n f\in\mathcal{C}$ for all $n\in\mathbb{N}$ we have $g+\alpha f\in\mathcal{C}\cup\{0\}$. } convex cone $\mathcal{C}$ satisfying $\mathcal{C} \cap - \mathcal{C} = \emptyset$, we define an
order relation by $f \preceq g$ if and only if $g - f \in \mathcal{C} \cup \{0\}$. We can then define
a projective metric by
\begin{equation}
\label{eq:H def}
\begin{split}
\bar\alpha(f,g) & = \sup \{ \lambda \in \mathbb{R}^+ : \lambda f \preceq g \} \\
\bar\beta(f,g) & = \inf \{ \mu \in \mathbb{R}^+ : g \preceq \mu f \} \\
\rho(f,g) & = \log \left( \frac{\bar\beta(f,g)}{\bar\alpha(f,g)} \right) .
\end{split}
\end{equation}
\subsection{ A cone of test functions}
For $W \in \mathcal{W}^s$, $\alpha \in (0, 1]$ and $a \in \mathbb{R}^+$, define a cone of test functions by
\[
\mathcal{D}_{a, \alpha}(W) = \left\{ \psi \in C^0(W) : \psi > 0 , \frac{\psi(x)}{\psi(y)} \le e^{a d(x,y)^\alpha} \right\},
\]
where $d(\cdot, \cdot)$ is the arclength distance along $W$.
The Hilbert metric associated with this cone and defined by \eqref{eq:H def}
depends on the constant $a$ and the exponent $\alpha$ determining the regularity of the functions.
For each such choice, the Hilbert metric has the following
convenient representation.
\begin{lemma}[{\cite[Lemma 2.2]{liv95}}]
\label{lem:H metric}
Choose $\alpha \in (0,1]$. For $\psi_1, \psi_2 \in \mathcal{D}_{a,\alpha}(W)$, the corresponding metric
$\rho_{W, a, \alpha}(\cdot, \cdot)$ is given by
\[
\rho_{W, a, \alpha}(\psi_1, \psi_2) = \log \left[ \sup_{x,y,u,v \in W}
\frac{e^{ad(x,y)^\alpha} \psi_1(x) - \psi_1(y)}{e^{a d(x,y)^\alpha} \psi_2(x) - \psi_2(y)}
\cdot \frac{e^{ad(u,v)^\alpha} \psi_2(u) - \psi_2(v)}{e^{a d(u,v)^\alpha} \psi_1(u) - \psi_1(v)} \right] .
\]
\end{lemma}
A corollary of this lemma is that $\mathcal{D}_{a,\alpha}(W)$ has finite diameter in $\mathcal{D}_{a,\beta}(W)$
if $\beta < \alpha$ and $|W| < 1$.
The next two lemmas are simple consequences of the regularity of functions in $\mathcal{D}_{a, \alpha}(W)$
for $W \in \mathcal{W}^s$. We denote by $m_W$ the measure induced by arclength along $W$.
\begin{lemma}
\label{lem:avg}
For any $\alpha \in (0,1]$ and $W \in \mathcal{W}^s$ with $|W| \in [\delta, 2 \delta]$, any
$\psi \in \mathcal{D}_{a, \alpha}(W)$ and
$x \in W$, we have
\[
\frac{\delta \psi(x)}{\int_W \psi \, dm_W} \le \frac{|W| \psi(x)}{\int_W \psi dm_W} \le e^{a|W|^\alpha} .
\]
\end{lemma}
\begin{proof}
The estimate is immediate since $\inf_{y \in W} \psi(y)\geq \psi(x)e^{-a|W|^\alpha}$.
\end{proof}
\begin{lemma}
\label{lem:pos}
Given $\alpha \in (0,1]$, $W \in \mathcal{W}^s$, $\psi_1, \psi_2 \in \mathcal{D}_{a, \alpha(W)}$ and $x, y \in W$,
\[
e^{- \rho_{W, a, \alpha}(\psi_1, \psi_2)} \le
\frac{\psi_1(x)\psi_2(y)}{\psi_2(x)\psi_1(y)} \le e^{\rho_{W, a, \alpha}(\psi_1, \psi_2)}
\]
\end{lemma}
\begin{proof}
According to \eqref{eq:H def}, we must have,
\[
\psi_2(x) - \bar\alpha \psi_1(x) \ge 0 \quad \forall x \in W \qquad \mbox{and} \qquad
\psi_2(y) - \bar\beta \psi_1(y) \le 0 \quad \forall y \in W .
\]
This in turn implies that
\[
\rho_{W, a, \alpha}(\psi_1, \psi_2) = \log \frac{\bar\beta(\psi_1, \psi_2)}{\bar\alpha(\psi_1, \psi_2)} \
\ge \log \left[ \frac{\psi_1(x)\psi_2(y)}{\psi_2(x)\psi_1(y)} \right]
\qquad \forall x, y \in W .
\]
\end{proof}
\subsection{Distances between curves and functions}
\label{sec:distances}
Due to the global stable cones $C^s$ defined in {\bf (H1)}, we may consider stable curves $W \in \mathcal{W}^s$
as graphs of $C^2$ functions over an interval $I_W$ in the $r$-coordinate:
\[
W = \{ G_W(r) = (r, \varphi_W(r)) : r \in I_W \} .
\]
Using this representation, we define a notion of distance between $W^1, W^2 \in \mathcal{W}^s$ by
\begin{equation}\label{eq:W-distance}
d_{\mathcal{W}^s}(W^1, W^2) = |\varphi_{W^1} - \varphi_{W^2}|_{C^1(I_{W^1} \cap I_{W^2})} +
|I_{W^1} \bigtriangleup I_{W^2}|,
\end{equation}
if $W^1$ and $W^2$ lie in the same homogeneity strip and $|I_{W^1} \cap I_{W^2}| > 0$;
otherwise, we set $d_{\mathcal{W}^s}(W^1, W^2) = \infty$. Note that $d_{\mathcal{W}^s}$ is not a metric,
but this is irrelevant for our purposes.
We will also find it necessary to compare between test functions on two different
stable curves. Given $W^1, W^2 \in \mathcal{W}^s$ with $d_{\mathcal{W}^s}(W^1, W^2) < \infty$,
and $\psi_i \in \mathcal{D}_{a, \beta}(W_i)$, define
\begin{equation}
\label{eq:psi-distance}
d_*(\psi_1, \psi_2) = | \psi_1 \circ G_{W^1} \|G_{W^1}'\|- \psi_2 \circ G_{W^2} \|G_{W^2}'\|\, |_{C^\beta(I_{W^1} \cap I_{W^2})},
\end{equation}
to be the (H\"older) distance between $\psi_1$ and $\psi_2$,
where $\| G_W' \| = \sqrt{1+ (d\varphi_W/dr)^2}$.
Also, by the bound $\bar B$ on the curvature of elements of $\mathcal{W}^s$, there exists
$B_*>0$ such that
\begin{equation}
\label{eq:2deriv}
B_* = \sup_{W \in \mathcal{W}^s} |\varphi_W''|_{C^0(W)} < \infty \, .
\end{equation}
\begin{remark}\label{rem:change-int}
Note that if $I_{W^1}=I_{W^2}$ and $d_*(\psi_1,\psi_2)=0$, then
\[
\int_{W^1}\psi_1 \, dm_{W_1} =\int_{W^2}\psi_2 \, dm_{W_2}.
\]
\end{remark}
\subsection{Definition of the cone}\label{sec:cone_dist}
In order to define a cone of functions adapted to our dynamics, we will fix the following
exponents, $\alpha, \beta, \gamma, q > 0$ and constant $a>1$ large enough. Choose $q \in (0, 1/2)$, $\beta < \alpha \le 1/3$
and finally $\gamma \le \min \{ \alpha - \beta, q \}$.
For a length scale $\delta \le \delta_0/3$, define
\[
\mathcal{W}^s_-(\delta) = \{ W \in \mathcal{W}^s : |W| \le 2 \delta \} \quad \mbox{and} \quad
\mathcal{W}^s(\delta) = \{ W \in \mathcal{W}^s : |W| \in [\delta, 2\delta] \} \, .
\]
Let $\mathcal{A}$ denote the set of functions on $M$ whose restriction to each $W \in \mathcal{W}^s$
is integrable with respect to the arclength measure $dm_W$. For $f \in \mathcal{A}$ define,
\begin{equation}
\label{eq:tri def}
{|\:\!\!|\:\!\!|} f {|\:\!\!|\:\!\!|}_+ = \sup_{\stackrel{\scriptstyle W \in \mathcal{W}^s(\delta)}{\psi \in \mathcal{D}_{a,\beta}(W)}} \frac{\int_W f \psi \, dm_W}{\int_W \psi \, dm_W} ,
\qquad \qquad
{|\:\!\!|\:\!\!|} f {|\:\!\!|\:\!\!|}_- = \inf_{\stackrel{\scriptstyle W \in \mathcal{W}^s(\delta)}{\psi \in \mathcal{D}_{a,\beta}(W)}} \frac{\int_W f \psi \, dm_W}{\int_W \psi \, dm_W} ,
\end{equation}
Note that if $f \in \mathcal{A}$, it must be that ${|\:\!\!|\:\!\!|} f {|\:\!\!|\:\!\!|}_- < \infty$.
Denote the average value of
$\psi$ on $W$ by $\fint_W \psi\, dm_W=\frac{1}{|W|}\int_W \psi \, dm_W$. Since all of our integrals in this section and the next
will be taken with respect to the arclength $dm_W$, to keep our notation concise, we will drop the measure from our
integral notation in what follows.
Now for $a, c, A, L >1$, and $\delta \in (0, \delta_0/3]$, define the cone
\begin{align}
\mathcal{C}_{c,A, L}(\delta) = \Big\{& f \in \mathcal{A}:
\qquad {|\:\!\!|\:\!\!|} f {|\:\!\!|\:\!\!|}_+\leq L{|\:\!\!|\:\!\!|} f {|\:\!\!|\:\!\!|}_- ;
\label{eq:cone 2} \\
& \sup_{W \in \mathcal{W}^s_-(\delta)} \sup_{\psi \in \mathcal{D}_{a, \beta}(W)} |W|^{-q}\frac{|\int_W f \psi|}{\fint_W\psi} \le A \delta^{1-q} {|\:\!\!|\:\!\!|} f {|\:\!\!|\:\!\!|}_- ;
\label{eq:cone 3} \\
&\forall\, W^1, W^2 \in \mathcal{W}^s_-(\delta): d_{\mathcal{W}^s}(W^1, W^2) \le \delta, \forall \psi_i \in \mathcal{D}_{a, \alpha}(W_i): d_*(\psi_1, \psi_2)=0, \nonumber\\
&\left|\frac{\int_{W^1} f \psi_1}{\fint_{W^1}\psi_1} - \frac{\int_{W^2} f \psi_2}{\fint_{W^2}\psi_2} \right|\leq
d_{\mathcal{W}^s}(W^1, W^2)^\gamma \, \delta^{1-\gamma} c A {|\:\!\!|\:\!\!|} f {|\:\!\!|\:\!\!|}_- \Big\} .
\label{eq:cone 5}
\end{align}
We write the constants $c, A, L$ explicitly as subscripts in our notation for the
cone since these will be the parameters which are contracted by the dynamics.
By contrast, the exponents $\alpha, \beta, \gamma, q$ are fixed and will not be altered by the dynamics, while the constant $a$, which will be chosen in Lemma \ref{lem:test contract}, will not appear directly in the contraction constant of the cone.
For convenience, we will require that $\delta_0$ is sufficiently small that
\begin{equation}
\label{eq:delta_0 condition}
e^{2 a \delta_0^\beta} \le 2 \, .
\end{equation}
This will imply similar bounds in terms of $\delta$ since $\delta \le \delta_0/3$.
\begin{remark}
As will become clear from our estimates in Sections~\ref{sec:cone} and \ref{sec:L contract}, in order to prove that the
parameters contract, we will need to choose $A$ large compared to $L$, and $c$ large compared to $A$.
This yields the compatible set of restrictions, $1< L < A < c$.
By contrast, the exponents are fixed by the regularity properties of the maps in question: $\alpha \le 1/3$ due to \eqref{eq:distortion},
and $\beta < \alpha$ so that $\mathcal{D}_{a, \beta}(W)$ has finite diameter in $\mathcal{D}_{a, \alpha}(W)$, while $\gamma \le \alpha - \beta$
is convenient to obtain the required contraction in Lemma~\ref{lem:compare}.
See Section \ref{sec:conditions} for all the conditions the constants must satisfy for Proposition~\ref{prop:almost}. Several further conditions are specified in Theorem~\ref{thm:cone contract} to prove the strict contraction of the cone.
\end{remark}
\begin{remark}\label{rem:A-L}
Note that condition \eqref{eq:cone 2} implies $(L-1){|\:\!\!|\:\!\!|} f{|\:\!\!|\:\!\!|}_-\geq {|\:\!\!|\:\!\!|} f{|\:\!\!|\:\!\!|}_+-{|\:\!\!|\:\!\!|} f{|\:\!\!|\:\!\!|}_-\geq 0$, hence for all $W \in \mathcal{W}^s(\delta), \psi \in \mathcal{D}_{a, \beta}(W)$,
\begin{equation}\label{eq:posi}
\int_W f \psi \, dm_W \ge {|\:\!\!|\:\!\!|} f{|\:\!\!|\:\!\!|}_-\int_W \psi \, dm_W\geq 0.
\end{equation}
In addition, condition \eqref{eq:cone 3} implies
\[
A {|\:\!\!|\:\!\!|} f {|\:\!\!|\:\!\!|}_-\geq \sup_{W \in \mathcal{W}_-^s (\delta)} \sup_{\psi \in \mathcal{D}_{a, \beta}(W)}\delta^{q-1} |W|^{1-q}\frac{|\int_W f \psi|}{\int_W\psi} \geq {|\:\!\!|\:\!\!|} f {|\:\!\!|\:\!\!|}_+.
\]
However condition \eqref{eq:cone 2} is not vacuous since we assume $A>L$.
\end{remark}
We will need the following lemma in Section~\ref{sec:mix}.
\begin{lemma}
\label{lem:b property}
For all $f \in \mathcal{C}_{c,A, L}(\delta)$, $W \in \mathcal{W}^s(\delta)$ and all $\psi_1, \psi_2 \in \mathcal{D}_{a, \beta}(W)$,
\[
\left|\frac{\int_W f \psi_1}{\fint_W\psi_1} - \frac{\int_W f \psi_2}{\fint_W\psi_2}\right|\leq
2 \delta L \rho_{W, a, \beta}(\psi_1, \psi_2) {|\:\!\!|\:\!\!|} f {|\:\!\!|\:\!\!|}_- \, .
\]
\end{lemma}
\begin{proof}
Let $f \in \mathcal{C}_{c, A, L}(\delta)$, $W \in \mathcal{W}^s(\delta)$ and $\psi_1, \psi_2 \in \mathcal{D}_{a, \beta}(W)$.
For each $\lambda, \mu>0$ such that $\lambda \psi_1\preceq \psi_2\preceq \mu\psi_1$, hence also $\lambda \psi_1\leq \psi_2\leq \mu\psi_1$, we have
\[
\frac{\int_W f \psi_2}{\fint_W\psi_2} = \frac{\lambda \int_W f \psi_1 + \int_W f (\psi_2 - \lambda \psi_1)}{\fint_W \psi_2} \geq \frac{\lambda\int_W f \psi_1}{\mu\fint_W \psi_1},
\]
where we have dropped the second term above due to \eqref{eq:posi} since $\psi_2 - \lambda \psi_1 \in \mathcal{D}_{a, \beta}(W)$.
Taking the sup on $\lambda$ and the inf on $\mu$, and recalling \eqref{eq:H def}, yields
\[
\frac{\int_W f \psi_1}{\fint_W\psi_1} - \frac{\int_W f \psi_2}{\fint_W\psi_2} \le \frac{\int_W f \psi_1}{\fint_W\psi_1} ( 1 - e^{-\rho_{W, a, \beta}(\psi_1, \psi_2)} )
\le \rho_{W, a, \beta} (\psi_1, \psi_2) \frac{\int_W f \psi_1}{\fint_W\psi_1} \, .
\]
Then, since $|W| \ge \delta$, we use \eqref{eq:cone 2} to estimate,
\[
\frac{\int_W f \psi_1}{\fint_W\psi_1} \le |W| {|\:\!\!|\:\!\!|} f {|\:\!\!|\:\!\!|}_+ \le 2\delta L {|\:\!\!|\:\!\!|} f {|\:\!\!|\:\!\!|}_- \, .
\]
Reversing the roles of $\psi_1$ and $\psi_2$ completes the
proof of the lemma.
\end{proof}
\section{Cone Estimates: Contraction of $c$ and $A$ }
\label{sec:cone}
In this section, fixing $\mathcal{F}(\tau_*, \mathcal{K}_*, E_*)$, we will prove the following proposition. Let $n_0 \ge 1$ be such that $A C_0 \theta_1^{n_0} \le 1/16$.
\begin{prop}
\label{prop:almost}
If the conditions on $\delta, n_0, a,c, A,L$ specified in Section \ref{sec:conditions} are satisfied, then there exists $\chi < 1$ such that for all $n \ge n_0$ and $\{ \iota_j \}_{j=1}^n \subset \mathcal{I}(\tau_*, \mathcal{K}_*, E_*)$,
\[
\mathcal{L}_n \mathcal{C}_{c,A,L}(\delta) \subseteq \mathcal{C}_{ \chi c, \chi A, 3 L}(\delta) \,.
\]
\end{prop}
Before proving Proposition \ref{prop:almost} we need some facts concerning the behaviour of the test functions under the dynamics.
\subsection{Contraction of test functions}
\label{sec:test}
For $\{ \iota_j \}_{j=1}^n \subset \mathcal{I}(\tau_*, \mathcal{K}_*, E_*)$, $W \in \mathcal{W}^s$, $\psi \in \mathcal{D}_{a, \beta}(W)$, and $W_i \in \mathcal{G}_n(W)$, define
\[
\widehat{T}_{n , W_i} \psi = \widehat{T}_{n, i} \psi := \psi \circ T_n J_{W_i}T_n.
\]
The following lemma is a consequence of {\bf (H1)}.
\begin{lemma}
\label{lem:test contract}
Let $n\geq 0$ be such that $C_1^{-1} \Lambda^{-\beta n}<1$, where $C_1 \le 1$ is from \eqref{eq:hyp}, and fix $a > (1- C_1^{-1} \Lambda^{-\beta n})^{-1}C_d \delta_0^{1/3 - \beta}$. For each $\beta \in (0, 1/3]$, there exist $\sigma, \bar\xi < 1$ such that for all $W \in \mathcal{W}^s$ and $W_i \in \mathcal{G}_n(W)$,
\begin{itemize}
\item[a)] $\widehat{T}_{n,i} (\mathcal{D}_{a, \beta}(W)) \subset \mathcal{D}_{\sigma a, \beta}(W_i)$;
\item[b)] $\rho_{W_i,a, \beta} (\widehat{T}_{n,i} \psi_1, \widehat{T}_{n,i} \psi_2) \le \bar\xi \rho_{W, a , \beta}(\psi_1, \psi_2)$
for all $\psi_1, \psi_2 \in \mathcal{D}_{a, \beta}(W)$.
\end{itemize}
\end{lemma}
\begin{proof}
(a) We need to measure the log-H\"older norm of $\widehat{T}_{n,i} \psi$ for $\psi \in \mathcal{D}_{a, \beta}(W)$.
For $x, y \in W_i$, we estimate,
\[
\frac{\widehat{T}_{n,i} \psi(x)}{\widehat{T}_{n,i} \psi(y)} = \frac{\psi(T_nx) J_{W_i}T_n(x)}{\psi(T_ny) J_{W_i}T_n(y)}
\le e^{a d(T_nx, T_ny)^\beta + C_d d(x,y)^{1/3}}
\le e^{(a C_1^{-\beta} \Lambda^{-\beta n} + C_d \delta_0^{1/3-\beta}) d(x,y)^{\beta}},
\]
where we have used \eqref{eq:hyp} and \eqref{eq:distortion} as well as the fact that $\beta \le 1/3$.
This proves the first statement of the lemma since $a C_1^{-1} \Lambda^{-\beta n} + C_d \delta_0^{1/3 - \beta} < a$.
\smallskip
\noindent
(b) Using Lemma~\ref{lem:H metric}, if $\psi_1, \psi_2 \in \mathcal{D}_{\sigma a, \beta}(W_i)$, then,
\begin{equation}
\label{eq:finite diam}
\begin{split}
\rho_{W_i,a,\beta}(\psi_1, \psi_2) & = \log \left[ \sup_{x,y,u,v \in W_i}
\frac{e^{ad(x,y)^\beta} \psi_1(x) - \psi_1(y)}{e^{a d(x,y)^\beta} \psi_2(x) - \psi_2(y)}
\cdot \frac{e^{ad(u,v)^\beta} \psi_2(u) - \psi_2(v)}{e^{a d(u,v)^\beta} \psi_1(u) - \psi_1(v)} \right] \\
& \le \log \left[ \sup_{x, y, u, v \in W} \frac{e^{(a+\sigma a)d(x,y)^\beta} - 1}{e^{(a-\sigma a)d(x,y)^\beta} -1}
\frac{e^{(a+\sigma a)d(u,v)^\beta} - 1}{e^{(a-\sigma a)d(u,v)^\beta} -1} \frac{\psi_1(y) \psi_2(v)}{\psi_2(y) \psi_1(u)} \right] \\
& \le \log \left[ \frac{(a+\sigma a)^2}{(a - \sigma a)^2} e^{2 a(1+\sigma) \delta_0^\beta} e^{2 a \delta_0^\beta} \right] =: K .
\end{split}
\end{equation}
Thus the diameter of $\mathcal{D}_{\sigma a, \beta}(W_i)$ is finite in $\mathcal{D}_{a, \beta}(W_i)$.
Part (b) of the lemma then follows from \cite[Theorem 1.1]{liv95}, with
$\bar\xi = \tanh (K/4) < 1$.
\end{proof}
\begin{cor}
\label{cor:contract}
Let $n_1$ denote the least positive integer satisfying $C_1^{-1} \Lambda^{-\beta n}<1$ and $a C_1^{-1} \Lambda^{-\beta n_1} + C_d \delta_0^{1/3 - \beta} < a$. Define $\xi = \bar\xi^{\frac{1}{2n_1}}<1$.
Then for $W \in \mathcal{W}^s$, $n \ge n_1$ and $W_i \in \mathcal{G}_n(W)$,
\[
\rho_{W_i, a, \beta} (\widehat{T}_{n,i} \psi_1, \widehat{T}_{n,i} \psi_2) \le \xi^n \rho_{W, a, \beta}(\psi_1, \psi_2) \qquad \mbox{for all $\psi_1, \psi_2 \in \mathcal{D}_{a, \beta}(W)$} .
\]
\end{cor}
\begin{proof}
The proof follows immediately from Lemma~\ref{lem:test contract} once we
decompose $n = kn_1 + r$, where $ r \in [0, n_1)$ and write
\[
\widehat{T}_{n, W_i} \psi = \widehat{T}_{n_1+r, W_i} \circ \widehat{T}_{n_1, T_{n_1+r}(W_i)}
\circ \widehat{T}_{n_1, T_{2n_1 + r}(W_i)} \circ \cdots \circ \widehat{T}_{n_1, T_{(k-1)n_1+r}(W_i)} \psi .
\]
Each of the operators $\widehat{T}_{n_1, T_{jn_1+r}(W_i)}$ satisfies Lemma~\ref{lem:test contract}
with the same $\sigma$ and $\bar\xi$. The corollary then follows
using the observation that $\bar\xi^{\lfloor n/n_1 \rfloor} \le \xi^n$, $\forall n \ge n_1$.
\end{proof}
It is important for what follows that the contractive factor
$\bar\xi<1$ is explicitly given in terms of the diameter $K$, which depends only on $a$ and
$\sigma$, and not on $\delta$. While $n_1$ depends on the parameter choice
$\beta$, it also is independent of $\delta$.
In what follows, we require $n_0 \ge n_1$ by definition, so that Lemma~\ref{lem:test contract} and Corollary~\ref{cor:contract} will hold for all $n \ge n_0$.
\subsection{Proof of Proposition~\ref{prop:almost}}
\label{sec:prop proof}
This section is devoted to the proof of Proposition~\ref{prop:almost}.
\subsubsection{Preliminary estimate on $L$}
Denote by $Sh_n(W; \delta)$ the
elements of $\mathcal{G}_n(W)$ of length less than $\delta$ and by $Lo_n(W; \delta)$ the elements of $\mathcal{G}_n(W)$ of length at least
$\delta$.
\begin{lemma}
\label{lem:first L}
Fix $\delta \in (0, \delta_0/3)$ so that $4A \delta \delta_0^{-1} \bar C_0 \le 1/4$, then, for all $f \in \mathcal{C}_{c,A,L}(\delta)$ and
$n \ge n_0$,
\[
{|\:\!\!|\:\!\!|} \mathcal{L}_n f {|\:\!\!|\:\!\!|}_+ \le \tfrac 32 {|\:\!\!|\:\!\!|} f {|\:\!\!|\:\!\!|}_+ \quad \mbox{and} \quad
{|\:\!\!|\:\!\!|} \mathcal{L}_n f {|\:\!\!|\:\!\!|}_- \ge \tfrac 12 {|\:\!\!|\:\!\!|} f {|\:\!\!|\:\!\!|}_- .
\]
\end{lemma}
\begin{proof}
Let $W \in \mathcal{W}^s(\delta)$, $\psi \in \mathcal{D}_{a,\beta}(W)$. Then,
\begin{equation}
\label{eq:first L}
\int_W \mathcal{L}_n f \, \psi = \sum_{W_i \in Lo_n(W; \delta)} \int_{W_i} f \, \psi \circ T_n \, J_{W_i}T_n
+ \sum_{W_i \in Sh_n(W; \delta)} \int_{W_i} f \, \psi \circ T_n \, J_{W_i}T_n .
\end{equation}
Now since $\psi \circ T_n J_{W_i}T_n \in \mathcal{D}_{a, \beta}(W_i)$ by Lemma~\ref{lem:test contract}, we
subdivide elements $W_i \in Lo_n(W; \delta)$ into curves
$U_\ell$ having length between $\delta$ and $2\delta$ and use the definition of ${|\:\!\!|\:\!\!|} f {|\:\!\!|\:\!\!|}_+$ on
each such curve to estimate,
\[
\int_{W_i} f \, \psi \circ T_n \, J_{W_i}T_n \le \sum_\ell {|\:\!\!|\:\!\!|} f {|\:\!\!|\:\!\!|}_+ \int_{U_\ell} \psi \circ T_n J_{V_j}T_n
= {|\:\!\!|\:\!\!|} f {|\:\!\!|\:\!\!|}_+ \int_{T_nW_i} \psi \, .
\]
To estimate the short pieces, we apply \eqref{eq:cone 3} and use
Lemma~\ref{lem:full growth}-(b) since $Sh_n(W;\delta) \subset \mathcal{G}_n(W)$.
\[
\begin{split}
\sum_{W_i \in Sh_n(W; \delta)} \int_{W_i} f \, \psi \circ T_n \, J_{W_i}T_n
& \le \sum_{W_i \in Sh_n(W; \delta)} {|\:\!\!|\:\!\!|} f {|\:\!\!|\:\!\!|}_- A |W_i|^q \delta^{1-q} \fint_{W_i} \psi \circ T_n \, J_{W_i}T_n \\
& \le \delta A {|\:\!\!|\:\!\!|} f {|\:\!\!|\:\!\!|}_- e^{a(2\delta)^\beta} \fint_W \psi \; (\bar C_0 \delta_0^{-1} |W| + C_0 \theta_1^n) .
\end{split}
\]
Putting these estimates together in \eqref{eq:first L} and using that $|W| \ge \delta$, we obtain,
\[
\begin{split}
\int_W \mathcal{L}_n f \, \psi & \le \sum_{W_i \in Lo_n(W; \delta)} {|\:\!\!|\:\!\!|} f {|\:\!\!|\:\!\!|}_+ \int_{T_nW_i} \psi +
A {|\:\!\!|\:\!\!|} f {|\:\!\!|\:\!\!|}_- e^{a(2\delta)^\beta} \int_W \psi \; (\bar C_0 \delta \delta_0^{-1} + C_0 \theta_1^n) \\
& \le {|\:\!\!|\:\!\!|} f {|\:\!\!|\:\!\!|}_+ \int_W \psi \; \left( 1 + A e^{a(2\delta)^\beta} (\delta \delta_0^{-1} \bar C_0 + C_0 \theta_1^n ) \right),
\end{split}
\]
where we have used Lemma~\ref{lem:avg}. Now \eqref{eq:delta_0 condition} implies $e^{a(2\delta)^\beta} \le 2$, and our choices of $n_0$ and $\delta$
imply $2A\max\{ \bar C_0 \delta \delta_0^{-1} , C_0 \theta_1^{n_0} \} \le 1/4$, which yields the
required estimate on ${|\:\!\!|\:\!\!|} \mathcal{L}_n f {|\:\!\!|\:\!\!|}_+$ for all $n \ge n_0$.
For the bound on ${|\:\!\!|\:\!\!|} \mathcal{L}_n f {|\:\!\!|\:\!\!|}_-$, we perform a similar estimate, except noting that
for $W_i \in Lo_n(W; \delta)$,
\[
\int_{W_i} f \, \psi \circ T_n \, J_{W_i}T_n \ge {|\:\!\!|\:\!\!|} f {|\:\!\!|\:\!\!|}_- \int_{T_nW_i} \psi,
\]
we follow \eqref{eq:first L} to estimate,
\[
\begin{split}
\int_W \mathcal{L}_n f \, \psi & \ge \sum_{W_i \in Lo_n(W; \delta)} {|\:\!\!|\:\!\!|} f {|\:\!\!|\:\!\!|}_- \int_{T_nW_i} \psi -
A {|\:\!\!|\:\!\!|} f {|\:\!\!|\:\!\!|}_- e^{a(2\delta)^\beta} \int_W \psi \; (\bar C_0 \delta \delta_0^{-1} + C_0 \theta_1^n) \\
& \ge {|\:\!\!|\:\!\!|} f {|\:\!\!|\:\!\!|}_- \int_W \psi \; \left( 1 - 2A e^{a(2\delta)^\beta} (\delta \delta_0^{-1} \bar C_0 + C_0 \theta_1^n )\right) \, .
\end{split}
\]
Again using our choice of $n_0$ and $\delta$, we have $4AC_0 \theta_1^n \le 1/4$
and $4A \delta \delta_0^{-1} \bar C_0 \le 1/4$, which yields
${|\:\!\!|\:\!\!|} \mathcal{L}_n f {|\:\!\!|\:\!\!|}_- \ge \frac 12 {|\:\!\!|\:\!\!|} f {|\:\!\!|\:\!\!|}_-$.
\end{proof}
In particular the above implies the estimate: for all $n \ge n_0$,
\begin{equation}\label{eq:no-expansion-L}
\frac{{|\:\!\!|\:\!\!|} \mathcal{L}_n f {|\:\!\!|\:\!\!|}_+}{{|\:\!\!|\:\!\!|} \mathcal{L}_n f {|\:\!\!|\:\!\!|}_-}\leq 3 \frac{{|\:\!\!|\:\!\!|} f {|\:\!\!|\:\!\!|}_+}{{|\:\!\!|\:\!\!|} f {|\:\!\!|\:\!\!|}_-}\leq 3L.
\end{equation}
\subsubsection{Contraction of the parameter $A$}
\label{sec:contraction-A}
We prove that the parameter $A$ contracts in \eqref{eq:cone 3}. Choose $f \in \mathcal{C}_{c, A, L}(\delta)$.
Let $W \in \mathcal{W}^s$ with $|W| \le 2\delta$, $\psi \in \mathcal{D}_{a,\beta}(W)$ and $x \in W$. From now on,
we will refer to $Lo_n(W; \delta)$ and $Sh_n(W; \delta)$ as simply
$Lo_n(W)$ and $Sh_n(W)$. We follow \eqref{eq:first L} to write
\[
\begin{split}
\left| \int_W \mathcal{L}_n f \, \psi \right| & \le \sum_{W_i \in Lo_n(W)} \int_{W_i} f \, \psi \circ T_n \, J_{W_i}T_n
+ \sum_{W_i \in Sh_n(W)} \left| \int_{W_i} f \, \psi \circ T_n \, J_{W_i}T_n \right| \\
& \le \sum_{W_i \in Lo_n(W)} {|\:\!\!|\:\!\!|} f {|\:\!\!|\:\!\!|}_+ \int_{W_i} \psi \circ T_n \, J_{W_i}T_n
+ \sum_{W_i \in Sh_n(W)} A \delta^{1-q} |W_i|^q {|\:\!\!|\:\!\!|} f {|\:\!\!|\:\!\!|}_- \fint_{W_i} \psi \circ T_n \,J_{W_i}T_n \\
& \le \sum_{W_i \in Lo_n(W)} {|\:\!\!|\:\!\!|} f {|\:\!\!|\:\!\!|}_- L \int_{T_nW_i} \psi + A \delta^{1-q} |W|^q {|\:\!\!|\:\!\!|} f {|\:\!\!|\:\!\!|}_- |\psi|_{C^0}
\sum_{W_i \in Sh_n(W)} \frac{|W_i|^q}{|W|^q} \frac{|T_nW_i|}{|W_i|} ,
\end{split}
\]
where in the second line we have used \eqref{eq:cone 3} for the sum on short pieces.
Since $|W| \le 2\delta$, the first sum above is bounded by
\[
{|\:\!\!|\:\!\!|} f {|\:\!\!|\:\!\!|}_- L |W| \fint_W \psi \le \| f {|\:\!\!|\:\!\!|}_- 2 L \delta^{1-q} |W|^q \fint_W \psi \, .
\]
For the sum on short pieces, we use Lemmas~\ref{lem:full growth}(b)
and a H\"older inequality to estimate
\[
\begin{split}
\sum_{W_i \in Sh_n(W)} \frac{|W_i|^q}{|W|^q} \frac{|T_nW_i|}{|W_i|}
& \le \left(\sum_{W_i \in Sh_n(W)} \frac{|T_nW_i|}{|W|} \right)^q
\left( \sum_{W_i \in Sh_n(W)} |J_{W_i}T_n|_{C^0(W_i)} \right)^{1-q} \\
& \le ( \bar C_0 \delta_0^{-1} |W| + C_0 \theta_1^n )^{1-q}
\end{split}
\]
Combining these two estimates with Lemma~\ref{lem:avg} yields,
\begin{equation}
\label{eq:A}
\frac{|\int_W \mathcal{L}_n f \, \psi|}{\fint_W \psi} \le A \delta^{1-q} |W|^q {|\:\!\!|\:\!\!|} f {|\:\!\!|\:\!\!|}_-
\left( 2 L A^{-1} + e^{a(2\delta)^\beta} ( \bar C_0 \delta_0^{-1} |W| + C_0 \theta_1^n )^{1-q} \right) .
\end{equation}
This contracts the parameter $A$ if $2L A^{-1} + e^{a(2\delta)^\beta} ( 2\bar C_0 \delta \delta_0^{-1} + C_0 \theta_1^n )^{1-q} <1$,
which we can achieve if $e^{a(2\delta)^\beta} \le 2$,
\begin{equation}
\label{eq:AL}
A>4L, \quad \mbox{ and } \quad
( 2\bar C_0 \delta \delta_0^{-1} + C_0 \theta_1^{n_0} )^{1-q} <1/4 \, .
\end{equation}
Remark that since $L \ge 1$, we have $A > 4$, and so according to the assumption
of Lemma~\ref{lem:first L}, $2 \bar C_0 \delta \delta_0^{-1} \le 1/32$. Moreover,
$C_0 \theta_1^{n_0} \le 1/64$ by choice of $n_0$, and since $1-q \ge 1/2$, the
second condition in \eqref{eq:AL} is always satisfied under the assumption of
Lemma~\ref{lem:first L}.
\subsubsection{Contraction of the parameter $c$}
\label{subsec:contract c}
Finally, we verify the contraction of $c$ via \eqref{eq:cone 5}.
Let $f \in \mathcal{C}_{c,A,L}(\delta)$ and $W^1, W^2 \in \mathcal{W}^s$ with $|W^k| \le 2 \delta$ and
$d_{\mathcal{W}^s}(W^1, W^2) \le \delta$. Take $\psi_k \in \mathcal{D}_{a, \alpha}(W^k)$ with
$d_*(\psi_1, \psi_2) = 0$.
Without loss of generality we can assume $|W^2|\geq |W^1|$ and $\fint_{W_1}\psi_1=1$.
Next, note that cone condition \eqref{eq:cone 3} implies (see Section \ref{sec:contraction-A})
\[
\left|\frac{\int_{W^1} \mathcal{L}_n f \, \psi_1}{\fint_{W^1}\psi_1} - \frac{\int_{W^2}\mathcal{L}_n f \, \psi_2}{\fint_{W^2}\psi_2} \right|\leq 2A\delta^{1-q}|W^2|^q{|\:\!\!|\:\!\!|} \mathcal{L}_n f{|\:\!\!|\:\!\!|}_-
\]
It follows that the contraction of the parameter $c$ is trivial for
$ |W^2|^q \leq \delta^{q-\gamma} \frac{d_{\mathcal{W}^s}(W^1,W^2)^{\gamma}c}4$.
Thus it suffices to consider the case
\begin{equation}
\label{eq:W2-long}
|W^2|^q \geq \delta^{q - \gamma} \frac{d_{\mathcal{W}^s}(W^1,W^2)^{\gamma}c}4 \, .
\end{equation}
We claim that \eqref{eq:W2-long} implies that $I_{W_1} \cap I_{W_2} \neq \emptyset$.
Define $C_s := \sqrt{1 + (\mathcal{K}_{\max} + \tau_{\min}^{-1})^2}$ to be the maximum absolute value of the slopes of curves in the
stable cone defined in \eqref{eq:hyp}. If $I_{W^1} \cap I_{W^2} = \emptyset$, then
$d_{\mathcal{W}^s}(W^1, W^2) = |I_{W^1} \bigtriangleup I_{W^2}| = |I_{W^1}| + |I_{W^2}|$, so it must be that
$|I_{W^2}| \le \delta$. Yet \eqref{eq:W2-long} implies that
\[
C_s |I_{W^2}| \ge |W^2| \ge \delta^{(q-\gamma)/q} d_{\mathcal{W}^s}(W^1, W^2)^{\gamma/q} (\tfrac{c}{4})^{1/q} \ge \delta^{(q-\gamma)/q} |I_{W^2}|^{\gamma/q} (\tfrac{c}{4})^{1/q},
\]
so we obtain the contradiction $|I_{W^2}| \ge \delta \cdot 2^{1/(q-\gamma)} $ provided
\begin{equation}
\label{eq:q-gamma1}
q > \gamma \; ; \; c \ge 8 C_s^q \, .
\end{equation}
Next, for any two manifolds $U^i\in\mathcal{W}^s_-(\delta)$ defined on the intervals $I_i$ with $J=I_1\cap I_2\neq\emptyset$, by the distance definition \eqref{eq:W-distance} we have,
\begin{equation}\label{eq:W-difference}
\begin{split}
|\,|U^1|-|U^2|\,|&\leq \int_J(\|G'_1\|-\|G'_2\|) dr +\sum_{i=1}^1\int_{I_i\setminus J}\|G'_i\| dr \\
&\leq \int_J \|G'_2-G'_1\| dr + C_s |I_1\Delta I_2| \leq (|U^1|+C_s)d_{\mathcal{W}^s}(U^1,U^2).
\end{split}
\end{equation}
Since $\fint_{W_1} \psi_1 = 1$, we have $|\psi_1|_\infty \le e^{a(2\delta)^\alpha}$. On the other hand, since
$I_{W^1} \cap I_{W^2} \neq \emptyset$ and
$d_*(\psi_1, \psi_2)=0$,
there must exist $r \in I_{W^1} \cap I_{W^2}$ such that $\psi_1 \circ G_{W^1}(r) \| G_{W^1}'(r) \| = \psi_2 \circ G_{W^2}(r) \| G_{W^2}'(r) \|$. Thus since,
\[
\begin{split}
\frac{\| G'_{W^1}(r) \| }{\| G'_{W^2}(r) \| } & = \sqrt{ \frac{1 + (\varphi_{W^1}'(r))^2}{ 1 + (\varphi_{W^2}'(r))^2 } }
= \sqrt{ 1 + \frac{(\varphi_{W^1}'(r) - \varphi_{W^2}'(r))( 2 \varphi_{W^2}'(r) + (\varphi_{W^1}'(r) - \varphi_{W^2}'(r)) )}{1 + (\varphi'_{W^2}(r))^2} } \\
& \le \sqrt{1 + d_{\mathcal{W}^s}(W^1, W^2) ( 2 + d_{\mathcal{W}^s}(W^1, W^2)) }
\le \sqrt{1 + 3 \delta} \le 2 \, ,
\end{split}
\]
where we use $\delta < 1$, we estimate,
\begin{equation}
\label{eq:psi2}
|\psi_2|_{\infty} \le 2e^{a(2\delta)^\alpha} |\psi_1|_\infty \le 2e^{2a(2\delta)^\alpha} \, .
\end{equation}
Then recalling Remark~\ref{rem:change-int}, it follows that
\[
\left|\int_{W^1}\psi_1 - \int_{W^2}\psi_2\right| \; \leq \; e^{a(2\delta)^\alpha} C_s |I_{W^1}\setminus I_{W^2}|
+ e^{2a(2\delta)^\alpha} 2C_s |I_{W^2}\setminus I_{W^1}|
\le 2C_s e^{2a(2\delta)^\alpha}d_{\mathcal{W}^s}(W^1,W^2).
\]
Putting this together with \eqref{eq:W-difference} and using $\int_{W_1} \psi_1 = |W_1|$, we estimate,
\begin{equation}
\label{eq:new-difference}
\begin{split}
\left| |W^2| - \int_{W^2} \psi_2 \right| \; & \le \; \left| |W^2| - |W^1| \right|
+ \left| \int_{W^1} \psi_1 - \int_{W^2} \psi_2 \right| \\
& \le \; (|W^1| + C_s ) d_{\mathcal{W}^s}(W^1, W^2) \le 6 C_s d_{\mathcal{W}^s}(W^1, W^2) \, ,
\end{split}
\end{equation}
where we have used \eqref{eq:delta_0 condition} and $\alpha>\beta$.
Hence, recalling Lemma~\ref{lem:first L} and \eqref{eq:A}, $d_{\mathcal{W}^s}(W^1,W^2)\leq \delta$ and using
\eqref{eq:W2-long}, \eqref{eq:q-gamma1} and \eqref{eq:new-difference}, we have
\begin{equation}
\label{eq:prepare-c}
\begin{split}
&\left|\frac{\int_{W^1} \mathcal{L}_n f \psi_1}{\fint_{W^1}\psi_1} - \frac{\int_{W^2}\mathcal{L}_n f \psi_2}{\fint_{W^2}\psi_2} \right|\leq \left|\int_{W^1} \mathcal{L}_n f \psi_1 - \int_{W^2} \mathcal{L}_n f \psi_2\right|
+\left|\int_{W^2} \mathcal{L}_n f \psi_2\right|\left|\frac{|W^2|}{\int_{W^2}\psi_2}-1\right|\\
&\leq\left|\int_{W^1} \mathcal{L}_n f \psi_1 - \int_{W^2} \mathcal{L}_n f \psi_2\right|
+A\left[\frac{\delta}{|W^2|}\right]^{1-q} \left| |W^2|-\int_{W^2}\psi_2\right| 2 {|\:\!\!|\:\!\!|} \mathcal{L}_n f{|\:\!\!|\:\!\!|}_-\\
&\leq\left|\int_{W^1} \mathcal{L}_n f \psi_1 - \int_{W^2} \mathcal{L}_n f \psi_2\right|
+ 2^{3-1/q} 3C_s^q A \delta^{1-\gamma} d_{\mathcal{W}^s}(W^1,W^2)^{\gamma}{|\:\!\!|\:\!\!|} \mathcal{L}^n f{|\:\!\!|\:\!\!|}_- \, .
\end{split}
\end{equation}
To conclude it suffices then to compare $\int_{W^1} \mathcal{L}^n f \, \psi_1$ and $\int_{W^2} \mathcal{L}^n f \, \psi_2$. To this end, define $\mathcal{G}_n^\delta(W^k)$ to be the $n$th generation homogeneous components of $T^{-n}W^k$ with long pieces subdivided to have length between $\delta$ and
$2 \delta$.
We split $\mathcal{G}_n^\delta(W^1)$ and $\mathcal{G}_n^\delta(W^2)$ into matched and unmatched pieces as follows.
On each curve $W^1_i \in \mathcal{G}_n^\delta(W^1)$, we place a foliation of vertical
line segments $\{ \ell_x \}_{x \in W^1_i}$
of length $C_1 \Lambda^{-n} d_{\mathcal{W}^s(W^1, W^2)}$.
Due to {\bf (H1)}, the images $T_i(\ell_x)$ are unstable curves for $i \ge 1$ and remain uniformly
transverse to the stable cone. Thus $T_n(\ell_x)$ undergoes the uniform expansion
given by \eqref{eq:hyp} and, if not cut by a singularity, will intersect both $W^1$ and $W^2$.
When these segments $T_n(\ell_x)$ survive uncut, we declare the subcurves $U^1_j$, $U^2_j$
connected by the original vertical segments $\ell_x$ to be `matched.'
Note that, by \cite[Proposition 4.47]{chernov book} there must exists two piecewise smooth curves in
$\mathcal{S}_{T_n}^{\mathbb{H}}$ that connect the boundaries of $U^1_j$ and $U^2_j$ forming a {\em rectangle} that does not
contain any element of $\mathcal{S}_{T_n}^{\mathbb{H}}$ in its interior.
All other subcurves we label
$V^1_j$, $V^2_j$ and declare them to be `unmatched.' It follows that there can be at most one
matched curve $U^k_j$ and two unmatched curves $V^k_j$ for each element
$W^k_i \in \mathcal{G}_n^\delta(W^k)$, $k=1, 2$.
Thus we have defined a composition $\mathcal{G}_n^\delta(W^k) = \cup_j U^k_j \cup \cup_j V^k_j$, such
that $U^1_j$ and $U^2_j$ are defined as the graphs of functions $G_{U^k_j}$ over the
same $r$-interval $I_j$ for each $j$.
Using this decomposition, writing $\widehat{T}_{n, U^k_j} \psi_k = \psi_k \circ T_n J_{U^k_j}T_n$
and similarly for $\widehat{T}_{n, V^k_j}\psi_k$,
we have
\begin{equation}
\label{eq:unstable split}
\begin{split}
\int_{W^k} \mathcal{L}_n f \, \psi_k = \sum_{j} \int_{U^k_j} f \, \widehat{T}_{n, U^k_j}\psi_k
+ \sum_j \int_{V^k_j} f \, \widehat{T}_{n, V^k_j} \psi_k.
\end{split}
\end{equation}
We estimate the contribution from unmatched pieces first. To do so, we group the $V^k_j$
as follows. We say $V^k_j$ is `created' at time $0 \le i \le n-1$ if $i$ is the smallest $t$ such
that either an endpoint of $T_{n-t}(V^k_j)$ is created by an intersection with $T_{\iota_{t+1}}(\mathcal{S}_0^\mathbb{H})$,
or $T_{n-t}(V^k_j)$ is
contained in a larger unmatched piece with this property (this second case can happen when both endpoints of
$V^k_j$ are created by subdivision of long pieces rather than cuts due to singularities).
Due to the uniform transversality of the stable cone with curves in $T_{\iota_{i+1}}(\mathcal{S}_0^\mathbb{H})$ as well
as the uniform transversality of the stable and unstable cones, we have
$|T_{n-i}V^k_j| \le \bar C_3 \Lambda^{-i} d_{\mathcal{W}^s}(W^1, W^2)$, for some constant $\bar C_3 > 0$.
Define $P(i) = \{ j : V^1_j \mbox{ created at time } i \}$.
Although we would like to change variables to estimate the contribution on the curves $T_{n-i}(V^1_j)$
for $j \in P(i)$, this is one time step before such cuts would be introduced according to our
definition of $\mathcal{G}_n^\delta(W)$, so Lemma~\ref{lem:full growth} would not apply since there may be
many such $T_{n-i}(V^1_j)$ for each $W^1_\ell \in \mathcal{G}_i^\delta(W^1)$. However, there can be at most
two curves $T_{n-i-1}(V^1_j)$, $j \in P(i)$, per element of $W^1_\ell \in \mathcal{G}_{i+1}^\delta(W^1)$, so we will change variables
to estimate the contribution from curves of the form $T_{n-i-1}(V^1_j)$ instead.
We have two cases.
\smallskip
\noindent
{\em Case 1. The curve in $T_{\iota_{i+1}}(\mathcal{S}_0^\mathbb{H})$ that creates $V^1_j$ at time $i$
is the preimage of the boundary of a homogeneity strip.} Then $T_{n-i-1}V^1_j$ still enjoys uniform transversality with the
boundary of the homogeneity strip and the unstable cone, and so
$|T_{n-i-1}V^1_j| \le \bar C_3 \Lambda^{-i-1} d_{\mathcal{W}^s}(W^1, W^2)$ as before.
\smallskip
\noindent
{\em Case 2. The curve in $T_{\iota_{i+1}}(\mathcal{S}_0^\mathbb{H})$ that creates $V^1_j$ at time $i$
is not the preimage of the boundary of a homogeneity strip.} Then $V^1_j$ undergoes bounded expansion from time
$n-i$ to time $n-i-1$. Thus $|T_{n-i-1}(V^1_j)| \le C \bar C_3 \Lambda^{-i} d_{\mathcal{W}^s}(W^1, W^2)$, where
$C>0$ depends only on our choice of $k_0$, the minimum index of homogeneity strips.
\smallskip
In either case, we conclude that $|T_{n-i-1}(V^1_j)| \le C_3 \Lambda^{-i} d_{\mathcal{W}^s}(W^1, W^2)$, for a uniform
constant $C_3>0$. Also, since $T_{n-i-1}(V^k_j)$ is contained in an element of $\mathcal{G}_{n-i-1}(W)$,
it follows that all such curves have length at most $2 \delta$, thus we may apply \eqref{eq:cone 3},
\[
\begin{split}
&\left| \sum_j \int_{V^1_j} f \, \widehat{T}_{n, V^1_j}n \psi_1 \right|
\le \sum_{i=0}^{n-1} \left| \sum_{j \in P(i)} \int_{T_{n-i-1}(V^1_j)} \mathcal{L}_{n-i-1} f \cdot \psi_1 \circ T_{i+1} \, J_{T_{n-i-1}(V^1_j)} T_{i+1} \right| \\
& \le \sum_{i=0}^{n-1} \sum_{j \in P(i)} A \delta^{1-q} |T_{n-i-1}(V^1_j)|^q {|\:\!\!|\:\!\!|} \mathcal{L}_{n-i-1}f {|\:\!\!|\:\!\!|}_-
|\psi_1|_{C^0(W^1)} |J_{T_{n-i-1}(V^1_j)}T_{i+1}|_{C^0(T_{n-i-1}(V^1_j))} \\
& \le \sum_{i=0}^{n-1} A \delta^{1-q} C_3^q \Lambda^{-iq} d_{\mathcal{W}^s}(W^1, W^2)^q
{|\:\!\!|\:\!\!|} \mathcal{L}^{n-i-1} f {|\:\!\!|\:\!\!|}_- \, (2 \bar C_0 + C_0 \theta_1^{i+1} ) | \psi_1|_{C^0(W^1)},
\end{split}
\]
where we have used Lemma~\ref{lem:full growth}-(b) for the sum over $j \in P(i)$ since there are at most two curve
$T_{n-i-1}(V^1_j)$ for each element $W^1_\ell \in \mathcal{G}_{i+1}^\delta(W)$.\footnote{ Notice that
since we subdivide curves in $\mathcal{G}_n^\delta(W)$ according to length $\delta$ and not
$\delta_0$, the estimate of Lemma~\ref{lem:full growth}(b) becomes
$\bar C_0 \delta^{-1} |W| + C_0 \theta_1^n \le 2 \bar C_0 + C_0 \theta_1^n$. }
Since $n \ge 2n_0$, we have either that $i+1 \ge n_0$ or $n - (i+1) \ge n_0$. In the former
case, ${|\:\!\!|\:\!\!|} \mathcal{L}_{n-i-1}f {|\:\!\!|\:\!\!|}_- \le 2 {|\:\!\!|\:\!\!|} \mathcal{L}_n f {|\:\!\!|\:\!\!|}_-$ by Lemma~\ref{lem:first L}. In the latter
case,
\begin{equation}
\label{eq:tri}
{|\:\!\!|\:\!\!|} \mathcal{L}_{n-i-1} f {|\:\!\!|\:\!\!|}_- \le {|\:\!\!|\:\!\!|} \mathcal{L}_{n-i-1} f {|\:\!\!|\:\!\!|}_+ \le \tfrac 32 {|\:\!\!|\:\!\!|} f {|\:\!\!|\:\!\!|}_+ \le \tfrac 32 \, L {|\:\!\!|\:\!\!|} f {|\:\!\!|\:\!\!|}_-
\le 3 L {|\:\!\!|\:\!\!|} \mathcal{L}_n f {|\:\!\!|\:\!\!|}_-,
\end{equation}
where we have used Lemma~\ref{lem:first L} twice, once on ${|\:\!\!|\:\!\!|} \mathcal{L}_{n-i-1}f {|\:\!\!|\:\!\!|}_+$ and
once on ${|\:\!\!|\:\!\!|} f {|\:\!\!|\:\!\!|}_-$. Since the latter estimate \eqref{eq:tri} is the larger of the two, we may use it
for all $i$.
Also, using the assumption that $d_{\mathcal{W}^s}(W^1, W^2) \le \delta $ and \eqref{eq:q-gamma1} yields,
\[
\delta^{1-q} d_{\mathcal{W}^s}(W^1, W^2)^{q}
\le \delta^{1 - \gamma} d_{\mathcal{W}^s}(W^1, W^2)^\gamma.
\]
Collecting these estimates and summing over the exponential factors yields (since the estimate for $V^2_j$ is the same),
\begin{equation}
\label{eq:V}
\sum_{j,k} \left| \int_{V^k_j} f \, \widehat{T}_{n, V^k_j} \psi_k \right|
\le C_4 AL \delta^{1-\gamma} d_{\mathcal{W}^s}(W^1, W^2)^\gamma {|\:\!\!|\:\!\!|} \mathcal{L}_n f {|\:\!\!|\:\!\!|}_- ,
\end{equation}
for some uniform constant $C_4$ depending only on $\mathcal{F}(\tau_*, \mathcal{K}_*, E_*)$ and not on the parameters of the cone.
Next, we estimate the contribution on matched pieces $U^k_j$. To do this, we will need to change
test functions on the relevant curves. Define the following functions on $U^1_j$,
\begin{equation}
\label{eq:switch}
\begin{split}
&\tilde{\psi}_2 = \psi_2 \circ T_n \circ G_{U^2_j} \circ G_{U^1_j}^{-1}\;;
\quad
\widetilde{J}_{U^2_j}T_n = J_{U^2_j}T_n \circ G_{U^2_j} \circ G_{U^1_j}^{-1}, \\
& \widetilde{T}_{n, U^2_j} (\psi_2) = \tilde{\psi}_2 \cdot \widetilde{J}_{U^2_j}T_n \frac{\|G_{U^2_j}'\| \circ G_{U^1_j}^{-1}}{\|G_{U^1_j}'\|\circ G_{U^1_j}^{-1}}.
\end{split}\end{equation}
Note that $d_*(\widehat{T}_{n, U^2_j} (\psi_2), \widetilde{T}_{n, U^2_j} (\psi_2))=0$ by construction.
Also we define
\begin{equation}\label{eq:psi-diff}
\begin{split}
&\psi^-_j=\min \big\{\widehat{T}_{n, U^1_j}( \psi_1),\widetilde{T}_{n, U^2_j}(\psi_2) \big\}\\
&\psi^\Delta_{1,j}=\widehat{T}_{n, U^1_j}( \psi_1)-\psi^-_j\;;\quad\psi^\Delta_{2,j}=\widetilde{T}_{n, U^2_j}(\psi_2) -\psi^-_j.
\end{split}
\end{equation}
We will need the following lemma to proceed.
\begin{lemma}
\label{lem:compare}
If $c> 4(1+M_0)^q$, $M_0$ is defined in \eqref{eq:M0}, then there exists $C_5 \ge 1$, independent of $n$, $W^1$ and $W^2$ satisfying \eqref{eq:W2-long}, such that for each $j$,
\begin{itemize}
\item[a)] $\displaystyle d_{\mathcal{W}^s}(U^1_j, U^2_j) \le C_5 \Lambda^{-n} d_{\mathcal{W}^s}(W^1, W^2) \,$ ;
\item[b)] $\displaystyle e^{-C_5 d_{\mathcal{W}^s}(W^1, W^2)^\alpha}\leq \frac{\widehat{T}_{n, U^1_j} \psi_1(x)}{\widetilde{T}_{n, U^2_j} \psi_2(x)} \le e^{C_5 d_{\mathcal{W}^s}(W^1, W^2)^\alpha}\quad \forall x \in U^1_j \,$ ;
\item[c)] setting $B= 8 \left[ C_5a^{-1}\right]^{\frac{\alpha-\beta}\alpha}d_{\mathcal{W}^s}(W^1, W^2)^{\alpha-\beta}$
we have $\psi^\Delta_{i,j}+B\psi_j^-\in\mathcal{D}_{a,\beta}(U^1_j)$, $i =1, 2$.
\end{itemize}
Moreover, $\widetilde{T}_{n, U^2_j} \psi_2$ and $\psi_j^-$ belong to $\mathcal{D}_{a,\alpha}(U^1_j)$.
\end{lemma}
We postpone the proof of the lemma and use it to conclude the estimates of this section.
For future use note that Lemma \ref{lem:compare}(b) implies
\begin{equation}\label{eq:deltapsi}
0\leq \psi^\Delta_{k,j}(x)\leq 2 C_5 d_{\mathcal{W}^s}(W^1, W^2)^\alpha \psi^-_j(x).
\end{equation}
\iffalse
{\color{magenta}
Assume that $|U^1_j|=r\leq \delta/2$ then also $|U^2_j|\leq \delta$ and hence, recalling \eqref{eq:cone 3},
\[
\begin{split}
\left|\int_{U^1_j} f \, \widehat{T}_{U^1_j} \psi_1 -\int_{U^2_j} f \, \widehat{T}_{U^2_j}\psi_2\right|&\leq \sum_{i=1}^2 A \delta^{1-q}{|\:\!\!|\:\!\!|} f {|\:\!\!|\:\!\!|}_-
|U^i_j|^q\fint_{U^i_j} \widehat{T}_{U^i_j} \psi_i \\
&\leq \sum_{i=1}^2 A \delta^{1-q}{|\:\!\!|\:\!\!|} f {|\:\!\!|\:\!\!|}_- |U^i_j|^q\|J_{U^i_j}T^n\|_\infty e^{2 a\delta^\alpha}\\
\end{split}
\]
Note that, remembering \eqref{eq:W2-long}, \eqref{eq:W-difference} and \eqref{eq:G-difference} imply
\[
\left|\fint_{W_2}\psi_2-1\right|\leq 4^{1+\frac 1q}\bar B\delta^{-1}d_{\mathcal{W}^s}(W^1,W^2)^{1-\frac{\gamma}q}.
\]
Thus cone condition \eqref{eq:cone 3}, Lemmas \ref{lem:contract}-(b), \ref{lem:compare}-(a) and equations \eqref{eq:W-difference}, \eqref{eq:q-gamma1} imply\footnote{ Note that, for matched manifolds, in \eqref{eq:W-difference} the term $\bar B$ is not present.}
\begin{equation}
\label{eq:short-c}
\begin{split}
\sum_{U^1_j\in Sh_n(W^1, r)}&\left|\int_{U^1_j} f \, \widehat{T}_{U^1_j} \psi_1 -\int_{U^2_j} f \, \widehat{T}_{U^2_j}\psi_2\right|\\
&\leq C_0\theta_0^nA\delta^{1-q}e^{2a^\beta \delta^\beta}\sum_{i=1}^2 |U^i_j|^{q}\fint_{W^i}\psi_i \;{|\:\!\!|\:\!\!|} f {|\:\!\!|\:\!\!|}_- \\
&\leq 2C_0 \theta_0^n A \delta^{1-q}r^q(1+4^{1+\frac 1q}\bar B\delta^{-1}d_{\mathcal{W}^s}(W^1,W^2)^{1-\frac{\gamma}q}){|\:\!\!|\:\!\!|} f {|\:\!\!|\:\!\!|}_- \\
&\leq \frac 12 A\delta c d_{\mathcal{W}^s}(W^1, W^2)^\gamma{|\:\!\!|\:\!\!|} f {|\:\!\!|\:\!\!|}_-,
\end{split}
\end{equation}
provided $r\leq \delta\left[\frac{c d_{\mathcal{W}^s}(W^1, W^2)^\gamma}{16 C_0\theta_0^n}\right]^{1/q}$.
It thus suffices that we consider the case $|U^1_j| \geq\delta\left[\frac{c d_{\mathcal{W}^s}(W^1, W^2)^\gamma}{32 C_0\theta_0^n}\right]^{1/q}=:r_*$.
}
\fi
Since $d_*(\widehat{T}_{n, U^2_j} \psi_2, \widetilde{T}_{n, U^2_j} \psi_2) =0$ by construction, and recalling
Remark \ref{rem:change-int}, Lemma \ref{lem:compare}(c), condition \eqref{eq:cone 3}, and \eqref{eq:psi-diff}, \eqref{eq:deltapsi},
\begin{equation}
\label{eq:c-decomposition}
\begin{split}
&\left|\int_{U^1_j} f \, \widehat{T}_{n, U^1_j} \psi_1 -\int_{U^2_j} f \, \widehat{T}_{n, U^2_j} \psi_2\right| \leq \left|\int_{U^1_j} f \, \widehat{T}_{n, U^1_j} \psi_1 -\int_{U^1_j} f \, \widetilde{T}_{n, U^2_j} \psi_2\right|\\
&+\left|\frac{\int_{U^1_j} f \, \widetilde{T}_{n, U^2_j} \psi_2}{\fint_{U^1_j} \widetilde{T}_{n, U^2_j} \psi_2} -\frac{\int_{U^2_j} f \, \widehat{T}_{n, U^2_j} \psi_2}{\fint_{U^2_j} \widehat{T}_{n, U^2_j} \psi_2}\right|\fint_{U^2_j} \widehat{T}_{n, U^2_j} \psi_2+\left|\frac{\int_{U^1_j} f \, \widetilde{T}_{n, U^2_j} \psi_2}{\fint_{U^1_j} \widetilde{T}_{n, U^2_j} \psi_2}\right|\,\left| \frac{|U^2_j|-|U^1_j|}{|U^1_j|}\right|\fint_{U^2_j} \widehat{T}_{n, U^2_j} \psi_2\\
&\leq A\delta^{1-q}|U^1_j|^q\frac{\left[\fint_{U^1_j}(\psi^\Delta_{1,j}+B\psi^-_j)+\fint_{U^1_j}(\psi^\Delta_{2,j}+B\psi^-_j)\right]}{ \fint_{U^2_j} \widehat{T}_{n, U^2_j} \psi_2 } \fint_{U^2_j} \widehat{T}_{n, U^2_j} \psi_2 {|\:\!\!|\:\!\!|} f {|\:\!\!|\:\!\!|}_- \\
&\quad +d_{\mathcal{W}^s}(U^1_j, U^2_j)^\gamma \delta^{1-\gamma} c A \fint_{U^2_j} \widehat{T}_{n, U^2_j} \psi_2 {|\:\!\!|\:\!\!|} f {|\:\!\!|\:\!\!|}_- \\
&\quad +A\delta^{1-q}|U^1_j|^q \left| \frac{|U^2_j|-|U^1_j|}{|U^1_j|}\right| \fint_{U^2_j} \widehat{T}_{n, U^2_j} \psi_2 {|\:\!\!|\:\!\!|} f {|\:\!\!|\:\!\!|}_- \, ,
\end{split}
\end{equation}
where for the first term, we have used that $|\widehat{T}_{n, U^1_j} \psi_1 - \widetilde{T}_{n, U^2_j} \psi_2 | = \psi_{1, j}^\Delta + \psi_{2,j}^\Delta$,
and for the second and third terms that $\widetilde{T}_{n, U^2_j}\psi_2 \in \mathcal{D}_{a,\alpha}(U^1_j)$ by Lemma~\ref{lem:compare}.
Then, recalling Lemma~\ref{lem:full growth}(b), \eqref{eq:psi2} and \eqref{eq:delta_0 condition}, we can estimate
\begin{equation}
\label{eq:summingU2}
\sum_j \fint_{U^2_j} \widehat{T}_{n, U^2_j}\psi_2\leq \sum_j \fint_{U^2_j} |J_{U_2^j}T_n |_\infty \psi_2\circ T_n
\leq (\bar C_0\delta^{-1} |W^2| + C_0\theta_1^n) 2 e^{2a(2\delta)^\alpha}\leq 24 \bar C_0 \, .
\end{equation}
Next, recalling \eqref{eq:W-difference}, we have\footnote{ Since the $U^k_j$ are vertically matched, the term on the right hand side of \eqref{eq:W-difference} proportional to $C_s$ is absent here.}
\[
|U^2_j| \le |U^1_j|(1 + d_{\mathcal{W}^s}(U^1_j, U^2_j)) \le 2|U^1_j|
\]
provided we impose
\begin{equation}\label{eq:c-cond}
C_5 \Lambda^{-n_0} \delta \le 1
\end{equation}
where $C_5$ is from Lemma~\ref{lem:compare}-(a) and $\Lambda$ is defined in \eqref{eq:hyp}.
Moreover, remembering the definition of $B$ in Lemma \ref{lem:compare}-(c) and equation \eqref{eq:deltapsi},
\begin{equation}\label{eq:Delta-psi-B}
\begin{split}
\fint_{U^1_j}(\psi^\Delta_{k,j}+B\psi^-_j)&\leq \fint_{U^1_j} 10 C_5 \widetilde{T}_{n, U^2_j}( \psi_2) d_{\mathcal{W}^s}(W^1,W^2)^\gamma\\
& \leq 10 C_5\frac{|U^2_j|}{|U^1_j|} \fint_{U^2_j}\widehat{T}_{n, U^2_j} (\psi_2) d_{\mathcal{W}^s}(W^1,W^2)^\gamma
\leq 20 C_5 \fint_{U^2_j}\widehat{T}_{n, U^2_j} (\psi_2) d_{\mathcal{W}^s}(W^1,W^2)^\gamma,
\end{split}
\end{equation}
where we have used the assumption $\alpha-\beta\geq \gamma$.
Again using \eqref{eq:W-difference} and Lemma \ref{lem:compare}-(a) we have
\begin{equation}
\label{eq:length-ratio}
\begin{split}
\left| \frac{|U^2_j|-|U^1_j|}{|U^1_j|^{1-q}}\right|&\leq d_{\mathcal{W}^s}(U^2_j, U^1_j)|U^1_j|^{q}\leq (2 \delta)^{q} C_5 \Lambda^{-n}d_{\mathcal{W}^s}(W^2, W^1).
\end{split}
\end{equation}
Inserting \eqref{eq:summingU2}, \eqref{eq:Delta-psi-B} and \eqref{eq:length-ratio} in \eqref{eq:c-decomposition} and
recalling Lemmas~\ref{lem:first L} and \ref{lem:compare}-(a) yields,
\begin{equation}\label{eq:use-in-4}
\begin{split}
& \sum_j \left|\int_{U^1_j} f \, \widehat{T}_{n, U^1_j} \psi_1 -\int_{U^2_j} f \, \widehat{T}_{n, U^2_j}\psi_2\right| \\
&\le
48 \bar C_0 A \delta^{1-\gamma} d_{\mathcal{W}^s}(W^1, W^2)^\gamma {|\:\!\!|\:\!\!|} \mathcal{L}_n f {|\:\!\!|\:\!\!|}_-
\left( 2^q 40 C_5 \delta^{q - \gamma}
+ c C_5 \Lambda^{-n \gamma}
+ 2^q C_5 \Lambda^{-n} \delta \right)
\end{split}
\end{equation}
Then using this estimate in \eqref{eq:prepare-c}, and recalling \eqref{eq:unstable split} and \eqref{eq:V} yields
\begin{equation}
\label{eq:switch test}
\begin{split}
& \left|\frac{\int_{W^1} \mathcal{L}_nf \, \psi_1}{\fint_{W^1}\psi_1} -\frac{\int_{W^2} \mathcal{L}_nf \, \psi_2}{\fint_{W^2}\psi_2}\right|\leq
\Big\{2^{3-1/q} 3C_s^q +C_4 L \\
& \qquad + 48 \bar C_0 \left( 2^q 40 C_5 \delta^{q - \gamma} + c C_5 \Lambda^{-n \gamma}
+ 2^q C_5 \Lambda^{-n} \delta \right) \Big\} A\delta^{1-\gamma} d_{\mathcal{W}^s}(W^1, W^2)^\gamma {|\:\!\!|\:\!\!|} \mathcal{L}_n f {|\:\!\!|\:\!\!|}_-
\end{split}
\end{equation}
which yields the wanted estimate, provided
\begin{equation}
\label{eq:c cond}
2^{3-1/q} C_s^q +C_4 L
+ 48 \bar C_0 \left( 2^q 40 C_5 \delta^{q-\gamma} + c C_5 \Lambda^{-n \gamma}
+ 2^q C_5 \Lambda^{-n} \delta \right) < c.
\end{equation}
\subsubsection{Proof of Lemma~\ref{lem:compare}}
\begin{proof}
(a) This is \cite[Lemma 3.3]{demzhang13}.
\smallskip
\noindent
(b) Recall that $U^k_j$ is defined as the graph of a function $G_{U^k_j}(r) = (r, \varphi_{U^k_j}(r))$, for $r \in I^k_j$, $k = 1,2$.
Due to the vertical matching, we have $I^1_j = I^2_j$.
Now for $x \in U^1_j$, let $r \in I^1_j$ be such that $G_{U^1_j}(r) = x$. Set
$\bar{x} = G_{U^2_j}(r)$ and note that $x$ and $\bar{x}$ lie on the same vertical line in $M$ since
$U^1_j$ and $U^2_j$ are matched. Thus by \eqref{eq:u dist},
\begin{equation}
\label{eq:M0}
\frac{J_{U^1_j}T_n(x)}{\widetilde{J}_{U^2_j}T_n(x)} = \frac{J_{U^1_j}T_n(x)}{J_{U^2_j}T_n(\bar{x})}
\le e^{C_d (d(T_nx, T_n\bar{x})^{1/3}+ \phi(x, \bar{x}))}
\le e^{C_d M_0 d_{\mathcal{W}^s}(W^1, W^2)^{1/3}},
\end{equation}
where $M_0$ is a constant depending only on the maximum and minimum slopes in $C^s$ and
$C^u$.
Next, for $x \in U^1_j$ consider
\[
\frac{\psi_1 \circ T_n(x)}{\tilde{\psi}_2(x)} \frac{\| G'_{U^1_j} \| \circ G_{U^1_j}^{-1}(x)}{\| G'_{U^2_j} \| \circ G_{U^1_j}^{-1}(x)}.
\]
Let $T_n(x)=(r,G_{W^1}(r))$ and $T_n(\bar x)=(\bar r,G_{W^2}(\bar r))$, then
\[
|r-\bar r|\leq M_0 d_{\mathcal{W}^s}(W^1,W^2) \, .
\]
If $r\in I_{W^2}$, then since $d_*(\psi_1, \psi_2)=0$,
\[
\frac{\psi_1 \circ G_{W^1}(r)}{\psi_2 \circ G_{W^2}(\bar r)} = \frac{\psi_1 \circ G_{W^1}(r)}{\psi_2 \circ G_{W^2}(r)} \frac{\psi_2 \circ G_{W^2}(r)}{\psi_2 \circ G_{W^2}(\bar r)} \leq
\frac{ \| G_{W^2}'(r) \|}{\| G_{W^1}'(r) \| } e^{a d(G_{W_1}(r), G_{W^2}(\bar r))^\alpha}.
\]
Next, since $\| G_{W^1}' - G_{W^2}' \| = |\varphi'_{W^1} - \varphi'_{W^2}|$ and $\| G_{W^k}' \| \ge 1$, we have
\[
\frac{ \| G_{W^2}'(r) \|}{\| G_{W^1}'(r) \| } \le e^{ \| G_{W^1}' - G_{W^2}' \| } \le e^{d_{\mathcal{W}^s}(W^1, W^2)} \, .
\]
Similarly, $\frac{\| G'_{U^1_j} \| \circ G_{U^1_j}^{-1}(x)}{\| G'_{U^2_j} \| \circ G_{U^1_j}^{-1}(x)} \le e^{d_{\mathcal{W}^s}(U^1_j, U^2_j)}$.
Hence, using part (a) of the lemma and assuming
\begin{equation}\label{eq:c-cond2}
C_5 n_0 \Lambda^{-n_0} \delta^{1-\alpha} \le 1,
\end{equation}
yields
\[
\frac{\psi_1 \circ T_n(x)}{\tilde{\psi}_2(x)} \frac{\| G'_{U^1_j} \| \circ G_{U^1_j}^{-1}(x)}{\| G'_{U^2_j} \| \circ G_{U^1_j}^{-1}(x)}
\le e^{(aM_0^\alpha+2) d_{\mathcal{W}^s}(W^1, W^2)^\alpha} \, .
\]
The same estimate holds if $\bar r \in I_{W^1}$. Otherwise it must be that
\[
|I_{W^1}\cap I_{W^2}|\leq M_0 d_{\mathcal{W}^s}(W^1,W^2)
\]
but then, since $|I_{W^1}\Delta I_{W^2}|\leq d_{\mathcal{W}^s}(W^1,W^2)$ we would have $|W^2|\leq (1+M_0) d_{\mathcal{W}^s}(W^1,W^2)$,
which violates \eqref{eq:W2-long} together with the assumption,
provided
\begin{equation}\label{eq:c-cond3}
c > 4(1+M_0)^q.
\end{equation}
The estimates with the opposite sign follow similarly. Putting together these
estimates yields part (b) of the lemma with $C_5 = M_0C_d \delta^{1/3 - \alpha} + aM_0^\alpha + 2$.
\smallskip
\noindent
(c) As noted in \eqref{eq:deltapsi}, by (b) it immediately follows that
\[
\left| \psi^\Delta_{i,j}(x)\right|\leq\left| \widehat{T}_{n, U^1_j} \psi_1(x)-\widetilde{T}_{n, U^2_j} \psi_2(x)\right|\leq
2 C_5 d_{\mathcal{W}^s}(W^1, W^2)^\alpha \psi_j^-(x).
\]
Next, for $x, y \in U^1_j$, let $\bar{x} = G_{U^2_j} \circ G_{U^1_j}^{-1}(x)$,
$\bar{y} = G_{U^2_j} \circ G_{U^1_j}^{-1}(y)$, and note these are well-defined due to the
vertical matching between $U^1_j$ and $U^2_j$.
Let $r = G_{U^1_j}^{-1}(x)$ and $s = G_{U^1_j}^{-1}(y)$. Recalling \eqref{eq:2deriv}, we have
\[
\frac{\| G'_{U^1_j}(r) \|}{\| G'_{U^1_j}(s) \|} \le e^{\| G'_{U^1_j}(r) - G'_{U^1_j}(s) \|} \le e^{B_* |r-s|} \le e^{B_* d(x,y)} \, ,
\]
and similarly for $\| G'_{U^2_j} \|$.
Using this estimate together with the proof of Lemma~\ref{lem:test contract}(a),
\begin{equation}
\label{eq:incone2}
\begin{split}
\frac{\widetilde{T}_{n, U^2_j} \psi_2(x)}{\widetilde{T}_{n, U^2_j} \psi_2(y)} & = \frac{\widehat{T}_{n, U^2_j} \psi_2(\bar{x})}{\widehat{T}_{n, U^2_j} \psi_2(\bar{y})}
\frac{ \| G'_{U^2_j} (r) \|}{\| G'_{U^1_j}(r)\| } \frac{\| G'_{U^1_j}(s) \|}{\| G'_{U^2_j}(s) \|} \\
& \le e^{(a C_1^{-1} \Lambda^{-\alpha n} + C_d (2\delta)^{1/3-\alpha}) d(\bar{x},\bar{y})^\alpha
+ 2B_* d(x,y)}\leq e^{ad(x,y)^\alpha},
\end{split}
\end{equation}
since $d(\bar{x}, \bar{y}) \le M_0 d(x,y)$ and provided
\begin{equation}\label{eq:cond-a}
(aC_1^{-1} \Lambda^{-\alpha n_0} + C_d (2\delta)^{1/3 - \alpha})M_0^\alpha
+ B_* (2\delta)^{1-\alpha} < a.
\end{equation}
To abbreviate what follows, let us denote $g_1 = \widehat{T}_{n, U^1_j} \psi_1$ and $g_2 = \widetilde{T}_{n, U^2_j} \psi_2$.
Then, given $x,y\in U^1_j$, we have $\psi_j^-(x)=g_{k(x)}$, $\psi_j^-(y)=g_{k(y)}$. If $k(x)=k(y)$, then,
by Lemma~\ref{lem:test contract}(a) and \eqref{eq:incone2},
\[
\frac{\psi_j^-(x)}{\psi_j^-(y)}=\frac{g_{k(y)}(x)}{g_{k(y)}(y)}\leq e^{a d(x,y)^\alpha} \, .
\]
If $k(x)\neq k(y)$, then without loss of generality, we can take $k(x)=1$ and $k(y)=2$. By definition, $g_1(x) \le g_2(x)$
and $g_2(y) \le g_1(y)$. Hence,
\[
e^{-ad(x,y)^\alpha} \le \frac{g_1(x)}{g_1(y)} \le \frac{\psi_j^-(x)}{\psi_j^-(y)}=\frac{g_1(x)}{g_2(y)} \le \frac{g_2(x)}{g_2(y)} \leq e^{a d(x, y)^\alpha} \, .
\]
It follows that $\psi_j^-\in\mathcal{D}_{a,\alpha}(U^1_j)$, and by \eqref{eq:incone2}, $\widetilde{T}_{n, U^2_j} \psi_2 \in \mathcal{D}_{a, \alpha}(U^1_j)$.
Then, for each $1>B\geq 2 C_5 d_{\mathcal{W}^s}(W^1, W^2)^\alpha$ and $x,y\in U^1_j$,
\[
\frac{\psi^\Delta_{i,j}(x)+B\psi_j^-(x)}{\psi^\Delta_{i,j}(y)+B\psi_j^-(y)}\leq \frac{(B+2 C_5 d_{\mathcal{W}^s}(W^1, W^2)^\alpha )\psi_j^-(x)}{(B-2 C_5 d_{\mathcal{W}^s}(W^1, W^2)^\alpha )\psi_j^-(y)}\leq e^{a d(x,y)^\alpha+ 4 B^{-1} C_5 d_{\mathcal{W}^s}(W^1, W^2)^\alpha}\leq e^{a d(x,y)^\beta}
\]
provided $8 B^{-1}C_5 d_{\mathcal{W}^s}(W^1, W^2)^\alpha\leq a d(x,y)^\beta$ and
\begin{equation}\label{eq:cond-c3}
(2\delta)^{\alpha-\beta}\leq \frac 12.
\end{equation}
It remains to consider the case $8 B^{-1}C_5 d_{\mathcal{W}^s}(W^1, W^2)^\alpha\geq a d(x,y)^\beta$.
Again we must split into two cases. If $k(x)=k(y)=k$, then, setting $\{\ell\}=\{1,2\}\setminus \{k\}$,
\begin{equation} \label{eq:aergh}
\begin{split}
\frac{\psi^\Delta_{\ell,j}(x)+B\psi_j^-(x)}{\psi^\Delta_{\ell,j}(y)+B\psi_j^-(y)}&\leq \frac{g_\ell(x)+(B-1)g_k(x)}{g_\ell(y)+(B-1)g_k(y)}
\leq \frac{e^{ a d(x,y)^\alpha} g_\ell(y)+e^{- a d(x,y)^\alpha} (B-1)g_k(y)}{g_\ell(y)+(B-1)g_k(y)}\\
&\leq e^{ a d(x,y)^\alpha}\left[1+\frac{2 a d(x,y)^\alpha}{B}\right]\leq e^{a [d(x,y)^{\alpha-\beta}(1+2B^{-1})] d(x,y)^\beta}
\leq e^{\frac a 2 d(x,y)^\beta}
\end{split}
\end{equation}
provided that
\[
d(x,y)^{\alpha-\beta}(1+2B^{-1})\leq 4 B^{-\frac\alpha\beta}\left[ 8 C_5 d_{\mathcal{W}^s}(W^1, W^2)^\alpha a^{-1}\right]^{\frac{\alpha-\beta}\beta}\leq \frac 12\,.
\]
That is,
\[
B\geq 8 \left[ C_5a^{-1}\right]^{\frac{\alpha-\beta}\alpha}d_{\mathcal{W}^s}(W^1, W^2)^{\alpha-\beta}.
\]
The second case is $k=k(x)\neq k(y)=\ell$. In this case, there must exist $\bar x \in [x,y]$ such that
$\psi_j^-(\bar{x}) = g_1(\bar{x}) = g_2(\bar{x})$. Then,
\[
\begin{split}
\frac{\psi^\Delta_{\ell,j}(x)+B\psi_j^-(x)}{\psi^\Delta_{\ell,j}(y)+B\psi_j^-(y)}
& = \frac{g_\ell(x)+(B-1)g_k(x)}{Bg_\ell(\bar x)} \frac{g_\ell(\bar x)+(B-1)g_k(\bar x)}{g_\ell(\bar{x})+(B-1)g_k(\bar{x})}
\leq e^{ a d(x,y)^\beta}
\end{split}
\]
by the estimate \eqref{eq:aergh}. A similar estimate holds for $\psi_{k,j}^\Delta$.
It follows that we can choose
\begin{equation}\label{eq:setB}
B= 8 \left[ C_5a^{-1}\right]^{\frac{\alpha-\beta}\alpha}d_{\mathcal{W}^s}(W^1, W^2)^{\alpha-\beta}
\end{equation}
and have $\psi^\Delta_{i,j}+B\psi_j^-\in\mathcal{D}_{a,\beta}(U^1_j)$.
\end{proof}
\subsection{Conditions on parameters }
\label{sec:conditions}
In this section, we collect the conditions imposed on the cone parameters during the proof
of Proposition~\ref{prop:almost}. Recall the conditions on the exponents
stated before the definition of $\mathcal{C}_{c,A,L}(\delta)$:
$\alpha \in (0, 1/3]$, $q \in (0,1/2)$, $\beta < \alpha$ and $\gamma \le \min \{ \alpha - \beta, q \}$.
From \eqref{eq:delta_0 condition} and Lemma~\ref{lem:first L} we require,
\[
e^{a (2\delta)^\beta} < e^{2a \delta_0^\beta} \le 2 \quad \mbox{and} \quad
4A \bar C_0 \delta \delta_0^{-1} \le 1/4 \, .
\]
From the proof of Lemma~\ref{lem:first L} and Lemma~\ref{lem:test contract}, we require
the following conditions on $n_0$,
\[
AC_0 \theta_1^{n_0} \le 1/16 \quad \mbox{and} \quad C_1^{-1} \Lambda^{-\beta n_0} < 1 \, .
\]
From Lemma~\ref{lem:test contract}, Corollary~\ref{cor:contract} and the proof of Lemma~\ref{lem:compare}, we require
\[
a > aC_1^{-1} \Lambda^{-\beta n_0} + C_d \delta_0^{1/3 - \beta} \quad \mbox{and} \quad
a > (aC_1^{-1} \Lambda^{-\alpha n_0} + C_d (2\delta)^{1/3 - \alpha})M_0^\alpha
+ B_* (2\delta)^{1-\alpha}
\]
(recall that we have chosen $n_0 \ge n_1$ after Corollary~\ref{cor:contract}).
From the bound on \eqref{eq:cone 3}, we require in \eqref{eq:AL},
\[
A > 4 L \, .
\]
For the contraction of $c$, we require (see \eqref{eq:q-gamma1}, the proof of Lemma \ref{lem:compare} and \eqref{eq:c cond})
\[
\begin{split}
& c > \max \left\{ 8 C_s^q , 4(1+M_0)^q \right\} \;;\quad C_5 \Lambda^{-n_0} \delta^{1-\alpha} \le 1 \; ;
\quad (2\delta)^{\alpha - \beta} \le \tfrac 12 \; ; \\
& 2^{3-1/q} 3C_s^q +C_4 L
+ 48 \bar C_0 \left( 2^q 40 C_5 \delta^{q-\gamma} + c C_5 \Lambda^{-n _0\gamma}
+ 2^q C_5 \Lambda^{-n_0} \delta \right) < c.
\end{split}
\]
Finally, in anticipation of \eqref{eq:delta cond}, we require,
\begin{equation}
\label{eq:cA}
cA > 2 C_s \, .
\end{equation}
These are all the conditions we shall place on the parameters for the cone, except for $\delta$, which
we will take as small as required for the mixing arguments of Section~\ref{sec:L contract}.
\section{Contraction of $L$ and Finite Diameter}
\label{sec:L contract}
Proposition~\ref{prop:almost} proves that the parameters $c$ and $A$ of the cone $\mathcal{C}_{c,A,L}(\delta)$
contract simply as a consequence of the uniform properties {\bf (H1)}-{\bf (H5)} for any sequence of maps
$(T_{\iota_j})_j \subset \mathcal{F}(\tau_*, \mathcal{K}_*, E_*)$. In this section, however, we will restrict our sequence of
maps to be drawn from a sufficiently small neighborhood of a single map $T_0 \in \mathcal{F}(\tau_*, \mathcal{K}_*, E_*)$
in order to use the uniform mixing properties maps $T$ close to $T_0$ to prove that the parameter $L$ also
contracts under the sequential dynamics.
This is done
in two steps. First, in Section~\ref{sec:scale}, we use a length scale $\delta_0 \ge \sqrt{\delta}$ and compare averages
on the two length scales, $\delta$ and $\delta_0$,
culminating in Proposition~\ref{prop:alternative}. This step does not yet require us to restrict our class of maps.
Second, in Section~\ref{sec:mix}, restricting our sequential system to a neighborhood of a fixed map $T_0$,
we obtain a bound on averages
in the length scale $\delta_0$ as expressed in Lemma~\ref{lem:match}. This leads to the strict contraction of $L$ established in
Theorem~\ref{thm:cone contract},
which proves the first statement of Theorem~\ref{thm:main}. We prove the second statement of Theorem~\ref{thm:main}
in Section~\ref{sec:diam}, showing that the cone $\mathcal{C}_{\chi c, \chi A, \chi L}(\delta)$ has finite diameter
in the cone $\mathcal{C}_{c,A,L}(\delta)$ (Proposition~\ref{prop:diameter}).
\subsection{Comparing averages on different length scales}
\label{sec:scale}
Recall the length scale $\delta_0$ from \eqref{eq:theta_1} and that $\delta < \delta_0/2$. We choose $\delta$ so that
$\delta \le \delta_0^2$.
Define
\[
{|\:\!\!|\:\!\!|} f {|\:\!\!|\:\!\!|}_+^0 = \sup_{\substack{W \in \mathcal{W}^s(\delta_0/2) \\ \psi \in \mathcal{D}_{a, \beta}(W) }}
\frac{\int_W f \, \psi \, dm_W}{\int_W \psi \, dm_W} ,
\qquad \qquad
{|\:\!\!|\:\!\!|} f {|\:\!\!|\:\!\!|}_-^0 = \inf_{\substack{W \in \mathcal{W}^s(\delta_0/2) \\ \psi \in \mathcal{D}_{a, \beta}(W) }}
\frac{\int_W f \, \psi \, dm_W}{\int_W \psi \, dm_W} .
\]
Recall that $\mathcal{W}^s(\delta_0/2)$ denotes those curves in $\mathcal{W}^s$ with length between
$\delta_0/2$ and $\delta_0$.
By subdividing curves of with length in $[\delta_0/2,\delta_0]$ into curves with length in $[\delta, 2\delta]$, we immediately deduce the relations,
\begin{equation}
\label{eq:diff scale}
{|\:\!\!|\:\!\!|} f {|\:\!\!|\:\!\!|}_- \le {|\:\!\!|\:\!\!|} f {|\:\!\!|\:\!\!|}_-^0 \le {|\:\!\!|\:\!\!|} f {|\:\!\!|\:\!\!|}_+^0 \le {|\:\!\!|\:\!\!|} f {|\:\!\!|\:\!\!|}_+ \, .
\end{equation}
\begin{lemma}
\label{lem:0 scale}
Recall $e^{a \delta_0^\beta} \le 2$ from \eqref{eq:delta_0 condition}
and $A\delta \le \delta_0/4$ from Lemma~\ref{lem:first L}.
For all $n \in\mathbb{N}$ and $\{ \iota_j \}_{j=1}^n \subset \mathcal{I}(\tau_*, \mathcal{K}_*, E_*)$,\footnote{The second inequality in \eqref{eq:upper 0} follows from equation \eqref{eq:theta_1}.}
\begin{eqnarray}
{|\:\!\!|\:\!\!|} \mathcal{L}_n f {|\:\!\!|\:\!\!|}_+^0 & \le& {|\:\!\!|\:\!\!|} f {|\:\!\!|\:\!\!|}_+^0 + 3 C_0 \sum_{i=1}^n \theta_1^i {|\:\!\!|\:\!\!|} f {|\:\!\!|\:\!\!|}_+
\le {|\:\!\!|\:\!\!|} f {|\:\!\!|\:\!\!|}_+^0 + \frac 14 {|\:\!\!|\:\!\!|} f {|\:\!\!|\:\!\!|}_+ \, , \label{eq:upper 0} \\
{|\:\!\!|\:\!\!|} \mathcal{L}_n f {|\:\!\!|\:\!\!|}_-^0 & \ge & \frac 34 {|\:\!\!|\:\!\!|} f {|\:\!\!|\:\!\!|}_-^0 \, . \label{eq:lower 0}
\end{eqnarray}
\end{lemma}
\begin{proof}
We prove \eqref{eq:upper 0} by induction on $n$. It holds trivially for $n = 0$. We assume the inequality
holds for $0 \le k \le n-1$ and prove the statement for $n$.
Let $W \in \mathcal{W}^s(\delta_0/2)$. Define $\widehat{L}_1(W)$ to be those elements of $\mathcal{G}_1(W)$
having length at least $\delta_0/2$. For $k > 1$, let $\widehat{L}_k(W)$ denote those curves of length
at least $\delta_0/2$ in $\mathcal{G}_k(W)$ that are not already contained in an element
of $\widehat{L}_i(W)$ for any $i = 1, \ldots, k-1$. For $V_j \in \widehat{L}_k(W)$, let
$P_k(j)$ be the collection of indices $i$ such that $W_i \in \mathcal{G}_n(W)$ satisfies
$T_{n-k}W_i \subset V_j$.
Denote by $\mathcal{I}^0_n(W)$ those indices $i$ for which $T_{n-k}W_i$ is never contained in an
element of $\mathcal{G}_k(W)$ of length at least $\delta_0/2$, $1 \le k \le n$, and
$\delta \le |W_i| < \delta_0/2$.
Let $\mathcal{I}_n(W)$ denote the remainder of the indices $i$ for curves in $\mathcal{G}_n(W)$, i.e.
those curves $W_i$ of length shorter than $\delta$ and for which $T_{n-k}W_i$ is not contained in
an element of $\mathcal{G}_k(W)$ of length at least $\delta_0/2$.
By construction, each $W_i \in \mathcal{G}_n(W)$
belongs to precisely one $P_k(j)$ or $\mathcal{I}^0_n(W)$ or $\mathcal{I}_n(W)$.
Now, for $\psi \in \mathcal{D}_{a, \beta}(W)$, note that
\[
\sum_{i \in P_k(j)}
\int_{W_i} f \, \psi \circ T_n \, J_{W_i}T_n
= \int_{V_j} \mathcal{L}_{n-k} f \, \psi \circ T_k \, J_{V_j}T_k .
\]
Using this equality, we estimate,
\[
\begin{split}
\int_W \mathcal{L}_n f \, \psi & = \sum_{k=1}^n \sum_{V_j \in \widehat{L}_k(W)} \int_{V_j} \mathcal{L}_{n-k} f \, \psi \circ T_k \, J_{V_j}T_k
\; \; + \; \; \sum_{i \in \mathcal{I}^0_n(W)} \int_{W_i} f \, \psi \circ T_n \, J_{W_i}T_n \\
& \qquad + \sum_{i \in \mathcal{I}_n(W)} \int_{W_i} f \, \psi \circ T_n \, J_{W_i}T_n \\
& \le \sum_{k=1}^n \sum_{V_j \in \widehat{L}_k(W)} {|\:\!\!|\:\!\!|} \mathcal{L}_{n-k} f {|\:\!\!|\:\!\!|}_+^0 \int_{V_j} \psi \circ T_k \, J_{V_j}T_k
\; \; + \; \; \sum_{i \in \mathcal{I}^0_n(W)} {|\:\!\!|\:\!\!|} f {|\:\!\!|\:\!\!|}_+ \int_{W_i} \psi \circ T_n \, J_{W_i}T_n \\
& \qquad + \sum_{i \in \mathcal{I}_n(W)} A \delta^{1-q} |W_i|^q {|\:\!\!|\:\!\!|} f {|\:\!\!|\:\!\!|}_- |\psi|_{C^0(W)}
|J_{W_i}T_n|_{C^0(W_i)} \\
& \le \sum_{k=1}^n \sum_{V_j \in \widehat{L}_k(W)} \Big({|\:\!\!|\:\!\!|} f {|\:\!\!|\:\!\!|}_+^0 + 3 \sum_{i=1}^{n-k} C_0 \theta_1^i {|\:\!\!|\:\!\!|} f {|\:\!\!|\:\!\!|}_+ \Big) \int_{T_kV_j} \psi \\
& \qquad + \sum_{i \in \mathcal{I}^0_n(W)} {|\:\!\!|\:\!\!|} f {|\:\!\!|\:\!\!|}_+ \frac{\delta_0}{2} |\psi|_{C^0(W)} |J_{W_i}T_n |_{C^0(W_i)} + A \frac{\delta}{\delta_0} \delta_0 |\psi|_{C^0(W)} {|\:\!\!|\:\!\!|} f {|\:\!\!|\:\!\!|}_+ C_0 \theta_1^n \\
& \le \int_W \psi \; \Big({|\:\!\!|\:\!\!|} f {|\:\!\!|\:\!\!|}_+^0 + 3 \sum_{i=1}^{n-1} C_0 \theta_1^i {|\:\!\!|\:\!\!|} f {|\:\!\!|\:\!\!|}_+ \Big)
+ \Big(1+2A \frac{\delta}{\delta_0}\Big) e^{a\delta_0^\beta} \int_W \psi \; {|\:\!\!|\:\!\!|} f {|\:\!\!|\:\!\!|}_+ C_0 \theta_1^n ,
\end{split}
\]
where for the second inequality we have used the inductive hypothesis, and for the
second and third we have used Lemmas ~\ref{lem:full growth}-(a) and \ref{lem:avg}.
This proves the required inequality if $\delta_0$ is small enough that $e^{a\delta_0^\beta} \le 2$
and $\delta$ is small enough that $A \delta \le \delta_0 /4$, both of which we have assumed.
We prove \eqref{eq:lower 0} similarly, although now the inductive hypothesis is
${|\:\!\!|\:\!\!|} \mathcal{L}_k f {|\:\!\!|\:\!\!|}_-^0 \ge (1 - 3\sum_{i=1}^{k} C_0 \theta_1^i)$ for each $k = 0, \ldots, n-1$.
We begin with
the same decompostion of $\mathcal{G}_n(W)$, although we simply drop the terms in $\mathcal{I}_n^0(W)$
since they are all positive.
\[
\begin{split}
\int_W \mathcal{L}_n f \, \psi & = \sum_{k=1}^n \sum_{V_j \in \widehat{L}_k(W)} \int_{V^j} \mathcal{L}_{n-k} f \, \psi \circ T_k \, J_{V_j}T_k
\; \; + \; \; \sum_{i \in \mathcal{I}^0_n(W)} \int_{W_i} f \, \psi \circ T_n \, J_{W_i}T_n \\
& \qquad + \sum_{i \in \mathcal{I}_n(W)} \int_{W_i} f \, \psi \circ T_n \, J_{W_i}T_n \\
& \ge \sum_{k=1}^n \sum_{V_j \in \widehat{L}_k(W)} {|\:\!\!|\:\!\!|} \mathcal{L}_{n-k} f {|\:\!\!|\:\!\!|}_-^0 \int_{V^j} \psi \circ T_k \, J_{V_j}T_k
- \sum_{i \in \mathcal{I}_n(W)} A \delta^{1-q} |W_i|^q {|\:\!\!|\:\!\!|} f {|\:\!\!|\:\!\!|}_- |\psi|_{C^0(W)}
|J_{W_i}T_n|_{C^0(W_i)} \\
& \ge \sum_{k=1}^n \sum_{V_j \in \widehat{L}_k(W)} \int_{T_kV_j} \psi \; {|\:\!\!|\:\!\!|} f {|\:\!\!|\:\!\!|}_-^0 \Big(1- 3 \sum_{i=1}^{n-k} C_0 \theta_0^i \Big)
- A \frac{\delta}{\delta_0} \delta_0 |\psi|_{C^0(W)} {|\:\!\!|\:\!\!|} f {|\:\!\!|\:\!\!|}_- C_0 \theta_1^n \\
& \ge \int_W \psi \; {|\:\!\!|\:\!\!|} f {|\:\!\!|\:\!\!|}_-^0 \Big(1 - 3 \sum_{i=1}^{n-1} C_0 \theta_1^i \Big)
- 2A \frac{\delta}{\delta_0} e^{a\delta_0^\beta} \int_W \psi \; {|\:\!\!|\:\!\!|} f {|\:\!\!|\:\!\!|}_-^0 C_0 \theta_1^n \\
& \quad - {|\:\!\!|\:\!\!|} f {|\:\!\!|\:\!\!|}_-^0 \Big(1 - 3 \sum_{i=1}^{n-1} C_0 \theta_1^i \Big) \sum_{i \in \mathcal{I}_n(W) \cup \mathcal{I}_n^0(W)} |W_i| |\psi|_{C^0(W)}
|J_{W_i}T_n|_{C^0(W_i)} \\
& \ge \int_W \psi \; {|\:\!\!|\:\!\!|} f {|\:\!\!|\:\!\!|}_-^0 \Big( 1 - 3 \sum_{i=1}^{n-1} C_0 \theta_1^i
- 2A \frac{\delta}{\delta_0} e^{a\delta_0^\beta} C_0 \theta_1^n
- e^{a\delta_0^\beta} C_0 \theta_1^n \Big) \, ,
\end{split}
\]
where again we have used Lemmas~\ref{lem:full growth}(a) and \ref{lem:avg} as well as the bound $ {|\:\!\!|\:\!\!|} f {|\:\!\!|\:\!\!|}_- \le {|\:\!\!|\:\!\!|} f {|\:\!\!|\:\!\!|}_-^0$.
This proves the inductive claim, and from this, \eqref{eq:lower 0} follows from \eqref{eq:theta_1}.
\end{proof}
Next, we have a partial converse of Lemma~\ref{lem:0 scale}.
\begin{lemma}
\label{lem:second scale}
For all $n \ge \frac{\log (8C_0(L \delta_0 \delta^{-1} + 2A))}{|\log \theta_1|}$ and $\{ \iota_j \}_{j=1}^n \subset \mathcal{I}(\tau_*, \mathcal{K}_*, E_*)$,
we have
\[
\begin{split}
{|\:\!\!|\:\!\!|} \mathcal{L}_n f {|\:\!\!|\:\!\!|}_+ & \le \max_{k = 0, \ldots n-1} {|\:\!\!|\:\!\!|} \mathcal{L}_k f {|\:\!\!|\:\!\!|}_+^0 + \frac 18 {|\:\!\!|\:\!\!|} f {|\:\!\!|\:\!\!|}_- \\
{|\:\!\!|\:\!\!|} \mathcal{L}_n f {|\:\!\!|\:\!\!|}_- & \ge \frac 34 \min_{k = 0, \ldots n-1} {|\:\!\!|\:\!\!|} \mathcal{L}_k f {|\:\!\!|\:\!\!|}_-^0 - \frac 18 {|\:\!\!|\:\!\!|} f {|\:\!\!|\:\!\!|}_-
\end{split}
\]
\end{lemma}
\begin{proof}
The proof follows along the lines of the proof of Lemma~\ref{lem:0 scale}, using the same decomposition into $\hat{L}_k(W)$,
$\mathcal{I}_n^0(W)$ and $\mathcal{I}_n(W)$, except that now we begin with $W \in \mathcal{W}^s(\delta)$ and $\psi \in \mathcal{D}_{a, \beta}(W)$.
We have,
\[
\begin{split}
\int_W \mathcal{L}_n f \, \psi &
\le \sum_{k=1}^n \sum_{V_j \in \widehat{L}_k(W)} {|\:\!\!|\:\!\!|} \mathcal{L}^{n-k} f {|\:\!\!|\:\!\!|}_+^0 \int_{V^j} \psi \circ T_k \, J_{V_j}T_k
\; \; + \; \; \sum_{i \in \mathcal{I}^0_n(W)} {|\:\!\!|\:\!\!|} f {|\:\!\!|\:\!\!|}_+ \int_{W_i} \psi \circ T_n \, J_{W_i}T_n \\
& \qquad + \sum_{i \in \mathcal{I}_n(W)} A \delta^{1-q} |W_i|^q {|\:\!\!|\:\!\!|} f {|\:\!\!|\:\!\!|}_- |\psi|_{C^0(W)}
|J_{W_i}T_n|_{C^0(W_i)} \\
& \le \int_{W} \psi \max_{k=0, \ldots n-1} {|\:\!\!|\:\!\!|} \mathcal{L}_k f {|\:\!\!|\:\!\!|}_+^0
+ {|\:\!\!|\:\!\!|} f {|\:\!\!|\:\!\!|}_+ C_0 \theta_1^n \frac{\delta_0}{ \delta} \int_W \psi + 2 A {|\:\!\!|\:\!\!|} f {|\:\!\!|\:\!\!|}_- C_0 \theta_1^n \int_W \psi \\
& \le \int_W \psi \; \Big( \max_{k=0, \ldots n-1} {|\:\!\!|\:\!\!|} \mathcal{L}_k f {|\:\!\!|\:\!\!|}_+^0 + {|\:\!\!|\:\!\!|} f {|\:\!\!|\:\!\!|}_- C_0 \theta_1^n (L \delta_0 \delta^{-1} + 2A) \Big) \, ,
\end{split}
\]
which proves the first inequality, given our assumed bound on $n$.
The second inequality follows similarly, again along the lines of Lemma~\ref{lem:0 scale}.
\[
\begin{split}
\int_W & \mathcal{L}_n f \, \psi
\ge \sum_{k=1}^n \sum_{V_j \in \widehat{L}_k(W)} {|\:\!\!|\:\!\!|} \mathcal{L}_{n-k} f {|\:\!\!|\:\!\!|}_-^0 \int_{V^j} \psi \circ T_k \, J_{V_j}T_k
- \sum_{i \in \mathcal{I}_n(W)} A \delta^{1-q} |W_i|^q {|\:\!\!|\:\!\!|} f {|\:\!\!|\:\!\!|}_- |\psi|_{C^0(W)}
|J_{W_i}T_n|_{C^0(W_i)} \\
& \ge \min_{k =0, \ldots n-1} {|\:\!\!|\:\!\!|} \mathcal{L}_k f {|\:\!\!|\:\!\!|}_-^0 \left( \int_W \psi - \sum_{i \in \mathcal{I}_n(W) \cup \mathcal{I}_n^0(W)} |W_i| |\psi|_{C^0(W)} |J_{W_i}T_n|_{C^0(W_i)} \right) - 2 A \int_W \psi \; {|\:\!\!|\:\!\!|} f {|\:\!\!|\:\!\!|}_- C_0 \theta_1^n \\
& \ge \int_W \psi \; \left( \min_{k = 0, \ldots n-1} {|\:\!\!|\:\!\!|} \mathcal{L}_k f {|\:\!\!|\:\!\!|}_-^0 ( 1 - \delta_0 \delta^{-1} C_0 \theta_1^n )
- 2A C_0 \theta_1^n {|\:\!\!|\:\!\!|} f {|\:\!\!|\:\!\!|}_- \right) \, ,
\end{split}
\]
and our bound on $n$ suffices to complete the proof of the lemma.
\end{proof}
Finally, we collect these estimates in the following proposition. Set
\begin{equation}\label{eq:N-}
N(\delta)^- = \frac{\log (8C_0(L \delta_0 \delta^{-1} + 2A))}{|\log \theta_1|},
\end{equation}
from Lemma~\ref{lem:second scale}.
\begin{prop}
\label{prop:alternative}
For all $n \ge N(\delta)^-$ and $\{ \iota_j \}_{j=1}^n \subset \mathcal{I}(\tau_*, \mathcal{K}_*, E_*)$, either,
\[
\frac{{|\:\!\!|\:\!\!|} \mathcal{L}_n f {|\:\!\!|\:\!\!|}_+}{{|\:\!\!|\:\!\!|} \mathcal{L}_n f {|\:\!\!|\:\!\!|}_-} \le \frac{8}{9} \frac{{|\:\!\!|\:\!\!|} f {|\:\!\!|\:\!\!|}_+}{{|\:\!\!|\:\!\!|} f {|\:\!\!|\:\!\!|}_-} \, ,
\]
or
\[
{|\:\!\!|\:\!\!|} \mathcal{L}_n f {|\:\!\!|\:\!\!|}_+ \le 8 {|\:\!\!|\:\!\!|} f {|\:\!\!|\:\!\!|}_+^0 \quad \mbox{and} \quad {|\:\!\!|\:\!\!|} \mathcal{L}_n f {|\:\!\!|\:\!\!|}_- \ge \frac{9}{20} {|\:\!\!|\:\!\!|} f {|\:\!\!|\:\!\!|}_-^0 \, .
\]
\end{prop}
\begin{proof}
Since $n \ge N(\delta)^- \ge n_0$, we may apply both Lemmas~\ref{lem:first L} and \ref{lem:second scale}. Now, by Lemma \ref{lem:second scale},
\[
{|\:\!\!|\:\!\!|} \mathcal{L}_n f {|\:\!\!|\:\!\!|}_- \ge \frac 34 \max_{k = 0, \ldots n-1} {|\:\!\!|\:\!\!|} \mathcal{L}_k f {|\:\!\!|\:\!\!|}_-^0 - \frac 18 {|\:\!\!|\:\!\!|} f {|\:\!\!|\:\!\!|}_-
\ge \frac{9}{16} {|\:\!\!|\:\!\!|} f {|\:\!\!|\:\!\!|}_-^0 - \frac 14 {|\:\!\!|\:\!\!|} \mathcal{L}_n f {|\:\!\!|\:\!\!|}_-\, ,
\]
applying Lemma~\ref{lem:0 scale} to the first term and Lemma~\ref{lem:first L} to the second. This yields immediately,
${|\:\!\!|\:\!\!|} \mathcal{L}_n f {|\:\!\!|\:\!\!|}_- \ge \frac{9}{20} {|\:\!\!|\:\!\!|} f {|\:\!\!|\:\!\!|}_-^0$, which is the final inequality in the statement of the lemma.
Now consider the following alternatives. If ${|\:\!\!|\:\!\!|} \mathcal{L}_n f {|\:\!\!|\:\!\!|}_+ \le \frac 25 {|\:\!\!|\:\!\!|} f {|\:\!\!|\:\!\!|}_+$, then
\[
\frac{{|\:\!\!|\:\!\!|} \mathcal{L}_n f {|\:\!\!|\:\!\!|}_+}{{|\:\!\!|\:\!\!|} \mathcal{L}_n f {|\:\!\!|\:\!\!|}_-} \le \frac{\frac 25 {|\:\!\!|\:\!\!|} f {|\:\!\!|\:\!\!|}_+}{\frac{9}{20} {|\:\!\!|\:\!\!|} f {|\:\!\!|\:\!\!|}_-^0}
\le \frac{8}{9} \frac{{|\:\!\!|\:\!\!|} f {|\:\!\!|\:\!\!|}_+}{{|\:\!\!|\:\!\!|} f {|\:\!\!|\:\!\!|}_-} \,
\]
proving the first alternative. On the other hand, if ${|\:\!\!|\:\!\!|} \mathcal{L}_n f {|\:\!\!|\:\!\!|}_+ \ge \frac 25 {|\:\!\!|\:\!\!|} f {|\:\!\!|\:\!\!|}_+$, then
using Lemmas~\ref{lem:second scale}, \ref{lem:0 scale} and \ref{lem:first L},
\[
\begin{split}
{|\:\!\!|\:\!\!|} \mathcal{L}_n f {|\:\!\!|\:\!\!|}_+ & \le \max_{k =0, \ldots n-1} {|\:\!\!|\:\!\!|} \mathcal{L}_k f {|\:\!\!|\:\!\!|}_+^0 + \frac 18 {|\:\!\!|\:\!\!|} f {|\:\!\!|\:\!\!|}_-
\le {|\:\!\!|\:\!\!|} f {|\:\!\!|\:\!\!|}_+^0 + \frac 14 {|\:\!\!|\:\!\!|} f {|\:\!\!|\:\!\!|}_+ + \frac 14 {|\:\!\!|\:\!\!|} \mathcal{L}_n f {|\:\!\!|\:\!\!|}_- \\
& \le {|\:\!\!|\:\!\!|} f {|\:\!\!|\:\!\!|}_+^0 + \frac 78 {|\:\!\!|\:\!\!|} \mathcal{L}_n f {|\:\!\!|\:\!\!|}_+ \, ,
\end{split}
\]
which yields the second alternative.
\end{proof}
\subsection{Mixing implies contraction of $L$}
\label{sec:mix}
The importance of Proposition~\ref{prop:alternative} is that either $L$ contracts within $N(\delta)^-$ iterates
or we can compare ratios of integrals on the length scale $\delta_0$ (which is fixed independently of $\delta$).
In the latter case we will use the uniform mixing property of maps $T \in \mathcal{F}(\tau_*, \mathcal{K}_*, E_*)$ in order to compare the value
of $\int_W \mathcal{L}_n f \psi$ for different $W$ of length approximately $\delta_0$.
To this end, we will define a Cantor set $R_*$ comprised of local stable and unstable manifolds of a certain length in order to make our comparison when curves cross this set.
We begin by recalling the open neighborhoods in $\mathcal{F}(\tau_*, \mathcal{K}_*, E_*)$ defined by \eqref{eq:close d 1}, using the distance $d_{\mathcal{F}}$ defined in
Section~\ref{sec:map distance}. Let $T \in \mathcal{F}(\tau_*, \mathcal{K}_*, E_*)$, and for
$0 < \kappa < \frac 12 \min \{ \tau_*, \mathcal{K}_*, E_* \}$, define
\begin{equation}
\label{eq:close d}
\mathcal{F}(T, \kappa) = \{ \widetilde{T} \in \mathcal{F}(\tau_*, \mathcal{K}_*, E_*) : d_{\mathcal{F}}(\widetilde{T}, T) < \kappa \} \, .
\end{equation}
Recall the index set corresponding to $\mathcal{F}(T, \kappa)$ is $\mathcal{I}(T, \kappa) \subset \mathcal{I}(\tau_*, \mathcal{K}_*, E_*)$.
Thus $\iota \in \mathcal{I}(T, \kappa)$ if and only if $T_\iota \in \mathcal{F}(T, \kappa)$. An important feature of the topology
induced by $d_{\mathcal{F}}$ is the following.
\begin{lemma}
\label{lem:compact F}
For any $\kappa >0$, the set $\mathcal{F}(\tau_*, \mathcal{K}_*, E_*)$ can be covered by finitely many sets
$\mathcal{F}(T, \kappa)$, $T \in \mathcal{F}(\tau_*, \mathcal{K}_*, E_*)$.
\end{lemma}
\begin{proof}
Recall that each $T \in \mathcal{F}(\tau_*, \mathcal{K}_*, E_*)$ is associated with a billiard table $Q \in \mathcal{Q}(\tau_*, \mathcal{K}_*, E_*)$.
Such billiard tables have exactly $K$ boundary curves with $C^3$ norm uniformly bounded by $E_*$.
Then since the torus is compact and the distance $ \mathbbm d (Q, \widetilde{Q})$ defined in
Section~\ref{sec:bill family} measures distance only in the $C^2$ norm, the set
$\mathcal{Q}(\tau_*, \mathcal{K}_*, E_*)$ is compact in the distance $\mathbbm d$. Thus for each $\varepsilon>0$, there exists
$N_\varepsilon \in \mathbb{N}$ and a set $\{ Q_{\iota_j} \}_{j =1}^{N_\varepsilon} \subset \mathcal{Q}(\tau_*, \mathcal{K}_*, E_*)$ such
that\footnote{Recall from Section~\ref{sec:bill family} that
$\mathcal{Q}(Q_{\iota_j}, E_*, \varepsilon) = \{ Q \in \mathcal{Q}(\frac 12 \tau_*, \frac 12 \mathcal{K}_*, E_*) : \mathbbm d(Q, Q_{\iota_j}) < \varepsilon \}$.}
$\cup_j \mathcal{Q}(Q_{\iota_j}, E_*, \varepsilon) \supset \mathcal{Q}(\tau_*, \mathcal{K}_*, E_*)$. By Proposition~\ref{prop:close maps},
if $Q_\iota \in \mathcal{Q}(Q_{\iota_j}, E_*, \varepsilon)$, then $d_{\mathcal{F}}(T_\iota, T_{\iota_j}) \le C\varepsilon^{1/3}$ for a uniform
constant $C>0$. Thus choosing $\varepsilon$ so that $C\varepsilon^{1/3} = \kappa$ gives us our required covering.
\end{proof}
\begin{remark}
The primary reason we restrict to $\widetilde{T} \in \mathcal{F}(T, \kappa)$ is to conclude Lemma~\ref{lem:proper cross}(b) for a fixed
time $n_*$ and rectangle $R_*$. This will enable us to make a type of `matching' argument
for our sequential system, the main comparison being established in Lemma~\ref{lem:match}.
The reader familiar with the subject will notice that the matching described here requires weaker properties
than the usual arguments
used in coupling. After stable curves are forced to cross a fixed rectangle by Lemma~\ref{lem:proper cross},
the `matched' pieces are not Cantor sets, but rather full curves. The cone technique thus enables us to bypass
the use of real stable/unstable manifolds used in classical coupling arguments for billiards (see \cite[Section~7]{chernov book},
and even the modified coupling developed for sequential systems which only couples for finite time along
approximate invariant manifolds, as in
\cite{young zhang}, both of which require a more delicate use of the structure of invariant manifolds.
\end{remark}
For a fixed $T \in \mathcal{F}(\tau_*, \mathcal{K}_*, E_*)$, we construct an approximate rectangle $D$ in $M$, contained in a single homogeneity strip, whose boundaries are comprised of two local
stable and two local unstable manifolds for $T$ as follows. Choose $\bar \delta_0>0$ and $x \in M$ such that dist$(T^{-n}x, \mathcal{S}_1^{\mathbb{H}}) \ge \bar \delta_0 \Lambda^{- |n|}$
for all $n \in \mathbb{Z}$. This implies that the homogenous local stable and unstable manifolds of $x$, $W^s_{\mathbb{H}}(x)$ and $W^u_{\mathbb{H}}(x)$, have length at least $\bar \delta_0$
on either side of $x$.
By the Sinai Theorem applied to homogeneous unstable manifolds (see, for example, \cite[Theorem~5.70]{chernov book}),
we may choose $\delta_0 < \bar \delta_0$ such that more than 9/10 of the measure of points
in $W^u_{\mathbb{H}}(x) \cap B_{\delta_0}(x)$ have homogeneous local stable manifolds longer than
$2\delta_0$ on both sides of $W^u_{\mathbb{H}}(x)$, and analogously for the points in
$W^s_{\mathbb{H}}(x) \cap B_{\delta_0}(x)$.
Let $D'_{2\delta_0}$ denote the minimal solid rectangle containing this set of stable and unstable
manifolds. Note that the stable and unstable diameters of $D'_{2\delta_0}$ have length at least $4\delta_0$.
There must exist a rectangle $D_{2\delta_0}$ fully crossing $D'_{2\delta_0}$ in the stable direction
and with boundary comprising two stable and two unstable manifolds, such that
the unstable diameter of $D_{2\delta_0}$ is between $\delta_0^4$ and $2\delta_0^4$ and the
set of local homogeneous stable and unstable manifolds fully crossing $D_{2\delta_0}$ comprise at least
$3/4$ of the measure of $D$ with respect to $\mu_{\mbox{\tiny SRB}}$; otherwise, at most $3/4$ of the measure
of $W^u_{\mathbb{H}}(x) \cap B_{\delta_0}(x)$ would have long stable manifolds on either side of
$W^u_{\mathbb{H}}(x)$, contradicting our choice of $\delta_0$.
Similarly, define $D_{\delta_0} \subset D_{2\delta_0}$ to have precisely the same stable boundaries, but
stable diameter $2 \delta_0$ rather than $4 \delta_0$, still centered at $x$.
Let $R_*^{\delta_0}$ denote the maximal set of homogeneous stable and unstable
manifolds in $D_{\delta_0}$ that fully cross $D_{\delta_0}$, and define $R_*^{2\delta_0}$ analogously.
By construction, $\mu_{\mbox{\tiny SRB}}(R_*^{\delta_0}) > (3/4) \mu_{\mbox{\tiny SRB}}(D_{\delta_0}) \approx \delta_0^5$.
Below, we denote $D_{\delta_0}$ by $D(R_*^{\delta_0})$ since it is the minimal solid rectangle that defines $R_*^{\delta_0}$.
We say that a stable curve $W \in \mathcal{W}^s$ {\em properly crosses} a Cantor rectangle $R$ (in the stable direction) if $W$ intersects the interior of
the solid rectangle $D(R)$, but does not terminate in
$D(R)$, and does not intersect the two stable manifolds contained in $\partial D(R)$.
\begin{lemma}
\label{lem:proper cross}
For $T \in \mathcal{F}(\tau_*, \mathcal{K}_*, E_*)$, let $R_*^{\delta_0} = R_*^{\delta_0}(T)$ be the Cantor rectangle constructed above.
\begin{itemize}
\item[a)] There exists $n_* \in \mathbb{N}$, depending only on $\delta_0$ and $\mathcal{F}(\tau_*, \mathcal{K}_*, E_*)$, such that for all $T \in \mathcal{F}(\tau_*, \mathcal{K}_*, E_*)$ and all $W \in \mathcal{W}^s$ with\footnote{Recall that
$\bar C_0$ is from Lemma~\ref{lem:full growth}.} $|W| \ge \delta_0/(6 \bar C_0)$, and all $n \ge n_*$,
$T^{-n}W$ contains a connected, homogeneous component that properly crosses $R_*^{\delta_0}(T)$.
\item[b)] There exists $\kappa>0$ such that for all $T \in \mathcal{F}(\tau_*, \mathcal{K}_*, E_*)$ and all $\{ \iota_j \}_{j=1}^{n_*} \subset \mathcal{I}(T, \kappa)$,
$T_{n_*}^{-1}W$ contains a connected, homogeneous component that properly crosses $R_*^{\delta_0}(T)$.
\end{itemize}
\end{lemma}
\begin{proof}
First we fix $T \in \mathcal{F}(\tau_*, \mathcal{K}_*, E_*)$ and prove items (a) and (b) of the lemma for this $T$, i.e. we demonstrate
that such an $n_*$ and $\kappa$ exist depending on $T$. Then we show how Lemma~\ref{lem:compact F}
implies that $n_*$ and $\kappa$ can be chosen uniformly for $T \in \mathcal{F}(\tau_*, \mathcal{K}_*, E_*)$.
\smallskip
\noindent
a) Fix $T \in \mathcal{F}(\tau_*, \mathcal{K}_*, E_*)$.
By \cite[Lemma~7.87]{chernov book}, there exist finitely many Cantor rectangles
$\mathcal{R}(\delta_0) = \{ R_1, \ldots, R_k \}$, with
$\mu_{\mbox{\tiny SRB}}(R_i) > 0$ for each $i$, such that any stable curve $W \in \mathcal{W}^s$ with $|W| \ge \delta_0/(6\bar C_0)$ properly crosses at least one of them.
Let $\varepsilon_{\mathcal{R}}$ to be the minimum length of an unstable manifold in $R_i$, for any $R_i \in \mathcal{R}(\delta_0)$.
Consider the solid rectangle $\bar D(R_*^{2\delta_0}) \subset D(R_*^{2\delta_0})$ which crosses
$D(R_*^{2\delta_0})$ fully in the stable direction, but
comprises the approximate middle $1/2$ of $D(R_*^{2\delta_0})$ in the unstable direction, with approximately $1/4$ of the unstable diameter of $D(R_*^{2\delta_0})$ on each side of $\bar D(R_*^{2\delta_0})$. Let $\bar R_*^{2\delta_0} := R_*^{2\delta_0} \cap \bar D(R_*^{2\delta_0})$ and note that $\mu_{\mbox{\tiny SRB}}(\bar R_*^{2\delta_0}) > 0$ since $\mu_{\mbox{\tiny SRB}}(R_*^{2\delta_0}) > (3/4) \mu_{\mbox{\tiny SRB}}(D)$ by construction.
Now given $W \in \mathcal{W}^s$ with $|W| \ge \delta_0/(6\bar C_0)$, let
$R_i \in \mathcal{R}(\delta_0)$ denote the Cantor rectangle which $W$ crosses properly.
By the mixing property of $T$, there exists $n_i^* > 0$ such that for all $n \ge n_i^*$, $T^n(\bar R_*^{2\delta_0}) \cap R_i \neq \emptyset$.
We may increase $n_i^*$ if necessary so
that $\Lambda^{n_i^*} \delta_0^4/8 \ge \varepsilon_{\mathcal{R}}$. We claim that $T^n(R_*^{2\delta_0})$ properly crosses
$R_i$ in the unstable direction for all $n \ge n_i^*$.
If not, then the unstable manifolds comprising $R_*^{2\delta_0}$ must be cut by a singularity curve in $\mathcal{S}_1^{\mathbb{H}}$ before time $n_i^*$ (since otherwise they would be longer than $2 \varepsilon_{\mathcal{R}}$ by choice of $n_i^*$), and the images of those unstable manifolds must terminate on
the unstable manifolds in $R_i$. But this implies that some unstable manifolds in $R_i$ will be cut under $T^{-n}$,
a contradiction.
Since $T^n(R_*^{2\delta_0})$ properly crosses $R_i$ in the unstable direction, it follows that $T^n(D(R_*^{2\delta_0}))$ contains a subinterval
of $W$ (here we use the fact that the stable manifolds of $R_*^{2\delta_0}$ cannot be cut under $T^n$, as well as that
the singularity curves of $T^n$ can only terminate on other elements of $\mathcal{S}_n^{\mathbb{H}}$ \cite[Proposition~4.47]{chernov book}), call it $V$. Thus $T^{-n}V$ properly crosses $R_*^{2\delta_0}$. Since $R_*^{\delta_0}$ has the
same stable boundaries as $R_*^{2\delta_0}$, but half the stable diameter, then $T^{-n}V$ also
properly crosses $R_*^{\delta_0}$, as required.
Since $\mathcal{R}(\delta_0)$ is finite, setting $n_* = \max_{1 \le i \le k} \{ n_i^* \} < \infty$ completes the proof of part (a)
with $n_* = n_*(T)$ depending on $T$.
\smallskip
\noindent
(b) In the proof of part (a), for $T \in \mathcal{F}(\tau_*, \mathcal{K}_*, E_*)$ we constructed a rectangle $\bar\mathcal{R}_*^{2\delta_0}$
and a time $n_*$ so that for
any $W \in \mathcal{W}^s$ and $n \ge n_*$, there exists $V \subset W$ such that $T^{-n}$ is smooth on $V$ and $T^{-n}V$ properly crosses $\bar\mathcal{R}_*^{2\delta_0}$. Now
for $\{ \iota_j \}_{j=1}^{n_*} \in \mathcal{I}(T, \kappa)$, condition (C1) in the definition of $d_{\mathcal{F}}$ guarantees
that $T_{n_*}^{-1}V$ is close to $T^{-n_*}V$ for $\kappa$ sufficiently small, except possibly when iterates land
in a neighborhood $N_\kappa(\mathcal{S}_{-1}^T \cup \mathcal{S}_{-1}^{T_{\iota_j}})$. But notice that for $T_\iota \in \mathcal{F}(T, \kappa)$,
the singularity sets $\mathcal{S}_{-1}^T$ and $\mathcal{S}_{-1}^{T_\iota}$ either differ by at most $C\kappa^{1/2}$ or new components
are formed in a $C\kappa^{1/2}$ neighborhood of $\mathcal{S}_0$, where $C$ depends only on $\mathcal{F}(\tau_*, \mathcal{K}_*, E_*)$. This can be proved as in \cite[Section~6.2]{demzhang13}.
Thus $T_{n_*}^{-1}V$ and $T^{-n_*}V$ can be cut differently only near their endpoints. By construction, since $\bar\mathcal{R}_*^{2\delta_0}$ has half the stable width and
twice the stable length as $\mathcal{R}_*^{\delta_0}$, then there exists $\kappa$, depending only on
$\delta_0$, such that $T_{n_*}^{-1}V$ properly crosses $\mathcal{R}_*^{\delta_0}$, as required.
\smallskip
Finally, we show how $n_*$ and $\kappa$ can be chosen uniformly in $\mathcal{F}(\tau_*, \mathcal{K}_*, E_*)$. For
each $T \in \mathcal{F}(\tau_*, \mathcal{K}_*, E_*)$, parts (a) and (b) yield $n_*(T)$ and $\kappa(T)$ with the stated properties.
Let $\varepsilon(T)$ be such that $C \varepsilon(T)^{1/3} = \frac{\kappa(T)}{2}$, where $C>0$ is from Proposition~\ref{prop:close maps}.
Then the set of open neighborhoods $\{ \mathcal{Q}(Q(T), E_*, \varepsilon(T)) \}_{T \in \mathcal{F}(\tau_*, \mathcal{K}_*, \varepsilon)}$ forms an open cover of $\mathcal{Q}(\tau_*, \mathcal{K}_*, E_*)$,
where $Q(T)$ is the billiard table associated with $T$.
By compactness (see the proof of Lemma~\ref{lem:compact F}) there exists a finite subcover
$\{ \mathcal{Q}(Q(T_{\iota_j}), E_*, \varepsilon(T_{\iota_j})) \}_{j=1}^{N_\varepsilon}$. Then by Proposition~\ref{prop:close maps},
for each $T \in \mathcal{F}(\tau_*, \mathcal{K}_*, E_*)$, there exists $j$ such that $T \in \mathcal{F}(T_{\iota_j}, \frac 12 \kappa(T_{\iota_j}) )$
and $\mathcal{F}(T, \frac 12 \kappa(T_{\iota_j}) \subset \mathcal{F}(T_{\iota_j}, \kappa(T_{\iota_j}))$. Thus
$n_*(T_{\iota_j})$ and $\frac 12 \kappa(T_{\iota_j})$ have the desired properties for this $T$. Thus
setting $n_* = \max_j n_*(T_{\iota_j})$ proves part (a) and $\kappa = \frac 12 \min_j \kappa(T_{\iota_j})$
proves part (b) of the lemma.
\end{proof}
From this point forward, we fix $T_0 \in \mathcal{F}(\tau_*, \mathcal{K}_*, E_*)$ and let $R_* = R_*^{\delta_0}(T_0)$
as constructed above. We will consider sequences $\{ \iota_j \}_j \subset \mathcal{I}(T_0, \kappa)$, where
$\kappa$ is from Lemma~\ref{lem:proper cross}(b), i.e. we will draw from maps $T \in \mathcal{F}(T_0, \kappa)$.
\begin{lemma}
\label{lem:close if cross}
Let $W^1, W^2 \in \mathcal{W}^s$, $n \ge 0$ and $\{ \iota_j \}_{j=1}^n \subset \mathcal{I}(T_0, \kappa)$.
Suppose $U_1 \in \mathcal{G}_n(W^1)$ and
$U_2 \in \mathcal{G}_n(W^2)$ properly cross $R_*$ and define $\bar U_i = U_i \cap D(R_*)$, $i=1,2$.
Then there exists $C_7 >0$, depending only on the maximum slope and maximum
curvature $\bar B$ of curves in $\mathcal{W}^s$, such that
$d_{\mathcal{W}^s}(\bar U_1, \bar U_2) \le C_7 \delta_0^2$.
\end{lemma}
\begin{proof}
Define a foliation of vertical line segments covering $D(R_*)$. Due to the uniform transversality
of the stable cone with the vertical direction, it is clear that the length of the segments connecting
$\bar U_1$ and $\bar U_2$ have length at most $C_3 \delta_0^4$, where $C_3>0$ depends only on the
maximum slope in $C^s(x)$. Moreover, the unmatched parts of $\bar U_1$ and $\bar U_2$
near the boundary of $D(R_*)$ also have length at most $C_3 \delta_0^4$.
Recalling the definition of $d_{\mathcal{W}^s}(\cdot, \cdot)$,
it remains to estimate the $C^1$ distance between the graphs of $\bar U_1$ and $\bar U_2$.
Denote by $\varphi_1(r)$ and $\varphi_2(r)$ the functions defining $\bar U_1$ and $\bar U_2$ on a common
interval $I = I_{\bar U_1} \cap I_{\bar U_2}$. Let $\varphi_i' = \frac{\varphi_i}{dr}$.
For $x \in \bar U_1$ over $I$, let $\bar x \in \bar U_2$
denote the point on the same vertical line segment as $x$.
Suppose there exists $x \in \bar U_1$ over $I$ such that $|\varphi_1'(r(x)) - \varphi_2'(r(\bar x))| > C \delta_0^2$
for some $C>0$, where $r(x)$ denotes the $r$-coordinate of $x = (r,\varphi)$.
Since the curvature of each $U_i$ is bounded by $\bar B$ by definition, we have
$|\varphi_i''| \le \bar B (1+(K_{\max} + \tau_{\min}^{-1})^2)^{3/2} =: \bar C_7$.
Now consider an interval $J \subset I$ of radius $\delta_0^2$ centered at $r(x)$.
Then $|\varphi_1'(r) - \varphi_1'(r(x))| \le \bar C_7 |r-r(x)|$ for all $r \in J$, and similarly for $\varphi_2'$.
Thus,
\[
|\varphi_1'(r) - \varphi_2'(r)| \ge C \delta_0^2 - 2 \bar C_7 \delta_0^2 = (C - 2\bar C_7) \delta_0^2 \; \mbox{ for all $r \in J$}.
\]
This in turn implies that there exists $r \in J$ such that $|\varphi_1(r) - \varphi_2(r) | \ge (C - 2\bar C_7) \delta_0^4$,
which is a contradiction if $C - 2\bar C_7 > C_3$. This proves the lemma with
$C_7 = 2 \bar C_7 + C_3$.
\end{proof}
Recall that by Lemma~\ref{lem:H metric}, for $W\in \mathcal{W}^s$ the cone $\mathcal{D}_{a, \alpha}(W)$ has finite diameter in $\mathcal{D}_{a, \beta}(W)$ for
$\alpha > \beta$, so that
\begin{equation}
\label{eq:D_0}
\rho_{W, a, \beta}(g_1, g_2) \le D_0 \quad \mbox{ for all $g_1, g_2 \in \mathcal{D}_{a, \alpha}(W)$}
\end{equation}
for some constant $D_0 > 0$ depending only on $a, \alpha$ and $\beta$. Without loss of generality,
we take $D_0 \ge 1$.
\begin{lemma}
\label{lem:match}
Suppose $W^1, W^2 \in \mathcal{W}^s$ with $|W^1|, |W^2| \in [\delta_0/3, \delta_0]$ and $d_{\mathcal{W}^s}(W^1, W^2) \le C_7\delta_0^2$. Assume
$\psi_\ell \in \mathcal{D}_{a,\alpha}(W^\ell)$ with $\int_{W^1} \psi_1 = \int_{W^2} \psi_2 =1$.
Recall that $\delta \le \delta_0^2$. Let $C>0$ be such that if $n \ge C \log(\delta_0/\delta)$
then $C_5 \Lambda^{-n} \le \delta/\delta_0^2$, where $C_5$ is from Lemma~\ref{lem:compare}.
For all $n$ such that $n \ge C \log (\delta_0/\delta) \ge 2n_0$ and all $\{ \iota_j \}_{j=1}^n \subset \mathcal{I}(T_0, \kappa)$,
we have
\[
\frac{\int_{W^1} \mathcal{L}_n f \, \psi_1}{\int_{W^2} \mathcal{L}_n f \, \psi_2} \le 2
\]
provided
\[
\left[\frac{2\bar C_0C_3C_7(3L A \delta^{1-q}\delta_0^{2q}
+ 3 L\delta_0^2)}{1 - \Lambda^{-q}} + 2\bar C_0 A \delta^{1-q} (2\delta^q + c\delta^{\gamma+q} + D_0 \delta^q + 3\delta_0^q) \right] 6e^{2a\delta_0^\alpha} \leq \delta_0.
\]
\end{lemma}
\begin{remark}
\label{rem:delta_0 ineq}
Since $\delta \le \delta_0^2$, the condition of Lemma~\ref{lem:match} will be satisfied if
\begin{equation}
\label{eq:delta_0 ineq}
\left[\frac{2\bar C_0C_3C_7 3L A \delta_0+ 3L\delta_0)}{1 - \Lambda^{-q}} + 2\bar C_0 A
\delta_0^{1-q} (2 \delta_0^q + c\delta_0^{2\gamma +q} + D_0 \delta_0^q + 3) \right] 6e^{2a\delta_0^\alpha} \leq 1.
\end{equation}
This will determine our choice of $\delta_0$.
\end{remark}
\begin{proof}
We will change variables to integrate on $T_n^{-1}W^\ell$, $\ell = 1, 2$. As in Section~\ref{subsec:contract c}, we split
$\mathcal{G}_n(W^\ell)$ into matched pieces $\{ U^\ell_j \}_j$ and unmatched pieces $\{ V^\ell_j \}_j$. Corresponding
matched pieces $U^1_j$ and $U^2_j$ are defined as graphs $G_{U^\ell_j}$
over the same $r$-interval $I_j$ and are connected by a foliation of vertical line segments. Following
\eqref{eq:unstable split}, we write,
\[
\int_{W^{\ell} }\mathcal{L}_n f \, \psi_{\ell} = \sum_j \int_{U^{\ell}_j} f \, \widehat{T}_{n,U^{\ell}_j} \psi_{\ell} + \sum_j \int_{V^{\ell}_j} f \, \widehat{T}_{n,V^{\ell}_j} \psi_{\ell},
\]
where $\widehat{T}_{n,U^\ell_j} \psi_\ell := \psi_\ell \circ T_n \, J_{U^\ell_j}T_n$, and
similarly for $\widehat{T}_{n,V^\ell_j} \psi_\ell$,
$\ell = 1, 2$.
We perform the estimate over unmatched pieces first, following the same argument as in Section~\ref{subsec:contract c}
to conclude that $|T_{n-i-1}V^1_j| \le C_3 \Lambda^{-i} d_{\mathcal{W}^s}(W^1, W^2) \le C_3 C_7 \Lambda^{-i} \delta_0^2$,
for any curve $V^1_j$ created at time $i$, $0 \le i \le n-1$.
Recalling the sets $P(i)$ from Section~\ref{subsec:contract c} of unmatched pieces created at time $i$, we split the
estimate into curves $P(i; S)$ if $|T_{n-i-1}V^1_j| < \delta$ and curves $P(i; L)$ if $|T_{n-i-1}V^1_j| \ge \delta$.
The estimate over short unmatched pieces is given by,
\begin{equation}
\label{eq:unmatched short}
\begin{split}
\sum_{i=0}^{n-1} \sum_{j \in P(i; S)} & \left| \int_{V^1_j} f \, \widehat{T}_{n,V^1_j} \psi_1 \right|
= \sum_{i=0}^{n-1} \sum_{j \in P(i; S)} \left| \int_{T_{n-i-1}V^1_j} \mathcal{L}_{n-i-1}f \cdot \psi_1 \circ T_{i+1} \, J_{T_{n-i-1}V^1_j}T_{i+1} \right| \\
& \le \sum_{i=0}^{n-1} \sum_{j \in P(i; S)} A \delta^{1-q} C_3^q \Lambda^{-iq} d_{\mathcal{W}^s}(W^1, W^2)^q {|\:\!\!|\:\!\!|} \mathcal{L}_{n-i-1} f {|\:\!\!|\:\!\!|}_- |\psi_1|_{C^0}
|J_{T_{n-i-1}V^1_j}T_{i+1}|_{C^0} \\
& \le \frac{\bar C_0 A}{1 - \Lambda^{-q}} C_3^q C_7^q \delta_0^{2q} 3L {|\:\!\!|\:\!\!|} \mathcal{L}_n f {|\:\!\!|\:\!\!|}_- \delta^{1-q} |\psi_1|_{C^0} \, ,
\end{split}
\end{equation}
where we have used Lemma~\ref{lem:full growth}-(b), $|W| \in [\delta_0/3, \delta_0]$, and
Remark~\ref{rem:improve} to estimate the sum over the Jacobians, as well as \eqref{eq:tri} to estimate ${|\:\!\!|\:\!\!|} \mathcal{L}_{n-i-1} f {|\:\!\!|\:\!\!|}_- \le 3L {|\:\!\!|\:\!\!|} \mathcal{L}_n f {|\:\!\!|\:\!\!|}_-$.
For the estimate over long pieces, we subdivide them into curves of length between $\delta$ and $2\delta$ and estimate them by ${|\:\!\!|\:\!\!|} \mathcal{L}_{n-i-1} f {|\:\!\!|\:\!\!|}_+$, then we recombine them to obtain,
\begin{equation}
\label{eq:unmatched long}
\begin{split}
\sum_{i=0}^{n-1} \sum_{j \in P(i; L)} &\left| \int_{V^1_j} f \, \widehat{T}_{n,V^1_j} \psi_1 \right|
= \sum_{i=0}^{n-1} \sum_{j \in P(i; L)} \left| \int_{T_{n-i-1}V^1_j} \mathcal{L}_{n-i-1}f \cdot \psi_1 \circ T_{i+1} \, J_{T_{n-i-1}V^1_j}T_{i+1} \right| \\
& \le \sum_{i=0}^{n-1} \sum_{j \in P(i; L)} {|\:\!\!|\:\!\!|} \mathcal{L}_{n-i-1} f {|\:\!\!|\:\!\!|}_+ \int_{T_{n-i-1}V^1_j} \psi_1 \circ T_{i+1} \,
J_{T_{n-i-1}V^1_j}T_{i+1} \\
& \le 3 L {|\:\!\!|\:\!\!|} \mathcal{L}_n f {|\:\!\!|\:\!\!|}_- \sum_{i=0}^{n-1} \sum_{j \in P(i; L)} |T_{n-i-1}V^1_j| |\psi_1|_{C^0} |J_{T_{n-i-1}V^1_j}T_{i+1}|_{C^0} \\
& \le \frac{C_3 C_7\bar C_0}{1 - \Lambda^{-1}} \delta_0^2 3 L {|\:\!\!|\:\!\!|} \mathcal{L}_n f {|\:\!\!|\:\!\!|}_-|\psi_1|_{C^0} \, ,
\end{split}
\end{equation}
where, in third line we used \eqref{eq:tri},
and in the fourth line, since $|W^1| \ge \delta_0/3$, we used Remark~\ref{rem:improve} to drop the second term in Lemma~\ref{lem:full growth}(b).
Next, we estimate the integrals over the matched pieces $U^1_j$. We argue as in
Section~\ref{subsec:contract c}, but our estimates here are somewhat simpler since we do not need to show that parameters
contract.
We first treat the matched short pieces with $|U^1_j| < \delta$ much as we did the unmatched ones.
By Lemma~\ref{lem:compare},
$d_{\mathcal{W}^s}(U^1_j, U^2_j) \le C_5 \Lambda^{-n} d_{\mathcal{W}^s}(W^1, W^2) \le \delta$,
since we have chosen $n \ge C \log (\delta_0/\delta)$. Thus if
$|U^1_j| < \delta$ then $|U^2_j| < 2\delta$, and the analogous fact holds for short
curves $|U^2_j| < \delta$.
With this perspective, we call $U^\ell_j$ short if either $|U^1_j| < \delta$ or $|U^2_j| < \delta$.
On short pieces, we apply \eqref{eq:cone 3}
\begin{equation}
\label{eq:match short}
\sum_{j \; \mbox{\scriptsize short}} \left| \int_{U^1_j} f \, \widehat{T}_{n,U^1_j} \psi_1 \right|
\le \sum_{j \; \mbox{\scriptsize short}} 2 A \delta {|\:\!\!|\:\!\!|} f {|\:\!\!|\:\!\!|}_- |\psi_1|_{C^0} |J_{U^1_j} T_n|_{C^0}
\le 4 A \delta {|\:\!\!|\:\!\!|} \mathcal{L}_n f {|\:\!\!|\:\!\!|}_- \bar C_0 |\psi_1|_{C^0} \, ,
\end{equation}
where we have again used Lemmas~\ref{lem:full growth}(b) and \ref{lem:first L} for the second inequality. Remark that the same argument holds for $W^2$ with test function $\psi_2$.
Finally, to estimate the integrals over matched curves with $|U^1_j|, |U^2_j| \ge \delta$ we
follow equation \eqref{eq:c-decomposition}, recalling \eqref{eq:switch}, although we no longer have Lemma~\ref{lem:compare}(c) at our disposal,
\begin{equation}
\label{eq:match decomp}
\begin{split}
&\left|\int_{U^1_j} f \, \widehat{T}_{n,U^1_j} \psi_1 -\int_{U^2_j} f \, \widehat{T}_{n,U^2_j} \psi_2\right|
\leq \left|\int_{U^1_j} f \, \widehat{T}_{n,U^1_j} \psi_1 -\int_{U^1_j} f \, \widetilde{T}_{n,U^2_j} \psi_2\right| \\
&+\left|\frac{\int_{U^1_j} f \, \widetilde{T}_{n,U^2_j} \psi_2}{\fint_{U^1_j} \widetilde{T}_{n,U^2_j} \psi_2} -\frac{\int_{U^2_j} f \, \widehat{T}_{n,U^2_j} \psi_2}{\fint_{U^2_j} \widehat{T}_{n,U^2_j} \psi_2}\right|\fint_{U^2_j} \widehat{T}_{n,U^2_j} \psi_2+\left|\frac{\int_{U^1_j} f \, \widetilde{T}_{n,U^2_j} \psi_2}{\fint_{U^1_j} \widetilde{T}_{n,U^2_j} \psi_2}\right|\,\left| \frac{|U^2_j|-|U^1_j|}{|U^1_j|}\right|\fint_{U^2_j} \widehat{T}_{n,U^2_j} \psi_2\\
&\leq \left|\int_{U^1_j} f \, \widehat{T}_{n,U^1_j} \psi_1 -\int_{U^1_j} f \, \widetilde{T}_{n,U^2_j} \psi_2\right|
+d_{\mathcal{W}^s}(U^1_j, U^2_j)^\gamma \delta^{1-\gamma} c A {|\:\!\!|\:\!\!|} f {|\:\!\!|\:\!\!|}_- |J_{U^2_j}T_n|_{C^0} |\psi_2|_{C^0} \\
&\quad +A\delta d_{\mathcal{W}^s}(U^1_j, U^2_j) {|\:\!\!|\:\!\!|} f {|\:\!\!|\:\!\!|}_- |J_{U^2_j}T_n|_{C^0} |\psi_2|_{C^0} \, ,
\end{split}
\end{equation}
where we have used \eqref{eq:length-ratio} to estimate $\left| \frac{|U^2_j|-|U^1_j|}{|U^1_j|}\right|$.
To estimate the first term on the right side above, we use \eqref{eq:cone 3} and Lemma~\ref{lem:b property},
\[
\begin{split}
\left|\int_{U^1_j} f \, \widehat{T}_{n,U^1_j} \psi_1 -\int_{U^1_j} f \, \widetilde{T}_{n,U^2_j} \psi_2\right|
& \le \left|\frac{\int_{U^1_j} f \, \widehat{T}_{n,U^1_j} \psi_1}{\fint_{U^1_j} \widehat{T}_{n,U^1_j} \psi_1} - \frac{\int_{U^1_j} f \, \widetilde{T}_{n,U^2_j} \psi_2}{\fint_{U^1_j} \widetilde{T}_{n,U^2_j} \psi_2} \right| \fint_{U^1_j} \widehat{T}_{n,U^1_j} \psi_1 \\
& \qquad + \frac{\int_{U^1_j} f \, \widetilde{T}_{n,U^2_j} \psi_2}{\fint_{U^1_j} \widetilde{T}_{n,U^2_j} \psi_2}
\left| \fint_{U^1_j} \widehat{T}_{n,U^1_j} \psi_1 - \fint_{U^1_j} \widetilde{T}_{n,U^2_j} \psi_2 \right| \\
& \le 2 \delta L \rho(\widehat{T}_{n,U^1_j} \psi_1, \widetilde{T}_{n,U^2_j} \psi_2) {|\:\!\!|\:\!\!|} f {|\:\!\!|\:\!\!|}_- |J_{U^1_j}T_n|_{C^0} |\psi_1|_{C_0} \\
& \qquad + A \delta^{1-q} \delta_0^q {|\:\!\!|\:\!\!|} f {|\:\!\!|\:\!\!|}_- \big( |J_{U^1_j}T_n|_{C^0} |\psi_1|_{C_0} + 2|J_{U^2_j}T_n|_{C^0} |\psi_2|_{C_0} \big) \, ,
\end{split}
\]
where we have used $|U^1_j| \le \delta_0$ in the last line.
We may apply \eqref{eq:D_0} since $\widehat{T}_{n,U^1_j} \psi_1, \widetilde{T}_{n,U^2_j} \psi_2 \in \mathcal{D}_{a, \alpha}(U^1_j)$ by
Lemma~\ref{lem:compare}. Now putting the above estimate together with \eqref{eq:match decomp},
recalling $d_{\mathcal{W}^s}(U^1_j, U^2_j) \le \delta$,
and using Lemma~\ref{lem:full growth}-(b) and Remark~\ref{rem:improve} as well as Lemma~\ref{lem:first L}, we sum over $j$ to obtain,
\begin{equation}
\label{eq:match long}
\begin{split}
\sum_{j \; \mbox{\scriptsize long}} & \left|\int_{U^1_j} f \, \widehat{T}_{n,U^1_j} \psi_1 -\int_{U^2_j} f \, \widehat{T}_{n,U^2_j} \psi_2\right| \\
& \qquad \le 2 A \delta^{1-q} {|\:\!\!|\:\!\!|} \mathcal{L}_n f {|\:\!\!|\:\!\!|}_- \bar C_0 \left( c \delta^{\gamma + q} + \delta^{1+q} + \frac{2LD_0 \delta^q}{A} +3 \delta_0^q \right) (|\psi_1|_{C^0} + |\psi_2|_{C^0}) .
\end{split}
\end{equation}
Collecting \eqref{eq:unmatched short}, \eqref{eq:unmatched long}, \eqref{eq:match short} and \eqref{eq:match long}, and recalling $D_0 \ge 1$ and $A > 4 L$, yields
\[
\begin{split}
\int_{W^1} \mathcal{L}_n f \, \psi_1 & \leq \;\frac{\bar C_0C_3C_7 (3L A \delta^{1-q}\delta_0^{2q}+3L\delta_0^2)}{1 - \Lambda^{-q}} {|\:\!\!|\:\!\!|} \mathcal{L}_n f {|\:\!\!|\:\!\!|}_- |\psi_1|_{C^0}
+ 4 \bar C_0 A \delta {|\:\!\!|\:\!\!|} \mathcal{L}_n f {|\:\!\!|\:\!\!|}_- |\psi_1|_{C^0} \\
& \qquad
+ \sum_j \int_{U^2_j} f \, \widehat{T}_{U^2_j}\psi_2 + 2 A \delta^{1-q} {|\:\!\!|\:\!\!|} \mathcal{L}_n f {|\:\!\!|\:\!\!|}_- \bar C_0 ( c \delta^{\gamma+ q} + D_0 \delta^q + 3\delta_0^q ) (|\psi_1|_{C^0} + |\psi_2|_{C^0}) \\
& \leq \; \left\{1+\Big[\frac{2 \bar C_0C_3C_7 (3L A \delta^{1-q}\delta_0^{2q}+3L\delta_0^2) }{1 - \Lambda^{-q}} \right. \\
& \qquad \left. + 2 \bar C_0 A \delta^{1-q} (2 \delta^q + c\delta^{\gamma+q} + D_0 \delta^q + 3 \delta_0^q ) \Big]\frac{|\psi_1|_{C^0} + |\psi_2|_{C^0}}{\int_{W^2}\psi_2}\right\} \int_{W^2} \mathcal{L}_n f \, \psi_2 \, .
\end{split}
\]
Now since $\int_{W^i} \psi_i = 1$, we have $e^{- a \delta_0^\alpha} \le |W^i| \psi_i \le e^{a \delta_0^\alpha}$. Thus since $|W^i| \ge \delta_0/3$,
\[
\frac{|\psi_1|_{C^0} + |\psi_2|_{C^0}}{\int_{W^2}\psi_2}
\le \frac{6}{\delta_0} e^{2a \delta_0^\alpha} \, ,
\]
which proves the Lemma.
\end{proof}
Our strategy will be the following. For $W^1$, $W^2 \in \mathcal{W}^s(\delta_0/2)$, $n$ sufficiently large and
$\{ \iota_j \}_{j=1}^n \subset \mathcal{I}(T_0, \kappa)$, we wish to compare
$\int_{W^1} \mathcal{L}_n f \, \psi_1$ with $\int_{W^2} \mathcal{L}_n f \, \psi_2$, where we normalize
$\int_{W^1} \psi_1 = \int_{W^2} \psi_2 = 1$. By Lemmas~\ref{lem:proper cross} and \ref{lem:close if cross},
we find $U^\ell_i \in \mathcal{G}_{n_*}(W^\ell)$, $\ell = 1, 2$, such that $U^\ell_i$ properly crosses $R_*$, and
$d_{\mathcal{W}^s}(\bar U^1_i, \bar U^2_i) \le C_7 \delta_0^2$, where $\bar U^\ell_i = U^\ell_i \cap D(R_*)$.
Next, for each $i$, we wish to compare $\int_{\bar U^1_i} \mathcal{L}_{n-n_*} f \, \widehat{T}_{n_*, U^1_i} \psi_1$
with $\int_{\bar U^2_i} \mathcal{L}_{n-n_*} f \, \widehat{T}_{n_*, U^2_i} \psi_2 $,
where, as usual,
$\widehat{T}_{n_*, U^\ell_i} \psi_\ell = \psi_\ell \circ T_{n_*} J_{U^\ell_i}T_{n_*}$.
However, the weights $\int_{\bar U^\ell_i} \widehat{T}_{n_*, U^\ell_i} \psi_\ell$ may be very different for $\ell = 1,2$ since
the stable Jacobians along the respective orbits before time $n_*$ may not be comparable. To remedy this,
we adopt the following strategy for matching integrals on curves.
For each curve $U^\ell_i \in \mathcal{G}_{n_*}(W)$ which properly crosses $R_*$, we redefine $\bar U^\ell_i$
to denote the middle third of $U^\ell_i \cap D(R_*)$. Let $M^\ell$ denote the index set of such $i$.
Let $p^\ell_i = \int_{\bar U^\ell_i} \widehat{T}_{n_*, U^\ell_i} \psi_\ell$, and let
$m_\ell = \sum_{i \in M^\ell} p^\ell_i$. Without loss of generality, assume $m_2 \ge m_1$.
We will match the integrals $\sum_{i \in M^1} \int_{\bar U^1_i} \mathcal{L}_{n-n_*}f \, \widehat{T}_{n_*, U^1_i} \psi_1$ with
$\sum_{j \in M^2} \frac{m_1}{m_2} \int_{\bar U^2_j} \mathcal{L}_{n-n_*}f \, \widehat{T}_{n_*, U^2_j} \psi_2$.
The remainder of the integrals
$\sum_{j \in M^2} \frac{m_2 - m_1}{m_2} \int_{\bar U^2_j} \mathcal{L}_{n-n_*}f \, \widehat{T}_{n_*, U^2_j} \psi_2$ as well as any unmatched
pieces (including the outer two-thirds of each $U^\ell_i$) we continue to iterate until such time as they can be matched
as the middle third of a curve that properly crosses $R_*$.
Set $\widehat{T}_{n_*, U^2_j} \tilde{\psi}_2 = \frac{m_1}{m_2} \widehat{T}_{n_*, U^2_j} \psi_2$, and
consider the following decomposition of the integrals we want to match,
\[
\sum_{\substack{i \in M^1 \\ j \in M^2}} \int_{\bar U^1_i} \mathcal{L}_{n-n_*}f \, \widehat{T}_{n_*, U^1_i} \psi_1 \, \frac{p_j^2}{m_2}
\quad \mbox{and} \quad
\sum_{\substack{i \in M^1 \\ j \in M^2}} \int_{\bar U^2_j} \mathcal{L}_{n-n_*}f \, \widehat{T}_{n_*, U^2_j} \tilde{\psi}_2 \, \frac{p_i^1}{m_1}
\]
For each pair $i,j$ in the first sum, the test function has integral weight $\frac{p^1_i p^2_j}{m_2}$, and the same is true for the
corresponding pair in the second sum. Thus these integrals are paired precisely according to the assumptions of
Lemma~\ref{lem:match}. It follows that if $n - n_* \ge C\log(\delta_0/\delta)$, then
\begin{equation}
\label{eq:match compare}
\begin{split}
\sum_{i \in M^1} \int_{\bar U^1_i} \mathcal{L}_{n-n_*}f \, & \widehat{T}_{n_*, U^1_i} \psi_1
= \sum_{\substack{i \in M^1 \\ j \in M^2}} \int_{\bar U^1_i} \mathcal{L}_{n-n_*}f \, \widehat{T}_{n_*, U^1_i}\psi_1 \frac{p_j^2}{m_2} \\
& \le 2 \sum_{\substack{i \in M^1 \\ j \in M^2}} \int_{\bar U^2_j} \mathcal{L}_{n-n_*}f \, \widehat{T}_{n_*, U^2_j} \tilde{\psi}_2 \, \frac{p_i^1}{m_1}
= 2 \sum_{j \in M^2} \int_{\bar U^2_j} \mathcal{L}_{n-n_*}f \, \widehat{T}_{n_*,U^2_j} \tilde{\psi}_2 \, .
\end{split}
\end{equation}
We want to repeat the above construction until most of the mass has been compared. To this end we
set up an inductive scheme.
Consider the family of curves $W^\ell_i \in \mathcal{G}_{n_*}(W^\ell)$ that have not been matched. Each
carries a test function $\psi_{\ell, i} := \widehat{T}_{n_*, W^\ell_i} \tilde{\psi}_\ell$. Renormalizing by a factor $\mathfrak{r}_{\ell,1} < 1$,
we have $\sum_i \int_{W^\ell_i} \psi_{\ell,i} = 1$.
\begin{defin}
\label{def:admit}
Given a countable collection of curves and test functions, $\mathfrak{F} = \{ W_i, \psi_i \}_i$, with $W_i \in \mathcal{W}^s$, $|W_i| \le \delta_0$,
$\psi_i \in \mathcal{D}_{a, \alpha}(W_i)$ and $\sum_i \int_{W_i} \psi_i = 1$, we call
$\mathfrak{F}$ an {\em admissible family} if
\begin{equation}
\label{eq:C* def}
\sum_i \fint_{W_i} \psi_i \le C_* \, , \quad \mbox{where $C_* := 3\bar C_0 \delta_0^{-1}$.}
\end{equation}
\end{defin}
Notice that any stable curve $W \in \mathcal{W}^s(\delta_0/2)$ together with test function $\psi \in \mathcal{D}_{a,\alpha}(W)$ normalized so that
$\int_W \psi = 1$ forms an admissible family since $|W| \ge \delta_0/2$.
The content of the following lemma is that an admissible family can be iterated and remain admissible; moreover,
a family with larger average integral in \eqref{eq:C* def} can be made admissible under iteration.
\begin{lemma}
\label{lem:propogate}
Let $\{ W_i, \psi_i \}_i$ be a countable collection of curves $W_i \in \mathcal{W}^s$, $|W_i| \le \delta_0$, with functions
$\psi_i \in \mathcal{D}_{a,\alpha}(W_i)$,
normalized so that $\sum_i p_i = 1$, where $p_i = \int_{W_i} \psi_i$.
Suppose that $\sum_i |W_i|^{-1} p_i = C_\sharp$.
Choose $n_\sharp \in \mathbb{N}$ so that $C_0 \theta_1^{n_\sharp} \frac{C_\sharp}{C_*} \le 1/6$.
Then for all $n \ge n_\sharp$, and all $\{ \iota_j \}_{j=1}^n \subset \mathcal{I}(T_0, \kappa)$,
the dynamically iterated family $\{ V^i_j \in \mathcal{G}_n(W_i), \widehat{T}_{n,V^i_j} \psi_i \}_{i,j}$
is admissible.
\end{lemma}
\begin{proof}
Setting $p^i_j = \int_{V^i_j} \widehat{T}_{n,V^i_j} \psi_i = \int_{V^i_j} \psi_i \circ T_n J_{V^i_j}T_n$, it is immediate that
$\sum_{i,j} p^i_j = 1$.
Now fix $W_i$ and consider $V^i_j \in \mathcal{G}_n(W_i)$. Then using Lemmas~\ref{lem:full growth} and \ref{lem:avg} we estimate,
\[
\begin{split}
\sum_j |V^i_j|^{-1} p^i_j & = \sum_j \fint_{V^i_j} \psi_i \circ T_n \, J_{V^i_j}T_n \le \sum_j |\psi_i|_{C^0(W_i)} |J_{V^i_j}T_n|_{C^0(V^i_j)} \\
& \le |\psi_i|_{C^0} ( \bar C_0 \delta_0^{-1} |W_i| + C_0 \theta_1^n )
\le \bar C_0 \delta_0^{-1} e^{a \delta_0^\alpha} p_i + C_0 \theta_1^n e^{a \delta_0^\alpha} |W_i|^{-1} p_i \, .
\end{split}
\]
Using that $e^{a \delta_0^\alpha} \le 2$, we sum over $i$ and use the assumption on the family $\{ W_i, \psi_i \}_i$ to obtain,
\begin{equation}
\label{eq:C* contract}
\sum_{i,j} \sum_j |V^i_j|^{-1} p^i_j
\le \sum_i \big( 2 \bar C_0 \delta_0^{-1} p_i + 2 C_0 \theta_1^n |W_i|^{-1} p_i \big)
\le 2 \bar C_0 \delta_0^{-1} + 2 C_0 \theta_1^n C_\sharp \, .
\end{equation}
Thus if $n \ge n_\sharp$, the above expression is bounded by $C_*$, as required.
\end{proof}
\begin{theorem}
\label{thm:cone contract}
Let $L \ge 60$. Suppose $a, c, A$ and $L$ satisfy the conditions of Section~\ref{sec:conditions}, and that in addition, $\delta \le \delta_0^2$ satisfy \eqref{eq:delta_0 ineq} and \eqref{eq:A-cond-s4}.
Then there exists $\chi < 1$ and $k_*\in\mathbb{N}$ such that if $n \in \mathbb{N}$ satisfies $n \ge N(\delta)^- + k_*n_*$,\footnote{ Recall that $n_*$ is defined in Lemma \ref{lem:proper cross} while $N(\delta)^-$ is defined in equation \eqref{eq:N-}.} with $k_*$ depending only on $\delta_0, L, n_*$ (see equation \eqref{eq:k*choice})
and $\{ \iota_j \}_{j=1}^n \subset \mathcal{I}(T_0, \kappa)$, then $\mathcal{L}_n \mathcal{C}_{ c, A, L}(\delta) \subset \mathcal{C}_{ \chi c, \chi A, \chi L}(\delta) $.
\end{theorem}
\begin{proof}
As before, we take $f \in \mathcal{C}_{c,A,L}(\delta)$,
$W^1$, $W^2 \in \mathcal{W}^s(\delta_0/2)$ and test functions $\psi_\ell \in \mathcal{D}_{a,\beta}(W^\ell)$ such that
$\int_{W^1} \psi_1 = \int_{W^2} \psi_2 = 1$. In order to iterate the matching argument described above,
we need upper and lower bounds on the amount of mass matched via the
process described by \eqref{eq:match compare}.
\medskip
{\em Upper Bound on Matching.}
By definition of $\bar U^\ell_i$, for each curve $U^\ell_i$ that properly crosses $R_*$ at time $n_*$,
at least 2/3 of the length of that curve remains not matched. Thus if $p_i = \int_{U^\ell_i} \widehat{T}_{n,U^\ell_i} \tilde{\psi}_i$, then at least
$(1- e^{a \delta_0^\alpha}/3) p_i$ remains unmatched. Using $e^{a \delta_0^\alpha} \le 2$, we conclude that at least
$(1/3) p_i$ of the mass remains unmatched.
Thus if $\mathfrak{r}$ denotes the total mass remaining after matching at time $n_*$, we have $\mathfrak{r} \ge 1/3$.
Renormalizing the family by $\mathfrak{r}$,
we have $\sum_i |W_i|^{-1} \frac{p_i}{\mathfrak{r}} \le 3 C_*$.
By the proof of Lemma~\ref{lem:propogate} with $C_\sharp = 3C_*$, we see that choosing
$n_\sharp$ such that
$6C_0 \theta_1^{n_\sharp} \le 1/3$, then the bound in \eqref{eq:C* contract} is less than $C_*$, and the family recovers its regularity
in the sense of Lemma~\ref{lem:propogate} after this number of iterates.
\medskip
{\em Lower Bound on Matching.} By definition of admissible family, for each $\varepsilon > 0$,
$\sum_{|W_i| < \varepsilon} p_i \le C_* \varepsilon$.
So choosing $\varepsilon = \delta_0 / (6\bar C_0)$, we have that
\[
\sum_{|W_i| \ge \delta_0/(6 \bar C_0)} p_i \ge \frac 12 \, .
\]
On each $W_i$ with $|W_i| \ge \delta_0/(6 \bar C_0)$, we have at least one $U^i_j \in \mathcal{G}_{n_*}(W_i)$ that properly
crosses $R_*$ by Lemma~\ref{lem:proper cross}. Then denoting by $\bar U^i_j$ the matched part (middle third)
of $U^i_j$, we have
\[
\begin{split}
\int_{\bar U^i_j} \widehat{T}_{n_*,U^i_j} \tilde{\psi}_i & = \int_{\bar U^i_j} \tilde{\psi}_i \circ T_{n_*} \, J_{U^i_j}T_{n_*}
\ge \tfrac{\delta_0}{3} \inf \tilde{\psi}_i \inf J_{U^i_j}T_{n_*} \\
& \ge \tfrac{1}{3} e^{-a \delta_0^\alpha} p_i e^{-C_d \delta_0^{1/3}} \frac{|T_{n*}U^i_j|}{|U^i_j|}
\ge \frac{1}{12} p_i \frac{C^{n_*} \delta_0^{(5/3)^{n_*}}}{\delta_0} =: \varepsilon_{n_*} p_i \, ,
\end{split}
\]
where we have used the fact that if $W \in \mathcal{W}^s$ and $T_{\iota_j}^{-1}W$ is a homogeneous stable curve, then
$|T_{\iota_j}^{-1}W| \le C^{-1} |W|^{3/5}$ for some constant $C>0$ by {\bf (H1)} (see, for example \cite[eq. (6.9)]{demzhang14}).
Thus a lower bound on the amount of mass coupled at time $n_*$ is $\frac{\varepsilon_{n_*}}{2} > 0$.
\medskip
We are finally ready to put these elements together. For $k_* \in \mathbb{N}$ and
$k = 1, \ldots k_*$, let $M^\ell(k)$ denote the index set of curves in
$\mathcal{G}_{kn_*}(W^\ell)$ which are matched at time $kn_*$. By choosing $\delta_0$ small, we can ensure
that $n_\sharp \le n_*$, where $n_\sharp$ from Lemma~\ref{lem:propogate} corresponds to $C_\sharp = 3 C_*$.
Thus the family of remaining curves is always admissible at time $kn_*$. Let $M^\ell(\sim)$ denote the index set of curves
that are not matched by time $k_*n_*$. We estimate using \eqref{eq:match compare} at each time $n=k n_*$,
\begin{equation}
\label{eq:gen match}
\begin{split}
\int_{W^1} \mathcal{L}_nf \, \psi_1 & = \sum_{k=1}^{k_*} \sum_{i \in M^1(k)} \int_{\bar U^1_i} \mathcal{L}_{n-kn_*} f \, \widehat{T}_{kn_*, U^1_i} \tilde{\psi}_1
+ \sum_{i \in M^1(\sim)} \int_{V^1_i} \mathcal{L}_{n-k_*n_*} f \, \widehat{T}_{k_*n_*, V^1_i} \tilde{\psi}_1 \\
& \le \sum_{k=1}^{k_*} \sum_{i \in M^2(k)} 2 \int_{\bar U^2_i} \mathcal{L}_{n-kn_*} f \, \widehat{T}_{kn_*, U^2_i} \tilde{\psi}_2
+ \sum_{i \in M^1(\sim)} \int_{V^1_i} \mathcal{L}_{n-k_*n_*} f \, \widehat{T}_{k_*n_*, V^1_i} \tilde{\psi}_1
\end{split}
\end{equation}
We estimate the sum over unmatched pieices $M^\ell(\sim)$ by splitting the estimate in curves longer than $\delta$, $M^\ell(\sim; Lo)$,
and curves shorter than $\delta$, $M^\ell(\sim; Sh)$.
\[
\begin{split}
\sum_{i \in M^\ell(\sim)} & \int_{V^\ell_i} \mathcal{L}_{n-k_*n_*} f \, \widehat{T}_{k_*n_*, V^\ell_i} \tilde{\psi}_\ell
= \sum_{i \in M^\ell(\sim; Lo)} \int_{V^\ell_i} \mathcal{L}_{n-k_*n_*} f \, \widehat{T}_{k_*n_*, V^\ell_i} \tilde{\psi}_\ell
+ \sum_{i \in M^\ell(\sim; Sh)} \int_{V^\ell_i} \mathcal{L}_{n-k_*n_*} f \, \widehat{T}_{k_*n_*, V^\ell_i} \tilde{\psi}_\ell \\
& \le \sum_{i \in M^\ell(\sim; Lo)} {|\:\!\!|\:\!\!|} \mathcal{L}_{n-k_*n_*} f {|\:\!\!|\:\!\!|}_+ \int_{V^\ell_i} \widehat{T}_{k_*n_*, V^\ell_i} \tilde{\psi}_\ell
+ \sum_{i \in M^\ell(\sim; Sh)} A {|\:\!\!|\:\!\!|} \mathcal{L}_{n-k_*n_*} f {|\:\!\!|\:\!\!|}_- \delta |\psi_\ell|_{C^0} |J_{V^\ell_i}T_{k_*n_*}|_{C^0} \\
& \le (1-\tfrac{\varepsilon_{n_*}}{2})^{k_*} 3L {|\:\!\!|\:\!\!|} \mathcal{L}_n f {|\:\!\!|\:\!\!|}_- + A 2 {|\:\!\!|\:\!\!|} \mathcal{L}_n f {|\:\!\!|\:\!\!|}_- \delta |\psi_\ell|_{C^0} \bar C_0 \, .
\end{split}
\]
where we have used \eqref{eq:tri} and the fact that $k_*n_* \ge n_0$. For the sum over long pieces, we used that the total mass of unmatched
pieces decays exponentially in $k$, while for the sum over short pieces,
we used
Lemma~\ref{lem:full growth} and Remark~\ref{rem:improve} to sum over the Jacobians since $|W^1| \ge \delta_0/2$.
Finally, since $|\psi_1|_{C^0} \le e^{a \delta_0^\alpha} \fint_{W^1} \psi_1 \le \frac{4}{\delta_0}$, we conclude,
\[
\begin{split}
\sum_{i \in M^1(\sim)} \left| \int_{V^1_i} \mathcal{L}_{n-k_*n_*} f \, \widehat{T}_{k_*n_*, V^1_i} \tilde{\psi}_1 \right|
& \le \left( 3L (1- \tfrac{\varepsilon_{n_*}}{2})^{k_*} + 8A \bar C_0 \tfrac{\delta}{\delta_0} \right) {|\:\!\!|\:\!\!|} \mathcal{L}_n f {|\:\!\!|\:\!\!|}_- \\
& \le \left( 3L (1- \tfrac{\varepsilon_{n_*}}{2})^{k_*} + 8A \bar C_0 \tfrac{\delta}{\delta_0} \right) \int_{W^2} \mathcal{L}_n f \, \psi_2 \, ,
\end{split}
\]
using the fact that $\int_{W^2} \psi_2 = 1$. A similar estimate holds for the sum over curves in $M^2(\sim)$.
Finally, we put together this estimate with \eqref{eq:gen match} to obtain,
\[
\begin{split}
\int_{W^1} \mathcal{L}_nf \, \psi_1
& \le \sum_{k=1}^{k_*} \sum_{i \in M^2(k)} 2 \int_{\bar U^2_i} \mathcal{L}_{n-kn_*} f \, \widehat{T}_{kn_*, U^2_i} \tilde{\psi}_2
+ \sum_{i \in M^1(\sim)} \int_{V^1_i} \mathcal{L}_{n-k_*n_*} f \, \widehat{T}_{k_*n_*, V^1_i} \tilde{\psi}_1 \\
& \le 2 \int_{W^2} \mathcal{L}^n f \, \psi_2
+ 2 \sum_{j \in M^2(\sim)} \left| \int_{V^2_j} \mathcal{L}_{n-k_*n_*} f \, \widehat{T}_{k_*n_*, V^2_j} \tilde{\psi}_2 \right| \\
& \qquad + \sum_{i \in M^1(\sim)} \left| \int_{V^1_i} \mathcal{L}_{n-k_*n_*} f \, \widehat{T}_{k_*n_*, V^1_i} \tilde{\psi}_1 \right| \\
& \le \int_{W^2} \mathcal{L}_n f \, \psi_2 \left( 2 + 3 \big(3L (1- \tfrac{\varepsilon_{n_*}}{2})^{k_*} + 8A \bar C_0 \tfrac{\delta}{\delta_0}\big) \right) \, .
\end{split}
\]
We choose $k_*$ such that
\begin{equation}\label{eq:k*choice}
3L(1 - \frac{\varepsilon_{n_*}}{2})^{k_*} < \frac 16.
\end{equation}
Note that this choice of $k_*$ depends only on $\delta_0$ via $\varepsilon_{n_*}$,
and not on $\delta$. Next, choose $\delta>0$ sufficiently small that
\begin{equation}\label{eq:A-cond-s4}
8A \bar C_0 \delta/\delta_0 < \frac 16.
\end{equation}
These choices imply that
\[
\int_{W^1} \mathcal{L}_n f \, \psi_1 \le 3 \int_{W^2} \mathcal{L}_n f \, \psi_2 \, .
\]
Finally we prove that the first alternative of Proposition~\ref{prop:alternative} must happen. Suppose the contrary.
Since this bound holds for all $W^1, W^2 \in \mathcal{W}^s(\delta_0/2)$ and test functions $\psi_1, \psi_2$ with $\int_{W^1} \psi_1 = \int_{W^2} \psi_2 = 1$, we conclude
that, for $k \ge k_*$ and $m\geq N(\delta)^-$,
\[
\frac{{|\:\!\!|\:\!\!|} \mathcal{L}_{kn_*+m} f {|\:\!\!|\:\!\!|}_+}{{|\:\!\!|\:\!\!|} \mathcal{L}_{kn_*+m} f {|\:\!\!|\:\!\!|}_- } \le \frac{160}{9} \frac{{|\:\!\!|\:\!\!|} \mathcal{L}_{kn_*} f {|\:\!\!|\:\!\!|}^0_+}{{|\:\!\!|\:\!\!|} \mathcal{L}_{kn_*} f {|\:\!\!|\:\!\!|}^0_- }
\le \frac{160}{3} \le \frac{8}{9} L \, ,
\]
if we choose $L \ge 60$.
\end{proof}
\subsection{Finite diameter}
\label{sec:diam}
In this section we prove the following proposition, which completes the proof of Theorem~\ref{thm:main}.
\begin{prop}
\label{prop:diameter}
For any $\chi \in\left(\max \{ \frac 12, \frac{1}{L}, \frac 1{\sqrt{A-1}} \} , 1\right)$, the cone $\mathcal{C}_{\chi c, \chi A, \chi L}(\delta)$ has diameter less than
$\Delta:= \log \left( \frac{(1+\chi)^2}{(1-\chi)^2} \chi L \right)<\infty$ in $\mathcal{C}_{c,A,L}(\delta)$,
assuming $\delta>0$ is sufficiently small to satisfy \eqref{eq:delta cond}.
\end{prop}
\begin{proof}
For brevity, we will denote $\mathcal{C} = \mathcal{C}_{c,A,L}(\delta)$ and $\mathcal{C}_\chi = \mathcal{C}_{ \chi c, \chi A, \chi L}(\delta)$.
For $f \in \mathcal{C}_\chi$, we will show that $\rho(f, 1) < \infty$, where $\rho$ denotes distance in the cone $\mathcal{C}$.
Fix $f \in \mathcal{C}_\chi$ throughout.
According to \eqref{eq:H def} if we find $\lambda>0$ such that $f-\lambda\succeq 0$, then $\bar \alpha(1,f)\geq \lambda$.
Notice that ${|\:\!\!|\:\!\!|} f- \lambda {|\:\!\!|\:\!\!|}_\pm = {|\:\!\!|\:\!\!|} f {|\:\!\!|\:\!\!|}_\pm - \lambda$. Hence $f - \lambda$ satisfies \eqref{eq:cone 2} if
\[
{|\:\!\!|\:\!\!|} f{|\:\!\!|\:\!\!|}_+-\lambda \le L ({|\:\!\!|\:\!\!|} f{|\:\!\!|\:\!\!|}_- - \lambda)\quad \Longleftarrow \quad \lambda \le \frac{L(1-\chi)}{L-1} {|\:\!\!|\:\!\!|} f{|\:\!\!|\:\!\!|}_-=: \bar \alpha_1 \, ,
\]
where we have used that $f \in \mathcal{C}_\chi$.
Similarly, $f - \lambda$ satisfies \eqref{eq:cone 3} if, for all $W \in \mathcal{W}^s_-(\delta)$ and $\psi\in \mathcal{D}_{a,\beta}(W)$,
\[
|W|^{-q}\frac{\left|\int_W f\psi-\lambda\int_W \psi \right|}{\fint_W\psi}\leq A\delta^{1-q}({|\:\!\!|\:\!\!|} f{|\:\!\!|\:\!\!|}_--\lambda)
\quad \Longleftarrow \quad \lambda\leq \frac{(1-\chi) A {|\:\!\!|\:\!\!|} f{|\:\!\!|\:\!\!|}_-}{A+1}=: \bar \alpha_2 \, .
\]
Next, notice that for any $\lambda \ge 0$, $W^1, W^2 \in \mathcal{W}^s_-(\delta)$ and
$\psi_\ell \in \mathcal{D}_{a,\alpha}(W^\ell)$,
\begin{equation}
\label{eq:c cancel}
\begin{split}
\left| \frac{\int_{W^1}(f - \lambda) \psi_1}{\fint_{W^1} \psi_1} \right. & \left. - \frac{\int_{W^2}(f-\lambda)\psi_2}{\fint_{W^2} \psi_2} \right|
= \left| \frac{\int_{W^1} f \, \psi_1}{\fint_{W^1} \psi_1} - \frac{\int_{W^2} f \, \psi_2}{\fint_{W^2} \psi_2}
- \lambda(|W^1|- |W^2|) \right| \\
& \le \chi^2 d_{\mathcal{W}^s}(W^1, W^2)^\gamma \delta^{1-\gamma} c A {|\:\!\!|\:\!\!|} f {|\:\!\!|\:\!\!|}_-
+ \lambda (\delta + C_s) d_{\mathcal{W}^s}(W^1, W^2) \, ,
\end{split}
\end{equation}
where we have used \eqref{eq:W-difference}, so that $f - \lambda$ satisfies \eqref{eq:cone 5} if
\[
\chi^2 d_{\mathcal{W}^s}(W^1, W^2)^\gamma \delta^{1-\gamma} c A {|\:\!\!|\:\!\!|} f {|\:\!\!|\:\!\!|}_-
+ \lambda (\delta + C_s) \delta^{ 1- \gamma} d_{\mathcal{W}^s}(W^1, W^2)^\gamma
\le d_{\mathcal{W}^s}(W^1, W^2)^\gamma {\delta^{1-\gamma} } c A ({|\:\!\!|\:\!\!|} f {|\:\!\!|\:\!\!|}_- - \lambda) \, .
\]
This occurs whenever
\[
\lambda \le \frac{cA {|\:\!\!|\:\!\!|} f {|\:\!\!|\:\!\!|}_- (1-\chi^2)}{\delta+C_s+ cA}
\quad \Longleftarrow \quad
\lambda \le (1-\chi) {|\:\!\!|\:\!\!|} f {|\:\!\!|\:\!\!|}_- =: \bar \alpha_ 3 \, ,
\]
provided that $\delta$ is chosen sufficiently small that
\begin{equation}
\label{eq:delta cond}
\delta + C_s \le \chi c A \, ,
\end{equation}
which is possible since $cA > 2C_s$ by \eqref{eq:cA} and $\chi > 1/2$.
Note that $\bar \alpha_2 \le \bar \alpha_3 \le \bar \alpha_1$, so that
$\bar\alpha_2 = \min_i \{ \bar \alpha_i \}$. Thus if $\lambda \le \bar \alpha_2$, then
$f - \lambda \in \mathcal{C}$, i.e.
$\bar \alpha(1, f) \ge \bar \alpha_2$.
Next, we proceed to estimate $\bar \beta(1,f)$ for $f \in \mathcal{C}_\chi$. If we find $\mu > 0$ such that $\mu - f \in \mathcal{C}$,
this will imply that $\bar \beta(1,f) \le \mu$. Remarking that
${|\:\!\!|\:\!\!|} \mu - f {|\:\!\!|\:\!\!|}_{\pm} = \mu - {|\:\!\!|\:\!\!|} f {|\:\!\!|\:\!\!|}_{\mp}$, we have that $\mu - f$ satisfies \eqref{eq:cone 2}
if
\[
\mu \ge \frac{L {|\:\!\!|\:\!\!|} f {|\:\!\!|\:\!\!|}_+ - {|\:\!\!|\:\!\!|} f {|\:\!\!|\:\!\!|}_-}{L-1} \quad \Longleftarrow \quad
\mu \ge \frac{L}{L-1} {|\:\!\!|\:\!\!|} f {|\:\!\!|\:\!\!|}_+ =: \bar \beta_1 \, ,
\]
while $\mu - f$ satisfies \eqref{eq:cone 3} if for all $W \in \mathcal{W}^s_-(\delta)$, $\psi \in \mathcal{D}_{a,\beta}(W)$,
\[
|W|^{-q} \frac{| \mu \int_W \psi - \int_W f \, \psi |}{\fint_W \psi} \le A \delta^{1-q}(\mu - {|\:\!\!|\:\!\!|} f {|\:\!\!|\:\!\!|}_+)
\quad \Longleftarrow \quad
\mu \ge \frac{(1 + \chi) A}{A-{2^{1-q} }} {|\:\!\!|\:\!\!|} f {|\:\!\!|\:\!\!|}_+ =: \bar \beta_2 \, .
\]
Finally, recalling \eqref{eq:c cancel} and again \eqref{eq:W-difference}, we have that
$\mu -f$ satisfies \eqref{eq:cone 5} whenever
\[
\chi^2 d_{\mathcal{W}^s}(W^1, W^2)^\gamma \delta^{1-\gamma} c A {|\:\!\!|\:\!\!|} f {|\:\!\!|\:\!\!|}_-
+ \mu (\delta + C_s) \delta^{ 1-\gamma} d_{\mathcal{W}^s}(W^1, W^2)^\gamma
\le d_{\mathcal{W}^s}(W^1, W^2)^\gamma \delta^{1-\gamma} c A (\mu - {|\:\!\!|\:\!\!|} f {|\:\!\!|\:\!\!|}_+) \, .
\]
This is implied by,
\[
\mu \ge \frac{cA (1 + \chi^2)}{cA - (\delta + C_s) } {|\:\!\!|\:\!\!|} f {|\:\!\!|\:\!\!|}_+
\quad \Longleftarrow \quad
\mu \ge \frac{1 + \chi^2}{1 - \chi} {|\:\!\!|\:\!\!|} f {|\:\!\!|\:\!\!|}_+ =: \bar \beta_3 \, ,
\]
where again we have assumed \eqref{eq:delta cond}.
Defining $\bar \beta = \max_i \{ \bar \beta_i \}$, it follows that if $\mu \ge \bar \beta$, then
$\mu - f \in \mathcal{C}$.
Thus $\bar \beta \ge \bar \beta(1, f)$.
Since $\chi > 1/L$ and $\chi^2 > 1/(A-1)$, it holds that
$\bar \beta_3 \ge \bar \beta_2 \ge \bar \beta_1$. Thus $\bar \beta = \bar \beta_3$.
Our assumption also implies $\chi > 1/A$, so that
$\bar \alpha_2 \ge \frac{1-\chi}{1+\chi} {|\:\!\!|\:\!\!|} f {|\:\!\!|\:\!\!|}_-$.
Finally, recalling \eqref{eq:H def}, we have
\[
\rho(1,f) = \log \left( \frac{\bar \beta(1,f)}{\bar \alpha(1,f)} \right)
\le \log \left( \frac{\bar \beta_3}{\bar \alpha_2} \right)
\le \log \left( \frac{\frac{1+\chi^2}{1-\chi}}{\frac{1-\chi}{1+\chi}} \frac{{|\:\!\!|\:\!\!|} f {|\:\!\!|\:\!\!|}_+}{{|\:\!\!|\:\!\!|} f {|\:\!\!|\:\!\!|}_-} \right)
\le \log \left( \frac{(1+\chi)^2}{(1-\chi)^2} \chi L \right) \, ,
\]
for all $f \in \mathcal{C}_\chi$, completing the proof of the proposition.
\end{proof}
\begin{remark} Note that, setting $\chi_*=\max \{ \frac 12, \frac{1}{L}, \frac 1{\sqrt{A-1}} \} $, for $\chi\leq \chi_*$ Proposition \ref{prop:diameter} implies only that the diameter of $\mathcal{C}_{\chi c, \chi A, \chi L}(\delta)\subset \mathcal{C}_{\chi_* c, \chi_* A, \chi_* L}(\delta)$, in $\mathcal{C}_{c,A,L}(\delta)$, is bounded by
$ \log \left( \frac{(1+\chi_*)^2}{(1-\chi_*)^2} \chi_* L \right)$. If needed, a more accurate formula can be easily obtained, but it would be more cumbersome.
\end{remark}
\section{Loss of Memory and Convergence to Equilibrium }
\label{sec:exp-mix}
In this section we show how Theorem~\ref{thm:main} (i.e. Theorem~\ref{thm:cone contract} and Proposition~\ref{prop:diameter} ) imply
the loss of memory and convergence to equilibrium stated in Theorems~\ref{thm:memory state} and \ref{thm:equi state}.
For a single map, the loss of memory is simply decay of correlations and the results are comparable to the ones obtained in \cite{demzhang11} since they apply to a similar (very) large class of observables (and possibly even distributions). Our loss of memory result is new for our class of billiards, although see Remark~\ref{rmk:stenlund} and \cite{young zhang}
for loss of memory in a related billiards model.
Before proving the main results of this section (Theorem~\ref{thm:memory} and Corollary \ref{cor:extend} prove
Theorem~\ref{thm:memory state} while Theorem~\ref{thm:equi} and Corollary~\ref{cor:extend} prove
Theorem~\ref{thm:equi state}), we
establish a key lemma that integration with respect to $\mu_{\mbox{\tiny SRB}}$ against suitable
test functions respects the ordering in our cone.
Recall the vector space of functions $\mathcal{A}$ defined in Section~\ref{sec:cone_dist}.
\begin{lemma}
\label{lem:order}
Let $\delta>0$ be small enough that
$2 C_\ell C_h (1+A) ( \delta^{4/3} + \delta^{1/3 + \beta} a \ell_{\max} ) < 1$,
where $C_\ell, C_h>0$ are from \eqref{eq:dominate} and $\ell_{\max}$ is the maximum
diameter of the connected components of $M$.
Suppose $\psi \in C^1(M)$ satisfies $2(2\delta)^{1-\beta} |\psi'|_{C^0(M)} \le a \min_M \psi$.
If $f, g \in \mathcal{A}$ with $f \preceq g$, then
$\int f \, \psi \, d\mu_{\mbox{\tiny SRB}} \le \int g \, \psi \, d\mu_{\mbox{\tiny SRB}}$.
\end{lemma}
\begin{proof}
Let $\psi_{\min} = \min_M \psi$. The assumption on $\psi$ implies that $\psi \in \mathcal{D}_{\frac{a}{2},\beta}(W)$ for each
$W \in \mathcal{W}^s_-(\delta)$ since,
\[
\left| \log \frac{\psi(x)}{\psi(y)} \right| \le
\frac{1}{\psi_{\min}} |\psi(x) - \psi(y)| \le \frac{|\psi'|_{C^0(M)}}{\psi_{\min}} d(x,y)
\le \frac{|\psi'|_{C^0(M)}}{\psi_{\min}} (2\delta)^{1-\beta} d(x,y)^\beta \, .
\]
Suppose $f, g \in \mathcal{A}$ satisfy $f \preceq g$. If $g-f=0$, then the lemma
holds trivially, so suppose instead that $g-f \in \mathcal{C}_{c,A,L}(\delta)$. Then according to
\eqref{eq:tri def} and \eqref{eq:cone 3}, for all $\psi \in \mathcal{D}_{a,\beta}(W)$,
\begin{eqnarray}
{|\:\!\!|\:\!\!|} g - f {|\:\!\!|\:\!\!|}_- \! \! \! \! \! & \! \! \int_W \psi \; \le \; \int_W (g-f) \psi \, dm_W \; \le \; {|\:\!\!|\:\!\!|} g-f {|\:\!\!|\:\!\!|}_+ \int_W \psi
& \quad \forall\, W \in \mathcal{W}^s(\delta)
\label{eq:long bound} \\
& \left| \int_W (g-f) \psi \, dm_W \right| \; \le \; {|\:\!\!|\:\!\!|} g-f {|\:\!\!|\:\!\!|}_- A \delta^{1-q}|W|^q \fint_W \psi
& \quad \forall\, W \in \mathcal{W}^s_-(\delta).
\label{eq:short bound}
\end{eqnarray}
Next, we disintegrate $\mu_{\mbox{\tiny SRB}}$ according to a smooth foliation of stable curves as follows.
Since the stable cones defined in {\bf (H1)} are globally constant and uniform in the family
$\mathcal{F}(\tau_*, \mathcal{K}_*, E_*)$, we fix a direction in the stable cone
and consider stable curves in the form of line segments with this slope. Let $k_\delta \ge k_0$
denote the minimal index $k$ of a homogeneity strip $\mathbb{H}_k$ such that the stable line segments
in $\mathbb{H}_k$ have length less than $\delta$. Due to the fact that the minimum
slope in the stable cone is $\mathcal{K}_{\min} > 0$, we have
\begin{equation}
\label{eq:k delta}
k_\delta = C_h \delta^{-1/3},
\end{equation}
for some constant $C_h >0$ independent of $\delta$.
Now for $k < k_\delta$,
we decompose $\mathbb{H}_k$ into horizontal bands $B_i$ such that every maximal line segment of the
chosen slope in $B_i$ has equal length between $\delta$ and $2\delta$. We do the
same on $M \setminus (\cup_{k \ge k_0} \mathbb{H}_k)$.
On each $B_i$, define
a foliation of such parallel line segments $\{ W_\xi \}_{\xi \in \Xi_i} \subset \mathcal{W}^s(\delta)$.
Using the smoothness of this foliation, we disintegrate $\mu_{\mbox{\tiny SRB}}$ into conditional
measures $\cos \varphi(x) dm_{W_\xi}$ on $W_\xi$
and a factor measure $\hat\mu$ on the index set $\Xi_i$. Note that our conditional
measures are not normalized - we include this factor in $\hat\mu$.
Finally, on each homogeneity strip $\mathbb{H}_k$, $k \ge k_\delta$, we carry out a similar
decomposition, but using homogeneous parallel line segments of maximal length in $\mathbb{H}_k$
(which are necessarily shorter than length $\delta$). We use the notation
$\{ W_\xi \}_{\xi \in \Xi_k} \subset \mathcal{W}^s_-(\delta)$ for the foliations in these homogeneity strips.
Note that in both cases, we have $\hat\mu(\Xi_i), \hat\mu(\Xi_k) \le C_\ell$, for some
constant $C_\ell$ depending only on the chosen slope and spacing of homogeneity strips.
Also, it follows as in \eqref{eq:distortion}, that for $x, y \in W \in \mathcal{W}^s_-(\delta)$,
\[
\log \frac{\cos \varphi(x)}{\cos \varphi(y)} \le C_d (2\delta)^{1/3-\beta} d(x,y)^\beta \, ,
\]
so that $\cos \varphi \in \mathcal{D}_{\frac{a}{2}, \beta}(W)$ by the assumption of
Lemma~\ref{lem:test contract}.
Thus $\psi \cos \varphi \in \mathcal{D}_{a, \beta}(W)$ for all $W \in \mathcal{W}^s_-(\delta)$.
Using this fact and our disintegration of $\mu_{\mbox{\tiny SRB}}$, we estimate the integral
applying \eqref{eq:long bound} on $\Xi_i$ and
\eqref{eq:short bound} on $\Xi_k$,
\begin{equation}
\label{eq:dominate}
\begin{split}
\int_M (g-f) & \psi \, d\mu_{\mbox{\tiny SRB}} = \sum_i \int_{\Xi_i} \int_{W_\xi} (g-f) \psi \cos \varphi \, dm_{W_\xi} d\hat\mu(\xi)
+ \sum_{k \ge k_\delta} \int_{\Xi_k} \int_{W_\xi} (g-f) \psi \cos \varphi \, dm_{W_\xi} d\hat\mu(\xi) \\
& \ge {|\:\!\!|\:\!\!|} g-f {|\:\!\!|\:\!\!|}_- \left(
\sum_i \int_{\Xi_i} \int_{W_\xi} \psi \cos \varphi \, dm_{W_\xi} d\hat\mu(\xi)
- A \delta \sum_{k \ge k_\delta} \int_{\Xi_k} \fint_{W_\xi} \psi \cos \varphi \, dm_{W_\xi} d\hat\mu(\xi) \right) \\
& \ge {|\:\!\!|\:\!\!|} g-f {|\:\!\!|\:\!\!|}_- \left( \psi_{\min} \mu_{\mbox{\tiny SRB}}(M \setminus (\cup_{k \ge k_\delta} \mathbb{H}_k)) - A \delta C_\ell
|\psi|_{C^0} \sum_{k \ge k_\delta} k^{-2} \right) \\
& \ge {|\:\!\!|\:\!\!|} g-f {|\:\!\!|\:\!\!|}_- \left( \psi_{\min}(1 - 2 C_\ell C_h \delta^{4/3}) - |\psi|_{C^0} A C_\ell C_h 2 \delta^{4/3} \right) \, ,
\end{split}
\end{equation}
where we have estimated $\sum_{k \ge k_\delta} k^{-2} \le 2 k_\delta^{-1}$ and
$\mu_{\mbox{\tiny SRB}}(\cup_{k \ge k_\delta} \mathbb{H}_k) \le 2 C_\ell C_h \delta^{4/3}$.
Now $|\psi|_{C^0} \le \psi_{\min} + \ell_{\max} |\psi'|_{C^0}$, where $\ell_{\max}$
is the maximum diameter of the connected components of $M$. Then by the
assumption on $\psi$, we have
\[
\begin{split}
2 C_\ell C_h (1+A) \delta^{4/3} |\psi|_{C^0} & \le 2 C_\ell C_h (1+A) \delta^{4/3} \psi_{\min} (1 + \ell_{\max} \tfrac a2 (2\delta)^{\beta-1} ) \\
& \le \psi_{\min} 2 C_\ell C_h (1+A) ( \delta^{4/3} + a \ell_{\max} \delta^{1/3 + \beta})
\le \psi_{\min} \, ,
\end{split}
\]
where for the last inequality we have used the assumption on $\delta$ in the statement of the
lemma. We conclude that the lower bound in \eqref{eq:dominate} cannot be less than 0.
\end{proof}
\begin{remark}
\label{rem:after order}
Lemma \ref{lem:order} implies there exists $\bar C \ge 1$ such that $\int_M f \, d\mu_{\mbox{\tiny SRB}} \ge \bar C^{-1} {|\:\!\!|\:\!\!|} f {|\:\!\!|\:\!\!|}_- > 0$ for all $f \in \mathcal{C}_{c,A,L}(\delta)$.
Using instead the upper bound in \eqref{eq:long bound} and following the estimate of \eqref{eq:dominate} yields,
\[
0 < \int_M f \psi \, d\mu_{\mbox{\tiny SRB}} \le {|\:\!\!|\:\!\!|} f {|\:\!\!|\:\!\!|}_+ C |\psi|_{C^0} \, ,
\]
for all $f \in \mathcal{C}_{c,A,L}(\delta)$ and $\psi$ as in the statement of Lemma~\ref{lem:order}.
This can be extended to all $\psi \in C^1(M)$ by defining $C_\psi$ as in \eqref{eq:C psi} below to conclude
\[
\int_M f \psi \, d\mu_{\mbox{\tiny SRB}} \le {|\:\!\!|\:\!\!|} f {|\:\!\!|\:\!\!|}_+ C | \psi |_{C^1} \, .
\]
\end{remark}
Loss of memory and convergence to equilibrium, including equidistribution, readily follow from the contraction in the
projective metric $\rho_{\mathcal{C}}(\cdot, \cdot)$ of the cone $\mathcal{C}_{c, A, L}(\delta)$. Set $\mu_{\mbox{\tiny SRB}}(f) = \int_M f \, d\mu_{\mbox{\tiny SRB}}$.
Recall $N_\mathcal{F} := N(\delta)^- + k_*n_*$ from Theorem~\ref{thm:cone contract} and the definition of
admissible sequence from Section~\ref{sec:main}:
A sequence $( \iota_j )_j$, $\iota_j \in \mathcal{I}(\tau_*, \mathcal{K}_*, E_*)$, is admissible if there exists a sequence $(N_k)_{k \ge 0}$
with $N_k \ge N_{\mathcal{F}}$, such that for
all $k \ge 0$ and $j \in [kN_k, (k+1)N_k-1]$,
$T_{\iota_j} \in \mathcal{F}(T_{\iota_{kN_k}}, \kappa)$.
That is, an admissible sequence remains
in a $\kappa$ neighborhood of $T_{\iota_{kN_k}}$ for $N_k \ge N_\mathcal{F}$ iterates at a time, but may undergo large
changes between such blocks.
Our first theorem concerns loss of memory for functions in our cone, both with respect to $\mu_{\mbox{\tiny SRB}}$ and with respect to the iteration of
individual stable curves, and does not use property {\bf (H5)}.
\begin{theorem}
\label{thm:memory}
Let $\delta>0$ satisfy the assumption of Lemma~\ref{lem:order}.
There exists $C>0$ and $\vartheta<1$ such that for all admissible sequences $(\iota_j)_j$, all $n \ge 0$, and all
$f, g \in \mathcal{C}_{c,A,L}(\delta)$ with $\int_M f \, d\mu_{\mbox{\tiny SRB}} = \int_M g \, d\mu_{\mbox{\tiny SRB}}$:
\begin{itemize}
\item[a)] For all all $W \in \mathcal{W}^s(\delta)$ and all $\psi \in C^1(W)$, we have
\[
\left| \fint_{W} \mathcal{L}_n f \, \psi \, dm_{W} - \fint_{W} \mathcal{L}_n g \, \psi \, dm_{W} \right|
\le C \vartheta^n \, |\psi|_{C^1} \min \{ {|\:\!\!|\:\!\!|} f {|\:\!\!|\:\!\!|}_+, {|\:\!\!|\:\!\!|} g {|\:\!\!|\:\!\!|}_+ \} \, ;
\]
\item[b)] For all $\psi \in C^1(M)$,
\begin{equation}
\label{eq:M conv}
\left| \int_M \mathcal{L}_n f \, \psi \, d\mu_{\mbox{\tiny SRB}} - \int_M \mathcal{L}_n g \, \psi \, d\mu_{\mbox{\tiny SRB}} \right|
\le C \vartheta^n |\psi|_{C^1(M)} \min \{ {|\:\!\!|\:\!\!|} f {|\:\!\!|\:\!\!|}_+ , {|\:\!\!|\:\!\!|} g {|\:\!\!|\:\!\!|}_+ \} \, .
\end{equation}
\end{itemize}
\end{theorem}
\begin{proof}
(a)
It is convenient to extend the definition of ${|\:\!\!|\:\!\!|} \cdot {|\:\!\!|\:\!\!|}_+ $ to all of $\mathcal{A}$ by
\[
{|\:\!\!|\:\!\!|} f {|\:\!\!|\:\!\!|}_+ = \sup_{\stackrel{\scriptstyle W \in \mathcal{W}^s(\delta)}{\psi \in \mathcal{D}_{a,\beta}(W)}} \frac{\left|\int_W f \psi \, dm_W\right|}{\int_W \psi \, dm_W}.
\]
Note that, with this definition, ${|\:\!\!|\:\!\!|} \cdot {|\:\!\!|\:\!\!|}_+ $ is an order-preserving semi-norm in $\mathcal{A}$.\footnote{ A semi-norm $\|\cdot\|$ is order preserving if $-g\preceq f\preceq g$ implies $\|f\|\leq \|g\|$. The space $\mathcal{A}$ is defined just before \eqref{eq:tri def}.}
Also $ \mu_{\mbox{\tiny SRB}}(f) := \int_M f \, d\mu_{\mbox{\tiny SRB}}$ is homogeneous and order preserving in $\mathcal{C}_{c,A,L}(\delta)$ by Lemma \ref{lem:order} applied to $\psi \equiv 1$.
Then \cite[Lemma 2.2]{LSV98} implies that, for all $f, g \in \mathcal{C}_{c,A,L}(\delta)$ with
$\mu_{\mbox{\tiny SRB}}(f)=\mu_{\mbox{\tiny SRB}}(g)$,\footnote{ \cite[Lemma 2.2]{LSV98} is stated for order preserving norms but its proof holds verbatim for order preserving semi-norms.}
\begin{equation}
\label{eq:adapted}
{|\:\!\!|\:\!\!|} \mathcal{L}_n f-\mathcal{L}_n g{|\:\!\!|\:\!\!|}_+\leq \left(e^{\rho_{\mathcal{C}}(\mathcal{L}_n f,\mathcal{L}_n g)}-1\right)\min \{{|\:\!\!|\:\!\!|} \mathcal{L}_n f {|\:\!\!|\:\!\!|}_+, {|\:\!\!|\:\!\!|} \mathcal{L}_n g {|\:\!\!|\:\!\!|}_+\}.
\end{equation}
Hence by\footnote{ We apply Theorem~\ref{thm:cone contract} to each block of $N_k$ iterates in the
admissible sequence.} Theorem~\ref{thm:cone contract}, Proposition~\ref{prop:diameter},
\cite[Theorem 2.1]{liv95} and Lemma~\ref{lem:first L},
there exists $C>0$ such that for all $n\geq N_\mathcal{F}:= N(\delta)^- + k_*n_*$,
\begin{equation}\label{eq:contract}
{|\:\!\!|\:\!\!|} \mathcal{L}_n f-\mathcal{L}_n g{|\:\!\!|\:\!\!|}_+\leq C \vartheta^n\min \{{|\:\!\!|\:\!\!|} f {|\:\!\!|\:\!\!|}_+, {|\:\!\!|\:\!\!|} g {|\:\!\!|\:\!\!|}_+\},
\end{equation}
where $\vartheta=\left[\tanh(\Delta/4)\right]^{1/n(\delta)}$. This proves (a) for any $W \in \mathcal{W}^s(\delta)$
and $\psi \in \mathcal{D}_{a,\beta}(W)$.
To extend this estimate
to more general $\psi \in C^1(W)$, define $\tilde{\psi} = \psi + C_\psi$, where
\begin{equation}
\label{eq:C psi}
C_\psi = |\psi_{\min}| + \tfrac{2}{a} |\psi'|_{C^0}(2\delta)^{1-\beta} \, .
\end{equation}
Then $\tilde{\psi}' = \psi'$ and $\min_W \tilde{\psi} \ge \frac{2}{a} |\tilde{\psi}'|_{C^0} (2\delta)^{1-\beta}$,
so that $\tilde{\psi} \in \mathcal{D}_{\frac{a}{2}, \beta}(W)$ by the proof of Lemma~\ref{lem:order}.
Then since also $C_\psi \in \mathcal{D}_{a, \beta}(W)$, the required estimate
follows by writing $\psi = \tilde{\psi} - C_\psi$ and using the triangle inequality.
\medskip
\noindent
(b) Following the same strategy as above, given $\psi \in C^1(M)$ satisfying
the assumption of Lemma~\ref{lem:order}, we define a pseudo-norm for $f \in \mathcal{A}$ by
\begin{equation}
\label{eq:psi seminorm}
\| f \|_\psi = \left| \int_M f \, \psi \, d\mu_{\mbox{\tiny SRB}} \right| \, .
\end{equation}
By Lemma~\ref{lem:order}, $\| \cdot \|_\psi$ is an order-preserving semi-norm, and as in \eqref{eq:adapted}, invoking
again \cite[Lemma~2.2]{LSV98}, Theorem~\ref{thm:cone contract}, Proposition~\ref{prop:diameter}
and \cite[Theorem~2.1]{liv95}, we have for $f, g \in \mathcal{C}_{c,A,L}(\delta)$ with
$\mu_{\mbox{\tiny SRB}}(f) = \mu_{\mbox{\tiny SRB}}(g)$ and $n \ge N_\mathcal{F}$,
\[
\| \mathcal{L}_n f - \mathcal{L}_n g \|_\psi \le C \vartheta^n \min \{ \| \mathcal{L}_n f \|_\psi , \| \mathcal{L}_n g \|_\psi \} \le C \vartheta^n |\psi|_{C^0} \min \{
{|\:\!\!|\:\!\!|} f {|\:\!\!|\:\!\!|}_+ , {|\:\!\!|\:\!\!|} g {|\:\!\!|\:\!\!|}_+ \} \, ,
\]
where we applied Remark~\ref{rem:after order}. This
proves (b) for $\psi$ satisfying the assumption of
Lemma~\ref{lem:order}. We extend to more general $\psi \in C^1(M)$ by defining
$\tilde{\psi} = \psi + C_\psi$, where $C_\psi$ is given by \eqref{eq:C psi}, and arguing as in the
proof of part (a).
\end{proof}
Since our maps all preserve $\mu_{\mbox{\tiny SRB}}$, the loss of memory also implies equidistribution of measures supported on stable curves and convergence to equilibrium, both of which are summarized in the following theorem.
\begin{theorem}
\label{thm:equi}
Let $\delta>0$ satisfy the assumption of Lemma~\ref{lem:order}.
There exists $C>0$ such that for all $n \ge 0$ and admissible sequences $(\iota_j)_j \subset \mathcal{I}(\tau_*, \mathcal{K}_*, E_*)$,
and all $f, g \in \mathcal{C}_{c,A,L}(\delta)$, with $\mu_{\mbox{\tiny SRB}}(f) = \mu_{\mbox{\tiny SRB}}(g)$:
\begin{itemize}
\item[a)] For all $W_1, W_2 \in \mathcal{W}^s(\delta)$ and all $\psi_i \in C^1(W_i)$ with $\fint_{W_1} \psi_1 = \fint_{W_2} \psi_2$, we have
\[
\left| \fint_{W_1} \mathcal{L}_n f \, \psi_1 \, dm_{W_1} - \fint_{W_2} \mathcal{L}_n g \, \psi_2 \, dm_{W_2} \right|
\le C \vartheta^n \, (|\psi_1|_{C^1} + |\psi_2|_{C^1} ) \mu_{\mbox{\tiny SRB}}(f) \, ;
\]
in particular, for all $W \in \mathcal{W}^s(\delta)$ and $\psi\in C^1(W)$,
\begin{equation}
\label{eq:avg conv}
\left| \fint_W \mathcal{L}_n f \, \psi \, dm_W - \mu_{\mbox{\tiny SRB}}(f) \fint_W \psi \, dm_W \right|
\le C \vartheta^n \, |\psi|_{C^1} \mu_{\mbox{\tiny SRB}}(f) \, ;
\end{equation}
\item[b)] for all $\psi \in C^1(M)$,
\[
\left| \int_M f \, \psi \circ T_n \, d\mu_{\mbox{\tiny SRB}} - \int_M f \, d\mu_{\mbox{\tiny SRB}} \int_M \psi \, d\mu_{\mbox{\tiny SRB}} \right|
\le C \vartheta^n |\psi|_{C^1(M)} \mu_{\mbox{\tiny SRB}}(f) \, .
\]
\end{itemize}
\end{theorem}
\begin{proof}
a) Since $\mathcal{L}_n 1 = 1$ and ${|\:\!\!|\:\!\!|} \mu_{\mbox{\tiny SRB}}(f) {|\:\!\!|\:\!\!|}_+ = \mu_{\mbox{\tiny SRB}}(f)$, applying \eqref{eq:contract} with $g = \mu_{\mbox{\tiny SRB}}(f)$ implies,
\begin{equation}
\label{eq:equi}
\begin{split}
\left| \fint_W \mathcal{L}_n f \, \psi \, dm_W - \mu_{\mbox{\tiny SRB}}(f) \fint_W \psi \right| &=
\fint_W \psi \left| \frac{\int_W \mathcal{L}_n f \, \psi \, dm_W}{\int_W\psi} - \frac{\int_W \mathcal{L}_n (\mu_{\mbox{\tiny SRB}}(f)) \, \psi}{\int_W\psi} \right|\\
&\le C \vartheta^n \, |\psi|_{C^0} \mu_{\mbox{\tiny SRB}}(f)\, ,
\end{split}
\end{equation}
which proves \eqref{eq:avg conv}
for $\psi \in \mathcal{D}_{a, \beta}(W)$. We extend this estimate
to more general $\psi \in C^1(W)$ by defining $\tilde{\psi} = \psi + C_\psi$ as in \eqref{eq:C psi}
and arguing as in the proof of Theorem~\ref{thm:memory}(a).
Finally, the first inequality of part (a) follows from an application of the triangle inequality.
\medskip
\noindent
b) Since $\mu_{\mbox{\tiny SRB}}$ is conformal with respect to $\mathcal{L}_{T}$ for each $T \in \mathcal{F}(\tau_*, \mathcal{K}_*, E_*)$,
and using that $\mathcal{L}_n 1 = 1$, we have
\[
\int_M f \, \psi \circ T_n \, d\mu_{\mbox{\tiny SRB}} - \int_M f \, d\mu_{\mbox{\tiny SRB}} \int_M \psi \, d\mu_{\mbox{\tiny SRB}}
= \int_M \mathcal{L}_n (f - \mu_{\mbox{\tiny SRB}}(f)) \, \psi \, d\mu_{\mbox{\tiny SRB}} \, .
\]
Thus applying \eqref{eq:M conv} to $g = \mu_{\mbox{\tiny SRB}}(f)$ proves part (b)
since ${|\:\!\!|\:\!\!|} \mu_{\mbox{\tiny SRB}}(f) {|\:\!\!|\:\!\!|}_+ = \mu_{\mbox{\tiny SRB}}(f)$.
\end{proof}
\begin{cor}
\label{cor:extend}
The convergence in Theorems~\ref{thm:memory} and \ref{thm:equi} extend to all $f, g \in C^1(M)$,
with $|f|_{C^1(M)}, |g|_{C^1(M)}$ on the right hand side.
\end{cor}
The proof of this corollary relies on the following lemma.
\begin{lemma}
\label{lem:dominate}
If $f \in C^1(M)$, then $\lambda + f \in \mathcal{C}_{c, A, L}(\delta)$ for any
\[
\lambda \geq \max \left\{ \frac{L+1}{L-1} |f|_{C^0} , \frac{A+2^{1-q}}{A- 2^{1-q}} |f|_{C^0},
\frac{cA + 8 C_s}{cA - 2C_s} |f|_{C^1}\right\} \, .
\]
\end{lemma}
\begin{proof}[Proof of Corollary~\ref{cor:extend}]
Let $f, g \in C^1(M)$ with $\mu_{\mbox{\tiny SRB}}(f) = \mu_{\mbox{\tiny SRB}}(g)$ and let $\psi \in C^1(M)$.
Let $\lambda_f, \lambda_g$ be the constants
from Lemma~\ref{lem:dominate} corresponding to $f$ and $g$, respectively, and set
$\lambda = \max \{ \lambda_f, \lambda_g \}$.
Then $f + \lambda, g + \lambda \in \mathcal{C}_{c,A,L}(\delta)$ and $\mu_{\mbox{\tiny SRB}}(f+\lambda) = \mu_{\mbox{\tiny SRB}}(g + \lambda)$, so that by Theorem~\ref{thm:memory}(b), for all $n \ge 0$,
\[
\begin{split}
\left| \int_M \mathcal{L}_n (f-g) \, \psi \, d\mu_{\mbox{\tiny SRB}} \right| & = \left| \int_M \mathcal{L}_n (f+\lambda-(g+\lambda)) \, \psi \, d\mu_{\mbox{\tiny SRB}} \right| \\
& \le C' \vartheta^n |\psi|_{C^1(M)} \max \{ |f|_{C^1(M)}, |g|_{C^1(M)} \} \, ,
\end{split}
\]
since ${|\:\!\!|\:\!\!|} f+ \lambda{|\:\!\!|\:\!\!|}_+ \le \lambda + |f|_{C^0}$, and by Lemma~\ref{lem:dominate},
$\lambda_f \ge C'' |f|_{C^1(M)}$, with analogous estimates for $g$. This proves the analogue of
part (b) Theorem~\ref{thm:memory} and the proof of part (a) follows similarly, replacing the integral over $M$
by the integral over $W \in \mathcal{W}^s$.
The extension of Theorem~\ref{thm:equi} to $f, g \in C^1(M)$ follows analogously,
replacing $f$ and $g$ in \eqref{eq:equi} with $f+\lambda$ and $g+\lambda$, respectively to prove
to prove the analogue of \eqref{eq:avg conv}, and then using the
triangle inequality to deduce the first inequality of part (a). Finally, part (b) follows immediately once
$f$ is replaced by $f+\lambda$ since $\int_M \psi \circ T_n \, d\mu_{\mbox{\tiny SRB}} = \int_M \psi \, d\mu_{\mbox{\tiny SRB}}$ due to
{\bf (H5)}.
\end{proof}
\begin{proof}[Proof of Lemma~\ref{lem:dominate}]
We must show that $\lambda + f$ satisfies conditions \eqref{eq:cone 2} - \eqref{eq:cone 5} in the
definition of $\mathcal{C}_{c,A,L}(\delta)$. Since
\begin{equation}
\label{eq:tri up/down}
{|\:\!\!|\:\!\!|} \lambda + f {|\:\!\!|\:\!\!|}_+ \le \lambda + | f |_{C^0}, \qquad \mbox{and} \qquad
{|\:\!\!|\:\!\!|} \lambda + f {|\:\!\!|\:\!\!|}_- \ge \lambda - |f|_{C^0} \, ,
\end{equation}
to guarantee \eqref{eq:cone 2}, we need
\[
\frac{\lambda + |f|_{C^0}}{\lambda - |f|_{C^0}} \le L
\qquad \impliedby \qquad
\lambda \ge |f|_{C^0} \frac{L+1}{L-1} \, .
\]
Next, to guarantee \eqref{eq:cone 3}, for $W \in \mathcal{W}^s_-(\delta)$, $\psi \in \mathcal{D}_{a, \beta}(W)$,
we need,
\[
|W|^{-q} \frac{\int_W (\lambda +f)\psi}{\fint_W \psi} \le A \delta^{1-q} (\lambda - |f|_{C^0})
\qquad \impliedby \qquad
\lambda \ge |f|_{C^0} \frac{A + 2^{1-q}}{A - 2^{1-q}} \, .
\]
Lastly, we need to show that \eqref{eq:cone 5} is satisfied. For this, we prove the claim:
\begin{equation}
\label{eq:C1 bound}
\left| \frac{\int_{W_1} f \psi_1}{\fint_{W_1} \psi_1 } - \frac{\int_{W_2} f \psi_2}{\fint_{W_2} \psi_2 } \right|
\le 8 C_s \delta^{1-\gamma} d_{\mathcal{W}^s}(W_1, W_2)^\gamma |f|_{C^1} \, ,
\end{equation}
for $W_1, W_2, \psi_1, \psi_2$ as in \eqref{eq:cone 5}. Recalling the notation
$W_k = \{ G_{W_k}(r) = (r, \varphi_{W_k}(r)) : r \in I_{W_k} \}$ for $k=1,2$ from Section~\ref{sec:distances}, we set
$\bar W_k = G_{W_k}(I_{W_1} \cap I_{W_2})$ and $W_k^c = W_k \setminus \bar W_k$.
As in Section~\ref{subsec:contract c}, we assume without loss of generality that $|W_2| \ge |W_1|$
and $\fint_{W_1} \psi_1 = 1$. Also, we may assume
$|W_2| \ge 2 C_s \delta^{1-\gamma} d_{\mathcal{W}^s}(W_1, W_2)^\gamma$;
otherwise, \eqref{eq:C1 bound} is trivially
bounded by $2 |W_2| |f|_{C^0} \le 4 C_s \delta^{1-\gamma} d_{\mathcal{W}^s}(W_1, W_2)^\gamma |f|_{C^0}$.
Next,
\begin{equation}
\label{eq:C1 split}
\left| \frac{\int_{W_1} f \psi_1}{\fint_{W_1} \psi_1 } - \frac{\int_{W_2} f \psi_2}{\fint_{W_2} \psi_2 } \right|
\le \left| \frac{\int_{\bar W_1} f \psi_1 - \int_{\bar W_2} f \psi_2}{\fint_{W_1} \psi_1 } \right|
+ \left| \frac{\int_{\bar W_2} f \psi_2}{\fint_{W_2} \psi_2 }
\left( \frac{\fint_{W_2} \psi_2}{\fint_{W_1} \psi_1 } - 1 \right) \right|
+ \sum_{k=1}^2 \frac{\big| \int_{W_k^c} f \psi_k \big|}{\fint_{W_k} \psi_k}
\end{equation}
To estimate the first term above, recalling \eqref{eq:psi-distance} and $d_*(\psi_1, \psi_2) = 0$, we have
for $r \in I_{W_1} \cap I_{W_2}$,
\[
\begin{split}
| & (f\psi_1) \circ G_{W_1}(r) \| G'_{W_1}(r) \| - (f\psi_2) \circ G_{W_2}(r) \| G'_{W_2}(r) \| | \\
& = \psi_1 \circ G_{W_1}(r) \| G'_{W_1}(r) \| | f \circ G_{W_1}(r) - f \circ G_{W_2}(r) |
\le \psi_1 \circ G_{W_1}(r) \| G'_{W_1}(r) \| | f' |_{C^0} d_{\mathcal{W}^s}(W_1, W_2) \, ,
\end{split}
\]
and integrating over $I_{W_1} \cap I_{W_2}$ yields,
\begin{equation}
\label{eq:C1 first}
\left| \frac{\int_{\bar W_1} f \psi_1 - \int_{\bar W_2} f \psi_2}{\fint_{W_1} \psi_1 } \right|
\le \frac{\int_{\bar W_1} \psi_1}{\fint_{W_1} \psi_1} | f' |_{C^0} d_{\mathcal{W}^s}(W_1, W_2)
\le 2\delta | f' |_{C^0} d_{\mathcal{W}^s}(W_1, W_2) \, .
\end{equation}
For the second term in \eqref{eq:C1 split}, note that our assumption $|W_2| \ge 2 C_s \delta^{1-\gamma} d_{\mathcal{W}^s}(W_1, W_2)^\gamma$ implies as in the estimate following \eqref{eq:W2-long}
that $I_{W_1} \cap I_{W_2} \neq \emptyset$. Thus we may apply \eqref{eq:psi2} and
\eqref{eq:new-difference} and use $\fint_{W_1} \psi_1 = 1$ to obtain,
\begin{equation}
\label{eq:C1 second}
\left| \frac{\int_{\bar W_2} f \psi_2}{\fint_{W_2} \psi_2 }
\left( \frac{\fint_{W_2} \psi_2}{\fint_{W_1} \psi_1 } - 1 \right) \right|
\le |f|_{C^0} \left| \int_{W_2} \psi_2 - |W_2| \right| \le |f|_{C^0} 6 C_s d_{\mathcal{W}^s}(W_1, W_2) \, .
\end{equation}
Finally, the third term in \eqref{eq:C1 split} can be estimated by
\[
\sum_k \frac{| \int_{W_k^c} f \psi_k |}{\fint_{W_k} \psi_k }
\le |f|_{C^0} e^{a(2\delta)^\alpha} \big( |W^c_1| + |W^c_2| \big) \le 2 C_s |f|_{C^0} d_{\mathcal{W}^s}(W_1, W_2) \, .
\]
Collecting this estimate together with \eqref{eq:C1 first} and \eqref{eq:C1 second}
in \eqref{eq:C1 split}, we obtain
\[
\left| \frac{\int_{W_1} f \psi_1}{\fint_{W_1} \psi_1 } - \frac{\int_{W_2} f \psi_2}{\fint_{W_2} \psi_2 } \right|
\le 8 C_s d_{\mathcal{W}^s}(W_1, W_2) |f|_{C^1} \, ,
\]
proving the bound in \eqref{eq:C1 bound} since $d_{\mathcal{W}^s}(W_1, W_2) \le \delta$.
With the claim proved, we proceed to verify \eqref{eq:cone 5}. Using \eqref{eq:W-difference} we estimate,
\[
\begin{split}
\left| \frac{\int_{W_1} ( f+ \lambda) \psi_1}{\fint_{W_1} \psi_1 } - \frac{\int_{W_2} (f + \lambda) \psi_2}{\fint_{W_2} \psi_2 } \right|
& \le \left| \frac{\int_{W_1} f \psi_1}{\fint_{W_1} \psi_1 } - \frac{\int_{W_2} f \psi_2}{\fint_{W_2} \psi_2 } \right| + \lambda | |W_1| - |W_2| | \\
& \le 8 C_s \delta^{1-\gamma} d_{\mathcal{W}^s}(W_1, W_2)^\gamma |f|_{C^1} + \lambda 2 C_s d_{\mathcal{W}^s}(W_1, W_2) \, .
\end{split}
\]
Thus \eqref{eq:cone 5} will be verified if
\[
8 C_s \delta^{1-\gamma} d_{\mathcal{W}^s}(W_1, W_2)^\gamma |f|_{C^1} + \lambda 2 C_s d_{\mathcal{W}^s}(W_1, W_2)
\le cA \delta^{1-\gamma} d_{\mathcal{W}^s}(W_1, W_2)^\gamma (\lambda - |f|_{C^0}) \, ,
\]
which is implied by the final condition on $\lambda$ in the statement of the Lemma since
$d_{\mathcal{W}^s}(W_1, W_2) \le \delta$ and $cA > 2C_s$ by \eqref{eq:cA}.
\end{proof}
\section{Applications}\label{sec:appl}
Suppose that we have a billiard table $Q = \mathbb{T}^2 \setminus \cup_i B_i$
and that the particle can escape from the table by entering certain
sets $\mathbb{G} \subset Q$, which we call {\em gates} or {\em holes},
but only at times $k N$ for some $N\in\mathbb{N}$.
One could easily consider also the case of $\mathbb{G} \subset Q\times S^1$ (i.e. some velocity directions are forbidden,
as studied in \cite{dem bill}), but we prefer to keep things simple.
In the literature, one often takes $N=1$, i.e. the particle can escape at each iterate of the map, but then the
holes are required to be very small, see for example \cite{DWY, dem inf, dem bill}.
By contrast, in this paper we will be interested in relatively large holes and so we will replace the
assumption of smallness with an assumption of occasional escape through possibly large sets. This
will facilitate the application of this method to two situations we have in mind: chaotic scattering (Section~\ref{sec:scattering})
and a random Lorentz gas (Section~\ref{sec:lorentz}).
We begin with the same setup as in Section~\ref{sec:bill family}, fixing $K$ numbers
$\ell_1, \ldots, \ell_K >0$ and identifying them as the arclengths of scatterers belonging to
$\mathcal{Q}(\tau_*, \mathcal{K}_*, E_*)$ for some fixed choice of $\tau_*, \mathcal{K}_*, E_* \in \mathbb{R}^+$.
As in Section~\ref{sec:uni hyp}, we fix an index set $\mathcal{I}(\tau_*, \mathcal{K}_*, E_*)$, identifying $\iota \in \mathcal{I}(\tau_*, \mathcal{K}_*, E_*)$ with a map $T_\iota \in \mathcal{F}(\tau_*, \mathcal{K}_*, E_*)$ induced by the table $Q_\iota \in \mathcal{Q}(\tau_*, \mathcal{K}_*, E_*)$.
A hole $\mathbb{G}_\iota \subset Q_\iota$ induces a hole $H_\iota \subset M$ in the phase space of the
billiard map $T_\iota$. We formulate here two abstract conditions on the set $H_\iota$, and then
provide examples of concrete, physically relevant situations which induce holes
satisfying our conditions in Section~\ref{sequential}.
\begin{itemize}
\item[(O1)] (Complexity) There exists $P_0 >0$ such that any stable curve of
length at most $\delta$ can be cut into at most $P_0$ pieces by $\partial H_\iota$, where $\delta$
is the length scale of the cone $\mathcal{C}_{c, A, L}(\delta)$.
\item[(O2)] (Uniform transversality) There exists $C_t > 0$ such that, for any stable curve
$W \in \mathcal{W}^s$ and $\varepsilon>0$, $m_W(N_\varepsilon(\partial H_\iota)) \le C_t \varepsilon$.
\end{itemize}
\begin{remark}
Assumption (O2) can be weakened to, e.g., $m_W(N_\varepsilon(\partial H_\iota)) \le C_t \varepsilon^{1/2}$,
but this would then require $d_{\mathcal{W}^s}(W^1, W^2) \le \delta^2$ in our definition of cone
condition \eqref{eq:cone 5}. Similar modifications are made to weaken the transversality assumption in
the Banach space setting, see for example \cite{demzhang14, dem bill}.
\end{remark}
For fixed $P_0, C_t >0$, we define $\mathcal{H}(P_0, C_t)$ to be the collection of holes $H_\iota$ satisfying
(O1) and and (O2) with respect to the given constants.
We let $\mbox{diam}^s(H_\iota)$ denote the maximal length of a stable curve in $H_\iota$, which
we call the {\em stable diameter}.
As in Section~\ref{sec:mix},
we fix $T_0 \in \mathcal{F}(\tau_*, \mathcal{K}_*, \tau_*)$ and consider sequences $\{ \iota_j \}_j \subset \mathcal{I}(T_0, \kappa)$,
where $\kappa >0$ is from Lemma~\ref{lem:proper cross}(b). Recalling \eqref{eq:close d},
this means we will initially consider sequential open systems comprised of maps $T \in \mathcal{F}(\tau_*, \mathcal{K}_*, E_*)$
with $d_{\mathcal{F}}(T, T_0) < \kappa$.
Denote by $\mathbbm{1}_{A}$ the characteristic function of the set $A$.
The relevant transfer operator for the open
sequential system (opening once every $N$ iterates) is given by $\mathcal{L}_{H_{\iota_N}, N}=\mathcal{L}_N\mathbbm{1}_{H_{\iota_N}^c}$, where $H_{\iota_N}^c$ denotes the
complement of $H_{\iota_N}$ in $M$, and $\mathcal{L}_N = \mathcal{L}_{T_{\iota_N}} \cdots \mathcal{L}_{T_{\iota_1}}$ is the usual transfer
operator for the $N$-step sequential dynamics.
The main objective is to control the action of the multiplication operator $\mathbbm{1}_{H_{\iota_N}^c}$
on the cone $\mathcal{C}_{c,A,L}(\delta)$.
\subsection{ Relatively small holes}
First we consider holes $H$ whose stable diameter is short compared to the length scale $\delta$.
\begin{lemma}\label{lem:case-a}
If $H \subset M$ satisfies (O1) and (O2), and if $\mbox{diam}^s(H) \le \delta \left[ \frac{1}{4P_0A } \right]^{1/q}$,
then
\[
\mathbbm{1}_{H^c}[\mathcal{C}_{c,A,L}(\delta)]\subset \mathcal{C}_{c',A',L'}(\delta),
\]
where
\[
\begin{split}
&L'=2 P_0^{1-q} e^{a(2\delta)^\beta} A \, , \quad
A'= 2 P_0^{1-q} e^{a(2\delta)^\beta} A \, , \\
&c'= P_0^q e^{a (2\delta)^\alpha} +
2 \left( 2^q \delta+ \tfrac 34 c \right) + 4 (P_0 + 2) P_0^{q-1} C_t^q \, .
\end{split}
\]
\end{lemma}
\begin{proof}
Letting $f \in \mathcal{C}_{c, A, L}(\delta)$, we must control the cone conditions one by one. We begin with \eqref{eq:cone 2}.
Given $W\in\mathcal{W}^s(\delta)$, let $\mathcal{G}_0$ denote the collection of connected curves in $W\setminus H$.
Then applying \eqref{eq:cone 3} to each $W' \in \mathcal{G}_0$,
for $\psi \in \mathcal{D}_{a, \beta}(W)$, we estimate
\begin{equation}\label{eq:appli1}
\begin{split}
\int_W(\mathbbm{1}_{H^c} f )\psi \, dm_W&= \sum_{W'\in\mathcal{G}_0}\int_{W'}f \psi \, dm_{W'}\\
&\leq \sum_{W'\in\mathcal{G}_0}\fint_{W'}\psi dm_{W'} |W'|^{q}A\delta^{1-q} {|\:\!\!|\:\!\!|} f{|\:\!\!|\:\!\!|}_-\\
&\leq P_0^{1-q} e^{a(2\delta)^\beta} A {|\:\!\!|\:\!\!|} f{|\:\!\!|\:\!\!|}_- \int_{W}\psi \, dm_{W}.
\end{split}
\end{equation}
On the other hand, if the collection of disjoint curves $\{W_i\}$ is such that $\cup_i W_i=W\cap H$,
\[
\begin{split}
\int_W(\mathbbm{1}_{H^c} f )\psi \, dm_W&= \int_{W} f \psi \, dm_{W} -\int_W(\mathbbm{1}_{H} f )\psi \, dm_W\\
&\geq {|\:\!\!|\:\!\!|} f{|\:\!\!|\:\!\!|}_-\int_{W} \psi \, dm_{W}-\sum_i |W_i|^qA\delta^{1-q}{|\:\!\!|\:\!\!|} f{|\:\!\!|\:\!\!|}_-\fint_{W_i} \psi \, dm_{W_i}\\
&\geq \left\{1- e^{a(2\delta)^\beta} A P_0 \delta^{-q} \mbox{diam}^s(H)^q \right\} {|\:\!\!|\:\!\!|} f{|\:\!\!|\:\!\!|}_- \int_{W}\psi \, dm_{W} \, .
\end{split}
\]
Hence, for $\mbox{diam}^s(H)$ small enough,
\begin{equation}
\label{eq:lower H}
{|\:\!\!|\:\!\!|} \mathbbm{1}_{H^c}f{|\:\!\!|\:\!\!|}_-\geq \frac 12 {|\:\!\!|\:\!\!|} f{|\:\!\!|\:\!\!|}_-.
\end{equation}
Accordingly, taking the supremum over $W,\psi$ in \eqref{eq:appli1},
\[
{|\:\!\!|\:\!\!|} \mathbbm{1}_{H^c} f{|\:\!\!|\:\!\!|}_+\leq 2 P_0^{1-q} e^{a(2\delta)^\beta} A {|\:\!\!|\:\!\!|} \mathbbm{1}_{H^c} f{|\:\!\!|\:\!\!|}_-=:L' {|\:\!\!|\:\!\!|} \mathbbm{1}_{H^c} f{|\:\!\!|\:\!\!|}_-
\]
Next, to verify \eqref{eq:cone 3}, if $W\in\mathcal{W}^s_-(\delta)$, then estimating
as in \eqref{eq:appli1},
\begin{equation}\label{eq:3H}
\begin{split}
\int_W(\mathbbm{1}_{H^c} f )\psi dm_W&=\sum_{W'\in\mathcal{G}_0}\int_{W'}f \psi dm_{W'}\\
&\leq\sum_{W'\in\mathcal{G}_0}e^{a(2\delta)^\beta} |W'|^{q}A\delta^{1-q} {|\:\!\!|\:\!\!|} f{|\:\!\!|\:\!\!|}_- \fint_{W}\psi dm_{W}\\
&\leq P_0^{1-q}|W|^q e^{a(2\delta)^\beta} A\delta^{1-q} {|\:\!\!|\:\!\!|} f{|\:\!\!|\:\!\!|}_- \fint_{W}\psi dm_{W}\\
&\leq 2 P_0^{1-q}|W|^q e^{a(2\delta)^\beta} A\delta^{1-q} {|\:\!\!|\:\!\!|}\mathbbm{1}_{H^c} f{|\:\!\!|\:\!\!|}_- \fint_{W}\psi dm_{W}\\
&=: A'|W|^q \delta^{1-q} {|\:\!\!|\:\!\!|}\mathbbm{1}_{H^c} f{|\:\!\!|\:\!\!|}_- \fint_{W}\psi dm_{W} \, ,
\end{split}
\end{equation}
where we have used \eqref{eq:lower H} for the third inequaity.
We are left with the last cone condition, \eqref{eq:cone 5}. We take $W^1, W^2 \in \mathcal{W}^s_-(\delta)$ with $d_{\mathcal{W}^s}(W^1, W^2) \le \delta $,
and $\psi_i \in \mathcal{D}_{a,\alpha}(W_i)$ with $d_*(\psi_1, \psi_2) = 0$.
As in Section~\ref{subsec:contract c}, we may assume w.l.o.g. that $|W^2|\geq |W^1|$ and $\fint_{W^1} \psi_1 = 1$. First of all note that, by condition \eqref{eq:cone 3} and our estimate above,
\[
\frac{\int_{W^k} \mathbbm{1}_{H^c} f\psi_k}{\fint_{W^k}\psi_k}\leq A' |W^k|^q\delta^{1-q}{|\:\!\!|\:\!\!|} \mathbbm{1}_{H^c} f{|\:\!\!|\:\!\!|}_-
\leq \frac 12 d_{\mathcal{W}^s}(W^1,W^2)^\gamma \delta^{1-\gamma} c A' {|\:\!\!|\:\!\!|} \mathbbm{1}_{H^c} f{|\:\!\!|\:\!\!|}_-,
\]
for $k=1,2$,
provided $ |W^2|^q \leq \delta^{q- \gamma} \frac c2 d_{\mathcal{W}^s}(W^1,W^2)^{\gamma}$.
Accordingly, it suffices to consider the case $|W^2|^q \geq \delta^{q-\gamma} \frac c2 d_{\mathcal{W}^s}(W^1,W^2)^{\gamma}$.
It follows from \eqref{eq:W-difference} that
$|W^1|^q \geq \frac 12 \delta^{q-\gamma} \frac c2 d_{\mathcal{W}^s}(W^1,W^2)^\gamma$, recalling that $d_{\mathcal{W}^s}(W^1, W^2) \le \delta$ and \eqref{eq:q-gamma1}.
By (O2), we may decompose $W^k \cap H^c$ into at most $P_0$ `matched' pieces
$W_j^k$ such that $d_{\mathcal{W}^s}(W_j^1,W_j^2)\leq d_{\mathcal{W}^s}(W^1,W^2)$ and $I_{W^1_j} = I_{W^2_j}$,
and at most $P_0+2$ `unmatched' pieces ${\overline{W}}^k_i$, which satisfy,
\[
|{\overline{W}}^k_i| \leq C_t d_{\mathcal{W}^s}(W^1,W^2).
\]
Then, using condition \eqref{eq:cone 3} and noticing that $d_*(\psi_1|_{W^1_j},\psi_2|_{W^2_j})=0$,
\begin{equation}
\label{eq:indicator c}
\begin{split}
&\left|\frac{\int_{W^1} \mathbbm{1}_{H^c} f \psi_1}{\fint_{W^1}\psi_1} - \frac{\int_{W^2} \mathbbm{1}_{H^c} f \psi_2}{\fint_{W^2}\psi_2} \right|\leq
\sum_j\left|\frac{\int_{W^1_j} f \psi_1}{\fint_{W^1}\psi_1} - \frac{\int_{W^2_j} f \psi_2}{\fint_{W^2}\psi_2} \right|+
\sum_{i,k} |{\overline{W}}^k_i|^q \delta^{1-q} A {|\:\!\!|\:\!\!|} f {|\:\!\!|\:\!\!|}_- e^{a(2\delta)^\alpha} \\
&\leq \; \sum_j \frac{\fint_{W^1_j} \psi_1}{\fint_{W^1}\psi_1}d_{\mathcal{W}^s}(W^1, W^2)^\gamma \delta^{1-\gamma} c A {|\:\!\!|\:\!\!|} f {|\:\!\!|\:\!\!|}_-
+\sum_j \left|\frac{\int_{W^2_j} f \psi_2}{\fint_{W^2}\psi_2}\left[ 1- \frac{\fint_{W^1_j} \psi_1\fint_{W^2} \psi_2}{\fint_{W_j^2} \psi_2\fint_{W^1}\psi_1}\right]\right|\\
&\phantom{\leq}
\quad + 8 (P_0+2) C_t^q d_{\mathcal{W}^s}(W^1,W^2)^{\gamma} \delta^{1-\gamma} A {|\:\!\!|\:\!\!|} \mathbbm{1}_{H^c} f{|\:\!\!|\:\!\!|}_-,
\end{split}
\end{equation}
using \eqref{eq:lower H}. Next, since $I_{W^1_j} = I_{W^2_j}$, recalling
Remark~\ref{rem:change-int} and \eqref{eq:W-difference} we have
$\int_{W^1_j} \psi_1 = \int_{W^2_j} \psi_2$ and\footnote{Since $I_{W^1_j} = I_{W^2_j}$,
the term on the right side of \eqref{eq:W-difference} proportional to $C_s$ is absent in this case.} $||W^1_j| - |W^2_j|| \le |W^1_j| d_{\mathcal{W}^s}(W^1, W^2)$.
Then applying \eqref{eq:cone 3} and recalling $\fint_{W_1} \psi_1 = 1$,
\begin{equation}
\label{eq:indicator split}
\begin{split}
& \left|\frac{\int_{W^2_j} f \psi_2}{\fint_{W^2}\psi_2} \left[ 1- \frac{\fint_{W^1_j} \psi_1\fint_{W^2} \psi_2}{\fint_{W_j^2} \psi_2\fint_{W^1}\psi_1}\right]\right|
\le A {|\:\!\!|\:\!\!|} f {|\:\!\!|\:\!\!|}_- \frac{\fint_{W^2_j} \psi_2}{\fint_{W^2} \psi_2} |W^2_j|^q \delta^{1-q}
\left| 1 - \frac{|W^2_j|}{|W^1_j|} \fint_{W^2} \psi_2 \right| \\
& \; \; \le A {|\:\!\!|\:\!\!|} f {|\:\!\!|\:\!\!|}_- e^{a (2\delta)^\alpha} \left( |W^2_j|^q \delta^{1-q} \left|1- \frac{|W^2_j|}{|W^1_j|} \right|
+ \frac{|W^2_j|^q}{|W^2|^q} \left(\frac{\delta}{|W^2|} \right)^{1-q} \left| |W^2| - \int_{W^2} \psi_2 \right| \frac{|W^2_j|}{|W^1_j|} \right) \\
& \; \; \le A {|\:\!\!|\:\!\!|} f {|\:\!\!|\:\!\!|}_- 2 \left( |W^2_j|^q \delta^{1-q} d_{\mathcal{W}^s}(W^1, W^2)
+ 2 \frac{|W^2_j|^q}{|W^2|^q} \left(\frac{\delta}{|W^2|} \right)^{1-q} \left| |W^2| - \int_{W^2} \psi_2 \right| \right).
\end{split}
\end{equation}
Next, recalling $|W^2| \ge \delta^{1 - \frac{\gamma}{q} } [c/2]^{\frac 1q} d_{\mathcal{W}^s}(W^1, W^2)^{\frac{\gamma}{q}}$
and using \eqref{eq:new-difference} yields,
\[
\begin{split}
\left(\frac{\delta}{|W^2|} \right)^{1-q} \left| |W^2| - \int_{W^2} \psi_2 \right|
& \le 6 C_s [2/c]^{\frac{1}{q}(1-q)} { \delta^{\frac{\gamma}{q} - \gamma} } d_{\mathcal{W}^s}(W^1, W^2)^{1+\gamma - \frac{\gamma}{q}} \\
& \le 4^{-\frac{1}{q} 6 c \delta^{1-\gamma} } d_{\mathcal{W}^s}(W^1, W^2)^\gamma \, ,
\end{split}
\]
where we have again used \eqref{eq:q-gamma1} and $d_{\mathcal{W}^s}(W^1, W^2) \le \delta$.
Using this estimate and the fact that $q \le 1/2$ in \eqref{eq:indicator split} and summing over $j$ yields,
\[
\begin{split}
\sum_j \left|\frac{\int_{W^2_j} f \psi_2}{\fint_{W^2}\psi_2} \left[ 1- \frac{\fint_{W^1_j} \psi_1\fint_{W^2} \psi_2}{\fint_{W_j^2} \psi_2\fint_{W^1}\psi_1}\right]\right|
& \le 2 A \delta^{1-\gamma} {|\:\!\!|\:\!\!|} f {|\:\!\!|\:\!\!|}_- d_{\mathcal{W}^s}(W^1, W^2)^\gamma \sum_j \delta^{1-q}
|W^2_j|^q + \frac 34 c \frac{|W^2_j|^q}{|W^2|^q} \\
& \le 2 A \delta^{1-\gamma} {|\:\!\!|\:\!\!|} f {|\:\!\!|\:\!\!|}_- d_{\mathcal{W}^s}(W^1, W^2)^\gamma P_0^{1-q} \left( 2^q \delta + \tfrac 34 c \right) \, .
\end{split}
\]
Finally, using this estimate in \eqref{eq:indicator c} concludes the proof of the lemma,
\[
\begin{split}
\left|\frac{\int_{W^1} \mathbbm{1}_{H^c} f \psi_1}{\fint_{W^1}\psi_1} - \frac{\int_{W^2} \mathbbm{1}_{H^c} f \psi_2}{\fint_{W^2}\psi_2} \right|
& \leq
d_{\mathcal{W}^s}(W^1, W^2)^\gamma \delta^{1-\gamma} A 2 P_0^{1-q} {|\:\!\!|\:\!\!|} \mathbbm{1}_{H^c} f {|\:\!\!|\:\!\!|}_-
\left( P_0^q e^{a (2\delta)^\alpha} \; + \right. \\
& \quad \left. + \; 2 \left( 2^q \delta + \tfrac 34 c \right) + 4 (P_0 + 2) P_0^{q-1} C_t^q \right) \, ,
\end{split}
\]
where we have again used \eqref{eq:lower H}.
\end{proof}
Remark that, by Theorem \ref{thm:cone contract}, we know that there exists
$N_{\mathcal{F}}\in\mathbb{N}$, $N_{\mathcal{F}}\leq k_*n_*+C_{\star}\ln\delta^{-1}$ where $n_*$, defined in Lemma \ref{lem:proper cross}, and $k_*$ from Theorem~\ref{thm:cone contract}, are uniform in $\mathcal{F}(\tau_*, \mathcal{K}_*, E_*)$, while $C_\star$ depends only on $c,A,L$, such that
$\mathcal{L}_{N_{\mathcal{F}}} \mathcal{C}_{c,A,L}(\delta) \subset \mathcal{C}_{\chi c, \chi A, \chi L}(\delta)$, for all $\{ \iota_j \}_{j=1}^{N_\mathcal{F}} \subset \mathcal{I}(T_0, \kappa)$.
\begin{prop}\label{prop:case-a}
For $n_\star\geq J N_{\mathcal{F}}$, with J depending only on $c,A,L, P_0,C_t$, if assumptions (O1) and (O2) are satisfied and $\mbox{diam}^s(H) \le \delta \left[ \frac{1}{4P_0A} \right]^{1/q}$, then,
for all $n\geq n_\star$, and all $\{ \iota_j \}_{j=1}^n \subset \mathcal{I}(T_0, \kappa)$,
$[\mathcal{L}_n\mathbbm{1}_{H^c}]\mathcal{C}_{c,A,L}(\delta)\subset \mathcal{C}_{\chi c,\chi A,\chi L}(\delta)$,
where $\mathcal{C}_{c,A,L}(\delta)$ is given in Theorem~\ref{thm:cone contract}.
\end{prop}
\begin{proof}
Define $N_{\mathcal{F}} = N(\delta)^- + k_*n_*$, where $N(\delta)^-$, $k_*$ and $n_*$
are defined in Theorem~\ref{thm:cone contract}. Then for $n = m N_{\mathcal{F}}$,
we may apply both Lemma~\ref{lem:case-a} and Theorem~\ref{thm:cone contract} to obtain,
\[
[\mathcal{L}_n \mathbbm{1}_{H^c}] \mathcal{C}_{c,A,L}(\delta) \subset \mathcal{L}_{mN_{\mathcal{F}}} \mathcal{C}_{c',A',L'}(\delta)
\le \mathcal{C}_{\chi^m c', \chi^m A', \chi^m L'}(\delta) \, ,
\]
for as long as $\chi^m c' > c$, $\chi^m A' > A$ and $\chi^m L' > L$. Letting $m_1$ denote
the least $m$ such that $\chi^m c' < c$, $\chi^m A' < A$ and $\chi^m L' < L$, and setting
$n_\star = ( m_1+1)N_{\mathcal{F}}$ produces the required contraction.
\end{proof}
\begin{remark}
Taking $\kappa=0$ we can also consider the case of a single map, $T_{\iota_j} = T_0$ for each $j$.
Then once we know the transfer operator for the open system acts as a strict contraction on the cone, it is straight
forward to
recover the usual full set of results for open systems with exponential escape, including a unique escape
rate and limiting conditional invariant measure for all elements of the cone. See Theorem~\ref{thm:open}
for an example.
\end{remark}
\subsection{ Large holes}
\label{sec:large}
The above pertains to relatively small holes. For many applications large holes must be considered.
To do so requires either a much closer look at the combinatorics of the trajectories or requiring the holes to open at even longer time intervals than what was needed before.
We will pursue the second, much easier, option with the intent to show that large holes are not
out of reach. To work with large holes it is convenient to weaken hypothesis (O1):
\begin{itemize}
\item[(O$1'$)] (Complexity) There exists $P_0 >0$ such that any stable curve of
length at most $\delta_0$ can be cut into at most $P_0$ pieces by $\partial H$.
\end{itemize}
When iterating $T^{-1}_nW$ for $W \in \mathcal{W}^s$, we will need to distinguish between elements of
$\mathcal{G}_n(W)$ which intersect $H$ and those that do not. Recall that $\mathcal{G}_n(W)$ subdivides
long homogeneous connected components of $T^{-1}_nW$ into curves of length between
$\delta_0$ and $\delta_0/3$. We let $\mathcal{G}_n^H(W)$ denote the connected components
of $W_i \cap H^c$, for $W_i \in \mathcal{G}_n(W)$, where $H^c = M \setminus H$. Following the notation of
Section~\ref{sec:prop proof}, let $Lo_n^H(W; \delta)$ denote those elements of $\mathcal{G}_n^H(W)$
having length at least $\delta$ and let $Sh_n^H(W; \delta)$ denote those elements having length
at most $\delta$.
Without the small hole condition, hypotheses (O$1'$) and (O2) are insufficient to prove Lemma~\ref{lem:case-a};
however, one can recover the results of Proposition \ref{prop:case-a} and its consequences
provided one is willing to wait for a longer time.
To prove the following result, we recall again the definition of
admissible sequence from Section~\ref{sec:main}.
Let $N_\mathcal{F} := N(\delta)^- + k_*n_*$ from Theorem~\ref{thm:cone contract} as before.
We call a sequence $( \iota_j )_j$, $\iota_j \in \mathcal{I}(\tau_*, \mathcal{K}_*, E_*)$, {\em admissible} if for
all $k \ge 0$ and $j \in [kN_\mathcal{F}, (k+1)N_\mathcal{F}-1]$,
$T_{\iota_j} \in \mathcal{F}(T_{\iota_{kN_\mathcal{F}}}, \kappa)$.
\begin{lemma}\label{lem:H3}
If (O1$'$) and (O2) are satisfied and $(i_j)_j$ is an admissible sequence, then for each $\delta>0$ small enough (depending on $\mu_{\mbox{\tiny SRB}}(H)$) there exists $n_\delta\in\mathbb{N}$, $n_\delta\leq C\ln\delta^{-1}$ for some constant $C>0$, such that for all $W\in\mathcal{W}^s(\delta)$ and $n\geq n_\delta$,
\[
\sum_{W'\in Lo^H_n(W; \, \delta)} |W|^{-1}\int_{W'} J_{W'}T_n {\; \geq\;} \frac 12(1 - \mu_{\mbox{\tiny SRB}}(H)) \, .
\]
\end{lemma}
\begin{proof}
Arguing exactly as in Lemma \ref{lem:case-a} it follows that if (O1$'$) and (O2) are satisfied, then there exists $c'\geq c,A'\geq A, L'\geq L$ such that $\mathbbm{1}_{H^c}+ 1 \in \mathcal{C}_{c',A',L'}(\delta)$.
Then by equation~\eqref{eq:avg conv} of Theorem~\ref{thm:equi} applied to this larger cone,
\[
\left|\fint_W \mathcal{L}_n (\mathbbm{1}_{H^c}) - (1-\mu_{\mbox{\tiny SRB}}(H) ) \right|=\left|\fint_W \mathcal{L}_n (\mathbbm{1}_{H^c}+1) - 2
+ \mu_{\mbox{\tiny SRB}}(H)\right|\leq C_H\vartheta^n \, .
\]
On the other hand, recalling Lemma \ref{lem:full growth},
\[
\begin{split}
\left| \fint_W \mathcal{L}_n (\mathbbm{1}_{H^c})-\sum_{W'\in Lo^H_n(W; \, \delta)} |W|^{-1}\int_{W'} J_{W'}T_n \right|&\leq \sum_{W'\in Sh^H_n(W; \, \delta)} |W|^{-1}\int_{W'} J_{W'}T_n\\
&\leq P_0(\bar C_0\delta_0^{-1}\delta+C_0\theta_1^n),
\end{split}
\]
which implies the Lemma.
\end{proof}
We are now able to state the analogue of Proposition \ref{prop:case-a} without the small hole condition. Note, however, that now $n_\star$ has a worse dependence on $\delta$ that we refrain from making explicit.
\begin{prop}\label{prop:case-b}
Under assumptions (O1$'$) and (O2), for each $\delta>0$ small enough there exists $c,A,L>0$,
$\chi\in (0,1)$ and $n_\star\in\mathbb{N}$ such that, for all admissible sequences $(\iota_j)_j$ and for all $n\geq n_\star$,
$[\mathcal{L}_n\mathbbm{1}_{H^c}]\mathcal{C}_{c,A,L}(\delta)\subset \mathcal{C}_{\chi c,\chi A,\chi L}(\delta)$.
\end{prop}
Before proving Proposition \ref{prop:case-b}, we state an auxiliary lemma, similar to Lemma~\ref{lem:case-a}.
\begin{lemma}
\label{lem:large hole}
There exists $\bar n_\delta > 0$ such that for $n \ge \bar n_\delta$ and all admissible sequences $(\iota_j)_j$, we have
$[\mathcal{L}_n\mathbbm{1}_{H^c}] \mathcal{C}_{c,A,L}(\delta) \subset \mathcal{C}_{c', A', L'}(\delta)$, where
\[
c' = cP_0, \quad A' = A \frac{6}{1-\mu_{\mbox{\tiny SRB}}(H)}, \quad \mbox{and} \quad L' = L \frac{9}{1-\mu_{\mbox{\tiny SRB}}(H)} \, .
\]
\end{lemma}
\begin{proof}[Proof of Proposition~\ref{prop:case-b}]
Letting $n = m N_{\mathcal{F}} + n_\delta$, with $N_{\mathcal{F}} = N(\delta)^- + k_*n_*$ as before,
we may apply both Lemma~\ref{lem:large hole} and Theorem~\ref{thm:cone contract} to obtain,
\[
[\mathcal{L}_n \mathbbm{1}_{H^c}] \mathcal{C}_{c,A,L}(\delta) \subset \mathcal{L}_{mN_{\mathcal{F}}} \mathcal{C}_{c',A',L'}(\delta)
\le \mathcal{C}_{\chi^m c', \chi^m A', \chi^m L'}(\delta) \, ,
\]
for as long as $\chi^m c' > c$, $\chi^m A' > A$ and $\chi^m L' > L$. Letting $m_1$ denote
the least $m$ such that $\chi^m c' < c$, $\chi^m A' < A$ and $\chi^m L' < L$, and setting
$n_\star = ( m_1+1)N_{\mathcal{F}} + n_\delta$ produces the required contraction.
\end{proof}
\begin{proof}[Proof of Lemma~\ref{lem:large hole}]
Let $n\geq n_\delta$ (from Lemma \ref{lem:H3}) and $f \in \mathcal{C}_{c, A, L}(\delta)$. For each $W\in \mathcal{W}^s(\delta)$ and $\psi \in \mathcal{D}_{a,\beta}(W)$, we have
\begin{equation}
\label{eq:H split}
\int_W\psi \, \mathcal{L}_n ( \mathbbm{1}_{H^c} f )=\sum_{W_i\in Lo^H_{n}(W; \delta)}\int_{W_i} \widehat{T}_{n,i} \psi \, f + \sum_{W_i \in Sh^H_{n}(W; \delta)}\int_{W_i } \widehat{T}_{n,i} \psi \, f ,
\end{equation}
where we are using the notation of Section~\ref{sec:test} for the test functions.
Since any element of $\mathcal{G}_n(W)$ may produce up to $P_0$ elements of
$Sh^H_{n}(W; \delta)$ according to assumption (O$1'$), we estimate
\[
\begin{split}
\int_W \psi \, \mathcal{L}_n ( \mathbbm{1}_{H^c} f ) & \le \sum_{W_i \in Lo_n^H(W; \delta)} {|\:\!\!|\:\!\!|} f {|\:\!\!|\:\!\!|}_+ \int_{T_nW_i} \psi +
A P_0 {|\:\!\!|\:\!\!|} f {|\:\!\!|\:\!\!|}_- e^{a(2\delta)^\beta} \int_W \psi \; (\bar C_0 \delta \delta_0^{-1} + C_0 \theta_1^n) \\
& \le {|\:\!\!|\:\!\!|} f {|\:\!\!|\:\!\!|}_+ \int_W \psi \; \left( 1 + A P_0e^{a(2\delta)^\beta} (\bar C_0 \delta \delta_0^{-1}
+ C_0 \theta_1^n) \right) \, ,
\end{split}
\]
where we have used $|W| \ge \delta$ and cone condition \eqref{eq:cone 3}, as well as
Lemma~\ref{lem:full growth}(b) to sum over elements of $Sh^H_n(W; \delta)$.
Analogously, using Lemma \ref{lem:H3},
\[
\begin{split}
\int_W \psi \, \mathcal{L}_n ( \mathbbm{1}_{H^c} f ) & \ge \sum_{W_i \in Lo_n^H(W; \delta)} {|\:\!\!|\:\!\!|} f {|\:\!\!|\:\!\!|}_- \int_{T_nW_i} \psi -
A P_0 {|\:\!\!|\:\!\!|} f {|\:\!\!|\:\!\!|}_- e^{a(2\delta)^\beta} \int_W \psi \; (\bar C_0 \delta \delta_0^{-1}+ C_0 \theta_1^n) \\
& \ge {|\:\!\!|\:\!\!|} f {|\:\!\!|\:\!\!|}_- \int_W \psi \; \left( \frac {e^{-a(2\delta)^\beta}}2(1-\mu_{\mbox{\tiny SRB}}(H))- A P_0e^{a(2\delta)^\beta} (\bar C_0 \delta \delta_0^{-1} + C_0 \theta_1^n ) \right) \, .
\end{split}
\]
Let $n_2$ be such that $2AP_0C_0\theta_1^{n_2}\leq \frac {1}{24} (1- \mu_{\mbox{\tiny SRB}}(H))$, then for $n\geq n_2$ and $\delta$ small enough we have
\begin{equation}
\label{eq:lower big H}
{|\:\!\!|\:\!\!|} \mathcal{L}_n(\mathbbm{1}_{H^c} f) {|\:\!\!|\:\!\!|}_- \ge {|\:\!\!|\:\!\!|} f {|\:\!\!|\:\!\!|}_- \; \frac{1}{6}(1-\mu_{\mbox{\tiny SRB}}(H)).
\end{equation}
Accordingly, for $n \ge \max\{n_2, n_\delta \} =: \bar n_\delta$ and $\delta$ small enough, we obtain
\begin{equation}
\label{eq:L bound}
\frac{{|\:\!\!|\:\!\!|} \mathcal{L}_n ( \mathbbm{1}_{H^c} f ) {|\:\!\!|\:\!\!|}_+ }{{|\:\!\!|\:\!\!|} \mathcal{L}_n ( \mathbbm{1}_{H^c} f ) {|\:\!\!|\:\!\!|}_-}
\le \frac{ \frac 32 {|\:\!\!|\:\!\!|} f {|\:\!\!|\:\!\!|}_+}{{|\:\!\!|\:\!\!|} f {|\:\!\!|\:\!\!|}_- (\frac 16(1-\mu_{\mbox{\tiny SRB}}(H))} \le \frac{9L}{1-\mu_{\mbox{\tiny SRB}}(H)} =: L' \, .
\end{equation}
The contraction of $A$ follows step-by-step from our estimates in
Section~\ref{sec:contraction-A}. Taking $W \in \mathcal{W}^s_-(\delta)$ and grouping terms as in \eqref{eq:H split} we treat both
long and short pieces precisely as in Section~\ref{sec:contraction-A} with the additional
observation that each element of $\mathcal{G}_n(W)$ produces at most $P_0$ elements
of $Sh_n^H(W; \delta)$ by assumption (O$1'$). Thus \eqref{eq:A} becomes,
\begin{equation}
\label{eq:A H}
\begin{split}
& \frac{|\int_W \psi \, \mathcal{L}_n (\mathbbm{1}_{H^c}f) |}{\fint_W \psi}
\le A \delta^{1-q} |W|^q {|\:\!\!|\:\!\!|} f {|\:\!\!|\:\!\!|}_-
\left( 2 L A^{-1} + P_0 e^{a(2\delta)^\beta} ( \bar C_0 \delta_0^{-1} |W| + C_0 \theta_1^n )^{1-q} \right)
\\
& \quad \le A \delta^{1-q}|W|^q {|\:\!\!|\:\!\!|} \mathcal{L}_n(\mathbbm{1}_{H^c}f) {|\:\!\!|\:\!\!|}_- \frac{6}{1-\mu_{\mbox{\tiny SRB}}(H)} =: A' \delta^{1-q}|W|^q {|\:\!\!|\:\!\!|} \mathcal{L}_n(\mathbbm{1}_{H^c}f) {|\:\!\!|\:\!\!|}_-\, ,
\end{split}
\end{equation}
where we have applied \eqref{eq:lower big H} and assumed $n \ge \max\{n_2, n_\delta \}$.
Finally, we show how the parameter $c$ contracts from cone condition \eqref{eq:cone 5}.
Following Section~\ref{subsec:contract c}, we take $W^1, W^2 \in \mathcal{W}^s_-(\delta)$ with
$d_{\mathcal{W}^s}(W^1, W^2) \le \delta$, and $\psi_k \in \mathcal{D}_{a,\alpha}(W^k)$ with
$d_*(\psi_1, \psi_2) = 0$. As before, we assume w.l.o.g.
that $|W^2|\geq |W^1|$ and $\fint_{W^1} \psi_1 = 1$.
We begin by recording that, by \eqref{eq:A H},
\[
\frac{\int_{W^k} \psi_k \, \mathcal{L}_n(\mathbbm{1}_{H^c} f)}{\fint_{W^k}\psi_k}\leq A' |W^k|^q\delta^{1-q}{|\:\!\!|\:\!\!|} \mathcal{L}_n (\mathbbm{1}_{H^c} f) {|\:\!\!|\:\!\!|}_-\leq \frac 12 d_{\mathcal{W}^s}(W^1,W^2)^\gamma
\delta^{1-\gamma} c A' {|\:\!\!|\:\!\!|} \mathbbm{1}_{H^c} f{|\:\!\!|\:\!\!|}_-,
\]
for $k=1,2$, provided $|W^2|^q \leq \delta^{q-\gamma} \frac c2 d_{\mathcal{W}^s}(W^1,W^2)^{\gamma}$.
Accordingly, it suffices to consider the case $|W^2|^q \geq \delta^{\gamma - q} \frac c2
d_{\mathcal{W}^s}(W^1,W^2)^{\gamma}$.
It follows from \eqref{eq:W-difference} that
$|W^1|^q \geq \frac 12 \delta^{q-\gamma} \frac c2 d_{\mathcal{W}^s}(W^1,W^2)^\gamma$, recalling that $d_{\mathcal{W}^s}(W^1, W^2) \le \delta$ and \eqref{eq:q-gamma1}.
Next, following \eqref{eq:prepare-c}, we decompose elements of $\mathcal{G}_n^H(W^k)$ into
matched and unmatched pieces, as in \eqref{eq:unstable split}. We estimate the
unmatched pieces precisely as in \eqref{eq:V}, noting that by (O$1'$) and the transversality
condition $(O2)$, each previously unmatched element of
$\mathcal{G}_n(W^k)$ may be subdivided into at most $P_0$ additional unmatched pieces $V^k_j$,
while each matched element may produce up to $P_0$ additional unmatched pieces
each having length at most,
\[
|V^k_j| \le C_t C_5 n\Lambda^n d_{\mathcal{W}^s}(W^1, W^2) \, ,
\]
by Lemma~\ref{lem:compare}(a). Thus,
\begin{equation}
\label{eq:V H}
\sum_{j,k} \left| \int_{V^k_j} f \, \widehat{T}_{n,V^k_j}\psi_k \right| \le \frac{9 P_0}{1-\mu_{\mbox{\tiny SRB}}(H)} C_4 A L
\delta^{1-\gamma} d_{\mathcal{W}^s}(W^1, W^2)^\gamma {|\:\!\!|\:\!\!|} \mathcal{L}_n(\mathbbm{1}_{H^c} f) {|\:\!\!|\:\!\!|}_- \, ,
\end{equation}
where we have used \eqref{eq:lower big H} in \eqref{eq:tri} to estimate
\begin{equation}
\label{eq:convert}
{|\:\!\!|\:\!\!|} \mathcal{L}_n f {|\:\!\!|\:\!\!|}_- \le {|\:\!\!|\:\!\!|} \mathcal{L}_n f {|\:\!\!|\:\!\!|}_+ \le \tfrac 32 {|\:\!\!|\:\!\!|} f {|\:\!\!|\:\!\!|}_+ \le \tfrac 32 L {|\:\!\!|\:\!\!|} f {|\:\!\!|\:\!\!|}_- \le \tfrac{9 L}{1-\mu_{\mbox{\tiny SRB}}(H)} {|\:\!\!|\:\!\!|} \mathcal{L}_n(\mathbbm{1}_{H^c} f) {|\:\!\!|\:\!\!|}_- \, .
\end{equation}
The estimate on matched pieces proceeds precisely as in \eqref{eq:c-decomposition}, and with an
additional factor of $P_0$ in \eqref{eq:summingU2}, we arrive at \eqref{eq:use-in-4},
again applying \eqref{eq:lower big H},
\[
\begin{split}
& \sum_j \left|\int_{U^1_j} f \, \widehat{T}_{n,U^1_j} \psi_1 -\int_{U^2_j} f \, \widehat{T}_{n,U^2_j}\psi_2\right| \\
&\le
\tfrac{6P_0}{1-\mu_{\mbox{\tiny SRB}}(H)} 24 \bar C_0 C_s A \delta^{1-\gamma} d_{\mathcal{W}^s}(W^1, W^2)^\gamma {|\:\!\!|\:\!\!|} \mathcal{L}_n(\mathbbm{1}_{H^c} f) {|\:\!\!|\:\!\!|}_- \left( 2^q 40 C_5 \delta^{q-\gamma}
+ c C_5 \Lambda^{-n \gamma}
+ 2^q C_5 \Lambda^{-n} \delta \right) .
\end{split}
\]
Combining this estimate together with \eqref{eq:V H} in \eqref{eq:prepare-c} (with $A'$ in place of
$A$ in \eqref{eq:prepare-c}), and recalling \eqref{eq:unstable split}, yields by \eqref{eq:switch test},
\[
\left|\frac{\int_{W^1} \mathcal{L}_nf \, \psi_1}{\fint_{W^1}\psi_1} -\frac{\int_{W^2} \mathcal{L}_nf \, \psi_2}{\fint_{W^2}\psi_2}\right|
\le \frac{6 P_0 }{1-\mu(H)} c A \delta^{1-\gamma} d_{\mathcal{W}^s}(W^1, W^2)^\gamma {|\:\!\!|\:\!\!|} \mathcal{L}_n(\mathbbm{1}_{H^c} f) {|\:\!\!|\:\!\!|}_- \, ,
\]
where we have applied \eqref{eq:c cond} to simplify the expression. Setting $c' = P_0 c$ and
recalling the definition of $A'$ from \eqref{eq:A H} completes the proof of the lemma.
\end{proof}
\subsection{ Loss of memory for sequential open billiards}
\label{sequential}
We conclude the section by illustrating several physically relevant models to which our results apply.
Admittedly, we cannot treat the most general cases, yet we believe the following shows convincingly that the
techniques developed here can be the basis of a general theory.
Dispersing billiards with small holes have been studied in \cite{DWY, dem inf, dem bill},
and results obtained regarding the existence and uniqueness of limiting distributions
in the form of SRB-like conditionally invariant measures, and singular invariant measures
supported on the survivor set. In the present context, we are interested in generalizing these
results to the non-stationary setting.
Analogous results for sequences of expanding maps with holes have been proved in
\cite{ott1, ott2}.
For concreteness, we give two example of physical holes that satisfy our hypotheses,
following
\cite{DWY, dem bill}.
\smallskip
\noindent
{\em Holes of Type I.} Let $\mathbb{G} \subset \partial Q$ be an arc in the boundary of one of the
scatterers. Trajectories of the billiard flow are absorbed when they collide with
$\mathbb{G}$. This induces a hole $H$ in the phase space $M$ of the billiard map of the
form $(a,b) \times [-\pi/2, \pi/2]$. Note that $\partial H$ consists of two vertical lines, so that
$H$ satisfies assumption (O2) since the vertical direction is uniformly transverse to the
stable cone, as well as assumptions (O1) and (O1$'$) with $P_0 = 3$.
\smallskip
\noindent
{\em Holes of Type II.} Let $\mathbb{G} \subset Q$ be an open convex set bounded away from
$\partial Q$ and having a $C^3$ boundary. Such a hole induces a hole $H$ in $M$
via its `forward shadow.'
We define $H$ to be the set of $(r,\varphi) \in M$ whose backward trajectory under the billiard flow enters
$\mathbb{G}$ before it collides with $\partial Q$. Thus points in $M$ which are about to enter
$\mathbb{G}$ before their next collision under the forward billiard flow are considered still in the open
system, while those points in $M$ which would have passed through $\mathbb{G}$ on the way
to their current collision are considered to have been absorbed by the hole.
With this definition, the geometry of $H$ is simple to state: if we view $\mathbb{G}$ as an additional
scatterer in $Q$, then $H$ is simply the image of $\mathbb{G}$ under the billiard map. Thus
$H$ will have connected components on each scatterer that has a line of sight to $\mathbb{G}$,
and $\partial H$ will comprise curves of the form $\mathcal{S}_0 \cup T(\mathcal{S}_0)$, which
are positively sloped curves, all uniformly transverse to the stable cone. Thus holes
of Type II satisfy (O2) as well as (O1) and (O1$'$) with $P_0=3$. (See the discussion
in \cite[Section~2.2]{dem bill}.)
\medskip
Still other holes are presented in \cite{dem bill} such as side pockets, or holes that depend
on both position and angle, which satisfy (O1), (O1$'$) and (O2), but for the sake of brevity, we do not
repeat those definitions here.
As noted, both holes of Type I and Type II satisfy (O1) and (O1$'$) with $P_0=3$. Moreover, holes
of Type I satisfy (O2) with $C_t$ depending only on the maximum slope of curves in the
stable cone, which is uniform in the family $\mathcal{F}(\tau_*, \mathcal{K}_*, E_*)$ according to {\bf (H1)}:
This (negative) slope is bounded below by $-\mathcal{K}_{\max} - \frac{1}{\tau_{\min}}$, so choosing
$C_t \ge \mathcal{K}_* + \tau_*^{-1}$ suffices. Since $\partial H$ for holes of Type II have positive
slope, the same choice of $C_t$ will suffice for such holes to satisfy (O2).
Fix $\mathcal{F}(\tau_*, \mathcal{K}_*, E_*)$ and recall that $\mathcal{H}(P_0, C_t)$ denotes the collection of holes in
$M$ satisfying (O1) or (O$1'$) and (O1) with the given constants $P_0$ and $C_t$.
We define a non-stationary open billiard by choosing an admissible sequence $(\iota_j )_j$, $\iota_j \in \mathcal{I}(\tau_*, \mathcal{K}_*, E_*)$ and a sequence of holes $H_k \in \mathcal{H}(P_0, C_t)$, $k \in \mathbb{Z}^+$.
Let $T_{j,i} = T_{\iota_j} \circ \cdots \circ T_{\iota_i}$.
For each $k$, the open system relative to $H_k$ is defined by
$\mathring{T}_k : (T_{(k+1)n_\star-1, kn_\star})^{-1}(M \setminus H_k) \to M \setminus H_k$, where
\[
\mathring{T}_k(x) = T_{\iota_{(k+1)n_\star -1}} \circ \cdots \circ T_{\iota_{k n_\star}} (x) \, \mbox{ for } \, x \in (T_{(k+1)n_\star-1, kn_\star})^{-1}(M \setminus H_k) \, ,
\]
and $n_\star$ is as in Proposition \ref{prop:case-a} or Proposition \ref{prop:case-b} depending on which hypotheses
are satisfied.
To concatenate these open maps into a sequential system, define
\[
\mathring{T}_{j,i}(x) = \mathring{T}_j \circ \cdots \circ \mathring{T}_i(x) \, \mbox{ for } \, x \in \cap_{l=1}^j \mathring{T}_i^{-1} \circ \cdots \circ \mathring{T}_l^{-1}(M \setminus H_l) \, ,
\]
thus we allow escape once every $n_\star$ iterates along the admissible sequence.
The transfer operator for the sequential open system is defined by
\begin{equation}\label{eq:sequential}
\mathring{\cL}_{j,i} f = \mathcal{L}_{T_{\iota_{(j+1)n_\star-1}} \circ \cdots T_{\iota_{jn_\star}} } \mathbbm{1}_{H_j^c} \cdots \mathcal{L}_{T_{\iota_{(i+1)n_\star-1}} \circ \cdots T_{\iota_{in_\star}} }
\mathbbm{1}_{H_i^c} f \, .
\end{equation}
We will be interested in the evolution of probability densities under the sequential system,
given by $\frac{\mathring{\cL}_{n,k} f}{ \int_M \mathring{\cL}_{n,k} f \, d\mu_{\mbox{\tiny SRB}} }$.
Note that if $f\in \mathcal{C}_{c,A,L}(\delta)$ then $\int_M \mathring{\cL}_{n,k} f > 0$ for each $n$ (thus the normalization is well defined).
When $f \ge 0$, this normalization coincides with the $L^1(\mu_{\mbox{\tiny SRB}})$ norm; however, we use the integral
rather than the $L^1$ norm as the normalization since the integral is order preserving with respect to our cone, while the $L^1$ norm is not.
We conclude the section with a result regarding exponential loss of memory for the sequence of open billiards.
\begin{theorem}
\label{thm:sequential}
Fix $\tau_*, \mathcal{K}_*>0$ and $E_* <\infty$, and let $a,c, A, L, \delta$ and $\delta_0$ satisfy the
conditions of Theorem~\ref{thm:cone contract} and Lemma~\ref{lem:order}. Let $P_0, C_t > 0$.
There exist $C>0$ and $\vartheta <1$ such that for all
admissible sequences $(\iota_j)_j \subset \mathcal{I}(\tau_*, \mathcal{K}_*, E_*)$, and all sequences
$(H_i)_i \subset \mathcal{H}(P_0, C_t)$,
for all $\psi \in C^1(M)$, for all $f, g \in \mathcal{C}_{c,A,L}(\delta)$, all $n\ge1$ and all $1 \le k \le n$,
\[
\left| \int_M \frac{\mathring{\cL}_{n,k} f}{ \mu_{\mbox{\tiny SRB}}( \mathring{\cL}_{n,k} f ) } \, \psi \, d\mu_{\mbox{\tiny SRB}}
- \int_M \frac{\mathring{\cL}_{n,k} g}{ \mu_{\mbox{\tiny SRB}} (\mathring{\cL}_{n,k} g )} \, \psi \, d\mu_{\mbox{\tiny SRB}} \right|
\le C L \vartheta^{n-k} |\psi|_{C^1(M)} \, .
\]
\end{theorem}
\begin{proof}
Remark that the constants appearing in Propositions \ref{prop:case-a} and \ref{prop:case-b} are uniform,
depending only on $\mathcal{F}(\tau_*, \mathcal{K}_*, E_*)$, $P_0$ and $C_t$.
Hence, if $f, g \in\mathcal{C}_{c,A,L}(\delta)$, then for each $k\leq n\in\mathbb{N}$, $\mathring{\cL}_{n,k} f$, $\mathring{\cL}_{n,k} g \in \mathcal{C}_{c,A,L}(\delta)$.
Since $\int_M \frac{\mathring{\cL}_{n,k} f}{ \mu_{\mbox{\tiny SRB}}(\mathring{\cL}_{n,k} f ) } d\mu_{\mbox{\tiny SRB}} = \int_M \frac{\mathring{\cL}_{n,k} g}{\mu_{\mbox{\tiny SRB}}( \mathring{\cL}_{n,k} g ) } d\mu_{\mbox{\tiny SRB}} =1$, the theorem follows arguing exactly as in the proof of Theorem \ref{thm:memory}(b), using again the order preserving
semi-norm $\| \cdot \|_\psi$, as well as the fact that by Remark~\ref{rem:after order},
\[
\frac{ \| \mathring{\cL}_{n,k} f \|_\psi }{ \mu_{\mbox{\tiny SRB}}( \mathring{\cL}_{n,k} f) } \le \bar C |\psi|_{C^1} \frac{{|\:\!\!|\:\!\!|} \mathring{\cL}_{n,k} f {|\:\!\!|\:\!\!|}_+}{{|\:\!\!|\:\!\!|} \mathring{\cL}_{n,k} f {|\:\!\!|\:\!\!|}_-} \le \bar C L |\psi|_{C^1} \, .
\]
When invoking \eqref{eq:adapted}, it holds that $\rho_{\mathcal{C}}( \mathring{\cL}_{n,k} f/ \mu_{\mbox{\tiny SRB}}( \mathring{\cL}_{n,k} f ), \mathring{\cL}_{n,k} g/ \mu_{\mbox{\tiny SRB}}( \mathring{\cL}_{n,k} g) ) = \rho_{\mathcal{C}} (\mathring{\cL}_{n,k} f, \mathring{\cL}_{n,k} g)$ due to the projective nature of the metric.
\end{proof}
Note that, by changing variables,
$\int_M \mathring{\cL}_{n,k} f \, \psi \, d\mu_{\mbox{\tiny SRB}} = \int_{\mathring{M}_{n,k}} f \, \psi \circ \mathring{T}_{n,k} \, d\mu_{\mbox{\tiny SRB}}$,
where $\mathring{M}_{n,k} = \cap_{i=k}^n \mathring{T}_k^{-1} \circ \cdots \circ \mathring{T}_i^{-1}(M \setminus H_i)$.
Thus the conclusion of the theorem is equivalent to the expression,
\[
\left| \frac{\int_{\mathring{M}_{n,k}} f \, \psi \circ \mathring{T}_{n,k} \, d\mu_{\mbox{\tiny SRB}}}{\int_{\mathring{M}_{n,k}} f \, d\mu_{\mbox{\tiny SRB}}}
- \frac{\int_{\mathring{M}_{n,k}} g \, \psi \circ \mathring{T}_{n,k} \, d\mu_{\mbox{\tiny SRB}}}{\int_{\mathring{M}_{n,k}} g \, d\mu_{\mbox{\tiny SRB}}} \right|
\le C L \vartheta^{n-k} |\psi|_{C^1(M)} \, .
\]
Next we show that sequential systems with holes allow us to begin investigating some physical problems that have attracted much attention: chaotic scattering and random Lorentz gasses.
\subsection{Chaotic scattering (boxed)}
\label{sec:scattering}
Consider a collection of strictly convex pairwise disjoint obstacles $\{B_i\}$ in $\mathbb{R}^2$ {\em for which the non-eclipsing condition may fail}.\footnote{ Remember that the {\em non-eclipsing condition} is the requirement that the convex hull of any two obstacles does not intersect any other obstacle.} Assume that there exists a closed rectangular box $R=[a,b]\times[c,d]$ such that if an obstacle does not intersect its boundary, then it is contained in the box. In addition, if an obstacle intersects the boundary of $R$, then it is symmetrical with respect to a reflection across all the linear pieces of the boundary which the obstacle intersects (see Figure 1 for a picture). Finally, we will assume a finite horizon condition on the cover $\widetilde Q$ defined after Remark~\ref{rem:whyB}.
\begin{remark}
The restriction regarding symmetrical reflections on the configuration of obstacles is necessary only because we did not develop the theory in the case of billiards in a polygonal box (see Remark \ref{rem:whyB} and the following text to see why this is relevant). Such an extension is not particularly difficult and should eventually be done. Other extensions that should be within reach of our technology are more general types of holes and billiards with corner points. Here, however, we are interested in presenting the basic ideas; addressing all possible situations would make our message harder to understand.
\end{remark}
\begin{figure}
\begin{centering}
\begin{tikzpicture}[scale=0.6]
\fill[gray!60, rotate=45] (6,3) ellipse (1 and 1.8);
\fill[gray!60, rotate=-45] (1,7) ellipse (1 and 2);
\fill[gray!60] (8,3) circle (1);
\fill[gray!60] (3,8) ellipse (1 and .5);
\fill[gray!60] (5,0) ellipse (.7 and 2);
\fill[gray!60] (2,4) ellipse (.5 and .3);
\fill[gray!60] (3,1.5) ellipse (.3 and .5);
\fill[gray!60] (0,0) circle (1.7);
\draw[dashed] (0,0)--(8,0);
\draw [dashed](0,0)--(0,8);
\draw[dashed](0,8)--(8,8);
\draw[dashed](8,8)--(8,0);
\draw (-8,2)--(-8,6);
\draw[->](-8,2)--(-6,2);
\draw[->](-8,3)--(-6,3);
\draw[->](-8,4)--(-6,4);
\draw[->](-8,5)--(-6,5);
\draw[->](-8,6)--(-6,6);
\node at (-7,1) {{\tiny Incoming particle beam}};
\end{tikzpicture}
\caption{Obstacle configuration for which the non-eclipse condition fails and the box $R$ (dashed line). }
\end{centering}
\end{figure}
\begin{lemma}\label{lem:escape}
If a particle exits $R$ at time $t_0\in\mathbb{R}$, then, in the time interval $(t_0,\infty)$, it will experience only a finite number of collisions and it will never enter $R$ again.
\end{lemma}
\begin{proof}
Recall that $R=[a,b]\times[c,d]$.
Of course, the lemma is trivially true if, after exiting $R$, the particle has no collisions. Let us immagine that the particle, after exiting from the vertical side $(b,c)-(b,d)$, collides instead with the obstacle $B_i$ at the point $p=(p_1,p_2)$. Note that $B_i$ must then intersect the same boundary, otherwise it would be situated to the left of the line $x=b$ and the particle could not collide since necessarily $p_1>b$. Our hypothesis that $B_i$ be symmetric with respect to reflection across $x=b$ implies that also $( 2b - p_1, p_2 )\in\partial B_i$. Thus, by the convexity of $B_i$, the horizontal segment joining $p$ and $(2b - p_1, p_2)$ is contained in $B_i$. This implies that, calling $\eta=(\eta_1,\eta_2)$ the normal to $\partial B_i$ in $p$, it must be $\eta_1\geq 0$. In addition, if $v=(v_1,v_2)$ denotes the particle's velocity just before collision, it must be that $v_1>0$ since the particle has crosses a vertical line to exit $B$. Finally, $\langle v,\eta\rangle\leq 0$, otherwise the particle would not collide with $B_i$. But since the velocity after collision is given by $v^+=v-\langle v,\eta\rangle\eta$, it follows $v^+_1=v_1-\langle v,\eta\rangle\eta_1\geq v_1$. That is, the particle cannot come back to the box $B$. Since all the obstacles are contained in a larger box $B_1$ and since there is a minimal distance between obstacles, the above also implies that the particle can have only finitely many collisions in the future. The other cases can be treated exactly in the same manner.
\end{proof}
\begin{remark}\label{rem:whyB}
We want to consider a scattering problem: the particles enter the box coming from far away and with random position and/or velocity, interact and, eventually, leave the box. The basic question is how long they stay in the box or, better, what is the probability that they stay in the box longer than some time $t$.
This is nothing other than an open billiard with holes. Unfortunately, the holes are large and our current theory allows us to deal with large holes only if enough hyperbolicity is present. To extend the result to systems with small hyperbolicity is a very important (and hard) problem as one needs to understand the combinatorics of the trajectories for long times.
An alternative is to study the scattering problem under the non-eclipsing condition, as done in
\cite{Mo91, LM96, Mo07}. Such an assumption avoids the technicalities associated with billiards
and results in Axiom A dynamics with a natural finite Markov partition for the collision map. Yet
the condition is artificial once there are more than 2 scatterers, hence the importance of developing an alternative approach.
\end{remark}
Given the above remark we modify the system in order to have the needed hyperbolicity. This is not completely satisfactory, yet it shows that our machinery can deal with large holes and illustrates exactly what further work is necessary to address the general case.
{\em Fixing $N$ sufficiently large, we suppose that when a particle enters the box, the boundaries of the box become reflecting and are transparent again only between the collisions $kN$ and $kN+1$, $k\in\mathbb{N}$, counting only collisions with the convex obstacles.}
More precisely, consider the billiard in $R$ with elastic reflection at $\partial R$. We call such a billiard table $Q$. Let $M= \big( \cup_i\partial B_i \cap R \big) \times [-\frac \pi 2,\frac \pi 2]$ be the Poincar\'e section,\footnote{ Recall that $\varphi \in [-\frac \pi 2,\frac \pi 2]$ is the angle made
by the post-collision velocity vector and the outward pointing normal to the boundary.} and consider the Poincar\'e map $T:M\to M$ describing the dynamics from one collision with a convex body to the next. Unfortunately, this is not a type of billiard that fits our assumptions since the table has corner points.
Yet, when the particle collides with $\partial R$ we can reflect the box and imagine that the particle continues in a straight line. Note that, by our hypothesis, the image of the obstacles that intersect the boundary are the obstacles themselves; this is the reason why we restrict the obstacle configuration. We can then reflect the box three times, say across its right and top sides and then once more to make a full rectangle with twice the width and height of $R$, and identify the opposite sides of this larger rectangle. In this way we obtain a torus $\mathbb{T}^2$ containing pairwise disjoint convex obstacles. Such a torus is covered by four copies of $R$, let us call them $\{R_i\}_{i=1}^4$. We call such a billiard
$\widetilde Q$, and we consider the Poincar\'e map $\widetilde T$ which maps from one collision with a convex body to the next, and denote its phase space by $\widetilde M = \cup_{i=1}^4 \widetilde{M}_i$.
{\em Our final assumption on the obstacle configuration is that $\widetilde Q$ is a Sinai billiard with finite horizon. }\\
Hence $\widetilde T : \widetilde M \circlearrowleft$ falls within the scope of our theory. By construction there is a map $\pi:\widetilde M \to M$ which sends the motion on the torus to the motion in the box. Indeed, if $\tilde x\in\widetilde M$ and $x=\pi(\tilde x)$, then $T^n(x)=\pi(\widetilde T^n(\tilde x))$, for all $n\in\mathbb{N}$.
We then consider the maps $\tilde S=\widetilde T^N$ and $S=T^N$, again $\pi(\widetilde S(\tilde x))=S(\pi(\tilde x))$.
Define also the projections $\tilde \pi_1 : \widetilde M \to \widetilde Q$ and $\pi_1 : M \to Q$, which map
a point in the Poincar\'e section to its position on the billiard table.
For $\tilde x \in \widetilde M$, let us call $\widetilde O(\tilde x)$ the straight trajectory in $\mathbb{T}^2$ between $\tilde \pi_1(\tilde x)$ and $\tilde \pi_1(\widetilde T(\tilde x))$, and setting $x = \pi(\tilde x)$, $O(x)$ the trajectory between $\pi_1(x)$ and $\pi_1(T((x))$. Note that the latter trajectory can consist of several straight segments joined at the boundary of $R$, where a reflection takes place. By construction, if $\widetilde O(\tilde x)$ intersects $m$ of the sets $\partial R_i$, then the trajectory
$O(x)$ experiences $m$ reflections with $\partial R$. Accordingly, we introduce, in our billiard system $(\widetilde M,\widetilde S)$, the following holes : $\widetilde H=\widetilde T \{\tilde x\in\widetilde M\;:\; \widetilde O(\tilde x) \cap(\cup_i \partial R_i)\neq \emptyset\}$ and set $H=\pi(\widetilde H)$.
The above makes precise the previous informal statement: the system $(M,S)$ with hole $H$, describes the dynamics of the billiard $(M,T)$ in which the particle can exit $R$ only at the times $kN$, $k\in\mathbb{Z}$.
The transfer operator associated with the open system $(M, S; H)$
is $\mathbbm{1}_{H^c} \mathcal{L}_S \mathbbm{1}_{H^c}$, yet since $(\mathbbm{1}_{H^c} \mathcal{L}_S \mathbbm{1}_{H^c})^n = \mathbbm{1}_{H^c} (\mathcal{L}_S \mathbbm{1}_{H^c})^n$, it is
equivalent to study the asymptotic properties of $\mathring{\cL}_S :=\mathcal{L}_S \mathbbm{1}_{H^c}$.
For a function $f : M \to \mathbb{C}$, we define its lift $\tilde{f} : \widetilde{M} \to \mathbb{C}$ by $\tilde{f} = f \circ \pi$. The
pointwise identity then follows,
\begin{equation}
\label{eq:ptwise}
\mathring{\cL}_{\tilde{S}} \tilde{f} := \mathcal{L}_{\tilde{S}} ( \mathbbm{1}_{\widetilde{H}^c} \tilde{f}) = \mathcal{L}_{\tilde{S}} ((\mathbbm{1}_{H^c} f) \circ \pi) = (\mathring{\cL}_S f) \circ \pi \, .
\end{equation}
While $\widetilde H$ is not exactly a hole of Type II, its boundary nevertheless comprises increasing curves since it is a forward image under the flow of a wave front with zero curvature (a segment of $\partial R_i$).
Hence condition (O$1'$) of Section~\ref{sec:large}
holds with $P_0 =3$
and condition (O2) holds with $C_t$ depending only on the uniform angle between stable and
unstable curves in $\widetilde M$.
Thus Proposition~\ref{prop:case-b} applies to $\mathring{\cL}_{\tilde{S}}$ with $n_\star$ depending on
$C_t$ and $P_0 = 3$.
In fact, our next result shows that also $\mathring{\cL}_S$ contracts $\mathcal{C}_{c, A, L}(\delta)$ on $M$.
\begin{prop}
\label{prop:project}
Let $n_\star \in \mathbb{N}$ be from Proposition~\ref{prop:case-b} corresponding to $P_0=3$ and $C_t>0$.
Then for each small enough $\delta>0$, there exist $c, A, L >0$, $\chi \in (0,1)$ such that
choosing $N \ge n_\star$, $\mathring{\cL}_S(\mathcal{C}_{c,A, L}(\delta)) \subset \mathcal{C}_{\chi c, \chi A, \chi L}(\delta)$, where
$S = T^N$.
\end{prop}
\begin{proof}
As already noted above, Proposition~\ref{prop:case-b} implies the existence of $\delta, c, A, L$ and $\chi$ such that
$\mathring{\cL}_{\tilde{S}}(\widetilde{\cC}_{c,A, L}(\delta)) \subset \widetilde{\cC}_{\chi c, \chi A, \chi L}(\delta)$ if we choose $N \ge n_\star$. Note that
the constant $C_t$ is the same on $\widetilde{M}$ and $M$. In fact the same choice of parameters for the
cone works for $\mathring{\cL}_S$.
For any stable curve $W$, $\pi^{-1}W = \cup_{i=1}^4 \widetilde{W}_i$ where each $\widetilde{W}_i$ is a stable curve satisfying
$\pi(\widetilde{W}_i) = W$. Since $\pi$ is invertible on each $\widetilde{M}_i$, we may define the restriction
$\pi_i = \pi|_{\widetilde{M}_i}$ such that $\pi_i^{-1}(W) = \widetilde{W}_i$. Conversely, the projection of any stable curve $\widetilde{W}$
in $\widetilde{M}$ is also a stable curve in $M$.
Since each $\pi_i$ is an isometry, and recalling \eqref{eq:ptwise}, for any stable curve $W \subset M$, each $f \in \mathcal{C}_{c,A,L}(\delta)$, and all $n \ge 0$,
\[
\int_{\widetilde{W}_i} \psi \circ \pi \, \mathring{\cL}_{\tilde{S}}^n \tilde{f} \, dm_{\widetilde{W}} = \int_W \psi \, \mathring{\cL}_S^n f \, dm_W, \quad \forall \; \psi \in C^0(\widetilde{W}),
\]
where $\tilde{f} = f \circ \pi$. Moreover, if $\psi \in \mathcal{D}_{a, \beta}(W)$, then $\psi \circ \pi \in \mathcal{D}_{a, \beta}(\widetilde{W}_i)$,
for each $i = 1, \ldots, 4$. This implies in particular that
${|\:\!\!|\:\!\!|} \mathring{\cL}_S^n f {|\:\!\!|\:\!\!|}_{\pm} = {|\:\!\!|\:\!\!|} \mathring{\cL}_{\tilde{S}}^n \tilde{f} {|\:\!\!|\:\!\!|}_{\pm}$
for all $n \ge 0$, and that $f \in \mathcal{C}_{c,A,L}(\delta)$ if and only if $\tilde{f} = f \circ \pi \in \widetilde{\cC}_{c,A,L}(\delta)$.
Consequently, $\mathring{\cL}_S f \in \mathcal{C}_{\chi c, \chi A, \chi L}(\delta)$ if and only if $\mathring{\cL}_{\tilde{S}} \tilde{f} \in \widetilde{\cC}_{\chi c, \chi A, \chi L}(\delta)$,
which proves the proposition.
\end{proof}
In contrast to the sequential systems studied in Section~\ref{sequential}, the open billiard in this section corresponds to
a fixed billiard map $T$ (and its lift $\widetilde T$). Thus we can expect the (normalized) iterates of $\mathring{\cL}_{S}$ to converge
to a type of equilibrium for the open system. Such an equilibrium is termed a limiting or physical conditionally invariant measure in the
literature, and often corresponds to a maximal eigenvalue for $\mathring{\cL}_{S}$ on a suitable function space.
Unfortunately, conditionally
invariant measures for open ergodic invertible systems are necessarily singular with respect to the invariant measure and
so will not be contained in our cone $\mathcal{C}_{c, A, L}(\delta)$. However, we will show that for our open billiard,
the limiting conditional invariant measure is contained in the completion of $\mathcal{C}_{c,A,L}(\delta)$ with respect
to the following norm.
\begin{defin}
Let $\mathbb{V} = \textrm{span} \big( \mathcal{C}_{c,A,L}(\delta) \big)$ For all $f \in \mathbb{V}$ we define
\[
\| f \|_\star = \inf \{ \lambda \ge 0 : - \lambda \preceq f \preceq \lambda \} \, .
\]
\end{defin}
\begin{lemma}
\label{lem:star}
The function $\|\cdot\|_\star$ has the following properties:
\begin{itemize}
\item[a)] The function $\| \cdot \|_\star$ is an order-preserving norm, that is: $- g \preceq f \preceq g$ implies
$\| f \|_\star \le \| g \|_\star$.
\item[b)] There exists $C>0$ such that for all $f \in \mathcal{C}_{c,A,L}(\delta)$ and $\psi \in C^1(M)$,
\[
\left| \int_{M} f \, \psi \, d\mu_{\mbox{\tiny SRB}} \right| \le C {|\:\!\!|\:\!\!|} f {|\:\!\!|\:\!\!|}_+ | \psi |_{C^1(M)} \le C \| f \|_{\star} | \psi |_{C^1(M)} \, .
\]
\end{itemize}
\end{lemma}
\begin{proof}
In this proof, for brevity we write $\mathcal{C}$ in place of $\mathcal{C}_{c,A,L}(\delta)$.
\smallskip
\noindent
a) First note that $\| f \|_\star < \infty$ for any $f \in \mathbb{V}$ by the proof of Proposition~\ref{prop:diameter}
since there for any $f \in \mathcal{C}$, we find $\lambda, \mu >0$ such that
$f - \lambda$ and $\mu - f$ belong to $\mathcal{C}$.
Next, if $\| f \|_\star = 0$, then there exists a sequence $\lambda_n \to 0$ such that $- \lambda_n \preceq f \preceq \lambda_n$, and so $\lambda_n + f, \lambda_n - f \in \mathcal{C})$ for each $n$. Since $\mathcal{C}$ is closed (see footnote~\ref{foo:close}), this yields
$f, -f \in \mathcal{C} \cup \{ 0 \}$ and so $f = 0$ since $\mathcal{C} \cap - \mathcal{C} = \emptyset$ by construction.
Since $f \preceq g$ is equivalent to $\nu f \preceq \nu g$ for $\nu\in\mathbb{R}_+$, it follows immediately that $\|\nu f\|_\star=\nu\| f \|_\star$.
To prove the triangle inequality, let $f, g\in\mathbb{V}$. For each $\varepsilon>0$, there exists
$a,b$, $a\leq \varepsilon+\| f\|_\star$, $b\leq \varepsilon+\|g\|_\star$ such that
$-a \preceq f\preceq a$ and $-b \preceq g\preceq b$. Then
\[
-(\| f\|_\star+\| g\|_\star+2\varepsilon) \preceq -(a+b)\preceq f+g\preceq a+b\leq \| f\|_\star+\| g\|_\star+2\varepsilon \, ,
\]
implies the triangle inequality by the arbitrariness of $\varepsilon$. We have thus proven that $\|\cdot\|_\star$ is a norm.
Next, suppose that $-g\preceq f\preceq g$ and let $b$ be as above. Then
\[
-\| g\|_\star - \varepsilon \preceq -b \preceq -g\preceq f\preceq g\preceq b \preceq \| g\|_\star + \varepsilon
\]
which implies $\| f\|_\star\leq \| g\|_\star$, again by the arbitrariness of $\varepsilon$. Hence, the norm is order preserving.
\smallskip
\noindent
b) The first inequality is contained in Remark~\ref{rem:after order}. For the second inequality, we will prove
that
\begin{equation}
\label{eq:major}
{|\:\!\!|\:\!\!|} f {|\:\!\!|\:\!\!|}_+ \le \| f \|_\star \qquad \mbox{for all $f \in \mathcal{C}$}.
\end{equation}
To see this, note that if $-\lambda \preceq f \preceq \lambda$, then ${|\:\!\!|\:\!\!|} \lambda - f {|\:\!\!|\:\!\!|}_- \ge 0$ by
Remark~\ref{rem:A-L}. Thus for any $\widetilde{W} \in \widetilde \mathcal{W}^s$ and $\psi \in \mathcal{D}_{a,\beta}(\widetilde{W})$,
\[
0 \le \frac{\int_{W} (\lambda - f) \, \psi}{\int_{W} \psi } \implies \frac{\int_{W} f \, \psi}{\int_{W} \psi} \le \lambda \, ,
\]
and taking suprema over $W$ and $\psi$ yields ${|\:\!\!|\:\!\!|} f {|\:\!\!|\:\!\!|}_+ \le \lambda$, which implies \eqref{eq:major}.
\end{proof}
We now define $\mathcal{C}_\star$ to be the completion of $\mathcal{C}_{c, A, L}(\delta)$ in the $\| \cdot \|_\star$ norm.
We remark that by Lemma~\ref{lem:star}(b), $\mathcal{C}_\star$ embeds naturally into $(C^1(M))'$, where $(C^1(M))'$ is the closure of $C^0(M)$
with respect to the norm
$\| f \|_{-1} = \sup_{| \psi |_{C^1}\leq1} \int_{M} f \psi \, d\mu_{\mbox{\tiny SRB}}$.
We shall show that the conditionally invariant measure for the open system $(M, T; H)$ belongs to $\mathcal{C}_\star$.
\begin{theorem}
\label{thm:open}
Let $(M, S; H)$ be as defined above, where $S = T^N$. If $N \ge n_\star$, where $n_\star$ is from Proposition~\ref{prop:case-b}, then:
\begin{itemize}
\item[a)] $\displaystyle h := \lim_{n \to \infty} \frac{\mathring{\cL}_S^n 1}{\mu_{\mbox{\tiny SRB}}( \mathring{\cL}_S^n 1 ) }$ is an element of $\mathcal{C}_\star$.
Moreover, $h$ is a nonnegative probability measure satisfying $\mathring{\cL}_S h = \nu h$ for some $\nu \in (0,1)$
such that
$$\log \nu = \lim_{n \to \infty} \frac 1n \log \mu_{\mbox{\tiny SRB}}(\cap_{i=0}^n S^{-i}(M \setminus H)) \, ,$$
i.e. $- \log \nu$ is the escape rate of the open system.
\item[b)] There exists $C>0$ and $\vartheta \in (0,1)$ such that for all $f \in \mathcal{C}_{c,A,L}(\delta)$ and $n \ge 0$,
\[
\left\| \frac{\mathring{\cL}_S^n f}{ \mu_{\mbox{\tiny SRB}}(\mathring{\cL}_S^n f)} - h \right\|_\star \le C \vartheta^n\, .
\]
In addition, there exists a linear functional $\ell : \mathcal{C}_{c, A, L}(\delta) \to \mathbb{R}$
such that for all $f \in \mathcal{C}_{c,A,L}(\delta)$, $\ell(f) > 0$ and
\[
\| \nu^{-n}\mathring{\cL}^n_{S} f- \ell( f) h \|_\star \leq C \vartheta^n \ell( f ) \| h \|_\star.
\]
The constant $C$ depends on $\mathcal{C}_{c,A,L}(\delta)$, but not on $f$.
\end{itemize}
\end{theorem}
\begin{remark}
(a) The conclusions of Theorem~\ref{thm:open} apply equally well to the open system $(\widetilde{M}, \tilde{S}; \widetilde{H})$.
\smallskip
\noindent
(b) By Lemma~\ref{lem:star}(b), the convergence in the $\| \cdot \|_\star$ norm given by Theorem~\ref{thm:open}(b)
implies convergence when integrated against smooth functions $\psi \in C^1(M)$. As usual, by standard approximation arguments, the same holds for H\"older functions.
\smallskip
\noindent
(c) Also by Lemma~\ref{lem:star}(b), the above convergence in $\| \cdot \|_\star$ implies leafwise convergence as well. First note that
for $W \in \mathcal{W}^s(\delta)$, each $f \in \mathcal{C}_{c,A,L}(\delta)$ induces a leafwise distribution on $W$ defined by
$f_W(\psi) = \int_W f \, \psi \, dm_W$, for $\psi \in \mathcal{D}_{a,\beta}(W)$. This extends by density to $f \in \mathcal{C}_\star$.
Since $h \in \mathcal{C}_\star$ by Theorem~\ref{thm:open}(a), let $h_W$ denote the leafwise measure induced by $h$ on $W \in \mathcal{W}^s(\delta)$.
Then by Lemma~\ref{lem:star}(b) and Theorem~\ref{thm:open}(b), there exists $C>0$ such that for all $n \ge 0$,
\[
\left| \frac{\int_W \mathring{\cL}_S^n f \, \psi \, dm_W}{\mu_{\mbox{\tiny SRB}}(\mathring{\cL}_S^n f)} - h_W(\psi) \right| \le C \delta^{-1} \vartheta^n \, ,
\quad \forall f \in \mathcal{C}_{c,A,L}(\delta) , \forall \psi \in C^\beta(W) \, ,
\]
and also,
\[
\left| \nu^{-n} \int_W \mathring{\cL}_S^n f \, \psi \, dm_W - \ell(f) h_W(\psi) \right| \le C \delta^{-1} \vartheta^n \ell(f) \, .
\]
In particular, the escape rate with respect to each $W \in \mathcal{W}^s(\delta)$ equals the escape rate with respect to
$\mu_{\mbox{\tiny SRB}}$.
\end{remark}
\begin{proof}[Proof of Theorem~\ref{thm:open}]
We argue as in the proof of Theorem~\ref{thm:memory}. Recalling that $\| \cdot \|_\star$ is an order-preserving
norm, we can apply \cite[Lemma 2.2]{LSV98}, taking the homogeneous function $\rho$ to also be $\| \cdot \|_\star$
and obtain that, as in \eqref{eq:adapted}, for all $f,g\in\mathcal{C}_{c,A,L}(\delta)$,
\begin{equation}
\label{eq:cauchy}
\left\| \frac{\mathring{\cL}^n_{S} f}{\| \mathring{\cL}^n_{S} f \|_\star}-\frac{\mathring{\cL}^n_{S} g}{\| \mathring{\cL}^n_{S} g\|_\star} \right\|_\star \leq C\vartheta^n \, ,
\end{equation}
since $\left\| \frac{\mathring{\cL}^n_{S} f}{ \| \mathring{\cL}^n_{S} f \|_\star } \right\|_* = 1$ and similarly for $g$.
This implies that $\left( \frac{\mathring{\cL}^n_{S} f}{\| \mathring{\cL}^n_{S} f \|_\star} \right)_{n \ge 0}$ is a Cauchy sequence in the $\| \cdot\|_\star$ norm,
and in addition, the limit is independent of $f$.
Hence, defining $h_0 = \lim_{n \to \infty} \frac{\mathring{\cL}_{S}^n 1}{\| \mathring{\cL}_{S}^n 1\|_\star}$, we have
$h_0 \in \mathcal{C}_\star$ with $\| h_0 \|_\star = 1$ such that\footnote{ Note that $\mathring{\cL}_{S}$ extends naturally to $(\mathcal{C}^1(M))'$
and therefore to $\mathcal{C}_\star$.} for all $\psi \in C^1(M)$,
\[
\int_{M} \mathring{\cL}_{S} h_0 \psi =\lim_{n\to\infty}\frac1{\| \mathring{\cL}^n_{S} 1\|_\star}\int_{\widetilde{M}} \mathring{\cL}^{n+1}_{S} 1\psi=\lim_{n\to\infty}\frac{\| \mathring{\cL}^{n+1}_{S} 1\|_\star}{\| \mathring{\cL}^n_{S}1\|_\star}\int_{M} h_0 \psi
=\| \mathring{\cL}_{S} h_0\|_\star \int_{M} h_0\psi=:\nu \int_{M} h_0\psi \, ,
\]
where all integrals are taken with respect to $\mu_{\mbox{\tiny SRB}}$. Thus, $\mathring{\cL}_{S} h_0=\nu h_0$.
Moreover, the definition of $h_0$ implies that,
\begin{equation}
\label{eq:h0}
| h_0(\psi) | \le |\psi|_{C^0} \lim_{n \to \infty} \frac{ \mu_{\mbox{\tiny SRB}}( \mathring{\cL}_{S}^n 1 )}{\| \mathring{\cL}_{S}^n 1\|_\star} = | \psi|_{C^0} h_0(1)
\, , \quad \forall \, \psi \in C^1(M) \, ,
\end{equation}
thus $h_0$ is a measure. In addition, by the positivity of $\mathring{\cL}_{S}$, $h_0$ is a nonnegative measure and
since $\| h_0 \|_\star = 1$, it must be that $h_0(1) \neq 0$. Thus we may renormalize and define
\[
h := \frac{1}{h_0(1)} h_0 \, .
\]
Then $\frac{\mathbbm{1}_{H^c} h}{h(H^c)}$ represents the limiting conditionally invariant probability measure for the open
system $(M, S; H)$. However, we will work with $h$ rather than its restriction to $H^c$ because $h$ contains information
about entry into $H$, which we will exploit in Proposition~\ref{prop:entry} below.
Due to the equality in \eqref{eq:h0}, $h$ has the alternative characterization,
\[
h = \lim_{n \to \infty} \frac{\mathring{\cL}_{S}^n 1}{\mu_{\mbox{\tiny SRB}}( \mathring{\cL}_{S}^n 1)} = \lim_{n \to \infty} \frac{\mathring{\cL}_{S}^n 1}{\mu_{\mbox{\tiny SRB}}( \mathring{M}^n)} \, ,
\]
as required for item (a) of the Theorem, where $\mathring{M}^n = \cap_{i=0}^n S^{-i} (M \setminus H)$ and
convergence is in the $\| \cdot \|_\star$ norm.
Remark that \eqref{eq:cauchy} implies $\frac{\mathring{\cL}_{S}^n f}{\| \mathring{\cL}_{S}^n f \|_\star}$ converges to $h_0$ at the
exponential rate $\vartheta^n$. Integrating this relation and using Lemma~\ref{lem:star}(b), we conclude that
in addition the normalization ratio $\frac{\mu_{\mbox{\tiny SRB}}( \mathring{\cL}_{S}^n f)}{\| \mathring{\cL}_{S}^n f \|_\star }$ converges to
$h_0(1)$ at the same exponential rate. Putting these two estimates together and using the
triangle inequality yields for all $n \ge 0$,
\[
\left\| \frac{\mathring{\cL}_{S}^n f}{\mu_{\mbox{\tiny SRB}}(\mathring{\cL}_{S}^n f)} - h \right\|_\star \le C \vartheta^n h_0(1)^{-1} \, , \quad \forall \, f \in \mathcal{C}_{c,A,L}(\delta) \, ,
\]
proving the first inequality of item (b).
Next, for each, $f \in \mathcal{C}_{c,A,L}(\delta)$ let
\begin{equation}\label{eq:conve_l}
\ell(f)=\limsup_{n\to\infty} \nu^{-n} \mu_{\mbox{\tiny SRB}}( \mathring{\cL}^n_{S} f ) \, .
\end{equation}
Note that $\ell$ is bounded, homogeneous of degree one and order preserving.
By Lemma~\ref{lem:star}(b), $\ell$ can be extended to $\mathcal{C}_\star$.
Since $\ell(h)=1$,
$\nu^{-n} \mathring{\cL}_{S}^n h = h$ and $\ell(\nu^{-n}\mathring{\cL}^n_{S} f)=\ell(f)$ we can apply, again,
\cite[Lemma 2.2]{LSV98}
as in \eqref{eq:adapted} to $f$ and $\ell(f)h$ and obtain
\begin{equation}
\label{eq:ell0-prop}
\| \nu^{-n}\mathring{\cL}^n_{S} f- h \ell( f) \|_\star = \nu^{-n} \| \mathring{\cL}^n_{S} f - \ell(f) \mathring{\cL}^n_{S} h \|_\star
\leq C \vartheta^n \ell( f ) \| h \|_\star \, ,
\end{equation}
proving the second inequality of item (b) of the Theorem. Note that \eqref{eq:ell0-prop} also implies
(integrating and applying Lemma~\ref{lem:star}(b) ) that the limsup in \eqref{eq:conve_l} is, in fact, a limit, and hence $\ell$ is linear. Remark that $\ell$ is also nonnegative for $f \in \mathcal{C}_{c,A,L}(\delta)$ by Remark~\ref{rem:after order}.
By definition, if $f \in \mathcal{C}_{c,A,L}(\delta)$ and $\lambda> \| f \|_\star$ then $\lambda + f, \lambda - f \in \mathcal{C}_{c,A,L}(\delta)$,
so that using the linearity and nonnegativity of $\ell$ yields,
\begin{equation}
\label{eq:dom ell}
- \lambda \ell(1) \le \ell(f) \le \lambda \ell(1) \, , \quad \forall \; f \in \mathcal{C}_{c,A,L}(\delta), \; \lambda > \| f \|_\star \, .
\end{equation}
Thus either $\ell(f) = 0$ for all $f \in \mathcal{C}_{c,A,L}(\delta)$ or $\ell(f) \neq 0$ for all $f \in \mathcal{C}_{c,AL}(\delta)$. But
if the first alternative holds, then by the continuity of $\ell$ with respect to the $\| \cdot \|_\star$ norm
(Lemma~\ref{lem:star}(b)), $\ell$ is identically 0 on $\mathcal{C}_\star$, which is a contradiction since $\ell(h) =1$.
Thus $\ell(f) > 0$ for all $f \in \mathcal{C}_{c,A,L}(\delta)$.
Finally, applying \eqref{eq:ell0-prop} to $f \equiv 1$ integrated with respect to $\mu_{\mbox{\tiny SRB}}$ and using again Lemma~\ref{lem:star}(b), we obtain
\[
| \nu^{-n} \mu_{\mbox{\tiny SRB}}(\mathring{M}^n) - \ell(1) | \le C \vartheta^n \ell(1) \| h \|_\star \, ,
\]
which in turn implies that $\log \nu = \lim_{n \to \infty} \frac 1n \log \mu_{\mbox{\tiny SRB}}(\mathring{M}^n)$ since $\ell(1) \neq 0$, as required for the
remaining item of part (a) of the Theorem. Note that $\nu \neq 0$ by Remark~\ref{rem:after order} and \eqref{eq:lower big H}, while $\nu \neq 1$ by monotonicity since the escape rate for this class of billiards is known
to be exponential for arbitrarily small holes \cite{DWY, dem bill}.
\end{proof}
We can use Theorem~\ref{thm:open} to obtain exit statistics from the open billiard in the plane.
As an example, for $\theta \in [0, 2\pi)$ let us define $H_\theta$ to be the set of $x \in H$ such that
the first intersection of $O(T^{-1} x)$ with $\partial R$ has velocity
making an angle of $\theta$ with the positive horizontal axis.
Note that $H_\theta$ is a finite union of unstable
curves since it is the image of a wave front with zero curvature moving with parallel velocities. The fact that $H_\theta$
comprises unstable curves is not altered by the fact that the flow in $R$ may reflect off of $\partial R$
several times before arriving at a scatterer because such collisions are neutral; also, since the corners
of $R$ are right angles, the flow remains continuous at these corner points.
If the incoming particles at time zero are distributed according to a measure with density $f \in\mathcal{C}_{c,A,L}(\delta)$, then the probability that a particle leaves the box at time $nN$ with a direction in the interval $\Theta=[\theta_1, \theta_2]$, call it $\mathbb{P}_f(x_n\in[\theta_1, \theta_2])$, can be expressed as
\begin{equation}
\label{eq:P def}
\mathbb{P}_f(x_n\in[\theta_1, \theta_2]) = \int_M \mathbbm{1}_{H_\Theta} \mathring{\cL}_S^n f \, d\mu_{\mbox{\tiny SRB}} \, ,
\end{equation}
where $H_\Theta := \cup_{\theta \in \Theta} H_\theta$. Although the boundary of $H_\Theta$ comprises
increasing (unstable) curves as already mentioned, the restriction on the angle may prevent $\partial H_\Theta$ from enjoying the property of continuation of singularities common to billiards. See Figure~\ref{fig:hole} (see also
\cite[Sect.~8.2.2]{dem bill} for other examples of holes without the continuation of singularities property).
Similarly, for $p \in \partial R$, define $H_p$ to be the set of $x \in H$ such that the last intersection of $O(T^{-1}x)$
with $\partial R$ is $p$. Then for an interval $P \subset \partial R$, we define $H_P = \cup_{p \in P} H_p$,
and $\int_M \mathbbm{1}_{H_P} \mathring{\cL}_S^n f$ denotes the probability that a particle leaves the box at time $nN$ through
the boundary interval $P$.
\begin{figure}[ht]
\begin{centering}
\begin{minipage}{.48 \linewidth}
\hspace{1.5cm}
\begin{tikzpicture}[scale=0.30]
\draw [dashed](0,3.5)--(0,-12.5);
\draw[very thick, dashed] (0,-3)--(0,-6);
\fill[gray!20!white] (0,3)--(0,-3) arc (-90:90:4 and 3)--cycle
\draw[very thick] (0,-3) arc (-90:90:4 and 3);
\fill[gray!20!white] (0,-6)--(0,-12) arc (-90:90:2 and 3)--cycle
\draw[very thick] (0,-12) arc (-90:90:2 and 3);
\fill[gray!20!white] (7.5,-4.6) circle (3.5)
\draw[very thick] (7.5,-4.6) circle (3.5);
\node at (1.7,0) {$B_1$};
\node at (7.2,-4.6) {$B_2$};
\node at (1,-9) {$B_3$};
\node at (-1.9, -4.5) {$\partial R$};
\node at (-0.7, -5.9) {\small $p$};
\node at (3, -13) {(a)};
\draw[thick, ->] (0,-5.9) -- (2.8, -5.62);
\draw[thick] (2.8, -5.62) -- (4.1, -5.49);
\draw[thick, ->] (0,-5.9) -- (2.8, -4.5);
\draw[thick] (2.8, -4.5) -- (4.1, -3.85);
\draw[thick] (0, -3.75) -- (5, -1.25);
\draw[thick, ->] (0, -3.75) -- (4, -1.75);
\draw[thick, ->] (0, -3.75) -- (2.9, -3.46);
\draw[thick] (2.9, -3.46) -- (4.3, -3.32);
\end{tikzpicture}
\end{minipage}
\begin{minipage}{.48 \linewidth}
\hspace{1.5cm}
\begin{tikzpicture}[scale = 0.5]
\draw[very thick] (2,11) -- (15,11);
\draw[very thick] (2,2) -- (15,2);
\draw[thick] (13,11) to[out=250, in=67.5] (12.2,9.2) to[out=242.5, in=65] (11,7) to [out=245, in=60] (9.8,5)
to [out=240, in=57.5] (8.8,3.3) to [out=237.5, in=55] (7.8,2);
\draw[thick] (10,11) to[out=220, in=35] (7.8,9.2) to[out=210, in=20] (5.7,8.2);
\draw[thick] (11.8,8.5) to[out=220, in=35] (9.6,6.7) to[out=210, in=20] (7.5,5.6) to [out=200, in=18] (6,5)
to [out=195, in=15] (3.5,4.5);
\draw[thick] (5.7,8.2) to[out=245, in=60] (5,6.7) to[out=240, in=50] (3.5,4.5);
\node at (0.5,2){$\varphi = - \frac{\pi}{2}$};
\node at (0.5,11){$\varphi = \frac{\pi}{2}$};
\node at (8.4,7.65){\small $H_{\Theta}$};
\node at (13, 7){\small $T\mathcal{S}_0(B_1)$};
\node at (7.5,9.7){\scriptsize $H_{\theta_2}$};
\node at (8.3,5.1){\scriptsize $H_{\theta_1}$};
\node at (4,6.5){\scriptsize $H_p$};
\node at (7,1) {(b)};
\end{tikzpicture}
\end{minipage}
\end{centering}
\caption{a) Sample rays with $\theta = \theta_1$ and $\theta=\theta_2$
striking the scatterer $B_2$. The point $p$ is the topmost point of $\partial B_3$.
b) Component of $H_\Theta$ on the scatterer $B_2$. In this configuration, $H_{\theta_1}$
intersects
the singularity curve $T\mathcal{S}_0$ coming from $B_1$ while $H_{\theta_2}$ reaches $\mathcal{S}_0$ directly; however,
the left boundary of $H_{\Theta}$ is an arc of $H_p$ and the continuation of singularities properties fails for a hole
of this type since $\theta_1 > 0$. }
\label{fig:hole}
\end{figure}
\begin{prop}
\label{prop:entry}
For any intervals of the form $\Theta = [\theta_1, \theta_2]$, or $P = [p_1, p_2]$, any $f \in C^1(M)$ with
$f \ge 0$ and
$\int f \, d\mu_{\mbox{\tiny SRB}} = 1$, and all $n \ge 0$, we have\footnote{ If instead $f \in \mathcal{C}_{c,A,L}(\delta)$, $f \ge 0$ and
$\int f \, d\mu_{\mbox{\tiny SRB}} =1$,
then $\| f \|_{C^1}$ can be dropped from the right hand side.}
\[
\begin{split}
&\mathbb{P}_{f} (x_n\in \Theta) =\nu^n h (\mathbbm{1}_{H_\Theta}) \ell(f)+ \|f\|_{C^1} \mathcal{O}\big(\nu^n\vartheta^{\frac{q}{q+1}n}\big) \, ,
\quad \mbox{and} \quad\\
&\mathbb{P}_{f} (x_n\in P) =\nu^n h (\mathbbm{1}_{H_P}) \ell(f)+ \|f\|_{C^1} \mathcal{O}\big(\nu^n\vartheta^{\frac{q}{q+1}n}\big) \, .
\end{split}
\]
\end{prop}
\begin{remark}
If $f \in \mathcal{C}_{c,A,L}(\delta)$, then $\ell(f) > 0$ by Theorem~\ref{thm:open}(b), and Proposition~\ref{prop:entry} provides a precise asymptotic
for the escape of particles through $H_{\Theta}$ and $H_P$. For more general $f \in C^1(M)$, it may be
that $\ell(f) =0$, in which case Proposition~\ref{prop:entry} merely gives an upper bound on the exit statistic compared
to the rate of escape given by $\nu$.
\end{remark}
\begin{proof}
We prove the statement for $\mathbbm{1}_{\Theta}$. The statement for $\mathbbm{1}_P$ is similar.
To start with we assume $f\in \mathcal{C}_{c,A,L}(\delta)$, and $f \ge 0$ with $\int f \, d\mu_{\mbox{\tiny SRB}} = 1$.
As already mentioned,
$\partial H_\Theta$ comprises finitely many unstable (increasing) curves in $M$ and so $H_\Theta$ satisfies
(O$1'$) and (O2) with $P_0=3$ and $C_t$ depending only on the uniform angle between the stable and unstable
curves. Since $\mathbbm{1}_{H_\Theta}$
is not in $C^1(M)$, we cannot apply Lemma~\ref{lem:star}(b) directly; we will use a mollification to bypass this
problem.
Let $\rho : \mathbb{R}^2 \to \mathbb{R}^2$ be a nonnegative, $C^\infty$ function supported in the unit disk with $\int \rho = 1$, and define
$\rho_\varepsilon(\cdot) = \varepsilon^{-2} \rho( \cdot \, \varepsilon^{-1} )$. For $\varepsilon > 0$, define the mollification,
\[
\psi_\varepsilon(x) = \int \mathbbm{1}_{H_\Theta}(y) \rho_\varepsilon(x-y) \, dy \qquad x \in M \, .
\]
We have $| \psi_\varepsilon|_{C_0} \le 1$ and $| \psi_\varepsilon' |_{C^0} \le C\varepsilon^{-1}$. Note that $\psi_\varepsilon = \mathbbm{1}_{H_\Theta}$
outside an $\varepsilon$-neighborhood of $\partial H_\theta$ (including $\mathcal{S}_0$). Letting $\tilde{\psi}_\varepsilon$ denote a $C^1$ function with $|\tilde{\psi}_\varepsilon|_{C^0} \le 1$, which is 1 on $N_\varepsilon(\partial H_\Theta)$ and 0 on $M \setminus N_{2 \varepsilon}(\partial H_\Theta)$,
we have $| \mathbbm{1}_{H_\Theta} - \psi_\varepsilon | \le \tilde{\psi}_\varepsilon$.
Due to (O2), for any
$W \in \mathcal{W}^s$ such that $W \cap N_\varepsilon(\partial H_{\Theta}) \neq \emptyset$,
using first the fact that $f \ge 0$ and then applying cone condition \eqref{eq:cone 3},
\begin{equation}
\label{eq:mollify}
\begin{split}
\int_W | \mathbbm{1}_{H_\Theta} - \psi_\varepsilon | \, \mathring{\cL}_S^n f \, dm_W
& \le \int_W \tilde{\psi}_\varepsilon \, \mathring{\cL}_S^n f \, dm_W \le \int_{W \cap N_{2\varepsilon}(\partial H_{\Theta})} \mathring{\cL}_S^n f \, dm_W \\
& \le 2^{1+q} A \delta^{1-q} C_t^q \varepsilon^q {|\:\!\!|\:\!\!|} \mathring{\cL}_S^n f {|\:\!\!|\:\!\!|}_- \, ,
\end{split}
\end{equation}
where we have used the fact that $W \cap N_{2\varepsilon}(\partial H_{\Theta})$ has at most 2 connected components
of length $2C_t \varepsilon$. Then integrating over $M$ and disintegrating $\mu_{\mbox{\tiny SRB}}$ as in the proof of Lemma~\ref{lem:order},
we obtain,
\begin{equation}
\label{eq:epsilon}
\int_M | \mathbbm{1}_{H_\Theta} - \psi_\varepsilon | \, \frac{\mathring{\cL}_S^n f}{\mu_{\mbox{\tiny SRB}}(\mathring{\cL}_S^n f)} \, d\mu_{\mbox{\tiny SRB}}
\le \int_M \tilde{\psi}_\varepsilon \, \frac{\mathring{\cL}_S^n f}{\mu_{\mbox{\tiny SRB}}(\mathring{\cL}_S^n f)} \, d\mu_{\mbox{\tiny SRB}} \le C \varepsilon^q \frac{ {|\:\!\!|\:\!\!|} \mathring{\cL}_S^n f {|\:\!\!|\:\!\!|}_-}{\mu_{\mbox{\tiny SRB}}(\mathring{\cL}_S^n f)} \, .
\end{equation}
By Remark~\ref{rem:after order}, $\mu_{\mbox{\tiny SRB}}(\mathring{\cL}_S^n f) \ge \bar C^{-1} {|\:\!\!|\:\!\!|} \mathring{\cL}_S^n f {|\:\!\!|\:\!\!|}_-$, so the bound is uniform in $n$.
Since $\tilde{\psi}_\varepsilon \in C^1(M)$ the bound carries over to $h(\tilde{\psi}_\varepsilon)$, and
since $h$ is a nonnegative measure, to $h(\mathbbm{1}_{H_\Theta} - \psi_\varepsilon)$. Thus for each $n \ge 0$ and $\varepsilon>0$,
\begin{equation}
\label{eq:cone asy}
\begin{split}
\int \mathbbm{1}_{H_\Theta} \, \mathring{\cL}_S^n f \, d\mu_{\mbox{\tiny SRB}}
& = \int ( \mathbbm{1}_{H_\Theta} - \psi_\varepsilon ) \, \mathring{\cL}_S^n f \, d\mu_{\mbox{\tiny SRB}}
+ \left( \int \psi_\varepsilon \, \mathring{\cL}_S^n f \, d\mu_{\mbox{\tiny SRB}} - \nu^n \ell(f) h(\psi_\varepsilon) \right) \\
& \qquad + \nu^n \ell(f) h(\psi_\varepsilon - \mathbbm{1}_{H_\Theta}) +
\nu^n \ell(f) h(\mathbbm{1}_{H_\theta}) \\
& = \mathcal{O}\big(\varepsilon^q \nu^n \ell(f)\big) + \mathcal{O}\big(|\psi_\varepsilon|_{C^1} \nu^n \vartheta^n \ell(f) \big) + \nu^n \ell(f) h(\mathbbm{1}_{H_\Theta}) \, ,
\end{split}
\end{equation}
where we have applied \eqref{eq:epsilon} to the first and third terms and
Theorem~\ref{thm:open}(b) and Lemma~\ref{lem:star}(b) to the second term.
Since $|\psi_\varepsilon|_{C^1} \le \varepsilon^{-1}$, choosing $\varepsilon = \vartheta^{n/(q+1)}$
yields the required estimate for $f\in \mathcal{C}_{c,A,L}(\delta)$.
To conclude, note that by Lemma~\ref{lem:dominate}, there exists $C_\flat>0$ such that, if $f\in C^1(M)$, then,
for each $\lambda\geq C_\flat \|f\|_{\mathcal{C}^1}$, $\lambda+f\in\mathcal{C}_{c,A,L}(\delta)$.
Hence, by the linearity of the integral, $\ell(f)$ as defined in \eqref{eq:conve_l} can be extended to $f \in C^1$
by $\ell(f) = \ell(\lambda + f) - \ell(\lambda)$, and the limsup is in fact a limit since since the limit exists for $\lambda + f, \lambda \in \mathcal{C}_{c,A,L}(\delta)$ (see \eqref{eq:ell0-prop} and following).
Now take $f \in C^1$ with $\int f \, d\mu_{\mbox{\tiny SRB}} =1$ and $\lambda \ge C_\flat \| f \|_{C^1}$ as above.
Then, necessarily $\lambda + f \ge 0$, and so recalling \eqref{eq:P def}, we have
\[
\begin{split}
\mathbb{P}_{\frac{\lambda+f}{1+\lambda}}(x_n\in \Theta)&=\int_M \mathbbm{1}_{H_\Theta} \mathring{\cL}_S^n { \left( \tfrac{\lambda+f}{1+\lambda} \right) }
=\frac{\lambda}{1+\lambda}\int_M \mathbbm{1}_{H_\Theta} \mathring{\cL}_S^n1+\frac{1}{1+\lambda}\int_M \mathbbm{1}_{H_\Theta} \mathring{\cL}_S^n f \\
&= \frac{\lambda}{1+\lambda}\mathbb{P}_{1}(x_n\in \Theta)+\frac{1}{1+\lambda}\mathbb{P}_{f}(x_n\in \Theta) .
\end{split}
\]
Hence by \eqref{eq:cone asy},
\[
\begin{split}
\mathbb{P}_{f}(x_n\in \Theta)&=(1+\lambda)\mathbb{P}_{\frac{\lambda+f}{1+\lambda}}(x_n\in \Theta)-\lambda \mathbb{P}_{1}(x_n\in \Theta)\\
&=\nu^n h (\mathbbm{1}_{H_{\Theta}}) \big(\lambda \ell(1)+\ell(f) \big)-\nu^n h (\mathbbm{1}_{H_{ \Theta}}) \lambda \ell(1)+ \lambda \mathcal{O}\big(\nu^n\vartheta^{\frac{q}{q+1}n}\big)\\
&=\nu^n h (\mathbbm{1}_{H_{ \Theta}}) \ell(f)+\|f\|_{\mathcal{C}^1}\mathcal{O}\big(\nu^n\vartheta^{\frac{q}{q+1}n}\big).
\end{split}
\]
\end{proof}
\subsection{Random Lorentz gas (lazy gates)}
\label{sec:lorentz}
Consider a Lorentz gas as described in \cite[Section 2]{AL}. That is, we have a lattice of cells of size one with
circular obstacles of fixed radius $r$ at their corners and a random obstacle $B(z)$ of fixed
radius $\rho$ and center in a set $\mathcal{O}$ at their interior.\footnote{ The assumption that all obstacles are circular is not essential and can be relaxed by requiring that the obstacles at the corners are symmetric with respect to reflections as described in Section~\ref{sec:scattering}. }
The central obstacle is small enough not to intersect with the other obstacles but large enough to prevent trajectories from crossing the cell without colliding with an obstacle.
We call the openings between different cells {\em gates}, see Figure 3b, and require that no trajectory can cross two
gates without making at least one collision with the obstacles. Thus we fix $r$ and $\rho$ satisfying\footnote{ Finite horizon requires $r \ge \frac{1}{1+\sqrt{2}}$, yet our added condition that a particle cannot cross diagonally from, say, $\hat R_1$ to $\hat R_2$ without making a collision requires further that $r \ge \frac 13$.}
the following conditions:
\begin{equation}
\label{eq:r rho}
\tfrac{1}{3} \le r < \tfrac 12 \, , \quad \mbox{and} \quad 1-2r < \rho < \tfrac{\sqrt{2}}{2} - r \, .
\end{equation}
With $r$ and $\rho$ fixed, the set of
possible configurations of the central obstacle are described by $\omega \in \Omega=\mathcal{O}^{\mathbb{Z}^2}$.
In order to ensure that particles cannot cross directly from $\hat R_1$ to $\hat R_3$ or from $\hat R_2$ to $\hat R_4$
without colliding with an obstacle, and to ensure a minimum distance between scatterers, we fix $\varepsilon_* > 0$ and require the center $c = (c_1, c_2)$ of the random obstacle
$B_\omega$, $\omega \in \Omega$,
(the central obstacle $C_5$ in Figure~3b) to satisfy,
\begin{equation}
\label{eq:c}
1 - (r+\rho - \varepsilon_*) \le c_1, c_2 \le r+\rho - \varepsilon_* \, .
\end{equation}
Note that \eqref{eq:r rho} and \eqref{eq:c} imply that all possible positions of the central scatterer $B_\omega$ result
in a billiard table with $\tau_{\min} \ge \tau_* := \min \{ \varepsilon_*, 1-2r \} > 0$.
On $\Omega$ the space of translations $\xi_z$, $z\in\mathbb{Z}^2$, acts naturally as $[\xi _z(\omega)]_x=\omega_{z+x}$, see Figure 3a. We assume that the obstacle configurations are described by a measure $\mathbb{P}_e$ which is ergodic with respect to the translations.
\begin{figure}[ht]\
\begin{minipage}{.48 \linewidth}
\hspace{1.5cm}
\begin{tikzpicture}[scale=0.40]
\fill[gray!20!white] (0,0) circle (1.5)
\draw (0,0) circle (1.5);
\fill[gray!20!white] (4,0) circle (1.5)
\draw (4,0) circle (1.5);
\fill[gray!20!white] (8,0) circle (1.5)
\draw (8,0) circle (1.5);
\fill[gray!20!white] (0,-4) circle (1.5)
\draw (0,-4) circle (1.5);
\fill[gray!20!white] (4,-4) circle (1.5)
\draw (4,-4) circle (1.5);
\fill[gray!20!white] (8,-4) circle (1.5)
\draw (8,-4) circle (1.5);
\fill[gray!20!white] (0,-8) circle (1.5)
\draw (0,-8) circle (1.5);
\fill[gray!20!white] (4,-8) circle (1.5)
\draw (4,-8) circle (1.5);
\fill[gray!20!white] (8,-8) circle (1.5)
\draw (8,-8) circle (1.5);
\fill[gray!20!white] (2.2,-2) circle (1)
\draw (2.2,-2) circle (1);
\node at (2.2,-2) {{\tiny$ B_\omega(a)$}};
\fill[gray!20!white] (6,-1.8) circle (1)
\draw (6,-1.8) circle (1);
\node at (6,-1.8) {{\tiny$ B_\omega(b)$}};
\fill[gray!20!white] (2,-6.3) circle (1)
\draw (2,-6.3) circle (1);
\node at (2,-6.3) {{\tiny$ B_\omega(0)$}};
\fill[gray!20!white] (6,-5.7) circle (1)
\draw (6,-5.7) circle (1);
\node at (6.1,-5.7) {{\tiny$ B_\omega(c)$}};
\node at (5,-11) {$a=(1,0); b=(1,1); c=(1,0)$};
\end{tikzpicture}
\end{minipage}
\begin{minipage}{.48 \linewidth}
\hspace{1.3cm}
\begin{tikzpicture}[scale=0.60]
\draw[dashed] (0,0)--(8,0);
\draw [dashed](0,0)--(0,-8);
\draw[dashed](0,-8)--(8,-8);
\draw[dashed](8,-8)--(8,0);
\draw[very thick, dashed] (3,0)--(5,0);
\draw[very thick, dashed] (3,-8)--(5,-8);
\draw[very thick, dashed] (0,-3)--(0,-5);
\draw[very thick, dashed] (8,-5)--(8,-3);
\fill[gray!20!white] (0,0)--(0,-3.5) arc (-90:0:3.5)--cycle
\draw[very thick] (0,-3.5) arc (-90:0:3.5);
\fill[gray!20!white] (0,-8)--(3.5,-8) arc (0:90:3.5)--cycle
\draw[very thick] (3.5,-8) arc (0:90:3.5);
\fill[gray!20!white] (8,-8)--(4.5,-8) arc (180:90:3.5)--cycle
\draw[very thick] (4.5,-8) arc (180:90:3.5);
\fill[gray!20!white] (8,0)--(4.5,0) arc (180:270:3.5)--cycle
\draw[very thick] (4.5,0) arc (180:270:3.5);
\fill[gray!20!white] (4.3,-4.4) circle (1.5)
\draw[very thick] (4.3,-4.4) circle (1.5);
\node at (1.7,-1.2) {$C_2$};
\node at (6.3,-1.2) {$C_1$};
\node at (1.7,-6.2) {$C_3$};
\node at (6.4,-6.2) {$C_4$};
\node at (4,-4) {$C_5$};
\node at (8.7,-4) {$\hat R_1$};
\node at (-.8,-4) {$\hat R_3$};
\node at (4,-.7) {$\hat R_2$};
\node at (4,-7.3) {$\hat R_4$};
\node at (.3,-1.5) {$r$};
\node at (5,-4.7) {$\rho$};
\draw (4.3,-4.4)--(5.8,-4.4);
\end{tikzpicture}
\end{minipage}
\\ \vskip.1cm
{ Fig 3a: {\it Configuration of random obstacles $B_\omega(z)$}\hskip 1.2cm Fig 3b: {\it Poincar\'e section $C_i$ and gates $\hat R_i$\hspace{.5cm}}}
\end{figure}
Exactly as in the section \ref{sec:scattering}, we assume that the gates are reflecting and become transparent only after $N$ collisions with the obstacles. Thus when the particle enters a cell it will stay in that cell for at least $N$ collisions
with the obstacles, hence the {\em lazy} adjective.
As described in section \ref{sec:scattering}, when the particle reflects against a gate one can reflect the table three times and see the flow (for the times at which the gates are closed) as a flow in a finite horizon Sinai billiard on the two torus.
Note that the Poincar\'e section $M=\cup_{i=1}^5 C_i\times[-\frac\pi 2,\frac \pi 2]$ in each cell is exactly the same for each $\omega$ and $z$
since the arclength of the boundary is always the same,
while the Poincar\'e map $T_z$ changes depending on the position of the central obstacle, see Figure 3b.
Let us call $\mathcal{F}(\tau_*)$ the collection of the different resulting billiard maps
corresponding to tables that maintain a minimum distance $\tau_*>0$ between obstacles, as required by
\eqref{eq:r rho} and \eqref{eq:c}.
(Note that the
parameters $\mathcal{K}_*$ and $E_*$ of Section~\ref{sequential} are fixed in this class once $r$ and $\rho$ are fixed.)
The only difference with Section~\ref{sec:scattering}, as far as the dynamics in a cell is concerned,
consists in the fact that we have to be more specific about which cell the particle enters,
as now exiting from one cell means entering into another.
Recalling the notation of Section~\ref{sec:scattering}, if we call $R(z)$ the cell at the position $z\in\mathbb{Z}^2$, then the
gates $\hat R_i$ are subsets of $\partial R(z)$. We denote by $\widetilde R(z)$
the lifted cell (viewed as a subset of $\mathbb{T}^2$) after reflecting $R(z)$ three times, and
by $(\widetilde{M}, \widetilde{T}_z)$ the corresponding billiard map. As before, the projection $\pi: \widetilde{M} \to M$ satisfies
$\pi \circ \widetilde{T} = T \circ \pi$.
Then the hole $\widetilde H(z)$ can be written as $\widetilde H(z)=\cup_{i=1}^4 \widetilde H_i(z)$, where
$\pi(\widetilde{H}_i(z))=: H_i(z)$ are the points $x \in M$ such that $O(T^{-1}x) \cap \partial R(z) \in \hat R_i$.\footnote{ The hole depends on the trajectory of $x$, which is different in different cells and hence depends on $z$,
while the gates $\hat R_i$ are independent of $z$.} Due to our assumption \eqref{eq:r rho}, this point of
intersection is unique for each $x$ since consecutive collisions with $\partial R$ cannot occur.
Then $H(z) = \pi(\widetilde{H}(z)) = \cup_{i=1}^4 H_i(z)$.
As discussed in Section~\ref{sec:scattering}, the holes are neither of Type I nor of Type II,
yet they satisfy (O1$'$) and (O2) with $P_0 = 3$ and $C_t$ depending only on the uniform angle
between stable and unstable cones for the induced billiard map.
Yet for our dynamics, when a particle changes cell at the $N$th collision, it is because after $N-1$ collisions,
that particle is in $G_i(z) := T_z^{-1}H_i(z)$, and in fact it will never reach $H_i(z)$. Unfortunately, the geometry
of $G(z) := \cup_{i=1}^4 G_i(z)$ is not convenient for our machinery, yet we will be able reconcile this difficulty after
defining the dynamics precisely as follows.
The phase space is $\mathbb{Z}^2\times M$. For $x \in M$, denote by $p(x)$ the position of $x$ in $R(z)$ and by
$\theta(x)$ the angle of its velocity with respect to the positive horizontal axis in $R(z)$.
We define
\[
w(z, x )=\begin{cases} 0=:w_0 &\text{ if } x \not \in G(z)\\
e_1=: w_1&\text{ if } x \in G_1(z)\\
e_2=:w_2&\text{ if } x \in G_2(z)\\
-e_1=:w_3&\text{ if } x \in G_3(z)\\
-e_2=:w_4&\text{ if } x \in G_4(z).
\end{cases}
\]
Also we set $\mathfrak{W}=\{w_0,\dots, w_4\}$.
If $ x \in G_i(z)$, then we call $\bar q(x)=(q,\theta) \in \hat R_i\times [0, 2\pi)$ the point
$\bar q$ such that $q=O(x)\cap \hat R_i$ and $\theta = \theta(x)$, i.e. without reflection at $\hat R_i$.
We then consider $\bar q$ as a point in the cell $z+w(z,x) = z + w_i$ and call $T_{z,i}(x)$ the post
collisional velocity at the next collision with an obstacle under the flow starting at $\bar q$.
Note that in the cell $R(z+w_i)$, $\bar q \in \hat R_{\bar i}$, where $\bar i = i + 2 \mbox{ (mod 4*)}$.\footnote{ By (mod 4*) we mean cyclic addition on 1, 2, 3, 4 rather than 0, 1, 2, 3.}
Thus if $\Phi_t^z$ denotes the flow in $R(z)$, then with this notation, $G_i(z)$ is the projection on $M$ of
$\hat R_i$ under the inverse flow $\Phi_{-t}^z$ while $H_{\bar i}(z+w(z,x))$ is the projection on $M$
of $\hat R_{\bar i}$ under the forward flow $\Phi_t^{z+w_i}$. Thus,
\begin{equation}
\label{eq:correct}
H_{\bar i}(z + w_i) = T_{z,i} G_i(z) \implies \mathbbm{1}_{G_i(z)} \circ T_{z,i}^{-1} = \mathbbm{1}_{H_{\bar i}(z+w_i)} \, ,
\end{equation}
which is a relation we shall use to control the action of the relevant transfer operators below.
Differing slightly from the previous section, here it is convenient to set $S_z=T_z^{N-1}$, and define
\[
F(z,x)=\begin{cases} (z, S_z\circ T_z(x))=:(z, \widehat S_z(x)) &\text{ if } x\not \in G(z)\\
(z+w(z,x), S_{z+w(z,x)}\circ T_{z,i}(p))=:(z+w(z,x), \widehat S_z(x))&\text{ if } x\in G_i(z).
\end{cases}
\]
We set $(z_n,x_n)=F^n(z,x)$ and we call $n$ the {\em macroscopic time}, which corresponds to $Nn$ collisions
with the obstacles.
The above corresponds to a dynamics in which when the particle enters a cell, it is
trapped in the cell for $N$ collisions with the obstacles; then the gates open and until the next collision the particle can change cells, after
which it is trapped again for $N$ collisions and so on.
We want to compute the probability that a particle visits the sets $G_{k_0}(z_0),\cdots G_{k_{n-1}}(z_{n-1})$, in this order, where we have set $G_0(z)=M\setminus \cup_{i=1}^4 G_i(z)$. Similarly, we define $H_0(z) = M \setminus \cup_{i=1}^4 H_i(z)$. This itinerary corresponds to a particle that at time $i$ changes its position in the lattice by
$w_{k_i}$.
Following the notation of \cite{AL}, we call $\mathbb{P}_\omega$ the probability distribution in the path space
$\mathfrak{W}^\mathbb{N}$ conditioned on the central obstacles being in the positions specified
by $\omega\in\Omega$.
Hence, if the particle starts from the cell $z_0 = (0,0)$ with $x$ distributed according to a probability distribution with smooth density $f \in \mathcal{C}_{c,A,L}(\delta)$, then we have\footnote{ Since $z_0 = (0,0)$, it is equivalent to
specify $z_1, \ldots z_n$ or $w_{k_0}, \ldots w_{k_{n-1}}$ since $w_{k_j}$ can be recovered as $w_{k_j} = z_{j+1} - z_j$.} $z_n= \sum_{k=0}^{n-1} w_{k_i}$ and, for each obstacle distribution $\omega\in \Omega$,
\begin{equation}\label{eq:path_prob}
\begin{split}
\mathbb{P}_\omega(z_0, z_1,\dots, z_n) =&\int_M f(x) \mathbbm{1}_{G_{k_0}(z_0)}(x)\mathbbm{1}_{G_{k_1}(z_1)}(\widehat S_{0}(x))\cdots \\
&\phantom{\int_M}
\cdots \mathbbm{1}_{G_{k_{n-1}}(z_{n-1})}(\widehat S_{z_{n-2}}\circ\cdots\circ \widehat S_{0}(x)) \, d\mu_{\mbox{\tiny SRB}}(x) \\
=&\int_M\mathring{\cL}_{G_{k_{n-1}}(z_{n-1})}\cdots \mathring{\cL}_{G_{k_0}(z_0)} f \, d\mu_{\mbox{\tiny SRB}}
\end{split}
\end{equation}
where $\mathring{\cL}_{G_{k_j}(z_j)} := \mathcal{L}_{T_{z_{j+1}}}^{N-1}\mathcal{L}_{T_{z_j,k_j}}\mathbbm{1}_{G_{k_j}(z_j)}$, and we have set $T_{z,0}:=T_z$. See \cite{AL} for more details. We will prove below that if $N$ is sufficiently large, then Theorem \ref{thm:sequential} applies to each
operator $\mathring{\cL}_{G_{k}}$. This suffices to obtain an exponential loss of memory property (the analogue of the result obtained for piecewise expanding maps in \cite[Theorem 6.1]{AL}), that is property {\bf Exp} in \cite[Section 4.1]{AL}.
This is the content of the following theorem.
\begin{theorem}
\label{thm:mixing}
There exist $C_*>0$ and $\vartheta\in (0,1)$ such that for $\mathbb{P}$-a.e. $\omega \in \Omega$,
if $x$ is distributed according to $f \in \mathcal{C}_{c,A,L}(\delta)$ with $\int_M f = 1$ and
$z_0=(0,0)$, for all $n > m \ge 0$ and
all $w \in \mathfrak{W}^\mathbb{N}$,
\begin{equation}\label{eq:gibbs}
\left| \mathbb{P}_{\omega}(w_{ k_n }\; | \; w_{ k_0} \ldots w_{ k_{n-1}}) - \mathbb{P}_{\xi_{z_{m}} \omega} ( w_{k_n} \; | \; w_{k_m} \ldots w_{k_{n-1}}) \right| \leq C_* \vartheta^{n-m}.
\end{equation}
\end{theorem}
\begin{proof}
Note that for $m \ge 0$, $\xi_{z_m}\omega$ sends the cell at $z_m$ to (0,0). Thus according to
equation \ref{eq:path_prob}, for $x$ distributed according to $f \in \mathcal{C}_{c,A,L}(\delta)$ with $z_0 = (0,0)$,
we have
\[
\mathbb{P}_{\xi_{z_m}\omega} (w_{k_m}, \ldots w_{k_n}) = \int_M \mathring{\cL}_{G_{k_n}(z_n)} \cdots \mathring{\cL}_{G_{k_m}(z_m)}f\, d\mu_{\mbox{\tiny SRB}}\, .
\]
As remarked earlier, the sets $G_i(z)$ do not satisfy assumption (O2) so that Proposition~\ref{prop:case-b} does not
apply directly. Yet, it follows from \eqref{eq:correct} that for $g \in \mathcal{C}_{c,A,L}(\delta)$,
\[
\mathring{\cL}_{G_{k_j}(z_j)} g = \mathcal{L}_{T_{z_{j+1}}}^{N-1}\mathcal{L}_{T_{z_j,k_j}}(\mathbbm{1}_{G_{k_j}(z_j)} g)
= \mathcal{L}_{T_{z_{j+1}}}^{N-1} \big( \mathbbm{1}_{H_{\bar k_j}(z_{j+1})} \mathcal{L}_{T_{z_j,k_j}}g \big) \, ,
\]
where, as before, $\bar k_j = k_j + 2$ (mod 4*). Then, just as in the proof of Proposition~\ref{prop:case-b},
it may be the case that $\mathcal{L}_{T_{z_j, k_j}}g$ is not in $\mathcal{C}_{c,A,L}(\delta)$. Yet,
it is immediate from our estimates in Section~\ref{sec:cone} that $\mathcal{L}_{T_{z_j, k_j}}g \in \mathcal{C}_{c',A', 3L}(\delta)$
for any billiard map $T_{z_j, k_j} \in \mathcal{F}(\tau_*)$ for some constants $c', A'$ depending only on $\mathcal{F}(\tau_*)$.
Since the sets $H_i(z)$ do satisfy (O$1')$ and (O2) with $P_0 = 3$ and $C_t$ depending only on the
angle between stable and unstable cones, which has a uniform minimum in the family $\mathcal{F}(\tau_*)$,
there exists $\chi<1$ and $N$ sufficiently large as in Proposition~\ref{prop:case-b} so that\footnote{ Here in fact our operators are of the form $\mathcal{L}^n\mathbbm{1}_H$ while in Proposition~\ref{prop:case-b} they have the form
$\mathcal{L}_n \mathbbm{1}_{H^c}$ for some set $H$. Yet, this is immaterial since the boundaries of $H$ and $H^c$ in $M$ are the
same so that (O$1'$) and (O2), and in particular Lemma~\ref{lem:H3}, apply equally well to both sets.}
$\big[ \mathcal{L}_{T_{z_{j+1}}}^{N-1} \mathbbm{1}_{H_{\bar k_j}(z_{j+1})} \big] \mathcal{C}_{c',A', 3L}(\delta) \subset \mathcal{C}_{\chi c, \chi A, \chi L}(\delta)$, and both $\chi$ and $N$ are independent of $z_{j+1}$ and $k_j$. This implies in particular that
\[
\mathring{\cL}_{G_i(z)} \mathcal{C}_{c,A,L}(\delta) \subset \mathcal{C}_{\chi c, \chi A, \chi L}(\delta) \qquad \mbox{for each $i$ and all $z \in \mathbb{Z}^2$.}
\]
Now as in the proof of Theorem~\ref{thm:memory}, using the fact that $\mu_{\mbox{\tiny SRB}}( \cdot )$ is homogeneous and
order preserving on $\mathcal{C}_{c,A,L}(\delta)$ and that $\mu_{\mbox{\tiny SRB}}(\bar \mathcal{L}_m f) = \mu_{\mbox{\tiny SRB}}(f) =1$, where
$\bar \mathcal{L}_m f = \frac{\mathring{\cL}_{G_{k_m-1}(z_{m-1})} \cdots \mathring{\cL}_{G_{k_0}(z_0)} f}{\int_M \mathring{\cL}_{G_{k_m-1}(z_{m-1})} \cdots \mathring{\cL}_{G_{k_0}(z_0)} f} \in \mathcal{C}_{c,A,L}(\delta)$, we estimate as in \eqref{eq:adapted} and \eqref{eq:contract},
\begin{equation}
\label{eq:integral contract}
\begin{split}
\int_M & \mathring{\cL}_{G_{k_{n-1}}(z_{n-1})} \cdots \mathring{\cL}_{G_{k_m}(z_m)} (f - \bar\mathcal{L}_m f) \, d\mu_{\mbox{\tiny SRB}} \\
& \le C \vartheta^{n-m} \min \left\{ \int_M \mathring{\cL}_{G_{k_{n-1}}(z_{n-1})} \cdots \mathring{\cL}_{G_{k_m}(z_m)} f,
\int_M \mathring{\cL}_{G_{k_{n-1}}(z_{n-1})} \cdots \mathring{\cL}_{G_{k_m}(z_m)} \bar\mathcal{L}_m f \right\} \, ,
\end{split}
\end{equation}
for some $\vartheta<1$ depending on the diameter of $\mathcal{C}_{\chi c, \chi A, \chi L}(\delta)$ in $\mathcal{C}_{c,A,L}(\delta)$.
Finally, the left hand side of \eqref{eq:gibbs} reads
\[
\begin{split}
&\left| \frac{\int_M\mathring{\cL}_{G_{k_{n-1}}(z_{n-1})}\cdots \mathring{\cL}_{G_{k_0}(z_0)} f}{\int_M\mathring{\cL}_{G_{k_{n-2}}(z_{n-2})}\cdots \mathring{\cL}_{G_{k_0}(z_0)}f}
-\frac{\int_M\mathring{\cL}_{G_{k_{n-1}}(z_{n-1})}\cdots \mathring{\cL}_{G_{k_m}(z_m)} f}{\int_M\mathring{\cL}_{G_{k_{n-2}}(z_{k_{n-2}})}\cdots \mathring{\cL}_{G_{k_m}(z_m)}f }\right| \\
& \le \left| \frac{\int_M\mathring{\cL}_{G_{k_{n-1}}(z_{n-1})}\cdots \mathring{\cL}_{G_{k_m}(z_m)} \bar \mathcal{L}_m f - \int_M\mathring{\cL}_{G_{k_{n-1}}(z_{n-1})}\cdots \mathring{\cL}_{G_{k_m}(z_m)} f }{\int_M\mathring{\cL}_{G_{k_{n-2}}(z_{n-2})}\cdots \mathring{\cL}_{G_{k_m}(z_m)} \bar\mathcal{L}_m f} \right| \\
& \qquad + \left| \frac{\int_M\mathring{\cL}_{G_{k_{n-1}}(z_{n-1})}\cdots \mathring{\cL}_{G_{k_m}(z_m)} f}{\int_M\mathring{\cL}_{G_{k_{n-2}}(z_{k_{n-2}})}\cdots \mathring{\cL}_{G_{k_m}(z_m)} \bar \mathcal{L}_m f } - \frac{\int_M\mathring{\cL}_{G_{k_{n-1}}(z_{n-1})}\cdots \mathring{\cL}_{G_{k_m}(z_m)} f}{\int_M\mathring{\cL}_{G_{k_{n-2}}(z_{k_{n-2}})}\cdots \mathring{\cL}_{G_{k_m}(z_m)}f }\right| \\
& \le C \vartheta^{n-m} + C \vartheta^{n-m-1} \, ,
\end{split}
\]
where we have applied \eqref{eq:integral contract} twice and used the fact that
$\frac{\int_M\mathring{\cL}_{G_{k_{n-1}}(z_{n-1})}\cdots \mathring{\cL}_{G_{k_m}(z_m)} g}{\int_M\mathring{\cL}_{G_{k_{n-2}}(z_{k_{n-2}})}\cdots \mathring{\cL}_{G_{k_m}(z_m)}g } \le 1$ for any $g \in \mathcal{C}_{c,A,L}(\delta)$.
\end{proof}
In particular, Theorem \ref{thm:mixing}, together with\footnote{ Remark that \cite[Theorem~6.4]{AL} requires
$\mu_{\mbox{\tiny SRB}}(G_i(z))$ to be the same for each $i$ and $z$, independently of $\omega$. This is precisely the case
here since $G_i(z)$ is defined as the projection of $\hat R_i$ under the inverse flow $\Phi^z_{-t}$, and
Leb$(\hat R_i \times [0, 2\pi))$ in the phase space of the flow is independent of $i$, while
$\mu_{\mbox{\tiny SRB}}$ is the projection onto $M$ of Lebesgue measure, which is invariant under the flow.}
\cite[Theorem~6.4]{AL},
implies that $\lim_{n\to \infty} \frac 1nz_n =0$ for $\mathbb{P}_e$ almost all $\omega$, that is, the walker has, $\mathbb{P}_e$-almost-surely, no drift. See \cite[Section~6]{AL} for details.\footnote{The arguments in \cite[Section~6]{AL} are developed for expanding maps, but the relevant parts apply verbatim to the present context.}
This latter fact could be deduced also from the ergodicity result in \cite[Theorem 5.4]{Lenci06}; however, Theorem \ref{thm:mixing} is much stronger (indeed, by \cite[Theorem 6.4]{AL}, it implies \cite[Theorem 5.4]{Lenci06}) since it proves some form of memory loss that is certainly not implied by ergodicity alone. It is therefore sensible to expect that more information on the random walk will follow from Theorem \ref{thm:mixing}, although this will require further work.
We conclude with a corollary of Theorem~\ref{thm:mixing} which implies the same exponential loss of
memory for particles distributed according to two different initial distributions. For $f \in \mathcal{C}_{c,A,L}(\delta)$,
let $\mathbb{P}_{\omega, f}( \cdot)$ denote the probability in the path space
$\mathfrak{W}^\mathbb{N}$ conditioned on the central obstacles being in position $\omega\in\Omega$ and with
$x$ initially distributed according to $f d\mu_{\mbox{\tiny SRB}}$.
\begin{cor}
There exist $C>0$ and $\vartheta\in (0,1)$ such that for all $f, g \in \mathcal{C}_{c,A,L}(\delta)$ with $\int_M f = \int_M g =1$
and $\mathbb{P}$-a.e. $\omega \in \Omega$, if
$z_0=(0,0)$, then for all $n \ge 0$ and
all $w \in \mathfrak{W}^\mathbb{N}$,
\[
\left| \mathbb{P}_{\omega, f}(w_{ k_n} \; | \; w_{ k_0} \ldots w_{ k_{n-1}}) - \mathbb{P}_{\omega, g} ( w_{k_n} \; | \; w_{k_0} \ldots w_{k_{n-1}}) \right| \leq C \vartheta^n.
\]
\end{cor}
\begin{proof}
The proof is the same as that of Theorem~\ref{thm:mixing} since \eqref{eq:integral contract} holds as well with
$\bar \mathcal{L}_m f$ replaced by $g$.
\end{proof}
\newpage
\small
|
0706.1385
|
\section{Introduction and preliminaries}
The notion of fuzzy sets was introduced by Zadeh \cite{Zad}. From then, various concepts
of fuzzy metric spaces were considered in \cite{Geo, Kal, Mih}. Many authors have studied fixed point theory in fuzzy metric spaces. The most interesting references are \cite{Fa1, Fa2, Gre, Had, Pap, Raz}. Some works on intuitionistic fuzzy metric/normed spaces has been carried out intensively in \cite{Raf, Sad}.
In the sequel, we shall adopt usual terminology, notation and convensions of fuzzy
metric spaces introduced by George and Veeramani \cite{Geo}.
\begin{definition}A binary operation $\ast \colon [0,1] \times [0,1]\to [0,1]$ is a continuous $t$-norm if $([0,1],\ast)$ is a topological monoid with unit 1 such that $a \ast b \leq c \ast d$ whenever $a \leq c$ and $b \leq d$ for $a,b,c,d \in [0,1]$.
\end{definition}
\begin{definition}A {\em fuzzy metric space}(briefly $FM$-space) is a triple $(X,M,\ast)$ where $X$ is an arbitrary set, $\ast$ is a continuous $t$-norm and $M\colon X\times X\times [0,+\infty]\to [0,1]$ is a (fuzzy) mapping satisfied the following conditions:\\
For all $x,y,z\in X$ and $s,t>0$,\\
(FM1) $M(x,y,0)=0$;\\
(FM2) $M(x,y,t)>0$;\\
(FM3) $M(x,y,t)=1$ if and only if $x=y$;\\
(FM4) $M(x,y,t)=M(y,x,t)$;\\
(FM5) $M(x,z,t+s)\geq M(x,y,t)\ast M(y,z,s)$;\\
(FM6) $M(x,y,\cdot)\colon (0,+\infty)\to [0,1]$ is continuous for all ${x,y} \in X$.
\end{definition}
\begin{lemma}$M(x,y,\cdot)$ is a nondecreasing function for all ${x,y}\in X$.
\end{lemma}
Let $U_M$ denote the $M$-uniformity, the uniformity generated by fuzzy metric $M$. Then the family
\[ \{U_{\epsilon,\lambda}\colon\epsilon > 0,\lambda \in (0,1)\}\] where
\[U_{\epsilon,\lambda}=\{(x,y)\in X\times X \colon M(x,y,\epsilon)>1-\lambda\},\]
and
\[\{U_{\lambda}\colon \lambda > 0\}\] where \[U_{\lambda}=\{(x,y)\in X \times X \colon M(x,y,\lambda)>1-\lambda\}\]
are bases for this uniformity.
\begin{remark}Every continuous $t$-norm $\ast$ satisfies \[\sup_{a<1}(a\ast a)=1\] to ensure the existence of the $M-uniformity$ on $X$.
\end{remark}
\medskip
\begin{lemma}\label{lem} If $\ast$ is a continuous $t$-norm, then if $r\in (0,1)$, there is a $s\in (0,1)$ such that $s\ast s\geq r$.
\end{lemma}
\medskip
\begin{definition}In $FM$-space $(X,M,\ast)$, the mapping $f\colon X\to X$ is said to be {\em fuzzy continuous} at $x_0$ if and only if for every $t>0$, there exists $s>0$ such that
\[M(x_{0},y,s)>1-s\Rightarrow M(fx_{0},fy,t)>1-t.\]
\end{definition}
\medskip
The mapping $f\colon X\to X$ is fuzzy continuous if and only if it is fuzzy continuous at every point in $X$.
\medskip
\begin{theorem}Let $(X,M,\ast)$ be a $FM$-space and $U_{M}$ be the $M$-uniformity induced by the fuzzy metric $M$. Then the sequence $\{x_{n}\}in X$ is said to be {\em fuzzy convergence\/} to $x\in X$ (in short, $x_{n}\to x$) if and only if \[\lim_{n\to\infty}M(x_{n},x,t)=1\] for all $t>0$.
\end{theorem}
\medskip
\begin{definition} A sequence $\{x_n\}$ in a $FM$-space $(X,M,\ast)$ is a fuzzy Cauchy sequence if and only if \[\lim_{n\to +\infty}M(x_{m},x_{n},t)=1\] for every $m,n,>0$ and $t>0$.
\end{definition}
\medskip
\begin{definition} The $FM$-space $(X,M,\ast)$ is said to be {\em fuzzy compact\/} if every sequence $\{x_{n}\}$ in $X$ has a subsequence $\{x_{n_k}\}$ such that $x_{n_k}\to x\in X$.
\end{definition}
\medskip
A fuzzy metric space in which every fuzzy Cauchy sequence is convergent is called a {\em complete fuzzy metric space\/}.
\medskip
\begin{definition} Let $(X,M,\ast)$ be a $FM$-space and $A$ be a nonempty subset of $X$. The fuzzy closure of $A$, denoted by $\overline A$ is the set
\[\overline A=\{y\in X\colon \exists x\in A, M(x,y,\epsilon)>1-\lambda, \epsilon >0, \lambda\in (0,1)\}\]
\end{definition}
\section{Main Results}
\noindent In our fixed point theorem we consider the FM-space $(X,M,\ast)$ endowed with the $M$-uniformity.
\begin{definition}\label{def1}Let $\Phi$ be the class of all mappings $\varphi\colon\mathbb R^{+}\to\mathbb R^{+}$ ($\mathbb R^{+} = [0,+\infty)$) with the following properties:
\begin{enumerate}
\item $\varphi$ nondecreasing,
\item $\varphi$ is right continuous,
\item $\lim_{n\to +\infty}\varphi ^{n}(t)=0$ for every $t>0$.
\end{enumerate}
\end{definition}
\medskip
\begin{remark}\label{rem1}(a) It is easy to see that under these conditions, the function $\varphi$ satisfies also $\varphi(t)< t$ for all $t>0$ and therefore $\varphi(0)= 0$.\\
(b) By property (iii), we mean that for every $\epsilon > 0$ and $\lambda\in (0,1)$ there exists an integer $N(\epsilon,\lambda)$ such that $\varphi^{n}(t) \leq min\{\epsilon,\lambda\}$ whenever $n \geq N(\epsilon,\lambda)$.
\end{remark}
\medskip
In \cite{Gol}, Golet has introduced $g$-contraction mappings in probabilistic metric spaces. In the following definition, we give the $g$-contraction in the fuzzy setting.
\medskip
\begin{definition}\label{def2}Let $f$ and $g$ are mappings defined on a $FM$-space $(X,M,\ast)$ and let suppose that $g$ is bijective. The mapping $f$ is called a fuzzy {\em $g$-contraction\/} if there exists a $k\in (0,1)$ such that for every ${x,y}\in X$, $t>0$,
\begin{eqnarray}\label{add1}
M(gx,gy,t)>1-t \Rightarrow M(fx,fy,kt)>1-kt.
\end{eqnarray}
\end{definition}
\medskip
By considering a mapping $\varphi \in \Phi$ as given in Definition ~\ref{def1}, we generalize the condition (\ref{add1}) in Definition ~\ref{def2} as follow.
\medskip
\begin{definition}\label{def3}
Let $(X,M,\ast)$ be a $FM$-space and $\varphi \in \Phi$. We say that the mapping $f\colon X \to X$ is a fuzzy {\em $(g,\varphi)$-contraction\/} if there exists a bijective function $g\colon X \to X$ such that for every ${x,y}\in X$ and for every $t>0$,
\begin{eqnarray}\label{add2}
M(gx,gy,t)>1-t \Rightarrow M(fx,fy,\varphi (t))>1- \varphi (t).
\end{eqnarray}
\end{definition}
\medskip
\noindent Note that, if $\varphi (t) = kt$ for $k \in (0,1)$, $t>0$, then the condition (\ref{add2}) is actually the fuzzy $g$-contraction due to Golet \cite{Gol}. If the function $g$ is identity function, then (\ref{add2}) represents fuzzy ($\varphi - H$)- contraction according to Mihet \cite{Mih}. Hence the $(g,\varphi)$-contraction generalizes the Golet and Mihet's contraction principles respectively in fuzzy metric spaces.
\medskip
The following lemma is reproduced from \cite{Gol} to suits our purposes in fuzzy metric spaces..
\medskip
\begin{lemma}\label{lem1}Let $g$ be an injective mapping on $(X,M,\ast)$.\\
(i) If $M^{g}(x,y,t)=M(gx,gy,t)$, then $(X,M^{g},\ast)$ is a fuzzy metric space.\\
(ii) If $X_{1}=g(X)$ and $(X,M,\ast)$ is a complete $FM$-space then $(X,M^{g},\ast)$
is also a complete $FM$-space.\\
(iii) If $(X_{1},M,\ast)$ is fuzzy compact then $(X,M^{g},\ast)$ is also fuzzy compact.
\end{lemma}
\begin{proof}The proof of (i) and (ii) are immediate. To prove (iii), let $\{x_{n}\}$ be a sequence in X. Then, for $u_{n}=gx_{n}$, $\{u_{n}\}$ is a sequence in $X_{1}$ for which we can find a convergent subsequence $\{u_{n_{k}}\}$, say $u_{n_{k}}\to u\in X_{1}$ for $k\to +\infty$. Suppose that the sequence $\{y_{n}\}$ and $y$ in $X$, and set $y_{n}=g^{-1}u_{n_{k}}$, and $y = g^{-1}u$. Then
\[ M^{g}(y_{n_{k}},y,t) = M(gy_{n_{k}},gy,t) = M(u_{n_{k}},u,t) \to 1\]
as $k \to +\infty$ for every $t>0$. This implies that $(X,M^{g},\ast)$ is fuzzy compact.
\end{proof}
\medskip
\begin{lemma}\label{lem2} If $f$ is a fuzzy $(g,\varphi)$-contraction. Then\\
(i) $f$ is a fuzzy (uniformly) continuous mapping on $(X,M^{g},\ast)$ with values in $(X,M,\ast)$.
(ii) $g^{-1}\circ f$ is a continuous mapping on $(X,M^{g},\ast)$ with values into itself.
\end{lemma}
\begin{proof}(i) Let $\{x_{n}\}$ a sequence in $X$ such that $x_{n}\to x$ in $X$. In $(X,M^{g},\ast)$, this implies that \[ \lim_{n\to +\infty}M^{g}(x_{n},x,t)=\lim_{n\to +\infty}M(gx_{n},gx,t)=1, \forall t>0.\]. By the fuzzy $(g,\varphi)$-contraction (\ref{add2}) it follows that \[ \lim_{n\to +\infty}M(fx_{n},fx,t)\geq \lim_{n\to +\infty}M(fx_{n},fx,\varphi (t))=1, \forall t>0.\]. This implies that $f$ is fuzzy continuous.\\
(ii) Note that, since
\[ \lim_{n\to +\infty}M(gg^{-1}fx_{n},gg^{-1}fx,t)= \lim_{n\to +\infty}M^{g}(g^{-1}fx_{n},g^{-1}fx,t)=1, \forall t>0,\]
this shows that the mapping $g^{-1} \circ f$ defined on $(X,M^{g},\ast)$ with values in itself is fuzzy continuous.
\end{proof}
\medskip
\begin{theorem}\label{main} Let $f$ and $g$ are two function defined on a complete $FM$-space $(X,M,\ast)$. If $g$ is bijective and $f$ is fuzzy $(g,\varphi)$-contraction, then there exists a unique coincidence point $z\in X$ such that $gz=fz$.
\end{theorem}
\begin{proof} It is obvious that $M^{g}(x,y,t)>1-t$ whenever $t>1$. Hence, we have $M(gx,gy,t)>1-t$. By condition (\ref{add2}), we have $M(fx,fy,\varphi (t))>1-\varphi (t)$. But,
\begin{eqnarray*}
M(fx,fy,\varphi (t)) & = & M(gg^{-1}fx,gg^{-1}fy,\varphi(t))\\
& = & M^{g}(g^{-1}fx,g^{-1}fy,\varphi(t))\\
& > & 1-\varphi(t).
\end{eqnarray*}
Now, by letting $h=g^{-1}f$, we have
\[ M^{g}(hx,hy,\varphi (t))>1-\varphi(t).\]
By condition (\ref{add2}), we have
\begin{eqnarray*}
M(fhx,fhy, \varphi^{2}(t)) &=& M(gg^{-1}fhx,gg^{-1}fhy,\varphi^{2}(t))\\
&=& M^{g}(g^{-1}fhx,g^{-1}fhy,\varphi^{2}(t))\\
&=& M^{g}(h^{2}x,h^{2}y,\varphi^{2}(t))\\
&>& 1-\varphi^{2}(t).
\end{eqnarray*}
By repeating this process, we have \[ M^{g}(h^{n}x,h^{n}y,\varphi^{n}(t))>1-\varphi^{n}(t).\]
Since $\lim_{n\to +\infty}\varphi^{n}(t)=0$, then for every $\epsilon >0$ and $\lambda \in (0,1)$, there exists a positive integer $N(\epsilon,\lambda)$ such that $\varphi^{n}(t)\leq min(\epsilon, \lambda)$, whenever $n\geq N(\epsilon,\lambda)$. Furthermore, since $M$ is nondecreasing we have
\[ M^{g}(h^{n}x,h^{n}y,\epsilon)\geq M^{g}(h^{n},h^{n},\varphi^{n}(t))>1-\varphi^{n}(t)>1-\lambda.\]
Let $x_{0}$ in $X$ to be fixed and let the sequence $\{x_{n}\}$ in $X$ defined recursively by $x_{n+1}=hx_{n}$, or equivalently by $gx_{n+1}=fx_{n}$. Now, consider $x=x_{p}$ and $y=x_{0}$, then from the above inequality we have
\[ M^{g}(x_{n+p},x_{n},\epsilon) = M^{g}(h^{n}x_{p},h^{n}x_{0},\epsilon)>1-\lambda,\]
for every $n\geq N(\epsilon,\lambda)$ and $p\geq 1$. Therefore $\{x_{n}\}$ is a fuzzy Cauchy sequence in $X$. Since $(X,M,\ast)$ is complete, by Lemma \ref{lem1}, $(X,M^{g},\ast)$ is complete and there is a point $z\in X$ such that $x_{n}\to z$ under $M^{g}$. Since by Lemma \ref{lem2}, $h$ is continuous, we have $z=hz$, i.e., $z=g^{-1}fz$, or equivalently $gz=fz$. For the uniqueness, assume $gw=fw$ for some $x\in X$. Then, for any $t>0$ and using (\ref{add2}) repeatedly, we can show that after $n$ iterates, we have
\[ M(gz,gw,t)>1-t \Rightarrow M(fz,fw,\varphi^{n}(t))>1-\varphi^{n}(t).\]
Thus, we have $\lim_{n\to +\infty}M(fz,fw,\varphi^{n}(t))=1$ which implies that $fz=fw$.
\end{proof}
\medskip
As an example, if we take in consideration that every metric space $(X,d)$ can be made into a fuzzy metric space $(X,M,\ast)$, in a natural way, by setting $M(x,y,t)= \frac{t}{t+d(x,y)}$, for every ${x,y}\in X$, $t>0$ and $a\ast b=ab$ for every $a,b\in [0,1]$, by Theorem ~\ref{main} one obtains the following fixed point theorem for mappings defined on metric spaces.
\begin{example}Let $f$ and $g$ are two mappings defined on a non-empty set $X$ with values in a complete metric space $(X,d)$, $g$ is bijective and $f$ is a $(g,\varphi)$-contraction, that is, there exists a $\varphi \in \Phi$ such that
\[ d(fx,fy) \leq \varphi(d(gx,gy)),\]
for every ${x,y}\in X$, then there exists a unique element $x^{*}\in X$ such that $fx^{*}=gx^{*}$.
\begin{proof} We suppose that $d(x,y)\in [0,1]$. If this is not true, we define the mapping $d_{1}(x,y)=1-e^{-d(x,y)}$, then the pair $(X,d_{1})$ is a metric space and the uniformities defined by the metrics $d$ and $d_{1}$ are equivalent.\\
Now, let suppose that $f$ is a fuzzy $(g,\varphi)$-contraction for some $\varphi\in \Phi$ on $(X,d)$, $t>0$, $M(gx,gy,t)>1-t$. Then we have $\frac{t}{t+d(gx,gy)}>1-t$. This implies $d(gx,gy)<t$ and consequently, we have, $\varphi(d(gx,gy))<\varphi(t)$ for all $t>0$. Thus, $d(fx,fy)<\varphi(t)$ which implies that $\frac{\varphi(t)}{\varphi(t)+d(fx,fy)}>1-\varphi(t)$, i.e., $M(fx,fy,\varphi(t))>1-\varphi(t)$. So, the mapping $f$ is a fuzzy $(g,\varphi)$-contraction defined on $(X,M,\ast)$ and the conclusion follows by the Theorem ~\ref{main}.
\end{proof}
\end{example}
\medskip
As a multivalued generalization of the notion of $g$-contraction (Definition \ref{def3}), we shall introduce the notion of a fuzzy $(g,\varphi)$-contraction where $\varphi \in \Phi$ for a multivalued mapping.
\medskip
Let $2^{X}$ be the family of all nonempty subsets of $X$ .
\begin{definition}\label{def4}Let $(X,M,\ast)$ be a $FM$-space, $A$ is a nonempty subset of $X$ and $T\colon A \to 2^{X}$. The mapping $T$ is called a fuzzy $(g,\varphi)$-contraction, where $\varphi \in \Phi$ if there exists a bijective function $g\colon X \to X$ such that for every ${x,y}\in X$ and every $t>0$,
\begin{eqnarray}
M^{g}(x,y,t)>1-t \Rightarrow \\
\nonumber\forall u \in (T\circ g)(x), \exists v\in (T\circ g)(y)\colon M(u,v,\varphi(t))>1-\varphi(t).
\end{eqnarray}
\end{definition}
\medskip
\begin{definition} Let $(X,M,\ast)$ be a $FM$-space, $A$ is a nonempty subset of $X$ and $T\colon A \to 2^{X}$. We say that $T$ is {\em weakly fuzzy demicompact\/} if for every sequence $\{x_{n}\}$ from $A$ such that $x_{n+1}\in Tx_{n}, n\in\mathbb{N}$ and \[\lim_{n\to+\infty}M(x_{n+1},x_{n},t)=1\]
for every $t>0$, there exists a fuzzy convergent subsequence $\{x_{n_{k}}\}$.
\end{definition}
\medskip
By $cl(X)$ we shall denote the family of all nonempty closed subsets of $X$.
\medskip
\begin{theorem}\label{main2}Let $(X,M,\ast)$ be a complete $FM$-space, $g\colon X \to X$ be a bijective function and $T\colon A \to cl(A)$ where $A\in cl(A)$ a fuzzy $(g,\varphi)$-contraction, where $\ varphi \in \Phi$. If $T$ is weakly fuzzy demicompact then there exists at least one element $x\in A$ such that $x\in Tx$.
\end{theorem}
\begin{proof}Let $x_{0}$, $x_{1}\in (T\circ g)(x_{0})$. Let $t>0$. Since $M^{g}(x_{1},x_{0},t)>0$, it follows that
\[M^{g}(x_{1},x_{0},t)>1-t.\]
The mapping $T$ is a fuzzy $(g,\varphi)$-contraction, therefore by Definition\ref{def4} there exists $x_{2}\in (T\circ g)(x_{1})$ such that
\[M(x_{2},x_{1},\varphi(t))>1-\varphi(t).\]
Since $M(x_{2},x_{1},\varphi(t))=M^{g}(g^{-1}x_{2},g^{-1}x_{1},\varphi(t))$, we have
\[ M^{g}(g^{-1}x_{2},g^{-1}x_{1},\varphi(t))>1-\varphi(t).\]
Similarly, it follows that there exists $x_{3}\in (T\circ g)(x_{2})$ such that
\[M(x_{3},x_{2},\varphi^{2}(t))>1-\varphi^{2}(t).\]
By repeating the above process, there exists $x_{n}\in (T\circ g)(x_{n-1})$ $(n\geq 4)$ such that
\[M(x_{n},x_{n-1},\varphi^{n-1}(t))=1-\varphi^{n-1}(t), \forall t>0.\]
By letting $n\to +\infty$, \[\lim_{n\to +\infty}M(x_{n},x_{n-1},\varphi^{n-1}(t))=1, \forall t>0.\]
Further, by Remark \ref{rem1}(ii), we have \[\lim_{n\to +\infty}M(x_{n},x_{n-1},\epsilon)=1, \forall t>0.\]
Since $T$ is weakly fuzzy demicompact from the above limit, there exists a convergent fuzzy subsequence $\{x_{n}\}$ such that $\lim_{k\to +\infty}x_{n_{k}}=x$ for some $x\in A$. Now, we show that $x\in (T\circ g)(x)$. Since $(T\circ g)(x)=\overline{(T\circ g)(x)}$, we shall prove that $x\in \overline{(T\circ g)(x)}$, i.e. for every $\epsilon >0$ and $\lambda\in (0,1)$, there exists $y\in (T\circ g)(x)$ such that \[M(x,y,\epsilon)>1-\lambda.\]
Note that, since $\ast$ is a continuous $t$-norm, by Lemma \ref{lem}, for $\lambda\in (0,1)$ there is a $\delta\in (0,1)$ such that \[(1-\delta)\ast (1-\delta)\geq 1-\lambda.\]
Further, if $\delta_{1}\in (0,1)$ is such that \[(1-\delta_{1})\ast (1-\delta_{1})\geq 1-\delta\]
and $\delta_{2}= min\{\delta,\delta_{1}\}$, we have
\begin{eqnarray*}
(1-\delta_{2})\ast [(1-\delta_{2})\ast (1-\delta_{2})] &\geq& (1-\delta)\ast [(1-\delta_{1})\ast (1-\delta_{1})]\\
&\geq& (1-\delta)\ast (1-\delta)\\
&>& 1-\delta.
\end{eqnarray*}
Since $\lim_{k\to +\infty}x_{n_{k}}=x$, there exists an integer $k_{1}$ such that \[M(x,x_{n},\epsilon/3)>1-\delta_{2}, \forall k\geq k_{1}.\].
Let $k_{2}$ be an integer such that
\[M(x_{n},x_{n+1},\epsilon/3)>1-\delta_{2}, \forall k\geq k_{2}.\]
Let $s>0$ be such that $\varphi(s)< min\{\epsilon/3,\delta_{2}\}$ and $k_{3}$ be an integer such that \[M^{g}(x_{n_{k}},x,s)>1-s, \forall k\geq k_{3}.\]
Since $T$ is $(g,\varphi)$-contraction there exists $y\in (T\circ g)(x)$ such that
\[M(x_{n_{k}+1},y,\varphi(s))>1-varphi(s),\]
and so
\[ M(x_{n_{k}+1},y,\epsilon /3)
\geq (M(x_{n_{k}+1},y,\varphi(s))
>1-\varphi(s)>1-\delta_{2}\]
for every $k\geq k_{3}$. If $k\geq max\{k_{1},k_{2},k_{3}\}$, we have
\begin{eqnarray*}
M(x,y,\epsilon) &\geq& M(x,x_{n_{k}},\epsilon /3)\ast (M(x_{n_{k}}, x_{n_{k}+1},\epsilon /3)\ast M(x_{n_{k}+1},y, \epsilon /3)\\
&\geq& (1-\delta_{2})\ast((1-\delta_{2})\ast (1-\delta_{2}))\\
&>& 1-\lambda.
\end{eqnarray*}
Hence $x\in\overline{(T\circ g)(x)}=(T\circ g)(x)$. The proof is completed.
\end{proof}
Note that, Theorem \ref{main2} above generalizes the theorem proved by Pap et al in \cite{Pap}.
\begin{remark}
We note that Theorem \ref{main} can be obtained in a more general setting, namely when $g$ does not satisfy the injective conditions. In this case we can use a pseudo-inverse of $g$ defined as a selector of the multi-valued inverse of $g$. We will consider this situation in our next paper.
\end{remark}
|
0706.2317
|
\section{Introduction}
The transitions \upsDec{n}{m} are of particular interest as probes of
heavy quark and low energy QCD systems. The large $b$~quark mass
causes the \ensuremath{b\bar{b}}\xspace bound state to have a very small radius
($\sim 1 $~GeV$^{-1}$) and to be
non-relativistic ($(v/c)^{2}\approx 0.1$).
This makes these transitions ideal to
study the process by which a pion pair is excited from the vacuum by
the gluon field. The transitions
among
the massive bound states
making up the \ups{n} family can be
calculated
in terms
of multipole moments of the chromo-dynamic field, providing simple
relative rate and transition rule predictions. The pion pair
excitation can be factored out and approximated separately. Most
recent theoretical work has concentrated on this latter aspect of the
decays.
The \ensuremath{\eta^{\prime} \rightarrow \eta \pi \pi}\xspace transition was the first decay of this form studied~\cite{London66},
followed some years later by the \ensuremath{\psi^{\prime} \rightarrow J/\psi \pi \pi}\xspace
transition~\cite{psiprime}. The $\eta'$
decay is only barely above threshold, and so the transition cannot
show significant structure.
Detailed study of the
kinematics confirmed this. In contrast to this, the $\psi'$ decay has
decay dynamics very different from a phase space distribution. The
di-pion invariant mass distribution of this decay shows strong
enhancement at larger values of ${\ensuremath{M_{\pi\pi}}}$. However, this is consistent
with the presence of only the simplest term in the general Lorentz
invariant amplitude derived from
PCAC
considerations \cite{Brown:1975dz,MBVoloshin:75}.
This is supported by the isotropic
decay angular distribution of the pions, implying a minimal $D$-wave
component.
Previous CLEO data have been used to study
\upsDec{n}{m}
transitions \cite{Butler:1993rq,Glenn:1998bd,Alexander:1998dq,Brock:pj},
with the \upsDec{2}{1} and \upsDec{3}{2} transitions following this same
pattern
in the di-pion invariant mass spectra as for the lighter mesons.
But the \upsDec{3}{1} transition has a second, strong rate enhancement near
the \ensuremath{\pi\pi}\xspace invariant mass threshold. This enhancement and the
accompanying depletion at intermediate invariant mass
are
inconsistent
with either pure phase space or the simple matrix element describing
the \ensuremath{\psi^{\prime} \rightarrow J/\psi \pi \pi}\xspace observations.
Either another term must be included in the
Lorentz invariant matrix element, or one must question the
applicability of PCAC to the pion excitation and the validity of
the multipole expansion of the \ensuremath{b\bar{b}}\xspace bound state.
Various mechanisms have been suggested to explain this anomaly, such as
(i) large contributions from final state interactions~\cite{BDM,CKK},
(ii) a $\sigma$ isoscalar resonance in the $\pi\pi$ system~\cite{KII,Uehara},
(iii) exotic $\Upsilon -\pi$ resonances~\cite{VoloshinJTEP, BDM, ABSZ,GSCP},
(iv) an {\it ad hoc} constant term in the amplitude~\cite{Moxhay},
(v) coupled channel effects~\cite{LipkinThuan, ZK},
(vi) $S-D$ mixing~\cite{CKK2},
and (vii) relativistic corrections~\cite{Voloshin:2006}.
More recent
experimental
analyses
with the very large
data sets accumulated by the $B$~factories at the $\ups{4}$ show
interesting behavior as well. Belle~\cite{Abe:2005bi} and
$B$a$B$ar~\cite{BaBar4S2S1S} do not see such anomalous
behavior in the \upsDec{4}{1} transition, but
$B$a$B$ar
does see such a double peaked
structure in the
\upsDec{4}{2} transition.
The shapes of the decay distributions originate in the details of the
excitation of the pion pair from the vacuum and the particular
projection of the initial state onto the final state. Hence, the
enhancement of the decay rate at low ${\ensuremath{M_{\pi\pi}}}$, thus far considered an
anomaly, is a good probe of the details of low energy QCD in the
transitions of the bound states and the excitation of light hadrons
from the vacuum.
The general matrix element constrained by
PCAC
was
derived by Brown and Cahn \cite{Brown:1975dz} and is further
constrained by treating the Upsilon transition
as
a multipole
expansion as derived by Gottfried \cite{Gottfried:1977gp},
Yan \cite{Yan:1980uh}, Voloshin and Zakharov \cite{Voloshin:1980zf},
and others. The general
transition amplitude is then given in non-relativistic form:
\begin{equation}
\clg{M} = \clg{A} \dpr{{\ensuremath{\epsilon'}}}{{\ensuremath{\epsilon}}} ( q^2 - 2 M_{\pi}^2 )\,+
\clg{B} \dpr{{\ensuremath{\epsilon'}}}{{\ensuremath{\epsilon}}} E_1 E_2 +
\clg{C} ( (\epsilon'\cdot q_1) (\epsilon\cdot q_2) +
(\epsilon'\cdot q_2) (\epsilon\cdot q_1) )~,
\label{Eq:BandCAmplitude}
\end{equation}
where ${\ensuremath{\epsilon'}}$ and ${\ensuremath{\epsilon}}$ are the polarization vectors of the parent and
final state Upsilons, and $q_{1,2}$ are
the
four-momenta
of the pions. In the first term, $q^2$ is the
invariant mass of the pion pair. The quantities $E_1$
and $E_2$ are the energies of the two pions in the parent rest frame,
essentially indistinguishable from the lab frame due to the large
masses of the Upsilons.\footnote {For transitions from the
$\Upsilon$(3S), the parent frame and lab frame are
virtually
identical. Even for $\upsDec{2}{1}$ transitions,
in which the $\Upsilon$(2S) comes from hadronic or electromagnetic
transitions from the $\Upsilon$(3S), the
parent's motion in the lab frame is unobservable other than in a small
broadening of recoil mass peak and a slight smearing of reconstructed
variables.}
The third, or ``${\cal C}$'' term in this
expression couples transitions via the chromo-magnetic moment of the
bound state $b$~quarks, hence requiring a spin flip. This is expected
to be
highly
suppressed by the large mass of the $b$~quark, so we expect only the
first two terms to contribute.
Neglecting the dependence on the parent and final state
Upsilon polarizations (which apply only to the
${\cal C}$-term), we have only
two degrees of freedom, the Dalitz variables $q^2 = M^2_{\pi\pi}$ and
$r^2 = M^2_{\Upsilon\pi}$.
In writing this amplitude, we have assumed the chiral limit, so that
a fourth term, $gM_{\pi}^{2}$, is taken to be
zero~\cite{MannelUrech, newVoloshin}.
The
expression in Eqn.~\ref{Eq:BandCAmplitude} can be made
fully Lorentz invariant by rewriting the energy product in the ${\cal B}$
term as
\begin{equation}
E_1 E_2 \approx [(P^{\prime}\cdot q_{1})(P\cdot q_{2}) + (P^{\prime}\cdot q_{2})(P\cdot q_{1})]/[2M_{\Upsilon '}M_{\Upsilon}],
\label{Eq:E1E2expansion}
\end{equation}
with ${\ensuremath{P'}}$ and ${\ensuremath{P}}$ being the initial state and final state $\Upsilon$
four-momenta.
The quantities $\clg{A}$, $\clg{B}$, and $\clg{C}$ are form factors
that depend on the detailed dynamics of the decay. They are in
principle functions of the Dalitz variables $q^2$ and $r^2$. However,
we expect them to vary on the scale of $\ensuremath{\Lambda_{\rm QCD}}\xspace$, which is
comparable to the total energy release of the decays, so to first
order we assume they are complex constants. Angular structure or ${\ensuremath{M_{\pi\pi}}}$
dependence beyond that indicated in the explicit amplitude,
Eqn.~\ref{Eq:BandCAmplitude}, would be an indication of the
non-constancy of these form factors, or alternately the breakdown of
the assumptions leading to Eqn.~\ref{Eq:BandCAmplitude}.
The di-pion transition can be interpreted as taking place in
sequential two-body decays through a fictitious intermediate state $X$
via the chain $\upsXDec{n}{m}$ and $X \rightarrow \pi \pi$ (see
Fig.~\ref{fig:upsDec}). In this
view we can define the helicity angle of the $X \rightarrow \pi \pi$
decay in the usual manner of the Jacob and Wick formalism. The polar
helicity angle is referred to as $\theta_{X}$.
Its cosine is used interchangeably
with the second Dalitz variable, $r^2$, as they are linearly related:
\begin{equation}
2 r^2 = 2 M_\pi^2 + M_{\Upsilon'}^2 + M_{\Upsilon}^2 - q^2
- \ensuremath{\cos \theta_X}\xspace \sqrt{ \frac{1}{q^2}
\left( q^2 - 4 M_\pi^2 \right)
\Lambda_3( M_{\Upsilon'}^2,
M_{\Upsilon}^2, q^2 ) }
\end{equation}
where $\Lambda_3( a, b, c ) = a^2 + b^2 + c^2 - 2 a b - 2 a c - 2 b
c$.
These variables ($r^{2}$ or $\ensuremath{\cos \theta_X}\xspace$) carry structure
from the second term in the
amplitude due to the following relation:
\begin{equation}
E_1 E_2 = \frac{1}{4}\left( ( E_1 + E_2 )^2 - \Delta E^{2}_{\rm max}
\cos^2 \theta_X \right)~,
\end{equation}
with $\Delta E \equiv E_{2} - E_{1}$.
Because the initial state and final state Upsilons are essentially at rest, the
energy sum $E_1 + E_2$ is
nearly
a constant and equal to the mass
difference between the Upsilons.
For the $\pi^{+}\pi^{-}$ final state, $\theta_{X}$ is defined as the
angle of the
positive pion, with
$-1 < \cos \theta_X < 1$; for the $\pi^{0}\pi^{0}$
final state, because
one cannot distinguish between the two neutral pions, we take
$0 < \cos \theta_X < 1$.
\begin{figure*}
\centerline{
\resizebox{0.80\textwidth}{!}{
\includegraphics{Figure1.eps}}
}
\caption[The decay process \upsDec{n}{m}]
{ (Left) The decay \upsDec{n}{m} follows the production
of an initial state labeled $\Upsilon'$ which decays to
an $\Upsilon \pi \pi$ state. In our analysis, the final state $\Upsilon$
decays to a lepton pair
whose momentum vectors are
very nearly back-to-back
due to the large energy release. (Center) The decay of
the initial state Upsilon is governed by two kinematic
variables, the Dalitz masses $M_{\Upsilon\pi}$ and
$M_{\pi\pi}$. (Right) Alternately one can think of the
$\pi \pi$ system as a composite, $X$, and study its structure
via the pion ``decay'' angles.}
\label{fig:upsDec}
\end{figure*}
Finding the presence of a non-zero $\clg{C}$ term would indicate the
breakdown of the multipole expansion, {\it i.e.,} of the
assumption that the pion pair excitation is independent of the Upsilon
transition process from $n^3S_1$ state to
$m^3S_1$, and that the spin flip of the $b$ quarks is
suppressed. However, finding a non-zero $\clg{C}$ term
could also be due to distortions of the
distribution not accountable for by using only the first two
terms with complex, but {\it constant}, coefficients ${\cal A}$ and ${\cal B}$.
\section{Data Sets and Event Selection}
Data were collected with the CLEO~III detector which is described in
detail
elsewhere~\cite{Viehhauser:2001ue,Peterson:2002sk,Artuso:2005dc}.
In this analysis we observe
$e^\pm$, $\mu^\pm$, $\pi^\pm$, and $\gamma$
particles in the final state, and so
use
both the tracking and
calorimetry information from the detector, as well as lepton
identification. Thus we
employ
global event, track, lepton,
shower, and neutral pion selection criteria, in addition to signal and
background identification criteria.
The data were taken while running
on the $\ups{3}$ resonance, subject to standard CLEO data quality
selections,
and represent an integrated luminosity of
\nUnits{1.14}{\ensuremath{{\rm fb}^{-1}}\xspace},
and an $\ups{3}$ production yield of
$(\nError{4.98}{0.01})\times10^6$. The $\ups{2}$ sample is
obtained by reconstruction of sequential decays, $\ups{3} \rightarrow
\ups{2}\,+\,{\rm anything}$, occurring in this sample. The $\ups{2}$
population of $(\nError{5.27}{0.40})\times10^5$
is estimated from the branching fraction \cite{PDG:2006} of
$\nError{10.6\%}{0.8\%}$ for the decay $\ups{3} \rightarrow
\ups{2}\,+\,{\rm anything}$, which is dominated by pion pair transitions and
sequential photon decays through the $\chi_b$(2P) states.
All integrals needed in the analysis (for evaluation of acceptances
and efficiencies) are calculated via the Monte Carlo method. Physics
event generation is performed
using the Lund Monte
Carlo~\cite{Sjostrand:2001yu} embedded in the CLEO physics Monte Carlo
{\tt QQ}~\cite{QQ}. The Lund event generator is used because it accurately accounts for the
physics of the QCD bound state production. The $\ups{3}$ produced in
the $e^+e^-$ collision is then decayed according to
standard
decay
tables and the detector response to the decay products is simulated
using
the physics simulation package
{\tt GEANT}~\cite{CERNLib:GEANT}.
In general, since all integrals are performed with respect to the
natural measure over phase space, only phase space decays need be
simulated. The decay amplitude is known exactly as a function of the
decay kinematics, so all inputs to the matrix element extraction
(other than acceptance and efficiency) are known to the precision of
detector reconstruction.
We select events containing two leptons ($\mu^+\mu^-$ or $e^+e^-$) and
two pions ($\pi^+\pi^-$ or $\pi^0\pi^0$).
All low momentum tracks are assumed to be pions, because
there is insufficient phase space for the production of
a pair of kaons in a transition among any two of the
three bound state Upsilons.
Electrons and muons are identified by their energy loss and
penetration depth in the detector as detailed below, and are required
to be consistent with originating from either an \ups{2} or an \ups{1}
decay. The pion candidates are constrained to come from a common point
at the beam location and the recoil mass
($M_{\rm rec}^2 = P_{\rm rec}\cdot P_{\rm rec}; P_{\rm rec} \equiv P_{\rm
beam} - q_1 - q_2$; see below) is used to identify the transition. The
lepton pair invariant mass spectra and the recoil mass spectra are
shown in Figs.~\ref{fig:lepMassSpectra}
and~\ref{fig:recMassSpectra}, respectively.
\begin{figure*}
\centerline{
\resizebox{0.66\textwidth}{!}{
\includegraphics{Figure2.eps}}
}
\caption[Di-lepton mass spectra, raw and with signal selections] {
Di-lepton invariant mass distributions for lepton
pairs; the abscissa is the di-lepton invariant mass,
showing peaks at the masses of the \ups{1} and \ups{2}
mesons. The hatching indicates the limits to the
invariant mass selection windows.
Candidates are
plotted after the signal selection described in
Section II.D. At
left are the di-muon candidates and at right the
di-electron candidates.}
\label{fig:lepMassSpectra}
\end{figure*}
\begin{figure*}
\centerline{
\resizebox{0.66\textwidth}{!}{
\includegraphics{Figure3.eps}}
}
\caption[Recoil mass spectra, raw and with signal selections] {
Recoil mass, $M_{\rm rec}$, distributions for all modes.
The upper plot is generated from neutral decays,
$\upsNDec{n}{m}$ and the lower from charged decays,
$\upsCDec{n}{m}$. The final signal selections
(track quality, pion quality, di-lepton mass, {\it etc.}) have
been applied. The peaks at the $\Upsilon$(1S) and
$\Upsilon$(2S) masses correspond to decays to these resonances
from an $\Upsilon$(3S) parent. The peaks at 9.8 GeV/$c^{2}$ are
from $\Upsilon$(2S)$ \to\Upsilon$(1S)$\pi\pi$ decays.
The hatching shows the bounds on the recoil
mass values for the three transitions.
See also the window
definitions in Tab.~\ref{tab:RecMassPeaks}. Yields are set to zero in
the regions that correspond neither to signal nor to
sidebands.}
\label{fig:recMassSpectra}
\end{figure*}
\subsection{Global Event Selection}
The data used in this analysis are required to have been
taken while running on the \ups{3} resonance energy.
Global event characteristics are used to preselect the events.
Excessive tracks or showers in an event can dramatically increase the
combinatoric background. To avoid this, reconstructed events are
selected subject to upper limits on number of charged particle tracks
and number of calorimeter showers. To establish
conservative
limits, signal
Monte Carlo is studied for \upsDec{2}{1}
transitions, which are the ``worst case'', in that extra
tracks and showers in these modes arise from the
initial transition from the \ups{3} to the \ups{2}. Neglecting stray
particles and secondary showers, there should be no more than four low
momentum charged particle tracks and no more than eight
electromagnetic showers in signal events. Comparison between data and
Monte Carlo show good agreement in the number of tracks and showers
found in the selected events.
\subsection{Selection of Final State Particles}
All
candidate charged tracks are required to satisfy
quality criteria.
They must:
\begin{list}{}{\leftmargin36.0pt \partopsep0.0pt
\parskip0.0pt \topsep2.0pt \itemindent0.0pt \parsep2.0pt
\itemsep0.0pt \labelsep2.0pt }
\item[-]{come from within \nUnits{5}{\ensuremath{{\rm cm}}\xspace} of the origin along the beam
axis (detector $\hat{z}$ axis)};
\item[-]{come within \nUnits{5}{\ensuremath{{\rm mm}}\xspace} of the beam axis (impact
parameter)};
\item[-]{have momentum less than the beam energy;}
\item[-]{have a good helix track fit, with $\chi^2$ per hit less than 20}.
\end{list}
These requirements are applied to all track candidates and are
augmented with identification criteria for leptons (see below) before
being accepted as decay candidates.
The charged transition pions frequently are of such low transverse
momentum that they make two or more semi-circular arcs in
the tracking volume. These ``excess'' tracks are removed by
comparing the helix
parameters, taking into account
the expected energy loss as these pions spiral through the
drift chamber.
Candidate muons and
electrons are required to have high momentum by requiring their
transverse momentum to be
$p_{T} > 1$ GeV/$c$,
which removes a large fraction of the events with non-leptonic
Upsilon decays.
Because
the leptons we seek originate from the decay of
objects
more massive
than \nUnits{9.4}{\ensuremath{{\rm GeV}/c^2}\xspace}, they pass this requirement
easily.
Muons are selected from among good tracks and
are
additionally required to
penetrate the muon chambers to a depth of at least
three
interaction
lengths. The ratio of energy deposition in the calorimeter to track
momentum must also be less than one half, $E/pc < 0.5$.
Electrons are selected from among good tracks and are additionally
required to have a ratio of energy deposited in the calorimeter to
track momentum $E/pc > 0.5$, as well as having
a profile of energy deposition consistent with that of an
electromagnetic shower
and a good spatial match between the shower and the
track. The $E/pc$ ratio selection is a very loose requirement added
only as a
precaution
against muons contaminating the electron sample.
The di-lepton mass is loosely required to
be that of the final state Upsilon being studied, as shown in
Fig.~\ref{fig:lepMassSpectra}. For the $\Upsilon$(1S) we require
$9.25 < M_{\ell\ell} < 9.75$ GeV/$c^{2}$, while for the
$\Upsilon$(2S) we demand $M_{\ell\ell} > 9.85$ GeV/$c^{2}$.
Due to the large widths of these invariant mass peaks, no
side band selection is
performed in this variable, but rather only in the recoil
mass distribution.
The $\pi^0$ candidates are reconstructed from photon
pairs. This begins by applying selection criteria to the showers.
To be considered a photon, a shower must:
\begin{list}{}{\leftmargin36.0pt \partopsep0.0pt
\parskip0.0pt \topsep2.0pt \itemindent0.0pt \parsep2.0pt
\itemsep0.0pt \labelsep2.0pt }
\item[-]{have energy greater than \nUnits{30}{\ensuremath{{\rm MeV}}\xspace}};
\item[-]{have a lateral shower profile consistent with that of a photon};
\item[-]{be inconsistent with the extrapolation of any track in the
detector};
\item[-]{not include noisy channels in the calorimeter};
\item[-]{not be in the overlap region between the barrel and endcap
calorimeter modules};
\item[-]{not be in the ring of crystals closest to the beam axis}.
\end{list}
Showers satisfying these selection criteria are considered to be
photons and are combined into $\pi^0$ candidates. Photon pairs are
required to have an invariant mass
within \nUnits{50}{\ensuremath{{\rm MeV}/c^2}\xspace} of the
nominal $\pi^0$ mass, $M_{\pi^{0}}$.
They are then required to fall within the
asymmetric window
\begin{equation}
-4 < \frac{M_{\gamma \gamma} - M_{\pi^0}}{\sigma_{\gamma \gamma}} < 3~.
\end{equation}
The photon-pair mass resolution, $\sigma_{\gamma \gamma}$, is typically
5-7 MeV/$c^{2}$.
Candidate photon pairs are then
kinematically constrained (subject to the measured
uncertainties on energies and shower spatial locations) to have an
invariant mass equal to
$M_{\pi^{0}}$.
To be used,
$\pi^0$ candidates are further required to have a
successful kinematic fit with confidence level (one degree of freedom)
greater than $0.1\%$.
\subsection{Recoil Mass and Signal and Background Regions}
We select events for each transition by cutting on the mass of the
system
recoiling against the two pions in
the
$\Upsilon^{\prime} \rightarrow\pi\pi + $''anything''
decay:
$M^2_{\rm rec} = M^2_{\Upsilon'} + q^2 -2 q\cdot P'$, where, as above,
$q = q_{1} + q_{2}$ and $P'$ is the Lorentz momentum of
the initial state Upsilon.
Given the large mass of the initial state Upsilon, the
dot product simplifies and the recoil mass can be well approximated by
$M^2_{\rm rec} \approx M^2_{\Upsilon'} + q^2 - 2 M_{\Upsilon'} (E_1 + E_2)$.
For the cascade decays, \upsDec{2}{1}, this is not quite correct
because the Lorentz momentum of the initial state Upsilon (the \ups{2}) is
not equal to the beam momentum. However, because the total momentum of
the pions is small and the initial state is approximately at rest, using the
incorrect momentum for the initial state does not significantly change the
recoil mass distribution other than to shift it by the difference
between \ups{3} and \ups{2} masses. Hence, we expect to find three
recoil mass peaks. The transitions originating from the \ups{3} will
generate recoil mass ($M_{\rm rec}$) peaks at the masses of the
\ups{1} and \ups{2}, while the \upsDec{2}{1} decays will yield a peak
at \nUnits{9.79}{\ensuremath{{\rm GeV}/c^2}\xspace}. These three peaks are clearly visible in
Fig.~\ref{fig:recMassSpectra}.
The recoil mass, $M_{\rm rec}$, is measured rather accurately, especially in the charged case, due to
the good resolution on the momenta of the low-momentum pions.
It is still quite good for the
neutral modes where the total pion momentum is given as the sum of
momenta of two $\pi^0$ candidates reconstructed from the calorimeter
showers.
\subsection{Signal and Background Selection}
The fit requires signal and background samples. They are determined
as a function of the recoil mass, $M_{\rm rec}$, only. The recoil
mass peak widths are determined from Monte Carlo with tight selections
on variables other than the recoil mass.
These widths
are then used to
determine mass windows to select events in both Monte Carlo and data
samples. The signal regions are defined as the range within three
times the peak width of the nominal recoil mass, while the backgrounds
are the regions from six to twelve times the peak width from the nominal
mass above and below the peak mass. The masses and widths used to
define these regions are listed in Table~\ref{tab:RecMassPeaks}. The
width of the recoil mass distribution in the decays \upsCDec{2}{1} is
roughly twice that of the direct decays. This is due to the boost of
the initial state Upsilon imparted in its production by the cascade from the
\ups{3}. The edges of the signal windows are indicated by the
hatching in Fig.~\ref{fig:recMassSpectra}. Note that
in Fig.~\ref{fig:recMassSpectra} the yield in the regions
not used for {\it either}
signal or background definition have been set to zero.
The Dalitz plot distributions for the selected data in
six
of the twelve final states are shown in
Figs.~\ref{fig:datax3mm} and ~\ref{fig:datax3ee}.
Comparison of
the $\pi^{0}\pi^{0}$
and $\pi^{+}\pi^{-}$ for the \upsDec{3}{1} shows the depletion
in charged particle efficiency at moderate di-pion invariant mass and
large $|{\rm cos}~\theta_{X}|$. Comparison of the charged modes for
\upsCDec{3}{1} and \upsCDec{2}{1} shows, in two dimensions, the obvious
disparity between the two distributions.
\begin{figure*}
\centerline{
\resizebox{0.90\textwidth}{!}{\includegraphics{Figure4.eps}}
}
\caption{
Candidate events that
passed all selection criteria,
and that have
the final state Upsilon decaying to $\mu^{+}\mu^{-}$. In the middle
is the decay $\upsCDec{3}{1}$. To the left is its neutral counterpart
$\upsNDec{3}{1}$.
To the right is the charged transition
$\upsCDec{2}{1}$, with the bulk of its distribution at large values of
dipion invariant mass. In each plot there are ten degrees of grey-scale
ranging from white (lowest occupancy per bin) to black (highest occupancy).}
\label{fig:datax3mm}
\end{figure*}
\begin{figure*}
\centerline{
\resizebox{0.90\textwidth}{!}{\includegraphics{Figure5.eps}}
}
\caption{
Candidate events that have
passed all selection criteria,
and that have
the final state Upsilon decaying to $e^{+}e^{-}$. As in the prior
plot, three transitions are, left to right, $\upsNDec{3}{1}$, $\upsCDec{3}{1}$,
and $\upsCDec{2}{1}$.}
\label{fig:datax3ee}
\end{figure*}
\begin{table*}
\centerline{
\begin{tabular}{lcccc} \hline \hline
Transition & Recoil Mass &
Width (Data) & Width (MC) &
Width (Cut) \\
& (\ensuremath{{\rm MeV}/c^2}\xspace) & (\ensuremath{{\rm MeV}/c^2}\xspace) & (\ensuremath{{\rm MeV}/c^2}\xspace) & (\ensuremath{{\rm MeV}/c^2}\xspace) \\
\hline
\upsCDec{3}{1} & 9~460.4 & 2.4 & 2.5 & 2.5 \\
\upsCDec{2}{1} & 9~792.4 & 5.0 & 5.0 & 5.0 \\
\upsCDec{3}{2} & 10~023.3 & 2.2 & 1.9 & 2.1 \\
\hline
\upsNDec{3}{1} & 9~460.4 & 15.0 & 12.7 & 13.8 \\
\upsNDec{2}{1} & 9~792.4 & 10.9 & 10.5 & 10.7 \\
\upsNDec{3}{2} & 10~023.3 & 3.4 & 3.4 & 3.4 \\
\hline \hline
\end{tabular}
}
\caption{ Recoil mass distribution central values and widths for
the signal and background selections used in the fit.
The central values and the widths agree well between data
and Monte Carlo. The signal windows are defined as the
region within three times the cut width (last column) of
the central mass and the background windows are defined
as the region from six to twelve cut widths from the
center on either side. The background subtraction is
only important for the cascade decays for which there is
a large contribution to the signal region from event
combinatorics. }
\label{tab:RecMassPeaks}
\end{table*}
\section{Matrix Element Fits}
\subsection{The Likelihood Fitter}
The binned likelihood fit to the kinematic distributions of the
$\Upsilon$(mS)$\rightarrow\Upsilon$(ns)$\pi\pi$ decays is designed to deal
correctly with the low bin yields expected from dividing approximately
2000 events over a two
dimensional space with
more than ten
bins per dimension. The general case of this problem is solved in
Ref.~\cite{Barlow:dm}. Specific details of our application of this technique,
including notes on variable smearing and background inclusion, are
found in the Appendix.
We fit the decay distributions to a product of the squared modulus of
the decay amplitude and the phase space density sculpted by the
detector acceptance. The matrix element has a known analytical form
(see Eqn. \ref{Eq:BandCAmplitude})
as a function of the form factors $\clg{A}$, $\clg{B}$, and $\clg{C}$,
which are taken as complex constants.
Its leading angular structure is known, and so long as the form
factors are known, too, the entire amplitude can be described exactly.
However, we cannot model the detector acceptance in analytic form,
so we approximate its effect via Monte Carlo integration.
We determine the integral of the phase space density in a bin in
($q^{2}, {\rm cos}~\theta_{X}$),
sculpted by acceptance and efficiency,
by counting Monte Carlo
events that pass the selection criteria and fall into that bin.
In Fig.~\ref{fig:eff} we show the two-dimensional phase space
after such sculpting. Note that while the overall efficiency for
the neutral final state is lower than for its charged counterpart,
the former is more uniform, particularly in the regions of intermediate
$M_{\pi\pi}$ and large $|{\rm cos}~\theta_{X}|$.
For
each bin of the observed distribution we predict the number of events
as a function of the matrix element parameters by multiplying the
Monte Carlo integral for that bin by the exactly calculated matrix
element value for that bin. This approach avoids generating Monte
Carlo integrated templates for each component of the angular
distribution and reduces the uncertainty due to finite Monte Carlo
sample size.
\begin{figure*}
\centerline{
\hspace*{\fill}
\resizebox{0.40\textwidth}{!}{\includegraphics{Figure6Left.eps}}
\hspace*{\fill}
\resizebox{0.40\textwidth}{!}{\includegraphics{Figure6Right.eps}}
}
\caption{ The efficiency-sculpted phase space
in the two-dimensional plane for
the transitions
$\Upsilon$(3S)$\to\Upsilon$(1S)$\pi^{+}\pi^{-}$ (left) and
$\Upsilon$(3S)$\to\Upsilon$(1S)$\pi^{0}\pi^{0}$ (right). Note that the
neutral final state has a more uniform efficiency, especially in the region
of moderate di-pion mass and large $|{\rm cos}(\theta_X)|$. }
\label{fig:eff}
\end{figure*}
\begin{figure*}
\centerline{
\resizebox{0.90\textwidth}{!}{\includegraphics{Figure7.eps}}
}
\caption{ The three functions used in the fit for the $\Upsilon$(3S)
decay to $\Upsilon$(1S)$\pi\pi$. From left to right these
are for the pure ${\cal A}$ term, the interference term, and the
pure ${\cal B}$ term. }
\label{fig:ME}
\end{figure*}
To fit the decay distribution we take the squared modulus of the decay
amplitude, Eqn.~\ref{Eq:BandCAmplitude}, and decompose it as a sum
of six functional forms each multiplied by one of $|\clg{A}|^2$,
$|\clg{B}|^2$, $|\clg{C}|^2$, $\Re(\clg{A}^*\clg{B})$,
$\Re(\clg{A}^*\clg{C})$, or $\Re(\clg{B}^*\clg{C})$. For
normalization, the
matrix element $\clg{A}$ is set to unity.
The functional forms ({\em e.g.,}\xspace $(q^2 - 2 M_\pi^2)^2$) depend on the Dalitz
variables and are pre-evaluated into templates over the Dalitz space.
The fitter then seeks the best fit as a function of the matrix
element ratios
$\clg{A}$, $\clg{B}$, and
$\clg{C}$. The input to the fitter consists of only the data,
background, and phase space Monte Carlo binned across the Dalitz plot,
and the component templates of the decay distribution derived from the
exact decay amplitude, but taking into account the kinematic smearing
and acceptance and efficiency effects due to reconstruction as
determined from the detector simulation.
The backgorund component is scaled by the ratio of the signal region width
(6 $\sigma$; see Section II.D) to the total backgorund sideband width
(nominally 12$\sigma$).
In Fig.~\ref{fig:ME}
we show the functional forms for $|\clg{A}|^2$, $\Re(\clg{A}^*\clg{B})$,
and $|\clg{B}|^2$ for the case of
$\Upsilon$(3S)$\to\Upsilon$(1S)$\pi\pi$.
In our experiment, the
complementarity of the neutral and charged final states is
particularly important in that the
rightmost of these (the form for $|\clg{B}|^2)$)
depletes the region for which the $\pi^{+}\pi^{-}$ channel
has falling efficiency.
Consistent results between the $\pi^{0}\pi^{0}$ and $\pi^{+}\pi^{-}$
transitions
gives us confidence that the simulation of this fall-off in efficiency
is reliable.
The matrix element extraction procedure is tested ``end-to-end''
by simulating signal
with known matrix elements in Monte Carlo and comparing the fit result
and its uncertainty with the known inputs. Samples of the same size
as the observed yield are generated and fit identically to the data.
The results yield standard normal distributions in the observed
uncertainty scaled residuals for widely distributed seed matrix
element values. This confirms the fitter is unbiased at the level of
precision to be expected from the sample size of the measurement.
\subsection{Fits with ${\cal C} = 0$}
The fits to the two dimensional distributions of ${\ensuremath{M_{\pi\pi}}}$ and $\ensuremath{\cos \theta_X}\xspace$
determine the matrix element $\ensuremath{\clg{B}/\clg{A}}\xspace$ and $\ensuremath{\clg{C}/\clg{A}}\xspace$. The extracted
values of $\Re(\ensuremath{\clg{B}/\clg{A}}\xspace)$ and $\Im(\ensuremath{\clg{B}/\clg{A}}\xspace)$ are summarized in
Table~\ref{tab:sep_sim_fit}, subject to the constraint that $\clg{C}
\equiv 0$. In that we only measure the cosine of the phase
difference between ${\cal B}$ and ${\cal A}$, $\Im(\ensuremath{\clg{B}/\clg{A}}\xspace)$ is only
known to within a sign.
The upper set of matrix elements are obtained from
independent fits to
ten individual decay modes; we cannot
individually fit the two modes associated with
$\Upsilon$(3S)$\to\Upsilon$(2S)$\pi^{+}\pi^{-}$ because of their
limited statistics. The lower
set of three are from the simultaneous fits of all final states for
each given Upsilon transition.
In the simultaneous fits the relative branching ratios between modes
are not constrained, but it is assumed that the di-pion excitation
dynamics is independent of the charge of the pion final state
(isospin symmetry) and thus the decay distributions should
be identical to within statistical fluctuations for all transitions
between the same Upsilon states. This assumption is supported by the
consistency among the matrix element values extracted independently,
as well as their consistency with the value extracted from the
simultaneous fit. In particular, the four final states studied for the
transition from $\Upsilon$(3S) to $\Upsilon$(1S) show excellent
agreement between the two lepton species and between charged and
neutral
pions.
\begin{table*}
\centerline{
\begin{tabular}{llcc}
\hline \hline
Individual Fits & & \ensuremath{\Re(\clg{B}/\clg{A})}\xspace & \ensuremath{\Im(\clg{B}/\clg{A})}\xspace~ \\
\hline
\upsCDec{3}{1}; & \ensuremath{\Upsilon \rightarrow \mu^+ \mu^-}\xspace & \nError{-2.514}{0.037} & \nError{\pm 1.164}{0.059} \\
& \ensuremath{\Upsilon \rightarrow e^+ e^-}\xspace & \nError{-2.527}{0.049} & \nError{\pm 1.180}{0.079} \\
\upsNDec{3}{1}; & \ensuremath{\Upsilon \rightarrow \mu^+ \mu^-}\xspace & \nError{-2.426}{0.085} & \nError{\pm 1.313}{0.159} \\
& \ensuremath{\Upsilon \rightarrow e^+ e^-}\xspace & \nError{-2.524}{0.093} & \nError{\pm 1.070}{0.153} \\
\hline
\upsCDec{2}{1}; & \ensuremath{\Upsilon \rightarrow \mu^+ \mu^-}\xspace & \nError{-0.656}{0.126} & \nError{\pm 0.431}{0.089} \\
& \ensuremath{\Upsilon \rightarrow e^+ e^-}\xspace & \nError{-0.689}{0.147} & \nError{\pm 0.425}{0.102} \\
\upsNDec{2}{1}; & \ensuremath{\Upsilon \rightarrow \mu^+ \mu^-}\xspace & \nError{-0.148}{0.280} & \nError{ ~~0.000}{1.655} \\
& \ensuremath{\Upsilon \rightarrow e^+ e^-}\xspace & \nError{-0.293}{0.330} & \nError{\pm 0.001}{1.130} \\
\hline
\upsNDec{3}{2}; & \ensuremath{\Upsilon \rightarrow \mu^+ \mu^-}\xspace & \nError{-0.283}{0.305} & \nError{\pm 0.001}{1.708} \\
& \ensuremath{\Upsilon \rightarrow e^+ e^-}\xspace & \nError{-0.583}{0.082} & \nError{\pm 0.003}{1.475} \\
\hline \hline
Simultaneous Fits & & \ensuremath{\Re(\clg{B}/\clg{A})}\xspace & \ensuremath{\Im(\clg{B}/\clg{A})}\xspace \\
\hline
\multicolumn{2}{l}{ \upsDec{3}{1} } & \nError{-2.523}{0.031} & \nError{\pm 1.189}{0.051} \\
\multicolumn{2}{l}{ \upsDec{2}{1} } & \nError{-0.753}{0.064} & \nError{\pm 0.000}{0.108} \\
\multicolumn{2}{l}{ \upsDec{3}{2} } & \nError{-0.395}{0.295} & \nError{\pm 0.001}{1.053} \\
\hline \hline
\end{tabular}
}
\caption{Fit results from $\upsDec{n}{m}$ transitions for
${\cal B}/{\cal A}$ with ${\cal C}$ set to zero. The upper
set of results is from individual fits to each separate
decay mode and the lower set of results is from
simultaneous fits to both lepton final states and both
pion charge modes. We cannot fit the \upsCDec{3}{2}
transitions,
individually in $e^{+}e^{-}$ and $\mu^{+}\mu^{-}$ or
combined, because of their limited statistics. In the
simultaneous fits the relative branching fractions are
allowed to float.
Note that we know the
value of the imaginary part of the ratio only to within a sign.}
\label{tab:sep_sim_fit}
\end{table*}
To study the fit quality
we
project the data and the expected decay distribution for the matrix
element value preferred by the fit onto the di-pion mass (${\ensuremath{M_{\pi\pi}}}$) and
di-pion
helicity angle ($\ensuremath{\cos \theta_X}\xspace$)
variables and calculate a $\chi^2$ for each projection.
To increase the bin contents we sum over lepton species but not over
pion charges. We expect the shapes for charged and neutral pions to
differ due to the rather different
efficiencies
for reconstruction
and resolutions,
as
well as the folding of the neutral angle in the fits.
Figure~\ref{fig:FQFWMC} presents plots of the
data overlaid with the fit results,
showing good qualitative agreement.
The
$\chi^2$ values
from these overlays, given in Table~\ref{tab:chisq},
are
acceptable, given the simplicity of the fitted matrix element.
\begin{table*}
\centerline{
\begin{tabular}{|c||l|l|l|l|}
\hline
Upsilon &
\multicolumn{2}{|c|}{$\pi^{+}\pi^{-}$} &
\multicolumn{2}{|c|}{$\pi^{0}\pi^{0}$}\\
\cline{2-5}
Transition & $\ensuremath{\cos \theta_X}\xspace$ & $M_{\pi\pi}$& $\ensuremath{\cos \theta_X}\xspace$ & $M_{\pi\pi}$ \\
\hline
$3S \to 1S$ & 33.2~(16)& 46.9~(32) & 4.3~(8) & 52.1~(32)\\
$2S \to 1S$ & 6.1~(10) & 22.7~(12) & 3.4~(5) & 13.7~(12)\\
$3S \to 2S$ & 7.1~(7) & 7.8~(6) & 7.4~(4) & 2.5~(7) \\
\hline
\end{tabular}
}
\caption{The figure of merit for each of the twelve projections in the
accompanying figure. For each projection we give the value of $\chi^{2}$
and, in parentheses, the number of bins used to calculate it.
Uncertainties
in the fit results due to limited simulation statistics are not included
in these calculations.}
\label{tab:chisq}
\end{table*}
\begin{figure*}
\centerline{
\hspace*{\fill}
\resizebox{0.70\textwidth}{!}{\includegraphics{Figure8.eps}}
\hspace*{\fill}
}
\caption{ Plots overlaying projections
of the data (points with error bars) and the fit result
(histograms)
onto the ${\ensuremath{M_{\pi\pi}}}$ and
$\ensuremath{\cos \theta_X}\xspace$ variables. The plots are summed over electrons
and muons, but are differentiated by pion charge. The
neutral modes (open symbols, dashed lines)
show only a positive distribution in
$\ensuremath{\cos \theta_X}\xspace$ because the two pions are indistinguishable. For the
charged modes (solid symbols, solid lines)
the angle is that of the $\pi^{+}$.
}
\label{fig:FQFWMC}
\end{figure*}
As a further fit quality test, we examine the two dimensional
distribution over the Dalitz variables of error-normalized
deviations.
The deviations,
$\delta_{i}$,
are the difference,
fit subtracted from the data,
divided
by the mutual uncertainty:
\begin{equation}
\delta_i = \frac{d_i - \tilde{d}_i}{\sigma_i},
\end{equation}
where
each
$\tilde{d}_{i}$
is the predicted decay population in bin $i$. The
bin-by-bin uncertainties, $\sigma_i$, are composed of the
uncertainty on the
data yield in the bin,
$\sigma_{d} = \sqrt{d_i}$,
and the uncertainty on the template function, dominated by the
fluctuation in the Monte Carlo phase space yield and proportional to
$1/\sqrt{a_i}$, where $a_i$ is the Monte Carlo phase space yield in
bin $i$. Hence,
$\sigma_i = \sqrt{d_i + \tilde{d}_{i}^{2}/a_i}$.
The bins for which $d_i = 0$ require special treatment,
and $\sigma_i$ is modified appropriately.
To minimize the effect of such bins with zero yield, we sum over muon and
electron final states. This takes a weighted average over the
distributions, rather than taking account of the differences between
the individual distributions and their individual template
predictions.
The deviations between the data and the fit templates, $\delta_{i}$,
are shown in
Fig.~\ref{fig:dev_yyy_xx} for the charged and neutral transitions
between $\Upsilon$(3S) and $\Upsilon$(1S).
No significant bunching
is observed that would indicate a bias. We neglect the small accumulations
in the areas of low tracking efficiency (at large $|\ensuremath{\cos \theta_X}\xspace|$ and
intermediate ${\ensuremath{M_{\pi\pi}}}$), probably attributable to the
Monte Carlo detector model not being
sufficiently accurate.
\begin{figure*}
\centerline{
\hspace*{\fill}
\resizebox{0.90\textwidth}{!}{\includegraphics{Figure9.eps}}
\hspace*{\fill}
}
\caption[Error Normalized Fit Deviations over $\ensuremath{\cos \theta_X}\xspace$ and ${\ensuremath{M_{\pi\pi}}}$]
{Plots of the bin by bin deviations of the data from the fit
templates normalized to the expected uncertainty on the bin
content for the transitions between $\Upsilon$(3S) and
$\Upsilon$(1S). The left plot is for the charged pion modes while
the right plot is for the neutral pion modes. The data are
summed over lepton species. No strong concentration of
deviations is apparent.}
\label{fig:dev_yyy_xx}
\end{figure*}
\subsection{Fits Including the Chromo-magnetic Term ${\cal C}$}
The fit results in Table~\ref{tab:sep_sim_fit} do not take into
account the possible presence of amplitude terms that come from
chromo-magnetic couplings, which would allow the additional $\clg{C}$
term to appear. This term is nearly degenerate with the $\clg{B}$
term, and fits allowing it to float show a strong covariance between
these two terms. This is caused by the similarity in structure of the
two terms; $\clg{B}$ accompanies a functional dependence $E_1 E_2$,
while $(\epsilon'\cdot q_{1,2}) (\epsilon\cdot q_{2,1})$ emphasizes the
regions of phase space in which the pion spatial momentum, and hence
also the energy, are large.
The low yield modes do not allow the
measurement of the term at all. We therefore only study it in the
$\upsDec{3}{1}$ transitions, and then only extract a value from the
simultaneous fit.
The covariance between $\clg{B}$ and
$\clg{C}$ for the \upsDec{3}{1} transition
is summarized in Fig.~\ref{fig:bc_fit_scan}, which shows
the variation of extracted $|\ensuremath{\clg{B}/\clg{A}}\xspace|$ with $|\ensuremath{\clg{C}/\clg{A}}\xspace|$, both as a fit
error ellipse, and as fit trials with $|\clg{C}|$ constrained to
different values.
The ellipse corresponding to one standard deviation from the best
fit gives a
value for
\upsDec{3}{1} of
$|\ensuremath{\clg{C}/\clg{A}}\xspace| = 0.45 \pm 0.18$, with the
uncertainty being purely the statistics of the fit.
The fit which includes real and imaginary
parts of $\ensuremath{\clg{C}/\clg{A}}\xspace$ shows an improvement over the one with
${\cal C}$ fixed at zero of $-2 \ln \clg{L} = 9.4$. Although this
implies a $\sim 3\sigma$ improvement in fit quality when ${\cal C}$ is
allowed to float, systematic uncertainties, which are
significant, have not yet been
taken into account.
With this extended fit the six projections of Fig.~\ref{fig:FQFWMC}
show no significant changes,
and
for the \upsDec{3}{1} transition
the best fit value
of $|\ensuremath{\clg{B}/\clg{A}}\xspace|$ changes minimally from 2.79 (${\cal C} = 0$)
to 2.89 (${\cal C}$ floating).
The phase of $\clg{B}$ with respect to $\clg{A}$, denoted
$\delta_{BA}$, changes little
(about 2 degrees)
from the 155 degrees of the
fit done with ${\cal C} = 0$.
The smallness of the effects is not surprising as
the shapes of the $\clg{B}$ and $\clg{C}$ components of the amplitude
are nearly degenerate.
A non-zero value of $|\ensuremath{\clg{C}/\clg{A}}\xspace|$ may be a consequence of statistical
fluctuations and small systematic biases or may be due to
$\clg{A}$ and $\clg{B}$ having some dependence on $q^{2}$ and/or
$r^{2}$, {\it i.e.,} not being complex constants.
\begin{figure*}
\centerline{
\resizebox{0.6\textwidth}{!}{\includegraphics{Figure10.eps}}
}
\caption{Variation of $\clg{B}$ with $\clg{C}$ magnitudes. The
points indicate the fit and error for $\clg{B}$ at fixed values
of $\clg{C}$. The ellipse indicates the one sigma bound on the
free fit, the axis of which agrees well with the point by point
fits. The bands indicate the one standard error limits on
$\clg{B}$ when $\clg{C}$ is fixed to zero.
}
\label{fig:bc_fit_scan}
\end{figure*}
\subsection{Partial Wave Decomposition}
Since the focus of this study is the decay dynamics of the di-pion
system it is useful to think about the spin structure of the di-pion
composite. The idea is to look for signatures of higher spin
resonances in the form factors $\clg{A}$ and $\clg{B}$. We must
account for the intrinsic spin structure of the Lorentz amplitude to
do this. We equate the Lorentz amplitude with the general partial wave
amplitude to relate the matrix elements.
The transition is of the form $\left<\Upsilon;X|\Upsilon'\right>$. If
the di-pion system has spin $J$ we have:
\begin{equation}
\left<1,m_{\Upsilon}; J_X, m_X|1,m_{\Upsilon'}\right>~.
\end{equation}
In that here we assume that only {\cal A} and {\cal B} are non-zero,
there is no change in the polarization from the initial state to
final state Upsilon; more general partial wave decompositions can also be
made \cite{CKK, newVoloshin}.
The
angular momentum projections are then
$m_{\Upsilon'} = m_{\Upsilon}$, and $m_X = 0$.
Hence the partial wave decomposition of the $X$ system can only have
$m = 0$ components. Since the pions are in an iso-singlet state,
their parities require their relative orbital angular momentum to be
even, and hence the orbital angular momentum between the final state
upsilon and the di-pion composite must also be even. We can only have
even partial waves in our decomposition:
\begin{equation}
\begin{array}{rcl}
\clg{M}_P &=& \clg{S}(q^2) Y^0_0 + \clg{D}(q^2) Y^0_2 \\
&=& \clg{S}(q^2) \frac{1}{\sqrt{4\pi}} +
\clg{D}(q^2) \sqrt{\frac{5}{4\pi}}\left(
\frac{3}{2} \cos^2 \theta_X - \frac{1}{2}\right).
\end{array}
\end{equation}
The functions $\clg{S}(q^2)$ and $\clg{D}(q^2)$ are composed of two
terms each, one from the $\clg{A}$ dependence and one from the
$\clg{B}$ dependence:
\begin{equation}
\clg{S}(q^2) = \clg{A}\clg{S}_{\clg{A}}(q^2) +
\clg{B}\clg{S}_{\clg{B}}(q^2),
\hspace*{5mm}{\rm and}\hspace*{5mm}
\clg{D}(q^2) = \clg{A}\clg{D}_{\clg{A}}(q^2) +
\clg{B}\clg{D}_{\clg{B}}(q^2).
\end{equation}
We here assume that there are no significant contributions from
partial waves higher
than $J = 2$. This will be true if there are no contributions from
variations of form factors over the Dalitz space. Higher $J$ terms
must originate from structure in the form factors $\clg{A}$ and
$\clg{B}$.
Equating the decay distributions (or equivalently, projecting inner
products over the angular space) yields the following forms:
\begin{equation}
{\clg{S}}_{\clg{A}}(q^2) = q^2 - 2 M_\pi^2~,
\hspace*{7mm}{\rm and}\hspace*{7mm}
{\clg{D}}_{\clg{A}}(q^2) = 0
\end{equation}
for a pure ``$\clg{A}$'' decay, and
\begin{widetext}
\begin{equation}
\begin{array}{rcl}
{\clg{S}}_{\clg{B}}(q^2) & = &
\frac{q^2 \left( {\left( M_{\Upsilon'}^2 -
M_{\Upsilon}^2 \right) }^2 +
\left( M_{\Upsilon'}^2 + M_{\Upsilon}^2 \right)
q^2 - 2 {q^2}^2 \right) +
2 M_{\pi}^2 \left( {M_{\Upsilon'}^2}^2 +
{\left( M_{\Upsilon}^2 - q^2 \right) }^2 -
2 M_{\Upsilon'}^2 \left( M_{\Upsilon}^2 + q^2
\right) \right) }{12
{\sqrt{M_{\Upsilon'}^2 M_{\Upsilon}^2}} q^2}~; \\
{\clg{D}}_{\clg{B}}(q^2) & = &
\frac{\left( 4 M_{\pi}^2 - q^2 \right)
\left( {M_{\Upsilon'}^2}^2 +
{\left( M_{\Upsilon}^2 - q^2 \right) }^2 -
2 M_{\Upsilon'}^2 \left( M_{\Upsilon}^2 + q^2 \right)
\right) }{12 {\sqrt{5}}
{\sqrt{M_{\Upsilon'}^2 M_{\Upsilon}^2}} q^2}
\end{array}
\end{equation}
\end{widetext}
for a pure ``$\clg{B}$''~decay. The overall amplitude is
\begin{equation}
\clg{M}_P = ( \clg{A}\,{\clg{S}}_{\clg{A}}(q^2) +
\clg{B}\,{\clg{S}}_{\clg{B}}(q^2) ) Y^0_0 +
( \clg{A}\,{\clg{D}}_{\clg{A}}(q^2) +
\clg{B}\,{\clg{D}}_{\clg{B}}(q^2) ) Y^0_2~,
\end{equation}
where it is implied that $Y^m_l$ is a function of the helicity angles
of the pseudo-decay $X \rightarrow \pi \pi$, $\theta_X$ and $\phi_X$
(although the latter variable plays no role in the description of this
decay, by the assumptions above). Interference between the $S$-wave
and $D$-wave components of the decay comes from the functions
$\clg{S}(q^2)$ and $\clg{D}(q^2)$ being complex valued. Though
$\clg{S}_{\clg{A},\clg{B}}(q^2)$ and $\clg{D}_{\clg{A},\clg{B}}(q^2)$
are real functions, $\clg{A}$ and $\clg{B}$ are complex coefficients
with nontrivial relative phase.
The structure of $S$ and $D$ components as functions of $q^2$ are
determined by the assumptions underlying the derivation of the general
Lorentz amplitude. The four functions from the pure~$\clg{A}$ and
pure~$\clg{B}$ components are sketched in Fig.~\ref{fig:pwsVsMpp}
together with the fractional $S$- and $D$-wave components in the
angular distribution (which can alternately be thought of as the
strengths of the $S$- and $D$-wave components), extracted from our fit
to $\upsDec{3}{1}$.
\begin{figure*}
\centerline{
\resizebox{0.8\textwidth}{!}{
\includegraphics{Figure11.eps}}
}
\caption[Partial Wave Components] {The left plot shows the amplitude
component functions ${\clg{S}}_{\clg{A}}$,
${\clg{S}}_{\clg{B}}$, ${\clg{D}}_{\clg{A}}$, and
${\clg{D}}_{\clg{B}}$ as a function of ${\ensuremath{M_{\pi\pi}}} \equiv
\sqrt{q^2}$. These are summed to obtain the total amplitude.
The partial rate to $S$-wave and $D$-wave components are shown
in the right plot for the \upsDec{3}{1} decay as determined
from the results of this analysis: $\ensuremath{\clg{B}/\clg{A}}\xspace = -2.52 + 1.19
i$. Note that the $D$-wave contribution is largest in the low
to intermediate range of $q^2$, and is suppressed at both
extrema by angular momentum barrier effects.
Note further
that this is not a resonance phenomenon
despite its shape in ${\ensuremath{M_{\pi\pi}}}$ and the changing angular
structure.}
\label{fig:pwsVsMpp}
\end{figure*}
This partial wave extraction becomes much more complex if the form
factors are assumed to be variable over the Dalitz space, for example
due to resonant structure/enhancement in the decay. This will
introduce higher powers of $\cos^2 \theta_X$ to the overall amplitude
and will need higher partial wave components to account for the
variation.
The presence of $D$-wave components in the angular
distribution of the decay is not in itself an indication of resonances
contributing, nor the presence of unaccounted-for physics.
The presence of
a $q^{2}$-dependent
$D$-wave
component could simply be a consequence of angular momentum
barriers in the three body phase space of the decay.
The data do not demand the introduction of a
$q^{2}$-dependent magnitude or phase for ${\cal B}/{\cal B}$.
These small $D$-wave components are consistent with those derived in
a recent paper by Voloshin~\cite{Voloshin:2006},
in which he emphasizes the importance
of relativistic and chromo-magnetic effects.
\section{Systematic Uncertainties}
We address three sources of systematic uncertainty in the measurements
of $\ensuremath{\clg{B}/\clg{A}}\xspace$ and $\ensuremath{\clg{C}/\clg{A}}\xspace$:
model dependence, detector efficiency and resolution, and
backgrounds.
In Sect. III we showed that our model provides a very good
description of the data in the $(q^{2},\ensuremath{\cos \theta_X}\xspace)$ plane and
that the
presence or absence of the chromo-magnetic coupled
term in the amplitude has little effect on $|\ensuremath{\clg{B}/\clg{A}}\xspace|$ and
$\delta_{BA}$.
Uncertainty in the estimation of the detector efficiency and
resolution contributes most significantly in the charged mode analyses
due to our limited knowledge of the tracking efficiency at very low
momentum. In that the low momentum region is precisely where the matrix
element has potential suppression in the $\clg{B}$ term, this can
potentially cause a significant bias.
To estimate this effect we use the full Monte Carlo simulation with
looser and tighter track reconstruction requirements to provide bounds
on the shape of the efficiency as a function of track curvature. We then
create a number of analytic functions that span these boundaries. Then
we use a toy Monte Carlo to simulate events with one of these
analytic functions and assume a different one for the reconstruction.
The variations in the fit results are conservatively
assumed to be one standard error uncertainties on the extracted
parameters.
The same process is repeated for the neutral modes, varying the
thresholds at which showers can be observed in the detector. This
obviously leads to a large variation in branching ratios from simple
inability to reconstruct the decays, but does not exhibit any
significant change in the shape of the efficiency function over the
measurement variables. This is to be expected since the $\pi^0$ decays
have largely flat acceptance over the kinematic range of these decay
modes.
We have evaluated the systematic errors associated with detector
resolution, and find them to be negligible in comparison with the
statistical errors from the fit and the other systematic errors
discussed here. The curvatures of the matrix element components
across the Dalitz plot
are all very much smaller than the variances of the reconstructed
measurement variables around their true values. No systematic
uncertainty is assigned to this source.
Background subtraction is only a source of bias if the upper and lower
sidebands in the recoil mass exhibit markedly different shapes or the
background is strongly peaked under the signal. In this case the
extrapolations of the background shape and magnitude under the peak
could be distorted.
We have redone the fits with the ratio of
the widths of sideband window to signal window
both doubled and halved, and with only using either the high-mass or low-mass
sideband.
The variations in the fit
are conservatively taken to represent one sigma
variations in the final result, and are given in the last
column of Table \ref{tab:full_result}.
Finally, the lepton reconstruction is capable of contributing bias
since all decay modes are fully reconstructed. However, the detector
response to leptons is sufficiently well measured in other analyses
that the detector simulation is much more precise than what is
required for this data set. The variation of the shapes is furthermore
only relevant for the final $\clg{C}$ term, which is dependent on the
lepton polar angle. With the exception of a small part of the
$\clg{C}$ terms there can be no effect due to lepton acceptance. We
estimate any systematic error associated with the lepton
reconstruction to be negligible.
The fit results combined with these systematic uncertainties are
summarized in Tables~\ref{tab:full_result}
and~\ref{tab:full_result_summary}. Since the magnitude $|\ensuremath{\clg{C}/\clg{A}}\xspace|$ in
the fit is only separated from zero by about one standard error and is
expected to be suppressed in the theoretical models, we set a limit
rather than claim observation of a non-zero value.
We set this limit by assuming the value of $\ensuremath{\clg{C}/\clg{A}}\xspace$ has a Gaussian
uncertainty in real and imaginary parts. We transform variables to
$|\ensuremath{\clg{C}/\clg{A}}\xspace|$ and $\arg(\ensuremath{\clg{C}/\clg{A}}\xspace)$, using the sum of the variances of
statistical and systematic origin as the overall variance.
We then find the 90\%
upper
limit
from the resulting
distribution as
\begin{equation}
\begin{array}{rclcll}
|\ensuremath{\clg{C}/\clg{A}}\xspace| & < & 1.09 & \hspace*{1cm} & \rm{at}~90\%~C.L.~. &
\end{array}
\end{equation}
\section{Summary and Acknowledgments}
We quote fit results for the three transitions from simultaneous fits
to the different decay modes with statistical and systematic
uncertainties in Table~\ref{tab:full_result_summary}. Only the simplest
features of the Brown and Cahn decay amplitude
(Eqn.~\ref{Eq:BandCAmplitude}) are included in our model, and the
fits account for the structure of the decay without introduction of
new physics or
contributions from
resonances.
The matrix elements are indicated as points in the complex plane in
Fig.~\ref{fig:matels_vs_plots}. For the ``anomalous''
$\upsDec{3}{1}$ transition we fit for the presence of the ``suppressed''
$\clg{C}$ term as a test for the breakdown of the underlying
assumptions leading to the standard matrix element.
This term is not significant when systematic errors are taken into account and
the quality of the fit to the data is good without it.
Therefore, we set an upper limit of
$|\ensuremath{\clg{C}/\clg{A}}\xspace| < 1.09$ at $90\%$ C.L..
We note in particular that the treatment of the di-pion transitions
via the full allowed matrix element under the assumptions
in Refs.~
\cite{Brown:1975dz,MBVoloshin:75,Gottfried:1977gp,Yan:1980uh,Voloshin:1980zf}
allows two matrix elements, only one of which has
traditionally been assumed to be non-zero. The description of the
$\upsDec{3}{1}$ transition di-pion mass and angular structure as
anomalous is only true in the limit of this assumption.
This analysis
shows in particular that the description of the decay process in terms
of the two favored amplitude terms,
with
complex
form factors constant over the Dalitz plane,
suffices to describe the decay distributions of
$\upsDec{3}{1}$, $\upsDec{3}{2}$, and $\upsDec{2}{1}$,
provided the form factors are allowed to vary with the transition.
For the $\Upsilon$(3S)$\to\Upsilon$(1S)$\pi\pi$ transition,
we find $|{\cal B}/{\cal A}|= 2.79 \pm 0.05$, which could imply
a large magnitude of ${\cal B}$ or a suppressed ${\cal A}$; recent
theoretical considerations~\cite{Voloshin:2006}
favor the latter interpretation.
While smaller than in the case of $\upsDec{3}{1}$, $|\ensuremath{\clg{B}/\clg{A}}\xspace|$ is
also determined to be non-zero for the case of $\upsDec{2}{1}$.
The large imaginary part of ${\cal B}/{\cal A}$ is
intriguing~\cite{newVoloshin}.
While there are not yet first principles predictions of the values of
the matrix elements of the decays studied here, this analysis does
provide complete measurements of the relative matrix element
magnitudes and phases that can serve as a point of comparison with
{\it ab initio} QCD calculations.
\begin{table*}
\centerline{
\begin{tabular}{llccccc}
\hline \hline
Fit, No $\clg{C}$ & & & stat. & effcy. ($\pi^\pm$) & effcy.($\pi^0$) & bg. sub. \\
\hline
\upsDec{3}{1}
& $\begin{array}{c} \ensuremath{\Re(\clg{B}/\clg{A})}\xspace \\ \ensuremath{\Im(\clg{B}/\clg{A})}\xspace \end{array}$
& $\begin{array}{r} -2.523 \\ \pm 1.189 \end{array}$
& $\begin{array}{l} \pm 0.031 \\ \pm 0.051 \end{array}$
& $\begin{array}{l} \pm 0.019 \\ \pm 0.026 \end{array}$
& $\begin{array}{l} \pm 0.011 \\ \pm 0.018 \end{array}$
& $\begin{array}{l} \pm 0.001 \\ \pm 0.015 \end{array}$ \\
\hline
\upsDec{2}{1}
& $\begin{array}{c} \ensuremath{\Re(\clg{B}/\clg{A})}\xspace \\ \ensuremath{\Im(\clg{B}/\clg{A})}\xspace \end{array}$
& $\begin{array}{r} -0.753 \\ 0.000 \end{array}$
& $\begin{array}{l} \pm 0.064 \\ \pm 0.108 \end{array}$
& $\begin{array}{l} \pm 0.059 \\ \pm 0.036 \end{array}$
& $\begin{array}{l} \pm 0.035 \\ \pm 0.012 \end{array}$
& $\begin{array}{l} \pm 0.112 \\ \pm 0.001 \end{array}$ \\
\hline
\upsDec{3}{2}
& $\begin{array}{c} \ensuremath{\Re(\clg{B}/\clg{A})}\xspace \\ \ensuremath{\Im(\clg{B}/\clg{A})}\xspace \end{array}$
& $\begin{array}{r} -0.395 \\ \pm 0.001 \end{array}$
& $\begin{array}{l} \pm 0.295 \\ \pm 1.053 \end{array}$
&
& $\begin{array}{r} \pm 0.025 \\ \pm 0.180 \end{array}$
& $\begin{array}{l} \pm 0.120 \\ \pm 0.001 \end{array}$ \\
\hline \hline
Fit, float $\clg{C}$ & & & stat. & effcy. ($\pi^\pm$) & effcy.($\pi^0$) & bg. sub. \\
\hline
\upsDec{3}{1}
& $\begin{array}{c} |\ensuremath{\clg{B}/\clg{A}}\xspace| \\ |\ensuremath{\clg{C}/\clg{A}}\xspace| \end{array}$
& $\begin{array}{r} 2.89 \\ 0.45 \end{array}$
& $\begin{array}{l} \pm 0.11 \\ \pm 0.18 \end{array}$
& $\begin{array}{l} \pm 0.19 \\ \pm 0.28 \end{array}$
& $\begin{array}{l} \pm 0.11 \\ \pm 0.20 \end{array}$
& $\begin{array}{l} \pm 0.027 \\ \pm 0.093 \end{array}$ \\
\hline \hline
\end{tabular}
}
\caption{Combined fit results for all transitions with statistical and
systematic uncertainties. The systematic uncertainties are in
order:
$\pi^\pm$ detection efficiency,
$\pi^0$ detection efficiency, and
background subtraction for the \upsDec{3}{2} transition.
The upper set of
results are for the fits assuming contributions to the amplitude
from only the $\clg{A}$ and $\clg{B}$ terms. The bottom two lines
are the fit results when the $\clg{C}$ term is allowed to be
non-zero. The imaginary part of the ratio has a two-fold ambiguity and
is only known to within a
sign. Note that for the transition $\upsDec{3}{2}$ we do
not have fits for the charged di-pion case.}
\label{tab:full_result}
\end{table*}
\begin{table*}
\centerline{
\begin{tabular}{llc}
\hline \hline
Fit, no $\clg{C}$, total error & & \\
\hline
\upsDec{3}{1}
& $\begin{array}{c} \ensuremath{\Re(\clg{B}/\clg{A})}\xspace \\
\ensuremath{\Im(\clg{B}/\clg{A})}\xspace \\
|\ensuremath{\clg{B}/\clg{A}}\xspace| \\
\delta_{BA} \end{array}$
& $\begin{array}{rcl} -2.52 & \pm & 0.04 \\
\pm 1.19 & \pm & 0.06 \\
2.79 & \pm & 0.05 \\
155(205) & \pm & 2 \end{array}$ \\
\hline
\upsDec{2}{1}
& $\begin{array}{c} \ensuremath{\Re(\clg{B}/\clg{A})}\xspace \\
\ensuremath{\Im(\clg{B}/\clg{A})}\xspace \\
|\ensuremath{\clg{B}/\clg{A}}\xspace| \\
\delta_{BA} \end{array}$
& $\begin{array}{rcl} -0.75 & \pm & 0.15 \\
0.00 & \pm & 0.11 \\
0.75 & \pm & 0.15 \\
180 & \pm & 9 \end{array}$ \\
\hline
\upsDec{3}{2}
& $\begin{array}{c} \ensuremath{\Re(\clg{B}/\clg{A})}\xspace \\ \ensuremath{\Im(\clg{B}/\clg{A})}\xspace \end{array}$
& $\begin{array}{rcl} -0.40 & \pm & 0.32 \\ 0.00 & \pm & 1.1 \end{array}$ \\
\hline \hline
Fit, float $\clg{C}$, total error & & \\
\hline
\upsDec{3}{1}
& $\begin{array}{c} |\ensuremath{\clg{B}/\clg{A}}\xspace| \\ |\ensuremath{\clg{C}/\clg{A}}\xspace| \end{array}$
& $\begin{array}{rcl} 2.89 & \pm & 0.25 \\ 0.45 & \pm & 0.40 \end{array}$ \\
\hline \hline
\end{tabular}
}
\caption{Fit results for all transitions with total uncertainties.
These numbers represent the final result of this analysis. In the
case of the magnitude ratio $|\ensuremath{\clg{C}/\clg{A}}\xspace|$, we also quote a limit as detailed
in the text. The phase angles are quoted in degrees, and have a two-fold
ambiguity of reflection in the real axis.}
\label{tab:full_result_summary}
\end{table*}
\begin{figure*}
\centerline{
\resizebox{0.5\textwidth}{!}{\includegraphics{Figure12.eps}}
}
\caption{
Complex values of matrix element ratio
$\clg{B}/\clg{A}$ from combined fits for the three
transitions under the assumption that ${\cal C} = 0$. Note the
two-fold ambiguity in the imaginary part.}
\label{fig:matels_vs_plots}
\end{figure*}
We gratefully acknowledge the effort of the CESR staff
in providing us with excellent luminosity and running conditions.
D.~Cronin-Hennessy and A.~Ryd thank the A.P.~Sloan Foundation.
This work was supported by the National Science Foundation,
the U.S. Department of Energy, and
the Natural Sciences and Engineering Research Council of Canada.
\newpage
\section{Appendix: Details of the Likelihood Fitter}
This appendix gives some details of our application of the
likelihood fitter.
Smearing due to reconstruction resolution adds a small variance to the
Poisson error on the Monte Carlo integral, but the smearing widths are
small compared to the scales over which the matrix element changes so
this additional variance is small. For any shape with an
approximately polynomial form at a point, the resolution is described
by convolving a Gaussian with the polynomial. As an example, we
assume a functional form $g^T = a + b\,x + c\,x^2$ and seek its
observed shape in terms of the observed variables, $g^O( x^O )$,
using a Gaussian transformation:
\begin{eqnarray}
g^O( x^O ) & = & \int dx^T \clg{G}( x^T - x^O | \mu \equiv 0, \sigma )
g^T( x^T ) \\
& = & \int dx^T \clg{G}( x^T - x^O | \mu \equiv 0, \sigma )
( a + b\,x^T + c\,( x^T )^2 ) \\
& = & ( a + c\,\sigma^2 ) + b\,x^O + c\,(x^O)^2
\end{eqnarray}
So long as
$\sigma^2 \ll a/c$,
{\em i.e.,}\xspace the resolution is small compared
to the curvature, the shape will not be materially changed. For the
angular dependence, which is quartic in $\ensuremath{\cos \theta_X}\xspace$ this means the
resolution need only be small compared to 1/2; the observed
resolutions are of the order of 5\% or less. In ${\ensuremath{M_{\pi\pi}}}$ the same
holds true, with the scale being given by the pion mass,
\nUnits{140}{\ensuremath{{\rm MeV}/c^2}\xspace}, and the observed resolutions being at worst
\nUnits{10}{\ensuremath{{\rm MeV}/c^2}\xspace}. The shape of the decay amplitude is not changed
significantly by these resolutions, but any residual effect is
included in the estimated tracking and shower systematic
uncertainties.
Our problem differs from that discussed in
Ref.~\cite{Barlow:dm} in that the templates do not have independent
Poisson fluctuations. The underlying phase space simulation has a
Poisson fluctuation, but the templates are known (very nearly) exactly
and uncertainties on them do not contribute to the overall likelihood
function.
In the absence of background this problem is solved as follows,
with each two-dimensional ($q^{2}$, cos$\theta_{X}$) bin denoted by subscript
$i$.
We compare the Monte Carlo simulated, acceptance and efficiency-corrected,
phase space distribution (with true and observed yields $A_i$
and $a_i$),
multiplied by the modulus squared of the amplitude, with the data
distribution (with true and observed yields $D_i$
and $d_i$).
Both distributions are subject to Poisson
fluctuation:
\begin{equation}
{\cal P}( d_i; D_i ) = \frac{e^{-D_i}{D_i}^{d_i}}{d_i!}
\hspace*{5mm}\mbox{and}\hspace*{5mm}
{\cal P}( a_i; A_i ) = \frac{e^{-A_i}{A_i}^{a_i}}{a_i!}.
\end{equation}
Bin-by-bin, the modulus squared of the decay amplitude appears in the exact
relation between the
true data
yields $D_i$ and the true phase
space yields $A_i$:
\begin{equation}
D_i = f_i( \alpha )\,A_i.
\end{equation}
The function $f_i$ represents the decay distribution ($|{\cal M}|^{2}$) in the kinematic
space bin $i$ as a function of $\alpha$, the decay parameters. In
this case $\alpha$ consists of real and imaginary parts of $\ensuremath{\clg{B}/\clg{A}}\xspace$ and
$\ensuremath{\clg{C}/\clg{A}}\xspace$.
The log likelihood used in this fit is then given by, summing over all the bins:
\begin{equation}
\ln {\cal L}(\alpha) = \sum_{i=1}^n \left(
d_i \ln{f_i(\alpha) A_i} - f_i(\alpha) A_i - \ln{d_i!}
+ a_i \ln{A_i} - A_i - \ln{a_i!} \right).
\end{equation}
The $A_i$ represent the phase space subject to efficiency and
acceptance effects and are uninteresting nuisance parameters
that can be eliminated by extremizing the likelihood with respect
to them. Proceeding
in analogy with the approach in \cite{Barlow:dm} we can find the
analytic extremum condition, solve for $A_i$
\begin{equation}
A_i = \frac{d_i + a_i}{f_i + 1}
\end{equation}
and substitute back into the likelihood function to give a reduced
likelihood:
\begin{equation}
\ln {\cal L}(\alpha) = \sum_{i=1}^n \left[ \rule{0pt}{16pt} d_i \ln f_i(\alpha)
- ( d_i+a_i ) \ln( 1 + f_i(\alpha) ) \right]+ {\rm const.}
\end{equation}
We then minimize $-2 \ln {\cal L}$ with respect to the fit parameters
$\alpha$ (occurring only in the coefficients $f_i$). This is
implemented using the CERN Library minimization package,
MINUIT~\cite{CERNLib:MINUIT}.
The full likelihood as used in the fit includes an extension of this
approach to account for background under the signal peaks. This
introduces additional parameters $B_i$ and $b_i$. These represent bin
by bin true and observed background yields. The $B_i$ are a second
set of nuisance parameters that are eliminated in the same way as were
the $A_i$ before. The resulting likelihood is significantly more
complicated in detail but not in principle. For brevity it is not
included here.
\newpage
|
0706.2492
|
\section{Introduction}
This paper is a continuation of Ref (\cite{AnSav06}), in which a
procedure was sketched for the construction of a
Positive-Operator-Valued Measure (POVM) for the time-of-arrival for
a particle described by a Hamiltonian $\hat{H}$. Here, we extend
this POVM to cover the case of particles tunneling through a
barrier. This procedure allows us to provide an unambiguous
determination for the tunneling-time as it can be measured in
time-of-arrival type of measurements.
Quantum tunneling refers to the escape of a particle from a region
through a potential barrier, whose peak corresponds to an energy higher
than that carried by the the particles. There are two important
questions (relevant to experiments) that can be asked in this
regard. The first is, how long does it take a particle to cross the
barrier (i.e. what is the tunneling time?). The second is, what is
the law that determines the rate of the particle's escape through
the barrier? In this paper, we develop a formalism that provides an
answer to these questions and we apply it to the first one. The
issue of the decay probability will be taken up in \cite{An07b}.
The issue of tunneling time has received substantial attention in
the literature, especially after the 1980's--see the reviews
\cite{reviews-tt, Sokol-book}. The reason is that there is an
abundance of candidates and a diversity of viewpoints with no clear
consensus. There are roughly three classes of approaches: (i) Wave
packet methods: one follows the particle's wave packet across the
barrier and determines the tunneling time through a "delay in
propagation" \cite{phasetime, wpm}, (ii) one defines suitable
variables for the particle's paths and one obtains a probability
distribution (or an average) for the transversal time corresponding
to each path. These paths can be constructed either through
path-integral methods \cite{Sokol-book, SokBa87, pathint}, through
Bohmian mechanics \cite{bohmian}, or through Wigner functions
\cite{wf}, and (iii) the use of an observable for time: this can
take the form of an additional variable playing the role of a clock
\cite{clock1, Larmor}, or of a formal time operator \cite{timeop}.
In general, these methods lead to inequivalent definitions and
values for the tunneling time.
\subsection{Our approach}
The basic feature of our approach to this problem is its operational
character. We identify the tunneling time by constructing
probabilities for the outcome of specific measurements. We assume
that the quantum system is prepared in an initial state $\psi(0)$,
which is localized in a region on one side of a potential barrier
that extends in a {\em microscopic} region. At the other side of the
barrier and a {\em macroscopic} distance $L$ away from
it\footnote{We explain in section 2.3 the sense in which we employ
the word "macroscopic".}, we place a particle detector, which
records the arrival of particles. Using an external clock to keep
track of the time $t$ for the recorder's clicks, we construct a
probability distribution $p(t)$ for the time of arrival. The fact
that the detector is a classical macroscopic object and that it lies
at a macroscopic distance away from the barrier allows one to state
(using classical language) that the detected particles must have
passed through the barrier (quantum effects like a particle crossing
the barrier and then backtracking are negligible). Hence, at the
observational level, the probability $p(t)$ contains all information
about the temporal behavior for the ensemble of particles: all
probabilistic quantities referring to tunneling can be reconstructed
from it.
With the considerations above, both
problems of determining the tunneling time and the escape
probability as a function of time (see \cite{An07b}) are mapped to
the single problem of determining the time-of-arrival at the
detector's location for an ensemble of particles described by the
wave function $\psi_0$ at $t= 0$ and evolving under a Hamiltonian
with a potential term. To solve this problem, we elaborate on the
result of \cite{AnSav06}, namely the construction of a Positive
Operator Valued Measure (POVM) for the time-of-arrival for particles
for a generic Hamiltonian $\hat{H}$--see \cite{Davies} and
\cite{BLM96} for definition, properties and interpretation of POVMs.
This POVM provides a unique
determination of the probability distribution $p(t)$ for the
time-of-arrival. It is important to emphasize that by construction
$p(t)$ is {\em linear} with respect to the initial density matrix,
positive-definite, normalized (when the alternative of
non-detection is also taken into account) and a genuine density with
respect to time.
Since our results depend on the POVM for the time-of-arrival
constructed in \cite{AnSav06}, we review here the basic physical
considerations involved in its construction. The technical aspect,
namely the construction of this POVM for the problem at hand is
undertaken in Sec. 2.
The POVM of \cite{AnSav06} involves no structures other than the
ones of standard quantum mechanics: the Hamiltonian, the initial
state and the location of the recording device. It also involves a
smearing function with respect to time, but we employ it in the
regime in which the results are independent of such a choice. The
first step in the derivation arises from the remark that the notion
of arrival-time is well-defined when one considers {\em histories}
for a physical system (both in classical and in quantum
probability). We assume that the detector lies at $x = L$ and that
the initial state is localized in the region $I = \{x, x < L\}$.
Moreover, we assume a discretization $t_0, t_1, t_2, \ldots, t_n$ of
a time interval $[0, T]$. One asks at any instant $t_i$ of time,
whether the particle lies in region I or in region $II = \{x, x>
L\}$. The set of all possible successive alternatives forms a
Boolean algebra. The key point is that one can construct a
subalgebra of events labeled by the time of first crossing (together
with the event of no crossing), namely by the first instant $t_i$
that the particle is found in region $II$. This implies that
propositions about the time-of-arrival have a well-defined algebraic
structure, which is compatible with the Hilbert space description of
quantum mechanics. The algebra of propositions for the
time-of-arrival is a special case of the so-called spacetime
coarse-grainings \cite{Har, scc} that have been studied within the
consistent histories approach to quantum mechanics \cite{Gri84,
Omn8894, GeHa9093, Har93a}.
The construction above takes place at the discrete-time level. One
should then implement the continuum limit within the quantum
mechanical formalism. The problem is that there is no proper
continuous limit if one works at the level of probabilities (for the
same mathematical reason that leads to the quantum Zeno effect).
However, {\em there is} a proper continuum limit for this algebra if
one works at the level of amplitudes. More specifically, one can
implement the continuous limit at the level of the decoherence
functional, an object introduced in the consistent histories
approach.
The decoherence functional is a hermitian, bilinear functional on
the space of histories that contains all probability and phase
information for the histories of the system\footnote{Alternatively,
it can be viewed as a generating functional for all possible
temporal correlation functions of the system \cite{Ana01, Ana03}.}.
The restriction of the decoherence functional to the algebra of
propositions about the time-of-arrival effectively yields a
hermitian function $\rho(t, t')$ which is a density with respect to
both of its arguments.
The decoherence functional contains sufficient information for the
construction of POVMs for measurements that involve variables that
refer to more than one instant of time. This has been established
for sequential measurements \cite{Ana06} and for time-extended
measurements \cite{AnSav07}. In these cases one can compare the
results to ones obtained from single-time quantum mechanics, but for
the time-of-arrival, there is no analogous construction without the
use of histories. Nonetheless, the method provides a definition of
POVM for the time-of-arrival through a suitable smearing of the
diagonal elements of the decoherence functional. For a free
particle, this reproduces Kijowski 's POVM \cite{Kij74} in the
appropriate regime.
The important point in the procedure above is that the POVM of
\cite{AnSav06} is valid for a generic Hamiltonian. The time
parameter entering the POVM is the external Newtonian time and the
identification of the time-of-arrival is done through purely
kinematical arguments. Hence, this result can also be applied to the
specific Hamiltonian operators that are relevant to tunneling. This
is the content of Sec. 2.
Summarizing, there are three basic features in our approach: a) the
reformulation of tunneling as a time-of-arrival problem, b) the use
of POVMs for the determination of the probabilities for the
tunneling particles, and c) the basic ideas of the histories
approach that enable us to construct a suitable POVM.
\subsection{Relation to other approaches}
There are some common points and some points of divergence with
previous work on the tunneling time issue. Yamada has employed the
decoherence functional showing that different definitions of
tunneling time correspond to different definitions of the
alternatives for the `paths' considered in the definition
\cite{Yam04}. The construction of the decoherence functional is
different from ours in one respect: the (coarse-grained) histories
we consider refer to the paths' first crossing of the
surface $x = L$, which lies a macroscopic distance away from the
barrier. In \cite{Yam04}, the histories refer to the crossing of
{\em the barrier} and the ambiguity in the definition of the
tunneling time reflects the inability to decide which of all
possible spacetime coarse-grainings provides the true measure of
tunneling time. This is due to the fact that quantum `mechanical'
paths may cross and then reenter the barrier region. In our case,
this is not an issue. The detector is far away from the barrier
region (at a macroscopic distance $L$) and the probability that a
particle crossing $x = L$ would ever backtrack to the barrier is
practically zero. Another difference is that Yamada argues within
the context of the decoherent histories programme that deals with
closed systems \cite{Yam99}. While we employ the methods and (many)
conceptual tools of consistent histories, our approach is strictly
operational within the Copenhagen interpretation. The probabilities
we construct refer to measurement outcomes in a statistical
ensemble. The decoherence functional is only used as a mathematical
object that allows us to construct a POVM and the particle crossing
of the surface $x = L$ is viewed as corresponding to an
(irreversible) act of measurement by a device located there.
The fact that the measurement of the particle takes place far away
from the barrier region suggests that our results should be
compatible with the asymptotic analysis of wave packets. Indeed, as
we shall see, our expression for tunneling time (whenever this can
be defined) corresponds to the classic Bohm-Wigner phase time
\cite{phasetime}. However, the methodology is different: we do not
identify time through the peak $x(t)$ of the wave-packet (or through
its center-of-mass), but the probability distribution for the
detection time is obtained from a POVM that is defined for all
possible initial states. Unlike time of detection, a sharp
definition for the tunneling time is only possible for initial
states characterized by a strong peak in their momentum
distribution. However, the generality of our construction allows us
to fully specify the limits in the definability of tunneling time.
From the technical point of view, our approach has more in common
with the second class of proposals we mentioned in the beginning:
time being identified at the kinematical level from the properties
of `paths'. In particular, the formalism bears substantial
resemblance to the Feynman path integral derivation of tunneling
times of Sokolovski and Baskin \cite{SokBa87}. However, our boundary
conditions are different, and more importantly the probabilities we
obtain arise from proper probability densities with respect to
time. While the time-averaged quantities in \cite{SokBa87} are
linear with respect to a restricted propagator, such propagators
appear in a quadratic form in our expression for the probability. It
was argued extensively in \cite{AnSav06} that this is necessary, in
order to obtain a genuine probability density in a way that respects
the convexity of the space of quantum states. The present
construction also shares these properties and this implies that the
issue of complex tunneling times does not arise.
\subsection{Our results }
In Sec.3, we apply the POVM we constructed in Sec. 2 to a simple
case of a particle in one-dimension. We consider a potential
barrier $V(x)$, which takes non-zero values only in a bounded
(microscopic) region of width $d$ around $x = 0$. A wave packet
approaches the barrier from the negative real axis, while the
detection of the particle takes place at $x = L >> a$.
We assume that the initial wave -function is well localized in
position and in momentum (e.g. a coherent state). In addition, we
require that $\sigma/k_0 << 1$, where $k_0$ the mean momentum and
$\sigma$ the momentum spread of the initial state. We then find (for
a rather general regime for the values of the parameters
characterizing the system) that the probability distribution for the
time of arrival is sharply peaked around a value $t_m$.
From this probability distribution, we identify the delay due to
the presence of the barrier as the difference between the time $t_m$
and the time it would take a classical particle of momentum $k_0$ to
travel from the center of the initial wavepacket to the location of
the detector. This `delay time' equals
\begin{eqnarray}
t_d = \frac{M}{k_0} \left(\frac{\partial T_k}{\partial k} \right)_{k
= k_0},
\end{eqnarray}
where $T_k$ is the transmission amplitude corresponding to the
potential $V(x)$. The delay time may be negative: the tunneling time
is obtained as the sum of $t_d$ with the time it would take the
particle to cross the barrier: it coincides with the classic phase
time.
Note that the delay and tunneling times defined this way are {\em
not} observables of the system: they cannot be defined for a
generic initial state, but only for states well localized in
momentum and for values of the parameters that lead to a probability
distribution $p(t)$ characterized by a sharp peak. In this case, one
can use classical arguments for their definition. However, if either
the initial distribution has a substantial momentum spread, or if
$p(t)$ exhibits a more complex structure, there is no unambiguous
way to define tunneling time. While the value for the
time-of-arrival is a genuine observable (a random variable on the
sample space of the POVM), the tunneling time as we define it here
requires the knowledge of the corresponding time-of-arrival for a
free particle: and this cannot be defined, unless the initial value
of momentum is known\footnote{Strictly speaking, the above
definition of tunneling time involves counterfactual reasoning.
However, in the operational setting we consider here this is not a
problem, as long as we keep in mind that the tunneling time
(whenever it can be defined) is a `property' of the ensemble of
detected particles and not of any individual one.}.
Hence, in this approach the tunneling time is a {\em parameter} of
the detection probability. It can only be identified for specific
initial states, and not for any state, because its definition
involves a correspondence argument to classical physics.
In Sec. 5, we propose a generalization of the results above that
leads to a definition of the tunneling time as a genuine random
variable. The idea is to consider a sequential measurement set-up:
the phase space properties of the particle are determined through an
unsharp phase space sampling before this attempts to cross the
barrier, {\em and then} the time-of-arrival for the particles that
crossed the barrier is measured. The sample space corresponding to
such sequential measurements accommodates the definition of the
tunneling time as a genuine quantum observable and it allows us to
construct a marginal POVM that provides its probabilities for a
generic initial state. In a specific regime, this POVM becomes
independent of the details of the first measurement: as such it
defines an ideal probability distribution for the delay and the
tunneling times: this distribution suggests a definition for a
delay-time and for a tunneling-time operator.
\section{The general probability measure}
In this section, we review the construction of the POVM for the
time-of-arrival in \cite{AnSav06}, and we extend it for the case
relevant to tunneling.
\subsection{The histories formalism}
The POVM of \cite{AnSav06} is constructed using some notions of
quantum mechanical histories, as they appear in the consistent
histories approach to quantum theory of Griffiths \cite{Gri84},
Omn\'es \cite{Omn8894}, Gell-Mann and Hartle \cite{GeHa9093,
Har93a}. We should note however that these objects are used in the
present context differently, namely in an operational approach to
quantum theory--see \cite{Ana03, Ana06}
A history intuitively corresponds to a sequence of properties of
the physical system at successive instants of time. A discrete-time
history $\alpha$ is then represented by a string $\hat{P}_{t_1},
\hat{P}_{t_2}, \ldots \hat{P}_{t_n}$ of projectors, each labeled by
an instant of time. From them, one can construct the class operator
\begin{equation}
\hat{C}_{\alpha} = \hat{U}^{\dagger}(t_1) \hat{P}_{t_1} \hat{U}(t_1)
\ldots \hat{U}^{\dagger} (t_n) \hat{P}_{t_n} \hat{U}(t_n)
\end{equation}
where $\hat{U}(s) = e^{-i\hat{H}s}$ is the time-evolution operator.
A candidate probability for the realisation of this history is
\begin{equation}
p(\alpha) = Tr \left( \hat{C}_{\alpha}^{\dagger}\hat{\rho}_0
\hat{C}_{\alpha} \right), \label{decfundef}
\end{equation}
where $\hat{\rho}_0$ is the density matrix describing the system at time $t = 0 $.
However, the expression above does not define a probability measure
in the space of all histories, because the Kolmogorov additivity
condition cannot
be satisfied: if $\alpha$ and $\beta$ are exclusive histories, and $\alpha \vee \beta$
denotes their conjunction as propositions, then it is not true that
\begin{equation}
p(\alpha \vee \beta ) = p(\alpha) + p(\beta) .
\end{equation}
The
histories formulation of quantum mechanics does not, therefore,
enjoy the status of a genuine
probability theory on the space of all histories.
However, an
additive probability measure {\it is} definable, when we restrict to
particular sets of histories.
These are called {\it consistent sets}. They are more conveniently
defined through the introduction of a new object: the decoherence
functional. This is a complex-valued function of a pair of histories
given by
\begin{equation}
d(\alpha, \beta) = Tr \left( \hat{C}_{\beta}^{\dagger} \hat{\rho}_0
\hat{C}_{\alpha} \right).
\end{equation}
A set of exclusive and exhaustive alternatives is called consistent,
if for all pairs of different histories $\alpha$ and $\beta$, we
have $
Re \; d(\alpha, \beta) = 0$.
In this case, one can use equation (\ref{decfundef}) to assign a
probability measure to this set.
\subsection{Time-of-arrival histories}
Using the histories formalism we construct a decoherence functional
for time-of-arrival histories with $N$ time steps $t_1, t_2, \ldots
t_N$(discrete-time). The reason for this construction is that the
decoherence functional has a good continuous time limit (unlike the
probabilities for histories).
We consider a particle in one dimension for concreteness, even
though the results obtained here only use abstract Hilbert space
operators and hold more generally. We split the line into the
interval $(- \infty, L]$ and the interval $[L, \infty)$. Let
$\hat{P}_-$ and $\hat{P}_+$ be the corresponding projectors. Our aim
is to identify histories that correspond to the statement that the
particle crossed from the $-$ region to the $+$ region during a
particular time step. If we assume that at $t = 0$ the particle
lies at the $-$ region then it is easy to verify that the history
\begin{eqnarray}
\alpha_m := (\hat{P}_-, t_1; \hat{P}_-, t_2; \ldots, \hat{P}_-, t_m;
\hat{P}_+, t_{m+1}; 1, t_{m+2}; \ldots 1, t_{N})
\end{eqnarray}
corresponds to the
proposition that the particle crossed $x=L$ for the first time
between the $m$-th and the $m+1$-th time step. The sequence
$\bar{\alpha} = (\hat{P}_-, t_1; \hat{P}_-, t_2; \ldots, \hat{P}_-,
t_{m}; \ldots \hat{P}_-, t_{N})$ corresponds to the proposition that
the particle did not cross $x = L$ within the $n$- time steps.
The set of histories $\alpha_m$ together with $\bar{\alpha}$ is
exhaustive and exclusive (a sublattice of the lattice of history
propositions)--see also \cite{Har, Hal95}. The decoherence
functional is then defined on this set of histories: it is a
hermitian bilinear functional on a sample space consisting of the
points $(t_1, \ldots, t_n)$ together with the point $N$
corresponding to no crossing
\begin{eqnarray}
d(t_n, t_m) &=& d(\alpha_n, \alpha_m) \\
d(N, t_n ) &=& d(\bar{\alpha}, \alpha_n) \\
d(N,N) &=& d(\bar{\alpha}, \bar{\alpha}).
\end{eqnarray}
We next consider
two discretisations $\{t_0 =0, t_1, t_2, \ldots t_N = T\}$ and $ \{t'_0
=0, t'_1, t'_2, \ldots t'_{N'} = T \}$ of the time interval $[0, T]$
with time-step $\delta t = T/N$, and $\delta t' = T/N'$. We
construct the decoherence functional $d([t, t+\delta t], [t',
t'+\delta t'])$, where $ n = t N/T$ and $m = t' N'/T$. This reads
\begin{eqnarray}
d([t, t+\delta t], [t', t'+\delta t']) = Tr \left( \hat{\rho}_0
[e^{i \hat{H} \delta t'} \hat{P}_-]^n e^{i \hat{H}\delta t'}
\hat{P}_+ \right. \nonumber \\
\left. \times e^{i \hat{H}(t'-t)} \hat{P}_+ e^{-i \hat{H} \delta t}
[\hat{P}_- e^{-i \hat{H} \delta t}]^m \right).
\end{eqnarray}
We then take the limit $N, N' \rightarrow \infty$, while keeping $t$
and $t'$ fixed. Assuming that $\rho_0$ lies within the range of
$\hat{P}_-$, i.e. $\hat{P}_- \hat{\rho}_0 \hat{P}_- = \hat{\rho}_0$
we obtain
\begin{eqnarray}
d([t, t+\delta t], [t', t'+\delta t']) = \delta t \delta t' Tr
\left( e^{i \hat{H}(t'-t)} \hat{P}_+ \hat{H} \hat{P}_- \hat{C}_{t}
\hat{\rho}_0 \hat{C}^{\dagger}_{t'} \hat{P}_- \hat{H} \hat{P}_+
\right),
\end{eqnarray}
where $\hat{C}_t = \lim_{n \rightarrow \infty} (\hat{P}_- e^{-i
\hat{H} t/n} \hat{P_-})^n$. Writing
\begin{eqnarray}
\rho(t,t') = Tr \left( e^{i \hat{H}(t'-t)} \hat{P}_+ \hat{H}
\hat{P}_- \hat{C}^{\dagger}_{t'} \hat{\rho}_0 \hat{C}_{t} \hat{P}_-
\hat{H} \hat{P}_+ \right) \label{densitydecf}
\end{eqnarray}
we see that the decoherence functional defines a complex-valued
density on $[0,T] \times [0,T]$. The additivity of the decoherence
functional (which reflects the additivity of quantum mechanical
amplitudes) allows us to obtain a continuum limit, something that
could not be done if we worked at the level of probabilities.
\subsection{The tunneling Hamiltonian}
For the simple case of a particle at a line with Hamiltonian
$\hat{H} = \frac{\hat{p}^2}{2 M} + V(\hat{x})$, where the potential
$V(x)$ is bounded from below, we employ a result in \cite{Har,
Facchi} that the restricted propagator $\hat{C}_t$ is obtained from
the Hamiltonian $\hat{H}$ by imposing Dirichlet boundary conditions
at $x = L$. If we also denote by $G_0(x,x'|t)$ the full propagator
in the position basis (corresponding to $e^{-i \hat{H}t}$), we
obtain
\begin{eqnarray}
\rho(t,t') &=& \frac{1}{4 M^2}\partial_x (\hat{C}_{t'} \psi_0)^*(L)
\partial_x(\hat{C}_t \psi_0)(L)
G_0(L,L|t-t') \label{basic}
\end{eqnarray}
where $\hat{\rho}_0 = | \psi_0 \rangle \langle \psi_0 |$, with
$\psi_0$ having support for $x < L$.
We now specialize to a case relevant for tunneling. We assume that
the potential is short-range: it is significantly different from
zero only in a region of width $d$ around $x = 0$. The distance $L$
is macroscopic, while $a$ is microscopic. This means that in the
neighborhood of $x = L$ the propagator is effectively that of a
free particle. Hence, we can substitute $G_0(L,L|t'-t)$ in Eq.
(\ref{basic}) with the corresponding expression for the free
particle
\begin{eqnarray}
G(L,L|t) = \left( \frac{M}{2 \pi i t} \right)^{1/2}. \label{freep}
\end{eqnarray}
The considerations above also specify the range of values for $L$
that are relevant to our problem. The first condition on $L$ is that
the propagator may be substituted by that of the free particle, as
in Eq. (\ref{freep}). The second is that $L$ is sufficiently far
away from the tunneling region so that the probability amplitude of
a particle backtracking to the barrier region from $L$ is
practically zero. Physically one expects that this is the case for
all initial states $\psi_0$ for which the position spread $\Delta
q(t)$ remains at all times much smaller than $L$. Clearly, with the
considerations above it is not necessary that $L$ is a macroscopic
distance in the literal sense of the word: the requirement that $L$
be macroscopic is a sufficient but not a necessary condition.
We next consider the Hamiltonian $\hat{H}_D$ that is obtained from
the original Hamiltonian $\hat{H}$ by imposing Dirichlet boundary
conditions at $x=L$. We distinguish two cases: (i) if $x$ takes
value in the half-line, the spectrum of $\hat{H}_D$ is expected to
be discrete; (ii) if $x$ takes values in the full real axis, at
least the positive energy spectrum will be continuous. (We restrict
to Hamiltonians having this property.) Either way, for $x
>> a$, $V(x) = 0$ and the solution of the Schr\"odinger equation
$\hat{H}_D \psi_E(x) = E \psi_E(x)$ with Dirichlet boundary
conditions is proportional to $\sin k(L -x)$, where $k =
(2ME)^{1/2}$. We choose to label the eigenstates of $\hat{H}_D$ by
$k$, namely we write $|k\rangle_D$ as a solution to the equation
\begin{eqnarray}
\hat{H} |k \rangle_D = \frac{k^2}{2M} | k \rangle_D,
\end{eqnarray}
with Dirichlet boundary conditions.
Normalizing $|k \rangle_D$ so that
\begin{eqnarray}
{}_D\langle k| k' \rangle_D = \delta (k, k'),
\end{eqnarray}
(and similarly in the discrete-spectrum case) we write
\begin{eqnarray}
\langle x|k\rangle_D = D_k \sin k (L-x),
\end{eqnarray}
where the form of the normalization factor $D_k$ is specified the
Hamiltonian's (generalized) eigenstates.
For the study of tunneling, we assume that the initial state of the
system has support only in the positive energy spectrum of
$\hat{H}$. Hence,
\begin{eqnarray}
\langle x|\hat{C}_t |\psi_0 \rangle = \sum_k e^{-ik^2t/2M} D_k \sin
k(L-x) c_k,
\end{eqnarray}
where $c_k = {}_D\langle k|\psi_0 \rangle$ and $\sum_k$ denotes the
integration with respect to the spectral measure of $\hat{H}_D$.
Substituting into Eq. (\ref{basic}), we obtain
\begin{eqnarray}
\rho(t,t') = \frac{1}{4M \sqrt{2 \pi i M (t- t')}} \sum_{kk'} D_k
D_{k'}^* c_k c^*_{k'} kk' e^{-i\frac{k^2t-k'^2t'}{2M}}. \label{ro}
\end{eqnarray}
\subsection{Construction of the POVM}
The decoherence functional contains sufficient information for the
construction of POVMs for the probabilities of measurement outcomes
for magnitudes that have an explicit time-dependence. In particular,
the probabilities for the measurement outcomes for single-time,
sequential and extended-in-time measurements (obtained through the
standard formalism) can be identified with suitable diagonal
elements of the decoherence functional--see \cite{Ana06,AnSav07}. In
other words, one can define POVMs by suitable smearing of the
decoherence functional and in the cases above, these POVMs coincide
with ones obtained from the standard methods in quantum measurement
theory. In the case of the time-of-arrival there is no analogous
expression obtained from standard methods. However, the smeared form
of the decoherence functional still defines a POVM, and the main
assumption in this paper is that this POVM yields the correct
probabilities.
With this assumption, we obtain the following probability density
for the time-interval $[0, T]$
\begin{eqnarray}
p^{\tau}(t) = \int_0^{T} ds \int_0^{T} ds' \sqrt{f^{\tau}(t,s)}
\sqrt{f^{\tau}(t,s')} \rho(s, s'), \label{ppp}
\end{eqnarray}
here $f_{\tau}(s, s')$ is a family of smeared delta functions
$f_{\tau}(s, s')$ characterized by the parameter $\tau$. The
functions $f_{\tau}$ satisfy the following property
\begin{eqnarray}
\int_0^T ds f^{\tau}(s, s') = \chi_{[0, T]}(s'),
\end{eqnarray}
where $\chi_{[0, T]}$ is the characteristic function of the interval
$[0, T]$: $\chi_{[0, T]}(s) = 1$ if $s \in [0, T]$, and $\chi_{[0,
T]} (s) = 0$ otherwise. The functions $f^{\tau}$ incorporate
specific features of the instrument that records particles crossing
the surface $x = L$.
Essentially, the key assumption in our approach (stated above) is
that the functions $f^{\tau}$ appearing in the definition of
(\ref{ppp}) are analogous to the smearing functions that appear in
the definition of POVMs for usual observables (i.e. ones other than
the time of arrival). In \cite{AnSav06}, we showed that this
assumption leads for the case of free particles to Kijowski's POVM
\cite{Kij74}.
The decoherence functional satisfies an hermiticity condition
$\rho(s, s') = \rho^*(s', s)$, which together with the positivity
condition for its diagonal elements
\begin{eqnarray}
\int_a^b ds \int_a^b \rho(s, s') \geq 0
\end{eqnarray}
guarantees that $p^{\tau}(t)$ is positive-definite for all values of
$t$.
The density (\ref{ppp}) is linear with respect to the initial
density matrix. Together with the probability of no-detection
\begin{eqnarray}
p^{\tau}(N) = 1 - \int_0^T ds p^{\tau}(s)
\end{eqnarray}
they define a POVM $\hat{\Pi}$ on the space $[0, T] \cup \{N\}$.
This POVM describes the time of detection of a particle by an
instrument located at $x = L$.
In this paper, we will be interested in taking $T \rightarrow
\infty$, i.e. taking $t \in [0, \infty)$. It is convenient to work
with Gaussian smearing functions
\begin{eqnarray}
f^{\tau}(s,s') = \frac{1}{\sqrt{2 \pi} \tau} e^{ - \frac{(s -
s')^2}{2 \tau^2}}. \label{Gauss}
\end{eqnarray}
However, these Gaussians are smeared delta-functions with respect to
the whole real axis and {\em not} with respect to the time-interval
$[0, \infty)$. To remedy this problem, we note that by Eq.
(\ref{ro}), $\rho(-s, -s') = \rho^*(s,s') = \rho(s', s)$ and that
the probability (\ref{ppp}) is symmetric to an exchange of $s$ and
$s'$. We also note that the contribution of terms that mix positive
and negative $s$ are significant only at times $|t|$ of order
$\tau$. Hence, the probability (\ref{ppp}) with the Gaussian
(\ref{Gauss}) is substituted in place of $f^{\tau}$ and the
integration limits taken from $- \infty$ to $\infty$, is twice the
probability density that is defined with an integral over the
positive half-axis. Hence, the use of Gaussian smearing functions
only involves dividing $p^{\tau}(t)$ in (\ref{ppp}) by a factor of 2
(for times $t>> \tau$).
Inserting (\ref{Gauss} into (\ref{ppp}), we change variables to $u =
\frac{1}{2} (s+s')$ and $v = s-s'$ noting that
\begin{eqnarray}
\sqrt{f^{\tau}(t,s)} \sqrt{f^{\tau}(t,s')} = f_{\tau} (u - t) e^{-
\frac{v^2}{8 \tau^2}}.
\end{eqnarray}
We substitute in the integration $f_{\tau}(u-t)$ with a delta
function $\delta(u-t)$. We then obtain
\begin{eqnarray}
p^{\tau}(t) = \frac{1}{8M \sqrt{2 \pi M }} \sum_{kk'} D_k D_{k'}^*
c_k c^*_{k'} kk' e^{-i\frac{k^2- k'^2}{2M}t} \; R\left(\frac{k^2 +
k'^2}{4M}\right), \label{fullppp}
\end{eqnarray}
where
\begin{eqnarray}
R(\epsilon) = \int_{-\infty}^{\infty} dv \frac{e^{-v^2/8\tau^2 - i
\epsilon v}}{\sqrt{iv}} = 2 \sqrt{\tau} \int_0^{\infty} dy
\frac{e^{-y^2/2} [\cos (2 \epsilon \tau y) + \sin (2 \epsilon \tau
y)]}{\sqrt{y}}.
\end{eqnarray}
At the limit of $\epsilon \tau >> 1$, i.e. if the detection time is
much larger than $\epsilon^{-1}$ \footnote{This condition is valid
if the mean energy of the initial state is much larger than the
energy uncertainty, and it is accurate for all times $t >> \tau$.}
\begin{eqnarray}
\int_0^{\infty}dy \; \frac{ e^{ - \frac{y^2}{2}} [ \cos (2 \epsilon
\tau y) + \sin(2 \epsilon \tau y)]}{\sqrt{y}} \simeq
\sqrt{\frac{\pi}{\epsilon \tau}}.
\end{eqnarray}
Hence, $R(\epsilon) = 2 \sqrt{\pi/\epsilon}$. It follows that
\begin{eqnarray}
p(t) = \frac{1}{ 2\sqrt{2 }M} \sum_{kk'} D_k D_{k}^* c_k c^*_{k'}
\frac{kk'}{\sqrt{k^2 +k'^2}} e^{-i \frac{k^2 - k'^2}{2M} t}.
\label{probab}
\end{eqnarray}
The probability for the time-of-arrival then becomes independent of
the parameter $\tau$, and it is expressed solely in terms of the
system's Hamiltonian, the initial state and the value of $L$.
Eq. (\ref{probab}) is
simplified if the spread $\Delta k$ of the initial state
$|\psi_0\rangle$ ( $\hat{k} = \sqrt{2M\hat{H}_D}$) is much smaller
than the corresponding mean value $\bar{k}$: in this case, $k^2 +
k'^2 \simeq 2 kk'$, hence
\begin{eqnarray}
p(t) = \left| \sum_k D_k c_k \sqrt{\frac{k}{4M}}
e^{-ik^2t/2M}\right|^2.
\end{eqnarray}
It was shown in \cite{AnSav06} that for the test case of a free
particle (, in which $D_k = (2 \pi)^{-1/2}$)
the probability distribution above reproduces the one of
Kijowski \cite{Kij74}.
\section{The detection probability}
In this section, we use the probability density (\ref{probab}) in a
specific context that allows us to determine a magnitude that
corresponds to the time the particle spends in the forbidden region.
In effect, we identify tunneling-time by the delay caused by the
presence of the barrier to the particles' time-of-arrival (see Sec.
4). This turns out to be the same definition as the one employed in
the methods involving the wave packet analysis. However, we do not
identify any specific features of the wave-packet (these objects
have no natural probabilistic or operational interpretation in
quantum mechanics), but we work directly at the level of measurement
outcomes, namely the probability distribution for the
time-of-arrival.
We consider the simplest possible example of a particle tunneling
through a potential barrier. We assume that the potential $V(x) \geq
0 $ takes non-zero values in a region of width $d$ around $x=0$. Let
$V_0$ be the maximum value of this potential. In classical mechanics
no particle with energy $E < V_0$ can cross the barrier, hence the
probability of detection at $x = L$ is zero at all times. We next
consider this problem in quantum theory.
Eq. (\ref{probab}) involves the eigenstates of the Hamiltonian with
Dirichlet boundary conditions at $x = L$. Since $x \in (-\infty,
\infty)$, the spectrum of the Dirichlet Hamiltonian is continuous.
The summation over $k$ is then substituted by an integral
$\int_0^{\infty} dk$.
The first step is to construct the generalized eigenstates of the
Hamiltonian with Dirichlet boundary conditions. To do so, we first
study the solutions to the Schr\"odinger equation
\begin{eqnarray}
-\frac{1}{2M} \partial^2_x u(x) + V(x) u(x) = \frac{k^2}{2M} u(x).
\end{eqnarray}
There are two linearly independent solutions for each value of $k$.
It will necessary to construct an orthonormal basis of generalized
eigenstates from these solutions. We pick one class of solutions
$u_k^+(x)$ that correspond to a particle propagating from $-\infty$
and scattering on the potential
\begin{eqnarray}
u_k^+(x) = \left\{ \begin{array}{c} A^+_k \left( e^{ikx} + R^+_k
e^{-ikx} \right) \; \; x < -d/2 \\ A_k^+ T^+_k e^{ikx}\; \; \; \; \;
\; \; \; \; \; \; \; \; \; \; x
>d/2
\end{array} \right.,
\end{eqnarray}
where $R_k^+$ and $T_k^+$ is the reflection and transmission
coefficient respectively, while $A_k^+$ is a normalization factor so
that $\int dx \bar{u}_k^+(x) u_k^+(x) = \delta(k-k')$. Let $u^-_k$
be a normalized linearly independent solution that satisfies $\int
dx \bar{u}_k^+(x) u^-_k(x) = 0$. Its form will be the following
\begin{eqnarray}
u_k^-(x) = \left\{ \begin{array}{c} A^-_k
\left( T^-_k e^{-ikx} + S_k e^{ikx} \right) \; \; x < -d/2 \\
A_k^- \left( e^{-ikx} + R^-_k e^{ikx} \right) \; \; x
>d/2
\end{array} \right.
\end{eqnarray}
Note that there is no reason for $u^-_k$ to have a physical
interpretation in terms of left-moving particles, and the labels
$T_k^-, R_k^-$ are chosen for convenience: they do not correspond to
a transmission and reflection coefficient of any short. We also note
that the coefficients in $u^+_k, u^-_k$ are not independent. For any
two solutions $\psi, \phi$ to the Schr\"odinger equation with the
same energy, the Wronskian $\psi' \phi - \phi' \psi$ must be
$x$-independent. This yields the following conditions
\begin{eqnarray}
T^+_k = T^-_k - S_k \bar{R}^+_k \\ \label{cond1}
S_k = \bar{T}^+_k R^-_k + T^-_k \bar{R}^+_k \\
|T^+_k|^2 + |R^+_k|^2 = 1 \\
|T^-_k|^2 + |R^-_k|^2 = 1 + |S_k|^2. \label{cond4}
\end{eqnarray}
To impose the Dirichlet boundary conditions on these solutions, we
take a linear combination $v_k(x)$ of $u_k^+(x)$ and $u_k^-(x)$ and
require that $v_k(L) = 0 $. This yields
\begin{eqnarray}
v_k(x) = C_k \left[ A_k^- (1 + R_k^- e^{2ikL}) u_k^+(x) - A_k^+
T_k^+ e^{2ikL} u_k^-(x) \right],
\end{eqnarray}
where
\begin{eqnarray}
C_k = \frac{1}{\sqrt{|A_k^-|^2 |1 + R_k^- e^{2ikL}|^2 + |A_k^+|^2
|T_k^+|^2}}
\end{eqnarray}
is a normalization constant chosen so that $\int dx \bar{v}_k(x)
v_{k'}(x) dx = \delta(k-k')$.
For $x > d/2$, we obtain
\begin{eqnarray}
v_k(x) = - 2i C_k A_k^- A_k^+ T^+_k e^{ikL} \sin k(L-x).
\end{eqnarray}
Hence,
\begin{eqnarray}
D_k = - 2i C_k A_k^- A_k^+ T^+_k e^{ikL}
\end{eqnarray}
We now consider a Gaussian initial state $\psi_0$ centered around
$x_0 < -d/2$ and having mean momentum $k_0 > 0$
\begin{eqnarray}
\psi_0(x) = \frac{1}{(2 \pi \delta^2)^{1/4}} e^{- \frac{(x-x_0)^2}{4
\delta^2} + i k_0 x}, \label{initial}
\end{eqnarray}
where $\delta$ is the spread in position and we assume that $ \delta
<< |x_0 + d/2|$ so that the initial state does not overlap with the
region where the potential is non-zero. In this region,
\begin{eqnarray}
v_k(x) = C_k A_k^- A_k^+ \left[ (1 + R_k^- e^{2ikL} - T_k^+ S_k
e^{2ikL}) e^{ikx} \right. \nonumber \\
\left. + (R_k^+ R_k^- e^{2ikL} - T_k^+ T_k^- e^{2ikL}) e^{-ikx}
\right].
\end{eqnarray}
The coefficients $c_k = {}_D\langle k| \psi_0 \rangle $ are then
given by
\begin{eqnarray}
c_k = \bar{C}_k \bar{A}_k^- \bar{A}_k^+ \left[ 1 + (\bar{R}_k^- -
\bar{T}_k^+ \bar{S}_k) e^{-2ikL} \right] \frac{1}{(2 \pi
\sigma^2)^{1/4}} e^{ - \frac{(k- k_0)^2}{4 \sigma^2} - ix_0 (k +
x_0)},
\end{eqnarray}
where we set $\sigma = (2 \delta)^{-1}$ the momentum spread. The
assumption that $\sigma/k_0 << 1$ allowed us to drop a term of
order $e^{-k_0^2/4\sigma^2}$.
The probability for the time-of-arrival at $x = L$ is then given by
$p(t) = |z(t)|^2$, where
\begin{eqnarray}
z(t) = \int_0^{\infty} dk \; B_k \; e^{ikL} \; \frac{1}{(2 \pi
\sigma^2)^{1/4}} e^{ - \frac{(k- k_0)^2}{4 \sigma^2} - ix_0 (k -
k_0)} \sqrt{\frac{k}{4M}} e^{-ik^2t/2M}. \label{z}
\end{eqnarray}
In (\ref{z}) we defined
\begin{eqnarray}
B_k = -2i \sqrt{2 \pi}|C_k|^2 |A_k^-|^2 |A_k^+|^2 \left[ 1 +
(\bar{R}_k^- - \bar{T}_k^+ \bar{S}_k) e^{-2ikL} \right] T_k^+.
\label{beta}
\end{eqnarray}
Since $\sigma/k_0 << 1$, we can expand $B_k$ around its value at $k
= k_0$. As a term $\sqrt{k}$ also appears in the integral outside
the exponential, we expand together
\begin{eqnarray}
\sqrt{k} B_k \simeq \sqrt{k_0} B_{k_0} e^{(\xi_{k_0} + i
\lambda_{k_0})(k-k_0)}, \label{expansion0}
\end{eqnarray}
where
\begin{eqnarray}
\xi_{k_0} = \frac{1}{2k_0} + \left(\frac{\partial \log
|B_k|}{\partial k}\right)_{k = k_0}\\
\lambda_{k_0} = \left(\frac{\partial \arg[B_k]}{\partial
k}\right)_{k = k_0}.
\end{eqnarray}
Within the same approximation, we take the limits of integration in
Eq. (\ref{z}) from $-\infty$ to $\infty$. We then obtain
\begin{eqnarray}
z(t) = B_{k_0} e^{ik_0L} \sqrt{\frac{k_0}{4M}} \frac{1}{(2 \pi
\sigma^2)^{1/4}} \nonumber \\
\times \int_{-\infty}^{\infty} dk e^{ - \frac{(k- k_0)^2}{4
\sigma^2} + i(|x_0| + L+ \lambda_{k_0} - i \xi_{k_0}) (k - k_0)}
e^{-ik^2t/2M}.
\end{eqnarray}
The expression above involves a standard Gaussian integral. Its
evaluation gives
\begin{eqnarray}
z(t) = B_{k_0} e^{-i k_0^2t/2M+ ik_0L} \sqrt{\frac{k_0}{4M}} \frac{
(8 \pi \sigma^2)^{1/4}}{\sqrt{1 + 2 i t \sigma^2/M}} \nonumber \\
\times \exp \left[ - \sigma^2 \frac{ (|x_0| + L + \lambda_{k_0} -
\frac{k_0t}{M} - i \xi_{k_0})^2}{1 + 2it \sigma^2/M} \right].
\end{eqnarray}
Hence,
\begin{eqnarray}
p(t) = |z(t)|^2 = |B_{k_0}|^2 e^{2 \sigma^2 \xi_{k_0}^2}
\frac{k_0}{4M} \sqrt{\frac{8 \pi \sigma^2}{1 + 4 t^2 \sigma^4/M^2}}
\nonumber \\
\times \exp \left\{ - \frac{2 k_0^2 \sigma^2/M^2}{1 + 4 t^2
\sigma^4/M^2} \left[(1 + 2 \xi_{k_0} \sigma^2/k_0) t - \frac{M(|x_0|
+ L + \lambda_{k_0})}{k_0}\right]^2 \right\} \label{p1}
\end{eqnarray}
This expression is the probability distribution for the
time-of-arrival, as it would be measured by a device located at
distance $L$ from the barrier. In the following section, we analyze
its properties: in particular, we identify the delay caused by the
presence of the barrier.
\section{Delay-time and tunneling time}
\subsection{The identification of delay time}
For a sufficiently monochromatic wave packet ($\sigma/k_0
\rightarrow 0$), we assume that $\xi_{k_0} \sigma^2/k_0 <<1$, hence
Eq. (\ref{p1}) yields
\begin{eqnarray}
p(t) = |z(t)|^2 = |B_{k_0}|^2 e^{2 \sigma^2 \xi_{k_0}^2}
\frac{k_0}{4M} \sqrt{\frac{8 \pi \sigma^2}{1 + 4 t^2 \sigma^4/M^2}}
\nonumber \\
\times \exp \left\{ - \frac{2 k_0^2 \sigma^2/M^2}{1 + 4 t^2
\sigma^4/M^2} \left[ t - \frac{M(|x_0| + L +
\lambda_{k_0})}{k_0}\right]^2 \right\} \label{p2}
\end{eqnarray}
The term $1 +4 t^2 \sigma^4/M^2$ corresponds to the spread in the
particle's wave function due to time evolution. Since we want a
configuration in which the determination of time is as sharp as
possible, we assume that the value of $\sigma$ is so small that this
spread is negligible at the time $t_m = \frac{M(|x_0| + L +
\lambda_{k_0})}{k_0}$, namely that $t_m^2 \sigma^2/M << 1$. Then we
obtain
\begin{eqnarray}
p(t) = |B_{k_0}|^2 e^{2 \sigma^2 \xi_{k_0}^2} \frac{k_0}{4M} \sqrt{8
\pi \sigma^2} \exp \left\{- \frac{2 k_0^2 \sigma^2}{M^2} \left[ t -
\frac{M(|x_0| + L + \lambda_{k_0})}{k_0}\right]^2 \right\}.
\label{p3}
\end{eqnarray}
Then $t = t_m$ is a sharp peak for the mean value of the
time-of-detection. A classical particle (or in quantum theory a
narrow wavepacket) that starts from $x_0$ with momentum $k_0$ {\em
in absence of the potential barrier} will arrive at $x = L$ at
(average) time $t_0 = M \frac{|x_0| + L}{k_0}$. Hence, the barrier
causes a `delay' $t_d = t_m - t_0$ to the time-of-arrival (of the
particles that are not reflected)
\begin{eqnarray}
t_d = M \lambda_{k_0}/k_0.
\end{eqnarray}
The presence of the barrier has increased the effective length that
has to be traversed by the particle by a factor of $\lambda_{k_0}$.
In fact, $\lambda_{k_0}$ may be negative: the time it takes the
particle to cross the forbidden region of the barrier is $ t_{tun} =
M (\lambda_{k_0} + d_{k_0})/k_0$, where $d_{k_0} = x_2(k_0) -
x_1(k_0) \geq 0$, where $x_{1,2}(k_0)$ are the points that
determine the forbidden region: they are respectively the lowest-
and highest-valued solutions of the equation $\frac{k_0^2}{2M} =
V(x)$. The total tunneling time has to be positive, but it is not
necessary that it is larger than the time $Md_{k_0}/k_0$ that the
forbidden region is traversed by a classical free particle.
We next calculate $\lambda_{k_0}$ in terms of the absorbtion and
reflection coefficients corresponding to the potential $V(x)$. From
Eq. (\ref{beta}) we see that the only term contributing to a phase
in $B_k$ is the product $\left[ 1 + f_k e^{-2ikL} \right] T_k^+ $,
where $f_k = (\bar{R}_k^- - \bar{T}_k^+ \bar{S}_k)$. We then obtain
\begin{eqnarray}
\lambda_{k_0} = Im \left( \frac{\partial \log T_k^+}{\partial k}
\right)_{k = k_0} + Im \frac{f'_{k_0} - 2iL f_{k_0}}{1 + f_k e^{-2i
k_0 L}} e^{-2i k_0L}. \label{lll}
\end{eqnarray}
The second term in the right-hand-side of (\ref{lll}) oscillates
very fast with $L$, because $L$ is much larger than the de-Broglie
wavelength $2 \pi/k_0$ of the particle. These oscillations are an
artifact of our modeling the detection process by a crossing of the
sharply defined surface $x = L$. In a realistic detection scheme the
particle detection cannot take place with an accuracy grater than
their de Broglie wavelength. For this reason, we can formally
average $L$ within a region of size $l <<L$.
Indeed, using a Gaussian smearing
function $\rho(L) = (\pi l^2)^{-1/2} e^{ - (L - L_0)^2/l^2}$ we
obtain a suppression factor of order $e^{ - k_0^2 l^2} << 1$ for the
oscillating terms.
Hence, the effective tunneling time is
\begin{eqnarray}
t_{tun} = \frac{M d_{k_0}}{k_0} + \frac{M}{k_0} Im \left(
\frac{\partial \log T_k^+}{\partial k} \right)_{k = k_0},
\end{eqnarray}
i.e. we recover the expression for the Bohm-Wigner phase time
\cite{phasetime}. It is important to emphasize that this derivation
did not employ any characteristics of the wave-packets (e.g. the
trajectory followed by their peak, or their `center-of-mass'). It is
a natural {\em operational} definition at the level of the
probability density that corresponds to the measurement outcomes.
Note that a precise treatment involves smearing the probability
function $p(t)$ of (\ref{p1}). The only $L$-dependent objects that
appear in this equation are the term $B_{k_0}$ and the Gaussian
exponential. If $\frac{1}{\sigma} >> l$, the effect of smearing is
to substitute $L$ by the mean value $L_0$: the expression is not
affected. The effect of smearing on $B_{k_0}$ is to suppress the
oscillations; it leads to an effective expression $\tilde{B}_{k_0}$
\begin{eqnarray}
\tilde{B}_{k_0} = -2i \sqrt{2 \pi} \frac{|A_{k_0}^-|^2
|A_{k_0}^+|^2}{|A_{k_0}^-|^2 (1 + |R_{k_0}^-|^2) + |A_{k_0}^+|^2
|T_{k_0}^+|^2} T_{k_0}^+. \label{btil}
\end{eqnarray}
Note that to a first (very rough) approximation, $|A_{k_0}^{\pm}|$
can be taken equal to $(2 \pi)^{-1/2}$, i.e. the value taken if the
contribution of the region with no zero potential is considered to
be negligible. Then
\begin{eqnarray}
\tilde{B}_{k_0} \simeq - \frac{i}{\sqrt{2 \pi}} T_{k_0}^+
\label{btil2}
\end{eqnarray}
Before continuing, we summarize the approximations involved in the
results we obtained in this section. Eq. (\ref{z}) only involves the
assumption that $\sigma/k_0 <<1$. Eq. (\ref{p1}) involves the
additional assumption that the function $\log B_k$ varies slowly
around $k = k_0$ so that it is sufficient to keep the first order in
its Taylor expansion. This approximation amounts to the condition
$|\frac{B_{k_0}''}{B_{k_0}'} - \frac{B_{k_0}'}{B_{k_0}}| \sigma
<<1$. Eq. (\ref{p2}) involves the additional assumption that
$\xi_{k_0} \sigma^2/k_0 << 1$. Finally, Eq. (\ref{p3}) involves the
assumption that $t_m^2 \sigma^2/M << 1$. This implies that $L$
cannot be too large, because the spread of the wave function due to
the free propagation will induce a large uncertainty in the
determination of tunneling time.
\subsection{Special cases}
\paragraph{Parity-invariant potentials.} The expression for the mode
functions and for $B_k$ simplifies greatly if the potential is
invariant under parity, namely if $V(x) = V(-x)$. This implies that
the eigenstate $u_k^-(x)$ can be identified with the parity
transform of $u_k^+(x)$. Hence $S_k = 0$, $T_k^+ = T_k^- := T_k$,
$R^+_k = R^-_k := R_k$ and $A_k^+ = A_k^- := A_k$. We then obtain,
\begin{eqnarray}
B_k = - 2i \sqrt{2 \pi}\frac{|A_k|^2}{|1 +R_k e^{2ikL}|^2 + |T_k|^2}
[1 + \bar{R}_k e^{-2ikL}] T_k
\end{eqnarray}
\paragraph{The square potential barrier} We apply our results to the simplest example of a square
potential barrier: $V(x) = V_0$ for $x \in [-d/2, d/2]$. Defining
$\gamma_k = \sqrt{2MV_0 - k^2}$, we obtain the following values for
the coefficients $T_k, R_k$
\begin{eqnarray}
T_k = \frac{2 k}{\gamma_k} e^{-ikd} \frac{2k \gamma_k [2 k \gamma_k
\cosh \gamma_k d - i (\gamma^2_k - k^2) \sinh \gamma_k d]}
{4 k^2 \gamma_k^2 + (\gamma_k^2 + k^2) \sinh^2 \gamma_k d} \\
R_k = -i e^{-ikd} \frac{(\gamma^2 + k^2) [2 k \gamma \cosh \gamma_k
d - i (\gamma_k^2 - k^2) \sinh \gamma_k d]}{4 k^2 \gamma_k^2 +
(\gamma_k^2 + k^2) \sinh^2 \gamma_k d} .
\end{eqnarray}
There are two limits, in which the results are particularly simple.
The limit of a long barrier $\gamma_k d >> 1$, for which
\begin{eqnarray}
T_k &\simeq& e^{-ikd} e^{-\gamma_k d} \frac{4 k
\gamma_k}{(\gamma_k^2 + k^2)^2} [2k \gamma_k - i (\gamma_k^2 - k^2)]
\label{tklong}
\\
R_k &\simeq& e^{-ikd} \frac{-(\gamma_k^2 - k^2) + ik\gamma_k}{4
\gamma_k^2 } \label{rklong}
\end{eqnarray}
In this limit, the parameter $\lambda_{k_0}$ is
\begin{eqnarray}
\lambda_{k_0} = -d + \frac{2}{\gamma_{k_0}} ,
\end{eqnarray}
i.e. it takes negative values (since $\gamma_{k_0} d >> 1$). The
tunneling time is therefore $t_{tun} = \frac{2M}{ \gamma_{k_0}
k_0}$.
The other limit is that of the delta function (very short) barrier.
It is obtained by letting $V_0 \rightarrow \infty$ and $d
\rightarrow 0$ such that $V_0 d$ is a constant (we denote this
constant as $\kappa/M$). At this limit, $\gamma_k d \simeq
\sqrt{\kappa d}$ and
\begin{eqnarray}
T_k = \frac{1}{1 + i \kappa/k} \label{tkdelta} \\
R_k = \frac{1}{1 + i k/\kappa}. \label{rkdelta}
\end{eqnarray}
Hence,
\begin{eqnarray}
\lambda_{k_0} = \frac{\kappa}{k_0^2 + \kappa^2}.
\end{eqnarray}
Since $d = 0$ the tunneling time is $t_{tun} = M
\frac{\kappa/k_0}{k_0^2 + \kappa^2}$.
\subsection{Comments}
\subsubsection{Domain of validity}
It is important to emphasize that our identification of a
tunneling time $t_{tun}$ relies on the fact that the probability of
detection has a unique sharp maximum at a specific moment of time.
This is only possible for specific initial states. For example, it
is easy to demonstrate that a superposition of Gaussians centered at
different values of momentum will lead to a probability distribution
with an oscillating behavior. While there is still a mean detection
time, we cannot read from it a time delay for the particle, because
the momentum uncertainty does not allow one to specify uniquely a
corresponding time for free particle evolution. Hence, the tunneling
time is not a proper observable (i.e. a random variable on the
sample space upon which the POVM is defined) in our description: it
is only a parameter that appears in the detection probability for a
class of initial states, which has an intuitive interpretation in
terms of classical concepts.
The fact that the concept of tunneling time has a restricted domain
of validity is highlighted by another point. We saw that for a long
square potential the tunneling time equals $t_{tun} = \frac{2M}{
\gamma_{k_0} k_0}$. If $d$ is very large, the condition
$\gamma_{k_0} d
>> 1$ can be satisfied even if $\gamma_{k_0}$ takes very small
values, i.e. if the particle's mean energy $\frac{k_0^2}{2M}$ is
very close to $V_0$. Hence, it is in principle possible to construct
configurations, in which $t_{tun}$ is arbitrarily small: the
effective `velocity' $d/t_{tun}$ in the crossing of the barrier is
then super-luminal. This is a well known effect in tunneling (the
Hartman effect \cite{Hartmann}). A full treatment in the present
context involves the consideration of relativistic systems--this we
will undertake in future work. Here, we only note that the regime of
very large values for $d /t_{tun}$, (very small values for
$\gamma_{k_0}$) is one for which the approximation involved in Eq.
(\ref{expansion0}) fails. The tunneling probability increases
rapidly in this regime and one would have to include further terms
in the expansion of $\log B_k$, which would lead to a substantially
deformed probability distribution $p(t)$ with no clear peak. The
definition of $t_{tun}$ would then be highly problematic, and so
would be the notion of a mean velocity in the tunneling region.
\subsubsection{Uncertainty in the specification of tunneling time}
The uncertainty in the determination of the peak in the probability
distribution (\ref{p3}) is $\frac{M}{k_0 \sigma}$. In order for the
delay time $\frac{M \lambda_{k_0}}{k_0}$ to be distinguishable (if
we ignore all other sources of uncertainty) it is necessary that
$\sigma |\lambda_{k_0}|
>> 1$. In order for the tunneling time to be
distinguishable, it is also necessary to take into account the
uncertainty in the quantity $\frac{M d_{k_0}}{k_0}$. To leading
order in $\sigma$ this equals $a_{k_0} \sigma$, where
\begin{eqnarray}
a_{k_0} = \frac{k_0}{M} \left(\frac{1}{V'[x_2(k_0)]} -
\frac{1}{V'[x_1(k_0)]} \right) - \frac{M d_{k_0}}{k_0^2}.
\end{eqnarray}
The overall uncertainty in the determination of the tunneling time
$t_{tun} = M(\lambda_{k_0} + d_{k_0})/k_0$ is of the order
\begin{eqnarray}
\frac{M }{k_0 \sigma} + |a_{k_0}| \sigma.
\end{eqnarray}
This expression is bounded from below by $2 \sqrt{M k_0 a_{k_0}}$.
Hence, a necessary condition for tunneling time $t_{tun}$ to be
distinguishable is
\begin{eqnarray}
t_{tun} >> \sqrt{M |a_{k_0}|/k_0}.
\end{eqnarray}
We note that for a parity symmetric potential $a_{k_0} = - M
d_{k_0}/k_0^2$, hence the condition becomes $t_{tun} >> M \sqrt{
d_{k_0}/k_0}$. For the long square barrier, this implies that
\begin{eqnarray}
\frac{ \gamma_{k_0}^2 d }{k_0} << 1.
\end{eqnarray}
This condition can only be satisfied if $\gamma_{k_0}/k_0 << 1$.
This is inadmissible, because the expansion (\ref{expansion0}) is
not adequate in this regime. Hence, for the long square barrier the
operational definition of the tunneling time is not meaningful. On
the other hand, there is no problem in the short barrier limit ($d
\rightarrow 0$).
\subsubsection{The dependence on $L$ }
Finally, we comment on the assumption that $L >>d$. The
consideration of a detector at a macroscopic distance away from the
barrier region greatly simplifies our results: it leads to an
expression for the tunneling time, which essentially coincides with
the results of the asymptotic analysis of the wave packets. This
assumption enters at two steps. First, in the construction of the
POVM, we assume that $L$ is sufficiently removed from the barrier
region, so that the value for the particle's propagator at $x = L$
can be substituted by the corresponding value for the free particle.
This condition is satisfied exactly if the corresponding Hamiltonian
has no (generalized) eigenstates with an asymptotic behavior that
does not correspond to that of a free particle (e.g. negative energy
states). This is the case we considered in this section. Hence, the
only place where the assumption of large $L$ enters in a non-trivial
way in the construction, is when we smear the probability
distribution in order to remove the contribution of the terms
oscillating as $e^{ik_0L}$. This implies that (at least formally),
the expression (\ref{lll}) for the parameter $\lambda_{k_0}$ is
valid for all values of $L$ such that the first condition stated
above holds. We therefore obtain an expression for the tunneling
time, even if the detector is located near the tunneling region.
Clearly, this will have a very sensitive dependence on $L$, because
the presence of the detector close to the barrier affects the
configuration of the system. Note however that this result is rather
formal, since it involves the idealization of the detection process
by the crossing of the sharply defined surface $x = L$. In a
realistic treatment the detailed physics of the detector are
expected to influence the tunnelling time.
For example, for a parity symmetric potential ($S_{k} = 0 $), we
obtain the following expression for the parameter $\lambda_{k_0}$,
\begin{eqnarray}
\lambda_{k_0} = \theta'_{k_0} \frac{r_{k_0}(2L + \theta'_{k_0}) [1
+ \cos (2 k_0 L +\theta_{k_0})] + r'_{k_0} \sin (2k_0 L +
\theta_{k_0})}{1 + r_{k_0}^2 + 2 r_{k_0} \cos (2 k_0 L +
\theta_{k_0})}, \; \; \; \; \;
\end{eqnarray}
where we wrote $R_{k} = r_k e^{i \theta_k}$ and the prime denotes
differentiation with respect to $k$.
\section{A POVM for the tunneling time through sequential
measurements}
We saw in the last section that the determination of tunneling time
through the time-of-arrival probability is only meaningful for a
specific class of initial states, because the delay time is not a
proper random variable on the sample space of the POVM. It depends
on the particle's initial momentum (and position) and as such it
cannot be inferred unless both the initial state and the detection
probability have very sharp maxima.
However, this problem can be alleviated if we make a change in the
experimental set-up, namely if we consider that a measurement of
momentum takes place before any recording of the time-of-arrival. In
effect, if one considers sequential measurements, it is possible to
construct a POVM for which the tunneling time is a genuine random
variable and no mixed classical-quantum arguments are needed for its
identification.
The procedure is the following. Let $\hat{Q}(x,k)$ be a POVM for
unsharp phase space measurements. Let us also assume that the
corresponding device is placed at the left-hand-side of the barrier;
we perform an unsharp phase space measurement to any particle that
moves towards the barrier that allows us to determine unsharp values
for its position $q$ and momentum $p$. The measurement is assumed
to be non-destructive, hence the particles continue their motion,
some of them cross the barrier and they are detected at distance $L$
away. In other words, we have a sequential measurement: first an
unsharp phase space measurement and then a time-of-arrival
measurement. For each particle, the outcomes of this sequential
measurement is encoded in the three numbers $(x, k, t)$ that span a
sample space $\Omega$.
The key point is that from the knowledge of $\hat{Q}$ and
$\hat{\Pi}$ (the time-of-arrival POVM), it is possible to construct
a POVM $\hat{E}$ on $\Omega$. The procedure is standard, see
\cite{Ana06} for a detailed analysis. The POVM $\hat{E}$ consists of
the positive operators
\begin{eqnarray}
\hat{E}(t, x, k) = \sqrt{\hat{Q}}(x, k) \hat{\Pi}(t)
\sqrt{\hat{Q}}(x, k),
\end{eqnarray}
and of the positive operator
\begin{eqnarray}
\hat{E}(N, x, k) = \sqrt{\hat{Q}}(x, k) \hat{\Pi}(N)
\sqrt{\hat{Q}}(x, k),
\end{eqnarray}
that corresponds to a phase space measurement and then no detection.
By construction it satisfies
\begin{eqnarray}
\int{dx dk}{2 \pi} \left(\int_0^{\infty} dt \hat{E}(t, x, k) +
\hat{E}(N, x, k) \right) = 1.
\end{eqnarray}
For an initial state $\hat{\rho}_0$, the joint probability density
on the sample space $\Omega$ is given by
\begin{eqnarray}
P(t, x, k) = Tr \left( \hat{\rho}_0 \hat{E}(t, x, k) \right).
\end{eqnarray}
The key benefit in the consideration of such a POVM is that the
delay-time
\begin{eqnarray}
t_d = t - \frac{M(L - x)}{k},
\end{eqnarray}
and the tunneling time
\begin{eqnarray}
t_{tun} = t_d + \frac{M d_k}{k},
\end{eqnarray}
are both random variables on the sample space $\Omega$. Hence, it is
possible to define a POVM on the space in which they take values. We
will do so after we construct explicitly the POVM $\hat{E}$.
We consider POVMs for the unsharp phase-space measurements of the
form
\begin{eqnarray}
\hat{Q}(x, k) = \int \frac{dk_0 dx_0}{2 \pi} f(x - x_0, k - k_0)
|x_0, k_0 \rangle \langle x_0, k_0|,
\end{eqnarray}
where $|x_0, k_0 \rangle$ is the coherent state (\ref{initial}), and
$f$ is a positive-valued function that determines the phase space
resolution of the apparatus. Since $\int \frac{dx dk}{2 \pi}
\hat{\Pi}(x, k) = 1$, it is necessary that the function $f$
satisfies
\begin{eqnarray}
\int \frac{dx dk}{2 \pi} f( x, k) = 1.
\end{eqnarray}
The minimum resolution measurements correspond to $f(x,k) = 2 \pi
\delta(x) \delta (k)$, in which case $\hat{Q}(x, k) = |x k \rangle
\langle x k|$. For simplicity, we will consider minimum resolution
measurements in what follows.
We obtain the following probability density on $\Omega$
\begin{eqnarray}
P(t, x, k) = \langle x k|\hat{\rho}|x k \rangle \langle x
k|\hat{\Pi}(t) | x k \rangle.
\end{eqnarray}
We note that $\langle x k|\hat{\Pi}(t) | x k \rangle$ equals the
probability density $p(t)$ of Eq. (\ref{p1}). We write this as
$p_{x, k}(t)$, in order to express its dependence on the initial
state [$k = k_0$ and $x = x_0$ in Eq. (\ref{p1})]. We then obtain
\begin{eqnarray}
P(t, x, k) = \langle x k|\hat{\rho}|x k \rangle p_{x,k}(t). \label{povmseq}
\end{eqnarray}
We next change variables in (\ref{povmseq}) from $t$ to the delay
time $t_d$. We note that on the full sample space, the relation
between $t_d$ and $t$ is not one-to-one. First, the random variable
$t_d$ takes values in the whole real axis, while $t$ only on the
positive real axis. It is therefore convenient to define the
probability $P(t, x, k)$ for $t$ running to all reals. This involves
defining $p_{x,k}(t)$ for all $t \in {\bf R}$; we saw in Sec. 2 that
this is obtained by doubling the values of $p_{x,k}(t)$ for $t \in
[0, \infty)$. With $t$ defined over all reals, we note that for each
value of $t$, one obtains the same value for $t_d$ {\em twice},
since $t_d$ is the same at points $(t, x, p)$ and $(t, 2L-x, -p)$.
We perform the change of variables taking the facts above into
account, and then we integrate over
$x$ and $k$, in order to obtain a
marginal probability distribution over $t_d$
\begin{eqnarray}
P_d(t_d) = 4 \int \frac{dx dk}{2 \pi} \langle x k|\hat{\rho}|x k
\rangle p_{x,k}(t_d + \frac{L - x}{k}).
\end{eqnarray}
The same procedure leads to a marginal probability distribution for
the tunneling time
\begin{eqnarray}
P_{tun}(t_{tun}) = 4 \int \frac{dx dk}{2 \pi} \langle x
k|\hat{\rho}|x k \rangle p_{x,k}(t_{tun} + \frac{L - x + d_k}{k}).
\end{eqnarray}
The two equations above are completely general, and they hold
without any approximations. They simplify significantly if we assume
that for all values of $k$ in the support of the initial state, the
following two conditions hold: (i) $p_{x, k}(t)$ is appreciably
different from zero only for times $t$ such that
$t^2
\sigma^2/M << 1$, and (ii) $\sigma \xi_k << 1$. The dependence on
$x$ of $p_{x,k}$ is then absorbed in the definition of the variable
$t_d$, and we obtain
\begin{eqnarray}
P_d(t_d) = \sqrt{8 \pi \sigma^2} \int dk \langle k|\hat{\rho_0} |k
\rangle |\tilde{B}_{k}|^2 \frac{|k|}{M} \exp \left\{- \frac{2 k^2
\sigma^2}{M^2} \left[ t_d - \frac{M \lambda_k}{k}\right]^2 \right\}.
\label{pd}
\end{eqnarray}
Similarly,
\begin{eqnarray}
P_{tun}(t_{tun} ) = \sqrt{8 \pi \sigma^2} \int dk \langle
k|\hat{\rho_0} |k \rangle |\tilde{B}_{k}|^2 \frac{|k|}{M} \exp
\left\{- \frac{2 k^2 \sigma^2}{M^2} \left[ t_{tun} - \frac{M
(\lambda_k + d_k)}{k}\right]^2 \right\}. \label{pt}
\end{eqnarray}
Note that neither $P_d$ nor $P_{tun}$ are normalized to unity. The
delay and tunneling times are only defined for the fraction of the
ensemble that corresponds to particles that have crossed the
barrier. To normalize it, we have to divide by the probability
corresponding to the detected particles $1 - Tr \left( \hat{\rho}_0
\hat{E}(N) \right)$.
Hence, we have constructed a positive definite probability density
for the delay and the tunneling times, which is valid for an
arbitrary initial state (with the restriction that its position
support lies on the left side of the barrier). This probability is
definable in the context of a sequential measurement: there is no
other way to define these times as quantum observables otherwise:
the definition in Sec.4 involved a mixture of quantum mechanics and
classical argumentation and was only meaningful for a specific class
of initial states. We have to keep in mind though that the
experimental set-up for which these probabilities are valid involves
keeping track of the phase space properties of {\em individual
particles} and then comparing them with the registered arrival time.
It requires relatively precise measurements at a microscopic scale,
and it cannot be implemented when working with particle beams.
We should also note that both probabilities $P_d$ and $P_{tun}$ are
contextual, i.e. they depend strongly on specific features of the
apparatus that performs the phase space sampling. They both have a
strong dependence on the parameter $\sigma$, which defines the
family of coherent states: in the present context $\sigma$ is the
inherent uncertainty in the specification of momentum\footnote{For
the contextuality of sequential measurements, see the extended
discussion in \cite{Ana06}}. At the limit $\sigma \rightarrow 0$,
both (\ref{pd}) and (\ref{pt}) vanish. There is, however, a limit in
which the results become $\sigma$-independent. If the initial state
has support on values of $k$, such that the mean of the Gaussian in
either probability density is much larger than its spread, then we
can approximate it by a delta function. This condition implies
\begin{eqnarray}
\sigma |\lambda_k| >> 1, \label{conditiond}
\end{eqnarray}
for (\ref{pd}) and
\begin{eqnarray}
\sigma (\lambda_k + d_k) >> 1 \label{conditiont}
\end{eqnarray}
for (\ref{pt}).
At these regimes, we obtain
\begin{eqnarray}
P_d(t_d) = 2\pi \int dk \langle k|\hat{\rho_0} |k \rangle
|\tilde{B}_{k}|^2 \delta (t_d - \frac{M \lambda_k }{k}), \label{pdid} \\
P_{tun}(t_{tun}) = 2 \pi \int dk \langle k|\hat{\rho_0} |k \rangle
|\tilde{B}_{k}|^2 \delta (t_{tun} - \frac{M (\lambda_k + d_k) }{k}).
\label{ptid}
\end{eqnarray}
In other words, the values of $P_d(t_d)$ and of $P_{tun}(t_{tun})$
are determined by the value of the probability distribution of the
initial's state momentum at values of $k$ that are solutions of the
algebraic equations $t_d = \frac{M \lambda_k }{k}$ and $t_{tun} =
\frac{M (\lambda_k + d_k) }{k}$ respectively. These expressions for
the probability distribution are independent of the detailed
characteristics of the phase space POVM: they only depend on the
initial state and on the characteristics of the
potential\footnote{Recall that by virtue of smearing the value of
$L$, there is no $L$-dependence in $\tilde{B_k}$; hence the marginal
probability distributions are also $L$-independent.}. They can
therefore be considered as {\em ideal} distributions of delay and
tunneling times respectively that exhibit little sensitivity to the
measurement scheme employed for their determination.
We can further simplify the expressions for $P_d$ and $P_{tun}$
using the estimation (\ref{btil2}) for $\tilde{B}_{k_0}$:
\begin{eqnarray}
P_d(t_d) = \int dk \langle k|\hat{\rho}_0 |k \rangle
|T_{k}|^2 \delta (t_d - \frac{M \lambda_k }{k}), \label{pdid2} \\
P_{tun}(t_{tun}) = \int dk \langle k|\hat{\rho}_0 |k \rangle
|T_{k}|^2 \delta (t_{tun} - \frac{M (\lambda_k + d_k) }{k}).
\label{ptid2}
\end{eqnarray}
In effect, the probability for $t_d$ and $t_{tun}$ are defined from
the corresponding values of the momentum distribution weighted by
the transmission probability. Defining the functions $F_d(k) :=
\frac{M \lambda_k }{k}$ and $F_{tun}(k) := \frac{M (\lambda_k + d_k)
}{k}$, we see that the probabilities (\ref{pdid2}-\ref{ptid2}) are
obtainable from the operators $\hat{T}_{d} = F_{d}(\hat{p})$ and
$\hat{T}_{tun} = F_{tun}(\hat{p})$ ($\hat{p}$ is the momentum
operator) when these act on the state
\begin{eqnarray}
\hat{\rho}_{cross} = \int dk |T_k|^2 \hat{P}_k \hat{\rho}_0
\hat{P}_k, \hspace{2cm} \hat{P}_k = |k \rangle \langle k|
\end{eqnarray}
that describes the sub-ensemble of particles that have crossed the
barrier. One could therefore call $\hat{T}_d$ and $\hat{T}_{tun}$
time-delay and tunneling-time operators respectively\footnote{There
is an ambiguity in their definition at $k = 0$. However, this does
not affect the probabilities (\ref{pdid2}) and (\ref{ptid2}),
because $|T_{k = 0}| = 0$.}.
We end this section, by examining the domain of validity of
conditions (\ref{conditiont}) and (\ref{conditiond}) for the square
potential barrier. At the large barrier limit, they read
\begin{eqnarray}
\sigma |-d + \frac{2}{\gamma_k}| >> 1 \\
\sigma/\gamma_{k} >> 1.
\end{eqnarray}
They are satisfied if the position $\sigma^{-1}$ spread of the
coherent states is much smaller than the effective lengths
corresponding to delay and tunneling time respectively. For the
delta function barrier, these conditions imply
\begin{eqnarray}
\sigma \frac{\kappa}{k^2 + \kappa^2} >> 1,
\end{eqnarray}
which is only possible if $\kappa$ is extremely small (a rather
unphysical case).
We see therefore that the ideal probability distributions
(\ref{pdid}) and (\ref{ptid}) can only be obtained if the initial
phase space measurement has a resolution for position substantially
smaller than the dimensions of the barrier. This is a type of
measurement that is not explicitly forbidden by quantum mechanics,
but clearly it would be extremely difficult to achieve in practice.
\section{Conclusions}
We reformulated tunneling as a problem in the determination of
probability for the time-of-arrival. This allowed us to identify the
classic Bohm-Wigner time as the most suitable measure for the
tunneling time. However, this identification only holds for a
specific class of initial states and potentials; in other regimes,
there is no operational definition of the concept. There is one way
to go around this problem by considering a sequential measurement
set-up: we first measure the phase space properties of the particles
(before they attempt to cross the barrier) {\em and then} we
determine their times-of-arrival. In this context, it is possible
to construct a probability measure for the tunneling time that is
valid for all initial states.
The key feature of our construction is that there is neither
interpretational nor probabilistic ambiguity. The probabilities we
derive are obtained through a POVM, hence they are always positive
and they respect the convexity of the space of quantum states. The
interpretation of these objects is concretely operational, in the
sense that it is tied to the statistics for the measurement of
particles' arrival times. Tunneling time is solely defined in terms
of the statistics of measurement outcomes.
In another paper \cite{An07b}, the POVM we constructed here will be
employed for the study of the decay probability of unstable states
through tunneling.
\section*{Acknowledgements}
N.S. acknowledges support from the EP/C517687 EPSRC grant.
|
0706.2185
|
\section{Introduction and Motivation}
Graphene is a two-dimensional (2D) allotrope of carbon on a honeycomb lattice
with one electron per $\pi$ orbital (half-filling). Its bare electronic
spectrum is described in terms of a linearly dispersing, massless, chiral
Dirac field ($\Psi_{\bf{p}}$). Since its isolation a few years ago \cite{Graphene} it was realized
that graphene displays a number of unique properties that are at odds with the standard
theory of metals. \cite{Graphene,Peres,AntonioRMP}
One of the most important unresolved questions in graphene
is the role of electron-electron interactions. \cite{Paco1,Paco2,EM,Barlas,Barlas2,NFL1,NFL2,Son,Joerg}
Even though, due to the vanishing of the density of states at the Fermi energy,
the electron-electron interactions are expected to remain unscreened and strong, it is not clear
what is their influence in the properties of graphene.
In the present work we study the influence of the electron-electron interactions
on the static dielectric function of graphene at half filling. We perform calculations to
one order beyond the conventional random phase approximation (RPA)
vacuum polarization bubble, by including self-energy and vertex corrections
into the polarization loop. Our main finding
is that vertex contributions become important in the coupling regime $\alpha \sim 1$,
which in turn means that other non-RPA contributions should also be included.
We were mainly motivated by the question whether the interactions can significantly
affect the screening properties. This issue is particularly relevant in graphene
for two reasons: (1.) The effective coupling constant $\alpha$
(see the precise definition below) in graphene is large $\alpha \sim 1$, and thus
interactions are expected to be generically important, and
(2.) Despite of the above, to the best of our knowledge, no clear signatures
of interaction effects have been observed so far in graphene.
For example measurements of the compressibility \cite{Yacoby} have not detected
electron correlation effects. In addition, screening of external
charged impurities introduced in graphene is also expected to
be sensitive to interaction effects, at least on theoretical
level, \cite{Impurity1,Impurity11,Impurity2,Impurity3} and could be relevant for interpretation
of recent experiments on charged impurity scattering.
\cite{ImpurityExp} It is thus
generally important to investigate the problem of how the correlations
affect the effective charge of the carriers in graphene, which is determined
by the vacuum polarization. We will assume that graphene at half filling
(i.e. when the chemical potential crosses the Dirac point)
remains a homogeneous gas of quasiparticles, which in itself is
not necessarily an innocent assumption due to the possibility of
puddles, ripples, etc. \cite{AntonioRMP} However we assume that the system
is homogeneous as the importance of the above effects is still unsettled.
Our starting point is the low-energy Hamiltonian of graphene which
can be written as (we use units such that $\hbar=1$),
\begin{equation}
\label{ham1}
H = \sum_{\bf{p}} \Psi_{\bf{p}}^{\dagger} ( v |{\bf p}| \hat{\sigma}_{\bf p} - \mu \hat{\sigma}_{0})
\Psi_{\bf{p}} + H_{I},
\end{equation}
where $v$ is the Fermi velocity, $\mu$ is the chemical potential away from half-filling,
$\hat{\sigma}_{0} = \hat{I}$ is the 2$\times$2 identity matrix,
$\hat{\sigma}_{\bf p} \equiv \hat{\sigma}\cdot{\bf p}/|{\bf p}| =
(\hat{\sigma}_{x} p_x + \hat{\sigma}_{y} p_y)|{\bf p}|^{-1} $,
and $\hat{\sigma}_{x},\hat{\sigma}_{y}$ are Pauli matrices.
The first term in the Hamiltonian (\ref{ham1}) reflects the effective Lorentz invariance that exists
in the non-interacting problem at low energies and gives rise to
bizarre electronic behavior analogous to the one found in quantum electrodynamics (QED). \cite{pw}
In Eq.(\ref{ham1}), $H_{I}$ is the electron-electron interaction,
\begin{equation}
\label{ham2}
H_{I} = \frac{1}{2} \sum_{\bf{p}} \hat{n}_{\bf p} V_{{\bf p}} \hat{n}_{\bf-p}, \
\ \hat{n}_{\bf p} \equiv \sum_{\bf{q}} \Psi_{\bf{q+p}}^{\dagger} \Psi_{\bf{q}},
\end{equation}
where
\begin{equation}
\label{Coulomb}
V_{{\bf p}}=\frac{2\pi e^{2}}{|{\bf p}|}
\end{equation}
is the Fourier transform of the Coulomb potential in 2D.
The relative strength of the Coulomb interactions to the kinetic energy
is determined by graphene's ``fine structure constant" $\alpha \equiv e^{2}/v$.
Unlike QED, the Dirac fermion velocity is much smaller than the speed of light, $c$, and hence the Coulomb
field can be treated as instantaneous ($v \approx 10^6 \mbox{m/s}$).
As a result, the Coulomb interaction
breaks the Lorentz invariance of the problem leading to fundamental differences between the graphene problem
and QED. From now on we absorb the dielectric constant of the
medium $\varepsilon$ into the definition of the effective charge $e$. For example in the typical case of
graphene on a SiO$_2$ substrate with dielectric constant $\varepsilon \approx 4$, we have
$e^{2} = 2e_{0}^{2}/(1+\varepsilon)$, where $e_{0}$ is the charge of the electron.
Keeping in mind that $e_{0}^{2}/v \approx 2.2$, one then finds
the coupling constant $\alpha \approx 0.9$. \cite{remark} Nevertheless, even in this situation
when the relation $\alpha \ll 1$ is not strictly satisfied, perturbation theory is expected
to give a good
indication for the behavior of physical quantities.
The rest of the paper is organized as follows. Section II deals
with corrections to the polarization loop arising from the dressing of the electron
propagators. In the Section III the true interaction (correlation) insertion, the vertex correction,
is examined. Section IV contains our conclusions.
\section{Self-energy corrections to the polarization}
We concentrate on the most interesting case of zero Fermi energy
($\mu \rightarrow 0$), when the low-energy physics is controlled by the
proximity to the Dirac point.
The free Dirac Green's function is
\begin{equation}
\label{gf1}
\hat{G}({\bf k},\omega) = \frac{1}{ \omega \hat{\sigma}_{0} - v|{\bf k}|\hat{\sigma}_{\bf k} +
i \hat{\sigma}_{0} 0^{+} {\mbox{sign}} (\omega)}.
\end{equation}
The interaction effects lead to the dressed Green's function
$\hat{G}^{-1} \rightarrow \hat{G}^{-1} - \hat{\Sigma} $, where the self-energy $\hat{\Sigma}$
is a sum of two terms with different matrix structure:
$\hat{\Sigma} = \hat{\Sigma}_{0} +{\bf{\hat{\Sigma}}}$,
$\hat{\Sigma}_{0} \propto \hat{\sigma}_{0}, {\bf{\hat{\Sigma}}} \propto \hat{\sigma}_{\bf k}$.
At Hartree-Fock (HF) level (first order in $\alpha$) a divergent contribution appears,
due to the long-range nature of the Coulomb interaction \cite{Paco1}
where $\Lambda \sim 1/a \gg k$ is an ultraviolet cutoff ($a$ is the lattice
spacing). One finds
\begin{equation}
\label{hf}
{\bf{\hat{\Sigma}}}^{(1)}(|{\bf k}|) = (\alpha/4) \ (v|{\bf k}|)
\hat{\sigma}_{\bf k} \ln(\Lambda/|{\bf k}|),
\end{equation}
which implies that the effective velocity changes
$v \rightarrow v(1 + (\alpha/4) \ln(\Lambda/|{\bf k}|))$, and grows without bound
at low energies $|{\bf k}|/\Lambda \rightarrow 0$.
This should in principle lead to anomalies in thermodynamic and spectral properties
of graphene. \cite{Paco1,vafek} From theoretical viewpoint, most importantly, the single
logarithmic behavior was found to persist to second order of perturbation
theory as well, \cite{EM} and consequently this is expected to be the case to all orders,
reflecting the fairly simple (at least at weak-coupling) renormalization structure of the theory.
We now turn to the calculation of the static polarization, $\Pi({\bf q})\equiv \Pi({\bf q},\omega=0)$.
The frequency variable in $\Pi({\bf q})$ is omitted from now on.
The bare polarization bubble (without any interaction lines in the loop) is known to be
\begin{eqnarray}
\Pi^{(0)}({\bf q}) &=& -i \sum_{\bf{k}} \int \frac{d\omega}{2\pi} {\mbox{Tr}} \{ \hat{G}({\bf k},\omega)
\hat{G}({\bf k + q},\omega)\} \nonumber \\
&& = - |{\bf q}|/(4v).
\end{eqnarray}
From now on the trace stands for summation over spin (s), valley (v) and pseudospin (Pauli matrix
$\sigma$) indices, i.e.
\begin{equation}
\label{trace}
{\mbox{Tr}} = \sum_{s,v} {\mbox{Tr}_{\sigma}} = 4 {\mbox{Tr}_{\sigma}}.
\end{equation}
The momentum sums are performed as $\sum_{\bf{k}}=\int d^{2}k/(2\pi)^{2}$.
Next, we calculate the bubble dressing due to the electron-electron interactions to
first order in $\alpha$. The two diagrams at this order are shown in Fig.~\ref{Fig1}.
We write the total polarization as
\begin{equation}
\Pi({\bf q}) = \Pi^{(0)}({\bf q}) + \Pi^{(1)}({\bf q}) + \Pi^{(2)}({\bf q}),
\end{equation}
where $\Pi^{(1)}$ and $\Pi^{(2)}$ stand for the contributions of Fig.~\ref{Fig1}(a) and Fig.~\ref{Fig1}(b),
respectively.
\begin{figure}[bt]
\centering
\includegraphics[height=88pt, keepaspectratio=true]{Fig1.eps}
\caption{First order interaction corrections to the polarization bubble: (a.) self-energy
correction, (b.) vertex correction. The explicit form of these diagrams is given by
equations (9) and (19).
The wavy line represents the Coulomb interaction, Eq.~(3).}
\label{Fig1}
\end{figure}
\noindent
The self-energy dressing of Fig.~\ref{Fig1}(a) can be written as
(the factor of $2$ originates from the two possible insertions)
\begin{eqnarray}
\label{pol1}
&\Pi^{(1)}({\bf q}) &= -2i \sum_{\bf{k}} \int \frac{d\omega}{2\pi} {\mbox{Tr}} \{ \hat{G}({\bf k},\omega)
\hat{G}({\bf k + q},\omega)\hat{G}({\bf k},\omega) \nonumber \\
&& \times \hat{\Sigma}({\bf k},\omega) \}.
\end{eqnarray}
At lowest order, the self-energy is simply the Hartree-Fock one, meaning that in (\ref{pol1})
we replace
\begin{equation}
\label{se}
\hat{\Sigma}({\bf k},\omega) \rightarrow
\hat{\Sigma}^{(1)}({\bf k}) = i \sum_{{\bf p}} \int \frac{d\omega_{1}}{2\pi} \hat{G}({\bf p},\omega_{1})
V_{\bf{k}-\bf{p}}.
\end{equation}
The large logarithm present in $\hat{\Sigma}^{(1)}({\bf k})$ at low momenta, Eq.~(\ref{hf}), is expected to
appear also in some form in $\Pi^{(1)}({\bf q})$.
Let us define the following quantities which appear in our results from now on
\begin{equation}
\label{notation}
\Delta(\bf{k},\bf{q}) \equiv 1 - \frac{\hat{\bf{k}}\cdot (\bf{k}+ \bf{q})}{ |\bf{k}+ \bf{q}|},
\ \ \ \hat{{\bf k}} \equiv {\bf k}/|{\bf k}|,
\end{equation}
\begin{equation}
E({\bf{k}},{\bf{q}}) \equiv v(|{\bf{k}}| + |\bf{k}+ \bf{q}|).
\end{equation}
After performing the frequency, and then momentum integrations in (\ref{pol1}),
and using the self-energy from (\ref{se}), we obtain
\begin{eqnarray}
\label{pol10}
&\Pi^{(1)}({\bf q}) &= 4 \sum_{\bf{k},\bf{p}} V_{\bf{k}-\bf{p}} (\hat{\bf{k}} \cdot \hat{\bf{p}})
\frac{\Delta({\bf k},{\bf q})}{[E({\bf k},{\bf q})]^{2}}
\nonumber \\
&& = \frac{\alpha}{16} \ \frac{|{\bf q}|}{v} \ \ln(\Lambda/|{\bf q}|), \ \ \Lambda/|{\bf q}| \gg 1.
\end{eqnarray}
This result means that the large logarithm in $\Pi^{(1)}({\bf q})$ simply reflects the
renormalization of the Fermi velocity, i.e. this divergence is not independent,
but can be simply reabsorbed into the velocity by replacing
$v \rightarrow v(1 + (\alpha/4) \ln(\Lambda/|{\bf q}|))$ in the one-loop result
$\Pi^{(0)}({\bf q}) = - |{\bf q}|/(4v)$. Due to the simple logarithmic structure of
the theory this is expected to hold to all orders of perturbation theory,
i.e. all self-energy corrections lead to a replacement of the coupling
$\alpha$ in all final expressions with the ``running" coupling $\alpha(q)$,
accounting for the velocity renormalization.
We therefore assume that the velocity renormalization procedure
is performed in all higher order diagrams.
At finite small chemical potential, $\mu \ll \Lambda$, which is the case in
any realistic experimental situation, the divergence is cut-off, $\ln(\Lambda/|{\bf q}|)
\rightarrow \ln(\Lambda/\mu)$. Due to the slow variation of the logarithmic function
or possible other factors (such as strong dielectric screening), no significant
variation of the velocity has been found in experiment. \cite{AntonioRMP}
An interesting effect, related to the interaction contribution $\Pi^{(1)}({\bf q})$,
Eq.~(\ref{pol10}), was recently
discussed in Ref.~\onlinecite{Impurity2} within the RG approach. Our calculation,
leading to Eq.~(\ref{pol10}), provides an explicit perturbative confirmation of the RG
results.
If we imagine an external Coulomb impurity with charge $Z>0$, probing the
polarization of the vacuum, then the induced charge density, in momentum space, is
$\rho_{ind}({\bf q}) = Z V_{{\bf q}}\Pi({\bf q})$. Here $\Pi({\bf q}) = \Pi^{(0)}({\bf q}) +
\Pi^{(1)}({\bf q})$. While the first term leads to induced charge $\rho_{ind}^{(0)}({\bf q})
= -Z\alpha (\pi/2)$, localized in real space at the impurity site and with a screening sign, the
interaction term $\rho_{ind}^{(1)}({\bf q}) = Z\alpha^{2} (\pi/8) \ln(\Lambda/|{\bf q}|)$
has an opposite sign and decays as an inverse power law ($1/r^{2}$, with logarithmic corrections).
This peculiar behavior simply reflects, however,
the renormalization (increase) of the Fermi velocity at low momenta, which
leads to suppression of screening at large distances.
\begin{figure}[bt]
\centering
\includegraphics[height=115pt, keepaspectratio=true]{Fig2.eps}
\caption{First order interaction correction to the vertex $\hat{\Gamma}(k;q)$,
given by Eq.~(\ref{vertex}).}
\label{Fig2}
\end{figure}
\noindent
\section{Vertex corrections to the polarization bubble}
We proceed with the calculation of the vertex correction in Fig.~\ref{Fig1}(b).
Before evaluating this expression, it is useful to examine the
(possible) singularity structure separately in the vertex operator,
shown in Fig.~\ref{Fig2}. For simplicity we use the notation
$\hat{\Gamma}(k;q)$ which stands for the more conventionally used
form $\hat{\Gamma}(k,k+q;q)$, where the variables denote both frequency and
momenta ($q$ is the bosonic momentum/frequency).
Since the Coulomb interaction is non-retarded, we have the simple expression
\begin{equation}
\label{vertex}
\hat{\Gamma}({\bf k};{\bf q}, \omega) = i \sum_{{\bf p}} \int \frac{d\omega_{1}}{2\pi}
\hat{G}({\bf p},\omega_{1}) \hat{G}({\bf p}+{\bf q},\omega_{1}+\omega) V_{\bf{k}-\bf{p}}.
\end{equation}
After evaluating the frequency integral, the result is a sum of an off-diagonal
and diagonal parts (with respect to the Pauli matrix indexes),
$\hat{\Gamma}=\hat{\Gamma}^{(o)} + \hat{\Gamma}^{(d)}$. More explicitly,
\begin{eqnarray}
\label{vertex1}
&\hat{\Gamma}^{(o)}({\bf k};{\bf q}, \omega)& \propto \sum_{{\bf p}} V_{\bf{k}-\bf{p}}
(\hat{\sigma}_{\bf p} - \hat{\sigma}_{\bf p + q}) \left ( \frac{1}{E(\bf{p},\bf{q}) + \omega}
\right. \nonumber \\
&& \left.
- \frac{1}{E(\bf{p},\bf{q}) - \omega} \right ).
\end{eqnarray}
We are interested only in the zero frequency limit (and only in the real part of $\hat{\Gamma}$,
since the imaginary part does not contribute to the polarization). In this case
the off-diagonal piece vanishes identically, $\hat{\Gamma}^{(o)}({\bf k};{\bf q}, \omega=0)=0$.
This is expected to be the case since the Coulomb interaction is diagonal and thus the vertex
can not generate a static contribution with a different matrix structure.
On the other hand the diagonal part is finite in the same limit
\begin{equation}
\label{vertex2}
\hat{\Gamma}^{(d)}({\bf k};{\bf q}, \omega=0) \propto \sum_{{\bf p}} V_{\bf{k}-\bf{p}}
(1-\hat{\sigma}_{\bf p}\hat{\sigma}_{\bf p + q}) \frac{1}{E(\bf{p},\bf{q})}.
\end{equation}
An explicit evaluation shows that $\hat{\Gamma}^{(d)}({\bf k};{\bf q}, \omega=0)$ does not
have any divergent contributions. For example at
$|{\bf k}| \ll |{\bf q}|$,
\begin{equation}
\hat{\Gamma}^{(d)}({\bf k};{\bf q},0) \propto \alpha \ {\mbox{const.}}, \ \ |{\bf k}| \ll |{\bf q}|,
\end{equation}
while in the opposite limit
\begin{equation}
\hat{\Gamma}^{(d)}({\bf k};{\bf q},0) \propto \alpha (|{\bf q}|/|{\bf k}|), \ \ |{\bf q}| \ll |{\bf k}|
\end{equation}
For our purposes the exact formulas are not important (we also do not show the dependence on the
angle between ${\bf k},{\bf q}$); our main conclusion at this stage is that
the vertex does not have any divergent parts. We have also examined diagrams of
higher order, such as ``ladder" and ``crossed" ladder vertex corrections, and have found that
all of them are finite.
Therefore the vertex insertions into the polarization function are expected to give a finite
contribution to that quantity, and below we evaluate the lowest order vertex correction numerically.
It is clear that a Ward identity relating divergent
contributions in the self-energy and in the vertex does not hold
here, unlike conventional QED where Lorentz (and gauge) invariance
guarantees cancellation between vertex and self-energy corrections, \cite{IZ}
and charge is renormalized only through simple polarization loops in the photon
propagator.
On the other hand in graphene, where the only non-trivially renormalized quantity is
the velocity $v$, both the polarization operator and the vertex operator do
not show any independent divergencies.
The diagram of Fig.~\ref{Fig1}(b) now reads
\begin{equation}
\label{pol2}
\Pi^{(2)}({\bf q}) = -i \sum_{\bf{k}} \! \int \! \frac{d\omega}{2\pi} {\mbox{Tr}} \{
\hat{G}({\bf k},\omega)\hat{\Gamma}({\bf k};{\bf q},0) \hat{G}({\bf k + q},\omega)
\},
\end{equation}
where the full expression for $\hat{\Gamma}$ from Eq.~(\ref{vertex}) should be used.
An explicit calculation, starting by evaluation of the energy integrations, leads to the result
\begin{equation}
\Pi^{(2)}({\bf q}) = - \frac{1}{4} {\mbox{Tr}} \sum_{{\bf k},{\bf p}} V_{{\bf k}-{\bf p}}
\frac{(1-\hat{\sigma}_{\bf p +q}\hat{\sigma}_{\bf p}) (1-\hat{\sigma}_{\bf k}\hat{\sigma}_{\bf
k + q})}{E({\bf k},{\bf q}) E({\bf p},{\bf q})}.
\end{equation}
Taking into account:
\begin{equation}
\hat{\sigma}_{\bf k}\hat{\sigma}_{\bf p} = \frac{1}{|{\bf k}||{\bf p}|} \left ({\bf k}.{\bf p} +
i\hat{\sigma}_{3} ({\bf k}\times {\bf p})_z \right ),
\end{equation}
where $({\bf p}\times {\bf q})$ stands for a vector product, we then arrive at the final formula
\begin{eqnarray}
\label{pol20}
&\Pi^{(2)}({\bf q})& = -2 \sum_{{\bf k},{\bf p}} \frac{V_{{\bf k}-{\bf p}}}{E({\bf k},{\bf q}) E({\bf p},{\bf q})}
\left \{ \Delta({\bf k},{\bf q})\Delta({\bf p},{\bf q}) \right. \nonumber \\
&& \left. + \ \frac{({\bf p}\times {\bf q})_{z}({\bf k}\times {\bf q})_{z}}{|{\bf p}||{\bf k}||{\bf p}+
{\bf q}||{\bf k}+{\bf q}|} \right \}.
\end{eqnarray}
It is clear on dimensional grounds that $\Pi^{(2)}({\bf q})$ is linear in $|{\bf q}|$. This is in fact
the case for polarization diagrams in all orders of perturbation theory.
The four-dimensional integrals, appearing in (\ref{pol20}), cannot be evaluated analytically.
We have found, as expected in light of our previous discussion of the vertex function,
that the expressions converge in the ultraviolet limit.
After computing the integrals numerically, we obtain the following result for
the combination $V_{{\bf q}}\Pi^{(2)}({\bf q})$, which appears in the dielectric function,
\begin{equation}
\frac{2\pi e^{2}}{|{\bf q}|}\Pi^{(2)}({\bf q}) = - 0.53 \alpha^{2} .
\end{equation}
Adding also the one-loop RPA result, we have finally (where ${\cal{E}}$ is the static
dielectric constant, defined by the formula below)
\begin{equation}
\label{charge}
V_{{\bf{q}}}^{{\mbox{eff}}} = \frac{V_{{\bf{q}}}}{1- V_{{\bf{q}}}\Pi({\bf q})} = \frac{V_{{\bf{q}}}}{{\cal{E}}},
\end{equation}
\begin{equation}
\label{diel}
{\cal{E}} = 1 + \frac{\pi}{2} \alpha + 0.53 \ \alpha^{2} +O(\alpha^{3}).
\end{equation}
We conclude that, at $\alpha \sim 1$, the vertex correction is more than $30\%$ of the
one-loop result. It also has a screening sign, i.e. it reduces
the effective charge. One also expects that finite contributions will appear
to all orders in $\alpha$. However, resummation of perturbation theory
by simple means seems impossible, as the contributions in question
are finite and accumulate over a wide range of momenta in the corresponding
diagrams (rather than within a specific integration window, from where divergent parts
typically originate, and thus can be easily collected).
Even though the vertex contribution is a sizable one, two remarks are in order:
(1.) It does not change drastically the structure of the theory, apart
from contributing towards further screening of the interactions.
(2.) The fact that perturbation theory is used with the intention of being
applied at a rather strong coupling is in itself questionable.
Nevertheless, perturbation theory provides a clear indication that a
significant contribution to screening exists beyond the conventional
one-loop RPA result. On the other hand in the weak-coupling regime, $\alpha \ll 1$,
RPA is parametrically well justified as far as the static polarization properties
are concerned (although the RPA is not justified for the self-energy. \cite{EM})
\section{Discussion and Conclusions}
It is also useful to compare our results to the situation in ordinary metals
with a finite Fermi surface. Certain approximations are typically used to
account for vertex corrections, such as the Hubbard form of the dielectric
function. When extrapolated to low momentum, the vertex contribution
tends to decrease the screening length, \cite{Mahan} i.e. it reduces further the range
of the interactions. Naturally in graphene, where the screening length is
infinite (for the case of zero chemical potential considered here),
the vertex correction affects directly the effective charge, without
changing the shape of the Coulomb potential.
Finally we mention two recent related works, discussing interaction effects,
that appeared while the present manuscript was being prepared.
In Ref.~\onlinecite{Mi2}, the effect of self-energy and vertex corrections
to lowest order ($\alpha$) on the minimal conductivity in graphene was discussed,
with the conclusion that the corrections is of order $1\%$.
Dynamical polarization properties were studied in Ref.~\onlinecite{Mi3},
where the vertex diagrams were found to have logarithmically singular
contributions near the threshold $\omega \sim vq$, leading to the possibility
of a plasmon mode.
In summary, we have shown that vertex corrections can have sizable effect
in the static vacuum polarization diagrams in the regime of strong coupling,
while for small coupling their importance diminishes parametrically.
The self-energy corrections are naturally absorbed into
the renormalization of the Fermi velocity.
The non-RPA vertex diagram at lowest order of perturbation theory
was found to decrease the effective charge, meaning that in principle
correlation effects at higher order must also be taken into account.
Thus the ultimate asymptotic behavior of the static polarization function for $\alpha \sim 1$
remains an open problem.
\begin{acknowledgments}
We are grateful to D. K. Campbell, R. Shankar, V. Pereira, O. Sushkov, and A. Polkovnikov
for many insightful discussions. B.U.
acknowledges CNPq, Brazil, for support under grant No.201007/2005-3.
\end{acknowledgments}
|
0706.0036
|
\section{Introduction}
One of the most profound concepts in string theory is the suggestive idea
that spacetime itself could be a mere emergent notion, a sort of effective
description of a more fundamental entity \cite{Seiberg}. This conception
relies on the existence of the duality symmetries of string theory, which
suggest that concepts such as the curvature and topology of the spacetime
might be only auxiliary notions. This idea is particularly realized by
examples that manifestly show the duality between string theory formulated
on curved backgrounds (e.g. black holes) and the theory in flat space but in
presence of tachyon-like potentials. This is the subject we will explore
here; and we will do this by studying the worldsheet description of the 2D
string theory in the black hole background (i.e. the gauged $SL(2,\mathbb{R}%
)_{k}/U(1)$ Wess-Zumino-Witten (WZW) model).
\subsubsection*{1.1 \ The subject}
The relation between string theory in the 2D black hole background and
Liouville-like conformal field theories representing \textquotedblleft
tachyon wall" potentials was extensively explored in the past. One of the
celebrated examples is the Mukhi-Vafa duality \cite{MV}, relating a twisted
version of the euclidean black hole to the $c=1$ matter coupled to 2D
gravity. The literature on the connection between the $c=1$ CFT and the
black hole CFT is actually quite rich; we should refer to the list of papers
\cite{22}-\cite{Sameer} and the references therein. Recently, a new relation
between the 2D string theory in the euclidean black hole background and a
deformation of the $c=1$ matter CFT has received remarkable attention: This
is the so-called Fateev-Zamolodchikov-Zamolodchikov conjecture (FZZ), which
states the equivalence between the black hole and the often called
sine-Liouville field theory \cite{FZZ,KKK}. In the last six years this FZZ
duality has been applied to study the spectrum and interactions of strings
in both the black hole geometry and the Anti-de Sitter space \cite%
{Notes,YoYu,Takayanagi2}; and the most important application of it was so
far the formulation of the matrix model for the two-dimensional black hole
\cite{KKK}. In fact, when one talks about the \textquotedblleft black hole
matrix model" one is actually referring to the matrix model for the
sine-Liouville deformation of the $c=1$ matter CFT, and thus the black hole
description in such a framework emerges through the FZZ correspondence. This
manifestly shows how useful the FZZ duality is in the context of string
theory.
Although at the beginning it appeared as a conjecture, a proof of the FZZ
duality was eventually given some years ago\footnote{%
More recently, after this paper was published, Y. Hikida and V. Schomerus
presented a proof of the FZZ conjecture \cite{HikidaSchomerus2}.}. This was
done in two steps: first, by proving the equivalence of the corresponding
N=2 supersymmetric extensions of both the 2D black hole $\sigma $-model and
the sine-Liouville theory \cite{HK}; and, secondly, by showing that the
fermionic parts of the N=2 theories eventually decouple, yielding the
bosonic duality as an hereditary property \cite{Maldacena}, see also \cite%
{IPT,IPT2}. This could be done because both sine-Liouville and the black
hole theory admit a natural\footnote{%
The 2D black hole can be realized by means of the Kazama-Suzuki construction
\cite{KS,KS2}, while the sine-Liouville theory can be seen as a sector of
the N=2 Liouville theory. The bosonic version of the FZZ duality can be seen
to arise by GKO quotienting the $U(1)$ R-symmetry of the N=2 version.}
embedding in N=2 theories, where the duality can be seen as a manifestation
of the mirror symmetry. However, one could be also interested in seeing
whether a proof of such a duality exists at the level of the bosonic theory
itself. In this paper we will show how such a duality can be actually proven
(at the level of the sphere topology) without resorting to arguments based
on supersymmetry but just making use of the conformal structure of the
theory.
\subsubsection*{1.2 \ The result}
We will show that any $N$-point correlation functions in the $SL(2,\mathbb{R}%
)_{k}/U(1)$ WZW ($\times $ $time$) on the sphere topology is equivalent to a
$N$-point correlation functions in a two-dimensional conformal field theory
that describes a linear dilaton $\sigma $-model perturbed by a tachyon-like
potential. This actually resembles the FZZ correspondence; however, instead
of considering a vortex perturbation with winding $|n|=1$ here we will
consider momentum modes of the sector $n=2$. To be precise, the theory we
will consider is defined by turning on the modes $\lambda _{n=2}\neq 0$ and $%
\lambda _{n=1}\neq 0$ in the following action
\begin{equation}
S=\frac{1}{4\pi }\int d^{2}z\left( \partial X\overline{\partial }X+\partial
\varphi \overline{\partial }\varphi -\frac{1}{2\sqrt{2}}\widehat{Q}R\varphi
+\sum_{n}\lambda _{n}e^{-\frac{\alpha _{n}}{\sqrt{2}}\varphi +in\sqrt{\frac{k%
}{2}}X}\right) \label{S}
\end{equation}%
where $\widehat{Q}=(k-2)^{-1/2}$ and $\alpha _{n}=\widehat{Q}(1+\sqrt{%
1+(kn^{2}-4)(k-2)})$. Namely, the perturbation we will consider is given by
the operator%
\begin{equation}
\mathcal{O}=\lambda _{1}e^{-\sqrt{\frac{k-2}{2}}\varphi +i\sqrt{\frac{k}{2}}%
X}+\lambda _{2}e^{-\sqrt{\frac{2}{k-2}}(k-1)\varphi +i\sqrt{2k}X},
\label{Pupapupapupa56}
\end{equation}%
where we denoted $X=X_{L}(z)+X_{R}(\overline{z})$, which has to be
distinguished from the T-dual direction $\widetilde{X}=X_{L}(z)-X_{R}(%
\overline{z})$. Operators $e^{-\frac{\alpha _{n}}{\sqrt{2}}\varphi +in\sqrt{%
\frac{k}{2}}X}$ are ($1,1$)-operators with respect to the stress-tensor of
the free theory%
\begin{equation}
T(z)=-\frac{1}{2}\left( \partial X\right) ^{2}-\frac{1}{2}\left( \partial
\varphi \right) ^{2}-\frac{\widehat{Q}}{\sqrt{2}}\partial ^{2}\varphi ,
\label{ThisCondition}
\end{equation}%
so that they represent marginal deformations of the linear dilaton theory.
However, it is worth pointing out that condition (\ref{ThisCondition}) is
not sufficient to affirm that the theory defined by action (\ref{S}) is
\textit{exactly} marginal. In general, proving a theory is an exact
conformal field theory is highly non-trivial. Nevertheless, there is strong
evidence that particular perturbations belonging to those in (\ref{S}) do
represent\footnote{%
One example of such a perturbation is sine-Liouville potential, which we
will discuss in section 3. Notice also that, at the critical value $k=9/4$,
the perturbations in (\ref{S}) are precisely those discussed in \cite{KKK}
in the context of matrix model.} CFTs.
Coefficients $\lambda _{n}$ in (\ref{S}) must satisfy the condition $\lambda
_{n}=\lambda _{-n}$ for the Lagrangian to be real, and thus the theory
results invariant under $X\rightarrow -X$. The scaling relations between
different couplings $\lambda _{n}$ are given by standard KPZ arguments \cite%
{KPZ,David,KPZ2}, being the scale of the theory governed by one of these
constants, analogously as to how the Liouville cosmological constant
introduces the scale in the $c=1$ matter CFT. The central charge of the
theory is then obtained from the operator product expansion of the
stress-tensor, yielding $c=2+6\widehat{Q}^{2}=2+\frac{6}{k-2}.$ Eventually,
we will be interested in adding a time-like free boson to the theory in
order to define a Lorentzian target space of the form $SL(2,\mathbb{R}%
)_{k}/U(1)\times time$, so the central charge will receive an additional
contribution $+1$ coming from the time $\mathbb{R}$ direction, yielding
\begin{equation}
c=3+6\widehat{Q}^{2}=3+\frac{6}{k-2}, \label{c}
\end{equation}%
while the stress-tensor will result supplemented by a term $+\frac{1}{2}%
\left( \partial T\right) ^{2}$. For practical purposes, this time-like
direction can be thought of as an auxiliary degree of freedom, and it does
not enter in the non-trivial part of the duality we want to discuss, being
coupled to the other directions just by the value of the central charge%
\footnote{%
In the case the theory corresponds to the product $SL(2,\mathbb{R}%
)/U(1)\times time$ the condition $c=26$ demands $k=52/23$. On the other
hand, if the space is just the coset $SL(2,\mathbb{R})/U(1)$ the
corresponding condition reads $k=9/4$.} $c$.
\subsubsection*{1.3 \ Outline}
The particular correspondence between the model (\ref{S}) and the 2D black
hole we will discuss turns out to be realized at the level of $N$-point
functions on the sphere topology, and corresponds to a twisted version of
the FZZ correspondence\footnote{%
In the sense that it involves a deformation of the sine-Liouville
interaction term in the action.}. Consequently, we will discuss the latter
first. While being similar, the duality we will discuss herein presents two
important differences with respects to the FZZ: The first difference is that
the new duality admits to be proven\footnote{%
cf. Ref. \cite{HikidaSchomerus2}.} in a relatively simple way without
resorting to arguments based on mirror symmetry of its supersymmetric
extension; secondly, it involves higher momentum modes ($n=2$) instead of
winding modes of the sector $n=1$. We will make the precise statement of the
new correspondence in section 4, where we also address its proof. The paper
is organized as follows: In section 2 we review some features of the
conformal field theories that play an important role in our work. First, we
review the computation of correlation functions in Liouville field theory
with the purpose of emphasizing some features and refer to the analogy with
the Liouville case whenever an illustrative example is needed. Secondly, we
discuss some general aspects of the 2D black hole $\sigma $-model. Once
these two CFTs are introduced, we discuss how correlation functions in both
theories are related through a formula recently proven by S. Ribault and J.
Teschner \cite{RT,R}. Their formula connects correlation functions in both
WZW and Liouville theory in a remarkably direct way \cite{S}, and it turns
out to be important for proving our result. In section 3 we briefly review
the FZZ dual for the 2D black hole; namely the sine-Liouville field theory.
In section 4 we introduce a \textquotedblleft twisted" version of the
sine-Liouville theory, and we show that such \textquotedblleft deformed"
sine-Liouville turns out to be a dual for the 2D black hole as well. A
crucial piece to show this new version of the duality is the
Ribault-Teschner formula mentioned above, for which we present a free field
realization that is eventually identified as being precisely the deformed
sine-Liouville model we want to study. Section 5 contains the conclusions.
\section{Conformal field theory}
To begin with, let us discuss some aspects of correlation functions in
Liouville field theory. The reason for doing this is that Liouville theory
is the prototypical example of non-compact conformal field theory \cite%
{Schomerus} and thus the techniques for computing correlation functions in
this model are analogous to those we will employ in the rest of the paper.
Moreover, the models we will consider here are actually deformations of the
Liouville theory coupled to a $c=1+1$ matter field, so that it is clearly
convenient to consider this model first.
\subsection{Liouville theory}
\subsubsection{Liouville field theory coupled to $c=1(+1)$ matter}
Liouville theory naturally arises in the formulation of the two-dimensional
quantum gravity and in the path integral quantization of string theory \cite%
{Polyakov}. This is a non-trivial conformal field theory \cite{Yu,Teschner}\
whose action reads%
\begin{equation}
S_{L}[\mu ]=\frac{1}{4\pi }\int d^{2}z\left( \partial \varphi \overline{%
\partial }\varphi +\frac{1}{2\sqrt{2}}QR\varphi +4\pi \mu e^{\sqrt{2}%
b\varphi }\right) \label{mancha}
\end{equation}%
where $\mu $ is a real positive parameter called \textquotedblleft the
Liouville cosmological constant". The background charge parameter takes the
value $Q=b+b^{-1}$ in order to make the Liouville barrier potential $\mu e^{%
\sqrt{2}b\varphi }$ to be a marginal operator. In the conformal gauge, the
linear dilaton term $QR\varphi $, which involves the two-dimensional Ricci
scalar $R,$ has to be understood as keeping track of the coupling with the
worldsheet curvature that receives a contribution coming from the point at
infinity. The theory is globally defined once one specifies the boundary
conditions, and this can be done by imposing the behavior $\varphi \sim -2%
\sqrt{2}Q\log |z|$ for large $|z|$, that is compatible with the spherical
topology. Under holomorphic transformations $z\rightarrow w$ Liouville field
transforms in a way that depends on $Q$, namely $\varphi \rightarrow \varphi
-\sqrt{2}Q\log |\frac{dw}{dz}|$. In this paper we will be interested in the
coupling of Liouville theory to a $U(1)$ boson field represented by an
additional $-\frac{1}{4\pi }\int d^{2}z\partial X\overline{\partial }X$
piece in the action (\ref{mancha}) above. Moreover, we can also include the
\textquotedblleft time" direction $\frac{1}{4\pi }\int d^{2}z\partial T%
\overline{\partial }T.$ Then, the central charge of the whole theory is
given by%
\begin{equation*}
c=2+c_{L}=3+6Q^{2},
\end{equation*}%
where $c_{L}$ refers to the Liouville central charge. Important objects of
the theory are the exponential vertex operators \cite{Teschnervertex}
\begin{equation*}
V_{\alpha }(z)\times e^{i\sqrt{2}p_{1}X(z)+i\sqrt{2}p_{0}T(z)}=e^{\sqrt{2}%
\alpha \varphi (z)+i\sqrt{2}p_{1}X(z)+i\sqrt{2}p_{0}T(z)},
\end{equation*}%
which turn out to be local operators of conformal dimension $h=\alpha
(Q-\alpha )+p_{1}^{2}-p_{0}^{2}$ with respect to the stress-tensor $T(z)$ of
the free theory,%
\begin{equation}
T(z)=\frac{1}{2}(\partial T)^{2}-\frac{1}{2}(\partial X)^{2}-\frac{1}{2}%
(\partial \varphi )^{2}+\frac{Q}{\sqrt{2}}\partial ^{2}\varphi . \label{T}
\end{equation}
Now, let us move on and discuss correlation functions.
\subsubsection{Liouville correlation functions}
The non-trivial part of correlation functions in the theory (\ref{T}) is
given by the Liouville correlation functions \cite{Teschner,ZZ,Zreloaded,DO}%
, and these are formally defined as follows%
\begin{equation*}
A_{(\alpha _{1},...\alpha _{N}|z_{1},...z_{N})}^{L}=\left\langle V_{\alpha
_{1}}(z_{1})...V_{\alpha _{N}}(z_{N})\right\rangle _{S_{L}[\mu ]}=\int
D\varphi e^{-S_{L}[\mu ]}\prod_{i=1}^{N}e^{\sqrt{2}\alpha _{i}\varphi
(z_{i})}
\end{equation*}%
and, on the spherical topology, these can be written by using that
\begin{align}
\left\langle \prod_{i=1}^{N}V_{\alpha _{i}}(z_{i})\right\rangle _{S_{L}[\mu
]}& =b^{-1}\mu ^{s}\Gamma (-s)\delta \left( s+b^{-1}(\alpha _{1}+\alpha
_{2}+...\alpha _{N})-1-b^{-2}\right) \times \notag \\
& \times \prod_{r=1}^{s}\int d^{2}w_{r}\left\langle \prod_{i=1}^{N}V_{\alpha
_{i}}(z_{i})\prod_{r=1}^{s}V_{b}(w_{r})\right\rangle _{S_{L}[\mu =0]},
\label{despues}
\end{align}%
namely,%
\begin{equation}
A_{(\alpha _{1},...\alpha _{N}|z_{1},...z_{N})}^{L}=b^{-1}\mu ^{s}\Gamma
(-s)\delta \left( s+b^{-1}(\alpha _{1}+\alpha _{2}+...\alpha
_{N})-1-b^{-2}\right) \times \notag
\end{equation}%
\begin{equation}
\times \prod_{r=1}^{s}\int d^{2}w_{r}\int D\varphi e^{-S_{L}[\mu
=0]}\prod_{i=1}^{N}e^{\sqrt{2}\alpha _{i}\varphi (z_{i})}\prod_{r=1}^{s}e^{%
\sqrt{2}b\varphi (w_{r})}. \label{da}
\end{equation}%
This permits to compute correlation functions by employing the standard
Gaussian measure and free field techniques. The overall factor $\Gamma (-s)$
and the $\delta $-function come from the integration over the zero-mode $%
\varphi _{0}$ of the Liouville field $\varphi $, and it also yields the
insertion of an specific amount, $s,$ of screening operators $V_{b}(w)$ in
the correlator. In deriving (\ref{da}), the identity $\mu ^{s}\Gamma
(-s)=\int dxx^{-1-s}e^{-\mu x}$ and the Gauss-Bonnet theorem were used to
find out the relation between $s$, $b$, and the momenta $\alpha _{i}$, which
for a manifold of generic genus $g$ and $N$ punctures would yield%
\begin{equation}
bs+\sum_{i=1}^{N}\alpha _{i}=Q(1-g). \label{pupo2}
\end{equation}%
So, the correlators can be computed through the Wick contraction of the $N+s$
operators by using the propagator $\left\langle \varphi (z_{1})\varphi
(z_{2})\right\rangle =-2\log |z_{1}-z_{2}|,$ which corresponds to the free
theory (\ref{T}) and yields the operator product expansion $e^{\alpha
_{1}\varphi (z_{1})}e^{\alpha _{2}\varphi (z_{2})}\sim
|z_{1}-z_{2}|^{-2\alpha _{1}\alpha _{2}}e^{(\alpha _{1}+\alpha _{2})\varphi
(z_{2})}+...$. In principle, this could be used to integrate the expression
for $A_{(\alpha _{1},...\alpha _{N}|z_{1},...z_{N})}^{L}$ explicitly.
Nevertheless, it is worth noticing that the expression (\ref{da}) can be
considered just formally since, in general, $s$ is not an integer number.
Hence, in order to compute generic correlation functions one has to deal
with the problem of making sense of such integral representation. With the
purpose of giving an example, let us describe below the computation of the
partition function on the sphere in detail. Such case corresponds to $g=0$
and $N=0$, and the number of screening operators to be integrated out turns
out to be $m=s-3=-2+b^{2}$. That is, in order to compute the genus zero
partition function we have to consider the correlation function of three
local operators $e^{\sqrt{2}b\varphi (z)}$ inserted at the points $%
z_{1}=0,z_{2}=1$ and $z_{3}=\infty $ to compensate the volume of the
conformal Killing group, $SL(2,\mathbb{C})$. This has to be distinguished
from the direct computation of the three-point function \cite{ZZ} of three
\textquotedblleft light" states $\alpha _{1}=\alpha _{2}=\alpha _{3}=b,$ as
we will discuss below.
\subsubsection{A working example: the spherical partition function}
Although it is usually said that string partition function on the spherical
topology vanishes, we know that this is not necessary the case when the
theory is formulated on non-trivial backgrounds. A classical example of this
is the two-dimensional string theory formulated in both tachyonic and
gravitational non-trivial backgrounds we will be discussing along this
paper. Such models admit a description in terms of the Liouville-type sigma
model actions, so that the computation of the corresponding genus zero
partition functions involves the computation of spherical partition function
of Liouville theory or some deformation of it. Here, we will describe a
remarkably simple calculation of the Liouville partition function on the
spherical topology by using the free field techniques. The free field
techniques to be employed here were developed so far by Dotsenko and Fateev
\cite{Fateev2,Dotsenko}, and by Goulian and Li \cite{GLi} (see also \cite%
{dot1,dot2,D}). The partition function $Z_{g=0}$ is then given by%
\begin{equation}
Z_{g=0}=\frac{\mu ^{m+3}}{b}\Gamma (-m-3)\lim_{z_{3}\rightarrow \infty
}|z_{3}|^{-4}\prod_{r=1}^{m}\int d^{2}w_{r}\int D\varphi e^{-S_{L}[\mu
=0]}e^{\sqrt{2}b\varphi (0)}e^{\sqrt{2}b\varphi (1)}e^{\sqrt{2}b\varphi
(z_{3})}\prod_{r=1}^{m}e^{\sqrt{2}b\varphi (w_{r})}. \label{empieza}
\end{equation}%
with $m=-2+b^{-2}$. According to the standard Wick rules, we can write%
\begin{equation*}
Z_{g=0}=b^{-1}\mu ^{3+m}\Gamma (-m-3)\prod_{r=1}^{m}\int d^{2}w_{r}\left(
\prod_{r=1}^{m}|w_{r}|^{4\rho }|1-w_{r}|^{4\rho
}\prod_{r<t}^{m-1,m}|w_{t}-w_{r}|^{4\rho }\right) .
\end{equation*}%
This can be explicitly solved for integer $m$ by using the Dotsenko-Fateev
integral formula worked out in reference \cite{Dotsenko}. Even though we are
interested in the case where $m$ is generic enough, and this can mean a
negative real number, we can assume that this is an integer positive number
through the integration and then try to analytically extend the final
expression accordingly. In this way, we get%
\begin{equation*}
Z_{g=0}=\frac{\mu ^{3+m}}{b}\Gamma (-m-3)\Gamma (m+1)\pi ^{m}\gamma
^{m}(1-\rho )\prod_{r=1}^{m}\gamma (r\rho )\prod_{r=0}^{m-1}\gamma
^{2}(1+(2+r)\rho )\gamma (-1-(3+r+m)\rho ).
\end{equation*}%
where, as usual, we denoted $\gamma (x)=\Gamma (x)/\Gamma (1-x)$; and we
also denoted $\rho =-b^{2}$ for notational convenience. Once again, this
expression only makes sense for $m$ being a positive integer number, so that
the non-trivial point here is that of performing analytic continuation. In
order to do this, we can rewrite the expression above by taking into account
that $\gamma (-1-(3+r+m)\rho )=\gamma (-(r+1)\rho ).$ So we can expand it as%
\begin{equation}
Z_{g=0}=\frac{\mu ^{3+m}}{b}\Gamma (-m-3)\Gamma (m+1)\pi ^{m}\gamma
^{m}(1-\rho )\prod_{r=1}^{m}\gamma (r\rho )\gamma (-r\rho
)\prod_{r=2}^{m+1}\gamma ^{2}(1+r\rho ). \label{introducidas}
\end{equation}%
Now, some simplifications are required. First, we can use that $%
m=-2+b^{-2}=-2-\rho ^{-1}$ and $1+r\rho =-(m+2-r)\rho $ to arrange the last
product. Then, we can rewrite the product as%
\begin{equation*}
\gamma (1+2\rho )\gamma (1+3\rho )...\gamma (1+m\rho )\gamma (1+(m+1)\rho
)=\gamma (-\rho )\gamma (-2\rho )...\gamma (-(m-1)\rho )\gamma (-m\rho ),
\end{equation*}%
that is%
\begin{equation*}
\prod_{r=2}^{m+1}\gamma (1+r\rho )=\prod_{r=1}^{m}\gamma (-r\rho ),
\end{equation*}%
and then use $\gamma (r\rho )\gamma (1-r\rho )=1$ to write%
\begin{equation*}
Z_{g=0}=b^{-1}\mu ^{Q/b}\Gamma (-m-3)\Gamma (m+1)\pi ^{m}\gamma ^{m}(1-\rho
)\gamma ^{2}(-\rho )(-1)^{m}\rho ^{-2m}\Gamma ^{-2}(m+1),
\end{equation*}%
where the identities $\gamma (x)\gamma (-x)=\gamma (x)/\gamma (1+x)=-x^{-2}$
were also used. Again, the properties of the $\gamma $-function can be used
to write $\gamma (2+\rho ^{-1})=-(1+\rho ^{-1})^{2}\gamma (1+\rho ^{-1}),$ $%
\gamma (1-\rho )=-\rho ^{2}\gamma (-\rho )$ and $\gamma (-1-\rho )=-(1+\rho
)^{-2}\gamma (-\rho )$. Then, once all is written in terms of $b,$ the
partition function reads\footnote{%
Notice that we have absorbed a factor $\sqrt{2}$ in the definition of the
measure of the path integral.}%
\begin{equation}
Z_{g=0}=\frac{(1-b^{2})\left( \pi \mu \gamma (b^{2})\right) ^{Q/b}}{\pi
^{3}Q\gamma (b^{2})\gamma (b^{-2})}. \label{ZZZ}
\end{equation}%
This is the exact result for the Liouville partition function on the
spherical topology, which turns out to be a non trivial function of $b$. It
oscillates with growing frequency and decreasing amplitude according $b^{2}$
approaches the values $b^{2}=0$ and $b^{2}=1$. One of the puzzling features
of the expression (\ref{ZZZ}) is the fact that it does not manifest the
self-duality that the Liouville theory seems to present under the
transformation $b\rightarrow 1/b$. In order to understand this point, it is
convenient to compare the direct computation of $Z_{g=0}$ we gave above with
the analogous computation of the Liouville structure constant (three-point
functions) $C(\alpha _{1},\alpha _{2},\alpha _{3})$ for the particular
configuration $\alpha _{1}=\alpha _{2}=\alpha _{3}=b$. The difference
between both calculations is given by the overall factor $\Gamma (-s)=\Gamma
(-m-3)$ in (\ref{empieza}). As mentioned, this factor comes from the
integration over the zero-mode of the field $\varphi $, but it can be also
thought of as coming from the combinatorial problem of permuting all the
screening operators. Actually, for integer $s$ this factor can be written as
$\Gamma (-s)=(-1)^{s}\Gamma (0)/s!$, where the divergent factor $\Gamma (0)$
keeps track of a divergence due to the non-compactness of the Liouville
direction. In fact, this yields the factorial $1/s!$ arising in the residue
corresponding to the poles of resonant correlators. On the other hand, in
the case of being computing the structure constant $C(b,b,b)$, unlike the
computation of $Z_{g=0}$, such overall factor should be $\Gamma (3-s)$
instead of $\Gamma (-s)$ since one has to divide by the permutation of $s-3$
screening charges. Hence, we have $C(b,b,b)/Z_{g=0}=\Gamma (3-s)/\Gamma
(-s)=-s!/(s-3)!=-(b^{-2}+1)b^{-2}(b^{-2}-1)$. This is precisely consistent
with the fact that $\frac{d^{3}Z}{d\mu ^{3}}=-C(b,b,b)\sim \mu ^{Q/b-3}$,
see Ref. \cite{Z}. Thus, this combinatorial problem appears as being the
origin of the breakdown of the Liouville self-duality at the level of the
partition function.
Now, let us move to study another CFT that is also a crucial piece in our
discussion: the CFT that describes the 2D black hole $\sigma $-model.
\subsection{String theory in the 2D black hole}
\subsubsection{The action and the semiclassical picture}
String theory in two dimensions presents very interesting properties that
make of it a fruitful ground to study features of its higher dimensional
analogues. One example is given by the 2D black hole solution discovered in
Refs. \cite{Witten,Maldal,Pakmandice}. This black hole solution is supported
by a dilaton configuration, and it turns out to be an exact conformal
background on which formulate string theory. In fact, the 2D black hole $%
\sigma $-model action corresponds to the gauged level-$k$ $SL(2,\mathbb{R}%
)_{k}/U(1)$ WZW theory \cite{Witten}. An excellent comprehensive review on
this model can be found in Ref. \cite{Persson}.
The worldsheet action for string theory in a two-dimensional metric-dilaton
background, once setting $\alpha ^{\prime }=2$, reads%
\begin{equation}
S_{P}=\frac{1}{4\pi }\int d^{2}z\left( G_{\mu \nu }(X)\partial X^{\mu }%
\overline{\partial }X^{\nu }+R\Phi (X)\right) , \label{Polyakov2}
\end{equation}%
where the indices $\mu ,\nu =\{1,D=2\}$ run over the two coordinates of the
target space, whose metric is $G_{\mu \nu }(X)$. This action is written in
the conformal gauge, so, as we discussed before, the dilaton term $R$%
\thinspace $\Phi (X)$ has to be understood as keeping track of the coupling
with the worldsheet curvature that receives a contribution coming from the
point at infinity. The vanishing of the one-loop $\beta $-functions demands $%
R_{\mu \nu }=\nabla _{\mu }\nabla _{\nu }\Phi $, with $R_{\mu \nu }$ being
now the Ricci tensor associated to the target space metric $G_{\mu \nu }$.
Since the 2D black hole string theory corresponds to the $SL(2,\mathbb{R}%
)_{k}/U(1)$ WZW model, it admits an exact algebraic description in terms of
the current conformal algebra of the WZW theory; and we will comment on this
in the following subsection. In the semiclassical limit, governed by the
large $k$ regime, the euclidean version of the background is described by
the following configurations for the metric $G_{\mu \nu }$ and the dilaton $%
\Phi $,%
\begin{equation*}
ds^{2}=k\left( dr^{2}+\tanh ^{2}r\ dX^{2}\right) ,\quad \quad \Phi (r)=\Phi
_{0}-2\log \left( \cosh r\right) .
\end{equation*}%
It is well known that the geometry of the euclidean black hole is that of a
semi-infinite cigar that asymptotically looks like a cylinder. The angular
coordinate of such cylinder is $X$, while the coordinate $r$ is the one that
goes along the cigar, running from $r=0$ (the tip of the cigar, where the
string theory is strongly coupled) to $r=\infty $ (where the string coupling
$e^{\Phi (r)}$ tends to zero). To get a semiclassical picture of this
geometry, let us consider the large $k$ regime and redefine the radial
coordinate as $\cosh ^{2}r=M^{-1}e^{\sqrt{2/k}\varphi }$. Then, in the large
$\varphi $ approximation, and by also rescaling the angular coordinate $X$
by a factor $\sqrt{2/k},$ the metric reads%
\begin{equation}
ds^{2}=2\left( 1+Me^{-\sqrt{2/k}\varphi }\right) d\varphi ^{2}+2\left(
1-Me^{-\sqrt{2/k}\varphi }\right) dX^{2}, \label{H}
\end{equation}%
that asymptotically looks like the cylinder of radius $R=\sqrt{k/2}$. The
parameter $M$ is related to the mass of the black hole, and it can be fixed
to any positive value by shifting $\varphi $. Considering finite-$k$
corrections leads to a shifting in $k$ and then the metric and the dilaton
result corrected. In such case, the dilaton reads%
\begin{equation*}
\Phi (\varphi )=\Phi _{0}-\log M+\sqrt{2}\widehat{Q}\varphi ,\qquad \widehat{%
Q}=(k-2)^{-1/2}.
\end{equation*}%
Thus, the 2D string theory in the euclidean black hole background can be
semiclassically described by a deformation of the linear dilaton theory
\begin{equation}
S_{0}=\frac{1}{4\pi }\int d^{2}z\left( \partial X\overline{\partial }%
X+\partial \varphi \overline{\partial }\varphi -\frac{1}{2\sqrt{2}}\widehat{Q%
}R\varphi \right) ; \label{S0}
\end{equation}%
and, according to (\ref{H}) and taking into account the finite-$k$
corrections, such \textquotedblleft deformation" corresponds to perturbing
the action (\ref{S0}) with the graviton-like operator \cite{24}%
\begin{equation}
\mathcal{O}=M\ \partial X\overline{\partial }X\ e^{-\sqrt{\frac{2}{k-2}}%
\varphi }; \label{graviton}
\end{equation}%
this is true up to a BRST-trivial\footnote{%
That means that it is pure gauge in the BRST cohomology.} operator of the
form $\delta \mathcal{O}\sim \partial \varphi \overline{\partial }\varphi \
e^{-\sqrt{\frac{2}{k-2}}\varphi }.$ In these terms, the theory can be in
principle solved (e.g. its correlation functions can be computed) by using
the free field approach and the Coulomb-like correlators $\left\langle
\varphi (z_{1})\varphi (z_{2})\right\rangle =\left\langle
X(z_{1})X(z_{2})\right\rangle =-2\log |z_{1}-z_{2}|$. Operator (\ref%
{graviton}) is usually called the \textquotedblleft black hole mass
operator". The inclusion of this operator in the action has to be thought of
as being valid in a semiclassical picture and can be shown to be equivalent
to the free field representation of the WZW model.
In the large $\varphi $ region of the space (where the theory turns out to
be weakly coupled) we have that the non-linear $\sigma $-model of strings in
the black hole seems to coincide with the action $S_{0}+\frac{1}{4\pi }\int
d^{2}z\ \mathcal{O}$. Furthermore, there is a way of seeing that operator (%
\ref{graviton}) actually describes the dilatonic black hole $\sigma $-model
beyond the semiclassical picture. To do so, it is necessary to argue that
such an action unambiguously describes the full theory beyond the weak limit
region \cite{Mukhi,MMP} and, for instance, reproduces the exact correlation
functions. This seems to be hard to be proven in general; nevertheless,
there is a nice way of showing that the perturbation (\ref{graviton})
corresponds to the theory on the black hole background. This relies on the
algebraic description of the $SL(2,\mathbb{R})_{k}/U(1)\times \mathbb{R}$
WZW theory and is quite direct: The point is that the action $S_{0}+\frac{1}{%
4\pi }\int d^{2}z\ \mathcal{O}$ , once supplemented with the BRST-trivial
operator $\delta \mathcal{O}$ and a free time-like boson $-\frac{1}{4\pi }%
\int d^{2}z\ \partial T\overline{\partial }T$, can be shown to be related to
the well known free field realization of the $SL(2,\mathbb{R})_{k}$ WZW
action through a $SO(2,1)$-boost given by\footnote{%
Please, do not mistake the time-like coordinate $T$ for the notation used
for the stress-tensor. Excuse us for this overlap in the notation.}
\begin{equation*}
T=i\sqrt{\frac{2}{k}}u-i\sqrt{\frac{k-2}{k}}\phi ,\ \ \ \ X=-\sqrt{\frac{k}{2%
}}v+i\frac{k-2}{\sqrt{2k}}u+i\sqrt{\frac{k-2}{k}}\phi ,\ \ \ \ \varphi =%
\sqrt{\frac{k-2}{2}}(u+iv)+\phi ,
\end{equation*}%
and the standard bosonization\footnote{%
It is usually convenient to use a different bosonization, expressing the
field $\beta $ as an exponential function. This would lead to a
Liouville-like interaction in the action.} $\gamma =e^{u+iv}$, $\beta
=i\partial ve^{-u-iv}$, with $\left\langle \beta (z_{1})\gamma
(z_{2})\right\rangle \sim (z_{1}-z_{2})^{-1},$ and with $\left\langle \phi
(z_{1})\phi (z_{2})\right\rangle =-2\log |z_{1}-z_{2}|,$ \cite{HHS}. In
fact, this leads to the Wakimoto free field description of the $SL(2,\mathbb{%
R})_{k}$ current algebra in terms of the linear dilaton field $\phi $ and
the $\beta ,\gamma $ ghost system \cite{Wakimoto}. In Wakimoto variables one
identifies the theory as being the WZW model formulated on $SL(2,\mathbb{R})$
with the elements of the group written in the Gauss parameterization. Then,
the coset theory $SL(2,\mathbb{R})_{k}/U(1)$ is obtained by simply taking
out the time-like direction $T$ which realizes the $U(1)$ current\footnote{%
Alternatively, an additional free boson, analogous to $X,$ can be added in
order to relize the gauging, see \cite{DVV,BB2} and referenctes therein.}
\begin{equation*}
J^{3}=\beta \gamma +\sqrt{\frac{k-2}{2}}\partial \phi =i\sqrt{\frac{k}{2}}%
\partial T;
\end{equation*}%
recall that this is a time-like direction so that the corresponding
correlator flips its sign and thus turns out to be $\left\langle
T(z_{1})T(z_{2})\right\rangle =+2\log |z_{1}-z_{2}|$.
On the other hand, let us mention that the dual theory (i.e. the
sine-Liouville theory) is also defined as a perturbation of (\ref{S0}); see (%
\ref{cos}) below. According to this picture, it is possible to consider FZZ
duality as a relation between different marginal deformations of the same
free linear dilaton background. This was the philosophy in Ref. \cite{MMP},
where the FZZ correspondence was seen from a generalized perspective,
considering it as an example of a set of connections existing between
different marginal deformations of (\ref{S0}). Here, we will be discussing a
similar correspondence; we will consider perturbations carrying momentum
modes $n=2$ of the tachyon potential and discuss how it describes $SL(2,%
\mathbb{R})_{k}/U(1)\times \mathbb{R}$ WZW correlation functions. We will
dedicate some effort to understand the relation between such $n=2$
perturbation and the standard FZZ duality (that involves $n=1$ modes). But,
first, let us continue our description of the theory in the black hole
background with appropriate detail.
\subsubsection{String spectrum in the 2D black hole and its relation to $%
AdS_{3}$ strings}
The spectrum of the 2D sting theory in the black hole background corresponds
to certain sector of the Hilbert space of the gauged $SL(2,\mathbb{R}%
)_{k}/U(1)$ WZW model, and is thus given in terms of certain representations
of $SL(2,\mathbb{R})_{k}\times \overline{SL}(2,\mathbb{R})_{k}$. The string
states are thus described by vectors $\left| \Phi _{j,m,\bar{m}}^{\omega
}\right\rangle $ which are associated to vertex operators $\Phi _{j,m,\bar{m}%
}^{\omega },$ where $j$, $m,$ and $\bar{m}$ are indices that label the
states of the representations of the group. In order to define the string
theory, it is necessary to identify which is the subset of representations
that have to be taken into account. Such a subset has to satisfy several
requirements\footnote{%
For an interesting discussion on non-compact conformal field theories see
\cite{Schomerus}.}. In the case of the free theory these requirements are
associated to the normalizability and unitarity of the string states. At the
level of the interacting theory, additional properties are requested, like
the closure of the fusion rules, the factorization properties of $N$-point
functions, etc.
The $SL(2,\mathbb{R})_{k}$ WZW model is behind the description of string
theory in both the 2D black hole background (through the coset construction)
and in $AdS_{3}$ space. These two models are closely related indeed, but
still different. In the case of the black hole, the states of the spectrum
are labeled by the index $j$ of the $SL(2,\mathbb{R})$ representations with
the indices $m$ and $\overline{m}$ falling in the lattice%
\begin{equation}
m-\overline{m}=n,\quad m+\overline{m}=-k\omega \label{u}
\end{equation}%
with $n$ and $\omega $ being integer numbers, and the conformal dimension of
the vertex operators is given by%
\begin{equation}
h=-\frac{j(j+1)}{k-2}+\frac{m^{2}}{k}. \label{harriba}
\end{equation}%
On the other hand, $AdS_{3}$ string theory can be described in terms of the
WZW model on the product between the coset $SL(2,\mathbb{R})_{k}/U(1)$ and a
time-like free boson \cite{HW}, so that the worldsheet theory turns out to
be the product between the time and the euclidean black hole. This can be
realized by adding the contribution\footnote{%
Besides, one can represent string theory in $AdS_{3}$ space in terms of the
Wakimoto free field realization mentioned above. In terms of these fields
the $AdS_{3}$ metric reads $ds^{2}=k\left( d\phi ^{2}+e^{2\phi }d\gamma d%
\overline{\gamma }\right) $.} $-\frac{1}{4\pi }\int d^{2}z\ \partial T%
\overline{\partial }T$ to the action (\ref{S0}) and by supplementing the
vertex operators with a factor $e^{i\sqrt{\frac{2}{k}}(m+\frac{k}{2}\omega
)T}$ that carries the charge under the field $T$. Thus, the vertex operators
on $AdS_{3}$ have conformal dimension given by%
\begin{equation}
h=-\frac{j(j+1)}{k-2}-m\omega -\frac{k}{4}\omega ^{2}, \label{h}
\end{equation}%
which corresponds to adding the conformal dimension $\delta h=-\frac{%
(m+k\omega /2)^{2}}{k}$ of the time-like part to the coset contribution (\ref%
{harriba}). In some sense, the string theory in the 2D black hole can be
thought of as having constrained the states of the theory in $AdS_{3}$ to
have vanishing bulk energy, $m+\overline{m}+k\omega =0$. In this way, one
has the theory on the background $time\times SL(2,\mathbb{R})_{k}/U(1)$\ as
an appropriate realization of sting theory in $AdS_{3}$ space \cite%
{MO1,GN2,GN3,GL}. However, before going deeper into the string
interpretation of the WZW model, some obstacles have to be overcame. In
fact, even in the case of the free string theory, the fact of considering
non-compact Lorentzian curved backgrounds is not trivial at all. The main
obstacle in constructing the space of states is the fact that, unlike what
happens in flat space, in curved space the Virasoro constraints are not
enough to decouple the negative-norm string states. In the early attempts
for constructing a consistent string theory in $AdS_{3}$, additional \textit{%
ad hoc} constraints were imposed on the vectors of the $SL(2,\mathbb{R})_{k}$
representations in order to decouple the ghosts. The vectors of $SL(2,%
\mathbb{R})$ representations are labeled by a pair of indices $j$ and $m$,
and thus such additional constraints (demanded as sufficient conditions for
unitarity) imply an upper bound for the index $j$ of certain
representations, and consequently an unnatural upper bound for the mass
spectrum. The modern approaches to the \textquotedblleft negative norm
states problem\textquotedblright\ also include such a kind of constraint on $%
j$, although this fact does not imply a bound on the mass spectrum as in the
old versions it did \cite{MO1}. The upper bound for the index $j$ of
discrete representations, often called \textquotedblleft unitarity
bound\textquotedblright , reads $1-k<2j<-1.$ In the case of Euclidean $%
AdS_{3}$, the spectrum of string theory is just given by the continuous
series of $SL(2,\mathbb{C})$, parameterized by the values $j=-\frac{1}{2}%
+i\lambda $ with $\lambda \in \mathbb{R}$ and by real $m$. On its turn, the
case of string theory in Lorentzian $AdS_{3}$ is richer and its spectrum is
composed by states belonging to both continuous $\mathcal{C}_{\lambda
}^{\alpha ,\omega }$ and discrete $\mathcal{D}_{j}^{\omega ,\pm }$ series.
The continuous series $\mathcal{C}_{\lambda }^{\alpha ,\omega }$ have states
with $j=-\frac{1}{2}+i\lambda $ with $\lambda \in \mathbb{R}$ and $m-\alpha
\in \mathbb{Z}$, with $\alpha \in \lbrack 0,1)\in \mathbb{R}$ (as in $SL(2,%
\mathbb{C})$, obviously). On the other hand, the states of discrete
representations $\mathcal{D^{\pm ,\omega }}_{j}$ satisfy $j=\pm m-n$ with $%
n\in \mathbb{Z}_{\geq 0}$. Other important ingredient for constructing the
Hilbert space is the index $\omega $ labeling the operators $\Phi _{j,m,\bar{%
m}}^{\omega }$. In the black hole background, $\omega $ turns out to be
given by (\ref{u}). In $AdS_{3}$, the quantum number $\omega $ is
independent of the bulk kinetic energy $m+\overline{m}$ and the bulk angular
momentum $m-\overline{m},$ contributing to the total energy as $m+\overline{m%
}+k\omega $. Then, the question arises as to how the index $\omega $ appears
in the Hilbert space of the $SL(2,\mathbb{R})_{k}$ WZW theory. The answer is
that in order to fully parameterize the spectrum in $AdS_{3}$ we have to
introduce the \textquotedblleft flowed\textquotedblright\ operators $\tilde{J%
}_{n}^{a}$ (with $a=3,-,+$) which are defined through the spectral flow
automorphism \cite{MO1}
\begin{equation}
J_{n}^{3}\rightarrow \tilde{J}_{n}^{3}=J_{n}^{3}-\frac{k}{2}\omega \delta
_{n,0},\quad J_{n}^{\pm }\rightarrow \tilde{J}_{n}^{\pm }=J_{n\pm \omega
}^{\pm } \label{arrova}
\end{equation}%
acting of the original $\hat{sl(2)}_{k}$ generators $J_{n}^{a}$, which
satisfy the Lie product that define the affine algebra%
\begin{equation}
\lbrack J_{n}^{-},J_{m}^{+}]=-2J_{n+m}^{3}+nk\delta _{n,-m},\qquad \lbrack
J_{n}^{3},J_{m}^{\pm }]=\pm J_{n+m}^{\pm },\qquad \lbrack
J_{n}^{3},J_{m}^{3}]=-n\frac{k}{2}\delta _{n,-m}. \label{elalgebra}
\end{equation}%
Then, states $\left| \Phi _{j,m,\bar{m}}^{\omega }\right\rangle $ belonging
to the \textit{flowed} discrete representations $\mathcal{D}_{j}^{\pm
,\omega }$ are those obeying\footnote{%
or analogous relations for the Weyl reflected representations, namely $%
j\rightarrow -1-j$.}
\begin{equation}
\tilde{J}_{0}^{\pm }\left| \Phi _{j,m,\bar{m}}^{\omega }\right\rangle =(\pm
j-m)\left| \Phi _{j,m\pm 1,\bar{m}}^{\omega }\right\rangle ,\quad \tilde{J}%
_{0}^{3}\left| \Phi _{j,m,\bar{m}}^{\omega }\right\rangle =m\left| \Phi
_{j,m,\bar{m}}^{\omega }\right\rangle \label{t2}
\end{equation}%
and being annihilated by the positive modes, namely
\begin{equation}
\tilde{J}_{n}^{a}\left| \Phi _{j,m,\bar{m}}^{\omega }\right\rangle =0\ ,\ \
\ n>0\ . \label{t3}
\end{equation}%
States with $m=\pm j$ represent highest (resp. lowest) weight states, while
primary states of the continuous representations $\mathcal{C}_{\lambda
}^{\alpha ,\omega }$ are annihilated by all the positive modes. On the other
hand, the excited states in the spectrum are defined by acting with the
negative modes $J_{-n}^{a}$ ($n\in \mathbb{Z}_{>0}$) on the Kac-Moody
primaries $\left| \Phi _{j,m,\bar{m}}^{\omega }\right\rangle $; these
negative modes play the role of creation operators (\textit{i.e.} creating
the string excitation). The \textquotedblleft flowed
states\textquotedblright\ (namely those being primary vectors with respect
to the $\tilde{J}_{n}^{a}$ defined with $|\omega |>1$) are not primary with
respect to the $\hat{sl(2)}_{k}$ algebra generated by $J_{n}^{a}$, and this
is clear from (\ref{arrova}). However, highest weight states in the series $%
\mathcal{D}_{j}^{+,\omega }$ are identified with lowest weight states of $%
\mathcal{D}_{-k/2-j}^{-,\omega }$, which means that spectral flow with $%
|\omega |=1$ is closed among certain subset of Kac-Moody primaries.
The states belonging to discrete representations have a discrete energy
spectrum and represent the quantum version of those string states that are
confined in the centre of $AdS_{3}$ space; these are called
\textquotedblleft short strings\textquotedblright\ and are the counterpart
of those states that are confined close to the tip of the cigar geometry. On
the other hand, the states of the continuous representations describe
massive \textquotedblleft long strings\textquotedblright\ that can escape to
the infinity, where the theory is weakly coupled. In the case of the 2D
black hole, the index $\omega $ of these long strings has a clear
interpretation as an \textquotedblleft asymptotically topological" degree of
freedom (is not a topological one though). Because of the euclidean black
hole has the geometry of a semi-infinite cigar and thus looks like a
cylinder very far from the tip, the states in the asymptotic region have a
winding number around such cylinder. However, this is not strictly a
cylinder but has topology $\mathbb{R}^{2}$ instead of $\mathbb{R}\times
S^{1} $, so that, as it happens in $AdS_{3}$, the winding number
conservation can be in principle violated. Of course, this feasibility of
violating $\omega $ is not evident from the background (\ref{S0})-(\ref%
{graviton}), which is reliable only far from the tip of the cigar, but the
phenomenon can occur when string interactions take place. Instead, in the
sine-Liouville theory, the violation of the winding number is understood in
a clear way, as due to the explicit dependence on the T-dual direction $%
\tilde{X}$. We will return to this point later. Now, let us discuss the
string interactions in the black hole geometry.
\subsubsection{String amplitudes and correlation functions in the $SL(2,%
\mathbb{R})_{k}$ WZW theory}
The string scattering amplitudes in the 2D black hole background are given
by (the integration over the inserting points of) correlation functions in
the $SL(2,\mathbb{R})_{k}$ WZW theory. The first exact computation of such
WZW three and two-point functions was performed by K. Becker and M. Becker
in Refs. \cite{B,BB2}, and it was subsequently extended and studied in
detail in Refs. \cite{T1}-\cite{T3} by J. Teschner. The interaction
processes of winding string states were studied later in \cite{MO3,GN3},
after J. Maldacena and H. Ooguri proposed the inclusion of spectral flowed
states in the spectrum of the theory \cite{MO1}. Moreover, several
formalisms were employed to study the correlators in this non-compact CFT
\cite{ADS3}-\cite{ADS3ultimo}. One of the most fruitful tools to work out
the functional form of these WZW correlators was the analogy between these
and Liouville correlators \cite{FZ,Teschner,Ponsot,ADS3}. Another useful
approach to compute the exact correlation functions is the free field
representation \cite{B,BB2,HOS,IOS,GN3,GN2,GL}, which for the WZW model
turns out to be similar to what we discussed for the Liouville theory. Let
us briefly describe how this \textquotedblleft free field computation" works
for the case of the two-point function: Consider the correlation functions
of exponential operators $\Phi _{j,m,\overline{m}}^{\omega }=e^{\sqrt{\frac{2%
}{k-2}}\widehat{j}\varphi -i\sqrt{\frac{2}{k}}mX-i\sqrt{\frac{2}{k}}(m+\frac{%
k}{2}\omega )T}$ (with $\widehat{j}=-1-j$) in the theory (\ref{S0})
perturbed by the operator (\ref{graviton}), namely%
\begin{equation}
\mathcal{O}=M\left( \sqrt{\frac{k-2}{2}}\partial \varphi +i\sqrt{\frac{k}{2}}%
\partial X\right) \left( \sqrt{\frac{k-2}{2}}\overline{\partial }\varphi +i%
\sqrt{\frac{k}{2}}\overline{\partial }X\right) e^{-\sqrt{\frac{2}{k-2}}%
\varphi }=M\ \beta \overline{\beta }e^{-\sqrt{\frac{2}{k-2}}\phi }.
\end{equation}%
Then, written in terms of the Wakimoto free fields\footnote{%
Please, do not mistake the Wakimoto field $\gamma $ (which is a local
function on the variable $z$) for the Euler $\gamma $-function introduced in
Eq. (\ref{introducidas}) (which is defined by $\gamma (x)=\Gamma (x)/\Gamma
(1-x)$). That is, the fields $\gamma $ in (\ref{FiRulete}) have to be
distinguished from the function $\gamma $ in (\ref{FiRulete2}). We preferred
to employ the standard notation here.} $\phi $, $\gamma $, and $\beta $,
such correlators read\footnote{%
In order to compare with the original computation in Ref. \cite{B} it is
necessary to consider the Weyl reflection $j\rightarrow -1-j$, which is a
symmetry of the formula for the conformal dimension, actually.}
\begin{equation*}
\left\langle \Phi _{j,m,\overline{m}}^{\omega }(z_{1})\Phi _{j,-m,-\overline{%
m}}^{-\omega }(z_{2})\right\rangle _{WZW}=\Gamma (-s)\delta
(s+2j+1)\prod_{r=2}^{s}\int d^{2}\omega _{r}\left\langle \gamma
^{-1-j-m}(z_{1})\overline{\gamma }^{-1-j-\overline{m}}(z_{1})\right. \times
\end{equation*}%
\begin{eqnarray}
&&\times \gamma ^{-1-j+m}(z_{2})\overline{\gamma }^{-1-j+\overline{m}%
}(z_{2})\beta (w_{1})\overline{\beta }(w_{1})\prod_{r=2}^{s}\left. \beta
(w_{r})\overline{\beta }(w_{r})\right\rangle \times \notag \\
&&\times \left\langle e^{-\sqrt{\frac{2}{k-2}}(j+1)\phi (z_{1})}e^{-\sqrt{%
\frac{2}{k-2}}(j+1)\phi (z_{2})}e^{-\sqrt{\frac{2}{k-2}}\phi (w_{1})}\right.
\prod_{r=2}^{s}\left. e^{-\sqrt{\frac{2}{k-2}}\phi (w_{r})}\right\rangle ,
\label{FiRulete}
\end{eqnarray}%
where the screening inserted at $w_{1}$ is then taken to be fixed at
infinity $w_{1}\rightarrow \infty $, while $z_{1}=0$ and $z_{2}=1$ as usual
(this is analogous to what we did when discussed the case of Liouville
partition function). It is easy to see that this can be solved by using the
(analytic extension of) Dotsenko-Fateev integrals, and one eventually finds%
\footnote{%
For instance, compare with formula (49) in Ref. \cite{GN3}, after the Weyl
reflection.}
\begin{equation*}
\left\langle \Phi _{j,m,\overline{m}}^{\omega }(0)\Phi _{j,-m,-\overline{m}%
}^{-\omega }(1)\right\rangle _{WZW}=-\frac{\Gamma (-j-m)\Gamma (-j+m)}{%
\Gamma (j+1+\overline{m})\Gamma (j+1-\overline{m})}\times .
\end{equation*}%
\begin{equation}
\times \left( -\pi M\gamma \left( \frac{1}{k-2}\right) \right) ^{-1-2j}\frac{%
\gamma (2j+2)}{k-2}\gamma \left( \frac{2j+1}{k-2}\right) , \label{FiRulete2}
\end{equation}%
where the $m$-dependent $\Gamma $-functions stand from the combinatorial
problem of counting the different ways of (Wick) contracting the $\gamma $%
-functions with the $\beta $-functions in (\ref{FiRulete}). Expression (\ref%
{FiRulete2}) is the so called $SL(2,\mathbb{R})_{k}$ WZW reflection
coefficient $\mathcal{R}_{k}(j,m)$ and corresponds to the exact results for
the two-point function. Notice that, in particular, (\ref{FiRulete2})
contains the factor $\gamma \left( \frac{2j+1}{k-2}\right) $ that keeps
track of finite-$k$ effects. Analogously, the expression of the three-point
functions $\left\langle \Phi _{j_{1},m_{1},\overline{m}_{1}}^{\omega
_{1}}(z_{1})\Phi _{j_{2},m_{2},\overline{m}_{2}}^{\omega _{2}}(z_{2})\Phi
_{j_{3},m_{3},\overline{m}_{3}}^{\omega _{3}}(z_{3})\right\rangle _{WZW}$
can be found by these means \cite{BB2}.
Also, some features of the four-point function are known, as the physical
interpretation of its divergences \cite{MO3}, and the crossing symmetry \cite%
{Teschner}. In fact, our understanding of correlation functions in both the
2D black hole and $AdS_{3}$ backgrounds has substantially increased
recently, and we have a relatively satisfactory understanding of these
observables. Nevertheless, some features remain still open questions: One
puzzle is the factorization properties of the generic four-point function
and the closure of the operator product expansion of unitary states.
Addressing these questions would require a deeper understanding of the
analytic structure of the four-point function. The general expression for
the $N$-point functions for $N>3$ is not known; however, a new insight about
its functional form appeared recently due to the discovery of a new relation
between these and analogous correlators in Liouville field theory \cite%
{S,RT,R}. This relation between WZW and Liouville correlators is one of the
key points for what we are going to study in this paper. Let us give some
details about it.
\subsection{A connection between Liouville and WZW correlation functions}
Let us comment on the particular connection that exists between the
correlation functions of the two conformal theories we discussed above;
namely, between Liouville and $SL(2,\mathbb{R})_{k}$ WZW correlation
functions. This relation is a result recently obtained by S. Ribault and J.
Teschner, who have found a direct way of connecting correlators in both $%
SL(2,\mathbb{C})_{k}/SU(2)$ WZW and Liouville conformal theories \cite{RT,R}%
. The formula they proved is an improved version of a previous result
obtained by A. Stoyanovsky some years ago \cite{S}. The Ribault-Teschner
formula (whose more general form was presented by Ribault in Ref. \cite{R})
connects the $N$-point tree-level scattering amplitudes in Euclidean $%
AdS_{3} $ string theory to certain subset of $N+M$-point functions in
Liouville field theory, where the relation between $N$ and $M$ is determined
by the winding number of the interacting strings. Even though this formula
was proven for the case of the Euclidean target space, it is likely that an
analytic continuation of it also holds for the Lorentzian model. The
Ribault-Teschner formula reads as follows: If $\Phi _{j,m,\bar{m}}^{\omega }$
represent the vertex operators in the WZW model, and $V_{\alpha }$ represent
the vertex operators of Liouville theory, then it turns out that
\begin{eqnarray}
\langle \prod_{i=1}^{N}\Phi _{j_{i},m_{i},\bar{m}_{i}}(z_{i})\rangle _{WZW}
&=&N_{k}(j_{1},...j_{N};m_{1},...m_{N})\prod_{r=1}^{M}\int d^{2}w_{r}\
F_{k}(z_{1},...z_{N};w_{1},...w_{M})\times \notag \\
&&\times \langle \prod_{t=1}^{N}V_{\alpha _{t}}(z_{t})\prod_{r=1}^{M}V_{-%
\frac{1}{2b}}(w_{r})\rangle _{S_{L}[\mu ]}\ , \label{rrtt}
\end{eqnarray}%
with the normalization factor given by
\begin{equation}
N_{k}(j_{1},...j_{N};m_{1},...m_{N})=\frac{2\pi ^{3-2N}b}{M!\ c_{k}^{M+2}}%
(\pi ^{2}\mu b^{-2})^{-s}\prod_{i=1}^{N}\frac{c_{k}\ \Gamma (-m_{i}-j_{i})}{%
\Gamma (1+j_{i}+\bar{m}_{i})} \label{N}
\end{equation}%
and the $z$-dependent function given by
\begin{eqnarray}
F_{k}(z_{1},...z_{N};w_{1},...w_{M}) &=&\frac{\prod_{1\leq
r<l}^{N}|z_{r}-z_{l}|^{k-2(m_{r}+m_{l}+\omega _{r}\omega _{l}k/2+\omega
_{l}m_{r}+\omega _{r}m_{l})}}{\prod_{1<r<l}^{M}|w_{r}-w_{l}|^{-k}%
\prod_{t=1}^{N}\prod_{r=1}^{M}|w_{r}-z_{t}|^{k-2m_{t}}}\times \notag \\
&&\times \frac{\prod_{1\leq r<l}^{N}(\bar{z}_{r}-\bar{z}_{l})^{m_{r}+m_{l}-%
\bar{m}_{r}-\bar{m}_{l}+\omega _{l}(m_{r}-\bar{m}_{r})+\omega _{r}(m_{l}-%
\bar{m}_{l})}}{\prod_{1<r<l}^{M}(\bar{w}_{r}-\bar{z}_{t})^{m_{t}-\bar{m}_{t}}%
}, \label{FF}
\end{eqnarray}%
and where the parameter $b$ of the Liouville theory is related to the
Kac-Moody level $k$ through $b^{-2}=k-2$. The quantum numbers of the states
of both conformal models are related ones to each others through the simple
relation $\alpha _{i}=bj_{i}+b+b^{-2}/2\ ,\ $with$\ i=1,2,...N.$ The factor $%
c_{k}$ in (\ref{N}) is a $k$-dependent ($j$-independent) normalization; see
\cite{R}. Furthermore, the following constraints also hold $m_{1}+...m_{N}=%
\bar{m}_{1}+...\bar{m}_{N}=\frac{k}{2}(N-M-2),$ $\omega _{1}+..\omega
_{N}=M+2-N$, $s=-b^{-1}(\alpha _{1}+...\alpha _{N})+b^{-2}\frac{M}{2}%
+1+b^{-2},$ where $s$ refers to the amount of screening operators $V_{b}=\mu
e^{\sqrt{2}b\varphi }$ to be included in the Liouville correlators in order
to get a non vanishing result, as in (\ref{pupo2}). Also notice that the
Liouville correlator in the r.h.s. of (\ref{rrtt}) contains $M$ degenerate
fields $V_{-1/2b}$ (i.e. states that contain null descendents in the
modulo), which have conformal dimension strictly lower than zero for
positive $b$. So that the formula (\ref{rrtt}) relates $N$-point functions
in the WZW theory to $M+N$-point functions in Liouville field theory.
Applications of (\ref{rrtt}) were discussed in \cite%
{Apl1,Apl2,YoYu,Takayanagi}, and ulterior generalizations were presented in
\cite{disc1,disc2,HikidaSchomerus1}. The way of proving (\ref{rrtt}) was
making use of the relation existing between solutions to the BPZ\
differential equations (satisfied by the Liouville correlation functions
involved in (\ref{rt}), \cite{BPZ}) and the generalized KZ differential
equation (satisfied by the WZW correlators \cite{KZ,R}). This remarkable
trick allowed to demonstrate the map between correlators in both theories
even though one does not know the generic form of such observables in any of
the two cases.
The dictionary given by the formula (\ref{rrtt}) will play a crucial role in
proving the correspondence between the 2D black hole and the flat tachyonic
background we are interested in. Conversely, our result can be seen as a
free field realization of the Ribault-Teschner formula (\ref{rrtt}). In
fact, in section 4 we will describe how (\ref{rrtt}) can be thought of as an
identity between the $SL(2,\mathbb{R})_{k}$ WZW theory and a CFT of the form
$Liouville\times U(1)\times \mathbb{R}$, for which the $U(1)$ dependences of
the correlators factorize out yielding the piece $%
F_{k}(z_{1},...z_{N};w_{1},...w_{M})$ in (\ref{FF}). In this realization,
the operators $V_{-1/2b}$ are seen as $M$ additional screening currents. The
details of this can be found in subsection 4.3. First, let us discuss the
FZZ duality.
\section{The FZZ dual for the 2D black hole}
In this section we will study the (standard) FZZ dual for the
two-dimensional black hole, namely the sine-Liouville field theory. In
particular, we will discuss it as an example of tachyon background for the
2D string theory.
\subsection{Tachyon-like backgrounds in 2D string theory}
Let us consider the non-linear $\sigma $-model on a generic curved target
space of metric $G_{\mu \nu },$ and in presence of both dilatonic $\Phi $
and tachyonic $\mathcal{T}$ backgrounds\footnote{%
As it is known, in two dimensions the expression \textquotedblleft
tachyonic" has to be understood just formally, since the tachyon is massless
in $D=2$; see \cite{fin} for an illustrative example.}. If we supplement the
worldsheet action (\ref{Polyakov2}) with the tachyonic term, the $\sigma $%
-model takes the form%
\begin{equation}
S_{P}=\frac{1}{4\pi }\int d^{2}z\left( G_{\mu \nu }(X)\partial X^{\mu }%
\overline{\partial }X^{\nu }+R\Phi (X)+\mathcal{T}(X)\right) ,
\label{Polyakov}
\end{equation}%
where, as before, $\mu ,\nu =\{1,2\};$ and where we adopt the convention $%
X^{1}\equiv X$ and $X^{2}\equiv \varphi $ representing the two coordinates
that parameterize the target space. Thus, conformal invariance at quantum
level demands the vanishing of the $\beta $-functions for the action (\ref%
{Polyakov}); and for the tachyon field, the one-loop linearized $\beta $%
-function reads \cite{Polchinski}%
\begin{equation}
\beta ^{\mathcal{T}}=-\nabla _{\mu }\nabla ^{\mu }\mathcal{T}+2\nabla _{\mu
}\Phi \nabla ^{\mu }\mathcal{T}-2\mathcal{T}=0, \label{labeta}
\end{equation}%
where higher powers of $\mathcal{T}$ were neglected. This equation, together
with the one-loop $\beta $-functions for the metric and the dilaton, admit
solutions of the form%
\begin{equation}
G_{\mu \nu }=\delta _{\mu \nu },\qquad \Phi (\varphi )=\frac{Q}{\sqrt{2}}%
\varphi ,\qquad \mathcal{T}(X,\varphi )=\sum_{n}\lambda _{n}\,e^{\sqrt{2}%
a_{n}\varphi +i\sqrt{2}b_{n}X}, \label{solutione}
\end{equation}%
with $a_{n}(Q-a_{n})+b_{n}^{2}=1$, $Q=2$. Here, coefficients $\lambda _{n}$
are real numbers that can be regarded as the Fourier modes of the tachyon
potential. The tachyon momenta $b_{n}$ are chosen to be consistent with the
compactification conditions for the $X$ direction; in particular, here we
will consider $b_{n}=n\sqrt{k}/2$, and the tachyon potential will be of the
Toda-like form%
\begin{equation}
\mathcal{T}(X,\varphi )=\sum_{n=-\infty }^{\infty }\lambda _{n}e^{\sqrt{2}%
(1\pm \sqrt{k}|n|/2)\varphi +i\sqrt{k/2}nX}; \label{torbellino2}
\end{equation}%
see (\ref{cos}) and (\ref{O}) below. Background (\ref{solutione}) is the
type of configuration we will deal with. A particular case of interest is
the sine-Liouville theory, which we discuss below.
\subsection{Sine-Liouville theory and the FZZ conjecture}
\subsubsection{Sine-Liouville field theory}
Sine-Liouville theory is a particular case of tachyon-like background, and,
according to the FZZ conjecture, this is dual to the 2D string theory on the
black hole spacetime. Sine-Liouville theory corresponds to perturb the free
action $S_{0}$ with the operator%
\begin{equation}
\mathcal{O}_{\widetilde{\lambda }_{-1}=\widetilde{\lambda }_{+1}=\lambda
}=4\lambda e^{-\sqrt{\frac{k-2}{2}}\varphi }\cos \left( \sqrt{k/2}\widetilde{%
X}\right) , \label{cos}
\end{equation}%
which is convenient to write as%
\begin{equation}
\mathcal{O}_{\widetilde{\lambda }_{-1}=\widetilde{\lambda }_{+1}=\lambda
}=2\lambda e^{-\sqrt{\frac{k-2}{2}}\varphi +i\sqrt{\frac{k}{2}}\widetilde{X}%
}+2\lambda e^{-\sqrt{\frac{k-2}{2}}\varphi -i\sqrt{\frac{k}{2}}\widetilde{X}%
}, \label{cos2}
\end{equation}%
where $\widetilde{X}=X_{L}(z)-X_{R}(\overline{z})$. The interaction term (%
\ref{cos}) resembles both the sine-Gordon and the Liouville field theories,
and this is the reason of the name of \textquotedblleft sine-Liouville".
Actually, this theory corresponds to the sine-Gordon model coupled to
two-dimensional gravity.
The sine-Liouville interaction (\ref{cos}) can be thought of as a particular
case of the action (\ref{S}) if the $X$ field is replace by its T-dual $%
\widetilde{X}$. It would correspond to the coupling $\widetilde{\lambda }%
_{n} $ $=$ $\lambda \left( \delta _{n+1}+\delta _{n-1}\right) $. This field
theory describes the phase of vortex condensation in the 2D string theory.
Unlike the euclidean black hole geometry, whose topology is $\mathbb{R}^{2}$%
, sine-Liouville theory is an interacting CFT formulated on the topology $%
\mathbb{R}\times S^{1}$. The distinct topologies arise because the angular
direction of the (simple connected) cigar is the one of the duality
transformation. Notice also that sine-Liouville interaction term is not
bounded from below, and this is ultimately related to the $\mathbb{R}^{2}$
topology of the cigar too.
Sine-Liouville theory and its relation to the 2D black hole have been
extensively studied in the last six years, and, as we mentioned, this has
led to the formulation of the matrix model for the black hole \cite{KKK}.
The matrix model turned out to be a very important tool for studying black
hole physics in string theory; in particular, it permitted to address the
question about the black hole formation in string theory \cite%
{MaldacenaStrominger}. Matrix model formulation also enabled to study the
integrability of the theory from a different point of view\footnote{%
The black hole turns out to be dual to the perturbed $c=1$ theory, and, on
the other hand, the $c=1$ theory perturbed by the vortex or tachyon
potential turns out to be integrable with the integrable structure described
in terms of the Toda hierarchy \cite{KKK}.}, and we emphasize that all this
was possible because of FZZ duality.
\subsubsection{The Fateev-Zamolodchikov-Zamolodchikov conjecture}
FZZ duality is a strong-weak duality. The semiclassical limit of
sine-Liouville theory corresponds to the limit $k\rightarrow 2,$ where the
black hole is highly curved. Conversely, the semiclassical limit of the
black hole theory corresponds to the large $k$ regime where the
sine-Liouville wave function is strongly suppressed in the $\varphi $
direction. Perhaps, the correct way of thinking FZZ duality is that the full
theory is actually described by both the WZW and sine-Liouville models, and
each of them dominates the dynamics of the theory in a different regime
(where the corresponding action is reliable as a good approximation).
However, it is worth mentioning that both theories have control on the
observables beyond the regime in which one would naively expect so. For
instance, even though one would expect the black hole $\sigma $-model action
to describe the theory only in the large $k$ regime, it turns out that the
Coulomb gas computation of correlation function using the screening operator
behaving like $\sim e^{-\sqrt{\frac{2}{k-2}}\varphi }$ do reproduce the
exact result, including finite-$k$ effects\footnote{%
We have exemplified this in the previous section by computing the two-point
function.} \cite{B,BB2,GN3}. Besides, the same feature occurs for the
computation in sine-Liouville theory \cite{FH}. This sourprising feature is
due to the analytic extension of the Coulomb gas type expressions, which is
powerful enough to reconstruct the exact expressions of the correlators.
This is precisely what permitted to perform consistency checks of the
conjecture. The interplay between perturbative poles and $k$-dependent poles
in correlation functions of both models was first discussed in \cite{KKK},
where it was shown that the poles of bulk amplitudes in sine-Liouville
precisely reproduce non-perturbative (finite-$k$ effects) poles of WZW
correlators\footnote{%
However, again, it is important to emphasize that such finite-$k$ poles can
be directly obtained by considering the perturbative action of the WZW
theory \cite{B}. For instance, in Ref. \cite{GN3} it was shown that the
computation in the WZW model involving operators behaving like $\sim e^{-%
\sqrt{2(k-2)}\varphi }$ exactly agree with those originally computed in \cite%
{B}, even though the dependence on $k$ is the opposite to the one appearing
in (\ref{graviton}).}.
Strong-weak FZZ correspondence turns out to be a very important piece for
our understanding of black hole physics in string theory. So, let us briefly
discuss how such correspondence works operatively. First, we present the
main ingredients: The sine-Liouville vertex operators we have to consider
are those of the form\footnote{%
Here, we are not explicitly writing the antiholomorphic contribution $e^{i%
\sqrt{2/k}\bar{m}X}$ for short; it has to be understood in all the formulae
below. Besides, let us notice that vertex (\ref{caritaredonda}) would
receive an extra piece $e^{i\sqrt{\frac{2}{k}}(m+k\omega /2)T+i\sqrt{\frac{2%
}{k}}(\bar{m}+k\omega /2)T}$ in the case that the theory one considers is
the product between the sine-Liouville action and the time direction $T$.}%
\begin{equation}
\mathcal{T}_{j,m,\overline{m}}=e^{\sqrt{\frac{2}{k-2}}j\varphi +i\sqrt{\frac{%
2}{k}}mX}. \label{caritaredonda}
\end{equation}%
The spectrum of the theory contains states obeying $m-\overline{m}=k\omega $
and $m+\overline{m}=n,$ with integers $n$ and $\omega $. Operators (\ref%
{caritaredonda}) have conformal dimension
\begin{equation*}
h=-\frac{j(j+1)}{k-2}+\frac{m^{2}}{k},
\end{equation*}%
and the coincidence with (\ref{harriba}) shows the convenience of this
notation. A crucial observation is that the sine-Liouville theory presents
symmetry under the $\widehat{sl}(2)_{k}$ affine algebra, and this can be
realized by free field techniques by defining \cite{Satohviejo}%
\begin{equation}
J^{\pm }(z)=\left( -i\sqrt{\frac{k}{2}}\partial X\pm \sqrt{\frac{k-2}{2}}%
\partial \varphi \right) e^{\mp i\sqrt{\frac{2}{k}}(T+X)},\qquad J^{3}(z)=i%
\sqrt{\frac{k}{2}}\partial T. \label{currents}
\end{equation}%
These currents satisfy the OPE
\begin{equation*}
J^{3}(z)J^{\pm }(w)=\pm \frac{1}{(z-w)}J^{\pm }(w)+...,\qquad
J^{3}(z)J^{3}(w)=-\frac{k/2}{(z-w)^{2}}+...,
\end{equation*}%
\begin{equation*}
J^{-}(z)J^{+}(w)=\frac{k}{(z-w)^{2}}-\frac{2}{(z-w)}J^{3}(w)+...,
\end{equation*}%
and thus realize (\ref{elalgebra}) by means of the relation $J_{n}^{a}=\frac{%
1}{2\pi i}\oint dz\ z^{-1-n}J^{a}(z)$. It is possible to verify that
sine-Liouville interaction commutes with these currents, namely that the
OPEs yield regular terms. This matching of symmetries is an important
necessary condition for the equivalence to the WZW theory. The next step
would be that of proposing a dictionary between observables: According to
FZZ\ prescription, operators (\ref{caritaredonda}) are associated in
one-to-one correspondence to those operators that expand $SL(2,\mathbb{R}%
)_{k}$ representations in the theory on the coset \cite{KKK}, namely%
\begin{equation*}
\mathcal{T}_{j,m,\overline{m}}\leftrightarrow \Phi _{j,m,\overline{m}},
\end{equation*}%
where $\Phi _{j,m,\overline{m}}$ are the vertex operators on the coset
theory $SL(2,\mathbb{R})_{k}/U(1)$, defined through their relation to the $%
SL(2,\mathbb{R})_{k}$ vertex, namely $\Phi _{j,m,\overline{m}}^{\omega
}=\Phi _{j,m,\overline{m}}\times e^{i\sqrt{\frac{2}{k}}(m+\frac{k}{2}\omega
)T}$. Then, once the sine-Liouville operators were introduced, one can
undertake the task of performing perturbative checks of the duality. To do
this, one should compare the analytic structure of correlation functions in
both conformal models; but, first, one has to know how to compute such
quantities. So, let us review the computation of correlators for the
sine-Liouville field theory.
\subsubsection{Correlation functions in sine-Liouville theory}
Correlation functions in the sine-Liouville theory are assumed to reproduce
the analytic structure of the WZW analogues. The former can be computed by
standard Coulomb gas techniques, and the precise prescription was studied in
Ref. \cite{FB,FH}. In the case $N\leq 3$ these correlators were explicitly
integrated. In general, $N$-point sine-Liouville amplitudes are expected to
exhibit poles at $s_{-}+s_{+}=\frac{2}{k-2}\left(
1+\sum_{i=1}^{N}j_{i}\right) $, where the residues turn out to be expressed
in terms of multiple integrals over the whole complex plane. These read
\begin{eqnarray}
A_{(j_{1},...j_{N}|z_{1},...z_{N})}^{sine-L} &=&\frac{\lambda ^{\frac{2}{k-2}%
(j_{1}+...j_{N}+1)}}{s_{-}!s_{+}!}\prod_{r=1}^{s_{+}}\int
d^{2}v_{r}\prod_{t=1}^{s_{-}}\int d^{2}v_{t}\left\langle \mathcal{T}%
_{j_{1},m_{1},\overline{m}_{1}}(z_{1})\mathcal{T}_{j_{2},m_{2},\overline{m}%
_{2}}(z_{2})...\right. \notag \\
&&...\mathcal{T}_{j_{N},m_{N},\overline{m}_{N}}(z_{N})\prod_{r=1}^{s_{+}}%
\mathcal{T}_{1-\frac{k}{2},\frac{k}{2},-\frac{k}{2}}(u_{r})%
\prod_{t=1}^{s_{-}}\left. \mathcal{T}_{1-\frac{k}{2},-\frac{k}{2},\frac{k}{2}%
}(v_{t})\right\rangle _{S_{[\lambda =0]}} \label{UUU}
\end{eqnarray}%
with $S_{[\lambda =0]}=S_{0}$, yielding%
\begin{equation*}
A_{(j_{1},...j_{N}|z_{1},...z_{N})}^{sine-L}=\frac{\lambda ^{\frac{2}{k-2}%
(j_{1}+...j_{N}+1)}}{\Gamma (s_{-}+1)\Gamma (s_{+}+1)}\prod_{a<b}^{N-1,N}%
\left| z_{a}-z_{b}\right| ^{-\frac{4j_{a}j_{b}}{k-2}}\left(
z_{a}-z_{b}\right) ^{\frac{2}{k}m_{a}m_{b}}\left( \bar{z}_{a}-\bar{z}%
_{b}\right) ^{\frac{2}{k}\bar{m}_{a}\bar{m}_{b}}\times
\end{equation*}%
\begin{eqnarray}
&&\ \times \prod_{r=1}^{s_{+}}\int d^{2}u_{r}\prod_{l=1}^{s_{-}}\int
d^{2}v_{l}\prod_{r<t}^{s_{+}-1,s_{+}}\left| u_{r}-u_{t}\right|
^{2}\prod_{l<t}^{s_{-}-1,s_{-}}\left| v_{t}-v_{s}\right|
^{2}\prod_{l=1}^{s_{-}}\prod_{r=1}^{s_{+}}\left| v_{l}-u_{r}\right|
^{2-2k}\times \notag \\
&&\ \times \prod_{a=1}^{N}\prod_{r=1}^{s_{+}}\left| z_{a}-u_{r}\right|
^{2(j_{a}+m_{a})}\left( \bar{z}_{a}-\bar{u}_{r}\right) ^{m_{a}-\overline{m}%
_{a}}\prod_{b=1}^{N}\prod_{l=1}^{s_{-}}\left| z_{b}-v_{l}\right|
^{2(j_{b}-m_{b})}\left( \bar{z}_{b}-\bar{v}_{l}\right) ^{m_{b}-\overline{m}%
_{b}}, \label{21}
\end{eqnarray}%
that follows from the Wick contraction of operators $\mathcal{T}_{j,m,%
\overline{m}}$ and $\mathcal{T}_{1-\frac{k}{2},\pm \frac{k}{2},\pm \frac{k}{2%
}}$. The poles that correspond to bulk amplitudes in sine-Liouville theory
can be shown to arise though the integration over the zero-mode of the field
$\varphi $ \cite{diFK}. In the case $N=3$ the pole structure of (\ref{21})
was shown to agree with that of the black hole theory, for which the finite-$%
k$ poles represent non-perturbative worldsheet effects. This non-trivial
matching between analytic structures was one of the strongest evidence in
favor of the FZZ conjecture at perturbative level \cite{KKK,FH,GL}. An
important piece of information is encoded in the fact that the
sine-Liouville correlators scale as $\lambda ^{\frac{2}{k-2}%
(j_{1}+j_{2}+...j_{N}+1)}$ while the black hole correlators scale as $M^{1+%
\widehat{j}_{1}+\widehat{j}_{2}+...\widehat{j}_{N}}$. In particular, it
tells us something about how the sine-Liouville correlators behave in the
large $k$ limit.
In Ref. \cite{FH} the authors translated the integrals $\prod_{r,l}\int
d^{2}u_{r}\int d^{2}v_{l}$ in (\ref{UUU}) into the product of contour
integrals. In this way, the integral representation above results described
by standard techniques developed in the context of rational conformal field
theory. Such techniques were used to evaluate the correlators to give a
formula for the contour integrals. The first step in the calculation is to
decompose the $u_{r}$ complex variables (resp. $v_{l}$) into two independent
real parameters (\textit{i.e.} the real and imaginary part of $u_{r}$) which
take values in the whole real line. Secondly, a Wick rotation for the
imaginary part of $(u_{r})$ has to be performed in order to introduce a
shifting parameter $\varepsilon $ which is ultimately used to elude the
poles in $z_{a}$. Then, the contours are taken in such a way that the poles
at $v_{r}\rightarrow z_{a}$ are avoided by considering the alternative order
with respect to this inserting points. The details of the prescription can
be found in the section 3 of Ref. \cite{FH}; see also Ref. \cite{FH2}.
\subsubsection{On the violation of the winding number conservation}
Now, let us return to the feature of the violation of winding conservation.
From the point of view of the sine-Liouville field theory the violation of
the total winding number in (\ref{UUU}) is given by $\sum_{a=1}^{N=3}\omega
_{a}=k^{-1}\sum_{a=1}^{N=3}(m_{a}+\overline{m}_{a})=s_{-}-s_{+}$ and comes
from the insertion of a different amount of screening operators $s_{-\text{ }%
}$ and $s_{+}$. It can be proven that for the three-point functions, the
winding can be violated up to $|\sum_{a=1}^{N=3}\omega _{a}|\leq N-2=1$ and,
presumably, this is the same for generic $N$. The key point for obtaining
such a constraint is noticing that the integrand that arises in the Coulomb
gas-like prescription contains contributions of the form%
\begin{equation*}
\int d^{2}v_{r}d^{2}v_{t}|v_{r}-v_{t}|^{2}...
\end{equation*}%
that come from the product expansion of two operators $\mathcal{T}_{1-\frac{k%
}{2},\pm \frac{k}{2},\pm \frac{k}{2}}$ inserted at the points $v_{r}$ and $%
v_{t}$ for $0\leq r,t\leq s_{-}$ (and the same for the points $u_{l}$ with $%
0\leq l\leq s_{+}$), and where the dots \textquotedblleft $...$%
\textquotedblright\ stand for \textquotedblleft other
dependences\textquotedblright\ on $v_{r}$ and $u_{l}$. As explained in \cite%
{FH}, the integral vanishes for certain alignments of contours\ due to the
fact that the exponent of $|v_{r}-v_{t}|$ is $+2$. Conversely, in the case
where such exponent is generic enough (let us say $2\rho $, following the
notation of \cite{FH}), the integral has a phase ambiguity due to the
multi-valuedness of $|v_{r}-v_{t}|^{2\rho }$ in the integrand. Then, those
integrals containing two contours of $v_{r}$ and $v_{t}$ just next to each
other vanish, and this precisely happens for all the contributions of those
correlators satisfying $|s_{+}-s_{-}|=|\sum_{a=1}^{N}\omega _{a}|>N-2$. This
led Fukuda and Hosomichi to prove that, for the three-point function, there
are only three terms that contribute: one with $\sum_{a=1}^{3}\omega _{a}=1$%
, a second with $\sum_{a=1}^{3}\omega _{a}=-1,$ and the conserving one, $%
\sum_{a=1}^{3}\omega _{a}=0$. A similar feature is exhibited in the
\textquotedblleft twisted" sine-Liouville model we will consider in the next
section, see \cite{0511252}.
On the other hand, one can wonder about how the violation of the winding
number conservation is seen from the point of view of the black hole theory,
where, unlike what happens in sine-Liouville theory, the action does not
seem to break the winding conservation. That it, even though the geometric
reason why the winding is not conserved in the cigar is quite clear, it is
not obvious how to understand such non-conservation in the calculation of
correlators. The answer to this puzzle was first given in Ref. \cite{FZZ},
and subsequently reviewed in \cite{MO3}. In fact, the computation of the
winding violating correlators in the WZW theory is far from being as simple
as in the case of sine-Liouville theory. In the WZW theory such computation
requires the insertion of one additional operator for each unit in which the
winding number is being violated. This additional operator is the often
called \textquotedblleft spectral flow operator" $\Phi _{-\frac{k}{2},\pm
\frac{k}{2},\pm \frac{k}{2}}^{1}$, and this is an auxiliary operator that
plays the role of changing (in one unit) the winding number $\omega $ of a
given $SL(2,\mathbb{R})$-state involved in the correlator. The spectral flow
operator corresponds to a conjugate representation of the identity operator,
so it has conformal dimension zero. For instance, the three-point scattering
amplitudes (violating winding in one unit) in the 2D black hole would be
actually given in terms of a four-point correlation functions involving a
fourth dimension-zero operator $\Phi _{-\frac{k}{2},\pm \frac{k}{2},\pm
\frac{k}{2}}^{1}$, after extracting the appropriate divergent factor coming
from the coincidence limit of spectral flow operator and the evaluation at $%
m=\bar{m}=\pm k/2;$ see \cite{MO3,YoYu} for the details.
Regarding the computation of correlation functions where the winding number
conservation is violated, let us mention that the most simple way of
computing such observables is that of making use of the twisted dual model
we will introduce in the next section. Perhaps this is the most useful
application of it, and we will comment on this feature later. Now, let us to
introduce the new dual model for the 2D black hole; which we will call the
\textquotedblleft twisted model" because it involves momentum modes of the
higher sector $n=2$.
\section{A twisted dual for the 2D black hole}
As said, we will now discuss an alternative dual description of string
theory in the 2D black hole ($\times $ $time$). First, we will introduce a
family of perturbations of the linear dilaton background (\ref{S0}) and, in
particular, we will introduce the perturbation that corresponds to the
twisted version of the sine-Liouville model which we want to relate to the
black hole $\sigma $-model. After doing this, we will make the precise
statement of such duality and show how to prove it by using the formula (\ref%
{rrtt}).
\subsection{Perturbations of higher winding and momentum modes}
\subsubsection{Momentum mode perturbations}
Let us begin by considering a rather general deformation of the theory (\ref%
{S0}), including higher modes of momentum and winding. The interaction term
in (\ref{S}) is given by the operator%
\begin{equation}
\mathcal{O}_{\lambda _{n}}=\sum_{n=-\infty }^{\infty }\lambda _{n}e^{-\frac{%
\alpha _{n}}{\sqrt{2}}\varphi +in\sqrt{\frac{k}{2}}X}, \label{O}
\end{equation}%
for which the condition $\lambda _{n}=\lambda _{-n}$ is required to be real.
Each term in this sum represents a marginal deformation of the linear
dilaton theory (\ref{S0}), and if the T-dual direction $\widetilde{X}$ is
considered instead of $X$ then this operator describes the sine-Liouville
field theory in the particular case $\widetilde{\lambda }_{n=1}=\widetilde{%
\lambda }_{n=-1}\neq 0$. The case $n=0$ is also included in the sum. In that
case the exponent is given by $\alpha _{n=0}=\frac{1+\sqrt{9-4k}}{\sqrt{k-2}}
$, so that it is real (represents a \textquotedblleft Liouville-like wall
potential") only for values $k\leq 9/4$. The value that saturates this
bound, $k=9/4,$ precisely corresponds to the black hole background, i.e. for
which the central charge of the coset $SL(2,\mathbb{R})_{k}/U(1)$ itself
turns out to be $26$. At $k=9/4$ the interaction term for $n=0$ turns out to
be $e^{-\sqrt{2}\varphi }$, i.e. the cosmological constant. We have to point
out that for $k=9/4$ the interaction (\ref{O}) agrees with the
two-dimensional string theory in an arbitrary winding background studied by
V. Kazakov, I. Kostov and D. Kutasov in\footnote{%
See formula (3.19) in Ref. \cite{KKK} and notice that the notation there
relates to the one employed here by $\varphi =\sqrt{2}\phi .$} Ref. \cite%
{KKK}. That is, for $k=9/4$ operator (\ref{O}) reads\footnote{%
Actually, the contribution $n=0$ at the point $k=9/4$ leads to the operator $%
\varphi e^{-\sqrt{2}\varphi }$ instead of $e^{-\sqrt{2}\varphi }$. This
comes from the fact that there are two possible values for $\alpha
_{n=0}=(1\pm \sqrt{9-4k})/\sqrt{k-2}$ which coincide (a resonance) in the
limit $k\rightarrow 9/4$ producing a degenerangy analogous to the case of
the Liouville cosmological term in the $b\rightarrow 1$ limit \cite{MTV}.
Also notice the difference between the signs of the exponents of (\ref%
{torbellino2}) and (\ref{torbellino1}); which is due to the sign of the
background charge in each case.}%
\begin{equation}
\mathcal{O}_{\lambda _{n}}=\lambda _{0}\varphi e^{-\sqrt{2}\varphi
}+\sum_{n\neq 0}\lambda _{n}e^{(|n|R-\sqrt{2})\varphi +inRX}.
\label{torbellino1}
\end{equation}%
with $R=\sqrt{k/2}=3/2\sqrt{2}$. The matrix model incorporating these
perturbations is constructed by implementing a deformed version of the Haar
measure on the $U(N)$ group manifold. The details of the matrix model
construction can be found in \cite{KKK}; here we will not discuss the
subject beyond the scope of the continuous limit.
\subsubsection{Adding vortex type perturbations}
It is instructive to explore other deformations. For instance, let us
consider the more general family%
\begin{equation}
\mathcal{O}_{\lambda _{n},\widetilde{\lambda }_{n}}=\sum_{n\neq 0}e^{-\frac{%
\alpha _{n}^{(-)}}{\sqrt{2}}\varphi }\left( \lambda _{n}^{(-)}e^{in\sqrt{%
\frac{k}{2}}X}+\widetilde{\lambda }_{n}^{(-)}e^{in\sqrt{\frac{k}{2}}%
\widetilde{X}}\right) +\sum_{n\neq 0}e^{-\frac{\alpha _{n}^{(+)}}{\sqrt{2}}%
\varphi }\left( \lambda _{n}^{(+)}e^{in\sqrt{\frac{k}{2}}X}+\widetilde{%
\lambda }_{n}^{(+)}e^{in\sqrt{\frac{k}{2}}\widetilde{X}}\right) ,
\label{pupo}
\end{equation}%
with $\alpha _{n}^{(\pm )}=\widehat{Q}(1\mp \sqrt{1+(kn^{2}-4)(k-2)}),$ so
that (\ref{O}) corresponds to the branch $\alpha _{n}^{(-)}$. In the 2D
black hole, couplings $\lambda _{n}$ turn the momentum modes on, while $%
\widetilde{\lambda }_{n}$ are the couplings of vortex operators turning
winding modes on, instead. Notice that perturbation (\ref{pupo})\ not only
includes the usual sine-Liouville interaction $\alpha _{\pm 1}^{(-)}$, but
also includes the dual sine-Liouville interaction introduced by A.
Mukherjee, S. Mukhi and A. Pakman in Ref. \cite{MMP} when the modes $\alpha
_{\pm 1}^{(+)}$ are considered\footnote{%
Notice that the notation in \cite{MMP} relates with ours here by $\varphi =-%
\sqrt{2}\phi .$}. Operators of the branches $\alpha _{n}^{(\pm )}$ have a
large $k$ behavior $\sim e^{\pm \sqrt{\frac{k}{2}}|n|\varphi }$, so that
only those of the branch $\alpha _{n}^{(-)}$ decrease for large $\varphi $
(where the theory is weakly coupled) in the black hole semiclassical limit $%
k\rightarrow \infty $. For our purpose, the interesting operators are those
having momentum $n=2$. In particular, here we are mainly interested in the
case $\alpha _{2}^{(-)}=\frac{2}{\sqrt{k-2}}(k-1)$; this is the one that
will enable us to present an alternative dual description for the 2D black
hole. For notational convenience, let us point out that operators with
momentum $\alpha _{n}^{(\pm )}$ can be written as
\begin{equation*}
\mathcal{T}_{j_{n}^{\pm },m_{n},m_{n}}=e^{-\frac{\alpha _{n}^{(\pm )}}{\sqrt{%
2}}\varphi +in\sqrt{\frac{k}{2}}X},\quad \mathcal{T}_{j_{n}^{\pm
},m_{n},-m_{n}}=e^{-\frac{\alpha _{n}^{(\pm )}}{\sqrt{2}}\varphi +in\sqrt{%
\frac{k}{2}}\widetilde{X}},
\end{equation*}%
so carrying momentum $j_{n}^{\pm }=-\frac{1}{2}\pm \frac{1}{2}\sqrt{%
1+(k-2)(kn^{2}-4)}$ and momentum or winding number $m_{n}\pm \bar{m}_{n}=kn$%
. Here we will discuss how certain correlation functions of the model
defined by the action (\ref{S}), when the momentum modes $n=2$ (represented
by operators $\mathcal{T}_{1-k,k,k}$) and $n=1$ (respectively represented by
$\mathcal{T}_{1-\frac{k}{2},\frac{k}{2},\frac{k}{2}}$) are turned on,
precisely agree with the correlation functions of the $SL(2,\mathbb{R})_{k}$
WZW model. So, now we are ready to make the main statement about the
correspondence and describe the precise prescription for computing the
correlators.
\subsection{Statement of the correspondence}
\subsubsection{Preliminary: some definitions}
We will discuss how the particular deformation of the family (\ref{O}) given
by $\lambda _{n}=\mu \delta _{n-2}+\lambda \delta _{n-1}$ is dual to the 2D
black hole, in a similar way as the sine-Liouville model is so. That is, by
\textquotedblleft dual" we mean that there exists a direct correspondence
between correlation functions of both CFTs at the level of the sphere
topology. Then, the interaction operator we will consider is
\begin{equation}
\mathcal{O}_{\lambda _{1}=\lambda ,\lambda _{2}=\mu }=\lambda e^{-\sqrt{%
\frac{k-2}{2}}\varphi +i\sqrt{\frac{k}{2}}X}+\mu e^{-\sqrt{\frac{2}{k-2}}%
(k-1)\varphi +i\sqrt{2k}X}, \label{Uf}
\end{equation}%
where the scaling relation between the coupling constants $\mu $ and $%
\lambda $ goes like $\lambda ^{2}=a_{k}\mu $ with a $k$-dependent
proportionality factor $a_{k}$ that will be specified below. In the large $k$
limit this operator behaves like
\begin{equation*}
\mathcal{O}_{\lambda _{1}=\lambda ,\lambda _{2}=\mu }\sim \lambda e^{-\sqrt{%
k/2}(\varphi -iX)}+\frac{1}{a_{k}}(\lambda e^{-\sqrt{k/2}(\varphi
-iX)})^{2}\sim \lambda e^{-\sqrt{k/2}(\varphi -iX)}+\frac{1}{a_{k}}\left(
\mathcal{O}_{\lambda _{1}=\lambda ,\lambda _{2}=0}\right) ^{2}.
\end{equation*}%
Also notice that
\begin{equation}
\mathcal{T}_{1-\frac{k}{2},\frac{k}{2},\frac{k}{2}}=e^{-\sqrt{\frac{k-2}{2}}%
\varphi +i\sqrt{\frac{k}{2}}X},\qquad \mathcal{T}_{1-k,k,k}=e^{-\sqrt{\frac{2%
}{k-2}}(k-1)\varphi +i\sqrt{2k}X}, \label{Considering}
\end{equation}%
so that
\begin{equation}
{\mathcal{O}}_{\lambda _{1}=\lambda ,\lambda _{2}=\mu }=\lambda \mathcal{T}%
_{1-\frac{k}{2},\frac{k}{2},\frac{k}{2}}+\frac{\lambda ^{2}}{a_{k}}\mathcal{T%
}_{1-k,k,k}.
\end{equation}%
Taking into account (\ref{caritaredonda}), we notice that the perturbation $%
\mathcal{T}_{1-\frac{k}{2},\frac{k}{2},\frac{k}{2}}$ corresponds to an
operator that satisfies the unitarity bound $1-k<2j<-1$ only for $k>3$,
while the operator $\mathcal{T}_{1-k,k,k}$ does not satisfy that bound for
any value of $k$ grater than $2$. With operators (\ref{Considering}), we
define the following correlation functions
\begin{equation*}
\left\langle \widetilde{\mathcal{T}}_{j_{1},m_{1},\overline{m}_{1}}(z_{1})...%
\widetilde{\mathcal{T}}_{j_{N},m_{N},\overline{m}_{N}}(z_{N})\right\rangle
_{S_{[\lambda ]}}=\frac{\Gamma (-s)}{b}\delta \left(
s+1+j_{1}+...j_{N}+M+(N-2-M)k/2\right) \times
\end{equation*}%
\begin{eqnarray}
&&\times \frac{1}{M!c_{k}^{M}}\delta \left( m_{1}+\overline{m}_{1}+...m_{N}+%
\overline{m}_{N}+k(M+2-N)\right) \delta _{m_{1}-\overline{m}_{1}+...+m_{N}-%
\overline{m}_{N}}\times \notag \\
&&\times \prod_{r=1}^{M}\int d^{2}w_{r}\prod_{t=1}^{s}\int
d^{2}v_{t}\left\langle \widetilde{\mathcal{T}}_{j_{1},m_{1},\overline{m}%
_{1}}(z_{1})...\widetilde{\mathcal{T}}_{j_{N},m_{N},\overline{m}%
_{N}}(z_{N})\prod_{r=1}^{M}\mathcal{T}_{1-\frac{k}{2},\frac{k}{2},\frac{k}{2}%
}(w_{r})\prod_{t=1}^{s}\mathcal{T}_{1-k,k,k}(v_{t})\right\rangle
_{S_{[\lambda =0]}}, \label{yesto}
\end{eqnarray}%
where $b^{-2}=k-2$, and where we fixed $a_{k}=c_{k}^{-2}$. The value of $%
\lambda $ was also fixed to a specific value. The vertex operators $%
\widetilde{\mathcal{T}}_{j,m,\overline{m}}$ appearing in this expression are
related to those introduced in (\ref{caritaredonda}) through
\begin{equation}
\widetilde{\mathcal{T}}_{j,m,\overline{m}}=\frac{c_{k}\Gamma (-m-j)}{\pi
^{2}\Gamma (1+j+\overline{m})}\mathcal{T}_{\widetilde{j},\widetilde{m},%
\widetilde{\overline{m}}}, \label{conjugate}
\end{equation}%
with
\begin{equation}
\widetilde{j}=-j(k-1)-m(k-2)-k/2,\quad \widetilde{m}=jk+m(k-1)+k/2,
\label{j}
\end{equation}%
and analogously\footnote{%
Notice that it means that now anti-holomorphic contribution comes with a
different index $\widetilde{\overline{j}}$.} for $\widetilde{\overline{m}}.$
Again, notice that in (\ref{yesto}) we already fixed the value of $\lambda $
to a specific value $c_{k},$ which is a $k$-dependent numerical factor that
is ultimately related to the one appearing in (\ref{rrtt}). Realization (\ref%
{yesto}) is similar to (\ref{despues}) in Liouville field theory and defines
the correlators we want to consider here. The overall factor $\frac{\Gamma
(-s)}{bM!c_{k}^{M}}$ and the $\delta $-functions are understood once the
prescription for inserting the screenings when computing correlators is
specified. These factors come from the integration over the zero-modes of
the fields. Besides, the condition $\sum_{i=1}^{N}(m_{i}-\overline{m}_{i})=0$
also holds. The conditions imposed by these $\delta $-functions are
equivalent to demanding $\sum_{i=1}^{N}(\widetilde{m}_{i}-\widetilde{%
\overline{m}}_{i})=0$ and $\sum_{i=1}^{N}(\widetilde{m}_{i}+\widetilde{%
\overline{m}}_{i})+k(M+2s)=0,$ being $M+s$ the total amount of screenings to
be inserted. We discuss our prescription for the insertion of the screening
charges below.
\subsubsection{A Coulomb gas-like prescription}
In this realization, the interaction operators $\mathcal{T}_{1-\frac{k}{2},%
\frac{k}{2},\frac{k}{2}}$ and $\mathcal{T}_{1-k,k,k}$ act in (\ref{yesto})
as screening operators, analogously to the computation of Liouville
correlation functions. Because of the $\delta $-functions appearing in (\ref%
{yesto}), the amount of these screening operators to be inserted turns out
to be given by%
\begin{equation}
s=1-N-\sum_{i=1}^{N}j_{i}+\frac{k-2}{2}(M+2-N),\quad
M=N-2+\sum_{i=1}^{N}\omega _{i}. \label{pupodos}
\end{equation}%
However, the statement is not complete unless one specifies how conditions (%
\ref{pupodos}) are to be satisfied. This is because in principle there is no
a unique way of choosing $s$ and $M$ in order to obey the first of the
charge symmetry conditions in (\ref{pupodos}). Thus, let us be precise about
the prescription to compute the r.h.s. of (\ref{yesto}): The prescription
adopted here is that $M$ represents a positive integer number of operators $%
\mathcal{T}_{1-\frac{k}{2},\frac{k}{2},\frac{k}{2}}$ to be inserted, and $M$
is actually fixed by the winding numbers $\omega _{i}$ of the $N$
interacting states. On the other hand, the amount $s$ of operators $\mathcal{%
T}_{1-k,k,k}$ is then appropriately chosen to make the r.h.s. of (\ref{yesto}%
) nonzero; and this is supposed to be the case even if (\ref{yesto}) has to
be analytically extended to non-integer values of $s$ (we already discussed
correlation functions with a non-integer amount of screening operators in
section 2). This is the set of correlation functions we will consider here;
and we emphasize that the equivalence between CFTs we will state in the
following subsection 4.2.3 has to be understood as holding only if the
prescription employed to compute the observables is the one we just gave in
this subsection. Now, once we precisely defined the correlators (\ref{yesto}%
), let us present the main assertion.
\subsubsection{Correspondence between correlation functions}
The statement is that the following identity between correlation functions
holds%
\begin{equation}
\left\langle \Phi _{j_{1},m_{1},\overline{m}_{1}}^{\omega
_{1}}(z_{1})...\Phi _{j_{N},m_{N},\overline{m}_{N}}^{\omega
_{N}}(z_{N})\right\rangle _{WZW}=\widehat{c}_{k}^{-2}\left\langle \widetilde{%
\mathcal{T}}_{j_{1},m_{1},\overline{m}_{1}}(z_{1})...\widetilde{\mathcal{T}}%
_{j_{N},m_{N},\overline{m}_{N}}(z_{N})\right\rangle _{S} \label{duality}
\end{equation}%
where $\widehat{c}_{k}^{2}$ is a numerical factor (independent of $N$) that
will be specified below, and where the correlators in the r.h.s. are given
by (\ref{yesto}) computed with the prescription specified above. The average
in the r.h.s. of (\ref{duality}) is taken w.r.t. the action (\ref{S}) with $%
\lambda _{n}=0$ for $n\neq 1$ and $n\neq 2$. Recall%
\begin{equation}
\widetilde{\mathcal{T}}_{j,m,\overline{m}}=\frac{c_{k}\Gamma (-m-j)}{\pi
^{2}\Gamma (1+j+\overline{m})}\mathcal{T}_{\widetilde{j},\widetilde{m},%
\widetilde{\overline{m}}}.
\end{equation}
Relation (\ref{duality}) reads
\begin{equation*}
\left\langle \Phi _{j_{1},m_{1},\overline{m}_{1}}^{\omega
_{1}}(z_{1})...\Phi _{j_{N},m_{N},\overline{m}_{N}}^{\omega
_{N}}(z_{N})\right\rangle _{WZW}=\frac{\Gamma (-s)}{\widehat{c}%
_{k}^{2}bM!c_{k}^{M-N}}\prod_{i=1}^{N}\frac{\Gamma (-m_{i}-j_{i})}{\Gamma
(1+j_{i}+\overline{m}_{i})}\delta _{m_{1}-\bar{m}_{1}+...+m_{N}-\bar{m}%
_{N}}\times
\end{equation*}%
\begin{equation*}
\times \delta \left( \sum_{i=1}^{N}(m_{i}+\overline{m}_{i})+(M+2-N)k\right)
\delta \left( s+1+\sum_{i=1}^{N}j_{i}+M+(N-2-M)k/2\right) \times
\end{equation*}%
\begin{equation}
\times \prod_{r=1}^{M}\int d^{2}w_{r}\prod_{t=1}^{s}\int
d^{2}v_{t}\left\langle \prod_{i=1}^{N}\mathcal{T}_{\widetilde{j}_{i},%
\widetilde{m}_{i},\widetilde{\overline{m}}_{i}}(z_{i})\prod_{r=1}^{M}%
\mathcal{T}_{1-\frac{k}{2},\frac{k}{2},\frac{k}{2}}(w_{r})\prod_{t=1}^{s}%
\mathcal{T}_{1-k,k,k}(v_{t})\right\rangle _{S_{[\lambda =0]}}.
\label{duality2}
\end{equation}%
This is the main result here. Equation (\ref{duality}) gives a realization
of any $N$-point function of the $SL(2,\mathbb{R})_{k}/U(1)$ WZW correlators
in terms of the analogous observables in the theory (\ref{S}) if the
perturbation is taken to be $\lambda _{n}=\mu \delta _{n-2}+\lambda \delta
_{n-1}$ (and choosing $\mu $ and $\lambda $ appropriately). The perturbation
involved in this realization corresponds to the operators $\mathcal{T}_{1-%
\frac{k}{2},\frac{k}{2},\frac{k}{2}}$ and $\mathcal{T}_{1-k,k,k},$ having
momentum modes $n=1$ and $n=2,$ respectively. This is different from the
standard FZZ duality, which corresponds to $\lambda _{n}=\widetilde{\lambda }%
\delta _{n-1}+\widetilde{\lambda }\delta _{n+1}$, instead. The perturbation
of the linear dilaton theory (\ref{S0}) with operators of different winding
numbers was also considered in Ref. \cite{MMP}, where it was suggested that
the multiply-wound tachyon operators are linked to the called higher-spin
black holes. It would be very interesting to understand the relation with
the realization of \cite{MMP} better and confirm such picture.
\subsubsection{Conjugate representations and spectral flow}
An interesting feature of the statement made above is that the r.h.s. of (%
\ref{duality}) involves \textquotedblleft conjugate operators" instead of
the ones introduced in (\ref{caritaredonda}). Ones are related to each
others by (\ref{j}), which represents a symmetry of the formula for the
conformal dimension (\ref{h}) (and not only for it, actually). Notice that,
in particular, we have
\begin{equation}
\widetilde{\mathcal{T}}_{1-\frac{k}{2},\frac{k}{2},\frac{k}{2}}\sim \mathcal{%
T}_{1-k,k,k},
\end{equation}%
and operators $\widetilde{\mathcal{T}}_{j,m,\overline{m}}$ and $\mathcal{T}%
_{j,m,\overline{m}}$ have exactly the same conformal dimension. Moreover,
the automorphism can be extended in order to be valid for the theory
formulated on the product $SL(2,\mathbb{R})_{k}/U(1)\times \mathbb{R} $ by
including the new winding number
\begin{equation}
\widetilde{\omega }=-\omega -1-2(j+m). \label{ww}
\end{equation}%
In such case, the operators $\widetilde{\mathcal{T}}_{j,m,\overline{m}}$ and
$\mathcal{T}_{j,m,\overline{m}}$, when both are extended by including the
time-like factor $e^{i\sqrt{\frac{2}{k}}(m+\frac{k}{2}\omega )T}$, satisfy%
\begin{equation*}
-\frac{j(j+1)}{k-2}-m\omega -\frac{k}{4}\omega ^{2}=-\frac{\widetilde{j}(%
\widetilde{j}+1)}{k-2}-\widetilde{m}\widetilde{\omega }-\frac{k}{4}%
\widetilde{\omega }^{2}
\end{equation*}%
and also have the same momentum under the $J^{3}$ current of the WZW model,
namely%
\begin{equation*}
m+\frac{k}{2}\omega =\widetilde{m}+\frac{k}{2}\widetilde{\omega }.
\end{equation*}
To understand the relation between $\widetilde{\mathcal{T}}_{j,m,\overline{m}%
}$ and $\mathcal{T}_{j,m,\overline{m}}$ in the algebraic framework, let us
comment on the $SL(2,\mathbb{R})_{k}$ representations again: As we said, the
principal continuous series $\mathcal{C}_{\lambda }^{\alpha ,\omega }$
correspond to $j=-\frac{1}{2}+i\lambda $ with $\lambda \in \mathbb{R}$ and
thus, through (\ref{j}), this results in the new values $j=-\frac{1}{2}+i%
\tilde{\lambda}-m(k-2)$ with $\tilde{\lambda}\in \mathbb{R}$, which only
belong to the continuous series if $m=0$. Besides, if we perform the change (%
\ref{j}) for generic $\tilde{\lambda}$, then $m$ turns out to be a non-real
number after of that. Then, the relation between $j,m$ and $\widetilde{j},%
\widetilde{m}$ can not be thought of as a simple identification between
states of different continuous representations but it does correspond to
different free field realizations (at least in what respects to the
continuous series $\mathcal{C}_{\lambda }^{\alpha ,\omega }$). On the other
hand, concerning the discrete representations, it is worth mentioning that
the quantity $j+m$ remains invariant under the involution (\ref{j}); though
it is not the case for the difference $j-m$ that, instead, remains invariant
under a $\mathbb{Z}_{2}$ reflected version of (\ref{j}). Then, unlike the
states of continuous representations, the transformation defined by (\ref{j}%
) and (\ref{ww}) is closed among certain subset of states of discrete
representations. This is because such transformation maps states of the
discrete series with $2(j+m)\in \mathbb{Z}$ among themselves. In particular,
the case $m+j=0$ corresponds to the well known identification between
discrete series $\mathcal{D}_{j}^{\pm ,\omega =0}$ and $\mathcal{D}%
_{-k/2-j}^{\mp ,\omega =\pm 1}$ since in that case (\ref{j}) and (\ref{ww})
reduce to $j\rightarrow -k/2-j,\ m\rightarrow k/2-m=k/2+j,\ \omega
\rightarrow -1-\omega $ (i.e. it includes\footnote{%
More precisely, the identification between Kac-Moody primary highest-weight
(lowets-weight) states that is induced by the sector $\omega =1$ of the
spectral flow coincides with a particular case of the identification given
by the symmetry (\ref{j}), (\ref{ww}).} such spectral flow transformation as
a particular case). Also notice that the condition $m-\bar{m} \in \mathbb{Z}$
is not preserved for generic values of $k$. The fixed points of (\ref{j})
describe a line in the space of representations, parameterized by $%
j+1/2=-m(k-2)/k$; in particular, a fixed point for generic $k$ corresponds
to $j=-1/2$ and $m=0 $, for which (\ref{ww}) reduces to $\omega \rightarrow
-\omega $. Also, in the tensionless limit $k\rightarrow 2$ transformation (%
\ref{j}) agrees with the Weyl reflection $j\rightarrow -1-j$. The relation
between quantum numbers manifested by (\ref{j}) permits to visualize the
relation between the vertex considered in our construction and those of
reference \cite{HHS}, and we emphasize that these correspond to two
different (alterative) representations of the vertex operators. The relation
between both is a kind of \textquotedblleft twisting\textquotedblright\ and
is presumably related to the representations studied in \cite{GL} for the
WZW theory. Certainly, the conjugate representations of the $SL(2,\mathbb{R}%
)_{k}$ vertex algebra do resemble the twisting (\ref{conjugate}), and it
seems to be a nice connection between correlators (\ref{yesto}) and the free
field representation studied in \cite{GN2,GN3,GL,HHS}. Conjugate
representations transform in a particular way under the Kac-Moody affine $%
\widehat{sl}(2)_{k} $ algebra, and are analogous to those introduced by
Dotsenko for the case of $\widehat{su}(2)_{k}$ in Refs. \cite{dot1,dot2}.
For the non-compact WZW these were first introduced in Ref. \cite{FZZ} to
describe winding violating amplitudes. Here, through Eq. (\ref{conjugate}),
these appear again (although we are referring to them as \textquotedblleft
twisted") within a similar context.
\subsubsection{Remark on the $\hat{sl}(2)_{k}$ affine symmetry}
To understand these twisted sectors better, let us make some remarks on the $%
sl(2)_{k}$ symmetry of the action (\ref{S}) when it is perturbed as we did
so. We are claiming (and we will prove in the following subsection) that
correlators (\ref{yesto}) do transform appropriately under the $sl(2)_{k}$
symmetry in order to describe the WZW correlators. However, even though one
can prove this a posteriori, the question arises as to why does it happen if
the operators $\mathcal{T}_{1-k,k,k}$ do not seem to commute with the $%
\widehat{sl}(2)_{k}$ currents though. To be precise, even though one
eventually proves that the free field representation employed here
transforms properly by construction (\textit{e.g.} it reproduces solutions
of the KZ equation), it is also true that this is not obvious because the
screening operators do not seem to commute with the free field
representation of the $sl(2)_{k}$ current algebra (\ref{currents}) as one
could naively expect. The explanation of this puzzling feature is that the
vertex operators $\widetilde{\mathcal{T}}_{j,m,\overline{m}}$ do not satisfy
the usual OPE with the $sl(2)_{k}$ currents either, and thus this restores
the symmetry. To see this explicitly, one has to consider the generators of
the affine algebra (\ref{currents}) and verify that those currents do not
have regular OPE with the operators $\mathcal{T}_{1-k,k,k}$. The remarkable
point is that this is precisely what makes the $SL(2,\mathbb{R})_{k}$ to be
restored: While these currents do not present regular OPE with the operator $%
\mathcal{T}_{1-k,k,k}$, these do not satisfy the usual OPE with the twisted
vertex operators $\widetilde{\mathcal{T}}_{j,m,\overline{m}}$ either; and
both facts seem to combine in such a way that render the set of observables (%
\ref{yesto}) $SL(2,\mathbb{R})_{k}$ invariant\footnote{%
G.G. thanks Yu Nakayama for addressing his attention to this remarkable
point.}. This depends on the presence of the normalization factor $\frac{%
\Gamma (-j-m)}{\Gamma (j+\overline{m}+1)}$ in (\ref{conjugate}), since its
presence is not innocuous for the transformation properties under the
generators $J_{n}^{\pm }$. This feature makes out of the correspondence (\ref%
{duality}) a non-trivial assertion, indeed.
\subsection{Proving the correspondence}
Here we will show that the formula (\ref{duality}) immediately follows from
the relation (\ref{rrtt}) between WZW and Liouville correlators. In order to
be concise, here we address the proof in two steps: First, we rewrite the
correlators (\ref{yesto}) and the operators involved there in a convenient
way\footnote{%
In Ref. \cite{Fateev} similar techniques were used to prove a different
(though related) correspondence: the one between Liouville and sine-Liouvile
correlation functions.}. The second step will be using the formula (\ref%
{rrtt}) to make contact with the WZW correlators.
\subsubsection{Step 1: Rewriting the correlators}
As we said, the proof of formula (\ref{duality}) directly follows from the
Stoyanovsky-Ribault-Teschner map (\ref{rrtt}) we discussed in section 2. In
order to make the proof simpler, let us begin by redefining fields as follows
\begin{equation}
\varphi (z)=(1-k)\widehat{\varphi }(z)+i\sqrt{k(k-2)}\widehat{X}(z),\quad
X(z)=i\sqrt{k(k-2)}\widehat{\varphi }(z)+(k-1)\widehat{X}(z). \label{newc}
\end{equation}%
That is
\begin{equation}
-\sqrt{\frac{k-2}{2}}\widehat{\varphi }+i\sqrt{\frac{k}{2}}\widehat{X}=-%
\sqrt{\frac{k-2}{2}}\varphi +i\sqrt{\frac{k}{2}}X \label{Argento2}
\end{equation}%
and%
\begin{equation}
-\frac{1}{\sqrt{k-2}}\varphi =\frac{k-1}{\sqrt{k-2}}\widehat{\varphi }-i%
\sqrt{k}\widehat{X}. \label{Argento1}
\end{equation}%
Also notice that it implies
\begin{equation}
\partial \varphi \overline{\partial }\varphi +\partial X\overline{\partial }%
X=\partial \widehat{\varphi }\overline{\partial }\widehat{\varphi }+\partial
\widehat{X}\overline{\partial }\widehat{X}; \label{Argento3}
\end{equation}%
so that the free field correlators are $\left\langle \widehat{\varphi }%
(z_{1})\widehat{\varphi }(z_{2})\right\rangle =\left\langle \widehat{X}%
(z_{1})\widehat{X}(z_{2})\right\rangle =-2\log |z_{1}-z_{2}|$. One can
wonder whether the field redefinition (\ref{newc}) is well defined or not
since it is a complex transformation and then both $\widehat{\varphi }$ and $%
\widehat{X}$ would acquire a non-real part. However, the correct way of
thinking this transformation is first considering a Wick rotation of the $X$
direction and then, after the transformation, Wick rotate $\widehat{X}$
back. It turns out to be a perfectly defined transformation for the Wick
rotated fields $iX$ and $i\widehat{X}$, which can be seen as real time-like
bosons. Transformation (\ref{newc}) is a $U(1,1)$ transformation, with
determinant $-1$. In fact, one can also turn it into a $SU(2)$-rotation by
supplementing (\ref{newc}) with a reflection $X\rightarrow -X$ (that is also
a symmetry of the theory). In that case, it is clear that (\ref{Argento1})
and (\ref{Argento3}) remain invariant, while (\ref{Argento2}) changes its
sign in the second term of the r.h.s.. So, in principle, it would be
possible to consider a $U(1,1)\times \overline{SU}(2)$ chiral transformation
(for the holomorphic part and the anti-holomorphic part, respectively) in
order to transform dependences on $X$ into dependences on $\widetilde{X}$.
In terms of these new fields $\widehat{\varphi }$ and $\widehat{X}$ one
finds that the linear dilaton theory defined by the action $S_{0}-\frac{1}{%
4\pi }\int d^{2}z\ \partial T\bar{\partial}T$ takes the form%
\begin{equation}
S=\frac{1}{4\pi }\int d^{2}z\left( -\partial T\bar{\partial}T+\partial
\widehat{X}\bar{\partial}\widehat{X}+\partial \widehat{\varphi }\bar{\partial%
}\widehat{\varphi }+\frac{1}{2\sqrt{2}}QR\widehat{\varphi }-\frac{i}{2}\sqrt{%
\frac{k}{2}}R\widehat{X}\right) \label{SA}
\end{equation}%
with $Q=b+b^{-1}$, and $b^{-2}=k-2$, so that $Q=\frac{k-1}{\sqrt{(k-2)}}%
=b+b^{-1}$ (cf. Eq. (\ref{T}) in section 2). That is, the background charge
operator $e^{-\sqrt{\frac{2}{k-2}}\varphi }$ transform through (\ref{j})
into a new background charge operator $e^{\sqrt{2}Q\varphi -i\sqrt{2k}X}$,
where $\tilde{j}=-1$, $\tilde{m}=0$ while $j=k-1$, $m=-k$. Consequently, the
stress-tensor reads \cite{0511252}
\begin{equation}
T(z)=\frac{1}{2}(\partial T)^{2}-\frac{1}{2}(\partial \widehat{X})^{2}-i%
\sqrt{\frac{k}{2}}\partial ^{2}\widehat{X}-\frac{1}{2}(\partial \widehat{%
\varphi })^{2}+\frac{k-1}{\sqrt{2(k-2)}}\partial ^{2}\widehat{\varphi },
\end{equation}%
and the dilaton now acquires a linear dependence on both directions $%
\widehat{X}$ and $\widehat{\varphi }$. This kind of CFT, representing $c<1$
matter coupled to perturbed 2D gravity, was recently discussed in Refs. \cite%
{Petkova,Petkova2,ZamolodchikovYang,BelavinZamolodchikov,Z}. It is possible
to verify that this stress-tensor leads to the appropriated central charge $%
c=3+\frac{6}{k-2},$ as it is of course expected. On the other hand, in terms
of the new fields the interaction (perturbation) term $\lambda \mathcal{T}%
_{1-\frac{k}{2},\frac{k}{2},\frac{k}{2}}(w_{r})+\mu \mathcal{T}%
_{1-k,k,k}(v_{t})$ takes the form%
\begin{equation}
\mathcal{O}_{\lambda _{1},\lambda _{2}}=c_{k}^{-1}e^{-\sqrt{\frac{k-2}{2}}%
\widehat{\varphi }+i\sqrt{\frac{k}{2}}\widehat{X}}+e^{\sqrt{\frac{2}{k-2}}%
\widehat{\varphi }}, \label{SB}
\end{equation}%
where we already fixed the scale $\mu $ to a specific value by shifting the
zero-mode of the Liouville field $\widehat{\varphi }$, and we also specified
the numerical factor $c_{k}$ as being the ratio between the couplings $\mu $
and $\lambda ^{2}$ in (\ref{Uf}). Notice that in these coordinates the
second term in the perturbation $\mathcal{O}_{\lambda _{2},\lambda _{1}}$
turns to be diagonalized (no dependence on $\widehat{X}$ arise there) and
agrees with the Liouville cosmological constant $\mu e^{\sqrt{2}b\widehat{%
\varphi }}$. On the other hand, the first term in (\ref{SB}) still has the
form of one of the two exponentials that form the cosine interaction (\ref%
{cos2})\ in sine-Liouville theory; this is due to (\ref{Argento2}). On the
other hand, the vertex operators in terms of $\widehat{X}$ and $\widehat{%
\varphi }$ take the form\footnote{%
Again, we are not explicitly writing the antiholomorphic contribution $e^{i%
\sqrt{\frac{2}{k}}\overline{m}\widehat{X}}$ for short. It has to be
understood in what follows.}%
\begin{equation}
\widetilde{\mathcal{T}}_{j,m,\overline{m}}=\frac{c_{k}\Gamma (-m-j)}{\pi
^{2}\Gamma (1+\overline{m}+j)}V_{\alpha }\times e^{i\sqrt{\frac{2}{k}}(m-%
\frac{k}{2})\widehat{X}+i\sqrt{\frac{2}{k}}(m+\frac{k}{2}\omega )T},
\label{oh}
\end{equation}%
with the Liouville field $V_{\alpha }=e^{\sqrt{2}\alpha \widehat{\varphi }}$%
, with $\alpha =bj+b+b^{-2}/2=b(j+k/2).$ Expanding the correlators we get
\begin{equation*}
\frac{1}{\hat{c}_{k}^{2}}\left\langle \widetilde{\mathcal{T}}_{j_{1},m_{1},%
\overline{m}_{1}}(z_{1})...\widetilde{\mathcal{T}}_{j_{N},m_{N},\overline{m}%
_{N}}(z_{N})\right\rangle _{S_{[\lambda ,\mu ]}}=\frac{\Gamma (-s)}{%
bM!c_{k}^{M}\widehat{c}_{k}^{2}}\delta _{m_{1}-\overline{m}_{1}+..m_{N}-%
\overline{m}_{N}}\times
\end{equation*}%
\begin{eqnarray*}
&&\times \delta \left( s+1+j_{1}+...j_{N}+M+(N-2-M)k/2\right) \delta \left(
m_{1}+\overline{m}_{1}+..m_{N}+\overline{m}_{N}+(N-2-M)k\right) \times \\
&&\times \prod_{r=1}^{M}\int d^{2}w_{r}\prod_{t=1}^{s}\int
d^{2}v_{t}\left\langle \prod_{i=1}^{N}\widetilde{\mathcal{T}}_{j_{i},m_{i},%
\overline{m}_{i}}(z_{i})\times \prod_{r=1}^{M}\mathcal{T}_{1-\frac{k}{2},%
\frac{k}{2},\frac{k}{2}}(w_{r})\prod_{t=1}^{s}\mathcal{T}_{1-k,k,k}(v_{t})%
\right\rangle _{S_{[\lambda =0,\mu =0]}}
\end{eqnarray*}%
and this can be written as
\begin{eqnarray*}
&=&\frac{1}{k\hat{c}_{k}^{2}}\delta \left( \omega _{1}+...\omega
_{N}+N-2-M\right) \prod_{a=1}^{N}\frac{c_{k}\ \Gamma (m_{a}-j_{a})}{\pi
^{2}\Gamma (1+j_{a}-\bar{m}_{a})}\left\langle \prod_{t=1}^{N}e^{i\sqrt{\frac{%
2}{k}}(m_{t}+\frac{k}{2}\omega _{t})T(z_{t})}\right\rangle _{S_{[\lambda
=0]}}\times \\
&&\times \frac{1}{M!c_{k}^{M}}\delta _{m_{1}-\overline{m}_{1}+...m_{N}-%
\overline{m}_{N}}\prod_{r=1}^{M}\int d^{2}w_{r}\ \left\langle
\prod_{t=1}^{N}e^{i\sqrt{\frac{2}{k}}(m_{t}-\frac{k}{2})\widehat{X}%
(z_{t})}\prod_{r=1}^{M}e^{i\sqrt{\frac{k}{2}}\widehat{X}(w_{r})}\right%
\rangle _{S_{[\lambda =0]}}\times
\end{eqnarray*}%
\begin{equation}
\times \frac{\Gamma (-s)}{b}\delta \left( s-1-\frac{2+M}{2b^{2}}+\frac{%
\alpha _{1}+...\alpha _{N}}{b}\right) \prod_{t=1}^{s}\int d^{2}v_{t}\
\left\langle \prod_{t=1}^{N}V_{\alpha _{t}}(z_{t})\prod_{r=1}^{M}V_{-\frac{1%
}{2b}}(w_{r})\prod_{t=1}^{s}V_{b}(v_{t})\right\rangle _{S_{L[\mu =0]}},
\label{dalegas}
\end{equation}%
where $S_{[\lambda =0]}$ here refers to the unperturbed action%
\begin{equation}
S_{[\lambda =0]}=\frac{1}{4\pi }\int d^{2}z\left( -\partial T\overline{%
\partial }T+\partial \widehat{X}\overline{\partial }\widehat{X}-\frac{i}{2}%
\sqrt{\frac{k}{2}}R\widehat{X}\right) . \label{uhyeste}
\end{equation}%
The third line in (\ref{dalegas}) turns out to be a $N+M$-point correlation
function in Liouville field theory (see Eq. (\ref{despues}) in section 2),
defined by the Liouville action%
\begin{equation*}
S_{L}[\mu ]=\frac{1}{4\pi }\int d^{2}z\left( \partial \widehat{\varphi }%
\overline{\partial }\widehat{\varphi }+\frac{1}{2\sqrt{2}}QR\widehat{\varphi
}+2\pi \mu e^{\sqrt{2}b\widehat{\varphi }}\right) ,
\end{equation*}%
with $Q=b+b^{-1}$. Recall that the parameter $b$ of the Liouville theory is
related to the Kac-Moody level $k$ through $b^{-2}=k-2$, while the quantum
numbers $\alpha _{i}$ are defined in terms of $j_{i}$ by $\alpha
_{i}=bj_{i}+b+b^{-2}/2,$ for$\ i=1,2,...N.$ After the Wick contraction, we
find
\begin{equation*}
\frac{\Gamma (-s)}{bM!c_{k}^{M}\widehat{c}_{k}^{2}}\prod_{r=1}^{M}\int
d^{2}w_{r}\prod_{t=1}^{s}\int d^{2}v_{t}\left\langle \widetilde{\mathcal{T}}%
_{j_{1},m_{1},\overline{m}_{1}}(z_{1})...\widetilde{\mathcal{T}}%
_{j_{N},m_{N},\overline{m}_{N}}(z_{N})\prod_{r=1}^{M}\mathcal{T}_{1-\frac{k}{%
2},\frac{k}{2},\frac{k}{2}}(w_{r})\prod_{t=1}^{s}\mathcal{T}%
_{1-k,k,k}(v_{t})\right\rangle _{S[\lambda =0]}=
\end{equation*}%
\begin{equation}
=N_{k}(j_{1},...j_{N};m_{1},...m_{N})\prod_{r=1}^{M}\int d^{2}w_{r}\
F_{k}(z_{1},...z_{N};w_{1},...w_{M})\langle \prod_{t=1}^{N}V_{\alpha
_{t}}(z_{t})\prod_{r=1}^{M}V_{-\frac{1}{2b}}(w_{r})\rangle _{S_{L}[\mu ]}\ ,
\label{rt}
\end{equation}%
where $\mu =b^{2}/\pi ^{2}$ and where, after fixing the value $\hat{c}%
_{k}^{2}=2c_{k}^{2}/b\pi ^{3}$, the normalization factor is
\begin{equation}
N_{k}(j_{1},...j_{N};m_{1},...m_{N})=\frac{2\pi ^{3-2N}b}{M!\ c_{k}^{M+2-N}}%
\prod_{i=1}^{N}\frac{\Gamma (-m_{i}-j_{i})}{\Gamma (1+j_{i}+\bar{m}_{i})}
\end{equation}%
and the function $F_{k}(z_{1},...z_{N};w_{1},...w_{M})$ is given by
\begin{eqnarray}
F_{k}(z_{1},...z_{N};w_{1},...w_{M}) &=&\frac{\prod_{1\leq
r<l}^{N}|z_{r}-z_{l}|^{k-2(m_{r}+m_{l}+\omega _{r}\omega _{l}k/2+\omega
_{l}m_{r}+\omega _{r}m_{l})}}{\prod_{1<r<l}^{M}|w_{r}-w_{l}|^{-k}%
\prod_{t=1}^{N}\prod_{r=1}^{M}|w_{r}-z_{t}|^{k-2m_{t}}}\times \notag \\
&&\times \frac{\prod_{1\leq r<l}^{N}(\bar{z}_{r}-\bar{z}_{l})^{m_{r}+m_{l}-%
\bar{m}_{r}-\bar{m}_{l}+\omega _{l}(m_{r}-\bar{m}_{r})+\omega _{r}(m_{l}-%
\bar{m}_{l})}}{\prod_{1<r<l}^{M}(\bar{w}_{r}-\bar{z}_{t})^{m_{t}-\bar{m}_{t}}%
}. \label{F}
\end{eqnarray}%
Remarkably, this has reproduced the r.h.s. of formula (\ref{rrtt}); cf. Eq. (%
\ref{FF}). Notice that the exponents of the differences $|z_{r}-z_{l}|$ in (%
\ref{F}) do depend on whether the theory is being formulated on the coset $%
SL(2,\mathbb{R})_{k}/U(1)$ or on its product with the time $T$. The vertex
operators (\ref{oh}) are the only fields that carry the $T$-dependences, so
that the rest of the OPEs are not affected.
According to (\ref{pupodos}), the amount of perturbations involved in (\ref%
{rt}) is constrained by the following conditions
\begin{equation}
\sum_{i=1}^{N}m_{i}=\sum_{i=1}^{N}\bar{m}_{i}=\frac{k}{2}(N-M-2),\quad
s=-b^{-1}\sum_{i=1}^{N}\alpha _{i}+b^{-2}\frac{M}{2}+1+b^{-2}\ , \label{ese}
\end{equation}%
where the number $s$ corresponds to the amount of screening operators $%
V_{b}=\mu e^{\sqrt{2}b\widehat{\varphi }}$ to be included in Liouville
correlators. The whole amount of vertex operators involved in the r.h.s. of (%
\ref{rt}) is then $N+M+s,$ and is related to the winding numbers of the
strings through
\begin{equation}
\sum_{i=1}^{N}\omega _{i}=M+2-N\geq -|N-2|. \label{selection}
\end{equation}%
Notice that the value of $\sum_{i=1}^{N}\omega _{i}$ can not be lower than $%
2-N$ if $M$ represents a positive integer number. Allowing negative values
of winding numbers requires the insertion of screening operators with $n=-1$
in addition to those of $n=+1$. $M$ runs between $0 $ and $N-2,$ which
implies that, according to the prescription given in subsection 4.2.2, the
absolute value of the violation of winding number conservation could not
exceed $N-2$. This is an interesting feature, and it is not trivial at all
to fully understand this bound. We can say here that it is closely related
to the $\widehat{sl}(2)_{k}$ symmetry of the theory, and we refer to the
appendix D of Ref. \cite{MO3} for a nice explanation. It is worth mentioning
that the selection rule for winding number violation (\ref{selection}) was
already part of the original FZZ conjecture \cite{FZZ}. A short note about
this rule can be also found in Ref. \cite{Fateev3}.
The formula (\ref{rt})-(\ref{F}), with the conditions (\ref{ese}), is the
main ingredient for proving (\ref{duality}). It only remains to argue that
the r.h.s. of (\ref{rt}) actually represents a WZW correlator; and,
actually, it can be already observed since it directly follows from the
formula (\ref{rrtt}). Indeed, the r.h.s. of (\ref{rt}) agrees with the
l.h.s. of (\ref{rrtt}) and this would complete the proof of (\ref{duality}).
Let us conclude the job by further commenting on it.
\subsubsection{Step 2: Realizing the Stoyanovsky-Ribault-Teschner map}
As we just mentioned, the last step in proving (\ref{duality}) is showing
that the r.h.s. of Eq. (\ref{rt}) precisely describes a WZW $N$-point
function, and, actually, this immediately follows from the main result of
Ref. \cite{R} (see formula (3.29) there, which we wrote in the Eq. (\ref%
{rrtt}) in section 2). Hence, we have managed to rewrote our result (\ref%
{duality}) in such a way that its proof turns out to be a direct consequence
of the observation made by S. Ribault in his paper \cite{R}, where he showed
that the l.h.s. of Eq. (\ref{rt}) is precisely equal to a correlation
function in the $SL(2,\mathbb{R})_{k}$ WZW model. Our achievement was to
prove that the auxiliary overall function $%
F_{k}(z_{1},...z_{N};w_{1},...w_{M})$ standing in the Ribault-Teschner
formula can be also thought of as coming from the correlation functions of
the linear dilaton CFT realized by the field $\widehat{X}$; namely%
\begin{equation}
F_{k}(z_{1},...z_{N};w_{1},...w_{M})=\left\langle \prod_{t=1}^{N}e^{i\sqrt{%
\frac{2}{k}}\left( (m_{t}-\frac{k}{2})\widehat{X}(z_{t})+(m_{t}+\frac{k}{2}%
\omega _{t})T(z_{t})\right) }\prod_{r=1}^{M}e^{i\sqrt{\frac{k}{2}}\widehat{X}%
(w_{r})}\right\rangle _{S_{[\lambda =0]}}. \label{torbellino}
\end{equation}%
That is, we showed how the Ribault-Teschner formula can be seen as an
identity between correlators of two different two-dimensional $\sigma $%
-models with three-dimensional target space each. While one of these is the $%
SL(2,\mathbb{R})_{k}$ WZW, the other is of the form%
\begin{equation}
Liouville\times {\mathcal{M}}_{k}\times time \label{cardinate}
\end{equation}%
of which Liouville theory is just a part. The $\mathbb{R}$ factor
corresponds to the time-like direction, parameterized by $T$. On the other
hand, the ${\mathcal{M}}_{k}$ factor is a $U(1)$\ direction, parameterized
by the field $\widehat{X}$, and describes a linear dilaton theory with
central charge $c=1-6k<1$. In fact, notice that the contribution of $%
\widehat{X}$ to the central charge is actually negative because $k>2$. The
field $\widehat{X}$ interacts with the Liouville field $\widehat{\varphi }$
through the tachyon-like potential, so that the first product in (\ref%
{cardinate}), unlike the second, is not a \textit{direct} product. The time
direction, instead, does not interact with the other fields, and it only
contributes to the total central charge and the conformal dimension of the
vertex operators.
The fact that a construction like (\ref{cardinate}) is possible is not a
minor detail: Realizing that the Ribault-Teschner formula (\ref{rrtt})
admits to be interpreted as the equivalence between these two CFTs demanded
not only the existence of a realization like (\ref{torbellino}), but also
demanded the contribution of the central charge coming from the $U(1)\times
\mathbb{R}$ part to agree with the difference between the Liouville central
charge $c_{L}=1+6Q^{2}$ and the $SL(2,\mathbb{R})$ WZW central charge $%
c_{SL(2)}=3+6\widehat{Q}^{2}=3k/(k-2)$, being reminded of $b^{-2}=k-2$.
Moreover, such value for the central charge of the CFT defined by fields $%
\widehat{X}$ and $T$ had to be consistent with the conformal dimension of
the fields in (\ref{torbellino}), leading to reproduce the formula for the
conformal dimension of the WZW vertex operators. Besides, another feature
that had to be explained was the presence of the $M$ additional fields $%
V_{-1/2b}$ arising in the r.h.s. of (\ref{rt}). Their presence is now
understood as follows: Since we know that the (the $M$-multiple integral of
the) product between the function $F_{k}(z_{1},...z_{N};w_{1},...w_{M})$ and
the $N+M$-point Liouville correlation function does satisfy the KZ\ equation
and so represents a correlation function in the WZW theory, it is then
expected that the Liouville degenerate fields $V_{-1/2b}$ arising there
admit to be expressed as a ($1,1$)-operator in a \textquotedblleft bigger"
theory with the form $Liouville\times CFT$ (i.e. the screening charge $%
V_{-1/2b}\times e^{i\sqrt{k/2}\widehat{X}}$ standing as the first term of
the r.h.s. of (\ref{SB})). That is, even though $V_{-1/2b}$ has dimension $%
h=-\frac{1}{2}-\frac{3}{4b^{2}}\neq 1$ with respect to the Liouville
stress-tensor, it does correspond to a ($1,1$)-operator\footnote{%
Even though the operator $V_{-1/2b}\times e^{i\sqrt{k/2}\widehat{X}}$ has
dimension $1$, it is not strictly correct to refer to it as a
\textquotedblleft screening" operator due to the remark on the $\widehat{sl}%
(2)_{k}$ transformation properties made in section 4.2.5.} $V_{-1/2b}\otimes
V_{CFT}$ with respect to the stress-tensor of the bigger model $%
Liouville\times CFT$. Of course, the theory also admits as a screening
operator the one that was already the screening for the \textquotedblleft
Liouville part of the theory", namely $V_{b}\otimes I$; so (\ref{SB}) can be
written as the sum of both, $\mathcal{O}_{\lambda _{1},\lambda
_{2}}=c_{k}^{-1}V_{-1/2b}e^{i\sqrt{k/2}\widehat{X}}+V_{b}$.
Notice that all the requirements mentioned above are actually obeyed by the
theory defined by the action (\ref{uhyeste}) perturbed by the operator (\ref%
{SB}). Hence, we have given a free field representation of the
Ribault-Teschner formula (\ref{rrtt}). Related to this, in Ref. \cite{R} it
was commented that a parafermionic realization of (\ref{rrtt}) is also
known, and the unpublished work by V. Fateev was referred. The parafermion
representation leads (see Eq. (3.31) in \cite{R}) to a formula similar to (%
\ref{rrtt}) provided the replacement of the factor $\prod_{1\leq
r<l}^{N}(z_{r}-z_{l})^{\frac{k}{2}-(m_{r}+m_{l}+\omega _{r}\omega _{l}\frac{k%
}{2}+\omega _{l}m_{r}+\omega _{r}m_{l})}(\overline{z}_{r}-\overline{z}_{l})^{%
\frac{k}{2}-(\overline{m}_{r}+\overline{m}_{l}+\omega _{r}\omega _{l}\frac{k%
}{2}+\omega _{l}\overline{m}_{r}+\omega _{r}\overline{m}_{l})}$ in (\ref{FF}%
) by a factor $\prod_{1\leq r<l}^{N}(z_{r}-z_{l})^{\frac{k}{2}+\frac{2}{k}%
m_{r}m_{l}-m_{r}-m_{l}}(\overline{z}_{r}-\overline{z}_{l})^{\frac{k}{2}+%
\frac{2}{k}\overline{m}_{r}\overline{m}_{l}-\overline{m}_{r}-\overline{m}%
_{l}},$ and notice that this is exactly what we find in our language (\ref%
{torbellino}) if we exclude the $T$ dependence in the vertex operators. This
realizes a correspondence like (\ref{rrtt}) but for the case of the coset $%
SL(2,\mathbb{R})_{k}/U(1)$. See the \textquotedblleft notes" at the end of
Ref. \cite{0511252} where the similarities with Fateev's work were already
mentioned. Besides, a realization of the Ribault-Teschner formula in terms
of Liouville times a $c<1$ matter CFT was independently presented by S.
Nakamura and V. Niarchos in Ref. \cite{Nuevo}. We would like to explore the
similarities between our realization and the one in that paper; we just
realized that the realization in \cite{Nuevo} does closely parallels ours.
Summarizing: because of Ribault-Teschner formula, it turns out that the
correlation function in the r.h.s. of (\ref{rt}) does correspond to the
string amplitude in the black hole ($\times time$) background, where the
winding number conservation is being violated in an amount $|N-2-M|$.\
Consequently, this implies that the l.h.s. of (\ref{rt}) do correspond to
WZW correlators as well, and this completes the proof of (\ref{duality}).
However, it has to be emphasized that the correspondence between BPZ and KZ
equations was proven for the Lorentzian theory, namely holding for
continuous representations. Thus, considering its validity beyond such
regime assumes a sort of analytic continuation. The convergence of integrals
in (\ref{rt}) is the subtle point here.
\subsection{A consistency check of the correspondence}
We have proven formula (\ref{duality}); this was first done in Ref. \cite%
{0511252}, but the order of the presentation was rather different there.
Formula (\ref{duality}) turns out to be a useful tool for computing
correlators in the WZW theory. A concise example was given in Ref. \cite%
{gaston}, where the free field representation in terms of the $%
Liouville\times U(1)\times time$ conformal field theory (\ref{SA})-(\ref{SB}%
) was employed to compute WZW three-point functions for the particular case
where the total winding number is violated in one unit. This quantity turns
out to be proportional to the Liouville correlator%
\begin{equation*}
\left\langle \Phi _{j_{1},m_{1},\overline{m}_{1}}^{\omega _{1}}(0)\Phi
_{j_{2},m_{2},\overline{m}_{2}}^{\omega _{2}}(1)\Phi _{j3,m_{3},\overline{m}%
_{3}}^{\omega _{3}}(\infty )\right\rangle _{WZW}\sim \prod_{i=1}^{3}\frac{%
\Gamma (-m_{i}-j_{i})}{\Gamma (j_{i}+1+\bar{m}_{i})}\prod_{t=1}^{s}\int
d^{2}v_{t}\left\langle e^{\sqrt{\frac{2}{k-2}}(j_{1}+1)\widehat{\varphi }%
(0)}\right. \times
\end{equation*}%
\begin{equation*}
\times e^{\sqrt{\frac{2}{k-2}}(j_{2}+1)\widehat{\varphi }(1)}e^{\sqrt{\frac{2%
}{k-2}}(j_{3}+1)\widehat{\varphi }(\infty )}\prod_{t=1}^{s}\left. e^{\sqrt{%
\frac{2}{k-2}}\widehat{\varphi }(v_{t})}\right\rangle _{S_{L}[\mu =0]}\delta
\left( s+j_{1}+j_{2}+j_{3}+1+\frac{k}{2}\right) ,
\end{equation*}%
and, up to an irrelevant $k$-dependent ($j$-$m$-independent) factor and
having fixed the value of the black hole mass, the final result reads%
\begin{equation*}
\left\langle \Phi _{j_{1},m_{1},\overline{m}_{1}}^{\omega _{1}}(0)\Phi
_{j_{2},m_{2},\overline{m}_{2}}^{\omega _{2}}(1)\Phi _{j3,m_{3},\overline{m}%
_{3}}^{\omega _{3}}(\infty )\right\rangle _{WZW}=\left( \pi \gamma \left(
\frac{1}{k-2}\right) \right) ^{-j_{1}-j_{2}-j_{3}-\frac{k}{2}%
-1}\prod_{i=1}^{3}\frac{\Gamma (-m_{i}-j_{i})}{\Gamma (j_{i}+1+\bar{m}_{i})}%
\times
\end{equation*}%
\begin{eqnarray}
&&\times \frac{G_{k}(j_{1}+j_{2}+j_{3}+\frac{k}{2})G_{k}(-j_{1}-j_{2}+j_{3}-%
\frac{k}{2})G_{k}(j_{1}-j_{2}-j_{3}-\frac{k}{2})G_{k}(1+j_{1}-j_{2}+j_{3}-%
\frac{k}{2})}{\gamma \left( -j_{1}-j_{2}-j_{3}-\frac{k}{2}\right) \gamma
\left( -\frac{2j_{2}+1}{k-2}\right)
G_{k}(-1)G_{k}(2j_{1}+1)G_{k}(1-k-2j_{2})G_{k}(2j_{3}+1)}\times \notag \\
&&\times \delta (m_{1}+m_{2}+m_{3}-k/2)\delta (\bar{m}_{1}+\bar{m}_{2}+\bar{m%
}_{3}-k/2)\delta (s+j_{1}+j_{2}+j_{3}+1+k/2). \label{result11}
\end{eqnarray}%
where the special function $G_{k}(x)$ is defined through%
\begin{equation*}
G_{k}(x)=(k-2)^{\frac{x(k-1-x)}{2(k-2)}}\Gamma _{2}(-x|1,k-2)\Gamma
_{2}(k-1+x|1,k-2),
\end{equation*}%
in terms of the Barnes function $\Gamma _{2}(x|1,y)$%
\begin{equation*}
\log \Gamma _{2}(x|1,y)=\lim_{\varepsilon \rightarrow 0}\frac{d}{%
d\varepsilon }\sum\limits_{n=0}^{\infty }\sum\limits_{m=0}^{\infty }\left(
(x+n+my)^{-\varepsilon }-(1-\delta _{n,0}\delta _{m,0})(n+my)^{-\varepsilon
}\right) ,
\end{equation*}%
where the presence of the factor $(1-\delta _{n,0}\delta _{m,0})$ in the
r.h.s. means that the sum in the second term does not take into account the
step $m=n=0$. Expression (\ref{result11}) does reproduce the exact result,
so that agrees with the result obtained in Refs. \cite{FZZ,MO3,YoPLB2005}.
The details of the computation can be found in \cite{gaston}. Rather than an
application, the calculation of (\ref{result11}) can be considered as a
consistency check of the representation (\ref{cardinate}) proposed here (and
in \cite{0511252})\ to represent WZW correlators. Besides, it also
represents an operative advantage since (unlike other free field
realizations for which the computation of violating winding three-point
function involves the additional spectral flow operator) this turns out to
be integrable in terms of the Dotsenko-Fateev type integrals (cf. the
calculations in Refs. \cite{GN3,MO3}). Nevertheless, it is worth pointing
out that the consistency check discussed here is the most simple
(non-trivial) computation one can do within this framework; this is because
it did not involve the degenerate Liouville fields $V_{-1/2b}$. Unlike, the
screening that we did use to realize (\ref{result11}) was the new one we
introduced; namely, the operator $\mathcal{T}_{1-k,k,k}=e^{\sqrt{\frac{2}{k-2%
}}\widehat{\varphi }}=e^{-\sqrt{\frac{2}{k-2}}(k-1)\varphi +i\sqrt{2k}X}$,
which represents a $n=2$ perturbation. A less trivial consistency check
would be that of trying to reproduce the winding-conservative WZW
three-point function in the often called $m$-basis, which would require to
make use of a non-trivial integral representation of the (hypergeometric)
special function of the kind studied in Refs. \cite{Satoh,HS,Lore}. Related
to this point, let us mention that explicit expressions for Liouville
four-point functions involving one degenerate state $V_{-\frac{1}{2b}}$ were
recently obtained \cite{Fateev2,BelavinZamolodchikov}. According to the
relation (\ref{rrtt}), these four-point functions are the ones representing
three-point functions that conservate the winding conservation in the WZW
side.
Other applications of the Stoyanovsky-Ribault-Teschner correspondence (\ref%
{rrtt}), (\ref{duality}), were early discussed in Ref. \cite%
{Apl1,Apl2,YoYu,Takayanagi}. In subsection 4.5.3, we will review one of the
observations made in \cite{YoYu}.
\subsection{Remarks}
\subsubsection{A comment on generalized minimal gravity}
Now, we would like to make a brief comment on the theory defined by the
action (\ref{SA}) and the perturbation (\ref{SB}); and let us focus our
attention on the two-dimensional sector corresponding to the fields $%
\widehat{\varphi }$ and $\widehat{X}$. Because of the field redefinitions (%
\ref{newc}), it turns out that the theory could be written as the Liouville
theory coupled to a $c<1$ CFT. Then, the natural question arises as to
whether such a $c<1$ model can be identified with one of the quoted minimal
models. As it is well known, the CFT minimal models are characterized by two
integers $p$ and $q$ which yield the value of the central charge, being $%
c=1-6(\beta ^{-1}-\beta )^{2}$ with rational\footnote{%
Besides, a generalized version of these CFTs can be considered, being valid
for generic values of $\beta ,$ \cite{Zreloaded}.} $\beta ^{2}=p/q$
satisfying $q>p$ (so that $\beta <1$). In our case, the value of the central
charge of the $c<1$ theory (corresponding to the part of the theory governed
by the field $\widehat{X}$) turns out to be $c=1-6k$ and, then, in order to
identify this with one of the minimal models we should demand $%
k=(p-q)^{2}/pq $ (that is $k=\left( \beta ^{-1}-\beta \right) ^{2}$) \cite%
{BPZ}. However, since we are interested in the whole range $k>2$, it turns
out that the condition $c=1-6(p-q)^{2}/pq$ is only consistent with
particular values of $k$. One example is precisely the model ($p=1,$ $q=4$)
which does correspond to $k=9/4,$ which is the value of $k$ for the 2D
theory on the coset. In such case, and taking into account that $k$ also
satisfies $k=2+b^{-2},$ we would have $\beta =b$ so that the theory
corresponds to the often called 2D minimal gravity (model that is supposed
to be exactly solved). For more general case, the 2D theory defined by the
fields $\widehat{\varphi }$ and $\widehat{X}$ can be regarded as the
Liouville theory coupled to a generalized minimal model (with
non-necessarily rational $\beta ^{2}$) perturbed by (\ref{SB}). Such
perturbation would then correspond to a Liouville-dressed operator in the
minimal model too. The operators of the minimal models admit a
representation in terms of the exponential form $\Phi _{mn}=e^{i\alpha _{mn}%
\widehat{X}},$ having conformal dimension $h_{mn}=\frac{1}{4}(m\beta
^{-1}-n\beta )^{2}-\frac{1}{4}(\beta ^{-1}-\beta )^{2}=\alpha _{mn}(\alpha
_{mn}+\beta -\beta ^{-1})$ for two positive integers $m$ and $n$; that is,
the momenta can take values $\alpha _{mn}=\frac{1}{2}(n-1)\beta -\frac{1}{2}%
(m-1)\beta ^{-1}$ or $\alpha _{mn}=\frac{1}{2}(m+1)\beta ^{-1}-\frac{1}{2}%
(n+1)\beta $. So, a perturbation operator with the form $e^{i\sqrt{k/2}n%
\widehat{X}+\sqrt{2}a_{n}\widehat{\varphi }}$ can be regarded as a dressed
field $\Phi _{n-1,n-1}$ of the minimal model ($p,q$) with $k=(p-q)^{2}/pq$.
In these terms, what we have proven is a correspondence between $N$-point
functions in the WZW theory and a subset of correlation functions of
perturbed Liouville gravity coupled to generalized minimal models.
\subsubsection{Duality between tachyon-like backgrounds}
By using the relation between Liouville correlators and WZW correlators we
wrote down identity (\ref{duality}). This gives a dual description for the
2D string theory in the black hole background. One of the questions that
arise is about the relation between (\ref{duality}) and the standard FZZ
correspondence. In fact, both models appear as alternative dual descriptions
of the WZW theory, so that we can use WZW correlators as an intermediate
step to eventually write the following seemly self-duality relation%
\begin{equation*}
\prod_{r=1}^{s_{+}}\int d^{2}u_{r}\prod_{t=1}^{s_{-}}\int
d^{2}v_{t}\left\langle \prod_{i=1}^{N}\mathcal{T}_{j_{i},m_{i},\overline{m}%
_{i}}(z_{i})\prod_{r=1}^{s_{+}}\mathcal{T}_{1-\frac{k}{2},-\frac{k}{2},\frac{%
k}{2}}(u_{r})\prod_{t=1}^{s_{-}}\mathcal{T}_{1-\frac{k}{2},\frac{k}{2},-%
\frac{k}{2}}(v_{t})\right\rangle _{S_{[\lambda =0]}}\sim
\end{equation*}%
\begin{equation}
\sim \prod_{r=1}^{\widetilde{s}_{+}}\int d^{2}u_{r}\prod_{t=1}^{\widetilde{s}%
_{++}}\int d^{2}\omega _{t}\left\langle \prod_{i=1}^{N}\widetilde{\mathcal{T}%
}_{j_{i},m_{i},\overline{m}_{i}}(z_{i})\prod_{r=1}^{\widetilde{s}_{+}}%
\mathcal{T}_{1-\frac{k}{2},\frac{k}{2},\frac{k}{2}}(u_{r})\prod_{t=1}^{%
\widetilde{s}_{++}}\mathcal{T}_{1-k,k,k}(\omega _{t})\right\rangle
_{S_{[\lambda =0]}}, \label{selfduality}
\end{equation}%
where the fearful symbol $\sim $ stands to make explicit the fact that this
identity depends on the details of how the FZZ conjecture relates the WZW
correlators to those of sine-Liouville theory\footnote{%
As far as we know, the checks of FZZ duality were performed by comparing the
analytic structures of both theories rather than verifying exact numerical
matching.}. This relation between correlators, realized by means of the
Coulomb gas realization, yields a non-trivial integral identity. On the
other hand, one can wonder whether (\ref{selfduality}) has to be referred as
a self-duality of sine-Liouville field theory or not. In fact, it merely
looks like a duality between two different deformations of the linear
dilaton theory (\ref{S0}) rather than a \textquotedblleft self-duality".
However, one can see that both sides in the identity above are in some sense
connected to sine-Liouville theory, and not only the left hand side.
Actually, the perturbation $\mathcal{T}_{1-k,k,k}$, that represents the
momentum $n=2$ operator, is connected to that of $n=1$ by the conjugation
relation (\ref{conjugate}). That is, while $j=1-m=1-k$ for the $n=2$
operator $\mathcal{T}_{1-k,k,k}$, the dual momenta (dual according to (\ref%
{j})) are\footnote{%
Strictly speaking, one has to consider the automorphism $m\rightarrow
\widetilde{m}=-jk-m(k-1)-k/2$ instead of (\ref{j}), which is a composition
with the reflection $m\rightarrow -m$.} $\widetilde{j}=1+\widetilde{m}=1-k/2$%
, and correspond to the momenta of the $n=-1$ operator $\mathcal{T}_{1-\frac{%
k}{2},-\frac{k}{2},-\frac{k}{2}}$. Thus, we could relate the correlators in
the l.h.s. of (\ref{selfduality}) to the following one\footnote{%
up to a $k$-dependent factor of the form $\left( b_{k}\right) ^{\widetilde{s}%
_{-}}$, with $b_{k}$ being independent on $j_{i},m_{i}$ and $\overline{m}%
_{i} $.}%
\begin{equation}
\sim \prod_{r=1}^{\widetilde{s}_{+}}\int d^{2}u_{r}\prod_{t=1}^{\widetilde{s}%
_{-}}\int d^{2}\omega _{t}\left\langle \prod_{i=1}^{N}\widetilde{\mathcal{T}}%
_{j_{i},m_{i},\overline{m}_{i}}(z_{i})\prod_{r=1}^{\widetilde{s}_{+}}%
\mathcal{T}_{1-\frac{k}{2},+\frac{k}{2},+\frac{k}{2}}(u_{r})\prod_{t=1}^{%
\widetilde{s}_{-}}\widetilde{\mathcal{T}}_{1-\frac{k}{2},-\frac{k}{2},-\frac{%
k}{2}}(\omega _{t})\right\rangle _{S_{[\lambda =0]}} \label{uhh}
\end{equation}%
with $\frac{k-2}{2}\widetilde{s}_{+}-\widetilde{s}_{-}-\frac{k}{2}\left(
N-2\right) =\frac{k-2}{2}\left( s_{+}+s_{-}\right) $ and $\widetilde{s}%
_{+}-(N-2)=s_{+}-s_{-}$. Hence, (\ref{selfduality}) turns out to be a
twisted version of the sine-Liouville model, i.e. can be written as in (\ref%
{uhh}). The presence of the tildes $\sim $ on the operators in (\ref{uhh})
gives rise to the expression \textquotedblleft twisted"; twisted in the
sense that (\ref{j}) is applied to the operators $\mathcal{T}_{1-\frac{k}{2}%
,-\frac{k}{2},-\frac{k}{2}}$ but is not applied to the operators $\mathcal{T}%
_{1-\frac{k}{2},+\frac{k}{2},+\frac{k}{2}}$. This kind of relation between
correlators (\ref{selfduality}) and (\ref{uhh}) is reminiscent of what
happens in the WZW theory, where standard and conjugate representations
stand as alternative realizations of the same correlation functions. Thus,
this suggests that (\ref{selfduality}) could be manifesting some kind of
self-duality relating two different realization of the same conformal theory%
\footnote{%
Let us also mention that another realization of the same correlators is
possible if one replace $\widetilde{\mathcal{T}}_{1-\frac{k}{2},-\frac{k}{2}%
,-\frac{k}{2}}\propto e^{-\sqrt{\frac{2}{k-2}}(k-1)\widehat{\varphi }+i\sqrt{%
2k}\widehat{X}}$ by its $k-2$ power $e^{-\sqrt{2(k-2)}(k-1)\widehat{\varphi }%
+i\sqrt{2k}(k-2)\widehat{X}}$. This is because of the Liouville self-duality
under $b\leftrightarrow b^{-1}$.}. Morally, the price to be paid to twist
(namely, to conjugate) the $N$ vertex operators $\mathcal{T}_{j_{i},m_{i},%
\overline{m}_{i}}$ in (\ref{selfduality}) is that of twisting the
left-handed screening operators $\mathcal{T}_{1-\frac{k}{2},-\frac{k}{2},-%
\frac{k}{2}}\rightarrow \widetilde{\mathcal{T}}_{1-\frac{k}{2},-\frac{k}{2},-%
\frac{k}{2}}\propto \mathcal{T}_{1-k,k,k}$, while keeping the right-handed $%
\mathcal{T}_{1-\frac{k}{2},+\frac{k}{2},+\frac{k}{2}}$ unchanged.
Consequently, the number of insertions changes from $s_{-}$ to $\widetilde{s}%
_{++}=\widetilde{s}_{-}$ (and also from $s_{+}$ to $\widetilde{s}_{+}$) by
keeping the formal relation $N-2=\widetilde{s}_{+}-s_{+}+s_{-}$ fixed.
Roughly speaking, the right hand side of (\ref{selfduality}) looks like a
\textquotedblleft half" of a sine-Liouville theory, because just one of the
two exponential operators $\mathcal{T}_{1-\frac{k}{2},\pm \frac{k}{2},\pm
\frac{k}{2}}$ that form the cosine interaction (\ref{cos2}) is present,
while the operators $\mathcal{T}_{1-k,k,k}$ seem to arise there for
compensating the conservation laws that make the correlator to be nonzero.
In the WZW theory, the analogue to the \textquotedblleft twisting" that
connects the operators $\widetilde{\mathcal{T}}_{j,m,\overline{m}}$ to
operators $\mathcal{T}_{j,m,\overline{m}}$ would be the relation existing
between conjugate and standard representations of the $\widehat{sl}(2)_{k}$
vertex algebra \cite{DVV,BK,GN2,GN3,GL}. The relation between
representations $\widetilde{\mathcal{T}}_{j,m,\overline{m}}$ and $\mathcal{T}%
_{j,m,\overline{m}}$ connects operators of the winding sector $n$ to those
of the sector $n+1$. Presumably, the twisted version of the FZZ duality we
presented in (\ref{duality}) can be extended in order to include higher
momentum and winding modes $n>2$. This would rise the obvious question as to
what would these twisted sectors be describing in terms of the black hole
picture. As it was pointed out in \cite{KKK}, if the $c=1$ theory is
perturbed by operators of the sector $n,$ then it behaves equivalently to
the theory compactified in a different\ radius $R/n$ and perturbed by the
sine-Liouville operators. In some sense, this is related to what was early
studied in Ref. \cite{Mukhi}. Nevertheless, the perturbation we considered
here presents operators of both sectors $n=1$ and $n=2$, so being a sort of
chirally twisted case. We would like to understand this deformations better.
Our hope is to make contact to the results of Refs. \cite{MMP} and \cite%
{Mukhi} in trying to answer this question, but this certainly requires
further study.
\subsubsection{The \thinspace $c\rightarrow 0$ limit of the $Liouville\times
U(1)\times \mathbb{R}$ model}
To conclude, we would like to discuss the particular limit where the central
charge of the model (\ref{cardinate}) vanishes. This was first studied in
Refs. \cite{Takayanagi2}\ and \cite{YoYu} (see also \cite{Nichols}-\cite%
{Nichols2}). This limit corresponds to $k\rightarrow 0,$ which, in fact, is
far from being well understood. Actually, one can rise several question
concerning whether in such a limit the CFT is well defined or not. However,
let us avoid these questions here and merely assume that such an extension
is admissible. In the limit $k\rightarrow 0$, the Liouville central charge
becomes $c_{L}=-2$ while the background charge for the field $\widehat{X}$
vanishes, so that the central charge for the $U(1)\times \mathbb{R}$ theory
(i.e. the fields $\widehat{X}$ and $T $) turns out to be $+2$. The
functional form of the correlation functions in the $k\rightarrow 0$ limit
requires a careful analysis because of subtle features arising through the
analytic continuation in the $b$ complex plane \cite{SchomerusII,Zreloaded}.
However, we can further speculate and assume for a while that an extension
of the correspondence (\ref{rrtt}) between WZW and Liouville theory still
holds at $k=0$. At this point, the sine-Liouville action actually coincides
with the Liouville action supplemented with that of a $c=1$ field $\widehat{X%
}$. This is because of the identification $b^{-2}=k-2$ and the fact that $Q=
- \widehat{Q}$ at the point $k=0$. Besides, at $k=0$ the sine-Liouville
interaction (\ref{cos}) does correspond to the Liouville cosmological
constant $e^{i\widehat{\varphi }/\sqrt{2}}$. This suggestive matching
between both actions can be tested at the level of correlation functions as
well. In fact, with the authors of \cite{Takayanagi2}, we could assume that
the FZZ conjecture is still valid in the limit $k\rightarrow 0$ and, then,
by invoking the Ribault-Teschner formula (\ref{rrtt}), eventually conclude
that the sine-Liouville correlators model coincide with the correlators of
the Liouville theory (times the free boson $\widehat{X}$) at $k=0$. To see
this, let us point out the following remarkable facts: First, notice that,
because we are taking a limit $R=\sqrt{k/2}$ going to zero (\textit{i.e.}
the asymptotic radius of the cigar), it is just enough to observe what
happens with the modes $m=\bar{m}=0$ on the cigar. From the point of view of
the T-dual model, the dual radius $\tilde{R}\sim 1/\sqrt{k}$ of the cylinder
goes to infinity and the states with finite momentum $p=\frac{m}{\sqrt{k}}$
(keeping $p$ fixed) decouple generating a $U(1)$ factor $\sim e^{i\sqrt{2}p%
\widehat{X}}$ in the correlation functions. Secondly, one can show (see \cite%
{YoYu}) that for $k=0$ the formula (\ref{rrtt}) reads
\begin{equation}
\left\langle \Phi _{j_{1},m_{1},\overline{m}_{1}}^{\omega
_{1}}(z_{1})...\Phi _{j_{N},m_{N},\overline{m}_{N}}^{\omega
_{N}}(z_{N})\right\rangle _{WZW}\sim \prod_{i=1}^{N}{\mathcal{R}}%
_{0}(j_{i},0)\ \langle V_{-\frac{i}{\sqrt{2}}j_{1}}(z_{1})...V_{-\frac{i}{%
\sqrt{2}}j_{N}}(z_{N})\rangle _{S_{L}[\mu ]}; \label{pros}
\end{equation}%
with $p_{1}+p_{2}+...p_{N}=\omega _{1}+\omega _{2}+...\omega _{N}=M-N+2=0$.
The function ${\mathcal{R}}_{k}(j,m)$ is the reflection coefficient of WZW
model, which is given by the two-point function (\ref{FiRulete2}). The
arising of these reflection coefficients (one for each vertex operator) is
ultimately attributed to the fact that the momenta of the WZW vertex
operators were the Weyl reflected $\widehat{j}_{i}=-1-j_{i}$ instead of $%
j_{i}$ (notice that the Liouville correlator in (\ref{pros}) scales like $%
\mu ^{\hat{j}_1+...\hat{j}_N+1}$). We also observe in (\ref{pros}) that,
besides the $s$ integrals over the screening insertions required in the
Liouville correlators, we implicitly have $M=N-2$ additional integrals over
the variables $v_{t}$ where $M$ operators $V_{-1/2b}(v_{t})$ are inserted.
This is consistent with what one would expect since $k=0$ implies $%
b^{2}=-1/2 $ and then the degenerate fields $V_{-1/2b}$ turn out to agree
with the screening operators $V_{b}$. Hence, at $k=0$ the integrals over
such variables $v_{t}$ are nothing more than screening insertions in
Liouville correlation functions\footnote{%
G.G. specially thanks Yu Nakayama for collaboration in this particular
computation. See Ref. \cite{YoYu}.}, and this is the reason why we did not
explicitly write them in (\ref{pros}). This shows that the Ribault-Teschner
formula turns out to be consistent with the FZZ conjecture. That is, at $k=0$
sine-Liouville agrees with the product between Liouville theory and a free $%
c=1$ boson, so that for the particular case $k=0$ equation (\ref{rrtt})
actually states the identity between $N$-point correlation functions in
sine-Liouville theory and $N$-point correlation functions in the 2D black
hole. Nevertheless, we should emphasize that all these digressions are
strongly based on the assumption that the CFT is still well defined in the
regime $k<2$ and, as far as we know, this is still far from being clear.
\section{Conclusions}
It is usually accepted that, probably, the FZZ duality is just an example of
a more general phenomenon which should be interesting to understand in a
deeper way \cite{KKK,MMP}. The purpose of this paper was precisely to
discuss an example of such kind of generalization. We studied a
correspondence between two-dimensional string theory in the euclidean black
hole ($\times time$) and a (higher mode) tachyon perturbation of a linear
dilaton background. Our main result is presented in Eq. (\ref{duality}).
The tachyon perturbation we considered here corresponds to momentum modes $%
n=1$ and $n=2,$ and so it can be considered as a kind of deformation of the
standard FZZ sine-Liouville theory. We argued that such a \textquotedblleft
deformation" (or \textquotedblleft twisting" in the sense of (\ref{j})) can
be thought of as a conjugate representation of the sine-Liouville
interaction term, presumably related to the conjugate representations of
operators in the WZW model \cite{GL}.
In section 4 we have given a dictionary that permits to express any $N$%
-point correlation function in the $SL(2,\mathbb{R})_{k}$ WZW model on the
sphere topology in terms of a correlation function in the tachyon perturbed
linear dilaton background, and we have given a precise prescription for
computing those correlators in the Coulomb gas approach. This correspondence
between correlators was proven by rewriting a nice formula worked out by S.
Ribault and J. Teschner in Refs. \cite{R,RT}, which directly follows from
the relation between the solutions of the KZ and the BPZ equations. Our
result (\ref{yesto}) realizes the general version of the formula proven in
\cite{R}. In fact, following \cite{0511252}, we showed that the auxiliary
overall function $F_{k}(z_{1},...z_{N};w_{1},...w_{M})$ standing in the
Ribault-Teschner formula (\ref{rrtt}) can be also seen as coming from the
correlation functions of a linear dilaton CFT perturbed by a tachyon-like
operator of higher ($n\geq 1$) momentum modes. Thus, the twisted dual we
discussed here turns out to be a free field realization of the
Ribault-Teschner formula. A remarkable feature of such realization is that
the $n=2$ mode perturbation $\mathcal{T}_{1-k,k,k}$ turns out to be related
to the sine-Liouville potential in the same way as to how the twisted
tachyon-like vertex operators $\widetilde{\mathcal{T}}_{j_{i},m_{i},%
\overline{m}_{i}}$ are related to the operators $\mathcal{T}_{j_{i},m_{i},%
\overline{m}_{i}}$ of the standard FZZ prescription. Both representations $%
\widetilde{\mathcal{T}}_{j_{i},m_{i},\overline{m}_{i}}$ and $\mathcal{T}%
_{j_{i},m_{i},\overline{m}_{i}}$ have the same eigenvalues under the Cartan $%
U(1)$ generator $J_{0}^{3}$ and the Virasoro-Casimir operator $L_{0}$.
Besides, it turns out that the fact that $\mathcal{T}_{j_{i},m_{i},\overline{%
m}_{i}}$ and $\widetilde{\mathcal{T}}_{j_{i},m_{i},\overline{m}_{i}}$
transform distinctly under the action of the $\widehat{sl}(2)_{k}$
generators $J_{n}^{\pm }$ combines with the fact that the $n=2$ operator $%
\mathcal{T}_{1-k,k,k}$ transforms non-trivially under those generators
either, and this makes the correlation functions (\ref{yesto})\ behave
properly under the $sl(2)_{k}$ algebra.
Tachyon-like perturbations of the linear dilaton background involving higher
winding modes were also studied recently by Mukherjee, Mukhi and Pakman in
Ref. \cite{MMP}, where they presented a generalized perspective of the FZZ
correspondence. One of the task for the future is to understand the relation
to \cite{MMP} better. Besides, the understanding of the connection of our
result to the standard FZZ correspondence also deserves more analysis.
Regarding this, we would like to conclude by mentioning that the idea of the
proof of (\ref{duality}) given in section 4 here could be actually adapted
to prove the standard FZZ duality (on the sphere) if one considers the
appropriated pieces in the literature. A key point in doing this would be a
result obtained some time ago by V. Fateev, who has found a very direct way
of showing the relation between correlation functions in both Liouville and
sine-Liouville theories \cite{Fateev}. Such connection, once combined with
the Ribault-Teschner formula \cite{RT,R}, would yield a proof of the FZZ
duality at the level of correlation functions on the sphere topology without
resorting to arguments based on supersymmetry.
\bigskip
This paper is an extended version of the authors' contribution to the XVIth
International Colloquium on Integrable System and Quantum Symmetries, held
in Prague, in June 2007. A brief version was published in Rep. Math. Phys.
\textbf{61}.2 (2008) 151-162. Besides, these notes are based on Refs. \cite%
{0511252,gaston}\ and summarize the contents of the seminars that one of the
authors has delivered at several institutions in the last year and a half.
G.G. would like to thank S. Murthy, Yu Nakayama, K. Narain, A. Pakman, S.
Ribault and V. Schomerus for conversations, for e-mail exchanges, and for
very important comments. He is also grateful to V. Fateev for sharing his
unpublished work \cite{Fateev}. The partial support of Universidad de Buenos
Aires, Agencia ANPCyT, and CONICET through grants UBACyT X861, PICT 34557,
PIP6160 is also acknowledged.
\bigskip
\textbf{Note:} After our paper appeared in arXives, the higher genus
generalization of the Ribault-Teschner formula was done in the very nice
paper \cite{HikidaSchomerus1}. There, Hikida and Schomerus derived a
generalization of (\ref{rrtt}) by using the path integral approach. One of
the fabulous applications of Hikida-Schomerus formula was that of giving a
proof of the FZZ duality conjecture \cite{HikidaSchomerus2}. A crucial step
in the proof giving in \cite{HikidaSchomerus2} is the result of Ref. \cite%
{0511252}, which we discussed here within a similar context.
The authors of \cite{HikidaSchomerus2} also asserted that the twisted dual
model described by the action (\ref{S}) perturbed with operator (\ref%
{Pupapupapupa56}) is actually not well defined. We understand they meant
that such an action cannot be taken literally for all purposes as it is not
weakly coupled. This is of course true, and this is why we emphasized in
subsection 4.2 that the twisted model we presented here makes sense only
when the appropriate prescription for computing correlattion functions is
considered. It is well known that free field realization and Coulomb
gas-like prescriptions work well in conformal models even when the
interaction terms are not necessarily well-defined perturbatively. This is,
for instance, the case of Liouville theory with the \textit{dual} screening
charge, and this is also the case of sine-Liouville theory, where the
Coulomb gas prescription is known to be suitable for computing three-point
correlators. Then, we understand the authors of \cite{HikidaSchomerus2}
would agree in that our twisted model can be consistently used to compute
correlation functions, even though writing its action could be misleading if
taken literally. After all, if appropriately used, this represents a free
field realization of identity (\ref{rrtt}).
G.G. thanks V. Schomerus for a conversation about this point.
|
0706.0281
|
\section*{Acknowledgment}
This work is supported by the Russian Foundation for Basic Research (Grant
\#\,05--02--16422).
\section*{References}
|
0706.2081
|
\section{Introduction}
The theory of transformations preserving different properties and
invariants dates back to the works by Frobenius, \cite{Fr}, Schur,
\cite{Sch}, and Dieudonn\'e, \cite{Die}, and is an intensively
developing part of algebra nowadays. A characterization of maps
preserving zeros of polynomials plays a central role in this area. The
most known results of this type were devoted to the characterization of
linear maps on matrix algebras, preserving the following polynomials:
$\mathfrak{p}(x)=x^k$, which correspond to nilpotent matrices,
see~\cite{BPW}, $\mathfrak{p}(x)=x^2-x$, whose zeros are idempotent
matrices, see~\cite{BP,CL,BS1}, potent matrices satisfying
$\mathfrak{p}(x)=x^k-x$, see~\cite{BS}, and matrices of finite order,
i.e., the zeros of $\mathfrak{p}(x)=x^k-1$, see~\cite{PD,GLS}. In 1980
Howard~\cite{howard} proved the general classification theorem for
bijective linear transformations on matrix algebras, preserving zeros
of a polynomial in one variable with at least two distinct zeros. This
result, together with the main theorem from~\cite{BPW}, provides the
complete characterization of bijective linear maps on matrix algebras
over algebraically closed fields, preserving zeros of polynomials in
one variable. Later in \cite{Li_Pierce} Li and Pierce investigated the
possibility to remove the invertibility assumption from Howard's
theorem and proved some related results.
In parallel since 1976, a question of characterizing linear
transformations preserving zeros of multilinear polynomials in several
noncommuting variables was considered. In particular, Watkins,
\cite{Watk}, characterized bijective linear transformations preserving
commutativity, i.e., zeros of the polynomial $\mathfrak{p}(x,y)=xy-yx$,
in \cite{Wong,Wong1} Wong classified operators preserving zero
products, i.e., zeros of $\mathfrak{p}(x,y)=xy$. During the last years
there was a big interest to this question, see
\cite{beasly_guterman_Lee_Song,beidar_lin,bresar,chan_li_sze,chebotar_fong_lee,fosner,petek_t,semrl:commutativity,zhao_hou}
and references therein. Also several related topics were intensively
investigated. Namely, additive transformations $\Phi$ on prime rings
satisfying the stronger condition $\mathfrak{p}(\Phi(x_1),\ldots,
\Phi(x_k))=\Phi(\mathfrak{p}(x_1,\ldots,x_k))$ for all $x_1,\ldots,
x_m$, were studied in~\cite{Beidar_Fong}. Linear maps that preserve
operators on infinitely dimensional algebras annihilated by a
polynomial in one variable, were treated
in~\cite{Hou_Hou,Semrl:Operator_Theory}. Bijective linear operators on
the matrix algebra preserving zeros of the involutory polynomial
$\mathfrak{p}(x,y)=xy-yx^*$ are classified in \cite{chebotar_fong_lee}.
{\em Additive\/} surjections on certain classes of algebras, preserving
zeros of $\mathfrak{p}(x)=x^2$ or preserving zeros of the Jordan
polynomial $\mathfrak{p}(x,y)=xy+yx$, were characterized
in~\cite[Theorem~4.1]{Chebotar_Ke_Lee}, \cite[Lemma 2.3]{hou_zhao}, and
in~\cite{zhao_hou}.
In spite of the constant interest to the characterization problems for
linear transformations, preserving zeros of matrix polynomials, there
were no general answers in the multivariable case, like the Howard's
theorem concerning linear preservers of zeros of polynomials in one
variable. In particular, all results were proved for some specific
polynomials.
In \cite{chebotar_fong_lee} Chebotar, Fong and Lee posed the question
about the general form of linear preservers for zeros of multivariable
polynomials explicitly as an open problem:
\medskip
{\bf Problem}. \cite[Problem 1.1]{chebotar_fong_lee} Let
$\mathfrak{p}(x_1,\dots,x_k)$ be a polynomial over a field ${\mathbb F}$ in
noncommuting indeterminates $x_1,\ldots, x_k$ of the degree $\deg
\mathfrak{p}>1$, and $\Phi: {\cal M}_n ({\mathbb F}) \to {\cal M}_n({\mathbb F})$ be a
linear map on the matrix algebra ${\cal M}_n ({\mathbb F})$. Suppose that
$\mathfrak{p}(A_1,\ldots, A_k )=0$ implies
$\mathfrak{p}(\Phi(A_1),\ldots , \Phi(A_k))=0$. Is it possible to
describe such $\Phi$?
\medskip
The authors of \cite{chebotar_fong_lee} have conjectured that if the
size of matrices, $n$, is big enough comparing with the number of
variables, $k$, and $\Phi$ is linear and bijective, then~$\Phi$ is a
sum of a scalar multiple of Jordan homomorphism and a transformation
that maps into the center of algebra.
The present paper is devoted to the solution of the above mentioned
problem for certain sufficiently large classes of polynomials of a
general type. Several remarks are in order. Firstly, our results are
non-linear in nature, we even do not assume additivity of a
transformation under consideration in advance. Secondly, the
transformation is not necessary required to be invertible and we only
assume that it is surjective. In addition we found some conditions
which replace the surjectivity assumption and also provide the examples
showing that the assumptions on the transformations, we have posed, are
indispensable. Moreover, the developed technique is characteristic free
and allows us to work without restrictions on the number of variables
of a polynomial. This is done by the exclusion of polynomials which do
not provide sufficient restrictions on the transformation to be
classified. For example, this is the case with polynomial identities of
${\cal M}_n({\mathbb F})$. Say, the polynomial
$\mathfrak{p}(x_1,\dots,x_{2n}):=\sum_{ \sigma\in{\cal
S}_{2n}}sign(\sigma)x_{\sigma(1)}\cdots x_{\sigma(2n)}$ is an identity
on ${\cal M}_n({\mathbb F})$ by the famous Amitsur-Levitzki's theorem,
see~\cite{Amitsur_Levitzki}. This polynomial vanishes on the whole
matrix algebra and therefore it gives no condition on~$\Phi$. Therefore
we divide our considerations in two parts. Firstly we consider the {\em
generic case\/}, where the sum of coefficients of a multilinear
polynomial ${\mathfrak p}$ is non-zero, and thus ${\mathfrak p}$ can
not be an identity in ${\cal M}_n({\mathbb F})$. Then we investigate the {\em
derogatory case\/}, where the sum of coefficients of ${\mathfrak p}$ is
zero and polynomial identities may appear.
\medskip
Throughout, $n\ge 3$ will be an integer and ${\cal M}_n({{\mathbb F}})$ will be
the algebra of $n\times n$-matrices over an arbitrary field ${{\mathbb F}}$.
Let $E_{ij}$ be its standard basis. Let $\mathrm{GL}_n({{\mathbb F}}) \subset
{\cal M}_n({{\mathbb F}})$ denote the group of invertible matrices, with
identity~$\mathop{\mathrm{Id}}\nolimits$. Let~${\cal I}^1\subseteq {\cal M}_n({\mathbb F})$ be the set of
all rank-one idempotents. Given a field
homomorphism~$\varphi:{\mathbb F}\to{\mathbb F}$ ({\small i.e., an additive and
multiplicative function on ${\mathbb F}$}), we let~$X^\varphi$ be a matrix,
obtained from~$X$ by applying~$\varphi$ entry-wise. In addition,
let~$X^{{\rm tr}\,}$ be the transposed matrix of~$X$. Matrices $P,Q\in {\cal
M}_n({\mathbb F})$ are called {\em orthogonal\/} to each other if $PQ=QP=0$.
Lastly, let~${\cal S}_k$ be the set of all permutations of the set
$\{1,\dots,k\}$.\bigskip
\medskip
\begin{de} Let $k\ge 2$. We say that a matrix $k$-tuple $(A_1,\ldots, A_k)$ {\em
is a zero of a homogeneous multilinear polynomial}
$${\mathfrak{p}}(x_1,\dots,x_k):=\sum_{\sigma\in{\cal S}_k}
\alpha_\sigma x_{\sigma(1)}\cdots x_{\sigma(k)} $$ if
$$\mathfrak{p}(A_1,\dots, A_k ):=\sum_{\sigma\in{\cal S}_k} \alpha_\sigma
A_{\sigma(1)} \cdots A_{\sigma(k)}=0.$$
The set of all such $k$-tuples will be denoted by~$\mathfrak{S}_{\mathfrak{p}}\subseteq {\cal
M}_n({\mathbb F})\times\dots\times {\cal M}_n({\mathbb F})$.
\end{de}
\begin{de}
Suppose~$\mathfrak{p}_1,\mathfrak{p}_2$ are two homogenous multilinear
polynomials. A transformation $\Phi:{\cal M}_n({\mathbb F})\to {\cal M}_n({\mathbb F})$
{\em maps the zeros of~$\mathfrak{p}_1$ to the zeros of
$\mathfrak{p}_2$} whenever the implication $\mathfrak{p}_1(A_1,\dots,
A_k )=0\Longrightarrow\mathfrak{p}_2(\Phi(A_1),\dots, \Phi(A_k) )=0$
holds ({\small equivalently, whenever $\Phi({\mathfrak{S}_{\mathfrak{p}}}_1)\subseteq
{\mathfrak{S}_{\mathfrak{p}}}_2$}). If~$\mathfrak{p}_1=\mathfrak{p}_2=:\mathfrak{p}$ then
$\Phi$ {\em preserves the zeros of~$\mathfrak{p}$}. In addition, if
$\mathfrak{p}(x,y)=xy-yx$ then $\Phi$ {\em preserves commutativity}.
\end{de}
\begin{de}A transformation $\Phi:{\cal M}_n({\mathbb F})\to {\cal M}_n({\mathbb F})$
{\em strongly maps the zeros of~$\mathfrak{p}_1$ to the zeros
of~$\mathfrak{p}_2$} whenever the equivalence
$\mathfrak{p}_1(A_1,\dots, A_k
)=0\Longleftrightarrow\mathfrak{p}_2(\Phi(A_1),\dots, \Phi(A_k) )=0$
holds. If $\mathfrak{p}_1=\mathfrak{p}_2=:\mathfrak{p}$ then $\Phi$
{\em strongly preserves the zeros of~$\mathfrak{p}$}.
\end{de}
\medskip
Our paper is organized as follows.
In Section~\ref{S4} we study the mappings between certain matrix
subspaces, including the map from the whole matrix algebra to itself,
which strongly preserve zeros of homogeneous multilinear polynomials
with nonzero sum of coefficients.
In Section~\ref{S3} we study the transformations that map zeros of a
homogeneous multilinear polynomials of arbitrary many variables with
zero sum of coefficients to zeros of another such polynomial. To avoid
the obstructions which come from the polynomial identities of matrix
algebra, it is necessary to restrict the set of permutations. The
general problem is then reduced to the already well-studied
commutativity preservers,
see~\cite{semrl:commutativity,fosner,bresar,petek_t,beidar_lin} for
their characterization. We remark that some ideas that we are using in
this section came from our recent paper~\cite{alieve_guterman_kuzma}.
Section~\ref{S5} contains a number of examples showing that our
assumption are indispensable without posing some additional conditions
on $\Phi$ or {on $\mathfrak{p}(x_1,\ldots,x_k)$}.
\section{\label{S4} Polynomials with non-vanishing sums of\\ coefficients}
In the present section we investigate surjective mappings on certain
matrix subspaces, in particular on the whole matrix space, that {\em
strongly} preserve zeros of a polynomial with non-vanishing sum of
coefficients.
We also refer to Chan, Li, and Sze~\cite{chan_li_sze}, where the zeros
of a polynomial $\mathfrak{p}(x,y):=xy$ were considered, and, similarly
to our results below, the nice structure was obtained solely on
matrices of rank-one. We will show in the last section that in a way
our results cannot be further improved, without imposing additional
hypothesis, say additivity of~$\Phi$.
However, if~$\Phi$ strongly preserves the zeros of a polynomial with at
least three variables, then we were able to deduce a structural result
holding {\em for all matrices} from the defining set of~$\Phi$.\bigskip
We now list the main results of the present section.
Let~$\mathfrak{D}_1\subseteq {\cal M}_n({\mathbb F})$ and $\mathfrak{D}_2\subseteq {\cal
M}_n({\mathbb F})$ be subsets {\em that contain all matrices of rank-one and
all idempotents of rank $n-1$\/}. Also in this section we assume that a
homogeneous multilinear polynomial
\begin{equation}\label{eq:poly-with-nonvanishing-sum}%
\mathfrak{p}(x_1,\dots ,x_k):=\sum_{\sigma\in{\cal S}_k} \alpha_\sigma
x_{\sigma(1)}\cdots x_{\sigma(k)};\qquad (\alpha_\sigma\in{\mathbb F})
\end{equation}
satisfies $\sum \alpha_\sigma\not=0$.\medskip
The most general form of the main result of this section is the following:
\begin{theorem}\label{thm:main-nonvanishing-sum}%
Let~${\mathbb F}$ be an arbitrary algebraically closed field. Assume~$n\ge 3$
and $k\ge 2$. If a surjection $\Phi:\mathfrak{D}_1\to\mathfrak{D}_2$ strongly
preserves the zeros of~$\mathfrak{p}(x_1, \ldots,x_k)$ then there
exists a field isomorphism $\varphi:{\mathbb F}\to{\mathbb F}$, a function
$\gamma:\mathfrak{D}_1\backslash\{0\}\to{\mathbb F}^\ast:={\mathbb F}\backslash\{0\}$, and an
invertible matrix~$T$ such that
\begin{itemize}
\item[{\rm (i)}]
$\Phi(A) = \gamma(A) \,
{T}A^{\varphi}{T}^{-1}$ for all rank-one
matrices~$A$, or
\item[{\rm (ii)}]
$\Phi(A) = \gamma(A) \,
{T}\left(A^{\varphi}\right) ^{{\rm tr}\,}{T}^{-1}$
for all rank-one matrices~$A$.
\end{itemize}
\end{theorem}
\begin{remark} {The {converse to Theorem~\ref{thm:main-nonvanishing-sum}}
does not hold without imposing additional constraints on
isomorphism~$\varphi$. Namely}, there are transformations of types (i)
and (ii) which do not preserve the zeros of~$\mathfrak{p}$. We refer
the reader to the last section for examples.
\end{remark}
When ${\rm char}\, {\mathbb F}\not=2$, and the dimension of matrices is $n\ge 4$, and
the polynomial has at least three variables we have a nice structural
result holding {\em for all matrices} from the defining set~$\mathfrak{D}_1$
of~$\Phi$. In particular, for all matrices if $\mathfrak{D}_1={\cal
M}_n({\mathbb F})$. We merely add scalar matrices to conclusions (i) and (ii)
of Theorem~\ref{thm:main-nonvanishing-sum}:
\begin{corollary} \label{cor:TAllMatr}%
Under the assumptions of Theorem~\ref{thm:main-nonvanishing-sum},
assume further ${\rm char}\, {\mathbb F}\not=2$, and $n\ge 4$, and \mbox{$k\ge 3$}.
Then,
\begin{itemize}
\item[{\rm (i)}]
$\Phi(A) = \gamma(A)\, {T}A^{\varphi}{T}^{-1}+\mu(A)\,\mathop{\mathrm{Id}}\nolimits$ for all
$A\in {\mathfrak{D}_1}$, or
\item[{\rm (ii)}]
$\Phi(A) = \gamma(A) \, {T}\left(A^{\varphi}\right)
^{{\rm tr}\,}{T}^{-1}+\mu(A)\,\mathop{\mathrm{Id}}\nolimits$ for all $A\in {\mathfrak{D}_1}$.
\end{itemize}
\end{corollary}
The situation is completely different when $k=2$, see
Example~\ref{exa:counter-axa-for-nonvanishing} below. However,
if~$\mathfrak{p}(x,y):=xy+yx$ is a polynomial of Jordan multiplication
we can still get the characterization for some special
matrices~$A\in\mathfrak{D}_1$.
\begin{corollary}\label{cor:corolary_to_Jordan_product}%
Under the assumptions of Theorem~\ref{thm:main-nonvanishing-sum},
assume further $\mathfrak{p}(x,y):=xy+yx$ and~${\rm char}\, {\mathbb F}\not=2$. Then, the
conclusions (i) and (ii) also hold for diagonalizable~$A\in\mathfrak{D}_1$,
with the spectrum, $\mathop{\mathrm{Sp}}(A)=\{\lambda, -\lambda\}$.
\end{corollary}
Moreover, we may remove the surjectively assumption from the
Theorem~\ref{thm:main-nonvanishing-sum}, at least for some polynomials:
\begin{corollary}\label{cor:nonsurjective}%
Under the assumptions of Theorem~\ref{thm:main-nonvanishing-sum},
suppose that a possibly nonsurjective~$\Phi:\mathfrak{D}_1\to\mathfrak{D}_2$ strongly
preserves the zeros of polynomial~$\mathfrak{p}$, defined in
Eq.~(\ref{eq:poly-with-nonvanishing-sum}), but such that the matrix
$$\mathrm{Cof}\,(\mathfrak{p}):=\begin{pmatrix}
\sum_{\sigma(1)=1}\alpha_\sigma &\sum_{\sigma(1)=2} \alpha_\sigma&\dots &\sum_{\sigma(1)=k}\alpha_\sigma\\
\sum_{\sigma(2)=1}\alpha_\sigma &\sum_{\sigma(2)=2} \alpha_\sigma&\dots &\sum_{\sigma(2)=k}\alpha_\sigma\\
\vdots & \vdots &\ddots &\vdots\\
\sum_{\sigma(k)=1}\alpha_\sigma &\sum_{\sigma(k)=2} \alpha_\sigma&\dots
&\sum_{\sigma(k)=k}\alpha_\sigma
\end{pmatrix}\in{\cal M}_k({\mathbb F})$$
is invertible. Then, the conclusions (i) and (ii) remain valid, with
the exception that a field homomorphism $\varphi:{\mathbb F}\to{\mathbb F}$ might be
nonsurjective.
\end{corollary}
\subsection{The proof of Theorem~\ref{thm:main-nonvanishing-sum}}
We first rewrite~$\mathfrak{p}$. Namely, its coefficients satisfy
$$\sum\alpha_\sigma=\sum_{\sigma(1)=1}\alpha_\sigma+\sum_{\sigma(2)=1}
\alpha_\sigma+\dots+ \sum_{\sigma(k)=1}\alpha_\sigma,$$
and since the left-hand side is nonzero, so must be at least one of
the summands on the right. Say,
$0\not=\sum_{\sigma(\imath_0)=1}\alpha_\sigma$. By
dividing~$\mathfrak{p}$ we may assume
$\sum_{\sigma(\imath_0)=1}\alpha_\sigma=1$. Moreover, we may also
assume $\imath_0=1$. Otherwise we would regard the polynomial
$\mathfrak{p}'(x_1,\dots,x_k):=\mathfrak{p}(x_{\tau(1)},\dots,x_{\tau(k)})$
for permutation~$\tau:=(1,\imath_0)$. Obviously,~$\Phi$ would still
strongly preserve the zeros of~$\mathfrak{p}'$. We can now
rewrite~$\mathfrak{p}$ into the form
\begin{equation}\label{eq:p}%
\mathfrak{p}(x_1,\dots,x_k):=\sum_{\sigma(1)=1}\alpha_\sigma
x_{\sigma(1)}\dots x_{\sigma(k)} +\sum_{\sigma(1)\not=1}\alpha_\sigma
x_{\sigma(1)}\dots x_{\sigma(k)};
\end{equation}
where $\sum_{\sigma(1)=1}\alpha_\sigma=1$, and where
$\xi:=\sum_{\sigma(1)\not=1}\alpha_\sigma\not=-1$. Similarly, there
exists $\jmath=\jmath_0$ such that
$$\xi_{\jmath_0k}:=\sum_{\sigma(\jmath_0)=k} \alpha_\sigma\not=0,$$ for
otherwise $0=\sum_{\sigma(k)=k} \alpha_\sigma+\sum_{\sigma(k-1)=k}
\alpha_\sigma+\dots +\sum_{\sigma(1)=k}
\alpha_\sigma=\sum\alpha_\sigma,$ a contradiction! In the sequel, we
will always use the equivalent form~(\ref{eq:p}) of
polynomial~$\mathfrak{p}$.
\bigskip
{It will be beneficial for our further considerations to associate with
each matrix~$A$ the two sets: $\Omega_{\bullet A}$ and
$\Omega_{A\bullet A}$, defined via the polynomial~$\mathfrak{p}$ by}
\begin{align}\label{eq:def_Omega_A}%
\Omega_{\bullet A}&:=\{X\in{\cal
M}_n({\mathbb F});\;\;\mathfrak{p}(X,A,\dots,A)=0\},\\
\Omega_{A\bullet A }&:=\{X\in{\cal
M}_n({\mathbb F});\;\;\mathfrak{p}(A,\dots,A,\fbox{$X$}_{\jmath_0},A,\dots,A)=0\},
\label{eq:def_Omega_A_1}
\end{align}
in the last equation,~$X$ is at the ${\jmath_0}$ position. Note that
$\Omega_{A\bullet A }=\Omega_{\bullet A}$ if~$\jmath_0=1$ ({\small say
for polynomial
$\mathfrak{p}(x_1,x_2):=x_1x_2-x_2x_1$}).\typeout{MyWarning: CHANGE on
22.2.07 input line \the\inputlineno.} Clearly,~$\Omega_{\bullet A}$ and
$\Omega_{A\bullet A }$ are both vector subspaces of~${\cal M}_n({\mathbb F})$.
Moreover,
we can rewrite the condition
$X\in\Omega_{\bullet A}$, respectively, $X\in\Omega_{A\bullet A}$, as
\begin{align}\label{eq:pom3}%
XA^{k-1}+(\beta_2\,AXA^{k-2}+\dots+\beta_{k}\,A^{k-1}X)&=0,
\quad\Bigl(\beta_i:=\sum_{\sigma(1)=i}\alpha_\sigma\Bigr)\\
\intertext{respectively,}
\xi_{\jmath_0 k}\,A^{k-1}X+(\tilde{\beta}_{k-1}\,A^{k-2}XA+\dots+
\tilde{\beta}_{1}\,XA^{k-1})&=0,\quad\Bigl(\tilde{\beta}_i:=
\sum_{\sigma(\jmath_0)=i}\alpha_\sigma\Bigr).\label{eq:pom3.1}%
\end{align}
In particular,~$\Omega_{\bullet A}$ equals the null space of the
elementary operator {$\mathbf{T}_{\bullet A}:{\cal M}_n({\mathbb F})\to{\cal
M}_n({\mathbb F})$}, defined by $X\mapsto
XA^{k-1}+(\beta_2\,AXA^{k-2}+\dots+\beta_{k}\,A^{k-1}X)$.
Similarly, $\Omega_{A\bullet A}$ equals the null space of of the
elementary operator {$\mathbf{T}_{A\bullet A} $}, defined by $X\mapsto
A^{k-1}X+(\hat{\beta}_{k-1}\,A^{k-2}XA+\dots+\hat{\beta}_{1}\,XA^{k-1})$,
where $\hat{\beta}_i:=\tilde{\beta}_i/\xi_{\jmath_0k}$.\bigskip
We proceed with a series of lemmas. The first lemma allows us to
compute the spectrum of elementary operators, in particular, of the
operators $\mathbf{T}_{\bullet A}$ and $\mathbf{T}_{A \bullet A}$. We
present the easy proof for the sake of convenience.
\begin{lemma}\label{lem:Sylvester--Rosenblum}%
Let ${\mathbb F}$ be an arbitrary field. Let $L\in{\cal M}_m({\mathbb F})$ and $M\in
{\cal M}_n({\mathbb F})$ be matrices with $\mathop{\mathrm{Sp}}(L)=\{\lambda\}$ and
$\mathop{\mathrm{Sp}}(M)=\{\mu\}$. Then, the spectrum of elementary operator
$\mathbf{T}_{L,M}:{\cal M}_{m\times n}({\mathbb F})\to{\cal M}_{m\times
n}({\mathbb F})$, defined by $ X\mapsto \beta_t L^tX+\beta_{t-1}
L^{t-1}XM+\dots+\beta_0 XM^t$ is a singleton:
$\mathop{\mathrm{Sp}}(\mathbf{T}_{L,M})=\{\beta_t \lambda^t+\beta_{t-1}
\lambda^{t-1}\mu+\dots+\beta_0\mu^t\}$.
\end{lemma}
\begin{proof}
Let us consider the decomposition~$L=S_L\hat{L}S_L^{-1}$,
where~$\hat{L}$ is the upper-triangular Jordan form of~$L$, and $M=S_M
\breve{M}S_M^{-1}$, where $ \breve{M}$ is lower-triangular Jordan form
of~$M$. It is well-known that the matrix representation of $X\mapsto
L^{t-k}XM^k$, relative to basis
$$E_{11},\dots,E_{n1},E_{12},\dots,E_{n2},\dots,E_{1n},\dots,E_{nn}$$
equals the tensor (Kronecker) product $(M^k)^{{\rm tr}\,}\otimes L^{t-k}$.
Moreover, the matrices $(M^k)^{{\rm tr}\,}\otimes L^{t-k}$; $(k=0,1,\dots)$
are simultaneously similar to~$(\breve{M}^k)^{{\rm tr}\,}\otimes
\hat{L}^{t-k}$ via similarity~$(S_M^{-1})^{{\rm tr}\,}\otimes S_L$.
Hence, the spectrum of $(M^k)^{{\rm tr}\,}\otimes L^{t-k}$ equals the spectrum
of $(\breve{M}^k)^{{\rm tr}\,}\otimes \hat{L}^{t-k}$. Now,
$(\breve{M}^k)^{{\rm tr}\,}$, as well as $\hat{L}^{t-k}$ are both
upper-triangular matrices, with $\mu^k$, respectively, $\lambda^{t-k}$
on the diagonal. Hence, their tensor product remains
upper-triangular, with~$\lambda^{t-k}\mu^k$ on the diagonal.
Consequently, $\mathbf{T}_{L,M}=\sum \beta_k (M^k)^{{\rm tr}\,}\otimes
L^{t-k}$ is similar to the upper-triangular matrix $\sum \beta_k
(\breve{M}^k)^{{\rm tr}\,}\otimes \hat{L}^{t-k}$, with the number $\sum\beta_k
\lambda^{t-k} \mu^k$ on main diagonal. This number is, hence, the only
eigenvalue of $\mathbf{T}_{L,M}$.
\end{proof}
We remark that over the field of complex numbers this fact follows from
the results of Lumer and Rosenblum~\cite{lumer_rosenblum} and
Curto~\cite{curto}, where it was proven in a different way for linear
operators on Hilbert spaces, possibly infinite dimensional. We also
remark that in the case $\mathbf{T}_{\bullet A}=A^{k-1}X+XA^{k-1}$ a
short proof of the corresponding result for Hilbert spaces is presented
by Bhatia and Rosenthal in~\cite[p.~2]{bhatia_rosenthal} and some
further properties of the spectrum can be found in~\cite{curto}, some
additional properties of $\mathbf{T}_{\bullet A}$ are investigated by
Chuai and Tian in~\cite{chuai_tian}.
\begin{lemma}\label{lem:A-circ-B=0<=>AB=0=BA}%
Let $\mu,\nu,\mu',\nu'\in{\mathbb F}$ be scalars, and let
$1+\mu+\nu\not=0$.
If $$PX+\mu\,PXP+\nu XP=0=XP+\mu'\,PXP+\nu' PX$$
holds for some idempotent~$P$ then $PX=0=XP$.
\end{lemma}
\begin{proof}
Postmultiply the equation $PX+\mu\,PXP+\nu XP=0$ with $P$ and subtract.
We derive $PX=PXP$. Likewise $PXP=XP$, from the second equation. Using
$PX=PXP=XP$ again in the first equation, we get $(1+\mu+\nu)PXP=0$, so
$XP=PXP=PX=0$.
\end{proof}
\begin{lemma}\label{lem:Phi(X)=0<==>X=0}%
Suppose $\mathfrak{D}\subseteq{ M}_n({\mathbb F})$ contains all matrices of rank-one.
Then, $A\in {\cal M}_n({\mathbb F})$ is a zero matrix if and only if
$\mathfrak{p}(A,X,\dots,X)=0$ for each~$X\in\mathfrak{D}$.
\end{lemma}
\begin{proof}
We prove only the nontrivial implication. Indeed,
substituting~$X:=E_{ii}$ we have
\begin{align*}
0&=\mathfrak{p}(A,E_{ii},\dots, E_{ii})=\sum_{\sigma(1)=1}\alpha_\sigma
AE_{ii}+ \sum_{\sigma(1)\not\in\{1,k\}}\alpha_\sigma
E_{ii}AE_{ii}+\sum_{\sigma(1)=k}\alpha_\sigma E_{ii}A\\
&= AE_{ii}+\alpha E_{ii}AE_{ii}+\beta_kE_{ii}A
\end{align*}
for $i=1,\ldots, n$.
Premultiply with idempotent~$E_{ii}$, and compare the two equations. We
get $A E_{ii}=E_{ii}AE_{ii}$. Hence, $0=AE_{ii}+(\alpha
AE_{ii}+\beta_{k} E_{ii}A)$. We may sum-up these $n$ equations to get
$0=A\mathop{\mathrm{Id}}\nolimits+(\alpha A\mathop{\mathrm{Id}}\nolimits+\beta_{k}\mathop{\mathrm{Id}}\nolimits A)=(1+\alpha +\beta_{k})A$, so~$A=0$.
\end{proof}
\begin{corollary}\label{cor:Phi-preserves-0}%
Let $\Phi$ satisfy conditions of
Theorem~\ref{thm:main-nonvanishing-sum}. Then, $0\in\mathfrak{D}_1$ if and
only if $0\in\mathfrak{D}_2$. Moreover, $\Phi(A)=0$ if and only if~$A=0$.
\end{corollary}
\begin{proof}
If~$A\not=0$ then, by Lemma~\ref{lem:Phi(X)=0<==>X=0}, there exists
some $X\in\mathfrak{D}_1$ with $\mathfrak{p}(A,X,\dots,X)\not=0$.
Consequently, $\mathfrak{p}(\Phi(A),\Phi(X),\dots,\Phi(X))\not=0$, and
so $\Phi(A)\not=0$. Hence, if~$0\not\in\mathfrak{D}_1$ then
$0\notin\mathop{\mathrm{Im}}\nolimits\Phi=\mathfrak{D}_2$. By surjectivity we likewise see
that~$\Phi(A)\not=0$ implies~$A\not=0$.
\end{proof}
We now characterize when two rank-one nilpotents are scalar multiple of
each other in terms of~$\Omega_{\bullet A}\cap\Omega_{A\bullet A}$,
i.e., in terms of the zeros of polynomial~$\mathfrak{p}$.
\begin{lemma}\label{lem:charact_(nilpot-of-rk1)_with_idempot}%
Let $n\ge 3$, let $\varphi:{\mathbb F}\to{\mathbb F}$ be a nonzero field homomorphism,
and let $N_1,N_2\in{\cal M}_n({\mathbb F})$ be rank-one nilpotents. Assume that
the following condition (i) is satisfied:
\begin{itemize}
\item[(i)] $N_1\in\Omega_{\bullet P}\cap\Omega_{P\bullet P}
\Longleftrightarrow N_2\in\Omega_{\bullet P^\varphi}\cap
\Omega_{P^\varphi\bullet P^\varphi}$ holds for every rank-one
idempotent~$P$.
\end{itemize}
Then $N_2=\lambda N_1^\varphi$ for some nonzero scalar~$\lambda\in
{\mathbb F}$.
\end{lemma}
\begin{proof}
Pick a similarity~$S$ such that $N_1 =SE_{12}S^{-1}$. Then, the
rank-one idempotents $P_3:=SE_{33}S^{-1},\dots, P_n:=S E_{nn}S^{-1}$,
and $P_{n+1}:=S(E_{n2}+E_{nn})S^{-1}$ are orthogonal to~$N_1$. Hence,
$N_1\in\Omega_{\bullet P_i}\cap\Omega_{P_i\bullet P_i}$, which by (i)
implies $N_2\in\Omega_{\bullet P_i^\varphi}\cap
\Omega_{P_i^\varphi\bullet P_i^\varphi}$ for $i=3,\ldots, (n+1)$. Using
the equivalent expressions~(\ref{eq:pom3})--(\ref{eq:pom3.1}), we can
easily rewrite this into
$$\begin{aligned}
0&=\bigl( (S^{-1})^\varphi N_2S^\varphi\bigr)\circ_\bullet E_{33} &
0&=\bigl( (S^{-1})^\varphi
N_2S^\varphi\bigr)\mathop{_\bullet\circ}E_{33}\\
\multispan4{\dotfill}\\
0&=\bigl((S^{-1})^\varphi N_2S^\varphi)\circ_\bullet E_{nn}
&
0&=\bigl((S^{-1})^\varphi N_2S^\varphi)\mathop{_\bullet\circ}E_{nn}\\
0&=\bigl( ( S^{-1})^\varphi N_2S^\varphi\bigr)\circ_\bullet
(E_{n2}+E_{nn}) & 0&=\bigl( ( S^{-1})^\varphi
N_2S^\varphi\bigr)\mathop{_\bullet\circ}(E_{n2}+E_{nn}),
\end{aligned}$$
where $X\circ_\bullet P:=XP+\beta PXP+\beta_k PX$, and where
$X\mathop{_\bullet\circ}P:=PX+\hat{\beta}\,PXP+ \hat{\beta}_{1}\,XP$
for scalars $\beta:=(\beta_{2}+\dots+\beta_{k-1})$,
$\hat{\beta}:=(\tilde{\beta}_{k-1}+\dots+\tilde{\beta}_{2})/\xi_{\jmath_0k}$,
and $\hat{\beta}_1:=\tilde{\beta}_1/\xi_{\jmath_0k}$.
By Lemma~\ref{lem:A-circ-B=0<=>AB=0=BA}, the above equations imply
orthogonality between the nilpotent $(S^{-1})^\varphi N_2S^\varphi$ and
idempotents $E_{33},\dots,E_{nn}, (E_{n2}+E_{nn})$. More precisely, the
first $(n-2)$ equalities give that $(S^{-1})^\varphi N_2S^\varphi$ can
be nonzero only in the upper-left $2\times 2$ block, while the last one
further yields $(S^{-1})^\varphi N_2S^\varphi=\varsigma E_{11}+\lambda
E_{12}$. Since~$N_2$ is nilpotent, $\varsigma=0$. Thus $N_2=\lambda
S^\varphi E_{12}(S^{-1})^\varphi= \lambda S^\varphi
E_{12}^\varphi(S^{-1})^\varphi =\lambda N_1^\varphi$, as desired.
\end{proof}
Similarly we can prove the following:
\begin{lemma}\label{lem:charact_(nilpot-of-rk1)_with_idempot'}%
Under the assumptions of
Lemma~\ref{lem:charact_(nilpot-of-rk1)_with_idempot}, suppose the
following condition (i') is satisfied:
\begin{itemize}
\item[(i')]$N_1\in\Omega_{\bullet P}\cap\Omega_{P\bullet P}
\Longleftrightarrow N_2\in\Omega_{\bullet (P^\varphi)^{{\rm tr}\,}}\cap
\Omega_{(P^\varphi)^{{\rm tr}\,}\bullet (P^\varphi)^{{\rm tr}\,}}$ holds for every
rank-one idempotent~$P$.
\end{itemize}
Then $N_2=\lambda (N_1^\varphi)^{{\rm tr}\,}$ for some nonzero
scalar~$\lambda\in {\mathbb F}$.
\end{lemma}
\begin{proof}
Similar to Lemma~\ref{lem:charact_(nilpot-of-rk1)_with_idempot}.
\end{proof}\bigskip
We next characterize scalar multiples of rank-one idempotents in terms
of~$\Omega_{\bullet A}\cap\Omega_{A\bullet A}$, i.e., in terms of the
zeros of polynomial~$\mathfrak{p}$. This is a chief Lemma in the proof
of Theorem~\ref{thm:main-nonvanishing-sum}.
\begin{lemma}\label{lem:charact_of_minimal_idempotents}%
Fix~$A\in{\cal M}_n({\mathbb F})$. Under the assumptions of
Theorem~\ref{thm:main-nonvanishing-sum}, {\em precisely one} of the
following three possibilities occurs for the set~$\Omega_{\bullet
A}\cap \Omega_{A\bullet A}$:
\begin{enumerate}
\renewcommand{\theenumi}{(\roman{enumi})}%
\item $\Omega_{\bullet A}\cap
\Omega_{A\bullet A}=\{0\}$.
\item $\Omega_{\bullet A}\cap
\Omega_{A\bullet A}$ contains a square-zero matrix of rank-one.
\item $\Omega_{\bullet A}\cap
\Omega_{A\bullet A}={\mathbb F}\,P$ where $P$ is a rank-one idempotent.
\end{enumerate}
\end{lemma}
\begin{proof}Obviously, the listed three possibilities are exclusive.
It, hence, remains to see that at least one of them does occur.
We will rely on the fact that $X\in\Omega_{\bullet A}\cap \Omega_{A\bullet A}$ is
equivalent to Eq.~(\ref{eq:pom3}) and Eq.~(\ref{eq:pom3.1}),
simultaneously.
Now, with the help of similarity we may assume
$A=C_{n_1}(\lambda_1)\oplus\dots\oplus C_{n_r}(\lambda_r)$ is already
in its Jordan block-diagonal form, where~$\lambda_1,\dots,\lambda_r$
are pairwise distinct eigenvalues of~$A$, and an $n_i\times n_i$ matrix
$C_{n_i}(\lambda_i)$ is a sum of all Jordan blocks that correspond to
eigenvalue~$\lambda_i$. We may decompose~$X=\bigl(X_{ij} \bigr)_{1\le
i,j,\le r}$ accordingly. Then,
$A^{s}=C_{n_1}(\lambda_1)^{s}\oplus\dots\oplus C_{n_r}(\lambda_r)^{s}$,
for $s=1,2,\dots,n$, are also block-diagonal, so the $(i,j)$-th block
of Eqs.~(\ref{eq:pom3})--(\ref{eq:pom3.1}) read
\begin{align}\label{eq:pom4}%
X_{ij}C_{n_j}(\lambda_j)^{k-1}+
\bigl(\beta_2\,C_{n_i}(\lambda_i)X_{ij}C_{n_j}(\lambda_j)^{k-2}+\dots+
\beta_{k}\,C_{n_i}(\lambda_i)^{k-1}X_{ij} \bigr)&=0
\intertext{respectively,}
\xi_{\jmath_0k}C_{n_i}(\lambda_i)^{k-1}X_{ij}\!+\!
\bigl(\tilde{\beta}_{k-1}\,C_{n_i}(\lambda_i)^{k-2}X_{ij}C_{n_j}(\lambda_j)+
\!\cdots\!+
\tilde{\beta}_{1}\,X_{ij}C_{n_j}(\lambda_j)^{k-1} \bigr)&\!=\!0\label{eq:pom4.1}%
\end{align}
Obviously, $\mathop{\mathrm{Sp}}\bigl(C_{n_i}(\lambda_i)^{s} \bigr)=\{\lambda_i^s\}$
for any integer~$s$. In view of Lemma~\ref{lem:Sylvester--Rosenblum} we
consequently introduce two polynomials
$$p_{\bullet
A}(\lambda,\mu):=\mu^{k-1}+(\beta_2\,\lambda\mu^{k-2}+\dots+\beta_{k}\,
\lambda^{k-1})\in{\mathbb F}[\lambda,\mu]$$
as well as its counterpart
$$p_{A\bullet A}(\lambda,\mu):=\xi_{\jmath_0k}\lambda^{k-1}+
(\tilde{\beta}_{k-1}\,\lambda^{k-2}\mu+\dots+\tilde{\beta}_{1}\,\mu^{k-1}),$$
and proceed in five steps:\medskip
\noindent {\bf Step 1}. Assume first that for no pair
$(\lambda_i,\lambda_j)\in\mathop{\mathrm{Sp}}(A)\times\mathop{\mathrm{Sp}}(A)$ we have simultaneously
$p_{\bullet
A}(\lambda_i,\lambda_j)=0=p_{A\bullet A}(\lambda_i,\lambda_j)$.
In this case, we note that the left-hand sides of each of the
Eqs.~(\ref{eq:pom4})--(\ref{eq:pom4.1}) define an elementary operator.
Therefore, Lemma~\ref{lem:Sylvester--Rosenblum}, with
$L:=C_{n_i}(\lambda_i)$ and $M:=C_{n_j}(\lambda_j)$ implies that their
spectra are equal to $\{p_{\bullet A}(\lambda_i,\lambda_j)\}$, and
$\{p_{A\bullet A}(\lambda_i,\lambda_j)\}$, respectively. At least one
does not contain zero, and therefore, the corresponding elementary
operator is invertible. The corresponding equation, on the other hand,
implies that $X_{ij}=0$. Hence, all blocks of~$X$ are zero. This
clearly demonstrates $\Omega_{\bullet A}\cap \Omega_{A\bullet A
}=\{0\}$, and we have condition~(i) satisfied.\medskip
\noindent{\bf Step 2}. Suppose next
$p_{\bullet A}(\lambda_i,\lambda_j)=0=p_{A\bullet
A}(\lambda_i,\lambda_j)$ for {\em distinct} $\lambda_i,\lambda_j$. For
simplicity, we assume $(i,j)=(1,2)$.
Here we consider the matrix~$X$ with all, but the $(1,2)$-th, blocks
zero. Then, by Eq.~(\ref{eq:pom4}), all blocks of
$\mathfrak{p}(X,A,\dots,A)$, but the $(1,2)$-th, are also zero. On the
other hand, its $(1,2)$ block equals to
\begin{align*}
\bigl(\mathfrak{p}(X,A,\dots,A)
\bigr)_{12}&=X_{12}C_{n_2}(\lambda_2)^{k-1}+\\
&\mbox{}+
\bigl(\beta_2\,C_{n_1}(\lambda_1)X_{12}C_{n_2}(\lambda_2)^{k-2}+\dots+
\beta_{k}\,C_{n_1}(\lambda_1)^{k-1}X_{12} \bigr)
\end{align*}
Now, write
$C_{n_1}(\lambda_1)=\lambda_1 \mathop{\mathrm{Id}}\nolimits_{n_1}+N_1$ and
$C_{n_2}(\lambda_2)=\lambda_2 \mathop{\mathrm{Id}}\nolimits_{n_2}+N_2$, for some upper-triangular
nilpotents~$N_1,N_2$. Next, consider an $n_1 \times n_2$ matrix
$\hat{X}_{12}$, with all entries, but the upper-right one, equal to
zero. It is easy to see that $\hat{X}_{12}N_2=0=N_1\hat{X}_{12}$, which
in turn, implies that the right side of the above equation simplifies
into $
\hat{X}_{12}\lambda_2^{k-1}+(\beta_2\,\lambda_1X_{12}\lambda_2^{k-2}+\dots+
\beta_{k}\,\lambda_1^{k-1}X_{12}
)=(\lambda_2^{k-1}+\beta_2\,\lambda_1\lambda_2^{k-2}+
\dots+\beta_{k}\lambda_1^{k-1})\hat{X}_{12}=p_{\bullet
A}(\lambda_1,\lambda_2)\hat{X}_{12}=0$.
Consequently, if a block-matrix $X\in {\cal M}_n({\mathbb F})$ has its
$(1,2)$-th block equal to $\hat{X}_{12}$ while the other blocks are
zero then it is a rank-one nilpotent, and
$\mathfrak{p}(X,A,\dots,A)=0$. {Similarly, by Eq.~(\ref{eq:pom4.1}), we
also infer
$\mathfrak{p}(A,\dots,A,\fbox{$X$}_{\jmath_0},A,\dots,A)=0$}.
Therefore, such~$X\in\Omega_{\bullet A}\cap \Omega_{ A\bullet A}$,
which guaranties the condition~(ii).\medskip
\noindent{\bf Step 3}. Suppose we are not under the conditions of
Step 2. We, thus, consider the case $p_{\bullet
A}(\lambda_i,\lambda_i)=0=p_{A\bullet A}(\lambda_i,\lambda_i)$ for some $i$.
Clearly, $p_{\bullet
A}(\lambda_i,\lambda_i)= \lambda_i^{k-1}(1+\xi)$ where
$\xi:=\beta_2+\dots+\beta_{k}$. However, $\xi\not=-1$, so $p_{\bullet
A}(\lambda_i,\lambda_i)=0$ implies $\lambda_i=0$. We now consider two
options, regarding the dimension of the corresponding block
$C_{n_i}(\lambda_i)=C_{n_i}(0)$.\medskip
\noindent{\bf Step 4}. Suppose $\lambda_i=0$ and the corresponding
block~$C_{n_i}(\lambda_i)$ has dimension~$n_i\ge 2$. For simplicity,
assume~$i=1$, so~$\lambda_1=0$.
Now, the first block of~$\mathfrak{p}(X,A,\dots,A)$ equals
$$X_{11}C_{n_1}(0)^{k-1}+
\bigl(\beta_2\,C_{n_1}(0)X_{11}C_{n_1}(0)^{k-2}+\dots+
\beta_{k}\,C_{n_1}(0)^{k-1}X_{11} \bigr)$$
Note that $C_{n_1}(0)$ is an upper-triangular
nilpotent. Since~$n_1\ge 2$, a straightforward computations show that
$E_{1n_1}C_{n_1}(0)^{k-1}+
\bigl(\beta_2\,C_{n_1}(0)E_{1n_1}C_{n_1}(0)^{k-1}+\dots+
\beta_{k}\,C_{n_1}(0)^{k-1}E_{1n_1} \bigr)=0$. Therefore,
$X:=E_{1n_1}\in\Omega_{\bullet A}$. {Similarly, we can also show that
$E_{1n_1}\in\Omega_{A\bullet A}$} and we are in the condition (ii)
again.\medskip
\noindent {\bf Step 5}. Finally, suppose we {\em are not} under the
conditions of Step 2 and there exists $i$ such that~$\lambda_i=0$ with
the corresponding block~$C_{n_i}(\lambda_i)=C_{n_i}(0)$ of
dimension~$n_i=1$. Again, for simplicity $i=1$, so that $A=0\oplus C$,
where~$C:= C_{n_2}(\lambda_2)\oplus\dots\oplus C_{n_r}(\lambda_r)$ is
an $(n-1) \times (n-1)$ matrix. Recall
that~$\lambda_2,\dots,\lambda_r\not=0$, {so~$C$ is invertible}.
It is straightforward to see that ${\mathbb F} E_{11}\subseteq \Omega_{\bullet
A}\cap \Omega_{A\bullet A}$ in this case. Let us show that {also} ${\mathbb F}
E_{11}\supseteq \Omega_{\bullet A}\cap \Omega_{A\bullet A}$.\medskip
Retaining the block structure of~$X$, the
Eqs.~(\ref{eq:pom4})--(\ref{eq:pom4.1}) simplify for the blocks in the
first row/column into $X_{1j} C_{n_j}(\lambda_j)^{k-1}=0$,
respectively, into $\xi_{\jmath_0k}\,C_{n_j}(\lambda_j)^{k-1}X_{j1}
=0$. Since $C_{n_j}(\lambda_j)$ are invertible for $j=2,\dots,r$, and
since $\xi_{\jmath_0k}\not=0$, we get $X_{1j}=0=X_{j1}$ whenever $j\ge
2$.
Consider also the block $X_{st}$ for $s,t\ge 2$. Now, if $s\not=t$, it
is impossible to have simultaneously $p_{\bullet
A}(\lambda_s,\lambda_t)=0=p_{ A\bullet A}(\lambda_s,\lambda_t)$, by
conditions of Step~2. This remains true if $s=t\ge 2$, for otherwise
$p_{\bullet A}(\lambda_s,\lambda_s)=0$, which would wrongly imply
$\lambda_s=0$. Then, however, we may copy the arguments from Step~1 to
deduce $X_{st}=0$. Therefore, the only possible nonzero block of~$X$ is
$X_{11}$, and so $X=\alpha E_{11}$ for some scalar~$\alpha$. Therefore,
$\Omega_{\bullet A}\cap\Omega_{A\bullet A }={\mathbb F} E_{11}$. This gives the
case~(iii) with $P=E_{11}$.
\end{proof}
Recall that ${\cal I}^1$ is the set of rank-one idempotents in ${\cal
M}_n({\mathbb F})$.
\begin{corollary}\label{cor:Phi-perserves-rk1-idempote-in-both-directions}%
Let conditions of Theorem~\ref{thm:main-nonvanishing-sum} be satisfied.
Then
\begin{itemize}
\item[(i)] $\Phi({\cal I}^1)\subseteq {\mathbb F} {\cal I}^1$.
\item[(ii)] If $\Phi(X)\in {\cal I}^1$ then $X=\alpha Y$ for certain
$\alpha\in {\mathbb F}\setminus\{0\}$ and $Y\in {\cal I}^1$.
\end{itemize}
\end{corollary}
\begin{proof}
{\bf Step 1}. Pick a rank-one idempotent~$P$. Then, $A:=\mathop{\mathrm{Id}}\nolimits-P$ is an
idempotent of rank $(n-1)$. Thus, $A\in\mathfrak{D}_1$. The direct
calculations show that $\Omega_{\bullet A}\cap \Omega_{A\bullet
A}=\Omega_{\bullet(\mathop{\mathrm{Id}}\nolimits-P)}\cap \Omega_{(\mathop{\mathrm{Id}}\nolimits-P)\bullet (\mathop{\mathrm{Id}}\nolimits-P)}$
consists precisely of those matrices~$X$ which satisfy
$$XA+\beta AXA+\beta_{k}
AX=0=\xi_{\jmath_0k}\,AX+\tilde{\beta} AXA+\tilde{\beta}_{1} XA;
\quad(\beta:=\beta_2+\dots+\beta_{k-1}).$$ By
Lemma~\ref{lem:A-circ-B=0<=>AB=0=BA}, $X$ is orthogonal to
idempotent~$A=\mathop{\mathrm{Id}}\nolimits-P$. Hence, $X=\lambda P$, and so, $\Omega_{\bullet
A}\cap \Omega_{A\bullet A}={\mathbb F}\,P$.\medskip
{\bf Step 2}. Since $\Phi$ preserves zeros of $\mathfrak{p}$, the first
step implies $\Phi(P)\in\Omega_{\bullet\Phi(A)}\cap
\Omega_{\Phi(A)\bullet \Phi(A)}$ . By
Corollary~\ref{cor:Phi-preserves-0},~$\Phi(P)\not=0$, so
$\Omega_{\bullet\Phi(A)}\cap \Omega_{\Phi(A)\bullet
\Phi(A)}\not=\{0\}$. By Lemma~\ref{lem:charact_of_minimal_idempotents}
it follows that $\Omega_{\bullet\Phi(A)}\cap \Omega_{\Phi(A)\bullet
\Phi(A)}$ either is equal to a scalar multiple of an idempotent or
contains a rank-one nilpotent. We assume erroneously the later, i.e.,
that there exists a rank-one nilpotent $Y\in\Omega_{\bullet\Phi(A)}\cap
\Omega_{\Phi(A)\bullet \Phi(A)}$. By the surjectivity of $\Phi$ it
follows that $Y=\Phi(X)$ for some~$X\in \mathfrak{D}_1$. Since $\Phi$
preserves zeros of $\mathfrak{p}$ strongly we have
$X\in(\Omega_{\bullet A}\cap\Omega_{A\bullet A })\cap\mathfrak{D}_1={\mathbb F} P\cap
\mathfrak{D}_1$.
By Corollary~\ref{cor:Phi-preserves-0}, $X\not=0$. On the other hand,
$Y^2=0$ since $Y$ is a nilpotent of rank-one. {Hence},
$\mathfrak{p}(Y,\dots, Y)=0$, so also
$0=\mathfrak{p}(X,\dots,X)=(1+\xi)X^k$. This is clearly a contradiction
since $X\in{\mathbb F} P$. Hence, $\Omega_{\bullet\Phi(A)}\cap
\Omega_{\Phi(A)\bullet \Phi(A)}$ contains no rank-one nilpotents.
{Therefore}, $\Phi(P)\in\Omega_{\bullet\Phi(A)}\cap
\Omega_{\Phi(A)\bullet \Phi(A)}={\mathbb F} Q$, for some rank-one
idempotent~$Q$. This proves~(i).\medskip
{\bf Step 3}. Conversely, if~$\Phi(X)$ is an idempotent of rank-one,
then ${\mathbb F}\,\Phi(X)=\Omega_{\bullet B}\cap\Omega_{B\bullet B}$ for
$B:=\mathop{\mathrm{Id}}\nolimits-\Phi(X)\in\mathfrak{D}_2$. Since $\Phi$ is surjective and preserves
zeros of $\mathfrak{p}$ strongly, we can prove that $X\in{\mathbb F}\, P$ for
some rank-one idempotent~$P$ in a similar way, as in the proof of Item
{(i)}.
\end{proof}
\begin{remark}\label{rem_redefined_Phi}%
Since the polynomial $\mathfrak{p}$ is homogeneous, the assumptions and
the conclusion of Theorem~\ref{thm:main-nonvanishing-sum} will not be
affected if we replace~$\Phi$ by a mapping $\hat \Phi:A\mapsto
\delta(A)\cdot\Phi(A)$, where~$\delta: {\mathfrak{D}_1}\to {\mathbb F}\backslash\{0\}$
is a scalar function. We may define~$\delta$ in such a way
that~$\hat{\Phi}$ preserves rank-one idempotents (i.e., $\hat
\Phi({\cal I}^1)\subseteq {\cal I}^1$), while~$\hat{\Phi}(A)=\Phi(A)$
for any other matrix from~$\mathfrak{D}_1$.
The redefined~$\hat{\Phi}$ may not be surjective, but we clearly have
${\cal I}^1\subseteq\mathop{\mathrm{Im}}\nolimits\hat{\Phi}$.
\end{remark}\bigskip
We are ready now to prove the main result.
\begin{proof}[Proof of Theorem~\ref{thm:main-nonvanishing-sum}]
We proceed in several steps.\medskip
{\bf Step 1}. {\em The transformation~$\hat\Phi$ preserves
orthogonality among rank-one idempotents.\/}
Indeed, suppose~$P,Q$ are two orthogonal rank-one idempotents. Then
$Q\in\Omega_{\bullet P}\cap\Omega_{P\bullet P}$. Therefore,
$\hat\Phi(Q)\in\Omega_{\bullet\hat\Phi(P)}\cap
\Omega_{\hat\Phi(P)\bullet \hat\Phi(P)}$, so that $X\hat\Phi(P)+\beta
\hat\Phi(P)X\hat\Phi(P)+\beta_{k} \hat\Phi(P)X=0
=\hat\Phi(P)X+\hat{\beta} \hat\Phi(P)X\hat\Phi(P)+\hat{\beta}_{1}
X\hat\Phi(P)$ for $X:=\hat\Phi(Q)$. The orthogonality between
$X=\hat\Phi(Q)$ and $\hat\Phi(P)$ now follows from
Lemma~\ref{lem:A-circ-B=0<=>AB=0=BA}.\medskip
{\bf Step 2.} {\em Let us show that $\hat\Phi$ is injective
transformation on the set of rank-one idempotents.\/}
We assume erroneously that~$\hat{\Phi}(P_1)=\hat{\Phi}(P_2)$ for some
{\em distinct} rank-one idempotents $P_1={\bf x}_1{\bf f}_1^{{\rm tr}\,}$ and
$P_2={\bf x}_2{\bf f}_2^{{\rm tr}\,}$. Then either ${\bf x}_1,{\bf x}_2$ are
linearly independent or ${\bf f}_1,{\bf f}_2$ are linearly independent
or both. Assume that ${\bf x}_1,{\bf x}_2$ are. Then, we can construct
a rank-one idempotent~$Q$ such that
$\mathfrak{p}(Q,P_1,\dots,P_1)=QP_1+\beta P_1QP_1+\beta_k P_1Q=0$,
but~$\mathfrak{p}(Q,P_2,\dots,P_2)=QP_2+\beta P_2QP_2+\beta_k
P_2Q\not=0$.
Indeed, we choose any nonzero ${\bf y}$ with
${\bf f}_1^{\rm tr}\,{\bf y}=0={\bf f}_2^{\rm tr}\,{\bf y}$.
Suppose first ${\bf y}=\mu_1 {\bf x}_1+\mu_2 {\bf x}_2$.
Then, $\mu_2\not=0$, since otherwise ${\bf y}={\mu_1} {\bf x}_1$,
contradicting ${\bf f}_1^{{\rm tr}\,}{\bf x}_1=1$, ${\bf f}_1^{{\rm tr}\,}{\bf y}=0$.
Now, as ${\bf x}_1,{\bf x}_2$ are linearly independent, we may choose
${\bf g}$ such that $ {\bf g}^{\rm tr}\,{\bf x}_1=0$ and ${\bf g}^{\rm tr}\,{\bf x}_2
=1/\mu_2$. Then ${\bf g}^{\rm tr}\,{\bf y}=1$. Now, if ${\bf y}, {\bf
x}_1,{\bf x}_2$ are linearly independent, we may choose ${\bf g}$ such
that ${\bf g}^{\rm tr}\,{\bf x}_1 =0$, ${\bf g}^{\rm tr}\,{\bf x}_2 =1$, ${\bf
g}^{\rm tr}\,{\bf y}=1$. In both cases, $Q:={\bf
y} {\bf g}^{{\rm tr}\,}$ is the required idempotent. \\
For the chosen $Q$ we have
$$0=\mathfrak{p}(\hat\Phi(Q),\hat\Phi(P_1),\dots,\hat\Phi
(P_1))=\mathfrak{p}(\hat\Phi(Q),\hat\Phi(P_2),\dots,\hat\Phi
(P_2))\not=0,$$ a contradiction.
On the other hand, if~${\bf f}_1,{\bf f}_2$ are independent, we can
similarly find~$Q$ with $\mathfrak{p}(P_1,Q,\dots,Q)=0$,
but~$\mathfrak{p}(P_2,Q,\dots,Q)\not=0$. As before, this leads to a
contradiction. {Indeed}, $\hat\Phi(P_1)\ne \hat\Phi(P_2)$.
\medskip
{\bf Step 3.} {\em Now we can characterize the action of $\hat\Phi$ on
the set of rank-one idempotents.\/}
Injective mappings on rank-one idempotents, which preserve
orthogonality are classified in
\v{S}emrl~\cite[Theorem~2.3]{semrl:commutativity} ({\small this result
is stated only for~${\mathbb F}=\mathbb{C}$, but it was already remarked by the
author that the proofs are valid for {any algebraically closed field}
}). It follows that there exists a field
homomorphism~$\varphi:{\mathbb F}\to{\mathbb F}$, and invertible matrix~$T$ such that
either $\hat\Phi(P)=TP^\varphi T^{-1}$ for every rank-one
idempotent~$P$, or else $\hat\Phi(P)=T(P^\varphi)^{{\rm tr}\,} T^{-1}$ for
every rank-one idempotent~$P$.\medskip
{\bf Step 4.} {\em Field homomorphism $\varphi$ is surjective.\/}
It suffices to see that the restriction of $\hat\Phi$ on the set of
rank-one idempotents, $\hat\Phi|_{{\cal I}^{1}}:{\cal I}^1\to{\cal
I}^1$, is surjective. Let us take any ${F}\in {\cal I}^1$. By
Remark~\ref{rem_redefined_Phi}, ${\cal I}^1\subseteq\mathop{\mathrm{Im}}\nolimits\hat\Phi$, thus
${F}=\hat\Phi(X)$ for certain $X\in \mathfrak{D}_1$.
Corollary~\ref{cor:Phi-perserves-rk1-idempote-in-both-directions} shows
that $X=\lambda P$ for some $P\in{\cal I}^1$. Now, choose pairwise
orthogonal $P_2,\dots,P_n\in{\cal I}^1$ that are also orthogonal
to~$P$. Clearly, $(\lambda P)\in\Omega_{\bullet P_i}\cap
\Omega_{P_i\bullet P_i}$, so also $\hat\Phi(\lambda
P)\in\Omega_{\bullet \hat\Phi(P_i)}\cap \Omega_{\hat\Phi(P_i)\bullet
\hat\Phi(P_i)}$. As in Step~1 we derive that ${F}=\hat\Phi(\lambda P)$
is orthogonal to~$\hat\Phi(P_2),\dots,\hat\Phi(P_n)$. On the other
hand, however, $\hat\Phi(P),\hat\Phi(P_2),\dots,\hat\Phi(P_n)$ are~$n$
pairwise orthogonal rank-one idempotents as well. This is possible only
when $\hat\Phi(P)={F}$, and the result follows.
\medskip
{\bf Step 5.} {\em The conclusion of
Theorem~\ref{thm:main-nonvanishing-sum} is valid for all {\em
non-nilpotent} rank-one matrices.\/}
Similar to Step 4 it can be shown that~$\hat\Phi(\lambda P) \in {\mathbb F}\,
\hat\Phi(P)$ for any $P\in{\cal I}^1$, i.e., there exists a
transformation $\delta' : {\mathfrak{D}_1 \to {\mathbb F}\backslash\{0\}}$ such that
$\hat\Phi(\lambda P)=\delta'(\lambda P)\hat\Phi(P)$ for all
$\lambda\in{\mathbb F}$, $P\in {\cal I}^1$.
\medskip
{\bf Step 6.} {\em $\hat\Phi$ preserves the set of rank-one
nilpotents.\/}
To see this, we choose any rank-one nilpotent~$N$. Using similarity, we
may assume~$N=E_{12}$. Then, we can find $n-2$ pairwise orthogonal
rank-one idempotents $E_{33},\dots,E_{nn}$ that are also orthogonal
to~$N$. Clearly, $N\in\Omega_{\bullet E_{ii}}\cap \Omega_{E_{ii}\bullet
E_{ii}}$, so also $\hat\Phi(N)\in\Omega_{\bullet \hat\Phi(E_{ii})}\cap
\Omega_{\hat\Phi(E_{ii})\bullet \hat\Phi(E_{ii})}$ for $n-2$ pairwise
orthogonal rank-one idempotents~$\hat\Phi(E_{ii})$, $i=3,\dots,n$.
Using similarity in the image space, we may
assume~$\hat\Phi(E_{ii})=E_{ii}$. Thus, we have
$\hat\Phi(N)\in\Omega_{\bullet E_{ii}}\cap \Omega_{E_{ii}\bullet
E_{ii}}$ for $i=3,\dots,n$. As in Step~1 we derive that
idempotents~$E_{ii}$ are orthogonal on $\hat\Phi(N)$ for $i=3,\dots,n$.
Consequently,~$\hat\Phi(N)$ could be nonzero only at the upper left $2
\times 2$ block. On the other hand,
$\mathfrak{p}(N,\dots,N)=(1+\xi)N^k=0$. Thus
also~$0=1/(1+\xi)\mathfrak{p}(\hat\Phi(N),\dots,\hat\Phi(N))$, i.e.,
$\hat\Phi(N)^k=0$. Hence,~$\hat\Phi(N)$ is a nonzero nilpotent and all
its non-zero elements are concentrated in the $2\times 2$ upper-left
block. Thus $\hat\Phi(N)$ is a nilpotent of rank-one.\medskip
{\bf Step 7.} {\em The end of the proof}.
Finally, consider the redefined
$\tilde{\Phi}:A\mapsto T^{-1}\hat\Phi (A)T$. By Step~3 either
$\tilde{\Phi}(P)\equiv P^\varphi$ for rank-one idempotents~$P$, or else
$\tilde{\Phi}(P)\equiv (P^\varphi)^{{\rm tr}\,}$ for rank-one idempotents~$P$.
Hence, applying~Lemma~\ref{lem:charact_(nilpot-of-rk1)_with_idempot}
({\small respectively,
Lemma~\ref{lem:charact_(nilpot-of-rk1)_with_idempot'}}) to rank-one
nilpotents $N_1:=N$ and $N_2:=\tilde{\Phi}(N)$ we obtain
$\tilde{\Phi}(N)=\delta''(N)\cdot N^\varphi$ for certain $\delta'':
{\cal M}_n({\mathbb F})\to {\mathbb F}$ ({\small respectively,
$\tilde{\Phi}(N)=\delta''(N)\cdot (N^\varphi)^{{\rm tr}\,}$}). Obviously, this
holds for any rank-one nilpotent~$N$.
\end{proof}
\subsection{Proof of Corollaries}
It will be beneficial to regard ${\mathbb F}^n$ as the space of matrices of
dimension $n$--by--$1$, i.e., {\em column vectors}. Thus, any rank--one
matrix~$A\in{\cal M}_n({\mathbb F})$ can be written as $A={\bf x}{\bf
f}^{{\rm tr}\,}$ for suitably chosen ${\bf x},{\bf f}\in{\mathbb F}^n$. Its trace
then equals $\mathrm{Tr}\, A={\bf f}^{{\rm tr}\,}{\bf x}$.
Now, to prove Corollary~\ref{cor:TAllMatr}, we will rely on the
folowing result due to Bre\v{s}ar and
\v{S}emrl~\cite[Thm.~2.4]{bresar_semrl}, which we state slightly
changed, recasting it into our framework:
\begin{lemma}\label{lem:bresar_semrl}%
Let ${\mathbb F}$ be an infinite field with ${\rm char}\, {\mathbb F}\not= 2$, and let
$R_1,R_2,R_3 \in{\cal M}_n({\mathbb F})$ be three matrices. Then (i) implies
(ii) below.
\begin{itemize}
\item[(i)] The vectors $R_1{\bf u}$, $R_2{\bf
u}$, and $R_3{\bf u}$ are linearly dependent for every ${\bf u}\in
{\mathbb F}^n$.
\item[(ii)]Either $R_1,R_2,R_3$ are linearly dependent, or there
exist ${\bf v},{\bf w},{\bf z} \in {\mathbb F}^n$ such that $\mathop{\mathrm{Im}}\nolimits R_i\subseteq
\mathop{\rm Lin}_{{\mathbb F}}\{{\bf v},{\bf w},{\bf z}\}$ for $i = 1,2,3$, or there exists
a rank-one idempotent $P\in{\cal M}_n({\mathbb F})$ such that
$$\dim\mathop{\rm Lin}\nolimits_{{\mathbb F}}\{(\mathop{\mathrm{Id}}\nolimits - P)R_1, \;(\mathop{\mathrm{Id}}\nolimits - P)R_2,\; (\mathop{\mathrm{Id}}\nolimits - P)R_3\} = 1.$$
\end{itemize}
\end{lemma}
With its help, the following generalization of
Lemma~\ref{lem:charact_(nilpot-of-rk1)_with_idempot} can be proven:
\begin{lemma}\label{lem:A=lambda*B_iff...}%
Let~$n\ge 4$ be an integer, let $A,B\in {\cal M}_n({\mathbb F})$ be nonzero
matrices, let $\varphi:{\mathbb F}\to{\mathbb F}$ be a nonzero field homomorphism, and
let~$\alpha,\beta\in{\mathbb F}$. Assume that the following condition (i) is
satisfied:
\begin{itemize}
\item[(i)] {$NAP+\alpha PAN=0\Longleftrightarrow
0=N^\varphi B P^\varphi+\beta P^\varphi B N^\varphi$ holds for every
rank-one idempotent~$P$ and every rank-one matrix~$N$
with~$PN=0=NP$.}
\end{itemize}
Then there exists $\gamma,\mu\in{\mathbb F}$ such that $B=\gamma
A^\varphi+\mu\mathop{\mathrm{Id}}\nolimits$.
\end{lemma}
\begin{proof}
Pick any nonzero vector ${\bf x}\in{\mathbb F}^n$ and assume erroneously that
the vector ${\bf b}:=B{\bf x}^\varphi\not\in\mathop{\rm Lin}_{{\mathbb F}}\{A^\varphi {\bf
x}^\varphi,{\bf x}^\varphi\}$. Denote ${\bf a}:=A{\bf x}$, and let
${\bf f}_1,\dots,{\bf f}_{l}$ be a basis of~$\{{\bf a},{\bf
x}\}^\bot:=\{{\bf f}\in{\mathbb F}^n;\;\; {\bf f}^{{\rm tr}\,}{\bf a}=0={\bf
f}^{{\rm tr}\,}{\bf x}\}$ ({\small here,~$l=n-2$ or $n-1$ if ${\bf x}$ and
${\bf a}$ are linear independent or not, correspondingly}). Since the
rank of a matrix equals the maximal size of its nonzero minors, the
vectors~${\bf f}_1^\varphi,\dots,{\bf f}_{l}^\varphi$ are also linearly
independent. Hence, they form a basis of~$\{{\bf a}^\varphi,{\bf
x}^\varphi\}^\bot$. Now,~${\bf b}\not\in\mathop{\rm Lin}_{{\mathbb F}}\{{\bf
a}^\varphi,{\bf x}^\varphi\}$, so~$({\bf f}_j^\varphi)^{{\rm tr}\,}{\bf
b}\not=0$ for at least one~$j$. Consequently, there exists ${\bf
f}={\bf f}_j$ such that
$${\bf f}^{{\rm tr}\,}{\bf x}=0={\bf f}^{{\rm tr}\,}A{\bf x}, \quad\hbox{and}\quad ({\bf
f}^\varphi)^{{\rm tr}\,}B{\bf x}^\varphi\not=0.$$
Since~$n > 2$ we can find~${\bf y}$
such that
\begin{equation} \label{Eq:x_y}
{\bf x}\not\in\mathop{\rm Lin}\nolimits_{{\mathbb F}}\{{\bf y},A{\bf y}\}.
\end{equation}
Indeed, write
${\mathbb F}^n=\mathop{\rm Lin}_{{\mathbb F}}\{{\bf x}\}\oplus M$. If $\mathop{\mathrm{Ker}}\nolimits (A|_M)\not=0$, then any
nonzero ${\bf y}\in\mathop{\mathrm{Ker}}\nolimits (A|_M)$ satisfies Eq.~(\ref{Eq:x_y}). Assume
now that $\mathop{\mathrm{Ker}}\nolimits (A|_M)=0$ and ${\bf x}\in\mathop{\rm Lin}_{{\mathbb F}}\{{\bf y},A{\bf y}\}$
for each~${\bf y}\in M$. Then $A|_M{\bf y}=\lambda_{\bf y}{\bf
x}+\mu_{\bf y}{\bf y}$. Since ${\bf x}\notin M$, we could deduce that
$\lambda_{\bf y}$ is a linear functional, on the space~$M$ with $\dim
M\ge 2$. Hence, $\lambda_{\bf y}=0$ for at least one nonzero ${\bf
y}={\bf y}_0\in M$. For this ${\bf y}_0$ we have $A{\bf y}_0=\mu_{{\bf
y}_0}{\bf y}_0$ and then $\mathop{\rm Lin}_{{\mathbb F}}\{{\bf y}_0,A{\bf y}_0\}={\mathbb F} {\bf
y}_0$. However, ${\bf x}\notin{\mathbb F} {\bf y}_0 \subset M$ --- a
contradiction.\medskip
Now, by~(\ref{Eq:x_y}), we may choose a vector~$\bf g$ with ${\bf
g}^{{\rm tr}\,}{\bf y}=0={\bf g}^{{\rm tr}\,} A{\bf y}$, but ${\bf g}^{{\rm tr}\,}{\bf
x}=1$. Then, $P:={\bf x}{\bf g}^{{\rm tr}\,}$ is an idempotent of rank-one,
and $N:={\bf y}{\bf f}^{{\rm tr}\,}$ is a matrix of rank-one, and we have
$PN=0=NP$. Moreover, $NAP+\alpha PAN= ({\bf f}^{{\rm tr}\,}A{\bf x})\,{\bf
y}{\bf g}^{{\rm tr}\,}+\alpha ({\bf g}^{{\rm tr}\,}A{\bf y})\,{\bf x}{\bf f}^{{\rm tr}\,}=
0+\alpha\cdot 0=0$. Consequently, the condition~(i) implies
\begin{align*}
0&= ( ({\bf f}^\varphi)^{{\rm tr}\,}B{\bf x}^\varphi)\cdot {\bf y}^\varphi
({\bf g}^\varphi)^{{\rm tr}\,}+
\beta\cdot (({\bf g}^\varphi)^{{\rm tr}\,} B{\bf y}^\varphi)\cdot {\bf x}^\varphi({\bf
f}^\varphi)^{{\rm tr}\,}.
\end{align*}
However, the first summand on the right is nonzero. Hence, the right
side is nonzero, since ${\bf y}^\varphi({\bf g}^\varphi)^{{\rm tr}\,} $ and
${\bf x}^\varphi({\bf f}^\varphi)^{{\rm tr}\,}$ are linearly independent
matrices ({\small namely, $({\bf g}^\varphi)^{{\rm tr}\,}{\bf x}^\varphi=1$,
while $({\bf f}^\varphi)^{{\rm tr}\,}{\bf x}^\varphi=0$}). This contradiction
establishes that
\begin{equation}\label{eq:Bx_in_(x,Ax)}%
B{\bf x}^\varphi\in\mathop{\rm Lin}\nolimits_{{\mathbb F}}\{A^\varphi{\bf x}^\varphi,{\bf
x}^\varphi\}\qquad \hbox{ for any ${\bf x}\in{\mathbb F}^n$.}
\end{equation} \medskip
By Eq.~(\ref{eq:Bx_in_(x,Ax)}), the vectors $B{\bf x}^\varphi, {\bf
x}^\varphi, A^\varphi {\bf x}^\varphi$ are always ${\mathbb F}$--linearly
dependent. Let us show that even more is true: indeed the matrices
$B,\mathop{\mathrm{Id}}\nolimits,A^\varphi$ are locally linearly dependent, i.e., for any ${\bf
z}\in {\mathbb F}^n$ the vectors $B{\bf z},{\bf z}, A^\varphi {\bf z}$ are
linearly dependent. To demonstrate this, we consider a matrix $\Xi({\bf
z}):=[B{\bf z},{\bf z},A^\varphi{\bf z}]$ with three columns. By
Eq.~(\ref{eq:Bx_in_(x,Ax)}), if ${\bf z}={\bf x}^\varphi$ for a certain
${\bf x}\in{\mathbb F}^n$, then all its $3$--by--$3$ minors must vanish.
Consider any such minor. It is a polynomial $q(z_1,\ldots, z_n)\in
{\mathbb F}[z_1,\ldots, z_n]$, where ${\bf z}=[z_1,\ldots, z_n]^{\rm tr}$. By
Eq.~(\ref{eq:Bx_in_(x,Ax)}) this polynomial vanishes identically
whenever the variables take the values from a subfield ${\cal
O}:=\varphi({\mathbb F})\subseteq{\mathbb F}$, i.e., for any values $\alpha_1, \ldots,
\alpha_n \in {\cal O}$ it holds that $q(\alpha_1,\ldots, \alpha_n)=0$.
Now, being algebraically closed,~${\mathbb F}$ and hence also ${\cal
O}=\varphi({\mathbb F})$ have infinitely many elements. It is easy to see then
that then,~$q$ is a zero polynomial. {\small For the sake of
completeness we sketch the proof here. We will write $q$ in the form
$$q(z_1,\dots,z_n)=a_n(z_1,\dots,z_{n-1})z_n^n
+\dots+a_1(z_1,\dots,z_{n-1})z_n+ a_0(z_1,\dots,z_{n-1}).$$ By the
assumptions, this vanishes whenever $z_1,\dots,z_n\in{\cal O}$. Now, at
each fixed $z_1,\dots,z_{n-1}\in{\cal O}$, this is a polynomial in only
one variable, $z_n$. However, it is zero for infinitely many values of
$z_n\in{\cal O}$. Hence, $x\mapsto q(z_1,\dots,z_{n-1},x)$ is a zero
polynomial for each fixed $(z_1,\dots,z_{n-1})\in {\cal O}^{n-1}$. That
is, all its coefficients, $a_i(z_1,\dots,z_{n-1})$ are zero for any
$(z_1,\dots,z_{n-1})\in{\cal O}^{n-1}$. It remains to show that
$a_i(z_1,\dots,z_{n-1})$ are identically zero, not only for any choice
of $z_1,\dots,z_{n-1}\in {\cal O}$, but also for any choice of
$z_1,\dots,z_{n-1}\in{\mathbb F}$. Now,
we may repeat the aforesaid procedure with each $a_i(z_1,\dots,z_{n-1})$:
Write it as
$$a_i(z_1,\dots,z_{n-1})=b_{im}(z_1,\dots,z_{n-2})z_{n-1}^m+\dots +
b_{i0}(z_1,\dots,z_{n-2})$$
and argue as before to deduce that $b_j(z_1,\dots,z_{n-2})$ vanishes
for any choice of $z_1,\dots,z_{n-2}\in{\cal O}$. Continuing in this
way we obtain at the end certain polynomials $c_l(z_1)\in{\mathbb F}[z_1]$
which are zero for any value $z_1\in {\cal O}$. It follows that
$c_l(z_1)$ is zero for infinitely many values of $z_1$, i.e., that
$c_l(z_1)$ is a zero polynomial. By the backward induction, we get that
all coefficients~$a_i(z_1,\dots,z_{n-1})$ are zero, i.e., that $q$ is
indeed a zero polynomial. }
Therefore, $q(z_1,\dots,z_n)=0$ holds for any $z_1,\ldots,z_n\in{\mathbb F}$.
We repeat this with all $3$--by--$3$ minors to deduce that ${\rm rk \,}\Xi({\bf
z})\le 2$ for any ${\bf z}\in{\mathbb F}^n$, as claimed.\medskip
Consequently, $(R_1,R_2,R_3):=(B,\mathop{\mathrm{Id}}\nolimits,A^\varphi)$ are locally linearly
dependent. We can now invoke Bre\v{s}ar and \v{S}emrl's theorem, see
Lemma~\ref{lem:bresar_semrl} in this text. Since~$R_2=\mathop{\mathrm{Id}}\nolimits$ and
$\dim(\mathop{\mathrm{Im}}\nolimits(\mathop{\mathrm{Id}}\nolimits))=n\gneq 3$, the only three possibilities left to
consider are (a) $A^\varphi=\lambda\mathop{\mathrm{Id}}\nolimits$, or (b)
$B=\gamma\,A^\varphi+\mu\mathop{\mathrm{Id}}\nolimits$, or
\begin{equation}
(\mathop{\mathrm{Id}}\nolimits-Q)B=\lambda(\mathop{\mathrm{Id}}\nolimits-Q)\mathop{\mathrm{Id}}\nolimits,\qquad
(\mathop{\mathrm{Id}}\nolimits-Q)A^\varphi=\lambda'(\mathop{\mathrm{Id}}\nolimits-Q)\mathop{\mathrm{Id}}\nolimits\tag{c}
\end{equation} for some rank-one idempotent~$Q$.
Under (a), $B$ must also be a scalar, in view of
Eq.~(\ref{eq:Bx_in_(x,Ax)}). So, under (a)--(b) we are done.\medskip
Consider lastly~(c). Decomposing $B=(\mathop{\mathrm{Id}}\nolimits-Q)B+QB$, and using Eq.~(c),
easily reveals $B=\lambda\mathop{\mathrm{Id}}\nolimits+Q(B-\lambda\mathop{\mathrm{Id}}\nolimits)=\lambda\mathop{\mathrm{Id}}\nolimits+\hat{\bf
u}\hat{\bf v}_B^{{\rm tr}\,}$ for some column vectors $\hat{\bf u},\hat{\bf
v}_B$. Likewise, $A^\varphi=\lambda'\mathop{\mathrm{Id}}\nolimits+\hat{\bf u}\hat{\bf
v}_A^{{\rm tr}\,}$. By passing the appropriate scalar to the other term (in
$\hat{\bf u}\hat{\bf v}_B^{{\rm tr}\,}$, $\hat{\bf u}\hat{\bf v}_A^{{\rm tr}\,}$), we
may assume that at least one entry of vector~$\hat{\bf u}$ equals~$1$.
Then, from $\lambda'\mathop{\mathrm{Id}}\nolimits+\hat{\bf u}\hat{\bf
v}_A^{{\rm tr}\,}=A^\varphi\in{\cal M}_n(\varphi({\mathbb F}))$ it follows that
$\hat{\bf u},\hat{\bf v}_A\in\varphi({\mathbb F}^n)$, and also
$\lambda'\in\varphi({\mathbb F})$. Let ${\bf u}$, ${\bf v}_A$, and $\lambda''$
be such that $\hat{\bf u}={\bf u}^{\varphi} $, $\hat{\bf v}_A={\bf
v}_A^{\varphi}$, and $\lambda'=\varphi(\lambda'')$. Then
$$A^\varphi=\varphi(\lambda'')\mathop{\mathrm{Id}}\nolimits+{\bf u}^\varphi({\bf
v}_A^\varphi)^{{\rm tr}\,} \quad\hbox{and}\quad B=\lambda\mathop{\mathrm{Id}}\nolimits+{\bf
u}^\varphi{\bf v}_B^{{\rm tr}\,}.$$
Now, if ${\bf v}_A^\varphi,{\bf v}_B$ are linearly dependent we are
done. Assume erroneously they are not. We first choose ${\bf v}$ such
that ${\bf v}^\varphi,{\bf v}_A^\varphi,{\bf v}_B$ are independent, and
at the same time, $({\bf v}^\varphi)^{{\rm tr}\,}{\bf u}^\varphi=1$ ({\small
Such a vector ${\bf v}$ exists since we can enlarge~${\bf v}_A$ with
${\bf v}_2,\dots,{\bf v}_{n} $ to a basis of~${\mathbb F}^n$, assuming ${\bf
v}_i^{{\rm tr}\,}{\bf u}=1$. Then, ${\bf v}_A^\varphi$, ${\bf
v}_2^\varphi,\dots,{\bf v}_n^\varphi$ is still a basis, so some ${\bf
v}^\varphi:={\bf v}_i^\varphi$ is independent of~${\bf
v}_A^\varphi,{\bf v}_B$}).
By the choice of ${\bf v}$, the vector ${\bf u}^\varphi\notin \{{\bf
v}^\varphi\}^\bot$. So much the more ${\bf u}^\varphi\notin \{{\bf
v}^\varphi,{\bf v}_A^\varphi\}^\bot$, so that $\mathop{\rm Lin}_{\mathbb F}\{{\bf
u}^\varphi\}\cap \{{\bf v}^\varphi,{\bf v}_A^\varphi\}^\bot=\{0\}$. We
next choose ${\bf w}$ with ${\bf w}^\varphi\in\{{\bf v}^\varphi,{\bf
v}_A^\varphi\}^\bot\backslash\{{\bf v}_B\}^\bot $ ({\small it is
possible since ${\bf v}^\varphi,{\bf v}_A^\varphi,{\bf v}_B$ are
independent .}) Hence, ${\bf w},{\bf u}$ are independent, so we can
choose ${\bf h}$ with ${\bf h}^{{\rm tr}\,}{\bf u}=0$ and ${\bf h}^{{\rm tr}\,}{\bf
w}:=1$. Lastly, choose nonzero ${\bf s}\in \{{\bf h}\}^\bot$.
We now form $N_1:={\bf s}{\bf v}^{{\rm tr}\,}$ and $P_1:={\bf w}{\bf
h}^{{\rm tr}\,}$. By its choice, ${\bf w}\in\{{\bf v},{\bf v}_A\}^\bot$ so it
follows $P_1N_1=0=N_1P_1$, and $P_1^2=P_1$. Moreover,
$N_1AP_1=0=P_1AN_1$. By assumptive condition~(i), we would have to have
$0=N_1^\varphi BP_1^\varphi+\beta P_1^\varphi B N_1^\varphi$. However,
it equals
$$
N_1^\varphi BP_1^\varphi+\beta P_1^\varphi B
N_1^\varphi=\underbrace{\bigl( ({\bf v}^\varphi)^{{\rm tr}\,}{\bf u}^\varphi
\bigr)}_{=1}\cdot\underbrace{({\bf v}_B^{{\rm tr}\,}{\bf
w}^\varphi)}_{\not=0}\cdot {\bf s}^\varphi({\bf
h}^\varphi)^{{\rm tr}\,}+\beta\cdot 0 \not=0.$$ This contradiction finally
establishes linear dependence between ${\bf v}_A^\varphi$ and ${\bf
v}_B$.
\end{proof}\bigskip
\begin{proof}[Proof of Corollary~\ref{cor:TAllMatr}]
Obviously,~$\Phi$ satisfies the hypothesis of
Theorem~\ref{thm:main-nonvanishing-sum}, hence also its conclusion.
That is, (i) and (ii) hold for rank-one matrices. Now, we may
replace~$\Phi$ by a mapping $X\mapsto \frac{1}{ \gamma(X)}\,
T^{-1}\Phi(X)T$ to achieve that either~$\Phi(X)\equiv X^\varphi$ holds
for all rank-one matrices, or else $\Phi(X)\equiv (X^\varphi)^{{\rm tr}\,}$
holds for all rank-one matrices. It remains to see that, modulo scalar
multiplication and scalar addition, same holds for~$A\in\mathfrak{D}_1$ of
rank~$\ge 2$.\medskip
Assume first $\Phi(X)\equiv X^\varphi$ for all~$X$ of rank-one, and
let~$A\in\mathfrak{D}_1$ be of rank $\ge 2$. Now, by definition, our
polynomials satisfy $\sum_{\sigma(1)=1}\alpha_\sigma=1$. Note that
$$1=\sum_{\sigma(1)=1}\alpha_\sigma=\sum_{\sigma(1)=1\atop\sigma(2)=2}
\alpha_\sigma+\sum_{\sigma(1)=1\atop\sigma(3)=2}\alpha_\sigma+\dots+\sum_{\sigma(1)=
1\atop\sigma(k)=2}\alpha_\sigma,$$ so at least one summand on the right
is nonzero. Say,
$\tau:=\sum_{\sigma(1)=1\atop\sigma(j'_0)=2}\alpha_\sigma\not=0$.
Having found~$j'_0$, we next pick any rank-one idempotent~$P$, and any
rank-one matrix~$N$ with~$PN=0=NP$. Consider now
$\mathfrak{p}(N,P,\dots,P, \fbox{$A$}_{j'_0},P,\dots,P)$ with matrix
$A$ at the position $j'_0$. An easy argument reveals that
\begin{align*}
\mathfrak{p}(N,P,\dots,P,
\fbox{$A$}_{j'_0},P,\dots,P)&=\sum_{\sigma(1)=1\atop\sigma(j'_0)=2}\alpha_\sigma
NAP+
\sum_{\sigma(1)=k\atop\sigma(j'_0)=k-1}\alpha_\sigma PAN+\\
&\hphantom{\sum_{\sigma(1)=1\atop\sigma(j'_0)=2}\alpha_\sigma
NAP}\mbox{}+
\sum_{\text{the rest permut.}}\alpha_\sigma\cdot 0\\[5mm]
&= \tau (NAP+\alpha PAN)
\end{align*}
where $\alpha:=1/\tau
\sum_{\sigma(1)=k\atop\sigma(j'_0)=k-1}\alpha_\sigma$. Similarly, the
value of
$\mathfrak{p}(\Phi(N),\Phi(P),\dots,\linebreak[3]\Phi(P),\fbox{$\Phi(A)$}_{j'_0},\Phi(P),\dots,\Phi(P))=
\mathfrak{p}(N^\varphi,
P^\varphi,\dots,P^\varphi,\fbox{$\Phi(A)$}_{j'_0},P^\varphi,\dots,P^\varphi)$
further equals $\tau(N^\varphi\Phi(A)P^\varphi+\alpha
P^\varphi\Phi(A)N^\varphi)$. Consequently,
$$ NAP+\alpha PAN =0
\Longleftrightarrow 0=N^\varphi\Phi(A)P^\varphi+\alpha
P^\varphi\Phi(A)N^\varphi,$$ so Lemma~\ref{lem:A=lambda*B_iff...}
gives~$\Phi(A)=\gamma(A) A^\varphi+\mu(A)\mathop{\mathrm{Id}}\nolimits$.\medskip
Assume lastly $\Phi(X)\equiv (X^\varphi)^{{\rm tr}\,}$ for all~$X$ of
rank-one. Pick rank-one~$N,P$, with~$P^2=P$ and~$PN=0=NP$. Also, pick
any~$A\in\mathfrak{D}_1$ and let~$B:=\Phi(A)$. We deduce, {similarly} as
before, that
\begin{equation}\label{eq:A^tr}%
\begin{aligned}
& &0&=NAP+\alpha PAN\\
&&&=\tfrac{1}{\tau}\,\mathfrak{p}(N,P,\dots,P,\fbox{$A$}_{j'_0},P,\dots,P)\qquad
\Longleftrightarrow\\
\Longleftrightarrow& & 0 &=\tfrac{1}{\tau}\,\mathfrak{p}(\Phi(N),\Phi(P),
\dots,\Phi(P),\fbox{$\Phi(A)$}_{j'_0},\Phi(P),\dots,\Phi(P))\\
&&&=\tfrac{1}{\tau}\,\mathfrak{p}\bigl((N^\varphi)^{{\rm tr}\,},(P^\varphi)^{{\rm tr}\,},
\dots,(P^\varphi)^{{\rm tr}\,},B,(P^\varphi)^{{\rm tr}\,},\dots,(P^\varphi)^{{\rm tr}\,}\bigr)\\
&&&=(N^\varphi)^{{\rm tr}\,}B(P^\varphi)^{{\rm tr}\,}+\alpha
(P^\varphi)^{{\rm tr}\,}B(N^\varphi)^{{\rm tr}\,}
\end{aligned}
\end{equation}
Choose~$(N,A,P):=(E_{22},E_{12},E_{11})$. Then, $PN=0=NP$ and
$B:=\Phi(A)=(E_{12}^\varphi)^{{\rm tr}\,}=E_{21}$. On one hand, this gives
$NAP+\alpha PAN=\alpha\,E_{12}$, and, on the other,
$(N^\varphi)^{{\rm tr}\,}B(P^\varphi)^{{\rm tr}\,}+\alpha
(P^\varphi)^{{\rm tr}\,}B(N^\varphi)^{{\rm tr}\,}=E_{21}$. Consequently, the
equivalence~(\ref{eq:A^tr}) reads $0=\alpha E_{12}\Longleftrightarrow
0=E_{21}$, and so $\alpha\not=0$.
We may now rewrite equivalence~(\ref{eq:A^tr}) into
$$NAP+\alpha PAN=0\Longleftrightarrow 0=\bigl(
N^\varphi B^{{\rm tr}\,}P^\varphi
\bigr)^{{\rm tr}\,}+\tfrac{1}{\alpha}\bigl(P^\varphi B^{{\rm tr}\,}N^\varphi
\bigr)^{{\rm tr}\,}.$$ The right side is further equivalent to $0= N^\varphi
B^{{\rm tr}\,}P^\varphi +\frac{1}{\alpha}P^\varphi B^{{\rm tr}\,}N^\varphi$. Hence,
Lemma~\ref{lem:A=lambda*B_iff...}
gives~$\Phi(A)^{{\rm tr}\,}=B^{{\rm tr}\,}=\gamma(A) A^\varphi+\mu(A)\mathop{\mathrm{Id}}\nolimits$.
\end{proof}\bigskip
\begin{proof}[Proof of Corollary~\ref{cor:corolary_to_Jordan_product}]
Assumption ${\rm char}\, {\mathbb F}\not=2$ ensures $\mathfrak{p}(x,y):=xy+yx$ is a
polynomial with nonvanishing sum of coefficients. Since both
mappings~$X\mapsto X^{\varphi}$ and~$X\mapsto X^{{\rm tr}\,}$ preserve the
zeros of~$\mathfrak{p}(x,y)$, we may replace~$\Phi$ by a mapping
$X\mapsto\bigl( \frac{1}{ \gamma(X)}\, T^{-1}\Phi(X)T
\bigr)^{\varphi^{-1}}$, respectively, by $X\mapsto \bigl(
\bigl(\frac{1}{ \gamma(X)}\,
T^{-1}\Phi(X)T\bigr)^{{\rm tr}\,}\bigr)^{\varphi^{-1}}$ to achieve that~$\Phi$
leaves fixed all rank-one matrices. It remains to see that
$\Phi(A)=\gamma(A) A$ holds also for diagonalizable matrices~$A$ with
the spectrum $\{\lambda,-\lambda\}$. In view of
Corollary~\ref{cor:Phi-preserves-0} we may assume
further~$A\not=0$.\medskip
Using a similarity~$S$, we may write $A=S\bigl( \lambda\mathop{\mathrm{Id}}\nolimits_{n_1}\oplus
(-\lambda)\mathop{\mathrm{Id}}\nolimits_{n_2}\bigr)S^{-1}$. It is easy to see
that $\Omega_A:=\{X\in{\cal M}_n({\mathbb F});\;\;XA+\,AX=0\}=S\left[\begin{smallmatrix} 0 & K\\
L & 0
\end{smallmatrix}\right] S^{-1}$ where $K,L$ are arbitrary matrices of an
appropriate size. Now, since~$\Phi$ fixes rank-one matrices, we have,
by the defining equation~(\ref{eq:poly-with-nonvanishing-sum}),
$X\Phi(A)+\Phi(A)X=0$
at least for each rank-one $X\in S\left[\begin{smallmatrix} 0 & K\\
L & 0
\end{smallmatrix}\right] S^{-1}$. Obviously, the set~$\Omega_{\Phi(A)}$ of all
matrices~$X$ with $X\Phi(A)+\Phi(A)X=0$ is a vector subspace of~${\cal
M}_n({\mathbb F})$, so actually
$$
\Omega_{\Phi(A)}\supseteq S\left[\begin{smallmatrix} 0 & K\\
L & 0
\end{smallmatrix}\right] S^{-1}.$$
We now write $\Phi(A)=S\left[\begin{smallmatrix} U & V\\
W & Z
\end{smallmatrix}\right]S^{-1}$, and solve the identity
$$\left[\begin{smallmatrix} U & V\\
W & Z
\end{smallmatrix}\right]\left[\begin{smallmatrix} 0 & K\\
L & 0
\end{smallmatrix}\right]+\left[\begin{smallmatrix} 0 & K\\
L & 0
\end{smallmatrix}\right]\left[\begin{smallmatrix} U & V\\
W & Z
\end{smallmatrix}\right]\equiv 0\qquad (\forall\,K,\,\forall\, L).$$
Straightforward calculations
give~$V=0=W$, and $V=\mu\mathop{\mathrm{Id}}\nolimits_{n_1}$, $Z=-\mu\mathop{\mathrm{Id}}\nolimits_{n_2}$. Thus,
$\Phi(A)=S\,{\rm diag}\,(\mu,-\mu)S^{-1}=\frac{\mu}{\lambda} \cdot A$.
\end{proof}\bigskip
\begin{proof}[Proof of Corollary~\ref{cor:nonsurjective}] We first
prove that~$\Phi$ maps rank-one idempotents into scalar multiples of
rank-one idempotents, and preserves their orthogonality.
Indeed, let $P_1,\ldots, P_n$ be the set of $n$ pairwise orthogonal
rank-one idempotents. Clearly then,
$\mathfrak{p}(P_i,P_j,\ldots,P_j)=0=\mathfrak{p}(P_j,P_i,P_j,
\ldots,P_j)=\dots=\mathfrak{p}(P_j,\ldots,P_j,P_i)$ for all $i\ne j$.
This implies a similar set of equations on their $\Phi$--images
$A_i:=\Phi(P_i)$ and~$A_j:=\Phi(P_j)$. We write them explicitly:
$$\begin{aligned}
\sum_{\sigma(1)=1}\alpha_\sigma A_i A_j^{k-1}&+\sum_{\sigma(1)=2}
\alpha_\sigma A_jA_i A_j^{k-2}&+\dots +\sum_{\sigma(1)=k}\alpha_\sigma
A_j^{k-1}A_i&=0\\
\sum_{\sigma(2)=1}\alpha_\sigma A_i A_j^{k-1}&+\sum_{\sigma(2)=2}
\alpha_\sigma A_jA_i A_j^{k-2}&+\dots +\sum_{\sigma(2)=k}\alpha_\sigma
A_j^{k-1}A_i&=0\\
\multispan4{\dotfill}\\
\sum_{\sigma(k)=1}\alpha_\sigma A_i
A_j^{k-1}&+\sum_{\sigma(k)=2}\alpha_\sigma
A_jA_iA_j^{k-2}&+\dots+\sum_{\sigma(k)=k}\alpha_\sigma A_j^{k-1}A_i&=0
\end{aligned}$$
These may be regarded as a system of~$k$ homogeneous linear equations
in `variables' $A_j^s A_iA_j^{k-1-s}$. By the assumptions, the
coefficient matrix is invertible, so the only solution is $A_j^s
A_iA_j^{k-1-s}=0$ for each~$s$. In particular,
$A_iA_j^{k-1}=A_j^{k-1}A_i=0$. That is,
\begin{equation}\label{eq:A_i*A_j=0}%
\mathop{\mathrm{Im}}\nolimits(A_j^{k-1})\subseteq \mathop{\mathrm{Ker}}\nolimits(A_i);\qquad (i\not= j).
\end{equation}
Moreover, $\mathfrak{p} (P_i,\ldots, P_i)=(1+\xi)P_i^k=(1+\xi)P_i\ne 0$
implies that $0\ne\mathfrak{p}(A_i,\ldots,A_i)=(1+\xi)A_i^k$, for any
$i$.\medskip
We can now follow the arguments from~\cite[Lemma~2.2]{chan_li_sze} of
Chan, Li, and Sze: Firstly, we claim that ${\rm rk \,}(A_i)=1$ for any $i$.
Suppose on a contrary that, say, ${\rm rk \,}(A_1)\ge 2$. Then,
$\dim\mathop{\mathrm{Ker}}\nolimits(A_1)<n-1$ and we deduce from~(\ref{eq:A_i*A_j=0}) that $\dim
(\mathop{\mathrm{Im}}\nolimits(A_2^{k-1}) +\ldots +\mathop{\mathrm{Im}}\nolimits (A_n^{k-1}))<n-1$. Hence, there exists $j$
such that
\begin{equation}\label{eq:sum}
\mathop{\mathrm{Im}}\nolimits(A_j^{k-1})\subseteq \sum\limits_{l=2,\ldots,n \atop l\ne j} \mathop{\mathrm{Im}}\nolimits
(A_l^{k-1}).
\end{equation}
Indeed, otherwise, by the induction, $\dim\bigl( \mathop{\mathrm{Im}}\nolimits(A_2^{k-1}) +\ldots
+\mathop{\mathrm{Im}}\nolimits (A_n^{k-1})\bigr)\ge n-1$, which is a contradiction. Again,
by~(\ref{eq:A_i*A_j=0}), the right hand side of Eq.~(\ref{eq:sum}) is
contained in $\mathop{\mathrm{Ker}}\nolimits(A_j)$. Thus $A^k_j=0$ which contradicts
$A^k_j=\mathfrak{p}(A_j,\dots,A_j)\not=0$ by the assumption (ii).
Therefore, ${\rm rk \,}(A_i)=1$ for any~$i$.
Since $A_i^k\ne 0$ there exist $\lambda_i\in {\mathbb F}\backslash\{0\}$ and
an idempotent matrix $Q_i$ of rank-one such that $A_i=\lambda_i Q_i$.
Lastly, it follows from $A_iA_j=\frac{1}{\lambda_j^{k-2}}
A_iA_j^{k-1}=0$ and $A_jA_i=\frac{1}{\lambda_i^{k-2}} A_jA_i^{k-1}=0$
that~$Q_i$ are pairwise orthogonal.\medskip
Thus, $\Phi$
maps orthogonal idempotents of rank-one into scalar multiples of
orthogonal idempotents of rank-one. We now redefine~$\Phi$ as in
Remark~\ref{rem_redefined_Phi}. The rest
--- with the sole exception of Step~4 --- follows directly the proof of
Theorem~\ref{thm:main-nonvanishing-sum}.
\end{proof}
\section{\label{S3} Polynomials with vanishing sums of coefficients}
\begin{de}\label{def:admissible}
Let~$k\ge 2$. A subset of nonidentical permutations $\Xi\subseteq{\cal
S}_k$ is called an {\em admissible subset\/} if the following two
conditions are satisfied:
\begin{itemize}
\item [(i)] There exists $t\in \{1,\ldots , k\}$ such that each
$\sigma\in\Xi$ fixes the first $t-1$ elements, but
$\sigma(t)\not=t$ ({\small note that $t<k$, otherwise,~$\sigma$ would have
to be identical permutation}).
\item[(ii)] There exist integers $w,u,v\in \{1,\ldots , k\}$, $u<v$, such that
$\sigma(w)=v$ and $\sigma(w+1)=u$ for each $\sigma\in \Xi$.
\end{itemize}
\end{de}
Note that in particular, $\sigma(w+1)<\sigma(w)$ for each $\sigma\in
\Xi$.
\begin{example}
We give two examples of admissible sets which are the most visualizing
on one hand and which show that there are many admissible sets among
the subsets of ${\cal S}_k$ on the other hand.
\begin{itemize}
\item $\Xi:=\{\sigma\}$ is admissible subset whenever~$\sigma$ is nonidentical.
\item An admissible subset is also the subset of all permutations from~${\cal
S}_k$ that swap $1$ and $2$ ({\small take $t:=1=:w$, $u:=1$, $v:=2$} in
Definition~\ref{def:admissible}).
\end{itemize}
It is not hard to see that the cardinality of an admissible set, with
$t=1$, i.e., which fixes no initial elements, is either $(k-2)!$ or
$(k-2)! - (k-3)!$, depending on choosing $w$ and $u,v$.
\end{example}
The main result of the present section is the following theorem. In
contrast to the Theorem~\ref{thm:main-nonvanishing-sum} we do not
assume that ${\mathbb F}$ is algebraically closed, and~$\Phi$ is a strong
preserver in this section.
\begin{theorem}\label{thm:main-on-vanishing-sum}%
Let ${\mathbb F}$ be an arbitrary field with more than 2 elements, let $n\ge
3$, $k\ge 2$ be integers, and let $\Xi\subset {\cal S}_k$ be a fixed
{\em admissible subset} of permutations. Suppose that
two given homogeneous multilinear polynomials
$$\mathfrak{p}_1(x_1,\dots ,x_k):=x_1\cdots x_k-\sum_{\sigma\in\Xi} \alpha_\sigma
x_{\sigma(1)}\cdots x_{\sigma(k)};\qquad (\alpha_\sigma\in{\mathbb F})$$
$$\mathfrak{p}_2(x_1,\dots ,x_k):=x_1\cdots x_k-\sum_{\sigma\in\Xi} \beta_\sigma
x_{\sigma(1)}\cdots x_{\sigma(k)};\qquad (\beta_\sigma\in{\mathbb F})$$ satisfy
$\sum_{\sigma\in\Xi} \alpha_\sigma=1=\sum_{\sigma\in\Xi} \beta_\sigma$.
Then, any bijection $\Phi:{\cal M}_n({\mathbb F})\to {\cal M}_n({\mathbb F})$ which
maps the zeros of~$\mathfrak{p}_1$ to the zeros of $\mathfrak{p}_2$
({\small i.e., $\Phi({\mathfrak{S}_{\mathfrak{p}}}_1)\subseteq {\mathfrak{S}_{\mathfrak{p}}}_2$}) preserves
commutativity.
\end{theorem}
\begin{remark}
In particular if $\mathfrak{p}_1 =\mathfrak{p}_2$ the result of
Theorem~\ref{thm:main-on-vanishing-sum} also holds.
\end{remark}
\begin{corollary} \label{CorZ_2}
Under the additional
requirement $\Phi(\mathop{\mathrm{Id}}\nolimits)\not=0$, the conclusion of
Theorem~\ref{thm:main-on-vanishing-sum} is valid for~${\mathbb F}={\mathbb Z}_2$ also.
\end{corollary}
The corollary below shows that the injectivity assumption on $\Phi$ can
be substituted by the requirement that $\Phi$ maps the zeros of
$\mathfrak{p}_1$ into the zeros of $\mathfrak{p}_2$
{\em strongly\/}.
\begin{corollary}\label{cor:zero-sum}%
In addition to the assumptions from
Theorem~\ref{thm:main-on-vanishing-sum}, suppose further that $k\ge 3$,
and that a ({\small possibly noninjective}) surjection~$\Phi:{\cal
M}_n({\mathbb F})\to{\cal M}_n({\mathbb F})$ strongly maps the zeros
of~$\mathfrak{p}_1$ to the zeros of~$\mathfrak{p}_2$.
{Then,~$\Phi(A)=0$ implies~$A=0$. Consequently,~$\Phi$ preserves
commutativity.}
\end{corollary}
The following remark provides the final forms of the transformations
satisfying conditions of Theorem~\ref{thm:main-on-vanishing-sum}.
\begin{remark}
A surjective {\em and additive} commutativity preservers are of the
form $\Phi(A)=\gamma\,TA^\varphi T^{-1}+f(A)\mathop{\mathrm{Id}}\nolimits$ or
$\Phi(A)=\gamma\,T(A^\varphi)^{{\rm tr}\,} T^{-1}+f(A)\mathop{\mathrm{Id}}\nolimits$. Here,
$\gamma\in{\mathbb F}$, $\phi:{\mathbb F}\to{\mathbb F}$ is a ring automorphism, and~$f:{\cal
M}_n({\mathbb F})\to{\mathbb F}$ an additive function; see Bre\v{s}ar~\cite{bresar},
Petek~\cite{petek_t}, and Beidar and Lin~\cite{beidar_lin}.
We refer to the works by \v{S}emrl~\cite{semrl:commutativity} and
Fo\v{s}ner~\cite{fosner} for a bijective, possibly non-additive
mappings, that {\em strongly preserve commutativity}. At least on the
subset of~${\cal M}_n({\mathbb C})$, consisting of those matrices whose Jordan
structure has only the cells of dimension at most two, they are of the
form $\Phi(A)=Tp_A(A^\varphi) T^{-1}$ or $\Phi(A)=Tp_A\bigl(
(A^\varphi)^{{\rm tr}\,} \bigr)T^{-1}$, where~$p_A$ is a certain polynomial
that depends on~$A$.
\end{remark}
\begin{remark}
The converse of Theorem~\ref{thm:main-on-vanishing-sum} does not hold.
Namely, there are many
polynomials~$\mathfrak{p}_1=\mathfrak{p}_2=:\mathfrak{p}$ and
commutativity preservers which {\em do not preserve the zeros
of~$\mathfrak{p}$}.
We refer the reader to the last section for
examples.
\end{remark}
\subsection{The proof of Theorem~\ref{thm:main-on-vanishing-sum}}
The proof will be given in a series of Lemmas.
We first recall some known results about rational forms for matrices
over an arbitrary field ${\mathbb F}$.
\begin{de}
A {\em companion matrix\/} of a monic polynomial
$$f(x)=x^m+a_{m-1}x^{m-1}+\ldots+ a_3x^3+a_2x^2+a_1x+a_0,$$ of degree
$m\ge 2$, is the matrix \begin{equation} \label{e0} C(f)=\left[
\begin{array}{ccccccl}
0&0&0&0&\ldots&0&-a_{0} \\
1&0&0&0&\ldots&0&-a_{1} \\
0&1&0&0&\ldots&0&-a_{2} \\
0&0&1&0&\ldots&0&-a_{3} \\
\vdots&\vdots&\vdots&\vdots&\ddots&\vdots&\vdots\\
0&0&0&0&\ldots&1&-a_{m-1} \end{array} \right]\in {\cal M}_{m}({\mathbb F}). \end{equation}
If $f(x)=x+a_0$ is of degree one we let $C(f):=-a_0$ be the
$1$--by--$1$ matrix, i.e., a scalar.
\end{de}
The following lemma is straightforward and well-known:
\begin{lemma} \label{L0}
The polynomial $f$ is a characteristic polynomial of its companion
matrix $C(f)$.
\end{lemma}
\begin{theorem} \label{T1} {\rm \cite[p. 144]{Grove}, \cite[Theorem 11.20]{RML}}
Any matrix $A\in {\cal M}_n({\mathbb F})$ is similar over ${\mathbb F}$ to a matrix
${\cal C}(A)=\bigoplus_j C(p_1^{e_{1j}})\oplus \ldots \oplus
\bigoplus_j C(p_k^{e_{kj}})$, where the~$p_i$ are distinct irreducible
factors of the characteristic polynomial $\chi_A(x)=\prod\limits_{{1\le
i\le k}\atop{1\le j\le k_i}} p_i(x)^{e_{ij}}$. The matrix ${\cal C}(A)$
is determined uniquely, up to the order of diagonal blocks $C(g_i)$.
\end{theorem}
\begin{de}
The matrix ${\cal C}(A)$ described in Theorem~\ref{T1} is called a {\em
primary rational form\/} of $A$.
\end{de}
In all statements till the end of this section we assume that
conditions of Theorem~\ref{thm:main-on-vanishing-sum} are satisfied.
\bigskip
\begin{lemma} \label{L1}
If~$\xi$ is nonzero scalar then the primary rational form of $\Phi(\xi
\mathop{\mathrm{Id}}\nolimits)$ does not contain non-zero nilpotent blocks.
\end{lemma}
\begin{proof} Pick a similarity $P\in \mathrm{GL}_n({\mathbb F})$ such that
$P^{-1}\Phi(\xi\mathop{\mathrm{Id}}\nolimits)P$ equals the primary rational form~${\cal
C}(\Phi(\xi\mathop{\mathrm{Id}}\nolimits))$ of Theorem~\ref{T1}. Now, if the claim is false, at
least one block of~${\cal C}(\Phi(\xi\mathop{\mathrm{Id}}\nolimits))$ is a nonzero nilpotent. For
simplicity, assume it is the first ({\small i.e.: the most
upper-left}) one. Therefore, it equals Eq.~(\ref{e0}), with zeros on
the last column. Then, with~$E:=PE_{11}P^{-1}$,
\begin{equation}\label{e1}%
{\Phi}(\xi\mathop{\mathrm{Id}}\nolimits)\,E=PE_{21}P^{-1},\quad\hbox{ and }\quad
E\,{\Phi}(\xi\mathop{\mathrm{Id}}\nolimits)=0.\end{equation} By surjectivity, $E={\Phi}(F)$ and
${\mathop{\mathrm{Id}}\nolimits}={\Phi}(J)$ for some $F,J\in {\cal M}_n({\mathbb F})$.
Pick an integer~$t\in \{1,\ldots,k\}$ from the
Definition~\ref{def:admissible} of admissible sequence. Note that
$t<\sigma(t)\le k$ for each $\sigma\in\Xi$. Consider now the following
matrix~$k$-tuple
$$\bigl(A_1:=J,\dots,A_{t-1}:=J,
{\mathbf A}_{t}:={\mathbf {(\xi\mathop{\mathrm{Id}}\nolimits)}},A_{t+1}:=F,\dots,A_{k}:=F
\bigr),$$
which lies in~${\mathfrak{S}_{\mathfrak{p}}}_1$, since each
$\sigma$ fixes the indices $\{1,\ldots,t-1\}$, and
$\sum\alpha_\sigma=1$. By the
assumptions,~$\Phi({\mathfrak{S}_{\mathfrak{p}}}_1)\subseteq{\mathfrak{S}_{\mathfrak{p}}}_2$, and we have
$$ \Phi(A_1)\cdots \Phi(A_k)=\sum_{\sigma\in\Xi} \bigl(\beta_\sigma
\Phi(A_{\sigma(1)})\cdots
\Phi(A_{\sigma(k)})\bigr),$$
i.e., \begin{equation} \label{e2}
\mathop{\mathrm{Id}}\nolimits^{t-1}\Phi(\xi\mathop{\mathrm{Id}}\nolimits)E^{k-t}=\sum_{\sigma\in\Xi}
(\beta_\sigma\mathop{\mathrm{Id}}\nolimits^{t-1}E^{g_\sigma}\,\Phi(\xi\mathop{\mathrm{Id}}\nolimits)E^{k-t-g_\sigma}), \end{equation}
where
$g_\sigma:=\sigma(t)-t>0$. However, the matrix $E$ is idempotent, so
$E^{k-t}=E=E^{g_\sigma}$, and it follows from~(\ref{e1}) that the left
hand side of the equality~(\ref{e2}) is equal to
${\Phi}(\xi\mathop{\mathrm{Id}}\nolimits)\,E=PE_{21}P^{-1}$, while the right hand side is equal
to 0, a contradiction. \end{proof}
\begin{lemma} \label{L2}
Suppose~${\mathbb F}\not={\mathbb Z}_2$. Then, there exists a nonzero scalar~$\xi$ such
that $\Phi(\xi\mathop{\mathrm{Id}}\nolimits)\in \mathrm{GL}_n({\mathbb F})$.
\end{lemma}
\begin{proof}Since~$\Phi$ is injective and the cardinality
$|{\mathbb F}\backslash\{0\}|\ge 2$, there
exists at least one nonzero scalar~$\xi$ such
that~$\Phi(\xi\mathop{\mathrm{Id}}\nolimits)\not=0$. As in the proof of Lemma~\ref{L1}, let~$t$
be fixed by Definition~\ref{def:admissible} let
$g_\sigma:=\sigma(t)-t>0$, and let $J:=\Phi^{-1}(\mathop{\mathrm{Id}}\nolimits)\in {\cal
M}_n({\mathbb F})$. Here, we consider the following matrix $k$-tuple:
$$\bigl(A_1:=J,\dots,A_{t-1}:=J,
{\mathbf A}_{t}:={\mathbf X},A_{t+1}:={(\xi \mathop{\mathrm{Id}}\nolimits)},\dots,A_{k}:={(\xi
\mathop{\mathrm{Id}}\nolimits)} \bigr).$$
Again, this $k$-tuple is in ${\mathfrak{S}_{\mathfrak{p}}}_1$ for an arbitrary
matrix~$X$. By the assumptions, $\Phi({\mathfrak{S}_{\mathfrak{p}}}_1)\subseteq{\mathfrak{S}_{\mathfrak{p}}}_2$, and we
have
\begin{equation}\label{eq:Phi(Id)-is-nonsingular}%
\mathop{\mathrm{Id}}\nolimits^{t-1}\Phi(X)\Phi(\xi \mathop{\mathrm{Id}}\nolimits)^{k-t}=
\sum_{\sigma\in\Xi}\bigl(\beta_\sigma \mathop{\mathrm{Id}}\nolimits^{t-1}\Phi(\xi \mathop{\mathrm{Id}}\nolimits)^{g_\sigma}
\Phi(X)\Phi(\xi \mathop{\mathrm{Id}}\nolimits)^{k-t-g_\sigma}\bigr),
\end{equation}
Let us assume that $\Phi(\xi \mathop{\mathrm{Id}}\nolimits)$ is singular.
Recall~$\Phi(\xi\mathop{\mathrm{Id}}\nolimits)\not=0$, so by Lemma~\ref{L1}, the primary rational
form, ${\cal C}(\Phi(\xi \mathop{\mathrm{Id}}\nolimits))$, contains at least one zero block and
at least one non-zero block. For simplicity, assume the first one is
zero, i.e., ${\cal C}(\Phi(\xi \mathop{\mathrm{Id}}\nolimits))=\mathbf{0}\oplus \mathbf{C}$,
where~$\mathbf{C}\not=0$ is a sum of all, but the first, blocks.
By Lemma~\ref{L1}, ${\cal C}(\Phi(\xi \mathop{\mathrm{Id}}\nolimits))^{k-t}=\mathbf{0}\oplus
\mathbf{C}^{k-t}\not=0$. Consequently, the matrix ${\cal C}(\Phi(\xi
\mathop{\mathrm{Id}}\nolimits))^{k-t}$ has a nonzero row, i.e., there exists $p$, $1\le p\le n$
such that {$E_{1p}\,{\cal C}(\Phi(\xi \mathop{\mathrm{Id}}\nolimits))^{k-t}\not=0$}. However,
note that ${\cal C}(\Phi(\xi \mathop{\mathrm{Id}}\nolimits))E_{1p}=0$, so also ${\cal C}(\Phi(\xi
\mathop{\mathrm{Id}}\nolimits))^{g}E_{1p}=0$ for all $g\in{\mathbb N}\backslash\{0\}$, in particular for
each $g:=g_\sigma$. Now, consider $P\in \mathrm{GL}_n({\mathbb F})$ such that $
\Phi(\xi \mathop{\mathrm{Id}}\nolimits)=P{\cal C}(\Phi(\xi \mathop{\mathrm{Id}}\nolimits))P^{-1}$, and choose a matrix~$X$
with $\Phi(X)=PE_{1p}P^{-1}$. For such~$X$, the left hand side
of~(\ref{eq:Phi(Id)-is-nonsingular}) is nonzero while the right hand
side is zero, a contradiction. \end{proof}
\begin{lemma} \label{L3}
$\Phi$ preserves commutativity.
\end{lemma}
\begin{proof} Without loss of generality we assume that $\xi=1$ in
Lemma~\ref{L2}, i.e., that~$\Phi(\mathop{\mathrm{Id}}\nolimits)\in \mathrm{GL}_n({\mathbb F})$ ---
otherwise, the bijection~$\Phi(\xi\cdot \llcorner\!\!\!\lrcorner)$ would be
considered instead of~$\Phi$.
By the definition of admissible sequence, there exists
an
integer $w$ such that $u\equiv\sigma(w+1) < \sigma(w)\equiv
v\;\forall\,\sigma\in\Xi$, we fix the smallest such index $w$.
Consider the following matrix $k$-tuple: \begin{equation}
\label{e3} (A_1:=\mathop{\mathrm{Id}}\nolimits,\ldots,A_{w-1}:=\mathop{\mathrm{Id}}\nolimits, {\mathbf A}_w:={\mathbf X},
{\mathbf A}_{w+1}:={\mathbf Y}, A_{w+2}:=\mathop{\mathrm{Id}}\nolimits,\ldots,A_k:=\mathop{\mathrm{Id}}\nolimits). \end{equation}
If $XY=YX$
then the $k$-tuple~(\ref{e3}) is in ${\mathfrak{S}_{\mathfrak{p}}}_1$. Thus,
\begin{equation}\label{e4}%
\begin{split}
\Phi(\mathop{\mathrm{Id}}\nolimits)^{w-1}\,\Phi(X)\Phi(Y)\,&\Phi(\mathop{\mathrm{Id}}\nolimits)^{k-w-1}= \\
&=\sum_{\sigma\in\Xi}\beta_\sigma \Phi(\mathop{\mathrm{Id}}\nolimits)^{u-1}\,\Phi(Y)\,
\Phi(\mathop{\mathrm{Id}}\nolimits)^{s}\,\Phi(X)\,\Phi(\mathop{\mathrm{Id}}\nolimits)^{k-v},
\end{split}\end{equation}
where ${s}=v-u-1\ge 0$. Note that by the definition of admissible
sequence $u$ and $v$ are independent of~$\sigma$ and thus the right
hand side is equal to
$\Phi(\mathop{\mathrm{Id}}\nolimits)^{u-1}\,\Phi(Y)\,\Phi(\mathop{\mathrm{Id}}\nolimits)^{s}\,\Phi(X)\,\Phi(\mathop{\mathrm{Id}}\nolimits)^{k-v}$.
Since~$\Phi(\mathop{\mathrm{Id}}\nolimits)$ is invertible, Eq.~(\ref{e4}) simplifies into:
\begin{equation}\label{e4.5}%
\Phi(X)\Phi(Y)= \Phi(\mathop{\mathrm{Id}}\nolimits)^{u-w}\,\Phi(Y)\,\Phi (\mathop{\mathrm{Id}}\nolimits)^s\,
\Phi(X)\,\Phi(\mathop{\mathrm{Id}}\nolimits)^{w-v+1}
\end{equation}
whenever $XY=YX$.
We
first claim that~$\Phi(\mathop{\mathrm{Id}}\nolimits)^{s}$ is a scalar matrix. Assume on the
contrary, and consider two cases:\medskip
{\bf Case 1}: The primary rational form~${\cal
C}(\Phi(\mathop{\mathrm{Id}}\nolimits)^{s})=P^{-1}\Phi(\mathop{\mathrm{Id}}\nolimits)^{s}P$ contains a block of
dimension~$\ge 2$. Again, for the sake of simplicity, we assume this
is the first block. Therefore, the~$(1,1)$-entry of~${\cal
C}(\Phi(\mathop{\mathrm{Id}}\nolimits)^{s})$ is zero. We would then let~$X=Y$ be such
that~$\Phi(X)=PE_{11}P^{-1}=\Phi(Y)$. This contradicts~(\ref{e4.5}),
since the left hand side would be $PE_{11}P^{-1}$, while the right
would be zero.\medskip
{\bf Case 2}: All blocks of ${\cal
C}(\Phi(\mathop{\mathrm{Id}}\nolimits)^{s})=P^{-1}\Phi(\mathop{\mathrm{Id}}\nolimits)^{s}P$ are one-dimensional, i.e.,
${\cal C}(\Phi(\mathop{\mathrm{Id}}\nolimits)^{s})={\rm diag}\,(d_1,\dots,d_n)$ is diagonal. Since
${\cal C}(\Phi(\mathop{\mathrm{Id}}\nolimits)^{s})$ is non-scalar, at least two diagonal entries
differ. For the sake of simplicity, assume~$d_1\not=d_2$. Note that by
Lemma~\ref{L2} the matrix~$\Phi(\mathop{\mathrm{Id}}\nolimits)^{s}$ is invertible,
so~$d_1\not=0$. Then, the
similarity by the matrix $S:=\left[\begin{smallmatrix} \frac{d_2}{d_1} & 1\\
1 & 1
\end{smallmatrix}\right]\oplus \mathop{\mathrm{Id}}\nolimits_{n-2}$ transforms this matrix into
$$S \,{\cal C}(\Phi(\mathop{\mathrm{Id}}\nolimits)^{s})\,S^{-1}=\begin{bmatrix}
0&d_2&\\
-d_1&d_1+d_2 &
\end{bmatrix}\oplus{\rm diag}\,(d_3,\dots,d_n)$$
with the zero $(1,1)$-entry. We can then reach a contradiction as in
Case~1.
Consequently, since~$\Phi(\mathop{\mathrm{Id}}\nolimits)$ is invertible we
have~$\Phi(\mathop{\mathrm{Id}}\nolimits)^s=\lambda \mathop{\mathrm{Id}}\nolimits\not=0$, and Eq.~(\ref{e4.5}) further
simplifies into:
\begin{equation}\label{eq:pom1}%
\Phi(X)\Phi(Y)=\lambda
\Phi(\mathop{\mathrm{Id}}\nolimits)^{u-w}\,\Phi(Y)\Phi(X)\,\Phi(\mathop{\mathrm{Id}}\nolimits)^{w-v+1}
\end{equation}
whenever $XY=YX$. Pick any distinct indices~$i,j\in\{1,\dots,n\}$.
Since~$n\ge 3$, there exists another one,
$k\in\{1,\dots,n\}\backslash\{i,j\}$. Now, by the surjectivity, we may
choose~$X=Y$ such that $\Phi(X)=\Phi(Y)=E_{ik}+E_{kj}$, which gives
$\Phi(X)\Phi(Y)=(E_{ik}+E_{kj})^2=E_{ij}=\Phi(Y)\Phi(X)$. We can
likewise find~$X=Y$ such that
$\Phi(X)\Phi(Y)=E_{ii}^2=E_{ii}=\Phi(Y)\Phi(X)$. Putting this into
Eq.~(\ref{eq:pom1}) we deduce that $E_{ij}=\lambda
\Phi(\mathop{\mathrm{Id}}\nolimits)^{u-v}\,E_{ij}\,\Phi(\mathop{\mathrm{Id}}\nolimits)^{w-v+1}$ for any indices~$(i,j)$.
Consequently,
$$A=\lambda
\Phi(\mathop{\mathrm{Id}}\nolimits)^{u-v}\,A\,\Phi(\mathop{\mathrm{Id}}\nolimits)^{w-v+1};\qquad \forall\,A\in{\cal
M}_n({\mathbb F}).$$
With~$A:=\mathop{\mathrm{Id}}\nolimits$ we have $\Phi(\mathop{\mathrm{Id}}\nolimits)^{u-v}\cdot
\Phi(\mathop{\mathrm{Id}}\nolimits)^{w-v+1}=1/\lambda \,\mathop{\mathrm{Id}}\nolimits$. Therefore, $AS=(\lambda S)A$ for
each~$A$, where $S:=\Phi(\mathop{\mathrm{Id}}\nolimits)^{u-v}$. A standard procedure
with~$A:=E_{ij}$ gives that~$S$ is scalar, and~$\lambda=1$. {Hence,
$\Phi(\mathop{\mathrm{Id}}\nolimits)^{w-v+1}=S^{-1}$ is also a scalar matrix}. Since $\lambda=1$,
this further reduces Eq.~(\ref{eq:pom1}) into the desired
$\Phi(X)\Phi(Y)=\Phi(Y)\Phi(X)$ whenever~$XY=YX$.
\end{proof}\bigskip
\subsection{Proof of Corollaries}
\begin{remark}\label{RKer}%
As it was seen in the proof of Theorem~\ref{thm:main-on-vanishing-sum},
we used the requirement of injectivity just once: in the proof of
Lemma~\ref{L2}. {Moreover, even there it suffices to have its curtailed
form, i.e., that} there exists $\xi\in {\mathbb F}$, $\xi\ne 0$, such that
$\Phi(\xi \mathop{\mathrm{Id}}\nolimits)\ne 0$. Thus the result of
Theorem~\ref{thm:main-on-vanishing-sum} holds under these, even more
general, conditions.
\end{remark}
\begin{proof}[Proof of Corollary~\ref{CorZ_2}]
{By the assumptions, $\Phi(\mathop{\mathrm{Id}}\nolimits)\ne 0$, so} the proof of Lemma~\ref{L2}
works by Remark~\ref{RKer}. It can be directly checked that the rest of
the proof of Theorem~\ref{thm:main-on-vanishing-sum} does not use the
assumptions on the ground field.
\end{proof}
\begin{proof}[Proof of Corollary~\ref{cor:zero-sum}]
{Suppose $\Phi(A)=0$ for some matrix~$A$, and} consider the following
matrix $k$-tuple
$$\bigl(X_1=\mathop{\mathrm{Id}}\nolimits,\dots,X_{w-1}=\mathop{\mathrm{Id}}\nolimits,{\bf X}_{w}={\bf A},{\bf
X}_{w+1}={\bf X},X_{w+2}=\mathop{\mathrm{Id}}\nolimits,\dots,X_k=\mathop{\mathrm{Id}}\nolimits\bigr).$$
This $k$-tuple is mapped
by~$\Phi$ into a $k$-tuple with~$w$-th member equal to
$\Phi(X_w)=\Phi(A)=0$. Therefore,
$(\Phi(X_1),\dots,\Phi(X_k))\in{\mathfrak{S}_{\mathfrak{p}}}_2$, so that also
$(X_1,\dots,X_k)=(\mathop{\mathrm{Id}}\nolimits,\dots,\mathop{\mathrm{Id}}\nolimits,X_w=A,X_{w+1}=X,\mathop{\mathrm{Id}}\nolimits,\dots,\mathop{\mathrm{Id}}\nolimits)\in{\mathfrak{S}_{\mathfrak{p}}}_1$,
for every choice of~$X$. Since~$\sigma(w+1)<\sigma(w)$ for every
admissible permutation, this further yields
$$\mathop{\mathrm{Id}}\nolimits^{w-1}AX\mathop{\mathrm{Id}}\nolimits^{k-w-1}=\sum_\sigma \alpha_\sigma\mathop{\mathrm{Id}}\nolimits^i X\mathop{\mathrm{Id}}\nolimits^j
A\mathop{\mathrm{Id}}\nolimits^l$$ for some integers~$i,j,l$. This immediately simplifies into
$AX=\sum\limits_\sigma \alpha_\sigma XA=XA$ for every~$X$. Hence,~$A$
has to be scalar.\medskip
It remains to show~$A=0$. Consider now the following $k$-tuple:
$X_{w}:=Y,X_{w+1}:=Z$, and $X_i:=A$ for the rest of indices ({\small
since~$k\ge 3$, at least one member equals~$A$}). As before we deduce
that this $k$-tuples is in~${\mathfrak{S}_{\mathfrak{p}}}_1$ for every choice of~$Y,Z$. Hence,
since~$A$ is scalar, and $\sigma(w+1)<\sigma(w)$,
$$A^{k-2} YZ=A^{k-2}\sum_{\sigma\in\Xi} \alpha_\sigma ZY=A^{k-2}ZY$$
for any choice of~$Y,Z$. This is possible only when~$A^{k-2}=0$,
i.e.,~$A=0$.\medskip
Therefore, nonzero scalar matrices are not annihilated by~$\Phi$, and
we can follow the proof of Theorem~\ref{thm:main-on-vanishing-sum} to
see that~$\Phi$ preserves commutativity.
\end{proof}
\section{\label{S5} Concluding remarks and examples}
This section mainly contains counterexamples to show that our results
cannot be further improved without imposing additional hypothesis.
{\em The following example shows that the inverse implication does not
hold in Theorem~\ref{thm:main-on-vanishing-sum}, namely there are
commutativity preserving mappings that do not preserve zeros of a fixed
polynomial.\/}
\begin{example}
Let $\Phi(A)=A+a_{12}\,{\rm Id}$ for all $A=[a_{ij}]\in {\cal
M}_n({\mathbb F})$. We consider the polynomial $\mathfrak{p}(x,y,z):=xyz-yxz$.
Then the triple $(x:=E_{11},\, y=z:=E_{12})$ is a zero of this
polynomial, but its image $(E_{11},\mathop{\mathrm{Id}}\nolimits+E_{12},\mathop{\mathrm{Id}}\nolimits+E_{12})$ is not a
zero of this polynomial.
\end{example}
{\em There exist ({\em even linear}!) transformation $\Phi$ which
strongly preserve commutativity, but have a nonzero kernel.\/}
\begin{example}
Let us consider $\Phi:{\cal M}_n({\mathbb F}) \to {\cal M}_n({\mathbb F})$ defined by
$\Phi(A):=A-\mathrm{Tr}(A)/n\, \mathop{\mathrm{Id}}\nolimits$. Then~$\Phi$ is linear, $\Phi$
strongly preserves commutativity, but $\Phi(\mathop{\mathrm{Id}}\nolimits)=0$.
\end{example}
{\em Corollary~\ref{cor:TAllMatr} is no longer true when~$k=2$, i.e.,
the transformation $\Phi$ may not be controllable on some large subsets
of $\mathfrak{D}_1$.\/}
\begin{example}\label{exa:counter-axa-for-nonvanishing}
Namely,
consider $\mathfrak{p}(x,y):=xy+yx$. Pick any $A\in\Theta:=\{A\in
{\cal M}_n({\mathbb F});\;\;0\not\in\mathop{\mathrm{Sp}}(A)+\mathop{\mathrm{Sp}}(A)\}$. Then, by
Sylvester--Rosenblum Theorem
({\small cf.\ the proof of Lemma~\ref{lem:Sylvester--Rosenblum}}), the
elementary operator $X\mapsto XA+AX$ is invertible, so
$\mathfrak{p}(X,A)=0$ if and only if~$X=0$. Hence, the restriction
of~$\Phi$ to the subset~$\Theta$
has no structure at all, i.e.,~$\Phi|_{\Theta}$ may arbitrarily permute
the elements of~$\Theta$, yet it still strongly preserves the zeros
of~$\mathfrak{p}(x,y)$. We remark that~$\Theta$ is a rather large
subset: when~${\mathbb F}={\mathbb C}$ it is nonempty and open in~${\cal M}_n({\mathbb C})$, by
continuity of the eigenvalues.
\end{example}
{\em The characterization of
Lemma~\ref{lem:charact_of_minimal_idempotents} is no longer valid
if~${\mathbb F}$ is not algebraically closed.\/}
\begin{example}
As a
counterexample, choose a polynomial~$\mathfrak{p}(x,y):=xy+yx$
and consider $A:=\left[\begin{smallmatrix}
0 & -3 \\
1 & 0
\end{smallmatrix}\right]\oplus 1\in{\cal M}_3({\mathbb R})$.
Then,~$\Omega_{\bullet A}=\Omega_{A\bullet A}=\{\left[\begin{smallmatrix}
-d & 3 c \\
c & d
\end{smallmatrix}\right]\oplus 0;\;\;c,d\in{\mathbb R}\}$, and it contains no
nontrivial square-zero matrices, nor it equals~${\mathbb R}\,P$ for
idempotent~$P$.
\end{example}
{\em The following example shows that there are field automorphism
which do not preserve zeros of matrix polynomials.\/}
\begin{example}
\label{ExPhi} Let us consider the polynomial $\mathfrak{p}=xy-i\,yx$
and automorphism $\varphi:{\mathbb C}\to {\mathbb C}$ which sends any
$x\in {\mathbb C}$ to its complex conjugated element $\overline x$.
Then consider $$ A:=\begin{bmatrix}
1 & 1 \\ -1 & 1 \end{bmatrix}\oplus \mathop{\mathrm{Id}}\nolimits_{n-2}\in {\cal M}_n(\mathbb C),\
B:=\begin{bmatrix} 1 & i \\ i & -1\end{bmatrix}\oplus \mathop{\mathrm{Id}}\nolimits_{n-2}\in
{\cal M}_n(\mathbb C).$$ The direct computations show that the matrices
$A,B$ are zeros of $\mathfrak{p}$ but their conjugated matrices
$A^{\varphi}, B^{\varphi}$ are not.
\end{example}
{\em Finally, the transposition transformation may not preserve zeros
of $\mathfrak{p}$ for some $\mathfrak{p}$.\/}
\begin{example}
\label{Ex_tr} Let us consider the polynomial $\mathfrak{p}=xy$ and the
matrices $A=E_{11}$, $B=E_{21}$. Then $\mathfrak{p}(A,B)=0$, however
$\mathfrak{p}(A^{\rm tr}\,,B^{\rm tr}\,)=E_{11}E_{12}=E_{12}\ne 0$.
\end{example}
{\bf Acknowledgments}. The authors are indebted to Professor Chi-Kwong
Li for inspiring conversations regarding the topic of Chapter~\ref{S4}.
|
1309.3614
|
\section{Introduction}
\label{Introduction}
Two-dimensional (2D) X-ray detectors are now widely used with synchrotron and laboratory sources to obtain powder diffraction data~\cite{hebob;b;tdxrd09,livet;aca07}. Their use results in experimental throughputs several orders of magnitude faster than point or linear detectors~\cite{chupa;jac03}. Another advantage is that a 2D diffraction pattern contains much more information than the diffraction pattern obtained by conventional powder diffraction~\cite{hebob;b;tdxrd09}, though this information is rarely fully utilized, and the 2D image is reduced to 1D diffraction pattern by integrating around the Debye-Scherrer rings~\cite{dinne;b;pdtap08}. Regardless, accurately estimating the uncertainties on data is crucial, not least since they are used as weights in least-squares estimates of fitting parameters, and optimizing an unweighted least-squares will not, in principle, result in the maximum likelihood solution~\cite{bevin;dreaps92}. Knowing the uncertainties on the diffraction data is also the starting point for estimating the precision of parameters in the model~\cite{princ;b;mticams04,pecha;b;fopdascom05}.
Generally, there are two difficulties in estimating the errors on points in the 1D diffraction pattern collected from 2D detectors. First is how to estimate the variance of the raw counts in each pixel of the detector. Second is how to handle and propagate error correlations coming from the fact that intensities in neighboring bins in the 1D pattern may not be statistically independent. The degree of statistical correlation between points depends sensitively on choices made during the data reduction process as we describe below. Error correlations are often ignored but can have a significant effect on uncertainty estimates on refined parameters in a Rietveld or PDF refinement.
Scattering is a quantum process and the counts of photons in a photon-counting detector follow a Poisson distribution which has a standard deviation of $\sigma_I \approx \sqrt{I}$~\cite{dinne;b;pdtap08}. However, most 2D detector technologies in wide use are integrating detectors, such as those based on CCDs or image plates, where the number of counts recorded in a pixel, after corrections for electronic noise and detector efficiencies, is proportional to, but not equal to, the number of detected photons and the uncertainties are therefore not simply the square root of the recorded counts. It is then not straightforward to estimate the uncertainties on raw intensities in a 2D image and they are often ignored.
If the detector gain is known, and it is assumed that non-Poissonian contributions to the noise (for example coming from shot noise in the electronics) are negligible, the uncertainties can be obtained by normalizing the intensities by the gain and taking the square root of the normalized counts as is done for photon counting detectors~\cite{hamme;hpr96,boldy;b;hpc10}. The challenge is to determine the detector gain in the particular experimental situation since it depends on X-ray energy and details of the readout.
A further complication comes from the fact that the 2D diffraction image is integrated, or averaged, around Debye-Scherrer rings to obtain the 1D diffraction pattern. If a pixel-splitting algorithm is used, this re-binning process may introduce statistical correlations between data in nearby points in the resulting 1D pattern. Accurate error estimations and error propagations should properly account for this, and in general it is necessary to propagate the full variance-covariance (VC) matrix, which quantifies not only the variance of the measured signal in each bin, but also the correlations between the errors on different points. Currently, most 2D integration software packages, such as Fit2D~\cite{hamme;esrf04}, Powder3d~\cite{hinri;zk06} and PyFAI~\cite{kieff;jpconfs13}, use the pixel-splitting algorithm as their default integration algorithm, which introduces statistical correlations between data in nearby bins in the 1D pattern even without subsequent rebinning or data resampling steps in the subsequent processing. As a result of all these issues, statistical uncertainties are rarely determined and propagated in powder data obtained from 2D detectors, which is a serious problem.
In this paper we assess different approaches for obtaining accurate uncertainty estimates on 1D diffraction patterns obtained from 2D data, including the degree of correlation in the errors quantified in the VC matrix. Different 2D to 1D reduction methods and different binning grids result in different levels of both uncertainties and error correlation between points in the 1D pattern. Currently, most data modeling programs, such as Rietveld refinement programs including GSAS~\cite{larso;unpub04}, FullProf~\cite{rodri;unpub90} and GSAS-II~\cite{toby;jac13} do not utilize the off-diagonal terms of the VC matrix, even when they are available. We describe protocols for obtaining accurate uncertainties and the full VC matrix on 1D powder patterns obtained from 2D diffraction data. We have also implemented these protocols in python based software modules and an open-source program called {\sc SrXplanar} (https://github.com/diffpy/diffpy.srxplanar) This could be used in the integration step for obtaining more accurate estimated uncertainties on data that will be used for Rietveld refinement programs and for other applications that utilize 2D detectors for powder diffraction such as small angle scattering~\cite{barba;acie06} and PDF analysis~\cite{chupa;jac03}. In practice, since most refinement programs cannot currently make use of the full VC matrix, we describe the best protocols for data processing to minimize statistical correlations in the 1D pattern.
\section{Theory}
\subsection{Overview of the process to integrate 2D images to 1D diffraction patterns}
The first step in the integration process is to calibrate the geometric parameters of the experiment such as incident beam center on the detector, sample-detector distance and tilt offsets of the detector. This is usually done by measuring the powder pattern from a calibration sample, such as National Institute of Science and Technology (NIST) standard silicon or ceria, where the structural parameters are known. Fitting routines for doing this are implemented in FIT2D~\cite{hamme;esrf04} and GSAS-II~\cite{toby;jac13}, for example. We have used Fit2D to obtain these geometric parameters for our images.
The next step is conversion of a 2D image of pixels into a 1D histogram of bins. The bins are typically on a $2\theta$-grid, where $2\theta$ is the scattering angle, or they may be on a reciprocal space grid such as $Q = 4\pi\sin\theta/\lambda$ or $s = \sin\theta/\lambda$, where $\lambda$ is the X-ray wavelength and $\theta$ is the Bragg angle which is half the scattering angle. The image integration process consists of taking the intensities in the detector pixels and assigning them to the correct bins of the 1D array with the correct normalization.
We consider specifically data from isotropically scattering samples such as powder diffraction data and small angle scattering from untextured powders. In this case it is necessary to azimuthally average the counts around the conic sections of constant $2\theta$, where we note that the conic sections are usually circles but may be distorted to ellipses, due to detector tilts, but in a known way allowing the correct constant-$2\theta$ integration to be carried out. Except for excluded or masked pixels, each pixel or part thereof, is collected into the 1D bins according to the position of the pixel, and the intensity in the bin is calculated as an average or weighted average (depending on the specific algorithm used in reduction) of the intensities of pixels overlapping that bin.
One approach is to assign pixel counts proportionally to the coverage of the pixel in the bin~\cite{hamme;hpr96,hebob;b;tdxrd09}. This method assumes the intensity function is smoothly varying and estimates the counts that actually fall into the corresponding bin range according to
\begin{equation}
\label{eq;binsplitintensity}
O_{i} = \frac{{\sum_{j=1}^{N_i}} a_{ij} R_{j}}{\sum_{j=1}^{N_i} a_{ij}},
\end{equation}
where $R_{j}$ is the number of counts in the $j$th pixel and the sum is taken over all pixels overlapping the $i$th bin, $a_{ij}$ is the weight factor, which is usually proportional to the coverage of pixel $j$ to bin $i$, and $N_i$ is the number of pixels overlapping the $i$th bin. By assuming that the measurement of each pixel is independent, which is not always true as we discuss below, an estimate of uncertainty on the counts in the $i$th bin may be approximately estimated as the properly weighted standard deviation of those values,
\begin{equation}
\label{eq;binsplituncertainty}
\sigma_{i} = \sqrt{\left(\frac{\sum_{j=1}^{N_i} a_{ij} R_{j}^{2}}{\sum_{j=1}^{N_i} a_{ij}} - O_{i}^{2}\right) \times \frac{N_i}{N_i-1}},
\end{equation}
though this is not expected to give an especially accurate estimate and is fraught with problems.
There may also be intensity corrections that should be applied before the integration, for example to correct for polarization and geometrical effects as well as corrections for detector dark-current, flat-field, spatial distortion, and so on, in addition to removal of masked, dead or saturated pixels. Discussion of these issues is beyond the scope of this paper, though it is assumed here that they have been correctly handled.
\subsection{Statistical correlations between points in the 1D pattern}
Correlations between data points in the 1D pattern have several sources, including the correlation between adjacent pixels in the 2D detector, the algorithm used in re-binning the 2D image to a 1D sequence of intensities, and any re-sampling process that rebins intensities on to different grids during processing, for example, a $2\theta$ to $Q$ conversion or rebinning onto a grid suitable for fast Fourier transformation (FFT).
\subsubsection{Statistical correlation between adjacent pixels in the image}
\label{sec;pixelpixelcorrelation}
Depending on the design of the detector and the experimental conditions, intensities recorded in nearby pixels on 2D detectors may not be statistically independent. The origin of the correlation is complex and quite dependent on the detector design. Detail discussion on it is beyond the scope of this paper. Some detectors are designed to minimize cross-talk between pixels, such as pixel-array detectors and micromachined scintillators, but in the cases in this study, we found pixel-pixel correlations to be quite significant, though it is often ignored.
Different images taken with an identical experimental setup and recorded with the same incident flux will give statistically independent estimates of the scattering intensity. Thus, we can study the statistical distribution of uncorrelated data by making use of multiple frames. Uncertainty on the raw counts can be estimated in a single pixel by considering that same pixel in multiple frames and determining the standard deviation of the measured counts between all frames. Since the frames are statistically independent, this will give an accurate estimate of the standard deviation of the underlying distribution of counts in that pixel. On the other hand, the intensity in the 1D pattern, obtained by integrating around the Debye-Scherrer rings, is influenced by the pixel-pixel correlation because neighboring pixels are often placed into the same 1D bin during the integration. In the results section we show that this effect is observed and may be large, indicating that pixel-to-pixel correlations were important in the case we studied, and should be taken into account in general.
If multiple frames are available, pixel-to-pixel correlations may be removed by making composite images by randomly selecting each pixel in the image from a different frame in the set of identical images. We show that when the images are randomly sampled in this way the correct standard deviation is obtained on the 1D bins.
\subsubsection{Correlations due to the 2D to 1D integration algorithm}
Even if we assume the correlation between adjacent pixels can be ignored or is removed by sampling, the pixel-splitting method will introduce error correlations between neighboring bins, since each pixel contributes to more than one bin in the 1D pattern. Although it can be turned off in Fit2D~\cite{hamme;esrf04} and a non-pixel-splitting algorithm is used in GSAS-II~\cite{toby;jac13}, pixel-splitting algorithms are currently the default in many 2D integration software packages and users should be aware of this issue.
\subsubsection{Correlations due to re-sampling process during processing}
It is sometimes required to re-sample the 1D diffraction intensities onto a different grid, for example, from a $2\theta$-grid to a $Q$-grid. Similar to re-binning that takes place during the 2D integration, re-sampling introduces significant error correlations unless the re-sampling is from a fine to a coarse grid which minimizes bin sharing. However, this is undesirable in most cases since information in the data is lost when re-sampled to a coarser grid, making this a bad tradeoff in most circumstances. We recommend a strategy that avoids rebinning by integrating the 2D image directly onto the final desired grid.
\subsection{Estimating uncertainties from 2D integrating detectors}
The first step in any error propagation process is to estimate the uncertainties on the raw data. This is already difficult for integrating detectors such as image plates, CCDs and related detector technologies, since the uncertainties are not simply the square root of the counts as in a photon counting detector. As we have discussed, the task is made more difficult due to error correlations between pixels in the image.
\subsubsection{Estimating uncertainties from multiple frames}
\label{sec;pixelSwitching}
We show below that directly calculating the standard deviation intensities after integration can give an overestimate due to pixel-pixel correlations. However, we can utilize the statistical independence of multiple frames to eliminate the effects of these correlations. This is done by making compound images of statistically independent pixels by randomly exchanging pixels between images. If a proper rebuild algorithm is used, which does not duplicate or drop pixels during the rebuilding process, the final intensity will not change since the 1D patterns are averaged in the last step; however, the standard deviation that is estimated before the pixel shuffling will be different if nearby pixels are correlated.
In practice, the algorithm we use for the frame resampling is to pick two frames from the set at random and randomly select 50~\% of their pixels to switch. This process is repeated many times, 5000 in this particular case, until each pixel in composed frame is randomly chosen from all frames at the same position. We refer to this as the pixel-switching method.
\subsubsection{Estimating the uncertainty on the counts in each 2D pixel of a single frame}
\label{sec;esdSingleFrame}
When multiple frames are measured, the above approach works. However, it is experimentally expensive and requires extra care in keeping identical experimental conditions between exposures, or perhaps multiple frames are not available or not numerous enough. In general, it is desirable to have a method to estimate uncertainties on points in the powder pattern from a single frame.
For a 2D integrating detector the readout raw counts $R$ of one pixel (after corrections for dark current, flat field, and so on) is proportional to the number of X-ray photons $N$ that impinge on that pixel, with the constant of proportionality being the detector gain $G$,
\begin{equation}
\label{eq;countsFromN}
R = G N.
\end{equation}
If we know $G$, we can calculate the uncertainty on the raw counts by assuming the intensity has a Poissonian distribution,
\begin{equation}
\label{eq;errorFromN}
\sigma_R = G \sigma_N \approx G \sqrt{N}.
\end{equation}
However, the detector gain is usually hard to determine and not the same for different experiments. Here we would like to estimate it from a region of the detector with a uniform intensity, i.e. a region without sharp diffraction features. To determine the gain we consider a set of pixels in this region and assume that the underlying photon intensity is invariant over the region. The actual counts will therefore represent the statistical distribution function of the counts and the standard deviation may be determined. The detector gain may then be determined by inverting Eq.~\ref{eq;errorFromN}. In detail, the uncertainty of pixel $i$ in this region of uniform intensity is given by
\begin{equation}
\sigma_{R_i} = \sqrt{\frac{1}{N-1} \sum_{m=1}^{N} \left( R_m-\bar{R} \right) ^{2}},
\end{equation}
where $N$ is the number of pixels in the set of pixels in the vicinity of $i$th pixel, and the summation is taken over all the pixels in the neighborhood set. Knowing the raw counts and the standard deviation of a pixel, we can calculate the detector gain, by combining Eq.~\ref{eq;countsFromN} and Eq.~\ref{eq;errorFromN},
\begin{equation}
\label{eq;detectorgain}
G = \frac{\sigma_{R}^{2}}{R}.
\end{equation}
With the further assumption that, after proper flat-field corrections, the detector gain is the same for each pixel in the detector, we obtain the gain for all pixels by averaging the gain calculated from different uniform regions of the detector. Once we have the detector gain, the uncertainty of raw counts on all pixels can be estimated with Eq.~\ref{eq;errorFromN}.
In this process, an implicit assumption is that the non-Poissonian contributions to the noise are negligible. This is a reasonable assumption if the pixels have enough counts. It may appear that the pixel-pixel correlations will render this approach problematic since this method assumes uncorrelated intensities between pixels. However, we will show later that it works quite well and this is a reasonable approximation if the regions selected are relatively low in counts, away from regions with sharp diffraction features.
\subsection{Integration method}
\label{sec:integrationMethod}
Correlations of uncertainties in the 1D diffraction pattern depend on the integration method with the pixel-splitting algorithm introducing correlations between bins. To avoid it, we consider a non-pixel-splitting method for the re-binning process where the entire content of a pixel is assigned to a single bin based on the position of the pixel center. Because every pixel has the same weight, the counts in the bin are calculated as the average of all the pixels contributing to it.
This method leads to a less smooth line-profile on a fine grid. However, in most cases the effects are relatively small, and from the perspective of the uncertainties there is a large advantage that no error correlations are introduced by this process. We recommend this method be used for obtaining the most accurate uncertainties on 1D patterns and the least biased and best estimates of uncertainties on refined parameters.
\subsection{Propagating the full variance-covariance matrix}
\label{sec;propagateVCmatrix}
The most robust method for propagating the errors is to propagate the full VC matrix through the full data analysis chain. Here we present the mathematical approach, which is also implemented in our software program for image integration, {\sc SrXplanar}.
We use the common approach of treating all data reduction steps, such as integration or re-sampling steps, as linear operations~\cite{princ;b;mticams04} and express them in matrix form which is easier to generalize. Then, the full VC matrix is propagated using
\begin{equation}
\mathbf{Cov}_{\mathbf{o}} = \mathbf{T} \ \mathbf{Cov}_{\mathbf{c}} \ \mathbf{T}^{T},
\end{equation}
where $\mathbf{Cov}_{\mathbf{c}}$ is the VC matrix of the input data, $\mathbf{T}$ is the transformation matrix, and $\mathbf{Cov}_{\mathbf{o}}$ is the VC matrix of the output data. Full details and derivation of the expression can be found in Appendix~\ref{app;vcmatrixPropagation}
\subsection{Refinement with full VC matrix}
We would like to test the effect on the estimated uncertainties on refined parameters of ignoring off-diagonal terms in the VC matrix. Refinement using correlated data was previously studied by David~\cite{david;jac04}. However, we are not aware of a Rietveld refinement package that can handle the full VC matrix. We have written a refinement program that fits a single Bragg peak with a Gaussian function. The data are low resolution data from a 2D detector and the Gaussian lineshape works adequately, though not perfectly as we describe in Appendix~\ref{app;refinevcmatrix}.
We should also point out that the uncertainty we estimated is actually a measurement of precision, that is, a measure of the width of the confidence interval that results from random fluctuations in the measurement process~\cite{hahn;b;itc05}, rather than the more interesting accuracy, which is a measure of trust in the region of the underlying correct value. However, assessing the accuracy requires the knowledge of systematic errors and deficiencies in the model used in the refinement, which is beyond the scope of this paper.
\section{Experiment}
\label{ExperimentMethod}
Diffraction data from a standard ceria sample and from KFe$_2$As$_2$ were collected at the 11-ID-C beamline at the Advanced Photon Source (APS) at Argonne National Laboratory, using the rapid acquisition pair distribution function (RA-PDF) technique \cite{chupa;jac03} with beam size $0.5\times 0.5$~mm$^{2}$, temperature 100~K and wavelength $\lambda = 0.10798$~\AA.
A 2D Perkin Elmer amorphous silicon detector was used in the experiments. Dark images, with the X-ray shutter closed, of the same length of time were collected for each exposed frame and subtracted from the image. The corrected raw counts were recorded in a tiff format file and integrated with {\sc SrXplanar}. A correction for the solid angle subtended by each pixel is made during the integration, as well as for beam polarization. The pixel size of the detector was $0.2\times 0.2$~mm$^{2}$, and the distance between the detector and sample was 391.12~mm, which was obtained using FIT2D from a calibration sample in the usual way. This pixel size results in a $2\theta$ spread from $0.03 ^{\circ}$ (for pixels near the center of the detector) to $0.02 ^{\circ}$ (pixels at detector edge). We use $0.03 ^{\circ}$ as the approximate $2\theta$ size of one pixel in our discussion. It should be noted that FIT2D uses the $2\theta$ value of the pixel size as the default integration interval for the 1D function though this will vary with the particular integration software used.
\section{Results and discussion}
\subsection{Correlation between adjacent pixels}
In this subsection we explore the different ways described above for determining estimated standard uncertainties (e.s.u.'s), which is actually the estimated predicted uncertainties, on our 1D datapoints from the 2D images. In so doing, we discover the presence of significant statistical correlations between neighboring pixels in the images.
In our dataset we have 50 2D powder diffraction images, or frames, measured serially from the same sample with roughly constant incident flux. Each frame consists of $2048 \times 2048$ pixels in a square array. We can consider that each frame is an independent measurement of the 2D diffraction pattern of the sample. Each of the 2D images is then integrated azimuthally around the powder rings to obtain a 1D powder diffraction pattern of around 2000 intensity vs. $2\theta$ bins. Roughly speaking we can take two approaches to determine the e.s.u.'s: estimate the uncertainty on each pixel and propagate these uncertainties to the 1D bins (estimate pixel and propagate (EPP) approach), or alternatively, to integrate the pixel intensities into the 1D pattern first and then estimate the e.s.u of each bin directly (estimate bin directly (EBD) approach). In the EPP approach, the matrix transformation method described in Sec.~\ref{sec;propagateVCmatrix} and Appendix~\ref{app;vcmatrixPropagation} is used to propagate the uncertainties from the image to the 1D pattern. The e.s.u.'s on the bins in the 1D pattern are plotted in Fig.~\ref{fig;stdmulone}(a) using the EPP approach and in (b) using the EBD approach.
\begin{figure}
\caption{(a) Standard uncertainties estimated on 1D integrated pattern using EPP approach (a) and EBD approach (b). Curves are calculated using EPP50 and EBD50 (green), EPP50PS and EBD50PS (blue), and EPP1 and EBD1 (red), as described in the text below.}
\label{fig;stdmulone}
\includegraphics[width=1.0\textwidth, keepaspectratio]{xrd_std_mul_one.eps}
\end{figure}
The green curves in Fig.~\ref{fig;stdmulone}(a) and (b) show the e.s.u.'s determined from the standard deviation of the observed intensities from the 50 independent measurements. In Fig.~\ref{fig;stdmulone}(a) this is determined on a pixel by pixel basis then propagated to the 1D pattern (we call this the EPP50 method) and in Fig.~\ref{fig;stdmulone}(b) each image is integrated to a 1D pattern and the standard deviation is determined on each bin after the integration (we call this the EBD50 approach). These should give the same result as each other, but comparison of the green curves in Fig.~\ref{fig;stdmulone}(a) and (b) shows that they do not. Much larger uncertainties are estimated from the same data using the EBD50 approach than the EPP50 approach. One explanation for this behavior would be if the intensities in neighboring pixels in a single image were significantly statistically correlated. Taking the standard deviation before and after integration into 1D bins would then give a different standard deviation if these correlations were not taken into account when we propagate the uncertainty from 2D data array to 1D bins. To test this idea we incorporated a pixel-switching (PS) step (described in Section~\ref{sec;pixelSwitching}) before the data integration. This mixes the pixels between the images resulting in neighboring pixels in the image which must be independent. The resulting e.s.u's are shown as the blue curves in Fig.~\ref{fig;stdmulone}(a) EPP50PS, and (b) EBD50PS. The blue and green curve in Fig.~\ref{fig;stdmulone}(a) must be the same as each other (the standard deviation is determined on precisely the same set of pixels) and they are. On the other hand, the EBD50PS approach (blue curve, Fig.~\ref{fig;stdmulone}(b)) now gives smaller e.s.u.'s that are in agreement with the EPP50 estimations.
To understand the slightly counterintuitive result that the EBD50 approach without pixel switching results in overestimated e.s.u's consider the following. Assume a correlation does exist in each individual frame, such that when the measured counts in one pixel fluctuate up from the true expected value, there is a greater probability that the counts in its adjacent pixel also fluctuate up. If both pixels are placed into the same 1D bin, the counts in that bin are fluctuated even higher than if the pixels were uncorrelated. A similar argument holds if the measured counts in the pixel fluctuate down, and the result is that the correlations amplify the fluctuations of the counts in the 1D bin.
This result shows that, in our case, significant statistical correlations exist between neighboring pixels in the images, i.e., for the Perkin-Elmer amorphous silicon detector we used, this effect is significant. It also shows that when estimating e.s.u.'s from the standard deviation of intensities from multiple images, an EPP approach should be used.
We now test whether accurate e.s.u.'s can be obtained using data from just a single image by using the single-frame method described in Section~\ref{sec;esdSingleFrame} (we call these EPP1 and EBD1, though in this case EPP1 and EBD1 actually amount to being same procedure). The Red curve in Fig.~\ref{fig;stdmulone}(a) and (b) shows the e.s.u's when the non-splitting integration is used. They are in good agreement with those obtained by the EPP50, EPP50PS, and EBD50PS method, which suggests that, at least for this detector, the single-frame method for estimating the e.s.u.'s is sufficiently accurate, as well as being much more convenient.
Given that we have shown that neighboring pixels are statistically correlated in our data, it is somewhat surprising that the single-frame estimate does work, since the detector gain is determined from the standard deviation of pixels in the same neighborhood in the image. The possible reason is that the pixel-pixel correlation is weak in the low intensity region without sharp diffraction features, so the detector gain estimated from the distribution of intensities in that region is correct. This is partially supported by the fact that both the EPP50 and EBD50 estimates are similar in regions of the pattern with low intensity diffuse scattering, which is precisely the regions used in the detector gain estimation. The single-frame method also relies on the assumption that, after the flat-field correction, the detector gain is the same for all pixels. To test this assumption, we calculated the detector gain of each pixel using Eq.~\ref{eq;detectorgain}, where $\sigma_{R}^{2}$ is calculated using intensities on the same pixel from 50 frames. We found that in the case we studied, the assumption of equal gain holds very well. The good agreement between the EPP50 and EPP1 estimates implies that the single-frame method produces accurate uncertainty estimates. To further verify this, we estimated uncertainties from several different single-frames in the set and compared them to each other. The results are very similar (not shown in the figure), further validating this approach.
\subsection{Splitting vs non-splitting integration method}
\label{sec;sp_vs_nonsp_method}
\subsubsection{Integrated intensity profile}
In Fig.~\ref{fig;smoothvsjagged} we compare the diffraction patterns obtained from the same image when the integration is carried out using the pixel-splitting and non-splitting method at two representative intervals, $\Delta 2\theta = 0.002^\circ$ and $\Delta 2\theta = 0.02^\circ$.
\begin{figure}
\caption{1D diffraction intensity integrated with: (a) $\Delta 2\theta = 0.002^\circ$ and (b) $\Delta 2\theta = 0.02^\circ$. We integrated the 2D diffraction image with the pixel-splitting (green) and non-pixel-splitting (blue) methods, respectively. The $2\theta$ value of the detector pixel size is equal to $0.03^\circ$ in this case.}
\label{fig;smoothvsjagged}
\includegraphics[width=1.0\textwidth, keepaspectratio]{xrd_smoothvsjagged.eps}
\end{figure}
The non-splitting method data have the same integrated peak intensity but appear less smooth, though slightly narrower. In fact, the pixel-splitting method is a {\it de facto} smoothing method, which gives smoother data at the expense of resolution. The basic assumption is that the diffraction intensity is uniformly distributed within each pixel, which makes the intensity change between bins smoothly due to the pixel sharing at bin edges. It also makes some pixels at bin edges in the peak center bin give part of their counts to the shoulder, which increases the intensity of shoulder and decreases the intensity of center resulting in a slight peak broadening.
In comparison, the non-splitting method assumes the entire intensity belongs to one bin even if part of that pixel falls into other bins. When data are binned on a very fine grid where the bin size is much smaller than the pixel size, the non-splitting method is under-sampled and some bins will be empty. However, there is little advantage to binning the data as finely as this since the resolution is limited by the pixel size at the very least, and in a well designed experiment the intrinsic resolution will be worse than the pixel width. When a large bin width is used, the number of pixels that cross the bin edge is small compared to the total number of pixels fully in the bin, and therefore the difference between two methods is small, though still significant when the bin width matches the pixel width, as evident in Fig.~\ref{fig;smoothvsjagged}.
Of course, the non-splitting method does not introduce any statistical correlations and is preferred for that reason.
\subsubsection{Propagating the full variance-covariance matrix}~
In Fig.~\ref{fig;vcmatrix} we show a false-color image of the VC matrix with the integration done in different ways.
\begin{figure}
\caption{VC matrix of the diffraction intensity integrated with: (a) non-splitting method and $\Delta 2\theta = 0.03^\circ$, (b) splitting method and $\Delta 2\theta = 0.03^\circ$, (c) splitting method and $\Delta 2\theta = 0.006^\circ$, and (d) splitting method and $\Delta 2\theta = 0.08^\circ$. The $2\theta$ value of the pixel size is equal to $0.03^\circ$ in this case. Only the non-splitting method gives a diagonal VC matrix.}
\label{fig;vcmatrix}
\includegraphics[width=1.0\textwidth, keepaspectratio]{xrd_vcmatrix.eps}
\end{figure}
As expected, the VC matrix obtained by the non-splitting integration method is diagonal, while that obtained by the splitting method is mostly diagonal but with a ridge along the diagonal. The width of the ridge depends on the bin width compared to the pixel size, with a broader ridge from the smaller bin-size, and a nearly diagonal ridge when the bin size is approximately equal to pixel size.
\subsection{Influence on the VC matrix of a re-sampling of the 1D pattern}
\label{sec;VCmatrix_resampling}
Here we study how re-sampling the 1D function onto a new grid affects the VC matrix. We tested two types of re-sampling steps which are common in data analysis. One is to re-sample the diffraction intensity from a regular $2\theta$-grid to a regular $Q$-space grid. The other is to re-sample the data on to another regular grid in the same integration space.
We started with the diffraction intensity initially integrated in $2\theta$-space with $\Delta 2\theta = 0.03^\circ$ and using the non-splitting method and the splitting method. The data were then re-sampled on to a $Q$-space grid of the same length, and alternatively onto another $2\theta$-space but with interval equal to $0.01^\circ$ and $0.05^\circ$. As a comparison, we have also integrated the 2D image directly onto the same $Q$-space grid and $2\theta$-grids. The results are shown in Fig.~\ref{fig;resample}.
Compared to data that were directly integrated, data that were re-sampled from a $2\theta$-grid to a $Q$-space grid, or to a finer grid, show increased statistical correlations, though we note that interpolation to a coarser grid does not induce significant additional correlations in the resulting data.
\begin{figure}
\caption{Selected range of VC matrix of diffraction intensity directly integrated or re-sampled to corresponding integration grid. The original intensity was integrated to a regular $2\theta$-grid with $\Delta 2\theta = 0.03 ^\circ$. The integration grids are: (first column, a, d, g, j) regular $Q$-space grid with same length of original intensity; (second column, b, e, h, k) regular $2\theta$-grid with $\Delta 2\theta = 0.01 ^\circ$; (third column, c, f, i, l) regular $2\theta$-grid with $\Delta 2\theta = 0.05 ^\circ$. The 2D diffraction pattern was (first row, a, b, c) directly integrated with non-splitting method, (second row, d, e, f) re-sampled from data integrated with non-splitting method, (third row, g, h, i) directly integrated with splitting method, (fourth row, j, k, l) re-sampled from data integrated with splitting method.}
\label{fig;resample}
\includegraphics[width=1.0\textwidth, keepaspectratio]{xrd_resample_one.eps}
\end{figure}
In summary, direct integration of the 2D data onto the final grid results in the smallest correlations and should be the best practice. Usually there is no need to re-sample if one carefully chooses the integration grid during 2D integration process. This result is not a surprise, but it is often not followed in practice and worth mentioning explicitly. It is especially relevant in measurements like small angle scattering or PDF where the subsequent data processing is conducted on a $Q$-grid. For Rietveld refinement, McCusker suggested the optimal binning for Rietveld is where there are 6-10 points over the FWHM of a peak~\cite{mccus;jac99}. If during processing it is determined that a new grid is needed, it is preferable that the software can reintegrate the original images onto the new grid.
\subsection{Structural refinements utilizing the full VC matrix}
Most powder diffraction data are modeled by some kind of peak fitting process and we would like to know the effects on refined parameters of neglecting the error correlations under discussion. Since no Rietveld programs available to us could handle the full VC matrix, we wrote our own simple fitting program and fit the second peak in a test data-set from a ceria sample ($1.80^{\circ} < 2 \theta < 2.15^{\circ}$) to a Gaussian shape function given by $y = \frac{A}{\sigma \sqrt{2 \pi}} \exp(-\frac{(x-x_0)^2}{2 \sigma ^2}) + B$, where $A$ and $x_0$ are the integrated intensity and peak position, respectively, and $B$ is a constant background. In the refinement, we refine $A$, $x_0$, $\sigma$ and $B$.
Here we discuss the effect on the values and estimated uncertainties of refined parameters due to different data reduction choices: (a) fitting data integrated with the non-splitting method (note that the VC matrix obtained by the non-splitting method is already diagonal), (b) fitting data integrated with the splitting-method but considering the full VC matrix in the fit, and (c) fitting data integrated with the splitting-method but only using the diagonal of the VC matrix (this approach is strictly incorrect but is the {\it de facto} current approach in most studies). We also extended the study to explore the effect on the values and estimated uncertainties of using a unit weight matrix in the least-squares equation instead of using the propagated statistical uncertainties. To this end, we explored (d) fitting data integrated with the non-splitting method but using a unity weight matrix and (e) as (d) except the fits were on data integrated with the splitting method. The uncertainties are estimated using Eq.~\ref{eq;covWithVC} and Eq.~\ref{eq;covWithoutVC}. In summary, we want to know the effect on the estimated uncertainties of ignoring off-diagonal covariance terms in the analysis (i.e., (c)) and the effect on the refined values themselves of using the wrong weights in the minimization (i.e., (d) and (e), and to a lesser extent (c))
The results of the fitting are shown in Table~\ref{tab;vcrefine}. We refine data that were processed with two integration intervals, an interval much smaller than the pixel size ($\Delta 2\theta = 0.13 \times \Delta 2\theta_0 = 0.004^{\circ}$) and an interval equal to the pixel size ($\Delta 2\theta = 1.0 \times \Delta 2\theta_0 = 0.03^{\circ}$). In the latter case the data correlations are much smaller as discussed earlier.
\begin{table}
\caption{Peak fit results. Refinements used (a) data integrated with the non-splitting method, (b) data integrated with the splitting method and considering the full VC matrix, (c) data integrated with the splitting method but ignoring the off-diagonal elements of the VC matrix, (d) data integrated with the non-splitting method and refined with a unity VC matrix, and (e) data integrated with the splitting method and refined with a unity VC matrix. $\Delta_1$ and $\Delta_2$ are uncertainties on the refined parameter estimated using Eq.~\ref{eq;covWithVC} and Eq.~\ref{eq;covWithoutVC}, respectively. Please refer to the main text for their meaning. For (d) and (e), $\Delta_1$ is not available since unity weight matrix was used in the refinement.}
\label{tab;vcrefine}
\centering
\begin{tabular}{lrrrrr}
\multicolumn{6}{l}{$\Delta 2\theta = 0.013 \times \Delta 2\theta_0 = 0.004^{\circ}$} \\
& (a) & (b) & (c) & (d) & (e) \\
\hline
\multicolumn{6}{l}{Peak position ($^{\circ}$)} \\
Value & 1.980729 & 1.980727 & 1.980725 & 1.980770 & 1.980754 \\
$\Delta_1$ & 0.000004 & 0.000004 & 0.000002 & - & - \\
$\Delta_2$ & 0.000109 & 0.000084 & 0.000078 & 0.000124 & 0.000087 \\
\hline
\multicolumn{6}{l}{Peak width ($^{\circ}$)} \\
Value & 0.030897 & 0.032103 & 0.032095 & 0.031032 & 0.032148 \\
$\Delta_1$ & 0.000003 & 0.000003 & 0.000002 & - & - \\
$\Delta_2$ & 0.000125 & 0.000097 & 0.000091 & 0.000124 & 0.000095 \\
\hline
\multicolumn{6}{l}{Peak intensity} \\
Value & 10977.4 & 10947.2 & 10994.1 & 11017.5 & 11012.6 \\
$\Delta_1$ & 1.5 & 1.4 & 0.5 & - & - \\
$\Delta_2$ & 47.0 & 35.0 & 33.0 & 46.0 & 32.0 \\
\hline
\multicolumn{6}{l}{Durbin-Watson $d$-statistic ($Q$=1.4)} \\
Value & 0.170 & 0.092 & 0.097 & 0.291 & 0.189 \\
\hline
\hline
& & & & & \\
\multicolumn{6}{l}{$\Delta 2\theta = 1.0 \times \Delta 2\theta_0 = 0.03^{\circ}$} \\
& (a) & (b) & (c) & (d) & (e) \\
\hline
\multicolumn{6}{l}{Peak position ($^{\circ}$)} \\
Value & 1.980709 & 1.980706 & 1.980700 & 1.980722 & 1.980731 \\
$\Delta_1$ & 0.000004 & 0.000004 & 0.000004 & - & - \\
$\Delta_2$ & 0.000203 & 0.000167 & 0.000167 & 0.000202 & 0.000166 \\
\hline
\multicolumn{6}{l}{Peak width ($^{\circ}$)} \\
Value & 0.031626 & 0.032784 & 0.032873 & 0.031637 & 0.032841 \\
$\Delta_1$ & 0.000003 & 0.000003 & 0.000003 & - & - \\
$\Delta_2$ & 0.000234 & 0.000195 & 0.000195 & 0.000233 & 0.000233 \\
\hline
\multicolumn{6}{l}{Peak intensity} \\
Value & 10932.2 & 10991.8 & 11003.0 & 10943.2 & 11006.4 \\
$\Delta_1$ & 1.5 & 1.5 & 1.2 & - & - \\
$\Delta_2$ & 85.0 & 70.0 & 70.0 & 126.0 & 99.0 \\
\hline
\multicolumn{6}{l}{Durbin-Watson $d$-statistic ($Q$=1.4)} \\
Value & 1.714 & 1.553 & 1.800 & 1.515 & 1.527 \\
\end{tabular}
\end{table}
\begin{figure}
\caption{Integrated intensity (blue), fitting results (red), and difference curve (green) of second peak integrated with $\Delta 2\theta = 0.004 ^\circ$. Refinement is performed on (a) data integrated with the non-splitting method, (b) data integrated with the splitting method and considering the full VC matrix. Part of the integrated intensity and the difference curve are zoomed 10x for better visualization}
\label{fig;fitting}
\includegraphics[width=1.0\textwidth, keepaspectratio]{xrd_fitting.eps}
\end{figure}
The uncertainties of refined parameters are calculated using two different metrics. $\Delta_1$, calculated using Eq.~\ref{eq;covWithVC}, only counts the contribution to the uncertainty due to statistical errors. On the contrary, $\Delta_2$, calculated using Eq.~\ref{eq;covWithoutVC}, is a less statistically justifiable metric but is widely used in refinement programs to estimate uncertainties. To some extent it takes into account model errors since it scales the estimated uncertainties by the residual between the experimental and calculated data.
For all refined parameters, $\Delta_1 $ is much smaller than $\Delta_2$ by one order of magnitude. This indicates, in our specific case, that model errors dominate the residual, i.e. inconsistencies between the calculated and data patterns due to deficiencies in the model are contributing much more to the residuals than the statistical errors. This is a common situation in many Rietveld refinements. However, the systematic error does not have the same effect on all the different refined parameters. For the peak position, when the statistical errors are correctly handled, i.e. method (a) and (b), or partially correctly handled, i.e. method (c), the refined results are very close to each other and all lie within $\Delta_1$ of each other. On the other hand, ignoring the statistical uncertainty in the refinement, i.e. method (d) and (e), leads to a significant deviation compared to (a-c). The deviation is much larger than $\Delta_1$ but smaller than $\Delta_2$. It implies that, in this case, the uncertainty of the peak position is dominated by the statistical uncertainties and the $\Delta_2$ uncertainty estimate grossly over-estimates the true uncertainty. This is because the peak in the data, whilst not pure Gaussian, is symmetric, and so fitting it with a symmetric function such as a Gaussian does not bias the refined value. This is supported by looking at the difference curve in Fig.~\ref{fig;fitting} which has a rather symmetric `w' shape indicating that the misfit is symmetric. On the other hand, the $\Delta_2$ estimation of the uncertainty does a much better job than the purely statistical $\Delta_1$ for the other refined parameters in the Gaussian. Thus, cases where the fit residual is dominated by model deficiencies, $\Delta_2$, are still a better measure of uncertainty than $\Delta_1$ although they cannot be relied on to give accurate uncertainty estimates on refined parameters.
In principle, the statistical errors have been correctly handled using both methods (a) and (b), which should therefore be in good agreement with each other for each bin-size. Indeed, the statistical uncertainties are in good agreement using the two methods. The agreement of (a) and (b) also implies that the splitting and non-splitting method carry the same amount of information when the full VC matrix is used in the analysis. On the other hand, when the off-diagonal terms in the VC matrix are ignored (the case (c)) the statistical uncertainties on refined parameters are consistently underestimated.
For the peak width, the situation is more complex, since the splitting method actually broadens the peaks and so it is not possible to compare the results of the splitting and non-splitting methods with each other, as discussed in Sec.~\ref{sec:integrationMethod}. This is clearly seen in Table~\ref{tab;vcrefine} where the splitting method results (b), (c) and (e), are all broader than the non-splitting method (a) and (d). The broadening is considerably larger even than the large $\Delta_2$ uncertainty estimates. This indicates that the smoothing effects of using the splitting method are much larger than the magnitude of systematic error caused by peak profile mismatch and should be therefore considered.
For the case of the peak intensity, we do not expect to see effects of the pixel-splitting protocol used since this preserves the integrated intensity. However, we see that the mismatch of the model peak profile to the data introduces large uncertainties on the peak intensity. The absolute difference between each value is $\sim 10\times$ larger than $\Delta_1$ though $\Delta_2$ seems to do a better job of estimating these uncertainties. Since the refined values were from the same original data-set, and it is not completely clear {\it a priori} which method will give the best estimates, the difference between the results from the different integration methods gives a measure of our actual uncertainty on the values of the refined parameters. In the case of the peak intensity, the errors coming from the inadequate model dominate the real uncertainties.
The serial correlation~\cite{hill1;jac87,andre;jac94,Berar;jac91} in each refinement was characterized using the Durbin-Watson $d$-statistic~\cite{durbi;biometrika50,durbi;biometrika51,durbi;biometrika71}, and the 0.1\% significance $Q$ value~\cite{theil;jasa61}. These results are presented in Table~\ref{tab;vcrefine}. Serial correlation is more significant when the fit is carried out on a fine grid as $d$ is much smaller than $Q$, but not significant when using a coarse gird. We also note that the non-splitting integration method gives less serial correlation than the splitting method, as its $d$ value is larger, which strengthens the point that the non-splitting method generates less correlated data, although serial correlation is only an indirect measurement of the correlation.
Given that most data analysis software programs, such as Rietveld refinement programs GSAS~\cite{larso;unpub04}, GSAS-II~\cite{toby;jac13} and FullProf~\cite{rodri;unpub90}, do not consider the full VC matrix it is important to minimize error correlations during data analysis since refinements on data that have been processed in a way to minimize the off-diagonal terms in the VC matrix will be the most accurate. In the future, refinement programs that can handle the full VC matrix, coupled with integration protocols that propagate the full VC matrix, may circumvent this issue, at the cost of increased computation time.
We should point out that although the deviation is smaller or comparable with $\Delta_2$, $\Delta_2$ does not give correct estimate for the statistical uncertainty, as is the case for the peak position, where the actual value fluctuations are much smaller. Due to the limitation of the least-square refinement method, it is hard to determine which method gives the correct refined values and uncertainty estimation. Further study, for example, using Bayesian methods may be required to obtain more reliable refinement results.
\section{Conclusions}
This paper discusses methods to extract reliable statistical uncertainties on points in a 1D powder diffraction pattern obtained from widely used 2D integrating detectors. It also explores the origin and extent of statistical correlations between points in the 1D diffraction pattern. The error correlations may be handled correctly by propagating the full VC matrix through the data analysis steps. A software program, {\sc SrXplanar}, is presented for azimuthally integrating 2D detector images while determining statistical uncertainties and the full VC matrix, and for propagating this to the final pattern. However, most modeling and fitting programs cannot utilize the information in the full VC matrix and so data processing steps that minimize error correlations are explored and an optimal protocol to minimize these correlations is presented. It is strongly suggested to use a non-pixel-splitting integration algorithm and to integrate data directly onto the final 1D grid that will be modeled or further processed. Although the effects of systematic error are larger than the statistical errors in the cases we considered, the true uncertainty may not be determined by systematic errors depending on model deficiencies. Using correct uncertainty information in refinements is important for obtaining correct uncertainty estimation. Failure to do so, for example by neglecting the off-diagonal terms of the VC matrix or fully ignoring the uncertainty information, may result in an underestimation of uncertainties on refined parameters. Estimating uncertainties using $\Delta_2$ defined in Eq.~\ref{eq;covWithoutVC} can account for some contributions of model errors to the uncertainty, but does not give accurate uncertainty estimates in all cases. Even when Eq.~\ref{eq;covWithoutVC} is used to estimate the uncertainties, it is recommended that the correct statistical weights are used in the least-squares equations during model minimization.
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1309.3608
|
\section{Introduction}
The adaptive finite element method plays an important role in the
numerical solution for partial differential equations
\cite{AinsworthOden00,Verfurth96}. The convergence and optimality of
the adaptive method have been much studied in recent years. For the
Poisson equation and its variants, the theory is well--developed
\cite{CarstensenHoppe05b,CasconKreuzerNochettoSiebert07,ChenHolstXu07,Dolfler96,MekchayNochetto05, MorinNochettoSiebert00,
Stevenson06,Stevenson05}.
However, for many other important
problems this is not the case. Among these under studied problems is the Stokes problem, the main subject of this paper.
The convergence analysis of the adaptive finite element method of the Poisson equation is based on the orthogonality property
\cite{CasconKreuzerNochettoSiebert07,Dolfler96,
MekchayNochetto05,MorinNochettoSiebert00}, such orthogonality can be weakened
to some quasi--orthogonality for the nonconforming and
mixed methods \cite{BeckerMao2008,BeckerMaoShi2010,CarstensenHoppe05a,
CarstensenHoppe05b,CarstensenRabus2010,ChenXuHoppe2010,ChenHolstXu07,HSX10,HuangHuangXu2010,MaoZhaoShi2010,Rabaus2010}.
The Stokes problem, as a saddle point problem with two variables
(velocity and pressure), lacks the usual orthogonality or
quasi--orthogonality that holds for the positive and definite problem. As a
result, it is not obvious how the technique for nonconforming and
mixed methods for the Poisson equation can be carried over to the
Stokes problem. Although the mixed formulation of the Poisson
equation is also a saddle point problem, analyses of this formulation's
convergence and optimality \cite{BeckerMao2008,CarstensenRabus2010,ChenHolstXu07} are not so
different from that for the primary formulation of the Poisson
equation. The reason is that only the stress variable, which can be decoupled from
the primary variable, needs to be involved in the analysis.
This is not, however, the case for the Stokes problem under
consideration here because the two variables, velocity and pressure,
are coupled and cannot be separated in analyses of the convergence and optimality. To circumvent this difficulty,
B\"{a}nsch, Morin, and Nochetto developed a modified adaptive
procedure in which the Uzawa algorithm on the continuous level is used as
the outer iteration
\cite{BanschMorinNochetto02,Kondratyuk06,KondratyukStevenson07}.
The optimality of the adaptive finite element method for the Poisson equation is analyzed based on
discrete reliability (see \cite{CasconKreuzerNochettoSiebert07,Stevenson06,Stevenson05} and the references therein).
Basically, we need one restriction operator
and one prolongation operator in order to analyze the discrete reliability. For the conforming method, a natural candidate
for the prolongation operator is the usual inclusion operator, and for the restriction operator a Scott--Zhang--type
can be used as it has both the local projection property and the global and uniform boundedness property. For the nonconforming method under consideration here, however,
it is a challenge to come up with a prolongation operator that has both the local projection property and the global and uniform boundedness property. For the nonconforming linear element method for the Poisson equation, such a difficulty can be circumvented
using the discrete Helmholtz decomposition \cite{BeckerMaoShi2010,Rabaus2010}. However, the Helmholtz decomposition seems not applicable for the problem under consideration because the existence of such a decomposition is unclear for the general case.
The first convergence and optimality analysis of a standard adaptive finite element
method for the Stokes problem was presented in a technical
report~\cite{HuXu2007} in 2007 by the authors of this paper. The analysis was based on some special relation
between the nonconforming $P_1$ element and the lowest Raviart--Thomas
element for the Stokes problem and one prolongation operator between
the discrete spaces.
But we later found a gap in our discrete reliability analysis
caused by the prolongation operator used therein.
A convergence and
optimality analysis was published in \cite{BeckerMao2011} in 2011; however, we also found
a gap in their analysis similar to that
in our earlier report \cite{HuXu2007} (see Appendix A for more details).
The present paper is an improved version of~\cite{HuXu2007} with simplified
and corrected
proofs. Its purpose is to provide a
rigorous analysis of the convergence and optimality of the adaptive
nonconforming linear element method for the Stokes problem. The main
idea is to establish the orthogonality or quasi--orthogonality of both
the velocity variable and the pressure variable. The nonconformity of the discrete velocity
space is the main difficulty in establishing the desired quasi--orthogonality property and
the discrete reliability estimate.
To overcome this difficulty we take two steps, (1) we establish
the quasi--orthogonality for both the velocity and pressure variables by using a special conservative
property of the nonconforming linear element, and (2) we introduce a new prolongation operator that has
both the projection property and the uniform boundedness property
for the discrete reliability analysis. To
analyze optimality within the standard nonlinear approximate class
\cite{CasconKreuzerNochettoSiebert07}, we define a new interpolation
operator to bound the consistency error and prove that the consistency error can be bounded by the approximation
error up to oscillation. This in fact implies that the nonlinear approximate class used in \cite{HuXu2007} is {\em equivalent}
to the standard nonlinear approximate class
\cite{CasconKreuzerNochettoSiebert07}. Finally, by introducing a new parameter-dependent error
estimator, we prove convergence and optimality estimates for the
Stokes problem.
The rest of the paper is organized as follows. In Section 2 we
present the Stokes problem and its nonconforming linear finite element
method, and recall a posteriori error estimate according to
\cite{ca04,CarstensenFunken01a,CarstensenHu07,DariDuranPadra95}. We prove the quasi--orthogonality in
Section 3 and then show the
reduction of some total error in Section 4 in terms of a new
parameter-dependent estimator. We introduce a new prolongation operator to
establish discrete reliability in Section 5. And, we show
optimality of the adaptive nonconforming linear element method in
Section 6.
\section{The adaptive nonconforming linear element}
Let us first introduce some notations. We use the standard gradient and divergence
operators $\nabla r:=(\partial r/\partial x\,,\partial r/\partial y)$ for a scalar function $r$, and $
\operatorname{div}\boldsymbol{\psi}:={\partial\psi_1}/{\partial x}+{\partial\psi_2}/{\partial y}$ for a vector
function $\boldsymbol{\psi}=(\psi_1,\psi_2)$. Given a polygonal
domain $\Omega\subset \mathcal{R}^2$ with the boundary $\partial\Omega$, we use
the standard notation for Sobolev spaces, such as $H^1(\Omega)$ and
$L^2(\Omega)$. We define
\begin{equation*}
\begin{split}
H_0^1(\Omega):=\{v\in H^1(\Omega), v=0 \text{ on } \partial \Omega\}\,, \text{ and }\\[1.0ex]
L_0^2(\Omega):=\{q\in L^2(\Omega), \int_{\Omega}qdx=0\}.
\end{split}
\end{equation*}
In addition, we denote $(\cdot, \cdot)_{L^2(\Omega)}$ as the
usual $L^2$ inner product of functions in the space $L^2(\Omega)$, and
$\|\cdot\|_{L^2(\Omega)}$ the $L^2$ norm.
Suppose that $\overline{\Omega}$ is covered exactly by a sequence of
shape--regular triangulations $\mathcal{T}_k$ ($k\geq 0$) consisting of triangles in
$2D$ (see \cite{CiaBook}), and that this sequence is produced by some
adaptive algorithm where $\mathcal{T}_k$ is some nested refinement of
$\mathcal{T}_{k-1}$ by the newest vertex bisection \cite{Stevenson06,Stevenson05}. Let $\mathcal{E}_k$ be the set of
all edges in $\mathcal{T}_k$; $\mathcal{E}_k(\Omega)$ the set of interior
edges; $\mathcal{E}(K)$ the set of edges of any given element $K$ in
$\mathcal{T}_k$; and $h_K=|K|^{1/2}$ the size of the element $K\in
\mathcal{T}_k$ where $|K|$ is the area of element $K$. $\omega_K$ is the
union of elements $K'\in \mathcal{T}_k$ that share an edge with $K$, and
$\omega_E$ is the union of elements that share a common edge
$E$. Given any edge $E\in\mathcal{E}_k(\Omega)$ with the length $h_E$, we
assign one fixed unit normal $\nu_E:=(\nu_1,\, \nu_2)$ and tangential
vector $\tau_E:=(-\nu_2,\,\nu_1)$. For $E$ on the boundary, we choose
$\nu_E:=\nu$, the unit outward normal to $\Omega$. Once $\nu_E$ and
$\tau_E$ are fixed on $E$, in relation to $\nu_E$ we define
the elements $K_{-}\in \mathcal{T}_k$ and $K_{+}\in \mathcal{T}_k$, with
$E=K_{+}\cap K_{-}$. Given $E\in\mathcal{E}_k(\Omega)$ and some
$\mathcal{R}^d$-valued function $v$ defined in $\Omega$, with $d=1,2$, we
denote $[v]:=(v|_{K_+})|_E-(v|_{K_-})|_E$ as the jump of $v$ across
$E$, where $v|_K$ is the restriction of $v$ on $K$ and $v|_E$ is the
restriction of $v$ on $E$.
\subsection{The Stokes problem and its nonconforming linear element }
The Stokes problem is defined as follows: Given $g\in L^2(\Omega)^2$, find $(u, p)\in V\times
Q:=(H_0^1(\Omega))^2\times L_0^2(\Omega)$ such that
\begin{equation}\label{eq7.1}
\begin{split}
&a(u,v)+b(v,p)+b(u,q)=(g,v)_{L^2(\Omega)} \text{ for any }(v,q)\in V\times
Q\,,
\end{split}
\end{equation}
where $u$ and $p$ are the velocity and pressure of the flow, respectively, and
\begin{equation}
a(u,v):=\mu(\nabla u,\nabla v)_{L^2(\Omega)} \text{ and } b(v,q):=(\operatorname{div} v,
q)_{L^2(\Omega)},
\end{equation}
where $\mu>0$ is the viscosity coefficient of the flow.
Given $\omega\subset \mathcal{R}^2$ and some integer $\ell$, denote $P_{\ell}(\omega)$ as the space
of polynomials of degree $\leq \ell$ over $\omega$. We define
\begin{equation*}
\begin{split}
V_k:&=\{v_k\in L^2(\Omega)^2, v_k|_{K}\in P_1(K)^2 \text{ for any }K\in
\mathcal{T}_k, \int_E[v_k]\,ds=0\\[1.0ex]
&\qquad\text{ for any }E\in\mathcal{E}_k(\Omega), \text{ and } \int_Ev_k\,ds=0
\text{
for any }E\in\mathcal{E}_k\cap\partial\Omega\}\,,\\[1.0ex]
Q_k:&=\{q_k\in Q, q_k|_K\in P_0(K) \text{ for any } K\in \mathcal{T}_k\}.
\end{split}
\end{equation*}
Since $V_k$ is not a subspace of $H^1(\Omega)^2$, the gradient and
divergence operators are defined element by element with respect to $\mathcal{T}_k$, and denoted by
$\nabla_k$ and $\operatorname{div}_k$.
Define the
piecewise smooth space
\begin{equation}
H^1(\mathcal{T}_k):=\{v\in L^2(\Omega), v|_K\in H^1(K) \text{ for any
}K\in\mathcal{T}_k\}\,.
\end{equation}
The discrete bilinear forms read
\begin{equation}
a_k(u,v):=\mu(\nabla_ku,\nabla_kv)_{L^2(\Omega)} \text{ and } b_k(v, q):=(\operatorname{div}_kv, q)_{L^2(\Omega)}
\end{equation}
for any $u, v\in (H^1(\mathcal{T}_k))^2, \text{ and } q\in Q$.
The nonconforming $P_1$ element, proposed in \cite{CroRav73}, for
the Stokes problem is as follows: Given $g\in
L^2(\Omega)^2$, find $(u_k,p_k)\in V_k\times Q_k$ such that
\begin{equation}\label{eq7.3}
\begin{split}
\begin{split}
&a_k(u_k,v)+b_k(v,p_k)+b_k(u_k,q)=(g,v)_{L^2(\Omega)} \text{ for any
}(v,q) \in V_k\times Q_k\,.
\end{split}
\end{split}
\end{equation}
Let $\id\in \mathcal{R}^{2\x2}$ be the identity matrix. Define
$$
\sigma_{k}:=\mu\nabla_ku_k+p_k\id.
$$
Then, we have
\begin{equation}
(\sigma_k, \nabla_kv_k)_{L^2(\Omega)}=(g, v_k)_{L^2(\Omega)} \text{ for any }v_k\in V_k.
\end{equation}
\subsection{The a posteriori error estimate} To recall the a posteriori error estimator of the nonconforming $P_1$ element, we define
the residual $\res_{k-1}(\cdot)$ by
\begin{equation}\label{Res4}
\res_{k-1}(v):=(g,v)_{L^2(\Omega)}-a_k(u_{k-1},v)-b_k(v,p_{k-1}) \text{ for any
}v\in H^1(\mathcal{T}_k)^2\,,
\end{equation}
with the solution $(u_{k-1}, p_{k-1})$ of \eqref{eq7.3} on the mesh $\mathcal{T}_{k-1}$,
which is a coarser and nested mesh of $\mathcal{T}_k$. It follows from the definition of $(u_{k-1}, p_{k-1})$ that
$$
\res_{k-1}(v_{k-1})=0 \text{ for any }v_{k-1}\in V_{k-1}.
$$
Given $K\in\mathcal{T}_k$, we define the element estimator
\begin{equation}
\eta_{K}(u_k, p_k):=h_K\|g\|_{L^2(K)}+(\sum\limits_{E\subset\partial
K}h_K\|[\nabla_ku_k \tau_E]\|_{L^2(E)}^2)^{1/2}.
\end{equation}
Given $S_k\subset\mathcal{T}_k$, we define the estimator over it by
\begin{equation}\label{eq7.12}
\eta^2(u_k, p_k, S_k):= \sum\limits_{K\in S_k}\eta_{K}^2(u_k, p_k).
\end{equation}
Given any $K\in \mathcal{T}_k$, denote $g_K$ as the $L^2$ projection of $g$
onto $P_{0}(K)$. We define the oscillation
\begin{equation}\label{eq5.1b}
\osc^2(g,\mathcal{T}_k):=\sum\limits_{K\in \mathcal{T}_k}h_K^2\|g-g_K\|_{L^2(K)}^2.
\end{equation}
The reliability and efficiency of the estimator $\eta(u_k,
p_k, \mathcal{T}_k)$ can be found in
\cite{ca04,CarstensenFunken01a,CarstensenHu07,DariDuranPadra95}, as stated in the following lemma.
\begin{lemma}\label{Theorem7.2} Let $(u,p)$ and $(u_k, p_k)$ be the solutions
of the Stokes problem
\eqref{eq7.1} and the discrete problem \eqref{eq7.3}, respectively. Then,
\begin{equation}\label{eq7.11}
\|\nabla_k(u-u_k)\|_{L^2(\Omega)}^2+\|p-p_k\|_{L^2(\Omega)}^2\lesssim
\eta^2(u_k, p_k, \mathcal{T}_k),
\end{equation}
\begin{equation}\label{eq7.11b}
\eta^2(u_k, p_k, \mathcal{T}_k)\lesssim
\|\nabla_k(u-u_k)\|_{L^2(\Omega)}^2+\|p-p_k\|_{L^2(\Omega)}^2+\osc^2(g,
\mathcal{T}_k).
\end{equation}
\end{lemma}
\begin{remark}
For the Stokes problem, the estimator usually involves the pressure approximation. For
the nonconforming $P_1$ element, as shown in the above lemma, we can decouple
the pressure from the velocity \cite{DariDuranPadra95}.
\end{remark}
Here and throughout the paper, we use the notations $\lesssim$
and $\approxeq$. When we write
$$
A_1 \lesssim B_1, \text{ and } A_2\approxeq B_2,
$$
possible constants $C_1$, $c_2$ and $C_2$ exist such that
$$
A_1 \leq C_1 B_1, \text{ and } c_2B_2\leq A_2\leq C_2 B_2.
$$
\subsection{The adaptive nonconforming finite element method}
The adaptive algorithm is defined as follows: Let
$\mathcal T_0$ be an initial shape--regular triangulation, a right--side $g\in L^2(\Omega)^2$, a tolerance $\epsilon$, and a parameter $0<\theta<1$.
\begin{algorithm}\label{Algorithm}
\noindent $[\mathcal{T}_N\,, u_N\,, p_N]$={\bf \small
ANFEM}$(\mathcal{T}_0, g, \epsilon, \theta)$
$\eta = \epsilon\,, k=0$
\smallskip
{\bf \small WHILE} $\eta \geq \epsilon$, {\bf \small DO}
\begin{enumerate}
\item Solve \eqref{eq7.3} on $\mathcal{T}_{k}$ to get the solution
$(u_{k}, p_{k})$.
\item Compute the error estimator $\eta=\eta(u_{k}, p_{k}, \mathcal{T}_{k})$.
\item Mark the minimal element set $\mathcal{M}_{k}$ such that
\begin{equation}\label{bulk}
\eta ^2(u_{k}, p_{k}, \mathcal{M}_{k})\geq \theta \, \eta ^2(u_{k}, p_{k}, \mathcal{T}_{k}).
\end{equation}
\item Refine each triangle $K \in \mathcal{M}_{k}$
by the newest vertex bisection to get $\mathcal{T}_{k+1}$ and set $k=:k+1$.
\end{enumerate}
{\bf \small END WHILE}
\smallskip
$\mathcal{T}_N=\mathcal{T}_k$.
\noindent {\bf \small END ANFEM}
\end{algorithm}
\section{Quasi--orthogonality}
The quasi--orthogonality property is the main
ingredient for the convergence analysis of the adaptive
nonconforming method under consideration. In this section we establish such a property by
exploring the conservative property of the nonconforming linear
element and by confirming that the stress is piecewise constant. To this
end, we define a canonical interpolation operator $\Pi_k$ for the
nonconforming space $V_k$
and a restriction operator $I_{k-1}$ from $V_k$ to the coarser space $V_{k-1}$.
Given $v\in V$, we define the interpolation $\Pi_kv\in V_k$ by
\begin{equation}\label{interpolation}
\int_E \Pi_kvds:=\int_E vds\text{ for any }E\in\mathcal{E}_k\,.
\end{equation}
In this paper, the above property is referred to as the conservative property. This property is crucial for the analysis herein.
A similar conservative property was first explored in \cite{HSX10} to analyze the quasi--orthogonality property of the Morley
element.
The interpolation admits the following estimate:
\begin{equation}\label{eq3.4a}
\|v-\Pi_kv\|_{L^2(K)}\lesssim h_K\|\nabla v\|_{L^2(K)}\text{ for any
}K\in\mathcal{T}_k\text{ and }v\in V\,.
\end{equation}
Given $v_k\in V_k$, we define the restriction interpolation $I_{k-1}v_k\in V_{k-1}$ by
\begin{equation}\label{Restriction}
\int_E I_{k-1}v_k ds:=\sum\limits_{l=1}^{\ell}\int_{E_l} v_k ds\,, E\in
\mathcal{E}_{k-1} \text{ with }E=E_1\cup E_2\cdots\cup E_{\ell}\text{
and }E_i\in\mathcal{E}_k\,.
\end{equation}
The properties of the restriction operator $I_{k-1}$ are summarized in
the following lemma.
\begin{lemma} Let the restriction operator $I_{k-1}$ be defined
in \eqref{Restriction}. Then,
\begin{equation}\label{eq3.4b}
I_{k-1}v_k=v_k \text{ for any } K\in \mathcal{T}_k\cap\mathcal{T}_{k-1}, v_k\in V_k\,,
\end{equation}
\begin{equation}\label{eq3.5b}
\|I_{k-1}v_k-v_k\|_{L^2(K)}\lesssim h_K\|\nabla_kv_k\|_{L^2(K)} \text{ for any
}K\in\mathcal{T}_{k-1}\backslash\mathcal{T}_k, v_k\in V_k\,.
\end{equation}
\end{lemma}
\begin{proof} The property \eqref{eq3.4b} directly follows from the definition of the
restriction interpolation. Only the estimate
\eqref{eq3.5b} needs to be proved.
In fact, both sides of \eqref{eq3.5b}
are semi-norms of the restriction $(V_k)_K$ of $V_k$ on $K$. If the
right--hand side vanishes for some $v\in (V_k)_K$, then $v_k$ is a
piecewise constant vector over $K$ with respect to $\mathcal{T}_k$. Given the average continuity of $v_k$ across the internal edges of $\mathcal{T}_k$,
it follows that $v_k$ is a constant vector on $K$. Therefore, the left--hand side also
vanishes for the same $v_k$. The desired result then follows a
scaling argument.
\end{proof}
\begin{remark} An alternative proof for the inequality \eqref{eq3.5b} follows the discrete Poincare inequality
established in \cite{Brenner2003} for the scalar function, which is further investigated in \cite{Rabaus2010}.
Notice that the positive constant of \eqref{eq3.5b} is independent of the ratio
\begin{equation}
\gamma:=\max\limits_{K\in\mathcal{T}_{k-1}\backslash\mathcal{T}_{k}}\max\limits_{\mathcal{T}_{k}\ni T\subset K}\frac{h_K}{h_T},
\end{equation}
see \cite[Lemma 4.1]{Rabaus2010}
for more details.
\end{remark}
\begin{lemma} Let $(u_{k-1}, p_{k-1})$ be the solution of the discrete
problem \eqref{eq7.3} on the mesh $\mathcal{T}_{k-1}$. It, therefore, holds that
\begin{equation}\label{estimateresidual}
|\res_{k-1}(v_k)|\lesssim
\big(\sum\limits_{K\in\mathcal{T}_{k-1}\backslash\mathcal{T}_k}h_K^2\|g\|_{L^2(K)}^2\big)^{1/2}\|\nabla_kv_k\|_{L^2(\Omega)}\text{
for any }v_k\in V_k.
\end{equation}
\end{lemma}
\begin{proof} For the reader's convenience, we recall the definition of the
residual as follows:
\begin{equation}\label{term-1}
\res_{k-1}(v_k)=(g,v_k)_{L^2(\Omega)}-(\sigma_{k-1},\nabla_kv_k)_{L^2(\Omega)}.
\end{equation}
To analyze the right-hand side of the above equation, we set
$v_{k-1}=I_{k-1}v_k$. As $\sigma_{k-1}$ is a piecewise constant tensor
with respect to the mesh $\mathcal{T}_{k-1}$, the definition of the
interpolation operator $I_{k-1}$ in \eqref{Restriction} leads to
\begin{equation}\label{term1}
\int_E (v_k-v_{k-1})\cdot \sigma_{k-1}\nu_Eds=0\text{ for any }E\in \mathcal{E}_{k-1}\,.
\end{equation}
For any $E\in \mathcal{E}_k$ that lies in the interior of some $K\in\mathcal{T}_{k-1}$,
the integral average of $v_k$ over
$E$ is continuous and $\sigma_{k-1}$ is a constant on $K$. Then,
\begin{equation}\label{term2}
\int_E [v_k-v_{k-1}]\cdot \sigma_{k-1}\nu_Eds=0.
\end{equation}
By integrating parts on the fine mesh $\mathcal{T}_k$ and using \eqref{term1} and \eqref{term2}, we get
\begin{equation}
(\nabla_k(v_k-v_{k-1}),\sigma_{k-1})_{L^2(\Omega)}=0.
\end{equation}
Inserting this identity into \eqref{term-1} and adopting the discrete
problem \eqref{eq7.3}, we employ properties
\eqref{eq3.4b} and \eqref{eq3.5b} of the interpolation operator
$I_{k-1}$ to derive
\begin{equation}
\begin{split}
|\res_{k-1}(v_k)|&=|(g,v_k-v_{k-1})_{L^2(\Omega)}|
\leq\sum\limits_{K\in\mathcal{T}_{k-1}\backslash\mathcal{T}_k}\|g\|_{L^2(K)}\|v_k-v_{k-1}\|_{L^2(K)}\\[0.5ex]
&\lesssim
\sum\limits_{K\in\mathcal{T}_{k-1}\backslash\mathcal{T}_k}h_K\|g\|_{L^2(K)}\|\nabla_kv_k\|_{L^2(K)}\,,
\end{split}
\end{equation}
which completes the proof.
\end{proof}
\begin{lemma}(Quasi-orthogonality of the velocity) \label{Lemma7.5}
Let $(u_k,p_k)$ and $(u_{k-1}, p_{k-1})$ be the discrete
solutions of \eqref{eq7.3} on $\mathcal{T}_k$ and $\mathcal{T}_{k-1}$, respectively.
Then,
\begin{equation*}
\begin{split}
|a_k(u-u_k,u_k-u_{k-1})|\lesssim \|\nabla_k(u-u_k)\|_{L^2(\Omega)}
\bigg(\sum\limits_{K\in\mathcal{T}_{k-1}\backslash\mathcal{T}_k}h_K^2\|g\|_{L^2(K)}^2\bigg)^{1/2}\,.
\end{split}
\end{equation*}
\end{lemma}
\begin{proof} The Stokes problem \eqref{eq7.1} and the discrete problem \eqref{eq7.3} give
\begin{equation}\label{eq7.26}
\begin{split}
a_k(u-u_k,u_k-u_{k-1}) =(\nabla_k(u-u_k), \sigma_k-\sigma_{k-1})_{L^2(\Omega)}.
\end{split}
\end{equation}
Given that $(\operatorname{div}_k(u-u_k), p_k-p_{k-1})_{L^2(\Omega)}=0$, let $v_k=\Pi_k(u-u_k)$.
And, $\sigma_k-\sigma_{k-1}$ is a piecewise
constant tensor with respect to the fine mesh $\mathcal{T}_k$; therefore, by the definition
of the interpolation operator $\Pi_k$ in \eqref{interpolation}, we
integrate by parts on $\mathcal{T}_k$ to obtain
\begin{equation}
(\nabla_k((u-u_k)-v_k), \sigma_k-\sigma_{k-1})_{L^2(\Omega)}=0.
\end{equation}
From the discrete problem \eqref{eq7.3}, we have
\begin{equation}\label{term0}
a_k(u-u_k,u_k-u_{k-1})=(g,v_k)_{L^2(\Omega)}-(\nabla_kv_k,\sigma_{k-1})_{L^2(\Omega)}=\res_{k-1}(v_k).
\end{equation}
The term on the right-hand side of the equation \eqref{term0} can be
estimated by the inequality \eqref{estimateresidual} as follows:
\begin{equation*}
\begin{split}
|\res_{k-1}(v_k)| &\lesssim
\sum\limits_{K\in\mathcal{T}_{k-1}\backslash\mathcal{T}_k}h_K\|g\|_{L^2(K)}\|\nabla_kv_k\|_{L^2(K)}\\[0.5ex]
&\lesssim
\sum\limits_{K\in\mathcal{T}_{k-1}\backslash\mathcal{T}_k}h_K\|g\|_{L^2(K)}\|\nabla_k(u-u_k)\|_{L^2(K)}\,,
\end{split}
\end{equation*}
which completes the proof.
\end{proof}
\begin{lemma}(Quasi--orthogonality of the pressure)\label{Lemma7.6}
Let $(u_k,p_k)$ and $(u_{k-1}, p_{k-1})$ be the discrete
solutions of \eqref{eq7.3} on $\mathcal{T}_k$ and $\mathcal{T}_{k-1}$, respectively.
Then,
\begin{equation}\label{eq7.36}
\begin{split}
&|(p-p_k, p_k-p_{k-1})_{L^2(\Omega)}|\\[0.3ex]
& \lesssim
\bigg(\big(\sum\limits_{K\in\mathcal{T}_{k-1}\backslash\mathcal{T}_k}
h_K^2\|g\|_{L^2(K)}^2\big)^{1/2}
+\|\nabla_k(u_k-u_{k-1})\|_{L^2(\Omega)}\bigg)\|p-p_k\|_{L^2(\Omega)}\,.
\end{split}
\end{equation}
\end{lemma}
\begin{remark} The quasi--orthogonality of the pressure herein is different
from those for the nonstandard method of the Poisson equation
\cite{CarstensenHoppe05a,CarstensenHoppe05b,ChenHolstXu07} by the fact that both
$\|\nabla_k(u_k-u_{k-1})\|_{L^2(\Omega)}$ and $\|p-p_k\|_{L^2(\Omega)}$ appear on the right--hand side of \eqref{eq7.36}.
\end{remark}
\begin{proof} Let $\Pi_{0, k}$ be the $L^2$ projection operator from $L^2_0(\Omega)$ onto $Q_k$.
It follows from the discrete inf-sup condition that there exists $v_k\in V_k$ with
\begin{equation}\label{eq7.37}
\operatorname{div}_k v_k=\Pi_{0, k}p-p_k, \text{ and
}\|\nabla_kv_k\|_{L^2(\Omega)}\lesssim \|\Pi_{0, k}p-p_k\|_{L^2(\Omega)}.
\end{equation}
Since $p_k-p_{k-1}\in Q_k$, it follows from the continuous problem
\eqref{eq7.1},
the discrete problem \eqref{eq7.3}, and the definition of the residual \eqref{Res4} that
\begin{equation*}
\begin{split}
(p-p_k, p_k-p_{k-1})_{L^2(\Omega)}
=(\operatorname{div}_k v_k, p_k-p_{k-1})_{L^2(\Omega)}
=\res_{k-1}(v_k)+a_k(u_{k-1}-u_k,v_k).
\end{split}
\end{equation*}
We use the estimates in \eqref{estimateresidual} and \eqref{eq7.37}
to get
\begin{equation*}
\begin{split}
&|(p-p_k, p_k-p_{k-1})_{L^2(\Omega)}|\\[0.5ex]
& \lesssim \bigg(\big(\sum\limits_{K\in\mathcal{T}_{k-1}\backslash\mathcal{T}_k}
h_K^2\|g\|_{L^2(K)}^2\big)^{1/2}+\|\nabla_k(u_k-u_{k-1})\|_{L^2(\Omega)}\bigg)\|p-p_k\|_{L^2(\Omega)},
\end{split}
\end{equation*}
which completes the proof.
\end{proof}
\section{The convergence of the ANFEM}
To prove the convergence of the adaptive algorithm, we first
prove the reduction of the error between the two nested meshes, $\mathcal{T}_k$
and $\mathcal{T}_{k-1}$, where $\mathcal{T}_k$ is the refinement of the coarser mesh
$\mathcal{T}_{k-1}$ with \eqref{bulk} by the newest vertex bisection. In order to control the volume part
$\sum\limits_{K\in\mathcal{T}_{k-1}\backslash\mathcal{T}_k}h_K^2\|g\|_{L^2(K)}^2$ appearing in Lemmas \ref{Lemma7.5} and \ref{Lemma7.6}, we introduce the following modified estimator:
\begin{equation}\label{eq7.53b}
\begin{split}
&\tilde{\eta}^2(u_{k-1}, p_{k-1},
\mathcal{T}_{k-1}):=\sum\limits_{K\in\mathcal{T}_{k-1}}\big(\beta_1
h_K^2\|g\|_{L^2(K)}^2+\eta_K^2(u_{k-1}, p_{k-1})\big)\,
\end{split}
\end{equation}
with the positive constant $\beta_1>0$ to be determined later. Note
that this modified estimator is introduced only for the convergence analysis and that
the final convergence and optimal complexity will be proved for Algorithm \ref{Algorithm}.
Note that the volume residual
$\sum\limits_{K\in\mathcal{T}_{k-1}} h_K^2\|g\|_{L^2(K)}^2$ does not contain the
unknowns. Hence, we add it to settle down the
lacking of the Galerkin--orthogonality or quasi--orthogonality. We stress that
the Galerkin--orthogonality or quasi--orthogonality is an essential ingredient for the convergence analysis
of the adaptive conforming, nonconforming, and mixed methods for the
Poisson-like problems \cite{CarstensenHoppe05a, CarstensenHoppe05b,CasconKreuzerNochettoSiebert07,ChenHolstXu07,Dolfler96,MekchayNochetto05,
MorinNochettoSiebert00}. This is
another reason that we need a modified estimator as in
\eqref{eq7.53b}.
We list three standard components for the convergence
analysis of the adaptive method, which can be proved by following
the arguments, for instance, in
\cite{CarstensenHoppe05b,CasconKreuzerNochettoSiebert07,Dolfler96}.
\begin{lemma}\label{Lemma5.3b} Let $\mathcal{T}_k$ be some refinement of $\mathcal{T}_{k-1}$ from Algorithm \ref{Algorithm},
then $\rho>0$ and a
positive constant $\beta\in(1-\rho\theta,1)$ exist,
such that
\begin{equation}\label{eq5.7}
{\eta}^2(u_{k-1},p_{k-1},\mathcal{T}_k)
\leq \beta {\eta}^2(u_{k-1},p_{k-1},\mathcal{T}_{k-1})+(1-\rho\theta-\beta){\eta}^2(u_{k-1}, p_{k-1}, \mathcal{T}_{k-1})\,.
\end{equation}\end{lemma}
\begin{proof} The result can be proved by following the idea in \cite{CarstensenHoppe05b,CasconKreuzerNochettoSiebert07,Dolfler96}.
The details are only given for the readers' convenience. In fact, we have
\begin{equation}
{\eta}^2(u_{k-1}, p_{k-1}, \mathcal{T}_k)={\eta}^2(u_{k-1}, p_{k-1}, \mathcal{T}_{k-1}\cap\mathcal{T}_k)+{\eta}^2(u_{k-1}, p_{k-1}, \mathcal{T}_k\backslash\mathcal{T}_{k-1}).
\end{equation}
For any $K\in \mathcal{T}_{k-1}\backslash\mathcal{T}_k$, we only need to consider the
case where $K$ is subdivided into $K_{1}\,,K_2\in\mathcal{T}_k$ with
$|K_1|=|K_2|=\frac{1}{2}|K|$. As $[\nabla_{k-1}u_{k-1} \tau_E]=0$ over the interior edge $E=K_1\cap K_2\in\mathcal{E}_k$, we have
\begin{equation}\label{eq4.5}
\begin{split}
&\sum\limits_{i=1}^2{\eta}_{K_i}^2(u_{k-1}, p_{k-1})\\
& :=\sum\limits_{i=1}^2\bigg(h_{K_i}\|g\|_{L^2(K_i)}+\bigg(\sum\limits_{\mathcal{E}_k\ni
E\subset\partial
K_i}h_{K_i}\|[\nabla_{k-1}u_{k-1} \tau_E]\|_{L^2(E)}^2\bigg)^{1/2}\bigg)^2\\[0.5ex]
& \leq\frac{1}{2^{1/2}} {\eta}^2_K(u_{k-1}, p_{k-1})\\
&:=\frac{1}{2^{1/2}}
\bigg( h_{K}\|g\|_{L^2(K)}+\bigg(\sum\limits_{\mathcal{E}_{k-1}\ni
E\subset\partial
K}h_{K}\|[\nabla_{k-1}u_{k-1}\tau_E]\|_{L^2(E)}^2\bigg)^{1/2}\bigg)^2\,.
\end{split}
\end{equation}
Consequently,
\begin{equation}
\begin{split}
\sum\limits_{K\in\mathcal{T}_{k-1}\backslash\mathcal{T}_k}
\sum\limits_{i=1}^2{\eta}_{K_i}^2(u_{k-1}, p_{k-1}) \leq
\frac{1}{2^{1/2}}{\eta}^2(u_{k-1}, p_{k-1}, \mathcal{T}_{k-1}\backslash\mathcal{T}_k)\,.
\end{split}
\end{equation}
Let $\rho=1-\frac{1}{2^{1/2}}$, therefore, we obtain
\begin{equation}
\begin{split}
{\eta}^2(u_{k-1}, p_{k-1}, \mathcal{T}_k) &\leq {\eta}^2(u_{k-1}, p_{k-1}, \mathcal{T}_{k-1})-\rho{\eta}^2(u_{k-1}, p_{k-1},
\mathcal{T}_{k-1}\backslash\mathcal{T}_k)\,.
\end{split}
\end{equation}
Choosing the positive parameter
$\beta$ with $1-\rho\theta<\beta<1$, we combine the above inequality and the bulk criterion \eqref{bulk} to
achieve the desired result.
\end{proof}
\begin{lemma}\label{rightcon}
Let $\mathcal{T}_k$ be some refinement of $\mathcal{T}_{k-1}$ produced in Algorithm \ref{Algorithm},
then there exists $\rho>0$ such that
\begin{equation}\label{righthandside}
\sum\limits_{K\in\mathcal{T}_k}h_K^{2}\|g\|_{L^2(K)}^2\leq
\sum\limits_{K\in\mathcal{T}_{k-1}}h_K^{2}\|g\|_{L^2(K)}^2-\rho\sum\limits_{K\in\mathcal{T}_{k-1}\backslash\mathcal{T}_k}h_K^{2}\|g\|_{L^2(K)}^2\,.
\end{equation}
\end{lemma}
\begin{proof} This can be proved by a similar argument proposed in the previous lemma.
\end{proof}
\begin{lemma}\label{continuity}(Continuity of the estimator) Let $u_k$ and $u_{k-1}$ be the solutions to the discrete problem \eqref{eq7.3}
on the meshes $\mathcal{T}_k$ and $\mathcal{T}_{k-1}$ obtained from Algorithm \ref{Algorithm}. Given any positive
constant $\epsilon$, there exists a positive constant
$\beta_2(\epsilon)$ dependent on $\epsilon$ such that
\begin{equation}\label{eq5.4}
{\eta}^2(u_k, p_k, \mathcal{T}_k)\leq (1+\epsilon) {\eta}^2(u_{k-1}, p_{k-1},
\mathcal{T}_k)
+\frac{1}{\beta_2(\epsilon)}\|\nabla_k(u_k-u_{k-1})\|_{L^2(\Omega)}^2\,.
\end{equation}
\end{lemma}
\begin{proof}
Given any $K\in\mathcal{T}_k$, it follows from the definitions of
${\eta}_K(u_k, p_k)$ and ${\eta}_K(u_{k-1}, p_{k-1})$ in \eqref{eq4.5} that
\begin{equation*}
\begin{split}
&\big |{\eta}_K(u_k, p_k)-{\eta}_K(u_{k-1}, p_{k-1})\big |\\
&=\bigg
|\bigg(\sum\limits_{\mathcal{E}_k\ni E\subset\partial K}h_K\|[\nabla_k
u_k\tau_E]\|_{L^2(E)}^2\bigg)^{1/2}
-\bigg(\sum\limits_{\mathcal{E}_k\ni
E \subset\partial K}h_K\|[\nabla_{k-1}
u_{k-1}\tau_E]\|_{L^2(E)}^2\bigg)^{1/2}\bigg |\\
&\leq \bigg(\sum\limits_{\mathcal{E}_k\ni
E \subset\partial K}h_K\|[\nabla_k
(u_k-u_{k-1})\tau_E]\|_{L^2(E)}^2 \bigg)^{1/2}.
\end{split}
\end{equation*}
Given $E\in\mathcal{E}_k$, let $K_1, K_2\in \mathcal{T}_k$ be the two elements that take $E$ as one edge. Then, we use the trace theorem and the fact
that $\nabla_k(u_k-u_{k-1})$ is a piecewise constant tensor to get
\begin{equation}
\begin{split}
&\|[\nabla_k
(u_k-u_{k-1})\tau_E]\|_{L^2(E)}\\
&\leq \|\nabla_k
(u_k-u_{k-1})\tau_E|_{K_1}\|_{L^2(E)}+\|\nabla_k
(u_k-u_{k-1})\tau_E|_{K_2}\|_{L^2(E)}\\[0.5ex]
&\lesssim h_K^{-1/2}\|\nabla_k
(u_k-u_{k-1})\|_{L^2(\omega_E)}\,,
\end{split}
\end{equation}
which gives
\begin{equation}
{\eta}_K(u_k, p_k)\leq {\eta}_K(u_{k-1}, p_{k-1})+ C_{Con} \|\nabla_k
(u_k-u_{k-1})\|_{L^2(\omega_K)},
\end{equation}
for some positive constant $C_{Con}$. Given any positive constant $\epsilon$, we apply the Young inequality to get
\begin{equation}
{\eta}^2_K(u_k, p_k)\leq (1+\epsilon) {\eta}_K^2(u_{k-1}, p_{k-1})+ \frac{C_{Con}^2(1+\epsilon)}{\epsilon} \|\nabla_k
(u_k-u_{k-1})\|_{L^2(\omega_K)}^2.
\end{equation}
A summation over all elements in $\mathcal{T}_k$ completes the proof with $\beta_2(\epsilon)=\frac{M\epsilon}{C_{Con}^2(1+\epsilon)}$,
where the positive constant $M$ depends on the finite overlapping of the patches $\omega_K$.
\end{proof}
In the following theorem, we prove the convergence of the adaptive
nonconforming finite element method for the Stokes problem. The main ingredients are the quasi--orthogonality
of both the velocity and the pressure in Lemmas \ref{Lemma7.5} and \ref{Lemma7.6}, and the relations of the estimators
between two the meshes $\mathcal{T}_k$ and $\mathcal{T}_{k-1}$ presented in Lemmas \ref{Lemma5.3b}--\ref{continuity}.
\begin{theorem}\label{Theorem7.8} Let $(u,p)$ and $(u_k,p_k)$
be the solutions of \eqref{eq7.1} and \eqref{eq7.3}.
Then $\gamma_1, \gamma_2, \beta_1>0$ and $0<\alpha<1$ exist, such that
\begin{equation}\label{eq7.44}
\begin{split}
&\|\nabla_k(u-u_k)\|_{L^2(\Omega)}^2+\gamma_1\|p-p_k\|_{L^2(\Omega)}^2+\gamma_2\tilde{\eta}^2(u_k,
p_k,
\mathcal{T}_k)\\[0.5ex]
& \leq \alpha\big(
\|\nabla_{k-1}(u-u_{k-1})\|_{L^2(\Omega)}^2+\gamma_1\|p-p_{k-1}\|_{L^2(\Omega)}^2+\gamma_2\tilde{\eta}^2(u_{k-1},
p_{k-1}, \mathcal{T}_{k-1})\big).
\end{split}
\end{equation}
\end{theorem}
\begin{proof}
First, we adopt the quasi--orthogonality of both the velocity and the pressure. Denote the multiplication constant in Lemma
\ref{Lemma7.5} by $C_{QOV}$. As
\begin{equation}
\begin{split}
\|\nabla_k(u-u_k)\|_{L^2(\Omega)}^2&=\|\nabla_k(u-u_{k-1}\|_{L^2(\Omega)}^2-\|\nabla_k(u_k-u_{k-1})\|_{L^2(\Omega)}^2\\
&\quad -2(\nabla_k(u-u_k), \nabla_k(u_k-u_{k-1}))_{L^2(\Omega)},
\end{split}
\end{equation}
it follows from the quasi--orthogonality of the velocity in Lemma
\ref{Lemma7.5} and the Young inequality that
\begin{equation}\label{first}
\begin{split}
&(1-\delta_1)\|\nabla_k(u-u_k)\|_{L^2(\Omega)}^2\\
&\leq
\|\nabla_{k-1}(u-u_{k-1})\|_{L^2(\Omega)}^2
-\|\nabla_k(u_k-u_{k-1})\|_{L^2(\Omega)}^2\\
&\quad +C_{1}(\delta_1)\sum\limits_{K\in\mathcal{T}_{k-1}\backslash\mathcal{T}_k}h_K^2\|g\|_{L^2(K)}^2,
\end{split}
\end{equation}
where $C_1(\delta_1)=\frac{C_{QOV}^2}{\delta_1}$ for any positive constant $0<\delta_1<1$. Denote the multiplication constant in Lemma \ref{Lemma7.6}
by $C_{QOP}$. From the quasi--orthogonality of the pressure proved in Lemma
\ref{Lemma7.6} and the Young inequality, we have
\begin{equation}\label{second}
\begin{split}
(1-\delta_2-\delta_3)\|p-p_k\|_{L^2(\Omega)}^2& \leq
\|p-p_{k-1}\|_{L^2(\Omega)}^2-\|p_k-p_{k-1}\|_{L^2(\Omega)}^2 \\[1.0ex]
&\quad
+\frac{1}{\beta_3(\delta_3)}\|\nabla_k(u_k-u_{k-1})\|_{L^2(\Omega)}^2\\[1.ex]
&\quad +C_{2}(\delta_2)
\sum\limits_{K\in\mathcal{T}_{k-1}\backslash\mathcal{T}_k}h_K^2\|g\|_{L^2(K)}^2;
\end{split}
\end{equation}
here $\beta_3(\delta_3)=\frac{\delta_3}{C_{QOP}^2}$ and
$C_2(\delta_2)=\frac{C_{QOP}^2}{\delta_2}$ for any constants $0<\delta_2, \delta_3<1$.
Then we multiply the inequality \eqref{first} by $\gamma_1>0$ and the inequality \eqref{second} by
$\gamma_2>0$ to obtain
\begin{equation}\label{third}
\begin{split}
&\gamma_1(1-\delta_1)\|\nabla_{k-1}(u-u_k)\|_{L^2(\Omega)}^2+\gamma_2(1-\delta_2-\delta_2)\|p-p_k\|_{L^2(\Omega)}^2\\[0.5ex]
& \leq
\gamma_1\|\nabla_{k-1}(u-u_{k-1})\|_{L^2(\Omega)}^2+\gamma_2\|p-p_{k-1}\|_{L^2(\Omega)}^2
-(\gamma_1-\frac{\gamma_2}{\beta_3(\delta_3)})\|\nabla_k(
u_k-u_{k-1})\|_{L^2(\Omega)}^2\\[0.5ex]
&\quad-\gamma_2\|p_k-p_{k-1}\|_{L^2(\Omega)}^2
+\big(\gamma_1C_{1}(\delta_1)+\gamma_2C_{2}(\delta_2)\big)
\sum\limits_{K\in\mathcal{T}_{k-1}\backslash\mathcal{T}_k}h_K^2\|g\|_{L^2(K)}^2.
\end{split}
\end{equation}
For the presentation, we introduce some short--hand notations
for any positive constants $\gamma_3, \gamma_4>0$:
\begin{equation*}
\begin{split}
\mathfrak{G}_k(u_k,p_k):&=\gamma_1(1-\delta_1)\|\nabla_{k-1}(u-u_k)\|_{L^2(\Omega)}^2
+\gamma_2(1-\delta_2-\delta_3)\|p-p_k\|_{L^2(\Omega)}^2\\[0.5ex]
&\quad+\gamma_3\eta^2(u_k, p_k, \mathcal{T}_k)
+\gamma_4\sum\limits_{K\in\mathcal{T}_k}h_K^2\|g\|_{L^2(K)}^2\,,
\end{split}
\end{equation*}
\begin{equation}
\begin{split}
\overline{\mathfrak{G}}_{k-1}(u_{k-1}, p_{k-1}):&=\gamma_1\|\nabla_{k-1}(u-u_{k-1})\|_{L^2(\Omega)}^2+\gamma_2\|p-p_{k-1}\|_{L^2(\Omega)}^2\\[0.5ex]
&\quad+\gamma_3\beta\eta^2(u_{k-1}, p_{k-1}, \mathcal{T}_{k-1})
+\gamma_4\sum\limits_{K\in\mathcal{T}_{k-1}}h_K^2\|g\|_{L^2(K)}^2\,.
\end{split}
\end{equation}
Second, we use the continuity of the estimators from Lemmas \ref{Lemma5.3b}--\ref{continuity} to cancel
both the term $\|\nabla_k(u_k-u_{k-1})\|_{L^2(\Omega)}$ and the volume estimator. In fact,
from \eqref{eq5.7} and \eqref{eq5.4}, we have
\begin{equation}
\begin{split}
\eta^2(u_k, p_k, \mathcal{T}_k)\leq \beta\eta^2(u_{k-1}, p_{k-1}, \mathcal{T}_{k-1})+\frac{1}{\beta_2(\epsilon)}\|\nabla_k(u_k-u_{k-1})\|_{L^2(\Omega)}^2\\
+\big((1-\rho\theta-\beta)(1+\epsilon)+\epsilon\beta\big){\eta}^2(u_{k-1},p_{k-1},\mathcal{T}_{k-1}).
\end{split}
\end{equation}
Then we combine the above inequality with the inequalities \eqref{third} and \eqref{righthandside}
to obtain
\begin{equation*}
\begin{split}
\mathfrak{G}_k(u_k,p_k) &\leq \overline{\mathfrak{G}}_{k-1}(u_{k-1},p_{k-1})
-\big(\gamma_1-\frac{\gamma_2}{\beta_3(\delta_3)}-\frac{\gamma_3}{\beta_2(\epsilon)}\big)\|\nabla_k(
u_k-u_{k-1})\|_{L^2(\Omega)}^2\\[0.5ex]
&\quad-\gamma_2\|p_k-p_{k-1}\|_{L^2(\Omega)}^2+\gamma_3\big((1-\rho\theta-\beta)(1+\epsilon)+\epsilon\beta\big){\eta}^2(u_{k-1},
p_{k-1},
\mathcal{T}_{k-1})\\[0.5ex]
&\quad+\big(\gamma_1C_{1}(\delta_1)+\gamma_2C_{2}(\delta_2)-\gamma_4\rho\big)
\sum\limits_{K\in\mathcal{T}_{k-1}\backslash\mathcal{T}_k}h_K^2\|g\|_{L^2(K)}^2.
\end{split}
\end{equation*}
It remains to prove that the positive constants $\delta_i, i=1, 2, 3$, $\gamma_i, i=1, 2, 3, 4$, $\epsilon$, $\beta$, and $\beta_1$ exist such that the contraction \eqref{eq7.44} holds for some constant $0<\alpha<1$. Further it is possible that the constant dependent on
the choices of the aforementioned parameters but independent of the meshsize $h$ and the level $k$. This will be achieved in the following three steps.
{\bf Step 1~} For the second, fourth, and fifth terms on the right-hand side of the above inequality to vanish, we set
\begin{equation}
\begin{split}
\gamma_2&=(\gamma_1-\frac{\gamma_3}{\beta_2(\epsilon)})\beta_3(\delta_3) \text{ with } \gamma_1>\frac{\gamma_3}{\beta_2(\epsilon)},\\[0.5ex]
\gamma_4&=(\gamma_1C_{1}(\delta_1)+\gamma_2C_{2}(\delta_2))/\rho,\\[0.5ex]
\beta&=(1-\rho\theta)(1+\epsilon).
\end{split}
\end{equation}
Note that $\gamma_2$, $\gamma_4$, and $\beta$ will be determined after $\delta_i, i=1, 2, 3$, $\gamma_1$, $\gamma_3$, and $\epsilon$ have been specified.
In the following, we assume that $\epsilon$ is fixed in such a way that $0<\beta<1$.
Also, we let $\gamma_1$ and $\gamma_3$ be fixed such that $\gamma_1>\frac{\gamma_3}{\beta_2(\epsilon)}$ and $\gamma_2>0$.
Hence, we have
\begin{equation*}
\mathfrak{G}_k(u_k,p_k) \leq \overline{\mathfrak{G}}_{k-1}(u_{k-1}, p_{k-1})\,.
\end{equation*} Let
the positive constant $\alpha$ with $\beta<\alpha<1$ be determined
later. We define
\begin{equation*}
\begin{split}
&\mathfrak{R}_{k-1}(u_{k-1},
p_{k-1})\\
&:=(1-\alpha(1-\delta_1))\gamma_1\|\nabla_{k-1}(u-u_{k-1})\|_{L^2(\Omega)}^2
+\gamma_2(1-\alpha(1-\delta_2-\delta_3))\|p-p_{k-1}\|_{L^2(\Omega)}^2\\[0.5ex]
&\quad+\gamma_3(\beta-\alpha){\eta}^2(u_{k-1}, p_{k-1}, \mathcal{T}_{k-1})
+\gamma_4(1-\alpha)\sum\limits_{K\in\mathcal{T}_{k-1}}h_K^{2}\|g\|_{L^2(K)}^2.
\end{split}
\end{equation*}
Then we perform the decomposition $\overline{\mathfrak{G}}_{k-1}(u_{k-1}, p_{k-1})=\alpha\mathfrak{G}_{k-1}(u_{k-1},p_{k-1})+\mathfrak{R}_{k-1}(u_{k-1}, p_{k-1})$ to
get
\begin{equation*}
\begin{split}
\mathfrak{G}_k(u_k,p_k)
&\leq\alpha\mathfrak{G}_{k-1}(u_{k-1},p_{k-1})+\mathfrak{R}_{k-1}(u_{k-1}, p_{k-1}).
\end{split}
\end{equation*}
{\bf Step 2~~}
Now we only need to show that it is possible to choose $\alpha<1$ such that $\mathfrak{R}_{k-1}(u_{k-1}, p_{k-1})\leq 0$.
This can be achieved by selecting parameters $\delta_i\,,i=1,2,3$. To this end, we recall the reliability of $\eta(u_{k-1}, p_{k-1}, \mathcal{T}_{k-1})$
in Lemma \ref{Theorem7.2} with the multiplication coefficient $C_{Rel}$:
\begin{equation}\label{eq5.8}
\|\nabla_{k-1}(u-u_{k-1})\|_{L^2(\Omega)}^2+\|p-p_{k-1}\|_{L^2(\Omega)}^2\leq
C_{Rel}{\eta}^2(u_{k-1}, p_{k-1}, \mathcal{T}_{k-1})\,.
\end{equation}
Further, we take $\delta_1=\delta_2+\delta_3$ with
$0<\delta_1<\min(
\frac{\gamma_3(1-\beta)}{C_{Rel}(\gamma_1+\gamma_2)},1)$. Then, we
take
$$\alpha:=\frac{(\gamma_1+\gamma_2)C_{Rel}+\gamma_3\beta+\gamma_4}
{(1-\delta_1)(\gamma_1+\gamma_2)C_{Rel}+\gamma_3+\gamma_4}.$$
It is straightforward to see that $\beta<\alpha< 1$. As
\begin{equation}
\sum\limits_{K\in\mathcal{T}_{k-1}}h_K^{2}\|g\|_{L^2(K)}^2 \leq {\eta}^2(u_{k-1},
p_{k-1}, \mathcal{T}_{k-1}),
\end{equation}
we obtain
\begin{equation*}
\begin{split}
&\mathfrak{R}_{k-1}(u_{k-1}, p_{k-1})\\
&\leq
\big((1-\alpha(1-\delta_1))(\gamma_1+\gamma_2)C_{Rel}+\gamma_3(\beta-\alpha)
+\gamma_4(1-\alpha)\big){\eta}^2(u_{k-1}, p_{k-1}, \mathcal{T}_{k-1})=0\,.
\end{split}
\end{equation*}
This proves that
\begin{equation*}
\begin{split}
\mathfrak{G}_k(u_k,p_k)
&\leq\alpha\mathfrak{G}_{k-1}(u_{k-1},p_{k-1}).
\end{split}
\end{equation*}
{\bf Step 3~~} Finally, we take $\beta_1:=\gamma_4/\gamma_3$ and
rearrange
$\gamma_2:=\gamma_2(1-\delta_2-\delta_3)/(1-\delta_1)\gamma_1$,
$\gamma_3:=\gamma_3/(1-\delta_1)\gamma_1$, which completes the proof.
\end{proof}
\section{The discrete reliability}
In this section, we prove the discrete reliability. The analysis needs
some prolongation operator from $V_k$ to $V_{k+\ell}$ with some integer $\ell\geq 1$. Some further notations are needed. Given
$E\in\mathcal{E}_{k+\ell}$, the edge patch $\omega_{E, k}$ of $E$ with respect to the
mesh $\mathcal{T}_k$ is defined as
\begin{equation}
\omega_{E, k}:=\{K\in\mathcal{T}_{k}, E\subset\partial K \text{ or } E \text{ lies in the interior of }K\}.
\end{equation}
Let $\xi_{E}=\card( \omega_{E, k})$. We define the prolongation
interpolation $I_{k+\ell}^{\prime}v_{k}\in V_{k+\ell}$ for any $v_k\in V_k$, as
\begin{equation}\label{eq5.13}
\int_E I_{k+\ell}^{\prime}v_k ds:=
\frac{1}{\xi_{E}}\sum\limits_{K\in \omega_{E, k}} \int_E (v_k|_{K})
ds\,\text{ for any } E\in\mathcal{E}_{k+\ell}\,.
\end{equation}
For the interpolation operator $I_{k+\ell}^{\prime}$, we have
\begin{equation}\label{eq5.3new}
I_{k+\ell}^{\prime}v_k=v_k\text{ for any }K\in\mathcal{T}_k\cap\mathcal{T}_{k+\ell}\text{ and
}v_k\in V_{k+\ell}.
\end{equation}
As we will see in Remark \ref{remark5.2} below, we cannot directly use the prolongation operator $I^{\prime}_{k+\ell}$ in the analysis of the discrete reliability.
An averaging operator is needed. Denote $\mathcal{N}_k$ as the set of internal vertexes of the mesh $\mathcal{T}_k$, and denote $S_k\subset H_0^1(\Omega)$ as the conforming linear element space over $\mathcal{T}_k$.
Given
$Z\in\mathcal{N}_{k}$, the nodal patch $\omega_{Z, k}$ is defined by
\begin{equation}
\omega_{Z, k}:=\{K\in\mathcal{T}_k, Z\in K\}.
\end{equation}
Denote $\phi_Z\in S_k$ as the canonical basis function associated to $Z$, which satisfies
$\phi(Z)=1$ and $\phi(Z^{\prime})=0$ for vertex $Z^{\prime}$ of $\mathcal{T}_k$ other than $Z$. We define
\begin{equation}
\mathcal{E}_Z:=\{E\in\mathcal{E}_k, Z\in\mathcal{N}_k \text{ is one end point of }E\}.
\end{equation}
The idea of \cite{Brenner2003} leads to the definition of the following averaging operator $\Pi: V_k\rightarrow (S_k)^2$:
\begin{equation}
\Pi v_k:=\sum\limits_{Z\in\mathcal{N}_k}v_Z\phi_Z \text{ for any }v_k\in V_k,
\end{equation}
where
\begin{equation}\label{equ5.7}
v_Z=\frac{1}{\xi_Z}\sum\limits_{K\in \omega_{Z, k}}(v_k|_K)(Z) \text{ with } \xi_{Z}=\card( \omega_{Z, k}).
\end{equation}
Given any $K\in\mathcal{T}_k$, we have
\begin{equation}\label{eq5.7new}
\begin{split}
&\|\Pi v_k-v_k\|_{L^2(K)}+h_K\|\nabla(\Pi v_k-v_k)\|_{L^2(K)}\\
&\lesssim h_K^{3/2}(\sum\limits_{T\in\mathcal{T}_k\& T\cap K\not=\emptyset}\sum\limits_{E\subset\partial T}\|[\nabla_k v_k\tau_E]\|_{L^2(E)}^2)^{1/2},
\end{split}
\end{equation}
for any $v_k\in V_k$, see \cite{Brenner2003} for the proof. Define
\begin{equation*}
\Omega_{\mathcal{R}}:=\textrm{interior}(\bigcup\{K: K\in\mathcal{T}_k\backslash\mathcal{T}_{k+\ell},\}),
\end{equation*}
and
\begin{equation*}
\Omega_{\mathcal{C}}:=\textrm{interior}(\bigcup\{K: K\in\mathcal{T}_k\cap\mathcal{T}_{k+\ell},\partial K\cap \partial \Omega_{\mathcal{R}}=\emptyset \}).
\end{equation*}
The main idea herein is to take the mixture of the prolongation operators $I_{k+\ell}^{\prime}$
and $\Pi$. More precisely, we use $\Pi$ in the region $\Omega_{\mathcal{R}}$ where the elements of $\mathcal{T}_k$ are refined and take $I_{k+\ell}^{\prime}$ in the region $\Omega_{\mathcal{C}}$, and we define some mixture in the layers between them. This leads to the prolongation
operator $J_{k+\ell}: V_k\rightarrow V_{k+\ell}$ as follows:
$$
{J}_{k+\ell}v_k:=\left\{\begin{array}{ll} \Pi v_k &\text{ on }\Omega_{\mathcal{R}},\\ v_k &\text{ on } \Omega_{\mathcal{C}},\\
v_{k+\ell, tr}&\text{ on }\Omega\backslash(\Omega_{\mathcal{R}}\cup \Omega_{\mathcal{C}}),\end{array}\right.
$$
where $v_{k+\ell, tr}$ is defined as
$$
\int_Ev_{k+\ell, tr}ds:=\left\{\begin{array}{ll} \int_E\Pi v_kds & \text{ for any } E\in \partial (\mathcal{T}_k\cap\mathcal{T}_{k+\ell})\\
\int_E v_k ds & \text{ otherwise }.
\end{array}\right.
$$
Define
$$
\mathcal{M}_{k, k+\ell}:=\{K\in\mathcal{T}_k, \partial K\cap \overline{\cup(\mathcal{T}_k\backslash\mathcal{T}_{k+\ell})}\not=\emptyset \}.
$$
\begin{remark} It follows immediately from regularity of the mesh $\mathcal{T}_k$ that
$$
\# \mathcal{M}_{k, k+\ell}\leq \kappa \# \mathcal{T}_k\backslash\mathcal{T}_{k+\ell}
$$
for a positive constant $\kappa\geq 1$ which is only dependent on the initial mesh $\mathcal{T}_0$.
\end{remark}
\begin{lemma}\label{Lemma 5.1} For any $v_k\in
V_k$, it holds that
\begin{equation}\label{eq7.72}
\|\nabla_{k+\ell}(J_{k+\ell}v_k-v_k)\|_{L^2(\Omega)}^2 \lesssim
\sum\limits_{K\in \mathcal{M}_{k, k+\ell}}\sum\limits_{E\subset \partial K}h_K\|[\nabla_k v_k\tau_E]\|_{L^2(E)}^2\,.
\end{equation}
\end{lemma}
\begin{proof} As $J_{k+\ell} v_k=\Pi v_k$ on $\Omega_{\mathcal{R}}$ and $J_{k+\ell}v_k=v_k$ on $\Omega_{\mathcal{C}}$,
from \eqref{eq5.3new} and \eqref{eq5.7new}, we
only need to estimate $\|\nabla(J_{k+\ell}v_k-v_k)\|_{L^2(K)}=\|\nabla(v_{k+\ell, tr}-v_k)\|_{L^2(K)}$ for $\mathcal{T}_{k}\ni K\subset \Omega\backslash(\Omega_{\mathcal{R}}\cup \Omega_{\mathcal{C}} )$.
Given $E\in\mathcal{E}_{k}$, let $\varphi_E$ be the canonical basis function of the nonconforming $P_1$ element on $\mathcal{T}_{k}$, which satisfies $\int_E\varphi_E ds=|E|$ and $\int_{E^{\prime}}\varphi_E ds=0$
for any $E^{\prime}\in\mathcal{E}_{k}$ other than $E$. A direct calculation yields
\begin{equation*}
\|\varphi_E\|_{L^2(\Omega)}+h_E\|\nabla_{k}\varphi_E\|_{L^2(\Omega)}\lesssim h_E.
\end{equation*}
Let $v_{E}^{\prime}:=\int_E v_{k+\ell, tr}|_Kds$ and $v_E:=\int_Ev_k|_Kds$; thus we have
\begin{equation}\label{eq5.10new}
\|\nabla(v_{k+\ell, tr}-v_k)\|_{L^2(K)}\lesssim \sum\limits_{E\subset\partial K}|v_{E}^{\prime}-v_E|/h_E.
\end{equation}
We only need to bound the terms $|v_E^{\prime}-v_E|$ for $E\subseteq\partial (\mathcal{T}_k\cap\mathcal{T}_{k+\ell})$.
We assume that $Z_1$ and $Z_2$ are two endpoints of $E$. Then, the trace of $v_k|_K$ on $E$ can be expressed as
\begin{equation}
v_k|_E=(v_k|_K)(Z_1)\phi_{Z_1}+(v_k|_K)(Z_2)\phi_{Z_2}.
\end{equation}
Note that
\begin{equation}
\Pi v_k|_E=v_{Z_1}\phi_{Z_1}+v_{Z_2}\phi_{Z_2}.
\end{equation}
We recall that $v_{Z_i}$ are defined in \eqref{equ5.7} and that $\phi_{Z_i}$ are the canonical basis functions
associated with vertexes $Z_i$ for the conforming linear element.
Therefore
\begin{equation}\label{eq5.12new}
\begin{split}
|v_{E}^{\prime}-v_E|&=|\int_E(\Pi v_k|_E-v_k|_E)ds|\\[0.5ex]
&=|\int_E ((v_{Z_1}-(v_k|_K)(Z_1))\phi_{Z_1}+(v_{Z_2}-(v_k|_K)(Z_2))\phi_{Z_2})ds|\\
&\lesssim h_E(\sum\limits_{i=1}^2\sum\limits_{E^{\prime}\in\mathcal{E}_{Z_i}}h_E^{\prime}\|[\nabla_kv_k\tau_{E^{\prime}}]\|_{L^2(E^{\prime})}^2)^{1/2}.
\end{split}
\end{equation}
By inserting the estimates of $|v_{E}^{\prime}-v_E|$ in \eqref{eq5.12new} into \eqref{eq5.10new}, we complete the proof.
\end{proof}
We define the ratio $\gamma$ as follows:
\begin{equation}\label{ratio}
\gamma:=\max\limits_{K\in\mathcal{T}_k\backslash\mathcal{T}_{k+\ell}}\max\limits_{\mathcal{T}_{k+\ell}\ni T\subset K}\frac{h_K}{h_T}
\end{equation}
\begin{lemma}\label{Lemma7.15} The following discrete reliability holds:
\begin{equation}\label{eq7.60}
\|\nabla_{k+\ell}( u_{k+\ell}-u_k)\|_{L^2(\Omega)}+\|p_{k+\ell}-p_k\|_{L^2(\Omega)} \lesssim
{\eta}(u_k, p_k, \mathcal{M}_{k, k+\ell})\,.
\end{equation}
\end{lemma}
\begin{remark}\label{remark5.2}
If we directly take the prolongation operator $I_{k+\ell}^{\prime}$ to analyze this discrete reliability, the constant
for the {\em established } discrete reliability will depend on the ratio $\gamma$ (see Appendix A for an example).
\end{remark}
\begin{proof} For any $v_{k+\ell}\in V_{k+\ell}$, we have the following
decomposition:
\begin{equation}\label{eq7.61}
\begin{split}
&\mu\|\nabla_{k+\ell}(u_{k+\ell}-u_k)\|_{L^2(\Omega)}^2\\
&=a_{k+\ell}(u_{k+\ell}-u_k, u_{k+\ell}-v_{k+\ell})+a_{k+\ell}(u_{k+\ell}-u_k, v_{k+\ell}-u_k)\,.
\end{split}
\end{equation}
We will first estimate the first term on the right--hand side of the
above equation. It follows the discrete problem \eqref{eq7.3} that
\begin{equation}\label{eq7.62}
a_{k+\ell}(u_{k+\ell}-u_k, u_{k+\ell}-v_{k+\ell})
=\res_k(u_{k+\ell}-v_{k+\ell})-b_{k+\ell}(u_{k+\ell}-v_{k+\ell},p_{k+\ell}-p_k)
\,.
\end{equation}
The first term on the right--hand side of \eqref{eq7.62} can be
bounded as in \eqref{estimateresidual}:
\begin{equation}\label{eq7.68}
\begin{split}
&|\res_k(u_{k+\ell}-v_{k+\ell})| \lesssim
(\sum\limits_{K\in \mathcal{M}_{k, k+\ell}}
h_K^2\|g\|_{L^2(K)}^2)^{1/2}\|\nabla_{k+\ell}(u_{k+\ell}-v_{k+\ell})\|_{L^2(\Omega)}\,.
\end{split}
\end{equation}
Now we turn to the second term on the right hand side of
\eqref{eq7.62}. Thanks to the discrete inf-sup condition, we use
the discrete problem \eqref{eq7.3} to get
\begin{equation}\label{eq7.69}
\begin{split}
\|p_{k+\ell}-p_k\|_{L^2(\Omega)}& \lesssim \sup\limits_{0\not=v_{k+\ell}\in
V_{k+\ell}}\frac{b_{k+\ell}(v_{k+\ell},p_{k+\ell}-p_k)}{\|\nabla_{k+\ell}v_{k+\ell}\|_{L^2(\Omega)}}\\[0.5ex]
&\lesssim \sup\limits_{0\not=v_{k+\ell}\in
V_{k+\ell}}\frac{\res_k(v_{k+\ell})}{\|\nabla_{k+\ell}v_{k+\ell}\|_{L^2(\Omega)}}
+\|\nabla_{k+\ell}(u_{k+\ell}-u_k)\|_{L^2(\Omega)}\,.
\end{split}
\end{equation}
An application of the Cauchy--Schwarz inequality leads to
\begin{equation}\label{eq7.70}
|b_{k+\ell}(u_{k+\ell}-v_{k+\ell},p_{k+\ell}-p_k)|
\leq \|p_{k+\ell}-p_k\|_{L^2(\Omega)}
\|\nabla_{k+\ell}(u_{k+\ell}-v_{k+\ell})\|_{L^2(\Omega)}\,.
\end{equation}
After inserting \eqref{eq7.62}, \eqref{eq7.68}, \eqref{eq7.69}, and
\eqref{eq7.70} into \eqref{eq7.61}, we use the triangle and
Young inequalities to derive
\begin{equation}\label{eq7.71}
\begin{split}
&\|\nabla_{k+\ell}(u_{k+\ell}-u_k)\|_{L^2(\Omega)}^2
+\|p_{k+\ell}-p_k\|_{L^2(\Omega)}^2\\[1.0ex]
& \lesssim \sum\limits_{K\in \mathcal{M}_{k, k+\ell}}
h_K^2\|g\|_{L^2(K)}^2 +\inf\limits_{v_{k+\ell}\in
V_{k+\ell}}\|\nabla_{k+\ell}(u_k-v_{k+\ell})\|_{L^2(\Omega)}^2\,.
\end{split}
\end{equation}
An application of \eqref{eq7.72} bounds the second term on the right--hand side of \eqref{eq7.71}. This completes the proof.
\end{proof}
With $\gamma_1$ from Theorem \ref{Theorem7.8}, we define the
following energy norm:
\begin{equation}\label{Norm3b}
|\hskip-0.8pt|\hskip-0.8pt| v,q|\hskip-0.8pt|\hskip-0.8pt|^2:=\|\nabla
v\|_{L^2(\Omega)}^2+\gamma_1\|q\|_{L^2(\Omega)}^2, \text{ for any }
(v,q)\in V\times Q.
\end{equation}
We denote its piecewise version by $|\hskip-0.8pt|\hskip-0.8pt|\cdot|\hskip-0.8pt|\hskip-0.8pt|_{k+\ell}$.
The following lemma gives links between the error reduction to the
bulk criterion.
\begin{lemma}\label{Theorem7.16}
Let $\mathcal{T}_{k+\ell}$ be the refinement of $\mathcal{T}_k$ with the following
reduction:
\begin{equation}\label{eq7.73}
\begin{split}
|\hskip-0.8pt|\hskip-0.8pt| {u}-u_{k+\ell}, {p}-p_{k+\ell}|\hskip-0.8pt|\hskip-0.8pt|^2_{k+\ell}+\gamma_2\osc^2(g,\mathcal{T}_{k+\ell})\\
\leq
\alpha^{\prime}\big( |\hskip-0.8pt|\hskip-0.8pt|
{u}-u_k, {p}-p_k|\hskip-0.8pt|\hskip-0.8pt|^2_k+\gamma_2\osc^2(g, \mathcal{T}_k)\big),
\end{split}
\end{equation}
with $0<\alpha^{\prime}<1$ and the positive constant $\gamma_2$ from
Theorem \ref{Theorem7.8}. There exists $0<\theta_{\ast}<1$ with
\begin{equation}\label{eq7.74}
\theta_{\ast}\eta^2(u_k, p_k,\mathcal{T}_k)\leq\eta^2(u_k,p_k,
{\mathcal{M}_{k, k+\ell}}).
\end{equation}
\end{lemma}
\begin{proof} It follows \eqref{eq7.73} and the definitions of the norms $|\hskip-0.8pt|\hskip-0.8pt|\cdot|\hskip-0.8pt|\hskip-0.8pt|_k$ and $|\hskip-0.8pt|\hskip-0.8pt|\cdot|\hskip-0.8pt|\hskip-0.8pt|_{k+\ell}$ that
\begin{equation*}
\begin{split}
& (1-\alpha^{\prime})(|\hskip-0.8pt|\hskip-0.8pt| {u}-u_k, {p}-p_k|\hskip-0.8pt|\hskip-0.8pt|^2_k+\gamma_2\osc^2(g,\mathcal{T}_k))\\[0.5ex]
&\leq|\hskip-0.8pt|\hskip-0.8pt|
{u}-u_k, {p}-p_k|\hskip-0.8pt|\hskip-0.8pt|^2_k+\gamma_2\osc^2(g,\mathcal{T}_k)-|\hskip-0.8pt|\hskip-0.8pt|
{u}-u_{k+\ell}, {p}-p_{k+\ell}|\hskip-0.8pt|\hskip-0.8pt|^2_{k+\ell}-\gamma_2\osc^2(g,\mathcal{T}_{k+\ell})\\[0.5ex]
&=\|\nabla_{k+\ell}( u_k-u_{k+\ell})\|_{L^2(\Omega)}^2
+\gamma_1\|p_k-p_{k+\ell}\|_{L^2(\Omega)}^2+\frac{2}{\mu}a_{k+\ell}( {u}-u_{k+\ell},
u_{k+\ell}-u_k)\\[0.5ex]
&\quad+2\gamma_1( {p}-p_{k+\ell},
p_{k+\ell}-p_k)_{L^2(\Omega)}+\gamma_2\osc^2(g,\mathcal{T}_k)-\gamma_2\osc^2(g,\mathcal{T}_{k+\ell})\\[0.5ex]
&=I_1+I_2+I_3+I_4+I_5.
\end{split}
\end{equation*}
The first two terms, $I_1$ and $I_2$, are estimated by the discrete
reliability in Lemma \ref{Lemma7.15},
\begin{equation}\label{eq5.14b}
\begin{split}
|\hskip-0.8pt|\hskip-0.8pt| u_{k+\ell}-u_k|\hskip-0.8pt|\hskip-0.8pt|_{k+\ell}^2+\gamma_1\|p_k-p_{k+\ell}\|_{L^2(\Omega)}^2\leq
C_{Drel} \eta^2(u_k,p_k, {\mathcal{M}_{k, k+\ell} }),
\end{split}
\end{equation}
where the coefficient $C_{Drel}$ is from Lemma \ref{Lemma7.15}. The third term $I_3$ can be estimated by the quasi--orthogonality
of the velocity in Lemma \ref{Lemma7.5}. In fact, let the multiplication constant therein be the coefficient $C_{QOV}$, so that we have
\begin{equation}\label{equ5.26}
\begin{split}
&|\frac{2}{\mu}a_{k+\ell}(u-u_{k+\ell},u_{k+\ell}-u_k)|\\
&\leq 2C_{QOV}\|\nabla_{k+\ell}(u-u_{k+\ell})\|_{L^2(\Omega)} \big(\sum\limits_{K\in
\mathcal{M}_{k, k+\ell} }h_K^{2}\|g\|_{L^2(K)}^2\big)^{1/2}\\[0.5ex]
&\leq
\frac{1-\alpha^{\prime}}{2}\|\nabla_{k+\ell}(u-u_{k+\ell})\|_{L^2(\Omega)}^2+\frac{2(C_{QOV})^2}{1-\alpha^{\prime}}\sum\limits_{K\in
\mathcal{M}_{k, k+\ell} }h_K^{2}\|g\|_{L^2(K)}^2.
\end{split}
\end{equation}
Next, we use the quasi--orthogonality of the pressure in Lemma \ref{Lemma7.6} to analyze the fourth term, $I_4$.
Denote the constant of Lemma \ref{Lemma7.6} by $C_{QOP}$, and we obtain
\begin{equation*}
\begin{split}
&|2\gamma_1(p-p_{k+\ell}, p_{k+\ell}-p_k)_{L^2(\Omega)}|\\[0.5ex]
& \leq 2\gamma_1C_{QOP}
\bigg(\big(\sum\limits_{K\in \mathcal{M}_{k, k+\ell}}
h_K^2\|g\|_{L^2(K)}^2\big)^{1/2}
+\|\nabla_{k+\ell}(u_{k+\ell}-u_k)\|_{L^2(\Omega)}\bigg)\|p-p_{k+\ell}\|_{L^2(\Omega)}\\[0.5ex]
& \leq\frac{2\gamma_1(C_{QOP})^2}{1-\alpha^{\prime}}\bigg(\big(\sum\limits_{K\in \mathcal{M}_{k, k+\ell}}
h_K^2\|g\|_{L^2(K)}^2\big)^{1/2}
+\|\nabla_{k+\ell}(u_{k+\ell}-u_k)\|_{L^2(\Omega)}\bigg)^2\\[0.5ex]
&\quad + \frac{1-\alpha^{\prime}}{2}\gamma_1\|p-p_{k+\ell}\|_{L^2(\Omega)}^2.
\end{split}
\end{equation*}
Hence it follows from \eqref{eq5.14b} that
\begin{equation}
\begin{split}
&|2\gamma_1(p-p_{k+\ell}, p_{k+\ell}-p_k)_{L^2(\Omega)}|\\[0.3ex]
&\leq \frac{1-\alpha^{\prime}}{2}\gamma_1\|p-p_{k+\ell}\|_{L^2(\Omega)}^2+
\frac{2\gamma_1(C_{QOP})^2(1+C_{Drel}^{1/2})^2}{1-\alpha^{\prime}}\eta^2(u_k,p_k, \mathcal{M}_{k, k+\ell}).
\end{split}
\end{equation}
A direct calculation leads to
\begin{equation}\label{eq5.27}
\gamma_2|\osc^2(f,\mathcal{T}_k)-\osc^2(f, \mathcal{T}_{k+\ell})|\leq \gamma_2\eta^2(u_k,
p_k, \mathcal{M}_{k, k+\ell})\,,
\end{equation}
we combine \eqref{eq5.14b}---\eqref{eq5.27}, and \eqref{eq7.73} with the efficiency of
the estimator, which proves the desired result by
the parameter
$$\theta_{\ast}=\frac{(1-\alpha^{\prime})^2C_{Eff}}{2(2(C_{QOV})^2+2\gamma_1(C_{QOP})^2(1+C_{Drel}^{1/2})^2
+(1-\alpha^{\prime})(C_{Drel}+\gamma_2))}\,,
$$
with the efficiency constant $C_{Eff}$ of the estimator $\eta(u_k,
p_k, \mathcal{T}_k)$ from Lemma \ref{Theorem7.2}.
\end{proof}
\section{The optimality of the ANFEM}\label{Sec8}
In this section, we address the optimality of
the adaptive nonconforming linear element method under
consideration. We need to control the consistency error $\consis(\sigma,\mathcal{T})$ defined by
\begin{equation}\label{eq7.54}
\consis(\sigma,\mathcal{T})=\sup\limits_{v_{\mathcal{T}}\in V_{\mathcal{T}}}\frac{(g,
v_{\mathcal{T}})_{L^2(\Omega)}-(\sigma, \nabla_{\mathcal{T}}
v_{\mathcal{T}})_{L^2(\Omega)}}{\|\nabla_{\mathcal{T}}v_{\mathcal{T}}\|_{L^2(\Omega)}}\quad \text{ with }
\sigma=\mu\nabla u+p\id\,,
\end{equation}
where $\mathcal{T}$ is some refinement of the initial mesh $\mathcal{T}_0$ by the newest vertex bisection.
The following conforming finite element space is needed:
\begin{equation}
P_3(\mathcal{T}):=\{v\in (H_0^1(\Omega))^2, v|_K\in (P_3(K))^2, \text{ for any }K\in\mathcal{T}\}.
\end{equation}
Then, there exists an interpolation operator $\Pi_{\mathcal{T}}: V_{\mathcal{T}}\rightarrow P_3(\mathcal{T})$ with
the following properties \cite[Lemma A.3]{GiraultRaviart1986}:
\begin{equation}\label{PropertyIn}
\begin{split}
&\int_E (v_{\mathcal{T}}-\Pi_{\mathcal{T}}v_{\mathcal{T}})\cdot c_Eds=0 \text{ for any }c_E\in (P_1(E))^2,\\
&\int_K (v_{\mathcal{T}}-\Pi_{\mathcal{T}}v_{\mathcal{T}}) dx=0,
\end{split}
\end{equation}
for any edge $E$ and element $K$ of $\mathcal{T}$. In addition, we have
\begin{equation}\label{PropertyIn2}
\|v_{\mathcal{T}}-\Pi_{\mathcal{T}}v_{\mathcal{T}}\|_{L^2(K)}+h_K\|\nabla\Pi_{\mathcal{T}}v_{\mathcal{T}}\|_{L^2(K)}\lesssim h_K \|\nabla_{\mathcal{T}}v_{\mathcal{T}}\|_{L^2(\omega_K)}.
\end{equation}
For any $s_{\mathcal{T}}\in V_{\mathcal{T}}$ and $q_{\mathcal{T}}\in Q_{\mathcal{T}}$, we define $\sigma_{\mathcal{T}}=\mu s_{\mathcal{T}}+q_{\mathcal{T}}$. The idea of \cite[Lemma 2.1]{Gudi2010}
leads to the following decomposition:
\begin{equation}
\begin{split}
&(g,
v_{\mathcal{T}})_{L^2(\Omega)}-(\sigma, \nabla_{\mathcal{T}}
v_{\mathcal{T}})_{L^2(\Omega)}\\
&=(g,
v_{\mathcal{T}}-\Pi_{\mathcal{T}}v_{\mathcal{T}})_{L^2(\Omega)}-(\sigma-\sigma_{\mathcal{T}}, \nabla_{\mathcal{T}}
(v_{\mathcal{T}}-\Pi_{\mathcal{T}}v_{\mathcal{T}}))_{L^2(\Omega)}\\
&\quad + (\sigma_{\mathcal{T}}, \nabla_{\mathcal{T}}
(v_{\mathcal{T}}-\Pi_{\mathcal{T}}v_{\mathcal{T}}))_{L^2(\Omega)}
\end{split}
\end{equation}
for any $v_{\mathcal{T}}\in V_{\mathcal{T}}$. By the properties \eqref{PropertyIn} and \eqref{PropertyIn2}, we obtain
\begin{equation}\label{consis}
\consis(\sigma,\mathcal{T})\lesssim \inf\limits_{(v_{\mathcal{T}},q_{\mathcal{T}})\in
V_{\mathcal{T}}\times Q_{\mathcal{T}}}|\hskip-0.8pt|\hskip-0.8pt| u-v_{\mathcal{T}},
p-q_{\mathcal{T}}|\hskip-0.8pt|\hskip-0.8pt|_{\mathcal{T}}+\osc(g,
\mathcal{T}).
\end{equation}
This implies that the nonlinear approximate class used in \cite{HuXu2007} is {\em equivalent } to the standard nonlinear
approximate class \cite{CasconKreuzerNochettoSiebert07}.
Hence, we can introduce the following semi-norm:
\begin{equation}\label{eq7.53}
\mathfrak{E}^2(N;u,p,
g):=\inf\limits_{\mathcal{T}\in\mathbb{T}_N}\big(\inf\limits_{(v_{\mathcal{T}},q_{\mathcal{T}})\in
V_{\mathcal{T}}\times Q_{\mathcal{T}}}|\hskip-0.8pt|\hskip-0.8pt| u-v_{\mathcal{T}},
p-q_{\mathcal{T}}|\hskip-0.8pt|\hskip-0.8pt|^2_{\mathcal{T}}+\gamma_2\osc^2(g,
\mathcal{T})\big)\,.
\end{equation}
Then the nonlinear approximate class $\mathbb{A}_s$ can be defined
by
\begin{equation}\label{eq7.55}
\mathbb{A}_s:=\{(u,p, g), |u,p,
g|_s:=\sup\limits_{N>N_0}N^{s}\mathfrak{E}(N;u,p, g)<+\infty\}.
\end{equation}
We must stress that this is the first time the standard nonlinear approximate class \cite{CasconKreuzerNochettoSiebert07}
has been used to analyze the adaptive nonconforming finite element method. In the relevant literature,
the discrete solution of the discrete problem has been used to define the nonlinear approximate class \cite{BeckerMaoShi2010,BeckerMao2011,MaoZhaoShi2010,Rabaus2010}. Let $(u_{\mathcal{T}}, p_{\mathcal{T}})$ be the approximation solution
of \eqref{eq7.3} on the mesh $\mathcal{T}$. It follows from the Strang Lemma \cite{CiaBook}
$$
|\hskip-0.8pt|\hskip-0.8pt| u-u_{\mathcal{T}}, p-p_{\mathcal{T}}|\hskip-0.8pt|\hskip-0.8pt|_{\mathcal{T}}\lesssim \inf\limits_{(v_{\mathcal{T}},q_{\mathcal{T}})\in
V_{\mathcal{T}}\times Q_{\mathcal{T}}}|\hskip-0.8pt|\hskip-0.8pt| u-v_{\mathcal{T}},
p-q_{\mathcal{T}}|\hskip-0.8pt|\hskip-0.8pt|_{\mathcal{T}}+\consis(\sigma,\mathcal{T}),
$$
and the following fact
$$
\inf\limits_{(v_{\mathcal{T}},q_{\mathcal{T}})\in
V_{\mathcal{T}}\times Q_{\mathcal{T}}}|\hskip-0.8pt|\hskip-0.8pt| u-v_{\mathcal{T}},
p-q_{\mathcal{T}}|\hskip-0.8pt|\hskip-0.8pt|_{\mathcal{T}}+\consis(\sigma,\mathcal{T})\lesssim |\hskip-0.8pt|\hskip-0.8pt| u-u_{\mathcal{T}}, p-p_{\mathcal{T}}|\hskip-0.8pt|\hskip-0.8pt|_{\mathcal{T}},
$$
that the nonlinear approximate class of \cite{BeckerMao2011} is equivalent to $\mathbb{A}_s$ of \eqref{eq7.55}.
A similar method herein proves that the nonlinear approximate class of \cite{BeckerMaoShi2010,MaoZhaoShi2010,Rabaus2010}
is equivalent to the standard nonlinear approximate class \cite{CasconKreuzerNochettoSiebert07}.
\begin{remark} After we submitted the revised version to the journal,
we learnt about that a different argument of \cite{CarstensenPeterseimSchedensack2012} shows that the nonlinear approximate class of \cite{BeckerMaoShi2010,MaoZhaoShi2010,Rabaus2010}
is equivalent to the standard nonlinear approximate class \cite{CasconKreuzerNochettoSiebert07}.
\end{remark}
Thanks to \eqref{consis}, we have
\begin{equation}\label{eq7.79}
\begin{split}
&|\hskip-0.8pt|\hskip-0.8pt| {u}-u_{k-1}, {p}-p_{k-1}|\hskip-0.8pt|\hskip-0.8pt|^2_{k-1}\\
& \lesssim \inf\limits_{(v_{k-1},
q_{k-1})\in V_{k-1}\times Q_{k-1}}|\hskip-0.8pt|\hskip-0.8pt| {u}-v_{k-1},
{p}-q_{k-1}|\hskip-0.8pt|\hskip-0.8pt|^2_{k-1}+\osc^2(g, \mathcal{T}_{k-1})\,.
\end{split}
\end{equation}
A straightforward investigation shows that if $\mathcal{T}_k$ is any
refinement of $\mathcal{T}_{k-1}$, then
it holds that
\begin{equation}\label{eq7.75}
\begin{split}
&\inf\limits_{(v_k, q_k)\in V_k\times Q_k}|\hskip-0.8pt|\hskip-0.8pt|
{u}-v_k,
{p}-q_k|\hskip-0.8pt|\hskip-0.8pt|^2_k+\gamma_2\osc^2(g, \mathcal{T}_k)\\[0.5ex]
&\qquad\leq C_{3}\big(\inf\limits_{(v_{k-1}, q_{k-1})\in V_{k-1}\times Q_{k-1}}|\hskip-0.8pt|\hskip-0.8pt|
{u}-v_{k-1}, {p}-q_{k-1}|\hskip-0.8pt|\hskip-0.8pt|^2_{k-1}+\gamma_2\osc^2(g, \mathcal{T}_{k-1})\big)\,.
\end{split}
\end{equation}
With these preparations, following
\cite{HSX10}, we have the following optimality:
\begin{theorem}\label{Theorem7.20}
Let
$( {u}, {p})$ be the solution of Problem \eqref{eq7.1}, and let
$(\mathcal{T}_k, V_k\times Q_k, (u_k,p_k))$ be the sequence of meshes, finite
element spaces, and discrete solutions produced by the adaptive
finite element methods. If $( {u}, {p}, g)\in \mathbb{A}_s$ with
$$\theta\leq\frac{C_{Eff}}{2(2(C_{QOV})^2+2\gamma_1(C_{QOP})^2(1+C_{Drel}^{1/2})^2
+C_{Drel}+\gamma_2)}\,.
$$
Then, it holds that
\begin{equation}\label{eq7.87}
|\hskip-0.8pt|\hskip-0.8pt| u-u_N, p-p_N|\hskip-0.8pt|\hskip-0.8pt|_N^2+\gamma_2\osc^2(g, \mathcal{T}_N)\lesssim |
{u},
{p}, g|_s^2(\#\mathcal{T}_N-\#\mathcal{T}_0)^{-2s}\,.
\end{equation}
\end{theorem}
|
2012.10038
|
\section{Introduction}
The asteroseismic scaling relations for red giants have so far proved to be an extremely useful tool to obtain stellar masses and radii. A critical issue associated with the scaling relations is that their limits are poorly understood \citep{hekker-2020-sc-review}. The intrinsic scatter of the scaling relations, originating from potential hidden dependencies not accounted for in the current relations, can cause a seemingly random fluctuation. Testing the intrinsic scatter of these relations is the aim of this paper.
The scaling relations rely on two characteristic frequencies in the power spectra of solar-like oscillations. The first one is \Dnu{}, the large separation of p modes, approximately proportional to the square root of mean density \citep{ulrich-1986-age}:
\begin{equation}\label{eq:sc-dnu}
\frac{\Delta\nu}{\Delta\nu_{\astrosun}} \approx \left(\frac{M}{M_{\astrosun}}\right)^{1/2} \left(\frac{R}{R_{\astrosun}}\right)^{-3/2}.
\end{equation}
The second is \numax{}, which is the frequency where the power of the oscillations is strongest. It relates to the surface properties $g/\sqrt{T_{\rm eff}}$ \citep{brown++1991-dection-procyon-scaling-relation,kjeldsen+1995-scaling-relations}:
\begin{equation}\label{eq:sc-numax}
\frac{\nu_{\rm max}}{\nu_{\rm max,\astrosun}} \approx \left(\frac{M}{M_{\astrosun}}\right) \left(\frac{R}{R_{\astrosun}}\right)^{-2} \left(\frac{T_{\rm eff}}{T_{\rm eff,\astrosun}}\right)^{-1/2}.
\end{equation}
Using these, the mass and radius can be determined if the effective temperature is known \citep{stello++2008-wire-kgiants,kallinger++2010-rg-corot-mass-radius}:
\begin{equation}\label{eq:sc-mass}
\frac{M}{M_{\astrosun}} \approx \left(\frac{\nu_{\rm max}}{\nu_{\rm max,\astrosun}}\right)^3 \left(\frac{\Delta\nu}{\Delta\nu_{\astrosun}}\right)^{-4} \left(\frac{T_{\rm eff}}{T_{\rm eff,\astrosun}}\right)^{3/2},
\end{equation}
\begin{equation}\label{eq:sc-radius}
\frac{R}{R_{\astrosun}} \approx \left(\frac{\nu_{\rm max}}{\nu_{\rm max,\astrosun}}\right) \left(\frac{\Delta\nu}{\Delta\nu_{\astrosun}}\right)^{-2} \left(\frac{T_{\rm eff}}{T_{\rm eff,\astrosun}}\right)^{1/2}.
\end{equation}
From a theoretical point of view, a more accurate value for \Dnu{} can be calculated from oscillation frequencies given a stellar model; thus it is possible to map the departure of Eq.~\ref{eq:sc-dnu}, as a function of [M/H], $M$, \teff{} and evolutionary state \citep{white++2011-asteroseismic-diagrams-cd-epsilon-deltaP-models,sharma++2016-population-rg-kepler,guggenberger++2016-metallicity-scaling-relation,rodrigues++2017-dpi-modelling,serenelli++2017-apokasc-dwarf-subgiant,pinsonneault++2018-apokasc}.
Improvements are seen when adopting this revised theoretical $\Delta\nu$ over the standard density scaling \citep[e.g.][]{brogaard++2018-accuracy-scaling-relation}. However, there are some degrees of uncertainty. \cite{jcd++2020-aarhus-rgb-osc} found a 0.2\% spread in the theoretical departure stemming from implementing the calculation with different codes, and the degree of model-dependency on physical processes has not been explored extensively.
The \numax{} scaling relation is much harder to assess theoretically because calculating \numax{} would require a detailed treatment of non-adiabatic processes, via either 1D or 3D stellar models \citep[e.g.][]{balmforth-1992-pulsation-stability-1-mode-thermodynamics,houdek++1999-amplitudes-solarlike,belkacem++2019-3d-rhd,zhou++2019-3d-amplitude-solar-oscillation}. Some works concluded a possible departure could correlate with, for example, the Mach number \citep{belkacem++2011-physics-under-numax-nuc}, magnetic activity \citep{jimenez++2011-nuac-sun-cycle} and mean molecular weight \citep{jimenez++2015-nuac-six-kepler,yildiz++2016-sc-gamma1,viani++2017-numax-sc-metal}. In general, it is still impossible to accurately predict \numax{} from theory.
Another way to test the scaling relations is by comparing with fundamental data from independent observations. This requires masses and radii obtained by other means, such as astrometric surveys, where radii are deduced using the Stefan-Boltzmann law, eclipsing binaries, where masses and radii are derived from dynamic modelling.
So far, the radii tests based on parallaxes suggest agreement within 4\% for stars smaller than 30\,\rsolar{} \citep{silvaaguirre++2012-seismic-parallax,huber++2017-seismic-radii-gaia,sahlholdt+2018-gaiadr2-sc-radius-dwarfs,hall++2019-rc-gaiadr2-seismo,khan++2019-gaiadr2-zero-point,zinn++2019-radius-sc}.
With 16 eclipsing binaries, \citet{gaulme++2016-eb-sc} found the asteroseismic masses and radii are systematically overestimated, by factors of 15\% and 5\%, respectively. This result is in disagreement with Gaia radii, possibly because the binary temperature is affected by blending \citep{huber++2017-seismic-radii-gaia,zinn++2019-radius-sc}.
Subsequent analyses indicate that the main source of departure could come from the \Dnu{} scaling relation \citep{brogaard++2018-accuracy-scaling-relation,sharma++2019-k2-hermes-age-metallicity-thick-disc}.
As we noted earlier, the random departures of the scaling relations can be associated with unaccounted factors,
for example, metallicity, rotation and magnetism, some of which are known to have a wide-ranging distribution among red giants \citep[e.g.][]{mosser++2012-rg-core-rotation-spin-down,stello-2016-supression-rg-kepler,ceillier++2017-surface-rotation-rg-kepler}.
They could be responsible for some intrinsic scatter in these rather simple relations
We propose a new approach to investigate the intrinsic scatter, based on two sharp features in the H--R diagram observed among the red giant population.
The first feature is the accumulation of stars at the bump of red-giant-branch (RGB). The second feature is the sharp edge formed by the zero-age sequence of core-helium-burning (HeB) stars.
These features were known before seismic observations became available.
The RGB bump is an evolutionary stage where a star ascending the RGB temporarily drops in luminosity before again ascending towards the tip of the RGB, causing a hump in the luminosity distribution. This feature is prominent in colour-magnitude diagrams of stellar clusters \citep{iben-1968-m15,king++1985-tuc47}. The luminosity drop takes place after the first dredge-up and is caused by a change in the composition profile near the hydrogen-burning shell, leading to a decrease in mean molecular weight outside the composition discontinuity point \citep{refsdal+1970-shell,jcd-2015-rgb-bump}. \kepler{} data show that this bump is also present in the distributions of \Dnu{} and \numax{} \citep{kallinger++2010-kepler-rg-4months,khan-2018-rgb-bump-constraints-envelope-overshooting}.
After reaching the tip of RGB, stars strongly decrease in luminosity and commence core helium burning, forming the red clump, also commonly recognised as the horizontal branch in metal-poor clusters \citep{cannon-1970-rc-discovery,girardi++2010-ngc419-src}. The low-luminosity edge defines the beginning of the red clump and secondary clump phase, which we we will refer to as the zero-age HeB (ZAHeB) phase. This feature is also imprinted on seismic observables \citep{kallinger++2010-kepler-rg-4months,huber++2010-800-rg-kepler,mosser++2010-rg-seismic-property-corot,yuj++2018-16000-rg}.
The fact that the seismic parameters (\Dnu{} and \numax{}) preserve these sharp features indicates that the seismic parameters must be tightly related to the fundamental stellar parameters. Put another way, if there were a large intrinsic scatter in the scaling relations, the features in the seismic diagrams would not be as sharp. Using this principle, we can quantify the limits on the intrinsic scatter in the scaling relations. That is the aim of this paper.
\begin{figure}
\includegraphics[width=\columnwidth]{figs/formalerrors.pdf}
\caption{Distributions of the \yu{} formal uncertainties of the \numax{} and \Dnu{} measurements.}
\label{fig:formal-error}
\end{figure}
\begin{figure*}
\includegraphics[width=\textwidth]{figs/diagram_rgb_lum.pdf}
\includegraphics[width=\textwidth]{figs/diagram_rgb_numax.pdf}
\includegraphics[width=\textwidth]{figs/diagram_rgb_dnu.pdf}
\includegraphics[width=\textwidth]{figs/diagram_rgb_mr.pdf}
\caption{$L$ vs. \teff{} (panels a--b), \numax{} vs. \teff{} (panels c--d),
\Dnu{} vs. \teff{} (panels e--f) and $R$ vs. $M$ (bottom g--h)
for RGB stars in the \apk{} sample (red) and the \gal{} sample (blue).
The RGB bumps were defined using the black straight fiducial lines. The grey-shaded areas denote the uncertainty of identifying the bump (see \ref{subsubsec:identify-features}).}
\label{fig:diagram-rgb}
\end{figure*}
\begin{figure}
\includegraphics[width=\columnwidth]{figs/rgbb_isochrone.pdf}
\caption{Radius vs. mass for RGB stars near the RGB bump, colour-coded by metallicity. The PARSEC and MIST isochrones predict different outcomes on the shape of the RGB bump. }
\label{fig:rgbb-isochrone}
\end{figure}
\section{Sample selection}
\label{sec:sample}
To create our sample we used the red giants observed by \kepler{},
with \Dnu{} and \numax{} measured by the SYD pipeline \citep{huber++2009-syd-pipeline,yuj++2018-16000-rg},
and classifications of evolutionary stage (RGB/HeB) from \citet{hon++2017-deep-learning-rc-rgb}.
We denote this sample as \yu{}, including 7543 HeB stars and 7534 RGB stars.
A subset of 2531 HeB and 3308 RGB stars with \teff{} and [M/H] from the APOKASC-2 catalog
\citep{pinsonneault++2018-apokasc} was also used, denoted as \apk{}.
In Fig.~\ref{fig:formal-error}, we show the distributions of the formal uncertainties
of \numax{} and \Dnu{} measured by \yu{}.
The \yu{} sample reports a typical formal uncertainty of
$2.1\%$ on \numax{} and $1.0\%$ on \Dnu{} in HeB stars,
and $0.95\%$ on \numax{} and $0.3\%$ on \Dnu{} in RGB stars.
To model the observed population, we used a synthetic sample produced by
\citet{sharma++2019-k2-hermes-age-metallicity-thick-disc} with a Galactic model,
\galaxia{} \citep{Sharma++2011-galaxia}.
Compared to a previous synthetic sample in \cite{sharma++2016-population-rg-kepler},
the synthetic sample we used in this work adds a metal-rich thick disc,
which improves the overall match with the \kepler{} observation
\citep{sharma++2019-k2-hermes-age-metallicity-thick-disc}.
Here we denote this sample as \gal{}.
The \gal{} simulated sample is about ten times larger than the \yu{} sample.
Each star in the simulated sample is associated with an initial mass, an age,
and a metallicity, sampled from a Galactic distribution function and passed
through a selection function tied to the \kepler{} mission.
Other fundamental stellar parameters (e.g. $M$, $R$ and \teff{}) were estimated
via two different sets of theoretical isochrones:
PARSEC \citep{marigo++2017-parsec} and
MIST \citep{Choi++2016-mist-1-solar-scaled-models}.
Both isochrones include some mass loss along the RGB,
using the \citet{reimers-1975-mass-loss} prescription
with an efficiency of $\eta_R=0.2$ (PARSEC) and $\eta_R=0.1$ (MIST),
consistent with the asteroseismology of open clusters
\citep{miglio++2012-mass-loss-ngc6791-ngc6819}.
The seismic parameters, \Dnu{} and \numax{}, were calculated through the scaling
relations (Eqs.~\ref{eq:sc-dnu} and~\ref{eq:sc-numax}) without any corrections.
By examining the sharpness of the two features discussed above, and comparing
the \galaxia{} simulation with the observations, we are able to draw conclusions
about the intrinsic scatter of the scaling relations.
\begin{figure}
\includegraphics[width=\columnwidth]{figs/hist_rgb_numax.pdf}
\includegraphics[width=\columnwidth]{figs/hist_rgb_dnu.pdf}
\includegraphics[width=\columnwidth]{figs/hist_rgb_mass.pdf}
\includegraphics[width=\columnwidth]{figs/hist_rgb_radius.pdf}
\caption{Distributions of distances to the bump features. The top two panels are measured in the \teff{}--\numax{} and \teff{}--\Dnu{} diagrams, and the bottom two panels are measured in the $M$--$R$ diagram. The \kepler{} (\apk{}) distributions are shown in red, fitted with a Gaussian model, denoted by the black dashed lines. The synthetic \gal{} samples are shown in blue. The grey-shaded areas denote the range used to compare the data. }
\label{fig:hist-rgb}
\end{figure}
\section{The RGB bump}
\label{sec:rgbb}
In this section, we look at the RGB bump.
In a traditional H--R diagram, the bump is tilted so that
the luminosity $L$ of the bump is a function
of \teff{} and its shape can be parameterized by stellar mass $M$,
using $L=L(M)$ and $T_{\rm eff}=T_{\rm eff} (M)$.
By introducing the \numax{} scaling relation, we can obtain
$\nu_{\rm max} \propto M L^{-1} T_{\rm eff} ^{7/2}$.
Therefore, a narrow bump in the $L$--\teff{} plane
will also show a bump due to this $M$ dependence in the \numax{}--\teff{} plane.
If the \numax{} scaling relation has some intrinsic scatter due to
other dependencies, such as metallicity, then the observed bump in
the \numax{}--\teff{} plane could be wider.
For \Dnu{}, the argument is similar.
Fig.~\ref{fig:diagram-rgb} shows the RGB bump for both \kepler{} and \galaxia{} samples.
Here we wish to model the width of the RGB bump. We will start by investigating the features in the \Dnu{}--\teff{} and \numax{}--\teff{} diagrams, and then in the $M$--$R$ diagram.
We further note that the width of RGB bump
strongly depends on how the physical processes are modelled in the isochrones.
This is illustrated in Fig.~\ref{fig:rgbb-isochrone},
where the shapes of the RGB bump predicted
by the two sets of isochrones are inconsistent.
The PARSEC models predict that the stellar radii at the RGB bump should decrease with masses for masses larger than $\sim$1.2 $M_{\astrosun}$.
However, the opposite is observed in the \kepler{} samples,
and this behaviour is correctly described by the MIST models.
It implies that the RGB bump may not serve as a useful diagnostic for the scaling relations.
We will examine this caveat more extensively in section~\ref{subsubsec:model-dependency}.
Nevertheless, here we still use the RGB bump to introduce our method and we analyse the \gal{} samples with the two isochrones separately.
\subsection{Modelling method}
\label{subsec:rgbb-method}
We used a forward-modelling approach by constructing synthetic
samples based on the \gal{} sample, and setting the intrinsic
scatter of the scaling relations, $\sigma$, as a free parameter.
The width of the bump was evaluated by measuring the distances
of model samples to the centre of the bump, and fitting their
distributions to the \apk{} sample.
The first step was to define the locations of the RGB bump
in the \apk{} and \gal{} samples
with straight lines in the \numax{}--\teff{}
and \Dnu{}--\teff{} diagrams, shown in Fig.~\ref{fig:diagram-rgb}.
We generated a synthetic population by adding random scatter
to the \gal{} sample.
Each physical quantity $x$ (one of \Dnu{}, \numax{}, $M$ or $R$)
for the $i$-th star in the sample was
\begin{equation} \label{eq:perturb} x_i = x_{{\textit{Galaxia}}, i} (1 + \sigma_{{\rm total},i}). \end{equation} %
The quantity $x_{{\textit{Galaxia}},i}$ is the physical value without any perturbation.
For $M$ and $R$, they were directly estimated from isochrones.
Note that $M$ is the actual mass rather than the initial mass.
Values for \Dnu{} and \numax{} were determined via scaling relations
(Eqs.~\ref{eq:sc-dnu} and~\ref{eq:sc-numax}) and further corrected using oscillation
frequencies (\Dnu{} in particular, see section~\ref{subsec:rgbb-seismo}).
We modelled the total scatter needed to reproduce the width of the RGB bump,
$\sigma_{{\rm total},i}$, which was drawn from a normal distribution with
a standard deviation $\sigma_{\rm total}$.
To account for the scatter induced by the formal uncertainties of the \Dnu{} and \numax{} measurements,
we modelled each quantity $x$ with
\begin{equation} \label{eq:perturb-sc} x_i = x_{{\textit{Galaxia}},i} (1 + \sigma_{x,i} + \sigma_{{\rm SR},i}), \end{equation} %
where $\sigma_{x,i}$ represents the fractional uncertainty of $x_{{\textit{Galaxia}},i}$,
and was drawn randomly from the \apk{} formal uncertainty distribution of RGB bump stars.
The intrinsic scatter in the scaling relation was modelled via $\sigma_{{\rm SR},i}$,
drawn from a normal distribution with a standard deviation $\sigma_{{\rm SR}}$.
We then calculated the distributions of distances to bump lines.
The bump lines, shown in Fig.~\ref{fig:diagram-rgb}, were picked
so that the distances to the line have the smallest standard deviation.
For \Dnu{} and \numax{}, we calculated the vertical distances
in the \Dnu{}--\teff{} and \numax{}--\teff{} diagrams, respectively.
For $M$ and $R$, we used the horizontal and vertical distances in the $M$--$R$ diagram.
This procedure allowed us to investigate the scatter in each relation
separately, because perturbing the horizontal value will not change the
vertical value, and vice versa.
In Fig.~\ref{fig:hist-rgb}, we plot the distributions of those distances
with two representative choices for $\sigma_{\rm{total}}$.
A larger value for $\sigma_{\rm{total}}$ flattens the hump, demonstrating the width
of the bump itself provides a measure of the intrinsic scatter in the scaling relations.
Next, we introduce our fitting strategy to enable the comparison,
which is to match the counts in each bin of the histograms.
We first identified a central region in the histograms of the
\apk{} sample by fitting a Gaussian profile plus a sloping straight line, illustrated
by the dashed curves in Fig.~\ref{fig:hist-rgb}.
The central region was defined to be a range centred around the Gaussian,
with a width of 6 times the Gaussian standard deviation.
In Fig.~\ref{fig:hist-rgb}, they are shown in grey-shaded areas.
In our fit described below, we matched the distributions in the central regions only.
Because the \gal{} sample is larger than the \apk{} sample,
we re-scaled the number of model samples by normalising according to
the \apk{} sample in the central region.
We also added a constant $c$ as a free parameter to the distance of
the model samples, in order to compensate for a possible offset of maxima,
which could originate from a bias in identifying the bump.
We optimised the likelihood function, assuming the distribution of
counts in each bin is set by Poisson statistics:
\begin{equation} \ln L = \sum_{m_j\neq 0}\left[ d_j \ln m_j - m_j - \ln(d_j!) \right], \end{equation}
where $d_j$ and $m_j$ are counts in the $j$-th bins of the \kepler{} and model distributions.
This fitting method is commonly used in population studies to constrain
the star formation history, initial mass function and binary properties
\citep[e.g.][]{dolphin-2002-smh-method,geha++2013-imf-dwarf-galaxy,el-badry-2019-twin-binary}.
The posterior distributions of parameters $c$ and $\sigma$ were sampled with
uninformative flat priors, using a Markov Chain Monte Carlo (MCMC) method.
We used 200 white walkers, burned-in for 500 steps to reach convergence and
then iterated for another 1000 steps.
The medians and 68\% credible uncertainties of the parameters were estimated from
the posteriors directly.
\begin{figure}
\includegraphics[width=\columnwidth]{figs/limits_rgb.pdf}
\caption{Intrinsic scatter of the scaling relations $\sigma_{\rm SR}$ (yellow) and total scatter $\sigma_{\rm total}$ (blue), derived using the width of the RGB bump.}
\label{fig:limits-rgb}
\end{figure}
\begin{figure*}
\includegraphics[width=\textwidth]{figs/diagram_heb_tnu.pdf}
\includegraphics[width=\textwidth]{figs/diagram_heb_mr.pdf}
\caption{\numax{}--\Dnu{} diagram (panels a--b) and $M$--$R$ diagram (panels c--d)
for HeB stars in the \yu{} sample (red) and the \gal{} sample (blue).
The zero-age HeB edges were defined using splines, shown as the black lines.
The grey-shaded areas denote the uncertainty of identifying the edges (see section~\ref{subsubsec:identify-features}).
The stars around $20$ $R_{\astrosun}$ in the \gal{} sample are at the asymptotic-giant-branch phase.
}
\label{fig:diagram-heb}
\end{figure*}
\subsection{Results}
\label{subsec:rgbb-seismo}
Our first step is to derive the total scatter responsible for the width of the RGB bump,
$\sigma_{\rm total}$, in Eq.~\ref{eq:perturb}.
We obtained
$0.61\%$ (\Dnu{}),
$2.89\%$ (\numax{}),
$4.05\%$ ($M$) and
$0.90\%$ ($R$) with PARSEC, and
$5.74\%\pm0.80\%$ (\Dnu{}),
$9.80\%\pm0.90\%$ (\numax{}),
$1.79\%\pm1.34\%$ ($M$) and
$2.75\%\pm0.88\%$ ($R$) with MIST.
Next we took the formal uncertainties of the \Dnu{} and \numax{} measurements into account
and obtained the limits on the intrinsic scatter
of the scaling relations, $\sigma_{\rm SR}$, in Eq.~\ref{eq:perturb-sc}.
With PARSEC, we obtained
$0.88\%$ (\Dnu{}),
$2.00\%$ (\numax{}),
$2.26\%$ ($M$) and
$0.60\%$ ($R$).
With MIST, we obtained
$5.97\%$ (\Dnu{}),
$9.76\%$ (\numax{}),
$1.89\%$ ($M$) and
$0.56\%$ ($R$).
These numbers are plotted in Fig.~\ref{fig:limits-rgb}. There is a huge difference between MIST and PARSEC. We will discuss it in section~\ref{subsubsec:model-dependency}.
\section{The ZAHeB edge}
\label{sec:zaheb}
Similar to the RGB bump, the zero-age sequence of HeB stars (ZAHeB) also forms a well-defined feature in the H--R diagram \citep{girardi++2010-ngc419-src,girardi-2016-rc-review}.
We note that the transition from the red clump (low-mass stars that ignite helium in a fully degenerate core)
to the secondary red clump (higher-mass stars that ignite helium in a partly or non-degenerate core)
is smooth and continuous \citep{girardi-2016-rc-review}.
Given the scaling relations, there should exist a close correlation between $\Delta\nu$ and $\nu_{\rm max}$ for the ZAHeB.
In Fig.~\ref{fig:diagram-heb}, we show the HeB stars in the \Dnu{}--\numax{} and $M$--$R$ diagrams.
The ZAHeB appears as a very sharp feature: all HeB stars are located at only one side of the ZAHeB, forming a remarkably sharp edge.
\footnote{It has not escaped our attention that Fig~\ref{fig:diagram-heb}(a) bears
a strong resemblance to the logo of a major footwear manufacturer.
We plan to investigate sponsorship opportunities.}
Now we use the sharpness of this edge to quantify the intrinsic scatter of the scaling relations.
\subsection{Modelling method}
\label{subsec:heb-method}
\begin{figure}
\includegraphics[width=\columnwidth]{figs/hist_heb_numax.pdf}
\includegraphics[width=\columnwidth]{figs/hist_heb_dnu.pdf}
\includegraphics[width=\columnwidth]{figs/hist_heb_mass.pdf}
\includegraphics[width=\columnwidth]{figs/hist_heb_radius.pdf}
\caption{Distributions of distances to the ZAHeB edges. The top two panels are measured in the \numax{}--\Dnu{} diagram, and the bottom two panels are measured in the $M$--$R$ diagram. The \kepler{} (\yu{}) distributions are shown in red, fitted with a half-Gaussian-half-Lorentzian model, denoted by the black dashed lines. The synthetic \gal{} samples are shown in blue. The grey-shaded areas denote the range used to compare the data. }
\label{fig:hist-heb}
\end{figure}
To measure the sharpness of the ZAHeB edge, we used a modelling method similar
to that for the RGB bump in section~\ref{subsec:rgbb-method}, but with three
important differences.
The first difference is related to defining the location of the feature.
For RGB stars, we used straight lines to denote the location of the bump.
For HeB stars, we used splines in the \Dnu{}--\numax{} and $M$--$R$ diagrams
to define the ZAHeB edges.
This is illustrated in Fig.~\ref{fig:diagram-heb}, where the edges are shown
as black lines.
The second difference is how we calculated the horizontal and vertical
distances to the ZAHeB edge.
In the \Dnu{}--\numax{} diagram, the stars below the lowest point of the edge
do not have a meaningful horizontal distance. We therefore excluded them for
the horizontal distance calculation.
The same strategy was also applied to all stars that lie on the left of
the leftmost point of the defined ZAHeB edge when calculating vertical distances.
Similarly, in the $M$--$R$ diagram, the stars above the highest point of
the ZAHeB edge were not considered in calculating horizontal distances.
In Fig.~\ref{fig:hist-heb}, we plot the distributions of those distances with
two $\sigma_{\rm total}$.
As for the RGB bump, we see that a larger scatter $\sigma_{\rm total}$ smooths the hump.
The third difference is that, in order to choose regions near each edge to compare,
we fitted a profile to the distributions for the \yu{} sample.
The profiles, shown as the dashed lines in Fig.~\ref{fig:hist-heb},
consisted of a half Gaussian (left) and a half Lorentzian (right).
The histogram region that we fitted was a range centred at the Gaussian's centre,
with a width of 6 times the Gaussian's standard deviation. These regions are
shown as grey-shaded areas.
\subsection{Results}
\label{subsec:heb-seismo}
\begin{figure}
\includegraphics[width=\columnwidth]{figs/limits_heb.pdf}
\caption{Intrinsic scatter of the scaling relations $\sigma_{\rm SR}$ (yellow) and total scatter $\sigma_{\rm total}$ (blue), derived using the sharpness of the ZAHeB edge.}
\label{fig:limits-heb}
\end{figure}
We measured the total scatter $\sigma_{\rm total}$ that contributes to the
broadening of the edges in the \numax{}--\Dnu{} and $M$--$R$ diagrams:
$1.25\%\pm0.05\%$ (\Dnu{}),
$2.23\%\pm0.12\%$ (\numax{}),
$9.10\%\pm0.50\%$ ($M$) and
$2.01\%\pm0.05\%$ ($R$) using the PARSEC models, and
$1.56\%\pm0.04\%$ (\Dnu{}),
$2.99\%\pm0.19\%$ (\numax{}),
$7.00\%\pm0.54\%$ ($M$) and
$2.29\%\pm0.07\%$ ($R$) using the MIST models.
These numbers are in general agreement with the formal uncertainties of \Dnu{} and \numax{}
reported by \yu{} for HeB stars (Fig.~\ref{fig:formal-error}), suggesting a main contribution to
the broadening of the ZAHeB edge.
Next, we tested whether we needed to add intrinsic scaling relation scatter to
the \yu{} measurement uncertainties in order to reproduce the sharpness of
the ZAHeB edge. We derived $\sigma_{\rm SR}$ with the PARSEC models:
$0.13\%\pm0.18\%$ (\Dnu{}),
$0.72\%\pm0.24\%$ (\numax{}),
$2.34\%\pm1.38\%$ ($M$) and
$0.22\%\pm0.12\%$ ($R$).
And with the MIST models we obtained
$0.89\%\pm0.11\%$ (\Dnu{}),
$1.52\%\pm0.09\%$ (\numax{}),
$0.28\%\pm0.32\%$ ($M$) and
$0.08\%\pm0.14\%$ ($R$).
These numbers are plotted in Fig.~\ref{fig:limits-heb}.
\section{Discussion}
\label{sec:disc}
\subsection{Assessing uncertainties}
\label{subsec:uncertainties}
\subsubsection{The uncertainty of modelling the stellar population}
\label{subsubsec:model-dependency}
Figs.~\ref{fig:limits-rgb} and~\ref{fig:limits-heb} present the total scatter $\sigma_{\rm total}$ and
the limits on the intrinsic scatter of the scaling relations
$\sigma_{\rm SR}$ derived under various assumptions.
A feature become immediately obvious: the results depend on how the synthetic stars are modelled.
We first discuss its impact on the RGB bump.
The input physics has significantly influenced the width of the RGB bump.
As we already illustrated in Fig.~\ref{fig:rgbb-isochrone},
the shapes of the RGB bump predicted by the two isochrones are inconsistent.
Furthermore, the PARSEC models predict a wider bump than the observation,
even when the quantities were not perturbed with any scatter.
In contrast, the MIST models present a much narrower bump,
and so a much larger scatter needs to be added to match the observed width.
For some cases in Fig.~\ref{fig:limits-rgb},
$\sigma_{\rm total}$ exceeds $\sigma_{\rm SR}$,
which is also a signature that the shape of RGB bump predicted by
models cannot properly match the observation.
For example, the \gal{} synthetic samples overestimate the number
of low-mass stars near $\sim1$ $M_{\astrosun}$, which can be seen from
the panel h of Fig.~\ref{fig:diagram-rgb}.
The mismatch was first discussed by \citet{sharma++2016-population-rg-kepler},
and \citet{sharma++2019-k2-hermes-age-metallicity-thick-disc} used
a metal-rich thick disc to ease the tension, but the inconsistency still exists.
The RGB bump is an important diagnostic for stellar physics.
\citet{jcd-2015-rgb-bump} linked the width of the RGB bump with
the magnitude of the hydrogen abundance discontinuity in the vicinity of the
hydrogen-burning shell, which depends on the evolution history.
The modelling of convection (e.g. mixing-length and overshoot) can also
have an impact on the location of the bump
(see \citealt{khan-2018-rgb-bump-constraints-envelope-overshooting} and references therein).
From the above discussion, we conclude that the RGB bump is not useful for our purpose,
unless an initial calibration of stellar models is properly done.
The calibration can be achieved
by matching luminosity distributions using benchmark data,
and then the feature can be compared in the seismic diagrams.
This is beyond the scope of this paper, and we defer it for future work.
Turning into the ZAHeB edge, we noticed the shape of the edges is also model-dependent.
A noticeable feature in Fig.~\ref{fig:diagram-heb} is that the mass limit of the
helium flash in models does not match with the observation.
The mass limit has been shown to be
dependent on the treatment of overshooting \citep{girardi-2016-rc-review},
which is often considered as a free parameter in stellar modelling.
Fig.~\ref{fig:diagram-heb} also shows a lack of low-mass HeB stars in the \gal{} sample,
likely because the synthetic sample
does not incorporate enough mass loss.
Despite these model uncertainties, we found they are less sensitive to
the values of $\sigma_{\rm SR}$ that we are interested in.
This means that using the ZAHeB edge
to put a limit on the intrinsic scatter of the scaling relations
is a realistic approach in the current work.
\subsubsection{The uncertainty of identifying the features}
\label{subsubsec:identify-features}
The chosen ZAHeB edges and RGB bumps in the \kepler{} samples
might deviate from their real positions.
Here we test its influence on the inferred $\sigma_{\rm SR}$ by
shifting the locations
in the observation samples.
We perturbed the points
used to define the splines (ZAHeB edge) and the straight lines (RGB bump)
with an amount of $s/\sqrt{N}$.
We took $s$ as the standard deviation of the Gaussian profiles fitted in
Figs.~\ref{fig:hist-rgb} and~\ref{fig:hist-heb}, and $N$ as the number of samples.
This perturbation is similar to the standard deviation of the sample mean,
and should provide a good approximation to the uncertainty of choosing the center of those features.
In Fig.~\ref{fig:diagram-rgb} and~\ref{fig:diagram-heb},
the grey-shaded areas show the amount of uncertainty.
We found the resulting $\sigma_{\rm SR}$ agrees with the reported values within
$0.06\%$ for \Dnu{}, $0.1\%$ for \numax{}, $1.8\%$ for $M$, and $0.7\%$ for $R$.
This result indicates that the uncertainty of identifying the features is much smaller
than $\sigma_{\rm total}$, but is on a similar level of $\sigma_{\rm SR}$.
In addition, we note that there is a selection effect
(will be shown in section~\ref{subsec:heb-method} and Fig.~\ref{fig:mass})
due to excluding HeB stars near the ZAHeB edge
when there were no horizontal or vertical distances.
For example, the obtained values
for the mass relation are only applicable to stars in the range of
$0.8$--$1.1$ $M_{\astrosun}$, so the derived numbers for
$\sigma_{\rm total}$ and $\sigma_{\rm SR}$ are the averages
for those specific subsamples, making the numbers between
each relations not directly comparable.
\subsubsection{The uncertainty of measuring \Dnu{} and \numax{}}
\label{subsubsec:syd-pip}
The limits we obtained for $\sigma_{\rm SR}$ depend on
how well the values for \Dnu{} and \numax{} are measured.
Up to now we focused our discussion using the SYD pipeline,
which measures \Dnu{} and \numax{} from a global fitting of the
power spectrum \citep{huber++2009-syd-pipeline,yuj++2018-16000-rg}.
Although the global fitting method is more common,
an alternative approach is to only use the radial mode frequencies
and avoid the effect from mixed modes. An example is the CAN pipeline
\citep{kallinger++2010-kepler-rg-4months},
which obtained a more precise measurements on \Dnu{} and \numax{}.
For stars near the ZAHeB edge, their typical formal uncertainties are
0.6\% for \numax{}, and 0.3\% for \Dnu{} \citep{pinsonneault++2018-apokasc}.
Using their reported uncertainties,
we show in Fig.~\ref{fig:diagram-heb},
that the values for $\sigma_{\rm SR}$ in the \Dnu{} and
\numax{} relations can greatly decrease.
If the values for \Dnu{} and \numax{} are measured in this way,
the scaling relations can have much smaller intrinsic scatter in principle.
However, we also found the intrinsic scatter in the $M$ and $R$ scaling relations
does not decrease accordingly, because the uncertainty of \teff{} still dominates.
In the rest of this paper, we continue our discussion using the SYD pipeline values.
\subsection{The intrinsic scatter of the scaling relations}
\label{subsec:scatter}
Based on the discussion in section~\ref{subsec:uncertainties},
we estimate the final values of the intrinsic scatter of the scaling relations, $\sigma_{\rm SR}$,
by averaging them from both RGB and HeB stars for the $M$ and $R$ relations,
but only HeB stars for the \Dnu{} and \numax{} relations,
because these values tend to show less severe dependencies on isochrones.
We conclude that the intrinsic scatter of the scaling relations have values of
$\sim0.5\%$ (\Dnu{}), $\sim1.1\%$ (\numax{}), $\sim1.7\%$ ($M$) and $\sim0.4\%$ ($R$),
for the SYD pipeline,
keeping in mind that the systematic uncertainty of our method is on a similar level.
The values of $\sigma_{\rm SR}$ are small in general,
suggesting the observational uncertainty typically exceeds the
intrinsic scatter of the scaling relations even with 4 yr of \kepler{} data for the SYD pipeline.
In our study, we separately located the ZAHeB edges in the \kepler{} and
\galaxia{} samples.
This means that any systematic offset in the scaling relations
(for example, using a different set of solar reference values)
would not be reflected in $\sigma_{\rm SR}$.
The intrinsic scatter in the scaling relation can still be small
compared to any systematic offset in the scaling relations.
\subsection{Correcting the scaling relations with theoretical models}
\label{subsec:heb-fdnu}
\begin{figure}
\includegraphics[width=\columnwidth]{figs/fdnu}
\caption{Distributions of the correction factor \fdnu{} for stars near the RGB bump and stars near the ZAHeB edge (grey-shaded area in Fig.~\ref{fig:hist-rgb} and~\ref{fig:hist-heb}) in both \kepler{} (red) and \galaxia{} (blue) samples.}
\label{fig:fdnu}
\end{figure}
It is interesting to test whether the common model-based
correction of \Dnu{} proposed by \citet{sharma++2016-population-rg-kepler}
can reduce the scatter in the scaling relations.
We calculated the departure of the \Dnu{} scaling relation, \fdnu{}, for
each star in both samples.
We implemented the corrected mass $M'=f_{\Delta\nu}^4 M$ and
radius $R'= f_{\Delta\nu}^2 R$ in the \kepler{} sample,
and the corrected p-mode separation $\Delta\nu' = f_{\Delta\nu} \Delta\nu$
in the synthetic sample.
These \Dnu{} corrections made little difference to our results,
for both RGB stars (Fig.~\ref{fig:limits-rgb}) and HeB stars
(Fig.~\ref{fig:limits-heb}).
The likely explanation is that \fdnu{} mainly corrects the
systematic offsets in the scaling relations, which affect the
location of the RGB bump and ZAHeB edge, but has a negligible
influence on the intrinsic scatter of the scaling
relations (Fig.~\ref{fig:fdnu}).
The standard deviation of \fdnu{} for stars near the ZAHeB
edge is below $0.5\%$, and that for stars near the RGB bump
is about $1.0\%$.
\subsection{The intrinsic scatter of the scaling relations as a function of mass and metallicity}
\label{subsec:heb-mass-metal}
\begin{figure}
\includegraphics[width=\columnwidth]{figs/mass.pdf}
\caption{Distances to the ZAHeB edge as a function of stellar mass for the \yu{} sample (grey points). The solid black line traces the median values of the distances in each mass bin. The light blue show the formal uncertainties of \Dnu{} and \numax{} reported by the SYD pipeline, and the dark blue regions show the intrinsic scatter of the scaling relations $\sigma_{\rm SR}$. The dashed black lines show the total scatter $\sigma_{\rm total}$. }
\label{fig:mass}
\end{figure}
\begin{figure}
\includegraphics[width=\columnwidth]{figs/feh.pdf}
\caption{Similar to Fig.~\ref{fig:mass} but shown as a function of metallicity, and restricted to the \apk{} sample.}
\label{fig:feh}
\end{figure}
We expect the intrinsic scatter in the scaling relations to be a
function of stellar mass and metallicity, as we see in \fdnu{}.
To test whether this dependence can be seen in our sample,
we used HeB stars and divided both the \kepler{} and \galaxia{} samples into bins
with equal widths in $M$ and [M/H], and repeated the exercise
in each bin.
We note that for \Dnu{}, \numax{} and $M$, we could only test
a limited range in mass, because some points do not have vertical
or horizontal distances.
To study the dependence on [M/H], we used the \apk{}
sample instead of the \yu{} sample, because the \apk{}
metallicities were derived from a single instrument.
In Fig.~\ref{fig:mass} and~\ref{fig:feh} we show
$\sigma_{\rm total}$ (dark blue regions) and $\sigma_{\rm SR}$
(dashed lines) as functions of $M$ and [M/H], respectively.
We find no obvious change in the spread of points for \Dnu{},
\numax{} and $M$, possibly due to a direct consequence of the method uncertainty
we claimed in section~\ref{subsubsec:identify-features}.
The data also suggest that a higher mass and
higher metallicity may result in a larger intrinsic scatter for
the radius scaling relation.
Whether this is a true statement can be found by populating more stars
in the high mass and high metallicity region with upcoming space missions.
\section{Conclusions}
\label{sec:conc}
In this paper, we used a forward-modelling approach to match the width of the RGB bump and the sharpness of the edge formed by ZAHeB stars. Matching the broadening of those features between the \kepler{} and \galaxia{} samples allowed us to constrain the intrinsic scatter of the asteroseismic scaling relations.
The main results are summarised in Figs.~\ref{fig:limits-rgb} and~\ref{fig:limits-heb}. We found that the observed broadening arises primarily from the measurement uncertainties of \Dnu{} and \numax{}. By taking into account the uncertainty reported by the SYD pipeline, the scaling relations have intrinsic scatter have values of $\sim0.5\%$ (\Dnu{}), $\sim1.1\%$ (\numax{}), $\sim1.7\%$ ($M$) and $\sim0.4\%$ ($R$). This confirms the remarkable constraining power of the scaling relations. The above numbers are appoximate, bacause the systematic uncertainties of our method arising from identifying the features is on a similar level. Although this result was obtained using stars in a limited parameter space, we expect they are applicable to a broader population spanning most low-mass red giants, provided they have similar surface properties.
Moreover, we demonstrate that using the theoretically corrected \Dnu{} does not reduce the scatter by a large amount. We also found a marginal dependence of the intrinsic scatter of the radius scaling relation on mass and metallicity. However these interpretations are limited by the systematic uncertainties of our method.
Future work could include using more data from both asteroseismology and spectroscopy to allow tests in more mass and metallicity bins, especially improving the constraints for secondary clump stars. Additionally, by considering the position of those features and matching the exact distributions of stellar parameters (instead of simply the distances to the edge), one could provide constraints on physical processes such as convection and mass loss, and potentially on the offset of the scaling relations.
\section*{Acknowledgements}
We thank the \emph{Kepler} Discovery mission funded by NASA’s Science Mission Directorate for the incredible quality of data.
We acknowledge funding from the Australian Research Council, and the Joint Research Fund in Astronomy (U2031203)
under cooperative agreement between the National Natural Science
Foundation of China (NSFC) and Chinese Academy of Sciences (CAS).
This work is made possible by the following open-source Python software: {\small Numpy} \citep{numpy}, {\small Scipy} \citep{scipy}, {\small Matplotlib} \citep{matplotlib}, {\small Corner} \citep{corner}, and {\small EMCEE} \citep{emcee}.
\section*{Data Availability}
The code repository for this work is available on Github.\footnote{https://github.com/parallelpro/nike}
The data sets will be shared on request to the corresponding author.
\bibliographystyle{mnras}
|
1904.02115
|
\section{Introduction} \label{sec:intro}
The stochastic gravitational wave (GW) background has been constrained by multiple observations. The cosmic microwave background (CMB) temperature and polarization observations have provided the tightest constraints on the GW background at very low frequencies, $f\alt 10^{-16}\,\Hz$ \cite{BKP,BKX,P18:main}. The upper bound on the amplitude of the primordial GW power spectrum has been translated to that on the stochastic GW background at scales larger than the angular resolution of CMB experiments, $k\alt 0.1\,\Mpc^{-1}$, which is equivalent to $f\alt 10^{-16}\,\Hz$ in frequency. Combining CMB and LSS data, the GW background is also constrained at higher frequencies, $f=10^{-16}-10^{-11}\,\Hz$, via the upper bound on the extra radiation at the time of CMB decoupling \cite{Smith:2006prl,Smith:2006prd,Pagano:2016}. Other astrophysical observations have put constraints on the GW background at roughly the same frequency range. References.~\cite{Gwinn:1996gv} and \cite{Darling:2018} use the motion (astrometry) of quasars and radio sources, respectively, to constrain the GW energy density. Reference~\cite{Titov:2011} derives the constraint on GWs from the secular aberration of the extragalatic radio sources caused by the rotation of the Solar System barycenter around the Galactic center. The stochastic GW background at higher frequencies, $f\agt 10^{-11}\,\Hz$, is also constrained by many other direct and indirect observations such as the Pulsar Timing Array \cite{Lasky:2016:PTA}, big bang nucleosynthesis (BBN) \cite{Henrot-Versille:2014jua}, and the Laser Interferometer Gravitational-Wave Observatory (LIGO) \cite{LIGO:2017:omegagw}. The absence of the primordial black hole also leads to the upper bound on the GW background at a broad range of frequencies \cite{Nakama:2016enz}.
In this paper, we revisit the CMB constraints on the energy density of the stochastic GW background at $k\alt 10\,\Mpc^{-1}$ based on the upper bound on the amplitude of the primordial tensor power spectrum. The GW constraints by the primordial tensor power spectrum have been discussed only at the CMB scales, $k\alt 0.1\,\Mpc^{-1}$ (see e.g. \cite{LIGO:2017:omegagw}). This is because a finite angular resolution of CMB maps limits the observable range of the CMB angular multipole to $\l\alt 1000$, and the observable scale is restricted to $k\simeq \l/\chi_*\alt 0.1\,\Mpc^{-1}$ where $\chi_*\sim 10^4$ is the comoving distance to the CMB last scattering surface. The CMB spectrum is most sensitive to the primordial GWs at $k\alt 0.01$ Mpc$^{-1}$ \cite{Hiramatsu:2018nfa}. However, the CMB data can be used to constrain the stochastic GW background at small scales compared to the CMB scale, $k\agt 10$ Mpc$^{-1}$. The CMB anisotropies and lensing at large-angular scales come from the GW perturbations at low redshifts. Although such contributions are very small, we find that the upper limits on the large-scale CMB fluctuations provide tighter constraints on the GW energy density at $k=0.1-10$ Mpc$^{-1}$ than those from other existing constraints at the same scales.
\begin{figure*}[tb]
\bc
\includegraphics[width=180mm,height=54mm,clip]{fig_aps.png}
\caption{
The angular spectra of the CMB temperature, $B$-mode and lensing curl-mode with varying the central frequency, $k_c$ (solid lines). The amplitude of the power spectra is given by $A_{\rm GW}=rA_s$ where the tensor-to-scalar ratio is chosen as the current best upper bound, $r=0.06$ \cite{BKX}, and the scalar amplitude at $k=0.05$ Mpc$^{-1}$ is consistent with the latest Planck cosmology, $A_s=2.1\times 10^{-9}$. The gray dashed lines in the temperature and $B$ mode spectra show the contributions from the scalar perturbations, while that in the curl-mode spectrum shows the reconstruction noise of the Planck observation. The black dashed lines show the inflationary GW contributions with $r=0.1$.
}
\label{fig:aps}
\ec
\end{figure*}
To constrain the energy density of the stochastic GW background using CMB angular spectra, we need to assume a cosmological model for the scalar perturbations, though the degeneracy between the cosmological parameters and GW energy density would be small due to the difference of the angular scale dependence. In this respect, the GW constraints obtained by the CMB angular spectra depend on the model of the scalar perturbations. A less model-independent way to constrain the GW energy density by CMB observations is to use the curl-mode of the CMB deflection angle which has been discussed in several papers \cite{Cooray:2005hm,Sarkar:2008ii,Namikawa:2014:gwcurl,Saga:2015}. In the standard cosmology, the curl-mode is consistent with $0$ within the current measurement error of Planck. In this paper, we use the curl-mode to constrain the GW energy density as a more robust way than using the CMB angular spectra.
This paper is organized as follows. In Sec.~\ref{sec:aps}, we begin by discussing the CMB power spectra generated by small-scale GWs, and see their typical behaviors at large angular scales. Then, Section~\ref{sec:method} describes the data and our method to derive the constraints on small-scale GWs. Section~\ref{sec:results} presents our main results, i.e., the upper bound on the energy density of stochastic GWs, together with future forecast. Finally, Sec.~\ref{sec:summary} summarizes our work.
\begin{figure*}[tb]
\bc
\includegraphics[width=180mm,height=54mm,clip]{fig_aps_limber.png}
\caption{
The angular spectra of the CMB temperature, $B$ mode and lensing curl-mode from a top-hat primordial GW power spectrum with $0.01\Mpc^{-1}\leq k\leq 0.018\Mpc^{-1}$ using the Limber approximation, $\l=k\chi$. The amplitude of the power spectra is the same as that in Fig.~\ref{fig:aps}. The vertical dashed lines show $\l=0.01\chi_*\,\Mpc^{-1}$ and $\l=0.018\chi_*\,\Mpc^{-1}$. For comparison, we also show the results without the Limber approximation (blue).
}
\label{fig:aps:limber}
\ec
\end{figure*}
\section{Angular spectrum} \label{sec:aps}
\begin{figure*}[tb]
\bc
\includegraphics[width=180mm,height=54mm,clip]{fig_aps_highk.png}
\caption{
Same as Fig.~\ref{fig:aps} but for $k_c>1$ Mpc$^{-1}$.
}
\label{fig:aps:highk}
\ec
\end{figure*}
CMB experiments observe the CMB temperature, $\Theta$, and Stokes Q/U maps at each pixel on the unit sphere. The Stokes Q/U maps are decomposed into the $E$/$B$ modes by the parity symmetry \cite{Kamionkowski:1996zd,Seljak:1996gy}. We then obtain the angular spectra of the temperature and $B$ modes by squaring the harmonic coefficients of the CMB anisotropies.
The CMB primary anisotropies are distorted by gravitational lensing from the large-scale structure \cite{Lewis:2006fu,Hanson:2009kr}. The lensing distortion is described by a remapping of the CMB fluctuations at the CMB last scattering by a deflection angle, $\bm{d}$. The lensing effect leads to mode coupling between different CMB multipoles \cite{Hanson:2009gu}. This mode couplings can be used to reconstruct a map of the curl-mode of the deflection angle, $\curl=(\star\bn)\cdot\bm{d}$, from an observed CMB map (e.g., \cite{Okamoto:2002ik,Namikawa:2011:curlrec}), where $\star$ is the rotation operator for a two dimensional vector \cite{Namikawa:2011:curlrec}. We then measure the curl-mode spectrum.
Given the initial dimensionless power spectrum of GWs, $\Delta_t(k)$, the CMB and curl-mode spectra are theoretically computed, with a help of the CMB Boltzmann code, as
\al{
C_\l^{XX} &= 4\pi \Int{}{\ln k}{} \Delta_t(k) \notag \\
\times & \Int{}{\chi}{} j_\l(k\chi) S^X(k,\chi)\Int{}{\chi'}{} j_\l(k\chi')S^X(k,\chi') \, \label{Eq:Cl}
}
with $X=\Theta$, $B$ or $\curl$. In this paper, the source functions, $S^X(k,\chi)$, and angular spectra $C_\l^{XX}$ are computed, modifying CAMB \cite{Lewis:1999bs}. To obtain a generic constraint on stochastic GWs in a rather model-independent manner, we consider a monochromatic GW given at the wavenumber, $k_{\rm c}$. The dimensionless power spectrum of GW is then given by
\al{
\Delta_t(k) = \begin{cases} A_{\rm GW}/(2\epsilon) & (|k/k_{\rm c}-1|\leq \epsilon) \\ 0 & (\text{otherwise}) \end{cases}
\,.
}
Here, $A_{\rm GW}$ is the amplitude of GW, and $\epsilon$ characterizes the width of the GW spectrum in wavenumber. We divide the wavenumber between $10^{-4}$Mpc$^{-1}\leq k\leq 10$Mpc$^{-1}$ into logarithmically equal $20$ bins. We checked that our result remains unchanged even if we change the bin number to a larger value.
Figure~\ref{fig:aps} shows the angular spectra of the CMB temperature (left), $B$-mode (middle), and curl-mode (right) for various wavenumber $k_{\rm c}$. The amplitude of the power spectra is given by $A_{\rm GW}=r\,A_s$, where the tensor-to-scalar ratio $r$ is set to be the current best upper bound, $r=0.06$ \cite{BKX}, with the scalar amplitude at $k=0.05$ Mpc$^{-1}$ determined by the latest Planck cosmology, $A_s=2.1\times 10^{-9}$ \cite{P18:main}. The large-scale Fourier modes ($k_c\ll 1$ Mpc$^{-1}$) contribute to the power spectrum at large scales. Similarly, the small-scale Fourier modes ($k_c\gg 1$ Mpc$^{-1}$) mostly generate the small-scale fluctuations. The large-scale fluctuations from such small-scale Fourier modes are typically small but their contribution is not exactly $0$.
To elucidate the low-$\l$ behaviors shown in Fig.~\ref{fig:aps}, one may compare the exact calculation with the Limber approximation. In the Limber approximation, the source function $S^X$ and power spectrum $\Delta_t$ in the integrand of \eq{Eq:Cl} are assumed to be a smooth function of $k$. Then, the integral convolving spherical Bessel function over $k$ exhibits a heavy cancellation, which results in a rather simplified form of the angular spectrum (see e.g. \cite{Loverde:2008,Lewis:2006fu});
\al{
C_\l^{XX} &\simeq \frac{2\pi^2}{\l^3}\Int{}{\chi}{}\chi\Delta_t\Bigl(k=\frac{\l}{\chi}\Bigr)\,\left[S^X\Bigl(k=\frac{\l}{\chi},\chi\Bigr)\right]^2
\, \label{Eq:Cl-Limber}
}
which tells us that the amplitude of $C_\l^{XX}$ is determined by the contribution of GWs at the wavenumber $k=\ell/\chi$ projected along the line-of-sight (i.e., $\chi$-integral). Strictly speaking, the above equation is inadequate in our case because the integrand contains the top-hat primordial power spectrum. Further, the tensor transfer function has oscillatory behaviors, which, combining with spherical Bessel function, leads to a rather nontrivial cancellation. Nevertheless, Eq.~\eqref{Eq:Cl-Limber} can be used for a qualitative understanding of the angular spectrum.
In Fig.~\ref{fig:aps:limber}, we specifically set the top-hat GW spectrum to the one centered at $k_{\rm c}=0.014\,\Mpc^{-1}$ with the width $\Delta k=0.008\,\Mpc^{-1}$, and plot the angular spectra with and without the Limber approximation. Then, in all cases, the results with Limber approximation exhibit a sharply peaked structure around $\ell_*\equiv k_{\rm c}\chi_*$, indicated by the two vertical dashed lines, where $\chi_*$ is the comoving distance to the last scattering surface of CMB. At higher multipoles of $\l>\l_*$, the amplitudes rapidly falls off, consistent with exact calculations. These behaviors basically come from the nature of photon radiative transfer encapsulated in the function $S^{\rm X}$. On the other hand, looking at the lower multipoles of $\l_*<\l$, while the Limber approximation predicts a rather suppressed $B$-mode spectrum that fails to reproduce the exact calculation, the predictions of the temperature and curl-mode spectra show a rather long tail, which qualitatively explains the behaviors in the exact calculations. Recall that in the Limber approximation, the top-hat GWs peaked at $k_{\rm c}$ can contribute to $C_\l$ only at the multipole $\l=k_{\rm c}\,\chi$, Fig.~\ref{fig:aps:limber} implies that the low-$\l$ behaviors of the exact calculation in the temperature and curl-mode spectrum mainly comes from the low-$z$ GW contributions (i.e., $\chi\lesssim\chi_*$), whereas a non-negligible amount of the high-$z$ GWs plays a role to determine the low-$\l$ amplitude of the $B$-mode spectrum.
Having confirmed the typical behaviors of the angular spectra, we further consider the small-scale GWs, and plot in Fig.~\ref{fig:aps:highk} the angular spectra for $k_{\rm c}>1$ Mpc$^{-1}$. The results are compared with the contributions from the scalar perturbations or the reconstruction noise. As we have seen in Figs.~\ref{fig:aps} and \ref{fig:aps:limber}, no appreciable low-$\l$ tail is developed for the $B$-mode spectrum, since the polarization is only generated at the reionization and recombination. Figure~\ref{fig:aps:highk} suggests that the $B$-mode constraint on the small-scale GWs is basically limited by the angular resolution of CMB observations. On the other hand, the temperature and curl-mode power spectra exhibit a low-$\l$ tail that is more prominent and is rather enhanced compared to the results in Fig.~\ref{fig:aps}. This implies that large angle CMB data can still give a meaningful constraint on small-scale GWs.
\section{Data and Method} \label{sec:method}
Here, we explain the data and the analysis method to constrain the stochastic GWs, particularly paying attention at small scales. In our analysis, we use the CMB temperature spectrum, $C_\l^{\Theta\T}$, measured by Planck \cite{P18:main}, the $B$-mode spectrum, $C_\l^{BB}$, by BICEP/Keck Array \cite{BKX}, and curl-mode spectrum, $C_L^{\curl\curl}$, by Planck \cite{P13:phi}. Note that the constraints from the temperature-$E$ cross and $E$-mode autospectra measured by Planck turn out to be statistically insignificant compared to that obtained by the temperature spectrum. In this paper, therefore, we only present the results from the temperature, $B$-mode and curl-mode spectra. We checked that our constraints remain unchanged even if we add other $B$-mode spectra measured by POLARBEAR \cite{PB17:BB} and SPTpol \cite{Keisler:2015hfa}.
Provided the data, the constraint on the amplitude of stochastic GW, $A_{\rm GW}$, assuming the monochromatic wave with wavenumber $k_{\rm c}$, is obtained by minimizing the likelihood function $\mC{L}$. We adopt here the Gaussian likelihood function,
\al{
-2\ln \mC{L} (A_{\rm GW}) = \sum_{b=1}^n \frac{[\hC^{XY}_b-C^{XY,{\rm fid}}_b(A_{\rm GW})]^2}{(\sigma^{XY}_b)^2}
\,, \label{Eq:likelihood}
}
where the subscript $b$ indicates the index of the multipole bins, and $n$ is the number of the multipole bins. The label $XY$ implies $\Theta\T$, $BB$ or $\curl\curl$. The power spectrum $\hC_b^{XY}$ is the measured binned spectrum, and $C_b^{XY,{\rm fid}}(A_{\rm GW})$ is the theoretical prediction having a specific GW amplitude $A_{\rm GW}$. Finally, $\sigma_b^{XY}$ is the measurement error of the angular spectrum provided by the CMB experiments described above. The multipole ranges used in the likelihood analysis are $2\leq\l\leq 2508$ for temperature, $37\leq\l\leq 332$ for $B$-mode, and $2\leq\l\leq2020$ for curl-mode spectra, respectively, at which the data are validated.
The observed temperature spectrum has contributions from both the scalar and tensor perturbations. In the $B$-mode spectrum, the gravitational lensing effect generates the $B$ mode even if there is only the scalar perturbation \cite{Zaldarriaga:1998ar}. Thus, we simultaneously need to model or constrain the non-GW contributions to constrain GWs in the angular spectra. In our analysis, we simply subtract the contributions of the scalar perturbations from the measured spectrum, assuming the Planck best-fit $\Lambda$CDM model, since the degeneracy between the GW amplitude and cosmological parameters would be small and does not increase the upper bound by more than an order of magnitude.
The upper bound on $A_{\rm GW}$ obtained from the above analysis is then translated to the GW fractional energy density defined as
\al{
\Omega_{\rm GW}(k) &\equiv \frac{1}{\rho\rom{c}}\D{\rho_{\rm GW}}{\ln k}
\biggl|_{\eta =\eta_0}
= \frac{\Delta_t(k)}{12H_0^2}
\left(\PD{T(k,\eta_0)}{\eta}\right)^2
\notag \\
&= \frac{(A_{\rm GW}/2\epsilon)}{12H_0^2}\left(\PD{T(k,\eta_0)}{\eta}\right)^2
\,,\label{eq:Omega_GW}
}
at $|k/k_{\rm c}-1|\leq \epsilon$ and $0$ otherwise, where $\rho\rom{c}$ is the critical density of the Universe, and $H_0$ is the Hubble parameter today. The time derivative of the GW transfer function is computed by CAMB assuming no neutrino anisotropic stress \cite{Lewis:1999bs}.
\section{Results} \label{sec:results}
\begin{figure}[tb]
\bc
\includegraphics[width=90mm,height=63mm,clip]{fig_omegagw.png}
\caption{
The current $95$\% upper bounds on the amplitude of the stochastic GW background at $k<10$ Mpc$^{-1}$ by the CMB observations. For reference, we show the energy density of the scale-invariant primordial GW with $r=0.06$ as a function of $k$ (black solid). The dashed lines are the expected upper bounds by a future CMB observation. The vertical dot-dashed line roughly corresponds to the maximum observable wavenumber determined by the resolution of CMB experiments.
}
\label{fig:omegagw:cmb}
\ec
\end{figure}
Figure~\ref{fig:omegagw:cmb} shows the $95$\% C.L. upper bounds on the GW amplitudes at each scale using the current best measurements of the CMB angular spectra. At $k=0.01$ Mpc$^{-1}$, the best constraints come from the $B$-mode spectrum measured by BICEP2/Keck Array. At smaller scales, $k\agt0.1$ Mpc$^{-1}$, however, the temperature spectrum obtained by Planck provides the constraints on the GW amplitudes comparable to that from the $B$-mode spectrum. At large scales, the constraint by the $B$-mode spectrum becomes very weak because the $B$-mode spectrum is not measured at $\l\alt 30$.
Figure~\ref{fig:omegagw:cmb} also plots the expected constraints on the GW spectrum amplitude by future CMB observations. In this forecast, we assume an idealistic future CMB experiment, i.e., a full sky observation, $1\mu$K-arcmin white noise with $1$ arcmin Gaussian beam. We also assume no foregrounds and $90$\% of the lensing $B$ mode is removed. Thus, the results provide the expected constraints in the most optimistic case. In the future, the constraints from the temperature spectrum are not significantly improved since the current temperature measurement is already dominated by the scalar perturbations at $\l<2000$. On the other hand, the $B$ mode and curl-mode are still dominated by the instrumental and reconstruction noise, respectively. The constraints are improved if the noise contribution is reduced in the future. The tightest constraints are obtained from the measurement of the $B$-mode spectrum.
\section{Summary and discussion} \label{sec:summary}
\begin{figure}[tb]
\bc
\includegraphics[width=90mm,height=63mm,clip]{fig_gwconst.png}
\caption{
Summary of the current status of the upper bounds on the energy density of the stochastic GW background. The thick red and blue shaded regions correspond to the constraints obtained in this paper using the CMB angular spectra and curl-mode of the gravitational lensing, respectively. We also show other constraints from the astrometry of radio sources (Astrometry) \cite{Darling:2018}, extra radiation before CMB decoupling (CMB+LSS+BBN) \cite{Pagano:2016,Kohri:2018awv}, Pulsar Timing (Pulsars) \cite{Lasky:2016:PTA}, and GW interferometers (aLIGO O1)\cite{LIGO:2017:omegagw}.
}
\label{fig:omegagw:all}
\ec
\end{figure}
In this paper, using the currently available CMB data, we presented robust constraints on stochastic GWs at Mpc scales, which are far below the resolution limit of CMB measurements. The key point is that even the short-wavelength GWs can still give an impact on the power spectrum of CMB anisotropies at large angular scales, to which we cannot apply the Limber approximation due to the heavily oscillatory behaviors of GWs. Using the available CMB data of temperature, $B$-mode polarization, and lensing from Planck and BICEP/Keck array, we find that currently both the temperature and $B$-mode data put the constraint almost at the same level at $k\gtrsim0.1$\,Mpc$^{-1}$, summarized as
\al{
\Omega_{\rm GW}\lesssim 10^{-12}\left(\frac{k}{0.1\,\mbox{Mpc}^{-1}}\right)^3 \,, \label{Eq:const}
}
which is compared with other constraints shown in Fig.~\ref{fig:omegagw:all}. Note that \eq{Eq:const} is, strictly speaking, valid at $k\lesssim10$\,Mpc$^{-1}$, at which we stop constraining GW due to the heavy oscillations of the spherical Bessel function and source function in the integrand of \eq{Eq:Cl}. However, it is expected from the behaviors seen in Fig.~\ref{fig:aps:highk} that the scaling given at \eq{Eq:const} still holds at $k\gtrsim10$\,Mpc$^{-1}$, thus giving a meaningful constraint complementary to those coming from astrometric observation and thermal history of the Universe.
In the future, a tighter constraint will be obtained from the $B$-mode polarization, and the expected constraint will be improved by several orders of magnitude. Note cautiously that the constraints coming from the temperature and $B$-mode data are model dependent in the sense that the constraints are obtained from the data subtracting the primary (temperature) or secondary ($B$-mode) signal created by the scalar perturbations, for which we assume $\Lambda$CDM model. In this respect, the curl-mode data give complementary information to the temperature and B-mode polarization, although the constraining power is significantly degraded.
Finally, the constraints obtained here are useful in narrowing the parameters of several models that can produce the GWs at Mpc scales. The early universe scenarios producing a blue-tilted stochastic GW are obviously interesting targets to constrain (e.g., \cite{Fujita:2018ehq}). Apart from these, the constraint on cosmic string models is also worth consideration. The stochastic GW spectrum produced by the cosmic strings depends significantly on the model parameters and assumptions as shown in Refs.~\cite{Sanidas:2012ee,Ringeval:2017eww,Blanco-Pillado:2017oxo}, and thus constraining the GWs over a very broad frequency range is important to pin down the cosmic string models. Another possible source of GWs at Mpc scales is the ultralight axion motivated by string theory. Reference ~\cite{Kitajima:2018zco} discusses the possibility that ultralight axion produces a sizable amount of GWs through the nonlinear scalar field interaction induced by parametric resonance, and the produced GWs can have a peak with broad distribution at Mpc scales if the axion mass is around $m\sim10^{-22}$\,eV. This mass scale lies at the scales for which the axion can put an interesting imprint on the small-scale structure \cite{Hu:2000:axion,Marsh:2013,Schive:2014dra,Schwabe:2016rze,Hui:2016:axion}. Several works also discuss generation of GWs at Mpc scales assuming that the primordial GW power spectrum has a sharp peak \cite{Saito:2008:pbh,Kohri:2018awv}. The constraint on stochastic GWs obtained in this paper certainly helps to exclude a part of parameter regions that can produce a large GW amplitude, although a detailed analysis is still necessary to get a constraint, using properly the shape of the predicted GW spectrum.
\begin{acknowledgments}
We thank Kazuyuki Akitsu, Kazunori Kohri, Toshiki Kurita, Blake Sherwin and Alexander van Engelen for helpful comments. T. N. acknowledges the support from the Ministry of Science and Technology (MOST), Taiwan, R.O.C. through the MOST research project grants (Grant No. 107-2112-M-002-002-MY3). This research used resources of the National Energy Research Scientific Computing Center, which is supported by the Office of Science of the U.S. Department of Energy under Contract No. DE-AC02-05CH11231. This work was supported in part by MEXT/JSPS KAKENHI Grants No. JP15H05899, No. JP17H06359 and No. JP16H03977 (A. T.), Grant No. 17K14304 (D. Y.), and Grant-in-Aid for JSPS fellows Grant No. 17J10553 (S. S.).
\end{acknowledgments}
|
1510.00012
|
\section{Introduction}
The discrete distribution, or discrete probability measure, is a well-adopted and succinct
way to summarize a batch of data. It often serves as a descriptor for
complex instances encountered in machine learning, {\it e.g.}, images,
sequences, and documents, where each instance itself is converted to a data collection instead of a vector.
In a variety of research areas including multimedia
retrieval and document analysis, the celebrated bag of
``words'' data model is intrinsically a discrete distribution.
The widely used normalized histogram is
a special case of discrete distributions with a fixed set of support points across the instances. In many applications involving moderate or high dimensions, the number of bins in the histograms is enormous because of the diversity across instances, while the histogram for any individual instance is highly sparse. This is frequently observed for
collections of images and text documents. The discrete distribution can function
as a sparse representation for the histogram, where support points and their
probabilities are specified in pairs, eliminating the
restriction of a fixed quantization codebook across instances.
The goal of this paper is to develop computationally efficient algorithms
for clustering discrete distributions under the Wasserstein distance.
The Wasserstein distance is a true metric for measures\cite{rachev1984}
and can be traced back to the mass transport problem proposed by Monge
in the 1780s and the relaxed formulation by Kantorovich in the 1940s~\cite{villani2003topics}.
Mallows\cite{mallows1972} used this distance to prove some
theoretical results in statistics, and thus the name Mallows
distance has been used by statisticians.
In computer science, it is better known as the
Earth Mover's Distance \cite{rubner2000earth}. In signal processing,
it is closely related to the Optimal Sub-Pattern Assignment (OSPA)
distance~\cite{schuhmacher2008consistent} used recently for
multi-target tracking~\cite{baum2015wasserstein}.
A more complete account on the history of this distance can be found in \cite{villani2008optimal}.
The Wasserstein distance is computationally expensive
because it has no closed form solution and
its worst time complexity is at least $O(m^3\log m)$ subject to
the number of support points in the distribution~\cite{orlin1993faster}.
We adopt the well-accepted criterion of minimizing the total within-cluster
variation under the Wasserstein distance, similarly as Lloyd's
K-means for vectors under the Euclidean distance. This clustering problem was originally explored by Li and Wang \cite{li2008real}, who coined the phrase {\em D2-clustering},
referring to both the optimization problem and their particular algorithm.
Motivations for using the Wasserstein
distance in practice are strongly argued by researchers (e.g. ~\cite{rubner2000earth,li2008real,pele2009fast,cuturi2013sinkhorn,cuturi2014fast})
and its theoretical significance in the optimal transport (OT) literature
(see the seminal books of Villani \cite{villani2008optimal,villani2003topics}).
D2-clustering computes a principled centroid to summarize each cluster of data.
The centroids computed from those clusters---also represented by discrete
distributions---are highly valuable for subsequent learning or data mining tasks.
Although D2-clustering holds much promise because of the advantages of the Wasserstein distance
and its kinship with K-means, the high computational complexity
has limited its applications in large-scale learning.
\subsection{Our Contributions}
We have developed an efficient and robust method for
optimizing the centroids in D2-clustering based on a modified version of the Bregman ADMM (B-ADMM).
The data setting in D2-clustering has subtle differences from those taken by the previous OT literature on computing barycenters.
As a result, the modified B-ADMM approach is validated to be currently the
most suitable (yet approximate) method for D2-clustering based on
comparisons with the other Wasserstein barycenter approaches adopted into the clustering process:
subgradient based method, ADMM, and entropy regularization.
The properties of the modified B-ADMM approach are thoroughly examined via experiments.
The new method called {\em AD2-clustering} is capable of handling a large number of instances if the support size of each instance is no more than several hundreds.
We have also developed a parallel algorithm for the modified B-ADMM method in
a multi-core environment with adequate scaling efficiency subject to hundreds of CPUs.
Finally, we tested our new algorithm on several large-scale real-world datasets,
revealing the high competitiveness of AD2-clustering.
\subsection{Related Work}
\noindent\textbf{Clustering distributions:}
Distribution clustering can be done subject to different affinity definitions. For example,
Bregman clustering pursues the minimal distortion between the cluster prototype---called the Bregman representative---and cluster members
according to a certain Bregman divergence~\cite{banerjee2005clustering}.
In comparison, D2-clustering is an extension of K-means to discrete distributions
under the Wasserstein distance~\cite{li2008real}, and the cluster prototype
is an approximate Wasserstein barycenter with sparse support.
In the D2-clustering framework, solving the cluster prototype or the centroid for discrete distributions under the
Wasserstein distance is computationally challenging \cite{cuturi2014fast,ye2014scaling,zhang2015parallel}.
In order to scale up the computation of D2-clustering,
a divide-and-conquer approach has been proposed~\cite{zhang2015parallel},
but the method is ad-hoc from optimization perspective.
A standard ADMM approach has also been explored~\cite{ye2014scaling},
but its efficiency is still inadequate for large datasets.
Although fast computation of Wasserstein distances has been much explored \cite{pele2009fast,cuturi2013sinkhorn,wang2014bregman}, how to perform top-down
clustering efficiently based on the distance has not.
We present below the related work and discuss their relationships with our current work. \\
\noindent\textbf{The true discrete Wasserstein barycenter and two approximation strategies:}
The centroid of a collection of distributions minimizing the average $p$th-order power of the $L_p$ Wasserstein distance is called Wasserstein barycenter~\cite{agueh2011barycenters}. In the D2-clustering algorithm~\cite{li2008real}, the 2nd order Wasserstein barycenter is simply referred to as a prototype or centroid; and is solved for the case of an unknown support with a pre-given cardinality.
The existence, uniqueness, regularity and other properties of the 2nd order Wasserstein barycenter have been established mathematically for continuous measures in the Euclidean space~\cite{agueh2011barycenters}. But the situation is more intricate
for discrete distributions, as will be explained later.
Given $N$ arbitrary discrete distributions each with $\bar{m}$ support points,
their true Wasserstein barycenter in theory can be solved via linear programming~\cite{agueh2011barycenters,anderes2015discrete}.
This is because the support points of the Wasserstein barycenter can only locate at a finite (yet huge) number of possible positions. However,
solving the true discrete barycenter quickly becomes intractable
even for a rather small number of distributions containing only 10 support points each.
An important theoretical progress has been made by
Anderes et al.~\cite{anderes2015discrete} who proved that
the actual support of a true barycenter of $N$ such distributions is extremely sparse, with cardinality $m$ no greater than $\bar{m}N$. However, the complexity of the problem is not reduced practically because so far there is no theoretically ensured way to sift out the optimal sparse locations. On the other hand, the new theory seems to backup the practice of assuming a pre-selected number of support
points in a barycenter as an approximation to the true solution.
To achieve good approximation, there are two computational strategies one can adopt in an optimization framework.
\begin{enumerate}
\item[(i)] Carefully select beforehand a large and representative set of support points as an approximation to the support of the true barycenter (e.g. by K-means).
\item[(ii)] Allow the support points in a barycenter to adjust positions at
every $\tau$ iterations.
\end{enumerate}
The first strategy of fixing the support of a barycenter can yield
adequate approximation quality in low dimensions (e.g. 1D/2D histogram data)~\cite{cuturi2014fast,benamou2014iterative}, but can face the challenge of exponentially growing support size when the dimension increases.
The second strategy allows one to use a possibly much smaller
number of support points in a barycenter to achieve the same
level of accuracy~\cite{li2008real,zhang2015parallel,ye2014scaling,cuturi2014fast}.
Because the time complexity per iteration of existing iterative methods is $O(\bar{m}m N)$,
a smaller $m$ can also save substantial computational time so as to trade in with the
extra amount of time $O(\bar{m} m d N / \tau)$ to recalculate the distance matrices.
In the extreme case when the barycenter support size is set to one ($m=1$),
D2-clustering reduces to K-means on the distribution means, which is a meaningful way of data reduction in its own right. Our experiments indicate that in practice
a large $m$ in D2-clustering is usually unnecessary (See section~\ref{sec:expr} for related discussions).
In applications on high-dimensional data, it is desirable to optimize the support points rather than fix them from the beginning. This however leads to a non-convex optimization problem. Our work aims at developing practical numerical methods. In particular, the method optimizes jointly the locations and weights of the support points in a single loop without resorting to a bi-level optimization reformulation, as was done in earlier work~\cite{li2008real,cuturi2014fast}.\\
\noindent\textbf{Solving discrete Wasserstein barycenter in different data settings:}
Recently, a series of works have been devoted to solving
the Wasserstein barycenter given a set of distributions (e.g. ~\cite{carlier2014numerical,cuturi2014fast,benamou2014iterative,ye2014scaling,cuturi2015smoothed}). How our method compares with the existing ones depends strongly on the specific data setting. We discuss the comparisons in details below and
motivate the use of our new method in D2-clustering.
In \cite{benamou2014iterative,cuturi2014fast,cuturi2013sinkhorn}, novel algorithms have been developed for solving the Wasserstein barycenter by adding an entropy regularization term on the optimal transport matching weights. The regularization is imposed on the transport weights, but not the barycenter distribution.
In particular, iterative Bregman projection (IBP)~\cite{benamou2014iterative}
can solve an approximation to the Wasserstein barycenter.
IBP is highly memory efficient for distributions with a shared support set (e.g. histogram data), with a memory complexity $O((m + \bar{m})N)$.
In comparison, our modified B-ADMM approach is of the same time complexity, but requires $O(m\bar{m}N)$ memory. If $N$ is large, memory constraints can limit our approach
to problems with relatively small $m$ or $\bar{m}$. Consequently, the second approximation strategy is crucial for reaching high accuracy by our approach, while the first strategy may not meet the memory constraint.
In the conventional OT literature, the emphasis is on computing the Wasserstein barycenter for a small number of
instances with dense representations
(e.g.~\cite{solomon2015convolutional,rabin2011wasserstein}); and IBP is more suitable.
But in many machine learning and signal processing applications, each instance is
represented by a discrete distribution with a sparse support set (aka, $\bar{m}$ is small).
The memory limitation of B-ADMM can be avoided via parallelization until the time budget is spent. Our focus is thus to achieve scalability in $N$. B-ADMM has important advantages over IBP that motivate the usage of the former in practice.
First of all, it is interesting to note that if the
distributions do not share the support set,
IBP~\cite{benamou2014iterative} has the same memory complexity
$O(m\bar{m}N)$ (for caching the distance matrix per instance) as our approach.
In addition, B-ADMM~\cite{wang2014bregman}, based on which our approach is developed, has the following advantages:
(1) It yields the exact OT and distance
in the limit of iterations. Note that the ADMM parameter
does not trade off the convergence rate. (2) It exhibits different convergence behaviors,
accommodating warm starts and early stops (to be illustrated later),
valuable traits for the task of D2-clustering. (3) It works well with single-precision floats, thus not pestered by the machine precision constraint. In contrast, this issue is serious for IBP, preventing it from
robustly outputting adequately accurate discrete
Wasserstein barycenters with sparse support (See~\cite{benamou2014iterative} and our experiments).%
\footnote{Here the ``adequately accurate'' means close to a local
minimizer of sum of the (squared) Wasserstein distances.}\\
\noindent\textbf{Optimization methods revisited:}
Our main algorithm is inspired by the B-ADMM algorithm of
Wang and Banerjee~\cite{wang2014bregman} for solving OT fast.
They developed the two-block version of
ADMM~\cite{boyd2011distributed} along with Bregman divergence to solve the OT problem
when the number of support points is extremely large.
Its algorithmic relation to IBP~\cite{benamou2014iterative} is discussed in Section~\ref{sec:compare}.
The OT problem at a moderate scale can in fact be efficiently handled
by state-of-the-art LP solvers~\cite{tang2013earth}.
As demonstrated by the line of work on solving the barycenter,
optimizing the Wasserstein barycenter is rather different from computing the distance.
Our modified B-ADMM algorithm is for solving the Wasserstein barycenter problem.
Naively adapting the B-ADMM to Wasserstein barycenter does not result in a proper algorithm.
The modification we made on B-ADMM is necessary.
Although the modified B-ADMM approach is not guaranteed to
converge to a local optimum, it often
yields a solution very close to the local optimum.
The new method is shown empirically to achieve higher accuracy than
IBP or its derivatives.
Finally, we note that although solving a single barycenter for a fixed set is
a key component in D2-clustering, the task of clustering per se bears some extra technical subtleties.
In a clustering setup, the partition of samples varies over the iterations,
and a sequence of Wasserstein barycenters are solved.
We found that the robustness with respect to the hyper-parameters in the optimization
algorithms is as important as the speed of solving one centroid because
it is impractical to tune these parameters over many iterations of different partitions. \\
\noindent\textbf{Outline:} The rest of the paper is organized as follows.
In Section~\ref{sec:d2_pre}, we define notations and provide preliminaries on
D2-clustering. In Section~\ref{sec:centroid}, we develop two scalable optimization approaches for the
Wasserstein barycenter problem and explain how they are embedded in the overall algorithm for D2-clustering.
In Section~\ref{sec:compare}, our main algorithm is compared with the IBP approach.
In Section~\ref{sec:impl}, several implementation issues including initialization, additional techniques for speed-up, and parallelization, are addressed. Comparisons of the algorithms in complexity/performance and guidelines for usage in practice are provided in Section~\ref{sec:compl}.
Section~\ref{sec:expr} reports experimental results. The numerical properties and computing performance of the algorithms are investigated; and clustering results on multiple datasets from different domains are compared with those obtained by widely used methods in the respective areas.
Finally, we conclude in Section~\ref{sec:concl} and discuss the limitations of the current work.
\section{Discrete Distribution Clustering}\label{sec:d2_pre}
Consider discrete distributions with finite support specified by a set of
support points and their associated probabilities, aka weights:
$$\{ ( w_{1} ,x_{1} ) , \ldots , ( w_{m} ,x_{m} ) \},$$
where $\sum_{i=1}^{m} w_{i} =1$ with $w_{i} \geqslant 0$, and $x_{i} \in
\mathbbm{M}$ for $i=1, \ldots ,m$.
Usually, $\mathbbm{M}=\mathbbm{R}^{d}$ is the
$d$-dimensional Euclidean space with the $L_p$ norm, and $x_i$'s are also called support vectors.
$\mathbbm{M}$ can also be a symbolic set provided with
symbol-to-symbol dissimilarity.
The Wasserstein distance between distributions $P^{( a )}=\{(w_i^{(a)},
x_i^{(a)}), i=1, ..., m_a\}$ and $P^{( b )}=\{(w_i^{(b)}, x_i^{(b)}),
i=1, ..., m_b\}$ is solved by the following linear programming (LP). For notation brevity, let $c(x_i^{(a)}, x_j^{(b)})=\|x_i^{(a)}-x_j^{(b)}\|_p^p$.
Define index set $\mathcal{I}_a=\{1, ..., m_a\}$ and $\mathcal{I}_b$
likewise. We define $\left(W_p( P^{( a )} ,P^{( b )})\right)^{p}:=$
\begin{equation}
\begin{array}{rl}
\min\limits_{\{\pi_{i,j} \geqslant 0\}} &
\sum\limits_{i\in\mathcal{I}_a,j\in\mathcal{I}_b} \pi_{i,j} c ( x_{i}^{( a )} ,x_{j}^{( b )} )\; , \\
\textrm{s.t.}\;\; & \sum_{i=1}^{m_a} \pi_{i,j} =w_{j}^{( b )} ,\; \forall
j\in\mathcal{I}_b\; \\ & \sum_{j=1}^{m_b}
\pi_{i,j} =w_{i}^{( a )} ,\; \forall i\in\mathcal{I}_a\;. \label{eq:primal}
\end{array}
\end{equation}
We call $\{\pi_{i,j}\}$ the {\em matching weights} between support points
$x_i^{(a)}$ and $x_j^{(b)}$ or the {\em optimal coupling} for $P^{(a)}$ and $P^{(b)}$. In D2-clustering, we use the $L_2$ Wasserstein distance. From now on, we will denote $W_2$ simply by $W$.
\begin{algorithm}[htp]
\caption{D2 Clustering~\cite{li2008real}}\label{alg:clustering}
\begin{algorithmic}[1]
\Procedure{D2Clustering}{$\{ P^{( k )} \}_{k=1}^{M}$, $K$}
\State Denote the label of each objects by $l^{( k )}$.
\State Initialize $K$ random centroid $\{ Q^{( i )} \}_{i=1}^{K}$.
\Repeat
\For{$k=1, \ldots ,M$}
\Comment{Assignment Step}
\State $l^{( k )} := \tmop{argmin}_{i} W ( Q^{( i )},P^{( k )} )$;
\EndFor
\For{$i=1, \ldots ,K$}
\Comment{Update Step}
\State$Q^{( i )} := \tmop{argmin}_{Q} \sum_{l^{( k )}
=i} W ( Q,P^{( k )} )$ (*)
\EndFor
\Until{the number of changes of $\{ l^{( k )} \}$ meets some stopping criterion}
\State \Return $\{ l^{( k )} \}_{k=1}^{M}$ and $\{ Q^{( i )} \}_{i=1}^{K}$.
\EndProcedure
\end{algorithmic}
\end{algorithm}
Consider a set of discrete distributions $\{P^{(k)}, k=1, ..., \bar{N}\}$,
where $P^{(k)}=\{(w_i^{(k)}, x_i^{(k)}), i=1, ..., m_k\}$. The goal of
D2-clustering is to find a set of centroid distributions $\{Q^{(i)}, i=1,
..., K\}$ such that the total within-cluster variation is minimized:
\[\displaystyle
\min_{\{Q^{(i)}\}}\sum_{k=1}^{\bar{N}}\min_{i=1, ..., K}W^2(Q^{(i)}, P^{(k)})\;
.\]
Similarly as in K-means, D2-clustering alternates the optimization of the
centroids $\{Q^{(i)}\}$ and the assignment of each instance to the nearest centroid, the iteration referred to as the {\em outer loop} (Algorithm~\ref{alg:clustering}).
The major computational challenge in the algorithm is to compute the optimal
centroid for each cluster. This also marks the main difference between
D2-clustering and K-means in which the optimal centroid is in a simple closed
form. The new scalable algorithms we develop here aim primarily at speeding
up this optimization step. For the clarity of presentation, we now focus
on this optimization problem and describe the notation below.
Suppose we have a set of discrete distributions
$\{ P^{( 1 )} , \ldots ,P^{( N )} \}$. $N$ is the sample size for computing one Wasserstein barycenter. We want to find a centroid $P^{} : \{
( w_{1} ,x_{1} ) , \ldots , ( w_{m} ,x_{m} ) \}$, such that
\begin{equation}
\min_{P} \frac{1}{N} \sum_{k=1}^{N} W^2 ( P,P^{( k )} ) \label{eq:centroid}
\end{equation}
with respect to the weights and support points of $P$. This is the
\textbf{main question} we tackle in this paper. There is an implicit layer of
optimization in {\eqref{eq:centroid}}---the computation of $W^2 (
P,P^{( k )} )$. The variables in optimization {\eqref{eq:centroid}} thus
include the weights in the centroid $\{ w_{i} \in \mathbbm{R}^{+} \}$, the
support points $\{ x_{i} \in \mathbbm{R}^{d} \}$, and the optimal coupling
between $P$ and $P^{( k )}$ for each $k$, denoted by $\{ \pi_{i,j}^{( k )} \}$ (see
Eq. {\eqref{eq:primal}}).
To solve {\eqref{eq:centroid}}, D2-clustering alternates the optimization of
$\{ w_{i} \}$ and $\{ \pi_{i,j}^{( k )}
\}$, $k=1, ..., N$, versus $\{ x_{i} \}$.
\begin{enumerate}
\item
$\Delta_k$ denotes a probability simplex of $k$ dimensions.
\item
$\mathbf{1}$ denotes a vector with all elements equal to one.
\item
$\tmmathbf{x} = ( x_{1} , \ldots ,x_{m} )
\in \mathbbm{R}_{d \times m}$,
$\tmmathbf{w} = ( w_{1} , \ldots ,w_{m} ) \in \Delta_{m}$.
\item
$\tmmathbf{x}^{(k)} = ( x^{(k)}_{1} , \ldots ,x^{(k)}_{m_k} )
\in \mathbbm{R}_{d \times m_k}$, $k=1, ..., N$.
\item
$\tmmathbf{w}^{(k)} = (w_1^{(k)},\ldots,w_{m_k}^{(k)})\in \Delta_{m_k}$
\item
$C(\tmmathbf{x}, \tmmathbf{x}^{(k)} ) = (\|x_i - x_j^{(k)}\|^2)_{i,j}\in \mathbb{R}_{m\times m_k}$
\item
$\tmmathbf{X}= ( \tmmathbf{x}^{( 1 )} , \ldots , \tmmathbf{x}^{( N
)} ) \in \mathbbm{R}_{d \times n}$, where $n=\sum\limits_{k=1}^{N} m_k$.
\item
$\Pi^{(k)} = ( \pi_{i,j}^{(k)} ) \in \mathbbm{R}_{m\times m_k}^{+}$, $k=1, ..., N$.
\item
$\Pi = ( {\Pi}^{( 1 )},\ldots , {\Pi}^{(N)} ) \in \mathbbm{R}^+_{m \times n}$.
\item
Index set $\mathcal{I}^{c}=\{1, ..., N\}$, $\mathcal{I}_k=\{1, ...,
m_k\}$, for\break $k\in \mathcal{I}^{c}$, and $\mathcal{I}'=\{1, ..., m\}$.
\end{enumerate}
With $\tmmathbf{w}$ and $\Pi$ fixed, the
cost function {\eqref{eq:centroid}}
is quadratic in terms of $\tmmathbf{x}$,
and the optimal $\tmmathbf{x}$ is solved by:
\begin{equation}
x_{i} := \frac{1}{N w_{i}} \sum_{k=1}^{N} \sum_{j=1}^{m_k}
\pi_{i,j}^{( k )} x_{j}^{( k )} , \;\; i\in\mathcal{I}' \, ,\label{eq:updatex}
\end{equation}
or we can write it in matrix form: $\tmmathbf{x} := \frac{1}{N} X \Pi^{T}
\tmop{diag} ( 1./\tmmathbf{w} )$.
However, with fixed $\tmmathbf{x}$, updating $\tmmathbf{w}$ and $\Pi$ is
challenging. D2-clustering solves a large LP
as follows:
\begin{eqnarray}
&\min\limits_{\Pi \in \mathbbm{R}^+_{m \times n}, \tmmathbf{w}\in \Delta_m} \sum\limits_{k=1}^{N} \langle C(\tmmathbf{x},\tmmathbf{x}^{(k)}), \Pi^{(k)}\rangle, \label{eq:fullbatch-lp}\\
\textrm{s.t.} & \mathbf{1}\cdot (\Pi^{(k)})^T =\tmmathbf{w}\;, \;\;
\mathbf{1}\cdot \Pi^{(k)} =\tmmathbf{w}^{(k)}, \forall k\in\mathcal{I}^c,\; & \nonumber
\end{eqnarray}
where the inner product $\langle A, B\rangle := \mbox{tr}(A B^t)$.
\begin{algorithm}[htp]
\caption{Centroid Update with Full-batch LP~\cite{li2008real,cuturi2014fast}}\label{alg:centroid0}
\begin{algorithmic}[1]
\Procedure{Centroid}{$\{ P^{( k )} \}_{k=1}^{N}$}
\Repeat
\State Updates $\{ x_{i} \}$ from Eq.~\eqref{eq:updatex};
\State Updates $\{ w_{i} \}$ from solving full-batch LP
{\eqref{eq:fullbatch-lp}};
\Until{$P$ converges}
\State \Return $P$
\EndProcedure
\end{algorithmic}
\end{algorithm}
By iteratively solving {\eqref{eq:updatex}} and {\eqref{eq:fullbatch-lp}},
referred to as the {\em inner loop},
the step of updating the cluster centroid in Algorithm~\ref{alg:clustering}
is fulfilled~\cite{li2008real,cuturi2014fast}.
We present the centroid update step in Algorithm~\ref{alg:centroid0}.
In summary, D2-clustering is given by Algorithm~\ref{alg:clustering} with
Algorithm~\ref{alg:centroid0} embedded as one key step.
The major difficulty in solving {\eqref{eq:fullbatch-lp}} is that a standard
LP solver typically has a polynomial complexity in terms of
the number of variables $m + \sum_{k=1}^N m_k m$, prohibiting its
scalability to a large number of discrete distributions in one cluster. When
the cluster size is small or moderate, say dozens, it is shown
that the standard LP solver can be faster than a scalable
algorithm~\cite{ye2014scaling}. However, when the cluster size grows, the
standard solver slows down quickly. This issue has been demonstrated by
multiple empirical studies~\cite{li2008real,ye2014scaling,zhang2015parallel}.
Our key observation is that in the update of a centroid distribution, the
variables in $\tmmathbf{w}$ are much more important than the matching weights
in $\Pi$ needed for computing the Wasserstein distances. The parameter
$\tmmathbf{w}$ is actually part of the output centroid, while $\Pi$ is not,
albeit accounting for the vast majority of the variables in
{\eqref{eq:fullbatch-lp}}. We also note that the solution to
{\eqref{eq:fullbatch-lp}} is not the end result but one round of centroid update in the outer
loop. It is thus adequate to have a sufficiently
accurate solution to {\eqref{eq:fullbatch-lp}}, motivating us to pursue
scalable methods such as ADMM, known to be fast for reaching the vicinity of
the optimal solution.
\section{Scalable Centroid Computation}\label{sec:centroid}
We propose algorithms scalable with large-scale datasets,
and compare their performance in terms of speed and memory.
They are (a) subgradient descent with $N$ mini-LP following
similar ideas of~\cite{cuturi2014fast} (included in Appendix~\ref{sec:descent}),
(b) standard ADMM with $N$ mini-QP, and (c) modified B-ADMM with closed
forms in each iteration of the inner loop.
The bottleneck in the computation of D2-clustering is the inner loop,
detailed in Algorithm \ref{alg:centroid0}. The approaches we develop
here all aim at fast solutions for the inner loop, that is, to improve
Algorithm \ref{alg:centroid0}. These new methods can reduce the computation
for centroid update to a comparable (or even lower) level as the label
assignment step, usually negligible in the original D2-clustering.
As a result, we also take measures to speed up the labeling step, with details provided
in Section \ref{sec:impl}.
\subsection{Alternating Direction Method of Multipliers}\label{sec:admm}
ADMM typically solves a problem with two sets of variables (in our case,
they are $\Pi$ and $\tmmathbf{w}$), which are only coupled in constraints,
while the objective function is separable across this splitting of the two sets (in our
case, $\tmmathbf{w}$ is not present in the objective function)~\cite{boyd2011distributed}.
Because problem {\eqref{eq:fullbatch-lp}} has multiple sets of constraints
including both equalities and inequalities, it is not a typical scenario to apply
ADMM. We propose to relax all equality constraints
$\sum_{l=1}^{m_k}\pi_{i,l}^{(k)}=w_{i}$, $\forall k\in\mathcal{I}^c$, $i\in\mathcal{I}'$ in {\eqref{eq:fullbatch-lp}} to their
corresponding augmented Lagrangians and
use the other constraints to determine a convex set for the parameters being optimized.
Let $\Lambda=(\lambda_{i,k})$, $i\in\mathcal{I}'$, $k\in\mathcal{I}^c$.
Let $\rho$ be a parameter to balance the objective function and the augmented
Lagrangians. Define
$\Delta_{\Pi} = \left\{ ( \pi_{i,j}^{( k )} ) : \sum_{i=1}^{m}
\pi_{i,j}^{( k )} =w_{j}^{( k )} , \pi_{i,j}^{( k )} \geqslant 0, k\in\mathcal{I}^c, i\in\mathcal{I}', j\in\mathcal{I}_k
\right\}$. Recall that $\Delta_m = \left\{ (
w_{1} , \ldots ,w_{m} ) | \sum_{i=1}^{m} w_{i} =1,w_{i} \geqslant 0
\right\}$.
As in the method of multipliers, we form the
scaled augmented Lagrangian $L \rho ( \Pi , \tmmathbf{w} , \Lambda )$ as follows
\begin{multline}
L_{\rho} ( \Pi , \tmmathbf{w} , \Lambda ) = \sum_{k=1}^{N} \langle C(\tmmathbf{x},\tmmathbf{x}^{(k)}), \Pi^{(k)}\rangle + \\ \rho
\sum_{\substack{i\in\mathcal{I}'\\k\in\mathcal{I}^c} } \lambda_{i,k}
\left( \sum_{j=1}^{m_k} \pi_{i,j}^{( k )} -w_{i} \right) +
\frac{\rho}{2} \sum_{\substack{i\in\mathcal{I}'\\k\in\mathcal{I}^c} } \left( \sum_{j=1}^{m_k} \pi_{i,j}^{( k )}
-w_{i} \right)^{2} \, .
\end{multline}
Problem
{\eqref{eq:fullbatch-lp}} can be solved using ADMM iteratively as follows.
\begin{eqnarray}
\Pi^{n+1} := \underset{\Pi \in \Delta_{\Pi}}{\tmop{argmin}}
L_{\rho} ( \Pi , \tmmathbf{w}^{n} , \Lambda^{n} ) \;, \qquad \quad \label{eq:admm0}\\
\tmmathbf{w}^{n+1} := \underset{\tmmathbf{w} \in
\Delta_m}{\tmop{argmin}} L_{\rho} ( \Pi^{n+1} , \tmmathbf{w} ,
\Lambda^{n} ) \;, \qquad \;\; \label{eq:admm1}\\
\lambda_{i,k}^{n+1} := \lambda_{i,k}^{n} + \sum_{j=1}^{m_k}
\pi_{i,j}^{( k ) ,n+1} -w_{i}^{n+1} \, , i\in\mathcal{I}', \, k\in\mathcal{I}^c \,. \label{eq:admm2}
\end{eqnarray}
Based on {\eqref{eq:admm0}}, $\Pi$ can be updated by updating $\Pi^{(k)}$, $k=1, ..., N$ separately. Comparing with the full batch LP in {\eqref{eq:fullbatch-lp}} which solves all $\Pi^{(k)}$, $k=1, ..., N$, together, ADMM solves instead $N$ disjoint constrained quadratic programming (QP). This is the key for achieving computational complexity linear in $N$, the main motivation for employing ADMM. Specifically, we solve {\eqref{eq:fullbatch-lp}} by solving {\eqref{eq:admmqp}} below for each $k=1, ..., N$:
\begin{equation}
\begin{array}{rl}
\min_{\pi_{i,j}^{( k )} \geqslant 0} & \langle C(\tmmathbf{x},\tmmathbf{x}^{(k)}), \Pi^{(k)}\rangle \\ & +
\dfrac{\rho}{2} \sum_{i=1}^{m} \left( \sum_{j=1}^{m_k} \pi_{i,j}^{( k )} -w_{i}^{n} + \lambda_{i,k}^{n} \right)^{2} \\
\textrm{s.t.} & \mathbf{1}\cdot \Pi^{( k )} =\tmmathbf{w}^{( k )}, k\in\mathcal{I}^c.
\end{array}\label{eq:admmqp}
\end{equation}
Since we need to solve small-size problem {\eqref{eq:admmqp}} in multiple
rounds, we prefer active set method with warm start. Define
$\tilde{w}_{i}^{(k),n+1} := \sum_{j=1}^{m_k} \pi_{i,j}^{( k )
,n+1} + \lambda_{i,k}^{n}$ for $i=1, ..., m$, $k=1, ..., N$. We can rewrite
step {\eqref{eq:admm1}} as
\begin{eqnarray}
\min_{\tmmathbf{w}\in \Delta_m} & \sum_{i=1}^{m}\sum_{k=1}^{N} ( \tilde{w}_{i}^{(k),n+1} -w_{i} )^{2} &
\nonumber
\label{eq:admmw}
\end{eqnarray}
We summarize the computation of the centroid distribution $P$ for distributions $P^{(k)}$, $k=1, ..., N$ in
Algorithm~\ref{alg:admmqp}. There are two hyper-parameters to choose: $\rho$ and the number of iterations $T_{admm}$.
We empirically select $\rho$ proportional to the averaged transportation costs:
\begin{equation}
\rho = \dfrac{\rho_0}{N n m} \sum_{k=1}^N\sum_{i\in \mathcal{I}'} \sum_{j\in \mathcal{I}_k} c(x_i, x_j^{(k)}) \;.
\label{eq:rho}
\end{equation}
Let us compare the computational efficiency of ADMM and the subgradient descent method.
In gradient descent based approaches, it is costly to choose an effective step-size along the descending
direction because at each search point, we need to solve $N$ LP --- an issue also discussed in~\cite{benamou2014iterative}.
ADMM solves $N$ QP sub-problems instead of LP.
The amount of computation in each sub-problem of ADMM is thus
usually higher and grows faster with the number of support
points in $P^{(k)}$'s. It is not clear whether
the increased complexity at each iteration of ADMM
is paid off by a better convergence rate (that is, a smaller number of iterations).
The computational limitation of ADMM caused by QP motivates us
to explore B-ADMM that avoids QP in each iteration.
\begin{algorithm}
\caption{Centroid Update with ADMM~\cite{ye2014scaling}}\label{alg:admmqp}
\begin{algorithmic}[1]
\Procedure{Centroid}{$\{ P^{( k )} \}_{k=1}^{N}$, $P$, $\Pi$}
\State Initialize $\Lambda^{0} =0$ and $\Pi^{0} := \Pi$.
\Repeat
\State Updates $\{ x_{i} \}$ from Eq.{\eqref{eq:updatex}};
\State Reset dual coordinates $\Lambda$ to zero;
\For {$iter = 1, \ldots, T_{admm}$}
\For{$k=1, \ldots ,N$}
\State Update $\{ \pi_{i,j} \}^{( k )}$ based on QP
Eq.{\eqref{eq:admmqp}};
\EndFor
\State Update $\{ w_{i} \}$ based on QP Eq.{\eqref{eq:admmw}};
\State Update $\Lambda$ based on Eq. \eqref{eq:admm2};
\EndFor
\Until $P$ converges
\State \Return $P$
\EndProcedure
\end{algorithmic}
\end{algorithm}
\subsection{Bregman ADMM}\label{sec:badmm}
Bregman ADMM (B-ADMM) replaces the quadratic augmented Lagrangians by the Bregman
divergence when updating the split variables~\cite{bregman1967relaxation}.
Similar ideas trace back at least to
early 1990s~\cite{censor1992proximal,eckstein1993nonlinear}.
We adapt the design in \cite{wang2014bregman,cuturi2013sinkhorn} for
solving the OT problem with a large
set of support points.
Consider two sets of variables $\Pi_{(k,1)}=(\pi_{i,j}^{( k,1 )})$,
$i\in\mathcal{I}'$, $j\in\mathcal{I}_k$, and
$\Pi_{(k,2)}=(\pi_{i,j}^{( k,2 )})$, $i\in\mathcal{I}'$, $j\in\mathcal{I}_k$, for $k=1, ..., N$ under the following constraints. Let
\begin{eqnarray}
\Delta_{k,1} := \left\{ \pi_{i,j}^{(k,1)} \geqslant 0:
\sum_{i=1}^{m} \pi_{i,j}^{(k,1)} =w_{j}^{( k )} ,j\in\mathcal{I}_k \right\}\;,\\
\Delta_{k,2} ( \tmmathbf{w} ) := \left\{ \pi_{i,j}^{(k,2)}
\geqslant 0 :\sum_{j=1}^{m_k} \pi_{i,j}^{(k,2)} =w_{i}
,i\in\mathcal{I}' \right\},
\end{eqnarray}
then
$\Pi^{( k,1 )} \in \Delta_{k,1}$ and $\Pi^{( k,2 )} \in \Delta_{k,2} ( \tmmathbf{w} )$.
We introduce some extra notations:
\begin{enumerate}
\item $\bar{\Pi}^{(1)}=\{\Pi^{(1,1)}, \Pi^{(2,1)}, \ldots, \Pi^{(N,1)}\}$,
\item $\bar{\Pi}^{(2)}=\{\Pi^{(1,2)}, \Pi^{(2,2)}, \ldots, \Pi^{(N,2)}\}$,
\item $\bar{\Pi}=\{\bar{\Pi}^{(1)}, \bar{\Pi}^{(2)}\}$,
\item $\Lambda =\{\Lambda^{(1)},\ldots,\Lambda^{(N)}\}$, where
$\Lambda^{(k)}=(\lambda_{i,j}^{(k)})$, $i\in\mathcal{I}'$, $j\in\mathcal{I}_k$,
is a $m\times m_k$ matrix.
\end{enumerate}
B-ADMM solves {\eqref{eq:fullbatch-lp}} by treating the augmented
Lagrangians conceptually as a designed divergence between $\Pi^{(k,1)}$ and $\Pi^{(k,2)}$, adapting to the updated variables. It restructures the original problem \eqref{eq:fullbatch-lp} as
\begin{eqnarray}
\underset{\bar{\Pi},\tmmathbf{w}}{\min} && \sum_{k=1}^{N} \langle C(\tmmathbf{x},\tmmathbf{x}^{(k)}), \Pi^{(k,1)}\rangle \\
\textrm{s.t.} && \tmmathbf w \in \Delta_m \nonumber \\
&& \Pi^{(k,1)}\in \Delta_{k,1}, \quad \Pi^{(k,2)}\in \Delta_{k,2}(\tmmathbf w),\;\; k=1,\ldots,N \nonumber \\
&& \Pi^{(k,1)} = \Pi^{(k,2)}, \;\; k=1,\ldots,N\;. \nonumber
\label{eq:badmm_prob}
\end{eqnarray}
Denote the dual variables $\Lambda^{(k)}=(\lambda_{i,j}^{(k)})$, $i\in\mathcal{I}'$, $j\in\mathcal{I}_k$, for $k=1, ..., N$.
Use $\tmop{KL} ( \cdot , \cdot )$ to denote the Kullback--Leibler divergence between two distributions.
The B-ADMM algorithm adds the augmented Lagrangians for the last
set of constraints in its updates, yielding the following equations.
\begin{eqnarray}
\bar{\Pi}^{( 1 ) ,n+1} := &&\underset{\{\Pi^{( k,1 )} \in
\Delta_{k,1}\}}{\tmop{argmin}} \sum_{k=1}^{N} \Bigg( \langle C(\tmmathbf{x},\tmmathbf{x}^{(k)}), \Pi^{(k,1)}\rangle \nonumber\\
&&\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! +
\langle \Lambda^{( k ) ,n}, \Pi^{( k,1 )}\rangle +
\rho \tmop{KL} ( \Pi^{( k,1 )} , \Pi^{( k,2 ) ,n} ) \Bigg),
\label{eq:badmm0}\\
\bar{\Pi}^{( 2 ) ,n+1} , \tmmathbf{w}^{n+1} := &&\!\!\!\!\!\!\!\!\!\!\!\!
\underset{\substack{\{\Pi^{( k,2 )}
\in \Delta_{k,1} ( \tmmathbf{w} )\} \\ \tmmathbf{w} \in
\Delta_m} }{\tmop{argmin}} \sum_{k=1}^{N} \Bigg( - \langle \Lambda^{( k ) ,n}
, \Pi^{( k,2 )} \rangle \nonumber \\
&& + \rho \tmop{KL} ( \Pi^{( k,2 )} , \Pi^{( k,1 ) ,n+1} ) \Bigg),\label{eq:badmm1}\\
\Lambda^{ n+1} := && \Lambda^{ n} + \rho ( \bar{\Pi}^{( 1 ) ,n+1} - \bar{\Pi}^{(2 ) ,n+1} ). \label{eq:badmm2}
\end{eqnarray}
We note that if $\tmmathbf{w}$ is fixed,
\eqref{eq:badmm0} and \eqref{eq:badmm1}
can be split by index $k=1, ..., N$, and have closed form solutions for each $k$. Let $eps$ be the floating-point tolerance (e.g. $10^{-16}$).
For any $i\in \mathcal{I}', \; j\in\mathcal{I}_k$,
\begin{eqnarray}
\tilde{\pi}_{i,j}^{( k,2 ) ,n} \!\!\!\!\!\!\!\! \!\!\!&&:= \pi_{i,j}^{( k,2 ) ,n} \exp\!\!
\left[ \frac{c \left( x_{i} ,x_{j}^{( k )}\! \right) \!+\! \lambda_{i,j}^{(k), n}}{-\rho}
\right]\!\!+\!eps \label{eq:badmmc0b} \\
\pi_{i,j}^{( k,1 ) ,n+1} \!\!\!\!\!\!\!\! \!\!\!&&:= \dfrac{\tilde{\pi}_{i,j}^{( k,2
) ,n}}{{\sum_{l=1}^{m}} \tilde{\pi}_{l,j}^{( k,2 ) ,n}}\cdot
w_{j}^{( k )} \label{eq:badmmc0} \\
\tilde{\pi}_{i,j}^{( k,1 ) ,n+1} \!\!\!\!\!\!\!\!\!\!\! &&:= \pi_{i,j}^{( k,1 ) ,n+1} \exp \left[
\frac{1}{\rho} \lambda_{i,j}^{(k), n} \right]\!+\!eps \label{eq:badmmc1b} \\
\pi_{i,j}^{( k,2 ) ,n+1} \!\!\!\!\!\!\!\!\!\!\! &&:= \dfrac{\tilde{\pi}_{i,j}^{( k,1 )
,n+1}}{{\sum_{l=1}^{m_k}} \tilde{\pi}_{i,l}^{( k,1 ) ,n+1}}\cdot
w_{i} \; . \label{eq:badmmc1}
\end{eqnarray}
Because we need to update $\tmmathbf{w}$ in each iteration, it is not so easy to solve
\eqref{eq:badmm1}. We consider decomposing \eqref{eq:badmm1} into two stages.
Observe that the minimum value of \eqref{eq:badmm1} under a given
$\tmmathbf{w}$ is
\begin{equation}
\min_{\tmmathbf w\in \Delta_m} \sum_{k=1}^{N} \sum_{i=1}^{m}
w_{i} \left[ \log ( w_{i} ) - \log \left(
\sum\nolimits_{j=1}^{m_k} \tilde{\pi}_{i,j}^{( k,1 ) ,n+1} \right) \right].
\label{eq:consensus0}
\end{equation}
The above term (a.k.a. the consensus operator) is minimized by
\begin{equation}
w_{i}^{n+1} \propto \left[ \prod\limits_{k=1}^{N} \left( \sum\limits_{j=1}^{m_k} \tilde{\pi}_{i,j}^{( k,1 ) ,n+1} \right) \right]^{{1}/{N}},\;\;
\sum\limits_{i=1}^{m} w_{i}^{n+1} = 1\;.
\end{equation}
However, the above equation is a geometric mean, which is numerically
unstable when $\sum_{j=1}^{m_k} \tilde{\pi}_{i,j}^{( k,1 ) ,n+1}
\rightarrow 0^{+}$ for some combination of $i$ and $k$.
Here, we employ a different technique. Let
\[
\tilde{w}_{i}^{( k,1 ) ,n+1} \propto
\sum_{j=1}^{m_k} \tilde{\pi}_{i,j}^{( k,1 ) ,n+1}, \;\; \textrm{s.t.}
\sum_{i=1}^{m}
\tilde{w}_{i}^{( k,1 ) ,n+1} =1.
\]
Let the distribution
$\tilde{\tmmathbf{w}}^{(k),n+1}= (\tilde{w}_{i}^{( k,1 ) ,n+1})_{i=1, ..., m} $.
Then Eq. {\eqref{eq:consensus0}} is equivalent to $\min_{\tmmathbf w\in
\Delta_m} \sum_{k=1}^{N}
\mbox{KL}({\tmmathbf{w}},\tilde{\tmmathbf{w}}^{(k),n+1})$. Essentially, a consensus
${\tmmathbf{w}}$ is sought to minimize the sum of KL divergence. In the same
spirit, we propose to find a consensus by changing the order of
${\tmmathbf{w}}$ and $\tilde{\tmmathbf{w}}^{(k),n+1}$ in the KL divergence:
$ \min_{\tmmathbf w\in \Delta_m}\sum\limits_{k=1}^{N}
\tmop{KL}(\tilde{\tmmathbf{w}}^{(k),n+1}, {\tmmathbf{w}})= $
\begin{equation}
\min_{\tmmathbf w\in \Delta_m}
\sum\limits_{k=1}^{N} \sum\limits_{i=1}^{m} \tilde{w}_{i}^{( k,1 ) ,n+1} ( \log ( \tilde{w}_{i}^{( k,1 ) ,n+1} ) - \log ( w_{i} ) )\; ,
\label{eq:consensus1}
\end{equation}
which again has a closed form solution:
\begin{equation}
\mbox{(R1)}: w_{i}^{n+1} \propto \frac 1N \sum\limits_{k=1}^{N} \tilde{w}_{i}^{( k,1 ) ,n+1} ,\;\;
\sum\limits_{i=1}^{m} w_{i}^{n+1} = 1\;.
\label{eq:badmmw}
\end{equation}
The solution of Eq.~\eqref{eq:consensus1} overcomes the numerical
instability. We will call this heuristic update rule as (R1),
which has been employed in the Bregman clustering method~\cite{banerjee2005clustering}.
In addition, a slightly different version of update rule can be
\begin{equation}
\mbox{(R2)}: \left(w_{i}^{n+1} \right)^{1/2}
\propto \frac 1N \sum\limits_{k=1}^{N} \left(\tilde{w}_{i}^{( k,1 ) ,n+1}\right)^{1/2} ,\;\;
\sum\limits_{i=1}^{m} w_{i}^{n+1} = 1\;.
\label{eq:badmmw2}
\end{equation}
In Section~\ref{sec:compare}, we conduct experiments for testing both (R1) and (R2).
We have tried other update rules, such as Fisher-Rao Riemannian center~\cite{srivastava2007riemannian}, and found that the
experimental results do not differ much in terms of the converged objective function.
It is worth mentioning that neither (R1) nor (R2) ensures the
convergence to a (local) minimum.
We summarize the B-ADMM approach in Algorithm~\ref{alg:badmm}. The
implementation involves one hyper-parameters $\rho$ (by default, $\tau=10$).
In our implementation, we choose $\rho$ relatively according to Eq.~\eqref{eq:rho}.
To the best of our knowledge, the convergence of B-ADMM has not been
proved for our formulation (even under fixed support points $\tmmathbf{x}$)
although this topic has been
pursued in recent literature~\cite{wang2014bregman}. In the general case of solving
Eq.~\eqref{eq:centroid}, the optimization of the cluster centroid is
non-convex because the support points are updated after B-ADMM is
applied to optimize the weights. In Section \ref{sec:convergence}, we empirically
test the convergence of the centroid optimization algorithm based on
B-ADMM. We found that B-ADMM usually converges quickly
to a moderate accuracy, making it preferable for D2-clustering.
In our implementation, we use a fixed number
of B-ADMM iterations (by default, 100) across multiple assignment-update rounds in D2-clustering.
\begin{algorithm}[htp]
\caption{Centroid Update with B-ADMM}
\label{alg:badmm}
\begin{algorithmic}[1]
\Procedure{Centroid}{$\{ P^{( k )} \}_{k=1}^{N}$, $P^{}$, $\Pi$}.
\State $\Lambda := 0$; $\bar{\Pi}^{( 2 ) ,0} := \Pi$.
\Repeat
\State Update $\tmmathbf{x}$ from Eq.{\eqref{eq:updatex}} per $\tau$ loops;
\For{$k=1, \ldots ,N$}
\State Update $\Pi^{( k,1 )}$ based on Eq.{\eqref{eq:badmmc0b} \eqref{eq:badmmc0}};
\State Update $\{\tilde{\pi}_{i,j}^{( k,1 )}\}$ based on Eq.{\eqref{eq:badmmc1b}};
\EndFor
\State Update $\tmmathbf{w}$ based on Eq.{\eqref{eq:badmmw}} or Eq.{\eqref{eq:badmmw2}};
\For{$k=1,\ldots, N$}
\State Update $\Pi^{( k,2 )}$ based on Eq.{\eqref{eq:badmmc1}};
\State $\Lambda^{( k )} := \Lambda^{( k )} + \rho ( \Pi^{( k,1)} - \Pi^{( k,2 )} )$;
\EndFor
\Until $P$ converges
\State \Return $P$
\EndProcedure
\end{algorithmic}
\end{algorithm}
\section{Comparison between B-ADMM and Iterative Bregman Projection}~\label{sec:compare}
Both the B-ADMM and IBP can be rephrased into two-step
iterative algorithms via mirror maps (in a similar way of
mirror prox~\cite{nemirovski2004prox} or mirror descent~\cite{beck2003mirror}).
One step is the free-space move in the dual space, and
the other is the Bregman projection (as used in IPFP) in the primal space.
Let $\Phi(\cdot)$ be the entropy function,
the mirror map used by IBP is $\Phi(\Pi)$, while the mirror map of B-ADMM is $$\Phi(\Pi^{(1)},\Pi^{(2)},\Lambda)=\Phi(\Pi^{(1)}) + \Phi(\Pi^{(2)})
+ \dfrac{\|\Lambda\|^2}{\rho^2},$$ where $\Lambda$ is the dual coordinate derived from
relaxing constraints $\Pi^{(1)}=\Pi^{(2)}$ to a saddle point reformulation.
IBP alternates the move
$-\left[\dfrac{C}{\varepsilon}\right]$ in the dual space and projection in $\Delta_1$ or $\Delta_2$ of the primal space.
In comparison, B-ADMM alternates the move
$$
-\begin{bmatrix}
\dfrac{C+\Lambda}{\rho}+\nabla\Phi(\Pi^{(1)})-\nabla\Phi(\Pi^{(2)})\\
-\dfrac{\Lambda}{\rho}+\nabla\Phi(\Pi^{(2)})-\nabla\Phi(\Pi^{(1)})\\
\rho (\Pi^{(1)}-\Pi^{(2)})
\end{bmatrix},
$$
and the projection in $\Delta_1\times\Delta_2\times \mathbb R_{m_1\times m_2}$.%
\footnote{The update is done in the Gauss-Seidel type, not in the usual Jacobi type.}
The convergence of B-ADMM is not evident from the
conventional optimization literature~\cite{bubeck2015convex} because the move is not monotonic (It is still monotonic for standard ADMM).
We conduct two pilot studies to compare B-ADMM and IBP in terms of the convergence behavior and the capacity to generate high quality Wasserstein barycenters with
sparse support in the sense of achieving the best value for the objective function.
In~\cite{benamou2014iterative}, their Figure 1 shows how their algorithm progressively shifts mass away from the diagonal over the iterations.
We adopt the same study here to visualize how the mass transport
between two 1D distributions evolves over the iterations, as shown in Fig.~\ref{fig:convergence}. The smoothing effect of the entropy regularization
diminishes as parameter $\varepsilon$ reduces.
When $\varepsilon=0.1/N$, the mass transport of IBP at 5000 iterations is close to
the unregularized solution albeit with noticeable difference. Setting $\varepsilon$ even smaller ($=0.04/N$)
introduces double-precision overflow. What we have observed about IBP is consistent with the previous remark that IBP was competitive
when the regularization term $\varepsilon$ is not too small~\cite{benamou2014iterative}.
In contrast, the output of mass transport by B-ADMM ($\rho_0=2$, the default setting)
at 5000 iteration is almost indiscernible from the unregularized solution by LP.
Unlike IBP, because B-ADMM does not require $\rho_0\!\to\!0$, no numerical difficulty has been encountered in practice.
In fact, the convergence speed of B-ADMM, as proved by~\cite{wang2014bregman},
does not depend on $\rho_0$.
For the second pilot study, we generated a set of 1000 discrete distributions, each with a sparse support set obtained by clustering pixel colors of images~\cite{li2008real} ($d=3$). The average number of support
points is around 6. Starting from the same initial estimate,
we calculate the approximate Wasserstein barycenter by different methods.
We obtained results for two cases:
barycenters with support size $m=6$ (the initial objective equal $1054.4$)
and barycenters with support size $m=60$ (the initial objective equal $989.7$).
The calculated barycenters are then evaluated by comparing the objective function (Eq.~\eqref{eq:centroid}) with that solved directly by the full batch LP (Algorithm~\ref{alg:centroid0})---the yardstick in the comparison.
The bare form of IBP treats the case of fixed locations for the support points, while B-ADMM does not constrain the locations. In order to compare the two methods on a common ground, we used the following experiments, referred to as two tracks. In the first track, the locations of
support points in the barycenters are fixed, while in
the second track, both locations and weights are optimized. To adopt
IBP in the second track, we experimented with two versions.
The first is to update the locations every $\tau$ iterations
without restarting IBP (V1), similarly as in our modified B-ADMM.
The second is to restart IBP every $\tau$ iterations as explained in~\cite{cuturi2014fast} (V2).
In MATLAB, we found IBP is about $10\times$ faster than B-ADMM per iteration. The difference is partly due to the fact we cannot implement a vectorization version for B-ADMM. By analyzing the actual operations in the two methods, we estimate IBP would be about $2-3\times$ faster than B-ADMM per iteration in C implementation. On the other hand, being fast in one iteration does not necessarily mean fast speed overall. To avoid apriori preference,
for the results reported in Table~\ref{table:compare}, we allocated $2,000$ iterations for the modified B-ADMM and $20,000$ iterations for IBP. We also provide the actual running time in the table.
Both IBP versions are tuned by hyper-parameter
$\tau \in \{100, 200, 400, 800, 1000, 2500, 4000, 5000, 10000\}$ and the best objective values are reported.
The results of the two tracks of experiments for support size $m=6, 60$ by LP, the modified B-ADMM (R1 and R2 as explained in Section \ref{sec:badmm}), and IBP (V1 and V2) are presented in Table~\ref{table:compare}. The performance is measured by the value achieved for the objective function. The results show that in both tracks, B-ADMM achieves lower values for the objective function than IBP and is quite close to LP. Moreover, there is no tuning of hyper-parameters in B-ADMM. For IBP, however, $\varepsilon_0$ influences the result considerably. In all the setups shown in Table~\ref{table:compare}, we could not push the objective function by IBP to the same level of B-ADMM before double-precision overflow arises. In most cases, the gap between B-ADMM and LP is much smaller than the gap between IBP and B-ADMM.
In comparison to the full LP approach, the modified B-ADMM does not yield the exact local minimum. This reminds us that the modified B-ADMM is still an approximation method, and it cannot fully replace
subgradient descent based methods that minimize the objective to a true local minimum.
\begin{figure}
\includegraphics[width=.5\textwidth,trim={0.5cm 2.4cm 4.85cm 2.5cm},clip]{compare_ot2.pdf}
\caption{Convergence behavior of IBP and B-ADMM
The solutions by IBP and B-ADMM are shown in pink, while the exact solution by linear programming is shown in green.
Images in the rows from top to bottom present results at different iterations
$\{1, 10, 50, 200, 1000, 5000\}$;
The left three columns are by IBP with $\varepsilon=\{0.1/N, 0.5/N, 2/N\}$,
where $[0,1]$ is discretized with $N=128$ uniformly spaced points. The last column is by B-ADMM.}\label{fig:convergence}
\end{figure}
\begin{table}
\begin{tabular}{lrcc}
\hline\hline
\multicolumn{4}{p{7cm}}{\textbf{Solved Wasserstein barycenter with a pre-fixed support} $(m=6) / (m=60)$}\\
method & iterations & seconds & obj. \\\hline
full LP & NA & - & 838.2 / 721.3 \\
Our approach (R1) & 2,000 & 3.02 / 32.5 & 841.3 / 724.0\\
Our approach (R2) & 2,000 & 3.02 / 35.0 & 839.1 / 721.9\\
IBP[19] $\varepsilon_0=0.2$ & 20,000 & 3.06 / 21.5 & 1040.0 / 1010.6 \\
IBP $\varepsilon_0=0.1$ & 20,000 & 3.06 / 22.0 & 1021.8 / 1032.1 \\
IBP $\varepsilon_0=0.02$ & 20,000 & 3.06 / 22.2 & 1005.1 / 1090.5 \\
IBP $\varepsilon_0=0.01$ & 20,000 & 3.06 / - & 1002.8 / overflow \\
IBP $\varepsilon_0\le 0.009$ & 20,000 & - / - & overflow / overflow \\\hline\hline
\multicolumn{4}{p{7cm}}{\textbf{Solved Wasserstein barycenter with an optimized support} $(m=6) / (m=60)$}\\
method & iterations & seconds & obj. \\\hline
full LP & 20 & - & 717.8 / 691.9 \\
Our approach (R1) & 2,000 & 3.08 / 35.1 & 720.8 / 693.1 \\
Our approach (R2) & 2,000 & 3.20 / 35.0 & 720.2 / 692.3 \\
IBP V1 $\varepsilon_0=0.2$ & 20,000 & 3.14 / 22.2 & 772.8 / 748.8\\
IBP V1 $\varepsilon_0=0.1$ & 20,000 & 3.20 / 22.1 & 729.3 / 700.1 \\
IBP V1 $\varepsilon_0=0.02$ & 20,000 & 2.98 / 21.7 & 743.8 / 694.9 \\
IBP V1 $\varepsilon_0\le 0.01$ & - & - / - & overflow / overflow\\
IBP V2 $\varepsilon_0=0.2$ & 20,000 & 2.99 / 21.8 & 770.7 / 731.6\\
IBP V2 $\varepsilon_0=0.1$ & 20,000 & 3.00 / 22.0 & 740.1 / 699.7 \\
IBP V2 $\varepsilon_0=0.02$ & 20,000 & 2.98 / 22.1 & 741.5 / 695.1\\
IBP V2 $\varepsilon_0\le 0.01$ & - & - / - & overflow / overflow \\\hline
\end{tabular}\\
\caption{Comparing the solutions of the Wasserstein barycenter by LP, modified B-ADMM (our approach) and IBP. The running time reported is based on MATLAB implementations.}~\label{table:compare}
\end{table}
\section{Algorithm Initialization and Implementation}\label{sec:impl}
In this section, we explain some specifics in the implementation of the
algorithms, such as initialization, warm-start in optimization, measures for further speed-up, and the method for parallelization.
The number of support vectors in the centroid distribution, denoted by $m$,
is set to the average number of support vectors in the distributions in the
corresponding cluster. To initialize a centroid, we select randomly a
distribution with at least $m$ support vectors from the cluster. If the
number of support vectors in the distribution is larger than $m$, we
will merge recursively a pair of support vectors according to an optimal criterion until the support size reaches $m$, similar as in linkage clustering. Consider a
chosen distribution $P=\{(w_1, x_1), ..., (w_m, x_m)\}$. We merge $x_i$ and
$x_j$ to $\bar{x}=(w_i x_i+w_jx_j)/\bar{w}$ , where $\bar{w}=w_i+w_j$
is the new weight for $\bar{x}$,
if $(i,j)$ solves
\begin{eqnarray}
\min_{i,j}{w_iw_j\|x_i-x_j\|^2}/({w_i+w_j})\, . \label{eq:merge}
\end{eqnarray}
Let the new distribution after one merge be $P'$. It is sensible to minimize the
Wasserstein distance between $P$ and $P'$ to decide which support vectors to
merge. We note that
\[
W^2(P, P')\leq w_i \|x_i-\bar{x}\|^2+w_i\|x_j-\bar{x}\|^2 \, .
\]
This upper bound is obtained by the transport mapping $x_i$ and $x_j$ exclusively to $\bar{x}$ and the other support vectors to themselves. To simplify computation, we instead minimize the upper bound, which is achieved by the $\bar{x}$ given above and by the
pair $(i,j)$ specified in Eq. (\ref{eq:merge}).
The B-ADMM method requires an initialization for $\Pi^{(k,2)}$, where
$k$ is the index for every cluster member, before starting the inner loops
(see Algorithm \ref{alg:badmm}). We use a warm-start for $\Pi^{(k,2)}$.
Specifically, for the members whose cluster labels are unchanged after the
most recent label assignment, $\Pi^{(k,2)}$ is initialized by its value
solved (and cached) in the previous round (with respect to the outer loop)
. Otherwise, we initialize $\Pi^{(k,2)}=(\pi_{i,j}^{(k,2)})$, $i=1, ..., m$,
$j=1, ..., m_k$ by
$\displaystyle \pi_{i,j}^{(k,2),0} := w_i w_j^{(k)}$.
This scheme of initialization is also applied in the first round of
iteration when class labels are assigned for the first
time and there exists no previous solution for this parameter.
At the relabeling step (that is, to assign data points to centroids after
centroids are updated), we need to compute $\bar{N}\cdot K$ Wasserstein
distances, where $\bar{N}$ is the data size and $K$ is the number of
centroids. This part of the computation, usually negligible in the original
D2-clustering, is a sizable cost in our new algorithms. To further boost the
scalability, we employ the technique of \cite{elkan2003using} to skip
unnecessary distance calculation by exploiting the triangle inequality of
a metric.
In our implementation, we
use a fixed number of iterations $\epsilon_i$ for all inner loops for
simplicity. It is not crucial to obtain highly accurate result for the inner loop because the partition will be changed by the outer loop. For B-ADMM, we found that setting $\epsilon_i$ to tens or a hundred
suffices. For subgradient descent and ADMM, an even smaller $\epsilon_i$ is
sufficient, {\it e.g.}, around or below ten. The number of iterations of the outer
loop $\epsilon_o$ is not fixed, but adaptively determined when a certain
termination criterion is met.
With an efficient serial implementation, our algorithms can be deployed to
handle moderate scale data on a single PC. We also implemented their
parallel versions which are scalable to a large data size and a large number
of clusters. We use the commercial solver provided by
Mosek\footnote{\url{https://www.mosek.com}}, which is
among the fastest LP/QP solvers available. In particular, Mosek provides
optimized simplex solver for transportation problems that fits our needs
well.
The algorithms we have developed here are all readily parallelizable by
adopting the Allreduce framework in MPI. In our implementation, we divide
data evenly into trunks and process each trunk at one processor. Each trunk
of data stay at the same processor during the whole program.
We can parallelize the algorithms simply by dividing
the data because in the centroid update step, the computation comprises
mainly separate per data point optimization problems. The main communication
cost is on synchronizing the update for centroids by the inner loop.
The synchronization time with equally partitioned data is negligible.
We experimented with discrete distributions over a vector space endowed with the
Euclidean distance as well as over a symbolic set. In the second case, a symbol to symbol distance matrix is provided. When applying D2-clustering to such data, the step of updating the support vectors can be skipped since the set of symbols is fixed. In some datasets, the support vectors in the distributions locate only on a pre-given grid. We can save memory in the implementation by storing the indices of the grid points rather than the direct vector values.
Although we assume each instance is a single distribution in all the previous discussion, it is straightforward to generalize to the case when an instance is an array of distributions (indeed the original setup of D2-clustering in \cite{li2008real}). For instance, a protein sequence can be characterized by three histograms over respectively amino acids, dipeptides, and tripeptides. This extension causes little extra work in the algorithms. When updating the cluster centroids, the distributions of different modalities can be processed separately, while in the update of cluster labels, the sum of squared Wasserstein distances for all the distributions is used as the combined distance.
\section{Complexity and Performance Comparisons}\label{sec:compl}
Recall some notations: $\bar{N}$ is the data size (total number of
distributions to be clustered); $d$ is the dimension of the support vectors;
$K$ is the number of clusters; and $\epsilon_i$ or $\epsilon_o$ is the number of iterations in the inner or outer loop.
Let $\bar{m}$ be the average number of support
vectors in each distribution in the training set and $m$ be the number of
support vectors in each centroid distribution ($\bar{m}=m$ in our setup).
In our implementation, to cut on the time of dynamic memory allocation,
we retain the memory for the matching weights between the support vectors of each distribution and its
corresponding centroid. Hence, the memory allocation is of order $O(\bar{N}\bar{m}m) + O(d\bar{N}\bar{m}+dKm)$.
For computational complexity, first consider the
time for assigning cluster labels in the outer loop. Without the
acceleration yielded from the triangle inequality, the complexity
is $O(\epsilon_o \bar{N} K l(\bar{m}m, d))$ where $l(\bar{m}m, d)$ is the average
time to solve the Wasserstein distance between distributions on a $d$
dimensional metric space. Empirically, we found that by
omitting unnecessary distance computation via
the triangle inequality, the
complexity is reduced roughly to $O(\epsilon_o(\bar{N} + K^2)l(\bar{m}m, d))$.
For the centroid update step, the time complexity of the serial
version of the ADMM method is $O(\epsilon_o \epsilon_i
\bar{N} m d) + O(T_{admm} \cdot \epsilon_o \epsilon_i \bar{N}
q(\bar{m} m,d))$, where $q(m'm,d)$ is the average time to solve QPs
(Eq~\eqref{eq:admmqp}).
The complexity of the serial B-ADMM
is $O(\epsilon_o \epsilon_i \bar{N} m d / \tau) + O( \epsilon_o \epsilon_i
\bar{N} \bar{m} m)$. Note that in the serial algorithms, the complexity for updating centroids
does not depend on $K$, but only on data size $\bar{N}$.
For the parallel versions of the algorithms, the communication load
per iteration in the inner loop is\break $O(T_{admm} K m d)$ for ADMM
and $O(K m (1 + d/\tau))$ for the B-ADMM.
Both analytical and empirical studies (Section \ref{sec:profile}) show
that the ADMM algorithm is significantly slower than the other two when the
data size is large due to the many constrained QP sub-problems required.
Although the theoretical properties of convergence are better understood for
ADMM, our experiments show that B-ADMM performs well consistently
in terms of both convergence and the quality of the clustering results.
Although the preference for B-ADMM is experimentally validated, given the lack of
strong theoretical results on its convergence, it is not clear-cut that B-ADMM
can always replace the alternatives. We were thus motivated to develop the
subgradient descent (in our supplement)
and standard ADMM algorithms to serve at least as yardsticks for comparison.
We provide the following guidelines on the usage of the algorithms.
\begin{itemize}
\item We recommend the modified B-ADMM
as the default data processing pipeline for its scalability,
stability, and fast performance. Large memory is assumed to be available under
the default setting.
\item It is known that ADMM type methods can approach the optimal solution quickly at
the beginning when the current solution is far from the optimum while the
convergence slows down substantially when the solution is in the proximity of
the optimum. Because we always reset the Lagrangian multipliers in B-ADMM
at the beginning of every round of the inner loop and a fixed number of iterations
are performed within the loop, our scheme does not pursue aggressively
high accuracy for the resulting centroids at every round.
However, if the need arises for highly accurate centroids,
we recommend the subgradient descent method that takes as initialization
the centroids first obtained by B-ADMM.
\end{itemize}
\section{Experiments}\label{sec:expr}
We have conducted experiments to examine the convergence of the algorithms, stability, computational/memory efficiency and scalability of the algorithms, and quality of the clustering results on large data from several domains.
\begin{table}[htp]
\centering
\caption{Datasets in the experiments. $\bar{N}$: data size, $d$: dimension of the support vectors ("symb" for symbolic data), $m$: number of support
vectors in a centroid, $K$: maximum number of clusters tested. An entry with the same value as in the previous row is indicated by "-". }
\begin{tabular}{c|cccc}
\hline
Data & $\bar{N}$ & $d$ & $m$ & $K$ \\ \hline
synthetic & 2,560,000 & $\ge$16 & $\ge$32 & 256 \\ \hline
image color & 5,000 & 3 & 8 & 10 \\
image texture & - & - & - & - \\ \hline
protein sequence 1-gram & 10,742 & symb. & 20 & 10 \\
protein sequence 3-gram & - & - & 32 & - \\
\hline
USPS digits & 11,000 & 2 & 80 & 360 \\ \hline
BBC news abstract & 2,225 & 300 & 16 & 15 \\
Wiki events abstract & 1,983 & 400 & 16 & 100 \\
20newsgroups GV & 18,774 & 300 & 64 & 40 \\
20newsgroups WV & - & 400 & 100 & - \\ \hline
\end{tabular}
\label{table:stat}
\end{table}
Table~\ref{table:stat} lists the basic information about the datasets used in our experiments. For the synthetic data, the support vectors are generated by sampling from a multivariate normal distribution and then adding a heavy-tailed noise from the student's t distribution. The probabilities on the support vectors are perturbed and normalized samples from Dirichlet distribution with symmetric prior. We omit details for lack of space. The synthetic data are only used to study the scalability of the algorithms. The image color or texture data are created from crawled general-purpose photographs. Local color or texture features around each pixel in an image are clustered (aka, quantized) to yield color or texture distributions. The protein sequence data are histograms over the amino acids (1-gram) and tripeptides (3-tuples, 3-gram).
The USPS digit images are treated as normalized histograms over the pixel locations covered by the digits,
where the support vector is the two dimensional coordinate of a
pixel and the weight corresponds to pixel intensity.
For the {\em 20newsgroups} data, we use the recommended ``bydate'' matlab version which includes 18,774 documents and 61,188 unique words. The two datasets, ``20 newsgroup GV'' and ``20newsgroup WV'' are created by characterizing the documents in different ways.
The ``BBC news abstract'' and ``Wiki events abstract'' datasets are
truncated versions of two document collections~\cite{greene2006practical,wu2015storybase}. These two sets of short documents retain only the title and the first sentence of each original post.
The purpose of using these severely cut documents is to investigate a more challenging setting for existing document or sentence analysis methods, where semantically related sentences are less likely to share the exact same words.
For example, ``NASA revealed its ambitions that humans can set foot on Mars''
and ``US is planning to send American astronauts to Red Planet'' describe the same event.
More details on the data are referred to Section \ref{sec:effect}.
\subsection{Convergence Analysis}\label{sec:convergence}
We empirically test the convergence and stability of the three approaches: modified B-ADMM, ADMM, and subgradient descent method, based on their sequential versions implemented in C. Four datasets are used in the test: protein sequence 1-gram, 3-gram data, and the image color and texture data. In summary, the experiments show that the modified B-ADMM
method has achieved the best numerical stability with respect to hyper-parameters while keeping a comparable
convergence rate as the subgradient descent method in terms of CPU time. Detailed results on the study of stability are provided in Appendix B for the lack of space. Despite of its popularity in large scale machine learning problems, by lifting $\bar{N}$ LPs to $\bar{N}$ QPs, the ADMM approach is much slower on large datasets than the other two.
We examine the convergence property of the B-ADMM approach for computing the centroid of a single cluster (the inner loop). In this experiment, a subset of image color or texture data with size $2000$ is used. For the two protein sequence datasets, the whole sets are used.
Fig.~\ref{fig:converge} shows the convergence analysis
results on the four datasets. The vertical axis in the plots in the first row of
Fig.~\ref{fig:converge} is the objective function of B-ADMM, given in
Eq.~\eqref{eq:badmm_prob}, but not the original objective function of
clustering in Eq.~\eqref{eq:centroid}. The running time is based on a single thread with 2.6
GHz Intel Core i7. The plots reveal two characteristics about the B-ADMM approach: 1)
The algorithm achieves consistent and comparable convergence rate under a wide
range of values for the hyper-parameter $\rho_0 \in \{0.5, 1.0, 2.0, 4.0,
8.0, 16.0\}$ and is numerically stable; 2) The effect of
the hyper-parameter on the decreasing ratio of the dual and primal residuals
follows similar patterns across the datasets.
\begin{figure*}[t]
\centering
\subfloat[image color]{\includegraphics[clip = true, viewport = 45 180 550 590, width=0.24\textwidth]{img/data-image-color+badmm.pdf}}~
\subfloat[image texture]{\includegraphics[clip = true, viewport = 45 180 550 590, width=0.24\textwidth]{img/data-image-texture+badmm.pdf}}~
\subfloat[prot. seq. 1-gram]{\includegraphics[clip = true, viewport = 45 180 550 590, width=0.24\textwidth]{img/data-protein-1gram+badmm.pdf}}~
\subfloat[prot. seq. 3-gram]{\includegraphics[clip = true, viewport = 45 180 550 590, width=0.24\textwidth]{img/data-protein-3gram+badmm.pdf}}\\
\subfloat[image color]{\includegraphics[clip = true, viewport = 45 180 550 590, width=0.24\textwidth]{img/data-image-color+badmm_res.pdf}}~
\subfloat[image texture]{\includegraphics[clip = true, viewport = 45 180 550 590, width=0.24\textwidth]{img/data-image-texture+badmm_res.pdf}}~
\subfloat[prot. seq. 1-gram]{\includegraphics[clip = true, viewport = 45 180 550 590, width=0.24\textwidth]{img/data-protein-1gram+badmm_res.pdf}}~
\subfloat[prot. seq. 3-gram]{\includegraphics[clip = true, viewport = 45 180 550 590, width=0.24\textwidth]{img/data-protein-3gram+badmm_res.pdf}}
\caption{Convergence analysis of the B-ADMM method for computing a single centroid based on four datasets. First row: objective function of B-ADMM based centroid computation with respect to CPU time; Second row: the trajectory of
dual residual vs. primal residual (in the negative log scale).
}
\label{fig:converge}
\end{figure*}
\subsection{Efficiency and Scalability}\label{sec:profile}
We now study the computational/memory efficiency and scalability of AD2-clustering with the
B-ADMM algorithm embedded for computing cluster centroids. We use the synthetic data that allow easy control over data size and other parameters in order to test their effects on the computational and memory load ({\it i.e.}, workload) of the algorithm.
We study the {\em scalability} of our parallel implementation on a cluster computer with distributed memory. Scalability here refers to the ability of a
parallel system to utilize an increasing number of processors.
\begin{table}[htp]\small
\centering
\caption{Scaling efficiency of AD2-clustering in parallel implementation.}
\begin{tabular}{c|ccccc}
\hline
\# processors & 32 & 64 & 128 & 256 & 512 \\\hline
SSE ($\%$) & 93.9 & 93.4 & 92.9 & 84.8 & 84.1 \\
WSE on $\bar{N}$ ($\%$) & 99 & 94.8 & 95.7 & 93.3 & 93.2\\
WSE on $m$ ($\%$) & 96.6 & 89.4 & 83.5 & 79.0 & - \\ \hline
\end{tabular}
\label{table:scale}
\end{table}
AD2-clustering can be both cpu-bound and memory-bound. Based on the observations from
the above serial experiments, we conducted three sets of experiments to
test the scalability of AD2-clustering
in a multi-core environment, specifically, strong scaling efficiency,
weak scaling efficiency with respect to $\bar{N}$ or $m$.
The configuration ensures that each iteration finishes within one hour
and the memory of a typical computer cluster is sufficient.
{\em Strong scaling efficiency (SSE)} is about the speed-up gained from using more and more processors when the problem is fixed in size. Ideally, the running time on parallel CPUs is the time on a single thread divided by the number of CPUs. In practice, such a reduction in time cannot be fully achieved due to communication between CPUs and time for synchronization. We thus measure SSE by the ratio between the ideal and the actual amount of time. We chose a moderate size problem
that can fit in the memory of a single machine (50GB):
$\bar{N}=250,000$, $d=16$, $m=64$, $k=16$. Table~\ref{table:scale} shows
the SSE values with the number of processors
ranging from 32 to 512. The results show that AD2-clustering scales well in SSE
when the number of processors is up to hundreds.
{\em Weak scaling efficiency (WSE)} measures how stable the real computation time
can be when proportionally more processors are used as the size of the problem grows.
We compute WSE with respect to both $\bar{N}$ and $m$.
Let $np$ be the number of processors. For WSE on $\bar{N}$,
we set $\bar{N}=5000 \cdot np$, $d=64$, $m=64$, and $K=64$ on each processor. The per-node memory is
roughly $1GB$. For WSE on $m$, we
set $\bar{N}=10,000$, $K=64$, $d=64$, and $m=32 \cdot \sqrt{np}$.
Table~\ref{table:scale} shows the values of WSE on $\bar{N}$ and $m$.
We can see that AD2-clustering also has good weak scalability, making it suitable for handling large scale data. In summary, our proposed method can be effectively accelerated with an increasing number of CPUs.
\subsection{Quality of Clustering Results}\label{sec:effect}
\textbf{Handwritten Digits:}
We conducted two experiments to
evaluate the results of AD2-clustering on USPS data, which contain
$1100\times 10$ instances ($1100$ per class).
First, we cluster the images at $K=30, 60, 120, 240$
and report in Figure~\ref{fig:usps-clu} the homogeneity versus completeness~\cite{rosenberg2007v} of the obtained clustering results.
We set $K$ to large values because clustering performed on such image data is often for the purpose of quantization where the number of clusters is much larger than the number of classes. In this case, homogeneity and completeness are more meaningful measures than the others used in the literature (several of which will be used later for the next two datasets). Roughly speaking, completeness measures how likely members of the same true class fall into the same cluster, while homogeneity measures how likely members of the same cluster belong to the same true class. By construction, the two measures have to be traded off.
We compared our method with Kmeans++~\cite{arthur2007k}. For this dataset, we found that Kmeans++, with more careful initialization, yields better results than the standard K-means. Their difference on the other datasets is negligible. Figure~\ref{fig:usps-clu} shows that AD2-clustering obviously outperforms Kmeans++ cross $K$'s.
Secondly, we tested AD2-clustering for quantization with the existence of
noise. In this experiment, we corrupted each sample by "blankout"---randomly deleting a percentage of pixels occupied by the digit (setting to zero
the weights of the corresponding bins), as is done in~\cite{globerson2006nightmare}. Then each class is randomly split into 800/300 training and test samples. Clustering
is performed on the 8000 training samples; and a class label is assigned
to each cluster by majority vote. In the testing phase, to classify an
instance, its nearest centroid is found and the class label of the
corresponding cluster is assigned. The test classification
error rates with respect to $K$ and the blankout rate are plotted in Figure~\ref{fig:usps-cla}. The comparison with Kmeans++ demonstrates that AD2-clustering performs consistently better, and the margin is remarkable when the number of clusters is large and the blankout rate is high.
\begin{figure}[htp]
\subfloat[Homogeneity vs. completeness]{
\includegraphics[ viewport = 45 180 550 590,clip,width=0.235\textwidth]{img/usps_clu.pdf}
\label{fig:usps-clu}
}
\subfloat[Test error rate vs. blankout rate]{
\includegraphics[ viewport = 45 180 550 590,clip,width=0.235\textwidth]{img/usps_cla.pdf}
\label{fig:usps-cla}
}
\caption{Comparisons between Kmeans++ and AD2-clustering on USPS dataset.
We empirically set the number of support vectors in the centroids
$m=80 (1-blankout\_rate)$.}
\end{figure}
\begin{table}[htp]
\caption{Compare clustering results of AD2-clustering and several baseline
methods using two versions of Bag-of-Words
representation for the 20newsgroups data. Top panel: the data are extracted
using the GV vocabulary; bottom panel: WV vocabulary. AD2-clustering is performed once on 16 cores with less than 5GB memory. Run-times of AD2-clustering are reported (along with the total number of iterations).}\label{tab:doc}
\begin{tabular}{c|cccccccccc}
\hline
GV & \multirow{2}{*}{tf-idf} & \multirow{2}{*}{LDA} & LDA & Avg. & \multirow{2}{*}{AD2} & \multirow{2}{*}{AD2} & \multirow{2}{*}{AD2}\\
Vocab. & & & Naive & vector & & & \\\hline
$K$ & 40 & 20 & 20 & 30 & 20 & 30 & 40 \\
AMI & 0.447 & 0.326 & 0.329 & 0.360 & 0.418 & \textbf{0.461} & 0.446 \\
ARI & 0.151 & 0.160 & 0.187 & 0.198 & 0.260 & 0.281 & \textbf{0.284} \\
hours &&&&& 5.8 & 7.5 & 10.4 \\
\# iter.&&&&& 44 & 45 & 61\\ \hline
\end{tabular}
\vskip 0.1in
\begin{tabular}{c|cccccccccc}
\hline
WV & \multirow{2}{*}{tf-idf} & \multirow{2}{*}{LDA} & LDA & Avg. & \multirow{2}{*}{AD2} & \multirow{2}{*}{AD2} & \multirow{2}{*}{AD2}\\
Vocab. & & & Naive & vector & & & \\\hline
$K$ & 20 & 25 & 20 & 20 & 20 & 30 & 40 \\
AMI & 0.432 & 0.336 & 0.345 & 0.398 & 0.476 & \textbf{0.477} & 0.455\\
ARI & 0.146 & 0.164 & 0.183 & 0.212 & \textbf{0.289} & 0.274 & 0.278 \\
hours &&&&& 10.0 & 11.3 & 17.1 \\
\# iter. &&&&& 28 & 29 & 36\\ \hline
\end{tabular}
\end{table}
\textbf{Documents as Bags of Word-vectors:}
The idea of treating each document as a bag of vectors has been explored
in previous work where a nearest neighbor classifier
is constructed using Wasserstein distance~\cite{wan2007novel,kusner2015word}.
One advantage of the Wasserstein distance is to account for the many-to-many mapping between
two sets of words.
However, clustering based on Wasserstein distance, especially the use of
Wasserstein barycenter, has not been explored in the literature of document analysis.
We have designed two kinds of experiments using different document data to assess the power of AD2-clustering.
To demonstrate the robustness of D2-clustering across different word embedding spaces,
we use 20newsgroups processed based on two pre-trained word embedding models.
We pre-processed the dataset by two steps:
remove stop words; remove other words that do not belong to a pre-selected
background vocabulary. In particular, two background vocabularies are tested:
English Gigaword-5 (denoted by GV) \cite{pennington2014glove} and
a Wikipedia dump with minimum word
count of 10 (denoted by WV) \cite{mikolov2013distributed}.
Omitting details due to lack of space, we validated that under the GV or WV vocabulary
information relevant to the class identities of the documents is almost intact.
The words in a document are then mapped to a vector space.
For GV vocabulary, the Glove mapping to a vector space of dimension $300$ is used~\cite{pennington2014glove}, while for WV, the Skip-gram model is used to train a mapping space of dimension $400$~\cite{mikolov2013distributed}. The frequencies on the words are adjusted by the popular scheme of tf-idf.
The number of different words in a document is bounded by
$m$ (its value in Table~\ref{table:stat}). If a document has more than
$m$ different words, some words are merged into hyper-words recursively until
reaching $m$, in the same manner as the greedy merging scheme used in
centroid initialization described in Section~\ref{sec:impl}.
We evaluate the clustering performance by two
widely used metrics: AMI\cite{vinh2010information} and
ARI\cite{rand1971objective,hubert1985comparing}. The baseline methods for
comparison include K-means on the raw tf-idf word frequencies, K-means on the LDA
topic proportional vectors\cite{hoffman2010online} (the number of LDA topics is chosen from $\{40,60,80,100\}$), K-means on the average word
vectors, and the naive way of treating the 20 LDA topics as clusters.
For each baseline method,
we tested the number of clusters $K\in\{10,15,20,25,30,40\}$ and report only the best performance for the baseline methods in Table~\ref{tab:doc}, while for AD2-clustering, $K=20, 30, 40$ are reported. Under any given setup of a baseline method, multiple runs were conducted with different initialization and the median value of the results was taken.
The experimental results show that AD2-clustering achieves the best
performance on the two datasets according to both AMI and ARI. Comparing with most
baseline methods, the boost in performance by AD2-clustering is substantial.
Furthermore, we also vary $m$ in the experimental setup of AD2-clustering. At $m=1$,
our method is exactly equivalent to K-means of the distribution means. We increased $m$ empirically
to see how the results improve with a larger $m$. We did not observe any further
performance improvement for $m\ge 64$.
We note that the high competitiveness of AD2-clustering
can be credited to (1) a reasonable word embedding model and (2) the bag-of-words model.
When the occurrence of words is sparse across documents, the semantic relatedness
between different words and their compositions in a document
plays a critical role in measuring the document
similarity.
In our next experiment, we study AD2-clustering for short documents,
a challenging setting for almost all existing methods based on the bag-of-words representation.
It shows that the performance boost of AD2-clustering is also substantial.
We use two datasets, one is called ``BBC news abstract'', and the other
``Wiki events abstract''. Each document is represented by only the title
and the first sentence from a news article or an event description.
Their word embedding models are same as the one used by the ``WV'' version in our previous experiment.
The ``BBC news'' dataset contains 5 news categories, and ``Wiki events'' dataset contains 54 events. In the supplement materials, the raw data we used are provided.
Clustering such short documents is more challenging due to the sparse nature of word occurrences.
As shown by Table~\ref{table:short}, in terms of generating clusters coherent with the labeled categories or events, methods which leverage either the bag-of-words model or the word embedding model (but not both) are outperformed by AD2-clustering which exploits both.
In addition, AD2-clustering is fast for those sparse support discrete distribution data. It takes only several minutes to finish the clustering in an 8-core machine.
To quantify the gain from employing an effective word embedding model, we also applied AD2-clustering to a random word embedding model, where a vector sampled
from a multivariate Gaussian with dimension 300 is used to represent a word in vocabulary. We find that the results are much worse
than those reported in Table~\ref{table:short} for AD2-clustering.
The best AMI for ``BBC news abstract'' is 0.187
and the best AMI for ``Wiki events abstract'' is 0.369, comparing respectively
with 0.759 and 0.545 obtained from a carefully trained word embedding model.
\begin{table}
\begin{tabular}{c|ccccc}\hline
& Tf-idf & LDA & NMF & Avg. vector & AD2 \\
BBC news abstract & 0.376 & 0.151 & 0.537 & 0.753 & 0.759 \\
Wiki events abstract & 0.448 & 0.280 & 0.395 & 0.312 & 0.545\\\hline
\end{tabular}
\caption{Best AMIs achieved by different methods on the two short document datasets.
NMF denotes for the non-negative matrix factorization method. }~\label{table:short}\vspace{-1cm}
\end{table}
\section{Conclusions and Future Work}\label{sec:concl}
Two first-order methods for clustering discrete distributions under the Wasserstein distance have been developed and empirically compared in terms of speed, stability, and convergence.
The experiments identified the modified B-ADMM method as most preferable for D2-clustering in an overall sense under common scenarios. The resulting clustering tool is easy to use, requiring no tuning of optimization parameters. We applied the tool to several real-world datasets, and evaluated the quality of the clustering results by comparing with the ground truth class labels. Experiments show that the accelerated D2-clustering often clearly outperforms other widely used methods in the respective domain of the data.
One limitation of our current work is that the theoretical convergence property of the modified B-ADMM algorithm is not well understood. In addition, to speed up D2-clustering, we have focused on scalability with respect to data size rather than the number of support points per distribution. Thus there is room for further improvement on the efficiency of D2-clustering. Given that the algorithms explored in this paper have different strengths, it is of interest to investigate in the future whether they can be integrated to yield even stronger algorithms.
|
2009.14008
|
\section{Introduction}
Quantum light sources including on-demand single-photon sources are promising candidates in numerous frontier photonic and quantum technologies\cite{Utzat_Science2019,Cosacchi_PRL2019,Krieg_Acs2018,OBrien_NatPh2009}.
Embracing both experimental and theoretical investigations, single-photon emitters (SPE) as intrinsic building blocks play an essential role in quantum
communication\cite{ZhangNatpho2019}, quantum computing\cite{OBrien_NatPh2009} and quantum information processing\cite{Takeda_APL2019}
In this area, a great deal of work has been done to characterize quantum light production using
nonlinear crystals\cite{Kwiat_PRL1997} or atomic systems\cite{Thompson_Science2006}, both of which suffer from
low photon emission rates and limited scalability. Moreover in a commonly used method of twin-state generation, spontaneous parametric down conversion\cite{Mosley_2008,Horn_PRL2012}, the efficiency is low and depends on an inherently in-determinant emission processes.
Semiconductor quantum dots as non-classical light emitters are of particular interest and more favorable for single photon production\cite{Heindel_Natcom2017,Somaschi_NatPho2016,Schweickert_AppPhyLette2017,Ding_PRL2016} due to their high compatibility with the current semiconductor technology. It has been proven that these excellent quantum emitters\cite{Heindel_Natcom2017,Somaschi_NatPho2016,Schweickert_AppPhyLette2017,Ding_PRL2016,Schulte_Science2015} can
produce single-photon states with high efficiency that are stimulated both by
optical\cite{Somaschi_NatPho2016} and electrical\cite{Schlehahn_APL2016} excitation.
Moreover, recent advances in fabrication techniques have paved the road for the production of ideal semiconductor single-photon sources\cite{Heindel_Natcom2017,Aharonovich_NatPhoton2016,Sapienza_Natcom2015,Gschrey_APL2013} showing that they possess significant capabilities for producing indistinguishable single-photons or entangled-photon pairs \cite{Senellart_NatNano2017,Aharonovich_NatPhoton2016}. However the implementation depends upon the scalability of quantum emitters (QEs), so
the on-demand generation of more complex photonic states is still a
challenging task.
Generally, the characteristic properties of an ideal single-photon source can be classified into three categories:
(i) single-photon purity in which the field does not accommodate more than one photon. This property is determined by the second-order intensity correlation function $g^{(2)}(0)$ through the Hanbury Brown and Twiss (HBT) experiment \cite{BROWN1956, Migdallbook}.
(ii) indistinguishability, that is related to adjusting the quantum interference of two single-photon wavepackets and
can be measured via Hong–Ou–Mandel (HOM) interference\cite{HOM1987,Martino_PRApp2014}.
(iii) brightness in which one measures the probability that each light pulse contains a single photon. This measurement gives us more insight into the information contained in the second-order coherence related to the photon number probabilities \cite{Migdallbook}.
These three properties have been defined differently depending on the scientific community, although the essential features behind them are the same: a single-photon source should generate light pulses with no more than one photon, the photon should be in a pure quantum state, and it should be generated as efficiently as feasible. All three of these essential features are determined by the second-order correlation function using different experiments.
In general correlation function measurements in the HBT experiment can be regarded as determining the probability of finding two or more photons in the same pulse. This is one of the parameters used to estimate the quality of the single photon source.
This is significant on the grounds that high single-photon purity assures the security of quantum communications and minimizes errors in quantum computation and simulation\cite{Broome_Science2013, Spagnolo_Natph2014}.
Bridging the concepts of an ideal SPE introduced here and the deficiency of scalable single-photon sources motivates the present work, in which we dig into the underlying mechanism for generation of close-to ideal twin-photon states.
As part of this, we map out the characteristic parameters relevant to cascade entangled photon emission and show that single-photons derived from pairs generated by cascade emission can be prepared directly in pure states. In addition, we determine the degree of purity via correlation functions and we provide a Schmidt number analysis. Through these studies, we address major obstacles in the advancement of quantum photonics.
In our theoretical development, the generation of correlated photon pairs in semiconductor emitters is assumed to take place through a biexciton-exciton radiative cascade\cite{Heindel_Natcom2017,Schweickert_AppPhyLette2017,Ding_PRL2016}. In this process, two electron–hole pairs form a biexciton state that radiatively decays with the emission of two photons as mediated by a single exciton
state being an intermediate. The purity of the system is then
limited by correlations within the photon pair as determined by the rates of decay of emission of the first and second photons.
\cite{Utzat_Science2019,Krieg_Acs2018,Loredana_NanoLetters2015}.
This analysis is based on the fact that the radiative biexciton cascade in a single semiconductor QD provides a source of entangled photons. \cite{Winik_PRB2017, Trotta_Nanolett2014}
Starting from the biexcitonic ground state of a pre-excited QD, the first electron-hole recombination leads to emission of one photon, and then the second electron of opposite spin recombines with a hole to give a second photon with opposite polarization. This results in anticorrelation in the polarization of the
emitted photons\cite{Solid-state_Nat2017,HighlyIndis_Nat2017,EP_Generation_2007,HighlyEPpairs_PRA2017}.
\begin{figure}
\includegraphics[clip,width=0.6\columnwidth]{Figs/4Level_pol1.png}
\caption{Schematic of radiative decay of the biexciton state $\ket{XX}$ in a typical(asymmetric) QD with fine-structure splitting $\Delta_{FSS}$. Here we assume the radiative decay of $\ket{XX}$
generates a pair of vertically or horizontally colinearly polarized photons; $\dfrac{1}{\sqrt{2}}(\ket{XX_{H}X_{H}}+\ket{XX_{V}X_{V}})$.}
\label{Fig:QDState}
\end{figure}
Our analysis begins with the theoretical prediction just described, but then we progress to a more sophisticated cascade model that is needed when generating entangled photons from semiconductor QDs due to asymmetry in the geometry of the QDs. This imperfectness induces splitting of the intermediate excitonic states, i.e., fine-structure splitting (FSS)
($\Delta_{FSS}$), which is modified as QD size varies.
This means that we are required to describe the QD biexciton cascade
using a four-level system composed of the biexciton state ($\ket{XX}$),
two bright intermediate exciton levels ($\ket{X_{H(V)}}$), and a ground state ($\ket{g}$) \cite{Cascade_Semicon_PRL2000}. Spontaneous decay of the biexciton state
to the ground state thus occurs via two intermediate exciton states leading to
the emission of pairs of photons through the transitions $\ket{XX}\rightarrow \ket{X_{H(V)}}$
and $\ket{X_{H(V)}}\rightarrow\ket{g}$ respectively (see \autoref{Fig:QDState}).
Thus the intermediate excitonic states lead to spin-dependent properties of the emissions.
With nonzero-FSS
\cite{Polarization-corre_PRB2002,EntangledBiexciton_PRB2003}, the degree of entanglement of the entangled polarization photon pairs is lower. However this can be modified using promising strategies that have been proposed in previous studies\cite{CascadeSemiconNat2006,ManipulatingFSS_APL2007,LowerBoundPRL2010,Entangled_AkopianPRL2006,Cavity-assisted_PRB2009,Pathak_PRB2009}.
\section*{Photon detection and quantum coherence functions in QEs}
The essence of the HBT experiment is to recognize when the detectors are recording a photocurrent, since the detectors use the photoelectric effect to make local field measurements.
For one detector, the photon counting rate is defined by a first-order correlation function \cite{Scullybook},
$
G^{(1)}(\textbf{r},t)
=\langle
E^{(-)}(\textbf{r},t)
E^{(+)}(\textbf{r},t)
\rangle
$ . Here the $E^{(+/-)}$'s are positive and negative frequency parts of the fields and the detector is positioned at $\bm{r}$.
For two photons and two detectors, the joint probability of observing one photoionization at
point $\bm{r}_{2}$ between $t_{2}$ and $t_{2}+dt_{2}$ and another one at point $\bm{r}_{1}$ between
$t_{1}$ and $t_{1}+dt_{1}$ with $t_{1}<t_{2}$ is governed by the second-order quantum mechanical correlation function which is measured in typical multi-photon counting
experiments.
In the first part of this work we assume that we have a perfect symmetric QD that can produce highly entangled photons in which the FSS is zero (degenerate intermediate states). All spin and polarization properties are therefore hidden in the notation that we use to describe the states.
In our cascade emission model, we define the almost perfectly symmetric excited QD as a three-level system where we use $\ket{e}$ and $\ket{m}$ instead of $\ket{XX}$ and $\ket{X_{H(V)}}$ for the excited and the intermediate states respectively. From \autoref{Fig:3Level}, we assume that the system is initially ($t = 0$) in the top level $\ket{e}$ with energy $\hbar(\omega_{\alpha}+\omega_{\beta})$ and width $\gamma_{\alpha}$. This means that the lifetimes of the two-photon excited state is $\gamma_{\alpha}^{-1}$.
\begin{figure}
\includegraphics[clip,width=0.6\columnwidth]{Figs/3Level_new.png}
\caption{ Symmetric (almost perfect) QD source: Three-level configuration used for observation of the two-photon cascade
emission. }
\label{Fig:3Level}
\end{figure}
The first spontaneous emission with frequency $\omega_{k}$ is associated with the transition from $\ket{e}$ to the intermediate state $\ket{m}$
and the second decay is to the ground state $\ket{g}$ via the emission of a photon of
frequency $\omega_{q}$.
It should be noted that when $\gamma_{\alpha}>\gamma_{\beta}$,
there will be some population growth in the state $\ket{m}$, but if $\gamma_{\alpha}\ll \gamma_{\beta}$, the state $\ket{m}$ lives for a short period of time and another photon is within a short time delay
emitted in the second emission. The latter situation (slow emission followed by fast emission) is the circumstance that we mainly focus on in this work.
\section*{Power Spectrum Analysis: First-order Correlation Function}
We assume that at time $t = 0$ the
emitter is in the excited state $\ket{e}$ and the field modes are in the vacuum
state $\ket{0}$. Given this, the state vector of the particle-field system at time $t$ is described by
\begin{eqnarray}
\begin{aligned}
\ket{\psi(t)}=
\eta_{e}(t)\ket{e,0}
+
\sum_{\textbf{k}}\eta_{m,\textbf{k}}(t)\ket{m,1_{\textbf{k}}}
+
\sum_{\textbf{k}\textbf{q}}\eta_{g,\textbf{kq}}(t)\ket{g,1_{\textbf{k}},1_{\textbf{q}}}
\label{state1}
\end{aligned}
\end{eqnarray}
where the symbol $\ket{1_{\textbf{k}},1_{\textbf{q}}}$ represents the tensor product $\ket{1_{\textbf{k}}} \otimes \ket{1_{\textbf{q}}}$
of two single photon states of frequency $\omega_{k(q)}$ of subsystem $\alpha(\beta)$ with amplitude of $\eta(\omega_{\textbf{k}},\omega_{\textbf{q}})\equiv \eta_{\textbf{kq}}$.
We determine the states of the particle and
radiation field as a function of time using the Weisskopf-Wigner approximation where the particle in an excited state decays to the ground state with a
characteristic lifetime but it does not make back and forth transitions.
From the Schr\"odinger equation we have
\begin{eqnarray}
\begin{aligned}
\ket{\dot{\psi}(t)}=
-\dfrac{i}{\hbar}\widehat{H}_{I}
\ket{\psi(t)}
\label{SEq}
\end{aligned}
\end{eqnarray}
Substituting \autoref{state1} in the Schr\"odinger equation, we arrive at the equations of motion for the amplitudes $\eta_{e}$, $\eta_{m,k}$ and $\eta_{g,k,q}$,
\begin{eqnarray}
\begin{aligned}
\dot{\eta}_{e}&=-i\sum_{\textbf{k}}g_{\alpha_{\textbf{k}}}\eta_{m,\textbf{k}}e^{i(\omega_{\alpha}-\omega_{k})t}\\
\dot{\eta}_{m,\textbf{k}}&=-ig_{\alpha_{\textbf{k}}}\eta_{e}e^{-i(\omega_{\alpha}-\omega_{k})t}
-i\sum_{\textbf{q}}
g_{\beta_{\textbf{q}}}\eta_{g,\textbf{kq}}e^{i(\omega_{\beta}-\omega_{q})t}
\\
\dot{\eta}_{g,\textbf{kq}}&=
-i
g_{\beta_{\textbf{q}}}\eta_{m,\textbf{k}}e^{-i(\omega_{\beta}-\omega_{q})t}
\label{eqmotion}
\end{aligned}
\end{eqnarray}
We assume that the modes of the field are closely spaced in frequency, so
we replace the summation over $\textbf{k}$ and $\textbf{q}$ by an integral, $\sum_{\textbf{k}}\rightarrow
2\dfrac{V}{(2\pi)^{3}}
\int^{2 \pi}_{0} d\phi
\int^{ \pi}_{0} \sin \theta d \theta
\int^{\infty}_{0} k^{2} dk$. Where V is the quantization volume.
The transition dipole; $e\bra{i}\textbf{r}\ket{j}=\boldsymbol\mu_{ij}$ and the radiative decay constants are then defined as
\begin{eqnarray}
\begin{aligned}
\gamma_{\alpha}&=\Gamma_{\alpha}/2= \dfrac{1}{4 \pi \epsilon_{0}}
\dfrac{4 \omega^{3}_{\alpha} \mu^{2}_{em}}{3\hbar c^{3}}\\
\gamma_{\beta}&=\Gamma_{\beta}/2= \dfrac{1}{4 \pi \epsilon_{0}}
\dfrac{4 \omega^{3}_{\beta} \mu^{2}_{mg}}{3\hbar c^{3}}
\end{aligned}
\end{eqnarray}
Here $g_{\alpha(\beta)}$ can be taken as a constant associated with the spontaneous emission rate \cite{Scullybook}.
We then carry out simple integration following \autoref{eqmotion} \cite{AvanakiJPCL2019}, and retrieve the probability amplitudes as
\begin{eqnarray}
\begin{aligned}
\eta_{m,\textbf{k}}(t)&=-g_{\alpha,\textbf{k}}
\dfrac
{e^{i(\omega_{k}-\omega_{\alpha})t-\gamma_{\alpha}t}-e^{-\gamma_{\beta}t}}
{(\omega_{k}-\omega_{\alpha})+i(\gamma_{\alpha}- \gamma_{\beta})}\\
\eta_{g,\textbf{kq}}(t)&=
\dfrac{g_{\alpha,\textbf{k}}g_{\beta,\textbf{q}}}{(\omega_{k}-\omega_{\alpha})+i(\gamma_{\alpha}-\gamma_{\beta})}
\left \{
\dfrac{1-e^{-\gamma_{\beta}t+i(\omega_{q}-\omega_{\beta})t}}{\omega_{q}-\omega_{\beta}+i\gamma_{\beta}} - \dfrac{1-e^{i(\omega_{k}+\omega_{q}-\omega_{\alpha}-\omega_{\beta})t-\gamma_{\alpha}t}}{\omega_{k}+\omega_{q}-\omega_{\alpha}-\omega_{\beta}+i\gamma_{\alpha}}
\right \}
\label{allcoeff}
\end{aligned}
\end{eqnarray}
The first emission arises from the transition from $\ket{e}$ to $\ket{m}$, at time $t$,
\begin{eqnarray}
\begin{aligned}
-\sum_{k}g_{\alpha}
\dfrac{\left \{
e^{i(\omega_{k}-\omega_{\alpha})t-\gamma_{\alpha}t}
-e^{-\gamma_{\beta}t}
\right \}}{(\omega_{k}-\omega_{\alpha})+i(\gamma_{\alpha}-\gamma_{\beta})}
\ket{m} \otimes \ket{1:\omega_{k},\alpha}
\end{aligned}
\end{eqnarray}
This photon has a Lorentzian distribution in frequency with the width $|\gamma_{\alpha}-\gamma_{\beta}|$. If $\gamma_{\alpha}\ll \gamma_{\beta}$, then the state does not
stay for a long time, the second photon is emitted
soon such that the state is given by
\begin{eqnarray}
\begin{aligned}
\sum_{k,q}
\dfrac{g_{\alpha}g_{\beta}}{(\omega_{k}-\omega_{\alpha})+i(\gamma_{\alpha}-\gamma_{\beta})}
\left \{
\dfrac{1-e^{-\gamma_{\beta}t+i(\omega_{q}-\omega_{\beta})t}}{\omega_{q}-\omega_{\beta}+i\gamma_{\beta}} - \dfrac{1-e^{i(\omega_{k}+\omega_{q}-\omega_{\alpha}-\omega_{\beta})t-\gamma_{\alpha}t}}{\omega_{k}+\omega_{q}-\omega_{\alpha}-\omega_{\beta}+i\gamma_{\alpha}}
\right \}\\
\ket{g}\otimes
\ket{1_{k},\alpha;1_{q},\beta}
\label{II_Cas}
\end{aligned}
\end{eqnarray}
Defining $\mathcal{N}_{EP}$ as the coefficient of the two-photon state ; $\mathcal{N}_{EP}=g_{\alpha}g_{\beta}=\dfrac{2c\sqrt{\gamma_{\alpha} \gamma_{\beta}}}{V^{1/3}}$, we may simplify the two-photon state in the long time limit (times long compared to the radiative decay, $t\gg \gamma_{\alpha}^{-1},\gamma_{\beta}^{-1} $):
\begin{eqnarray}
\begin{aligned}
\ket{2P_{cas}}&=\sum_{kq}\eta^{cas}_{\textbf{k},\textbf{q}} \ket{1_{k},\alpha;1_{q},\beta}
\\
\eta^{cas}_{\textbf{k},\textbf{q}}&=
\dfrac{\mathcal{N}_{EP}}{
{(\omega_{q}-\omega_{\beta}+i\gamma_{\beta}) (\omega_{k}+\omega_{q}-\omega_{\alpha}-\omega_{\beta}+i\gamma_{\alpha}})}
\label{II_Cas1L}
\end{aligned}
\end{eqnarray}
It should be noted that in this limit, both $\eta_{m,\textbf{k}}(t)$ and $\eta_{e}(t)$ are zero and only $\eta^{cas}_{\textbf{k},\textbf{q}}\equiv\eta_{g,\textbf{k},\textbf{q}}(\infty)$ survives which is known as the "joint spectral amplitude".
In order to visualize the spectra of the emitted photons, here we look into the one-photon correlation function $G^{(1)}(\tau)$, and its normalized counterpart $g^{(1)}(\tau )$, which gives the degree of first-order temporal coherence between the emission
fields at time $t$ and $t+\tau$ and takes the values $0 \le |g^{(1)}(\tau )| \le 1$ for all light sources \cite{Loudonbook, Scullybook}.
\begin{eqnarray}
\begin{aligned}
G^{(1)}_{m}(\tau)
&=\langle
E^{(-)}_{m}(t)
E^{(+)}_{m}(t+\tau)
\rangle\\
g^{(1)}_{m}(\tau)
&=\dfrac{\langle
E^{(-)}_{m}(t)
E^{(+)}_{m}(t+\tau)
\rangle}
{\langle E^{(-)}_{m}(t)E^{(+)}_{m}(t)\rangle}
\quad \quad m=\alpha, \beta
\end{aligned}
\end{eqnarray}
In the limit when the line width ($\gamma_{\alpha(\beta)}$) goes to zero, the light field is perfectly coherent and $g
^{(1)}(\infty) = 1$. Using the above expression, the first-order correlation for a linear polarized field $
E^{(+)}(\textbf{r},t)=\sum_{k} \hat a_\textbf{k}
e^{-i \omega_{k} t+i\textbf{k}\cdot \textbf{r}}$ is obtained as;
\begin{eqnarray}
\begin{aligned}
G^{(1)}_{\alpha}(\tau)=e^{i\omega_{\alpha}\tau -(\gamma_{\alpha}+\gamma_{\beta})|\tau|},\quad \quad \quad
G^{(1)}_{\beta}(\tau)=e^{i\omega_{\beta}\tau -\gamma_{\beta}|\tau|}
\end{aligned}
\end{eqnarray}
However, an important property of the first-order correlation function is that it forms a Fourier transform pair with the power spectrum expressed as: $S(\omega)=\dfrac{1}{\pi} Re \int^{\infty} _{0}d \tau G^{(1)}(\tau) e^{-i\omega \tau}$.
The spectrum is obtained by performing a photon number measurement for a specific mode on a given state, i.e. for a given two-photon state density $\rho$, its spectrum is given by $ S(\omega)=\text{Tr}[\hat a^{\dagger}(\omega)\hat a(\omega)\rho]$.
Using \autoref{allcoeff}, we arrive at
\begin{eqnarray}
\begin{aligned}
G^{(1)}_{\alpha}(\omega) & = \sum_{k}|\eta_{m}(\omega,\omega_{k})|^{2}
\\
G^{(1)}_{\beta}(\omega) &= \sum_{k}|\eta_{g}(\omega_{k},\omega)|^{2}
\end{aligned}
\end{eqnarray}
Hence, the distribution of the emitted photons in the frequency domain is the power spectrum, expressed as
\begin{eqnarray}
\begin{aligned}
S_{\alpha}(\omega)&
=
\dfrac{2c}{V^{1/3}}
\dfrac{\gamma_{\alpha}+\gamma_{\beta}}{(\omega-\omega_{\alpha})^{2}+(\gamma_{\alpha}+\gamma_{\beta})^{2}}
\\
S_{\beta}(\omega)&
=
\dfrac{2c}{V^{1/3}}
\dfrac{\gamma_{\beta}}{(\omega-\omega_{\beta})^{2}+\gamma_{\beta}^{2}}
\end{aligned}
\end{eqnarray}
\begin{figure*}[h]
\begin{center}
\includegraphics[clip,width=0.6\columnwidth]{Figs/Spec.png}
\caption{
Total spectral density in cascade fluorescence (\autoref{SpecTot}). In these set of calculations we assume that $\gamma_{\alpha}=0.005~\text{GHz}$, $\omega_{\alpha}=1.5~\text{GHz}$ and $\omega_{\beta}=3.5~\text{GHz}$, as described in the text. Different colors show how the spectra and density change as the ratio of $\Gamma_{\beta \alpha}=\dfrac{\gamma_{\beta}}{\gamma_{\alpha}}$ varies. }
\label{Fig:Density_Spec}
\end{center}
\end{figure*}
We see that the
distribution $S_{\alpha}(\omega)$ associated with the first photon is given by a Lorentzian centered at $\omega_{\alpha}$ and the width at
half-maximum is the sum of the natural widths; $2(\gamma_{\alpha}+\gamma_{\beta})$. In the same fashion, the power spectrum of the second photon is given by $S_{\beta}(\omega)$, leading to a Lorentzian curve of width at half-maximum $2\gamma_{\beta}$, localized around $\omega_{\beta}$. These bandwidths are measures of
the coherence time of the emissions.
Note that for a general pure two-photon state $ \ket{2P_{cas}}=\sum_{kq}\eta(\omega_{k},\omega_{q}) \ket{1:\omega_{k},\alpha;1:\omega_{q},\beta}$, we define the spectral density based on summing up the lineshapes of the photons:
$S_{2EP}(\omega)=\sum_{k} |\eta(\omega_{k},\omega)|^{2}+|\eta(\omega,\omega_{k})|^{2}$.
This means, the emitted photon power spectrum is only determined by the diagonal elements of the corresponding density matrix. This also indicates that the spectrum of a pure two-photon state is the same as that from the corresponding diagonal density matrix.
Therefore the total spectral density is a blend of two Lorentzian functions with central frequencies $\omega_{\alpha}$, $\omega_{\beta}$
and widths $\gamma_{\alpha}+\gamma_{\beta}$ and $\gamma_{\beta}$ respectively.
\begin{eqnarray}
\begin{aligned}
S_{2EP}(\omega)=
\dfrac{2c}{V^{1/3}}
\Big[
\dfrac{\gamma_{\alpha}+\gamma_{\beta}}{(\omega-\omega_{\alpha})^{2}+(\gamma_{\beta}+\gamma_{\alpha})^{2}}
+
\dfrac{\gamma_{\beta}}{(\omega-\omega_{\beta})^{2}+\gamma_{\beta}^{2}}
\Big]
\label{SpecTot}
\end{aligned}
\end{eqnarray}
In \autoref{Fig:Density_Spec}, we plot the spectral density for cascade fluorescence (right panel).
Different colors show how the spectra change as the ratio of $\gamma_{\beta}/\gamma_{\alpha}$ varies.
The structure of the emission spectrum and the tail behavior of the density is in agreement with a previous work \cite{2018acsomega}. The bimodal shape of the spectrum originates from overlap of the two emissions as can be explained based on the contribution of each state according to its density of states.
In this figure, and elsewhere in this paper, we performed our calculations using parameters in the same range as polarization-entangled
photon pairs from a biexciton cascade from a single InAs QD embedded in a GaAs/AlAs planar microcavity \cite{CascadeSemiconNJPhys2006,CascadeSemiconNat2006}. In those experiments the pair of entangled photon emissions are at $1.398$~eV and $1.42$~eV. The following references \cite{Polariton-mediatedNat2014,Ebbesen_Angewandte2016,Ebbesen_Angewandte2017,Solid-state_Nat2017} are also relevant.
\section*{Second-order Correlation Function}
Now consider the HBT experiment with a multiphoton source, where we look for the rate of coincidences in the photon-count rates using two detectors. This rate is governed by a second-order correlation function defined as,
\begin{eqnarray}
\begin{aligned}
G^{(2)}(\bm{r},\bm{r'},t,t') =
\bra{\psi}
E^{(-)}(\bm{r},t)
E^{(-)}(\bm{r'},t')
E^{(+)}(\bm{r'},t')
E^{(+)}(\bm{r},t)
\ket{\psi}
\end{aligned}
\end{eqnarray}
in which the normalized form is
\begin{eqnarray}
\begin{aligned}
g^{(2)}(\bm{r},\bm{r'},t,t') =\dfrac{
\langle
E^{(-)}(\bm{r},t)
E^{(-)}(\bm{r'},t')
E^{(+)}(\bm{r'},t')
E^{(+)}(\bm{r},t)
\rangle
}
{
\langle
E^{(-)}(\bm{r},t)E^{(+)}(\bm{r},t)
\rangle
\langle
E^{(-)}(\bm{r'},t')E^{(+)}(\bm{r'},t')
\rangle
}
\label{g2origin}
\end{aligned}
\end{eqnarray}
Here $\ket{\psi}=\sum_{\bm{kq}}\ket{1_{\bm{k}}, 1_{\bm{q}}}$ is the two photon state, and the correlation function refers to detection of photon $\bm{k}$ (at $\bm{r}, t$) followed by detection of photon $\bm{q}$
(at $\bm{r'}, t'$).
In this work, the second-order
correlation function for two-photon emission from cascade emitters can be recast as
\begin{eqnarray}
\begin{aligned}
G^{(2)}(t,t') &=
\bra{2P_{cas}}
E^{(-)}_{\alpha}(t)
E^{(-)}_\beta(t')
E^{(+)}_\beta(t')
E^{(+)}_{\alpha}(t)
\ket{2P_{cas}}\\
&=\sum_{ \{n\} }
\bra{2P_{cas}}
E^{(-)}_{\alpha}(t)E^{(-)}_{\beta}(t')\ket{ \{n\} }
\bra{\{n\}}
E^{(+)}_{\beta}(t')E^{(+)}_{\alpha}(t)
\ket{2P_{cas}}\\
& =
\bra{2P_{cas}}
E^{(-)}_{\alpha}(t)E^{(-)}_{\beta}(t') \ket{0}
\bra{0}
E^{(+)}_{\beta}(t')E^{(+)}_{\alpha}(t)
\ket{2P_{cas}}
\\
&=\Psi^{*(2)}(t,t') \Psi^{(2)}(t,t')
\label{G2}
\end{aligned}
\end{eqnarray}
Where we defined $\Psi^{(2)}(t,t') \equiv \bra{0}
E^{(+)}_{\alpha}(t)E^{(+)}_{\beta}(t')
\ket{2P_{cas}}$. Note here that a complete set of states; ($\sum_{\{n\}}\ket{\{n\}}
\bra{\{n\}}=1$) is included.
Since our two-photon state consists of $\ket{1_{k}, 1_{q}}$ and is annihilated by $E^{(+)}(t)E^{(+)}(t')$,
only the
$\ket{0}
\bra{0}$ term survives, making the final form of \autoref{G2} relatively simple. (It is also true that only the vacuum level persists at long times in the complete wavefunction).
Making use of the two-photon state introduced earlier in \autoref{II_Cas1L} for the detection at times $t$ and $t+\tau$, we then arrive at\cite{Scullybook}
\begin{eqnarray}
\begin{aligned}
\Psi^{(2)}(t,t+\tau)
&\equiv
\bra{0}
E^{(+)}_{\alpha}(t)E^{(+)}_{\beta}(t+\tau)
\ket{2P_{cas}}\\
&=\sum_{kq}\eta_{cas} e^{-i\omega_{k}t-i\omega_{q}(t+\tau)}
\\
&=
-\dfrac{V^{1/3}\sqrt{\gamma_{\alpha}\gamma_{\beta}}}{c}
e^{-(i\omega_{\alpha}+i\omega_{\beta}+\gamma_{\alpha})t}
\Theta(t)
e^{-(i\omega_{\beta}+\gamma_{\beta})\tau}
\Theta(\tau)
\label{psi2ttau}
\end{aligned}
\end{eqnarray}
Here $\Theta(t)$ is a unit step function. If we are considering the HBT experimental setup with two detectors ($D_{1}$ and $D_{2}$) for the measurements, the first term in the above expression indicates that the $\omega_{\alpha}$ photon goes to the $D_{1}$ detector and the second photon $\omega_{\beta}$ to $D_{2}$. This amplitude should be added to the vice-versa situation to determine the total amplitude, where the latter has the same form here since both detectors are assumed to be located the same distance from the QD source.
Substituting the final expression from \autoref{psi2ttau} into \autoref{G2} and ultimately back into \autoref{g2origin}, we see in our bipartite system, the correlations
between parts can be determined with the help of the following (normalized) cross temporal correlation function,
\begin{eqnarray}
\begin{aligned}
g^{(2)}_{\times}(t,t+\tau) &=
\dfrac{\bra{2P_{cas}}
E^{(-)}_{\alpha}(t) E^{(-)}_{\beta}(t+\tau)
E^{(+)}_{\beta}(t+\tau)
E^{(+)}_{\alpha}(t)
\ket{2P_{cas}}}
{\bra{2P_{cas}}
E^{(-)}_{\alpha}(t)
E^{(+)}_{\alpha}(t)
\ket{2P_{cas}}
\bra{2P_{cas}}
E^{(-)}_{\beta}(t+\tau)
E^{(+)}_{\beta}(t+\tau)
\ket{2P_{cas}}
}
\\
&\approx
\dfrac{V^{2/3}}{c^{2}}
\dfrac{{\gamma_{\alpha}\gamma_{\beta}}}{\pi^{2}}
(\dfrac{\gamma_{\beta}}{\gamma_{\alpha}}-1)
\Big[
\Theta(t) e^{-2\gamma_{\alpha}t}
\Theta(\tau) e^{-2\gamma_{\beta}\tau }
+
\Theta(-\tau)e^{2(\gamma_{\beta}-\gamma_{\alpha})\tau }
\Theta(t+\tau) e^{-2\gamma_{\alpha}t}
\Big]
\label{g2xtimetau}
\end{aligned}
\end{eqnarray}
With regards to the use of entangled photons in quantum photonics, there are always some concerns about pure and reproducible entangled photon generation if the QD is degraded or if the QD is re-excited after the entangled photons are emitted or
background photons are present. Therefore the purity of the single-photon source is critical for high fidelity QD-photon entanglement, and this generally can be evaluated through the HBT setup, where the following experimentally relevant cross-correlation function is measured,
\begin{eqnarray}
\begin{aligned}
g^{(2)}_{\times}(\tau) &=\langle g^{(2)}_{\times}(t,t+\tau)\rangle_{t}=\lim_{T\to\infty}\int^{T}_{-T} g^{(2)}_{\times}(t,t+\tau) dt
\end{aligned}
\end{eqnarray}
Here the total detection time $T$ is taken to be long compared to
the single photon pulsewidth ($T\rightarrow \infty$).
The above formulation
involves calculating the normalized time dependent second-order correlation function after integrating for a long enough time, $\langle g^{(2)}_{\times}(t,t+\tau)\rangle_{t}$ \cite{Ripka_science2018}. The purity of the system
as a single-photon source is then extracted from $g^{(2)}_{\times}(\tau=0)$ \cite{Crocker_Express2019,Silva_SciRep2016,Kiraz_PRA2004}. For our three-level model of Cascade emission the second-order correlation function is found to be
\begin{eqnarray}
\begin{aligned}
g^{(2)}_{\times}(\tau)
&\approx
\dfrac{V^{1/3}}{c}
\dfrac{{\gamma_{\beta}}}{\pi}
(\dfrac{\gamma_{\beta}}{\gamma_{\alpha}}-1)
\Big[\Theta(\tau) e^{-2\gamma_{\beta}\tau }+\Theta(-\tau) e^{2\gamma_{\beta}\tau }
\Big]
\end{aligned}
\end{eqnarray}
Here we see that $g^{(2)}_{\times}(\tau)$ decays exponentially with $\lvert\tau\rvert$, which makes sense given that the second photon is emitted very shortly after the first. Also, $g^{(2)}_{\times}(\tau=0)$ depends on the radiative decay rates of the two emissions, which is in agreement with the previous studies \citep{Ahn_OptExpress2020,Ripka_science2018,Hamsen_Natphys2018}, and the quantization volume also plays a role \cite{Ripka_science2018}. When $\gamma_{\alpha}\ll \gamma_{\beta} $, $g^{(2)}_{\times}(0) >0$, and a positive pure cross correlation is found.
Generally, the area of the peak; $g^{(2)}_{\times}(\tau)$ around
$\tau\sim 0$ (0th peak) gives the normalized coincidence detection probability when two photons are incident in different
inputs of the beam splitter in the HBT experimental setup. Only in the limit $\gamma_{\alpha}= \gamma_{\beta} $ does this area go to zero. This makes sense as in this limit there is no entanglement (see later discussion of Schmidt numbers).
In this derivation, we have assumed that a light pulse interacts with the system to produce the initial biexciton excited state, and then this decays with no correlation to its initial preparation. However in the actual experimental setup a series of pulses is typically used to excite the system, and there may exist some amplitude from a previous pulse that has not decayed to zero when the next pulse arrives. This leads to a series of peaks in the $g^{(2)}_{\times}(\tau)$ function, but this is not important in the present study. We also note that the inclusion of coherence between the excitation and emission steps will lead to a more complex peak at $\tau=0$, as has often been discussed for other emitters \cite{Crocker_Express2019,Silva_SciRep2016,Kiraz_PRA2004, Yu-Ming_Nat2016}.
\begin{figure}
\centering
\includegraphics[width=.49\textwidth]{Figs/gx2_wk_wq_new.png}
\includegraphics[width=.47\textwidth]{Figs/gx2_t_tau2.png}
\caption{Left: Second-order frequency cross correlation function $g^{(2)}_{\times}(\omega, \omega')$ for the cascade emission process in the frequency domain (logarithmic scale).
Right: $g^{(2)}_{\times}(t,t+\tau)$, the normalized cross temporal correlation function. Here $\gamma_{\beta}/\gamma_{\alpha}=40$, $\gamma_{\alpha}=0.005~\text{GHz}$, $\omega_{\alpha}=1.5~\text{GHz}$ and $\omega_{\beta}=3.5~\text{GHz}$. }
\label{fig:g2xwt}
\end{figure}
As we did in the time domain, in order to fully describe the correlations between two emissions, at different frequencies, $ \omega$ and $ \omega'$, one would have to compute a double Fourier transform according to the cross-correlation definition;
$g^{(2)}_{\times}(\omega, \omega')=\dfrac{1}{\pi^{2}} \Re \int^{\infty} _{-\infty}\int^{\infty} _{-\infty}\,dt \,dt' e^{-i\omega t}e^{-i\omega' t'}g^{(2)}_{\times}(t,t') $.
This function provides a measure of resemblance of the two photons as a function of the frequency displacement of one relative to the other. Applying the cross-correlation definition to \autoref{g2xtimetau}, we obtain:
\begin{eqnarray}
\begin{aligned}
g^{(2)}_{\times}(\omega,\omega')
&\approx
\dfrac{{\gamma_{\alpha}\gamma_{\beta}}}{\pi^{2}}
(\dfrac{\gamma_{\beta}}{\gamma_{\alpha}}-1)
\dfrac{
4\gamma_{\alpha}\gamma_{\beta}+ (\omega+\omega'-\omega_{\alpha}-\omega_{\beta})(\omega'-\omega_{\beta})
}
{[(\omega+\omega'-\omega_{\alpha}-\omega_{\beta})^{2}+4\gamma_{\alpha}^{2}][(\omega'-\omega_{\beta})^{2}+4\gamma_{\beta}^{2}]}
\label{g2xwt}
\end{aligned}
\end{eqnarray}
This function is useful as it determines the width of the frequency anticorrelation associated with the two emitted photons.
In Fig.~(\ref{fig:g2xwt}), the $g^{(2)}_{\times}$ function in both time (right) and frequency (left) domains is depicted.
In the left plot, one observes that the function
has significant values only on the anti-diagonal, along the line $\omega+\omega'=\omega_{\alpha}+\omega_{\beta}$. This implies that the corresponding states exhibit strong frequency anticorrelation. The width of the anti-diagonal which gives the characteristic width of
the frequency anticorrelation,
is equal to $\gamma_{\alpha}$.
In the right panel of Fig.~(\ref{fig:g2xwt}), $g^{(2)}_{\times}$ from \autoref{g2xtimetau} is plotted versus $t$ for different values of time delay $\tau$ and with $\gamma_{\alpha} <\gamma_{\beta}$. The illustration shows the expected exponential decay of $g^{(2)}_{\times}$ with $t$ as determined by the $\gamma_{\alpha}$ rate, and also that there is exponential decay as a function of $\tau$ as determined by $\gamma_{\beta}$. Note that
negative anti-correlation is obtained when $\gamma_{\alpha} >\gamma_{\beta}$, which is consistent with the recent study \cite{Ahn_OptExpress2020}.
\section*{Heralded Single Photons}
So far we have studied the properties of emitted entangled photons and the role of the relevant spectral parameters in the emission spectrum. In this section, we focus on the influence of frequency correlation on the purity of
heralded single photons that are derived from the two photon state. In the frequency dependence of the general state from a cascade emitter,
the two-photon component
can be obtained from \autoref{II_Cas1L} which represents a pure state. Here the joint spectral amplitude generally contains correlations between
frequencies of the sibling photons. As a result of this combination of purity and correlation,
$\ket{2P_{cas}}$ is entangled in the frequency of the two product photons. The purity of either heralded single photon that originates from $\ket{2P_{cas}}$ can then be determined from the
density matrix. It is worth recalling that, this property is $inversely$ related to the degree of the entanglement of our two-photon state.
For a bipartite two-photon source, the density matrix (from \autoref{II_Cas1L}) reads as
\begin{eqnarray}
\begin{aligned}
\rho^{cas}
&=
\ket{2P_{cas}}\bra{2P_{cas}}=\sum_{\textbf{kq}}
|\eta^{cas}_{\textbf{k},\textbf{q}}|^{2} \ket{1_{\textbf{k}},\alpha;1_{\textbf{q}},\beta}
\bra{1_{\textbf{k}},\alpha;1_{\textbf{q}},\beta}
\label{rate}
\end{aligned}
\end{eqnarray}
where $|\eta^{cas}_{\textbf{k},\textbf{q}}|^{2}$ is the joint spectral probability density.
The purity of either heralded single photon derived from $\ket{2P_{cas}}$ is related to the two reduced density operators of the partner photons, given by:
\begin{eqnarray}
\begin{aligned}
\rho_{\alpha}=\text{Tr}_{\beta}\rho= \sum_{k} \xi_{k} \ket{1:\omega_{k}, \alpha}\bra{1:\omega_{k}, \alpha},
\quad
\quad
\text{where}
\quad
\xi_{k}=\sum_{q}\eta_{kq}
\\
\rho_{\beta}=\text{Tr}_{\alpha}\rho= \sum_{q} \zeta_{q} \ket{1:\omega_{q}, \beta}\bra{1:\omega_{q}, \beta},
\quad
\quad
\text{where}
\quad
\zeta_{q}=\sum_{k}\eta_{kq}.
\end{aligned}
\end{eqnarray}
The purity of the individual photons is then determined using
\begin{eqnarray}
\begin{aligned}
\mathcal{P}_{\alpha}=\text{Tr}(\rho^{2}_{\alpha}) ,
\quad
\quad
\mathcal{P}_{\beta}=\text{Tr}(\rho^{2}_{\beta}).
\label{purity1}
\end{aligned}
\end{eqnarray}
To see how the spectral correlations in $\ket{2P_{cas}}$ are involved in the purity of the heralded single photons, we examine the Schmidt decomposition of the joint two-photon state. Schmidt decomposition is a characteristic method for describing a bipartite system in terms of a complete set of basis states. Through this decomposition, one can calculate the Schmidt number which defines the “degree” of entanglement of the two-photon state. In this decomposition, \autoref{II_Cas1L} becomes\cite{Eberly_PRL2000,Eberly2006,Chen2017}
\begin{eqnarray}
\begin{aligned}
\ket{2P_{cas}}= \sum_{k}\sqrt{\lambda_{k}}
\ket{\phi^{\alpha}_{k}}
\otimes
\ket{\phi^{\beta}_{k}},
\quad\quad
\text{where}
\quad
\bra{\phi^{\mu}_{k}}\ket{\phi^{\mu}_{q}}
=\delta_{kq},
\quad
\text{and}
\quad
\sum_{k} \lambda_{k}=1.
\label{decompose1}
\end{aligned}
\end{eqnarray}
The orthonormal basis states $\ket{\phi^{\mu}_{k(q)}}$ ($\mu=\alpha, \beta$ ) are known as Schmidt modes which can be thought of as the basic building
blocks of entanglement in the sense that if the first
photon is described by a function $\ket{\phi^{\alpha}_{k}}$,
we know with certainty that its second sibling is determined
by the corresponding function $\ket{\phi^{\beta}_{k}}$.
Note that each set depends only on one subsystem of $\ket{2P_{cas}}$ and each pair of modes is weighted by its Schmidt magnitude, $\lambda_{k}$. The number of mandatory non-zero components in the sum needed to construct the $\ket{2P_{cas}}$ state (in terms of its Schmidt modes), indicates the effective number of modes that are correlated, while the homogeneity in the distribution of coefficients is determined by the
Schmidt number, $\kappa$, expressed as
\begin{eqnarray}
\begin{aligned}
\kappa=\dfrac{1}{\text{Tr}(\rho^{2}_{\alpha})}
=
\dfrac{1}{\text{Tr}(\rho^{2}_{\beta})}
=
\dfrac{1}{\sum_{k=1}^{N}\lambda^{2}_{k}}
\label{Schnumb1}
\end{aligned}
\end{eqnarray}
Comparing \autoref{Schnumb1} and \autoref{purity1}, we realize that the purity of both reduced states is equal to the sum of the squares of the Schmidt coefficients and thus the inverse of the Schmidt number; $\mathcal{P}_{\alpha(\beta)}= 1/ \kappa$.
From the experimental point of view, this is relevant to the expected number of required modes.
It should also be pointed out that the number of non-zero Schmidt
coefficients in the sum is called the Schmidt rank, or sometimes the dimensionality of the entanglement, as it represents the minimum local Hilbert space dimension required to correctly represent correlations of the quantum state \cite{Chen2017}.
\section*{Schmidt Analysis}
We now provide an analytical determination of the Schmidt number for the cascade emitter. Different numerical and analytical frameworks for the Schmidt decomposition of paired photons have been proposed \cite{Eberly_PRL2000,Lamata_2005, Eberly2006,Chen2017} which we can use to advantage for characterizing separability/purity of our bipartite two-photon state.
To describe the cascade emission presented by $\ket{2P_{cas},\alpha \beta}=\sum_{k,q}\eta(\omega_{k},\omega_{q})\ket{1:\omega_{k},\alpha;1:\omega_{q},\beta}$, we introduce $
\ket{\psi_{k}}\propto \sum_{m}\eta(\omega_{m},\omega_{k}) \ket{1:\omega_{m}}
$ as a basis set in which we should find the normalization factor first ( i.e. $\bra{\psi_{j}}\ket{\psi_{k}}=\sum_{m}\eta(\omega_{m},\omega_{k})\eta^{*}(\omega_{m},\omega_{j})$). We then construct the reduced density matrix.
In our mathematical approach, we approximate the normalized $\ket{\psi_{k}}$ as a piece-wise state $\ket{\phi_{s}}$ by discretizing $\omega_{k}$ into a small interval of $2 \gamma_{\alpha}$, i.e. $\omega_{k}= 2 N_{s}\gamma_{\alpha}$ where $N_{s}$ is an integer.
The normalized basis set is therefore defined as
\begin{eqnarray}
\begin{aligned}
\ket{\phi_{s}}&\approx
\sqrt{\dfrac{\pi}{\gamma_{\beta}}
\dfrac {(2\gamma_{\alpha} N_{s}-\omega_{\beta})^{2}+\gamma_{\beta}^{2} }
{4\gamma_{\alpha}^{2}}}
\int^{2\gamma_{\alpha} (N_{s}+1)}_{2\gamma_{\alpha} N_{s}}\sum_{m}\eta(\omega_{m},\omega)d\omega \ket{1:\omega_{m}} \quad \quad N_{s}=\omega_{k}/2\gamma_{\alpha}
\end{aligned}
\end{eqnarray}
Here we have approximated the sum over 'm' as $\sum_{m} \leftrightarrow \int \dfrac{L}{2\pi c}d \omega_{m}$ and used Cauchy's integral theorem. Then the orthonormal relations read as $ \bra{\phi_{r}}\ket{\phi_{s}}
\approx \delta_{rs}$ and we obtain the reduced density matrix within a very good approximation as (see more details in SI).
\begin{eqnarray}
\begin{aligned}
\rho^{red}
&=\sum_{N_{s}}\dfrac{\gamma_{\beta}}{\pi}
\dfrac{2\gamma_{\alpha}}
{(2\gamma_{\alpha} N_{s}-\omega_{\beta})^{2}+\gamma_{\beta}^{2} }
\ket{\phi_{s}}\bra{\phi_{s}}
\end{aligned}
\end{eqnarray}
Accordingly the “Schmidt coefficients” $\{ \sqrt{\lambda_{s}}\}$ are given by the square roots of the eigenvalues of the reduced density matrix, $\sqrt{\lambda_{s}}
\approx
\sqrt{\dfrac{\gamma_{\beta}}{\pi}
\dfrac{2\gamma_{\alpha}}
{(2\gamma_{\alpha} N_{s}-\omega_{\beta})^{2}+\gamma_{\beta}^{2} }
}$
and we can simply show that $\sum_{s} \lambda_{s}\approx 1$.
Generally finding an analytical expression for Schmidt modes and Schmidt number is tricky and complicated. However using the purity definition which is the inverse of the Schmidt number, it is possible to obtain the Schmidt number with no further approximation.
Thus the Schmidt number of the bipartite two photon
state generated by cascade emission can be obtained from \autoref{Schnumb1}.
The resulting formula after some lengthy algebra is:
\begin{eqnarray}
\begin{aligned}
\kappa=\dfrac{1}{\sum_{mn}|\sum_{k}\eta_{km}\eta^{*}_{kn}|^{2}}=1+\dfrac{\gamma_{\beta}}{\gamma_{\alpha}}
\end{aligned}
\end{eqnarray}
We see that the minimum Schmidt number from this analysis falls at unity
corresponding to the limit $\gamma_{\alpha}\gg \gamma_{\beta}$. For this situation, the population
of the doubly excited state in the three-level QD source will decay to the intermediate state within a very short time, and then the transition from intermediate level to ground state occurs on a longer time scale. Therefore, the total energy of the two-photon state varies significantly as a function of time, and there is weak frequency anti-correlation. Also, if $\gamma_{\alpha}$ is extremely large, de-excitation from the top state occurs instantaneously after coupling to the radiation field is turned on, and the second photon is uncorrelated from the first.
If $\gamma_{\alpha}$ is very small compared with $\gamma_{\beta}$, then the second photon is emitted soon after the first photon emission, and the fluctuations in the total energy of the two-photon state will be negligible. This implies that the two photons have strong frequency anti-correlation, which is consistent with our earlier analysis. In the extreme case where $\gamma_{\alpha}$ is close to zero, the photon pairs are fully frequency-anticorrelated, the expression for $\eta_{kq}$ breaks down into $\eta(\omega_{k},\omega-\omega_{k})$ , and the state is clearly in the form of a Schmidt decomposition.
\begin{figure}
\centering
\includegraphics[width=1.05
\textwidth]{Figs/JointSpect_NoPol_k.png}\\
\caption{Joint spectral density (JSD) of QD emitter in three-level model. JSD profile goes from
a broad linewidth ($\gamma_{\alpha}$) to being
symmetric along the digonal while the ratio of $\Gamma_{\beta \alpha}=\gamma_{\beta}/\gamma_{\alpha}$ is increased. Here we assume that $\gamma_{\alpha}=0.005~\text{GHz}$, $\omega_{\alpha}=1.5~\text{GHz}$ and $\omega_{\beta}=3.5~\text{GHz}$.
}
\label{fig:JointSpect}
\end{figure}
Due to the flatness of the distribution of Schmidt coefficients, the corresponding Schmidt number can be very large. From this we conclude that a larger ratio of $\gamma_{\beta}/\gamma_{\alpha}$ gives rise to better frequency correlations i.e. stronger entanglement of the total state.
In contrast to the high entanglement case, to collect heralded single photons in a pure state, one must ensure that the relevant parameters of the system closely meet the condition; $\gamma_{\alpha}\gg \gamma_{\beta}$. This also means that to separate the entangled photons into two single photons, one should reduce
correlations such that the Schmidt number has a low value. For the cascade source we consider, the only way this can be done is to make the joint spectral density given by $|\eta^{cas}_{\textbf{k},\textbf{q}}|^{2}$ be factorable.
We plot the Joint Spectral Density (JSD) in \autoref{fig:JointSpect}. This characterizes the joint spectrum of the two photons, and it can be manipulated
via the emission bandwidths and by other relevant parameters of the two photon state.
Experimentaly this can be done by measuring the JSD profile with tunable narrow band filters \cite{Valencia_PRL2007} in a same manner as HOM quantum interference quantifies the two-photon coherence bandwidth and
the indistinguishability of the photon pair. This is a common method for heralding the signal and idler photons in a parametric down conversion experiment.
The JSDs in \autoref{fig:JointSpect} illustrate how the frequency correlation is related to the degree of entanglement of states of the two-photons when the ratio of $\gamma_{\beta}/\gamma_{\alpha}$ is changed. Here the vertical and horizontal axes show the frequencies of the first and second emissions. The direct consequences of the entanglement are seen when the Schmidt number is decreased (going from the left to the right) over the range $\kappa=21-2$.
The results indicate that the probability of frequency correlation is highest in a very short range close to the transition frequencies, $\omega_{\alpha}$ and $\omega_{\beta}$. Noticeably we see that the distribution is highly aligned with the anti-diagonal wherein $\omega_{\alpha}+\omega_{\beta}= \omega_{k}+\omega_{q}$ when $\kappa$ is high. As $\kappa$ is decreased there is a broadening of the distribution centered on $\omega_{q}=\omega_{\beta}$.
Also, the correlation intensity is reduced as the JSD profile goes from being closely aligned along the anti-diagonal to a broadened line shape. We may conclude that the more asymmetric and spread-out is the spectral density, the less entangled the photons are.
The images in figure \autoref{fig:JointSpect} also zoom in for a smaller frequency range around the anti-diagonal, and paying closer attention to the middle part of these distributions in which the intensity is very high.
Overall, we recognize that a biphoton state can be ideally suited for generating heralded pure-state single photons when the side lobes that hinder the generation of the entangled photon, symmetrically are enhanced.
Although the JSD is very informative about the properties of the states, the JSD alone is insufficient to conclude that they are frequency entangled \cite{Chen2017}.
\section*{Polarization Effects in Entangled Photons}
\begin{figure*}[h]
\begin{center}
\includegraphics[clip,width=0.95\columnwidth]{Figs/4Level_pol.png}
\caption{ Entangled photon generation from biexciton cascade emission: The final two-photon state is created by sequential emission of two photons (XX and X) separated by a short time delay $\tau$. The resulting state is a superposition of horizontal (H) and vertical (V) polarization states. }
\label{Fig:QD_FSS2}
\end{center}
\end{figure*}
As we explained in the introduction section,
it is more appropriate if we define a QD biexciton cascade
using a four-level system composed of a biexciton state (XX),
two bright intermediate exciton levels ($X_{H(V)}$) with different polarizations(either horizontal (H) or vertical (V)) and a ground state (g).
Therefore, the decay proceeds via one of two paths (See \autoref{Fig:QD_FSS2}).
Here we assume that the system is initially
in a superposition of the biexciton-exciton photonic states. After emitting the first photon, it evolves to the exciton (X) state in which the degeneracy is split. The quantum dot remains in a superposition of $X_{H}$ and $X_{V}$ for a time delay $\tau_{e}$, during which a phase difference develops due to the fine structure splitting $\Delta_{\text{FSS}}$ between different exciton states. Finally, the exciton photon $X_{H(V)}$ with the same polarization as the first biexciton photon is emitted, and the QD goes back to the ground state.
The system is now found in a superposition of orthogonally polarized photon pair states, with a phase between them that is characterized by the time delay $\tau_{e}$
(generally in the order of tens of ps).
If we denote the state of the photon in each emission by the corresponding state of the QD,
including for the polarization effect, our cascade two-photon emission wavefunction in \autoref{II_Cas1L} is modified to:
\begin{eqnarray}
\begin{aligned}
\ket{2P_{cas}}&= \dfrac{1}{\sqrt{2}}\Big(
\sum_{p,r}\eta^{(H)}_{p,r}\ket{XX_{H}X_{H}} +
\sum_{q,s}\eta^{(V)}_{q,s}\ket{XX_{V}X_{V}}
\Big) \quad \quad\text{for}\quad \Delta_{\text{FSS}}=0
\\
\ket{2P_{cas}}&= \dfrac{1}{\sqrt{2}}\Big(
\sum_{p,r}\eta^{(H)}_{p,r} \ket{XX_{H}X_{H}} +
\sum_{q,s}\eta^{(V)}_{q,s}e^{i\Delta_{\text{FSS}} \tau_{e}/\hbar} \ket{XX_{V}X_{V}}
\Big)
\quad \quad \text{for}\quad \Delta_{\text{FSS}}\neq 0
\label{II_Cas1}
\end{aligned}
\end{eqnarray}
Here
\begin{eqnarray}
\begin{aligned}
\eta^{(H)}_{p,r}&=
\dfrac{\mathcal{N}_{1}}{
{(\omega_{r}-\omega_{\beta_{1}}+i\gamma_{\beta_{1}}) (\omega_{p}+\omega_{r}-\omega_{\alpha_{1}}-\omega_{\beta_{1}}+i\gamma_{\alpha_{1}}})}\\
\eta^{(V)}_{q,s}&=
\dfrac{\mathcal{N}_{2}}{
{(\omega_{s}-\omega_{\beta_{2}}+i\gamma_{\beta_{2}}) (\omega_{q}+\omega_{s}-\omega_{\alpha_{2}}-\omega_{\beta_{2}}+i\gamma_{\alpha_{2}}})}
\end{aligned}
\end{eqnarray}
So the joint spectral density is defined as
\begin{eqnarray}
\begin{aligned}
JS_{H-V}&=|\eta^{(H)}_{p,r}+ e^{i\Delta_{\text{FSS}} \tau_{e}/\hbar}\eta^{(V)}_{q,s}|^{2}
\label{JSI_Pol}
\end{aligned}
\end{eqnarray}
\begin{figure}
\centering
\includegraphics[width=1.05\textwidth]{Figs/JointSpect_Pol_FulNew2.png}
\caption{Joint spectral density in cascade emission from typical QD. The polarization of states is included. Different values of $\Gamma_{\beta \alpha}=\dfrac{\gamma_{\beta}}{\gamma_{\alpha}}$ and the phase $\phi=\Delta_{\text{FSS}} \tau_{e}/\hbar=\pi/4$ are used.}
\label{fig:JointSpect_FSS2}
\end{figure}
With this perspective, we have more degrees of freedom for choosing the relevant parameters of the system to control purity of the output photons.
Studies have shown that the maximum entanglement is obtained when the $\omega_{\alpha_{i}}=\omega_{\beta_{j}}$ and $\gamma_{\alpha_{i}}=\gamma_{\beta_{j}}$ ( where $i,j=1,2$) \cite{Hudson_PRL2007, Stevenson_PRL2008,Trotta_Nanolett2014,Winik_PRB2017}. We are mostly interested in predicting the degree of the entanglement by looking at the joint spectral density plotted in \autoref{fig:JointSpect_FSS2}.
Note that the horizontal axis is relative to the frequency of $X_{H}(_{V})$
in the excitonic transitions and the vertical axis is relevant to the frequency of
$XX_{H}(_{V})$ in the biexcitonic transitions.
Compared to the three-level model discussed earlier, here the model is closer to reality and we see more details in the JSD plots. More importantly we see the star shaped emission pattern of exciton and biexciton more explicitly at different frequencies when $\Gamma_{\beta \alpha}$ is higher.
As this ratio becomes smaller from left to right, the probability density becomes more circular in a narrow domain of frequency around $\omega_{\alpha}$ and $\omega_{\beta}$ which assures better separation (greater purity) in the photon production.
Note that the broken symmetry on the right hand plot can be improved by optimizing geometry of the QD experimentally \cite{CascadeSemiconNat2006,ManipulatingFSS_APL2007,LowerBoundPRL2010,Entangled_AkopianPRL2006,Cavity-assisted_PRB2009,Pathak_PRB2009} and removing the phase term, $\phi=\Delta_{\text{FSS}} \tau/\hbar$.
Full analysis and more details of other properties of cascade emitters with this model will be reported in our future work.
\section*{Conclusion}
In conclusion, we theoretically studied the underlying mechanism of entangled two-photon generation in semiconductor QD emitters including use of these emitters as an on-demand single photon source. We developed analytical expressions for the characteristic parameters associated with the first- and second- order correlation function, and the Schmidt number of the entangled cascade emission.
We extended our model by including for the effects of polarization and fine structure splitting, and the emission delay of the exciton
relative to the biexciton. The extended model broadens our vision to see the capacity of other relevant parameters for the practical application of semiconductor quantum dot emitters as single source emitters and offers more details about the underlying mechanism and purity properties of entangled photon production. Although we have only investigated this effect in the joint spectral density, it is straightforward to include other properties as well.
The theoretical studies and the analysis here provides guidelines for the experimental design and engineering of on-demand single photon source applications as diverse as quantum computing and quantum information.
\begin{suppinfo}
The analytical derivations of Schmidt number for cascade emission is explained here.
\end{suppinfo}
\begin{acknowledgement}
This work was supported by the U.S. National Science Foundation under Grant No. CHE-1760537. This research was supported in part through the computational resources and staff contributions provided for the Quest high performance computing facility at Northwestern University which is jointly supported by the Office of the Provost, the Office for Research, and Northwestern University Information Technology.
\end{acknowledgement}
|
2009.14040
|
\subsection{Big systems}
What are the implications of the statement that a system is ``big''? Firstly, some concepts that suit ``small'' systems do not suit large systems. One of the most obvious of these concepts is the assumption of global states and steps that update global states \cite{reisig2020component_models}. Global states and steps adequately describe, for example, the behaviour of s small digital circuit. To describe the behaviour of stakeholders of a business as a sequence of global steps, is, however, conceptually not adequate. In a big system, e.g. a business, cause and effect of a step are locally confined; and this confinement is essential to understand behaviour. As another specific concept, a big system requires conventions to confine validity of names, i.e.\ to avoid globally valid names, with a few exceptions such as URLs.
In \textsc{Heraklit}, single behaviours (runs, executions) of a subsystem can be represented by means of states and steps that are global only within the subsystem. Upon composing two such systems, those local states and steps are not necessarily embedded into global states and steps of the composed system. Instead, single behaviours of the composed system are represented without assuming global states and steps. Local names of a subsystem are confined to the subsystem and its direct neighboring subsystems.
\subsection {Composition of systems}
Every ``big'' real life system is composed from subsystems that are mutually related: they may exchange messages or jointly execute activities. The composition of subsystems is particularly challenging, as the subsystems themselves may be represented in different languages, modelling methods, etc.\cite{frank2014:modeling} Modelling techniques for such systems must provide means to compose models of subsystems. Many modelling techniques provide such means; they all come with specific, frequently parameterized composition operators, concentrating on special ways to exchange data, e.g.\ synchronously or asynchronously. A ``big'' system, composed from many systems $S_1, \ldots , S_n$, is favorably written
\begin{equation}
S = S_1\bullet \dots \bullet S_n,
\label{equ:composition}
\end{equation}
with ``$\bullet$'' being any version of a composition operator. This bracket free notation requires that the composition operator is associative, i.e.\ that for any three models $R$, $S$, $T$ hold: $(R \bullet S) \bullet T = R \bullet (S \bullet T)$. Typical examples for the notation \eqref{equ:composition} include supply chains, sequences of production machines in a factory, etc. Associativity of composition is rarely discussed explicitly, but frequently assumed without saying \cite{reisig2019associative}.
\textsc{Heraklit} offers a simple, universally employable and associative composition operator. In \textsc{Heraklit}, the diversity of specific, parameterized composition operators is expressed by help of \textit{adapters}: Specific aspects and properties of the composition $R \bullet S$ of two models $R$ and $S$ are formulated in an adapter $A$, such that $R \bullet A \bullet S$ expresses the wanted properties. The advantages of this concept are obvious: One technical composition operator fits all content-wise requirements, adapters can themselves be composed, etc.
\subsection{Abstraction and refinement}
A number of general principles has been proposed in literature, to adequately cover the abstraction and refinement of systems. In particular, it is most useful to start out with an abstract specification and to refine it systematically, such that properties of the refined system imply the relevant properties of the abstract system. Vice versa, a given system may be abstracted, yielding a more compact version.
Abstraction and refinement should harmonize with the composition. To refine a part $T$ of a system $S$, one would partition $S$ into $T$ and the environment of $T$, and then refine $T$. The remaining subsystems in the environment of $T$ should not be affected by this procedure. Systems on different refinement levels should be composable; an overall concept of hierarchy levels for subsystems should not be required. \textsc{Heraklit} suggests concepts for refinement and abstraction that respect these requirements.
\subsection{Modelling of data and things equally}
In a big system, data, physical items, algorithms, activities of persons, steps of organizations, etc., are entangled. They must be modelled by similar means that differentiate between them only in pragmatical aspects: data can be generated, deleted, transformed into different representations, manipulated by computers, copied, updated, composed, etc. Physical items behave differently: A physical item always occupies a distinguished place in space. In models, one frequently does not want to distinguish
``equal'' items explicitly; their number matters.
\subsection{Behaviour}
The behaviour of a large system is composed of single actions. An action updates some local state components. It is up to the modeler to embed local state components into more global views, if wanted. For a really large system, a single execution (run) should not be represented as a sequence of actions (though one may argue that all behaviour occurs along a global time scale). Independence of actions should explicitly be represented and not be spoiled by representing them in an arbitrary order.
\textsc{Heraklit} suggests to base the description of behaviour on Petri nets with data carrying tokens \cite{reisig2013understanding}.
This choice is motivated by multiple aspects:
\begin{itemize}
\item Petri nets can easily be specialized to include interfaces: Just select some places, transitions, and even arcs to serve as interface elements.
\item The composition of Petri nets with interfaces is again a Petri net with interfaces.
\item Petri nets suggest the notion of concurrent runs that partially order actions of a run, thus, avoiding them to be mapped onto a global time scale.
\end{itemize}
\subsection{Describing systems on a schematic level}
Data, real life items, as well as entire systems must be describable on an abstract, schematic level. In particular, it must be possible to describe just the existence of data, items, functions, etc., without any concrete description of how they look like, how many of them there are, etc. On this schematic level, it should be possible to describe activities in systems, e.g.\ the principles of executing a client’s order of an enterprise. A concrete enterprise is then an instantiation of the schema.
\textsc{Heraklit} provides techniques to model such schemata, and to characterize concrete enterprises as instantiations of such a schema. Here, we adapt notions such as structures, signatures and instantiations of signatures, that are well-known from first order logic and algebraic specifications. (Technically, a signature is just a set of sorted symbols for sets, constants, and functions. An instantiation interprets these symbols consistently). We extend signatures by requirements to exclude ``unwanted'' instantiations, in the spirit of specification languages such as the Z language.
Signatures and their instantiations can naturally be transferred to define Petri net schemata – we call them \textsc{Heraklit} schemata. Such a schema can be instantiated in different ways; each instantiation results in a concrete Petri net. This concept is useful to model, for example, not just a distinguished business, but a class of businesses that all follow the same business rules. Hence, \textsc{Heraklit} strongly supports the idea of reference modelling, a core topic of business informatics \cite{rehse2019:situation_ref_mod}.
\subsection{Verification}
The notion of correctness has many implications for big systems. Some ideal properties of a big system can be composed of corresponding properties of the component systems. Not all relevant properties can formally be captured, yet they deserve a proper framework to reason about them. Particularly interesting are methods to prove properties at run-time.
\textsc{Heraklit} integrates a number of formal and semi-formal verification techniques to support structured arguments about the correct behaviour of modules.
\section{Modules and their composition}
\subsection{Modules}
In Section 1 we discussed a number of principles that are inevitable for modelling big systems: no globally effective structures, associative composition of models of any two systems, composition must be compatible with abstraction, modelling of data and real items, modelling of behaviour, parameterized models. Now we must model systems in such a way that all these principles are met.
We start out with the obvious observation that a real system in general consists of interdependent subsystems. This paves the way for the central notion of \textsc{Heraklit}-modules: A \textsc{Heraklit} module is a model, graphically depicted as a rectangle, with two decisive components:
\begin{itemize}
\item Its inner: this may be any kind of graph or text. Three variants are frequent: (a) the inner consists only of the name of the module, (b) it consists of (connected) submodules, (c) it describes dynamic behaviour.
\item Its surface: this consists of gates, each gate is labelled, i.e.\ inscribed by a symbol. The gates of the surface are arranged on the surface of the module’s rectangle. Alternatively, each gate is represented as a line, linking the module’s rectangle with the gate’s label.
\end{itemize}
The following Fig.~\ref{fig:modules} shows typical \textsc{Heraklit} modules.
\subsection{Composition of modules}
Composing two modules $A$ and $B$ follows a simple idea: two equally labelled gates of $A$ and $B$ are “glued” and turned into an inner element of the module $A\bullet B$. However, in this simple version, the composition is fundamentally flawed: Upon composing three or more modules, the order of composition matters: for three modules $A$, $B$, and $C$, the two modules $(A \bullet B) \bullet C$ and $A \bullet (B \bullet C)$ differ from one another. In technical terms: this version of composition is not associative. But associativity is a central requirement, as discussed in Chapter 1.2.
To solve this problem, we return to modules shaped $S = S_1 \bullet \dots \bullet S_n$. As discussed in Sec. 1.2: each module $S_i$ generally has a left and a right neighbor ($S_0$ has no left, $S_n$ has no right neighbor). $S$ is composed by composing $S_{i-1}$ with $S_i$ (for $i = 2, \dots,n$). In the real world, systems frequently exhibit
this kind of structure, physically or conceptually.
Therefore, \textsc{Heraklit} partitions the surface of a module $L$ into its left and right interface, written $^\ast L$ and $L ^\ast$, resp. To compose two modules $L$ and $M$, equally labelled gates of $L ^\ast$ and $^\ast M$ are glued and turn into inner elements of $L \bullet M$. The remaining elements of $L ^\ast $ go to $(L \bullet M) ^\ast$ (together with $M^ \ast $), and the remaining elements of $^\ast M$ go to $^\ast (L \bullet M)$ (together with $ ^\ast L$). Most important: A general theorem guarantees that this kind of composition is associative \cite{reisig2019associative}.
\section{Case study: a service system}
\subsection{The different modules of the system}
Today, many organizations offer a complex service portfolio for their customers or clients \cite{boehmann2014service,chesbrough2006manifesto}. Typical examples are banking or financial services, insurance services, legal services, and the medical or health services offered by a hospital or a medical center.
\begin{figure}[htb]
\centering
\includegraphics[width=0.66\linewidth]{figures/signatur.png}
\caption{Signature of the service system}
\label{fig:signature}
\end{figure}
Here, we model the organization of such a service system, serving clients, customers, or patients that want confidential consultation about particular services or a particular treatment, provided by experts.
Fig.~\ref{fig:signature} shows the signature of the system: there are five sorts of elements in a service system, indicated by $C$, $E$, $R$, $A$, and $S$. Their intuitive meaning is indicated in italic. In a concrete service system, there are sets of experts, available consulting rooms, and admins, symbolically represented by $EX$, $RO$ and $AD$. Their type is $P(E)$, $P(R)$, and $P(A)$, resp., with $P(\cdot)$ standing for
``powerset''. Furthermore, we need a function symbol $f$ and five variables, one for each basic sort. An \textit{instantiation} assigns each basic sort an arbitrarily chosen concrete set, each constant symbol a set of elements of the indicated sort, and $f$ a function that assigns each service the set of experts that offer consultations for this service.
Fig.~\ref{fig:clients} shows a module that represents the behaviour of clients: For every instantiation of the variables $c$ and $s$ by a client and a service, resp., transition $a$ is enabled. Transition $a$ represents the policy that any client may enter the service system with any kind of wish for consultation for a service $s$. Hence, place $A$ may eventually hold any number of tokens, with each token consisting of a client and a service. Transition $b$ indicates the service systems’s help desk, accepting each client’s wishes and asking them to wait at place $B$. There, a client will eventually receive a message either at place $C$ or at place $D$. A message at place $C$ indicates that no expert is available; so the client leaves the service system along transition $c$. A message at place $D$ indicates that the client should proceed to the consulting room named or numbered $r$. The client will do so along transition $d$ and arrow $E$. He will later on return along arrow $F$ and leave the service system by transition $e$.
The module in Fig.~\ref{fig:experts} represents the behaviour of the service system’s experts. There is a set of experts, depicted as $EX$, fixed when the schema is instantiated, and initially represented as unengaged at place $G$. One might expect this to be expressed by the symbol $EX$ at place $G$. However, this would indicate one token at place $G$. This is not what we want: we want each single expert to be represented as a token. This is achieved by means of the function $elm$: Applied to a token that represents a set $M$, $elm(M)$ returns each element of $M$ as a token. For an expert $e$, the message $(e,r)$ arriving at place $H$ indicates that $e$ must go to consulting room $r$, due to transition $f$ and arc $I$. He will eventually return along arc $J$, release room $r$, and will be again unengaged at place $G$.
The module in Fig.~\ref{fig:rooms} shows the consulting rooms: A client $c$ and an expert $e$ arriving at room $r$ along the arcs $E$ and $I$, resp., start their consultation by transition $h$, end it by $i$, and leave the room by arcs $F$ and $J$.
\begin{figure}
\centering
\subcaptionbox{clients\label{fig:clients}}
{\includegraphics[width=0.50\textwidth]{figures/clients.png}}
\subcaptionbox{experts\label{fig:experts}}
{\includegraphics[width=0.375\textwidth]{figures/experts.png}}
\subcaptionbox{admin\label{fig:admin}}
{\includegraphics[width=0.75\textwidth]{figures/admin.png}}
\subcaptionbox{consulting rooms\label{fig:rooms}}
{\includegraphics[width=0.75\textwidth]{figures/consulting_rooms.png}}
\caption{The four modules of the system}\label{fig:modules}
\end{figure}
\begin{figure}[H]
\centering
\includegraphics[width=1\textheight, angle=90]{figures/overall_model.png}
\caption{Overall model of a service system}
\label{fig:overall}
\end{figure}
The behaviour of clients, experts, and the consulting rooms must be properly synchronized. The admin module of Fig.~\ref{fig:admin} organizes this. Place $P$ initially contains each admin as a token (we employ again the function $elm$ as explained above for the experts). An admin $a$ engages with a client $c$ and their request for an expert for service $s$, along transition $b$. A token $(a,c,s)$ on place $Q$ then continues either along transition $k$ or transition $j$. Transition $j$ requires an expert $e$ on place $R$, such that $e$ offers the service $s$. The inscription of $j$ indicates this requirement. $R$ always contains a “digital twin” for each expert that is not engaged with a client. The place $S$ always contains a digital twin of each empty consulting room. Hence, transition $j$ is enabled with proper instantiations of all five variables $a,c,s,e,$ and $r$. The occurrence of $j$ then renders the admin $a$ available in $P$ for new clients, sends messages to the client $c$, and the expert $e$ to proceed to room $r$, and moves the digital twin of $e$ to place $T$. This way, the digital twin of each expert $e$ is either a token in $R$ or in $T$. With $e$ in $T$, the expert $e$ eventually indicates by transition $g$ that they finished their consultation and they release the room $r$. Finally, transition $k$ manages the case where for a token $(a,c,s)$ no expert for service $s$ is available in $R$. As discussed above, the digital twin of each such expert is a token in $T$. Hence, all tokens in the set $f(s)$ of experts for $s$ are in $T$. This is “tested” by means of the loop between $k$ and $T$. Occurrence of $k$ then renders the admin $a$ available in $P$ for new clients and sends a corresponding message to the client $c$.
Notice the subtle treatment of experts and rooms as a scarce resource: If no corresponding expert is available, a client is turned away, as it may take too long until an expert for $s$ is available. But if no room is available, the client is just waiting as long as one room will be available.
\subsection{Overall model and abstract composition}
Fig.~\ref{fig:overall} finally “glues” the four modules into one big module. In \textsc{Heraklit}, this can just be written as: $ clients \bullet admin \bullet consulting \ rooms \bullet experts.
$
Similarly, it is possible to construct an abstract composition of the system. Fig.~\ref{fig:abstract_composition} depicts such a composition of the four abstract modules by using the abstraction operator $[\cdot]$, which deletes the inner structure of a module. Formally written as: $ [clients] \bullet [admin] \bullet [consulting \ rooms] \bullet [experts].
$
\begin{figure}[htb]
\centering
\includegraphics[width=0.6\textwidth]{figures/abstract_composition.png}
\caption{Abstract composition of the overall model}
\label{fig:abstract_composition}
\end{figure}
\section{Related work}
Modelling is typically understood as an interdisciplinary field that is used in many different disciplines as a method or instrument to capture knowledge or to assist other (research) actions \cite{frank2014:modeling,beverungen2020bpm}. As we discuss above, \textsc{Heraklit} mainly does not invent new modeling concepts but integrates proven and well-known modelling approaches. Compared to other integrated approaches which currently dominate the modelling practice, e.g. BPMN, \textsc{Heraklit} provides integrated means to descrive model structure, data, and behaviour. In the central concept of a module, \textsc{Heraklit} combines three proven, intuitively easy to understand, and mathematically sound concepts that have been used for the specification of systems in the past:
1. Abstract data types and algebraic specifications for the formulation of concrete and abstract data: since the 1970s such specifications have been used, built into specification languages, and often used for (domain-specific) modelling. The book \cite{sanella20212algebraic} presents systematically the theoretical foundations and some applications of algebraic specifications. Abstract state machines \cite{gurevich1993algebras} also belong to this context.
2. Petri nets for formulating dynamic behaviour: \textsc{Heraklit} uses the central ideas of Petri nets. A step of a system, especially a large system, has locally limited causes and effects. This allows processes to be described without having to use global states and globally effective steps. This concept from the early 1960s \cite{petri1962diss} was generalized at the beginning of the 1980s with predicate logic and \textit{colored marks} \cite{genrich1979predicatenets,jensen1982colourednets}. The connection with algebraic specifications is established by \cite{reisig1991algebraic}. \textsc{Heraklit} adds two decisive aspects to this view: uninterpreted constant symbols for sets in places that use the $elm$ function to hold instantiations with many possible initial marks, and the $elm$ function as an inscription for an arrow to describe flexible mark flow.
3. The composition calculus for structuring large systems: this calculus with its widely applicable associative composition operator is the most recent contribution to the foundations of \textsc{Heraklit}. The obvious idea, often discussed in the literature, of modeling composition as a fusion of the interface elements of modules is supplemented by the distinction of left and right interface elements, and composition $A \bullet B$ as a fusion of right interface elements of $A$ with left interface elements of $B$. According to \cite{reisig2019associative,reisig2020component_models}, this composition is associative (as opposed to the naive fusion of interface elements); it also has a number of other useful properties. In particular, this composition is compatible with refinement/coarseness and with individual (distributed) runs.
These three theoretical principles harmonize with each other and generate further \textit{best practice} concepts that contribute to a methodical approach to modeling with \textsc{Heraklit}, and which will only be touched upon in this paper. On the down-side, industrially mature modelling tools for \textsc{Heraklit} are still under development.
\section{Conclusions}
The presented case study clearly demonstrates how \textsc{Heraklit} provides an integrated view on structure, data, and locally defined behaviour. Hence, \textsc{Heraklit} covers all central aspects of every computer-integrated system. Such a description can be used for different purposes, e.g.\ business process management, service engineering, software analysis, design, verification, and development. The used techniques are well-known but combined in a novel and innovative way.
By providing such an integrated method for system specification, \textsc{Heraklit} paves the way for many important innovations which are currently so much in need \cite{beverungen2020bpm,houy2010:bpm_large}. In particular, we like to introduce the idea of \textit{Systems Mining}.
While Data Mining and Process Mining \cite{aalst2012process_mining} exploit the knowledge implicitly represented in data tuples and event sequences, respectively, Systems Mining is able to analyze the structure, data, and behaviour of a system. For such analysis, \textsc{Heraklit} provides the necessary techniques to specify all essential characteristics of a system. The observed structure of the system can be represented as modules, the observed data is captured by both concrete and abstract data structures, and the observed behaviour is specified as (distributed) runs. Based on such a powerful framework, Systems Mining provide a much richer picture of and deeper insights into big systems.
The presented case study of a service system illustrates powerful possibilities. Based on these \textsc{Heraklit} models, Systems Mining can answer a wide spectrum of interesting questions: (1) Do typical communications patterns between the modules of the system exist? (2) Which services are often requested by customers? (3) Do customers follow particular patterns for requesting services? (4) Which particular service requests and assignments of experts and rooms typically cause long waiting times for a customer? (5) Are there particular behaviour patterns and service requests which typically cause customers to leave the service system without getting a service or treatment?
Such questions and many more can easily be specified with \textsc{Heraklit}. Additionally, \textsc{Heraklit} provides a richer foundation for predictive and prescriptive process management as well as deeper insights for explaining process behaviour \cite{evermann2017prediction,mehdiyev2020explanation}. Hence, \textsc{Heraklit} lays the foundation for the next step after Process Mining.
\bibliographystyle{splncs04}
|
1111.2382
|
\section{Thermodynamics of spin transport}
Eq.~(6) of the main text suggests that spin flow to and from excited magnon states vanishes when there is no thermal or spin gradient, i.e., when $\beta_{L}=\beta_{R}$ and $\Delta\mu=\mu_{L}$.
However, when either of these conditions is not met, $j_{\rm ex}\neq0$
and spin (as well as energy) is transported across the insulator/conductor interface. In a steady state (i.e., zero spin current), in normal phase with thermal bias, $\beta_{R}-\beta_{L}\neq0$, a spin chemical potential difference $\delta\mu=\Delta\mu-\mu_{L}$ develops to oppose it:
\[
\delta\mu \approx\left(\beta_{R}-\beta_{L}\right)\frac{\int_{\epsilon_{\rm gs}}^{\infty} d\epsilon\left( \epsilon-\bar{\mu} \right) \left(\epsilon-\Delta\mu\right)n'_{\mathrm{B}}\left(\bar{\beta}\left(\epsilon-\bar{\mu}\right) \right)}{\bar{\beta} \int_{\epsilon_{\rm gs}}^{\infty} d\epsilon \left(\epsilon-\Delta\mu\right)n'_{\mathrm{B}}\left(\bar{\beta}\left(\epsilon -\bar{\mu}\right) \right)}\,,\]
where $\bar{\beta}\equiv \left(\beta_{L}+\beta_{R} \right)/2$, $\bar{\mu}\equiv\left(\mu_L+\Delta \mu \right)/2$, $n'_B=\partial_\epsilon n_B (\epsilon)$, and the thermodynamic biases are assumed to be small (i.e. $ \beta_R-\beta_L \ll \bar{\beta}, \delta \mu\ll\bar{\mu}$).
On the other hand, the condensed spin current $j_{\rm gs}$
is independent of both $T_L$ and $T_R$, and, provided $\epsilon_{\rm gs}>\Delta\mu$,
always carries spin away from the conductor, irrespective of the temperature
gradient between the two systems. The explanation for this behavior
can be understood as follows. Consider a single tunneling event involving
the creation (destruction) of a ground-state magnon ($\Delta N_{{\rm gs}}=\pm1$)
and the corresponding creation of a down-(up-)spin electron-hole excitation
in the conductor ($\Delta N_{R}=-\Delta N_{\rm gs}$), which we call process A (B) in Fig.~\ref{Excluded}. The entropy change in the insulator
associated with either process vanishes when the magnons form a BEC,
so that the entropy change of the whole system is just $dS_{R}$, which
can be found by enforcing energy conservation:\[
\Delta S_{\mathrm{tot}}=\Delta S_{R}=\frac{1}{T_{R}}\left(\epsilon_{\rm gs}-\Delta\mu\right)\Delta N_{R}\,.\]
Thus, process B (A) is favored ($\Delta N_{R}\gtrless0$) for tunneling events involving ground-state
magnons when $\epsilon_{\rm gs}\gtrless \Delta\mu$, in agreement with Eq.~(4) of the main text. Put differently, if $\Delta\mu=0$ the phase space of the conductor is unaffected with either the introduction of an up-spin excitation or the introduction of a down-spin excitation. However, process A requires the conductor to surrender an energy quantum $\epsilon_{\mathrm{gs}}$ to the insulator, whereas process B means a net gain in energy for the conductor; the overall entropy gain in the conductor (and therefore the entire system) is thus greater for process B than A. The zero-temperature version of this explanation is presented graphically in Fig.~\ref{Excluded}.
\begin{figure}[h]
\includegraphics[width=0.55\linewidth,clip=]{fig4}
\caption{A down electron may relax the ferromagnetic insulator, carrying away the excess energy away in a scattering state above the Fermi surface $\epsilon_{F}$ (process B). An incident up electron on the Fermi surface, however, cannot transfer up spin to the insulator magnetization (process A), since such an energy-preserving process would raise the energy of the magnet, lowering that of the electron and therefore landing it below the Fermi surface, which is Pauli blockaded. Process B therefore dominates, and the insulator magnetization relaxes towards the easy axis.}
\label{Excluded}
\end{figure}
\section{Detection of phase transition}
The BEC-normal phase transition presents some of the most interesting
physics of the system, yet as can be seen from Fig.~3 of the main text, it
is difficult to discern from the total magnetization of the insulator alone: Whereas for fixed $T_L$ the density of excited magnons $n_{\rm ex}$ plateaus as $z_L\rightarrow1$, the rate of change of the total number of magnons $\dot{n}_L=\dot{n}_{\rm ex}+\dot{n}_{\rm gs}$ remains always continuous function of time. The transition can, however, be observed by Brillioun
light scattering, wherein the scattered light intensity scales quadratically with the lateral junction size if the ground-state condensate is indeed coherent (see Ref.~[4] of the main text).
Alternatively, electron spin resonance (or, for that matter, any spectroscopic probe of a coherent microwave radiation) can provide clear evidence of the presence of quasiequilibrated Bose-Einstein condensation of magnons. Consider a test-particle electron at a fixed distance $\mathbf{r}$ from the magnetic insulator. Provided that the electron experiences the insulator as a single quantum magnetic moment $\hat{\mathbf{m}}$, one may neglect details involving spatial fluctuations of the magnetization and allow the two systems to interact via dipole-dipole coupling; the Hamiltonian describing the interaction is therefore of the form:
\[\hat{H}_{\rm d-d}=\sum_{i,j=x,y,z}\hat{m}_{i} T_{ij}\hat{\sigma}_{j}\,,\]
where $T_{ij}$ is a tensor that depends on $\mathbf{r}$ and $\hat{\sigma}$ is the electron spin operator. Supposing the electron, subjected to a strong applied magnetic field in the $z$ direction, begins in the state $\left|\uparrow\right\rangle$, the probability that quantum fluctuations in the magnetization $\hat{\mathbf{m}}$ spin flip the electron is, to lowest order in $T_{ij}$,
\[P_{\uparrow\rightarrow\downarrow}(t)=\int_{0}^{t}dt'\int_{0}^{t}dt''\sum_{iji'j'}T_{ij}T_{i'j'}\left\langle \hat{m}_{i}\left(t'\right)\hat{m}_{i'}\left(t''\right)\right\rangle\left\langle \uparrow\right|\hat{\sigma}_{j}\left|\downarrow\right\rangle \left\langle \downarrow\right|\hat{\sigma}_{j'}\left|\uparrow\right\rangle e^{i\omega_{z}\left(t'-t''\right)}\,,\]
where $\omega_z$ is the electronic Larmor frequency in the applied magnetic field. Choosing our coordinate system to coincide with the eigenbasis of $T_{ij}$ and for simplicity asserting cylindrical symmetry around the $z$ axis (so that $T_{xx}=T_{yy}=T_{\perp}$), the transition probability becomes
\[P_{\uparrow\rightarrow\downarrow}\left(t\right)=\int_{0}^{t}dt'\int_{0}^{t}dt''T_{\perp}^2 \left\langle \hat{m}^{-}\left(t'\right)\hat{m}^{+}\left(t''\right)\right\rangle e^{i\omega_{z}\left(t'-t''\right)}=tT_\perp^2S_{-+}(\omega_z)\,,\]
where $S_{-+}(\omega)=\int dte^{i\omega t}\langle\hat{m}^-(t)\hat{m}^+(0)\rangle\propto N_{\rm gs}$ is the spectral density of magnetic oscillations in a steady state. The transition rate is thus proportional to $N_{\rm gs}$, which scales linearly with the lateral dimensions of our junction in BEC phase and is size independent in normal phase. This simple treatment is pertinent to the case when the magnons condense at $\mathbf{q}=0$. Otherwise (as is the case in YIG, for example) one needs to come up with means to couple coherently to magnetic fluctuations at a finite $\mathbf{q}$ (perhaps using some form of grating).
\end{document}
|
2109.06534
|
\section{Introduction}\label{Intro}
The problem of dark matter has a long history \cite{1}.
As was found by J.~Oort in 1932 when studying stellar motion in the neighborhood
of a galaxy, the galaxy mass must be well over than that of its visible
constituents \cite{2}. A year later, F.~Zwicky \cite{3} applied the virial theorem
to the Coma cluster of galaxies in order to determine its mass. The obtained mass
value turned out to be much larger than that found from the number of observed
galaxies belonging to the Coma cluster multiplied by their mean mass.
An excess mass, which reveals itself only gravitationally, received the name
{\it dark matter}.
According to current concepts, the dark matter contributes approximately 27\%
to the energy of the Universe although its physical nature remains unknown.
There are many approaches to this problem based on the role of some hypothetical
particles, such as axions, arions, massive neutrinos, weakly interacting massive
particles (WIMP), barionic dark matter, modified gravity etc.
(see \cite{1,4,5,6,7,8} for a review).
The model of dark matter which finds a support from astrophysics and cosmology
is referred to as {\it cold dark matter}.
According to this model, dark matter consists of light hypothetical particles which
are produced in the early Universe and become nonrelativistic already at the first
stages of its evolution. The most popular particle of this kind is an axion, i.e.,
a pseudoscalar Nambu-Goldstone boson introduced to solve the problem of strong
CP violation in Quantum Chromodynamics (QCD) \cite{9,10,11}.
It has been known that the gauge invariant QCD vacuum depends on an angle $\theta$,
and this dependence violates the CP invariance of QCD. However, experiment says
that strong interactions are CP invariant and the electric dipole moment of a neutron
is equal to zero up to a high degree of accuracy. An introduction of the Peccei-Quinn
symmetry and axions, which are connected with its violation, helps to solve this
problem.
Axions and other axionlike particles, which arise in many extensions to the Standard
Model, can interact both with photons and with fermions (electrons and nucleons).
These interactions are used for an axion search and for constraining axion masses and
coupling constants from observations of numerous astrophysical and cosmological
processes, as well as from various laboratory experiments (see reviews
\cite{1,4,5,6,7,8,12,13,14,15,16,17,18,19,20} of already obtained bounds on the axion
mass and coupling constants to photons, electrons, and nucleons).
Prof.~Yuri~N.~Gnedin obtained many important results investigating the interaction
of dark matter axions with photons in astrophysics and cosmology. He proposed
\cite{21} to employ
the polarimetric methods for a search for axions and arions (i.e., the axions of
zero mass \cite{22}) in the emission from pulsars, X-ray binaries with low-mass
components, and magnetic white dwarfs. For this purpose, it was suggested to use
the conversion process of photons into axions in the magnetic field of compact stars
and in the interstellar and intergalactic space (i.e., the Primakoff effect).
Next, Prof.\ Gnedin demonstrated an appearance of the striking feature in
the polarized
light of quasistellar objects due to the resonance magnetic conversion of photons into
massless axions \cite{23}.
Using these results, Prof.~Yu.~N.~Gnedin organized the axion search by the 6-m
telescope at the Special Astronomical Observatory in Russia. Both the Primakoff effect
and the inverse process of an axion decay into two photons were searched in the
integalactic light of clusters of galaxies and in the brightness of night sky due to
axions in the halo of our Galaxy \cite{24}. Although no evidences of axions were
found, it was possible to find the upper limit on the photon-to-axion coupling
constant from the polarimetric observations of magnetic chemically peculiar stars of
spectral type A possessing strong hydrogen Balmer absorption lines \cite{24}.
The above results, as well as the ground-based cavity experiments searching for
galactic axions, searches of an axion decay in the galactic and extragalactic light,
for the solar and stellar axions, and the obtained limits on the coupling constant
of axions to photons, were discussed in the review \cite{25}.
In the further research of dark matter axions, Yu.~N.~Gnedin and his collaborators
analyzed \cite{26} the intermediate results of PVLAS experiment interpreted
\cite{27} as arising due to a conversion of photons into axions with a coupling
constant to photons of the order of $4\times 10^{-6}~\mbox{GeV}^{-1}$.
By considering the astrophysical and cosmological constraints, they have shown
\cite{26} that this result is in contradiction with the data on stellar evolution
that exclude the standard model of QCD axions.
Using the cosmic orientation of the electric field vectors of polarized radiation
from distant quasars, Yu.~N.~Gnedin and his collaborators placed rather strong
limit on the coupling constant of axions to electric field \cite{28}.
Numerous results related to the processes of axion decay into two photons, the
transformation of photons into axions in the magnetic fields of stars and of
interstellar or intergalactic media, and the transformation of axions generated
in the cores of stars into X-ray photons were discussed in the review \cite{29}.
It has been known that the coupling constant of axionlike particles to fermions
can be constrained in the laboratory experiments on measuring the Casimir force
between two closely spaced test bodies. This force is caused by the zero-point and
thermal fluctuations of the electromagnetic field. It acts between any material
surfaces --- metallic, dielectric or semiconductor \cite{30,31}.
A constraint on the electron-arion coupling constant from old measurements of the
Casimir force \cite{31a} was obtained in \cite{31b}.
The competitive constraints on the coupling constants of axionlike particle to
nucleons from different experiments on measuring the Casimir interaction were
obtained in \cite{32,32a,33,34,35,36,37,38}. All experiments used for obtaining
these constraints have been performed in the separation range below $1~\mu$m.
Starting from 1982 \cite{39}, measurements of the van der Waals and Casimir forces
were also used for constraining the Yukawa-type corrections to Newton's law of
gravitation. These corrections arise due to an exchange of light scalar particles
between atoms of two closely spaced macrobodies \cite{40} and in the
extra-dimensional unification schemes with a low-energy compactification scale
\cite{41,42,43,44}. A review of the most precise measurements of the Casimir
interaction and constraints on non-Newtonian gravity obtained from them can be
found in \cite{20,45}.
In this paper, we obtain new constraints on the coupling constant of axionlike
particles to nucleons and on the Yukawa-type corrections to Newtonian gravity
following from recent experiment on measuring the
differential Casimir force between two
Au-coated bodies spaced at separations from 0.2 to $8~\mu$m \cite{46}.
This experiment was performed by Prof.~R.~S.~Decca by means of a
micromechanical torsional oscillator. The differential Casimir force was
measured between an Au-coated sapphire sphere and the top and bottom of
Au-coated deep Si trenches. The measurement results were compared with the
exact theory using the scattering approach and found to be
in good agreement with it over
the entire measurement range with no fitting parameters under a condition that
the relaxation properties of conduction electrons are not included in
computations. Another theoretical approach, which takes into account the
relaxation properties of conduction electrons, was excluded by the measurement
data over the range of separations from 0.2 to $4.8~\mu$m \cite{46}.
We calculate additional forces arising in the experimental configuration due
to an exchange of two axionlike particles between nucleons of the test bodies,
as well as due to the Yukawa-type correction to Newton's gravitational potential.
Taking into account that no extra force was observed, the constraints on the
masses and coupling constants of axions and on the strength and interaction
range of the Yukawa interaction were found from the extent of agreement between
the measured and calculated Casimir forces.
According to our results, the obtained constraints on axionlike particles are
up to a factor of 4 stronger than those following from other measurements of the
Casimir force. The new constraints on the Yukawa-type interaction are up to a
factor of 24 stronger than those obtained previously from measuring the Casimir force.
Stronger constraints on an axion \cite{35} and non-Newtonian gravity \cite{47}
were obtained only from the experiment by R.~S.~Decca \cite{47} where the Casimir
force was completely nullified.
The paper is organized as follows. In Section 2, we consider the effective
potentials due to an exchange of pseudoscalar and scalar particles. Section 3
is devoted to brief discussion of the recent experiment on measuring
the differential Casimir
force in the micrometer range. In Section 4, we calculate the additional force
due to an exchange of two axionlike particles between nucleons and obtain
constraints on the axion mass and coupling constant. The constraints on the
Yukawa-type correction to Newtonian gravity are found in Section 5.
In Sections 6 and 7, the reader will find the discussion of the obtained results
and our conclusions, respectively.
We use the system of units with $\hbar=c=1$.
\section{Effective Potentials due to Exchange of Pseudoscalar and
Scalar Particles}
\newcommand{\ri}{{\rm i}}
An interaction of the field of originally introduced QCD axions $a(x)$
\cite{9,10,11}, which describes the Nambu-Goldstone bosons, with the
fermionic field $\psi(x)$ is given by the pseudovector Lagrangian
density \cite{12,15}
\begin{equation}
{\cal L}_{\rm pv}(x)=\frac{g}{2m_a}\bar{\psi}(x)\gamma_5\gamma_{\mu}
\psi(x)\partial^{\mu}a(x),
\label{eq1}
\end{equation}
\noindent
where $m_a$ is the mass of an axion, $\gamma_n$ with $n=0,\,1,\,2,\,3,\,4,\,5$
are the Dirac matrices, and $g$ is the dimensionless coupling constant of the
axions to fermionic particles of spin 1/2 (in our case a proton or a neutron).
The axionlike particles introduced in Grand Unified Theories (GUT) interact
with fermions by means of the pseudoscalar Lagrangian
density \cite{12,15,48}
\begin{equation}
{\cal L}_{\rm ps}(x)=-\ri{g}\bar{\psi}(x)\gamma_5\psi(x)a(x).
\label{eq2}
\end{equation}
On a tree level the Lagrangian densities (\ref{eq1}) and (\ref{eq2}) result
in the same action and common effective potential due to an exchange of one
axion or axionlike particle between two nucleons of mass $m$ spaced at a
distance $r=|\mbox{\boldmath$\vec{r}$}_1-\mbox{\boldmath$\vec{r}$}_2|$ \cite{49,50}
\begin{eqnarray}
&&
V_{an}(r;\mbox{\boldmath$\vec\sigma$}_1,\mbox{\boldmath$\vec\sigma$}_2)=
\frac{g^2}{16\pi m^2}\left[
(\mbox{\boldmath$\vec\sigma$}_1\cdot\mbox{\boldmath$\hat{r}$})
(\mbox{\boldmath$\vec\sigma$}_2\cdot\mbox{\boldmath$\hat{r}$})
\left(\frac{m_a^2}{r}+\frac{3m_a}{r^2}+\frac{3}{r^3}\right)\right.
\nonumber \\
&&~~~~~~~~~~~~~~~~
\left.
-(\mbox{\boldmath$\vec\sigma$}_1\cdot\mbox{\boldmath$\vec\sigma$}_2)
\left(\frac{m_a}{r^2}+\frac{1}{r^3}\right)\right],
\label{eq3}
\end{eqnarray}
\noindent
where $\mbox{\boldmath$\hat{r}$}=(\mbox{\boldmath$\vec{r}$}_1-\mbox{\boldmath$\vec{r}$}_2)/r$
is the unit vector directed along the line connecting the two nucleons and
$\mbox{\boldmath$\vec\sigma$}_1,\,\mbox{\boldmath$\vec\sigma$}_2$ are their spins.
Here, we assume equal the coupling constants of an axion to a neutron and
a proton and notate $m$ their mean mass.
Taking into account that the effective potential (\ref{eq3}) is spin-dependent,
the resulting additional force averages to zero when integrated over the
volumes of unpolarized test bodies used in experiments on measuring the
Casimir force. Therefore, the process of one-axion exchange cannot be used
for the axion search in measurements of the Casimir force performed up to
date \cite{31,51} (the proposed measurement of the Casimir force between
two test bodies possessing the polarization of nuclear spins \cite{52}
is, however, quite promising for this purpose).
A process of the two-axion exchange between two nucleons deserves a particular
attention. If the pseudovector Lagrangian density (\ref{eq1}) is used, the
respective effective potential is still unknown. This is because the actual
interaction constant $g/(2m_a)$ is dimensional, and the resulting quantum
field theory is nonrenormalizable (the current status of this problem is
reflected in \cite{53}). However, in the case of the pseudoscalar
Lagrangian density (\ref{eq2}) describing the interaction of axionlike
particles with fermions, the effective potential due to two-axion exchange
is spin-independent and takes the following simple form \cite{40,54,55}:
\begin{equation}
V_{aan}(r)=-\frac{g^4}{32\pi^3m^2}\,\frac{m_a}{r^2} K_1(2m_ar),
\label{eq4}
\end{equation}
\noindent
where $K_1(z)$ is the modified Bessel function of the second kind.
The effective potential (\ref{eq4}) can be summed over all pairs of nucleons
belonging to the test bodies $V_1$ and $V_2$ leading to their total
interaction energy
\begin{equation}
U_{aan}(z)=-\frac{m_ag^4}{32\pi^3m^2}n_1n_2
\int_{V_1}d\,\mbox{\boldmath$\vec{r}$}_1\int_{V_2}d\,\mbox{\boldmath$\vec{r}$}_2
\frac{K_1(2m_ar)}{r^2},
\label{eq5}
\end{equation}
\noindent
where $n_1$ and $n_2$ are the numbers of nucleons per unit volume
of the first and second test bodies, respectively, and $z$ is the closest
separation distance between them. Finally, from (\ref{eq5}) one arrives to
the additional force acting between the test bodies due to the two-axion
exchange
\begin{equation}
F_{aan}(z)=-\frac{\partial U_{aan}(z)}{\partial z}.
\label{eq6}
\end{equation}
Equations (\ref{eq5}) and (\ref{eq6}) can be used for the search of axionlike
particles in experiments on measuring the Casimir force and for constraining their
parameters.
Similar situation takes place for the Yukawa-type corrections to Newton's
gravitational law which arises due to an exchange of one scalar particle of mass
$m_s$ between two pointlike particles (atoms or nucleons) with masses $m_1$ and
$m_2$ separated by a distance $r$. It is conventional to notate the coupling
constant of the Yukawa interaction as $\alpha G$, where $G$ is the Newtonian
gravitational constant and $\alpha$ is the proper dimensionless Yukawa strength.
Then, the Yukawa-type correction to Newtonian gravity takes the form
\begin{equation}
V_{\rm Yu}(r)=-\frac{Gm_1m_2}{r}\alpha e^{-r/\lambda},
\label{eq7}
\end{equation}
\noindent
where the interaction range $\lambda=1/m_s$ has the meaning of the Compton
wavelength of a scalar particle. As was mentioned in Section 1, the same correction
to Newton's gravitational potential also arises in the extra-dimensional physics
with a low-energy compactification scale \cite{41,42,43,44}.
The Yukawa-type interaction energy between two macrobodies $V_1$ and $V_2$ arising
due to the potential (\ref{eq7}) is given by
\begin{equation}
U_{\rm Yu}(z)=-\alpha G\rho_1\rho_2
\int_{V_1}d\,\mbox{\boldmath$\vec{r}$}_1\int_{V_2}d\,\mbox{\boldmath$\vec{r}$}_2
\, \frac{e^{-r/\lambda}}{r},
\label{eq8}
\end{equation}
\noindent
where $\rho_1$ and $\rho_2$ are the mass densities
of the first and second test bodies, respectively. In this case,
the additional force acting between two test bodies is equal to
\begin{equation}
F_{\rm Yu}(z)=-\frac{\partial U_{\rm Yu}(z)}{\partial z}.
\label{eq9}
\end{equation}
Both hypothetical forces (\ref{eq6}) and (\ref{eq9}) act on the background of
the Casimir force measured at separations below a few micrometers.
Note that the Newtonian gravitational force
\begin{equation}
F_{\rm gr}(z)= G\rho_1\rho_2\frac{\partial}{\partial z}
\int_{V_1}d\,\mbox{\boldmath$\vec{r}$}_1\int_{V_2}d\,\mbox{\boldmath$\vec{r}$}_2
\, \frac{1}{r}
\label{eq10}
\end{equation}
\noindent
calculated within the same range of separations is much less than the error
in measuring the Casimir force and can be neglected (see below).
\section{Measurements of the Casimir Force in the Micrometer
Separation Range}
All experiments on measuring the Casimir force between two macrobodies used for
obtaining constraints on axionlike particles \cite{32a,33,34,35,36,37,38}
were performed at separations below a micrometer. A single direct measurement
of the Casimir force between two macrobodies at separations up to $8~\mu$m
was reported quite recently and compared with theory with no fitting parameters
\cite{46}. Below we briefly elucidate the main features of this experiment needed
for obtaining new constraints on the axionlike particles and Yukawa-type corrections
to Newtonian gravity.
In the experiment \cite{46}, the micromechanical torsional oscillator was used to
measure the differential Casimir force between an Au-coated sapphire sphere of
$R=149.7~\mu$m radius and the top and bottom of Au-coated rectangular silicon trenches
of $H=50~\mu$m depth. So large depth of the trenches was chosen in order the
Casimir force between a sphere and a trench bottom (and all the more additional
hypothetical forces) be equal to zero. This means that the actually measured Casimir
force $F_{\rm C}(z)$ acts between a sphere and a trench top. In so doing, however,
the differential measurement scheme used allowed reaching rather low total
experimental error equal to $\Delta F_{\rm C}=2.2~$fN at the separation distance
of $z=3~\mu$m. Similar schemes of differential force measurements were previously
used in \cite{47,58,56,57}.
The thicknesses of Au coatings on the sphere and the trench surfaces equal to
$d_{\rm Au}^{(s)}=250~$nm and $d_{\rm Au}^{(t)}=150~$nm, respectively, were thick
enough in order Au-coated test bodies could be considered as made from solid Au
when calculating the Casimir force. For technological reasons, before depositing
Au coatings, the sapphire sphere and silicon trench surfaces were also covered
with Cr layers of thickness $d_{\rm Cr}=10~$nm. These layers do not influence
the Casimir force but should be accounted for in calculations of the additional
forces due to two-axion exchange and the Yukawa-type potential in Sections 4 and
5, respectively.
The Casimir force between an Au-coated sphere and an Au-coated trench top was measured
over the separation region $0.2~\mu\mbox{m}\leqslant z\leqslant 8~\mu$m and compared
with exact theory developed in the sphere-plate geometry using the scattering approach
in the plane-wave representation \cite{59,60,61} and the derivation expansion
\cite{62,63,64,65}.
In doing so the contribution of patch potentials to the measured signal was
characterized by Kelvin probe microscopy. An impact of surface roughness was taken
into account perturbatively \cite{31,51,65a,65b} and found to be negligible.
It was shown \cite{46} that the measurement data are in agreement
with theory to within the total experimental error $\Delta F_{\rm C}$ over the entire
measurement range if the relaxation properties of conduction electrons are not included in computations. An inclusion of the relaxation of conduction electrons (i.e., using the
dissipative Drude model at low frequencies) results in strong contradiction between
experiment and theory over the range of separations from 0.2 to $4.8~\mu$m.
These results are in line with previous precise experiments on measuring the Casimir
interaction at shorter sphere-plate separations
\cite{57,66,67,68,69,70,71,72,73,74,75,76} and discussions of the so-called Casimir
puzzle \cite{76a,76b,76c} (see also recent approaches to the resolution of this
problem \cite{76d,76e,76f}).
It is necessary to stress that calculation of the Casimir force between the first test
body (sphere) and the flat top of rectangular trenches in \cite{46} was based on first
principles of quantum electrodynamics at nonzero temperature and did not use any
approximate methods, such as the proximity force approximation applied in the most
of previously performed experiments \cite{31,51}, or fitting parameters.
As to the total experimental errors $\Delta F_{\rm C}$, they were found in \cite{46}
at the 95\% confidence level as a combination of both random and systematic errors.
Notice that, according to \cite{ad1,ad2}, one and the same experiment cannot be
used to exclude an alternative model and to constrain the fundamental forces.
This claim was made in 2011 when some of the background effects in the
Casimir force (such as the role of patch potentials) and theoretical
uncertainties (such as an error in the proximity force approximation) were
not completely settled. As was, however, immediately objected in the literature
\cite{ad3,ad4}, such a difference between the excluded and confirmed models of the
Casimir force is of quite another form than a correction to the fundamental force.
This fact allows to constrain the parameters of the latter. After
the seminal experiment by R. S. Decca performed in 2016 \cite{57}, where the
theoretical predictions of two models for the Casimir force differ by up to
a factor of 1000 and one of them was conclusively rejected whereas another one was
confirmed, it became amply
clear that a comparison with the confirmed model can be reliably used for
constraining the fundamental
forces in all subsequent experiments.
When discussing the measure of agreement between experiment and theory, it is necessary
also to take into account the contribution of differential Newtonian gravitational
force between the sphere and the top and bottom of rectangular trenches.
In \cite{46} this force was assumed to be negligibly small. Taking into account that
in Sections 4 and 5 the measure of agreement between experiment and theory is used
for constraining the parameters of axionlike particles and non-Newtonian gravity,
below we estimate the role of Newton's gravitational force more precisely.
The second test body, which is concentrically covered by the rectangular trenches, is
a $D=25.4~$mm diameter Si disc (schematic of the experimental setup is shown
in Figure 1 of \cite{46}).
As mentioned above, both the test
bodies are coated by layers of Cr and Au. In our estimation of the upper bound
for the Newton's gravitational force in this experiment, we assume that both the sphere
and the disc of diameter $D$ and thickness $H$, where $H$ is the trench depth, are
all-gold. By choosing the disc thickness equal to $H$, we disregard the Si substrate
underlying trenches because it does not contribute to the differential gravitational
force between the top and bottom of the trenches (unlike the Casimir and additional
hypothetical forces, the more long-range gravitational force from the trench bottom
cannot be considered equal to zero).
Taking into account that the disc radius $D/2$ is much larger that the sphere
radius $R$, the gravitational force between them is given by \cite{77}
\begin{equation}
F_{\rm gr}=-\frac{8\pi^2}{3}G\rho_{\rm Au}^2HR^3,
\label{eq11}
\end{equation}
\noindent
where $\rho_{\rm Au}=19.28~\mbox{g/cm}^3$ is the density of gold.
Substituting the values of all parameters in (\ref{eq11}), one obtains
$F_{\rm gr}=0.11~$fN. It is seen that the upper bound for the gravitational
force is much less that the experimental error in measuring the Casimir force
which is equal to $\Delta F_{\rm C}=2.2~$fN at $z=3~\mu$m (the variations of
gravitational force depending on whether the sphere is above the top or bottom of
the trenches are well below 0.11~fN). This confirms that the contribution of
Newton's gravitational force in this experiment is very small and can be
neglected.
\section{Constraints on axionlike particles}
The results of experiment \cite{46} on measuring the Casimir force in the
micrometer seperation range allow one to obtain new constraints on the mass of
axionlike particles and their coupling constants to nucleons. As explained in
Section 2, measurements of the Casimir force between unpolarized test bodies can
be used for constraining the parameters of axionlike particles whose interaction
with fermions is described by the Lagrangian density (\ref{eq2}).
For this purpose, one should use the process of two-axion exchange between
nucleons of the laboratory test bodies which leads to the effective potential
(\ref{eq4}) and force (\ref{eq6}). Thus, it is necessary to calculate this force
in the experimental configuration of \cite{46}.
According to Section 3, this configuration reduces to a sapphire (Al$_2$O$_3$)
sphere of radius $R$ (the sapphire density is
$\rho_{{\rm Al}_2{\rm O}_3}=4.1~\mbox{g/cm}^{3}$) coated by the layers of Cr
and Au of thicknesses $d_{\rm Cr}$ and $d_{\rm Au}^{(s)}$, respectively,
interacting with a Si plate ($\rho_{{\rm Si}}=2.33~\mbox{g/cm}^{3}$) whose
thickness exceeds $H$ and can be considered as infinitely large when
calculating the hypothetical interactions rapidly decreasing with separation.
The plate is also coated by the layer of Cr of thickness $d_{\rm Cr}$ and
by an external layer of Au of thickness $d_{\rm Au}^{(t)}$ indicated in
Section 3.
The additional force arising in this configuration due to two-axion exchange
can be calculated by (\ref{eq5}) and (\ref{eq6}). The results of this
calculation for homogeneous a sphere and a plate are presented in \cite{38}.
An impact of two metallic layers is easily taken into account in perfect
analogy to \cite{35,37}.
Finally, the additional force due to two-axion exchange in the configuration
of experiment \cite{46} takes the form
\begin{equation}
F_{aan}(z)=-\frac{\pi}{2m_am^2m_{\rm H}^2}\int_1^{\infty}\!\!du
\frac{\sqrt{u^2-1}}{u^2}e^{-2m_auz}X_t(m_au)X_s(m_au),
\label{eq12}
\end{equation}
\noindent
where $m_{\rm H}$ is the mass of atomic hydrogen and the following expressions
for the functions $X_t(x)$ and $X_s(x)$ are obtained:
\begin{eqnarray}
&&
X_t(x)=C_{\rm Au}\Bigl(1-e^{-2xd_{\rm Au}^{(t)}}\Bigr)+
C_{\rm Cr}e^{-2xd_{\rm Au}^{(t)}}\Bigl(1-e^{-2xd_{\rm Cr}}\Bigr)
+C_{\rm Si}e^{-2x(d_{\rm Au}^{(t)}+d_{\rm Cr})},
\nonumber\\
&&
X_s(x)=C_{\rm Au}\Bigl[\Phi(R,x)-e^{-2xd_{\rm Au}^{(s)}}
\Phi(R-d_{\rm Au}^{(s)},x)\Bigr]
\nonumber\\
&&~~~~~~~~~~
+
C_{\rm Cr}e^{-2xd_{\rm Au}^{(s)}}\Bigl[\Phi(R-d_{\rm Au}^{(s)},x)-
e^{-2xd_{\rm Cr}}\Phi(R-d_{\rm Au}^{(s)}-d_{\rm Cr},x)\Bigr]
\nonumber\\
&&~~~~~~~~~~
+C_{{\rm Al}_2{\rm O}_3}e^{-2x(d_{\rm Au}^{(s)}+d_{\rm Cr})}
\Phi(R-d_{\rm Au}^{(s)}-d_{\rm Cr},x),
\label{eq13}
\end{eqnarray}
\noindent
where the function $\Phi$ is defined as
\begin{equation}
\Phi(r,x)=r-\frac{1}{2x}+e^{-4rx}\left(r+\frac{1}{2x}\right).
\label{eq14}
\end{equation}
The numerical coefficients $C$ in (\ref{eq13}) are specific for any material.
They are calculated by the following equation:
\begin{equation}
C=\rho\frac{g^2}{4\pi}\left(\frac{Z}{\mu}+\frac{N}{\mu}\right),
\label{eq14a}
\end{equation}
\noindent
where $Z$ and $N$ are the number of protons and the mean number of
neutrons in an atomic nucleus of a given substance, and $\mu=M/m_{\rm H}$
with $M$ being the mean atomic or molecular mass.
The values of (\ref{eq14a}) for Au, Cr, Si, and Al$_2$O$_3$, i.e.,
$C_{\rm Au}$, $C_{\rm Cr}$, $C_{\rm Si}$, and $C_{{\rm Al}_2{\rm O}_3}$,
are proportional to $g^2$ with the factors found using the densities of
Au, Si, and Al$_2$O$_3$ indicated above and $\rho_{\rm Cr}=7.15~\mbox{g/cm}^3$.
The values of $Z/\mu$ for Au, Cr, Si, and Al$_2$O$_3$ are equal to
0.40422, 0.46518, 0.50238, and 0.49422, respectively, and of $N/\mu$ to
0.60378, 0.54379, 0.50628, and 0.51412 for the same respective materials
\cite{40}.
Taking into consideration that in the experiment \cite{46} no additional
interaction was observed within the total experimental error
$\Delta F_{\rm C}(z)$, the interaction (\ref{eq12}) should satisfy the
inequality
\begin{equation}
|F_{aan}(z)|\leqslant\Delta F_{\rm C}(z).
\label{eq15}
\end{equation}
\noindent
Using equations (\ref{eq12})--(\ref{eq14}), one can constrain
from (\ref{eq15}) the possible
values of the axionlike particles mass $m_a$ and their coupling
constant to nucleons $g$.
The numerical analysis of (\ref{eq15}) with account of (\ref{eq12})--(\ref{eq14})
and the values of $\Delta F_{\rm C}$ at different $z$ \cite{46} shows that the
strongest constraints on $m_a$ and $g$ follow at $z=3~\mu$m where
$\Delta F_{\rm C}=2.2~$fN (see Section 4). These constraints are presented by the
line labeled ``new" in Figure~1. For comparison purposes, in the same figure
we show the constraints obtained earlier from the Cavendish-type experiment
\cite{78} (line ``gr$_1$" \cite{79}), from measuring the smallest gravitational
forces using the torsional oscillator \cite{80,81,82} (line ``gr$_2$"
\cite{53}), from measuring the effective Casimir pressure \cite{68,69}
(line 1 \cite{32a} ),
from the experiment using a beam of molecular hydrogen \cite{83} (line
``H$_2$" \cite{84}), and from the differential measurement where the Casimir
force was completely nullified \cite{58} (the dashed line \cite{35}).
Note that the figure field above each line is excluded and below each line
is allowed by the results of respective experiment.
\begin{figure}[!b]
\centering
\vspace*{-8.2cm}
\hspace*{0.cm}\includegraphics[width=20 cm]{figAx-1.ps}
\vspace*{-11.5cm}
\caption{The strongest constraints on the coupling constant of axionlike
particles to nucleons obtained from different experiments. Lines labeled
``gr$_1$" and ``gr$_2$" are found from the Cavendish-type experiments
and from measuring the minimum gravitational forces, respectively.
Lines labeled ``new" and ``H$_2$" follow from the recent experiment on
measuring the Casimir force in the micrometer separation range and from
the experiment using a beam of molecular hydrogen. Line 1 and the
dashed line are obtained from measuring the effective Casimir pressure and
from the experiment nullifying the Casimir force. The regions above each line
are excluded and below --- are allowed.
\label{fg1}}
\end{figure}
As is seen in Figure~1, in the region of axion masses from 10 to 74~meV
the constraints on $g$ obtained from the recent experiment \cite{46}
(line ``new") are stronger than those following from measuring the effective
Casimir pressure (line 1) and the smallest gravitational forces
(line ``gr$_2$"). The maximum strengthening up to a factor of 4 is reached
for the axion mass $m_a\approx 16~$meV. Thus, in the region of axion masses
indicated above, this experiment leads to stronger constraints than the
previous experiments on measuring the Casimir force \cite{20,45}.
All the more strong constraints shown by the dashed line follow only
from the experiment \cite{58} where the Casimir force was completely
nullified.
\section{Constraints on non-Newtonian gravity}
As was mentioned in Section1, the results of experiment \cite{46} on measuring
the Casimir force in the micrometer separation range can be also used for
constraining the parameters of Yukawa-type corrections to Newtonian gravity.
For this purpose, one should calculate the Yukawa-type force (\ref{eq8}),
(\ref{eq9}) in the experimental configuration using its geometrical
characteristics and densities of all constituting materials presented in
Sections 3 and 4.
The calculation results using (\ref{eq8}) and
(\ref{eq9}) for a homogeneous sphere above a homogeneous plate are
presented in \cite{38}. Here, we modify them in perfect analogy to \cite{35,37}
for taking into account the contributions of Cr and Au layers covering both
a sphere and a trench in this experiment. The result is
\begin{equation}
F_{\rm Yu}(z)=-4\pi^2G\alpha\lambda^3e^{-z/\lambda}Y_t(\lambda)Y_s(\lambda),
\label{eq16}
\end{equation}
\noindent
where
\begin{eqnarray}
&&
Y_t(\lambda)=\rho_{\rm Au}\Bigl(1-e^{-d_{\rm Au}^{(t)}/\lambda}\Bigr)+
\rho_{\rm Cr}e^{-d_{\rm Au}^{(t)}/\lambda}\Bigl(1-e^{-d_{\rm Cr}/\lambda}\Bigr)
+\rho_{\rm Si}e^{-(d_{\rm Au}^{(t)}+d_{\rm Cr})/\lambda},
\nonumber\\
&&
Y_s(\lambda)=\rho_{\rm Au}\Bigl[\Psi(R,\lambda)-e^{-d_{\rm Au}^{(s)}/\lambda}
\Psi(R-d_{\rm Au}^{(s)},\lambda)\Bigr]
\nonumber\\
&&~~~~~~~~~~
+
\rho_{\rm Cr}e^{-d_{\rm Au}^{(s)}/\lambda}\Bigl[\Psi(R-d_{\rm Au}^{(s)},\lambda)-
e^{-d_{\rm Cr}/\lambda}\Psi(R-d_{\rm Au}^{(s)}-d_{\rm Cr},\lambda)\Bigr]
\nonumber\\
&&~~~~~~~~~~
+\rho_{{\rm Al}_2{\rm O}_3}e^{-(d_{\rm Au}^{(s)}+d_{\rm Cr})/\lambda}
\Psi(R-d_{\rm Au}^{(s)}-d_{\rm Cr},\lambda),
\label{eq17}
\end{eqnarray}
\noindent
and the function $\Psi$ is defined as
\begin{equation}
\Psi(r,\lambda)=r-\lambda+(r+\lambda)e^{-2r/\lambda}.
\label{eq18}
\end{equation}
By virtue of the fact that the Yukawa-type force (\ref{eq16}) was not
observed within the limits of the measurement error, it should satisfy
the inequality
\begin{equation}
|F_{\rm Yu}(z)|\leqslant\Delta F_{\rm C}(z).
\label{eq19}
\end{equation}
\noindent
Similar to the case of an additional interaction due to two-axion exchange
between nucleons, the strongest constraints on the parameters of the
Yukawa-type interaction $\alpha$ and $\lambda$ follow from (\ref{eq19})
at $z=3~\mu$m where $\Delta F_{\rm C}=2.2~$fN.
\begin{figure}[!t]
\centering
\vspace*{-8.2cm}
\hspace*{0.cm}\includegraphics[width=20 cm]{figAx-2.ps}
\vspace*{-11.5cm}
\caption{The strongest constraints on the interaction constant and
interaction range of the Yukawa-type interaction obtained from different
experiments. Lines 1, 2, and 3 are found from measuring the effective Casimir
pressure between two parallel plates, normal, and lateral Casimir forces
between the sinusoidally corrugated surfaces of a sphere and a plate,
respectively. Line labeled ``n" is found from the experiments on neutron
scattering. Lines labeled ``new" and 4 follow from the recent experiment on
measuring the Casimir force in the micrometer separation range and from the
experiment using a torsion pendulum. Line labeled ``gr" and the dashed line
are obtained from the Cavendish-type experiments and from the experiment
nullifying the Casimir force. The regions above each line are excluded
and below each line --- are allowed.
\label{fg2}}
\end{figure}
The obtained constraints are shown by the line labeled ``new" in
Figure~2. For comparison purposes, in this figure we also show the
constraints following from measuring of the effective Casimir pressure
\cite{68,69} (line 1), of the normal Casimir force between sinusoidally
corrugated surfaces of a sphere and a plate at different orientation
angles of corrugations \cite{85,86} (line 2 \cite{87}), and of the lateral
Casimir force force between sinusoidally
corrugated surfaces of a sphere and a plate \cite{88,89} (line 3 \cite{90}).
Note that somewhat stronger constraints than those shown by the line 3 were
obtained \cite{91} from the experiment on measuring the Casimir force
between crossed cylinders \cite{92} on an undefined confidence level
\cite{31,51}. However, the strongest constraints on the parameters of
Yukawa-type interaction in the region below $\lambda=10^{-8}~$m follow
from the experiments on neutron scattering. They are shown by the line
labeled ``n" \cite{93,94}.
Similar to Figure 1, the regions above each line are excluded
by the results of respective experiment, whereas the regions below each line
are allowed.
We continue description of Figure 2 in the region of larger $\lambda$.
The constraints shown by the line 4 are obtained from measuring the Casimir
force by means of the torsion pendulum \cite{95}. The line labeled ``gr"
indicates constraints on the Yukawa-type interaction following from the
Cavendish-type experiments \cite{78,79,96}. As to the dashed line, which
covers the largest interaction range, it is obtained \cite{58} from the
differential force measurement where the Casimir force was completely
nullified (compare with the dashed line in Figure 1 found from the same
experiment).
As is seen in Figure 2, the constraints labeled ``new" are stronger than
the constraints of lines 1 and ``gr" following from measuring the effective
Casimir pressure and from the Cavendish-type experiments over the
interaction range from 550~nm to $4.4~\mu$m. The maximum strengthening by
up to a factor of 24 is reached at $\lambda=3.1~\mu$m. The obtained
constraints are weaker only as compared to those following from the
experiment where the Casimir force was nullified \cite{58}.
\section{Discussion}
In this article, we have considered the problems of dark matter axions,
non-Newtonian gravity and constraints on them. As discussed in Section 1,
axions and axionlike particles have gained wide recognition as the most
probable constituents of dark matter. An active search for axions using
their interactions with photons, electrons, and nucleons is under way in
many laboratories all over the world. A major contribution to the investigation
of interactions between axions and photons in different astrophysical
processes have been made by Prof.~Yu.~N.~Gnedin who suggested several
prospective possibilities for observation of axionlike particles and
constraining their parameters.
Here, we obtain new constraints on the coupling constants of axionlike
particles to nucleons which follow from the recently performed measurement
of the differential Casimir force between Au-coated surfaces of the sphere
and top and bottom of rectangular trenches \cite{46}. The differential
character of this experiment allowed reaching a very high precision and
obtaining the meaningful data up to a very large separation distance of
$8~\mu$m. The measure of agreement between the obtained data and the
theoretical predictions based on first principles of quantum electrodynamics
at nonzero temperature allowed to find rather strong constraints on the
axionlike particles and non-Newtonian gravity of Yukawa type.
The obtained constraints on the coupling constants of axionlike particles
to nucleons are stronger by up to a factor of 4 than the previously known
ones derived \cite{53} from the gravitational experiments and from measuring
the effective Casimir pressure \cite{33,68,69}. This strengthening holds in
the range of axion masses $m_a$ from 10 to 74~meV. We have also shown that
the same experiment on measuring the differential Casimir force in the
micrometer separation range \cite{46} results in up to a factor of 24 stronger
constraints on the interaction constant of Yukawa-type interactions as compared
to the ones found previously from measuring the effective Casimir pressure
\cite{68,69}, an experiment using the torsion pendulum \cite{95}, and the
Cavendish-type experiments \cite{78,79,96}. In this case the strengthening holds
in the interaction range from $\lambda=550~$nm to $\lambda=3.1~\mu$m.
Although the obtained constraints are not the strongest ones (the strongest
constraints on both the axionlike particles and the Yukawa-type interaction
in the interaction ranges indicated above were obtained from the experiment
\cite{58} where the Casimir force was nullified), they complement the results
found from previous measurements of the Casimir interaction and can be
considered as their additional confirmation.
The obtained results fall into intensive studies of axionlike particles,
non-Newtonian gravity and constraints on their parameters. In addition to the
literature already discussed in Section 1, it is pertinent to mention a
haloscope search for dark matter axions which excludes some range of the
axion-photon couplings in models of invisible axions \cite{97} and another
haloscope experiment for the search of galactic axions using a
superconducting resonant cavity \cite{98}. The first results of the promising
experiment for searching the dark matter axions with masses below $1~\mu$eV
are reported in \cite{99}. Two more haloscope experiments are performed for
the search of dark matter axions using their interaction with electronic
spins \cite{100} and photons \cite{101}.
A few prospective experiments for constraining the parameters of axionlike
particles and non-Newtonian gravity are suggested in the literature but not
yet performed. Here, one should mention an experiment on measuring the
Casimir pressure between parallel plates at up to $25-30~\mu$m separations
(Cannex) \cite{102,103,103a,104}. An approach for searching dark matter
axions with $m_a<1~\mu$eV using a superconducting radio frequency cavity
is proposed in \cite{105}. Several possibilities for probing the
non-Newtonian gravity in a submillimeter interaction range by means of
temporal lensing \cite{106}, molecular spectroscopy \cite{107}, and neutron
interferometry \cite{107,108} are also discussed. Finally, very recently
the possibility to detect the axion-nucleon interaction in the Casimir-less
regime by means of levitated system was proposed \cite{109}. According to
the authors, their approach gives a possibility to strengthen the current
constraints on $g$ by several orders of magnitude in the wide region of
axion masses from $10^{-10}~$eV to 10~eV.
\section{Conclusions}
To conclude, the search for dark matter axions, non-Newtonian gravity and
constraints on their parameters is a multidisciplinary problem of the
elementary particle physics, quantum field theory, gravitational theory,
astrophysics and cosmology. At the moment neither axions nor corrections
to Newton's gravitational law, other than that predicted by the General
Relativity theory, are observed, but more and more stringent constraints
on them are obtained. Keeping in mind that there are very serious reasons
for a creation of axions at the very early stages of the Universe
evolution and for existence of
deviations from the Newton law of gravitation at very
short separations, as predicted by the extended Standard Model,
Supersymmetry, Supergravity and String theory, one may hope that these
predictions will find experimental confirmation in the not too far
distant future.
\funding{
This work was supported by the Peter the Great
Saint Petersburg Polytechnic
University in the framework of the Russian state assignment for basic research
(project N FSEG-2020-0024).
V.M.M.~was also partially funded by the Russian Foundation for Basic Research grant number
19-02-00453 A.
}
\acknowledgments{
V.M.M.~is grateful for partial support by the Russian Government Program of Competitive
Growth of Kazan Federal University. }
\reftitle{References}
|
2301.00639
|
\section{Introduction}
The theory of marginally outer trapped surfaces has played an important role in several areas of mathematical general relativity, for example, in proofs of the spacetime positive mass theorem (\sl{e.g.} \cite{EHLS,LeeLesUng}) and in results on the topology of black holes (\sl{e.g.} \cite{GalMots2,GalSch}). In \cite{GalMots2}, a local MOTS rigidity result was obtained, which implies that an outermost MOTS (\sl{e.g.} the surface of a black hole) in an initial data set satisfying the dominant energy condition ($\mu\ge|J|$) is positive Yamabe, \sl{i.e.} admits a metric of positive scalar curvature. This in turn leads to well-known restrictions on the topology of $3$-dimensional outermost MOTS. Such results extend to the spacetime setting well-known results concerning Riemannian manifolds of nonnegative scalar curvature.
In \cite{EicGalMen}, the authors, together with M.~Eichmair, obtained, among other results, a global version of the local MOTS rigidity result in \cite{GalMots2}, which, in particular, does not require a weakly outermost condition; see \cite[Theorem~1.2]{EicGalMen}. This result was motivated in part by J.~Lohkamp's approach to the spacetime positive mass theorem in \cite{Lohkamp2016}. It implies, in dimensions $3\le n\le 7$, Lohkamp's result on the nonexistence of `$\mu-|J|>0$ islands', \cite[Theorem~2]{Lohkamp2016}. Theorem~1.2 in \cite{EicGalMen} has also been applied to obtain a positive mass theorem for asymptotically hyperbolic manifolds with boundary; see \cite{ChruGal}. This theorem will be a useful tool in the present work as well.
In this paper, we present some further initial data rigidity results for compact initial data sets, in both the boundary and no boundary cases. In \cite{GalMen}, the authors considered $3$-dimensional initial data sets containing spherical MOTS. It was shown, roughly speaking, that in a matter-filled spacetime, perhaps with positive cosmological constant, a stable marginally outer trapped $2$-sphere must satisfy a certain area inequality; namely, its area must be bounded above by $4\pi/c$, where $c > 0$ is a lower bound on a natural energy-momentum term. We then established rigidity results for stable, or weakly outermost, marginally outer trapped $2$-spheres when this bound is achieved. In particular, we prove a local splitting result, \cite[Theorem~3.2]{GalMen}, that extends to the spacetime setting a result of H.~Bray, S.~Brendle, and A.~Neves~\cite{BraBreNev} concerning area minimizing $2$-spheres in Riemannian $3$-manifolds with positive scalar curvature. These spacetime results have interesting connections to the Vaidya and Nariai spacetimes \cite{GalMen}.
One of the main aims of the present work is to obtain a global version of \cite[Theorem~3.2]{GalMen}; see Theorem~\ref{thm:main.boundary} in Section~\ref{sec:boundary.cases} for a statement. The proof makes use of certain techniques introduced in \cite{EicGalMen}. In this work, we have also been led to consider certain variations of \cite[Theorem~5.2]{EicGalMen}; see Theorems~\ref{thm:Brane.1}~and~\ref{thm:Brane.2} in Section~\ref{sec:boundary.cases}. Here, it becomes useful to consider the so-called `brane action', as well as the area functional. These results are then used to examine the question of the existence of MOTS in closed (compact without boundary) initial data sets in Section~\ref{sec:closed.cases}. The relationship to known spacetimes is also discussed.
The paper is organized as follows: in Section~\ref{sec:preliminaries}, we review some background material on MOTS; in Section~\ref{sec:boundary.cases}, we state and prove several global rigidity results for compact-with-boundary initial data sets; and, in Section~\ref{sec:closed.cases}, we apply the results obtained in Section~\ref{sec:boundary.cases} to prove some global rigidity statements for closed initial data sets. In Section~\ref{sec:closed.cases}, we also give various examples in order to illustrate the results presented in this paper.
\medskip
\paragraph{\bf{Acknowledgements.}} The work of GJG was partially supported by the Simons Foundation, under Award No. 850541. The work of AM was partially supported by the Conselho Nacional de Desenvolvimento Científico e Tecnológico - CNPq, Brazil (Grant 305710/2020-6), the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES, Brazil (CAPES-COFECUB 88887.143161/2017-0), and the Fundação de Amparo à Pesquisa do Estado de Alagoas - FAPEAL, Brazil (Process E:60030.0000002254/2022). The authors would like to thank Ken Baker and Da Rong Cheng for helpful comments. They would also like to thank Christina Sormani for a comment that motivated the results in Section~\ref{sec:closed.cases}.
\section{Preliminaries}\label{sec:preliminaries}
All manifolds in this paper are assumed to be connected and orientable except otherwise stated.
An \sl{initial data set} $(M,g,K)$ consists of a Riemannian manifold $(M,g)$ with boundary $\partial M$ (possibly $\partial M=\varnothing$) and a symmetric $(0,2)$-tensor $K$ on $M$.
Let $(M,g,K)$ be an initial data set.
The \sl{local energy density} $\mu$ and the \sl{local current density} $J$ of $(M,g,K)$ are given by
\begin{align*}
\mu=\frac{1}{2}(S-|K|^2+(\tr K)^2)\quad\mbox{and}\quad J=\div(K-(\tr K)g),
\end{align*}
where $S$ is the scalar curvature of $(M,g)$. We say that $(M,g,K)$ satisfies the \sl{dominant energy condition} (DEC for short) if
\begin{align*}
\mu\ge|J|\quad\mbox{on}\quad M.
\end{align*}
Consider a closed embedded hypersurface $\Sigma\subset M$. Since, by assumption, $\Sigma$ and $M$ are orientable, we can choose a unit normal field $\nu$ on $\Sigma$. If $\Sigma$ separates $M$, by convention, we say that $\nu$ points to the outside of $\Sigma$.
The \sl{null second fundamental forms} $\chi^+,\chi^-$ of $\Sigma$ in $(M,g,K)$ with respect to $\nu$ are given by
\begin{align*}
\chi^+=K|_\Sigma+A\quad\mbox{and}\quad\chi^-=K|_\Sigma-A,
\end{align*}
where $A$ is the second fundamental form of $\Sigma$ in $(M,g)$ with respect to $\nu$. More precisely,
\begin{align*}
A(X,Y)=g(\nabla_X\nu,Y)\quad\mbox{for}\quad X,Y\in\mathfrak{X}(\Sigma),
\end{align*}
where $\nabla$ is the Levi-Civita connection of $(M,g)$.
The \sl{null expansion scalars} $\theta^+,\theta^-$ of $\Sigma$ in $(M,g,K)$ with respect to $\nu$ are given by
\begin{align}\label{eq:MOTS.def}
\theta^+=\tr_\Sigma(K)+H\quad\mbox{and}\quad\theta^-=\tr_\Sigma(K)-H,
\end{align}
where $H=\tr A$ is the mean curvature of $\Sigma$ in $(M,g)$ with respect to $\nu$. Observe that $\theta^\pm=\tr\chi^\pm$.
R.~Penrose introduced the now famous notion of a \sl{trapped surface}, when both $\theta^+$ and $\theta^-$ are negative. Restricting to one side, we say that $\Sigma$ is \sl{outer trapped} if $\theta^+<0$, \sl{weakly outer trapped} if $\theta^+\le0$, and \sl{marginally outer trapped} if $\theta^+=0$. In the latter case, we refer to $\Sigma$ as a \sl{marginally outer trapped surface} (MOTS for short).
Assume now that $\Sigma$ is a MOTS in $(M,g,K)$, with respect to a unit normal $\nu$, that is a boundary in $M$. More precisely, assume that $\nu$ points towards a top-dimensional submanifold $M^+\subset M$ such that $\partial M^+=\Sigma\sqcup S$, where $S$ (possibly $S=\varnothing$) is a union of components of $\partial M$ (in particular, if $\Sigma$ separates $M$). We think of $M^+$ as the region outside of $\Sigma$. Then we say that $\Sigma$ is \sl{outermost} (resp. \sl{weakly outermost}) if there is no closed embedded hypersurface in $M^+$ with $\theta^+\le0$ (resp. $\theta^+<0$) that is homologous to and different from $\Sigma$. The notions of \sl{locally weakly outermost} and \sl{locally outermost} MOTS can be given in an analogous way.
\begin{rmk}
It is important to mention that initial data sets arise naturally in general relativity. In fact, let $M$ be a spacelike hypersurface in a \sl{spacetime}, \sl{i.e.} a time-oriented Lorentzian manifold, $(\bar N,\bar h)$. Let $g$ be the Riemannian metric on $M$ induced from $\bar h$ and $K$ be the second fundamental form of $M$ in $(\bar N,\bar h)$ with respect to the future-pointing timelike unit normal $u$ on $M$. Then $(M,g,K)$ is an initial data set. As before, let $\Sigma$ be a closed embedded hypersurface in $M$. In this setting, $\chi^+$ and $\chi^-$ are the \sl{null second fundamental forms} of $\Sigma$ in $(\bar N,\bar h)$ with respect to the \sl{null normal fields}
\begin{align*}
\ell^+=u|_\Sigma+\nu\quad\mbox{and}\quad\ell^-=u|_\Sigma-\nu,
\end{align*}
respectively. Observe that $\theta^\pm=\div_\Sigma\ell^\pm$. Physically, $\theta^+$ (resp. $\theta^-$) measures the divergence of the outward pointing (resp. inward pointing) light rays emanating from $\Sigma$.
\end{rmk}
An initial data set $(M,g,K)$ is said to be \sl{time-symmetric} or \sl{Riemannian} if $K=0$. In this case, a MOTS in $(M,g,K)$ is nothing but a minimal hypersurface in $(M,g)$. Moreover, the energy condition $\mu-|J|\ge c$, for some constant $c$, reduces to the requirement on the scalar curvature $S\ge2c$. Quite generally, marginally outer trapped surfaces share many properties with minimal hypersurfaces, which they generalize; see \sl{e.g.} the survey article \cite{AndEicMet}.
As in the minimal hypersurfaces case, an important notion for the theory of MOTS is the notion of \sl{stability} introduced, in the context of MOTS, by L.~Andersson, M.~Mars, and W.~Simon~\cite{AndMarSim05,AndMarSim08}, which we now recall.
Let $\Sigma$ be a MOTS in $(M,g,K)$ with respect to $\nu$. Consider a normal variation of $\Sigma$ in $M$, \sl{i.e.} a variation $t\rightarrow\Sigma_t$ of $\Sigma=\Sigma_0$ with variation vector field $\frac{\partial}{\partial t}|_{t=0}=\phi\,\nu$, $\phi\in C^\infty(\Sigma)$. Let $\theta^\pm(t)$ denote the null expansion scalars of $\Sigma_t$ with respect to $\nu_t$, $\nu=\nu_t|_{t=0}$. Computations as in \cite[p.~2]{AndMarSim05} or \cite[p.~861]{AndMarSim08} give,
\begin{align}\label{eq:first.theta}
\frac{\partial\theta^+}{\partial t}\Big|_{t=0}=L\phi,
\end{align}
where
\begin{align*}
L\phi=-\Delta\phi+2\langle X,\nabla\phi\rangle+(Q+\div X-|X|^2)\phi
\end{align*}
and
\begin{align*}
Q=\frac{1}{2}S_\Sigma-(\mu+J(\nu))-\frac{1}{2}|\chi^+|^2.
\end{align*}
Here, $\Delta$ is the negative semi-definite Laplace-Beltrami operator, $\nabla$ the gradient, $\div$ the divergence, and $S_\Sigma$ the scalar curvature of $\Sigma$ with respect to the induced metric $\langle\,\cdot\,,\,\cdot\,\rangle$. Moreover, $X$ is the tangent vector field on $\Sigma$ that is dual to the 1-form $K(\nu,\,\cdot\,)|_\Sigma$.
It is possible to prove (see \cite[Lemma~4.1]{AndMarSim08}) that $L$ has a real eigenvalue $\lambda_1=\lambda_1(L)$, called the \sl{principal eigenvalue} of $L$, such that $\Re\lambda\ge\lambda_1$ for any other complex eigenvalue $\lambda$. Furthermore, the corresponding eigenfunction $\phi_1$, $L\phi_1=\lambda_1\phi_1$, is unique up to a multiplicative constant and can be chosen to be real and everywhere positive.
Then a MOTS $\Sigma$ is said to be \sl{stable} if $\lambda_1(L)\ge0$. This is equivalent to the existence of a positive function $\phi\in C^\infty(\Sigma)$ such that $L\phi\ge0$. It follows directly from \eqref{eq:first.theta} with $\phi=\phi_1$ that every locally weakly outermost (in particular, locally outermost) MOTS is stable.
Observe that in the Riemannian case, $L$ reduces to the classical stability operator, also known as the Jacobi operator, for minimal hypersurfaces. As such, in the literature, $L$ is known as the \sl{MOTS stability operator} or the \sl{stability operator for MOTS}.
The study of rigidity results for minimal surfaces in Riemannian manifolds with a lower scalar curvature bound has been, and continues to be, an active area of research. From the point of view of initial data sets, these are time-symmetric results, as noted above. It has been of interest to extend some of these results to general initial data sets. In the context of general relativity, black hole horizons within initial data sets are often modeled by MOTS, and, in particular, minimal surfaces in the time-symmetric case. These rigidity results often shed light on properties of spacetimes with black holes, as noted in the introduction.
The next proposition and theorem extend to the general non-time-symmetric setting some results of Bray, Brendle, and Neves \cite{BraBreNev}.
\begin{prop}[Infinitesimal rigidity, \cite{GalMen}]\label{prop:infinitesimal.rigidity}
Let $\Sigma$ be a stable MOTS in a $3$-dimensional initial data set $(M,g,K)$ with respect to a unit normal field $\nu$. Suppose there exists a constant $c>0$ such that $\mu+J(\nu)\ge c$ on $\Sigma$. Then the area of $\Sigma$ satisfies,
\begin{align*}
\mathcal{A}(\Sigma)\le\frac{4\pi}{c}.
\end{align*}
Moreover, if $\mathcal{A}(\Sigma)=4\pi/c$, then the following hold:
\begin{enumerate}
\item[(a)] $\Sigma$ is a round $2$-sphere with Gaussian curvature $\kappa_\Sigma=c$,
\item[(b)] the second fundamental form $\chi^+$ of $\Sigma$ with respect to $\nu$ vanishes, and
\item[(c)] $\mu+J(\nu)=c$ on $\Sigma$.
\end{enumerate}
\end{prop}
The proposition above is used in the proof of the following local splitting theorem. But, before stating the next result, which is also used in the proof of Theorem~\ref{thm:main.boundary}, let us remember the notion of an \sl{area minimizing surface}.
With respect to a fixed Riemannian metric $g$ on a $3$-dimensional manifold $M$, a closed embedded surface $\Sigma\subset M$ is said to be \sl{area minimizing} if $\Sigma$ is of least area in its homology class in $M$, that is, $\mathcal{A}(\Sigma)\le\mathcal{A}(\Sigma')$ for any closed embedded surface $\Sigma'$ that is homologous to $\Sigma$ in $M$. In this case, we also say that $\Sigma$ \sl{minimizes area}. Similarly, $\Sigma$ is said to be \sl{locally area minimizing} if $\mathcal{A}(\Sigma)\le\mathcal{A}(\Sigma')$ for any such $\Sigma'$ in a neighborhood of $\Sigma$ in $M$.
\begin{thm}[Local splitting, \cite{GalMen}]\label{thm:local.splitting}
Let $(M,g,K)$ be a $3$-dimensional initial data set with boundary. Suppose that $(M,g,K)$ satisfies the energy condition $\mu-|J|\ge c$ for some constant $c>0$. Let $\Sigma_0$ be a closed connected component of $\partial M$ such that the following conditions hold:
\begin{enumerate}
\item $\Sigma_0$ is a MOTS with respect to the normal that points into $M$ and
\item $\Sigma_0$ is locally weakly outermost and locally area minimizing.
\end{enumerate}
Then $\Sigma_0$ is topologically $S^2$ and its area satisfies,
\begin{align*}
\mathcal{A}(\Sigma_0)\le\frac{4\pi}{c}.
\end{align*}
Furthermore, if $\mathcal{A}(\Sigma_0)=4\pi/c$, then a collar neighborhood $U$ of $\Sigma$ in $M$ is such that:
\begin{enumerate}
\item[(a)] $(U,g)$ is isometric to $([0,\delta)\times\Sigma_0,dt^2+g_0)$ for some $\delta>0$, where $g_0$ - the induced metric on $\Sigma_0$ - has constant Gaussian curvature $\kappa_{\Sigma_0}=c$,
\item[(b)] $K=a\,dt^2$ on $U$, where $a\in C^\infty(U)$ depends only on $t\in[0,\delta)$, and
\item[(c)] $\mu=c$ and $J=0$ on $U$.
\end{enumerate}
\end{thm}
This theorem extends to the general non-time-symmetric setting the local rigidity statements in \cite{BraBreNev}. The local rigidity obtained in \cite{BraBreNev} is then used to obtain a global rigidity result; see \cite[Proposition~11]{BraBreNev}. In Theorem~\ref{thm:main.boundary} in the next section, we obtain a global version of Theorem~\ref{thm:local.splitting}. A key improvement in this global rigidity result is that it does not require the `weakly outermost' assumption, and hence parallels somewhat more closely the global result in \cite{BraBreNev}.
Now, we recall two topological concepts that are important for our purposes; see also \cite{EicGalMen}.
We say that $M$ satisfies the \sl{homotopy condition} with respect to $\Sigma\subset M$ provided there exists a continuous map $\rho:M\to\Sigma$ such that $\rho\circ i:\Sigma\to\Sigma$ is homotopic to~$\id_\Sigma$, where $i:\Sigma\hookrightarrow M$ is the inclusion map (for example, if $\Sigma$ is a retract of $M$).
On the other hand, a closed not necessarily connected manifold $N$ of dimension~$m$ is said to satisfy the \sl{cohomology condition} if there are $m$ classes $\omega_1,\ldots,\omega_m$ in the first cohomology group $H^1(N)$, with integer coefficients, whose cup product
\begin{align*}
\omega_1\smile\cdots\smile\omega_m\in H^m(N)
\end{align*}
is nontrivial. For example, the $m$-torus $T^m=S^1\times\cdots\times S^1$ satisfies the cohomology condition. More generally, the connected sums $T^m\,\sharp\,Q$ satisfy the cohomology condition for any closed $m$-manifolds $Q$. A version of this condition is considered in \cite[Theorem~5.2]{SY2017}. Here, we are using the form of the condition as it appears in \cite[Theorem~2.28]{Lee}. A manifold $N$ satisfying this cohomology condition has a component that does not carry a metric of positive scalar curvature; see the discussion in \cite{Lee}.
We will make use of the following theorem (mentioned in the introduction) in several situations.
\begin{thm}[{\cite[Theorem~1.2]{EicGalMen}}]\label{thm:EicGalMen}
Let $(M,g,K)$ be an $n$-dimensional, $3\le n\le 7$, compact-with-boundary initial data set. Suppose that $(M,g,K)$ satisfies the dominant energy condition, $\mu\ge|J|$. Suppose also that the boundary can be expressed as a disjoint union $\partial M=\Sigma_0\cup S$ of nonempty unions of components such that the following conditions hold:
\begin{enumerate}
\item[(1)] $\theta^+\le0$ on $\Sigma_0$ with respect to the normal that points into $M$,
\item[(2)] $\theta^+\ge0$ on $S$ with respect to the normal that points out of $M$,
\item[(3)] $M$ satisfies the homotopy condition with respect to $\Sigma_0$, and
\item[(4)] $\Sigma_0$ satisfies the cohomology condition.
\end{enumerate}
Then the following hold:
\begin{enumerate}
\item[(a)] $M\cong[0,\ell]\times\Sigma_0$ for some $\ell>0$.
\item[] Let $\Sigma_t\cong\{t\}\times\Sigma_0$ with unit normal $\nu_t$ in direction of the foliation.
\item[(b)] $\chi^+=0$ on $\Sigma_t$ for every $t\in[0,\ell]$.
\item[(c)] $\Sigma_t$ is a flat $(n-1)$-torus with respect to the induced metric for every $t\in[0,\ell]$.
\item[(d)] $\mu+J(\nu_t)=0$ on $\Sigma_t$ for every $t\in[0,\ell]$. In particular, $\mu=|J|$ on $M$.
\end{enumerate}
\end{thm}
The following is the basic existence result for MOTS due to L.~Andersson and J.~Metzger in $3$-dimensions, and M.~Eichmair in dimensions $3\le n\le 7$. It is used in the proof of Theorem~\ref{thm:main.boundary}, and is the source of the dimension restriction appearing in various results discussed herein.
\begin{thm}[Existence of MOTS, \cite{AndMet,Eic09,Eic10}]\label{thm:MOTS.existence}
Let $(M,g,K)$ be an $n$-dimensional, $3\le n\le 7$, compact-with-boundary initial data set. Suppose that the boundary can be expressed as a disjoint union $\partial M=\Sigma_{\inn}\cup\Sigma_{\out}$, where $\Sigma_{\inn}$ and $\Sigma_{\out}$ are nonempty unions of components of $\partial M$ with $\theta^+\le 0$ on $\Sigma_{\inn}$ with respect to the normal pointing into $M$ and $\theta^+>0$ on $\Sigma_{\out}$ with respect to the normal pointing out of $M$. Then there is an outermost MOTS in $(M,g,K)$ that is homologous to $\Sigma_{\out}$.
\end{thm}
\section{The compact-with-boundary cases}\label{sec:boundary.cases}
In this section, we obtain several global initial data results. These results, in turn, will be applied in the next section to the case that the initial data manifold is closed. The first is a global version of Theorem~\ref{thm:local.splitting}; see the comments above, after the statement of Theorem~\ref{thm:local.splitting}.
\begin{thm}\label{thm:main.boundary}
Let $(M,g,K)$ be a $3$-dimensional compact-with-boundary initial data set. Suppose that $(M,g,K)$ satisfies the energy condition $\mu-|J|\ge c$ for some constant $c>0$. Suppose also that the boundary can be expressed as a disjoint union $\partial M=\Sigma_0\cup S$ of nonempty unions of components such that the following conditions hold:
\begin{enumerate}
\item[(1)] $\theta^+\le0$ on $\Sigma_0$ with respect to the normal that points into $M$,
\item[(2)] $\theta^+\ge0$ on $S$ with respect to the normal that points out of $M$,
\item[(3)] $M$ satisfies the homotopy condition with respect to $\Sigma_0$,
\item[(4)] the relative homology group $H_2(M,\Sigma_0)$ vanishes, and
\item[(5)] $\Sigma_0$ minimizes area.
\end{enumerate}
Then $\Sigma_0$ is topologically $S^2$ and its area satisfies,
\begin{align*}
\mathcal{A}(\Sigma_0)\le\frac{4\pi}{c}.
\end{align*}
Moreover, if $\mathcal{A}(\Sigma_0)=4\pi/c$, then the following hold:
\begin{enumerate}
\item[(a)] $(M,g)$ is isometric to $([0,\ell]\times\Sigma_0,dt^2+g_0)$ for some $\ell>0$, where $g_0$ - the induced metric on $\Sigma_0$ - has constant Gaussian curvature $\kappa_{\Sigma_0}=c$,
\item[(b)] $K=a\,dt^2$ on $M$, where $a\in C^\infty(M)$ depends only on $t\in[0,\ell]$, and
\item[(c)] $\mu=c$ and $J=0$ on $M$.
\end{enumerate}
\end{thm}
\begin{proof}
First, observe that $\Sigma_0$ is connected, since $M$ is connected and satisfies the homotopy condition with respect to $\Sigma_0$.
If $\Sigma_0$ is not homeomorphic to $S^2$, then $\Sigma_0$ is homeomorphic to $T^2\,\sharp\,Q$ for some closed orientable surface $Q$. In particular, $\Sigma_0$ satisfies the cohomology condition and so Theorem~\ref{thm:EicGalMen} applies to $(M,g,K)$. Therefore, $0=\mu-|J|\ge c$ on $M$, which is a contradiction. Then $\Sigma_0$ is topologically $S^2$.
{\bf Claim:} $\Sigma_0$ is a weakly outermost MOTS in $(M,g,K)$ of area $\mathcal{A}(\Sigma_0)=4\pi/c$ unless $\mathcal{A}(\Sigma_0)<4\pi/c$.
Assume that $\mathcal{A}(\Sigma_0)\ge4\pi/c$.
If $\theta_K^+\le0$ is not identically zero on $\Sigma_0$, it follows from \cite[Lemma~5.2]{AndMet} that there is a surface $\Sigma\subset M$ - obtained by a small perturbation of $\Sigma_0$ into $M$ - such that $\theta_K^+<0$ on $\Sigma$ with respect to the normal pointing away from $\Sigma_0$. Let $W$ be the connected compact region bounded by $\Sigma$ and $S$ in $M$. Observe that $\theta_{-K}^+\le0$ on $S$ with respect to the normal that points into $W$ and $\theta_{-K}^+>0$ on $\Sigma$ with respect to the normal that points out of $W$. Applying the MOTS existence theorem (Theorem~\ref{thm:MOTS.existence}), we obtain an outermost MOTS $\Sigma'$ in $(W,g,-K)$ that is homologous to and disjoint from $\Sigma$. Clearly, $\Sigma'$ is homologous to $\Sigma_0$ in $M$.
Without loss of generality, we may assume that each connected component of $\Sigma'$ is homologically nontrivial in $M$. Also, because $H_2(M,\Sigma_0)=0$, $\Sigma'$ is connected.
Since we are assuming that $\Sigma_0$ minimizes area in its homology class, we have
\begin{align*}
\frac{4\pi}{c}\le\mathcal{A}(\Sigma_0)\le\mathcal{A}(\Sigma').
\end{align*}
On the other hand, because $\Sigma'$ is an outermost MOTS in $(W,g,-K)$, in particular stable, the infinitesimal rigidity (Proposition~\ref{prop:infinitesimal.rigidity}) gives that $\mathcal{A}(\Sigma')=4\pi/c$. Therefore, $\Sigma'$ is an area minimizing outermost MOTS in $(W,g,-K)$ of area $\mathcal{A}(\Sigma')=4\pi/c$ and then the local splitting theorem (Theorem~\ref{thm:local.splitting}) applies so that an outer neighborhood of $\Sigma'$ in $W$ is foliated by MOTS, which is a contradiction.
This proves that $\Sigma_0$ is a MOTS in $(M,g,K)$.
Now, we claim that $\Sigma_0$ is weakly outermost in $(M,g,K)$. If not, there is a surface $\Sigma$ that is homologous to $\Sigma_0$ in $M$ and such that $\theta_K^+<0$ on it. Perturbing $\Sigma$ a bit, we may assume that $\Sigma\cap\Sigma_0=\varnothing$. Also, by the strong maximum principle as in \sl{e.g.} \cite[Proposition~2.4]{AndMet} or \cite[Proposition~3.1]{AshGal}, $\Sigma\cap S=\varnothing$.
As before, without loss of generality, we may assume that each connected component of $\Sigma$ is homologically nontrivial in $M$ and, in particular, $\Sigma$ is connected. Let $W$ be the region in $M$ bounded by $\Sigma$ and $S$. Arguing with $(W,g,-K)$ as above, we have a contradiction. Thus $\Sigma_0$ is weakly outermost.
We have then proved that, if $\mathcal{A}(\Sigma_0)\ge4\pi/c$, then $\Sigma_0$ is a weakly outermost MOTS in $(M,g,K)$. In this case, by the infinitesimal rigidity, $\mathcal{A}(\Sigma_0)=4\pi/c$.
This finishes the proof of the Claim.
We have then obtained that $\Sigma_0$ is homeomorphic to $S^2$ and its area satisfies $\mathcal{A}(\Sigma_0)\le4\pi/c$. Furthermore, if $\mathcal{A}(\Sigma_0)=4\pi/c$, then $\Sigma_0$ is an area minimizing weakly outermost MOTS in $(M,g,K)$. In this case, by the local splitting theorem, there is a collar neighborhood $U\cong[0,\delta)\times\Sigma_0$ of $\Sigma_0$ in $M$ such that conclusions (a), (b), and (c) of the theorem hold on $U$. Clearly, $\Sigma_t\cong\{t\}\times\Sigma_0$ converges to a closed embedded MOTS $\Sigma_\delta$ of area $4\pi/c$ as $t\nearrow\delta$. If $\Sigma_\delta\cap S\neq\varnothing$, by the strong maximum principle, $\Sigma_\delta=S$. If $\Sigma_\delta\cap S=\varnothing$, we can replace $\Sigma_0$ by $\Sigma_\delta$ and $M$ by the complement of $U$ and run the process again. The result then follows by a continuity argument.
\end{proof}
The next two theorems make use of the notion of \sl{$(n-1)$-convexity} of a symmetric $(0,2)$-tensor. Imposing such convexity leads to stronger rigidity.
We say that a symmetric $(0,2)$-tensor $P$ on $(M,g)$ is \sl{$(n-1)$-convex} if, at every point $p\in M$, the sum of the smallest $(n-1)$ eigenvalues of $P$ with respect to $g$ is nonnegative (in particular, if $P$ is positive semi-definite). This is equivalent to the trace of $P$ with respect to any $(n-1)$-dimensional linear subspace of $T_pM$ being nonnegative, for every $p\in M$. In particular, if $P$ is $(n-1)$-convex, then $\tr_\Sigma P\ge0$ for every hypersurface $\Sigma\subset M$. This convexity condition has been used by the second-named author in \cite{Men} and by the authors, together with M.~Eichmair, in \cite{EicGalMen} in related contexts.
Let $(M,g,K)$ be as in Theorem~\ref{thm:EicGalMen}, and let $\Sigma$ be a closed embedded hypersurface homologous to $\Sigma_0$.
The next theorem makes use of the functional,
\begin{align*}
\mathcal{B}_\epsilon(\Sigma)=\mathcal{A}(\Sigma)-(n-1)\,\epsilon\,\mathcal{V}(\Sigma),\quad\epsilon=0,1,
\end{align*}
where $\mathcal{A}(\Sigma)$ is the area of $\Sigma$ and $\mathcal{V}(\Sigma)$ is the volume of the region bounded by $\Sigma$ and $\Sigma_0$. In the case $\epsilon=0$, we are just talking about the area functional. In the case $\epsilon=1$, we are talking about the functional associated with hypersurfaces of constant mean curvature $n-1$, sometimes referred to as the \sl{brane action} and denoted by $\mathcal{B}$.
The following theorem extends in a couple of directions Theorem~5.2 in \cite{EicGalMen}.
\begin{thm}\label{thm:Brane.1}
Let $(M,g,K)$ be as in Theorem~\ref{thm:EicGalMen}. Assume that
\begin{enumerate}
\item[(i)] $K+\epsilon\,g$ is $(n-1)$-convex, where $\epsilon=0$ or $\epsilon=1$, and
\item[(ii)] $\Sigma_0$ and $S$ are such that $\mathcal{B}_\epsilon(\Sigma_0)\le\mathcal{B}_\epsilon(S)$.
\end{enumerate}
Then the following hold:
\begin{enumerate}
\item[(a)] $\Sigma_0$ is a flat $(n-1)$-torus with respect to the induced metric $g_0$,
\item[(b)] $(M,g)$ is isometric to $([0,\ell]\times\Sigma_0,dt^2+e^{2\,\epsilon\,t}g_0)$ for some $\ell>0$,
\item[(c)] $K=(1-\epsilon)a-\epsilon\,g$ on $M$, where $a\in C^\infty(M)$ depends only on $t\in[0,\ell]$, and
\item[(d)] $\mu=0$ and $J=0$ on $M$.
\end{enumerate}
\end{thm}
The convexity assumption holds if, in particular, $K$ satisfies, $K\ge-\epsilon\,g$. In the case $\epsilon = 0$, this would apply to cosmological models that are expanding to the future (in all directions).
\begin{proof}
By Theorem~\ref{thm:EicGalMen},
\begin{enumerate}
\item[-] $M\cong[0,\ell]\times\Sigma_0$ for some $\ell>0$, and
\item[-] each leaf $\Sigma_t\cong\{t\}\times\Sigma_0$ is a MOTS with respect to the unit normal $\nu_t$ in direction of the foliation.
\end{enumerate}
On the other hand, since $K+\epsilon\,g$ is $(n-1)$-convex, we have
\begin{align}\label{eq:aux.1}
H(t)-(n-1)\,\epsilon\le H(t)+\tr_{\Sigma_t}K=0,
\end{align}
where $H(t)=\div_{\Sigma_t}\nu_t$ is the mean curvature of $\Sigma_t$.
Now, the first variation of $\mathcal{B}_\epsilon$ gives that
\begin{align}\label{eq:aux.2}
\frac{d}{dt}\mathcal{B}_\epsilon(\Sigma_t)=\int_{\Sigma_t}\phi\,(H(t)-(n-1)\,\epsilon)\,d\Sigma_t\le0,
\end{align}
where $\phi=\langle\nu_t,\partial_t\rangle$ is the lapse function of the foliation. Therefore,
$\mathcal{B}_\epsilon(t)=\mathcal{B}_\epsilon(\Sigma_t)$ is a nonincreasing function on $[0,\ell]$ satisfying $\mathcal{B}_\epsilon(0)\le \mathcal{B}_\epsilon(\ell)$. Thus $\mathcal{B}_\epsilon(t)=\mathcal{B}_\epsilon(\Sigma_t)$ is constant. Inequalities \eqref{eq:aux.1} and \eqref{eq:aux.2} give that $H(t)=(n-1)\,\epsilon=-\tr_{\Sigma_t}K$ for all $t\in[0,\ell]$. In particular, $\theta^-=-2\,(n-1)\,\epsilon$ on $\Sigma_\ell=S$.
The result then follows directly from \cite[Theorem~1.3]{EicGalMen} (observe that our sign convention in the definition of $\theta^-$ in this work is the opposite of that one in \cite{EicGalMen}).
\end{proof}
In the next theorem we consider $\mathcal{B}=\mathcal{B}_1$ under a modified convexity condition.
\begin{thm}\label{thm:Brane.2}
Let $(M,g,K)$ be as in Theorem~\ref{thm:EicGalMen}. Assume that $-(K+g)$ is $(n-1)$-convex. Then $\mathcal{B}(\Sigma_0)\le\mathcal{B}(S)$. Moreover, if equality holds, we have the following:
\begin{enumerate}
\item [(a)] $(M,g)$ is isometric to $([0,\ell]\times\Sigma_0,dt^2+g_t)$ for some $\ell>0$, where $g_t$ is the induced metric on $\Sigma_t\cong\{t\}\times\Sigma_0$.
\item[(b)] Each $\Sigma_t$ is a flat $(n-1)$-torus with respect to $g_t$ and has constant mean curvature $H(t)=n-1$.
\item[(c)] The scalar curvature of $(M,g)$ satisfies $S\le-n(n-1)$. If equality holds, $(M,g)$ is isometric to $([0,\ell]\times\Sigma_0,dt^2+e^{2\,t}g_0)$.
\item[(d)] For each $t\in[0,\ell]$, $\mu+J(\nu_t)=0$ on $\Sigma_t$. In particular, $\mu=|J|$ on $M$.
\item[(e)] $\tr K\le-n$ on $M$. If equality holds, $K=-g$, $S=-n(n-1)$, $\mu=0$, and $J=0$ on $M$.
\end{enumerate}
\end{thm}
The convexity assumption holds if, in particular, $K$ satisfies, $K\le-g$. If one views $K$ as being defined with respect to the past directed unit normal, this would apply to cosmological models that are strongly contracting to the past, \sl{e.g.} that begin with a `big bang'.
\begin{proof}
By Theorem~\ref{thm:EicGalMen},
\begin{itemize}
\item[-] $M\cong[0,\ell]\times\Sigma_0$ with
\begin{align}\label{eq:metric}
g=\phi^2dt^2+g_t,
\end{align}
where $g_t$ is the induced metric on $\Sigma_t\cong\{t\}\times\Sigma_0$.
\item[-] Each $(\Sigma_t,g_t)$ is a flat $(n-1)$-torus.
\item[-] Every leaf $\Sigma_t$ is a MOTS in $(M,g,K)$. In fact,
\begin{align*}
0=\chi^+(t)=A(t)+K|_{\Sigma_t},
\end{align*}
where $A(t)$ is the second fundamental form of $\Sigma_t$ computed with respect to the unit normal $\nu_t$ in direction of the foliation.
\item[-] For each $t\in[0,\ell]$, $\mu+J(\nu_t)=0$ on $\Sigma_t$. In particular, $\mu=|J|$ on $M$.
\end{itemize}
Now, since $-(K+g)$ is $(n-1)$-convex, we have
\begin{align}\label{eq:aux.3}
H(t)-(n-1)\ge H(t)+\tr_{\Sigma_t}K=0,
\end{align}
where $H(t)=\tr A(t)$ is the mean curvature of $\Sigma_t$. Then the first variation of $\mathcal{B}$ gives that
\begin{align}\label{eq:aux.4}
\frac{d}{dt}\mathcal{B}(\Sigma_t)=\int_{\Sigma_t}\phi\,(H(t)-(n-1))\,d\Sigma_t\ge0.
\end{align}
Therefore, $\mathcal{B}(t)=\mathcal{B}(\Sigma_t)$ is a nondecreasing function defined on $[0,\ell]$. In particular, $\mathcal{B}(0)\le\mathcal{B}(\ell)$, that is, $\mathcal{B}(\Sigma_0)\le\mathcal{B}(S)$.
If $\mathcal{B}(\Sigma_0)=\mathcal{B}(S)$, then $\mathcal{B}(t)=\mathcal{B}(\Sigma_t)$ is constant. Therefore, inequalities \eqref{eq:aux.3} and~\eqref{eq:aux.4} imply that $H(t)=n-1=-\tr_{\Sigma_t}K$ for all $t\in[0,\ell]$.
Now, fix $t\in[0,\ell]$, $p\in\Sigma_t$, and let $\{e_1,\ldots,e_{n-1}\}$ be an orthonormal basis for $T_p\Sigma_t$. Define
\begin{align*}
\eta(s)=\cos s\cdot e_{n-1}+\sin s\cdot\nu_t,\quad s\in\mathbb{R},
\end{align*}
and let $\pi(s)$ be the $(n-1)$-dimensional linear subspace of $T_pM$ generated by
\begin{align*}
\{e_1,\ldots,e_{n-2},\eta(s)\}.
\end{align*}
Since $-(K+g)$ is $(n-1)$-convex and $\tr_{\Sigma_t}K=-(n-1)$, we have
\begin{align*}
f(s)\coloneqq\tr_{\pi(s)}K\le-(n-1)\quad\mbox{and}\quad f(0)=\tr_{\Sigma_t}K=-(n-1).
\end{align*}
Therefore, $s=0$ is a critical point of $f(s)$. Observing that
\begin{align*}
f(s)=\sum_{i=1}^{n-2}K(e_i,e_i)+K(\eta(s),\eta(s)),
\end{align*}
we obtain,
\begin{align*}
0=f'(0)=2K(\eta'(0),\eta(0))=2K(\nu_t,e_{n-1}).
\end{align*}
Analogously, $K(\nu_t,e_i)=0$ for $i=1,\ldots,n-2$. This gives that $X^\flat=K(\nu_t,\,\cdot\,)|_{\Sigma_t}=0$ for all $t\in[0,\ell]$.
On the other hand, the first variation of $\theta^+(t)=0$ reads as
\begin{align*}
\frac{\partial\theta^+}{\partial t}=-\Delta\phi+2\langle X,\nabla\phi\rangle+(Q+\div X-|X|^2)\phi=-\Delta\phi+Q\phi,
\end{align*}
where
\begin{align*}
Q=\frac{1}{2}S_{\Sigma_t}-(\mu+J(\nu_t))-\frac{1}{2}|\chi^+(t)|^2=0.
\end{align*}
Thus $\Delta\phi=0$ on $\Sigma_t$ and then $\phi=\phi(t)$ is constant on $\Sigma_t$ for each $t\in[0,\ell]$. Hence, by a simple change of variable in \eqref{eq:metric}, we have
\begin{align}\label{eq:metric.2}
g=dt^2+g_t.
\end{align}
In particular, the $t$-lines are geodesics. Hence, along each leaf $\Sigma=\Sigma_t$, $H=H(t)$ satisfies the scalar Riccati equation,
\begin{align*}
\frac{\partial H}{\partial t}=-\Ric(\partial_t,\partial_t)-|A|^2,
\end{align*}
which, since $H(t)=n-1$, implies,
\begin{align*}
\Ric(\partial_t,\partial_t)+|A|^2=0.
\end{align*}
By the Gauss equation, we have the standard rewriting of the left-hand side in the above equation,
\begin{align*}
\Ric(\partial_t,\partial_t)+|A|^2=\frac12(S-S_{\Sigma}+|A|^2+H^2).
\end{align*}
Hence, since $S_{\Sigma}=0$, we have,
\begin{align}\label{eq:aux.6}
S=-|A|^2-H^2\le-\frac{H^2}{n-1}-H^2=-n(n-1),
\end{align}
which establishes the inequality part in (c). If equality holds, then $|A(t)|^2=n-1$, which, together with $H(t)=n-1$, implies that each $\Sigma_t$ is umbilic; in fact, $A(t)=g_t$. Using this in \eqref{eq:metric.2} easily implies the isometry part in (c).
Since $-(K+g)$ is $(n-1)$-convex, it is not difficult to see that $\tr K\le -n$. In fact, if $\{e_1,\ldots,e_n\}$ is an orthonormal basis for $T_pM$, $p\in M$, then
\begin{align}\label{eq:aux.5}
(n-1)\tr K=\sum_{i=1}^n\sum_{j\neq i}K(e_j,e_j)\le-\sum_{i=1}^n(n-1)=-n(n-1),
\end{align}
that is, $\tr K\le-n$. If $\tr K=-n$, it follows from \eqref{eq:aux.5} that
\begin{align*}
\sum_{j\neq i}K(e_j,e_j)=-(n-1)\quad\mbox{for each}\quad i=1,\ldots,n.
\end{align*}
Therefore,
\begin{align*}
-n=\tr K=K(e_i,e_i)+\sum_{j\neq i}K(e_j,e_j)=K(e_i,e_i)-(n-1),
\end{align*}
that is, $K(e_i,e_i)=-1$ for each $i=1,\ldots,n$. Since $\{e_1,\ldots,e_n\}$ is arbitrary, we have $K=-g$. Thus, using that $A(t)=-K|_{\Sigma_t}=g|_{\Sigma_t}$ in \eqref{eq:aux.6}, we obtain
\begin{align*}
S=-|A(t)|^2-|H(t)|^2=-n(n-1).
\end{align*}
Finally,
\begin{align*}
\mu=\frac{1}{2}(S-|K|^2+(\tr K)^2)=0\quad\mbox{and}\quad J=\div(K-(\tr K)g)=0.
\end{align*}
\end{proof}
\section{Applications: closed cases}\label{sec:closed.cases}
In this section, we wish to apply the results of the previous section to initial data manifolds that are closed (compact without boundary). These results naturally relate to cosmological (\sl{i.e.} spatially closed) spacetimes. We'll illustrate the results with various examples.
\subsection{The spherical case}\label{sphericalcase}
In this section, we want to apply Theorem~\ref{thm:main.boundary} to the case that $M$ is closed.
Let $M$ be an $n$-dimensional closed manifold. Suppose the $(n-1)$-th homology group $H_{n-1}(M)$ is nontrivial. Any nontrivial element of $H_{n-1}(M)$ gives rise to a smooth closed embedded non-separating orientable hypersurface $\Sigma\subset M$. In particular, $\Sigma$ is \sl{two-sided} in $M$, \sl{i.e.} there is an embedding $F:[-1,1]\times\Sigma\to M$ such that $F(0,p)=p$ for each $p\in\Sigma$. Let $U$ denote the open set $F((-1,1)\times\Sigma)\subset M$. We say that $M$ is \sl{retractable with respect to $\Sigma$} if $M\setminus U$ retracts onto some component of $\partial U$. If we consider a Riemannian metric $g$ on $M$, given a unit normal field $\nu$ on $\Sigma$ with respect to $g$, we say that $M$ is \sl{retractable with respect to $\Sigma$ towards $\nu$} if $M\setminus U$ retracts onto the component of $\partial U$ towards which $\nu$ points.
An obvious situation where this occurs is when $M$ is of the form $M=S^1\times Q$, with $Q$ closed. Then $M$ is retractable with respect to $\Sigma=\{x\}\times Q$, $x\in S^1$. Another situation of interest is when $M$ is of the form $M=T^n\,\sharp\,Q$. View $T^n$ as an $n$-dimensional cube with opposite boundary faces identified. To obtain $M$, we may assume the connected sum takes place in a bounded open set $U$ inside the cube. Let $\Sigma$ be an $(n-1)$-torus parallel to one of the faces away from the set $U$. Then $M$ is retractable with respect to $\Sigma$. More generally, if $M$ is retractable with respect to $\Sigma$, then so is $M\,\sharp\,Q$, with $Q$ closed, provided the connect sum takes place away from $\Sigma$.
\begin{thm}\label{thm:cor.boundary}
Let $(M,g,K)$ be a $3$-dimensional closed initial data set satisfying the energy condition $\mu-|J|\ge c$ for some constant $c>0$. Suppose that $(M,g,K)$ admits a MOTS $\Sigma$, with respect to a unit normal field $\nu$, such that the following conditions hold:
\begin{enumerate}
\item[(I)] $M$ is retractable with respect to $\Sigma$ towards $\nu$,
\item[(II)] the homology group $H_2(M)$ is generated by the class of $\Sigma$, and
\item[(III)] $\Sigma$ minimizes area.
\end{enumerate}
Then $\Sigma$ is topologically $S^2$ and its area satisfies,
\begin{align}\label{ineqA}
\mathcal{A}(\Sigma)\le\frac{4\pi}{c}.
\end{align}
Moreover, if $\mathcal{A}(\Sigma)=4\pi/c$, then the following hold:
\begin{enumerate}
\item[(a')] $(M,g)$ is isometric to $[0,\ell]\times\Sigma/{\sim}$ endowed with the induced metric from the product $([0,\ell]\times\Sigma,dt^2+h)$, where `\,$\sim$' means that $\{0\}\times\Sigma$ and $\{\ell\}\times\Sigma$ are suitably identified and $h$ - the induced metric on $\Sigma$ - has constant Gaussian curvature $\kappa_\Sigma=c$,
\item[(b')] $K=a\,dt^2$ on $M$, where $a\in C^\infty(M)$ depends only on $t$, and
\item[(c')] $\mu=c$ and $J=0$ on $M$.
\end{enumerate}
\end{thm}
\begin{proof}
First, observe that, by making a `cut' along $\Sigma$, we obtain a $3$-dimensional compact manifold $M'$ with two boundary components, say $\Sigma_0$ and $S$. Also, the initial data $(g,K)$ on $M$ gives rise to data $(g',K')$ on $M'$ in the natural way. The boundary components $\Sigma_0$ and $S$ are both isometric to $\Sigma$ with respect to the corresponding induced metrics.
Now, consider the initial data set $(M',g',K')$. Observe that the boundary components $\Sigma_0$ and $S$ of $M'$ can be chosen in such a way that conditions (1)-(5) of Theorem~\ref{thm:main.boundary} are satisfied. In fact,
\begin{enumerate}
\item $\Sigma_0$ is a MOTS with respect to the normal that points into $M'$,
\item $S$ is a MOTS with respect to the normal that points out of $M'$,
\item $M'$ satisfies the homotopy condition with respect to $\Sigma_0$, since $M$ is retractable with respect to $\Sigma$ towards $\nu$,
\item the relative homology group $H_2(M',\Sigma_0)$ vanishes, since $H_2(M)$ is generated by the class of $\Sigma$, and
\item $\Sigma_0$ minimizes area in $(M',g')$ as $\Sigma$ minimizes area in $(M,g)$.
\end{enumerate}
Conditions (1) and (2) above follow from the fact of $\Sigma$ being a MOTS in $(M,g,K)$ with respect to $\nu$ and the choice of $\Sigma_0$ and $S$. Therefore, by Theorem~\ref{thm:main.boundary}, $\Sigma_0$ is topologically $S^2$ and its area satisfies $\mathcal{A}(\Sigma_0)\le4\pi/c$. The same conclusions hold for~$\Sigma$. Moreover, if $\mathcal{A}(\Sigma)=4\pi/c$, that is, $\mathcal{A}(\Sigma_0)=4\pi/c$, then conclusions (a)-(c) of Theorem~\ref{thm:main.boundary} hold for $(M',g',K')$ and thus $(M,g,K)$ satisfies (a')-(c').
\end{proof}
\begin{rmk}
Initial data sets satisfying the assumptions of Theorem~\ref{thm:cor.boundary} arise naturally in the Nariai spacetime. The Nariai spacetime is a solution to the vacuum Einstein equations with positive cosmological constant, $\Lambda>0$. It is a metric product of $2$-dimensional de Sitter space $dS_2$ and $S^2$,
\begin{align*}
\bar N=(\mathbb{R}\times S^1)\times S^2,\quad\bar h=-dt^2+a^2\cosh^2(t/a)\,d\chi^2+a^2d\Omega^2,
\end{align*}
where $a=\frac{1}{\sqrt{\Lambda}}$. As described in \cite{Bousso, BousHawk}, the Nariai spacetime is an interesting limit of Schwarzschild-de Sitter space, as the size of the black hole increases and its area approaches the upper bound in \eqref{ineqA}, with $c=\Lambda$.
Under the transformation, $\cosh(t/a)=\sec\tau$, the metric $\bar h$ becomes,
\begin{align*}
\bar h=\frac{a^2}{\cos^2(\tau)}\left(-d\tau^2+d\chi^2\right)+a^2d\Omega^2,
\end{align*}
where $\tau$ is in the range, $-\frac{\pi}{2}<\tau<\frac{\pi}{2}$. With this change of time coordinate, we see that $dS_2$ is locally conformal to the Minkowski plane. A Penrose type diagram for $(\bar N,\bar h)$ is depicted in Figure~\ref{dsfig1}. Each point in the diagram represents a round $2$-sphere of radius $a$. In the diagram, $M=\Gamma\times S^2$, where $\Gamma$ is a smooth spacelike graph over the circle: $\tau=0$, $0\le\chi\le2\pi$ in $dS_2$. Taking $\Sigma$ to be the $2$-sphere intersection of $M$ with the totally geodesic null hypersurface $H$, one easily verifies that $(M,g,K)$, where $g$ is the induced metric and $K$ is the second fundamental form of $M$, respectively, satisfies the assumptions of Theorem~\ref{thm:cor.boundary}, with equality in \eqref{ineqA}. We note that there are initial data sets in (spatially closed) Schwarzschild-de Sitter that satisfy all the assumptions of Theorem~\ref{thm:cor.boundary}, except for equality in \eqref{ineqA}.
\end{rmk}
\begin{figure}[ht]
\begin{center}
\mbox{
\includegraphics[width=3.2in]{dsfig.pdf}
}
\end{center}
\caption{Nariai spacetime.}
\label{dsfig1}
\end{figure}
\subsection{The non-spherical case}\label{sec:nonsphere}
We now consider applications of Theorems \ref{thm:EicGalMen}, \ref{thm:Brane.1} and \ref{thm:Brane.2} to closed initial data sets satisfying the DEC. As a consequence, we obtain results concerning the existence and rigidity of MOTS with nontrivial (\sl{e.g.} toroidal) topology in the cosmological setting.
We first consider an example. Let $(\bar N,\bar h)$ be the FLRW spacetime,
\begin{align*}
\bar N=(0,a)\times M,\quad\bar h=-dt^2+g_t,
\end{align*}
where $g_t=G^2(t)\,d\Omega^2$ and $(M,d\Omega^2)$ is the unit $3$-sphere. For each $t\in(0,a)$, consider the initial data $(M_t=\{t\}\times M,g_t,K_t)$, where $K_t$, the second fundamental form, is given by
\begin{align*}
K_t=\frac{\dot{G}(t)}{G(t)}\,g_t.
\end{align*}
In particular, either $K_t$ or $-K_t$ is $2$-convex, depending on the sign of $\dot{G}(t)$. One easily verifies that the DEC holds (strictly) for any choice of scale factor $G(t)$.
For each $t\in(0,a)$, it is easy to see that $(M_t,g_t,K_t)$ contains a spherical MOTS. Indeed, the latitudinal $2$-spheres take on all mean curvature values between $-\infty$ and $+\infty$. Choose the latitudinal $2$-sphere $\Sigma_t$ such that its mean curvature satisfies
\begin{align}\label{motscondition}
H_t=-\tr_{\Sigma_t}K_t=-2\frac{\dot{G}(t)}{G(t)}.
\end{align}
Then, by \eqref{eq:MOTS.def}, $\Sigma_t$ is a MOTS, $\theta^+_t=0$.
In fact, it is also the case that $(M_t,g_t,K_t)$ contains a toroidal MOTS. Here, we rely on the one-parameter family of Clifford tori $T_r$ in the unit $3$-sphere $S^3$. By identifying $S^3$ with the unit sphere centered at the origin in $\mathbb{R}^4$, $T_r$, $0<r<1$, is defined as
\begin{align*}
T_r=\left\{(x,y,u,v)\in S^3:x^2+y^2=r^2,\,u^2+v^2=1-r^2\right\}.
\end{align*}
The `standard' Clifford torus is obtained by setting $r = \frac1{\sqrt{2}}$. An elementary computation shows that each $T_r$ has constant mean curvature (see \cite{Kit}),
\begin{align*}
H_r=\frac{1-2r^2}{r\sqrt{1-r^2}}.
\end{align*}
In particular, the Clifford tori take on all mean curvature values between $-\infty$ and~$+\infty$. Thus, arguing as above in the sphere case, there exists an embedded torus $\Sigma_t$ in $(M_t,g_t,K_t)$ satisfying \eqref{motscondition}, which hence is a MOTS.
One can modify the initial data set $(M_t,g_t,K_t)$ by adding a handle from one side of the torus $\Sigma_t$ to the other, \sl{à la} Gromov-Lawson \cite{GL}, so that $\Sigma_t$ is no longer homologically trivial, and such that the DEC still holds. However, the resulting initial data manifold won't be retractable with respect to $\Sigma_t$, as follows from the next theorem.
\begin{thm}\label{thm:cor.boundary.2}
Let $(M,g,K)$ be an $n$-dimensional, $3\le n\le 7$, closed initial data set satisfying the DEC, $\mu\ge|J|$. Suppose that $(M,g,K)$ admits a MOTS $\Sigma$, with respect to a unit normal field $\nu$, such that the following conditions hold:
\begin{enumerate}
\item[(I)] $M$ is retractable with respect to $\Sigma$ towards $\nu$ and
\item[(II)] $\Sigma$ satisfies the cohomology condition.
\end{enumerate}
Then $\chi^+=0$ on $\Sigma$ and $\Sigma$ is a flat $(n-1)$-torus with respect to the induced metric. Moreover, the following hold:
\begin{enumerate}
\item[(a')] $M\setminus\Sigma\cong(0,\ell)\times\Sigma$ for some $\ell>0$.
\item[] Let $\Sigma_t\cong\{t\}\times\Sigma$ with unit normal $\nu_t$ in direction of the foliation.
\item[(b')] $\chi^+=0$ on $\Sigma_t$ for every $t\in(0,\ell)$.
\item[(c')] $\Sigma_t$ is a flat $(n-1)$-torus with respect to the induced metric for every $t\in(0,\ell)$.
\item[(d')] $\mu+J(\nu_t)=0$ on $\Sigma_t$ for every $t\in(0,\ell)$. In particular, $\mu=|J|$ on $M$.
\end{enumerate}
If we assume further that $K$ is $(n-1)$-convex, we also have:
\begin{enumerate}
\item[(e')] $(M,g)$ is isometric to $[0,\ell]\times\Sigma/{\sim}$ endowed with the induced metric from the product $([0,\ell]\times\Sigma,dt^2+h)$, where $h$ is the induced metric on $\Sigma$. In particular, $(M,g)$ is flat.
\item[(f')] $K=a\,dt^2$, where $a\in C^\infty(M)$ depends only on $t$.
\item[(g')] $\mu=0$ and $J=0$ on $M$.
\end{enumerate}
\end{thm}
\begin{proof}
As in the proof of Theorem~\ref{thm:cor.boundary}, let $(M',g',K')$ be the initial data set derived from $(M,g,K)$ - by making a `cut' along $\Sigma$ - with two boundary components, $\Sigma_0$ and $S$, both isometric to $\Sigma$, such that $\Sigma_0$ is a MOTS with respect to the normal that points into $M'$ and $S$ is a MOTS with respect to the normal that points out of~$M'$.
It is not difficult to see that $(M',g',K')$ satisfies all the assumptions of Theorem~\ref{thm:EicGalMen} and then all its conclusions. Thus $\Sigma$ is a flat $(n-1)$-torus with $\chi^+=0$ on it and conclusions (a')-(d') of the theorem hold.
If $K$ is $(n-1)$-convex, since $\mathcal{A}(\Sigma_0)=\mathcal{A}(S)$, it follows that $(M',g',K')$ satisfies all the hypotheses of Theorem~\ref{thm:Brane.1} for $\epsilon=0$. Conclusions (e')-(g') then follow.
\end{proof}
\begin{rmk}
It follows, for example, that in a $4$-dimensional spacetime which satisfies the DEC {\it strictly} and which has toroidal Cauchy surfaces, there cannot be any homologically nontrivial toroidal MOTS in any Cauchy surface. This applies, in particular, to the time slices in the toroidal ($k=0$) FLRW spacetimes, that satisfy the Einstein equations with dust (zero-pressure perfect fluid) source.
In view of property (g'), to find initial data sets satisfying the assumptions of Theorem~\ref{thm:cor.boundary.2}, one should perhaps consider vacuum spacetimes. A well-known class of examples are the toroidal Kasner spacetimes,
\begin{align*}
\bar N=(0,\infty)\times M,\quad\bar h=-dt^2+t^{2p_1}dx^2+t^{2p_2}dy^2+t^{2p_3}dz^2,
\end{align*}
where $x,y,z$ are to be understood as periodic coordinates, and where $p_1\le p_2\le p_3$ must satisfy,
\begin{align*}
p_1+p_2+p_3=1\quad\text{and}\quad p_1^2+p_2^2+p_3^2=1.
\end{align*}
Let $M_0$ be the $t=1$ time slice, with metric $g$ and second fundamental form $K$ induced from $(\bar N,\bar h)$. It is not hard to show that in order for $K$ to be $2$-convex, one must have, $p_1=p_2=0$ and $p_3=1$, so that $\bar h$ becomes,
\begin{align*}
\bar h=-dt^2+dx^2+dy^2+t^2dz^2.
\end{align*}
This is an exceptional Kasner spacetime, known as `flat Kasner'. It is locally isometric to Minkowski space. Taking $\Sigma$ to be the torus $t=1$, $z=z_0$, we see that $M_0$ satisfies the assumptions of Theorem~\ref{thm:cor.boundary.2}, including the $2$-convexity assumption.
We mention one further example which illustrates a certain flexibility in initial data sets satisfying (I) and (II), but not the convexity condition. It's a small modification of Example 4.2 in \cite{EicGalMen}.
Let $\mathbb{R}^3_1$ be Minkowski space with standard coordinates $t,x,y,z$. Consider the box $\mathcal{B}=\{(x,y,z):0\le x\le 1,0\le y\le 1,0\le z\le 1\}$ in the $t=0$ slice. Let $f:\mathcal{B}\to\mathbb{R}$ be a smooth function that vanishes near the boundary of $\mathcal{B}$ and whose graph is spacelike in $\mathbb{R}^3_1$. By identifying opposite sides of the box, we obtain an initial data set $(M,g,K)$ with $M\cong T^3$, where $M$ is given by the graph of $f$, and where $g$ and $K$ are induced from the graph of $f$. Let $\Sigma$ be the intersection of $M$ with the null hyperplane $t=z-\frac12$; see Figure~\ref{dsfig2}. Because the null hyperplane is totally geodesic, $\Sigma$ is necessarily a MOTS. It follows that $(M,g,K)$ satisfies (I) and (II) with respect to $\Sigma$. Note also that $(M,g,K)$ satisfies the DEC; in fact, because it essentially sits in Minkowski space, it is a vacuum initial data set, $\mu=0$, $J=0$. Hence, $(M,g,K)$ satisfies all the assumptions of Theorem~\ref{thm:cor.boundary.2}, except, in general, the convexity condition on $K$. The foliation by MOTS guaranteed by properties (a')-(d') comes from intersecting $M$ with the null hyperplanes $t=z+c$. That these properties hold may be understood in terms of special features of totally geodesic null hypersurfaces.
\end{rmk}
\begin{figure}[ht]
\begin{center}
\mbox{
\includegraphics[width=4in]{closedIDS.pdf}
}
\end{center}
\caption{Initial data set satisfying (I) and (II) of Theorem~\ref{thm:cor.boundary.2}.}
\label{dsfig2}
\end{figure}
\begin{rmk}\label{rem:LohPMT}
Finally, we mention a connection to the spacetime positive mass theorem, specifically the approach taken by Lohkamp~\cite{Lohkamp2016}, from a perspective slightly different from the discussion in \cite{EicGalMen}. Lohkamp reduces the proof to a stand alone result, namely the nonexistence of `$\mu-|J|>0$ islands', see \cite[Theorem~2]{Lohkamp2016}. By a standard compactification (which Lohkamp also considers), the setting of Theorem~2 immediately gives an initial data set satisfying the DEC, with initial data manifold $M\cong T^n\,\sharp\,Q$, $Q$ closed, and a toroidal MOTS $\Sigma$, such that $M$ is retractable with respect to $\Sigma$ (see the discussion at the beginning of Section \ref{sphericalcase}). Theorem~\ref{thm:cor.boundary.2} then yields that $\mu=|J|$ (among other things), which implies Lohkamp's no $\mu-|J|>0$ islands result in dimensions $3\le n\le 7$.
\end{rmk}
Lastly, we consider the following consequence of Theorem~\ref{thm:Brane.2}.
\begin{cor}
Let $(M,g,K)$ be an $n$-dimensional, $3\le n\le 7$, closed initial data set satisfying the DEC, $\mu\ge|J|$. Assume that $-(K+g)$ is $(n-1)$-convex. Then $(M,g,K)$ cannot satisfy conditions {\rm (I)-(II)} of Theorem~\ref{thm:cor.boundary.2}.
\end{cor}
Examples like those discussed at the beginning of Section~\ref{sec:nonsphere} show that, while the conditions (I) and (II) can't be simultaneously satisfied, either one can be.
\begin{proof}
Let $(M',g',K')$ be as in the proof of Theorem~\ref{thm:cor.boundary.2}. If $-(K+g)$ is $(n-1)$-convex, in particular, $-(K'+g')$ is $(n-1)$-convex, it follows from the first part of Theorem~\ref{thm:Brane.2} that $\mathcal{B}(\Sigma_0)\le\mathcal{B}(S)$, which is a contradiction, because
\begin{align*}
\mathcal{B}(\Sigma_0)=\mathcal{A}(\Sigma_0)=\mathcal{A}(S)>\mathcal{B}(S).
\end{align*}
\end{proof}
\bibliographystyle{amsplain}
|
2206.12267
|
\section{Introduction} \label{sec:intro}
Genome-wide association studies (GWAS) have led to the identification of hundreds of common genetic variants, or single nucleotide polymorphisms (SNPs), associated with complex traits~\citep{Visscher2017} and are typically conducted by testing association on each SNP independently. However, these studies are plagued with the multiple testing burden that limits discovery of potentially important predictors. Moreover, GWAS have brought to light the problem of missing heritability, that is, identified variants only explain a low fraction of the total observed variability for traits under study~\citep{Manolio2009}. Multivariable regression methods, on the other hand, simultaneously fit many SNPs in a single model and are exempt from the multiple testing burden. In both simulations and analysis of high-dimensional data, sparse regularized logistic models have shown to achieve lower false-positive rates and higher precision than methods based on univariable GWAS summary statistics in case-control studies~\citep{Hoggart2008, Priv2019}. Contrary to univariable methods which implicitly assume that SNPs are independent, a regularized model makes use of the linkage disequilibrium (LD) structure between different loci, assigning weights to SNPs based on their relative importance after accounting for all other SNPs already in the model.
Confounding due to population structure or subject relatedness is another major issue in genetic association studies. Modern large scale cohorts will often include participants from different ethnic groups as well as admixed individuals, that is, subjects with individual-specific proportions of ancestries, or individuals with known or unknown familial relatedness, defined as cryptic relatedness~\citep{Sul2018}. Confounding comes from the fact that allele frequencies can differ greatly between individuals who do not share similar ancestry. When ignored, population structure and subject relatedness can decrease power and lead to spurious associations~\citep{Astle2009, Price2010}. Common practice is still to drop samples by applying filters for relatedness or genetic ancestry, which can result in decreasing the sample size by nearly 30\%~\citep{Loh2018} in the full UK Biobank data set~\citep{Bycroft2018}.
Principal component analysis (PCA) can control for the confounding effect due to population structure by including the top eigenvectors of a genetic similarity matrix (GSM) as fixed effects in the regression model~\citep{Price2006}.
With admixture and population structure being low dimensional fixed-effects processes, they can correctly be accounted for by using a relatively small number of PCs (e.g. 10)~\citep{Astle2009, Novembre2008}. However, using too few PCs can result in residual bias leading to false positives, while adding too many PCs as covariates can lead to a loss of efficiency~\citep{Zhao2018}. Alternatively, using mixed models (MMs), one can model population structure and/or closer relatedness by including a polygenic random effect with variance-covariance structure proportional to the GSM~\citep{Yu2005}. Indeed, kinship is a high-dimensional process, such that it cannot be fully captured by a few PCs~\citep{Hoffman2013}. Thus, it would require the inclusion of too many PCs as covariates, relative to the dimension of the sample size. Hence, while both PCA and MMs share the same underlying model, MMs are more robust in the sense that they do not require distinguishing between the different types of confounders~\citep{Price2010}. Moreover, MMs alleviate the need to evaluate the optimal number of PCs to retain in the model as fixed effects.
Several authors have proposed to combine penalized quasi-likelihood (PQL) estimation with sparsity inducing regularization to perform selection of fixed and/or random effects in generalized linear mixed model (GLMMs)~\citep{grollVariableSelectionGeneralized2014, huiJointSelectionMixed2017}. However, none of these methods are currently scalable for modern large-scale genome-wide data, nor can they directly incorporate relatedness structure through the use of a kinship matrix. Indeed,
the computational efficiency of recent multivariable methods for high-dimensional MMs rely on performing a spectral decomposition of the covariance matrix to rotate the phenotype and design matrix such that the transformed data become uncorrelated~\citep{Bhatnagar2020, Rakitsch2012}. These methods are typically restricted to linear models since in GLMMs, it is no longer possible to perform a single spectral decomposition to rotate the phenotype and design matrix, as the covariance matrix depends on the sample weights which in turn depend on the estimated regression coefficients that are being iteratively updated. This limits the application of high-dimensional MMs to analysis of binary traits in genetic association studies.
In this paper, we introduce a new method called \texttt{pglmm} that allows to simultaneously select variables and estimate their effects, accounting for between-individual correlations and binary nature of the trait. We develop a scalable algorithm based on PQL estimation which makes it possible, for the first time, to fit penalized GLMMs on high-dimensional GWAS data. To speedup computation, we estimate the variance components and dispersion parameter of the model under the null hypothesis of no genetic effect. Secondly, we use an upper-bound for the inverse variance-covariance matrix in order to perform a single spectral decomposition of the GSM and greatly reduce memory usage. Finally, we implement an efficient block coordinate descent algorithm in order to find the optimal estimates for the fixed and random effects parameters. Our method is implemented in an open source \texttt{Julia} programming language ~\citep{bezanson2017julia} package called \texttt{PenalizedGLMM.jl} and freely available at \url{https://github.com/julstpierre/PenalizedGLMM}.
The rest of this paper is structured as follows. In Section \ref{sec:methods} we present our model and describe the block coordinate gradient descent algorithm that is used to estimate the model parameters. We also discuss several approaches to select the optimal tuning parameter in regularized models, and we detail how predictions are obtained in GLMs with PC adjustment versus our proposed mixed model. In Section \ref{sec:results}, we show through simulations that both LMM and logistic model with PC adjustment fail to correctly select important predictors and estimate their effects when the dimensionality of the kinship matrix is high. Further, we demonstrate through the analysis of two polygenic binary traits in the UKBB data that our method achieves higher predictive performance, while also selecting consistently fewer predictors than a logistic lasso with PC adjustment. We finish with a discussion of our results, some limitations and future directions in Section \ref{sec:discussion}.
\section{Methods}\label{sec:methods}
\subsection{Model}\label{subsec:model}
We consider the following GLMM
\begin{align}\label{eq:model}
g(\mu_i) = \eta_i = \bm{X_}i\bm\alpha +\bm{G_}i\bm{\gamma} + b_i,
\end{align}
for $i=1,..,n$, where $\mu_i=\mathbb{E}(y_{i}=1 | \bm{X}_i, \bm{G}_i, b_i)$, $\bm{X}_i$ is a $1\times m$ row vector of covariates for subject $i$, $\bm\alpha$ is a $m \times 1$ column vector of fixed covariate effects including the intercept, $\bm{G}_i$ is a $1 \times p$ row vector of genotypes for subject $i$ taking values $\{0,1,2\}$ as the number of copies of the minor allele, and $\bm\gamma$ is a $p \times 1$ column vector of fixed additive genotype effects. We assume that $\bm{b}=(b_1,...,b_n)^\intercal \sim \mathcal{N}(0, \sum_{s=1}^S \tau_s \bm{V}_s)$ is an $n \times 1$ column vector of random effects, $\bm{\tau}=(\tau_1,...,\tau_S)^\intercal$ are variance component parameters, and $\bm{V}_s$ are known $n\times n$ relatedness matrices. The phenotypes $y_i$ are assumed to be conditionally independent and identically distributed given $(\bm{X}_i, \bm{G}_i, \bm{b})$ and follow any exponential family distribution with canonical link function $g(\cdot)$, mean $\mathbb{E}(y_i | \bm{b}) =\mu_i$ and variance $\text{Var}(y_i| \bm{b}) = \phi a_i^{-1} \nu(\mu_i),$ where $\phi$ is a dispersion parameter, $a_i$ are known weights and $\nu(\cdot)$ is the variance function. In order to estimate the parameters of interest and perform variable selection, we need to use an approximation method to obtain a closed analytical form for the marginal likelihood of model \eqref{eq:model}. Following the derivation of~\citet{Chen2016}, we propose to fit \eqref{eq:model} using a penalized quasi-likelihood (PQL) method, from where the log integrated quasi-likelihood function is equal to
\begin{align}\label{eq:A3}
ql(\bm{\alpha}, \bm{\gamma}, \phi, \bm{\tau}) = -\frac{1}{2}\text{log}\left|\sum_{s=1}^S\tau_s\bm{V}_s\bm{W} + \bm{I}_n\right| + \sum_{i=1}^n ql_i(\bm{\alpha}, \bm{\gamma}|\bm{\tilde{b}}) - \frac{1}{2}\bm{\tilde{b}}^\intercal \left(\sum_{s=1}^S \tau_s\bm{V}_s\right)^{-1}\bm{\tilde{b}},
\end{align}
where $\bm{W} = \textrm{diag}\left\{ \frac{a_i}{\phi\nu(\mu_i)[g'(\mu_i)^2]}\right\}$ is a diagonal matrix containing weights for each observation, $ql_i(\bm{\alpha,\gamma}|\bm{b}) = \int_{y_i}^{\mu_i}\frac{a_i(y_i-\mu)}{\phi\nu(\mu)} d\mu$ is the quasi-likelihood for the $ith$ individual given the random effects $\bm b$, and $\tilde{\bm{b}}$ is the solution which maximizes \eqref{eq:A3}.
In typical genome-wide studies, the number of predictors is much greater than the number of observations ($p > n$), and the parameter vector $\bm{\gamma}$ becomes underdetermined when modelling all SNPs jointly. Thus, we propose to add a lasso regularization term~\citep{lasso} to the negative quasi-likelihood function in \eqref{eq:A3} to seek a sparse subset of $\bm{\gamma}$ that gives an adequate fit to the data. Because $ql(\bm{\alpha}, \bm{\gamma}, \phi, \bm{\tau})$ is a non-convex loss function, we propose a two-step estimation method to reduce the computational complexity. First, we obtain the variance component estimates $\hat{\phi}$ and $\bm{\hat{\tau}}$ under the null hypothesis of no genetic effect ($\bm{\gamma} = \bm{0}$) using the AI-REML algorithm~\citep{Gilmour1995, Chen2016} detailed in Appendix A of the supplementary material. Assuming that the weights in $\bm{W}$ vary slowly with the conditional mean, we drop the first term in \eqref{eq:A3}~\citep{breslowApproximateInferenceGeneralized1993} and define the following objective function which we seek to minimize with respect to $(\bm{\alpha}, \bm{\gamma}, \tilde{\bm{b}})$:
\begin{align}\label{eq:objfunc}
(\hat{\bm{\alpha}}, \hat{\bm{\gamma}}, \hat{\bm{b}}) &= \underset{\bm{\alpha}, \bm{\gamma}, \tilde{\bm{b}}}{\text{argmin }} Q_{\lambda}(\bm{\alpha}, \bm{\gamma}, \tilde{\bm{b}}), \nonumber \\
Q_{\lambda}(\bm{\alpha}, \bm{\gamma}, \tilde{\bm{b}}) &= -\sum_{i=1}^n ql_i(\bm{\alpha}, \bm{\gamma}|\bm{\tilde{b}}) + \frac{1}{2}\bm{\tilde{b}}^\intercal\left(\sum_{s=1}^S \hat{\tau}_s\bm{V}_s\right)^{-1}\bm{\tilde{b}} + \lambda\sum_j v_j|\gamma_j| \nonumber \\
&:= -\ell_{PQL}(\bm{\alpha}, \bm{\gamma}, \hat\phi, \hat{\bm{\tau}}) +\lambda\sum_j v_j|\gamma_j|,
\end{align}
where $\lambda$ is a nonnegative regularization parameter, and $v_j$ is a penalty factor for the $j^{th}$ predictor.
In Appendix B, we detail our proposed general purpose block coordinate gradient descent algorithm (CGD) to solve \eqref{eq:objfunc} and obtain regularized PQL estimates for $\bm{\alpha},\bm{\gamma}$ and $\tilde{\bm{b}}$. Briefly, our algorithm is equivalent to iteratively solve the two penalized generalized least squares (GLS) $$\underset{\tilde{\bm{b}}}{\textrm{argmin}}\left(\tilde{\bm{Y}} - \tilde{\bm{X}}{\bm{\beta}} - \tilde{\bm{b}}\right)^\intercal\bm{W}^{-1}\left(\tilde{\bm{Y}} - \tilde{\bm{X}}{\bm{\beta}} - \tilde{\bm{b}}\right) + \tilde{\bm{b}}^\intercal\left(\sum_{s=1}^S \hat{\tau}_s\bm{V}_s\right)^{-1}\tilde{\bm{b}},$$ and $$\underset{\bm{\beta}}{\textrm{argmin}}\left(\tilde{\bm{Y}} - \tilde{\bm{X}}\bm{\beta}\right)^\intercal\bm{\Sigma}^{-1}\left(\tilde{\bm{Y}} - \tilde{\bm{X}}\bm{\beta}\right) + \lambda\sum_j v_j|\beta_j|,$$ where $\bm{\Sigma}=\bm{W}^{-1} + \sum_{s=1}^S \hat{\tau}_s\bm{V}_s$ is the covariance matrix of the working response vector $\tilde{\bm{Y}}$, $\tilde{\bm{X}}=\left[\bm{X};\ \bm{G}\right]$ and $\bm{\beta}=(\bm{\alpha}^\intercal, \bm{\gamma}^\intercal)^\intercal $. We use the spectral decomposition of $\bm{\Sigma}$ to rotate $\tilde{\bm{Y}}$, $\tilde{\bm{X}}$ and $\tilde{\bm{b}}$ such that the
transformed data become uncorrelated. For binary data, because the covariance matrix $\bm{\Sigma}$ depends on the sample weights $\bm{W}$, we use an upper-bound on $\bm{\Sigma}^{-1}$ to ensure a single spectral decomposition is performed~\citep{bohningMonotonicityQuadraticapproximationAlgorithms1988}. By cycling through the coordinates and minimizing the objective function with respect to one parameter at a time, $\tilde{\bm{b}}$ can be estimated by fitting a generalized ridge-like model with a diagonal penalty matrix equal to the inverse of the eigenvalues of $\sum_{s=1}^S \hat{\tau}_s\bm{V}_s$. Then, conditional on $\tilde{\bm{b}}$, ${\bm{\beta}}$ is estimated by solving a weighed least squares (WLS) with a lasso regularization term. All calculations and algorithmic steps are detailed in Appendix B.
\subsection{Model selection}\label{subsec:modelselection}
Approaches to selecting the optimal tuning parameter in regularized models are of primary interest since in real data analysis, the underlying true model is unknown. A popular strategy is to select the value of the tuning parameter that minimizes out-of-sample prediction error, e.g., cross-validation (CV), which is asymptotically equivalent to the Akaike information criterion (AIC) \citep{Akaike1998,Yang2005}. While being conceptually attractive, CV becomes computationally expensive for very high-dimensional data. Moreover, in studies where the proportion of related subjects is important, either by known or cryptic relatedness, the CV prediction error is no longer an unbiased estimator of the generalization error~\citep{Rabinowicz2020}. Through simulation studies and real data analysis, ~\citet{Wang2020} found that LD and minor allele frequencies (MAF) differences between ancestries could explain between 70 and 80\% of the loss of relative accuracy of European-based prediction models in African ancestry for traits like body mass index and type 2 diabetes. Thus, there is no clear approach to how multiple admixed and/or similar populations should be split when using CV to minimize out-of-sample prediction error.
Alternatively, we can use the generalized information criterion (GIC) to choose the optimal value of the tuning parameter $\lambda$, defined as
\begin{align}\label{eq:GIC}
\textrm{GIC}_{\lambda} = -2 \ell_{PQL} + a_n \cdot \hat{df}_{\lambda},
\end{align}
where $\ell_{PQL}$ is defined in \eqref{eq:objfunc}, and $\hat{df}_{\lambda}=|\{1 \le k \le p : \hat{\beta}_k \ne 0 \}| + \textrm{dim}(\hat{\bm{\tau}})$ is the number of nonzero fixed-effects coefficients~\citep{Zou2007} plus the number of variance components. Special cases of the GIC include AIC ($a_n=2$) and the Bayesian information criterion (BIC)~\citep{Schwarz1978} ($a_n=\text{log}(n)$).
\subsection{Prediction}\label{subsec:prediction}
It is often of interest in genetic association studies to make predictions on a new set of individuals, e.g., the genetic risk of developing a disease for a binary response or the expected outcome in the case of a continuous response. In what follows, we compare how predictions are obtained in \texttt{pglmm} versus a GLM with PC adjustment.
\subsubsection{\texttt{pglmm}}\label{subsubsec:mixed}
For the sake of comparison with the GLM with PC adjustment, we suppose a sampling design where a single variance component is needed such that $\bm{b}\sim\mathcal{N}(\bm{0}, \tau_1\bm{V_1})$ where $\bm{V_1}$ is the GSM between $n$ subjects that are used to fit the GLMM \eqref{eq:model}. We iteratively fit on a training set the working linear mixed model $$\tilde{\bm{Y}} = \tilde{\bm{X}}\bm{\beta} + \bm{\bm{b}} + \bm{\epsilon},$$ where $\bm{\epsilon}=g'(\bm{\mu})(\bm{y}-\bm{\mu}) \sim \mathcal{N}(0, \bm{W}^{-1})$. Let $\tilde{\bm{Y}}_s$ be the latent working vector in a set of individuals with predictor set $\tilde{\bm{X}}_s$ that were not used in the model training, $n_s$ denote the number of observations in the testing set and $n$ the number of observations in the training set. Similar to~\citep{Bhatnagar2020}, we assume that the marginal joint distribution of $\tilde{\bm{Y}}_s$ and $\tilde{\bm{Y}}$ is multivariate Normal :
\begin{align*}
\begin{bmatrix} \tilde{\bm{Y}}_s \\ \tilde{\bm{Y}}\end{bmatrix} \sim \mathcal{N}\left(\begin{bmatrix} \tilde{\bm{X}}_s\bm{\beta} \\ \tilde{\bm{X}}\bm{\beta}\end{bmatrix},\begin{bmatrix} \bm{\Sigma}_{11} & \bm{\Sigma}_{12} \\ \bm{\Sigma}_{21} & \bm{\Sigma}_{22} \end{bmatrix}\right),
\end{align*}
where $\bm{\Sigma}_{12}=\tau_1\bm{V}_{12}$ and $\bm{V}_{12}$ is the $n_s \times n$ GSM between the testing and training individuals. It follows from standard normal theory that
\begin{align*}
\tilde{\bm{Y}}_s|\tilde{\bm{Y}}, \phi,\tau_1, \bm{\beta}, \tilde{\bm{X}}, \tilde{\bm{X}}_s \sim \mathcal{N}\left(\tilde{\bm{X}}_s\bm{\beta} + \bm{\Sigma}_{12}\bm{\Sigma}_{22}^{-1}(\tilde{\bm{Y}}-\tilde{\bm{X}}\bm{\beta}), \bm{\Sigma}_{11}-\bm{\Sigma}_{12}\bm{\Sigma}_{22}^{-1}\bm{\Sigma}_{21}\right).
\end{align*}
The estimated mean response $\hat{{\bm{\mu}}}_s$ for the testing set is given by
\begin{align}\label{eq:predmix}
g^{-1}\left(\mathbb{E}[\tilde{\bm{Y}}_s|\tilde{\bm{Y}}, \hat\phi,\hat\tau_1, \hat{\bm{\beta}}, \tilde{\bm{X}}, \tilde{\bm{X}}_s]\right) &= g^{-1}\left(\tilde{\bm{X}}_s\hat{\bm{\beta}} + \bm{\Sigma}_{12}\bm{\Sigma}_{22}^{-1}(\tilde{\bm{Y}}-\tilde{\bm{X}}\hat{\bm{\beta}})\right) \nonumber \\
&= g^{-1}\left(\tilde{\bm{X}}_s\hat{\bm{\beta}} + \hat\tau_1\bm{V}_{12}\left(\bm{W}^{-1} + \hat\tau_1\bm{V}_1\right)^{-1}(\tilde{\bm{Y}}-\tilde{\bm{X}}\hat{\bm{\beta}})\right) \nonumber \\
&= g^{-1}\left(\tilde{\bm{X}}_s\hat{\bm{\beta}} + \bm{V}_{12}\bm{U}\left(\frac{1}{\hat\tau_1}\bm{D} + \tilde{\bm{U}}^\intercal\bm{W}\tilde{\bm{U}}\right)^{-1}\tilde{\bm{U}}^\intercal\bm{W}(\tilde{\bm{Y}}-\tilde{\bm{X}}\hat{\bm{\beta}})\right),
\end{align}
where $g(\cdot)$ is a link function and $\tilde{\bm{U}} = \bm{UD}$ is the $n \times n$ matrix of PCs obtained from the spectral decomposition of the GSM for training subjects.
\subsubsection{GLM with PC adjustment}\label{subsubsec:glmPC}
Another approach to control for population structure and/or subjects relatedness is to use the first $r$ columns of $\tilde{\bm{U}}$ as unpenalized fixed effects covariates. This leads to the following GLM
\begin{align*}
g(\bm{\mu}) = \tilde{\bm{X}}\bm{\beta} + \tilde{\bm{U}}_r\bm{\delta},
\end{align*}
where $\tilde{\bm{X}}=[\bm{X}; \bm{G}]$ is the $n \times (m+p)$ design matrix for non-genetic and genetic predictors, $\bm{\beta} \in \mathbb{R}^p$ is the corresponding sparse vector of fixed effects, $\tilde{\bm{U}}_r$ is the $n \times r$ design matrix for the first $r$ PCs and $\delta \in \mathbb{R}^r$ is the corresponding vector of fixed effects. Letting $\tilde{\bm{Y}} = \tilde{\bm{X}}\bm{\beta} + \tilde{\bm{U}}_r\bm{\delta} + g'(\bm{\mu})(\bm{y-\mu})$ be the working response vector, one can show that
\begin{align}\label{eq:glspc}
\hat{\bm{\delta}} = \left(\tilde{\bm{U}}_r^\intercal\bm{W}\tilde{\bm{U}}_r\right)^{-1}\tilde{\bm{U}}_r^\intercal\bm{W}\left(\tilde{\bm{Y}} - \tilde{\bm{X}}\hat{\bm{\beta}}\right),
\end{align}
where $\bm{W}$ is the diagonal matrix of GLM weights. Let $\bm{V}_{12}$ be the $n_s \times n$ GSM between a testing set of $n_s$ individuals and $n$ training individuals such that the projected PCs on the testing subjects are equal to $\bm{V}_{12}\bm{U}_{r}$. Then, the estimated mean response $\hat{{\bm{\mu}}}_s$ for the testing set is given by
\begin{align}\label{eq:predpca}
\hat{\bm{\mu}}_s = g^{-1}\left(\tilde{\bm{X}_s}\hat{\bm{\beta}} + \bm{V}_{12}\bm{U}_{r}\hat{\bm{\delta}}\right) = g^{-1}\left(\tilde{\bm{X}_s}\hat{\bm{\beta}} + \bm{V}_{12}\bm{U}_{r}\left(\tilde{\bm{U}}_r^\intercal\bm{W}\tilde{\bm{U}}_r\right)^{-1}\tilde{\bm{U}}_r^\intercal\bm{W}\left(\tilde{\bm{Y}} - \tilde{\bm{X}}\hat{\bm{\beta}}\right)\right).
\end{align}
By comparing \eqref{eq:predmix} and \eqref{eq:predpca}, we see that both GLM with PC adjustment and \texttt{pglmm} use a projection of the training PCs on the testing set to predict new responses, but with different coefficients for the projected PCs. For the former, the estimated coefficients for the first $r$ projected PCs in \eqref{eq:glspc} are obtained by iteratively solving generalized least squares (GLS) on the partial working residuals $\tilde{\bm{Y}} - \tilde{\bm{X}}\hat{\bm{\beta}}$. For \texttt{pglmm}, the estimated coefficients for all projected PCs are also obtained by iteratively solving GLS on the partial working residuals $\tilde{\bm{Y}} - \tilde{\bm{X}}\hat{\bm{\beta}}$, with an extra ridge penalty for each coefficient that is equal to $\hat{\tau_1}^{-1}\Lambda_i$ with $\Lambda_i$ the $i^{th}$ eigenvalue of $\bm{V}$ that is associated with the $i^{th}$ PC.
From a Bayesian point of view, the fixed effect GLM assumes that each of the $r$ selected PCs have equal prior probability, while the remaining $n-r$ components are given a prior with unit mass at zero~\citep{Astle2009}. In contrast, the MM puts a Gaussian prior on regression coefficients with variances proportional to the corresponding eigenvalues. This implies that the PCs with the largest eigenvalues have a higher prior probability of explaining the phenotype. Hence, \texttt{pglmm} shrinks PCs coefficients in a smooth way, while the fixed effect GLM uses a tresholding approach; the first $r$ predictors with larger eigenvalues are kept intact, and the others are completely removed. This implies that the confounding effect from population structure and/or relatedness on the phenotype is fully captured by the first $r$ PCs. As we show in simulations results, departure from this assumption leads to less accurate coefficients estimation, lower prediction accuracy and higher variance in predictions.
\subsection{Simulation design}\label{subsec:simdesign}
We evaluated the performance of our proposed method against that of a lasso LMM, using the \texttt{R} package \texttt{ggmix}~\citep{ggmix}, and a logistic lasso, using the \texttt{Julia} package \texttt{GLMNet} which wraps the \texttt{Fortran} code from the original \texttt{R} package \texttt{glmnet}~\citep{glmnet}. We compared \texttt{glmnet} when we included or not the first 10 PCs in the model (\texttt{glmnetPC}). We performed a total of 50 replications for each simulation scenario, drawing anew genotypes and simulated traits. Values for all simulation parameters are presented in Table \ref{tab:1}.
\subsubsection{Simulated genotype from the admixture model}
In the first scenario, we studied the performance of all methods for different population structures by simulating random genotypes from the BN-PSD admixture model for 10 or 20 subpopulations with 1D geography or independent subpopulations using the \texttt{bnpsd} package in \texttt{R}~\citep{Ochoa2016_1, Ochoa2016_2}. Sample size was set to $n=2500$. We simulated $p$ candidate SNPs and randomly selected $p\times c$ to be causal. The kinship matrix $\bm{V}$ and PCs were calculated using a set of $50,000$ additional simulated SNPs. We simulated covariates for age and sex using Normal and Binomial distributions, respectively.
\subsubsection{Real genotypes from the UK Biobank data}
In the second scenario, we compared the performance of all methods when a high proportion of related individuals are present, using real genotype data from the UK Biobank. We retained a total of 6731 subjects of White British ancestry having estimated 1st, 2nd or 3rd degree relationships with at least one other individual. We sampled $p$ candidate SNPS among all chromosomes and randomly selected $p\times c$ to be causal. We used PCs as provided with the data set. These were computed using a set of unrelated samples and high quality markers pruned to minimise LD~\citep{Bycroft2018}. Then, all subjects were projected onto the principal components using the corresponding loadings. Since the markers that were used to compute the PCs were potentially sampled as candidate causal markers in our simulations, we included all candidate SNPs in the set of markers used for calculating the kinship matrix $\bm{V}$. We simulated age using a Normal distribution and used the sex covariate as provided with the data.
\subsubsection{Simulation model}
The number of candidate SNPs and fraction of causal SNPs were set to $p=5000$ and $c=0.01$ respectively. Let $S$ be the set of candidate causal SNPs, with $|S|=p\times c$, then the causal SNPs fixed effects $\beta_j$ were generated from a Gaussian distribution $\mathcal{N}(0,h^2_g\sigma^2/|S|)$, where $h^2_g$ is the fraction of variance on the logit scale that is due to total additive genetic fixed effects. That is, we assumed the candidate causal markers explained a fraction of the total polygenic heritability, and the rest was explained by a random polygenic effect $b\sim\mathcal{N}(0,h^2_b\sigma^2\bm{V})$. We simulated a SNR equal to 1 for the fixed genetic effects ($h^2_g=50\%$) under strong random polygenic effects ($h^2_b=40\%$). Finally, we simulated a binary phenotype using a logistic link function
\begin{align}\label{eq:simlogit1}
\text{logit}(\pi) = \text{logit}(\pi_0) -\text{log}(1.3)\times Sex+\text{log}(1.05)Age/10+ \sum_{j\in S}\beta_j\cdot \widetilde{G}_{j} + b,
\end{align}
where the parameter $\pi_0$ was chosen to specify the prevalence under the null, and $\widetilde{G}_{j}$ is the $j^{th}$ column of the standardized genotype matrix $\tilde{g}_{ij}=(g_{ij}-2p_i)/\sqrt{2p_i(1-p_i)}$ and $p_i$ is the MAF.
\subsubsection{Metric of comparison}\label{subsubsec:metrics}
For each replication, subjects were partitioned into training and test sets using an 80/20 ratio, ensuring all related individuals were assigned into the same set in the second scenario. Variable selection and coefficient estimation were performed on training subjects for all methods. We compared each method at a fixed number of predictors, ranging from 5 to 50 which corresponds to the number of true causal SNPs. Comparisons were based on three criteria: the ability to retrieve the causal predictors, measured by the true positive rate ${\textrm{TPR} = |\{1\le k \le p : \hat{\beta}_k \ne 0 \cap {\beta}_k \ne 0\}|/|\{1\le k \le p : \hat{\beta}_k \ne 0\}|}$; the ability to accurately estimate coefficients, measured by the root mean squared error ${\textrm{RMSE} = \sqrt{\frac{1}{p}\sum_{k=1}^p (\hat{\beta}_k - \beta_k)^2}}$; and the ability to predict outcomes in the test sets, measured by the area under the roc curve (AUC).
In addition, we evaluated the performance of our proposed method when using either AIC, BIC or cross-validation as model selection criteria, rather than fixing the number of predictors in the model. For this, subjects from the real UKBB data were randomly split into training (40\%), validation (30\%) and test (30\%) sets, again ensuring all related individuals were assigned into the same set. For cross-validation, the full lasso solution path was fitted on the training set, and the regularization parameter was obtained on the model which maximized AUC on the validation set. We compared methods performance on the basis of TPR, AUC on the test sets and RMSE. Additionally, we compared each model selection approach on the total number of predictors selected and on the model precision, which is defined as the proportion of selected predictors that are true positives.
\subsection{Real data application} \label{subsec:realdata}
We used the real UK Biobank data set presented in Section \ref{subsec:simdesign} to illustrate the potential advantages of \texttt{pglmm} over logistic lasso with PC adjustment for constructing a PRS on two highly heritable binary traits, asthma and high cholesterol, in a set of related individuals. Asthma is a common respiratory disease characterized by inflammation and partial obstruction of the bronchi in the lungs that results in difficulty breathing~\citep{Anderson2008}. High cholesterol can form plaques and fatty deposits on the walls of the arteries, and thus prevent the blood to circulate to the heart and brain. It is one of the main controllable risk factors for coronary artery disease, heart attack and stroke~\citep{kathiresan_polymorphisms_2008}.
After filtering for SNPs with missing rate greater than $0.01$, MAF above $0.05$ and a p-value for the Hardy–Weinberg exact test above $10^{-6}$, a total of 320K genotyped SNPs were remaining. To better understand the contribution of the PRS for predicting asthma and high cholesterol, we fitted for each trait a null model with only age, sex, genotyping array and the first 10 PCs as main effects. For our proposed \texttt{pglmm} method, we did not include any PC since kinship is accounted for by a random effect. Finally, we compared with a logistic lasso in which the top 10 PCs were included as unpenalized covariates in addition to age, sex and genotyping array. To find the optimal regularization parameter for both methods, we split the subjects in training (60\%), validation (20\%) and test (20\%) sets for a total of 40 times. For each replication, the full lasso solution path was fitted on the training set, and the regularization parameter was obtained on the model which maximized AUC on the validation set. We compared mean prediction accuracy on the test sets as well as the median number of predictors included in all models. Finally, we also compared our method's performance when the best model was chosen using BIC on the training fit.
\section{Results}\label{sec:results}
\subsection{Simulation results from the admixture model} \label{subsec:simresultsadmix}
Results for selection of important predictors in the first simulation scenario, as measured by the mean TPR in 50 replications, are presented in Figure \ref{fig:tpr_sim}. For both 1D linear admixture and independent subpopulations, \texttt{glmnet} without PC adjustment failed to retrieve causal markers compared to all other methods. This is expected under population stratification; SNPs that differ in frequency between subpopulations are identified as important predictors because prevalence is not constant across each group. When the first 10 PCs were added as unpenalized covariates, \texttt{glmnetPC}'s ability to select causal predictors was lesser to that of \texttt{pglmm} and \texttt{ggmix} for the 20 independent subpopulations. In this case, the rank of the GSM used to infer the PCs is close to 20, and including only 10 PCs in the model does not correctly capture the confounding structure. Because there is less overlap between subpopulations in the admixture data compared to the independent populations~\citep{Reisetter2021}, a greater proportion of the simulated polygenic random effect is explained by the GSM and including only 10 PCs is enough to correct for confounding even when $K=20$ (bottom-left panel of Figure \ref{fig:tpr_sim}). On the other hand, including a random effect with variance-covariance structure proportional to the GSM correctly adjusts for population structure in all scenarios while alleviating the burden of choosing the right number of fixed predictors to include in the model. Even though \texttt{ggmix} assumes a standard LMM for the binary trait, it was able to identify causal markers at the same rate as \texttt{pglmm}.
Results for estimation of SNP effects as measured by the mean RMSE in 50 replications are presented in Figure \ref{fig:rmse_sim}. Results are consistent with TPR results in that \texttt{glmnet} without PC adjustment performed poorly in all scenarios, while \texttt{pglmm} outperformed all other methods for the 20 independent subpopulations and performed comparably with \texttt{glmnetPC} for all other settings. As expected, \texttt{ggmix} had higher RMSE compared to \texttt{pglmm} and \texttt{glmnetPC}. Thus, even though \texttt{ggmix} was able to identify causal markers at the same rate as other methods that accounted for the binary nature of the response, resulting estimates for the SNP effects were not accurate.
For both 1D linear admixture and independent subpopulations, \texttt{ggmix} and \texttt{glmnet} had poor predictive performance for $K=10$ and $K=20$, as reported in Figure \ref{fig:auc_sim}. Also, the predictive performance of \texttt{glmnetPC} was greatly reduced when $K=20$ for both admixture and independent populations. In the case of the admixture data, the RMSE for estimation of SNP effects was comparable for \texttt{glmnetPC} and \texttt{pglmm}. This means that the observed discrepancy in predictive accuracy is due to the difference in how each method handle the confounding effects. Using only 10 PCs as fixed effects when $K=20$ likely results in overfitted coefficients, which may potentially decrease prediction accuracy and increase variance of predictions in independent subjects. By using a ridge-like estimator for the random effects, $\texttt{pglmm}$ is less likely to overfit the confounding effects compared to \texttt{glmnetPC}. This is supported by the results of Table \ref{tab:3}, where the relative decrease in AUC standard deviation for the predictions obtained by $\texttt{pglmm}$ could be as high as $16\%$ for $K=20$ subpopulations.
\subsection{Simulation results from real genotype data} \label{subsec:simresultsreal}
Results for selection of important predictors, estimation of SNPs effects and prediction accuracy in the second simulation scenario are presented in Figure \ref{fig:ukbb}. We compared the ability of \texttt{glmnetPC} and \texttt{pglmm} to adjust for potential confounding stemming from subjects relatedness. Both methods' ability to retrieve important predictors were comparable as measured by mean TPR, with \texttt{pglmm} having a slight advantage. In terms of predictor effect estimation, \texttt{pglmm} had lower reported mean RMSE. Furthermore, \texttt{pglmm} outperformed \texttt{glmnetPC} when making predictions in independent test sets. Once again, this is explained by the fact that \texttt{pglmm} uses a random effect parameterized by the $n-$dimensional kinship matrix, which we have shown in Section \ref{subsec:prediction} to be equivalent to include all PCs as predictors in the model and shrink their coefficients proportionally to their relative importance in a smooth way. On the other hand, for \texttt{glmnetPC}, only the first $10$ PCs with larger eigenvalues are kept intact, and the others are completely removed. As the confounding effect from relatedness on the phenotype can not be fully captured by using only the first $10$ PCs, prediction accuracy is greatly reduced.
\subsection{Model selection} \label{subsec:simresultsmodel}
Boxplots of the model selection \sout{simulations} results are presented in Figure \ref{fig:model_ukbb}. As expected, BIC tended to choose sparser models with very high precision values, compared to AIC and CV which tended to select larger models with negligibly higher prediction performance. Thus, using BIC as a model selection criteria resulted in trading a bit of prediction accuracy for a large boost in the model precision. In many situations where it is of interest to identify a smaller set containing the most important predictors, BIC should be preferred over AIC and CV. Moreover, BIC alleviates the computational challenge of performing out-of-sample predictions, which includes identifying pedigrees to ensure independence between training, validation and testing sets.
\subsection{PRS for the UK Biobank} \label{subsec:realdataresults}
Results for asthma and high cholesterol PRSs are summarized in Table \ref{tab:2}. For asthma, \texttt{pglmm} with either BIC or CV as model selection criteria performed better than \texttt{glmnetPC} and the null model with covariates only when comparing mean AUC on the test sets. The median number of predictors selected by \texttt{glmnetPC} was four times higher than for \texttt{glmnet} when using CV for both methods. Moreover, the variability in predictors selected was more important for \texttt{glmnetPC}, as reported by an IQR value equal to 486, compared to 145 for our method. \texttt{pglmm} with BIC selected 1 predictor (IQR: 1) compared to 16 (IQR: 145) for \texttt{pglmm} with CV. This is consistent with our simulation results showing that BIC results in sparser models with comparable predictive power. For high cholesterol, very few genetic predictors were selected by all models, which suggests that it may not be a highly polygenic trait. In fact, using only the non-genetic covariates and first 10 PCs resulted in the best model \sout{for high cholesterol} based on mean test sets AUC.
\section{Discussion}\label{sec:discussion}
We have introduced a new method called \texttt{pglmm} based on regularized PQL estimation, for selecting important predictors and estimating their effects in high-dimensional GWAS data, accounting for population structure, close relatedness and binary nature of the trait. Through a variety of simulations, using both simulated and real genotype data, we showed that \texttt{pglmm} was markedly better than a logistic lasso with PC adjustment when the number of subpopulations was greater than the number of PCs included, or when a high proportion of related subjects were present. We also showed that a lasso LMM was unable to estimate predictor effects with accuracy for binary responses, which greatly decreased its predictive performance. Performance assessment was based on TPR of selected predictors, RMSE of estimated effects and AUC of predictions. These results strongly advocate for using methods that explicitly account for the binary nature of the trait while effectively controlling for population structure and relatedness in genetic studies.
When the dimensionality of the confounding structure was low, we showed that \texttt{pglmm} was equivalent to logistic lasso with PC adjustement. Hence, adjusting a GLM with PCA is at best equivalent to \texttt{pglmm}, but with the additional burden of selecting an appropriate number of PCs to retain in the model. Estimating the dimensionality of real datasets, and thus the number of PCs to include as fixed effects in a regression model can reveal to be challenging because estimated eigenvalues have biased distributions~\citep{Yao2022}. Another strategy involves selecting the appropriate number of PCs based on the Tracy-Widom test~\citep{Tracy1994}. However, it is known that this test tends to select a very large number of PCs~\citep{Lin2011}, causing convergence problems when fitting too many predictors. On the other hand, modeling the population structure by using a random polygenic effect correctly accounts for low and high-dimensional confounding structures, while only fitting one extra variance component parameter.
We used real genotype data from the UK Biobank to simulate binary responses and showed that BIC effectively selected sparser models with very high precision and prediction accuracy, compared to AIC and CV. Using the same data set, we illustrated the potential advantages of \texttt{pglmm} over a logistic lasso with PC adjustment for constructing a PRS on two highly heritable binary traits in a set of related individuals. Results showed that \texttt{pglmm} had higher predictive performance for asthma, while also selecting consistently fewer predictors as reported by median and IQR values.
In this study, we focused solely on the lasso as a regularization penalty for the genetic markers effects. However, it is known that estimated effects by lasso will have large biases because the resulting shrinkage is constant irrespective of the magnitude of the effects. Alternative regularization like the Smoothly Clipped Absolute Deviation (SCAD)~\citep{Fan2001} penalty function should be explored. Although, we note that the number of nonzero coefficients in the SCAD estimates is no longer an unbiased estimate of its degrees of freedom. Other alternatives include implementation of the relaxed lasso, which has shown to produce sparser models with equal or lower prediction loss than the regular lasso estimator for high-dimensional data~\citep{Meinshausen2007}. It would also be of interest to explore if tuning of the generalized ridge penalty term on the random effects that arises from the PQL loss could result in better predictive performance.
A limitation of \texttt{pglmm} compared to a logistic lasso with PC adjustment is the computational cost of performing multiple matrix calculations that comes from the estimation of variance components under the null. Indeed, at each iteration, we perform a matrix inversion based on Cholesky decomposition with complexity of $O(n^3)$ and matrix multiplications with complexity of $O(mn^2 + S^2 n^2 + p^2n)$, where $n$ is the sample size, $m$ is the number of non-genetic covariates, and $S$ is the number of variance components. Then, we need to perform a spectral decomposition of the covariance matrix with a computation time $O(n^3)$. These computations become prohibitive for large cohorts such as the full UK Biobank with a total of 500$K$ samples. A solution to explore to increase computation speed and decrease memory usage would be the use of conjugate gradient methods with a diagonal preconditioner matrix, as proposed by~\citet{zhouEfficientlyControllingCasecontrol2018}.
Finally, we can take advantage of the fact that it is possible to allow for multiple random effects to account for complex sampling designs and extend \texttt{pglmm} to a wide variety of models. For example, building a PRS for a bivariate binary trait, explicitly accounting for the shared causal pathways of many diseases or complex traits. Moreover, \texttt{pglmm} could be used in models where there is interest in selecting over fixed genetic and gene-environment interaction (GEI) effects. Due to the hierarchical structure between the main genetic and GEI effects, we will have to consider using a lasso for hierarchical structures~\citep{Zemlianskaia2022}.
\section*{Software}\label{sec:software}
Our Julia package called \texttt{PenalizedGLMM} is available on Github \url{https://github.com/julstpierre/PenalizedGLMM}.
\section*{Funding}
This work was supported by the Fonds de recherche Québec-Santé [267074 to K.O.]; and the Natural Sciences and Engineering Research Council of Canada [RGPIN-2019-06727 to K.O., RGPIN-2020-05133 to S.B.].
\section*{Acknowledgments}
This research has been conducted using the UK Biobank Resource under Application Number 20802. This study was enabled in part by support provided by Calcul Québec (\url{https://www.calculquebec.ca/}) and Compute Canada (\url{https://www.computecanada. ca/}). We thank the UK Biobank and all participants for providing information.
\section*{Data availability statement}\label{sec:dataavail}
Simulated data are available on Github \url{https://github.com/julstpierre/PenalizedGLMM/data}. UK Biobank data are available via application directly to UK Biobank (\url{https://www.ukbiobank.ac.uk/enable-your-research}). The current study was conducted under UK Biobank application number 20802.
\noindent{\it Conflict of Interest}: None declared.
\bibliographystyle{apalike}
\subsection*{Appendix A. Estimation of Variance Component Parameters} \label{subsec:varcomp}
We fit the generalized linear mixed model (GLMM) \eqref{eq:model} under the assumption of no genetic association to estimate the variance components and dispersion parameters. We detail in this Appendix the AI-REML algorithm as derived by~\citet{Chen2016}.
If $\phi$ and $\bm\tau$ are known, we jointly choose $\hat{\bm{\alpha}}(\phi, \bm{\tau})$, $\hat{\bm{\gamma}}(\phi, \bm{\tau})$ and $\hat{\bm{b}}(\phi, \bm{\tau})$ to minimize \eqref{eq:A3}, then $\hat{\bm{b}}(\phi, \bm{\tau}) = \tilde{\bm{b}}(\hat{\bm{\alpha}}(\phi, \bm{\tau}), \hat{\bm{\gamma}}(\phi, \bm{\tau}))$ because $\tilde{\bm{b}}$ maximizes $f(\bm{b})$ for given $(\bm{\alpha}, \bm{\gamma})$. Assuming that the weights in $\bm{W}$ vary slowly with the
conditional mean, the derivatives of \eqref{eq:A3} at $\bm{\gamma}=0$ with respect to $(\bm{\alpha}, \bm{b})$ are given by
\begin{align}\label{eq:A4_}
\frac{\partial ql(\bm{\alpha}, \bm{\gamma}=0, \phi, \bm{\tau} )}{\partial\bm{\alpha}} &= -\sum_{i=1}^n \frac{a_i(y_i - \mu_i)}{\phi \nu(\mu_i)}\frac{1}{g'(\mu_i)}\bm{X_}i^\intercal = -\bm{X}^\intercal\bm{W}\Delta(\bm{y}-\bm{\mu}), \nonumber \\
\frac{\partial ql(\bm{\alpha}, \bm{\gamma}=0, \phi, \bm{\tau})}{\partial\bm{b}} &=
-\sum_{i=1}^n \frac{a_i(y_i - \mu_i)}{\phi \nu(\mu_i)}\frac{1}{g'(\mu_i)}\bm{Z_}i^\intercal + \left( \sum_{s=1}^S\tau_s\bm{V}_s \right)^{-1}\bm{b} = \left( \sum_{s=1}^S\tau_s\bm{V}_s \right)^{-1}\bm{b} - \bm{W}\Delta(\bm{y} - \bm{\mu}), \nonumber
\end{align}
where $\Delta = \textrm{diag}(g'(\mu_i))$ and $\bm{Z}_i$ is a $n\times 1$ vector of indicators such that $b_i=\bm{Z}_i\bm{b}$. Defining the working vector $\bm{\tilde{Y}}$ with elements $\tilde{Y_i} = \eta_i + g'(\mu_i)(y_i - \mu_i)$, the solution of
\begin{gather*}
\begin{cases}
\qquad \bm{X}^\intercal\bm{W}\Delta(\bm{y}-\bm{\mu}) =0 \\
\bm{W}\Delta(\bm{y} - \bm{\mu}) = \left( \sum_{s=1}^S\tau_s\bm{V}_s \right)^{-1}\bm{b}
\end{cases}
\end{gather*}
can be written as the solution to the system
\begin{align*}
\begin{bmatrix}
\bm{X^}\intercal\bm{W}\bm{X} &
\bm{X^}\intercal\bm{W} \\
\bm{W}\bm{X} &
\left( \sum_{s=1}^S\tau_s\bm{V}_s \right)^{-1} + \bm{W}
\end{bmatrix}
\begin{bmatrix}
\bm{\alpha} \\
\bm{b}
\end{bmatrix}
=
\begin{bmatrix}
\bm{X^}\intercal\bm{W}\bm{\tilde{Y}}\\
\bm{W}\bm{\tilde{Y}}
\end{bmatrix}.
\end{align*}
Let $\bm{\Sigma} = \bm{W}^{-1} + \sum_{s=1}^S\tau_s\bm{V}_s$, $\bm{P}=\bm{\Sigma}^{-1} - \bm{\Sigma}^{-1}\bm{X}\left(\bm{X}^\intercal\bm{\Sigma}^{-1}\bm{X}\right)^{-1}\bm{X}^\intercal\bm{\Sigma}^{-1}$,
then
\begin{align*}
&\begin{cases}
\hat{\bm{\alpha}} = \left(\bm{X^}\intercal\bm{\Sigma}^{-1}\bm{X}\right)^{-1}\bm{X}^{\intercal}\bm{\Sigma}^{-1} \bm{\tilde{Y}} \\
\hat{\bm{b}}=\left( \sum_{s=1}^S\tau_s\bm{V}_s \right)\bm{\Sigma}^{-1}\left(\bm{\tilde{Y}}-\bm{X}\hat{\bm{\alpha}}\right)
\end{cases}.
\end{align*}
Of note, we have that
\begin{align*}
\bm{\tilde{Y}}-\hat{\bm{\eta}}
&=\bm{\tilde{Y}}-\bm{X}\hat{\bm{\alpha}} - \hat{\bm{b}} \nonumber \\
&= \left\{\bm{I} - \left(\sum_{s=1}^S\tau_s\bm{V}_s\right) \bm{\Sigma}^{-1} \right\}\left(\bm{\tilde{Y}}-\bm{X}\hat{\bm{\alpha}} \right) \nonumber\\
&= \bm{W}^{-1}\bm{\Sigma}^{-1}\left(\bm{\tilde{Y}}-\bm{X}\hat{\bm{\alpha}} \right) \nonumber\\
&= \bm{W}^{-1}\bm{P}\bm{\tilde{Y}}.
\end{align*}
The log integrated quasi-likelihood function in \eqref{eq:A3} evaluated at $(\hat{\bm{\alpha}}, \bm{\gamma}=0, \phi, \bm{\tau})$ becomes
\begin{align*}
ql(\hat{\bm{\alpha}}(\phi, \bm{\tau}), \bm{\gamma}=0, \phi, \bm{\tau}) = &-\frac{1}{2}\text{log}\left| \sum_{s=1}^S\tau_s\bm{V}_s\bm{W} + \bm{I}\right| -\frac{1}{2}\sum_{i=1}^n\frac{a_i(y_i-\hat\mu_i)^2}{\phi\nu(\hat\mu_i)} - \frac{1}{2}\hat{\bm{b}}^\intercal\left( \sum_{s=1}^S\tau_s\bm{V}_s \right) \bm{\Sigma}^{-1}\hat{\bm{b}} \\
=&-\frac{1}{2}\text{log}\left|\bm{\Sigma}\bm{W}\right| -\frac{1}{2}(\bm{\tilde{Y}} - \hat{\bm{\eta}})^\intercal\bm{W}(\bm{\tilde{Y}} - \hat{\bm{\eta}}) \\
& - \frac{1}{2}\left(\bm{\tilde{Y}}-\bm{X}\hat{\bm{\alpha}}\right)^\intercal\bm{\Sigma}^{-1}\left( \sum_{s=1}^S\tau_s\bm{V}_s \right) \bm{\Sigma}^{-1}\left(\bm{\tilde{Y}}-\bm{X}\hat{\bm{\alpha}}\right) \\
=&-\frac{1}{2}\text{log}\left|\bm{W}\right| -\frac{1}{2}\text{log}\left|\bm{\Sigma}\right| -\frac{1}{2}\bm{\tilde{Y}}^{\intercal}\bm{P}\bm{W}^{-1}\bm{P}\bm{\tilde{Y}} \\
& - \frac{1}{2}\bm{\tilde{Y}}^{\intercal}\bm{P}\left( \sum_{s=1}^S\tau_s\bm{V}_s \right)\bm{P}\bm{\tilde{Y}} \\
&=c -\frac{1}{2}\text{log}\left|\bm{\Sigma}\right| -\frac{1}{2}\bm{\tilde{Y}}^{\intercal}\bm{P}\bm{\Sigma}\bm{P}\bm{\tilde{Y}} \\
&=c -\frac{1}{2}\text{log}\left|\bm{\Sigma}\right| -\frac{1}{2}\bm{\tilde{Y}}^{\intercal}\bm{P}\bm{\tilde{Y}}.
\end{align*}
Similarly, the restricted maximum likelihood (REML) version is
\begin{align*}
ql_R(\hat{\bm{\alpha}}(\phi, \bm{\tau}), \bm{\gamma}=0, \phi, \bm{\tau})=
c_R &-\frac{1}{2}\text{log}\left|\bm{\Sigma}\right| -\frac{1}{2}\text{log}\left| \bm{X}^\intercal\bm{\Sigma}^{-1}\bm{X}\right| -\frac{1}{2}\bm{\tilde{Y}}^{\intercal}\bm{P}\bm{\tilde{Y}}.
\end{align*}
We need to maximize ${ql_R(\hat{\bm{\alpha}}(\phi, \bm{\tau}), \bm{\gamma}=0, \phi, \bm{\tau})}$ with respect to $\phi, \bm{\tau}$. Let $\bm{V}_0 = \textrm{diag}\{a_i^{-1}\nu(\mu_i)[g'(\mu_i)^2]\} = \phi^{-1}\bm{W}^{-1}$, then $\bm{\Sigma} = \phi\bm{V}_0 + \sum_{s=1}^S \tau_s\bm{V}_s$, and the first derivatives of ${ql_R(\hat{\bm{\alpha}}(\phi, \bm{\tau}), \bm{\gamma}=0, \phi, \bm{\tau})}$ with respect to $\phi$ and $\tau_s$ are
\begin{align}
\frac{\partial ql_R(\hat{\bm{\alpha}}(\phi, \bm{\tau}), \bm{\gamma}=0, \phi, \bm{\tau})}{\partial \phi} &= \frac{1}{2}\left\{\bm{\tilde{Y}}^\intercal\bm{P}\bm{V}_0\bm{P}\bm{\tilde{Y}} -tr(\bm{P}\bm{V}_0) \right\} \label{eq:partialderiv1}\\
\frac{\partial ql_R(\hat{\bm{\alpha}}(\phi, \bm{\tau}), \bm{\gamma}=0, \phi, \bm{\tau})}{\partial {\tau_s}} &= \frac{1}{2}\left\{\bm{\tilde{Y}}^\intercal\bm{P}\bm{V}_s\bm{P}\bm{\tilde{Y}} -tr(\bm{P}\bm{V}_s) \right\}, \label{eq:partialderiv2}
\end{align}
since one can show that
\begin{align*}
\frac{\partial\bm{P}}{\partial \phi}=-\bm{P}\bm{V}_0\bm{P}, \qquad \frac{\partial\bm{P}}{\partial \tau_s}=-\bm{P}\bm{V}_s\bm{P}.
\end{align*}
$\hat{\phi}$ and $\hat{\bm{\tau}}$ are estimated by finding the solutions of \eqref{eq:partialderiv1} and \eqref{eq:partialderiv2} equal to zero. Let $\bm{\theta}=(\phi, \bm{\tau})$, and recall that in the REML iterative process, $\hat{\bm{\theta}}$ at the (i+1)th iteraton is updated by $\bm{\hat\theta}^{(i+1)} = \bm{\hat\theta}^{(i)} + J(\bm{\hat\theta}^{(i)})^{-1} S(\bm{\hat\theta}^{(i)})$, where $S(\bm{\theta})=\frac{\partial ql_{R}(\bm{\theta})}{\partial\bm{\theta}}$ and $J(\bm{\theta})=-\frac{\partial^2 ql_{R}(\bm\theta{})}{\partial \bm{\theta}^2}$. The elements of the observed information matrix $J(\bm{\theta})$ are
\begin{align*}
& -\frac{\partial^2 ql_R(\hat{\bm{\alpha}}(\phi, \bm{\tau}), \bm{\gamma}=0, \phi, \bm{\tau})}{\partial \phi^2} = \bm{\tilde{Y}}^\intercal\bm{P}\bm{V}_0\bm{P}\bm{V}_0\bm{P}\bm{\tilde{Y}} - \frac{1}{2}tr(\bm{P}\bm{V}_0\bm{P}\bm{V}_0) \\
& -\frac{\partial^2 ql_R(\hat{\bm{\alpha}}(\phi, \bm{\tau}), \bm{\gamma}=0, \phi, \bm{\tau})}{\partial \phi \partial \tau_s} = \bm{\tilde{Y}}^\intercal\bm{P}\bm{V}_0\bm{P}\bm{V}_s\bm{P}\bm{\tilde{Y}} - \frac{1}{2}tr(\bm{P}\bm{V}_0\bm{P}\bm{V}_s) \\
& -\frac{\partial^2 ql_R(\hat{\bm{\alpha}}(\phi, \bm{\tau}), \bm{\gamma}=0, \phi, \bm{\tau})}{\partial \tau_l \partial \tau_s} = \bm{\tilde{Y}}^\intercal\bm{P}\bm{V}_l\bm{P}\bm{V}_s\bm{P}\bm{\tilde{Y}} - \frac{1}{2}tr(\bm{P}\bm{V}_l\bm{P}\bm{V}_s).
\end{align*}
The elements of the expected information matrix are
\begin{align*}
& E \left(-\frac{\partial^2 ql_R(\hat{\bm{\alpha}}(\phi, \bm{\tau}), \bm{\gamma}=0, \phi, \bm{\tau})}{\partial \phi^2}\right) = \frac{1}{2}tr(\bm{P}\bm{V}_0\bm{P}\bm{V}_0)\\
& E \left(-\frac{\partial^2 ql_R(\hat{\bm{\alpha}}(\phi, \bm{\tau}), \bm{\gamma}=0, \phi, \bm{\tau})}{\partial \phi \partial \tau_s} \right)= \frac{1}{2}tr(\bm{P}\bm{V}_0\bm{P}\bm{V}_s)\\
& E \left(-\frac{\partial^2 ql_R(\hat{\bm{\alpha}}(\phi, \bm{\tau}), \bm{\gamma}=0, \phi, \bm{\tau})}{\partial \tau_l \partial \tau_s} \right)= \frac{1}{2}tr(\bm{P}\bm{V}_l\bm{P}\bm{V}_s).
\end{align*}
The average information matrix $\bm{AI}$ is defined as the average of the observed information $J(\bm{\theta})$ and the expected information
\begin{align*}
\bm{AI}_{\phi\phi} &= \frac{1}{2}\bm{\tilde{Y}}^\intercal\bm{P}\bm{V}_0\bm{P}\bm{V}_0\bm{P}\bm{\tilde{Y}}, \\
\bm{AI}_{\phi\tau_s} &= \frac{1}{2}\bm{\tilde{Y}}^\intercal\bm{P}\bm{V}_0\bm{P}\bm{V}_s\bm{P}\bm{\tilde{Y}}, \\
\bm{AI}_{\tau_s\tau_l} &= \frac{1}{2}\bm{\tilde{Y}}^\intercal\bm{P}\bm{V}_s\bm{P}\bm{V}_l\bm{P}\bm{\tilde{Y}}.
\end{align*}
Let $\bm{\theta}$ be the variance component and dispersion parameters to estimate, that is when $\phi \ne 1$, $\bm{\theta}=(\phi, \bm{\tau}),$ and $\bm{AI}$ is a $(S + 1) \times (S+1)$ matrix. For binary data, $\phi=1$, $\bm{\theta}=\bm{\tau},$ and $\bm{AI}$ is a $S \times S$ matrix containing only $\bm{AI}_{\tau_s\tau_l}$. We use the following algorithm to estimate $\bm{\theta}$, $\bm{\alpha}$ and $\bm{b}$:
\begin{algorithm}[H]
\begin{enumerate}
\item \textit{Initialization} \\ Fit a generalized linear model with $\bm{\tau}=0 \text{ and get } \hat{\bm{\alpha}}^{(0)} \text{ and working vector } \bm{\tilde{Y}}^{(0)}$; \\
Use $\bm{\theta}^{(0)} = Var(\bm{\tilde{Y}}^{(0)})/S \text{ (if }\phi = 1\text{) or }\bm{\theta}^{(0)}=Var(\bm{\tilde{Y}}^{(0)})/(S+1) \text{ (if }\phi \ne 1)$ as the initial value of $\bm{\theta}$; \\
For each $s=0,1,...,S, \text{ update } \bm{\theta} \text{ using } \theta_s^{(1)}= \theta_s^{(0)}+2n^{-1}\{\theta_s^{(0)}\}^2(\partial ql_{R}(\bm{\theta}^{(0)})/\partial\theta_s)$; \\
\item \textit{Iteration} \\
\For{$t=1,2,...,$ until convergence}{
\qquad Update $\bm{\theta}^{(t+1)}=\bm{\theta}^{(t)} + \{\bm{AI}^{(t)}\}^{-1}(\partial ql_{R}(\bm{\theta}^{(t)})/\partial\bm{\theta})$; \\
\qquad Calculate $\hat{\bm{\alpha}}^{(t+1)} \text{ and }\hat{\bm{b}}^{(t+1)} \text{ using } \bm{\tilde{Y}}^{(t)} \text{ and } \bm{\theta}^{(t+1)}$; \\
\qquad Update $\bm{\tilde{Y}}^{(t+1)} \text{ using } \hat{\bm{\alpha}}^{(t+1)} \text{ and } \hat{\bm{b}}^{(t+1)}$;
}
\end{enumerate}
${}$ \\
Convergence is defined using $2\textrm{ max}\{|\hat{\bm{\alpha}}^{(t)} - \hat{\bm{\alpha}}^{(t-1)}| / (|\hat{\bm{\alpha}}^{(t)}| + |\hat{\bm{\alpha}}^{(t-1)}|), |\hat{\bm{\theta}}^{(t)} - \hat{\bm{\theta}}^{(t-1)}|/(|\hat{\bm{\theta}}^{(t)}| + |\hat{\bm{\theta}}^{(t-1)}|)\} \le \text{tolerance}$
\caption{AI-REML algorithm}
\end{algorithm} \leavevmode\newline
\subsection*{Appendix B. Block Coordinate Descent for PQL Regularized Parameters} \label{subsec:}
Assuming that the variance components and dispersion parameters are known, we fit the full GLMM \eqref{eq:model} with lasso regularization on ${\bm{\beta}}=( {\bm{\alpha}}^\intercal , {\bm{\gamma}}^\intercal)^\intercal$ to obtain PQL regularized estimates for $\bm{\beta}$ and $\bm{b}$. At each iteration, we cycle through the coordinates and minimize the objective function \eqref{eq:objfunc} with respect to one coordinate only. Suppose we have estimates $\tilde{\bm{\beta}}$ and we wish to partially optimize \eqref{eq:objfunc} with respect to $\tilde{\bm{b}}$. The derivative at $\bm{\beta}=\tilde{\bm{\beta}}$ is given by
\begin{align}\label{eq:A4}
\frac{\partial Q_{\lambda}(\bm{\beta}, \bm{b})}{\partial\bm{b}}| _{\bm{\beta}=\tilde{\bm{\beta}}} &=
-\sum_{i=1}^n \frac{a_i(y_i - \mu_i)}{\hat\phi \nu(\mu_i)}\frac{1}{g'(\mu_i)}\bm{Z_}i^\intercal + \left( \sum_{s=1}^S\hat\tau_s\bm{V}_s \right)^{-1}\bm{b} \nonumber \\ &
= - {\bm{W}}\Delta(\bm{y} - \bm{\mu}) + \left( \sum_{s=1}^S\hat\tau_s\bm{V}_s \right)^{-1}\bm{b},
\end{align}
where $\Delta = \textrm{diag}(g'(\mu_i))$ and $\bm{Z}_i$ is a $n\times 1$ vector of indicators such that $b_i=\bm{Z}_i\bm{b}$. Defining the working vector $\bm{\tilde{Y}}$ with elements $\tilde{Y_i} = \eta_i + g'(\mu_i)(y_i - \mu_i)$, the solution of \eqref{eq:A4} is equal to
\begin{align}\label{eq:bsolution}
\hat{\bm{b}}=\left( \sum_{s=1}^S\hat\tau_s\bm{V}_s \right){\bm{\Sigma}}^{-1}\left(\bm{\tilde{Y}}-\tilde{\bm{X}}\tilde{\bm{\beta}}\right),
\end{align}
where $\tilde{\bm{X}}=\left[\bm{X} \quad \bm{G}\right]$ and ${\bm{\Sigma}} = {\bm{W}}^{-1} + \sum_{s=1}^S\hat\tau_s\bm{V}_s$. Let $\sum_{s=1}^S\hat{\tau}_s \bm{V}_s = \bm{U}\bm{D}\bm{U}^\intercal$ be the associated eigen-spectral decomposition of the variance-covariance matrix of $\bm{b}$, where $\bm{U}_{n\times n}$ is an orthonormal matrix of eigenvectors and $\bm{D}_{n\times n}$ is a diagonal matrix of eigenvalues, such that \eqref{eq:bsolution} can be rewritten as
\begin{align}
\label{eq:bsolution2}
\hat{\bm{b}} = \bm{U}\left(\bm{D}^{-1} + \bm{U}^\intercal\bm{W}\bm{U}\right)^{-1}\bm{U}^\intercal\bm{W}\left(\tilde{\bm{Y}} - \tilde{\bm{X}}\tilde{\bm{\beta}}\right).
\end{align}
By rotating the random effect $\bm{\delta} = \bm{U}^\intercal\bm{b}$, we have that \eqref{eq:bsolution2} is equivalent to solving the following generalized ridge regression problem
\begin{align*}
\hat{\bm{\delta}} = \underset{\bm{\delta}}{\textrm{argmin }}\left(\tilde{\bm{Y}} - \tilde{\bm{X}}\tilde{\bm{\beta}} - \bm{U\delta}\right)^\intercal\bm{W}^{-1}\left(\tilde{\bm{Y}} - \tilde{\bm{X}}\tilde{\bm{\beta}} - \bm{U\delta}\right) + \bm{\delta}^\intercal\bm{D}^{-1}\bm{\delta}.
\end{align*}
Consider now a coordinate descent step for $\bm{\beta}$. That is, suppose we have updates $\tilde{\bm{b}}$ and $\tilde{\bm{\beta}}_l$ for $l \ne j$, and we wish to partially optimize with respect to $\beta_j$. We would like to compute the gradient at $\beta_j = \tilde{\beta_j}$, which only exists if $\tilde\beta_j \ne 0$. If $\tilde\beta_j > 0$, then
\begin{align}\label{eq:appxB1}
\frac{\partial Q_{\lambda}(\bm{\beta}, \bm{b})}{\partial\beta_j}| _{(\bm{b}, \bm{\beta})=(\tilde{\bm{b}}, \tilde{\bm{\beta}})} &=
-\sum_{i=1}^n \frac{a_i(y_i - \mu_i)}{\hat\phi \nu(\mu_i)}\frac{1}{g'(\mu_i)}\tilde{X}_{ij} + \lambda v_j = -\tilde{\bm{X}}_{j}^\intercal{\bm{W}}\Delta(\bm{y} - \bm{\mu}) + \lambda v_j,
\end{align}
where $\tilde{\bm{X}}_j$ is a $n\times 1$ column vector for predictor $j$.
Recall that we defined the working vector $\tilde{\bm{Y}}$ such that $\bm{y}-\bm{\mu} = {\bm{\Delta}}^{-1}(\tilde{\bm{Y}} - \tilde{\bm{X}}\tilde{\bm{\beta}} - \tilde{\bm{b}})$. Thus, plugging $\tilde{\bm{b}}$ from \eqref{eq:bsolution} and solving \eqref{eq:appxB1} leads to
\begin{gather*}
-\tilde{\bm{X}}_{j}^\intercal{\bm{W}}\left(\bm{\tilde{Y}}-\tilde{\bm{X}}\tilde{\bm{\beta}} - \tilde{\bm{b}}\right) + \lambda v_j = 0 \\
\Longleftrightarrow -\tilde{\bm{X}}_{j}^\intercal{\bm{W}}\left(\bm{I} - \left(\sum_{s=1}^S\hat\tau_s\bm{V}_s\right){\bm{\Sigma}}^{-1}\right)\left(\bm{\tilde{Y}}-\tilde{\bm{X}}\tilde{\bm{\beta}}\right) + \lambda v_j = 0 \\
\Longleftrightarrow -\tilde{\bm{X}}_{j}^\intercal{\bm{\Sigma}}^{-1}\left(\bm{\tilde{Y}}-\tilde{\bm{X}}\tilde{\bm{\beta}}\right) + \lambda v_j = 0,
\end{gather*}
since ${\bm{\Sigma}} = \bm{W}^{-1} + \sum_{s=1}^S\hat{\tau}_s\bm{V}_s.$ Finally, isolating $\hat{\beta}_j$ yields
\begin{align}\label{eq:appxB2}
\hat{\beta}_j &= \frac{\tilde{\bm{X}_j}^\intercal{\bm{\Sigma}}^{-1}\left(\tilde{\bm{Y}}-\sum_{l\ne j}\tilde{\bm{X}}_l\tilde{\beta_l}\right) - \lambda v_j }{\tilde{\bm{X}}_j^\intercal{\bm{\Sigma}}^{-1}\tilde{\bm{X}}_j}.
\end{align}
Because the weights $\bm{W}$ are being updated at every iteration, the solution in \eqref{eq:appxB2} requires inverting a different variance-covariance matrix ${\bm{\Sigma}}$ each time we update our estimate for ${\beta_j}$. In modern large-scale genetics data sets, the sample size $n$ can be very large, thus we want to avoid costly matrices inversions. As detailed in Appendix C, we instead replace the inverse variance-covariance matrix $\bm{\Sigma}^{-1}$ in \eqref{eq:appxB2} by an upper-bound $\tilde{\bm{\Sigma}}^{-1}$~\citep{bohningMonotonicityQuadraticapproximationAlgorithms1988} and rotate the response vector $\tilde{\bm{Y}}$ and design matrix $\tilde{\bm{X}}$ by the eigenvectors of $\sum_{s=1}^S\hat{\tau}_s\bm{V}_s$, which yields
\begin{align*}
\hat{\beta}_j &= \frac{\tilde{\bm{X}_j}^\intercal\tilde{\bm{\Sigma}}^{-1}\left(\tilde{\bm{Y}}-\sum_{l\ne j}\tilde{\bm{X}}_l\tilde{\beta_l}\right) - \lambda v_j }{\tilde{\bm{X}}_j^\intercal\tilde{\bm{\Sigma}}^{-1}\tilde{\bm{X}}_j} \nonumber \\
&= \frac{\tilde{\bm{X}_j}^\intercal\left(c\bm{I}_n + \bm{UDU}^{\intercal}\right)^{-1}\left(\tilde{\bm{Y}}-\sum_{l\ne j}\tilde{\bm{X}}_l\tilde{\beta_l}\right) - \lambda v_j }{\tilde{\bm{X}}_j^\intercal\left(c\bm{I}_n + \bm{UDU}^{\intercal}\right)^{-1}\tilde{\bm{X}}_j} \nonumber \\
&= \frac{\tilde{\bm{X}_j}^\intercal\bm{U}\left(c\bm{I}_n + \bm{D}\right)^{-1}\bm{U}^{\intercal}\left(\tilde{\bm{Y}}-\sum_{l\ne j}\tilde{\bm{X}}_l\tilde{\beta_l}\right) - \lambda v_j }{\tilde{\bm{X}}_j^\intercal\bm{U}\left(c\bm{I}_n + \bm{D}\right)^{-1}\bm{U}^{\intercal}\tilde{\bm{X}}_j} \nonumber \\
&= \frac{\sum_{i=1}^n\frac{1}{c + \Lambda_i}\tilde{X}_{ij}^*\left(\tilde{Y}_i^*-\sum_{l\ne j}\tilde{X}_{il}^*\tilde{\bm{\beta}_l}\right)- \lambda v_j}{\sum_{i=1}^n\frac{1}{c + \Lambda_i}\tilde{X}_{ij}^{*^2}},
\end{align*}
where $c=4$ for binary responses, and $c=\hat\phi$ otherwise, $\Lambda_i$ are the eigenvalues of $\sum_{s=1}^S\hat{\tau}_s\bm{V}_s$, $\tilde{\bm{Y}}^* = \bm{U}^\intercal\tilde{\bm{Y}}$ and $\tilde{\bm{X}}^* = \bm{U}^\intercal\tilde{\bm{X}}$. By proceeding in a similar way for $\tilde{\beta}_j < 0$, one can show \citep{friedmanPathwiseCoordinateOptimization2007} that the coordinate-wise update for $\beta_j$ has the form
\begin{align}\label{eq:betasolution}
\hat{\beta}_j
&= \frac{S\left(\sum_{i=1}^n\frac{1}{c + \Lambda_i}\tilde{X}_{ij}^*\left(\tilde{Y}_i^*-\sum_{l\ne j}\tilde{X}_{il}^*\tilde{\bm{\beta}_l}\right), \lambda v_j \right)}{\sum_{i=1}^n\frac{1}{c + \Lambda_i}\tilde{X}_{ij}^{*^2}},
\end{align}
where $S(z, \gamma)$ is the soft-thresholding operator:
\begin{align*}
\textrm{sign}(z)(|z| - \gamma)_{+} =
\begin{cases}
z - \gamma & \text{if } z>0 \text{ and } \gamma < |z| \\
z + \gamma & \text{if } z<0 \text{ and } \gamma < |z| \\
0 & \text{if } \gamma \ge |z|.
\end{cases}
\end{align*}
The updates estimate for $\bm{b}$ can also be simplified by replacing the inverse variance-covariance matrix $\bm{\Sigma}^{-1}$ by $\tilde{\bm{\Sigma}}^{-1}$ in \eqref{eq:bsolution}, which yields
\begin{align*}
\hat{\bm{b}} = \bm{U}\left(c\bm{D}^{-1} + \bm{I}_n\right)^{-1}\left(\tilde{\bm{Y}}^* - \tilde{\bm{X}}^*\tilde{\bm{\beta}}\right).
\end{align*}
This leads to the following updates estimate for $\bm{\eta}$
\begin{align}\label{eq:eta}
\hat{\bm{\eta}}
&=\bm{\tilde{Y}}- \left(\tilde{\bm{Y}} - \bm{X}\hat{\bm{\beta}} - \hat{\bm{b}}\right) \nonumber \\
&= \tilde{\bm{Y}} - \bm{U}\left\{\bm{I}_n -(c\bm{D}^{-1} + \bm{I}_n)^{-1} \right\}\left(\bm{\tilde{Y}}^*-\tilde{\bm{X}}^*\hat{\bm{\beta}} \right) \nonumber\\
&= \tilde{\bm{Y}} - \bm{U}\left(c^{-1}\bm{D} + \bm{I}_n\right)^{-1}\left(\bm{\tilde{Y}}^*-\tilde{\bm{X}}^*\hat{\bm{\beta}} \right).
\end{align}
The block coordinate descent algorithm to obtain regularized PQL estimates for $\bm{\beta}=(\bm{\alpha}^\intercal, \bm{\gamma}^\intercal)^\intercal$ and ${\bm{b}}$ is as follows:
\begin{algorithm}[H]
\begin{enumerate}
\item \textit{Initialization} \\ Set $\hat{\bm{\beta}}^{(0)} = (\hat{\bm{\alpha}}^\intercal, \bm{0}^\intercal) \text{ and } \hat{\bm{b}}^{(0)} = \hat{\bm{b}}$ , where $\hat{\bm{\alpha}},\hat{\bm{b}}$ are the estimates from the AI-REML algorithm; \\
Calculate $\hat{\bm{\eta}}^{(0)} = \tilde{\bm{X}}\hat{\bm{\beta}}^{(0)} + \hat{\bm{b}}^{0} \text{, } \tilde{\bm{Y}}^{*(0)} = \bm{U}^\intercal\bm{\tilde{Y}}^{(0)} \text{ and } \tilde{\bm{X}}^{*}=\bm{U}^\intercal\tilde{\bm{X}}$;
\item \textit{Iteration} \\
\For{$\lambda = \lambda_{max}$ to $\lambda_{min}$}{
\qquad \For{$t=1,2,...,$ until outer-loop convergence}{
\qquad \For{$j=1,...,m+p$}{\qquad Calculate $$\hat{\bm{\beta}}_j^{(t)}=\frac{S\left(\sum_{i=1}^n\frac{1}{c + \Lambda_i}\tilde{X}_{ij}^*\left(\tilde{Y}_i^{*(t-1)}-\sum_{l\ne j}\tilde{X}_{il}^*\hat{\bm{\beta}_l}^{(t-1)}\right), \lambda v_j \right)}{\sum_{i=1}^n\frac{1}{c + \Lambda_i}\tilde{X}_{ij}^{*^2}},$$ \qquad until inner-loop convergence;
}
\qquad Calculate $\hat{\bm{\eta}}^{(t)} = \tilde{\bm{Y}}^{(t-1)} - \bm{U}\left(c^{-1}\bm{D} + \bm{I}_n\right)^{-1}\left(\bm{\tilde{Y}}^{*(t-1)}-\tilde{\bm{X}}^*\hat{\bm{\beta}}^{(t)} \right)$; \\
\qquad $\text{Update }\bm{\tilde{Y}}^{(t)} \text{ and } \bm{\tilde{Y}}^{*(t)} \text{ using } \hat{\bm{\eta}}^{(t)}$;
}
\quad Set $\hat{\bm{\beta}}^{(0)} = \hat{\bm{\beta}}^{(t)}$, $\bm{\tilde{Y}}^{(0)} = \bm{\tilde{Y}}^{(t)} \text{ and } \bm{\tilde{Y}}^{*(0)} = \bm{\tilde{Y}}^{*(t)}$ as warm starts for next $\lambda$;
}
\end{enumerate}
\caption{Block coordinate descent for regularized PQL estimation}
\end{algorithm} \leavevmode\newline
For inner-loop convergence, we use the same criteria as~\citet{friedmanPathwiseCoordinateOptimization2007}, that is after a complete cycle of coordinate descent we look at $$\max_j\Delta_j = \max_j\sum_{i=1}^n \frac{1}{c+\Lambda_i}\tilde{X}_{ij}^{*2}(\hat{\beta}_j^{(t-1)} - \hat{\beta}_j^{(t)})^2,$$ which measures the maximum weighted sum of squares of changes in fitted values for all coefficients. If $\max_j\Delta_j$ is smaller than tolerance, we stop the coordinate descent loop. For outer-loop convergence, we calculate the fractional change in the loss function $-l_{PQL}(\bm{\alpha}, \bm{\gamma}, \hat\phi, \hat{\bm{\tau}})$ and declare convergence if its value is smaller than tolerance.
\subsection*{Appendix C. Upper-bound for the inverse variance-covariance matrix}
We want to show that the inverse of the variance-covariance matrix $\bm{\Sigma} = \bm{W}^{-1} + \sum_{s=1}^S\tau_s \bm{V}_s$ is upper bounded by a positive definite matrix $\tilde{\bm{\Sigma}}^{-1}$ that does not depend on the sample weights $\bm{W}$, that is, $$\bm{\Sigma}^{-1} = \left( \bm{W}^{-1} + \sum_{s=1}^S\tau_s \bm{V}_s \right)^{-1} \preceq \tilde{\bm{\Sigma}}^{-1},$$ where $\bm{W}^{-1} = \textrm{diag}\left\{ \frac{a_i}{\phi\nu(\mu_i)[g'(\mu_i)^2]}\right\}^{-1}$ and $\sum_{s=1}^S\tau_s \bm{V}_s$ are $n \times n$ positive definite matrices.
For binary responses with logistic link and variance function equal to $\nu(\mu_i)=\mu_i(1-\mu_i)$, we have that $\bm{W}^{-1} \succeq 4 \bm{I}$. It follows that for any $h \in \mathbb{R}^n$,
\begin{align*}
h^\intercal\left( \bm{W}^{-1} + \sum_{s=1}^S\tau_s \bm{V}_s \right)h &> h^\intercal\left( 4\bm{I} + \sum_{s=1}^S\tau_s \bm{V}_s \right)h.
\end{align*}
Let $\bm{A} = \bm{W}^{-1} + \sum_{s=1}^S\tau_s \bm{V}_s$ and $\bm{B}=4\bm{I} + \sum_{s=1}^S\tau_s \bm{V}_s$ such that the previous inequality can be rewritten as $$h^\intercal \bm{A} h > h^\intercal \bm{B} h.$$ Let $\lambda^A,\lambda^B$ be the eigenvalues for $\bm{A}$ and $\bm{B}$ respectively, then
\begin{align*}
h^\intercal \bm{A} h \ge \lambda^A_{min} & > \lambda_{max}^B \ge h^\intercal \bm{B} h > 0 \\
\Longleftrightarrow \frac{1}{ \lambda_{max}^B} &> \frac{1}{ \lambda_{min}^A} \\
\Longleftrightarrow \lambda_{min}^{B^{-1}} &> \lambda_{max}^{A^{-1}} \\
\Longleftrightarrow h^\intercal\left( \bm{W}^{-1} + \sum_{s=1}^S\tau_s \bm{V}_s \right)^{-1}h &< h^\intercal\left( 4\bm{I} + \sum_{s=1}^S\tau_s \bm{V}_s \right)^{-1}h.
\end{align*}
Thus, we have $$\bm{\Sigma}^{-1} = \left( \bm{W}^{-1} + \sum_{s=1}^S\tau_s \bm{V}_s \right)^{-1} \preceq \left( 4\bm{I} + \sum_{s=1}^S\tau_s \bm{V}_s \right)^{-1} \equiv \tilde{\bm{\Sigma}}^{-1}.$$
\bibliographystyle{apalike}
|
1202.1970
|
\section{Introduction}
Homogeneous turbulence submitted to distortions such as solid body rotation,
stratification, or the Lorentz force in the MHD context, exhibit axisymmetric statistics,
as a clear departure from isotropy. These anisotropic effects that arise
due to modified dynamics or energy exchange do not fall within the classical
description of turbulence in Kolmogorov's theory.
The following general questions, pertaining to the understanding of anisotropic homogeneous
turbulence, can therefore be raised:
\begin{enumerate}
\item How can we characterize the
anisotropy of the flow, at one or two point, in physical or spectral statistical descriptions?
\item Are passage relations available between: (a) the two-point
statistics of structure functions, necessarily
measured in physical space in experiments, and (b) kinetic energy and transfer spectra?
(Which can be obtained
for high Reynolds number turbulence from statistical models.)
\item How can we
interpret the modified dynamics of anisotropic turbulence? (It can
be observed
in terms of anomalous scalings of
energy spectra or energy
transfer spectra readily available in two-point statistical models.)
\item Can we exploit some features
easily accessed in spectral space to refine the phenomenological analysis
of the dynamics of turbulence in physical space?
\item Do Kolmogorov scalings available for high Reynolds number isotropic turbulence
apply to isotropically integrated statistics of anisotropic turbulence?
\item What level of complexity do we have to introduce to go beyond the
mere isotropic description, \textit{e.g.} starting with axisymmetric flows?
\end{enumerate}
As a bootstrap, we start, hereafter in this introductory section, by reviewing a few of the
existing works related to these issues, and introduce some of the
existing results and formalism available in the statistical characterization
of axisymmetric turbulence.
In section~\ref{sec1}, we describe the relationships
available for isotropic turbulence for second- and third-order statistics
appearing in both Lin's and K\'arm\'an-Howarth's equations, and suggest to extend
the description of the velocity increment statistics to the anisotropic
case, and to relate them to the modified dynamics in axisymmetric homogeneous turbulence,
especially energy and transfer spectra. Results for the latter, in the case of
stably stratified or rotating turbulence, are provided in section~\ref{edqnmres}. We
choose to use an anisotropic two-point statistical model, described
shortly in section~\ref{secedqnm}, since it permits to reach higher
Reynolds number and smoother statistics than direct numerical simulation.
Assuming isotropized passage relations, we then derive second- and
third-order velocity structure functions, presented in section~\ref{physres},
and discuss the results against existing scalings for isotropic turbulence.
The spectral formalism for homogeneous
turbulence allows to remove altogether the role of pressure, and hence permits
a highly refined analysis of the dynamics of anisotropic turbulence, and
on its sources; for instance
from the role of inertial waves interactions in rotating turbulence, or for the
dynamics of stably stratified turbulence in which the transfers are split
between internal waves and potential vorticity interactions.
Even in flows which are thought to be isotropic, in experimental or numerical
realizations, a degree of anisotropy may be hidden, depending on how
the characterization of the isotropy of the flow has been done:
the isotropy of
\textit{rms} velocity components or of integral length scales only characterizes
the large scales, whereas vorticity or dissipation can be used for the
smaller ones. The computation of anisotropic spectra
allows to quantify such scale-dependent level of isotropy, or anisotropy,
and to isolate the most important energy transfer contributions.
Alternately, the statistics in physical space
do not easily permit the removal of pressure from balance equations, and
thus prevent from a tractable access to modal decompositions (\textit{e.g.} the
toroidal/poloidal decomposition). It however applies to locally inhomogeneous
flows, which is not easily accessible to spectral analysis.
We therefore
emphasize the importance of both spectral and physical viewpoints, and
the fact that they are \textit{complementary}, which we expose and discuss in this paper,
taking rotating or stably stratified turbulence as supporting evidences.
In what follows in this section, we review a few studies which have
been devoted to the characterization of anisotropy in turbulence,
from the point of view of directional statistics in physical space --- including
both anisotropy of velocity components or dependence on the separation vector ---
and studies about the anisotropic spectral scalings in anisotropic turbulence,
related to the dynamics of the flow.
If initially isotropic, homogeneous turbulence can be rendered anisotropic by introducing
an external distortion on the flow: solid body rotation is present for instance
in geophysical flows and acts through the Coriolis force,
as well as a buoyancy force in density- or temperature- stratified layers.
In conducting fluids, the action of an external magnetic field also modifies
the symmetries by means of the Lorentz force. In these three examples, the
intensity of the corresponding force depends on the orientation of the
fluid motion with respect
to either the rotation axis, the gravity axis, the background magnetic field
axis. Wave propagation can also be present, \textit{e.g.} inertio-gravity waves or Alfv\'en
waves. They provide an anisotropic way of redistributing energy in terms of scale
and direction, such that the dynamics of energy exchange in the turbulence is
strongly modified. We focus here on the effects of solid body rotation and
stable density stratification on homogeneous turbulence.
Studies about anisotropic turbulence were proposed and stemmed from
various preoccupations. The theory of axisymmetric turbulence was
regarded as a logical extension to the theory of isotropic turbulence,
and was developed by Batchelor~\cite{batchelor46} --- in which parallel
and perpendicular directions (with respect to the axis of symmetry)
were distinguished to
express statistical quantities, \textit{e.g.}
the dissipation --- and by~Chandrasekhar \cite{chandrasekhar50} in the 50s.
In these studies, the imposed symmetries
do not account for the possible presence of helicity, unlike the extension proposed by
Lindborg about the kinematics of axisymmetric turbulence \cite{lindborg95}.
More recently, some exact vectorial laws were also proposed by Galtier for rotating homogeneous
turbulence, exposing the need for a transverse/longitudinal components
distinction (with respect to the separation vector)
when computing velocity structure functions \cite{galtier09}.
An extensive discussion and review of the anisotropy in turbulent flows was proposed
by Biferale \& Procaccia \cite{biferale_anisotropy_2005} using the
symmetry group analysis (SO(3) for Navier-Stokes equations). These authors
discuss Kolmogorov's theory and how to relate anisotropic flows to the
$n$-th order structure functions, especially looking at the 4/5 law for
the third-order one, which includes the isotropy hypothesis.
On the one hand, the structure-function approach has recently been used for
an experimental characterization of rotating turbulence dynamics by
Lamriben \textit{et al.}
\cite{lamriben2011}, and for mesoscale turbulence in the atmosphere by Lindborg
\& Cho \cite{LindborgCho},
both works including a discussion of the level of anisotropy to account for, as
a departure to classical Kolmogorov scalings of velocity increments moments.
In an experimental study concerning a turbulent jet, Xu \& Antonia \cite{xu_antonia07} emphasize the
importance of discriminating between the longitudinal and transverse velocity
components when expressing the
structure function in axisymmetric turbulence, and Oyewola
\textit{et al.} use structure functions to describe the anisotropy of the small scales in a
turbulent boundary layer \cite{oyewola}.
In situ experiments by Kurien \textit{et al.} in the atmospheric
boundary layer also permitted to exhibit the different scalings of the structure
functions, distinguishing between longitudinal and transverse increments, by
separating the lowest order anisotropy contributions thanks to the SO(3)
symmetry group. \cite{kurien2000}
On the other hand, several efforts were devoted to establishing
the scalings that should apply to two-point velocity correlation spectra
depending on the perpendicular $k_\perp$ or the parallel $k_\parallel$
wavenumber components with respect to the axis of symmetry.
In the geophysical context, one may retrieve atmospheric spectra as in rotating strongly
stratified turbulence. At small scales (several kilometers),
Kolmogorov $k_\perp^{-5/3}$ scaling
is retrieved, whereas at large scales (several hundreds of kilometers), depending on the
velocity component considered the scaling is either $k_\perp^{-3}$ (zonal wind)
or $k_\perp^{-2}$ (meridional wind). \cite{lindborg06} In the context of conducting fluids, plasmas and astrophysics,
Galtier has also proposed several spectral scalings with spectra of the form
$k_\perp^{-\alpha}k_\parallel^{-\beta}$ to account for anisotropy in magnetohydrodynamic
turbulence \cite{galtier2005}. However, the physical arguments available for discussing the anisotropy of conducting fluid submitted to the Lorentz force are not available for
disentangling the intricate nonlinear dynamics of rotating turbulence, and, to some
extent of stably stratified turbulence.
In most of the above-mentioned works, spectra are used to characterize the cascade
of energy and the dynamics of anisotropic turbulence, or structure functions compared
with the isotropic theory scalings. Very few works are devoted to the relationship
between the two formalisms, and the current paper is a first attempt at providing
\textit{quantitative} information on how anisotropic spectra dynamics can be used
to interpret physical space velocity increment statistics.
Thus, taking into account the modifications of the turbulence
structure and dynamics due to external effects can be done at different levels.
The statistical description of the velocity field
can be done at a two-point level in physical space by the second-order
velocity correlation tensor
$R_{ij}(\bm{r})=\langle u_i(\bm{x}) u_j(\bm{x}+\bm{r}\rangle$
where $\bm{r}$ is the separation vector. Fourier-transforming this
tensor yields the spectrum tensor
$\Phi_{ij}(\bm{k})={1}/(8\pi)^3\int R_{ij}(\bm{r})\exp(-\mathrm{i}\bm{k}\cdot\bm{r})\mathrm{d}\bm{r}$,
whose trace is $\Phi_{ii}(\bm{k})=E(k)/2\pi k^2$ in isotropic turbulence, with $k=|\bm{k}|$ and $E(k)$
is the kinetic energy spectrum. Kolmogorov theory supports the scaling $E(k) \sim k^{-5/3}$ in isotropic
turbulence at high Reynolds number.
However, external distortions render the turbulent flow statistics dependent
on the orientation of $\bm{r}$ in physical space, \textit{e.g.} for the two-point correlations
$R_{ij}(\bm{r})$. In isotropic turbulence, this tensor depends only
on two scalar function, namely the longitudinal two-point correlation function $f(r)$,
and the transverse one $g(r)$ \cite{batchelor53}, and is expanded over
the tensors $\delta_{ij}$ and $r_ir_j/r^2$. In axisymmetric turbulence,
which is the case we consider here---where the axis of symmetry $\bm{n}$ is borne
by the rotation vector in rotating turbulence or the axis of gravity in stably stratified
turbulence---,
$R_{ij}(\bm{r})$ depends on four scalars and requires the addition of
the two expansion tensors $r_in_j/r$ and $n_in_j$ \cite{batchelor46}.
The anisotropic dependence of the two-point correlation
tensor in turn translates
into a dependence of the corresponding spectra $\Phi_{ij}(\bm{k})$
on the orientation $\theta$ of the wavevector $\bm{k}$ with respect to
the axis of symmetry $\bm{n}$ (the latter is the same as in physical space).
The spectra can similarly also be decomposed in a general way over four
scalars (see \textit{e.g.} \cite{cambon97} and \cite{sagaut08}).
This correspondence between the physical space two-point statistical formalism
of turbulence and the spectral description not only applies to second-order correlation tensors
and spectra, but it also extends to third-order statistics, and moreover
to velocity structure functions. The latter are of great use in Kolmogorov's
description of isotropic turbulence, and are described \textit{e.g.}
in \cite{monin} and in \cite{frisch}. Considering the velocity
increment $\delta\bm{u}=\bm{u}(\bm{x}+\bm{r})-\bm{u}(\bm{x})$, in
the isotropic context one commonly computes the $n^{th}$-order structure
function $\langle(\delta u)^n\rangle$ based on the longitudinal projection $u$
of the velocity vector onto the separation vector $\bm{r}$. The brackets
indicate averaging using statistical homogeneity. Obviously, when
turbulence is anisotropic, one should also distinguish different
components in the structure functions and different orientations
of $\bm{r}$, although most of the theoretical
results are available only in the isotropic context.
One of them is
the exact
Kolmogorov four-fifth law wherein the third-order structure function
$D_{LLL}(r)=\langle(\delta u)^3\rangle=-4/5\varepsilon r$, valid for
$r$ in the inertial subrange, where $\varepsilon$ is the averaged
dissipation \cite{k41a}, or, in a different form proposed by Antonia \textit{et al.}~\cite{antonia97}:
\begin{equation}
\langle\delta u (\delta u_j\delta u_j)\rangle=-\frac{4}{3}\varepsilon r .
\label{anton}
\end{equation}
where repeated indices imply summation.
In the following, we investigate how anisotropic turbulence can recover
the above scalings, using the two-point statistical EDQNM model.
The question about the convergence of the statistics of the structure functions
to the 4/5 limit has been addressed before. Antonia \& Burattini studied this limit
at increasing Reynolds numbers \cite{antonia06}, and it appears that the convergence
is very slow, and may not be yet reached in the existing experiments. However, this
low-Reynolds number effects has to be separated from the anisotropic effect
in non isotropic flows. The leading order small-scales K41 scalings may be recovered
in anisotropic flows, provided suitable filtering has been applied to the
data, as shown with DNS fields by Taylor \textit{et al.} \cite{taylorkurien}.
In the following, the interpretation of the scaling laws provided by EDQNM
results therefore mixes the isotropic small-scales turbulent
behavior with the anisotropic contributions, since no such separation as
in Taylor \textit{et al.} has been made.
\section{The link between spectral dynamics and velocity structure functions}
\label{sec1}
The above-mentioned structure functions provide a scale-dependent statistical characterization
of turbulence with a two-point separation. It is clear that they are related
to the two-point correlation functions, \textit{and their spectral
counterparts}, such that second-order velocity structure functions
are related to two-point velocity correlation spectra, and third-order structure functions to
energy transfer spectra (triple velocity correlations at two points).
In the anisotropic context, and
when dealing with statistically axisymmetric turbulence, the orientation $\theta$ of the
Fourier vector $\bm{k}$ with respect to the axis of symmetry borne by
$\bm{n}$, has to be taken into
account, so that an additional dependence has to be introduced in the energy spectrum: $E(k,\theta)$.
Integrating the latter over $\theta\in [-\pi,\pi]$ of course provides an equivalent spherically
averaged $E(k)$.
At this stage, the question of the resulting inertial scaling of the spectrum is raised.
The kinetic energy spectrum $E(k)$ and the nonlinear energy transfer spectrum $T(k)$ appear in
the Lin equation as
\begin{equation}
\partial_t E(k)+2\nu k^2 E(k)=T(k) \label{eqLin}
\end{equation}
in the isotropic case, where $\nu$ is the viscosity, such that the total
dissipation is $\varepsilon=2\nu\int k^2 E(k)\textrm{d}k$.
In the rotating case for instance, phase-scrambling anisotropically changes
the shape of the energy exchange term $T(k,\theta)$ in an equation
equivalent to~(\ref{eqLin}) (see next equation \ref{eqneuf}). The transfer
contains third-order correlation terms, and is thus related to the third-order
statistics that are used to quantify nonlinearity such as the skewness of
velocity gradient.
Regarding second-order moments, one can obtain second-order structure
functions starting with the two-point correlation spectra.
For instance,
\begin{equation}
\langle \left( \delta u_i \delta u_i \right) \rangle = 2 \int^{\infty}_0 \left(1 -
\frac{\sin kr}{kr} \right) E(k) \mathrm{d}k\ .
\label{eqqdeux}
\end{equation}
where repeated indices are summed, contains information
from all the velocity components.
Equation~(\ref{eqLin}) therefore relates the evolution of second-order
statistics---the energy spectrum---to third-order ones---the transfer.
The analogous in physical space of~(\ref{eqLin}) is
the K\'arm\'an-Howarth equation
\begin{equation}
\partial_t\left(u'^2 f\right)=\left(\partial_r+\frac{4}{r}\right)\left[R_{LL,L}(r,t)
+2\nu\partial_r\left(u'^2 f\right)\right] \label{eqKH}
\end{equation}
in which $f(r)$ is the longitudinal two-point correlation function;
the longitudinal two-point third-order correlation function is
$R_{LL,L}(r,t)=\langle u_i(\bm{x},t) u_m(\bm{x},t) u_i(\bm{x}+\bm{r},t) \rangle {r_m}/{r}$,
$r=|\bm{r}|$, and $(3/2)u'^2$ is the total kinetic energy. Upon examining~(\ref{eqLin}) and~(\ref{eqKH})
together, it is clear that if the dynamics of turbulence
is anisotropic, one might expect not only a different equilibrium between the
dissipative and nonlinear related terms of these equations, and
a directional dependence with $\bm{k}$ of the Lin equation
for $E(\bm{k})$ transposed into anisotropy with respect to $\mathbf{r}$ in the
K\'arm\'an-Howarth equation.
The derivation of the latter equation relies on assumption of isotropy,
but the general K\'arm\'an-Howarth-Monin equation, which contains
the same terms that depend on the separation \textit{vector} $\mathbf{r}$
can be derived as (see Frisch \cite{frisch}, p. 78 for a derivation,
and \textit{e.g. } \cite{lamriben2011}):
\begin{equation}
\frac{1}{2}\frac{\partial }{\partial t}
\langle\mathbf{u}(\mathbf{x})\mathbf{u}(\mathbf{x+r}\rangle
=\frac{1}{4}\mathbf{\nabla}\langle\delta\mathbf{u}(\mathbf{r})|\delta\mathbf{u}(\mathbf{r})|^2\rangle
+\nu \nabla^2 \langle\mathbf{u}(\mathbf{x})\mathbf{u}(\mathbf{x+r}\rangle
\end{equation}
The dependence of the velocity,
or velocity increments $\delta\bm{u}$, on the
direction of $\bm{r}$ with respect to $\bm{n}$,
is therefore an important parameter in axisymmetric turbulence statistics.
For isotropic turbulence, Kolmogorov's theory (1941) only considers a scalar
longitudinal
increment $\delta u$,
provides the scaling
$\langle(\delta u)^n\rangle\sim \left(\varepsilon r\right)^{n/3}$, and allows
to
draw from equation~(\ref{eqKH}) a simplified
relationship between second- and third-order structure functions
$$
\langle\left(\delta u\right)^3\rangle=-\frac{4}{5}\varepsilon r+6\nu\partial_r\langle(\delta u)^2\rangle.
$$
which yields, at infinite Reynolds number,
the famous four fifths law
\begin{equation}
\langle\left(\delta u\right)^3\rangle=-({4}/{5})\varepsilon r ,
\label{eqdix}
\end{equation}
or equation~(\ref{anton}). \cite{antonia97}
A useful relationship can be established between third-order spectral statistics \textit{i.e.} the
nonlinear energy transfer $T(k)$, and the third-order structure function :
\begin{equation}
\langle\delta u\left(\delta q\right)^2\rangle=4r\int_0^{\infty}g(kr)T(k)\mathrm{d}k \label{thstr}
\end{equation}
where $g(kr)=(\sin kr - kr\cos kr)/(kr)^3$ and $(\delta q)^2=(\delta u_i\delta u_i)$. It shows the direct link of the third-order
structure function on the dynamics of turbulence.
Formulas similar to equations (\ref{eqqdeux}) and (\ref{thstr})
rigorously derived for anisotropic turbulence are still not available
in the general case, but, in the rotating case, the isotropic relationships apply,
since the Coriolis force is not a production term in the balance equations,
and drops out altogether in the isotropized balance equations,
both in physical and spectral spaces.
In general anisotropic turbulence, the
analogous of~(\ref{thstr}) implies a vector dependence of the structure functions
on $\bm{r}$. It also shows that a dynamics modified by an external distortion applied to homogeneous
turbulence translates immediately in a modification of the structure functions.
It is therefore interesting to discuss the applicability of Kolmogorov scalings in
flows that contain some anisotropy. Attempts at such scalings for
rotating turbulence are proposed by Galtier \cite{galtier09}, or by Lindborg \&
Cho using atmospheric data. \cite{LindborgCho}
In the axisymmetric case, an extensive work is
in progress~\cite{ANISO}.
\section{Statistical model for the spectral anisotropy of axisymmetric turbulence}
\label{secedqnm}
We study a simplified case which nonetheless captures some important
anisotropic physical mechanisms in \textit{e.g.} geophysical flows: stable
stratification and rotation, taken into account in the
Boussinesq system of equations for an incompressible fluid:
\def\vu{\bm{u}}
\def\vn{\bm{n}}
\def\vnab{\bm{\nabla}}
\begin{eqnarray}
\label{eq:realnavier1}
{\partial_t \vu} + \vu \cdot \vnab \vu - \nu \nabla^2 \vu &=&
- \vnab p
- 2 \Omega \; \vn \times \vu + b \; \vn, \\
\label{eq:buoyancy}
{\partial_t b} + \vu \cdot \vnab b - \chi \nabla^2 b &=&
-N^2 \vn \cdot \vu,\\
\label{eq:incomp}
\vnab \cdot \vu &=& 0,
\end{eqnarray}
where $b$ is the buoyancy field,
associated with fluid density fluctuations around background
density augmented by mean density gradient.
$N$ is the Brunt-V\"ais\"al\"a (buoyancy) frequency
and $\Omega$ the rotational frequency.
The corresponding
terms, Coriolis force and buoyancy force, act on the
velocity field linearly and compete against the
nonlinear advection term. In this study, the separate
effects of either one is considered (although the
combination of both characterizes some atmospheric
or oceanic flows, with a ratio $\alpha=2\Omega/N\sim 1/10$).
In rotating or stratified turbulence, the dynamics
of the flow is superimposed with wave-like motion that
transfers energy differently from the classical turbulent
dynamics. In the linear limit, at large $N$ or $\Omega$,
one recovers wave turbulence, as a soup of superimposed
inertial waves or internal gravity waves.
For instance, ``turbulence'' and ``wave'' like
dynamics may be defined by splitting the velocity field
in the eigenmodes of the linearized
system~(\ref{eq:realnavier1})--(\ref{eq:incomp}), so that
the total turbulent energy of a rotating and stratified
flow can be divided into a
``vortex'' mode and a ``wave'' mode.
The linear evolution of the wave mode is
governed by he dispersion relation
$\sigma(\bm{k})=N \sin \theta$ for internal
waves and $\sigma(\bm{k})=2\Omega \cos \theta$ for inertial waves,
which depend on $\theta$, the polar angle with the vertical, whence
a directional dependence of the turbulent motion.
In order to study relatively high Reynolds number turbulence,
we introduce a statistical model which describes the evolution
of the two-point correlation spectra, whose dynamical equations
derive from~(\ref{eq:realnavier1})--(\ref{eq:incomp}), introducing
a closure known as EDQNM. The Eddy-Damped Quasi-Normal Markovian
model for isotropic turbulence was studied by several
authors (see \textit{e.g.} Orszag \cite{ORSZAG71}) decades
ago, although it has recently regained interest for its
ability to reach very high Reynolds numbers
as demonstrated by the spectrum we computed with EDQNM, shown on figure~\ref{fig5}.
(Similar
figures are presented in \cite{bos_dynamics_2006}.)
The more advanced version developed for rotating or
stably stratified turbulence
(denoted EDQNM$_2$ since it differs significantly from the model
for isotropic turbulence)
reflects more accurately the wave dynamics and is capable of taking into
account anisotropic features. In the limit of very strong
rotation, for instance, it becomes identical to the
wave turbulence closures described by Zakharov \cite{zakharov_kolmogorov_1992},
although in this book, Zakharov \textit{et al.} do not consider anisotropic dispersion
relations as for instance inertial waves.
An extensive description of the EDQNM$_2$ closure model
is provided in \cite{godeferd2003, cambon97}, as well
as comparisons with Direct Numerical Simulations (DNS)
and Large Eddy Simulations (LES) which show a very good agreement and
illustrate the capability of the model to accurately represent
anisotropic turbulence.
We find that, among other closures, only the EDQNM$_2$ model for closing
the spectral transfer terms (the equivalent to $T$ in equation~\ref{eqLin})
was capable of predicting all the anisotropic
features observed in the DNS and LES results.
The model equations for the generalized
transfer terms, \textit{e.g.} $T^e(\bm{k})$ for the directional
kinetic energy spectrum, involve sums of eight contributions (according to
polarities of triadic interactions), weighted
by the rotation-dependent factor (or Brunt-V\"ais\"al\"a frequency in the stratified
case):
\begin{equation}
T^e=\sum_{\epsilon,\epsilon',\epsilon''=\pm 1}
\int_{\bm{k}+\bm{p}+\bm{q}=0}
\frac{\mathbf{\mathsf{S^e_{(QN)}}}}{\vartheta_{kpq}^{\epsilon,\epsilon',\epsilon''}}
\mathrm{d}^3\bm{p}
\label{eqneuf}
\end{equation}
The numerator of the integrand takes into account
the quasi-normal expansion for non-isotropic turbulence and is closed in terms
of quadratic combinations of the
two-point correlation spectra,
whereas the denominator involves viscous and eddy-damping effects through
\begin{equation}
\vartheta_{kpq}^{\epsilon,\epsilon',\epsilon''}=
\vartheta_{kpq}(1+2\mathsf{i}\vartheta_{kpq}
\Omega(\epsilon k_{\parallel}/k+\epsilon' p_{\parallel}/p+\epsilon'' q_{\parallel}/q)) \label{eqdamp}
\end{equation}
for a triad of wavevectors $\mathbf{k}+\mathbf{p}+\mathbf{q}=0$ with
polarities $\{\epsilon,\epsilon',\epsilon''\}=\{\pm 1\}^3$,
thus accounting for the explicit linear rotation effects on triple correlation through the phases.
Thus, the classical timescale $\vartheta_{kpq}$ in the isotropic non-rotating case is
replaced by the preceding triadic complex timescales.
As already mentioned, the above expression is consistent
with wave turbulence results~\cite{BENNEY-SAFFMAN,bellet_wave_2006}.
The isotropic EDQNM model consists only of the Lin equations (\ref{eqLin})
in which the transfer term is closed using double products of energy
spectra $E(k)$, and where the only dependence variable is the wavenumber $k$.
\section{Second- and third-order structure functions derived from spectral statistics}
The EDQNM$_2$ model for rotation and stable stratification consists of equations
for the two-point correlation spectra similar to (\ref{eqLin}) in which the nonlinear transfer
spectra in the right-hand-side are replaced by their closure. In the model
for rotating turbulence, four coupled equations are solved for the
energy density spectrum $e(\mathbf{k})$, the helicity $h(\mathbf{k})$
and the polarization spectra $Z(\mathbf{k})$, each of which is expressed at each discretized spectral point
in the polar-spherical representation of wavevectors $\bm{k}$, that is as functions
of discretized $(k,\theta)$:
\begin{eqnarray}
\left({\partial \over \partial t} + 2 \nu k^2\right)e &=& T^e \nonumber \\
\left({\partial \over \partial t} + 2 \nu k^2 + 4{\rm i}\Omega {k_3
\over k} \right) Z &=& T^z \label {eq12} \\
\left({\partial \over \partial t} + 2 \nu k^2\right)h & =& T^h \nonumber
\end{eqnarray}
The transfer spectra are also discretized in the same way.
For stratified
turbulence, the density spectrum of rescaled
potential energy $\Phi_3(\mathbf{k})$ and the cross-correlation spectrum for density and
velocity $\Psi(\mathbf{k})$ are involved
\begin{eqnarray}
\left(\frac{\partial}{\partial t} + 2\nu k^2\right) \Phi_1 &=& T^1 \nonumber \\
\left(\frac{\partial}{\partial t} + 2\nu k^2\right) \left(\Phi_2+\Phi_3\right) &=& T^{2}+T^{3} \label{eqstrat}\\
\left(\frac{\partial}{\partial t} + 2\nu k^2 -2\mathrm{i}N\sin\theta_k \right)
\left( \Phi_2-\Phi_3+\mathrm{i}\Psi_R \right) &=& T^2-T^3+IT^{\Psi_R} \nonumber
\end{eqnarray}
along with the kinetic energy spectrum decomposed in poloidal and toroidal
spectra $\Phi_2(\mathbf{k})$ and $\Phi_1(\mathbf{k})$.
Equations (\ref{eq12}) and (\ref{eqstrat})
are therefore generalized versions of the Lin equation (\ref{eqLin}) and are exact
equations, unless a closure such as equation (\ref{eqneuf}) is applied.
Moreover, in both systems (\ref{eq12}) and (\ref{eqstrat}), the decomposition of the spectra
is optimized to yield simple and meaningful equations: the $Z$ equation in (\ref{eq12})
contains the oscillatory part due to the effect of rotation, and the last equation
of (\ref{eqstrat}) represents oscillations due to the kinetic/potential energy
exchange, mediated by the flux $\Psi$ produced by buoyancy.
The complexity of these systems of equations and anisotropic
damping (\ref{eqdamp}) provided by the EDQNM$_2$ model
needs to be contrasted with the simplicity of the EDQNM model for $E(k)$
whose dynamical equation is only (\ref{eqLin}).
The corresponding set of time-dependent differential equations are advanced in
time starting from initial conditions which correspond to a narrow-band distribution
of kinetic energy centered about a prescribed peak. The spectra then evolve into
developing an inertial subrange and a dissipative one. From this time on,
turbulence evolves in a self-similar decay.
The evolved rotating turbulence spectrum presented on figure \ref{fig5} corresponds to
a Reynolds number $\textit{Re}=7000$ and the Rossby number is $\textit{Ro}=u/(\Omega L)=0.04$ ($u$
is the \textit{rms} velocity, $L$ the integral scale and $\Omega$ the rotation rate)
although the figures at initial time were higher, since they usually decay in time.
For the stratified turbulence spectrum, $\textit{Re}=1000$ and the Froude number
is $\textit{Fr}=u/(NL)=0.09$, where $N$ is the Brunt-V\"ais\"al\"a frequency which characterizes
the intensity of the vertical density gradient. Note that the Reynolds number is lower
in the stratified simulation, since a part of the dynamics is contained in the
potential energy stored by density fluctuations around the mean density.
\subsection{Quick comments on the spectral statistics and the dynamics of anisotropic turbulence}
\label{edqnmres}
The developed spectra for isotropic, rotating, and stably stratified turbulence are
plotted on figure \ref{fig5}. Note that spectra of anisotropic turbulence,
in our model, but also in DNS or in experimental measurements, depend on
the orientation of the wavevector, in agreement with two-point correlation
functions which depend on the orientation of the separation vector. The relevant
parameter is the angle $\theta$ with the axis of symmetry.
We have purposefully chosen an very high Reynolds number for the
isotropic spectrum for three reasons: (a) it demonstrates the capacity
of statistical EDQNM closures to reach Reynolds numbers closer to
those
observed in geophysical or astrophysical contexts than is possible with DNS; (b) the anisotropic
EDQNM$_2$ closure requires additional computational effort, and
therefore the inertial ranges observed on the corresponding
spectra of figure \ref{fig5} span roughly two decades and are reduced
with respect to the isotropic case; however, equivalent
DNS would require an order of magnitude larger computational effort; (c)
although the isotropic EDQNM spectrum exhibits six decades of
inertial subrange,
the corresponding kinetic energy transfer plotted on figure \ref{fig7}
only exhibits a narrow plateau at zero (if at all) between the energy-losing large scales
and the energy-gaining small dissipative scales, also corresponding to
constant downscale spectral energy flux.
One should also note that statistical two-point models are by essence
better adapted to produce smooth spectral statistics than Direct Numerical Simulations.
First, the discretization of the spectral space can be adapted easily when
solving EDQNM equations, by adjusting both the wavenumber $k$ distribution
and its orientations, with the possibility of accumulating grid points towards
the equatorial or the polar directions. By construction, DNS is constrained to
a Cartesian uniform grid discretization of Fourier components.
Moreover, the recent experiment by Lamriben \textit{et al.} \cite{lamriben2011}
has shown that ensemble averaging requested to compute smooth third-order statistics
--- namely third-order structure functions or energy transfers ---
required hundreds of realizations, an issue which also applies to DNS realizations.
Since the statistical two-point model equations already concern the statistics
of the second- and third-order moments, explicit ensemble averaging is
self-contained in the model (as observed in the following smooth figures of results produced
by EDQNM).
The structure of decaying rotating turbulence has been described extensively from
experimental \cite{MOISY}, theoretical, and numerical results
\cite{cambon97,yoshimatsu_columnar_2011}.
Inertial wave turbulence was studied by Galtier \cite{galtierwave03}, and
Mininni \& Pouquet investigated energy and helicity spectra of rotating turbulence \cite{mininni09}, with different spectral scaling for each. The dynamics and
modified energy cascade produce an anisotropic structure which was
simulated with high resolution DNS also by Morinishi \textit{et al.}
\cite{morinishi_dynamics_2001}. Anomalous scaling of structure functions
in rotating turbulence also permitted Seiwert \textit{et al.} to investigate
the intermittency of rotating turbulence \cite{seiwert}.
The
interpretation of the dynamics leading to rotating homogeneous
turbulence structuration is still debated;
the two main viewpoints rely either on linear timescales of inertial waves, with the Coriolis
force acting on locally inhomogeneous structures that emit waves \cite{staplehurst08}, or
on a long-term nonlinear effect \cite{cambon97}. It is not the object of this paper to
reconcile these viewpoints, but, in short, both agree that in rotating turbulence
vortices are elongated along the axis of rotation.
The corresponding dynamics is that of preferential energy transfer towards
motion close to two-dimensional, although complete two-dimensionalization is not expected.
It also means that energy accumulates in orientation dependent kinetic energy
spectra $E(k,\theta)$ in the vicinity of the two-dimensional manifold $\theta\simeq\pi/2$ which is the neighborhood
of horizontal wavevectors in spectral space. This is confirmed by the angular
dependent energy spectrum plotted on the top panel of figure \ref{fig6}, which also compares well
with spectra processed from DNS of decaying rotating turbulence. Note that
the power-law scaling of the corresponding spectrum in figure \ref{fig5} is the
result of the spherical averaging over the directional spectra of figure \ref{fig6}, with
no single identical power-law applying to each individual one.
The bottom panel of figure \ref{fig6} presents a tentative measure of the anisotropy
in the spectra, by subtracting to the directional spectral the averaged spectrum. The curves
present only the limit spectra, equatorial or polar; also note that the anisotropy is \textit{reversed} between the rotating and the stratified cases, with more energy in the polar
direction (along the symmetry axis) in stratified turbulence, whereas the accumulation
of spectral energy in rotating turbulence is in the vicinity of the equatorial (perpendicular
to the symmetry axis) direction.
For the stably stratified case, the spectral structuration is reversed, such that
the energy accumulates in spectra of vertical $\bm{k}$ ($\theta\simeq 0$) as shown on figure
\ref{fig6} (see also \textit{e.g.} Lindborg \cite{lindborg06}). Again, the anisotropic statistical model compares extremely well
with experimental wind-tunnel observations and DNS \cite{godeferd2003}. These
comparisons not only concern dynamical quantities, such as energy or dissipation,
but also directional integral length scales, and the most refined possible comparison
between directional spectra $E(k,\theta)$, that is for every direction of every scale. In both the stratified
and the rotating cases, EDQNM$_2$ model predictions are very close to the
same spectral statistics extracted from DNS fields, down to the most refined
ones.
In physical space, the structuration of stably stratified turbulence
corresponds to a layering of the flow, with
strong horizontal motion, and vanishing vertical velocity, although
the flow retains a strong vertical \textit{variability} due to the presence
of large vertical gradients $\partial/\partial z$. \cite{yoon_evolution_1990,praud_decaying_2005}
The anisotropy observed in the spectra of figure \ref{fig6} is the departure
of each spectrum $E(k,\theta)$ around the average shown on figure \ref{fig5}.
Since the plot is in logscale, the clear difference between horizontal
wavenumber spectra and vertical ones demonstrates the large ratio between the
energy in vertical and horizontal motion. Moreover, one observes that in the
stratified turbulence spectrum the anisotropy is largest at the larger, energy-containing,
scales, although it extends to the middle of the inertial sub-range; in the
rotating turbulence spectrum, the anisotropy extends throughout the inertial
range down to the smallest scales. Of course, this observation depends
on the values of the Rossby and Froude numbers as well as on the Reynolds number.
The rotation (or stratification) timescale has to be compared to the
local timescale of a given turbulent structure associated with wavenumber
$k$ in order to assess whether its dynamics may be affected by the Coriolis
or the Boussinesq forces (for instance by computing a Zeman scale $(\Omega^3/\epsilon)^{-1/2}$ \cite{MININNI-ROSENBERG-POUQUET}, where $\epsilon$ is the dissipation, or similarly
an Ozmidov scale
for stratified turbulence by replacing $\Omega$ by $N$ \cite{Ozmidov}).
It nonetheless demonstrates that there exist parameter ranges at which turbulence
may be strongly affected by external distortions, and its structure and dynamics
can depart significantly from that of isotropic turbulence.
In addition to providing the second-order statistics corresponding to the two-point
energy spectra discussed above, the closure provides quantitative
information on third-order two-point correlation spectra, \textit{i.e.} the
kinetic energy transfer spectra shown
on figure \ref{fig7}. Apart from the Reynolds number difference,
the spherically averaged transfer spectra of anisotropic turbulence are
similar to the isotropic turbulence transfer. Note that we present
spectra of the kinetic energy transfer $T^e(k)$, which is only a part
of the energy transfers occurring in stratified turbulence, since there is
also coupling with the potential energy mode. As observed in the isotropic
case, in the anisotropic cases energy is drawn from the large scales and injected in
the small scales in
a classical forward cascade. In the mean time, if one observes the
orientation-dependent transfer spectra (not presented here), one also notices a re-distribution
of energy among different orientations of wavevectors, even at constant
wavenumber, \textit{i.e.} without interscale exchange.
\begin{figure}
\centering
\unitlength 1mm
\begin{picture}(50,70)
\put(-15,5){\includegraphics[width=0.5\textwidth]{e}}
\put(50,0){$k\eta$}
\put(-5,33){\rotatebox{90}{$E(k)$}}
\end{picture}
\caption{\label{fig5} Solid line: Kinetic energy spectrum $E(k)$ of isotropic turbulence
at $\textit{Re}^L=26\times 10^6$ given by the EDQNM closure model. Dot-dashed line: spherically accumulated
kinetic energy spectrum given by
the EDQNM$_2$ model for rotating turbulence. Dashed line: spectrum given
by the closure for stratified turbulence. The inertial range $k^{-5/3}$ power-law is also
shown on the figure, corresponding to the scaling of the isotropic spectrum. The spectra for rotating and stratified turbulence both scale like $k^{-1.9}$. The wavenumber $k$ is normalized by the inverse of the Kolmogorov
length scale $\eta$.}
\end{figure}
\begin{figure}
\centering
\unitlength 1mm
\begin{picture}(50,70)
\put(-15,5){\includegraphics[width=0.5\textwidth]{te}}
\put(50,0){$k\eta$}
\put(-5,33){\rotatebox{90}{$T(k)$, $T^e(k)$}}
\end{picture}
\caption{\label{fig7} Solid line: Kinetic energy transfer spectrum $T(k)$ of isotropic turbulence
given by the EDQNM closure model (scaled by $10^2$ to render it visible). Dot-dashed line: spherically accumulated
kinetic energy transfer spectrum $T^e(k)$ given by
the EDQNM$_2$ model for rotating turbulence. Dashed line: transfer spectrum given
by the closure for stratified turbulence. }
\end{figure}
\begin{figure}
\centering
\unitlength 1mm
\begin{picture}(50,70)
\put(-15,5){\includegraphics[width=0.5\textwidth]{ea}}
\put(50,0){$k\eta$}
\put(-5,33){\rotatebox{90}{$E(k,\theta)$}}
\end{picture}
\centering
\unitlength 1mm
\begin{picture}(50,70)
\put(-15,5){\includegraphics[width=0.5\textwidth]{estratrot}}
\put(50,0){$k\eta$}
\put(-5,33){\rotatebox{90}{$E(k,\theta=\pi/2,0)-E(k)$}}
\end{picture}
\caption{\label{fig6} Top panel: Orientation-dependent kinetic energy spectra $E(k,\theta)$ (
seen to be monotonically varying with $\theta$). Solid
lines: results from EDQNM$_2$ model for stratified turbulence; dashes: results from
the EDQNM$_2$ model for rotating turbulence, shifted down one decade. For each case,
a subset of the discretized orientations $\theta$ are presented. Spectra
corresponding to horizontal wavevectors are indicated by black dots.
Bottom panel: A measure of the anisotropy showing the difference between the polar and equatorial spectra with the spherically averaged one. Solid lines: EDQNM$_2$ result for stratified turbulence, dashed lines for the rotating results.
}
\end{figure}
\subsection{Structure functions}
\label{physres}
The above observations of second- and third-order correlations spectra
have a direct impact on the second- and third-order structure functions
of turbulence, according to the duality of spectral and physical space.
Some passage relations between the two spaces have been
presented in section \ref{sec1}. We now present the results obtained by using
these relations applied to the EDQNM closure results for isotropic and
anisotropic turbulence. It is however not possible to derive any $n^{th}$-order
structure function from spectral information contained in two-point statistical
closures, precisely because the spectral information is restricted to
$n^{th}$-order correlations at \textit{two} points. A case-by-case inspection is thus required.
The relation (\ref{eqqdeux}) linking the kinetic energy spectrum to the
second-order structure function is applied to the results obtained by EDQNM,
and also to the anisotropic turbulence results.
The numerical resolution of the isotropic EDQNM model directly provides
discretized distribution of $E(k)$ and $T(k)$. For the EDQNM$_2$, the spherically
averaged spectra $E(k)$ and $T(k)$ are obtained from the anisotropic
spectral data $E(k,\theta)$ and $T^e(k,\theta)$, by averaging over $\theta$.
Once these data are obtained, a simple numerical quadrature of the integrals in
equations (\ref{eqqdeux}) and (\ref{thstr}) is used to obtain $\langle\left( \delta u_i \delta u_i \right) \rangle$ and $\langle\delta u\left(\delta q\right)^2\rangle$ (likewise for $\langle(\delta u)^3\rangle$ with an integral not recalled here).
The corresponding structure
function is normalized by $(\varepsilon r)^{2/3}$ and plotted against
$r/\eta$ ($\eta$ is the Kolmogorov length scale) on figure \ref{fig1}, showing the
scaling which is obtained for isotropic turbulence, and illustrating the advantage
brought by achieving a high value of the Reynolds number. For rotating turbulence and
for stably stratified turbulence, the corresponding values of the Reynolds number
are lower, and one has to deal with a narrower curve. It is clear, however, that the
isotropic scaling is far from being applicable to the anisotropic turbulence structure functions,
since, here, if we compute $\textrm{max}(\langle \left( \delta u_i \delta u_i \right) \rangle/(\varepsilon r)^{2/3})$, one obtains 3.4 for the isotropic case, 4.2 for the stratified
case, 6.6 for the rotating case.
This is also observed on the curves for the third-order structure functions
plotted on figures \ref{fig2} and \ref{fig3} in two different forms. Figure \ref{fig2}
shows the averaged cubed longitudinal increment $\langle(\delta u)^3\rangle$ divided
by the behavior proposed in Kolmogorov theory: $-(\varepsilon r)$. The expected proportionality
constant $4/5$ is clearly recovered by the EDQNM closure transfer processed through
equation (\ref{eqdix}), with a nice plateau tangential to the $y=4/5$ horizontal line.
It should be noted that even at this very high value of the Reynolds number the
extent of the plateau is less than a decade in scale. This may explain part of the
early controversies about the precise value of the proportionality constant obtained
from experimental measurements. The same considerations apply for the other form of the
third-order structure function proposed by \cite{antonia97}, $\langle\delta u (\delta u_j\delta u_j)\rangle$, with a proportionality constant $4/3$ with $\varepsilon r$, plotted
on figure \ref{fig3} using equation (\ref{anton}). In both plots, the scaling is obviously
not the right one for
rotating turbulence or for stratified turbulence, although the order
of magnitude of the peaks of the structure functions for the anisotropic
turbulence cases are consistent with the isotropic
turbulence one. ($\textrm{max}(\langle\delta u (\delta u_j\delta u_j)\rangle)/(\varepsilon r))$
is marginally lower than 0.8 in the isotropic case, 0.39 for stratified results and 1.02 for
the rotating results.)
In rotating turbulence, the energy cascade is known to be reduced
by the presence of rotation, so that the dissipation is lower than for an equivalent
Reynolds number isotropic turbulence dynamics. This can explain the fact that
the peaks of $-\langle\delta u (\delta u_j\delta u_j)\rangle/(\varepsilon r)$ or of
$-\langle(\delta u)^3\rangle/(\varepsilon r)$ are higher than in isotropic turbulence.
In addition, one observes on figures \ref{fig2} and \ref{fig3} a negative excursion of the third-order
structure function in the very large scales, which could be attributed to a trend towards a
reverse cascade, although this has to be confirmed by other simulations.
In the stratified case, the trend is reversed, an effect which could be attributed
to augmented dissipation. The analysis is subtle here, since there is not only a
cascade of energy due to classical nonlinear turbulent interaction, but there is also
a coupling between the kinetic energy and the potential energy. It seems that the
balance between energy taken away from the velocity field for creating density
fluctuations is more than compensated by the increased dissipation from the
added scalar cascade. An attempt at including the part of dissipation arising
in the potential energy balance fails at correcting enough the scaling of the
third-order structure functions of stratified turbulence in figures \ref{fig2} and \ref{fig3}.
Finally, in order to address the Reynolds number dependence issue for
the third-order structure function convergence, we computed an isotropic turbulence
EDQNM result for a lower Reynolds number than proposed above, at $\textit{Re}=2600$,
of the same order of magnitude as those for the anisotropic EDQNM$_2$ data.
The solid curve of figure \ref{fig2} shows the corresponding result, which underestimates
significantly the 4/5 constant. The departure amplitude is similar to that of the anisotropic
data, although we believe it is of different nature. Following the analysis by Taylor \textit{et al.} \cite{taylorkurien}, there are therefore two sources of departure to the K41 scaling, one
due to low Reynolds number effect, the other to anisotropy. Here, for the anisotropic
runs, anisotropy is responsible for the departure from the asymptotic large Reynolds number
scaling, since for rotating turbulence it leads to an \textit{overestimate}, whereas for
stratified turbulence it is \textit{underestimated}. Also, the measures presented by
Antonia \& Burattini for isotropic turbulence exhibit a uniform convergence from \textit{below} the value 4/5 \cite{antonia06}. Thus, although there may be a finite
Reynolds number effect, we believe that, to first order, the difference with the
high Reynolds number isotropic scaling is due to anisotropy. This argument is
also supported by the curve for the second-order structure function in isotropic turbulence
at the lower Reynolds number $\textit{Re}=2600$ presented on figure~\ref{fig1}. It
shows that although the asymptotic large Reynolds number
scaling of is not yet reached, the maximum of
$\langle (\delta u_i\delta u_i)\rangle / (\varepsilon r)^{2/3}$ is within 30\%
of the asymptotic value, whereas the peak for rotating turbulence is twice
this value. The shape of the spectra and the difference in the downscale cascading rate
in the anisotropic runs with respect to the isotropic one can only explain this
departure.
\section{Conclusion and perspectives}
We have described in this work the spectral statistics of anisotropic turbulence,
with spectra of second- and third-order two-point correlations. These kinetic
energy spectra and kinetic energy transfer spectra are the result of the spherical
averaging of spectra of correlations taken with separations
along different angles with respect to the axis of symmetry, or in other words
for different orientations of the wavevectors. Not only these angle-dependent
spectra correspond to variable intensities of the energy, but their inertial range
scaling, when averaged, produces the power-law scaling of the overall kinetic energy spectrum.
Results obtained by a two-point statistical closure of EDQNM type show that,
at high Reynolds number, the resulting anisotropy can be strong, and corresponds
to a non classical structuration of the flow. The cases of strongly rotating turbulence
and strongly stratified turbulence were investigated, with respectively an elongation
of the vortices along the rotation axis or a horizontal layering.
The velocity components are consequently of variable magnitude in the vertical
and horizontal directions, so that an impact on the velocity second-order structure functions
is expected.
The access to the detailed energy transfers also permits the computation
of third-order structure functions, and the comparison with exact
scalings obtained for isotropic turbulence. As expected, the constants
obtained are different, although thanks to the relationships presented
in section \ref{sec1} one is able to closely related the anisotropic
spectral characterization to the structure functions, thus to link the dynamics
of energy cascade and inter-orientation transfer to the statistical
moments in physical space.
The departure from K41 theory for anisotropic
turbulence statistics of the second- and third-order structure
functions shows that
the K\`arm\`an-Howarth equation from which theoretical scalings are derived
exhibits a modified equilibrium when the structure of turbulence is anisotropic.
The correspondence with the Lin equation shows that the dynamics of anisotropic
turbulence is also modified, and we are able to trace the origin of the
modified energy spectra scalings to the transfer terms. Since these transfer
terms are accessible by spectral theory, this may be a way to improve
the prediction of the third-order structure function moment in rotating or
stratified turbulence, using two-point statistical closures.
Of course, this calls for additional work and further developments, including the derivation
of relationships equivalent to (\ref{eqqdeux}) and (\ref{anton}) but considering
the specific components of velocity parallel or orthogonal to the axis of symmetry.
In so doing, one will highlight the value of the anisotropic EDQNM$_2$ closure which
not only allows to model high Reynolds number turbulence but also provides
quantitative information on the anisotropy of statistics.
\medskip
This work is funded by the French \textit{Agence Nationale de la Recherche} under grant number
ANISO-340803. The reviewing work and suggestions of the Referees is also thankfully acknowledged.
\begin{figure}
\centering
\unitlength 1mm
\begin{picture}(50,70)
\put(-15,5){\includegraphics[width=0.5\textwidth]{deltaq2_Nov2011}}
\put(50,0){$r/\eta$}
\put(-5,33){\rotatebox{90}{$\langle (\delta u_i\delta u_i)\rangle / (\varepsilon r)^{2/3}$}}
\end{picture}
\caption{\label{fig1} Second-order structure function $\langle(\delta u_i\delta u_i)\rangle$ computed from (\ref{eqqdeux}) normalized
by $(\varepsilon r)^{2/3}$ as a function of the separation $r$ normalized by the Kolmogorov
length scale $\eta$.
Open squares: results from the EDQNM model for isotropic turbulence;
Filled squares: results from EDQNM$_2$ model for stratified turbulence;
Filled circles: results from EDQNM$_2$ model for rotating turbulence.
The dotted line presents isotropic turbulence EDQNM results at $\textit{Re}=2600$.}
\end{figure}
\begin{figure}
\centering
\unitlength 1mm
\begin{picture}(50,70)
\put(-15,5){\includegraphics[width=0.5\textwidth]{deltauL3_new}}
\put(50,0){$r/\eta$}
\put(-5,33){\rotatebox{90}{$-\langle(\delta u)^3\rangle/(\varepsilon r)$}}
\end{picture}
\caption{\label{fig2} Third-order structure function of the longitudinal
velocity increment $\langle(\delta u)^3\rangle$ normalized by $-(\varepsilon r)$ as a function
of the separation normalized by the Kolmogorov length scale. Same symbol convention as
in figure \ref{fig1}. The horizontal line exhibits the $4/5$ scaling. The dotted line presents isotropic turbulence EDQNM results at $\textit{Re}=2600$.}
\end{figure}
\begin{figure}
\centering
\unitlength 1mm
\begin{picture}(50,70)
\put(-15,5){\includegraphics[width=0.5\textwidth]{deltauLdeltaq2_new}}
\put(50,0){$r/\eta$}
\put(-5,33){\rotatebox{90}{$-\langle\delta u (\delta u_j\delta u_j)\rangle/(\varepsilon r)$}}
\end{picture}
\caption{\label{fig3} Third-order structure function $\langle\delta u (\delta u_j\delta u_j)\rangle$
as proposed by \cite{antonia97} normalized by $-(\varepsilon r)$ as a function
of the separation normalized by the Kolmogorov length scale. Same symbol convention as
in figure \ref{fig1}. The horizontal line exhibits the $4/3$ scaling.}
\end{figure}
\bibliographystyle{elsarticle_num}
|
1202.2103
|
\section{Introduction}
Let $S_i$ ($i=1,...,n$) be operators acting on a Hilbert space $\H$. The $n$-tuple $S=[S_1\,\cdots\,S_n]$ is said to be isometric
if it is an isometry from $\H^n$ to $\H$, equivalently,
\[
S_i^*S_j=\delta_{i,j}I\quad (i,j=1,...,n).
\]
On one hand, it is well known that the C*-algebra generated
by an isometric $n$-tuple $S$ is isomorphic to the Cuntz algebra $\O_n$ if $\sum_iS_iS_i^*=I$,
and isomorphic to the Cuntz-Toeplitz algebra ${\mathcal{E}}_n$ if $\sum_iS_iS_i^*<I$.
On the other hand, the unital norm closed (non-selfadjoint) operator algebra generated by $S$
is completely isometrically isomorphic to Popescu's non-commutative disk algebra ${\mathcal{A}}_n$ (cf. \cite{Pop96}).
Because of those rigidities,
Davidson and Pitts initiated the study of free semigroup algebras in \cite{DavPit99}
in order to study the fine structure of isometric $n$-tuples.
Since then, free semigroup algebras have attracted a great deal of attention.
See, for example, \cite{Dav2, Dav1, DavKatPit, DKS, DLP, DavPit, DavPit98, DavPit99,
DavWri, DavYang, Ken0, Ken}.
It should be mentioned that the structure of isometric tuples has been completely described
by Kennedy in \cite{Ken} very recently. It turns out that an isometric tuple has a higher-dimensional Lebesgue-von Neumann-Wold decomposition.
A free semigroup algebra ${\mathfrak{S}}$ is the unital $\textsc{wot}$-closed (non-selfadjoint) operator algebra generated by
an isometric tuple.
The prototypical example of free semigroup algebras is the non-commutative analytic Toeplitz algebra $\L_n$ generated
by the left regular representation of the free semigroup $\mathbb{F}_n^+$ with $n$ generators.
The algebra ${\L_n}$ was introduced in \cite{Pop91} by Popescu in connection with a non-commutative von Neumann inequality, and
plays a prominent role in the theory of free semigroup algebras.
It is singled out to name a class of free semigroup algebras analytic:
A free semigroup algebra ${\mathfrak{S}}$ generated by $S$ is said to be analytic if it is algebraically isomorphic to $\L_n$.
Analyticity plays a fundamental role in the existence of wandering vectors. Recall that a wandering vector for
an isometric tuple $S$ acting on $\H$ is a unit vector $\xi$ in $\H$ such that the set $\{S_w\xi:w\in\mathbb{F}_n^+\}$ is orthonormal.
Here $S_w=S_{i_1}\cdots S_{i_k}$ for a word $w=i_1\cdots i_k$ in $\mathbb{F}_n^+$.
It was conjectured in \cite{DavKatPit} that every
analytic free semigroup algebra has wandering vectors. This conjecture has been settled in \cite{Ken0} recently.
It is shown there that a free semigroup algebra either has a wandering vector, or is
a von Neumann algebra.
The non-commutative disk algebra ${\mathcal{A}}_n$ is the unital norm closed algebra generated by
the left regular representation of $\mathbb{F}_n^+$.
It was also introduced by Popescu in \cite{Pop91}.
He showed in \cite{Pop96} that the unital norm closed algebra ${\mathfrak{A}}_n$ generated by an arbitrary isometric $n$-tuple
is completely isometrically isomorphic to ${\mathcal{A}}_n$.
The algebra ${\mathcal{A}}_n$ also plays an important role in studying free semigroup algebras.
The free semigroup algebra ${\mathfrak{S}}$ generated by an isometric $n$-tuple $S$ is said to be absolutely continuous
if the representation of ${\mathcal{A}}_n$ induced by $S$ can be extended to a weak*-continuous representation of
$\L_n$. This notion was introduced in \cite{DLP} in order to obtain a natural analogue of
the measure-theoretic notion of absolute continuity, and played a central role there.
From the definitions and \cite[Theorem 1.1]{DavKatPit} (cf. Theorem \ref{T:DKP} below), one sees that analyticity
implies absolute continuity. The converse fails in the classical case
(i.e., when $n=1$). Refer to, for example, \cite{Ken, Wer}. However it holds true in higher-dimensional
cases \cite{Ken}.
Let ${\mathfrak{S}}$ be an analytic free semigroup algebra generated by an isometric $n$-tuple.
The main purpose of this paper is to explore richer structures of
${\mathfrak{S}}$ and its predual ${\mathfrak{S}}_*$ (recall from \cite{DavWri} that ${\mathfrak{S}}$ has a unique predual), so that
one could study those objects from other points of view. This could lead us to understand them better and deeper.
More precisely, we show that both ${\mathfrak{S}}$ and ${\mathfrak{S}}_*$ are Hopf algebras (Theorems \ref{T:abscon} and \ref{T:Conv}).
Moreover, the structures of ${\mathfrak{S}}$ and ${\mathfrak{S}}_*$ are closely connected with each other.
We prove that there is a bijection between
the set of completely bounded representations of ${\mathfrak{S}}_*$ and
the set of corepresentations of ${\mathfrak{S}}$ (Theorem \ref{T:corep}),
and that the following duality result holds true:
${\mathfrak{S}}$ can be recovered from the coefficient operators of completely bounded representations of ${\mathfrak{S}}_*$ (Theorem \ref{T:dual}).
Furthermore, as an amusing application of Theorem \ref{T:corep},
we show that the (Gelfand) spectrum of ${\mathfrak{S}}_*$ is precisely $\mathbb{F}_n^+$ (Theorem \ref{T:char}).
Rather surprisingly, according to the author's best knowledge,
the main results of this paper seem new even in the classical case,
that is, when ${\mathfrak{S}}$ is the analytic Toeplitz algebra.
It should be pointed out that this work connects with harmonic analysis.
It is shown in \cite{DavPit99, DavWri} that $\L_n$ has a unique predual
\[
{\L_n}_*=\{[\xi\eta^*]: \xi,\eta\in\ell^2(\Fn)\},
\]
where $[\xi\eta^*]$ is the rank one linear functional given by $[\xi\eta^*](A)=(A\xi, \eta)$ for all $A\in\L_n$.
If we regard $\L_n$ as a non-selfadjoint analogue of the group von Neumann algebra $\text{vN}({\mathbb{F}}_n)$ of the free group ${\mathbb{F}}_n$
with $n$ generators, then ${\L_n}_*$ is analogous to the Fourier algebra $A({\mathbb{F}}_n)$,
the predual of $\text{vN}({\mathbb{F}}_n)$. See, for example, \cite{Eym} for more information about Fourier algebras.
So one could think of ${\L_n}_*$ as a sort
of ``Fourier algebra of the free semigroup $\mathbb{F}_n^+$''.
To seek an algebra structure for ${\L_n}_*$ is actually the starting motivation of this work.
This point of view, in my opinion, could be fruitful in further studying free semigroup algebras.
In this paper, we shall see certain analogy between ${\L_n}_*$
and $A({\mathbb{F}}_n)$.
The rest of the paper is organized as follows. In Section \ref{S:Pre}, some preliminaries on free semigroup algebras,
dual algebras and operator theory are given. Since the non-commutative analytic algebras play a very
crucial role in our paper, we take a closer look at them and prove two main results
in Section \ref{S:Ln}. Firstly, we prove that $(\L_m\overline\otimes \L_n)_*$ and ${\L_m}_*\widehat\otimes {\L_n}_*$ are completely
isomorphically isomorphic. This isomorphism will be used to prove that ${\L_{n}}_*$ is a completely contractive Banach algebra.
Secondly, it is shown that for any $k\ge 1$, the isometric $n$-tuple $[L_1^{\otimes k}\, \cdots \, L_n^{\otimes k}]$
induced from the left regular representation of $\mathbb{F}_n^+$
is analytic (actually, more can be said). We will apply this result to construct a Hopf algebra structure for an arbitrary analytic free semigroup ${\mathfrak{S}}$.
In Section \ref{S:AnaHop}, we show that
every analytic free semigroup algebra ${\mathfrak{S}}$ is a Hopf dual algebra, while in Section \ref{S:HCA}, we prove that its predual ${\mathfrak{S}}_*$
is a Hopf convolution dual algebra. Those structures between ${\mathfrak{S}}$ and ${\mathfrak{S}}_*$ are tied by completely bounded representations
of ${\mathfrak{S}}_*$. In Section \ref{S:cbrep}, we show that, on one hand,
there is a one-to-one correspondence between the set of completely bounded representations of ${\mathfrak{S}}_*$ and
the set of corepresentations of ${\mathfrak{S}}$; and
on the other hand, ${\mathfrak{S}}$ can be recovered from the coefficient operators of completely bounded representations
of ${\mathfrak{S}}_*$. Moreover, it is shown that the (Gelfand) spectrum of ${\mathfrak{S}}_*$ is $\mathbb{F}_n^+$ if ${\mathfrak{S}}$ is
generated by an analytic isometric $n$-tuple.
\section{Preliminaries}
\label{S:Pre}
In this section, we will give some background which will be used later. Also we take this opportunity to
fix our notation.
\subsection*{Free semigroup algebras}
The material in this subsection is mainly from \cite{Dav2} and the references therein.
A \textit{free semigroup algebra} ${\mathfrak{S}}$ is the unital $\textsc{wot}$-closed operator algebra generated by
an isometric tuple.
Let us start with a distinguished representative of the class of those algebras --
the non-commutative analytic Toeplitz algebra $\L_n$.
Let $\mathbb{F}_n^+$ be the unital free semigroup
with the unit $\varnothing$ (the empty word), which is generated by the non-commutative symbols $1,...,n$. We form the Fock space
$\H_n:=\ell^2(\Fn)$ with the orthonormal basis $\{\xi_w: w\in\mathbb{F}_n^+\}$. Consider the left regular representation $L$
of $\mathbb{F}_n^+$: For $i=1,...,n$,
\[
L_i(\xi_w)=\xi_{iw} \quad (w\in \mathbb{F}_n^+).
\]
It is easy to see that $L=[L_1\,\cdots\,L_n]$ is isometric.
Then the free semigroup algebra generated by $L$ is denoted as $\L_n$,
which is known as the \textit{non-commutative analytic Toeplitz algebra}.
The case of $n=1$ yields the $\textsc{wot}$-closed algebra generated by the unilateral shift,
and so reproduces the (classical) analytic Toeplitz algebra.
Similarly, one can define the right regular representation $R$ of $\mathbb{F}_n^+$: For $i=1,...,n$,
\[
R_i(\xi_w)=\xi_{wi}\quad (w\in \mathbb{F}_n^+).
\]
Then by ${\mathcal{R}}_n$ we mean the free semigroup algebra generated by the isometric $n$-tuple $R=[R_1\,\cdots\, R_n]$.
It is easy to see that $\L_n$ and ${\mathcal{R}}_n$ are unitarily equivalent.
Moreover, it turns out that $\L_n$ and ${\mathcal{R}}_n$ are the commutants of each other:
${\mathcal{R}}_n=\L_n'$ and $\L_n={\mathcal{R}}_n'$.
As shown in \cite{DavPit}, every element $A\in\L_n$ is uniquely determined by its ``Fourier expansion''
\[
A\sim\sum_{w\in\mathbb{F}_n^+}a_wL_w
\]
in the sense that
\[
A\xi_w=\sum_{w\in\mathbb{F}_n^+}a_w\xi_w\quad (w\in\mathbb{F}_n^+),
\]
where the ``$w$-th Fourier coefficient'' is given by $a_w=(A\xi_\varnothing,\xi_w)$.
It is often useful to heuristically view $A$ as its Fourier expansion.
For $k\ge 1$, the $k$-th Ces\'aro sum of the Fourier series of $A$ is defined by
\[
\Sigma_k(A)=\sum_{|w|<k}(1-\frac{|w|}{k})a_wL_w,
\]
where $|w|$ is the length of the word $w$ in $\mathbb{F}_n^+$.
The sequence $\{\Sigma_k(A)\}_k$
converges to $A$ in the strong operator topology.
Another important operator algebra naturally associated with the left regular representation $L$ of $\mathbb{F}_n^+$ is
the \textit{non-commutative disk algebra} ${\mathcal{A}}_n$, which, by definition, is the unital norm closed algebra generated by $L$.
Clearly, ${\mathcal{A}}_n$ is weak*-dense in $\L_n$.
The case of $n=1$ yields the (classical) disk algebra.
The algebra ${\mathcal{A}}_n$ first appeared in \cite{Pop89}, which is related to multivariable non-commutative dilation theory.
It was shown in \cite{Pop96} that the unital norm closed operator algebra generated by an arbitrary isometric $n$-tuple
is completely isometrically isomorphic to ${\mathcal{A}}_n$.
Two classes of free semigroup algebras are particularly important and they are defined in terms of $\L_n$ and ${\mathcal{A}}_n$:
\begin{defn} (\cite{DavKatPit, DLP, Ken})
Let ${\mathfrak{S}}$ be a free semigroup algebra generated by an isometric $n$-tuple $S$.
\begin{itemize}
\item[(i)] ${\mathfrak{S}}$ is said to be \textit{analytic} if it is algebraically isomorphic to $\L_n$.
\item[(ii)] ${\mathfrak{S}}$ is said to be \textit{absolutely continuous}
if the representation of ${\mathcal{A}}_n$ induced by $S$ can be extended to a weak*-continuous representation of
$\L_n$.
\end{itemize}
Sometimes, we also call the isometric tuple $S$ \textit{analytic} in (i) and \textit{absolute continuous} in (ii), respectively.
\end{defn}
In \cite{DavKatPit}, analytic free semigroup algebras are called \textit{type L}. The notion used here follows \cite{Ken}.
Analyticity plays a fundamental role in the existence of wandering vectors.
In \cite{DavKatPit} it was conjectured that every
analytic free semigroup algebra has wandering vectors. Recently, this conjecture has been settled in \cite{Ken0}.
It is shown there that a free semigroup algebra either has a wandering vector, or is
a von Neumann algebra.
Motivated by the classical case, the notion of absolute continuity was introduced by Davidson-Li-Pitts in \cite{DLP},
and provided a major device there. By Theorem \ref{T:DKP} below and the definitions, analyticity implies absolute continuity.
It turns out that these two notions coincide in higher-dimensional cases \cite{Ken}.
However, this is not true in the classical case (cf. \cite{Ken, Wer}).
In what follows, we record some results of analytic free semigroup algebras for later reference.
Recall that a weak*-closed operator algebra ${\mathcal{A}}$ on $\H$ has \textit{property ${\mathbb{A}}_1(1)$}, if
given a weak*-continuous linear functional $\tau$ on ${\mathfrak{S}}$ and $\epsilon>0$, there
are vectors $x,y\in \H$ such that $\|x\| \|y\|\le \|\tau\|+\epsilon$ and $\tau=[xy^*]$.
In particular, the weak* and weak topologies on such an algebra coincide.
\begin{thm} \rm{(\cite{DavKatPit})}
\label{T:DKP}
If ${\mathfrak{S}}$ is an analytic free semigroup algebra generated by an isometric $n$-tuple $S=[S_1\, \cdots \, S_n]$,
then there is a canonical weak*-homeomorphic and completely isometric isomorphism
$\phi$ from ${\mathfrak{S}}$ onto $\L_n$, which maps $S_i$ to $L_i$ for $i=1,...,n$.
\end{thm}
\begin{thm}
\label{T:ww} \rm{(\cite{Ber1} for $n=1$ and \cite{Ken} for $n\ge 2$)}
Any analytic free semigroup algebra ${\mathfrak{S}}$ has property ${\mathbb{A}}_1(1)$.
\end{thm}
\subsection*{Dual algebras and operator spaces}
The main sources of this subsection are \cite{ER, ER1, Ruan}.
In this paper, a \textit{dual algebra} is a \textit{unital} weak*-closed subalgebra of ${\mathcal{B}}(\H)$ for some Hilbert space $\H$.
Usually, this is called a (unital) \textit{concrete dual algebra}.
There is a characterization for abstract dual algebras, and
it turns out that every abstract dual algebra has a concrete realization.
We will not need those facts in this paper.
Observe that free semigroup algebras are dual algebras.
Let ${\mathcal{A}}\subseteq{\mathcal{B}}(\H)$ be a dual algebra. Then ${\mathcal{A}}$ has the standard predual ${\mathcal{B}}(\H)_*/{{\mathcal{A}}_\perp}$, where,
as usual, ${\mathcal{B}}(\H)_*$ is the space of all bounded weak*-continuous
linear functionals on ${\mathcal{B}}(\H)$, and ${\mathcal{A}}_\perp$ is the preannihilator of ${\mathcal{A}}$ in ${\mathcal{B}}(\H)_*$.
Of course, ${\mathcal{A}}$ may have more than one predual. We will use the notation ${{\mathcal{A}}_\H}_*$, or simply ${\mathcal{A}}_*$ if the context is clear, to denote
its standard predual. This will not cause any confusion in the context of
free semigroup algebras because of the uniqueness of their preduals (cf. \cite{DavWri}).
It is very useful to notice that dual algebras are intimately connected with the theory of operator spaces.
Let $V$ be an operator space. Then its dual $V^*$ is
also an operator space, which is called the \textit{operator space dual} of $V$.
An operator space $W$ is said to be a \textit{dual operator space} if $W$ is completely isometrically isomorphic to $V^*$ for some operator space $V$. In this case, we also say that $V$ is an \textit{operator predual} of $W$.
The normal spatial tensor product of dual operator spaces (resp. algebras) is again a dual operator space (resp. algebra).
Any dual algebra ${\mathcal{A}}$ is a dual operator space.
Furthermore, ${\mathcal{A}}_*$ inherits a natural operator space structure
from ${\mathcal{A}}^*$, and so ${\mathcal{A}}_*$ itself is an operator space.
In fact, this paper heavily relies on the operator space structure of ${\mathcal{A}}_*$.
Given dual operator spaces $V^*$ and $W^*$ in ${\mathcal{B}}(\H)$ and ${\mathcal{B}}({\mathcal{K}})$, respectively,
the \textit{normal Fubini tensor product} $V^*\overline\otimes_{\mathcal{F}} W^*$ is defined by
\begin{align*}
V^*\overline\otimes_{\mathcal{F}} W^*=\{A\in{\mathcal{B}}(\H\otimes {\mathcal{K}}): \ & (\omega_1\otimes {\operatorname{id}})(A)\in W^*, ({\operatorname{id}}\otimes \omega_2)(A)\in V^*\\
&\mbox{for all}\ \omega_1\in {\mathcal{B}}(\H)_*, \ \omega_2\in{\mathcal{B}}({\mathcal{K}})_*\},
\end{align*}
where $\omega_1\otimes {\operatorname{id}}$ and ${\operatorname{id}}\otimes \omega_2$ are the right and left slice mappings determined
by $\omega_1$ and $\omega_2$, respectively.
Let $V^*\overline\otimes W^*$ and $V\widehat\otimes W$ stand for the normal spatial tensor product of $V^*$ and $W^*$,
and the operator projective tensor product of $V$ and $W$, respectively.
Since $\omega_1\otimes {\operatorname{id}}$ and ${\operatorname{id}}\otimes \omega_2$
are weak*-continuous, we have $V^*\overline\otimes W^*\subseteq V^*\overline\otimes_{\mathcal{F}} W^*$.
The following result characterizes when they are equal.
\begin{thm}
\label{T:ERR}
\rm{(\cite{ER1, Ruan})}
(i) We have the following weak*-homeomorphic completely isometric isomorphism:
\[
V^*\overline\otimes_{\mathcal{F}} W^*\cong(V\widehat\otimes W)^*.
\]
(ii) $V^*\overline\otimes_{\mathcal{F}} W^* = V^*\overline\otimes W^*$ if and only if $(V^*\overline\otimes W^*)_*\cong V\widehat\otimes W$.
\end{thm}
\smallskip
\subsection*{Notation}
For a Hilbert space $\H$, we use $\H^k$ and $\H^{\otimes k}$ to denote the direct sum of $k$ copies of $\H$ and
the tensor product of $k$ copies of $\H$, respectively. For $A\in{\mathcal{B}}(\H)$, by $A^{(k)}\in{\mathcal{B}}(\H^{k})$ and $A^{\otimes k}\in{\mathcal{B}}(\H^{\otimes k})$,
we mean the direct sum of $k$ copies of $A$ acting on $\H^k$ and the tensor product of $k$ copies of $A$ acting on $\H^{\otimes k}$, respectively.
Let $S=[S_1\,\cdots\,S_n]$ be an arbitrary isometric $n$-tuple acting on $\H$.
We think of $S$ either as $n$ isometries acting on the common Hilbert space $\H$ or as an isometry from $\H^n$ to $\H$.
Also, for $k\ge 1$, we set
$S^{\otimes k}= [S_1^{\otimes k}\,\cdots \, S_n^{\otimes k}]$.
Let ${\mathcal{A}}$ and ${\mathcal{B}}$ be two dual algebras. We use
${\mathcal{A}}\overline\otimes{\mathcal{B}}$ to denote the normal spatial tensor product of ${\mathcal{A}}$ and ${\mathcal{B}}$.
\section{$\L_n$ revisited}
\label{S:Ln}
In this section, we will take closer look at the non-commutative analytic Toeplitz algebras, and
prove some results which are vital in constructing Hopf algebra structures of an analytic free semigroup algebra
and its predual.
By $\L_m\overline\otimes^{\rm w} \L_n$, we mean the
$\textsc{wot}$-closed algebra generated by spatial tensor product of $\L_m$ and $\L_n$.
The notation ${\mathcal{R}}_m\overline\otimes^{\rm w} {\mathcal{R}}_n$ is defined similarly.
It is easy to see that ${\mathcal{R}}_m\overline\otimes^{\rm{w}} {\mathcal{R}}_n\subseteq (\L_m\overline\otimes^{\rm{w}} \L_n)'$.
Then, by \cite[Theorem 4.3]{Berc}, $\L_m\overline\otimes^{\rm{w}} \L_n$
has property ${\mathbb{A}}_1(1)$ if $m>1$ or $n>1$, so that the weak* and $\textsc{wot}$ topologies on it coincide.
This is also the case if $m=n=1$ (cf., e.g., \cite{BerWes}).
Thus $\L_m\overline\otimes^{\rm{w}} \L_n=\L_m\overline\otimes\L_n$ for all $m,n\in{\mathbb{N}}$. By \cite{KriPow},
$\L_m\overline\otimes\L_n$ can be identified with the higher rank analytic Toeplitz operator algebra $\L_\Lambda$
associated with a simple rank 2 graph $\Lambda$.
Using a general result on rank 2 graphs there, one has the following result (refer to \cite[Section 1 and Section 3]{KriPow}).
\begin{lem}
\label{L:com}
$
(\L_m\overline\otimes \L_n)'={\mathcal{R}}_m\overline\otimes {\mathcal{R}}_n \quad\mbox{and}\quad (\L_m\overline\otimes \L_n)''=\L_m\overline\otimes \L_n.
$
\end{lem}
In what follows, we identify the standard predual of $\L_m\overline\otimes \L_n$.
\begin{prop}
\label{P:pretenpro}
We have the following completely isometric isomorphism
\begin{align}
\label{E:lmln}
(\L_m\overline\otimes \L_n)_* & \cong {\L_m}_*\widehat\otimes {\L_n}_*.
\end{align}
\end{prop}
\begin{proof}
Although \cite[Theorem 7.2.4]{ER} handles the selfadjoint dual algebras (i.e., von Neumann algebras),
the proof of our proposition follows the same line there. For self-containedness, it is also included here.
Clearly,
it suffices to show that $\L_m\overline\otimes_{\mathcal{F}} \L_n\subseteq \L_m \overline\otimes \L_n$.
By Lemma \ref{L:com}, we just need to check that each $T\in \L_m\overline\otimes_{\mathcal{F}} \L_n$ commutes with all operators in the
commutant $(\L_m \overline\otimes \L_n)'={\mathcal{R}}_m\overline\otimes {\mathcal{R}}_n$.
Thus it is sufficient to show that
\[
T(I\otimes B)=(I\otimes B)T \quad (B\in{\mathcal{R}}_n)
\]
and
\[
T(A\otimes I)=(A\otimes I)T\quad (A\in{\mathcal{R}}_m).
\]
For the first identity, notice the following identification (\cite{ER})
\begin{align*}
\pi: {\mathcal{B}}(\H_m \otimes \H_n)&\cong {\mathcal{C}}{\mathcal{B}}({\mathcal{B}}(\H_m)_*, {\mathcal{B}}(\H_n))\\
X&\mapsto \pi(X): \omega_1\mapsto (\omega_1\otimes {\operatorname{id}})(X).
\end{align*}
So it suffices to show that
\[
(\omega_1\otimes {\operatorname{id}})(T(I\otimes B))=(\omega_1\otimes {\operatorname{id}})((I\otimes B)T)
\]
for all $w_1\in {{\mathcal{B}}(\H_m)}_*$, namely,
\[
(\omega_1\otimes {\operatorname{id}})(T)B=B(\omega_1\otimes {\operatorname{id}})(T).
\]
The above identity holds true since
$T\in \L_m\overline\otimes_{\mathcal{F}} \L_n$ implies that $(\omega_1\otimes {\operatorname{id}})(T)\in \L_n$.
The second identity can be proved similarly.
\end{proof}
Before giving our next result, let us recall that, for $1\le k \le \infty$, an isometric $n$-tuple $S$
is said to be \textit{pure of multiplicity $k$} if $S$ is unitarily equivalent to $L^{(k)}$.
In particular, any pure isometric tuple is analytic.
Also, a subspace ${\mathcal{W}}$ is \textit{wandering for an isometric $n$-tuple $S$} if the subspaces $\{S_u{\mathcal{W}} : u\in \mathbb{F}_n^+\}$ are pairwise orthogonal.
\begin{prop}
\label{P:LiLi}
Let $L$ be the left regular representation of $\mathbb{F}_n^+$. Then
for any $k>1$, $L^{\otimes k}$ is a pure isometric $n$-tuple with multiplicity $\infty$.
\end{prop}
\begin{proof}\footnote{I am indebted to Adam Fuller for showing me this proof. }
Obviously, $L^{\otimes k}$ is an isometric $n$-tuple.
In what follows, we prove that $L^{\otimes k}$ is pure of multiplicity $\infty$.
Set
\[
{\mathcal{K}}=\overline\operatorname{span}\{\xi_{u_1}\otimes\cdots\otimes \xi_{u_k}: u_1,...,u_n \mbox{ have no common prefix}\}.
\]
That is, ${\mathcal{K}}$ is spanned by
the basis vectors $\xi_{u_1}\otimes\cdots\otimes \xi_{u_k}$, where
$u_i=\varnothing$ for \textit{some} $i\in\{1,...,k\}$,
or $u_i=k_iu_i'$ with $k_i\ne k_j$ for \textit{some} $i\ne j$.
It is easy to check that ${\mathcal{K}}$ is wandering for $L^{\otimes k}$. Suppose now $\xi_{u_1}\otimes\cdots\otimes \xi_{u_k}$
is an arbitrary basis vector for $\H_n^{\otimes k}$ with $\xi_{u_1}\otimes\cdots\otimes \xi_{u_k}\not\in {\mathcal{K}}$.
One can choose a word $w\in \mathbb{F}_n^+$ of maximal
length such that $u_i=wu_i'$ for some $u_i'\in\mathbb{F}_n^+$ $(i=1,...,k)$. Notice that $w\neq \varnothing$ as $\xi_{u_1}\otimes\cdots\otimes \xi_{u_k}\not\in {\mathcal{K}}$.
On the other hand, since $w$ is maximal, it follows that $\xi_{u_1}'\otimes\cdots\otimes \xi_{u_k}'\in {\mathcal{K}}$. Clearly,
\[
\xi_{u_1}\otimes\cdots\otimes \xi_{u_k}=L_w^{\otimes k}(\xi_{u_1}'\otimes\cdots\otimes \xi_{u_k}').
\]
Thus, for any basis vector $\xi_{u_1}\otimes\cdots\otimes \xi_{u_k}\in \H_n^{\otimes k}$,
we have
\[
\xi_{u_1}\otimes\cdots\otimes \xi_{u_k}\in\bigoplus_{w\in\mathbb{F}_n^+}L_w^{\otimes k}\, {\mathcal{K}},
\]
and so
\[
\H_n^{\otimes k}=\bigoplus_{w\in\mathbb{F}_n^+} L_w^{\otimes k}\,{\mathcal{K}}.
\]
Obviously, $\dim{\mathcal{K}}=\infty$ as $k>1$. Therefore, it follows from \cite{Pop89} that $L^{\otimes k}$ is a pure isometric tuple with multiplicity $\infty$.
\end{proof}
As an immediately consequence of the above proposition, we have that $L^{\otimes k}$ is analytic for all $k\in{\mathbb{N}}$.
\section{Analytic free semigroup algebras are Hopf algebras}
\label{S:AnaHop}
Following Effros-Ruan in \cite{ER1}, we define the term Hopf algebras from analysts' point of view as follows.
A \textit{Hopf algebra} $({\mathcal{A}}, m, \Delta)$ consists of a linear
space ${\mathcal{A}}$ with norms or matrix norms, an associative bilinear multiplication $m: {\mathcal{A}}\times {\mathcal{A}}\to {\mathcal{A}}$,
and a coassociative comultiplication $\Delta: A\to {\mathcal{A}}\widetilde{\otimes} {\mathcal{A}}$, where
$\widetilde\otimes$ is a suitable tensor product, and $\Delta$ is an algebra homomorphism. The maps
are assumed to be bounded in some appropriate sense.
In this paper, we are interested in two classes of Hopf algebras -- Hopf dual algebras and Hopf convolution dual algebras, where
the latter is induced from the former. The former is investigated in this section, and the latter will
be studied in next section. Some definitions are first.
\begin{defn}
\label{D:coalg}
(i) A dual algebra ${\mathcal{A}}$ is said to be a \textit{Hopf dual algebra} if there is an injective unital
weak*-continuous completely contractive
homomorphism $\Delta:{\mathcal{A}}\to {\mathcal{A}}\overline\otimes {\mathcal{A}}$
such that it is also coassociative:
\[
({\operatorname{id}}\otimes \Delta)\Delta=(\Delta\otimes {\operatorname{id}})\Delta.
\]
That is, the following diagram commutes:
\[
\begin{xy}
(0,20)*+{{{\mathcal{A}}}}="a"; (35,20)*+{{{\mathcal{A}}\overline\otimes{\mathcal{A}}}}="b";%
(0,0)*+{{{\mathcal{A}}}\overline\otimes{{\mathcal{A}}}}="c"; (35,0)*+{{\mathcal{A}}\overline\otimes{\mathcal{A}}\overline\otimes{\mathcal{A}}.}="d";%
{\ar@{->} "a";"b"};?*!/_2mm/{\Delta};
{\ar@{->} "a";"c"};?*!/^/{\Delta};
{\ar@{->} "b";"d"};?*!/_7mm/{\Delta\otimes{\operatorname{id}}};
{\ar@{->} "c";"d"};?*!/^2mm/{{\operatorname{id}}\otimes\Delta},
\end{xy}
\]
The homomorphism $\Delta$ is called a \textit{comultiplication}
or \textit{coproduct} on ${\mathcal{A}}$.
(ii) A Hopf dual algebra is \textit{counital} if there is a non-zero weak*-continuous homomorphism $\epsilon:{\mathcal{A}}\to{\mathbb{C}}$ such that
\[
(\epsilon\otimes{\operatorname{id}})\circ\Delta={\operatorname{id}}=({\operatorname{id}}\otimes\epsilon)\circ\Delta.
\]
The homomorphism $\epsilon$ is
called a \textit{counit} of ${\mathcal{A}}$.
(iii) A Hopf dual algebra ${\mathcal{A}}$ is said to be \textit{cocommutative} if
$\sigma\circ \Delta=\Delta$, where $\sigma$ is the flip map $a\otimes b\mapsto b\otimes a$.
\end{defn}
We shall use the pair $({\mathcal{A}}, \Delta)$, or $({\mathcal{A}}, \Delta_{\mathcal{A}})$ if ${\mathcal{A}}$ needs to be stressed, or simply ${\mathcal{A}}$ if the context is clear,
to denote the Hopf dual algebra ${\mathcal{A}}$ with the comultiplication $\Delta$.
\begin{defn}
\label{D:mor}
A \textit{morphism of two Hopf dual algebras} $({\mathcal{A}}, \Delta_{\mathcal{A}})$ and $({\mathcal{B}}, \Delta_{\mathcal{B}})$ is a unital weak*-continuous
completely contractive homomorphism $\pi:{\mathcal{A}}\to {\mathcal{B}}$ that makes $\Delta_{\mathcal{A}}$ and $\Delta_{\mathcal{B}}$ compatible in the following sense:
\[
\Delta_{\mathcal{B}}\circ \pi=(\pi \otimes \pi)\circ \Delta_{\mathcal{A}}.
\]
If $\pi$ is weak*-weak* homeomorphic and completely isometrical isomorphic, then ${\mathcal{A}}$ and ${\mathcal{B}}$ are said to be \textit{isomorphic as dual Hopf algebras}, and denoted
as $({\mathcal{A}},\Delta_{\mathcal{A}})\cong({\mathcal{B}},\Delta_{\mathcal{B}})$.
\end{defn}
\begin{defn}
\label{D:int}
Let $({\mathcal{A}}, \Delta)$ be a Hopf dual algebra. A unital weak*-continuous linear functional $\varphi:{\mathcal{A}}\to {\mathbb{C}}$ is
called a \textit{left} (resp. \textit{right}) \textit{integral} on $({\mathcal{A}},\Delta)$ if it is
left-invariant (resp. right-invariant), that is,
\[
({\operatorname{id}}\otimes \varphi)(\Delta(a))=\varphi(a)I\quad (\mbox{resp. }(\varphi\otimes {\operatorname{id}})(\Delta(a))=\varphi(a)I).
\]
A left and right integral is briefly called an \textit{integral}.
\end{defn}
Some remarks for the above definitions are in order.
\begin{rem}
\label{R:sim1}
Note that if ${\mathcal{A}}$ in Definition \ref{D:coalg} is selfadjoint (i.e., ${\mathcal{A}}$ is a von Neumann algebra), then $({\mathcal{A}}, \Delta)$ is nothing but a Hopf
von Neumann algebra (\cite{Enock}). This is due to the fact that a unital completely contractive homomorphism between two
C*-algebras is automatically a *-homomorphism (\cite{Pau}).
So, as one expects, the notions of Hopf dual and Hopf von Neumann algebras coincide in the selfadjoint case.
Then, in this case, the notion of morphisms in Definition \ref{D:mor} is the same as the one for Hopf von Neumann algebras.
\end{rem}
\begin{rem}
\label{R:sim3}
All tensor maps involved in Definitions \ref{D:coalg}, \ref{D:mor} and \ref{D:int}
have unique weak*-continuous extensions. For example, $\Delta\otimes{\operatorname{id}}$ has a unique weak*-continuous extension,
still denoted by $\Delta\otimes{\operatorname{id}}$, which maps ${\mathcal{A}}\overline\otimes{\mathcal{A}}$ to ${\mathcal{A}}\overline\otimes{\mathcal{A}}\overline\otimes{\mathcal{A}}$.
This can be seen, e.g., from \cite{ER1} (also cf. the proof of Theorem \ref{T:abscon} below).
\end{rem}
\begin{rem}
\label{R:sim2}
Notice that integrals in Definition \ref{D:int} are normalized.
It is not hard to check that integrals of a Hopf dual algebra are unique.
If ${\mathcal{A}}$ is a von Neumann algebra, then an integral is nothing but a Haar state.
\end{rem}
We are now ready to give the first main theorem in this section. It seems to have been overlooked in the classical case.
\begin{thm}
\label{T:coalg}
$(\L_n,\Delta)$ is a cocommutative Hopf dual algebra with the integral $\varphi_0=[\xi_\varnothing\xi_\varnothing^*]$,
where $\Delta$ is determined by $L_i\mapsto L_i\otimes L_i$ for $i=1,...,n$.
\end{thm}
\begin{proof}
Since $L^{\otimes 2}$ is an isometric $n$-tuple,
there is a representation
$\Delta$ of the non-commutative disk algebra ${\mathcal{A}}_n$ determined by
\[
\Delta: {\mathcal{A}}_n\to \L_n\overline\otimes \L_n, \quad L_i\mapsto L_i\otimes L_i \quad (i=1,...,n).
\]
By Proposition \ref{P:LiLi}, $L^{\otimes 2}$ is analytic and so absolutely continuous. Thus
$\Delta$ can be extended to a weak*-continuous representation of $\L_n$, which is still denoted as $\Delta$ by
abusing notation:
\[
\Delta: \L_n\to \L_n\overline\otimes \L_n, \quad L_i\mapsto L_i\otimes L_i\quad (i=1,...,n).
\]
Let $A\sim\sum_wa_wL_w$ be an arbitrary element in $\L_n$. Then
\begin{align}
\nonumber
(\Delta\otimes{\operatorname{id}})\circ\Delta(A)
&=\Delta\otimes{\operatorname{id}}(\Delta(\textsc{sot}\!\!-\!\!\lim_k(\Sigma_k(A))))\\
\nonumber
&=\Delta\otimes{\operatorname{id}}(\Delta(\textsc{wot}\!\!-\!\!\lim_k(\Sigma_k(A))))\\
\nonumber
&=\Delta\otimes{\operatorname{id}}(\Delta(\mbox{weak*}\!\!-\!\!\lim_k(\Sigma_k(A))))\\
\nonumber
&=\mbox{weak*}\!\!-\!\!\lim_k\Delta\otimes{\operatorname{id}}(\Delta(\Sigma_k(A)))\\
\nonumber
&=\mbox{weak*}\!\!-\!\!\lim_k\sum_{|w|<k}(1-\frac{|w|}{k})a_w\Delta\otimes{\operatorname{id}}(L_w\otimes L_w)\\
\nonumber
&=\mbox{weak*}\!\!-\!\!\lim_k\sum_{|w|<k}(1-\frac{|w|}{k})a_w(L_w\otimes L_w\otimes L_w)\\
\label{E:oper}
&=\textsc{wot}\!\!-\!\!\lim_k\sum_{|w|<k}(1-\frac{|w|}{k})a_w(L_w\otimes L_w\otimes L_w),
\end{align}
where the above third ``='' uses the fact that $\textsc{wot}$ and weak* topologies coincide on $\L_n$ by Theorem \ref{T:ww},
the fourth ``='' is from the weak*-continuity of $\Delta\otimes {\operatorname{id}} $ and $\Delta$,
and the last second one comes from the definition of $\Delta$.
Some obviously slight changes of the above proof yield
\[
(\Delta\otimes{\operatorname{id}})\circ\Delta(A)
=\textsc{wot}\!\!-\!\!\lim_k\sum_{|w|<k}(1-\frac{|w|}{k})a_w(L_w\otimes L_w\otimes L_w).
\]
Thus $\Delta$ is coassociative.
Similarly, one can show the cocommutativity of $\L_n$.
Thus $\L_n$ is a cocommutative Hopf dual algebra.
Finally, we verify that $\varphi_0=[\xi_\varnothing \xi_\varnothing^*]$ is the integral of $\L_n$.
Clearly $\varphi_0$ is a weak*-continuous linear functional and $\varphi_0(I)=1$.
Then using some calculations similar to the above, one obtains
\[
(\varphi_0\otimes {\operatorname{id}})\circ \Delta(A)=\varphi_0(A)I=({\operatorname{id}}\otimes \varphi_0)\circ \Delta(A)
\]
for all $A\in \L_n$. Therefore, $\varphi_0$ is the integral on $\L_n$.
\end{proof}
\begin{rem}
(i) By Proposition \ref{P:LiLi}, $L^{\otimes 3}$ is an analytic isometric $n$-tuple.
From the above proof, one actually obtains that the map
$(\Delta\otimes{\operatorname{id}})\circ\Delta$, or $({\operatorname{id}}\otimes\Delta)\circ\Delta$,
is nothing but the representation of $\L_n$ extended from that of ${\mathcal{A}}_n$ induced by $L^{\otimes 3}$:
\begin{align*}
(\Delta\otimes{\operatorname{id}})\circ\Delta=({\operatorname{id}}\otimes\Delta)\circ\Delta: \L_n&\to \L_n\overline\otimes \L_n\overline\otimes\L_n\\
L_i&\mapsto L_i^{\otimes 3}\quad (i=1,...,n).
\end{align*}
Also, the operator in \eqref{E:oper}
is the one with the Fourier expansion
$\sum_w a_w(L_w\otimes L_w\otimes L_w)$.
(Refer to the proof of Theorem \ref{T:char} below for some related details.)
(ii) It is worth noticing that, in the free group case with $n\ge 2$, the above integral $\varphi_0$ is the canonical faithful trace of the
group von Neumann algebra $\text{vN}({\mathbb{F}}_n)$, which is a II$_1$ factor.
\end{rem}
\medskip
One can now generalize the above result to an arbitrary analytic isometric tuple.
\begin{thm}
\label{T:abscon}
Suppose that ${\mathfrak{S}}$ is an analytic free semigroup algebra generated by an isometric $n$-tuple $S=[S_1\, \cdots \, S_n]$. Then
$({\mathfrak{S}},\Delta_{\mathfrak{S}})$ is a cocommutative Hopf dual algebra with an integral,
where $\Delta_{\mathfrak{S}}$ maps $S_i$ to $S_i\otimes S_i$ for all $1\le i\le n$.
\end{thm}
\begin{proof}
Let $\phi:{\mathfrak{S}}\to \L_n$ be the canonical weak*-homeomorphic completely isometric isomorphism.
From \cite[Corollary 4.1.9]{ER}, we have that
$\phi_*:{\L_n}_*\to {\mathfrak{S}}_*$ is completely isometrically isomorphic, and so is
$\phi_*\otimes \phi_*: {\L_n}_*\widehat\otimes {\L_n}_*\to {\mathfrak{S}}_*\widehat\otimes{\mathfrak{S}}_*$
by \cite[Proposition 7.1.7]{ER}.
Applying \cite[Corollary 4.1.9]{ER} again gives that
$(\phi_*\otimes \phi_*)^*:({\mathfrak{S}}_*\widehat\otimes{\mathfrak{S}}_*)^*\to ({\L_n}_*\widehat\otimes {\L_n}_*)^*$
is a weak*-homeomorphic completely isometric isomorphism.
Hence $(\phi_*\otimes \phi_*)^*$ is a weak*-homeomorphic completely isometric isomorphism
between ${\mathfrak{S}}\overline\otimes_{\mathcal{F}}{\mathfrak{S}}$ and $\L_n\overline\otimes_{\mathcal{F}}\L_n$ by Theorem \ref{T:ERR}.
But it follows from Theorem \ref{T:ERR} and Proposition \ref{P:pretenpro} that $\L_n\overline\otimes\L_n=\L_n\overline\otimes_{\mathcal{F}}\L_n$.
Hence one has
\begin{align}
\label{E:SS}
{\mathfrak{S}}\overline\otimes{\mathfrak{S}}={\mathfrak{S}}\overline\otimes_{\mathcal{F}}{\mathfrak{S}}.
\end{align}
Let $\Phi$ be the inverse of $(\phi_*\otimes \phi_*)^*$ (which is actually the extension of $\phi^{-1}\otimes\phi^{-1}$).
Then $\Phi$ is a weak*-homeomorphic completely isomorphism from $\L_n\overline\otimes\L_n$ onto ${\mathfrak{S}}\overline\otimes{\mathfrak{S}}$.
Set $\Delta_{\mathfrak{S}}:=\Phi\circ\Delta\circ\phi$, where $\Delta$ is the comultiplication on $\L_n$ given in Theorem \ref{T:coalg}.
By \cite{DavYang}, $\Delta$ is weak*-continuous completely isometrically homomorphic.
Then $\Delta_{\mathfrak{S}}: {\mathfrak{S}}\to {\mathfrak{S}}\overline\otimes{\mathfrak{S}}$ is a unital weak*-continuous completely isometric homomorphism. Also,
from the above analysis, $\Delta_{\mathfrak{S}}$ maps $S_i$ to $S_i\otimes S_i$
($i=1,..., n$). Then, as in the proof of Theorem \ref{T:coalg}, it is not hard to check the coassociativity of $\Delta_{\mathfrak{S}}$
and the cocommutativity of ${\mathfrak{S}}$. Therefore, ${\mathfrak{S}}$ is a cocommutative Hopf dual algebra.
Let $\epsilon=\varphi_0\circ \phi$. Then one can easily verify that $\epsilon$ is
an integral of ${\mathfrak{S}}$.
\end{proof}
As a byproduct of the above proof, one can construct analytic isometric tuples from a given one as follows.
\begin{cor}
\label{C:S2abs}
Suppose that $S$ is an analytic isometric tuple.
Then so is $S^{\otimes k}$ for any $k\ge 1$.
\end{cor}
\begin{proof}
Let $k\ge 1$ and $S$ be an analytic isometric $n$-tuple.
As obtaining $\Phi$ in the proof of Theorem \ref{T:abscon}, one has that
there is a weak*-homeomorphic completely isometric isomorphism
$\widetilde \Phi$ between $\overline\otimes_{i=1}^k \L_n$ and
$\overline\otimes_{i=1}^k{\mathfrak{S}}$, the normal spatial tensor product of $k$ copies of ${\mathfrak{S}}$.
Let $\pi: \L_n\to \overline\otimes_{i=1}^k \L_n$ be the weak*-continuous, completely isometric homomorphism determined by Proposition \ref{P:LiLi}.
Then the composition $\widetilde\Phi\circ\pi$ is the weak*-continuous extension of the representation of ${\mathcal{A}}_n$ induced by $S^{\otimes k}$.
Therefore, $S^{\otimes k}$ is analytic.
\end{proof}
Suppose that $S$ is an analytic isometric $n$-tuple. Then ``$\L_n = {\mathfrak{S}}$'' as operator algebras.
As before, one can check that the canonical homomorphism $\sigma: {\mathfrak{S}}\to \L_n$ makes the comultiplications $\Delta$ of $\L_n$ and $\Delta_{\mathfrak{S}}$
of ${\mathfrak{S}}$ compatible: $\Delta\circ\phi=(\phi\otimes \phi)\circ \Delta_{\mathfrak{S}}$. Thus, ``$\L_n = {\mathfrak{S}}$'' as Hopf dual algebras.
We record this as a corollary.
\begin{cor}
\label{C:mor}
Suppose that ${\mathfrak{S}}$ is an analytic free semigroup algebra generated by an isometric $n$-tuple. Then
$(\L_n,\Delta)\cong ({\mathfrak{S}},\Delta_{\mathfrak{S}})$.
\end{cor}
\section{Hopf Convolution Dual Algebras}
\label{S:HCA}
Let ${\mathfrak{S}}$ be an analytic free semigroup algebra generated by an isometric $n$-tuple. In this section,
we shall show that its predual ${\mathfrak{S}}_*$ is also a Hopf algebra.
Naturally, ${\mathfrak{S}}_*$ and ${\L_n}_*$ are canonically isomorphic as Hopf algebras.
Analogous to the Fourier algebra $A({\mathbb{F}}_n)$, the algebra ${\L_n}_*$ is non-unital.
In order to equip ${\mathfrak{S}}_*$ with an algebra structure,
let $m_{{\mathfrak{S}}_*}=(\Delta_{\mathfrak{S}})_*$, where $\Delta_{\mathfrak{S}}$ is the comultiplication of ${\mathfrak{S}}$
constructed in the previous section. Then it follows essentially from Proposition \ref{P:pretenpro} and Theorem \ref{T:abscon}
that $m_{{\mathfrak{S}}_*}$ gives a completely contractive
multiplication of ${\mathfrak{S}}_*$. Thus ${\mathfrak{S}}_*$ is a completely contractive Banach algebra.
To construct a coalgebra structure of ${\mathfrak{S}}_*$, we follow the same line with Effros-Ruan \cite[Section 7]{ER1}.
The idea is sketched as follows. In order to endow ${\mathfrak{S}}_*$ with a comultiplication $\Delta_{{\mathfrak{S}}_*}$,
one needs to use the normal Haagerup tensor product
${\mathfrak{S}}\stackrel{\sigma\rm h}\otimes {\mathfrak{S}}$, because it is suitable for linearizing the multiplication $m$ on ${\mathfrak{S}}$ in the following sense:
The multiplication $m: {\mathfrak{S}}\times{\mathfrak{S}}\to{\mathfrak{S}}$ extends uniquely to a weak*-continuous
\textit{completely contractive} map $m_{\mathfrak{S}}: {\mathfrak{S}}\stackrel{\rm{\sigma}h}\otimes{\mathfrak{S}}\to {\mathfrak{S}}$.
Then it turns out that its preadjoint $(m_{{\mathfrak{S}}})_*$ produces a comultiplication of ${\mathfrak{S}}_*$.
To prove that it is indeed an \textit{algebra homomorphism}, we need two important ingredients:
(i) a Shuffle Theorem,
and (ii) $({\mathfrak{S}}\overline\otimes{\mathfrak{S}})_*\cong {\mathfrak{S}}_*\widehat\otimes{\mathfrak{S}}_*$.
Fortunately, we have both (i) and (ii): (i) holds true ``automatically'' because the Shuffle Theorem \cite[Theorem 6.1]{ER1}
holds true for
\textit{arbitrary} operator spaces,
and (ii) can be proved based on some results obtained in previous sections.
Recall that, given operator spaces $V$ and $W$, the normal and extended Haagerup tensor products are connected by
$V^*\stackrel{\sigma\rm h}\otimes W^*=(V\stackrel{\rm{eh}}\otimes W)^*$.
Also, a complex algebra ${\mathcal{A}}$ is called a \textit{completely contractive Banach algebra}
if ${\mathcal{A}}$ is an operator space and the multiplication $m:{\mathcal{A}}\times{\mathcal{A}}\to{\mathcal{A}}$
is a completely contractive bilinear mapping, namely, it determines a completely contractive linear mapping $m:{\mathcal{A}}\widehat\otimes{\mathcal{A}}\to{\mathcal{A}}$.
\begin{thm}
\label{T:Conv}
Suppose that ${\mathfrak{S}}$ is an analytic free semigroup algebra. Then
$({\mathfrak{S}}_*, m_{{\mathfrak{S}}_*}, \Delta_{{\mathfrak{S}}_*})$ is a Hopf algebra, where
the multiplication $m_{{\mathfrak{S}}_*}: {\mathfrak{S}}_*\widehat\otimes{\mathfrak{S}}_*\to {\mathfrak{S}}_*$
and the comultiplication $\Delta_{{\mathfrak{S}}_*}: {\mathfrak{S}}_*\to {\mathfrak{S}}_*\stackrel{\rm{eh}}\otimes{\mathfrak{S}}_*$
are completely contractive, and they are given by
$m_{{\mathfrak{S}}_*}=(\Delta_{\mathfrak{S}})_*$
and
$\Delta_{{\mathfrak{S}}_*}=(m_{\mathfrak{S}})_*$, respectively.
\end{thm}
\begin{proof}
By Proposition \ref{P:pretenpro} and the identity \eqref{E:SS}, one has
\begin{align}
\label{E:Stenpro}
({\mathfrak{S}}\overline\otimes{\mathfrak{S}})_* \cong {\mathfrak{S}}_*\widehat\otimes{\mathfrak{S}}_*.
\end{align}
Since the comultiplication $\Delta_{\mathfrak{S}}: {\mathfrak{S}}\to {\mathfrak{S}}\overline\otimes{\mathfrak{S}}$ is
a weak*-continuous complete isometry (from the proof of Theorem \ref{T:abscon}), it follows from \cite[Corollary 4.1.9]{ER} that
the preadjoint $(\Delta_{\mathfrak{S}})_*$
is a completely quotient mapping from ${{\mathfrak{S}}}_*\widehat\otimes {{\mathfrak{S}}}_*$ to
${{\mathfrak{S}}}_*$. Hence $({\mathfrak{S}}_*, (\Delta_{\mathfrak{S}})_*)$ is a completely contractive Banach algebra.
More precisely, for any $\varphi, \psi\in{{\mathfrak{S}}}_*$, the multiplication
$\varphi\ast\psi:=(\Delta_{\mathfrak{S}})_*(\varphi\otimes \psi)$ is defined by
\[
\varphi\ast\psi(x)=\varphi\otimes \psi(\Delta_{\mathfrak{S}}(x)) \quad (x\in {\mathfrak{S}}).
\]
It is now straightforward to check that $\varphi\ast\psi=\psi\ast\varphi$, implying the commutativity of $({\mathfrak{S}}_*,\ast)$.
Therefore $({{\mathfrak{S}}}_*, (\Delta_{\mathfrak{S}})_*)$ is a
completely contractive commutative Banach algebra.
Using the isomorphism
\eqref{E:Stenpro}
and the Shuffle Theorem \cite[Theorem 6.1]{ER1},
one can obtain that $(m_{\mathfrak{S}})_*$ is a completely contractive (algebra) homomorphism.
The proof is completely similar to that of \cite[Theorem 7.1]{ER1} (also refer to the discussion preceding the statement of this theorem),
and so it is omitted here.
Therefore $({\mathfrak{S}}_*,(\Delta_{\mathfrak{S}})_*, (m_{\mathfrak{S}})_*)$ is a Hopf algebra.
\end{proof}
Applying Theorems \ref{T:coalg} and \ref{T:Conv} to the case of $n=1$, we immediately obtain the following result in the classical case,
which seems new. Recall that
$(H^\infty)_*=L_1({\mathbb{T}})/H_1(0)$, where $H_1(0)=zH_1({\mathbb{T}})$ (\cite{Koo}).
\begin{cor}
$H^\infty$ and $L_1({\mathbb{T}})/H_1(0)$ are Hopf algebras.
\end{cor}
Following Effros-Ruan \cite{ER1}, Hopf algebras obtained from Theorem \ref{T:Conv} are called \textit{Hopf convolution dual algebras}.
Of course, counits, morphisms and integrals can be defined as those in Definitions \ref{D:coalg}, \ref{D:mor} and \ref{D:int},
but without the weak*-continuity requirement.
\medskip
Since ${\mathbb{F}}_n$ is a free group, it is well known that the Fourier algebra $A({\mathbb{F}}_n)$ is not unital.
Actually, for $n\ge 2$, more is true: $A({\mathbb{F}}_n)$ does not have bounded approximate identities.
In our case, so far we can show that ${\L_n}_*$ is not unital. Before proving this result, let us first recall from
\cite{Dav2, DavPit} that the set $M({\L_n})$ of
$\textsc{wot}$-continuous multiplicative linear functionals consists of those linear functionals having the form
$\varphi_\lambda=[\nu_\lambda\nu_\lambda^*]$, where $\lambda\in{\mathbb{B}}_n$ (the open unit ball of ${\mathbb{C}}^n$) and
$\nu_\lambda$ is a certain unit vector in $\H_n$ (whose precise definition is not important for us and so is omitted here).
It is known that $\varphi_\lambda(p(L))=p(\lambda)$ for any polynomial
$p=\sum_wa_ww$ in the semigroup algebra ${\mathbb{C}}\mathbb{F}_n^+$.
\begin{prop}
\label{P:nonuni}
If ${\mathfrak{S}}$ is an analytic free semigroup algebra, then ${\mathfrak{S}}_*$ is non-unital.
\end{prop}
\begin{proof}
It is not hard to see that it suffices to show that
${\mathfrak{S}}$ is not counital.
To the contrary, assume that it has a counit $\epsilon_{\mathfrak{S}}$. Then
$\epsilon:=\phi^{-1}\circ \epsilon_{\mathfrak{S}}$ is a counit of $\L_n$. So $\epsilon=\varphi_\lambda$ for some $\lambda\in{\mathbb{B}}_n$.
From the identity required in the definition of a counit, it is easy to check that $\epsilon(L_w)=1$ for all $w\in\mathbb{F}_n^+$. However,
as mentioned above,
$\varphi_\lambda(L_w)=w(\lambda)$,
where $w(\lambda)=\lambda_1^{k_1}\cdots\lambda_n^{k_n}$ for
$w=i_1^{k_1}\cdots i_n^{k_n}$. This particularly forces $\lambda_i=1$ for all $i=1,...,n$.
Obviously, this is impossible as $\lambda\in{\mathbb{B}}_n$.
\end{proof}
It is worthy to notice that $M({\L_n})$ is also closed under the multiplication $\ast$ of ${\L_n}_*$
and has an involution $\dag$.
In fact, for $\varphi_\lambda,\ \varphi_\mu\in \{\varphi_\nu: \nu\in{\mathbb{B}}_n\}$, then
$\varphi_\lambda \ast \varphi_\mu=\varphi_{\lambda*\mu}$, where $\lambda*\mu=(\lambda_1\mu_1,...,\lambda_n\mu_n)$
for $\lambda=(\lambda_1,...,\lambda_n)$ and $\mu=(\mu_1,...,\mu_n)$ in ${\mathbb{B}}_n$.
Also, $\dag$ can be defined by $\varphi_\lambda^\dag=\varphi_{\overline\lambda}$.
\medskip
If ${\mathfrak{S}}$ is an analytic free semigroup algebra generated by an isometric $n$-tuple,
then ${\mathfrak{S}}$ and $\L_n$ are completely isometrically isomorphic. So ${\mathfrak{S}}_*\cong {\L_n}_*$ as operator spaces.
The following result tells us this is also the case as Hopf convolution algebras.
\begin{prop}
Suppose that ${\mathfrak{S}}$ is an analytic free semigroup algebra generated by an isometric $n$-tuple.
Then ${\mathfrak{S}}_*\cong {\L_n}_*$ as Hopf convolution algebras.
\end{prop}
\begin{proof}
For brevity, we use $*$ for the multiplication of ${\L_n}_*$ and that of ${\mathfrak{S}}_*$ constructed above.
It follows from \cite[Corollary 4.1.9]{ER} that $\phi_*:{\L_n}_*\to {\mathfrak{S}}_*$ is completely isometric.
We now show that it is also an algebra homomorphism,
namely, $\phi_*(f\ast g)=\phi_*(f) \ast \phi_*(g)$ for all $f,g\in{\L_n}_*$.
Indeed, we have
\begin{align*}
\phi_*(f\ast g)
&=(f\ast g)\circ\phi=(f\otimes g)\circ\Delta\circ \phi\\
&=(f\otimes g)\circ\Phi\circ\Delta_{\mathfrak{S}} \\
&=(f\otimes g)\circ(\phi\otimes\phi)\circ\Delta_{\mathfrak{S}} \\
&=(\phi_*\otimes \phi_*)\circ(f\otimes g)\circ\Delta_{\mathfrak{S}} \\
&=(\phi_*(f)\otimes \phi_*(g))\circ\Delta_{\mathfrak{S}} \\
&=\phi_*(f)\ast\phi_*(g).
\end{align*}
Also, it is easy to check that $(\phi_*)^{-1}=(\phi^{-1})_*$.
Therefore, $\phi_*$ is an algebra isomorphism.
In order to show that $\phi_*$ is compatible with the comultiplications on ${\L_n}_*$ and ${\mathfrak{S}}_*$,
consider the following commuting diagram
\[
\begin{xy}
(0,20)*+{{{\mathfrak{S}}}}="a"; (40,20)*+{{\L_n}}="b";%
(0,0)*+{{{\mathfrak{S}}}\stackrel{\rm{\sigma h}}\otimes{{\mathfrak{S}}}}="c"; (40,0)*+{{\L_n}\stackrel{\rm{\sigma h}}\otimes{\L_n}}="d";%
{\ar@{->} "a";"b"};?*!/_2mm/{\phi};
{\ar@{->} "c";"a"};?*!/_5mm/{m_{{\mathfrak{S}}}};
{\ar@{->} "d";"b"};?*!/^5mm/{m_{\L_n}};
{\ar@{->} "c";"d"};?*!/^2mm/{\phi\otimes\phi},
\end{xy}
\]
and then take its preadjoint
\[
\begin{xy}
(0,20)*+{{{\mathfrak{S}}}_*}="a"; (40,20)*+{{\L_n}_*}="b";%
(0,0)*+{{{\mathfrak{S}}}_*\stackrel{\rm{eh}}\otimes{{\mathfrak{S}}}_*}="c"; (40,0)*+{{\L_n}_*\stackrel{\rm{eh}}\otimes{{\L_n}_*}}="d";%
{\ar@{<-} "a";"b"}?*!/_2mm/{\phi_*};
{\ar "a";"c"};{\ar "b";"d"};%
{\ar "a";"c"};?*!/^5mm/{\Delta_{{{\mathfrak{S}}_*}}};
{\ar@{->} "b";"d"};?*!/_5mm/{\Delta_{{L_n}_*}};
{\ar@{<-} "c";"d"};?*!/^4mm/{\phi_*\stackrel{\rm{eh}}\otimes\phi_*};
\end{xy}.
\]
\end{proof}
\section{Completely Bounded Representations of ${\mathfrak{S}}_*$}
\label{S:cbrep}
Let ${\mathfrak{S}}$ be an analytic free semigroup algebra.
In this section, we shall study completely bounded representations of ${\mathfrak{S}}_*$.
Through those representations, ${\mathfrak{S}}$ and ${\mathfrak{S}}_*$ are closely connected with each other.
On one hand, we show that there is a bijection between
the set of completely bounded representations of ${\mathfrak{S}}_*$ and the set of corepresentations of ${\mathfrak{S}}$. On the other hand,
${\mathfrak{S}}$ can be recovered by completely bounded representations of ${\mathfrak{S}}_*$:
The algebra induced from the coefficient operators of completely bounded representations
of ${\mathfrak{S}}_*$ is precisely ${\mathfrak{S}}$. This gives us a new duality between ${\mathfrak{S}}_*$ and ${\mathfrak{S}}$.
As an amusing application of the former bijection, we prove that the spectrum of ${\mathfrak{S}}_*$, as a
commutative Banach algebra, is precisely $\mathbb{F}_n^+$.
In order to achieve the latter duality, one needs to
introduce the notion of tensor products of completely bounded representations of ${\mathfrak{S}}_*$.
Fortunately, for this one can borrow it from \cite[Section 7]{ER1}. Actually, the entire subsection on tensor products
is from there. It should be mentioned that we have the equality ${\mathcal{C}}({\mathfrak{S}}_*)={\mathfrak{S}}$ as Theorem \ref{T:dual},
rather than just an inclusion ${\mathcal{C}}({\mathfrak{S}}_*)\subseteq{\mathfrak{S}}$ in \cite[Theorem 7.2]{ER1}, because we are dealing with only
``Fourier algebras'', rather than general Hopf convolution dual algebras.
\subsection*{Corepresentations of ${\mathfrak{S}}$}
We begin this subsection with the following definition.
\begin{defn}
\label{D:corep}
Let $({\mathcal{A}}, \Delta)$ be a Hopf dual algebra acting on $\H$.
A \textit{corepresentation} of $({\mathcal{A}},\Delta)$ on a Hilbert space ${\mathcal{K}}$ is an (arbitrary) operator $V\in {\mathcal{A}}\, \ol\otimes\, {\mathcal{B}}({\mathcal{K}})$ satisfying
\[
V_{1,3}V_{2,3}= (\Delta\otimes {\operatorname{id}})(V).
\]
Here $V_{1,3}$ and $V_{2,3}$ are the standard \textit{leg notation} (cf. \cite{BS}): $V_{1,3}$ is a linear operator on the Hilbert space
$\H\otimes\H\otimes{\mathcal{K}}$, which acts as $V$ on the first and third tensor factors and as the identity on the second one.
The notation $V_{2,3}$ is defined similarly.
\end{defn}
Before stating our first main theorem in this section, we need the following lemma, which
is probably a folklore. Here we give a direct proof based on special properties of
$\L_n$.
\begin{lem}
\label{L:LnBK}
Suppose that ${\mathfrak{S}}$ is an analytic free semigroup algebra. Then
we have a completely isometric isomorphism
\[
{\mathcal{C}}{\mathcal{B}}({{\mathfrak{S}}}_*, {\mathcal{B}}({\mathcal{K}})) \cong {\mathfrak{S}} \overline\otimes {\mathcal{B}}({\mathcal{K}}).
\]
\end{lem}
\begin{proof}
We first claim that this lemma holds true when ${\mathfrak{S}}=\L_n$:
\begin{align}
\label{E:Lncor}
{\mathcal{C}}{\mathcal{B}}({\L_n}_*, {\mathcal{B}}({\mathcal{K}}))\cong\L_n\overline\otimes {\mathcal{B}}({\mathcal{K}}).
\end{align}
To this end, observe that
\[
(\L_n\overline\otimes {\mathcal{B}}({\mathcal{K}}))'={\mathcal{R}}_n\overline\otimes {\mathbb{C}} I
\quad \mbox{and}\quad
(\L_n\overline\otimes {\mathcal{B}}({\mathcal{K}}))''=\L_n\overline\otimes {\mathcal{B}}({\mathcal{K}}).
\]
Then, applying some obvious modifications to the proof of Proposition \ref{P:pretenpro}, one can show that
$({\L_n}_*\widehat\otimes {\mathcal{B}}({\mathcal{K}})_*)^*$ and $\L_n\overline\otimes {\mathcal{B}}({\mathcal{K}})$
are completely isometrically isomorphic.
On the other hand, there is a natural completely isometric isomorphism
$({\L_n}_*\widehat\otimes {\mathcal{B}}({\mathcal{K}})_*)^*\cong{\mathcal{C}}{\mathcal{B}}({\L_n}_*, {\mathcal{B}}({\mathcal{K}}))$ by \cite[Corollary 7.1.5]{ER}.
This proves our claim.
Now suppose ${\mathfrak{S}}$ is an arbitrary analytic free semigroup algebra. Using a proof completely similar
to that of Theorem \ref{T:abscon}, we obtain that
\[
({{\mathfrak{S}}}_*\widehat\otimes {\mathcal{B}}({\mathcal{K}})_*)^*\cong({\L_n}_*\widehat\otimes {\mathcal{B}}({\mathcal{K}})_*)^*
\]
and
\[
{\mathfrak{S}}\overline\otimes {\mathcal{B}}({\mathcal{K}}) \cong \L_n\overline\otimes {\mathcal{B}}({\mathcal{K}}).
\]
Therefore
\begin{align*}
{\mathcal{C}}{\mathcal{B}}({{\mathfrak{S}}}_*, {\mathcal{B}}({\mathcal{K}}))
& \cong({{\mathfrak{S}}}_*\widehat\otimes {\mathcal{B}}({\mathcal{K}})_*)^*\cong({\L_n}_*\widehat\otimes {\mathcal{B}}({\mathcal{K}})_*)^*\\
& \cong{\mathcal{C}}{\mathcal{B}}({\L_n}_*, {\mathcal{B}}({\mathcal{K}}))\cong\L_n\overline\otimes {\mathcal{B}}({\mathcal{K}}) \\
& \cong {\mathfrak{S}} \overline\otimes {\mathcal{B}}({\mathcal{K}}),
\end{align*}
where the first $\cong$ is from \cite[Corollary 7.1.5]{ER}, while the fourth one is from \eqref{E:Lncor}.
\end{proof}
\begin{thm}
\label{T:corep}
Suppose that ${\mathfrak{S}}$ is an analytic free semigroup algebra acting on a Hilbert space $\H$. Then
there is a bijection between the set of all corepresentations of ${\mathfrak{S}}$
and the set of all completely bounded representations
of ${\mathfrak{S}}_*$.
More precisely, if $V$ is a corepresentation of ${\mathfrak{S}}$ on ${\mathcal{K}}$, then the corresponding representation $\pi_V$ of ${{\mathfrak{S}}}_*$ is given by
\begin{align}
\label{E:piV}
\pi_V(\varphi)=(\varphi\otimes {\operatorname{id}})(V) \quad (\varphi\in{\mathfrak{S}}_*);
\end{align}
if $\pi$ is a completely bounded representation of ${\mathfrak{S}}_*$ on ${\mathcal{K}}$, then the
corresponding corepresentation $V_\pi$ of ${\mathfrak{S}}$ is given by
\begin{align}
\label{E:Vpi}
(V_\pi(\xi\otimes x), \eta\otimes y)=(\pi([\xi\eta^*]) x,y)
\end{align}
for all $\xi,\eta\in\H$, $x,y\in{\mathcal{K}}$.
\end{thm}
\begin{proof}
The following calculations show that, for a corepresentation $V$ of ${\mathfrak{S}}$, the map $\pi_V$ defined in \eqref{E:piV}
indeed yields a representation of ${\mathfrak{S}}_*$ on ${\mathcal{K}}$: For $\varphi\in{\mathfrak{S}}_*$, we have
\begin{align*}
(\varphi\otimes {\operatorname{id}})(V)(\psi\otimes {\operatorname{id}})(V)
&=(\varphi\otimes\psi\otimes {\operatorname{id}})(V_{1,3}V_{2,3})\\
&=(\varphi\otimes\psi\otimes {\operatorname{id}})((\Delta_{\mathfrak{S}}\otimes {\operatorname{id}})(V))\\
&=\varphi\ast \psi(V).
\end{align*}
The fact that $\pi_V$ is completely bounded follows from the completely isometric isomorphism in Lemma \ref{L:LnBK}.
For $\pi\in {\mathcal{C}}{\mathcal{B}}({\mathfrak{S}}_*, {\mathcal{B}}({\mathcal{K}}))$, the identification given in Lemma \ref{L:LnBK} determines an operator
$V_\pi\in {\mathfrak{S}}\overline\otimes {\mathcal{B}}({\mathcal{K}})$ via $\pi(\varphi)=(\varphi\otimes {\operatorname{id}})(V_\pi)$ for all $\varphi\in{\mathfrak{S}}_*$. Reversing the above proof,
we see that $V_\pi$ is a corepresentation of ${\mathfrak{S}}$.
In order to get \eqref{E:Vpi}, it suffices to make use of Theorem \ref{T:ww} stating that ${\mathfrak{S}}$ has property ${\mathbb{A}}_1(1)$.
\end{proof}
It is time to look at some examples.
\begin{eg}
\label{Eg:W}
Let $W\in{\mathcal{B}}(\H_n\otimes\H_n)$ be defined by $W(\xi_u\otimes \xi_v)=\xi_{vu}\otimes \xi_v$. Then
one can show that $W$ is a proper isometry and intertwines $L_w\otimes L_w$ and ${\operatorname{id}}\otimes L_w$:
\[
(L_w\otimes L_w)W=W({\operatorname{id}}\otimes L_w)\quad \mbox{for all}\quad w\in \mathbb{F}_n^+.
\]
Also one can check that $W$ is a corepresentation of $\L_n$. In fact, checking on the basis vectors, it is easy to see that
it satisfies the identity required in Definition \ref{D:corep}. In order to check $W\in \L_n\overline\otimes {\mathcal{B}}(\H_n)$, it suffices to
check that $W$ commutes each element in the commutant $(\L_n\overline\otimes {\mathcal{B}}(\H_n))'={\mathcal{R}}_n\otimes {\mathbb{C}} I$.
For $R_u\in{\mathcal{R}}_n$ ($u\in \mathbb{F}_n^+)$, then
a simple calculation shows that $W(R_u\otimes {\operatorname{id}})=(R_u\otimes {\operatorname{id}})W$. We are done.
It is not hard to check that the completely bounded representation $\pi_W$ of ${\L_n}_*$ corresponding to the above $W$ in Theorem \ref{T:corep}
is nothing but the left multiplication representation:
\[
\pi_W(\varphi)\xi_u=\varphi(L_u)\xi_u \quad (\varphi\in {\L_n}_*).
\]
\end{eg}
Two points are worth being mentioned here. Firstly, in the group case, the above corepresentation $W$ is known as the fundamental operator
in the theory of Hopf von Neumann algebras.
Secondly, in spite of the fact that the above $W$ is a proper isometry, the tuples
$[L_1\otimes L_1\,\cdots\, L_n\otimes L_n]$ and $[{\operatorname{id}} \otimes L_1\,\cdots\, {\operatorname{id}}\otimes L_n]$ are unitarily equivalent
because both of them are pure of multiplicity $\infty$ (cf. \cite{Pop89} and Section \ref{S:Ln}).
\begin{eg}
\label{Eg:Lw}
Fix $w\in \mathbb{F}_n^+$. Let $\rho_w: {\L_n}_*\to {\mathbb{C}}$ be defined by $\rho_w(\varphi)=\varphi(L_w)$. Then it is a character
of ${\L_n}_*$. In fact, for all
$\varphi,\psi\in {\L_n}_*$, we have
\begin{align*}
\rho_w(\varphi\ast\psi)
&=\varphi\ast \psi(L_w)=\varphi\otimes\psi(\Delta(L_w))\\
&=\varphi\otimes\psi(L_w\otimes L_w)=\varphi(L_w)\psi(L_w)\\
&=\rho_w(\varphi)\rho_w(\psi).
\end{align*}
In particular, it is a completely contractive representation.
It is easy to check that its corresponding corepresentation $V_{\rho_w}$ of $\L_n$ in Theorem \ref{T:corep}
is given by
\[
V_{\rho_w}=L_w\otimes {\operatorname{id}}_{\mathbb{C}}\cong L_w.
\]
\end{eg}
In what follows, as an amusing application of Theorem \ref{T:corep}, we identify the spectrum of ${\mathfrak{S}}_*$.
The proof also takes advantage of the useful idea of Fourier expansions of operators in $\L_n$ and $\L_n\overline\otimes\L_n$.
\begin{thm}
\label{T:char}
Let ${\mathfrak{S}}$ be an analytic free semigroup algebra generated by an isometric $n$-tuple. Then
the (Gelfand) spectrum of ${\mathfrak{S}}_*$ is $\mathbb{F}_n^+$.
\end{thm}
\begin{proof}
Obviously, it suffices to show that the spectrum $\Sigma_{{\L_n}_*}$ of ${\L_n}_*$ is $\mathbb{F}_n^+$ .
By Example \ref{Eg:Lw} above, one has $\mathbb{F}_n^+\subseteq \Sigma_{{\L_n}_*}$.
In order to prove $\Sigma_{{\L_n}_*}\subseteq \mathbb{F}_n^+$, we first
observe that every operator $A\in \L_n\overline\otimes \L_n$ is completely determined by the image $A(\xi_\varnothing\otimes\xi_\varnothing)$ of
the ``vacuum vector'' $\xi_\varnothing\otimes \xi_\varnothing$ under $A$.
Indeed, assume that
\[
A(\xi_\varnothing\otimes \xi_\varnothing)=\sum_{u,v\in\mathbb{F}_n^+}a_{u,v}\, \xi_u\otimes \xi_v
\]
with $a_{u,v}\in{\mathbb{C}}$.
Then for all $\alpha,\beta\in\mathbb{F}_n^+$, one has
\begin{align*}
A(\xi_\alpha\otimes \xi_\beta)
&=AR_{ \alpha}\otimes R_{\beta}(\xi_\varnothing\otimes \xi_\varnothing)\\
&=R_{\alpha}\otimes R_{\beta}A(\xi_\varnothing\otimes \xi_\varnothing)\\
&=R_{ \alpha}\otimes R_{\beta}(\sum_{u,v}a_{u,v}\, \xi_u\otimes \xi_v)\\
&=\sum_{u,v}a_{u,v}\, \xi_{u\widetilde\alpha}\otimes \xi_{v\widetilde\beta},
\end{align*}
where $\widetilde\alpha$ is the word in $\mathbb{F}_n^+$ obtained by reversing the order of $\alpha$,
and similarly for $\widetilde\beta$.
So similar to $\L_n$, we heuristically view $A$ as its Fourier expansion:
\[
A\sim \sum_{u,v\in\mathbb{F}_n^+}a_{u,v}\, L_u\otimes L_v,
\]
where the $(u,v)$-th Fourier coefficient is given by $a_{u,v}=(A(\xi_{\varnothing}\otimes\xi_{\varnothing}),\xi_{u}\otimes\xi_{v})$.
Now assume that $\varphi: {\L_n}_*\to {\mathbb{C}}$ is a character of ${\L_n}_*$. Let $c_{1,1}^\varphi\in\L_n$ be the coefficient operator defined
by
\[
\langle f, c_{1,1}^\varphi\rangle = \varphi(f) \quad (f\in{\L_n}_*).
\]
Since $\varphi$ is a character, it is automatically a completely bounded representation. By Theorem \ref{T:corep}, we have that
$V:=c_{1,1}^\varphi\in\L_n$ is a corepresentation of $\L_n$. That is,
\begin{align}
\label{E:V}
V_{1,3}V_{2,3}= (\Delta\otimes {\operatorname{id}}_{\mathbb{C}})(V).
\end{align}
Since $V\in\L_n$, one can assume that it has the following Fourier expansion:
\[
V\sim\sum_{w\in\mathbb{F}_n^+}a_w\,L_w.
\]
Then it follows from \eqref{E:V} that
\begin{align}
\label{E:AW}
\sum_w a_wL_w\otimes \sum_w a_wL_w\ \sim\ \sum_w a_w(L_w\otimes L_w)\in\L_{n}\overline\otimes\L_n.
\end{align}
Then taking the $(w, w)$-th Fourier coefficients of both sides of \eqref{E:AW} gives
\begin{align*}
&((\sum_w a_wL_w\otimes \sum_w a_wL_w)(\xi_\varnothing\otimes\xi_\varnothing), \xi_w\otimes\xi_w)\\
&=(\sum_w a_w(L_w\otimes L_w)(\xi_\varnothing\otimes\xi_\varnothing), \xi_w\otimes\xi_w).
\end{align*}
This implies $a_w=0$ or $1$ for every $w\in\mathbb{F}_n^+$. On one hand, there is a word $w\in\mathbb{F}_n^+$ such that $a_w\ne 0$ as $\varphi\ne 0$ implies $V\ne 0$.
On the other hand, suppose that
there are $u\ne v\in\mathbb{F}_n^+$ such that both $a_u$ and $a_v$ are non-zero. Then
$a_u=a_v=1$.
Taking the $(u, v)$-th Fourier coefficients of both sides of \eqref{E:AW}, we have
\begin{align*}
1=a_ua_v&=((\sum_w a_wL_w\otimes \sum_w a_wL_w)(\xi_\varnothing\otimes\xi_\varnothing), \xi_u\otimes\xi_v)\\
&=(\sum_w a_w(L_w\otimes L_w)(\xi_\varnothing\otimes\xi_\varnothing), \xi_u\otimes\xi_v)\\
&=a_u\delta_{u,v}=0.
\end{align*}
This is an obvious contradiction. So $V=L_w$ for some $w\in\mathbb{F}_n^+$.
Therefore, $\Sigma_{{\L_n}_*}\subseteq \mathbb{F}_n^+$.
\end{proof}
\subsection*{Tensor products of completely bounded representations of ${\mathfrak{S}}_*$}
This subsection borrows from \cite[Section 7]{ER1}, and is rather sketched.
Refer to \cite[Section 7]{ER1} for details.
Suppose that $\pi_i: {\mathfrak{S}}_*\to {\mathcal{B}}(\H_i)$ ($i=1,2$) are two completely bounded representations of ${\mathfrak{S}}_*$.
Define their multiplication $\pi_1\times \pi_2$ as the composition of the following mappings
\begin{align*}
{\mathfrak{S}}_*\stackrel{\Delta_{{\mathfrak{S}}_*}}\longrightarrow {\mathfrak{S}}_*\stackrel{\rm{eh}}\otimes{\mathfrak{S}}_*\stackrel{\pi_1\otimes\pi_2}\longrightarrow &{\mathcal{B}}(\H_1)\stackrel{\rm{eh}}\otimes{\mathcal{B}}(\H_2)\\
\subseteq &{\mathcal{B}}(\H_1)\stackrel{\sigma\rm{h}}\otimes{\mathcal{B}}(\H_2)\stackrel{\theta}\to {\mathcal{B}}(\H_1\otimes\H_2),
\end{align*}
where $\theta$ is determined by the product of
\[
{\mathcal{B}}(\H_1)\to {\mathcal{B}}(\H_1\otimes\H_2): S\mapsto S\otimes I_{\H_2}
\]
and
\[
{\mathcal{B}}(\H_2)\to {\mathcal{B}}(\H_1\otimes\H_2): T\mapsto I_{\H_1}\otimes T.
\]
Let
\[
{\mathcal{C}}({\mathfrak{S}}_*)=\left\{c_{\xi,\eta}^\pi: c_{\xi,\eta}^\pi(f)=(\pi(f)\xi, \eta), \ f\in{\mathfrak{S}}_*,\, \xi, \, \eta\in\H_\pi\right\}
\]
be the set of coefficient operators of completely bounded representations of ${\mathfrak{S}}_*$.
Then we have the following duality result.
\begin{thm}
\label{T:dual}
Suppose that ${\mathfrak{S}}$ is an analytic free semigroup algebra.
Then ${\mathcal{C}}({\mathfrak{S}}_*)={\mathfrak{S}}$.
\end{thm}
\begin{proof}
This directly follows from \cite[Theorem 7.2]{ER1} and Example \ref{Eg:Lw}.
\end{proof}
\medskip
Let us end this paper with two remarks.
Firstly, most of the results on Hopf convolution dual algebras of this paper
hold true for more general ones, e.g., those induced from Hopf dual algebras
${\mathcal{A}}$ with the property $({\mathcal{A}}\overline\otimes{\mathcal{A}})_*\cong{\mathcal{A}}_*\widehat\otimes{\mathcal{A}}_*$.
Secondly, as we have seen from above, there is a certain analogy between the Fourier algebra $A({\mathbb{F}}_n)$ and ${\L_n}_*$.
So, basically, for whatever property $A({\mathbb{F}}_n)$ has, one could ask if it holds true for ${\L_n}_*$ as long as it makes sense.
This and more will be investigated in the future.
\medskip
\noindent
\textbf{Acknowledgements.} I would like to thank my colleague Prof. Zhiguo Hu for several useful conversations
at the early stage of this work, and Prof. Matthew Kennedy for some useful discussions after
I gave a talk at the CMS 2011 Winter Meeting, in which some results of this paper were
presented. Thanks also go to Prof. Laurent Marcoux for showing me some references.
|
1202.1867
|
\section{Introduction}
\label{sec:intro}
The spontaneous breakdown of chiral symmetry plays a central role
in the spectrum of light hadrons. Since it is an intrinsically
non-perturbative phenomenon, the only way to study it from the
first principles of QCD is via the lattice regularization. Yet,
already many years ago Nielsen and Ninomiya proved that a
translationally invariant, local lattice formulation of the QCD
Dirac operator $D$, retaining chiral symmetry in the massless
limit, and with the correct number of physical fermionic degrees
of freedom, is forbidden~\cite{Nielsen_Ninomiya}. This no-go
theorem can be circumvented, by constructing lattice fermions
satisfying a \emph{modified} form of chiral
symmetry~\cite{latticechiralsymmetry}, and obeying the Ginsparg-
Wilson relation~\cite{Ginsparg:1981bj}. Although explicit
formulations of lattice Ginsparg-Wilson fermions are
known~\cite{overlap_DW_perfect}, currently their practical use in
realistic, large-scale lattice QCD simulations is still limited,
due to the high computational overhead.
The most widely-used lattice discretizations of the Dirac
operator are either based on the addition of a second-derivative
term to the kinetic part of the quark
action~\cite{Wilson_fermions} to remove (or ``quench'') the
unphysical ``doubler'' modes in the continuum limit by giving
them a mass ${\cal O}(a^{-1})$, or on a site-dependent spin
diagonalization, which leads to the so-called staggered
formulation~\cite{Kogut:1974ag}. The former approach introduces
an explicit breaking of chiral symmetry, and, as a consequence,
an additive renormalization of the quark mass, which has to be
fine-tuned. In contrast, the staggered operator preserves a
remnant of chiral symmetry (sufficient to forbid additive mass
renormalization), and leads to a reduction of the matrix size.
However, the staggered formulation only removes part of the
unphysical modes, reducing the number of quark species in four
($d$) spacetime dimensions from 16 ($2^d$) down to four
($2^{d/2}$) ``tastes'', which become
degenerate~\cite{Follana:2005km} (and consistent with the
properties related to the global symmetries of the continuum
Dirac operator~\cite{Bruckmann:2008xr}) in the $a \to 0$ limit.
In order to simulate QCD with two light fermions, one then has to
apply the so-called ``rooting trick'', which has been a subject
of debate for the last few years~\cite{rooting_saga}.
Some recent works have discussed the idea of using a staggered
kernel with a \emph{taste-dependent} mass term to obtain two (or
one~\cite{deForcrand:2011ak, Hoelbling:2010jw}) massless fermion
species. Such formulation, which is one of the various approaches
aiming at minimally doubled
fermions~\cite{minimallydoubledfermions}, could combine the
advantages of the overlap construction with the computational
efficiency of a staggered kernel. Furthermore, this
formulation appears to be particularly attractive from the point
of view of topological properties~\cite{staggeredoverlap1}.
Using a staggered operator with a ``flavored'' mass term as the
kernel in an overlap construction is a very appealing idea, but
the properties of such operators (with various taste-dependent
mass terms) are interesting on their own. In fact, while the
overlap construction completely removes the need for fine tuning
to achieve massless fermions, it still leads to a considerable
computational overhead. In contrast, using a staggered operator
with taste-dependent mass \emph{\`a la} Wilson requires fine
tuning to obtain exactly massless modes, but, by
virtue of the reduced size of the operator, may still be a
computationally competitive alternative to the usual Wilson
discretization, while avoiding the rooting prescription.
This motivation led us to address a numerical investigation of
different operators of this type that we present here
(preliminary results have appeared in \cite{deForcrand:2011ak}).
In the following, we present a systematic classification of the
possible taste-dependent mass terms, discuss their analytical
features in the free theory, and then move on to the interacting
case that we study via numerical simulations. We perform an
elementary measurement of the pion mass on a set of quenched
configurations, and verify the expected PCAC behaviour as one
approaches the chiral limit. In an Appendix, we also explore the
properties of the staggered overlap operator proposed in
\cite{staggeredoverlap1}, in comparison with the usual overlap
based on the Wilson kernel. In particular, we compare the
locality of the operators, and the computational cost of applying
them to a vector and of solving for the quark propagator.
The structure of this paper is as follows. First, in
sec.~\ref{sec:theoretical} we recall theoretical aspects of
the construction of taste-dependent mass terms, and discuss their
spectral structure in the free field case. Then, we address the
interacting case, presenting our numerical studies in
sec.~\ref{sec:numerical}. We summarize our findings and
discuss their implications for possible future, large-scale
applications of these operators in sec.~\ref{sec:conclusions}. Finally,
in the appendix~\ref{appendix}, we report on our study of an overlap
operator based on a staggered kernel, as proposed in ref.~\cite{staggeredoverlap1}.
\section{Theoretical formulation and general features}
\label{sec:theoretical}
The staggered operator~\cite{Kogut:1974ag}
\begin{equation}
\label{dks}
D_\text{KS} = \frac{1}{2a} \sum_{\mu=1}^{d} \eta_\mu \left( V_\mu - V_\mu^\dagger \right)
\end{equation}
with $\eta_\mu(x)=(-1)^{\sum_{\nu<\mu} x_\nu}$ and $\left( V_\mu \right)_{x,y}
= U_\mu(x) \delta_{x+a\hat\mu, y}$, is a computationally very
efficient way to discretize the massless QCD Dirac operator on a
$d$-dimensional Euclidean hypercubic lattice of spacing $a$. This
operator is invariant under a global $U(1)$ symmetry, which can
be interpreted as a remnant of chiral symmetry: in fact, $D_\text{KS}$
anticommutes with the operator $\Gamma_{55}$ defined by $\left(
\Gamma_{55}\right)_{x,y}=(-1)^{\sum_{\nu=1}^d x_\nu}$. In the
free theory, one can easily see that in four dimensions the
operator $\Gamma_{55}$ has $\gamma_5 \otimes \gamma_5$ structure
in spin-taste space~\cite{spinflavorinterpretation}. The
construction of $D_\text{KS}$ is based on a local spin diagonalization,
which, for the four-dimensional case, allows one to reduce the
number of fermion components by a factor of $4$ with respect to the
naive operator, and yields four tastes in the continuum limit.
The degeneracy between these four tastes is explicitly broken by
gauge interactions at finite lattice spacing $a$, but is
recovered in the continuum limit $a \to 0$.
Recently, various works explored the idea of using staggered
operators with taste-dependent
mass terms~\cite{staggeredoverlap1, deForcrand:2011ak, Hoelbling:2010jw}. Following, e.g., the discussion in the classic
paper by Golterman and Smit~\cite{Golterman:1984cy}, the possible
matrix structures (in taste space) for a mass term can be
classified as
\begin{itemize}
\item $\mathbf{1}$ (``0-link''),
of the form $\delta_{x,y}$
\item $\gamma_\alpha$ (``1-link''),
involving a sum of terms, each containing 1 link $U_\mu$
\item $\sigma_{\alpha\beta}$ (``2-link''),
involving a sum of terms, each containing 2 links $U_\mu U_\nu$
\item $\gamma_5 \gamma_\alpha$ (``3-link''),
involving a sum of terms, each containing 3 links $U_\mu U_\nu U_\rho$
\item $\gamma_5$ (``4-link''),
involving a sum of terms, each containing 4 links $U_\mu U_\nu U_\rho U_\sigma$
\end{itemize}
It is highly desirable to preserve the symmetry
$\Gamma_{55} D \Gamma_{55} = D^\dagger$, because it guarantees
that $\det D$ is real, and non-negative (in the absence of real negative eigenvalues), thus avoiding a ``sign
problem'' in the measure~\cite{signproblem}. This symmetry is
satisfied only if the hermitian mass term connects sites of the
same parity. Thus, we do not consider the 1-link or a 3-link mass
terms further.
This leaves three possibilities: 0-, 2- and 4-link mass
terms.\footnote{It is also possible to consider the above matrix
possibilities with an extra factor
$\Gamma_{55}$~\cite{Golterman:1984cy}. In that case, $\gamma_5$-
hermitian mass terms are obtained in the 0-, 1- and 3-link cases.
However, we did not find a continuum-like dispersion relation for
the real modes in any of these cases.}
The 0-link mass term corresponds to the usual staggered operator,
with a taste-independent bare mass
\begin{equation}
\label{plainvanillastaggeredmass}
D_0 = D_\text{KS} + m .
\end{equation}
The staggered operator with a 2-link mass term, which was
discussed in refs.~\cite{deForcrand:2011ak, Hoelbling:2010jw},
can be written in the form
\begin{equation}
\label{twolinkkernel}
D_2 = D_\text{KS} + \frac{\rho}{\sqrt{3}}\left(M_{12}+M_{13}+M_{14}+M_{23}-M_{24}+M_{34}\right),
\end{equation}
where (following the notation of ref.~\cite{Hoelbling:2010jw})
\begin{equation}
M_{\mu\nu}=i \eta_{\mu \nu} C_{\mu\nu},
\end{equation}
\begin{equation}
(\eta_{\mu\nu})_{x,y}=-(\eta_{\nu\mu})_{x,y}=(-1)^{\sum_{i=\mu+1}^{\nu} x_i}\delta_{x,y}\textrm{, for $\mu < \nu$},
\end{equation}
\begin{equation}
C_{\mu\nu}=\frac{1}{2}\left(C_\mu C_\nu+C_\nu C_\mu\right),
\end{equation}
\begin{equation}
\label{Cmudefinition}
C_\mu = \frac{1}{2} \left( V_\mu + V_\mu^\dagger \right).
\end{equation}
Finally, a staggered operator featuring a mass term with
$\gamma_5$ structure in taste space~\cite{staggeredoverlap1} can
be written as
\begin{equation}
\label{fourlinkkernel}
D_4 = D_\text{KS} - \frac{\rho}{a}\Gamma_{55}\Gamma_{5},
\end{equation}
with
\begin{equation}
\Gamma_{5}=\eta_5 C,
\end{equation}
where
\begin{equation}
\label{eta5_definition}
\eta_5 (x) =\prod_{\mu=1}^4 \eta_\mu (x),
\end{equation}
while $C$ is the average of four-link parallel transporters
joining sites at opposite corners of the elementary lattice
hypercubes
\begin{equation}
\label{C_definition}
C = \frac{1}{4!} \sum_{\mbox{\tiny{perm}}} C_\mu C_\nu C_\rho C_\sigma.
\end{equation}
Note that the mass term appearing on the r.h.s. of
eq.~(\ref{fourlinkkernel}) is Hermitean and commutes with
$\Gamma_{55}$.
\begin{figure}
\centerline{
\hspace*{-3.0cm}
\includegraphics[width=0.73\textwidth]{free_stag.pdf}
\hspace*{-1.9cm}
\includegraphics[width=0.43\textwidth,viewport=-1 -10 495 402]{HoelblingBatman.pdf}
\hspace*{0.5cm}
\includegraphics[width=0.295\textwidth,clip=true]{StaggeredBatman.pdf}
}
\caption{Left panel: Spectrum of $D_\text{KS}$ in the free limit.
Central panel: Free spectrum of operator $D_2$ (eq.(\ref{twolinkkernel})), which includes a
taste-dependent mass term with tensor-like structure in taste
space (i.e., a 2-link mass term). Right panel: Free spectrum of
operator $D_4$ (eq.(\ref{fourlinkkernel})), which includes a taste-dependent mass term with
$\gamma_5$ structure in taste space (i.e., a 4-link mass term).}
\label{fig:freespectra}
\end{figure}
To understand the properties of these three different types of
operators it is instructive to start by discussing their spectra
in the free limit. The three panels in Fig.~\ref{fig:freespectra}
show the structure of the spectrum of eigenvalues for $D_\text{KS}$ (for
$D_0$, the spectrum is just trivially shifted by $m$), for $D_2$,
and for $D_4$ in the non-interacting case.
In the free limit the eigenvalues of $D_0$ on a
lattice with $N_\mu$ sites along the $\mu$ direction read
\begin{equation}
\label{D0_free_spectrum}
\lambda = m \pm i \sqrt{\sum_{i=1}^d \sin^2 p_\mu }, \qquad \mbox{with: }\;\; p_\mu=\frac{2\pi}{N_\mu} ( k_\mu + \varepsilon_\mu) , \qquad k_\mu \in \{ 0, 1, 2, \dots , L_\mu/2-1 \},
\end{equation}
with eight degenerate eigenvalues of both signs, and where $\varepsilon_\mu=0$ ($1/2$) if the fermionic field satisfies
(anti-)periodic boundary conditions along the $\mu$ direction.
For $D_2$ the free eigenvalues take the form (for $\rho = 1$)
\begin{equation}
\label{D2_free_spectrum_1}
\lambda_1 = \pm \sqrt{ A_1 - p^2 \pm 2 i \sqrt{A_1 p^2} },
\end{equation}
and
\begin{equation}
\label{D2_free_spectrum_2}
\lambda_2 = \pm \sqrt{ A_2 - p^2 \pm 2 i \sqrt{A_2 p^2} }
\end{equation}
in which the $\pm$ signs are chosen independently and the eigenvalues are doubly degenerate, and having
defined
\begin{equation}
p^2 = \sum_{\mu=1}^4 \sin^2 p_\mu,
\end{equation}
\begin{equation}
A_1 = \frac{ c_1^2 c_2^2 + c_1^2 c_3^2+ c_1^2 c_4^2 +
c_2^2 c_3^2 + c_2^2 c_4^2 + c_3^2 c_4^2}{3} - 2 c
= 0 + \mathcal{O}(a^2),
\end{equation}
\begin{equation}
A_2 = \frac{ c_1^2 c_2^2 + c_1^2 c_3^2+ c_1^2 c_4^2 +
c_2^2 c_3^2 + c_2^2 c_4^2 + c_3^2 c_4^2}{3} + 2 c
= 4 + \mathcal{O}(a^4),
\end{equation}
where $c_\mu = \cos p_\mu$, and $c = c_1 c_2 c_3 c_4$.
Expanding for small momenta gives
\begin{equation}
\lambda_1 = \pm \sqrt{ - p^2 } = \pm i p, \qquad
\lambda_2 = \pm 2 \sqrt{ 1 \pm i p } = \pm 2 \pm i p,
\end{equation}
so that at low momenta, the eigenmodes corresponding to
$\lambda_2$ get a mass of $\pm 2$, while the eigenmodes
corresponding to $\lambda_1$ are massless.
Finally, the free spectrum of $D_4$ reads:
\begin{equation}
\label{D4_free_spectrum}
\lambda_1 = - c \frac{\rho}{a} \pm i \sqrt{p^2}, \quad
\lambda_2 = + c \frac{\rho}{a} \pm i \sqrt{p^2}, \quad
\end{equation}
Note that, in the continuum limit, the point where the spectrum
of the $D_\text{KS}$ operator intersects the real axis corresponds to
four massless modes. By contrast, $D_2$ leads to one mode in each
of the two intersections away from the origin, and two at the
origin. Finally, for $D_4$ one obtains two modes at each of the
two intersections of the spectrum with the real axis.
The taste chirality of the eigenmodes $\Psi$ of $D_4$ and $D_2$, is given
by $(\bar\Psi \Gamma_{55} \Gamma_5 \Psi)$,
where $\Gamma_{55}=\gamma_5\otimes\gamma_5$ exactly, and
$\Gamma_5 = \gamma_5\otimes {\bf 1} + {\cal O}(a)$ in spin $\otimes$ taste.
The taste chirality of the eigenmodes corresponding to eigenvalues
$\lambda_1$ is $c$, while that of the $\lambda_2$-eigenvectors is $-c$.
This is also depicted in Fig.~\ref{lip};
one can see that the taste chirality of the real eigenmodes is $\pm 1$, and is the same ($+1$ or $-1$)
for all modes in a given branch of the $D_4$ or $D_2$ spectrum. The implications of a well-defined
taste chirality have been stressed in \cite{staggeredoverlap2}: if $\Gamma_{55} \Gamma_5 \approx \pm 1$,
then $\Gamma_{55} \approx \pm \Gamma_5$, so that the {\em spin} chirality of the real eigenmodes can
be probed by $\Gamma_{55}$. This is the reason why the index theorem applies to $D_2$ and $D_4$, while
it does not for $D_\text{KS}$ (where both the $\pm 1$ taste chiralities lay on the same single branch.)
\begin{figure}
\centerline{
\includegraphics[width=0.50\textwidth]{psibarG5G55psiv2.pdf}
\includegraphics[width=0.50\textwidth]{two_link_free_withG5.pdf}
}
\label{lip}
\caption{(Left) Taste chirality properties of the $D_2$ and $D_4$ eigenvectors, as a
function of (the component of minimum modulus of) their momentum,
in the free limit. On an infinite lattice, the eigenvectors
associated with real eigenvalues have vanishing momentum and a
well-defined taste chirality $\pm 1$. For eigenmodes corresponding to
eigenvalue $\lambda_1$, the taste chirality becomes $+1$, while for
$\lambda_2$-eigenmodes the taste chirality approaches $-1$. (Right)
The taste chiralities of the eigenmodes of the $D_2$ operator; the size
of the points corresponds to the magnitude of $c$, while the color
indicates the sign: blue for $+c$, red for $-c$.}
\end{figure}
A shift of the spectra by a real value can thus lead to chiral
low-momentum zero modes in each branch, and hence to the possibility
of constructing an appropriate index. A common way to
study the index consists of looking at the flow of eigenvalues
$\lambda(m)$ of:
\begin{equation}
H(m) = \gamma_5 (D + m).
\label{standardflow}
\end{equation}
In general, if $(D + m)$ has a zero-mode $|\Psi_0\rangle$ for
$m=m_0$, then, correspondingly, $H$ has a vanishing eigenvalue
$\lambda(m_0)=0$. With a small perturbation of $m$ away from $m_0$,
i.e. $m=m_0+\delta m$, at leading order the eigenvalues get
displaced by an amount $\langle\Psi_0| \gamma_5 (m-m_0) |
\Psi_0\rangle$, namely one finds \emph{crossings} $\lambda(m)=
\pm (m-m_0)$, if $|\Psi_0\rangle$ is a chiral mode:
$\langle\Psi_0| \gamma_5 |\Psi_0\rangle = \pm 1$.
As pointed out in \cite{Adams:2011xf},
the saturation of $(\bar\Psi \Gamma_{55} \Gamma_5 \Psi)$ at value $\pm 1$
discussed above allows us to trade $\Gamma_5$ for $\Gamma_{55}$ and use the latter in
eq.(\ref{standardflow}).
An alternative
way to look at the spectral flow was proposed in
ref.~\cite{staggeredoverlap1} for the $D_4$ operator, by studying
the eigenvalues of\footnote{Actually, Ref.~\cite{staggeredoverlap1} proposed to consider the
spectral flow of $(i D_\text{KS} - \frac{\rho}{a} \Gamma_5)$. As recognized in \cite{staggeredoverlap2},
that operator is the same as eq.(\ref{Adamsflow}) up to a redefinition of the $\eta_\mu$ phase factors.}
\begin{equation}
\hat{H}(\rho) = \Gamma_{55} D_\text{KS} - \frac{\rho}{a} \Gamma_{5} .
\label{Adamsflow}
\end{equation}
Fig.~3 displays a comparison of the
two different ways to define the spectral flow for the $D_4$
operator (see \cite{Follana:2011kh} for a recent related study): the plots in the top row show the flow of eigenvalues
of $\hat{H}$ as a function of $\rho$ (eq.(\ref{Adamsflow})),
whereas those in the bottom row refer to the ``standard''
definition of the flow, using eq.~(\ref{standardflow}). In each
row, the left panel displays the results from a cold (i.e., free)
configuration on a lattice of size $16^3 \times 32$, while the
central panel is obtained from a cooled configuration of
topological charge $Q=1$ on a lattice of size $8^4$, and finally
the right panel displays the results from a ``rough'' (i.e.,
non-cooled) quenched $Q=-1$ configuration at $\beta=6/g^2=6$, on a
lattice of size $12^4$. In the latter case, the comparison of the
two flow definitions shows that, with the standard definition,
the region around the real axis is populated by a large number of
eigenvalues, preventing one from identifying the crossing with
accuracy.
\begin{figure}
\centerline{
\includegraphics[width=0.39\textwidth]{flow_cf1632_cold.pdf} \hfill
\includegraphics[width=0.39\textwidth]{flow_cf_84_Q1.pdf} \hfill
\includegraphics[width=0.39\textwidth]{flow_cf124_b60.pdf}
}
\centerline{
\includegraphics[width=0.39\textwidth]{flow_cf1632_cold_Adams_classic.pdf} \hfill
\includegraphics[width=0.39\textwidth]{flow_cf_84_Q1_Adams_classic.pdf} \hfill
\includegraphics[width=0.39\textwidth]{flow_cf124_b60_Adams_classic.pdf}
}
\label{Adams_original_classic}
\caption{Comparison of the spectral flow for the $D_4$ operator
as obtained from the eigenvalues of the operators defined in
eq.~(\protect\ref{Adamsflow}) (top row panels) and
eq.~(\protect\ref{standardflow}) (bottom panels). The three plots
(from left to right) in each row show, respectively, the
eigenvalues of $\tilde{H}$ (or $H$) as a function of $\rho/a$ (or
$m$) from a free configuration on a lattice of size $16^3 \times
32$, from a cooled configuration of topological charge $Q=1$ on a
lattice of size $8^4$, and from a non-cooled $Q=-1$ gauge
configuration at $\beta=6$, on a lattice of size $12^4$.}
\end{figure}
Next, it is interesting to compare the identification of the
index, using the spectral flow defined from
eq.~(\ref{standardflow}), for staggered fermions with a taste-dependent mass term, and for conventional Wilson fermions. This
is shown in Fig.~\ref{flowcomparison}: the left, central and
right plot in each row show the spectral flow for $D_4$, $D_2$
and a standard Wilson operator, respectively, while the three
different rows, from top to bottom, refer to a cold
configuration, to a cooled $Q=1$ configuration, and to a non-
cooled $Q=-1$ quenched configuration at $\beta=6$. It is
interesting to observe that, as expected, the spectral flow on a
cooled instanton configuration clearly reveals $N_f \times Q$
crossings. However, one already sees that in the plots of the
$\beta=6$ configuration the gap tends to close. This is
especially the case for the $D_4$ operator, and is related to the
properties that will be discussed in Section~\ref{sec:numerical}.
\begin{figure}
\centerline{\includegraphics[width=0.39\textwidth]{flow_cf1632_cold_Adams_classic.pdf} \hfill
\includegraphics[width=0.39\textwidth]{flow_cf1632_cold_2link.pdf} \hfill
\includegraphics[width=0.39\textwidth]{flow_cf1632_cold_Neuberger.pdf}
}
\centerline{
\includegraphics[width=0.39\textwidth]{flow_cf_84_Q1_Adams_classic.pdf} \hfill
\includegraphics[width=0.39\textwidth]{flow_84_Q1_2link.pdf} \hfill
\includegraphics[width=0.39\textwidth]{flow_cf_84_Q1_Neuberger.pdf}
}
\centerline{
\includegraphics[width=0.39\textwidth]{flow_cf124_b60_Adams_classic.pdf} \hfill
\includegraphics[width=0.39\textwidth]{flow_cf124_b60_2link.pdf} \hfill
\includegraphics[width=0.39\textwidth]{flow_cf124_b60_Neuberger.pdf}
}
\caption{Spectral flows for $D_4$ (left), $D_2$ (center) and a
standard Wilson operator (right), on a cold configuration (top),
on a cooled $Q=1$ configuration (middle), and on a non-cooled
$Q=-1$ quenched configuration at $\beta=6$ (bottom).
The solid blue lines in the top row show analytic results.}
\label{flowcomparison}
\end{figure}
The overall message that can already be drawn from these
observations (before addressing a full-fledged numerical
investigation) is that the gauge field fluctuations in
interacting configurations reduce the width of the gap in the
spectrum, and blur the distinction between light modes and
doublers.
\section{Numerical investigation on interacting configurations}
\label{sec:numerical}
As we showed in the previous section, the fluctuations in typical
interacting configurations lead to a filling of the gap in the
spectral flow for the various lattice Dirac operators that we are
considering, making a proper identification of the index
difficult. A related effect can also be seen directly in the
spectra of the operators: the panels in
Fig.~\ref{Adams_vs_Hoelbling} show a comparison of the spectra of
$D_4$ (top row) and $D_2$ (bottom row) in the free case (left),
and in interacting configurations at $\beta=6$ (central panels, in which different values of $\rho/a$ or $m$ are used) and at
$\beta=5.8$ (right). The figure shows evidence for the superior
robustness of lattice fermions based on the $D_2$ operator, over
$D_4$: at $\beta=5.8$ for example, a gap remains clearly visible
for $D_2$, while it has all but disappeared for $D_4$.
This can be understood from the fact that, since $D_4$ involves
4-link parallel transporters, it is more sensitive to the gauge
field fluctuations in interacting configurations than $D_2$ which
involves 2-link transporters only\footnote{Note that the same reason also
explains the fact that the chirality of near-zero modes of the ordinary
staggered operator is typically small~\cite{deForcrand:1998ng}.}.
\begin{figure*}
\centerline{\includegraphics[width=0.275\textwidth,clip=true]{StaggeredBatman.pdf} \hfill
\includegraphics[width=0.265\textwidth]{Adams_b60_resize.jpg} \hfill
\includegraphics[width=0.265\textwidth,clip=true]{Adams_b60_vary_rho_resize2.jpg} \hfill
\includegraphics[width=0.265\textwidth,clip=true]{Adams_b58_resize.jpg}
}
\centerline{
\includegraphics[width=0.235\textwidth,viewport=-1 5 495 402]{HoelblingBatman.pdf}\hspace{0.3cm}
\includegraphics[width=0.248\textwidth]{2-link_b60.jpg}\hspace{0.3cm}
\includegraphics[width=0.248\textwidth,clip=true]{2-link_b60_Q0.jpg}\hspace{0.3cm}
\includegraphics[width=0.248\textwidth,clip=true]{2-link_b58.jpg}
}
\caption{Spectra of $D_4$ (top) and $D_2$ (bottom) on different
types of configurations. As compared to the free case (left), the
gap in the spectrum of eigenvalues of $D_4$ on interacting
configurations tends to close more rapidly than in the case of
$D_2$. The second and third plot in each row are obtained from
quenched configurations at $\beta=6$ (in the third plot on the
top row, symbols of different colors correspond to different
values of $\rho/a$). Finally, the plots on the right are obtained
from a coarser lattice, at $\beta=5.8$ (roundoff errors cause some
breaking of the complex conjugation symmetry of the spectrum).}
\label{Adams_vs_Hoelbling}
\end{figure*}
However, for practical applications in large-scale simulations,
it is important to remark that, as usual, the effect of gauge
fluctuations can be considerably reduced through some suitably
optimized smearing procedure.
Next, we considered the effectiveness of these operators for
spectroscopy calculations. To this end, we performed a simple
test, by studying the mass $m_{PS}$ of the lightest meson in the
pseudoscalar channel (the pion). We computed the quark propagator
$G(x,y,z,t)$ from a point source, on quenched configurations at
$\beta=6$ on a lattice of size $16^3 \times 32$, then we
evaluated the $\vec{p}=\vec{0}$ correlation function
\begin{equation}
C(t) = \sum_{xyz} G(x,y,z,t) \Gamma_{55} G(x,y,z,t)^\dagger
\Gamma_{55} = \sum_{xyz} |G(x,y,z,t)|^2,
\end{equation}
and extracted $a m_{PS}$ searching for the large-time plateau in
the effective mass plot, as a function of $t$. Monitoring the
behavior of $(a m_{PS})^2$ as a function of $(a m)$, one can
study the partially conserved axial current and the issues
related to mass renormalization.
Fig.~\ref{pion_free_conf} shows the correlators obtained on a
free configuration, for $D_0$ (left panel), for $D_2$ (central
panel) and for $D_4$ (right panel). As expected, the $D_2$
operator leads to a massless pion for both $am=0$ and $am=2$.
\begin{figure*}
\centerline{
\includegraphics[width=0.41\textwidth]{KS_free_pion_prop.pdf} \hfill
\includegraphics[width=0.41\textwidth]{2link_free_pion_prop.pdf} \hfill
\includegraphics[width=0.41\textwidth]{4link_free_pion_prop.pdf}
}
\caption{The exponential decay of the correlation function
associated with the lightest meson in the pseudoscalar channel
for $D_0$ (left), for $D_2$ (center) and $D_4$ (right panel) on a
free configuration, for different values of the bare quark mass.}
\label{pion_free_conf}
\end{figure*}
For an interacting configuration (at $\beta=6$), the comparison
between $D_2$ and $D_4$ shown in Fig.~\ref{pion_interacting_conf}
reveals that for $D_2$ one obtains a massless pion at
approximately $am\sim 1.15$, while for $D_4$ the same happens for
$am \sim 0.25$. Comparing these numbers with the values of the
bare masses corresponding to a massless pseudoscalar state in the
free limit (1 and 2 respectively), these results give an
indication that the mass renormalization is more pronounced for
$D_4$ than for $D_2$. Quantitatively, one can observe that the
renormalization factor grows exponentially with the length of the
parallel transporters used: $(0.25/1)^{1/4} \sim (1.15/2)^{1/2}$,
in agreement with the fact that $D_4$ involves 4-link terms, as
opposed to $D_2$, in which the mass term is constructed from 2-
link terms.
Remarkably, with the $D_2$ operator, the pion mass shows a
square-root behaviour of three different kinds: one can approach
the critical bare quark mass $a m_0 \sim 1.15$ from the left
or from the right, i.e. from the inside or the outside of the
$D_2$ spectrum (the behaviour is square-root-like even though the theory
describes one flavour only -- it is caused by the approach to the Aoki phase).
In addition, one can also approach the other critical quark
mass $a m = 0$, corresponding to the central branch of the
spectrum, which remains zero as in the free case by symmetry of
the average spectrum. The transition from one branch to another seems rather
abrupt, and the scaling of the pion mass can be observed over a
broad range of quark masses approaching zero.
The lesson is that $D_2$ may provide a cost-effective way to
simulate $N_f=2$ light quark species, {\em without} fine-tuning
of the bare quark mass to approach the chiral limit.
\begin{figure*}
\centerline{\includegraphics[width=0.58\textwidth]{2link_mpi_vs_mq_b60.pdf} \hfill
\includegraphics[width=0.58\textwidth]{4link_mpi_vs_mq_b60.pdf}
}
\caption{Tuning of the bare quark mass to obtain a light pion:
the two plots show the pion mass as a function of the bare quark
mass, for $D_2$ (left panel) and $D_4$ (right panel) at
$\beta=6$.}
\label{pion_interacting_conf}
\end{figure*}
\section{Conclusions}
\label{sec:conclusions}
In this work, we performed a numerical study of staggered
Dirac operators with a taste-dependent mass term. We restricted
our attention to operators including mass terms with tensor or
pseudoscalar structure in taste space: their $\gamma_5$-hermiticity
properties are such, that their eigenvalues come in
complex conjugate pairs (as is the case for the usual staggered
Dirac operator), leading to a real fermionic determinant, which
is non-negative in the absence of negative real eigenvalues.
Such operators were proposed by
Adams~\cite{staggeredoverlap1,staggeredoverlap2} and by
Hoelbling~\cite{Hoelbling:2010jw}. We compared their properties
both in the free limit and on interacting configurations at
typical values of the gauge coupling.
Our results show that these operators can indeed be used to
separate the low-lying modes and reduce the number of tastes, in
a way characterized by well-defined topological
properties. Our study of the spectral flow reveals that, for the
4-link operator (with a taste-pseudoscalar mass term), the gap in
the eigenvalue spectrum tends to close rather early, obstructing
an easy identification of the eigenvalue crossings, which are
related to the index. As one might have expected, the 2-link
operator shows markedly more robustness to gauge fluctuations.
We also performed an elementary study of pion propagators,
which shows that the lightest meson is rather easy to isolate
without explicitly disentangling spin and taste degrees of
freedom. Approaching the chiral limit requires in general the
fine-tuning of an additive mass term, as for Wilson fermions. One
important exception occurs for the 2-link operator: if one
chooses the middle branch of the spectrum, one can study a theory
with two tastes, where the additive mass renormalization vanishes
due to the symmetry of the spectrum. Therefore, no fine-tuning is
needed.
Although the 2-link operator was designed to produce a
single taste (with a fine-tuned additive mass), it may well be
that its most promising use is to simulate two tastes without
additive mass renormalization.
Note that the heavy doubler modes do not completely decouple
in that situation. In the background of a topological charge $Q$,
they contribute real eigenvalues $\sim (+1/a)^Q$ and $(-1/a)^Q$,
making the determinant negative when $Q$ is odd. The $\theta$-parameter
is thus equal to $\pi$. This sign $(-1)^Q$ should be removed by hand
(or simply ignored) in order to simulate the $\theta=0$ theory.
Finally, we studied the properties of an overlap operator
with a $D_4$ kernel (see Appendix). We found that its locality properties are
similar to those of the operator based on a Wilson kernel. As it
concerns the computational cost for a quark propagator
calculation, we found that, in the free limit or on very smooth
gauge configurations, the inversion of the operator based on a
kernel with a four-link mass term is almost one order of
magnitude faster than using an overlap with Wilson kernel.
However, we also observed a significant loss of efficiency on
interacting (quenched) configurations at $\beta=6$, where the
operator with the $D_4$ kernel is only approximately twice as
fast as that with a Wilson kernel. The reason for this can
probably be traced back to the fact that the four-link
transporters in the mass term are more sensitive to the effect of
the fluctuations in gauge configurations on coarser lattices.
Our crude assessment indicates that this new, staggered, overlap
operator does not bring a major computational advantage over a
Wilson kernel, while producing two degenerate flavors, but
without the full $SU(2)$ flavor symmetry.
Two copies of an overlap operator with a kernel based on
Hoelbling's 2-link operator would give more flexibility, e.g.
that of simulating two flavors with unequal masses, for a similar
computer effort.
{\bf Note added:}
After this paper was completed, a difficulty with the Hoelbling
operator $D_2$ eq.(\ref{twolinkkernel}) was pointed out by Steve Sharpe,
and clarified by David Adams, during the Yukawa Institute Workshop
``New Types of Fermions on the Lattice''. It appears that the Hoelbling
operator lacks sufficient rotational symmetry, so that fine-tuned
Wilson loop counterterms will presumably be needed to maintain hypercubic
rotational symmetry in unquenched simulations. Adams' operator $D_4$
eq.(\ref{fourlinkkernel}) does not suffer from this problem.
\section*{Acknowledgements}
This research was supported by the Natural Sciences and
Engineering Research Council of Canada, by the Academy of
Finland, project 1134018, and in part by the National Science
Foundation under Grant No. PHY11-25915. Ph.~de~F. thanks the
Yukawa Institute for Theoretical Physics, Kyoto, Japan, for
hospitality. Ph.~de~F. and M.~P. gratefully acknowledge the Kavli
Institute for Theoretical Physics in Santa Barbara, USA, for
support and hospitality during the ``Novel Numerical Methods for
Strongly Coupled Quantum Field Theory and Quantum Gravity''
program, during which part of this work was done. We thank
D.~H.~Adams, M.~Creutz, S.~D\"urr, C.~Hoelbling, S.~Kim, T.~Kimura, T.~Misumi,
A.~Ohnishi, S.~Sharpe and all participants of the Yukawa Institute Workshop
``New Types of Fermions on the Lattice'' for discussions.
|
1205.2068
|
\section{Introduction}
Equilibration can be defined as the evolution of a
system out of equilibrium towards a stationary state in the long-time limit.
For quantum systems, the question arises how equilibration is possible in spite of the linear and unitary time evolution,
how the stationary state depends on the initial conditions,
and to which extent it can be described as a thermal state.
General arguments relate equilibration to dephasing of quantum states~\cite{BS08,LPSW09,Rei08,Rei10,Yuk11}.
Starting from the expansion of an initial state $|\psi(0)\rangle = \sum_{n=1}^N \psi_n |n\rangle$
in the eigenstates $|n\rangle$ of the Hamiltonian $H = \sum_{n=1}^N E_n |n\rangle \langle n|$,
the time evolution of an operator expectation value is given by
\begin{eqnarray}
\langle A(t) \rangle &=& \langle \psi(t) | A | \psi(t) \rangle \nonumber\\
&=& \sum_{m,n=1}^N \psi_m^* \psi_n \, \mathrm e^{\mathrm i(E_m-E_n)t} \langle m | A |n \rangle \;.
\end{eqnarray}
In the thermodynamic limit \mbox{$N \to \infty$}
we can expect that only diagonal terms $m=n$ survive for $t \to \infty$,
such that the long-time limit of the expectation value is
\begin{equation}\label{GeneralA}
\lim_{t \to \pm \infty} \langle A(t) \rangle \simeq \mathrm{tr} \big[ \rho_\infty A \big] \quad (N \to \infty) \;,
\end{equation}
with the density matrix $\rho_\infty = \sum_{n=1}^N |\psi_n|^2 |n\rangle \langle n|$.
This argument can be justified with the Riemann-Lebesgue lemma~\cite{Koe89} that states
\begin{equation}\label{ftprop}
\lim_{t \to \pm \infty} \int_{-\infty}^\infty f(\omega) \, \mathrm e^{\mathrm i \omega t} \, {\mathrm d} \omega = 0
\end{equation}
for any integrable function $f(\omega)$ (here: the density of states $D(\omega)$).
Although this argument explains the origin of equilibration,
not much is learned about the properties of the stationary state $\rho_\infty$.
Especially the question of thermalization is left open.
In this paper we study equilibration and thermalization of dissipative quantum harmonic oscillators, using the standard model of a central oscillator coupled to a harmonic oscillator bath. For this example we can determine the stationary state $\rho_\infty$ explicitly and analyze its dependence on the initial conditions completely.
Crucially, we allow for arbitrary non-thermal bath preparations in our study.
Thermalization is subject to additional conditions in this more general situation, and we show how the temperature of the asymptotic stationary state is obtained from the initial energy distribution of the oscillator bath rather than from the initial bath temperature.
We also include the case study of an interaction quench in an infinite harmonic chain, where undamped oscillations can prevent equilibration at strong damping.
The dissipative quantum harmonic oscillator is studied extensively in the literature~\cite{HR85,FLC85,FLC88,FC07},
covering such diverse topics as Brownian motion~\cite{Ull66a,Ull66b,Ull66c,Ull66d,Gra06}, quantum fluctuations~\cite{NG02}, driven dissipative systems~\cite{ZH95},
entanglement~\cite{JB04}, the existence of local temperatures~\cite{HMH04}, or the second law of thermodynamics~\cite{KM10}.
Reviews are given, e.g., in~\cite{GSI88,Wei99,FRH11}.
With an exact solution this model is also an important example for the derivation of master equations~\cite{UZ89,HPZ92,KG97,BP02}, the discussion of fundamental statistical relations such as
fluctuation-dissipation theorems~\cite{CHT11}
and their connection to detailed balance and the Kubo-Martin-Schwinger condition~\cite{FH12},
or for the assessment of numerical methods that provide a perspective for non-linear models~\cite{TRJF98,TRH00}.
It appears, however, that the questions addressed here have not been previously analyzed in detail, especially not for non-thermal bath preparations.
To obtain our results we proceed as follows.
After introduction of the model in Sec.~\ref{sec:model},
we construct the exact solution for non-thermal initial states in Sec.~\ref{sec:EOM},
including the propagating function in Sec.~\ref{sec:prop}.
Further details, including the extension to driven oscillators, are given in App.~\ref{app:Deriv} and App.~\ref{app:Prop}.
The central results for equilibration and thermalization are formulated in Sec.~\ref{sec:Equil}.
We discuss these results for the example of an infinite chain of harmonic oscillators in Sec.~\ref{sec:Chain},
before we conclude in Sec.~\ref{sec:Summary}.
\section{\label{sec:model}The model}
The Hamiltonian for the dissipative quantum harmonic oscillator,
\begin{equation}\label{Ham}
H = H_S + H_B + H_{SB} \;,
\end{equation}
is the sum of the contribution of the central oscillator,
\begin{equation}\label{HamS}
H_S = \frac 1 2 \Big[ P^2 + \Omega^2 Q^2 \Big] \;,
\end{equation}
the contribution of the harmonic oscillator bath,
\begin{equation}\label{HB}
H_B = \frac 1 2 \sum_{\nu=1}^N \Big[ P^2_\nu + \omega_\nu^2 Q_\nu^2 \Big] \;,
\end{equation}
and the linear interaction term
\begin{equation} \label{HSB}
H_{SB} = Q \sum_{\nu=1}^N \lambda_\nu Q_\nu \;.
\end{equation}
In these expressions, $Q_\nu$, $P_\nu$ are position and momentum operators
with canonical commutation relations, e.g. $[Q_\mu,P_\nu]=\mathrm i \delta_{\mu \nu}$.
Summations over Greek indices, used for bath oscillator operators $Q_\nu$, $P_\nu$,
run from $1, \dots, N$.
We suppress an index for the central oscillator operators.
The size of the coupling constants $\lambda_\nu$ is restricted by the
positivity condition
\begin{equation}\label{pos}
\Omega^2 - \sum_{\nu=1}^N \frac{\lambda_\nu^2}{\omega_\nu^2} \geq 0 \;.
\end{equation}
It guarantees that the normal modes of the total Hamiltonian $H$ have real frequencies,
such that $H$ is bounded from below~\cite{Ull66a}.
A positive Hamiltonian can always be obtained through addition of the term
$(1/2) \sum_{\nu=1}^N (\lambda_\nu/\omega_\nu)^2 Q^2$,
which leads to renormalization of the central oscillator frequency~\cite{Wei99}.
We prefer the present form of the Hamiltonian since it allows for a more natural treatment of the harmonic chain in Sec.~\ref{sec:Chain}.
Of primary interest to us is the central oscillator density matrix
\begin{equation}
\rho_S(t) = \mathrm{tr}_B [ \exp(-\mathrm i H t) \rho(0) \exp( \mathrm i H t) ] \;,
\end{equation}
which is obtained from the initial state $\rho(0)$ through propagation with the total Hamiltonian $H$ and subsequent evaluation of the partial trace $\mathrm{tr}_B$ over the bath degrees of freedom.
A natural choice for $\rho(0)$ are factorizing initial conditions
\begin{equation}\label{RhoInitial}
\rho(0) = \rho_S(0) \otimes \rho_B(0) \;,
\end{equation}
which correspond to the picture that at $t=0$ the previously isolated central oscillator is brought into contact with the oscillator bath.
The restriction to factorizing initial conditions is not essential for the following derivations,
especially not for the long-time limit in Sec.~\ref{sec:Equil},
but it is a natural assumption that simplifies the presentation. For example, mixed central/bath oscillator terms drop out of the expressions for the central oscillator variance (see Sec.~\ref{sec:ExpValues}).
\section{\label{sec:EOM}Solution of the dissipative quantum oscillator for general initial conditions}
The central oscillator density matrix $\rho_S(t)$ can be obtained in various ways,
e.g. through transformation of $H$ to normal modes~\cite{Ull66a,HR85}
or by using path integrals~\cite{GWT84,GSI88} based on the Feynman-Vernon influence functional formalism~\cite{FV63,CL83a,CL83b}.
The arguably simplest approach is the direct solution of the Heisenberg equations of motion for
the operators $Q(t)$, $P(t)$, which reduces to the solution of a classical equation of motion.
The initial conditions $\rho_S(0)$ and $\rho_B(0)$ enter only the evaluation of central oscillator expectation values, such that we can allow for general initial bath states.
The full solution is then given by the propagating function.
\subsection{Reduction to classical equation of motion}
As further detailed in App.~\ref{app:Deriv}, the central piece of information is the solution
$u(t) \in \mathbb{R}$ of the classical equation of motion
\begin{equation}\label{uhom}
\ddot{u}(t) = - \Omega^2 u(t) + \int_{0}^{t} K(t-\tau) u(\tau) \, {\mathrm d}\tau \;,
\end{equation}
which is subject to the conditions
\begin{enumerate}
\item
$u(t)$ solves Eq.~\eqref{uhom} for $t>0$,
\item
$u(t)=0$ for $t<0$,
\item
the initial conditions are $u(0)=0$, $\dot{u}(0)=1$.
\end{enumerate}
We here introduced the damping kernel
\begin{equation}\label{DampKern}
K(t) = \sum\limits_{\nu=1}^N \frac{\lambda_\nu^2}{\omega_\nu} \sin \omega_\nu t \;.
\end{equation}
The function $u(t)$ can be calculated as the Fourier transform~\cite{Dav02}
\begin{equation}\label{UFromFourier2}
u(t) = \frac{2}{\pi} \int_0^\infty \sin \omega t \Im F(\omega + \mathrm i 0^+) \, {\mathrm d}\omega
\end{equation}
of the function
\begin{equation}
F(z) = \Big( \Omega^2 - z^2 + \sum_{\nu=1}^N \frac{\lambda_\nu^2}{z^2 - \omega_\nu^2} \Big)^{-1} \; ,
\end{equation}
writing $F(\omega + \mathrm i 0^+) = \lim_{\eta \to 0, \eta>0} F(\omega + \mathrm i \eta)$.
We note that the positivity condition~\eqref{pos} implies that the poles of $F(z)$ occur on the real axis, such that $u(t)$ is a quasiperiodic function for finite $N$ while $u(t) \to 0$ for $t \to \infty$ is possible in the thermodynamic limit $N \to \infty$.
An explicit example for the computation of $u(t)$ is given for the harmonic chain in Sec.~\ref{sec:Chain} (see Eq.~\eqref{UChain}).
To proceed, we introduce the partial Fourier transforms
\begin{equation}\label{UFour}
\tilde{u}(t, \omega) = \mathrm e^{\mathrm i t \omega} \int_0^t u(\tau) \, \mathrm e^{- \mathrm i \omega \tau} \, {\mathrm d}\tau \;,
\end{equation}
\begin{equation}\label{UFourDot}
\tilde{v}(t, \omega) = \mathrm e^{\mathrm i t \omega} \int_0^t \dot{u} (\tau) \, \mathrm e^{- \mathrm i \omega \tau} \, {\mathrm d}\tau
= u(t) + \mathrm i \omega \tilde{u}(t, \omega) \;,
\end{equation}
and define the matrices
\begin{equation}\label{UMat}
{\mathbf U}(t) = \begin{pmatrix} U_{QQ}(t) & U_{QP}(t) \\ U_{PQ}(t) & U_{PP}(t) \end{pmatrix} = \begin{pmatrix} \dot{u}(t) & u(t) \\ \ddot{u}(t) & \dot{u}(t) \end{pmatrix} \;,
\end{equation}
\begin{equation}\label{UMatOm}
{\mathbf U}(t, \omega) = \begin{pmatrix} \Re \tilde{u}(t, \omega) & \dfrac{\Im \tilde{u}(t, \omega)}{\omega} \\[2ex] \Re \tilde{v}(t, \omega) & \dfrac{\Im \tilde{v}(t, \omega)}{\omega} \end{pmatrix} \;.
\end{equation}
We now obtain the central oscillator operators from the matrix equation
\begin{equation}\label{QSolMat}
\begin{pmatrix} Q(t) \\ P(t) \end{pmatrix}
= \mathbf{U}(t) \begin{pmatrix} Q(0) \\ P(0) \end{pmatrix}
- \sum_{\nu=1}^N \lambda_\nu \mathbf{U}(t,\omega_\nu) \begin{pmatrix} Q_\nu(0) \\ P_\nu(0) \end{pmatrix} \;.
\end{equation}
\subsection{\label{sec:ExpValues}Central oscillator expectation values}
Eq.~\eqref{QSolMat} gives the operators $Q(t)$, $P(t)$ as linear combinations of the operators $Q(0), P(0)$ and $Q_\nu(0), P_\nu(0)$.
This allows us to express central oscillator expectation values for $t\ge 0$
in terms of the initial expectation values at $t=0$.
The linear expectation values are given by the matrix equation
\begin{equation}\label{X}
{\mathbf X}(t) \equiv \begin{pmatrix} \langle Q(t) \rangle \\[0.5ex] \langle P(t) \rangle \end{pmatrix} = {\mathbf U}(t) {\mathbf X}(0) + {\mathbf I}(t) \,,
\end{equation}
with the same shape as Eq.~\eqref{QSolMat}.
In addition to the initial expectation values $\mathbf{X}(0)$ it contains the contribution
\begin{equation}\label{I}
{\mathbf I}(t) = \begin{pmatrix} I_Q(t) \\ I_P(t) \end{pmatrix} = - \sum_{\nu=1}^N \lambda_\nu {\mathbf U}(t, \omega_\nu) \breve{\mathbf X}_\nu \,,
\end{equation}
where we mark the initial bath expectation values
\begin{equation}\label{XBathInitial}
\breve{\mathbf X}_\nu = \begin{pmatrix} \langle Q_\nu(0) \rangle \\ \langle P_\nu(0) \rangle \end{pmatrix}
\end{equation}
with a breve $\breve{\phantom{x}}$ as a notational convention.
Note that if $\breve{{\mathbf X}}_\nu \equiv 0$, e.g. for a thermal bath, the `noise term' $\mathbf{I}(t)$ vanishes. Then, position $\langle Q(t) \rangle$ and momentum $\langle P(t) \rangle$ of the central oscillator follow the classical equation of motion~\eqref{uhom}.
For the quadratic expectation values we define the variance of operators $A$, $B$ as
\begin{equation}
\Sigma_{AB} = \frac{1}{2} \langle A B + B A \rangle - \langle A \rangle \langle B \rangle \;,
\end{equation}
which simplifies to $\Sigma_{AA} = \langle A^2 \rangle -\langle A \rangle^2$ for $A=B$,
and write $\Sigma_{AB}(t) = \Sigma_{A(t) B(t)}$.
We combine the central oscillator variances into the real symmetric matrix
\begin{equation}\label{SigmaMatrix}
\mathbf{\Sigma}(t) = \begin{pmatrix} \Sigma_{Q Q}(t) & \Sigma_{Q P}(t) \\[1ex] \Sigma_{Q P}(t) & \Sigma_{P P}(t) \end{pmatrix} \;,
\end{equation}
and denote the initial bath variances with the matrix
\begin{equation}\label{Sigma_numu}
\breve{\mathbf{\Sigma}}_{\nu\mu} = \begin{pmatrix} \Sigma_{Q_\nu Q_\mu}(0) & \Sigma_{Q_\nu P_\mu}(0) \\[1ex] \Sigma_{Q_\mu P_\nu}(0) & \Sigma_{P_\nu P_\mu}(0)
\end{pmatrix} \;.
\end{equation}
Note the index swap in the off-diagonal elements,
and recall that mixed central oscillator/bath variances such as $\Sigma_{Q Q_\nu}$ vanish for our choice~\eqref{RhoInitial} of factorizing initial conditions.
We now obtain with Eq.~\eqref{QSolMat} the matrix equation
\begin{equation}\label{Sigma}
\mathbf{\Sigma}(t) = {\mathbf U}(t) \mathbf{\Sigma}(0) {\mathbf U}^T(t) + {\mathbf C}(t) \;.
\end{equation}
Similar to Eq.~\eqref{X}, the first term results from the time evolution of the central oscillator according to the classical equation of motion~\eqref{uhom},
and appears in the same form for an isolated oscillator.
Only the second term
\begin{eqnarray}\label{C}
{\mathbf C}(t) &=& \begin{pmatrix} C_{QQ}(t) & C_{QP}(t) \\ C_{QP}(t) & C_{PP}(t) \end{pmatrix} \nonumber\\
&=& \sum_{\nu, \mu=1}^N \lambda_\nu \lambda_\mu {\mathbf U}(t, \omega_\nu) \breve{\mathbf{\Sigma}}_{\nu\mu} {\mathbf U}^T(t, \omega_\mu)
\end{eqnarray}
depends on the initial bath oscillator variances $\breve{\mathbf{\Sigma}}_{\nu\mu}$.
Mixed terms in $\mathbf{U}(t)$, $\mathbf{U}(t,\omega_\nu)$ do not appear for factorizing initial conditions.
\subsection{\label{sec:TDLimit}The thermodynamic limit}
Because $u(t)$ is a quasi-periodic function for a finite number $N$ of bath oscillators,
equilibration becomes possible only in the thermodynamic limit $N \to \infty$.
We assume that for $N \to \infty$ the density of states
\begin{equation}\label{D}
D(\omega) = \frac{1}{N} \sum_{\nu=1}^N \delta(\omega - \omega_\nu)
\end{equation}
converges to a continuous function.
Note that $D(\omega)=0$ for $\omega<0$ since the bath oscillator frequencies are positive.
The coupling constants
appear in the damping kernel $K(t)$ and in Eq.~\eqref{uhom}
as $\lambda_\nu^2$, and must thus scale as $N^{-1/2}$.
We assume that
\begin{equation} \label{LambdaCont}
\lambda_\nu = \lambda(\omega_\nu) / \sqrt{N}
\end{equation}
with a continuous function $\lambda(\omega)$, and introduce the bath spectral function
\begin{equation}\label{GammaCont}
\gamma(\omega) = D(\omega) \frac{\lambda(\omega)^2}{\omega} \,,
\end{equation}
with $\gamma(\omega)=0$ for $\omega < 0$.
The damping kernel is now given as
\begin{equation}\label{Kcont}
K(t) = \int_0^\infty \gamma(\omega) \sin \omega t \, {\mathrm d}\omega \,,
\end{equation}
and the positivity condition reads
\begin{equation}\label{PosCont}
\Omega^2 \ge \int_0^\infty \frac{\gamma(\omega)}{\omega} \, {\mathrm d}\omega \;.
\end{equation}
The function $F(z)$ in Eq.~\eqref{UFromFourier2} for $u(t)$ can be written as
\begin{equation}\label{FCont}
F(z) = \Big( \Omega^2 - z^2 + \int_0^\infty \frac{\omega \gamma(\omega)}{z^2 - \omega^2} \, {\mathrm d}\omega \Big)^{-1} \;.
\end{equation}
Under mild assumptions, the evaluation of the $\omega$-integral in this equation is possible by contour integration and results in
\begin{equation}\label{FAnalytic}
F(z) = \Big( \Omega^2 - z^2 + \Gamma(z) \Big)^{-1}
\end{equation}
for $\Im z > 0$,
where the complex function $\Gamma(z)$ with
$\gamma( \omega) = \mp (2/\pi) \Im \Gamma( \pm \omega + \mathrm i 0^+)$ is the analytic continuation of $\gamma(\omega)$ into the upper half of the complex plane (see Sec.~\ref{sec:Chain} for an example). For future use in Sec.~\ref{sec:Equil} we note the relation
$\gamma(\omega) |F(\omega)|^2 = (2/\pi) \Im F(\omega + \mathrm i 0^+)$ that follows from this representation.
The analytic properties of $F(z)$ determine the behavior of $u(t)$ in the long-time limit, which is essential for equilibration (see condition (E0) in Sec.~\ref{sec:Equil}): It is $u(t) \to 0$ for $t \to \infty$ if and only if $F(z)$ has no isolated poles.
The linear expectation values $\breve{\mathbf{X}}_\nu$ enter Eq.~\eqref{I} with the prefactors $\lambda_\nu \propto N^{-1/2}$.
To obtain a finite result for the sum over $N$ terms,
also $\breve{\mathbf{X}}_\nu$ has to scale as $N^{-1/2}$, which leads to the ansatz
\begin{equation}
\breve{{\mathbf X}}_\nu = \frac{1}{\sqrt{N}} \breve{{\mathbf X}}(\omega_\nu)
\end{equation}
with a continuous vector-valued function $\breve{\mathbf{X}}(\omega)$.
Then, Eq.~\eqref{I} becomes
\begin{equation}\label{ICont}
{\mathbf I}(t) = - \int_0^\infty D(\omega) \lambda(\omega) {\mathbf U}(t, \omega)\breve{{\mathbf X}}(\omega) \, {\mathrm d}\omega \;.
\end{equation}
The variances $\breve{\mathbf{\Sigma}}_{\nu \mu}$ enter the sum in Eq.~\eqref{C} with the prefactors $\lambda_\nu \lambda_\mu \propto N^{-1}$.
We must now distinguish between the $N^2$ off-diagonal terms $\nu \ne \mu$,
which require an additional $1/N$ prefactor for convergence,
and the $N$ diagonal terms $\nu = \mu$.
Therefore, we make the ansatz
\begin{equation}\label{SigmaAnsatz}
\breve{\mathbf{\Sigma}}_{\nu \mu} = \frac{1}{N} \breve{\mathbf{\Sigma}}^{(2)}(\omega_\nu,\omega_\mu)
+ \breve{\mathbf{\Sigma}}^{(1)}(\omega_\nu) \delta_{\nu \mu}
\end{equation}
with continuous matrix-valued functions $\breve{\mathbf{\Sigma}}^{(2)}(\omega_1,\omega_2)$ and $\breve{\mathbf{\Sigma}}^{(1)}(\omega)$.
Then, $\mathbf{C}(t)$ from Eq.~\eqref{C} is the sum of
the off-diagonal term
\begin{eqnarray}\label{CContOD}
{\mathbf C}^{(2)}(t) &=& \iint_{0}^{\infty} D(\omega_1) D(\omega_2) \lambda(\omega_1) \lambda(\omega_2) \nonumber\\
&&\times {\mathbf U}(t, \omega_1) \breve{\mathbf{\Sigma}}^{(2)}(\omega_1,\omega_2) {\mathbf U}^T(t, \omega_2) \, {\mathrm d}\omega_1 \, {\mathrm d}\omega_2 \qquad
\end{eqnarray}
and the diagonal term
\begin{equation}\label{CCont}
{\mathbf C}^{(1)}(t) = \int_0^\infty \omega \gamma(\omega) {\mathbf U}(t, \omega) \breve{\mathbf{\Sigma}}^{(1)}(\omega) {\mathbf U}^T(t, \omega) \, {\mathrm d}\omega \,.
\end{equation}
If the initial bath state is uncorrelated,
such as for a thermal bath or a general product state $\rho_B(0) = \rho_B^1(0) \otimes \cdots \otimes \rho_B^N(0)$,
the off-diagonal term $\mathbf{C}^{(2)}(t)$ vanishes.
When we construct the propagating function in the next subsection,
we will conveniently assume that the initial bath state $\rho_B(0)$ is a Gaussian state.
For the long-time limit, the situation of interest here,
this assumption can be justified in the thermodynamic limit on general grounds~\cite{CDEO08,CE10}.
The principal mechanism is illustrated with counting arguments of the following kind:
Consider an uncorrelated bath state, where only $N$ diagonal terms contribute in any sum over the bath oscillators. If we consider a higher order cumulant of bath operators, say $Q_3(\nu) = \langle Q_\nu^3 \rangle - 3 \langle Q_\nu^2\rangle \langle Q_\nu \rangle + 2 \langle Q_\nu \rangle^3$ as mentioned before, it appears with a prefactor $\lambda_\nu^3 \propto N^{-3/2}$.
Therefore, the total contribution of these cumulants scales as $N \times N^{-3/2} = N^{-1/2}$
and vanishes in the limit $N \to \infty$.
Similar counting arguments can be given for cumulants involving two or more bath oscillators in the presence of correlations.
Because higher order cumulants vanish and only linear and quadratic bath expectation values survive
the $N \to \infty$ and $t \to \infty$ limit, we can treat the bath state as Gaussian in any calculation of the central oscillator density matrix.
For the formulation and proof of a strict result, which is involved even under some simplifying assumptions, see~\cite{CE10}.
\subsection{\label{sec:prop}The propagating function}
Knowledge of the expectation values $\mathbf X(t)$, $\mathbf \Sigma(t)$ does not suffice to obtain the central oscillator density matrix $\rho_S(t)$, unless we
restrict ourselves completely to Gaussian oscillator states (cf. Eq.~\eqref{GaussianWigner} below).
Otherwise, the general solution is given by the propagating function $J(\cdot)$ that, in position representation, expresses the density matrix $\rho_S(q,q',t) = \langle q|\rho_S(t) |q' \rangle$ for $t \ge 0$ as
\begin{equation}\label{JDef}
\rho_S(q_f, q_f', t) = \iint_{-\infty}^\infty \! J(q_f, q_f', q_i, q_i', t) \rho_S(q_i, q_i', 0) \, {\mathrm d} q_i \, {\mathrm d} q_i' \;.
\end{equation}
This expression must hold for all $\rho_S(0)$ and $t \ge 0$, and a fixed initial bath state $\rho_B(0)$.
The propagating function can be calculated using path integrals
and the result for a thermal bath is given, e.g., in~\cite{GSI88}.
Within our approach it is more natural to construct the propagating function
directly, using only that an initial Gaussian state of the joint central/bath oscillator system remains a Gaussian state during time evolution with the bilinear Hamiltonian $H$.
With respect to the final remarks in Sec.~\ref{sec:TDLimit}, we assume a Gaussian bath state $\rho_B(0)$.
We can then consider the most general ansatz for $J(\cdot)$ that maps an initial Gaussian state $\rho_S(0)$ in Eq.~\eqref{JDef}
onto a Gaussian state $\rho_S(t)$ for $t \ge 0$,
and will find that the parameters of this ansatz are fully specified through the linear maps~\eqref{X},~\eqref{Sigma} of $\mathbf X(t)$, $\mathbf \Sigma(t)$.
The result is valid for arbitrary $\rho_S(0)$ in Eq.~\eqref{JDef},
but we do not need to consider non-Gaussian $\rho_S(t)$ explicitly.
To translate this argument into equations we work with the Wigner function~\cite{Wig32,Schl01}
\begin{equation}
W(q,p,t) = \frac{1}{2\pi} \int_{-\infty}^\infty \rho_S \Big(q+\frac{s}{2},q-\frac{s}{2},t \Big) \, \mathrm e^{-\mathrm i p s} \, {\mathrm d} s
\end{equation}
instead of the density matrix $\rho_S(q,q',t)$ in position representation
(see also Refs.~\cite{CRV03,FRH11} for a related calculation).
The propagating function $J_W(\tilde{\mathbf{x}},\mathbf{x},t) = J_W(\tilde{q},\tilde{p},q,p,t)$ is defined by the relation
\begin{equation}\label{JMap}
W(\tilde {\mathbf x},t) = \int_{{\mathbb{R}^2}} J_W(\tilde {\mathbf x}, \mathbf x, t) W(\mathbf x,0) \, {\mathrm d} \mathbf x \;,
\end{equation}
where we write $W(\mathbf{x},t) = W(q,p,t)$ with
$\mathbf{x} = (q, p)^T$ and ${\mathrm d} \mathbf x = {\mathrm d} q \, {\mathrm d} p$ for abbreviation.
Note that $W(\mathbf x,t)$ and $J_W(\tilde {\mathbf x}, \mathbf x,t)$ are real functions.
A Gaussian state to given $\mathbf X(t)$, $\mathbf \Sigma(t)$ has the Wigner function
\begin{equation}\label{GaussianWigner}
W_g(\mathbf{x},t) = \frac{\exp \big[ - \frac{1}{2} (\mathbf{x} - \mathbf{X}(t)) \cdot \mathbf{\Sigma}^{-1}(t) (\mathbf{x} - \mathbf{X}(t))
\big] }{2 \pi \sqrt{\det \mathbf{\Sigma}(t)}} \;,
\end{equation}
and the most general expression for $J_W(\cdot)$ that respects this structure is an exponential function of the 14 linear and quadratic terms in the coordinates $q,p, \tilde{q}, \tilde{p}$.
The normalization $\int_{\mathbb{R}^2} W(\mathbf x,t) {\mathrm d}\mathbf x =1 $ of Wigner functions implies the condition
\begin{equation}\label{JNorm}
\int_{\mathbb{R}^2} J_W(\tilde {\mathbf x}, \mathbf x,t) \, {\mathrm d} \tilde{ \mathbf x} = 1
\end{equation}
on the propagating function, which fixes the prefactors of the 5 terms $q^2, p^2, qp, q, p$ in the initial coordinates.
This leaves $9$ free parameters that have to be fixed in accordance with the linear transformations
~\eqref{X},~\eqref{Sigma} of expectation values.
The final result is
\begin{widetext}
\begin{equation}\label{JW}
J_W(\tilde{\mathbf{x}},\mathbf{x},t) = \frac{\exp \Big[ - \dfrac{1}{2} \big( \tilde{\mathbf{x}} - \mathbf{U}(t) \mathbf{x} - \mathbf{I}(t) \big) \cdot \mathbf{C}^{-1}(t)
\big( \tilde{\mathbf{x}} - \mathbf{U}(t) \mathbf{x} - \mathbf{I}(t) \big) \Big] }
{2 \pi \sqrt {\det \mathbf{C}(t)}} \;,
\end{equation}
\end{widetext}
where the $4+3+2=9$ parameters are the entries of the $2 \times 2$ matrix $\mathbf{U}(t)$ from Eq.~\eqref{UMat}, the symmetric and positive definite $2 \times 2$ matrix $\mathbf{C}$(t) from Eq.~\eqref{C}, and the two-dimensional vector $\mathbf{I}(t)$ from Eq.~\eqref{I}.
In order to check that this expression indeed reproduces the transformations~\eqref{X},~\eqref{Sigma}, we can express the expectation values at $t \ge 0$ in terms of those at $t=0$ through the evaluation of simple Gaussian integrals.
To give an example, it is
\begin{eqnarray}\label{QTrans1}
{\langle Q(t) \rangle} &=& \int_{\mathbb{R}^2} \tilde{q} \, W(\tilde {\mathbf x},t) \, {\mathrm d} \tilde{ \mathbf x} \nonumber\\
&=& \int_{\mathbb{R}^4} \tilde{q} J_W(\tilde {\mathbf x}, \mathbf x,t) W(\mathbf x,0) \, {\mathrm d} \tilde {\mathbf x } \, {\mathrm d} \mathbf x \;.
\end{eqnarray}
The integral of $\tilde{q} J_W(\tilde {\mathbf x}, \mathbf x,t)$
over $\tilde {\mathbf x}$ is a Gaussian integral with a linear term,
and gives
\begin{equation}\label{QTrans2}
\int_{\mathbb{R}^2} \tilde{q} J_W(\tilde { \mathbf{x}}, \mathbf x,t) \, {\mathrm d} \tilde {\mathbf x} =
U_{QQ}(t) q + U_{QP}(t) p + I_Q(t) \;.
\end{equation}
The final integration over $\mathbf x$ in Eq.~\eqref{QTrans1},
which now involves the right hand side of~\eqref{QTrans2},
generates the initial expectation values $\langle Q(0) \rangle$, $\langle P(0) \rangle$.
Therefore, we obtain the relation
$\langle Q(t) \rangle = U_{QQ}(t) \langle Q(0) \rangle + U_{QP}(t) \langle P(0) \rangle + I_Q(t) = \dot{u}(t) \langle Q(0) \rangle + u(t) \langle P(0) \rangle + I_Q(t)$ in accordance with Eq.~\eqref{X}.
Following this recipe, we find that the given expression~\eqref{JW} for the propagating function $J_W(\cdot)$ reproduces the entire transformations~\eqref{X},~\eqref{Sigma} of the expectation values $\mathbf X(t)$, $\mathbf \Sigma(t)$, as we required.
If $\mathbf{C}(t) \to 0$, we get a representation of the distribution $\delta(\tilde{\mathbf{x}}-\mathbf{U}(t) \mathbf{x} - \mathbf{I}(t))$ from Eq.~\eqref{JW}.
In particular for $t=0$,
where $\mathbf{U}(0) = 1$, $\mathbf{I}(0)=0$ in addition to $\mathbf{C}(0)=0$,
we have the correct result $J_W(\tilde{\mathbf{x}},\mathbf{x},0)=\delta(\tilde{\mathbf{x}}-\mathbf{x})$ in Eq.~\eqref{JMap}.
We note that the conveniently simple derivation of $J_W(\cdot)$ relies on the use of Wigner functions.
Of course, the expressions for $\rho_S(q_f, q_f',t)$ in position representation often reported in the literature can be recovered from Eq.~\eqref{JW} (see App.~\ref{app:Prop}).
\section{\label{sec:Equil}Equilibration and thermalization}
The results from the previous section allow us to study the behavior of the central oscillator density matrix $\rho_S(t)$ in the long-time limit $t \to \infty$.
We can classify the behavior according to the general criteria of equilibration and thermalization.
Equilibration means convergence to a stationary state
as expressed in the two conditions
\begin{itemize}
\item[(E1)] the central oscillator density matrix $\rho_S(t)$ converges for $t \to \infty$,
\item[(E2)] the stationary state $\rho_S^\infty=\lim_{t \to \infty} \rho_S(t)$ is independent of $\rho_S(0)$ \;.
\end{itemize}
Note that $\rho^\infty_S$ will depend on the initial bath state $\rho_B(0)$.
Note further that the above definition of equilibration does not distinguish between stationary equilibrium states and stationary non-equilibrium states with finite heat flows. The latter cannot arise for a single bath with continuous initial conditions as in Eq.~\eqref{SigmaAnsatz} such that condition (E1) is sufficient for the present study.
Equilibration (E1) implies convergence of central oscillator expectation values for $t \to \infty$.
This, in turn, requires convergence of the matrix $\mathbf U(t)$ in Eqs.~\eqref{X},~\eqref{Sigma}.
Because the only stationary solution of the homogeneous differential Eq.~\eqref{uhom} is $u(t) \equiv 0$,
convergence of $\mathbf U(t)$ is equivalent to $\mathbf U(t) \to 0$ or $u(t) \to 0$ for $t \to \infty$.
Therefore, we assume in this section the condition
\begin{itemize}
\item[(E0)] $u(t) \to 0$ for $t \to \infty$
\end{itemize}
as the prerequisite for equilibration (E1).
Under this assumption, we will be able to show convergence of expectation values and, building on this result, convergence of the central oscillator density matrix.
In the weak damping limit,
condition (E0) is equivalent to $\gamma(\Omega)>0$ (taking the thermodynamic limit for granted).
This expresses the basic fact that equilibration occurs through energy exchange with the environment, which is not possible for an isolated oscillator with $\gamma(\Omega)=0$.
We note that a small value of $\gamma(\Omega)$ can result in long transients
that prevent equilibration over the observation time.
Thermalization additionally requires that the stationary state $\rho_S^\infty$ is a thermal state,
and we have the three increasingly stronger properties
\begin{itemize}
\item[(T1)] the stationary state $\rho_S^\infty$ is a thermal state,
\item[(T2)] the stationary state is a thermal state $\rho_S^\infty \propto \mathrm e^{-H_S/T_\infty}$ of the central oscillator,
\item[(T3)] the temperature $T_\infty$ of the stationary thermal state $\rho_S^\infty$ is independent of the central oscillator frequency.
\end{itemize}
We will see that the stationary state is always Gaussian, which implies property (T1). Property (T2) reduces to an equipartition condition on the central oscillator variances that determine the Gaussian state, while property (T3) leads to a strong condition on the initial bath state.
\subsection{Expectation values in the long-time limit}
The assumption $\mathbf U(t) \to 0$ for $t \to \infty$ implies that
the terms $\mathbf{U}(t) \mathbf{X}(0)$ in Eq.~\eqref{X}
and $\mathbf{U}(t) \mathbf{\Sigma}(0) \mathbf{U}^T(t)$ in Eq.~\eqref{Sigma}
drop out of the expressions for $\mathbf{X}(t)$ and $\mathbf\Sigma(t)$ in the long-time limit.
Only the terms $\mathbf{I}(t)$ and $\mathbf{C}(t)$, which depend exclusively on the initial bath preparation, can survive the $t \to \infty$ limit:
All information about the initial central oscillator state is lost.
We can not immediately draw a conclusion about the long-time behavior because the functions $\tilde{u}(t,\omega)$, $\tilde{v}(t,\omega)$ from Eqs.~\eqref{UFour},~\eqref{UFourDot} do not converge for $t\to \infty$.
Instead, we note that $\tilde{u}(t,\omega)$ behaves asymptotically as
\begin{equation}\label{UAsym}
\tilde{u}_{as}(t,\omega) \simeq \mathrm e^{\mathrm i \omega t} \int_0^\infty u(\tau) \, \mathrm e^{-\mathrm i \omega \tau} \, {\mathrm d} \tau \qquad (t \to \infty) \;.
\end{equation}
Similarly, it follows $\tilde{v}(t,\omega) \simeq \mathrm i \omega \tilde{u}_{as}(t,\omega)$ for $t \to \infty$ from Eq.~\eqref{UFourDot}.
Consequently, the matrix $\mathbf{U}(t,\omega)$ behaves asymptotically as
\begin{equation}
{\mathbf U}(t, \omega) \simeq \begin{pmatrix} \Re \tilde{u}_{as}(t,\omega) & \dfrac{ \Im \tilde{u}_{as}(t,\omega)}{\omega} \\[2ex] -\omega \Im \tilde{u}_{as}(t,\omega) & \Re \tilde{u}_{as}(t,\omega) \end{pmatrix} \quad (t \to \infty) \;,
\end{equation}
and remains oscillating for $t \to \infty$ even if $u(t) \to 0$.
The contributions to the term $\mathbf{I}(t)$ in Eq.~\eqref{ICont}, say to $\langle Q(t)\rangle$, are of the form
\begin{equation}
- \Re \int_0^\infty D(\omega) \lambda(\omega) \tilde{u}_{as}(t,\omega) \breve{X}_Q(\omega) \, {\mathrm d} \omega \;.
\end{equation}
The integrand depends on $t$ through the factor $\mathrm e^{\mathrm i \omega t}$ from Eq.~\eqref{UAsym},
such that the integral is the Fourier transform
of an integrable (by assumption even continuous) function of $\omega$.
If we recall the Riemann-Lebesgue lemma~\eqref{ftprop}
we see that $\mathbf{I}(t) \to 0$ for $t \to \infty$.
Altogether, it follows that
the position and momentum expectation values vanish
in the long-time limit, i.e. $\mathbf{X}(t) \to 0$ for $t \to \infty$.
For the variances,
a finite contribution can survive the $t \to \infty$ limit
because the squares of the matrix elements of $\mathbf{U}(t,\omega)$ occur in $\mathbf{C}(t)$.
For example, the diagonal term $\mathbf{C}^{(1)}(t)$ from Eq.~\eqref{CCont} contributes to $\Sigma_{QQ}(t)$ the integral
\begin{equation}\label{C1Explicit}
C^{(1)}_{QQ}(t) = \int_0^\infty \omega \gamma(\omega) \, c_{QQ}(t,\omega) \, {\mathrm d} \omega
\end{equation}
of the function
\begin{eqnarray}\label{C1QQ}
c_{QQ}(t,\omega) &=& [\Re \tilde{u}(t,\omega)]^2 \, \breve{\Sigma}_{QQ}^{(1)}(\omega) \nonumber\\
&&+ \frac{2 [\Re \tilde{u}(t,\omega)] [\Im \tilde{u}(t,\omega)] }{\omega} \breve{\Sigma}_{QP}^{(1)}(\omega) \nonumber\\
&&+ \frac{[\Im \tilde{u}(t,\omega)]^2}{\omega^2} \breve{\Sigma}_{PP}^{(1)}(\omega) \;.
\end{eqnarray}
Here we write, using the notation from Eq.~\eqref{SigmaMatrix},
\begin{equation}
{\mathbf C}^{(1)}(t) = \begin{pmatrix} C^{(1)}_{Q Q}(t) & C_{Q P}^{(1)}(t) \\[1ex] C_{Q P}^{(1)}(t) & C_{P P}^{(1)}(t) \end{pmatrix}
\end{equation}
for the matrix elements of $ {\mathbf C}^{(1)}(t)$ and
\begin{equation}
\breve{\mathbf \Sigma}^{(1)}(\omega) = \begin{pmatrix} \breve \Sigma^{(1)}_{Q Q}(\omega) & \breve \Sigma_{Q P}^{(1)}(\omega) \\[1ex] \breve \Sigma_{Q P}^{(1)}(\omega) & \breve \Sigma_{P P}^{(1)}(\omega) \end{pmatrix}
\end{equation}
for the matrix elements of $ \breve{\mathbf \Sigma}^{(1)}(\omega)$ from Eq.~\eqref{SigmaAnsatz}.
The contribution from the first term in $c_{QQ}(t,\omega)$ is
\begin{equation}
\int_0^\infty \omega \gamma(\omega) [\Re \tilde{u}_{as}(t,\omega)]^2 \breve\Sigma^{(1)}_{QQ}(\omega) \, {\mathrm d}\omega \;.
\end{equation}
If we expand the square $ [\Re \tilde{u}_{as}(t,\omega)]^2$ according to
\begin{equation}\label{exprcompl}
[\Re \mathrm e^{\mathrm i \omega t} z]^2 = \frac{|z|^2}{2} + \frac{z_r^2-z_i^2}{2} \cos 2 \omega t - z_r z_i \sin 2 \omega t \,,
\end{equation}
for a complex number $z$ with $z_r = \Re z$, $z_i = \Im z$,
we see that in the above integral a contribution $|\tilde{u}_{as}(t,\omega)|^2/2$ remains finite for $t \to \infty$, while the oscillatory terms with $\cos 2 \omega t$, $\sin 2 \omega t$ vanish according to the Riemann-Lebesgue lemma~\eqref{ftprop}.
Similar expressions are obtained for the remaining terms in $\mathbf{C}^{(1)}(t)$.
The off-diagonal term $\mathbf{C}^{(2)}(t)$ from Eq.~\eqref{CContOD} is given by a double Fourier integral
and contains only oscillatory terms in the two frequencies $\omega_1$, $\omega_2$.
Therefore, $\mathbf{C}^{(2)}(t) \to 0$ for $t \to \infty$.
We can now collect the finite contributions from the different terms in $\mathbf{C}^{(1)}(t)$, to find that the central oscillator variances converge to stationary values
$\mathbf{\Sigma}^\infty = \lim_{t \to \infty} \mathbf\Sigma(t)$
in the long-time limit.
They are given by
\begin{equation}\label{CQQLong}
\Sigma^\infty_{QQ} = \int_0^\infty \gamma(\omega) \left|\int_0^\infty \mathrm e^{\mathrm i \tau \omega} u(\tau) \, {\mathrm d}\tau \right|^2 \frac{\breve{\mathcal{E}}(\omega)}{\omega} \, {\mathrm d} \omega \,,
\end{equation}
\begin{equation}\label{CPPLong}
\Sigma^\infty_{PP} = \int_0^\infty \gamma(\omega) \left|\int_0^\infty \mathrm e^{\mathrm i \tau \omega} u(\tau) \, {\mathrm d}\tau \right|^2 \omega \breve{\mathcal{E}}(\omega) \, {\mathrm d} \omega \,,
\end{equation}
\begin{equation}\label{CQPLong}
\Sigma^\infty_{QP} = 0 \,,
\end{equation}
where
\begin{equation}\label{ECal}
\breve{\mathcal{E}}(\omega) = \frac{1}{2} \left(\omega^2 \breve{\Sigma}^{(1)}_{QQ}(\omega) + \breve{\Sigma}_{PP}^{(1)}(\omega) \right) \;.
\end{equation}
Comparison with Eqs.~\eqref{UFromFourier2},~\eqref{FAnalytic}
gives the alternative expressions
\begin{equation}\label{CQQLongF}
\Sigma^\infty_{QQ} = \frac{2}{\pi} \int_0^\infty \Im F(\omega + \mathrm i 0^+) \,\frac{\breve{\mathcal{E}}(\omega)}{\omega} \, {\mathrm d}\omega \,,
\end{equation}
\begin{equation}\label{CPPLongF}
\Sigma^\infty_{PP} = \frac{2}{\pi} \int_0^\infty \Im F(\omega + \mathrm i 0^+) \, \omega \breve{\mathcal{E}}(\omega) \, {\mathrm d}\omega \,.
\end{equation}
Recall that $F(\omega+\mathrm i 0^+)$ is a continuous function according to our assumption $u(t) \to 0$.
As noted before, the values $\mathbf{\Sigma}^\infty$ are independent of the initial central oscillator state. Furthermore, the initial bath state $\rho_B(0)$ occurs only through the frequency-resolved
energy distribution $\breve{\mathcal{E}}(\omega)$.
In particular,
the known equations for thermal baths~\cite{HR85}
are recovered whenever $\breve{\mathcal{E}}(\omega) = E(T,\omega)$,
where
\begin{equation}\label{thermprop}
E(T,\Omega) = \frac{\Omega}{2} \coth \frac{\Omega}{2T}
\end{equation}
is the energy of a thermal oscillator at temperature $T$.
Because there are no separate conditions
on the two functions $\breve{\Sigma}^{(1)}_{QQ}(\omega)$, $\breve{\Sigma}^{(1)}_{PP}(\omega)$,
thermalization is possible also in non-thermal environments (see below).
Eqs.~\eqref{CQQLong}--\eqref{CQPLong} follow directly if we assume a thermal bath from the outset, with initial conditions
$\omega^2 \breve{\Sigma}^{(1)}_{QQ} (\omega) = \breve{\Sigma}^{(1)}_{PP} (\omega) = E(T,\omega)$ and $\breve{\Sigma}^{(1)}_{QP}(\omega)=0$.
Equipartition of energy
allows us to combine the terms in Eq.~\eqref{C1QQ} to $c_{QQ}(t,\omega)=|\tilde u (t,\omega)|^2 E(T,\omega)/\omega^2$, which depends only on the modulus of $\tilde{u}(t,\omega)$.
We can then drop the exponential factor $\mathrm e^{\mathrm i \omega t}$ from Eq.~\eqref{UAsym},
and convergence of $\mathbf \Sigma(t)$ is evident. This short cut is not available in the general case.
\subsection{Equilibration of the central oscillator}
If the initial bath state $\rho_B(0)$ and the central oscillator state $\rho_S(0)$ are both Gaussian states,
the central oscillator density matrix $\rho_S(t)$ is Gaussian for all $t \ge 0$.
Then, $\rho_S(t)$ is completely determined by the values of $\mathbf X(t)$, $\mathbf \Sigma(t)$, and their convergence suffices to establish equilibration (E1), and also (E2), in this case.
Otherwise, for non-Gaussian initial states $\rho_S(0)$,
we can use the propagating function
$J_W(\tilde{\mathbf{x}},\mathbf{x},t)$ from Eq.~\eqref{JW} to find $\rho_S(t)$ for $t \to \infty$.
Recall that according to Sec.~\ref{sec:TDLimit} we can assume that the initial bath state
is Gaussian in the thermodynamic limit, which allows for the construction given in Sec.~\ref{sec:prop}.
Equilibration follows now from the observation that $J_W(\tilde{\mathbf{x}},\mathbf{x},t)$
converges for $t \to \infty$ whenever $\mathbf X(t)$, $\mathbf \Sigma(t)$ converge.
The long-time limit
\begin{equation}\label{JWLong}
J^\infty_W(\tilde{\mathbf{x}}) = \lim_{t \to \infty} J_W(\tilde{\mathbf{x}},\mathbf{x},t) =
\frac{\exp \Big[\! - \dfrac{1}{2} \tilde{\mathbf{x}} \cdot (\mathbf{\Sigma}^\infty)^{-1}
\tilde{\mathbf{x}} \Big] }
{2 \pi \sqrt {\det \mathbf{\Sigma}^\infty}} \;.
\end{equation}
is obtained
through substitution of $\lim_{t\to\infty} \mathbf I(t)= 0$ and
$\lim_{t\to\infty} \mathbf C(t)= \mathbf \Sigma^\infty$ from Eqs.~\eqref{CQQLong}---\eqref{CQPLong}.
Because $\mathbf U (t) \to 0$, the result
does not depend on $\mathbf{x}$.
The long-time limit of the Wigner function $W_S^\infty(\mathbf x)=\lim_{t \to \infty} W_S(\mathbf x,t)$ follows immediately with Eq.~\eqref{JMap}:
The integration over $\mathbf x$ in the resulting expression
\begin{equation}
W_S^\infty(\tilde { \mathbf x}) = \int_{\mathbb{R}^2} J_W^\infty(\tilde {\mathbf x}) W_S(\mathbf x,0) \, {\mathrm d} \mathbf x = J_W^\infty(\tilde {\mathbf x})
\end{equation}
evaluates to one because $W_S(\mathbf x,0)$ is normalized,
such that $W_S^\infty(\mathbf x)$ is equal to $J^\infty_W(\mathbf{x})$.
In other words, the stationary state $\rho_S^\infty$ is a Gaussian state~\eqref{GaussianWigner} with parameters $\mathbf X = 0$, $\mathbf{\Sigma} = \mathbf{\Sigma}^\infty$.
These parameters
depend on the initial bath state according to Eqs.~\eqref{CQQLong}, \eqref{CPPLong}, but they are independent from the initial central oscillator state.
This proves equilibration (E1) and (E2) for general initial central oscillator states.
In particular, the stationary state is Gaussian also for non-Gaussian initial states.
We note that the propagating function in position representation does not converge in the long-time limit (cf. App.~\ref{app:Prop}), which prevents an equally simple argument.
\subsection{Thermalization of the central oscillator}
Because the stationary state $\rho_S^\infty$ in the long-time limit is a Gaussian state for which only $\Sigma^\infty_{QQ}$, $\Sigma^\infty_{PP}$ are non-zero,
it can always be interpreted as the thermal equilibrium state of some harmonic oscillator.
This establishes the weakest thermalization property (T1).
The effective oscillator frequency $\Omega_\infty$ and temperature $T_\infty$ associated with
$\rho_S^\infty$
are
\begin{equation}\label{OmTLong}
\Omega^2_\infty = \frac{\Sigma^\infty_{PP}}{\Sigma^\infty_{QQ}} \;, \qquad T_\infty = \frac{\Omega_\infty}{2} \mathrm{arcoth}^{-1} \Big[ 2 \sqrt{ \Sigma^\infty_{QQ} \Sigma^\infty_{PP} } \, \Big] \,.
\end{equation}
Generally, $\Omega_\infty$
is not equal to the central oscillator frequency $\Omega$
such that the stronger property (T2) is not fulfilled.
By Eq.~\eqref{OmTLong}, the condition $\Omega_\infty = \Omega$ is equivalent to
equipartition of kinetic and potential energy $ \langle P^2\rangle = \Sigma^\infty_{PP} = \Omega^2 \Sigma^\infty_{QQ} =\Omega^2 \langle Q^2\rangle$.
The violation of this condition arises from
the integrations over $\omega$ in Eqs.~\eqref{CQQLong},~\eqref{CPPLong} or~\eqref{CQQLongF},~\eqref{CPPLongF}, which cover a finite energy range and include values $\omega \ne \Omega$.
Quantum corrections of this type are characteristic for strong damping~\cite{HR85}.
Equipartition of energy is achieved in the limit of weak damping ($\gamma(\Omega) \to 0$, when according to Eq.~\eqref{FAnalytic}
the function $(2/\pi) \Im F(\omega + \mathrm i 0^+)$ in Eqs.~\eqref{CQQLongF},~\eqref{CPPLongF}
converges to $2 \delta(\omega^2 - \Omega^2)= (\delta(\omega+ \Omega) + \delta(\omega-\Omega))/\Omega$.
Therefore, the values
$\Omega^2 \Sigma_{QQ}^\infty = \Sigma_{PP}^\infty = \breve{\mathcal{E}}(\Omega)$
are obtained.
This establishes the stronger thermalization property (T2) in the weak damping limit.
For these values of $ \Sigma_{QQ}^\infty, \Sigma_{PP}^\infty$ it is (cf. Eq.~\eqref{thermprop})
\begin{equation}\label{TWC}
\Omega^{(\mathrm{WD})}_\infty = \Omega \;, \qquad T^{(\mathrm{WD})}_\infty(\Omega) = \frac{\Omega}{2} \mathrm{arcoth}^{-1} \, \frac{2 \breve{\mathcal{E}}(\Omega)}{\Omega} \,,
\end{equation}
such that the stationary state $\rho^\infty_S$ is a thermal equilibrium state of the central oscillator.
The temperature $T_\infty(\Omega)$ is determined by the energy $\breve{\mathcal{E}}(\Omega)$ of the bath oscillator at frequency $\Omega$ in the initial state.
Note that the assumption (E0) implies $\gamma(\Omega) \ne 0$ and $D(\omega) \ne 0$, such that the value of $\breve{\mathcal{E}}(\Omega)$ is defined.
In particular, $\breve{\mathcal{E}}(\Omega) \ge \Omega/2$ and the argument of $\mathrm{arcoth}(\cdot)$ is equal to or greater than one.
Still, the asymptotic temperature $T_\infty=T^{(\mathrm{WD})}_\infty(\Omega)$ from Eq.~\eqref{TWC}
is a function of $\Omega$.
The functional dependence is determined by the choice of $\breve{\mathcal{E}}(\omega)$.
If we demand, for the strongest thermalization property (T3), that $T_\infty$ is independent of $\Omega$
we have to solve Eq.~\eqref{TWC}
to obtain the condition
\begin{equation}\label{ThermAtWC}
\breve{\mathcal{E}}(\omega) = \frac{\omega}{2} \coth \frac{\omega}{2T_\infty} \;.
\end{equation}
Note that this is a condition on the particular combination $\breve{\mathcal{E}}(\omega)$ of initial bath variances $\breve{\Sigma}^{(1)}_{QQ}(\omega)$, $\breve{\Sigma}^{(1)}_{PP}(\omega)$, and not on the individual functions.
Therefore, any initial bath preparation with $ \breve{\mathcal{E}}(\omega) = E(T_0,\omega)$ results in the same stationary states as the thermal bath at temperature $T_0$.
One example for this additional freedom is the choice
\begin{equation}\label{NonThermalThermalization}
\breve{\Sigma}^{(1)}_{QQ}(\omega)= \frac{\coth (\omega/2 T_0) - 1/2}{\omega} \;, \qquad \breve{\Sigma}_{PP}^{(1)}(\omega) = \omega/2 \;,
\end{equation}
and arbitrary $\breve{\Sigma}_{QP}^{(1)}(\omega)$.
It can be realized, e.g., by superposition of coherent oscillator states at different positions.
This initial bath state is not a thermal state for $T_0 > 0$, in particular it violates equipartition of energy $\omega^2 \breve{\Sigma}^{(1)}_{QQ}(\omega) = \breve{\Sigma}_{PP}^{(1)}(\omega)$.
But since $\breve{\mathcal{E}}(\omega) = E(T_0,\omega)$ we find that the stationary central oscillator state $\rho_S^\infty$ is identical to that obtained with a thermal bath at temperature $T_0$:
Thermalization is well possible in non-thermal environments, even those far from thermal equilibrium.
\subsection{\label{sec:EquilWithList}Summary}
In summary, we have a hierarchy of conditions for equilibration and thermalization:
\begin{enumerate}\itemindent6ex\hangindent6ex
\item[(E1), (E2)] the central oscillator equilibrates whenever $u(t) \to 0$ for $t \to \infty$,
\item[(T1)] the stationary state is always a Gaussian and thermal state,
\item[(T2)] equipartition of kinetic and potential energy occurs precisely at weak damping,
\item[(T3)] the asymptotic temperature $T_\infty$ is independent of the central oscillator frequency under the additional condition~\eqref{ThermAtWC} on $\breve{\mathcal{E}}(\omega)$.
\end{enumerate}
It is a special feature of linear systems such as the one studied here
that equilibration depends only on the asymptotic behavior of the solution $u(t)$ of a classical equation of motion~\eqref{uhom}.
Another feature is that the stationary state always is Gaussian such that equilibration implies thermalization, albeit only in the weak sense of property (T1).
We noted earlier that in the situation studied here, with coupling to a single bath, a stationary state does not admit finite heat flows as would become possible for several baths with different preparations $\breve {\mathcal E}(\omega)$. Therefore, conditions (E1), (E2) capture the standard notion of thermodynamic equilibrium.
We note that a consistent definition of thermalization requires the strong property (T3).
Suppose we deal with two central oscillators with frequencies $\Omega_1 \ne \Omega_2$.
In the weak damping limit, the stationary state is the product state of two independent thermal states with respective temperatures $T_\infty(\Omega_1)$ and $T_\infty(\Omega_2)$.
Such a state is only a thermal state of the combined system comprising the two oscillators
if $T_\infty(\Omega_1)=T_\infty(\Omega_2)$.
Therefore, thermalization of multiple oscillators, already in the weak sense (T1), requires the strong property (T3) and thus condition~\eqref{ThermAtWC}
(but recall that this condition can be fulfilled also for non-thermal environments as in Eq.~\eqref{NonThermalThermalization}).
\section{\label{sec:Chain}The infinite harmonic chain}
\begin{figure}
\includegraphics[width=\linewidth]{Fig1}
\caption{\label{fig:Chain}(Color online) Sketch of the infinite harmonic chain as defined in Eqs.~\eqref{ChainHB},~\eqref{ChainHSB}.}
\end{figure}
As an example for equilibration in a non-thermal environment
we consider an infinite chain of harmonic oscillators (see Fig.~\ref{fig:Chain}).
Oscillators in the right ($n \ge1 $) and left ($n \le -1$) half of the chain, with frequency $\Omega_b$, are coupled to their neighbors ($n \pm 1$) with spring constant $k_b$.
They form the harmonic oscillator bath for the central oscillator at $n=0$,
with oscillator frequency $\Omega$ and coupling $k$ to the oscillators at $n=\pm 1$.
For $\Omega=\Omega_b$ and $k=k_b$ we obtain a homogeneous, translational invariant chain.
Related examples have been studied in numerous publications, see e.g.~\cite{Ull66c,Rub60,RH69,Aga71,HRN71,TS94,TY94,DR09}.
The behavior for thermal initial conditions, e.g. in a homogeneous chain~\cite{Rub60} or a chain with a single heavy mass~\cite{Ull66c}, is well understood.
Equilibration in a harmonic chain with non-thermal initial conditions as discussed in Refs.~\cite{RH69,Aga71} can be expressed in terms of our conditions from Sec.~\ref{sec:EquilWithList}.
General arguments for the appearance of Gaussian states in the long-time limit are given in~\cite{TS94,CE10}.
Still, a satisfactory and explicit analysis of equilibration and thermalization of the simple chain in non-thermal environments is missing.
Some studies assume too quickly that equilibration implies thermalization, in the sense of our condition (T1), failing to note, e.g., that the appearance of Gaussian states is the general behavior of linear systems and unrelated to thermalization as expressed by condition (T3).
According definitions of `temperature' have to be taken with care.
In addition we must carefully analyze the role of undamped oscillatory behavior that prevents equilibration and, therefore, thermalization.
\subsection{Mapping onto the central oscillator model}
To address the harmonic chain within the formalism from Secs.~\ref{sec:model}---\ref{sec:Equil} we must transform the Hamilton operator
\begin{equation}\label{ChainHB}
H_B = \frac{1}{2} \sum_{n=1}^\infty \Big[ p_n^2 + \Omega_b^2 q_n^2 \Big] - k_b \sum_{n=1}^\infty q_n q_{n+1}
\end{equation}
for the harmonic oscillator bath (with operators $q_n$, $p_n$ for the oscillator at site $n \ne 0$) to normal modes.
The same transformation has to be applied to the operator
$k q_1$ in the coupling term
\begin{equation}\label{ChainHSB}
H_{SB} = - k \, Q ( q_1 + q_{-1} )
\end{equation}
between the central oscillator and the chain oscillators at $n = \pm 1$.
It suffices to treat one of the two half-infinite chains explicitly, say the right chain $n \ge 1$ as in Eq.~\eqref{ChainHB},
and include a factor of two in $\gamma(\omega)$ to account for the left chain $n \le -1$.
Note that in doing so we implicitly assume identical initial conditions for both sides of the chain and thus exclude the possibility of stationary non-equilibrium states with finite heat flow between the right and left half-infinite chain.
The normal modes of $H_B$ are the standing wave solutions
$f_\nu(n) \propto \sin \big( \frac{\pi \nu n}{N+1} \big)$ (for a finite chain of length $N$),
and after a few lines of calculation we obtain the spectral function
\begin{equation}\label{SpecChain}
\gamma(\omega) = \frac{2}{\pi} \frac{k^2}{k_b^2} \sqrt{4 k_b^2 - \big( \Omega_b^2 - \omega^2 \big)^2} \qquad \mathrm{for} \; |\Omega_b^2-\omega^2| < 2 k_b
\end{equation}
in the thermodynamic limit $N \to \infty$.
It is $\gamma(\omega)=0$ for $|\Omega_b^2-\omega^2| > 2 k_b$,
and we impose the positivity condition $\Omega_b^2 \ge 2 k_b \ge 0$
to exclude negative frequencies of the bath.
To proceed it is convenient to introduce the dimensionless model parameters
\begin{equation}\label{NormalizedParams}
\kappa_b = \frac{2 k_b}{\Omega_b^2} \;, \qquad
\kappa = \frac{2 k}{\Omega_b^2} \;, \qquad
\Omega_r = \frac{\Omega}{\Omega_b} \;,
\end{equation}
and to use the normalized quantities
\begin{equation}
\bar{\omega} = \frac{\omega}{\Omega_b} \;, \qquad
\bar{t} = t \Omega_b \;, \qquad
\bar{u}(\bar{t}) = \Omega_b u(\bar{t}) \;.
\end{equation}
Note that $0 \le \kappa_b \le 1$.
\subsection{Conditions for equilibration in the harmonic chain}
As discussed in Sec.~\ref{sec:Equil}, equilibration depends entirely on the decay of the function $u(t)$ for $t \to \infty$, and thus on the absence of poles in $F(z)$ from Eq.~\eqref{FCont}.
To obtain $F(\omega)$, we use the representation~\eqref{FAnalytic} with the complex function
\begin{equation}
\Gamma(z) = \frac{k^2}{k_b^2} \Big( z^2 - \Omega_b^2 \mp \sqrt{(\Omega_b^2 - z^2)^2 - 4 k_b^2} \, \Big) \;,
\end{equation}
where the branch cut of the root must be chosen along the positive real axis,
and the minus (plus) sign applies for $\Re z > 0$ ($\Re z < 0$).
Note that the positivity condition~\eqref{pos}, which can now be rewritten as $\Omega^2 +\Gamma(\mathrm i 0^+) \ge 0$, requires that
\begin{equation}\label{PosForChain}
\Omega_r^2 \ge \frac{\kappa^2}{\kappa_b^2} \left( 1 - \sqrt{1 - \kappa_b^2} \right)\;.
\end{equation}
\begin{figure}
\includegraphics[width=0.7\linewidth]{Fig2}
\caption{\label{fig:GamZ} (Color online) Real (dashed curve) and imaginary (solid curve) part of $\Gamma(\omega + \mathrm i 0^+)$ for $\kappa_b=1/2$ and $\omega > 0$.
For $\bar{\omega}^2 = \{ 1-\kappa_b , 1 , 1+\kappa_b \}$ the function value is
$\{-1, -\mathrm i, 1\} \times (\kappa \Omega_b)^2/\kappa_b$, respectively.}
\end{figure}
Before we can determine the function $u(t)$ with Eq.~\eqref{UFromFourier2}
we must consider the possibility of isolated poles of $F(z)$.
According to Eq.~\eqref{FAnalytic} we have to compare the functions $\omega^2 - \Omega^2$ and $\Re \Gamma(\omega + \mathrm i 0^+)$ in regions where $\Im \Gamma(\omega + \mathrm i 0^+) = 0$.
From the qualitative behavior of $\Gamma(\omega + \mathrm i 0^+)$, shown in Fig.~\ref{fig:GamZ},
we deduce that isolated poles of $F(z)$ do not exist if and only if
the inequalities
\begin{equation}\label{NoPoles}
1 - \frac{\kappa_b^2 - \kappa^2}{\kappa_b} \le \Omega_r^2 \le 1 + \frac{\kappa_b^2 - \kappa^2}{\kappa_b}
\end{equation}
are fulfilled.
The first inequality excludes poles in the interval $\bar{\omega}^2 < 1 - \kappa_b$,
the second inequality in the interval $\bar{\omega}^2 > 1 + \kappa_b$.
Another more fundamental restriction is the positivity condition~\eqref{PosForChain}, which is however less restrictive than the present condition.
\begin{figure}
\includegraphics[width=0.7\linewidth]{Fig3}
\caption{\label{fig:params} (Color online) Diagram of the admissible parameter space for equilibration according to condition~\eqref{NoPoles}.
The white triangular region above the solid black lines is the maximal set of allowed parameter combinations.
Outside of this region an isolated pole exists even in the weak damping limit $\kappa \to 0$.
For $\kappa > 0$, the region of admissible parameters shrinks as depicted by the dashed black curves.
The parameter combinations of homogeneous chains ($\Omega_r=1$) corresponds to the cusps $\kappa_b=\kappa$ of the curves, marked with red dots.
The parameter combinations of chains with a single heavy mass~\cite{Ull66c}
correspond to the intersections of the curves with the $\kappa_b=1$ line at $\Omega_r=\kappa$,
marked with green squares.
At these points, condition~\eqref{NoPoles} coincides with the positivity condition~\eqref{PosForChain}.}
\end{figure}
The admissible parameter combinations for equilibration of the harmonic chain that follow from condition~\eqref{NoPoles} are depicted in Fig.~\ref{fig:params}.
We note the basic restrictions
\begin{equation}\label{NoPolesWC}
\kappa \le \kappa_b \qquad \rm{and} \qquad |1-\Omega_r^2| \le \kappa_b \;.
\end{equation}
The second inequality guarantees that the central oscillator frequency $\Omega_r$ lies within the interval $\bar{\omega} \in [\sqrt{1-\kappa_b},\sqrt{1+\kappa_b}]$ where $\gamma(\bar{\omega})>0$.
If this is fulfilled, equilibration is always possible for sufficiently small $\kappa$.
Since $\kappa_b \le 1$, it restricts the admissible parameters to the rectangle $(\kappa_b, \Omega_r^2) \in [0,1]\times [0,2]$.
Condition~\eqref{NoPoles} is always fulfilled for the homogeneous chain
(and we note that $\kappa=\kappa_b$ requires $\Omega_r=1$).
The chain studied by Ullersma corresponds to parameters $\kappa_b=1$ and $\Omega_r = \kappa$ ($\Omega_r^2$ equals the mass ratio $\mu$ in~\cite{Ull66c}).
Condition~\eqref{NoPoles} is fulfilled if $\Omega_r \le 1$, i.e. only for a heavy mass.
Both examples lie on the boundary of the admissible parameter space,
with one or two of the inequalities in~\eqref{NoPoles} becoming equalities.
\subsection{Dynamical evolution of the harmonic chain}
Depending on parameters, the harmonic chain features rich dynamical behavior.
For parameter combinations that fulfill condition~\eqref{NoPoles}
the explicit result for $u(t)$ from Eq.~\eqref{UFromFourier2} reads
\begin{eqnarray}\label{UChain}
\bar{u}(\bar{t}) &=& \frac{2 \kappa^2}{\pi} \int_{\sqrt{1 - \kappa_b}}^{\sqrt{1+ \kappa_b}} \sin \bar{\omega} \bar{t} \, \sqrt{\kappa_b^2 - (1 - \bar{\omega}^2)^2} \nonumber\\
&&\times \frac{1}{\kappa_b^2 (\bar{\omega}^2-\Omega_r^2)^2 - 2 \kappa^2 (\bar{\omega}^2-1)(\bar{\omega}^2-\Omega_r^2)
+ \kappa^4} \, {\mathrm d}\bar{\omega} \;. \nonumber\\
\end{eqnarray}
For parameter combinations violating condition~\eqref{NoPoles}
isolated poles of $F(z)$ occur and additional (undamped) sine functions $\xi_i \sin \bar \Omega_i t$ must be added to this expression. According to Eq.~\eqref{FAnalytic}, the poles of $F(z)$ are the solutions of $\Omega^2 - \Omega_i^2 + \Gamma(\Omega_i) = 0$,
which gives a quadratic equation for the harmonic chain such that zero, one, or two (positive) poles are possible.
\begin{figure}
\includegraphics[width=0.8\linewidth]{Fig4}
\caption{\label{fig:paramsK2} (Color online) The $(\Omega_r^2,\kappa_b)$ parameter space of the infinite harmonic chain for $\kappa = 1/2$.
The solid/dotted black curves give the boundary of the two regions
defined by each of the inequalities from condition~\eqref{NoPoles}.
The central oscillator equilibrates for parameters in the unshaded region above the solid curve, where the condition is fulfilled (note that $\kappa_b \le 1$).
For parameters lying between the solid and dotted curves one of the two inequalities is violated and a single isolated pole of $F(z)$ exists.
Below the dotted curve $F(z)$ has two isolated poles.
The dashed region to the left indicates where the positivity condition~\eqref{PosForChain} is violated, but such parameters already violate the first inequality in~\eqref{NoPoles}.
Both conditions coincide at $\kappa_b=1$, $\Omega_r=\kappa$.
The dashed red/green lines indicate the path followed in the next Fig.~\ref{fig:isomod},
the crosses marked a---d indicate the parameters used in Fig.~\ref{fig:ut}.
}
\end{figure}
\begin{figure}
\includegraphics[width=0.8\linewidth]{Fig5}
\caption{\label{fig:isomod} (Color online) Position $\bar{\Omega}_i$ and total weight $\xi$ of isolated poles of $F(z)$.
We set $\kappa=1/2$ and change $\Omega_r$, $\kappa_b$ along the dashed path from Fig.~\ref{fig:paramsK2}, i.e.
from $\Omega_r=0$, $\kappa_b=0.2$ to $\Omega_r=2$, $\kappa_b=0.8$.
The position of the poles is compared to the continuum of bath modes
in the interval $\bar{\omega}^2 \in [1-\kappa_b,1+\kappa_b]$,
filling the grey area around $\bar\Omega^2_i=1$ in the plot.
Between the two vertical dashed lines at $\Omega_r=1$, $\kappa_b=1/2$ (left)
and $\kappa_b=0.8$, $\Omega_r=1.4875$ (right)
no poles exist in agreement with condition~\eqref{NoPoles}.
For $\Omega_r^2 \lessapprox 0.12628$, in the dashed region to the left, the positivity condition~\eqref{PosForChain} is violated and one $\bar{\Omega}_i^2$ becomes negative.
}
\end{figure}
As an example let us consider the case $\kappa=1/2$.
The restrictions on the parameters arising from the positivity condition~\eqref{PosForChain} and the stronger condition~\eqref{NoPoles} are summarized in Fig.~\ref{fig:paramsK2}.
We now follow the dashed path in this figure and plot the position $\bar \Omega_{1/2}^2$ of isolated poles and their total weight $\xi=\xi_1+\xi_2$ in Fig.~\ref{fig:isomod}.
Only for parameter combinations in the white unshaded area in Fig.~\ref{fig:paramsK2},
which corresponds to the part between the dashed vertical lines in Fig.~\ref{fig:isomod},
condition~\eqref{NoPoles} is fulfilled.
Accordingly, only panel (c) in Fig.~\ref{fig:ut} (the parameter combination ``c'' in Figs.~\ref{fig:paramsK2},~\ref{fig:isomod})
shows a situation where $\bar{u}(\bar{t}) \to 0$ for $t \to \infty$.
Otherwise, one (parameter combination ``d'') or two (``a'', ``b'') isolated poles exist if one or both inequalities from~\eqref{NoPoles} are violated. Then, the amplitude of oscillations in $\bar{u}(\bar{t})$ remains finite in the long-time limit.
\begin{figure}[ht]
\includegraphics[width=0.48\linewidth]{Fig6a}
\hfill
\includegraphics[width=0.48\linewidth]{Fig6b} \\[1ex]
\includegraphics[width=0.48\linewidth]{Fig6c}
\hfill
\includegraphics[width=0.48\linewidth]{Fig6d}
\caption{\label{fig:ut}Function $\bar u(\bar{t})$ for the harmonic chain with $\kappa=1/2$.
The parameters from panels (a)--(d) correspond to the crosses in Figs.~\ref{fig:paramsK2},~\ref{fig:isomod}.
They are:
(a) $\kappa_b=0.2$, $\Omega_r^2=0.4$ (two poles $\bar{\Omega}_1=0.48$, $\bar{\Omega}_2=1.10$, $\xi_1=0.82$, $\xi_2=0.10$),
(b) $\kappa_b=0.4$, $\Omega_r^2=1$ (two poles $\bar{\Omega}_1=0.76$, $\bar{\Omega}_2=1.20$, $\xi_1=0.26$, $\xi_2=0.26$),
(c) $\kappa_b=0.6$, $\Omega_r^2=1$ (no pole),
(d) $\kappa_b=0.8$, $\Omega_r^2=1.6$ (one pole $\bar{\Omega}_1=1.35$, $\xi_1=0.50$).
}
\end{figure}
\begin{figure}[ht]
\includegraphics[width=0.48\linewidth]{Fig7a}
\hfill
\includegraphics[width=0.48\linewidth]{Fig7b}
\caption{\label{fig:ut2} (Color online) Function $\bar u(\bar{t})$ for the inhomogeneous (left panel,
with $\Omega_r=1$, $\kappa_b=0.5$, $\kappa=0.1$)
and homogeneous (right panel,
with $\Omega_r=1$, $\kappa_b=0.1$, $\kappa=0.1$) harmonic chain at weak damping.
The dashed red curves indicate the exponential decay from Eq.~\eqref{ChainOhm} (left panel) and the
asymptotic decay $\propto 1/\sqrt{\bar{t}}$ of the Bessel function from Eq.~\eqref{HomChainBessel} (right panel).}
\end{figure}
For strong damping situations ($\kappa \sim 1$) shown in Fig.~\ref{fig:ut}
the function $u(t)$ deviates significantly from an exponentially decaying function, even in the absence of poles (panel (c)).
Exponential decay occurs
only for weak damping $\kappa \ll \kappa_b$.
For $|\Omega_r^2-1| \ll \kappa_b$ we have
\begin{equation}\label{ChainOhm}
\bar{u}(\bar{t}) = \frac{\sin( \Omega_r \bar{t} \,)}{\Omega_r} \, \exp \Big(\! - \! \frac{\kappa^2}{2 \kappa_b} \bar{t} \Big) \qquad (\kappa \ll 1) \;,
\end{equation}
as plotted in Fig.~\ref{fig:ut2} (left panel).
Note that the case $\gamma(\Omega) = 0$,
with an undamped sine function in the weak damping limit, is excluded by the second inequality in Eq.~\eqref{NoPolesWC}.
For the homogeneous chain, with $\Omega_r=1$ and $\kappa_b=\kappa$,
the weak damping limit gives a different result.
Since $\kappa_b = \kappa$,
the width of the continuum of bath states shrinks to zero for $\kappa \to 0$
such that we do not obtain exponential decay of $u(t)$.
Instead, it is
\begin{equation}\label{HomChainBessel}
\bar{u}(\bar{t}) =J_0\! \left(\!\frac{\kappa \bar{t}}{2} \right) \, \sin \bar{t} \qquad (\kappa \ll 1, \, \mathrm{hom. \; chain})
\end{equation}
with the Bessel function $J_0(x)$ (cf. Refs.~\cite{Rub60,RH69,Aga71}).
According to condition~\eqref{NoPoles} isolated poles of $F(z)$ cannot occur in this situation.
From the asymptotic behavior of the Bessel function we find that
here $\bar u(\bar t)$ decays only as $2 (\pi \kappa \bar{t})^{-1/2}$ for $\bar{t} \gg 1$,
as shown in the right panel of Fig.~\ref{fig:ut2}.
Exponential decay in the weak damping limit is only achieved if the coupling $\kappa$ of the central oscillator to the chain becomes small also in comparison to the width ($\sim \kappa_b$) of the continuum of bath states.
\subsection{Thermalization after a quench}
According to the previous discussion,
the central oscillator in the harmonic chain
equilibrates precisely for parameter combinations that fulfill condition~\eqref{NoPoles}.
We now study, under these conditions, thermalization after a quench that generates a non-thermal environment for the central oscillator (cf. Eq.~\eqref{InitChainT2} below).
\subsubsection{Initial conditions generated by the quench}
We imagine that for $t < 0$ all oscillators are decoupled ($\kappa = \kappa_b=0$) and in thermal equilibrium at temperature $T_0$.
Every oscillator has the same variance
\begin{equation}\label{ChainInitial}
\Omega_b^2 \breve{\Sigma}_{qq}(n) = \breve{\Sigma}_{pp}(n) = E(T_0, \Omega_b) \;,
\end{equation}
and we do not need to specify further initial expectation values if we are only interested in the stationary state in the long-time limit.
At $t=0$ we quench the system by cranking up the coupling to finite values $\kappa, \kappa_b > 0$. Since $\breve{\Sigma}_{qq}(n)$, $\breve{\Sigma}_{pp}(n)$ do not depend on $n$,
transformation to the normal modes of the bath results in constant functions
\begin{equation}
\Omega_b^2 \breve{\Sigma}^{(1)}_{QQ}(\omega) = \breve{\Sigma}^{(1)}_{PP}(\omega) = E(T_0, \Omega_b)
\end{equation}
for the initial bath variances at $t=0$.
The initial bath state is uncorrelated with $\breve{\mathbf{\Sigma}}^{(2)}(\omega_1, \omega_2) = 0$.
According to Sec.~\ref{sec:Equil}, the stationary state in the long-time limit
depends only on the frequency-resolved energy $\breve{\mathcal{E}}(\omega)$ of the initial bath state,
which for the present example is given by the function
\begin{equation}\label{InitChainT2}
\breve{\mathcal{E}}(\omega) = \frac{1+(\omega/\Omega_b)^2}{2} E(T_0,\Omega_b) \;.
\end{equation}
This function acquires a dependence on $\omega$ through the dispersion of the bath modes after the quench,
but it does not fulfill Eq.~\eqref{ThermAtWC}.
We thus see that the thermal equilibrium state of uncoupled oscillators before the quench corresponds to a non-thermal state of the coupled chain of oscillators after the quench.
According to condition (T3) from Sec.~\ref{sec:EquilWithList} we expect that the temperature $T_\infty$ of the stationary state depends on the central oscillator frequency $\Omega_r$ even at weak coupling.
From Eqs.~\eqref{CQQLong}--\eqref{CQPLong} or Eqs.~\eqref{CQQLongF},~\eqref{CPPLongF}, the variances in the long-time limit are obtained as
\begin{equation}\label{SigQQChain}
{\Sigma}_{QQ}^\infty = \frac{1}{2 \Omega_b^2} \left( 1+ \frac{1}{ \Omega_r^2 - \frac{\textstyle \kappa^2}{\textstyle \kappa_b^2} \left(1 - \sqrt{1- \kappa_b^2} \right)} \right) E(T_0,\Omega_b) \;,
\end{equation}
\begin{equation}\label{SigPPChain}
{\Sigma}_{PP}^\infty = \frac{1}{2} \left( 1+ \Omega_r^2 \right) E(T_0, \Omega_b) \;.
\end{equation}
We will give further results using normalized quantities
\begin{equation}
\bar{\Omega}_\infty = \Omega_\infty/\Omega_b \;,
\quad
\bar{T}_\infty = T_\infty / \Omega_b \;,
\quad
\bar{T}_0 = T_0 / \Omega_b \;.
\end{equation}
choosing $\Omega_b$ as the unit of energy.
\subsubsection{Thermalization (T2)}
We recall that according to property (T1) the stationary state is always a thermal state of some harmonic oscillator Hamiltonian, such that we should check the stronger property (T2).
From Eq.~\eqref{OmTLong}, the effective frequency associated with the stationary state is
\begin{equation}\label{ChainOm}
\frac{\bar \Omega_\infty^2}{\Omega_r^2} = \frac{ \Omega_r^2 + 1}{ \Omega_r^2 + \Big[ 1 - \frac{\textstyle \kappa^2}{\textstyle \Omega_r^2 \kappa_b^2} \left(1 - \sqrt{1- \kappa_b^2} \right) \Big]^{-1} } \;.
\end{equation}
We observe that equipartition of energy, i.e. $\bar \Omega_\infty = \Omega_r$, can be achieved only in the weak damping limit $\kappa \to 0$.
For $\kappa > 0$, it is always $\bar \Omega_\infty < \Omega_r$.
This confirms the conditions given for property (T2) in Sec.~\ref{sec:EquilWithList}.
\subsubsection{Thermalization (T3)}
For weak damping, Eqs.~\eqref{SigQQChain},~\eqref{SigPPChain} simplify to
\begin{equation}
\Omega^2 \Sigma_{QQ}^\infty = \Sigma_{PP}^\infty = \frac{1}{2} \left( 1 + \Omega_r^2 \right) E(T_0,\Omega_b) \qquad (\mathrm{for} \; \kappa \to 0) \;.
\end{equation}
Equipartition of energy in the stationary state is evident, and the thermalization (T2) property fulfilled.
To check property (T3), we calculate the temperature
\begin{equation}\label{TChainWC2}
\frac{2 \bar T_\infty(\Omega)}{\Omega_r} = \mathrm{arcoth}^{-1} \Bigg[ \frac{1}{2} \Big( \Omega_r + \frac{1}{\Omega_r} \Big) \coth \Big( \frac{1}{2 \bar T_0} \Big) \Bigg]
\end{equation}
of the stationary state with Eq.~\eqref{OmTLong} or
the weak damping result~\eqref{TWC}.
We see that $\bar T_\infty(\Omega_r)$ depends explicitly on the central oscillator frequency $\Omega_r$,
as depicted in Fig.~\ref{fig:TChain}.
It is $\bar T_\infty = \bar T_0$ only for $\Omega_r=1$.
As discussed before,
this results from the fact that $\breve{\mathcal{E}}(\omega)$ after the quench violates condition~\eqref{ThermAtWC}.
\begin{figure}
\includegraphics[width=0.8\linewidth]{Fig8}
\caption{\label{fig:TChain} (Color online) Temperature $\bar{T}_\infty$ of the stationary thermal state at weak damping as given in Eq.~\eqref{TChainWC2}.
It is shown as a function of $\Omega_r$ for different temperatures $\bar{T}_0 = 0.2, \dots, 0.8 $ of the initial state, as indicated.
Note that $\bar{T}_\infty$ does not depend on $\kappa_b$, but the admissible values of $\Omega_r$ for which equilibration occurs are restricted by the second condition in Eq.~\eqref{NoPolesWC} (see also Fig.~\ref{fig:params}).
In particular, it must be $0 \le \Omega_r^2 \le 2$.
}
\end{figure}
We note that $\kappa_b$ does not appear in Eq.~\eqref{TChainWC2}.
In the present example the value of $\kappa_b$ only determines
the admissible values of $\Omega_r$ that lead to equilibration,
as given by the second inequality in Eq.~\eqref{NoPolesWC}.
Once equilibration has been observed,
the temperature of the stationary state at weak damping depends only on the value of $\breve{\mathcal{E}}(\Omega)$ not on
the functional dependence of the spectral function $\gamma(\omega)$.
\subsubsection{The homogeneous chain}
\begin{figure}
\includegraphics[width=0.8\linewidth]{Fig9}
\caption{\label{fig:HomChain} (Color online) Frequency $\bar{\Omega}_\infty$ (dashed curve) and temperature $\bar T_\infty$ (solid curves)
for the homogeneous chain, from Eqs.~\eqref{HomChain1},~\eqref{HomChain2}
and shown as a function of $\kappa$.
The temperature curves are plotted for $ \bar T_0 = 0,0.2,0.5,1$ as indicated.}
\end{figure}
For the homogeneous chain with $\kappa_b = \kappa$, $\Omega_r = 1$ Eqs.~\eqref{SigQQChain},~\eqref{SigPPChain} simplify to
\begin{equation}\label{SigQQHomChain}
\Sigma_{QQ}^\infty = \frac{1}{2 \Omega_b^2} \left( 1 + \frac{1}{ \sqrt{1 - \kappa^2}} \right) E(T_0,\Omega_b) \;,
\end{equation}
\begin{equation}\label{SigPPHomChain}
\Sigma_{PP}^\infty = E(T_0,\Omega_b) \;.
\end{equation}
Equipartition of energy is violated
for any $\kappa > 0$, such that the effective frequency
\begin{equation}\label{HomChain1}
\bar\Omega_\infty^2 = \frac{2}{1+(1-\kappa^2)^{-1/2} }
\end{equation}
associated with the stationary state deviates from the central oscillator frequency
(it is always $\bar{\Omega}_\infty \le 1$).
The temperature of the stationary state is
\begin{equation}\label{HomChain2}
\frac{2 \bar T_\infty}{\bar \Omega_\infty} = \mathrm{arcoth}^{-1} \left[ \coth \Big( \frac{1}{2 \bar T_0} \Big) \sqrt{\frac{1+ (1-\kappa^2) ^{-1/2}}{2}} \, \right] \;.
\end{equation}
It is $\bar T_\infty > \bar T_0$ for $\kappa > 0$,
for example $\bar T_\infty \to 1/2$ for $\kappa \to 1$ and $\bar T_0 \to 0$ (see Fig.~\ref{fig:HomChain}).
The situation simplifies again in the weak damping limit $\kappa \to 0$,
where we recover from Eqs.~\eqref{SigQQHomChain},~\eqref{SigPPHomChain} the equilibration/thermalization result for the homogeneous chain formulated in Refs.~\cite{RH69,Aga71}:
At weak damping the central oscillator evolves into a stationary thermal state,
with equipartition of energy $\Omega^2 \Sigma_{QQ}^\infty = \Sigma_{PP}^\infty = E(T_0,\Omega_b)$.
Because of translational invariance this statement applies to every chain oscillator.
We note, however, that thermalization of the homogeneous chain is not perfect.
As discussed in Sec.~\ref{sec:EquilWithList},
observation of a single oscillator in the homogeneous chain
is not sufficient to establish thermalization of the entire chain.
Thermalization fails for a finite chain segment consisting of two or more oscillators, because property (T3) is not fulfilled as seen in Eq.~\eqref{TChainWC2}.
Note that there is no possibility to check property (T3) directly for the homogeneous chain ($\Omega_r=1$ is fixed here), such that results restricted to this situation have to be interpreted carefully~\cite{RH69,Aga71}.
\section{\label{sec:Summary}Conclusion}
Our study of the dissipative quantum harmonic oscillator
addresses equilibration and thermalization in non-thermal environments.
Equilibration is the generic behavior, which is prevented only in situations where the classical oscillator equation of motion
possesses undamped oscillatory solutions.
The infinite harmonic chain is an example for this behavior.
Thermalization of the central oscillator depends on additional conditions.
Just as for thermal environments,
equipartition of energy requires the weak damping limit but is independent of the precise initial conditions.
The asymptotic temperature $T_\infty$ is obtained from the energy distribution $\breve{\mathcal{E}}(\omega)$ in the initial bath state, and generally depends on the central oscillator frequency $\Omega$.
If we demand that $T_\infty$ is independent of $\Omega$, another condition on $\breve{\mathcal{E}}(\omega)$ follows.
This condition is essential for simultaneous thermalization of several oscillators,
when a thermal state of the combined system is obtained only if the
same asymptotic temperature is assumed by each oscillator.
Part of the behavior discussed here generalizes to systems with non-linear interactions.
First, we note that equilibration is possible although the linear system is integrable.
Equilibration occurs because, in a rough sense, the reduced density matrix of the central oscillator involves an average over conserved quantities of the joint oscillator-bath system.
In other words, equilibration of small systems embedded in a large environment
does not require ergodicity.
Second, because of the linearity and unitarity of quantum mechanical time evolution the stationary state depends explicitly on the initial (bath) state.
But already for the linear system some properties,
such as equipartition of energy, are independent of the initial conditions.
Furthermore, the stationary state depends only
on the energy distribution $\breve{\mathcal{E}}(\omega)$ in the initial bath state.
Effectively, information is lost in the long-time limit and thermalization is possible for a large class of (non-thermal) initial states.
We did neither discuss the generalization of the fluctuation-dissipation relation to the present non-thermal setting, nor the role of stationary non-equilibrium states with finite heat flow that would require coupling to at least two baths with different preparations.
Multi-time correlations functions can be computed within the present formalism, which will allow for the analysis of both issues in the future.
\begin{acknowledgments}
The authors wish to acknowledge helpful discussions with M. Cramer, G.-L. Ingold, and M. Thorwart.
This work was supported by Deutsche Forschungsgemeinschaft through SFB 652 (B5)
and AL1317/1-2.
\end{acknowledgments}
|
1811.01488
|
\section{Introduction}
Understanding how the hyperons undergo changes in nuclear matter is a
very important issue in contemporary nuclear physics. In particular,
it is of great interest to see how the hyperons are related to
in-medium kaon properties at low densities and
how they can be changed in higher densities that can be found in the
interior
of neutron stars.~\cite{Gal:2016boi,Lattimer:2015nhk}.
In the present contribution, we will discuss a recent work on the
hyperon properties in nuclear matter, which was carried out in a
simple but plausible framework of a chiral soliton approach to
nonzero density phenomena in the SU(3) sector~\cite{Hong:2018sqa}.
Previously, a similar approach was developed in the non-strangeness
sector to study various phenomena in medium (for example, see
Ref.~\cite{Yakhshiev:2013eya} and references therein) and the results
were in qualitative agreement with those from other different
approaches. In Ref.~\cite{Hong:2018sqa}, we extended the work of
Ref.~\cite{Yakhshiev:2013eya} to the SU(3) including the hyperons.
We discuss the main results and significance of the work.
\section{The model}
The Lagrangian of the present model written by the following
form~\cite{Hong:2018sqa}
\begin{eqnarray}
\mathcal{L}=&-\frac{F_\pi^2}{16}\alpha_2^t(\rho)
{\rm Tr} L_0L_0+\frac{F_\pi^2}{16}\alpha_2^s(\rho){\rm Tr} L_iL_i
-\frac{\alpha_4^t(\rho)}{16e^2}
{\rm Tr}[L_0,L_i]^2 +\frac{\alpha_4^s(\rho)}{32e^2}{\rm Tr}[L_i,L_j]^2\cr
& +\frac{F_\pi^2}{16}\alpha_{\chi SB}(\rho){\rm Tr}
\mathcal{M}(U+U^\dagger-2)+ \mathcal{L}_{WZ},
\label{ModLag}
\end{eqnarray}
where $L_\mu=U^\dagger\partial_\mu U$ and $U(\bm{x},t)$ is a chiral
field in SU(3). The Wess-Zumino term~\cite{Wess:1971yu}
$\mathcal{L}_{\mathrm{WZ}}$ in the Lagrangian constrains the soliton
to be identified as a baryon and is expressed by a five-dimensional
integral over a disk $D$
\begin{eqnarray}
S_{\rm WZ} = -\frac{iN_c}{240\pi^2} \int_{D} d^5 \vec x\,
\epsilon^{\mu\nu\alpha\beta\gamma}
{\rm Tr}(L_\mu L_\nu L_\alpha L_\beta L_\gamma).
\end{eqnarray}
Here $\epsilon^{\mu\nu\alpha\beta\gamma}$ is the totally antisymmetric
tensor defined as $\epsilon^{01234}=1$ and $N_c=3$ is the number of
colors. The values of input parameters are defined in free space:
$F_\pi=108.783$\,MeV denotes the pion decay constant, $e=4.854$
represents the Skyrme parameter, the masses of the $\pi$ and $K$
mesons are given respectively as $m_\pi=134.976$\,MeV and
$m_K=495$\,MeV, and the mass matrix of the pseudo-Nambu-Goldstone
bosons $\mathcal{M}$ has the diagonal form
$\mathcal{M}=(m_\pi^2,m_\pi^2,m_K^2)$. The density-dependent
functions $\alpha_2^t(\rho)$, $\alpha_2^s(\rho)$, $\alpha_4^t(\rho)$,
$\alpha_4^s(\rho)$ and $\alpha_{\chi SB}(\rho)$ reflect the changes
of the meson properties in nuclear medium. In an approximation of
homogeneous infinite nuclear matter they are expressed in terms
of the three linear density-dependent functions
$f_{i}(\rho)=1+C_i\rho,\,(i=1,2,3)$. The numerical values of $C_i$ are
fixed to be $C_1=-0.279$, $C_2=0.737 $ and $C_3=1.782$, respectively.
They reproduce very well the equations of state (EoS)
for symmetric nuclear matter near the normal nuclear matter density
$\rho_0$ and
at higher densities that may exist in the interior of a neutron star.
The medium modification of the kaon properties is achieved by
considering the following scheme
\begin{eqnarray}
F_\pi m_K\rightarrow F_K^* m_K^*=F_\pi m_K(1-C\rho/\rho_0)
\label{comKprop}
\end{eqnarray}
and can be explained in terms of the alteration of the kaon decay
constant and/or of the kaon mass in nuclear environment.
The quantization of the model is performed by considering the
time-dependent rigid rotation of a static soliton
\begin{equation}
U(\bm{r},t)=\mathcal{A}(t)U_0(\bm{r})\mathcal{A}(t)^\dagger,
\end{equation}
where $U_0(\bm{r})$ denotes the static SU(3) chiral soliton with
trivial embedding. The time-dependent rotational matrix
$\mathcal{A}(t)$ is decomposed
\begin{eqnarray}
\mathcal{A}(t)&=\left(\begin{array}{cc}
A(t)&0\\
0^\dagger&1\end{array}\right)S(t),
\end{eqnarray}
in terms of the SU(2) isospin rotation
$A(t)=k_0(t){\bf 1}+i \sum_{a=1}^3\tau_a k_a(t)$ and
fluctuations into the strangeness sector given by the
matrix $S(t)=\exp\left\{i\sum_{p=4}^7k_p \lambda_p\right\}$.
Here $\tau_{1,2,3}$ denote the Pauli matrices, whereas $\lambda_p$
stand for the strange part of the SU(3) Gell-Mann matrices.
The time-dependent functions $k_a(t)$ $(a=0,1,2,\dots,7)$ represent
arbitrary collective coordinates. The more details of the approach
can be found in Ref.~\cite{Hong:2018sqa}.
\section{Results and discussions}
All model parameters in free space and in nuclear matter, except for
the parameter $C$ in Eq.~(\ref{comKprop}), are fixed
in the SU(2) sector. The only remaining parameter $C$ could be fixed
by data on kaon-nucleus scattering and kaonic atoms. However, in the
present work we carry out a qualitative analysis of the
effects in the baryonic sector due to the modification of the kaon
properties in nuclear medium. Consequently, we discuss the density
dependence of the mass splittings among the various baryon multiplet
members. In our calculation, the parameter value $C=0$ corresponds to
the case when the properties of kaon will not change in nuclear matter
whereas a nonzero value of the parameter $C\neq 0$ indicates that the
mass and/or kaon dynamics is alters in a dense nuclear environment.
The results show that in general the masses of the baryon
octet tend to decrease in nuclear matter. Only $\Sigma$
showed a different tendency if the parameter value is set to be $C=0$.
In the case of $C=0.2$, $m_\Sigma$ also tends to decrease as
the density of nuclear matter increases~\cite{Hong:2018sqa}.
In comparison, the results from SU(3) chiral effective field
theory~\cite{Petschauer:2015nea} show that $m_{\Lambda}^*$ is
decreased by about 17~\% at normal nuclear matter density
$\rho_0$. The $\Xi$ hyperon is
behaved in a similar manner. At $\rho_0$ the change in the mass of
$\Xi^*$ was about 6~\% and 16~\% for the corresponding parameter
values $C=0$ and $C=0.2$, respectively.
The masses of the baryon decuplet increase in general as $\rho$
increases. Changes are dramatic for $C=0$ while for $C=0.2$ they are
less changeable.
We present the density dependence of the mass splittings among the
multiplet members in Figs.~\ref{Fig1} and \ref{Fig2}.
\begin{figure}[th]
\includegraphics[width=0.45\textwidth]{fig1a.eps}
\includegraphics[width=0.45\textwidth]{fig1b.eps}
\caption{ (Color online.) Density dependence of the mass splittings
among the baryon octet members. The mass splittings in nuclear matter
are normalized to the corresponding free space mass
splittings. The left and right panels in the figure corresponds to the
results with $C=0$ and $C=0.2$, respectively.
}
\label{Fig1}
\end{figure}
\begin{figure}[th]
\includegraphics[width=0.45\textwidth]{fig2a.eps}
\includegraphics[width=0.45\textwidth]{fig2b.eps}
\caption{ (Color online.) Density dependence of the mass splittings
among the baryon decuplet members.
Notations are the same as in Fig.~\ref{Fig1}.
}
\label{Fig2}
\end{figure}
Figure~\ref{Fig1} shows the density dependence of the mass splittings
among the baryon octet members while Fig.~\ref{Fig2} depicts the results
corresponding to the mass splittings among the decuplet members.
All the mass splittings in nuclear matter are normalized
to the vaues of the corresponding ones in free space.
The left and right panels in the figures illustrate the results with
two different values of parameter $C$, respectively.
It is interesting to see that except $m^*_{\Sigma}-m^*_{\Lambda}$
all the mass splittings tend to decrease
up to $(1.5-2)\rho_0$. This behavior can be explained
in terms of the density-dependent functionals $\omega^*_-$ and $c^*$
entering into the mass formula (see Eq.\,(36) in
Ref.~\cite{Hong:2018sqa}). The first functional describes the
fluctuations in the strangeness direction and comes into play for the
mass splitting formula between the same strangeness members while all
other mass splittings presented in the figures depend linearly on
$\omega_-^*$.
This indicates that at large densities the fluctuations in strangeness
direction gets weaker. From the figures one concludes also that
at large densities SU(3) flavor symmetry tends to be restored.
The work is supported by Basic Science Research Program through the
National Research Foundation (NRF) of Korea funded by the
Korean government (Ministry of Education, Science and Technology,
MEST), Grant No. 2016R1D1A1B03935053 (UY) and Grant
No. NRF-2018R1A2B2001752 (HChK).
|
2102.12816
|
\section{Introduction}
\subsection{Background and Motivation}\label{bnm}
\IEEEPARstart{I}{n} recent years, decentralized optimization problems have been extensively investigated in different research fields, such as distributed control of multi-robot systems, decentralized regularization problems with massive data sets, and economic dispatch problems in power systems \cite{SC20,margellos2017distributed,bai2017distributed}.
In those problems, there are two main categories of how information and agent actions are managed: synchronous and asynchronous. In a synchronous system, certain global clock for agent interactions and activations is established to ensure the correctness of optimization result \cite{arjomandi1983efficiency}. However, in many decentralized systems, there is no such a guarantee. The reasons mainly lie in the following two aspects.
\begin{enumerate}
\item {\em{Asynchronous Activations:}} In some multi-agent systems, each agent may only be keen on his own updates regardless of the process of others. Such an action pattern may cause an asynchronous computation environment. For example, some agents with higher computation capacity may take more actions during a given time horizon without ``waiting for'' the slow ones \cite{tseng1991rate}.
\item {\em{Communication Delays:}} In synchronous networks, the agents are assumed to access the up-to-date information without any packet loss. This settlement requires an efficient communication process or reserving a ``zone'' between two successive updates for the data transmission. However, in large-scale decentralized systems, complete synchronization of communications may be costly if the delay is large and computational frequency is high \cite{hannah2018unbounded}.
\end{enumerate}
In addition, we consider a composite optimization problem with coupling constraints, where the objective function is separable and composed of smooth and possibly non-smooth components. The concerned problem structure arises from various fields, such as logistic regression, boosting, and support vector machines \cite{kleinbaum2002logistic,schapire2013boosting,hearst1998support}. Observing that proximal gradient method takes the advantage of some simple-structured composite functions and is usually numerically more stable than the subgradient counterpart \cite{bertsekas2011incremental}, in this paper, we aim to develop a decentralized proximal gradient based algorithm for solving the composite optimization problem in an asynchronous\footnote[1]{In this paper, ``asynchronous'' may be referred to as ``partially asynchronous'' which could be different from ``fully asynchronous'' studied in some works (e.g., \cite{baudet1978asynchronous}) without more clarification for briefness. The former definition may contain some mild assumptions on the asynchrony, e.g., bounded communication delays.} network.
\subsection{Literature Review}\label{881}
Proximal gradient method is related to the proximal minimization algorithm which was studied in early works \cite{martinet1970regularisation,rockafellar1976monotone}. By this method, a broad class of composite optimization problems with simple-structured objective functions can be solved efficiently \cite{bao2012real,chen2012smoothing,banert2019general}.
\cite{aybat2017distributed,hong2017stochastic,li2019decentralized} further studied the decentralized realization of proximal gradient based algorithms.
Decentralized proximal gradient methods dealing with global linear constraints were discussed in \cite{li2017convergence,wang2021distributed,ma2016alternating,jiang2012inexact}. \cite{beck2009fast,chen2012fast,li2015accelerated} present some accelerated versions of proximal gradient method. Different from the existing works, we will show that by our proposed penalty based Asyn-PPG algorithm, a class of composite optimization problems with coupling constraints can be solved asynchronously in the proposed SAN, which enriches the exiting proximal gradient methods and applications.
To deal with the asynchrony of multi-agent networks, existing works usually capture two factors: asynchronous action clocks and unreliable communications \cite{tsitsiklis1986distributed}. In those problems, the decentralized algorithms are built upon stochastic or deterministic settings depending on whether the probability distribution of the asynchronous factors is utilized. Among those works, stochastic optimization based models and algorithms are fruitful \cite{xu2017convergence,notarnicola2016asynchronous,bastianello2020asynchronous,mansoori2019fast,wei20131}. For instance,
in \cite{xu2017convergence}, an asynchronous distributed gradient method was proposed for solving a consensus optimization problem by considering random communications and updates. The authors of \cite{notarnicola2016asynchronous} proposed a randomized dual proximal gradient method, where the agents execute node-based or edge-based asynchronous updates activated by local timers. An asynchronous relaxed ADMM algorithm was proposed in \cite{bastianello2020asynchronous} for solving a distributed optimization problem with asynchronous actions and random communication failures.
All the optimization algorithms in \cite{xu2017convergence,wei20131,notarnicola2016asynchronous,bastianello2020asynchronous,mansoori2019fast} require the probability distribution of asynchronous factors to establish the parameters of algorithms and characterize convergence properties. However, in practical applications, the probability distributions may be difficult to acquire and would cause inaccuracy issues in the result due to the limited historical data \cite{bertsekas2002introduction}. To overcome those drawbacks, some works leveraging on deterministic analysis arose in the recent few decades \cite{bertsekas2011incremental,chazan1969chaotic,bertsekas1989parallel,zhou2018distributed,kibardin1980decomposition,nedich2001distributed,hong2017distributed,kumar2016asynchronous,tian2020achieving,hale2017asynchronous,cannelli2020asynchronous}. For instance, in \cite{chazan1969chaotic}, a chaotic relaxation method was studied for solving a quadratic minimization problem by allowing for both asynchronous actions and communication delays, which can be viewed as a prototype of a class of asynchronous problems. The authors of \cite{bertsekas1989parallel} further investigated the asynchronous updates and communication delays in a routing problem in data networks based on deterministic relaxations. The authors of \cite{zhou2018distributed} proposed an m-PAPG algorithm in asynchronous networks by employing proximal gradient method in machine learning problems with a periodically linear convergence guarantee.
Another line of asynchronous optimizations with deterministic analysis focuses on incremental (sub)gradient algorithms, which can be traced back to \cite{kibardin1980decomposition}. In more recent works, a wider range of asynchronous factors have been explored. For example, in \cite{nedich2001distributed}, a cluster of processors compute the subgradient of their local objective functions triggered by asynchronous action clocks. Then, a master processor acquires all the available but possibly outdated subgradients and updates its state for the subsequent round. The author of \cite{bertsekas2011incremental} proposed an incremental proximal method, which admits a fixed step-size compared with the diminishing step-size of the corresponding subgradient counterpart.
The author of \cite{hong2017distributed} introduced an ADMM based incremental method for asynchronous non-convex optimization problems.
However, the results in \cite{bertsekas2011incremental,chazan1969chaotic,bertsekas1989parallel,zhou2018distributed,kibardin1980decomposition,nedich2001distributed,hong2017distributed,kumar2016asynchronous,tian2020achieving,hale2017asynchronous,cannelli2020asynchronous} are limited to either smooth individual objective functions or uncoupled constraints. In addition, the incremental (sub)gradient methods require certain fusion node to update the full system-wide variables continuously.
More detailed comparisons with the aforementioned works are listed as follows. (i) Different from \cite{xu2017convergence,wei20131,notarnicola2016asynchronous,bastianello2020asynchronous,mansoori2019fast}, in this work, the probability distribution of the asynchronous factors in the network is not required, which overcomes the previously discussed drawbacks of stochastic optimizations. (ii) In terms of the mathematical problem setup, the proposed Asyn-PPG algorithm can handle the non-smoothness of all the individual objective functions, which is not considered in \cite{xu2017convergence,chazan1969chaotic,bertsekas1989parallel,hong2017distributed,kumar2016asynchronous,mansoori2019fast,tian2020achieving,hale2017asynchronous}\footnote[2]{In \cite{hong2017distributed,kumar2016asynchronous}, the objective function of some distributed nodes is assumed to be smooth.}. The algorithms proposed in \cite{bertsekas2011incremental,chazan1969chaotic,kibardin1980decomposition,zhou2018distributed,nedich2001distributed,cannelli2020asynchronous} cannot address coupling constraints, and it is unclear whether the particularly concerned (consensus) constraints in \cite{notarnicola2016asynchronous,xu2017convergence,mansoori2019fast,bastianello2020asynchronous,hong2017distributed,tian2020achieving,bertsekas1989parallel,kumar2016asynchronous} can be extended to general linear equality/inequality constraints. In addition, the algorithms proposed in \cite{xu2017convergence,chazan1969chaotic,tian2020achieving,mansoori2019fast} cannot handle local constraints in their problems.
We hereby summarize the contributions of this work as follows.
\begin{itemize}
\item A penalty based Asyn-PPG algorithm is proposed for solving a linearly constrained composite optimization problem in a partially asynchronous network. More precisely, we take the local/global constraints, non-smoothness of objective functions, asynchronous updates, and communication delays into account simultaneously with deterministic analysis, which, to the best knowledge of the authors, hasn't been addressed in the existing research works (e.g., \cite{xu2017convergence,wei20131,notarnicola2016asynchronous,bastianello2020asynchronous,mansoori2019fast,bertsekas2011incremental,chazan1969chaotic,kibardin1980decomposition,zhou2018distributed,nedich2001distributed,tian2020achieving,cannelli2020asynchronous,hong2017distributed,bertsekas1989parallel,kumar2016asynchronous,hale2017asynchronous}) and, hence, can adapt to more complicated optimization problems in asynchronous networks with deterministic convergence result.
\item An SAN model is established by splitting the whole time domain into sequential time slots. In this model, all the agents are allowed to execute multiple updates asynchronously in each slot. Moreover, the agents only access the state of others at the beginning of each slot, which alleviates the intensive message exchanges in the network. In addition, the proposed interaction mechanism allows for communication delays among the agents, which are not considered in \cite{bastianello2020asynchronous,xu2017convergence,wei20131,notarnicola2016asynchronous},
and can also relieve the overload of certain central node as discussed in \cite{bertsekas2011incremental,nedich2001distributed,hong2017distributed,kibardin1980decomposition,hale2017asynchronous}.
\item By the proposed Asyn-PPG algorithm, a periodic convergence rate $\mathcal{O}(\frac{1}{K})$ can be guaranteed with the coefficient of penalties synchronized at the end of each slot. The feasibility of the Asyn-PPG algorithm is verified by solving a distributed least absolute shrinkage and selection operator (LASSO) problem and a social welfare optimization problem in the electricity market.
\end{itemize}
The rest of this paper is organized as follows. {Section \ref{se2}} includes some frequently used notations and definitions in this work. {Section \ref{se3}} formulates the considered optimization problem. Basic definitions and assumptions of the SAN are provided therein. {Section \ref{se4}} presents the proposed Asyn-PPG algorithm and relevant propositions to be used in the subsequent analysis. In {Section \ref{se5}}, the main theorems on the convergence analysis of the Asyn-PPG algorithm are provided. {Section \ref{se6}} verifies the feasibility of the Asyn-PPG algorithm by two motivating applications. {Section \ref{se7}} concludes this paper.
\section{Preliminaries}\label{se2}
In the following, we present some preliminaries on notations, graph theory, and proximal mapping to be used throughout this work.
\subsection{Notations}
Let $\mid \mathcal{A}\mid$ be the size of set $\mathcal{A}$. $\mathbb{N}$ and $\mathbb{N}_+$ denote the non-negative integer space and positive integer space, respectively. $\mathbb{R}_{\succeq \mathbf{u}}^n$ denotes the $n$-dimensional Euclidian space with each element larger than or equal to the corresponding element in $\mathbf{u}$. $\parallel \cdot\parallel_1$ and $\parallel \cdot \parallel$ denote the $l_1$ and $l_2$-norms, respectively. $\langle \cdot, \cdot \rangle$ is an inner product operator. $\otimes$ is the Kronecker product operator. $\mathbf{0}$ and $\mathbf{1}$ denote the column vectors with all elements being 0 and 1, respectively. $\mathbf{I}_n$ and $\mathbf{O}_{m \times n}$ denote the $n$-dimensional identity matrix and $(m \times n)$-dimensional zero matrix, respectively. $\mathbf{relint}\mathcal{A}$ represents the relative interior of set $\mathcal{A}$.
\subsection{Graph Theory}
A multi-agent network can be described by an undirected graph ${\mathcal{G}}= \{{\mathcal{V}},{\mathcal{E}}\}$, which is composed of the set of vertices ${\mathcal{V}} = \{1,2,...,N \}$ and set of edges ${\mathcal{E}} \subseteq \{ (i,j)| i,j \in \mathcal{V} \hbox{ and } i \neq j\}$ with $(i,j) \in \mathcal{E}$ an unordered pair. A graph $\mathcal{G}$ is said connected if there exists at least one path between any two distinct vertices. A graph $\mathcal{G}$ is said fully connected if any two distinct vertices are connected by a unique edge. ${\mathcal{V}}_i = \{ j | (i,j) \in \mathcal{E}\}$ denotes the set of the neighbours of agent $i$. Let ${\mathbf{L}} \in \mathbb{R}^{N \times N}$ denote the Laplacian matrix of ${\mathcal{G}}$. Let $l_{ij}$ be the element at the cross of the $i$th row and $j$th column of ${\mathbf{L}}$. Thus, $l_{ij} = -1$ if $(i,j) \in {\mathcal{E}}$, $l_{ii} = \mid {\mathcal{V}}_i \mid$, and $l_{ij} = 0$ otherwise, $i,j \in {\mathcal{V}}$ \cite{chung1997spectral}.
\subsection{Proximal Mapping}
A proximal mapping of a closed, proper, convex function $\zeta: \mathbb{R}^n \rightarrow (-\infty,+\infty]$ is defined by
\begin{align}\label{d1}
\mathrm{prox}^a_{\zeta} (\mathbf{u})= \arg \min \limits_{\mathbf{v} \in \mathbb{R}^n} ( \zeta(\mathbf{v}) + \frac{1}{2a} \parallel \mathbf{v} - \mathbf{u}\parallel^2 ),
\end{align}
with step-size ${a}>0$ \cite{parikh2014proximal}.
\section{Problem Formulation and Network Modeling}\label{se3}
The considered mathematical problem and the proposed network model are presented in this section.
\subsection{The Optimization Problem}
In this paper, we consider a multi-agent network ${\mathcal{G}}=\{{\mathcal{V}},{\mathcal{E}}\}$. $f_i:\mathbb{R}^M \rightarrow (-\infty,+\infty]$ and $h_i: \mathbb{R}^M \rightarrow (-\infty,+\infty]$ are private objective functions of agent $i$, where $f_i$ is smooth and $h_i$ is possibly non-smooth, $i \in \mathcal{V}$. $\mathbf{x}_i= (x_{i1},...,x_{iM})^{\top} \in \mathbb{R}^M$ is the strategy vector of agent $i$, and $\mathbf{x}= (\mathbf{x}^{\top}_1,...,\mathbf{x}^{\top}_{N})^{\top} \in \mathbb{R}^{M N}$ is the collection of all strategy vectors. A linearly constrained optimization problem of $\mathcal{V}$ can be formulated as
\begin{align}\label{}
\hbox{\textbf{(P1)}}: \quad
\min \limits_{\mathbf{x}} \quad & {F}(\mathbf{x}) = \sum_{i \in \mathcal{V}} (f_i(\mathbf{x}_i) + h_i(\mathbf{x}_i)) \nonumber \\
\hbox{subject to} \quad & \mathbf{A} \mathbf{x} = \mathbf{0}, \label{4}
\end{align}
where $\mathbf{A} \in \mathbb{R}^{B \times NM}$.
For the convenience of the rest discussion, we define $f(\mathbf{x})= \sum_{i\in \mathcal{V}}f_i(\mathbf{x}_i)$, $h(\mathbf{x})= \sum_{i\in \mathcal{V}}h_i(\mathbf{x}_i)$, and $ {F}_i(\mathbf{x}_i) = f_i(\mathbf{x}_i) + h_i(\mathbf{x}_i)$. Let $\mathbf{A}_i \in \mathbb{R}^{B \times M}$ be the $i$th column sub-block of $\mathbf{A}$, i.e., $\mathbf{A}=(\mathbf{A}_1,...,\mathbf{A}_i,...,\mathbf{A}_{N})$. Let $\mathbf{W} = \mathbf{A}^{\top} \mathbf{A} \in \mathbb{R}^{M N \times M{N}}$ and $\mathbf{W}_{ij}$ be the $(i,j)$th $(M \times M)$-dimensional sub-block of $\mathbf{W}$. Define $\mathbf{W}_i= (\mathbf{W}_{i1},...,\mathbf{W}_{iN}) = \mathbf{A}_i^{\top} \mathbf{A} \in \mathbb{R}^{M \times M{N}}$.
\begin{Assumption}\label{a0}
(Connectivity) $\mathcal{G}$ is undirected and fully connected.\footnote[3]{Strictly speaking, in this work, the requirement on the connectivity of the graph depends on how the individual variables are coupled in (\ref{4}). In some specific problems, $\mathcal{G}$ is not necessarily fully connected (see an example in Section \ref{rm2}).}
\end{Assumption}
\begin{Assumption}\label{a1}
({Convexity}) $f_i$ is proper, $L_i$-Lipschitz continuously differentiable and $\mu_i$-strongly convex, $L_i>0$, $\mu_i > 0$; $h_i$ is proper, convex and possibly non-smooth, $i\in \mathcal{V}$.
\end{Assumption}
The assumptions in Assumption \ref{a1} are widely used in composite optimization problems \cite{florea2020generalized,beck2014fast,notarnicola2016asynchronous}.
\begin{Assumption}\label{a1-1}
(Constraint Qualification \cite{boyd2004convex}) There exists an $\breve{\mathbf{x}} \in \mathbf{relint} \mathcal{D}$ such that $\mathbf{A} \breve{\mathbf{x}} = \mathbf{0}$, where $\mathcal{D}$ is the domain of $F(\mathbf{x})$.
\end{Assumption}
\begin{Remark}\label{re12}
Problem (P1) defines a prototype of a class of optimization problems. One may consider an optimization problem with local convex constraint $\mathbf{x}_i \in \Omega_i$ and coupling inequality constraint $\mathbf{A} \mathbf{x} + \mathbf{b} \preceq \mathbf{0}$ by introducing slack variables and indicator functions into Problem (P1) \cite{boyd2004convex}, which gives
\begin{align}\label{}
\hbox{\textbf{(P1+)}}: \min \limits_{\mathbf{x}_i, \mathbf{y}, \forall i \in \mathcal{V}} \quad & \sum_{i \in \mathcal{V}} (f_i(\mathbf{x}_i) + h_i(\mathbf{x}_i) + \mathbb{I}_{\Omega_i} (\mathbf{x}_{i})) + \mathbb{I}_{\mathbb{R}^B_{\succeq \mathbf{b}}} (\mathbf{y}) \nonumber \\
\hbox{subject to} \quad & \mathbf{A} \mathbf{x} + \mathbf{y} = \mathbf{0},
\end{align}
where $\Omega_i \subseteq \mathbb{R}^{M}$ is non-empty, convex and closed, $\mathbf{y} \in \mathbb{R}^B$ is a slack variable, and
\begin{align}\label{}
& \mathbb{I}_{\Omega_i} (\mathbf{x}_{i}) = \left\{
\begin{array}{cc}
0 & \mathbf{x}_{i} \in \Omega_i, \\
+ \infty & \hbox{otherwise,}
\end{array} \right. \\
& \mathbb{I}_{\mathbb{R}^B_{\succeq \mathbf{b}}} (\mathbf{y}) = \left\{
\begin{array}{cc}
0 & \mathbf{y} \in \mathbb{R}^B_{\succeq \mathbf{b}}, \\
+ \infty & \hbox{otherwise.}
\end{array} \right.
\end{align}
To realize decentralized computations, $\mathbf{y}$ can be decomposed and assigned to each of the agents. Since $\mathbb{I}_{\Omega_i}$ and $\mathbb{I}_{\mathbb{R}^B_{\succeq \mathbf{b}}}$ are proper and convex, the structure of Problem (P1+) is consistent with that of Problem (P1).
\end{Remark}
\subsection{Characterization of Optimal Solution}
By recalling Problem (P1), we define Lagrangian function
\begin{align}
{\mathcal{L}}({\mathbf{x}}, \bm{\lambda})= & {F}(\mathbf{x}) + \langle \bm{\lambda}, \mathbf{A}\mathbf{x} \rangle,
\end{align}
where $\bm{\lambda} \in \mathbb{R}^{B}$ is the Lagrangian multiplier vector. Let $\mathcal{X}$ be the set of the saddle points of ${\mathcal{L}}({\mathbf{x}}, \bm{\lambda})$. Then, any saddle point $({\mathbf{x}}^*,{\bm{\lambda}}^*) \in \mathcal{X}$ can be characterized by \cite{boyd2004convex}
\begin{align}\label{5}
({\mathbf{x}}^*,{\bm{\lambda}}^*) = \arg \max_{\bm{\lambda}} \min_{\mathbf{{x}}} {\mathcal{L}}({\mathbf{x}},\bm{\lambda}),
\end{align}
where ${\mathbf{x}}^*=((\mathbf{x}^*_1)^{\top},...,(\mathbf{x}^*_{N})^{\top})^{\top}$ and $\bm{\lambda}^* = (\lambda^*_1,...,\lambda^*_B)^{\top}$. Then, $\forall \mathbf{x} \in \mathbb{R}^{M{N}}$, we have
\begin{align}\label{}
& {F}(\mathbf{x}) + \langle \bm{\lambda}^*, \mathbf{A}\mathbf{x}\rangle - {F}(\mathbf{x}^*) - \langle \bm{\lambda}^*, \mathbf{A}\mathbf{x}^* \rangle \geq 0. \nonumber
\end{align}
With the fact $ \mathbf{A}\mathbf{x}^* = \mathbf{0}$, we can obtain
\begin{align}\label{sd1}
& {F}(\mathbf{x}) + \langle \bm{\lambda}^*, \mathbf{A}\mathbf{x}\rangle - {F}(\mathbf{x}^*) \geq 0.
\end{align}
\subsection{Slot-based Asynchronous Network}
Regarding the asynchrony issues outlined in Section \ref{bnm}, we propose an SAN model which consists of the following two key features.
\begin{enumerate}
\item The whole time domain is split into sequential time slots and the agents are permitted to execute multiple updates in each slot. There is no restriction on which time instant should be taken, which enables the agents to act asynchronously.
\item All the agents can access the information of others in the previous slot at the beginning of the current slot, but the accessed state information may not be the latest depending on how large the communication delay of the network is.
\end{enumerate}
For practical implementation, the proposed SAN model is promising to be applied in some time-slot based problems, such as bidding and auctions in the electricity market and task scheduling problems in multi-processor systems \cite{david2000strategic,andersson2010implementing}.
The detailed mathematical descriptions of SAN are presented as follows. We let $\mathcal{T}= \{0,1,2,...\}$ be the collection of the whole discrete-time instants and $\mathcal{M}= \{t_m \}_{m\in \mathbb{N}} \subseteq \mathcal{T}$ be the sequence of the boundary of successive time slots. $\mathcal{T}_i \subseteq \mathcal{T}$ is the action clock of agent $i$. Slot $m$ is defined as the time interval $[t_m,t_{m+1})$.
\begin{Assumption}\label{a4-1}
(Uniform Slot Width) The width of slots is uniformly set as $H$, i.e., $t_{m+1} - t_{m} = H$, $H \in \mathbb{N}_+$, $m\in \mathbb{N}$.
\end{Assumption}
\begin{Assumption}\label{a4}
(Frequent Update) Each agent performs at least one update within $[t_m,t_{m+1})$, i.e., $\mathcal{T}_i \cap [t_m,t_{m+1}) \neq \emptyset$, $\forall i \in \mathcal{V}$, $m \in \mathbb{N}$.
\end{Assumption}
The update frequency of agent $i$ in slot $m$ is defined by $P_{i,m}$, i.e., $P_{i,m} = \mid \mathcal{T}_i \cap [t_m,t_{m+1}) \mid \in [1,H]$. Define $\mathcal{P}^m_i= \{1,2,...,P_{i,m} \}$, $i\in \mathcal{V}$, $m \in \mathbb{N}$. Let $t_m^{(n)} \in \mathcal{T}$ denote the instant of the $n$th update in slot $m$. For the mathematical derivation purpose, we let
\begin{align}
& t_{m}^{(P_{i,m}+1)} = t_{m+1}^{(1)}, \label{pan2} \\
& t_{m+1}^{(0)} = t_{m}^{(P_{i,m})}. \label{pan3}
\end{align}
(\ref{pan2}) and (\ref{pan3}) are the direct extensions of the action indexes between two sequential slots. That is, the $1$st action instant in slot $m+1$ is equivalent to the $(P_{i,m}+1)$th action instant in slot $m$; the $0$th action instant in slot $m+1$ is equivalent to the $P_{i,m}$th action instant in slot $m$.
\begin{Proposition}\label{pp10}
In the proposed SAN, $\forall i \in \mathcal{V}$, $m \in \mathbb{N}$, we have the following inequality:
\begin{align}
& t_{m}^{(P_{i,m})} \leq t_{m+1}-1 < t_{m+1} \leq t_{m+1}^{(1)}. \label{pan1}
\end{align}
\end{Proposition}
\begin{proof}
Note that $t_{m}^{(P_{i,m})}$ and $t_{m+1}^{(1)}$ are the last update instant in $[t_m,t_{m+1})$ and the first update instant in $[t_{m+1},t_{m+2})$ of agent $i$, respectively. Therefore, the validation of (\ref{pan1}) is straightforward.
\end{proof}
\begin{Assumption}\label{as1}
(Information Exchange) Each agent always knows the latest information of itself, but the state information of others can only be accessed at the beginning of each slot, i.e., $t_m$, $\forall m \in \mathbb{N}$.
\end{Assumption}
Assumption \ref{as1} enables the agents to only communicate at the instants in $\mathcal{M}$, which can relieve the intensive information exchanges in the network. However, due to communication delays, in slot $m$, certain agent $i$ may not access the latest information of agent $j$ at time $t_m$, i.e., $\mathbf{x}_j(t_m)$, $j \in \mathcal{V} \setminus \{i\}$, but a possibly delayed version $\mathbf{x}_j(\tau(t_m))$ with $\tau(t_m) \leq t_m$, $\tau(t_m) \in \mathcal{T}$. $\mathbf{x}_j(\tau(t_m)) \neq \mathbf{x}_j(t_m)$ means that agent $j$ performs update(s) within $[\tau(t_m),t_m)$. Therefore, the full state information available at instant $t_{m}$ may not be $\mathbf{x}(t_m)$ but a delayed version $\mathbf{x}^{\mathrm{d}}(t_m)= ((\mathbf{x}_1^{\mathrm{d}})^{\top}(t_m),...,(\mathbf{x}_{N}^{\mathrm{d}})^{\top}(t_m))^{\top} = (\mathbf{x}^{\top}_1(\tau (t_m)),..., \mathbf{x}^{\top}_{N}(\tau (t_m)))^{\top} \in \mathbb{R}^{M{N}}$.\footnote[4]{In slot $m$, the time instant of the historical state, i.e., $\tau(t_{m})$, is identical, which means the communication delay is uniform for the all the agents (as discussed in \cite{wang2013consensus}) in certain slot and can be varying in different slots.}
\begin{Assumption}\label{a5}
({Bounded Delay}) The communication delays in the network are upper bounded by $D \in \mathbb{N}_+$ with $D \leq H$, i.e., $t_m- \tau(t_m) \leq D$, $\forall m \in \mathbb{N}$.
\end{Assumption}
In slot $m$, the historical state of agent $i$ can be alternatively defined by $ \mathbf{x}_i(t^{(n_{i,m})}_m) = \mathbf{x}_i^{\mathrm{d}}(t_{m+1})$, where $t^{(n_{i,m})}_m$ is the largest integer no greater than $\tau(t_{m+1})$ in set $\mathcal{T}_i$, and $n_{i,m} \in \mathbb{N}_+$ is the index of the update. Then, the number of updates within $[t^{(n_{i,m})}_m, t^{(P_{i,m})}_m]$ should be no greater than the number of instants in $[\tau(t_{m+1}), t_{m+1})$, i.e.,
\begin{equation}\label{6}
P_{i,m} - n_{i,m} \leq t_{m+1} -1 - \tau(t_{m+1}) \leq D -1.
\end{equation}
The relationship among $\mathcal{T}$, $\mathcal{T}_i$ and delay in slot $m$ is illustrated in Fig. \ref{1234}.
\begin{figure}[htbp]
\centering
\includegraphics[width=7cm]{1234}\\
\caption{An illustration of the relationship among $\mathcal{T}$, $\mathcal{T}_i$ and delay in slot $m$. In this example, $P_{i,m} - n_{i,m}=2$ and $t_{m+1} - \tau(t_{m+1})=4$, which satisfies (\ref{6}).}\label{1234}
\end{figure}
\section{Asynchronous Penalized Proximal Gradient Algorithm}\label{se4}
Based on the SAN model, the Asyn-PPG algorithm is designed in this section.
Let $\{\alpha_i(t^{(n)}_m)_{>0}\}_{n \in \mathcal{P}^m_i}$ and $\{\eta_i(t^{(n)}_m)_{>0}\}_{n \in \mathcal{P}^m_i}$ be two sequences assigned to agent $i$ in slot $m$. In addition, we introduce a sequence $\{\alpha_i(t_{m+1}-1) \}_{m \in \mathbb{N}}$ and a scalar $\beta > 0$, where $\alpha_i(t_{m+1}-1)$ is the value of $\alpha_i$ at time instant $t_{m+1}-1$. Then, by considering the overall action/non-action instants, the updating law of the agents is given in Algorithm \ref{a1x}.\footnote[5]{We assume that the Asyn-PPG algorithm starts from slot $1$ by viewing the states in slot $0$ as historical data.}
\begin{algorithm}
\caption{Asynchronous Penalized Proximal Gradient Algorithm}\label{a1x}
\begin{algorithmic}[1]
\State Initialize $\mathbf{x}_i(t^{(1)}_1)$, $\mathbf{x}^{\mathrm{d}}(t_1)$, $\forall i \in \mathcal{V}$.
\State For all $t \in \mathcal{T}, i \in \mathcal{V}, n \in \mathcal{P}^m_i, m \in \mathbb{N}_+$,
\State if $t \in \mathcal{T}_i\cap [t_m,t_{m+1})$, then
\State $\quad t_m^{(n)} \leftarrow t$,
\State $\quad$update parameters: $\alpha_i(t^{(n)}_m)$, $\alpha_i(t_{m+1}-1)$, $\eta_i(t^{(n)}_m)$,
\State $\quad$update state:
\begin{align}\label{}
& \mathbf{x}_i(t^{(n)}_{m}+1) = \mathrm{prox}_{h_i}^{\eta_i(t^{(n)}_m)} (\mathbf{x}_i(t^{(n)}_m) - \eta_i(t^{(n)}_m) \nonumber \\
& \quad \quad \cdot (\nabla f_i(\mathbf{x}_i(t^{(n)}_m)) + \frac{\beta \mathbf{W}_i}{\alpha_i (t_{m+1}-1)}\mathbf{x}^{\mathrm{d}}(t_m))); \nonumber
\end{align}
\State if $ t \in [t_m,t_{m+1}) \hbox{ \& }t \notin \mathcal{T}_i$, then
\State $\quad$ $\mathbf{x}_i(t+1) = \mathbf{x}_i(t)$.
\State Stop under certain convergence criterion.
\end{algorithmic}
\end{algorithm}
Note that $\mathbf{W}_i\mathbf{x}^{\mathrm{d}}(t_m) = \mathbf{A}^{\top}_i \mathbf{A} \mathbf{x}^{\mathrm{d}}(t_m)$. Hence, $\frac{\beta \mathbf{W}_i\mathbf{x}^{\mathrm{d}}(t_m)}{\alpha_i (t_{m+1}-1)}$ can be viewed as a violation penalty of a ``delayed'' global constraint $\mathbf{A} \mathbf{x}^{\mathrm{d}} (t_m)= \mathbf{0}$, which is a variant of the penalty method studied in \cite{li2017convergence}.
Algorithm \ref{a1x} provides a basic framework for solving the proposed optimization problem in the SAN. An illustrative state updating process by Asyn-PPG algorithm in a 3-agent SAN is shown in Fig. \ref{p3}.
\begin{figure}[htbp]
\centering
\includegraphics[height=7cm,width=10cm]{123}\\
\caption{An illustrative updating process of the Asyn-PPG algorithm in a 3-agent SAN. In this example, the state of the agents evolves from $(\mathbf{x}_1(t^{(1)}_1),\mathbf{x}_2(t^{(1)}_1),\mathbf{x}_3(t^{(1)}_1))$ to $(\mathbf{x}_1(t^{(1)}_2 +1 ),\mathbf{x}_2(t^{(1)}_2+1),\mathbf{x}_3(t^{(1)}_2+1))$ with the historical state provided at the beginning of each time slot. This updating process is parallel but asynchronous due to the arbitrarily determined action instants of the agents. Specifically, to compute $\mathbf{x}_3(t^{(1)}_2+1)$, the state information available for agent 3 in slot 2 is $(\mathbf{x}_1(t^{(1)}_1+1),\mathbf{x}_2(t^{(1)}_1+1),\mathbf{x}_3(t^{(2)}_1+1))$ rather than $(\mathbf{x}_1(t^{(2)}_1+1),\mathbf{x}_2(t^{(2)}_1+1),\mathbf{x}_3(t^{(2)}_1+1))$ since the action instants of $\mathbf{x}_1(t^{(1)}_1+1) \rightarrow \mathbf{x}_1(t^{(2)}_1+1)$ and $\mathbf{x}_2(t^{(1)}_1+1) \rightarrow \mathbf{x}_2(t^{(2)}_1+1)$ are too close to $t_2$, and therefore, $\mathbf{x}_1(t^{(2)}_1+1)$ and $\mathbf{x}_2(t^{(2)}_1+1)$ can not reach agent 3 by $t_2$ due to the communication delays in the network.}\label{p3}
\end{figure}
Based on Asyn-PPG algorithm, we have the following two propositions.
\begin{Proposition}\label{pp1}
({Equivalent Representation A}) By Algorithm \ref{a1x}, $\forall i \in \mathcal{V}$, $n \in \mathcal{P}^m_i$, $m \in \mathbb{N}$, we have
\begin{subequations}
\begin{align}\label{}
& \mathbf{x}_i (t^{(1)}_{m+1}) = \mathbf{x}_i (t^{(P_{i,m}+1)}_{m}), \label{e1} \\
& \alpha_i(t^{(P_{i,m})}_{m})=\alpha_i(t^{(0)}_{m+1}), \label{e6} \\
& \eta_i(t^{(P_{i,m})}_{m})= \eta_i(t^{(0)}_{m+1}), \label{e7} \\
& \mathbf{x}_i (t^{(n)}_{m}+1) = \mathbf{x}_i (t^{(n+1)}_{m}), \label{e0} \\
& \mathbf{x}_i(t_{m+1}) = \mathbf{x}_i (t^{(P_{i,m}+1)}_{m}), \label{e2} \\
& \alpha_i(t_{m}^{(P_{i,m})}) = \alpha_i(t_{m}^{(P_{i,m}+1)} -1), \label{e4} \\
& \alpha_i(t_{m+1}-1)= \alpha_i(t_{m}^{(P_{i,m}+1)} -1), \label{e4+1} \\
& \eta_i(t_{m}^{(P_{i,m})}) = \eta_i(t_{m}^{(P_{i,m}+1)} -1), \label{e5} \\
& \eta_i(t_{m+1}-1) = \eta_i(t_{m}^{(P_{i,m}+1)} -1). \label{e5+1}
\end{align}
\end{subequations}
\end{Proposition}
\begin{proof}
See Appendix \ref{pp1p}.
\end{proof}
\begin{Proposition}\label{la2}
By Algorithm \ref{a1x}, $\forall m \in \mathbb{N}$, we have
\begin{align}
& \parallel \mathbf{x} (t_{m+1}) - \mathbf{x}^{\mathrm{d}}(t_{m+1}) \parallel^2 \nonumber \\
& \leq \sum_{i \in \mathcal{V}}\sum_{n = 1}^{P_{i,m}} D \parallel \mathbf{x}_i(t^{(n+1)}_{m}) - \mathbf{x}_i(t^{(n)}_{m}) \parallel^2, \label{l1}\\
& \parallel \mathbf{x} (t_{m+1}) - \mathbf{x}(t_m) \parallel^2 \nonumber \\
& \leq \sum_{i \in \mathcal{V}}\sum_{n = 1}^{P_{i,m}} H \parallel \mathbf{x}_i(t^{(n+1)}_{m}) - \mathbf{x}_i(t^{(n)}_{m}) \parallel^2. \label{l2}
\end{align}
\end{Proposition}
\begin{proof}
See Appendix \ref{la2p}.
\end{proof}
\section{Main Result}\label{se5}
In this section, we will establish the parameters of the Asyn-PPG algorithm for solving Problem (P1) in the SAN.
\subsection{Determination of Parameters}
In Algorithm \ref{a1x}, the penalty coefficient $\frac{\beta}{\alpha_i (t_{m+1}-1)}$ is designed to be increased steadily with $m \rightarrow + \infty$, which can speed up convergence rate compared with the corresponding fixed penalty method. The updating law of sequence $\{\alpha_i(t)_{>0}\}_{t\in \mathcal{T}_i}$ for agent $i$ is designed as
\begin{equation}\label{th12+1}
\frac{1- {\theta_i(t^{(n)}_m)}}{\alpha_i(t^{(n)}_m)} = \frac{1}{\alpha_i(t^{(n-1)}_m)},
\end{equation}
and sequence $\{\eta_i(t)_{>0}\}_{t\in \mathcal{T}_i}$ is decided by
\begin{equation}\label{th14+1}
\frac{\theta_i(t^{(n)}_m) - \theta_i(t^{(n)}_m)\eta_i(t^{(n)}_m)\mu_i}{\eta_i(t^{(n)}_m)\alpha_i(t^{(n)}_m)} \leq \frac{\theta_i(t^{(n-1)}_{m})}{\eta_i(t^{(n-1)}_{m})\alpha_i(t^{(n-1)}_{m})},
\end{equation}
with $\theta_i(t^{(n)}_m) \in (0,1)$, $\forall i \in \mathcal{V}$, $m \in \mathbb{N}_+$, $n \in \mathcal{P}^m_i$.
\begin{Proposition}\label{pr1}
({Strictly Decreasing}) Given that $\alpha_i(t^{(0)}_1) > 0$, the sequence $\{ \alpha_i(t)\}_{t \in \mathcal{T}_i}$ generated by (\ref{th12+1}) is strictly decreasing with $t \rightarrow + \infty$, $\forall i \in \mathcal{V}$.
\end{Proposition}
\begin{proof}
The validation of proposition \ref{pr1} is straightforward with $\theta_i(t^{(n)}_m) \in (0,1)$ in (\ref{th12+1}) and relation (\ref{e6}).
\end{proof}
\begin{Proposition}\label{pp2}
({Equivalent Representation B}) By Algorithm \ref{a1x} and (\ref{th12+1}), $\forall i \in \mathcal{V}$, $m \in \mathbb{N}$, we have
\begin{subequations}
\begin{align}\label{}
& \theta_i(t^{(P_{i,m})}_{m})= \theta_i(t^{(0)}_{m+1}), \label{e9} \\
& \theta_i(t_{m}^{(P_{i,m})}) = \theta_i(t_{m}^{(P_{i,m}+1)} -1), \label{e8} \\
& \theta_i(t_{m+1}-1)= \theta_i(t_{m}^{(P_{i,m}+1)} -1). \label{e8+1}
\end{align}
\end{subequations}
\end{Proposition}
\begin{proof}
Note that by (\ref{th12+1}), the values of $\theta_i$ and $\alpha_i$ are updated simultaneously at any instant in $\mathcal{T}_i$ after the initialization of $\alpha_i$. Hence, by recalling equivalent representations (\ref{e6}), (\ref{e4}) and (\ref{e4+1}), (\ref{e9})-(\ref{e8+1}) can be verified.
\end{proof}
Before more detailed discussions on the updating law of $\alpha_i$ and $\theta_i$, we introduce the following definition.
\begin{Definition}\label{dd1}
(Synchronization of $\{\alpha_i(t_{m}-1) \}_{m \in \mathbb{N}_+}$) In the SAN, sequence $\{\alpha_i(t_{m}-1) \}_{m \in \mathbb{N}_+}$ is synchronized if
\begin{align}\label{tha1}
& \alpha_1(t_{m}-1)= ... = \alpha_i(t_{m}-1)= ... = \alpha_{N}(t_{m}-1).
\end{align}
\end{Definition}
Under condition (\ref{tha1}), we further define a common sequence $\{\alpha(t_{m}-1) \}_{m \in \mathbb{N}_+}$ with
\begin{align}\label{cc}
\alpha(t_{m}-1) =\alpha_i(t_{m}-1)
\end{align}
for convenience purpose, $\forall i \in \mathcal{V}$.
The synchronization strategy for $\{\alpha_i(t_{m}-1) \}_{m \in \mathbb{N}_+}$ is not unique. One feasible realization is provided as follows.
\begin{Lemma}\label{lm3}
Let (\ref{th12+1}) hold. Let
\begin{align}\label{1}
{\alpha_1(t^{(P_{1,0})}_{0})}= ... = {\alpha_{N}(t^{(P_{{N},0})}_{0})}
\end{align}
and
\begin{align}\label{lp1}
\frac{\theta_i(t^{(n)}_m)}{\alpha_i(t^{(n)}_m)} = \frac{1}{P_{i,m}},
\end{align}
$\forall i \in \mathcal{V}$, $m \in \mathbb{N}_+$, $n \in \mathcal{P}^m_i$. Then, we have (\ref{tha1}),
\begin{align}
& \alpha(t_m-1)=\frac{\alpha(t_{1}-1)}{(m-1)\alpha(t_{1}-1)+1}, \label{17-1} \\
& \frac{1}{\alpha_i(t_m^{(n)})} = \frac{n}{P_{i,m}} + \frac{1}{\alpha(t_1 - 1)} + m-1, \label{17-2} \\
& \frac{\alpha_i(t_m^{(n)})}{\alpha(t_{m+2}-1)} \in ( 1,\Pi], \quad \Pi = \frac{2\alpha(t_1-1) + 1}{\frac{1}{H}\alpha(t_1-1) + 1}. \label{17-5}
\end{align}
\end{Lemma}
\begin{proof}
See Appendix \ref{lm3p}.
\end{proof}
\begin{Lemma}\label{lpl}
Let (\ref{lp1}) hold. Let
\begin{align}\label{llp}
& \frac{\eta_i(t_{m}-1)}{\eta_j(t_{m}-1)} = \frac{P_{j,m-1}}{P_{i,m-1}},
\end{align}
$\forall i,j \in \mathcal{V}$, $m \in \mathbb{N}_+$. Then,
\begin{align}\label{cx}
\frac{\theta_1(t_{m}-1)} {\alpha_1(t_{m}-1)\eta_1(t_{m}-1)} & = ... = \frac{\theta_i(t_{m}-1)} {\alpha_i(t_{m}-1)\eta_i(t_{m}-1)} \nonumber \\
= ... = & \frac{\theta_{N}(t_{m}-1)} {\alpha_{N}(t_{m}-1)\eta_{N}(t_{m}-1)}.
\end{align}
\end{Lemma}
\begin{proof}
See Appendix \ref{lplp}.
\end{proof}
Under condition (\ref{cx}), we define a common sequence $\{ \Xi_m \}_{m \in \mathbb{N}_+}$ with
\begin{align}\label{271}
\Xi_m = \frac{\theta_i(t_{m}-1)} {\alpha_i(t_{m}-1)\eta_i(t_{m}-1)}
\end{align}
for convenience purpose, $\forall i \in \mathcal{V}$.
\begin{Remark}
{Lemmas \ref{lm3}} and {\ref{lpl}} imply that the determination of $\alpha_i$, $\theta_i$, and $\eta_i$ requires some slot-wide knowledge of the actions, i.e., $P_{i,m}$, which is realizable when each agent knows the update frequency of itself.
\end{Remark}
\subsection{Convergence Analysis}
Based on the previous discussions, we are ready to provide the main results as follows.
\begin{Lemma}\label{z}
In the proposed SAN, suppose that {Assumptions \ref{a0}} to {\ref{a5}}, (\ref{th12+1}), and (\ref{th14+1}) hold. Then, by Algorithm \ref{a1x}, for any $({\mathbf{x}}^*,{\bm{\lambda}}^*) \in \mathcal{X}$, $i\in \mathcal{V}$, $m \in \mathbb{N}_+$, $n \in \mathcal{P}^m_i$, we have
\begin{align}\label{zt2}
&\frac{1}{\alpha_i(t_{m+1}-1)}({F}_i(\mathbf{x}_i(t_{m+1})) - {F}_i(\mathbf{x}_i^*) + \langle \bm{\lambda}^*, \mathbf{A}_i\mathbf{x}_i(t_{m+1})\rangle) \nonumber \\
& - \frac{1}{\alpha_i(t_{m}-1)}({F}_i(\mathbf{x}_i(t_{m})) - {F}_i(\mathbf{x}_i^*) + \langle \bm{\lambda}^*, \mathbf{A}_i\mathbf{x}_i(t_{m})\rangle) \nonumber \\
\leq & \frac{1}{\beta} \langle \bm{\lambda}^*- \frac{\beta \mathbf{A}\mathbf{x}^{\mathrm{d}}(t_m)}{\alpha_i(t_{m+1}-1)}, \frac{\beta\mathbf{A}_i\mathbf{x}_i(t_{m+1})}{\alpha_i(t_{m+1}-1)} - \frac{\beta\mathbf{A}_i\mathbf{x}_i(t_{m})}{\alpha_i(t_m-1)} \rangle \nonumber \\
& + \sum_{n=1}^{P_{i,m}} \frac{1}{2\alpha_i(t^{(n)}_m)}(L_i- \frac{2-\theta_i(t^{(n)}_m)}{\eta_i(t^{(n)}_m)} ) \parallel \mathbf{x}_i(t_m^{(n+1)}) \nonumber \\
& -\mathbf{x}_i(t_m^{(n)}) \parallel^2 + \sum_{n=1}^{P_{i,m}} \frac{\theta_i(t_m^{(n)})}{\alpha_i(t_m^{(n)})} \langle \frac{\beta \mathbf{A}\mathbf{x}^{\mathrm{d}}(t_m)}{\alpha_i(t_{m+1}-1)}, \mathbf{A}_i \mathbf{x}_i^* \rangle \nonumber \\
& + \frac{\theta_i(t_{m}-1)}{2\alpha_i(t_{m}-1)\eta_i(t_{m}-1)} \parallel \mathbf{x}_i^*-\mathbf{x}_i(t_{m}) \parallel^2 \nonumber \\
& - \frac{\theta_i(t_{m+1}-1)}{2\alpha_i(t_{m+1}-1)\eta_i(t_{m+1}-1)} \parallel \mathbf{x}_i^*-\mathbf{x}_i(t_{m+1}) \parallel^2.
\end{align}
\end{Lemma}
\begin{proof}
See Appendix \ref{z1}.
\end{proof}
{Lemma \ref{z}} provides a basic result for further convergence analysis. It can be seen that, in the proposed SAN, the state of agent $i$ is decided by its own parameters $\alpha_i$, $\theta_i$ and $\eta_i$, which are further decided by the action instants in $\mathcal{T}_i$. By the parameter settings in Lemmas \ref{lm3} and \ref{lpl}, we have the following theorem.
\begin{Theorem}\label{th1}
In the proposed SAN, suppose that {Assumptions \ref{a0}} to {\ref{a5}}, (\ref{th12+1}), (\ref{1}), and (\ref{lp1}) hold. Choose an $\eta_i(t^{(n)}_m)$ such that (\ref{th14+1}), (\ref{llp}), and
\begin{align}\label{th13}
& \frac{1}{\eta_i(t^{(n)}_m)} \geq L_i + \frac{2 (H+D)\beta \Pi \parallel \mathbf{A} \parallel^2}{\alpha(t_{m+2}-1)}
\end{align}
hold, $\forall i \in \mathcal{V}$, $ m \in \mathbb{N}_+$, $n \in \mathcal{P}^m_i$. Then, by Algorithm \ref{a1x}, for certain $K \in \mathbb{N}_+$ and any $({\mathbf{x}}^*,{\bm{\lambda}}^*) \in \mathcal{X}$, we have
\begin{align}\label{}
& \mid F(\mathbf{x}(t_{K+1})) - F(\mathbf{x}^*) \mid \nonumber \\
& \quad \quad \quad \leq ( \Delta_1 + \Delta_2 \parallel \bm{\lambda}^* \parallel) \alpha(t_{K+1}-1), \label{f2} \\
& \parallel \mathbf{A}\mathbf{x} (t_{K+1}) \parallel \leq \Delta_2 \alpha (t_{K+1}-1), \label{f3}
\end{align}
where
\begin{align}\label{}
\Delta_1= & \frac{1}{\alpha(t_{1}-1)}({F}(\mathbf{x}(t_{1})) - {F}(\mathbf{x}^*) + \langle \bm{\lambda}^*, \mathbf{A}\mathbf{x}(t_{1})\rangle) \nonumber \\
& + \frac{1}{2\beta } \parallel \frac{\beta\mathbf{A}\mathbf{x}(t_1)}{\alpha(t_1-1)} - \bm{\lambda}^* \parallel^2 + \frac{\Xi_1}{2} \parallel \mathbf{x}^*-\mathbf{x}(t_1) \parallel^2 \nonumber \\
& + \sum_{i\in \mathcal{V}}\sum_{n=1}^{P_{i,0}} \frac{D\beta \parallel \mathbf{A} \parallel^2}{\alpha^2(t_{2}-1)} \parallel \mathbf{x}_i(t_{0}^{(n+1)}) - \mathbf{x}_i(t_{0}^{(n)}) \parallel^2, \label{c1} \\
\Delta_2= & \frac{\sqrt{2\beta \Delta_1} + \parallel \bm{\lambda}^* \parallel }{\beta}. \label{c2}
\end{align}
\end{Theorem}
\begin{proof}
See Appendix \ref{tp1}.
\end{proof}
\begin{Remark}\label{re1}
{Theorem \ref{th1}} provides a sufficient condition of the convergence of the Asyn-PPG algorithm, which is characterized by the initial state of all the time slots and results in a periodic convergence result with period length $H$ (see some similar periodic convergence results in \cite{tseng1991rate,zhou2018distributed}).
\end{Remark}
To achieve the result of {Theorem \ref{th1}}, we need to choose a suitable $\eta_i(t^{(n)}_m)$ which is located in the space determined by (\ref{th14+1}), (\ref{llp}) and (\ref{th13}) adaptively. In the following, we investigate the step-size $\eta_i(t^{(n)}_m)$ in the form of
\begin{align}\label{16}
& \frac{1}{\eta_i(t^{(n)}_m)} = P _{i,m} ( Q_{m} + \frac{2(H+D)\beta \Pi \parallel \mathbf{A} \parallel^2}{\alpha_i(t_{m+2}-1)} ),
\end{align}
where $Q_{m}$, $\alpha_i(t_{m+2}-1)$ and $\beta$ are to be determined ($P_{i,0}$ and $Q_0$ are defined to initialize $\eta_i$).
\begin{Lemma}\label{l5}
Suppose that (\ref{th12+1}), (\ref{1}), and (\ref{lp1}) hold. Let the step-size be in the form of (\ref{16}) and $Q_{m} \geq L^{\mathrm{g}}$ with $L^{\mathrm{g}} = \max_{j \in \mathcal{V}} L_j$. Then, (\ref{llp}) and (\ref{th13}) hold. In addition, $\forall i \in \mathcal{V}$, $m \in \mathbb{N}_+$, $n \in \mathcal{P}^m_i$, we have
\begin{align}\label{151}
& \frac{\theta_i(t^{(n)}_m)}{\alpha_i(t^{(n)}_m)\eta_i(t^{(n)}_m)} - \frac{\theta_i(t^{(n-1)}_{m})} {\alpha_i(t^{(n-1)}_{m})\eta_i(t^{(n-1)}_{m})} \nonumber \\
& \leq \max \{ 0,Q_m - Q_{m-1} \nonumber \\
& \quad \quad \quad \quad \quad \quad + 2(H+D)\beta \Pi \parallel \mathbf{A} \parallel^2\}.
\end{align}
\end{Lemma}
\begin{proof}
See Appendix \ref{l5p}.
\end{proof}
\begin{Theorem}\label{la1}
In the proposed SAN, suppose that {Assumptions \ref{a0}} to {\ref{a5}}, (\ref{th12+1}), (\ref{1}), and (\ref{lp1}) hold and $\parallel \mathbf{A} \parallel \neq 0$. The step-size is selected based on (\ref{16}). Then, by Algorithm \ref{a1x}, given that
1) there exist a $K \in \mathbb{N}_+$ and an $\epsilon>0$, such that
\begin{align}\label{27}
K \geq \frac{1}{\epsilon} - \frac{1}{\alpha(t_{1}-1)},
\end{align}
2) there exists a $Q_{m}$, such that
\begin{align}
& Q_{m} \geq L^{\mathrm{g}}, \label{33}\\
& Q_{m} - Q_{m-1} < \frac{\mu}{H}, \label{35}
\end{align}
with $\mu=\min_{j \in \mathcal{V}} \mu_j$, and
3) $\beta$ is chosen as
\begin{align}\label{32-1}
\beta \in (0, & \frac{ \frac{\mu}{H} - \max_{k \in \mathbb{N}_+} (Q_{k} - Q_{k-1})}{2(H+D) \Pi \parallel \mathbf{A} \parallel^2}],
\end{align}
for any $({\mathbf{x}}^*,{\bm{\lambda}}^*) \in \mathcal{X}$, we have
\begin{align}
& \mid F(\mathbf{x}(t_{K+1})) - F(\mathbf{x}^*) \mid \leq \epsilon (\Delta_1 + \Delta_2 \parallel \bm{\lambda}^* \parallel), \label{22+1} \\
& \parallel \mathbf{A}\mathbf{x}(t_{K+1}) \parallel \leq \epsilon \Delta_2,\label{22+2}
\end{align}
where $\Delta_1$ and $\Delta_2$ take on the forms of (\ref{c1}) and (\ref{c2}), respectively.
Moreover, from a start position to $\mathbf{x}(t_{K+1})$, the convergence rate is given by
\begin{align}
& \mid F(\mathbf{x}(t_{K+1})) - F(\mathbf{x}^*) \mid \leq \mathcal{O}(\frac{1}{K}), \label{23} \\ & \parallel \mathbf{A}\mathbf{x}(t_{K+1}) \parallel \leq \mathcal{O}(\frac{1}{K}). \label{24}
\end{align}
\end{Theorem}
\begin{proof}
See Appendix \ref{tp2}.
\end{proof}
\begin{Remark}
To determine $Q_m$ by (\ref{33}) and (\ref{35}), one can choose a uniform $Q_0 = ... = Q_m = ... \geq L^{\mathrm{g}}$, $m \in \mathbb{N}$, such that (\ref{33}) and (\ref{35}) hold at all times and $ \beta \in (0, \frac{ \mu}{2H(H+D) \Pi \parallel \mathbf{A} \parallel^2}]$. Alternatively, a varying $Q_m$ means that, in slot $m$, one can choose $Q_m \in [L^{\mathrm{g}} , Q_{m-1} + \frac{\mu}{H})$, which is non-empty if $Q_{m-1} \geq L^{\mathrm{g}}$. That means, given that $Q_0 \geq L^{\mathrm{g}}$, $Q_m$ can be determined by (\ref{33}) and (\ref{35}) throughout the whole process. In the trivial case that $\parallel \mathbf{A} \parallel = 0$, as seen from Algorithm \ref{a1x}, $\beta$ can be chosen in $\mathbb{R}$.
\end{Remark}
\subsection{Distributed Realization of Asyn-PPG Algorithm}\label{rm2}
In some large-scale distributed networks, directly implementing Algorithm \ref{a1x} can be restrictive in the sense that each agent needs to access the full state information, which can be unrealizable if the communication networks are not fully connected \cite{pang2019randomized}. To overcome this issue, a promising solution is to establish a central server responsible for collecting, storing and distributing the necessary information of the system (as discussed in \cite{zhou2018distributed,ho2013more,li2014scaling}), which can also effectively relieve the storage burden of the historical data for the agents. In such a system, each agent pushes its state information, e.g., $\mathbf{x}_i(t)$, into the server and pulls the historical information, e.g., $\mathbf{x}^{\mathrm{d}}(t_m)$, from the server due to the delays between the agent side and the server.
As another distributed realization, we consider a composite objective function ${F}(\mathbf{x})=\sum_{i \in \mathcal{V}} {F}_i(\mathbf{x}_i)$ without any coupling constraint, where the agents aim to achieve an agreement on the optimal solution to $\min_{\mathbf{x}} {F}(\mathbf{x})$ by optimizing private functions ${F}_i(\mathbf{x}_i)$, $\forall i\in \mathcal{V}$. To this end, we can apply graph theory and consensus protocol $\mathbf{x}_1 = ... = \mathbf{x}_{N}$ if $\mathcal{G}$ is connected. In this case, $\mathbf{A}$ can be designed as ($\mathbf{A}_{ei}$ is the $(e,i)$th sub-block of $\mathbf{A}$)
\begin{align}\label{}
\mathbf{A}_{e \in {\mathcal{E}},i \in \mathcal{V}} = \left\{\begin{array}{cc}
\mathbf{I}_M & \hbox{if }e = (i,j) \hbox{ $\&$ } i < j, \\
-\mathbf{I}_M & \hbox{if }e = (i,j) \hbox{ $\&$ } i > j, \\
\mathbf{O}_{M \times M} & \hbox{otherwise,}
\end{array}
\right.
\end{align}
which is an augmented incidence matrix of $\mathcal{G}$ \cite{dimarogonas2010stability}. It can be checked that the consensus of $\mathbf{x}$ can be defined by $\mathbf{A}\mathbf{x}=\mathbf{0}$. Then, we can have $\mathbf{W} = \mathbf{A}^{\top} \mathbf{A} = \mathbf{L} \otimes \mathbf{I}_M \in \mathbb{R}^{NM \times NM }$ with $\mathbf{W}_{ij} \neq \mathbf{O}_{M \times M}$ only if $(i,j) \in \mathcal{E}$ or $i = j$ \cite{sun2017distributed}. Hence, the updating of $\mathbf{x}_i$ in $\mathcal{T}_i$ can be written as
\begin{align}\label{}
\mathbf{x}_i(t^{(n)}_{m}+1) = & \mathrm{prox}_{h_i}^{\eta_i(t^{(n)}_m)} (\mathbf{x}_i(t^{(n)}_m) - \eta_i(t^{(n)}_m) (\nabla f_i(\mathbf{x}_i(t^{(n)}_m)) \nonumber \\
+ & \frac{\beta }{\alpha_i (t_{m+1}-1)} \sum_{j\in \mathcal{V}_i \cup \{i\} }\mathbf{W}_{ij}\mathbf{x}_j^{\mathrm{d}}(t_m))),
\end{align}
which means the agent only needs to access the delayed information from neighbours. See an example in simulation A.
\section{Numerical Simulation}\label{se6}
In this section, we discuss two motivating applications of the proposed Asyn-PPG algorithm.
\subsection{Consensus Based Distributed LASSO Problem}\label{ss1}
\begin{figure}[htbp]
\centering
\includegraphics[height=3.5cm,width=4.5cm]{5agent}\\
\caption{Communication typology of the 5-agent SAN.}\label{gr}
\end{figure}
In this subsection, the feasibility of the Asyn-PPG algorithm will be demonstrated by solving a consensus based distributed LASSO problem in a connected and undirected 5-agent SAN ${\mathcal{G}} = \{{\mathcal{V}}, {\mathcal{E}} \}$. The communication topology is designed in Fig. \ref{gr}.
In this problem, the global cost function is considered as $\widetilde{F}_A(\tilde{\mathbf{x}})= \frac{1}{2} \sum_{i \in \mathcal{V}} \parallel \mathbf{P}_i\tilde{\mathbf{x}} - \mathbf{q}_i \parallel^2 + \varrho \parallel \tilde{\mathbf{x}} \parallel_1$, $\tilde{\mathbf{x}} \in \mathbb{R}^5$, $\varrho >0$. To realize a consensus based distributed computation fashion, inspired by \cite{bazerque2010distributed}, the local cost function of agent $i$ is designed as $F_{A,i} (\mathbf{x}_i) = \frac{1}{2} \parallel \mathbf{P}_i\mathbf{x}_i - \mathbf{q}_i \parallel^2 + \frac{\varrho}{|\mathcal{V}|} \parallel \mathbf{x}_i \parallel_1$, $\mathbf{x}_i \in \mathbb{R}^5$. The idea of generating the data follows the method introduced in \cite{parikh2014proximal}. Firstly, we generate a $(5 \times 5)$-dimensional matrix $\mathbf{P}'_i$, where each element is generated by a normal distribution $\mathcal{N}(0,1)$. Then, normalize the columns of $\mathbf{P}'_i$ to have $\mathbf{P}_i \in \mathbb{R}^{5 \times 5}$. $\mathbf{q}_i$ is generated by $\mathbf{q}_i=\mathbf{P}_i\hat{\mathbf{x}}_i+\bm{\delta}_i$, where $\hat{\mathbf{x}}_i \in \mathbb{R}^{5}$ is certain given vector and $\bm{\delta}_i \sim \mathcal{N}(\mathbf{0},10^{-3}\mathbf{I}_5)$ is an additive noise, $\forall i \in {\mathcal{V}}$.
Then, the consensus based distributed LASSO problem can be formulated as the following linearly constrained optimization problem:
\begin{align}\label{sta}
\hbox{\textbf{(P2)}}: \quad \min_{\mathbf{x}} \quad & F_A(\mathbf{x})=\sum_{i \in \mathcal{V}} F_{A,i} (\mathbf{x}_i) \nonumber \\
\hbox{subject to} \quad & \mathbf{M} \mathbf{x}=\mathbf{0},
\end{align}
where $ \mathbf{M}$ is generated by the method introduced in section \ref{rm2}, i.e.,
\begin{align}\label{}
\mathbf{M} = \left(
\begin{array}{ccccc}
\mathbf{I}_5 & -\mathbf{I}_5 & \mathbf{O}_{5 \times 5} & \mathbf{O}_{5 \times 5} & \mathbf{O}_{5 \times 5} \\
\mathbf{O}_{5 \times 5} & \mathbf{I}_5 & -\mathbf{I}_5 & \mathbf{O}_{5 \times 5} & \mathbf{O}_{5 \times 5} \\
\mathbf{O}_{5 \times 5} & \mathbf{O}_{5 \times 5} & \mathbf{I}_5 & -\mathbf{I}_5 & \mathbf{O}_{5 \times 5} \\
\mathbf{I}_5 & \mathbf{O}_{5 \times 5} & \mathbf{O}_{5 \times 5} & -\mathbf{I}_5 & \mathbf{O}_{5 \times 5} \\
\mathbf{O}_{5 \times 5} & \mathbf{O}_{5 \times 5} & \mathbf{O}_{5 \times 5} & \mathbf{I}_5 & -\mathbf{I}_5 \\
\end{array}
\right) ,
\end{align}
and $\mathbf{x}=(\mathbf{x}^{\top}_1,\mathbf{x}^{\top}_2,...,\mathbf{x}^{\top}_{5})^{\top}$.
\subsubsection{{Simulation Setup}}
The width of time slots is set as $H=10$ and the upper bound of communication delays is set as $D=2$. To represent the ``worst delays'', we let $\tau (t_m) = t_m- D$, $\forall m \in \mathbb{N}_+$. In slot $m$, the update frequency of agent $i$ is chosen from $P_{i,m} \in \{1,2,...,H\}$, and the action instants are randomly determined. $\varrho$ is set as $10$. Other settings for $\alpha_i$, $\eta_i$ and $\beta$ are consistent with the conditions specified in {Theorem 2}, $i \in \mathcal{V}$. To show the dynamics of the convergence error, we let $\mathbf{x}^*$ be the optimal solution to Problem (P2) and define $\gamma_A(t)= | F_A(\mathbf{x}(t)) - F_A(\mathbf{x}^*) |$, $t \in \mathcal{T}$.
\subsubsection{{Simulation Result}}
By Algorithm \ref{a1x}, the simulation result is shown in Figs. \ref{g2}-(a) to \ref{g2}-(c).
The action clock of the agents is depicted in Fig. \ref{g2}-(a).
By performing Algorithm \ref{a1x}, Fig. \ref{g2}-(b) shows the dynamics of decision variables of all the agents. It can be noted that all the trajectories of the agents converge to $\mathbf{x}^*$. The dynamics of $\gamma_A(t)$ is shown in Fig. \ref{g2}-(c).
\begin{figure*}
\centering
\subfigure[Action clock of the agents. ``1'' represents ``action'' and ``0'' represents ``non-action''.]{
\begin{minipage}[t]{0.33\linewidth}
\centering
\includegraphics[width=6.5cm]{po10}
\end{minipage
\subfigure[Dynamics of the state of the agents.]{
\begin{minipage}[t]{0.33\linewidth}
\centering
\includegraphics[width=6.5cm]{po11}
\end{minipage
\subfigure[Dynamics of convergence error $\gamma_A(t)$.]{
\begin{minipage}[t]{0.33\linewidth}
\centering
\includegraphics[width=6.5cm]{po12}
\end{minipage
\caption{Result of Simulation A with $H=10$ and $D=2$.}\label{g2}
\end{figure*}
\begin{figure*}
\centering
\subfigure[Action clock of UCs and users. ``1'' represents ``action'' and ``0'' represents ``non-action''.]{
\begin{minipage}[t]{0.33\linewidth}
\centering
\includegraphics[width=6.5cm]{TD=5}
\end{minipage
\subfigure[Dynamics of the state of UCs and users.]{
\begin{minipage}[t]{0.33\linewidth}
\centering
\includegraphics[width=6.5cm]{D=5}
\end{minipage
\subfigure[Dynamics of convergence error $\gamma_B(t)$.]{
\begin{minipage}[t]{0.33\linewidth}
\centering
\includegraphics[width=6.5cm]{D1=5}
\end{minipage
\caption{Result of Simulation B with $H=15$ and $D=5$.}\label{g3}
\end{figure*}
\subsection{Social Welfare Optimization Problem in Electricity Market}
In this subsection, we verify the feasibility of our proposed Asyn-PPG algorithm by solving a social welfare optimization problem in the electricity market with 2 utility companies (UCs) and 3 users.
The social welfare optimization problem is formulated as
\begin{align}\label{}
\textbf{(P3)}: \quad \min_{\mathbf{x}} \quad & \sum_{i \in \mathcal{V}_{{\textrm{UC}}}} C_i(x^{{\textrm{UC}}}_{i}) - \sum_{j \in \mathcal{V}_{\textrm{user}}} U_j(x^{\textrm{user}}_{j}) \nonumber\\
\hbox{subject to} \quad & \sum_{i \in \mathcal{V}_{{\textrm{UC}}}} x^{{\textrm{UC}}}_{i} = \sum_{j \in \mathcal{V}_{\textrm{user}}} x^{\textrm{user}}_{j}, \label{s1} \\
& x^{{\textrm{UC}}}_{i} \in [0,x^{{\textrm{UC}}}_{i,\textrm{max}}], \quad \forall i \in \mathcal{V}_{{\textrm{UC}}} \\
& x^{\textrm{user}}_{j} \in [0,x^{\textrm{user}}_{j,\textrm{max}}], \quad \forall j \in \mathcal{V}_{\textrm{user}}
\end{align}
where $\mathcal{V}_{{\textrm{UC}}}$ and $\mathcal{V}_{\textrm{user}}$ are the sets of UCs and users, respectively. $\mathbf{x} = (x^{{\textrm{UC}}}_{1},...,x^{{\textrm{UC}}}_{|\mathcal{V}_{{\textrm{UC}}}|}, x^{\textrm{user}}_{1},...,x^{\textrm{user}}_{|\mathcal{V}_{\textrm{user}}|})^{\top}$ with $x^{{\textrm{UC}}}_{i}$ and $x^{\textrm{user}}_{j}$ being the quantities of energy generation and consumption of UC $i$ and user $j$, respectively. $C_i(x^{{\textrm{UC}}}_{i})$ is the cost function of UC $i$ and $U_j(x^{\textrm{user}}_{j})$ is the utility function of user $j$, $i \in \mathcal{V}_{{\textrm{UC}}}$, $j \in \mathcal{V}_{\textrm{user}}$. Constraint (\ref{s1}) ensures the supply-demand balance in the market. $x^{{\textrm{UC}}}_{i,\textrm{max}}>0$ and $x^{\textrm{user}}_{j,\textrm{max}}>0$ are the upper bounds of $x^{{\textrm{UC}}}_{i}$ and $x^{\textrm{user}}_{j}$, respectively.
The detailed expressions of $C_i(x^{{\textrm{UC}}}_{i})$ and $U_j(x^{\textrm{user}}_{j})$ are designed as \cite{pourbabak2017novel}
\begin{align}\label{}
& C_i(x^{{\textrm{UC}}}_{i}) = \kappa_i (x^{{\textrm{UC}}}_{i})^2 + \xi_i x^{{\textrm{UC}}}_{i} +\varpi_i, \nonumber \\
& U_j(x^{\textrm{user}}_{j}) =
\left\{\begin{array}{ll}
\nu_j x^{\textrm{user}}_{j} - \varsigma_j (x^{\textrm{user}}_{j})^2, & x^{\textrm{user}}_{j} \leq \frac{\nu_j}{2\varsigma_j} \\
\frac{\nu_j^2}{4 \varsigma_j}, & x^{\textrm{user}}_{j} > \frac{\nu_j}{2\varsigma_j} \nonumber
\end{array}
\right.
\end{align}
respectively, where $\kappa_i,\xi_i,\varpi_i,\nu_j,\varsigma_j$ are all parameters, $\forall i \in \mathcal{V}_{{\textrm{UC}}}$, $\forall j \in \mathcal{V}_{\textrm{user}}$.
Note that the structure of Problem (P3) can be modified into that of (P1) with the method introduced in Remark \ref{re12}.
By some direct calculations, the optimal solution to Problem (P3) can be obtained as $\mathbf{x}^* = (0, 179.1, 55.51, 65.84, 57.75)^{\top}$. Define $F_B(\mathbf{x})= \sum_{i \in \mathcal{V}_{{\textrm{UC}}}} C_i(x^{{\textrm{UC}}}_{i}) - \sum_{j \in \mathcal{V}_{\textrm{user}}} U_j(x^{\textrm{user}}_{j})$ and $\gamma_B(t) = |F_B(\mathbf{x}(t))-F_B(\mathbf{x}^*)|$, $t \in \mathcal{T}$.
\subsubsection{{Simulation Setup}}
The parameters of this simulation are listed in Table I \cite{pourbabak2017novel}. The width of slots and the upper bound of communication delays are set as $H=15$ and $D=5$, respectively. In addition, to test the performance of the Asyn-PPG algorithm with large heterogeneity of the update frequencies, the percentages of action instants of UC 1, UC 2, user 1, user 2, and user 3 are set around $80\%$, $20\%$, $100\%$, $50\%$, and $70\%$, respectively.
\subsubsection{{Simulation Result}}
The simulation result is shown in Figs. \ref{g3}-(a) to \ref{g3}-(c). Fig. \ref{g3}-(a) shows the action clock of UCs and users. Fig. \ref{g3}-(b) shows the dynamics of the decision variables of them. The dynamics of convergence error is shown in Fig. \ref{g3}-(c). It can be seen that their states converge to the optimal solution $\mathbf{x}^*$. Notably, due to the local constraints on the variables, the optimal supply quantities of UC 1 and UC 2 reach the lower and upper bounds, respectively, and other variables converge to interior optimal positions.
\begin{table}\label{t2}
\caption{Parameters of UCs and users}
\label{tab2}
\begin{center}
\begin{tabular}{cccccccc}
\bottomrule
& \multicolumn{4}{l}{$ \quad \quad \quad \quad \quad$ \quad UCs} & \multicolumn{3}{l}{$\quad \quad \quad$ Users} \\
\hline
$i/j$ & $\kappa_i$ & $\xi_i$ & $\varpi_i$ & $x^{\textrm{UC}}_{i,\textrm{max}}$ & $\nu_j$ & $\varsigma_j$ & $x^{\textrm{user}}_{j,\textrm{max}}$ \\
\hline
1 & 0.0031& 8.71 & 0 & 113.23 & 17.17 & 0.0935 & 91.79 \\
2 & 0.0074 & 3.53 & 0 & 179.1 & 12.28 & 0.0417 & 147.29 \\
3 & - & - & - & - & 18.42 & 0.1007 & 91.41 \\
\bottomrule
\end{tabular}
\end{center}
\end{table}
\section{Conclusion}\label{se7}
In this work, we proposed an Asyn-PPG algorithm for solving a linearly constrained composite optimization problem in a multi-agent network. An SAN model was established where the agents are allowed to update asynchronously with possibly outdated information of other agents. Under such a framework, a periodic convergence with rate $\mathcal{O}(\frac{1}{K})$ is achieved. As the main feature, the theoretical analysis of the Asyn-PPG algorithm is based on deterministic derivation, which is advantageous over the stochastic method which relies on the acquisition of large-scale historical data. The distributed realization of Asyn-PPG algorithm in some specific networks and problems was also discussed.
|
2010.07261
|
\section{Introduction}
Enabling chatbots to indulge in engaging conversations requires massive datasets of human-human conversations \cite{ritter2011data,sordoni2015neural,vinyals2015neural,zhang2018personalizing,zhang2019dialogpt}.
Training such dialog agents requires substantial time and effort expended in the collection of adequate number of high quality conversation samples.
\citet{hancock2019learning} alleviate this problem by introducing a self-feeding chatbot which can directly learn from user interactions.
This chatbot requests users to provide natural language feedback when the users are dissatisfied with its response.
\citet{hancock2019learning} treat this feedback as a gold response to the wrong turn and use it as an additional training sample to improve the chatbot.
\begin{figure}
\centering
\includegraphics[width=\columnwidth]{EMNLP-Chatbot-camera-ready.pdf}
\caption{
When the bot provides a poor response to the question posed by the user, the bot requests natural language feedback.
We use the conversation context and the feedback to construct a plausible response to the user query and use it as an additional training sample to improve the chatbot.
}
\label{fig:interface}
\vspace{-4mm}
\end{figure}
Although natural language feedback is cheap to collect from a chatbot's end-users, most often, feedback cannot be used directly as a training sample since feedback is usually not the answer itself, but simply contains hints to the answer.
\Cref{tab:response_samples} shows some feedback text samples.
Naive modification of feedback using heuristics like regular expressions would lead to generic responses that are ineffective in improving the dialog ability of chatbots \cite{li2016diversity}.
Additionally, writing an exhaustive set of regular expression rules is time consuming and requires extensive analysis of the data.
Annotating data to convert feedback text to natural response is also expensive and defeats the purpose of learning from feedback text.
\begin{table}[t]
\centering
\footnotesize
\begin{tabular}{|l|}
\hline
you could say hey, i’m 30. how old are you? \\ \hline
yes, i play battlefield would be a great answer \\ \hline
tell me what your favorite breakfast food is \\ \hline
answer the question about having children! \\ \hline
\end{tabular}
\caption{\label{tab:response_samples} Samples of feedback to the chatbot. These contain hints to the answer but they are not the answers themselves.}
\vspace{-4mm}
\end{table}
In this work, we propose a generative adversarial setup for converting such noisy feedback instances into natural, human-like responses that provide better training signals for the dialog agents.
\Cref{fig:interface} gives a bird's-eye view of our problem.
We frame this problem as a variant of text style transfer where the generator is tasked with making the feedback resemble the optimal response to the user's previous utterance and the discriminator is a classifier that distinguishes whether a given response is feedback or natural.
Our main contributions are the following:
\begin{itemize}
\item We introduce \textsc{Feed2Resp}\xspace, a text style transfer system that converts feedback to natural responses without full supervision, thus generating additional training samples (\Cref{sec:feedresp}).
\item We show that the training on \textsc{Feed2Resp}\xspace modified responses leads to improved accuracy of chatbots (\Cref{sec:experiments}).
Our results also reveal that training naively on feedback doesn't help when the original chatbot is already a strong model, whereas \textsc{Feed2Resp}\xspace also helps strong models.
\end{itemize}
\section{Feedback to Natural Response Model}
\label{sec:feedresp}
\citet{hancock2019learning} introduce a novel variant of a self-feeding chatbot in which the dialogue agent is equipped with the capability of extracting new training samples while in conversation with humans after deployment (Figure 1). The agent also employs a satisfaction module which is trained to predict how satisfied the partner is with the responses it provides. When the chatbot is engaged in a conversation where the predicted satisfaction is below a defined threshold(usually 0.5), a feedback loop is triggered where the agent requests feedback from the human user on what should have been the response. The agent then utilizes the feedback text as the target response in new training examples for the primary dialogue ranking task. \citet{hancock2019learning} show that this cost-efficient method of extracting new examples improves the chatbot's dialogue abilities. In this work, we show that naive use of the collected feedback is not necessarily a good technique and instead, we propose an approach to better utilize the collected feedback samples.
We pose the problem of converting feedback to resemble natural response as a text style transfer problem. We observe that feedback is more instructional and judgemental, whereas natural response is direct (answering questions) and engaging (asking questions, contains humor).
We naturalize the feedback to a response and use it as an additional training sample to improve the chatbot.
A fully supervised approach to convert feedback to natural response is infeasible as we do not have paired (feedback $\leftrightarrow$ response) examples and thus we adopt an adversarial setup.
We utilize a GAN \citep{gangoodfellow} formulation where the generator modifies the feedback's style to make it seem part of a natural conversation, and in turn fool the discriminator which knows how to distinguish natural responses and feedback.
Our model, \textsc{Feed2Resp}\xspace, is shown in \Cref{fig:a}.
\begin{figure}[tp]
\centering
\includegraphics[width=\columnwidth]{Feed2Resp}
\caption{\textsc{Feed2Resp}\xspace Architecture}
\label{fig:a}
\vspace{-4mm}
\end{figure}
\subsection{Adversarial Setup}
Given an input sentence $\mathbf{x}$ (feedback or natural response) with source style $\mathbf{s}$, conversation history $\mathbf{h}$ and target style $\widehat{\mathbf{s}}$, the generator performs the mapping
\vspace{-2mm}
\begin{equation}
g_{\theta}:(\mathbf{x}, \mathbf{h}, \widehat{\mathbf{s}})\mapsto\widehat{\mathbf{y}}
\vspace{-2mm}
\end{equation}
Here $\widehat{\mathbf{y}}$ is the rewrite of $\mathbf{x}$ into style $\widehat{\mathbf{s}}$.
It is often the case that feedback and desired responses share many words (see \Cref{tab:pred-examples}).
We use BART encoder-decoder initialized with pretrained weights as our generator since its denoising objective helps in copying from the input while also producing realistic sentences \cite{lewis2019bart}.
We additionally pretrain our model under the summarization setting to extract only the response when presented with conversation history and response.
This helps maintain brevity while still integrating details from the context in the response.
The discriminator is a transformer encoder network that learns to distinguish the style of feedback and natural responses.
Given an input text $\mathbf{x}$ and conversation history $\mathbf{h}$, it predicts the style class $\mathbf{c}$ of $\mathbf{x}$.
Formally, it is defined as follows:
\vspace{-4mm}
\begin{equation}
d_{\phi}:(\mathbf{x}, \mathbf{h}) \mapsto\mathbf{c}
\vspace{-2mm}
\end{equation}
\subsection{\textsc{Feed2Resp}\xspace Learning}
We train \textsc{Feed2Resp}\xspace on three main objectives that help the model to reconstruct sentences when the style is not changed, change its style meaningfully and distinguish different styles.
These objectives are shown to work well in other style transfer scenarios \cite{dai2019style}.
\paragraph{Self reconstruction objective}
For the scenario where the target style is the same as the source style, we train the generator to reconstruct the sentence given as input.
Considering the input sentence as $\mathbf{x}$, the source and the target style as $\mathbf{s}$, we minimize the negative log-likelihood loss to generate the same sentence $\mathbf{x}$ as output
\begin{equation}
{Loss}_{self}(\theta)=-\log p_{\theta}(\widehat{\mathbf{y}}=\mathbf{x} | \mathbf{x}, \mathbf{h}, \mathbf{s})
\end{equation}
\paragraph{Cycle consistency objective}
Taking inspiration from Cycle GAN \cite{zhu2017unpaired}, we introduce a cycle consistency constraint to ensure that the model learns to preserve the meaning when it modifies the style of the original sentence.
We first transform $\mathbf{x}$ to style $\widehat{\mathbf{s}}$ to produce $\widehat{\mathbf{y}}$, i.e., $g_{\theta}(\mathbf{x}, \mathbf{h}, \widehat{\mathbf{s}})$.
Subsequently, we feed as input $\widehat{\mathbf{y}}$ with the target style as ${\mathbf{s}}$ and the model is trained to reconstruct the original sentence $\mathbf{x}$.
We minimize the negative log-likelihood loss which is given by,
\begin{equation}
{Loss}_{cycle}(\theta)=- \log p_{\theta}\left(\mathbf{y}=\mathbf{x} | g_{\theta}(\mathbf{x}, \mathbf{h}, \widehat{\mathbf{s}}), \mathbf{h}, \mathbf{s}\right)
\end{equation}
\paragraph{Style modification objective}
To ensure that the style of an input sentence $\mathbf{x}$ is changed to match the target one $\widehat{\mathbf{s}}$, we use the discriminator's confidence as training signal.
The generator wants to maximize the probability of the discriminator to classify transformed input to the target style, and therefore, we use the negative log-likelihood of the discriminator as our loss.
\begin{equation}
{Loss}_{style}(\theta)=-p_{\phi}\left(\mathbf{c}= \widehat{\mathbf{s}} | g_{\theta}(\mathbf{x}, \mathbf{h}, \widehat{\mathbf{s}}\right))
\end{equation}
\subsection{End-to-end training}
The discrete nature of sampling and non-differentiability of the argmax operator prevents gradient backpropogation.
Following \citet{dai2019style}, we consider the softmax distribution produced by the generator, $g_{\theta}$ as the `soft' generated sentence and use it as input for further downstream networks to maintain differentiability.
\section{Experimental Setup}
\label{sec:experimental-details}
In \textsc{Feed2Resp}\xspace, the optimizer for both the generator and discriminator is AdamW. The learning rate of generator is 5e-6 while the learning rate of discriminator is 1e-4. The discriminator uses 4 stacked transformer layers and 4 attention heads. The token embedding size, style embedding size, positional embedding size and hidden size are all 256. For the BART \cite{lewis2019bart} generator, we use the implementation from HuggingFace \cite{Wolf2019HuggingFacesTS} and initialize the model with pretrained weights from the CNN/Daily Mail summarization task. Due to the characteristics of human response(refer Appendix \ref{dataset-statistics}), we limit the length of text generation to a maximum of 50 words and impose a repetition penalty of 2.0 to improve diversity of output.\bigbreak
While evaluating the effectiveness of the modified feedback responses, we use two implementations of dialog agents provided by ParlAI \cite{miller2017parlai}, \textsc{BiEncoder}\xspace and \textsc{PolyEncoder}\xspace. \textsc{BiEncoder}\xspace has two transformer layers and 2 attention heads. The optimizer is Adamax with learning rate of 0.0025. \textsc{PolyEncoder}\xspace uses 12 transformer layers and 12 attentions heads. The optimizer is Adamax with learning rate of 5e-05.\bigbreak
The hyperparmeters for the best performing model are arrived at by random sampling and subsequently verifying the outputs using human evaluation to rate the outputs from the style transfer task. The entire list of hyper-parameters is listed in the Table~\ref{tab:hyperparameters}.
\section{Experiments}
\label{sec:experiments}
Our goal is to test whether feedback helps improve the chatbot.
To do this, we compare models trained on conversational data with and without feedback data.
Below we describe the chatbot evaluation setting, our datasets, the main models and different settings of these models with and without feedback.
\subsection{Chatbot evaluation task and metrics}
Following \citet{hancock2019learning}, we choose PersonaChat \cite{zhang2018personalizing} as the main evaluation dataset.
This dataset consists of human-human conversations collected using crowdsourcing where each crowdworker takes a persona.
Since persona representation is a challenging research problem on its own, \citeauthor{hancock2019learning} ignore the persona and just use the conversations to train chatbots and we follow the same approach.
At test time, the model is presented the conversation history and 20 candidate responses and the model has to pick the correct response. Thus, we use HITS@1/20 metric for evaluation.
\subsection{Feedback data}
We use the feedback data collected by \citet{hancock2019learning} as this removes orthogonal factors such as differences in chatbot interfaces and annotation framework etc. which are not the focus of this work.
\citeauthor{hancock2019learning} collected this feedback by deploying bi-encoder chatbots (Section ~\ref{sec:models}) trained on varying levels of training data and making it converse with crowdworkers.
Whenever the bot's response is not satisfactory, natural language feedback is collected from the crowdworker.
The data thus collected contains 60k human-bot turns, of which the last turn is always the feedback.
\subsection{Chatbot Models}
\label{sec:models}
Given the conversation history and several candidate responses, the chatbot is trained to rank the correct candidate on the top.
We use the following models as our chatbots.
\vspace{0.2em}
\noindent \textbf{\textsc{BiEncoder}\xspace} \cite{hancock2019learning,humeau2020poly} contains two transformers, one for summarizing the conversation history and the other to summarize candidate responses to embeddings.
The response with highest similarity is taken as the best candidate response.
\vspace{0.2em}
\noindent \textbf{\textsc{PolyEncoder}\xspace} \cite{humeau2020poly} summarizes a context and candidate responses into several embeddings.
In order to contextualize context and candidates together, it performs a cross-encoder attention on the summary embeddings and scores each candidate.
\subsection{Feedback-based Models}
\label{sec:feedback_models}
We train and test the above models in the following settings.
\vspace{0.2em}
\noindent \textbf{\textsc{NoFeedback}\xspace}: The model is trained only on human conversations.
\vspace{0.2em}
\noindent \textbf{\textsc{Feedback}\xspace}: We train on the combination of human conversations and unmodified feedback data.
This setting is similar to \citet{hancock2019learning}.
\vspace{0.2em}
\noindent \textbf{\textsc{Heuristic}\xspace}: We design and use six regular expression rules based on the frequent patterns in the data that convert feedback to plausible dialog responses (see \Cref{app:regex-rules})
and train the chatbot models on human conversations along with the modified feedback.
\vspace{0.2em}
\noindent \textbf{\textsc{Feed2Resp}\xspace}: We use our main model (\Cref{sec:feedresp}) to modify feedback to natural responses and train the chatbot models on modified feedback along with human conversations.
\begin{table}[tp]
\footnotesize
\begin{center}
\begin{tabular}{lcc}
\toprule
{\bf Model} & {\bf Development} & {\bf Test}\\
\midrule
\multicolumn{3}{c}{\textsc{BiEncoder}\xspace chatbot} \\
\midrule
\bf \textsc{NoFeedback}\xspace & 49.03 (0.66) & 49.49 (0.49) \\
\bf \textsc{Feedback}\xspace & 49.27 (1.06) & 49.97 (1.30) \\
\bf \textsc{Heuristic}\xspace & 48.85 (0.70) & 49.85 (0.72) \\
\bf \textsc{Feed2Resp}\xspace & 50.84 (0.50) & 51.32 (0.43) \\
\midrule
\multicolumn{3}{c}{\textsc{PolyEncoder}\xspace chatbot} \\
\midrule
\bf \textsc{NoFeedback}\xspace & 73.35 (0.70) & 69.94 (0.37) \\
\bf \textsc{Feedback}\xspace & 72.63 (0.14) & 68.48 (0.64) \\
\bf \textsc{Heuristic}\xspace & 72.65(0.35) & 68.83(0.31) \\
\bf \textsc{Feed2Resp}\xspace & \bf 78.14 (0.40) & \bf 75.96 (0.80) \\
\bottomrule
\end{tabular}
\end{center}
\caption{Hits@1/20 of models on \textsc{PersonaChat}\xspace.
Naive and heuristic use of feedback results in marginal improvement or hurts performance, whereas \textsc{Feed2Resp}\xspace modified feedback gives large improvements.
The variances across three different runs are also shown.}
\label{tab:results}
\vspace{-2mm}
\end{table}
\section{Results and Discussion}
The experimental details of the model variants are described in \Cref{sec:experimental-details}.
\Cref{tab:results} shows the average HITS@1/20 of all models on the \textsc{PersonaChat}\xspace validation and test sets over 3 runs.
We were able to replicate results of \citet{hancock2019learning} which show that \textsc{BiEncoder}\xspace performance improves slightly (+0.48 on test) when \textsc{Feedback}\xspace is used.
\textsc{Heuristic}\xspace edits to feedback don't help while
\textsc{Feed2Resp}\xspace responses improve the results higher than \textsc{Feedback}\xspace and also have less variance.
Coming to \textsc{PolyEncoder}\xspace, it is a much stronger chatbot than \textsc{BiEncoder}\xspace.
We see that naive use of \textsc{Feedback}\xspace or \textsc{Heuristic}\xspace deteriorates the performance of \textsc{PolyEncoder}\xspace while \textsc{Feed2Resp}\xspace emerges a clear winner with +6.0 point improvement on the test set over \textsc{NoFeedback}\xspace.
\begin{table}[tp]
\footnotesize
\begin{tabular}{p{4.5cm}ll}
\toprule
Example & Freq. & Acc. \\
\midrule
\multicolumn{3}{c}{Modification type: Rewrite} \\
\midrule
F: tell me about your favorite show & 18.5\% & 81\% \\
F2R: I love watching TV shows and sitcoms like friends \\
\midrule
\multicolumn{3}{c}{Modification type: Remove} \\
\midrule
F: you could’ve said, yes the sugar cinnamon kind is my favorite & 40\% & 68.7\% \\
F2R: yes the sugar cinnamon kind is my favorite \\
\midrule
\multicolumn{3}{c}{Modification type: Retain} \\
\midrule
F: the temperature is hot & 41.5\% & 74.6\% \\
F2R: the weather is hot \\
\bottomrule
\end{tabular}
\caption{Statistics of different modification types based on 200 random feedback texts.
F stands for feedback, and F2R is the response of \textsc{Feed2Resp}\xspace model.
Freq. indicates the frequency of the modification type, and Acc. the accuracy of \textsc{Feed2Resp}\xspace on each type. \Cref{app:feed2resp examples} lists additional examples of modified feedback responses.}
\label{tab:ling-phenomena}
\vspace{-2mm}
\end{table}
\begin{figure*}
\centering
\includegraphics[width=\textwidth]{feedback_classification.png}
\caption{Attention maps for Feedback responses. Words such such \textit{you should have}, \textit{you could have}, \textit{tell me} heavily influence the discriminator to classify it as feedback and hence the generator learns to remove such words to fool the discriminator. Darker shades of green mean higher attention scores and shades of red mean lower attention scores.}
\label{fig:my_label}
\end{figure*}
\paragraph{Feed2Resp analysis}
We randomly sample 200 feedback responses from \textsc{Feed2Resp}\xspace to determine the kind of modifications the model performs (\Cref{tab:ling-phenomena}).
We observe three main types of modifications --- \textit{Rewrite, Retain and Remove}.
\textsc{Rewrite}\xspace is when the feedback implies an hint to the answer but not the answer itself.
\textsc{Remove}\xspace is when the feedback contains the answer with extraneous words that have to be removed.
\textsc{Retain}\xspace are cases where the model copies or paraphrases the feedback.
Among these, \textsc{Remove}\xspace has the lowest accuracy of modification.
Upon inspection, we find that these are the cases which require multiple removals.
For example, for \textit{You should reply with either yes or no}, the model predicts \textit{yes or no} together instead of either one of them.
Additionally, we visualize the attention maps of the discriminator to observe which words contribute most to the classification decision of the discriminator (\Cref{fig:my_label}). The discriminator learns to distinguish feedback from normal dialog responses due to the presence of sequences like \textit{you could have}, \textit{you should have}, \textit{tell me}, etc.
Thus the generator learns to remove such extraneous sequences and make the feedback seem like plausible responses.
We present a sample of modified outputs of \textsc{Feed2Resp}\xspace in \Cref{app:feed2resp examples}.
\section{Conclusion}
In this work, we show that while chatbots can be improved using natural language feedback, converting feedback to natural responses that fit in the conversation outperform the naive usage of feedback.
We presented \textsc{Feed2Resp}\xspace, a generative adversarial model, that converts feedback to natural responses without requiring manually annotated parallel data.
Our results show that \textsc{Feed2Resp}\xspace results in a 6~point improvement for the \textsc{PolyEncoder}\xspace chatbot, an already powerful dialog ranking agent.
This is a strong result as HITS@1/20 is a tough metric to improve upon \citep{hancock2019learning}.
Our work joins the class of models that use natural language feedback to improve different tasks, e.g., image captioning \cite{NIPS2017_7092}, classification \cite{srivastava2017joint,hancock-2018-training,murty2020expbert}.
While these methods use feedback for reward shaping or feature extraction, we use feedback to produce correct response using adversarial learning.
We pose this problem as a style transfer problem inspired from the style transfer literature \citep{shen2017style,xu2018unpaired,li2018delete,lample2019,dai2019style}.
While these focus on studying the stylistic attributes of sentences, e.g, sentiment, we explore this problem in the context of improving chatbots.
\section{Acknowledgements}
We thank Yue Dong for her multiple helpful discussions during the course of this project. We also thank Sandeep Subramanian for his insightful guidance at a crucial stage of this work. This research was enabled in part by computations support provided by Compute Canada (www.computecanada.ca).
The last author is supported by the NSERC Discovery Grant on \textit{Robust conversational models for accessing the world's knowledge}.
\bibliographystyle{acl_natbib}
|
2206.04498
|
\section{Introduction}
Graph signal processing (GSP) has attracted increased attention as it allows us to capture complex correlations in many practical problems. Signals observed in many applications can be modeled as graph signals. Examples include photographs, CMOS sensor images, and readings from sensor networks. GSP has been applied to various problems, including signal recovery, prediction, sampling, and anomaly detection \cite{Shuman2013, San13, San14, Gad14, Moura2015, Don16, Egi17, Isu17, Isu17b, Ort18, JiTay:J19}.
Given a graph $G$, a \emph{graph signal} on $G$ assigns a value to each node of $G$, resulting in a vector of dimension equal to the size (number of nodes) of the graph. A graph signal is indexed not by time but by the nodes of the graph, called the \emph{graph domain}, capturing irregular domains. A fundamental philosophy of GSP is to perform an orthogonal transformation of the graph domain, called a \emph{graph Fourier transform}. The new domain is usually called the \emph{frequency domain}, analogous to classical Fourier theory. The components of a signal represented in the frequency domain are its \emph{Fourier coefficients}. As in classical signal processing, a graph signal is analyzed by inspecting its Fourier coefficients. An orthogonal transformation is usually obtained via a \emph{graph shift operator} $A$, such as the adjacency matrix or Laplacian of $G$. Under favorable conditions, such as $G$ being undirected, $A$ has an orthonormal eigenbasis and gives rise to the desired transformation. The frequencies are ordered according to size of the eigenvalues.
Basic GSP requires full knowledge of the graph, such as to perform graph Fourier transforms. Such a requirement has shortcomings. For example, graphs can be used to model very large networks such as social networks. Gathering data and network information as well as performing signal processing as above can be costly and time consuming. Moreover, the framework is highly centralized, giving rise to issues such as data privacy.
On the other hand, some GSP methods do not require the global approach. To give an example, the concept of smoothness corresponds to signals having Fourier coefficients concentrated in the low frequencies. However, to leverage smoothness, one can optimize the quadratic form of total signal variation, which can be done in a decentralized manner. Considerations as such lead to the topic of distributed GSP, including the following efforts. Signal recovery is considered in \cite{Che15, Wan16}. The authors of \cite{Dor15} investigate distributed sparse signal representation with graph spectral dictionaries. Signal reconstruction based on node sampling is studied in \cite{Lor17}. In \cite{Jia19}, the authors consider distributed construction of filter banks. \cite{Ren21} proposes and analyzes a communication efficient distributed optimization framework for general non-convex non-smooth signal processing and machine learning problems under an asynchronous protocol.
A closely related area of research is distributed optimization. This is a well established field dating back to as early as 1980s \cite{Tsi84,Tsi86, Tsi89}. Though we are not able to give an overview of such a vast topic here, recent survey article \cite{Yan19} contains comprehensive discussions of historical developments and recent advances in distributed optimization. The key component is an iterative procedure that repeats the following: each member of the distributed system solves its own problem using information gathered from its pre-determined ``neighbors'' and share its findings with the neighbors. Theoretical results focus on convergence of such a procedure to the optimal solution. Tables 1--4 in \cite{Yan19} list all the key features of a large collection of algorithms proposed in the literature.
In terms of methods, the above mentioned works on distributed GSP have spirit common to distributed optimization. In this paper, we are going to consider a different approach. We realize that it is possible that an iterative procedure might be avoided if we tweak the information sharing step with neighbors. Instead of sharing numerical values, we propose sharing elements from a coherent system of vector bundles. The primary example is a smooth function on a domain $D$, whose local incarnation at every point of $D$ is a system of polynomials (by Taylor expansion). The advantage is that more information is contained in such a mathematical object and we only have to deal with each player of the system once. In addition, the framework allows us to leverage numerical invariants such as dimension of manifolds to analyze the problem solving procedure. Thus it is convenient for us to answer theoretical questions such as solubility of a given problem without running the algorithm. In this work, we do not claim that our approach is superior to classical approaches, but we want to provide an alternative point of view, potentially increasing the domain of applicability of GSP.
The rest of the paper is organized as follows. We formulate the distributed GSP problem in \cref{sec:pro}. In \cref{sec:ner}, we formalize message passing in our setup. In particular, we define what an abstract message is. Moreover, to study solubility questions later on, we introduce the notion of ``local solution''. To handle the abstract notion of message, we need tools from the theory of differentiable manifolds. In \cref{sec:jet}, we introduce jet spaces and Whitney topologies, which are particularly relevant to our discussions. In \cref{sec:sub}, we discuss Morse functions and real analytic functions as the candidates of messages. Both enjoy nice properties that make the procedure of message passing well-behaved. In \cref{sec:sol}, we make use of the theory developed so far to analyze the solubility questions of distributed GSP problems. We propose how to implement our method in practise and present simulation results in \cref{sec:app}, and conclude in \cref{sec:con}.
\section{Problem formulation} \label{sec:pro}
In this section, we describe the setup of the main problem to be studied in the paper, including the notation used throughout the paper.
Let $G=(V,E)$ be an undirected graph, where $V$ is the vertex set and $E$ is the set of edges. Let $t$ be a positive integer, and let $G_i = (V_i,E_i)$, $i\in\{1,\ldots,t\}$ be $t$ subgraphs of $G$ such that $V = \cup_{i=1}^{t} V_i$. For each $i\in\{1,\ldots,t\}$, let $S_i\subset V_i\backslash (\cup_{j\neq i}V_j)$ be a set of nodes contained exclusively in $V_i$ (cf.\ \cref{defn:ato} below). Hence, the $S_i$, $i\in\{1,\ldots,t\}$ are disjoint.
Each node in $S_i$ is understood to be \emph{observable} in the sense that for any signal $x$ on $G_i$, its restriction to $S_i$, denoted $x|_{S_i}$, are known in $G_i$ for the purpose of certain tasks (defined and illustrated by examples below). Such a restriction $x|_{S_i}$ is called an \emph{observation}. Let $c_i=|S_i|$ denote the size of $S_i$ and $S=\cup_{i=1}^{t} S_i$ denote their union, with size $c=|S|=\sum_{i=1}^{t}c_i$ (by disjointness of the $S_i$).
For each $i\in\{1,\ldots,t\}$, recall that graph signals on $G_i$ are functions on the discrete $V_i$, which can be identified with $\mathbb{R}^{|V_i|}$. Let $f_i: \mathbb{R}^{|V_i|} \to \mathbb{R}$ be a convex function on the space of graph signals on $G_i$. Each $f_i$ can be extended to the space of signals on $G$ by composing $f_i$ with the projection $\mathbb{R}^{|V|} \to \mathbb{R}^{|V_i|}$. Let $f: \mathbb{R}^{|V|} \to \mathbb{R}$ denote the sum of these extensions: $f = \sum_{i=1}^{t}f_i$. We are interested in minimizing $f$ and solving certain problems associated with the minimizer. There are global as well as local versions of these problems, formally introduced next.
Recall that a manifold $\mathcal{M}$ of dimension $d$ is a Hausdorff topological space that is locally Euclidean of dimension $d$, i.e., every point has an open neighborhood homeomorphic to an open subset of $\mathbb{R}^d$. In Appendix~\ref{sec:fun}, we give a self-contained introduction to the fundamentals of differentiable manifolds.
\begin{Definition} \label{defn:ato}
A \emph{task on $G$} is a function $\tau: \mathbb{R}^{|V|} \to \mathcal{M}$ where $\mathcal{M}$ is a manifold.
For $U \subset \mathbb{R}^{|S|}$, any graph signal $s \in U$ and minimizer $\hat{x}_s = \argmin_{x|_{S}=s} f(x)$, if $\tau(\hat{x}_s) \in \mathcal{M}$ is well-defined (see explanation below), then we write $\Phi_{\tau}(s)=\tau(\hat{x}_s)$ for the composition of $\tau$ and $\hat{x}_s$. The function $\Phi_{\tau}: U \to \mathcal{M}$ is called the \emph{global problem}.
\end{Definition}
In the definition of $\Phi_{\tau}$, we require $\tau(\hat{x}_s)$ to be well-defined in the following sense. Recognizing that in general $\hat{x}_s$ is not unique, $\tau(\hat{x}_s)$ is \emph{well-defined} if and only if $\tau$ is constant on the set of minimizers $x$ of $f$ subject to $x|_{S}=s$. In particular, if for any $s\in \mathbb{R}^{|S|}$, $f$ has a unique minimizer $x$ such that $x|_{S}=s$, then the domain of $\Phi_{\tau}$ is the full space $\mathbb{R}^{|S|}$. A global problem $\Phi_{\tau}$ is said to be continuous, smooth, etc., if it bears the stated property. Next, we give two examples to illustrate the definitions above.
\begin{Example} \label{eg:ltt}
\begin{enumerate}[1)]
\item As a first example, let the task $\tau$ be the same as $f: \mathbb{R}^{|V|} \to \mathbb{R}$. Then the global problem $\Phi_{f}$ is well-defined on all of $\mathbb{R}^{|S|}$; i.e., the domain of $\Phi_{f}$ is $\mathbb{R}^{|S|}$.
\item \label{it:ite} In this second example, we introduce \emph{distributed sampling}, which will be revisited later. Let $z_1,\ldots, z_k \in \mathbb{R}^{|V|}$ be $k$ linearly independent graph signals on $G$ (so $k\leq|V|$). For each $i\in\{1,\ldots,t\}$ and $j\in\{1,\ldots,k\}$, $z_j$ restricted to $V_i$ is denoted ${z_j}|_{V_i}$. Let the function $f_i$ on $G_i$ be given by
\begin{align*}
f_i(x) = \min_{(r_1,\ldots,r_k)\in \mathbb{R}^k}
\norm{\sum_{j=1}^{k}r_j{z_j}|_{V_i} -x}^2,
\end{align*}
which is the square distance between $x\in\mathbb{R}^{|V_i|}$ and its orthogonal projection onto the span of $z_1|_{V_i},\ldots, z_k|_{V_i}$.
Next, consider the task $\tau: \mathbb{R}^{|V|} \to \mathbb{R}^k$ defined by
\begin{align*}
\tau(x) = \argmin_{(r_1,\ldots,r_k)\in \mathbb{R}^k} \norm{\sum_{j=1}^{k}r_j{z_j} -x}^2,
\end{align*}
which is the unique orthogonal resolution of $x$ onto the span of $z_1,\ldots,z_k$. In this case, the codomain manifold $\mathcal{M}$ of $\tau$ is $\mathbb{R}^k$.
For further simplification, let $t=2$ and assume that the two subgraphs $G_1$ and $G_2$ of $G$ have nonempty intersection, i.e., $V_1\cap V_2\neq\emptyset$.
Suppose that $x$ is a linear combinations of $z_1,\ldots, z_k$. Then, clearly $f_1(x)=f_2(x)=0$. So $\min f = 0$, where $f=f_1+f_2$, and $f_1$ and $f_2$ have been extended to $\mathbb{R}^{|V|}$.
We now explore conditions under which $\Phi_{\tau}$ is well defined, i.e., for any $s\in\mathbb{R}^{|S|}$, $\tau$ is constant on the set of minimizers of $f$ that agree with $s$ on $S$. To be clear, to say that a signal $x\in\mathbb{R}^{|V|}$ agrees with $s\in\mathbb{R}^{|S|}$ on $S$ means that $x|_{S}=s$. For $\tau$ to be constant on the set of such minimizers, the argmin in the definition of $\tau(x)$ must be the same regardless of the minimizer $x$.
For any minimizer $x$ of $f$, the two vectors of coefficients $(r_1,\ldots,r_k)$ in the definitions of $f_1(x)$ and $f_2(x)$ above are not necessarily the same. So, there are $2k$ decision variables (coefficients). Moreover, these $2k$ coefficients satisfy the following:
\begin{enumerate}[(a)]
\item The two linear combinations of $z_1,\ldots, z_k$ with the two vectors of $k$ coefficients are two signals $s_1$ and $s_2$ that agree on $S$; i.e., their restrictions to $S$ are the same: $s_1|_{S} = s_2|_{S}$.
\item The signals $s_1$ and $s_2$ also agree on $V_1\cap V_2$, i.e., $s_1|_{V_1\cap V_2}=s_2|_{V_2\cap V_2}$.
\end{enumerate}
To ensure uniqueness of the argmin in the definition of $\tau(x)$ as explained above, we expect that $2k = |V_1\cap V_2| + c$.
\end{enumerate}
\end{Example}
\section{Message passing} \label{sec:ner}
\subsection{Nerve skeleton and message passing}
We want to make use of the message passing paradigm. It is a way of information transfer across a network, and there is nothing new about the process. In this section, we adopt message passing in our setup. Given $G$ and subgraphs $G_i$, $i\in\{1,\ldots,t\}$, we may construct the nerve skeleton $N$ to package information regarding pairwise intersections between different $G_i$'s as follows. Our goal is to solve a global problem in a distributed way. Therefore, to solve the problem on $G_i$ for some $i$, one gathers information from the rest of the graph at the intersections of $G_i$ with the other subgraphs. This motivates us the consider the following.
\begin{Definition}
The \emph{nerve skeleton}\footnote{There is a more general construction called nerve construction giving a simplicial complex. Here we just use its $1$-skeleton, and hence call the resulting graph the nerve skeleton.} $N_G=(V_{N_G}, E_{N_G})$ is an undirected graph of size $t$. Each vertex $g_i$ of $V_{N_G} = \{g_1,\ldots, g_t\}$ corresponds to the subgraph $G_i$. A pair $(g_i,g_j)$ with $i\neq j$ is an edge of $E_{N_G}$ if and only if $G_i\cap G_j \neq \emptyset$.
\end{Definition}
To perform message passing later on, it can be convenient to work with trees (e.g., \cite{Shah11}). Therefore, we are interested in spanning trees of $N$.
\begin{Definition} \label{defn:stn}
Suppose $T\subset N_G$ is a spanning tree. Its complement $T^c$ is the closure of $N_G\backslash T$. If $g \in V_{N_G}$ is considered as a root, then $T_g$ is the unique directed tree on $T$ such that each edge is directed towards $g$, i.e., the head of an edge $(g_i,g_j)$ is $g_j$ if $g_j$ is below $g_i$, where we say that $g_j$ is below $g_i$ if $g_i$ is on the path connecting $g$ and $g_j$.
\end{Definition}
Fixing a root $g$, we use the directed tree $T_g$ to perform message passing, while $T^c$ is used to keep track of connections between intersecting subgraphs not captured by $T$. We now start to discuss message passing.
On $T_g$, consider a directed edge $(g_i,g_j)$ on $T_g$ such that $g_i$ is the tail and $g_j$ is the head, i.e., the direction goes from $g_i$ to $g_j$. Suppose at $g_i$, there is a convex multi-variable function $h_i$. We would like categorize the input variables of the function $h$ as follows:
\begin{enumerate}[1)]
\item $s_i$: this set of multi-variables corresponds to signals on $S_i$.
\item $x_i$: this set of multi-variables corresponds to signals on the union of the set $X_i$ of nodes $V_i\cap V_j$ and $\cup_{(g_i,g_k)\in T^c}(V_i\cap V_k)$.
\item $y_i$: this set of multi-variables corresponds to signals on the nodes $Y_i$ belonging only to $g_j$ below $g_i$. In particular, $Y_i$ includes nodes contained exclusively in $V_i$.
\item $z_i$: this set includes all the remaining variables, use $Z_i$ to denote those coordinates.
\end{enumerate}
The \emph{message associated with $h_i$ along the edge $(g_i,g_j)$} is
\begin{align}
\tilde{h}_i(x_i,z_i) = \min_{y_i} h_i(s_i,x_i,y_i,z_i): \mathbb{R}^{|X_i|+|Z_i|} \to \mathbb{R}.
\end{align}
To explain the domain, we notice that $s_i$ are in fact fixed because we can take observations on $S_i$. Hence, the message $\tilde{h}_i$ is a function on the variables $x_i$ and $z_i$. According to such a definition, a message is not merely a number or a vector, but instead it is a function.
Before describing message passing, we state the following {\bf conventions}, given a function $f: \mathbb{R}^a \to \mathbb{R}$, for convenience: $f$ gives rise to a function, also denoted by $f$, as $f: \mathbb{R}^{a+b} \to \mathbb{R}$, by composing $f$ with the projection $\mathbb{R}^{a+b} \to \mathbb{R}^a$.
We are ready to describe the \emph{message passing} on $T_g$ with root $g$ as the following procedure:
\begin{enumerate}[S1]
\item \label{it:sft} Starting from the leaves of $T_g$, each leaf node $g_i$ pass the messages $\tilde{f}_i$ associated with $f_i$ to its only immediate neighbor.
\item For each node $g_j$ other than $g$, once it receives messages from all edges with $g_j$ as the head node, sum them up and $f_j$ to obtain $h_j$.
\item The message $\tilde{h}_j$ associated with $h_j$ is passed to $g_k$, along the unique edge $(g_j,g_k)$ such that $g_k$ is the head.
\item \label{it:tpt} The procedure terminates at $g$. Take the sum over all the messages received at $g$ including the original convex function at $g$. The resulting function is denoted by $h_g$.
\end{enumerate}
For later use, we formally extract the key ingredients of the above procedure as follows.
\begin{Definition} \label{defn:wct}
We call the collection of functions $\tilde{f}_i, \tilde{h}_j$ and $h_g$ in \ref{it:sft}$-$\ref{it:tpt} as \emph{messages of the message passing on $T_g$}. The function $h_g$ is the \emph{aggregated message along $T_g$}.
\end{Definition}
We notice that according to the definition, each node is associate with exactly one function as its message.
\begin{Example} \label{eg:cts}
For illustration, consider the situation given in \cref{fig:dsp1}. At the top of the figure, we use the Venn diagram to describe the intersection properties of the $3$ subgraphs $G_1, G_2$ and $G_3$. The capital letters label the nodes in respective regions. The nerve skeleton is the complete graph on $3$ nodes as shown on the bottom left. We consider two different spanning trees $T_1$ and $T_2$ shown on the bottom right. Either $T_1^c$ or $T_2^c$ is the single dashed edge.
For perform message passing on both $T_1$ and $T_2$ with $g_1$ as the root. Let the resulting directed trees be $T_{1,g_1}$ and $T_{2,g_1}$. On $T_{1,g_1}$, messages $\tilde{f}_2(x_2,x_3) = \min_{y_2} f_2(x_2,x_3,y_2,s_2)$ and $\tilde{f}_3(x_1,x_3) = \min_{y_3} f_3(x_1,x_3,y_3,s_3)$ are passed from $g_2, g_3$ to $g_1$ concurrently. At $g_1$, to minimize the aggregated message, we have: $\min_{x_1,x_2,x_3,y_1} f_1 + \tilde{f}_2 + \tilde{f}_3$.
On $T_{2,g_1}$, we first pass the message $\tilde{f}_2(x_2,x_3) = \min_{y_2}f_2(x_2,x_3,y_2,s_2)$ from $g_2$ to $g_3$. At $g_3$, we form the new function $h_3 = f_3 + \tilde{f}_2$, and the message \begin{align*}\tilde{h}_3(x_1,x_2) = \min_{y_3,x_3}(\min_{y_2}f_2(x_2,x_3,y_2,s_2)+ f_3(x_1,x_3,y_3,s_3))\end{align*} is subsequently passed to $g_1$. Finally at $g_1$, minimizing the aggregated message is $\min_{x_1,x_2,y_1} f_1 +\tilde{h}_3$.
\end{Example}
\begin{figure}
\centering
\includegraphics[width=0.3\columnwidth]{dsp1}
\caption{Setup for message passing of \cref{eg:cts}.} \label{fig:dsp1}
\end{figure}
We end this subsection by giving a formal definition of abstract message. Readers unfamiliar with the terminologies may ignore this part, as in the paper we work exclusively with the explicit examples of functions. The terms in the definition will take their concrete forms in \cref{sec:jet}.
\begin{Definition}
An \emph{abstract message} is an element in the limit of a sequence of vector bundles over a manifold $M$.
\end{Definition}
We now provide some insights in additional to the concrete examples at hand. Loosely speaking, a vector bundle is a parametrized family of vectors. It contains information about the parameter space, while we can still perform algebraic operations such as taking sums. Therefore, it is a natural choice if we want to generalize numerical or vectorial information. On the other hand, we may also need to work with infinite dimension objects, hence the necessity to consider the limit of a sequence of finite dimensional objects.
\subsection{Local solution}
In \cref{defn:ato}, we have introduced the notion of global problem $\Phi_{\tau}: U \to \mathcal{M}$ given a task $\tau: \mathbb{R}^{|V|} \to \mathcal{M}$, where $U\subset \mathbb{R}^{|S|}$ is the domain of $\Phi_{\tau}$. To understand what we expect from a distributed approach, we fix a spanning tree $T$ of the nerve skeleton $N_G$.
\begin{Definition} \label{defn:f1w}
For $i\in\{1,\ldots,t\}$, we say that the global problem $\Phi_{\tau}$ can be \emph{solved at $g_i$ via message passing along $T$} if the following holds. Let $h_{g_i}$ be the aggregated message along $T_{g_i}$ with domain $D_i$. There is $\tau_i: D_i \to \mathcal{M}$ such that: for $\hat{x_i} \in \argmin_{x\in D_i} h_{g_i}(x)$, we have $\Phi_{\tau}(s) = \tau_i(\hat{x_i})$.
$\Phi_{\tau}$ can be \emph{solved locally via message passing along $T$} if it can be solved at $g_i$ for every $i\in\{1,\ldots,t\}$.
\end{Definition}
We summarize this definition in the following commutative diagram (with $\hat{x} \in \argmin_x f(x)$):
\[
\begin{tikzcd}[every arrow/.style={draw,mapsto}]
\{f_1,\ldots,f_t;s\} \arrow{d} \arrow{r} & h_{g_i} \arrow{r} & \hat{x_i} \arrow{d}{\tau_i} \\ \{f,s\} \arrow{r}
& \hat{x} \arrow{r}{\tau} & \Phi_{\tau}(s).
\end{tikzcd}
\]
In the diagram, the top route is via message passing while the bottom route is via solving the global problem. Local solubility requires the existence of the right vertical map $\tau_i$ such that both routes have a ``common destination''.
To give an example, the following result is a generalization of \cref{eg:cts}, which is essentially due to re-arrangement of ordering in taking $\min$. In fact, this result is key in motivating us considering messages as functions. It also gives a concrete example of local solubility to a global problem with minimum requirements on $f_i$, $i\in\{1,\ldots,t\}$. General study of local solubility shall be contained in \cref{sec:sol} below.
\begin{Proposition} \label{prop:ief}
Suppose $\tau = f: \mathbb{R}^{|V|} \to \mathbb{R}$. Then for any $T$ and $i\in\{1,\ldots,t\}$, $\Phi_f$ can be solved locally via message passing along $T$.
\end{Proposition}
\begin{IEEEproof}
We first make some remarks regarding message passing. In general, a sum of convex functions is convex. Minimizing a convex function over a subset of variables is convex on the remaining variable. Therefore, each message is a well defined convex function, as long as so does each $f_i$, $i\in\{1,\ldots,t\}$.
We prove the result by induction on the size of $T_{g}$. The result is trivially true if $T_{g}$ is a single node. Suppose that $T_g$ contains a leaf node $g_i$, and it is connected by a directed edge $(g_i,g_j)$ to $g_j$. Let $g_{i,1}, \ldots, g_{i,m}$ be the neighbors of $g_i$ in $T^c$, i.e., $\{g_j, g_{i,1}, \ldots, g_{i,m}\}$ are all the neighbors of $g_i$ in $N_G$. During the message passing from $g_i$ to $g_j$, the variables of $f_i$ are re-grouped into $s_i$, $x_i$, $y_i$ and $z_i$, where $x_i, z_i$ accounts for the nodes $V_i \cap (V_j \cup_{i=1}^{m}V_{i,l})$. The message being passed to $V_j$ is thus $\tilde{f_i}(x_i,z_i) = \min_{y_i}f_i(s_i,x_i,y_i,z_i)$.
For the global problem $\Phi_f$, we want to minimize $f = (f_1 + \ldots + f_{i-1} + f_{i+1} +\ldots + f_t) + f_i$, written as $h+f_i$. We now re-group the variables of $f$ as $x_i,y_i,z_i$ and $r$ where the variables $r$ are disjoint from $x_i,y_i$ and $z_i$. Hence $\Phi_f(s_1,\ldots,s_t) = \min_{x_i,y_i,z_i,r} h(x_i,z_i,r) + f_i(s_i,x_i,y_i,z_i)$. Notice here $s_1,\ldots, s_t$ are fixed numbers at observable nodes. However, $y_i$ is not involved in $h$. Therefore, $\Phi_f(s_1,\ldots, s_t) = \min_{x_i,z_i,r}h(x_i,z_i,r) + \tilde{f}_i(x_i,z_i)$. The right-hand-side is a global problem on the subtree of $T_g$ removing $g_i$, which can be solved locally at $g$ by the induction hypothesis.
\end{IEEEproof}
We end this section by considering the following example where local solution does not exist.
\begin{Example} \label{eg:tiac}
This is a continuation of \cref{eg:ltt}~\ref{it:ite} and we use the setup, such as choices of $f_i$, $i\in\{1,\ldots,t\}$, stated over there. To be concrete, we assume $t=2$ and the graph $G$ with $2$ subgraphs $G_1=(V_1,E_1)$ and $G_2=(V_2,E_2)$ is shown in \figref{fig:dsp4}. Hence $V_1=\{s_1,s_2,y_1,x\}$, $V_1=\{s_3,s_4,y_2,x\}$ and the intersection $V_1\cap V_2$ contains only $x$.
To describe $f_1,f_2$ as in \cref{eg:ltt}~\ref{it:ite}, we specify $k=3$. We arrange the vectors $v_1,v_2,v_3$ according to $s_1,s_2,y_1,x,s_3,s_4,y_2$ as:
\begin{align*}
v_1 = (1, 0, 1, 0, 1, 0, 1)', v_2 =(0, 1, 0, 0, 0, 1, 0)',
v_3 = (1, 0, 0, 1, 1, 0, 0)'.
\end{align*}
Let the task be $\tau: \mathbb{R}^7 \to \mathbb{R}, (s_1,s_2,y_1,x,s_3,s_4,y_2) \to y_1-y_2$.
For the global problem $\Phi_{\tau}$, it is in fact smooth as one may verify that $\Phi_{\tau}(s) = s_1-s_3$ for any observation $s = (s_1,s_2,s_3,s_4)$. However, it cannot be solved locally via message passing. To see this, one can show that $\tilde{f}_i$ is the constant $0$ function. One can directly verify this or observe that the components of $v_1,v_2,v_3$ corresponding to $s_1,s_2,x$ are linearly independent. Therefore, the aggregated message at $g_2$ (corresponding to $G_2$) is just $f_2$ itself. Given observations $s_3,s_4$, $\min_{x,y_2} f_2(x,s_3,s_4,y_2)=0$. To minimize $f_2$, the value $y_2$ can be any number. Therefore, we cannot find the $\tau_i$ required in \cref{defn:f1w}.
To give another perspective, as the message is the constant $0$ function, i.e., all the coefficients in the Taylor expansion are $0$. Information on both $s_1,s_2$ are lost, and we have lost ``two degrees of freedom''. Therefore, we are not able to solve the problem at $G_2$. We shall formalize such a point of view in subsequent sections.
\end{Example}
\begin{figure}
\centering
\includegraphics[width=0.4\columnwidth]{dsp4}
\caption{$G_1$ is the square and $G_2$ contains a single edge. They intersect at a single node.} \label{fig:dsp4}
\end{figure}
\section{Jet spaces and Whitney topologies} \label{sec:jet}
In \cref{sec:pro}, we cast the global problem as a function $\Phi_{\tau}: \mathbb{R}^{|S|} \supset U \to M$. We are interested in whether the global problem can be solved locally via message passing in \cref{sec:ner}. During message passing, a message is also an element of a function space. Instead of considering function spaces as sets, which do not have enough structures, our discussions shall revolve around topological structures of smooth function spaces in this section. A self-contained account on background materials is given in Appendix~\ref{sec:fun}.
Recall that a (real) multi-variable function is \emph{smooth} if its partial derivatives of any order exist. For a multi-variable smooth $f$, according to the Taylor's theorem, we can always approximate $f$ around any point by using a polynomial. On the other hand, polynomials of a bounded degree form a finite dimensional vector space. These observations prompts the following \cite{Gol73}.
\begin{Definition}
Let $M,N$ be open subsets of real vector spaces and $C^{\infty}(M,N)$ be the space of smooth functions from $M$ to $N$, i.e., each component is a smooth function on $M$. The \emph{$k$-jet space} $J_p^k(M,N)$ at $p \in M$ is the equivalent classes of $f, g \in C^{\infty}(M,N)$ with: $f \sim g$ if $f$ and $g$ have the same partial derivatives up to $k$-th order at $p$. By convention, equality on the $0$-th order partial derivative means $f(p)=g(p)$. The equivalence class of $f$ is denoted by $f_p$, and the resulting quotient map is $\pi_p^k: C^{\infty}(M,N) \to J_p^k(M,N)$, i.e., $\pi_p^k(f) = f_p$.
\end{Definition}
Apparently, each class of $J_p^k(M,N)$ has a unique representation which is a degree $k$-polynomial. If $M$ is an open subset of $\mathbb{R}^n$ and $N = \mathbb{R}^m$, there is a canonical isomorphism from $J_p^k(M,N)$ to the space of polynomials up to degree $k$: $P^k_{m,n} = (\mathbb{R}[x_1,\ldots,x_n]/(x_1,\ldots,x_n)^{k+1})^m$, the latter is a finite dimensional vector space parametrized by polynomial coefficients. For some simple examples: $\log(1-x) = -x - x^2/2 - x^3/3$ and $\sin(x) = x - x^3/6$ in $J_0^3(\mathbb{R},\mathbb{R})$.
For each $k\geq 0$, we have the obvious quotient map $J_p^{k+1}(M,N) \to J_p^k(M,N)$, which is also denoted by $\pi_p^{k}$ for convenience (as it is also a quotient map to $J_p^k(M,N)$).
If we let $p$ vary, then $J^k(M,N)$ is defined as the disjoint union of $J_p^k(M,N)$, namely, $J^k(M,N) = \{(p,f), p\in M, f\in J_p^k(M,N)\}$. As a manifold, $J^k(M,N)$ is identified with the Cartesian product $M \times P^k_{m,n}$. We remark that if $M$ and $N$ are general manifolds, we can still define $J^k(M,N)$ as a manifold by applying the above construction locally using coordinate maps. However, the total space $J^k(M,N)$ is no longer a product space in general. We shall not use the general construction in the sequel and details can be found in \cite{Gol73} Section 2.
We now use these jet spaces to give $C^{\infty}(M,N)$ topologies, called the \emph{Whitney topologies} \cite{Gol73}. For each $k$ and $f \in C^{\infty}(M,N)$, we have map $\Pi^k(f): M \to J^k(M,N): \Pi^k(f)(p) = (p,f_p)$. Hence, we have the set $\Pi^k(f)(M) \subset J^k(M,N)$
\begin{align*}
\Pi^k(f)(M) = \{(p,f_p) \mid p \in M\}.
\end{align*}
On the other hand, we also have the quotient maps, whose notation inherits from $\pi_p^k$, as $\pi^k: J^{k+1}(M,N) \to J^k(M,N), (p,f) \to (p,f_p)$. For illustration, a summary is given in \figref{fig:dsp2}.
\begin{figure}
\centering
\includegraphics[width=0.75\columnwidth]{dsp2}
\caption{Summary of the relations among the smooth function space and various jet spaces.} \label{fig:dsp2}
\end{figure}
\begin{Definition} \label{def:tbo}
The basis of the \emph{Whitney $C^k$-topology} on $C^{\infty}(M,N)$ is given by: $S^k(U) = \{f \in C^{\infty}(M,N) \mid \Pi^k(f)(M)\subset U\}$, where $U$ is open in $J^k(M,N)$. This means that open sets in $C^{\infty}(M,N)$ are arbitrary unions of sets of the form $S^k(U)$.
Let $W_k$ be the set of open sets given by the $C^k$-topology. The Whitney $C^{\infty}$-topology has basis $W = \cup_{k\geq 0} W_k$.
\end{Definition}
To give an intuition of the Whitney topologies, a sequence of maps $f_n \in C^{\infty}(M,N), n>0$ converges to $f$ in the $C^k$-topology if and only if the following hods: there is a compact subset $K$ of $M$ such that $\Pi^k(f_n): M \to J^k(M,N), n>0$ converges uniformly to $\Pi^k(f)$ on $K$ and all but finitely many $f_n$ agrees with $f$ outside $K$.
Back to the message passing paradigm, we are interested in maps such as $\alpha: U \to C^{\infty}(M,N)$, where $U$ is an open subset of a Euclidean space $\mathbb{R}^d$ and $N = \mathbb{R}$. In turn, composing with $\pi_p^k: C^{\infty}(M,N) \to J_p^k(M,N), p\in M, k\geq 0$ leads to a map $\alpha_p^k: U \to J_p^k(M,N)$, which depends on both $p\in M$ and $k$. To obtain a map independent of $p$, we enlarge the domain and consider
\begin{align*}
\alpha^k: M\times U \to J^k(M,N), (p,s) \mapsto (p, \alpha_p^k(s)).
\end{align*}
The maps $\alpha^k$ are consistent in the sense that $\alpha^k = \pi^k\circ \alpha^{k+1}$. The construction gives the map $\alpha^k$ between differentiable manifolds $M\times U$ and $J^k(M,N)$. With these maps for different $k$, we want to give $\alpha$ numerical invariants to measure the size of its image.
\begin{Definition} \label{def:sfe}
Suppose for each $k$, there are dense open subsets such $U_k$ of $M \times U$ such that the image $\alpha(U_k)$ of $\alpha^k: U_k \to J^k(M,N)$ is a submanifold of $J^k(M,N)$. Then define $b_{\alpha}^k = \dim (\alpha^k(U_k)) - \dim M$, and $b_{\alpha} = \sup_{k\geq 0} b_{\alpha}^k$.
\end{Definition}
We use a simple example to illustrate how one may compute something such as $\dim (\alpha^k(U_k))$. Suppose $r_1, r_2$ are two parameters. Then functions $\{(r_1+r_2)^2x^2+(r_1+r_2)x+1 \mid r_1,r_2\in \mathbb{R}\}$ is $1$ dimensional as the coefficient of the degree $1$ term determined uniquely the polynomial function. On the other hand, $\{r_1^2x^2+(r_1+r_2)x+1 \mid r_1,r_2\in \mathbb{R}\}$ is $2$ dimensional as we need to know both the degree $1$ and degree $2$ coefficients to know the polynomial function from the set.
\begin{Proposition} \label{prop:fos}
For open subset $U$ of $\mathbb{R}^d$ and $\alpha: U \to C^{\infty}(M,N)$, we assume that $b_{\alpha}^k,k\geq 0$ are well-defined and let $U_k, k\geq 0$ be given as in \cref{def:sfe}. Then the following holds:
\begin{enumerate}[1)]
\item If for each $k$, $\alpha^k$ is a submersion at some $p\in U_k$, then $b_{\alpha} \leq d$.
\item \label{it:tia} Suppose in addition that $\alpha^k$ is a submersion on $U_k$. There is $k_0$ such that for all $k\geq k_0$, $\pi^k: J^{k+1}(M,N) \to J^k(M,N)$ restricts to a diffeomorphism on a dense open subset of $\alpha^{k+1}(U_{k+1})$ to a dense open subset of $\alpha^k(U_k)$. In particular, $ b_{\alpha} = b_{\alpha}^k$ for each $k\geq k_0$.
\end{enumerate}
\end{Proposition}
\begin{IEEEproof}
For 1), as $\alpha^k: U_k \to J^k(M,N)$ is a submersion at $p$, it induces a surjective linear transformation on the tangent spaces of $U_k$ and $\alpha(U_k)$ at $p$ and $\alpha^k(p)$, respectively. We have $\dim (\alpha^k(U_k)) \leq \dim U_k = d+\dim M$. This shows that $b_{\alpha}^k \leq d$ and hence $b_{\alpha}\leq d$.
To prove 2), as the map $\pi^k: J^{k+1}(M,N) \to J^k(M,N)$ is onto, we notice that $b_{\alpha}^k$ is a bounded (by $d$) and non-decreasing sequence, when $k\to \infty$. By the monotone convergence theorem, $\lim_{k\to \infty} b_{\alpha}^k$ exists and equals to $b_{\alpha}$. As each $b_{\alpha}^k$ is an integer, there is $k_0$ such that $b_{\alpha}^k = b_{\alpha}$ for $k\geq k_0$.
Now for $k\geq k_0$, let $U_{k,k+1} = U_k\cap U_{k+1}$, which is again open dense. Then we have the following relation among various maps and spaces:
\begin{align*}
\alpha^k: U_{k,k+1} \stackrel{\alpha^{k+1}}{\to} J^{k+1}(M,N) \stackrel{\pi^k}{\to} J^k(M,N).
\end{align*}
In particular, $\pi^k$ induces a differentiable map from $\alpha^{k+1}(U_{k,k+1})$ to $\alpha^k(U_{k,k+1})$, with both having the same dimension $b_{\alpha}$. On the other hand, the differential $d \pi^k$ is every surjective. Therefore, $d \pi^k$ must be invertible when restricted $\alpha^{k+1}(U_{k,k+1})$. By the inverse function theorem, $\alpha^{k+1}(U_{k,k+1})$ is diffeomorphic to $\alpha^k(U_{k,k+1})$.
\end{IEEEproof}
\section{Messages passing for subfamilies of functions} \label{sec:sub}
The smooth function spaces discussed in \cref{sec:jet} is the playground for us to perform analysis. However, smooth convex functions in general does not behave well under message passing. To analyze local solubility of global problems, we need additional regularities on the functions. On the other hand, we also want the subfamilies contain most familiar functions. In the following, we introduce Morse functions and real analytic functions.
\subsection{Morse functions}
\begin{Definition}
Suppose $M$ is a differentiable manifold and $f:M \to \mathbb{R}$ is smooth. If at $p\in M$, $df_p=0$, then $p$ is called a \emph{critical point} and $f(p)$ is the \emph{critical value}.
The \emph{Hessian matrix} is $(\partial^2 f/\partial x_i\partial x_j)_{i,j}$. A critical point $p$ is called \emph{non-degenerate} if the Hessian is non-singular at $p$.
The function $f$ is called a \emph{Morse function} if it has no degenerate critical points.
\end{Definition}
A Morse function $f$ enjoys many nice properties \cite{Mil63}. For example if $M$ is an open subset of $\mathbb{R}^d$, then in an open neighborhood of any critical point $p$, $f$ takes the form $f(p) - x_1^2-\ldots -x_b^2 + x_{b+1}^2 +\ldots + x_d^2$. As a consequence, all the critical points of $f$ are isolated. In particular, if $f$ is also convex, it has a unique minimal point, i.e., $f$ is strictly convex.
Though it may seen from the definition that Morse functions consist of a restricted subfamily of all smooth functions. However, it is known that they in fact form an open dense subset of all smooth functions, under the $C^2$-topology given in \cref{def:tbo}. This means that Morse functions are omnipresent.
For later use, we make the following observation regarding Morse functions.
\begin{Lemma} \label{lem:iia}
Suppose $f(x,y): \mathbb{R}^{m_x+m_y} \to \mathbb{R}$ is a smooth, convex function on sets of multi-variables $x=(x_j)_{1\leq j\leq m_x}$ and $y = (y_k)_{1\leq k\leq m_y}$ of sizes $m_x$ and $m_y$ respectively.
\begin{enumerate}[1)]
\item If $f_{y_0}(x) = f(x,y_0)$ is a Morse function on $x$, then $\tilde{f}(y) = \min_{x \in \mathbb{R}^{m_x}}f(x,y)$ is a smooth function on an open neighborhood $U_{y_0} \subset \mathbb{R}^{m_y}$ containing $y_0$.
\item \label{it:tso} The set of $y$ such that $f_y=f(\cdot, y)$ being Morse forms an open subset of $\mathbb{R}^{m_y}$.
\end{enumerate}
\end{Lemma}
\begin{IEEEproof}
For 1), to find $\tilde{f}(y)$, we need to solve \begin{align}\frac{\partial f}{\partial x}(x,y) = 0.\label{eq:par}\end{align} If $f_{y_0}(x)$ is a Morse function, then the Hessian of $f_{y_0}$ at $x$ satisfying \cref{eq:par} is non-singular. By the implicit function theorem (c.f.\ \cref{thm:ift}), there is an open subset $U_{y_0}$ containing $y_0$, such that $g(y) = x$ with $(x,y)$ solving Equation~(\ref{eq:par}), is smooth in $y \in U_{y_0}$. Therefore, $\tilde{f}(y) = f(g(y))$ is a smooth function for $y\in U_{y_0}$.
For 2), by 1) and convexity, if $y \in U_{y_0}$, then $f_{y}(x)$ has a unique global minimum. Therefore, it has a unique isolated critical point, which must be non-degenerate. Hence, $f_y$ is a Morse function.
\end{IEEEproof}
The upshot of this result is that under favorable conditions, such as being Morse on a subset of variables, messages are smooth functions. More concretely, if $g_i$ is a leaf node of a directed tree $T_g$ and $f_i$ satisfies the condition of \cref{lem:iia}, in the initial step of message passing, we obtain a map from $\mathbb{R}^{|S_i|} \to C^{\infty}(M,\mathbb{R})$, where $M$ is a Euclidean space.
\subsection{Analytic functions}
Another important subfamily of smooth functions are the analytic functions. The subfamily includes many familiar ones such as the polynomials, exponential functions and the trigonometric functions. Formally, a function $f$ is \emph{analytic} in a connected open subset $D$ of a Euclidean space if for each $x\in D$, there is an open neighborhood $D_x\subset D$ of $x$ such that $f$ agrees with its Taylor series expansion about $x$ for any other point in $D_x$. The following observation related to our theme is essentially the identity theorem of analytic functions.
\begin{Lemma} \label{lem:sui}
Suppose $U$ is a connected open subset of $\mathbb{R}^d$ and $\alpha: U \to C^{\infty}(M,N)$ satisfies \cref{prop:fos}~\ref{it:tia} and $k\geq k_0$ as in there. If the image $Im(\alpha)$ contains only analytic functions, then the map $M \times Im(\alpha) \to J^k(M,N), (p,f) \mapsto (p,f_p)$ for $p\in M, f\in Im(\alpha)$ is injective.
\end{Lemma}
\begin{IEEEproof}
Suppose for analytic functions $f_1,f_2 \in Im(\alpha)$ and $p_1,p_2\in M$, we have $(p_1,f_{1,p})=(p_2,f_{2,p}) \in J^k(M,N)$. Then, $p_1=p_2$ and $f_1$ and $f_2$ have the same partial derivative at $p_1$ up to $k$-th order. As $k\geq k_0$, by \cref{prop:fos}, all the partial derivatives of $f_1$ and $f_2$ at $p_1$ are the same. Therefore, $f_1=f_2$ by the identity theorem \cite{Fri12}.
\end{IEEEproof}
\subsection{Types of message passing}
We come back to message passing in this subsection. Though it is known that Morse functions is dense under $C^2$-topology, the same does not hold for the subspace of smooth convex functions. On the other hand, for optimization, it is favorable to work with convex functions. Therefore, we need a weak notion of ``density'' to deal with the above quagmire.
Recall that in \cref{sec:jet}, for each $f\in C^{\infty}(M,N)$, we introduce $\Pi^k(f): M \to J^k(M,N)$. For subset $K\subset M$ and open subset $U \subset J^k(M,N)$, we use $S^k(K,U)$ to denote the $f \in C^{\infty}(M,N)$ such that $\Pi^k(f)(K) \subset U$.
For example, the basis of the $C^k$-topology consists of sets $S^k(U)= S^k(M,U)$. On the other hand, if $K$ is compact, then $S^k(K,U)$ is related to the compact open topology on the functions from $M$ to $J^k(M,N)$. The notion about being ``dense'' is given in a more general form as follows.
\begin{Definition} \label{def:fkf}
Fix $k\geq 0$ and Euclidean spaces $M_i$, $i\in\{1,\ldots,t\}$ as well as $N$. Suppose we are given $W_1\subset W_2 \subset \prod_{i=1}^{t}C^{\infty}(M_i,N)$ and $K = \prod_{i=1}^{t} K_i$ with $K_i\subset M_i$.
We say that $W_1$ is dense in $W_2$ w.r.t.\ $K$ if the following holds: for every $U = \prod_{i=1}^{t}U_i$ with $U_i$ in $J^2(M_i,N)$ and $(f_i)_{i\in\{1,\ldots,t\}} \in W_2$ with $f_i \in S^k(U_i)=S^k(M_i,U_i)$, there is an $(h_i)_{i\in\{1,\ldots,t\}} \in W_1$ such that $h_i \in S^k(K_i,U_i)$.
\end{Definition}
For the rest of this section, we work solely with $k=2$. We first consider $t=1$, namely on a single function space $C^{\infty}(M,N)$. Under this definition, dense w.r.t.\ $M$ agrees with the usual notion of dense subset. In general, it is easy to see that if $K_1\subset K_2$, then dense w.r.t.\ $K_2$ implies dense w.r.t.\ $K_1$.
\begin{Proposition} \label{prop:lmm}
Let $M = \mathbb{R}^{m}, m= m_x+m_y$ and $\Delta(M)$ be the space of smooth convex function on multi-variables $x=(x_j)_{1\leq j\leq m_x},y = (y_k)_{1\leq k\leq m_y}$. Define $\Omega(M)$ to be the set of $f \in \Delta(M)$, such that $f_y = f(\cdot,y)$ (on $x$) is Morse for all $y$ in some dense open subset $W_f \subset M_y = \mathbb{R}^{m_y}$. Then for any compact subset $K$ of $M$, $\Omega(M)$ is dense w.r.t.\ $K$ in $\Delta(M)$.
\end{Proposition}
\begin{IEEEproof}
We first make some general observations regarding the $S^2(K,U)$, with $K$ compact in $M$ and $U$ open in $J^2(M,\mathbb{R})$ containing $\bar{0} = \{(p,0), p\in M\}$. As $J^2(M,\mathbb{R})$ is homeomorphic to $M \times P^2_{m,1}$, for each $p\in M$, we can always find a open set $U_p \in M$ and an open ball $B_p \in P^2_{m,1}$ centered at $0$ with radius $r_p>0$ such that $U_p\times B_p \subset U$. Here, we take note of the fact that the Euclidean space structure of $P^2_{m,1}$ is given by the polynomial coefficients. Therefore, $\cup_{p\in M} U_p\times B_p \subset U$. On the other hand, $K\subset \cup_{p\in K}U_p$. As $K$ is compact, it has a finite subcover, i.e., $K\subset \cup_{i=1}^{l}U_{p_i}$. Let $r = \min \{r_{p_i},i=1,\ldots,l\}$ and $B$ be the open ball in $P^2_{m,1}$ centered at $0$ with radius $r$. Then $K\times B \subset U$.
The upshot of the discussion is that: any function of the form $h(x,y) = \sum_{1\leq j\leq m_x} a_jx_j^2 + \sum_{1\leq k\leq m_y} b_ky_k^2 \in \Delta(M)$ belongs to $S^2(K,U)$ as long as the non-negative coefficients $a_j,1\leq j\leq m_x$ and $b_k, 1\leq k\leq m_y$ are sufficiently small. This is because for $a_j$'s and $b_k$'s sufficiently small, the coefficients of the Taylor expansion of $h$ belongs to $B$ for each $p \in K$.
We now choose a countable dense subset $\{y_i \in M_y, i\geq 1\}$. By \cref{lem:iia}~\ref{it:tso}, it suffices to show that for any $f \in \Delta(M)$ and any base open neighborhood $S^2(U)$ of the $0$ function, there is an $h\in \Delta(M)\cap S^2(K,U)$ such that $(f+h)_{y_i}$ is Morse for each $i\geq 1$. Here we use the observation that: adding $f$ translates an open neighborhood of $0$ translates to an open neighborhood of $f$, and $f+h$ is viewed as a small perturbation of $f$.
Suppose $h(x,y) = \sum_{1\leq j\leq m_x} a_jx_j^2$ with positive coefficients. Then the tuples $(a_j)_{1\leq j\leq m_x}$ and $i\geq 1$ such that $(f+h)_{y_i}$ is not Morse has Lebesgue measure zero. This is because the sum is not Morse only if some $a_j$ cancels with coefficient of $x_j^2$ in the expansion of $f$ at $y_i$. Such a collection of $(a_j)_{1\leq j\leq m_x}$ has measure zero in $\mathbb{R}^{m_x}$. The set $\{y_i\in \mathbb{R}^{m_y}\}_{i\geq 1}$ is countable. Therefore, there is always $(a_j)_{1\leq j\leq m_x}$, with each component as small as we wish, such that $(f+g)_{y_i}$ is Morse for each $i\geq 1$.
\end{IEEEproof}
We now consider message passing on a directed spanning tree $T_g$ discussed in \cref{sec:ner}. For each $i\in\{1,\ldots,t\}$, we start with a smooth convex functions $f_i \in C^{\infty}(M_i, \mathbb{R})$ where $M_i = \mathbb{R}^{|V_i|}$. The most desirable scenario for us to perform analysis is when all the messages of message passing (\cref{defn:wct}) on $T_g$ are smooth (resp.\ Morse) functions on dense open subsets of the domains, called \emph{smooth (resp.\ Morse) message passing}. Apparently, a Morse message passing is always a smooth message passing. We now discuss how likely they are.
We notice that the tuple of functions $(f_i)_{i\in\{1,\ldots,t\}}$ belongs to the product space $\Delta = \prod_{i=1}^{t}\Delta(M_i)$. Let $\Omega \subset \Delta$ consist of tuples $(f_i)_{i\in\{1,\ldots,t\}}$ admitting Morse message passing on $T_g$, for any spanning tree $T$ of the nerve skeleton and $g\in G$.
\begin{Theorem} \label{thm:fac}
For any compact subsets $K_i \subset M_i$, $i\in\{1,\ldots,t\}$, we have that $\Omega$ is dense in $\Delta$ w.r.t.\ $K = \prod_{i=1}^{t}K_i$.
\end{Theorem}
\begin{IEEEproof}
The strategy of the proof is similar to that of \cref{prop:lmm}. Namely, we want to modify each $f_i$ by adding a degree $2$ polynomial $q_i$ on the variables of $f_i$, with small positive coefficients. In order to do so, we need to examine the conditions for Morse message passing. As in \cref{prop:lmm}, we want to show that each choice of spanning tree $T$, root node $g$, being Morse message passing on $T_g$ prohibits at most a measure zero set of choices for coefficients. Once this is shown, the rest follows the same argument as in \cref{prop:lmm}.
There are only finitely many choices for $T$ and $g$. We only need to show the above holds for any fixed $T$ and $g$. For any node $g_i$, let $h_i$ denote the function such that its associated message $\tilde{h}_i$ is one of those in \cref{defn:wct}. By definition, $f_i$ is a summand $h_i$. To obtain $\tilde{h}_i$, we need to optimize over the variables associated with $g_j$ below $g_i$ on $T_g$ (c.f.\ \cref{defn:stn}). More precisely, there are two types of such variables: $y$ associated with nodes contained in $G_i$, and $y'$ associated with nodes outside $G_i$.
To ensure $\tilde{h}_i$ Morse on a dense open subset of its domain, we need to add a positive definite quadratic $q_y$ on $y$ to $f_i$. If a subset of variables $z$ of $y'$ are associated with nodes in $G_j$. We add a quadratic $q_z$ on $z$ to $f_j$, with exactly any one chosen $G_j$ to avoid repetition. It is important to notice that during the message passing until the current stage with $g_i$, we have not performed any optimization over any subset of variables of $y$ and $y'$. Therefore, no quadratic on $z$ is added to $G_j$ until the current stage. As a consequence, to guarantee $\tilde{h}_i$ is Morse, we only need to avoid a measure zero set on the coefficients of quadratic functions $q_y$ and $q_z$'s.
In subsequent steps of message passing, we do not need to optimize over the variables $y$ and $y'$. Therefore, there are no additional conditions we need to impose on the above mentioned quadratic functions, and this completes the proof.
\end{IEEEproof}
From the proofs, we see that we may modify each of the $f_i$, $i\in\{1,\ldots,t\}$ by adding a positive definite quadratic function on the variables with ``small coefficients'' such that the resulting functions permit smooth (Morse) message passing. Such a procedure could be understood as \emph{regularization}. Moreover, the coefficients can be chosen (uniformly) randomly within a prescribed small domain at $0$.
\section{Solubility results} \label{sec:sol}
In this section, we discuss results on the solubility of a global problem (c.f.\ \cref{sec:pro}) in a distributed way via message passing (c.f.\ \cref{sec:ner}). We give conditions on both the global problem can or cannot be solved locally.
Recall that a global problem takes the form $\Phi_{\tau}: U \to \mathbb{R}^{|V|} \stackrel{\tau}{\to} \mathcal{M}$ (c.f.\ \cref{sec:pro}) for some manifold $\mathcal{M}$. To properly state the results, we assume that $U$ is a connected open subset of $\mathbb{R}^c$ and $\Phi_{\tau}$ is a \emph{smooth surjection}, $c = \sum_{i=1}^{t}c_i$. As in our setup, $c$ is the size of the nodes in $G$ where observation can be made. We remark that requiring $\Phi_{\tau}$ being surjective is not restrictive, for otherwise, we may just consider $\Phi_{\tau}$ as a map from $U$ to its image as long as the image is a manifold.
\subsection{Individual message passing step} \label{sec:imp}
In this subsection, we examine closely each individual message passing step. For each $k$, there are three spaces involved in the discussion, namely $U$, $J^k(M,N)$ and $C^{\infty}(M,N)$. The first two are manifolds, while $C^{\infty}(M,N)$ in general is not a manifold. Therefore, we want to pass the study of message passing to jet spaces. Suppose $\alpha: U \to C^{\infty}(M,N)$ is given. It induces $\alpha^k: M\times U \to J^k(M,N), (p,s) \mapsto (p,\alpha^k_p(s))$ where $\alpha^k_p(s) = \pi_p^k\Big(\alpha(s) \Big)$ (c.f.\ \cref{sec:jet}). In other words, $\alpha^k$ is the composition $M\times U \stackrel{Id\times \alpha}{\to} M\times C^{\infty}(M,N) \stackrel{\pi^k}{\to} J^k(M,N) $. We make the following assumptions:
\begin{enumerate}[1)]
\item \cref{prop:fos}~\ref{it:tia} holds and let $k_0$ be as defined there. Moreover, $k\geq k_0$.
\item The map $\pi^k: \alpha^k(M \times U) \to J^k(M,N), (p, \alpha(s)) \mapsto \Big(p, \pi^k_p(\alpha(s))\Big)$ is injective.
\end{enumerate}
As we have seen in \cref{lem:sui}, the second assumption holds if $\alpha(U)$ contains only analytic functions.
Consider a single instance in message passing $\mathscr{P}: f \mapsto \tilde{f}$. It extends to a map on pairs $(f,p), p\in M$ by $\mathscr{P}((f,p)) = (\tilde{f},p')$ with $p'$ is the projection of $p$ to the domain $M'$ of $\tilde{f}$. Suppose we consider a subset $\tilde{M}$ of $M\times C^{\infty}(M,N)$ containing $Im(Id\times \alpha)$, such that $\mathscr{P}(\tilde{M}) = \tilde{M'}\subset M'\times C^{\infty}(M',N)$. We have the following diagram of maps:
\[
\begin{tikzcd}
M\times U \arrow{r}{Id\times \alpha} & \tilde{M} \arrow{r}{\mathscr{P}} \arrow[swap]{d}{\pi^k} & \tilde{M'} \arrow{d}{\pi^k} \\
& \pi^k(\tilde{M}) \arrow[dashed]{r}{\bar{\mathscr{P}}} & \pi^k(\tilde{M'}),
\end{tikzcd}
\]
where $\pi^k(\tilde{M})$ and $\pi^k(\tilde{M'})$ belong to $J^k(M,N)$ and $J^k(M',N)$ respectively. Then we can find a set map $\bar{\mathscr{P}}$ (the dashed arrow) from $\pi^k(\tilde{M})$ to $\pi^k(\tilde{M'})$ making the diagram commute, i.e., $\bar{\mathscr{P}} \circ \pi^k = \pi^k \circ \mathscr{P}$. This is because our injectivity assumption on $\pi^k$ guarantees that $\pi^k: \tilde{M} \to \pi^k(\tilde{M})$ is a bijection.
On the other hand, we may also view $Id\times \alpha: M\times U \to M\times C^{\infty}(M,N)$ with a different perspective. Here, we may ignore the first component, being the identity. For each $s\in U$, $\alpha(s): M \to N$. Therefore equivalently, we may interpret this as $F_{\alpha}: M\times U \to N$ by $F_{\alpha}(p,s) = \alpha(s)(p)$.
\begin{Lemma}
Write the components of $p\in M$ as $p=(x,y)$ such that domain of $M'$ is on $x$. Assume that the following holds:
\begin{enumerate}[1)]
\item \label{it:fmt} $F_{\alpha}: M\times U \to N$ is smooth and Morse on the joint variables $(x,s)$.
\item $\pi^k$ is injective on $\tilde{M}$.
\end{enumerate}
Then $\pi^k \circ \mathscr{P} \circ \alpha: M\times U \to \pi(\tilde{M'})$ is smooth. Moreover, if $\pi^k\circ \alpha: M\times U \to \pi^k(\tilde{M})$ is a submersion, then $\bar{\mathscr{P}}$ is smooth.
\end{Lemma}
Notice that $\mathscr{P}:\tilde{M} \to \tilde{M'}$, where $\tilde{M} \subset M \times C^{\infty}(M,N)$ and $\tilde{M'} \subset M' \times C^{\infty}(M',N)$. Therefore, $\mathscr{P}$ essentially has two components.
\begin{IEEEproof}
The lemma essentially follows from the implicit function theorem as in \cref{lem:iia}. Given a pair $(p,s) \in M\times U$, its image under $\alpha$ is the pair $(p, f)$, where $f(x,y) = F_{\alpha}(x,y,s)$. Write the variables of $f$ as $p=(x,y)$. The map $\mathscr{P}$ projects the component $p=(x,y)$ to $p'=x$, which is clearly smooth. We want to show that the partial order derivatives of $\tilde{f}(x) = \min_{y}F_{\alpha}(x,y,s)$ is smooth on $x$ and $s$. By the Morse condition on $F_{\alpha}$, $\tilde{f}(x) = F_{\alpha}(x,g(x,s),s)$ for smooth function $g$. The partial derivatives of $\tilde{f}$ is a polynomial of those of $F_{\alpha}$ evaluated at $(x,g(x,s),s)$ and those of $g$ evaluated at $(x,s)$. Hence, $\tilde{f}$ is smooth on $x$ and $s$ and so are its partial derivatives.
If $\pi^k\circ \alpha$ is a submersion, it is locally a coordinate projection. Moreover, as $\bar{\mathscr{P}}\circ \pi^k\circ \alpha = \pi^k\circ \mathscr{P} \circ \alpha$ is smooth, so is $\bar{\mathscr{P}}$.
\end{IEEEproof}
If we put each individual message passing step together, we obtain a global picture as illustrated in \figref{fig:dsp3}. The main point is that it can usually be difficult or in-explicit to work with smooth functions. However, under favorable conditions, message passing can be viewed as a procedure on the jet spaces, which are Euclidean spaces or more generally manifolds. In doing so, we replace studying functions by studying its derivatives up to certain fixed order. An important advantage is we can now use simple numerical invariants such as dimensions.
\begin{figure}
\centering
\includegraphics[width=0.5\columnwidth]{mp}
\caption{In the top layer, we have the message passing, the spaces are smooth function spaces. In the bottom layer, we have the jet spaces. As we have seen, message passing induces maps in the bottom layer. Study the maps in the bottom layer can be easier as they are between manifolds.} \label{fig:dsp3}
\end{figure}
\subsection{Message passing in its entirety}
In this subsection, we are going to state and prove the main result on solubility of a global problem $\Phi_{\tau}$ via message passing.
\begin{Theorem} \label{thm:ltb}
\begin{enumerate}[1)]
Let $T$ be a spanning tree of $N_G$ and $g_k\in V_{N_G}$.
\item \label{it:sfa} Suppose $f_1,\ldots, f_t$ admits a Morse message passing. Let $h_k$ be the aggregated message along $T_{g_k}$ (c.f.\ \cref{defn:wct}). Denote the domain of $f=\sum_{i=1}^{t}f_i$ and $h_k$ be $D_f$ and $D_{h_k}$ respectively. Then there is an open dense subset $U$ of $\mathbb{R}^{|S|}, S = \cup_{i=1}^{t}S_i$ such that for any $s=(s_i)_{i=1}^{t}\in U$
\begin{align*}
\hat{x} = \argmin_{x \in D_f, x|_S=s} f(x), \text{ and } \hat{y_k} = \argmin_{y \in D_{h_k}, y|_{S_k} = s_k} h_k(y)
\end{align*}
depend smoothly on $s$ and $\hat{y_k} = \hat{x}|_{V_k}$.
\item \label{it:fga} For $g_i$ a leaf of $T$ connected to $g_j$, let $M$ be the domain of the message $\hat{f_i}$ from $g_i$ to $g_j$ and $U$ be a dense open subset of $\mathbb{R}^{|S|}$ such that $\hat{f_i}$ is smooth for any $s\in U$. Denote by $\alpha: U \to C^{\infty}(M,\mathbb{R})$. Consider $k_0$ and $b_{\alpha}$ as defined in \cref{def:sfe}, we assume $k\geq k_0$ and $\alpha^k$ is a submersion. If $|S_i|-b_{\alpha} > |S|-\dim \mathcal{M}$, then there does not exist local solution via message passing along $T$.
Moreover, if the condition is verified for one spanning tree $T$, then the (insolubility) conclusion holds for any spanning tree of $N_G$.
\end{enumerate}
\end{Theorem}
\begin{IEEEproof}
\begin{enumerate}[1)]
\item As we assume that $f_1,\ldots,f_t$ admits a Morse message passing, both $f$ and $h_k$ are smooth Morse functions. Therefore, $\hat{x}$ and $\hat{y_k}$ are uniquely determined. We have shown in \cref{prop:ief} that both $f$ and $h_k$ have the same global minimum. Hence, $\hat{y_k} = \hat{x}|_{V_k}$. It suffices to see that $\hat{x}$ depends smoothly $s$, which follows by applying the implicit function theorem to $f$.
\item If $M$ is the domain of the message from $g_i$ to $g_j$, then we modify the global problem by adding in the identity map on a factor of $M$ as $Id_M\times \Phi_{\tau}: M\times U \to M\times \mathcal{M}$, which remains to be continuous and surjective. Here, $Id_M$ is the identity map on $M$. Suppose under the given conditions that the global problem $\Phi_{\tau}$ has a local solution via message passing along $T$. Then the modified global task $Id_M\times \Phi_{\tau}$ can be decomposed as $M\times U \stackrel{Id_M\times \alpha}{\to} M\times C^{\infty}(M,\mathbb{R})\times U'\to M\times \mathcal{M}$, where $U'$ are on $S\backslash S_i$. The existence of the map $M\times C^{\infty}(M,\mathbb{R})\times U'\to M\times \mathcal{M}$ is due to the assumption on the existence of local solution. Let $\tilde{M}$ be the image of the map $\alpha$.
Consider the diagram of maps
\[
\begin{tikzcd}
M\times U \arrow{r}{Id_M\times \alpha} & \tilde{M} \arrow{r} \arrow[swap]{d}{\pi^k\times Id_{U'}} & M\times \mathcal{M} \\
& \pi^k(\tilde{M})\times U' \arrow[dashed]{ur}{\Phi'}.
\end{tikzcd}
\]
As $\pi^k$ is bijective, there is $\Phi':\pi^k(\tilde{M})\times U' \to M\times \mathcal{M}$ making the triangle in the above diagram commute. By our assumptions, $(\pi^k\times Id_{U'})\circ (Id_M\times \alpha)$ is a submersion and $\pi^k$ is a bijection. In particular, $(\pi^k\times Id_{U'})\circ (Id_M\times \alpha)$ is locally a coordinate projection, and $\Phi'$ is differentiable as the top row is the smooth surjection $\Phi_{\tau}$. By Sard's theorem (\cref{thm:sard}), we can find $(p,s) \in M\times U$ such that $Id_M\times \Phi_{\tau}$ is a submersion at $(p,s)$. In view of $b_{\alpha} = \dim(\pi^k(\tilde{M}))-\dim M$ and $\dim U' = |S|-|S_i|$, if $|S_i|-b_{\alpha} > |S|-\dim \mathcal{M}$, we have $\dim(\pi^k(\tilde{M})\times U') < \dim(M\times \mathcal{M})$. This gives a contradiction as $d\Phi_{\tau}$ is surjective at $(p,s)$, while $d\Phi'$ cannot be surjective from a smaller space to a larger space.
The same argument applies to any $T$ as the existence of the diagram does not depend on $T$.
\end{enumerate}
\end{IEEEproof}
Let us summarize what we have found in the results. First of all, \cref{thm:ltb}~\ref{it:sfa} states that when we have a Morse message passing, then the optimizer of the aggregated message agrees exactly with the restriction of the optimizer of the original global problem. If this result is viewed together with \cref{thm:fac}, then we can always add a quadratic regularization term to $f_i$, $i\in\{1,\ldots,t\}$ for the purpose. Of course, we have to pay the price that the solution is not exactly the same as the intended one. On the other hand, \cref{thm:ltb}~\ref{it:fga} gives an explicit condition that local solubility is impossible. The condition is relatively easy to verify as we only need to look at the partial order derivatives of the original functions $f_i$, $i\in\{1,\ldots,t\}$.
\begin{Example}
We revisit \cref{eg:tiac} to demonstrate the idea of \cref{thm:ltb}~\ref{it:fga}. In that example, $\mathcal{M} = \mathbb{R}$ and its dimension is $1$. The set $S=\{s_1,s_2,s_3,s_4\}$ has size $4$. Hence, $|S| - \dim \mathcal{M}=3$. On the other hand, $|S_1| = 0$. To compute $b_{\alpha}$, as we have seen in \cref{eg:tiac}, the message $\tilde{f}_1$ from $g_1$ to $g_2$ the constant $0$ function, and hence the dimension of the image of $\pi^k$ is $0$. Moreover, the domain of $f_1$ is $2$ dimensional. Therefore, $b_\alpha = 0-2=-2$. Now, we compare that $|S_i|-b_{\alpha}=4>3 = |S|-\dim \mathcal{M}$. By \cref{thm:ltb}~\ref{it:fga}, the global problem cannot be solved locally via message passing.
\end{Example}
\section{Approximated message passing} \label{sec:app}
So far, we have been mainly focused on theoretical aspects of the message passing scheme. One of the key ingredients is the message, which takes the form $h(x) = \min_{y\in D} h(x,y)$ over variable $y$ over a domain $D$. In practice, it is usually not possible to write down an explicit analytic expression of $h$. In this section, we propose to use a neural network to approximate such a message $h$.
\begin{figure}
\centering
\includegraphics[width=0.5\columnwidth]{dsp3}
\caption{An illustration of MLP.} \label{fig:dsp5}
\end{figure}
Recall that a multilayer perceptron (MLP) (illustrated in \figref{fig:dsp5}) is a function that consists of a concatenation of (hidden) layers, each consist of a composition of a linear transformation and non-linear activation such as the rectifier linear unit (ReLU). For each layer, a finite set of learnable parameters dictates the linear transformation. It is known that a MLP with one hidden layer is enough to approximate any continuous function \cite{Hor89}. Based on this, we propose the \emph{approximated message passing algorithm} as:
\begin{enumerate}[S1']
\item Construct $N_G$ and fix a spanning tree $T$.
\item For each $g$, form the directed tree $T_g$.
\item Perform message passing following the procedure described in \cref{sec:ner}, with the following modification.
\begin{enumerate}[s1]
\item \label{it:sng} Suppose node $g_j$ is to pass $\tilde{h}_j: D \to \mathbb{R}$ to its neighbor $g_k$. Node $g_j$ will randomly choose samples $x_1,\ldots, x_m \in D$ and compute $y_i = \tilde{h}_j(x_i)$ for each $i\in\{1,\ldots,m\}$. The pairs $\{(x_i,y_i)_{i\in\{1,\ldots,m\}}\}$ are passed to $g_k$.
\item Using the received samples $\{(x_i,y_i)_{i\in\{1,\ldots,m\}}\}$, node $g_k$ learns an MLP $\bar{h}_j$ as an approximation of $\tilde{h}_j$.
\item One sums up all the approximated messages at $g_k$ and obtain $\tilde{h}_k$. Step~\ref{it:sng} is repeated at $g_k$.
\end{enumerate}
\item The aggregated message $h_g$ at $g$ along $T_g$ is optimized locally at $g$.
\end{enumerate}
It is interesting to notice that in the procedure described above, there is only one exchange of information along each edge. This is radically different from many distributed algorithms involving numerous information exchange until convergence.
For the rest of the section, we present some numerical examples based on the approximated message passing algorithm.
\begin{figure}
\centering
\includegraphics[width=0.5\columnwidth]{test1}
\caption{The nerve skeleton.} \label{fig:test1}
\end{figure}
\begin{table}[!htb]
\caption{A summary of the intersection statistics} \label{tab:1}
\centering
\scalebox{1.2}{
\begin{tabular}{|l|c|c|c|c|}
\hline
\emph{Subgraph} & \emph{$|X_i|$} & \emph{$|Y_i|$} & \emph{$|S_i|$} &\emph{$|V_i|$} \\
\hline \hline
$G_1$ & 4 & 6 & 12 & 22\\
\hline
$G_2$ & 12 & 4 & 10 & 26\\
\hline
$G_3$ & 6 & 6 & 14 & 26\\
\hline
$G_4$ & 6 & 8 & 12 & 26\\
\hline
$G_5$ & 12 & 6 & 10 & 28\\
\hline
$G_6$ & 14 & 8 & 6 & 28\\
\hline
$G_7$ & 10 & 4 & 12 & 26\\
\hline
$G_8$ & 8 & 2 & 14 & 24\\
\hline
$G_9$ & 6 & 6 & 12 & 24\\
\hline
$G_{10}$ & 10 & 6 & 10 & 26\\
\hline
$G_{11}$ & 4 & 8 & 13 & 25\\
\hline
$G_{12}$ & 4 & 8 & 14 & 26\\
\hline
\end{tabular}}
\end{table}
\begin{table}[!htb]
\caption{Simulation results ($\%$).} \label{tab:2}
\centering
\scalebox{1.2}{
\begin{tabular}{|l||c|c|c|c|c|c|}
\hline
\emph{$k$} & 25 & 30 & 35 & 40 & 45 & 50\\
\hline
\emph{R($\%$)} & 4.93 & 5.65 & 3.86 & 4.26 & 6.16 & 4.14\\
\hline
\end{tabular}}
\end{table}
We simulate the distributed sampling in \cref{eg:ltt}~\ref{it:ite}, and follows entirely the setup described in the example. While we are not able to draw the entire graph $G$, we present the nerve skeleton in \figref{fig:test1}. From the figure, we see that there are $12$ subgraphs $G_i=(V_i,E_i)$, $i\in\{1,\ldots,12\}$ of $G$, with $G_i$ corresponds to the node $g_i$. The intersection properties of $G_i$ are given in \cref{tab:1} and \figref{fig:test1}. Recall the notations, for $i\in\{1,\ldots,12\}$, $S_i$ and $Y_i$ are nodes contained exclusively in $V_i$, while $S_i$ are nodes where observations are available. $X_i$ are the nodes in the intersection of $G_i$ with other subgraphs.
For each $i\in\{1,\ldots,12\}$, the function $f_i$ on $G_i$ is
\begin{align*}
f_i(x) = \min_{r_1,\ldots,r_k} \norm{\sum_{1\leq j\leq k}r_j{z_j}|_{V_i} -x}^2.
\end{align*}
as described in \cref{eg:ltt}\ref{it:ite}. In the simulations, we randomly generated the vectors $z_j, 1\leq j\leq k$. For the parameter $k$, we test the performance of approximated message passing with $k=25,30,\ldots, 50$. For large $k$, each graph has much more unknowns than observable nodes.
We follow the steps of approximated message passing algorithm to get the final approximation of the global optimization. For each simulation, we measure the performance by calculating error between ground truth and estimated optimal value of $f$, and then take the ratio $R$ between error and ground truth. For each $k$, multiple simulations are performed, and the average results are shown in \cref{tab:2}. From the results, we see that although we do not have a perfect result, the approximated message passing yields reasonably good results even when $k$ is large, as our theory expects.
During message passing, each time we transfer $80$ samples from a node to its neighbor, i.e., $m=80$ in Step~\ref{it:sng}. We further investigate the performance for different $m$ (with fixed choice $k=50$). The results are shown in \figref{fig:dsp6}. We see that the performance starts to show steady improvements from $m=50$ onward, until having the reasonable performance when $m=80$.
\begin{figure}
\centering
\includegraphics[width=0.5\columnwidth, trim={0 6cm 0 6cm},clip]{dsp5.pdf}
\caption{Plot of the error $R$ against the number of samples $m$. In this plot, $R$ being $80\%$ in fact means that the error is larger than $80\%$ and the performance is unstable.} \label{fig:dsp6}
\end{figure}
\section{Conclusions} \label{sec:con}
In this paper, we propose a framework on distributed graph signal processing. We employ the approach of message passing, and introduce the concept of abstract messages. Our approach provides an alternative point of view of distributed graph signal processing as compare with classical approaches. Moreover, the framework is convenient to analyze theoretical questions such as solubility of distributed problems. Though our work is mainly theoretical, we still present numerical findings to verify the theory. While the topic is distributed GSP, there are still centralized components. Moreover, as we introduce a new notion of message, privacy can be an important concern. In the future, it could be an central topic to investigate.
\appendices
\section{Fundamentals of differentiable manifolds} \label{sec:fun}
In this appendix, we give a self-contained introduction to fundamentals of differentiable manifolds. We shall highlight the results needed for the paper. Readers can further consult textbooks on manifold theory (e.g., \cite{War83}) for thorough discussions of the theory. We start by defining what a topological space is.
\begin{Definition}
A \emph{topology} on a set $X$ is a collection $\mathcal{T}$ of subsets of $X$ s.t.:
\begin{enumerate}[1)]
\item $\emptyset, X \in \mathcal{T}$.
\item The union of the elements of any subcollection of $\mathcal{T}$ is in $\mathcal{T}$, i.e., $\mathcal{T}$ is preserved under arbitrary union.
\item The intersection of the elements of any \emph{finite} subcollection of $\mathcal{T}$ is in $\mathcal{T}$, i.e., $\mathcal{T}$ is preserved under finite intersection.
\end{enumerate}
$X$ with $\mathcal{T}$ is called a \emph{topological space}, and elements of $\mathcal{T}$ are called \emph{open subsets} of $X$.
\end{Definition}
Primary examples of topological spaces include discrete set where each point is open, and Euclidean spaces $\mathbb{R}^n$ where open sets are unions of open balls. In general, a subset $Y$ of $X$ is also a topological space, with an open takes the form $Y\cap U$ where $U$ is open in $X$.
Not every subset of $X$ is open, but there can be other interesting families. A subset $C$ of $X$ is closed if its complement is open. A subset $D$ is dense in $X$ if it has non-empty intersection with any open subset of $X$. Another key notion is \emph{compact set}: $K$ is compact if a cover of $K$ by open sets (i.e., $K$ is contained in their union) has a finite subcover.
There are many other quantifiers we can associate with topological spaces. We mention two of them here. $X$ is said to be \emph{connected} if it cannot be decomposed into the union of two disjoint non-empty open subsets. It is called \emph{Hausdorff} if for any $x\neq y \in X$, there are disjoint open subset $U_x,U_y$ such that $x\in U_x$ and $y\in U_y$, i.e., one can separate points by using open sets.
Topological spaces are related by \emph{continuous functions}: $f: X\to Y$ is continuous if the inverse image of any open set is open. A continuous function $f$ is a \emph{homeomorphism} if it has a continuous inverse.
In addition to being topological spaces related by continuous functions, Euclidean spaces carry more structures, namely differentiable structures. Moreover, we may glue together Euclidean spaces to form more general spaces. We next discuss these ideas and put them together leading to differentiable manifolds.
\begin{Definition} \label{def: fka}
For $k\geq 0$, a function $f: \mathbb{R}^n \to \mathbb{R}$ belongs to $C^k(\mathbb{R}^n,\mathbb{R})$ if $f$ has all continuous partial derivatives up to $k$-th order. The space of smooth functions $C^{\infty}(\mathbb{R}^n,\mathbb{R})$ is $\cap_{k\geq 0} C^{k}(\mathbb{R}^n,\mathbb{R})$, i.e., $f$ is smooth if all partial derivatives exist.
For $f: \mathbb{R}^n \to \mathbb{R}^m$ and $k\geq 0$ or $k=\infty$, it belongs to $C^k(\mathbb{R}^n, \mathbb{R}^m)$ or smooth if each of the $m$-components belongs to $C^k(\mathbb{R}^n,\mathbb{R})$.
\end{Definition}
Taking partial derivative of $f$ at $p$ depends only the value of $f$ at any open set containing $p$. Therefore, the same definition can be used without change if the domain $\mathbb{R}^n$ is replaced by any open subset $U$. If the domain and codomain are clear from context, we may abbreviate $C^k$ or $C^{\infty}$ for convenience. By convention, a function with empty domain belongs to $C^{\infty}$.
A smooth function $f$ is called a \emph{diffeomorphism} if it has a smooth inverse function. We give a more general version of this notion later on once we have introduced differentiable manifolds. Key results regarding differentiable functions are the inverse function theorem and the implicit function theorem \cite{War83, Kau11}.
\begin{Theorem}[The inverse function theorem]
Let $U \subset \mathbb{R}^d$ be open, and let $f: U \to \mathbb{R}^d$ be $C^{\infty}$. If the Jacobian matrix $$\{\frac{\partial r_i\circ f}{\partial r_j}\}_{1\leq i,j\leq d}$$ is non-singular at $r_0\in U$, then there is an open set $V$ with $r_0\in V\subset U$, such that $f(V)$ is open and $f: V\to f(V)$ is a diffeomorphism.
\end{Theorem}
\begin{Theorem}[The implicit function theorem] \label{thm:ift}
Let $U \subset \mathbb{R}^{c-d}\times \mathbb{R}^d$ be open, and let $f: U \to \mathbb{R}^d$ be $C^{\infty}$. We denote the canonical coordinate system on $\mathbb{R}^{c-d}\times \mathbb{R}^d$ by $(r_1,\ldots, r_{c-d}, s_1,\ldots, s_d)$. Suppose that at the point $(r_0,s_0) \in U$, $f(r_0,s_0)=0$; and that the matrix $$Df(r_0,s_0) = \{\frac{\partial f_i}{\partial s_j}|_{(r_0,s_0)}\}_{i,j=1,\ldots, d}$$ is non-singular. Then there exists an open neighborhood $V$ of $r_0$ in $\mathbb{R}^{c-d}$ and an open neighborhood $W$ of $s_0$ in $\mathbb{R}^d$ s.t. $V\times W \subset U$, and there exists a $C^{\infty}$ map $g: V\to W$ s.t.\ for each $(p,q) \in V\times W$: $f(p,q)=0$ iff $q = g(p)$.
\end{Theorem}
A \emph{locally Euclidean space} $M$ of dimension $d$ is a Hausdorff topological space $M$ for which each point has a neighborhood homeomorphic to an open subset of Euclidean space $\mathbb{R}^d$.
A \emph{differentiable structure} $\mathcal{F}$ on $M$ of class $C^k$ is a collection of coordinate systems $\{(U_{\alpha},\psi_{\alpha})\mid \alpha \in A\}$ with index set $A$ such that the following holds:
\begin{enumerate}[1)]
\item Each $U_{\alpha}$ is open in $M$ and $M = \cup_{\alpha\in A}U_{\alpha}$.
\item $\psi_{\alpha}: U_{\alpha} \to V_{\alpha}$ is a homeomorphism, i.e., continuous bijection with a continuous inverse, from $U_{\alpha}$ to an open subset $V_{\alpha}$ of $\mathbb{R}^d$. Each $\psi_{\alpha}$ is called a coordinate map.
\item If $U_{\alpha}\cap U_{\beta} \neq \emptyset$, then $\psi_{\alpha} \circ \psi_{\beta}^{-1}: W_{\beta} \to V_{\alpha} \in C^k(\mathbb{R}^d,\mathbb{R}^d)$, where $W_{\beta} = \psi_{\beta}(U_{\alpha}\cap U_{\beta})$.
\end{enumerate}
$M$ with such an differentiable structure $\mathcal{F}$ is called \emph{a differentiable manifold of class $C^k$}. We usually consider $k=\infty$ and just call such an $M$ a differentiable manifold.
The coordinate maps allows us to match open sets of $M$ with those of $\mathbb{R}^d$. As a consequence, we can talk about differentiability of functions on $M$ by pulling back them to a Euclidean space using one of the coordinate maps.
For example, for an open subset $U\subset M$, $f: U\to \mathbb{R}$ is a $C^{\infty}$ function or smooth on $U$ if $f\circ \psi_{\alpha}^{-1}$ is $C^{\infty}$ for each coordinate map $\psi_{\alpha}, \alpha \in A$.
For differentiable manifolds $M$ and $N$, a continuous map $f: M \to N$ is said to be differentiable of class $C^{\infty}$ or smooth if $\gamma_{\beta} \circ f \circ \psi_{\alpha}^{-1}$ is $C^{\infty}$ for each coordinate $\psi_{\alpha}$ on $M$ and $\gamma_{\beta}$ on $N$. Such a map $f$ is a \emph{diffeomorphism} if $f$ is bijective and $f^{-1}$ is also smooth.
For each point $p \in M$ contained in $U_{\alpha}$ with coordinate map $\psi_{\alpha}$, a \emph{tangent vector} is a linear map on the space of smooth functions satisfying the Leibniz rule. The tangent space $T_pM$ is the vector space of tangent vectors spanned by the ``partial derivatives'' $\frac{\partial}{\partial x_i }|_p$:
\begin{align*}
\frac{\partial}{\partial x_i}|_p(f) = \frac{\partial (f\circ \psi_{\alpha}^{-1})}{\partial r_i}|_{\psi_{\alpha}(p)},\ i\in\{1,\ldots,d\},
\end{align*}
where $f \in C^{\infty}(M,\mathbb{R})$ and $r_i$ is the $i$-th coordinate function of $\mathbb{R}^d$.
Let $f: M \to N$ be a smooth function and $p\in M$. The \emph{differential of $f$} at $p$ is the linear map $df_p: T_pM \to T_{f(p)}N$ by $$df_p(v)(h) = v(h\circ f),$$ for $v \in T_pM$ and $g \in C^{\infty}(N,\mathbb{R})$. Locally at $p$, the differential $df_p$ gives a linear approximation of $f$. Many properties of $f$ can studied through $df_p$, which is more accessible being a linear transformation. To demonstrate, we define what immersion, imbedding and submersions are.
The smooth function $f$ is called an \emph{immersion} if $df_p$ is non-singular for every $p\in M$. The manifold $M$ with $f$ is a \emph{submanifold} of $N$ if $f$ is both injective and an immersion. One should take note here that we are not making a repetition of conditions as being being an immersion refers to injectivity of the differential at each point, but not the injectivity of $f$ itself. An even stronger notion is that of \emph{imbedding}: $M$ is a submanifold and $f:M \to f(M)$ is a homeomorphism. These notions are all nonequivalent, and \cite{War83} 1.28 contains concrete examples for this. On the other hand, $f$ is a \emph{submersion} if $df_p$ is surjective for every $p\in M$. A consequence of being submersion is that for every $p\in M$, there is an open neighborhood $U_p$ of $p$ diffeomorphic to a Euclidean open set such that $f$ restricts to a coordinate projection on $U_p$. In particular, $f$ is an open map from $f$ to $f(M)$, i.e., the image of open set is open. Though not every smooth surjective map is a submersion (e.g., $f: \mathbb{R} \to \mathbb{R}, x \to x^3$ is not a submersion at the single point $x=0$), Sard's theorem \cite{Sar42} states that for most points $f$ is a submersion. More precisely, we have the following special case of Sard's theorem:
\begin{Theorem} \label{thm:sard}
Let $f: M \to N$ be a smooth surjection between differentiable manifolds. If $X\subset M$ consists points where $f$ is not a submersion, then $f(X)$ has Lebesgue measure $0$ in $N$.
\end{Theorem}
\bibliographystyle{IEEEtran}
\section{Introduction}
Graph signal processing (GSP) has attracted increased attention as it allows us to capture complex correlations in many practical problems. Signals observed in many applications can be modeled as graph signals. Examples include photographs, CMOS sensor images, and readings from sensor networks. GSP has been applied to various problems, including signal recovery, prediction, sampling, and anomaly detection \cite{Shuman2013, San13, San14, Gad14, Moura2015, Don16, Egi17, Isu17, Isu17b, Ort18, JiTay:J19}.
Given a graph $G$, a \emph{graph signal} on $G$ assigns a value to each node of $G$, resulting in a vector of dimension equal to the size (number of nodes) of the graph. A graph signal is indexed not by time but by the nodes of the graph, called the \emph{graph domain}, capturing irregular domains. A fundamental philosophy of GSP is to perform an orthogonal transformation of the graph domain, called a \emph{graph Fourier transform}. The new domain is usually called the \emph{frequency domain}, analogous to classical Fourier theory. The components of a signal represented in the frequency domain are its \emph{Fourier coefficients}. As in classical signal processing, a graph signal is analyzed by inspecting its Fourier coefficients. An orthogonal transformation is usually obtained via a \emph{graph shift operator} $A$, such as the adjacency matrix or Laplacian of $G$. Under favorable conditions, such as $G$ being undirected, $A$ has an orthonormal eigenbasis and gives rise to the desired transformation. The frequencies are ordered according to size of the eigenvalues.
Basic GSP requires full knowledge of the graph, such as to perform graph Fourier transforms. Such a requirement has shortcomings. For example, graphs can be used to model very large networks such as social networks. Gathering data and network information as well as performing signal processing as above can be costly and time consuming. Moreover, the framework is highly centralized, giving rise to issues such as data privacy.
On the other hand, some GSP methods do not require the global approach. To give an example, the concept of smoothness corresponds to signals having Fourier coefficients concentrated in the low frequencies. However, to leverage smoothness, one can optimize the quadratic form of total signal variation, which can be done in a decentralized manner. Considerations as such lead to the topic of distributed GSP, including the following efforts. Signal recovery is considered in \cite{Che15, Wan16}. The authors of \cite{Dor15} investigate distributed sparse signal representation with graph spectral dictionaries. Signal reconstruction based on node sampling is studied in \cite{Lor17}. In \cite{Jia19}, the authors consider distributed construction of filter banks. \cite{Ren21} proposes and analyzes a communication efficient distributed optimization framework for general non-convex non-smooth signal processing and machine learning problems under an asynchronous protocol.
A closely related area of research is distributed optimization. This is a well established field dating back to as early as 1980s \cite{Tsi84,Tsi86, Tsi89}. Though we are not able to give an overview of such a vast topic here, recent survey article \cite{Yan19} contains comprehensive discussions of historical developments and recent advances in distributed optimization. The key component is an iterative procedure that repeats the following: each member of the distributed system solves its own problem using information gathered from its pre-determined ``neighbors'' and share its findings with the neighbors. Theoretical results focus on convergence of such a procedure to the optimal solution. Tables 1--4 in \cite{Yan19} list all the key features of a large collection of algorithms proposed in the literature.
In terms of methods, the above mentioned works on distributed GSP have spirit common to distributed optimization. In this paper, we are going to consider a different approach. We realize that it is possible that an iterative procedure might be avoided if we tweak the information sharing step with neighbors. Instead of sharing numerical values, we propose sharing elements from a coherent system of vector bundles. The primary example is a smooth function on a domain $D$, whose local incarnation at every point of $D$ is a system of polynomials (by Taylor expansion). The advantage is that more information is contained in such a mathematical object and we only have to deal with each player of the system once. In addition, the framework allows us to leverage numerical invariants such as dimension of manifolds to analyze the problem solving procedure. Thus it is convenient for us to answer theoretical questions such as solubility of a given problem without running the algorithm. In this work, we do not claim that our approach is superior to classical approaches, but we want to provide an alternative point of view, potentially increasing the domain of applicability of GSP.
The rest of the paper is organized as follows. We formulate the distributed GSP problem in \cref{sec:pro}. In \cref{sec:ner}, we formalize message passing in our setup. In particular, we define what an abstract message is. Moreover, to study solubility questions later on, we introduce the notion of ``local solution''. To handle the abstract notion of message, we need tools from the theory of differentiable manifolds. In \cref{sec:jet}, we introduce jet spaces and Whitney topologies, which are particularly relevant to our discussions. In \cref{sec:sub}, we discuss Morse functions and real analytic functions as the candidates of messages. Both enjoy nice properties that make the procedure of message passing well-behaved. In \cref{sec:sol}, we make use of the theory developed so far to analyze the solubility questions of distributed GSP problems. We propose how to implement our method in practise and present simulation results in \cref{sec:app}, and conclude in \cref{sec:con}.
\section{Problem formulation} \label{sec:pro}
In this section, we describe the setup of the main problem to be studied in the paper, including the notation used throughout the paper.
Let $G=(V,E)$ be an undirected graph, where $V$ is the vertex set and $E$ is the set of edges. Let $t$ be a positive integer, and let $G_i = (V_i,E_i)$, $i\in\{1,\ldots,t\}$ be $t$ subgraphs of $G$ such that $V = \cup_{i=1}^{t} V_i$. For each $i\in\{1,\ldots,t\}$, let $S_i\subset V_i\backslash (\cup_{j\neq i}V_j)$ be a set of nodes contained exclusively in $V_i$ (cf.\ \cref{defn:ato} below). Hence, the $S_i$, $i\in\{1,\ldots,t\}$ are disjoint.
Each node in $S_i$ is understood to be \emph{observable} in the sense that for any signal $x$ on $G_i$, its restriction to $S_i$, denoted $x|_{S_i}$, are known in $G_i$ for the purpose of certain tasks (defined and illustrated by examples below). Such a restriction $x|_{S_i}$ is called an \emph{observation}. Let $c_i=|S_i|$ denote the size of $S_i$ and $S=\cup_{i=1}^{t} S_i$ denote their union, with size $c=|S|=\sum_{i=1}^{t}c_i$ (by disjointness of the $S_i$).
For each $i\in\{1,\ldots,t\}$, recall that graph signals on $G_i$ are functions on the discrete $V_i$, which can be identified with $\mathbb{R}^{|V_i|}$. Let $f_i: \mathbb{R}^{|V_i|} \to \mathbb{R}$ be a convex function on the space of graph signals on $G_i$. Each $f_i$ can be extended to the space of signals on $G$ by composing $f_i$ with the projection $\mathbb{R}^{|V|} \to \mathbb{R}^{|V_i|}$. Let $f: \mathbb{R}^{|V|} \to \mathbb{R}$ denote the sum of these extensions: $f = \sum_{i=1}^{t}f_i$. We are interested in minimizing $f$ and solving certain problems associated with the minimizer. There are global as well as local versions of these problems, formally introduced next.
Recall that a manifold $\mathcal{M}$ of dimension $d$ is a Hausdorff topological space that is locally Euclidean of dimension $d$, i.e., every point has an open neighborhood homeomorphic to an open subset of $\mathbb{R}^d$. In Appendix~\ref{sec:fun}, we give a self-contained introduction to the fundamentals of differentiable manifolds.
\begin{Definition} \label{defn:ato}
A \emph{task on $G$} is a function $\tau: \mathbb{R}^{|V|} \to \mathcal{M}$ where $\mathcal{M}$ is a manifold.
For $U \subset \mathbb{R}^{|S|}$, any graph signal $s \in U$ and minimizer $\hat{x}_s = \argmin_{x|_{S}=s} f(x)$, if $\tau(\hat{x}_s) \in \mathcal{M}$ is well-defined (see explanation below), then we write $\Phi_{\tau}(s)=\tau(\hat{x}_s)$ for the composition of $\tau$ and $\hat{x}_s$. The function $\Phi_{\tau}: U \to \mathcal{M}$ is called the \emph{global problem}.
\end{Definition}
In the definition of $\Phi_{\tau}$, we require $\tau(\hat{x}_s)$ to be well-defined in the following sense. Recognizing that in general $\hat{x}_s$ is not unique, $\tau(\hat{x}_s)$ is \emph{well-defined} if and only if $\tau$ is constant on the set of minimizers $x$ of $f$ subject to $x|_{S}=s$. In particular, if for any $s\in \mathbb{R}^{|S|}$, $f$ has a unique minimizer $x$ such that $x|_{S}=s$, then the domain of $\Phi_{\tau}$ is the full space $\mathbb{R}^{|S|}$. A global problem $\Phi_{\tau}$ is said to be continuous, smooth, etc., if it bears the stated property. Next, we give two examples to illustrate the definitions above.
\begin{Example} \label{eg:ltt}
\begin{enumerate}[1)]
\item As a first example, let the task $\tau$ be the same as $f: \mathbb{R}^{|V|} \to \mathbb{R}$. Then the global problem $\Phi_{f}$ is well-defined on all of $\mathbb{R}^{|S|}$; i.e., the domain of $\Phi_{f}$ is $\mathbb{R}^{|S|}$.
\item \label{it:ite} In this second example, we introduce \emph{distributed sampling}, which will be revisited later. Let $z_1,\ldots, z_k \in \mathbb{R}^{|V|}$ be $k$ linearly independent graph signals on $G$ (so $k\leq|V|$). For each $i\in\{1,\ldots,t\}$ and $j\in\{1,\ldots,k\}$, $z_j$ restricted to $V_i$ is denoted ${z_j}|_{V_i}$. Let the function $f_i$ on $G_i$ be given by
\begin{align*}
f_i(x) = \min_{(r_1,\ldots,r_k)\in \mathbb{R}^k}
\norm{\sum_{j=1}^{k}r_j{z_j}|_{V_i} -x}^2,
\end{align*}
which is the square distance between $x\in\mathbb{R}^{|V_i|}$ and its orthogonal projection onto the span of $z_1|_{V_i},\ldots, z_k|_{V_i}$.
Next, consider the task $\tau: \mathbb{R}^{|V|} \to \mathbb{R}^k$ defined by
\begin{align*}
\tau(x) = \argmin_{(r_1,\ldots,r_k)\in \mathbb{R}^k} \norm{\sum_{j=1}^{k}r_j{z_j} -x}^2,
\end{align*}
which is the unique orthogonal resolution of $x$ onto the span of $z_1,\ldots,z_k$. In this case, the codomain manifold $\mathcal{M}$ of $\tau$ is $\mathbb{R}^k$.
For further simplification, let $t=2$ and assume that the two subgraphs $G_1$ and $G_2$ of $G$ have nonempty intersection, i.e., $V_1\cap V_2\neq\emptyset$.
Suppose that $x$ is a linear combinations of $z_1,\ldots, z_k$. Then, clearly $f_1(x)=f_2(x)=0$. So $\min f = 0$, where $f=f_1+f_2$, and $f_1$ and $f_2$ have been extended to $\mathbb{R}^{|V|}$.
We now explore conditions under which $\Phi_{\tau}$ is well defined, i.e., for any $s\in\mathbb{R}^{|S|}$, $\tau$ is constant on the set of minimizers of $f$ that agree with $s$ on $S$. To be clear, to say that a signal $x\in\mathbb{R}^{|V|}$ agrees with $s\in\mathbb{R}^{|S|}$ on $S$ means that $x|_{S}=s$. For $\tau$ to be constant on the set of such minimizers, the argmin in the definition of $\tau(x)$ must be the same regardless of the minimizer $x$.
For any minimizer $x$ of $f$, the two vectors of coefficients $(r_1,\ldots,r_k)$ in the definitions of $f_1(x)$ and $f_2(x)$ above are not necessarily the same. So, there are $2k$ decision variables (coefficients). Moreover, these $2k$ coefficients satisfy the following:
\begin{enumerate}[(a)]
\item The two linear combinations of $z_1,\ldots, z_k$ with the two vectors of $k$ coefficients are two signals $s_1$ and $s_2$ that agree on $S$; i.e., their restrictions to $S$ are the same: $s_1|_{S} = s_2|_{S}$.
\item The signals $s_1$ and $s_2$ also agree on $V_1\cap V_2$, i.e., $s_1|_{V_1\cap V_2}=s_2|_{V_2\cap V_2}$.
\end{enumerate}
To ensure uniqueness of the argmin in the definition of $\tau(x)$ as explained above, we expect that $2k = |V_1\cap V_2| + c$.
\end{enumerate}
\end{Example}
\section{Message passing} \label{sec:ner}
\subsection{Nerve skeleton and message passing}
We want to make use of the message passing paradigm. It is a way of information transfer across a network, and there is nothing new about the process. In this section, we adopt message passing in our setup. Given $G$ and subgraphs $G_i$, $i\in\{1,\ldots,t\}$, we may construct the nerve skeleton $N$ to package information regarding pairwise intersections between different $G_i$'s as follows. Our goal is to solve a global problem in a distributed way. Therefore, to solve the problem on $G_i$ for some $i$, one gathers information from the rest of the graph at the intersections of $G_i$ with the other subgraphs. This motivates us the consider the following.
\begin{Definition}
The \emph{nerve skeleton}\footnote{There is a more general construction called nerve construction giving a simplicial complex. Here we just use its $1$-skeleton, and hence call the resulting graph the nerve skeleton.} $N_G=(V_{N_G}, E_{N_G})$ is an undirected graph of size $t$. Each vertex $g_i$ of $V_{N_G} = \{g_1,\ldots, g_t\}$ corresponds to the subgraph $G_i$. A pair $(g_i,g_j)$ with $i\neq j$ is an edge of $E_{N_G}$ if and only if $G_i\cap G_j \neq \emptyset$.
\end{Definition}
To perform message passing later on, it can be convenient to work with trees (e.g., \cite{Shah11}). Therefore, we are interested in spanning trees of $N$.
\begin{Definition} \label{defn:stn}
Suppose $T\subset N_G$ is a spanning tree. Its complement $T^c$ is the closure of $N_G\backslash T$. If $g \in V_{N_G}$ is considered as a root, then $T_g$ is the unique directed tree on $T$ such that each edge is directed towards $g$, i.e., the head of an edge $(g_i,g_j)$ is $g_j$ if $g_j$ is below $g_i$, where we say that $g_j$ is below $g_i$ if $g_i$ is on the path connecting $g$ and $g_j$.
\end{Definition}
Fixing a root $g$, we use the directed tree $T_g$ to perform message passing, while $T^c$ is used to keep track of connections between intersecting subgraphs not captured by $T$. We now start to discuss message passing.
On $T_g$, consider a directed edge $(g_i,g_j)$ on $T_g$ such that $g_i$ is the tail and $g_j$ is the head, i.e., the direction goes from $g_i$ to $g_j$. Suppose at $g_i$, there is a convex multi-variable function $h_i$. We would like categorize the input variables of the function $h$ as follows:
\begin{enumerate}[1)]
\item $s_i$: this set of multi-variables corresponds to signals on $S_i$.
\item $x_i$: this set of multi-variables corresponds to signals on the union of the set $X_i$ of nodes $V_i\cap V_j$ and $\cup_{(g_i,g_k)\in T^c}(V_i\cap V_k)$.
\item $y_i$: this set of multi-variables corresponds to signals on the nodes $Y_i$ belonging only to $g_j$ below $g_i$. In particular, $Y_i$ includes nodes contained exclusively in $V_i$.
\item $z_i$: this set includes all the remaining variables, use $Z_i$ to denote those coordinates.
\end{enumerate}
The \emph{message associated with $h_i$ along the edge $(g_i,g_j)$} is
\begin{align}
\tilde{h}_i(x_i,z_i) = \min_{y_i} h_i(s_i,x_i,y_i,z_i): \mathbb{R}^{|X_i|+|Z_i|} \to \mathbb{R}.
\end{align}
To explain the domain, we notice that $s_i$ are in fact fixed because we can take observations on $S_i$. Hence, the message $\tilde{h}_i$ is a function on the variables $x_i$ and $z_i$. According to such a definition, a message is not merely a number or a vector, but instead it is a function.
Before describing message passing, we state the following {\bf conventions}, given a function $f: \mathbb{R}^a \to \mathbb{R}$, for convenience: $f$ gives rise to a function, also denoted by $f$, as $f: \mathbb{R}^{a+b} \to \mathbb{R}$, by composing $f$ with the projection $\mathbb{R}^{a+b} \to \mathbb{R}^a$.
We are ready to describe the \emph{message passing} on $T_g$ with root $g$ as the following procedure:
\begin{enumerate}[S1]
\item \label{it:sft} Starting from the leaves of $T_g$, each leaf node $g_i$ pass the messages $\tilde{f}_i$ associated with $f_i$ to its only immediate neighbor.
\item For each node $g_j$ other than $g$, once it receives messages from all edges with $g_j$ as the head node, sum them up and $f_j$ to obtain $h_j$.
\item The message $\tilde{h}_j$ associated with $h_j$ is passed to $g_k$, along the unique edge $(g_j,g_k)$ such that $g_k$ is the head.
\item \label{it:tpt} The procedure terminates at $g$. Take the sum over all the messages received at $g$ including the original convex function at $g$. The resulting function is denoted by $h_g$.
\end{enumerate}
For later use, we formally extract the key ingredients of the above procedure as follows.
\begin{Definition} \label{defn:wct}
We call the collection of functions $\tilde{f}_i, \tilde{h}_j$ and $h_g$ in \ref{it:sft}$-$\ref{it:tpt} as \emph{messages of the message passing on $T_g$}. The function $h_g$ is the \emph{aggregated message along $T_g$}.
\end{Definition}
We notice that according to the definition, each node is associate with exactly one function as its message.
\begin{Example} \label{eg:cts}
For illustration, consider the situation given in \cref{fig:dsp1}. At the top of the figure, we use the Venn diagram to describe the intersection properties of the $3$ subgraphs $G_1, G_2$ and $G_3$. The capital letters label the nodes in respective regions. The nerve skeleton is the complete graph on $3$ nodes as shown on the bottom left. We consider two different spanning trees $T_1$ and $T_2$ shown on the bottom right. Either $T_1^c$ or $T_2^c$ is the single dashed edge.
For perform message passing on both $T_1$ and $T_2$ with $g_1$ as the root. Let the resulting directed trees be $T_{1,g_1}$ and $T_{2,g_1}$. On $T_{1,g_1}$, messages $\tilde{f}_2(x_2,x_3) = \min_{y_2} f_2(x_2,x_3,y_2,s_2)$ and $\tilde{f}_3(x_1,x_3) = \min_{y_3} f_3(x_1,x_3,y_3,s_3)$ are passed from $g_2, g_3$ to $g_1$ concurrently. At $g_1$, to minimize the aggregated message, we have: $\min_{x_1,x_2,x_3,y_1} f_1 + \tilde{f}_2 + \tilde{f}_3$.
On $T_{2,g_1}$, we first pass the message $\tilde{f}_2(x_2,x_3) = \min_{y_2}f_2(x_2,x_3,y_2,s_2)$ from $g_2$ to $g_3$. At $g_3$, we form the new function $h_3 = f_3 + \tilde{f}_2$, and the message \begin{align*}\tilde{h}_3(x_1,x_2) = \min_{y_3,x_3}(\min_{y_2}f_2(x_2,x_3,y_2,s_2)+ f_3(x_1,x_3,y_3,s_3))\end{align*} is subsequently passed to $g_1$. Finally at $g_1$, minimizing the aggregated message is $\min_{x_1,x_2,y_1} f_1 +\tilde{h}_3$.
\end{Example}
\begin{figure}
\centering
\includegraphics[width=0.3\columnwidth]{dsp1}
\caption{Setup for message passing of \cref{eg:cts}.} \label{fig:dsp1}
\end{figure}
We end this subsection by giving a formal definition of abstract message. Readers unfamiliar with the terminologies may ignore this part, as in the paper we work exclusively with the explicit examples of functions. The terms in the definition will take their concrete forms in \cref{sec:jet}.
\begin{Definition}
An \emph{abstract message} is an element in the limit of a sequence of vector bundles over a manifold $M$.
\end{Definition}
We now provide some insights in additional to the concrete examples at hand. Loosely speaking, a vector bundle is a parametrized family of vectors. It contains information about the parameter space, while we can still perform algebraic operations such as taking sums. Therefore, it is a natural choice if we want to generalize numerical or vectorial information. On the other hand, we may also need to work with infinite dimension objects, hence the necessity to consider the limit of a sequence of finite dimensional objects.
\subsection{Local solution}
In \cref{defn:ato}, we have introduced the notion of global problem $\Phi_{\tau}: U \to \mathcal{M}$ given a task $\tau: \mathbb{R}^{|V|} \to \mathcal{M}$, where $U\subset \mathbb{R}^{|S|}$ is the domain of $\Phi_{\tau}$. To understand what we expect from a distributed approach, we fix a spanning tree $T$ of the nerve skeleton $N_G$.
\begin{Definition} \label{defn:f1w}
For $i\in\{1,\ldots,t\}$, we say that the global problem $\Phi_{\tau}$ can be \emph{solved at $g_i$ via message passing along $T$} if the following holds. Let $h_{g_i}$ be the aggregated message along $T_{g_i}$ with domain $D_i$. There is $\tau_i: D_i \to \mathcal{M}$ such that: for $\hat{x_i} \in \argmin_{x\in D_i} h_{g_i}(x)$, we have $\Phi_{\tau}(s) = \tau_i(\hat{x_i})$.
$\Phi_{\tau}$ can be \emph{solved locally via message passing along $T$} if it can be solved at $g_i$ for every $i\in\{1,\ldots,t\}$.
\end{Definition}
We summarize this definition in the following commutative diagram (with $\hat{x} \in \argmin_x f(x)$):
\[
\begin{tikzcd}[every arrow/.style={draw,mapsto}]
\{f_1,\ldots,f_t;s\} \arrow{d} \arrow{r} & h_{g_i} \arrow{r} & \hat{x_i} \arrow{d}{\tau_i} \\ \{f,s\} \arrow{r}
& \hat{x} \arrow{r}{\tau} & \Phi_{\tau}(s).
\end{tikzcd}
\]
In the diagram, the top route is via message passing while the bottom route is via solving the global problem. Local solubility requires the existence of the right vertical map $\tau_i$ such that both routes have a ``common destination''.
To give an example, the following result is a generalization of \cref{eg:cts}, which is essentially due to re-arrangement of ordering in taking $\min$. In fact, this result is key in motivating us considering messages as functions. It also gives a concrete example of local solubility to a global problem with minimum requirements on $f_i$, $i\in\{1,\ldots,t\}$. General study of local solubility shall be contained in \cref{sec:sol} below.
\begin{Proposition} \label{prop:ief}
Suppose $\tau = f: \mathbb{R}^{|V|} \to \mathbb{R}$. Then for any $T$ and $i\in\{1,\ldots,t\}$, $\Phi_f$ can be solved locally via message passing along $T$.
\end{Proposition}
\begin{IEEEproof}
We first make some remarks regarding message passing. In general, a sum of convex functions is convex. Minimizing a convex function over a subset of variables is convex on the remaining variable. Therefore, each message is a well defined convex function, as long as so does each $f_i$, $i\in\{1,\ldots,t\}$.
We prove the result by induction on the size of $T_{g}$. The result is trivially true if $T_{g}$ is a single node. Suppose that $T_g$ contains a leaf node $g_i$, and it is connected by a directed edge $(g_i,g_j)$ to $g_j$. Let $g_{i,1}, \ldots, g_{i,m}$ be the neighbors of $g_i$ in $T^c$, i.e., $\{g_j, g_{i,1}, \ldots, g_{i,m}\}$ are all the neighbors of $g_i$ in $N_G$. During the message passing from $g_i$ to $g_j$, the variables of $f_i$ are re-grouped into $s_i$, $x_i$, $y_i$ and $z_i$, where $x_i, z_i$ accounts for the nodes $V_i \cap (V_j \cup_{i=1}^{m}V_{i,l})$. The message being passed to $V_j$ is thus $\tilde{f_i}(x_i,z_i) = \min_{y_i}f_i(s_i,x_i,y_i,z_i)$.
For the global problem $\Phi_f$, we want to minimize $f = (f_1 + \ldots + f_{i-1} + f_{i+1} +\ldots + f_t) + f_i$, written as $h+f_i$. We now re-group the variables of $f$ as $x_i,y_i,z_i$ and $r$ where the variables $r$ are disjoint from $x_i,y_i$ and $z_i$. Hence $\Phi_f(s_1,\ldots,s_t) = \min_{x_i,y_i,z_i,r} h(x_i,z_i,r) + f_i(s_i,x_i,y_i,z_i)$. Notice here $s_1,\ldots, s_t$ are fixed numbers at observable nodes. However, $y_i$ is not involved in $h$. Therefore, $\Phi_f(s_1,\ldots, s_t) = \min_{x_i,z_i,r}h(x_i,z_i,r) + \tilde{f}_i(x_i,z_i)$. The right-hand-side is a global problem on the subtree of $T_g$ removing $g_i$, which can be solved locally at $g$ by the induction hypothesis.
\end{IEEEproof}
We end this section by considering the following example where local solution does not exist.
\begin{Example} \label{eg:tiac}
This is a continuation of \cref{eg:ltt}~\ref{it:ite} and we use the setup, such as choices of $f_i$, $i\in\{1,\ldots,t\}$, stated over there. To be concrete, we assume $t=2$ and the graph $G$ with $2$ subgraphs $G_1=(V_1,E_1)$ and $G_2=(V_2,E_2)$ is shown in \figref{fig:dsp4}. Hence $V_1=\{s_1,s_2,y_1,x\}$, $V_1=\{s_3,s_4,y_2,x\}$ and the intersection $V_1\cap V_2$ contains only $x$.
To describe $f_1,f_2$ as in \cref{eg:ltt}~\ref{it:ite}, we specify $k=3$. We arrange the vectors $v_1,v_2,v_3$ according to $s_1,s_2,y_1,x,s_3,s_4,y_2$ as:
\begin{align*}
v_1 = (1, 0, 1, 0, 1, 0, 1)', v_2 =(0, 1, 0, 0, 0, 1, 0)',
v_3 = (1, 0, 0, 1, 1, 0, 0)'.
\end{align*}
Let the task be $\tau: \mathbb{R}^7 \to \mathbb{R}, (s_1,s_2,y_1,x,s_3,s_4,y_2) \to y_1-y_2$.
For the global problem $\Phi_{\tau}$, it is in fact smooth as one may verify that $\Phi_{\tau}(s) = s_1-s_3$ for any observation $s = (s_1,s_2,s_3,s_4)$. However, it cannot be solved locally via message passing. To see this, one can show that $\tilde{f}_i$ is the constant $0$ function. One can directly verify this or observe that the components of $v_1,v_2,v_3$ corresponding to $s_1,s_2,x$ are linearly independent. Therefore, the aggregated message at $g_2$ (corresponding to $G_2$) is just $f_2$ itself. Given observations $s_3,s_4$, $\min_{x,y_2} f_2(x,s_3,s_4,y_2)=0$. To minimize $f_2$, the value $y_2$ can be any number. Therefore, we cannot find the $\tau_i$ required in \cref{defn:f1w}.
To give another perspective, as the message is the constant $0$ function, i.e., all the coefficients in the Taylor expansion are $0$. Information on both $s_1,s_2$ are lost, and we have lost ``two degrees of freedom''. Therefore, we are not able to solve the problem at $G_2$. We shall formalize such a point of view in subsequent sections.
\end{Example}
\begin{figure}
\centering
\includegraphics[width=0.4\columnwidth]{dsp4}
\caption{$G_1$ is the square and $G_2$ contains a single edge. They intersect at a single node.} \label{fig:dsp4}
\end{figure}
\section{Jet spaces and Whitney topologies} \label{sec:jet}
In \cref{sec:pro}, we cast the global problem as a function $\Phi_{\tau}: \mathbb{R}^{|S|} \supset U \to M$. We are interested in whether the global problem can be solved locally via message passing in \cref{sec:ner}. During message passing, a message is also an element of a function space. Instead of considering function spaces as sets, which do not have enough structures, our discussions shall revolve around topological structures of smooth function spaces in this section. A self-contained account on background materials is given in Appendix~\ref{sec:fun}.
Recall that a (real) multi-variable function is \emph{smooth} if its partial derivatives of any order exist. For a multi-variable smooth $f$, according to the Taylor's theorem, we can always approximate $f$ around any point by using a polynomial. On the other hand, polynomials of a bounded degree form a finite dimensional vector space. These observations prompts the following \cite{Gol73}.
\begin{Definition}
Let $M,N$ be open subsets of real vector spaces and $C^{\infty}(M,N)$ be the space of smooth functions from $M$ to $N$, i.e., each component is a smooth function on $M$. The \emph{$k$-jet space} $J_p^k(M,N)$ at $p \in M$ is the equivalent classes of $f, g \in C^{\infty}(M,N)$ with: $f \sim g$ if $f$ and $g$ have the same partial derivatives up to $k$-th order at $p$. By convention, equality on the $0$-th order partial derivative means $f(p)=g(p)$. The equivalence class of $f$ is denoted by $f_p$, and the resulting quotient map is $\pi_p^k: C^{\infty}(M,N) \to J_p^k(M,N)$, i.e., $\pi_p^k(f) = f_p$.
\end{Definition}
Apparently, each class of $J_p^k(M,N)$ has a unique representation which is a degree $k$-polynomial. If $M$ is an open subset of $\mathbb{R}^n$ and $N = \mathbb{R}^m$, there is a canonical isomorphism from $J_p^k(M,N)$ to the space of polynomials up to degree $k$: $P^k_{m,n} = (\mathbb{R}[x_1,\ldots,x_n]/(x_1,\ldots,x_n)^{k+1})^m$, the latter is a finite dimensional vector space parametrized by polynomial coefficients. For some simple examples: $\log(1-x) = -x - x^2/2 - x^3/3$ and $\sin(x) = x - x^3/6$ in $J_0^3(\mathbb{R},\mathbb{R})$.
For each $k\geq 0$, we have the obvious quotient map $J_p^{k+1}(M,N) \to J_p^k(M,N)$, which is also denoted by $\pi_p^{k}$ for convenience (as it is also a quotient map to $J_p^k(M,N)$).
If we let $p$ vary, then $J^k(M,N)$ is defined as the disjoint union of $J_p^k(M,N)$, namely, $J^k(M,N) = \{(p,f), p\in M, f\in J_p^k(M,N)\}$. As a manifold, $J^k(M,N)$ is identified with the Cartesian product $M \times P^k_{m,n}$. We remark that if $M$ and $N$ are general manifolds, we can still define $J^k(M,N)$ as a manifold by applying the above construction locally using coordinate maps. However, the total space $J^k(M,N)$ is no longer a product space in general. We shall not use the general construction in the sequel and details can be found in \cite{Gol73} Section 2.
We now use these jet spaces to give $C^{\infty}(M,N)$ topologies, called the \emph{Whitney topologies} \cite{Gol73}. For each $k$ and $f \in C^{\infty}(M,N)$, we have map $\Pi^k(f): M \to J^k(M,N): \Pi^k(f)(p) = (p,f_p)$. Hence, we have the set $\Pi^k(f)(M) \subset J^k(M,N)$
\begin{align*}
\Pi^k(f)(M) = \{(p,f_p) \mid p \in M\}.
\end{align*}
On the other hand, we also have the quotient maps, whose notation inherits from $\pi_p^k$, as $\pi^k: J^{k+1}(M,N) \to J^k(M,N), (p,f) \to (p,f_p)$. For illustration, a summary is given in \figref{fig:dsp2}.
\begin{figure}
\centering
\includegraphics[width=0.75\columnwidth]{dsp2}
\caption{Summary of the relations among the smooth function space and various jet spaces.} \label{fig:dsp2}
\end{figure}
\begin{Definition} \label{def:tbo}
The basis of the \emph{Whitney $C^k$-topology} on $C^{\infty}(M,N)$ is given by: $S^k(U) = \{f \in C^{\infty}(M,N) \mid \Pi^k(f)(M)\subset U\}$, where $U$ is open in $J^k(M,N)$. This means that open sets in $C^{\infty}(M,N)$ are arbitrary unions of sets of the form $S^k(U)$.
Let $W_k$ be the set of open sets given by the $C^k$-topology. The Whitney $C^{\infty}$-topology has basis $W = \cup_{k\geq 0} W_k$.
\end{Definition}
To give an intuition of the Whitney topologies, a sequence of maps $f_n \in C^{\infty}(M,N), n>0$ converges to $f$ in the $C^k$-topology if and only if the following hods: there is a compact subset $K$ of $M$ such that $\Pi^k(f_n): M \to J^k(M,N), n>0$ converges uniformly to $\Pi^k(f)$ on $K$ and all but finitely many $f_n$ agrees with $f$ outside $K$.
Back to the message passing paradigm, we are interested in maps such as $\alpha: U \to C^{\infty}(M,N)$, where $U$ is an open subset of a Euclidean space $\mathbb{R}^d$ and $N = \mathbb{R}$. In turn, composing with $\pi_p^k: C^{\infty}(M,N) \to J_p^k(M,N), p\in M, k\geq 0$ leads to a map $\alpha_p^k: U \to J_p^k(M,N)$, which depends on both $p\in M$ and $k$. To obtain a map independent of $p$, we enlarge the domain and consider
\begin{align*}
\alpha^k: M\times U \to J^k(M,N), (p,s) \mapsto (p, \alpha_p^k(s)).
\end{align*}
The maps $\alpha^k$ are consistent in the sense that $\alpha^k = \pi^k\circ \alpha^{k+1}$. The construction gives the map $\alpha^k$ between differentiable manifolds $M\times U$ and $J^k(M,N)$. With these maps for different $k$, we want to give $\alpha$ numerical invariants to measure the size of its image.
\begin{Definition} \label{def:sfe}
Suppose for each $k$, there are dense open subsets such $U_k$ of $M \times U$ such that the image $\alpha(U_k)$ of $\alpha^k: U_k \to J^k(M,N)$ is a submanifold of $J^k(M,N)$. Then define $b_{\alpha}^k = \dim (\alpha^k(U_k)) - \dim M$, and $b_{\alpha} = \sup_{k\geq 0} b_{\alpha}^k$.
\end{Definition}
We use a simple example to illustrate how one may compute something such as $\dim (\alpha^k(U_k))$. Suppose $r_1, r_2$ are two parameters. Then functions $\{(r_1+r_2)^2x^2+(r_1+r_2)x+1 \mid r_1,r_2\in \mathbb{R}\}$ is $1$ dimensional as the coefficient of the degree $1$ term determined uniquely the polynomial function. On the other hand, $\{r_1^2x^2+(r_1+r_2)x+1 \mid r_1,r_2\in \mathbb{R}\}$ is $2$ dimensional as we need to know both the degree $1$ and degree $2$ coefficients to know the polynomial function from the set.
\begin{Proposition} \label{prop:fos}
For open subset $U$ of $\mathbb{R}^d$ and $\alpha: U \to C^{\infty}(M,N)$, we assume that $b_{\alpha}^k,k\geq 0$ are well-defined and let $U_k, k\geq 0$ be given as in \cref{def:sfe}. Then the following holds:
\begin{enumerate}[1)]
\item If for each $k$, $\alpha^k$ is a submersion at some $p\in U_k$, then $b_{\alpha} \leq d$.
\item \label{it:tia} Suppose in addition that $\alpha^k$ is a submersion on $U_k$. There is $k_0$ such that for all $k\geq k_0$, $\pi^k: J^{k+1}(M,N) \to J^k(M,N)$ restricts to a diffeomorphism on a dense open subset of $\alpha^{k+1}(U_{k+1})$ to a dense open subset of $\alpha^k(U_k)$. In particular, $ b_{\alpha} = b_{\alpha}^k$ for each $k\geq k_0$.
\end{enumerate}
\end{Proposition}
\begin{IEEEproof}
For 1), as $\alpha^k: U_k \to J^k(M,N)$ is a submersion at $p$, it induces a surjective linear transformation on the tangent spaces of $U_k$ and $\alpha(U_k)$ at $p$ and $\alpha^k(p)$, respectively. We have $\dim (\alpha^k(U_k)) \leq \dim U_k = d+\dim M$. This shows that $b_{\alpha}^k \leq d$ and hence $b_{\alpha}\leq d$.
To prove 2), as the map $\pi^k: J^{k+1}(M,N) \to J^k(M,N)$ is onto, we notice that $b_{\alpha}^k$ is a bounded (by $d$) and non-decreasing sequence, when $k\to \infty$. By the monotone convergence theorem, $\lim_{k\to \infty} b_{\alpha}^k$ exists and equals to $b_{\alpha}$. As each $b_{\alpha}^k$ is an integer, there is $k_0$ such that $b_{\alpha}^k = b_{\alpha}$ for $k\geq k_0$.
Now for $k\geq k_0$, let $U_{k,k+1} = U_k\cap U_{k+1}$, which is again open dense. Then we have the following relation among various maps and spaces:
\begin{align*}
\alpha^k: U_{k,k+1} \stackrel{\alpha^{k+1}}{\to} J^{k+1}(M,N) \stackrel{\pi^k}{\to} J^k(M,N).
\end{align*}
In particular, $\pi^k$ induces a differentiable map from $\alpha^{k+1}(U_{k,k+1})$ to $\alpha^k(U_{k,k+1})$, with both having the same dimension $b_{\alpha}$. On the other hand, the differential $d \pi^k$ is every surjective. Therefore, $d \pi^k$ must be invertible when restricted $\alpha^{k+1}(U_{k,k+1})$. By the inverse function theorem, $\alpha^{k+1}(U_{k,k+1})$ is diffeomorphic to $\alpha^k(U_{k,k+1})$.
\end{IEEEproof}
\section{Messages passing for subfamilies of functions} \label{sec:sub}
The smooth function spaces discussed in \cref{sec:jet} is the playground for us to perform analysis. However, smooth convex functions in general does not behave well under message passing. To analyze local solubility of global problems, we need additional regularities on the functions. On the other hand, we also want the subfamilies contain most familiar functions. In the following, we introduce Morse functions and real analytic functions.
\subsection{Morse functions}
\begin{Definition}
Suppose $M$ is a differentiable manifold and $f:M \to \mathbb{R}$ is smooth. If at $p\in M$, $df_p=0$, then $p$ is called a \emph{critical point} and $f(p)$ is the \emph{critical value}.
The \emph{Hessian matrix} is $(\partial^2 f/\partial x_i\partial x_j)_{i,j}$. A critical point $p$ is called \emph{non-degenerate} if the Hessian is non-singular at $p$.
The function $f$ is called a \emph{Morse function} if it has no degenerate critical points.
\end{Definition}
A Morse function $f$ enjoys many nice properties \cite{Mil63}. For example if $M$ is an open subset of $\mathbb{R}^d$, then in an open neighborhood of any critical point $p$, $f$ takes the form $f(p) - x_1^2-\ldots -x_b^2 + x_{b+1}^2 +\ldots + x_d^2$. As a consequence, all the critical points of $f$ are isolated. In particular, if $f$ is also convex, it has a unique minimal point, i.e., $f$ is strictly convex.
Though it may seen from the definition that Morse functions consist of a restricted subfamily of all smooth functions. However, it is known that they in fact form an open dense subset of all smooth functions, under the $C^2$-topology given in \cref{def:tbo}. This means that Morse functions are omnipresent.
For later use, we make the following observation regarding Morse functions.
\begin{Lemma} \label{lem:iia}
Suppose $f(x,y): \mathbb{R}^{m_x+m_y} \to \mathbb{R}$ is a smooth, convex function on sets of multi-variables $x=(x_j)_{1\leq j\leq m_x}$ and $y = (y_k)_{1\leq k\leq m_y}$ of sizes $m_x$ and $m_y$ respectively.
\begin{enumerate}[1)]
\item If $f_{y_0}(x) = f(x,y_0)$ is a Morse function on $x$, then $\tilde{f}(y) = \min_{x \in \mathbb{R}^{m_x}}f(x,y)$ is a smooth function on an open neighborhood $U_{y_0} \subset \mathbb{R}^{m_y}$ containing $y_0$.
\item \label{it:tso} The set of $y$ such that $f_y=f(\cdot, y)$ being Morse forms an open subset of $\mathbb{R}^{m_y}$.
\end{enumerate}
\end{Lemma}
\begin{IEEEproof}
For 1), to find $\tilde{f}(y)$, we need to solve \begin{align}\frac{\partial f}{\partial x}(x,y) = 0.\label{eq:par}\end{align} If $f_{y_0}(x)$ is a Morse function, then the Hessian of $f_{y_0}$ at $x$ satisfying \cref{eq:par} is non-singular. By the implicit function theorem (c.f.\ \cref{thm:ift}), there is an open subset $U_{y_0}$ containing $y_0$, such that $g(y) = x$ with $(x,y)$ solving Equation~(\ref{eq:par}), is smooth in $y \in U_{y_0}$. Therefore, $\tilde{f}(y) = f(g(y))$ is a smooth function for $y\in U_{y_0}$.
For 2), by 1) and convexity, if $y \in U_{y_0}$, then $f_{y}(x)$ has a unique global minimum. Therefore, it has a unique isolated critical point, which must be non-degenerate. Hence, $f_y$ is a Morse function.
\end{IEEEproof}
The upshot of this result is that under favorable conditions, such as being Morse on a subset of variables, messages are smooth functions. More concretely, if $g_i$ is a leaf node of a directed tree $T_g$ and $f_i$ satisfies the condition of \cref{lem:iia}, in the initial step of message passing, we obtain a map from $\mathbb{R}^{|S_i|} \to C^{\infty}(M,\mathbb{R})$, where $M$ is a Euclidean space.
\subsection{Analytic functions}
Another important subfamily of smooth functions are the analytic functions. The subfamily includes many familiar ones such as the polynomials, exponential functions and the trigonometric functions. Formally, a function $f$ is \emph{analytic} in a connected open subset $D$ of a Euclidean space if for each $x\in D$, there is an open neighborhood $D_x\subset D$ of $x$ such that $f$ agrees with its Taylor series expansion about $x$ for any other point in $D_x$. The following observation related to our theme is essentially the identity theorem of analytic functions.
\begin{Lemma} \label{lem:sui}
Suppose $U$ is a connected open subset of $\mathbb{R}^d$ and $\alpha: U \to C^{\infty}(M,N)$ satisfies \cref{prop:fos}~\ref{it:tia} and $k\geq k_0$ as in there. If the image $Im(\alpha)$ contains only analytic functions, then the map $M \times Im(\alpha) \to J^k(M,N), (p,f) \mapsto (p,f_p)$ for $p\in M, f\in Im(\alpha)$ is injective.
\end{Lemma}
\begin{IEEEproof}
Suppose for analytic functions $f_1,f_2 \in Im(\alpha)$ and $p_1,p_2\in M$, we have $(p_1,f_{1,p})=(p_2,f_{2,p}) \in J^k(M,N)$. Then, $p_1=p_2$ and $f_1$ and $f_2$ have the same partial derivative at $p_1$ up to $k$-th order. As $k\geq k_0$, by \cref{prop:fos}, all the partial derivatives of $f_1$ and $f_2$ at $p_1$ are the same. Therefore, $f_1=f_2$ by the identity theorem \cite{Fri12}.
\end{IEEEproof}
\subsection{Types of message passing}
We come back to message passing in this subsection. Though it is known that Morse functions is dense under $C^2$-topology, the same does not hold for the subspace of smooth convex functions. On the other hand, for optimization, it is favorable to work with convex functions. Therefore, we need a weak notion of ``density'' to deal with the above quagmire.
Recall that in \cref{sec:jet}, for each $f\in C^{\infty}(M,N)$, we introduce $\Pi^k(f): M \to J^k(M,N)$. For subset $K\subset M$ and open subset $U \subset J^k(M,N)$, we use $S^k(K,U)$ to denote the $f \in C^{\infty}(M,N)$ such that $\Pi^k(f)(K) \subset U$.
For example, the basis of the $C^k$-topology consists of sets $S^k(U)= S^k(M,U)$. On the other hand, if $K$ is compact, then $S^k(K,U)$ is related to the compact open topology on the functions from $M$ to $J^k(M,N)$. The notion about being ``dense'' is given in a more general form as follows.
\begin{Definition} \label{def:fkf}
Fix $k\geq 0$ and Euclidean spaces $M_i$, $i\in\{1,\ldots,t\}$ as well as $N$. Suppose we are given $W_1\subset W_2 \subset \prod_{i=1}^{t}C^{\infty}(M_i,N)$ and $K = \prod_{i=1}^{t} K_i$ with $K_i\subset M_i$.
We say that $W_1$ is dense in $W_2$ w.r.t.\ $K$ if the following holds: for every $U = \prod_{i=1}^{t}U_i$ with $U_i$ in $J^2(M_i,N)$ and $(f_i)_{i\in\{1,\ldots,t\}} \in W_2$ with $f_i \in S^k(U_i)=S^k(M_i,U_i)$, there is an $(h_i)_{i\in\{1,\ldots,t\}} \in W_1$ such that $h_i \in S^k(K_i,U_i)$.
\end{Definition}
For the rest of this section, we work solely with $k=2$. We first consider $t=1$, namely on a single function space $C^{\infty}(M,N)$. Under this definition, dense w.r.t.\ $M$ agrees with the usual notion of dense subset. In general, it is easy to see that if $K_1\subset K_2$, then dense w.r.t.\ $K_2$ implies dense w.r.t.\ $K_1$.
\begin{Proposition} \label{prop:lmm}
Let $M = \mathbb{R}^{m}, m= m_x+m_y$ and $\Delta(M)$ be the space of smooth convex function on multi-variables $x=(x_j)_{1\leq j\leq m_x},y = (y_k)_{1\leq k\leq m_y}$. Define $\Omega(M)$ to be the set of $f \in \Delta(M)$, such that $f_y = f(\cdot,y)$ (on $x$) is Morse for all $y$ in some dense open subset $W_f \subset M_y = \mathbb{R}^{m_y}$. Then for any compact subset $K$ of $M$, $\Omega(M)$ is dense w.r.t.\ $K$ in $\Delta(M)$.
\end{Proposition}
\begin{IEEEproof}
We first make some general observations regarding the $S^2(K,U)$, with $K$ compact in $M$ and $U$ open in $J^2(M,\mathbb{R})$ containing $\bar{0} = \{(p,0), p\in M\}$. As $J^2(M,\mathbb{R})$ is homeomorphic to $M \times P^2_{m,1}$, for each $p\in M$, we can always find a open set $U_p \in M$ and an open ball $B_p \in P^2_{m,1}$ centered at $0$ with radius $r_p>0$ such that $U_p\times B_p \subset U$. Here, we take note of the fact that the Euclidean space structure of $P^2_{m,1}$ is given by the polynomial coefficients. Therefore, $\cup_{p\in M} U_p\times B_p \subset U$. On the other hand, $K\subset \cup_{p\in K}U_p$. As $K$ is compact, it has a finite subcover, i.e., $K\subset \cup_{i=1}^{l}U_{p_i}$. Let $r = \min \{r_{p_i},i=1,\ldots,l\}$ and $B$ be the open ball in $P^2_{m,1}$ centered at $0$ with radius $r$. Then $K\times B \subset U$.
The upshot of the discussion is that: any function of the form $h(x,y) = \sum_{1\leq j\leq m_x} a_jx_j^2 + \sum_{1\leq k\leq m_y} b_ky_k^2 \in \Delta(M)$ belongs to $S^2(K,U)$ as long as the non-negative coefficients $a_j,1\leq j\leq m_x$ and $b_k, 1\leq k\leq m_y$ are sufficiently small. This is because for $a_j$'s and $b_k$'s sufficiently small, the coefficients of the Taylor expansion of $h$ belongs to $B$ for each $p \in K$.
We now choose a countable dense subset $\{y_i \in M_y, i\geq 1\}$. By \cref{lem:iia}~\ref{it:tso}, it suffices to show that for any $f \in \Delta(M)$ and any base open neighborhood $S^2(U)$ of the $0$ function, there is an $h\in \Delta(M)\cap S^2(K,U)$ such that $(f+h)_{y_i}$ is Morse for each $i\geq 1$. Here we use the observation that: adding $f$ translates an open neighborhood of $0$ translates to an open neighborhood of $f$, and $f+h$ is viewed as a small perturbation of $f$.
Suppose $h(x,y) = \sum_{1\leq j\leq m_x} a_jx_j^2$ with positive coefficients. Then the tuples $(a_j)_{1\leq j\leq m_x}$ and $i\geq 1$ such that $(f+h)_{y_i}$ is not Morse has Lebesgue measure zero. This is because the sum is not Morse only if some $a_j$ cancels with coefficient of $x_j^2$ in the expansion of $f$ at $y_i$. Such a collection of $(a_j)_{1\leq j\leq m_x}$ has measure zero in $\mathbb{R}^{m_x}$. The set $\{y_i\in \mathbb{R}^{m_y}\}_{i\geq 1}$ is countable. Therefore, there is always $(a_j)_{1\leq j\leq m_x}$, with each component as small as we wish, such that $(f+g)_{y_i}$ is Morse for each $i\geq 1$.
\end{IEEEproof}
We now consider message passing on a directed spanning tree $T_g$ discussed in \cref{sec:ner}. For each $i\in\{1,\ldots,t\}$, we start with a smooth convex functions $f_i \in C^{\infty}(M_i, \mathbb{R})$ where $M_i = \mathbb{R}^{|V_i|}$. The most desirable scenario for us to perform analysis is when all the messages of message passing (\cref{defn:wct}) on $T_g$ are smooth (resp.\ Morse) functions on dense open subsets of the domains, called \emph{smooth (resp.\ Morse) message passing}. Apparently, a Morse message passing is always a smooth message passing. We now discuss how likely they are.
We notice that the tuple of functions $(f_i)_{i\in\{1,\ldots,t\}}$ belongs to the product space $\Delta = \prod_{i=1}^{t}\Delta(M_i)$. Let $\Omega \subset \Delta$ consist of tuples $(f_i)_{i\in\{1,\ldots,t\}}$ admitting Morse message passing on $T_g$, for any spanning tree $T$ of the nerve skeleton and $g\in G$.
\begin{Theorem} \label{thm:fac}
For any compact subsets $K_i \subset M_i$, $i\in\{1,\ldots,t\}$, we have that $\Omega$ is dense in $\Delta$ w.r.t.\ $K = \prod_{i=1}^{t}K_i$.
\end{Theorem}
\begin{IEEEproof}
The strategy of the proof is similar to that of \cref{prop:lmm}. Namely, we want to modify each $f_i$ by adding a degree $2$ polynomial $q_i$ on the variables of $f_i$, with small positive coefficients. In order to do so, we need to examine the conditions for Morse message passing. As in \cref{prop:lmm}, we want to show that each choice of spanning tree $T$, root node $g$, being Morse message passing on $T_g$ prohibits at most a measure zero set of choices for coefficients. Once this is shown, the rest follows the same argument as in \cref{prop:lmm}.
There are only finitely many choices for $T$ and $g$. We only need to show the above holds for any fixed $T$ and $g$. For any node $g_i$, let $h_i$ denote the function such that its associated message $\tilde{h}_i$ is one of those in \cref{defn:wct}. By definition, $f_i$ is a summand $h_i$. To obtain $\tilde{h}_i$, we need to optimize over the variables associated with $g_j$ below $g_i$ on $T_g$ (c.f.\ \cref{defn:stn}). More precisely, there are two types of such variables: $y$ associated with nodes contained in $G_i$, and $y'$ associated with nodes outside $G_i$.
To ensure $\tilde{h}_i$ Morse on a dense open subset of its domain, we need to add a positive definite quadratic $q_y$ on $y$ to $f_i$. If a subset of variables $z$ of $y'$ are associated with nodes in $G_j$. We add a quadratic $q_z$ on $z$ to $f_j$, with exactly any one chosen $G_j$ to avoid repetition. It is important to notice that during the message passing until the current stage with $g_i$, we have not performed any optimization over any subset of variables of $y$ and $y'$. Therefore, no quadratic on $z$ is added to $G_j$ until the current stage. As a consequence, to guarantee $\tilde{h}_i$ is Morse, we only need to avoid a measure zero set on the coefficients of quadratic functions $q_y$ and $q_z$'s.
In subsequent steps of message passing, we do not need to optimize over the variables $y$ and $y'$. Therefore, there are no additional conditions we need to impose on the above mentioned quadratic functions, and this completes the proof.
\end{IEEEproof}
From the proofs, we see that we may modify each of the $f_i$, $i\in\{1,\ldots,t\}$ by adding a positive definite quadratic function on the variables with ``small coefficients'' such that the resulting functions permit smooth (Morse) message passing. Such a procedure could be understood as \emph{regularization}. Moreover, the coefficients can be chosen (uniformly) randomly within a prescribed small domain at $0$.
\section{Solubility results} \label{sec:sol}
In this section, we discuss results on the solubility of a global problem (c.f.\ \cref{sec:pro}) in a distributed way via message passing (c.f.\ \cref{sec:ner}). We give conditions on both the global problem can or cannot be solved locally.
Recall that a global problem takes the form $\Phi_{\tau}: U \to \mathbb{R}^{|V|} \stackrel{\tau}{\to} \mathcal{M}$ (c.f.\ \cref{sec:pro}) for some manifold $\mathcal{M}$. To properly state the results, we assume that $U$ is a connected open subset of $\mathbb{R}^c$ and $\Phi_{\tau}$ is a \emph{smooth surjection}, $c = \sum_{i=1}^{t}c_i$. As in our setup, $c$ is the size of the nodes in $G$ where observation can be made. We remark that requiring $\Phi_{\tau}$ being surjective is not restrictive, for otherwise, we may just consider $\Phi_{\tau}$ as a map from $U$ to its image as long as the image is a manifold.
\subsection{Individual message passing step} \label{sec:imp}
In this subsection, we examine closely each individual message passing step. For each $k$, there are three spaces involved in the discussion, namely $U$, $J^k(M,N)$ and $C^{\infty}(M,N)$. The first two are manifolds, while $C^{\infty}(M,N)$ in general is not a manifold. Therefore, we want to pass the study of message passing to jet spaces. Suppose $\alpha: U \to C^{\infty}(M,N)$ is given. It induces $\alpha^k: M\times U \to J^k(M,N), (p,s) \mapsto (p,\alpha^k_p(s))$ where $\alpha^k_p(s) = \pi_p^k\Big(\alpha(s) \Big)$ (c.f.\ \cref{sec:jet}). In other words, $\alpha^k$ is the composition $M\times U \stackrel{Id\times \alpha}{\to} M\times C^{\infty}(M,N) \stackrel{\pi^k}{\to} J^k(M,N) $. We make the following assumptions:
\begin{enumerate}[1)]
\item \cref{prop:fos}~\ref{it:tia} holds and let $k_0$ be as defined there. Moreover, $k\geq k_0$.
\item The map $\pi^k: \alpha^k(M \times U) \to J^k(M,N), (p, \alpha(s)) \mapsto \Big(p, \pi^k_p(\alpha(s))\Big)$ is injective.
\end{enumerate}
As we have seen in \cref{lem:sui}, the second assumption holds if $\alpha(U)$ contains only analytic functions.
Consider a single instance in message passing $\mathscr{P}: f \mapsto \tilde{f}$. It extends to a map on pairs $(f,p), p\in M$ by $\mathscr{P}((f,p)) = (\tilde{f},p')$ with $p'$ is the projection of $p$ to the domain $M'$ of $\tilde{f}$. Suppose we consider a subset $\tilde{M}$ of $M\times C^{\infty}(M,N)$ containing $Im(Id\times \alpha)$, such that $\mathscr{P}(\tilde{M}) = \tilde{M'}\subset M'\times C^{\infty}(M',N)$. We have the following diagram of maps:
\[
\begin{tikzcd}
M\times U \arrow{r}{Id\times \alpha} & \tilde{M} \arrow{r}{\mathscr{P}} \arrow[swap]{d}{\pi^k} & \tilde{M'} \arrow{d}{\pi^k} \\
& \pi^k(\tilde{M}) \arrow[dashed]{r}{\bar{\mathscr{P}}} & \pi^k(\tilde{M'}),
\end{tikzcd}
\]
where $\pi^k(\tilde{M})$ and $\pi^k(\tilde{M'})$ belong to $J^k(M,N)$ and $J^k(M',N)$ respectively. Then we can find a set map $\bar{\mathscr{P}}$ (the dashed arrow) from $\pi^k(\tilde{M})$ to $\pi^k(\tilde{M'})$ making the diagram commute, i.e., $\bar{\mathscr{P}} \circ \pi^k = \pi^k \circ \mathscr{P}$. This is because our injectivity assumption on $\pi^k$ guarantees that $\pi^k: \tilde{M} \to \pi^k(\tilde{M})$ is a bijection.
On the other hand, we may also view $Id\times \alpha: M\times U \to M\times C^{\infty}(M,N)$ with a different perspective. Here, we may ignore the first component, being the identity. For each $s\in U$, $\alpha(s): M \to N$. Therefore equivalently, we may interpret this as $F_{\alpha}: M\times U \to N$ by $F_{\alpha}(p,s) = \alpha(s)(p)$.
\begin{Lemma}
Write the components of $p\in M$ as $p=(x,y)$ such that domain of $M'$ is on $x$. Assume that the following holds:
\begin{enumerate}[1)]
\item \label{it:fmt} $F_{\alpha}: M\times U \to N$ is smooth and Morse on the joint variables $(x,s)$.
\item $\pi^k$ is injective on $\tilde{M}$.
\end{enumerate}
Then $\pi^k \circ \mathscr{P} \circ \alpha: M\times U \to \pi(\tilde{M'})$ is smooth. Moreover, if $\pi^k\circ \alpha: M\times U \to \pi^k(\tilde{M})$ is a submersion, then $\bar{\mathscr{P}}$ is smooth.
\end{Lemma}
Notice that $\mathscr{P}:\tilde{M} \to \tilde{M'}$, where $\tilde{M} \subset M \times C^{\infty}(M,N)$ and $\tilde{M'} \subset M' \times C^{\infty}(M',N)$. Therefore, $\mathscr{P}$ essentially has two components.
\begin{IEEEproof}
The lemma essentially follows from the implicit function theorem as in \cref{lem:iia}. Given a pair $(p,s) \in M\times U$, its image under $\alpha$ is the pair $(p, f)$, where $f(x,y) = F_{\alpha}(x,y,s)$. Write the variables of $f$ as $p=(x,y)$. The map $\mathscr{P}$ projects the component $p=(x,y)$ to $p'=x$, which is clearly smooth. We want to show that the partial order derivatives of $\tilde{f}(x) = \min_{y}F_{\alpha}(x,y,s)$ is smooth on $x$ and $s$. By the Morse condition on $F_{\alpha}$, $\tilde{f}(x) = F_{\alpha}(x,g(x,s),s)$ for smooth function $g$. The partial derivatives of $\tilde{f}$ is a polynomial of those of $F_{\alpha}$ evaluated at $(x,g(x,s),s)$ and those of $g$ evaluated at $(x,s)$. Hence, $\tilde{f}$ is smooth on $x$ and $s$ and so are its partial derivatives.
If $\pi^k\circ \alpha$ is a submersion, it is locally a coordinate projection. Moreover, as $\bar{\mathscr{P}}\circ \pi^k\circ \alpha = \pi^k\circ \mathscr{P} \circ \alpha$ is smooth, so is $\bar{\mathscr{P}}$.
\end{IEEEproof}
If we put each individual message passing step together, we obtain a global picture as illustrated in \figref{fig:dsp3}. The main point is that it can usually be difficult or in-explicit to work with smooth functions. However, under favorable conditions, message passing can be viewed as a procedure on the jet spaces, which are Euclidean spaces or more generally manifolds. In doing so, we replace studying functions by studying its derivatives up to certain fixed order. An important advantage is we can now use simple numerical invariants such as dimensions.
\begin{figure}
\centering
\includegraphics[width=0.5\columnwidth]{mp}
\caption{In the top layer, we have the message passing, the spaces are smooth function spaces. In the bottom layer, we have the jet spaces. As we have seen, message passing induces maps in the bottom layer. Study the maps in the bottom layer can be easier as they are between manifolds.} \label{fig:dsp3}
\end{figure}
\subsection{Message passing in its entirety}
In this subsection, we are going to state and prove the main result on solubility of a global problem $\Phi_{\tau}$ via message passing.
\begin{Theorem} \label{thm:ltb}
\begin{enumerate}[1)]
Let $T$ be a spanning tree of $N_G$ and $g_k\in V_{N_G}$.
\item \label{it:sfa} Suppose $f_1,\ldots, f_t$ admits a Morse message passing. Let $h_k$ be the aggregated message along $T_{g_k}$ (c.f.\ \cref{defn:wct}). Denote the domain of $f=\sum_{i=1}^{t}f_i$ and $h_k$ be $D_f$ and $D_{h_k}$ respectively. Then there is an open dense subset $U$ of $\mathbb{R}^{|S|}, S = \cup_{i=1}^{t}S_i$ such that for any $s=(s_i)_{i=1}^{t}\in U$
\begin{align*}
\hat{x} = \argmin_{x \in D_f, x|_S=s} f(x), \text{ and } \hat{y_k} = \argmin_{y \in D_{h_k}, y|_{S_k} = s_k} h_k(y)
\end{align*}
depend smoothly on $s$ and $\hat{y_k} = \hat{x}|_{V_k}$.
\item \label{it:fga} For $g_i$ a leaf of $T$ connected to $g_j$, let $M$ be the domain of the message $\hat{f_i}$ from $g_i$ to $g_j$ and $U$ be a dense open subset of $\mathbb{R}^{|S|}$ such that $\hat{f_i}$ is smooth for any $s\in U$. Denote by $\alpha: U \to C^{\infty}(M,\mathbb{R})$. Consider $k_0$ and $b_{\alpha}$ as defined in \cref{def:sfe}, we assume $k\geq k_0$ and $\alpha^k$ is a submersion. If $|S_i|-b_{\alpha} > |S|-\dim \mathcal{M}$, then there does not exist local solution via message passing along $T$.
Moreover, if the condition is verified for one spanning tree $T$, then the (insolubility) conclusion holds for any spanning tree of $N_G$.
\end{enumerate}
\end{Theorem}
\begin{IEEEproof}
\begin{enumerate}[1)]
\item As we assume that $f_1,\ldots,f_t$ admits a Morse message passing, both $f$ and $h_k$ are smooth Morse functions. Therefore, $\hat{x}$ and $\hat{y_k}$ are uniquely determined. We have shown in \cref{prop:ief} that both $f$ and $h_k$ have the same global minimum. Hence, $\hat{y_k} = \hat{x}|_{V_k}$. It suffices to see that $\hat{x}$ depends smoothly $s$, which follows by applying the implicit function theorem to $f$.
\item If $M$ is the domain of the message from $g_i$ to $g_j$, then we modify the global problem by adding in the identity map on a factor of $M$ as $Id_M\times \Phi_{\tau}: M\times U \to M\times \mathcal{M}$, which remains to be continuous and surjective. Here, $Id_M$ is the identity map on $M$. Suppose under the given conditions that the global problem $\Phi_{\tau}$ has a local solution via message passing along $T$. Then the modified global task $Id_M\times \Phi_{\tau}$ can be decomposed as $M\times U \stackrel{Id_M\times \alpha}{\to} M\times C^{\infty}(M,\mathbb{R})\times U'\to M\times \mathcal{M}$, where $U'$ are on $S\backslash S_i$. The existence of the map $M\times C^{\infty}(M,\mathbb{R})\times U'\to M\times \mathcal{M}$ is due to the assumption on the existence of local solution. Let $\tilde{M}$ be the image of the map $\alpha$.
Consider the diagram of maps
\[
\begin{tikzcd}
M\times U \arrow{r}{Id_M\times \alpha} & \tilde{M} \arrow{r} \arrow[swap]{d}{\pi^k\times Id_{U'}} & M\times \mathcal{M} \\
& \pi^k(\tilde{M})\times U' \arrow[dashed]{ur}{\Phi'}.
\end{tikzcd}
\]
As $\pi^k$ is bijective, there is $\Phi':\pi^k(\tilde{M})\times U' \to M\times \mathcal{M}$ making the triangle in the above diagram commute. By our assumptions, $(\pi^k\times Id_{U'})\circ (Id_M\times \alpha)$ is a submersion and $\pi^k$ is a bijection. In particular, $(\pi^k\times Id_{U'})\circ (Id_M\times \alpha)$ is locally a coordinate projection, and $\Phi'$ is differentiable as the top row is the smooth surjection $\Phi_{\tau}$. By Sard's theorem (\cref{thm:sard}), we can find $(p,s) \in M\times U$ such that $Id_M\times \Phi_{\tau}$ is a submersion at $(p,s)$. In view of $b_{\alpha} = \dim(\pi^k(\tilde{M}))-\dim M$ and $\dim U' = |S|-|S_i|$, if $|S_i|-b_{\alpha} > |S|-\dim \mathcal{M}$, we have $\dim(\pi^k(\tilde{M})\times U') < \dim(M\times \mathcal{M})$. This gives a contradiction as $d\Phi_{\tau}$ is surjective at $(p,s)$, while $d\Phi'$ cannot be surjective from a smaller space to a larger space.
The same argument applies to any $T$ as the existence of the diagram does not depend on $T$.
\end{enumerate}
\end{IEEEproof}
Let us summarize what we have found in the results. First of all, \cref{thm:ltb}~\ref{it:sfa} states that when we have a Morse message passing, then the optimizer of the aggregated message agrees exactly with the restriction of the optimizer of the original global problem. If this result is viewed together with \cref{thm:fac}, then we can always add a quadratic regularization term to $f_i$, $i\in\{1,\ldots,t\}$ for the purpose. Of course, we have to pay the price that the solution is not exactly the same as the intended one. On the other hand, \cref{thm:ltb}~\ref{it:fga} gives an explicit condition that local solubility is impossible. The condition is relatively easy to verify as we only need to look at the partial order derivatives of the original functions $f_i$, $i\in\{1,\ldots,t\}$.
\begin{Example}
We revisit \cref{eg:tiac} to demonstrate the idea of \cref{thm:ltb}~\ref{it:fga}. In that example, $\mathcal{M} = \mathbb{R}$ and its dimension is $1$. The set $S=\{s_1,s_2,s_3,s_4\}$ has size $4$. Hence, $|S| - \dim \mathcal{M}=3$. On the other hand, $|S_1| = 0$. To compute $b_{\alpha}$, as we have seen in \cref{eg:tiac}, the message $\tilde{f}_1$ from $g_1$ to $g_2$ the constant $0$ function, and hence the dimension of the image of $\pi^k$ is $0$. Moreover, the domain of $f_1$ is $2$ dimensional. Therefore, $b_\alpha = 0-2=-2$. Now, we compare that $|S_i|-b_{\alpha}=4>3 = |S|-\dim \mathcal{M}$. By \cref{thm:ltb}~\ref{it:fga}, the global problem cannot be solved locally via message passing.
\end{Example}
\section{Approximated message passing} \label{sec:app}
So far, we have been mainly focused on theoretical aspects of the message passing scheme. One of the key ingredients is the message, which takes the form $h(x) = \min_{y\in D} h(x,y)$ over variable $y$ over a domain $D$. In practice, it is usually not possible to write down an explicit analytic expression of $h$. In this section, we propose to use a neural network to approximate such a message $h$.
\begin{figure}
\centering
\includegraphics[width=0.5\columnwidth]{dsp3}
\caption{An illustration of MLP.} \label{fig:dsp5}
\end{figure}
Recall that a multilayer perceptron (MLP) (illustrated in \figref{fig:dsp5}) is a function that consists of a concatenation of (hidden) layers, each consist of a composition of a linear transformation and non-linear activation such as the rectifier linear unit (ReLU). For each layer, a finite set of learnable parameters dictates the linear transformation. It is known that a MLP with one hidden layer is enough to approximate any continuous function \cite{Hor89}. Based on this, we propose the \emph{approximated message passing algorithm} as:
\begin{enumerate}[S1']
\item Construct $N_G$ and fix a spanning tree $T$.
\item For each $g$, form the directed tree $T_g$.
\item Perform message passing following the procedure described in \cref{sec:ner}, with the following modification.
\begin{enumerate}[s1]
\item \label{it:sng} Suppose node $g_j$ is to pass $\tilde{h}_j: D \to \mathbb{R}$ to its neighbor $g_k$. Node $g_j$ will randomly choose samples $x_1,\ldots, x_m \in D$ and compute $y_i = \tilde{h}_j(x_i)$ for each $i\in\{1,\ldots,m\}$. The pairs $\{(x_i,y_i)_{i\in\{1,\ldots,m\}}\}$ are passed to $g_k$.
\item Using the received samples $\{(x_i,y_i)_{i\in\{1,\ldots,m\}}\}$, node $g_k$ learns an MLP $\bar{h}_j$ as an approximation of $\tilde{h}_j$.
\item One sums up all the approximated messages at $g_k$ and obtain $\tilde{h}_k$. Step~\ref{it:sng} is repeated at $g_k$.
\end{enumerate}
\item The aggregated message $h_g$ at $g$ along $T_g$ is optimized locally at $g$.
\end{enumerate}
It is interesting to notice that in the procedure described above, there is only one exchange of information along each edge. This is radically different from many distributed algorithms involving numerous information exchange until convergence.
For the rest of the section, we present some numerical examples based on the approximated message passing algorithm.
\begin{figure}
\centering
\includegraphics[width=0.5\columnwidth]{test1}
\caption{The nerve skeleton.} \label{fig:test1}
\end{figure}
\begin{table}[!htb]
\caption{A summary of the intersection statistics} \label{tab:1}
\centering
\scalebox{1.2}{
\begin{tabular}{|l|c|c|c|c|}
\hline
\emph{Subgraph} & \emph{$|X_i|$} & \emph{$|Y_i|$} & \emph{$|S_i|$} &\emph{$|V_i|$} \\
\hline \hline
$G_1$ & 4 & 6 & 12 & 22\\
\hline
$G_2$ & 12 & 4 & 10 & 26\\
\hline
$G_3$ & 6 & 6 & 14 & 26\\
\hline
$G_4$ & 6 & 8 & 12 & 26\\
\hline
$G_5$ & 12 & 6 & 10 & 28\\
\hline
$G_6$ & 14 & 8 & 6 & 28\\
\hline
$G_7$ & 10 & 4 & 12 & 26\\
\hline
$G_8$ & 8 & 2 & 14 & 24\\
\hline
$G_9$ & 6 & 6 & 12 & 24\\
\hline
$G_{10}$ & 10 & 6 & 10 & 26\\
\hline
$G_{11}$ & 4 & 8 & 13 & 25\\
\hline
$G_{12}$ & 4 & 8 & 14 & 26\\
\hline
\end{tabular}}
\end{table}
\begin{table}[!htb]
\caption{Simulation results ($\%$).} \label{tab:2}
\centering
\scalebox{1.2}{
\begin{tabular}{|l||c|c|c|c|c|c|}
\hline
\emph{$k$} & 25 & 30 & 35 & 40 & 45 & 50\\
\hline
\emph{R($\%$)} & 4.93 & 5.65 & 3.86 & 4.26 & 6.16 & 4.14\\
\hline
\end{tabular}}
\end{table}
We simulate the distributed sampling in \cref{eg:ltt}~\ref{it:ite}, and follows entirely the setup described in the example. While we are not able to draw the entire graph $G$, we present the nerve skeleton in \figref{fig:test1}. From the figure, we see that there are $12$ subgraphs $G_i=(V_i,E_i)$, $i\in\{1,\ldots,12\}$ of $G$, with $G_i$ corresponds to the node $g_i$. The intersection properties of $G_i$ are given in \cref{tab:1} and \figref{fig:test1}. Recall the notations, for $i\in\{1,\ldots,12\}$, $S_i$ and $Y_i$ are nodes contained exclusively in $V_i$, while $S_i$ are nodes where observations are available. $X_i$ are the nodes in the intersection of $G_i$ with other subgraphs.
For each $i\in\{1,\ldots,12\}$, the function $f_i$ on $G_i$ is
\begin{align*}
f_i(x) = \min_{r_1,\ldots,r_k} \norm{\sum_{1\leq j\leq k}r_j{z_j}|_{V_i} -x}^2.
\end{align*}
as described in \cref{eg:ltt}\ref{it:ite}. In the simulations, we randomly generated the vectors $z_j, 1\leq j\leq k$. For the parameter $k$, we test the performance of approximated message passing with $k=25,30,\ldots, 50$. For large $k$, each graph has much more unknowns than observable nodes.
We follow the steps of approximated message passing algorithm to get the final approximation of the global optimization. For each simulation, we measure the performance by calculating error between ground truth and estimated optimal value of $f$, and then take the ratio $R$ between error and ground truth. For each $k$, multiple simulations are performed, and the average results are shown in \cref{tab:2}. From the results, we see that although we do not have a perfect result, the approximated message passing yields reasonably good results even when $k$ is large, as our theory expects.
During message passing, each time we transfer $80$ samples from a node to its neighbor, i.e., $m=80$ in Step~\ref{it:sng}. We further investigate the performance for different $m$ (with fixed choice $k=50$). The results are shown in \figref{fig:dsp6}. We see that the performance starts to show steady improvements from $m=50$ onward, until having the reasonable performance when $m=80$.
\begin{figure}
\centering
\includegraphics[width=0.5\columnwidth, trim={0 6cm 0 6cm},clip]{dsp5.pdf}
\caption{Plot of the error $R$ against the number of samples $m$. In this plot, $R$ being $80\%$ in fact means that the error is larger than $80\%$ and the performance is unstable.} \label{fig:dsp6}
\end{figure}
\section{Conclusions} \label{sec:con}
In this paper, we propose a framework on distributed graph signal processing. We employ the approach of message passing, and introduce the concept of abstract messages. Our approach provides an alternative point of view of distributed graph signal processing as compare with classical approaches. Moreover, the framework is convenient to analyze theoretical questions such as solubility of distributed problems. Though our work is mainly theoretical, we still present numerical findings to verify the theory. While the topic is distributed GSP, there are still centralized components. Moreover, as we introduce a new notion of message, privacy can be an important concern. In the future, it could be an central topic to investigate.
\appendices
\section{Fundamentals of differentiable manifolds} \label{sec:fun}
In this appendix, we give a self-contained introduction to fundamentals of differentiable manifolds. We shall highlight the results needed for the paper. Readers can further consult textbooks on manifold theory (e.g., \cite{War83}) for thorough discussions of the theory. We start by defining what a topological space is.
\begin{Definition}
A \emph{topology} on a set $X$ is a collection $\mathcal{T}$ of subsets of $X$ s.t.:
\begin{enumerate}[1)]
\item $\emptyset, X \in \mathcal{T}$.
\item The union of the elements of any subcollection of $\mathcal{T}$ is in $\mathcal{T}$, i.e., $\mathcal{T}$ is preserved under arbitrary union.
\item The intersection of the elements of any \emph{finite} subcollection of $\mathcal{T}$ is in $\mathcal{T}$, i.e., $\mathcal{T}$ is preserved under finite intersection.
\end{enumerate}
$X$ with $\mathcal{T}$ is called a \emph{topological space}, and elements of $\mathcal{T}$ are called \emph{open subsets} of $X$.
\end{Definition}
Primary examples of topological spaces include discrete set where each point is open, and Euclidean spaces $\mathbb{R}^n$ where open sets are unions of open balls. In general, a subset $Y$ of $X$ is also a topological space, with an open takes the form $Y\cap U$ where $U$ is open in $X$.
Not every subset of $X$ is open, but there can be other interesting families. A subset $C$ of $X$ is closed if its complement is open. A subset $D$ is dense in $X$ if it has non-empty intersection with any open subset of $X$. Another key notion is \emph{compact set}: $K$ is compact if a cover of $K$ by open sets (i.e., $K$ is contained in their union) has a finite subcover.
There are many other quantifiers we can associate with topological spaces. We mention two of them here. $X$ is said to be \emph{connected} if it cannot be decomposed into the union of two disjoint non-empty open subsets. It is called \emph{Hausdorff} if for any $x\neq y \in X$, there are disjoint open subset $U_x,U_y$ such that $x\in U_x$ and $y\in U_y$, i.e., one can separate points by using open sets.
Topological spaces are related by \emph{continuous functions}: $f: X\to Y$ is continuous if the inverse image of any open set is open. A continuous function $f$ is a \emph{homeomorphism} if it has a continuous inverse.
In addition to being topological spaces related by continuous functions, Euclidean spaces carry more structures, namely differentiable structures. Moreover, we may glue together Euclidean spaces to form more general spaces. We next discuss these ideas and put them together leading to differentiable manifolds.
\begin{Definition} \label{def: fka}
For $k\geq 0$, a function $f: \mathbb{R}^n \to \mathbb{R}$ belongs to $C^k(\mathbb{R}^n,\mathbb{R})$ if $f$ has all continuous partial derivatives up to $k$-th order. The space of smooth functions $C^{\infty}(\mathbb{R}^n,\mathbb{R})$ is $\cap_{k\geq 0} C^{k}(\mathbb{R}^n,\mathbb{R})$, i.e., $f$ is smooth if all partial derivatives exist.
For $f: \mathbb{R}^n \to \mathbb{R}^m$ and $k\geq 0$ or $k=\infty$, it belongs to $C^k(\mathbb{R}^n, \mathbb{R}^m)$ or smooth if each of the $m$-components belongs to $C^k(\mathbb{R}^n,\mathbb{R})$.
\end{Definition}
Taking partial derivative of $f$ at $p$ depends only the value of $f$ at any open set containing $p$. Therefore, the same definition can be used without change if the domain $\mathbb{R}^n$ is replaced by any open subset $U$. If the domain and codomain are clear from context, we may abbreviate $C^k$ or $C^{\infty}$ for convenience. By convention, a function with empty domain belongs to $C^{\infty}$.
A smooth function $f$ is called a \emph{diffeomorphism} if it has a smooth inverse function. We give a more general version of this notion later on once we have introduced differentiable manifolds. Key results regarding differentiable functions are the inverse function theorem and the implicit function theorem \cite{War83, Kau11}.
\begin{Theorem}[The inverse function theorem]
Let $U \subset \mathbb{R}^d$ be open, and let $f: U \to \mathbb{R}^d$ be $C^{\infty}$. If the Jacobian matrix $$\{\frac{\partial r_i\circ f}{\partial r_j}\}_{1\leq i,j\leq d}$$ is non-singular at $r_0\in U$, then there is an open set $V$ with $r_0\in V\subset U$, such that $f(V)$ is open and $f: V\to f(V)$ is a diffeomorphism.
\end{Theorem}
\begin{Theorem}[The implicit function theorem] \label{thm:ift}
Let $U \subset \mathbb{R}^{c-d}\times \mathbb{R}^d$ be open, and let $f: U \to \mathbb{R}^d$ be $C^{\infty}$. We denote the canonical coordinate system on $\mathbb{R}^{c-d}\times \mathbb{R}^d$ by $(r_1,\ldots, r_{c-d}, s_1,\ldots, s_d)$. Suppose that at the point $(r_0,s_0) \in U$, $f(r_0,s_0)=0$; and that the matrix $$Df(r_0,s_0) = \{\frac{\partial f_i}{\partial s_j}|_{(r_0,s_0)}\}_{i,j=1,\ldots, d}$$ is non-singular. Then there exists an open neighborhood $V$ of $r_0$ in $\mathbb{R}^{c-d}$ and an open neighborhood $W$ of $s_0$ in $\mathbb{R}^d$ s.t. $V\times W \subset U$, and there exists a $C^{\infty}$ map $g: V\to W$ s.t.\ for each $(p,q) \in V\times W$: $f(p,q)=0$ iff $q = g(p)$.
\end{Theorem}
A \emph{locally Euclidean space} $M$ of dimension $d$ is a Hausdorff topological space $M$ for which each point has a neighborhood homeomorphic to an open subset of Euclidean space $\mathbb{R}^d$.
A \emph{differentiable structure} $\mathcal{F}$ on $M$ of class $C^k$ is a collection of coordinate systems $\{(U_{\alpha},\psi_{\alpha})\mid \alpha \in A\}$ with index set $A$ such that the following holds:
\begin{enumerate}[1)]
\item Each $U_{\alpha}$ is open in $M$ and $M = \cup_{\alpha\in A}U_{\alpha}$.
\item $\psi_{\alpha}: U_{\alpha} \to V_{\alpha}$ is a homeomorphism, i.e., continuous bijection with a continuous inverse, from $U_{\alpha}$ to an open subset $V_{\alpha}$ of $\mathbb{R}^d$. Each $\psi_{\alpha}$ is called a coordinate map.
\item If $U_{\alpha}\cap U_{\beta} \neq \emptyset$, then $\psi_{\alpha} \circ \psi_{\beta}^{-1}: W_{\beta} \to V_{\alpha} \in C^k(\mathbb{R}^d,\mathbb{R}^d)$, where $W_{\beta} = \psi_{\beta}(U_{\alpha}\cap U_{\beta})$.
\end{enumerate}
$M$ with such an differentiable structure $\mathcal{F}$ is called \emph{a differentiable manifold of class $C^k$}. We usually consider $k=\infty$ and just call such an $M$ a differentiable manifold.
The coordinate maps allows us to match open sets of $M$ with those of $\mathbb{R}^d$. As a consequence, we can talk about differentiability of functions on $M$ by pulling back them to a Euclidean space using one of the coordinate maps.
For example, for an open subset $U\subset M$, $f: U\to \mathbb{R}$ is a $C^{\infty}$ function or smooth on $U$ if $f\circ \psi_{\alpha}^{-1}$ is $C^{\infty}$ for each coordinate map $\psi_{\alpha}, \alpha \in A$.
For differentiable manifolds $M$ and $N$, a continuous map $f: M \to N$ is said to be differentiable of class $C^{\infty}$ or smooth if $\gamma_{\beta} \circ f \circ \psi_{\alpha}^{-1}$ is $C^{\infty}$ for each coordinate $\psi_{\alpha}$ on $M$ and $\gamma_{\beta}$ on $N$. Such a map $f$ is a \emph{diffeomorphism} if $f$ is bijective and $f^{-1}$ is also smooth.
For each point $p \in M$ contained in $U_{\alpha}$ with coordinate map $\psi_{\alpha}$, a \emph{tangent vector} is a linear map on the space of smooth functions satisfying the Leibniz rule. The tangent space $T_pM$ is the vector space of tangent vectors spanned by the ``partial derivatives'' $\frac{\partial}{\partial x_i }|_p$:
\begin{align*}
\frac{\partial}{\partial x_i}|_p(f) = \frac{\partial (f\circ \psi_{\alpha}^{-1})}{\partial r_i}|_{\psi_{\alpha}(p)},\ i\in\{1,\ldots,d\},
\end{align*}
where $f \in C^{\infty}(M,\mathbb{R})$ and $r_i$ is the $i$-th coordinate function of $\mathbb{R}^d$.
Let $f: M \to N$ be a smooth function and $p\in M$. The \emph{differential of $f$} at $p$ is the linear map $df_p: T_pM \to T_{f(p)}N$ by $$df_p(v)(h) = v(h\circ f),$$ for $v \in T_pM$ and $g \in C^{\infty}(N,\mathbb{R})$. Locally at $p$, the differential $df_p$ gives a linear approximation of $f$. Many properties of $f$ can studied through $df_p$, which is more accessible being a linear transformation. To demonstrate, we define what immersion, imbedding and submersions are.
The smooth function $f$ is called an \emph{immersion} if $df_p$ is non-singular for every $p\in M$. The manifold $M$ with $f$ is a \emph{submanifold} of $N$ if $f$ is both injective and an immersion. One should take note here that we are not making a repetition of conditions as being being an immersion refers to injectivity of the differential at each point, but not the injectivity of $f$ itself. An even stronger notion is that of \emph{imbedding}: $M$ is a submanifold and $f:M \to f(M)$ is a homeomorphism. These notions are all nonequivalent, and \cite{War83} 1.28 contains concrete examples for this. On the other hand, $f$ is a \emph{submersion} if $df_p$ is surjective for every $p\in M$. A consequence of being submersion is that for every $p\in M$, there is an open neighborhood $U_p$ of $p$ diffeomorphic to a Euclidean open set such that $f$ restricts to a coordinate projection on $U_p$. In particular, $f$ is an open map from $f$ to $f(M)$, i.e., the image of open set is open. Though not every smooth surjective map is a submersion (e.g., $f: \mathbb{R} \to \mathbb{R}, x \to x^3$ is not a submersion at the single point $x=0$), Sard's theorem \cite{Sar42} states that for most points $f$ is a submersion. More precisely, we have the following special case of Sard's theorem:
\begin{Theorem} \label{thm:sard}
Let $f: M \to N$ be a smooth surjection between differentiable manifolds. If $X\subset M$ consists points where $f$ is not a submersion, then $f(X)$ has Lebesgue measure $0$ in $N$.
\end{Theorem}
\bibliographystyle{IEEEtran}
|
2206.04509
|
\section{Introduction}
Let $\mathbb{Z}\subset \mathbb{R}\subset \mathbb{C}$ be the set of
integers, real and complex numbers, respectively. Throughout,
$\mathfrak{g}=\mathfrak{g}(A)$ stands for a complex finite-dimensional
simple or an affine Kac--Moody Lie algebra -- accordingly, in short we
say $\mathfrak{g}$ is of finite or affine type -- corresponding to a
Cartan matrix $A$. We fix for $\mathfrak{g}$: a Cartan subalgebra
$\mathfrak{h}$, triangular decomposition
$\mathfrak{n}^-\oplus\mathfrak{h}\oplus\mathfrak{n}^+$, root system
$\Delta\subset \mathfrak{h}^*$, set of simple roots $\Pi=\{\alpha_i\ | \
i\in \mathcal{I}\}$ in $\Delta$ and simple co-roots
$\Pi^{\vee}=\{\alpha_i^{\vee}\ |\ i\in\mathcal{I}\}\subset\mathfrak{h}$.
Here $\mathcal{I}$ is the (fixed) common indexing set for the simple
roots, simple co-roots, rows/columns of $A$, and the set of nodes in the
Dynkin diagram of $\mathfrak{g}$. We say $\Delta$ is of finite or affine
type depending on the type of $\mathfrak{g}$.
This note showcases in finite and affine types, some of the main results
in \cite{MBWWM} and \cite{WFHMRS}, which study the root systems and
weights of arbitrary highest weight modules (see Section~\ref{S3} for the
definition) over $\mathfrak{g}$.
The results stated below were shown more generally over all
Kac--Moody algebras in the above two papers. These are inspired by and
build on the works of Chari, Khare, and their co-authors
\cite{Chari-Greenstein}, \cite{Dolbin}, \cite{KhareAdv},
\cite{KhareJ.Alg}, \cite{Ridenour} etc.
\subsection{Parabolic partial sum property and its applications}
Begin by recalling the well-known property of root systems $\Delta$, the
\textit{partial sum property (PSP)}: every root in $\Delta$ is an ordered
sum of simple roots such that all partial sums are also roots. In this
paper, we give a novel ``parabolic-generalization'', termed the
\textit{parabolic partial sum property} -- or \textit{parabolic PSP} --
which we state formally in the next section. It has many applications to
representation theory and algebraic combinatorics; this note records
three of them. Informally, the parabolic PSP says that given a nonempty
set $S \subseteq \Pi$ of simple roots, every positive root $\beta$
involving some simple roots from $S$ is an ordered sum of positive roots,
such that each root involves exactly one simple root from $S$, and all
partial sums are also roots. Observe, the case $S = \Pi$ is the ``usual''
PSP.
The parabolic PSP was originally a question of Khare, whose motivation
was to obtain a ``minimal description'' for the weights of \textit{all}
non-integrable simple highest weight $\mathfrak{g}$-modules, and more
generally, all parabolic Verma modules -- thus, the label
``parabolic''.
The simple highest weight modules over $\mathfrak{g}$ are crucial in
representation theory, combinatorics, and physics among other areas.
These modules form the building blocks of, and thereby pave ways for the
study of, modules central to Lie theory. Their weights and characters,
particularly those of \textit{integrable} (simple) highest weight
$\mathfrak{g}$-modules, have rich combinatorial properties and numerous
applications. For instance, when $\mathfrak{g}$ is of finite
type $A$, these characters are precisely the Schur polynomials.
Recall, the simple highest weight $\mathfrak{g}$-modules are indexed by
their highest weights $\lambda\in\mathfrak{h}^*$; correspondingly, let
$L(\lambda)$ denote the simple module. For a weight module $V$ --
meaning, $V$ is a direct sum of its weight spaces/the simultaneous
eigenspaces of $V$ for the action of $\mathfrak{h}$ -- let $\weight V$
denote the set of weights of $V$.
The weights and characters of the integrable simple modules $L(\lambda)$
-- for all \textit{dominant integral} $\lambda\in\mathfrak{h}^*$ -- were
well understood decades ago. However, for non-integrable simple
$\mathfrak{g}$-modules $L(\lambda)$, even their sets of weights seem to
have not been known until 2016 \cite{KhareJ.Alg}. In this paper, Khare
showed three formulas for the weights of a large class of highest weight
$\mathfrak{g}$-modules, including all modules $L(\lambda)$.
Dhillon--Khare \cite{KhareAdv} extended these formulas to hold over all
Kac--Moody $\mathfrak{g}$. Here we focus on one of these three formulas,
the Minkowski difference formula \eqref{Minkowski difference formula for
wt L(lambda)}. This uses the $\mathbb{Z}_{\geq 0}$-cone of
the positive roots lying outside a subroot system
$\Delta_{J_\lambda}$. Finding minimal generating sets for these cones
would yield a minimal description for the weights of simple highest
weight $\mathfrak{g}$-modules. In this note we prove the parabolic PSP,
thereby solving the minimal description problem.
As a second application, the analysis and understanding of the above
minimal generating sets and the parabolic PSP, were fruitful in showing
several results about weights of arbitrary highest weight
$\mathfrak{g}$-modules in \cite{MBWWM} and \cite{WFHMRS}. Namely, in
\cite{MBWWM} we obtain a Minkowski difference formula for the set of
weights of \textit{every} highest weight $\mathfrak{g}$-module, which
looks similar to those shown by Dhillon--Khare for simple modules. This
note discusses that result, as well as the ones from \cite{WFHMRS}
explained in the next subsection.
\subsection{Weak faces of root systems and weights}
The parabolic partial sum property and its two applications (above)
occupy the first half of this note. The second half is a third
application: classifying and determining two combinatorial subsets of
$\weight V$ and its convex hull, for arbitrary highest weight
$\mathfrak{g}$-modules $V$. These are the \textit{weak faces} and the
\textit{212-closed subsets} (see \cite{KhareJ.Alg} for the choice of
these names),
and they arise from the combinatorics of root systems $\Delta$, and of
subsets that maximize linear functionals over $\Delta$. Such subsets
were studied by Chari and her co-authors in \cite{Chari-Greenstein},
\cite{Dolbin}
and used to show several interesting results in representation theory.
These include constructing Koszul algebras, obtaining character formulas
for the specialization at $q=1$ of Kirillov--Reshetikhin modules over
untwisted quantum affine algebras $U_q(\widehat{\mathfrak{g}})$, and
constructing irreducible ad-nilpotent ideals in parabolic subalgebras of
$\mathfrak{g}$. For a detailed overview of these motivations,
see \cite{KhareJ.Alg} and \cite{WFHMRS}.
Khare and Ridenour \cite{Ridenour} extended the results of
\cite{Chari-Greenstein} from maximizer subsets of $\Delta$ to the weights
falling on the faces of \textit{Weyl polytopes}, i.e., the shapes $\conv
(\weight L(\lambda))$ for all dominant integral
$\lambda\in\mathfrak{h}^*$. Even more generally, Khare \cite{KhareJ.Alg}
considered $\conv (\weight V)$ for highest weight modules $V$
with arbitrary highest weight $\lambda \in \mathfrak{h}^*$,
and introduced:
\begin{definition}\label{weak face defiinition}
Let $\mathbb{A}$ be a fixed non-trivial additive subgroup of
$(\mathbb{R},+)$, and $\emptyset\neq Y\subseteq X$ be two subsets of a
real vector space. Define $\mathbb{A}_{\geq 0}:=\mathbb{A}\cap
[0,\infty)$.
\begin{enumerate}
\item $Y$ is said to be a \textbf{\textit{weak}-$\mathbb{A}$-\textit{face of}} $X$ if
\[
\begin{rcases*}
\sum\limits_{i=1}^{n}r_iy_i=\sum\limits_{j=1}^{m}t_jx_j\text{ and }\sum\limits_{i=1}^{n}r_i=\sum\limits_{j=1}^{m}t_j >0\qquad
\text{for }m,n\in\mathbb{N},\\
y_i\in Y,\text{ }
x_j\in X,\text{ }r_i,t_j\in \mathbb{A}_{\geq 0}\text{ }\forall\text{ }1\leq i\leq n,1\leq j\leq m
\end{rcases*}
\implies\text{ }x_j\in Y\ \forall t_j\neq 0.
\]
By \textit{weak faces} of $X$ we mean the collection of all weak-$\mathbb{A}$-faces of $X$ for all additive subgroups $\{0\}\subsetneqq\mathbb{A}\subseteq (\mathbb{R},+)$.
\item $Y$ is said to be a \textbf{212-\textit{closed subset of}}
\big(or \textbf{212-\textit{closed in}}\big) $X$ if
\[
(y_1)+(y_2)=(x_1)+(x_2)\ \ \text{for some }\ y_1,y_2\in Y\text{ and }x_1,x_2\in X\quad\implies\quad x_1,x_2\in Y.
\]
\end{enumerate}
\end{definition}
\begin{remark}\label{weak face is 212-closed}
For any pair of subsets $Y \subseteq X$ of a real vector space, and all
non-trivial $\mathbb{A} \subseteq (\mathbb{R},+)$, it can be checked by
the definitions that each part below implies the next: (1) $Y$ maximizes
a linear functional $\psi$ on $X$, i.e., $\psi(x)\leq \psi(y)$ for all
$x\in X$ and $y\in Y$. (2)~$Y$ is a weak-$\mathbb{R}$-face of $X$. (3)
$Y$ is a weak-$\mathbb{A}$-face of $X$. (4) $Y$ is 212-closed in $X$.
\end{remark}
For a subset $X\neq\emptyset$ of a real vector space, weak faces of $X$
generalize the ``classical'' faces of the convex hull $\conv (X)$.
Namely, when $\conv (X)$ is a polyhedron, \cite{Ridenour} shows that weak
faces of $X$ are precisely the elements of $X$ that fall on faces of
$\conv(X)$ -- i.e., conditions (1)--(3) in Remark~\ref{weak face is
212-closed} are equivalent. Our goal is to show that the same holds for
the even weaker notion (a priori) of (4) 212-closed subsets, when $X$ is
a distinguished set in Lie theory and algebraic combinatorics, i.e., of
the form $\Delta$, $\Delta \sqcup \{ 0 \}$, $\weight V$, $\conv(\weight
V)$.
\begin{remark}\label{212-closed interpretation}
There is a natural combinatorial interpretation for ``212-closed subsets
$Y \subseteq X$'' of a real vector space $\mathbb{V}$ when $X$ is the set
of lattice points in a lattice polytope -- i.e., $X$ is the intersection
of $\conv(X)$ with a lattice in $\mathbb{V}$ that contains the vertices
of $\conv(X)$. (We explain the connection to $X = \weight L(\lambda)$ in
Remark \ref{lattice polytope discussion}.) Suppose $Y$ denotes a subset
of ``colored'' (or in a contemporary spirit, ``infected'') lattice points
in $X$, with the property that if $y \in Y$ is the average of two points
in $X$, then the ``color'' or ``infection'' spreads to both points
from~$y$. (More precisely, if two pairs of -- not necessarily distinct --
points have the same average, and one pair is colored, then the color
spreads to the other pair.) We would like to understand the extent to
which the spread happens. A ``continuous'' variant works with the entire
convex hull itself, rather than the lattice points in $X$.
\end{remark}
Khare \cite{KhareJ.Alg} classified the weak faces and 212-closed subsets for $X=\weight V$, for $V$ from a large class of highest weight $\mathfrak{g}$-modules including all simple highest weight modules.
He showed that these two classes of subsets are equal, and they coincide with the sets of weights falling on the faces of the convex hull of weights. The interesting part here is:
\begin{remark}\label{lattice polytope discussion}
$\conv (\weight L(\lambda))$ is a convex polytope, and moreover,
a lattice polytope as in Remark~\ref{212-closed interpretation}; see
\eqref{Bump's question}, which explains the latter part. Khare's
(partial) results and ours below show that for \textit{all} such lattice
polytopes $\conv (\weight L(\lambda))$, the weak faces and 212-closed
subsets of $X$ are the same, and these two classes of subsets are
equivalent to the faces of $\conv(X)$. This equivalence is striking in
view of the ``minimality'' in the definition of 212-closed subsets,
particularly in contrast to the definition of (weak) faces. Furthermore,
it naturally raises the question of exploring for general lattice
polytopes, the extent to which these results hold; particularly, the
equivalence of these three notions.
\end{remark}
Recently in \cite{WFHMRS}, we have shown all of the (partial) results of
Khare \cite{KhareJ.Alg}, for important sets in Lie theory: $X= \Delta$ or
$\Delta\sqcup\{0\}=\weight \mathfrak{g}$, or $\weight V\text{ or
}\conv(\weight V)$ (the $\mathbb{R}$-\textit{convex hull} of $\weight V$)
for any highest weight module $V$, over any Kac--Moody $\mathfrak{g}$.
More precisely, for $X=\weight V$ (or $\conv (\weight V)$), we completely
classify these two classes of subsets of $X$ over any Kac--Moody algebra
$\mathfrak{g}$. More strongly, we show their equivalence with the sets of
weights on the faces (respectively, with the faces) of $\conv (\weight
V)$. For $X=\Delta\sqcup\{0\}$ or $X=\Delta$, we show the analogous
results, and that these two classes of subsets of $X$ are equal, except
in two interesting cases where $X=\Delta$ is of type either $A_2$ or
$\widehat{A_2}$. In this note, we discuss these results of \cite{WFHMRS}
for $\mathfrak{g}$ of finite and affine types.
\section{Parabolic partial sum property}
Throughout, $\Delta$ is the root system of $\mathfrak{g}$ which is either
finite-dimensional simple or an affine Kac--Moody Lie algebra;
accordingly, we say $\Delta$ is of finite type or affine type,
respectively. $\Delta^+$ denotes the set of positive roots in $\Delta$.
Recall $\mathfrak{g}$ has the root space decomposition
$\mathfrak{h}\oplus \bigoplus_{\beta\in\Delta}\mathfrak{g}_{\beta}$,
where $\mathfrak{g}_{\beta} :=\{x\in\mathfrak{g}\ | \ h x=
\langle\beta,h\rangle x\ \forall \ h\in\mathfrak{h}\}$ is the root space
corresponding to $\beta$; here $\langle\beta,h\rangle$ denotes the
evaluation of $\beta\in\mathfrak{h}^*$ at $h\in\mathfrak{h}$. We begin by
developing some notation needed to state the parabolic partial sum
property.
Let $\mathcal{I}$ index the simple roots in $\Delta^+ \subset
\Delta$.
For $\emptyset\neq I\subseteq \mathcal{I}$ we define a special (for this entire paper) subset $\Delta_{I,1}$ of positive roots, and for a vector $x=\sum_{i\in\mathcal{I}} c_i\alpha_i$ for $c_i\in\mathbb{C}$ we define two important height functions, as follows:
\begin{equation}\label{unit I-height roots}
\height(x):=\sum_{i\in\mathcal{I}}c_i, \quad\text{and}\quad
\height_I(x):=\sum_{i\in I} c_i.\qquad\quad
\Delta_{I,1}:=\left\{\beta\in\Delta\ |\ \height_I(\beta)=1\right\}\
\subseteq \Delta^+. \end{equation}
The subsets $\Delta_{I,1}$ form the minimal generating sets for the cones in weights of simple highest weight modules mentioned in the introduction:
\begin{theorem}[{\bf Parabolic Partial Sum Property}]\label{P-psp}
Let $\Delta$ be a finite or an affine root system, and fix $\emptyset\neq
I\subseteq \mathcal{I}$. Suppose $\beta$ is a positive root with
$m=\height_I(\beta)>0$. Then
\[
\text{there exist roots }\ \ \gamma_1,\ldots,\gamma_m\in\Delta_{I,1}\ \
\text{ such that }\ \ \beta=\sum_{j=1}^m\gamma_j\ \ \text{ and }\ \
\sum_{j=1}^i\gamma_j\in\Delta^+\ \forall \ 1\leq i\leq m.
\]
In other words, every root with positive $I$-height is an ordered sum of
roots, each with unit $I$-height, such that each partial sum of that
ordered sum is also a root.
\end{theorem}
When $I=\mathcal{I}$, this is precisely the usual PSP. We
next present another example:
\begin{example}\label{type A_6 example}
Let $\mathfrak{g}$ be of type $A_6$ ($7\times 7$ trace zero matrices
over $\mathbb{C}$), $\mathcal{I}=\{1,2,3,4,5,6\}$, and fix $I=\{2,4,5\}$. The Dynkin diagram for $\mathfrak{g}$ (with nodes from $I$ boxed) is:
\begin{center}
\begin{dynkinDiagram}[
*/.append style={ultra thick, blue color},edge length=0.5cm,edge/.style={black,very thick}, labels={\textcolor{black}{\textbf{1}},\boxed{\textcolor{black}{\textbf{2}}},\textcolor{black}{\textbf{3}},\boxed{\textcolor{black}{\textbf{4}}},\boxed{\textcolor{black}{\textbf{5}}},\textcolor{black}{\textbf{6}}},scale=2]{A}{******}
\end{dynkinDiagram}
\end{center}
Recall, the roots in $\Delta$ are precisely $\sum_{i\in T}\alpha_i$,
where $T \subset \mathcal{I}$ has consecutive indices.
Now,
\[
\Delta_{I,1}=\big\{\alpha_2,\ \ \alpha_1+\alpha_2,\ \ \alpha_2+\alpha_3,\
\ \alpha_1+\alpha_2+\alpha_3,\ \
\alpha_4,\ \ \alpha_3+\alpha_4,\ \ \alpha_5,\ \ \alpha_5+\alpha_6\big\}.
\]
Let $\beta=\sum_{i=1}^6\alpha_i$ denote the highest root in $\Delta$.
Check that $\height(\beta)=6$ and $\height_I(\beta)=3$. Set
$\gamma_1=\alpha_1+\alpha_2$, $\gamma_2=\alpha_3+\alpha_4$ and
$\gamma_3=\alpha_5+\alpha_6$. Observe that
$\gamma_1+\gamma_2+\gamma_3=\beta$, and moreover, both $\gamma_1$ and
$\gamma_1+\gamma_2$ are roots.
\end{example}
\begin{proof}[\textnormal{\textbf{Sketch of proof for parabolic PSP}}]
Fix $\emptyset\neq I\subseteq \mathcal{I}$ and a root $\beta$ with
$m=\height_I(\beta)>0$. The parabolic PSP trivially holds for $\beta$ if
$\height_I(\beta)=1$; hence, we now assume $\height_I(\beta)>1$. The
parabolic PSP is shown by applying the following stronger, structural
result:
\begin{theorem}\label{Lie words theorem}
Let $\mathfrak{g}$ be of finite or affine type. Fix $\emptyset\neq
I\subseteq \mathcal{I}$, and $\beta\in \Delta^+$ with
$m := \height_I(\beta)>0$. Then the root space $\mathfrak{g}_{\beta}$ is
spanned by the right normed Lie words of the form:
\begin{equation}\label{right normed Lie words}
\big[e_{\gamma_m},[\cdots, [e_{\gamma_2},e_{\gamma_1}]\cdots]\big]\
\text{ where }\gamma_t\in\Delta_{I,1},\
e_{\gamma_t}\in\mathfrak{g}_{\gamma_t}, \ \forall\ 1\leq t\leq m,\
\text{and }\sum_{t=1}^m \gamma_t=\beta.
\end{equation}
\end{theorem}
In fact, a stronger result (in \cite{MBWWM}) shows the parabolic PSP to
the best possible extent, and moreover at the level of \textit{Lie
words}, via proving Theorem \ref{Lie words theorem} for \textit{any}
general Lie algebra (over any field) graded over \textit{any} free
abelian semigroup.
Indeed, by (the stronger analogue of) Theorem \ref{Lie words theorem},
let the Lie word in \eqref{right normed Lie words} be non-zero. Then all
its right normed Lie sub-words -- namely, $\big[e_{\gamma_i},[\cdots,
[e_{\gamma_2},e_{\gamma_1}]\cdots\big]$ $\forall$ $1\leq i \leq m$ -- are
non-zero. This immediately implies that each partial sum of
$\sum_{t=1}^m\gamma_t$ is a root, proving the parabolic PSP.
Finally, the proof for Theorem~\ref{Lie words theorem} involves structure
theory ideas, and goes as follows. Fix a \textit{right normed Lie word}
$0\neq x=\big[e_{i_n},[\cdots,[e_{i_2},e_{i_1}]\cdots]\big]$ in
$\mathfrak{g}_{\beta}$, where $e_{i_t}$ is a (simple) root vector in the
simple root space $\mathfrak{g}_{\alpha_{i_t}}$ $\forall$ $1\leq t\leq n$
and $\sum_{t=1}^n\alpha_{i_t}=\beta$. Recall, every root space of
$\mathfrak{g}$ is spanned by such right normed Lie words. Inducting on
$n$, we can show that $x$ can be written as a linear combination of the
Lie words as in \eqref{right normed Lie words}. More precisely, we assume
that $\big[e_{i_{n-1}},[\cdots,[e_{i_2},e_{i_1}]\cdots\big]$ is in the
span of Lie words as in \eqref{right normed Lie words}, and then use the
Jacobi identity to take $e_{i_n}$ inside these new Lie words if
$\height_I(\alpha_{i_n})=0$.
\end{proof}
Alternately for $\mathfrak{g}$ of finite type, the parabolic PSP is
implied by:
\begin{lemma}
Assume that $\mathfrak{g}$ is of finite type, and $\emptyset\neq
I\subseteq \mathcal{I}$. Suppose $\beta\in\Delta^+$ is such that
$\height_I(\beta)>1$. Then there exists a root $\gamma\in\Delta_{I,1}$
such that $\beta-\gamma\in\Delta^+$.
\end{lemma}
\begin{proof}
We reach a contradiction, working with
$\beta$ of least height such that the lemma fails
for $\beta$. Recall, $\beta$ is a sum of
simple roots, and the Killing form on $\mathfrak{h}$, hence on
$\mathfrak{h}^*$, is positive definite, i.e.\ $(x,x)>0$ $\forall$
$x\in\mathfrak{h}^*$. Thus, fix a simple root $\alpha$ such that
$(\beta,\alpha)>0$. Now \cite[Lemma 9.4]{Humphreys} implies
$\beta-\alpha$ is a root. The assumption on $\beta$ forces
$\height_I(\alpha)=0$, so $\height_I(\beta-\alpha)=\height_I(\beta)>1$.
Since $\height(\beta-\alpha)<\height(\beta)$, we have a root
$\eta\in\Delta_{I,1}$ such that $\beta-\alpha-\eta\in\Delta^+$. Now
applying \cite[Lemma 1.1(iii)]{Dolbin} for $\beta-\alpha-\eta$, $\eta$
$\alpha$ (in place of $\alpha,\beta$, $\gamma$, respectively), we have
either: (1)~$(\beta-\alpha-\eta)+(\alpha)=\beta-\eta$ is a root, or (2)
$\alpha+\eta$ is root \big(in which case
$\alpha+\eta\in\Delta_{I,1}$\big). Both of these cases contradict the
choice of $\beta$.
\end{proof}
\section{Minkowski difference formulas for weights}\label{S3}
In this section, we state and discuss our minimal description result for
$\weight L(\lambda)$ for all weights $\lambda$ $\in\mathfrak{h}^*$, and a
Minkowski difference formula for the set of weights of an arbitrary
highest weight $\mathfrak{g}$-module, using the parabolic PSP. We need
the following notation.
For any subset $S\neq \emptyset$ of a real vector space, let
$\mathbb{Z}S$ (or $\mathbb{Z}_{\geq 0}S$) comprise the set of
$\mathbb{Z}$-linear (or $\mathbb{Z}_{\geq 0}$-linear) combinations of
elements of $S$. Let $\conv (S)$ denote the $\mathbb{R}$-convex hull of
$S$. Recall, $\mathcal{I}$ is the set of nodes in the Dynkin diagram of
$\mathfrak{g}$. Set $I^c := \mathcal{I}\setminus I$ for $I\subseteq
\mathcal{I}$. Now for $i \in \mathcal{I}$, let $s_i$ denote the simple
reflection about the hyperplane perpendicular to the simple root
$\alpha_i$. These generate the Weyl group $W$ of $\mathfrak{g}$. Let
$e_i, f_i,\alpha_i^{\vee}$ $\forall$ $i\in\mathcal{I}$ be the Chevalley
generators of $\mathfrak{g}$.
We now fix $\emptyset\neq J\subseteq \mathcal{I}$ and define the
parabolic analogues. Define $\mathfrak{g}_J$ to be the Lie subalgebra of
$\mathfrak{g}$ generated by $e_j,f_j, \alpha_j^{\vee}$ $\forall$ $j\in
J$. Since $\mathfrak{g}$ is of finite or affine type, recall that
$\mathfrak{g}_J$ is always semisimple for $J \subsetneq \mathcal{I}$. The
subalgebra $\mathfrak{g}_J$ corresponds to the Cartan matrix $A_{J\times
J}$. We denote the Cartan subalgebra, root system, the (fixed) simple
roots and simple co-roots of $\mathfrak{g}_J$ by $\mathfrak{h}_J$,
$\Delta_J$, $\Pi_J:=\{\alpha_j\}_{j\in J}$ and $\Pi_J^{\vee}$,
respectively; note that $\Delta_J=\Delta\cap \mathbb{Z}\Pi_J$. The
parabolic subgroup of $W$ generated by the simple reflections
$\{s_j\}_{j\in J}$ corresponding to $J$, is the Weyl group of
$\mathfrak{g}_J$; it is denoted by $W_J$.
Fix $\lambda\in\mathfrak{h}^*$ for this section. Recall, $\lambda$ is \textit{integral}, respectively \textit{dominant}, if $\langle\lambda,\alpha_i^{\vee}\rangle\in\mathbb{Z}$, respectively $\langle\lambda,\alpha_i^{\vee}\rangle\in\mathbb{R}_{\geq 0}$, for all $i\in\mathcal{I}$; $\lambda$ is \textit{dominant integral} if it is both. For a $\mathfrak{g}$-module $V$, recall the definitions of the $\lambda$-weight space and the set of weights of $V$:
\[
V_{\lambda}:=\{v\in M\text{ }|\text{ }h\cdot v= \langle\lambda,h\rangle v\text{ }\forall\text{ }h\in \mathfrak{h}\}\quad\text{and}\quad \weight V:=\{\mu\in\mathfrak{h}^*\text{ }|\text{ }V_{\mu}\neq\{0\}\}.
\]
$V$ is said to be a highest weight module of highest weight $\lambda$, if
there exists a vector $0\neq v\in V$ such that (1) $v$ generates $V$ over
$\mathfrak{g}$, (2) $hv=\langle\lambda,h\rangle v$ $\forall$
$h\in\mathfrak{h}$ (so $v\in V_{\lambda}\neq\{0\}$), and (3) $e_iv=0$
$\forall$ $i\in\mathcal{I}$. We call such a $v$ to be a highest weight
vector of $V$ and it is unique up to scalars. We denote the simple
highest weight module over $\mathfrak{g}_J$ with highest weight $\lambda$
\big(or rather $\lambda$ restricted to $\mathfrak{h}_J$\big) by
$L_J(\lambda)$. In this section, we deal with an important subset $
J_{\lambda}\ :=\ \{j\in \mathcal{I}\ |\
\langle\lambda,\alpha_j^{\vee}\rangle\in \mathbb{Z}_{\geq 0} \}$ of
$\mathcal{I}$
-- the \textit{integrability} of $\lambda$ (or of
$L(\lambda)$).
\begin{remark}
$L_{J_{\lambda}}(\lambda)$ is the integrable highest weight
$\mathfrak{g}_{J_{\lambda}}$-module corresponding to $\lambda$; i.e.,
$f_j$ acts nilpotently on each vector in $L_{J_{\lambda}}(\lambda)$
(nilpotently on all of $L_{J_{\lambda}}(\lambda)$ when $\mathfrak{g}$ is
of finite type) $\forall$ $j\in J_{\lambda}$. So, we know well the set
$\weight L_{J_{\lambda}}(\lambda)$. It (or equivalently, the set of nodes
$J_{\lambda}$) determines $\weight L(\lambda)$, as shown by Khare
\cite{KhareJ.Alg} and Dhillon--Khare \cite{KhareAdv}:
\begin{align}\label{Minkowski difference formula for wt L(lambda)}
\weight L(\lambda)\ =\ & \ \weight L_{J_{\lambda}}(\lambda)\ -\ \mathbb{Z}_{\geq 0}\left(\Delta^+\setminus \Delta_{J_{\lambda}}^+\right).\\
\weight L(\lambda) \ = \ & \ \conv\left(\weight L(\lambda)\right) \cap \left(\lambda-\mathbb{Z}_{\geq 0}\Pi\right).\label{Bump's question}
\end{align}
\end{remark}
\noindent Observe by formula \eqref{Minkowski difference formula for wt
L(lambda)} that $\conv (\weight L(\lambda))$ equals the Minkowski
difference between $\conv\big(\weight L_{J_{\lambda}}(\lambda)\big)$ and
the real cone $\mathbb{R}_{\geq 0}\big(\Delta^+\setminus
\Delta^+_{J_{\lambda}}\big)$. Formula \eqref{Bump's question}
affirmatively answers a question of Daniel Bump on recovering weights
from their convex hulls for these simple $\mathfrak{g}$-modules. Formula
\eqref{Minkowski difference formula for wt L(lambda)} is the Minkowski
difference formula mentioned in the introduction, expressing $\weight
L(\lambda)$ in terms of two well-understood subsets. The parabolic
partial sum property was posed in order to determine the minimal
generators for the (non-negative integer) cone in \eqref{Minkowski
difference formula for wt L(lambda)}, which is generated by all the
positive roots outside $\Delta^+_{J_{\lambda}}$; these are found in the
next theorem. Recall the definition of $\Delta_{J_{\lambda}^c, \ 1}$ from
\eqref{unit I-height roots}.
\begin{theorem}
The cone $\mathbb{Z}_{\geq 0}
\left(\Delta^+\setminus\Delta_{J_{\lambda}}^+\right)$ is minimally
generated (over $\mathbb{Z}_{\geq 0}$) by $\Delta_{J_{\lambda}^c, \ 1}$,
which is always finite if $\Delta$ is of finite or affine type.
Therefore, we have the following minimal description:
\begin{equation}
\weight L(\lambda)\ =\ \weight L_{J_{\lambda}}(\lambda)-\mathbb{Z}_{\geq
0}\Delta_{J_{\lambda}^c,\ 1}, \qquad \forall \lambda \in \mathfrak{h}^*.
\end{equation}
\end{theorem}
\begin{proof}[Sketch of proof]
Notice that $\Delta_{J_{\lambda}^c,\ 1}\subseteq \Delta^+\setminus
\Delta_{J_{\lambda}}^+$. Conversely, if $\beta\in
\Delta^+\setminus\Delta^+_{J_{\lambda}}$, then
$\height_{J_{\lambda}^c}(\beta)>0$. Now the partial sum property implies
$\beta\in \mathbb{Z}_{\geq 0}\Delta_{J_{\lambda}^c,\ 1}$. Therefore,
$\mathbb{Z}_{\geq 0}\left(\Delta^+\setminus
\Delta_{J_{\lambda}}^+\right)=\mathbb{Z}_{\geq 0}\Delta_{J_{\lambda}^c, \
1}$. It remains to show the minimality -- or irredundancy --
of $\Delta_{J_{\lambda}^c,\ 1}$. Suppose there is a root
$\gamma\in\Delta_{J_{\lambda}^c,\ 1}$ such that
$\gamma=\sum_{i=1}^n\gamma_i$ for some roots $\gamma_1,\ldots,\gamma_n\in
\Delta^+\setminus \Delta_{J_{\lambda}}^+$. Comparing the
$J_{\lambda}^c$-heights on both sides, it follows that $n=1$ and
$\gamma_1=\gamma$.
\end{proof}
Next, we give our formula for the weights of an arbitrary highest weight
$\mathfrak{g}$-module $V$ of highest weight $\lambda\in\mathfrak{h}^*$.
We define for convenience $\weight_{J_{\lambda}} V:=
(\lambda-\mathbb{Z}_{\geq 0}\Pi_{J_{\lambda}})\cap \weight V$, which is
the set of weights of the $\mathfrak{g}_{J_{\lambda}}$-module generated
by a highest weight vector in $V$.
\begin{theorem}\label{wt V formula theorem}
Let $\lambda\in\mathfrak{h}^*$, and $V$ be a highest weight $\mathfrak{g}$-module with highest weight $\lambda$. Then
\begin{equation}\label{Minkowski difference formula for wt V}
\weight V\ =\ \weight _{J_{\lambda}}V-\mathbb{Z}_{\geq 0}\Delta_{J_{\lambda}^c,\ 1}\ =\ \weight _{J_{\lambda}}V-\mathbb{Z}_{\geq 0}(\Delta^+\setminus \Delta_{J_{\lambda}}^+).
\end{equation}
\end{theorem}
\begin{remark}
When $V=L(\lambda)$, \eqref{Minkowski difference formula for wt V} is the same as the formula \eqref{Minkowski difference formula for wt L(lambda)} by Dhillon and Khare, as $\weight_{J_{\lambda}}L(\lambda)=\weight L_{J_{\lambda}}(\lambda)$.
The significance of the formula \eqref{Minkowski difference formula for wt V} is as follows:
\noindent The cone in it is well understood, and the unknown part is
$\weight_{J_{\lambda}}V$.
In view of this, if we find the sets of weights of highest weight $\mathfrak{g}$-modules with dominant integral highest weights, then we have the formulas for weights of all highest weight $\mathfrak{g}$-modules.
\end{remark}
\begin{proof}[Sketch of proof of Theorem \ref{wt V formula theorem}]
The inclusion $\weight V\subseteq \weight
_{J_{\lambda}}V-\mathbb{Z}_{\geq 0}\Delta_{J_{\lambda}^c,\ 1}$ follows by
\cite[Corollary 3.2(b)]{MBWWM}, which involves the
Poincar\'{e}--Birkhoff--Witt theorem and the parabolic PSP.
To show the reverse inclusion in the previous
sentence, recall that \cite[Lemma 4.2]{MBWWM} says $\weight
_{J_{\lambda}}V-\mathbb{Z}_{\geq 0}\Pi_{J_{\lambda}^c}\subseteq \weight
V$. Using this inclusion as the base step, for any
$\gamma_1,\ldots,\gamma_n\in\Delta_{J_{\lambda}^c,\ 1}$, we induct on
$\height_{ J_{\lambda}^c}\left(\sum_{i=1}^n\gamma_i\right)$ to prove:
$\weight_{J_{\lambda}}V-\mathbb{Z}_{\geq 0}\sum_{i=1}^n\gamma_i\
\subseteq \weight V$, which finishes the proof. Showing this crucial step
is tremendously simplified if one works with the generating
set $\Delta_{J_{\lambda}^c,\ 1}$ for $\mathbb{Z}_{\geq
0}\left(\Delta^+\setminus \Delta^+_{J_{\lambda}}\right)$ instead of
$\Delta^+\setminus \Delta^+_{J_{\lambda}}$.
\end{proof}
\section{Weak faces and 212-closed subsets}
Recall the definitions of weak faces and 212-closed subsets from
Definition \ref{weak face defiinition}. Throughout, $\mathfrak{g}$ (or
equivalently, $\Delta$) is of finite or affine type, $\mathbb{A}\neq
\{0\}$ is a fixed arbitrary additive subgroup of $\mathbb{R}$, and $V$ is
an arbitrary highest weight $\mathfrak{g}$-module of highest weight
$\lambda\in\mathfrak{h}^*$. The symbol $Y$ always denotes a weak face or
a 212-closed subset of $X$, where $X=\Delta$ or $\Delta\sqcup\{0\}$ or
$\weight V$ or $\conv(\weight V)$. The main results of this section are
as follows. Theorem \ref{weak faces of roots main theorem} completely
determines as well as discusses the equivalence of the 212-closed subsets
and weak faces both for $X=\Delta$ and $X=\Delta\sqcup\{0\}$. It turns
out that $X=\Delta$ of types $A_2$ or (affine) $\widehat{A_2}$ are the
only two exceptions for which these two classes of subsets are not equal.
In Theorem \ref{weak faces of weights main theorem}: 1) we show for
$X=\weight V$ that these two classes of subsets are the same as the sets
of weights falling on the faces of $\conv(\weight V)$, and 2) for
$X=\conv(\weight V)$ these two notions are the same as the usual faces.
Let us first understand a geometric interpretation of 212-closedness.
\begin{remark}\label{geometric interpretation of 212-closedness}
In the definition of a 212-closed set, consider the equality $(y_1) +
(y_2) = (x_1) + (x_2)$ for $y_1,y_2\in Y$ and $x_1,x_2\in X$ implying
$x_1,x_2\in Y$. This equality arises notably when: (1) $y_1=y_2$ is the
midpoint between $x_1$ and $x_2$. (2) $x_1=x_2$ is the midpoint between
$y_1$ and $y_2$. (3) $y_1, y_2, x_1, x_2$ form vertices of a
parallelogram with $y_1, y_2$ (similarly, $x_1, x_2$) on a diagonal.
\end{remark}
Using this remark, we now look at the 212-closed subsets of $\Delta$ for
the example below:
\begin{example}\label{212-closed subsets in type A_2 example}
Let $\Delta$ be of type $A_2$, with the simple roots $\alpha_1$ and
$\alpha_2$. $\Delta=\{\pm \alpha_1,\ \pm\alpha_2$,
$\pm(\alpha_1+\alpha_2) \}$ is the set of vertices of the hexagon in
Figure \eqref{A_2 root system} below, where $\alpha_1=(1,0)$ and
$\alpha_2=(-1/2, \sqrt{3}/2)$. The figure in $\eqref{A_2 root polytope}$
is the $\mathbb{R}$-convex hull of $\Delta$.
\begin{figure}[h]
\centering
\begin{subfigure}{.4\linewidth}
\begin{tikzpicture}[scale=1.5]
\draw
(cos 0, sin 0) --
(cos 60, sin 60) --
(cos 120, sin 120) --
(cos 180, sin 180) --
(cos 240, sin 240) --
(cos 300, sin 300) --
cycle;
\filldraw[black](cos 0,sin 0) circle (2pt) node[anchor=west]{$(1,0)$};
\filldraw[black](cos 60,sin 60) circle (2pt) node[anchor=west]{$\left(\frac{1}{2},\frac{\sqrt{3}}{2}\right)$};
\draw (0,0) node{(0,0)};
\filldraw[black](cos 120,sin 120) circle (2pt) node[anchor=east]{$\left(-\frac{1}{2},\frac{\sqrt{3}}{2}\right)$\hspace{3pt}};
\filldraw[black](cos 180,sin 180) circle (2pt) node[anchor=east]{$(-1,0)$};
\filldraw[black](cos 240,sin 240) circle (2pt) node[anchor=east]{$\left(-\frac{1}{2},-\frac{\sqrt{3}}{2}\right)$};
\filldraw[black](cos 300,sin 300) circle (2pt) node[anchor=west]{$\left(\frac{1}{2},-\frac{\sqrt{3}}{2}\right)$};
\end{tikzpicture}
\caption{Type $A_2$ root system}\label{A_2 root system}
\end{subfigure}
\begin{subfigure}{.4\linewidth}
\begin{tikzpicture}[scale=1.5]
\filldraw[fill=blue!70]
(cos 0, sin 0) --
(cos 60, sin 60) --
(cos 120, sin 120) --
(cos 180, sin 180) --
(cos 240, sin 240) --
(cos 300, sin 300) --
cycle;
\filldraw[black](cos 0,sin 0) circle (2pt) node[anchor=west]{$\alpha_1$};
\filldraw[black](cos 60,sin 60) circle (2pt) node[anchor=west]{$\alpha_1+\alpha_2$};
\draw (0,0) node{(0,0)};
\filldraw[black](cos 120,sin 120) circle (2pt) node[anchor=east]{$\alpha_2$};
\filldraw[black](cos 180,sin 180) circle (2pt) node[anchor=east]{$-\alpha_1$};
\filldraw[black](cos 240,sin 240) circle (2pt) node[anchor=east]{$-\alpha_1-\alpha_2$};
\filldraw[black](cos 300,sin 300) circle (2pt) node[anchor=west]{$-\alpha_2$};
\end{tikzpicture}
\caption{Type $A_2$ root polytope}\label{A_2 root polytope}
\end{subfigure}
\caption{The $A_2$ case}
\end{figure}
$\Delta$ is 212-closed in itself trivially. By Remark \ref{geometric
interpretation of 212-closedness}(1), since no root lies between
two others, all singletons in $\Delta$ are 212-closed.
By Remark \ref{geometric
interpretation of 212-closedness} the following are all the rest of the 212-closed subsets in $\Delta$:
(i) non-antipodal pairs of points;
(ii) any subset of 3 vertices, such that in their induced subgraph none
of them are isolated, or all of them are isolated.
Next about the 212-closed subsets of $\Delta\sqcup\{0\}$, once again by
Remark \ref{geometric
interpretation of 212-closedness} points (1) and (3), if $Y$ 212-closed in $\Delta\sqcup\{0\}$ is such that $Y$ contains $0$ or a
pair of non-adjacent vertices, then $Y=\Delta\sqcup\{0\}$.
Theorem \ref{weak faces of weights main theorem}(b), and the formulas
given for the faces and for weights falling on the faces \eqref{standard
face}, can be verified/understood with the aid of the convex set in
Figure \eqref{A_2 root polytope}, which equals
$\conv(\weight L(\alpha_1+\alpha_2))$.
\end{example}
We now generalize below some of the notable observations in this example.
\begin{note}\label{all 212-closed sets are proper}
For $X=\Delta$ or $\Delta\sqcup\{0\}$, throughout, every weak face and
every 212-closed subset $Y$ of $X$ will be assumed to be a proper subset
of $X$. This is to overcome the obstruction to our uniform comparisons
of these notions: $\Delta$ is 212-closed (also a weak-$\mathbb{A}$-face
for any $\mathbb{A}$) in $\Delta$, but not in $\Delta\sqcup\{0\}$, since
if $Y=\Delta$ is 212-closed in $\Delta\sqcup\{0\}$, then
\begin{equation}\label{Delta is not 212-closed in Delta U 0}
\text{for any }\xi\in Y=\Delta,\quad (\xi)+(-\xi)=2(0)\quad \implies 0\in \Delta \ \Rightarrow\!\Leftarrow.
\end{equation}
\end{note}
\begin{remark}\label{0 not in 212-closed subsets}
Let $Y$ be 212-closed in $X=\Delta$ or $\Delta\sqcup\{0\}$. Since
$Y\subsetneqq X$, observe via the reversed equation before the
implication in \eqref{Delta is not 212-closed in Delta U 0} that $0\notin
Y$. This has two further implications:
(1) For any root $\xi$, both $\xi$ and $-\xi$ cannot belong to $Y$.
(2) Every 212-closed subset of $\Delta\sqcup\{0\}$ is contained in
$\Delta$, and therefore (by the definition) it is 212-closed in $\Delta$.
These assertions hold equally well for weak faces, since they are
212-closed.
\end{remark}
Let $\mathfrak{g}$ be of finite type, with highest long root $\theta$. As
$\mathfrak{g}$ and $L(\theta)$ are isomorphic,~$\weight \mathfrak{g}$ =
$\Delta\sqcup\{0\}=\weight L(\theta)$. So \cite[Theorem C]{KhareJ.Alg},
which finds and shows the equivalence of all the weak faces and
212-closed subsets of $X=\weight L(\lambda)$ $\forall$
$\lambda\in\mathfrak{h}^*$, applied for $\lambda=\theta$~yields:
\begin{proposition}
For $\mathfrak{g}$ of finite type, the following classes of subsets are
all the same: 1) 212-closed subsets of $\Delta\sqcup\{0\}$, 2) weak faces
of $\Delta\sqcup\{0\}$, 3) the sets of weights falling on the faces of
$\conv(\Delta\sqcup\{0\})$, 4) maximizer subsets for linear functionals
on $\Delta\sqcup\{0\}$. These are precisely:
\begin{equation}\label{standard face}
w\left[\big(\theta-\mathbb{Z}_{\geq 0}\Pi_I\big)\cap \weight
L(\theta)\right]\quad\quad \text{for all }w\in W\text{ and }I\subsetneqq
\mathcal{I}.
\end{equation}
\end{proposition}
\begin{remark}
Recall, the convex hulls of the subsets in \eqref{standard face} are all
the faces of the root polytope $\conv(\Delta)$ from Borel--Tits
\cite{Borel}, Satake \cite{Satake}, Vinberg \cite{Vinberg}; for instance
see \cite[Theorem 2.17]{KhareJ.Alg} which quotes this result. Let
$\omega_i$ $\forall$ $i\in\mathcal{I}$ be the fundamental dominant
integral weights in $\mathfrak{h}^*$. The set in \eqref{standard face}
maximizes the linear functional $\left(w\sum_{j\in I^c} \omega_j\
,-\right)$. So by Remark \ref{weak face is 212-closed}, it is a
weak-$\mathbb{A}$-face (also 212-closed) in $\Delta\sqcup\{0\}$ for every
$\emptyset\neq \mathbb{A}\subseteq (\mathbb{R},+)$.
\end{remark}
Now assume that $\Delta$ is of affine type and
$\mathcal{I}=\{0,1,\ldots,\ell\}$, where $\ell$ is the rank of $A$ or
$\Delta$. A root $\beta$ is real if $(\beta,\beta)>0$, and imaginary if
$(\beta,\beta)=0$; every root is either real or imaginary. $W\Pi$ (the
$W$ orbit of all the simple roots) are all the real roots. The imaginary
roots are $\{n\delta\ |\ n\in\mathbb{Z}\setminus\{0\}\}$, where $\delta$
is the smallest positive imaginary root. In a 212-closed subset of
$\Delta$, every root is real by the remark below.
\begin{remark}
Let $Y$ be 212-closed in $\Delta$ of affine type and $\eta\in\Delta$
imaginary. Then $\eta\notin Y$:
\begin{equation}
\text{If }\eta\in Y,\quad \text{then }\ 2(\eta)=(3\eta)+(-\eta) \ \text{
implies }\ \pm \eta\in Y \ \ \Rightarrow\!\Leftarrow\ \text{(by Remark
\ref{0 not in 212-closed subsets})}.
\end{equation}
\end{remark}
Recall from Tables Aff 1--Aff 3 and Subsection 6.3 of Kac's
book~\cite{Kac}: (1)~the subroot system $\mathring{\Delta}$ generated by
$\{\alpha_1,\ldots,\alpha_{\ell}\}$ is of finite type; (2) the roots in
$\Delta$ are explicitly describable in terms of the roots in
$\mathring{\Delta}$. Recall, there are at most two lengths in a finite
type root system. Let $\mathring{\Delta}_s$ and $\mathring{\Delta}_l$
denote the set of roots in $\mathring{\Delta}$ of the shortest and
longest lengths, respectively. Note,
$\mathring{\Delta}_s=\mathring{\Delta}_l$ when $\mathring{\Delta}$ is of
(simply laced) types $A_n,D_n,E_6,E_7,E_8$.
With the above preliminaries and observations, we are ready to state our
next main theorem, with one last notation:
For $\Delta$ of type $X_{\ell}^{(r)}$, $r\in\{1,2,3\}$ -- see
\cite[Tables Aff 1--Aff 3]{Kac} -- and $Y\subseteq \Delta\sqcup\{0\}$, define
\begin{equation}\label{E2.4}
Y_s:=\begin{cases}
(Y\cap\mathring{\Delta}_s)+\mathbb{Z}\delta &\text{if } Y\cap \mathring{\Delta}_s\neq\emptyset,\\
\emptyset &\text{if }Y\cap\mathring{\Delta}_s=\emptyset,
\end{cases}\quad Y_l:=\begin{cases}
(Y\cap\mathring{\Delta}_l)+r\mathbb{Z}\delta &\text{if } Y\cap \mathring{\Delta}_l\neq\emptyset,\\
\emptyset &\text{if }Y\cap\mathring{\Delta}_l=\emptyset.
\end{cases}
\end{equation}
\begin{theorem}\label{weak faces of roots main theorem}
\begin{itemize}
\item[(a)] For $\mathfrak{g} \neq A_2$ of finite type, the following classes of subsets are all the same: 212-closed subsets of $\Delta$, 212-closed subsets of $\Delta\sqcup\{0\}$, weak faces of $\Delta$, weak faces of $\Delta\sqcup\{0\}$, and the subsets in \eqref{standard face}.
\item[(b)] For $\mathfrak{g}$ of type $A_2$, the following classes of subsets are all the same: 212-closed subsets of $\Delta\sqcup\{0\}$, weak faces of $\Delta$, weak faces of $\Delta\sqcup\{0\}$, and the subsets in \eqref{standard face}. All of these subsets and the following additional ones, are all the 212-closed subsets of $\Delta$. \begin{equation}
\label{non standard 212-closed subsets}
W\text{-conjugates of}:\ \Pi=\{\alpha_1,\alpha_2\},\
\Delta^+=\{\alpha_1,\alpha_2,\alpha_2+\alpha_1\},\
\{\alpha_1,\alpha_2,-\alpha_2-\alpha_1\}.\end{equation}
\item[(c)] Assume that $\mathfrak{g}$ is of affine type. Then the following two statements 1) and 2) are equivalent:
\begin{itemize}
\item[1)] $Y$ is a 212-closed subset of $\Delta$
\big(respectively, of $\Delta\sqcup\{0\}$\big).
\item[2)] $Y=Z_s\cup Z_l$ for some 212-closed subset $Z$
of $\mathring{\Delta}$ \big(respectively, of
$\mathring{\Delta}\sqcup\{0\}$\big).
\end{itemize}
For $\mathfrak{g}$ not of type $\widehat{A_2}$ (so
$\mathring{\Delta}$ is not of type $A_2$) -- similar to part
(a) -- we have the equality of the four classes: 212-closed
subsets and weak faces of both $\Delta$ and $\Delta\sqcup\{0\}$.
When $\mathfrak{g}$ is of type $\widehat{A_2}$
($\mathring{\Delta}$ is of type $A_2$) -- as in part (b) -- these
classes other than the 212-closed subsets of $\Delta$ are the
same. The additional 212-closed subsets of $\Delta$ correspond to
those of $\mathring{\Delta}$ in \eqref{non standard 212-closed
subsets}.
\end{itemize}
\end{theorem}
\begin{proof}[Ideas in proof]
In \cite{WFHMRS}, each part of the theorem is shown in cases. Part (a)
for types $A_1$, $B_2$ and $G_2$, as well as part (b), can be verified
alternately via looking at their pictures/plots and the observations
similar to those in Example \ref{212-closed subsets in type A_2 example}.
In particular, in this manner, for $\Delta$ of type $A_2$, the
disjointedness of the two lists in \eqref{standard face} and \eqref{non
standard 212-closed subsets} can be verified. If $Y$ belongs to the list
in \eqref{non standard 212-closed subsets}, then $Y$ is not a weak face
of $\Delta$ of type $A_2$, because
\[
\text{for any } \mathbb{A}\subseteq (\mathbb{R},+)\text{ and }0<
a\in\mathbb{A},\quad a(\alpha_1)+a(\alpha_2) = a(\alpha_1+\alpha_2)+(0)\
\implies 0\in Y,
\]
which is a contradiction. The proof in \cite{WFHMRS} of part (c), where
$\Delta$ is of affine type, runs over four steps. In all of them, we
heavily use the description of the roots given by the well-known result
\cite[Proposition 6.3]{Kac}. This description and the definition of
212-closed sets immediately give the equivalence of points 1) and 2) in
part (c). This close relation between the 212-closed subsets of $\Delta$
and $\mathring{\Delta}$ leads to the equivalences of the various classes
of sets in part (c).
\end{proof}
We conclude with our final main theorem for the set of weights and their
convex hull, for an arbitrary highest weight $\mathfrak{g}$-module $V$.
When $V=L(\lambda)$, part (a) of our theorem below recovers \cite[Theorem
C]{KhareJ.Alg} of Khare. We define the \textit{integrability} of $V$ to
be $I_V:=\{i\in \mathcal{I}\ |\ f_i \text{ acts nilpotently on each
vector in }V\text{ or equivalently on}\ V_{\lambda}\}$.
Recall, $V$ is an integrable $\mathfrak{g}_{I_V}$-module, and so $\weight
V$ and $\conv(\weight V)$ are $W_{I_V}$-invariant. $I_V$ determines
$\weight V$ to a significant extent, and $\conv(\weight V)$ completely,
by \cite{KhareJ.Alg} and \cite{KhareAdv}.
\begin{theorem} \label{weak faces of weights main theorem}
Let $V$ be an arbitrary highest weight $\mathfrak{g}$-module of
highest weight $\lambda\in\mathfrak{h}^*$. Then:
\begin{itemize}
\item[(a)] The following classes of subsets are equal:
(1)~Weak faces of $\weight V$.
(2)~212-closed subsets of $\weight V$.
(3)~The subsets
\begin{equation}\label{standard faces of wt V}
w[(\lambda-\mathbb{Z}_{\geq 0}\Pi_I)\cap\weight V]\qquad \text{for all }\ w \in W_{I_V}\ \text{ and }\ I\subseteq \mathcal{I}.
\end{equation}
\item[(b)] The following classes of subsets of $X =
\conv_{\mathbb{R}} (\weight V)$ are equal:
(1)~Exposed faces, i.e., the maximizer subsets of $X$ with
respect to linear functionals.
(2)~Weak faces of $X$.
(3)~212-closed subsets of $X$.
(4)~The convex hulls of the subsets in \eqref{standard faces of wt V}.
\end{itemize}
\end{theorem}
|
0907.1317
|
\section{Model}
The original Vicsek model is extended to incorporate the influence of the earlier and later-coming
leaders: a small proportion of the whole group of $N$-individuals is
given a preferred motion direction representing, for example, the
direction to a known food resource or a migration target, or the
faithfulness for one political belief or commodity brand.
In our model there are three types of
individuals: i) $N_r$ earlier-coming leaders moving rightwards,
namely, $\mathcal{N}_r$, whose dynamics are
$x_{r_i}(t+1)=x_{r_i}(t)+v\measuredangle 0^o, ~i=1,\cdots, N_r$; ii)
$N_l$ minority later-coming leaders moving leftwards, namely,
$\mathcal{N}_l$, whose dynamics are
$x_{l_i}(t+1)=x_{l_i}(t)+v\measuredangle 180^o, ~i=1,\cdots, N_l$
with $N_l\leq N_r$; and iii) $N_f$ uninformed individuals, namely,
$\mathcal{N}_f$, whose dynamics are updated by $x_{f_i}
(t+1)=x_{f_i}(t)+v\measuredangle \theta_{f_i}(t)$, $i=1,\cdots,N_f$,
with $N_f= N-N_r-N_l$. Here, $x_i$ denotes the position of
individual $i$, and the leaders move without being affected the
others. $\mathcal{N}_f$ are naive and have no preference in
any particular direction but just follow the average directions
$\theta_{f_i}(t)$ of their neighbors, and cannot differentiate the
leaders and the followers. Note that, at the beginning, there are
just $\mathcal{N}_r$ and $\mathcal{N}_f$ and once the
orientation of $\mathcal{N}_f$ has completely aligned to
$\mathcal{N}_r$, $\mathcal{N}_l$ appears to compete with
$\mathcal{N}_r$ to reverse the orientation of $\mathcal{N}_f$.
The velocity of the $f_i$-th follower, i.e. $v_{f_i} (t)$, has a
constant speed $v$ and a direction $ \theta
(t+1)=\left<\theta\left(t\right)\right>_r$, where
$\left<\theta\left(t\right)\right>_r$ denotes the average direction
of individuals within a circle of radius $r$ surrounding individual
$i$ (including itself) \cite{vi95}.
This kind of aligning mechanism can
nicely mimic the local dynamics of ``\emph{go
with the stream}" in both bio-groups and human society. To
focus on the effects of the leaders, we do not consider the influence of
the external noise.
Here, without loss of generality, we set $N=500$ and $L=10$ with
periodic boundary conditions, and $r=1$, $v=0.03$ as Ref.
\cite{vi95}.
The global orientation of the whole group is defined as normalized steady-state
alignment index $V_m=1-\theta_a/(\pi /2)$, where $\theta_a$ denotes
the steady-state direction of the whole group, thus the values $1$,
$-1$ and $0$ of $V_m$ mean that the whole group is completely following
$\mathcal{N}_r$, $\mathcal{N}_l$ and no preference in direction,
respectively.
\begin{figure}[htp]
\centering
\resizebox{5.0cm}{!}{\includegraphics[width=12.0cm]{factors.eps}}
\caption{Illustration of effective leadership factors
\emph{\textbf{F1}} and \emph{\textbf{F2}} by normal length $\eta$
and clustering factor $\sigma$, whose definitions will be given
later. Here, $\| x \|_{2}=\sqrt{x^{T}x}$.}
\label{fig: factors}
\end{figure}
Now recall the key problem this Letter addresses: \emph{is it
possible for $\mathcal{N}_l$ to defeat $\mathcal{N}_r$?}
Questions closely relevant to this issue have already kindled up the
interests of not only physicists and biologists but also social
scientists and marketing researchers for years \cite{co05,su08}. In
these studies, it is generally drawn that the followers would be apt
to follow the majority leaders since in the alignment models
each individual follows the average direction of its
neighbors. Then, it can be seen that, if the majority and the
minority have the same geographical distribution pattern, the
majority has either larger influence area or higher particle density
which intensifies their leadership. Thereby, it is natural to
deduce that decisive parameter(s) for effective leadership may be
not the absolute number of the leaders but the following two key,
but competing factors: \emph{\textbf{F1}}) \emph{Effective Range}:
to distribute leaders' influence to as many followers as possible
within a given time; \emph{\textbf{F2)}} \emph{Persuasive
Intensity}: to be sufficiently persuasive to govern the followers
they can influence with high density. In other words, minority leaders may defeat the
majority ones provided that they have better \emph{\textbf{F1}} or
\emph{\textbf{F2}} or both, and our work will verify such a
hypothesis. In our scenario given above, \emph{\textbf{F1}} and
\emph{\textbf{F2}} can be quantified by the normal length $\eta$ and
clustering factor $\sigma$, as shown in Fig.~\ref{fig: factors}.
Obviously, these two factors are however somewhat contradictive
since \emph{\textbf{F1}} requires the leaders to distributed
sparsely into the followers' region while \emph{\textbf{F2}} favors
highly condensed leader groups, and this contradiction constitutes
the main challenge of the problem.
\begin{figure}[htp]
\centering
\resizebox{8.3cm}{!}{\includegraphics[width=5.0cm]{pattern.eps}}
\caption{(Color online) Typical distribution patterns of leaders. }
\label{fig: pattern}
\end{figure}
To address this interesting question, we have implemented
simulations with $\mathcal{N}_r$ distributed in the $L\times L$
square (Fig.~\ref{fig: pattern}(a)), as considered in most previous works.
We then examine the effectiveness of $\mathcal{N}_l$ leader group on
various distribution patterns, such as the $L\times L$ square, $L\times L$ diagonal,
$L\times 1$ vertical, $1\times L$ horizontal and $1\times 1$ dotted
regions as shown in Figs.~\ref{fig: pattern}(a)--(e), respectively.
According to the two factors \emph{\textbf{F1}} and
\emph{\textbf{F2}} of effective leadership and taking into
consideration of the moving directions of $\mathcal{N}_l$ and
$\mathcal{N}_r$, one can expect that $L\times 1$ outperforms
$L\times L$ because they have the identical \emph{\textbf{F1}} but
$L\times 1$ favors \emph{\textbf{F2}}. Analogously, $L\times 1$ can
also be expected to be superior to $1\times L$ since they have the same
\emph{\textbf{F2}} but $L\times 1$ have better \emph{\textbf{F1}}.
However, it is difficult to compare $L\times 1$ and $1\times 1$,
since $L\times 1$ has better \emph{\textbf{F1}} while $1\times 1$
greatly favors \emph{\textbf{F2}}. Thus, one has to resort to
numerical simulations to reveal more concrete rules behind.
\begin{figure}[htp]
\centering
\resizebox{7.0cm}{!}{\includegraphics[width=12.0cm]{5key.eps}}
\vspace*{-2.1cm} \caption{Orientation reversion of the mass
followers under the later-coming leaders $\mathcal{N}_l$ obeying 5
typical different distribution patterns given in Fig.~\ref{fig:
pattern}. The follower group $\mathcal{N}_f$ has completely aligned
to the leaders $\mathcal{N}_r$ when $\mathcal{N}_l$ sets in at
$t=0$.
Here, $N_r=N_l=10$.
Each point is an average over $1000$ independent
runs for this and the following figures.}
\label{fig: distribution}
\end{figure}
Four more interesting and concrete phenomena are
observed from Fig.~\ref{fig: distribution}: i) $L\times 1$ vertical
and $1\times 1$ dotted patterns (Figs.~\ref{fig: pattern}(c) and
(e)) can defeat the earlier-coming $L\times L$ square pattern and
reverse the orientation of the followers, while $1\times L$
horizontal, $L\times L$ square and $L\times L$ diagonal patterns
(Figs.~\ref{fig: pattern}(a), (b) and (d)) not. ii) $L\times 1$
vertical and $1\times L$ horizontal patterns are the most and least
effective ones, respectively; iii) $L\times L$ diagonal distribution
is a little bit better than the $L\times L$ square distribution; iv)
$1\times 1$ dotted pattern is the second most effective one (just
below $L\times 1$). Inspiringly, for $\mathcal{N}_l$ adopting a
dotted distribution $1\times 1$, it takes
considerable running steps to propagate its influence to remote
followers, so that the converging time is much longer.
These simulation results help us understand more deeply
the nature of \emph{\textbf{F1}}and \emph{\textbf{F2}}.
Specifically, in our scenario,\emph{\textbf{F1}} has been realized
by spreading $\mathcal{N}_l$ out sufficiently \emph{perpendicularly} to
their movement direction, while a condensed distribution corresponding to
favorable \textbf{\emph{F2}} could first slave the followers locally and then
propagate the influence to the whole population.
\begin{figure}[htp]
\centering
\begin{tabular}{cc}
\hspace*{-0.3cm}
\resizebox{4.8cm}{!}{\includegraphics[width=12.0cm]{Fig4_Index.eps}}
& \hspace*{-0.8cm}
\resizebox{4.8cm}{!}{\includegraphics[width=12.0cm]{Fig4_Fit_Real.eps}}
\end{tabular}
\vspace*{-0.6cm}\caption{(Color online) Effective leadership for $\mathcal{N}_l$ vs
$\mathcal{N}_r$ leaders with different distribution patterns $L\times 1$ vertical and
$L\times L$ square, respectively.
(a) $V_m$ from the model simulations. (b) $V_{m1}$ from Eq. (1).
}
\label{fig: following the majority or minority}
\end{figure}
To this end, one can be delighted to infer that even if
$\mathcal{N}_f$ have been completely dominated by $\mathcal{N}_r$,
it is highly possible for the minority $\mathcal{N}_l$ to reverse
the followers' opinion with their better distribution pattern. Indeed, as
demonstrated clearly in Fig.~\ref{fig: following the majority or minority} (a),
in a large region above the white dashed line where $N_l<N_r$, the whole
group reverses the orientation from $V_m=1$ to $V_m=-1$ (blue region) to
follow the leadership of the minority $\mathcal{N}_l$ which has a better distribution pattern
($L\times 1$ vertical) compared to the majority ($L \times L$ square).
Similar results are observed for the other favorable $1 \times 1$ dot pattern of $N_l$.
The numerical simulations suggest that the two factors
\textbf{\emph{F1}} (influencing area) and \emph{\textbf{F2}}
(clustering intensity) determining the leadership performance can be
quantified as below. As shown in Fig.~\ref{fig:
factors}, \emph{\textbf{F1}} can be represented by the length of
the leaders' distribution region perpendicular to the movement
direction of the leader, namely the \emph{normal length} $\eta$. The
clustering intensity \emph{\textbf{F2}} can be characterized by the reciprocal of the average geographical distance among the leaders,
namely the \emph{clustering factor} $\sigma$.
We find that for the number of leaders $N_l$,
the average geographical distance between the first $2N_l $ nearest pairs of leaders
can sensitively distinguish various patterns discussed in Fig. 2 and Fig. 5.
More precisely, for the $N_l$ leaders,
$\sigma_{l}=1/(\frac{1}{2N_{l}}\sum_{i,j\in \mathcal{N}_{l},~j\neq
i,\|x_i-x_j\|_{2} \leq\bar{d}_{2N_{l}}}\|x_i-x_j\|_{2})$. Here,
$\bar{d}_{2N_{l}}$ denotes the geographical distance between the
$2N_{l}$-th nearest pair of the leader group. The same formula hold
for $\sigma_r$ of the leader group $\mathcal{N}_r$.
\begin{figure}[htp]
\centering
\begin{tabular}{cc}
\hspace*{-0.3cm}
\resizebox{4.5cm}{!}{\includegraphics[width=10.0cm]{Fig5a.eps}}
& \hspace*{-0.2cm}
\resizebox{4.5cm}{!}{\includegraphics[width=10.0cm]{Fig5b.eps}}
\end{tabular}
\caption{Roles of clustering factor ratio $R_{\sigma}$ and normal
length ratio $R_{\eta}$ on the orientation of the followers. (a)
Effect of $R_{\sigma}$; (b) Combined effects of both $R_{\sigma}$
and $R_{\eta}$. $V_{m1}$ in Eq. (1) is compared to $V_m$ from the
Vicsek model simulations. }
\label{five}
\end{figure}
Now with these factors $\eta_{l,r}$ and $\sigma_{l,r}$ (normal lengths and clustering factors of
$\mathcal{N}_l$ and $\mathcal{N}_r$, respectively), one is ready to make a concrete comparison
between different leaderships. To quantify the effective
leadership using these factors, we further define an influencing region
ratio $R_{\eta}=\eta_l/\eta_r$ and a clustering intensity ratio
$R_{\sigma}=\sigma_l/\sigma_r$.
Remarkably, we find that the effective leadership can be
reasonable predicted solely by these parameters of the leader groups.
We let $\mathcal{N}_r$ randomly
distribute in the unbiased $L\times L$ square (Fig.~\ref{fig:
pattern}(a)), and let $\mathcal{N}_l$ (with
$N_l=N_r=10$) randomly distribute in $L\times L$, $L\times (L-1)$,
$\cdots$, $L\times 1$ rectangular regions, respectively, and hence
$\eta_r=\eta_l=L$ ( $R_{\eta}=1$) while $\sigma_l$ is increasing
monotonously. As shown in Fig.~\ref{five}(a), $V_m$ drops with
increasing $R_{\sigma}$ until asymptotically approaching a
saturation value of $-1$, indicating dominant leadership of $\mathcal{N}_l$.
To investigate the
role of $R_{\eta}$ and $R_{\sigma}$ simultaneously,
we let $\mathcal{N}_l$ (with $N_l=N_r=10$) randomly
distribute in $L\times L$, $(L-1)\times (L-1)$, $\cdots$, $1\times
1$ square regions, respectively, which implies that $\eta_l$ is falling
whilst $\sigma_l$ is rising along this distribution sequence.
Fig.~\ref{five}(b) shows that $V_m$
reduces with increasing $R_{\eta}$ and $R_{\sigma}$.
As a consequence, one can confidently draw
that the minority later-coming leaders do have the potential to
reverse the followers if only they have larger value of $\eta$ or
$\sigma$ or both.
The observation from these intensive simulations suggest that
we could determine effective leadership for the two competing leader groups
solely basing on specific combinations of the geometrical parameters
$\eta$ and $\sigma$. In fact, taking into consideration of the maximal and minimal saturation
values $1$ and $-1$ of $V_m$ and $V_m(1,1)=0$, we hereby
propose a discriminant index $V_{m1}(R_{\eta},R_{\sigma})$ by
\begin{equation}\label{eq: index new}
V_{m1}(R_{\eta},R_{\sigma}) =w_1 \tanh[\gamma(1-R_{\sigma})]+w_2
\tanh(1-R_{\eta}).
\end{equation}
Here, $\gamma$ is used to adjust the origin-traversing slope of
$\tanh(\cdot)$ function, which endows $V_{m1}$ an essential degree
of freedom. According to our extensive numerical simulations in Fig. 5,
$\gamma\in [1.3,1.6]$ yields satisfactory approximation
performance. Thereby, without loss of generality, we set
$\gamma=1.5$, and then apply Least Square Estimation to identify
$w_1=1.0$, $w_2=0.2$ commonly for all the results in Fig. 5.
Note that this index is self-consistent in
the sense that: i) the maximal and minimal saturation values keep at
$1$ and $-1$ for the feasible ranges of $R_{\sigma}$ and $R_{\eta}$,
and ii) if either value of $R_{\sigma}$ and $R_{\eta}$ is 1 then
$V_{m1}$ will be merely determined by the other one.
It is important to note that the definition of the clustering factor $\sigma$ naturally
takes the effect of the number of leaders $N_l$ and $N_r$ into account.
Remarkably, Eq. (1) with the same parameter $\gamma=1.5, w_1=1.0$ and $w_2=0.2$
can account for the effective leadership for fixed patterns, but varying numbers
$N_l$ and $N_r$ ( Fig.~\ref{fig: following the majority or
minority}(b)). A comparison of the Figs.~\ref{fig: following the majority or
minority} (a) and (b) shows clearly that $V_{m1}$ basing on the geometrical characterization
of the distribution patterns of the leaders can very nicely predict the effective
leadership $V_m$ in the model .
In summary, uncovering the nature of effective leadership is of great theoretical
and practical significance.
We have shown that later-coming leaders,
even in quantitative minority, have the potential to defeat the
earlier-coming dominating ones, if only the former obeys a better
distribution pattern. A better distribution pattern has larger
influential region and greater clustering factor, which can equip
the leaders with the capability of influencing more followers in a
given period and strengthening the persuasion power on the followers
as well. Intriguing enough, the mechanism underlying
such an apparent ``following the minority'' in the whole
group is due to the scheme of ``following the majority'' locally.
Moreover, we have demonstrated that an index merely basing on the
geometrical parameters of the distribution patterns of the leaders can provide
nice prediction of the effective leadership in competition.
With this index
one can quantify the
advantage of one leader group over another
so as to design an economical
way for the later-coming leaders to defeat the majority
earlier-coming ones.
Our simulations on the other two more sophisticated models,
the Couzin's three-sphere model \cite{co03a} and the alignment
model \cite{co05}, strongly support our conclusion on
the effective leadership mechanism.
Our investigation has launched a new exploration on the essential rules
that govern leadership potentials.
Motivated by both natural and social systems,
our findings have potential industrial and social
applications as well. The results are valuable to explain how the migrating
birds or foraging insects deviate to a new promising direction at a
low cost of additional leadership. The findings are also helpful to endow the
newborn corporations with some economic strategy to compete with
their dominating opponents in marketing competitions. Moreover,
industrial multi-agent systems can be expected to benefit from this
work to improve its adaptability to a new environment.
The work is partially supported by the National Natural Science
Foundation of China (NNSFC, Grant No. 60704041) and the
Research Fund for the Doctoral Program of Higher Education (RFDP, Grant No. 20070487090) (HTZ), the NNSFC (Grant No. 10635040) (TZ) and
by the Hong Kong Baptist University (CSZ).
|
0907.0968
|
\section{Introduction}
MAGIC are two imaging atmospheric Cherenkov telescopes for gamma-ray astronomy located at the \textit{Observatorio del Roque de los Muchachos} (European Northern Observatory, La Palma island, Spain). The production of a large amount of data during normal operation is inherent to the observational technique. The storage and processing of these data is a technical challenge which the MAGIC collaboration has solved by profitting from infrastructures like those developed for LHC experiments.\\
During the last years, the MAGIC groups at IFAE and UAB (Barcelona) and UCM (Madrid) have set up, in collaboration with PIC, the MAGIC Data Center. The facility became operational in February 2007 and, as of now, is equipped with the needed storage resources and computing capabilities to process the data from the first telescope and make them available to the MAGIC Collaboration. However, as MAGIC II is expected to start generating data this year, we expect the data volume to be increased by a factor of 3 with respect to the present single telescope situation. In consequence, we will increase the capabilities of the Data Center by providing it with the needed hardware and human resources to make it able to centralise the storage and analysis of the data. The main goals of this extension are to allow fast massive (re-)processings of all stored data and to support data analysis for all MAGIC collaborators.\\
In what follows we will describe the present status of the Data Center and the foreseen upgrades required to deal with the data flow from the two-telescopes system. This will lead us to provide additional services useful for the collaboration and also for the astrophysical community.\\
\begin{table*}[th]
\begin{center}
\small
\begin{tabular}{lrrrr}
\hline \hline
Telescope system &\bf MAGIC\,I&\bf MAGIC\,II &\bf MAGIC\,I+II\\
\hline
\# of pixels &577 &1081 &1658 \\
\# of samples &50 &50 &50 \\
Bytes per saple &2 &2 &2 \\
Event size (kByte) &60.7 &110.0 &170.7 \\
Event rate (Hz) &350 &350 &350 \\
RAW data volume (MByte/s) &20.8 &37.6 &58.4 \\
RAW data volume (GByte/h) &73.0 &132.1 &205.1 \\
Observation time per year (h) &1500 &1500 &1500 \\
\bf RAW data volume (TByte/yr) &106.9 &193.6 &300.5\\
Gzip compress factor on RAW data &0.3 &0.3 &0.3 \\
\bf Gzipped RAW data volume (TByte/yr) &32.1 &58.1 &90.2\\
Calibration reduction factor &0.034 &0.034 &0.034 \\
\bf CALIB data volume (TByte/yr) &3.6 &6.6 &10.2 \\
Star reduction factor &0.0047 &0.0026 &0.0033 \\
Star data volume (TByte/yr) &0.5 &0.5 &1.0 \\
\bf REDUCED data volume (TByte/yr) & --- &--- &1.1 \\
\hline \hline
\end{tabular}
\caption{Data volume for the different phases of the MAGIC telescope's analysis chain}
\label{tab:volume}
\end{center}
\end{table*}
\begin{table*}[!h]
\begin{center}
\small
\begin{tabular}{lccc}
\hline \hline
Phase & Program & Input file & Output file \\
\hline
Data acquisition & DAQ & --- & RAW \\
Calibration & callisto & RAW & CALIB \\
Reduction & star+superstar+melibea & CALIB & REDUCED \\
High level analysis & others & REDUCED & others \\
\hline \hline
\end{tabular}
\caption{The phases of MAGIC standard analysis, with their associated standard programs and input/output data types}
\label{tab:analysis}
\end{center}
\end{table*}
\normalsize
\section{MAGIC data: production and analysis}
The MAGIC telescopes have in their focal plane a camera segmented into a number \textit{c} of different pixels (each equipped with a photo-multiplier), whose signals are digitized by the DAQ system~\cite{signalreco}. Currently, MAGIC I is in service as a single telescope with a camera of 577 pixels. MAGIC II is under commissioning, and has an improved camera with 1039 pixels for regular operation plus 42 additional pixels equiped with experimental high quantum efficiency photodetectors for test purposes~\cite{hpd}\cite{camera}. \\
For every trigger, the single pixel signal is sampled \textit{s} times. Each sample is digitized with 12 bit precision and the resulting values stored in 2 byte fields. The information is then saved into a RAW data file. The size of a MAGIC RAW event is given by \textit{h} + 2byte\textit{$\cdot$s$\cdot$c}, where \textit{h} is a fixed-size (4.5 kByte) header describing the event. The event rate depends on the observation conditions and trigger configuration, and can range between 200 and 700 Hz. The average event rate during Observation Periods 67-73 (May-Dec 2008) was 350Hz. We will use this value to compute the data volume and storage needs summarized in Table~\ref{tab:volume}. \\
RAW event files are processed using the MAGIC standard Analysis and Reconstruction Software (MARS)~\cite{Abe}. The first step is a program dubbed \textit{callisto}, which calibrates the Cherenkov pulse's intensity and arrival time, producing a so-called CALIB data file. This part of the analysis is the most CPU demanding, therefore CALIB data files are saved before further data processing. The rest of the analysis chain consists of a set of executables taking as input the output of the previous program in the chain: \textit{star} computes the parameters describing the Cherenkov images of the individual telescope~\cite{hillas}\cite{timinganl}; \textit{superstar} merges the information of a given shower from the two telescopes; and \textit{melibea} computes the estimated energy, arrival direction and the so-called \textit{hadronness} (a parameter used for gamma/hadron discrimination~\cite{rf}). The output from \textit{star} and \textit{melibea} will be referred to as different steps REDUCED data files, and is also stored permanently.\\
Estimations of the data volume at the different stages of the analysis chain, and for the different telescope configurations are also shown in Table~\ref{tab:volume}, The different phases of the standard analysis, the name of the standard programs and the input and output file formats are summarized in Table~\ref{tab:analysis}.\\
A diferent route is followed by the Monte-Carlo simulated events that are used for the estimation of the energy and \textit{hadronness}. In this case, instead of RAW files, atmospheric particle showers are generated using CORSIKA and then digested in two steps (\textit{reflector} and \textit{camera}) that finally produce data-like files that are calibrated and reduced with the same programs used for real data. The parameters of the detector simulation are adapted to the telescope performance in different observation periods and configurations. Therefore, several versions of the Monte-Carlo library are provided at the Data Center. Presently, the simulated events are generated at the INFN Padova and Udine, and the resulting files require 10TByte of disk space.\\
\section{Description of the Data Center services}
\label{sec:needs}
Currently, the MAGIC Data Center takes care of the following tasks:
\begin{itemize}
\item Data transfer from La Palma to PIC, via internet and tapes
\item Data storage on tapes and disk at PIC
\item Data access at all data processing levels (RAW, CALIB, REDUCED) for all MAGIC collaborators
\item Real-time, automatic analysis of the data, processed with MAGIC standard software
\item Reanalysis of all stored data in case of software updates and bugfixes
\item MAGIC data base
\item Software repository and bug tracker
\item Storage of the data quality control files
\end{itemize}
During normal data taking on the site, RAW data are stored into disk, and later recorded to tape. The tapes are then sent from La Palma to PIC via airmail, since currently there is not enough Internet bandwidth between the island and Europe to support the transfer of the files.\\
The computing system on the site also performs the so-called \textit{OnSite}~\cite{Igor} analysis, by which RAW data are processed, producing CALIB files and REDUCED files right after the data taking. These files are indeed transferred to PIC via Internet, together with some log files generated by the subsystems of the telescope.\\
Currently, all tapes received at PIC are downloaded to a buffer disk and then written back to tape grouped by source and observation night in a single file (an \textit{ISO volume}) that can be mounted as an external unit in the file system. This procedure is obsolete and recently has been found more optimal to store the files directly in tapes.\\
The data are organized in a data base and served to the collaboration through a user interface machine and a web site. Until late 2008 these data were in a NFS file system, but currently, all data are being migrated to a new GRID-based file system (dCACHE) that allows more transparent access (in the sense of making no difference between tape and disk storage) to any level of RAW and processed data. It is planned that in July 2009 the NFS will be finally dismantled and all applications and data access will be based on GRID.\\
The reduction of the files that arrive via Internet is triggered by scripts that run automatically every few minutes and notice when transfers have successfully finished. When this happens, batch jobs are generated and submitted to the GRID with a specific configuration that ensures that they will run at PIC (the only Computing Element where MARS is currently installed). This set of scripts can also be used to massively reprocess all the data stored at PIC in case that a bug is found in the software or some improvement in the analysis makes it worth. Massive operations of this kind have been performed twice in 2008 and once more in 2009. For the last one, we have estimated that we used 115 days of CPU time in two weeks. This means that we have the capability to run \textit{star} on one year of data in about 8 days. It is worth to mention that this peak processing rate would have never been achieved with just the minimum number of CPU cores that the MAGIC project is granted at PIC according to its share. In fact, we should always have access to at least 7 cores any time, but in periods of low usage from other experiments we have got up to 10 times these resources.\\
Finally the Data Center also provides a ``Concurrent Version Server'' (\textit{CVS}) for the software development and the Daily Check, which generates a daily report on the data quality and the telescope stability. \\
\section{Future plans for the Data Center}
In a near future we want to provide additional services to allow a more agile analysis by any MAGIC collaborator and also easier access of anyone to published data. For this we intend to:
\begin{itemize}
\item Extend the automatic data reduction up to \textit{melibea}
\item Provide resources and tools for high level analyses by any MAGIC collaborator
\item Open the MAGIC public data to the whole scientific community by linking it to the European Virtual Observatory
\end{itemize}
Currently, the high level analysis, starting from Melibea, is carried out by analysers that select a MC-gamma sample and a real data sample fitting well the observational conditions of the analyzed data. These samples are used in the training of the multidimensional technique of selection of gamma-like events (the Random Forest method~\cite{rf}). We intend to automatize also this part of the analysis at PIC in the near future, making the task of the analyzer simpler.\\
Also the computing power of PIC can be more widely exploited by opening the job submission to the rest of the collaboration. The already working roles of the GRID scheme allow to assign priorities to the CPU farm users according to their duties, securing that the official data reduction is not delayed.\\
Finally, we intend to establish a link with the European Virtual Observatory in order to share potentially interesting data for the astrophysical community. For this purpose a software that will translate ROOT information in the widely used format in astronomy - FITS is currently being developed.\\
\section{Conclusion}
The MAGIC Data Center based at PIC is already providing quality services to the MAGIC collaboration, exploiting when possible the extra resources that a GRID-based infrastructure implies.\\
The success of the two year experience as official Data Center makes us push for the extension of the current facilities. We hope this will improve the access to the data by the MAGIC analyzers, and will make it more transparent for interested astrophysicists outside the collaboration.\\
|
0907.0314
|
\section{Introduction}
Tropical algebra (also known as max-plus algebra) is the linear
algebra of the real numbers augmented with $-\infty$ when equipped
with the binary operations of addition and maximum. Interest in this
branch of mathematics is motivated by a wide range of applications
in numerous subject areas including combinatorial optimisation and
scheduling problems \cite{butkovic}, analysis of discrete event
systems \cite{maxplus}, control theory \cite{cohen}, formal language
and automata theory \cite{pin, simon}, phylogenetics
\cite{eriksson}, statistical inference \cite{pachter}, algebraic
geometry \cite{bergman, mikhalkin, richter} and
combinatorial/geometric group theory \cite{bieri}. Tropical algebra
and many of its basic properties have been independently
rediscovered many times by researchers in these fields. The first
detailed axiomatic study of ``max-plus algebra" was conducted by
Cuninghame-Green \cite{cuninghame} and this theory has been
developed further by a number of researchers (see \cite{baccelli,
heidergott} for surveys).
Many problems arising from application areas are naturally expressed
as tropical matrix algebra problems, and much of the theory of
tropical algebra is concerned with matrices. An important aspect is
the algebraic structure of tropical matrices under multiplication;
many authors have proved a number of interesting \textit{ad hoc}
results (see for example \cite{dalessandro, gaubert, pin, simon})
but so far there has been no systematic study in this area. This
surprising omission is due largely to the difficulty, both
conceptual and technical, of the subject. Even the case of $2 \times
2$ matrices, which is the main object of study in this paper,
demonstrates a number of interesting phenomena. We believe
that the development of a coherent and comprehensive theory of
tropical matrix semigroups of arbitrary finite dimension is a major
challenge.
The aim of this paper is to initiate the systematic study of the
semigroup-theoretic structure of tropical matrices under
multiplication, by considering the most natural starting
point: the monoid of all $2 \times 2$ tropical matrices. We give a
complete geometric description of Green's relations in this
semigroup, from which we are also able to deduce that the semigroup
is regular, and to describe all of its maximal subgroups. Since
conducting this research, we have learned that an independent study of
some of these topics has recently been conducted by Izhakian and
Margolis \cite{izhakiantalk}.
In addition to this introduction, this paper comprises three sections.
In Section~\ref{sec_prelim} we give a brief expository introduction to
the tropical semiring and tropical matrix algebra, including a summary
of known results about tropical matrix semigroups. Section~\ref{sec_green} is devoted to an
examination of the ideal structure of the monoid of all $2 \times 2$
tropical matrices, obtaining in particular geometric descriptions of Green's
relations $\mathcal{L}$, $\mathcal{R}$, $\mathcal{H}$, $\mathcal{D}$ and
$\mathcal{J}$, and of the associated partial orders.
Finally, in Section~\ref{sec_idpt} we consider the idempotent elements
of this monoid; combined with the results of the previous section, this
allows us to prove that the monoid is \textit{regular}, and to describe
completely its maximal subgroups.
\section{Preliminaries}\label{sec_prelim}
Let $\bar{\mathbb{R}} = \mathbb{R} \cup \{-\infty\}$. We extend the
addition and order on $\mathbb{R}$ to $\bar{\mathbb{R}}$ in
the obvious way, and define
operations multiplication $\otimes$ and addition $\oplus$ on $\bar{\mathbb{R}}$
by $a \otimes b = a + b$ and $a \oplus b = \max\{a, b\}$ for all
$a, b \in \bar{\mathbb{R}}$. Then $\bar{\mathbb{R}}$ is a
semiring with multiplicative identity $0$ and additive identity
$-\infty$. In fact, $\bar{\mathbb{R}}$ is an idempotent semifield,
since
$a \oplus a = a$ for all $a \in \bar{\mathbb{R}}$ and $a \otimes -a
= 0$ for all $a \in \mathbb{R}$. We call $(\bar{\mathbb{R}},
\otimes, \oplus )$ the \emph{tropical semiring}; some authors refer to
it as the \textit{max-plus} semiring.
For each positive integer $n$ let $M_n(\bar{\mathbb{R}})$ denote the
set of $n \times n$ matrices with entries in $\bar{\mathbb{R}}$. The
$\otimes$ and $\oplus$ operations on $\barr$ induce corresponding
operations on $M_n(\bar{\mathbb{R}})$ in the obvious way. Indeed, if
$A, B \in M_n(\bar{\mathbb{R}})$ then we have
\begin{eqnarray*}
(A \otimes B)_{ij} &=& \bigoplus_{k=1}^{n} A_{ik} \otimes B_{kj}, \text{ and }\\
(A \oplus B)_{ij} &=& A_{ij} \oplus B_{ij},
\end{eqnarray*}
for all $1 \leq i, j \leq n$, where $X_{i,j}$ denotes the $(i,j)$th
entry of the matrix $X$. For brevity, we shall usually write
$AB$ in place of $A \otimes B$ for a product of matrices. It is then
easy to check that $M_n(\bar{\mathbb{R}})$ is an idempotent
semiring, with multiplicative identity
\[
\left(
\begin{array}{ccccc}
0&-\infty &\cdots& -\infty\\
-\infty&0 & \ddots &\vdots\\
\vdots&\ddots &\ddots &-\infty\\
-\infty&\cdots &-\infty &0\\
\end{array}\right)
\]
and additive identity
\[
\left(
\begin{array}{ccc}
-\infty&\cdots& -\infty\\
\vdots& \ddots&\vdots\\
-\infty&\cdots &-\infty\\
\end{array}\right).
\]
We call $(M_n(\bar{\mathbb{R}}), \otimes, \oplus)$ the \emph{$n
\times n$ tropical matrix semiring}. The main object of study in this
paper is the multiplicative monoid of this semiring, which we shall
refer to simply as $M_n(\bar{\mathbb{R}}), \otimes$.
We summarise some known results about this semigroup. It is readily
verified (see for example \cite{ellis}) that the invertible elements
of $\msn$ (the \textit{units} in
the terminology of ring theory or semigroup theory) are exactly the
\textit{monomial matrices}, that is, matrices with exactly one entry
in each row and column not equal to $-\infty$. It follows easily that
the group of units in $\msn$ is isomorphic to the permutation wreath
product $\mathbb{R} \wr (S_n, \lbrace 1, \dots, n \rbrace)$ of the additive
group of real numbers with the symmetric group on $n$ points.
It is known \cite{dalessandro} that the semigroup $\msn$ is
\textit{weakly permutable}, in the sense that there is a positive
integer $k$ such that every sequence of $k$ elements admits two
distinct permutations such that the corresponding products of
elements are equal in the semigroup. It is clear from the definition
that weak permutability is inherited by subsemigroups. It is also
known \cite{blyth,curzio} that a group is weakly permutable if and
only if it has an abelian subgroup of finite index. It follows
that every subgroup of $\msn$ (including those whose identity
element is an idempotent other than the identity of $\msn$) has an
abelian subgroup of finite index. Moreover, it is also shown in
\cite{dalessandro} that finitely generated subsemigroups of $\msn$
have polynomial growth.
The semigroup $M_n(\bar{\mathbb{R}})$ acts naturally on the left and
right of the space of $n$-vectors over $\barr$, known as
\textit{affine tropical $n$-space}. Notice that a tropical multiple
of a vector $(x_1, \ldots, x_n) \in \bar{\mathbb{R}}^n$ has the form
$(x_1+\lambda, \ldots, x_n+\lambda)$ for some $\lambda \in
\bar{\mathbb{R}}$. From affine tropical $n$-space we obtain
\textit{projective tropical $(n-1)$-space} by discarding the zero
vector $(-\infty,
\ldots, -\infty)$ and identifying two non-zero vectors if
one is a tropical multiple of the other.
We can represent affine tropical $2$-space (or the \textit{tropical
plane}) pictorially as a quadrant of the Euclidean plane with two
sets of axes as shown in Figure \ref{axes}. The set of tropical
multiples of $v \in \barr^2$ is then equal to the line of gradient 1
which passes through $v$, as shown in Figure~\ref{scaling}; notice that this
line includes the zero vector. Vector addition in $\barr^2$ may
also be described pictorially as follows. For $u, v \in \barr^2$ the
sum $u \oplus v$ is given by the upper right-most vertex of the
unique rectangle with $u$ and $v$ as vertices and edges parallel to
the axes, see Figure~\ref{addition}. Note that the sides of this
rectangle may have infinite length.
\begin{figure}[htp]
\begin{center}
\includegraphics[scale=0.5]{dot.eps}\caption{The tropical axes.}\label{axes}
\end{center}
\end{figure}
\begin{figure}[ht]
\begin{center}\psfrag{a}{u}\psfrag{b}{$u\oplus v$ }\psfrag{c}{$v$}
\psfrag{d}{\hspace{-0.7cm}$\lambda \otimes v$}\psfrag{e}{$v$}
\subfigure[{Tropical vector scaling in $\barr^2$.
}]{\includegraphics[scale=0.5]{twoDots.eps}\label{scaling}}\hspace{1cm}
\subfigure[{Tropical vector addition in
$\barr^2$.}]{\includegraphics[scale=0.5]{threeDots.eps}\label{addition}}
\caption{Tropical linear combinations of
vectors.}\label{lincomb}\end{center}
\end{figure}
\textit{Projective} tropical $1$-space can be conveniently identified with
the two point compactification of the real line
$$\hatr = \mathbb{R} \cup \lbrace -\infty, \infty \rbrace$$
via the map which takes the equivalence class of a non-zero vector
$(a,b) \in \barr^2$ to $b-a$ if $a$ and $b$ are real, $\infty$ if $a
= -\infty$ and $-\infty$ if $b = -\infty$\footnote{In fact, if we
extend subtraction in the obvious way to $\bar{\mathbb{R}} \times
\bar{\mathbb{R}} \setminus \lbrace (-\infty, -\infty) \rbrace$,
we have that the projection of $(a,b)$
corresponds to $b-a$ for all non-zero points $(a,b)$.}. In
pictorial terms, the image of a point $(a,b)$ with real coordinates
under this projection may be thought of as the intercept of the line
of gradient $1$ through the point $(a,b)$ with the
vertical\footnote{The choice of the vertical axis here is
of course arbitrary. One could instead take signed perpendicular distance
of the given line from the point $(0,0)$; this is arguably
conceptually cleaner but makes no practical difference and
introduces an extra factor of $\sqrt{2}$ into computations.} axis
through $(0,0)$.
\section{Green's Relations}\label{sec_green}
We begin by briefly recalling the definitions of a number of binary
relations which are used to analyse the structure of a monoid. For
further reference and examples we refer the reader to \cite{clifford}.
Let $S$ be a monoid and let $A, B \in S$.
We define a binary relation $\leqr$ on $S$ by $A \leqr B$ exactly if
$AS \subseteq BS$, or equivalently, if $A = BX$ for some $X \in S$.
Similarly, we define $A \leql B$ if $SA \subseteq SB$, and $A \leqj
B$ if $SAS \subseteq SBS$. The relations $\leqr$, $\leql$ and
$\leqj$ are \textit{preorders} (reflexive, transitive binary
relations) on the monoid $S$.
Next, we define a binary relation $\mathcal{R}$ on $S$ by
$A\mathcal{R}B$ if $A$ and $B$ generate the same principal right
ideal in $S$, or equivalently, if $A \leqr B$ and $B \leqr A$.
Similarly, we define $A\mathcal{L}B$ if $A$ and $B$ generate the
same principal left ideal in $S$, and $A \mathcal{J} B$ if $A$ and
$B$ generate the same principal two-sided ideal in $S$. The
relations $\mathcal{R}$, $\mathcal{L}$ and $\mathcal{J}$ are all
equivalence relations. In fact they are the largest equivalence
relations contained in the preorders $\leqr$, $\leql$ and $\leqj$
respectively, from which it follows that these preorders induce
partial orders on the equivalence classes of the respective
equivalence relations.
We let $\mathcal{H}$ denote the intersection $\mathcal{L} \cap
\mathcal{R}$, and $\mathcal{D}$ be the intersection of all
equivalence relations containing $\mathcal{L}$ and $\mathcal{R}$.
Both are equivalence relations, and it is well known and easy to
show that we have $A \mathcal{D} B$ if and only if there exists $Z
\in S$ such that $A \mathcal{R} Z$ and $Z \mathcal{L} B$.
We shall also need some basic ideas from tropical geometry. For each
positive integer $k$ we define a \emph{($k$-generated)
convex cone} in $\bar{\mathbb{R}}^n$ to be a non-empty set which is
the set of all tropical linear combinations of vectors from some given subset (of cardinality $k$
or less) of $\bar{\mathbb{R}}^n$. Convex cones are the tropical analogue of
linear subspaces in classical linear algebra. However, we shall
refrain from terming them \textit{(tropical linear) subspaces},
since this term is generally applied to a distinct concept which in
tropical geometry plays the role of affine linear subspaces in
classical algebraic geometry \cite{develin}.
Since convex cones are closed under scaling, each convex cone $V$ in
affine tropical $n$-space is naturally associated with a subset in
projective $(n-1)$-space, which we call the \textit{projectivisation} of
$V$. We define a \textit{($k$-generated) convex set} in projective
tropical $(n-1)$-space to be the projectivisation of a ($k$-generated)
convex cone in affine tropical $n$-space. In the case that $n=2$, so that
the projective space is $\hatr$, it is easily seen that the only convex
sets are the empty set, the singleton sets and \textit{intervals} (open, closed,
half-open and half-closed) where the latter are defined in the obvious
way using the order on $\hatr$. The $2$-generated convex sets are the
empty set, singleton sets, and \textit{closed} intervals of $\hatr$; we
call these the \textit{closed} convex sets.
Now let $A \in \msn$. We define the \textit{column
space} $C(A)$ of $A$ to be the convex cone which is the set of tropical
linear combinations of the columns of $A$.
We shall also be interested in the projectivisation of $C(A)$, which
we call the \textit{projective column space} of $A$ and denote
$\prjc{A}$. Dually, the \textit{row space} $R(A)$ of $A$ is the
convex cone given by the set of tropical linear combinations of the
rows of $A$, and its projectivisation is called the
\textit{projective row space} of $A$, denoted $\prjr{A}$.
The following characterisation of the $\mathcal{R}$ and
$\mathcal{L}$ preorders is well known at least in the
case of matrices over fields (see for example
\cite[Lemma~2.1]{okninski}) and extends without difficulty to matrices
over the tropical semiring. For completeness, we include a brief proof.
\begin{lemma}\label{lemma_rclasses}
Let $A, B \in M_n(\bar{\mathbb{R}})$. Then the following are
equivalent:
\begin{itemize}
\item[(i)] $A \leqr B$ [respectively, $A \leql B$];
\item[(ii)] $C(A) \subseteq C(B)$ [respectively, $R(A) \subseteq R(B)$] in affine tropical $n$-space;
\item[(iii)] $\prjc{A} \subseteq \prjc{B}$ [respectively, $\prjr{A} \subseteq \prjr{B}$] in projective tropical $(n-1)$-space.
\end{itemize}
\end{lemma}
\begin{proof}
We prove the equivalence of the statements involving $\leqr$ and
column spaces, the equivalence of the statements involving $\leql$
and row spaces being dual. The equivalence of (ii) and (iii)
follows from the fact that convex cones, and hence column spaces, are closed
under scaling, so it will suffice to show that (i) and (ii) are equivalent.
If (i) holds, that is, $A \leqr B$, then by
definition there is a matrix $X \in \msn$ such that $BX = A$. Now, since
the columns of $BX$ are contained in $C(B)$ it follows that
$C(BX) = C(A) \subseteq C(B)$ so that (ii) holds.
Conversely, suppose that (ii) holds. Since the tropical semiring has a
multiplicative identity, the columns of $A$ are contained in $C(A)$, and
hence in $C(B)$. Thus, every column of $A$ can be written as a
linear combination of the columns of $B$, which means exactly that there
exists $X \in \msn$ such that $A = BX$. Thus (i) holds.
\end{proof}
\begin{corollary}\label{cor_rclasses}
Let $A, B \in M_n(\bar{\mathbb{R}})$. Then the following are
equivalent:
\begin{itemize}
\item[(i)] $A \mathcal{R} B$ [respectively, $A \mathcal{L} B$];
\item[(ii)] $C(A) = C(B)$ [respectively, $R(A) = R(B)$] in affine tropical $n$-space;
\item[(iii)] $\prjc{A} = \prjc{B}$ [respectively, $\prjr{A} = \prjr{B}$] in projective tropical $(n-1)$-space.
\end{itemize}
\end{corollary}
By Corollary~\ref{cor_rclasses}, the $\mathcal{R}$-classes
of $\mstwo$ are in a natural bijective correspondence with the $2$-generated
tropical convex cones in the tropical plane, and hence also with the
with the closed convex sets
in $\hatr$. For such set $M \subseteq \hatr$ we
denote by $R_M$ the corresponding $\mathcal{R}$-class. Since $\hatr$ with
the obvious topology is homeomorphic to the closed unit interval, and
the closed intervals are definable topologically, combining with
Lemma~\ref{lemma_rclasses} yields the following natural description
of the natural partial order on the $\mathcal{R}$-classes, or equivalently,
on the intersection lattice of principal right ideals.
\begin{corollary}
The lattices of principal right ideals and of principal left ideals in
$\mstwo$ are isomorphic to the intersection lattice generated by the
closed subintervals of the closed unit interval.
\end{corollary}
It follows easily from the description of tropical vector scaling
and addition given in Section~\ref{sec_prelim} that the
$2$-generated convex cones in the affine tropical plane can take $8$
essentially distinct forms. Figure~\ref{fig:cones} shows these in
affine space, the captions giving the associated subsets of projective
space $\hatr$.
\begin{figure}[ht]
\begin{center}
\psfrag{g}{\hspace{-0.2cm}$y$}
\psfrag{f}{\hspace{-0.2cm}$y$}\psfrag{i}{\hspace{-0.2cm}$y$}\psfrag{h}{$x$}\psfrag{j}{\hspace{0.1cm}$y$}
\subfigure[$\emptyset$]{\includegraphics[scale=0.25]{dot.eps}\label{emptyset}}\hspace{1cm}
\subfigure[$\lbrace -\infty
\rbrace$]{\includegraphics[scale=0.25]{infinityDown.eps}\label{minusinfty}}
\hspace{1cm} \subfigure[$\lbrace y
\rbrace$]{\includegraphics[scale=0.25]{lineAndDot.eps}\label{k}}
\hspace{1cm} \subfigure[$\lbrace \infty
\rbrace$]{\includegraphics[scale=0.25]{infinityLeft.eps}\label{infty}}
\hspace{1cm} \subfigure[${\left[-\infty,
y\right]}$]{\includegraphics[scale=0.25]{halfPlane1.eps}\label{minusinftyk}}
\hspace{1cm}
\subfigure[${\left[x,y\right]}$]{\includegraphics[scale=0.25]{strip.eps}\label{jk}}
\hspace{1cm} \subfigure[${\left[y,
\infty\right]}$]{\includegraphics[scale=0.25]{halfPlane2.eps}\label{kinfty}}
\hspace{1cm} \subfigure[${\left[-\infty,
\infty\right]}$]{\includegraphics[scale=0.25]{greySquare.eps}\label{hatr}}
\caption{The $2$-generated tropical convex cones of
$\bar{\mathbb{R}}^2$, which correspond to closed intervals in
$\hatr$ and to the $\mathcal{R}$-classes of
$M_2(\bar{\mathbb{R}})$.}\label{fig:cones}\end{center}
\end{figure}
Using the geometric description of tropical vector operations given
in Figure~\ref{lincomb}, it is easily seen that for a non-zero
matrix
$$A = \left( \begin{array}{c c} a & b \\ c&d\\ \end{array} \right)$$
the (affine) column space $C(A)$ is exactly the region of the
quadrant bounded by the lines
$$\lbrace (a+\lambda, c+\lambda) \mid \lambda \in \barr \rbrace \text{ and } \lbrace (b+\lambda, d+\lambda) \mid \lambda \in \barr \rbrace.$$
If $A$ has a zero column, $a=c=-\infty$, say, then the projective
column space of $A$ is the singleton $\{d-b\}$ (using the natural
extension of substraction to $\barr \times \barr \setminus \lbrace (-\infty,-\infty) \rbrace$
as described in Section~\ref{sec_prelim}).
Otherwise, the projective
column space of $A$ is the closed interval (or singleton if $c-a = d-b$)
with endpoints
$c-a$ and $d-b$. Explicit descriptions of the $\mathcal{R}$-classes as sets of
matrices are given in Figure~\ref{fig:rclass}.
\begin{figure}[ht]
\begin{tabular}{l l l l}
$R_\emptyset$ &=& $\left\{
\left( \begin{array}{c c} -\infty & -\infty \\
-\infty&-\infty\\
\end{array}
\right)\right\}$,\\
\vspace{-0.3cm}\\ $R_{\lbrace -\infty \rbrace}$ &=& $\left\{
\left( \begin{array}{c c} a& b \\
-\infty&-\infty\\
\end{array}
\right)\mid a, b \in \bar{\mathbb{R}}, a \oplus b \in \mathbb{R}\right\}$,\\
\vspace{-0.3cm}\\
$R_{\lbrace y \rbrace}$ &=& $\left\{
\left( \begin{array}{c c} a & b \\
a+y&b+y\\
\end{array}
\right)\mid a, b \in \bar{\mathbb{R}}, a \oplus b \in \mathbb{R}\right\}$,\\
\vspace{-0.3cm}\\
$R_{\lbrace \infty \rbrace}$ &=& $\left\{
\left( \begin{array}{c c} -\infty &-\infty \\
a&b\\
\end{array}
\right)\mid a, b \in \bar{\mathbb{R}}, a \oplus b \in \mathbb{R}\right\}$,\\
\vspace{-0.3cm}\\
$R_{[-\infty, y]}$ &=& $\left\{\left( \begin{array}{c c} a &b \\
-\infty&b+y\\
\end{array}
\right),
\left( \begin{array}{c c} b & a \\
b+y&-\infty\\
\end{array}
\right)\mid a, b \in \mathbb{R}\right\}$,\\
\vspace{-0.3cm}\\
$R_{[x,y]}$ &=& $\left\{\left( \begin{array}{c c} a &b \\
a+x&b+y\\
\end{array}
\right),
\left( \begin{array}{c c} b & a \\
b+y&a+x\\
\end{array}
\right)\mid a, b \in \mathbb{R}\right\}$,\\
\vspace{-0.3cm}\\
$R_{[y,\infty]}$ &=& $\left\{\left( \begin{array}{c c} -\infty&a \\
b&a+y\\
\end{array}
\right),
\left( \begin{array}{c c} a & -\infty \\
a+y&b\\
\end{array}
\right)\mid a, b \in \mathbb{R}\right\}$,\\
\vspace{-0.3cm}\\
$R_{\hatr}$ &=& $\left\{\left( \begin{array}{c c} a &-\infty \\
-\infty&b\\
\end{array}
\right),
\left( \begin{array}{c c} -\infty & a \\
b&-\infty\\
\end{array}
\right)\mid a, b \in \mathbb{R}\right\}$.\\
\end{tabular}
\caption{The $\mathcal{R}$-classes of $M_2(\bar{\mathbb{R}})$. The
parameters $x$ and $y$ run through all values in $\mathbb{R}$ with
$x<y$.}\label{fig:rclass}
\end{figure}
For $U \subseteq M_2(\bar{\mathbb{R}})$ we define the transpose of
$U$ to be the set $U^T$ of all transposes of matrices in $U$, $ U^T
= \{ A^T :A \in U\}$. It follows easily from
Corollary~\ref{cor_rclasses} that each $\mathcal{L}$-class is the
transpose of an $\mathcal{R}$-class; for each closed convex
subset $M$ of $\hatr$ we therefore define $L_M = R_M^T$.
Our next objective is to describe the $\mathcal{D}$ and
$\mathcal{J}$ relations and the $\mathcal{J}$-preorder on $\mstwo$.
Recall that every $\mathcal{D}$-class and every $\mathcal{J}$-class
is a union of $\mathcal{R}$-classes, and that the
$\mathcal{R}$-class of a matrix is determined by its projective
column space. It therefore follows that the $\mathcal{D}$ and
$\mathcal{J}$ relations can be described in terms of projective
column spaces (or symmetrically, of projective row spaces).
To obtain such a description, we consider the natural distance function
$\delta : \hatr \times \hatr \to \mathbb{R} \cup \lbrace \infty
\rbrace$ defined by
$$\delta(x,y) = \begin{cases}
|y-x| & \text{ if } x, y \in \mathbb{R} \\
0 & \text{ if } x = y = -\infty \text{ or } x = y = \infty \\
\infty & \text{ otherwise.}
\end{cases}$$
The function $\delta$ satisfies $\delta(x,y) = 0$ if and only if $x
= y$. It is also symmetric and satisfies a triangle inequality when
the usual order on $\mathbb{R}$ is extended to $\mathbb{R} \cup
\lbrace \infty \rbrace$ in the obvious way. It is thus a metric,
except that it may take the value $\infty$, and so induces obvious
notions of \textit{isometric embedding} and \textit{isometry}
between subsets of $\hatr$. For $M, N \subseteq \hatr$ we write $M
\cong N$ to denote that $M$ and $N$ are isometric. Note that we do
\textit{not} require isometries or isometric embeddings to preserve
the orientation of $\hatr$, so for example $[-\infty, 0] \cong [0, \infty]$.
We define the \textit{diameter} $d(S)$ of a subset $S \subseteq \hatr$ (or of an
isometry type of subsets of $\hatr$) to be
$$d(S) = \sup_{x,y \in S} \delta(x,y)$$
where of course $0$ is the supremum of the empty set, and $\infty$ the supremum of
any set not bounded above by a real number.
We shall be
particularly interested in isometries and isometric embeddings
between closed convex subsets of $\hatr$, where a simple
combinatorial characterisation applies.
It is readily verified that
two distinct such sets are isometric if and only if (i) they are
both singletons, (ii) they are both closed intervals of
the same finite diameter, or (iii) they are both closed intervals with one real
endpoint and one endpoint at $\infty$ or $-\infty$. It is also easy to
check that isometric embedding induces a partial order on the closed
convex subsets (the only non-trivial part of this claim being that the order
is antisymmetric, that is, that two such sets which embed isometrically into
each other are necessarily isometric).
\begin{proposition}\label{prop_duality}
Let $A \in \mstwo$. Then $\prjc{A} \cong \prjr{A}$.
\end{proposition}
\begin{proof}
We proceed by case analysis, considering each possible form of
$\prjc{A}$. If $\prjc{A} = \emptyset$ then $A$ is the zero matrix so
$\prjr{A} = \emptyset$. If $\prjc{A} = \hatr$ then $A$ is a unit and
so $\prjr{X} = \hatr$.
If $\prjc{A} = \lbrace y \rbrace$ is a singleton then $A \in
R_{\lbrace y \rbrace}$ for some $y \in \hatr$. By reference to
Figure~\ref{fig:rclass} we see that $A$ has at least one non-zero
row $(a,b)$. It is then easy to verify (for example, by locating $A^T$
in Figure~\ref{fig:rclass}) that in each case $A^T \in
R_{\lbrace b-a \rbrace}$, where we again using the extended subtraction.
defined in Section~\ref{sec_prelim}. Thus, $PR(A) = PC(A^T) = \lbrace b-a \rbrace$
is isometric to $PC(A)$.
If $\prjc{A} = [x,y]$ is a closed interval with real endpoints then
using Figure~\ref{fig:rclass} once again we see that either
$$A = \twomat{a}{b}{a+x}{b+y} \text{ or } A = \twomat{b}{a}{b+y}{a+x},$$
where $a,b \in \mathbb{R}$. In the former case we have
$$A^T = \twomat{a}{(a+x)}{a+(b-a)}{(a+x)+(b-a+y-x)}$$
from which it follows that $A^T \in R_{[b-a,b-a+y-x]}$ and $\prjr{A}
= [b-a,b-a+y-x]$ is again a closed interval of diameter $y-x$ and hence
isometric to $\prjc{A}$. The latter case is similar, as are the
cases where one end of the interval is $\infty$ or $-\infty$.
\end{proof}
\begin{proposition}\label{prop_intersect}
Let $M$ and $N$ be closed convex subsets in $\hatr$, and suppose
$M \cong N$. Then there exists a matrix $Z \in \mstwo$ such that
$\prjc{Z} = M$ and $\prjr{Z} = N$
\end{proposition}
\begin{proof}
Once again, the proof is by case analysis with reference to
Figure~\ref{fig:rclass}. If $M = \emptyset$ then $N = \emptyset$ and
it suffices to take $Z$ to be the zero matrix, while if $M = \hatr$
then $N = \hatr$ and we may take $Z$ to be the identity matrix.
Suppose now that $M = \lbrace x \rbrace$ is a singleton (with $x \in
\hatr$ either real or infinite). Then $N = \lbrace y \rbrace$ must
be a singleton too and by reference to Figure~\ref{fig:rclass} it is
seen that the matrices
$$A = \twomat{0}{y}{x}{x+y} \mbox{and } B = \twomat{-(x+y)}{-x}{-y}{0}$$
satisfy $A \in R_{\lbrace x \rbrace}$, $A^T \in R_{\lbrace y
\rbrace}$, for $x,y \neq \infty$ and $B \in R_{\lbrace x \rbrace}$,
$B^T \in R_{\lbrace y \rbrace}$, for $x,y \neq -\infty$. Similarly,
the matrices
$$X =\twomat{-\infty}{-\infty}{0}{-\infty}, Y =\twomat{-\infty}{0}{-\infty}{-\infty}$$
satisfy $X \in R_{\lbrace \infty \rbrace}$, $X^T \in R_{\lbrace
-\infty \rbrace}$ and $Y \in R_{\lbrace -\infty \rbrace}$, $Y^T \in
R_{\lbrace \infty \rbrace}$. Thus, for every pair $(x,y) \in \hatr
\times \hatr$ there exists a matrix $Z$ satisfying $\prjc{Z} =
\lbrace x \rbrace$ and $\prjr{Z} = \prjc{Z^T} = \lbrace y \rbrace$
as required.
Next suppose $M = [x,y]$ is an interval with real endpoints. Then $N
= [w,z]$ must be an interval with real endpoints satisfying $z-w =
y-x$ so that $w+y = x+z$. Now consider the matrix
$$Z = \twomat{0}{w}{x}{w+y} = \twomat{0}{w}{x}{x+z}.$$
Referring once more to Figure~\ref{fig:rclass} we see that $Z \in
R_{[x,y]}$ while $Z^T \in R_{[w,z]}$ so that $\prjc{Z} = M$
and $\prjr{Z} = \prjc{Z^T} = N$ as required.
Now consider the case that $M = [-\infty, y]$ with $y$ real. Then
either $N = [-\infty, z]$ with $z$ real, or $N = [x, \infty]$ with
$x$ real. In the former case it suffices to take the matrix
$$Z = \twomat{0}{z}{y}{-\infty},$$
while in the latter case one considers
$$Z = \twomat{0}{x}{-\infty}{x+y}.$$
In both cases, reference to Figure~\ref{fig:rclass} once more establishes
that the given matrix has the correct column and row spaces.
Finally, an argument entirely similar to the previous one applies in
the case that $M = [y, \infty]$ with $y$ real, and hence completes
the proof.
\end{proof}
\begin{theorem}\label{thm_jorder}
Let $A,B \in \mstwo$. Then the following are equivalent:
\begin{itemize}
\item[(i)] $A \leqj B$;
\item[(ii)] $\prjc{A}$ embeds isometrically in $\prjc{B}$;
\item[(iii)] $\prjr{A}$ embeds isometrically in $\prjr{B}$.
\end{itemize}
\end{theorem}
\begin{proof}
The equivalence of (ii) and (iii) follows from Proposition~\ref{prop_duality}.
Suppose next that (i) holds, and let $X, Y \in \mstwo$ be such that
$A = XBY$. Then $A = XBY \leqr XB$ so by Lemma~\ref{lemma_rclasses},
$\prjc{A} \subseteq \prjc{XB}$, and in particular $\prjc{A}$ embeds
isometrically in $\prjc{XB}$. Similarly, $XB \leql B$ so by
Lemma~\ref{lemma_rclasses}, $\prjr{XB}$ embeds isometrically in
$\prjr{B}$. Now, by Proposition~\ref{prop_duality}, $\prjc{XB} \cong
\prjr{XB}$ and $\prjr{B} \cong \prjc{B}$, so by transitivity of
isometric embedding we conclude that $\prjc{A}$ embeds isometrically
in $\prjc{B}$ and (ii) holds.
Finally, suppose (ii) holds.
Let $M \subseteq \prjc{B}$ be the image of an isometric embedding of
$\prjc{A}$ into $\prjc{B}$. Then $M$ is clearly a closed
convex set isometric to $\prjc{A}$ which by
Proposition~\ref{prop_duality} is also isometric to $\prjr{A}$.
Hence, by Proposition~\ref{prop_intersect}, there is a matrix $Z \in
\mstwo$ such that $\prjc{Z} = M \subseteq \prjc{B}$ and $\prjr{Z} =
\prjr{A}$. But now by Corollary~\ref{cor_rclasses} and
Lemma~\ref{lemma_rclasses} we have $A \mathcal{L} Z$ and $Z \leqr
B$, from which it follows that $A \leqj B$.
\end{proof}
\begin{theorem}\label{thm_dj}
Let $A, B \in \mstwo$. Then the following are equivalent:
\begin{itemize}
\item[(i)] $A \mathcal{D} B$;
\item[(ii)] $A \mathcal{J} B$;
\item[(iii)] $\prjc{A} \cong \prjc{B}$;
\item[(iv)] $\prjr{A} \cong \prjr{B}$.
\end{itemize}
\end{theorem}
\begin{proof}
The equivalence of (iii) and (iv) follows from Proposition~\ref{prop_duality}.
That (i) implies (ii) follows from general facts about semigroups, while
the fact that (ii) implies (iii) is a corollary of Theorem~\ref{thm_jorder}.
Finally, if (iii) holds then by Proposition~\ref{prop_duality} we
have $\prjr{A} \cong \prjc{A} \cong \prjc{B}$, so by
Proposition~\ref{prop_intersect} there is a matrix $Z \in \mstwo$
such that $\prjc{Z} = \prjc{B}$ and $\prjr{Z} = \prjr{A}$. By
Corollary~\ref{cor_rclasses} it follows that $B \mathcal{R} Z$ and
$Z \mathcal{L} A$. Since $\mathcal{D}$ is an equivalence relation
containing $\mathcal{L}$ and $\mathcal{R}$ we conclude that $X
\mathcal{D} Y$ so that (i) holds.
\end{proof}
Theorems~\ref{thm_jorder} and \ref{thm_dj} allows us to deduce a great
deal about the two-sided ideal structure of $\mstwo$. An immediate
corollary is a description of the lattice order on the two-sided principal
ideals (or equivalently, on the $\mathcal{J}$-classes).
\begin{corollary}
The lattice of principal two-sided ideals in $\mstwo$ is isomorphic
to the lattice of isometry types of closed convex subsets of $\hatr$
under the partial order given by isometric embedding.
\end{corollary}
We now turn our attention to non-principal ideals, which it transpires
can also be characterized by convex sets in $\hatr$. Let $\mathbb{S}$ be
the set of convex sets in $\hatr$ consisting of all the closed convex
sets, all the open intervals of finite diameter,
and the open interval $(-\infty, \infty)$. Note that we exclude the
half-infinite open intervals. Once again, it is easily seen
that isometric embedding induces a partial order on the isometry types of
sets in $\mathbb{S}$. Note also that no two isometry types of sets in
$\mathbb{S}$ admit
isometric embeddings of exactly the same collection of closed convex
sets.
\begin{theorem}\label{thm_allideals}
Let $I$ be an ideal of $\mstwo$. Then there exists a subset $I' \in \mathbb{S}$
such that for all $X \in \mstwo$ we have $X \in I$ if
and only if the projective column space of $X$ embeds isometrically
into $I'$. Moreover, the set $I'$ is unique up to isometry.
\end{theorem}
\begin{proof}
Let $I$ be an ideal of $\mstwo$, and let $T$ be the set of all isometry
types of closed convex sets in $\hatr$ which arise as
projective column spaces (or equivalently, projective row spaces) of
matrices in $I$. If $T$ has a maximal element under the isometric embedding
order, then it follows from Theorem~\ref{thm_jorder} that it suffices to take
$I'$ to be this convex set.
Suppose, then, that $T$ has no maximal element. Then clearly it cannot
contain the isometry type of a convex set of infinite diameter (since
there are only finitely many such up to isometry, and they are above all
other convex sets in the isometric embedding order), but must contain
infinitely many intervals of finite diameter. If the diameters of these
intervals are bounded above by a real number, then we let $w$ be the supremum of
the diameters. Since $T$ has no maximal element, this supremum is not
attained in $T$. It follows from Theorem~\ref{thm_jorder} that a matrix lies
in $I$ if and only if its projective column space has diameter strictly less than
$w$. This is the case exactly if the projective column space embeds isometrically
in an open interval of diameter $w$, so it suffices to take $I'$ to be
such an interval.
On the other hand, if the diameters of the intervals are not bounded
above then, by Theorem~\ref{thm_jorder} again, we see that $I$ contains
every matrix with projective row space of finite diameter, and it follows
that we may take $I'$ to be the open interval $(-\infty, \infty)$.
Finally, the uniqueness up to isometry of $I'$ follows from Theorem~\ref{thm_jorder} and
the fact that no two distinct isometry types of sets in $\mathbb{S}$ embed exactly
the same closed convex sets.
\end{proof}
For $I$ an ideal of $\mstwo$, we denote by $S(I)$ the unique convex
subset $S(I) \in \mathbb{S}$ such that $I$ consists of those matrices
with projective column space which embeds isometrically in $S(I)$.
\begin{corollary}\label{cor_totalorder}
The two-sided ideals of $\mstwo$ are totally ordered under inclusion.
\end{corollary}
\begin{proof}
This follows immediately from Theorem~\ref{thm_allideals}, and the obvious
fact that the isometry types of sets in $\mathbb{S}$ are totally ordered under isometric
embedding.
\end{proof}
\begin{corollary}\label{cor_closedprincipalfg}
Let $I$ be an ideal in $\mstwo$. Then the following are equivalent:
\begin{itemize}
\item[(i)] $S(I)$ is closed;
\item[(ii)] $I$ is principal;
\item[(iii)] $I$ is finitely generated.
\end{itemize}
\end{corollary}
\begin{proof}
By Proposition~\ref{prop_intersect}, every closed convex set is the
projective column space of some matrix in $\mstwo$, so that (i) implies
(ii) follows from Theorem~\ref{thm_jorder}.
That (ii) implies (iii) is by definition. Finally, suppose (iii) holds
let $G$ be a finite generating set for $I$, and let $S = \lbrace PC(X) \mid X \in G \rbrace$.
By Corollary~\ref{cor_totalorder}, $S$ is totally ordered under
isometric embedding, and since it is finite, it must contain a maximum
element. This maximum element is a closed convex set, and an easy argument now
shows that it must be equal to $S(I)$.
\end{proof}
The equivalence of (ii) and (iii) in Corollary~\ref{cor_closedprincipalfg}
may be viewed as a manifestation of the fact that every finitely generated
tropical convex cone in $\barr^2$ is closed.
\begin{corollary}
Every ideal in $\mstwo$ is either principal, or the difference between
a principal ideal and its generating $\mathcal{J}$-class.
\end{corollary}
\begin{proof}
Let $I$ be an ideal and consider the convex set $S(I) \in \mathbb{S}$.
If $S(I)$ is
closed then by Corollary~\ref{cor_closedprincipalfg}, $I$ is principal.
Otherwise, $S(I)$ is an open interval. Let $J$ be the the smallest closed
interval in $\hatr$ containing $S(I)$. Clearly, a given closed interval
$K$ embeds isometrically into $S(I)$ if and only if it embeds isometrically
into $J$ but is not isometric to $J$. Hence, by Theorems~\ref{thm_jorder} and
\ref{thm_dj}, a matrix is in $I$ if and only if it lies in the ideal
corresponding to $J$ (which by Corollary~\ref{cor_closedprincipalfg} is
principal) but not in the $\mathcal{J}$-class corresponding to $J$.
\end{proof}
\section{Idempotents and Subgroups}\label{sec_idpt}
Our aim in this section is to identify the idempotent elements of
$\mstwo$, and draw some conclusions about both its
semigroup-theoretic structure and its maximal subgroups.
Recall that an element $e$ in a semigroup is called
\textit{idempotent} if $e^2 = e$.
\begin{proposition}\label{prop_idpt}
The idempotents of $\mstwo$ are exactly the matrices of the form
\[
\left( \begin{array}{c c} 0 & x \\
y&x+y\\
\end{array}
\right), \;\; \left( \begin{array}{c c} 0 & x \\
y&0\\
\end{array}
\right),\;\; \left( \begin{array}{c c} x+y & x \\
y&0\\
\end{array}
\right) \;\; \mbox{and} \;\;
\left( \begin{array}{c c} -\infty & -\infty \\
-\infty&-\infty\\
\end{array}
\right)
\]
where $x, y \in \bar{\mathbb{R}}$ with $x+y \leq 0$.
\end{proposition}
\begin{proof}
It is readily verified by direct computation that these matrices are
idempotent. Conversely, suppose that
\[
\left( \begin{array}{c c} a & b \\
c&d\\
\end{array}
\right) \ \left( \begin{array}{c c} a & b \\
c&d\\
\end{array}
\right) = \left( \begin{array}{c c} a & b \\
c&d\\
\end{array}
\right).
\]
Then we have
\begin{eqnarray}
\max(a+a, b+c)= a,&&\;\;\max(b+c, d+d)= d,\label{diag}\\
\max(a+c, c+d)= c,&& \;\;\max(a+b, b+d)= b\label{offdiag},
\end{eqnarray}
giving $-\infty \leq a,d \leq 0$.
First suppose that $a < 0$. Then by \eqref{diag} we must have that
$a=b+c$. If $d = 0$ then we have a matrix of the form
$$\left( \begin{array}{c c} b+c & b \\
c&0\\
\end{array}
\right),$$ where $-\infty \leq b+c = a < 0$. On the other hand, if $d \neq 0$
then by \eqref{diag} we must have the zero matrix.
Next suppose that $a=0$. By \eqref{diag} we see that $b+c \leq 0$
and either $d=0$ or $d=b+c$, giving matrices of the form
$$\left( \begin{array}{c c} 0 & b \\
c&0\\
\end{array}
\right)\mbox{ and } \left( \begin{array}{c c} 0 & b \\
c&b+c\\
\end{array}
\right)$$ where $-\infty \leq b+c \leq 0$, respectively.
\end{proof}
While the purely computational approach to finding idempotents employed
in the proof of Proposition~\ref{prop_idpt} is straightforward in the
$2 \times 2$ case, it is conceptually unenlightening and quickly becomes
intractable in higher dimensions.
In any semigroup of functions, the idempotents are exactly the
\textit{projections}, that is, those functions which fix their
images pointwise. In $\mstwo$, then, an idempotent element is a
matrix which (viewed as acting from the left on column vectors)
fixes the tropical convex cone generated by its own columns.
Figure~5 illustrates the geometric action of
some typical idempotents. In higher dimensions, the complex structure
of tropical convex cones \cite{develin} makes it a delicate
task to locate the idempotents by geometric arguments, but nevertheless we
believe that only this approach is feasible.
\begin{figure}[ht]
\begin{center}\psfrag{a}{u}\psfrag{b}{$u\oplus v$ }\psfrag{c}{$v$}
\psfrag{d}{\hspace{-0.7cm}$\lambda \otimes v$}\psfrag{e}{$v$}
\subfigure[{A projection onto the column space corresponding to
$R_{\lbrace y
\rbrace}$.}]{\includegraphics[scale=0.5]{arrowLines.eps}\label{line}}\hspace
{1cm}
\subfigure[{A projection onto the column space corresponding to
$R_{[x,y]}$.}]{\includegraphics[scale=0.5]{arrowStrip.eps}\label{strip}}
\caption{Examples of projections.}\label{projections}\end{center}
\begin{picture}(1,10)
\put(115,202){$y$} \put(95,154){$-x$}\put(283,209){$y$}
\put(283,184){$x$}
\end{picture}
\label{fig:projections}
\end{figure}
Cross-referencing Proposition~\ref{prop_idpt} with
Figure~\ref{fig:rclass}, we quickly see that $\mstwo$ has an idempotent
in every $\mathcal{R}$-class. Recall that a semigroup $S$ is called
\textit{regular} if for every element $X \in S$ there is an element
$Y \in S$ such that $XYX = X$ (\textit{von Neumann regularity} in
the terminology of ring theory). It is well known that a semigroup
is regular if and only if every $\mathcal{R}$-class contains an
idempotent, so we have established the following.
\begin{theorem}
The semigroup $\mstwo$ of all $2 \times 2$ tropical matrices is regular.
\end{theorem}
We now turn our attention to maximal subgroups of $\mstwo$.
It is a foundational result of semigroup theory (see for example \cite{clifford})
that every subgroup of a semigroup lies in a unique maximal subgroup, and that
the maximal subgroups are exactly the $\mathcal{H}$-classes of idempotent
elements. We thus begin by describing those $\mathcal{H}$-classes which
contain idempotents.
\begin{theorem}\label{thm_idpthclasses}
Let $M$ and $N$ be closed convex subsets of $\hatr$. Then the $\mathcal{H}$-class $R_M \cap
L_N$ contains an idempotent if and only if one of the
following conditions holds:
\begin{itemize}
\item[(i)] $M=\{x\}$ and $N=\{y\}$ with $\lbrace x,y \rbrace \neq \lbrace -\infty,\infty \rbrace$;
\item[(ii)] $M = -N = \lbrace -x \mid x \in N \rbrace$ where $|N| \neq1$.
\end{itemize}
\end{theorem}
\begin{proof}
Suppose first that $R_M \cap L_N$ contains an idempotent $E$. Then
$E$ must have one of the four forms given by
Proposition~\ref{prop_idpt}. Clearly if $E$ is the zero matrix then $M =
N = \emptyset$ and (ii) holds. If $E$ has the form
$\twomat{0}{x}{y}{x+y}$ for $x, y \in \barr$ with $x+y \leq 0$, then
it is readily verified that $PC(E) = \lbrace y \rbrace$ and $PR(E) =
\lbrace x \rbrace$ and hence (i) holds. An entirely similar argument
holds if $E$ has the form $\twomat{x+y}{x}{y}{0}$, where this time
$PC(E) = \lbrace -x \rbrace$ and $PR(E) = \lbrace -y \rbrace$.
Finally, if $E$ has the form $\twomat{0}{x}{y}{0}$ with $x+y \leq 0$
then a simple computation shows that $PC(E) = [y,-x]$ and $PR(E) =
[x,-y]$ so that once again (ii) holds.
Conversely, suppose (i) holds, say $M = \lbrace x \rbrace$ and $N =
\lbrace y \rbrace$, where $\lbrace x, y \rbrace \neq \lbrace -\infty, \infty \rbrace$
so that $x+y$ is well-defined. If $x+y \leq 0$
then $x,y \neq \infty$ and the matrix $\twomat{0}{y}{x}{x+y}$ is an
idempotent by Proposition~\ref{prop_idpt}, and is easily seen (by
computing the projective row and column spaces) to lie in the
claimed $\mathcal{H}$-class. On the other hand, if $x+y \geq 0$ then
$x,y \neq -\infty$, so we have $-x, -y \in \barr$ and $(-x) + (-y)
\leq 0$. It follows by Proposition~\ref{prop_idpt} that the matrix
$\twomat{-x-y}{-y}{-x}{0}$ is idempotent and once again it is easily
verified that it lies in $R_M \cap L_N$.
Finally, suppose (ii) holds. If $M$ is empty then so is $N$, and the
zero matrix is an idempotent in $R_M \cap L_N$. Suppose, then, that
$M$ is a closed interval $[x,y]$ with $x,y \in \hatr$ and $x \leq
y$. Then $y \neq -\infty$ so $-y$ is well-defined, and $x+(-y) < 0$.
Hence, by Proposition~\ref{prop_idpt}, the
matrix $\twomat{0}{-y}{x}{0}$ is idempotent. Once more, it is
straightforward to verify that this matrix lies in $R_M \cap L_N$.
\end{proof}
Having ascertained which $\mathcal{H}$-classes are maximal subgroups, it
remains to identify the algebraic structure of each.
\begin{theorem}
The maximal subgroups in the $\mathcal{D}$-class of elements with
row and column space isometric to a closed convex subset $M
\subseteq \hatr$ are isomorphic to:
\begin{itemize}
\item[(i)] the trivial group, if $M = \emptyset$;
\item[(ii)] the additive group $\mathbb{R}$ of real numbers, if $M$ is a
point, or an interval with precisely one real endpoint;
\item[(iii)] the direct product $\mathbb{R} \times S_2$, if $M$ is an interval
with two real endpoints;
\item[(iv)] the wreath product $\mathbb{R} \wr S_2$, if $M = \hatr$.
\end{itemize}
\end{theorem}
\begin{proof}
If $M = \emptyset$ then the only matrix in $R_M \cap L_M$ is the
zero matrix, so this $\mathcal{H}$-class is isomorphic to the
trivial group.
Now suppose $M = \lbrace x \rbrace$ is a singleton. Since maximal
subgroups in a $\mathcal{D}$-class are always isomorphic, by
Theorem~\ref{thm_dj} it suffices to consider the case that $M =
\lbrace -\infty \rbrace$. By Theorem~\ref{thm_idpthclasses}, $R_M
\cap L_M$ contains an idempotent. Reference to
Figure~\ref{fig:rclass} shows that
$$R_M \cap L_M = \lbrace W_a \mid a \in \mathbb{R} \rbrace$$
where
$$W_a = \twomat{a}{-\infty}{-\infty}{-\infty}.$$
Direct calculation shows that $W_a W_b = W_{a+b}$ for all $a, b \in
\mathbb{R}$ so that $R_M \cap L_M$ is isomorphic to the additive
group $\mathbb{R}$ as required.
Next suppose $M = [x,y]$ is an interval with distinct real
endpoints, so that $x-y < 0$. Then by Theorem~\ref{thm_idpthclasses}, setting $N = -M =
[-y,-x]$ we have that the $\mathcal{H}$-class $R_M \cap L_N$
contains an idempotent. A direct computation using Figure~\ref{fig:rclass} shows that
$$R_M \cap L_N = \lbrace X_a, Y_a \mid a \in \mathbb{R} \rbrace$$
where
$$X_a = \twomat{a}{a-y}{a+x}{a} \text{ and } Y_a = \twomat{a}{a-x}{a+y}{a}.$$
Simple calculation, recalling the fact that $x-y < 0$, shows that $X_a X_b = X_{a+b}$, $X_a Y_b = Y_b
X_a = Y_{a+b}$ and $Y_a Y_b = X_{a+b+(y-x)}$ for all $a, b \in
\mathbb{R}$. We deduce that $X_0$ is idempotent and hence is the
identity of $R_M \cap L_N$ and that the $X_a$'s form a central
subgroup isomorphic to the real numbers.
Moreover, choosing $z = (x-y)/2$ we see that
$(Y_z)^2 = X_0$ and every element $Y_b$ can be written in the form
$Y_z X_a$ for some $a \in \mathbb{R}$.
We have shown that $R_M \cap L_N$ is the product of commuting
subgroups with trivial intersection, one of them isomorphic to
$\mathbb{R}$ and the other to $S_2$. It follows that the subgroup is
isomorphic to $\mathbb{R} \times S_2$, as claimed.
Now suppose $M$ is an interval with one real and one infinite
endpoint. By Theorem~\ref{thm_dj} we may assume that $M = [x,
\infty]$. Set $N = -M = [-\infty, -x]$. Then by
Theorem~\ref{thm_idpthclasses} we have that $R_M \cap L_N$ contains
an idempotent. Another reference to
Figure~\ref{fig:rclass} reveals that
$$R_M \cap L_N = \lbrace Z_a \mid a \in \mathbb{R} \rbrace$$
where
$$Z_a = \twomat{a}{-\infty}{a+x}{a}.$$
Once again, we find that $Z_a Z_b = Z_{a+b}$ so that $R_M \cap L_N$
is isomorphic to the additive group $\mathbb{R}$
Finally, if $M = \hatr$ then we also have $N = \hatr$, and $R_M \cap L_N$
is the group of units. We remarked in Section~\ref{sec_prelim} that it is
known that the group of units of $\msn$ is isomorphic to the permutation
group wreath product $\mathbb{R} \wr (S_n, \lbrace 1, \dots, n \rbrace)$. In
the case $n=2$, since the right translation action of $S_2$ on itself is
isomorphic to its standard action on $\lbrace 1, 2 \rbrace$, the group of units is
also isomorphic to the wreath product $\mathbb{R} \wr S_2$ of abstract groups.
\end{proof}
We remarked in Section~\ref{sec_prelim} that it is known that every group
admitting a faithful representation by finite dimensional tropical matrices
has an abelian subgroup of finite index \cite{dalessandro}. In the case of
groups admitting faithful $2 \times 2$ tropical matrix representations, we
can now be rather more precise.
\begin{corollary}
Every group admitting a faithful representation by $2 \times 2$
tropical matrices is either torsion-free abelian or has a torsion-free
abelian subgroup of index $2$.
\end{corollary}
In general, we conjecture that a group admitting a faithful representation
by $n \times n$ tropical matrices must have a torsion free abelian subgroup
of index at most $n!$.
\section*{Acknowledgements}
The authors thank the participants of the
\textit{Manchester Tropical Mathematics Reading Group}, and the organisers,
speakers and participants of the \textit{First de Br\'un Workshop on Computational Algebra}
(held at the National University of Ireland, Galway in 2008) for their
help with learning the background required for this study.
They also thank Claas R\"over for some helpful conversations, and
Gemma Lloyd for her assistance with the diagrams in this paper.
|
0907.0919
|
\section{}
\section{Introduction}
The Monte Carlo Renormalization Group (MCRG) methods, based on
Wilson's renormalization group theory, were developed and used extensively
in the 1980's to study the critical properties of spin and gauge models
\cite{Swendsen:1979gn,Swendsen:1981rb,Swendsen:1984vu,Hasenfratz:1984bx,Hasenfratz:1984hx,Bowler:1984hv,Hasenfratz:1984ju,Hasenfratz:1987zi,Decker:1987mu,Gupta:1984gq,Patel:1984dw}.
The 2-lattice matching MCRG proved to be particularly useful to calculate
the $\beta$ function of asymptotically free theories, like quenched
QCD \cite{Bowler:1984hv,Hasenfratz:1984bx,Hasenfratz:1984hx}. The approach has all
but been forgotten in the last 20 years as lattice QCD calculations
focused on spectral and other experimentally measurable quantities.
Lately there has been increased interest in beyond-QCD lattice models
\cite{Catterall:2007yx,Appelquist:2007hu,Shamir:2008pb,Deuzeman:2008sc,DelDebbio:2008zf,
Catterall:2008qk,Fodor:2008hm,DelDebbio:2008tv,DeGrand:2008kx,Hietanen:2008mr,
Fodor:2008hn,Appelquist:2009ty,Hietanen:2009az,Deuzeman:2009mh}
as they could describe strongly coupled beyond-Standard Model physics
\cite{Hill:2002ap,Banks:1981nn,Sannino:2004qp,Dietrich:2006cm}.
Ref. \cite{Fleming:2008gy} is a good summary of the issues and
recent lattice results.
A basic discussion of the physical picture with some remarks on expectations for lattice simulations
were presented in Ref. \cite{DeGrand:2009mt}. In this paper I follow the Wilson renormalization
group RG language used in \cite{DeGrand:2009mt}.
SU(3) gauge models with $N_f$ fermions in the fundamental representation can have very different phase structure depending on the number of fermions \cite{Banks:1981nn}. If $N_f>16$ asymptotic freedom is lost, the gauge coupling is irrelevant and the continuum theory is free. For small $N_f$ the $g=0$, $m=0$ Gaussian fixed point (GFP) is the only critical fixed point (FP). The theory is asymptotically free, confining and chirally broken. Somewhere around $N_f\approx10$ the gauge coupling develops a new FP at $g^*\ne0$ \cite{Caswell:1974gg,Dietrich:2006cm}. At $g^*$ the gauge coupling is irrelevant, it is an infrared FP (IRFP). The continuum limit defined in the basin of attraction of this IRFP is neither confining not chirally broken; it is conformal when $m=0$. This conformal phase is expected to exits all the way to $N_f=16$. Identifying the lower end of the conformal window and the critical properties of the IRFP are the main issues of recent lattice simulations.
The MCRG
method was designed to answer these kind of questions and in this
paper I present the first such study in $N_{f}=4$ and $N_{f}=16$
flavor SU(3) theories. I also investigate the pure gauge SU(3) model
where it is possible to do high statistics, large volume simulations.
I have chosen these models as the expected phase structure
is rather well known, so I can use them to calibrate and test the method.
My eventual goal is to extend these studies to other flavor numbers or fermions in
different representations.
Since MCRG has been used very little in the last 20 years, I devote Sect.\ref{sect:MCRG} to
the basic description of the 2-lattice matching MCRG. The method allows the determination
of a sequence of couplings $\beta_0,\beta_1,...\beta_n,...$ with lattice spacings
that differ by a factor of s between consecutive points,
$a(\beta_n)=a(\beta_{n-1})/s$. $s$ is the scale change of the RG transformation, $s=2$ in this study.
This sequence is analogous to the step scaling function defined in the Schrodinger functional (SF) method
\cite{Luscher:1992zx,Luscher:1993gh,Capitani:1998mq},
but in MCRG it is defined through the bare couplings.
To emphasize this difference I will use the notation
$s_b(\beta_n;s)=\beta_n-\beta_{n-1}$ for the bare step scaling function instead of the more traditional $\s(u;s)$ used in the SF approach.
The
sequence $\beta_0,\beta_1,...\beta_n,...$ can be used to determine the renormalized
running coupling in theories that are governed by the GFP
if at the weak coupling end of the chain a renormalized coupling, like the SF $\bar{g}^2$,
is calculated and connected to a continuum regularization scheme, while at the strong coupling end some physical quantity is used to
determine the lattice scale.
I do not pursue this calculation here, though I will compare results for $s_b(\beta;s=2)$ from SF and MCRG in Sect.
\ref{sect:pure_SU3}.
The rest of the paper is organized as follows. Sect. \ref{sect:PT} summarizes the perturbative picture of these many fermion theories.
Sect. \ref{sect:MCRG} describes the 2-lattice matching method and defines the RG block transformation used in this work. The numerical simulations and results are discussed in Sect. \ref{sect:simulations}. The technical aspects of MCRG are described in detail for the pure gauge SU(3) theory in Sect.\ref{sect:pure_SU3} as it serves to justify the approach used for the $N_f=4$ and $N_f=16$ models in Sects. \ref{sect:nf4} and \ref{sect:nf16}. I use nHYP smeared staggered fermions in this work. I present some basic properties of 4 flavor staggered fermions, together with the MCRG calculation of the bare step scaling function in Sect. \ref{sect:nf4}. The existence of an IRFP requires the re-evaluation of the MCRG method. This, together with preliminary results for the scaling exponent of the mass in the $N_f=16$ model are presented in Sect. \ref{sect:nf16}.
\section{The perturbative picture \label{sect:PT}}
Before discussing the MCRG method and the numerical results I briefly summarize the perturbative picture.
The universal 2-loop $\beta$ function for SU(3) gauge with $N_f$ fermions in the fundamental representation is
\begin{equation}
\beta(g^{2})=\frac{dg^{2}}{d\log(\mu^{2})}=
\frac{b_{1}}{16\pi^{2}}g^{4}+\frac{b_{2}}{(16\pi^{2})^{2}}g^{6}+\dots\,,
\label{eq:betaQCD}
\end{equation}
\begin{eqnarray}
b_{1} & = & -11 + \frac{2}{3} N_f\,, \nonumber \\
b_{2} & = & -102+\frac{38}{3} N_f\,. \nonumber
\end{eqnarray}
For $N_f<16.5$ the 1-loop coefficient $b_1$ is negative, the gauge coupling is relevant at the $g=0$ Gaussian FP,
the theory is asymptotically free. Dimensional transmutation is responsible for mass generation. The energy scale
changes by a factor of 2 between couplings $g_1$ and $g_2$ if
\begin{equation}
\rm{ln}(2) = -\,\int_{g_1}^{g_2} \frac{g}{\beta(g^2)} dg \,.
\label{eq:sb_int}
\end{equation}
At one loop level this leads to a constant shift in $\beta=6/g^2$ and the bare step scaling function is
\begin{equation}
s_b(\beta_1;s=2) = \beta_1-\beta_2 = -\,\frac{3 \,\rm{ln}(2)}{4\p^2} b_1\,\,\,\,\,\,
(\rm{1-loop}).
\label{eq:db_pert}
\end{equation}
For small fermion numbers
the higher order terms are small, the $\beta$ function is expected to remain negative. Lattice simulations indicate that for
$N_f\le8$ the system is confining and chiral symmetry is spontaneously broken. For $N_f>8$ the 2-loop $\beta$ function
develops a zero at $g^*\ne0$ Banks-Zaks FP\cite{Banks:1981nn}.
At this new FP $g$ is irrelevant, it is an IRFP for the gauge coupling. The infinite cut-off limit in the vicinity
of $g^*$ is conformal.
When the perturbatively predicted $g^*$ is large, higher order or non-perturbative effects can destroy the existence of the IRFP. Analytical considerations and numerical simulations suggest that the bottom of the conformal window is around $N_f\approx10$
\cite{Caswell:1974gg,Dietrich:2006cm}. At $N_f=16$, the largest flavor number that is
still asymptotically free, the Banks-Zaks FP occours at a small value $g^*\sim \e$, perturbation theory could correctly describe the conformal phase.
At the IRFP there is only one relevant operator, the mass. Its scaling dimension (critical exponent) is close to its engineering one $y_m\sim 1 + \cal{O}(\e)$, while the scaling dimension of the gauge coupling is $y_g\sim -\e^2$. The slope of the $\beta$ function at $g^*$ predicts the exponent
\begin{eqnarray}
\beta(g^2) & = &-y_g (g^2-g^{*2}) + {\cal O} ((g^2-g^{*2})^2) \,,\\
y_g & = &-\frac{b_1^2}{b_2}\,.
\label{eq:gamma_g}
\end{eqnarray}
Eq. \ref{eq:sb_int} now gives
\begin{eqnarray}
g_1^2-g^{*2} &=& (g_2^2-g^{*2}) 2^{-2y_g}\,,\\
s_b(\beta_1;s=2) &=& \beta_1-\beta_2 = (\beta_2-\beta^*)(2^{-2y_g}-1)\,,
\label{eq:db_IRFP}
\end{eqnarray}
if $\beta_1-\beta^*\,,\,\beta_2-\beta^* \ll 1$.
For 16 flavors perturbatively $y_g\approx -0.01$, the gauge coupling is almost marginal. For smaller $N_f$ or higher representation fermions
$|y_g|$ can be larger, though both numerical and analytical considerations find that $|y_g|$ remains small even at the bottom of the conformal window\cite{Gardi:1998ch,Appelquist:2009ty,DeGrand:2009mt}.
The mass is a relevant operator both at the GFP and at the IRFP, with critical value $m^*=0$. Under a scale change $s=2$ it changes as
\begin{equation}
m_1=m_2 2^{-y_m}\,,
\label{eq:mass_scale}
\end{equation}
where $1/y_m=\nu$ is the critical index of the mass.
\section{The MCRG method \label{sect:MCRG}}
The Wilson RG description of statistical systems is a very effective
approach to describe the phase diagram, calculate critical indices,
and in case of lattice discretized quantum field theories, understand
the infinite cut-off continuum limit of these models.
There are many books and review articles written about the subject. I do not attempt
to explain Wilson RG here, I only summarize the main points. Two reviews that could be useful for
other parts of this paper are Refs. \cite{Hasenfratz:1984ju,DeGrand:2009mt}.
In the inherently
non-perturbative Wilson RG approach one considers the evolution
of all the possible couplings under an RG transformation
that preserves the internal symmetries of the system but integrates
out the cut-off level UV modes. The fixed points of the transformation
are characterized by the number of relevant operators,
i.e. couplings with positive scaling dimensions that flow away from the FP.
Irrelevant couplings have negative scaling dimensions and they flow towards the
FP. The IR values of irrelevant operators
are independent of their UV values. Continuum (or infinite
cut-off) limits can be defined by tuning the relevant couplings
towards the FP, thus controlling their IR value. The number of relevant operators
and their speed along the RG flow lines are universal,
related to the infrared properties of the underlying continuum limit.
On the other hand the location of the FP is not physical, in fact different
RG transformations have different fixed points.
In quantum field theories the best understood fixed points are at vanishing couplings
(Gaussian FPs) as they can be treated perturbatively. For example the GFP of the 4
dimensional SU(3) pure gauge model has one relevant operator, the
gauge coupling, and no other FP of the model is known to exist. The
Gaussian FP of 2-flavor QCD has two relevant operators, the mass and
the gauge coupling. Gauge theories with many flavors can develop, in addition to the GFP, a
new fixed point where only the mass is relevant (Banks-Zaks infrared
fixed point) \cite{Banks:1981nn}. These new FPs are rarely in the perturbative region and to study their existence
and properties is the main motivation for this paper.
\subsection{The 2-lattice matching MCRG method}
Consider a $d$-dimensional lattice model with action $S(K_{i})$.
$\{K_{i}\}$ denotes the set of all possible couplings, though in
a typical lattice simulation only a few of them are non-zero. The
system is characterized by one or more length scales, like the correlation
length $\xi$, inverse quark masses, etc. In numerical simulations we always
deal with finite volume and for now I assume a hypercubic geometry with
linear size $L=\hat{L}a$. The first step of a real space renormalization
group block transformation is to define block variables.
These new variables are defined as some kind of local average of
the original lattice variables and for a scale $s>1$ transformation
they live on an $\hat{L}/s$ lattice. By integrating out the original
variables while keeping the block variables fixed one removes the
ultraviolet fluctuations below the length scale $sa$. The action
that describes the dynamics of the block variables is usually much
more complicated than the original one, but if $s$ is much smaller
than the lattice correlation length, the long distance infrared properties
of the system are unchanged. After repeated block transformation steps
the blocked actions describe a flow line in the multi-dimensional action
space
\begin{equation}
\{K_{i}\}\equiv\{K_{i}^{(0)}\}\to\{K_{i}^{(1)}\}\to\{K_{i}^{(2)}\}\to....\,,
\end{equation}
where $\{K_{i}^{(n)}\}$ denotes the couplings after $n$ blocking
steps. While the physical correlation length is unchanged, the lattice correlation
length after $n$ blocking steps is
\begin{eqnarray}
\xi^{(n)} & = & s^{-n}\xi^{(0)}\,.
\label{eq:corr_length}
\end{eqnarray}
The RG can have fixed points only when $\xi=\infty$ (critical) or
$\xi=0$ (trivial). We are, of course, interested in the former one.
Near the critical fixed point the linearized RG transformation predicts
the scaling operators and their corresponding scaling dimensions.
\begin{figure}
\includegraphics[width=0.65\textwidth,clip]{rg_flow.eps}
\caption{Sketch of the RG flow around a FP with one relevant operator. The coupling pair $(K,K')$
indicates matched couplings whose correlation length differ by a factor of $s$. \label{fig:RG-flow}}
\end{figure}
It is easy to visualize the renormalization group flow lines when
there is only one relevant coupling at the fixed point, as illustrated
in Fig. \ref{fig:RG-flow}. The sketch depicts the flow lines in the
parameter space $\{K_{0},K_{1}\}$, where for simplicity I assume
that the critical surface is at $K_{0}=0$. Flow lines starting near
the critical surface approach the fixed point in the irrelevant direction(s)
but flow away in the relevant one. After a few RG steps the irrelevant
operators die out and the flow follows the unique renormalized trajectory
(RT), independent of the original couplings. If we can identify
two sets of couplings, $\{K_{i}\}$ and $\{K_{i}^{\prime}\}$, that
end up at the same point along the RT after repeated blocking steps,
we can conclude that their correlation lengths are identical. If they end
up at the same point along the RT but one requires one less blocking
steps to do so, according to Eq. \ref{eq:corr_length} their lattice correlation lengths
differ by a factor of $s$. This is also illustrated in Fig. \ref{fig:RG-flow}.
From $K'$ one needs 3 while from $K$ one needs 4 RG steps to reach
the same point of the RT (up to small corrections in the irrelevant
direction(s)), therefore $\xi'=\xi/s$. This gives the bare step scaling
function with scale change $s$. The two lattice matching \cite{Hasenfratz:1984hx,Hasenfratz:1984bx} is
a numerical method to identify $(K,K')$ pairs.
In order to identify a pair of couplings $(K,K^{\prime})$ with $\xi'=\xi/s$
we have to show that after $n$ and $(n-1)$ blocking steps their
actions are identical, $S(K_{i}^{(n)})=S(K_{i}^{\prime(n-1)})$. It
is quite difficult to calculate the blocked action, but fortunately
we do not need to know the actions explicitly to shows that they are
identical. It is sufficient to show that the expectation values of
every operator measured on configurations generated with one or the
other action are identical. Furthermore it is possible to create a configuration
ensemble with Boltzman weight of an RG blocked action by generating
an ensemble with the original action and blocking the configurations
themselves \cite{Swendsen:1979gn}. This suggests the following procedure for the 2-lattice
matching:
\begin{enumerate}
\item Generate a configuration ensemble of size $L^{d}$ with action $S(K)$.
Block each configuration $n$ times and measure a set of expectation
values on the resulting $(L/s^{n})^{d}$ set.
\item Generate configurations of size $(L/s)^{d}$ with action $S(K')$,
where $K'$ is a trial coupling. Block each configuration $n-1$
times and measure the same expectation values on the resulting $(L/s^{n})^{d}$
set. Compare the results with that obtained in step 1. and tune the
coupling $K'$ such that the expectation values agree.
\end{enumerate}
A few basic comments are in order:
\begin{itemize}
\item [{a)}] Since we always compare measurements on the same lattice size, the
finite volume corrections are minimal and even very small lattices
can be used.
\item[{b)}] It is not necessary to work
on lattices that are larger than the correlation length of the system,
nor does it matter if we are in the confined or deconfined phase of
the system.
\item [{c)}] If the flow lines follow the unique RT, even one operator
expectation value is sufficient to find the matching coupling, all
other operators should give the same prediction. In practice we can
do only a few blocking steps and the flow lines might not reach the
RT. That will be reflected by different operators predicting different
matching values. The spread of these predictions measure
the goodness of the matching. Increasing the number of blocking steps
improves the matching, and when the RT is reached, consequetive blocking
steps predict the same matching couplings.
\item [{d)}] The location of the fixed point and its renormalized trajectory
in the irrelevant directions depend on the block transformation. Block
transformations that have free parameters can be optimized so their
RT is reached fast and the matching is reliable after a very few RG steps.
This optimization proved essential in previous applications \cite{Hasenfratz:1984bx,Hasenfratz:1984hx,Bowler:1984hv,Hasenfratz:1987zi,DeGrand:1995ji}.
\item [{e)}] Since we can match by comparing local operators, the statistical
accuracy is usually acceptable even with small configuration sets.
\end{itemize}
If the FP has two relevant operators, the matching proceeds similarly
but one has to tune 2 operators. In practice this is much more difficult
than the tuning of a simple coupling. It is frequently easier to fix one of the relevant couplings to its FP value and proceed with the matching in the second relevant coupling as described above.
I will illustrate the above points in Sect.\ref{sect:simulations}.
\subsection{The renormalization group block transformation}
I chose a scale $s=2$ block transformation, similar to what was used
in Refs. \cite{Swendsen:1981rb,Hasenfratz:1984hx}
\begin{equation}
V_{n,\mu}={\rm Proj[}(1-\alpha)U_{n,\mu}U_{n+\mu,\mu}+\frac{\alpha}{6}\sum_{\nu\ne\mu}U_{n,\nu}U_{n+\nu,\mu}U_{n+\mu+\nu,\mu}U_{n+2\mu,\nu}^{\dagger}]\,,\label{eq:block-trans}
\end{equation}
where ${\rm Proj}$ indicates projection to $SU(3)$. The parameter
$\alpha$ is arbitrary and can be used to optimize the blocking. The
block transformation used in Refs. \cite{Hasenfratz:1984hx,Bowler:1984hv,DeGrand:1995ji} had
$\alpha$ fixed, $1-\alpha=\alpha/6$, but instead of projecting to SU(3) the blocked
link was allowed to fluctuate around $V_{n,\mu}$, depending on a free parameter.
In my experience the two block transformations are very similar.
In principle one can define an RG transformation for fermions as well.
However it is easier to do the RG transformation after the fermions
are integrated out, i.e. when the action depends on the gauge fields only.
The role of the parameter $\alpha$ is to optimize the block transformation.
While the critical surface of a system is well defined, the location
of the fixed point itself is not physical, it can be changed by changing
the RG transformation. It is important to optimize the blocking so
its FP and RT can be reached in a few steps. The optimal blocking is characterized
by
\begin{enumerate}
\item Consistent matching between the different operators: along the RT all
expectation values should agree on the matched configuration sets. Any
deviation is a measure that the RT has not been reached.
\item Consecutive blocking steps should give the same matching coupling.
When they predict different values, one can try to extrapolate to
the FP using the first non-leading critical exponent.
\end{enumerate}
In the next Section I will show that both of the above conditions can be
satisfied in numerical simulations if the blocking parameter is optimized.
\section{Simulations \label{sect:simulations}}
\subsection{SU(3) pure gauge theory \label{sect:pure_SU3}}
At the Gaussian $g=0$ FP of the pure gauge SU(3) model the gauge coupling is relevant, the
theory is asymptotically free. According to Eq. \ref{eq:db_pert} 1-loop perturbation theory predicts that the bare step scaling function is constant, independent of the
gauge coupling. The 2-loop corrections are small in a wide range of coupling, $s_b^{(\rm{pert})}\approx0.59$ is a good approximation.
The bare step scaling function was studied in Refs.\cite{Bowler:1984hv,Hasenfratz:1984bx,Hasenfratz:1984hx,Gupta:1984gq}
with the 2-lattice matching method. Here I repeat some of those
calculations with a different block transformation and extend them
to larger volumes and statistics. Where they overlap, the results I present below are consistent with the original calculations.
This section mainly serves as a test of the method.
I generated 200-300 independent configurations at several coupling values
with the Wilson plaquette gauge action
and calculated the bare step scaling function $s_b(\beta;s=2)$
matching $32^{4}$ volumes
on to $16^{4}$ , and also $16^{4}$ volumes to $8^{4}$.
The $32^{4}$ volume can be blocked up
to 4 times and compared to the $16^{4}$ volume that is blocked
up to 3 times. At each blocking level I measured 5 operators: the plaquette,
the 3 6-link loops and a randomly chosen 8-link loop.
Figure \ref{fig:plaq_match_nf0} illustrates the 2-lattice MCRG.
The plot shows the matching of the plaquette
with the $s=2$ renormalization group transformation of Eq. \ref{eq:block-trans} and blocking parameter $\alpha=0.65$.
The $32^{4}$ volume simulations
were done at $\beta=7.0$, and the cyan, blue and red
symbols are the values of the blocked plaquette after 2, 3 and 4
blocking steps. The solid curves interpolate the plaquette values, measured at many couplings on $16^4$ volumes, after
1, 2 and 3 blocking steps. The $32^{4}$
data match the $16^{4}$ values at $\beta'=6.49$ for all blocking levels.
The final blocked volume is $2^{4}$, but finite size
effects are minimal as one always compares observables on the same
volume.
\begin{figure}
\includegraphics[width=0.65\textwidth]{plaq_match_nf0.eps}
\caption{The matching of the plaquette for pure gauge SU(3) theory. The simulations
were done on $32^{4}$ volumes at $\beta=7.0$ (symbols) and $16^{4}$
volumes at many coupling values (solid interpolating lines). The configurations
were blocked with $s=2$, $\alpha=0.65$ parameter block transformation 2(1) (cyan), 3(2) (blue) and 4(3)
times (red). \label{fig:plaq_match_nf0}}
\end{figure}
\begin{figure}
\includegraphics[width=1\textwidth,clip]{sd_nf0.eps}
\caption{Matching at $\beta=7.0$ from $32^4$ to $16^4$ lattices. The matching values
$\Delta\beta$
as the function of the blocking parameter $\alpha$ are shown for the 5 different
operators measured. Left panel: blocking level $n_b=2(1)$; middle panel $n_b=3(2)$;
right panel $n_b=4(3)$. }
\label{fig:sd_nf0}
\end{figure}
\begin{figure}
\includegraphics[width=0.65\textwidth,clip]{deltabeta_vs_alpha_b7.0_nf0.eps}
\caption{Matching at $\beta=7.0$ from $32^4$ to $16^4$ lattices. The average matching values
$\Delta\beta$
as the function of the blocking parameter for the last 3 blocking levels are shown. Note that the "error bars"
denote the spread of the predictions for the 5 different
operators used and thus represent systematical errors.
}
\label{fig:deltabeta_vs_alpha_b7.0_nf0}
\end{figure}
The matching can be repeated with different operators and RG transformations. Figure \ref{fig:sd_nf0} shows
the difference between the matched couplings,
\begin{equation}
\Delta\beta = \beta -\beta'\,,
\end{equation}
as the function of the blocking parameter $\alpha$ for the 5 different operators at the last 3 blocking levels for $\beta=7.0$.
The plots show two trends. First, the spread of the predicted $\Delta\beta$ values from the different operators decrease with increasing blocking level, signaling that the RG flow lines are approaching the RT of the block transformation. Second, the dependence of $\Delta\beta$ on the blocking parameter decreases with increasing blocking levels suggesting a unique value for $\Delta\beta$ in the $n_b\to\infty$ limit.
Figure \ref{fig:deltabeta_vs_alpha_b7.0_nf0}
summarizes the plots of Figure \ref{fig:sd_nf0}. The average matching $\Delta\beta$ values are plotted as the function of $\alpha$ for the last three blocking levels. The ``error
bars'' show the standard deviation (spread) of the predicted values, therefore they
represent the systematic errors
of the matching procedure. The statistical errors are small,
comparable to the systematic errors only at the last blocking level
at the best matching around $\alpha=0.65$.
The 3 different blocking levels converge around $\alpha=0.65$, the same value where the spread
from the different operators is minimal, predicting the relation between the lattice spacings $a(\beta'=6.485) = 2 \, a(\beta=7.0)$.
In the $n_b\to\infty$ limit the quantity $\Delta\beta(\beta)=\beta-\beta'$
is the bare step scaling function $s_b(\beta;s=2)$, analogue to the
step scaling function of the renormalized coupling used in the SF formalism.
In the following I will use the intersection
of the last two blocking levels to identify $s_b(\beta)$ as it is usually less sensitive
to the statistical errors than the spread of the individual operators.
\begin{table}
\begin{tabular}{|c|c|c|c|c|}
\hline
$\beta_{\hat{L}}$ & $\alpha_{\rm{opt}}$ &
$s_b,\,\hat{L}=32$ &
$s_b,\,\hat{L}=32$ &
$s_b,\,\hat{L}=16$ \tabularnewline
& & $n_{b}=3(2)$ & $n_{b}=4(3)$ & $n_{b}=3(2)$ \tabularnewline
\hline
\hline
6.0 & 0.71 & & & 0.365(4) \tabularnewline
\hline
6.2 & 0.72 & & & 0.410(7) \tabularnewline
\hline
6.4 & 0.71 & 0.451(12) & 0.448(10) & 0.468(16) \tabularnewline
\hline
6.6 & 0.69 & 0.488(15) & 0.483(5) & 0.496(13) \tabularnewline
\hline
6.8 & 0.66 & & & 0.511(19) \tabularnewline
\hline
7.0 & 0.66 & 0.517(27) & 0.515(6) & 0.516(10) \tabularnewline
\hline
7.2 & 0.63 & & & 0.536(26) \tabularnewline
\hline
7.4 & 0.61 & 0.548(38) & 0.571(6) & 0.575(42) \tabularnewline
\hline
7.8 &0.60 & 0.558(34) & 0.575(5) & 0.573(42) \tabularnewline
\hline
\end{tabular}
\caption{The bare step scaling function for the pure gauge SU(3) system.
The second column list the optimal blocking parameter. The third and fourth columns are results from simulations on $32^{4}$
volumes matched to $16^{4}$ after 3(2) and 4(3)
blocking steps. The last column shows results from $16^{4}$ volumes
matched to $8^{4}$ after 3(2) blocking steps. }
\label{tab:pure_gauge}
\end{table}
Table \ref{tab:pure_gauge} summarizes
the results at different couplings and volumes, together with the optimal blocking parameters $\alpha$. The data indicate consistency
between the different volumes and increasing blocking levels. This observation will be important
in the study of the $N_f=4$ and 16 systems where matching is done on $16^4\to8^4$ lattices only.
The uncertainty of the predictions from the $16^4\to8^4$ are considerably larger than from the larger
volumes. This is not statistical, rather reflects the fact that the systematical errors of the matching after 3(2)
blocking steps are larger than after one more blocking level. It is possible that a different block transformation
would give better matching. I have not been able to modify the scale $s=2$ transformation to make it better. Adding HYP smearing\cite{Hasenfratz:2001hp}
before constructing the blocked links pulls the RT closer, but at the same time reduces the dependence of
the expectation values on the couplings, thus increasing the errors. It would be worthwhile to try combining the block transformation with
a simple APE smearing, or use a variation of the scale $s=\sqrt{3}$ transformation of Refs.\cite{Patel:1984dw,Gupta:1984gq} that would allow more
blocking steps from the same lattice size.
\begin{figure}
\begin{center}
\includegraphics[width=0.65\textwidth,clip]{db_nf0.eps}
\end{center}
\caption{The bare step scaling function $s_b(\beta;s=2)$ for the pure gauge SU(3) system
as predicted by different methods. The 1-loop perturbative prediction is $s_b^{(\rm{pert})}(s=2)=0.59$.
\label{fig:db_nf0}}
\end{figure}
The quantity $s_b(\beta;s)$ is the bare step
scaling function for scale change $s=2$. One can predict its value using physical observables like the Sommer
parameter $r_0$\cite{Sommer:1993ce} or the critical temperature $T_c$ . The SF calculation or the recently proposed
new method to calculate the renormalized coupling based on Wilson loop matching (WL) \cite{Bilgici:2009kh} can also predict $s_b(\beta)$.
In Figure \ref{fig:db_nf0} I compare the MCRG result for the bare step scaling function with predictions from
other methods. Note that I show errors only for the MCRG results. I used the interpolating formula from Ref.
\cite{Necco:2001xg} to find $(\beta,\beta')$ pairs where $r_0/a$ differ by a factor of 2, while for $T_c$ I used
the $N_T=8$ and 4, and $N_T=12$ and 6 transition temperatures from Ref. \cite{Boyd:1996bx}.
In case of the SF and WL calculations I attempted to find matching $(\beta,\beta ')$ pairs by identifying bare couplings where the renormalized SF couplings are related as $\bar g^2(\hat L) = \bar g'^2(2\hat L)$. In principle this relation should be taken in the $\hat{L} \to \infty$ limit but the numerical data do not show significant finite volume effects.
The predictions in Figure \ref{fig:db_nf0} use data from the 1-loop
improved SF \cite{Luscher:1993gh}.
The bare couplings used in the 2-loop improved SF paper do not match close enough to use
them in this analysis \cite{Capitani:1998mq}.
All the above calculations use the Wilson plaquette gauge action,
so in the scaling regime they should give the same prediction. It is very satisfying to see the agreement between
MCRG, $r_0$ and $T_c$ even at relatively strong couplings. In the range $\beta \in (6.0,7.0)$ the predicted
values differ considerably from the 2-loop perturbative results.
It is difficult to measure $r_0$ or $T_c$ at much finer lattice spacings and show perturbative scaling for them.
On the other hand both
the SF, WL and MCRG methods approach $s_b^{(\rm{pert})}$, but the latter one only at $\beta\ge7.0$. The relatively large difference between the SF
and $r_0$ data was discussed and analyzed in Ref. \cite{Necco:2001xg} where it was also noted that the 2-loop improved SF
shows significantly smaller scaling violations relative to $r_0$.
Based on the results presented in this section the 2-lattice MCRG matching method could be competitive with other methods in determining the running coupling of asymptotically free theories when it is combined with an independent definition of the renormalized coupling in the weak coupling regime.
\subsection{$N_f=4$ flavor model \label{sect:nf4}}
The 4 flavor SU(3) gauge theory is expected to be confining and chirally broken even at large gauge couplings.
At the Gaussian $g=0,\,m=0$ FP both the mass and the gauge coupling are relevant operators, the
model is asymptotically free.
Perturbation theory predicts $s_b^{(\rm{pert})}(s=2)=0.45$.
The results I present here were obtained using nHYP smeared staggered fermions \cite{Hasenfratz:2007rf}.
I chose nHYP smearing as it significantly reduces taste breaking of staggered fermions and therefore
even in strong coupling has manageable lattice artifacts.
\subsubsection{The nHYP staggered action \label{sect:nHYP_action}}
Very little is known about the 4-flavor system with nHYP or HYP smeared fermions.
The finite temperature phase transition of the HYP smeared model
was studied with the partial-global Monte Carlo update in Ref. \cite{Hasenfratz:2001ef}. The phase transition in the chiral limit is expected to be first order,
most likely extending to finite mass before turning into a crossover at large masses.
Simulations with thin link staggered fermions have confirmed this, finding a strong discontinuity even at fairly large quark masses.
The conclusion of Ref. \cite{Hasenfratz:2001ef} was quite different: we found no signal for discontinuity, the
phase transition appeared to be a crossover both for $N_T=4$ and 6 even at fairly
small masses. The updating technique used in Ref. \cite{Hasenfratz:2001ef} was not efficient enough to pursue much larger volumes, and
we did not continue our investigation of the $N_f=4$ system.
\begin{figure}[htbp]
\centering
\includegraphics[width=1\textwidth,clip]{nf4_NT4_0.04.eps}
\caption{ The condensate and the disconnected chiral condensate on $8^3\times4$ volumes at $m=0.04$ in the $N_f=4$ theory. }
\label{fig:nf4_NT4}
\end{figure}
\begin{figure}[htbp]
\centering
\includegraphics[width=1\textwidth,clip]{nf4_NT6_0.05.eps}
\caption{ The condensate and the disconnected chiral condensate on $12^3\times6$ volumes at $m=0.05$ in the $N_f=4$ theory. }
\label{fig:nf4_NT6}
\end{figure}
The nHYP smeared action is nearly identical to the HYP smeared one, but nHYP is differentiable and the efficient molecular dynamics
update can be used with it \cite{Hasenfratz:2007rf}.
I have confirmed the raw
data of Ref. \cite{Hasenfratz:2001ef} with the nHYP action, and extended it further
toward the strong coupling region. Figure \ref{fig:nf4_NT4} shows the condensate
$\langle \bar{\psi}\psi\rangle$ and the disconnected chiral susceptibility \cite{Bernard:1996zw}
\begin{equation}
\chi_{\rm{disc}}=\langle\langle\bar{\psi}\psi\rangle^2_{\rm{conf}}\rangle_U - \langle\langle\bar{\psi}\psi\rangle_{\rm{conf}}\rangle^2_U\,
\end{equation}
on $8^3\times4$ lattices at $m=0.04$. The condensate is almost identical to
Figure 5 of \cite{Hasenfratz:2001ef}, a smooth function of the gauge
coupling with no obvious sign of discontinuity. The disconnected chiral susceptibility has a strong peak at $\beta=4.4$,
signaling a crossover. In contrast,
the data with thin link staggered fermions at $N_T=4$ show a discontinuity in the condensate $\Delta\langle \bar{\psi}\psi\rangle\approx 0.2$. Figure
\ref{fig:nf4_NT6} is the same as Figure \ref{fig:nf4_NT4} but on $12^3\times 6$ lattices at $m=0.05$. Again, the condensate is smooth, the susceptibility suggests a crossover around $\beta=5.0$. Obviously much more work is needed to determine the transition temperatures and the order of the phase transition accurately. Larger spatial volumes might sharpen the transition, but in any case the endpoint of the first order line
with nHYP fermions occurs at much smaller masses than with the thin link action.
To set the scale I measured the static potential on $16^4$ lattices used later in the MCRG study. I found $r_0/a = 5.8(3)$ at $\beta=5.4$, $m=0.01$. Ref. \cite{Hasenfratz:2001ef} quotes $r_0/a=3.34(7)$ at $\beta=5.2$, $m=0.04$ and $r_0/a=2.2(1)$ at $\beta=5.0$, $m=0.10$.
\subsubsection{MCRG matching}
\begin{figure}[htbp]
\centering
\includegraphics[width=0.65\textwidth,clip]{plaq_match_nf4_0.75.eps}
\caption{Matching of the plaquette in the $N_f=4$ model. The blue and red symbols show the
plaquette value after 2 and 3 blocking steps on the $16^4$ volumes at
$\beta=6.0$, $m=0.01$. The lines interpolate the plaquette
on the $8^4$ volumes at several couplings and $m=0.015$ (dashed) and $m=0.025$ (dotted)
after 1 and 2 blocking levels. The black line is the unblocked plaquette on the $8^4$ lattices.
The dashed and dotted curves are barely distinguishable. }
\label{fig:plaq_match_nf4}
\end{figure}
In principle MCRG matching could be done similarly to the pure gauge SU(3) model,
but since the fermionic model has 2 relevant operators, the matching requires tuning in both the gauge coupling and the mass .
This is a considerably harder numerical task than a single parameter matching. One can reduce this
complication by setting one of the relevant couplings to its critical value, since then only the other
coupling has to be matched. The critical value of the mass is $m=0$. Simulations in small volumes are
possible even with vanishing mass. In addition the dependence of the local observables used in the matching
is so weak on the mass that a small mass in the simulations is also acceptable. Setting the mass to zero or to a small value allows
matching in the gauge coupling only and the bare step scaling function can be calculated in the same way
as for the pure gauge system.
To calculate the step scaling function I considered $16^4\to8^4$ matching at several
gauge couplings between $\beta\in(5.4,8.0)$ (see Table \ref{tab:NF4}). All the $16^4$ configurations except $\beta=5.4$ are in the deconfined, chirally symmetric phase, but that does not matter for the MCRG matching method.
I have generated 100-150 configurations on the larger volumes and $\approx300$
configurations on the smaller ones, separated by 10 molecular dynamics trajectories. I have used the
same 5 operators in the matching as in the pure gauge SU(3) system.
\begin{figure}
\includegraphics[width=0.65\textwidth,clip]{deltabeta_nf4_6.0.eps}
\caption{Matching at $\beta=6.0$ from $16^4$ to $8^4$ lattices in the $N_f=4$ model. The matching values
$\Delta\beta$
as the function of the blocking parameter for the last 2 blocking levels are shown. Note that the error bars
denote the spread of the predictions for the 5 different
operators used and thus represent systematical errors.
}
\label{fig:deltabeta_nf4}
\end{figure}
\begin{table}
\begin{tabular}{|c|c|c|}
\hline
$\beta_{16}$ & $\alpha_{\rm{opt}}$ & $s_b(s=2)$ \tabularnewline
\hline
\hline
5.4 & 0.81 &0.388(10) \tabularnewline
\hline
5.6 & 0.81 &0.330(13) \tabularnewline
\hline
5.8 & 0.77 &0.335(20) \tabularnewline
\hline
6.0 & 0.76 & 0.303(25) \tabularnewline
\hline
6.4 & 0.67 & 0.400(29) \tabularnewline
\hline
6.8 & 0.67 & 0.365(42) \tabularnewline
\hline
7.2 & 0.63 & 0.404(39) \tabularnewline
\hline
8.0 & 0.59 &0.470(53) \tabularnewline
\hline
\end{tabular}
\caption{The bare step scaling function
$s_b(s=2)$ for the 4-flavor simulation. The second column lists the optimal blocking parameter $\alpha$.\label{tab:Nf4}}
\label{tab:NF4}
\end{table}
\begin{figure}[htbp]
\centering
\includegraphics[width=0.65\textwidth,clip]{db_nf4.eps}
\caption{The bare step scaling function for 4 flavor SU(3) theory. The dashed line indicates the 1-loop perturbative value.}
\label{fig:db_nf4}
\end{figure}
On the $16^4$ lattices I chose $m=0.01$. If the critical exponent for the mass were its engineering dimension, the matching
mass on the smaller volume would be $m=0.02$. I generated $8^4$ lattices with $m=0.015$ and 0.025 to bracket this value and
to check for any dependence on the mass. The matching for the plaquette is shown in Figure \ref{fig:plaq_match_nf4}. The
blue and red symbols represent the plaquette value after 2 and 3 blocking steps on the $16^4$ lattice at $\beta=6.0$ while the dashed and dotted lines interpolate the 1 and 2 times blocked plaquette values on the $8^4$ lattices with $m=0.015$ and $m=0.025$. For completeness I also show the unblocked plaquette on the $8^4$ volumes (black line) though there is no consistent matching at this level.
The dotted and dashed curves are barely distinguishable. None of the other observables show any dependence on the mass even after 3
blocking steps beyond the fairly small statistical errors of the simulations, implying that the system indeed can be considered
critical in the mass.
The dependence of the matched values on the blocking parameter $\alpha$ is similar to the pure gauge case. The analogue of Figure \ref{fig:deltabeta_vs_alpha_b7.0_nf0} is Figure \ref{fig:deltabeta_nf4}. Since for $16^4\to8^4$ matching only two blocking levels can be used, one has only two sets of predictions, $n_b=2(1)$ and $n_b=3(2)$. In Sect. \ref{sect:pure_SU3}, Table \ref{tab:pure_gauge} I found that one
can safely identify the optimal blocking parameter and matching value from the intersection of
these blocking levels even on $16^4\to8^4$ matching.
Table \ref{tab:NF4} and Figure \ref{fig:db_nf4} summarize the simulation results.
The step scaling function $s_b(s=2)$ is consistent with asymptotic freedom and approaches the 2-loop perturbative prediction
at weak couplings.
Just like in the pure gauge system, the data in Table \ref{tab:NF4} could be used to determine the running coupling of the
4-flavor model. To do so one needs to calculate a renormalized coupling at $\beta=8.0$ and connect it through perturbation
theory to a continuum scheme. The change of the lattice scale between $\beta=5.4$ and $\beta=8.0$ can be determined form the data, while
the lattice spacing can be obtained by measuring a physical quantity at some strong coupling. Improved block transformation or larger volumes
would allow a more precise determination of $s_b(\beta)$. Higher statistics especially at the larger $\beta$ values, would also help.
\subsection{$N_{f}=16$ flavor model \label{sect:nf16}}
The 16 flavor SU(3) model is still asymptotically free, but
2-loop perturbation theory predicts the emergence of an IRFP
at weak gauge coupling \cite{Banks:1981nn}. Continuum limits can be defined both at the Gaussian
FP and at the new IRFP. In the former case both the gauge coupling and the mass has to be tuned
towards the FP, in the latter one the gauge coupling is irrelevant, only the mass has to be tuned to $m=0$.
There is no confinement or spontaneous chiral symmetry breaking
in the weak coupling phase, the continuum massless theory at the IRFP is conformal. The conformal phase was first identified in Ref. \cite{Heller:1997vh}.
It is generally believed that in the strong coupling the lattice model is confining and chirally
broken, so there has to be a (bulk) transition separating the strong
and weak coupling phases.
Since the bulk transition is a lattice artifact, it is probably not
associated with critical behavior or continuum quantum field theory.
Most likely it is a first order phase transition at $m=0$ and it might extend to $m>0$ before
turning into a crossover \cite{Damgaard:1997ut,DeGrand:2009mt}. One should mention that some numerical results indicate that
this confining phase might not even exists \cite{Iwasaki:2003de}.
\subsubsection{MCRG around an infrared fixed point}
Assuming that the simulations are done in the conformal phase, the behavior
we expect from MCRG depends on whether we study the critical ($m=0$) or the $m\ne 0$ phases:
\begin{itemize}
\item
On the $m=0$ critical surface at very weak coupling the system is in the attractive region of the
Gaussian FP. 2-lattice matching could reveal the running of the gauge coupling, $s_b^{(\rm{pert})}=0.016$
from the 1-loop $\beta$ function, though
numerical simulations probably will not be close enough to the Gaussian FP to see this behavior. It is much more likely
that the RG flow will be determined by the Banks-Zaks IRFP. At this FP all operators are irrelevant when $m=0$. The 2-lattice matching is designed
to map out the flow speed in the relevant direction, so we have to re-evaluate the method when there is no relevant operator.
When all operators are irrelevant, they all flow into the FP according to their scaling dimensions. In the $n_b\to\infty$ limit
all expectation values approach their FP value independent of the bare couplings, so matching is meaningless. At finite $n_b$
the RG flow can pick up the "least irrelevant" operator. If there is one operator with a nearly zero
scaling dimension matching can make sense: after a few RG steps all other operators are already in the FP, so the flow line
follows that single operator. According to Eq. \ref{eq:gamma_g} the scaling
exponent of the gauge coupling for the $N_f=16$ flavor theory at the Banks-Zaks FP is small, $\gamma_g= -0.01$, it
is almost a marginal operator. The scaling exponents of the other irrelevant operators are likely close to their engineering dimensions, starting
at $\gamma\approx-2$, so these operators will die out much faster than the gauge coupling.
The picture we expect in the 2-lattice matching is now clear. Since the gauge coupling is nearly marginal, the matching will follow its flow. $s_b(\beta)$ is given by
Eq. \ref{eq:db_IRFP} and for a marginal operator $s_b(\beta)=0$ near the FP.
For an almost marginal operator $\Delta\beta=\beta-\beta'$
can be either positive or negative, depending on whether the FP of the actual RG transformation is at smaller or larger gauge coupling.
The evolution of the blocked operators can also signal the IRFP. As the RG flow approaches the IRFP all expectation values approach their
IRFP value. On the other hand if the GFP controls the system the flow lines follow the RT and run into the trivial $\beta=0$ FP where all local expectation values vanish. This difference is one of the strongest signal for the existence of an IRFP in the MCRG method.
\item
At finite mass the RG transformation is dominated by the flow of the relevant mass operator. However the nearly marginal gauge coupling can still have a strong influence on the flow, the situation is more like matching 2 relevant operators than matching a single one.
The easiest way to deal with this is to set the gauge coupling to its FP value (i.e. to the value that corresponds to the IRFP of the RG transformation used) and match in the mass only. This matching predicts the scaling dimension (or critical exponent) of the mass.
While the IRFP of the RG transformation depends on the blocking parameter, the exponent itself is independent of both $\alpha$ and the gauge coupling $\beta$.
\end{itemize}
\subsubsection{Numerical simulations and results}
I concentrate on the renormalization group properties of the model
in the massless limit and show only preliminary results at finite mass.
I did 2-lattice matching on $16^4\to8^4$ lattices using nHYP smeared staggered fermions \cite{Hasenfratz:2007rf}, and as in the $N_f=4$ case, the simulations were done with a small mass, $m=0.01$ on the $16^4$ and $m=0.02$ on the $8^4$ lattices. I have collected 100-150 configurations on the larger volumes, $\sim200$ on the smaller ones, separated by 10 molecular dynamics steps.
On the $8^4$ lattices I covered the coupling range $\beta\in(2.4,8)$, trying to identify the bulk transition to the strong coupling phase. The data show no sign of a phase transition, though for $\beta<4.0$
the condensate starts increasing slowly, suggesting the development of spontaneous chiral symmetry breaking. It is likely that the bulk transition exists only at very small, possibly vanishing, mass. This
does not contradict the results of Ref. \cite{Damgaard:1997ut} where a strong first order bulk transition was observed with
$N_f=16$ fermions.
As we learned from the finite temperature investigation with $N_f=4$ flavors in Sect. \ref{sect:nHYP_action},
smearing the fermionic action can soften or wash away first order phase transitions, and that might be the situation here as well.
\begin{figure}
\begin{center}
\includegraphics[width=0.65\textwidth,clip]{plaq_vs_mass_5.8_0.75.eps}
\end{center}
\caption{The dependence of the plaquette on the mass after 1 (red diamonds and solid line) and 2 (blue diamonds and solid line)
blocking steps on $8^4$ volumes at $\beta=5.8$. The bursts at $m=0.15$ show the plaquette after 2 and 3 blocking steps starting form $16^4$ volumes at the same $\beta$ value. The dashed lines indicate matching in the mass.
The blocking parameter is $\alpha=0.75$, close to the optimal value at $\beta=5.8$. }
\label{fig:plaq_vs_mass}
\end{figure}
\begin{figure}
\begin{center}
\includegraphics[width=0.65\textwidth,clip]{plaq_match_nf16_0.78.eps}
\end{center}
\caption{The matching of the plaquette for $N_f=16$ flavor. The individual
data points are at $\beta=5.6$ on $16^4$ lattices after 2 and 3 levels of blocking.
They are compared to the 1 and 2 times blocked values as measured on the $8^4$ lattices
(solid lines).
The matching values $\Delta\beta\approx 0$ indicate a
nearly marginal flow. Note that the plaquette increases with $n_b$ implying that the RG flow is to an IRFP and not the $\beta=0$ trivial FP. }
\label{fig:nf16_plaq_match}
\end{figure}
Just like in the $N_f=4$ case the mass dependence of the measured operators is weak for small $m$.
Figure \ref{fig:plaq_vs_mass} shows the plaquette blocked 1 and 2 times on the $8^4$ configurations at $\beta=5.8$ at different masses.
The solid lines are simple spline extrapolations. Within errors there is no mass dependence up to about $m\le0.05$ for the plaquette.
Other observables are similar, so for the MCRG matching the $m=0.01-0.02$ data set can be considered critical in the mass.
Figure \ref{fig:nf16_plaq_match} is the analogue of Figures \ref{fig:plaq_match_nf0} and \ref{fig:plaq_match_nf4}, showing the
matching of the plaquette. The data points are at $\beta=5.6$ on $16^4$ lattices after 2 and 3 levels of blocking.
They are compared to the 1 and 2 times blocked values as measured on the $8^4$ lattices. The block transformation is
with parameter
$\alpha=0.78$, optimal for $\beta=5.6$ and matching is consistent for both blocking levels with $\Delta\beta= 0$.
From the matching value alone it is not possible to distinguish a marginally relevant flow (like at the GFP) from an
almost marginal irrelevant flow (expected at the IRFP). Comparing Figure \ref{fig:nf16_plaq_match}
to the $N_f=4$ or pure gauge Figures \ref{fig:plaq_match_nf0} and \ref{fig:plaq_match_nf4}
reveals an important difference. When the flow is governed
by the GFP the flow lines follow the RT towards the trivial $\beta=0$ FP. In the $n_b\to\infty$ limit all expectation values vanish. In Figures \ref{fig:plaq_match_nf0} and \ref{fig:plaq_match_nf4} the plaquette indeed decreases with increasing blocking levels.
On the other hand when the flow
lines approach an IRFP all expectation values take the value at the FP. The plaquette in Figure \ref{fig:nf16_plaq_match} increases with
increasing blocking steps for $\beta \ge 5.4$, indicating that the flow lines are not running towards the $\beta=0$ FP. Eventually all expectation values should become independent of the blocking steps and $\beta$. This trend is not obvious from the figure for two main reasons. The first is
that finite volume effects are considerable on $2^4$ lattices, the second that
with only 2 or 3 blocking steps
one approaches the FP only if the RG transformation is optimized. Nevertheless the fact that with optimal blocking the blocked operators increase with $n_b$ already signals
that the flow runs toward an IRFP.
Other operators show similar behavior at $\beta\ge 5.6$ but the results are quite different at stronger gauge couplings. Figure \ref{fig:nf16_plaq_match} shows that the blocked plaquette values cross around $\beta=5.4$ and they decreases with $n_b$ for $\beta<5.4$. The location of the crossing depends on the blocking parameter but for the $N_f=16$ flavor model it never drops below $\beta=5.4$. It appears that the flow is running towards the $\beta=0$ FP when $\beta<5.4$ and to the IRFP when $\beta>5.4$.
\begin{figure}
\begin{center}
\includegraphics[width=1\textwidth,clip]{deltabeta_nf16.eps}
\end{center}
\caption{Matching at $\beta=5.4$ (left panel) and $\beta=6.6$ (right panel) from $16^4$ to $8^4$ lattices in the $N_f=16$ flavor theory. }
\label{fig:nf16_db_a}
\end{figure}
\begin{table}
\begin{tabular}{|c|c|c|}
\hline
$\beta_{16}$ & $\alpha_{\rm{opt}}$ & $s_b(s=2)$ \tabularnewline
\hline
\hline
5.4 & & none \tabularnewline
\hline
5.6 & 0.78 & 0.00(3) \tabularnewline
\hline
5.8 & 0.78 & -0.12(4) \tabularnewline
\hline
6.2 & 0.66 & 0.09(5) \tabularnewline
\hline
6.6 & 0.67 & -0.02(4) \tabularnewline
\hline
\end{tabular}
\caption{The parameters and MCRG results of the $N_{f}=16$ flavor simulations. The second column lists the optimal blocking parameter $\alpha$. \label{tab:nf16}}
\end{table}
\begin{figure}
\begin{center}
\includegraphics[width=0.65\textwidth,clip]{m1_vs_m2.eps}
\end{center}
\caption{Matched $(m_1,m_2)$ pairs in the $N_f=16$ system. Red diamond: $\beta=5.8$, blue bursts: $\beta=6.6$. The linear fit predicts
$y_m=1.02(7)$. }
\label{fig:m1_vs_m2}
\end{figure}
\begin{table}
\begin{tabular}{|c|c|c|c|}
\hline
$\beta$ & $\alpha_{\rm{opt}}$ & $m_1$ & $m_2$ \tabularnewline
\hline
\hline
5.8 & 0.75 & 0.10 & 0.17(3) \tabularnewline
\hline
5.8 & 0.75 & 0.15 & 0.31(2) \tabularnewline
\hline
6.6 &0.68 & 0.07 & 0.16(3) \tabularnewline
\hline
6.6 & 0.68 & 0.10 & 0.20(4) \tabularnewline
\hline
6.6 & 0.68 & 0.15 & 0.33(4) \tabularnewline
\hline
\end{tabular}
\caption{Matched mass pairs in the $N_f=16$ system. \label{tab:mass_match}}
\end{table}
The 2-lattice matching is also different below and above $\beta=5.4$. Figure \ref{fig:nf16_db_a} shows $\Delta\beta$ as the function of the blocking parameter $\alpha$ for $\beta=5.4$ and $\beta=6.6$. This figure is the analogue of Figure \ref{fig:deltabeta_vs_alpha_b7.0_nf0}. As the left panel of Figure \ref{fig:nf16_db_a} shows the $n_b=2(1)$ and 3(2) blocking levels
get close but do not actually converge at $\beta=5.4$, there is no consistent matching. The situation is similar, even more enhanced, at stronger couplings.
I have not observed this kind of behavior either with $N_f=0$ or 4, though I had simulations at even stronger couplings
there.
While it is
possible that higher blocking levels would predict matching and the expectation values eventually approach their FP value,
it is more likely that we see the effect of a nearby bulk transition and the strong coupling confining region beyond it.
Apparently $\beta\le5.4$ is not governed by the IRFP. The situation is entirely different at $\beta=6.6$ where predictions from the two different blocking levels converge around $\alpha=0.675$ predicting $\Delta\beta=-0.022(44)$ (right panel of Figure \ref{fig:nf16_db_a}). Results are similar at other $\beta > 5.4$ couplings, as summarized in Table \ref{tab:nf16}. The optimal block transformation predicts $\Delta\beta\approx0$ for all coupling values, in agreement with the expectations, i.e. that the RG flows are governed by an almost marginal operator. Combining this with the observation that with optimal blocking parameter the expectation values of the blocked operators increase leads to the conclusion that the
$\beta\in(5.6,6.6)$ coupling range in the massless limit is governed by an IRFP.
The mass is a relevant operator at this IRFP, therefore it should scale according to Eq. \ref{eq:mass_scale} under an $s=2$ RG transformation.
One can calculate the exponent $y_m$ by identifying matched $(m_1,m_2)$ mass values. The gauge coupling is irrelevant, any
$\beta$ in the attractive basin of the IRFP could be chosen for this. Since the gauge coupling is nearly marginal it is best to use the same coupling at both mass values.
For matching one can use the same operators as before or add others that are more sensitive to the ferminons. I believe that direct
fermionic observables would allow more precise matching at smaller quark masses, but as Figure \ref{fig:plaq_vs_mass} illustrates the gauge observables also work. In fact Figure \ref{fig:plaq_vs_mass} already shows a matched mass pair. At $\beta=5.8$ with $\alpha=0.75$ RG transformation the $m_1=0.15$ mass on the $16^4$ configurations match $m_2=0.318(6)$ on the $8^4$ configurations after 2(1) blocking steps, while the matching mass is $m_2=0.297(10)$ after 3(2) blocking steps. These values are for the plaquette but other observables give similar values, predicting $m_2=0.31(3)$ at the optimal $\alpha=0.75$ parameter. I have preliminary data at $\beta=5.8$ and 6.6 at a couple of mass values. The matching $(m_1,m_2)$ pairs are listed in Table \ref{tab:mass_match}. The fit according to Eq. \ref{eq:mass_scale} predicts $y_m=1.02(7) $ as shown in Figure \ref{fig:m1_vs_m2}
The scaling dimension of the mass is very close, within errors undistinguishable, form it engineering dimension. This is not unexpected as the IRFP is at weak coupling. In a recent publication $y_m=1.5$ was predicted for sextet fermions \cite{DeGrand:2009et}.
It would be interesting to study the sextet model, or the $N_f=12$ model, where the $y_m$ might be significantly different from 1.
\section{Conclusion}
Renormalization group methods have been designed to study the critical behavior of statistical systems. In this paper I have shown that
they are equally suitable to study the renormalization group structure of quantum field theories. I have used a numerical Monte Carlo Renormalization Group method to calculate the
step scaling function and critical exponent of SU(3) gauge theories with $N_f=0$, 4 and 16 flavors. I chose these fairly well understood systems
as my goal was to test the method before using it in more relevant simulations. The paper is fairly pedagogical, explaining in detail the 2-lattice matching MCRG method.
In the $N_f=0$ case I demonstrated that the bare step scaling function predicted by the 2-lattice matching method is consistent with the more traditional
Schrodinger functional results. It is also consistent with results obtained from the scaling of the $r_0$ parameter and the finite temperature phase transition
even at strong gauge coupling, suggesting scaling, though not 2-loop perturbative scaling there.
The $N_f=4$ and 16 flavor simulations were done using nHYP smeared staggered fermions. I chose nHYP smearing because its
highly improved taste symmetry.
In the $N_f=4$ flavor case I briefly studied the finite temperature phase transition to develop a feel for the parameters of the model.
I calculated the step scaling function at vanishing quark mass and showed that the 2-lattice method
works equally well in the fermionic system.
The $N_f=16$ flavor model brings in new challenges as it is governed by an infrared fixed point at finite gauge coupling. At this IRFP the
scaling dimension of the gauge coupling is small, it is an almost marginal operator. Accordingly the gauge coupling runs very slowly. The
measured step scaling function is consistent with an almost marginal operator, but on its own it cannot predict if the gauge coupling is relevant or irrelevant.
I argued that the evolution of blocked operators signal if the RG flow is towards an IRFP or to the $\beta=0$ trivial FP. For the $N_f=16$ model the flow clearly indicates an IRFP.
Finally I presented preliminary measurements for the scaling dimension of the mass. I found $y_m=1.02(7)$, undistinguishable from the
free field exponent. This is not surprising for the $N_f=16$ theory.
\section{Acknowledgment }
I thank T. DeGrand and J. Kuti for discussions and T. DeGrand for his help with modifying the staggered nHYP code for many fermions.
This research was partially supported by the US Dept. of Energy.
\bibliographystyle{apsrev}
|
astro-ph/0512163
|
\section{Introduction}
In cold dark matter cosmology, the initially smooth distribution of
matter in the universe is expected to collapse into a complex network
of filaments and voids, structures which have been termed the ``cosmic
web''. The filamentary distribution of galaxies in the nearby universe
has been revealed in detail by recent large galaxy redshift surveys
such as the 2dFGRS (Colless et al. 2001, Baugh et al. 2004), the
Sloan Digital Sky Survey (SDSS, Stoughton et al. 2002, Doroshkevich
et al. 2004) and the 2$\mu m$ All Sky Survey (2MASS, Maller et al.
2002). Numerical simulations successfully reproduce this network
(Jenkins et al. 1998; Colberg et al. 2004) and indicate that galaxies
are only the tip of the iceberg in this cosmic web (Katz et al. 1996;
Miralda-Escud\'{e} et al. 1996). Hydrodynamic simulations suggest
that at the present epoch, in addition to dark matter and galaxies,
the filaments are also composed of a mixture of cool, photoionised gas
(the low$-z$ remnants of the \mbox{Ly$\alpha$}\relax\ forest) and a shock heated,
low-density gaseous phase at temperatures between $10^5$~K and
$10^7$~K that contains most of the baryonic mass, the ``warm-hot''
intergalactic medium (WHIM, Cen \& Ostriker 1999; Dav\'{e} et
al. 1999).
Observational constraints on the physical conditions, distribution,a
nd metal enrichment of gas in the low-redshift cosmic web are
currently quite limited. The existence of the WHIM appears to be a
robust prediction of cosmological simulations (Dav\'e et al. 2001).
Thus, observational efforts are increasingly being invested in the
search for WHIM gas and, more generally, the gaseous filamentary
structures predicted by the models. Large-scale gaseous filaments have
been detected in X-ray emission (Wang et al. 1997; Scharf et al. 2000;
Tittley \& Henriksen 2001; Rines et al 2001). However, X-ray emission
studies with current facilities predominantly reveal gas which is
hotter and denser than the WHIM; this X-ray emitting gas is not
expected to contain a substantial portion of the present-epoch baryons
(Dav\'{e} et al. 2001). The most promising method for observing the
WHIM in the near term is to search for UV (\ion{O}{6}, \ion{Ne}{8})
and X-ray (\ion{O}{7}, \ion{O}{8}, \ion{Ne}{9}) absorption lines due
to WHIM gas in the spectra of background QSOs/AGNs (Tripp et al.
2000, 2001; Savage et al. 2002,2005; Nicastro et al. 2002; Bergeron
et al. 2002; Richter et al. 2004; Sembach et al. 2004; Prochaska et
al. 2004; Danforth \& Shull 2005). While absorption lines provide a
sensitive and powerful probe of the WHIM, the pencil-beam nature of
the measurement along a sight line provides little information on the
context of the absorption, e.g., whether the lines arise in an
individual galaxy disk/halo, a galaxy group, or lower-density regions
of a large-scale filament or void.
Thus, to understand the nature of highly ionised absorbers at low
redshifts, several groups are pursuing deep galaxy redshift surveys
and observations of QSOs behind well-defined galaxy groups or
clusters. For example, to study gas located in large-scale filaments,
Bregman et al. (2004) have searched for absorption lines indicative of
the WHIM in regions between galaxy clusters/superclusters and have
identified some candidates. In this paper, we carry out a similar
search as part of a broader program that combines a large {\it HST}
survey of low$-z$ \ion{O}{6} absorption systems observed on sight
lines to low$-z$ quasars (Tripp et al. 2004) and a ground based survey
to measure the redshifts and properties of the galaxies foreground to
the background QSOs. The ground based survey is done in two steps:
first, multi-band (U,B,V,R and I) imagery is obtained to identify the
galaxies and to estimate their photometric redshifts. Then,
spectroscopic redshifts are obtained for the galaxies that are
potentially (according to the photometric redhshifts) at lower
redshift that the background object. As part of the large {\it HST}
survey, we have observed the quasar HS0624+6907 ($z_{\rm QSO}$ =
0.3700) with the E140M echelle mode of the Space Telescope Imaging
Spectrograph (STIS) on board the {\it Hubble Space Telescope}. We have
also obtained multiband images and spectroscopic redshifts of galaxies
in the \mbox{HS~0624+6907}\relax field. The sight line to \mbox{HS~0624+6907}\relax passes by several
foreground Abell clusters (\S~\ref{sec:abell_clusters}) and provides
an opportunity to search for gas in large-scale filaments. We shall
show that gas (absorption systems) and galaxies are detected at the
redshifts of the structures delineated by the Abell clusters in this
direction. While the absorbing gas is intergalactic, and it is likely
that we are probing gas in cosmic web filaments, the properties of
these absorbers are surprising. Instead of low-metallicity WHIM gas,
we predominantly find cool, photoionised, and high-metallicity gas in
these large-scale structures.
This paper is organized as follows. The observations and data
reduction procedures are described in \S2, including {\it HST}/STIS
and {\it Far Ultraviolet Spectroscopic Explorer} ({\it FUSE}) observations as well
as ground-based imaging and galaxy redshift measurements. In \S3, we
present information on the foreground environments probed by the
\mbox{HS~0624+6907}\relax sight line, derived from the literature on Abell clusters and
from our new galaxy redshift survey. The absorption-line measurement
methods are described in \S4, and we investigate the physical state
and metallicity of the absorbers in \S5. Section 6 reviews the
properties of the full sample of Ly$\alpha$ lines derived from the
STIS spectrum with emphasis on the search for broad Ly$\alpha$ lines.
Section 7 discusses the implications of this study, and we summarize
our conclusions in \S8. Throughout this paper, we use the following
cosmological parameters: $h_{70}=H_0/70$~\mbox{km~s$^{-1}$}\relax, $\Omega_m=0.3$ and
$\Lambda_o=0.7$.
\section{Observations}
\subsection{Ultraviolet QSO Spectroscopy}
\mbox{HS~0624+6907}\relax\ was observed with STIS on 2 Jan. 2002 and 23-24 Feb. 2002 as part of
a Cycle 10 {\it HST} observing program (ID=9184) . The echelle
spectrograph was used with the E140M grating which provides a
resolution of 7~\mbox{km~s$^{-1}$}\relax\ FWHM and covers the 1150$-$1730~\AA\ range with
only a few small gaps between orders at wavelengths greater than
1630~\AA. The $0\farcs 2 \times 0\farcs 06$ entrance aperture was
used to minimize the effect of the wings of the line spread function.
The total exposure time was 61.95~ksec. The data were reduced as
described in Tripp et al. (2001) using the STIS Team version of
CALSTIS at the Goddard Space Flight Center. The final signal-to-noise
(S/N) per resolution element is 3 at 1150~\AA, increases
linearly to 14 at 1340\AA\ and then decreases to 7 at
1730~\AA. For further information on the design and performance of
STIS, see Woodgate et al. (1998) and Kimble et al. (1998).
HS0624+6907 was also observed by the {\it FUSE}\ PI Team on several
occasions between 1999 November and 2002 February (Program IDs
P1071001, P1071002, S6011201, and S6011202) . {\it FUSE}\ records spectra
with four independent spectrographs (``channels''), two with SiC
coatings for coverage of the 905$-$1105 \AA\ wavelength range, and two
with LiF coatings optimized to cover 1100$-$1187 \AA\ (see Moos et al.
2000,2002 for details about {\it FUSE}\ design and performance). The
spectrograph resolutions range from 20$-$30 km s$^{-1}$ (FWHM). For
HS0624+6907, the total integration time in the LiF1 channel was 110
ksec; the other channels had somewhat lower integration times due to
channel coalignment problems during some of the observations. We have
retrieved the {\it FUSE}\ spectra from the archive and have reduced the
data using CALFUSE version 2.4.0 as described in Tripp et al.
(2005). Because the spectra in the individual channels have modest S/N
ratios, we have aligned and combined all available LiF channels to
form the final spectra that we used for our measurements (we find that
combining all available LiF data does not degrade the spectral
resolution). For the spectral range uniquely covered by the SiC
channels, we used only the SiC2a data. Finally, we compared absorption
lines of comparable strength (e.g., \ion{Fe}{2} $\lambda$1144.94
vs. \ion{Fe}{2} $\lambda$1608.45) observed by {\it FUSE} and STIS in
order to align the {\it FUSE} spectrum with the STIS spectrum and
thereby correct the wavelength zero point of the {\it FUSE}\ data.
\subsection{Optical Galaxy Imaging and Spectroscopy}\label{sec:optgal}
One of the primary goals of our low$-z$ QSO absorption line program is to study the
connections between galaxies and absorption systems. These
studies require good imaging (for galaxy target selection and information on
individual galaxies of interest) followed by optical spectroscopy for
accurate redshift measurements. To initiate the galaxy-absorber study
toward \mbox{HS~0624+6907}\relax , we first obtained a $10' \times10'$ mosaic of images
centered on the QSO with SPICam on the Apache Point Observatory (APO)
3.5m telescope on 2002 October 5. Subsequently, we obtained images of
a larger field in better seeing with the NOAO 8k$\times$8k CCD
mosaic camera (MOSA, Muller et al. 1998), on the Kitt Peak National
Observatory (KPNO) 4m telescope. The SPICam images were used to select
targets for the first spectroscopic observing run, but thereafter we
only used the better-quality MOSA images.
\begin{table*}
\begin{minipage}{150mm}
\caption{Optical Imaging Observations of \mbox{HS~0624+6907}\relax \label{tab:obslog}}
\begin{tabular}{@{}lccccc}
\hline
& \multicolumn{5}{c}{Filters} \\
& \colhead{\it U} & \colhead{\it B} & \colhead{\it V} & \colhead{\it R} & \colhead{\it I}\\
\hline
UT Observation Date & 2003 Jan. 30 & 2003 Jan. 30 & 2003 Jan. 29 & 2003 Jan. 29 & 2003
Jan. 29 \\
Integration Time (s)& 7$\times$900 & 6$\times$400 & 5$\times$360 & 5$\times$360 & 5$\times$600 \\
\hline
\end{tabular}
\end{minipage}
\end{table*}
\mbox{HS~0624+6907}\relax was observed with MOSA on the 4m on 2003 January
29-30. The field of view is $36'\times36'$ with a scale of
$0\farcs26$/pixel. As summarized in Table~\ref{tab:obslog}, images
were recorded in $U, B, V, R,$ and $I$ with a standard dithering
pattern for filling in gaps between the CCDs and for rejection of
cosmic rays. Photometric standard stars from Landolt (1992) were also
observed at regular intervals. During these observations, the seeing
ranged from $1\farcs$0 to 1$\farcs$3. The data were reduced with the
IRAF\footnote{IRAF is distributed by the National Optical Astronomy
Observatories, which are operated by the Association of Universities
for Research in Astronomy, Inc., under cooperative agreement with the
National Science Foundation.} software package {\tt MSCRED} following
standard procedures. The final R-band MOSA image of \mbox{HS~0624+6907}\relax is shown
in Figures~\ref{fig:field}~and~\ref{fig:fieldzoom}.
Galaxy targets for follow-up spectroscopy were selected from the
images using the SExtractor software package (Bertin \& Arnouts 1996). Redshifts of 29
galaxies were obtained using the Double Imaging Spectrograph (DIS) on
the APO 3.5~m telescope on the following dates: 2002 November 12, 2003
January 29$-$31, 2003 April 03, 2003 April 21, and 2003 December 25.
Spectra were recorded using a single 1.5 arcsec wide slit with total
exposure times ranging from 360 to 1800 s per object. The data were
processed in the conventional manner, and were wavelength calibrated
using helium-neon-argon arc-lamp exposures. Small zero-point offsets
in wavelength were applied as needed, after comparing observed skyline
wavelengths with their rest values. The spectra were typically
recorded at resolutions of $\sim 7-8$~\AA\ FWHM.
\begin{figure*}
\begin{center}
\resizebox{1.0\hsize}{!}{\includegraphics{figure_01.ps}}
\caption{KPNO 4m {\it R}-band image of the field around
\mbox{HS~0624+6907}\relax. Galaxies are labeled with their ID names (from
Table~\ref{tab:spec_red}) and spectroscopic redshifts when available.
Theses redshift labels are colour-coded as follows: red for
\zbtw{0.062}{0.066} (10 galaxies), blue for \zbtw{0.073}{0.077} (7
galaxies), green for \zbtw{0.100}{0.114} (6 galaxies) and black for
the other redshifts. A close up of the 10'$\times$10' region centered
on the QSO (dashed box) is shown in Figure
\ref{fig:fieldzoom}.\label{fig:field}}
\end{center}
\end{figure*}
\begin{figure*}
\begin{center}
\resizebox{1.0\hsize}{!}{\includegraphics{figure_02.ps}}
\caption{Close-up of the KPNO image shown in Figure
\ref{fig:field}. This portion of the image shows the $10' \times10'$
region centered on the QSO and is labeled and colour-coded as described
in Figure~\ref{fig:field}. At redshift $z\simeq$ 0.065, 0.076 and 0.11, $10'$ corresponds to 750, 860~kpc and 1.2~Mpc respectively.}\label{fig:fieldzoom}
\end{center}
\end{figure*}
{
\def~{\phn}
\begin{table*}
\begin{minipage}{150mm}
\caption{Spectroscopic Redshifts of Galaxies in the Field of HS 0624+6907\label{tab:spec_red}}
\renewcommand{\footnoterule}{}
\begin{tabular}{llccccccccc}
& & \multicolumn{2}{c}{Coordinate (J2000)} & & \colhead{$\theta$\tablenotetext{a}{Angular separation to the HS~0624+6907 sightline.}} &
\colhead{$\rho$\tablenotetext{b}{Projected distance to the HS~0624+6907 sightline.} } &&&&\\
\colhead{\#} & \colhead{ID} & \colhead{RA} & \colhead{DEC} & \colhead{$z$} &\colhead{($'$)} &\colhead{({\mbox{$h_{70}^{-1}\,{\rm Mpc}$}})} &\colhead{$R$}&\colhead{$V$}&\colhead{$U-V$}&\colhead{$M_R$}
\\\hline
& QSO & 06:30:02.50 & 69:05:03.99 & 0.3700 & ~0.0 & 0.000 & 13.8\tablenotetext{c}{Due to saturation of the CCD, the magnitudes of the quasar could be underestimated.} & 14.2\tablenotemark{c} & 0.1\tablenotemark{c} & -27.7\tablenotemark{c}\\
01 & SE12 & 06:30:41.70 & 68:58:32.71 & 0.0327 & ~7.4 & 0.290 & 16.6 & 17.8 & 1.0 & -19.2\\
02 & SE3 & 06:30:55.32 & 69:02:41.99 & 0.0424 & ~5.3 & 0.265 & 19.6 & 20.5 & 0.9 & -16.7\\
03 & NE1 & 06:30:56.14 & 69:08:00.90 & 0.0547 & ~5.6 & 0.358 & 15.9 & 17.0 & 0.9 & -21.0\\
04 & NW2 & 06:29:46.66 & 69:08:03.59 & 0.0560 & ~3.3 & 0.216 & 18.6 & 19.5 & 0.9 & -18.4\\
05 & SE4 & 06:30:33.00 & 68:53:02.00 & 0.0622 & 12.3 & 0.887 & 16.8 & 17.9 & 0.9 & -20.5\\
06 & SE5 & 06:32:55.20 & 68:56:59.99 & 0.0637 & 17.4 & 1.282 & 16.8 & 18.3 & 1.3 & -20.5\\
07 & SE8 & 06:31:01.79 & 68:57:35.89 & 0.0638 & ~9.2 & 0.675 & 16.0 & 17.1 & 0.9 & -21.2\\
08 & SE1 & 06:30:11.22 & 69:02:09.61 & 0.0640 & ~3.0 & 0.222 & 18.8 & 19.4 & 0.7 & -18.5\\
09 & SW3 & 06:29:07.80 & 69:03:32.01 & 0.0650 & ~5.1 & 0.384 & 17.4 & 18.3 & 0.9 & -20.0\\
10 & SE13 & 06:30:58.30 & 69:04:34.11 & 0.0650 & ~5.0 & 0.375 & 16.9 & 18.5 & 1.3 & -20.4\\
11 & SE6 & 06:32:50.70 & 68:56:03.00 & 0.0652 & 17.6 & 1.318 & 15.9 & 17.3 & 1.2 & -21.5\\
12 & NE3 & 06:30:21.40 & 69:05:39.70 & 0.0655 & ~1.8 & 0.135 & 16.6 & 18.1 & 1.3 & -20.8\\
13 & NW11 & 06:29:23.48 & 69:22:43.29 & 0.0660 & 18.0 & 1.367 & 16.3 & 17.9 & 1.3 & -21.1\\
14 & SE7 & 06:32:49.20 & 68:56:00.39 & 0.0664 & 17.5 & 1.334 & 17.0 & 18.0 & 0.7 & -20.4\\
15 & NE2 & 06:32:25.55 & 69:20:05.81 & 0.0733 & 19.7 & 1.646 & 16.5 & 18.0 & 1.3 & -21.1\\
16 & NW1 & 06:29:43.65 & 69:09:35.33 & 0.0760 & ~4.8 & 0.417 & 15.9 & 17.3 & 1.1 & -21.8\\
17 & SW2 & 06:28:33.03 & 68:59:26.30 & 0.0763 & ~9.8 & 0.849 & 16.6 & 18.3 & 1.3 & -21.1\\
18 & SE9 & 06:30:14.81 & 68:49:44.79 & 0.0764 & 15.4 & 1.334 & 16.4 & 18.0 & 1.3 & -21.3\\
19 & NW12 & 06:29:11.59 & 69:07:07.89 & 0.0764 & ~5.0 & 0.433 & 16.4 & 18.0 & 1.4 & -21.3\\
20 & SE10 & 06:32:52.40 & 68:57:59.01 & 0.0764 & 16.8 & 1.457 & 16.5 & 18.1 & 1.3 & -21.2\\
21 & NW3 & 06:29:53.77 & 69:08:20.51 & 0.0766 & ~3.4 & 0.293 & 18.3 & 19.6 & 1.1 & -19.4\\
22 & SW1 & 06:29:33.24 & 69:05:01.00 & 0.0903 & ~2.6 & 0.264 & 17.1 & 18.2 & 0.9 & -21.0\\
23 & SE11 & 06:30:06.84 & 68:52:22.20 & 0.1001 & 12.7 & 1.407 & 16.5 & 18.1 & 1.2 & -21.9\\
24 & NW7 & 06:26:43.70 & 69:14:06.91 & 0.1009 & 19.9 & 2.215 & 16.7 & 18.2 & 1.4 & -21.6\\
25 & NW9 & 06:29:03.89 & 69:17:33.40 & 0.1108 & 13.5 & 1.639 & 16.4 & 18.2 & 1.3 & -22.2\\
26 & NW4a & 06:29:35.43 & 69:07:25.80 & 0.1125 & ~3.4 & 0.415 & 18.5 & 20.2 & 1.2 & -20.1\\
27 & NW8 & 06:28:29.39 & 69:17:28.89 & 0.1126 & 14.9 & 1.832 & 16.7 & 18.5 & 1.3 & -21.9\\
28 & NW10 & 06:29:05.25 & 69:17:46.69 & 0.1129 & 13.7 & 1.685 & 18.1 & 19.9 & 1.3 & -20.5\\
29 & NW6 & 06:29:35.30 & 69:09:44.99 & 0.1429 & ~5.3 & 0.794 & 19.4 & 20.4 & 0.8 & -19.8\\
30 & SE2 & 06:30:44.35 & 69:01:08.09 & 0.1664 & ~5.4 & 0.927 & 19.3 & 20.8 & 1.0 & -20.2\\
31 & NW5 & 06:29:43.52 & 69:09:19.01 & 0.2061 & ~4.6 & 0.927 & 18.4 & 20.3 & 1.4 & -21.7\\
32 & NW4b & 06:29:35.62 & 69:07:37.91 & 0.3008 & ~3.5 & 0.940 & 21.0 & 22.2 & 0.4 & -19.9
\\\hline
\end{tabular}
\vspace{-0.6cm}
\end{minipage}
\end{table*}
}
The redshift measurements were made following the procedure
described by Jenkins~et~al. (2003). We used the IRAF routine FXCOR to
cross-correlate the galaxy spectra with that of the radial velocity
standard HD~182572. In general we only used the blue channel DIS data
for the cross-correlation, where the 4000~\AA\ break and stellar
absorption lines were most apparent. Red channel data were usually
used to identify and measure the wavelengths of redshifted emission
lines ([O~{\sc iii}], H$\beta$, H$\alpha$, etc.) when present. The
galaxy redshifts obtained in this way are summarized in
Table~\ref{tab:spec_red} and are accurate to between 70 and 170~\mbox{km~s$^{-1}$}\relax (which corresponds to a sight line distance displacement uncertainty of 1.0 to 2.4~Mpc for an unperturbed Hubble flow).
We also observed three galaxies with the Echellette Spectrometer and
Imager (ESI; Sheinis et al. 2001) on the 10m Keck II telescope on the
nights of 2004 September 10 and 11 during morning twilight. We
observed galaxy NE3 (see Table~\ref{tab:spec_red}) in echellette mode
with the 0.5$''$ slit which provides $\approx 30 $~\mbox{km~s$^{-1}$}\relax\ spectral
resolution (FWHM). The fainter SE13 and SW3 galaxies were observed in
low dispersion mode using a 1$''$ slit which affords $R \sim 2000$ at
$\lambda = 5000$~\AA. The exposures were flat fielded and wavelength
calibrated with the ESIRedux package (Prochaska et al. 2003%
\footnote{http://www.ucolick.org/$\sim$xavier/ESI/index.html}). The
NE3 redshift was derived from the centroids of the high-resolution
\ion{Na}{1} and H$\alpha$ absorption lines, and the redshift
uncertainty is $\sim$30 km s$^{-1}$. For SE13 and SW3, redshifts were
measured by fitting \ion{Na}{1}, H$\beta$, and \ion{Ca}{2} H and K,
and the uncertainties are $\sim$150 km s$^{-1}$.
The completeness of our galaxy redshift survey (i.e., the percentage
of targets brighter than a given magnitude in the SExtractor galaxy
catalog with good spectroscopic redshifts) is graphically summarized
in Figure~\ref{fig:completeness} as a function of limiting $R$
magnitude and angular separation from the sight line ($\Delta
\theta$). In the $10'\times10'$ region centered on \mbox{HS~0624+6907}\relax , we have
measured spectroscopic redshifts for all galaxies brighter than $R$ =
19.0, and the survey is $\approx 60$ per cent complete for $R < 20$. As we
shall see, there is a prominent cluster of absorption lines in the
\mbox{HS~0624+6907}\relax spectrum at $z \approx$ 0.0635; at this redshift, 5$'$
corresponds to a projected distance of 367 $h_{70}^{-1}$ kpc, and $R$
= 19.0 corresponds to $M=-18.3$ or $L = 0.1 L*$ (taking $M_{R}^{*} =
-21.0$ from Lin et al. 1996). For comparison, the Large Magellanic cloud has a magnitude equal to $M_R=-17$ or $L=0.02 L*$. At this redshift, we have good
completeness even for low luminosity galaxies. At larger radii, a substantial
number of bright galaxies are found, and our redshift survey is
shallower. Nevertheless, within a 10' radius circle, our survey is still
60 per cent complete for galaxies brighter than $R < 19$.
\begin{figure}
\resizebox{1.0\hsize}{!}{\includegraphics{figure_03.ps}}
\caption{Completeness of the spectroscopic galaxy redshift
measurements as a function of limiting $R$ magnitude for targets within
an angular separation $\Delta \theta$ = 5, 10, and 16$'$ as indicated
in the key. The key also shows in parentheses the projected physical
distances corresponding to these $\Delta \theta$ values at $z =
0.0635$.\label{fig:completeness}}
\end{figure}
\section{Absorber Environment}\label{sec:environment}
Using information gleaned from the literature in combination with our
galaxy redshift survey, we can identify several large-scale structures
that are pierced by the \mbox{HS~0624+6907}\relax sight line. In this section we comment
on these structures including nearby Abell clusters
(\S\ref{sec:abell_clusters}) as well as smaller (and closer) galaxy
groups (\S\ref{sec:close_groups}).
\subsection{Nearby Abell Clusters and Large-Scale Structure}\label{sec:abell_clusters}
Clusters are clustered and often reveal even larger cosmic structures,
i.e., superclusters (Einasto et al. 2001 and references therein). In
cosmological simulations, clusters are found at the nodes where
large-scale filaments connect. To test the fidelity of cosmological
simulations, which are now being used in a wide variety of
astrophysical analyses, it is important to search for observational
evidence of the expected {\it gaseous} filaments feeding into clusters and to
measure the properties of the filaments. The sight line to \mbox{HS~0624+6907}\relax
passes through a region of relatively high Abell cluster density and
is well-suited for investigation of this topic.
\begin{figure}
\begin{center}
\resizebox{1.0\hsize}{!}{\includegraphics{figure_04.ps}}
\caption{Position of the Abell clusters around the sight line of
HS0624$+$6907. The size of the circles indicates the relative
population richness of the cluster. If known, the redshift of the
Abell cluster is indicated in parentheses. The shaded regions schematically indicate
the possible large scale structures traced by the Abell clusters at
\zapp{0.077} and \zapp{0.110}. The dashed rectangle corresponds to the
limits of the KPNO image and the star to the position of the quasar.
The 1, 2, 3 and 4 degrees angular separation to the quasar are shown by
the dotted circles. At redshift z$\simeq$ 0.076 and 0.110, 2 degrees correspond to 10 and 14~Mpc respectively.}\label{fig:abell_map}
\end{center}
\end{figure}
\fig{fig:abell_map} shows the positions of Abell clusters around
the sight line to \mbox{HS~0624+6907}\relax, including the
cluster richness class and spectroscopic redshift (when available from
the literature). The density of Abell clusters in this region is
relatively high compared to the vicinity of the other clusters in the
Abell catalog: the number of Abell clusters within 2$^{\rm o}$
(3$^{\rm o}$) of \abell{557} (the cluster closest to the \mbox{HS~0624+6907}\relax sight
line) is 2 (1.6) times larger than the average number within 2$^{\rm
o}$ (3$^{\rm o}$) of all Abell clusters. Einasto et al. (2001) have
identified two superclusters in the direction of \mbox{HS~0624+6907}\relax. Their
supercluster SCL71 (at $z \approx 0.110$) includes
\abell{554},\abell{557}, \abell{561}, \abell{562}, and \abell{565}
while SCL72 (at $z \approx 0.077$) includes \abell{559} and
\abell{564}. However, \abell{557} and \abell{561} do not have
spectroscopic redshifts, and based on our spectroscopic redshifts in
the field of HS0624+6907/Abell557 (see Table~\ref{tab:spec_red}), it
appears likely that the visually identified \abell{557} is a false
cluster due to the superposition of galaxy groups at several different
redshifts. To be conservative, we only use clusters with
spectroscopic redshifts to identify large-scale structures. The
clusters at \zapp{0.077} (\abell{564} and \abell{559}) are separated by
4.7~\mbox{$h_{70}^{-1}\,{\rm Mpc}$}\ from each other, and the clusters at \zapp{0.110} ( \abell{565}, \abell{562}
and \abell{554}) are separated by 3.9 and 8.7~\mbox{$h_{70}^{-1}\,{\rm Mpc}$}. According to
Colberg et al. (2004), in cosmological simulations, more than 85 per cent of
the clusters with a separation lower than 10~\mbox{$h_{70}^{-1}\,{\rm Mpc}$}\ are connected
with a filament. We will show in subsequent sections that both
absorption lines in the spectrum of \mbox{HS~0624+6907}\relax and galaxies close to the
sight line are detected at the redshifts of both of these Abell
cluster structures, which indicates that gaseous filaments
connect into the clusters.
\subsection{Individual Galaxies and Groups}\label{sec:close_groups}
In this section we offer
some brief comments about specific galaxies and galaxy groups close to the sight line of \mbox{HS~0624+6907}\relax, as revealed by our optical spectroscopy.
We place these observations in the context of the Abell clusters
described in the previous section. We plot in
\fig{fig:zgaldist} the redshift distribution of the galaxies from
Table~\ref{tab:spec_red}. From this figure we can identify three
galaxy groups: two galaxy groups appear to be present at redshifts
similar to those of the Abell clusters, i.e, at $z\sim0.077$ (7
galaxies) and $z\sim0.11$ (6 galaxies). This indicates that
the filament of galaxies connecting \abell{559} to \abell{564} must
extend more then 3 degrees (15~\mbox{$h_{70}^{-1}\,{\rm Mpc}$}) west from \abell{559} and
structure linking \abell{562} and \abell{554} likely extends by at least
3 degrees (22~\mbox{$h_{70}^{-1}\,{\rm Mpc}$}) in order to cross the QSO sight line. However,
the most prominent group in \fig{fig:zgaldist} includes 10 galaxies at
\zapp{0.064}, which does not match up with any Abell cluster with a
known spectroscopic redshift. To show the spatial distribution of
galaxies in the three prominent groups in the MOSA field,
\fig{fig:gal_colour} provides a colour-coded map of projected spatial
coordinates of the galaxies. We see that the group at $z \approx
0.064$ is mostly southeast of the sight line while the galaxies
associated with Abell 554/562/565 at \zapp{0.11} are predominantly
northwest of \mbox{HS~0624+6907}\relax . The galaxies associated with the Abell 559/564
\zapp{0.07} supercluster appear to extend from the southwest across
the sight line to the northeast.
We note that spectroscopic redshifts are not
available for several of the clusters shown in Fig.
\ref{fig:abell_map}, including the one that is closest to the sight
line, \abell{557}. However, Abell clusters are visually identified
without the aid of spectroscopy, and it can be seen from
Figures~\ref{fig:field} and \ref{fig:gal_colour} that several discrete
groups are found at the location of \abell{557}. It is likely that
\abell{557} is not a true cluster but rather is the superposition of
several groups in projection.
\begin{figure}
\begin{center}
\resizebox{1.0\hsize}{!}{\includegraphics{figure_05.ps}}
\caption{Histogram of galaxy redshifts in the field of \mbox{HS~0624+6907}\relax. \ion{H}{1}
Lyman-$\alpha$ systems are indicated on the top of the figure with a
line height proportional to the absorption line rest equivalent width. Also
reported on the figure are the known redshifts of the Abell
clusters shown in Figure \ref{fig:abell_map}. }\label{fig:zgaldist}
\end{center}
\end{figure}
\begin{figure}
\begin{center}
\resizebox{1.0\hsize}{!}{\includegraphics{figure_06.ps}}
\caption{Map of the galaxy distribution around the quasar \mbox{HS~0624+6907}\relax\ in
projected physical coordinates (north is up and east is to the
left). The QSO is at (x,y) = (0,0), and the galaxies with
spectroscopic redshifts (see Table~\ref{tab:spec_red}) are shown with
stars. The symbol size is inversely proportional to redshift. Some
galaxies are colour-coded by redshift in ranges of $z$ which correspond to
peaks in the redshift distribution shown in Figure~\ref{fig:zgaldist} or the redshifts of nearby Abell clusters (Figure~\ref{fig:abell_map}). Distances of 0.1,
0.5, 1.0 and 2.0~\mbox{$h_{70}^{-1}\,{\rm Mpc}$}\ to the quasar are indicated by dotted
circles. }\label{fig:gal_colour}
\end{center}
\end{figure}
Is gas also present in these large-scale cosmic filaments? To address
this question, we searched the spectrum of \mbox{HS~0624+6907}\relax for any absorption
counterparts at the redshifts of galaxies and galaxy structures near
the QSO sight line. The redshifts of the \ion{H}{1} \mbox{Ly$\alpha$}\relax\ systems that we have
identified and measured (see \S\ref{sec:absline}) in the spectrum of
\mbox{HS~0624+6907}\relax are plotted at the top of \fig{fig:zgaldist}; the length of
the line is proportional to the rest equivalent width of the \mbox{Ly$\alpha$}\relax\ line. This search has
revealed three interesting results: First, when a galaxy is located at
an impact parameter $\rho \lesssim 500 $~\mbox{$h_{70}^{-1}\,{\rm kpc}$}\ from the sight
line, \mbox{Ly$\alpha$}\relax\ absorption is almost always found within a few hundred
\mbox{km~s$^{-1}$}\relax\ of the galaxy redshift (compare Table~\ref{tab:spec_red} to
Table~\ref{tab:lyalist}), consistent with the findings of previous
studies (e.g., Lanzetta et al. 1995; Tripp et al. 1998; Impey et
al. 1999; Chen et al. 2001; Bowen et al. 2002; Penton et
al. 2002). Second, strong absorption is clearly detected at the
redshift of the $z\sim0.07$ Abell 564/559 supercluster. This absorption system is
detected in \mbox{Ly$\alpha$}\relax , \mbox{Ly$\beta $}\relax , and the \ion{C}{4} $\lambda \lambda$
1548.20, 1550.78 doublet (\S\ref{ss:z076}), and the absorption
redshift is very similar to that of \abell{559}. Weak \mbox{Ly$\alpha$}\relax\
absorption is also detected at $z_{\rm abs}$ = 0.10822, which is
within 500 \mbox{km~s$^{-1}$}\relax\ of the Abell 554/562/565 filament. Evidently, and not
surprisingly, gas is also found in the filaments that feed into the
clusters near \mbox{HS~0624+6907}\relax. Third, Figure~\ref{fig:zgaldist} qualitatively
indicates that the strongest \mbox{Ly$\alpha$}\relax\ lines are situated in the regions
of highest galaxy density, which is similar to the conclusions of
Bowen et al. (2002) and C{\^ o}t\'e et al. (2005).
Could these \mbox{Ly$\alpha$}\relax\ absorbers simply arise in the halos of individual
galaxies? As we show in the next section, the strong \ion{H}{1} system
at \zabsapp{0.064} is comprised of a large number of components spread
over 1000 \mbox{km~s$^{-1}$}\relax. Such kinematics are unprecedented in single galaxy
halos. In addition, we find no obvious pattern that shows a connection
between individual \mbox{Ly$\alpha$}\relax\ lines and individual galaxies in this
complex. In the case of this strong \ion{H}{1} system at
\zabsapp{0.064}, the closest observed galaxy to the sight line around
this redshift has $\rho = 135\,h_{70}^{-1}{\rm kpc} $ (NE3 in table
\ref{tab:spec_red}). However, a closer and fainter galaxy could have
been missed by the spectroscopic survey. Using photometric redshifts
(measured as described in Chen et al. 2003) to cull the distant
background galaxies with photometric redshifts $> 0.2$, we find that
there are only four galaxies closer to the sight line than NE3 that
could be near \zapp{0.064}. These four objects are only candidates
since photometric redshifts have substantial uncertainties. However,
if we assume the redshift of these candidates to be $z=0.064$, the
closest one to \mbox{HS~0624+6907}\relax has still a large impact parameter $\rho =
85\,h_{70}^{-1}$~kpc. In the case of \mbox{Ly$\alpha$}\relax\ at $z_{\rm abs}$ = 0.07573,
the closest galaxy in projection is NW3 at $\rho =
293\,h_{70}^{-1}$~kpc. It is conceivable that this absorption
originates in the large halo of this particular galaxy, but we note
that three galaxies are found at $\rho \leq 500\,h_{70}^{-1}$~kpc at
this $z$, and many other origins are possible (e.g., intragroup gas or
tidally stripped debris). Finally, the absorption at $z_{\rm abs} =
0.10822$ is at a substantial distance (415~\mbox{$h_{70}^{-1}\,{\rm kpc}$}) from the nearest
known galaxy (NW4a) and is unlikely to be halo gas associated with
that object.
\section{Absorption Line Measurements}\label{sec:absline}
We now turn to the absorption-line measurements. As discussed in the
previous section, Figure~\ref{fig:zgaldist} compares the distributions
of galaxies and Ly$\alpha$ lines in the direction of \mbox{HS~0624+6907}\relax. We
have measured the column densities and Doppler parameters of the
Ly$\alpha$ lines in the spectrum of \mbox{HS~0624+6907}\relax using the Voigt profile
decomposition software VPFIT (see Webb 1987\footnote{http://www.ast.cam.ac.uk/$\sim$rfc/vpfit.html}). Table~\ref{tab:lyalist}
summarizes the \ion{H}{1} equivalent widths, column densities, and
$b-$values measured in this fashion (some of the lines are strongly
saturated and consequently Voigt profile fitting does not provide
reliable measurements; we discuss our treatment of these cases below).
A particularly dramatic cluster of Ly$\alpha$ lines is evident at
$z_{\rm abs} \approx$ 0.0635 in Figure~\ref{fig:zgaldist}. The
portion of the STIS spectrum of HS0624+6907 covering this Ly$\alpha$
cluster is shown in Figure~\ref{fig:hi00635}. To avoid confusion with
galaxy clusters, we will hereafter refer to this group of \mbox{Ly$\alpha$}\relax\ lines
as a \mbox{Ly$\alpha$}\relax\ ``complex''. This complex contains at least 13 \ion{H}{1}
components spread over a velocity range of 1000 km s$^{-1}$. We will
refer to the strongest Ly$\alpha$ absorption in
Figure~\ref{fig:hi00635} at $z_{\rm abs}$ = 0.06352 as ``component
A''. Component A is detected in absorption in the \ion{H}{1}
Ly$\alpha$, Ly$\beta$, and Ly$\gamma$ transitions as well as the
\ion{Si}{3} $\lambda$1206.50, \ion{Si}{4} $\lambda \lambda$1393.76,
1402.77, and \ion{C}{4} $\lambda \lambda$1548.20, 1550.78 lines. Low
ionisation stages such as \ion{O}{1}, \ion{C}{2}, and \ion{Si}{2} are
not detected at the redshift of Component A or at the redshifts of any
of the other components evident in Figure~\ref{fig:hi00635}. The
\ion{O}{6} doublet at the redshifts of the Ly$\alpha$ cluster in falls
in a region that is partially blocked by Galactic H$_{2}$ and
\ion{Fe}{2} lines. Nevertheless, much of the region is free from
blending, and we find no evidence for \ion{O}{6} absorption. We also
do not see the \ion{N}{5} doublet.
\begin{table*}
\begin{minipage}{150mm}
\caption{Equivalent Widths, Column Densities, and Doppler Parameters of \ion{H}{1} Ly$\alpha$ Lines in the Spectrum of HS 0624$+$6907\label{tab:lyalist}}
\begin{tabular}{lccl|lccl}
& $W_{obs}$ & $\log({\rm N})$ & $b$ & & $W_{obs}$ & $\log({\rm N})$ & $b$ \\
$z$ & (m\AA) & (\mbox{cm$^{-2}$}\relax) & (\mbox{km~s$^{-1}$}\relax) & $z$ & (m\AA) & (\mbox{cm$^{-2}$}\relax) & (\mbox{km~s$^{-1}$}\relax)
\\\hline
0.017553$\pm$~1.0e-5 & ~45$\pm$10 & 12.96$\pm$0.05 & ~29$\pm$~4.3 &0.207540$\pm$~0.5e-5 & 150$\pm$~9 & 13.48$\pm$0.02 & ~27$\pm$~1.5 \\
0.030651$\pm$~0.4e-5 & ~99$\pm$~9 & 13.36$\pm$0.03 & ~22$\pm$~1.7 &0.213232$\pm$~1.6e-5 & ~98$\pm$14 & 13.22$\pm$0.05 & ~45$\pm$~5.6 \\
0.041156$\pm$~0.8e-5 & 104$\pm$11 & 13.33$\pm$0.03 & ~41$\pm$~3.0 &0.219900$\pm$~2.3e-5 & 143$\pm$15 & 13.39$\pm$0.05 & ~60$\pm$~8.6 \\
0.053942$\pm$~0.6e-5 & ~85$\pm$~7 & 13.26$\pm$0.04 & ~24$\pm$~2.3 &0.223290$\pm$~0.3e-5 & 256$\pm$12 & 13.86$\pm$0.02 & ~25$\pm$~0.9 \\
0.054367$\pm$~4.1e-5$^{a}$ & ~65$\pm$13 & 13.09$\pm$0.11 & ~60$\pm$19.2$^b$ &0.232305$\pm$~2.8e-5$^a$ & 125$\pm$13 & 13.33$\pm$0.08 & ~44$\pm$~7.7$^b$ \\
0.054829$^{a,c}$ & 458$\pm$10 & $\sim$14.5$^c$ & ~$\sim$35$^c$ &0.232547$\pm$~2.3e-5$^a$ & ~44$\pm$10 & 12.86$\pm$0.21 & ~24$\pm$~7.3 \\
0.055153$\pm$~7.8e-5$^a$ & 237$\pm$14 & 13.68$\pm$0.17 & ~84$\pm$30.7$^b$ &0.240599$\pm$~0.6e-5 & 110$\pm$10 & 13.33$\pm$0.04 & ~20$\pm$~2.0 \\
0.061879$\pm$~0.4e-5 & 184$\pm$~6 & 13.77$\pm$0.03 & ~21$\pm$~1.4 &0.252251$\pm$~1.2e-5 & ~55$\pm$11 & 12.96$\pm$0.06 & ~24$\pm$~4.2 \\
0.062014$\pm$~1.0e-5 & ~21$\pm$~4 & 12.63$\pm$0.17 & ~~8$\pm$~4.7 &0.268559$\pm$~2.1e-5 & ~68$\pm$14 & 13.03$\pm$0.05 & ~51$\pm$~7.2 \\
0.062150$\pm$~1.5e-5$^a$ & ~13$\pm$~4 & 12.41$\pm$0.22 & ~10$\pm$~7.9 &0.272240$\pm$~0.6e-5 & ~37$\pm$~8 & 12.80$\pm$0.06 & ~12$\pm$~2.2 \\
0.062343$\pm$~0.8e-5$^a$ & 128$\pm$~7 & 13.45$\pm$0.05 & ~30$\pm$~4.0 &0.279771$\pm$~1.7e-5$^a$ & 174$\pm$13 & 13.50$\pm$0.06 & ~34$\pm$~4.9 \\
0.062647$\pm$~2.5e-5$^a$ & 101$\pm$~7 & 13.31$\pm$0.14 & ~35$\pm$12.3 &0.280171$\pm$~0.7e-5$^a$ & 576$\pm$15 & 14.32$\pm$0.02 & ~43$\pm$~1.9$^b$ \\
0.062762$\pm$~0.7e-5$^a$ & ~39$\pm$~3 & 12.95$\pm$0.28 & ~~8$\pm$~3.7 &0.295307$\pm$~0.7e-5 & 309$\pm$15 & 13.80$\pm$0.02 & ~42$\pm$~2.0 \\
0.062850$\pm$~1.2e-5$^a$ & 110$\pm$~5 & 13.42$\pm$0.14 & ~20$\pm$~7.0 &0.296607$\pm$~0.9e-5 & 203$\pm$18 & 13.54$\pm$0.02 & ~52$\pm$~2.9 \\
0.063037$\pm$~1.4e-5$^a$ & 101$\pm$~6 & 13.33$\pm$0.13 & ~27$\pm$~8.8 &0.308991$\pm$~0.6e-5 & 167$\pm$12 & 13.49$\pm$0.03 & ~28$\pm$~1.8 \\
0.063456$\pm$~1.6e-5$^a$ & 569$\pm$~9 & 14.46$\pm$0.30 & ~48$\pm$~8.4$^b$ &0.309909$\pm$~5.5e-5$^a$ & 246$\pm$18 & 13.61$\pm$0.10 & ~66$\pm$12.3$^b$ \\
0.063481$\pm$~1.6e-5$^a$ & 443$\pm$~6 & 15.27$\pm$0.13 & ~24$\pm$~5.5 &0.310454$\pm$~8.0e-5$^a$ & 170$\pm$17 & 13.43$\pm$0.33 & ~62$\pm$40.3$^b$ \\
0.063620$\pm$~2.7e-5$^a$ & 153$\pm$~4 & 14.29$\pm$0.38 & ~10$\pm$~5.6 &0.310881$\pm$14.4e-5 & ~88$\pm$16 & 13.13$\pm$0.43 & ~51$\pm$27.7 \\
0.064753$\pm$~0.9e-5$^a$ & 257$\pm$~9 & 13.87$\pm$0.04 & ~33$\pm$~3.0 &0.312802$\pm$~4.4e-5 & 257$\pm$18 & 13.65$\pm$0.10 & ~54$\pm$~9.3 \\
0.065016$\pm$~0.8e-5$^a$ & 282$\pm$~8 & 13.97$\pm$0.04 & ~31$\pm$~2.7 &0.313028$\pm$~1.6e-5 & ~72$\pm$10 & 13.09$\pm$0.24 & ~17$\pm$~6.8 \\
0.075731$\pm$~0.2e-5 & 292$\pm$~8 & 14.18$\pm$0.03 & ~24$\pm$~0.8 &0.313261$\pm$~4.7e-5 & 244$\pm$19 & 13.62$\pm$0.10 & ~55$\pm$10.9 \\
0.090228$\pm$~4.2e-5 & 106$\pm$12 & 13.29$\pm$0.08 & ~76$\pm$13.7 &0.317901$\pm$~1.2e-5 & 139$\pm$18 & 13.37$\pm$0.04 & ~34$\pm$~3.6 \\
0.130757$\pm$~1.0e-5 & 114$\pm$~9 & 13.34$\pm$0.04 & ~34$\pm$~3.6 &0.320889$\pm$~0.4e-5 & 349$\pm$14 & 13.97$\pm$0.02 & ~31$\pm$~1.2 \\
0.135966$\pm$~3.9e-5 & 119$\pm$11 & 13.33$\pm$0.10 & ~57$\pm$10.7 &0.327245$\pm$~5.0e-5$^a$ & 316$\pm$21 & 13.73$\pm$0.32 & ~69$\pm$15.6$^b$ \\
0.160541$\pm$~5.0e-5$^a$ & ~69$\pm$~7 & 13.08$\pm$0.21 & ~34$\pm$10.3 &0.327721$\pm$38.7e-5$^a$ & 264$\pm$26 & 13.61$\pm$0.43 & 115$\pm$62.1$^b$ \\
0.160744$\pm$~1.0e-5$^a$ & 200$\pm$~7 & 13.66$\pm$0.05 & ~30$\pm$~2.4 &0.332674$\pm$~1.1e-5 & 202$\pm$18 & 13.55$\pm$0.04 & ~38$\pm$~3.4 \\
0.199750$\pm$~0.6e-5 & ~87$\pm$~9 & 13.24$\pm$0.05 & ~17$\pm$~2.0 &0.339759$\pm$~0.3e-5 & 647$\pm$13 & 14.45$\pm$0.03 & ~42$\pm$~1.3 \\
0.199946$\pm$~1.2e-5$^a$ & ~83$\pm$11 & 13.17$\pm$0.06 & ~26$\pm$~4.6 &0.346824$\pm$~0.6e-5 & 221$\pm$16 & 13.59$\pm$0.02 & ~39$\pm$~1.9 \\
0.204831$\pm$~0.3e-5 & 208$\pm$~9 & 13.72$\pm$0.02 & ~24$\pm$~1.0 &0.348645$\pm$~0.9e-5 & ~40$\pm$10 & 12.78$\pm$0.06 & ~18$\pm$~3.0 \\
0.205326$\pm$~0.2e-5 & 322$\pm$~8 & 14.12$\pm$0.03 & ~25$\pm$~0.8
\\\hline
\end{tabular}
\vspace{-0.4cm}
\footnotetext[1]{Due to blending, the uncertainties in
the line parameters could be larger than the formal uncertainties
estimated by VPFIT that are listed in this table.}
\footnotetext[2]{This line is part of a multicomponent fit, i.e., the
fit is not a single-component fit to an isolated line. Consequently,
there is a greater chance that the large Doppler parameter could be
due to a blend of several narrow components instead of thermal
broadening.}
\footnotetext[3]{For most high-$N$(H~I) lines, we have
fitted multiple Lyman series lines so that the fits are constrained by
adequately weak, unsaturated lines. However, in this case, all
available lines are strongly saturated, and consequently the line
parameters are highly uncertain.}
\end{minipage}
\renewcommand{\thefootnote}{\arabic{footnote}}
\end{table*}
\begin{figure}
\begin{center}
\resizebox{1.0\hsize}{!}{\rotatebox{-90}{\includegraphics{figure_07.ps}}}
\caption{STIS spectrum of \mbox{HS~0624+6907}\relax between 1288~\AA\ and 1298~\AA. All
the absorption lines shown are due to a \ion{H}{1} \mbox{Ly$\alpha$}\relax\ complex
covering a velocity range $\simeq$1000~km~s$^{-1}$. The velocity of
the upper axis is relative to \mbox{Ly$\alpha$}\relax\ at $z=0.06352$.
The tick marks show the galaxy redshifts from our
spectroscopic survey (\S\ref{sec:optgal}). Apart from the
identification name, the labels indicate the redshifts of
the galaxies and their projected distances (in \mbox{$h_{70}^{-1}\,{\rm Mpc}$}) from the QSO sight
line, respectively.\label{fig:hi00635}}
\end{center}
\end{figure}
Figure~\ref{fig:syst0063} compares the absorption profiles of the
Ly$\alpha$, Ly$\beta$, Ly$\gamma$, \ion{Si}{3}, \ion{Si}{4}, and
\ion{C}{4} lines at $z_{\rm abs}$ = 0.06352, and
Table~\ref{tab:cldnlist0064} lists the equivalent widths, column
densities, and $b-$values of the metals detected at this redshift as
well as upper limits on undetected species of interest. Both
Voigt-profile fitting and direct integration of the ``apparent column
density'' profile (see Savage \& Sembach 1991; Jenkins 1996) were used
to estimate the metal column densities. These independent methods are
generally in good agreement for the metal lines, which do not appear
to be strongly affected by unresolved saturation. The \ion{Si}{3}
$\lambda 1206.50$ absorption at $z_{\rm abs} =$ 0.06352 is slightly
blended with a strong \ion{H}{1} line at $z_{\rm abs} =$ 0.05486 (see
Figure~\ref{fig:syst0063}). However, the metal lines at this redshift
have a distinctive two-component profile (see
Figures~\ref{fig:syst0063} and \ref{fig:civ064}), and the \ion{Si}{3}
profile shape is in good agreement with those of the \ion{C}{4} and
\ion{Si}{4} lines. This indicates that (unrelated) blended \mbox{Ly$\alpha$}\relax\ from
$z_{\rm abs}$ = 0.05486 contributes little optical depth to the
wavelength range where the \ion{Si}{3} absorption occurs. We fitted
the \ionl{Si}{3}{1206} and \ion{H}{1} \mbox{Ly$\alpha$}\relax\ at $z_{\rm abs}=$0.05486 simultaneously, assuming
all of the \mbox{Ly$\alpha$}\relax\ components are centered shortward of the \ion{Si}{3}
line. The resulting joint fit is shown in Figure~\ref{fig:syst0063}.
\begin{figure}
\begin{center}
\resizebox{1.0\hsize}{!}{\includegraphics{figure_08.ps}}
\caption{Transitions observed at $z_{\rm abs}$ = 0.06352
in the spectrum of \mbox{HS~0624+6907}\relax plotted in the absorber frame ($v = 0$ km
s$^{-1}$ at $z_{\rm abs}$ = 0.06352). The dashed line is centered on
``component A'', the strongest \cion{H}{1} component in the Ly$\alpha$
cluster. The solid line shows Voigt profiles fitted to lines at this
redshift; the dotted line indicates fits to unrelated lines at other
redshifts. These unrelated lines were fitted in order to deblend the
features from the \cion{Si}{3} $\lambda 1206.50$
transition.}\label{fig:syst0063}
\end{center}
\end{figure}
\begin{figure}
\begin{center}
\resizebox{1.0\hsize}{!}{\includegraphics{figure_09.ps}}
\caption{Contour maps of the equivalent width of \ion{H}{1} \mbox{Ly$\alpha$}\relax\ (in
red), \mbox{Ly$\beta $}\relax\ (in green) and \mbox{Ly$\gamma$}\relax\ (in blue) as a function of the column
density $N$ and the Doppler parameter $b$. For each transition, the
contours correspond to $W_{r}\pm1\sigma$ (solid lines) and
$W_{r}\pm2\sigma$ (dashed lines). Only the values of $b$ and $N$ in the
shaded regions can reproduce the observed equivalent widths for the
three transitions. }\label{fig:cog}
\end{center}
\end{figure}
The profile parameters derived for most of the \mbox{Ly$\alpha$}\relax\ lines in the $z =
0.0635$ cluster are reasonably well-constrained. Some of the
components are strongly blended and are consequently more uncertain
than the formal profile-fitting error bars indicate; these are marked
in Table~\ref{tab:lyalist}. Component A was also difficult to measure
for a different reason: the three usable Lyman series lines (Ly$\alpha$,
Ly$\beta$ and Ly$\gamma$) are all saturated, and consequently Voigt
profile fitting did not provide good constraints for the determination
of the component A \ion{H}{1} column densities. As shown in
Figure~\ref{fig:cog}, using a single-component curve of growth and the
observed equivalent widths of Ly$\alpha, \beta$, and $\gamma$, we
find that the component A \ion{H}{1} absorption lines can be
reproduced by two distinct sets of values for the \ion{H}{1} column
density and the Doppler parameter. One of the two sets implies
N(\ion{H}{1})$\,\sim10^{17.4}\;\mbox{cm$^{-2}$}\relax$, which should produce a strong
absorption edge characteristic of a Lyman Limit System (LLS) at an
observed wavelength of 970~\AA. This wavelength region is covered by
the {\it FUSE}\ spectrum in the SiC2a channel and is shown in
Figure~\ref{fig:lls_rutr}. The S/N of the SiC2a channel is low but is
adequate to constrain $N$(\ion{H}{1}). The
optical depth $\tau(\lambda)$ of the Lyman limit absorption and the
\ion{H}{1} column density are approximately related by
\begin{equation}
N({\rm H~I}) = \frac{\tau (\lambda)}{\sigma (\lambda)} = 1.6 \times
10^{17}\left( \frac{912 \rm\AA}{\lambda}\right)^3\tau(\lambda)\;\;{\rm atoms \;\; cm}^{-2}\label{eq:nhills}
\end{equation}
\noindent where $\sigma (\lambda$) is the absorption cross section and
$\lambda$ is the rest-frame wavelength. The SiC2a spectrum does not
show any compelling evidence of a Lyman limit edge at the expected
wavelength, but the continuum placement is somewhat uncertain and
because of this, a small Lyman limit decrement could be present.
Based on the small depth of the flux decrement at $\lambda _{\rm obs}$
= 970 \AA , we derive a $3\sigma$ upper limit of $N$(\ion{H}{1}) $\leq
10^{16.1}$ cm$^{-2}$ (upper black solid curve in Figure
\ref{fig:lls_rutr}). We also show in Figure~\ref{fig:lls_rutr} the
Lyman limit absorption expected for $N$(\ion{H}{1}) = $10^{16.7}$
cm$^{-2}$ (lower black curve), which is too strong with our adopted
continuum placement. The absence of a strong Lyman limit edge rules
out the higher \ion{H}{1} column density of 10$^{17.5}$~cm$^{-2}$
predicted by the curve of growth shown in Figure~\ref{fig:cog}. The
lower $N$(\ion{H}{1}) derived from the curve of growth
(10$^{15.4}$~cm$^{-2}$) is consistent with the lack of a Lyman limit
edge. To be conservative, we present below the metallicities derived
both from the upper limit [$N$(\ion{H}{1}) $\leq 10^{16.1}$] and from
the somewhat lower best value from the curve of growth shown in
Figure~\ref{fig:cog}.
\begin{figure}
\begin{center}
\resizebox{1.0\hsize}{!}{\rotatebox{-90}{\includegraphics{figure_10.ps}}}
\caption{Portion of the \mbox{HS~0624+6907}\relax {\it FUSE}\ spectrum. The data come from the
SiC2A channel and are rebinned over 10 pixels. The crossed circles
indicate the position of the telluric lines and the black triangles
show the position of five strong H$_2$ absorption features. The
dashed curve shows the adopted continuum placement, and the solid
curves represent two models of LLS absorption with
$N$(\ion{H}{1})=10$^{16.7}\;\mbox{cm$^{-2}$}\relax$ (lower curve) and $10^{16.1}\;\mbox{cm$^{-2}$}\relax$
(upper curve). }\label{fig:lls_rutr}
\end{center}
\end{figure}
\section{Ionisation and Metallicity}%
We next examine the physical conditions and metal enrichment of the
absorption systems implied by the column densities and the Doppler
parameters obtained from Voigt profile fitting. We concentrate on the
absorbers at $z_{\rm abs}$=0.06352 and 0.0757 because these systems show
metal absorption and can be associated with nearby galaxies/structures
as discussed in \S\ref{sec:environment}. To derive abundances from
the detected metals in these systems (\ion{Si}{3}, \ion{Si}{4}, and
\ion{C}{4}), we must apply ionisation corrections, which depend on the
ionisation mechanism and physical conditions of the gas. We will show
that the gas is predominantly photoionised, and that the implied
metallicities are relatively high.
To investigate the absorber ionisation corrections and metallicities,
we employ CLOUDY photoionisation models (v96, Ferland et al. 1998) as
described in Tripp et al. (2003). In these models, the absorbers are
approximated as constant density, plane-parallel gas slabs with a
thickness that reproduces the observed \ion{H}{1} column density. The
gas in the cloud is photoionised by the UV background from quasars at
$z\simeq0.06$. We used the UV background spectrum shape computed by
Haardt \& Madau (1996) with the intensity normalized to
$J_\nu=1\times10^{-23}\;{\rm ergs\;s^{-1}\;cm^{-2}\;Hz^{-1}}$ at 1
Rydberg. This value is consistent with theoretical and observational
constraints (Shull et al. 1999; Weymann et al. 2001; Dav\'e \& Tripp
2001, and references therein). With the models required to match the
observed $N$(\ion{H}{1}), the predicted metal column densities
depend mainly on the ionisation parameter $U$ ($\equiv$ ionising
photon density/total hydrogen number density), the overall
metallicity,\footnote{In this paper, we express metallicities using
the usual logarithmic notation, [X/Y] = log ($N$(X)/$N$(Y)) - log
(X/Y)$_{\odot}$.} and the assumed relative abundances of the
metals. We assume solar relative abundances, and we adopt recent
revisions reported by Allende Prieto et al. (2001,2002) and Holweger
(2001) for oxygen, carbon, and silicon, respectively.
The high-ion column densities predicted by the photoionisation models
depend on the assumed UV background shape. In this paper, we primarily
use the UV background shape computed by Haardt \& Madau (1996), but we
note that other assessments of the UV background (e.g., Madau, Haardt,
\& Rees 1999; Shull et al. 1999) adopt a somewhat steeper EUV spectral
index for quasars, which changes the CLOUDY ionisation patterns for a
given metallicity and ionisation parameter. We investigate how these
UV background uncertainties affect our results by modeling our systems
with both UV background shapes (see below).
\subsection{$z_{\rm abs} = 0.06352$}\label{ss:z064}
{
\def~{\phn}
\def\footnote{Rest wavelengths and $f$-values are from Morton (1991).}{\footnote{Rest wavelengths and $f$-values are from Morton (1991).}}
\def\footnote{\parbox[t]{0.9\hsize}{Sum of the column densities obtained from Voigt profile fitting of the two evident components in the system.}}{\footnote{\parbox[t]{0.9\hsize}{Sum of the column densities obtained from Voigt profile fitting of the two evident components in the system.}}}
\def\footnote{Integrated apparent column density $N_a=\int N_a(v){\rm d}v$.}{\footnote{Integrated apparent column density $N_a=\int N_a(v){\rm d}v$.}}
\begin{table}
\caption{Equivalents width and column densities of lines at $z\approx0.06352$\label{tab:cldnlist0064}}
\begin{minipage}{\hsize}
\begin{tabular}{@{}l@{ }c@{ }c@{ }ccc@{}}
& $\lambda_0$ & & $W_{obs}$ & & \\
Species & (\AA) & $\log\,f\lambda_0$\footnote{Rest wavelengths and $f$-values are from Morton (1991).} & (m\AA) & $\log\,N_{tot}$\footnote{\parbox[t]{0.9\hsize}{Sum of the column densities obtained from Voigt profile fitting of the two evident components in the system.}} & $\log\,N_{a}$\footnote{Integrated apparent column density $N_a=\int N_a(v){\rm d}v$.}
\\\hline
H~\textsc{i} & 1215.670 & 2.704 & 603$\pm$~9 & 15.37$^{+0.10\mathstrut}_{-0.16}$ & \nodata \\
& 1025.722 & 1.909 & 349$\pm$21 & & \nodata\\
& ~972.537 & 1.450 & 302$\pm$22 & & \nodata\\
O \textsc{i} & 1302.168 & 1.804 & ~~6$\pm$~9 & & $<13.1$\\
O \textsc{vi} & 1037.617 & 1.836 & ~28$\pm$17 & & $<13.6$\\
C \textsc{ii} & 1334.532 & 2.232 & 104$\pm$~9 & & $<14.3$\\
C \textsc{iv} & 1548.204 & 2.470 & 143$\pm$~9 & 13.67$\pm0.04$ & 13.61$\pm$0.03\\
& 1550.781 & 2.169 & ~90$\pm$10 & & 13.63$\pm$0.05\\
Si \textsc{ii} & 1260.422 & 3.104 & ~23$\pm$12 & & $<12.8$\\
Si \textsc{iii}& 1206.500 & 3.304 & 151$\pm$10 & 13.02$^{+0.24}_{-0.21\mathstrut}$ & 13.08$\pm$0.03\\
Si \textsc{iv} & 1393.760 & 2.855 & ~76$\pm$10 & 13.05$^{+0.16}_{-0.12\mathstrut}$ & 12.95$\pm$0.06\\
& 1402.773 & 2.554 & ~49$\pm$11 & & 13.05$\pm$0.10\\
N \textsc{v} & 1242.804 & 1.988 & ~23$\pm$10 & & $<13.8$
\\\hline
\end{tabular}
\vspace{-6mm}
\end{minipage}
\end{table}
}
As discussed above, the \ion{H}{1} absorption lines detected at
$z_{\rm abs} = 0.0635$ are spread over a wide velocity range of
$\Delta v=1000$~\mbox{km~s$^{-1}$}\relax\ (see Figure~\ref{fig:hi00635}). From the
velocity centroids of the 13 fitted Voigt profiles and by using the biweight statistic as described in Beers et al. (1990), we estimate that
the line-of-sight velocity dispersion of this \ion{H}{1} absorption
complex is $\sigma_v=265$~\mbox{km~s$^{-1}$}\relax. The velocity dispersion of
galaxies near this redshift is comparable to this value though substantially more uncertain (due to the larger uncertainties in the galaxy redshifts). This velocity dispersion is comparable to those observed in elliptical-rich galaxy groups (e.g. Zabludoff \& Mulchaey 1998),
which is interesting because elliptical-rich groups often show diffuse
X-ray emission (Mulchaey \& Zabludoff 1998) indicative of hot gas in
the intragroup medium. However, we will show in \S
\ref{sec:discussion} that the available information suggests that this
group is predominantly composed of late-type spiral and S0 galaxies.
Spiral-rich groups are much fainter in X-rays but could still contain
hot intragroup gas if the gas is somewhat cooler ($10^{5} - 10^{6}$ K)
or has a much lower density than that found in elliptical-rich groups
(Mulchaey 2000). However, we argue that most of the gas in the \mbox{Ly$\alpha$}\relax\
complex at $z = 0.0635$ is unlikely to be hot, collisionally ionised
gas for several reasons:
First, the \ion{H}{1} lines in the \mbox{Ly$\alpha$}\relax\ complex are generally too
narrow. If the line broadening is dominated by thermal motions, then
the Doppler parameter is directly related to the gas temperature, $b =
\sqrt{2kT/m} = 0.129\sqrt{T/A}$, where $m$ is the mass, $A$ is the
atomic mass number, and the numerical coefficient is for $b$ in km
s$^{-1}$ and $T$ in K. Since other factors such as turbulence and
multiple components can contribute to the line broadening, $b-$values
provide only upper limits on the temperature. Applying this equation
to the \mbox{Ly$\alpha$}\relax\ line $b-$values from Table~\ref{tab:lyalist}, we find
that most of the \ion{H}{1} lines in the $z = 0.0635$ complex indicate
that $T \ll 10^{5}$~K, which is colder than expected for the diffuse
intragroup medium based on observed group velocity dispersions, even
in spiral-rich groups (e.g., Mulchaey et al. 1996). In a complex
cluster of \mbox{Ly$\alpha$}\relax\ lines, it is easy to hide a broad \mbox{Ly$\alpha$}\relax\ component
indicative of hot gas (see, e.g., Figure 6 in Tripp \& Savage 2000),
so the narrow \mbox{Ly$\alpha$}\relax\ lines do not preclude the presence of hot gas, but
they do indicate that many cool clouds are present in the intragroup
medium.
Second, the metal line profiles in component A favor cool,
photoionised gas. If component A metal lines were to arise in gas
in collisional ionisation equilibrium (CIE), the
$N$(\ion{C}{4})/$N$(\ion{Si}{4}) and $N$(\ion{Si}{4})/$N$(\ion{Si}{3})
column density ratios (integrated across both components seen in these species, see Table \ref{tab:cldnlist0064}) would require a gas temperature $T \approx 10^{4.9}$~K (Sutherland \& Dopita 1993). However
the \ion{C}{4} component at $v=-35$~km~s$^{-1}$ is rather narrow. To show this, Figure \ref{fig:civ064} plots an expanded view of the \ion{C}{4} doublet. We see that the $v=-35$~km~s$^{-1}$ is marginally resolved at the STIS E140M resolution of $\sim7$~km~s$^{-1}$. Voigt profile fitting for this component formally yields $b \ =\;4\pm2$~\mbox{km~s$^{-1}$}\relax, which is significantly lower than the $b-$value implied by the CIE temperature, i.e., $b \approx 10.5$
km s$^{-1}$. The stronger component at $v = 0$ km s$^{-1}$ is broader
(see Figure~\ref{fig:civ064}), but the $N$(\ion{C}{4})/$N$(\ion{Si}{4}) and
$N$(\ion{Si}{4})/$N$(\ion{Si}{3}) ratios are similar in the components at $v =
-35$ and 0 km s$^{-1}$, and we expect the ionisation mechanism and
physical conditions to be similar in both components.
\begin{figure}
\begin{center}
\resizebox{1.0\hsize}{!}{\includegraphics{figure_11.ps}}
\caption{Expanded view of the continuum-normalized \ion{C}{4} $\lambda
\lambda$1548.20, 1550.78 doublet at $z_{\rm abs} = 0.06352$, plotted
vs. absorber velocity. In this figure, the data are binned 2 pixels
into 1 (i.e., $\approx$ 7 km s$^{-1}$ pixels). The component at $-35$
km s$^{-1}$ is formally found to have $b = 4\pm 2$ km s$^{-1}$, which
is only marginally resolved at STIS E140M resolution. For clarity,
the \ionl{C}{4}{1550.78} profile is shifted upward by
0.5 flux units.\label{fig:civ064}}
\end{center}
\end{figure}
\begin{figure}
\begin{center}
\resizebox{1.0\hsize}{!}{\includegraphics{figure_12.ps}}
\caption{Predicted column densities, calculated with a CLOUDY
photoionisation model as described in the text, of \ion{O}{6} (small
circles), \ion{C}{4} (small squares), \ion{Si}{4} (small crosses), and
\ion{Si}{3} (small triangles) as a function of the ionisation
parameter $U$ (lower axis) and hydrogen number density $n_{\rm H}$
(upper axis) with $\log N$(\ion{H}{1})=15.37 and nearly solar
metallicity ([M/H] = $-0.05$). The larger markers with 1$\sigma$ error
bars are the observed \ion{C}{4}, \ion{Si}{4}, and \ion{Si}{3} column
densities measured from the STIS spectrum, plotted with the same
symbols at the best-fitting value of $U$. The 3$\sigma$ upper limit on
$N$(\ion{O}{6}) from the {\it FUSE}\ data is indicated with a large open
circle.}\label{fig:cloudym}
\end{center}
\end{figure}
Third, CLOUDY models photoionised by the UV background from QSOs are
fully consistent with the measured \ion{Si}{3}, \ion{Si}{4}, and
\ion{C}{4} column densities (and upper limits on undetected species)
at $z_{\rm abs}$ = 0.06352. Figure~\ref{fig:cloudym} shows the
relevant metal column densities predicted by CLOUDY models (small
symbols connected with solid lines) with log $N$(\ion{H}{1}) = 15.37
and [M/H] = $-0.05$ compared to the observed column densities (large
symbols). We can see that the metal column densities are in agreement
(within the 1$\sigma$ observational uncertainties) with this model at
log $U \approx -2.5$ (log $n_{\rm H} \approx -3.9$).
The narrow \ion{H}{1} and \ion{C}{4} components at this redshift could
still arise in shock-heated material if they originate in gas that is
not in ionisation equilibrium. Many papers have considered the
properties of gas that is initially shock-heated to some high
temperature and then cools more rapidly than it can recombine (e.g.,
Shapiro \& Moore 1976; Edgar \& Chevalier 1986).
However, if component A metal lines were to arise in gas in such a state, according to both computations from Shapiro \& Moore (1996) and Schmutzler \& Tscharnuter (1992), the $N$(\ion{Si}{3})/$N$(\ion{Si}{4}) column density ratio would require a gas temperature similar ($T \approx 10^{4.9}$~K) to the one found for the collisional ionisation equilibrium hypothesis. Moreover, assuming solar abundances, the predicted \ion{O}{6} column density at this temperature is always higher than our observed upper limit ($\sim$5 times higher in the Schmutzler model). Finally, the $N$(\ion{C}{4})/$N$(\ion{Si}{4}) column density ratio implies an even higher temperature than 10$^{4.9}$~K (2.5 times higherr in the Shapiro model). Because of these points, this non-equilibrium cooling gas scenario seems unlikely
to apply to the $z_{\rm abs} = 0.06352$ absorber toward \mbox{HS~0624+6907}\relax.
The CLOUDY modeling has some other interesting implications in
addition to the basic conclusion that the gas is photoionised. For
example, { the photoionisation model indicates that the absorber has a
relatively high metallicity of $Z\simeq0.9Z_\odot$ even though we have found no luminous
galaxies within $\rho\leq135$~\mbox{$h_{70}^{-1}\,{\rm kpc}$}.} A similarly high
metallicity ([O/H]$\simeq-0.2$) was recently reported by Jenkins et al. (2005) for a LLS
in the spectrum of PHL1811, but that system is much closer in
projection to a luminous galaxy. If we adopt the more conservative
upper limit on $N$(\ion{H}{1}) from the absence of a Lyman limit edge
($N$(\ion{H}{1})$=10^{16.1}$~\mbox{cm$^{-2}$}\relax, see Figure~\ref{fig:lls_rutr}) instead of the curve-of-growth
\ion{H}{1} column, we obtain [M/H] $\geq -0.75$. This lower limit is
still substantially higher than metallicities typically observed in
analogous absorbers at higher redshifts (e.g., Schaye et al. 2003) and
is comparable to abundances seen in high-velocity clouds near the
Milky Way (e.g., Sembach et al. 2001,2004; Collins, Shull, \& Giroux
2003; Tripp et al. 2003; Ganguly et al. 2005; Fox et al. 2005).
To derive confidence limits on parameters extracted from our CLOUDY
models, for combinations of metallicity $Z$ and ionisation parameter
$U$ we calculated the $\chi^{2}$ statistic,
\begin{equation}
\chi^2(Z,U) = \sum_i\Big(\frac{N_{i,obs} -
N_{i,model}(Z,U)}{\sigma(N_{i,obs})}\Big)^2,
\end{equation}
where $N$ indicates column density and the sum is over the three ions
\ion{Si}{3}, \ion{Si}{4}, and \ion{C}{4}. With the minimum $\chi^{2}$
obtained at [M/H] = $-0.05$ and log $U = -2.5$, we evaluated
confidence limits by finding parameters that increased $\chi ^{2}$ by
the amount appropriate for a given confidence level (see Lampton,
Margon, \& Bowyer 1976; Press et al. 1992). In this way, we find
[M/H] = $-0.05\pm 0.4$ at the 2$\sigma$ confidence level. Of course,
these confidence limits do not fully reflect potential sources of
systematic error such as uncertainties in the shape of the ionising
flux field or accuracy of the atomic data incorporated into
CLOUDY.
When we used the steeper UV background shape (e.g. Madau, Haardt, \& Rees 1999; Shull et al. 1999), the observations are still consistent with the CLOUDY model for a lower metallicity of [M/H] = $-0.24$ and a larger ionisation parameter log $U = -2.4$.
We can also place constraints on physical quantities such as the
absorber size (the length of the path through the absorbing region)
and the thermal gas pressure (but see the caveats discussed in \S 5 of
Tripp et al. 2005). Figure~\ref{fig:absorbersize} shows confidence
interval contours for the absorber size $L$ and thermal pressure $P/k$
implied by the photoionisation model. The best fit implies that $L
\approx$ 3.5 kpc. If spherical, the baryonic mass of this cloud would
be $\approx 10^{5} M_{\odot}$. However, we can see from
Figure~\ref{fig:absorbersize} that the model allows a large range for
$L$ at the 2$\sigma$ level. The low thermal pressure implied is also
notable. When the steeper UV background shape is used for the CLOUDY model, the predicted pressure is lower by a factor 1.5 and the absorber size increases to $\simeq$8~kpc.%
The range of pressures within the contours in
Figure~\ref{fig:absorbersize} is several orders of magnitude lower
than the gas pressure measured in the disk of the Milky Way (see
Jenkins \& Tripp 2001) and is even lower than pressures measured in
HVCs in the Milky Way halo (e.g. Wakker, Oosterloo, \& Putman 2002;
Fox et al. 2005). However, Sembach et al. (1995, 1999) found similar pressure for CIV HVCs surrounding the Milky-Way with somewhat lower metallicity.
Moreover, pressures this low are predicted in some
theoretical models of galactic halos (Wolfire et al. 1995). Finally,
the derived pressure depends on the intensity used to normalize
the ionising flux field (see Tripp et al. 2005) and both the particle density and
the pressure could be higher if the radiation field is brighter than
we assumed.
\begin{figure}
\begin{center}
\resizebox{1.0\hsize}{!}{\includegraphics{figure_13.ps}}
\caption{The thermal pressure $P/k$ of the absorber vs. its size $L$
parameter contours at 60, 90 and 99 per cent confidence levels in the
joint-fit with the CLOUDY photoionisation modelling. The solid and
the dotted black lines indicate for each parameter the best estimation
and the 68.7 per cent confidence interval
respectively.\label{fig:absorbersize}}
\end{center}
\end{figure}
\subsection{$z_{\rm abs} = 0.07573$}\label{ss:z076}
As discussed in \S \ref{sec:environment}, the absorption lines
detected at $z_{\rm abs}$ = 0.07573 occur in a large-scale structure
connected to \abell{559} and \abell{564}. Only \ion{H}{1} Ly$\alpha$,
Ly$\beta$ and the \ion{C}{4} $\lambda \lambda$1548.20,1550.78 doublet are
clearly detected at this redshift; the profiles of these absorption
lines are shown in Figure~\ref{fig:syst0076}. The weakness of
Ly$\beta$ and the absence of Ly$\gamma$ suggest that the \ion{H}{1}
lines are not badly affected by unresolved saturation, and profile
fitting measurements are robust. Likewise, comparison of the
\ion{C}{4} apparent column density profiles shows no evidence of
unresolved saturation. We have fitted these lines with only one
component. The results of the fit are listed in Table
\ref{tab:cldnlist076}. The width of the \ion{H}{1} line ($b=24.6$~\mbox{km~s$^{-1}$}\relax) indicates a
temperature for the gas lower than $10^{4.5}$~K, which again favors a
photoionisation process. No \ion{O}{6} is evident at this redshift,
but several strong unrelated lines of various elements are found close to the expected wavelength of
the \ion{O}{6} doublet, and these lines might mask weak \ion{O}{6}
absorption.
Despite the fact that \ion{C}{4} is the only metal detected in this
system, we can nevertheless place an interesting lower limit on the
absorber metallicity. The carbon abundance can be expressed as [C/H]=
log[\Nion{C}{4}/\Nion{H}{1}] + log[\fcion{H}{1}/\fcion{C}{4}] - log (C/H)$_\odot$, where \fcion{H}{1} and \fcion{C}{4} are the ion fractions of
\ion{H}{1} and \ion{C}{4}, respectively\footnote{We can derive a lower limit because \fcion{H}{1}/\fcion{C}{4} has a minimum value or, put another way, \Nion{C}{4} has a maximum value for any \Nion{H}{1} and [C/H] combination. The maximum \Nion{C}{4} is not evident in Figure~\ref{fig:cloudym} because it occurs at a higher value of $U$ than the range shown.}. With the \ion{H}{1} column
from the Ly$\alpha$+Ly$\beta$ fit (Table \ref{tab:cldnlist076}), and
again assuming that the gas is photoionised by the UV background from
QSOs (Haardt \& Madau 1996), we find that
log[\fcion{H}{1}/\fcion{C}{4}]$\geq-3.2$, and therefore
$[{\rm C}/{\rm H}]>-0.6$ in the \zabs{0.07573} absorber.
Once again, this metallicity lower limit is relatively high despite
the fact that no luminous galaxies have been found near the sight line
(the closest galaxy is NW3 at $\rho \ = 293 h_{70}^{-1}$ kpc, see
Table~\ref{tab:spec_red}).
\begin{figure}
\begin{center}
\resizebox{1.0\hsize}{!}{\includegraphics{figure_14.ps}}
\caption{Transitions observed at $z_{\rm abs}$ = 0.07573 in the
spectrum of \mbox{HS~0624+6907}\relax including \ion{H}{1} Ly$\alpha$, Ly$\beta$, and
the \ion{C}{4} doublet. The continuum-normalized profiles are plotted
vs. velocity in the absorber frame ($v$ = 0 km s$^{-1}$ at $z_{\rm
abs}$ = 0.07573). }\label{fig:syst0076}
\end{center}
\end{figure}
{
\def~{\phn}
\begin{table}
\begin{minipage}{\hsize}
\caption{Equivalents width and column densities of detected lines at $z = 0.07573$ \label{tab:cldnlist076}}
\begin{tabular}{@{}l@{}c@{\hspace{3ex}}c@{\hspace{3ex}}c@{\hspace{3ex}}c@{\hspace{3ex}}c@{}}
\colhead{} & \colhead{$\lambda_0$} & \colhead{$W_{obs}$} & \colhead{} & \colhead{$b$} & \colhead{}\\
\colhead{Species} & \colhead{(\AA)} & \colhead{(m\AA)} & \colhead{log $N_{tot}$} & \colhead{\mbox{km~s$^{-1}$}\relax} & \colhead{log $N_{a}$}
\\\hline
H~\textsc{i} & 1215.670 & 309$\pm$10 & 14.18$\pm$0.04 & 24.6$\pm$1.0 & 14.06$\pm$0.03\\
& 1025.722 & 108$\pm$21 & & & 14.28$\pm$0.08\\
O~\textsc{vi} & 1037.617 & ~~0$\pm$19 & & & $<13.9$\\
C~\textsc{iv} & 1548.204 & ~51$\pm$~7 & 13.18$\pm$0.04 & 13.1$\pm$1.5 & 13.24$\pm$0.04\\
& 1550.781 & ~27$\pm$~8 & & & 12.09$\pm$0.22\\
Si~\textsc{iii}& 1206.500 & ~~0$\pm$10 & & & $<12.1$\\
Si~\textsc{iv} & 1393.760 & ~17$\pm$14 & & & $<12.8$\\
N~\textsc{v} & 1238.821 & ~25$\pm$~9 & & & $<13.4$
\\\hline
\end{tabular}
\end{minipage}
\end{table}
}
\subsection{$z_{\rm abs} = 0.113$}\label{ss:z113}
In \S \ref{sec:environment} we noted that the clusters \abell{554},
\labell{562}, and \labell{565} indicate that a large-scale
filament/supercluster is foreground to \mbox{HS~0624+6907}\relax. Our galaxy
redshift survey has also revealed four galaxies close to \mbox{HS~0624+6907}\relax at
the redshift of the Abell 554/562/565 structure (see
Figure~\ref{fig:field} and Table~\ref{tab:spec_red}), which suggests
that the large-scale filament extends across the \mbox{HS~0624+6907}\relax
field. However, we only find a couple of weak Ly$\alpha$ lines near
the redshift of this structure at redshifts
substantially offset from those of the galaxies and Abell
clusters. It is possible that the Ly$\alpha$ lines are weak/absent
because the gas in the filament is so hot that the \ion{H}{1} ion
fraction makes the Ly$\alpha$ line undetectable, but we also note that
the nearest galaxies are farther from the sight line in this case
($\rho \geq 1.2 h_{70}^{-1}$ Mpc) than in the structures at $z = $
0.064\ and 0.077\ discussed above, so it is also possible that the sight
line does not penetrate the part of the dark matter filament where the
potential is deep enough to accumulate gas and galaxies (see
discussion in Bowen et al. 2002).
\section{Comparison to Other Ly$\alpha$ Absorbers}
How do the properties of the Ly$\alpha$ absorbers at 0.064 , 0.077 , and
0.110\ compare to the other Ly$\alpha$ lines in the \mbox{HS~0624+6907}\relax spectrum
(and in other sight lines)? We have found that the systems at 0.064\
and 0.077\ arise in photoionised cool gas; is this true of the majority
of the Ly$\alpha$ lines in the spectrum? In particular, do we find
Ly$\alpha$ lines that arise in hot gas? Richter et al. (2004, 2005)
and Sembach et al. (2004) have recently identified a population of
broad Ly$\alpha$ lines (BLAs) with $b >$ 40 km s$^{-1}$ in the spectra
of several low$-z$ QSOs (PG0953+415, PG1116+215, PG1259+593, and
H1821+643). Williger et al. (2005) similarly find a substantial
number of BLAs in the spectrum of PKS0405-123. Bowen et al. (2002) have also identified some BLA candidates using somewhat lower resolution data. If these lines are
mainly broadened by thermal motions, then they trace the warm-hot IGM,
and moreover, they in this case would contain a substantial portion of
the baryons in the Universe at the present epoch (see Richter et
al. 2004, 2005; Sembach et al. 2004). Based on simulations, Richter
et al. (2005) and Williger et al. (2005) find that some of the BLAs
are not predominantly thermally broadened but instead are due to line
blends that are difficult to recognize at the S/N afforded by typical
STIS echelle spectra. However, Richter et al. (2005) conclude that
approximately 50 per cent of the BLAs are mainly thermally broadened, and
some high S/N examples in the above papers are remarkably smooth and
broad and appear to entirely consistent with a single broad Gaussian
(see, e.g., Figures 4 and 5 in Richter et al. 2005).
\begin{figure}
\begin{center}
\resizebox{1.0\hsize}{!}{\includegraphics{figure_15.ps}}
\caption{Doppler parameters $b$ as a function of the \ion{H}{1} column
density $N$ for the Ly$\alpha$ lines detected in the spectrum of
\mbox{HS~0624+6907}\relax (solid circles, see Table~\ref{tab:lyalist}). The Ly$\alpha$
lines detected in the complex at $z$ = 0.064\ are shown with stars. For
comparison, we also plot the measurements from Sembach et al. (2004)
and Richter et al. (2004) from the PG1116+215 and PG1259+593 sight
lines (open squares). The solid line is the relation between $b$ and
$N$ for a Ly$\alpha$ absorption line with central optical depth = 0.1;
this represents a detection threshold for these data. The dotted line
shows the minimum $b-$value as a function of $N$(\ion{H}{1}) predicted
by equation 5 of Dav\'{e} \& Tripp (2001), which is based on the
hydrodynamic cosmological simulations of Dav\'{e} et al. (1999). Lines
above the horizontal dashed line have $T > 10^{5}$ K if the lines are
predominantly thermally broadened.}\label{fig:larg}
\end{center}
\end{figure}
In this paper, in addition to using a different sight line, we have
employed methods that are independent from (e.g., using different
software) the techniques used in the papers above for continuum
normalization, line detection, and profile fitting. Consequently, we
have an opportunity to independently check the BLA findings reported
in these papers. For the \mbox{HS~0624+6907}\relax sight line, we find that the mean
$b-$value for all Ly$\alpha$ lines is $<b>$ = 37 km s$^{-1}$, and the
median $b_{m} = 33$ km s$^{-1}$. However, as noted in
Table~\ref{tab:lyalist}, some of the Ly$\alpha$ lines are
significantly blended, and the line parameters are accordingly
uncertain. If we exclude these uncertain blended cases, we find $<b>
= 34$ km s$^{-1}$ and $b_{m} = 30$ km s$^{-1}$. These ensemble
$b-$values are in reasonable agreement with the previous
high-resolution Ly$\alpha$ studies.
In Figure~\ref{fig:larg} we compare our measurements of the \ion{H}{1}
$b-$values and column densities from Table~\ref{tab:lyalist} to the
measurements reported by Richter et al. (2004) and Sembach et
al. (2004). The solid line indicates $b$ vs. $N$(\ion{H}{1}) for a
Gaussian line with central optical depth = 0.1. This is effectively a
detection threshold; lines that have ($b,N$) combinations to the left
of this line are not likely to be detected. The dotted line in
Figure~\ref{fig:larg} shows the minimum $b-$value as a function of
$N$(\ion{H}{1}) predicted by Dav\'{e} \& Tripp (2001, see their
equation 5) from the hydrodynamic cosmological simulations of Dav\'{e}
et al. (1999). This predicted lower envelope appears to be in
reasonable agreement with the observed lower envelope for the three
sight lines shown in the figure.
From Figure~\ref{fig:larg}, we see that the ($b,N$) distribution that
we have obtained from the \mbox{HS~0624+6907}\relax sight line appears to be generally
similar to those obtained by Sembach et al. (2004) and Richter et
al. (2004). Sembach et al. and Richter et al. did find more extremely
broad Ly$\alpha$ lines ($b >$ 80 km s$^{-1}$) than we have been able
to positively identify in the \mbox{HS~0624+6907}\relax spectrum. This may be partly
due to signal-to-noise differences -- the data employed by Richter et
al. and Sembach et al. have higher S/N -- because such broad and
shallow lines are difficult to detect in the \mbox{HS~0624+6907}\relax data. For $40 <
b < 80$ km s$^{-1}$, the different sight lines appear to be in broad
agreement. Excluding lines within 5000 km s$^{-1}$ of the QSO
redshift,\footnote{Lines within 5000 km s$^{-1}$ of the QSO redshift
can arise in intrinsic gas ejected by the QSO, and these lines can be
rather broad (see, e.g., Yuan et al. 2002), even if not part of a
full-blown broad absorption line outflow.} the \mbox{HS~0624+6907}\relax spectrum can be
used to search for BLAs between $z_{\rm min} = 0.004$ and $z_{\rm max}
= 0.347$. Accounting for regions in which broad lines could have been
masked by IGM or ISM lines, we obtain a blocking-corrected total
redshift path $\Delta z = 0.329$. With 21 Ly$\alpha$ lines in the
sample with $b >$ 40 km s$^{-1}$, we thus obtain $dN/dz$(BLA) =
64$\pm$16. This is somewhat larger than the values reported by Sembach
et al. (2004) and Richter et al. (2004,2005). However, Richter et
al. have excluded BLAs that are located in complex blends on the
grounds that these cases are more likely to be affected by non-thermal
broadening. If we follow the same procedure, we must reject 10 BLAs
(see Table~\ref{tab:lyalist}); the remaining 11 BLAs would then imply
$dN/dz$(BLA) = 33$\pm$10. Using equations 1, 5, and 6 from Sembach et
al. (2004), but adjusted for the somewhat different cosmological
parameters assumed in this paper, we find that our full sample implies
that the BLA baryonic content is $\Omega_{b}$(BLA) = 0.017
$h_{70}^{-1}$ (in the usual notation, i.e., $\Omega =
\rho/\rho_{c}$). This high value probably substantially overestimates
the BLA baryonic content, largely because of false BLAs that arise
from blends. If we exclude BLAs that are located in complex blends,
this drops to $\Omega_{b}$(BLA) = 0.0036 $h_{70}^{-1}$, which is
similar to values obtained by Richter et al. and Sembach et al. The
uncertainties in $\Omega_{b}$(BLA) due to, e.g., lines that are broad
due to blends or other non-thermal broadening mechanisms, are large
and currently difficult to assess (see discussion in Richter et
al. 2005). However, these initial calculations suggest that BLAs may
harbor an important quantity of baryons. With future UV spectrographs,
it would be valuable to obtain high-resolution spectra with
substantially better S/N in order to accurately assess the baryonic
content of BLAs as part of the general census of ordinary matter in
the nearby Universe.
\section{Discussion}\label{sec:discussion}
We have acquired detailed information about the abundances, physical
conditions, and galaxy proximity of absorption systems in the
direction of \mbox{HS~0624+6907}\relax. What are the implications of these measurements
for broader questions of galaxy evolution and cosmology? The
processes that add gas to galaxies (e.g. accretion) and remove gas from galaxies
(e.g., winds, dynamical stripping) can have profound effects on galaxy
evolution, and the ``feedback'' of matter and energy from galaxies
into the IGM is now believed to play an important
role in shaping structures that subsequently grow out of the IGM (Voit G. M., 2005). The quantity and implications of the $10^{5} - 10^{7}$ K WHIM gas is a topic of
particular interest currently. The galaxies and absorption systems in
the direction of \mbox{HS~0624+6907}\relax, particularly the galaxy group and Ly$\alpha$
complex at $z$ = 0.064 , have some interesting, and perhaps surprising,
implications regarding these questions, which we now discuss.
\begin{figure}
\begin{center}
\resizebox{1.0\hsize}{!}{\includegraphics{figure_16.ps}}
\caption{Images of galaxies found in the $z$ = 0.064\ group in the
field of \mbox{HS~0624+6907}\relax, recorded in the $R$ band with the KPNO 4m MOSA camera.
Each box in the montage
spans $50''\times 50''$, and the galaxy name from
Table~\ref{tab:spec_red} is listed in the lower left corner. The
galaxy is in the center of each panel. SE6 and SE7 are in close
proximity; SE6 is the more extended spiral galaxy.}\label{fig:galpics}
\end{center}
\end{figure}
ROSAT observations of diffuse X-ray emission have established that
galaxy groups that are dominated by early-type galaxies often contain
diffuse, hot intragroup gas (Mulchaey 2000, and references
therein). Based on the observed relation between intragroup gas
temperature and velocity dispersion $\sigma$ in X-ray bright groups
($T \propto \sigma ^{2}$) and the fact that spiral-rich groups have
lower velocity dispersions than elliptical-rich groups, Mulchaey et
al. (1996) have hypothesized that spiral-rich groups might have
somewhat cooler intragroup media that could give rise to QSO
absorption lines at WHIM temperatures (e.g., \ion{O}{6}).
However, the galaxy group at $z$ = 0.064\ appears to have properties
that are not consistent with the elliptical-rich groups detected with
ROSAT nor with the idea that spiral-rich groups contain warm-hot
intragroup gas. It is unclear if the galaxy group at $z$ = 0.064\ is a
spiral-rich group. Figure~\ref{fig:galpics} shows $R$ images from the
MOSA data of the 10 galaxies that we have found in this group. Most of
the galaxies in the group show evidence of disks and bulges (both in
the direct images and in radial brightness profiles). We find from
visual inspection that 4-5 of the 10 galaxies have indications of
spiral structure (SW3, SE1, SE8, SE6, and possibly SE4). However, the
remaining galaxies could be early-type S0 galaxies, and therefore the
early-type fraction might be comparable to groups that show diffuse
X-ray emission (see, e.g, Figure 7 in Zabludoff \& Mulchaey 1998). The
NE3, SE13, and SE5 galaxies, which morphologically appear to be
early-type galaxies, have colours and magnitudes consistent with the
``red sequence'' colour-magnitude relation observed in clusters (e.g.,
Bower, Lucey, \& Ellis 1992; McIntosh et al. 2005); these galaxies are
likely S0s (the other 7 galaxies have blue colours characteristic of
late types). The velocity dispersion of the group at $z$ = 0.064 ,
albeit uncertain, is more comparable to those of elliptical-dominated
groups than spiral-rich groups (Mulchaey et al. 1996; Zabludoff \&
Mulchaey 1998).
Regardless of whether the group is elliptical- or spiral-rich, it is
surprising that we find a large number of cool, photoionised clouds in
the intragroup medium (\S \ref{ss:z064}). In the hot intragroup
medium of an elliptical-rich group, \ion{H}{1} lines should be
extremely broad and weak, but instead we find strong, narrow lines
(see Figure~\ref{fig:hi00635}). Even in the cooler gas predicted to
be found in late-type dominated groups, the \ion{H}{1} lines should be
broader. We could entertain models of cooling intragroup gas, but in
such models \ion{O}{6} is expected to be stronger. Likewise, if the
intragroup gas is a multiphase medium with cooler clouds (which cause
the \ion{H}{1} absorption lines) embedded in a hotter phase, then we
might expect to detect \ion{O}{6} from the interface between the
phases (Fox et al. 2005), unless conduction is somehow suppressed.
The lack of evidence of hot gas leads us to question whether this
group is a bound, virialized system. An alternative possibility is
that our sight line passes along the long axis of a large-scale
filamentary structure in the cosmic web. In this case, the projection
of the galaxies and Ly$\alpha$ clouds along the sight line could give
a false impression of a group in which hot gas would be
expected. However, in cosmological simulations of large-scale
filaments, WHIM gas is expected to be widespread at the present epoch,
even in modest-overdensity regions (see, e.g., Figure 4 in Cen \&
Ostriker 1999b), so it is interesting that we find a substantial
number of cool clouds at $z \approx$ 0.064 , somewhat contrary to
theoretical expectations. As noted above, our data do not preclude the
presence of WHIM gas at $z \approx$ 0.064 , but we find no clear
evidence of it. Other sight lines show similar clusters of Ly$\alpha$
lines, e.g., the Ly$\alpha$ complex at $z_{abs} \approx$ 0.057 toward
PKS2155-304 (Shull et al. 1998; Shull, Tumlinson, \& Giroux 2003) or
the Ly$\alpha$ lines at $z_{abs} \approx$ 0.121 toward H1821+643
(Tripp et al. 2001). However, unlike the \mbox{HS~0624+6907}\relax Ly$\alpha$ complex,
the PKS2155-304 and H1821+643 examples both show evidence of warm-hot
intragroup gas. To test whether the observations and simulations are
in accord, it would be useful to assess the frequency and physical
properties of these Ly$\alpha$ complexes in cosmological simulations
for comparison with the observations.
It is also interesting that the two systems for which we have obtained
abundance constraints (at $z$ = 0.064\ and 0.077 ) both indicate
relatively high metallicities, but both of these systems are at least
100 kpc away (in projection) from the nearest known galaxy. This
naturally raises a question: how did gas that is so far from a galaxy
attain such a high metallicity? The gas could have been driven out of
a galaxy by a galactic wind; some wind models predict that the
outflowing material will have a high metallicity (Mac Low \& Ferrara
1999), The difficulty with this interpretation is that winds from
nearby galaxies are usually observed to contain substantial amounts of
hot gas (e.g., Strickland et al. 2004), which seems to be inconsistent
with the absorption line properties as we have discussed.
A more likely explanation is that the high-metallicity gas we have
detected in absorption has been tidally stripped out of one of the
nearby galaxies. There are indications that tidal stripping could be a
more gentle process for removing gas from galaxies, and a tidally
stripped origin can therefore more easily accommodate the observed
low-ionisation state of the gas. For example, in the direction of
NGC3783, the Galactic high-velocity cloud (HVC) at $v_{\odot} =$ 247
\mbox{km~s$^{-1}$}\relax\ is now recognized to be tidally stripped debris from the
SMC. While this tidally stripped material shows a wide array of
low-ionisation absorption lines, it has little or no associated
high-ion absorption (Lu et al. 1994; Sembach et al. 2001a). Moreover,
the tidally-stripped HVC contains molecular hydrogen, which Sembach et
al. argue formed in the SMC and survived the rigors of tidal stripping
(as opposed to forming in situ in the stream). Both the absence of
high ions and the survival of H$_{2}$ suggest that the stripping
process did not substantially ionise and heat this HVC. Several
galaxies are close enough to the \mbox{HS~0624+6907}\relax sight line to be plausible
sources of the gas in a tidal stripping scenario. One of the nearby
galaxies, SE1, has a distorted spiral morphology. This galaxy is a
plausible source of tidally stripped matter.
\section{Summary}
We have presented a study of absorption-line systems in the direction
of \mbox{HS~0624+6907}\relax using a combination of high-resolution UV spectra obtained
with {\it HST}/STIS and {\it FUSE}\ plus ground-based imaging and
spectroscopy of galaxies within $\sim$30' of the sight line. In
addition to presented the basic measurements and ancillary
information, we have reported the following findings:
1. There are several Abell galaxy clusters in the foreground of
\mbox{HS~0624+6907}\relax, including two clusters at $z \approx$ 0.077 (\abell{559} and
\labell{564}) and three at $z \approx$ 0.110 (\abell{554}, \labell{562}, and
\labell{565}). These clusters trace large-scale dark matter structures,
i.e., superclusters or filaments of the ``cosmic web''. Our galaxy
redshift survey has revealed galaxies at these supercluster
redshifts in the immediate vicinity of the \mbox{HS~0624+6907}\relax sight line, and
therefore our QSO spectra provide an opportunity to study the gas
in large-scale intergalactic filaments. The most prominent group of
galaxies found in our galaxy redshift survey is at $z \approx$
0.064\ and is not associated with an Abell cluster with a
spectroscopic redshift from the literature. However, \abell{557}, for
which no spectroscopic redshift has been reported, is at least
partly due to the galaxy group at $z \approx$ 0.064 .
2. The two strongest Ly$\alpha$ absorption systems at $z < 0.2$ arise
in galaxy groups at $z = $ 0.064\ and 0.077 . The Ly$\alpha$
absorption at 0.064\ is particularly dramatic: at this redshift, we
find 13 Ly$\alpha$ lines spread over a velocity range of 1000 km
s$^{-1}$ with a line-of-sight velocity dispersion of 265 km
s$^{-1}$. The second-strongest system at 0.077\ is associated with
the Abell 559/564 large-scale structure, and this indicates that a
filament containing gas and galaxies feeds into the Abell 559/564
supercluster.
3. Analysis of the Ly$\alpha$ absorption-line complex at $z =$ 0.064\
provides strong evidence that the gas is photoionised and
relatively cool; we find no compelling evidence of warm-hot gas in
this large-scale filament. We detect \ion{Si}{3}, \ion{Si}{4}, and
\ion{C}{4} in the strongest component in this Ly$\alpha$ complex,
and photoionisation models indicate that the gas metallicity is
high, [M/H] = $-0.05 \pm 0.3$. This is somewhat surprising because
we do not find any luminous galaxies close to the sight line; the
closest galaxy is at a projected distance $\rho = 135 h_{70}^{-1}$
kpc. The Ly$\alpha$ system at 0.077\ is only detected in the
\ion{C}{4} doublet, but nevertheless we find a similar result: the
lower limit on the metallicity is relatively high ([C/H] $> -0.6$)
while the nearest galaxy is at $\rho = 293 h_{70}^{-1}$ kpc.
4. We have compared the distribution of Ly$\alpha$ Doppler parameters
and \ion{H}{1} column densities to high-resolution measurements
obtained from other sight lines, and we find good agreement. We
find that the number of broad Ly$\alpha$ absorbers with $b >$ 40 km
s$^{-1}$ per unit redshift is in agreement with results recently
reported by Richter et al. (2004,2005) and Sembach et al. (2004).
The baryonic content of these broad \ion{H}{1} lines is still
highly uncertain and requires confirmation with higher S/N data,
but it is probable that some of these broad line arise in warm-hot
gas and contain an important portion of the baryons in the nearby
universe. We also find that the lower bound on $b$
vs. $N$(\ion{H}{1}) is in agreement with predictions from
cosmological simulations.
5. The absence of warm-hot gas in the galaxy group/Ly$\alpha$ complex
at $z =$ 0.064\ is difficult to reconcile with X-ray observations of
bound galaxy groups. It seems more likely that this in this case
we are viewing a large-scale cosmic web filament along its long
axis. The filament contains a mix of early- and late-type galaxies
and many cool, photoionised clouds.
6. The high-metallicity, cool cloud at $z_{\rm abs}$ = 0.06352 is
probably tidally stripped material. This origin can explain the
high metallicity and the lack of hot gas. One of the nearby
galaxies has a disturbed morphology consistent with this
hypothesis.
Similar clusters of Ly$\alpha$ lines have been observed in other sight
lines, and additional examples are likely to be found as we continue
to analyse STIS data. We look forward to comparisons of these
observations to predictions from cosmological simulations and other
theoretical work in order to better understand the processes that
affect the evolution of galaxies and the intergalactic medium.
\section*{Acknowledgments}
We thank Dan McIntosh and Neal Katz for useful discussions. The STIS
observations of HS0624+6907 were obtained for {\it HST} program 9184
with financial support through NASA grant HST GO-9184.08-A from the
Space Telescope Science Institute. This research was also supported in
part by NASA through Long-Term Space Astrophysics grant NNG
04GG73G. The {\it FUSE}\ data were obtained by the PI team of the
NASA-CNES-CSA {\it FUSE}\ project, which is operated by Johns Hopkins
University with financial support through NASA contract NAS
5-32985. This research has made use of the NASA/IPAC Extragalactic
Database (NED), which is operated by the Jet Propulsion Laboratory,
California Institute of Technology, under contract with NASA.
|
math/0512500
|
\section{Introduction}
\subsection{History of the problem}
The theory of dynamical quantum groups is nowadays a well established part of mathematics,
see for example the review by P. Etingof \cite{E}. This theory originated from the notion of Dynamical Yang-Baxter equation which arose in the work of Gervais-Neveu on Liouville theory
\cite{GN} and was formalized first by G. Felder \cite{Fe} who also understood its
relation with IRF statistical models. The work \cite{ABB} gives some connections between Dynamical Yang-Baxter Equation and various models in mathematical physics.
\medskip
In 1991, O. Babelon \cite{Bab} found a universal explicit solution $F(x)$ to
the Quantum Dynamical Yang-Baxter equation in the case where $\mathfrak{g}=sl(2)$. He obtained this solution $F(x)$ by showing that it is a Quantum Dynamical coBoundary, i.e. $F(x)=\Delta(M(x)) M_2(x)^{-1}(M_1(xq^{h_2}))^{-1}$. The question whether this work can be generalized to any finite dimensional simple Lie algebra is an important problem which has, until now, received only uncomplete solution.
It has been shown in \cite{ABRR} (see also \cite{JKOS}) that, in the case of any finite dimensional simple Lie algebra $\mathfrak{g}$, the standard solution $F(x)$ of the Dynamical Cocycle Equation of weight zero satisfying an additional condition (of triangularity type) can be obtained as the unique solution of a linear equation now called the ABRR equation.
It implies that $F(x)$ can be expressed as an explicit infinite product which converges in any finite dimensional representation.
It remained nevertheless to study whether $F(x)$ is a dynamical coboundary and, if it is the case, to construct explicitely $M(x)\in U_q(\mathfrak{g})$.
For $\mathfrak{g}=sl(2)$, we have shown \cite{BR2} that $M(x)$ can be written as a simple infinite product which simplifies greatly the solution given in \cite{Bab}.
Concerning the more general case $\mathfrak{g}=sl(n+1)$, the first hint
appears in the article \cite{CG} on the study of Toda field theory:
the authors of \cite{CG} proved in particular that, in the fundamental
representation of $sl(n+1)$, the standard solution of the dynamical
Yang-Baxter equation (computed first in \cite{BG}) can be dynamically gauged
through a matrix $M(x)$ to a constant solution of the Yang-Baxter Equation
which is non standard in the case where $n\geq 2$ and is now called the
``Cremmer-Gervais's'' solution. The expression of $M(x)^{-1}$ is especially
simple: it is a Vandermonde matrix.
In the classification of Belavin and Drinfeld \cite{BD}, in which all the non-skewsymmetric classical r-matrices for simple Lie algebras are, up to an isomorphism, classified by a combinatorial object called the Belavin-Drinfeld triple, the Cremmer-Gervais's solution is associated to a particular Belavin-Triple of $sl(n+1)$ known as the shift $\tau$.
Moreover, it was proved in \cite{BDF} that it is only for $\mathfrak{g}=sl(n+1)$ that the standard solution of the dynamical Yang-Baxter equation can be dynamically gauged to a constant solution of Yang-Baxter Equation which, in that case, is a quantization of the r-matrix associated to the shift $\tau.$
Later, O. Schiffman \cite{Sc} generalized the notion of Belavin-Drinfeld triple and provided a classification of classical dynamical r-matrices up to isomorphism through the use of generalized Belavin-Drinfeld triple.
Then, P. Etingof, T. Schedler and O. Schiffmann \cite{ESS} managed to quantize explicitely all the previous generalized Belavin-Drinfeld triple. In particular, they obtained a universal expression for the twist $J\in U_q(sl(n+1))^{\otimes 2}$ associated to the shift.
The universal Cremmer-Gervais's solution is therefore $R^{J}=J_{21}^{-1}R_{12}J_{12}.$
\medskip
On the basis of all these works a natural problem to address is to construct a universal coboundary element in $U_q(\mathfrak{g})$ for $\mathfrak{g}=sl(n+1)$
, i.e. to solve the equation
\begin{equation}
F(x)=\Delta(M(x)){ J} M_2(x)^{-1}(M_1(xq^{h_2}))^{-1}
\end{equation}
where $F(x)$ is the standard solution of the dynamical cocycle equation in
$U_q(sl(n+1))^{\otimes 2}$ and $J$ is the universal twist associated to the shift.
We will solve this problem in the present work.
\subsection{Description and content of the present work}
Section~\ref{sectionQUEA} contains results (old and new) on quantum groups. We recall the definition and
properties of $U_q(\mathfrak g)$ and give emphasis on the quantum Weyl group and the
explicit expressions for the standard $R$-matrix.
We then recall the definition of a Belavin-Drinfeld triple and analyze the
$sl(n+1)$ case with the special triple known as shift.
We give in this case a direct construction of the universal twist as a simple finite
product of $q$-exponentials. Note that this construction is purely combinatorial and does not rely on the result of \cite{ESS}.
In Section~\ref{sectionDQG} we recall some results on dynamical quantum groups. We present the quantum dynamical cocycle equation, the linear equation
and give a summary of the results of Etingof, Schedler, Schiffmann \cite{ESS}.
We then formulate precisely the Dynamical coBoundary Equation and recall the known results in the fundamental representation of
$sl(n+1).$
Section~\ref{sectionDBP} is the core of our work. We introduce the notion of primitive loop which is constructed
from any solution of the Dynamical coBoundary Equation. This primitive loop satisfies a reflection equation with the corresponding $R$-matrix being of Cremmer-Gervais's type.
We then study the Gauss decomposition of $M(x)$ and shows that additional properties satisfied by these objects implies the
Dynamical coBoundary Equation.
Section~\ref{sectionConstr} is devoted to the explicit construction of $M(x)$ in the $U_q(sl(n+1))$ case as an infinite product converging
in every finite dimensional representation.
In Section~\ref{sectionQWG} we analyze the relations between $M(x)$ and the dynamical quantum Weyl group.
Section~\ref{sectionQRA} contains a construction of $M(x)$ through the
representation theory of non-standard reflection algebras, and in particular
through what we call the primitive representations.
\section{Results on Quantum Universal Envelopping Algebras}
\label{sectionQUEA}
\subsection{Basic Results on Quantized Simple Lie Algebras}
Let $\mathfrak{g}$ be a finite dimensional simple complex Lie algebra.
We denote by ${\mathfrak h}$ a Cartan subalgebra and by $r$ the rank of
$\mathfrak{g}.$
Let $\Phi$ be the set of roots, $(\alpha_{i},i=1,\dots,r)$ a choice of simple
roots and $\Phi^+$ the corresponding set of positive roots.
Let $(,)$ be a non zero $ad$-invariant symmetric bilinear form on
$\mathfrak{g}$. Its restriction on $\mathfrak{h}$ is non degenerate and
therefore induces a non degenerate symmetric bilinear form on
$\mathfrak{h}^*$ that we still denote $(,)$.
We denote by $\Omega$ the element in $S^2(\mathfrak{g})^{\mathfrak g}$
(the vector space of symmetric elements of $\mathfrak{g}^{\otimes 2}$ invariant
under the adjoint action) associated to
$(,)$, and by $\Omega_{\mathfrak h}$ its projection on
$\mathfrak{h}^{\otimes 2}.$
Let $A=(a_{ij})$ be the Cartan matrix of $\mathfrak{g}$, with elements
$a_{ij}=2\frac{(\alpha_i,\alpha_j)}{(\alpha_i,\alpha_i)}$.
There exists a unique collection of coprime positive integers $d_i$ such
that $d_i a_{ij}=a_{ji}d_j$.
$(,)$ is then uniquely defined by imposing that $(\alpha_i,\alpha_i)=2 d_i.$
If $\alpha\in\mathfrak{h}^{\star}$, we define the element
$t_{\alpha}\in \mathfrak{h}$ such that
$(t_{\alpha},h)=\alpha(h)$, $\forall h\in \mathfrak{h}$,
and denote $\mathfrak{h}_{\alpha}={\mathbb C}t_{\alpha}.$
To each root $\alpha$ we associate the coroot $\alpha^{\vee}=
\frac{2}{(\alpha,\alpha)}\alpha$ and denote
$h_{\alpha}=t_{\alpha^{\vee}},$ therefore
$ h_{\alpha_i}=\frac{1}{d_i}t_{\alpha_i}.$
Let $\lambda^{\alpha_1},\ldots,\lambda^{\alpha_r} \in \mathfrak{h}^{\star}$
be the set of fundamental weights, i.e. $\lambda^{\alpha_i}(h_{\alpha_j})=\delta^i_j.$
We will also denote by $\zeta^{\alpha_i}=t_{\lambda^{\alpha_i}}\in \mathfrak{h}$.
\bigskip
Let us now define the Hopf algebra $U_q(\mathfrak{g}).$
We will assume that $q$ is a complex number with $0<\vert q\vert <1.$
We define for each root $q_{\alpha}=q^{(\alpha,\alpha)/2},$
as well as $q_{i}=q_{\alpha_{i}}$,
and we denote $[z]_q=\frac{q^z-q^{-z}}{q-q^{-1}},\ z\in {\mathbb C}.$
$U_q(\mathfrak{g})$ is the unital associative algebra generated by
$e_{\alpha_1},\ldots,e_{\alpha_r},f_{\alpha_1},\ldots,f_{\alpha_r}$ and
$q^h,h\in \mathfrak{h}$, with defining relations:
\begin{align}
& q^h q^{h'}=q^{h+h'},\quad
q^h e_{\alpha_i} q^{-h}=q^{\alpha_i(h)}e_{\alpha_i},\quad
q^h f_{\alpha_i} q^{-h}=q^{-\alpha_i(h)}f_{\alpha_i},\quad
\forall h,h'\in \mathfrak{h},
\\
& [e_{\alpha_i},f_{\alpha_j}]=\delta_{ij}
\frac{q_i^{h_{\alpha_i}}-q_i^{-h_{\alpha_i}}}{q_i-q_i^{-1}},
\\
& \sum_{k=0}^{1-a_{ij}}\frac{(-1)^k}{[k]_{q_i}!\, [1-a_{ij}-k]_{q_i}!}
\,e_{\alpha_i}^{1-a_{ij}-k}\,e_{\alpha_j}\,e_{\alpha_i}^k=0,
\label{serree}\\
& \sum_{k=0}^{1-a_{ij}}\frac{(-1)^k}{[k]_{q_i}!\, [1-a_{ij}-k]_{q_i}!}
\,f_{\alpha_i}^{1-a_{ij}-k}\,f_{\alpha_j}\,f_{\alpha_i}^k=0.
\label{serref}
\end{align}
$U_q(\mathfrak{g})$ is a Hopf algebra with coproduct:
\begin{equation}
\Delta(q^h)=q^h\otimes q^h,
\quad
\Delta(e_{\alpha_i})= e_{\alpha_i}\otimes q_i^{h_{\alpha_i}}
+1\otimes e_{\alpha_i},
\quad
\Delta(f_{\alpha_i})=f_{\alpha_i}\otimes 1
+q_i^{-h_{\alpha_i}}\otimes f_{\alpha_i}.\label{coproductsln}
\end{equation}
Let us now define different notions associated to the polarisation of
$U_q({\mathfrak g}).$
We denote by $U_q(\fB_+)$ (resp. $U_q(\fB_-)$) the
algebra generated by $q^h$, $h\in\fH$, $e_{\alpha_i}$, $i=1,\dots,r$
(resp. $q^h$, $f_{\alpha_i}$, $i=1,\dots,r$),
and by $\fU_q(\fN_+)$ (resp. $\fU_q(\fN_-)$) the subalgebra of
$U_q(\mathfrak{g})$ generated
by $e_{\alpha_i},\ i=1,\dots,r$ (resp. $f_{\alpha_i},\ i=1,\dots,r$).
We have $U_q(\fB_+)=\fU_q(\fN_+)\otimes U_q({\mathfrak h})$ as a vector space
as well as $U_q(\fB_-)= U_q({\mathfrak h})\otimes \fU_q(\fN_-)$.
We denote by $\iota_{\pm}:U_q(\fB_{\pm})\rightarrow U_q({\mathfrak h})$
the projections on the zero-weight subspaces. $\iota_{\pm}$ are morphisms of algebra and we define the ideals $U^{\pm}_q({\mathfrak g})=ker \iota_{\pm}$. Endly, we denote by
${ C}_q({\mathfrak h})$ the centralizer in $U_q(\mathfrak{g})$
of the subalgebra
$U_q({\mathfrak h})$, i.e. the subalgebra of zero-weight elements of
$U_q(\mathfrak{g}).$
\bigskip
We now define a completion of $U_q(\mathfrak g)$ which enables us to define
elements (such as $R$) which are expressed in $U_q(\mathfrak{g})$ as an
infinite series or infinite product, but which evaluation in each finite
dimensional representation is well defined.
We denote ${\bf Rep}_{U_q(\mathfrak{g})}$ the category of finite dimensional
representations of $U_q(\mathfrak{g})$,
its objects are finite dimensional $U_q(\mathfrak{g})$-modules and
arrows are interwiners.
We define ${ \bf Vect}$ to be the category of vector spaces.
There is a forgetful functor ${\bf U}:{\bf Rep}_{U_q(\mathfrak{g})}\rightarrow
{ \bf Vect}.$
We now define ${\rm End}(\bf{U})$ to be the set of natural transformations from
${\bf U}$ to ${\bf U}$ which preserve the addition,
i.e. an element $a\in {\rm End}({\bf U})$ is a family of endomorphisms
$a_V\in {\rm End}(V)$ such that:
\begin{alignat}{2}
&\text{(naturality)} &\quad
&\text{for all } f:V\rightarrow W \text{ intertwiner then }
a_W\circ f=f\circ a_V,
\label{naturality}\\
& \text{(additivity)} &\quad
&a_{V\oplus W}=a_{V}\oplus a_{W}\label{additivity}.
\end{alignat}
${\rm End}(\bf{U})$ is naturally endowed with a structure of algebra.
We have a canonical homomorphism of algebra
$U_q(\mathfrak{g})\rightarrow {\rm End}(\bf{U})$ which associates to each element
$a\in U_q(\mathfrak{g}) $ the natural transformation whose $V$ component is
the element $\pi_V(a).$ We denote $(U_q(\mathfrak{g}))^c={\rm End}(\bf{U})$ this completion.
We can extend this construction to define a completion of
$U_q(\mathfrak{g})^{\otimes n}$ as follows.
We define $ {\bf U}^{\otimes n}$ to be the functor from the category
${\bf Rep}_{U_q(\mathfrak{g})}^{\times n} $ to
${ \bf Vect}$ which associates to an n-uplet $(V_1,\ldots,V_n)$ the vector space
$\otimes_{i=1}^n V_i$.
We define ${\rm End}({\bf U}^{\otimes n})$ to be the set of natural transformations
from ${\bf U}^{\otimes n}$ to ${\bf U}^{\otimes n}$ which are additive in
each entry.
An element $a$ in ${\rm End}({\bf U}^{\otimes n})$ is therefore a family of
endomorphisms $a_{V_1,\ldots,V_n}\in {\rm End}(\otimes_{i=1}^n V_i)$ satisfying the axioms of naturality
and additivity in each entry.
${\rm End}({ \bf U}^{\otimes n})$ is naturally endowed with a structure of algebra and we denote as well
$(U_q(\mathfrak{g})^{\otimes n})^c={\rm End}({ \bf U}^{\otimes n}).$
The coproduct of $U_q(\mathfrak{g})$ therefore defines a morphism of algebras
from $(U_q(\mathfrak{g}))^c $ to $(U_q(\mathfrak{g})^{\otimes 2})^c$ which associates to the
element $a\in (U_q(\mathfrak{g}))^c $ the element
$\Delta(a)$
where $\Delta(a)_{V,W}=a_{V\otimes W}.$
We now define the corresponding completions of $U_q^{\pm}(\mathfrak g),\ U_q(\mathfrak{h}),\
{ C}_q({\mathfrak h}).$
Let $V$ be a finite dimensional $U_q(\mathfrak{g})$-module, it is $\mathfrak{h}$ semisimple and we have $V=\bigoplus_{\lambda\in {\mathfrak h}^*}V[\lambda].$
An element $a\in {\rm End}(V)$ is said strictly upper triangular if
$aV[\lambda]\subset \oplus_{\lambda'> \lambda} V[\lambda'].$
An element $a\in {\rm End}(V)$ is said to be zero-weight if $aV[\lambda]\subset V[\lambda].$ If $a$ is zero-weight and such that the restriction of $a$ to $V[\lambda]$ is proportional to ${\rm id}_{V[\lambda]},$ $a$ is said to be diagonal.
We define $(U_q^{+}(\mathfrak{g}))^c$ as being the subspace of elements $a\in (U_q(\mathfrak{g}))^c$ such that $a_V$ is strictly upper triangular for all finite dimensional module $V.$
The analog definition holds for strictly lower triangular and this notion defines the subspace $(U_q^{-}(\mathfrak{g}))^c.$
Note that $(U_q^{\pm}(\mathfrak{g}))^c$ are subalgebras (without unit) of
$(U_q(\mathfrak{g}))^c$ and that the canonical homomorphisms can be restricted to homomorphisms of algebras
$U_q^{\pm}(\mathfrak{g})\rightarrow (U_q^{\pm}(\mathfrak{g}))^c.$
We define $(U_q(\mathfrak{h}))^c$ as being the
subalgebra of elements $a\in (U_q(\mathfrak{g}))^c$ such that $a_V$ is diagonal for all finite dimensional module $V.$
We also define $({ C}_q({\mathfrak h}))^c$ the subalgebra of elements $a\in (U_q(\mathfrak{g}))^c$ such that $a_V$ is zero-weight for all finite dimensional module $V.$
As a result we define the subalgebras $(U_q^{}(\mathfrak{b}_\pm))^c=(U_q(\mathfrak{h}))^c\oplus(U_q^{\pm}(\mathfrak{g}))^c.$
\bigskip
$U_q({\mathfrak g})$ is a quasitriangular Hopf algebra with an $R$-matrix
$R\in (U_q(\mathfrak{b}_+)\otimes U_q(\mathfrak{b}_-))^c$,
called the standard $R$-matrix,
which satisfies the following quasitriangularity axioms:
\begin{align}
&(\Delta \otimes id)(R^{(\pm)})=R^{(\pm)}_{13}\,R^{(\pm)}_{23},
\qquad
(id \otimes \Delta)(R^{(\pm)})=R^{(\pm)}_{13}\,R^{(\pm)}_{12},
\label{quasitriangularity}\\
&R^{(\pm)}\, \Delta(a) =\Delta'(a)\, R^{(\pm)},\
\forall a\in U_q(\mathfrak{g}).
\label{quasitriangularity2}
\end{align}
Here we have used the notation $R^{(+)}=R_{12},\ R^{(-)}=R_{21}^{-1}$.
The explicit expression of $R$ in terms of the root system will be recalled
further (see Eq.~\eqref{R=KR}).
Moreover $U_{q}({\mathfrak g})$ is a ribbon Hopf algebra, which means
that it exists an invertible element $v\in(U_q(\mathfrak{g}))^c $ such that
\begin{align}
&\mbox{$v$ is a central element,}\nonumber\\
& v^2=uS(u), \quad \epsilon(v)=1, \quad S(v)=v, \label{v}\\
&\Delta(v)=(R_{21}R_{12})^{-1}(v\otimes v).\label{propertyv}
\end{align}
Here, $u$ is the element $u=\sum_i S(b_i)a_i\in (U_q(\mathfrak{g}))^c$,
where $S$ is the antipode
and $R=\sum_i a_i \otimes b_i$. It satisfies the properties
\begin{align}
&S^2(x)=uxu^{-1},\ \forall x \in U_{q}{(\mathfrak{g})},\\
&\Delta(u)=(R_{21}R_{12})^{-1}(u\otimes u).\label{propertyu}
\end{align}
In this framework, the element $\mu=uv^{-1}$ is a group-like element that we
choose as follows:
\begin{equation*}
\mu=q^{2 t_\rho}=q^{2\sum_i \zeta^{\alpha_i}}
\quad \text{with}\quad \rho=\frac{1}{2}\sum_{\alpha \in \Phi^+} \alpha.
\end{equation*}
As a result, in a representation $\pi$ of highest weight $\lambda$,
$v$ is constant and takes the value $q^{-(\lambda,\lambda+2\rho)}$.
In the case where ${\mathfrak g}=sl(n+1)$, the fundamental representation
(also called the vector representation) is denoted
$\stackrel{f}{\pi}: U_q(\mathfrak{g})\rightarrow
\mathrm{Mat}_{n+1}(\mathbb{C})$
and we have
$\stackrel{f}{\pi}(h_{\alpha_i})=E_{i,i}-E_{i+1,i+1}$,
$\stackrel{f}{\pi}(e_{\alpha_i})=E_{i,i+1}$,
$\stackrel{f}{\pi}(f_{\alpha_i})=
E_{i+1,i},$ where $E_{i,j}$ is the basis of elementary matrices of
$\mathrm{Mat}_{n+1}(\mathbb{C}).$
The explicit value of $v$ in the fundamental representation is given by
$\stackrel{f}{\pi}(v)=q^{-\frac{n(n+2)}{n+1}} 1.$
\bigskip
Let us now review the explicit construction of the $R$-matrix using the
quantum Weyl group.
In order to obtain simple formulas we need to introduce the $q$-exponential
function.
The $q$-exponential is the meromorphic function
\begin{equation}
e_q^{z}=\sum_{n=0}^{+\infty}\frac{z^n}{(n)_q!}
=\frac{1}{((1-q^2)z;q^2)_{\infty}},
\quad z\in \mathbb{C} \label{defqexp}
\end{equation}
with the notation $(z)_q=q^{z-1}[z]_q=\frac{1-q^{2z}}{1-q^2}.$
Elementary properties of the $q$-exponential function, useful to derive
combinatorially the properties of $R$-matrices and the quantum Weyl elements
in the spirit of \cite{KT}, are now recalled.
We have, for any $z\in {\mathbb C}$,
\begin{align}
&e_q^{z}\;e_{q^{-1}}^{-z}=1,\label{expinverse}\\
&e_q^{q^2z}=e_q^{z}\;(1+(q^2-1)z),\label{expshift}
\end{align}
and for any elements $x,y$
\begin{align}
&e_q^{x}\;y\;e_{q^{-1}}^{-x}=y+\sum_{k=1}^{+\infty}\frac{1}{(k)_q!}
[x,[\cdots,[x,y]]_{q^2}\cdots]_{q^{2k-2}},
\end{align}
where $[x,y]_{q^{2m}}=xy-q^{2m}yx.$
For any $x,y$ such that $xy=q^2yx,$
\begin{align}
&e_q^{x+y}=e_q^{y}\;e_q^{x},\label{expsomme}\\
&e_q^{x}\; e_q^y=e_q^y \; e_q^{(1-q^{-2})xy}\;e_q^{x}\label{expproduit}.
\end{align}
In the case where $\mathfrak{g}=sl(2)$, we have $R=K \hat{R}$ where
$K=q^{\frac{h\otimes h}{2}}$ and $\hat{R}=e_{q^{-1}}^{(q-q^{-1})e\otimes f}$.
The quantum Weyl group of $U_q(sl(2))$ is formed by the element
$\hat{w}\in (U_q(sl(2)))^c$ defined as:
\begin{align}
{\hat w}&=e_{q^{-1}}^{{f}}\, q^{-\frac{h^2}{4}}\,
e_{q^{-1}}^{{-e}}\, q^{-\frac{h^2}{4}}\,
e_{q^{-1}}^{{f}}\, q^{-\frac{h}{2}}\\
&=
e_{q^{-1}}^{{-e}}\, q^{-\frac{h^2}{4}}\,
e_{q^{-1}}^{{f}}\, q^{-\frac{h^2}{4}}\,
e_{q^{-1}}^{{-e}}\, q^{-\frac{h}{2}}.\label{saitow}
\end{align}
The quantum Weyl group element $\hat{w}$ satisfies the two
identities:
\begin{align}
&\Delta(\hat{w})=\hat{R}^{-1}(\hat{w}\otimes\hat{w}),\\
&\hat{w}^2=q^{\frac{h^2}{2}}\xi v,
\end{align}
where $\xi\in (U_q(sl(2)))^c $ is a central group element
which value in each
irreducible finite dimensional representation of dimension $k$ is $(-1)^{k-1}.$
In the general case, let $W$ be the Weyl group associated to
the root system $\Phi$. For each root $\alpha$, let $s_{\alpha}\in W$ be
its associated reflection.
For any two distinct nodes $i,j$ of the Dynkin diagram we define an integer
$m_{ij}$ by $m_{ij}=2,3,4,6$ respectively if $a_{ij}a_{ji}=0,1,2,3.$
The defining relations of $W$ are:
$s_{\alpha_i}^2=1,\ (s_{\alpha_i}s_{\alpha_j})^{m_{ij}}=1.$
The braid group $\cal{B}$ associated to $W$ is the group generated by
$\sigma_1,\ldots,\sigma_r$ and satisfying the braid relations
\begin{equation}
[\sigma_i\sigma_j]^{m_{ij}/2}=[\sigma_j\sigma_i]^{m_{ij}/2}
\label{braidrelations},
\end{equation}
where we have used the following notation:
if $a,b$ are two elements of a group we define $[ab]^{n/2}$ for $n\geq 0$ to
be the element $(ab)^{n/2}$ if $n$ is even and $(ab)^{(n-1)/2}a$ if $n$
is odd.
To each simple root $\alpha$ we associate an element $\hat{w}_{\alpha}$ as
follows:
if $U_q(sl(2))_{\alpha}$ is the Hopf subalgebra of $U_q(\mathfrak g)$
generated by $e_{\alpha},\ f_{\alpha},\ q^{h}\ (h\in \mathfrak{h}_{\alpha})$,
we have $U_q(sl(2))_{\alpha}=U_{q_{\alpha}}(sl(2))$ as a Hopf algebra,
and therefore we can construct the element $\hat{w}_{\alpha}$ of
$(U_q(sl(2))_{\alpha})^c$ by the same procedure as (\ref{saitow}).
The elements $\hat{w}_{\alpha_i}$ satisfy the braid relations
({\ref{braidrelations}}).
One therefore obtains a morphism from the braid group $\cal B$ to the group
of invertible elements of $U_q(\mathfrak g)$ which is called the
quantum Weyl group \cite{KR,LS}.
Let $w=s_{\alpha_{i_1}}\ldots s_{\alpha_{i_k}}$ be a reduced expression of an
element $w\in W$, then the element $\hat{w}=
\hat{w}_{\alpha_{i_1}}\ldots\hat{w}_{\alpha_{i_k}}$ does not depend on the
choice of the reduced expression.
One can therefore associate to $w$ the automorphism $T_{w}$ of
$U_q(\mathfrak{g})$ defined as
$T_{w}(a)=\hat{w}a{\hat{w}}^{-1}.$
Let $w_0$ be the longest element of $W$ and let
$w_0=s_{\alpha_{i_1}}\ldots s_{\alpha_{i_p}}$ be a reduced expression.
We have $p=\vert \Phi^+ \vert.$
The set $\{\alpha_{i_1},s_{\alpha_{i_1}}(\alpha_{i_2}),\ldots ,
s_{\alpha_{i_1}}\ldots s_{\alpha_{i_{p-1}}}(\alpha_{i_p})\}$ contains every
positive root exactly once and defines therefore an order on the set of
positive root by:
\begin{equation}
\alpha_{i_1}<s_{\alpha_{i_1}}(\alpha_{i_2})<\cdots <
s_{\alpha_{i_1}}\ldots s_{\alpha_{i_{p-1}}}(\alpha_{i_p}).
\end{equation}
This ordering of positive roots is a normal ordering in the sense of
V. Tolstoy \cite{KT}.
Using this ordering, one can now express the standard $R$-matrix of
$U_q(\mathfrak{g})$ in a similar way as in the $U_q(sl(2))$ case.
Indeed, for $\alpha\in \Phi^+$, there exists a unique $k$ such that
$\alpha=s_{\alpha_{i_1}}\ldots s_{\alpha_{i_{k-1}}}(\alpha_{i_k})$,
which enables us to define
$e_{\alpha}=T_{\alpha_{i_1}}\ldots T_{\alpha_{i_{k-1}}}(e_{i_k})$ as well as
$f_{\alpha}=T_{\alpha_{i_1}}\ldots T_{\alpha_{i_{k-1}}}(f_{i_k})$.
The algebra generated by
$e_{\alpha},\ f_{\alpha},\ q^h\ (h\in \mathfrak{h}_{\alpha})$ is
$U_{q_{\alpha}}(sl(2))$. We therefore define
\begin{equation}
\hat{R}_{\alpha}=e_{q_{\alpha}^{-1}}^{(q_{\alpha}-q_{\alpha}^{-1})
e_{\alpha}\otimes f_{\alpha}}.
\end{equation}
The standard $R$-matrix of $U_q(\mathfrak{g})$ is expressed as:
\begin{equation}
R=K{\widehat{R}} \ \mbox{ where }\
K=\prod_{j=1}^r q^{h_{\alpha_j}\otimes\zeta^{\alpha_j}}\ \mbox{ and }\
{\widehat{R}}=\prod_{\alpha\in \Phi^+}^{>}{\widehat{R}}_{\alpha},
\label{R=KR}
\end{equation}
Its associated classical r-matrix is
\begin{equation}
r=\frac{1 }{2}\Omega_{\mathfrak{h}}+
\sum_{\alpha\in \Phi^+} \frac{(\alpha,\alpha)}{2} e_{\alpha} \otimes f_{\alpha}.
\end {equation}
An important result states that the value of the standard $R$-matrix
\eqref{R=KR} is independent of
the choice of the reduced expression of $w_{0}.$
Moreover, we have by construction
\begin{equation}
\Delta(\hat{w}_{0})={\widehat{R}}^{-1}(\hat{w}_{0}\otimes \hat{w}_{0}).
\end{equation}
If we define $\omega=q^{-\frac{h_{\alpha_i}\zeta^{\alpha_i}}{2}}\hat{w}_{0}$,
this element satisfies:
\begin{align}
&\Delta(\omega)=R^{-1} \omega_1 \omega_2\label{deltaw2}.
\end{align}
Let us finally precise simpler conventions in the case
$\mathfrak{g}=sl(n+1).$
We will use the shortened
notations $h_{(i)}=h_{\alpha_i},\ \zeta^{(i)}=\zeta^{\alpha_i},\
e_{(i)}= e_{\alpha_i},\ f_{(i)}=f_{\alpha_i}$, as well as
$w_{\alpha_i}=w_{(i)}.$
We will choose the following reduced expression of $w_{0}$:
\begin{equation}
w_{0}= w_{(1)}(w_{(2)}w_{(1)})\dots (w_{(n)}\dots w_{(1)}),
\end{equation}
which implies the following ordering on roots:
\begin{multline}
\alpha_1<\alpha_1+\alpha_2<\alpha_2<\cdots\\
\cdots<\alpha_{n-1}< \alpha_1+\alpha_2+\cdots +\alpha_n
<\alpha_2+\cdots+\alpha_n<\cdots<\alpha_{n-1}+\alpha_n<\alpha_n.
\end{multline}
If $1\leq i\leq j\leq n$, we define the positive root
$\alpha_{ij}=\sum_{k=i}^j\alpha_k.$
We can therefore construct a Cartan-Weyl basis in two different ways.
The first one has already been explained, we denote $e_{\alpha_{ij}},
f_{\alpha_{ij}}$ the corresponding elements.
The second one uses the inductive algorithm of \cite{KT}.
We denote by $e_{(ij)},f_{(ij)}$ the corresponding elements.
We have $e_{\alpha_{ij}}=(-1)^{i-j}e_{(ij)}$,
$f_{\alpha_{ij}}=(-1)^{i-j}f_{(ij)}.$
In the fundamental representation, we have
$\stackrel{f}{\pi}(e_{(ij)})=E_{i,j+1}$,
$\stackrel{f}{\pi}(f_{(ij)})=E_{j+1,i}$, and the explicit expression
$\mathbf{R}=(\stackrel{f}{\pi}\otimes \stackrel{f}{\pi})(R)$ of the standard
$R$-matrix is given by
\begin{equation}
\mathbf{R}= q^{-\frac{1}{n+1}}
\bigg\{q\sum_{i=1}^{n+1} E_{ii}\otimes E_{ii}+
\sum_{1\leq i\not=j\leq n+1} \!\!\! E_{ii}\otimes E_{jj}
+(q-q^{-1}) \!\!\! \sum_{1\leq i<j\leq n+1} \!\!\! E_{ij}\otimes E_{ji}\bigg\}.
\end{equation}
\subsection{Belavin-Drinfeld Triples and Cremmer-Gervais $R$-matrices}
We recall here the notion of Belavin-Drinfeld triple and study the
$sl(n+1)$ case with a special attention to the shift and to the universal construction of the Cremmer-Gervais's solution.
A Belavin-Drinfeld triple \cite{BD} for a simple Lie algebra $\mathfrak{g}$ is a triple $(\Gamma_1,\Gamma_2, T)$ where $\Gamma_1,\Gamma_2$ are subsets of the Dynkin diagram $\Gamma $ of $\mathfrak{g}$ and $T:\Gamma_1\rightarrow \Gamma_2$ is an isomorphism which preserves the inner product and satisfies the nilpotency condition: if
$\alpha\in \Gamma_1$ then there exists $k$ such that $T^{k-1}(\alpha)\in \Gamma_1$ but $T^{k}(\alpha)\notin \Gamma_1.$
We extend $T$ to a Lie algebra homomorphism $T:\mathfrak{n}_+ \rightarrow \mathfrak{n}_+$ by setting on simple root elements $T(e_{\alpha})=e_{T(\alpha)}$ for $\alpha\in \Gamma_1$, and zero otherwise.
Any solution $\bf{r}$ to the CYBE satisfying ${ \bf r}+{ \bf r}_{21}=\Omega$ is equivalent, under an automorphism of $\mathfrak{g}$, to a solution of the form
\begin{equation}
r_{T,s}=r-s+\sum_{\alpha\in \Phi^+}\sum_{l=1}^{+\infty}\frac{(\alpha,\alpha)}{2}
T^l(e_{\alpha})\wedge f_{\alpha},
\end{equation}
where $s\in \bigwedge^2 \mathfrak{h}$ is a solution of the affine equations:
\begin{equation}
2((\alpha-T\alpha)\otimes id)(s)=((\alpha+T\alpha)\otimes id)(\Omega_{\mathfrak{h}}).
\end{equation}
Given a Belavin-Drinfeld triple, the affine space of the $s$ satisfying the previous equations is of dimension $k(k-1)/2$ where
$k=\vert \Gamma\setminus \Gamma_1\vert$.
In the present work, we are mainly concerned with the case where ${\mathfrak g}=sl(n+1)$, and with the following Belavin-Drinfeld triple, known as the shift:
\begin{align}
&\Gamma=\{\alpha_1,\ldots,\alpha_{n}\},
\quad
\Gamma_1=\{\alpha_2,\ldots,\alpha_{n}\},
\quad
\Gamma_2=\{\alpha_1,\ldots,\alpha_{n-1}\},
\nonumber\\
&\tau:\ \Gamma_1\rightarrow \Gamma_2,\ \alpha_i\mapsto\alpha_{i-1}.
\label{belavintriple}
\end{align}
Since in this case $k=1$, this Belavin-Drinfeld triple selects a unique $s\in \bigwedge^2 \mathfrak{h}$ which is
\begin{equation}
s=-\frac{1}{2}\sum_{j=1}^{n-1}\zeta^{(j)}\wedge \zeta^{(j+1)}.\label{sCG}
\end{equation}
The quantization of the corresponding $r$-matrix in the fundamental representation is known as the Cremmer-Gervais's solution \cite{CG,BG}.
A universal construction has been given in \cite{KM} in the case $\mathfrak{g}=sl(3)$, whereas the complete understanding of the explicit expression of the quantization of any solution of Belavin-Drinfeld type has been given in \cite{ESS}.
We will not follow here this last result, and will instead construct directly the quantization of the $r$-matrix associated to the shift.
The expression that we will obtain is simpler than the one obtained by the method of \cite{ESS}.
Let us first recall the notion of cocycle.
An invertible element $J\in (U_q(\mathfrak{g})^{\otimes 2})^c$ satisfying
\begin{equation}
(\Delta \otimes id)(J)J_{12}=(id \otimes \Delta)(J)J_{23},\label{cocycle1}
\end{equation}
is called a cocycle.
If $J$ is a cocycle, $(U_q(\mathfrak{g}))^c$ can be endowed with a new
quasitriangular Hopf algebra structure with twisted coproduct
$\Delta^{J}(.)={ J}_{12}^{-1}\Delta(.){ J}_{12}$,
and twisted $R$-matrix $R^{J}={ J}_{21}^{-1}R{ J}_{12}.$
We define $R^{ J(+)}=R^{J},$ $R^{J(-)}=(R_{21}^{J})^{-1}.$
We will now construct the universal twist associated to the shift triple.
We define two morphisms of algebras:
\begin{alignat}{2}
\tau :\
&U_q({\mathfrak n}^+) \rightarrow U_q({\mathfrak n}^+) & &
\nonumber\\
&e_{(i)}\mapsto e_{(i-1)},\ \forall i=2,\ldots n, &
\qquad &e_{(1)}\mapsto 0,
\label{deftau}\\
\tilde{\tau} :\
&U_q({\mathfrak n}^-) \rightarrow U_q({\mathfrak n}^-) & &
\nonumber\\
&f_{(i)}\mapsto f_{(i+1)},\ \forall i=1,\ldots n-1, &
\qquad &f_{(n)}\mapsto 0.
\label{deftautilde}
\end{alignat}
\begin{rem}\label{rem-notations}
In order to simplify expressions we will use the notations $e_{(0)}=\zeta^{(0)}=f_{(n+1)}=\zeta^{(n+1)}=0$.
\end{rem}
We have the following result:
\Theorem{\label{cocycleBCG}}{
For ${\mathfrak g}=sl(n+1)$, a solution $J$ of the cocycle equation
\begin{equation}
(\Delta \otimes id)(J)\,J_{12}=(id \otimes \Delta)(J)\,J_{23},
\label{cocycle}
\end{equation}
associated to the shift $\tau$ is given by:
\begin{equation}
J=\prod_{k=1}^{+\infty}J^{[k]} \qquad \text{with}\quad
J^{[k]}=W^{[k]}\;\widehat{J}{}^{[k]},
\end{equation}
where, $\forall k \in {\mathbb N}^*$,
\begin{align}
&\widehat{J}{}^{[k]}=(\tau^{k} \otimes id)(\widehat{R})
\quad\in
\left( 1\otimes1+(U^+_q({\mathfrak g})\otimes U^-_q({\mathfrak g}))^c\right),
\\
&W^{[k]}_{12}=S^{[k]}(S^{[k+1]})^{-1},
\qquad
S^{[k]}=q^{\sum_{i=1}^{n-k} \zeta^{(i)} \otimes
\zeta^{(i+k)}}.
\end{align}
This solution will be called {\em Cremmer--Gervais cocycle}
and $R^J=J^{-1}_{21}RJ_{12}$ will be called {\em Cremmer--Gervais $R-$matrix}.
}
\begin{rem}
Due to the nilpotency of $\tau$ all the products are actually finite. More precisely, $\widehat{J}{}^{[k]}=S^{[k]}=1 \otimes 1$, $\forall k \geq n$.
\end{rem}
\begin{rem}
For any $m$ such that $0\leq m\leq k$, one has also $\widehat{J}{}^{[k]}=(\tau^{k-m} \otimes \tilde{\tau}{}^{m})(\widehat{R})$,
the resulting expression being independent of $m$.
\end{rem}
\medskip\noindent {\sf{\underline{Proof}:\ }}
Let us give a direct proof of the cocycle identity for $J$ in this framework.
First, let us remark the following elementary results deduced from the properties of the coproduct:
\begin{equation}
(\Delta \otimes id)(S^{[k]})=S_{13}^{[k]}\, S_{23}^{[k]},\qquad
(id \otimes \Delta)(S^{[k]})=S_{13}^{[k]}\, S_{12}^{[k]},
\label{prop11}
\end{equation}
and the properties of the $R$-matrix using $\tau$:
\begin{align}
&(\Delta \otimes id)(\widehat{J}{}^{[k]})
=\widehat{J}{}^{[0,k]}_{1(2\mid 3}\,\widehat{J}{}^{[k]}_{23},
\label{propJ1}\\
&(id \otimes \Delta)(\widehat{J}{}^{[k]})
=\widehat{J}{}^{[k,0]}_{1\mid 2)3}\,\widehat{J}{}^{[k]}_{12},
\label{propJ2}\\
&\widehat{J}{}^{[k]}_{12} \, \widehat{J}{}^{[k,m]}_{1 (2\mid 3}\,
\widehat{J}{}^{[m]}_{23} = \widehat{J}{}^{[m]}_{23} \,
\widehat{J}{}^{[k,m]}_{1 \mid 2) 3}\, \widehat{J}{}^{[k]}_{12},
\label{propJ3}
\end{align}
where we have defined
\begin{align}
&\widehat{J}{}^{[k,m]}_{1(2\mid 3}
=(\tau^{k} \otimes id \otimes\tilde{\tau}{}^{m})
(K^{-1}_{23}\widehat{R}_{13}\,K_{23}),
\label{defJ1}\\
&\widehat{J}{}^{[k,m]}_{1\mid 2) 3}
=(\tau^{k} \otimes id \otimes \tilde{\tau}{}^{m})
(K^{-1}_{12}\widehat{R}_{13}\,K_{12}).
\label{defJ2}
\end{align}
These elements are well defined because $K^{-1}_{23}\widehat{R}_{13}K_{23}$ and
$K^{-1}_{12}\widehat{R}_{13}K_{12}$ belong to
$(U_q({\mathfrak n}^+)\otimes U_q({\mathfrak h})\otimes U_q({\mathfrak n}^-))^c.$
Using the explicit values of $W,J$ and the properties of $\tau$, one can also prove that:
\begin{align}
&[J^{[n-i]}_{23},J^{[j]}_{12}]=0,\quad\forall \;i,j\; / \;1 \leq i \leq j \leq n-1,
\label{prop1}\\
&[\widehat{J}{}^{[k]}_{12},W^{[k+m]}_{13}W^{[m]}_{23}]=[\widehat{J}{}^{[k]}_{23},W^{[k+m]}_{13}W^{[m]}_{12}]=0,
\;\forall k,m \in\{1,\ldots,n\!-\!1 \}\; /\; k\!+\!m \leq n\!-\!1,
\label{prop2}\\
&W^{[m+1]}_{23}\widehat{J}{}^{[l-m-1,m+1]}_{1(2\mid 3}(W^{[m+1]}_{23})^{-1}=W^{[l-m]}_{12}\widehat{J}{}^{[l-m,m]}_{1\mid 2) 3}(W^{[l-m]}_{12})^{-1},\;\forall 0\leq m < l \leq n\!-\!1.
\label{prop3}
\end{align}
\eqref{prop1} is an immediate consequence of the fact that $J^{[i]}_{12}\in A^+_{i} \otimes A^-_{i}$, where $A^+_{i}$ (resp. $A^-_{i}$) is the subalgebra of $U_q({\mathfrak g})$ generated by $q^{\zeta^{(k)}},e_{(k)},\;k=1,\ldots,n-i$ (resp. generated by $q^{\zeta^{(k)}},f_{(k)},\;k=i+1,\ldots,n$).
In order to check \eqref{prop2} and \eqref{prop3}, we use the following notation:
\begin{equation}
W^{[k]}_{12}=\prod_{{i,j\in \{1,\cdots,n\}}} q^{\varepsilon^{k}_{i,j}\,
\zeta^{(i)} \otimes \zeta^{(j)}},
\end{equation}
with
\begin{equation}
\varepsilon^{(m)}_{i,j}=\left\{ \begin{array}{lll}
1\qquad\! \text{if} \quad i,j,m\in \{1,\ldots,n-1\},\quad j=m+i,\\
-1\quad \text{if} \quad i,j,m\in \{1,\ldots,n-1\},\quad j=m+i+1,\\
0 \qquad\! \text{otherwise.}
\end{array}\right.
\end{equation}
\eqref{prop2} follows immediately from the fact that
\begin{align}
&\varepsilon^{m}_{i,j}=\varepsilon^{m-k}_{i+k,j},\;\; \forall i,j,k,m \in \{1,\ldots,n\!-\!1\}\;/\; i\!+\!k \leq n\!-\!1,1 \leq m\!-\!k,
\label{identiteepsilon1}\\
&\varepsilon^{m}_{i,j}=\varepsilon^{m-k}_{i,j-k},\;\;\forall i,j,k,m \in \{1,\ldots,n\!-\!1\}\;/\; 1 \leq m\!-\!k,1 \leq j\!-\!k,
\label{identiteepsilon2}
\end{align}
and \eqref{prop3} is equivalent to
\begin{equation}
h_{(p-m-1)}+h_{(p-m)}=\varepsilon^{m+1}_{i,p}\,\zeta^{(i)}
+\varepsilon^{l-m}_{p-l,j}\,\zeta^{(j)},
\quad\forall l,m,p\; /\;
1 \leq m\!+\!1 \!\leq\! l \!\leq \! p\!-\!1 \!\leq \!n\!-\!1,
\label{identiteepsilon3}
\end{equation}
which is satisfied due to the relation
\begin{equation*}
h_{(i)}=2\zeta^{(i)}-\zeta^{(i-1)}-\zeta^{(i+1)}.
\end{equation*}
We can now prove the cocycle identity for $J$ by recursion. Indeed, using
properties \eqref{prop11}--\eqref{prop3}, we deduce easily the following
recursion relation, proved in Appendix~\ref{sec-lemmas}
(Lemma~\ref{appendlemma1}):
\begin{multline*}
\prod_{k=p}^{n-1}\left\{ W^{[k]}_{13}\, W^{[k-p+1]}_{23}\,
\widehat{J}{}^{[p-1,k-p+1]}_{1(2\mid 3}\,
\widehat{J}^{[k-p+1]}_{23} \right\}
\Big\{ \prod_{k=p}^{n-1}J^{[k]}_{12} \Big\}\\
=(id \otimes \Delta)(J^{[p]})
\prod_{k=p+1}^{n-1}\left\{ W^{[k]}_{13}\, W^{[k-p]}_{23}\,
\widehat{J}{}^{[p,k-p]}_{1 (2 \mid 3}\,
\widehat{J}^{[k-p]}_{23}\right\}
\Big\{ \prod_{k=p+1}^{n-1} J^{[k]}_{12} \Big\}
J^{[n-p]}_{23},
\end{multline*}
and having remarked that
\begin{align*}
&(\Delta \otimes id)(J) \, J_{12}
=\prod_{k=1}^{n-1}\left\{ W^{[k]}_{13} \, W^{[k]}_{23}
\widehat{J}{}^{[0,k]}_{1(2\mid 3}\,
\widehat{J}^{[k]}_{23} \right\}
\Big\{\prod_{k=1}^{n-1}J^{[k]}_{12}\Big\},\\
&(id \otimes \Delta)(J)\, J_{23}
=\prod_{k=1}^{n-1}(id \otimes \Delta)(J^{[k]})
\;\prod_{k=1}^{n-1}J^{[k]}_{23},
\end{align*}
we conclude the proof of the cocycle identity verified by $J.$
\qed
In the following, we extend the morphisms of algebras $\tau$ and $\tilde\tau$
to $U_q(\mathfrak{b}^+)$ and $U_q(\mathfrak{b}^-)$ respectively
as
\begin{alignat}{2}
&\tau(\zeta^{(i)})=\zeta^{(i-1)},\ i=2,\ldots,n,
& \qquad &\tau(\zeta^{(1)})=0,
\label{deftau1}\\
&\tilde\tau(\zeta^{(i)})=\zeta^{(i+1)},\ i=1,\ldots,n-1,
&\qquad &\tilde\tau(\zeta^{(n)})=0.
\label{deftautilde1}
\end{alignat}
Note that, with this definition, $\tau$ and $\tilde\tau$ are not
morphisms of Hopf algebras and are different from the extension of \cite{ESS}.
Indeed, their action on the coproduct is given as
\begin{alignat}{2}
&(\tau\otimes\tau)\left(\Delta(a)\right) = q^{\zeta^{(n-1)}\otimes\zeta^{(n)}}
\,\Delta\left(\tau(a)\right)\, q^{-\zeta^{(n-1)}\otimes\zeta^{(n)}},
\quad
& &\forall a \in U_q(\mathfrak{b}^+),
\label{deltatau}\\
&(\tilde\tau\otimes\tilde\tau)\left(\Delta(a)\right) =
q^{\zeta^{(1)}\otimes\zeta^{(2)}}
\,\Delta\left(\tilde\tau(a)\right)\, q^{-\zeta^{(1)}\otimes\zeta^{(2)}},
& &\forall a \in U_q(\mathfrak{b}^-).
\label{deltatautilde}
\end{alignat}
Using this definition, we have
\begin{equation}\label{Stau}
S^{[k]}=(\tau^k\otimes id)(S^{[0]}), \qquad\text{with}\quad
S^{[0]}=q^{\sum_{i=1}^n \zeta^{(i)}\otimes\zeta^{(i)}}.
\end{equation}
\bigskip
In the fundamental representation,
we denote by $\mathbf{R}^J$ the explicit $(n+1)^2\times (n+1)^2$
Cremmer-Gervais's solution of the Quantum Yang-Baxter Equation associated to
the previous twist. It is given by
\begin{equation}\label{RJ-fund}
{\mathbf R}^J= (D\otimes D)\; \widetilde{\mathbf{R}}^J\; (D\otimes D)^{-1},
\end{equation}
where
\begin{align}
&D=\sum_i\; q^{\frac{i^2-3i}{2(n+1)}}\; E_{i,i},\\
&\widetilde{\mathbf{R}}^J=q^{-\frac{1}{n+1}}\bigg\{\, q\, \sum_{r,s} \; q^{\frac{2(r-s)}{n+1}} \; E_{r,r}\otimes E_{s,s}
\nonumber\\
&\hspace{4.5cm}
+(q-q^{-1})\sum_{i,j,k}\; q^{\frac{2(i-k)}{n+1}}\; \eta(i,j,k)\;
E_{i,j+i-k}\otimes E_{j,k} \bigg\},
\end{align}
with
\begin{equation}
\eta(i,j,k)=\left\{ \begin{array}{lll}
1\qquad\! \text{if} \quad i\le k<j,\\
-1\quad \text{if} \quad j\le k<i,\\
0 \qquad\! \text{otherwise.}
\end{array}\right.
\end{equation}
\begin{rem}
$R^J$ is not $\mathfrak{h}$-invariant, but is $t_\rho$-invariant.
This implies that $\mathbf{R}^J$ is homogeneous in the sense that
$(\mathbf{R}^J)_{ij}^{kl}\ne 0$ only if $i+j=k+l$.
\end{rem}
\section{Results on Dynamical Quantum Groups}
\label{sectionDQG}
\subsection{Quantum Dynamical Yang-Baxter Equation}
We first begin with a formulation of the dynamical Yang--Baxter equation which
is not sufficiently mathematically precise for our future purposes.
\Definition{Quantum Dynamical Yang-Baxter Equation (Formal)}{A universal
solution of the {\em Quantum Dynamical Yang--Baxter
Equation (QDYBE)}, also known as
{\em Gervais--Neveu--Felder
equation}, is a map $R:\mathbb{C}^r\rightarrow
U_q(\mathfrak{g})^{\otimes 2}$ such that
$R(x)$ is $\mathfrak{h}$-invariant
and
\begin{equation}
R_{12}(x)R_{13}(xq^{h_2})R_{23}(x)=
R_{23}(xq^{h_1})R_{13}(x)R_{12}(xq^{h_3}) \label{sYB1}
\end{equation}
where we have denoted
$xq^{h}=(x_1q^{h_{\alpha_1}},\ldots,x_r
q^{h_{\alpha_r}})$.}
This is sufficient for formal manipulations but it is not enough precise in
the sense that the standard universal solution of the dynamical Yang-Baxter
equation is such that $R(x)$ is an infinite series in $U_q(\mathfrak{g})$ with
coefficients being rational function of $x_1,\ldots,x_r$ with coefficients in
$U_q(\mathfrak{h}).$
As a result we can extend the construction of
${\rm End}({\bf U})$ and $ {\rm End}({\bf U}^{\otimes 2})$ as follows.
Let $A$ be a unital algebra over the complex field, we define the
functor $A\otimes {\bf U}:{\bf Rep}_{U_q(\mathfrak{g})}\rightarrow
A\otimes{\bf Vect}$, where $A\otimes{\bf Vect}$ is the category which
objects are $A\otimes V$ where $V$ is a vector space and the maps are
${\rm id}_A\otimes\phi$ where $\phi$ is a linear map between vector spaces.
We can define the functor $A\otimes {\bf U}$ which associates to each finite dimensional
module $V$ the vector space $A\otimes V$ and to each interwiner $\phi$ the map ${\rm id}_A\otimes \phi.$
We define
${\rm End}(A\otimes {\bf U})$ to be the set of additive natural transformations
between the functors
$A\otimes {\bf U}$ and $A\otimes {\bf U}.$ An element of
${\rm End}(A\otimes {\bf U})$ is a family
$a_V\in A\otimes {\rm End}(V)$ such that it satisfies the naturality and
additivity condition.
We can define similarly ${\rm End}(A\otimes {\bf U}^{\otimes n}).$
${\rm End}(A\otimes {\bf U}^{\otimes n})$ is
naturally endowed with a structure of algebra.
We will denote $(A\otimes U_q(\mathfrak g)^{\otimes n})^c={\rm End}(A\otimes {\bf U}^{\otimes n}).$
We could similarly have defined
$( U_q(\mathfrak g)^{\otimes n}\otimes A)^c.$
\Definition{Quantum Dynamical Yang-Baxter Equation (Precise)}{A universal
solution of the {\em Quantum Dynamical Yang--Baxter
Equation (QDYBE)}, also known as
{\em Gervais--Neveu--Felder
equation}, is an $\mathfrak{h}$-invariant element $R(x)$ of
$(\mathbb{C} (x_1,\ldots,x_r)\otimes U_q(\mathfrak g)^{\otimes 2})^c$ satisfying
\begin{equation}
R_{UV}(x)R_{UW}(xq^{h_V})R_{VW}(x)=
R_{VW}(xq^{h_U})R_{UW}(x)R_{UV}(xq^{h_W}) \label{sYB}
\end{equation}
where we have denoted
$xq^{h}=(x_1q^{h_{\alpha_1}},\ldots,x_r q^{h_{\alpha_r}})$ and where
$U,V,W$ are any finite dimensional $U_q(\mathfrak g)$-modules.}
If moreover $R(x)$ belongs to
$(\mathbb{C} (x_1^2,\ldots,x_r^2)\otimes U_q(\mathfrak g)^{\otimes 2})^c$
it is called $x^2$-rational.
\bigskip
We will often study the particular case where $\mathfrak{g}=sl(n+1).$
\Definition{}{
Let $\stackrel{f}{\pi}$ be the fundamental representation of
${\mathfrak g}=U_q(sl(n+1)).$ Once a $n$-uplet $(x_1,\ldots,x_{n})$ of
complex numbers is given, we will denote
\begin{equation}
\stackrel{f}{\pi}\bigg(\prod_{i=1}^n x_i^{2\zeta_{\alpha_i}}\bigg)
=\mathrm{diag}(\nu_1,\ldots,\nu_{n+1}).\label{deflambda}
\end{equation}
As a result we have $\prod_{i=1}^{n+1} \nu_i=1$ and $x^2_i=\nu_i\nu_{i+1}^{-1}.$
}
Therefore, in the $sl(n+1)$ case, we can define a notion of regularity as
follows:
Let $\nu_1,\ldots,\nu_n$ be $n$ indeterminates and define $\nu_{n+1}$ by
$\prod_{i=1}^{n+1} \nu_i=1$ and $x^2_i=\nu_i\nu_{i+1}^{-1}$,
for $i=1,\ldots,n.$
An element $a\in
(\mathbb{C} (\nu_1,\ldots,\nu_n)\otimes U_q(\mathfrak g)^{\otimes n})^c$ will be called
{\em $\nu$-rational}.
An element $a\in (\mathbb{C} (x_1^2,\ldots,x_n^2)\otimes U_q(\mathfrak g)^{\otimes n})^c $ will be
called {\em $x^2$-rational}.
Note that obviously an $x^2$-rational element is also $\nu$-rational.
This distinction is important because in the $U_q(sl(n+1))$ case the standard
solution of QDYBE is $x^2$-rational, whereas the universal solution of the
dynamical coboundary equation is (almost) $\nu$-rational.
\subsection{Quantum Dynamical coCycles}
The first to understand universal aspects of the dynamical Yang-Baxter equation
was O. Babelon in his work on quantum Liouville on a lattice \cite{Bab}.
There he introduced the notion of {\em Quantum Dynamical coCycle}
$F(x)\in (\mathbb{C}(x^2)\otimes U_q(sl(2))^{\otimes 2})^c$ and gave an exact formula
for $F(x)$ expressed as a series.
More generally, in $U_q(\mathfrak{g})^{\otimes 2}$,
a universal solution of QDYBE equation can be obtained from a
solution of the {\em Quantum Dynamical coCycle Equation (QDCE)}
(\ref{eq:s-cocycle}) in the following sense:
\Theorem{Quantum Dynamical Cocycle Equation\label{TheoremQDCE}}{
If $F(x)\in (\mathbb{C}(x_1^2,\ldots,x_r^2)\otimes U_q(\mathfrak{g})^{\otimes 2})^c $ is an
$x^2$-rational map such that
\begin{enumerate}
\item $F(x)$ is invertible
\item $F(x)$ is $\mathfrak{h}$-invariant , i.e.
\begin{equation}
[F_{12}(x),h\otimes 1+1\otimes h]=0,\ \forall h\in \mathfrak{h},
\label{hinvariance}
\end{equation}
\item $F(x)$ satisfies the {\em Quantum Dynamical coCycle Equation (QDCE)},
\begin{equation}
(\Delta\otimes {\rm id})(F(x))\; F_{12}(xq^{h_3})=
({\rm id}\otimes \Delta)(F(x))\; F_{23}(x),
\label{eq:s-cocycle}
\end{equation}
\end{enumerate}
then
\begin{equation}
R(x)=F_{21}(x)^{-1}R_{12}\,F_{12}(x)
\label{R=FRF}
\end{equation}
satisfies the universal QDYBE \eqref{sYB} where
($R$ is the standard universal $R-$matrix in $(U_q(\mathfrak{g})^{\otimes 2})^c$ defined by (\ref{R=KR}) ).
}
Although simpler than the QDYBE equation, (\ref{eq:s-cocycle}) is difficult
to solve directly.
However, it is possible to determine its general solutions through
an auxiliary linear equation, the
{\em Arnaudon-Buffenoir-Ragoucy-Roche Equation (ABRR)}
(see \cite{ABRR}, but it was first remarked in \cite{BR1}):
\Theorem{ABRR Equation\label{TheoremABRR}}{
We define $B(x)\in ({\mathbb A}_r(x_1,\ldots,x_r)\otimes U_q(\mathfrak{h}))^c$ by
\begin{equation}\label{Bx}
B(x)=\prod_{j=1}^r
x_{j}^{2\zeta^{\alpha_j}}q^{h_{\alpha_j}\zeta^{\alpha_j}},
\end{equation}
where ${\mathbb A}_r(x_1,\ldots,x_r)=
\otimes_{i=1}^r{\mathbb A}[x_i]$ with ${\mathfrak A}[x]$ being the algebra
generated by $x^{\alpha},\ \alpha\in {\mathbb R}$, with the relations
$x^{\alpha+\alpha'}=x^{\alpha}x^{\alpha'}.$\\
An element $F(x)\in ({\mathbb C}(x_1^2,\ldots,x_r^2)\otimes U_q(\mathfrak{g})^{\otimes 2})^c$ satisfies the {\em ABRR Equation}
if and only if
\begin{equation}
F_{12}(x)\, B_{2}(x)={\widehat{R}}_{12}^{-1}\,B_{2}(x)\,F_{12}(x).
\label{lineareq}
\end{equation}
The importance of the ABRR equation comes from the following theorem \cite{ABRR}:\\
Under the hypothesis
\begin{align*}
&
(F(x) - 1\otimes 1)\; \in (\mathbb{C}(x_1^2,\ldots,x_r^2) \otimes U_q ({\mathfrak g})\otimes U^-_q ({\mathfrak g}))^c,
\end{align*}
there exists a unique solution of Equation (\ref{lineareq}).
This solution is invertible, $\mathfrak{h}-$ invariant and satisfies the {\em QDCE} (\ref{eq:s-cocycle}). It is called the standard solution of the {\em QDCE}. \\
Let $V$ be a finite dimensional $U_q(\mathfrak g)$-module, there exists a positive number $c_V$ such that, if $x_1,\ldots,x_r\in {\mathbb C}$ with $\vert x_i\vert< c_V$, the infinite product $\prod_{k=0}^{+\infty}\big( B_{2}^{-k-1}(x)\, {\widehat{R}}_{12}\,
B_{2}^{k+1}(x)\big)$ is convergent when represented on $V$.
It satisfies moreover the $ABRR$ Equation and belongs to $1\otimes 1+(\mathbb{C}(x_1^2,\ldots,x_r^2) \otimes U_q^+ ({\mathfrak g})\otimes U^-_q ({\mathfrak g}))^c$. As a result, the standard solution $F(x)$ of the {\em QDCE} is given by:
\begin{equation}
F(x)=\prod_{k=0}^{+\infty}\Big( B_{2}^{-k-1}(x)\, {\widehat{R}}_{12}\,
B_{2}^{k+1}(x)\Big).
\label{Fprod}
\end{equation}
}
\begin{rem}
$B(x)$ satisfies the following useful relations:
\begin{align}
&R_{12}(x)\,B_{2}(x)=B_{2}(x)\,K_{12}^2 \,R_{21}(x)^{-1},
\label{Rlinear1}\\
&\Delta(B(x))=B_1(x)\,B_2(x)\,K^2=B_1(xq^{h_2})\,B_2(x)
=B_1(x)\,B_2(xq^{h_1}).
\label{DeltaB}
\end{align}
\end{rem}
\bigskip
In the fundamental representation $\stackrel{f}{\pi}$ of $U_q(sl(n+1))$,
let us denote by $\mathbf{F}(x)$ the explicit expression of the quantum
dynamical cocycle \eqref{Fprod}. It is given by
\begin{equation}\label{Fx-fund}
\mathbf{F}(x)=1\otimes 1 - (q-q^{-1}) \sum_{i<j}
\Big(1-\frac{\nu_j}{\nu_i}\Big)^{-1} E_{i,j}\otimes E_{j,i}.
\end{equation}
The corresponding expression \eqref{R=FRF} of the standard dynamical
$R$-matrix for $sl(n+1)$ in the fundamental representation, denoted
${\mathbf R}(x)$,
is then
\begin{multline}\label{Rx-fund}
{\mathbf R}(x)=q^{-\frac{1}{n+1}} \bigg\{
q\sum_{i}E_{i,i}\otimes E_{i,i}
+ \sum_{i \not= j}E_{i,i}\otimes E_{j,j}
+(q-q^{-1})\sum_{i\not= j} \Big(1-\frac{\nu_i}{\nu_j}\Big)^{-1}
E_{i,j}\otimes E_{j,i}\\
- (q-q^{-1})^2\sum_{i>j}
\frac{\nu_i}{\nu_j}\Big(1-\frac{\nu_i}{\nu_j}\Big)^{-2}
E_{i,i}\otimes E_{j,j}\bigg\}.
\end{multline}
\bigskip
The construction of Theorem \ref{TheoremABRR} has been used by P. Etingof,
T. Schedler and O. Schiffmann \cite{ESS} to build the quantization
of $r$-matrices associated to any Belavin-Drinfeld triple.
Indeed, for any Belavin-Drinfeld triple $T$, they have constructed
explicitely a twist $J^{T}\in (U_q({\mathfrak g})^{\otimes 2})^c$
which satisfies
\begin{equation}
(\Delta \otimes id)(J^{T})\,J^{T}_{12}
=(id \otimes \Delta)(J^{T})\,J^{T}_{23},
\label{cocycle2}
\end{equation}
such that $(J^{T}_{21})^{-1}RJ^{T}_{12}$ is a solution of the Yang-Baxter
equation and is a quantization of the classical $r^{{}_{T}}$ matrix
associated to $T$.
The general expression for $J^{T}$ was obtained through a nice use of dynamical
quantum groups and of a modification of the ABRR equation.
In general, for any finite dimensional simple Lie algebra ${\mathfrak g}$ and
any Belavin-Drinfeld triple $T$, $J{}^{T}$ is expressed as a finite product of
explicit invertible elements as
\begin{align}
J{}^{T}=S{}^{T}\widehat{J}{}^{T}\qquad\text{with}\quad
S^{T}\in (U_q({\mathfrak h})^{\otimes 2})^c,\quad
\widehat{J}{}^{T}\in 1\otimes1
+ \left(U^+_q({\mathfrak g})\otimes U^-_q({\mathfrak g})\right)^c.
\end{align}
The reader is invited to read \cite{ESS} for a general construction of
$S{}^{T}$ and $\widehat{J}{}^{T}$ using the ABRR identity.
The result of \cite{ESS} is completely general. However, for reasons explained
below, we will only be interested in the case where
${\mathfrak g}=sl(n+1),$ $n\in {\mathbb N}^*$, and where $T=\tau$ is the
Cremmer-Gervais triple.
The construction of $J^{\tau}$ described in the last section is therefore enough for our
purpose.
\subsection{Quantum Dynamical coBoundary Problem}
We have seen in the last section that Theorem \ref{TheoremABRR} provides a way
to solve the QDCE \eqref{eq:s-cocycle}.
However in \cite{Bab}, for the $U_q(sl(2))$ case, O. Babelon used a
different approach: he noticed that $F(x)$ is a quantum dynamical
coboundary, i.e. that there exists an explicit invertible element
$M(x)\in (\mathbb{C}(x)\otimes U_q(sl(2)))^c$ solution of the following Dynamical coBoundary
Equation
:
\begin{equation}
F(x)=\Delta(M(x))\; M_2(x)^{-1}\,(M_1(xq^{h_2}))^{-1}.
\end{equation}
Note that in the $U_q(sl(2))$ this is a particular case of \eqref{coboundaryM} since ${J}$ is merely equal
to $1.$
More generally, we have the following property which is formal at this stage because we do not
precise the analytic property of $M(x)$:
\Theorem{Quantum Dynamical coBoundary equation (QDBE) (Formal)\label{coboundcocycl}}{
Let $J$ be a given cocycle in $U_q({\mathfrak g})^{\otimes 2}$,
if there exists a map $M$ from ${\mathbb C}^r$ to $U_q({\mathfrak g}),$
such that $M(x)$ is invertible for any $x$ where it exists,
and such that the map $F$ defined by
\begin{equation}
F(x)=\Delta(M(x))\;{ J}\; M_2(x)^{-1}\,(M_1(xq^{h_2}))^{-1}
\label{coboundaryM}
\end{equation}
verifies
\begin{equation}\label{hinv}
[F_{12}(x),h\otimes 1+1\otimes h]=0,\quad\forall h\in \mathfrak{h},
\end{equation}
then $F$ is a solution of the Quantum Dynamical coCycle Equation
(\ref{eq:s-cocycle}) (see \cite{EN} for details).
As a result, the corresponding quantum dynamical $R$-matrix \eqref{R=FRF}
can be obtained in terms of $M$ as
\begin{equation}
R(x)= M_2(xq^{h_1})\, M_1(x)\;R^{J}\;M_2(x)^{-1}\,M_1(xq^{h_2})^{-1}.
\label{RMM=MMR}
\end{equation}
}
\vspace{-0.5cm}
\medskip\noindent {\sf{\underline{Proof}:\ }}
If $F$ satisfies (\ref{coboundaryM}) and (\ref{hinv}), we have,
using the cocycle equation \eqref{cocycle}:
\begin{align*}
&(\Delta \otimes id)(F(x)^{-1})\,(id \otimes \Delta)(F(x))\\
&\qquad =\Delta_{12}(M(xq^{h_3}))\,M_3(x)\,
(\Delta \otimes id)(J^{-1})\,(id \otimes \Delta)(J)\,
\Delta_{23}(M(x)^{-1})\,M_1(xq^{h_2+h_3})^{-1}\\
&\qquad =\Delta_{12}(M(xq^{h_3}))\,M_3(x)\,J_{12}\,J_{23}^{-1}\,
\Delta_{23}(M(x)^{-1})\,M_1(xq^{h_2+h_3})^{-1}\\
&\qquad =\Delta_{12}(M(xq^{h_3}))\,J_{12}\,M_{2}(xq^{h_3})^{-1}\,
M_{1}(xq^{h_2+h_3})^{-1}\, M_{1}(xq^{h_2+h_3})\\
&\hspace{4.9cm}\times M_{2}(xq^{h_3})\,M_3(x)\,J_{23}^{-1}\,
\Delta_{23}(M(x)^{-1})\,M_1(xq^{h_2+h_3})^{-1}\\
&\qquad =F_{12}(xq^{h_3})\, M_{1}(xq^{h_2+h_3})\,F_{23}(x)^{-1}\,
M_1(xq^{h_2+h_3})^{-1}\\
&\qquad=F_{12}(xq^{h_3})\, F_{23}(x)^{-1},
\end{align*}
and thus $F$ is a solution of \eqref{eq:s-cocycle}.
\eqref{RMM=MMR} follows directly from \eqref{R=FRF} and \eqref{coboundaryM}
using \eqref{quasitriangularity2}.
\qed
Since the article of O. Babelon on the $sl(2)$ case,
the theory of dynamical quantum groups
has been the subject of numerous works, and is now well
understood for $U_q({\mathfrak g})$ where ${\mathfrak g}$ is a Kac-Moody
algebra of affine or finite type.
However, quite surprisingly, only little progress has been made
concerning the dynamical coboundary equation (see however
the articles \cite{BBB},\cite{BR2},\cite{EN},\cite{St}).
The first results in this subject concerning higher rank cases
have been obtained by Bilal-Gervais in \cite{BG}, where they have
found the expression for $R(x)$ in the fundamental representation of
$sl(n+1)$. Cremmer-Gervais \cite{CG} have then shown that,
in the fundamental representation of $sl(n+1)$,
it is possible to absorb the dynamical dependence of $R(x)$ through a
dynamical gauge transformation.
More precisely, we have the following theorem:
\Theorem{\cite{CG},\cite{BDF},\cite{Hodges}\label{lemme1}}{
In the fundamental representation of $U_q({sl(n+1)})$,
the expression $\mathbf{R}(x)$ \eqref{Rx-fund} of the standard dynamical
$R$-matrix is related to the expression $\mathbf{R}^J$ \eqref{RJ-fund}
of the Cremmer-Gervais $R$-matrix as,
\begin{equation}\label{RMM-fund}
\mathbf{R}(x)= {\mathbf M}_2(xq^{h_1}) {\mathbf M}_1(x)\;
{\mathbf R}^{J}\;{\mathbf M}_2(x)^{-1}{\mathbf M}_1(xq^{h_2})^{-1},
\end{equation}
where the $(n+1)\times (n+1)$ matrix ${\mathbf M}(x)$ is given by
\begin{equation}\label{Minv-fund}
{\mathbf M}(x)^{-1}=D\;\mathcal{V}(x)\;{\mathcal U}(x)D^{-1}
\end{equation}
in term of the Vandermonde matrix
\begin{equation*}
\mathcal{V}(x)=\sum_{i,j}\;\nu_j^{i-1}\; E_{i,j},
\end{equation*}
and of the following diagonal matrices
\begin{align*}
&D=\sum_i\, q^{\frac{i^2-3i}{2(n+1)}}\, E_{i,i},\\
&\mathcal{U}(x)=\sum_i \frac{\nu_i^{-i+1}\prod_{k=2}^{n+1}\nu_k^{-\frac{1}{2}(\delta_{i\leq k-1}-\frac{k-1}{n+1})}q^{\frac{1}{2}(\delta_{i\leq k-1}-\frac{k-1}{n+1})^2}}{\prod_{r=i+1}^{n+1}(1-\frac{\nu_i}{\nu_r})}E_{i,i}.
\end{align*}
}
\vspace{-0.5cm}
\medskip\noindent {\sf{\underline{Proof}:\ }}
{}From \eqref{Rx-fund}, it is easy to check that
\begin{equation}
\mathbf{R}(x)=\mathcal{U}_2(x\, q^{h_{1}})^{-1}\,\mathcal{U}_1(x)^{-1}\;
\widetilde{\mathbf{R}}(x)\;\mathcal{U}_2(x) \, \mathcal{U}_1(x\, q^{h_{2}}),
\end{equation}
where
\begin{multline}
\widetilde{\mathbf{R}}(x)=q^{-\frac{1}{n+1}} \bigg\{
q\sum_{i}E_{i,i}\otimes E_{i,i}
+\sum_{i \not= j} \Big(q-q^{-1}\frac{\nu_i}{\nu_j}\Big)
\Big(1-\frac{\nu_i}{\nu_j}\Big)^{-1}
E_{i,i}\otimes E_{j,j}\\
+(q-q^{-1})\sum_{i\not= j} \Big(1-\frac{\nu_i}{\nu_j}\Big)^{-1}
E_{i,j}\otimes E_{j,i}\bigg\}.
\end{multline}
Note that this expression is the one that can be found in \cite{BG,ABB}
Therefore, it remains to show that
\begin{equation}
\widetilde{\mathbf{R}}(x)= \mathcal{V}_2(xq^{h_1})^{-1} \, \mathcal{V}_1(x)^{-1}\;\widetilde{\mathbf R}^{J}\; \mathcal{V}_2(x)\, \mathcal{V}_1(xq^{h_2}),
\end{equation}%
which is proved in \cite{CG,BDF,Hodges}.
\qed
The following theorem \cite{BDF} shows that it is only in the case where $\mathfrak{g}=sl(n+1)$ that the coboundary equation can be eventually solved.
\Theorem{\cite{BDF}\label{lemme2}}{
Let $\mathfrak{g}$ be a finite dimensional simple Lie algebra, and let
$R:\mathbb{C}^r\rightarrow U_q(\mathfrak{g})^{\otimes 2}$ be the standard
solution \eqref{R=FRF} of the QDYBE.
If $\mathfrak{g}$ is not of $A$-type, it is not possible to find any pair
$(R^J,M)$, with $M:\mathbb{C}^r\rightarrow U_q(\mathfrak{g})^{\otimes 2}$ and
$R^J$ a solution of the (non-dynamical) QYBE, such that \eqref{RMM=MMR} is
satisfied. If $\mathfrak{g}$ is of $A$-type and if such a pair exists, then
$R^J$ can be expanded as $R^J=1+\hbar\, r_J+o(\hbar)$ where $r_J$ coincides, up to an automorphism of the Lie algebra, with $r_{\tau,s}$ associated to the shift.
}
\medskip\noindent {\sf{\underline{Proof}:\ }}
We refer the reader to \cite{BDF} for the proof of this theorem.
For completeness we have given an explanation of this proof in Appendix~\ref{sec-BDF}.
\qed
Therefore, quite surprisingly, except for $n=1,$ ${\mathbf R}^{J}$ is not the Drinfeld's solution of the Yang-Baxter equation and is not of
zero weight, but is instead the Cremmer-Gervais $R$-matrix.
We will therefore assume in the rest of this section that $\mathfrak{g}=sl(n+1).$
\bigskip
{}From the expression of ${\bf M}(x)$ in the fundamental representation one sees, because of the expression of ${\cal U}$, that ${\bf M}(x)$ lies in
${\mathbb C}(\tilde{\nu_1},\ldots,\tilde{\nu_n})\otimes {\rm End}(\stackrel{f}{V})$, where $\nu_i=\tilde{\nu_i}^{2(n+1)}.$
The next section and the explicit form of ${\bf M}(x)$ motivates the following definition:
An element $a\in ({\mathbb C}(\tilde{\nu_1},\ldots,\tilde{\nu_n})\otimes U_q(\mathfrak{g}))^c$ is said to be {\em almost $\nu$-rational} if there exists an invertible element $b\in ({\mathbb C}(\tilde{\nu_1},\ldots,\tilde{\nu_n})\otimes U_q(\mathfrak{h}))^c$ such that
\begin{enumerate}
\item $b^{-1}a$ is $\nu$-rational,
\item $\Delta(b)\, b_2(x)^{-1}b_1(xq^{h_2})^{-1}$ is $\nu$-rational,
\end{enumerate}
where $\nu_i=\tilde{\nu_i}^{2(n+1)},\ i=1,\ldots,n+1$, and
$\tilde{\nu}_1\ldots\tilde{\nu}_{n+1}=1.$
For ${\mathfrak g}=sl(n+1)$ and for the Cremmer-Gervais cocycle $J\in (U_q(\mathfrak{g})^{\otimes 2})^c$, we are now ready to address the following precise problems which will be solved in this paper:
\Problem{Weak Quantum Dynamical coBoundary Problem (WQDBP)\label{WP}}{
Find an almost $\nu$-rational invertible element
${\cal M}\in ({\mathbb C}(\tilde{\nu_1},\ldots,\tilde{\nu_n})\otimes U_q({\mathfrak g}))^c$ such that
\begin{equation}\label{eq1-pb1}
\mathcal{F}(x)=\Delta(\mathcal{M}(x))\, J \, \mathcal{M}_2(x)^{-1}
\mathcal{M}_1(xq^{h_2})^{-1}
\end{equation}
is a zero-weight solution of the QDCE \eqref{eq:s-cocycle}
and such that
\begin{equation}\label{eq2-pb1}
{\cal R}(x)={\cal F}_{21}(x)^{-1} R_{12} \, {\cal F}_{12}(x)
\end{equation}
satisfies the following linear equation
\begin{equation}
{\cal R}_{12}(x)\, B_2(x)=B_2(x)\, K_{12}^2\, {\cal R}_{21}(x)^{-1},
\label{Rlinear}
\end{equation}
where $B(x)$ is defined by
Eq.~\eqref{Bx}.
}
Note that, since ${\cal M}$ is almost $\nu$-rational and $ \mathcal{F}(x)$ is $\mathfrak{h}$-invariant, $ \mathcal{F}(x)$ is $\nu$-rational.
\Problem{Strong Quantum Dynamical coBoundary Problem (SQDBP)\label{SP}}{
Find an almost $\nu$-rational invertible element $M(x)\in ({\mathbb C}(\tilde{\nu_1},\ldots,\tilde{\nu_n})\otimes U_q({\mathfrak g}))^c$ solution of the Weak Quantum Dynamical coBoundary Problem such that
${\cal F}(x)$ is equal to the standard solution $F(x)$ of the QDCE.
}
\begin{rem}
It would have been better to denote $M(\tilde{\nu})$ such an element,
but for notational reasons we prefer to call it $M(x)$. This should cause no confusion.
\end{rem}
\section{Solving the Quantum Dynamical coBoundary problem}
\label{sectionDBP}
In this section we assume that ${\mathfrak g}=sl(n+1)$, and
$J$ will denote the Cremmer-Gervais cocycle defined in
Theorem~\ref{cocycleBCG}. $F(x)$ and $R(x)$
will respectively denote the standard solutions of zero weight
(\ref{Fprod}) and (\ref{R=FRF}) of the equations
(\ref{lineareq},\ref{eq:s-cocycle}) and (\ref{sYB}) respectively.
We present here some general results concerning the weak and strong QDBP.
Our aim is to identify and construct elementary objects
obeying simple algebraic rules which will be the building blocks of the solutions
of these problems.
We first propose, in Section \ref{sec-prim-loop}, a procedure to solve
the WQDBP using the notion of {\em primitive loop}.
The study of these primitive loops does not however
enable us to solve the SQDBP.
Therefore, in Section \ref{sec-GaussQDC}, we introduce a different approach,
based on the Gauss decomposition of $\mathcal{M}(x)$,
which leads to the solution of the SQDBP.
\subsection{Primitive loops}\label{sec-prim-loop}
In order to study the properties of the solutions of the WQDBP, let us
first introduce the notion of {\em primitive loop}, defined as follows:
\Definition{Primitive loop }{
For any solution ${\cal M}(x)$ of the WQDBP, we define an element ${\cal P}(x) \in ({\mathbb C}(\nu_1,\ldots,\nu_n)\otimes U_q({\mathfrak g}))^c $ by:
\begin{equation}
{\cal P}(x)=v\,{\cal M}(x)^{-1}B(x)\,{\cal M}(x).\label{PdeXsl2}
\end{equation}
${\cal P}$ will be called the {\em primitive loop} associated to ${\cal M}(x).$
}
Note that, because ${\cal M}(x)$ is almost $\nu$-rational, ${\cal P}(x)$ is $\nu$-rational.
A primitive loop satisfies various properties and verifies a {\em reflection equation}
(see \eqref{echangeP} below) which is
related to the notion of {\em reflection algebra} that will be introduced in
Section~\ref{sectionQRA}. More precisely,
\Proposition{Properties of the primitive loop}{
Let ${\cal P}(x)$ be a primitive loop associated to a solution ${\cal M}(x)$
of the WQDBP. We have the following relations:
\begin{align}
& R^J_{12}\,{\cal P}_{2}(x)\,R^J_{21}={\cal M}_1(x)^{-1}
{\cal P}_{2}(xq^{h_1})\,{\cal M}_1(x), \label{linearPdeX}\\
& \Delta^J({\cal P}(x))=(R^J_{12})^{-1}{\cal P}_1(x)\,R^J_{12}
\,{\cal P}_{2}(x). \label{deltaPdeX}
\end{align}
As a consequence, ${\cal P}(x)$ satisfy the reflection equation:
\begin{align}
&R^J_{21}\,{\cal P}_1(x)\,R^J_{12}\,{\cal P}_{2}(x)
={\cal P}_{2}(x)\,R^J_{21}\,{\cal P}_1(x)\,R^J_{12}.
\label{echangeP}
\end{align}
}
\vspace{-0.5cm}
\Proof{Eq.\eqref{linearPdeX} follows immediatly from the linear
equation (\ref{Rlinear}) and from the relation (\ref{RMM=MMR}),
itself inherited from \eqref{eq1-pb1}-\eqref{eq2-pb1} using
\eqref{quasitriangularity2}.
Then, using successively (\ref{PdeXsl2}), (\ref{eq1-pb1}) and
\eqref{propertyv}, the zero-weight property and \eqref{DeltaB}, and
\eqref{linearPdeX},
we have:
\begin{align*}
\Delta^J({\cal P}(x))
&=J^{-1}\,\Delta(v)\,\Delta({\cal M}(x)^{-1})\,\Delta(B(x))\,
\Delta({\cal M}(x))\,J\\
&=(R^J_{21}R^J_{12})^{-1} v_1v_2\;{\cal M}_2(x)^{-1}
{\cal M}_1(xq^{h_2})^{-1}{\cal F}_{12}(x)^{-1}\nonumber\\
&\hspace{5.8cm}\times\Delta(B(x))\;{\cal F}_{12}(x)\,{\cal M}_1(xq^{h_2})\,
{\cal M}_2(x)\\
&=(R^J_{21}R^J_{12})^{-1} v_1v_2\;{\cal M}_2(x)^{-1}
{\cal M}_1(xq^{h_2})^{-1}B_1(xq^{h_2})\,B_2(x)\,
{\cal M}_1(xq^{h_2})\,{\cal M}_2(x)\\
&=(R^J_{21}R^J_{12})^{-1} {\cal M}_2(x)^{-1}{\cal P}_1(xq^{h_2})\,
{\cal M}_2(x)\,{\cal P}_2(x)\\
&=(R^J_{21}R^J_{12})^{-1}R^J_{21}\,{\cal P}_{1}(x)\,R^J_{12}\,
{\cal P}_2(x)\\
&=(R^J_{12})^{-1}{\cal P}_1(x)\,R^J_{12}\,{\cal P}_{2}(x),
\end{align*}
which concludes the proof of \eqref{deltaPdeX}.
The relation (\ref{echangeP})
is a direct consequence of (\ref{deltaPdeX})
using (\ref{quasitriangularity2}).}
\begin{rem}
If ${\cal M}(x)$ is a solution of the WQDBP and if ${\cal P}(x)$ is the associated primitive loop, then, in the fundamental representation of $U_q(sl(n+1))$,
one has
$\mathrm{tr}(\stackrel{f}{\pi}({\cal P}(x))=
\mathrm{tr}(\stackrel{f}{\pi}(vB(x))=q^{-n}(\nu_1+\cdots+\nu_{n+1}).$
\end{rem}
\begin{rem}
If $M(x)$ is a solution of the SQDBP such that its expression (\ref{Minv-fund}) in the fundamental
representation of $U_q(sl(n+1))$ is given by ${\mathbf M}(x)$,
the associated primitive loop ${ P}(x)$ can also be computed in the fundamental representation and its explicit expression is given by
\begin{equation}
{\bf P}(x)=
\stackrel{f}{\pi}({ P}(x))=D\; q^{-n}\Big\{ \sum_{j=1}^n E_{j,j+1}
+\sum_{k=0}^{n}(-1)^{n-k}{\cal S}_{n+1-k}(x)\; E_{n+1,k+1}\Big\} \;D^{-1},
\label{Pinfund}
\end{equation}
where, for $1\leq m\leq n+1$, ${\cal S}_{m}(x)$ denotes the symmetric
polynomial in $\nu_1,\ldots,\nu_{n+1}$ defined as
${\cal S}_{m}(x)=\sum_{1\leq i_1 < \cdots <i_m \leq n+1}\prod_{k=1}^{m}\nu_{i_k}.$
Note that ${\cal S}_{n+1}(x)=\nu_1\ldots\nu_{n+1}=1.$
\end{rem}
\bigskip
We now prove a sufficient condition for a given ${\cal M}$ to be a solution of the WQDBP.
\Proposition{\label{theoreme}}{
Let ${\cal M}$ be an almost $\nu$-rational element of $(\mathbb{C}(\tilde{\nu}_1,\ldots,\tilde{\nu}_n)\otimes U_q(\mathfrak g))^c $, and let us define
${\cal P}(x)\in (\mathbb{C}(\nu_1,\ldots,\nu_n)\otimes U_q(\mathfrak g))^c $ as
\begin{align}
&{\cal P}(x)=v{\cal M}(x)^{-1}B(x)\,{\cal M}(x).
\label{diagonalisation}
\end{align}
If ${\cal P}$ satisfies the property
\begin{align}
& R^J_{12}\,{\cal P}_{2}(x)\,R^J_{21}={\cal M}_1(x)^{-1}
{\cal P}_{2}(xq^{h_1})\,{\cal M}_1(x),
\label{linearPdeX2}
\end{align}
then ${\cal F}(x)=\Delta({\cal M}(x))\,J\,{\cal M}_2(x)^{-1}
{\cal M}_1(xq^{h_2})^{-1}$
is of weight zero and is $\nu$-rational. \\
As a result, ${\cal F}$ is a solution of
the QDCE (\ref{eq:s-cocycle}) and
${\cal R}(x)={\cal F}_{21}(x)^{-1}R_{12}\,{\cal F}_{12}(x)$
obeys the QDYBE (\ref{sYB}).
${\cal R}(x)$ satisfies the linear equation (\ref{Rlinear}) and therefore
${\cal M}(x)$ is a solution of the WQDBP.
}
\medskip\noindent {\sf{\underline{Proof}:\ }}
The zero-weight property of ${\cal F}(x)$ is obtained from the
following computation, using \eqref{diagonalisation}, \eqref{linearPdeX2}
and the quasitriangularity of $( U_q(\mathfrak{g}),\Delta^J,R^J)$:
\begin{align*}
{\cal P}_{3}(xq^{h_1+h_2}&) \left\{
\Delta_{12}({\cal M}(x))\,J_{12}\,{\cal M}_2(x)^{-1}
{\cal M}_1(xq^{h_2})^{-1} \right\}
\\
&=J_{12}\,
(\Delta^J \otimes id)({\cal M}_1(x)\,R^J_{12}\,{\cal P}_{2}(x)\,R^J_{21})\;
{\cal M}_2(x)^{-1}{\cal M}_1(xq^{h_2})^{-1}
\\
&=\Delta_{12}({\cal M}(x))\,J_{12}\,R^J_{13}\,R^J_{23}\,{\cal P}_{3}(x)\,
R^J_{32}\,R^J_{31}\,{\cal M}_2(x)^{-1}{\cal M}_1(xq^{h_2})^{-1}
\\
&=\Delta_{12}({\cal M}(x))\,J_{12}\,R^J_{13}\, {\cal M}_2(x)^{-1}
{\cal P}_{3}(xq^{h_2})\,R^J_{31}\,{\cal M}_1(xq^{h_2})^{-1}
\\
&=\left\{\Delta_{12}({\cal M}(x))\,J_{12}\, {\cal M}_2(x)^{-1}
{\cal M}_1(xq^{h_2})^{-1}\right\} {\cal P}_{3}(xq^{h_2+h_1}).
\end{align*}
By representing the third space on the fundamental representation and by tracing on it, we obtain that ${\cal F}$ commutes with ${\cal S}_1(xq^{h_1+h_2}).$
We now prove that this condition implies that ${\cal F}$ is of zero weight.
Let $V,W$ be two finite dimensional $U_q(\mathfrak g)$-modules,
${\cal F}_{VW}\in {\mathbb C}(\tilde{\nu}_1,\ldots,\tilde{\nu}_n)\otimes {\rm End}(V\otimes W).$
We can decompose
$V\otimes W=\oplus_{\lambda\in \mathfrak{h}^*}(V\otimes W)[\lambda].$
Let us denote by $P_{\lambda}$ the projection on $(V\otimes W)[\lambda]$ and
let ${\cal F}_{\lambda,\mu}=P_{\lambda} {\cal F}_{VW}P_{\mu}.$
The previous condition implies that ${\cal F}_{\lambda,\mu}(a_\lambda-a_\mu)=0$, where
\begin{equation}
a_{\lambda}={\cal S}_1(xq^{h(\lambda)})=
\sum_{i=1}^{n+1}\nu_i(x) q^{2(\zeta^{(i)}-\zeta^{(i-1)})(\lambda)}.
\end{equation}
When $\lambda\not=\mu,\ a_\lambda-a_\mu\not=0,$ and therefore, since ${\cal F}_{\lambda,\mu}\in {\mathbb C}(\tilde{\nu}_1,\ldots,\tilde{\nu}_n)\otimes {\rm End}(V\otimes W),$ we obtain that ${\cal F}_{\lambda,\mu}=0$ which is equivalent to the zero-weight condition.
As a result,
${\cal F}$ is a solution of
the QDCE (\ref{eq:s-cocycle}) due to Theorem \ref{coboundcocycl},
and thus $\mathcal{R}(x)$ satisfies the QDYBE \eqref{sYB} due to
Theorem \ref{TheoremQDCE}. It obviously verifies
\eqref{RMM=MMR}, and
the fact that it satisfies (\ref{Rlinear}) follows immediately from
(\ref{diagonalisation},\ref{linearPdeX2}).
\qed
\begin{rem}
The previous propositions show that the primitive loop element is of fundamental importance for solving the WQDBP and that the analog of the ABRR equation for ${\cal M}(x)$ is Eq.~(\ref{linearPdeX}), a (major) difference being that ${\cal P}(x)$ is also not known.
\end{rem}
\begin{rem}
The previous proposition leads naturally to an algorithmic approach to the
computation of the universal $M(x).$
Indeed, the explicit value of ${\bf M}(x)$ being known in the fundamental
representation, we can view the system of $(n+1)^2$ equations
\begin{equation}
\big(\stackrel{f}{\pi} \otimes id\big)
\big( R^J_{12}{\cal P}_{2}(x)R^J_{21}\big)
={\bf M}_1(x)^{-1}{\cal P}_{2}(xq^{h_1}){\bf M}_1(x),
\end{equation}
with moreover $\stackrel{f}{\pi}({\cal P})$ given by
${\bf P}(x)=v_f{\bf M}(x)^{-1} {\bf B}(x)\,{\bf M}(x),$
as a system of universal equations fixing ${\cal P}(x)$ up to a
central element which can be
determined using the relation
\begin{equation}
\Delta^J({\cal P}(x))=(R^J_{12})^{-1}{\cal P}_1(x)\,R^J_{12}\,
{\cal P}_{2}(x).
\label{deltaPdeX2}
\end{equation}
The universal expression of ${\cal P}(x)$ being known, the equation
\begin{equation}
{\cal M}(x){\cal P}(x)=vB(x){\cal M}(x)
\end{equation}
is then a universal linear relation
fixing ${\cal M}(x)$ up to a left-multiplication by an element of
$({\mathbb C}(\nu_1,\ldots,\nu_n)\otimes {C}_q({\mathfrak h}))^c.$
In order to obtain a generic solution of the WQDBP,
it is sufficient to show that this expression verifies universally
(\ref{linearPdeX2}).
This is the path that we had initially followed in order to obtain explicit universal expressions for ${\cal P}(x)$ and ${\cal M}(x)$
such as those presented in the next section.
However, although the properties of the primitive loop ${\cal P}(x)$
associated to ${\cal M}(x)$ are powerful tools to solve very explicitely
the WQDBP, these relations are not sufficient to ensure that
${\cal M}(x)$ is a solution of the SQDBP.
More precisely, let ${\cal M}^{(1)}(x)$ be a solution of
(\ref{diagonalisation}), (\ref{linearPdeX2}) and (\ref{deltaPdeX2}), and let
$u(x)\in ({\mathbb C}(\nu_1,\ldots,\nu_n)\otimes {C}_q({\mathfrak h}))^c$,
we define ${\cal M}^{(2)}(x)=u(x){\cal M}^{(1)}(x).$
Let us denote ${\cal P}^{(1)}(x),\ {\cal P}^{(2)}(x)$ the primitive loops
corresponding respectively to ${\cal M}^{(1)}(x),\ {\cal M}^{(2)}(x)$.
We have ${\cal P}^{(1)}(x)={\cal P}^{(2)}(x),$ and
${\cal M}^{(2)}(x)$ is also a solution of (\ref{diagonalisation}),
(\ref{linearPdeX2}) and (\ref{deltaPdeX2}).
Nevertheless, in general, the corresponding
${\cal F}^{(1)}(x)$ and ${\cal F}^{(2)}(x)$ are different. This shows that
solutions of the WQDBP are in general not solutions of
the SQDBP.
In the next section we will solve this problem and
obtain sufficient conditions on
${\cal M}(x)$ solution of the WQDBP to ensure that it is also a solution to
the SQDBP.
\end{rem}
\subsection{Gauss Decomposition of Quantum Dynamical coBoundary}
\label{sec-GaussQDC}
We propose here a new approach to construct the solutions of the SQDBP,
based on the study of some fundamental building blocks entering the Gauss
decomposition of $M(x)$. We will prove in this section the
following theorem:
\Theorem{\label{maintheorem}}{
Let ${ \cal M}^{(0)}\in
({\mathbb C}(\tilde{\nu}_1,\ldots,\tilde{\nu}_n)\otimes U_q({\mathfrak h}))^c$ and ${\mathfrak C}^{[\pm]}\in 1\oplus \left({\mathbb C}(\nu_1,\ldots,\nu_n)
\otimes U^{\pm}_q({\mathfrak g})\right)^c$.
Let ${\cal M}^{(\pm)}\in 1\oplus \left({\mathbb C}(\nu_1,\ldots,\nu_n)
\otimes U^{\pm}_q({\mathfrak g})\right)^c$
be given by
\begin{align}
{\cal M}^{(\pm)}(x)=\prod_{k=1}^{+\infty} {\mathfrak C}^{[\pm k]}(x)^{\pm 1},
\quad \text{with}\quad
&{\mathfrak C}^{[+ k]}(x)= \tau^{k-1}\left({\mathfrak C}^{[+]}(x)\right),
\nonumber\\
&{\mathfrak C}^{[- k]}(x)=B(x)^{-k}\,{\mathfrak C}^{[-]}(x)\, B(x)^{k} .
\label{MdeXprod2}
\end{align}
Because $\tau$ is nilpotent, the product defining ${\cal M}^{(+)}(x)$ is finite.
We define ${ \cal M}\in \left({\mathbb C}(\tilde{\nu}_1,\ldots,\tilde{\nu}_n)\otimes U_q({\mathfrak g})\right)^c$ as
\begin{equation}\label{MdeXprod}
{ \cal M}(x)= { \cal M}^{(0)}(x)\, {\cal M}^{(-)}(x)^{-1}
{ \cal M}^{(+)}(x).
\end{equation}
The following algebraic relations on ${\cal M}^{(0)}$ and
${\mathfrak C}^{[\pm]} $ are sufficient conditions to ensure that
${\cal M}(x)$ is a solution of the SQDBP:
\begin{align}
&\Delta({\cal M}^{(0)}(x))\,S^{[1]}_{12}\,{\cal M}^{(0)}_2(x)^{-1}
{\cal M}^{(0)}_1(xq^{h_2})^{-1}=1 \otimes 1,
\label{axiomABRR0}\\
&K_{12}^{-1}\,\Delta({\mathfrak C}^{[\pm]}(x))\,K_{12}=
\big\{S^{[1]}_{21}\,{\mathfrak C}^{[\pm]}_{1}(x)\, (S^{[1]}_{21})^{-1}\big\}\,
K^{\mp 1}_{12}\,
\big\{S^{[1]}_{12}\,{\mathfrak C}^{[\pm]}_{2}(x)\, (S^{[1]}_{12})^{-1}\big\}\,
K^{\pm 1}_{12},
\label{axiomABRR1d}\\
&{\mathfrak C}^{[\pm]}_{1}(xq^{h_2})=
\big\{(S^{[1]}_{12})^{-1}S^{[1]}_{21}\,K_{12}\big\}\,
{\mathfrak C}^{[\pm]}_{1}(x)\,
\big\{K_{12}^{-1}(S^{[1]}_{21})^{-1}S^{[1]}_{12}\big\},
\label{axiomABRR1}\\
&{\mathfrak C}^{[-]}_2(x)\, {\mathfrak C}^{[+]}_1(xq^{h_2}) \,
\big\{ B_2(x)\, (S^{[2]}_{12})^{-1}\widehat{J}{}^{[1]}_{12}\,
S^{[2]}_{12} B_2(x)^{-1}\big\}\nonumber\\
&\hspace{6.5cm}=\big\{ (S^{[1]}_{12})^{-1} \widehat{R}_{12}\,
S^{[1]}_{12}\big\}\,
{\mathfrak C}^{[+]}_1(xq^{h_2})\, {\mathfrak C}^{[-]}_2(x).
\label{axiomABRR2prime}
\end{align}
}
\begin{rem}
In the next section we will find explicit solutions to these
sufficient algebraic equations.
\end{rem}
The proof of this theorem decomposes in three lemmas.
The first lemma contains elementary results on $\mathcal{M}^{(+)}$
and $\mathcal{M}^{(-)}$:
\Lemma{\label{firstresults}}{
The infinite products (\ref{MdeXprod2}) define elements $\mathcal{M}^{(+)}$ and $\mathcal{M}^{(-)}$ belonging to
$1\oplus ({\mathbb C}(\nu_1,\ldots,\nu_n)\otimes U^{\pm}_q(\mathfrak{g}))^c.$
If Equations \eqref{axiomABRR1d}, \eqref{axiomABRR1} and
\eqref{axiomABRR2prime} are satisfied, the element
$U(x)\in ({\mathbb C}(\nu_1,\ldots,\nu_n)\otimes
U_q({\mathfrak g})^{\otimes 2})^c$ defined as
\begin{equation}
U(x)= \Delta({\cal M}^{(+)}(x)) J {\cal M}^{(+)}_2(x)^{-1},
\end{equation}
can also be written as
\begin{equation}
U_{12}(x)=S^{[1]}_{12}
\prod_{k=1}^{n}\left(
{\mathfrak C}^{[+k]}_1(xq^{h_2})\, (S^{[k+1]}_{12})^{-1}
\widehat{J}{}^{[k]}_{12}\,S^{[k+1]}_{12} \right) ,\label{Uprod}
\end{equation}
and therefore belongs to $({\mathbb C}(\nu_1,\ldots,\nu_n)\otimes U_q(\mathfrak{b}^+)\otimes U_q(\mathfrak{b}^-))^c.$
It satisfies the properties
\begin{align}
&
(id \otimes \iota_{-})(U_{12}(x))
= S^{[1]}_{12}\,{\cal M}_1^{(+)}(xq^{h_2}),
\label{axiomABRR3}\\
&{\mathfrak C}^{[-]}_2(x)\, B_2(x)\, (S^{[1]}_{12})^{-1}U_{12}(x)
= (S^{[1]}_{12})^{-1}\widehat{R}_{12}\, U_{12}(x)\,
{\mathfrak C}^{[-]}_2(x)\, B_2(x) .
\label{axiomABRR2bis}
\end{align}
Moreover, $\mathcal{M}^{(-)}$ satisfies the following relations:
\begin{align}
&{\mathfrak C}^{[-]}(x)\, B(x)\, {\cal M}^{(-)}(x)
= {\cal M}^{(-)}(x)\, B(x),
\label{BM-T}\\
&B_2(x)\,K_{12}\,\Delta( {\cal M}^{(-)}(x)^{-1})
=\Delta'( {\cal M}^{(-)}(x)^{-1})\,S_{12}^{[1]}\, K_{12}\,
{\mathfrak C}^{[-]}_2(x)\, B_2(x)\,(S_{12}^{[1]})^{-1},
\label{BdeltaMV}\\
&(id \otimes \iota_{-})(\Delta({\cal M}^{(-)}(x)^{-1}))
=S^{[1]}_{12}\,{\cal M}_1^{(-)}(xq^{h_2})^{-1}\,
(S^{[1]}_{12})^{-1}.
\label{axiomABRR4}
\end{align}
}
\vspace{-0.5cm}
\medskip\noindent {\sf{\underline{Proof}:\ }}
\eqref{Uprod} is shown in Appendix~\ref{sec-lemmas}
(see Lemma \ref{appendlemma2}).
\eqref{axiomABRR3} can then be derived from (\ref{Uprod}) using the fact that
$\widehat{J}{}^{[k]}$ belongs to
$\left( 1\otimes 1+(U^+_q({\mathfrak g})\otimes U^-_q({\mathfrak g}))^c\right)$.
\eqref{axiomABRR2bis}
can be proved as follows:
first, acting by $\tau^{k-1}$ on the first space of (\ref{axiomABRR2prime}),
we obtain
\begin{multline*}
{\mathfrak C}^{[-]}_2(x)\, {\mathfrak C}^{[+ k]}_1(xq^{h_2})\,
\big\{ B_2(x)\, (S^{[k+1]}_{12})^{-1}\widehat{J}{}^{[k]}_{12}\,
S^{[k+1]}_{12}\, B_2(x)^{-1}\big\} \\
= \big\{ (S^{[k]}_{12})^{-1} \widehat{J}{}^{[k-1]}_{12}\, S^{[k]}_{12}
\big\}\,
{\mathfrak C}^{[+ k]}_1(xq^{h_2})\, {\mathfrak C}^{[-]}_2(x),
\end{multline*}
then, using (\ref{Uprod}), we conclude the proof of (\ref{axiomABRR2bis})
by recursion.
\eqref{BM-T} follows directly from the definition of
${\cal M}^{(-)}(x)$, and
\eqref{BdeltaMV} follows directly from the definition of
${\cal M}^{(-)}(x)$ and from condition \eqref{axiomABRR1d}.
Finally, from \eqref{axiomABRR1d}, \eqref{axiomABRR1} and \eqref{DeltaB},
we have
\begin{multline*}
\Delta({\cal M}^{(-)}(x)^{-1})
=\prod_{k=+\infty}^{1}\big\{ B_1(xq^{h_2})^{-k} B_2(x)^{-k}
S^{[1]}_{12}\,{\mathfrak C}_1^{[-]}(xq^{h_2})\,
K_{12}^2\,\\
\times{\mathfrak C}_2^{[-]}(x)\,(S^{[1]}_{12})^{-1}
K_{12}^{-2} B_2(x)^{k}\,B_1(xq^{h_2})^{k}\big\},
\end{multline*}
which directly implies \eqref{axiomABRR4}.
\qed
\Lemma{\label{lemme-ABRR}}{
Let us suppose that ${\cal M}$ is defined as in Theorem~\ref{maintheorem} and assume that the hypothesis of Theorem~\ref{maintheorem} are satisfied.
Then ${\cal F}(x)$ defined as
${\cal F}(x)=\Delta({\cal M}(x))\,J\,{\cal M}_2(x)^{-1}
{\cal M}_1(xq^{h_2})^{-1}$ belongs to
$ 1 \otimes 1 + \left({\mathbb C}(\tilde{\nu}_1,\ldots,\tilde{\nu}_n)\otimes U_q({\mathfrak g})\otimes U^-_q({\mathfrak g})\right)^c$ and satisfies
the ABRR identity \eqref{lineareq}.
}
\medskip\noindent {\sf{\underline{Proof}:\ }}
Since, from Lemma~\ref{firstresults}, $U(x)$ belongs to
$({\mathbb C}(\nu_1,\ldots,\nu_n)\otimes U_q(\mathfrak{g})\otimes U_q(\mathfrak{b}^-))^c$,
we have ${\cal F}(x)\in ({\mathbb C}(\nu_1,\ldots,\nu_n)\otimes U_q(\mathfrak{g})\otimes U_q(\mathfrak{b}^-))^c.$
We can therefore define $(id\otimes \iota_-)({\cal F}(x))$.
{}From the relations \eqref{axiomABRR3}, \eqref{axiomABRR4}
and the identity \eqref{axiomABRR0}, we deduce that
$(id \otimes \iota_-)({\cal F}(x))=1 \otimes 1$,
which means that ${\cal F}(x)\in
1 \otimes 1 + \big({\mathbb C}(\tilde{\nu}_1,\ldots,\tilde{\nu}_n)\otimes U_q({\mathfrak g})\otimes U^-_q({\mathfrak g})\big)^c.$
The fact that $\mathcal{F}$ satisfies the ABRR relation
can be proved as follows.
Using respectively (\ref{MdeXprod}), (\ref{BdeltaMV}), the quasitriangularity
property \eqref{quasitriangularity2}, (\ref{axiomABRR2bis}) and
(\ref{BM-T}), we have
\begin{align*}
{\widehat{R}}{}^{-1}B_2(x)\,{\cal F}_{12}(x)
&=R^{-1} K_{12}\, B_2(x)\, \Delta({\cal M}^{(0)}(x))\,
\Delta( {\cal M}^{(-)}(x)^{-1})\, U_{12}(x)\nonumber\\
&\hspace{3cm} \times
{\cal M}_2^{(-)}(x)\, {\cal M}_2^{(0)}(x)^{-1}{\cal M}_1(xq^{h_2})^{-1}\\
&= \Delta({\cal M}^{(0)}(x))\, R^{-1} \Delta'( {\cal M}^{(-)}(x)^{-1})\,
S_{12}^{[1]}\, K_{12}\,\nonumber\\
&\hspace{3cm} \times
\big\{ {\mathfrak C}^{[-]}_2(x)\, B_2(x)\,(S_{12}^{[1]})^{-1} U_{12}(x)\big\}
\nonumber\\
&\hspace{3cm} \times
{\cal M}_2^{(-)}(x)\, {\cal M}_2^{(0)}(x)^{-1}{\cal M}_1(xq^{h_2})^{-1}\\
&= \Delta({\cal M}^{(0)}(x))\,\Delta( {\cal M}^{(-)}(x)^{-1})\,U_{12}(x)
\nonumber\\
&\hspace{3cm} \times
\big\{{\mathfrak C}^{[-]}_2(x)\, B_2(x)\,
{\cal M}_2^{(-)}(x)\big\}
{\cal M}_2^{(0)}(x)^{-1}{\cal M}_1(xq^{h_2})^{-1}\\
&=\Delta({\cal M}(x))\,J{\cal M}_2(x)^{-1} B_2(x)\, {\cal M}_1(xq^{h_2})^{-1}
\\
&={\cal F}_{12}(x)\, B_2(x),
\end{align*}
which ends the proof of the ABRR identity for ${\cal F}(x).$
\qed
Finally, we have the following uniqueness lemma:
\Lemma{\label{uniqueness}}{
Let ${\cal F}\in\left( 1 \otimes 1
+({\mathbb C}(\tilde{\nu}_1,\ldots,\tilde{\nu}_n)\otimes U_q({\mathfrak g})\otimes U^-_q({\mathfrak g}))^c\right)$ and assume that ${\cal F}(x)$ is a solution of the ABRR equation, then ${\cal F}(x)$ is equal to the standard
solution $F(x)$
\eqref{Fprod} of the QDCE.
}
\medskip\noindent {\sf{\underline{Proof}:\ }}
${\cal F}(x)$ and $F(x)$ being both in
$\left(1 \otimes 1 + ({\mathbb C}(\tilde{\nu}_1,\ldots,\tilde{\nu}_n)\otimes U_q({\mathfrak g})\otimes U^-_q({\mathfrak g}))^c\right)$,
we define $Y=F^{-1}(x){\cal F}(x)-1\otimes 1$.
Then $Y \in ({\mathbb C}(\tilde{\nu}_1,\ldots,\tilde{\nu}_n)\otimes U_q({\mathfrak g})\otimes U^-_q({\mathfrak g}))^c.$
${\cal F}(x)$ and $F(x)$ being both solutions of the ABRR identity,
we also have $[Y(x),B_2(x)]=0.$
Let $V,W$ be finite dimensional $U_q(\mathfrak{g})$-modules. We can decompose
$W=\oplus_{\lambda\in \mathfrak{h}^*}W[\lambda]$,
and consider $P_\lambda$ the associated projection on $W[\lambda]$.
We define
$Y_{\lambda,\mu}=({\rm id}\otimes P_{\lambda})Y_{V,W}({\rm id}\otimes P_{\mu}).$
Then, the fact that $[Y(x),B_2(x)]=0$ implies that
$(b(\lambda)-b(\mu)) Y_{\lambda,\mu}=0$ with
$B(x)_{\vert V[\lambda]}=b(\lambda){\rm id}_{V[\lambda]}.$
Since $Y_{V,W}$ is strictly lower triangular on $W$, the only possible nonzero $Y_{\lambda,\mu}$ are associated to
$\lambda<\mu.$
In this case the rational fraction $(b(\lambda)-b(\mu))\not=0$ and therefore
$Y_{\lambda,\mu}=0.$
As a result, $Y=0.$
\qed
Theorem~\ref{maintheorem} is a direct consequence of Lemma~\ref{lemme-ABRR}
and Lemma~\ref{uniqueness}.
\bigskip
We now give a direct derivation of a weaker result but which proof is interesting in itself, namely that, under the hypothesis of Theorem~\ref{maintheorem}, $\mathcal{M}(x)$ is a solution of the WQDBP.
\Lemma{\label{lemme-Mweak}}{
If ${\cal M}(x)$ is defined as in
Theorem~\ref{maintheorem},
then ${\cal F}(x)=\Delta({\cal M}(x))\,J\,{\cal M}_2(x)^{-1}
{\cal M}_1(xq^{h_2})^{-1}$ is a $\nu$-rational zero-weight solution of
the QDCE.
As a result, ${\cal M}(x)$ is also a solution of the WQDBP.
}
\medskip\noindent {\sf{\underline{Proof}:\ }}
Let us consider ${\cal P}(x)=v {\cal M}(x)^{-1} B(x)\,{\cal M}(x)$ and show
that the hypothesis of Proposition~\ref{theoreme} are satisfied.
First, from (\ref{MdeXprod}) and (\ref{BM-T}), we have
\begin{align}
{\cal P}(x)=v{\cal M}^{(+)}(x)^{-1} {\mathfrak C}^{[-]}(x)\, B(x)\,
{\cal M}^{(+)}(x).\label{PM+M-}
\end{align}
In order to show that $\mathcal{P}$ satisfies the linear
equation (\ref{linearPdeX2}),
we consider the quantity
\begin{equation*}
{\cal X}_{12}(x)={\cal M}^{(+)}_1(x)\, R^J_{12}\,{\cal P}_{2}(x)\,
R^J_{21} \, {\cal M}^{(+)}_1(x)^{-1}
\end{equation*}
and observe, using successively (\ref{PM+M-}), the definition of $U(x)$
and Eq.~(\ref{axiomABRR2bis}) of Lemma~\ref{firstresults}, that
%
\begin{align*}
{\cal X}_{12}(x)
&=v_2 \,U_{21}(x)^{-1} R_{12}\, U_{12}(x)\,
{\mathfrak C}_2^{[-]}(x)\, B_2(x)\, U_{12}(x)^{-1} R_{21}\, U_{21}(x)\\
&=v_2\, U_{21}(x)^{-1} K_{12}\, S^{[1]}_{12}\,\mathfrak{C}_2^{[-]}(x)\,B_2(x)\,
(S^{[1]}_{12})^{-1} R_{21}\,U_{21}(x).
\end{align*}
Note that, from \eqref{axiomABRR3},
${\cal X}_{12}(x)\in (U_q({\mathfrak b}^-)\otimes U_q({\mathfrak g}))^c$.
Then, using successively (\ref{axiomABRR2bis}), (\ref{axiomABRR1d}) and
\eqref{DeltaB}, the quasitriangularity property (\ref{quasitriangularity2}),
(\ref{axiomABRR1d}) and \eqref{DeltaB} again, and finally
(\ref{axiomABRR2bis}), we have
\begin{align*}
\big\{ {\mathfrak C}^{[-]}_1(x)\, B_1(x) \big\}\, {\cal X}_{12}(x)
&=v_2 \, \big\{ {\mathfrak C}^{[-]}_1(x)\, B_1(x)\, U_{21}(x)^{-1} \big\}
\\
&\hspace{2cm}\times
K_{12}\,S^{[1]}_{12}\,\mathfrak{C}_2^{[-]}(x)\,B_2(x) \,
(S^{[1]}_{12})^{-1} R_{21}\,U_{21}(x)\\
&=v_2 \, U_{21}(x)^{-1} \widehat{R}^{-1}_{21} \\
&\hspace{2cm}\times\big\{
S^{[1]}_{21}\,\mathfrak{C}_1^{[-]}(x)\,(S^{[1]}_{21})^{-1} K_{12}\,
S^{[1]}_{12}\,\mathfrak{C}_2^{[-]}(x)\,(S^{[1]}_{12})^{-1} K_{12}^{-1}
\big\}\\
&\hspace{2cm}\times \big\{B_1(x)\, B_2(x)\, K^2\}\,\widehat{R}_{21}\,U_{21}(x)
\\
&=v_2\, U_{21}(x)^{-1} R^{-1}_{21} \Delta({\mathfrak C}^{[-]}(x))\,
\Delta(B(x))\, R_{21}\, U_{21}(x)
\\
&=v_2\, U_{21}(x)^{-1} \Delta'({\mathfrak C}^{[-]}(x))\, \Delta(B(x))\,U_{21}(x)
\\
&=v_2\, U_{21}(x)^{-1}\, K_{12}\,S^{[1]}_{12}\,\mathfrak{C}_2^{[-]}(x)\,
(S^{[1]}_{12})^{-1} B_2(x)\, K_{12}\,\\
&\hspace{2cm}\times \big\{
S^{[1]}_{21}\,\mathfrak{C}_1^{[-]}(x)\,B_1(x)\,(S^{[1]}_{21})^{-1}
U_{21}(x)\big\}\\
&=\big\{v_2\, U_{21}(x)^{-1} K_{12}\, S^{[1]}_{12}\,\mathfrak{C}_2^{[-]}(x)\,
B_2(x)\,
(S^{[1]}_{12})^{-1} R_{21}\,U_{21}(x)\big\}\,\\
&\hspace{2cm}\times
\big\{{\mathfrak C}^{[-]}_1(x)\, B_1(x)\big\}\\
&={\cal X}_{12}(x)\, \big\{ {\mathfrak C}^{[-]}_1(x)\, B_1(x)\big\} .
\end{align*}
%
As a consequence, if we denote
\begin{align*}
{\cal M}^{(-)\;[N]}(x)&=\prod_{k=1}^{N}
\Big\{ B(x)^{-k}{\mathfrak C}^{[-]}(x)^{-1}B(x)^{k} \Big\}
\end{align*}
we obtain, using (\ref{axiomABRR1}),
\begin{align*}
{\cal M}^{(-)\;[N]}_1(x)^{-1}
{\cal X}_{12}(x)\,{\cal M}^{(-)\;[N]}_1(x)
&=B_1(x)^{-N}{\cal X}_{12}(x)\, B_1(x)^{N}\\
&=v_2\, \big\{ B_1(x)^{-N}
U_{21}(x)\, B_1(x)^{N} \big\}^{-1} S^{[1]}_{21}\,
{\mathfrak C}_2^{[-]}(xq^{h_1})\,\\
&\hspace{1.8cm}\times
B_2(xq^{h_1})\,(S_{21}^{[1]})^{-1}
\big\{ B_1(x)^{-N}\widehat{R}_{21}\, B_1(x)^{N} \big\}\,\\
&\hspace{1.8cm}\times
\big\{ B_1(x)^{-N} U_{21}(x)\,B_1(x)^{N} \big\}.
\end{align*}
Using the fact that
\begin{align*}
\forall \xi \in U_q({\mathfrak b}^-),\
\text{lim}_{N \rightarrow +\infty}\big\{B(x)^{-N} \xi\, B(x)^{N}\big\}
=\iota_-(\xi),
\end{align*}
(which is shown in each finite dimensional $U_q(\mathfrak g)$ module),
as well as the fact that $\widehat{R}_{12}\in
\big(1\otimes1 +U^+_q({\mathfrak g})\otimes U^-_q({\mathfrak g})\big)$
and the property (\ref{axiomABRR3}), we obtain
\begin{align*}
{\cal M}_1(x)\, R^J_{12}\,{\cal P}_{2}(x)\,R^J_{21}\,{\cal M}_1(x)^{-1}
&=v_2\, {\cal M}^{(+)}_2(xq^{h_1})^{-1} {\mathfrak C}_2^{[-]}(xq^{h_1})\,
B_2(xq^{h_1})\,{\cal M}^{(+)}_2(xq^{h_1})\\
&={\cal P}_2(xq^{h_1}),
\end{align*}
which concludes the proof of (\ref{linearPdeX2}).
${\cal M}(x)$ being almost $\nu$-rational, the hypothesis of Proposition~\ref{theoreme} are satisfied and $\mathcal{F}$ is
a $\nu$-rational zero-weight solution of the QDCE. Therefore $\mathcal{M}$ is
a solution of the WQDBP.
\qed
\begin{rem} Trivial Gauge transformations.
\par\noindent
It is important to remark that the solutions to the SQDBP and WQDBP we found
along the previous construction are by no means unique.
Indeed, let $M(x)$ be a solution of the SQDBP associated to the standard
solution $F(x)$ of the QDCE and to a given cocycle $J.$
Let $u$ be a dynamical group like element
$u \in ({\mathbb C}(\tilde{\nu}_1,\ldots,\tilde{\nu}_n)\otimes U_q({\mathfrak h}))^c$, i.e. verifying $\Delta(u(x))=u_1(xq^{h_2})u_2(x),$ and let
$y\in (U_q({\mathfrak g}))^c,$ then the element $u(x)M(x)y^{-1}$ is a solution of the QDBE
associated to the standard solution $F(x)$ of the QDCE and to
the cocycle $\Delta(y)Jy_2^{-1}y_1^{-1}.$
It is an interesting problem, not adressed here, to find the entire set of solutions of the SQDBP up to these transformations.
\end{rem}
\section{Explicit construction of Quantum Dynamical Coboundaries for
$U_q(sl(n+1))$}
\label{sectionConstr}
\subsection{The $U_q(sl(2))$ case}
In this case the following result holds, which gives a new derivation of the result of \cite{BR2}.
\Theorem{\label{th-case1}}{
In the $U_q(sl(2))$ case , a solution ${\cal M}(x)$ to the SQDBP is given by (\ref{MdeXprod}) and
(\ref{MdeXprod2}) with
\begin{align}
&{\cal M}^{(0)}(x)=1,\qquad
{\mathfrak C}^{[+]}(x)=e_{q^{-1}}^{- x e},\qquad
{\mathfrak C}^{[-]}(x)^{-1}= e_{q^{-1}}^{(xq^{h+1})^{-1}f}.
\label{MdeXsl2}
\end{align}
}
\vspace{-0.5cm}
\medskip\noindent {\sf{\underline{Proof}:\ }}
Equations (\ref{axiomABRR1d},\ref{axiomABRR1},\ref{axiomABRR2prime}) are obtained from the properties (\ref{expsomme},\ref{expproduit}) of the $q$-exponential.
\qed
\subsection{General solution of the SQDBP for $U_q(sl(n+1))$}
\Theorem{\label{theo-Mexpr}}{
A solution ${\cal M}(x)$ of the SQDBP for $U_q(sl(n+1))$ ($n\geq 1$)
is given by the infinite product
\begin{align}
&{\cal M}(x)= {\cal M}^{(0)}(x)\
\prod_{k=+\infty}^{1}\Big(B(x)^{-k}{\mathfrak C}^{[-]}(x)\,B(x)^{k}\Big)\
\prod_{k=0}^{+\infty} \tau^{k}\big({\mathfrak C}^{[+]}(x)\big),
\end{align}
with
\begin{align}
&{\cal M}^{(0)}(x)= \prod_{k=1}^{n} \nu_{k+1}^{\frac{1}{2}\zeta^{(k)}}
q^{-\frac{1}{2}(\zeta^{(k)})^2},
\nonumber\\
&{\mathfrak C}^{[+]}(x)=
\prod_{k=1}^{n}e_{q^{-1}}^{-\nu^{-1}_{k+1}\,q^{\zeta^{(k-1)}}e_{(k)}},
\nonumber\\
&{\mathfrak C}^{[-]}(x)^{-1}
\prod_{k=1}^{n}e_{q^{-1}}^{\nu_{k+1}\,q^{-\zeta^{(k-1)}-h_{(k)}-1}f_{(k)}}.
\label{MdeXsln}
\end{align}
}
\begin{rem}
One may wonder why the expression of ${\cal M}^{(0)}(x)$ that one obtains here in the $n=1$ case is not equal to $1$ as in Theorem~\ref{th-case1}. Actually,
since in this case $J=1,$ one can also choose the simpler solution ${\cal M}^{(0)}(x)=1$, which gives the result of the previous section.
\end{rem}
We first begin by proving a lemma interesting in itself.
\Lemma{}{Let ${\mathfrak C}^{[\pm]}$ solutions to the equations
(\ref{axiomABRR1d},\ref{axiomABRR1}) we define
\begin{align*}
&{\cal W}_{12}(x)= {\mathfrak C}^{[+]}_1(xq^{h_2})\,
\{B_{2}(x)\,(S^{[2]}_{12})^{-1}\widehat{J}^{[1]}_{12}\,
S^{[2]}_{12}\,B_{2}(x)^{-1}\}\,
{\mathfrak C}_2^{[-]}(x)^{-1}, \\
&\widetilde{\cal W}_{12}(x)=
{\mathfrak C}^{[-]}_2(x)^{-1}
\{(S^{[1]}_{12})^{-1}\widehat{R}_{12}\,S^{[1]}_{12}\}\,
{\mathfrak C}_1^{[+]}(xq^{h_2}).
\end{align*}
These elements satisfy the relations:
\begin{align*}
&(id \otimes \Delta)({\cal W}_{12} (x))
= K_{23}\, S^{[1]}_{32}\, {\cal W}_{13}(x q^{h_2})\,
K^{-1}_{23}\,(S^{[1]}_{32})^{-1}\,
\\
&\hspace{5cm}\times{\mathfrak C}^{[+]}_1(xq^{h_2+h_3})^{-1}
S^{[1]}_{23}\,
{\cal W}_{12}(x q^{h_3})\,(S^{[1]}_{23})^{-1},\\
&(id \otimes \Delta)(\widetilde{\cal W}_{12}(x))
= K_{23}\, S^{[1]}_{32}\, \widetilde{\cal W}_{13}(x q^{h_2})\,
K^{-1}_{23}\,(S^{[1]}_{32})^{-1}\,
\\
&\hspace{5cm}\times{\mathfrak C}^{[+]}_1(xq^{h_2+h_3})^{-1}
S^{[1]}_{23}\,
\widetilde{\cal W}_{12}(x q^{h_3})\,(S^{[1]}_{23})^{-1},
\end{align*}
as well as
\begin{align*}
&(\Delta \otimes id)({\cal W}_{12} (x))=
K_{12}\,S^{[1]}_{21}\,{\cal W}_{13} (x)\, K^{-1}_{12} (S^{[1]}_{21})^{-1}
{\mathfrak C}_3^{[-]}(x)\, S^{[1]}_{12}\,{\cal W}_{23} (x)\,
(S^{[1]}_{12})^{-1},\\
&( \Delta \otimes id)(\widetilde{\cal W}_{12}(x))=
K_{12}\,S^{[1]}_{21}\,\widetilde{\cal W}_{13} (x)\,
K^{-1}_{12} (S^{[1]}_{21})^{-1}
{\mathfrak C}_3^{[-]}(x)\, S^{[1]}_{12}\,\widetilde{\cal W}_{23} (x)\,
(S^{[1]}_{12})^{-1},
\end{align*}
}
\medskip\noindent {\sf{\underline{Proof}:\ }}
This is easy to show using only only (\ref{axiomABRR1d}), (\ref{axiomABRR1}), the definition of the
cocycle in Theorem~\ref{cocycleBCG} and the quasitriangularity properties of
the $R$-matrix.
We now give the proof of the theorem.
\medskip\noindent {\sf{\underline{Proof}:\ }}
As $\mathcal{M}$ is of the form \eqref{MdeXprod}, \eqref{MdeXprod2}, with
${\mathfrak C}^{[\pm]} $ being $\nu$-rational, it is enough to show that
the hypothesis of Theorem~\ref{maintheorem} are verified, i.e. that
\eqref{axiomABRR0}, \eqref{axiomABRR1d}, \eqref{axiomABRR1} and
\eqref{axiomABRR2prime} are satisfied.
Equations (\ref{axiomABRR0}) and (\ref{axiomABRR1}) are trivial to check
using the elementary property
\begin{align}
\nu_k(xq^{h})=\nu_k(x)\,
q^{2(\zeta^{(k)}-\zeta^{(k-1)})},
\quad \forall k=1,\ldots,n+1.
\end{align}
Equation (\ref{axiomABRR1d}) can be easily deduced from property
(\ref{expsomme}) and from the following commutation properties,
verified for all $i,j$ satisfying $2 \leq j+1 \leq i \leq n$,
\begin{align*}
& [ q^{-\zeta^{(i-1)}-2h_{(i)}} \otimes q^{-\zeta^{(i-1)}-h_{(i)}}f_{(i)}
\ ,\
q^{-\zeta^{(j-1)}-h_{(j)}}f_{(j)} \otimes
q^{-\zeta^{(j-1)}-h_{(j)}} ]=0,
\\
&[ q^{\zeta^{(i-1)}}e_{(i)} \otimes q^{\zeta^{(i-1)}+h_{(i)}}
\ ,\
q^{\zeta^{(j-1)}} \otimes q^{\zeta^{(j-1)}}e_{(j)} ]
=0.
\end{align*}
Equation (\ref{axiomABRR2prime}) is slightly more difficult to show.
Although our proof is a bit unsatisfactory, we have chosen a method preventing us
to enter too deeply in the combinatorics of $q$-exponentials.
Our proof consists in two steps: first, we show that this relation holds in
the fundamental representation, and then that it can be obtained in other
representations by fusion from the fundamental representation.
The fact that \eqref{axiomABRR2prime} is satisfied in the fundamental
representation, i.e. that
\begin{equation*}
(\stackrel{f}{\pi}\otimes \stackrel{f}{\pi})
({\cal W}_{12}-\widetilde{\cal W}_{12})(x)=0,
\end{equation*}
is proved by an explicit computation in Appendix~\ref{sec-lemmas}
(Lemma \ref{appendlemma3}).
Let us now study the fusion properties of this relation. This is the consequence of the previous lemma and from it we obtain:
if
$(\stackrel{\Lambda_i}{\pi},\stackrel{\Lambda_i}{V})$, $i=1,2,3$, are
representations of $U_q({\mathfrak g})$, %
\begin{align*}
( \stackrel{\Lambda_1}{\pi}\otimes \stackrel{\Lambda_i}{\pi})
({\cal W}_{12}-\widetilde{\cal W}_{12})(x)=0,\ i=2,3,\qquad
\text{implies}\qquad
( \stackrel{\Lambda_1}{\pi}\otimes \stackrel{\Lambda}{\pi})
({\cal W}_{12}-\widetilde{\cal W}_{12})(x)=0,
\end{align*}
for any submodule
$\stackrel{\Lambda}{V}$ of $\stackrel{\Lambda_2}{V}\otimes \stackrel{\Lambda_3}{V}.$
From this lemma we obtain that:
\begin{align*}
(\stackrel{\Lambda_i}{\pi} \otimes \stackrel{\Lambda_1}{\pi})
({\cal W}_{12}-\widetilde{\cal W}_{12})(x)=0,\ i=2,3,\qquad
\text{implies}\qquad
(\stackrel{\Lambda}{\pi} \otimes \stackrel{\Lambda_1}{\pi})
({\cal W}_{12}-\widetilde{\cal W}_{12})(x)=0
\end{align*}
for any submodule
$\stackrel{\Lambda}{V}$ of $\stackrel{\Lambda_2}{V}\otimes \stackrel{\Lambda_3}{V}.$
It is a basic result that any irreducible module of
$U_q(sl(n+1))$ is obtained as a submodule of some tensor power of the fundamental
representation. Therefore,
$(\stackrel{\Lambda_1}{\pi}\otimes \stackrel{\Lambda_2}{\pi})
({\cal W}_{12}-\widetilde{\cal W}_{12})(x)=0$ for any couple
$(\stackrel{\Lambda_i}{\pi},\stackrel{\Lambda_i}{V})$, $i=1,2$,
of irreducible representations of $U_q( sl(n+1)).$
This concludes the proof.
\qed
\begin{rem}
Note that Lemma~\ref{lemme-Mweak}, Lemma~\ref{lemme-ABRR}, the uniqueness
condition of Lemma~\ref{uniqueness} and the theorem give a new proof of the fact that
$F(x)$ is a zero-weight solution of the QDCE in the $U_q(sl(n+1))$ case.
\end{rem}
\begin{rem}
It is easy to compute the expression of ${\cal M}^{(\pm)}(x),{\cal M}^{(0)}(x),
{\cal M}(x)$ in the fundamental representation from the explicit universal expression given by theorem \ref{theo-Mexpr}. One obtains:
\begin{eqnarray*}
&&\stackrel{f}{\pi}({\cal M}^{(+)}(x))=1+\sum_{1\leq i<j\leq n+1}a_{ij}(-1)^{i-j}
q^{-\frac{(j-i)(j+i-3)}{2(n+1)}}E_{i,j} \\
&&\stackrel{f}{\pi}(({\cal M}^{(+)}(x))^{-1})=1+\sum_{1\leq i<j\leq n+1}b_{ij}
q^{-\frac{(j-i)(j+i-3)}{2(n+1)}}E_{i,j} \\
&&\text{with}\;\; a_{ij}=\sum_{i<a_1<...<a_{j-i}\leq n+1}
\nu_{a_1}^{-1}\cdots \nu_{a_{j-i}}^{-1}\;\;,\; b_{ij}=\sum_{j<a_1<...<a_{j-i}
\leq n+1} \nu_{a_1}^{-1}\cdots \nu_{a_{j-i}}^{-1}.
\end{eqnarray*}
One also has:
\begin{eqnarray*}
&&\stackrel{f}{\pi}({\cal M}^{(-)}(x))=1+\sum_{1\leq i<j\leq n+1}c_{ij}
q^{\frac{(j-i)(j+i-3)}{2(n+1)}}E_{j,i} \\
&&\stackrel{f}{\pi}(({\cal M}^{(-)}(x))^{-1})=1+\sum_{1\leq i>j\leq n+1}d_{ij}
q^{\frac{(i-j)(j+i-3)}{2(n+1)}}E_{i,j} \\
&&\text{with}\;\; c_{ij}=\frac{\nu_i^{j-i}}{\prod_{r=i+1}^j (1-\nu_i \nu_r^{-1})}\;\; ,\;
d_{ij}=
\frac{\nu_i^{j-i}}{\prod_{r=j}^{i-1} (1-\nu_i \nu_r^{-1})}.
\end{eqnarray*}
Using these results an explicit computation shows that:
\begin{equation*}
\stackrel{f}{\pi}(({\cal M}(x)^{-1})=D\;\mathcal{V}(x)\;{\mathcal U}(x)D^{-1}
\end{equation*}
with $D,\mathcal{V}(x), {\mathcal U}(x)$ defined in Theorem \ref{lemme1}.
\end{rem}
\section{Quantum Dynamical coBoundaries and Quantum Weyl Group}
\label{sectionQWG}
\subsection{ Primitive Loops and Quantum Coxeter Element }
Let $\mathfrak{g}$ be a finite dimensional simple Lie algebra, and $W$ its
associated Weyl group.
We define the shifted action of $W$ on $\mathfrak{h}$ by
$w.\lambda= w(\lambda+\rho)-\rho.$
We will denote $w.x$ the corresponding action on the variable
$x_i,\ i=1,\ldots,r.$
In the case where $\mathfrak{g}=sl(n+1)$, we can identify the Weyl group with
the permutation group $S_{n+1}.$
Its shifted action on $\nu_1,\ldots,\nu_{n+1}$ is given by
$w.(\nu_1,\ldots,\nu_{n+1})=(\nu_{w(1)},\ldots,\nu_{w({n+1})}).$
We will assume in this subsection that $\mathfrak{g}=sl(n+1)$ and that $M(x)$ is the solution to the SQDBP defined in Theorem \ref{theo-Mexpr}.
We will study the behaviour of ${ P}(x),\ M(x)$ under the shifted action of the Weyl group.
\Proposition{}{
The primitive loop is invariant under the shifted action of the Weyl group,
i.e.
${ P}(w.x)={ P}(x), \forall w\in W.$}
\medskip\noindent {\sf{\underline{Proof}:\ }}
Because of the fusion property (\ref{deltaPdeX}), it is sufficient to prove that ${ P}(x)$ is invariant in the fundamental representation. We have computed ${ P}(x)$ in the fundamental representation using the explicit form of $M(x)$, it is given by (\ref{Pinfund}) and depends on $\nu_i$ only through the symmetric polynomials.
As a result, ${ P}(x)$ is invariant under the shifted action of $W.$
\qed
An interesting question is what is the explicit expression of
${ P}(x)$? From equation (\ref{PM+M-},\ref{MdeXsln}), $P(x)$ can be expressed as a finite product of $q$-exponential.
However, on this expression, the invariance of $P(x)$ under the shifted action of the Weyl group is not explicit.
We can simplify the expression of $P(x)$ by using the quantum Weyl group.
Indeed, in the $sl(2)$ case, we have the property:
\Proposition{Expression of $P(x)$ for $sl(2).$}{
\vspace{-0.5cm}
\begin{align}
{P}(x)&= v \; e_{q}^{xe}\; e_{q}^{-(xq^{h+1})^{-1}f}\; B(x)\;e_{q^{-1}}^{-xe}
\label{explicitPdeX2}\\
&=\omega\;e_{q^{-1}}^{-xe}\;e_{q^{-1}}^{-x^{-1}e}.
\label{explicitPdeX1}
\end{align}}
\vspace{-0.5cm}
\medskip\noindent {\sf{\underline{Proof}:\ }} Formula (\ref{explicitPdeX1})
is deduced from (\ref{explicitPdeX2}) using (\ref{saitow}). This last formula is explicitely symmetric under the exchange of $x$ and $x^{-1}.$\qed
This result can be generalized as follows:
\Theorem{}{${P}(x)$ satisfies
\begin{align}
{P}(x)&= v \hat{w}_C^{-1}\; Q(x),
\end{align}
where $Q\in (\mathbb{C}(\nu_1,\ldots,\nu_{n})\otimes U_q({\mathfrak b}^+))^c$
is invariant under the shifted action of the Weyl group and $\hat{w}_C$ is a
quantum analog of the Coxeter element
$\hat{w}_C= \prod_{i=1}^{n}\hat{w}_{(i)}.$}
\medskip\noindent {\sf{\underline{Proof}:\ }}
{}From the expression (\ref{PM+M-}), we obtain that
\begin{equation*}
P(x)=v\tau(M^{(+)-1}(x)) (\prod_{k=n}^1{ E}^{[k]}(x))
(\prod_{k=n}^1{ F}^{[k]}(x)) B(x)\ M^{(+)}(x),
\end{equation*}
with
\begin{equation*}
{ E}^{[k]}(x)=e_{q}^{\nu^{-1}_{k+1}\,q^{\zeta^{(k-1)}}e_{(k)}},\qquad
{ F}^{[k]}(x)=e_{q}^{-\nu_{k+1}\,q^{-\zeta^{(k-1)}-h_{(k)}-1}f_{(k)}}.
\end{equation*}
As a result,
\begin{equation*}
P(x)=v\tau(M^{(+)-1}(x))\prod_{k=n}^1({ E}^{[k]}(x){ F}^{[k]}(x))B(x) M^{(+)}(x).
\end{equation*}
Using Saito's formula \eqref{saitow}, we have
\begin{equation*}
q^{-\frac{h}{2}}\hat{w}^{-1}e_{q^{-1}}^{-e}q^{-\frac{h^2}{2}}
=e_q^{e}\cdot e_q^{(-q^{-1-h}f)},
\end{equation*}
which implies that
\begin{equation*}
{ E}^{[k]}(x)\; { F}^{[k]}(x)=G_k\; \hat{w}_{(k)}^{-1}\;
e_{q^{-1}}^{-e_{(k)}}\; H_k
\end{equation*}
with $G_k=q^{-\frac{h_{(k)}}{2}}\nu_{k+1}^{-\frac{h_{(k)}}{2}}
q^{\frac{h_{(k)}\zeta^{(k-1)}}{2}}$ and
$ H_k^{-1}=G_k q^{\frac{h_{(k)}^2}{2}}q^{\frac{h_{(k)}}{2}}.$
We therefore have proven that
\begin{equation*}
P(x)=v\tau(M^{(+)-1})\prod_{k=n}^1(G_k \hat{w}_{(k)}^{-1}
e_{q^{-1}}^{-e_{(k)}} H_k)B(x) M^{(+)}(x),
\end{equation*}
and it remains to move all the $ \hat{w}_{(k)}^{-1}$ on the left.
Using the fact that $s_{\alpha_1}\ldots s_{\alpha_k}(\alpha_{k+1})$
is a positive root,
we obtain that
$(\prod_{p=1}^k\hat{w}_{(p)})e_{(k)} (\prod_{p=1}^k\hat{w}_{(p)})^{-1} \in
( U_q(\mathfrak{b}_+))^c.$
Because $\hat{w}_C e_{k}\hat{w}_C^{-1}\in (U_q(\mathfrak{b}^+))^c,$ for $1\leq k\leq n-1,$
we obtain $\hat{w}_C \tau (M^{(+)-1})\hat{w}_C^{-1}\in
(U_q(\mathfrak{b}^+))^c.$ This finishes the proof.
\qed
\begin{rem}
An interesting question would be to find the simplest exact form of $Q(x)$ expressed as a finite product of $q$-exponentials when $n\geq 2,$ exhibiting invariance under the shifted action of the Weyl group.
\end{rem}
\subsection{Dynamical coBoundary and Dynamical Quantum Weyl Group}
We show in this section that the shifted action of $W$ on $M(x)$ is controlled via a dynamical quantum Weyl group \cite{EV}.
Although the notion of dynamical Weyl group can be defined for any simple Lie algebra, we will assume here that $\mathfrak{g}=sl(n+1)$.
We first recall that the dynamical Weyl group controls the shifted action of
$W$ on $F(x)$, where $F(x)$ denotes the standard solution to the QDCE.
Etingof and Varchenko have constructed a map
$A:W\rightarrow ({\mathbb C}(\nu_1,\ldots,\nu_n)\otimes U_q(\mathfrak g))^c,\
w\mapsto A_w(x),$
satisfying the following properties:
\begin{align}
& \hat{w} A_w(x) \;\;\text{is a zero-weight element}, \label{dynWeyl1}\\
& \Delta A_{w}(x)\, F(x)=F(w.x)\, (A_w)_{2}(x)\,(A_w)_{1}(xq^{h_2}),
\label{dynWeyl2}\\
& A_{ww'}(x)=A_{w}(w'.x)\, A_{w'}(x),\ \forall w,w' \in W.\label{dynWeyl3}
\end{align}
The third property can be reformulated as $A_w(x)$ being a one $W$-cocycle
taking values in
$({\mathbb C}(\nu_1,\ldots,\nu_n)\otimes U_q(\mathfrak{g}))^c.$
We have the following proposition:
\Proposition{}{Let $M$ denote the solution of the SQDBP defined in
Theorem ~\ref{theo-Mexpr}.
We can define a map
$\tilde{A}:W\rightarrow ({\mathbb C}(\nu_1,\ldots,\nu_n)\otimes
U_q(sl(n+1)))^c,\ w\mapsto\tilde{A}_w(x)$ by
\begin{equation}
\tilde{A}_w(x)=M(w.x)\,M(x)^{-1}.\label{A=M}
\end{equation}
This map satisfies all the properties (\ref{dynWeyl1},\ref{dynWeyl2},\ref{dynWeyl3}).
}
\medskip\noindent {\sf{\underline{Proof}:\ }}
We first show that $\hat{w}
\tilde{A}_w(x)$ is a zero-weight element. We have:
\begin{align*}
\tilde{A}_w(x)_1\, P_2(xq^{h_1})
&=M_1(w.x)\, M_1(x)^{-1}{ P_2 }(xq^{h_1}) \\
&=M_1(w.x)\, R_{12}^J\, { P}_2(x)\, R_{21}^J\, M_1(x)^{-1} \\
&=M_1(w.x)\, R_{12}^J\, { P}_2(w.x)\, R_{21}^J\, M_1(x)^{-1} \\
&=M_1(w.x)\, M_1(w.x)^{-1}{ P}_2((w.x)q^{h_1})\, M_1(w.x)\, M_1(x)^{-1} \\
&={ P}_2((w.x)q^{h_1})\, \tilde{A}_w(x)_{1} \\
&={ P}_2(xq^{w(h)_1})\, \tilde{A}_w(x)_{1}.
\end{align*}
This shows that $(\hat{w}
\tilde{A}_w(x))_1$ commutes with ${ P}_2(xq^{h_1})$. Therefore, using the
same argument as in Proposition \ref{theoreme}, we obtain that $\hat{w}
\tilde{A}_w(x)$ is a zero-weight element.
We now prove that ${\tilde A}_w(x)$ satisfies the property (\ref{dynWeyl2}).
The dynamical coboundary equation implies that
\begin{equation*}
F(w.x)=\Delta M(w.x)\, J\, M_2(w.x)^{-1} M_1(w.x q^{h_2})^{-1}.
\end{equation*}
Therefore,
\begin{align*}
\Delta M(x)^{-1}\, F(x)&=J\, M_2(x)^{-1} M_1(x q^{h_2})^{-1 }\\
&=\Delta M(w.x)^{-1}\, F(w.x)\, M_1(w.x q^{h_2})\, M_2(w.x)\, M_2(x)^{-1}
M_1(x q^{h_2})\\
&=\Delta M(w.x)^{-1}\, F(w.x)\, M_2(w.x)\, M_2(x)^{-1} M_1((w.x) q^{wh_2})\,
M_1(x q^{h_2})\\
&=\Delta M(w.x)^{-1}\, F(w.x)\, \tilde{A}_w(x)_{2}\, \tilde{A}_w(xq^{h_2})_{1},
\end{align*}
which ends the proof of the proposition.
\qed
One can precise the relation between $A$ and $\tilde{A}.$
\Proposition{}{
If $A,A'$ are two maps $W\rightarrow ({\mathbb C}(\nu_1,\ldots,\nu_n)\otimes U_q(\mathfrak g))^c$ satisfying the axioms $(\ref{dynWeyl1},
\ref{dynWeyl2}, \ref{dynWeyl3}),$
they are related as $A'_w=A_w Y_w$, where $Y$ is a map
$Y: W\rightarrow ({\mathbb C}(\nu_1,\ldots,\nu_n)\otimes U_q(\mathfrak h))^c$ which is a one-cocycle, i.e.
\begin{equation}
Y_{ww'}(x)= \hat{w}'(Y_{w}(w'.x))\hat{w}'{}^{-1}\ Y_{w'}(x),
\end{equation}
and such that each $Y_w(x)$ is a dynamical group like element, i.e.
\begin{equation}
\Delta(Y_{w}(x))=Y_w(x)_2 Y_w(xq^{h_2})_1.
\end{equation}
}
\vspace{-0.7cm}
\medskip\noindent {\sf{\underline{Proof}:\ }}
We define $Y_{w}=A_{w}^{-1}A_{w}'=(\hat{w}A_{w})^{-1}(\hat{w}A_{w}')$ which is
zero-weight.
We have,
\begin{equation}
F(x)^{-1}\Delta(Y_{w}(x))\, F_{12}(x)=A_{w}(xq^{h_2})_1^{-1}\,
Y_{w}(x)_2 \,
A_{w}'(xq^{h_2})_1=Y_w(x)_2\, Y_w(xq^{h_2})_1.
\end{equation}
We now apply the Lemma 2.15 of Etingof-Varchenko \cite{EV2} and we obtain that
$Y_{w}(x)$ lies in $({\mathbb C}(\nu_1,\ldots,\nu_n)\otimes U_q(\mathfrak h))^c.$
This ends the proof.
\qed
\begin{rem} It would be interesting to precise the value $Y_w(x)$ relating $A_w$ the dynamical quantum Weyl group of Etingof-Varchenko to $\tilde{A}_w(x)$ defined by (\ref{A=M}).
\end{rem}
\begin{rem}The solutions of (\ref{dynWeyl1},\ref{dynWeyl2},\ref{dynWeyl3})
exist for all simple Lie algebra $\mathfrak{g}$, however we give a solution of these equations as a coboundary of the form (\ref{A=M}) only in the $sl(n+1)$ case.
It would be an interesting problem to study if the dynamical Weyl group of Etingof-Varchenko can be written as $A_w(x)=M(w.x)M(x)^{-1}Y_w(x)$ with $Y_w(x)$ satisfying properties of the previous proposition and $M:\mathbb {C}^r\rightarrow U_q(\mathfrak{g}).$ In the negative this would provide an alternative proof of the Balog-D\c abrowski-Feh\'er Theorem.
\end{rem}
\section{Quantum reflection algebras}
\label{sectionQRA}
\subsection{Definitions, properties and primitive representations}
In this section we define and study the quantum reflection algebra
${\cal L}_q({\mathfrak g},J)$ associated to any cocycle
$J\in (U_q(\mathfrak{g}))^{\otimes 2}$.
Although there exists a morphism of algebra
${\cal L}_q({\mathfrak g},J)\rightarrow U_q(\mathfrak g)$, this morphism of algebra is not an isomorphism.
In particular, in the case where $\mathfrak{g}=sl(n+1)$ and $J=J^{\tau}$,
${\cal L}_q( sl(n+1),J)$ admits one-dimensional representations which do not
extend to $U_q(sl(n+1))$-representations.
We call them primitive representations and study their relation with the primitive loop.
\Definition{Quantum Reflection Algebra}{
For any simple Lie algebra $\mathfrak{g}$ and any cocycle $J\in (U_q(\mathfrak{g}))^{\otimes 2}$, we define the algebra ${\cal L}_q({\mathfrak g},J)$ as being the associative algebra generated by the components ${\mathfrak P}_ {V}$ for all finite dimensional module $V$ of the element ${\mathfrak P} \in ( U_q({\mathfrak g})\otimes
{\cal L}_q({\mathfrak g},J))^c $ with the relations:
\begin{equation}
(\Delta^{J} \otimes id)({\mathfrak P})=(R^{J}_{12})^{-1} {\mathfrak P}_1 R^{J}_{12} {\mathfrak P}_{2},
\;\;\;\;\;\;\;\;(\varepsilon \otimes id)({\mathfrak P})=1.\label{quantummoduli}
\end{equation}
}
Let $(V,\pi)$ be a finite dimensional $U_q(\mathfrak g)$-module, we denote
$\stackrel{\pi}{U} =(\pi\otimes {\rm id})(\mathfrak P)\in \mathrm{End}(V)\otimes {\cal L}_q({\mathfrak g},J).$
These matrices satisfy the following reflection equations for all finite dimensional $U_q(\mathfrak g)$ modules $V,V':$
\begin{equation}
{ R}^{J}_{21} \stackrel{\pi}{U}_1 {R}^{J}_{12} \stackrel{\pi'}{U}_2
=\stackrel{\pi'}{U}_2 { R}^{J}_{21} \stackrel{\pi}{U}_1 { R}^{J}_{12}.\label{echangeloop}
\end{equation}
Reflection algebras satisfy the following property:
\Proposition{}{The map $\kappa:{\cal L}_q({\mathfrak g},J)\rightarrow U_q(\mathfrak{g})$ defined by
\begin{equation}
({\rm id}\otimes \kappa)(\mathfrak{P})=R^{J(-)-1}R^{J}
\end{equation}
is a morphism of algebra.
}
\medskip\noindent {\sf{\underline{Proof}:\ }} It follows from the quasitriangularity of $R^J.$
\qed
This map is usually thought as being an isomorphism of algebra but this is not true, it is true only if we localize certain elements of
${\cal L}_q({\mathfrak g},J).$ This aspect will be central in the rest of this section.
${\cal L}_q({\mathfrak g},J)$ is not a Hopf algebra but it is naturally endowed with a structure of left $U_q({\mathfrak g})$-comodule algebra.
\Proposition{}{The map $\sigma:{\cal L}_q({\mathfrak g},J) \rightarrow
U_q({\mathfrak g}) \otimes {\cal L}_q({\mathfrak g},J)$ defined by
\begin{equation}
({\rm id}\otimes \sigma)(\mathfrak{P})=R^{J(-)-1}_{12}\mathfrak{P}_{13}R^{J}_{12}
\end{equation}
is a morphism of algebra and is a left $U_q(\mathfrak g)$-coaction with the coproduct being $\Delta^J.$}
\medskip\noindent {\sf{\underline{Proof}:\ }} Trivial.
\qed
Although this map does not define a structure of Hopf algebra on ${\cal L}_q({\mathfrak g},J)$ we still have the following commutative diagram:
\begin{equation}
(\kappa\otimes {\rm id})\sigma=\Delta^J\kappa.
\end{equation}
A natural problem is the study of the representation theory of
${\cal L}_q({\mathfrak g},J).$
Because of the two previous propositions we have the following general properties:
\begin{enumerate}
\item a representation $\pi$ of $U_q(\mathfrak g)$ defines a representation $\pi\circ \kappa$ of $ {\cal L}_q({\mathfrak g},J).$
\item if $(V,\pi)$ is a representation of $U_q(\mathfrak{g})$ and
$(W,\omega)$ is a representation of $ {\cal L}_q({\mathfrak g},J),$ one can define the tensor product $\pi\hat{\otimes}\omega=(\pi\otimes \omega)\sigma$ representation of $ {\cal L}_q({\mathfrak g},J)$ acting on $V\otimes W.$
\end{enumerate}
As an example we first classify the irreducible finite dimensional representations of ${\cal L}_q(sl(2))$ with $J=1.$
We first define a simpler presentation of ${\cal L}_q(sl(2))$ using the fundamental representation of $U_q(sl(2)).$
${\cal L}_q(sl(2))$ is generated
by the matrix elements
$a=\stackrel{f}{U}{}^1_1,b=\stackrel{f}{U}{}^1_2,c=\stackrel{f}{U}{}^2_1,d=\stackrel{f}{U}{}^2_2$ with relations (\ref{quantummoduli})
which can be explicited respectively as (\ref{echangeloop}) with an additional relation:
\begin{align*}
&ac=q^2 ca \;\;\;\;\;\;\;\;\;\;\;\;ba=q^2 ab\;\;\; \;\;\;\;\;\;\;\;\;bc-cb=(1-q^{-2})a(d-a)\\
&cd-dc=(1-q^{-2})ca \;\;\;\;\;\;\;\;\;\;\;\;db-bd= (1-q^{-2})ab\;\;\;\;\;\;\;\;\;\;\;\;ad=da\\
&\mbox{\rm and}\;\;\;\;\;\;\;\;\;\;\;\;ad-q^2 cb=1.
\end{align*}
The center of ${\cal L}_q(sl(2))$ is the polynomial algebra generated by
$z=q^{-1}a+qd.$ \\
We denote $\hat{\cal L}_q(sl(2))$ the algebra ${\cal L}_q(sl(2))$ localized in $a$ and we denote $a^{-1}$ the inverse of $a$.
$\hat{\cal L}_q(sl(2))$ is now isomorphic to $U_q(sl(2))$ through the explicit homomorphism
\begin{equation}
\rho: U_q(sl(2)) \longrightarrow {\hat{\cal L}}_q(sl(2)),
\;\;\;\;\;\;\;\;\rho(q^h)=a,\;\;\;\rho(e)=\frac{c}{1-q^{-2}}, \;\;\;\rho(f)=\frac{a^{-1}b}{q-q^{-1}},
\end{equation}
which satisfies
$\rho \circ \kappa =id_{\mid {\cal L}_q(sl(2))}.$\\
It is easy to classify the irreducible representations of ${\cal L}_q(sl(2)).$
\Proposition{}{The irreducible representations of ${\cal L}_q(sl(2))$ consists in: \\
-The representation $\pi\circ \kappa$ where $\pi$ is an irreducible
representation of $U_q(sl(2)).$ These representations extends to
$\hat{\cal L}_q(sl(2)).$\\
-The one-dimensional representations ${\cal E}_{x,\alpha}$ with $x,\alpha\in \mathbb C^*,$ defined as:
\begin{equation}
{\cal E}_{x,\alpha}(a)=0\;\;\;\;\;{\cal E}_{x,\alpha}(d)=q^{-1}(x+x^{-1})\;\;\;\;\;
{\cal E}_{x,\alpha}(b)=q^{-1}\alpha\;\;\;\;\;{\cal E}_{x,\alpha}(c)=-q^{-1}\alpha^{-1}.
\;\;\;\;\;
\end{equation} Note that ${\cal E}_{x,\alpha}(z)=(x+x^{-1})$ and ${\cal E}_{x,\alpha}(\stackrel{f}{U})=D_{\alpha}{\bf P}(x)D_{\alpha}^{-1}$ with $D_{\alpha}^2=
diag (\alpha,\alpha^{-1}).$
}
\medskip\noindent {\sf{\underline{Proof}:\ }}
Let $(\Pi, V)$ be a finite dimensional irreducible ${\cal L}_q(sl(2))$ module. $z$ being central is represented by a complex number
$z\in {\mathbb C}.$
If $\Pi(a)$ is invertible one obtains a representation of $\hat{\cal L}_q(sl(2))\simeq U_q(sl(2))$ which from the classification of irreducible representations of $U_q(sl(2))$ when $q$ is not a root of unit is of the form $\Pi=\pi\circ \kappa.$
If not, we define $W=ker (\Pi(a))$, it is a submodule and the restriction of $a,b,c,d$ to $W$ satisfies:
\begin{equation}
a=0,\quad bc=cb,\quad -q^2bc=1,\quad d=q^{-1}z.
\end{equation}
This is an abelian algebra and therefore the irreducible finite dimesional representations are of dimension one. Therefore $W=V$ and is one dimensional. The representation is therefore of the type ${\cal E}_{x,\alpha}.$
\qed
We will call the family of one-dimensional representations of
${\cal L}_q(sl(2))$ for which $a=0$ {\it primitive representations.}
\begin{rem}
The study of the representation theory of the algebra ${\cal L}_q(\mathfrak g, J)$ is an interesting problem that we will only look at in the case where $\mathfrak{g}=sl(n+1)$ and $J=J^\tau.$
\end{rem}
The primitive representation in the $sl(2)$ case appears first in \cite{KSS}
The classification of all characters, i.e. all one dimensional representations, of
${\cal L}_q(sl(n+1), J)$ for $J=1$ has been obtained in \cite{M}.
\begin{rem}
Finite dimensional representations of reflection algebras are not completely reducible. An example is given for ${\cal L}_q(sl(2))$ by \cite{Sa}.
\end{rem}
In the next section we will study the generalisation of these primitive
representations to the case of ${\cal L}_q(sl(n+1),J^{\tau}).$
We will study the decomposition of the tensor product of an irreducible representation $\pi$ of $U_q(sl(n+1))$ with an irreducible representation $\omega$ of
${\cal L}_q(sl(n+1),J^{\tau}).$
When the irreducible representation $\omega=\pi'\circ\kappa$, where $\pi'$ is a representation of $U_q(sl(n+1)),$ the intertwining map is governed by ordinary Clebsch-Gordan map of $U_q(sl(n+1))$ and there is nothing new, but
when $\omega$ is a primitive representation the intertwining map is entirely governed by the coboundary element evaluated in the representation $\pi.$
We can also invert this process and define the coboundary in
the representation $\pi$ as being the intertwining map.
We now assume that $\mathfrak{g}=sl(n+1)$ and $J=J^\tau.$
In the first part of this section we assume that $M(x)$ is the solution of the SQDBP with associated primitive loop ${ P}(x).$
\Definition{Primitive Representations }{
The primitive representation ${\cal E}$ of ${\cal L}_q(sl(n+1),J^{\tau})$ is the representation of ${\cal L}_q(sl(n+1),J^{\tau})$ with values in ${\mathbb C}[\nu_1,\nu_1^{-1},\ldots,\nu_r,\nu_r^{-1}] $
defined by
\begin{equation}
(id \otimes {\cal E}_x)({\mathfrak P})={P}(x)
\end{equation}
We will abusively denote ${\cal E}_x$ this representation.}
\medskip\noindent {\sf{\underline{Proof}:\ }}
Trivial.
\qed
Of course if we fix $\nu_1,\ldots,\nu_r$ to non zero complex numbers, we obtain characters of ${\cal L}_q(sl(n+1),J^{\tau}).$
We have the following proposition
\Proposition{Intertwining map and Coboundary }{\label{PropIntMapCobound}
Let $(V,\pi)$ be a finite dimensional representation of $U_q(\mathfrak g)$, the following decomposition property holds:
\begin{equation}
\pi\hat{\otimes} {\cal E}_x=\bigoplus_{\lambda\in {\mathfrak h}^* }
{\cal E}_{xq^\lambda}^{\oplus m_\lambda}\;\;\; \text{where}\;\;
m_\lambda=dim V[\lambda],
\end{equation}
and $M(x)_V$ is an intertwining map between these representations.
}
\medskip\noindent {\sf{\underline{Proof}:\ }}
${\cal E}_x$ acts on the module $\mathbb C$ generated by $1,$ we will show that if $v\in V[\lambda]$ then the action of $\pi\hat{\otimes} {\cal E}_x $ on $w=M(x)_V^{-1}v\otimes 1$ is
${\cal E}_{xq^\lambda}.$
Indeed
\begin{align*}
(\pi\hat{\otimes} {\cal E}_x)(\stackrel{\pi'}{U}{}^a_b)(w\otimes 1)
&=\pi((\stackrel{\pi'\;.}{R}{}^{J(-)-1}){}^{a}_m
(\stackrel{\pi'\;.}{R}{}^{J(+)}){}^{n}_b)w\otimes
\stackrel{\pi'}{P}(x){}^m_n\;1\\
&=(\pi'\otimes id)(R^{J(-)-1}_{12}P_1(x)R^{J(+)}_{12})^a_b.(w\otimes 1)
\\
&=(M_2(x)^{-1}P_1(xq^{h_2})M_2(x))^{a}_b M_2(x)^{-1} (v\otimes 1)\\
&=(M_2(x)^{-1}P_1(xq^{\lambda}))^{a}_b(v\otimes 1)
=P(xq^{\lambda})^{a}_b (w\otimes 1).
\end{align*}
The proposition follows.
\qed
\subsection{A representation framework for weak solution of dynamical coBoundary elements }
In this subsection we give hints towards a purely representation theoretical approach of QDBE. Because we already have obtained an explicit universal solution of this problem in its strongest formulation, the aim of this subsection is not to give completely rigorous reconstruction of $M(x).$ We would like to emphasize the puzzling fact that $M(x)$ and $F(x)$ can be constructed solely with interwining operators involving primitive representations of loop algebras and finite dimensional representations of $U_q(\mathfrak g)$ whereas in the work of \cite{EV} $F(x)$ is built using intertwining operators between Verma modules and finite dimensional representations of $U_q(\mathfrak g).$
${\cal L}_q(sl(n+1),J^{\tau})$ can be presented in term of matrix
elements of $\stackrel{f}{U}.$
We use the results of \cite{GPS} and denote ${\check{\bf R}}^{J}=
{\bf R}^{J}_{12}P_{12},$ where $P:V_f^{\otimes 2}\rightarrow V_f^{\otimes 2}$ is the permutation operator.
Because $\check{{\bf R}}$ satisfies the Hecke symmetry it is also true for ${\check{\bf R}}^{J}.$
As a result all the constructions of \cite{GPS} concerning the $Tr_q$ and $det_q$ can be applied.
We denote $A^{(k)}$ the antisymmetrizer associated to ${\check{\bf R}}$ acting on $V_f^{\otimes k},$ it is denoted $P_{-}^k$ in \cite{GPS}.
Similarly we can define $A_J^{(k)}$ the antisymmetrizer associated to ${\check{\bf R}}^J$ acting on $V_f^{\otimes k}.$
${\check{\bf R}}$ and ${\check{\bf R}}^J$ are both Hecke symmetry of rank $n+1$, i.e
$A^{(n+2)}=A_J^{(n+2)}=0.$
${\cal L}_q(sl(n+1),J^{\tau})$ is isomorphic to the algebra
$A(sl(n+1),J)$ generated by the matrix elements of
$U\in End(V_f)\otimes A(sl(n+1),J)$ with relations
\begin{align}
&{ \bf R}^{J}_{21} U_1 {\bf R}^{J}_{12} U_2
=U_2 {\bf R}^{J}_{21} U_1 {\bf R}^{J}_{12}\\
&det_q(U)=det_q({\bf P}(x))
\end{align}
where
$det_q(U)$ is the central element defined by
\begin{equation}
det_q(U)=tr_{1\cdots n+1}(A_{J}^{(n+1)}(U_1
{\check{\bf R}}^J_{12}{\check{\bf R}}^J_{23}... {\check{\bf R}}^J_{n,n+1})^{n+1}).
\end{equation}
The isomorphism is obtained by identifying $\stackrel{f}{U}$ and $U.$
\\
Note that $det_q({\bf P}(x))=q^{-n(n+1)}.$
Let ${\bf M}(x)$ be a matrix such that (\ref{RMM-fund}) is satisfied and define
${\bf P}(x)=v{\bf M}(x)^{-1}{\bf B}(x){\bf M}(x).$
By construction we have
\begin{align}
{ \bf R}^{J}_{21} {\bf P}(x)_1 {\bf R}^{J}_{12} {\bf P}(x)_2
&= {\bf P}(x)_2{\bf R}^{J}_{21} {\bf P}(x)_1 {\bf R}^{J}_{12},\\
{ \bf R}^{J}_{21} {\bf P}(x)_1 {\bf R}^{J}_{12}
&= {\bf M}(x)^{-1}_2 {\bf P}(xq^{h_2})_1 {\bf M}(x)_2.\label{PM}
\end{align}
As a result we still define the primitive representation ${\cal E}$ of $ A(sl(n+1),J)$ as the representation of $ A(sl(n+1),J)$ with values in ${\mathbb C}[\nu_1,\nu_1^{-1},\ldots,\nu_r,\nu_r^{-1}] $
defined by
\begin{equation}
(id \otimes {\cal E}_x)(U)={\bf P}(x).
\end{equation}
Let $(V,\pi)$ be any finite dimensional representation of $U_q(sl(n+1))$, we want to show the following decomposition property:
\begin{equation}
\pi\hat{\otimes} {\cal E}_x=\bigoplus_{\lambda\in {\mathfrak h}^* }
{\cal E}_{xq^\lambda}^{\oplus m_\lambda}\;\;\; \text{where}\;\;
m_\lambda=dim V[\lambda]\label{reduc}
\end{equation}
and will denote ${\cal M}_V(x)$ any intertwiner between these representations.
As a consequence of (\ref{PM}), the same proof as Proposition \ref{PropIntMapCobound} shows that
\begin{equation}
\pi_f\hat{\otimes} {\cal E}_x=\bigoplus_{\lambda\in {\mathfrak h}^* }
{\cal E}_{xq^\lambda}^{\oplus m_\lambda}\;\;\; \text{where}\;\;
m_\lambda=dim V_f[\lambda].
\end{equation}
Hence (\ref{reduc}) holds also for $\pi_f^{\otimes p} \circ ( \Delta^J)^{(p)},$ i.e
\begin{equation}(\pi_f)^{\otimes_{_{J}} p}\hat{\otimes} {\cal E}_x=\bigoplus_{\lambda\in {\mathfrak h}^* }
{\cal E}_{xq^\lambda}^{\oplus m_\lambda}\text{where}\;\;
m_\lambda=dim V_f^{\otimes_{_{J}} p}[\lambda].
\end{equation}
Let $H_p(q)$ be the $A_p$-Hecke algebra generated by $\sigma_1,...,\sigma_{p-1}.$
If $W$ is an irreducible submodule of $\pi_f^{\otimes p}$ there exists an idempotent element
${\cal Y}_W={\cal Y}_W(\sigma_1,...,\sigma_{p-1})\in H_p(q)$ such that
${\cal Y}_W(\check{\bf R}_{12},...,\check{\bf R}_{p-1,p})$ is the projector on the submodule $W.$
As a result if $W$ is the submodule of $\pi_f^{\otimes_{_{J}} p}$ the associated projector is
$J_{1,...,p}^{-1}{\cal Y}_WJ_{1,...,p}=
{\cal Y}_W(\check{\bf R}^J_{12},...,\check{\bf R}^J_{p-1,p})={\cal Y}_{W}^J$ where
$(\Delta^J)^{(p)}(a)=J_{1,...,p}^{-1}\Delta^{(p)}(a)J_{1,...,p}.$
It remains to show that
\begin{equation}({\cal Y}_{W}^J (\pi_f)^{\otimes_{_{J}} p})\hat{\otimes} {\cal E}_x=\bigoplus_{\lambda\in {\mathfrak h}^* }
{\cal E}_{xq^\lambda}^{\oplus m_\lambda}\text{where}\;\;
m_\lambda=dim({\cal Y}_{W}^J V_f^{\otimes_{_{J}} p})[\lambda].
\end{equation}
It is straightforward to verify, using (\ref{RMM-fund}), the following properties:
\begin{eqnarray}
{\cal Y}_{W}^J {\bf M}_{1,..,p}(x)^{-1}&=& {\bf M}_{1,..,p}(x)^{-1}{\cal Y}_{W}(\check{\bf R}_{12}(x),...,\check{\bf R}_{p-1,p}(x))\\
&=& {\bf M}_{1,..,p}(x)^{-1}F_{1,..,p}(x)^{-1}
{\cal Y}_W(\check{\bf R}_{12},...,\check{\bf R}_{p-1,p})F_{1,..,p}(x)
\end{eqnarray}
where \begin{eqnarray*} {\bf M}_{1,..,p}(x)&=& {\bf M}_1(xq^{h_1+\cdots +h_{p}})\cdots {\bf M}_{p-1}(xq^{h_p})
{\bf M}_p(x)\\
F_{1,..,p}(x)&=&{\bf F}_{(1...p-1,p)}(x){\bf F}_{(1...p-2,p-1)}(xq^{h_p})...{\bf F}_{1,2}(xq^{h_p+...+h_3}),
\end{eqnarray*}
with ${F}_{(1...k,k+1)}=(\Delta^{(k)}\otimes id)(F(x)),$
which concludes the proof. \\
Having defined ${\cal M}_V(x)$ for $V$ simple we straightforwardly define ${\cal M}_W(x)$ for $W$ semi-simple. Let $(\pi,V),(\pi',W)$ be two representations of $U_q(sl(n+1)).$ Using decomposition property \eqref{reduc} we obtain the property $$J^{-1}_{V,W}{\cal M}_{V\otimes W}(x)^{-1}\;(V\otimes_{{}_{J}}W)[\lambda]=({\cal M}_{V}(xq^{h_W}){\cal M}_{W}(x))^{-1}\;(V\otimes_{{}_{J}}W)[\lambda].$$ As a result, ${\cal F}_{V,W}(x)$ defined by $${\cal F}_{V,W}(x)={\cal M}_{V\otimes W}(x)J_{V,W}({\cal M}_{V}(xq^{h_W}){\cal M}_{W}(x))^{-1}$$ is of zero weight. The family of intertwing maps ${\cal M}_V(x)$ defines therefore a solution of the WQDBP.
The previous framework shows that one can obtain a definition of
${\cal M}_V(x)$ in a purely representation theoretical setting and that
${\cal M}$ is a solution of the WQDBP.
We think that this method can be further pursued to obtain a purely representation theoretical approach to the SQDBP. As an example, we give in Appendix~\ref{sec-constrM} some remarks concerning this point in the rank one case, the higher rank case being still unclear for us.
\section{Conclusion}
\label{sectionConcl}
In this work we have given a universal explicit solution of the Quantum Dynamical coBoundary Equation. This was obtained through the use of the primitive loop, which study led us to this solution.
However many points are still unclear to us.
The first one concerns the Balog-D\c abrowski-Feh\'er Theorem. Although the
result is unquestionable, the proof seems unnatural. The fact that it selects precisely $sl(n+1)$ and the
Cremmer-Gervais 's $r$-matrix still remains unclear.
The second one comes from the possible various generalizations. We have studied here the QDBE in the case where $F(x)$ is the standard solution and $\mathfrak{h}$ is the Cartan sub-algebra, but we could imagine considering also the case when $F(x)$ is associated to a generalized Belavin-Drinfeld triple of the type considered by O. Schiffmann \cite{Sc}, or generalizing this equation to the non abelian case. These problems are still completely open.
As a third point, one may also wonder whether one can generalize the straightforward proof of
Appendix~\ref{sec-constrM}, presented here in the $U_q(sl(2))$ case, to higher rank.
Finally, we would like to mention that the coboundary equation originates in the IRF-Vertex transform \cite{Bax}, and all the tools are now present for the construction of a universal IRF-Vertex transform in the quantum affine case. This universal coboundary element will relate the face type twistors and the vertex type twistors of elliptic quantum algebras of \cite{JKOS}.
\newpage
\section{Appendix}
\subsection{The Balog-D\c abrowski-Feh\'er Theorem}
\label{sec-BDF}
We give here elements of the proof of Theorem \ref{lemme2} following the
arguments of \cite{BDF}.
\medskip
For a finite dimensional simple Lie algebra $\mathfrak{g}$,
let $R(x)$ be the universal standard solution of the QDYBE
and assume that there exist $M(x)$, $R^J$ such that the equation
(\ref{RMM=MMR}) holds.
We fix $x$ and expand each factor of \eqref{RMM=MMR} in term of $\hbar$,
with $q=e^{\frac{\hbar}{2}}$.
We have
$R(x)=1+\hbar\, r(x)+o(\hbar)$, where
\begin{equation}
r(x)=\frac{1}{2}\Omega_{\mathfrak h}+\sum_{\alpha\in \Phi}
r_{\alpha}(x)\, e_\alpha\otimes f_{\alpha}
\quad\text{with}\quad
r_{\alpha}(x)=\frac{(\alpha,\alpha)}{2}
\Big(1-\prod_j x_j^{2\alpha(\zeta^{\alpha_j})}\Big)^{-1}.
\end{equation}
As a result, $R^J=1+\hbar\, r_J+o(\hbar)$, and the linear term $r_J$ is given
through \eqref{RMM=MMR} in terms of $r(x)$ as
\begin{equation}
r_{J}=M_1(x)^{-1}M_2(x)^{-1}\Big(r(x)+\frac{1}{2}\sum_{j=1}^r A_j(x)\wedge h_{\alpha_j}\Big)\,M_1(x)\, M_2(x),
\end{equation}
where $A=A_i dx^i$ is a flat connection defined as
$A_i=x_i (\partial_{i}M)M^{-1}\in \mathfrak{g}$.
The condition $\partial_{i}r_{J}=0$ can then be expressed only in term of
$r(x)$ and of the connection $A$, and reads:
\begin{equation}\label{cond-rJ}
x_i\partial_i\Big(r(x)+\frac{1}{2}\sum_{j=1}^r A_j\wedge h_{\alpha_j}\Big)+
\Big[r(x)
+\frac{1}{2}\sum_{j=1}^r A_j\wedge h_{\alpha_j}, A_i\otimes 1+1\otimes A_i
\Big]=0.
\end{equation}
\bigskip
Balog, D\c abrowski and Feh\'er have shown that the set of flat connections
satisfying this equation is empty when $\mathfrak{g}$ does not belong to the
$A_n$ series.
In order to prove this result, we decompose $A$ on the root subspaces as
\begin{equation}\label{Aroot}
A_j=\sum_{i=1}^r A_j^i \, h_{\alpha_i}
+\sum_{\alpha\in \Phi}A_j^{\alpha} \, e_\alpha.
\end{equation}
The differential equations satisfied by $A$ are the flatness condition
$D_A A=0$ and the equation $D_A(r(x)+\frac{1}{2}\sum_{j=1}^r A_j\wedge h_j)=0$
(Eq. \eqref{cond-rJ}),
which give respectively, when projected on the root subspaces,
\begin{align}
&x_i\partial_i A_j^m -x_j\partial_j A_i^m
-\frac{2}{(\alpha,\alpha)}\sum_{\alpha\in \Phi}
A_i^\alpha A_j^{-\alpha} \alpha(\zeta^{\alpha_m})=0,
\label{flatness1}\displaybreak[0]\\
&x_i\partial_i A_j^\alpha-x_j\partial_j A_i^\alpha
-\sum_{\substack{\beta,\gamma \\ \beta+\gamma=\alpha}}
N_{\beta\gamma}^{\alpha}A_i^\beta A_j^\gamma
-A_j^\alpha\sum_{n}A_i^n \alpha(h_{\alpha_n})
+A_i^\alpha\sum_n A_j^n \alpha(h_{\alpha_n})=0,
\label{flatness2}\displaybreak[1]\\
&x_i\partial_i ( A_n^m - A_m^n )
+ \frac{2}{(\alpha,\alpha)}\sum_{\alpha\in \Phi} \big[
A_n^\alpha A_i^{-\alpha} \alpha(\zeta^{\alpha_m})
-A_m^\alpha A_i^{-\alpha} \alpha(\zeta^{\alpha_n}) \big] =0,
\label{DAR1}\displaybreak[0]\\
&\frac{1}{2} x_i \partial_i A_n^\alpha
+\Big(\frac{1}{2} -\frac{2}{(\alpha,\alpha)}
r_\alpha(x)\Big) A_i^\alpha \alpha(\zeta^{\alpha_n})
\nonumber\\
&\hspace{2cm}
+\sum_{\substack{\beta,\gamma \\ \beta+\gamma=\alpha}}
N_{\alpha\beta}^{\gamma}A_n^\beta A_i^\gamma
+\frac{1}{2}\sum_{m} (A_n^m A_i^\alpha -A_n^\alpha A_i^m -A_m^n A_i^\alpha)
\alpha(h_{\alpha_m})=0,
\label{DAR2}\displaybreak[0]\\
&x_i\partial_i r_{\alpha}
-\frac{1}{2}A_i^\alpha \sum_j A_j^{-\alpha} \alpha(h_{\alpha_j})
-\frac{1}{2}A_i^{-\alpha} \sum_j A_j^{\alpha} \alpha(h_{\alpha_j})=0,
\label{DAR3}\displaybreak[0]\\
&A_i^\alpha \sum_j A_j^\beta \alpha (h_{\alpha_j})
-A_i^\beta \sum_j A_j^\alpha \beta(h_{\alpha_j})
-2\Big(r_\alpha(x)-\frac{(\alpha,\alpha)}{(\beta,\beta)}r_{-\beta}(x)\!\Big)
N_{-\alpha,\alpha+\beta}^{\beta}A_{i}^{\alpha+\beta}=0,
\label{DAR4}
\end{align}
for $\alpha \ne -\beta$, where we have denoted
$[e_{\alpha}, e_{\beta}]=N_{\alpha,\beta}^{\alpha+\beta}e_{\alpha+\beta}.$
Combining equations (\ref{flatness1},\ref{DAR1}) on the one hand, and
equations (\ref{flatness2},\ref{DAR2}) on the other hand,
one obtains respectively the following linear
equations for $A_i^n$ and $A_i^\alpha$:
\begin{align}
&x_m\partial_m A^n_i-x_n\partial_n A^m_i=0,
\label{rotA=0}\\
&x_n\partial_n A_i^\alpha
+\alpha(\zeta^{\alpha_n})F_{\alpha}(x)A_i^\alpha
-A_i^\alpha\sum_m A_m^n\alpha(h_{\alpha_m})=0,
\label{diffeqAalpha}
\end{align}
where $F_\alpha(x)=-(1+\prod_j x_j^{2\alpha(\zeta^{\alpha_j})})
(1-\prod_j x_j^{2\alpha(\zeta^{\alpha_j})})^{-1}.$
The general solution of the equation (\ref{rotA=0}) is
\begin{equation}\label{A-der-phi}
A_i^n=x_n\partial_n \phi_i,
\end{equation}
where $\phi_i$ are arbitrary functions.
The general solution of the equation (\ref{diffeqAalpha}) is then
\begin{equation}\label{A-alpha}
A_i^\alpha=C_i^\alpha\, (1+F_\alpha)\,\prod_{j}x_j^{-\alpha(\zeta^{\alpha_j})}
\exp\Big(\sum_m \phi_m \alpha(h_{\alpha_m})\Big),
\end{equation}
in terms of some (so far arbitrary) constants $C_i^{\alpha}$.
Let us define the weight $C^{\alpha}$ by $C^{\alpha}=\sum_i C_i^{\alpha} \alpha_i^{\vee},$
the equations (\ref{DAR3},\ref{DAR4}) become algebraic equations:
\begin{align}
&\frac{(\alpha,\alpha)}{2}\alpha +C^\alpha
(C^{-\alpha},\alpha)+C^{-\alpha}(C^\alpha, \alpha)=0,\label{DAR3'}\\
&C^\alpha (C^\beta, \alpha)-C^\beta (C^\alpha, \beta)
+C^{\alpha+\beta}N^{\alpha,\beta}_{\alpha+\beta}=0,\label{DAR4'}
\end{align}
where we have denoted $N^{\alpha,\beta}_{\alpha+\beta}
=\frac{(\alpha,\alpha)}{2}N^{\beta}_{-\alpha,\alpha+\beta}.$
The first equation is uniquely solved by decomposing $C^{\alpha}=
\frac{c(\alpha)}{2}(\alpha+K^\alpha)$ for $\alpha>0$ and $K^\alpha \perp \alpha.$
As a result we obtain that $C^{-\alpha}=\frac{1}{2c(\alpha)}(-\alpha+K^\alpha)$ for $\alpha>0.$
It remains to show that the set of equations (\ref{DAR4'}) rules out all the simple Lie algebras except the $A_n$ series.
Proving this property is simplified by the following observation:
if (\ref{DAR4'}) admits a solution $(C^\alpha)$ for a Lie algebra associated
to the Dynkin diagram $D$ labelling the simple roots $\alpha_1,\ldots,\alpha_r$,
and if $D'$ is a connected subdiagram of $D$ associated to the roots
$\alpha_{j},$ $j\in D'\subset D=\{1,\ldots, r\}$, then the orthogonal projection
of $(C^\alpha)$ on the vector space generated by $\alpha_{j}$, $j\in D'$,
is a solution of (\ref{DAR4'}) for the Lie algebra generated by $D'$.
As a result, one obtains that it is sufficient to show that the solution of (\ref{DAR4'}) is empty for $D_4$, $B_2$ and $G_2$ for ruling out all but the
$A_n$ series.
The next observation comes from the theorem that if $\alpha,\beta$ are roots
such that $\alpha+\beta$ and $\alpha-\beta$ are simultaneously non roots,
then $(\alpha,\beta)=0.$
As a result, in this case, by combining the equations (\ref{DAR4'}) for
$\alpha+\beta$ and $\alpha-\beta$, one obtains that $K^\alpha\perp \beta.$
As a result, we obtains that $K^\alpha\perp V_\alpha$ where
$V_{\alpha}=\mathbb{C}\alpha+\sum_{\beta\in \Phi,\alpha\pm \beta\notin \Phi}\mathbb{C}\beta.$
We trivially have $V_{w\alpha}=w V_{\alpha}$ where $w$ is any element of the Weyl group.
As a result, $\mathrm{codim} (V_{\alpha})$ only depends on the length of $\alpha.$
An elementary analysis ot the root system of $D_4$ and $G_2$ proves that
$\mathrm{codim} (V_{\alpha})=0$ for all roots.
Therefore $K_{\alpha}=0$, but the corresponding $C^{\alpha}$ is not a solution of (\ref{DAR4'}).
In the case of $B_2$ one obtains that $\mathrm{codim} (V_{\alpha})=0$ if
$\alpha$ is long and $\mathrm{codim} (V_{\alpha})=1$ if $\alpha$ is short.
The explicit study of the system (\ref{DAR4'}) shows that once again the set of solutions is empty.
This concludes the proof that the coboundary equation can admit solutions only
in the case where $\mathfrak{g}=sl(n+1)$.
\bigskip
We will now show that, in the case where $\mathfrak{g}=sl(n+1)$,
such a solution $r_J$ is unique up to an automorphism.
As we know that $r_{\tau,s}$, with $s$ defined as (\ref{sCG}), is
a solution of the coboundary equation in this case
(this is a direct consequence of Theorem~\ref{lemme1} in the fundamental
representation, and of Theorem~\ref{theo-Mexpr} at the universal level),
it means that, for any solution $r_J$ of the coboundary equation,
there exists an automorphism $\phi$ of $sl(n+1)$ such that
$r_J=(\phi\otimes \phi)(r_{\tau,s})$.
In order to prove this uniqueness property, let us caracterise completely all
the possible solutions for the connection $A$ in the $sl(n+1)$ case.
In this case, positive roots are of the form $u_i-u_j$, $i<j$, where
$u_1,\ldots,u_{n+1}$ are orthonormal vectors. From the previous
considerations, $K^{u_i-u_j}$ should be simultaneously orthogonal to
$u_i-u_j$ and to all $u_k-u_l$ such that $\{k,l\}\cap \{i,j\}=\emptyset$.
It is therefore of the form
\begin{equation*}
K^{u_i-u_j}=\epsilon_{ij}\Big(u_i+u_j-\frac{2}{n+1}\sum_{k=1}^{n+1}u_k\Big),
\end{equation*}
and it can be shown, using Eq.\eqref{DAR4'}, that all the constants
$\epsilon_{ij}$ are equal to some common value $\epsilon\in\{+1,-1\}$.
Furthermore, still
from Eq.\eqref{DAR4'}, the constants $c(\alpha)$ associated to positive roots
have to satisfy
\begin{equation*}
c(u_i-u_j)\, c(u_j-u_k)=\epsilon\, c(u_i-u_k),\quad \text{if\ } i<j<k,
\end{equation*}
which means that there exists some constants $c_1,\ldots,c_n$ such that,
for all positive roots $\alpha$, we have
$c(\alpha)=\epsilon\, \exp\big(\sum_m c_m \alpha(h_{\alpha_m})\big)$.
On the other hand, pluging \eqref{A-der-phi} and \eqref{A-alpha} into
\eqref{flatness1}, we obtain the following conditions on the functions
$\phi_i$:
\begin{equation*}
\phi_j=\frac{1}{2}\sum_{\alpha\in\Phi^+} K_j^\alpha
\log\Big(\prod_l x_l^{\alpha(\zeta^{\alpha_l})}
-\prod_l x_l^{-\alpha(\zeta^{\alpha_l})}\Big)
+x_j \partial_j\psi,
\end{equation*}
where $\psi$ is an arbitrary function. Note at this stage that we may as well
fix $c(\alpha)=\epsilon$ and absorb the arbitrariness of this constant in $\psi$.
Let us now prove that a modification of the function
$\psi$ does not affect $r_J$.
Let us denote by $A_{(\psi)}$, $M_{(\psi)}$, $(r_J)_{(\psi)}$
(respectively by $A_{(0)}$, $M_{(0)}$, $(r_J)_{(0)}$ )
the connection, the coboundary and the corresponding $r$-matrix associated
to a given $\psi$ (respectively to $\psi=0$).
We have:
\begin{align}
& M_{(\psi)}= M_{(0)}\cdot g_\psi,\\
& (A_{(\psi)})_j= x_j \partial_j g_\psi \cdot g_\psi^{-1}
+ g_\psi\cdot (A_{(0)})_j\cdot g_\psi^{-1},\label{Apsi}
\end{align}
where $g_\psi=\exp (\sum_m x_m \partial_m\psi \; h_{\alpha_m} )$.
Note that, due to the specific form of $g_\psi$, only the second term in
\eqref{Apsi} gives a non-zero contribution to
$(A_{(\psi)})_j\wedge h_{\alpha_j}$,
and therefore,
\begin{align}
(r_J)_{(\psi)}
& = (M_{(0)})_1^{-1} (M_{(0)})_2^{-1}
(g_\psi)_1^{-1} (g_\psi)_2^{-1}
\Big( r(x)
+\frac{1}{2}\sum_{j=1}^n (g_\psi \,(A_{(0)})_j\, g_\psi^{-1})
\wedge h_{\alpha_j}
\Big) \nonumber\\
&\hspace{7cm}\times
(g_\psi)_1\,(g_\psi)_2\, (M_{(0)})_1\, (M_{(0)})_2\nonumber\\
&= (M_{(0)})_1^{-1} (M_{(0)})_2^{-1}
\Big( r(x)+\frac{1}{2}\sum_{j=1}^n (A_{(0)})_j
\wedge h_{\alpha_j}
\Big)\, (M_{(0)})_1\, (M_{(0)})_2=(r_J)_{(0)}.
\end{align}
Finally, the only arbitrariness in $r_J$ is due to gauge transformations of
the form
\begin{align*}
&M(x) \rightarrow M(x)\cdot u,\\
&r_J \rightarrow (u\otimes u)^{-1} r_J \, (u\otimes u),
\quad u\in\mathfrak{g},
\end{align*}
and to the automorphism $\alpha\rightarrow -\alpha,\ \forall\alpha\in\Phi$,
corresponding to the change $\epsilon\rightarrow -\epsilon$.
Thus, the solution $r_J$ is unique up to an automorphism of $sl(n+1)$,
which concludes the proof.
\newpage
\subsection{Miscellaneous lemmas}
\label{sec-lemmas}
\Lemma{\label{appendlemma1}}{
Under the hypothesis of Theorem~\ref{cocycleBCG}, we have,
for all $p=1,\ldots,n-1$,
\begin{multline*}
\prod_{k=p}^{n-1}\left\{ W^{[k]}_{13}\, W^{[k-p+1]}_{23}\,
\widehat{J}{}^{[p-1,k-p+1]}_{1(2\mid 3} \,
\widehat{J}^{[k-p+1]}_{23} \right\}
\Big\{\prod_{k=p}^{n-1}J^{[k]}_{12}\Big\}
\\
=(id \otimes \Delta)(J^{[p]}) \;
\prod_{k=p+1}^{n-1}\Big\{ W^{[k]}_{13}\, W^{[k-p]}_{23} \,
\widehat{J}{}^{[p,k-p]}_{1 (2 \mid 3} \,
\widehat{J}^{[k-p]}_{23} \Big\}
\Big\{ \prod_{k=p+1}^{n-1} J^{[k]}_{12} \Big\}\, J^{[n-p]}_{23},
\end{multline*}
with $\widehat{J}{}^{[k,m]}_{1 (2 \mid 3}$ defined as in \eqref{defJ1}.
}
\medskip\noindent {\sf{\underline{Proof}:\ }}
Reorganising the factors in the product and using successively
\eqref{prop1}, \eqref{prop2},
\eqref{prop3}, \eqref{propJ3}, \eqref{prop11} and \eqref{propJ2}, we can
reexpress the left hand side as
\begin{align*}
LHS
&=W^{[p]}_{13} \left(W^{[1]}_{23}\,\widehat{J}{}^{[p-1,1]}_{1(2\mid 3}\,
(W^{[1]}_{23})^{-1} \right)\\
&\quad\times
\prod_{k=p+1}^{n-1}\Big\{ J^{[k-p]}_{23}\, W^{[k]}_{13}\,
\Big( W^{[k-p+1]}_{23}\,\widehat{J}{}^{[p-1,k-p+1]}_{1(2\mid 3}\,
(W^{[k-p+1]}_{23})^{-1} \Big) \Big\}\, J^{[n-p]}_{23}\,
\Big\{ \prod_{k=p}^{n-1}J^{[k]}_{12} \Big\},
\\
&=W^{[p]}_{13} \left(W^{[1]}_{23}\,\widehat{J}{}^{[p-1,1]}_{1(2\mid 3}\,
(W^{[1]}_{23})^{-1} \right)\,
\prod_{k=p+1}^{n-1}\Big\{ J^{[k-p]}_{23}\, W^{[k]}_{13}\, \\
&\quad\times
\Big( W^{[k-p+1]}_{23}\,\widehat{J}{}^{[p-1,k-p+1]}_{1(2\mid 3}\,
(W^{[k-p+1]}_{23})^{-1} \Big) \Big\}\, J^{[p]}_{12} \,
\Big\{ \prod_{k=p+1}^{n-1}J^{[k]}_{12} \Big\}\, J^{[n-p]}_{23},
\\
&=W^{[p]}_{13}\,W^{[p]}_{12} \left(
(W^{[p]}_{12})^{-1}\,W^{[1]}_{23}\,\widehat{J}{}^{[p-1,1]}_{1(2\mid 3}\,
(W^{[1]}_{23})^{-1} W^{[p]}_{12} \right)\,
\prod_{k=p+1}^{n-1} \Big\{(W^{[p]}_{12})^{-1} J^{[k-p]}_{23}\,
W^{[p]}_{12}\,W^{[k]}_{13} \\
&\quad\times
\left( W^{[k-p+1]}_{23}\,(W^{[p]}_{12})^{-1}
\widehat{J}{}^{[p-1,k-p+1]}_{1(2\mid 3}\, W^{[p]}_{12}\,
(W^{[k-p+1]}_{23})^{-1} \right) \Big\}\,
\widehat{J}{}^{[p]}_{12} \,
\Big\{ \prod_{k=p+1}^{n-1}J^{[k]}_{12} \Big\}\, J^{[n-p]}_{23},
\\
&=W^{[p]}_{13}\, W^{[p]}_{12} \left((W^{[p]}_{12})^{-1} W^{[1]}_{23}\,
\widehat{J}{}^{[p-1,1]}_{1(2\mid 3}\, (W^{[1]}_{23})^{-1} W^{[p]}_{12}
\right)\\
&\quad\times
\prod_{k=p+1}^{n-1}\left\{ W^{[k]}_{13} \,J^{[k-p]}_{23}
\left( (W^{[p]}_{12})^{-1} W^{[k-p+1]}_{23}\,
\widehat{J}{}^{[p-1,k-p+1]}_{1(2\mid 3}\, (W^{[k-p+1]}_{23})^{-1}
W^{[p]}_{12} \right) \right\}\,\\
&\quad\times \widehat{J}{}^{[p]}_{12}\,
\Big\{ \prod_{k=p+1}^{n-1}J^{[k]}_{12} \Big\}\, J^{[n-p]}_{23},
\\
&=W^{[p]}_{13}\,W^{[p]}_{12}\, \widehat{J}{}^{[p,0]}_{1 \mid 2) 3} \,
\prod_{k=p+1}^{n-1}\left\{ W^{[k]}_{13}\,W^{[ k-p]}_{23}\,
\widehat{J}{}^{[k-p]}_{23}\, \widehat{J}{}^{[p,k-p]}_{1 \mid 2) 3} \right\}
\, \widehat{J}{}^{[p]}_{12}\,
\Big\{ \prod_{k=p+1}^{n-1} J^{[k]}_{12} \Big\} \,J^{[n-p]}_{23},
\\
&=W^{[p]}_{13}\, W^{[p]}_{12}\, \widehat{J}{}^{[p,0]}_{1 \mid 2) 3} \,
\widehat{J}{}^{[p]}_{12}
\prod_{k=p+1}^{n-1}\left\{ W^{[k]}_{13}\, W^{[k-p]}_{23} \,
\widehat{J}{}^{[p,k-p]}_{1 ( 2 \mid 3}\, \widehat{J}{}^{[k-p]}_{23}\right\}
\Big\{ \prod_{k=p+1}^{n-1} J^{[k]}_{12} \Big\} \, J^{[n-p]}_{23},
\\
&=RHS,
\end{align*}
which concludes the proof.
\qed
\bigskip
\Lemma{\label{appendlemma2}}{
With the hypothesis of Theorem \ref{maintheorem}, we have
\begin{align}
U_{12}(x)=S^{[1]}_{12}\prod_{k=1}^{n}\left(
{\mathfrak C}^{[+k]}_1(xq^{h_2})\,
(S^{[k+1]}_{12})^{-1}\,\widehat{J}{}^{[k]}_{12}\,S^{[k+1]}_{12} \right) ,
\end{align}
where $U_{12}(x)=\Delta\big(\mathcal{M}^{(+)}(x)\big)\, J\,
\mathcal{M}^{(+)}_2(x){}^{-1}$.
}
\medskip\noindent {\sf{\underline{Proof}:\ }}
Let us first mention some useful relations which can be derived from the properties of $\tau$ and $S^{[k]}$:
\begin{align}
&\Delta({\mathfrak C}^{[+k]}(x))=S^{[1]}_{21}\,K_{12}\;
{\mathfrak C}^{[+k]}_1(x)\;(S^{[1]}_{21})^{-1} K^{-1}_{12}\,
S^{[1]}_{12}\;{\mathfrak C}^{[+k]}_2(x)\;(S^{[1]}_{12})^{-1},
\label{Delta-plus-k}\\
&(S^{[1]}_{12})^{-1} K_{12}\, S^{[1]}_{21}
=q^{2\zeta^{(n)}\otimes \zeta^{(n)}}
(\tau \otimes id)\big( (S^{[1]}_{12})^{-1} K^{-1}_{12} S^{[1]}_{21}\big),
\label{tau-cart}\\
&(\tau^p\otimes id)(R)=q^{\zeta^{(n-p+1)}\otimes \zeta^{(n)}}\,
(S^{[p]})^{2} \,(S^{[p+1]})^{-1} (S^{[p-1]})^{-1}
\widehat{J}{}^{[p]},\quad\forall p\ge 1,
\label{tauR}\\
&{\mathfrak C}^{[+k]}_1(xq^{h_2})
=(\tau^{k-1} \otimes id)
\big(K_{12}\,(S^{[1]}_{12})^{-1} S^{[1]}_{21}\,
{\mathfrak C}^{[+]}_1(x)\, K_{12}^{-1} S^{[1]}_{12}\,
(S^{[1]}_{21})^{-1}\big)
\nonumber\\
&\phantom{{\mathfrak C}^{[+k]}_1(xq^{h_2})}
=(S_{12}^{[k]})^{-2} (S_{12}^{[k-1]})^{2} \;
{\mathfrak C}^{[+k]}_1(x)\;
(S_{12}^{[k]})^2 \,(S_{12}^{[k-1]})^{-2},\quad\forall k\geq 2.
\label{cpluskshift}
\end{align}
Using \eqref{Delta-plus-k}, reorganising the factors in the product and using
\eqref{tau-cart}, we can rewrite $\Delta\big(\mathcal{M}^{(+)}(x)\big)$
as
\begin{align}
\Delta\big(\mathcal{M}^{(+)}(x)\big)
&=\prod_{k=1}^n \Delta\big(\mathfrak{C}^{[+k]}(x)\big),\nonumber\\
&= K_{12}\, S_{21}^{[1]}\;\mathfrak{C}^{[+]}_1(x)\; (S^{[0]})^{-1}\,
q^{-\zeta^{(n)}\otimes\zeta^{(n)}}
\nonumber\\
&\hspace{2cm}\times (\tau\otimes id)\bigg(
\prod_{k=1}^n \Delta'\big(\mathfrak{C}^{[+k]}(x)\big)\bigg)\
(S^{[0]})^{-1} S_{12}^{[1]}\, q^{\zeta^{(n)}\otimes\zeta^{(n)}}.
\label{prod1}
\end{align}
On the other hand, from \eqref{tauR}, $J$ can be expressed as
\begin{align}
J &= \prod_{p=1}^n \big\{ S^{[p]}\,(S^{[p+1]})^{-1}\widehat{J}{}^{[p]}\big\},
\nonumber\\
&= q^{-\zeta^{(n)}\otimes\zeta^{(n)}} S^{[0]}\, (S^{[1]})^{-1}\;
\prod_{p=1}^{n-1} \Big\{ (\tau^p\otimes id)(R)\;
q^{-\zeta^{(n-p)}\otimes\zeta^{(n)}}\, S^{[p]}\, (S^{[p+1]})^{-1}\Big\}
.
\label{prod2}
\end{align}
Therefore,
\begin{multline}
\Delta\big(\mathcal{M}^{(+)}(x)\big)\, J
=
S^{[1]}\,\mathfrak{C}^{[+]}_1(x q^{h_2})\, S^{[0]}\,(S^{[1]})^{-2}\,
q^{-\zeta^{(n)}\otimes\zeta^{(n)}}
\ (\tau\otimes id)\bigg(
\prod_{k=1}^n \Delta'\big(\mathfrak{C}^{[+k]}(x)\big)\bigg)\\
\times \prod_{p=1}^{n-1} \Big\{ (\tau^p\otimes id)(R) \
q^{-\zeta^{(n-p)}\otimes\zeta^{(n)}}\, S^{[p]}\, (S^{[p+1]})^{-1}\Big\}
,
\label{prod3}
\end{multline}
where we have also used \eqref{axiomABRR1} to reexpress the first factor.
To reorganise factors in the product, we use recursively the following relation, derived from the quasitriangularity
property \eqref{quasitriangularity2}:
\begin{align*}
(\tau^p\otimes id)\bigg(
\prod_{k=1}^n \Delta'\big(\mathfrak{C}^{[+k]}(x)\big)\bigg)\
(\tau^p\otimes id)(R)
&= (\tau^p\otimes id)(R)\ (\tau^p\otimes id)\bigg(
\prod_{k=1}^n \Delta\big(\mathfrak{C}^{[+k]}(x)\big)\bigg),
\end{align*}
which, using \eqref{prod1} and \eqref{tauR}, can be rewritten as,
\begin{multline}
(\tau^p\otimes id)\bigg(
\prod_{k=1}^n \Delta'\big(\mathfrak{C}^{[+k]}(x)\big)\bigg)\
\Big\{(\tau^p\otimes id)(R)\ q^{\zeta^{(n-p)}\otimes\zeta^{(n)}}\,
S^{[p]}\, (S^{[p+1]})^{-1}\Big\}\\
= \Big\{q^{\zeta^{(n-p+1)}\otimes \zeta^{(n)}}\,
(S^{[p]})^{2} \,(S^{[p+1]})^{-1} (S^{[p-1]})^{-1}
\widehat{J}{}^{[p]}\, (S^{[p]})^{2}\, (S^{[p+1]})^{-1} \\
\times
\mathfrak{C}^{[+(p+1)]}_1(x)\;
(S^{[p]})^{-1}\, q^{-\zeta^{(n-p)}\otimes\zeta^{(n)}}\Big\}\
(\tau^{p+1}\otimes id)\bigg(
\prod_{k=1}^n \Delta'\big(\mathfrak{C}^{[+k]}(x)\big)\bigg).
\label{prod4}
\end{multline}
This gives us
\begin{align}
\Delta\big(\mathcal{M}^{(+)}(x)\big)\, J
&= S^{[1]}\,\mathfrak{C}^{[+]}_1(x\, q^{h_2})\,(S^{[2]})^{-1}
\widehat{J}{}^{[1]}\nonumber\\
&\quad\times
\prod_{p=2}^n\Big\{(S^{[p-1]})^{2}\, (S^{[p]})^{-1}
\mathfrak{C}^{[+p]}_1(x)\;(S^{[p]})^{2}\,(S^{[p-1]})^{-2} (S^{[p+1]})^{-1}
\widehat{J}{}^{[p]}\Big\}\nonumber\\
&\quad\times
(\tau^{n}\otimes id)\bigg(
\prod_{k=1}^n \Delta'\big(\mathfrak{C}^{[+k]}(x)\big)\bigg)\
,
\nonumber\\
&=\prod_{p=1}^n\Big\{S^{[p]}\,\mathfrak{C}^{[+p]}_1(x\, q^{h_2})\,
(S^{[p+1]})^{-1}\widehat{J}{}^{[p]}\Big\}\
\prod_{k=1}^n \mathfrak{C}^{[+k]}_2(x),
\label{prod5}
\end{align}
where we have used \eqref{cpluskshift} and the nilpotency of $\tau$.
We finally multiply this expression by $\mathcal{M}_2^{(+)}(x)^{-1}$,
and this concludes the proof.
\qed
\bigskip
\Lemma{\label{appendlemma3}}{
Under the hypothesis of Theorem~\ref{theo-Mexpr}, we have
\begin{align*}
(\stackrel{f}{\pi}\otimes \stackrel{f}{\pi})
({\cal W}_{12}-\widetilde{\cal W}_{12})(x)=0,
\end{align*}
with
\begin{align*}
&{\cal W}_{12}(x)= {\mathfrak C}^{[+]}_1(xq^{h_2})\,
\{B_{2}(x)\,(S^{[2]}_{12})^{-1}\widehat{J}^{[1]}_{12}\,
S^{[2]}_{12}\,B_{2}(x)^{-1}\}\,
{\mathfrak C}_2^{[-]}(x)^{-1}, \\
&\widetilde{\cal W}_{12}(x)=
{\mathfrak C}^{[-]}_2(x)^{-1}
\{(S^{[1]}_{12})^{-1}\widehat{R}_{12}\,S^{[1]}_{12}\}\,
{\mathfrak C}_1^{[+]}(xq^{h_2}).
\end{align*}
}
\medskip\noindent {\sf{\underline{Proof}:\ }}
A direct computation leads to
\begin{align*}
&(\stackrel{f}{\pi}\otimes \stackrel{f}{\pi})
((S^{[1]}_{12})^{-1} \widehat{R}_{12}\,S^{[1]}_{12})
= 1 \otimes 1+ (1-q^{-2})\sum_{i<j}q^{\frac{2(i-j)}{(n+1)}}\
E_{i,j}\otimes E_{j,i},\\
&(\stackrel{f}{\pi}\otimes \stackrel{f}{\pi})
(B_{2}(x)\,(S^{[2]}_{12})^{-1} \widehat{J}^{[1]}_{12}\,S^{[2]}_{12}\,
B_{2}(x)^{-1})
= 1 \otimes 1\\
&\hspace{6.5cm}+ (1-q^{-2})\sum_{i<j}q^{\frac{3(j-i)}{(n+1)}}\,
\frac{\nu_{j+1}}{\nu_{i+1}}\ E_{i,j}\otimes E_{j+1,i+1},\\
&(\stackrel{f}{\pi}\otimes \stackrel{f}{\pi})
({\mathfrak C}^{[-]}_2(x)^{-1})
=1 \otimes1+\sum_{k=1}^{n} q^{\frac{k-1}{n+1}}\, \nu_{k+1}
\ 1 \otimes E_{k+1,k},\\
&(\stackrel{f}{\pi}\otimes \stackrel{f}{\pi})
({\mathfrak C}^{[+]}_1(xq^{h_2}))=1 \otimes1+\sum_{i<j}\sum_k (-1)^{i-j}
q^{-\frac{(j-i)(i+j-7)}{2(n+1)}-2 \delta_{i<k \leq j}}\,
\prod_{l=i+1}^j\nu^{-1}_{l}\ E_{i,j}\otimes E_{kk}.
\end{align*}
Using these intermediary results, it is then
straightforward to check the announced result.
\qed
\subsection{A shortcut construction of $M(x)$ in the $U_q(sl(2))$ case }
\label{sec-constrM}
A simpler alternative construction of $M(x)$ solution of the SQDBP can be done in the rank one case. This suggests a deeper relation between Primitive Representations of reflection algebras and Dynamical coboundaries but we unfortunately have not been able to generalize the present method for $U_q(sl(n+1))$ with $n\geq 2.$ The proof is however so simple that we have not resisted to include it here.
\Lemma{}{
Let us define a map $G:\mathbb{C}\rightarrow U_q(sl(2))^{\otimes 2}$
by
$G_{12}(x)=B_1(x)^{\frac{1}{2}}R_{12}(x)K_{12}^{-1}B_1(x)^{-\frac{1}{2}},$ we have the following
dynamical quasitriangularity property
\begin{equation}
(\Delta\otimes id)(G(x))=F_{12}(x)G_{13}(xq^{h_2})G_{23}(x)F_{12}(xq^{h_3})^{-1}.
\end{equation}
Moreover $G(x)$ is an element of ${\cal L}_q(sl(2))^{\otimes 2}.$ As a result we have also $F_{12}(xq^{h_3}) \in {\cal L}_q(sl(2))^{\otimes 3},$ and $(id \otimes id \otimes {\cal E})(F_{12}(xq^{h_3}))=1 \otimes 1$ for any primitive representation ${\cal E}$ of ${\cal L}_q(sl(2)).$
}
\Proof{It is straightforward, using the quasitriangularity property (\ref{quasitriangularity}) for $R$, the dynamical cocycle equation (\ref{eq:s-cocycle}) and the zero weight property of $F(x),$ to obtain that $G(x)$ satisfies the dynamical quasitriangularity equation.
{}From the explicit expression of $R(x)$ and from the isomorphism between
$U_q(sl(2))$ and $\hat{{\cal L}}_q(sl(2))$ we have the property
$G(x)\in {\cal L}_q(sl(2))^{\otimes 2}.$ (This is the step we are not able to generalize in the higher rank case).
The other properties are trivial.
}
\Proposition{}{Due to the previous lemmas, for any primitive representation
${\cal E}$ of ${\cal L}_q(sl(2)),$ it makes sense to
define the following map from ${\mathbb C}$ to $ {\cal L}_q(sl(2))$ by
\begin{equation}
M^{({\cal E})}(x)=(id \otimes {\cal E})(G(x)).\label{M=EG}
\end{equation}
For any ${\cal E},$ $M^{({\cal E})}(x)$ verifies the QDBE. Its explicit expression is given by (\ref{MdeXsl2}) up to trivial gauge transformations.
}
\Proof{
Using previous lemmas and the fact that ${\cal E}$ is a morphism, we have
\begin{align*}
\Delta(M^{({\cal E})}(x))&=(\Delta \otimes {\cal E})(G(x))=(id \otimes id \otimes {\cal E})
\left( F_{12}(x)G_{13}(xq^{h_2})G_{23}(x)F_{12}(xq^{h_3})^{-1} \right)\\
&=F_{12}(x)M^{({\cal E})}_{1}(xq^{h_2})M^{({\cal E})}_{2}(x)
\end{align*}
which concludes the proof of the dynamical coboundary equation. The explicit expression (\ref{MdeXsl2})
is recovered from (\ref{M=EG}) by a trivial computation.
}
\newpage
|
math/0512038
|
\section{Introduction}
Let $M$ be a {\it hyperbolic\/} 3-manifold, by which we shall mean a
compact, connected, orientable 3-manifold such that $M$ with its
boundary tori removed admits a complete hyperbolic structure with
totally geodesic boundary, and suppose that $M$ has a torus boundary
component $T_0$. If $r$ is a slope on $T_0$ then $M(r)$ will denote
the 3-manifold obtained by $r$-Dehn filling on $M$, i.e.\ attaching a
solid torus $V_r$ to $M$ along $T_0$ in such a way that $r$ bounds a
disk in $V_r$. The Dehn filling $M(r)$ and the slope $r$ are said to
be {\it exceptional\/} if $M(r)$ is either reducible,
$\partial$-reducible, annular, toroidal, or a small Seifert fiber space.
Modulo the Geometrization Conjecture, the manifold $M(r)$ is
hyperbolic if and only if $M(r)$ is not exceptional.
Thurston's Hyperbolic Dehn Surgery Theorem asserts that there are only
finitely many exceptional Dehn fillings on each torus boundary
component of $M$. It is known that if $r,s$ are both exceptional then
the geometric intersection number $\Delta = \Delta(r,s)$, also known
as the {\it distance\/} between $r$ and $s$, is small. In fact, the
least upper bounds for $\Delta$ have been determined for all cases
where neither $M(r)$ nor $M(s)$ is a small Seifert fiber space, by the
work of many people. See [GW2] and the references therein.
For toroidal fillings, it was shown by Gordon [Go] that if $r,s$ are
toroidal slopes then $\Delta(r,s) \leq 8$, and moreover there are
exactly two manifolds $M$ with $\Delta=8$, one with $\Delta=7$, and
one with $\Delta=6$. In this paper we classify all the hyperbolic
3-manifolds which admit two toroidal Dehn fillings with $\Delta=4$ or
$5$.
Already when $\Delta=5$ there are infinitely many such manifolds. To
see this, let $M$ be the exterior of the Whitehead sister link, also
known as the $(-2,3,8)$-pretzel link. The boundary of $M$ consists of
two tori $T_0$ and $T_1$, and there are slopes $r,s$ on $T_0$ with
$\Delta(r,s) = 5$ such that the Dehn filled manifolds $M(r) = M(r,*)$,
$M(s) = M(s,*)$ are toroidal; see for example [GW3]. Now for
infinitely many slopes $t$ on $T_1$, $M_t =M(*,t)$ will be hyperbolic
and $M_t(r) = M(r)(t)$, $M_t(s) = M(s)(t)$ will be toroidal. In this
way we get infinitely many hyperbolic 3-manifolds with boundary a
single torus having two toroidal fillings at distance $5$. We shall
show that, modulo this phenomenon, there are only finitely many $M$
with two toroidal fillings at distance $4$ or $5$, and explicitly
identify them. Define two triples $(N_1, r_1, s_1)$ and $(N_2, r_2,
s_2)$ to be {\it equivalent}, denoted by $(N_1, r_1, s_1) \cong (N_2,
r_2, s_2)$, if there is a homeomorphism from $N_1$ to $N_2$ which
sends the boundary slopes $(r_1, s_1)$ to $(r_2, s_2)$ or $(s_2,
r_2)$.
\begin{thm} There exist $14$ $3$-manifolds $M_i$, $1\leq i\leq 14$,
such that
(1) $M_i$ is hyperbolic, $1\leq i\leq 14$;
(2) $\partial M_i$ consists of two tori $T_0, T_1$ if $i \in
\{1,2,3,14\}$, and a single torus $T_0$ otherwise;
(3) there are slopes $r_i, s_i$ on the boundary component $T_0$ of
$M_i$ such that $M(r_i)$ and $M(s_i)$ are toroidal, where
$\Delta(r_i,s_i) = 4$ if $i \in \{1,2,4,6,9,13,14\}$, and
$\Delta(r_i,s_i) = 5$ if $i \in \{3,5,7,8,10,11,12\}$;
(4) if $M$ is a hyperbolic 3-manifold with toroidal Dehn fillings
$M(r), M(s)$ where $\Delta(r,s) = 4$ or $5$, then $(M,r,s)$ is
equivalent either to $(M_i,r_i,s_i)$ for some $1\leq i \leq 14$, or to
$(M_i(t),r_i,s_i)$ where $i \in \{1,2,3,14\}$ and $t$ is a slope on
the boundary component $T_1$ of $M_i$.
\end{thm}
\noindent {\bf Proof.} The manifolds $M_i$ are defined in Definition 21.3. (1) is
Theorem 23.14. (2) follows from the definition. (3) and (4) follow
from Theorem 21.4. \quad $\Box$ \medskip
\begin{remark}
Part (4) in Theorem 1.1 is still true if
the hyperbolicity is replaced by the assumption that $M$ is compact,
connected, orientable, irreducible, atoroidal, and non Seifert
fibered, in other words, $M$ may be annular or $\partial$-reducible but
not Seifert fibered.
\end{remark}
\noindent {\bf Proof.} First assume $M$ is $\partial$-irreducible. Then any essential
annulus must have at least one boundary component on a non-toroidal
boundary component of $M$ as otherwise $M$ would be either toroidal or
Seifert fibered. Attaching a hyperbolic manifold $X$ to each
non-toroidal boundary component of $M$ will produce a hyperbolic
manifold $M'$, and we may choose $X$ so that $M'$ has more than three
boundary components. One can show that the $M'(r_i)$ are still
toroidal for $i=1,2$, which is a contradiction to Theorem 21.4. Now
assume $M$ is $\partial$-reducible. Then $M$ is obtained by attaching
1-handles to the boundary of a manifold $M''$. If a 1-handle is
attached to a toroidal component $T$ of $\partial M''$ then either $M'' =
T \times I$, which is impossible because $M(r_i)$ would be a
handlebody and hence atoroidal, or $T$ would be an essential torus in
$M$, contradicting the assumption. It follows that $M''$ has a higher
genus boundary component. One can check that the $M''(r_i)$ are still
toroidal, which leads to a contradiction to Theorem 21.4 as above.
\quad $\Box$ \medskip
The manifolds $M_1, M_2$ and $M_3$ were discussed in [GW1]; $M_i$,
$i=1,2,3$ is the exterior of a link $L_i$ in $S^3$, where $L_1$ is the
Whitehead link, $L_2$ is the $2$-bridge link associated to the
rational number $3/10$, and $L_3$ is the Whitehead sister link. See
Figure 24.1. The other $M_i$ can be built using intersection graphs
on tori, see Definition 21.3 for more details. For $i \neq 4,5$, each
$M_i$ can also be described as a double branched cover of a tangle
$Q_i = (W_i, K_i)$, where $W_i$ is a 3-ball for $i=6,...,13$, and a
once punctured 3-ball for $i=1,2,3,14$. This is done in [GW1] for
$i=1,2,3$, and in Section 22 for the other cases. See Lemma 22.2.
Some results on the case $\Delta=5$ have been independently obtained
by Teragaito [T2]. He obtains a finite set of pairs of intersection
graphs of punctured tori at distance $5$ which must contain all the
pairs of graphs that arise from two toroidal fillings on a hyperbolic
3-manifold at distance 5. One of his pairs produces a non-hyperbolic
manifold while the others correspond to the manifolds in our list for
$\Delta=5$.
We remark that the related problem of determining all hyperbolic
$3$-manifolds with two Dehn fillings at distance at least $4$ that
yield manifolds containing Klein bottles has been solved by Lee [L1,
L2] (see also [MaS]).
Since $M_i$ has more than one boundary component only when $i \in
\{1,2,3,14\}$, we have the following corollary to Theorem 1.1
(together with [Go]), which in the case $\Delta = 5$ is due to Lee
[L1]. Note that all boundary components of the manifolds are tori.
\begin{cor} Let $M$ be a hyperbolic 3-manifold with more than one
boundary component, having toroidal Dehn fillings $M(r), M(s)$ with
$\Delta = \Delta(r,s) \geq 4$. Then each boundary component of $M$
is a torus, and either
(1) $\Delta = 4$ and $(M,r,s) \cong (M_i,r_i,s_i)$ for $i \in
\{1,2,14\}$, or
(2) $\Delta = 5$ and $(M,r,s) \cong (M_3,r_3,s_3)$.
\end{cor}
In [GW1] and [GW3] it is shown that if $M$ is a hyperbolic 3-manifold
with fillings $M(r)$ and $M(s)$, one of which is annular and the other
either toroidal or annular, then either $(M,r,s) \cong (M_i,r_i,s_i)$ for $i
\in \{1,2,3\}$, or $\Delta(r,s) \leq 3$. It is also known that if $M(r)$
contains an essential sphere or disk, and $M(s)$ contains an essential
sphere, disk, annulus or torus, then $\Delta(r,s) \leq 3$; see [GW2] and
the references listed there. Corollary 1.3 then gives
\begin{cor} Let $M$ be a hyperbolic 3-manifold with a torus boundary
component $T_0$ and at least one other boundary component. Let $r,s$
be exceptional slopes on $T_0$. Then either $(M,r,s) \cong
(M_i,r_i,s_i)$ for $i \in \{1,2,3,14\}$, or $\Delta(r,s) \leq 3$.
\end{cor}
A pair $(M, T_0)$ is called a {\it large manifold\/} if $T_0$ is a
torus on the boundary of the 3-manifold $M$ and $H_2(M, \partial M - T_0)
\neq 0$ (see [Wu3]). Teragaito [T2] proved that there is no large
hyperbolic manifold $M$ admitting two toroidal fillings of distance at
least 5. The following corollary clarifies the case of distance $4$.
\bigskip
\noindent {\bf Theorem 22.3} \; {\em
Suppose $(M,T_0)$ is a large manifold and $M$ is hyperbolic and
contains two toroidal slopes $r_1, r_2$ on $T_0$ with $\Delta(r_1,
r_2) \geq 4$. Then $M$ is the Whitehead link exterior, and
$\Delta(r_1, r_2)=4$.
} \bigskip
Theorem 1.1 gives information about toroidal Dehn surgeries on
hyperbolic knots in $S^3$. It follows from [Go] that the only such
knot with two toroidal surgeries at distance $> 5$ is the figure eight
knot, for which the $4$ and $-4$ surgeries are toroidal. Teragaito has
shown [T1] that the only hyperbolic knots with two toroidal surgeries
at distance $5$ are the Eudave-Mu\~noz knots $k(2,-1,n,0)$, $n \neq
1$. We can now determine the knots with toroidal surgeries at
distance $4$. Denote by $L_i = K'_i \cup K''_i$ the link in Figure
24.1(i), where $K'_i$ is the component on the left. Denote by
$L_i(n)$ the knot obtained from $K''_i$ by $1/n$ surgery on $K'_i$.
One can check that $L_3(n)$ is the same as the Eudave-Mu\~noz knot
$k(3,1,-n,0)$ in [Eu, Figure 25], which is the mirror-image of
$k(2,-1,1+n,0)$ [Eu, Proposition 1.4].
\bigskip
\noindent
{\bf Theorem 24.4} \;
{\em Suppose $K$ is a hyperbolic knot in $S^3$ admitting two
toroidal surgeries $K(r_1), K(r_2)$ with $\Delta(r_1, r_2) \geq 4$.
Then $(K, r_1, r_2)$ is equivalent to one of the following, where $n$
is an integer.
(1) $K = L_1(n)$, $r_1 = 0$, $r_2 = 4$.
(2) $K = L_2(n)$, $r_1 = 2-9n$, $r_2 = -2-9n$.
(3) $K = L_3(n)$, $r_1 = -9 - 25n$, $r_2 = -(13/2) - 25n$.
(4) $K$ is the Figure 8 knot, $r_1 = 4$, $r_2 = -4$.
}
\bigskip
The only hyperbolic knots known to have more than two toroidal
surgeries are the figure eight knot and the $(-2,3,7)$-pretzel knot,
with toroidal slopes $\{-4,0,4\}$ and $\{16,37/2,20\}$ respectively.
This led Eudave-Mu\~noz [Eu] to conjecture that a hyperbolic knot in
$S^3$ has at most three toroidal surgeries. Teragaito [T1] showed
that there can be at most five toroidal surgeries. Theorem 1.1 and
[T1, Corollary 1.2] lead to the following improvement.
\bigskip
\noindent
{\bf Corollary 24.5} \;
{\em A hyperbolic knot in $S^3$ has at most four toroidal surgeries. If
there are four, then they are consecutive integers.}
\bigskip
Here is a sketch of the proof of Theorem 1.1. A toroidal Dehn filling
$M(r)$ on a hyperbolic $3$-manifold $M$ gives rise to an essential
punctured torus $F$ in $M$ whose boundary consists of $n>0$ circles of
slope $r$ on $T_0$, where the capped-off surface $\hat F$ is an
essential torus in $M(r)$. Hence, in the usual way (see Section 2),
two toroidal fillings $M(r_1), M(r_2)$ give rise to a pair of
intersection graphs $\Gamma_1, \Gamma_2$ on the tori $\hat F_1, \hat
F_2$, with $n_1, n_2$ vertices respectively. The proof consists of a
detailed analysis of the possible pairs of intersection graphs with
$\Delta(r_1,r_2)=4$ or $5$, using Scharlemann cycles and other tools
developed in earlier works in this area. This enables us to eliminate
all but $17$ pairs of graphs. As is usual in this kind of setting,
the permissible graphs all have small numbers of vertices. Eleven of
the pairs correspond to the manifolds $M_i$, $4\leq i\leq 14$. We
show that any of the remaining pairs must correspond to a pair of
fillings on $M_i$ or $M_i(t)$ for $i \in \{1, 2, 3\}$.
Here is a more detailed summary of the organization of the paper.
Section 2 contains the basic definitions and some preliminary lemmas.
In Sections 3-5 we deal with the generic case $n_1, n_2 > 4$,
ultimately showing (Proposition 5.11) that this case cannot occur.
More specifically, Section 3 shows that the reduced positive graph
$\hat \Gamma_a^+$ of $\Gamma_a$ (see Section 2 for definitions) has no interior
vertices, and this is strengthened in Section 4 to showing that each
component of $\hat \Gamma_a^+$ must be one of the 11 graphs in Figure 4.2. These
are ruled out one by one in Section 5. In Sections 6-11 we consider
the case where some $n_a=4$. Section 6 discusses the situation where
the graph $\Gamma_a$ is {\it kleinian}; this arises when the torus $\hat
F_a$ is the boundary of a regular neighborhood of a Klein bottle in
$M(r_a)$. (The results here are also used in the discussion of the
case $n_1, n_2 \leq 2$.) Sections 7, 8 and 9 show that if $n_a = 4$
and $\Gamma_b$ is non-positive then $n_b\leq 4$. Section 10 shows that if
$\Gamma_1$ and $\Gamma_2$ are both non-positive then $n_1=n_2=4$ is
impossible. Section 11 shows (Proposition 11.9) that if $\Gamma_b$ is
positive then there are exactly two pairs of graphs, one with $n_b=2$,
the other with $n_b=1$. These give the manifolds $M_4$ and $M_5$
respectively. If we suppose $n_a \leq n_b$, it now easily follows
(Proposition 11.10) that $n_a\leq 2$.
In Sections 12-16 we deal with the case $n_a\leq 2$, $n_b\geq 3$. The
conclusion (Proposition 16.8) is that here there are exactly six pairs
of graphs. Two of these are the ones described in Section 11, and the
four new pairs give the manifolds $M_6, M_7, M_8$ and $M_9$. More
precisely, in Section 12 we rule out the case where $\Gamma_b$ is positive,
and in Sections 13 and 14 we consider the case where $n_b>4$ and both
graphs $\Gamma_1$ and $\Gamma_2$ are non-positive. It turns out that
here there is exactly one pair of graphs (Proposition 14.7),
corresponding to the manifold $M_6$. We may now assume that $n_b = 3$
or $4$. Section 15 establishes some notation and elementary properties
for graphs with $n_a\leq 2$. In Section 16 we show that if $\Gamma_1$
and $\Gamma_2$ are non-positive then $n_b=3$ is impossible and if
$n_b=4$ then there are exactly three examples, $M_7$, $M_8$ and $M_9$.
Sections 17-20 deal with the remaining cases where both $n_1$ and
$n_2$ are $\leq 2$. In Section 17 we introduce an equivalence
relation, {\it equidistance}, on the set of edges of a graph $\Gamma_a$,
and show that, under the natural bijection between the edges of
$\Gamma_1$ and $\Gamma_2$, the two graphs induce the same equivalence
relation. This gives a convenient way of ruling out certain pairs of
graphs. Section 18 considers the case $n_a=2$ and $n_b=1$, and shows
that here there are exactly three examples. Section 19 considers the
case $n_1=n_2=2$, $\Gamma_b$ positive, showing that there are two
examples. Finally, in Section 20 we consider the case $n_1=n_2=2$,
$\Gamma_1$ and $\Gamma_2$ both non-positive, and show that there are
exactly six pairs of graphs in this case. The final list of all 11
possible pairs of graphs with $n_1, n_2 \leq 2$ is given in
Proposition 20.4. Five of these correspond to the manifolds $M_{10},
M_{11}, M_{12}, M_{13}$ and $M_{14}$.
The remaining six pairs of graphs in Proposition 20.4 have the
property that one of the graphs has a non-disk face. In Section 21 we
show (Lemma 21.2), using the classification of toroidal/annular and
annular/annular fillings at distance $\geq 4$ given in [GW1] and
[GW3], that in this case the manifold $M$ is either $M_1, M_2$ or
$M_3$, or is obtained from one of those by Dehn filling along one of
the boundary components.
In Section 22 we show how the manifolds $M_i$, $6\leq i\leq 14$ may be
realized as double branched covers. Using this, in Section 23 we show
that the manifolds $M_i$ are hyperbolic. Finally, in Section 24 we
give the applications to toroidal surgeries on knots in $S^3$.
\section{Preliminary Lemmas}
Throughout this paper, we will fix a hyperbolic 3-manifold $M$, with a
torus $T_0$ as a boundary component. A compact surface properly
embedded in $M$ is {\it essential\/} if it is $\pi_1$-injective, and
is not boundary parallel. We use $a, b$ to denote the numbers $1$ or
$2$, with the convention that if they both appear in a statement then
$\{a, b\} = \{1,2\}$.
A slope on $T_0$ is a {\it toroidal slope\/} if $M(r_{a})$ is
toroidal. Let $r_{a}$ be a toroidal slope on $T_0$. Denote by
$\Delta = \Delta(r_1, r_2)$ the minimal geometric intersection number
between $r_1$ and $r_2$. When $\Delta > 5$ the manifolds $M$ have been
determined in [Go]. We will always assume that $\Delta = 4$ or $5$.
Let $\hat F_a$ be an essential torus in $M(r_a)$, and let $F_a = \hat
F_a \cap M$. If $M(r_a)$ is reducible then by [Wu1] and [Oh] we would
have $\Delta \leq 3$, which is a contradiction. Therefore both
$M(r_a)$ are irreducible.
Let $n_{a}$ be the number of boundary components of $F_{a}$ on $T_0$.
Choose $\hat F_{a}$ in $M(r_{a})$ so that $n_{a}$ is minimal among
all essential tori in $M(r_{a})$. Minimizing the number of
components of $F_1 \cap F_2$ by an isotopy, we may assume that
$F_1\cap F_2$ consists of arcs and circles which are essential on both
$F_{a}$. Denote by $J_{a}$ the attached solid torus in $M(r_{a})$,
and by $u_i$ ($i=1,...,n_{a}$) the components of $\hat F_{a}\cap
J_{a}$, which are all disks, labeled successively when traveling
along $J_{a}$. Similarly let $v_j$ be the disk components of $\hat F_b\cap
J_b$. Let $\Gamma_{a}$ be the graph on $\hat F_{a}$ with the
$u_i$'s as (fat) vertices, and the arc components of $F_1\cap F_2$ as
edges. Similarly for $\Gamma_b$. The minimality of the number of
components in $F_1\cap F_2$ and the minimality of $n_{a}$ imply that
$\Gamma_{a}$ has no trivial loops, and that each disk face of
$\Gamma_{a}$ in $\hat F_{a}$ has interior disjoint from $F_{b}$.
If $e$ is an edge of $\Gamma_{a}$ with an endpoint $x$ on a fat
vertex $u_i$, then $x$ is labeled $j$ if $x$ is in $u_i\cap v_j$. In
this case $e$ is called a {\it $j$-edge} in $\Gamma_a$, and an $i$-edge in
$\Gamma_b$. Labels in $\Gamma_a$ are considered as mod $n_b$ integers; in
particular, $n_b +1 = 1$. When going around $\partial u_i$, the labels of
the endpoints of edges appear as $1, 2, \ldots, n_b$ repeated $\Delta$
times. Label the endpoints of edges in $\Gamma_b$ similarly.
Each vertex of $\Gamma_{a}$ is given a sign according to whether
$J_{a}$ passes $\hat F_{a}$ from the positive side or negative side
at this vertex. Two vertices of $\Gamma_{a}$ are {\it parallel\/} if
they have the same sign, otherwise they are {\it antiparallel.} Note
that if $\hat F_{a}$ is a separating surface, then $n_{a}$ is even,
and $v_i, v_j$ are parallel if and only if $i, j$ have the same
parity. We use $val(v, G)$ to denote the valence of a vertex $v$ in a
graph $G$. If $G$ is clear from the context, we simply denote it by
$val(v)$.
When considering each family of parallel edges of $\Gamma_{a}$ as a
single edge $\hat e$, we get the {\it reduced graph\/}
$\hat{\Gamma}_{a}$ on $\hat F_{a}$. It has the same vertices as
$\Gamma_{a}$. Each edge of $\hat \Gamma_a$ represents a family of parallel
edges in $\Gamma_a$. We shall often refer to a family of parallel edges as
simply a {\it family}.
\begin{defn}
(1) An edge of $\Gamma_{a}$ is a {\it positive edge\/} if it
connects parallel vertices. Otherwise it is a {\it negative edge}.
(2) The graph $\Gamma_a$ is {\it positive\/} if all its vertices are
parallel, otherwise it is {\it non-positive}.
p\end{defn}
We use $\Gamma_{a}^+$ (resp.\ $\Gamma_{a}^-$) to denote the
subgraph of $\Gamma_{a}$ whose vertices are the vertices of
$\Gamma_{a}$ and whose edges are the positive (resp.\ negative)
edges of $\Gamma_{a}$. Similarly for $\hat{\Gamma}_{a}^+$
and $\hat{\Gamma}_{a}^-$.
A cycle in $\Gamma_{a}$ consisting of positive edges is a {\it
Scharlemann cycle\/} if it bounds a disk with interior disjoint from
the graph, and all the edges in the cycle have the same pair of labels
$\{i, i+1\}$ at their two endpoints, called the {\it label pair\/} of
the Scharlemann cycle. A Scharlemann cycle containing only two edges
is called a {\it Scharlemann bigon.} A Scharlemann cycle with label
pair, say, $\{1,2\}$ will also be called a $(12)$-Scharlemann cycle.
If $\Gamma_b$ contains a Scharlemann cycle with label pair $\{i,i\pm 1\}$,
we shall sometimes abuse terminology and say that the vertex $u_i$ of
$\Gamma_a$ is a {\it label of a Scharlemann cycle}. An {\it extended
Scharlemann cycle\/} is a cycle of edges $\{e_1, ..., e_k\}$ such
that there is a Scharlemann cycle $\{e'_1, ..., e'_k\}$ with $e_i$
parallel and adjacent to $e'_i$ and $e_i \neq e'_j$, $1\leq i,j \leq
k$. If $\{e_1, ..., e_k\}$ is a Scharlemann cycle in $\Gamma_a$ then the
subgraph of $\Gamma_b$ consisting of these edges and their vertices is
called a {\it Scharlemann cocycle}.
A subgraph $G$ of a graph $\Gamma$ on a surface $F$ is {\it
essential\/} if it is not contained in a disk in $F$. The following
lemma contains some common properties of the graphs $\Gamma_{a}$.
It can be found in [GW1, Lemma 2.2].
\begin{lemma}
(1) {\rm (The Parity Rule)} An edge $e$ is a positive edge in
$\Gamma_1$ if and only if it is a negative edge in $\Gamma_2$.
(2) A pair of edges cannot be parallel on both $\Gamma_1$ and
$\Gamma_2$.
(3) If $\Gamma_{a}$ has a set of $n_{b}$ parallel negative
edges, then on $\Gamma_{b}$ they form mutually disjoint essential
cycles of equal length.
(4) If $\Gamma_{a}$ has a Scharlemann cycle, then $\hat F_{b}$ is
separating. In particular, $\Gamma_b$ has the same number of positive and
negative vertices, so $n_{b}$ is even, and two vertices $v_i, v_j$
of $\Gamma_b$ are parallel if and only if $i, j$ have the same parity.
(5) If $\Gamma_{a}$ has a Scharlemann cycle $\{e_1,\ldots, e_k\}$,
then the corresponding Scharlemann cocycle on $\Gamma_{b}$ is
essential.
(6) If $n_{b} > 2$, then $\Gamma_{a}$ contains no extended
Scharlemann cycle.
\end{lemma}
Let $\hat e$ be a collection of parallel negative edges on
$\Gamma_{b}$, oriented from $v_1$ to $v_2$. Then $\hat e$ defines a
permutation $\varphi: \{1, \ldots, n_{a}\} \to \{1, \ldots,
n_{a}\}$, such that an edge $e$ in $\hat e$ has label $k$ at $v_1$ if
and only if it has label $\varphi(k)$ at $v_2$. Call $\varphi$ the
{\it transition function associated to $\hat e$}. Define the {\it
transition number\/} to be the mod $n_a$ integer $s = s(\hat e)$
such that $\varphi(k) = k + s$. If we reverse the orientation of
$\hat e$ then the transition function is $\varphi^{-1}$, and the
transition number is $-s$; hence if $\hat e$ is unoriented then
$\varphi$ is well defined up to inversion, and $s(\hat e)$ is well
defined up to sign.
\begin{lemma}
(1) If a family of parallel negative edges in
$\Gamma_a$ contains more than $n_b$ edges (in particular, if the
family contains 3 edge endpoints with the same label), then $\Gamma_b$
is positive, and the transition function associated to this family is
transitive.
(2) If $\Gamma_a$ contains two Scharlemann cycles with disjoint label
pairs $\{i, i+1\}$ and $\{j, j+1\}$, then $i\equiv j$ mod 2.
(3) If $n_b > 2$ then a family of parallel positive edges in
$\Gamma_a$ contains at most $n_b/2 + 2$ edges, and if it does contain
$n_b/2 +2$ edges, then $n_b \equiv 0$ mod 4.
(4) $\Gamma_a$ has at most four labels of Scharlemann cycles, at most two
for each sign.
(5) A loop edge $e$ and a non-loop edge $e'$ on $\Gamma_a$ cannot be
parallel on $\Gamma_b$.
(6) If $n_b \geq 4$ then $\Gamma_a$ contains at most $2n_b$ parallel
negative edges.
\end{lemma}
\noindent {\bf Proof.}
(1) This is obvious if $n_b \leq 2$, and it can be found in [GW1,
Lemma 2.3] if $n_b > 2$.
(2) and (3) are basically Lemmas 1.7 and 1.4 of [Wu1]. If $\Gamma_a$ has
$n_b/2 + 2$ parallel positive edges then the two outermost pairs form
two Scharlemann bigons. One can then check the labels of these
Scharlemann bigons and use (2) to show that $n_b \equiv 0$ mod 4.
(4) If $\Gamma_a$ has more than four labels of Scharlemann cycles, then
either one can find two Scharlemann cycles with disjoint label pairs
$\{i, i+1\}$ and $\{j, j+1\}$ such that $i - j \equiv 1$ mod 2, which
is a contradiction to (2), or one can find three Scharlemann cycles
with mutually disjoint label pairs, in which case one can replace
$\hat F_a$ by another essential torus to reduce $n_a$ and get a
contradiction. See [Wu1, Lemma 1.10].
If $\Gamma_a$ has three positive labels of Scharlemann cycles $u_{i_j}$
then it has negative labels of Scharlemann cycles $u_{i_j +
\epsilon_j}$ for some $\epsilon_j = \pm 1$, which cannot all be the
same, hence $\Gamma_a$ has at least 5 labels of Scharlemann cycles,
contradicting the above.
(5) Since $e$ is positive on $\Gamma_a$, it is negative in $\Gamma_b$. If $e$
has endpoints on $u_i$ in $\Gamma_a$ then on $\Gamma_b$ its two endpoints are
both labeled $i$, hence the corresponding transition number is $0$, so
any edge $e'$ parallel to $e$ on $\Gamma_b$ must also have the same label
at its two endpoints, which implies that $e'$ is a loop on $\Gamma_a$.
(6) This is [Go, Corollary 5.5].
\quad $\Box$ \medskip
\begin{lemma}
If a label $i$ appears twice among the endpoints
of a family $\hat e$ of parallel positive edges in $\Gamma_{a}$, then
$i$ is a label of a Scharlemann bigon in $\hat e$. In particular, if
$\hat e$ has more than $n_{b}/2$ edges, then it contains a
Scharlemann bigon.
\end{lemma}
\noindent {\bf Proof.} Since the edges are positive, by the parity rule $i$ cannot
appear at both endpoints of a single edge in this family. Let $e_1,
e_2, \ldots, e_k$ be consecutive edges of $\hat e$ such that $e_1$ and
$e_k$ have $i$ as a label. Now $k$ must be even, otherwise the edge
$e_{(k+1)/2}$ would have the same label at its two endpoints. If
$k\geq 4$ and $n_b>2$ one can see that these edges contain an extended
Scharlemann cycle, which contradicts Lemma 2.2(6). Therefore $k=2$ or
$n_b=2$, in which case $e_1, e_2$ form a Scharlemann bigon with $i$ as
a label.
If $\hat e$ has more than $n_{b}/2$ edges, then it has more than
$n_{b}$ endpoints, so some label must appear twice.
\quad $\Box$ \medskip
\begin{lemma}
$\hat \Gamma_a$ contains at most $3n_a$ edges.
\end{lemma}
\noindent {\bf Proof.} Let $V,E,F$ be the number of vertices, edges and disk faces of
$\hat \Gamma_a$. Then $V - E + F \geq 0$ (the inequality may be strict if
there are some non-disk faces.) Each face of $\hat \Gamma_a$ has at least
three edges, hence we have $ 3F \leq 2E$. Solving those two
inequalities gives $E \leq 3V$. \quad $\Box$ \medskip
\begin{lemma} If $n_b> 4$ then the vertices of $\Gamma_a$ cannot all
be parallel. \end{lemma}
\noindent {\bf Proof.} By Lemma 2.5 the reduced graph $\hat \Gamma_a$ has at most $3n_{a}$
edges. For any $i$, since $v_i$ on $\Gamma_b$ has valence at least
$4n_{a}$, there are $4n_{a}$ $i$-edges on $\Gamma_a$, hence two of them
must be parallel, so $i$ is a label of a Scharlemann cycle. Since
there are at most $4$ such labels (Lemma 2.3(4)), we would have
$n_{b} \leq 4$, contradicting the assumption. \quad $\Box$ \medskip
A vertex $v$ of a graph is a {\it full vertex\/} if all edges incident
to it are positive.
\begin{lemma} Suppose $n_b > 4$. Then
(1) a family of parallel negative edges in $\Gamma_b$ contains at most
$n_{a}$ edges, hence any label $i$ appears at most twice among the
endpoints of such a family;
(2) two families of positive edges in $\Gamma_a$ adjacent at a vertex
contain at most $n_b + 2$ edges; and
(3) three families of positive edges in $\Gamma_a$ adjacent at a vertex
contain at most $2n_b$ edges, and if there are $2n_b$ then $n_b = 6$.
\end{lemma}
\noindent {\bf Proof.} (1) If a family of parallel negative edges on $\Gamma_b$ contains
more than $n_a$ edges then by Lemma 2.3(1) all vertices of $\Gamma_a$ are
parallel, which contradicts Lemma 2.6.
If $i$ appears three times among the endpoints of a family of parallel
negative edges in $\Gamma_b$ then this family would contain more than
$n_{a}$ edges, which is a contradiction.
(2) By Lemma 2.3(3) a family of parallel positive edges contains $r
\leq n_b/2 + 2$ edges. If two adjacent families $\hat e_1, \hat e_2$
contain more than $n_b + 2$ edges, then one of them, say $\hat e_1$,
has $n_b/2+2$ edges while the other one has either $n_b/2+1$ or
$n_b/2+2$ edges. Now $\hat e_1$ contains two Scharlemann bigons,
which must appear on the two sides of the family because there is no
extended Scharlemann cycle. There is also at least one Scharlemann
bigon in $\hat e_2$. Examining the labels of these Scharlemann bigons
we can see that they contain at least $5$ labels, which contradicts
Lemma 2.3(4).
(3) Assume the three families contain $r \geq 2n_b$ edges. Then one of
the families contains more than $n_b/2$ edges, so by Lemma 2.2(4) $n_b$
is even. By (2) two adjacent families of parallel edges contain at
most $n_b + 2$ edges, while by Lemma 2.3(3) the other family has at
most $n_b/2+2$ edges, so we have $2n_b \leq r \leq (n_b+2) + (n_b/2+2)$,
which gives $n_b \leq 8$.
If $n_b = 8$ then the above inequalities force the three families to
have $6,4,6$ edges, and we see that all 8 labels appear as labels of
Scharlemann bigons, which contradicts Lemma 2.3(4). So we must have
$n_b = 6$. By Lemma 2.3(3) we have $2n_b \leq r \leq 3(n_b/2 + 1) =
12 = 2n_b$. Hence $r = 2n_b$. \quad $\Box$ \medskip
\begin{lemma} If a vertex $u_i$ of $\Gamma_a$ is incident to more
than $n_b$ negative edges, then $\Gamma_b$ has a Scharlemann cycle.
\end{lemma}
\noindent {\bf Proof.} In this case there are $n_b + 1$ positive $i$-edges in $\Gamma_b$,
which cut the surface $F_b$ into faces, at least one of which is a disk
face in the sense that it is a topological disk whose interior
contains no vertices of $\Gamma_b$. Hence the subgraph of $\Gamma_b$ consisting
of these edges is a $x$-edge cycle in the sense of Hayashi-Motegi [HM,
Page 4468]. By [HM, Proposition 5.1] a disk face of this $x$-edge
cycle contains a disk face of a Scharlemann cycle. \quad $\Box$ \medskip
Consider a graph $G$ on a closed surface $F$, and assume that $G$ has
no isolated vertex. If the vertices of $G$ have been assigned $\pm$
signs (for example $\hat \Gamma_a^+$), let $X$ be the union of $G$ and all its
faces $\sigma$ such that all vertices on $\partial \sigma$ have the same
sign, otherwise let $X$ be the union of $G$ and all its disk faces. A
vertex $v$ of $G$ is an {\it interior vertex\/} if it lies in the
interior of $X$. A vertex $v$ of $G$ is a {\it cut vertex\/} if a
regular neighborhood of $v$ in $X$ with $v$ removed is not connected.
A vertex $v$ of $G$ is a {\it boundary vertex\/} if it is not an
interior or cut vertex. Note that if $G = \hat \Gamma_a^+$ then an interior
vertex is a full vertex. Alternatively, let $\delta(v)$ be the number
of corners around $v$ which lie in $X$. Then $v$ is an interior
vertex if $\delta(v) = val(v, G)$, a boundary vertex if $\delta(v) =
val(v, G) - 1$, and a cut vertex if $\delta(v) \leq val(v,G) - 2$.
Given a graph $G$ on a surface $D$, let $c_i(G)$ be the number of
boundary vertices of $G$ with valence $i$. Define
\begin{eqnarray*}
\varphi(G) & = & 6c_0(G) + 3 c_1(G) + 2 c_2(G) + c_3(G) \\
\psi(G) & = & c_0(G) + c_1(G) + c_2(G) + c_3(G).
\end{eqnarray*}
Note that $\psi(G)$ is the number of boundary vertices of $G$ with
valence at most 3.
\begin{lemma} Let $G$ be a connected reduced graph in a disk
$D$ such that any interior vertex of $G$ has valence at least $6$.
Then $\varphi(G) \geq 6$. Moreover, if $G$ is not homeomorphic to an
arc or a single point then $\psi(G) \geq 3$. \end{lemma}
\noindent {\bf Proof.} Let $X$ be the union of $G$ and all its disk faces. The result
is obviously true if $G$ is a tree. So we assume that $G$ has some
disk faces.
First assume that $X$ has no cut vertex, so it is a disk, and $c_0(G)
= c_1(G) = 0$. The double of $G$ along $\partial X$ is then a graph
$\tilde G$ on the double of $X$, which is a sphere. Note that the
valence of a vertex $v$ of $\tilde G$ is either at least $6$, or it is
$2$ or $4$ when $v$ is a boundary vertex of $G$ with valence $2$ or
$3$, respectively. Since each face has at least three edges, an
Euler characteristic argument gives $$
2 = V - E + F \leq V - \frac 13
E = \sum_i (1 - \frac 16 val(v_i, \tilde G)) \leq \frac 23 c_2(G) +
\frac 13 c_3(G)$$
Therefore $\varphi(G) = 2c_2(G) + c_3(G) \geq 6$.
Since $c_0(G) = c_1(G) = 0$, we also have $\psi(G) = c_2(G) + c_3(G)
\geq \frac 12 \varphi(G) \geq 3$.
Now assume that $X$ has a cut vertex $v$. Since $G$ is connected and
contained in a disk, $X$ is simply connected, so we can write $X = X_1
\cup X_2$, where $X_i$ are subcomplexes of $X$ such that $X_1 \cap X_2
= v$, and $G_i = G \cap X_i$ are nontrivial connected subgraphs of
$G$. The valence of $v$ in $G_i$ is at least $1$, so its
contribution to $\varphi(G_i)$ is at most 3. Hence by induction we
have
$$\varphi(G) \geq (\varphi(G_1)-3) + (\varphi(G_2)-3) \geq 6.$$
By assumption $X$ is not homeomorphic to an arc, so at least one of
the $X_i$, say $X_1$, is not homeomorphic to an arc, and the other one
has at least 2 boundary vertices of valence at most 3, whether it is
homeomorphic to an arc or not. Hence
$$
\psi(G) \geq \psi(G_1) + \psi(G_2) - 2 \geq 3 + 2 - 2 = 3.
$$
\quad $\Box$ \medskip
\begin{lemma} Let $G$ be a reduced graph on a torus $T$ with
no interior or isolated vertex. Let $V$ and $E$ be the number of
vertices and edges of $G$, and let $k$ be the number of boundary
vertices of $G$.
(1) $k \geq E-V$, and equality holds if and only if all disk face of
$G$ are triangles, all non-disk faces are annuli, and each cut vertex
has exactly two corners on annular faces.
(2) $G$ has at most $2V$ edges.
\end{lemma}
\noindent {\bf Proof.} (1) Let $D$ be the number of disk faces of $G$. Then $0 =
\chi(T) \leq V - E + D$, and equality holds if and only if all
non-disk faces are annuli. Thus $D \geq E - V$. For each vertex $u$
of $G$, let $\delta(u)$ be the number of corners of disk faces
incident to $u$. Then $\sum_u val(u) = 2E$, and $\sum_u \delta(u)
\geq 3D \geq 3(E -V)$. Since there is no isolated or interior vertex,
we have $val(u) - \delta(u) \geq 1$, and equality holds if and only if
$u$ is a boundary vertex. Let $p$ be the number of non-boundary
vertices. Then $$
p \leq \sum_u(val(u) - \delta(u) - 1) \leq 2E -
3(E-V) - V = 2V - E.$$
It follows that the number of boundary vertices
is $k = V - p \geq E - V$, and equality holds if and only if (i)
$V-E+D=0$, i.e.\ non-disk faces are annuli, (ii) $\sum_u \delta(u) =
3D$, so all disk faces are triangles, and (iii) $val(u) - \delta(u) -
1 = 1$ for any cut vertex, i.e.\ each cut vertex has exactly two
corners not on disk faces.
(2) Since the number of boundary vertices is at most $V$, by (1) we
have $V \geq k \geq E - V$, hence $E \leq 2V$.
\quad $\Box$ \medskip
\begin{lemma} Suppose all interior vertices of $\hat \Gamma_a^+$ have
valence at least 6, and all boundary vertices of $\hat \Gamma_a^+$ have valence
at least 4. Let $G$ be a component of $\hat \Gamma_a^+$. Then either (i) $G$
is topologically an essential circle on the torus $\hat F_a$, or (ii)
$G$ has no cut vertex, all interior vertices of $G$ are of valence
exactly 6, and all boundary vertices of $G$ are of valence exactly
4. \end{lemma}
\noindent {\bf Proof.} Let $X$ be the union of $G$ and all its disk faces. If $X$ is
the whole torus then all vertices are interior vertices, and an easy
Euler characteristic argument shows that all vertices must be of
valence 6, so (ii) follows. Also, by Lemma 2.9 $X$ is not in a disk
in $\hat F_a$ as otherwise $G$ would have a boundary vertex of valence at
most 3. Therefore we may assume that $X$ has the homotopy type of
a circle.
First assume that $X$ has a cut vertex $v$. Recall that $X$ is
homotopy equivalent to a circle, so if $X - v$ is not connected, then
$v$ cuts off a subcomplex $W$ of $X$ which lies in a disk in $\hat
F_a$. By Lemma 2.9 the graph $G \cap W$ has at least two boundary
vertices of valence at most 3, hence at least one such vertex $v'$
other than $v$, which contradicts the assumption because $v'$ is then
a boundary vertex of $\hat \Gamma_a^+$ of valence at most 3. Therefore we
may assume that $X - v$ is connected. Since $X$ has the homotopy type
of a circle, $X$ cut at $v$ is a simply connected planar complex $W$,
and $X$ is obtained by identifying exactly two points of $W$. Let
$G'$ be the corresponding graph on $W$. We may assume that $X$ is not
a circle as otherwise (i) is true. Thus $W$ is not homeomorphic to an
arc. Therefore by Lemma 2.9 we have $\psi(G') \geq 3$, hence $G'$ has
at least one boundary vertex $v'$ of valence at most 3 which is not
identified to $v$ in $G$. By definition $v'$ is a boundary vertex of
$\hat \Gamma_a^+$ of valence at most 3, which is a contradiction. This
completes the proof that $X$ has no cut vertex.
We may now assume that $X$ is an annulus, so all vertices of $G$ are
either interior vertices of valence at least 6 in the interior of $X$,
or boundary vertices of valence at least 4 on $\partial X$.
Consider the double $G''$ of $G$ on the double of $X$ along $\partial X$.
Since each boundary vertex of $G$ of valence $k$ gives rise to a
vertex of valence $2k - 2$ in $G''$, we see that $G''$ is a reduced
graph on a torus such that all of its vertices have valence at
least 6. An Euler characteristic argument shows that all vertices of
$G''$ must have valence exactly 6, hence (ii) follows.
\quad $\Box$ \medskip
\begin{lemma} If $M(r_a)$ contains a Klein bottle $K$, then
(1) $T = \partial N(K)$ is an essential torus in $M(r_a)$; and
(2) $K$ intersects the core $K_a$ of the Dehn filling solid torus
at no less than $n_a/2$ points. \end{lemma}
\noindent {\bf Proof.} $T$ bounds a twisted $I$-bundle over the Klein bottle $N(K)$ on
one side. Since $M(r_a)$ is assumed irreducible, if $T$ is
compressible on the other side then $M(r_a)$ is a Seifert fiber space
over a sphere with (at most) three singular fibers of indices
$(2,2,p)$ for some $p$, and if $T$ is boundary parallel then $M(r_a)$
is a twisted $I$-bundle over the Klein bottle. Either case
contradicts the assumption that $M(r_a)$ is toroidal. Therefore $T$
is an essential torus. If $|K \cap K_a| < n_a/2$ then $T$ would
intersect $K_a$ in less than $n_a$ points, contradicting the choice of
$n_a$. \quad $\Box$ \medskip
\begin{lemma} Suppose $n_a>2$, and $\Gamma_b$ has both a
$12$-Scharlemann bigon $e_1 \cup e_2$ and a $23$-Scharlemann bigons
$e_3 \cup e_4$. If $e_1 \cup e_2$ and $e_3 \cup e_4$ are isotopic on
$\hat F_a$, then the disk face $D$ they bound on $\hat F_a$ contains
at least $(n_a/2)-1$ vertices in its interior. \end{lemma}
\noindent {\bf Proof.} Let $m$ be the number of vertices in the interior of $D$. Let
$D_1, D_2$ be the disk faces of $(12)$- and $(23)$-Scharlemann bigons
in $\Gamma_b$. Shrinking the Dehn filling solid torus of $M(r_a)$ to its
core $K_a$, the union $D_1 \cup D_2 \cup D$ is a Klein bottle $Q$ in
$M(r_a)$. A regular neighborhood of $Q$ intersects $K_a$ at an arc
from $u_1$ to $u_2$ then to $u_3$, and one arc for each vertex of
$\Gamma_a$ in the interior of $D$. Hence $Q$ can be perturbed to intersect
$K_a$ at $1+m$ points. By Lemma 2.12(2) we have $m+1 \geq n_a/2$,
hence the result follows. \quad $\Box$ \medskip
An edge $e$ of $\Gamma_a$ is a {\it co-loop\/} edge if it has the same
label on its two endpoints, in other words, it is a loop on the other
graph $\Gamma_b$. Given a codimension 1 manifold $X$ in a manifold $Y$,
use $Y|X$ to denote the manifold obtained by cutting $Y$ along
$X$.
\begin{lemma} Let $\hat e$ be a family of negative edges in
$\Gamma_a$. Let $G$ be the subgraph of $\Gamma_b$ consisting of the edges of
$\hat e$ and their vertices.
(1) Each cycle component of $G$ is an essential loop on $\hat F_b$.
(2) (The 3-Cycle Lemma.) $G$ cannot contain three disjoint cycles; in
particular, $\Gamma_a$ cannot have three parallel co-loop edges.
(3) (The 2-Cycle Lemma.) If $\Gamma_b$ is positive then $G$ cannot contain
two disjoint cycles; in particular, $\Gamma_a$ cannot have two parallel
co-loop edges.
\end{lemma}
\noindent {\bf Proof.} (1) Assume to the contrary that some cycle component of $G$ is
inessential on $\hat F_b$. Let $D$ be a disk bounded by an innermost
cycle component of $G$, and let $D'$ be the bigon disks on $F_a$
between edges of $\hat e$. Let $V_b$ be the Dehn filling solid torus
in $M(r_b)$. Then a regular neighborhood $W$ of $D \cup V_b \cup D'$
is a solid torus containing the core of $V_b$ as a cable knot winding
along the longitude at least twice. See the proof of [GLi,
Proposition 1.3]. In this case $W \cap M$ is a cable space, which is
a contradiction to the assumption that $M$ is a hyperbolic manifold.
(2) Let $\hat e = e_1 \cup ... \cup e_k$, oriented
consistently, with tails at $u'$ and heads at $u''$ on $\Gamma_a$. Let $s$
be the transition number of $\hat e$. We may assume that $e_i$ has
label $i$ at its tail, so it has label $i+s$ at its head. Let $D_j$
be the bigon on $\hat F_a$ between $e_j$ and $e_{j+1}$.
If $k>n_b$ then by Lemma 2.3(1) the transition function associated
with $\hat e$ has only one orbit, hence we may assume $k\leq n_b$. On
$\Gamma_b$ these edges form disjoint cycles and chains. Assume there are
at least three cycles. Then $e_1, e_2, e_3$ belong to three distinct
cycles $C_1, C_2, C_3$. Thus for $i=1,2,3$, $$C_i = e_i \cup e_{i+s}
\cup ... \cup e_{i+(p-1)s}$$
is an oriented cycle on $\hat F_b$ for some fixed $p$. By (1) these
are essential loops on $\hat F_b$, so they are parallel as unoriented
loops.
Each bigon $D_{1+js}$ gives a parallelism between an edge of $C_1$ and
an edge of $C_2$, hence when shrinking the Dehn filling solid torus
$V_b$ to its core knot $K_b$, the union $A_1 = \cup D_{1+js}$ is an
annulus in $M(r_b)$ with $\partial A_1 = C_1 \cup C_2$. Similarly, $A_2 =
\cup D_{2+js}$ is an annulus in $M(r_b)$ with $\partial A_2 = C_2 \cup
C_3$. These $A_i$ are essential in $M(r_b)|\hat F_b$, the manifold
obtained from $M(r_b)$ by cutting along $\hat F_b$, otherwise $K_b$
would be isotopic to a curve having fewer intersections with $\hat
F_b$.
Let $A'_1, A'_2, A'_3$ be the annuli $\hat F_b | (C_1 \cup C_2 \cup
C_3)$, with $\partial A'_i = C_i \cup C_{i+1}$ (subscripts mod 3.) Let
$m_i$ be the number of times that $K_b$ intersects the interior of
$A'_i$. Then
$$\sum m_i + 3p = n_b$$
The annulus $A_i$ is said to be of type I if a regular neighborhood of
$\partial A_i$ lies on the same side of $\hat F_b$, otherwise it is of
type II. Note that if $\hat F_b$ is separating then $A_i$ must be of
type I. There are several possibilities. In each case one can find
an essential torus $T'$ in $M(r_b)$ which has fewer intersections with
$K_b$. This will contradict the choice of $\hat F_b$ and complete
the proof of (1).
\smallskip
Case 1. {\it $C_2$ is anti-parallel to both $C_1$ and $C_3$. }
In this case each $T_i = A_i \cup A'_i$ is a Klein bottle for $i=1,2$,
which can be perturbed to intersect $K_b$ at $p + m_i$ points. Since
$\sum m_i + 3p = n_b$, either $T_1$ or $T_2$ can be perturbed to
intersect $K_b$ at fewer than $n_b/2$ points, contradicting Lemma 2.12.
\smallskip
Case 2. {\it $C_2$ is anti-parallel to $C_1$, say, and parallel to the
other cycle $C_3$.}
Let $T_1 = A_1 \cup A'_1$ and $T_2 = A_1 \cup A_2 \cup A'_3$. Then
$T_i$ are Klein bottles, and they can be perturbed to intersect $K_b$
at $p+m_1$ and $m_3$ points, respectively. One of these contradicts
Lemma 2.12.
\smallskip
Case 3. {\it $C_2$ is parallel to both $C_1$ and $C_3$.}
If one of the $A_i$, say $A_1$, is of type II, then $T_1 = A_1 \cup
A'_1$ is a non-separating torus (because it can be perturbed to
intersect $\hat F_b$ transversely at a single circle), and it
intersects $K_b$ at $p+m_i < n_b$ points. Since $M(r_b)$ is
irreducible, $T_1$ is incompressible and hence essential, which
contradicts the choice of $\hat F_b$.
If both $A_i$ are of type I then one can show that $A_1 \cup A_2 \cup
A'_3$ is an essential torus $T$ which can be perturbed to intersect
$K_b$ in $m_3 + p < n_b$ points. The proof is standard: The torus
$\hat F_b$ and the annuli $A_1, A_2$ cut $M(r_b)$ into a manifold
whose boundary contains four tori $T_1 = A_1 \cup A'_1$, $T_2 = A_1
\cup A'_2 \cup A'_3$, $T_3 = A_2 \cup A'_2$, and $T_4 = A_2 \cup A'_1
\cup A'_3$. Each of these tori $T_i$ can be perturbed to have fewer
than $n_b$ intersections with the knot $K_b$, and hence bounds a
manifold $W_i$ which is either a solid torus or a $T^2 \times I$
between $T_i$ and a component of $\partial M_(r_b)$. Moreover, if $W_i$
is a solid torus $W_i$ then the annulus $T_i \cap \hat F_b$ is
essential on $\partial W_i$ in the sense that it is neither meridional nor
longitudinal (otherwise $\hat F_b$ would be compressible or could be
isotoped to have fewer intersections with $K_b$). Now we have $M(r_b)
= (W_1 \cup W_4) \cup (W_2 \cup W_3) = W' \cup W''$, with $W' \cap
W''$ a torus $T = A_1 \cup A_2 \cup A'_3$ which can be perturbed to
intersect $K_b$ at $m_3 + p < n_b$ points. Since $W' = W_1 \cup
_{A'_1} W_4$ and $A'_1$ is essential in both $W_1$ and $W_4$, $T$ is
incompressible and not boundary parallel in $W'$; similarly for $W''$.
It follows that $T$ is a contradiction to the choice of $\hat F_b$.
(3) The proof of this part is much simpler. Let $A_1, A'_1$ be as
above, and let $A''_1$ be the complement of $A'_1$ on $\hat F_b$. If
$C_1, C_2$ are parallel then $A_1 \cup A'_1$ is a nonseparating torus
in $M(r_b)$ which can be perturbed to intersect $K_b$ less than $n_b$
times, contradicting the choice of $\hat F_b$. If $C_1, C_2$ are
anti-parallel then $A_1 \cup A'_1$ and $A_1 \cup A''_1$ are Klein
bottles, which can be perturbed to intersect $K_b$ at a total of
$n-2p$ points, where $p$ is the number of vertices in $C_i$; hence one
of those will intersect $K_b$ less than $n_b/2$ times, which
contradicts Lemma 2.12. \quad $\Box$ \medskip
When studying Dehn surgery via intersection graphs, we usually fix the
surfaces $F_1, F_2$, and hence the graphs $\Gamma_1, \Gamma_2$ are
also fixed. The following technique will allow us to modify the
surfaces and hence the graphs in certain situation. Lemma 2.15 will
be used in the proofs of Lemmas 12.16 and 19.6.
Consider two surfaces $F_1, F_2$ in a 3-manifold $M$ with boundary
slopes $r_1, r_2$ respectively and suppose they intersect minimally.
Let $\Gamma_a, \Gamma_b$ be the intersection graphs on $\hat F_1, \hat F_2$,
respectively. Let $\alpha$ be a proper arc on a disk face $D$ of
$\Gamma_a$ with boundary on edges of $\Gamma_a$. Then one can replace two small
arcs of $\Gamma_a$ centered at $\partial \alpha$ by two parallel copies of
$\alpha$ to obtain a new graph $\Gamma_a'$, called the graph obtained from
$\Gamma_a$ by {\it surgery along $\alpha$}.
\bigskip
\leavevmode
\centerline{\epsfbox{Figure2.1.eps}}
\bigskip
\centerline{Figure 2.1}
\bigskip
A face $D'$ of $\Gamma_a$ is called a {\it coupling face\/} to another face
$D$ of $\Gamma_a$ along an edge $e_1$ of $D$ if $D'$ has an edge $e_2$ such
that $e_1, e_2$ are adjacent parallel edges on $\Gamma_b$, and the
neighborhoods in $D$ and $D'$ of the $e_i$'s lie (locally) on the same
side of $\hat F_b$. Note that this is independent of whether $\hat
F_b$ is orientable or separating in $M$. See Figure 2.1. By
definition $D$ has no coupling face along $e_1$ if $e_1$ has no
parallel edge on $\Gamma_b$, one coupling face along $e_1$ if $e_1$ has
some parallel edges and is a border edge of the family, and two
otherwise. A 4-gon face $D$ of $\Gamma_a$ looks like a ``saddle surface''
in $M | F_b$. In general it is not possible to push the saddle up or
down to change the intersection graph. However, if some coupling face
to an edge of $D$ is a bigon then this is possible. See Figure 2.2.
More explicitly, we have the following lemma.
\bigskip
\leavevmode
\centerline{\epsfbox{Figure2.2.eps}}
\bigskip
\centerline{Figure 2.2}
\bigskip
\begin{lemma} Let $\Gamma_1, \Gamma_2$ be a pair of
intersection graphs. Let $Q$ be a face of $\Gamma_a$, and let $e$ be an
edge on $\partial Q$. Let $\alpha$ be an arc on $Q$ with boundary in the
interior of edges of $\Gamma_a$, cutting off a disk $B_1$ containing $e$
and exactly two corners of $Q$. If some coupling face $Q'$ of $Q$
along $e$ is a bigon, then $F_a$ can be isotoped so that the new
intersection graph $\Gamma_a'$ is obtained from $\Gamma_a$ by surgery along
$\alpha$. \end{lemma}
\noindent {\bf Proof.} Cut $M$ along $F_b$. Then the face $Q$ is as shown in Figure
2.2. Let $B_2$ be the bigon in $\Gamma_b$ between $e$ and the edge $e'$ on
$Q'$. After shrinking the Dehn filling solid torus $V_b$ to its core
knot $K_b$, the union $B_1 \cup B_2 \cup Q'$ is a disk $Q''$ with
boundary the union of $\alpha$ and an arc on $\hat F_b$. Pushing
$Q''$ off $K_b$ gives a disk $P$ in $M$ which has boundary the union
of $\alpha$ and an arc on $F_b$, and has interior disjoint from $F_b
\cup F_a$. Therefore we can isotope $F_a$ through this disk $P$ to
get a new surface $F'_a$. It is clear that the new intersection graph
$\Gamma_a'$ is obtained from $\Gamma_a$ by surgery along $\alpha$. \quad $\Box$ \medskip
Let $u$ be a vertex of $\Gamma_a$, and $P, Q$ two edge endpoints on
$\partial u$. Let $I$ be the interval on $\partial u$ from $P$ to $Q$ along
the direction induced by the orientation of $u$. The edge endpoints
of $\Gamma_a$ cut $I$ into $k$ subintervals for some $k$. Then the {\it
distance\/} from $P$ to $Q$ on $\partial u$ is defined as $d_u(P, Q) =
k$. Some times we also use $d_{\Gamma_a}(P,Q)$ to denote $d_u(P,Q)$.
If $P, Q$ are the only edge endpoints of $e_1, e_2$ on $\partial u$,
respectively, then we define $d_u(e_1, e_2) = d_u(P,Q)$. Notice that
if the valence of $u$ is $m$, then $d_u(Q, P) = m - d_u(P, Q)$.
The following lemma can be found in [Go].
\begin{lemma} {\rm [Go, Lemma 2.4]} \quad
(i) Suppose $P, Q \in \partial u_i \cap \partial v_k$ and $R, S \in \partial u_j
\cap \partial v_l$. If $d_{u_i}(P, Q) = d_{u_j}(R,S)$ then $d_{v_k}(P,Q)
= d_{v_l}(R,S)$.
(ii) Suppose that $P \in u_i \cap v_k$, $Q \in u_i \cap v_l$, $R \in
u_j \cap v_k$, and $S \in u_j\cap v_l$. If $d_{u_i}(P, Q) =
d_{u_j}(R, S)$, then $e_{v_k}(P,R) = d_{v_l}(Q,S)$.
\end{lemma}
Suppose two edges $e_1, e_2$ of $\Gamma_a$ connect the same pair of
vertices $u_i, u_j$. Let $p_k, q_k$ be the endpoints of $e_k$ on
$u_i, u_j$, respectively, $k=1,2$. Then $e_1, e_2$ are {\it
equidistant\/} if $d_{u_i}(p_1, p_2) =d_{u_j}(q_2, q_1)$. (Note
that the orders of the edge endpoints have been reversed.) Thus for
example a pair of parallel positive edges is always equidistant, but a
pair of parallel negative edges is not unless their distance is
exactly half of the valence of the vertices.
Note that when $u_i \neq u_j$ the above equation can be written as
$d_{u_i}(e_1, e_2) = d_{u_j}(e_2, e_1)$. When $u_i = u_j$,
$d_{u_i}(e_1, e_2)$ is not defined, and there are two choices for the
pair $p_k, q_k$, but one can check that whether the equality
$d_{u_i}(p_1, p_2) =d_{u_j}(q_2, q_1)$ holds is independent of the
choice of $p_i, q_i$.
The following lemma is called the {\it Equidistance Lemma}. It
follows from Lemma 2.16, and can also be found in [GW1].
\begin{lemma} {\rm [GW1, Lemma 2.8]} \quad Let $e_1, e_2$ be a
pair of edges with $\partial e_1 = \partial e_2$ in both $\Gamma_1$ and
$\Gamma_2$. Then $e_1, e_2$ are equidistant in $\Gamma_1$ if and only
if they are equidistant in $\Gamma_2$. \end{lemma}
Given two oriented slopes $r_1, r_2$ on $T_0$, choose an oriented
meridian-longitude pair $m,l$ on the torus $T_0$ so that $r_1 = m$,
then the slope $r_2$ is homologous to $Jm+\Delta l$ for some mod
$\Delta$ integer $J = J(r_1, r_2)$, called the {\it jumping number\/}
between $r_1, r_2$. Note that if $\Delta = 4$, then $J =\pm 1$, and
if $\Delta = 5$, then $J = \pm 1$ or $\pm 2$. The following lemma is
call the {\it Jumping Lemma} and can be found in [GW1].
\begin{lemma} {\rm [GW1, Lemma 2.10]} \quad Let $P_1, \ldots,
P_{\Delta}$ be the points of $\partial u_i \cap \partial v_j$, labeled
successively on $\partial u_i$. Let $J = J(r_1, r_2)$ be the jumping
number of $r_1, r_2$. Then on $v_j$ these points appear in the order
of $P_J, P_{2J},\ldots, P_{\Delta J}$. In particular, they appear
successively as $P_1,...,P_{\Delta}$ along some direction of $\partial
v_j$ if and only if $J = \pm 1$.
\end{lemma}
\begin{lemma} Let $e_1 \cup ... \cup e_p$ and $e'_1 \cup ...
\cup e'_q$ be two sets of parallel edges on $\Gamma_a$. Suppose $e_1$ is
parallel to $e'_1$ and $e_p$ parallel to $e'_q$ on $\Gamma_b$. Then $p =
q$. \end{lemma}
\noindent {\bf Proof.} Let $D_1, D_2, D_3, D_4$ be the disks realizing the
parallelisms of $e_1 \cup e_p$ and $e'_1 \cup e'_q$ on $\Gamma_a$, and $e_1
\cup e'_1$ and $e_p \cup e'_q$ on $\Gamma_b$. Then the union $A = D_1 \cup
... \cup D_4$ is a M\"obius band or annulus in $M$ with boundary on
$T_0$. (It is embedded in $M$, otherwise there is a pair of edges
parallel in both graphs, contradicting Lemma 2.2(2).) If $A$ is a
M\"obius band then it is already a contradiction to the hyperbolicity of
$M$. If $A$ is an annulus and $p \neq q$ then a boundary component
$c$ of $A$ has intersection number $p-q \neq 0$ with $\cup \partial v_i$
and hence is an essential curve on $T_0$. Since $e_1$ is an essential
arc on both $A$ and $F_1$ and $F_1$ is boundary incompressible, $A$
cannot be boundary parallel. It follows that $A$ is an essential
annulus in $M$, which again contradicts the assumption that $M$ is
hyperbolic. \quad $\Box$ \medskip
\begin{lemma} Suppose $\Gamma_b$ is positive, $n_b \geq 3$, and
$\Gamma_a$ contains bigons $e_1 \cup e_2$ and $e'_1 \cup e'_2$, such that
$e_1,e'_1$ have label pair $\{i, j\}$ and $e_2,e'_2$ have label pair
$\{i+1, j+1\}$, where $j \neq i$. Let $C_1 = e_1 \cup e'_1$ and $C_2
= e_2 \cup e'_2$ be the loops on $\hat F_b$. If $C_1$ is essential on
$\hat F_b$ then $C_2$ is essential on $\hat F_b$ and not homotopic to
$C_1$. \end{lemma}
\noindent {\bf Proof.}
Let $B$ and $B'$ be the bigon faces bounded by $e_1 \cup e_2$ and
$e'_1 \cup e'_2$, respectively. Shrinking the Dehn filling solid
torus to the core knot $K_b$, the union $B \cup B'$ becomes an annulus
$A_1$ in $M(r_b)$ with boundary $C_1 \cup C_2$. Since $\hat F_b$ is
incompressible and $C_1$ is essential on $\hat F_b$, it follows that
$C_2$ must also be essential on $\hat F_b$.
Now assume $C_i$ are essential and homotopic on $\hat F_b$.
Since $i\neq j$ and $n_b>2$, $C_1, C_2$ have at most one
vertex in common. If $C_1, C_2$ are disjoint, let $A_2$ be an annulus
on $\hat F_b$ bounded by $C_1 \cup C_2$. If $C_1, C_2$ has a common
vertex $v_{i+1} = v_j$, let $A_2$ be the disk face of $C_1 \cup C_2$
in $\hat F_b$, which will be considered as a degenerate annulus as it
can be obtained from an annulus by pinching an essential arc to a
point. Let $A'_2$ be the closure of $\hat F_b - A_2$. Let $m$ and
$m'$ be the number of vertices in the interior of $A_2$ and $A'_2$,
respectively. Then $n_b = m+m'+k$, where $k$ is the number of vertices
on $C_1 \cup C_2$, i.e., $k=4$ if $C_1 \cap C_2 = \emptyset$, and
$k=3$ otherwise.
First consider the case that $C_1 \cap C_2 = \emptyset$. Orient $C_1,
C_2$ so that they are parallel on the annulus $A_1$. If they are also
parallel on $\hat F_b$ then $A_1 \cup A_2$ is a nonseparating torus
which can be perturbed to intersect $K_b$ at $m+2<n$ points, which is
a contradiction. If they are anti-parallel then $A_1 \cup A_2$ and
$A_1 \cup A'_2$ are Klein bottles which can be perturbed to intersect
$K_b$ at $m$ and $m'$ points, respectively. Since at least one of $m,
m'$ is less than $n_b/2$, this contradicts Lemma 2.12.
The case that $C_1 \cap C_2 \neq \emptyset$ is similar. If $C_1, C_2$
are parallel then $A_1 \cup A_2$ is a torus and can be perturbed to
intersect $K_b$ at $m+1<n$ points; if they are anti-parallel then $A_1
\cup A_2$ and $A_1 \cup A'_2$ can be perturbed to be Klein bottles
intersecting $K_b$ at $m+1$ and $m'$ points, respectively, which leads
to contradictions as above because $m+1+m'<n_b$ implies either
$2(m+1)<n_b$ or $2m' < n_b$.
\quad $\Box$ \medskip
A triple of edge endpoints $(p_1, p_2, p_3)$ on $\Gamma_b$ is {\it
positive\/} if they appear on the boundary of the same vertex $v_i$,
and in this order on $\partial v_i$ along the orientation of $\partial v_i$.
Note that this is true if and only if $d_{v_i}(p_1, p_2) +
d_{v_i}(p_2, p_3) = d_{v_i}(p_1, p_3)$.
\begin{lemma} (1) Suppose $(p_1,p_2, p_3)$ is a positive
triple on $\Gamma_b$. Let $k$ be a fixed integer and let $p'_i$ be edge
endpoints such that $d_{\Gamma_a}(p_i, p'_i) = k$ for all $i$. Then
$(p'_1, p'_2, p'_3)$ is also a positive triple on $\Gamma_b$.
(2) Let $e_1 \cup ... \cup e_r$ be a set of parallel negative edges
with end vertices $u_1, u_2$ in $\Gamma_a$. Let $u(p) \in \{u_1,u_2\}$ for
$p=1,2,3$, and let $e_j(u(p))$ be the endpoint of $e_j$ at $u(p)$. If
$(e_i(u(1)), e_j(u(2)), e_k(u(3)))$ is a positive triple and $i, j, k
\leq r-t$, then $(e_{i+t}(u(1)), e_{j+t}(u(2)), e_{k+t}(u(3)))$ is
also a positive triple. \end{lemma}
\noindent {\bf Proof.} (1) Geometrically this is obvious: Flowing on $T_0$ along $\partial
F_a$ moves the first triple to the second triple, hence the
orientations of the components of $\partial F_b$ containing these triples
are the same on $T_0$.
Alternatively one may use Lemma 2.16(ii) to prove the result. Since
$d_{\Gamma_a}(p_i, p'_i) = d_{\Gamma_a}(p_j, p'_j) = k$ for all $i,j$, by Lemma
2.16(ii) we have $d_{\Gamma_b}(p_i, p_j) = d_{\Gamma_b}(p'_i, p'_j)$ for all
$i,j$. Therefore $d_{\Gamma_b}(p_1, p_2) + d_{\Gamma_b}(p_2, p_3) =
d_{\Gamma_b}(p_1, p_3)$ if and only if $d_{\Gamma_b}(p'_1, p'_2) + d_{\Gamma_b}(p'_2,
p'_3) = d_{\Gamma_b}(p'_1, p'_3)$.
(2) This is a special case of (1) because
\begin{eqnarray*}
d_{\Gamma_a}(e_i(u(1)), e_{i+t}(u(1))) & = & d_{\Gamma_a}(e_j(u(2)), e_{j+t}(u(2))) \\
& = & d_{\Gamma_a}(e_k(u(3)), e_{k+t}(u(3))) = t \qquad
\end{eqnarray*}
\quad $\Box$ \medskip
\begin{lemma} Suppose $\Gamma_b$ is positive and $n = n_b \geq 3$.
(1) Suppose $\hat e \supset e_1 \cup ... \cup e_{n+2}$, and the
transition number $s = 1$. Let $A$ be the annulus obtained by cutting
$\hat F_b$ along the cycle $e_1 \cup ... \cup e_n$. Then the
edges $e_{n+1}, e_{n+2}$ lie in $A$ as shown in Figure 2.3, up to
reflection along the center circle of the annulus.
(2) If $s_1 = 1$, $n=3$ and $\hat e_1$ contains $6$ edges $e_1 \cup
... \cup e_6$ then the edges are as shown in Figure 2.4.
(3) Any family of parallel negative edges in $\Gamma_a$ contains at most
$2n$ edges.
\end{lemma}
\noindent {\bf Proof.} (1) Let $u, u'$ be the end vertices of $\hat e$ in $\Gamma_a$.
Orient $e_i$ from $u$ to $u'$ and assume without loss of generality
that $e_i$ has label $i$ at its tail $e_i(t)$ in $\Gamma_a$. Since $s =
1$, the head of $e_i$, denoted by $e_i(h)$, has label $i+1$ in $\Gamma_a$.
The edges $e_1, ..., e_n$ form an essential loop on the torus $\hat
F_b$. Cutting $\hat F_b$ along this loop produces an annulus, as
shown in Figure 2.3.
Up to reflection along the center circle of the annulus we may
assume that the edge $e_{n+1}$ appears in this annulus as shown in
Figure 2.3. We need to prove that $e_{n+2}$ appears in $\Gamma_b$ as shown
in the figure.
\bigskip
\leavevmode
\centerline{\epsfbox{Figure2.3.eps}}
\bigskip
\centerline{Figure 2.3}
\bigskip
Since $\Gamma_b$ is positive, we may assume that all vertices on Figure 2.3
are oriented counterclockwise. Note that $(e_1(h), e_{n+1}(h),
e_2(t))$ is a positive triple on $\Gamma_b$. By Lemma 2.21(2) the triple
$(e_2(h), e_{n+2}(h), e_3(t))$ is also a positive triple. This
determines the location of the head of $e_{n+2}$, as shown in Figure
2.3. Applying Lemma 2.20 to $e_1 \cup e_2$ and $e_{n+1} \cup
e_{n+2}$, we see that the loop $e_2 \cup e_{n+2}$ is essential and not
homotopic to $e_1 \cup e_{n+1}$ on $\hat F_b$, so these two loops must
intersect transversely at the common vertex $v_2$ on $\hat F_b$.
Hence the edge $e_{n+2}$ must appear as shown in Figure 2.3.
(2) By (1) the first 5 edges must be as shown in Figure 2.4. These
cut the torus into a 3-gon and a 7-gon. The edge $e_6$ is not
parallel to the other $e_i$'s on $\Gamma_b$ and hence must lie in the
7-gon, connecting $v_3$ to $v_1$. For the same reason as above,
$(e_3(h), e_6(h), e_4(t))$ is a positive triple on $\partial v_1$, hence
the head of $e_6$ must be in the corner on $\partial v_1$ from $e_1(t)$ to
$e_4(t)$ because the corner from $e_3(h)$ to $e_1(t)$ lies in the
3-gon. Similarly, since $(e_1(h), e_5(t), e_4(h))$ is a positive
triple on Figure 2.4, by Lemma 2.21(2) $(e_2(h), e_6(t), e_5(h))$ is
also a positive triple, which determines the position of the tail of
$e_6$. Therefore $e_6$ must be as shown in Figure 2.4.
(3) This follows from [Go, Corollary 5.5] when $n\geq 4$. Now assume
$n=3$ and suppose there exist $2n+1 = 7$ parallel edges $e_1 \cup ...
\cup e_7$ on $\Gamma_a$. By the 3-Cycle Lemma 2.14(2) we may assume that the
transition number $s \neq 0$. Since $n= 3$, we may assume without
loss of generality that $s = 1$, hence by (2) the subgraph of $\Gamma_b$
consisting of the edges $e_1 \cup ... \cup e_6$ is as shown in Figure
2.4. By the same argument as above, $(e_4(h), e_7(h), e_5(t))$ and
$(e_3(h), e_7(t), e_6(h))$ are positive triples on $\partial v_2$ and
$\partial v_1$, respectively. Since $e_7$ must lie in the $6$-gon face
$D$ in Figure 2.4, this is possible only if $e_7$ is parallel to
$e_1$, which is a contradiction to Lemma 2.2(2). \quad $\Box$ \medskip
\bigskip
\leavevmode
\centerline{\epsfbox{Figure2.4.eps}}
\bigskip
\centerline{Figure 2.4}
\bigskip
\begin{lemma} Let $sign(v)$ be the sign of a vertex $v$, and
define $p_a$ to be the sum of the signs of the vertices of $\Gamma_a$.
Then either $p_1=0$ or $p_2=0$. In particular, $n_1, n_2$ cannot both
be odd.
\end{lemma}
\noindent {\bf Proof.}
For each edge endpoint $c$ on $u_i \cap v_j$, define $\text{\rm sign}(c) =
\text{\rm sign}(u_i)\; \text{\rm sign}(v_j)$. Then the parity rule says that the
two endpoints of an edge $e$ have different sign. Summing over all
edge endpoints on $\Gamma_a$ gives
$$0 = \sum_{i,j} \Delta (\text{\rm sign}(u_i)\; \text{\rm sign}(v_j)) =
\Delta \sum_i \text{\rm sign}(u_i)\; \sum_j \text{\rm sign}(v_j) = \Delta p_1 p_2
$$
hence either $p_1 = 0$ or $p_2 = 0$.
\quad $\Box$ \medskip
\section{$\hat \Gamma_a^+$ has no interior vertex}
In this section we will show that if $n_{b} > 4$ then the graph $\Gamma_a$
does not have interior vertices; in particular the vertices of $\Gamma_a$
cannot all be parallel. Recall that we have assumed that $\Delta \geq 4$.
\begin{lemma} If $n_{b} > 4$ then $\hat \Gamma_a$ has no full vertex of
valence at most 6. \end{lemma}
\noindent {\bf Proof.}
First assume $n_b \neq 6$. Then by Lemma 2.7(3), three adjacent
families of positive edges in $\Gamma_a$ contain at most $2n_b -1$ edges,
hence if $\Gamma_a$ has a full vertex of valence at most 6 then
$$ 4n_b \leq \Delta n_b \leq 2(2n_b - 1) = 4n_b - 2,$$
a contradiction.
So suppose $n_b = 6$, and let $u_1$ be a full vertex of $\hat \Gamma_a$ of
valence at most $6$. By Lemma 2.3(3), each family of edges incident
to $u_1$ contains at most $4$ edges. Since there are at least $24$
edges in at most $6$ families, there must be exactly 6 families, each
containing exactly 4 edges. Each family contains a Scharlemann bigon,
so there are six Scharlemann bigons at the vertex $u_1$. Since there
are no extended Scharlemann cycles, each Scharlemann bigon appears at
one end of a family of parallel edges. Thus by examining the labels
around the vertex $u_1$, one can see that if one Scharlemann bigon has
label pair $\{1,2\}$ then the others must have label pair $\{1,2\}$,
$\{3,4\}$ or $\{5,6\}$, and that at least two pairs do occur as label
pairs of Scharlemann bigons. On the other hand, by Lemma 2.3(4) all
three pairs cannot appear as label pairs of Scharlemann bigons.
Hence, without loss of generality, there are incident to $u_1$ at
least three $(12)$-Scharlemann bigons and a $(34)$-Scharlemann bigon.
Since on $\Gamma_b$ the edges of the $(34)$-Scharlemann bigon form an
essential loop on $\hat \Gamma_b$, there are at most two edges of $\hat
F_b$ joining $v_1$ to $v_2$. Since the three $(12)$-Scharlemann bigons
give rise to six negative $1$-edges of $\Gamma_b$ joining $v_1$ to $v_2$,
three of these must be parallel, contradicting Lemma 2.7(1).
\quad $\Box$ \medskip
\begin{lemma} If $n_{b} > 4$ then $\hat \Gamma_a^+$ has no interior
vertices. \end{lemma}
\noindent {\bf Proof.} This follows from Lemma 3.1 if $n_a \leq 2$ because in this
case either $\hat \Gamma_a^+ = \hat \Gamma_a$ and there is a full vertex of valence at
most $6$, or $n_a = 2$ and there is no interior vertex. Therefore we
may assume that $n_a \geq 3$.
Suppose to the contrary that $\hat \Gamma_a^+$ has an interior vertex $u_i$. By
Lemma 3.1 all interior vertices of $\hat \Gamma_a^+$ have valence at least 7,
hence we can apply Lemma 2.11 to conclude that $\hat \Gamma_a^+$ has a boundary
vertex $u_1$ of valence at most 3.
By Lemma 2.7(3) the three families of adjacent positive edges at $u_1$
contain at most $2n_b$ edges, hence there are $2n_b$ adjacent negative
edges. On $\Gamma_b$ this implies that each vertex $v_j$ is incident to
two positive edges with label $1$ at $v_j$, which cannot be parallel
as otherwise there would be at least $n_a+1 > n_a/2 + 2$ parallel
positive edges, contradicting Lemma 2.3(3). Therefore the reduced
graph $\hat \Gamma_b$ contains at least $n_b$ positive edges. On the other
hand, the existence of an interior vertex in $\hat \Gamma_a^+$ implies that
$\hat \Gamma_b$ contains at least $2n_b$ negative edges, as shown in the proof
of Lemma 3.1. Since $\hat \Gamma_b$ has at most $3n_b$ edges (Lemma 2.5), it
must have exactly $n_b$ positive edges and $2n_b$ negative edges.
Since we have shown above that each vertex in $\hat \Gamma_b$ is incident to at
least two positive edges, it follows that it is incident to exactly
two positive edges.
We claim that a family of parallel positive edges in $\Gamma_b$ contains at
most $n_a/2$ edges. If such a family contains more than $n_a/2$
edges, then there is a Scharlemann bigon on one side of the family,
and by looking at the labels one can see that all labels appear among
the endpoints of this family, which is impossible because $u_i$ being
an interior vertex in $\hat \Gamma_a^+$ implies that all edges in $\Gamma_b$ with $i$
as a label are negative.
Since each vertex $v_j$ is incident to two families of positive edges,
each containing at most $n_a/2$ edges, we see that $v_j$ is incident
to at least $3n_a$ negative edges. By Lemmas 3.1 and 2.5 we see that
$\hat \Gamma_a$ has less than $3n_a$ positive edges, hence two of the negative
edges incident to $v_j$ are parallel in $\Gamma_a$, so $j$ is a label of a
Scharlemann bigon in $\Gamma_a$. Since this is true for all vertices in
$\Gamma_b$, by Lemma 2.3(4) we have $n_b \leq 4$, which is a contradiction.
\quad $\Box$ \medskip
\section{Possible components of $\hat \Gamma_a^+$ }
\begin{lemma} Suppose $\hat \Gamma_a^+$ has no isolated vertex or
interior vertex. If some $v_i$ of $\Gamma_b$ is incident to more than
$2n_{a}$ negative edges in $\Gamma_b$, or if $n_a > 4$ and $v_i$ is
incident to at most two families of positive edges in $\Gamma_b$, then $i$
is a label of a Scharlemann bigon in $\Gamma_a$. \end{lemma}
\noindent {\bf Proof.} If $v_i$ is a vertex of $\Gamma_b$ incident to more than $2n_{a}$
negative edges then by Lemma 2.10(2) two of them are parallel in
$\Gamma_a$, so by Lemma 2.4 they form a Scharlemann bigon, hence $i$ is the
label of a Scharlemann bigon in $\Gamma_a$. If $n_a>4$ and $v_i$ is
incident to two families of positive edges in $\Gamma_b$ then by Lemma
2.3(3) each family contains less than $n_{a}$ edges, hence $v_i$ is
incident to more than $2n_{a}$ negative edges and the result follows
from the above. \quad $\Box$ \medskip
In the rest of this section we assume $n_{a} > 4$ for $a = 1,2$, and
$\Delta \geq 4$. By Lemma 3.2 $\hat \Gamma_a^+$ has no interior vertices. We
will show that each component of $\hat \Gamma_a^+$ must be one of the 11 graphs
in Figure 4.2.
\begin{lemma} No vertex $u$ of $\hat \Gamma_a$ is incident to at most
four positive edges and at most one negative edge. \end{lemma}
\noindent {\bf Proof.} By Lemmas 2.7(1) and 2.7(2) a family of negative edges contains
at most $n_{b}$ edges, and four adjacent families of positive edges
contain at most $2(n_{b} + 2) = 2n_{b} + 4$ edges. Since $\Gamma_a$ has
at least $4n_{b}$ edges incident to $u$, we would have $n_{b} \leq
4$, which is a contradiction to our assumption. \quad $\Box$ \medskip
\begin{lemma} Suppose $u_i$ is incident to at most three
positive edges in $\hat \Gamma_a$, and if there are three then two of them are
adjacent. Then $i$ is a label of a Scharlemann bigon in $\Gamma_b$.
\end{lemma}
\noindent {\bf Proof.} In this case each label appears at the endpoint of some
negative edge at $u_i$, so $\hat \Gamma_b^+$ has no isolated vertex. By Lemma
4.1 the result is true if $u_i$ is incident to more than $2n_{b}$
negative edges. So we assume that $u_i$ is incident to no more than
$2n_{b}$ negative edges, and hence at least $2n_{b}$ positive edges.
By Lemma 2.7(2) the two adjacent families of positive edges contain at
most $n_{b}+2$ edges, while the other positive family contains no more
than $n_{b}/2+2$ edges. Thus $(n_{b}+2) + (n_{b}/2+2) \geq 2n_{b}$, which
gives $n_{b} \leq 8$. Since one of the positive families contains more
than $n_{b}/2$ edges, it contains a Scharlemann bigon; by Lemma 2.2(4)
$n_{b}$ must be even, so $n_{b} = 8$ or $6$. Using the above inequality
and the fact that when $n_{b} = 6$ each positive family contains at most
4 edges (Lemma 2.3(3)), we see that $u_i$ is incident to exactly
$2n_{b}$ positive edges and $2n_{b}$ negative edges. Dually, this implies
that in $\Gamma_b$ there are exactly $2n_{b}$ positive $i$-edges and $2n_{b}$
negative $i$-edges. (As always, an edge with both endpoints labeled
$i$ is counted twice.)
If $i$ is not a label of a Scharlemann bigon in $\Gamma_b$ then the $2n_{b}$
positive $i$-edges in $\Gamma_b$ are mutually nonparallel, so $\hat \Gamma_b$ has at
least $2n_{b}$ positive edges. By Lemma 2.5 the reduced graph $\hat \Gamma_b$
has no more than $3n_{b}$ edges, so it has at most $n_{b}$ negative edges.
On the other hand, by Lemma 2.7(1) each family of parallel negative
edges in $\Gamma_b$ has at most two endpoints labeled $i$; since there are
$2n_{b}$ such endpoints, $\Gamma_b$ must have at least $n_{b}$ families of
negative edges. It follows that $\Gamma_b$ has exactly $n_{b}$ families of
negative edges, each having exactly two endpoints labeled $i$.
Suppose $n_{b} = 6$. Then there are 12 edges in the three families
incident to $u_i$, and by Lemma 2.3(3) each family contains at most
four edges, hence each family contains exactly four edges. If some of
these edges are loops, then there are four loops and four non-loop
edges. No loop can be parallel to a non-loop edge in $\Gamma_b$
since otherwise the label $i$ would appear three times among a set of
parallel edges in $\Gamma_b$. It follows that all the 8 positive edges
incident to $u_i$ are mutually nonparallel in $\Gamma_b$, so the reduced
graph $\hat \Gamma_b$ would have at least 8 negative edges, which is a
contradiction as we have shown above that $\hat \Gamma_b$ has exactly $n_b = 6$
negative edges. Hence we can assume there is no loop based at $u_i$.
Note that a family of four parallel edges in $\Gamma_a$ contains a
Scharlemann bigon. If the label pair of the Scharlemann bigon is
$\{j, j+1\}$, then these two labels appear twice among the endpoints
of this family, and each of the other four labels appears exactly
once. By Lemma 2.3(4) at most four labels are the labels of some
Scharlemann bigons in $\Gamma_b$, so there is some $k$ which is not a label
of a Scharlemann bigon and hence appears exactly three times among the
endpoints of the positive edges incident to $u_i$. Dually, this
implies that some negative edge in $\hat \Gamma_b$ contains only one $i$-edge,
which is a contradiction as we have shown above that each negative
edge in $\hat \Gamma_b$ must contain exactly two negative $i$-edges.
The proof for $n_{b}=8$ is similar. In this case the numbers of edges
in the three positive families incident to $u_i$ are either $(6, 5,
5)$ or $(6, 6, 4)$. Using the fact that there are at most four labels
of Scharlemann cycles one can show that in either case some label
appears three times among the endpoints of these edges, which would
lead to a contradiction as above.
\quad $\Box$ \medskip
\begin{lemma} No vertex $u_i$ is incident to at most
one edge in $\hat \Gamma_a^+$.
\end{lemma}
\noindent {\bf Proof.} By Lemma 3.2 there are no interior vertices, hence by Lemma 2.11
either (i) $\hat \Gamma_b^+$ has a circle component, or (ii) $\hat \Gamma_b^+$ has a
boundary vertex of valence at most 3, or (iii) all vertices of $\hat \Gamma_b^+$
are boundary vertices of valence 4.
In case (i) a vertex $v_j$ on the circle component is incident to at
most two positive edges with label $i$ at $v_j$, hence dually there
are at most two negative edges with label $j$ at $u_i$, and hence at
least $\Delta - 2 \geq 2$ positive edges with label $j$ at $u_i$,
which is impossible because $u_i$ is incident to at most one family of
positive edges and by Lemma 2.3(3) such a family contains at most one
edge with label $j$ at $u_i$.
The proof for case (ii) is similar because by Lemma 2.7(3) a valence 3
boundary vertex $v_j$ of $\hat \Gamma_b^+$ is incident to at most $2n_a$
positive edges of $\Gamma_b$ and hence at most two positive edges with
label $i$ at $v_j$.
In case (iii), since $u_i$ is incident to at most $n_b/2+2 < n_b$
positive edges, there is a label $j$ such that all four edges with
label $j$ at $u_i$ are negative. Dually $v_j$ has four positive
$i$-edges. Since it is a boundary vertex, it is incident to at least
$3n_a + 1$ positive edges. On the other hand, since $v_j$ has valence
4 in $\hat \Gamma_b^+$, by Lemma 2.7(2) it has at most $2(n_a + 2) < 3n_a$
positive edges, a contradiction.
\quad $\Box$ \medskip
\begin{cor} Each component of $\hat \Gamma_a^+$ is contained in an
essential annulus but not a disk on $\hat F_a$.
\end{cor}
\noindent {\bf Proof.} By Lemma 2.6 $\hat \Gamma_a^+$ has at least two components, so if the
result is not true then one can find a disk $D$ on $\hat F_a$ such
that $D \cap \hat \Gamma_a^+$ is a component $G$ of $\hat \Gamma_a^+$. By Lemma 4.4 $G$
is not an arc, so by Lemma 2.9 it has at least three boundary vertices
of valence at most 3. By Lemma 4.3 these vertices are labels of
Scharlemann cycles in $\Gamma_b$, which is a contradiction because by Lemma
2.3(4) $\Gamma_a$ contains at most two labels of Scharlemann cycles of each
sign. \quad $\Box$ \medskip
Let $G$ be a component of $\hat \Gamma_a^+$ contained in the interior of an
essential annulus $A$ on $\hat F_a$. By Corollary 4.5, $G$ is not
contained in a disk, hence it contains some cycles which are
topologically essential simple closed curves on $\hat F_a$, and all
such cycles are isotopic to the core of $A$. We call such a cycle an
{\it essential cycle\/} on $G$. Note that a cycle may have more than
two edges incident to a vertex, but an essential cycle does not. An
essential cycle $C$ of $G$ is {\it outermost on $A$\/} if all
essential cycles of $G$ lie in one component of $A|C$. By cutting and
pasting one can see that outermost essential cycles always exist, and
there are at most two of them, which we denote by $C_l$ and $C_r$,
called the {\it leftmost cycle\/} and the {\it rightmost cycle},
respectively. Let $A_l^l$ and $A_l^r$ be the components of $A|C_l$,
called the {\it left annulus\/} and the {\it right annulus\/} of
$C_l$, respectively, labeled so that $A_l^l$ contains no essential
cycles of $G$ other than $C_l$. Similarly for $A_r^l$ and $A_r^r$,
where the right annulus $A_r^r$ of $C_r$ is the one that contains no
essential cycles other than $C_r$.
\begin{lemma} The interiors of $A_l^l$ and $A_r^r$ do not
intersect $G$.
\end{lemma}
\noindent {\bf Proof.} Assuming the contrary, let $G'$ be the closure of a component
of $G \cap A_l^l$. Since $G$ is connected, $G'$ must intersect $C_l$
at some vertex $v$, but it cannot intersect $C_l$ at more than one
vertex, as otherwise the union of an arc in $G'$ and an arc on $C_l$
would be an essential cycle in $A_l^l$ other than $C_l$, contradicting
the definitions of leftmost cycle and its left annulus. For the same
reason, $G'$ contains no essential cycles, hence it lies on a disk $D$
in $A_l^l$. By Lemma 4.4 $G$ has no vertex of valence 1, so $G'$ is
not homeomorphic to an arc. By Lemma 2.9 $G'$ has at least three
boundary vertices of valence at most 3. Let $v^1$ and $v^2$ be
such vertices other than $v$. They are boundary vertices of $G$ lying
in the interior of $A_l^l$ with valence at most $3$, and $v^i \neq v$.
By Lemma 4.3, for $i=1,2$ there is a Scharlemann bigon $\{e^i_1,
e^i_2\}$ on $\Gamma_b$ with $v^i$ as a label, and by Lemma 2.2(5) $C_i =
e^i_1 \cup e^i_2$ is an essential curve on $\hat F_{a}$ containing
$v^i$. Since $v^2$ is a boundary vertex of $G'$, it is not a cut
vertex, hence there is an arc $C'$ on $G'$ connecting $v^1$ to $v$
which is disjoint from $v^2$. Now the union $C_1 \cup C' \cup C_l$
cuts $\hat F_{a}$ into an annulus and a disk $D$ containing $v^2$ in its
interior, so the cycle $C_2$ is also contained in the disk $D$, which
is a contradiction to the fact that $C_2$ is topologically an
essential curve on $\hat F_{a}$. \quad $\Box$ \medskip
Lemma 4.6 shows that $G$ is contained in the region $R$ between $C_l$
and $C_r$. Since $G$ has no interior vertices, all its vertices are on
$C_l \cup C_r$. If $C_l$ is disjoint from $C_r$ then $R$ is an
annulus, and if $C_l = C_r$ then $R = C_l = C_r$ is a circle. In the
generic case we have $C_l \cap C_r = E_1 \cup ... \cup E_k$, where
each $E_i$ is either a vertex or an arc. The region $R$ is then a
union of these $E_i$ and some disks $D_1, ..., D_k$, such that $\partial
D_i$ is the union of two arcs, one in each of $C_r$ and $C_l$. When
$k=1$ and $E_1 = v$ is a vertex, $D_1$ is a disk with a pair of
boundary points identified to the single point $v$. Note that a
vertex of $G$ is a boundary vertex if and only if it is on $C_l \cup
C_r - C_l \cap C_r$.
\begin{lemma} Let $C = C_l$ or $C_r$.
(1) If $C$ has a boundary vertex $u_i$ of valence at most $3$ then it
has no other boundary vertex of valence at most $4$.
(2) If $C$ has a boundary vertex $u_i$ of valence $2$ then it has no
other boundary vertex. \end{lemma}
\noindent {\bf Proof.} (1) By Lemma 4.3, $i$ is a label of a Scharlemann bigon in
$\Gamma_b$. On $\Gamma_a$ the edges of this Scharlemann bigon form a cycle $C'$
containing $u_i$ and another vertex $u_k$. By Lemma 2.2(5) $C'$ is
topologically an essential circle on the torus $\hat F_a$. Since
$u_i$ is a boundary vertex, one can see that $C'$ is topologically
isotopic to $C$. By Lemma 4.6 applied to $G$ and to the component of
$\hat \Gamma_a^+$ containing $u_k$, there are no other vertices of $\Gamma_a$ between
$C'$ and $C$. Hence any boundary vertex $u_j \neq u_i$ on $C$ is
incident to at at most one family of parallel negative edges,
connecting it to $u_k$. The result now follows from Lemma 4.2.
(2) Note that since $u_i$ is a boundary vertex, the edges of any
Scharlemann bigon on $\Gamma_b$ with $i$ as a label must connect $u_i$ to
the same vertex $u_k$ on $\Gamma_a$, so there are at most $n_{b}$ such
bigons because there are only two edges on $\hat \Gamma_a$ connecting $u_i$ to
$u_k$, each representing a family of at most $n_{b}$ edges. By Lemma
2.7(2) $u_i$ is incident to at most $n_{b}+2$ positive edges, hence
at least $3n_{b} - 2$ negative edges. If a pair of these edges are
parallel on $\Gamma_b$ then they form a Scharlemann bigon. Hence by the
above we see that there are at most $n_{b}$ pairs of such edges. It
follows that $\hat \Gamma_b$ has at least $3n_{b} -2 - n_{b} = 2n_{b} - 2$
positive edges.
If $u_j$ is a boundary vertex of $C$ other than $u_i$ then as in the
proof of (1) it is incident to at most one family of negative edges,
so it has at least $3n_{b}$ positive edges. Since no three of those
are parallel on $\Gamma_b$, we see that $\hat \Gamma_b$ has at least $3n_{b}/2$
negative edges, so $\hat \Gamma_b$ would have a total of at least $2n_{b} -
2 + 3n_{b}/2 > 3n_{b}$ edges, contradicting Lemma 2.5. \quad $\Box$ \medskip
Now suppose $C_l \cap C_r \neq \emptyset$, and $C_l \neq C_r$. Then
the region $R$ between $C_l$ and $C_r$ can be cut along vertices of
$C_l \cap C_r$ to obtain a set of disks, and possibly some arcs. Let
$D$ be such a disk. If $C_l \cap C_r$ is a single vertex $v$ then $D$
is obtained by cutting $R$ along $v$, in which case we use $D \cap G$
to denote the graph on $D$ obtained by cutting $G$ along $v$.
\bigskip
\leavevmode
\centerline{\epsfbox{Figure4.1.eps}}
\bigskip
\centerline{Figure 4.1}
\bigskip
\begin{lemma}
$G' = D \cap G$ is one of the four graphs in Figure 4.1. \end{lemma}
\noindent {\bf Proof.} Let $v', v''$ be the vertices of $G'$ lying on both $C_l$ and
$C_r$. (Note that they are distinct vertices on $G'$ but may be
identified to a single vertex on $G$.) These vertices divide $\partial
D$ into two arcs $E_1$ and $E_2$, with $E_1 \subset C_l$ and $E_2
\subset C_r$.
By Lemma 4.7, each $E_j$ contains at most one vertex of valence at most
3 in its interior. Therefore, if $D$ contains at most one
interior edge then $G'$ has at most four vertices, so it is one of the
four graphs in Figure 4.1. We need to show that $D$ cannot have more
than one interior edge.
First suppose there is an interior edge $e$ of $G'$ which has both
endpoints on $E_1$. We may choose $e$ to be outermost in the sense
that there is an arc $E'$ on $E_1$ with $\partial E' = \partial e$, and there
is no edge of $G'$ inside the disk bounded by $E' \cup e$. Since $G'$
has no parallel edges, there must be a vertex $v$ in the interior of
$E'$, which has valence 2. By Lemma 4.7(2), in this case $C_l$ has no
other boundary vertices, so $E_1$ has no vertex other than $v$ in its
interior; in particular, $e$ must have its endpoints on $v'$ and
$v''$. This implies that all interior edges have both endpoints on
$E_2$, and by the same argument as above we see that $E_2$ has
exactly one vertex in its interior, and all edges must have endpoints
on $v'$ and $v''$. Since $G'$ has no parallel edges, it can have at
most one edge connecting $v'$ to $v''$, and we are done.
We can now assume that every interior edge of $G'$ has one endpoint in
the interior of each $E_i$. Let $G''$ be the union of the interior
edges. The above implies that $G''$ cannot have a cycle, so it is a
union of several trees with endpoints in the interiors of $E_1$ and
$E_2$. A vertex of valence 1 in $G''$ is a vertex of valence 3 in
$G'$, and by Lemma 4.7(1) there is at most one such for each $E_i$.
Therefore $G''$ is a chain, with two vertices of valence 1 and $k\geq
0$ vertices of valence 2, so $G'$ has one vertex of valence 3 on each
$E_i$, and $k$ vertices of valence 4. Note that these are boundary
vertices. However, by Lemma 4.7(1), if $G$ has a vertex of valence
$3$ on $C_l$ then it has no boundary vertex of valence at most $4$ on
$C_l$, and similarly for $C_r$. It follows that $k=0$, which again
implies that $G'$ has only one interior edge. \quad $\Box$ \medskip
\bigskip
\leavevmode
\centerline{\epsfbox{Figure4.2.eps}}
\bigskip
\centerline{Figure 4.2}
\bigskip
\begin{lemma} If $G$ is a component of $\hat \Gamma_a$
and $C_l \cap C_r \neq \emptyset$, then $G$ is one of the graphs in
Figure 4.2 (1) -- (8).
\end{lemma}
\noindent {\bf Proof.} If $C_l = C_r$ then $G$ is a simple cycle, in which case each
vertex has valence 2 and hence is a label of a Scharlemann bigon by
Lemma 4.3. By Lemma 2.3(4), $G$ has at most two such vertices, hence
$G$ is the graph in Figure 4.2(1) or (2).
Suppose $C_l \neq C_r$ and $C_l \cap C_r \neq \emptyset$. We call the
endpoints of $C_l \cap C_r$ {\it breaking points} of $G$, which cut the
region $R$ between $C_l$ and $C_r$ into several disks $D_1, ..., D_k$
and possibly some arcs. By Lemma 4.8 each $G_i = D_i \cap G$ is one
of the graphs in Figure 4.1. We say that $G_i$ is of type ($j$) if it
is the graph in Figure 4.1(j). Since $G$ can have at most two
boundary vertices of valence at most three, we see that either $k=1$,
or $k=2$ and both $G_i$ are of type (1).
First assume that $k=2$ and $G_1, G_2$ are of type (1). By Lemma 4.7
the two boundary vertices of $G_i$ must be one on each of $C_l,
C_r$. If the component of $C_l \cap C_r$ containing a breaking point
$v'$ on $G_i$ is an arc instead of a vertex, then $v'$ would be a
vertex of $\hat \Gamma_b$ which is incident to three positive edges, two of
which are adjacent, in which case by Lemma 4.3 $v'$ is a label of a
Scharlemann bigon in $\Gamma_b$. Since $G$ contains no more than two
Scharlemann bigon labels, this cannot happen. It follows that $G$ is
the graph shown in Figure 4.2(6).
We can now assume $k=1$. For the same reason as above, we see that if
$G_1$ is of type (1), (2) or (3), then $G$ is as shown in Figure 4.2(3),
(5) or (4), respectively. If $G_1$ is of type (4), the breaking vertices
may be incident to an edge in $C_l \cap C_r$, so $G$ is the graph in
Figure 4.2(7) or (8). \quad $\Box$ \medskip
\begin{lemma} If $G$ is a component of $\hat \Gamma_a^+$ and $C_l \cap
C_r = \emptyset$, then $G$ is one of the three graphs in Figure 4.2
(9), (10), or (11). \end{lemma}
\noindent {\bf Proof.} Note that in this case all vertices on $C_l$ and $C_r$ are
boundary vertices. If $C_l$ has a vertex of valence 2 then by Lemma
4.7 it has no other vertices, in which case $C_l$ is a loop and we have
$G = C_l$, so $C_l = C_r$, a contradiction. Therefore $C_l$ and $C_r$
have no vertices of valence 2, hence all vertices of $G$ have valence
at least 3.
Doubling the annulus and calculating Euler characteristic, we see that
$$\sum (4 - {\text val}(v_i)) \geq 0.$$
By Lemma 4.7 $G$ has at most two vertices of valence 3.
First assume that $G$ has two vertices $v_1, v_2$ of valence 3. By
Lemma 4.7 $v_1, v_2$ cannot both be on $C_l$ or $C_r$, hence each of
$C_l$ and $C_r$ contains exactly one vertex of valence 3. By Lemma
4.7 they cannot contain vertices of valence 4. By the above formula
$G$ has either (i) no other vertex, or (ii) one other vertex with
valence 6 or 5, or (iii) two other vertices, both having valence 5.
One can check that in Case (i) the graph is that of Figure 4.2(9),
Case (ii) does not happen, and in Case (iii) the graph is the one in
Figure 4.2(11).
If $G$ contains only one vertex $v$ of valence 3, then by the above
formula it contains at least one vertex of valence 5, and all other
vertices are of valence 4. If $C_l$ contains $v$ then by Lemma 4.7 it
contains no vertices of valence 4. Since each edge of $G$ must have
one endpoint on each of $C_l$ and $C_r$, we see that $C_l$ must
contain a vertex of valence 5, and $C_r$ contains exactly two
vertices, each of valence 4. One can check that there is no reduced
graph satisfying these conditions.
Now assume that $G$ has no vertices of valence 3. Then by the above
formula all vertices of $G$ are of valence 4. Since $G$ has no
parallel edges, all edges of $G - C_l \cup C_r$ must connect $C_r$ to
$C_l$, so the graph $G$ is completely determined by the number of
vertices $k$ on $C_r$, which must be the same as that on $C_l$.
Denote such a graph by $G_k$. When $k = 1$, the graph $G = G_1$ is
shown in Figure 4.2(10). We need to show that $k>1$ does not happen.
Suppose $k>1$. By Lemma 4.2 each vertex on $C_l$ is incident to at
least two negative edges in $\hat \Gamma_a$. Let $G'$ be the component of
$\hat \Gamma_a^+$ adjacent to $C_l$, and let $C'_r$ be the outermost cycle of
$G'$ adjacent to $C_l$. If $C'_r$ has only one vertex then two
negative edges based at some vertex $v_i$ on $C_l$ form an essential
loop on the annulus between $C'_r$ and $C_l$, so there is only one
negative edge of $\hat \Gamma_a$ incident to any other vertex on $C_l$, which
is a contradiction. Similarly if some vertex on $C'_r$ is a boundary
vertex of valence at most 3 then by Lemma 4.3 it is a label of a
Scharlemann bigon, which is again a contradiction because the two
edges of the Scharlemann bigon would form an essential loop as above.
This rules out the possibility of $G'$ being a graph in Figure 4.2 (1)
or (3) -- (11). If $G'$ is the one in Figure 4.2(2) then by Lemma 4.3
both of its vertices are labels of Scharlemann bigons. The edges of
these two Scharlemann bigons form two cycles, which cannot be on the
same side of $G'$ as otherwise one of them would lie on a disk, which
contradicts Lemma 2.2(5). Hence one of the pairs of edges connect a
vertex of $G'$ to a vertex of $C_l$, which is again a contradiction.
It now follows that if some component of $\hat \Gamma_a^+$ is a $G_k$ for $k\geq
2$, then so are all the other components. Moreover, none of the
vertices is a label of a Scharlemann cycle as otherwise some vertex
would be incident to a single negative edge in $\hat \Gamma_a$, which would
contradict Lemma 4.2. Hence $\Gamma_b$ has no Scharlemann cycles. On the
other hand, by Lemma 2.7(2) the four families of positive edges at a
vertex $v_i$ of $G_k$ contain at most $2n_{b}+4$ edges, so $v_i$ is
incident to at least $\Delta n_{b} - (2n_{b}+2) > n_{b}$ negative
edges. By Lemma 2.8 this implies that $\Gamma_b$ does have a Scharlemann
cycle, which is a contradiction. \quad $\Box$ \medskip
\begin{cor} Suppose $\Delta \geq 4$, and $n_a > 4$
for $a=1,2$. Then
(1) each component of $\hat \Gamma_a^+$ is one of the 11 graphs in Figure 4.2;
and
(2) each $\Gamma_a$ contains a Scharlemann cycle, hence $\hat F_{b}$ is
separating, and $n_{b}$ is even for $b = 1,2$.
\end{cor}
\noindent {\bf Proof.} (1) This follows from Lemmas 4.9 and 4.10.
(2) By (1), $\hat \Gamma_a^+$ contains either a vertex $v$ of valence 2 or a
boundary vertex of valence at most 4. In the first case the result
follows from Lemma 4.3. In the second case by Lemma 2.7(2) $v$ is
incident to at most $2n_{b} + 4$ positive edges, hence at least $2n_{b}-4
> n_{b}$ negative edges, so by Lemma 2.8 $\Gamma_a$ has a Scharlemann cycle.
\quad $\Box$ \medskip
\section {The case $n_1, n_2 > 4$ }
In this section we will complete the proof that the generic case $n_1,
n_2 > 4$ cannot happen. We assume throughout the rest of the section
that $n_1, n_2 > 4$. Let $G$ be a component of $\hat \Gamma_a^+$. By Corollary
4.11 $G$ is one of the graphs in Figure 4.2. We need to rule out all
these possibilities. Recall that a component of $\hat \Gamma_a^+$ is of type
(k) if it is the graph in Figure 4.2(k).
Here is a sketch of the proof. We first show (Lemma 5.4) that $\hat \Gamma_a^+$
cannot have two boundary vertices of valence 2, hence no component of
$\hat \Gamma_a^+$ is of type (5)--(8). Types (3) and (11) will be ruled out in
Lemmas 5.6 and 5.7, so we are left with types (1), (2), (4), (9) and
(10). Lemma 5.8 will show that each vertex of a type (10) component
is a label of Scharlemann cycle, which implies that all vertices of
$\hat \Gamma_a^+$ are labels of Scharlemann cycles, except the valence 4 vertex
in a type (4) component. Since $\hat \Gamma_a^+$ has at most two Scharlemann
labels of each sign, we see that each $\hat \Gamma_a^+$ is a union of two type
(4) components. This will be ruled out in Lemma 5.10, completing the
proof of the theorem.
Each vertex $u_i$ in $\Gamma_a$ has $\Delta$ edge endpoints labeled $j$.
Define $\sigma(u_i, v_j)$ to be the number of those on positive edges
minus the number of those on negative edges. In other words, it is
the sum of the signs of the edges with an endpoint labeled $j$ at
$u_i$.
Define a vertex $u$ of $\hat \Gamma_a^+$ to be {\it small\/} if it is either of
valence 2 or is a boundary vertex of valence 3. Note that a component
of type (1) or (3) in Figure 4.2 has one small vertex, a component of
type (10) has no small vertex, and all others have two small vertices.
\begin{lemma} (1) $\sigma(u_i, v_j) = - \sigma(v_j, u_i)$.
(2) If $v_j$ is a small vertex in $\hat \Gamma_b^+$ then $\sigma(u_i, v_j) \geq
0$ for all $i$.
(3) If $\hat \Gamma_a^+$ has a boundary vertex $u_i$ of valence 2, then
$\sigma(u_i, v_j) < 0$ for all but at most two $j$, at most one
for each sign.
(4) If $\hat \Gamma_a^+$ has a boundary vertex of valence 2, then $\hat \Gamma_b^+$ has at
most one small vertex of each sign.
\end{lemma}
\noindent {\bf Proof.}
(1) This follows from the parity rule Lemma 2.2(1).
(2) If $v_j$ has valence 2 in $\hat \Gamma_b^+$ then each label $i$ appears at
most twice among the positive edge endpoints. If $v_j$ is a boundary
vertex of valence 3 in $\hat \Gamma_b^+$ then by Lemma 2.7(3) it is incident to
at most $2n_a$ adjacent positive edges in $\Gamma_b$, hence again each $i$
appears at most twice among the positive edge endpoints. Since $\Delta
\geq 4$, the result follows.
(3) If $u_i$ is a boundary vertex of valence 2 then by Lemma 2.7(2)
there are at most $n_b + 2$ adjacent positive edges, so at most two
labels appear more than once among the positive edge endpoints, and if
there are two then they are adjacent, so there is only one for each
sign.
(4) This follows immediately from (2) and (3).
\quad $\Box$ \medskip
\begin{lemma} Suppose $\hat \Gamma_a^+$ has a boundary vertex $u_i$ of
valence 2. Then all components of $\hat \Gamma_b^+$ are of type (1), (3) or
(10). Moreover, for each sign there is at most one component with
vertices of that sign which is of type (1) or (3).
\end{lemma}
\noindent {\bf Proof.} This follows immediately from Lemma 5.1(4) and the fact that
a component of type (1) or (3) has one small vertex, a component of
type (10) has no small vertex, and all others have two small vertices.
\quad $\Box$ \medskip
\begin{lemma} Let $v_j$ be a vertex of a type (10) component
$G$ of $\hat \Gamma_b^+$.
(1) $v_j$ is incident to at most $2n_a+2$ positive edges in $\Gamma_b$.
(2) $\sigma(u_i, v_j) \geq 0$ for all but at most two $u_i$, one for
each sign.
\end{lemma}
\noindent {\bf Proof.} (1) By Lemma 2.7(2) the four families of adjacent parallel positive
edges incident to $v_j$ contain $m \leq 2(n_a+2)$ edges. If $m >
2n_a+2$, then in particular one of the families contains more than
$n_a/2$ edges, so it contains a Scharlemann bigon. By Lemma 2.2(4) and
Lemma 2.2(1) the labels at the endpoints of a loop at $v_j$ must have
different parity, which rules out the possibility $m=2n_a+3$.
Hence $m=2n_a+4$. Note that in this case there are at least 4
parallel loops $\{e_1, ..., e_4\}$, where $e_1$ is the outermost edge
on the annulus containing $G$. By looking at the labels at the
endpoints of these loops, we see that $e_2, e_3$ form a Scharlemann
bigon, which contradicts Lemma 2.2(6) because $\{e_1, e_4\}$ is then
an extended Scharlemann cycle.
(2) Since $v_j$ is a boundary vertex of $G$, the positive edges
incident to $v_j$ are adjacent. Therefore (1) implies that
$\sigma(v_j, u_i) \leq 0$ for all but at most two $i$, hence by Lemma
5.1(1) we have $\sigma(u_i, v_j) \geq 0$ for all but at most two
$u_i$, and if there are two such $u_i$ then they are of opposite sign.
\quad $\Box$ \medskip
\begin{lemma} $\hat \Gamma_a^+$ cannot have two parallel boundary
vertices of valence 2; in particular, no component $G$ of $\hat \Gamma_a^+$ is
of type (5), (6), (7) or (8). \end{lemma}
\noindent {\bf Proof.} Suppose to the contrary that $\hat \Gamma_a^+$ has two boundary vertices
$u_{i_1}, u_{i_2}$ of valence 2, and of the same sign. By Lemma 5.2,
each component $G'$ of $\hat \Gamma_b^+$ is of type (1), (3) or (10). If $G'$
is of type (3) then it has a boundary vertex of valence 2, so
applying Lemma 5.2 to this vertex (with $\hat \Gamma_a^+$ and $\hat \Gamma_b^+$ switched),
we see that $G$ must be of type (1), (3) or (10), which is a
contradiction. Therefore $G'$ must be of type (1) or (10).
By Lemma 5.1(3), $\sigma(u_{i_1}, v_k) < 0$ for all but at most two
$v_k$. Similarly for $\sigma(u_{i_2}, v_k)$. Since $n_b > 4$, there
is a vertex $v'$ such that $\sigma(u_r, v') < 0$ for both $r=i_1,
i_2$. On the other hand, if $v'$ is on a component $G'$ and if $G'$
is of type (1) then by Lemma 5.1(2) we have $\sigma(u_r, v') = -
\sigma(v', u_r) \geq 0$ for all $u_r$, while if $G'$ is of type (10)
then Lemma 5.3(2) says $\sigma(u_r, v') \geq 0$ for either $r=i_1$ or
$i_2$ because $u_{i_1}$ and $u_{i_2}$ are of the same sign. This is a
contradiction. \quad $\Box$ \medskip
Note that a vertex $u$ on a component $G$ of $\hat \Gamma_a^+$ is a boundary
vertex if it lies on one outermost essential cycle $C_1$ of $G$ but
not the other one. In this case there is a unique component $G'$ of
$\hat \Gamma_a^+$ and a unique outermost essential cycle $C_2$ on $G'$ such that
$C_1 \cup C_2$ bounds an annulus on $\hat F_a$ whose interior contains
no vertex of $\Gamma_a$. We say that $G'$ and $C_2$ are {\it adjacent\/}
to $u$.
\begin{lemma} Let $u_i$ be a vertex on a type (10) component
$G$ of $\hat \Gamma_a^+$. If $u_i$ is not a label of a Scharlemann cycle in
$\Gamma_b$, then
(i) the component $G'$ of $\hat \Gamma_a^+$ adjacent to $u_i$ is of
type (1), (3) or (10);
(ii) $u_i$ is incident to exactly $2n_{b}-2$ negative edges; and
(iii) $\hat \Gamma_b^+$ has only two components, each of type (4) or (11).
\end{lemma}
\noindent {\bf Proof.} We assume that $u_i$ is not a label of a Scharlemann cycle.
Let $G'$ and $C$ be the component and outermost cycle adjacent to
$u_i$. If $C$ has a boundary vertex $u_j$ of valence at most 3,
then by Lemma 4.3 $u_j$ is a label of a Scharlemann cycle. Since
$u_j$ is a boundary vertex and there is no vertex between $C$ and the
outermost cycle on $G$ containing $u_i$, the edges of the above
Scharlemann cycle must connect $u_j$ to $u_i$, hence $u_i$ is also a
label of the Scharlemann cycle, which is a contradiction. Also, if
$G'$ is of type (2) then by Lemma 4.1 each of its vertices is a label
of a Scharlemann cycle. Recall that the edges of a Scharlemann cycle
in $\Gamma_b$ cannot lie in a disk on $\hat F_{a}$, hence the edges of one
of the Scharlemann cycles must connect a vertex on $C$ to $u_i$, which
again is a contradiction. Therefore $C$ does not have a boundary
vertex of valence at most 3, and it is not on a type (2)
component. Examining the graphs in Figure 4.2, we see that $G'$ must
be of type (1), (3) or (10). Moreover, if it is of type (3) then $C$
is the loop there. In any case, $C$ contains only one vertex.
Let $t$ be the number of negative edges incident to $u_i$. Since $C$
has only one vertex $u_j$, $u_i$ is incident to at most two families
of negative edges $\hat e_1, \hat e_2$, all connecting $u_i$ to $u_j$,
so by Lemma 2.7(1) $t \leq 2n_{b}$. On the other hand, by Lemma 5.3
$u_i$ is incident to at most $2n_{b} + 2$ positive edges, so $t \geq
2n_{b} -2$. Therefore we have $2n_{b} \geq t \geq 2n_{b} -2$.
First assume $t=2n_{b}$. Then each of $\hat e_1$ and $\hat e_2$
contains exactly $n_{b}$ edges. Since $u_i$ is not a label of
Scharlemann cycle, by Lemma 2.4 these $2n_{b}$ edges are mutually
non-parallel on $\Gamma_b$, hence $\hat \Gamma_b^+$ has at least $2n_{b}$ edges. On
the other hand, by Lemma 4.1 it cannot have more than $2n_{b}$ such
edges, hence $\hat \Gamma_b^+$ has exactly $2n_{b}$ edges, each containing exactly
one edge in $\hat e_1 \cup \hat e_2$. Counting the number of edges on
each graph in Figure 4.2, we see that each component of $\hat \Gamma_b^+$ must
be of type (10) or (11). Also, a component of type (11) has a vertex
$v_k$ of valence 5 in $\hat \Gamma_b^+$, so the above implies that the label $k$
appears 5 times among the endpoints of edges in $\hat e_1 \cup \hat
e_2$, which is absurd. This rules out the possibility for a component
to be of type (11). Now notice that these two families of $n_{b}$
parallel edges have the same transition function, hence if some edge
has the same labels on its two endpoints, then they all do. It
follows that no component can be of type (10) because it has both loop
and non-loop edges. This completes the proof for the case
$t=2n_{b}$.
If $t = 2n_{b}-1$ then one of $\hat e_1, \hat e_2$ contains $n_{b}$
edges and the other contains $n_{b}-1$ edges. Examining the labels at
the endpoints of these edges we see that if an edge in $\hat e_1$ has
labels of the same parity at its two endpoints then an edges in $\hat
e_2$ would have labels of different parities at its endpoints, and
vice versa. This contradicts the parity rule (Lemma 2.2(1)).
We can now assume $t = 2n_{b} -2$. Without loss of generality we may
assume that the labels of the endpoints of $\hat e_1 \cup \hat e_2$
appear as $1,2,...,n_{b},1,..., n_{b}-2$ on $\partial u_i$ when traveling
clockwise, and we assume that the first $n_{b}$ are endpoints of $\hat
e_1$. (The other cases are similar.) Let $e^k_p$ ($k=1,2$) be the
edge in $\hat e_k$ with label $p$ at $u_i$, and assume that the label
of $e^1_1$ on $u_j$ is $1+r$ for some $r$. Then one can check that
the label of $e^1_p$ on $u_j$ is $p+r$, and the label of $e^2_p$ on
$u_j$ is $p+r+2$. (All labels are integers mod $n_{b}$.) Hence for any
$p$ between $3$ and $n_{b}$, the edges $e^1_p$ and $e^2_{p-2}$ have the
same label $p+r$ at $u_j$. On $\Gamma_b$ this implies that there are two
positive edges, connecting $v_p$ to $v_{p+r}$ and $v_{p+r}$ to
$v_{p-2}$, so $v_p$ are $v_{p-2}$ are in the same component of
$\hat \Gamma_b^+$. Since this is true for all $p$ between $3$ and $n_{b}$, it
follows that $\hat \Gamma_b^+$ has only two components.
By Lemmas 2.8 and 2.2(4) $\hat \Gamma_b^+$ has the same number of positive
vertices and negative vertices, hence each component $G$ has at least
three vertices. This rules out the possibility for $G$ to be of type
(1), (2), (3), (9) or (10). Combined with Lemme 5.4 we see that each
component of $\hat \Gamma_b^+$ is of type (4) or (11). \quad $\Box$ \medskip
\begin{lemma} No component of $\hat \Gamma_a^+$ is of type (3).
\end{lemma}
\noindent {\bf Proof.} By Lemma 5.2 if $\hat \Gamma_a^+$ has a component of type (3) then each
component of $\hat \Gamma_b^+$ is of type (1), (3) or (10), and there is at most
one component of type (1) or (3) for each sign. Since $n_b > 4$ and a
component of type (1) or (3) has at most 2 vertices, there is at least
one component $G$ of $\hat \Gamma_b^+$ of type (10) and at least one other
component $G'$ of the same sign. On the other hand, by Lemma 5.5 each
vertex of $G$ is a label of a Scharlemann cycle, and by Lemmas 4.1 and
4.3 at least one vertex of $G'$ is a label of a Scharlemann cycle, so
there are at least three labels of Scharlemann cycles of the same
sign, contradicting Lemma 2.3(4). \quad $\Box$ \medskip
\begin{lemma} No component of $\hat \Gamma_a^+$ is of type (11).
\end{lemma}
\noindent {\bf Proof.} An outermost cycle on a component $G$ of type (11) contains two
parallel vertices $u_i$ and $u_j$, where $u_i$ is of valence 3 and
hence the label of a Scharlemann bigon (Lemma 4.3), and $u_j$ has
valence 5. If $\{e_1, e_2\}$ is a Scharlemann bigon on $\Gamma_b$ with
label pair $\{i, i+1\}$, say, then on $\Gamma_a$ these edges form an
essential curve containing the vertices $u_i$ and $u_{i+1}$, which
separates $u_j$ from all other vertices of opposite sign, hence all
negative edges incident to $u_j$ have their other endpoints on
$u_{i+1}$, and they are all parallel. Thus $u_j$ has at most $n_{b}$
negative edges, and hence at least $3n_{b}$ adjacent positive edges. In
particular, each label appears at least three times among endpoints of
positive edges at $u_j$. Dually, each vertex $v_k$ in $\Gamma_b$ is
incident to at least three negative edges labeled $j$ at $v_k$. If
$v_k$ is a boundary vertex, then this implies that it is incident to
at least $2n_{a}+1$ negative edges, so by Lemma 4.1 it is a label of a
Scharlemann cycle.
By Lemmas 5.4 and 5.6 a component of $\hat \Gamma_b^+$ is of type (1), (2), (4),
(9), (10) or (11). By the above and Lemma 4.1 all vertices of $\Gamma_b$
except those with valence 4 in type (4) components are labels of
Scharlemann cycles. Since $n_{b}>4$ and there are at most two
Scharlemann labels for each sign, we see that $\hat \Gamma_b^+$ has only two
components, each of type (4), so $n_{b} = 6$, and $\hat \Gamma_b^+$ has 10
positive edges. By Lemma 2.5 $\hat \Gamma_b$ has at most $3n_{b} - 10 = 8$
negative edges. On the other hand, we have shown that $u_j$ in $\Gamma_a$
is incident to at least $3n_{b} = 18$ positive edges; since no three of
them are parallel in $\Gamma_b$, $\hat \Gamma_b$ has at least $18/2 = 9$ negative
edges, which is a contradiction. \quad $\Box$ \medskip
\begin{lemma} Each vertex of a type (10) component of $\hat \Gamma_a^+$
is a label of a Scharlemann bigon.
\end{lemma}
\noindent {\bf Proof.} Suppose that a vertex $u_i$ of a type (10) component of $\hat \Gamma_a^+$
is not a label of a Scharlemann bigon. By Lemmas 5.5 and 5.7 $\hat \Gamma_b^+$
is a union of two type (4) components, so $n_b=6$, $\hat \Gamma_b$ has 10
positive edges, and no more than $3n_{b} - 10 = 8$ negative edges.
By Lemma 5.5(ii) $u_i$ is incident to $(\Delta-2)n_b+2 = 6\Delta - 10$
positive edges (loops counted twice). By Lemma 2.7(1) no three of
these are parallel in $\Gamma_b$, hence they represent at least $3\Delta -
5$ negative edges in $\hat \Gamma_b$. Therefore $\Delta = 4$, and we have at
least $7$ negative edges in $\hat \Gamma_b$. We need to find two more to get a
contradiction.
By Lemma 5.5(i) and Lemma 5.6 the component $G$ of $\hat \Gamma_a^+$ adjacent to
$u_i$ is of type (1) or (10), so the outermost cycle of $G$ adjacent
to $u_i$ has a single vertex $u_j$ and a single edge $E_0$. We claim
that $E_0$ contains at least two edges of $\Gamma_b$.
If $G$ is of type (10), then $u_j$ is incident to four families of
positive edges in $\Gamma_a$, with a total of $2n_b + 2 = 14$ edges, where
loops are counted twice. By Lemma 2.3(3) each family contains no more
than 4 edges, so the loop edge $E_0$ contains at least $(14 - 2 \times
4)/2 = 3$ edges of $\Gamma_b$. If $G$ is of type (1) then since no three
negative edges incident to $u_j$ are parallel in $\Gamma_b$, and since
$\hat \Gamma_b^+$ has only 10 edges, we see that $u_j$ is incident to at most 20
negative edges, hence $E_0$ contains at least $(24-20)/2=2$ edges.
This completes the proof of the above claim.
Let $e'_1, e'_2$ be the two edges in $E_0$ closest to $u_i$. By Lemma
2.2(2) they are not parallel on $\Gamma_b$. We claim that on $\Gamma_b$ neither
of them is parallel to any edge incident to $u_i$, hence $\hat \Gamma_b$
contains at least $7+2=9$ negative edges. This will be a
contradiction as we have shown above that $\hat \Gamma_b$ has at most 8
negative edges.
By Lemma 5.5(ii) there are exactly $2n_b - 2 = 10$ negative edges
$e_1, ..., e_{10}$ connecting $u_i$ to $u_j$. Without loss of
generality we may assume that the sequence of labels of the endpoints
of these edges at $u_i$ is $1,...,6,1,...,4$, counting clockwise, and
the labels of their endpoints at $u_j$ are $r+2, r+3, ..., r-1$,
counting counterclockwise. Thus $\{e'_1, e'_2\}$ is a Scharlemann
bigon with label pair $\{r, r+1\}$.
Since $e'_i$ is a loop, by Lemma 2.3(5) if it is parallel in $\Gamma_b$ to
an edge $e$ incident to $u_i$ then $e$ is also a loop. Note that
$e'_i$ and $e$ must have the same label pair. Let $E_3$ be the loop
of $\hat \Gamma_a$ based at $u_i$. It has at most four edges $e''_1, e''_2,
e''_3, e''_4$, with label pairs $\{5,6\}, \{6,5\}, \{1, 4\}, \{2,3\}$,
respectively. By Lemma 2.3(2) we have $\{r, r+1\} \neq \{2, 3\}$,
hence if $e'_i$ is parallel to some $e''_j$ then $\{r, r+1\} =
\{5,6\}$, so $r = 5$, and hence the label sequence of the above
negative edges at $u_j$ is also $1, ..., 6, 1, ..., 4$.
The 10 edges $e_1, ..., e_{10}$ are divided into two families $E_1,
E_2$. Since $|E_i| \leq 6$, we have $|E_1| = 4$, 5, or 6. If
$|E_1|=5$ then the edge $e_1$ would have label $1$ at $u_i$ and label
$6$ at $u_j$. Since $v_1$ and $v_6$ on $\Gamma_b$ are antiparallel, this
is impossible by the parity rule. If $|E_1| = 4$ then $e_1$ has the
same label $1$ at its two endpoints, which contradicts the fact that
$\Gamma_b$ has no loop. Similarly if $|E_1|=6$ then $e_{7}$ has the same
label $1$ at its two endpoints, which is again a contradiction. This
completes the proof of the Lemma. \quad $\Box$ \medskip
\begin{lemma} Each $\hat \Gamma_a^+$ is a union of two type (4)
components. \end{lemma}
\noindent {\bf Proof.} By Lemmas 5.4, 5.6 and 5.7, each component $G$ of $\hat \Gamma_a^+$ is of
type (1), (2), (4), (9) or (10). By Lemmas 4.1, 4.3 and 5.8, we see
that all vertices $u_i$ of $G$ are labels of Scharlemann bigons,
unless $G$ is of type (4) and $u_i$ is the vertex of valence 4 in $G$.
Since $n_{a}>4$ and $\hat \Gamma_a^+$ has at most two vertices which are labels of
Scharlemann bigons for each sign, we see that $\hat \Gamma_a^+$ consists of exactly two
components, each of type (4). \quad $\Box$ \medskip
\bigskip
\leavevmode
\centerline{\epsfbox{Figure5.1.eps}}
\bigskip
\centerline{Figure 5.1}
\bigskip
\begin{lemma} One of the $\hat \Gamma_a^+$ is not a union of two type (4)
components. \end{lemma}
\noindent {\bf Proof.} Assume that each $\hat \Gamma_a^+$ is a union of two type (4) components.
Each vertex of $\Gamma_a$ has valence $\Delta n_b \geq 24$, hence $\Gamma_a$ has
at least 72 edges. Since a positive edge in $\Gamma_a$ is a negative
edge in $\Gamma_b$, we may assume that $\Gamma_1$ has no more negative
edges than positive edges, so $\Gamma^+_1$ has at least 36 positive
edges. Thus one component $G$ of $\Gamma^+_1$ has at least 18 edges.
Denote by $\hat G$ the reduced graph of $G$. It is of type (4), so it
is obtained from the graph in Figure 5.1(a) by identifying the top and
bottom vertices.
Let $E_1, ..., E_5$ be the edges of $\hat G$. Denote by $|E_i|$ the
number of edges of $G$ in $E_i$, and call it the {\it weight\/} of
$E_i$. By Lemma 2.3(3), each $|E_i| \leq 4$. Since $G$ has at least
18 edges, up to relabeling the weights of the edges are at least
$(4,4,4,4,2)$ or $(4,4,4,3,3)$.
Let $D$ be a triangle face of $\hat G$, and let $E_1$, $E_2, E_3$
be the edges of $D$. We will also use $D$ to denote the
corresponding triangle face in $G$. If $|E_i| = 4$ then by Lemma 2.4
$E_i$ contains a Scharlemann bigon, which must be at one end of the
family of parallel edges in $E_i$. We say that the Scharlemann bigon
in $E_i$ is {\it adjacent to $D$\/} if one of its edges is on the
boundary of $D$.
\bigskip
\leavevmode
\centerline{\epsfbox{Figure5.2.eps}}
\bigskip
\centerline{Figure 5.2}
\bigskip
\noindent
{\bf Sublemma} {\it If $|E_1| = |E_2| = 4$, then (i) $|E_3|=2$, and
(ii) exactly one of $E_1$ and $E_2$ has its Scharlemann bigon adjacent
to $D$. }
\bigskip
\noindent {\bf Proof.} Let $V_1$ be the fat vertex incident to both $E_1$ and $E_2$.
Without loss of generality we may assume that the labels on $\partial V_1$
are as shown in Figure 5.2(a), where $E_1$ is the upper right family of
edges. Note that the positions of the Scharlemann bigons in $E_1,
E_2$ determine the labels on $\partial V_2$ and $\partial V_3$.
If both $E_1$ and $E_2$ have their Scharlemann bigons adjacent to $D$,
then the labels are as shown in Figure 5.2(a), in which case we have
three Scharlemann bigons with disjoint label pairs, contradicting
Lemma 2.3(4). If both Scharlemann bigons of $E_1, E_2$ are
non-adjacent to $D$, then the labels are as shown in Figure 5.2(b), in
which case the edges adjacent to those of $D$ form an extended
Scharlemann cycle, which contradicts Lemma 2.2(6). This proves (ii).
We may now assume without loss of generality that the Scharlemann
bigon of $E_1$ is adjacent to $D$ while that of $E_2$ is not
adjacent to $D$. See Figure 5.2(c). In this case the label pair
of the Scharlemann bigon in $E_1$ is $\{3,4\}$. If $|E_3| \geq 3$
then $E_3$ contains a Scharlemann bigon with label pair $\{6, 1\}$.
This contradicts Lemma 2.3(2), completing the proof of the sublemma.
\quad $\Box$ \medskip
If the weights of the $E_i$ are $(4,4,4,3,3)$, or if the weights are
$(4,4,4,4,2)$ and the horizontal edge in Figure 5.1(a) has weight 4,
then the boundary edges of one of the triangles in Figure 5.1(a) have
weights $(4,4,3)$ or $(4,4,4)$, which contradicts the sublemma.
Therefore the edges of $G$ are exactly as shown in Figure 5.1(b). As
in the proof of the sublemma, we may assume that the labels at the
three vertices in the upper triangle of $G$ are as shown in Figure
5.1(b). The Scharlemann bigons in the upper triangle have label pairs
$\{3,4\}$ and $\{1,2\}$, hence by Lemma 2.3(4) $G$ cannot have a
Scharlemann bigon on label pair $\{5,6\}$. Therefore the labels of the
endpoints of the lower-right edges must be as shown in Figure 5.1(b).
This determines the labels at the lower vertex. But then neither
Scharlemann bigon in the lower triangle is adjacent to the triangle,
contradicting the sublemma. \quad $\Box$ \medskip
\begin{prop} The case that both $n_1, n_2 > 4$ is
impossible. \end{prop}
\noindent {\bf Proof.}
This follows from the contradiction between Lemma 5.9 and Lemma 5.10.
\quad $\Box$ \medskip
\section {Kleinian graphs }
In Sections 6 -- 11 we will improve Proposition 5.11 to show that $n_i
\leq 2$ for $i=1$ or $2$. For the most part we will assume that $n_a
= 4$. In this section we prove some useful lemmas. In particular,
Lemmas 6.2 -- 6.5 study kleinian graphs. Lemma 6.2 gives basic
properties of kleinian graphs, which will also be used later in
studying the case $n_a = 2$.
\begin{defn} The graph $\Gamma_a$ is said to be {\it
kleinian\/} if $\hat F_a$ bounds a twisted $I$-bundle over the Klein
bottle $N(K)$ such that each component of $N(K) \cap V_a$ is a $D^2
\times I$, and each component of $N(K) \cap F_b$ is a bigon.
\end{defn}
By Lemma 2.12, if $M(r_a)$ contains a Klein bottle $K$ intersecting
$K_a$ at $n_a/2$ points then $\partial N(K)$ is an essential torus
intersecting $K_a$ at $n_a$ points, hence in this case we may assume
that $\hat F_a = \partial N(K)$, where $N(K)$ is a small regular
neighborhood of $K$; in particular, $\Gamma_a$ is kleinian. In this case
$N(K)$ is called the {\it black region}, and all faces of $\Gamma_b$ lying
in this region are called {\it black faces}, and the others {\it white
faces}. We assume that the vertices of $\Gamma_a$ have been labeled so
that $u_{2i-1} \cup u_{2i}$ lie on the same component of $V_a \cap
N(K)$. The following lemma lists the main properties of kleinian
graphs.
\begin{lemma} Suppose $\Gamma_a$ is kleinian. Then
(1) each black face of $\Gamma_b$ is a bigon;
(2) each family of parallel edges in $\Gamma_b$ contains an even number of edges;
(3) $\Gamma_b$ has no white Scharlemann disk, hence any Scharlemann cycle
of $\Gamma_b$ has label pair $\{k, k+1\}$ with $k$ odd;
(4) there is a free involution of $\hat F_a$, which preserves $\Gamma_a$,
sending $u_{2i-1}$ to $u_{2i}$ and preserving the labels of edge
endpoints. \end{lemma}
\noindent {\bf Proof.} (1) follows from the definition. (2) follows from (1) because
if there is a family containing an odd number of edges then one side
of that family would be adjacent to a black face, which is not a
bigon.
(3) Each edge of a white face is adjacent to a black bigon, so if
there is a white Scharlemann disk then the edges of the Scharlemann
cycle and the adjacent edges would form an extended Scharlemann cycle,
which would be a contradiction to Lemma 2.2(6).
(4) We may assume that the Dehn filling solid torus $V_a$ and the
surface $F_b$ intersect $N(K)$ in $I$-fibers. Thus the involution of
$\hat F_a$ obtained by mapping each point to the other end of the
$I$-fiber gives rise to the required involution of $\Gamma_a$. \quad $\Box$ \medskip
\begin{lemma} Suppose $n_a = 4$. Then $\Gamma_a$ is kleinian if
each vertex of $\Gamma_a$ is a label of a Scharlemann bigon in $\Gamma_b$.
\end{lemma}
\noindent {\bf Proof.} Without loss of generality we may assume that $\Gamma_b$ has a
$(12)$ Scharlemann bigon. By assumption there is a Scharlemann bigon
with $3$ as a label. If there is no $(34)$ Scharlemann bigon
then this Scharlemann bigon must have label pair $(23)$. Similarly
the Scharlemann bigon with $4$ as a label must have label pair
$(14)$. We may therefore relabel the vertices of $\Gamma_a$ so that the
label pairs of the above Scharlemann bigons are $(12)$ and $(34)$
respectively.
Shrinking the Dehn filling solid torus to its core, the Scharlemann
bigons become M\"obius bands $B_{12}$ and $B_{34}$ in $M(r_a)$. The
union of these M\"obius bands, together with an annulus on $\hat F_a$,
becomes a Klein bottle which can be perturbed to intersect the core of
the Dehn filling solid torus at $2=n_a/2$ points. By the convention
after Definition 6.1, $\hat F_a$ should have been chosen so that $\Gamma_a$ is
kleinian.
\quad $\Box$ \medskip
\begin{lemma} Suppose $n_a=4$. Then $\Gamma_a$ is kleinian if one
of the following holds.
(1) $\Gamma_b$ has a family of 4 parallel positive edges.
(2) $\Gamma_b$ is positive.
(3) $\hat \Gamma_b^+$ has a full vertex $v_j$ of valence at most 7.
(4) $\hat \Gamma_b^+$ contains 4 adjacent families of positive edges with a
total of at least 12 edges.
\end{lemma}
\noindent {\bf Proof.} (1) Each label appears exactly twice among the edge endpoints
of a family of four parallel positive edges, hence by Lemma 2.4 it is a
label of a Scharlemann bigon.
(2) If $\Gamma_b$ is positive then every vertex $u_i$ of $\Gamma_a$ is incident
to at least $4n_b$ negative edges, two of which must be parallel in
$\Gamma_b$ because by Lemma 2.5 $\hat \Gamma_b$ contains at most $3n_b$ edges.
Hence by Lemma 2.4 these two edges form a Scharlemann bigon with $i$ as
a label. Since this is true for all $i$, $\Gamma_a$ is kleinian by Lemma
6.3.
(3) Consider the subgraph $G$ of $\hat \Gamma_a$ consisting of negative edges.
Then the signs of the vertices around the boundary of a face of $G$
alternate, hence each face has an even number of edges. Using an Euler
characteristic argument one can show that $G$ contains at most $2n_a =
8$ edges. By (2) we may assume $\Gamma_b$ is not positive, so by Lemma
2.3(1) no 3 $j$-edges are parallel on $\Gamma_a$, hence $G$ has exactly 8
negative edges, each containing exactly 2 $j$-edges, with one $j$
label at each ending vertex. Since each vertex $u_i$ has 4
$j$-labels, we see that $u_i$ is incident to exactly 8 $j$-edges, two
of which must be parallel in $\Gamma_b$ because $val(v_j, \hat \Gamma_b^+) \leq 7$. By
Lemma 2.4 they form a Scharlemann bigon with $i$ as one of its labels.
(4) By (1) we may assume that each family contains exactly 3 edges, so
the labels at the endpoints of the middle edge in each family are the
labels of a Scharlemann bigon. It is easy to see that the 4 endpoints
of the middle edges at the vertex are mutually distinct, hence include
all labels.
\quad $\Box$ \medskip
\bigskip
\leavevmode
\centerline{\epsfbox{Figure6.1.eps}}
\bigskip
\centerline{Figure 6.1}
\bigskip
\begin{lemma} Suppose $n_a = 4$. Let $e_1\cup e_2 \cup e_3
\cup e_4$ and $e'_1\cup e'_2 \cup e'_3 \cup e'_4$ be two families of
parallel edges in $\Gamma_b$ as shown in Figure 6.1. Then $e_i$ is
parallel to $e'_i$ on $\Gamma_a$ for all $i$. \end{lemma}
\noindent {\bf Proof.} Since $e_1\cup e_2$ and $e_3\cup e_4$ form two disjoint
essential cycles on $\hat F_a$ by Lemma 2.2(5), any $(12)$-edge must
be parallel to $e_1$ or $e_2$ and any $(34)$-edge parallel to $e_3$ or
$e_4$ on $\Gamma_a$. Note also that if $e_1$ is parallel to $e'_2$ on
$\Gamma_a$ then $e_2$ must be parallel to $e'_1$ (instead of $e'_2$) on
$\Gamma_a$ as otherwise $e_1, e_2$ would be parallel on both graphs.
Therefore if the result is not true then either $e_1$ is parallel to
$e'_2$ or $e_4$ is parallel to $e'_3$, so there is a subset $e'_r \cup
... \cup e'_s$ of the second family containing less than $4$ edges,
such that $e'_r \cup e_1$ and $e'_s \cup e_4$ are parallel pairs on
$\Gamma_a$. This contradicts Lemma 2.19
\quad $\Box$ \medskip
\begin{lemma} Suppose $n_a=4$ and $\Gamma_b$ is non-positive.
(1) No vertex $v_j$ of $\Gamma_b$ can have two families of 4 positive edges
with the same label sequence on $\partial v_j$. In particular, $v_j$
cannot have two adjacent families of 4 positive edges.
(2) If $\Gamma_a$ is kleinian, then two adjacent families of positive edges
of $\Gamma_b$ contain at most 6 edges, three contain at most 10, and four
contain at most 12.
(3) A full vertex of $\hat \Gamma_b^+$ has valence at least 6.
\end{lemma}
\noindent {\bf Proof.} (1) If there are two families of 4 positive edges with the same
label sequence on $\partial v_j$ then by Lemma 6.5 the two starting edges
$e_1, e'_1$ of these families will be parallel in $\Gamma_a$. If $e_1,
e'_1$ have label $i$ at $v_j$ then on $\Gamma_a$ they have the same label
$j$ at $u_i$, so there are $n_b +1$ parallel negative edges at $u_i$,
and hence by Lemma 2.3(1) $\Gamma_b$ would be positive, a contradiction.
(2) By Lemma 6.2(2) the number of edges in each family of positive
edges is either 2 or 4, so by (1) two adjacent families contain a
total of at most 6 edges. The other two cases follow from this.
(3) Otherwise by Lemma 6.4(3) $\Gamma_b$ is kleinian, so the weight of each
positive family of $\Gamma_b$ is either 2 or 4. If some full vertex $v_i$
has valence $5$ or less in $\hat \Gamma_b^+$ then it has two adjacent edge of
weight 4, contradicting (1).
\quad $\Box$ \medskip
A bigon is called a {\it non-Scharlemann bigon\/} if it is not a
Scharlemann bigon.
\begin{lemma} Suppose $n_a = 4$ and $\Gamma_a$ is kleinian.
(1) Exactly one edge on the boundary of a triangle face of $\hat \Gamma_b^+$
represents a non-Scharlemann bigon. Each of the other two represents
either a Scharlemann bigon or a union of two Scharlemann bigons.
(2) If some vertex $v_i$ is incident to two edges of weight 4 in
$\hat \Gamma_b^+$ then any other edge of $\hat \Gamma_b^+$ incident to $v_i$ represents a
non-Scharlemann bigon.
\end{lemma}
\noindent {\bf Proof.} (1) Let $\hat e_1, \hat e_2, \hat e_3$ be the edges of a
triangle face $\delta$ of $\hat \Gamma_b^+$. By Lemma 6.2(2) each edge of
$\hat \Gamma_b^+$ represents 2 or 4 edges. From the labeling of the edges
around $\delta$ one can see that there are exactly one or three $\hat
e_i$ which are neither a Scharlemann bigon nor a union of two
Scharlemann bigons. If there are three then they form an extended
Scharlemann cycle, which is impossible by Lemma 2.2(6). Hence there
must be exactly one such $\hat e_i$.
(2) Otherwise $v_i$ would be incident to 5 Scharlemann bigons, three
of which have the same label pair, say $\{1,2\}$. Then on $\Gamma_a$ there
are six $i$-edges connecting $u_1$ to $u_2$, which form at most two
families because there is a Scharlemann cocycle containing $u_3,
u_4$. It follows there there are three $i$ labels at the endpoints of
a family, so it contains more than $n_b$ edges, contradicting Lemma
2.3(1).
\quad $\Box$ \medskip
Suppose $\Delta = 4$. Then a label $j$ is a {\it jumping label\/} at
$u_i$ if the signs of the four $j$-edges incident to $u_i$ alternate.
\begin{lemma} Suppose $\Delta = 4$. Then a label $i$ is a
jumping label at $v_j$ if and only if $j$ is a jumping label at $u_i$.
In particular, if $v_j$ is a boundary vertex of $\hat \Gamma_b^+$ then $j$ is
not a jumping label at any $u_i$. \end{lemma}
\noindent {\bf Proof.} This follows from the Jumping Lemma 2.18. Let $x_1, ...,
x_4$ be the four points of $u_i \cap v_j$. Since $\Delta = 4$, the
jumping number must be $\pm 1$. Therefore they appear in this order
on both $\partial u_i$ and $\partial v_j$, appropriately oriented. If $j$
is a jumping label at $u_i$ then we may assume $x_1, x_3$ are positive
edge endpoints and $x_2, x_4$ are negative edge endpoints on $\partial
u_i$, which by the parity rule implies that $x_1, x_3$ are negative
edge endpoints and $x_2, x_4$ are positive edge endpoints on $\partial
v_j$, hence $i$ is a jumping label at $v_j$. \quad $\Box$ \medskip
\begin{lemma} Suppose $n_a=4$, $n_b\geq 4$, and $\Gamma_a$ is
non-positive. Then $\hat F_a$ is separating. In particular, $u_1$ is
parallel to $u_3$ and antiparallel to $u_2$ and $u_4$.
\end{lemma}
\noindent {\bf Proof.} The result follows from Lemmas 2.8 and 2.2(4) if some vertex
$u_i$ is incident to more than $n_b$ negative edges. In particular,
since each family of positive edges contain no more than $n_b$ edges,
the result is true if $val(u_i, \hat \Gamma_a^+) \leq 2$ for some $i$. Hence we
may assume that $val(u_i, \hat \Gamma_a^+) > 2$ for all $i$. One can check that
in this case each component of $\hat \Gamma_a^+$ must be as shown in Figure
4.2(9), (10) or (11). In each case $\hat \Gamma_a^+$ has a boundary vertex
$u_i$ of valence at most 4, so if $n_b > 4$ then by Lemma 2.7(b) the 4
families of positive edges contain at most $2(n_b+2) < 3n_b$ edges,
hence $u_i$ is incident to more than $n_b$ negative edges and the
result follows. Similarly if
$\hat \Gamma_a^+$ has a boundary vertex of valence at most $3$ then by Lemma
2.7(c) it is incident to at most $2n_b$ positive edges in $\Gamma_a$ and the
result follows. Therefore we may assume $n_b = 4$.
A vertex $u_i$ on a component in Figure 4.2 (9) or (10) is a boundary
vertex of valence 3 or 4 in $\hat \Gamma_a^+$, so by Lemma 2.7(b)--(c) it is
incident to less than $3n_b$ positive edges, and hence more than $n_b$
negative edges, unless $n_b = 4$ and $val(u_i, \hat \Gamma_a^+) = 4$. In
particular $\hat \Gamma_a^+$ must be of type (10) in Figure 4.2. In this last
case by Lemma 6.4(4) $\Gamma_b$ is kleinian, so by Lemma 6.2(2) each family
of positive edges of $\Gamma_a$ contains either 2 or 4 edges. Since there
is a total of at least 12 edges and by Lemma 6.6(2) two adjacent
families contain at most 6 edges, the weights of the four edges of
$\hat \Gamma_a^+$ incident to $u_i$ must be $(4,2,4,2)$ successively. However
since the first and the last belong to a loop in $\hat \Gamma_a^+$, their
weights must be the same, which is a contradiction. \quad $\Box$ \medskip
\section {If $n_a=4$, $n_b \geq 4$ and $\hat \Gamma_a^+$ has a small
component then $\Gamma_a$ is kleinian.
}
A component of $\hat \Gamma_a^+$ is {\it small \/} if it has at most two edges;
otherwise it is {\it large\/}. In this section we will show that if
$n_a=4$, $n_b \geq 4$ and $\hat \Gamma_a^+$ has a small component then $\Gamma_a$ is
kleinian. It is easy to see that the assumption implies that either
$val(u_1, \hat \Gamma_a^+) \leq 1$, or $val(u_1,\hat \Gamma_a^+)=val(u_3, \hat \Gamma_a^+) = 2$ up
to relabeling. (See the proof of Proposition 7.6.) The two cases are
handled in Lemmas 7.3 and 7.5, respectively.
\begin{lemma} Suppose $\Gamma_a$ contains a loop edge at $u_3$.
Then $\Gamma_b$ cannot contain both $(12)$- and $(14)$-Scharlemann bigons.
\end{lemma}
\noindent {\bf Proof.} The loop $e$ at $u_3$ must be essential, otherwise it would
bound some disk containing some vertex and hence one of the
Scharlemann cocycles in its interior, which contradicts Lemma 2.2(5).
Now the $(12)$- and $(14)$-Scharlemann bigons in $\Gamma_b$ form two
essential cycles in $\Gamma_a$ disjoint from $e$, so they must be isotopic
on $\hat F_a$, bounding a disk face containing no vertices of $\Gamma_a$ in
its interior. This is a contradiction to Lemma 2.13. \quad $\Box$ \medskip
\begin{lemma} Suppose $n_a = 4$ and $n_b \geq 4$. If $val(u_1,
\hat \Gamma_a^+) \leq 1$ and $\hat \Gamma_b^+$ has a boundary vertex $v_j$ of valence at
most 3, then $\Gamma_a$ is kleinian. \end{lemma}
\noindent {\bf Proof.} By Lemma 6.9 $u_1$ is parallel to $u_3$ and antiparallel to
$u_2, u_4$. Since $u_1$ is incident to at most 1 family of positive
edges, it is incident to at least three negative $j$-edges at $u_1$,
so $v_j$ has at least three positive edge endpoints labeled 1. Hence
$v_j$ being a boundary vertex implies that it has at least 9 positive
edges. If $v_j$ is incident to 10 or more positive edges of $\Gamma_b$
then it has a family of 4 parallel positive edges and hence $\Gamma_a$ is
kleinian. Therefore we may assume that it has exactly 9 positive
edges, divided into three families of parallel edges, each family
containing exactly three edges. See Figure 7.1.
\bigskip
\leavevmode
\centerline{\epsfbox{Figure7.1.eps}}
\bigskip
\centerline{Figure 7.1}
\bigskip
Since $n_a = 4$, each of these families contains a Scharlemann bigon,
so the labels of the middle edge in the family are labels of a
Scharlemann bigon. It follows that $1,2,4$ are labels of Scharlemann
bigons. Thus if the result is not true then $3$ is not a label of
Scharlemann bigon, and $\Gamma_b$ contains both $(12)$- and
$(14)$-Scharlemann bigons.
There are 7 adjacent negative edges at $v_j$, so three of
then have labels $1$ or $3$ at $v_j$. These cannot all be parallel in
$\Gamma_a$ as otherwise there would be three $j$-edges in a family and
hence the family would contain more than $n_b$ edges, contradicting
Lemma 2.3(1) and the fact that $\Gamma_b$ is not positive. On $\hat \Gamma_a^+$ this
implies that there are at least two edges with endpoints on $\{u_1,
u_3\}$, hence $val(u_1, \hat \Gamma_a^+) \leq 1$ implies that there is a loop
$\hat e$ based at $u_3$. Since $\Gamma_b$ contains both $(12)$- and
$(14)$-Scharlemann bigons, this is a contradiction to Lemma 7.1.
\quad $\Box$ \medskip
\begin{lemma} Suppose $n_a = 4$ and $n_b \geq 4$. If $val(u_i,
\hat \Gamma_a^+) \leq 1$ for some $i$ then $\Gamma_a$ is kleinian. \end{lemma}
\noindent {\bf Proof.} If $\Gamma_b$ is positive then $\Gamma_a$ is kleinian by Lemma 6.4(2).
Therefore we may assume that $\Gamma_b$ is non-positive. By Lemmas 2.3(3)
and 2.7(1) each family of parallel edges in $\Gamma_a$ contains at most
$n_b$ edges. Also, notice that since $u_i$ is incident to more than
$n_b$ negative edges, by Lemmas 2.8 and 2.2(4) the surface $\hat F_a$
is separating, hence $u_i$ is parallel to $u_j$ if and only if $i$ and
$j$ have the same parity.
Without loss of generality we may assume that $val(u_1, \hat \Gamma_a^+)\leq 1$.
Assume $\Gamma_a$ is not kleinian. Then by Lemma 7.2 $\hat \Gamma_b^+$ has no
boundary vertex of valence $3$, and by Lemma 6.4(3) it has no interior
vertex of valence at most 7. Also, each vertex $v_j$ of $\hat \Gamma_b^+$ has
valence at least 3 because it is incident to at least three positive
edges with label $1$ at $v_j$, which by Lemma 2.3(3) must be mutually
non-parallel. Therefore by Lemma 2.11 all vertices of $\hat \Gamma_b^+$ are
boundary vertices of valence 4.
If $n_b > 4$ then by Lemma 2.3(3) the family of positive edges at
$u_1$ contains at most $n_b/2 + 2 < n_b$ edges, so some $v_j$ is
incident to 4 positive edges with label $1$ at $v_j$, which implies
that $v_j$ has at least 13 positive edges in four families, so one of
the families contains 4 edges and hence $\Gamma_a$ is kleinian by Lemma
6.4(1). Similarly if $\Delta > 4$ then $\Gamma_a$ is kleinian.
\bigskip
\leavevmode
\centerline{\epsfbox{Figure7.2.eps}}
\bigskip
\centerline{Figure 7.2}
\bigskip
Now suppose $\Delta = n_b = 4$. Then $val(v_j, \hat \Gamma_b^+)=4$ for all $j$
implies that each component of $\hat \Gamma_b^+$ has two loops and two non-loop
edges, as shown in Figure 7.2(a). By the parity rule a loop based at
$v_j$ has labels of different parity on its two endpoints, hence one
sees that the number of positive edge endpoints of $\Gamma_b$ at each $v_j$
is even. By Lemma 6.4(4) we may assume that $v_j$ has less than 12
positive edges, hence the above implies that each $v_j$ is incident to
exactly 10 positive edges. If some $v_j$ is incident to only one loop
in $\Gamma_b$ then each of the non-loop family incident to $v_j$ contains 4
edges and we are done. If some $v_j$ is incident to two parallel
loops in $\Gamma_b$ then they form a Scharlemann bigon with label pair
$\{1,2\}$, say. Each of the two non-loop families contains three edges,
hence the middle edge endpoint is a label of a Scharlemann bigon.
Examining the labeling we see that all labels are Scharlemann bigon
labels.
We now assume that each $v_j$ is incident to three parallel loop
edges. See Figure 7.2(b). The two outermost loops form a Scharlemann
bigon with $1$ as one of its labels. There are 6 adjacent negative
edges at $v_1$, so three of then have labels $1$ or $3$ at $v_1$. By
the same argument as in the last paragraph of the proof of Lemma 7.2
we may assume that the two Scharlemann bigons at $v_1$ and $v_3$ have
the same label pair $(12)$. The labeling of edge endpoints around
$v_1$ and $v_3$ in a component of $\Gamma_b^+$ is now as shown in Figure
7.2(b).
Because of the parity rule, the 4 non-loop edges cannot be divided
into a family of 1 and another family of 3 edges, so they must form
two pairs of parallel edges. From the labeling in Figure 7.2(b) one
can see that they form two Scharlemann bigons with label pairs
$\{2,3\}$ and $\{4,1\}$, respectively. The result now follows from
Lemma 6.3.
\quad $\Box$ \medskip
We now assume that $val(u_1, \hat \Gamma_a^+) = val(u_3, \hat \Gamma_a^+) = 2$. Then
$\hat \Gamma_a^+$ contains either a cycle $C$ containing both $u_1, u_3$, or
it has two cycle components $C, C'$ containing $u_1, u_3$,
respectively.
\begin{lemma} If $val(u_i, \hat \Gamma_a^+) = 2$ then $i$ is a label of
a Scharlemann bigon in $\Gamma_b$.
\end{lemma}
\noindent {\bf Proof.} Let $k$ be the number of interior vertices in $\hat \Gamma_b^+$. Let $m$
be the number of edges in $\hat \Gamma_b^+$. We claim that $m_1 \leq 2n_b+k$.
Formally adding edges to $\hat \Gamma_b^+$ if necessary we may assume that any face
$A$ between two adjacent components of $\hat \Gamma_b^+$ is an annulus. It is
easy to see that if $\partial A$ contains $p$ vertices then we can add $p$
edges to make each face on $A$ a triangle. Therefore we can add at
least $n_b - k$ edges to $\hat \Gamma_b^+$ to create a graph $G$ on the torus
$\hat F_b$ whose faces are all triangles. By an Euler characteristic
argument we see that $G$ has $3n_b$ edges, hence $\hat \Gamma_b^+$ has at most
$3n_b - (n_b - k) = 2n_b + k$ edges, and the claim follows.
Now let $m'$ be the number of negative edges of $\Gamma_a$ incident to
$u_i$. Note that if $m < m'$ then two negative edges at $u_i$ are
parallel in $\Gamma_b$ and we are done. By Lemma 2.3(3) each positive
family $\hat e$ in $\Gamma_a$ contains at most $(n/2) + 2$ edges.
Moreover, if $k>0$ then some label does not appear on endpoints of
edges in $\hat e$, so $\hat e$ has at most $n_b/2$ edges. Since $u_i$
is incident to two families of positive edges, we have $m' \geq 4n_b -
2(n_b/2) = 3n_b$ if $k>0$, and $m' \geq 4n_b - 2(n_b/2 + 2) = 3n_b -
4$ if $k=0$. Since $m\leq 2n_b + k$, we have $m < m'$ (and hence $i$
is a label of a Scharlemann bigon in $\Gamma_b$), unless $k=0$, $n_b = 4$
and $m = m' = 8$.
In this last case ($k=0$, $n_b = 4$ and $m = m' = 8$), all vertices of
$\hat \Gamma_b^+$ are boundary vertices, hence by Lemma 6.8 there is no jumping
label at $u_i$. On the other hand, since $m' = 8$, each positive
family at $u_i$ have 4 edges, so the two positive families cannot be
adjacent by Lemma 6.6(2); hence there is a label $j$ such that the
two negative edges labeled $j$ at $u_i$ are separated by the two
positive edges labeled $j$ at $u_i$, so $j$ is a jumping label at
$u_i$, which is a contradiction. \quad $\Box$ \medskip
\begin{lemma} If $val(u_1, \hat \Gamma_a^+) = val(u_3, \hat \Gamma_a^+)=2$ then
$\Gamma_a$ is kleinian. \end{lemma}
\noindent {\bf Proof.} By Lemma 7.4 $u_1, u_3$ are labels of Scharlemann bigons. If
some vertex, say $u_4$, is not a label of Scharlemann bigon then there
must be $(12)$- and $(23)$-Scharlemann bigons in $\Gamma_b$. By Lemma 7.4
we have $val(u_4, \hat \Gamma_a^+) > 2$, so there is a loop edge $e$ of
$\hat \Gamma_a^+$ based at $u_4$. This is a contradiction to Lemma 7.1 (with
labels permuted).
\quad $\Box$ \medskip
\begin{prop} If $\hat \Gamma_a^+$ has a small component then (1)
$\Gamma_a$ is kleinian, and (2) $\hat \Gamma_a^+$ has at most 4 edges. \end{prop}
\noindent {\bf Proof.} Let $G$ be a small component of $\hat \Gamma_a^+$. If $G$ contains only
one vertex $u_1$ and two edges then it cuts the torus into a disk
containing the other three vertices. It is easy to see that in this
case there is a vertex of valence at most 2 in $\hat \Gamma_a^+$, which by Lemma
2.3(3) is incident to at most $2n_b$ edges, hence at least $2n_b$
negative edges. By Lemma 2.8 $\Gamma_b$ has a Scharlemann cycle, so the
surface $\hat F_a$ is separating. Therefore $u_3$ is parallel to
$u_1$ and is antiparallel to $u_2$ and $u_4$. It follows that $u_3$
is incident to no positive edges, so by Lemma 7.3 $\Gamma_a$ is kleinian.
If $G$ is not as above then either it contains a vertex of valence at
most 1, or it is a cycle, in which case (1) follows from Lemmas 7.3
and 7.5.
Since $\Gamma_a$ is kleinian, by Lemma 6.2(4) there is a free involution of
$\Gamma_a$ sending $u_i$ to $u_{i+1}$, hence the number of edges ending at
$\{u_2, u_4\}$ is the same as the number of edges ending at $\{u_1,
u_3\}$, which is at most two in all cases discussed above. Hence (2)
follows. \quad $\Box$ \medskip
\section {If $n_a=4$, $n_b \geq 4$ and $\Gamma_b$ is non-positive then
$\hat \Gamma_a^+$ has no small component
}
Denote by $X$ the union of $\hat \Gamma_b^+$ and all its disk faces.
\begin{lemma} Suppose $n_a=4$, $n_b \geq 4$, $\Gamma_b$ is
non-positive, and $\hat \Gamma_a^+$ has a small component. Then
(1) each vertex of $\Gamma_b$ is incident to at most 8 negative edges;
(2) if $v_j$ is incident to more than 4 negative edges then $j$ is a
label of a Scharlemann bigon;
(3) if $v_j$ is a boundary vertex of valence 3 in $\hat \Gamma_b^+$ then it is
incident to either 6 or 8 negative edges, and $j$ is a label of a
Scharlemann bigon;
(4) $val(v_j, \hat \Gamma_b^+) \geq 3$ if $v_j$ is a boundary vertex, and $\geq
2$ otherwise;
(5) each component of $X$ is either (a) a cyclic union of disks
and (possibly) arcs, or (b) a cycle, or (c) an annulus.
\end{lemma}
\noindent {\bf Proof.}
Since $\hat \Gamma_a^+$ has a small component, by Proposition 7.6 $\Gamma_a$ is
kleinian, and $\hat \Gamma_a^+$ has at most 4 edges.
(1) If $v_j$ is incident to 9 negative edges then three of them are
parallel on $\Gamma_a$ because $\hat \Gamma_a^+$ has at most four edges, which
contradicts Lemma 2.3(1).
(2) If $v_j$ is incident to 5 negative edges then two of them form a
Scharlemann bigon in $\Gamma_a$ because $\hat \Gamma_a^+$ has only four edges by
Proposition 7.6.
(3) Since $\Gamma_a$ is kleinian, by Lemma 6.2(2) $v_j$ is incident to an
even number of negative edges. Each family of positive edges contains
at most four edges, and by Lemma 6.6 two adjacent families contain at
most 6 edges, hence the three positive families at $v_j$ contain at
most 10 edges. The result now follows from (1) and (2).
(4) By (1) $v_j$ is incident to at least 8 positive edges, which are
divided into at least two families, and if two then they cannot be
adjacent by Lemma 6.6(1).
(5) If a component of $X$ is contained in a disk then by Lemma 2.9 it
would have either a boundary vertex of valence at most 2, which is
impossible by (4), or six boundary vertices of valence 3, which is a
contradiction because by (3) each such vertex is a label of
Scharlemann bigon while by Lemma 2.3(4) $\hat \Gamma_b^+$ has at most two labels
of Scharlemann bigons for each sign. Therefore no component of $X$ is
contained in a disk on the torus $\hat F_b$. Since $\Gamma_b$ is not
positive, this implies that each component of $X$ is contained in an
annulus but not a disk on $\hat F_b$.
If there is a sub-disk $D$ of $X$ such that $D \cap \overline{X-D}$ is a
single point $v$ then either $\hat \Gamma_b^+ \cap D$ contains a boundary vertex
of valence 2 other than $v$, or 3 boundary vertices of valence 3 other
than $v$, which again leads to a contradiction as above. \quad $\Box$ \medskip
Let $X_1$ be a component of $X$, and let $v_1$ be a boundary vertex on
the left cycle $C_l$ of $X_1$, as defined in Section 4. Then there
is another component $X_2$ of $X$ such that the annulus $A$ between
$C_l$ and the right cycle $C'_r$ of $X_2$ has interior disjoint from
$\hat \Gamma_b^+$. Denote by $m_j$ the number of negative edges incident to
$v_j$, and by $m'= m'_1$ the number of negative edges on $A$ which are
not incident to $v_1$.
\begin{lemma} Suppose $n_a=4$, $n_b \geq 4$, $\Gamma_b$ is
non-positive, and $\hat \Gamma_a^+$ has a small component. Let $v_1$ be a
boundary vertex of $X_1$ with $m_1 > 4$. Then
(1) $v_1$ is a label of a Scharlemann bigon;
(2) $m' = 0$ if $m_1 = 8$;
(3) $m' \leq 2$ if $m_1 > 4$;
(4) $C_l$ contains no other boundary vertices of valence at most 4.
\end{lemma}
\noindent {\bf Proof.} (1) Since $\hat \Gamma_a^+$ has only four edges, two of the negative
edges at $v_1$ form a Scharlemann bigon on $\Gamma_a$.
(2) If $m_1 = 8$ then since $\hat \Gamma_a^+$ has only 4 edges, the 8 negative
edges at $v_1$ form 4 Scharlemann cocycles, which must all go to the
same vertex $v_2$ on $C'_r$ because $v_1$ is a boundary vertex and the
cocycles are essential loops. These cocycles separate $C_l$ from
$C'_r$, hence all negative edges in $A$ incident to a vertex of $C_l -
v_1$ must have the other endpoint on $v_2$. On the other hand, by
Lemma 8.1(1) $v_2$ is incident to at most 8 negative edges, and by the
above all of them must connect $v_2$ to $v_1$. Hence $m' = 0$.
(3) By Proposition 7.6 $\Gamma_a$ is kleinian, so by Lemma 6.2(2) $m_1$ is
even; hence by (2) we may assume that $m_1 = 6$. Since $\hat \Gamma_a^+$ has
only 4 edges, the 6 negative edges incident to $v_1$ contain at least
2 Scharlemann cocycles, which connect $v_1$ to some $v_2$ on $C'_r$.
If $v_2$ is incident to $8$ negative edges in $A$ then as in (2) these
edges form 4 Scharlemann cocycles, which must all connect to the same
vertex $v_1$ and hence $m' = 0$. By Lemma 6.2(2) each family of
parallel edges in $\Gamma_b$ has an even number of edges, so $v_2$ cannot
be incident to 7 negative edges in $A$. If $v_2$ is incident to 6 or
less negative edges in $A$ then by the above 4 of them connect to
$v_1$, so there are at most 2 connecting to $C_l - v_1$, hence $m'
\leq 2$.
(4) A boundary vertex on $C_l - v_1$ of valence at most $4$ in $\hat \Gamma_b^+$
is incident to at most 12 positive edges by Lemma 6.6(2), and hence at
least 4 negative edges, which must lie in $A$ because it is a boundary
vertex on $C_l$. This is contradicts (3). \quad $\Box$ \medskip
\begin{lemma} Suppose $n_a=4$, $n_b \geq 4$, $\Gamma_b$ is
non-positive, and $\hat \Gamma_a^+$ has a small component. If a component $X_1$
of $X$ contains a boundary vertex $v_1$ of valence 3, then $X_1$ is an
annulus containing exactly two vertices, both of which are of valence
3 and are labels of Scharlemann bigons. \end{lemma}
\noindent {\bf Proof.} By Lemma 6.6(2) $v_1$ is incident to at least 6 negative edges.
Consider the three possible types of $X_1$ in Lemma 8.1(5). It cannot
be a cycle because it has a boundary vertex $v_1$. If $X_1$ is an
annulus or a cyclic union of disks and arcs then by Lemma 8.2(4),
$C_l-v_1$ has no boundary vertex of valence at most 4 in $\hat \Gamma_b^+$,
which implies that there is a boundary vertex $v_3$ of valence 3 on
the right circle $C_r$ of $X_1$, hence for the same reason $C_r - v_3$
contains no boundary vertex of valence at most 4. By Lemma 8.2(3)
there are at most 4 negative edges incident to $C_l \cup C_r - \{v_1,
v_3\}$, so there is no (non-boundary) vertex of valence 2 on $X_1$.
Thus either
(i) $X_1$ is an annulus containing only the two vertices $v_1$ and
$v_3$; or
(ii) $X_1$ is an annulus containing exactly four vertices and the
other two are boundary vertices of valence 5; or
(iii) $X_1$ is as in Figure 8.1 (a) or (b).
Case (i) gives the conclusion of the lemma because, as in the proof of
Lemma 8.2, a boundary vertex of valence 3 must be a label of a
Scharlemann bigon. We need to show that (ii) and (iii) are
impossible.
\bigskip
\leavevmode
\centerline{\epsfbox{Figure8.1.eps}}
\bigskip
\centerline{Figure 8.1}
\bigskip
In case (ii), let $v_5, v_7$ be the boundary vertices of valance 5 on
$X_1$, with $v_5 \subset C_l$. Note that all faces of $X_1$ are
triangles. If $v_1$ had 8 negative edges then by Lemma 8.2(2) $v_5$
would have no negative edge, which is impossible by Lemma 6.4(3).
Therefore $v_1$ has exactly 6 negative edges and 10 positive edges, so
by Lemma 6.6(1) the weights of the edges of $X_1$ incident to $v_1$
must be $(4,2,4)$. Now the middle edge of $X_1$ at $v_5$ has weight 2
and the two boundary edges have weight $4$, so again by Lemma 6.6(1)
the weights around $v_5$ must be $(4,2,2,2,4)$. By Lemma 6.7(2) the
edges of weight 2 must be non-Scharlemann. This is a contradiction
because a triangle with a corner at $v_5$ bounded by two weight 2
edges has two non-Scharlemann bigons on its boundary while by Lemma
6.7(1) it has only one.
In case (iii), we assume $X_1$ is as in Figure 8.1(a). The other case
is similar. Let $v_1, v_3$ be the vertices of valence $3$ in the
figure, and let $v_5$ be the other vertex. If $m_1 = 8$ then by Lemma
8.2(2)$v_5$ is incident to no negative edges on the side of $C_l$, and at
most 2 negative edges on the side of $C_r$. Therefore the four
positive edges of $\hat \Gamma_b^+$ at $v_5$ are adjacent to each other,
representing a total of $14$ edges. It follows that there are two
adjacent families of positive edges, each containing 4 edges, which is
a contradiction to Lemma 6.6(1). If $m_1 = m_3 = 6$ then each of
$v_1$ and $v_3$ is incident to 10 positive edges, so by Lemma 6.6(1)
the weights of the edges of $\hat \Gamma_b^+$ at $v_1$ and $v_3$ are $(4,2,4)$,
in which case $v_5$ again has two adjacent families of 4 positive
edges each, contradicting Lemma 6.6(1). \quad $\Box$ \medskip
\begin{lemma} Suppose $n_a=4$, $n_b \geq 4$, $\Gamma_b$ is
non-positive, and $\hat \Gamma_a^+$ has a small component. If a component $X_1$
of $X$ does not contain a boundary vertex of valence 3, then
(1) $X_1$ is either a cycle or an annulus containing exactly two
vertices; and
(2) all vertices of $X_1$ are labels of Scharlemann bigons.
\end{lemma}
\noindent {\bf Proof.} By Lemma 6.6(3) all interior vertices of $X$ have valence at
least 6. Since $X_1$ has no boundary vertex of valence 3, by Lemma
2.11 it is either a cycle, or an annulus with all interior vertices of
valence 6, all boundary vertices of valence 4, and all faces
triangles. The result is true when $X_1$ is a cycle because any
vertex of valence 2 has more than 4 negative edges and hence is a
label of Scharlemann cycle. Therefore we assume that $X_1$ is an
annulus. If $X_1$ has an interior vertex $v_1$ then by Lemmas 6.6(4)
the weights of the edges of $X_1$ around $v_1$ must be $(4,2,4,2,2,2)$
and any edge of weight 2 represents a non-Scharlemann bigon. Thus the
triangle with a corner at $v_1$ bounded by two weight 2 edges has the
property that it has at least two edges representing non-Scharlemann
bigons, which is a contradiction to Lemma 6.7(1). Therefore $X_1$ has
no interior vertex.
First assume that some vertex $v_1$ on the left cycle $C_l$ of $X_1$
is incident to less than 12 positive edges, and hence more than 4
negative edges. Since all vertices of $X_1$ are of valence 4 in
$\hat \Gamma_b^+$, by Lemma 8.2(3) $C_l$ has no other vertices. In this case
$C_r$ also contains only a single vertex $v_3$. By Lemma 6.6(2) $v_3$
cannot have more than 12 positive edges, and if 12 then the weights
around it are $(4,2,4,2)$. However this cannot happen because the
first and the last numbers are the weights of the loop edge at $v_3$
and hence must be the same. Therefore $v_3$ has less than 12 positive
edges. Now by Lemma 8.1(2) $v_3$ and $v_1$ are labels of Scharlemann
bigons, and the result follows.
Now assume that each vertex of $X_1$ is incident to 12 positive edges.
This implies that the weights of the edges of $X_1$ around each vertex
are either $(4,2,4,2)$ or $(2,4,2,4)$, so any pair of adjacent edges
have different weights. However, this is impossible because two of
the three edges of a triangle face must have the same weight.
\quad $\Box$ \medskip
\begin{lemma} Suppose $n_a=4$, $n_b \geq 4$, $\Gamma_b$ is
non-positive, and $\hat \Gamma_a^+$ has a small component. Then
(1) $n_b = 4$.
(2) $val(u_i, \hat \Gamma_a^+) \geq 2$ for all $i$.
(3) Each component of $\hat \Gamma_a^+$ is a loop.
\end{lemma}
\noindent {\bf Proof.} (1) By Lemmas 8.3 and 8.4 each vertex of $\Gamma_b$ is a label of a
Scharlemann bigon, and by Lemma 2.3(4) there are at most 4 such
labels. Hence $n_b = 4$.
(2) Suppose $u_1$ is incident to at most one edge of $\hat \Gamma_a^+$. By
Lemmas 8.3 and 8.4 each component $G$ of $\hat \Gamma_b^+$ consists of either
(i) a cycle, or (ii) two loops and one non-loop edge, or (iii) two
loops and two non-loop edges. Since there are at least 3 negative
$j$-labels at $u_1$, there are at least 3 positive $1$-labels at each
$v_j$, hence (i) cannot happen. Moreover, since each $v_j$ is a
boundary vertex containing at least three positive $1$-labels, it has
more than 8 positive edges.
Suppose $G$ is of type (ii). Then the label $1$ appears three times
on positive edge endpoints around each of the two vertices of $G$,
hence it appears a total of $6$ times among the three families of
positive edges in $G$, so by Lemmas 2.4 and 6.2(3) there is a
$(12)$-Scharlemann bigon among each of these families. Since a loop
and a non-loop edge cannot be parallel in $\Gamma_a$ (Lemma 2.3(5)), these
represents at least four edges of $\hat \Gamma_a$ connecting $u_1$ to $u_2$,
which cut the torus $\hat F_a$ into two disks. On the other hand, by
the parity rule a loop at a vertex $v$ of $G$ must have labels of
different parity on its two endpoints, so the total number of positive
edges at $v$ is at least 10, divided into three families, hence one of
the families has four parallel edges, which contains a
$(34)$-Scharlemann bigon, giving a pair of edges on $\Gamma_a$ lying in the
interior of the disks above which must therefore be parallel. This is
a contradiction to the fact that a Scharlemann cocycle is essential
(Lemma 2.2(5)).
Now suppose $G$ is of type (iii). If some $v_j$ is incident to $12$
positive edges then by Lemma 6.6(1) the weights of the positive edges
around $v_j$ are $(4,2,4,2)$, which is impossible because the first
and the last weights are for a loop and hence must be the same. Since
$v_j$ is incident to more than $8$ positive edges, it is incident to
exactly 10 positive edges, so the weights are $(2,4,2,2)$ or
$(2,2,4,2)$ around each vertex. The two loops must be
$(12)$-Scharlemann bigons in order for each vertex to have 3 edge
endpoints labeled $1$. This completely determines the labeling of the
edge endpoints up to symmetry. Examining the labeling one can see
that the family of 4 parallel edges form an extended Scharlemann
cycle, which is a contradiction to Lemma 2.2(6).
(3) By Proposition 7.6 $\Gamma_a$ is kleinian, hence the torus $\hat F_a$
is separating, so each edge of $\hat \Gamma_a^+$ has endpoints on vertices whose
subscripts have the same parity. By (2) a small component of $\hat \Gamma_a^+$
must be a loop $C$. Let $u_1$ be a vertex of $C$. If $C$ does not
contain $u_3$ then the component of $\hat \Gamma_a^+$ containing $u_3$ must also
be a loop because it contains no other vertices, and it cannot contain
more than one edge as otherwise some component of $\hat \Gamma_a^+$ would lie in
a disk and hence would have a vertex of valence at most 1,
contradicting (2). Thus the graph $G$ consisting of $u_1, u_3$ and
all edges with endpoints on them is either one loop or two disjoint
loops. By Lemma 6.2(4) there is a involution of $\hat \Gamma_a^+$ mapping $u_1$
and $u_3$ to $u_2$ and $u_4$ respectively, hence it maps $G$ to $\hat \Gamma_a^+
- G$. Therefore the components in $\hat \Gamma_a^+ - G$ are also loops.
\quad $\Box$ \medskip
\begin{prop} Suppose $n_a=4$, $n_b \geq 4$, and $\Gamma_b$ is
non-positive. Then $\hat \Gamma_a^+$ has no small component. \end{prop}
\noindent {\bf Proof.} Suppose to the contrary that $\hat \Gamma_a^+$ has a small component. By
Lemma 8.5 we have $n_b = 4$ and each component of $\hat \Gamma_a^+$ is a loop.
Thus each $u_i$ is incident to at most $8$ positive edges, so $\Gamma_a$
has no more positive edges than negative edges.
By Lemmas 8.3 and 8.4, each component $X_1$ of $X$ is either a circle
or an annulus containing two vertices of $\hat \Gamma_b^+$. First assume that
$X_1$ is a circle, so it is a small component of $\hat \Gamma_b^+$. Applying
Proposition 7.6 and Lemma 8.5 with $\Gamma_a$ and $\Gamma_b$ reversed, we see
that $\Gamma_b$ is kleinian, and all components of $\hat \Gamma_b^+$ are also
circles, hence $\Gamma_b$ also has the property that it has no more
positive edges than negative edges. Applying the parity rule we see
that both graphs have the same number of positive edges and negative
edges. In particular, each family of positive edges contains exactly
4 edges, which by Lemma 6.2(3) must consist of a (12)-Scharlemann bigon
and a (34)-Scharlemann bigon. Dually it implies that all negative
edges connect $u_1$ to $u_2$ or $u_3$ to $u_4$, so there are 4
families of negative edges, each containing exactly 4 edges. Now the
two positive families at $u_1$ contain 4 edges each, and, whether
separated by the two negative families or not, their endpoints at
$u_1$ have the same label sequence. This contradicts Lemma 6.6(1).
We may now assume that $X$ consists of two annular components $X_1,
X_2$, each containing two vertices. Assume $v_1, v_3 \in X_1$. As in
the last paragraph of the proof of Lemma 8.5(2), in this case each
vertex of $\Gamma_b$ is incident to 8 or 10 positive edges, therefore by
Lemma 8.1(2) each vertex is a label of Scharlemann bigon, hence $\Gamma_b$
is also kleinian.
If $val(v_1, \Gamma_b^+) = 10$ then $v_1$ is incident to 6 negative edges,
which are divided into two families of parallel edges on the annulus
bounded by the loops at $v_1$ and $v_2$. Since $\Gamma_b$ is kleinian,
each family contains an even number of edges, hence the number of
edges in these two families are $4$ and $2$, respectively. Examining
the labels at the endpoints of these edges, we see that two edges with
the same label $i$ at $v_1$ have different labels at $v_2$. On
$\Gamma_a$ this means that there are both loop and non-loop positive edges
incident to $u_i$, which is a contradiction to the fact that $\hat \Gamma_a^+$
consists of cycles only.
We have shown that $val(v_j, \Gamma_b^+) = 8$ for all $j$. Thus there are
16 positive edges on $\Gamma_a$, so each of the two positive families
incident to $u_1$ contains 4 edges. Since all vertices of $\Gamma_b$ are
boundary vertices, there is no jumping label on any vertex of $\Gamma_b$,
hence by Lemma 6.8 there is no jumping label at $u_1$, so the two
families of positive edges must be adjacent. This is a contradiction
to Lemma 6.6(2). \quad $\Box$ \medskip
\section {If $\Gamma_b$ is non-positive and $n_a=4$ then $n_b \leq 4$
}
Note that if $\Gamma_a$ is positive then each vertex of $\Gamma_b$ is a label of
a Scharlemann bigon and hence $n_b \leq 4$ by Lemma 2.3(4). By
Proposition 8.6 the statement in the title is true if $\hat \Gamma_a^+$ has a small
component. Therefore we may assume that $\hat \Gamma_a^+$ consists of two large
components $G_1$ and $G_2$, each of which must be one of the graphs of
type (3), (9) or (10) in Figure 4.2. As before, denote by $X_a$ the
union of $\hat \Gamma_a^+$ and all its disk faces, and by $X_i$ the components
of $X_a$ containing $G_i$, $i=1,2$. Denote $n = n_b$.
\begin{lemma} Suppose that $\Gamma_b$ is non-positive, $n_a=4$ and
$n > 4$. Then $\hat \Gamma_b^+$ contains no interior vertex.
\end{lemma}
\noindent {\bf Proof.} Otherwise $\Gamma_b$ has a vertex $v_i$ which is incident to
positive edges only. By Lemma 2.3(1) no three of these edges are
parallel on $\Gamma_a$, so $\hat \Gamma_a$ contains at least $\Delta n_a / 2 \geq 8$
negative edges, and hence at most $3n_a - 8 = 4$ positive edges by
Lemma 2.5, so $\hat \Gamma_a^+$ has a small component, contradicting our
assumption. \quad $\Box$ \medskip
\begin{lemma} Suppose that $\Gamma_b$ is non-positive, $n_a=4$ and
$n > 4$. Suppose $X_a$ is a disjoint union of two annuli. Let $G$
be the subgraph of $\Gamma_b$ consisting of positive $1$-edges and all
vertices. Then $G$ cannot have two triangle faces $D_1, D_2$ with an
edge in common. \end{lemma}
\noindent {\bf Proof.} Since $X_a$ is a disjoint union of two annuli, all negative
edges of $\Gamma_a$ incident to $u_1$ must have the other endpoint on the
same vertex, say $u_2$, and vice versa. On $G$ this means that every
edge has label pair $(12)$, and all positive edges with an endpoint
labeled $1$ or $2$ are in $G$. Thus no edge in the interior of $D_i$
has label $1$ or $2$ at any of its endpoints. Up to symmetry the
labels on the boundary of the two triangles must be as shown in Figure
9.1. Since the labels 3 and 4 must appear between two label 1 at a
vertex, one of the triangles, say $D_1$, must contain some $(34)$
edges. Since all the vertices are parallel, one can see that the
labels $3$ and $4$ appear at each corner of $D_1$, hence there are
three edges inside of $D_1$. Since there is no trivial loop, they
must form an extended Scharlemann cycle, contradicting Lemma 2.2(6).
\quad $\Box$ \medskip
\bigskip
\leavevmode
\centerline{\epsfbox{Figure9.1.eps}}
\bigskip
\centerline{Figure 9.1}
\bigskip
\begin{lemma} Suppose that $\Gamma_b$ is non-positive, $n_a=4$ and
$n > 4$. If $X_a$ is a disjoint union of two annuli then $\Gamma_a$ is
kleinian. \end{lemma}
\noindent {\bf Proof.} By Lemma 6.3 it suffices to show that each vertex $u_i$ of
$\Gamma_a$ is a label of a Scharlemann bigon. Let $t_i$ be the number of
negative edges at $u_i$. Since there are at most two families of
negative edges incident to $u_i$, we have $t_i \leq 2n$ by Lemma
2.3(1). On the other hand, since $u_i$ is a boundary vertex of
valence at most 4 in $\hat \Gamma_a^+$, by Lemma 2.7 it is incident to at most
$2(n+2) = 2n + 4$ positive edges. If there are $2n+4$ then $u_i$ is
incident to at least four loops and the four outermost loops form an
extended Scharlemann cycle, which is impossible by Lemma 2.2(6). Also
by the parity rule the loops have labels of different parity at its
two endpoints, hence the number of positive edges at $u_i$ must be
even. It follows that $u_i$ is incident to either $2n$ or $2n + 2$
positive edges, hence $t_i = 2n$ or $2n-2$.
Assume that $u_1$ is not a label of a Scharlemann bigon. Then the
negative edges at $u_1$ are mutually non-parallel positive edges in
$\Gamma_b$, hence $\hat \Gamma_b^+$ has at least $2n - 2$ edges. Denote by $G$ the
subgraph of $\Gamma_b$ consisting of positive $1$-edges. Let $Y$ be the
union of $G$ and its disk faces, and let $k$ be the number of boundary
vertices of $G$.
First assume $t_1 = 2n$. By Lemma 9.1 $G$ has no interior vertex, and
clearly it has no isolated vertex, so we can apply Lemma 2.10(1) to
conclude that $k \geq t_1 - n = n$. Since $G$ only has $n$ vertices,
we must have $k=n$, so all vertices of $G$ are boundary vertices, and
hence there is no cut vertex. In this case $Y$ contains exactly $n$
boundary edges, so it has at least one (actually $n$) interior edge
$e$. Since equality holds for the above inequality, by Lemma 2.10(1)
all faces of $Y$ are triangles. Therefore $e$ is the common edge of
two adjacent triangle faces, which is a contradiction to Lemma 9.2.
Therefore this case is impossible.
Now assume $t_1 = 2n-2$. In this case the two outermost loops at
$u_1$ form a Scharlemann bigon, so by Lemma 2.2(4) $\hat F_b$ is
separating, hence two vertices of $\Gamma_b$ are parallel if and only if
they have the same parity. Therefore we can define $G_1$ (resp.\
$G_2$) to be the union of the components of $G$ containing $v_i$ with
odd (resp.\ even) $i$. Similarly for $Y_1$ and $Y_2$. Then $G_1$
contains all the negative edges at $u_1$ with odd labels, and $G_2$
those with even labels. Therefore each $G_i$ contains exactly $n - 1$
edges.
The $2n + 2$ positive edges at $u_1$ form at least $n+1$ negative
edges in $\hat \Gamma_b$ because any family of $\Gamma_b$ contains at most 2 such
edges. Hence $\hat \Gamma_b$ contains at least $(n+1) + (2n - 2) = 3n - 1$
edges. Since a reduced graph on a torus contains at most $3n$ edges
(Lemma 2.5), we may add at most one edge to make the faces of the
graph all triangles. Hence $\hat \Gamma_b$ has at most one 4-gon and all other
faces are triangles. In particular, one of the $G_i$, say $G_1$, has
the property that all its faces are triangles.
Let $V$ and $E$ be the number of vertices and edges of $G_1$, and let
$E_b, V_b$ be the number of non-interior edges and boundary vertices,
respectively. Note that $V-V_b$ is the number of cut vertices, and
$E-E_b$ is the number of interior edges. We have shown that $V=n/2$
and $E=n-1$.
By Lemma 2.10(1) we have $V_b \geq E-V = (n-1) - n/2$, hence
$G$ has $V - V_b \leq V -(E-V) = (n/2) - (n-1-n/2) = 1$ cut vertex.
If there is no cut vertex then the number of non-interior edges is the
same as the number of vertices $V$, i.e.\ $E_b = V$. If it has a cut
vertex $v$ then the equality $V_b = E - V$ holds, so by Lemma 2.10(1),
$v$ has exactly two corners not on disk faces, which implies $v$ is
incident to at most 4 non-interior edges, while every other vertex is
incident to exactly two non-interior edges, hence $E_b \leq V+1$. In
either case we have $E - E_b \geq E - (V+1) = (n-1)-(n/2+1) \geq 1$,
so $G$ has at least one interior edge $e$. Since all faces of $Y_1$
are triangles, $e$ is incident to two triangle faces of $G_1$, which
is a contradiction to Lemma 9.2. \quad $\Box$ \medskip
\begin{lemma} Suppose that $\Gamma_b$ is non-positive, $n_a=4$ and
$n > 4$. Then $X_a$ is not a disjoint union of two annuli.
\end{lemma}
\noindent {\bf Proof.}
Assume to the contrary that $X_a$ is a union of two annuli. Let $t_i$
be the number of negative edges incident to $u_i$. As in the proof of
Lemma 9.3, we have $t_i = 2n-2$ or $2n$.
First assume that $t_1 = 2n-2$. Let $\hat e_1, \hat e_2$ be the two
families of edges in $\Gamma_a$ connecting $u_1$ to $u_2$. Note that a
$(12)$-Scharlemann bigon in $\Gamma_b$ must have one edge in each of $\hat
e_1$ and $\hat e_2$. By Lemma 9.3 $\Gamma_a$ is kleinian, so all
$(12)$-edges belong to Scharlemann bigons in $\Gamma_b$, hence each edge
$e_i$ in $\hat e_1$ is parallel in $\Gamma_b$ to an edge $e_i'$ in $\hat
e_2$, and the label of $e_i$ at $u_1$ is the same as that of $e_i'$ at
$u_2$. In particular, $\hat e_1$ and $\hat e_2$ have the same number
of edges, hence each contains exactly $n-1$ edges. Without loss of
generality we may assume that the label $n$ does not appear at the
endpoints at $u_1$ of edges in $\hat e_1$. By the above, $n$ does not
appear at the endpoints at $u_2$ of edges of $\hat e_2$, hence the
labels must be as in Figure 9.2. However, in this case the edge
labeled $1$ at $u_1$ has its other endpoint labeled $n$, which is
a contradiction to the parity rule.
\bigskip
\leavevmode
\centerline{\epsfbox{Figure9.2.eps}}
\bigskip
\centerline{Figure 9.2}
\bigskip
We now assume $t_1 = 2n$. Then the two families of negative edges
from $u_1$ to $u_2$ have the same transition function $\varphi$.
Since there is a $(12)$-Scharlemann bigon, $\varphi^2 = id$, so the
length of each $\varphi$-cycle is $1$ or $2$. Since $n>4$, it follows
that the edges of $\hat e_1$ form at least 3 cycles on $\Gamma_b$, which is
a contradiction to Lemma 2.14(2).
\quad $\Box$ \medskip
\begin{lemma} Suppose that $\Gamma_b$ is non-positive, $n_a=4$ and
$n > 4$. If $G_1$ is of type (3), then $G_2$ is also of type (3),
and the two loops of $\hat \Gamma_a^+$ do not separate the two vertices $u_3,
u_4$ which are not on the loops. \end{lemma}
\noindent {\bf Proof.} Let $u_3$ be the vertex of $G_1$ which has valence 2. If $G_2$
is not of type (3), or if the two loops $\hat e_1 \cup \hat e_2$ of
$\hat \Gamma_a^+$ separates $u_3$ and $u_4$ then all negative edges incident to
$u_3$ have their other endpoint on the same vertex $u_2$ of $G_2$, and
there are only two such families. Hence $u_3$ is incident to only
four families of parallel edges, so by Lemmas 2.3(1) and 2.3(3) we
must have $n = 4$ and each family contains exactly four edges.
Since the two families of positive edges are adjacent, this
contradicts Lemma 6.6(1). \quad $\Box$ \medskip
\begin{lemma} Suppose that $\Gamma_b$ is non-positive, $n_a=4$ and
$n > 4$. Then $\hat \Gamma_a^+$ cannot be a union of two type (3) components.
\end{lemma}
\noindent {\bf Proof.} Suppose that $\hat \Gamma_a^+$ is a union of two type (3) components, so
$\hat \Gamma_a$ has 6 positive edges $\hat e_1, ..., \hat e_6$. By Lemma 9.5
the two loops do not separate the two vertices which are not on the
loops, so the edges appear as in Figure 9.3. By Lemmas 2.3(1), 2.3(3)
and 6.6(1) each vertex has valence at least 5, so there is one edge
$\hat e_{7}$ from $u_2$ to $u_3$, two edges $\hat e_8, \hat e_9$ from
$u_1$ to $u_2$, and one edge $\hat e_{10}$ from $u_1$ to $u_4$. There
are one or two edges $\hat e_{11}$ and $\hat e_{12}$ connecting $u_3$
to $u_4$. See Figure 9.3.
\bigskip
\leavevmode
\centerline{\epsfbox{Figure9.3.eps}}
\bigskip
\centerline{Figure 9.3}
\bigskip
Denote by $w_i$ the number of edges in $\hat e_i$. By Lemma 2.7(2) the
two positive edges $\hat e_2, \hat e_3$ contain at most $n + 2$
edges, and the three negative families at $u_1$ contains at most $3n$
edges, hence $\Delta = 4$. Since each family contains at most $n$
edges, we have the following inequalities.
\begin{eqnarray}
& n \leq w_2 + w_3 \leq n+2 \\
& 3n-2 \leq w_8 + w_9 + w_{10} \leq 3n \\
& n-2 \leq w_i \leq n \qquad \text{for $i=7,8,9,10$}
\end{eqnarray}
Since $u_1$ has at least $n$ adjacent positive edge endpoints and $n$
adjacent negative edge endpoints, each vertex of $\Gamma_b$ is incident to
a positive edge and a negative edge, hence $\hat \Gamma_b^+$ has no isolated or
interior vertex.
\medskip
Claim 1. {\it $\hat \Gamma_b^+$ has at least $2n -2$ edges, hence $\Gamma_a$
has at most two jumping labels.}
Let $e \in \hat e_8$, and $e' \in \hat e_{10}$. Then $e$ and $e'$ are
not parallel in $\Gamma_b$ because if they were then they would form a
Scharlemann bigon and hence have the same label pair on their
endpoints, which is not the case because their label pairs are $\{1,
2\}$ and $\{1,4\}$ respectively. Therefore the number of edges in
$\hat \Gamma_b^+$ is at least
$$w_8 + w_{10} \geq 3n-2 - w_9 \geq 2n - 2$$
Since $\hat \Gamma_b^+$ has no interior or isolated vertex, by Lemma 2.10(1) it
has at most $2n - (2n - 2) = 2$ non-boundary vertices. Since
these are the only vertices containing jumping labels, by Lemma 2.18
they are the only possible jumping labels of $\Gamma_a$.
\medskip
Claim 2. {\it $w_{11} + w_{12} \geq 2n - 2$.}
Note that since $w_9, w_{10} \leq n$, we have
\begin{eqnarray*}
8n & = val(u_1,\Gamma_a) + val(u_2,\Gamma_a) \\
& = (w_2+w_3+ w_{10}) + (w_5+w_6+w_7) + 2 (w_8 + w_9)\\
& \leq (w_2+w_3+w_7) + (w_5+w_6+w_{10}) + 4n
\end{eqnarray*}
Thus either $w_2 + w_3 + w_7 \geq 2n$ or $w_5 + w_6 + w_{10} \geq
2n$. Because of symmetry we may assume
\begin{eqnarray}
w_2 + w_3 + w_7 \geq 2n
\end{eqnarray}
Divide the edge endpoints on $\partial u_3$ into $P_1, P_2, P_3, P_4$, as
shown in Figure 9.4. Denote by $k_i$ the number of edge endpoints in
$P_i$. A label that appears twice in one of the $P_i$ will be called
a {\it repeated label.\/} Note that if $P_i$ contains a repeated label
then $k_i > n$. Note also that a non-jumping label is a repeated
label. Thus by Claim 1 there are at least $n-2$ repeated labels among
all the $P_i$. Since $k_1 = w_7 \leq n$, there is no repeated label
in $P_1$.
\bigskip
\leavevmode
\centerline{\epsfbox{Figure9.4.eps}}
\bigskip
\centerline{Figure 9.4}
\bigskip
First assume that $k_3 > n$. If both $k_2, k_4 \leq n$ then all
repeated labels are in $P_3$, so $k_3 \geq 2n -2$ and we are done.
Because of symmetry we may now assume to the contrary that $k_2 > n$.
Thus one of $\hat e_1, \hat e_3$ contains more than $n/2$ edges,
so by Lemmas 2.4 and 2.2(4) $n$ is even, hence $n\geq 6$. Note that
in this case $k_4 \leq n$, as otherwise we would have $4n = \sum k_i
\geq (n-2) + 3(n+1) > 4n$, a contradiction. Thus $P_2, P_3$ contain
all the repeated labels and there are at least $n-2$ of them, so $k_2
+ k_3 \geq 3n-2$. Since $k_2 = w_1 + w_3 \geq n+1$ and $w_3 \leq
(n/2)+2$, we have $w_1 \geq (n/2)-1$, so $w_3-w_1 \leq 3$. By
equation (4) above, we have
\begin{eqnarray*}
4n & = val(u_3, \Gamma_a) = w_7 + k_2 + k_3 + (w_1 + w_2) \\
& = (k_2 + k_3) + (w_7 + w_3 + w_2) - (w_3 - w_1) \\
& \geq (3n-2) + 2n - 3 = 5n - 5 > 4n \end{eqnarray*}
This is a contradiction, which completes the proof for the case
$k_3>n$.
We now assume $k_3 \leq n$. Then each non-jumping label is a repeated
label in either $P_2$ or $P_4$. By Lemma 2.7(2) we have $k_2, k_4
\leq n+2$, so each of $P_2, P_4$ contains at most $2$ repeated labels.
Since there are at least $n-2$ repeated labels, we have $n-2 \leq 4$,
i.e., $n\leq 6$. As above, $P_2$ or $P_4$ containing a repeated label
implies that $n$ is even, so $n = 6$. In this case $k_2 = k_4 = 8$,
so by Lemma 2.3(3) each of $\hat e_1, \hat e_2, \hat e_3$ contains
exactly 4 edges.
By equation (3) above we have $ n-2 \leq k_1 = w_7 \leq n$. If $k_1 =
n-1$ or $n-2$ then the labels that do not appear in $P_1$ are repeated
labels of both $P_2$ and $P_4$, so there are at most 3 distinct
repeated labels, and hence at least $n-3 = 3$ jumping labels,
contradicting Claim 1. If $k_1 = n$ then $k_1 + k_2 + k_4 = 3n+4$, so
the endpoints of the 4 edges of $\hat e_1$ at $P_2$ have the same
label sequence as those of $\hat e_1$ at $P_4$. Hence these edges
form an extended Scharlemann cycle, which is impossible by Lemma
2.2(6).
\medskip
Claim 3. {\it $\Gamma_a$ is kleinian.}
Clearly each of $u_1, u_2$ is incident to more than $2n$ edges. Since
$w_7 + w_8 + w_9 = 4n - (w_5 + w_6) \geq 3n-2$, we have $w_7 \geq
n-2$. By Claim 2 $w_{11}+w_{12} \geq 2n-2$, hence $w_{11}+w_{12}+w_7
\geq 3n-4 > 2n$, so $u_3$ is also incident to more than $n$ negative
edges. Similarly for $u_4$. By Lemmas 9.1 and 2.10(2) $\hat \Gamma_b^+$ has at
most $2n$ edges, so two of the negative edges at $u_i$ are parallel on
$\Gamma_b$, hence $u_i$ is a label of Scharlemann bigon. Since this is
true for all $i$, $\Gamma_a$ is kleinian.
\medskip
Claim 4. $w_{11} + w_{12} = 2n - 2$.
Since $\Gamma_a$ is kleinian, by Lemma 6.2(4) there is a free involution
$\eta$ of $\Gamma_a$ mapping $u_1$ to $u_2$ and $u_3$ to $u_4$. Thus
$\eta$ must map $\hat e_{10}$ to $\hat e_7$ and hence $w_7 =
w_{10}$. We now have $w_2 + w_3 + w_7 = w_2 + w_3 + w_{10} \geq 4n -
w_8 - w_9 \geq 2n$. Since $w_1 \geq 1$, we see that $w_{11} + w_{12}
= 4n - (2w_1 + w_2 + w_3 + w_7) \leq 2n - 2$. The result now
follows from Claim 2.
\medskip
The involution $\eta$ maps $\hat e_{11}$ to $\hat e_{12}$. Hence
Claim 4 implies that $w_{11} = w_{12} = n - 1$. Since $\eta$ is
label preserving, the label sequence of $\hat e_{11} \cup \hat e_{12}$
on $\partial u_3$ is the same as that on $\partial u_4$, so we may assume
without loss of generality that the label sequences are as shown in
Figure 9.2. One can see that in this case the transition function of
$\hat e_{11}$ defined in Section 2 is transitive, which implies that
all vertices of $\Gamma_b$ are parallel, contradicting the assumption.
This completes the proof of the lemma. \quad $\Box$ \medskip
\begin{prop} Suppose $n_a = 4$ and $\Gamma_b$ is
non-positive. Then $n_b \leq 4$. \end{prop}
\noindent {\bf Proof.} Consider $\hat \Gamma_a^+$. If $\hat \Gamma_a^+$ has a small component then by
Proposition 8.6 we have $n_b \leq 4$. If $\hat \Gamma_a^+$ has no small
component then the component $G$ containing $u_1$ must also contain
$u_3$, and it is either of type (3), (9) or (10). The result follows
from Lemma 9.4 if both components are of type (9) or (10), and from
Lemmas 9.5 and 9.6 if at least one component is of type (3).
\quad $\Box$ \medskip
\section {The case $n_1 = n_2 = 4$ and $\Gamma_1, \Gamma_2$
non-positive }
In this section we assume that $n_1 = n_2 = 4$ and $\Gamma_a$ is
non-positive for $a=1,2$. We will show that this case cannot happen.
Denote by $X_a$ the union of $\hat \Gamma_a^+$ and all its disk faces. By
Theorem 8.6 $\hat \Gamma_a^+$ has no small component, so each component of
$\hat \Gamma_a^+$ is of type (3), (9) or (10) in Figure 4.2.
\begin{lemma} Suppose $n_1 = n_2 = 4$, and both $\Gamma_1$ and
$\Gamma_2$ are non-positive. Then at least one component of $\hat
\Gamma^+_1$ or $\hat \Gamma^+_2$ is of type (3). \end{lemma}
\noindent {\bf Proof.} Suppose to the contrary that all components of $\hat
\Gamma^+_1$ and $\hat \Gamma^+_2$ are of type (9) or (10). Then each
component of $X_a$ is an annulus, hence any vertex $u_i$ of $\Gamma_a$ is
incident to at most 2 families of negative edges. By Lemma 2.3(1)
each negative family contains at most 4 edges, so $u_i$ is incident to
at most 8 negative edges, and hence the number of negative edges is no
more than the number of positive edges in $\Gamma_a$. Since this is true
for $a=1,2$ and since a positive edge in one graph is a negative edge
in the other, the numbers of positive and negative edges of $\Gamma_a$ must
be the same, hence each vertex must be incident to exactly 8 positive
and 8 negative edges, so each negative family contains exactly 4
edges. Since $\Gamma_b$ contains a loop, one of the negative families in
$\Gamma_a$ contains a co-loop and hence is a set of 4 parallel co-loops,
which is a contradiction to the 3-Cycle Lemma 2.14(2).
\quad $\Box$ \medskip
\begin{lemma} Suppose $n_1 = n_2 = 4$, and both $\Gamma_1$ and
$\Gamma_1$ are non-positive. If both components of $\hat \Gamma_a^+$ are of
type (3), and no component of $\hat \Gamma_b^+$ is of type (3), then $\Gamma_a$ is
kleinian. \end{lemma}
\noindent {\bf Proof.} Note that in this case all vertices of $\Gamma_b$ are boundary
vertices. Let $u_1, u_2$ be the vertices of valence $2$ in $\hat \Gamma_a^+$.
By Lemma 6.6(1), $u_1$ is incident to fewer than $8$ positive edges,
hence there are three negative edges in $\Gamma_a$ incident to $u_1$ having
the same label $j$ at $u_1$. On $\Gamma_b$ this implies that the vertex
$v_j$ is incident to at least three positive edges with label $1$ at
$v_j$; since $v_j$ is a boundary vertex, it is incident to at least 9
positive edges. Since a loop at $v_j$ must have labels of different
parity on its two endpoints, we see that $val(v_j, \Gamma_b^+)$ is even. If
$val(v_j, \Gamma_b^+) = 12$ then by Lemma 6.4(4) $\Gamma_a$ is kleinian. Hence
we may assume that $val(v_j, \Gamma_b^+) = 10$. By Lemma 6.4(1) we may
assume that each family of positive edges at $v_j$ has at most 3
edges. This implies that $v_j$ is incident to 2 or 3 loops.
Examining the labels we see that the two outermost loops form a
Scharlemann bigon, with $1$ as a label. For the same reason, $2$ is a
label of a Scharlemann bigon in $\Gamma_b$. If there is no
$(12)$-Scharlemann bigon then there must be $(14)$- and
$(23)$-Scharlemann bigon, so $\Gamma_a$ is kleinian and we are done.
Therefore we may assume that $\Gamma_b$ contains a $(12)$-Scharlemann
bigon.
The $(12)$-Scharlemann bigon and $\hat \Gamma_a^+$ cuts $\hat F_a$ into faces.
There is now only one edge class in these faces which connects $u_1$ to
$u_4$, hence by Lemma 2.2(5) $\Gamma_b$ contains no $(14)$-Scharlemann
bigon. Similarly there is no $(23)$-Scharlemann bigon. It follows
that all Scharlemann bigons of $\Gamma_b$ have label pair $(12)$. In
particular, the two outermost loops at $v_j$ must form a
$(12)$-Scharlemann bigon.
We have shown above that the vertex $v_j$ has 2 or 3 loops. If it has
2 loops then the weights of the positive families at $v_j$ are
$(2,3,3,2)$, and the middle label of a family of $3$ is a label of
Scharlemann bigon, which implies that both $3$ and $4$ are labels of
Scharlemann bigons, which is a contradiction. Hence $v_j$ has exactly
3 loops $e_1, e_2, e_3$, and 4 non-loop edges divided to 2
non-Scharlemann bigons $e_4\cup e_5$ and $e_6 \cup e_7$.
As shown above, $e_1 \cup e_2$ is a (12)-Scharlemann bigon, so up to
symmetry we may assume that the edges $e_4, e_5$ have labels $4$ and
$1$ at $v_j$. Since these edges do not form a Scharlemann bigon, the
labels at their other endpoints must be $3$ and $2$ respectively, so
$e_5$ is a $(12)$-edge, which must be parallel on $\Gamma_a$ to one of the
two $(12)$-loops $e_1, e_2$ because $\Gamma_a$ has only two families
connecting $u_1$ to $u_2$. This is a contradiction because by Lemma
2.3(5) a loop and a non-loop edge cannot be parallel on $\Gamma_a$.
\quad $\Box$ \medskip
\begin{lemma} Suppose $n_1 = n_2 = 4$, and both $\Gamma_1$ and
$\Gamma_2$ are non-positive. Then all components of $\hat \Gamma^+_1$
and $\hat \Gamma^+_2$ are of type (3), and $\hat \Gamma_1$ and $\hat
\Gamma_2$ are subgraphs of the graph shown in Figure 10.1.
\end{lemma}
\noindent {\bf Proof.} By Lemma 10.1 we may assume that $\hat \Gamma_a^+$ has a component $C$ of
type (3). Let $u_1$ be the valence $2$ vertex in $C$. If $u_1$ is
incident to at most $4$ families of parallel edges then each family
contains exactly 4 edges, but since the two positive families are
adjacent, this would be a contradiction to Lemma 6.6(1). Therefore
$u_1$ is incident to at least 5 families of edges. Note that if the
other component $C'$ of $\hat \Gamma_a^+$ is not of type (3), or if it is of
type (3) but the loop of $C'$ separates $u_1$ from the valence 2
vertex $u_2$ of $C'$ then $u_1$ would have only two families of
negative edges, which is a contradiction. It follows that $C'$ is
also of type (3), so the graph $\hat \Gamma_a$ is a subgraph of that in Figure
10.1.
For the same reason if some component of $\hat \Gamma_b^+$ is of type (3) then
$\hat \Gamma_b$ is a subgraph of that in Figure 10.1 and we are done.
Therefore we may assume that no component of $\hat \Gamma_b^+$ is of type (3).
By Lemma 10.2 $\Gamma_a$ is kleinian.
Consider the edges $\hat e_1, ..., \hat e_6$ of $\hat \Gamma_a$ incident to
$u_3$. See Figure 10.1. Let $p_i$ be the weight of $\hat e_i$. By
Lemmas 6.2(2) and 2.3(1), $p_i$ is even and at most 4. Note that
$p_1, p_2, p_3$ are non-zero, so by Lemma
6.6(1) we know that $p_1+p_2$ and $p_1 + p_3$ are between 4 and 6. If
$p_4=0$ then $u_2$ would have valence 4 in $\hat \Gamma_a$, which would lead to
a contradiction as above. Hence $p_4 > 0$. If $p_5+p_6 = 0$ then
$val(u_3, \hat \Gamma_a^+)=5$, so either $p_1=4$ or $p_2=p_3=p_4 = 4$; either case
contradicts Lemma 6.6(1). Therefore $p_5 + p_6 \geq 2$.
One can now check, from the labeling around the boundary of $u_3$,
that if $p_4 = 2$ then both labels at the edge endpoints of $\hat e_4$
at $u_3$ are jumping labels at $u_3$, and if $p_4 = 4$ then all labels
at the endpoints of $\hat e_5 \cup \hat e_6$ at $u_3$ are jumping
labels. This is a contradiction because all vertices of $\hat \Gamma_b^+$ are
boundary vertices and hence by Lemma 6.8 $u_3$ should have no jumping
label.
\quad $\Box$ \medskip
\bigskip
\leavevmode
\centerline{\epsfbox{Figure10.1.eps}}
\bigskip
\centerline{Figure 10.1}
\bigskip
By Lemma 10.3 we may assume that both $\hat \Gamma_1, \hat
\Gamma_2$ are subgraphs of that in Figure 10.1. Label the edges of
$\hat \Gamma_a$ as in Figure 10.1, and let $p_i$ be the weight of $\hat e_i$.
Label $\hat \Gamma_b$ similarly using $\hat e'_i$.
\begin{lemma} $\Delta = 4$, $p_2 + p_3 = 6$, $p_8=p_4=2$,
and $p_7=p_9=4$. Moreover, edges in $\hat e_4 \cup \hat e_8$ are
co-loops, while those in $\hat e_7 \cup \hat e_9$ are not.
\end{lemma}
\noindent {\bf Proof.} If $p_2+p_3 > 6$ then one of the $\hat e_2, \hat e_3$ contains
4 edges, so by Lemma 6.4(1) $\Gamma_a$ is kleinian. By Lemma 6.2(2) the
$p_i$ are even, so $p_2+p_3>6$ implies that $p_2=p_3 = 4$, which is a
contradiction to Lemma 6.6(2). Therefore we have $p_2 + p_3 \leq 6$.
Since each $p_i \leq 4$, by counting edges at $u_1$ we have
$\Delta=4$, and $p_7 + p_8 + p_9 \geq 10$.
Recall that if a pair of negative edges of $\Gamma_a$ incident to $u_i$ are
parallel in $\Gamma_b$ then they form a Scharlemann bigon in $\Gamma_b$, hence
must have the same label pair in $\Gamma_b$, so they are incident to the
same pair of vertices in $\Gamma_a$. Therefore no edge in $\hat e_7$ or
$\hat e_9$ is parallel in $\Gamma_b$ to an edge in $\hat e_8$. Since
$\hat \Gamma_b^+$ has at most 6 edges, we have $\max \{p_7, p_9\} + p_8 \leq 6$.
Since $p_i \leq 4$, this gives $p_7 + p_8 + p_9 \leq 10$, hence $p_2 +
p_3 = \Delta n_b - (p_7 + p_8 + p_9) \geq 6$. Together with the
inequalities above, we have $p_2 + p_3 = 6$, $p_7 + p_8 + p_9 = 10$.
Since $\max \{p_7, p_9\} + p_8 \leq 6$ and $p_i \leq 4$, this holds
only if $p_8=2$ and $p_7 = p_9 = 4$. Similarly $p_4 = 2$.
We have shown that edges in $\hat e_7 \cup \hat e_8$ belong to
distinct families in $\hat \Gamma_b^+$. Since $\hat \Gamma_b^+$ has at most 6 edges,
$\hat e_7 \cup \hat e_8$ represent all edges in $\hat \Gamma_b^+$. If some edge
in $\hat e_i$ is a co-loop then all of them are. Therefore the edges
in $\hat e_7$ cannot be co-loops because $\hat \Gamma_b^+$ has only two loops.
It follows that the edges in $\hat e_8$ must be co-loops. Similarly,
edges of $\hat e_4$ are co-loops, and those in $\hat e_9$ are not.
\quad $\Box$ \medskip
\begin{prop} Suppose both $\Gamma_1$ and $\Gamma_2$ are
non-positive, and $n_a = 4$. Then $n_b < 4$. \end{prop}
\noindent {\bf Proof.} By Proposition 9.7 we have $n_b \leq 4$. Assume to the
contrary that $n_b = 4$. Since the two edges in $\hat e_8$ are
co-loops, they have labels $3,4$ at $u_1$. Consider the three
negative edges $e_7, e_8, e_9$ such that $e_i \in \hat e_i$, and they
all have label $3$ at $u_1$. In $\Gamma_b$ these are $1$-edges at $v_3$.
Since $e_8$ is a loop, it belongs to $\hat e'_1$. The other two edges
are non-loop positive edges on $\Gamma_b$, so they belong to $\hat e'_2
\cup \hat e'_3$. Applying Lemma 10.4 to $\Gamma_b$, we see that $\hat e'_2
\cup \hat e'_3 \cup \hat e'_4$ contains $8$ edges, so the two
edges $e_7, e_9$ are adjacent among the four edges labeled $1$ at
$v_3$ in $\Gamma_b$. Since they are not adjacent among the four edges
labeled $3$ at $u_1$ in $\Gamma_a$, this is a contradiction to the Jumping
Lemma 2.18. \quad $\Box$ \medskip
\section {The case $n_a = 4$, and $\Gamma_b$ positive
}
In this section we assume that $n_a = 4$ and $\Gamma_b$ is positive. We
will determine all the possible graphs for this case. Recall from
Lemma 6.4(2) that in this case $\Gamma_a$ is kleinian, so the weights of
edges of $\hat \Gamma_b$ are all even.
\begin{lemma} Suppose $n_a = 4$ and $\Gamma_b$ is positive.
(1) Two families of 4 parallel edges with the same label sequence at a
given vertex $v_j$ of $\Gamma_b$ connect $v_j$ to the same vertex $v_k$.
(2) There are at most three families of 4 parallel edges with the same
label sequence at any vertex $v_j$, and if $n_b > 2$ then there are at
most two such.
(3) if $\Delta = 4$ then $val(v_j, \hat \Gamma_b) \geq 5$ for all $j$;
(4) if $\Delta = 5$ then $val(v_j, \hat \Gamma_b) \geq 6$ for all $j$;
(5) two weight 4 edges $\hat e_1, \hat e_2$ of $\hat \Gamma_b$ adjacent at a
vertex $v_j$ form an essential loop on $\hat F_b$. \end{lemma}
\noindent {\bf Proof.}
(1) If there are two families of 4 parallel edges with the same label
sequence $1,2,3,4$ at $v_j$ then by Lemma 6.5 the initial edges
$e_1, e'_1$ of the two families are parallel in $\Gamma_a$, with the same
label $j$ at the vertex $u_1$, hence the other endpoints of $e_1$
and $e'_1$ must also have the same label $k$, which implies that in
$\Gamma_b$ the two families have the same endpoints.
(2) If there were four then the leading edges of the $(12)$
Scharlemann bigons in these families are parallel in $\Gamma_a$, so they
belong to a family of at least $3n_b+1$ parallel edges connecting $u_1$
to $u_2$. Since $\hat \Gamma_b$ has at most $3n_b$ edges by Lemma 2.5, two of
these edges would be parallel on both graphs, which is a contradiction
to Lemma 2.2(2).
If $n_b > 2$ and there are three families of 4 parallel edges with the
same label sequence at $v_j$ then as above there would be a family of
$2n_b+1$ parallel edges in $\Gamma_a$, which contradicts Lemma 2.22(3).
(3) and (4) follow immediately from (2).
(5) By (1) these two edges have their other endpoints at the same
vertex $v_k$, hence form a loop $C=\hat e_1 \cup \hat e_2$ on $\hat
F_b$. If $C$ is not essential then we can choose $C$ to be an
innermost such cycle. $C$ bounds a disk $D$ on $\hat F_b$, which must
contain some vertex because $\hat \Gamma_b$ is reduced. If some vertex in the
interior of $D$ has valence 5 then it is incident to two adjacent
weight 4 edges, which would form another inessential loop,
contradicting the choice of $C$. Hence all vertices in the interior
of $D$ have valence at least 6. By Lemma 2.9 in this case there
should be at least three vertices on $\partial D$, which is a
contradiction. \quad $\Box$ \medskip
$\hat F_a$ separates $M(r_a) = M \cup V_a$ into the {\it black\/} and
{\it white\/} sides $X_B$ and $X_W$. Since $\Gamma_b$ is kleinian (Lemma
6.4(2)), the black side $X_B$ is a twisted $I$-bundle over the Klein
bottle. A face of $\Gamma_b$ is white if it lies in the white region
$X_W$, otherwise it is black. In the next two lemmas we assume that
$\Gamma_b$ contains a white bigon and a white 3-gon as in Figure 11.1.
\bigskip
\leavevmode
\centerline{\epsfbox{Figure11.1.eps}}
\bigskip
\centerline{Figure 11.1}
\bigskip
\begin{lemma} Suppose $n_a = 4$ and $\Gamma_b$ is positive.
Suppose $\Gamma_b$ contains a white bigon $A$ and a white 3-gon $B$ as in
Figure 11.1. Then up to homeomorphism of $\hat F_a$, the edges of $A$
and $B$ appear on $\hat F_a$ as shown in Figure 11.2.
\end{lemma}
\noindent {\bf Proof.} Let $V_{23}$ and $V_{41}$ be the components of $V_a \cap X_W$
that run between $u_2, u_3$ and $u_4,u_1$, respectively. Let $Y$ be a
regular neighborhood of $\hat F_a \cup V_{23} \cup V_{41} \cup A \cup
B$. Then $\partial Y = \hat F_a \cup T$, where $T$ is a torus in $M$, and
hence either $X_W = Y$, or $X_W$ is the union of $Y$ and a solid torus
along $T$.
Take a regular neighborhood $D$ of $e_1 \cup e_2 \cup f_3$ as
``base point'' for $\pi_1(\hat F_a) \cong \Bbb Z \times \Bbb Z$ and for
$\pi_1(X_W)$. (See Figure 11.2). The cores of the 1-handles $V_{23}$
and $V_{41}$ represents elements $x, y$ respectively of $\pi_1(X_W)$,
and $\pi_1(X_W)$ is generated by $\Pi = \pi_1(\hat F_a)$ together with
$x$ and $y$. Note that since $\hat F_a$ is essential in $M(r_a)$,
$\Pi$ is a proper subgroup of $\pi_1(X_W)$.
\bigskip
\leavevmode
\centerline{\epsfbox{Figure11.2.eps}}
\bigskip
\centerline{Figure 11.2}
\bigskip
The bigon $A$ gives the relation $xy = 1$, so the $23$-corners and the
$41$-corner of $\partial B$ represent $x$ and $x^{-1}$ respectively. The
$32$-edge on $\partial B$ lies in $D$ and hence represents $1$. If one of
the other two edges $f_1, f_2$ of $B$ represents $1 \in \pi_1(\hat
F_a)$, then $B$ gives a relation of the form $x = \gamma$ for some
$\gamma \in \pi_1(\hat F_a)$, hence $\pi_1(X_W) = \Pi$, a
contradiction. Therefore the edge $f_1$ is as shown in Figure 11.2,
and represents a nontrivial element $\alpha \in \pi_1(\hat F_a)$, when
oriented from $u_1$ to $u_2$. Similarly, the edge $f_2$ represents a
non-trivial element $\gamma$, say, of $\pi_1(\hat F_a)$, when oriented
from $u_3$ to $u_4$. Since $e_1 \cup f_1$ and $e_2\cup f_2$ are
disjoint, we must have $\gamma = \alpha$ or $\alpha^{-1}$. The union
$V_{23} \cup V_{41} \cup N(A)$ is a single $1$-handle attached to
$\hat F_a$, and $\partial B$ is a simple closed curve on $\hat F_a \cup
\partial H$. One can see that $\alpha x^2 \alpha^{-1} x^{-1}$ cannot be
realized by a simple closed curve. It follows that $\gamma = \alpha$,
so $f_2$ appears on $\hat F_a$ as in Figure 11.2. \quad $\Box$ \medskip
\bigskip
\leavevmode
\centerline{\epsfbox{Figure11.3.eps}}
\bigskip
\centerline{Figure 11.3}
\bigskip
If we orient an edge $e$ of $\Gamma_b$ then the corresponding oriented edge
of $\Gamma_a$ represents an element $\gamma$ of $\pi_1(\hat F_a)$, and we
will label the edge $e$ with $\gamma$. Thus the edge labels of the
bigon $A$ are both $1$, and the edge labels of the 3-gon $B$ are as in
Figure 11.3(a). Note that any $12$-edge, oriented from $1$ to $2$, or
any $34$-edge, oriented from $3$ to $4$, has label $1$ or $\alpha$.
Also $\pi_1 (\hat F_a)$ has a basis $\{\alpha, \beta\}$, where $\beta$
is represented by an arc joining $u_1$ and $u_4$, disjoint from $\partial
A$ and $\partial B$, as shown in Figure 11.4.
\bigskip
\leavevmode
\centerline{\epsfbox{Figure11.4.eps}}
\bigskip
\centerline{Figure 11.4}
\bigskip
\begin{lemma} Suppose $n_a = 4$ and $\Gamma_b$ is positive.
Suppose $\Gamma_b$ contains a white bigon $A$ and a white 3-gon $B$ as in
Figure 11.1.
(1) All edges of white bigons have label $1$.
(2) $\beta$ can be chosen so that any 3-gon must have edge labels as
shown in Figure 11.3(a) or (b). \end{lemma}
\noindent {\bf Proof.} (1) Since the edges $e_1 \cup f_1$ and $e_2 \cup f_2$ on $\partial
A$ and $\partial B$ form two parallel essential circles on $\hat F_a$, any
$12$-edge of $\Gamma_b$ must be parallel to either $e_1$ or $f_1$, and any
$34$-edge of $\Gamma_b$ must be parallel to either $e_2$ or $f_2$.
Let $A'$ be another white bigon. Applying Lemma 11.2 to $A'$ and $B$
gives that the $12$-edge of $A'$ and the $12$-edge $f_1$ of $B$ are
not parallel, and similarly the $34$-edge of $A'$ and the $34$-edge
$f_2$ of $B$ are not parallel. Hence both edges of $A'$ are labeled
$1$ since they must be parallel to $e_1$ and $e_2$, respectively.
(2) Since $\Gamma_b$ has no extended Scharlemann cycle, each triangle face
$B'$ has either one or two $23$-corners. If $B'$ has two $23$-corners
then applying Lemma 11.2 to $A$ and $B'$ shows that the $12$-edge of
$B'$ is not parallel to $e_1$, so it must be parallel to $f_1$ and
hence is labeled $\alpha$, as in Figure 11.3(a). Similarly the
$34$-edge of $B'$ is also labeled $\alpha$. If the $32$-edge of $B'$
is labeled $\gamma$ then $B$ and $B'$ together give the relation
$\gamma = 1$, so $\gamma = 1$, as in Figure 11.3(a).
If $B'$ has only one $23$-corner, let $f'_1, f'_2, f'_3$ be the $12$-,
$34$- and $14$-edges of $B'$, respectively. Applying Lemma 11.2 to
$A$ and $B'$ gives that $f'_1$ is not parallel to $e_1$, so by the
above it must be parallel to $f_1$ and hence is labeled $\alpha$.
Similarly $f'_2$ is parallel to $f_2$ and is also labeled $\alpha$.
The two loops $e_1 \cup f_1$ and $e_2 \cup f_2$ cut $\hat F_a$ into
two annuli $A_L$ and $A_R$, where $A_R$ contains $f_3$; see Figure
11.4. If the $14$-edge $f'_3$ of $B'$ lies in $A_R$ then it is
labeled $\alpha$, and $B'$ gives the relation $x^{-1} \alpha x \alpha
x^{-1} \alpha = 1$. It is easy to see that, together with the
relation $x^2 \alpha x^{-1} \alpha = 1$ coming from $B$, this implies
$a = x^2$, so $x^5=1$, and hence $\alpha^5 = 1$, which is a
contradiction to the fact that $\alpha^5$ is a nontrivial element in
$\pi_1(\hat F_a)$ and hence is nontrivial in $\pi_1(X_W)$. Therefore
$f'_3$ lies in $A_L$, as shown in Figure 11.4. Let $\beta$ be the
corresponding element of $\pi_1(\hat F_a)$; then the edge labels of
$B'$ are as in Figure 11.3(b). If $B''$ is any other $3$-gon with one
$23$-corner then the argument above in the case of $3$-gons with two
$23$-corners, using $A$ and the present $B'$, shows that the edge
labels of $B''$ are also as shown in Figure 11.3(b). \quad $\Box$ \medskip
\bigskip
\leavevmode
\centerline{\epsfbox{Figure11.5.eps}}
\bigskip
\centerline{Figure 11.5}
\bigskip
\begin{cor} Suppose $n_a = 4$ and $\Gamma_b$ is positive.
(1) Let $G$ be the subgraph of $\Gamma_a$ consisting of edges of white
bigons and white 3-gons on $\Gamma_b$. Then the reduced graph $\hat G$ is
a subgraph of that in Figure 11.5.
(2) If $\Gamma_b$ contains a white bigon then it cannot contain a (black)
Scharlemann bigon which is flanked on each side by a (white) 3-gon.
(3) If $\Gamma_b$ contains a white bigon then it cannot contain three
3-gons occurring as consecutive white faces at a vertex.
\end{cor}
\noindent {\bf Proof.}
(1) This follows immediately from Lemma 11.3.
(2) The edges of a (black) Scharlemann bigon are either $(12)$- or
$(34)$-edges, so by Lemma 11.3(2) both edges of the Scharlemann bigon
are labeled $\alpha$ and hence are parallel on $\Gamma_a$, contradicting
Lemma 2.2(2).
(3) By (2) the two black bigons between the white 3-gons are
$(12,34)$-bigons, as shown in Figure 11.6. But then the middle
3-gon would be a white Scharlemann cycle, contradicting Lemma 6.2(3).
\quad $\Box$ \medskip
\bigskip
\leavevmode
\centerline{\epsfbox{Figure11.6.eps}}
\bigskip
\centerline{Figure 11.6}
\bigskip
Let $G$ be a reduced graph on a torus $T$ with disk faces. One can
endow $T$ with a singular Euclidean structure by letting each edge
have length 1 and each $n$-gon face a regular Euclidean $n$-gon. The
cone angle $\theta(v)$ at a vertex $v$ of $G$ is the sum of the angles
of the corners incident to $v$. Such a structure is {\it
hyperbolic\/} if $\theta(v) \geq 2\pi$ for all $v$, and $\theta(v) >
2\pi$ for some $v$. The following lemma says that no singular
Euclidean structure on $T$ is hyperbolic.
\begin{lemma} Let $G$ and $\theta(v)$ be defined as above.
Then either $\theta(v) < 2\pi$ for some $v$, or $\theta(v) = 2\pi$ for
all $v$. \end{lemma}
\noindent {\bf Proof.} Denote by $V, E, F$ the numbers of vertices, edges and faces of
$G$, respectively. Let $\theta(c)$ be the angle at a corner $c$ of
the graph. If $\sigma$ is a face of $G$, denote by $|\sigma|$ the
number of edges of $\sigma$. Since $\sigma$ is a regular
$|\sigma|$-gon, for each corner $c \in \sigma$ we have $\theta(c) =
\pi(1 - 2/|\sigma|)$. In the following, the first sum is over all
vertices $v$ of $G$, and the second is over all corners $c$. Grouping
corners by faces $\sigma$, we get
\begin{eqnarray*}
\sum_v \theta(v) & = \sum_c \theta(c) = \sum_{\sigma} \sum_{c \in \sigma} \theta(c)
= \sum_{\sigma} \sum_{c \in \sigma} \pi ( 1 - \frac{2}{|\sigma|}) \\
& = \pi (\sum_{\sigma} |\sigma| - \sum_{\sigma} \sum_{c \in \sigma} \frac 2{|\sigma|})
= \pi (2E - 2F) = 2\pi(E - F) = 2 \pi V.
\end{eqnarray*}
Therefore $ \sum_v (2\pi - \theta(v)) = 0$, and the result follows.
\quad $\Box$ \medskip
\begin{lemma} Let $G$ be a reduced graph on a torus $T$ such
that $val(v) \geq 5$ for all $v$. Then either
(1) there exists a vertex of valence 5 with at least four 3-gons
incident; or
(2) there exists a vertex of valence 6 and all vertices of valence 6
have all incident faces 3 gons; or
(3) all faces of $G$ are 3-gons or 4-gons, and every vertex has
valence 5 and has exactly three 3-gons incident.
\end{lemma}
\noindent {\bf Proof.} We have $\theta(v) > 2\pi$ if $val(v)>6$, $\theta(v)
\geq 6 \times \pi/3 = 2\pi$ if $val(v) = 6$.
Assuming (1) is not true, then we also have
$$\theta(v) \geq 3\times \frac{\pi}3 + 2 \times \frac{\pi}{4} = 2\pi$$
if $val(v)=5$. Thus there is no vertex with cone angle $\theta(v) <
2\pi$, so by Lemma 11.5 we see that $\theta(v) = 2\pi$ for all $v \in
G$, hence there is no vertex of valence more than 6, all faces
incident to vertices of valence 6 are 3-gons, and exactly 3 faces
incident to a vertex of valence 5 are 3-gons and the other two are
4-gons. Therefore either (2) or (3) holds.
\quad $\Box$ \medskip
\bigskip
\leavevmode
\centerline{\epsfbox{Figure11.7.eps}}
\bigskip
\centerline{Figure 11.7}
\bigskip
\begin{lemma} Suppose $n_a = 4$, $n_b > 2$, and $\Gamma_b$ is
positive. Then no vertex $v_j$ of $\hat \Gamma_b$ with $val(v_j)=5$ has four
corners belonging to 3-gons. \end{lemma}
\noindent {\bf Proof.} Let $v_j$ be a vertex of $\hat \Gamma_b$ with $val(v_j)=5$. By Lemma
11.1(4) we must have $\Delta = 4$. The weights of the edges at $v_j$
are $4,4,4,2,2$. By Lemma 11.1(2) the three weight 4 edges are not
consecutive, hence the order around $v_j$ is $(4,4,2,4,2)$. Label
these edges by $\hat e_1, ..., \hat e_5$, respectively, where $\hat
e_3, \hat e_5$ have weight 2. By Lemma 11.1(5) the two edges $\hat
e_1, \hat e_2$ form an essential cycle on the torus $\hat F_b$, hence
the graph looks like that in Figure 11.7.
Let $c$ be the corner at $v_k$ between these two edges $\hat e_1, \hat
e_2$, as shown in Figure 11.7. We claim that $c$ contains no other
edge endpoint. Let $e$ and $e'$ be the edges in $\hat e_1$ and $\hat
e_2$ with label $1$ at $v_j$. Let $P,Q$ be the endpoints of $e,e'$ at
$v_j$, and let $R,S$ be the endpoints of $e,e'$ at $v_k$,
respectively. By Lemma 6.5 these edges are parallel on $\Gamma_a$, so they
connect the same pair of vertices $u_1, u_r$ for some $r$. On $\Gamma_b$
this means that $P,Q$ have the same label $1$, and $R,S$ have the same
label $r$.
Since $e,e'$ are parallel negative edges on $\Gamma_a$, we have
$d_{u_1}(P,Q) = d_{u_r}(R,S)$, therefore the four points $P,Q,R,S$
satisfy the assumptions of Lemma 2.16(1), hence by the lemma we have
$d_{v_j}(P,Q) = d_{v_k}(R,S)$. Without loss of generality assume that
the orientations on $\partial v_j, \partial v_k$ are counterclockwise on
Figure 11.7. Then one can see that $d_{v_j}(P,Q) = 4$, hence
$d_{v_k}(R,S) = 4$, which implies that there are only 3 edge endpoints
from the endpoint of $e$ to that of $e'$ on $\partial v_k$, so there is no
edge endpoint in the corner $c$ in Figure 11.7. This proves the
claim.
Label the corners at $v_j$ as shown in Figure 11.7. The above implies
that the corner $p$ and $s$ belong to the same face $\sigma$, so if
$v_j$ is incident to at least four 3-gons then $\sigma$ must be a
3-gon, hence $\hat e_3 = \hat e_5$ is a loop. Now the corners $q$ and
$r$ belong to the same face $\sigma'$, which cannot be a 3-gon, hence
the result follows.
\quad $\Box$ \medskip
\begin{lemma} Suppose $n_a = 4$, $n_b > 2$, and $\Gamma_b$ is
positive. Suppose $\Delta = 4$.
(1) All faces of $\hat \Gamma_b$ are 3-gons or 4-gons, every vertex has valence
5 and has exactly three 3-gons incident, and the weight sequence of
the edges incident to the vertex is $(4,4,2,4,2)$. In particular,
$n_b$ is even.
(2) The two weight 2 edges at any vertex form a loop, which is
incident to a 3-gon whose other two edges are of weight 4.
(3) Each edge in a weight 2 family of $\Gamma_b$ has label pair $(23)$ or
$(14)$. \end{lemma}
\noindent {\bf Proof.} (1) By Lemma 11.6 and Lemma 11.1(3) and (4), $\hat \Gamma_b$ is one of
the three types stated there. Lemma 11.7 shows that $\hat \Gamma_b$ cannot be
of type (1). If $\hat \Gamma_b$ has a vertex of valence 6 then the weights are
$4,4,2,2,2,2$, hence there are two consecutive edges of weight 2. By
Corollary 11.4(3) the three faces incident to these two edges cannot
all be 3-gons, hence $\hat \Gamma_b$ cannot be of type (2) in Lemma 11.6. It
follows that $\hat \Gamma_b$ is of type (3) in Lemma 11.6, so the weights
of the edges at every vertex of $\hat \Gamma_b$ are $4,4,4,2,2$. Thus the
number of weight 4 edge endpoints in $\hat \Gamma_b$ is $3n_b$, which must be
even, hence $n_b$ is even. By Lemma 11.1(2) the three weight 4 edges
cannot all have the same label sequence, hence the weight sequence is
$(4,4,2,4,2)$ at each vertex.
(2) By Lemma 11.1(5) the two adjacent weight 4 edges at $v_j$ connects
$v_j$ to a vertex $v_k$ and form an essential loop on $\hat F_b$. The
other weight 4 edge at $v_j$ connect to some vertex $v_r$, whose two
other weight 4 edges connect to another vertex $v_s$ and form an
essential loop. These five weight 4 edges cut off a 6-gon containing
the four weight 2 edges at $v_j$ and $v_r$. The 6-gon cannot contain
any vertex in its interior because each vertex is incident to two
weight 4 edges forming an essential cycle on $\hat F_b$. Therefore
the four weight 2 edges at $v_j$ and $v_r$ form two loops.
(3) The loop $\hat e$ of $\hat \Gamma_b$ at $v_j$ cuts off a 3-gon in the 6-gon
above. Let $e_1$ be the edge of $\hat e$ which is on the boundary of
a 3-gon face $\sigma$ of $\Gamma_b$. Then the other two edges of $\sigma$
belong to families of 4 edges and hence must have label pair $(12)$
and $(34)$ respectively, so the labels on $\partial \sigma$ are as shown
in Figure 11.3, and $e_1$ has label pair $(23)$ or $(14)$. Since the
other loop edge of $\hat e$ is parallel to $e_1$, it has label pair
$(14)$ or $(23)$, respectively. \quad $\Box$ \medskip
\bigskip
\leavevmode
\centerline{\epsfbox{Figure11.8.eps}}
\bigskip
\centerline{Figure 11.8}
\bigskip
\begin{prop} Suppose $n_a = 4$ and $\Gamma_b$ is positive.
(1) If $\Delta = 4$ then $n_b = 2$, and the graphs are as shown in
Figure 11.9.
(2) If $\Delta = 5$ then $n_b = 1$, and the graphs are as
shown in Figure 11.10.
\end{prop}
\noindent {\bf Proof.} (1) Put $n=n_b$. If $n = 1$ then the assumption $\Delta=4$
implies that the weights of the edges of $\hat \Gamma_b$ are either $(4,4,0)$
or $(4,2,2)$. In either case a family of four edges form an extended
Scharlemann cycle, which is impossible.
Now assume $n > 2$. Let $C = \hat e_1 \cup \hat e_2$ be the two edges
in $\hat \Gamma_a$ connecting $u_1$ to $u_2$. Since $\hat F_a$ contains both
$(12)$- and $(34)$-Scharlemann cocycles, any edge of $\Gamma_b$ with label
pair $(12)$ is parallel to an edge of $C$ on $\Gamma_a$. By Lemma 11.8(1)
each vertex of $\hat \Gamma_b$ has valence $5$ and hence is incident to 3
weight 4 edges, so $\hat \Gamma_b$ has $3n/2$ weight 4 edges. Each weight 4
edge contributes 2 edges to $C$, one for each $\hat e_i$, and by Lemma
11.8(3) no weight 2 edge of $\hat \Gamma_b$ contributes to $C$. Thus each
$\hat e_i$ represents exactly $3n/2$ edges, and each label $j$ appears
exactly three times among the edge endpoints of $\hat e_1 \cup \hat
e_2$ at $u_1$. Hence if the edge endpoints of $\hat e_1$ at $u_1$ are
labeled $1, ..., n, 1, ..., r$, then the labels of those in $\hat e_2$
must be $r+1, ..., n, 1, ..., n$, where $r=n/2$. It follows that the
$n$ edge endpoints at $u_1$ that do not belong to $C$ must be on one
side of $C$, so up to relabeling we may assume that the labels at
$u_1$ are as shown in Figure 11.8(a). Let $\varphi$ be the involution
on $\hat F_a$ given by Lemma 6.2(4). Then $\varphi$ maps $\hat e_1$
to $\hat e_2$ and is label preserving, so the labels at $u_2$ must be
as shown in Figure 11.8(b). Now the transition function of $\hat e_1$
maps $1$ to $r+1$, which has period $2$. Since $n > 2$, this function
is not transitive, contradicting Lemma 2.3(1).
We have shown that $n=2$. The graph $\hat \Gamma_b$ is now a subgraph of that
in Figure 13.1, with vertices labeled $v_1, v_2$ instead of $u_1,
u_2$. Let $w_i$ be the weight of $\hat e_i$. By Lemma 6.4(2) $\Gamma_a$
is kleinian, and by Lemma 6.2(2) the $w_i$ are all even. By Lemma
11.1(3) we have $w_5 > 0$. If $w_5 = 4$ then $\hat e_5$ containing no
extended Scharlemann bigon implies that either $w_1 + w_2 = 6$ or $w_3
+ w_4 = 6$, so $w_i =4$ for some $i\leq 4$, in which case $\hat e_i$
contains a $(12)$-Scharlemann bigon, whose edges, by the above, must
be parallel in $\Gamma_a$ to the $(12)$-edges of $\hat e_5$, which is a
contradiction to Lemma 2.3(5). Therefore we must have $w_5 = 2$. For
the same reason, the two loops in $\hat e_5$ cannot be a Scharlemann
bigon, so we must have $w_1 = w_2 = 4$ and $w_3+w_4=4$ up to symmetry.
Let $e_1\cup ... \cup e_4$ and $e'_1 \cup ... \cup e'_4$ be the edges
in $\hat e_1, \hat e_2$ respectively, such that $e_i, e'_i$ have label
$i$ at $v_1$. By Lemma 6.5 $e_i, e'_i$ are parallel on $\Gamma_a$, with
the same label $1$ at $u_i$. Therefore there is another edge between
them, which must belong to $\hat e_3 \cup \hat e_4$. If $w_3=w_4 =2$
then one can check that these edges would have label pairs $(14)$ and
$(23)$, which is a contradiction. Therefore we may assume $w_3=4$ and
$w_4=0$. The graph $\Gamma_b$ is now as shown in Figure 11.9(b).
As shown above, there are 12 edges on $\Gamma_b$ with label pair $(12)$ or
$(34)$, divided into 4 families of 3 edges each on $\Gamma_a$. Label the
edges of $\Gamma_b$ as in the figure. Up to symmetry we may assume the
edge $A$ is as shown in Figure 11.9(a). Since $\Delta=4$, we may
assume that the jumping number $J=1$. The 1-edges around $v_1$ appear
in the order $A,E,G,P$. By Lemma 6.5 $G,P$ are parallel on $\Gamma_a$.
This determines the position of these edges as well as the orientation
of $u_1$. The $2$-edges at $v_2$ appear in the order $E,R,G,P$,
and $E$ on $\Gamma_a$ has already been determined above, hence the position
of $R,G,P$ must appear around $\partial u_2$ are shown in the figure.
Other edges on $\Gamma_a$ can be determined similarly.
\bigskip
\leavevmode
\centerline{\epsfbox{Figure11.9.eps}}
\bigskip
\centerline{Figure 11.9}
\bigskip
(2) The proof for $\Delta = 5$ is similar but simpler. In this case
by Lemma 11.1(4) each vertex $v_j$ of $\hat \Gamma_b$ has valence 6, and the
edges at $v_j$ have weights $4,4,4,4,2,2$. Thus all white faces are
bigons and 3-gons, so by Corollary 11.4(1) $\hat \Gamma_a$ is a subgraph of
that in Figure 11.5. By Lemma 11.1(2) the two weight 2 edges at any
vertex are non-adjacent, thus any edge $e$ in a weight 2 family is on
the boundary of a 3-gon whose other two edges have label pairs $(12)$
and $(34)$, so the label pair of $e$ must be $(14)$ or $(23)$ and
hence $e$ is not a vertical edge in Figure 11.5. On the other hand,
each weight 4 edge of $\hat \Gamma_b$ contributes one edge to each vertical
family in Figure 11.5, hence each vertical edge has weight exactly
$2n_b$. As above, one can show that the transition function defined
by a family of vertical edges is the identity function, which by Lemma
2.3(1) implies that $n_b = 1$, so the graph $\Gamma_b$ must be as shown in
Figure 11.10(b). By the above discussion, $\Gamma_a$ is as shown in Figure
11.10(a). Label edges as in Figure 11.10(b). By Lemma 6.5 the edges
$A,H$ are parallel on $\Gamma_b$, hence we may assume the jumping number
$J=1$. One can now easily determine the labels of the edges of $\Gamma_a$.
\quad $\Box$ \medskip
\bigskip
\leavevmode
\centerline{\epsfbox{Figure11.10.eps}}
\bigskip
\centerline{Figure 11.10}
\bigskip
\begin{prop} Suppose $n_a \leq n_b$. Then $n_a \leq 2$.
\end{prop}
\noindent {\bf Proof.} By Proposition 5.11 we have $n_a \leq 4$. Assume $n_a=4 \leq
n_b$. Then by Proposition 11.9, $\Gamma_b$ is non-positive. Therefore, by
Proposition 9.7, $n_a=n_b=4$, and, by Proposition 11.9 again, $\Gamma_a$ is
also non-positive, contradicting Proposition 10.5.
Suppose $n_a = 3$. Then $\hat F_a$ is non-separating. If $\Gamma_a$ is
non-positive then some vertex $u_1$, say, has different sign to the
other two vertices $u_2, u_3$. One of the vertices has valence at
most two in $\hat \Gamma_a^+$, so it is incident to at most $2n_b$ positive
edges, and hence at least $2n_b$ negative edges. By Lemma 2.8 this
implies that $\Gamma_b$ has a Scharlemann cycle, so by Lemma 2.2(4) $\hat
F_a$ is separating, which is a contradiction. If $\Gamma_a$ is positive
then by Lemma 3.1 we have $n_b \leq 4$. By Lemma 2.23 $n_1, n_2$
cannot both be odd, hence $n_a=3$ implies that $n_b$ is even, so we
must have $n_b = 4$. Now applying Proposition 11.9 with $n_a, n_b$
reversed, we get $n_a \leq 2$, which is a contradiction. \quad $\Box$ \medskip
\section {The case $n_a=2$, $n_b \geq 3$, and $\Gamma_b$ positive
}
The next few sections deal with the case that $n_a=2$ and $n_b \geq
3$. The main result of this part is Proposition 16.8, which shows
that there are only a few possibilities for the graphs $\Gamma_a, \Gamma_b$.
Throughout this section we will assume that $n_a=2$, $n_b \geq 3$ and
$\Gamma_b$ is positive. We will show that this case is impossible. To
simplify notation, denote $n_b$ by $n$. Note that $\hat \Gamma_a$ has at most
four edges $\hat e_1, ..., \hat e_4$, all connecting $u_1$ to $u_2$.
We will always assume that the first edge of $\hat e_1$ has label $1$
at $u_1$. Write $\Gamma_a = (a_1, a_2, a_3, a_4)$, where $a_i$ is the
weight of $\hat e_i$. Let $s_i = s(\hat e_i)$ be the transition
number of $\hat e_i$ from $u_1$ to $u_2$. In the following lemma the
subscripts are mod $4$ integers.
\begin{lemma}
(1) $s_{i+1} \equiv s_i - a_i - a_{i+1}$ mod $n$;
(2) $s_{i+2} \equiv s_i - a_{i+1} + a_{i+3}$ mod $n$. In particular,
$s_i \equiv s_{i+2}$ if and only if $a_{i+1} \equiv a_{i+3}$ mod $n$.
\end{lemma}
\noindent {\bf Proof.} (1) Orient edges from $u_1$ to $u_2$, and denote by $e(h),
e(t)$ the head and tail of an edge $e$. Let $e, e'$ be the first edge
of $\hat e_i, \hat e_{i+1}$, respectively. Let $x$ be the label of
$e(t)$, and $y$ the label of $e'(h)$. Then traveling from $e(t)$ to
$e'(t)$ on $\partial u_1$ then to $e'(h)$ through $e'$ gives $y \equiv x+
a_i + s_{i+1}$ mod $n$, while traveling through $e$ to $e(h)$ then
along $\partial u_2$ to $e'(h)$ gives $y \equiv x + s_i - a_{i+1}$ mod
$n$. Hence $s_{i+1} \equiv s_i - a_i - a_{i+1}$ mod $n$.
(2) Applying (1) twice gives $s_{i+2} \equiv s_i - a_i - a_{i+1} -
a_{i+1} - a_{i+2} \equiv s_i - a_{i+1} + a_{i+3}$ mod $n$. \quad $\Box$ \medskip
\begin{lemma} Let $e, e'$ be edges of $\Gamma_b$ joining a pair of
distinct vertices, such that $e \cup e'$ is null-homotopic in $\hat
F_b$. If $e$ belongs to a family of at least $n$ parallel edges in
$\Gamma_a$ then $e$ and $e'$ are parallel on $\Gamma_b$.
\end{lemma}
\noindent {\bf Proof.}
Let $D$ be the disk in $\hat F_b$ bounded by $e \cup e'$. The family
of $n$ parallel edges of $\Gamma_a$ containing $e$ gives a set of essential
loops on $\hat F_b$, corresponding to the orbits of the associated
permutation. It follows that $D$ contains no vertices in its
interior, and hence $e$ and $e'$ are parallel on $\Gamma_b$.
\quad $\Box$ \medskip
\begin{lemma} Suppose $\Gamma_a$ contains bigons $e_1 \cup e_2$
and $e'_1 \cup e'_2$, such that $e_1,e'_1$ have label $i$ at $u_1$ and
$j$ at $u_2$, and $e_2,e'_2$ have label $i+1$ at $u_1$ and $j+1$ at
$u_2$. Suppose either
(i) $j \neq i \pm 1$, or
(ii) $\Gamma_a$ contains a pair of edges $e_3, e'_3$ with the same
label pair $(r, s)$, such that $r,s \notin \{i, i+1, j, j+1\}$ and
$C_3 = e_3 \cup e'_3$ form an essential loop on $\hat F_b$.
Then $C_1 = e_1 \cup e'_1$ and $C_2 = e_2 \cup e'_2$ are inessential
on $\hat F_b$.
\end{lemma}
\noindent {\bf Proof.}
(i) In this case $C_1 \cap C_2 = \emptyset$, so they cannot be
essential and yet non-homotopic on $\hat F_b$, hence by Lemma
2.20 they must be inessential.
(ii) In this case $C_1, C_2$ lie in the interior of the annulus
obtained from $\hat F_b$ by cutting along $C_3$, which again implies
that $C_1, C_2$ cannot be essential and yet non-homotopic on $\hat
F_b$.
\quad $\Box$ \medskip
An edge is a {\it border edge\/} if it is the first or last edge in a
family of parallel edges.
\begin{lemma} Suppose $s_k \neq \pm 1$. Then
(1) $a_k \leq n+1$, and
(2) if $a_k = n+1$ and $e'$ is an edge of $\hat e_j$ which has the
same label pair as that of a border edge $e_1$ of $\hat e_k$, then
$e'$ is a border edge.
\end{lemma}
\noindent {\bf Proof.}
(1) Assume $a_k \geq n+2$. Label the first $n+2$ edges of $\hat e_k$
successively as $e_1, e_2, ..., e_n, e_{n+1}, e_{n+2}$. Let $e'_i =
e_{n+i}$. Since $s_k
\neq \pm 1$, $e_1, e_2, e'_1, e'_2$ satisfy Condition (i) in
Lemma 12.3, so $e_1 \cup e'_1$ is an inessential loop on $\hat
F_b$. By Lemma 12.2 this implies that $e_1$ and $e'_1$ are parallel
on $\Gamma_b$, and hence parallel on both graphs, which is a contradiction.
(2) If $e'$ is not a border edge then the bigon $e_1 \cup e_2$ and
one of the two bigons containing $e'$ satisfy the assumption of
Lemma 12.3(i), hence $e'$ is parallel to $e_1$ on $\Gamma_b$.
Similarly, using the bigon between $e_n, e_{n+1}$ and the other bigon
containing $e'$ one can show that $e'$ is also parallel to $e_{n+1}$
on $\Gamma_b$, hence $e_1, e_{n+1}$ are parallel on both $\Gamma_a$ and $\Gamma_b$,
which is again a contradiction.
\quad $\Box$ \medskip
\begin{lemma} Let $\hat e, \hat e'$ be families of at least
$n$ parallel edges in $\Gamma_a$, and let $i,j,k,l \in \Bbb Z_n$ be
distinct. Then $\Gamma_a$ cannot contain both
(i) $ij$-edges $e_1, e_2, e_3$ with $e_1, e_2 \in \hat e$ and $e_3$
non-equidistant with $e_1, e_2$; and
(ii) $kl$-edges $e'_1, e'_2 \in \hat e'$.
\end{lemma}
\noindent {\bf Proof.}
The edges $e_1, e_2, e_3$ are pairwise non-parallel in $\Gamma_b$. Since
$e_1, e_2 \in \hat e$, no pair of $e_1, e_2, e_3$ cobounds a disk in
$\hat F_b$ by Lemma 12.2. Hence $e'_1, e'_2$ cobound a disk in $\hat
F_b$. Since $e'_1 \in E'$, $e'_1, e'_2$ are parallel on $\Gamma_b$ by
Lemma 12.2. This contradicts Lemma 2.2(2).
\quad $\Box$ \medskip
Two families $\hat e, \hat e'$ of $\Gamma_a$ are {\it A-conjugate\/} it
there are $e \in \hat e$ and $e' \in \hat e'$ such that they are
anti-parallel on $\Gamma_b$ when oriented on $\Gamma_a$ from $u_1$ to $u_2$.
They are {\it P-conjugate\/} if the $e, e'$ above are parallel on
$\Gamma_b$ as oriented edges. They are {\it conjugate\/} if they are
either A-conjugate or P-conjugate.
\begin{lemma} (1) There exist $\hat e_i, \hat e_j$ on $\hat \Gamma_a$
which are A-conjugate. Moreover, if $a_i<(\Delta-3)n$ or
$a_j<(\Delta-3)n$ then there is another such pair. (The two pairs may
have one family in common.)
(2) If $\hat e_i, \hat e_j$ are A-conjugate then $s_i \equiv - s_j$
mod $n$; moreover, if $a_i \geq n$ or $a_j \geq n$ then $s_i \not
\equiv s_j$ mod $n$. \end{lemma}
\noindent {\bf Proof.} (1) Since there are $\Delta n$ edges while $\hat \Gamma_b$ has at most
$3n$ edges, $\Gamma_b$ has at least $(\Delta - 3)n$ bigons. The two edges
of a bigon in $\Gamma_b$ belong to a pair of A-conjugate families $\hat
e_i, \hat e_j$ on $\Gamma_a$. If $a_i<(\Delta-3)n$ or $a_j<(\Delta-3)n$
then these families cannot contain all the bigons on $\Gamma_b$, hence
there must be another A-conjugate pair.
(2) If $\hat e_i, \hat e_j$ are A-conjugate then by definition
there exist $e \in \hat e_i$ and $e' \in \hat e_j$ which are
anti-parallel on $\Gamma_b$, hence the label of $e'$ at $u_2$ is the same
as that of $e$ at $u_1$, and vice versa. Therefore $s_i \equiv -s_j$
mod $n$.
If we also have $s_i \equiv s_j$ mod $n$ then $s_i=0$ or $n/2$, which
is a contradiction to the 2-Cycle Lemma 2.14(3).
\quad $\Box$ \medskip
\begin{lemma} Let $\hat e = e_1 \cup ... \cup e_p$ and $\hat
e' = e'_1 \cup ... \cup e'_q$ be two families of $\Gamma_a$, where $p\leq
q$.
(1) If $\hat e$ and $\hat e'$ are conjugate then $p\equiv q$ mod $2$.
(2) If $\hat e$ and $\hat e'$ are conjugate and $q\geq p$ then each
edge $e_r$ is parallel to the edge $e'_{r+c}$, where $c=(q-p)/2$;
hence the set of edges in $\hat e'$ which are parallel to those in
$\hat e$ lie exactly in the middle of $\hat e'$.
(3) If $p+q \equiv 0$ mod $2n$ and $\hat e, \hat e'$ are adjacent on
$\hat \Gamma_a$ then they are not A-conjugate.
(4) If $p+q \equiv 0$ mod $2n$, $\hat e, \hat e'$ are adjacent on
$\hat \Gamma_a$, and $J\neq \pm 1$ then they are not conjugate.
\end{lemma}
\noindent {\bf Proof.} (1) By definition there are edges $e_i, e'_j$ which are
parallel in $\Gamma_b$.
First consider the case that $\hat e, \hat e'$ are adjacent.
Denote by $e(k)$ the endpoints of $e$ at $u_k$. We may assume that
the first edge $e'_1$ of $\hat e'$ is adjacent to the last edge $e_p$
of $\hat e$ on $\partial u_1$. Then the distance from $e_i(1)$ to
$e'_j(1)$ is
\begin{eqnarray}
d_{u_1}(e_i, e'_j) = j + p - i
\end{eqnarray}
On $\partial u_2$
$e'_q(2)$ is adjacent to $e_1(2)$, so we have
\begin{eqnarray}
d_{u_2}(e'_j, e_i) = i + q - j
\end{eqnarray}
Since $e_i, e'_j$ are parallel positive edges on $\Gamma_b$, they
are equidistant, hence by Lemma 2.17 we have $d_{u_1}(e_i, e'_j) =
d_{u_2}(e'_j, e_i)$, which gives
\begin{eqnarray}
2(j-i) = q-p
\end{eqnarray}
and
\begin{eqnarray}
2d = p+q
\end{eqnarray}
where $d = d_{u_1}(e_i, e'_j) = d_{u_2}(e'_j, e_i)$. Equation (C)
gives $q-p \equiv 0$ mod $2$.
Now suppose $\hat e, \hat e'$ are not adjacent. Let $\hat e''$ be the
family whose endpoints on $\partial u_1$ are between $e_p(1)$ and
$e'_1(1)$. Then on $\partial u_2$ the endpoints of $\hat e''$ are also
exactly the ones between $e'_q(2)$ and $e_1(2)$. Thus if $\hat e''$
has $k$ edges then the equations (5) and (6) above become $d = j + k
+ p - i$ and $d = i + k + q - j$. Therefore again we have $2(j-i) =
q-p$, and the result follows.
(2) From equation (7) we have $j = i + (q-p)/2$. If $i>1$ then
the above and the condition $q\geq p$ imply that $j>1$. Since $e_i$
is parallel to $e'_j$ on $\Gamma_b$, by Lemma 2.20 applied to the bigons
$e_{i-1}\cup e_i$ and $e'_{j-1} \cup e'_j$, the loop $e_{i-1} \cup
e'_{j-1}$ is null-homotopic on $\hat F_b$, hence by Lemma 12.2
$e_{i-1}$ is parallel to $e'_{j-1}$ on $\Gamma_b$. Similarly, if $i<p$
then the edge $e_{i+1}$ is parallel to $e'_{j+1}$ on $\Gamma_b$. By
induction it follows that every edge $e_k$ in $\hat e$ is parallel to
the edge $e'_{k+(q-p)/2}$.
(3) Assume without loss of generality that $\hat e = \hat e_1$ and
$\hat e' = \hat e_2$. If they are A-conjugate then the label of
$e'_j$ at $u_1$ is the same as the label of $e_i$ at $u_2$, so $d =
d_{u_1} (e_i, e'_j) \equiv s_1$ mod $n$. Hence equation (8) and the
assumption $p+q\equiv 0$ mod $2n$ gives $s_1 \equiv d \equiv
0$ mod $n$. Since $n\geq 3$, this is a contradiction to Lemma 2.14(2).
(4) By (3) $\hat e, \hat e'$ are not A-conjugate. Assume they are
P-conjugate.
If $p+q=4n$ then by Lemma 2.22(3) we have $p=q=2n$, and by (2) each
$e_i$ of $\hat e_1$ is parallel to the corresponding edge $e'_i$ of
$\hat e_2$ on $\Gamma_b$ for $i=1,...,2n$. Since $\hat e, \hat e'$ are not
A-conjugate, $e_i, e'_i$ are parallel as oriented edges, with
orientation from $u_1$ to $u_2$. Hence there is another edge $e''_i$
between them, which cannot belong to $\hat e \cup \hat e'$ as
otherwise there would be two edges parallel on both graphs,
contradicting Lemma 2.2(2). This gives at least $6n$ edges on $\Gamma_a$,
which is a contradiction.
Now assume $p+q=2n$. Let $e, e'$ be the edges of $\hat e, \hat e'$
which are parallel on $\Gamma_b$ as oriented edges, so they have the same
label $k$ at $u_1$ for some $k$. The condition $p+q=2n$ implies that
$e, e'$ are adjacent among edges labeled $k$ at $u_1$. Since $J \neq
\pm 1$, they are non-adjacent on $\Gamma_b$ among edges labeled $1$ at
$v_k$, hence they belong to a family of at least 5 parallel edges,
which is a contradiction to Lemma 2.2(2) because $\hat \Gamma_a$ has at most 4
edges. \quad $\Box$ \medskip
\begin{lemma} If the jumping number $J=\pm 1$ (in particular
if $\Delta = 4$), then $\Gamma_a$ has at most $n+1$ parallel edges.
\end{lemma}
\noindent {\bf Proof.} Suppose for contradiction that $\hat e_1$, say, contains edges
$e_1, ..., e_{n+2}$. By Lemma 12.4(1) we may assume that $s_1 = 1$,
so the label sequences of these edges are $(1,2,..., n, 1,2)$ at
$u_1$, and $(2,3,...,n,1,2,3)$ at $u_2$. By Lemma 2.22(1) we may
assume that the subgraph of $\Gamma_b$ consisting of these edges is as
shown in Figure 2.3. Up to symmetry we may assume that the
orientation of $\partial v_i$ is counterclockwise on Figure 2.3.
Orient edges from $u_1$ to $u_2$. Denote by $h_i, t_i$ the head and
tail of $e_i$, respectively. For $i>1$, $h_{i-1}$ and $t_i$ both have
label $i$ on $\Gamma_a$, so they are on $\partial v_i$. Define $d_i =
d_{v_i}(t_i, h_{i-1})$, where $i = 2, ..., n+2$. Note that $d_i = 1$
implies that the corner from $t_i$ to $h_{i-1}$ on $\partial v_i$ contains
no edge endpoint.
\medskip
CLAIM 1. {\it $d_i = d_j$ for $2 \leq i,j \leq n+2$.}
\medskip
\noindent {\bf Proof.} Isotoping on $T_0$ along the positive direction of $\partial u_i$
moves $h_1$ to $h_2$ and $t_2$ to $t_3$, so the distance on $\partial v_2$
from $h_1$ to $t_2$ should be the same as that on $\partial v_3$ from
$h_2$ to $t_3$, i.e., $d_2 = d_3$. (Alternatively one may apply Lemma
2.16 to obtain the result.) Similarly we have $d_i = d_{i+1}$
for $2\leq i \leq n+1$.
\quad $\Box$ \medskip
\medskip
CLAIM 2. {\it $d_i = 1$ for $2 \leq i \leq n+2$.}
\medskip
\noindent {\bf Proof.} By assumption we have $J = \pm 1$, so either $d_{v_2}(h_1,
h_{n+1}) = 2$ (when $J=1$), or $d_{v_1}(t_{n+1}, t_1) = 2$ (when
$J=-1$). In the first case, from Figure 2.3 we see that the tail
of $e_{n+2}$ is the only edge endpoint at the corner from $h_1$ to
$h_{n+1}$, hence $d_{n+2} = 1$. Similarly in the second case the head
of $e_n$ is the only edge endpoint on $\partial v_1$ from $t_{n+1}$ to
$t_1$, hence $d_{n+1} = d_{v_1}(t_{n+1}, h_n) = 1$. In either case by
Claim 1 we have $d_i = 1$ for all $i$ between $2$ and $n+2$.
\quad $\Box$ \medskip
Let $D$ be the disk face indicated in Figure 2.3. Claim 2 shows that all
corners of $D$ except $c_1, c_2, c_3$ shown in the figure contain no
edge endpoints.
When $J=1$, we have $d_{v_1}(t_1, t_{n+1}) = 2$, so there is one edge
endpoint in $c_1$. Similarly there is one edge endpoint in $c_3$.
Since $d_{v_2}(t_{n+2}, t_2) = 2\Delta - 2 \geq 6$, there are at least
4 edge endpoints in $c_2$. Thus there would be some trivial loops
based at $v_2$, contradicting the assumption that $\Gamma_b$ has no trivial
loops.
When $J = -1$, $d_{v_1}(t_1, t_{n+1}) = d_{v_3}(h_2, h_{n+2}) =
2\Delta - 2 \geq 6$, and $d_{v_2}(t_{n+2}, t_2) = 2$, so there are at
least 5 edge endpoints in each of $c_1$ and $c_3$, and no edge
endpoints in $c_2$. It follows that $D$ contains at least 5 interior
edges, all parallel to each other, two of which would then be parallel
on both graphs, contradicting Lemma 2.2(2). \quad $\Box$ \medskip
\begin{lemma} $\Gamma_a$ has at most $n+2$ parallel edges.
\end{lemma}
\noindent {\bf Proof.} Assume to the contrary that $\hat e_1 \supset e_1 \cup ... \cup
e_{n+3}$. By Lemma 12.4(1) we may assume without loss of generality
that $s_1 = 1$. By Lemma 2.22(1) the first $n+2$ edges appear in
$\Gamma_b$ as shown in Figure 2.3
First assume $n\geq 4$. Orient edges of $\Gamma_a$ from $u_1$ to $u_2$, and
denote by $e(h), e(t)$ the head and tail of an edge $e$, respectively.
From Figure 2.3 we see that $(e_2(h), e_{n+2}(h), e_3(t))$ is a
positive triple, hence by Lemma 2.21(2) the triple $(e_3(h), e_{n+3}(h),
e_4(t))$ is also positive, so the head of $e_{n+3}$ lies on
the inner circle in Figure 2.3. Note that $e_{n+2}$ shields this edge
endpoint from the outside circle of the annulus in Figure 2.3, hence
the tail of $e_{n+3}$ also lies in the inner circle in the figure,
therefore $e_{n+3}$ is parallel to $e_3$ on $\Gamma_b$, which is a
contradiction as they cannot be parallel on both graphs.
Now consider the case $n=3$. By Lemma 2.22(3) we may assume $a_i \leq
6$, and $a_1 = n+3 = 6$. By Lemma 12.8 we may assume that $\Delta=5$
and the jumping number $J \neq \pm 1$. Also, $a_j \neq 5$, otherwise
by Lemma 12.7(1) the 11 edges in $\hat e_1 \cup \hat e_j$ would be
mutually non-parallel on $\Gamma_b$, contradicting the fact that $\hat \Gamma_b$ has
at most $3n = 9$ edges (Lemma 2.5). One can now check that the
following are the only possible values of $(a_1, a_2, a_3, a_4)$ up to
symmetry, where $*$ indicates any possible value. Let $s=s_1$. Then
the other $s_i$ can be calculated using Lemma 12.1. The second
quadruple in the following list indicates the values of $(s_1, s_2,
s_3, s_4)$.
$$
\begin{array}{lll}
(1) & \qquad (6,6,*,*) & \qquad (s,\; s,\; *,\; *) \\
(2) & \qquad (6,1,6,2) & \qquad (s,\; s-1,\; s+1,\; s-1) \\
(3) & \qquad (6,1,4,4) & \qquad (s,\; s-1,\; s, \; s+1) \\
(4) & \qquad (6,4,3,2) & \qquad (s,\; s-1,\; s+1,\; s-1) \\
(5) & \qquad (6,4,1,4) & \qquad (s,\; s-1,\; s,\; s+1) \\
(6) & \qquad (6,3,2,4) & \qquad (s,\; s,\; s+1,\; s+1 ) \\
(7) & \qquad (6,3,4,2) & \qquad (s,\; s,\; s-1,\; s-1) \\
(8) & \qquad (6,3,3,3) & \qquad (s,\; s,\; s,\; s) \end{array}
$$
In case (1) by Lemma 12.7(4) the 12 edges in $\hat e_1 \cup \hat e_2$
are mutually non-parallel on $\Gamma_b$, which is impossible because $\hat \Gamma_b$
contains at most $3n = 9$ edges. Case (8) is impossible by Lemma
12.6. Also, Lemma 2.14(3) implies that $s_i \not \equiv 0$ mod $3$
if $a_i \geq 2$, which can be applied to exclude cases (2), (4) and
(5).
In case (6) and (7), by Lemma 12.7(1) $\hat e_2$ is not conjugate to
$\hat e_1, \hat e_3$ or $\hat e_4$, so $\hat e_1 \cup \hat e_2$
represents all $9$ edges in $\hat \Gamma_b$, hence each of $\hat e_3$ and $\hat
e_4$ must be conjugate to $\hat e_1$. By Lemma 12.7(2) the two middle
edges of $\hat e_1$ are parallel to the middle edges in each of $\hat
e_3$ and $\hat e_4$, so $\hat e_3, \hat e_4$ are conjugate. Since
$a_3+a_4 = 6 = 2n$ and $J\neq \pm 1$, this is a contradiction to Lemma
12.7(4).
In case (3), by Lemma 2.14(3) we have $s=1$. Since $a_1 + a_3 = a_1 +
a_4 = 10$ while $\hat \Gamma_b$ has at most 9 edges, each of $\hat e_3, \hat
e_4$ must have an edge parallel to some edge of $\hat e_1$ on $\Gamma_b$.
By Lemma 12.7(2) this implies that each edge of $\hat e_3 \cup \hat
e_4$ is parallel to one of the 4 middle edges in $\hat e_1$. Note
that the edge $e'$ in $\hat e_2$ is a loop based at $v_1$ in $\Gamma_b$,
which cannot be parallel to any other edge on $\Gamma_b$. Therefore $\Gamma_b$
has exactly 7 families. Moreover, if we let $e_1, e_6$ be the first
and last edges in $\hat e_1$ then each of $e_1, e_6, e'$ forms a
single family.
Now consider the graph in Figure 2.4. Clearly there is only one
possible position for $e'$, which has exactly one endpoint on the
corner from the tail of $e_1$ to the head of $e_6$. By the above
there are no other edge endpoints in this corner, which is a
contradiction because the label of the tail of $e_1$ is $1$ while the
label of the head of $e_6$ is $2$, so the number of edge endpoints
between them must be even.
\quad $\Box$ \medskip
\begin{lemma} Suppose $n\geq 4$. Then $\Delta = 4$ and
$\Gamma_a$ has at most $n+1$ parallel edges.
\end{lemma}
\noindent {\bf Proof.} We need only show that $\Delta=4$. The second statement will
then follow from Lemma 12.8.
Suppose to the contrary that $\Delta = 5$. First assume that
$a_i < n+2$ for all $i$. Then $\Delta n = 5n \leq 4(n+1)$, so $n=4$,
and $\Gamma_a = (5,5,5,5)$. By Lemma 2.3(1) $s_1$ is coprime with $n=4$,
so we may assume without loss of generality that $s_1 = 1$. Thus the
label sequences of $\hat e_1$ are $(1,2,3,4,1)$ at $u_1$ and
$(2,3,4,1,2)$ at $u_2$. One can check that the label sequences of
$\hat e_3$ are $(3,4,1,2,3)$ at $u_1$, and $(4,1,2,3,4)$ at $u_2$.
This contradicts Lemma 12.5 with $e_1, e_2$ the two $12$-edges in
$\hat e_1$, $e_3$ the $12$-edge in $\hat e_3$, and $e'_1, e'_2$ the
two $34$-edges in $\hat e_3$.
We may now assume without loss of generality that $a_1 > n+1$. By
Lemma 12.9 we must have $a_1 = n+2$. By Lemma 12.4(1) we have $s_1 =
\pm 1$.
\medskip
CLAIM 1. {\it $a_2, a_4 \leq n+1$}.
\medskip
\noindent {\bf Proof.} Suppose $a_2 = n+2$. Then $s_2 = \pm 1$ by Lemma 12.4(1). Also
by Lemma 12.1 we have
$$ s_1 - s_2 \equiv a_1 + a_2 \equiv 4 \qquad \text{mod $n$}
$$
Hence either $n=4$ and $s_1 = s_2$ ($=1$ say), or $n=6$, $s_1 = -1$
and $s_2 = 1$. In either case one can check that there is a pair of
parallel $12$-edges $e_1, e_2$ in $\hat e_1$, a $12$-edge $e_3$ in
$\hat e_2$ which is not equidistant to $e_1, e_2$, and a pair of
parallel $34$-edges $e'_1, e'_2$ in $\hat e_2$. This is a
contradiction to Lemma 12.5.
Hence $a_2 \leq n+1$. A symmetric argument shows that $a_4 \leq n+1$.
\quad $\Box$ \medskip
We now have $5n \leq 2(n+2) + 2(n+1)$, giving $n\leq 6$. Also if
$n=6$ then $\Gamma_a = (8,7,8,7)$.
\medskip
CLAIM 2. {$n = 5$.}
\medskip
\noindent {\bf Proof.} Otherwise we have $n=4$ or $6$. If $a_2 = n+1$ then Lemma 12.1
gives $s_2 = s_1 - a_1 - a_2 = \pm 1 - (n+1) - (n+2) \equiv 0$ mod
$2$, which is a contradiction to the fact that the transition function
of a family of more than $n$ edges must be transitive (Lemma 2.3(1)).
Therefore we have $a_2 \leq n$. Similarly for $a_4$. This rules out
the case $n=6$.
When $n=4$ we must have $\Gamma_a = (6,4,6,4)$. Assume without loss of
generality that $s_1 = 1$. We now apply Lemma 12.5 with $e_1, e_2$
the $12$-edges in $\hat e_1$, $e_3$ the $12$-edge in $\hat e_3$, and
$e'_1, e'_2$ the $34$-edges in $\hat e_3$.
\quad $\Box$ \medskip
\medskip
CLAIM 3. {If $n = 5$ then $a_3 \neq 7$.}
\medskip
\noindent {\bf Proof.} Otherwise by Claim 1 we have $(a_2, a_4) = (6,5)$ or $(5,6)$,
so $a_4 - a_2 = \pm 1$. We may assume that $s_1 = 1$. By Lemma 12.1
we have
$$ s_3 \equiv s_1 - a_2 + a_4 = 1 \mp 1 = \text{ $0$ or $2$ mod $5$}
$$
which contradicts the fact that $s_3 = \pm 1$ mod $n$ (Lemma 12.4(1)).
\quad $\Box$ \medskip
The only possibility left is that $n = 5$ and $\Gamma_a = (7,6,6,6)$.
We may assume $s_1 = 1$. Then this can be ruled out by applying Lemma
12.5 with $e_1, e_2$ the $12$-edges in $\hat e_1$, $e_3$ the $12$-edge
in $\hat e_3$, and $e'_1, e'_2$ the $45$-edges in $\hat e_3$.
\quad $\Box$ \medskip
\begin{lemma}
(a) $\Gamma_a$ has at most $n+1$ parallel edges.
(b) $\Delta = 4$.
\end{lemma}
\noindent {\bf Proof.} (a) This follows from Lemmas 12.8 and 12.10 if either $J=\pm
1$, or $\Delta=4$, or $n\geq 4$. Hence we may assume that $\Delta=5$,
$J\neq \pm 1$, and $n = 3$. By Lemma 12.9 we have $a_i \leq n+2 = 5$.
Thus the possible values of $(a_1, a_2, a_3, a_4)$ are given below.
The second quadruple gives $(s_1, s_2, s_3, s_4)$, calculated as
functions of $s = s_1$, using Lemma 12.1.
$$
\begin{array}{lll}
(1) & \qquad (5,5,5,0) & \qquad (s,\; s-1,\; s+1,\; - ) \\
(2) & \qquad (5,5,4,1) & \qquad (s,\; s-1,\; s-1,\; s ) \\
(3) & \qquad (5,3,2,5) & \qquad (s,\; s+1,\; s-1,\; s+1) \\
(4) & \qquad (5,4,5,1) & \qquad (s,\; s, \; s, \; s ) \\
(5) & \qquad (5,4,4,2) & \qquad (s,\; s, \; s+1,\; s+1) \\
(6) & \qquad (5,4,3,3) & \qquad (s,\; s, \; s-1,\; s-1) \\
(7) & \qquad (5,4,2,4) & \qquad (s,\; s, \; s ,\; s ) \\
(8) & \qquad (5,3,5,2) & \qquad (s,\; s+1,\; s-1,\; s+1) \\
(9) & \qquad (5,3,4,3) & \qquad (s,\; s+1,\; s , \; s-1)
\end{array}
$$
Cases (1), (3), (8) and (9) are impossible because there is an $i$ such
that $a_i\geq 2$ and $s_i = 0$, contradicting Lemma 2.14(3). Cases (4)
and (7) contradict Lemma 12.6.
In case (2), by Lemma 12.7(1) the edges in $\hat e_3$ are not parallel
to those in $\hat e_1 \cup \hat e_4$ on $\Gamma_b$, and by Lemma 12.7(4)
the edge in $\hat e_4$ is not parallel to those in $\hat e_1$. Thus
the 10 edges in $\hat e_1 \cup \hat e_3 \cup \hat e_4$ are mutually
non-parallel on $\Gamma_b$, contradicting the fact that $\hat \Gamma_b$ has at most
$3n$ edges (Lemma 2.5). Similarly, in case (5) the edges in $\hat e_1
\cup \hat e_3 \cup \hat e_4$ are mutually non-parallel on $\Gamma_b$, and
in case (6) the edges in $\hat e_2 \cup \hat e_3 \cup \hat e_4$ are
mutually non-parallel on $\Gamma_b$, which lead to the same contradiction.
(b) Assume $\Delta=5$. By Lemma 12.10 we have $n=3$, and by (a) we
have $a_i \leq 4$, hence the weights of $\hat e_i$ must be $(4,4,4,3)$
up to symmetry, and the transition numbers are $(s, s+1, s-1, s+1)$.
This is a contradiction to Lemma 2.14(2) because one of the families
has $s_i = 0$ and hence is a set of co-loops. \quad $\Box$ \medskip
\begin{lemma} Suppose $\Delta = 4$. Let $e, e'$ be edges of
$\Gamma_a$ with label $i$ at vertex $u_1$ and $j$ at $u_2$, $i\neq j$,
where the $i$-labels of $e, e'$ at $u_1$ are not adjacent among all
$i$-labels at $u_1$. Suppose also that $e$ belongs to a family of at
least $n$ parallel edges of $\Gamma_a$. Then $e\cup e'$ forms an essential
loop on the torus $\hat F_b$. \end{lemma}
\noindent {\bf Proof.} Note that in this case the jumping number $J=\pm 1$, so the
assumption that the $i$-labels of $e, e'$ at $u_1$ are not adjacent
implies that the $1$-labels of $e_1, e'_1$ at the vertex $v_i$ in
$\Gamma_b$ are not adjacent among all $1$-labels. By assumption $e$
belongs to a family of at least $n$ parallel edges of $\Gamma_a$, so if
$e\cup e'$ is inessential on $\hat F_b$ then by Lemma 12.2 $e_1$ and
$e'_1$ are parallel on $\Gamma_b$, which gives rise to at least $5$
parallel edges in $\Gamma_b$, contradicting Lemma 2.2(2) because $\hat \Gamma_a$ has
at most $4$ edges. \quad $\Box$ \medskip
Up to symmetry we may assume that $a_1 \geq a_3$, $a_2 \geq a_4$, and
$a_1 + a_3 \geq a_2 + a_4$. Since $a_i \leq n+1$, the possibilities
for $\Gamma_a$ are listed below. The second quadruple indicates the values
of $s_i$, calculated in terms of $s=s_1$ using Lemma 12.1.
$$
\begin{array}{lll}
(1) & \quad (n+1,\; n+1,\; n+1,\; n-3) & \qquad (s,\;s-2,\;s-4,\;s-2) \\
(2) & \quad (n+1,\; n+1,\; n,\; n-2) & \qquad (s,\; s-2,\; s-3,\; s-1) \\
(3) & \quad (n+1,\; n+1,\; n-1,\; n-1) & \qquad (s,\; s-2,\; s-2,\; s) \\
(4) & \quad (n+1,\; n,\; n+1,\; n-2) & \qquad (s,\; s-1,\; s-2,\; s-1) \\
(5) & \quad (n+1,\; n,\; n,\; n-1) & \qquad (s,\; s-1,\; s-1,\; s) \\
(6) & \quad (n+1,\; n,\; n-1,\; n) & \qquad (s,\; s-1,\; s,\; s+1) \\
(7) & \quad (n+1,\; n-1,\; n+1,\; n-1) & \qquad (s,\;s,\;s,\;s) \\
(8) & \quad (n,\; n,\; n,\; n) & \qquad (s,\;s,\;s,\;s)
\end{array}
$$
\begin{lemma} Cases (4), (5), (6), (7), (8) are impossible.
\end{lemma}
\noindent {\bf Proof.} In case (4) $\hat e_1, \hat e_3$ are not A-conjugate to $\hat
e_2, \hat e_4$ by Lemma 12.7(1). Since $a_4 < n$, by Lemma 12.6(1)
$\hat e_2, \hat e_4$ cannot be the only A-conjugate pair, hence $\hat
e_1$ must be A-conjugate to $\hat e_3$. Since they have the same
number of edges, by Lemma 17.2 the first edge $e$ of $\hat e_1$ is
parallel to the first edge $e'$ of $\hat e_3$. Since $e, e'$ have
labels $1,2$ at $u_1$, respectively, the label of $e$ at $u_2$ is $2$,
hence $s=1$. Now $\hat e_2$ is a family of at least 3 co-loops,
contradicting Lemma 2.14(2).
In case (5), by Lemma 12.7(1) $\hat e_1$ can only be conjugate to
$\hat e_4$ and $\hat e_2$ to $\hat e_3$, but since $a_4<n$, by Lemma
12.6(1) $\hat e_2$ must be A-conjugate to $\hat e_3$. Since $a_2+a_3
= 2n$, this is a contradiction to Lemma 12.7(3).
For the same reason, in case (6) $\hat e_2$ must be A-conjugate to
$\hat e_4$. By Lemma 12.7(2) the first edge $e_1$ of $\hat e_2$ must
be parallel to the first edge $e'_1$ of $\hat e_4$. Examining the
labeling we see that they have labels $2$ and $1$ at $u_1$,
respectively, so the label of $e_1$ at $u_2$ is $1$, hence $s_2 =
-1$. It follows that $s_1=s=0$, which is a contradiction to Lemma
2.3(1).
Cases (7) and (8) are impossible by Lemma 12.6.
\quad $\Box$ \medskip
\begin{lemma} Case (1) is impossible.
\end{lemma}
\noindent {\bf Proof.}
Since $a_4<n$, by Lemma 12.6(1) two of the first three families are
$A$-conjugate. Up to symmetry we may assume that $\hat e_1$ is
$A$-conjugate to $\hat e_2$ or $\hat e_3$. By Lemma 12.7(2) the first
edges of the above conjugate pair must have the same label pair.
Examining the labels of these edges on $u_1$ we see that $s=1$ if
$\hat e_1$ is $A$-conjugate to $\hat e_2$, and $s=2$ if $\hat e_1$ is
$A$-conjugate to $\hat e_3$. The second case cannot happen because
then $\hat e_2$ would be a set of at least 3 co-loops, contradicting
Lemma 2.14(2).
The graph $\Gamma_a$ is now shown in Figure 12.1. If $n\geq 5$ then there
are bigons $e_1 \cup e_2$ in $\hat e_2$ and $e'_1, e'_2$ in $\hat e_4$
with labels $4,5$ at $u_1$ and $5,6$ at $u_2$ ($6=1$ when $n=5$).
Note also that there is a pair of parallel $23$-edges $e_3\cup e'_3$
in $\hat e_2$. By Lemma 12.3(ii), these conditions imply that $e_1
\cup e'_1$ is inessential on $\hat F_b$, which contradicts Lemma
12.12.
When $n=4$, there is a pair of $14$-edges $e_1, e_2$ in $\hat e_1$,
a $14$-edge $e_3$ in $\hat e_4$, and a pair of $23$-edges in $\hat
e_2$. Note that $e_3$ is not equidistant to $e_1, e_2$. This leads
to a contradiction to Lemma 12.5.
When $n=3$, $s_i=0$ for some $i=1,2,3$, so one of the first three
families contains 4 co-loop edges, which is a contradiction
to the 3-Cycle Lemma.
\quad $\Box$ \medskip
\bigskip
\leavevmode
\centerline{\epsfbox{Figure12.1.eps}}
\bigskip
\centerline{Figure 12.1}
\bigskip
\begin{lemma} Case (2) is impossible.
\end{lemma}
\noindent {\bf Proof.} In this case $a_1 \equiv a_2 \not \equiv a_3 \equiv a_4$ mod
$2$, so by Lemma 12.7(1) no edge in $\hat e_1 \cup \hat e_2$ is
parallel to an edge in $\hat e_3 \cup \hat e_4$. Since $\Gamma_b$ contains
at least $n$ bigons while $\hat e_3 \cup \hat e_4$ contributes
at most $a_4 = n-2$ bigons on $\Gamma_b$, it follows that some edge in
$\hat e_2$ is parallel to an edge in $\hat e_1$ on $\Gamma_b$. Since $a_1
= a_2$, by Lemma 12.7(2) this implies that the first edge $e_1$ of
$\hat e_1$ is parallel to the first edge $e'_1$ of $\hat e_2$ on
$\Gamma_b$. In particular, they must have the same label pair. Since
$e_1$ has label $1$ at $u_1$ and $e'_1$ has label $2$ at $u_1$, we see
that $e_1$ has label $2$ at $u_2$, hence $s=1$. Since $s_4 = s-1 =
0$, this is a contradiction to Lemma 2.14(3) unless $a_4 = n-2 <
2$, i.e.\ $n \leq 3$.
Now suppose $n=3$. Let $e_1, e_2$ be the two $12$-edges in $\hat
e_1$. Note that there is a $12$-edge $e_3$ in $\hat e_3$, which by
the above is not parallel to any edge in $\hat e_1$, hence $e_1, e_2,
e_3$ cut $\hat F_b$ into a disk. Now $\hat e_4$ is a loop based at
$v_3$ in $\Gamma_b$, so it must be a trivial loop. This is a contradiction
because $\Gamma_b$ contains no trivial loop.
\quad $\Box$ \medskip
\begin{lemma} Case (3) is impossible.
\end{lemma}
\noindent {\bf Proof.} We claim that $s=1$. By Lemma 12.6(1) one of $\hat e_1, \hat
e_2$ is A-conjugate to some other $\hat e_j$. Because of symmetry we
may assume that $\hat e_1$ is conjugate to some $\hat e_j$. If $j=2$
then by Lemma 12.7(2) the first edge $e_1$ of $\hat e_1$ is parallel
on $\Gamma_b$ to the first edge $e'_1$ of $\hat e_2$, which has label $2$
at $u_1$, hence $e_1$ has label $2$ at $u_2$, so $s = s_1 = 1$.
Similarly, if $j=3$ then by Lemma 12.7(2), $a_1=n+1$ and $a_3=n-1$
implies that the second edge $e_2$ of $\hat e_1$ is parallel on $\Gamma_b$
to the first edge of $\hat e_3$, which has label $3$ at $u_1$, hence
$e_2$ has label pair $(23)$, which again implies that $s=1$. The case
$j=4$ is impossible by Lemma 12.6(2). The graph $\Gamma_a$ is now shown in
Figure 12.2.
\bigskip
\leavevmode
\centerline{\epsfbox{Figure12.2.eps}}
\bigskip
\centerline{Figure 12.2}
\bigskip
There are four edges $e'_1, ..., e'_4$ with label pair $(2,3)$, where
$e'_i \in \hat e_i$. One can check on Figure 12.2 that they are all
equidistant to each other. We claim that they are all parallel in
$\Gamma_b$.
The first $n$ edges of $\hat e_1$ form a loop $C$ on $\hat F_b$. Let
$a_1, a_2, a_3$ be the first three edges of $\hat e_1$, oriented from
$u_1$ to $u_2$, and let $a_i(t), a_i(h)$ be the tail and head of
$a_i$, respectively. Then as in the proof of Lemma 2.22(1), one can
show that $d_{v_2}(a_1(h), a_2(t)) = d_{v_3}(a_2(h), a_3(t))$. In
other words, the corners at $v_2, v_3$ on one side of the above loop
contain the same number of edge endpoints. Since $e'_2$ is
equidistant to $e'_1 = a_2$, we have $d_{v_2}(a_2(t), e'_2(h)) =
d_{v_3}(e'_2(t), a_2(h))$, hence the two endpoints of $e'_2$ lie on
the same side of the loop $C$. It follows that $e'_2$ is parallel to
$e'_1$. Similarly, $e'_3, e'_4$ are also parallel to $e'_1$. This
proves the claim above.
Among the four parallel edges $e'_1,..., e'_4$, at least one of $e'_3,
e'_4$ is adjacent to $e'_1$ or $e'_2$ on $\Gamma_b$. Because of symmetry
we may assume without loss of generality that $e'_3$ is adjacent to
$e'_1$ or $e'_2$. Relabel it as $e_3$.
Note that $e_3$ is a border edge. There is a face $D$ of $\Gamma_a$ with
$\partial D = e_1 \cup e_2 \cup e_3 \cup e_4$, see Figure 12.3. Let
$\alpha$ be the arc in $D$ connecting the middle points of $e_2, e_4$.
Since $e_3 = e'_3$ is parallel and adjacent to $e'_1$ or $e'_2$ and
$e'_1, e'_2$ are non-border edges in $\Gamma_a$, the face $D$ has a bigon
as a coupling face. (See Section 2 for definition.) It follows from
Lemma 2.15 that the surface $F_a$ can be isotoped rel $\partial$ so that
the new intersection graph $\Gamma_a'$ is obtained from $\Gamma_a$ by deleting
$e_2, e_4$ and replacing them with two edges parallel to $e_1, e_3$
respectively. The first family of $\Gamma_a'$ has $n+2$ edges, which is a
contradiction to Lemma 12.8. \quad $\Box$ \medskip
\bigskip
\leavevmode
\centerline{\epsfbox{Figure12.3.eps}}
\bigskip
\centerline{Figure 12.3}
\bigskip
\begin{prop} The case that $n_a=2$, $n=n_b \geq 3$ and
$\Gamma_b$ positive, is impossible.
\end{prop}
\noindent {\bf Proof.}
We have shown that $\Gamma_a$ has 8 possibilities. These have
been ruled out in Lemmas 12.13 -- 12.16.
\quad $\Box$ \medskip
\section {The case $n_a = 2$, $n_b > 4$, $\Gamma_1, \Gamma_2$
non-positive, and $\text{max}(w_1 + w_2,\,\, w_3 + w_4) = 2n_b-2$
}
Suppose $n_a \leq 2$ and $n=n_b \geq 4$. In Section 12 it has been
shown that $\Gamma_b$ cannot be positive. In sections 13--16 we will
discuss the case that $\Gamma_b$ is non-positive. The result will be given
in Propositions 14.7 and 16.8.
As before, we will use $n$ to denote $n_b$.
\begin{lemma} Suppose $n_a = 2$, $n \geq 4$, and $\Gamma_1,
\Gamma_2$ are non-positive.
(1) The reduced graph $\hat \Gamma_a$ is a subgraph of the graph shown in
Figure 13.1.
(2) Let $w_i$ be the weight of $\hat e_i$. Then up to relabeling we
may assume $w_3 + w_4 \leq w_1 + w_2$, and $w_1 + w_2 = 2n -2$ or
$2n$. \end{lemma}
\bigskip
\leavevmode
\centerline{\epsfbox{Figure13.1.eps}}
\bigskip
\centerline{Figure 13.1}
\bigskip
\noindent {\bf Proof.} (1) First note that the number of loops in $\Gamma_a$ at $u_1$ is the
same as that at $u_2$, because they have the same valence and the same
number of non-loop edges. If $\hat \Gamma_a$ has two loops based at $u_1$ then
they cut the torus into a disk, so there is no loop at $u_2$, which
would be a contradiction to the above. Therefore $\hat \Gamma_a$ has at most
one loop edge at each vertex. If there is no loop at $u_i$ then
$\hat \Gamma_a$ has at most four edges connecting $u_1$ to $u_2$. If there is
one loop of $\hat \Gamma_a$ at each $u_i$ then these cut the torus into two
annuli, each containing at most two edges of $\hat \Gamma_a$. In either case
$\hat \Gamma_a$ is a subgraph of that in Figure 13.1.
(2) Up to relabeling we may assume that $w_1 + w_2 \geq w_3 + w_4$.
Since $\Gamma_b$ is non-positive, by Lemma 2.3(1) we have $w_i \leq n$,
hence $w_1 + w_2 \leq 2n$.
Assume $w_1 + w_2 = 2n - k$ and $k>0$. Then $w_5 \geq (4n - (w_1 +
... + w_4))/2 \geq k$. Note that the set of $k$ edges $e_1 \cup ...
\cup e_k$ of $\hat e_5$ adjacent to $\hat e_1 \cup \hat e_2$ has the
same set of labels at each of its two ends. Hence by Lemma 2.4 $\hat
e_5$ contains a Scharlemann bigon, so by Lemma 2.2(4) and the parity
rule $k$ must be even. If $k\geq 4$ then $e_1 \cup ... \cup e_k$
contains an extended Scharlemann bigon, which is a contradiction to
Lemma 2.2(6). Hence $k=2$. \quad $\Box$ \medskip
If $\Gamma_b$ has a Scharlemann cycle then by Lemma 2.2(4) the surface $\hat
F_a$ is separating, cutting $M(r_a)$ into a black region and a white
region. Two Scharlemann cycles of $\Gamma_b$ have the same color if the
disks they bound lie in the same region.
\begin{lemma} Suppose $n_a=2$ and $n \geq 1$.
(1) If $e_1 \cup e_2$ and $e'_1 \cup e'_2$ are two Scharlemann bigons
of $\Gamma_b$ of the same color, then either (i) up to relabeling $e_i$ is
parallel to $e'_i$ on $\Gamma_a$ for $i=1,2$, or (ii) $\Gamma_a$ is kleinian,
and the four edges $e_1, e_2, e'_1, e'_2$ are mutually non-parallel on
$\Gamma_a$.
(2) If $\Gamma_b$ has four parallel positive edges then $\Gamma_a$ is kleinian.
\end{lemma}
\noindent {\bf Proof.} (1) If the four edges are in two families of $\Gamma_a$ then (i)
holds. If they are in three families, i.e., $e_1$ is parallel to
$e'_1$ but $e_2$ is not parallel to $e'_2$, then the nontrivial loop
$e_2 \cup e'_2$ on $\hat F_a$ is homotopic in $M(r_a)$ to the trivial
loop $e_1 \cup e'_1$, which contradicts the incompressibility of $\hat
F_a$.
Now assume that they are mutually non-parallel. Let $G$ be the
subgraph of $\Gamma_a$ consisting of these four edges and the two vertices
of $\Gamma_a$. Let $B$ be the side of $\hat F_a$ which contains the two
Scharlemann bigons. Shrinking the Dehn filling solid torus $V_a$ to
its core $K_a$ and cutting $B$ along $\hat F_a$ and then along the two
Scharlemann bigons, we obtain a manifold whose boundary consists of
the two disk faces of $G$ and two copies of the two Scharlemann
bigons, which is a sphere, so by the irreducibility of $M(r_a)$ it
bounds a 3-ball. It follows that $B$ is a twisted $I$-bundle over a
Klein bottle $K$, and $K$ intersects $K_a$ at a single point.
Therefore by Lemma 2.12 $\Gamma_a$ is kleinian.
(2) If $\Gamma_b$ has four parallel positive edges then they form two
Scharlemann bigons of the same color. By Lemma 2.2(2) no two of these
edges are parallel on $\Gamma_a$, hence by (1) $\Gamma_a$ is kleinian.
\quad $\Box$ \medskip
In the remainder of this section we assume that $\Gamma_a, \Gamma_b$ are
non-positive, $n_a = 2$, $n = n_b > 4$, and $\text{max}(w_1 +
w_2,\,\, w_3 + w_4) = 2n-2$. We may assume without loss of generality
that $w_3 + w_4 \leq w_1 + w_2 = 2n-2$. Since $w_1 + ... + w_4 +
2w_5 = \Delta n \geq 4n$, we have $w_5 = w_6 \geq 2$. Let $\alpha_1
\cup \alpha_2$ (resp.\ $\beta_1 \cup \beta_2$) be the two edges of
$\hat e_5$ (resp.\ $\hat e_6$) adjacent to $\hat e_1 \cup \hat e_2$.
Note that these are Scharlemann bigons, hence $\hat F_b$ is
separating, and $n$ is even. Without loss of generality we may assume
that $\alpha_1 \cup \alpha_2$ is a $(12)$-Scharlemann bigon. Assume
that $\beta_1 \cup \beta_2$ is a $(k,k+1)$-Scharlemann bigon.
\begin{lemma} (1) $k$ is even if and only if $w_2 = n-1$.
(2) $\{1, 2\} \cap \{k, k+1\} = \emptyset$.
\end{lemma}
\noindent {\bf Proof.} (1) This follows from the parity rule. Orient $u_1$
counterclockwise and $u_2$ clockwise in Figure 13.1. If $w_2 = n-1$
then the first edge of $\hat e_2$ has label $2$ at $u_1$ and $k+2$ at
$u_2$, so by the parity rule $k$ must be even. Similarly if $w_2 = n$
or $n-2$ then $k$ is odd.
(2) If $k = 1$ then by (1) we have $w_2 = n$ or $n-2$. In the first
case the edges of $\hat e_1$ would be co-loops, while in the second
case the edges of $\hat e_2$ would be co-loops. If $k = 2$ then
$w_1=w_2 = n-1$ and the edges of $\hat e_1$ are co-loops. Similarly
if $k=n$ then the edges of $\hat e_2$ are co-loops. Since $n-2>2$,
all cases contradict Lemma 2.14(2) because the above would imply that
there are at least three parallel co-loop edges. \quad $\Box$ \medskip
\begin{lemma} Suppose $w_1 = w_2 = n-1$. Then for $i=1,2$, the
edges of $\hat e_i$ on $\Gamma_b$ form a cycle $C_i$ and a chain $C'_i$
disjoint from $C_i$. Moreover, the vertices of $C_1$ ($C'_2$) are the
set of $v_j$ with $j$ odd, while the vertices of $C_2$ ($C'_1$) are the
set of $v_j$ with $j$ even. The cycles $C_1, C_2$ are essential on
$\hat \Gamma_b$.
\end{lemma}
\noindent {\bf Proof.} Let $\varphi_i$ be the transition function of $\hat e_i$. Let
$h$ be the number of orbits of $\varphi_i$. Since $\hat e_i$ has
$n-1$ edges, all but one component of the subgraph of $\Gamma_b$ consisting
of edges of $\hat e_i$ are cycles. Therefore $h-1 \leq 2$ by Lemma
2.14(2). Note also that each orbit contains the same number ($n/h$)
of vertices. Since $\Gamma_a$ has a Scharlemann bigon, $\hat F_b$ is
separating and the number of positive vertices of $\Gamma_b$ is the same as
that of negative vertices, hence the number of orbits $h$ is even, so
we must have $h=2$. Hence $\hat e_i$ forms exactly one cycle
component $C_i$ and one non-cycle component $C'_i$ on $\Gamma_b$. Since
each odd number appears twice at the endpoints of $\hat e_1$, $C_1$
contains $v_j$ with $j$ odd, and $C'_1$ contains those with $j$ even.
For the same reason the edges of $\hat e_2$ form a cycle $C_2$ and a
chain $C'_2$. Since $n/2$ edges of $\hat e_2$ have even labels at
$u_1$, $C_2$ must contain $v_j$ with $j$ even, while $C'_2$ contains
$v_j$ with $j$ odd. It follows that $C_1 \cap C_2 = \emptyset$.
\quad $\Box$ \medskip
When $w_1 = n-2$ and $w_2 = n$, the edges of $\hat e_2$ form exactly
two cycles $C_1$ and $C_2$ on $\Gamma_b$, essential on $\hat F_b$, where
the vertices of $C_1$ ($C_2$) are the $v_j$ with $j$ odd (even). This
is because by Lemma 2.14(2) they cannot form more than two cycles,
while $\Gamma_b$ being non-positive implies that $\hat e_2$ cannot form
only one cycle. When $w_1 = w_2 = n-1$, let $C_1, C_2$ be the cycles
given in Lemma 13.4. In either case, let $A_1, A_2$ be the annuli
obtained by cutting $\hat F_b$ along $C_1 \cup C_2$. Consider the
cycles $\alpha = \alpha_1 \cup \alpha_2$ and $\beta = \beta_1 \cup
\beta_2$ on $\Gamma_b$. Note that either $\alpha$ and $\beta$ are in
different $A_i$, or each of them has exactly one edge in each $A_i$.
We say that $\alpha, \beta$ are {\it transverse to\/} $C_i$ in the
second case.
\begin{lemma} The cycles $\alpha, \beta$ are disjoint, and
transverse to $C_i$. \end{lemma}
\noindent {\bf Proof.} The first statement follows from Lemma 13.3(2), so we need only
show that $\alpha, \beta$ are transverse to $C_i$.
First assume $\Delta = 5$. Then $w_5 = \frac 12 (\Delta n -
(w_1 + ... + w_4)) \geq \frac n2 + 2$. By Lemma 2.3(3) we also have
$w_5 \leq \frac n2 + 2$, hence $w_5 = \frac n2 + 2$, in which case the
two outermost bigons of the family $\hat e_5$ are Scharlemann bigons,
with label pair $(12)$ and $(r+1, r+2)$, respectively, where $r=n/2$.
By Lemma 2.3(4) the label pair of $\beta_1 \cup \beta_2$ must be
either $(1,2)$ or $(r+1, r+2)$, and by Lemma 13.3 it cannot be the
former. Therefore it must be $(r+1, r+2)$.
If $\alpha$ is not transverse to $C_i$, then it is an essential cycle
in one of the annuli, say $A_1$, obtained by cutting $\hat F_b$ along
$C_1 \cup C_2$, so $\beta$ must be an essential cycle in the other
annulus $A_2$. The two cycles $\alpha$ and $\beta$ separate the
vertices of $C_1$ from $C_2$, except $v_1, v_2, v_{r+1}$ and $v_{r+2}$
which lie on $\alpha \cup \beta$. On the other hand, the edge $e$ in
$\hat e_5$ adjacent to $\alpha_2$ has label pair $(3,n)$, so there is
an edge on $\Gamma_b$ connecting $v_3$ to $v_n$. Since $n$ is even, the
vertices $v_3, v_n$ belong to different $C_i$, but since $n>4$,
neither $3$ nor $n$ belongs to the set $\{1,2,r+1, r+2\}$, which is a
contradiction.
Now assume $\Delta = 4$. In this case the jumping number $J(r_a, r_b)
= \pm 1$. Consider the two negative edges $e', e''$ of $\Gamma_a$ with
label $2$ at $u_1$. Note that their endpoints at $u_1$ are separated
by the label 2 endpoints of $\alpha_1, \alpha_2$, hence by the Jumping
Lemma, on $\Gamma_b$ the endpoints of $e', e''$ at $v_2$ are separated by
those of $\alpha$; in other words, $e', e''$ are on different sides of
the cycle $\alpha$. Assume that $v_2 \in C_2$ is positive. If
$\alpha$ is not transverse to $C_2$ then all positive edges at $v_2$
must be on one side of $\alpha$ because the other side is shielded by
the cycle $C_1$, which contains only negative vertices. This is a
contradiction. Therefore $\alpha$, and hence $\beta$, must be
transverse to $C_i$. \quad $\Box$ \medskip
\begin{lemma} Each edge of $\hat e_1 \cup \hat e_2$ is either
on $C_1 \cup C_2$ or parallel to an edge of $C_1 \cup C_2$ on $\Gamma_b$.
\end{lemma}
\noindent {\bf Proof.} Let $C_1, C_2$ and $\alpha, \beta$ be as above. By definition
$C_2$ consists of the edges in $\hat e_2$ with even labels. Let
$C'_1$ be the edges of $\hat e_1$ with even labels. Because of
symmetry it suffices to show that each edge of $C'_1$ is parallel to
an edge in $C_2$.
Note that $\alpha \cap C_2 = v_2$. Let $v_t = \beta \cap C_2$. (Thus
$t$ is the even label of the Scharlemann bigon $\beta_1 \cup \beta_2$
in $\Gamma_a$.) Since $w_1+w_2 = 2n-2$ and the edges adjacent to $\hat e_1
\cup \hat e_2$ on $\Gamma_a$ are the $(12)$-Scharlemann bigon $\alpha_1
\cup \alpha_2$ and the $(t,t+1)$- or $(t-1,t)$-Scharlemann bigon
$\beta_1 \cup \beta_2$, we see that $2, t$ are the only even labels
appear three times among the endpoints of edges in $\hat e_1 \cup \hat
e_2$, hence on $\Gamma_b$ the edges of $C'_1$ form a chain with endpoints
at $v_2, v_t$, and possibly some cycle components. Therefore $C'_1 -
v_2 \cup v_t$ is disjoint from $\alpha \cup \beta \cup C_1$, hence
lies in the interior of the two disks obtained by cutting $\hat F_b$
along $\alpha \cup \beta \cup C_1$. By Lemma 2.14(1) this implies
that $C'_1$ has no cycle component, and hence is a chain. Since
$C'_1$ contains all vertices of $C_2$, this also implies that one
component of $C_2 - v_2 \cup v_t$ contains no vertices of $\Gamma_b$; in
other words, the two vertices $v_2, v_t$ are adjacent on $C_2$.
Let $q, p$ be the transition number of $\hat e_1, \hat e_2$,
respectively. Since $C_2$ has an edge connecting $v_2$ to $v_t$, we
have $p \equiv \pm (t-2)$ mod $n$. Since $C'_1$ is a chain of length
$(n/2)-1$ connecting $v_2, v_t$, we have $((n/2) - 1) q \equiv \pm
(t-2)$ mod $n$. An edge of $C'_1$ has even labels on both endpoints,
so $q$ is even, hence $((n/2) -1)q \equiv -q$ mod $n$. It follows
that $p \equiv \pm q$ mod $n$, which implies that each edge $e'$ of
$C'_1$ has its endpoints on adjacent vertices of $C_2$. Let $e$ be
the edge of $C_2$ connecting these two vertices. Since $e'$ has
interior disjoint from $\alpha \cup \beta \cup C_1$, it must be
parallel to $e$. \quad $\Box$ \medskip
\begin{prop} The case that $\Gamma_a, \Gamma_b$ are non-positive,
$n_a = 2$, $n = n_b > 4$, and $w_3 + w_4 \leq w_1 + w_2 = 2n-2$, is
impossible. \end{prop}
\noindent {\bf Proof.} First assume that $w_1 = n-2$ and $w_2 = n$. By Lemma 13.3, the
label pair of $\beta$ is $(k, k+1)$, where $k$ is odd and $k\neq 1$.
If $k=n-1$ then the edges of $\hat e_2$ are co-loops, which
contradicts Lemma 2.14(2). Therefore $n-3 \geq k \geq 3$.
Since the label sequence of $\hat e_1$ at $u_2$ is $k+2, ..., n, 1,
..., k-1$, the above implies that there are adjacent edges $e'_1, e'_n
\in \hat e_1$ with labels $1$ and $n$ at $u_2$, respectively. By
Lemma 13.6 each edge of $\hat e_1$ is parallel to some edge of $\hat
e_2$ on $\Gamma_b$, hence the transition function $\psi_1$ of $\hat e_1$ is
either equal to $\psi_2$ of $\hat e_2$, or $\psi_2^{-1}$, but since
the two edges of $\hat e_1 \cup \hat e_2$ with label $n$ at $u_1$ have
labels $k+1$ and $k-1$ respectively at $u_2$, the first case is
impossible, hence $\psi_1 = \psi_2^{-1}$. Let $\hat e_2 = e_1 \cup
... \cup e_n$, where $e_i$ has label $i$ at $u_1$. Since $e_1$ is the
only edge of $\hat e_2$ with label $1$ at $u_1$, it must be the one
that is parallel to $e'_1$ on $\Gamma_b$. Similarly, $e_n$ is parallel to
$e'_n$ on $\Gamma_b$. This is a contradiction to Lemma 2.19.
Now assume that $w_1 = w_2 = n-1$. As above, let $\hat e_2 = e_2 \cup
... \cup e_n$, where $e_i$ has label $i$ at $u_1$. The label sequence
of $\hat e_1$ at $u_2$ is $k+1, k+2, ..., n, 1, ..., k-1$. By Lemma
13.3, $k$ is even, and $\{1,2\} \cap \{k, k+1\} = \emptyset$, so $n-2
\geq k \geq 4$. It follows that there are three consecutive edges
$e'_n, e'_1, e'_2$ of $\hat e_1$ such that $e'_i$ has label $i$ at
$u_2$. For the same reason as above, $e'_2$ is parallel to $e_2$ and
$e'_n$ is parallel to $e_n$ on $\Gamma_b$. Since the number of edges
between $e'_n$ and $e'_2$ is 1 while the number of edges between $e_2$
and $e_n$ is $n-3 > 1$ on $\Gamma_a$, this is a contradiction to Lemma
2.19. \quad $\Box$ \medskip
\section {The case $n_a = 2$, $n_b > 4$, $\Gamma_1, \Gamma_2$
non-positive, and $w_1 = w_2 = n_b$ }
In this section we consider the case that $n_a = 2$, $n = n_b > 4$,
$\Gamma_1, \Gamma_2$ non-positive, and $w_1 = w_2 = n$. We will also
assume without loss of generality that $w_3 \geq w_4$. Let $\hat e_1
= e_1 \cup ... \cup e_{n}$, $\hat e_2 = e'_1 \cup ... \cup e'_{n}$,
and assume that $e_i, e'_i$ have label $i$ at $u_1$.
Let $r$ be such that the label of the endpoint of $e_1$ on $\partial u_2$
is $r+1$. One can check that both $e_i, e'_i$ have label $r+i$ at
$\partial u_2$.
Since $\Gamma_b$ is non-positive, the vertices of $\Gamma_b$ cannot all be
parallel, so the edges of $\hat e_1$ form at least two cycles on $\Gamma_b$.
By Lemma 2.14(2) they form exactly two cycles $C_1 \cup C_2$ on $\Gamma_b$.
\begin{lemma} $\Gamma_a$ is not kleinian. In particular, $\Gamma_b$
cannot contain four parallel positive edges.
\end{lemma}
\noindent {\bf Proof.} If $\Gamma_a$ is kleinian then by Lemma 6.2(4) there is a free
orientation reversing involution $\phi$ of $(\hat F_a, \Gamma_a)$, which
maps $u_1$ to $u_2$, and is label preserving. If there is no loop on
$\Gamma_a$ (i.e.\ $w_5 = w_6 = 0$), then $\Delta = 4$ and $w_i = n$ for all
$i$, so the label sequences of $\hat e_i$ at $u_1$ are all the same.
The above implies that the label sequences of $\hat e_i$ at $u_2$ are
also the same as those at $u_1$, so the transition function $\varphi$
defined by $\hat e_i$ is the identity map and hence all edges of
$\Gamma_a$ are co-loops, contradicting the 3-Cycle Lemma 2.14(2).
Now assume $w_5 = w_6 > 0$. Then $\phi$ maps $\hat e_1 \cup\hat e_2$
to either $\hat e_1 \cup \hat e_2$ or $\hat e_3 \cup \hat e_4$. In
the first case since $\phi$ is label preserving and orientation
reversing on the torus, the label sequence of $\hat e_1$ at $u_2$ is
the same as that of $\hat e_1$ at $u_1$, hence all edges of $\hat e_1$
are co-loops and we have a contradiction to Lemma 2.14(2). In the
second case $w_3 = w_4 = w_1 = w_2 = n$, so $\Delta = 5$ and $w_5 =
w_6 = n/2$. We have assumed that $\hat e_1$ has label sequence
$1,2,...,n$ at $u_1$. so $\hat e_3$ has the same label sequence at
$u_2$. Since $w_5 = n/2$, the label sequence of $\hat e_1$ at $u_2$
is $(k+1, k+2, ...n, 1, ...,k)$, where $k = n/2$. Therefore $\phi$ is
of period 2, so it has $n/2 > 2$ orbits, which again contradicts Lemma
2.14(2).
The second statement follows from the above and Lemma 13.2(2).
\quad $\Box$ \medskip
\begin{lemma} The edges $e_i, e'_i$ are parallel on $\Gamma_b$.
\end{lemma}
\noindent {\bf Proof.} The cycles $C_1 \cup C_2$ defined at the beginning of the
section cut the torus $\hat F_b$ into two annuli $A_1, A_2$. Each
$e'_i$ lies in one of the $A_j$ and has the same endpoints as $e_i$,
so if it is not parallel to $e_i$ and $e_i \subset C_1$ then it is
parallel to $C_1 - e_i$. There are at most two such $e'_i$ for $C_1$,
one in each $A_j$. Since $n > 4$, $C_1$ contains at least three
edges, hence there exists some $e'_j$ parallel to $e_j \subset C_1$.
Assume $e_i$ is not parallel to $e'_i$, and let $e_j, e'_j$ be
parallel on $\hat F_b$, which exist by the above. Let $D$ (resp.
$D'$) be the disk on $F_a$ that realizes the parallelism between $e_i,
e_j$ (resp.\ $e'_i, e'_j$), and let $D''$ be the disk between $e_j$
and $e'_j$ on $F_b$. Shrinking $V_b$ to its core $K_b$, $B = D \cup
D' \cup D''$ becomes a disk in $M(r_b)$ with $\partial B = e_i \cup
e'_i$, which contradicts the fact that $\hat F_b$ is incompressible in
$M(r_b)$.
\quad $\Box$ \medskip
\bigskip
\leavevmode
\centerline{\epsfbox{Figure14.1.eps}}
\bigskip
\centerline{Figure 14.1}
\bigskip
\begin{lemma} $w_3 + w_4 \neq n$.
\end{lemma}
\noindent {\bf Proof.} Assume to the contrary that $w_3 + w_4 = n$. We have $\Delta =
4$ as otherwise there would be $n$ parallel positive edges in $\hat
e_5$, contradicting Lemma 2.3(3). Now $w_5 = w_6 = n/2$, so the graph
$\Gamma_a$ is as shown in Figure 14.1, where $k=n/2$. Since $\Delta = 4$, we
may assume that the jumping number is $1$.
Let $i$ be a label such that $1 \leq i \leq k$, so it appears on the
top of the vertex $u_1$ in Figure 14.1. Consider the vertex $v_i$ of
$\Gamma_b$, see Figure 14.2. By Lemma 14.2 $e_i$ of $\hat e_1$ is parallel
to $e'_i$ of $\hat e_2$. Since $e_i$ and $e'_i$ have the same label
$1$ at $v_i$, there is an edge of $\hat e_3 \cup \hat e_4$ between
them. Similarly there are parallel edges $e_j, e'_j$ with label $2$
at $v_i$, and there is another edge between them. See Figure 14.2.
From the labeling we see that the two negative edges at $v_i$
(corresponding to loops in $\Gamma_a$) must be adjacent to each other.
\bigskip
\leavevmode
\centerline{\epsfbox{Figure14.2.eps}}
\bigskip
\centerline{Figure 14.2}
\bigskip
On $\partial u_1$ the $i$-labels appear on endpoints of edges in the order
of $A, B, C, D$, where $A \in e_i$, $B \in e'_i$, $C$ is a loop in
$\hat e_5$, and $D \in \hat e_3 \cup \hat e_4$. Since the jumping
number is $1$, the $1$-labels at $v_i$ also appear in the same order.
In Figure 14.2 this implies that the negative edge $C$ appears on the
top of the vertex.
Now consider the four edges labeled $2$ at $v_i$, denoted by $E, F, G,
H$, where $E$ is the negative edge, which is uniquely determined.
Since $F, G$ are parallel positive edges on $\Gamma_b$, they are the $e_j,
e'_j$ given above, belonging to $\hat e_1 \cup \hat e_2$. On $\partial
u_2$ this implies that the endpoint of the loop $E$ labeled $i$
appears on the top of $\partial u_2$ in Figure 14.1. Since $i$ is any
label between $1$ and $n/2$, it follows that the labels on the top of
$\partial u_2$ must be $1,2,...,k$, so the integer $r$ in the figure
satisfies $r = k$. However, in this case the edges of $\hat e_1$
would form cycles of length 2 in $\Gamma_b$, which is a contradiction to
Lemma 2.14(2). \quad $\Box$ \medskip
\begin{lemma} $w_3 = n$, and $0 < w_4 < n$. Moreover, an edge
$e''$ of $\hat e_3$ with label $j$ at $u_2$ is parallel to the edges
$e_j$ and $e'_j$ .
\end{lemma}
\noindent {\bf Proof.} We have assumed $w_4 \leq w_3\leq n$. If $w_4 = n$ then the
argument of Lemma 14.2 applied to $\hat e_3, \hat e_4$ shows that each
edge of $\hat e_3$ is parallel to exactly one edge of $\hat e_4$. On
the other hand, since the two parallel edges $e_i, e'_i$ in the proof
of Lemma 14.2 have the same label $1$ at the vertex $v_i$, there must
be another edge $e''_i$ in $\hat e_3 \cup \hat e_4$ between $e_i$ and
$e'_i$. Together with the other edge in $\hat e_3 \cup \hat e_4$
which is parallel to $e''_i$, we get four parallel positive edges in
$\Gamma_b$, which contradicts Lemma 14.1. Therefore $w_4 < n$.
Recall from Lemma 14.2 that the edges $e_i$ and $e'_i$ are parallel in
$\Gamma_b$, with the same label $1$ at $u_i$, so there must be another edge
$e''_i \in \hat e_3 \cup \hat e_4$ between them. Note also that if
$e_i$ has label $i+r$ at $u_2$ then $e''_i$ has the property that it
has label $i$ at $u_2$ and $i+r$ at $u_1$. This is true for all $i$,
so either $\hat e_3$ and $\hat e_4$ have the same transition function,
or these $e''_i$ all belong to the same family. The first case
happens only if $w_3 + w_4 \equiv 0$ mod $n$, which is impossible
because by Lemma 14.3 we have $w_3 + w_4 \neq n$, while $w_3 \leq n$
and by the above we have $w_4 < n$. Therefore all the $e''_i$ belong
to $\hat e_3$. Since $w_3 \leq n$, this implies that $w_3 = n$.
Again by Lemma 14.3 we have $w_4 \neq 0$. \quad $\Box$ \medskip
\bigskip
\leavevmode
\centerline{\epsfbox{Figure14.3.eps}}
\bigskip
\centerline{Figure 14.3}
\bigskip
\begin{lemma}
The label sequence of $\hat e_3$ is $1,2,...,n$ at $u_2$, and
$1+r, 2+r, ..., n, 1, ..., r$ at $u_1$. The labels of $\Gamma_a$ are as
shown in Figure 14.3.
\end{lemma}
\noindent {\bf Proof.} First assume that the label sequence of $\hat e_3$ at $u_2$ is
not $1,2,..., n$. Then there is a pair of adjacent parallel edges
$e''_{n}, e''_1$ with label $n$ and $1$ at $u_2$, respectively. By
Lemma 14.4 $e_1 \cup e''_1$ and $e_n \cup e''_n$ are parallel pairs on
$\Gamma_b$. Since $e''_1, e''_n$ are adjacent on $\Gamma_a$ while $e_1, e_n$
are not, this is a contradiction to Lemma 2.19. Therefore the label
sequence of $\hat e_3$ at $u_2$ must be $1,2,..., n$.
Since by Lemma 14.4 the edge $e''_i$ connects $v_i$ to $v_{i+r}$ with
label $2$ at $v_i$ and $1$ at $v_{i+r}$, we see that on $\Gamma_a$ it has
label $i$ at $u_2$ and $i+r$ at $u_1$, hence the label sequence of
$\hat e_3$ at $u_1$ is $r+1, ..., n, 1, ..., r$. The labels of $\hat
e_1, \hat e_2, \hat e_3$ determine those of the loops, and hence those
of $\hat e_4$. Therefore $\Gamma_a$ must be as shown in Figure 14.3.
\quad $\Box$ \medskip
\begin{lemma} (1) The jumping number $J = \pm 1$.
(2) Orient the negative edges of $\Gamma_a$ from $u_1$ to
$u_2$. Then on $\Gamma_b$ the edges of $\hat e_1$ form two essential
cycles of opposite orientation on $\hat F_b$. \end{lemma}
\noindent {\bf Proof.} (1) Since $\Delta =
4$ or $5$, the jumping number is either $\pm 1$ or $\pm 2$. Let $e_i,
e'_i$ be the edges of $\hat e_1, \hat e_2$, respectively, with label
$i$ at $u_1$. If $J = \pm 2$ then these edges are not adjacent among
the $1$-edges at $v_i$ in $\Gamma_b$. Since by Lemma 14.2 they are parallel
in $\Gamma_b$, there would be more than $2n_a = 4$ parallel edges in $\Gamma_b$,
which contradicts Lemma 2.2(2). Therefore $J = \pm 1$. Changing the
orientation of $\hat F_b$ if necessary, we may assume that $J = 1$.
(2) Now let $C_1, C_2$ be the cycles on $\Gamma_b$ consisting of edges of
$\hat e_1$. We need to show that they are of opposite orientation.
Let $a_1, a_2, \bar a, a_3$ be the edges with label $1$ at $u_1$,
where $a_i \in \hat e_i$ for $i=1,2,3$, and $\bar a \in \hat e_5$.
Note that they appear in this order on $\partial u_1$.
Since $J=1$, they also appear in this order on $\partial v_1$ in $\Gamma_b$,
see Figure 14.4. By the proof of Lemma 14.3 we see that $a_3$ is in
the middle of a pair of parallel positive edges incident to $v_1$,
which is not parallel to $a_1, a_2$, hence the orientation of $C_1$
must be as shown in Figure 14.4, where $C_1$ is represented by the
lower level chain.
\bigskip
\leavevmode
\centerline{\epsfbox{Figure14.4.eps}}
\bigskip
\centerline{Figure 14.4}
\bigskip
Now consider the edges labeled $n$ at $u_1$. There are 5 of them if
$\Delta = 5$, but we only consider $\bar a$ and the edges $a'_1, a'_2,
a'_3$, where $a'_i \in \hat e_i$. The order of the label $n$
endpoints of these edges on $\partial u_1$ is $a'_3, \bar a, a'_1, a'_2$,
while the orientation of $v_n$ is opposite to that of $v_1$.
Therefore these edges appear on $\partial v_n$ as shown in Figure 14.4.
We see that $C_1, C_2$ are of opposite orientation on $\hat F_b$.
\quad $\Box$ \medskip
\bigskip
\leavevmode
\centerline{\epsfbox{Figure14.5.eps}}
\bigskip
\centerline{Figure 14.5}
\bigskip
\begin{prop} Suppose $n_a = 2$, $n > 4$, $\Gamma_a, \Gamma_b$ are
non-positive, and $w_1 = w_2 = n$.
(1) On $\Gamma_b$ each edge of $\hat e_4$ connects a pair of adjacent
vertices of some $C_i$, but is not parallel to an edge of $C_i$.
(2) Two adjacent edges of $\hat e_4$ lie in different annuli of $\hat
F_b - \cup C_i$.
(3) $w_4 = w_5 = w_6 = 2$, $\Delta = 4$, and $n = 6$.
(4) The graphs $\Gamma_a, \Gamma_b$ and their edge correspondence are as shown
in Figure 14.5, where $e_i$ (resp.\ $e'_i$) is the edge in $\hat e_1$
(resp.\ $\hat e_2$) with label $i$ at $u_1$, and the edge between
$e_i, e'_i$ is the edge of $\hat e_3$ with label $i$ at $u_2$.
\end{prop}
\noindent {\bf Proof.} (1) From Figure 14.3 we see that an edge $e$ of $\hat e_4$ with
label $i$ at $u_1$ has label $i+r$ at $u_2$. Since the transition
function of $\hat e_1$ also maps $i$ to $i+r$, $v_i$ and $v_{i+r}$ are
connected by the edge $e_i$ of $\hat e_1$, and hence are adjacent on
one of the cycles $C_j$. This proves the first part of (1). By
Lemmas 14.2 and 14.4 each edge $e_i$ of $C_j$ is parallel to an edge
$e'_i$ in $\hat e_2$ and an edge $e''_i$ in $\hat e_3$, so if $e$ is
parallel to $e_i$ then there would be four parallel positive edges in
$\Gamma_b$, which would contradict Lemma 14.1.
(2) By Lemma 14.6(2) the two cycles $C_1, C_2$ have opposite
orientations. Without loss of generality we may assume that the
orientations of $C_i$ are as shown in Figure 14.5. Recall that $C_1$
is the cycle containing the vertex $v_1$. By Lemma 14.6(1) we may
assume without loss of generality that the jumping number of the
graphs is $1$. Let $e$ be an edge of $\hat e_4$ with label $k$ at
$u_1$, and let $e_k, e'_k, e''_k$ be the edges of $\hat e_1, \hat e_2,
\hat e_3$ with label $k$ at $u_1$. Then the endpoints of these edges
appear at $\partial u_1$ in the order $e_k, e'_k, e''_k, e$, so on
$\partial v_k$ they appear in the same order. If $v_k$ is in $C_1$ then
the orientation of $C_1$ points to the left and the orientation of
$v_k$ is clockwise, so $e$ is in the annulus below $C_1$. If $v_k$ is
in $C_2$ then the orientation of $C_1$ points to the right and the
orientation of $v_k$ is counterclockwise, so again $e$ is in the
annulus below $C_1$. Since the labels of adjacent edges of $\hat e_4$
belong to different $C_i$ in $\Gamma_b$, it follows from the above that
they are in different annuli of $\hat F_b - \cup C_i$.
(3) Since each $C_i$ contains $n/2 > 2$ vertices, there cannot be two
edges on the same side of $C_i$ connecting two different pairs of
adjacent vertices and yet not parallel to an edge of $C_i$. Hence by
(1) and (2) $\hat e_4$ contains at most two edges. By Lemma 14.4 $w_4
> 0$, and from the labeling in Figure 14.3 we see that $w_4$ is even.
Therefore $w_4 = 2$.
If $\Delta = 5$ then the loop family of $\Gamma_a$ at $u_1$ contains $n-1$
edges. This contradicts Lemma 2.3(3) for $n>4$. Hence $\Delta =
4$.
Let $e$ be an edge of $\hat e_4$ with endpoints on $v_i$ and $v_j$ in
$C_2$, lying on the annulus $A$ below $C_2$. By (1) it is not
parallel to the edge on $C_2$ connecting $v_i, v_j$, so on $A$ it
separates $C_1$ from other vertices of $C_2$, hence there is no edge
in $A$ connecting $C_1$ to vertices of $C_2$ except possibly $v_i$ and
$v_j$. By Lemmas 14.2 and 14.4 there are three parallel edges for
each edge of $C_i$. Together with $e$, they contribute $7$ edge
endpoints to each of $v_i$ and $v_j$, therefore $\Delta = 4$ implies
that there are at most two edges in $A$ connecting $C_1$ to $C_2$, one
for each of $v_i, v_j$. Note that these correspond to loop edges in
$\Gamma_a$. Therefore the two annuli give rise to at most 4 loops in
$\Gamma_a$, so $w_5 = w_6 \leq 2$. Since $n>4$ and $2w_5 + w_4 = (\Delta -
3)n = n$, it follows that $n=6$, and $w_5=w_6 = 2$.
(4) By Lemma 14.5 $\Gamma_a$ is the graph in Figure 14.3. We have $w_4 =
w_5 = w_6 = 2$ and $w_1=w_2=w_3 = n = 6$, hence $\Gamma_a$ is as shown in
Figure 14.5(a).
The edges in $\hat e_2, \hat e_3$ are parallel to those in $\hat e_1$,
as shown in Lemmas 14.2 and 14.5, therefore they form families of
three parallel edges, as shown in Figure 14.5(b). Orientations are
from $u_1$ to $u_2$ on $\Gamma_a$, so the tails of these edges are labeled
$1$ and the heads labeled $2$ on $\Gamma_b$. The two edges in $\hat e_4$
connect $v_4, v_6$ and $v_3, v_5$ respectively, and by (1) and (2)
they are not parallel to edges in $C_i$ and lie in different annuli of
$\hat F_b - C_1\cup C_2$, hence we may assume that they look like that
in Figure 14.5(b). The four edges in $\hat e_5$ and $\hat e_6$ are
now determined by the labeling of the edges and the vertices on
$\Gamma_b$. The labeling of the weight 3 families in $\Gamma_b$ are determined
by the single edges and the assumption that the jumping number is $1$.
\quad $\Box$ \medskip
\section {$\Gamma_a$ with $n_a \leq 2$ }
The next few sections deal with the case that $n_a \leq 2$ and $n_b
\leq 4$. In this section we set up notation and give some
preliminary results.
We use $G = (b_1, b_2, b_3)$ to denote a graph $G$ on a torus with one
vertex and three families of edges weighted $b_1, b_2, b_3$.
Similarly, denote by $G = (\rho; a_1, ..., a_4)$ a graph $G$ on a
torus which has two vertices, two families of loops of weight $\rho$,
and four families of edges $\hat e_i$ with weight sequence $a_1, ...,
a_4$ around the vertices. It is possible that $\rho$ and some of the
$a_i$ may be zero. When $\rho = 0$ we will simply write $G = (a_1,
..., a_4)$. Note that the weight sequence is defined up to cyclic
rotation and reversal of order. When $\rho=0$, any weight $0$ can be
moved around without changing the graph, hence $(2,2,0,0)$ is
equivalent to $(2,0,2,0)$, but $(1;2,0,2,0)$ is different from
$(1;2,2,0,0)$ and $(3,1,3,1)$ is different from $(3,3,1,1)$. When it
is necessary to indicate whether the vertices of $G$ are parallel or
antiparallel, we write $G =+(\rho; a_1, ..., a_4)$ if the vertices of
$G$ are parallel, and $G=-(\rho; a_1, ..., a_4)$ otherwise.
If $n_a = 2$ then $\Gamma_a$ is of the form $(\rho; a_1, ..., a_4)$. Note
that if $e$ is co-loop then all edges parallel to $e$ are. Hence we
may define an edge $\hat e$ in $\hat \Gamma_a$ to be co-loop if one (and hence
all) of its edges is co-loop. Define $\epsilon_i = 0$ if $\hat e_i$ is a
co-loop, and $\epsilon_i=1$ otherwise. Note that if $n_b =2$ then
$\epsilon_i$ measures the difference between the labels at the two
endpoints of an edge in $\hat e_i$, so it is actually the same as the
transition number defined in Section 2.
\begin{lemma} {\bf (The Congruence Lemma.)} Suppose $n_a =
2$. Let $\hat e_i, \hat e_j$ be edges in $\hat \Gamma_a$ with the endpoints on the
same pair of vertices $u_1, u_2$. Let $a_k$ be the weight of $\hat e_k$.
(1) If $\hat \Gamma_a$ has no loops and $a_i, a_j\neq 0$ then $a_i +
\epsilon_i \equiv a_j + \epsilon_j$ mod $2$. In other words, $a_i
\equiv a_j$ mod $2$ if and only if $\hat e_i$ and $\hat e_j$ are both
co-loop or both non co-loop.
(2) If $\hat \Gamma_a$ has loops and $a_i, a_j \neq 0$, then $a_i \equiv
a_j$ mod $2$.
(3) If $\hat \Gamma_a$ has loops and the endpoints of $\hat e_i, \hat e_j$ at $u_1$
are on the same side of the loop at $u_1$ then $a_i \equiv a_j$
mod $2$. \end{lemma}
\noindent {\bf Proof.} (1) Delete edges of $\hat \Gamma_a$ with zero weight. We need only
prove the statement for adjacent edges $\hat e_1, \hat e_2$ of $\hat \Gamma_a$ with
non-zero weight. Let $e_1, ..., e_{a_1}$ and $e'_1, ..., e'_{a_2}$
be the edges in $\hat e_1$ and $\hat e_2$, respectively, so that $e'_1$ is
adjacent to $e_{a_1}$ on $\partial u_1$. Then $e'_{a_2}$ is adjacent to
$e_1$ on $\partial u_2$. Without loss of generality assume that the label
of $e_i$ at $u_1$ is $i$. (Since $n_b=2$, all labels of endpoints of
$e_i, e'_j$ are mod 2 integers.) Then the label of $e_1$ at $u_2$ is
$1+\epsilon_1$. On the other hand, the label of $e'_1$ at $u_1$ is
$a_1 + 1$, so the label of $e'_{a_2}$ at $u_1$ is $a_1 + a_2$, and
the label of $e'_{a_2}$ at $u_2$ is $a_1 + a_2 + \epsilon_2$. See
Figure 15.1. Since $e'_{a_2}$ is adjacent to $e_1$ on $u_2$, the
label of $e_1$ on $u_2$ is $a_1 + a_2 + \epsilon_2 + 1$. These two
equations give
$$
1 + \epsilon_1 \equiv a_1 + a_2 + \epsilon_2 + 1 \qquad \text{ mod
} 2$$
It follows that $a_1 \equiv a_2$ if and only if $\epsilon_1
\equiv \epsilon_2$ mod $2$.
\bigskip
\leavevmode
\centerline{\epsfbox{Figure15.1.eps}}
\bigskip
\centerline{Figure 15.1}
\bigskip
(2) Again we need only prove the statement for adjacent edges $\hat e_1,
\hat e_2$ with non-zero weight. Since $\Gamma_a$ has loops, the two vertices
of $\hat \Gamma_b$ must be antiparallel. If the two vertices of $\Gamma_a$ are
parallel then $\hat e_1, \hat e_2$ are both positive on $\Gamma_a$ and hence both
negative on $\Gamma_b$, so they are both non co-loops. Similarly if the
two vertices of $\Gamma_a$ are antiparallel then $\hat e_1, \hat e_2$ are both
co-loops. Therefore $\epsilon_1 = \epsilon_2$ in either case. Note
that the endpoints of $\hat e_1, \hat e_2$ are on the same side of the loop
at $u_1$ if and only if their other endpoints are on the same side of
the loop at $u_2$. Since the number of loops at the two vertices are
the same, the distance between the endpoints of $e_{a_1}$ and $e'_1$
on $\partial u_1$ is the same as that of $e'_{a_2}$ and $e_1$ on $\partial
u_2$, hence the above argument can be modified to show that if $a_1,
a_2 \neq 0$ then $a_1 \equiv a_2$ mod $2$. More explicitly, if
there is no loop between the endpoints of $\hat e_1, \hat e_2$ then the
above argument follows verbatim, while if there are $k$ loops between
them then the endpoint $e'_{a_2}$ is $a_1 + a_2 + k$ at $u_1$, and
$a_1 + a_2 + k + \epsilon_2$ at $u_2$, and we have
$a_1 + a_2 + k + \epsilon_2 + k + 1 = 1 + \epsilon_1$, hence the
result follows because $\epsilon_1 = \epsilon_2$ mod $2$.
(3) This follows from (2) if $a_i, a_j \neq 0$. If $a_i=0$ and
$\hat e_j$ is on the same side of the loop as $\hat e_1$ (which is empty),
then since a loop has different labels on its two
endpoints, the number of edges in $\hat e_2$ must be even, hence $a_2
\equiv a_1 = 0$ mod $2$. \quad $\Box$ \medskip
\begin{lemma} Suppose $\Gamma_b$ is positive, and contains a black
bigon $e_1\cup e_2$ and a white bigon $e'_1 \cup e'_2$. Then on $\Gamma_a$
the four edges $e_1, e_2, e'_1, e'_2$ cannot be contained in two
families of parallel edges. \end{lemma}
\noindent {\bf Proof.} Recall that no two edges are parallel on both graphs, so if the
lemma is not true then we may assume that $e_i$ is parallel to $e'_i$
on $\Gamma_a$. Let $B_i$ be the disk on $F_a$ realizing the parallelism,
and let $D, D'$ be the bigon on $F_b$ bounded by $e_1\cup e_2$ and
$e'_1 \cup e'_2$, respectively. Then $A = D \cup D' \cup B_1 \cup
B_2$ is either a M\"obius band or an annulus. The first case
contradicts the fact that a hyperbolic manifold $M$ contains no M\"obius
bands. In the second case $A$ contains a single white bigon and hence
each of its boundary components intersects a curve of slope $r_a$
transversely at a single point. Since $e_1$ is an essential arc on
both $F_a$ and $A$, $A$ cannot be boundary parallel, and hence is
essential in $M$, which is again a contradiction to the hyperbolicity
of $M$. \quad $\Box$ \medskip
\section {The case $n_a = 2$, $n_b=3$ or $4$, and $\Gamma_1,
\Gamma_2$ non-positive }
Throughout this section we assume that $n_a = 2$, $n_b = 3$ or $4$,
and both $\Gamma_1, \Gamma_2$ are non-positive. We will show that in
this case there are only three possibilities for the pair $(\Gamma_a,
\Gamma_b)$, given in Figures 16.6, 16.8 and 16.9. The following lemma
rules out the possibility that $n_b = 3$.
\begin{lemma} The case $n_a=2$, $n_b=3$ and $\Gamma_a, \Gamma_b$
non-positive, is impossible. \end{lemma}
\noindent {\bf Proof.} The graph $\Gamma_a$ contains at most one loop at each vertex as
otherwise it would contain a Scharlemann bigon, which contradicts
Lemma 2.2(4) because $n_b = 3$ implies that $\hat F_b$ is
non-separating. There are at most four families of edges on $\Gamma_a$
connecting $u_1$ to $u_2$, containing a total of at least $\Delta n_b
- 2 \geq 10$ edges, hence there is a family containing 3 edges $e_1
\cup e_2 \cup e_3$. These are positive edges in $\Gamma_b$, and we may
assume that $e_i$ has label $i$ at $u_1$. Since one of the vertices
of $\Gamma_b$, say $v_1$, is anti-parallel to the other two vertices, the
edge $e_1$ is a loop on $\Gamma_b$, so its label on $u_2$ is also $1$.
Since $u_1, u_2$ are antiparallel, we see that the label of $e_i$ at
$u_2$ is $i$ for $i=1,2,3$, hence they are all co-loop edges on $\Gamma_a$.
This is a contradiction to the 3-Cycle Lemma 2.14(2). \quad $\Box$ \medskip
We will assume in the remainder of this section that $n_b = 4$. By
Lemma 13.1 the graph $\hat \Gamma_a$ is as shown in Figure 13.1. Note that
$\hat e_1, \hat e_2$ are on the same side of the loop at each $u_i$.
Denote by $w_i$ the weight of $\hat e_i$, and put $\lambda = w_5 =
w_6$. Then we can denote $\Gamma_a$ by $(\lambda; w_1, w_2, w_3, w_4)$,
and by Lemma 13.1(2) we may assume that $w_3 + w_4 \leq w_1 + w_2 = 6$
or $8$. By Lemmas 2.3(1) and 2.3(3) we have $\lambda, w_i \leq 4$.
Also, counting the number of edges incident to $u_i$ gives $$
\sum_{i=1}^4 w_i + 2\lambda = 4\Delta $$
\begin{lemma} (1) If $w_i \geq 3$ then $s_i = 2$, where $s_i$
is the transition number of $\hat e_i$.
(2) $v_1$ is parallel to $v_3$ and antiparallel to $v_2$ and $v_4$.
(3) $(w_1, w_2)$ and $(w_3, w_4)$ cannot be $(3,2)$, $(3,3)$ or
$(3,4)$.
\end{lemma}
\noindent {\bf Proof.} (1) Let $s_i$ be the transition number of $\hat e_i$. By the
3-Cycle Lemma (2.14(2)) we have $s_i \neq 0$. If $s_1 = \pm 1$ then
all vertices of $\Gamma_b$ would be parallel, which is a contradiction to
the assumption that $\Gamma_b$ is non-positive. Since $n_b = 4$, the only
remaining possibility is that $s_i = 2$.
(2) If $\lambda \geq 3$ then $\Gamma_a$ contains a Scharlemann cycle among
the loops, so $\hat F_b$ is separating and the result follows. If
$\lambda \leq 2$ then the equation $\sum w_i + 2\lambda = 4 \Delta$
gives $w_i \geq 3$ for some $i$. By (1) and the parity rule, $v_j$ is
parallel to $v_{j+2}$, hence the result follows because $\Gamma_b$ is
non-positive.
(3) Assume $w_1 = 3$. By the equation above, $\lambda > 0$, hence by
Lemma 15.1 $w_2$ is odd. The transition function of $\hat e_1$ is
given by (1), and it will determine that of $\hat e_2$. If $w_2 = 3$
then one can check that the transition function of $\hat e_2$ would
map $j$ to $j$, which would be a contradiction to (1). \quad $\Box$ \medskip
\begin{lemma} $\lambda \geq 2$.
\end{lemma}
\noindent {\bf Proof.} First assume $\lambda = 0$. Then $\Gamma_a = (0;4,4,4,4)$. By
Lemma 16.2(1) all edges of $\Gamma_a$ have label pair $(1,3)$ or $(2,4)$,
see Figure 16.1. Thus $\hat \Gamma_b$ is a union of two cycles, hence all edges
from $v_1$ to $v_3$ in $\Gamma_b$ are equidistant. Since two of these
edges are in $\hat e_1$ and are not equidistant on $\Gamma_a$, this is a
contradiction to the Equidistance Lemma 2.17.
\bigskip
\leavevmode
\centerline{\epsfbox{Figure16.1.eps}}
\bigskip
\centerline{Figure 16.1}
\bigskip
If $\lambda = 1$, then $w_j \geq 2$ for $j=1,...,4$, so $w_i \neq 3$
by Lemma 16.2(3). Hence $\Gamma_a = (1;4,4,4,2)$. One can check that the
one of the families of weight 4 would have the same label on the two
endpoints of any of its edges, which is a contradiction to Lemma
16.2(1). \quad $\Box$ \medskip
\begin{lemma} Suppose $w_i = w_j = 4$ and $\hat e_i = e_1 \cup
... \cup e_4$ and $\hat e_j = e'_1 \cup ... \cup e'_4$ satisfy
(i) they have the same label sequence at $u_1$, and (ii)
$e_1$ is equidistant to $e'_1$ on $\Gamma_a$. Then there exist at least 4
non co-loop edges in the other two non-loop families of $\Gamma_a$.
\end{lemma}
\noindent {\bf Proof.} The graph $\Gamma_a$ is as shown in Figure 16.2 for the case $(i,j)
= (1,2)$. (The proof works in all cases.) Note that $e_1$ being
equidistant to $e'_1$ implies that $e_k$ is equidistant to $e'_k$ for
$k=1,2,3,4$. We may assume that the label sequence of $\hat e_i$ and
$\hat e_j$ is $1,2,3,4$ at $u_1$. By Lemma 16.2(1) the four edges
$e_1 \cup ... \cup e_4$ form two essential cycles on $\Gamma_b$, so any
edge on $\Gamma_b$ with endpoints $v_1, v_3$ must be parallel to $e_1$ or
$e_3$. In particular, the edge $e'_1$ has label pair $(1,3)$ and
hence must be parallel to either $e_1$ or $e_3$. Note that two
parallel positive edges are equidistant. Since $e'_1$ is equidistant
to $e_1$ and $e_1$ is not equidistant to $e_3$ on $\Gamma_a$, it follows
that $e'_1$ is not equidistant to $e_3$ on $\Gamma_a$, therefore by the
Equidistance Lemma and the above we see that $e'_1$ must be parallel
to $e_1$ on $\Gamma_b$. Similarly each $e'_k$ is parallel to $e_k$ on
$\Gamma_b$. Since $e'_k$ and $e_k$ have the same label $k$ at $u_1$ on
$\Gamma_a$, they have the same label $1$ at $v_k$ in $\Gamma_b$, so there must
be another edge $e''_k$ between them. By the above $e''_k$ cannot be
in $\hat e_i \cup \hat e_j$, hence they belong to the other two
families of non-loop edges in $\Gamma_a$, and the result follows.
\quad $\Box$ \medskip
\bigskip
\leavevmode
\centerline{\epsfbox{Figure16.2.eps}}
\bigskip
\centerline{Figure 16.2}
\bigskip
\begin{lemma} $\lambda = 3$ is impossible.
\end{lemma}
\noindent {\bf Proof.}
Suppose $\lambda = 3$. Using the Congruence Lemma (Lemma 15.1) and
Lemma 16.2(3) one can show that $\Gamma_a$ has the following possibilities.
(1) $\Delta = 5$, $(3;4,4,4,2)$;
(2) $\Delta = 4$, $(3;4,4,2,0)$;
(3) $\Delta = 4$, $(3;4,2,4,0)$;
(4) $\Delta = 4$, $(3;4,2,2,2)$.
In each case, the family of $\hat e_1$ has weight $4$. We assume that
its label sequence at $u_1$ is $1,2,3,4$. Then by Lemma 16.2(1) its
label sequence at $u_2$ is $3,4,1,2$, which then completely determines
the labels of $\Gamma_a$. One can check that in case (1) and (3) the
family $\hat e_3$ gives $4$ parallel co-loops, which is a
contradiction to the 3-Cycle Lemma (Lemma 2.14(2)). Case (2) is
impossible by Lemma 16.4.
It remains to consider case (4). The graph $\Gamma_a$ is shown in Figure
16.3. The third edge $A$ of $\hat e_1$ and the second edge $B$ of
$\hat e_3$ in the figure both have label pair $(1,3)$. As in the
proof of Lemma 16.4, this implies that they are parallel on $\Gamma_b$.
Since $\hat \Gamma_a$ has at most 4 negative edges and at most 2 positive
edges, by Lemma 2.2(2) $\Gamma_b$ cannot have more than $2n_a = 4$ parallel
edges, so the endpoints of $A$ and $B$ at $v_3$ are adjacent among the
four edge endpoints labeled $1$ at $v_3$. Since $\Delta =4$, the
jumping number is $\pm 1$, so the endpoints of $A, B$ at $u_1$ in
$\Gamma_a$ are also adjacent among the four edge endpoints labeled $3$ at
$u_1$. This is a contradiction because this is not the case in Figure
16.3. \quad $\Box$ \medskip
\bigskip
\leavevmode
\centerline{\epsfbox{Figure16.3.eps}}
\bigskip
\centerline{Figure 16.3}
\bigskip
\begin{lemma} If $\lambda = 4$, then $\Gamma_a = (4; 4,2,4,2)$, and
the graphs $\Gamma_a, \Gamma_b$ and their edge correspondence are as shown in
Figure 16.6. \end{lemma}
\noindent {\bf Proof.} Since $\Gamma_a$ does not contain an extended Scharlemann cycle, by
considering the labels at the endpoints of the four loops at $u_1$ we
see that $w_1 + w_2 \equiv w_3 + w_4 \equiv 2$ (mod $4$). This,
together with Lemmas 15.1 and 16.2(3), give the following
possibilities for $\Gamma_a$.
(1) $\Delta = 5$, $\Gamma_a = (4; 4,2,4,2)$;
(2) $\Delta = 5$, $\Gamma_a = (4; 4,2,2,4)$;
(3) $\Delta = 4$, $\Gamma_a = (4; 4,2,2,0)$.
We shall show that (2) and (3) are impossible, and (1) gives the
example in Figure 16.6.
Case (2) can be excluded by Lemma 16.4. The graph $\Gamma_a$ is shown in
Figure 16.4. Note that the corresponding edges of the two non-loop
families of weight 4 are equidistant in $\Gamma_a$, and they have the same
label sequence at $u_1$. Since the other two non-loop families of
$\Gamma_a$ consist of co-loops, this is a contradiction to Lemma 16.4.
\bigskip
\leavevmode
\centerline{\epsfbox{Figure16.4.eps}}
\bigskip
\centerline{Figure 16.4}
\bigskip
The graph for case (3) is shown in Figure 16.5. Note that there is a
loop in $\Gamma_b$ based at each vertex $v_i$, so two edges connecting
$v_i$ to different vertices must be on different sides of the loop.
Consider the four edges with label $3$ at $u_1$, indicated by $A, B,
C, D$ in Figure 16.5. Note that they appear in this order on $\partial
u_1$. Since $\Delta = 4$, the jumping number is $\pm 1$, so they must
also appear in such an order on $\partial v_3$ in $\Gamma_b$.
On the other hand, since $A$ connects $v_3$ to $v_1$ while
$B, D$ connect $v_3$ to $v_4$, $A$ must be on a different side of the
loop $C$ at $v_3$ than $B, D$. Hence when traveling around $\partial v_3$
in a certain direction the four edges appear in the order $A,C,B,D$
or $A,C,D,B$. This is a contradiction. Therefore Case (3) is
impossible.
\bigskip
\leavevmode
\centerline{\epsfbox{Figure16.5.eps}}
\bigskip
\centerline{Figure 16.5}
\bigskip
In case (1), the graph $\Gamma_a$ is shown in Figure 16.6(a). By the same
argument as above, we see that the edges $B \cup E$ and $A \cup C$
must be on different sides of the loop $D$ in $\Gamma_b$. Therefore $B, E$
are adjacent among the 5 edges labeled $1$ at $v_3$. Since they are
not adjacent among the $3$-edges at $u_1$, the jumping number must be
$\pm 2$. This completely determines the edges around the vertex $v_3$
up to symmetry, which in turn determine the edges at adjacent vertices
$v_1, v_3$ and then the edges at $v_2$. The graph $\Gamma_b$ is shown in
Figure 16.6(b).
\quad $\Box$ \medskip
\bigskip
\leavevmode
\centerline{\epsfbox{Figure16.6.eps}}
\bigskip
\centerline{Figure 16.6}
\bigskip
\begin{lemma} If $\lambda = 2$, then either $\Delta=5$ and
$\Gamma_a = (2;4,4,4,4)$, or $\Delta = 4$ and $\Gamma_a = (2;4,4,4,0)$. The
graphs $\Gamma_a, \Gamma_b$ are as shown in Figures 16.8 and 16.9.
\end{lemma}
\noindent {\bf Proof.}
Here the possibilities for $\Gamma_a$ are
(1) $\Delta = 5$, $\Gamma_a = (2;4,4,4,4)$;
(2) $\Delta = 4$, $\Gamma_a = (2;4,4,4,0)$;
(3) $\Delta = 4$, $\Gamma_a = (2;4,4,2,2)$;
(4) $\Delta = 4$, $\Gamma_a = (2;4,2,4,2)$;
(5) $\Delta = 4$, $\Gamma_a = (2;4,2,2,4)$.
The graphs in cases (3) -- (5) are shown in Figure 16.7 (a) -- (c).
In cases (3) and (4) the corresponding edges in the two weight 4
families are equidistant, and the other two non-loop families are
co-loops. Therefore these cases are impossible by Lemma 16.4. In
case (5) there are loops at $v_1$ and $v_2$ in $\Gamma_b$, and there is a
(34)-Scharlemann bigon in $\Gamma_a$ which forms another essential cycle
$C$ in $\Gamma_b$. Consider the two edges of $\Gamma_a$ with label $3$ at $u_1$
and label $1$ at $u_2$. On $\Gamma_b$ these edges connect $v_3$ and $v_1$,
and therefore must lie on the same side of $C$. Hence they are
adjacent among the four edges labeled $1$ at $v_3$ because the other
two edges connect $v_3$ to $v_4$. Since $\Delta = 4$, the jumping
number must be $\pm 1$, so these edges are also adjacent among the
four edges with label $3$ at $u_1$, which is a contradiction because
on Figure 16.7(c) the two edges with label $3$ at $u_1$ and $1$ at
$u_2$ are not adjacent among the four edges labeled $3$ at $u_1$.
Therefore (5) is also impossible.
\bigskip
\leavevmode
\centerline{\epsfbox{Figure16.7.eps}}
\bigskip
\centerline{Figure 16.7}
\bigskip
In case (1) the graph $\Gamma_a$ is shown in Figure 16.8(a). Label the
edges as in the figure, and orient non-loop edges of $\Gamma_a$ from $u_1$
to $u_2$. As in the proof of Lemma 16.4, the $i$-th edge $e_i$ in
$\hat e_1$ must be parallel to the $i$-th edge $e'_i$ in $\hat e_2$ on
$\Gamma_b$, and there is an edge of $\hat e_3 \cup \hat e_4$ between them
because $e_i, e'_i$ both have label $1$ at $v_i$. For the same reason
the $i$-th edge of $\hat e_3$ is parallel to the $i$-th edge of $\hat
e_4$, hence the positive edges of $\Gamma_b$ form four families of weight
4. The two edges $e_i, e'_i$ are adjacent among the five edges
labeled $1$ at $v_i$ in $\Gamma_b$, hence the jumping number $J = \pm 1$.
Reversing the orientation of the vertices of $\Gamma_b$ if necessary we may
assume $J=1$. We may also assume that the vertices $v_1, v_3$ are
oriented counterclockwise and $v_2, v_4$ clockwise, otherwise we may
look at $\hat F_b$ from the other side.
Since $\Gamma_b$ contains 4 parallel positive edges, by Lemma 13.2(2) $\Gamma_a$
is kleinian, so the weight of edges of $\hat \Gamma_b$ are all even. There are
only two $(14)$-edges $K, W$ in $\Gamma_a$, so they must be parallel in
$\Gamma_b$. They may appear in the order $(K,W)$ or $(W,K)$ on $\partial
v_1$, but there is a homeomorphism of $(\hat F_a, \Gamma_a)$ which is label
preserving, interchanging $u_1, u_2$ and mapping $K$ to $W$, hence up
to symmetry we may assume that the order is $(K,W)$. Thus up to
symmetry we may assume that $K$ and $W$ appear in $\Gamma_b$ as shown in
Figure 16.8(b). This, together with the orientation of the vertices
and the fact that $J=1$, completely determines the edges around $v_1$
and $v_4$, and then the edges around $v_2$ and $v_3$. See Figure
16.8(b).
\bigskip
\leavevmode
\centerline{\epsfbox{Figure16.8.eps}}
\bigskip
\centerline{Figure 16.8}
\bigskip
The graph $\Gamma_a$ in case (2) is shown in Figure 16.9(a). As above, one
can show that each edge $e_i$ in $\hat e_1$ is parallel to $e'_i \in
\hat e_2$ and $e''_i \in \hat e_3$, where $e_i, e'_i$ have label $i$ at
$u_1$ and $e''_i$ has label $i$ at $u_2$. Orient $v_i$ as above. Up
to symmetry we may assume $J=1$, and $A,E$ on $\Gamma_b$ are as shown in
Figure 16.9(b). This determines $P$ and the position of $K$ at $\partial
v_1$, and hence the labels of the $2$-edges at $v_1$. The $4$-labels
at $u_1$ appear in the order $K,D,H,N$, so on $\Gamma_b$ they appear in
this order around $v_4$, clockwise, hence $D,H$ must be to the right
of $v_4$ in the figure. This also determines the $2$-edges at $v_4$.
In particular, the edges $K$ and $S$ must be non-parallel. The
remaining two edges $R$ and $L$ can be determined similarly, using
labels at $v_2$ and $v_3$. See Figure 16.9(b). \quad $\Box$ \medskip
\bigskip
\leavevmode
\centerline{\epsfbox{Figure16.9.eps}}
\bigskip
\centerline{Figure 16.9}
\bigskip
\begin{prop} Suppose $n_a \leq 2$ and $n_b \geq 3$. Then
$\Gamma_a, \Gamma_b$ and their edge correspondence are given in Figure 11.9,
11.10, 14.5, 16.6, 16.8 or 16.9.
\end{prop}
\noindent {\bf Proof.} First assume that $\Gamma_a$ is positive. Then $n_b \leq 4$ by
Lemma 3.2. By Lemma 2.23 $n_b$ must be even, hence our assumption
implies that $n_b = 4$. By Proposition 11.10 the graphs are as shown
in Figure 11.9 or 11.10.
Now assume $\Gamma_a$ is non-positive. Then we have $n_a = 2$. The case
that $\Gamma_b$ is positive has been ruled out by Proposition 12.17. Hence
$\Gamma_a, \Gamma_b$ are both non-positive. By Lemma 16.1 $n_b$ cannot be $3$.
By Proposition 14.7 if $n_b > 4$ then $\Gamma_a, \Gamma_b$ are given in Figure
14.5. Finally if $n_b = 4$. Then Lemma 16.3 and 16.5 says that
$\lambda = 4$ or $2$, which are covered by Lemmas 16.6 and 16.7,
respectively, showing that if $\lambda=4$ then the graphs are in
Figure 16.6, and if $\lambda = 2$ then the graphs are the pair in
Figure 16.8 or 16.9. \quad $\Box$ \medskip
\section {Equidistance classes }
The next few sections deal with the case that $n_i \leq 2$ for
$i=1,2$. In this section we introduce the concept of equidistance
classes. The main properties are given in Lemmas 17.1 and 17.2, which
will be used extensively in the next few sections.
Define a relation on the set of edges $E_a$ of $\Gamma_a$ such that $e_1
\sim e_2$ if and only if (i) they have the same label pair, (ii) they
have the same endpoint vertices, and (iii) they are equidistant.
\begin{lemma} This is an equivalence relation.
\end{lemma}
\noindent {\bf Proof.} We need only show that condition (iii) is transitive, i.e.\ if
$e_1, e_2, e_3$ are edges on a graph $\Gamma$ such that $e_1, e_2$ and
$e_2, e_3$ are equidistant pairs, then $e_1, e_3$ are equidistant.
By definition we have $d_{u_1}(e_1, e_2) = d_{u_2}(e_2, e_1)$, and
$d_{u_1}(e_2, e_3) = d_{u_2}(e_3, e_2)$, hence $d_{u_1}(e_1, e_3) =
d_{u_1}(e_1, e_2) + d_{u_1}(e_2, e_3) = d_{u_2}(e_2, e_1) +
d_{u_2}(e_3, e_2) = d_{u_2}(e_3, e_1)$. This completes the proof.
\quad $\Box$ \medskip
We will call this equivalence relation the {\it ED relation}. An
equivalence class is then called an {\it ED class}, and the number of
ED classes is called the {\it ED number\/} of $\Gamma_a$, denoted by
$\eta_a = \eta(\Gamma_a)$. We can then define $D_a = D(\Gamma_a) = (c_1, ...,
c_{\eta_a})$, where $c_i$ are the number of edges of the equivalence
classes, ordered lexicographically.
\begin{lemma} Let $\Gamma_a, \Gamma_b$ be intersection graphs. Then
the edge correspondence between the graphs induces a one to one
correspondence between the ED classes of $\Gamma_a$ and $\Gamma_b$; in
particular $\eta(\Gamma_a) = \eta(\Gamma_b)$, and $D(\Gamma_a) = D(\Gamma_b)$.
\end{lemma}
\noindent {\bf Proof.} Note that $e_1, e_2$ satisfy (i) on $\Gamma_a$ if and only if they
satisfy (ii) on $\Gamma_b$. The Equidistance Lemma 2.17 now says that a
pair of edges are equivalent on $\Gamma_a$ if and only if they are
equivalent on $\Gamma_b$. \quad $\Box$ \medskip
\bigskip
\leavevmode
\centerline{\epsfbox{Figure17.1.eps}}
\bigskip
\centerline{Figure 17.1}
\bigskip
\noindent
{\bf Example 17.3} (1) Consider a graph $\Gamma_a = +(3,3,1,1)$ and
assume $n_b = 2$, see Figure 17.1(a). In general if $n_b =2$ then all
parallel positive edges are in the same ED class because they have the
same label pairs and they are equidistant. One can check that
non-parallel edges are not equidistant. (For example, let $u_1$ be
the top vertex, $u_2$ the bottom vertex, and let $e_1, e_2$ be as
shown in the figure; then $d_{u_1}(e_1, e_2) = 4 \neq 2 = d_{u_2}(e_2,
e_1)$.) Hence $D(\Gamma_a) = (3,3,1,1)$. Compare this with $\hat \Gamma_a =
+(3,1,3,1)$, in which case the two families of 3 edges are
equidistant, and the other two families of weight 1 are equidistant,
hence $D(+(3,1,3,1)) = (6,2)$.
(2) Consider $\Gamma_a = -(3,3,1,1)$ or $-(3,1,3,1)$, and suppose that the
edges of $\Gamma_a$ are not co-loops (hence conditions (i) and (ii) in the
definition of ED equivalence are satisfied), see Figure 17.1(b) and
(c). Equidistant edges are indicated in the figure by different kind
of lines. We can see that $D(-(3,3,1,1)) = D(-(3,1,3,1) = (4,2,2)$.
(3) When $\Gamma_a = -(4,2,2,0)$, each of the middle edges of the family of
4 is equidistant to one edge in each of the two weight 2 families, and
the other two edges of the weight 4 families are not equidistant to
any other edges. Hence $D(-(4,2,2,0)) = (3,3,1,1)$
(4) Suppose $\Gamma_a = +(4,2,2,0)$ and all edges have label pair
$(12)$. Then one can check that each family of parallel edges forms an
ED class, hence $D(\Gamma_a) = (4,2,2)$.
(5) Suppose $\Gamma_a = +(2,2,2,2)$ and all edges have label pair $(12)$.
Then one can show that the first family is equidistant to the third
family, but not to the adjacent families. Hence $D(\Gamma_a) = (4,4)$.
(6) Similarly if $\Gamma_a = +(4,4,0,0)$ and all edges have the same label
pair then $D(\Gamma_a) = (8)$.
\medskip
\section {The case $n_b = 1$ and $n_a = 2$}
\begin{lemma} Suppose $n_a = 2$ and $n_b = 1$. Then
one of the following holds.
(1) $\Gamma_a = -(1,1,1,1)$ and $\Gamma_b = (4,0,0)$.
(2) $\Gamma_a = -(2,2,0,0)$ and $\Gamma_b = (2,2,0)$.
(3) $\Gamma_a = -(2,1,1,1)$ and $\Gamma_b = (3,1,1)$. The graphs $\Gamma_a, \Gamma_b$ and
their edge correspondence are given in Figure 18.2.
\end{lemma}
\noindent {\bf Proof.} In this case $\hat \Gamma_b$ has a single vertex, and $\hat \Gamma_a$ has two
vertices of opposite orientation and has no loops. Hence we have $\Gamma_a
= (a_1, ..., a_4)$, and $\Gamma_b = (b_1, b_2, b_3)$. We have $b_1 + b_2 +
b_3 = \Delta$. If $b_i, b_j$ are non-zero and $b_i+b_j$ is odd then
one can check that one of the $\hat e_i, \hat e_j$ is a family of
co-loops, which is a contradiction to the parity rule. Hence $b_i
\equiv b_j$ mod $2$ for all $b_i, b_j$ non-zero. Thus if $\Delta= 5$
then up to symmetry we have $\Gamma_b = (3,1,1)$, and if $\Delta = 4$ then
$\Gamma_b = (4,0,0)$ or $(2,2,0)$.
If $\Gamma_b = (4,0,0)$ then the four parallel edges are mutually
non-parallel on $\Gamma_a$, hence $\Gamma_a = -(1,1,1,1)$.
If $\Gamma_b = (2,2,0)$, one can check that edges in different families are
not equidistant, hence $D(\Gamma_b) = (2,2)$. Since each pair of parallel
edges contributes one edge to each of two families in $\Gamma_a$, we have
$\Gamma_a = -(2,2,0,0)$, $-(2,1,1,0)$ or $-(1,1,1,1)$. When
$\Gamma_a=-(2,1,1,0)$ the two single edges are equidistant, while each of
the two parallel edges form an ED class, so $D(\Gamma_a) = (2,1,1) \neq
D(\Gamma_b)$. Also, when $\Gamma_a=-(1,1,1,1)$ we have $D(\Gamma_a) = (4)$.
Therefore in this case we have $\Gamma_a = -(2,2,0,0)$.
No suppose $\Gamma_b = (3,1,1)$. In this case the three parallel edges are
equidistant, and each of the other two edges is not equidistant to any
other edges. Hence $D(\Gamma_b) = (3,1,1)$. Since the three parallel
edges in $\Gamma_b$ are mutually non-parallel on $\Gamma_a$, $\hat \Gamma_a$ has at least
three edges. One can show that $D(-(2,2,1,0)) = (2,2,1) \neq D(\Gamma_b)$,
hence $\hat \Gamma_a \neq -(2,2,1,0)$. Therefore $\Gamma_a = -(3,1,1,0)$ or
$-(2,1,1,1)$.
In the case that $\Gamma_a = -(3,1,1,0)$ and $\Gamma_b = (3,1,1)$, the graphs
are as shown in Figure 18.1. The three parallel edges $B,C,E$ are
equidistant, hence they represent the two weight 1 edges $\hat e_2,
\hat e_3$ and the middle edge of the weight 3 edge $\hat e_1$, so the
other two edges $A, D$ must be as shown in Figure 18.1(a) up to
symmetry. Since they are non-adjacent at $u_1$ and their label $1$
endpoints are non-adjacent among the five label $1$ edge endpoints at
$v_1$ in $\Gamma_b$, the jumping number must be $\pm 1$. This determines
the edge correspondence between $\Gamma_a$ and $\Gamma_b$, as shown in Figure
18.1.
The torus $\hat F_a$ cuts $M(r_a)$ into two components. Let $W$ be
the one containing the bigon $\alpha$ on $F_b$ bounded by $B\cup E$
and the 3-gon $\beta$ bounded by $A\cup C\cup D$. It can be
constructed by attaching a 1-handle representing part of the Dehn
filling solid torus, then two 2-handles represented by $\alpha,
\beta$, then a 3-cell. The fundamental group of $W$ is generated by
the horizontal circle $x$ and the vertical circle $y$ shown in the
figure, and the $1$-handle $z$ from $u_2$ to $u_1$. On the boundary
of $\alpha, \beta$, $A, B, C, D, E$ represent $1,x,xy,1,1$,
respectively, and each corner represents $z$, hence $\alpha, \beta$
give the relations $zzx = 1$ and $zxyzz = 1$, respectively. Solving
these in $x$ and $y$ shows that $\pi_1(W) = \Bbb Z$, generated by
$z$. It follows that $\hat F_a$ is not $\pi_1$-injective in $W$, and
hence is compressible. This is a contradiction.
\bigskip
\leavevmode
\centerline{\epsfbox{Figure18.1.eps}}
\bigskip
\centerline{Figure 18.1}
\bigskip
We now have $\Gamma_a = -(2,1,1,1)$ and $\Gamma_b = (3,1,1)$. The three
parallel edges $B,C,E$ are equidistant, hence on $\Gamma_a$ they are the
single edges because they are equidistant to each other but not to the
edges in the weight 2 family. Since the edge endpoints of these are
consecutive on $\partial v_1$ while the $1$-label endpoint of $E$ at $v_1$
is not adjacent to that of either $B$ or $C$, the jumping number must
be $\pm 2$. This determines the correspondence of the edges up to
symmetry, see Figure 18.2. \quad $\Box$ \medskip
\bigskip
\leavevmode
\centerline{\epsfbox{Figure18.2.eps}}
\bigskip
\centerline{Figure 18.2}
\bigskip
\section {The case $n_1 = n_2 = 2$ and $\Gamma_b$ positive
}
In this section we assume that $n_1 = n_2 = 2$ and $\Gamma_b$ is positive.
Then no edge of $\hat \Gamma_a$ is a loop, hence $\Gamma_a = -(a_1, ... a_4)$, and
$\Gamma_b = +(\rho; b_1, ..., b_4)$.
When $\rho \neq 0$ we may rearrange the $a_i$ to write $\Gamma_a =
-(r_1, ..., r_p \, |\, s_1, ..., s_q)$, where $r_i$ are the weights of
the co-loop edges, and $s_j$ are the weights of the non co-loop
edges.
\begin{lemma} Suppose $n_1 = n_2 = 2$ and $\Gamma_b$ is positive.
(1) All non-zero $b_i$ are of the same parity, all non-zero $r_j$ are
of the same parity, all non-zero $s_k$ are of the same parity, and the
non-zero $r_j$ and $s_k$ are of opposite parity.
(2) $r_i \leq 2$, $s_j \leq 4$, and $\rho + b_k \leq 4$.
(3) $2 \rho + \sum b_i = 2 \Delta$, and $\sum r_i + \sum s_j = 2 \Delta$.
\end{lemma}
\noindent {\bf Proof.}
(1) This follows from the Congruence Lemma 15.1.
(2) Since $\hat \Gamma_b$ has at most two loops, each co-loop family of $\Gamma_a$
contains at most two edges, hence $r_i \leq 2$. Similarly, since
$\hat \Gamma_b$ has at most four non-loop edges, $s_j \leq 4$. On $\Gamma_b$,
$\rho$ is the number of edges in a loop family, which is no more than
$p$, the number of co-loop edges in $\hat \Gamma_a$. Similarly, $b_k$ is no
more than $q$, the number of non co-loop edges in $\hat \Gamma_a$. Since
$\hat \Gamma_a$ has at most 4 edges, we have $\rho + b_k \leq 4$.
(3) This follows from the fact that each vertex of $\Gamma_a$ or $\Gamma_b$
has valence $2 \Delta$.
\quad $\Box$ \medskip
\begin{lemma} Suppose $n_1 = n_2 = 2$ and $\Gamma_b$ is positive.
If $\rho=4$ then $\Gamma_a = -(2,2,2,2)$, and $\Gamma_b = +(4; 0, 0, 0, 0)$.
\end{lemma}
\noindent {\bf Proof.} Since $\rho + b_j \leq 4$ (Lemma 19.1(2)), we have $b_i = 0$
for all $i$, hence from Lemma 19.1(3) we have $\Delta = 4$. Thus
$\hat \Gamma_b$ is a union of two disjoint loops, each representing a family of
four edges. Since each family of four parallel edges in $\hat \Gamma_b$
contributes one edge to each family in $\Gamma_a$, we have $\Gamma_a =
-(2,2,2,2)$. \quad $\Box$ \medskip
\begin{lemma}
Suppose $n_1 = n_2 = 2$ and $\Gamma_b$ is positive. Then $\rho \neq 3$.
\end{lemma}
\noindent {\bf Proof.} Suppose $\rho = 3$. The three loops in a family represent
different classes on $\hat \Gamma_a$, so $\hat \Gamma_a$ has at least three co-loop
edges. Since $\Gamma_b$ has some non-loop edges, $\hat \Gamma_a$ has at least one
non co-loop edge. It follows that $\hat \Gamma_a$ has exactly three co-loop
edges, so $\Gamma_a = -(2,2,2\, |\, s_1)$. Since $\sum r_i + \sum s_j =
2\Delta$ is even, $s_1$ is even, which contradicts Lemma 19.1(1).
\quad $\Box$ \medskip
\begin{lemma}
Suppose $n_1 = n_2 = 2$ and $\Gamma_b$ is positive. Then $\rho \neq 2$.
\end{lemma}
\noindent {\bf Proof.} On $\hat \Gamma_a$ there are non-co-loop edges, so there are at most
three co-loop edges, but since $r_i \leq 2$ and $\sum r_i = 4$ and the
$r_i$'s are of the same parity, there must be exactly two co-loop
edges. Hence $\Gamma_a = -(2,2 \, |\, s_1, s_2)$. By the Congruence
Lemma, the $s_i$ are odd, hence either $\Delta = 5$ and $\Gamma_a = -(2,2\,
|\, 3,3)$, or $\Delta = 4$ and $\Gamma_a = -(2,2\, |\, 3,1)$.
If $\Delta=4$ and $\Gamma_a = -(2,2\, |\, 3,1)$, then from Lemma 19.1 we
have $b_i \leq 2$, $\sum b_i = 4$, and $b_i \equiv b_j$ mod $2$ if
$b_i, b_j \neq 0$. These conditions give $\Gamma_b = +(2; 1,1,1,1)$,
$+(2; 2,2,0,0)$ or $+(2;2,0,2,0)$. One can check that in the first
two cases the four non-loop edges of $\Gamma_b$ form two equidistance
classes of 2 edges each, so $D_b = (2,2,2,2)$, and in the third case
the four non-loop edges are all equidistant to each other, so $D_b =
(4,2,2)$. On the other hand, the three parallel edges of $\Gamma_a$ belong
to distinct classes, and there are at least two co-loop classes, hence
$\eta(\Gamma_a) \geq 5$. This is a contradiction to Lemma 17.2.
If $\Delta=5$ and $\Gamma_a = -(2,2\, |\, 3,3)$, then from Lemma 19.1 we
have $b_i \leq 4-\rho = 2$, $\sum b_i = 6$, and $b_i \equiv b_j$
mod $2$ if $b_i, b_j \neq 0$, so we must have $\Gamma_b = +(2; 2, 2, 2,
0)$. Depending on the weight sequence of the edges of $\hat \Gamma_a$, we have
$\Gamma_a = -(3,2,3,2)$ or $-(3,3,2,2)$. If $\Gamma_a = -(3,2,3,2)$ then from
the labeling one can see that the two edges with both endpoints
labeled $1$ are not equidistant. Since these are parallel loops at
$v_1$ of $\Gamma_b$, they are equidistant on $\Gamma_b$, which is a
contradiction to the Equidistance Lemma 2.17. Therefore $\Gamma_a \neq
(-3,2,3,2)$.
Now suppose $\Gamma_a = -(3,3,2,2)$, and $\Gamma_b = +(2;2,2,2,0)$. Then the
graphs are as shown in Figure 19.1. Consider the edges $A,B,C,D,E$ with
label $1$ at $u_1$ of $\Gamma_a$. These correspond to the 5 edges with
label $1$ at $v_1$ of $\Gamma_b$. Note that on $\Gamma_a$ $D,E$ are co-loop
edges, hence on $\Gamma_b$ they are the two loops at $u_1$. Since their
endpoints with label $1$ are not adjacent among the $1$-edges at $v_1$
in $\Gamma_b$, the jumping number must be $\pm 2$, so among these edges in
$\Gamma_b$, the edge $B$ is the one in $\Gamma_b$ which is adjacent to both $D$
and $E$ at $v_1$, as shown in the Figure. Now consider the five edges
labeled $2$ at $u_2$. Note that they appear in the order $CABFG$.
Using the same argument as above we see that the edge $A$ is the one
adjacent to both $F$ and $G$, so we would have $A=B$, which is a
contradiction.
\quad $\Box$ \medskip
\bigskip
\leavevmode
\centerline{\epsfbox{Figure19.1.eps}}
\bigskip
\centerline{Figure 19.1}
\bigskip
\begin{lemma}
Suppose $n_1 = n_2 = 2$ and $\Gamma_b$ is positive. Then $\rho \neq 1$.
\end{lemma}
\noindent {\bf Proof.} Suppose $\rho = 1$. Then $\Gamma_a$ has two co-loop edges, hence
$\Gamma_a = -(1,1\, |\, s_1, s_2)$ or $-(2 \, |\, s_1, s_2, s_3)$.
If $\Delta = 5$ then the second case does not happen since by the
Congruence Lemma $s_i$ would be $0$ or odd and $\sum s_i = 8$, which
would give $s_i > 4$ for some $i$, contradicting Lemma 19.1(2).
Therefore $\Gamma_a = -(1,1 \, |\, 4,4)$. Since each weight 4 family
contributes one edge to each non-loop family of $\Gamma_b$, we have $\Gamma_b =
+(1; 2,2,2,2)$. Now the graph $\hat \Gamma_b$ contains both black and white
bigons, whose edges all belong to the two weight 4 families on $\Gamma_a$.
This is a contradiction to Lemma 15.2.
If $\Delta = 4$ then $\Gamma_a = -(1,1\, |\, 4,2)$ or $-(2\, |\, 3,3,0)$.
The first case cannot happen because the $\hat e_i$ of weight 4
contributes one edge to each family in $\hat \Gamma_b$, while the edge of
weight 2 contributes one edge to each of two families, so $\Gamma_b = +(1;
2,2,1,1)$, which contradicts the Congruence Lemma. In the second case
for the same reason above we must have $\Gamma_b = +(1; 2,2,2,0)$. Again
there are black and white bigons, which contradicts Lemma 15.2 because
on $\Gamma_a$ the edges of these bigons all belong to the two weight 3
families. \quad $\Box$ \medskip
\begin{lemma}
Suppose $n_1 = n_2 = 2$ and $\Gamma_b$ is positive. If $\rho = 0$ then
$\Delta = 5$.
\end{lemma}
\noindent {\bf Proof.} In this case there is no loop on either graph, hence by Lemma
19.1(1) all non-zero $a_i$ have the same parity, and all non-zero
$b_j$ have the same parity. Any two edges connect the same pair of
vertices and have the same pair of labels on their two endpoints,
hence by definition they are ED equivalent if and only if they are
equidistant.
Assume $\Delta = 4$. By the Congruence Lemma each of $\hat \Gamma_a$ and $\Gamma_b$
is of type $(4,4,0,0)$, $(2,2,2,2)$, $(4,2,2,0)$, $(3,1,3,1)$, or
$(3,3,1,1)$. Let $e_1 \cup e_2$ be a bigon on $\Gamma_b$. Then $e_1$ and
$e_2$ are equidistant on $\Gamma_b$, so by Lemma 2.17 they form an
equidistant pair on $\Gamma_a$. Note that since $e_1$ and $e_2$ are not
loops on $\Gamma_b$, on $\Gamma_a$ they have different labels on $u_1$. On the
other hand, one can check that if $\Gamma_a = -(4,4,0,0)$ or $-(2,2,2,2)$
then an equidistant pair $e_1, e_2$ on $\Gamma_a$ must have the same label
on $u_1$, which is a contradiction. Therefore $\Gamma_a =-(4,2,2,0)$,
$-(3,3,1,1)$ or $-(3,1,3,1)$. (Note that the above argument does not
apply to $\hat \Gamma_b$ since a pair of parallel edges on $\Gamma_a$ is not an
equidistant pair.) We will rule these out one by one.
\medskip
CLAIM 1. {\it The case $\Gamma_a = - (4,2,2,0)$ is impossible.}
If $\Gamma_a = -(4,2,2,0)$ then $b_i \neq 0$ for all $i$, hence $\Gamma_b =
+(2,2,2,2)$ or $+(3,1,3,1)$, or $+(3,3,1,1)$. In the first case all
black (say) faces of $\Gamma_b$ are bigons, so Lemma 13.2(2) implies that
$\Gamma_b$ is kleinian because $\hat \Gamma_a$ has more than two edges. Since
$\hat \Gamma_a$ has a single edge of weight 4, it will be fixed by the free
involution given in Lemma 6.2(4), which is absurd. By Example 17.3 we
have $D(-(4,2,2,0)) = (3,3,1,1)$, and $D(+(3,1,3,1)) = (6,2)$, hence
$\Gamma_b \neq +(3,1,3,1)$. It follows that $\Gamma_b = +(3,3,1,1)$. The
graphs are shown in Figure 19.2. Each of $A, D$ on $\Gamma_a$ forms an
equidistance class, hence they are the single edges on $\Gamma_b$. Up to
symmetry we may assume that $A, D$ are as shown in Figure 19.2(b).
This and the jumping number $J$ determines the edge correspondence of
the graphs. The case that $J=1$ is shown in the figure. When $J=-1$
the edges $G, E$ would be equidistant on $\Gamma_a$ but not on $\Gamma_b$, which
is impossible.
\bigskip
\leavevmode
\centerline{\epsfbox{Figure19.2.eps}}
\bigskip
\centerline{Figure 19.2}
\bigskip
Let $P_1, P_2$ be the bigon disks on $F_a$ bounded by $A\cup B$
and $C\cup D$, respectively. Then the union of $P_1, P_2$ and two
disks on $T_0$ form an annulus $Q$. More explicitly, let $a_1$
(resp.\ $b_1$) be the arc on $\partial u_1$ (resp.\ $\partial u_2$) from the
endpoint of $A$ to that of $B$, $a_2$ (resp.\ $b_2$) the arc on $\partial
u_1$ (resp.\ $\partial u_2$) from the endpoint of $C$ to that of $D$, $a_3$
(resp.\ $b_3$) on $\partial v_1$ (resp.\ $\partial v_2$) from $A$ to $C$, and
$a_4$ (resp.\ $b_4$) on $\partial v_2$ (resp.\ $\partial v_1$) from $B$ to
$D$. Then $a_1 \cup ... \cup a_4$ (resp.\ $b_1 \cup ... \cup b_4$)
bounds a disk $P_3$ (resp.\ $P_4$) on the boundary torus $T_0$. Now
$Q = P_1 \cup ... \cup P_4$ is an annulus in $M$.
Note that $\partial Q$ consists of two simple closed curves $\partial_1 = A
\cup C \cup a_3 \cup b_3$ and $\partial_2 = B \cup D \cup a_4 \cup b_4$.
Orient $A, B$ to point from $u_1$ to $u_2$ on $\Gamma_a$. This determines
the orientation of $\partial_i$. Note that they are parallel on the
annulus $Q$. On $\Gamma_b$ the orientations of $A, B$ are from label $1$
to label $2$, as shown in the figure. This determines the orientation
of $C, D$. It is important to see that $\partial_1, \partial_2$ are parallel
as oriented curves on $\hat F_b$. Let $Q'$ be an annulus on $\hat
F_b$ with $\partial Q' = \partial_1 \cup \partial_2$. Then $Q \cup Q'$ is a
non-separating torus (not a Klein bottle!) in $M(r_b)$ intersecting
the Dehn filling solid torus at a single meridian disk, which
contradicts the choice of $\hat F_b$. Therefore this case is
impossible.
\medskip
CLAIM 2. {\it The case $\Gamma_a = - (3,1,3,1)$ is impossible. If
$\Gamma_a = -(3,3,1,1)$ then $\Gamma_b = +(4,2,2,0)$.}
Now suppose $\Gamma_a = -(3,3,1,1)$ or $-(3,1,3,1)$ and $\Gamma_b = +(4,4,0,0)$,
$+(4,2,2,0)$, $+(2,2,2,2)$, $+(3,3,1,1)$ or $+(3,1,3,1)$. By Example
17.3 we have $D(-(3,3,1,1)) = -(3,1,3,1) = (4,2,2)$. On the other
hand, by Example 17.3 we also have $D(+(4,4,0,0)) = (8)$,
$D(+(4,2,2,0)) = (4,2,2)$, $D(+(2,2,2,2)) = (4,4)$, $D(+(3,3,1,1)) =
(3,3,1,1)$, and $D(+(3,1,3,1)) = (6,2)$. Therefore by Lemma 17.2 in
this case we must have $\hat \Gamma_b = +(4,2,2,0)$. If $\hat \Gamma_a = -(3,1,3,1)$
then the four edges in the same ED class all have label $2$ (say) at
$u_1$, which means that on $\Gamma_b$ they all have label $1$ at $v_2$, so
they cannot be the four parallel edges in $+(4,2,2,0)$. Therefore
$\hat \Gamma_a \neq -(3,1,3,1)$.
\medskip
CLAIM 3. {\it The case $\hat \Gamma_a = -(3,3,1,1)$ is impossible.}
By Claim 2 we have $\hat \Gamma_b = +(4,2,2,0)$. The graphs are as shown in
Figure 19.3. While the graphs are similar to that in Figure 19.2, the
argument is necessarily different because the orientation of the
vertices of $\Gamma_b$ here are parallel while that of $\Gamma_a$ in Figure 19.2
are antiparallel. One can check that up to symmetry the edge
correspondence must be as shown in the figure.
We would like to apply Lemma 2.15 to get a contradiction. To do that,
let $Q$ be the face of $\Gamma_a$ bounded by $A\cup B\cup E \cup H$. The
edge $B$ is parallel to $C$ on $\Gamma_b$, and $C$ is a non-border edge on
$\Gamma_a$, hence one of the bigons $C\cup H$ or $C\cup F$ is a coupling
face $Q'$ of $Q$ along the edge $B$. By Lemma 2.15 there is a rel
$\partial$ isotopy of $F_a$ such that the new intersection graph $\Gamma_a'$ is
obtained from $\Gamma_a$ by deleting $A$ and $E$ and adding two edges
parallel to $B$ and $F$, respectively. It follows that $\Gamma_a' = -
(4,2,2,0)$. This is impossible by Claim 1. Therefore the case $\hat \Gamma_a
= -(3,3,1,1)$ is also impossible.
\quad $\Box$ \medskip
\bigskip
\leavevmode
\centerline{\epsfbox{Figure19.3.eps}}
\bigskip
\centerline{Figure 19.3}
\bigskip
\begin{lemma}
Suppose $n_1 = n_2 = 2$ and $\Gamma_b$ is positive. If $\rho = 0$ then
$\Gamma_a = -(3,3,3,1)$ and $\Gamma_b = +(3,3,3,1)$. The graphs $\Gamma_a, \Gamma_b$
and their edge correspondence are shown in Figure 19.4.
\end{lemma}
\noindent {\bf Proof.} By Lemma 15.1 all non-zero $a_i$ have the same parity, and all
non-zero $b_j$ have the same parity. By Lemma 19.6 we have $\Delta =
5$, so each of $\Gamma_a$ and $\Gamma_b$ is of type $(4,4,2,0)$, $(4,2,2,2)$
or $(3,3,3,1)$. If some $b_i=4$ then by Lemma 13.2(2) $\Gamma_a$ is
kleinian, but since each of the above type has an edge whose weight is
non-zero and different from the others, it must be mapped to itself by
the involution in Lemma 6.2(4), which is a contradiction because it is
supposed to be a free involution on $\hat F_a$. It follows that $\Gamma_b
= +(3,3,3,1)$. Direct calculation gives $D(+(3,3,3,1)) = (4,3,3)$,
$D(-(4,4,2,0)) = (3,3,2,2)$, $D(-(4,2,2,2)) = (4,4,1,1)$, and
$D(-(3,3,3,1)) = (4,3,3)$. Hence by Lemma 17.2 we have $\Gamma_a =
-(3,3,3,1)$.
\bigskip
\leavevmode
\centerline{\epsfbox{Figure19.4.eps}}
\bigskip
\centerline{Figure 19.4}
\bigskip
The graphs $\Gamma_a, \Gamma_b$ are as shown in Figure 19.4. Label the edges of
$\Gamma_a$ as in the figure. Relabeling the vertices of $\Gamma_b$ if
necessary, we may assume that the labels of edges of $\Gamma_a$ at $u_1$
are as shown. Since $\Gamma_b$ has no loops, each edge of $\Gamma_a$ has
different labels on its two endpoints, which determines the labels at
$u_2$. One can check that $\Gamma_a$ has three equidistance classes $c_1 =
\{B,E,G,I\}$, $c_2 = \{A,D,H\}$ and $c_3 = \{C,F,J\}$. Since $\Gamma_b$ is
positive, each family belongs to an equidistance class; moreover, one
can check that the single edge is equidistant to the non-adjacent
family of weight 3, which we will denote $\hat e_1$. Therefore these
must belong to $c_1$. On $\Gamma_a$, $B$ has label $1$ at $u_2$, so on
$\Gamma_b$ $B$ has label $2$ at $v_1$. It follows that $B$ is the middle
edge in $\hat e_1$. This determines the labels on $v_1$ and $v_2$.
Now the endpoints of $E,G,I \in c_1$ are adjacent on $\partial u_1$ among
edge endpoints labeled $1$, but they are not all adjacent on $\partial
v_1$ because the single edge is not adjacent to those in $\hat e_1$
among edges with label $1$ at $v_1$ in $\Gamma_b$. Therefore the jumping
number $J$ cannot be $\pm 1$, so $J=\pm 2$. Reversing the orientation
of $v_1, v_2$ if necessary we may assume that $J = 2$. Thus the edges
$E,G,I$ in $\Gamma_b$ must be as shown. The other edges are now determined
by this information. For example, the edges with label $2$ at $u_1$
appear in the order $B,D,F,H,J$, so on $\Gamma_b$ the edges with label
$1$ at $v_2$ appear in the order $B,H,D,J,F$. \quad $\Box$ \medskip
\section {The case $n_1 = n_2 = 2$ and both $\Gamma_1, \Gamma_2$
non-positive }
In this section we assume that $n_1 = n_2 = 2$ and both $\Gamma_1,
\Gamma_2$ are non-positive. Let $\Gamma_a = (\rho_a; a_1, ..., a_4)$, and
$\Gamma_b = (\rho_b; b_1, ..., b_4)$. Without loss of generality we may
assume that $\rho_b \geq \rho_a$.
\begin{lemma} Suppose $n_1 = n_2 = 2$, and $\Gamma_1,
\Gamma_2$ are non-positive.
(1) $\Delta/2 \leq \rho_b \leq 4$.
(2) $2\rho_a + \sum a_i = 2\Delta$, and $2\rho_b + \sum b_i =
2\Delta$.
(3) $a_i, b_i \leq 2$.
\end{lemma}
\noindent {\bf Proof.} (1) Since a loop in $\Gamma_a$ corresponds to a non-loop in $\Gamma_b$
and vice versa, we have $\rho_a + \rho_b = \Delta$. We have assumed
$\rho_b \geq \rho_a$, so $\rho_b \geq \Delta/2$. Since no two edges
are parallel on both graphs and $\hat \Gamma_a$ has at most four non-loop
edges, we also have $\rho_b \leq 4$.
(2) This follows from the fact that the valence of a vertex in $\Gamma_a$
or $\Gamma_b$ is $2 \Delta$.
(3) Since $\Gamma_a$ and $\Gamma_b$ are non-positive, a non-loop edge in $\Gamma_a$
is a loop in $\Gamma_b$, hence there are at most two edges in each non-loop
family of $\Gamma_a$, i.e.\ $a_i \leq 2$. Similarly for $b_i$.
\quad $\Box$ \medskip
\bigskip
\leavevmode
\centerline{\epsfbox{Figure20.1.eps}}
\bigskip
\centerline{Figure 20.1}
\bigskip
\begin{lemma} Suppose $n_1 = n_2 = 2$, $\Delta = 5$, and
$\Gamma_1, \Gamma_2$ are non-positive. Then $\Gamma_a = -(2;2,2,2,0)$ and
$\Gamma_b = -(3;1,1,1,1)$ or $-(3;2,2,0,0)$. \end{lemma}
\noindent {\bf Proof.} By Lemma 20.1(1) we have $\rho_b = 3$ or $4$. If $\rho_b = 4$
then each loop family contributes one edge to each non-loop family of
$\Gamma_a$, hence $\Gamma_a = -(1;2,2,2,2)$. The four loops $e_1, e_2, e_3,
e_4$ at $v_1$ are equidistant to each other; on the other hand, from
Figure 20.1 one can see that $e_i$ is equidistant to $e_j$ on $\Gamma_a$ if
and only if $e_i$ and $e_j$ are on the same side of the loop at $u_1$.
This is a contradiction. Therefore this case cannot happen.
Now assume $\rho_b = 3$. Then by the Congruence Lemma we have $\Gamma_b =
-(3;2,0,2,0)$, $-(3;2,2,0,0)$ or $-(3;1,1,1,1)$. Since $\rho_a =
\Delta - \rho_b = 2$, we have $\sum a_i = 10 - 2 \rho_a = 6$. By
Lemma 20.1 we have $a_i \leq 2$, therefore by the Congruence Lemma we
must have $\Gamma_a = -(2;2,2,2,0)$. The first case for $\Gamma_b$ above cannot
happen because the two non-loop $1$-edges are not equidistant in $\Gamma_b$
while as parallel loops on $\Gamma_a$ they are equidistant on $\Gamma_a$.
Therefore $\Gamma_b = -(3;2,2,0,0)$ or $-(3;1,1,1,1)$.
\quad $\Box$ \medskip
\bigskip
\leavevmode
\centerline{\epsfbox{Figure20.2.eps}}
\bigskip
\centerline{Figure 20.2}
\bigskip
\begin{lemma} Suppose $n_1 = n_2 = 2$, $\Delta = 4$, and
$\Gamma_1, \Gamma_2$ are non-positive. Then one of the following
holds.
(1) $\Gamma_a = -(2,2,2,2)$ and $\Gamma_b = -(4;0,0,0,0)$.
(2) Both $\Gamma_1$ and $\Gamma_2$ are of type $-(2;1,1,1,1)$.
(3) Both $\Gamma_1$ and $\Gamma_2$ are of type $-(2;2,0,2,0)$.
(4) Both $\Gamma_1$ and $\Gamma_2$ are of type $-(2;2,2,0,0)$.
\end{lemma}
\noindent {\bf Proof.} If $\rho_b = 4$ then $\Gamma_b = -(4;0,0,0,0)$ and each loop family
contributes one edge to each family of $\Gamma_a$, hence $\Gamma_a =
-(2,2,2,2)$.
If $\rho_b = 3$, then by the Congruence Lemma 15.1 we have $\Gamma_a = -(1;
2,2,2,0)$. Let $e_1, e_2, e_3$ be the three loops at $v_1$. As
parallel positive edges, they are equidistant on $\Gamma_b$. On $\Gamma_a$ they
are as shown in Figure 20.2. One can check that $e_1$ is equidistant
to $e_2$ but not $e_3$, which is a contradiction to Lemma 2.17.
Therefore $\rho_b \neq 3$.
When $\rho_b=\rho_a = 2$, by the Congruence Lemma each of $\Gamma_a$ and
$\Gamma_b$ is of type $-(2;1,1,1,1)$ or $-(2;2,2,0,0)$ or $-(2;2,0,2,0)$.
We are done if both $\Gamma_a, \Gamma_b$ are of the same type.
If $\Gamma_a = -(2;2,2,0,0)$ then the two non-loop edges with label $1$ at
both endpoints are adjacent among the four edges labeled $1$ at $u_1$,
hence on $\Gamma_b$ the two loops at $u_1$ are adjacent among the four
edges with label $1$ at $v_1$, which implies that $\Gamma_b$ cannot be
$-(2;2,0,2,0)$ or $-(2;1,1,1,1)$.
It remains to rule out the possibility that $\Gamma_a = -(2;1,1,1,1)$ and
$\Gamma_b = -(2;2,0,2,0)$. In this case the graphs are as shown in Figure
20.3.
Label the edges of $\Gamma_a$ as in the figure. We want to show that this
determines the labels of the edges of $\Gamma_b$ up to symmetry. Since
$\Delta=4$, by changing the orientation of $\hat F_b$ if necessary we
may assume that the jumping number is $1$. The $1$-edges at $u_1$ are
in the order $A,B,C,D$, so these labels appear in this order at
$v_1$ on $\Gamma_b$. The order of the $1$-edges at $u_2$ is $A,X,C,Y$, so
the $2$-edges at $v_1$ are also in this order, which determines the
edges $X,Y$ on $\Gamma_b$. Finally, the order of the $2$-edges at $u_1$
determines the edges $E,F$ in $\Gamma_b$. Hence the labels of the graphs
are as shown in Figure 20.3.
One way to see that these graphs are not realizable is to consider the
annulus $A$ from $\partial v_1$ to $\partial v_2$ along the positive
orientation, draw the segments of $\partial u_1, \partial u_2$ on this annulus
and check that these arcs must intersect on $A$, which contradicts the
fact that $\partial u_1, \partial u_2$ are parallel curves on the torus $T_0$.
Here is another way. Consider the endpoints of the edges $D,X$,
labeled $a,b,c,d$ on the two graphs. We have $$d_{v_1}(a,c) =
d_{v_2}(b,d) = 1$$
so by Lemma 2.16 (applied with $u_i, u_j, v_k, v_l$
replaced by $v_1, v_2, u_1, u_2$ and $P,Q,R,S$ replaced by $a,c,b,d$),
we should have $$d_{u_1}(a,b) = d_{u_2}(c,d)$$
However, on $\Gamma_a$ we
have $d_{u_1}(a,b) = 5$ while $d_{u_2}(c,d) = 3$, which is a
contradiction. \quad $\Box$ \medskip
\bigskip
\leavevmode
\centerline{\epsfbox{Figure20.3.eps}}
\bigskip
\centerline{Figure 20.3}
\bigskip
\begin{prop} Suppose $n_a, n_b \leq 2$. Then up to
symmetry $\Gamma_a$ and $\Gamma_b$ are one of the following pairs.
$$
\begin{array}{lll}
(1) \qquad \qquad \qquad & -(1,1,1,1) \qquad \qquad \qquad &(4,0,0) \\
(2) \qquad \qquad \qquad & -(2,2,0,0) & (2,2,0) \\
(3) \qquad \qquad \qquad & -(2,1,1,1) & (3,1,1) \\
(4) \qquad \qquad \qquad & -(2,2,2,2) &+(4;0,0,0,0) \\
(5) \qquad \qquad \qquad & -(3,3,3,1) &+(3,3,3,1) \\
(6) \qquad \qquad \qquad & -(2;2,2,2,0) &-(3;1,1,1,1) \\
(7) \qquad \qquad \qquad & -(2;2,2,2,0) &-(3;2,2,0,0) \\
(8) \qquad \qquad \qquad & -(2,2,2,2) &-(4;0,0,0,0) \\
(9) \qquad \qquad \qquad & -(2;1,1,1,1) &-(2;1,1,1,1) \\
(10) \qquad \qquad \qquad & -(2;2,0,2,0) &-(2;2,0,2,0) \\
(11) \qquad \qquad \qquad & -(2;2,2,0,0) &-(2;2,2,0,0)
\end{array}
$$
\end{prop}
\noindent {\bf Proof.} This follows from the lemmas in Sections 18--20. More
precisely, the case $n_b = 1$ is done in Lemma 18.1, which gives
(1)--(2) above; the case $n_a=n_b=2$ and $\Gamma_b$ positive is discussed
in Lemmas 19.2--19.7 according to different numbers of loops on $\Gamma_b$,
which gives (3)--(5); the case $n_a=n_b=2$ with both graphs
non-positive is discussed in Lemmas 20.2--20.3, with the possibilities
listed in (6)--(11). \quad $\Box$ \medskip
\begin{prop} For each of the cases (3), (5), (6), (9)
and (10) of Proposition 20.4, the correspondence between edges of
$\Gamma_a, \Gamma_b$ is unique up to symmetry, and is shown in Figures 18.2,
19.4, 20.4, 20.5 and 20.6, respectively. \end{prop}
\noindent {\bf Proof.}
For cases (3) and (5) this follows from Lemmas 18.1 and 19.7.
In case (6) we have $\Gamma_a = -(2;2,2,2,0)$ and $\Gamma_b = -(3;1,1,1,1)$.
The graphs $\Gamma_a, \Gamma_b$ are as shown in Figure 20.4. Label the edges of
$\Gamma_b$ as shown in the figure. By symmetry we may assume that the
labels on the edge endpoints of $\Gamma_b$ are as in the figure. Also up to
symmetry of $\Gamma_b$ on the torus $\hat F_b$ we may assume that the
labels on $v_1$ are as in the figure.
The label 1 endpoints of $A,B,C$ are non-adjacent among the $1$-labels
on $\partial v_1$. These are non-loops on $\Gamma_a$, and one in each family,
hence their endpoints at $u_1$ are also non-adjacent among endpoints
labeled 1. This forces the jumping number $J$ to be $\pm 1$. Now on
$\Gamma_a$ the edge $A$ must be as shown. It is easy to see that this
determines the labels on the other edges in $\Gamma_a$.
\bigskip
\leavevmode
\centerline{\epsfbox{Figure20.4.eps}}
\bigskip
\centerline{Figure 20.4}
\bigskip
In case (9) we have $\Gamma_a = -(2;1,1,1,1)$, $\Gamma_b = -(2;1,1,1,1)$, and
$\Delta = 4$, so we may assume $J=1$. Label edge endpoints and edges
of $\Gamma_a$ as in the figure. Using symmetry we may assume $A$ to be any
one of the two non-loop edges labeled $1$ at $v_1$. Then this
determines the labels on the other edges. See Figure 20.5.
The determination of the edge correspondence for case (10) is similar.
The graphs are shown in Figure 20.6. \quad $\Box$ \medskip
\bigskip
\leavevmode
\centerline{\epsfbox{Figure20.5.eps}}
\bigskip
\centerline{Figure 20.5}
\bigskip
\bigskip
\leavevmode
\centerline{\epsfbox{Figure20.6.eps}}
\bigskip
\centerline{Figure 20.6}
\bigskip
\section {The main theorems }
Suppose $M$ is a hyperbolic manifold admitting two toroidal Dehn
fillings $M(r_1), M(r_2)$. Let $F_a$ be essential punctured tori in
$M$ such that $\partial F_a$ consists of a minimal number of copies of
$r_a$, and $F_1$ intersects $F_2$ minimally. Let $X(F_1, F_2)$ be
obtained from $N(F_1 \cup F_2 \cup T_0)$ by capping off its 2-sphere
boundary components with 3-balls. We will use $X(r_1, r_2)$ to denote
any $X(F_1, F_2)$ above with $\partial F_a$ of slope $r_a$, and call it a
{\it core\/} of $M$ with respect to the toroidal slopes $r_1, r_2$.
Note that $X(r_1, r_2)$ may not be unique.
\begin{lemma} Suppose $M$ is a hyperbolic manifold
admitting two toroidal Dehn fillings $M(r_1), M(r_2)$ of distance $4$
or $5$. Then each of $\partial X(r_1, r_2)$ and $\partial M$ is a union of
tori.
\end{lemma}
\noindent {\bf Proof.} By the result of the previous sections we see that $\Gamma_a, \Gamma_b$
are either the graphs in Figures 11.9, 11.10, 14.5, 16.6, 16.8, 16.9,
or one of the pairs given in Proposition 20.4.
In all figures except 11.9 and 11.10, $\Gamma_a$ has two vertices and they
have opposite signs. Now $X(F_1, F_2)$ can be constructed by adding
thickened faces of $\Gamma_b$ to $N(F_a \cup T_0)$, which has two boundary
components of genus 2. It is easy to check that in all cases $\Gamma_b$
has at least one disk face on each side of $F_a$. The boundary of a
disk face of $\Gamma_b$ is always an essential curve on $F_a \cup T_0$.
Adding a 2-handle corresponding to a disk face will change a genus 2
boundary component to one or two tori. It follows that the boundary
of $X(F_1, F_2)$ is a union of tori. Since $M$ is irreducible and
atoroidal, each torus boundary component of $X(F_1, F_2)$ either is
boundary parallel, or bounds a solid torus. Therefore $\partial M$ is
also a union of tori.
The proof for Figures 11.9 and 11.10 is similar. In these cases $\Gamma_a$
has 4 vertices, so $N(F_a \cup T_0)$ has two boundary components $S_i$
of genus 3. It suffices to find two faces on each side of $F_a$ whose
boundary curves give rise to non-parallel and non-separating curves on
$S_i$. For 11.9 one can check that the bigons on $F_b$ bounded by the
edges $K\cup J$ and $H\cup G$ are on the same side of $F_a$ and give
non-parallel boundary curves on $S_1$, say, while the bigon bounded by
the edges $J\cup H$ and the $3$-gon bounded by the edges $K \cup L
\cup R$ give non-parallel non-separating curves on $S_2$. Hence the
result follows. For 11.10, use the bigons bounded by $E \cup
F$ and $G \cup H$ on one side, and the bigon $F \cup G$ and the 3-gon
$E \cup K \cup A$ on the other side.
\quad $\Box$ \medskip
Consider the three manifolds $M_1, M_2, M_3$ in [GW1, Theorem 1.1].
More explicitly, $M_1$ is the exterior of the Whitehead link, $M_2$ is
the exterior of the 2-bridge link associated to the rational number
$3/10$, and $M_3$ is the exterior of the $(-2, 3,8)$ pretzel link,
also known as the Whitehead sister link. Each of these manifolds
admits two Dehn fillings $M_i(r_1)$ and $M_i(r_2)$, both toroidal and
annular, with $\Delta = 4$ for $i=1,2$, and $\Delta = 5$ for $i=3$.
Let $T_0$ be the Dehn filling component of $\partial M_i$, and let $T_1$
be the other component of $\partial M_i$. Then for all except a few
slopes $s$ on $T_1$, $M' = M_i(s)$ is a hyperbolic manifold, and it
admits two toroidal Dehn fillings $M'(r_1), M'(r_2)$ of distance $4$
or $5$. The following lemma shows that several of the cases in
Proposition 20.4 can only be realized by these manifolds.
\begin{lemma} Suppose $\Gamma_a$ has a non-disk face. Then $M =
M_i$ or $M_i(s)$ for some $i=1,2,3$ and slope $s$ on $T_1$, and the
toroidal slopes $r_1, r_2$ are the same as the toroidal/annular slopes
given in [GW1, Theorem 1.1].
\end{lemma}
\noindent {\bf Proof.} Let $K$ be a curve on $F_a$ which is essential on $\hat F_a$
and disjoint from $F_a \cap F_b$. Consider the manifold $X = M-{\rm Int}
N(K)$. If $X$ is hyperbolic then $\hat A = \hat F_a - {\rm Int} N(K)$ is
an essential annulus in $X(r_a)$ and $\hat F_b$ is an essential torus
in $X(r_b)$, so by [GW1, Theorem 1.1] $X = M_i$ for some $i=1,2,3$,
and we are done. Hence we may assume that $X$ is non-hyperbolic. $X$
is irreducible as otherwise there would be an essential sphere $S$ in
$X$ bounding a 3-ball in $M$ containing $K$, which would be a
contradiction to the fact that $K$ is an essential curve on $\hat
F_a$. Also $X$ cannot be a Seifert fibered manifold as otherwise $M =
X \cup N(K)$ would be non-hyperbolic. Since by Lemma 21.1 $\partial M$ is
a union of tori, the above implies that $X$ must be toroidal.
Since $M$ is atoroidal, an essential torus $T$ in $X$ must be
separating. Let $T_0$ be the Dehn filling torus component of $M$, and
let $T_1 = \partial N(K)$. Recall that $M|T$ denotes the manifold
obtained by cutting $M$ along $T$. Let $V = V_T$ and $W = W_T$ be the
components of $M|T$, where $W$ is the component containing $T_0$.
Among all essential tori in $X$, choose $T$ so that (a) if there is
some $T$ in $X$ such that $T_1 \subset W_T$, choose $T$ so that $V_T$
contains no essential torus; (b) if every essential torus in $X$
separates $T_0$ from $T_1$, choose $T$ such that $W_T$ contains no
essential torus.
Since $M$ is atoroidal, $T$ is inessential in $M$, hence $V$ is either
(i) a solid torus, or (ii) $T^2 \times I$, or (iii) a 3-ball with a
knotted hole. Note that in the first two cases $V$ must contain the
curve $K$. Let $N = V - {\rm Int} N(K)$ in the first two cases, and $N =
V$ in the last case. Let $C = T \cap F_a$. Using a standard cut and
past argument we may assume that each component of $C$ is essential on
both $T$ and $F_a$. In case (iii) let $D$ be a compressing disk of
$T$ in $W$.
\medskip
{\bf Claim 1.} {\it $C \neq \emptyset$.}
\medskip
\noindent {\bf Proof.} If $C = \emptyset$ then $F_a$ lies in $W$, which is
impossible in cases (i) or (ii) because the curve $K$ on $F_a$
lies in $V$. In case (iii) $D \cap F_a$ is a set of circles and
one can use the incompressibility of $F_a$ in $W$ to isotop $F_a$ so
that it is disjoint from $D$. But then $D$ is disjoint from $K$, so
$T$ would be compressible in $X$, which is a contradiction.
\quad $\Box$ \medskip
\medskip
{\bf Claim 2.} {\it $C$ is a set of essential curves on $\hat F_a$
parallel to $K$.}
\medskip
\noindent {\bf Proof.} Since $C$ is disjoint from $K$, we need only show that each
component $\alpha$ of $C$ is an essential curve on $\hat F_a$. Assume
to the contrary that $\alpha$ bounds a disk $E$ on $\hat F_a$ and is
innermost on $\hat F_a$. Then $E$ must contain some boundary
component of $F_a$, hence $E \subset W(r_a)$. In case (i) $V \cup
N(E)$ is either a 3-ball, or a punctured lens space or $S^1 \times
S^2$, containing the curve $K$, contradicting the fact that $\hat F_a$
is incompressible and $M(r_a)$ irreducible. In case (ii) $V \cup
N(E)$ is a punctured solid torus, so the irreducibility of $M(r_a)$
implies that $M(r_a)$ is a solid torus, which is absurd because it is
supposed to be toroidal. In case (iii), for homological reasons $\partial
E$ and $\partial D$ must be homotopic on $T$, hence $\partial E$ is
null-homotopic in $M$, which contradicts the facts that $C$ is
essential on $F_a$ and $F_a$ is incompressible in $M$. \quad $\Box$ \medskip
\medskip {\bf Claim 3.} {\it Case (iii) cannot happen, i.e.\
$V$ is not a 3-ball with a knotted hole.}
\medskip
\noindent {\bf Proof.} We have shown that all components of $C$ are essential curves
on $\hat F_a$ parallel to $K$, and $C \neq \emptyset$. Let $\alpha$ be
a component of $C$. Then $K$ is isotopic to $\alpha$ in $M(r_a)$,
but since $\alpha \subset T$ lies in the 3-ball $V \cup N(D)$,
$\alpha$, and hence $K$, is null-homotopic in $M(r_a)$, which
contradicts the fact that $\hat F_a$ is incompressible in $M(r_a)$.
\quad $\Box$ \medskip
\medskip {\bf Claim 4.} {\it $W$ is hyperbolic.}
\medskip
\noindent {\bf Proof.} Clearly $W$ is irreducible (since $X$ is) and not a Seifert
fibered space (since $M$ is hyperbolic). Suppose $W$ contains an
essential torus $T'$. By Claim 3 we see that $T'$ cannot be of type
(iii), so it must be of type (i) or (ii), which, by our choice of $T$,
implies that every essential torus in $X$ separates $T_0$ from $T_1$.
By the choice of $T$, $W$ must be atoroidal. \quad $\Box$ \medskip
We now continue with the proof of Lemma 21.2. Let $A$ be a component of
$F_a \cap W$ which contains some boundary components of $F_a$. By
Claims 1 and 2, the corresponding component $\hat A$ of $\hat F_a \cap
W(r_a)$ is an annulus in $W(r_a)$, which is incompressible because
$\hat F_a$ is incompressible, and not boundary parallel because
otherwise $\hat F_a$ would be isotopic to a torus with fewer
intersections with the Dehn filling solid torus. Therefore $W(r_a)$
is annular.
Let $P$ be the component of $F_a \cap V$ containing $K$, and let
$\beta$ be a component of $P \cap T$. Note that $P$ is an annulus.
Since $F_b$ is disjoint from $K$, it can be isotoped to be disjoint
from $P$, hence after isotopy we may assume that $F_b \cap T$ and $F_a
\cap T$ are all parallel to $\beta$ and hence mutually disjoint. If
$F_b \cap T = \emptyset$ then $\hat F_b$ is an essential torus in
$W(r_b)$, and if $F_b \cap T \neq \emptyset$ then as above, a
component of $\hat F_b \cap W(r_b)$ which intersects the Dehn filling
solid torus is an essential annulus in $W(r_b)$, hence $W(r_b)$ is
either toroidal or annular. Using Theorem 1.1 of [GW1] in the first
case and Theorem 1.1 of [GW3] in the second case, we see that $W =
M_i$ for $i=1$, $2$, or $3$.
By Claim 3 $V$ is either a solid torus or $T^2 \times I$. In the
first case $M = M_i(s)$ for some $s$ on $T_1 = \partial V$, and in the
second case $M = M_i$.
\quad $\Box$ \medskip
\begin{defn}
(1) Define a set of triples $(M_i, r'_i, r''_i)$ as follows. For
$i=1,2,3$, $(M_i,r'_i,r''_i)$ are the manifolds and the
toroidal/annular slopes given in Theorem 1.1 of [GW1]. $M_4, ...,
M_{14}$ are the manifolds $X(F_1, F_2)$ corresponding to the
intersection graphs given in Figures 11.9, 11.10, 14.5, 16.6, 16.8,
16.9, 18.2, 19.4, 20.4, 20.5 and 20.6, and $r'_i,r''_i$ are the
boundary slopes of the corresponding surfaces $F_1, F_2$.
(2) Two triples $(M, r', r'')$ and $(N, s', s'')$ are {\it
equivalent}, denoted by $(M,r',r'') \cong (N,s',s'')$, if there is a
homeomorphism from the 3-manifold $M$ to $N$ which sends the boundary
slopes $(r', r'')$ to $(s', s'')$ or $(s'', s')$.
\end{defn}
The following theorem shows that if a hyperbolic manifold $M$ admits
two toroidal Dehn fillings along slopes $r_1, r_2$ of distance 4 or 5
then $(M, r_1, r_2)$ is either one of these triples, or obtained from
such an $M_i$ by Dehn filling on $\partial M_i - T_0$.
\begin{thm}
Let $M$ be a hyperbolic 3-manifold admitting two toroidal Dehn
fillings $M(r_1), M(r_2)$ with $\Delta(r_1, r_2) = 4$ or $5$. Let
$n_a$ be the minimal number of intersections between essential tori
and the Dehn filling solid torus in $M(r_a)$. Assume $n_a \leq
n_b$. Let $(M_i, r'_i, r''_i)$ be the manifolds defined above, and
let $T_0$ be the boundary component of $M_i$ containing $r'_i,
r''_i$. Then
(1) $n_a \leq 2$, $n_b \leq 6$;
(2) either $(M,r_1,r_2) \cong (M_i, r'_i, r''_i)$ for some $i=1,...,14$,
or $(M,r_1,r_2) \cong (M_i(s), r'_i,r''_i)$, where $i\in
\{1,2,3,14\}$ and $s$ is a slope on $T_1 = \partial M_i - T_0$; and
(3) $i \in \{1,2,4,6,9,13,14\}$ if $\Delta = 4$, and $i \in
\{3,5,7,8,10,11,12\}$ if $\Delta = 5$.
\end{thm}
\noindent {\bf Proof.} This is a summary of the results in the previous sections.
Assume $n_a \leq n_b$. Then by Proposition 11.10 we have $n_a \leq 2$.
By Proposition 16.8 if $n_b \geq 3$ then $X$ is one of those in Figure
11.9, 11.10, 14.5, 16.6, 16.8 or 16.9.
We may now assume $n_a, n_b \leq 2$. Then by Proposition 20.4 $\Gamma_a,
\Gamma_b$ is one of the 11 pairs listed there. One can check that all but
cases (3), (5), (6), (9), (10) have the property that one of $\hat
F_a, \hat F_b$ contains a non-disk face, so by Lemma 21.2 the triple
$(M,r_1,r_2)$ is $(M_i,r'_i, r''_i)$ for some $i=1,2,3$. Finally, by
Proposition 20.5 the graphs of the above cases are given in Figures
18.2, 19.4, 20.4, 20.5 and 20.6.
(3) follows by counting $\Delta$ for the graph pairs of each of the
manifolds listed in (2). \quad $\Box$ \medskip
\section {The construction of $M_i$ as a double branched cover}
The first three of the 14 manifolds $M_i$ have already been identified
as the exteriors of links in $S^3$. See [GW1]. The links are shown in
Figure 24.1 Besides $M_4$ and $M_5$, the other nine manifolds $M_6,
..., M_{14}$ have the property that $\Gamma_a$ is a graph on $\hat F_a$
with two vertices of opposite signs. In this section we will
construct, for each $i=6,...,14$, a tangle $Q_i = (W_i, K_i)$, where
$W_i$ is a 3-ball for $i=6,...,13$, and an $S^2 \times I$ for $i=14$,
such that $M_i$ is the double branched cover of $W_i$ with branch set
$K_i$. It is well known that once we have such a presentation then
the Dehn filling $M_i(r)$ will be the double branched cover of
$Q_i(r)$, where $Q_i(r)$ is obtained by attaching a rational tangle of
slope $r$ to $Q_i$, with coordinates properly chosen.
Here is a sketch of the construction. Assume $\Gamma_a$ is non-positive
and $n_a = 2$, and suppose there is an orientation-preserving
involution $\alpha_1$ on $F_a$ which maps $\partial u_1$ to $\partial u_2$ and
preserves $\Gamma_a$. The restriction of $\alpha_1$ on $\partial F_a$ extends
to an involution $\alpha_2$ on $T_0$ which has four fixed points, and
it preserves the curves $\partial F_b$ on $T_0$. Thus $\alpha = \alpha_1
\cup \alpha_2$ is an involution on $F_a \cup T_0$, which has eight
fixed points, four on each of $F_a$ and $T_0$. Since $\alpha$
preserves $\Gamma_a \cup \partial F_b$, it extends over each disk face of $F_b$
to give an involution on $F_b$. One can now further extend the
involution $\alpha$ from $F_a \cup F_b \cup T_0$ to a regular
neighborhood $Y$ of $F_a \cup F_b \cup T_0$. For $i\geq 6$, $M_i$ is
obtained by capping off spherical boundary components of $Y$ by
3-balls, hence $\alpha$ extends to an involution of $M_i$. Clearly
the quotient of $N(F_a \cup T_0)$ is a twice punctured 3-ball $W_i$.
After attaching 2-handles corresponding to faces of $F_b$ and some
3-balls we see that $W_i$ is a punctured 3-ball. From the
construction below we will see that $W_i$ is a 3-ball when $i = 6,
..., 13$, and an $S^2 \times I$ when $i=14$. Denote by $K_i$ the
branch set of $\alpha$ in $W_i$. Then $Q_i = (W_i, K_i)$ is the
tangle corresponding to the manifold $M_i$, and $M_i$ is the double
branched cover of $Q_i$ in the sense that it is the double branched
cover of $W_i$ with branch set $K_i$. Attaching a rational tangle of
slope $t$ to $T_i$, we obtain a new tangle $Q_i(t)$ whose double
branched cover is $M(r)$ for some slope $r$ on $T_0 \subset \partial M$.
This makes it possible to see the essential torus in $M_i(r_a)$ as a
lifting of some surface in $Q_i(t)$.
To illustrate this procedure, we give below a step by step
construction of the tangle $Q_6 = (W_6, K_6)$ for the manifold $M_6$
corresponding to the graphs in Figure 14.5. The constructions for the
other manifolds are similar.
Denote by $N(C)$ a regular neighborhood of a set $C$ in a 3-manifold,
and by $I$ the interval $[-1, 1]$.
\medskip
STEP 1. {\it Identify $[N(F_a \cup T_0)/\alpha] - D_2 \times I$ with
$S^2 \times I$.}
\medskip
Recall that $\alpha$ has four branch points on each of $T_0$ and
$F_a$, so $T_0/\alpha = S$ is a 2-sphere, and $F_a/\alpha = D_1$ is a
disk. Let $D_2$ be a small disk in the interior of $D_1$, disjoint
from $\Gamma_b/\alpha$ and the branch points of $\alpha$. Then $A_1 = D_1
- {\rm Int} (D_2)$ is a collar of $\partial D_1$. Therefore $N( (F_a \cup
T_0)/\alpha)$ can be written as $$(S \times I) \cup (A_1 \times I)
\cup (D_2 \times I)$$
Note that $A_1 \times I$ is a collar of the
attaching annulus $\partial D_1 \times I$, hence $X = (S\times I) \cup
(A_1 \times I)$ is homeomorphic to $S^2 \times I$.
One boundary component of $X = S^2 \times I$ is $\partial_- X = (T_0\times
\{-1\})/\alpha$, and the other boundary component $\partial_+ X$ can be
written as $D_+ \cup D_- \cup A$, where the two disks $D_+ \cup D_- =
\partial X \cap (S \times 1)$ lift to two annuli on $T_0 \times 1$, and
$A$ is the annulus $\partial X \cap (A_1 \times I)$. We identify $X$ with
$(\Bbb R^2 \cup \{\infty\}) \times I$, so that the disks $D_{\pm}$ are
identified with the squares $I \times [\pm 2, \pm 4]$ on the plane $P
= \Bbb R^2 \times 1$ on $\partial X$, the annulus $A$ is the closure of
$P- D_+ \cup D_-$, and the core $c_0$ of $A$ is identified with the
closure of the $x$-axis of $P$. See Figure 22.1. (Not drawn to
scale.)
\bigskip
\leavevmode
\centerline{\epsfbox{Figure22.1.eps}}
\bigskip
\centerline{Figure 22.1}
\bigskip
The branch set of $\alpha$ now consists of eight arcs. Four of them
come from the fixed points of $\alpha$ on $T_0 \times I$, and are of
type $p_i \times I$ in $X$, where $p_1, p_2 \in D_+$, and $p_3, p_4
\in D_-$. These will be represented by four dots $p_1, ..., p_4$ on
$P$, two in each of $D_{\pm}$, as shown in Figure 22.1. The other
four branch arcs of $\alpha$ are of type $a_i = q_i \times I \subset A
\times I$, where $q_1, ..., q_4$ are the branch points of $\alpha$ in
$A = D_1 - {\rm Int} (D_2)$. Note that $a_i$ has both endpoints on $S^2
\times 1$. We may assume that these project to four vertical arcs on
the annulus $A$ in $P$ above, and we may arrange so that the endpoints
of these arcs have $y$-coordinates $\pm 1$ on the plane $P$. See
Figure 22.1. Denote by $a_i(1)$ and $a_i(-1)$ the endpoints of $a_i$
with $y$ coordinates $1$ and $-1$, respectively.
STEP 2. {\it Draw the arcs $G' = (\Gamma_a \times \{\pm 1\})/\alpha$ on
$P$, with edges and edge endpoints labeled.}
\medskip
The graph $\Gamma_a \times 1$ on $F_a \times 1$ projects to a set of arcs
$E$ on $D_1 \times 1$. We may choose the disk $D_2$ above to be
disjoint from $E$. Then $E$ lies in the annulus $A_+ = A \times 1$.
If a family $\hat e_i$ has $2k$ edges then they project to $k$ edges
on $A_+$ with endpoints on $\partial D_+$, each circling around the branch
point $a_i(1)$. If $\hat e_i$ has $2k+1$ edges then the quotient is a
set of $k$ edges as above together with an edge connecting a point on
$\partial D_+$ to $a_i(1)$. Up to isotopy we may assume that all edge
endpoints of $E$ on $\partial D_+$ lie on the horizontal line $y=2$ on
$P$. Similarly the projection of $\Gamma_a \times (-1)$ is a set of arcs
on the annulus $A_- = A \times (-1)$, which is the mirror image of
the arcs $(\Gamma_a \times 1)/\alpha$ along the circle $c_0$ on $P$.
Denote by $G'$ the set of arcs above.
For the graph $\Gamma_a$ in Figure 14.5(a), the edges in $G'$ are shown in
Figure 22.2. The edges are labeled by the corresponding edges in
$\Gamma_a$. (We only show a few of the labels in the figure; the others
should be easy to identify.) Each edge in $G$ is the image of two
edges in $\Gamma_a$, hence it has two labels. (Note that if one of the
families has an odd number of edges then the middle one projects to an
arc in $G'$ with a single label.) All arcs appear in the region $I
\times [-2,2]$. The top and bottom lines in the figure represent arcs
on $\partial D_{\pm}$. Note that each edge endpoint on $\partial D_+$
corresponds to one edge endpoint on each of $\partial u_1$ and $\partial u_2$.
The labels on the top and bottom lines correspond to the labels on
$\partial u_1$ in Figure 14.5(a).
\bigskip
\leavevmode
\centerline{\epsfbox{Figure22.2.eps}}
\bigskip
\centerline{Figure 22.2}
\bigskip
STEP 3. {\it Add arcs $G'' = (\partial F_b \times 1)/\alpha$ on $P$ to
obtain $G = G' \cup G''$.}
\medskip
Recall that the preimage of $D_{\pm}$ are two annuli $A_{\pm}$ on $T_0
\times 1$. The curves $G'' = (\partial F_b \times 1 \cap A_{\pm})/\alpha$
is now a set of arcs in $D_{\pm}$, with $\partial G''$ the union of $\partial
G'$ and possibly some of the branch points $p_i$ in $D_{\pm}$. We
need to determine how the endpoints of $G'$ are connected by the edges
of $G''$.
Consider the circle $\partial v_6$ in Figure 14.5(b). We may assume that
the segments on $\partial v_6$ from label 1 to label 2 (in the
counterclockwise direction) project to arcs in $D_+$ while those from
label $2$ to label $1$ project to arcs in $D_-$. Consider the arc
$\beta$ on $\partial v_6$ from the tail of $e_6$ to the head of $e''_6$.
(Recall that $e''_6$ is the edge between $e_6$ and $e'_6$ in Figure
14.5(a)). Note that the tail of $e_6$ projects to the endpoint of
$e_1=e_6$ with label $6$ on $\partial D_+$ in Figure 22.2. The other
endpoint $q$ of $\beta$ is the head of $e''_6$, which lies on $\partial
u_2$. Since the labels in Figure 22.2 are the ones corresponding to
those on $\partial u_1$ in Figure 14.5(a), we have to find the
corresponding point on $\partial u_1$ in order to determine the position
of the edge endpoint $q$ on $\partial D_+$. On Figure 14.5(a) the
involution $\alpha$ restricted to $\partial u_2$ is a vertical
translation, which maps the head of $e''_6$ (i.e.\ the edge in $\hat
e_3$ labeled $6$ at $u_2$) to the tail of $e''_1$, which has label $3$
at $u_1$. It follows that $q$ is the endpoint of $e''_1=e''_6$ in
Figure 22.2 with label $3$ at $\partial D_+$. The two endpoints of
$\beta$ are represented by the two dots on the top line in Figure
22.2. Similarly, let $\beta'$ be the arc on $\partial v_6$ from the head
of $e''_6$ to the tail of $e'_6$. Then it is an arc in $D_-$ with
endpoints on the dots at the bottom line in Figure 22.2.
The arcs $G''$ in $D_{\pm}$ are parallel to each other, and they are
non-trivial in the sense that none of them cuts off a disk in
$D_{\pm}$ that does not contain a branch point of $\alpha$. Therefore
the above information completely determines the arcs $G''$ as well as
the branch points $p_1, ..., p_4$ of $\alpha$. (Note that if the
number of edge endpoints between the dots is odd then the middle arc
will have an endpoint on a branch point $p_i$.) The graph $G = G'
\cup G''$ is now shown in Figure 22.3.
\bigskip
\leavevmode
\centerline{\epsfbox{Figure22.3.eps}}
\bigskip
\centerline{Figure 22.3}
\bigskip
STEP 4. {\it Construct the tangle $Q_i$.}
\medskip
Each component $c \neq c_0$ of $G$ lifts to curves on $F_b$ bounding
disk faces $\sigma$ of $\Gamma_b$. The quotient of $\sigma \times I$ is
either a 2-handle attached to $c$ if $c$ is a circle, or a 3-ball
attached to a neighborhood of $c$ if $c$ is an arc. Examining the
branch set of $\alpha$ in $\sigma \times I$ gives the following
procedure. We use $X$ to denote the initial manifold at the beginning
of each step below. In particular, $X = S^2 \times I$ before the
first step.
(1) If $c$ is an arc then $X \cup (\sigma \times I)/\alpha$ is
homeomorphic to $X$. The new branch set is obtained by adding a
trivial arc in $(\sigma\times I)/\alpha$ joining the two endpoints of
$c$. Therefore we can simply modify the branch set of $\alpha$ by
pushing $c$ into the interior of $X$.
(2) If $c \neq c_0$ is a circle component bounding a disk $D_1$ on
$\partial X$ containing no branch point of $\alpha$, then attaching a
2-handle along $c$ creates a 2-sphere boundary component, which must
bound a 3-ball in $M_i/\sigma$. Thus after attaching the 2-handle and
the 3-ball the manifold is homeomorphic to $X$, and the homeomorphism
maps the new branch set to the old one. Therefore in this case we
can simply delete the curve $c$ from $G$.
(3) If $c \neq c_0$ is a circle component bounding a disk $D_1$ on
$\partial X$ containing one branch point of $\alpha$, then $c$ lifts to a
circle on the boundary of a face $\sigma$ of $\Gamma_b$, which necessarily
contains a fixed point of $\alpha$. Hence the cocore of the
corresponding 2-handle is a branch arc of $\alpha$. The 2-sphere
boundary component created after attaching the 2-handle contains two
branch points of $\alpha$, hence bounds a 3-ball containing a trivial
arc as branch set of $\alpha$. Thus after attaching the 2-handle and
the 3-ball the manifold is homeomorphic to $X$, and the branch set of
$\alpha$ has not changed. As in Case (2), we will simply delete the
curve $c$ from $G$ in this case.
(4) If a circle component $c \neq c_0$ of $G$ bounds a disk $D_1$
containing exactly two branch points of $\alpha$, then after attaching
a 2-handle and a 3-cell, the manifold is homeomorphic to $X$, and the
branch set of $\alpha$ is obtained by adding a trivial arc in the
3-cell joining the two branch points of $\alpha$ in $D_1$. Therefore
in this case we will add an arc in $D_1$ joining the two branch points
of $\alpha$, push the arc into the interior of $X$ as branch set of
$\alpha$, and then delete the curve $c$.
(5) If $c$ is a circle component of $G$ bounding a disk $D_1$
containing $k > 2$ branch points of $\alpha$, simply attach a
2-handle along $c$. If $k$ is odd, add an arc in the center of the
2-handle to the branch set of $\alpha$.
(6) Finally, attach a 2-handle along $c_0$, fill each 2-sphere
boundary component containing at most 2 branch points with a 3-ball,
and add a trivial arc in the 3-ball to the branch set if the 2-sphere
contains exactly two branch points. If the 2-sphere contains four branch
point, shrink it by an isotopy to a small sphere, which projects to a
small disk on the diagram, with four branch arcs attached. (This
happens only for $Q_{14}$. See Figure 22.13.) This completes the
construction of the tangle $Q_i$.
For $M_6$, the above procedure produces the tangle $Q_6 = (W_6, K_6)$
in Figure 22.4(a), where $K_6$ should be considered as a tangle lying
in the half space $Q_6$ (including $\infty$) in front of the
blackboard. The four boundary points of $K_6$ lie on the blackboard,
which is the boundary of $W_6$.
\bigskip
\leavevmode
\centerline{\epsfbox{Figure22.4.eps}}
\bigskip
\centerline{Figure 22.4}
\bigskip
STEP 5. {\it Find the tangles $Q_i(t_a) = M_i(r_a)/\alpha$.}
\medskip
There is one branch point $p_i$ in each quadrant of $P' = \Bbb R^2$.
Let $m, l$ be curves on $T_0$ that project to the $y$-axis union
$\infty$ and the $x$-axis union $\infty$ on $P'$, respectively. This
sets up coordinate systems on $T_0$ and $P'$. For $t=p/q$ a rational
number or $\infty$, let $M_i(t)$ denote the Dehn filling along slope
$pm + ql$, and $Q_i(t)$ denote the tangle obtained by attaching a
rational tangle of slope $t$ to $P'$. In other words, $Q_i(t)$ is
obtained by attaching a 3-ball to $Q_i$ on $P'$, and adding two arcs
on $P'$ connecting the branch points of $\alpha$, which lift to curves
of slope $t$ on $T_0$. Since the attached rational tangle lifts to a
solid torus with meridional slope $t$ on $T_0$, $M_i(t)$ is the double
branched cover of $Q_i(t)$.
By construction $\partial F_a$ projects to the $x$-axis, hence $M_i(r_1) =
M_i(0)$. The slope $r_2$ can be obtained by connecting the curves
$G''$ in $D_{\pm}$ by vertical arcs in $A = P' - \cup D_{\pm}$. For
$M_6$, one can check that the slope $r_2 = 4$.
\bigskip
Denote by $T(a_1, a_2)$ a Montesinos tangle which is the sum of two
rational tangles of slopes $1/a_1$ and $1/a_2$, respectively, where
$a_1, a_2$ are integers. Denote by $T(a_1, b_1; a_2, b_2)$ the
collection of pairs $(S^3, L)$ which can be obtained by gluing two
tangles $T(a_i, b_i)$ along their boundary. Denote by
$X(a_1,a_2)$ the collection of Seifert fiber spaces with orbifold a
disk with two cone points $c_1, c_2$ of index $a_1$ and $a_2$, i.e.\
the cone angle at $c_i$ is $2\pi/a_i$. Note that the double branched
cover of $T(a_1, a_2)$ is in $X(a_1, a_2)$. Denote by
$X(a_1,b_1;a_2,b_2)$ the collection of graph manifolds which are the
union of two manifolds $X_1, X_2$ glued along their boundary, where
$X_i \in X(a_i, b_i)$.
Denote by $K_{p/q}$ the two bridge knot or link associated to the
rational number $p/q$. Denote by $C(p_1, q_1; p_2, q_2)$ the link
obtained by replacing each component $K_i$ of a Hopf link by its
$(p_i, q_i)$ cable $K'_i$, where $q_i$ is the number of times $K'_i$
winds around $K_i$. Denote by $Y(p_1,q_1;p_2,q_2)$ the double
branched cover of $S^3$ with branch set $C(p_1, q_1; p_2, q_2)$.
Denote by $C(C; p,q)$ the link obtained by replacing one component
$K_1$ of a Hopf link by a Whitehead knot in the solid torus $N(K_1)$,
and the other component $K_2$ by a $(p,q)$ cable of $K_2$. Let
$Y(C;p,q)$ be the double branched cover of $S^3$ with branch set
$C(C;p,q)$. Denote by $Z$ the double branched cover of $S^3$ with
branch set the 2-string cable of the trefoil knot shown in Figure
22.12(d).
If $Q_i(r) = (S^3, L)$ then we will sometimes simply write $Q_i(r) =
L$.
\begin{lemma} (1) $Q_6(0) \in T(2,6; 2,3)$, as shown in Figure
22.4(b).
(2) $Q_6(4) = C(3,1; 2,5)$, as shown in Figure 22.5(b).
(3) $Q_6(\infty) = K_{9/2}$.
\end{lemma}
\noindent {\bf Proof.} (1) The tangle $Q_6(0) = (S^3,L)$ is shown in Figure 22.4(b).
A horizontal line at the middle of the diagram corresponds to a
2-sphere $S$ which cuts the link $L$ into two Montesinos tangles
$T(2,6)$ and $T(2,3)$.
(2) The tangle $Q_6(4)$ is shown in Figure 22.5(a), which can be
isotoped to that in Figure 22.5(b). One can see that it is the link
$C(3,1;2,5)$ in $S^3$.
(3) The tangle $Q_6(\infty)$ is shown in Figure 22.5(c). One can
check that it is isotopic to the knot $K_{2/9}$ in Figure 22.5(d).
\quad $\Box$ \medskip
\bigskip
\leavevmode
\centerline{\epsfbox{Figure22.5.eps}}
\bigskip
\centerline{Figure 22.5}
\bigskip
\begin{lemma} (1) For $i=6,...,14$, each $M_i$ is the double branched
cover of a tangle $Q_i = (W_i, K_i)$, where $Q_i$ is shown in Figure
22.4(a) for $i=6$, and in Figure 22.i(b) (with dotted lines removed)
when $i>6$.
(2) Each $M_i$ ($i=6, ..., 13$) admits a lens space surgery
$M_i(r_3)$. For each $i$, let $r_1, r_2$ be the slopes $r'_i,
r''_i$ given in Definition 21.3. Then the manifolds $M_i(r_1)$,
$M_i(r_2)$ and $M_i(r_3)$ are given in the following table.
$$
\begin{array}{lll} M_6(0) \in
X(2,6;2,3) \quad & M_6(4) = Y(3,1;5,2)
\quad & M_6(\infty) = L(9,2) \\
M_7(0) \in X(2,3;3,3) \quad & M_7(-5/2) \in X(2,3;2,2)
\quad & M_7(\infty) = L(20,9) \\
M_8(0) \in X(2,2;2,6) \quad & M_8(-5/4) = Y(3,1;2,5)
\quad & M_8(-1) = L(4,1) \\
M_9(0) \in X(2,3;2,3) \quad & M_9(-4/3) = Y(3,1;2,4)
\quad & M_9(-1) = L(8,3) \\
M_{10}(0) \in X(2,3;2,3) \quad & M_{10}(-5/2) = Y(C;2,1)
\quad & M_{10}(\infty) = L(14,3) \\
M_{11}(0) \in X(2,4;2,4) \quad & M_{11}(-5/2) = Y(C; 2,1)
\quad & M_{11}(\infty) = L(24,5) \\
M_{12}(0) \in X(2,3;2,3) \quad & M_{12}(5) = Y(3,1;2,3)
\quad & M_{12}(\infty) = L(3,1) \\
M_{13}(0) \in X(2,3;2,3) \quad & M_{13}(4) = Z \quad &
M_{13}(\infty) = L(4,1)
\end{array}
$$
\end{lemma}
\noindent {\bf Proof.} The result for $M_6$ follows from Lemma 22.1 because $M_6(r)$
is a branched cover of $Q_6(r)$. The proof for the other cases are
similar. Each $M_i(r)$ is the double branched cover of $Q_i(r)$ and
the tangle $Q_i(r)$ is a link $L$ in $S^3$. More explicitly, Figure
22.i(a) shows the curves $G = G' \cup G''$ in Step 3 of the above
construction; Figure 22.i(b) gives the tangle $Q_i$ as well as
$Q_i(r_1)$, which is obtained by attaching a $0$-tangle (the two
horizontal dotted lines) to $Q_i$; Figure 22.i(c) gives $Q_i(r_2)$,
which is simplified to that in Figure 22.i(d); $Q_i(r_3)$ is in Figure
22.i(e), which is simplified to that in Figure 22.i(f) for some $i$.
(The figures are numbered so that Figure 22.i corresponds to the
manifold $M_i$ for $i\geq 7$. Note that there is no Figure 22.6.)
The manifold $M_{14}(r_3)$ is the double branched cover of
$Q_{14}(r_3) = T(2,2)$ in Figure 22.14(e), and hence is a twisted
$I$-bundle over the Klein bottle.
\quad $\Box$ \medskip
\bigskip
\leavevmode
\centerline{\epsfbox{Figure22.7.eps}}
\bigskip
\centerline{Figure 22.7}
\bigskip
\bigskip
\leavevmode
\centerline{\epsfbox{Figure22.8.eps}}
\bigskip
\centerline{Figure 22.8}
\bigskip
\bigskip
\leavevmode
\centerline{\epsfbox{Figure22.9.eps}}
\bigskip
\centerline{Figure 22.9}
\bigskip
\bigskip
\leavevmode
\centerline{\epsfbox{Figure22.10.eps}}
\bigskip
\centerline{Figure 22.10}
\bigskip
\bigskip
\leavevmode
\centerline{\epsfbox{Figure22.11.eps}}
\bigskip
\centerline{Figure 22.11}
\bigskip
\bigskip
\leavevmode
\centerline{\epsfbox{Figure22.12.eps}}
\bigskip
\centerline{Figure 22.12}
\bigskip
\bigskip
\leavevmode
\centerline{\epsfbox{Figure22.13.eps}}
\bigskip
\centerline{Figure 22.13}
\bigskip
\bigskip
\leavevmode
\centerline{\epsfbox{Figure22.14.eps}}
\bigskip
\centerline{Figure 22.14}
\bigskip
Recall that a manifold $M$ with a fixed torus $T_0 \subset \partial M$ is
large if $H_2(M, \partial M - T_0) \neq 0$. Teragaito [T2] proved that
there is no large hyperbolic manifold $M$ admitting two toroidal
fillings of distance at least 5. The following result shows that
there is only one such manifold for $\Delta=4$.
\begin{thm}
Suppose $(M,T_0)$ is a large manifold and $M$ is hyperbolic and
contains two toroidal slopes $r_1, r_2$ on $T_0$ with $\Delta(r_1,
r_2) \geq 4$. Then $M$ is the Whitehead link exterior, and
$\Delta(r_1, r_2)=4$.
\end{thm}
\noindent {\bf Proof.} Let $r$ be a slope on $T_0$, $V_r$ the Dehn filling solid torus
in $M(r)$, and $K_r$ the core of $V_r$. By duality we have $H_2(M,
\partial M-T_0) \cong H^1(M, T_0)$, which is isomorphic to the free part
of $H_1(M, T_0)$. Also, $$H_1(M, T_0) \cong H_1(M(r), V_r) \cong
H_1(M(r)) / H_1(K_r).$$
Put $G(M,r) = H_1(M(r)) / H_1(K_r)$. Then we need only show that
$G(M_i,r)$ is a (possibly trivial) torsion group for $i=2,...,14$ and
$r$ some slope on $T_0$.
For $i=2,3$, $M_i$ is the exterior of a closed braid $K_i$ in a solid
torus $V$. Let $r$ be the meridian slope of $K_i$. Then $G(M_i, r) =
\Bbb Z_p$, where $p$ is the winding number of $K_i$ in $V$.
For $i=6,...,13$, by Lemma 22.2 $M_i$ has a lens space filling
$M_i(r_3)$. Therefore $G(M_i, r_3)$ is a quotient of the finite cyclic
group $H_1(M_i(r_3))$ and hence is a torsion group. Similarly for the
four manifolds in [Go] with toroidal slopes of distance at least $6$.
For $i=14$, take a regular neighborhood of $u_1 \cup u_2 \cup D$ on
$\hat F_a$ as a base point. See Figure 20.6(a). Then
$H_1(M_{14}(r_a))$ is generated by $x,y,s_1,s_2$, where $x$ is the
element of $H_1(\hat F_a)$ represented by the edges $C$ on Figure
20.6(a), oriented from the label $2$ endpoint to the label $1$
endpoint, $y$ is represented by $B$, oriented from $u_1$ to $u_2$, and
$s_i$ by the part of the core of the Dehn filling solid torus running
from $u_i$ to $u_{i+1}$ with respect to the orientation of $\partial F_b$.
Then the bigons $B\cup D$, $C\cup E$ and the 4-gon bounded by $C\cup
D\cup E \cup Y$ on $F_b$ give relations $2s_1 - y = 0$, $2x = 0$, and
$y + 2x = 0$. The other faces of $\Gamma_b$ are parallel to these.
To calculate $G(M_{14}, r_a) = H_1(M_{14}(r_a)) / H_1(K_a)$ we further
add the relation $s_1 + s_2 =0$. One can now check that $G(M_{14},
r_a) = \Bbb Z_2 \oplus \Bbb Z_2$, and the result follows.
For $i=4$, choose a regular neighborhood of $v_1 \cup v_2 \cup J$ in
Figure 11.9(b) as a base point. Then $H_1(M_4(r_b))$ is generated by
$x,y,s_1,s_2$, where $x,y$ are represented by the edges $L,C$ in
Figure 11.9(b), oriented from $v_1$ to $v_2$, and $s_i$ by the part of
the core of the Dehn filling solid torus from $v_i$ to $v_{i+1}$. The
faces bounded by $L\cup C$, $C\cup K$ and $Q\cup K \cup M \cup A$ give
the relations $y - s_1 + x + s_2 = 0$, $s_1 - x - s_2 = 0$, and $s_2 -
s_1 + y = 0$. Together with the relation $s_1+s_2=0$ from $H_1(K_b)=0$,
these give $G(M_4, r_b) = \Bbb Z_2$.
For $i=5$, $H_1(M_5(r_b))$ is generated by $x,y,s$, where $x,y$ are
represented by edges $E$ and $C$ on Figure 11.10(b), oriented from
label $3$ to label $4$, and $s$ is represented by the core of the Dehn
filling solid torus. Then the bigon $A\cup H$ and the annulus bounded
by $A\cup G \cup C \cup E$ on Figure 11.10(a) containing $J$ give the
relations $x+y=0$ and $2x - 2y = 0$. Adding the relation
$s=0$ gives $G(M_5, r_b) = \Bbb Z_4$. \quad $\Box$ \medskip
\section {The manifolds $M_i$ are hyperbolic}
The manifolds $M_1, M_2, M_3$ in Definition 21.3 are known to be
hyperbolic, see [GW1, Theorem 1.1]. In this section we will show that
the other 11 manifolds $M_i$ in Definition 21.3 are also hyperbolic. See
Theorem 23.14 below.
A knot $K$ in a solid torus $V$ is a $(p,q)$ knot if it is isotopic to
a $(p,q)$ curve on $\partial V$ with respect to some longitude-meridian
pair on $\partial V$. In particular, the winding number of $K$ in $V$ is
$p$.
\begin{lemma} (1) If $i\in \{6,...,13\}$ and $j=1$ or $i\in
\{6,7,8,9,12,13\}$ and $j=2$, then $M_i(r_j)$ contains an
essential torus $T$ cutting it into two Seifert fiber spaces
$E_1, E_2$.
(2) For $i=10, 11$, $M_i(r_2)$ contains a non-separating essential
torus cutting $M_i(r_2)$ into a Seifert fiber space whose orbifold is
an annulus with a cone point of index 2.
(3) For $i=6,...,13$ and $j=1,2$, $M_i(r_j)$ is irreducible, and
contains no hyperbolic submanifold bounded by an incompressible torus.
\end{lemma}
\noindent {\bf Proof.} (1) By Lemma 22.2, for $i=6,...,13$, $M_i(r_1)$ is of type
$X(a_1, b_1; a_2, b_2)$, which is the union of two Seifert fiber
spaces of types $X(a_1, b_a)$ and $X(a_2, b_2)$, hence the result is
true for $M_i(r_1)$. Similarly it is true for $M_7(r_2)$.
For $i=13$, $Q_i(r_2) = (S^3, L)$, where $L$ is the link in Figure
22.13(d), which consists of two parallel copies of the trefoil knot.
The two components of $L$ bound an annulus $A$. Cutting $S^3$ along
$A$ gives the trefoil knot exterior $E$, and $A$ becomes a torus $T$.
The double branched cover of $Q_{13}(r_2)$ is obtained by gluing two
copies of $E$ along $T$. Hence the result is true because $E$ is a
Seifert fiber space and $T$ is incompressible in $E$.
For $i=6,8,9,12$, $Q_i(r_2) = (S^3, L)$ is of type $C(p_1,q_1;
p_2,q_2)$, so there is a torus $T'$ cutting $S^3$ into two solid tori
$V_1, V_2$, such that each $L_j = L \cap V_j$ is a $(p_j,q_j)$ knot in
$V_j$ for some $p_j>1$. Note also that in these cases at least one of
the $p_j$ is odd, which implies that $T'$ lifts to a single torus $T$,
cutting $M_i(r_2)$ into two components $W_1, W_2$, such that $W_j$ is
a double branched cover of $(V_j, L_j)$. The $(p_j,q_j)$ fibration of
$V_j$ now lifts to a Seifert fibration of $W_j$, hence the result
follows.
(2) For $i=10,11$, $Q_i(r_2) = (S^3, L)$, and there is a torus $T'$
cutting $S^3$ into two solid tori $V_1, V_2$, such that $L_1 = V_1
\cap L$ is a $(2,1)$ knot, and $L_2 = V_2 \cap L$ is a Whitehead knot
in the solid torus $V_2$. Since both winding numbers of $L_j$ are
even, $T'$ lifts to two tori in $M_i(r_2)$. Let $W_i$ be the lifting
of $V_i$. A meridian disk of $V_1$ lifts to an annulus in $W_1$,
hence $W_1$ is a $T^2 \times I$ (not a twisted $I$-bundle over the
Klein bottle because $\partial W_1$ has two components). Let $T$ be the
core of this $T^2 \times I$. Then it cuts $M_i(r_2)$ into the
manifold $W_2$. We need to show that $W_2$ is Seifert fibered.
Let $D$ be a meridian disk of $V_2$ which intersects $L_2$ at two
points. Then $(V_2, L_2) = (B_1, L'_1) \cup (B_2, L'_2)$, where $B_1
= N(D)$, $B_2$ is the closure of $V_2 - B_1$, and $L'_k = L_2 \cap
B_k$. Note that each $L'_k$ is a trivial tangle in $B_k$, hence its
double branched cover $V'_k$ is a solid torus. One can check that
each component of $V'_1 \cap V'_2$ is a longitudinal annulus on $\partial
V'_1$, and it is an annulus on $\partial V'_2$ with winding number $2$ in
$V'_2$. Therefore $W_2 = V'_1 \cup V'_2$ is a Seifert fiber space
whose orbifold is the union of a $D^2$ and a $D^2(2)$ glued along two
boundary arcs. Since $W_2$ has two torus boundary components, the
orbifold must be an annulus with a single cone point of index $2$.
(3) Let $T$ be the essential torus in $M_i(r_j)$ given in the above
proof. Then it cuts $M_i(r_j)$ into one or two bounded Seifert fiber
spaces, which are irreducible. Since $T$ is incompressible,
$M_i(r_j)$ is also irreducible. The second statement follows from the
fact that the JSJ (Jaco-Shalen-Johannson) decomposition of an
irreducible closed 3-manifold is unique. \quad $\Box$ \medskip
A $(p',q')$ knot $K$ in a solid torus $V$ is also called a $0$-bridge
knot. In this case there is an essential annulus in $V - {\rm Int} N(K)$
with one boundary component in each of $\partial V$ and $\partial N(K)$. This
defines a longitude $l$ for $K$, which is unique if $K$ is not the
core of $V$. A $(p,q)$ cable of a 0-bridge knot $K$ is a knot on
$\partial N(K)$ which represents $pl + qm$ in $H_1(\partial N(K))$, where $m$
is a meridian of $K$. We refer the readers to [Ga1] for the definition
of a $1$-bridge braid in $V$.
\begin{lemma} Suppose $X$ is an irreducible,
$\partial$-irreducible, compact, orientable $3$-manifold with $\partial X =
T_1 \cup T_0$ a pair of tori. Let $r_1, r_2$ be distinct slopes on
$T_0$ such that $X(r_1)$, $X(r_2)$ are both $\partial$-reducible. Let
$K_a$ be the core of the Dehn filling solid torus in $X(r_a)$. Then
one of the following holds, up to relabeling of $r_i$.
(1) Each $X(r_a)$ is a solid torus, $K_a$ is a $0$- or $1$-bridge
braid in $X(r_a)$, and $\Delta(r_1, r_2) = 1$ if it is not a
$0$-bridge knot.
(2) $X(r_1)$ is a solid torus, and $X(r_2) = (S^1 \times D^2) \#
L(p,q)$ with $p\geq 2$. $K_1$ is a $(p,q)$ cable of a $(p',q')$ knot
in $X(r_1)$, and $r_2$ is the cabling slope of $K_1$ in $X(r_1)$.
Moreover, if $m_a$ is the slope on $T_1$ bounding a disk in $X(r_a)$,
then $\Delta(m_1, m_2) = pp'$. \end{lemma}
\noindent {\bf Proof.} If both $X(r_1), X(r_2)$ are irreducible then they are solid
tori and (1) holds by [Ga1, Theorem 1.1] and [Ga2, Lemma 3.2]. Now
assume $X(r_2)$ is reducible. Then by [Sch, Theorem 6.1] $K_1$ is a
$(p,q)$ cable of some knot $K'$ in $X(r_1)$ with respect to some
meridian-longitude pair $(m,l)$ of $K'$, and $r_2$ is the cabling
slope. In this case $X(r_2)$ is a connected sum $W_1 \cup L(p,q)$,
where $W_1$ is obtained by surgery on the knot $K'$ in $X(r_a)$ along
the cabling slope $r' = pl+qm$. Denote by $m'$ the meridian slope of
$K'$. Then $\Delta(m', r') = p > 1$.
Denote by $K'(s)$ the manifold obtained by $s$-surgery on $K'$ in $V =
X(r_1)$. The assumption on $X$ implies that $T_1$ is incompressible
in $V-K'$, and $V-K'$ is irreducible. By the above, $T_1$ is
compressible in both $K'(m') = V$ and $K'(r') = W_1$, and
$\Delta(m',r')>1$, hence by [Wu2, Theorem 1.1] and [CGLS, Theorem
2.4.3], either $V-{\rm Int} N(K') = T^2 \times I$, or there is an annulus
$A$ in $V-{\rm Int} N(K')$ with one boundary component on $T_1$ and another
boundary component a curve of slope $r$ on $T' = \partial N(K')$,
satisfying $\Delta(r, m') = \Delta(r, r') = 1$. In either case $K'$
is isotopic to a curve on $T_1$ and hence is a $0$-bridge knot. Since
$V-K$ is irreducible, this implies that $V$ is also irreducible.
Therefore $V = X(r_1)$ is a solid torus, $K'$ is a $(p',q')$ knot in
$V$ for some $(p',q')$, and $r$ is the cabling slope of $K'$ when
$p'>1$.
If $p'=1$, i.e.\ $V-{\rm Int} N(K') = T^2 \times I$, then $K$ is the
$(p,q)$ cable of the core $K'$ of $V$, and $r_2$ is the cabling slope
of $K$. We have $X(r_2) = V \# L(p,q)$, and the slope $m_2$ on $T_1$
which bounds a disk in $X(r_2)$ is the $(p,q)$ curve on $T_1$, hence
$\Delta(m_1, m_2) = p = pp'$.
Now assume $p' > 1$. Choose the longitude $l$ of $K'$ to be the
cabling slope $r$ of $K'$ given above. Since $r' = pr+qm$, the
equation $\Delta(r, r')=1$ above implies that $q=\pm 1$. Reversing
the orientation of $m'$ if necessary we may assume $q=1$. Hence $K$
is a $(p,1)$ cable of a $(p',q')$ knot in $V$, and $r_2$ is the
cabling slope. It is easy to see that the meridian slopes $m_a$ of
$X(r_a)$ satisfy $\Delta(m_1, m_2) = p p'$, and the result
follows. \quad $\Box$ \medskip
\begin{lemma} Let $i=6,...,13$. Let $\alpha$ be the covering
transformation of the double branched cover $M_i \to Q_i$.
(1) $M_i$ is irreducible, not Seifert fibered, and contains no
non-separating torus.
(2) If $M_i$ is not hyperbolic then it contains a separating essential
torus $T$ such that $T$ is $\alpha$-equivariant, and the component $W$
of $M_i|T$ which does not contain $T_0$ is either Seifert fibered or
hyperbolic. \end{lemma}
\noindent {\bf Proof.} (1) If $M_i$ is reducible then the summand which does not
contain $T_0$ is a summand of $M_i(r_j)$ for all $r_j$, but since
$M_i(r_3)$ is a lens space while $M_i(r_1)$ does not have a lens space
summand, this is impossible.
By Lemma 22.2 $M_i(r_3)$ is a lens space, so if $M_i$ is Seifert
fibered then the orbifold of $M_i$ is a disk with two cone points,
hence $M_i(r_1)$ is either a connected sum of two lens spaces or a
Seifert fibered space with orbifold a sphere with at most three cone
points. This is impossible because by Lemma 22.2 $M_i(r_1)$ is of
type $X(a_1, b_1; a_2, b_2)$ with some $a_i$ or $b_i$ greater than
$2$, which is irreducible and contains a separating essential torus,
at least one side of which is not an $I$-bundle.
If $M_i$ contains a non-separating torus then the lens space
$M_i(r_3)$ would contain a non-separating surface, which is absurd.
(2) If $M_i$ is non-hyperbolic then by (1) it has a non-trivial JSJ
decomposition. By [MeS] we may choose the JSJ decomposition surfaces
$F$ to be $\alpha$-equivariant. If we define a graph $G$ with the
components of $M_i|F$ as vertices and the components of $F$ as edges
connecting adjacent components of $M_i|F$, then the fact that $M_i$
contains no non-separating torus implies that $G$ is a tree. Let $T$
be a component of $F$ corresponding to an arc incident to a vertex $v$
of valence $1$ in $G$. Then $T$ bounds the manifold $W$ corresponding
to $v$, which by definition of the JSJ decomposition is either Seifert
fibered or hyperbolic. \quad $\Box$ \medskip
\begin{lemma} Suppose $M_i$ is non-hyperbolic and let $T$ be
the essential torus in $M_i$ given in Lemma 23.3(2). Let $X, W$ be
the components of $M_i | T$, where $X \supset T_0$. If $T$ is
compressible in $M_i(r_a)$ for some $a=1,2$, then both $X(r_a)$ and
$X(r_3)$ are solid tori, and $W$ is hyperbolic.
\end{lemma}
\noindent {\bf Proof.} $T$ is compressible in $X(r_3)$ because $M_i(r_3)$ is a lens
space. By assumption $T$ is also compressible in $M_i(r_a)$. Since
$M_i(r_a)$ contains no lens space summand, by Lemma 23.2 either both
$X(r_a)$ and $X(r_3)$ are solid tori, or $X(r_a)$ is a solid torus and
$X(r_3) = (S^1 \times D^2) \# L(p,q)$ for some $p>1$. We need to show
that the second case is impossible.
Let $m_j$ be the slope on $T$ which bounds a disk in $X(r_j)$, $j=a,
3$. Then $M_i(r_3) = W(m_3) \# L(p,q)$. Since $M_i(r_3)$ is a lens
space $L(p,q)$, we have $W(m_3) = S^3$, so $W$ is the exterior of a
knot in $S^3$. If $W$ is Seifert fibered then it is the exterior of a
torus knot, so $M_i(r_a) = W(m_a)$ is obtained by Dehn surgery on a
torus knot in $S^3$ and hence is atoroidal, which contradicts Lemma
23.1. Since by definition $W$ is Seifert fibered or hyperbolic, this
implies that $W$ is hyperbolic. Note that by Lemma 22.2 $p \geq 3$
for all $i \in \{6,...,13\}$. By Lemma 23.2(2) we have $\Delta(m_a,
m_3) \geq p \geq 3$. Since $W(m_a) = M_i(r_a)$ is toroidal and
$W(m_3) = S^3$, this is a contradiction to [GLu, Theorem 1.1], which
says that only integral or half integral surgeries on hyperbolic knots
in $S^3$ can produce toroidal manifolds. This completes the proof
that both $X(r_a)$ and $X(r_3)$ are solid tori.
If $W$ is not hyperbolic then by definition $W$ is Seifert fibered.
By the above $X(r_3)$ is a solid torus. Let $m_3$ be a meridian slope
of $X(r_3)$. Then $M_i(r_3) = W(m_3)$, so $M_i(r_3)$ being a lens
space implies that the orbifold of $W$ is a disk with two cone points,
in which case $M_i(r_a) = W(m_a)$ is either a connected sum of two
lens spaces or a Seifert fiber space with orbifold a sphere with at
most three cone points. In the first case $M_i(r_a)$ contains no
essential torus, while in the second case the only possible essential
torus in $M_i(r_a)$ is a horizontal torus cutting the manifold into a
$T^2 \times I$, or two twisted $I$-bundles over the Klein bottle.
This is a contradiction because by Lemma 23.1 $M_i(r_a)$ contains an
essential torus cutting it into either a Seifert fiber space with
orbifold an annulus with a cone point of index 2, or two Seifert fiber
spaces, at least one of which is not a twisted $I$-bundle over the
Klein bottle. \quad $\Box$ \medskip
\begin{lemma} The torus $T$ in Lemma 23.3(2) is incompressible
in both $M_i(r_1)$ and $M_i(r_2)$. \end{lemma}
\noindent {\bf Proof.} First assume that $T$ is compressible in both $X(r_1)$ and
$X(r_2)$. By Lemma 23.4 $X(r_j)$ is a solid torus for $j=1,2,3$.
Since $\Delta(r_1,r_2)>1$, by Lemma 23.2 we see that $X$ is the
exterior of a $(p,q)$ knot in a solid torus. Since $T$ is not
boundary parallel, $p>1$. Let $r$ be the cabling slope on $T_0$.
Since $X(r_j)$ is a solid torus, we have $r_j \neq r$. Therefore by
[CGLS, Theorem 2.4.3] we must have $\Delta(r, r_j) = 1$ for $i=1,2,3$.
By Lemma 23.2 one can check that $\Delta(r_1, r_3) = 1$ and
$\Delta(r_2,r_3) \leq 2$. Since $\Delta(r_1, r_2) \geq 4$, this is a
contradiction because any three slopes $r_1, r_2, r_3$ with distance
$1$ from a given slope $r$ have the property that $\Delta(r_a, r_b) +
\Delta(r_b, r_c) = \Delta(r_a, r_c)$ for some permutation $(a,b,c)$ of
$(1,2,3)$.
Now assume that $T$ is compressible in $M_i(r_1)$, say. By Lemma 23.4
$W$ is hyperbolic. On the other hand, by the above $T$ is
incompressible in $M_i(r_2)$, so $W$ is a submanifold in $M_i(r_2)$
bounded by an incompressible torus, hence by Lemma 23.1(3) it is
non-hyperbolic, which is a contradiction. \quad $\Box$ \medskip
\begin{lemma} Let $T(a_1, b_1; a_2, b_2) = (S^3, L)$, where
$a_i, b_i \geq 2$. If at least one of $a_1, b_1, a_2, b_2$ is
greater than $2$ then the exterior of $L$ is atoroidal, and there is
no M\"obius band $F$ in $S^3$ with $F \cap L$ a component of $L$.
\end{lemma}
\noindent {\bf Proof.} Denote by $T(a)$ a rational tangle with slope $1/a$, where $a$
is an integer. Given a tangle $\tau=(B^3, \tau)$, denote $B^3 - {\rm Int}
N(\tau)$ by $E$ or $E(\tau)$, and call it the tangle space of $\tau$.
Since $T(a)$ is a trivial tangle in the sense that $\tau$ is rel
$\partial$ isotopic to arcs on $\partial B^3$, the tangle space $E(a)$ is
atoroidal, and any incompressible annulus in $B^3-\tau$ is trivial in
the sense that it is either parallel to an annulus on $\partial
(B^3-\tau)$ or cuts off a $D^2 \times I$ in $B^3$ with $\tau \cap (D^2
\times I)$ a core arc.
The tangle space $E(r_1, r_2)$ of a Montesinos tangle $T(r_1, r_2)$ is
obtained by gluing $E(r_1), E(r_2)$ along a twice punctured disk $P =
E(r_1) \cap E(r_2)$. The above implies that $E(r_1, r_2)$ is always
atoroidal. If $A$ is an essential annulus in $E(r_1, r_2)$ with
minimal intersection with $P$, then an innermost circle outermost arc
argument shows that $A$ intersects $P$ in essential arcs or circles in
$A$. If the intersection is a set of circles then each component of
$A \cap E(r_i)$ is a set of trivial annuli, which implies that $A$ is
also trivial. If each component of $A \cap P$ is an essential arc
then each component of $A \cap E(r_i)$ is a bigon in the sense that it
is a disk intersecting $P$ in two arcs, which implies that $r_i = 2$
for $i=1,2$. Therefore $E(r_1, r_2)$ contains no essential annulus
unless $r_1 = r_2 = 1/2$ mod $1$.
By definition $T(a_1, b_1; a_2, b_2)$ is the union of two Montesinos
tangles $T(a_i, b_i)$. If the tangle space of $T(a_1, b_1; a_2, b_2)$
is toroidal then either one of the $T(a_i, b_i)$ is toroidal or they
are both annular. By the above neither case is possible if at least
one of the $a_1, b_1, a_2, b_2$ is greater than $2$.
The proof for a M\"obius band is similar. If $F$ is a M\"obius band
in $S^3$ bounded by a component of $L$ and has interior disjoint from
$L$ then after cutting along the surface $P_1 = E(a_1, b_1) \cap
E(a_2, b_2)$ it either lies in one of the $E(a_i, b_i)$ or intersects
each in bigons. One can show that the first case is impossible, and
in the second case $a_i = b_i = 2$ for $i=1,2$. \quad $\Box$ \medskip
\begin{lemma} $M_i$ is hyperbolic for $i=6$ or $8\leq i \leq 13$.
\end{lemma}
\noindent {\bf Proof.} Let $T$ be the $\alpha$-equivariant essential torus in $M_i$
given in Lemma 23.3(2). By Lemma 23.5 $T$ is incompressible in both
$M_i(r_a)$, $a=1,2$. Since $T$ is $\alpha$-equivariant, its image $F$
in $Q_i = M_i/\alpha$ is a 2-dimensional orbifold with zero orbifold
Euler characteristic (see [Sct] for definition), and all the cone
points have indices $2$. Hence it is $T^2$, $K^2$, $P^2(2,2)$,
$S^2(2,2,2,2)$, $A^2$, $M^2$, or $D^2(2,2)$, where the surfaces are
torus, Klein bottle, projective plane, sphere, annulus, M\"obius band
and disk, and the numbers indicate the indices of the cone points.
Note that in the last three cases the boundary of the surface is part
of the branch set of $\alpha$. Since $T$ is incompressible in
$M_i(r_a)$, $F$ is incompressible in $Q_i(r_a)$ in the sense that if
some simple loop on $F$ bounds a disk in $Q_i(r_a)$ intersecting the
branch set at most once then it bounds such a disk on $F$. We need to
show that for each type of surface above there is some $a=1,2$ such
that no such incompressible 2-dimensional orbifold exists in
$Q_i(r_a)$.
We have $Q_i = (B^3, K_i)$, where $B^3 = M_i/\alpha$ is a 3-ball and
$K_i$ is the branch set of $\alpha$. Since $F$ lies in $B^3$, it
cannot be $K^2$ or $P^2$. For all $i$ one can check that the branch
set $K_i$ of $\alpha$ in $Q_i$ contains at most one closed circle,
hence the case $A^2$ is also impossible.
By Lemma 22.2, $Q_i(r_1) = T(a_1, b_1; a_2, b_2)$ for some $a_j, b_j
\geq 2$, and $(a_j, b_j) \neq (2,2)$ for some $j$. Therefore by Lemma
23.6 we have $F \neq T^2, M^2$ for $i=6,...,13$ because there is no
such surfaces in $Q_i(r_1)$. It remains to show that $F \neq D^2(2,2)$
or $S^2(2,2,2,2)$.
For $i=13$, $Q_i(r_2) = (S^3, L)$, where $L$ consists of two parallel
copies of a trefoil knot $K$. Since each component of $L$ is
non-trivial in $S^3$, $F \neq D^2(2,2)$ in this case. Suppose $F =
S^2(2,2,2,2)$, and let $V$ be a regular neighborhood of the trefoil
knot containing $L$, intersecting $F$ minimally. Then $F \cap V \neq
\emptyset$, and $F$ is not contained in $V$ as otherwise one can show
that $F-L$ would be compressible. Therefore $F\cap V$ is a union of
two meridian disks, and $F \cap S^3 - {\rm Int} V$ is an essential annulus
in $S^3 - {\rm Int} V$. Since $S^3 - {\rm Int} V$ contains no essential annulus
with the meridian of $V$ as boundary slope, this is a contradiction.
The proofs for the cases $i \in \{6,8,9,10,11,12\}$ are similar. In
these cases $Q_i(r_2) = (S^3, L)$, and there is a torus $T'$ cutting
$S^3$ into two solid tori $V_1, V_2$, each containing some components
of $L$. One can check that no component $L'$ of $L$ bounds a disk
intersecting $L-L'$ at two points, so $F \neq D^2(2,2)$. If
$F=S^2(2,2,2,2)$ then either $F$ lies in one of the $V_j$, or it
intersects one of the $V_j$ in two meridional disks and the other
$V_k$ in an essential annulus with boundary slope the meridional slope
of $V_j$. Neither case is possible for the $Q_i(r_2)$ listed in Lemma
22.2. \quad $\Box$ \medskip
\begin{lemma} $M_7$ is hyperbolic.
\end{lemma}
\noindent {\bf Proof.} By Lemma 22.2 we have $M_7(r_1) \in X(3,3;2,3)$ and $M_7(r_2)
\in X(2,2;2,3)$. Consider the tangle decomposition sphere $P_a$ of
the orbifold $Q_7(r_a)$, $a=1,2$, which corresponds to a horizontal
plane in Figure 22.7(b), (d) respectively. It lifts to an essential
torus $T_a$ in $M_7(r_a)$.
Each side of $P_a$ is a Montesinos tangle of type $T(r_1, r_2)$, which
is the sum of two rational tangles over a disk $D$. The boundary of
$D$ determines the fibration of the double branched cover $X(r_1,
r_2)$ of $T(r_1, r_2)$, which has a unique Seifert fibration unless
$r_1 = r_2 = 2$, in which case the closed circle in the tangle is
isotopic (without crossing the arcs) to a curve on the punctured
sphere, which lifts to a fiber in the other fibration of $X(r_1,r_2)$.
It is easy to check from Figures 22.7(b) and (d) that the fiber curves
from the two sides of $P_a$ do not match, so $M_7(r_a)$ is not a
Seifert fiber space. Since each side of $T_a$ is a small Seifert
fiber space with orbifold a disk with two cone points, it follows that
$M_7(r_a)$ contains no other essential torus.
Suppose $M_7$ is non-hyperbolic and let $T$ be the essential torus in
$M_7$ given by Lemma 23.3. By Lemma 23.5 it is incompressible in both
$M_7(r_a)$, therefore by the uniqueness of $T_a$ above we see that $T
= T_a$ in $M_7(r_a)$ up to isotopy. As before, denote by $W$ and $X$
the components of $M_7|T$, with $X \supset T_0$. Then $W$ is the
manifold on one side of $T_a$ in $M_7(r_a)$. Therefore we must have
$W = X(2,3)$, so $X(r_1) = X(3,3)$ and $X(r_2) = X(2,2)$. We will
show that this is impossible.
Let $Y$ be the component of the JSJ decomposition of $X$ that contains
$T$. Then $Y$ is either hyperbolic or Seifert fibered. There are
three cases.
Case 1. {\it $T_0 \subset \partial Y$ and $Y$ is Seifert fibered.\/} By
Lemma 23.5 $T$ is incompressible in $Y(r_a)$ for $a=1,2$, so $r_a$ is
not the fiber slope on $T_0$. Hence the Seifert fibration extends
over $Y(r_1)$ and $Y(r_2)$. In this case $\partial Y - T_0$ is
incompressible in $Y(r_a)$. Since $X(r_a)$ is atoroidal, either
$Y(r_1) \cong Y(r_2) \cong T^2 \times I$, or $Y = X$. In the first
case we have $X(r_1) = X(r_2)$, which is a contradiction because
$X(r_1) = X(3,3) \not \cong X(2,2) = X(r_2)$. In the second case,
Since $X(r_1)$ has orbifold $D^2(3,3)$, the orbifold of $X$ must be
$A^2(3,3)$ or $A^2(3)$. On the other hand, since $X(r_2)$ has
orbifold $D^2(2,2)$, the orbifold of $X$ must be $A^2(2,2)$ or
$A^2(2)$, which contradicts the fact that Seifert fibrations
for these manifolds are unique.
Case 2. {\it $T_0 \subset \partial Y$ and $Y$ is hyperbolic.\/} If $\partial
Y$ has more than two boundary components then the fact that $X(r_a)$
is atoroidal implies that $Y(r_a)$ is either $\partial$-reducible, or a
$T^2 \times I$. If $\partial Y = T \cup T_0$ then $Y=X$ and by assumption
both $Y(r_a) = X(r_a)$ are annular and atoroidal. In either case
$Y(r_a)$ is either $\partial$-reducible or annular and atoroidal. Since
$\Delta(r_1, r_2) = 5$ and $Y$ is hyperbolic, this is a contradiction
to [Wu2, Theorem 1.1] if both $Y(r_a)$ are $\partial$-reducible, to [GW2]
if one of them is $\partial$-reducible and the other is annular, and to
[GW3] if both $Y(r_a)$ are annular and atoroidal. (The main theorem
of [GW3] said that if $\Delta = 5$ then $Y$ and $r_1, r_2$ are listed
in one of the three possibilities in [GW3, Theorem 1.1], but in that
case both $Y(r_a)$ are toroidal.)
Case 3. {\it $T_0 \not \subset \partial Y$.\/} Let $X_1$ be the component
of $X|(\partial Y)$ containing $T_0$, and let $X_2$ be the closure of $X -
X_1$, so $X = X_1 \cup X_2$. Since $M_i$ contains no non-separating
torus (Lemma 23.3), $T_1 = X_1 \cap X_2$ is a single torus. Since
$X(r_a)$ is atoroidal, $T_1$ must be compressible in $X_1(r_a)$ for
$a=1,2$. Thus Lemma 23.2 and the fact that $X(r_a)$ contains no lens
space summand implies that $X_1(r_a)$ is a solid torus for $a=1,2$, as
in Lemma 23.2(1); moreover, since $\Delta(r_1, r_2) > 1$, by Lemma
23.2(1) $X_1$ is a $(p,q)$ cable space, and by [CGLS, Theorem 2.4.3]
$\Delta(r, r_a) = 1$ for $r$ the cabling slope. It is easy to see
that the meridian slopes $m_a$ of $X_1(r_a)$ satisfies $\Delta(m_1,
m_2) = p \Delta(r_1,r_2) \geq 5$. Now $X(r_a) = X_2(m_a)$, so by
Cases 1 and 2 above applied to $X_2$ we see that this case is also
impossible. \quad $\Box$ \medskip
Denote by $c\cdot d$ the minimal intersection number between the two
isotopy classes of simple closed curves on a surface represented by
$c$ and $d$, respectively. If $\varphi: F\to F$ is a homeomorphism
and $K$ a curve on the surface $F$, then $K$ is said to be {\it
$\varphi$-full\/} if for any essential curve $c$ on $F$ there is
some $i$ such that $c \cdot \varphi^i(K) \neq 0$.
If $K$ is a knot in a 3-manifold $Y$ with a preferred
meridian-longitude, denote by $Y(K,p/q)$ the manifold obtained from
$Y$ by $p/q$ surgery on $K$. Let $X = F \times I/\psi$ be an
$F$-bundle over $S^1$ with gluing map $\psi$, let $F_t = F \times t$,
$t \in I = [0,1]$, and let $K$ be an essential curve on $F_{1/2}$.
Then there is a preferred meridian-longitude pair $(m,l)$ on $\partial
N(K)$, with $l$ the slope of $F_{1/2} \cap \partial N(K)$.
\begin{lemma} Let $X = F \times I/\psi$. Let $\eta: F\times I \to
F_0$ be the projection, $\varphi = \eta\circ \psi$, and $K$ an
essential curve on $F_{1/2}$. If $\eta(K) \subset F_0$ is
$\varphi$-full and $q>1$, then $X(K, p/q)$ is hyperbolic.
\end{lemma}
\noindent {\bf Proof.} Let $A_i$ be an annulus in $X$ with $\partial A_i = K \cup K_i$,
where $i=0,1$ and $K_i \subset F_i$. Let $V_i$ be a regular
neighborhood of $A_i$. Put $Y = F \times I$. Then $Y = V_1 \cup W$,
where $W$ is homeomorphic to $F \times I$, and $V_1 \cap W$ is an
annulus $A'$. After $p/q$ surgery on $K$ we have $Y(K, p/q) = V_1(K,
p/q) \cup W$. Note that $V(K, p/q)$ is a solid torus with $A'$ an
annulus on $\partial V_1(K, p/q)$ running $q$ times along the longitude.
By an innermost circle outermost arc argument one can show that $Y(K,
p/q)$ is irreducible, $\partial$-irreducible, atoroidal, and any essential
annulus $A_2$ can be isotoped to be disjoint from $K_1$, i.e.\ $\partial
A_2 \cdot K_1 = 0$. Moreover, if $A_2$ has at least one boundary
component on $F_1$ then $A_2$ is either vertical in the sense
that it is isotopic to $c \times I \subset F\times I$ for some curve
$c \subset F$, or isotopic to $A'$ and hence has both boundary curves
parallel to $K_1$. Similarly, using $A_0$ and $V_0$ one can show that
$\partial A_2 \cdot K_0 = 0$, and if $A_2$ is not vertical and $A_2 \cap
F_0 \neq \emptyset$ then it has both boundary curves parallel to
$K_0$.
The above facts imply that $X(K, p/q) = Y(K, p/q)/\psi$ is
irreducible. Since the non-separating surface $F_0$ cuts $X(K, p/q)$
into $Y(K, p/q)$, which is not an $I$-bundle, we see that $X(K, p/q)$
is not Seifert fibered. It remains to show that $X(K, p/q)$ is
atoroidal.
If $T$ is an essential torus in $X(K, p/q)$ then it can be isotoped so
that $T \cap Y(K, p/q) = Q$ is a set of essential annuli. Let
$C_i = Q \cap F_i$. We claim that for any curve $c \subset
C_0$, $\varphi(c)$ is isotopic to a curve in $C_0$.
We have $\psi(C_0) = C_1$, so $\psi(c) \subset C_1$. If $\psi(c)$
belongs to a vertical annulus $Q_1$ then $\varphi(c) = \eta(\psi(c))
\cong Q_1 \cap F_0 \subset C_0$. If $\psi(c)$ belongs to a
non-vertical annulus then by the property proved above, $\psi(c)$ is
isotopic to $K_1$, so $\varphi(c) \cong \eta(K_1) = K_0$. Note that
if $Q$ has a non-vertical component with boundary on $F_1$ then the
fact that $C_0, C_1$ have the same number of components implies that
there is also a non-vertical component $Q_0$ with boundary on $F_0$,
and we have shown that each component $c'$ of $\partial Q_0$ is isotopic
to $K_0$, so $\varphi(c) \cong c' \subset \partial Q_0 \subset C_0$. This
completes the proof of the claim.
Let $c$ be a component of $C_0$. We have shown above that $c \cdot
K_0 = 0$ for any $c \subset C_0$. Applying the above to
$\varphi^{-1}$ we see that there is a curve $c' \subset C_0$ such that
$\varphi(c') \cong c$. By induction we have $c \cdot \varphi^i(K_0) =
c' \cdot \varphi^{i-1}(K_0) = 0$ for all $i$, which is a contradiction
to the assumption that $K_0 \cong \eta(K)$ is $\varphi$-full and hence
$c \cdot \varphi^i(K_0) \neq 0$ for some $i$. \quad $\Box$ \medskip
\begin{lemma} The manifold $M_5$ is hyperbolic.
\end{lemma}
\noindent {\bf Proof.} Let $W = M_5 | F_b$, let $F_+, F_-$ be the two copies of $F_b$
in $W$, and let $A$ be the annulus $T_0 | \partial F_b$. Then $W$ is
obtained from $Y = F_+ \cup F_- \cup A$ by attaching faces of $\Gamma_a$
and then some 3-cells.
Two faces of $\Gamma_a$ are {\it parallel\/} if their boundary curves are
parallel on $Y$. Since parallel faces cobound a 3-cell in $W$, we
need only attach one such face among a set of parallel faces. From
Figure 11.10 one can check that the four bigons are parallel faces,
and the two 6-gons are parallel to each other. Therefore $W$ is
obtained from $Y$ by attaching one bigon $\sigma_1$, one 6-gon
$\sigma_2$ and then a 3-cell. Let $\sigma_1$ be the bigon on Figure
11.10 between the edges $B$ and $G$, and assume that the edge $B
\subset F_+$.
Cutting $W$ along $\sigma_1$, we obtain a manifold $W_1$ with boundary
a torus, and it contains the $6$-gon $\sigma_2$. Therefore it is a
solid torus such that the remnant of $F_+$, denoted by $F'_+$, runs
along the longitude three times. If we replace $\sigma_2$ and the
attached 3-cell by a solid torus $J$ with meridian intersecting $F'_+$
in one essential arc then $W_1$ becomes a $F'_+ \times I$ and $W$
becomes $X = F_+ \times I$. Therefore $W = X(K, p/q)$, where $K$ is
the core of $J$, and $q=3$. Let $\psi: F_- \to F_+$ be the gluing
map, $\eta: F_+ \times I \to F_-$ the projection, and $\varphi = \eta
\circ \psi$. By Lemma 23.9 we need only show that the curve $K_-$ on
$F_-$ isotopic to $K$ is $\varphi$-full.
In $M_5$ the bigon $\sigma_1$ has boundary edges $B \cup G$ on $F_b$,
as shown in Figure 11.10(b). Suppose $B \subset F_+$ and $G\subset
F_-$ when we consider $\sigma_1$ as a bigon in $F_+\times I$. Then
$\psi$ maps the curve $B$ on $F_-$ to the curve $B$ on $F_+$, which is
mapped to $G$ on $F_-$ by $\eta$. Therefore $\varphi: F_- \to F_-$
maps $B$ to $G$. Since $F_- | B$ is an annulus and $B$ is disjoint
from the curve $K_-$ above, this determines $K_-$. Also
$\varphi(K_-)$ is the curve on $F_-$ disjoint from $\varphi(B) = G$,
so $K_-$ intersects $\varphi(K_-)$ transversely at a single point,
cutting $F_-$ into an annulus. Therefore $K_-$ is $\varphi$-full, and
the result follows. \quad $\Box$ \medskip
\begin{lemma} The manifold $M_4$ is hyperbolic.
\end{lemma}
\noindent {\bf Proof.} The proof is similar to that of Lemma 23.10. In this case $W =
M_4 | F_b$ is obtained from $F_+ \cup F_- \cup A_1 \cup A_2$ by
attaching two bigons $\sigma_1, \sigma_2$ and one $4$-gon $\sigma_3$,
so $W = X(K,p/q)$ with $q=2$, where $X = F_+ \times I$ and $K$ is
disjoint from $\sigma_1, \sigma_2$. Choose $\sigma_1, \sigma_2$ to be
the bigons in Figure 11.9(a) bounded by $H\cup E$ and $E\cup N$,
respectively. Then $F_-$ can be identified with $F_b$, and $\sigma_1,
\sigma_2$ intersects $F_-$ in the edges $E$ and $N$, respectively.
These cut $F_-$ into an annulus containing the curve $K_-$ isotopic to
the knot $K$ in $X = F_+ \times I$. The map $\varphi: F_- \to F_-$
maps the edges $E$ and $N$ in Figure 11.9(b) to $H$ and $E$,
respectively, so $\varphi(K_-)$ is the curve in the annulus $F_- |
(H\cup E)$. The curves $K_-$ and $\varphi(K_-)$ intersect transversely
at a single point, cutting $F_-$ into a neighborhood of $\partial F_-$,
hence $K_-$ is $\varphi$-full, and $M_4$ is hyperbolic by Lemma 23.9.
\quad $\Box$ \medskip
\begin{lemma} Let $F$ be a closed orientable surface of genus 2, and
let $\alpha, \beta$ be two non-separating simple closed curves on
$F$, intersecting minimally, cutting $F$ into disks. Let $X$ be
obtained from $F\times I$ ($I=[0,1]$) by attaching a 2-handle along
$\alpha \times \{0\}$. Identify $F$ with $F \times \{1\} \subset
F\times I$, and let $T = \partial X - F$. Then
(1) a compressing disk $D$ of $F$ intersects $\beta$ at least 3
times; and
(2) an incompressible annulus $A$ in $X$ with $\partial A \subset F$ and
$\partial A \cap \beta = \emptyset$ is boundary parallel.
\end{lemma}
\noindent {\bf Proof.} (1) Let $E$ be the disk in $X$ bounded by $\alpha \times 1$,
cutting $X$ into $X' = T \times I$. Note that $X$ is a compression
body, and $\{E \}$ is the unique (up to isotopy) complete disk system
for $X$.
By assumption $\alpha \cup \beta$ cuts $F$ into disks, hence $|\alpha
\cap \beta| \geq 3$. Since $\alpha$ intersects $\beta$ minimally, we
may choose a hyperbolic structure on the surface $F$ so that $\alpha,
\beta$ are geodesics. Let $D$ be a compressing disk for $F$ in $X$.
Up to isotopy we may assume that $\gamma = \partial D$ is a geodesic or a
slight push off of a geodesic if it is isotopic to $\alpha$ or
$\beta$. Then both $|\gamma \cap \alpha|$ and $|\gamma \cap \beta|$
are minimal up to isotopy.
We may assume that $D \cap E$ consists of arcs. If $D \cap E \neq
\emptyset$, by taking an arc that is outermost on $D$, surgering $E$
along the corresponding outermost disk in $D$, and discarding one of
the resulting components, we get a new disk $E'$ having fewer
intersections with $D$, such that $\{E'\}$ is a complete disk system
for $X$. Since $\{E\}$ is the unique complete disk system for $X$,
$E'$ is isotopic to $E$. Since $|\partial E' \cap \partial D| < |\partial E \cap
\partial D| = |\alpha \cap \gamma|$ and $\partial E'$ is isotopic to $\partial E$,
this is a contradiction to the fact that $|\gamma \cap \alpha|$ is
minimal up to isotopy. Therefore $D\cap E = \emptyset$. Hence $D$
either (a) is parallel to $E$, or (b) cuts off a solid torus
containing $E$.
In case (a) $\partial D$ is a parallel copy of $\alpha$, so $|\partial D \cap
\beta| = |\alpha \cap \beta| \geq 3$ and we are done. In case (b),
let $F_1$ be the punctured torus on $F$ bounded by $\gamma$ which does
not contain $\alpha$. If $|\gamma \cap \beta| \leq 2$ then $F_1$
contains at most one arc of $\beta$, so it contains an essential loop
disjoint from $\alpha \cup \beta$, which is a contradiction to the
assumption that $\alpha \cup \beta$ cuts $F$ into disks.
(2) Let $A$ be an incompressible annulus in $X$ with $\partial A \subset F
- \beta$. We may assume that $\alpha, \beta$ are hyperbolic
geodesics, and each component of $\partial A$ is either a geodesic, or a
slight push off of a geodesic if it is parallel to $\alpha$, $\beta$
or another component of $\partial A$. Thus both $|\partial A \cap \alpha|$
and $|\partial A \cap \beta|$ are minimal; in particular, $\partial A \cap
\beta = \emptyset$. As in (1), this implies that $A \cap E$ consists
of essential arcs on $A$. If $A \cap E = \emptyset$ then $\partial A$
lies in $F |(\alpha \cup \beta)$, but since $A$ is incompressible
while each component of $F |(\alpha \cup \beta)$ is a disk, this is
impossible. Therefore we may assume that $A \cap E$ is a non-empty
set $C$ of essential arcs on $A$.
Let $B_i$ be a component of $A|C$. Then $B_i$ is a disk in $X' =
T\times I$, so $\partial B_i$ is a trivial loop on $T'$, bounding a disk
$B'_i$ on $T'$. Let $E_1, E_2$ be the two copies of $E$ on $T'$. If
$B'_i \cap (E_1 \cup E_2)$ is a single disk then one can use a disk
component of $B'_i \cap F$ to isotope $A$ to reduce $|\partial A \cap \partial
E| = |\partial A \cap \alpha|$, which is a contradiction to the minimality
of $|\partial A \cap \alpha|$. Therefore $B'_i \cap (E_1 \cup E_2)$
consists of two disks, and $B' \cap F$ is a single disk $P_i$. One
can check that $\cup P_i$ is an annulus on $F$ parallel to $A$. \quad $\Box$ \medskip
\begin{lemma}
The manifold $M_{14}$ is hyperbolic.
\end{lemma}
\noindent {\bf Proof.} Cutting $M_{14}$ along the surface $F_b$, we obtain two
manifolds $X_1, X_2$, where $X_1$ is the one containing the four bigon
faces of $F_a$, and $X_2$ contains the two 4-gon faces of $F_a$. Let
$\sigma_1, \sigma_2$ be the bigons on $F_a$ bounded by the edges
$E\cup F$ and $B\cup Y$ respectively in Figure 20.6, and let
$\sigma_3$ be the 4-gon bounded by the edges $B \cup C \cup Y \cup F$.
Note that any other face of $F_a$ is parallel in $X_i$ to one of
these.
Let $A_i = X_i \cap T_0$. Then $X_1$ is obtained from the genus 2
surface $F_b \cup A_1$ by attaching $\sigma_1, \sigma_2$ and then a
3-cell, hence it is a handlebody of genus 2 because $\partial \sigma_1,
\partial \sigma_2$ are disjoint nonparallel nonseparating curves on
$F_b\cup A_1$. The core of $A_1$ is a curve on $\partial X_1$ such that
after attaching a 2-handle to $X_1$ along $A_1$ we get the manifold on
the side of $\hat F_b$ which contains no torus boundary component,
hence from Figure 22.14(b) we see that it is the double branched cover
of a Montesinos tangle $T(2,2)$, which is a twisted $I$-bundle over
the Klein bottle. This implies that the surface $F_b = \partial X_1 -
A_1$ is incompressible in $X_1$.
Now consider $X_2$. Let $F$ be the genus 2 surface $F_b \cup A_2$,
$\alpha$ the boundary of $\sigma_3$, and $\beta$ the core of $A_2$.
Then $\alpha$ intersects $\beta$ minimally at four points. From
Figure 20.6(b) we see that the edges $B,C,Y,F$ cut the surface $F_b$
into two disks, hence $\alpha \cup \beta$ cuts $F$ into disks. $X_2$
is obtained from $F\times I$ by attaching a 2-handle along the curve
$\alpha \times \{0\}$. Therefore it satisfies the conditions of Lemma
23.12. In particular, $F_b$ is incompressible in $X_2$.
Since $X_1$ is a handlebody and $X_2$ is a compression body, they are
irreducible and atoroidal. Since $M_{14}$ is obtained by gluing $X_1,
X_2$ along the incompressible surface $F_b$, $M_{14}$ is also
irreducible. It is well known that an incompressible surface in a
Seifert fiber space is either vertical, and therefore an annulus or
torus, or horizontal, in which case it intersects all boundary
components. Since the surface $F_b$ satisfies neither condition, we
see that $M_{14}$ is not Seifert fibered. It remains to show that
$M_{14}$ is atoroidal.
Assume $M_{14}$ is toroidal and let $T_1$ be an essential torus in
$M_{14}$ intersecting $F_b$ minimally. Since $X_i$ is atoroidal,
$T_1$ intersects $X_i$ in incompressible annuli. A component $A'_2$
of $T_1 \cap X_2$ is an incompressible annulus in $X_2$ disjoint from
$\beta$, hence by Lemma 23.12 it is parallel to an annulus $A''$ on
$\partial X_2$. If $A'' \subset F_b$ then $T_1$ can be isotoped to reduce
$|T_1 \cap F_b|$, which is a contradiction to the minimality
assumption. Therefore $A'' \supset \beta$ and hence $A'' \supset
A_2$, so each component of $\partial A''$ is parallel to a component of
$\partial F_b$. Since this is true for all components of $T_1 \cap X_2$,
we see that each component of $T_1 \cap F_b$ is parallel to a
component of $\partial F_b$.
Now let $A'_1$ be a component of $T_1 \cap X_1$. By the above, the
two boundary components of $A'_1$ are parallel on $\partial X_1$. Since
$X_1$ is a handlebody, $A'_1$ is parallel to an annulus $A''_2$ on
$\partial X_1$. For the same reason as above, it must contain the annulus
$A_1$. This is true for all components of $T_1 \cap X_1$. Let $A'_i$
be a component of $T_1 \cap X_i$ which is closest to $A_i$. Then
$\partial A'_1 = \partial A'_2$, hence $T_1 = A'_1 \cup A'_2$. It follows
that $T_1$ is parallel to $T_0$, contradicting the assumption that
$T_1$ is essential in $M_{14}$. \quad $\Box$ \medskip
\begin{thm} The manifolds $M_i$ in Definition 21.3 are all
hyperbolic. \end{thm}
\noindent {\bf Proof.} This follows from [GW1, Theorem 1.1] for $i=1,2,3$, and from
Lemmas 23.7, 23.8, 23.10, 23.11 and 23.13 for $i>3$. \quad $\Box$ \medskip
\section {Toroidal surgery on knots in $S^3$}
Recall that each of the manifolds $M_1, M_2, M_3$ admits two toroidal
Dehn fillings $r'_i, r''_i$ on a torus boundary component $T_0$ with
distance $4$ or $5$. These are the exteriors of the links $L_1, L_2,
L_3$ in Figure 24.1. Let $L_i = K'_i \cup K''_i$, where $K'_i$ is the
left component of $L_i$. Let $T_1 = \partial N(K'_i)$, and $T_0 = \partial
N(K''_i)$.
\bigskip
\leavevmode
\centerline{\epsfbox{Figure24.1.eps}}
\bigskip
\centerline{Figure 24.1}
\bigskip
Each $M_i$ has a pair of toroidal slopes $r'_i, r''_i$ on $T_0$.
These are given in [GW1, Theorem 7.5] and shown in Figures 7.2, 7.4
and 7.5 of [GW1].
\begin{lemma}
With respect to the preferred meridian-longitude pair of $K''_i$,
the slopes $r'_i, r''_i$ are given as follows, up to relabeling.
(1) $r'_1 = 0$ and $r''_1 = 4$.
(2) $r'_2 = -2$ and $r''_2 = 2$.
(3) $r'_3 = -9$ and $r''_3 = -23/2$.
\end{lemma}
\noindent {\bf Proof.} (1) This is basically proved in [GW1, Lemma 7.1]. It was shown
that $M_1$ is the double branched cover of the tangle $Q_1$ in Figure
7.2(c) of [GW1]. Let $m$ be the meridian, $l$ the preferred
longitude, and $l'$ the blackboard longitude of the diagram of $K''_1$
in [GW1, Figure 7.2(a)]. Calculating the linking number of $l'$ with
$K''_1$ in Figure 7.2(a) we see that $l' = 2m + l$. Let $\eta: M_1
\to Q_1$ be the branched covering map. If $r$ is a slope on $T_0$
then $\eta(r)$ is a curve of a certain slope on the inside boundary
sphere, which will be denoted by a number in $\Bbb Q_0 = \Bbb Q \cup
\{\infty\}$. One can check that $\eta(m) = 0/1$, and $\eta(l') =
1/0$. The two toroidal slopes $r'_1, r''_1$ map to slopes $-1/2$ and
$1/2$, as shown in Figure 7.2(d) and (e) of [GW1]. We have
$\varphi(-2m+l') = (-2\times 0 + 1)/(-2 \times 1 + 0) = -1/2$, and
$\varphi(2m+l') = (2\times 0 + 1)/(2 \times 1 + 0) = 1/2$. Therefore
$r'_1 = -2m + l' = -2m + (2m + l) = l$ and $r''_1 = 2m + l' = 4m + l$.
(2) This is similar to (1), using [GW1, Figure 7.4] instead. We have
$\eta(m) = 0/1$, $\eta(l') = 1/2$, $\eta(r'_2) = 1/0$, $\eta(r''_2) =
1/4$, and $l = l'$. Therefore $r'_2 = -2m + l$ and $r''_2 = 2m + l$.
(3) Use 7.5(k) to denote [GW1, Figure 7.5(k)]. 7.5(a) shows that
$l' = l - 6m$. A careful tracking of $l'$ during the modification
from 7.5(a) to 7.5(b) then to 7.5(c) shows that $\eta(m) = 0/1$ and
$\eta(l') = 1/3$. From 7.5(c) and 7.5(e) we see that $\eta(r'_3) =
1/0$ and $\eta(r''_3) = 2/5$. Therefore $r'_3 = -3m + l' = -9m + l$,
and $r''_3 = -m + 2l' = -m + 2(l-6m) = -13m + 2l$.
\quad $\Box$ \medskip
\begin{lemma}
Suppose $K$ is a hyperbolic knot in $S^3$ admitting two toroidal
Dehn surgeries $K(r_1), K(r_2)$ with $\Delta(r_1, r_2) = 4$ or $5$.
Then there is an $i \in \{1,2,3\}$ and a slope $s$ on $T_1$ of
$M_i$ such that $(E(K), r_1, r_2) \cong (M_i(s), r'_i, r''_i)$.
\end{lemma}
\noindent {\bf Proof.} Let $F_a$ be an essential punctured torus in $M_K = S^3 - {\rm Int}
N(K)$ such that $\hat F_a$ is an essential torus in $K(r_a)$, chosen
so that $|\partial F_a|$ is minimal. By Theorem 21.4 the triple $(E(K),
r_1, r_2)$ is equivalent to either $(M_i, r'_i, r''_i)$ with $1\leq
i\leq 14$, or to $(M_i(s), r'_i, r''_i)$ for some $i=1,2,3,14$.
Therefore we need only show that the manifold $M_i$ ($i=4,...,14$) is
not the exterior of a knot or link in $S^3$.
When $i=4$, the surface $F_b$ has two boundary circles on $T_0$ with
the same orientation. Let $A$ be an annulus on $T_0$ connecting these
two boundary components. Then $F_b \cup A$ is a non-orientable closed
surface in $M_4$. It follows that $M_4$ cannot be the exterior of a
knot in $S^3$ because $S^3$ contains no embedded non-orientable
surface.
For $i=5$, let $V_b$ be the Dehn filling solid torus of $M_5(r_b)$.
Then the $\Bbb Z_2$ homology group $H$ of $V_b \cup F_b$ is generated
by $\alpha$, $x$ and $y$, where $\alpha$ is the core of $V_b$, and $x,
y$ are represented by the edges $A$ and $E$ in Figure 11.10(b),
respectively. A bigon in Figure 11.10(a) gives the relation $x=y$.
Consider the quotient group $H'$ obtained from $H$ by identifying $x$
with $y$. Then $H' = \Bbb Z_2 \oplus \Bbb Z_2$ is generated by $\alpha$
and $x$. Each corner of Figure 11.10(a) represents the element
$\alpha$, and each edge represents $x$ in $H'$. Since each face in
Figure 11.10(a) has an even number of edges and an even number of
corners on its boundary, it represents $0$ in $H'$. Therefore
$$H_1(M_5(r_b), \Bbb Z_2) = H_1 (V_b \cup F_b \cup F_a, \Bbb Z_2) = H' =
\Bbb Z_2 \oplus \Bbb Z_2.$$
Since the $\Bbb Z_2$ homology of any manifold obtained by Dehn surgery
on a knot in $S^3$ is either trivial or $\Bbb Z_2$, it follows that
$M_5$ is not a knot exterior.
Now assume that $M_i$ is the exterior of a knot $K$ in $S^3$ for some
$i\geq 6$. By Theorem 23.14 $K$ is hyperbolic. Put $\Bbb Q_0 = \Bbb
Q \cup \{\infty \}$. A number in $\Bbb Q_0$ is represented by $p/q$,
where $p,q$ are coprime integers, and $q\geq 0$. Given a
meridian-longitude pair $(m,l)$ and $r = p/q \in \Bbb Q_0$, denote by
$K(r)$ the manifold obtained by surgery on $K$ along the slope
$pm+ql$. There is a one to one correspondence $\eta: \Bbb Q_0 \to
\Bbb Q_0$ such that $\Delta(\eta(r), \eta(s)) = \Delta(r, s)$,
and $K(\eta(r))$ is the double branched cover of $Q_i(r)$, which is
the manifold $X_i(r)$ given in Lemma 22.2. Since $K(\eta(r_3))$ is a
lens space, by the Cyclic Surgery Theorem [CGLS, p.237] the slope
$\eta(r_3)$ is an integer slope with respect to the preferred
meridian-longitude of $K$. To simplify the calculation, let $l =
\eta(r_3)$.
By [GLu, Theorem 1.1] $\eta(r_1)$ and $\eta(r_2)$ are integer or half
integer slopes. Suppose $\eta(r_i) = p_i/q_i$. Then $p_3/q_3 = 0/1$.
By the above we have $q_i = 1$ or $2$ for $i=1,2$. By Lemma 22.2,
$|p_1| = \Delta(r_1, r_3) = 1$, and $|p_2| = \Delta(r_2, r_3) \leq 2$.
If $|p_2| = 1$ then $4 \leq \Delta(r_1, r_2) = \Delta(\eta(r_1),
\eta(r_2)) = |p_1q_2 - p_2q_1|$ implies that $q_1 = q_2 = 2$. This is
a contradiction to [GWZ, Theorem 1], which says that a hyperbolic knot
in $S^3$ admits at most one non-integral toroidal surgery.
We now have $|p_2|=2$, so $\eta(r_2) = p_2/q_2 = \pm 2/1$. Since
$\Delta(r_1, r_2) = |p_1q_2 - p_2q_1| = |\pm 1 - (\pm 2)q_1| \geq 4$,
we must have $p_1/q_1 = \mp 1/2$, and $\Delta = 5$. From Lemma 22.2
we see that for $i\in \{6,...,13\}$, the only $M_i$ satisfying
$\Delta(r_2, r_3) = 2$ and $\Delta(r_1, r_2)=5$ are the ones with
$i=7$, $10$ or $11$.
Consider the case $i=10$. Let $r_0$ be the slope such that
$\eta(r_0)$ is the meridian slope $1/0$. Then we have $\Delta(r_0,
r_i) = \Delta(\eta(r_0), \eta(r_i)) = \Delta(1/0, p_i/q_i) = q_i$.
Therefore by the above we have $\Delta(r_0, r_i) = q_i = 2, 1, 1$ for
$i=1,2,3$, respectively. By Lemma 22.2 we have $r_1 = 0/1$, $r_2 =
-5/2$ and $r_3 = 1/0$. Let $r_0 = p'/q'$. Then we have
\begin{eqnarray*}
& & \Delta(r_0, r_1) = |p'| = 2 \\
& & \Delta(r_0, r_2) = |2p'+5q'| = 1 \\
& & \Delta(r_0, r_3) = q' = 1
\end{eqnarray*}
These equations have a unique solution $r_0 = -2/1$. One can check
that $Q_{10}(-2/1)$ is the 2-bridge knot $K_{2/7}$, so its double
branched cover is $L(7,2) \neq S^3$, which is a contradiction.
The tangles $Q_7$, $Q_{11}$ and $Q_{14}$ in Figures 22.7, 22.11 and
22.14 have a circle component. If $M_i$ is a knot exterior in
$S^3$ then there is a slope $r$ such that the double branched cover of
$Q_i(r)$ is $S^3$. Since each of $Q_7(r)$ and $Q_{11}(r)$ has at
least two components, its double branched cover has nontrivial $\Bbb
Z_2$ homology [Sa, Sublemma 15.4], so $M_7$ and $M_{11}$ are not knot
exteriors in $S^3$. Similarly $M_{14}$ is not the exterior of a link
in $S^3$.
\quad $\Box$ \medskip
\begin{lemma} (1) Let $i\in \{1,2,3\}$. If $r_1, r_2$ are toroidal
slopes of $M_i$ on $T_0$ with $\Delta(r_1, r_2) \geq 4$, then
$\{r_1, r_2\} = \{r'_i, r''_i\}$.
(2) The slope $-7$ is a solid torus filling slope on $T_0$ of $M_3$,
and there is an orientation preserving homeomorphism of $M_3$ which
interchanges the two solid torus filling slopes $\{1/0, -7/1\}$ and
the two toroidal slopes $\{-9, -13/2\}$.
\end{lemma}
\noindent {\bf Proof.} Since $M_1, M_2, M_3, M_{14}$ are the only ones in Definition
21.3 with two boundary components, by Theorem 21.4 $(M_i, r_1, r_2)$
is equivalent to one of the $(M_j, r'_j, r''_j)$ with $j=1,2,3,14$.
Since $M_1,M_2,M_3$ are link complements in $S^3$ and by Lemma 24.2
$M_{14}$ is not, we have $j\neq 14$. Computing $H_1(M_j, T_1)$ shows
$H_1(M_1, T_0) = \Bbb Z$, $H_1(M_2, T_0) = \Bbb Z_3$, and $H_1(M_1,
T_0) = \Bbb Z_5$, hence we must have $j=i$.
By definition there is a homeomorphism $\varphi: (M_i, r_1, r_2) \to
(M_i, r'_i, r''_i)$, up to relabeling of $r_1, r_2$. For $i=1,2$, by
[Ga1] and [Be] the knot $K''_i$ has no nontrivial solid torus surgery,
hence $\varphi(m) = m$, where $m$ is a meridian of $K''_i$. Since
$\Delta(m, r'_i) = \Delta(m, r''_i) = 1$, by the homeomorphism we also
have $\Delta(m, r_1) = \Delta(m, r_2) = 1$, so $r_1, r_2$ are also
integer slopes. It follows that if $\{r_1, r_2\} \neq \{r'_1,
r''_1\}$ then there is a pair of toroidal slopes with distance at
least 5. Since $M_1, M_2$ is not homeomorphic to $M_3$, this is a
contradiction to Theorem 21.4 and [Go].
Now suppose $i=3$. By an isotopy one can deform the tangle in [GW1,
Figure 7.5(c)], which is shown in Figure 24.2(a), to the one in Figure
24.2(b), which is invariant under the $\pi$ rotation $\psi$ along the
forward slash diagonal. The $1/0$ slope in Figure 24.2(b)
corresponds to the $1/2$ slope in Figure 24.2(a), which, by the
proof of Lemma 24.1(3), lifts to the slope $-7m + l$ on $T_0$. The
two toroidal slopes $1/0$ and $5/2$ for the tangle in Figure 22.2(a)
correspond to the slopes $-1/2$ and $2$ in Figure 24.2(b), which are
interchanged by $\psi$. It follows that $\psi$ lifts to an
orientation preserving homeomorphism $\psi': M_3 \to M_3$, which
interchanges the two solid torus filling slopes $\{1/0, -7/1\}$ and
the two toroidal slopes $\{-9, -13/2\}$. In fact, $\psi'$ is
represented by the matrix
$$A = \left(
\array{rr}
7 & 1 \\
-1 & 0
\endarray
\right)
$$
in the sense that if $A(p,q)^t = (p',q')^t$ (where $B^t$ denotes the
transpose of the matrix $B$) then $\psi'(pm + ql) = p'm + q'l$.
Solid torus surgeries on knots in a solid torus have been completely
classified by Gabai [Ga1] and Berge [Be]. It was shown that there is
only one knot admitting two nontrivial solid torus surgeries, which is
a 7-braid. Since $K''_3$ is a 5-braid, we see that $m = 1/0$ and $m'
= -7/1$ are the only solid torus filling slopes on $T_0$. Therefore
the homeomorphism $\varphi: (M_3, r'_3, r''_3) \to (M_3, r_1, r_2)$
must map the set of two curves $\{m, m'\}$ to itself, possibly with
the orientation of one or both of the curves reversed. If $\varphi$
preserves the orientation of $m'$ and reverses the orientation of $m$
then $\varphi(r'_3) = 9/1$ would also be a toroidal slope, which is a
contradiction to [Go] because $\Delta(-9/1, 9/1) = 18 > 8$. Similarly
$\varphi$ cannot preserve the orientation of $m$ while reversing the
orientation of $m'$. Therefore $\varphi$ is orientation preserving
and its induced map on the set of slopes on $T_0$ is either the
identity map, which fixes $\{r'_3, r''_3\}$, or the same as that
induced by $\psi'$ above, which interchanges $\{r'_3, r''_3\}$.
\quad $\Box$ \medskip
\bigskip
\leavevmode
\centerline{\epsfbox{Figure24.2.eps}}
\bigskip
\centerline{Figure 24.2}
\bigskip
\begin{thm} Suppose $K$ is a hyperbolic knot in $S^3$ admitting two
toroidal surgeries $K(r_1), K(r_2)$ with $\Delta(r_1, r_2) \geq 4$.
Then $(K, r_1, r_2)$ is equivalent to one of the following, where $n$
is an integer.
(1) $K = L_1(n)$, $r_1 = 0$, $r_2 = 4$.
(2) $K = L_2(n)$, $r_1 = 2-9n$, $r_2 = -2-9n$.
(3) $K = L_3(n)$, $r_1 = -9 - 25n$, $r_2 = -(13/2) - 25n$.
(4) $K$ is the Figure 8 knot, $r_1 = 4$, $r_2 = -4$.
\end{thm}
\noindent {\bf Proof.} By [Go] the Figure 8 knot $L_1(-1)$ is the only hyperbolic knot
in $S^3$ admitting two toroidal surgeries of distance at least 6, so
we assume $\Delta = 4$ or $5$. By Lemma 24.2 there is a homeomorphism
$\varphi: (M_i(s), \{r'_i, r''_i\}) \cong (E(K), \{r_1, r_2\})$ for some
$i=1,2,3$ and $s \subset T_1$. We only need to show that $s = 1/n$
because the slopes $r_i$ can then be calculated using Lemma 24.1 and
the Kirby calculus [Ro, p.267].
If Dehn filling on $T_1$ of $\partial M_i$ along slope $s$ produces a knot
exterior $E(K) = M_i(s)$, then the meridian-longitude of $K$ may be
different from that of $K''_i$ on $T_0$. We use $(m'',l'')$ (resp.\
$(m,l)$) to denote a meridian-longitude pair of $K''_i$ (resp.\ $K$) in
$S^3$.
\medskip
\noindent
Claim 1. {\it If $E(K) = M_1(s)$ for some $s$ on $T_1$ of $\partial M_1$
then $s=1/n$.}
\medskip
Since the linking number between the two components of $L_1$ is $0$, a
$p/q$ Dehn filling on $T_1$ produces a manifold $M_1(p/q)$ with
$H_1(M_1(p/q), \Bbb Z) = \Bbb Z \oplus \Bbb Z_p$, hence $M_1(p/q)$ is
a knot complement only if $|p| = 1$. It follows that $K = L_1(n)$,
where $n=qp$.
\medskip
\noindent
Claim 2. {\it If $E(K) = M_2(s)$ for some $s$ on $T_1$ of $\partial M_2$
then $s=1/n$ for some $n$.}
\medskip
As before, let $L_2 = K'_i \cup K''_i$. Let $M = E(K)$. Assume $s
= p/q$ and $|p|>1$. We have $K(m'') = L_2(s,m'') = K'_i(s) = L(p,q)$.
Therefore by the Cyclic Surgery Theorem [CGLS], $m''$ is an integer
slope with respect to $(m,l)$, say $m'' = am + l$. By [GLu] the
toroidal slopes $r_1, r_2$ of $K$ are integer or half integer slopes
with respect to $(m,l)$. Recall that $\varphi(r'_2) = r_1$ and
$\varphi(r''_2) = r_2$. Since $m''$ is an integer slope, $r_1, r_2$
cannot both be integer slopes, otherwise $4 = \Delta(r_1, r_2) \leq
\Delta(r_1, m'') + \Delta(m'', r_2) = 2$, which is a contradiction.
Also by [GWZ] they cannot both be half integer slopes.
Now assume $r_1$ is an integer slope and $r_2$ is a half integer
slope with respect to $(m,l)$. Since $m''$ is an integer slope, we
may choose $l=m''$. Then $r_1 = p_1 m + l$ and $r_2 = p_2 m + 2l$,
so $\Delta(r_1, m'') = \Delta(r_2, m'') = 1$ implies $p_1, p_2 = \pm
1$. But then $\Delta(r_1, r_2) = |2p_1 - p_2| \leq 3$, a
contradiction.
\medskip
\noindent
Claim 3. {\it If $E(K) = M_3(s)$ then there is an integer $n$ and a
homeomorphism $\eta: (M_3(1/n), \{r'_3, r''_3\}) \to (E(K), \{r_1,
r_2\})$.}
\medskip
By Lemma 24.3 $M_3(-7)$ is a solid torus, and the meridian
slope $m''$ and the slope $r=-7$ are the only solid torus filling
slopes on $T_0$. If $\varphi(m'') = m$ then $S^3 = K(m'') = K'_i(s)$
implies that $s=1/n$ for some $n$, so $\eta = \varphi$ is the required
map. If $\varphi(r) = m$, let $\psi$ be the orientation preserving
homeomorphism of $M_3$ given in Lemma 24.3, which maps $m''$ to $r$. By
Lemma 24.3 $\psi$ interchanges the slopes $r'_3, r''_3$. Let $s' =
\psi^{-1}(s)$. Then $\varphi \circ \psi :(M_3(s'), r''_3, r'_3) \cong
(E(K), r_1, r_2)$ maps $m''$ to $m$. As above this implies that $s' =
1/n$, hence $\eta = \varphi\circ \psi$ is the required map.
We now assume that $\varphi(m'') \neq m$ and $\varphi(r) \neq m$.
Note that $K(m'')$ and $K(r)$ can be obtained from the solid tori
$M_3(m'')$ and $M_3(r)$ by $s$ filling on $T_1$, so they have cyclic
$\pi_1$, hence by [CGLS] $r, m''$ are integer slopes of $K$. Choose
$l = m''$. Since $\Delta(m'', r)=1$, we may assume $r = 1/1$ up to
rechoosing the orientation of $l$. The toroidal slopes $r'_3 = -9$
satisfy $\Delta(r'_3, m'') = 1$ and $\Delta(r'_3, r) = 2$, which
implies $r'_3 = -1/1$ or $1/3$ with respect to $(m,l)$. The second is
impossible by [GLu]. Similarly the fact that $\Delta(r''_3, m'') = 2$
and $\Delta(r''_3, r) = 1$ implies that $r''_3 = 2$ with respect to
$(m,l)$. But then we have $5 = \Delta(r'_3, r''_3) = \Delta(-1, 2) =
3$, a contradiction.
\quad $\Box$ \medskip
\begin{cor}
A hyperbolic knot $K$ in $S^3$ has at most four toroidal surgeries.
If there are four, then they are consecutive integers.
\end{cor}
\noindent {\bf Proof.} By [GLu] a toroidal slope of $K$ must be integer or half
integer, and if it is a half integer then $K$ is a Eudave-Mu\~noz
knot. By [T1, Corollary 1.2], if $K$ is a Eudave-Mu\~noz knot then it
has at most three toroidal slopes, hence the result is true if $K$ has
a half integer toroidal slope. Therefore we may assume that all
toroidal slopes of $K$ are integer slopes. The result follows if
$\Delta(r,s) < 4$ for all pairs of toroidal slopes $(r,s)$ of $K$.
Therefore by Theorem 24.4 we need only show that if $K$ is either
$L_1(n)$ or $L_2(n)$ for some $n$ then $K$ has at most three integer
toroidal slopes.
If $K$ is the knot $L_1(n)$ in Theorem 24.4(1) then by [BW] it has
exactly two toroidal slopes unless it is the Figure 8 knot, which has
three toroidal slopes.
Now consider a knot $K = L_2(n)$ in Theorem 24.4(2) and let $r$ be an
integral toroidal slope of $K$ other than $r_1, r_2$ in the Theorem.
Since $\Delta(r_i, r)\leq 4$, $r$ must be between $r_1$ and $r_2$.
Denote by $M_2(p/q)$ the $p/q$ filling on $T_0$ with respect to the
preferred meridian-longitude pair of $L_2$. By the proof of Lemma
24.1(2), $M_2(-1)$ is the double branched cover of $Q_2(1)$. Using
the tangle in [GW1, Figure 7.4(c)] one can check that $Q_2(1)$ is a
Montesinos tangle $T(1/2, -2/5)$, therefore $M_2(-1)$ is a small
Seifert fiber space with orbifold $D^2(2,5)$. Since $L_2(n)(-1-9n)$
is obtained from $M_2(-1)$ by Dehn filling on $T_1$ and contains no
non-separating surface, it is atoroidal. Because of symmetry ($L_2$
is amphicheiral), $M_2(1)$ is homeomorphic to $M_2(-1)$, so
$L_2(n)(1-9n)$ is also atoroidal. It follows that the only possible
integer toroidal slopes of $L_2(n)$ are $j - 9n$ for $j=-2, 0, 2$.
This completes the proof. (Actually it can be shown that $-9n$ is not
a toroidal slope of $L_2(n)$ either, so it has at most two integer
toroidal slopes.) \quad $\Box$ \medskip
The following corollary is an immediate consequence of Theorem 24.4.
\begin{cor} Let $K$ be a hyperbolic knot in $S^3$ which admits two
toroidal surgeries along slopes $r_1, r_2$, and $\Delta =
\Delta(r_1, r_2) \geq 4$. Then one of the $r_i$ is an integer, and
the other one is an integer if $\Delta \neq 5$, and a half integer
if $\Delta = 5$.
\end{cor}
Although there are infinitely many hyperbolic 3-manifolds $M$ with
toroidal fillings $M(r), M(s)$ at distance 4 or 5, we have shown that
they all come from finitely many cores $X(r,s)$ as defined in Section
21.
\begin{qtn}
Are there only finitely many cores $X(r,s)$ of toroidal Dehn
fillings on hyperbolic 3-manifolds with $\Delta(r,s) = 3$?
$\Delta(r,s) = 2$?
\end{qtn}
We observe that the answer to Question 24.7 in the case
$\Delta(r,s)=1$ is almost certainly `no'; here is an outline of an
argument. Let $N$ be a closed irreducible $3$-manifold with a unique
incompressible torus $T$ up to isotopy. Let $F$ be a once-punctured
torus, regarded as a disk with two bands. It is intuitively clear
that, for any positive integer $n$, by tangling the bands in a
sufficiently complicated fashion we can construct an embedding $F_n$
of $F$ in $N$ so that if $K_n = \partial F_n$, then $N-K_n$ is hyperbolic,
and $K_n$ cannot be isotoped to meet $T$ in fewer than $n$ points. Let
$M_n = N-{\rm Int} N(K_n)$, and let $r,s$ on $\partial M_n$ be the meridian of
$K_n$ and the longitudinal slope defined by $F_n$, respectively. Then
$\Delta(r,s)=1$, $M_n(r) = N$ is toroidal by definition, and $M_n(s)$
contains the non-separating torus $\hat F_n = F_n \cup D$, where $D$
is a meridian disk of $V_s$. Hence, if we make sure that $M_n(s)$ does
not contain a non-separating sphere, then $M_n(s)$ is also toroidal.
Since the number of intersections of $K_n$ with $T$ is at least $n$,
the triples $(M_n,r,s)$ cannot all come from only finitely many cores.
|
math/0512227
|
\section*{Introduction}
The direct sum of the Solomon-Tits algebras of type $A_n$,
or twisted descent algebra, has been shown in~\cite{patsch06}
to carry a rich algebraic structure which extends and generalizes
the structures on the classical descent algebra (the direct sum
of the Solomon algebras of type $A_n$). From a combinatorial
point of view, moving from the classical to the twisted descent
algebra means moving from the combinatorics of \emph{compositions}
(sequences of integers) to the combinatorics of
\emph{set compositions} (sequences of mutually disjoint sets).
The purpose of the present article is to pursue further the
study of the algebraic structures associated to set compositions,
or twisted descents, and related objects.
Fields of application of the theory include, among others,
the geometry of Coxeter complexes of type $A_n$, the internal
structure of twisted Hopf algebras, Markov chains associated to
hyperplane arrangements, and Barratt's twisted Lie algebra
structures in homotopy theory. We refer to~\cite{patsch06} for
a survey of the history of the subject, and further details on
its various fields of application.
We first show, in Section~\ref{S1}, that the natural basis of
the Solomon-Tits algebra of type $A_n$ (or, equivalently,
the set of faces of the hyperplane arrangement of type $A_n$,
the set of set compositions of $\{1,\ldots,n\}$, or the set of
cosets of standard parabolic subgroups in the symmetric
group $S_n$) is in bijection with the set of increasing planar
rooted trees with $n$ branchings. The result is new, to our
best knowledge, although it appears to be a very natural
extension of the classical bijection between permutations
and increasing planar binary trees (see, for
example,~\cite{raw86,stan86} and the appendix of~\cite{loday01}),
and of the mapping from
set compositions to planar rooted trees introduced
in~\cite{tonks97} and further studied in~\cite{chapoton00}.
This provides another link between
combinatorial structures, Hopf algebras and trees. This domain
has received a considerable attention recently, due
in particular to the discovery of
its role in the understanding of high energy physics through
the seminal work of Connes and Kreimer on Feynman graphs
and Zimmermann's renormalization formula~\cite{conkre98,grafig04}.
Other approaches and other problems have also enlightened the
explanatory power of this link. These influential
contributions include Chapoton's work on Hopf algebras,
trees and the geometry of Coxeter complexes~\cite{chapoton00},
Loday's and Ronco's work on planar binary trees and
operads~\cite{lodron98}, and the work of Brouder-Frabetti
on planar binary trees and QED~\cite{broufra03}.
Set compositions of finite sets of positive integers are also
in $1$-$1$ correspondence with monomials in non-commuting
variables $x_1,x_2,\ldots$. This connects our work with the
theory of word quasi-symmetric functions, $\mathsf{WQsym}$,
and the theory of quasi-symmetric functions in non-commuting
variables, $\mathsf{NCQsym}$, recently introduced
by Hivert et al. (see, for example,~\cite{novthibon06})
and Bergeron et al. (see, for example,~\cite{bergeron1,bergeron2}), respectively.
In Sections~\ref{S2} and~\ref{S3}, we revisit the twisted Hopf
algebra structure on the twisted descent algebra introduced in~\cite{patsch06} and start to analyze the algebraic
implications of our combinatorial result. We study in detail
the two Hopf algebra structures on increasing planar rooted trees
which are induced by this twisted Hopf algebra, by means of the symmetrisation and cosymmetrisation processes
of~\cite{patreu04,stover93}.
One of these Hopf algebras is neither commutative nor
cocommutative. This Hopf algebra turns out to be dual to the
algebra $\mathsf{NCQsym}$, thereby revealing a different approach to
the algebra of quasi-symmetric functions in non-commuting
variables. This observation has a striking consequence.
Namely, the classical triple at the heart of Lie theory (Solomon
algebra of type $A_n$; descent algebra (the direct sum of Solomon
algebras); quasi-symmetric functions) lifts to the world of
twisted objects as a triple (Solomon-Tits algebra of type $A_n$;
twisted descent algebra; quasi-symmetric functions in
non-commuting variables). The link between the first two objects
was investigated in detail in~\cite{patsch06}, whereas the
duality properties between the twisted descent algebra and the
algebra of quasi-symmetric functions in non-commuting variables
are a by-product of our considerations in Section~\ref{S2}
of the present article. This emphasizes, once again, the
conclusion drawn from~\cite{patreu04,patsch06,sch05}
that the twisted descent algebra is the ``natural framework''
to lift classical algebraic and combinatorial structures
(compositions, descents, shuffles, free Lie algebras, and so on)
to the enriched setting of set compositions, tensor species,
Barratt's free twisted Lie algebras, and so on.
The other Hopf algebra structure provides the twisted descent
algebra (or, equivalently, the linear span of increasing planar
rooted trees), with the structure of an enveloping algebra.
This is the Hopf algebra we will be mainly interested in,
in view of its rich algebraic and combinatorial structure. We
show, for example,
that the twisted descent algebra is a free associative algebra
with generators in bijection with so-called balanced increasing
rooted trees, and furthermore, naturally, the enveloping algebra
of a free Lie algebra that we describe explicitly.
In final Section~\ref{S4}, we study Hopf subalgebras of the
twisted descent algebra. The relationships of the noncommutative
noncocommutative Hopf algebra structure on set compositions to
its most remarkable Hopf subalgebras appear quite simple and
natural. For example, whereas Chapoton's Hopf
algebra~\cite{chapoton00} is related to the Malvenuto--Reutenauer
Hopf algebra by means of a sub-quotient construction, we can
show that the natural embedding of the free twisted associative
algebra on one generator into the twisted descent algebra
implies that the (noncommutative noncocommutative) Hopf
algebra we consider contains Malvenuto--Reutenauer's (and
therefore all its Hopf subalgebras) as proper Hopf subalgebras.
However, our main concern is, once again, the other structure,
that is, the cocommutative case. We show that the direct sum of
the symmetric
group algebras, when provided with the enveloping algebra
structure introduced in~\cite[Section~6]{patreu04}, embeds as
an enveloping algebra in the twisted descent algebra. We recover
in particular, as a corollary of our results on the twisted
descent algebra, Theorem~21 in~\cite{patreu04}, stating that the
Lie algebra of primitive elements in this direct sum is a free
Lie algebra.
Finally, we also introduce enveloping algebra structures on
planar trees and planar binary trees
which seem to be new. We show that these
enveloping algebras embed into the enveloping algebra of
increasing planar rooted trees as well.
\section{Set compositions and planar rooted trees}
\label{S1}
We give here a $1$-$1$ correspondence between set compositions
and increasing planar rooted trees which extends a construction
of~\cite{tonks97} (see also~\cite{chapoton00}).
Our bijection also extends the classical correspondence between increasing planar binary rooted trees and permutations.
Let $n$ be a non-negative integer and set $[n]:=\{1,2,\ldots,n\}$. A \emph{set composition} of $[n]$ of length $k$ is a $k$-tuple
$P=(P_1,\ldots,P_k)$ of mutually disjoint non-empty subsets
$P_1,\ldots,P_k$ of $[n]$ such that $P_1\cup\ldots\cup P_k=[n]$.
There is an obvious $1$-$1$ correspondence between surjective maps
$\varphi:[n]\to[k]$ and set compositions $P$ of length $k$ of $[n]$
which assigns to any such $\varphi$ the $k$-tuple
$P=(\varphi^{-1}(1),\varphi^{-1}(2),\ldots,\varphi^{-1}(k))$. For
example, if $n=3$, $k=2$ and $\varphi(1)=1=\varphi(3)$,
$\varphi(2)=2$, then $P=(13,2)$. Note that we dropped several commas
and curly brackets in $P$.
In what follows, a (planar, rooted) \emph{tree} is a finite planar
non-empty oriented connected graph $T$ without loops such that any
vertex of $T$ has at least two incoming edges and exactly one outgoing edge. In illustrations, the
root appears at the bottom, the leaves appear at the top, and the
orientation is dropped with the understanding that all
edges are oriented from top to bottom. Three trees
$T_0$, $T_1$, $T_2$ with two vertices and, respectively,
three, four and three leaves are displayed in
Figure~\ref{fig01}.
\begin{figure}[h]
\hspace*{\fill}
$T_0=\,$\raisebox{%
-4.5mm}{\includegraphics[width=12mm]{figure01a.eps}}
\hspace*{\fill}
$T_1=\,$\raisebox{%
-4.5mm}{\includegraphics[width=12mm]{figure01b.eps}}
\hspace*{\fill}
$T_2=\,$\raisebox{%
-4.5mm}{\includegraphics[width=12mm]{figure01c.eps}}
\hspace*{\fill}
\caption{\it Three trees.}
\label{fig01}
\end{figure}
The \emph{trivial tree}
\,\raisebox{-1pt}{%
\includegraphics[width=1.3pt]{trivial-tree.eps}}\,
without vertex will be denoted by $\varepsilon$.
If $m\ge 1$, then the \emph{wedge} of $m+1$ trees $T_0,T_1,\ldots,T_m$ is obtained by
joining the roots of $T_0,\ldots, T_m$ to a new vertex and
creating a new root. It is denoted by
$\bigvee(T_0,T_1,\ldots,T_m)$. For example, the wedge of the trees
$T_0$, $T_1$, $T_2$ in Figure~\ref{fig01}
is given in Figure~\ref{fig02}.
\begin{figure}[h]
\hspace*{\fill}
$\bigvee(T_0,T_1,T_2)=$
\raisebox{-10mm}{%
\includegraphics[width=40mm]{figure02.eps}}
\hspace*{\fill}
\caption{\it The wedge of three trees.}
\label{fig02}
\end{figure}
Any tree $T\neq\varepsilon$ can be written uniquely as the
wedge $\bigvee(T_0,T_1,\ldots,T_m)$ of certain subtrees of~$T$.
We define the \emph{$m$-corolla} $C_m$
by $C_m=\bigvee(\underbrace{\varepsilon,\varepsilon,\ldots,\varepsilon}_{m})$. Hence $C_m$ is
the unique tree with one vertex and $m$ leaves. For example,
$C_4=\raisebox{-3mm}{\includegraphics[width=12mm]{b4.eps}}$.
Tonks~\cite{tonks97} and Chapoton~\cite{chapoton00} studied
a surjective map which assigns to each set composition $P$ a
tree $T$.
We will consider additional structure on trees which will allow us to turn
this surjection into a bijection.
A \emph{branching} $b$ of a tree $T$ is a subgraph of $T$
isomorphic to
$C_2=\raisebox{-2mm}{\includegraphics{b1.eps}}$.
Three branchings $b_x,b_y,b_z$ are indicated in
Figure~\ref{fig03}.
\begin{figure}[h]
\hspace*{\fill}
\includegraphics[width=40mm]{figure03.eps}
\hspace*{\fill}
\caption{\it Three branchings.}
\label{fig03}
\end{figure}
We denote the set of all branchings of $T$ by $B(T)$, and its
cardinality by $b(T)$. The set $B(T)$ admits a ``left-to-right''
order $\preceq$ in a natural way: if $T$ is the $m+1$-corolla $C_{m+1}$ for some $m\ge 0$, then $T$ has $m$ branchings $b_1,b_2,\ldots,b_m$ (labelled from left to right)
and we define $b_1\preceq b_2\preceq\cdots\preceq b_m$;
see Figure~\ref{fig04}.
\begin{figure}[h]
\hspace*{\fill}
$C_{m+1}=$
\raisebox{-7mm}{%
\includegraphics[width=50mm]{figure04.eps}}
\hspace*{\fill}
\caption{\it The branchings of the $m+1$-corolla.}
\label{fig04}
\end{figure}
If, more generally, $T$ is the wedge of (possibly non-empty) trees
$T_0,\ldots,T_m$,
we proceed by induction and extend the orders on
$B(T_0),\ldots,B(T_m)$ to an order on $B(T)$ by setting
$$
b_i\prec b\prec b_{i+1}
$$
for all $i\in\{0,1,\ldots,m\}$ and $b\in B(T_i)$, where $T_i$ is
embedded in $T$. (The term $b_i$ on the left, respectively, $b_{i+1}$
on the right, does not appear when $i=0$, respectively, when $i=m$.)
For example, for the branchings indicated in Figure~\ref{fig03}, we
obtain $b_x\preceq b_y\preceq b_z$.
The order $\preceq$ induces a \emph{natural labelling} of the
branchings of $T$, namely the order-preserving map $([n],\le)\to
(B(T),\preceq)$, where $n=b(T)$ and $\le$ denotes the usual order
on $[n]$. The natural labelling of the tree in
Figure~\ref{fig03} is illustrated in Figure~\ref{fig05}.
\begin{figure}[h]
\hspace*{\fill}
\includegraphics[width=40mm]{figure05.eps}
\hspace*{\fill}
\caption{\it Natural labelling of the branchings.}
\label{fig05}
\end{figure}
A \emph{root branching} of $T$ is a branching $b\in B(T)$ which
contains the root. The tree in Figure~\ref{fig05}, for example, has
two root branchings, with natural labels $3$ and $8$.
With $T=\bigvee(T_0,\ldots,T_m)$ as above,
the natural label of a root branching $b_i$ is
$$
x_i=i+\sum_{j=0}^{i-1} b(T_j)
$$
for all $i\in[m]$. Furthermore, if $i\in[m]\cup\{0\}$,
$\tilde{b}\in B(T_i)$ has natural label $j$ in $T_i$ and
$b$ is the branching of $T$ corresponding to $\tilde{b}$
(via the embedding of $T_i$ in $T$), then $b$ has natural
label $x_i+j$ in $T$, where $x_0:=0$.
A \emph{level function} on a tree $T$ is a surjective map
$\lambda:V\to X$ such
that $\lambda$ is strictly increasing along each path connecting
a leaf of $T$ with the root of $T$. Here $V$ is the set of
vertices of $T$ and $X$ is a totally ordered set.
Such a level function is said to be \emph{standard}
if $X=[k]$ (with the usual order), for some non-negative
integer $k$. A tree provided with a standard level function is
an \emph{increasing tree}.
Three standard level functions are illustrated in
Figure~\ref{fig06}.
\begin{figure}[h]
\hspace*{\fill}
\includegraphics[width=22.5mm]{figure06a.eps}
\hspace*{\fill}
\hspace*{1ex}
\includegraphics[width=22.5mm]{figure06b.eps}
\hspace*{\fill}
\hspace*{1ex}
\includegraphics[width=22.5mm]{figure06c.eps}
\hspace*{\fill}
\caption{\it Standard level functions.}
\label{fig06}
\end{figure}
It is easy to see that there are no other standard level
functions on this tree.
Suppose $X$ has order $k$ and $\iota:X\to[k]$ is the
order-preserving bijection. It is clear that
any level function $\lambda:V\to X$ on $T$
yields the standard level function $\iota\circ\lambda$
on $T$. We refer to this standard level function as
the \emph{standardization} of $\lambda$. It will be
advantageous at a later stage to consider arbitrary ordered
sets $X$ in the definition of a level function.
When illustrating an increasing tree, it is more convenient to draw each vertex $v$
at the level $\lambda(v)$ (where levels increase from top to bottom)
rather than to label $v$ by $\lambda(v)$; see Figure~\ref{fig07}.
\begin{figure}[h]
\hspace*{\fill}
\includegraphics[width=100mm]{figure07.eps}
\hspace*{\fill}
\caption{\it Increasing trees.}
\label{fig07}
\end{figure}
(We have added here the natural labels of the branchings in
each case for purposes which will be clear later.)
Some geometrical properties of increasing trees can be described
in terms of natural labels and levels. For example, the vertices
of two branchings $b$ and $b'$ of an increasing tree $T$ with
natural labels $i<i'$ at levels $l<l'$ (or $l>l'$, respectively)
belong to a common path joining a leaf of $T$ to the root of $T$
if and only if the vertices of all the branchings $d$ with labels
between $i$ and $i'$ have levels (strictly) greater than $l$
(or $l'$, respectively).
If $\lambda:V\to X$ is a level function on $T$ and $b$ is
a branching of $T$ with vertex $v$, we assign to $b$ the
level $\lambda(v)$. Let
$n=b(T)$, and suppose that $X=[k]$ for some non-negative
integer~$k$. Then the composite of the natural labelling of $B(T)$
with the standard level function $\lambda$ (extended to $B(T)$ in the way described) yields a
surjective map $[n]\to[k]$ or, equivalently, a set composition
$$
\sigma(T,\lambda)=P=(P_1,\ldots,P_k).
$$
More explicitly, $P_i$
consists of the natural labels of all branchings of $T$ at
level~$i$, for all $i\in[k]$.
The set compositions arising in this way from the increasing trees given in Figure~\ref{fig07} are, respectively,
$(1,34,25)$, $(134,25)$ and $(34,1,25)$.
\begin{theorem} \label{bijection}
The map $(T,\lambda)\mapsto \sigma(T,\lambda)$ is a bijection from
the set of all increasing trees with $n$ branchings onto the
set of all set compositions of $[n]$.
\end{theorem}
By restriction, we obtain a correspondence between increasing
binary trees and permutations, identified with set compositions
of $[n]$ of length $n$.
Note that, in this correspondence, a permutation $\pi$
is associated to the binary tree which is classically
associated to $\pi^{-1}$; see~\cite{raw86,stan86}
and the appendix of~\cite{loday01}.
In order to prove the theorem, we give an inductive description
of the inverse map $\tau$ of $\sigma$. For this purpose, it is
convenient to identify a standard level function
$\lambda:B(T)\to[k]$
on a tree~$T$ with $n$ branchings with the corresponding map
$\varphi:[n]\to[k]$,
by means of the natural labelling of the branchings.
Let $P=(P_1,\ldots,P_k)$ be any set composition of $[n]$,
and denote the corresponding surjective map from $[n]$ to $[k]$
by $\varphi$.
Define $T_1$ to be the 2-corolla with level function
$\lambda_1=\varphi|_{[1]}$, the restriction of $\varphi$
to $[1]$.
Let $i\in[n-1]$, and assume inductively that a tree $T_i$ has
been constructed with level function $\lambda_i=\varphi|_{[i]}$.
Then define $(T_{i+1},\lambda_{i+1})$
to be the (unique) increasing tree
with $i+1$ branchings obtained by adding to $(T_i,\lambda_i)$
a branching on the right at level $\varphi(i+1)$.
After $n$ steps, we arrive at a tree $T_n$ with $n$ branchings
and standard level function $\lambda_n=\varphi$. We set
$$
\tau(P)
:=
(T_n,\lambda_n).
$$
It is immediate from the definitions that $\tau$
is a left and a right inverse of $\sigma$. This proves the
theorem.
The construction is best understood through an example.
The case where $P=(26,34,1,5)$ is illustrated in
Figure~\ref{fig08}.
\begin{figure}[h]
\hspace*{\fill}
\includegraphics[width=108mm]{figure08.eps}
\hspace*{\fill}
\caption{\it Construction of the increasing tree corresponding
to $(26,34,1,5)$.}
\label{fig08}
\end{figure}
The composite of $\tau$ with the forgetful map from increasing
trees to trees can be shown to agree with Chapoton's map $\Psi$
(see~\cite[pp.267]{chapoton00}), although their definitions might look different at first sight. The proof is left to the reader.
To conclude this section, we observe that the construction
of $\tau(P)$ applies, more generally, to an arbitrary surjective
map $\varphi$ from a finite ordered set $A$ onto an ordered
set $X$ (instead of the surjection $\varphi:[n]\to[k]$
corresponding to $P$). As a result, we obtain a tree with
branchings labeled by $A$ and levels drawn from $X$.
In particular, we can assign an increasing tree $\tau(Q)$
with labels in an ordered set $A$ to a set composition $Q$ of
length $k$ of an arbitrary finite ordered set $A$ (since
such a set composition corresponds to a surjective
map $\varphi:A\to[k]$).
If $P=(P_1,\ldots,P_k)$ is a set composition of $[n]$ and $A$ is
a subset of $[n]$, then $Q:=(P_1\cap A,\ldots,P_k\cap A)^\#$
is a set composition of $A$, where the upper index $\#$ indicates that empty sets are deleted. In geometric terms, the increasing
tree $\tau(Q)$ is then obtained from the increasing tree
$(T,\lambda):=\tau(P)$ by ``contracting'' in a certain way all
branchings of $T$ with natural labels not contained in~$A$
(and keeping the levels of branchings with labels in $A$).
Accordingly, $\tau(Q)$ is called the
\emph{contraction of $(T,\lambda)$ relative to $A$},
or \emph{$A$-contraction of $(T,\lambda)$}.
An example is displayed in Figure~\ref{fig09}, where
$(T,\lambda)$ is the increasing tree corresponding to
the set composition $P=(26,34,1,5)$ and $A=\{1,2,4,6\}$.
\begin{figure}[h]
\hspace*{\fill}
\raisebox{-17.5mm}{%
\includegraphics[width=35mm]{figure09a.eps}}
\qquad
$\longrightarrow$
\qquad
\raisebox{-8.75mm}{%
\includegraphics[width=22.75mm]{figure09b.eps}}
\hspace*{\fill}
\caption{\it The contraction of $(26,34,1,5)$ relative to $A=\{1,2,4,6\}$.}
\label{fig09}
\end{figure}
The notion of contraction will play a vital role in our
constructions of coproducts on the linear span of increasing trees
in the sections that follow.
\section{The twisted descent algebra as a Hopf algebra}
\label{S2}
Let $\mathbb{Z}$ and $\mathbb{N}$ denote the sets of all integers and of all
positive integers, respectively, and set $\mathbb{N}_0:=\mathbb{N}\cup\{0\}$.
The free twisted descent algebra $\mathcal{T}$ has $\mathbb{Z}$-linear basis
the set of all set compositions of finite subsets of $\mathbb{N}$.
Equivalently, by Theorem~\ref{bijection}, we can take the
set of all increasing trees (with natural labels drawn from $\mathbb{N}$)
as a basis of $\mathcal{T}$.
The algebra $\mathcal{T}$ is a \emph{twisted Hopf algebra}.
It was shown in~\cite{patreu04}, for arbitrary twisted
Hopf algebras $(H,\ast,\delta)$, that the twisted product
$\ast$ induces two (ordinary) products on $H$
and that the twisted coproduct $\delta$ induces two
(ordinary) coproducts on $H$. These turn $H$ into an
ordinary Hopf algebra in two different ways.
In this section, we will make explicit these Hopf algebra
structures on $\mathcal{T}$ and thereby explore the algebraic
implications of the definitions given in~\cite{patsch06}
for the study of set compositions and increasing trees.
In particular, we will show that one of our constructions
recovers the graded dual of the algebra of quasi-symmetric
functions in non-commuting variables considered
in~\cite{bergeron2} (see Lemmas~\ref{3.4} and~\ref{3.7}).
For notational brevity, we refer to the free twisted descent
algebra of~\cite{patsch06} simply as ``the twisted descent
algebra''.
All graded vector spaces considered here are connected:
their degree~$0$ component is naturally isomorphic to the
ground ring. Hence the two notions of bialgebra and of Hopf
algebra coincide on these spaces, and there is no need to
specify the antipode.
Let us recall some definitions. More details and references
on twisted algebraic structures can be found
in~\cite{patreu04,patsch06,sch05}.
A \emph{tensor species} is a functor from the category of finite sets and set isomorphisms $\mathsf{Fin}$ to the category $\mathsf{Mod}$ of vector spaces over a field or modules over a commutative ring.
Unless otherwise specified, we will work over $\mathbb{Z}$, so that
$\mathsf{Mod}$ is the category of abelian groups. For convenience,
we will also assume that the finite sets we consider (and
therefore the objects in $\mathsf{Fin}$) are subsets of $\mathbb{N}$.
The category $\mathsf{Sp}$ of tensor species is a linear symmetric
monoidal category for the tensor product defined by:
$$
(F\otimes G)(S)
:=
\bigoplus\limits_{T\coprod U=S}F(T)\otimes G(U)
$$
for all $F,G\in \mathsf{Sp}$, $S\in \mathsf{Fin}$.
Here $T\coprod U=S$ means that $S$ is the \emph{disjoint} union
of $T$ and $U$.
Let $\mathsf{Comp}_S$ denote the set of set compositions of $S$,
for all $S\in\mathsf{Fin}$. As a tensor species, the twisted descent
algebra $\mathcal{T}$ can be identified with the linearization of the
set composition functor:
$$
\mathcal{T}(S):=\mathbb{Z}[\mathsf{Comp}_S].
$$
We also write $\mathcal{T}_S$ instead of $\mathcal{T}(S)$.
The twisted descent algebra carries two products:
the internal or composition product $\circ$, and the
external or convolution product $\ast$. It also carries
a coproduct $\delta$. All three structures are induced by
the natural action of set compositions on twisted Hopf
algebras~\cite{patsch06}. They are defined as follows.
\begin{definition}
The $\mathsf{Fin}$-graded components $\mathcal{T}_S$ of $\mathcal{T}$ are associative
unital algebras for the \emph{composition product $\circ$},
defined by:
\begin{eqnarray*}
\lefteqn{%
(P_1,\ldots,P_k)\circ (Q_1,\ldots,Q_l)}\\[1mm]
& := &
(P_1\cap Q_1,\ldots,P_1\cap Q_l,
\quad\ldots\quad,
P_k\cap Q_1,\ldots,P_k\cap Q_l)^{\#}
\end{eqnarray*}
for all $(P_1,\ldots,P_k),(Q_1,\ldots,Q_l)\in \mathsf{Comp}_S$.
As in Section~\ref{S1}, the upper index $\#$ indicates that
empty sets are deleted.
\end{definition}
The algebra $(\mathcal{T}_{[n]},\circ)$ is now widely referred to as
the ``Solomon--Tits algebra'', a terminology introduced
in the preprint version of~\cite{patsch06}. The connections
between Tits's seminal ideas (which ultimately led to the
definition of the product $\circ$) and graded Hopf algebraic
structures were also first emphasized in~\cite{patsch06} and
have since been subject to increasing interest; see, for
example,~\cite{bergeron2} and~\cite{novthibon06}. We do not
study here the internal product and refer to~\cite{sch05}
for detailed structure results on the algebra~$(\mathcal{T}_S,\circ)$.
Recall that a \emph{twisted} associative algebra is an
associative algebra in the symmetric monoidal category of
tensor species. Twisted versions of coassociative coalgebra,
bialgebra, and so on, are defined in the same way.
\begin{definition}
The functor $\mathcal{T}$ is provided with the structure of an
associative unital twisted algebra by the
\emph{convolution product $\ast$}, defined by:
$$
(P_1,\ldots,P_k)\ast (Q_1,\ldots,Q_l)
:=
(P_1,\ldots,P_k,Q_1,\ldots,Q_l)
$$
for all $S,T\in \mathsf{Fin}$ with $S\cap T=\emptyset$
and all $(P_1,\ldots,P_k)\in\mathsf{Comp}_S$, $(Q_1,\ldots,Q_l)\in \mathsf{Comp}_T$.
The identity element in $(\mathcal{T},\ast)$ is the empty
tuple~$\emptyset$.
The functor $\mathcal{T}$ is also provided with the structure of a
coassociative cocommutative counital twisted coalgebra
by the coproduct $\delta$, defined by:
$$
\delta (P_1,\ldots,P_k)
:=
\sum_{Q_i\coprod R_i=P_i}
(Q_1,\ldots,Q_k)^\# \otimes (R_1,\ldots,R_k)^\#
$$
for all $S\in \mathsf{Fin}$, $(P_1,\ldots,P_k)\in\mathsf{Comp}_S$.
For example,
\begin{eqnarray*}
\lefteqn{%
\delta (14,7)
=
(14,7)\otimes \emptyset
+
(14)\otimes (7)
+
(1,7)\otimes (4)}\\[1mm]
&&
+
(4,7)\otimes (1)
+
(1)\otimes (4,7)
+
(4)\otimes (1,7)
+
(7)\otimes (14)
+
\emptyset\otimes (14,7).
\end{eqnarray*}
\end{definition}
Note that
$
\delta(P)
\in
(\mathcal{T}\otimes\mathcal{T})[S]
=
\bigoplus\limits_{A\coprod B=S} \mathcal{T}_A\otimes\mathcal{T}_B
$
for all $P\in\mathsf{Comp}_S$.
We write $\delta_{A,B}$ for the component of the image
of $\delta$ in $\mathcal{T}_A\otimes\mathcal{T}_B$, so that, for example, $\delta_{\{1,7\},\{4\}}(14,7)=(1,7)\otimes (4)$.
The following is an immediate consequence of~\cite{patsch06}.
\begin{prop}
The triple $(\mathcal{T},\ast,\delta)$ is a cocommutative twisted
Hopf algebra.
\end{prop}
Whenever $S,T\in\mathsf{Fin}$ have the same
cardinality $n$, the unique order-preserving bijection $S\to T$
induces a linear isomorphism $\mathsf{is}_{S,T}:\mathcal{T}_S\to \mathcal{T}_T$ in
an obvious way. If, in particular, $T=[n]$, then we write
$\mathcal{T}_n:=\mathcal{T}_{[n]}$ and $\mathsf{is}_S$ for the isomorphism from $\mathcal{T}_S$
onto $\mathcal{T}_n$.
We will now use the isomorphisms $\mathsf{is}_{S,T}$ to describe
the algebra and coalgebra structures on
the twisted descent algebra which arise from the general
constructions of~\cite{patreu04}. For this purpose, let us
introduce the graded vector space
$$
\mathcal{T}_\bullet :=\bigoplus\limits_{n\in \mathbb{N}_0} \mathcal{T}_n\,.
$$
\begin{lem} \label{3.4}
The vector space $\mathcal{T}_\bullet$ is a graded coassociative
counital coalgebra with respect to the
\emph{restricted coproduct $\overline{\delta}$},
defined on the $n$th component by
$$
\overline{\delta}
:=
\bigoplus_{p+q=n}
(\mathcal{T}_p\otimes \mathsf{is}_{p+[q]})\circ \delta_{[p],p+[q]},
$$
where we write $\mathcal{T}_p$ for the identity on $\mathcal{T}_p$ and
$p+[q]$ for $\{p+1,\ldots,p+q\}$.
\end{lem}
This is a direct consequence of the definition of a
twisted coalgebra.
For any $A\subseteq [n]$ and any set composition
$P=(P_1,\ldots,P_k)$ of $[n]$, we write
$$
P|_A
:=
\mathsf{is}_A\Big((P_1\cap A,\ldots,P_k\cap A)^{\#}\Big)
$$
so that, for example,
$
(35,62,1,47)|_{\{1,3,5,7\}}
=
\mathsf{is}_{\{1,3,5,7\}}((35,\emptyset ,1,7)^\# )
=
(23,1,4)
$.
Then, in particular, we have
$$
\overline{\delta}(P)
=
\sum_{p+q=n} P|_{[p]}\otimes P|_{p+[q]}
$$
for all $n\in\mathbb{N}_0$, $P\in\mathsf{Comp}_{[n]}$.
A comparison with~\cite[Eq.~(21)]{bergeron2} now shows that
the linear map
$W:(\mathcal{T}_\bullet,\overline{\delta})\to(\mathsf{NCQsym}^*,\Delta^*)$,
defined by $P\mapsto W_P$, is an isomorphism of coalgebras.
The restricted coproduct translates naturally into the
language of trees, by means of Theorem~\ref{bijection}.
If $p+q=n$, then the map
$(\mathcal{T}_p\otimes \mathsf{is}_{p+[q]})\circ \delta_{[p],p+[q]}$
sends an increasing tree $T$ to $T_1\otimes T_2$, where
$T_1$ is the $[p]$-contraction of $T$ and $T_2$
is the (standardization of) the $(p+[q])$-contraction of $T$,
as defined at the end of Section~\ref{S1}.
Consider, for example, the tree $T$ displayed in
Figure~\ref{fig05}, interpreted as an increasing tree
with branchings $6$ and $7$ at level $1$, branchings $1$
and $5$ at level $2$, and so on.
The $(5,4)$ component of $\overline{\delta}(T)$ is given
in Figure~\ref{fig10}.
\begin{figure}[h]
\begin{eqnarray*}
\lefteqn{%
(\mathcal{T}_5\otimes \mathsf{is}_{5+[4]})\circ \delta_{[5],5+[4]}
\left(
\raisebox{ -13.3mm}{%
\includegraphics[width=37.5mm]{figure05.eps}}
\right)}\hspace*{38mm}\\[6mm]
& = &
\raisebox{-8.5mm}{%
\includegraphics[width=33.3mm]{figure10b.eps}}
\quad
\otimes
\quad
\raisebox{-8.5mm}{%
\includegraphics[width=21.6mm]{figure10c.eps}}
\end{eqnarray*}
\caption{\it %
The $(5,4)$ component of the restricted coproduct
$\overline{\delta}$.}
\label{fig10}
\end{figure}
\begin{lem}
The vector space $\mathcal{T}_\bullet$ is a graded coassociative
cocommutative counital coalgebra with respect to the
\emph{cosymmetrized coproduct $\hat{\delta}$},
defined on the $n$th component by
$$
\hat{\delta}
:=
\bigoplus_{A\coprod B=[n]}
(\mathsf{is}_A\otimes \mathsf{is}_B)\circ \delta_{A,B}\,.
$$
\end{lem}
This follows from~\cite[pp.207]{patreu04}. Equivalently, we have
$$
\hat{\delta}(P)
=
\sum_{A\coprod B} P|_A\otimes P|_B
$$
for all $n\in\mathbb{N}_0$, $P\in\mathsf{Comp}_{[n]}$.
The cosymmetrized coproduct on increasing trees can also be
described in terms of the contraction process introduced in
Section~\ref{S1}. If $T$ is an increasing tree with $n$ branchings
and $A\coprod B=[n]$, then the $(A,B)$ component of
$\hat{\delta}(T)$ reads $T_A\otimes T_B$, where $T_A$
and $T_B$ are the contractions of $T$ relative to $A$
and $B$, respectively.
For example, if $T$ is the increasing tree displayed in
Figure~\ref{fig05} again and $A=\{1,2,5,8\}$, $B=\{3,4,6,7,9\}$,
then the $(A,B)$ component of $\hat{\delta}(T)$ is displayed in
Figure~\ref{fig11}.
\begin{figure}[h]
\begin{eqnarray*}
\lefteqn{%
(\mathsf{is}_A\otimes \mathsf{is}_B)\circ \delta_{A,B}
\left(
\raisebox{-13.3mm}{%
\includegraphics[width=37.5mm]{figure05.eps}}
\right)}\hspace*{46mm}\\[6mm]
& = &
\raisebox{-11.6mm}{%
\includegraphics[width=25mm]{figure11b.eps}}
\quad
\otimes
\quad
\raisebox{-11.6mm}{%
\includegraphics[width=17.5mm]{figure11c.eps}}
\end{eqnarray*}
\caption{\newline\it %
The $(\{1,2,5,8\},\{3,4,6,7,9\})$ component of the
cosymmetrized coproduct $\hat{\delta}$.}
\label{fig11}
\end{figure}
\begin{lem} \label{lem10}
The vector space $\mathcal{T}_\bullet$ is a graded
associative unital algebra with respect to the
\emph{restricted product $\overline{\ast}$},
defined on $\mathcal{T}_p\otimes\mathcal{T}_q$ by
$$
\overline{\ast}:=\ast\circ (\mathcal{T}_p\otimes \mathsf{is}_{[q],p+[q]})
$$
for all $p,q\in\mathbb{N}_0$.
\end{lem}
This is again a direct consequence of the definition of
a twisted algebra. In terms of trees, the restricted product
is obtained by grafting an increasing tree $T_1$ with $p$
branchings on the left-most leaf of an increasing tree $T_2$
with $q$ branchings, resulting in a new tree $T$ with $p+q$
branchings. The level function on $T$ is obtained by keeping
the levels of $T_1$ (viewed now as a subtree of $T$) and
adding the root level $m$ of $T_1$ to all levels of $T_2$
(viewed also as a subtree of $T$). An example is given in
Figure~\ref{fig12}.
\begin{figure}[h]
$$
\raisebox{-6.6mm}{\includegraphics[width=13.3mm]{figure12a.eps}}
\qquad
\overline{\ast}
\quad
\raisebox{-6.6mm}{\includegraphics[width=29mm]{figure12b.eps}}
\quad
=
\quad
\raisebox{-6.6mm}{\includegraphics[width=41.6mm]{figure12c.eps}}
$$
\caption{\it %
The restricted product $\overline{\ast}$.}
\label{fig12}
\end{figure}
\begin{lem} \label{3.7}
The vector space $\mathcal{T}_\bullet$ is a graded
associative unital algebra with respect to
\emph{symmetrized product $\hat{\ast}$},
defined on $\mathcal{T}_p\otimes\mathcal{T}_q$ by
$$
\hat{\ast}
:=
\sum_{A\coprod B=[p+q],\,|A|=p,\,|B|=q}
\ast\circ (\mathsf{is}_{[p],A}\otimes \mathsf{is}_{[q],B})
$$
for all $p,q\in\mathbb{N}_0$.
\end{lem}
This construction is dual to the construction of $\hat{\delta}$
(see~\cite[pp. 212]{patreu04}).
A comparison with~\cite[Eq.~(20)]{bergeron2} shows that
the map $W$ considered after Lemma~\ref{3.4} is also an
isomorphism of algebras from $(\mathcal{T}_\bullet,\hat{\ast})$
onto $\mathsf{NCQsym}^*$.
The symmetrized product is slightly more difficult to describe
in terms of increasing trees and should be thought of as the
right notion of ``shuffle product'' for increasing trees.
We give an example in Figure~\ref{fig13}.
\begin{figure}[h]
\begin{eqnarray*}
\lefteqn{%
\raisebox{-6.6mm}{%
\includegraphics[width=13.3mm]{figure13a.eps}}
\qquad
\hat{\ast}
\quad
\raisebox{-6.6mm}{%
\includegraphics[width=41mm]{figure13b.eps}}}
\hspace*{32mm}\\[5mm]
& = &
\cdots
+
\quad
\raisebox{-10mm}{\includegraphics[width=48mm]{figure13c.eps}}
\quad
+
\cdots
\end{eqnarray*}
\caption{\newline\it %
The $(\{2,4,6\},\{1,3,5,7,8\})$ summand
of the symmetrized product $\hat{\ast}$.}
\label{fig13}
\end{figure}
\begin{theorem}
The triple $(\mathcal{T}_\bullet,\overline\ast ,\hat\delta )$
is a graded connected cocommutative Hopf algebra.
The triple $(\mathcal{T}_\bullet,\hat\ast ,\overline\delta )$
is a graded connected Hopf algebra and isomorphic to
the graded dual of the Hopf algebra of quasi-symmetric
functions in non-commuting variables, $\mathsf{NCQsym}^*$,
considered in~\cite{bergeron2}.
\end{theorem}
This follows from~\cite[Section~3]{patreu04}
and the remarks after Lemmas~\ref{3.4} and~\ref{3.7}.
\section{Freeness and enveloping algebra}
\label{S3}
In this section, we study the algebra structures on $\mathcal{T}_\bullet$
given by the reduced product $\overline{\ast}$ and the
symmetrized product $\hat{\ast}$.
Both algebras turn out to be free, and we will specify
a set of free generators.
With an eye on the Cartier-Milnor-Moore
theorem~\cite{milmoo65} (see~\cite{pat94} for a modern
combinatorial proof), we will then show that the cocommutative
Hopf algebra $(\mathcal{T}_\bullet,\overline{\ast},\hat{\delta})$
is in fact the enveloping algebra of a free Lie algebra.
Let $n\in\mathbb{N}_0$ and set $\mathsf{Comp}_n=\mathsf{Comp}_{[n]}$.
We say that a set composition $(P_1,\ldots,P_k)\in\mathsf{Comp}_n$
is \emph{reduced} if there is no pair $(a,m)$ with $a<k$
and $m<n$ such that $\bigcup\limits_{i=1}^a P_i=[m]$.
This corresponds to the notion of \emph{balanced tree},
where an increasing tree $T$ with $n$ branchings and $k$
levels is said to be \emph{balanced} if no pair $(a,m)$
with $a<k$ exists such that the branchings of $T$ with
levels in $[a]$ are naturally labeled by the elements
of $[m]$.
It is clear that each set composition $P$ can be written
uniquely as a product
$P=P^{(1)}\overline{\ast}\cdots\overline{\ast}P^{(m)}$
of reduced set compositions
$P^{(1)},\ldots,P^{(m)}$, where $\overline{\ast}$ is the
restricted product defined in Lemma~\ref{lem10}.
Equivalently, each increasing tree factorizes uniquely
as a restricted product of balanced increasing trees.
This implies:
\begin{prop} \label{free-reduced}
$(\mathcal{T}_\bullet,\overline{\ast})$ is a free associative
algebra, freely generated by the set of reduced set
compositions in $\mathsf{Comp}$.
\end{prop}
Here we write
$
\mathsf{Comp}
=
\bigcup\limits_{n\in\mathbb{N}_0} \mathsf{Comp}_n
$
for the set of all set compositions of initial subsets of
$\mathbb{N}$.
For the next result, we work over the rational
number field $\mathbb{Q}$ and consider the vector space
$\mathcal{T}_\bullet^\mathbb{Q}=\mathbb{Q}\otimes_\mathbb{Z} \mathcal{T}_\bullet$
with $\mathbb{Q}$-basis in bijection with $\mathsf{Comp}$.
\begin{cor} \label{envelope}
The graded connected cocommutative Hopf algebra
$(\mathcal{T}_\bullet^\mathbb{Q} ,\overline\ast ,\hat\delta )$
is the enveloping algebra of a free Lie algebra whose
set of generators is naturally in bijection with the
set of reduced set compositions or, equivalently, of
balanced increasing trees.
\end{cor}
\begin{proof}
This follows from~\cite[Lemma~22]{patreu04} since
all graded components of $\mathcal{T}_\bullet^\mathbb{Q}$ have finite dimension.
\end{proof}
The generators of this free Lie algebra can be computed
explicitly, using the techniques of~\cite{pat94} (see
also~\cite{pat93}). Let us write $\mathsf{Red}$ for the set of reduced
set compositions in $\mathsf{Comp}$ and $e^1$ for the logarithm of
the identity of $\mathcal{T}_\bullet$ in the convolution algebra of
linear endomorphisms of~$\mathcal{T}_\bullet$. Then the elements
$e^1(R)$, $R\in\mathsf{Red}$, form a set of free generators for
the primitive Lie algebra of
$(\mathcal{T}_\bullet^\mathbb{Q} ,\overline\ast ,\hat\delta )$.
Further details can be found in the proof
of~\cite[Lemma~22]{patreu04}.
In concluding this section, we note that
Proposition~\ref{free-reduced} holds for the symmetrized algebra
$(\mathcal{T}_\bullet,\hat{\ast})$ as well. This was shown by
Bergeron-Zabrocki~\cite{bergeron2} and
Novelli-Thibon~\cite{novthibon06}, in the dual setting of
quasi-symmetric functions in non-commuting variables.
In our approach it follows from Proposition~\ref{free-reduced},
by means of a standard triangularity argument: there is
a strict total ordering $\ll$ on set compositions such that,
for all $P\in\mathsf{Comp}_p$, $Q\in\mathsf{Comp}_q$,
\begin{equation} \label{triangle}
P\mathop{\hat{\ast}} Q
\in
P\mathop{\overline{\ast}} Q
+
\mathsf{span}_\mathbb{Z}\{\,U\in \mathsf{Comp}_{p+q}\,|\,P\mathop{\overline{\ast}} Q\ll U\,\}.
\end{equation}
We give the details below, for the sake of completeness.
The order $\ll$ is defined as follows.
Consider the set $\overline{\mathbb{N}}$ of the positive integers
together with the comma symbol: $\overline{\mathbb{N}}:=\mathbb{N}\cup \{,\}$.
We extend the natural order on $\mathbb{N}$ to $\overline{\mathbb{N}}$
by putting $,<1$.
Any set composition of $[n]$ can be viewed as a word over
the alphabet $\overline{\mathbb{N}}$. For example, the set composition $(145,26,3)$ translates into the word $145,26,3$.
If $A,B\subseteq\mathbb{N}$, $P$ is a set composition of $A$
and $Q$ is a set composition of $B$, we set
$$
P\ll Q
$$
if $|A|<|B|$, or if $|A|=|B|$ and $P$ is smaller than $Q$
with respect to the lexicographic order on words over the
alphabet $\overline{\mathbb{N}}$. For example, we have
$(245,169)\ll (245,178)$
because $6<7$,
and
$(23,4569)\ll (234,1,9,5)$
because $,<4$.
We claim that~\eqref{triangle} holds.
We start with following lemma which is clear from the definition.
\begin{lem} \label{congruence}
Suppose $P$, $Q$, $R$, $S$ are set compositions of $A$,
$B$, $C$, $D$, respectively, and that
$A\cap C=\emptyset=B\cap D$.
If $|A|=|B|$, then $P\ll Q$ implies that $P\ast R\ll Q\ast S$.
\end{lem}
To see~\eqref{triangle} now, note first that
$P\mathop{\hat{\ast}} Q-P\mathop{\overline{\ast}} Q$
is equal to the sum of all set compositions
$$
U=\mathsf{is}_{[p],A}(P)\ast\mathsf{is}_{[q],B}(Q),
$$
where $A$, $B$ are such that
$A\coprod B=[p+q]$, $|A|=p$ and $A\neq[p]$.
Hence, by Lemma~\ref{congruence}, it suffices to show that
\begin{equation} \label{helper}
P\ll \mathsf{is}_{[p],A}(P)
\end{equation}
for all subsets $A$ of $[p+q]$ of order $p$ with $A\neq[p]$.
Suppose $P=(P_1,\ldots,P_k)$, and let $\iota:[p]\to A$ be the
order-preserving bijection. Then
$\mathsf{is}_{[p],A}(P)=(\iota(P_1),\ldots,\iota(P_k))$.
Hence, if $i\in[k]$ is minimal with $P_i\neq\iota(P_i)$
and $j\in P_i$ is minimal with $j\neq\iota(j)$,
then $j<\iota(j)$ because $\iota$ preserves the orders
on $[p]$ and $A$. This implies~\eqref{helper}, hence
also~\eqref{triangle}.
\begin{theorem} \label{free-sym}
$(\mathcal{T}_\bullet,\hat{\ast})$ is a free associative
algebra, freely generated by the set of reduced set
compositions in $\mathsf{Comp}$.
\end{theorem}
\begin{proof}
We need to show that the symmetrized products
$P^{(1)}\mathop{\hat{\ast}}\cdots\mathop{\hat{\ast}} P^{(m)}$
of reduced set compositions $P^{(1)},\ldots,P^{(m)}$
in $\mathsf{Comp}$ form a $\mathbb{Z}$-basis of~$\mathcal{T}_\bullet$.
From Lemma~\ref{congruence} and~\eqref{triangle}
it follows that such a product is contained in
$$
P^{(1)}\mathop{\overline{\ast}}\cdots\mathop{\overline{\ast}} P^{(m)}
+
\mathsf{span}_\mathbb{Z}\{%
\,U\in \mathsf{Comp}\,|\,P^{(1)}\mathop{\overline{\ast}}\cdots\mathop{\overline{\ast}} P^{(m)}\ll U\,\}.
$$
Hence the claim follows from Proposition~\ref{free-reduced},
because the reduced products
$P^{(1)}\mathop{\overline{\ast}}\cdots\mathop{\overline{\ast}} P^{(m)}$
form a $\mathbb{Z}$-basis of $\mathcal{T}_\bullet$.
\end{proof}
\section{Enveloping algebras and trees}
\label{S4}
In this section we consider a number of subspaces of
the twisted descent algebra (up to natural identifications),
such as the direct sum
$\SS_\bullet :=\bigoplus\limits_{n\in\mathbb{N}_0}\mathbb{Z}[S_n]$ of the symmetric group algebras or the linear span of (planar rooted)
trees. We will show how the
Hopf algebra structures on set compositions and increasing trees
studied in the previous sections restrict to these subspaces.
In this way, we will obtain various algebraic structures.
Some of them are known: the Malvenuto-Reutenauer Hopf algebra of permutations, and the enveloping algebra structure on $\SS_\bullet$ introduced in~\cite{patreu04};
others seem to be new.
As far as we can say, there is no obvious connection between
our Hopf algebras of trees
and
the Hopf algebra structures on trees and forests appearing in renormalization theory (see, for
example,~\cite{broufra03,conkre98,grafig04}).
First of all, we show that the enveloping algebra
with underlying graded vector space $\SS_\bullet$,
as defined in~\cite{patreu04}, is a
sub-enveloping algebra of the enveloping algebra of twisted
descents
$(\mathcal{T}_\bullet,\overline{\ast},\hat{\delta})$
considered in Corollary~\ref{envelope} (up to an anti-involution).
Second, the Malvenuto-Reutenauer Hopf algebra is a
Hopf subalgebra of the (non-cocommutative) Hopf algebra of
twisted descents $(\mathcal{T}_\bullet,\hat{\ast},\overline{\delta})$.
Third, we turn to (planar rooted) trees and show that the corresponding graded vector space can be provided with an enveloping algebra structure. It embeds, as an enveloping algebra, into the enveloping algebra of increasing trees.
Finally, we show that a stronger result holds for (planar rooted) binary trees, whose associated sub-enveloping algebra of the enveloping algebra of trees also embeds into the enveloping algebra associated to~$\SS_\bullet$.
These various embeddings of enveloping algebras are
induced by embeddings of sets of combinatorial objects (trees,
set compositions, and so on) and illustrated in
Figure~\ref{diagram}.
Note that the embeddings are not canonical; they will be defined
below.
\begin{figure}[h]
\hspace*{-8pt}
$
\xymatrix@C=8ex@R=8ex{%
\mybox{Planar rooted trees}
\ar@{^{(}->}[r]%
&
\mybox{%
\begin{minipage}{29ex}
\begin{center}
Increasing\\
planar rooted trees\\
=\\
set compositions
\end{center}
\end{minipage}}%
\ \\
\mybox{Planar rooted binary trees}
\ar@{^{(}->}[u]%
\ar@{^{(}->}[r]%
&
\mybox{Permutations}
\ar@{^{(}->}[u]%
}
$
\caption{\it Diagram of embeddings.}
\label{diagram}
\end{figure}
As far as the space $\SS_\bullet$ is concerned, all our results build on the functorial superstructure associated to the
corresponding tensor species~$\SS$.
This tensor species is defined on objects by
$$
\SS[T]:=\mathbb{Z}[\mathsf{Aut}_{\mathsf{Fin}}(T)]
$$
for all $T\in\mathsf{Fin}$. If $T=\{t_1,\ldots,t_k\}\subseteq \mathbb{N}$
such that $t_1<\cdots<t_k$, we can think of any
bijection $\sigma\in\mathsf{Aut}_\mathsf{Fin}(T)$ as the
ordered sequence $(\sigma (t_1),\ldots,\sigma (t_k))$.
The action of a bijection $\phi$ from $T$ to $S$ is then given
by
$$
\phi (\sigma (t_1),\ldots,\sigma (t_k))
:=
(\phi\circ \sigma (t_1),\ldots,\phi\circ\sigma (t_k)).
$$
A twisted Hopf algebra structure on $\SS$ can be defined
as follows.
If
$S=\{s_1,\ldots,s_k\},T=\{t_1,\ldots,t_l\}\subseteq\mathbb{N}$
such that $S\cap T=\emptyset$
and
$\alpha\in \mathsf{Aut}_{\mathsf{Fin}}(S)$, $\beta\in\mathsf{Aut}_{\mathsf{Fin}}(T)$,
then
$$
(\alpha (s_1),\ldots,\alpha (s_k))
\times
(\beta (t_1),\ldots,\beta (t_l))
:=
(\alpha (s_1),\ldots,\alpha (s_k),\beta (t_1),\ldots,\beta (t_l)).
$$
Furthermore, if $\sigma\in\mathsf{Aut}_\mathsf{Fin}(S\coprod T)$, then
$$
\delta_{S,T}(\sigma):=\sigma|_S\otimes\sigma|_T\,.
$$
Here we write $\sigma|_S$ for the subsequence of $\sigma$
associated to the elements of $S$. For example,
$(3,5,2,4,1)|_{\{1,3,5\}}=(3,5,1)$.
The following lemma is a direct consequence of the definitions.
\begin{lem} \label{twisted-emb}
The canonical embeddings of the symmetric groups $S_n$ into
$\mathsf{Comp}_n$, defined by
\begin{equation} \label{perm-emb}
\sigma
\longmapsto
(\sigma (1),\ldots,\sigma (n))
\end{equation}
for all $n\in\mathbb{N}_0$ and $\sigma\in S_n$, yield an embedding
of the twisted Hopf algebra $\SS$ into the twisted Hopf
algebra $\mathcal{T}$ of set compositions.
\end{lem}
Recall that to each twisted Hopf algebra is associated a
symmetrized Hopf algebra and a cosymmetrized Hopf algebra.
The latter is an enveloping algebra (that is, a graded connected cocommutative Hopf algebra) if the twisted Hopf algebra is cocommutative.
Since
these constructions are natural in a functorial sense, an embedding of twisted Hopf algebras
induces embeddings of the associated symmetrized and
cosymmetrized Hopf algebras.
The cosymmetrized Hopf algebra stucture on $\SS_\bullet$
associated with the twisted Hopf algebra $\SS$
is defined as follows.
\begin{definition}
The graded vector space $\SS_\bullet$ is an enveloping algebra with respect
to the usual concatenation product $\times$ and
the cosymmetrized coproduct $\hat\delta$.
These are defined by
$$
(\alpha\times\beta )(i)
:=
\left\{\begin{array}{ll}
\alpha(i), & \mbox{ if $i\le n$,}\\[1mm]
n+\beta (i-n), & \mbox{ if $i>n$,}
\end{array}\right.
$$
for all $n,m\in\mathbb{N}_0$, $\alpha\in S_n$,
$\beta\in S_m$, $i\in[n+m]$,
and
$$
\hat{\delta}(\alpha)
:=
\sum\limits_{S\coprod T=[n]}
\mathsf{is}_S(\alpha|_S)\otimes \mathsf{is}_T(\alpha|_T)
$$
for all $n\in\mathbb{N}_0$, $\alpha\in S_n$.
\end{definition}
Here $\mathsf{is}_S$ and $\mathsf{is}_T$ are the standardization maps considered
earlier so
that, for example, $\mathsf{is}_{\{1,3,5\}}((3,5,2,4,1)|_{\{1,3,5\}})=(2,3,1)$.
A detailed description of the Lie algebra of primitive elements
associated with this enveloping algebra is given
in~\cite{patreu04}.
The symmetrized Hopf algebra stucture on $\SS_\bullet$
associated with the twisted Hopf algebra $\SS$
yields the Malvenuto-Reutenauer algebra. We recall
its definition.
\begin{definition}
The \emph{Malvenuto-Reutenauer algebra} $\SS_\bullet$ is a
graded connected Hopf algebra with respect to the
convolution product $\ast$ and the restricted coproduct
$\overline{\delta}$. These are defined by
$$
\alpha\ast\beta
:=
q_{(n,m)}\cdot (\alpha\times \beta )
$$
for all $n,m\in\mathbb{N}_0$, $\alpha\in S_n$, $\beta\in S_m$,
and
$$
\overline{\delta}(\alpha)
:=
\sum\limits_{i=0}^n
\alpha|_{[i]}
\otimes
\mathsf{is}_{\{i+1,\ldots,n\}}(\alpha|_{\{i+1,\ldots,n\}})
$$
for all $n\in\mathbb{N}_0$, $\alpha\in S_n$.
\end{definition}
Here we write $q_{(n,m)}$ for the sum in $\mathbb{Z}[S_{n+m}]$
of all permutations $\pi\in S_{n+m}$ such that
$\pi(1)<\cdots<\pi(n)$ and $\pi(n+1)<\cdots<\pi(n+m)$.
Note that these definitions agree with the structures studied
in~\cite{patreu04} only up to the involution
$\mathsf{inv}:\SS_\bullet\to\SS_\bullet$
which maps any permutation to its inverse;
for, in that article, twisted bialgebras were studied from
Barratt's point of view~\cite{barratt77} (that is, by
considering \emph{right} modules over symmetric groups or,
equivalently, by considering \emph{contravariant} functors
from the category of finite sets and bijections), whereas
here and in~\cite{patsch06}, twisted bialgebras have been
studied from Joyal's point of view~\cite{joyal86}
(that is, by considering
\emph{left} modules over symmetric groups or, equivalently, by
considering \emph{covariant} functors). As usual, one can move from one
point of view to the other using the map $\mathsf{inv}$.
Details on the two point of views and their relative behaviours
can be found in the first section of~\cite{patreu04}.
From Lemma~\ref{twisted-emb}, we can now deduce without further
ado:
\begin{theorem} \label{ord-emb}
The canonical embedding of $\SS_\bullet$ into $\mathcal{T}_\bullet$
given by~\eqref{perm-emb} is an embedding of enveloping
algebras
$
(\SS_\bullet,\overline{\ast},\hat{\delta})
\to
(\mathcal{T}_\bullet,\overline{\ast},\hat{\delta})
$.
It is also an embedding of the Malvenuto-Reutenauer Hopf
algebra into $(\mathcal{T}_\bullet,\hat{\ast},\overline{\delta})$.
\end{theorem}
We now turn to the enveloping algebra structures on trees and
binary trees and consider $\mathcal{T}_\bullet$ as the linear span
of all increasing trees. Recall that the product of two increasing
trees in the twisted algebra of increasing trees is the grafting
of the first tree on the left most leaf of the second. If we
assume, for simplicity, that the first tree has levels
$1,\ldots,n$ and the second tree has non-standard levels $n+1,\ldots,n+m$,
then the level of each branching in the product is the same
before and after the grafting.
The same operation (grafting on the left most
leaf) defines an associative product on the linear span of
(planar rooted) trees. The forgetful map
$$
\mathsf{Fgt}:\mathcal{T}_\bullet\to \overline{\mathcal{T}}_\bullet
$$
from increasing trees to trees is clearly an algebra map,
where we write
$$
\overline{\mathcal{T}}_\bullet
=
\bigoplus\limits_{n\in\mathbb{N}_0}\overline{\mathcal{T}}_n
$$
for the graded vector space with basis the set of trees,
graded by the number of branchings.
We claim that $\mathsf{Fgt}$ has a section $\mathsf{Inc}$ in the category of graded
associative algebras with identity.
Let $T$ be any non-empty planar rooted tree with $n$ branchings.
Then $T$ can be written uniquely as a wedge
$T=\bigvee (T_0,\ldots,T_m)$.
Let $b_1,\ldots,b_m$ denote the natural labels of the root
branchings of $T$. We assume inductively that an increasing tree
$(T_i,\lambda_i)=\mathsf{Inc}(T_i)$
has been defined for all $0\le i\le m$.
A level function $\lambda$ on $T$ can then be defined by
requiring that:
\begin{itemize}
\item[(i)]
$(T_i,\lambda_i)$ is the $S_i$-contraction of
$(T,\lambda)$ for all $0\leq i\leq m$,
where $S_i$ is the set of branchings
of $T$ which belong to $T_i$ (embedded in $T$).
\item[(ii)]
The levels on $T_i$ (embedded in $T$) are strictly
less than the levels on $T_j$ (embedded in $T$), for
all $0\leq i<j\leq m$.
\end{itemize}
We set $\mathsf{Inc}(T):=(T,\lambda)$ and observe:
\begin{lem} \label{lev}
The map $\mathsf{Inc}$ from trees to increasing trees is a section
of the forgetful map $\mathsf{Fgt}$. It defines an embedding of
algebras
$
(\overline{\mathcal{T}}_\bullet,\overline{\ast})
\to
(\mathcal{T}_\bullet,\overline{\ast})
$
where $\overline{\ast}$ is the left grafting product
on $\overline{\mathcal{T}}_\bullet$ (by slight abuse of notation)
and the restricted product on $\mathcal{T}_\bullet$.
\end{lem}
The proof is geometrically straightforward and left to the reader.
Any increasing tree in the image of $\mathsf{Inc}$ is called
\emph{left increasing}. Due to the recursive definition of $\mathsf{Inc}$,
left increasing trees can be characterized as follows.
\begin{lem} \label{inc-char}
Let $(T,\lambda)$ be an increasing tree. Then $(T,\lambda)$
is left increasing if and only if, for any branchings $b$
and $b'$ of $T$, we have $\lambda(b)<\lambda(b')$
whenever $b$ is to the left of $b'$ in $T$ and the associated
vertices $v$, $v'$ do not lie on a common path connecting a
leaf of $T$ with the root of $T$.
\end{lem}
\begin{cor}
Any contraction of a left increasing tree is left increasing.
\end{cor}
\begin{proof}
It is enough to understand how all three notions occuring
in Lemma~\ref{inc-char}
(to lie on a common path, to lie further to the left, to have a
smaller level) behave with respect to the contraction process.
Let $(T',\lambda')$ be a contraction of a left increasing
tree $(T,\lambda)$.
Let $b$, $b'$ be branchings of $T'$, and denote the corresponding
branchings of $T$ by $\tilde{b}$ and $\tilde{b}'$, respectively.
The contraction process is an order preserving map with
respect to the natural labelling of branchings. This
follows directly from its recursive left-to-right definition.
In particular,
$b$ is to the left of $b'$ in $T'$ if and only if $\tilde{b}$
is to the left of $\tilde{b}'$ in $T$.
Furthermore, we have $\lambda'(b)<\lambda'(b')$ if and
only if $\lambda(\tilde{b})<\lambda(\tilde{b}')$ since
$\lambda'$ is the standardization of the restriction
of $\lambda$ to the set of branchings of $T'$.
Let us assume that $b$ is to the left of $b'$. As mentioned
in Section~\ref{S1}, the vertices associated to $\tilde{b}$
and $\tilde{b}'$ lie on a common path from a leaf to the root
of $T$ if and only if either
$\lambda(\tilde{b})<\lambda(\tilde{b}')$
and for any branching $k$ between $\tilde{b}$ and $\tilde{b}'$
in the
left-to-right ordering $\lambda(k)>\lambda(\tilde{b})$, or
$\lambda(\tilde{b})>\lambda(\tilde{b}')$ and, with the same
notation, $\lambda(k)>\lambda(\tilde{b}')$. Since the same
characterization holds for $b$ and $b'$ in $T'$, and since levels
(up to standardization) and the left-to-right ordering are
preserved by the contraction process, it follows that the
property of lying on a common path from a leaf to the root is
preserved by the contraction $T\mapsto T'$.
In particular, if $b$ and $b'$ do not lie on a common path,
the same property is true for $\tilde{b}$ and $\tilde{b}'$.
Since $T$ is left increasing, we get
$\lambda(\tilde{b})<\lambda(\tilde{b}')$
and
$\lambda(b)<\lambda(b')$,
which concludes the proof.
\end{proof}
As a consequence of the preceding result,
the cosymmetrized coproduct $\hat{\delta}$
restricts to a coproduct on the $\mathbb{Z}$-linear span of left
increasing trees, $\mathsf{Inc}(\overline{\mathcal{T}}_\bullet)$, in $\mathcal{T}_\bullet$.
Combined with Lemma~\ref{lev}, this gives:
\begin{theorem} \label{inc-tree}
The linear span of left increasing trees (or, equivalently,
the linear span of trees) is a Hopf subalgebra of the
cocommutative Hopf algebra of increasing trees.
In particular, this Hopf algebra of trees is an enveloping
algebra. It is free as an algebra, and the enveloping algebra
of a free Lie algebra.
\end{theorem}
To conclude, we observe that the left increasing tree
$\mathsf{Inc}(T)$ corresponding to a \emph{binary} tree $T$ is
characterized by the property that it has a single grafting
at each level.
This property is also preserved by the contraction process.
Furthermore, the set composition $(\sigma(1),\ldots,\sigma(n))$
corresponding to a permutation $\sigma\in S_n$ can be
characterized in the same way.
Hence we get from Theorem~\ref{inc-tree}:
\begin{cor}
The linear span of (planar rooted) binary trees is
naturally embedded in $\SS_\bullet$ and $\mathcal{T}_\bullet$
as an enveloping algebra, and is the enveloping algebra
of a free Lie algebra.
\end{cor}
As far as we can say, there is no direct connection between
our Hopf algebra of planar binary trees and
Loday--Ronco's~\cite{lodron98}.
|
astro-ph/0512448
|
\section{Introduction}\label{sec:introduction}
In the standard model of accretion disks, turbulent viscosity plays an
important role in bringing material inward and transporting angular
momentum outward \citep[see][]{Frank2002}. At the same time, viscous
dissipation converts gravitational potential energy to thermal energy
and heats up the disk, which then radiates away this energy as thermal
emission. In most applications of accretion disks around central
objects, as in, e.g., X-ray binaries, the self-gravity of the flow is
negligible compared to that of the central object. However, there are
disk-like systems, such as active galactic nuclei as well as
protostellar and protoplanetary disks, where the effects of
self-gravity change not only the properties of angular momentum
transport \citep{Boss1998, Balbus1999, Mejia2005}, but also the energy
balance equation \citep{Bertin1997,Bertin1999,Bertin2001}, which
affect the global structure of the disk.
Besides the angular momentum transport problem, self-gravitating disks
are also very important in studying star and planet formation.
Indeed, gravitational instabilities in protoplanetary disks have been
proposed as viable planet formation mechanisms. Although there has
been a large amount of work done on gravitational instabilities in
these system \citep[see, for example,][and references
therein]{Pickett2003,Mejia2005}, it still remains an open question
whether the fragmentation by gravitational instabilities can produce
bound planetary objects, or if reservoirs of small solid cores are
needed for the accretion mechanism to lead to rapid planet formation.
In the first paper in this series (Chan, Psaltis, \& \"Ozel 2005), we
presented a pseudo-spectral method for solving the equations that
describe the evolution of two dimensional, viscous hydrodynamic flows.
There, we addressed issues related to the implementation of boundary
conditions, spectral filtering, and time-stepping in spectral methods,
and verified our algorithm using a suite of hydrodynamic test
problems. In this second paper of the series, we present our
implementation of a Poisson solver that allows us to take into account
the effects of self-gravity of the flow.
Spectral methods are particularly suitable for incorporating the
effects of self-gravity. Because Poisson's equation is linear, we can
pre-compute the numerical solution for the gravitational potential for
each individual mode of the density, thus reducing substantially the
computational cost of the method. In fact, Fourier methods have been
used extensively in analytical studies of gravitational potentials in
systems with periodic boundary conditions \citep{Binney1987}. On the
other hand, in numerical studies of self-gravitating disks, the
strengths of spectral methods have only been partially incorporated.
Hybrid hydrodynamic algorithms with self-gravity have been developed
in such a way that modified spectral methods are used only for solving
Poissson's equation for the gravitational field, whereas the
hydrodynamic parts are still treated with finite difference schemes.
For example, \citet{Boss1992} describe a spherical harmonic
decomposition method and a second-order scheme in the radial direction
to solve Poisson's equation, whereas \citet{Myhill1993} use a modified
Fourier method to find the gravitational potential of an isolated
distribution of sources. Both algorithms employ explicit second-order
finite difference methods to advance the hydrodynamic equations.
\citet{Pickett1998,Pickett2000} describe an implementation of an
algorithm that uses Fourier decomposition in the azimuthal direction
of cylindrical coordinate to solve Poisson's equation together with a
von Neumann \& Richtmeyer AV scheme for the hydrodynamics. Some other
examples can be found in Grandclement et al. (2001), Broderick \&
Rathore (2004), and Dimmelmeier et al. (2005). The main advantage of
using hybrid methods is that one can employ currently available
hydrodynamic algorithms based on finite difference schemes. However,
a hybrid method does not exploit the high order of the spectral
algorithm, because the hydrodynamic difference schemes typically have
a much lower order compared to that of the Poisson solver. Contrary
to these efforts, our algorithm uses a spectral decomposition method
for solving both the hydrodynamic and Poisson's equation, providing a
consistent treatment of the whole problem. To our knowledge, this is
the first time that spectral methods have been used in studying
astrophysical disks with self gravity.
By construction, there is an ambiguity in the definition of the
gravitational field in two-dimensional problems. We can assume either
that the density profile is independent of the third coordinate (i.e.,
an infinite cylinder) or that it is a delta (or any other
predetermined) function along the third coordinate (i.e., an
infinitesimally thin disk); the resulting gravitational field on the
two-dimensional domain of solution will be different in the two
cases. For example, the gravitational potential of a ``point source''
on the two-dimensional domain of solution will be proportional to
$\log(r)$ in the first case and to $1/r$ in the second, where $r$ is
the distance from the source. In order to consider both geometries,
here we describe two different approaches to computing the
gravitational field of a two-dimensional hydrodynamic flow with
pseudo-spectral methods. When the flow has the geometry of an infinite
cylinder, we use a standard two-dimensional pseudo-spectral Poisson
solver, which has been proven to be numerically stable and accurate.
When the flow has the geometry of an infinitesimally thin disk, we
perform a direct integration of the Green's function for the
gravitational potential, following the work of Cohl \& Tohline (1999).
In the following section, we present our assumptions and equations.
In \S\ref{sec:numerical_methods}, we discuss the details of our
numerical methods, include both the standard Poisson solver and the
Green's function integrator. Next, we present a series of tests in
\S\ref{sec:tests}, to verify our algorithm. Finally, we apply our
method to a numerical study of Toomre's stability criterion of
self-gravitating disks in \S5.
\section{Equations and Assumptions} \label{sec:equations_assumptions}
We consider two-dimensional, viscous, compressible flows. In this
second paper, we include self-gravity and continue to neglect the
magnetic fields of the flows. The hydrodynamic equations, therefore,
contain the continuity equation
\begin{equation}
\frac{\partial\Sigma}{\partial t} + \nabla\cdot(\Sigma\mathbf{v}) = 0, \label{eq:continuity}
\end{equation}
the Navier-Stokes equation
\begin{equation}
\Sigma\frac{\partial\mathbf{v}}{\partial t} + \Sigma(\mathbf{v}\cdot\nabla)\mathbf{v}
= -\nabla P + \nabla\mathbf\tau + \Sigma\;\mathbf{g}, \label{eq:navier_stokes}
\end{equation}
and the energy equation
\begin{equation}
\frac{\partial E}{\partial t} + \nabla\cdot(E\mathbf{v}) = - P\,\nabla\cdot\mathbf{v} + \Phi - \nabla\cdot\mathbf{q}
- \nabla\cdot\mathbf{F} - 2F_z. \label{eq:energy}
\end{equation}
We denote by $\Sigma$ the height-integrated density, by $\mathbf{v}$
the fluid velocity, and by $E$ the thermal energy. In the
Navier-Stokes equation, $P$ is the height-integrated pressure,
$\mathbf\tau$ is the viscosity tensor, and $\mathbf{g}$ is the
gravitational acceleration. We use $\Phi$ to denote the viscous
dissipation rate, $\mathbf{q}$ to denote the heat flux vector, and
$\mathbf{F}$ to denote the radiation flux on the $r$-$\phi$ plane.
The last term in the heat equation, $2F_z$, takes into account the
radiation losses in the vertical direction. The analytical forms of
the various physical quantities in
equations~(\ref{eq:continuity})--(\ref{eq:energy}) are given in
\citet{Chan2005}, except for the gravitational acceleration
$\mathbf{g}$, for which we need to introduce a new equation.
In Newtonian gravity, the gravitational field $\mathbf g$ is
conservative, so we can define the gravitational potential $\Psi$ by
\begin{equation}
\mathbf{g} \equiv -\nabla\Psi.
\end{equation}
The gravitational potential associated with the (three-dimensional)
mass density, $\rho$, is given by the volume integral
\begin{equation}
\Psi(t,\mathbf{x}) = -G\int\frac{\rho(t,\mathbf{x'})}{|\mathbf{x} - \mathbf{x'}|}d^3x' \label{eq:integrator_3d_total}
\end{equation}
over all space, where $G$ is the gravitational constant. Rewriting
equation~(\ref{eq:integrator_3d_total}) in differential form, we
obtain Poisson's equation
\begin{equation}
\nabla^2\Psi = 4\pi G\rho,
\end{equation}
with $\Psi$ satisfying the boundary condition $\Psi(t,\infty) = 0$ at
all times. When simulating astrophysical flows, the computational
domain $\mathcal{D}^{(3)}$ is usually finite. Based on its linearity,
we can decompose the Poisson equation into two parts
\begin{equation}
\nabla^2\Psi_\mathrm{int} = 4\pi G\rho_\mathrm{int}, \label{eq:possion_int}
\end{equation}
and
\begin{equation}
\nabla^2\Psi_\mathrm{ext} = 4\pi G\rho_\mathrm{ext},
\end{equation}
where $\rho_\mathrm{int}$ denotes the mass density within the
computational domain, which in our case is the flow density, and
$\rho_\mathrm{ext}$ refers to external sources such as a central
object and/or a companion star. The gravitational field is then given
by
\begin{equation}
\mathbf{g} = \mathbf{g}_\mathrm{int} + \mathbf{g}_\mathrm{ext} = -\nabla(\Psi_\mathrm{int} + \Psi_\mathrm{ext}).
\label{eq:total_g}
\end{equation}
In the astrophysical context of interest here, $\Psi_\mathrm{ext}$ is
usually generated by a set of spherical objects. Hence the external
gravitational potential is given by
\begin{equation}
\Psi_\mathrm{ext}(t,\mathbf x) = \sum_i \frac{GM_i}{|\mathbf x - \mathbf x_i(t)|},
\end{equation}
where $M_i$ and $\mathbf x_i(t)$ are the mass and positions of the
corresponding objects. Regarding the self-gravity of the flow, solving
equation~(\ref{eq:possion_int}) within $\mathcal{D}^{(3)}$ is
equivalent to computing the integral
\begin{equation}
\Psi_\mathrm{int}(t,\mathbf{x}) = -G\int_{\mathcal{D}^{(3)}}
\frac{\rho_\mathrm{int}(t,\mathbf{x'})}{|\mathbf{x} - \mathbf{x'}|}d^3x'. \label{eq:integrator_3d}
\end{equation}
Once $\Psi_\mathrm{ext}$ and $\Psi_\mathrm{int}$ are obtained, we can
then use equation~(\ref{eq:total_g}) to obtain the total gravitational
field and subsequently use it in both the Navier-Stokes equation and
in integrating the trajectories $\mathbf x_i(t)$ for the external
objects. Although for our test problems we assume that the central
object does not move and solve the hydrodynamic equations in a fixed
reference frame, it is trivial to generalize our algorithm to co-moving
coordinates.
There are two different approaches to reducing the above
three-dimensional formalism to two dimensions, depending on the
physical problem under study. The first one assumes that the density
is independent of the vertical coordinate $z$, i.e.,
$\rho_\mathrm{int}(t,r,\phi,z) \equiv \rho_\mathrm{ind}(t,r,\phi)$. In
this case, we define $\psi(t,r,\phi) \equiv
\Psi_\mathrm{int}(t,r,\phi,z)$ and we are left with a two-dimensional
problem. We obtain the gravitational potential by solving the
two-dimensional Poisson's equation \begin{equation}
\nabla^2\psi \equiv \left(\frac{\partial^2}{\partial r^2} + \frac{\partial}{r\partial r}
+ \frac{\partial^2}{r^2\partial\phi^2}\right)\psi = 4\pi G\rho_\mathrm{ind}
\end{equation}
\citep[This approach is used in ZEUS-2D, see][for details.]{Stone1992}
The second approach assumes that the vertical structure of the density
is described by some function $Z(r,z)$ that is independent of $t$ and
$\phi$, i.e., $\rho_\mathrm{int}(t,r,\phi,z) \equiv
\Sigma(t,r,\phi)Z(r,z)$. We are interested in the gravitational
potential on the $z=0$ plane, so that we solve for $\psi(t,r,\phi) =
\Psi_\mathrm{int}(t,r,\phi, z=0)$. This is a
``pseudo-two-dimensional'' problem, where the potential is not
described by a two-dimensional Poisson equation. In order to compute
the potential properly, the easiest method, in this case, is to
integrate directly the equation
\begin{equation}
\psi(t,r,\phi) = \int_{\mathcal{D}^{(2)}}\mathcal{G}(r,\phi;r',\phi')\Sigma(t,r',\phi')r'dr'd\phi',
\label{eq:integrator_2d}
\end{equation}
where $\mathcal{D}^{(2)}$ denotes the two-dimensional computational
domain and
\begin{equation}
\mathcal{G}(r,\phi;r',\phi') \equiv -G\int_{-\infty}^{\infty}
\frac{Z(r',z')dz'}{\sqrt{r^2 + r'^2 - 2rr'\cos(\phi-\phi') + z'^2}} \label{eq:green_2d}
\end{equation}
is the ``modified Green's function'' for our problem.
\section{Numerical Methods}\label{sec:numerical_methods}
In this section, we describe the numerical methods we use to solve the
two classes of self-gravity problems. As in the first paper
\citep{Chan2005}, we use pseudo-spectral methods, in which we expand
all functions in series by choosing the truncated functions to agree
with the approximated functions exactly at the grid points.
We use the modified Chebyshev collocation method along the radial
direction and the Fourier collocation method along the azimuthal
direction. Let $N+1$ and $M$ be the number of collocation points in
the radial and azimuthal direction, respectively. For every function
$f(r,\phi)$, we use the subscript $m$ to indicate the Fourier
coefficients
\begin{equation}
f(r,\phi) = \sum_{m=-M/2}^{M/2-1} \hat f_m(r)e^{im\phi}
\end{equation}
and the subscript $n$ to indicate the Chebyshev-Fourier coefficients
\begin{equation}
f(r,\phi) = \sum_{n=0}^N \sum_{m=-M/2}^{M/2-1} \check f_{nm}T_n(\bar r)e^{im\phi},
\end{equation}
where $\bar r \in [-1,1]$ denotes the standardized coordinate \citep[see][for notation]{Chan2005}.
We implement a standard two-dimensional pseudo-spectral Poisson
solver, which we describe briefly in \S\ref{sec:poisson_solver}
\citep[a good introduction is available in][]{Trefethen2000}. For the
direct integrator, we describe in \S\ref{sec:direct_integrator} how to
generalize the compact cylindrical Green's function expansion
\citep{Cohl1999} with pseudo-spectral methods. Both methods use
pre-computed matrices to speed up the algorithms. The run-time
computational cost for both methods are of order $\mathcal{O}(N^2M +
NM\log_2M)$.
\subsection{Two-Dimensional Pseudo-Spectral Poisson Solver} \label{sec:poisson_solver}
We describe here a two-dimensional, pseudo-spectral Poisson solver.
We use the fact that the basis polynomials in the $\phi$ direction,
$e^{im\phi}$, are also eigenfunctions of the operator
$\partial^2/\partial\phi^2$ in order to split the Poisson's equation
to
\begin{equation}
\Delta_m\hat\psi_m \equiv \left(\frac{\partial^2}{\partial r^2} + \frac{\partial}{r\partial r}
- \frac{m^2}{r^2}\right)\hat\psi_m = \hat{f}_m \label{eq:possion_m}
\end{equation}
for $m = 0,1,\dots,M/2$, where we have set $f \equiv 4\pi
G\rho_\mathrm{ind}$. Note that, in discrete Fourier transforms, $\hat
f_0$ and $\hat f_{M/2}$ are real but the other coefficients are
complex, so there are, in total, $M$ independent equations. We use
$\Delta_m$ to denote the differential operator for each $m$. In the
case of the $r-$direction, the Chebyshev polynomials are not
eigenfunctions of $\Delta_m$. Nevertheless, because $\Delta_m$ is
time independent and linear, we can write $\Delta_m$ in matrix
representation and pre-compute its inverse $\Delta_m^{-1}$ with an
$\mathcal{O}(N^2)$ solver for each $m$.
Let $\bar r$ be the standardized coordinate of Chebyshev polynomials
and $r = g(\bar r)$ be the mapped (physical) coordinate
\citep[see][for more details]{Chan2005}. It is well known that the
spectral Chebyshev derivative in the standardized coordinate can be
written in matrix representation as
\begin{equation}
\bar D_{ij} = \left(\begin{array}{c|ccc|c}
\frac{2N^2+1}{6} & \cdots & 2\frac{(-1)^j}{1-x_j} & \cdots & \frac{1}{2}(-1)N \\\hline
\vdots & \ddots & & \frac{(-1)^{i+j}}{x_i-x_j} & \vdots \\
-\frac{1}{2}\frac{(-1)^i}{1-x_i} & & \frac{-x_j}{2(1-x_j^2)} & & -\frac{1}{2}\frac{(-1)^{N+i}}{1+x_i} \\
\vdots & \frac{(-1)^{i+j}}{x_i-x_j} & & \ddots & \vdots \\\hline
-\frac{1}{2}(-1)N & \cdots & -2\frac{(-1)^{N+j}}{1+x_j} & \cdots & -\frac{2N^2+1}{6}
\end{array}\right),
\end{equation}
which is a $(N+1)\times(N+1)$ matrix \citep[discussion of computing
this matrix accurately can be found in][]{Baltensperger2003}. The
derivative in the mapped coordinate can be obtained by the chain rule
\begin{equation}
\frac{\partial}{\partial r} = \frac{d\bar r}{dr}\frac{\partial}{\partial\bar r}
= \frac{1}{dg/d\bar r}\frac{\partial}{\partial r},
\end{equation}
so that the derivative matrix in the mapped coordinate is given by the
product
\begin{equation}
D_{ij} \equiv \frac{1}{(dg/d\bar r)|_{\bar r_i}}\bar D_{ij}.
\end{equation}
With this we are able to write
\begin{equation}
\Delta_{m,ij}
= \sum_k D_{ik}D_{kj} + \sum_k \frac{\delta_{ik}}{x_i}D_{kj} - \frac{\delta_{ij}}{x_i^2}m^2.
\end{equation}
The matrix $\Delta_{m,ij}$ is, of course, singular because a unique
solution to $\psi_m$ does not exist until the boundary conditions are
given.
An important case is the vanishing boundary condition
$\hat\psi_m(r_{\min}) = \hat\psi_m(r_{\max}) = 0$. Let
$\tilde\Delta_{m,ij}$ be the $(N-1)\times(N-1)$ sub-matrix of
$\Delta_{m,ij}$ generated by removing the first and last rows and
columns. Imposing the vanishing boundary condition to
$\Delta_{m,ij}^{-1}$ is then equivalent to finding the inverse of the
$\tilde\Delta_{m,ij}$ and filling back the first and last rows and
columns with zeros, i.e.,
\begin{equation}
\Delta^{-1(0)}_{m,ij} = \left(\begin{array}{c|ccc|c}
0 & 0 & \cdots & 0 & 0 \\ \hline
0 & & & & 0 \\
\vdots & & \tilde\Delta_{m,ij}^{-1}& & \vdots \\
0 & & & & 0 \\ \hline
0 & 0 & \cdots & 0 & 0
\end{array}\right).
\end{equation}
We can then calculate
\begin{equation}
\hat\psi^{(0)}_m(r_i) = \sum_{j=0}^N \Delta_{m,ij}^{-1(0)}\hat f_m(r_j), \label{eq:matrix_product}
\end{equation}
where the superscript $(0)$ indicates that the solutions satisfy the
homogeneous boundary conditions for all $m$. Taking the inverse
Fourier transform along the $\phi$-direction, we then obtain the
solution $\psi^{(0)}(r,\phi)$ which satisfies the vanishing boundary
conditions.
In order to apply more generic boundary conditions, we first look for
solutions $\hat\psi^{(1)}_m(r)$ that satisfy the homogeneous
differential equation
\begin{equation}
\Delta_m \hat\psi^{(1)}_m = 0 \label{eq:possion_homogenous}
\end{equation}
with the proper boundary conditions. Then it is clear that the sum
$\hat\psi^{(0)}_m + \hat\psi^{(1)}_m$ satisfies
\begin{equation}
\Delta_m\left(\hat\psi^{(0)}_m + \hat\psi^{(1)}_m\right) = \hat f_m
\end{equation}
with the same boundary conditions. The solutions to
equation~(\ref{eq:possion_homogenous}) are
\begin{equation}
\hat\psi^{(1)}_m(r) = \left\{\begin{array}{lll}
C_m\ln r + D_m & , & m = 0 \\
C_m r^{m} + D_m r^{-m} & , & m = 1,2,\dots,M/2
\end{array}\right. \label{eq:boundary}
\end{equation}
where $C_m$ and $D_m$ are complex constants, which are fixed by the
boundary conditions. In general, these boundary conditions depend on
azimuth, i.e.,
\begin{eqnarray}
\psi(r_{\min},\phi) & = & \beta_\mathrm{in} (\phi), \\
\psi(r_{\max},\phi) & = & \beta_\mathrm{out}(\phi).
\end{eqnarray}
We first take the Fourier transform of $\beta_\mathrm{in}(\phi)$ and
$\beta_\mathrm{out}(\phi)$, we solve for $C_m$ and $D_m$ using
\begin{eqnarray}
\hat\beta_{\mathrm{in},m} & = & \left\{\begin{array}{lll}
C_m\ln r_{\min} + D_m & , & m = 0 \\
C_m r_{\min}^m + D_m r_{\min}^{-m} & , & m = 1,2,\dots,M/2
\end{array}\right. \label{eq:boundary_in}\\
\hat\beta_{\mathrm{out},m} & = & \left\{\begin{array}{lll}
C_m\ln r_{\min} + D_m & , & m = 0 \\
C_m r_{\min}^m + D_m r_{\min}^{-m} & , & m = 1,2,\dots,M/2,
\end{array}\right. \label{eq:boundary_out}
\end{eqnarray}
we add these terms to $\hat\psi^{(0)}_{m}$, and finally take the
inverse Fourier transform.
We summarize the two-dimensional pseudo-spectral Poisson solver using
the following steps:
\begin{enumerate}
\item We take the Fourier transform of each physical quantity $f$
along the $\phi$-direction and obtain $\hat f_m$ by a fast
Fourier transform, which is of order $\mathcal{O}(NM\log_2
M)$.
\item We then compute the matrix product~(\ref{eq:matrix_product})
for each $m$, which is of order $\mathcal{O}(N^2M)$.
\item We compute $\hat\beta_m$ by a fast Fourier transform, which is
of order $\mathcal{O}(M\log_2 M)$.
\item We then solve for $C_m$ and $D_m$ using
equations~(\ref{eq:boundary_in}) and (\ref{eq:boundary_out}),
which is of order $\mathcal{O}(M)$.
\item We impose the boundary conditions using
equation~(\ref{eq:boundary}), for each $m$, and obtain
$\hat\psi_m(r_k)$, which is of order $\mathcal{O}(NM)$.
\item Finally, we take the inverse transform of $\hat\psi_m$ to
obtain the potential $\psi$, which is of order
$\mathcal{O}(NM\log_2 M)$.
\end{enumerate}
Therefore, the overall computational cost is $\mathcal{O}(N^2M +
NM\log_2M)$.
\subsection{Two-Dimensional Gravity Integrator} \label{sec:direct_integrator}
\begin{figure*}
\plotone{Iknm.eps}
\caption{Plots of the quantity $\mathcal{I}_m(\bar r';r)$ for the
first six values of $m$, when $N = 33$, $M = 64$, and
$Z(r,z) = \delta(z)$.} \label{fig:Iknm}
\end{figure*}
As described in \S\ref{sec:equations_assumptions}, we assume in this
case that the vertical structure of the density is time-independent
and has cylindrical symmetry. Hence we can write
\begin{equation}
\rho_\mathrm{int}(t,r,\phi,z) = \Sigma(t,r,\phi)Z(r,z), \label{eq:restriction}
\end{equation}
where we normalize $Z(r,z)$ along the $z$ direction to unity, i.e.,
\begin{equation}
\int_{-\infty}^\infty Z(r,z) dz = 1,
\end{equation}
so that $\Sigma$ has the physical meaning of a height-integrated
density.
We present here a new numerical method to compute the
integral~(\ref{eq:integrator_2d}) efficiently and accurately. Our
method is motivated by \citet{Cohl1999}, who showed that the
three-dimensional Green's function in cylindrical coordinates can be
expanded in the compact form
\begin{equation}
\frac{1}{|\mathbf{x} - \mathbf{x'}|}
\equiv \frac{1}{\pi\sqrt{rr'}}\sum_{m=-\infty}^{\infty}e^{im(\phi-\phi')}Q_{m-1/2}(\chi), \label{eq:cohl_expansion}
\end{equation}
where $\chi\equiv[r^2 + r'^2 + (z-z')^2]/2rr'$ and $Q_{m-1/2}(\chi)$
are the half-integer degree Legendre functions. Note that the $\phi$
and $\phi'$ dependence appears only in the exponential
$e^{im(\phi-\phi')}$. This can be done because the Green's function
itself can be written as a function of $\Delta\phi = \phi-\phi'$ so
that $Q_{m-1/2}(\chi)/\pi\sqrt{rr'}$ is simply the Fourier transform
of $|\mathbf{x} - \mathbf{x'}|^{-1}$ with respect to $\Delta\phi$.
This property is still true for the modified Green's function because
$Z$ is $\phi$-independent:
\begin{equation}
\mathcal{G}(r,\phi\;;r',\phi') = \mathcal{G}(r,r',\Delta\phi).
\end{equation}
Expanding it in Fourier series, i.e.,
\begin{equation}
\mathcal{G}(r,\phi;r',\phi') = \sum_{m=-\infty}^\infty\hat\mathcal{G}_m(r,r')e^{im(\phi-\phi')},
\end{equation}
and substituting it into equation~(\ref{eq:integrator_2d}), we obtain
\begin{equation}
\psi(r,\phi) = \int_{\mathcal{D}^{(2)}}\sum_{m=-\infty}^{\infty}\hat\mathcal{G}_m(r,r')e^{im(\phi-\phi')}
\Sigma(r',\phi')r'dr'd\phi'. \label{eq:integrator_2d_fourier}
\end{equation}
Note that the integration over $\phi'$,
\begin{equation}
\int_0^{2\pi}d\phi'\Sigma(r',\phi')e^{-im\phi'} = 2\pi\Sigma_m(r'),
\end{equation}
is the Fourier transform of $\Sigma(r',\phi')$ in the azimuthal
direction. Therefore, we can rewrite
equation~(\ref{eq:integrator_2d_fourier}) as
\begin{equation}
\psi(r,\phi) = \sum_{m=-\infty}^{\infty}\int_{r_{\min}}^{r_{\max}}\hat\mathcal{G}_m(r,r')
2\pi\hat\Sigma_m(r')e^{im\phi}r'dr'.
\end{equation}
Taking the Fourier transform of the whole equation again with respect
to $\phi$, we obtain a one-dimensional integral for each value of $m$
\begin{equation}
\hat\psi_m(r) = \int_{r_{\min}}^{r_{\max}}\hat\mathcal{G}_m(r,r')2\pi\hat\Sigma_m(r')r'dr'.
\label{eq:integrator_fourier_transformed}
\end{equation}
We now expand $\hat\Sigma_m(r')$ in Chebyshev polynomials, i.e.,
\begin{equation}
\hat\Sigma_m(r') = \sum_{n=0}^\infty\check\Sigma_{nm}T_n(\bar r').
\end{equation}
Recalling that $\bar r'$ is the standardized coordinate, we let $r' =
g(\bar r')$ be some coordinate mapping.
Equation~(\ref{eq:integrator_fourier_transformed}) can then be
rewritten as
\begin{equation}
\hat\psi_m(r) = \sum_{n=0}^\infty\check\Sigma_{nm}\int_{r_{\min}}^{r_{\max}}
\hat\mathcal{G}_m(r,r')2\pi T_n(\bar r')r'dr'.
\end{equation}
Because the orthogonality condition of Chebyshev polynomials has a
weight function $(1-\bar r'^2)^{-1/2}$, i.e.,
\begin{equation}
\langle T_l, T_n \rangle
= \int_{-1}^{1}\frac{T_l(\bar r')T_n(\bar r')}{\sqrt{1 - \bar r'^2}}d\bar r'
= \frac{\pi}{c_n}\;\delta_{ln},
\end{equation}
where $c_0 = 2$ and $c_n = 1$ for $n = 1, 2, \dots$, it is not
difficult to see that, by defining
\begin{equation}
\hat\mathcal{I}_m(\bar r';r) = \hat\mathcal{G}_m\left[r,g(\bar r')\right]
\pi\sqrt{1-\bar r'^2}\;\frac{dg^2}{d\bar r'},
\end{equation}
we can write
\begin{eqnarray}
\hat\psi_m(r) & = & \sum_{n=0}^{\infty}\check\Sigma_{nm}
\int_{-1}^{1}\frac{\hat\mathcal{I}_m(\bar r';r)T_n(\bar r')}{\sqrt{1 - \bar r'^2}}d\bar r' \nonumber\\
& = & \sum_{n=0}^{\infty}\check\Sigma_{nm}\langle\hat\mathcal{I}_m, T_n \rangle \nonumber\\
& = & \sum_{n=0}^{\infty}\frac{\check\Sigma_{nm}\check\mathcal{I}_{nm}(r)}{c_n},
\label{eq:integrator_exact}
\end{eqnarray}
where $\check\mathcal{I}_{nm}(r)$ is the $n$-th Chebyshev coefficient
of $\hat\mathcal{I}_m(\bar r';r)$ with respect to $\bar r'$.
Up to this point, although the function $\Sigma(r',\phi')$ has been
written in Chebyshev-Fourier series, we have assumed that the series
is infinite and hence the formalism is exact. In order to perform it
numerically, we have to discretize
equation~(\ref{eq:integrator_exact}).
For the azimuthal direction, which is periodic, we simply replace the
continuous Fourier transform by a discrete Fourier transform and use
the proper normalization, which does not change
equation~(\ref{eq:integrator_exact}) but only limits the index $m$ to
the range $-M/2$ to $M/2-1$. For the $r$-direction, we truncate the
infinite sum at a finite number of terms, $N$. Therefore,
equation~(\ref{eq:integrator_exact}) becomes
\begin{equation}
\hat\psi_m(r_k) = \sum_{n=0}^{N}\frac{\check\Sigma_{nm}\check\mathcal{I}_{nm}(r_k)}{c_n},
\end{equation}
where $r_k$ denotes the collocation points in the $r$-direction.
Once we obtain $\check\mathcal{I}_{nm}(r_k)$ for $n = 0, 1, \dots, N$,
the calculation of the gravitational potential is trivial. However,
note that $Q_{m-1/2}(\chi)$ is related to the complete elliptic
integral, which is singular when $r' = r_k$. In general, this
difficulty arises because of the singularity of
$|\mathbf{x}-\mathbf{x}'|^{-1}$. In order to avoid the singularity,
we use the ``Chebyshev-roots grid'':
\begin{equation}
\bar r'_j = \cos\left[\frac{(2j + 1)\pi}{2N}\right],\ \ \ 0 \le j < N
\end{equation}
so that $\hat\mathcal{I}_m(\bar r'_j;r_k)$ is finite for all $m$ and
$j$. In Figure~\ref{fig:Iknm}, we plot $\hat\mathcal{I}_m(\bar r';r)$
for the first six values of $m$ for $N = 33$, $M = 64$, and $Z(r,z) =
\delta(z)$, which is the case of a very thin disk.
The Chebyshev coefficients $\check\mathcal{I}_{nm}$ are now well
defined as
\begin{equation}
\hat\mathcal{I}_m(\bar r'_j) = \sum_{n=0}^{N-1}\check\mathcal{I}_{nm}T_n(\bar r'_j)
= \sum_{n=0}^{N-1}\check\mathcal{I}_{nm}\cos\left[\frac{\pi n(2j+1)}{2N}\right].
\end{equation}
Note that $\check\mathcal{I}_{nm}$ cannot be computed by a standard
discrete cosine transform. It is related to the discrete Fourier
transform with different parity properties. This non-standard cosine
transform is known as type-II discrete cosine transform~\citep[see,
e.g.][p.41]{Frigo2003}. The computational order is still
$\mathcal{O}(N\log_2N)$. The only potential inconsistency about this
method is that there are only $N$ collocation points, instead of
$N+1$. Nevertheless, because the Chebyshev coefficients converge
exponentially for smooth functions, $\Sigma_{Nm}$ is assumed to be
very small. The artificial vanishing of $\check\mathcal{I}_{Nm}$ is,
therefore, negligible in the final solution.
In our implementation, we pre-compute the operator
$\check\mathcal{I}_{km}(r_k)$ to speed up the algorithm. The steps to
setup the gravity integrator can then be summarized by the following:
\begin{enumerate}
\item We first take the discrete Fourier transform of
$\Sigma(r_k,\phi_j)$ along the $\phi$-direction and obtain
$\hat\Sigma_m(r_l)$, which is of order $\mathcal{O}(NM\log_2
M)$.
\item We then calculate the matrix product $\hat\psi_m(r_k) =
\sum_{l=0}^{N}\hat\Sigma_m(r_l)
\check\mathcal{I}_{ml}(r_k)/c_n$ for each $m$, which is of
order $\mathcal{O}(N^2M)$.
\item Finally, we take the inverse Fourier transform of
$\hat\psi_m(r_k)$ and obtain the restricted potential
$\psi(r_k,\phi_j)$, which is of order $\mathcal{O}(NM\log_2
M)$.
\end{enumerate}
The overall computational cost is $\mathcal{O}(N^2M + NM\log_2M)$.
\subsection{Coupling to Hydrodynamic Equations}
We evolve the hydrodynamic equations following \citet{Chan2005}, i.e.,
using a low-storage, third-order explicit Runge-Kutta method. The
only difference is that, at the beginning of every time step, we
update the gravitational potential.
\section{Code Verification}\label{sec:tests}
In \citet{Chan2005}, we verified the hydrodynamic part of our
algorithm. In this second paper, we will first carry out a few
time-independent tests for both the Poisson solver and the direct
gravity integrator. We use for each test an analytic
density-potential pair and show that the numerical solutions agree
with the analytic expressions. We also perform some time-dependent
tests, which demonstrate that the gravity solver couples to the
hydrodynamic equation correctly.
\subsection{Time Independent Tests of Poisson Solver}
\label{sec:poisson}
\begin{figure*}
\plotone{poisson.eps}
\caption{Grayscale plots of the analytical function
$f_\mathrm{ana}$, the numerical potential
$\psi_\mathrm{n}$, and the difference $\epsilon$ between
the numerical solution and the analytical solution for the
test problem discussed in \S\ref{sec:poisson}. In all
plots, darker shades correspond to larger magnitude.}
\label{fig:poisson}
\end{figure*}
The simplest way to test the Poisson solver is to compare some
numerical potential $\psi_\mathrm{num}$ obtained by the Poisson solver
to its analytical expression $\psi_\mathrm{ana}$. The tests in this
subsection are done in the computational domain $\mathcal{D}^{(2)} =
[0.2,1.8]\times[-\pi,\pi)$. We consider a very simple potential
\begin{equation}
\psi_\mathrm{ana}(r,\phi) = \frac{1}{3}\left[r^2 - \sigma\left(1.82r - \frac{0.0648}{r}\right)\right]\sin\phi,
\end{equation}
where $\sigma$ is some arbitrary parameter. We chose this potential
because its Laplacian takes the form
\begin{equation}
f_\mathrm{ana}(r,\phi) = \nabla^2\psi_\mathrm{ana} = \sin\phi,
\end{equation}
which is independent of $\sigma$. Hence, the parameter $\sigma$
forces our solution to satisfy different boundary conditions.
We choose three different boundary conditions to test the Poisson
solver. The first one is
\begin{equation}
\left\{\begin{array}{rcl}
\psi_\mathrm{ana}(r_{\min}, \phi) & = & \frac{1}{3}r_{\min}^2\sin\phi \\
\psi_\mathrm{ana}(r_{\max}, \phi) & = & \frac{1}{3}r_{\max}^2\sin\phi
\end{array}\right.\;,
\end{equation}
which corresponds to $\sigma = 0$. For the second one, we choose the
vanishing boundary condition
\begin{equation}
\left\{\begin{array}{rcl}
\psi_\mathrm{ana}(r_{\min}, \phi) & = & 0 \\
\psi_\mathrm{ana}(r_{\max}, \phi) & = & 0
\end{array}\right.\;,
\end{equation}
which corresponds to $\sigma = 1$. Finally, we also choose
\begin{equation}
\left\{\begin{array}{rcl}
\psi_\mathrm{ana}(r_{\min}, \phi) & = & \frac{1}{3}(r_{\min}^2 - 3.64r_{\min} + 0.1296/r_{\min})\sin\phi \\
\psi_\mathrm{ana}(r_{\max}, \phi) & = & \frac{1}{3}(r_{\max}^2 - 3.64r_{\max} + 0.1296/r_{\max})\sin\phi
\end{array}\right.\;,
\end{equation}
which corresponds to $\sigma = 2$. In Figure~\ref{fig:poisson}, we
show the contour of $f_\mathrm{ana}$, $\psi_\mathrm{num}$, and the
difference, $\epsilon = |\psi_\mathrm{num} - \psi_\mathrm{ana}|$, for
each choice of $\sigma$. The contour lines for $\psi_\mathrm{num}$
show clearly how the boundary conditions change the solution. The
numerical results agree very well with the analytical solution and the
difference is only of order $10^{-15}$, which is the machine accuracy.
\subsection{Time Independent Tests of Gravity Integrator} \label{sec:int}
\begin{figure*}
\plotone{exponential_disks.eps}
\caption{Grayscale plots of the density $\Sigma$, the potential
$\psi$, and the difference $\epsilon$ between the
analytical and the numerical values of the gravitational
potential for a selection of infinitesimally thin
exponential disks (\S\ref{sec:int}). The top panel
corresponds to $\sigma=0.025$, the middle panel corresponds
to $\sigma=0.05$, and the bottom panel corresponds to
$\sigma=0.1$.}\label{fig:exponential_disks}
\end{figure*}
\begin{figure*}
\plotone{gaussian_spheres.eps}
\caption{Grayscale plots of the density $\Sigma$, the potential
$\psi$, and the difference $\epsilon$ between the
analytical and the numerical value of the gravitational
potential of a collection of three-dimensional Gaussian
spheres (\S\ref{sec:int}). The top panel corresponds to
$\sigma=0.05$, the middle panel corresponds to
$\sigma=0.1$, and the bottom panel corresponds to
$\sigma=0.2$.}\label{fig:gaussian_spheres}
\end{figure*}
To test the gravity integrator, we consider an infinitesimally thin
disk, with an exponentially decaying surface density, i.e.,
\begin{equation}
\rho(r,\phi,z) = \Sigma(r,\phi)\delta(z) = \Sigma_0 e^{-r/\sigma}\delta(z).
\end{equation}
The corresponding potential on the $z = 0$ plane is given by
\begin{equation}
\psi(r,\phi) = -\pi G\Sigma_0 r[I_0(y)K_1(y) - I_1(y)K_0(y)]
\end{equation}
where $y \equiv r/2\sigma$ \citep{Binney1987}. The functions $I_n(y)$
and $K_n(y)$ are the modified Bessel functions of the first and second
kinds, respectively. The modified Green's function for this problem
is simply
\begin{equation}
\mathcal{G}(r,\phi,r',\phi') = \frac{-G}{\sqrt{r^2 + r'^2 - 2rr'\cos(\phi-\phi')}}. \label{eq:green_delta}
\end{equation}
In order to test our gravity integrator for non-axisymmetric problems,
we place three disks in the computational domain with each disk
centered at some $(r_i,\phi_i)$. Let
\begin{equation}
R_i \equiv \sqrt{r^2 + r_i^2 - 2rr_i\cos(\phi-\phi_i)} \label{eq:define_R}
\end{equation}
be the distance between $(r_i,\phi_i)$ and $(r,\phi)$. Normalizing
the total mass for each disk to unity, we have
\begin{equation}
\Sigma_{(r_i,\phi_i)}(r,\phi) = \frac{1}{2\pi\sigma^2}\exp\left(-\frac{R_i}{\sigma}\right).
\end{equation}
The corresponding potential is
\begin{equation}
\psi_{(r_i,\phi_i)}(r,\phi) = -\frac{G}{\sigma}y_i[I_0(y_i)K_1(y_i) - I_1(y_i)K_0(y_i)]\;,
\end{equation}
where $y_i \equiv R_i/2\sigma$.
Because the potential is singular, we put the center of each disk off
grid. Specifically, we choose the coordinates $(0.9,\pi/4)$,
$(0.9,\pi)$, and $(1,-\pi/3)$. We also change the total mass of each
disk, in order to avoid cancellation of errors by symmetry, by
multiplying them by different constants so that
\begin{equation}
\Sigma_\mathrm{ana}(r,\phi)
= \Sigma_{(0.9,\pi/4)}(r,\phi) + \frac{1}{2}\Sigma_{(0.9,\pi)}(r,\phi) + 2\Sigma_{(1,-\pi/3)}(r,\phi).
\end{equation}
We repeated this test for different choices of $\sigma$, namely,
$\sigma = 0.025$, $\sigma = 0.05$, and $\sigma = 0.1$. In
Figure~\ref{fig:exponential_disks}, we show the grayscale plots for
$\Sigma_\mathrm{ana}$, $\psi_\mathrm{num}$, and the fractional error
$\epsilon = |\psi_\mathrm{num}/\psi_\mathrm{ana} - 1|$, for each value
of $\sigma$. The contour lines in the error plots show that, at the
smooth density region, the error is of order $10^{-5}$. The maximum
errors appear around the density peak, where the analytic potential is
singular.
Besides being able to handle a $\delta$-function in the $z-$direction,
our gravity integrator is able to solve problems with other
time-independent and $\phi$-independent vertical structures. For
example, we consider the Gaussian sphere given by
\begin{equation}
\rho(r,\phi,z) = \frac{M}{(2\pi\sigma^2)^{3/2}}
\exp\left[-\frac{r^2+z^2}{2\sigma^2}\right]\;,
\label{eq:3d_gaussian}
\end{equation}
which is normalized so that the total mass is $M$. Here $\sigma$ is a
parameter that controls the spread of the mass in each sphere. As
$\sigma\rightarrow0$, the Gaussian sphere approaches a point source.
If we have a collection of such spheres with the same $\sigma$, the
vertical structure can be factored out of the equations.
Using Gauss' law, we find that the potential on the $z=0$ plane is
given by
\begin{equation}
\psi(r,\phi) = -\frac{1}{r}\mathrm{erf}\left(\frac{r}{\sqrt{2}\sigma}\right)\;,
\end{equation}
where $\mathrm{erf}(x)$ is the error function
\begin{equation}
\mathrm{erf}(x) = \frac{2}{\sqrt{\pi}}\int_0^x e^{-x'^2}dx'.
\end{equation}
For a collection of Gaussian spheres centered on the $z = 0$ plane
with the same $\sigma$, we can factor out the $z$-dependence, i.e.
\begin{equation}
\rho(r,\phi,z) = \sum_i\rho_{(r_i,\phi_i)}(r,\phi,z) = Z(r,z)\sum_i\Sigma_{(r_i,\phi_i)}(r,\phi)\;,
\end{equation}
where we have used the same notation as in the previous subsection to
indicate that the center of each sphere is located at $(r_i,\phi_i)$.
The normalized vertical structure is given by
\begin{equation}
Z(r,z) = \frac{1}{\sqrt{2\pi\sigma^2}}\exp\left(-\frac{z^2}{2\sigma^2}\right).
\end{equation}
The surface density for each Gaussian sphere is, therefore,
\begin{equation}
\Sigma_{(r_i,\phi_i)}(r,\phi) = \frac{1}{2\pi\sigma^2}\exp\left(-\frac{R_i^2}{2\sigma^2}\right),
\end{equation}
where $R_i$ is defined in equation~(\ref{eq:define_R}).
The modified Green's function for our problem is
\begin{eqnarray}
\mathcal{G}(r,\phi;r',\phi') & = & \int_{-\infty}^{\infty}
\frac{-Z(r',z')dz'}{\sqrt{r^2 + r'^2 - 2rr'\cos(\phi-\phi')+z'^2}} \nonumber\\
& = & -\frac{e^{R^2/4}K_0(R^2/4)}{\sqrt{2\pi\sigma^2}}\;,
\label{eq:green_3d_gaussian}
\end{eqnarray}
where $K_0$ denotes the modified Bessel function of the second kind
and we have set $G = 1$ for simplicity. We use the modified Green's
function described in equation~(\ref{eq:green_3d_gaussian}) to obtain
the corresponding $\check{\mathcal{I}}_{nm}(r_k)$. Then we follow the
algorithm described in \S\ref{sec:direct_integrator} to perform the
calculation for the gravitational field for three Gaussian spheres
located at $(r,\phi) = (1,0)$, $(0.75, 2\pi/3)$, and $(1.25, -2\pi/3)$
with a total mass of $2$, $1/2$, and $1$, respectively, i.e.,
\begin{equation}
\Sigma_\mathrm{ana}(r,\phi)
= 2\Sigma_{(1,0)}(r,\phi) + \frac{1}{2}\Sigma_{(0.9,3\pi/4)}(r,\phi) + \Sigma_{(1,-\pi/2)}(r,\phi)
\end{equation}
In Figure~\ref{fig:gaussian_spheres}, we show the numerical solutions
for $\sigma = 0.05$, $\sigma = 0.1$, and $\sigma = 0.2$. The maximum
numerical error is less than $0.5\%$.
\subsection{Free Fall of a Dust Ring Under Self-Gravity}
\begin{figure}
\plotone{free_fall.eps}
\caption{The radial density profile of the numerical solution for a
free-falling dust ring at $\phi = 0$. The solid lines
represent the solution using our spectral algorithm and the
open circles represent the solution using a simple N-body
code.} \label{fig:free_fall}
\end{figure}
\begin{figure*}
\plotone{orbiting_cylinders.eps}
\caption{Grayscale plots of the density $\Sigma$ of the two orbiting
cylinders and vectors plots of the velocity fields at
different times in the simulation. In all plots, darker
colors correspond to larger magnitudes.}
\label{fig:orbiting_cylinders}
\end{figure*}
In order to perform a dynamic test of our spectral self-gravity
solver, we follow the free-fall of a self-gravitating dust ring. The
initial condition of this problem is the same as in \S4.1 of
\citet{Chan2005}, i.e.,
\begin{equation}
\rho(r,\phi,z) = \rho(r,\phi) = \exp[-20(r-1)^2],
\end{equation}
and the initial velocity is zero. The only difference is that, this
time, we explicitly calculate the self-gravity of the flow instead of
using the gravitational field provided by a central object. We set
the gravitational constant to $G = 1$ and let the ring free fall. We
neglect pressure and viscosity and set the resolution to $513\times
64$ in order to resolve the sharp peak in the density profile that
appears at late times.
In order to test our implementation of the spectral method and in the
absence of an analytic solution to the problem we wrote a very simple
N-body algorithm. We placed 100,000 particles in the computational
domain based on the initial density of the ring. Then we computed the
gravitational interactions between each particle pair directly and
integrated their trajectories.
In Figure~\ref{fig:free_fall}, we plot the density profiles of the
numerical solution of the free falling dust ring at $\phi = 0$ using
the spectral algorithm and the N-body method. Different lines
represent the solution from our pseudo-spectral algorithm at different
times and the open circles represent the solution from the particle
method. The two solutions agree very well.
\subsection{Orbiting Cylinders}
\begin{figure}
\plotone{orbiting_l.eps}
\caption{The total angular momentum of the orbiting cylinders
(normalized initially to unity) as function of time. The
angular momentum is conserved to better than 2\% over $t =
100$, which corresponds to about 16 periods or $\sim
5\times 10^4$ timesteps.}\label{fig:orbiting_separation}
\end{figure}
We adopt this test from \citet{Fryer2005}, who simulated two spheres
orbiting around each other with smooth particle hydrodynamics.
Because our algorithm is two-dimensional, we assume that all variable
are independent of $z$ and modify the problem to two infinite
cylinders orbiting around each other.
Because the region outside the cylinders has very low density, small
fluctuations of the solution in this low-density region result in a
large error in the pressure force. In order to resolve the numerical
instability that arises from this effect, we evolve the whole set of
hydrodynamic equations including the energy equation and introduce a
small background term in both the continuity equation and the energy
equation. This background needs to be small compared to the physical
properties of the cylinders so that it does not affect the orbital
motion significantly, while at the same time it needs to be large
enough so it can screen out the errors at the low density region in
our algorithm. Then, a strong spectral filter is applied to the
velocity field to reduce numerical instabilities. The spectral
filters for density and energy, on the other hand, are relatively weak
in order to conserve mass and total energy.
We consider a Gaussian density distribution for each cylinder
\begin{equation}
\Sigma(r,\phi) = \frac{e^{-r^2/2\sigma^2}}{2\pi\sigma^2}.
\end{equation}
In order to find the initial condition for the energy equation, we
first need to solve for the gravitational potential using the Neumann
boundary condition $\partial\psi/\partial r = 0$ at the origin. The
second boundary condition is not important here because it only shifts
the potential by a constant, which does not affect the gravitational
field. Setting the arbitrary constant for the second boundary
condition to zero, we obtain
\begin{equation}
\psi = G\left[2\ln r - \mathrm{Ei}\left(-\frac{r^2}{2\sigma^2}\right)\right],
\end{equation}
where $\mathrm{Ei}(x)$ is the exponential integral function defined by
\begin{equation}
\mathrm{Ei}(x) = -\int_x^\infty \frac{e^{-x}}{x'}dx'\;.
\end{equation}
The corresponding gravitational field of this potential is
\begin{equation}
g(r,\phi) = \frac{2G}{r}\left(e^{-r^2/2\sigma^2}-1\right).
\end{equation}
Therefore, we require the pressure to be
\begin{equation}
P(r,\phi) \equiv \int \Sigma g\;dr = \frac{G}{2\pi\sigma^2}
\left[\mathrm{Ei}(-r^2/\sigma^2) - \mathrm{Ei}(-r^2/2\sigma^2)\right]
\end{equation}
in order to balance self-gravity. Note that we again set the
integration constant to zero in the pressure. The modified Green's
function for this problem is
\begin{equation}
\mathcal{G}(r,\phi,r',\phi') =
G\ln\left[r^2 + r'^2 - 2rr'\cos(\phi-\phi')\right]\;.
\end{equation}
In our simulation we have two Gaussian cylinders centered at
$(r_i,\phi_i)$. Therefore, we set
\begin{equation}
\Sigma_{(r_i,\phi_i)}(r,\phi) = \frac{e^{-R_i^2/2\sigma^2}}{2\pi\sigma^2}
\end{equation}
and
\begin{equation}
P_{(r_i,\phi_i)}(r,\phi) = \frac{G}{2\pi\sigma^2}
\left[\mathrm{Ei}(-R_i^2/\sigma^2) - \mathrm{Ei}(-R_i^2/2\sigma^2)\right],
\end{equation}
where $R_i$ is defined in equation~(\ref{eq:define_R}). Because the
pressure is singular when $R_i = 0$, we choose the initial position of
the cylinders $(r_i,\phi_i)$ off grid. In particular, we also choose
the background density such that the area between the cylinders
contains only $1\%$ of the total mass in the whole system. Because
the area of our domain is $\pi(1.8^2 - 0.2^2) = 3.2\pi$, we set
\begin{equation}
\Sigma(r,\phi) = 0.99\left[\Sigma_{(1,10^{-3})}(r,\phi) + \Sigma_{(1,\pi + 10^{-3})}(r,\phi)\right]
+ \frac{0.02}{3.2\pi}
\end{equation}
and
\begin{equation}
P(r,\phi) = P_{(1,10^{-3})}(r,\phi) + P_{(1,\pi + 10^{-3})}(r,\phi) + \frac{0.02}{3.2\pi}.
\end{equation}
The initial energy is then chosen using the equation $E = 3P/2$ that
is based on an ideal gas law. Setting $G = 1$, the angular velocity
of the cylinder needed to balance the gravitational force is equal to
unity. Hence, we set the initial velocities to
\begin{equation}
v_r = 0, \ \ \ v_\phi = r.
\end{equation}
We use $\sigma = 0.1$ and evolve the simulation to a dimensionless
time $t = 100$, which is equal to about 16 complete orbital rotations.
In Figure~\ref{fig:orbiting_cylinders}, we show the gray-scale plots
of the density as well as the velocity field for different times in
the simulation. In Figure~\ref{fig:orbiting_separation}, we plot the
angular momentum as a function of time. Our algorithm is able to
conserve angular momentum to better than 2\% for up to about 16
complete orbiting motion, which corresponds to $\sim 5\times 10^4$
timesteps. The 2\% dissipation of angular momentum is mostly due to
the strong spectral filtering, which is equivalent to artificial
viscosity, and the deformation of the two cylinders during the
simulation.
\section{The Stability of Self-Gravitating Hydrodynamic Disks}\label{sec:application}
\begin{figure}
\plotone{toomre.eps}
\caption{The minimum value of the Toomre parameter $\min(Q)$
anywhere in the disk, for different values of the
polytropic constant $K$. The three different lines
correspond to different values of the polytropic index
$\Gamma$. When $\min(Q) < 1$, i.e. in the shaded region,
the linear analysis suggests that the disk is unstable.
Filled circles denote simulations in which the disk was
unstable, whereas open circles denote simulations in which
the disk was stable.}\label{fig:toomre}
\end{figure}
\begin{figure*}
\plotone{q2_kmax.eps}
\caption{The Toomre parameter $Q^2$ as a function of radius $r$ for
solutions with different values of the polytropic constant
$K$. The dark gray region corresponds to $Q^2 < 0$, for
which no steady-state rotating solution exists; the light
gray region corresponds to $0 < Q^2 < 1$ which is Toomre
unstable; the white region corresponds to $Q^2>1$, which is
Toomre stable. The contour lines correspond to the size of
the minimum unstable wavelength according to the linear
mode analysis.} \label{fig:q2_kmax}
\end{figure*}
Pseudo-spectral methods effectively evolve in time the hydrodynamic
equations for each mode independently. For this reason, they are
ideal tools for confirming the results of linear mode analysis and for
extending them to the non-linear regime. As a case study, we address
here numerically the stability of self-gravitating
infinitesimally-thin disks.
We consider self-gravitating gas disks with a polytropic equation of state
\begin{equation}
P = K\Sigma^\Gamma, \label{eq:polytropic}
\end{equation}
where $K$ is the polytropic constant and $\Gamma$ is the polytropic
index. The disks are locally stable to all non-axisymmetric
perturbations \citep{Goldreich1965a,Goldreich1965b,Julian1966}. For
axisymmetric perturbations, linear-mode analysis of the hydrodynamic
equations results in the dispersion relation \citep[][]{Binney1987}
\begin{equation}
\omega^2 = \kappa^2 - 2\pi G\Sigma|k| + k^2c_\mathrm{s}^2,
\label{eq:dissipation}
\end{equation}
where we use $c_\mathrm{s}$ to denote the sound speed of the gas,
$\kappa$ to denote the epicyclic frequency in the disk, and $\omega$
and $k$ to denote the mode frequency and wavenumber, respectively.
The disk is locally stable if $\omega^2 > 0$. The condition for all
modes to be stable is given by Toomre's stability criterion
\citep{Safronov1960, Toomre1964},
\begin{equation}
Q \equiv \frac{c_\mathrm{s}\kappa}{\pi G\Sigma} > 1\;. \label{eq:toomre}
\end{equation}
Although criterion~(\ref{eq:toomre}) is only a local condition, it is
also sufficient for global stability of axisymmetric modes with $kr
\gg 1$. At the critical value $Q = 1$, the disk can be globally
unstable to non-axisymmetric modes. As $Q$ increases, the disk
becomes globally stable to all non-axisymmetric modes. However, for
very large values of $Q$, pressure dominates the self-gravity effects
and the disk becomes globally unstable to non-axisymmetric
perturbations again. Our goal in this section is to study numerically
the stability of self-gravitating disks in situations where the
unstable wavelengths are larger than the characteristic length scales
in the disks.
We simulate the time evolution of disks that obey Plummer's density
model~\citep[see][chapter~2]{Binney1987}
\begin{equation}
\Sigma(r) = \frac{M_\mathrm{d}\sigma}{2\pi(r^2 + \sigma^2)^{3/2}}\;,
\end{equation}
where $M_\mathrm{d}$ stands for the initial disk mass and $\sigma$ is
a parameter that controls the central concentration of the disk
density. The corresponding gravitational potential (at the $z=0$
plane) is given by
\begin{equation}
\psi(r) = - \frac{GM_\mathrm{d}}{\sqrt{r^2 + \sigma^2}}.
\end{equation}
We set the initial radial velocities to zero, i.e., $v_r = 0$, and
choose the azimuthal velocities, $v_\phi$, so as to balance the
pressure and self-gravity of the flow, i.e.,
\begin{equation}
\frac{v_\phi^2}{r} = \frac{2\pi G\Sigma r}{\sigma} - \frac{3K\Gamma\Sigma^{\Gamma-1}r}{r^2+\sigma^2}.
\end{equation}
Although Plummer's model assumes that the density is a
$\delta$-function along the $z$-direction, its corresponding scale
height $H = rc_\mathrm{s}/v_\phi$ is given by
\begin{equation}
\frac{H}{r} = \frac{c_\mathrm{s}}{v_\phi} = \left(\frac{2\pi Gr^2}{K\Gamma\Sigma^{\Gamma-2}\sigma}
- \frac{3r^2}{r^2 + \sigma^2}\right)^{-1/2}.
\end{equation}
We choose $129\times64$ grid points and perform the simulations in the
domain $[0.005,5]\times[-\pi,\pi)$ with $\sigma = 1$. We set the
total disk mass to $M_\mathrm{d} = 1$.
Figure~\ref{fig:toomre} a parameter study of the dependence of the
stability of the disk on the polytropic index $\Gamma$ and the
constant $K$. We plot the minimum value of the Toomre parameter
$\min(Q)$ anywhere in the disk against $K$. The three different lines
correspond to different values of the polytropic index $\Gamma$.
According to linear analysis, when $\min(Q) < 1$, there is some region
in the disk where the disk becomes unstable. We present the results
of our simulation on the same plot, with filled circles denoting the
solutions that become unstable and with open circles the solutions
that remained stable during the course of the simulations.
For small values of the polytropic index ($\Gamma\le 4/3$) or for
small values of the polytropic constant ($K<0.2$), our numerical
simulations agree with the predictions of the linear mode analysis.
The small disagreements near the $Q=1$ separatrix come from two facts:
that, (i) we have a finite domain of integration on which we have to
impose boundary conditions, and (ii) we cannot calculate the evolution
of the instabilities and the perturbations in the gravitational field
from matter outside the boundaries. Regarding point (i), we have used
the absorbing boundary condition discussed in the first paper in the
series \citep[see][]{Chan2005}, which reduce the reflection
significantly. However, a very small but finite amount of the energy
of the wave is still trapped in the domain, which causes the
instability. As a result, when the gravitational potential deviates
from Plummer's model because of the finite size of the domain, the
disk close to the outer boundaries becomes unstable.
A striking difference, however, appears when the polytropic index and
the polytropic constant become larger: our numerical solutions are
always stable, contrary to the predictions of the linear mode
analysis. The reason for this disagreement between the linear mode
analysis and our numerical simulations becomes evident in
Figure~\ref{fig:q2_kmax}, where we plot Toomre's parameter $Q^2$ as
function of radius $r$ for different values of the polytropic constant
$K$. When $Q^2 < 0$, there is no initial value of the azimuthal
velocity $v_\phi$ that satisfies the steady-state equation for radial
force balance; we shade this region using dark gray. For $0 < Q^2 <
1$, the disk is unstable according to Toomre's criterion; we shade
this region using light gray. Finally, when $Q^2 > 1$ the disk is
linearly stable; we leave this region white. We also plot in
Figure~\ref{fig:q2_kmax} contours that correspond to the minimum
unstable wavelength, in the unstable region, computed by
\begin{equation}
\frac{2\pi}{\lambda_{\min}} = k_{\max} = \frac{\pi\Sigma}{c_\mathrm{s}}\left(1 + \sqrt{1 - Q^2}\right)
\end{equation}
Figure~\ref{fig:q2_kmax} shows that when $\Gamma > 4/3$ and for large
values of the polytropic constant $K$, the minimum unstable wavelength
is larger than the extent of the region in which the disk is unstable.
For example, for $\Gamma = 4.5/3$ and for $K \in [0.42, 0.56]$, the
unstable region only extends to $r \in [0,1.5]$. However, in that
same region, the minimum unstable wavelength is $\lambda_{\min}
\approx 3$, in the appropriate units. Because the unstable
wavelengths are larger than the characteristic length scale in the
system (which can be defined here as the extent of the region in which
$0<Q<1$), the approximations involved in the linear mode analysis are
not justified. Indeed, the numerical simulations show that the
unstable modes cannot grow and the disk is stable.
\section{Conclusions}
Problems that involve self-gravity are usually time-consuming tasks in
computational physics. In standard finite difference methods,
Poisson's equation is solved as the steady state of a diffusion
equation (using relaxation and over-relaxation methods) and has to be
recalculated together with the hydrodynamic equations at every
timestep. Although hybrid algorithms have been developed which use
high-order methods to solve Poisson's equation, finite difference
schemes are still used for evolving the hydrodynamic equations. The
resulting inconsistency in the order of differencing the hydrodynamic
equations and Poisson's equations either significantly reduces the
accuracy of the Poisson solver, or requires the extra complication of
interpolating the gravitational filed at each grid point. More
importantly, existing two-dimensional hybrid algorithms can only
address a limited number of self-gravitating problems because the
solutions to the two-dimensional and three-dimensional Poisson's
equation are fundamentally different (see \S\ref{sec:introduction} and
\S\ref{sec:equations_assumptions}).
In this paper, we presented two different approaches to using
pseudo-spectral methods to solve self-gravity problems. For the first
approach, we described the implementation of a standard
pseudo-spectral Poisson solver that solves the two-dimensional
Poisson's equations to machine accuracy. Instead of solving a
diffusion equation, spectral methods allowed us to invert the Poisson
operator in spectral space, making the algorithm fast and accurate.
For the second approach, we investigated a fast gravity integrator for
disks-like flows with known, time-independent vertical structures.
This algorithm allowed us to study the evolution of flows with finite,
but not only infinitesimal, thickness. This improvement allows
two-dimensional algorithms to solve a whole new class of problems. We
demonstrated here the ability of our algorithm to compute properly and
efficiently the gravitational potential of flattened flows using
different test problems. Even for the high resolution that
corresponds to $129\times256$ collocation points, our Poisson solvers
and integrator use less than 10\% of the total computational time.
We also explored how to extend Toomre's stability criterion to
self-gravitating disks for which the characteristic properties of the
flows change over a length scale that is shorter than the minimum
unstable wavelength. Based on our simulations, we find that for
Plummer's density model, if the disk is hot and has a polytropic index
$\Gamma > 4/3$, all oscillatory modes in the disk are stable, contrary
to the predictions of the analytic calculation.
\acknowledgements
We thank an anonymous referee for constructive comments that improved
the clarity of our paper. C.-K.\,C. and D.\,P.\ also acknowledge
support from the NASA ATP grant NAG-513374.
|
2202.12318
|
\section{Introduction}
The study of random quotients of hyperbolic groups goes back to the origins of geometric group theory. Indeed, Gromov posed a question concerning Property (T) in such quotients in his original monograph \cite[$\S$4.5C]{Gromov_hyperbolic}. This question was later refined by Ollivier in \cite[$\S$IV.c.]{Ollivier_Jan_Invitation}. Let $G=\langle S\;\vert \; T\rangle $ be a finite presentation of a non-elementary hyperbolic group with growth rate $\mathfrak{h},$ so that letting $S_{l}(G)$ be the sphere of radius $l$ in $G$, we have $\vert S_{l}(G)\vert =\eta(1+o_{l}(1))\exp\{\mathfrak{h}l\}$ for some constant $\eta=\eta (G) >0$ \cite{Cannon_combinatorial_structure_hyperbolic_groups} (see e.g. \cite{calegari2013ergodic} for an overview). Since hyperbolic groups are coarse-geometric objects, it is most natural to study quotients at length `near' $l$; see e.g. \cite[\S 1.2C]{Ollivier_Jan_Invitation}.
\begin{question*}[Gromov--Ollivier]
Does a random quotient of a non-elementary hyperbolic group $G$ at density $d>1\slash 3$ have Property (T) with probability tending to $1$ as $l$ tends to infinity?
\end{question*}
The above question was answered in the case of free groups by a major result of \.{Z}uk \cite{Zuk} (c.f. \cite{kotowski, DRUTU_Mackay, ashcroft2021property}). This work provided a large class of hyperbolic groups with (T), a previously rare phenomenon. However, one of the major advances of geometric group theory when compared to classical combinatorial group theory is the ability to finely control the quotients of hyperbolic groups, as opposed to understanding only the quotients of free groups. It is therefore still of great interest to answer the Gromov--Ollivier question; we prove the following, answering the Gromov--Ollivier question in the affirmative.
\begin{theoremalph}\label{mainthm: property t in random quotients of hyperbolic groups}
Let $G=\langle S\;\vert \; T\rangle $ be a non-elementary hyperbolic group with growth rate $\mathfrak{h}.$ Let $\omega(l)\leq \log\log l$ be any slow-growing function. Let $d>1\slash 3$ and for each $l\geq 1$ uniformly randomly select a set $\mathcal{R}_{l}\subseteq Ann_{l,\omega}(G)$ of size $ \exp\{\mathfrak{h}ld\}$. The group $G\slash \langle\langle \mathcal{R}_{l}\rangle\rangle$ has Property (T) with probability tending to $1$ as $l$ tends to infinity.
\end{theoremalph}
A function $\omega$ is \emph{slow-growing} if both $\omega(n+1)\slash \omega(n)\rightarrow 1$ and $\omega (n) \rightarrow \infty$ as $n\rightarrow \infty.$ The main difficulty in the Gromov--Ollivier question stems from the fact that the Markov chain, $M$, parametrizing the automatic structure for $G$ will often not be connected, let alone ergodic. To circumvent this, we use a specific connected component $C^{max}$ of $M$ that arises in the work of \cite{gekhtman2018counting}. If we assume that $C^{max}$ is aperiodic, we may prove the following, stronger, theorem.
\begin{theoremalph}\label{mainthm: property t in random quotients of hyperbolic groups aperiodic case}
Let $G=\langle S\;\vert \; T\rangle $ be a non-elementary hyperbolic group with growth rate $\mathfrak{h}.$ Let $d>1\slash 3$ and for each $l\geq 1$ uniformly randomly select a set $\mathcal{R}_{l}\subseteq Ann_{l,2}(G)$ of size $\exp\{\mathfrak{h}ld\}$. If $C^{max}$ is aperiodic, then the group $G\slash \langle\langle \mathcal{R}_{l}\rangle\rangle$ has Property (T) with probability tending to $1$ as $l$ tends to infinity.
\end{theoremalph}
Clearly, Theorems \ref{mainthm: property t in random quotients of hyperbolic groups} and \ref{mainthm: property t in random quotients of hyperbolic groups aperiodic case} hold for any constant multiple of $\exp\{\mathfrak{h}ld\}$.
Since the Markov chain $M$ is connected and aperiodic for free groups, Theorem \ref{mainthm: property t in random quotients of hyperbolic groups aperiodic case} strengthens the result of \cite{Zuk, kotowski} (c.f. \cite{DRUTU_Mackay}), which considers random quotients of free groups, i.e. the Gromov--Ollivier question for free groups only. \footnote{Ollivier's question for free groups concerns random quotients at length $l$ and discusses elements of length $l\slash 3$, so clearly requires $l$ to be divisible by $3$. There is no distinction in the literature between the case of random quotients at exactly length $3l$ versus quotients at length near $l$, with function $\omega (l)=2$. It should be possible to extend Theorem \ref{mainthm: property t in random quotients of hyperbolic groups aperiodic case} to choices of random subsets $R_{l}\subseteq S_{l}(G)$ in a manner similar to the extension of \.{Z}uk's theorem in \cite{ashcroft2021property}. Due to the length of this paper we have decided to omit this extension.}
Studying the random quotients of a general hyperbolic group is much more complicated than studying the random quotients of a free group. One of the main difficulties is that the sphere of radius $l$ in $F_{n}$ is far more easily understood than that of a more general hyperbolic group. This difficulty occurs as the Markov chain parameterizing geodesics is irreducible for free groups, but will often be reducible for hyperbolic groups. Detailed study of this Markov chain can be found in \cite{gekhtman2018counting} and \cite{gekhtman2020central}. A further difficulty of the Gromov--Ollivier question stems from the fact that we are quotienting by elements of the group, rather than words in the generating set. In the latter case, this would give us a quotient of a random quotient of $F_{n}$, and so we would be able to apply the work of \.{Z}uk \cite{Zuk} (c.f. \cite{kotowski}) to deduce (T). However, if the group $G$ has cogrowth $\xi$ (which necessarily satisfies $1\geq \xi\geq 1\slash 2$), and $d>1-\xi$, then such a random quotient is trivial with high probability \cite[Theorem 2]{olliviersharp}. We could therefore meet situations where $1\slash 3<1-\xi$, and so all such quotients would be trivial with high probability.
Importantly, the \emph{only} requirement we place on $G$ is that it is non-elementary hyperbolic. In particular, $G$ is allowed to contain torsion. We believe this to be the only result in the literature that considers the random quotients of \emph{any} non-elementary hyperbolic group. To our knowledge, the following two results are the only prior results in the literature studying random quotients of a class of non-elementary hyperbolic groups that is not limited to contain only free groups. In \cite{olliviersharp}, Ollivier considered quotienting non-elementary hyperbolic groups with `harmless torsion' by a set of $ \exp\{\mathfrak{h}ld\}$ words chosen randomly with respect to a sequence of measures $\{\mu_{l}\}_{l}$. In the case that the quotients are taken with respect to the uniform measure on elements of $S_{l}(G)$, then such a quotient is infinite, torsion-free, and hyperbolic with probability tending to $1$ if $d<1\slash 2$ and is trivial if $d>1\slash 2$ \cite[Theorem 3]{olliviersharp} (this is a specific case of the far more general result proved in the same paper; see \cite[Theorem 13]{olliviersharp}). Let us suppose instead that $G=\pi_{1}(X)$ for a non-positively curved cube complex, and take a random quotient of $G$ by a uniformly random set of $ \exp\{\mathfrak{h}ld\}$ words in $S_{l}(G)$. There exists a $d_{cub}=d_{cub}(\langle S\;\vert\;T\rangle)$ such that for $d<d_{cub}$, a random quotient at density $d$ is also the fundamental group of a non-positively curved cube complex with probability tending to $1$ \cite{futer2021cubulating}. Of course, Futer--Wise's theorem cannot hold for all hyperbolic groups $G$. If $G$ has (T), then this is inherited by all quotients of $G$, so that quotients of $G$ are the fundamental group of a non-positively curved cube complex if and only if they are finite. There are, to our knowledge, no previous results on Property (T) in random quotients of a hyperbolic group that is not the free group $F_{n}$.
The density bound in Theorem \ref{mainthm: property t in random quotients of hyperbolic groups} is clearly not optimal for every hyperbolic group $G$. If the group $G$ has $(T)$, then so does any of its quotients. We note that by Ollivier, for a group $G$ with harmless torsion, and for $d<1\slash 2$, the above quotients are also infinite, non-elementary hyperbolic, and torsion free with high probability \cite{olliviersharp}. Therefore, Theorem \ref{mainthm: property t in random quotients of hyperbolic groups}, coupled with \cite{olliviersharp}, provides an interesting class of group presentations with Property (T). However, we cannot apply the work of \cite{olliviersharp} in the case that $G$ has harmful torsion. The example provided by Ollivier of the failure of \cite[Theorem 4]{olliviersharp} is not enlightening for our purposes, since the quotient is not taken with a uniform selection of words; the chosen measures $\mu_{l}$ are non-uniform. This leads us to a natural question.
\begin{question}
Let $G$ be a non-elementary hyperbolic group with harmful torsion in the sense of Ollivier, and let $1\slash 3 <d<1\slash 2$. Let $\Gamma_{l}:=G\slash \langle \langle R_{l}\rangle\rangle$ be a sequence of quotients as in Theorem \ref{mainthm: property t in random quotients of hyperbolic groups} at density $d$ (so that has $\Gamma_{l}$ has (T) almost surely). Is $\Gamma_{l}$ almost surely infinite? Is $\Gamma_{l}$ almost surely hyperbolic?
\end{question}
The classical \emph{density model} of random groups is obtained by taking $G$ to be the free group $F_{n}=\langle a_{1},\hdots ,a_{n}\;\vert\;\rangle$ and restricting relators to be cyclically reduced words. It is typical when working with models of randomness to consider asymptotics. There are many theorems concerning the properties of random groups in the density model; all of the following statements hold with probability tending to $1$. Firstly, a random group in the density model at density $d<1\slash 2$ is infinite, non-elementary hyperbolic, and torsion free \cite{gromovasymptotic} (c.f. \cite{olliviersharp}). Groups in the density model are virtually special for $d<1\slash 6$ \cite{Ollivier-Wise, Agol13} and contain a free codimension-$1$ subgroup for $d<3\slash 14$ \cite{montee2021random} (c.f. \cite{mackay_przytycki2015balanced}). If we instead consider Property (T), then a random group in the density model has Property (T) at $d>1\slash 3$ \cite{Zuk,kotowski,ashcroft2021property} (c.f. \cite{DRUTU_Mackay}). There is a similar model (the \emph{k-gonal model}) to the density model where we keep $l$ fixed and let $n$ tend to infinity. There are results concerning hyperbolicity \cite{Zuk,odrsquaremodel,ashcroftrandom}, cubulation \cite{duong,odrcubulatingsquare,odrzygozdz2019bent}, and Property (T) \cite{Zuk,kotowski,antoniuktriangle,odrzygozdz2019bent,Montee_prop_t,ashcroft2021property} in this model.
\subsection{The structure of the paper}
The paper is structured as follows.
In Section \ref{sec: graph defs}, we remind the reader of some definitions relating to graphs and spectral graph theory. In Section \ref{sec: a spectral criterion for Property (T)}, we introduce a new spectral criterion for Property (T), which we obtain by an application of the work of Ozawa \cite{ozawapropertyt}. This new spectral criterion relates to an easily understood graph, $\Upsilon$, constructed from a set of relators. The edges are added to this graph independently, and so we can model $\Upsilon$ by a suitable random graph. We split each relator $r=uvw$ into $3$ almost-equal parts and add edges between $u$ and $w^{-1}$ in $\Upsilon$. The main advantage to this is that we do not need $uw$ to be geodesic. If we were to use \.{Z}uk's criterion, then we would also add the edges $(v,u^{-1})$ and $(w,v^{-1})$. Since these edges require $uv$ and $vw$ to be geodesic, this could effectively decompose $\Upsilon$ into strongly connected `cliques', with distinct cliques only connected by few edges. This could potentially decrease the first eigenvalue of $\Upsilon$ below $1\slash 2$ (the value required by \.{Z}uk's criterion \cite{zuk1996}).
The spectral criterion we introduce requires two objects to be analysed. Firstly, we are required to find a language of geodesics with certain properties. We use a result of \cite{gekhtman2018counting} on the dynamical structure of geodesics in hyperbolic groups in Section \ref{sec: counting geodesics in hyperbolic groups}, which allows us to study a set of geodesic words. Section \ref{appendix: Spectral theory of restricted graphs} studies the eigenvalues of certain random graphs that model the graph $\Upsilon$ appearing in our spectral criterion. Section \ref{sec: Property (T) in quotients of hyperbolic groups} then uses these results to provide a proof of Theorem \ref{mainthm: property t in random quotients of hyperbolic groups}. We then discuss how to adapt the proof strategy in the case that $C^{max}$ is aperiodic in order to prove Theorem \ref{mainthm: property t in random quotients of hyperbolic groups aperiodic case}.
\subsection{Some notation}
We now briefly discuss some notation and assumptions. We will fix an ordering on $S$ and choose the automatic structure for $G$ of shortlex geodesics, i.e. each element of $G$ is represented by a unique word of the same length. Let $\mathcal{L}^{geo}$ be the regular prefix-closed language of lexicographically first geodesics. For each element $g$ of $G$, we identify it with the unique word $w_{g}\in \mathcal{L}^{geo}$ that evaluates to $g$. In particular, by $S_{l}(G)$ we are referring to the elements in the sphere of radius $l$ in $G$, as well as the unique elements of $\mathcal{L}^{geo}$ evaluating to them.
We are dealing with asymptotics, and so we frequently arrive at situations where $m$ is some parameter tending to infinity that is required to be an integer. If $m$ is not integer, then we will implicitly replace it by $\lfloor m\rfloor$. Since we are dealing with asymptotics, this will not affect any of our arguments.
\begin{definition}
Given $m_{1}:\mathbb{N}\rightarrow \mathbb{N}$ a function such that $m_{1}(m)\rightarrow\infty$ as $m\rightarrow\infty$, we write $m_{2}=m_{2}(m_{1})$ to mean that $m_{2}(m)=f(m_{1}(m))$ for some function $f$, and $m_{2}(m_{1}(m))\rightarrow\infty$ as $m\rightarrow\infty$, i.e. $m_{2}$ only depends on $m_{1}$, and tends to infinity as $m_{1}$ tends to infinity.\end{definition}
The following are standard.
\begin{definition}
Let $f,g:\mathbb{N}\rightarrow \mathbb{R}_{+}$ be two functions. We write $f=o(g)$ if $f(m)\slash g(m)\rightarrow 0$ as $m\rightarrow\infty$,
$f=O(g)$ if there exists a constant $N\geq 0$ and $M\geq 1$ such that $f(m)\leq N g(m)$ for all $m\geq M$, and $f=\Omega (g)$ if $g=o(f)$. Finally, $f=\Theta (g)$ if $f(m)\slash g(m)\rightarrow c\in (0,\infty)$ as $m\rightarrow\infty.$
We write $f=o_{m}(g)$ etc to indicate that the variable name is $m$. Typically we will deal with functions $m_{2}=m_{2}(m_{1})$, and $f=f(m_{1},m_{2})$. We will write $f=o_{m_{1}}(g)$ etc to mean that the function $f'(m_{1})=f(m_{1},m_{2}(m_{1}))=o_{m_{1}}(g'(m_{1})),$ where $g'(m_{1})=g(m_{1},m_{2}(m_{1}))$.
\end{definition}
\begin{definition}
Let $\mathcal{M}(m)$ be some model of random groups (or graphs) depending on a parameter $m$, and let $\mathcal{P}$ be a property of groups (or graphs). We say that $\mathcal{P}$ holds \emph{asymptotically almost surely with} $m$ (\emph{a.a.s.}$(m)$) if
$$\lim_{m\rightarrow\infty}\mathbb{P}(G\sim \mathcal{M}(m)\mbox{ has }\mathcal{P})=1.$$
Again, we will regularly have to deal with cases where $m_{2}=m_{2}(m_{1})$ is fixed, $\mathcal{M}(m_{1},m_{2})$ is some model of random groups (or graphs) depending on parameters $m_{1}$ and $m_{2}$, and $\mathcal{P}$ is a property of groups (or graphs). We say that $\mathcal{P}$ holds \emph{asymptotically almost surely with} $(m_{1})$ (\emph{a.a.s.}$(m_{1})$) if
$$\lim_{m_{1}\rightarrow\infty }\mathbb{P}(G\sim \mathcal{M}(m_{1},m_{2}(m_{1}))\mbox{ has }\mathcal{P})=1.$$
\end{definition}
Finally, we will often be working with bipartite graphs. The vertex partition of a bipartite graph $\Sigma$ will always be written $V(\Sigma)=V_{1}(\Sigma)\sqcup V_{2}(\Sigma)$.
\section*{Acknowledgements}
I would like to thank Emmanuel Breuillard for discussions on random groups, as well as introducing me to the matrix Bernstein inequality and its application to random graphs, which has been of great use. I would like to thank Tom Hutchcroft for advice on references for Markov chains. I am grateful to Daniel Groves for conversations regarding the ergodic theory of encodings of geodesics in hyperbolic groups. I would like to thank Sam Taylor for pointing out that Lemma \ref{lem: Cmax gens finite index} was a result already present in the literature, as well as pointing out a mistake in the original proof of Theorem \ref{mainthm: property t in random quotients of hyperbolic groups}, which lead to the corrected statement and proof of Theorem \ref{mainthm: property t in random quotients of hyperbolic groups}, as well as the introduction of Theorem \ref{mainthm: property t in random quotients of hyperbolic groups aperiodic case}. As always, I would like to thank Henry Wilton for his advice and mathematical conversations, as well as comments on an earlier draft of this paper.
\section{Graphs and eigenvalues}\label{sec: graph defs}
Let $\Sigma=(V,E)$ be a graph (we allow multiple edges between vertices, as well as loops). Given two graphs $\Sigma_{1}=(V_{1},E_{1})$ and $\Sigma_{2}=(V_{2},E_{2})$, we define the \emph{union} of $\Sigma_{1}$ and $\Sigma_{2}$ as the graph $\Sigma_{1}\cup \Sigma_{2}:=(V_{1}\cup V_{2},E_{1}\sqcup E_{2})$. In particular, we take the union of vertices, and the disjoint union of edge sets (i.e. we assume that different graphs have disjoint edge sets).
Let $\Sigma=(V,E)$ be a graph with vertex set $V=\{v_{1},\hdots ,v_{m}\}$. The \emph{adjacency matrix} of $\Sigma$, $A(\Sigma)$, is the $m\times m$ matrix with $A(\Sigma)_{i,j}$ defined to be the number of edges between $v_{i}$ and $v_{j}$. The \emph{degree matrix} of $\Sigma$, $D(\Sigma)$, is the diagonal matrix with entries $D(\Sigma)_{i,i}=deg(v_{i})$. The \emph{normalised Laplacian} of $\Sigma$, $L(\Sigma)$, is defined by $$L(\Sigma)=I-D^{-1\slash 2}AD^{-1\slash 2}.$$
We note that $L(\Sigma)$ is symmetric positive semi-definite, with eigenvalues $$0\leq \lambda_{0}(L(\Sigma))\leq \lambda_{1}(L(\Sigma))\leq \hdots \leq \lambda_{m-1}(L(\Sigma))\leq 2.$$ For $i=1,\hdots, m$, we define $\lambda_{i}(\Sigma):=\lambda_{i}(L(\Sigma))$.
We can also define $\mathcal{L}(\Sigma)=I-D(\Sigma)^{-1}A(\Sigma).$ The matrices $L(\Sigma)$ and $\mathcal{L}(\Sigma)$ are similar, so that $\lambda_{i}(L(\Sigma))=\lambda_{i}(\mathcal{L}(\Sigma))$ for each $i$. We will switch between $L(\Sigma)$ and $\mathcal{L}(\Sigma)$ depending on which best suits our purpose.
If $M$ is a symmetric real $m\times m$ matrix, then $M$ has real eigenvalues, which we order by $\lambda_{0}(M)\leq \lambda_{1}(M)\leq \hdots \leq \lambda_{m-1}(M)$. We define the reverse ordering of eigenvalues $\mu_{1}(M)\geq \mu_{2}(M)\geq \hdots\geq \mu_{m}(M)$, i.e. $\mu_{i}(M)=\lambda_{m-i}(M)$. Therefore we may also define $\mu_{i}(\Sigma)=\mu_{i}(L(\Sigma)).$
The reason we introduce this ordering is that $\lambda_{i}(\Sigma)$ has a close connection to $\mu_{i}(A(\Sigma)).$
\begin{remark}\label{rmk: switching between eigenvalues of graphs}
Let $M$ be a symmetric $m\times m$ matrix. For $i=1,\hdots ,m:$ $$\mu_{i}(-M)=-\mu_{m+1-i}(M).$$
Let $\Sigma$ be an undirected graph. For $i=0,\hdots , \vert V(\Sigma)\vert -1$:
$$\lambda_{i}(\Sigma)=1-\mu_{i+1}(D(\Sigma)^{-1\slash 2}A(\Sigma)D(\Sigma)^{-1\slash 2})=1-\mu_{i+1}(D(\Sigma)^{-1}A(\Sigma)).$$\end{remark}
\begin{remark}
If $\Sigma$ is instead a directed graph, there is an analogous definition of $A(\Sigma)$, $D(\Sigma)$, $L(\Sigma)$, and $\lambda_{i}(\Sigma)$. In particular, $A(\Sigma)_{i,j}$ is equal to the number of directed edges starting at $v_{i}$ and ending at $v_{j}$. Furthermore $D(\Sigma)$ is the diagonal matrix with $D(\Sigma)_{i,i}=deg(v_{i})$ (i.e. the number of directed edges starting at $v_{i}$).
\end{remark}
We note the following well known lemma that allows us to find an upper bound for $\vert \mu_{i}(A(\Sigma))\vert.$
\begin{lemma}\label{lem: max eigenvalue of adjacency matrix}
If $\Sigma$ is an undirected graph, then
$$\max_{i}\vert \mu_{i}(A(\Sigma))\vert\leq \max_{v\in V(\Sigma)}deg(v).$$ If $\Sigma$ is bipartite, then
$$\max_{i}\vert \mu_{i}(A(\Sigma))\vert\leq \max_{\substack{v\in V_{1}(\Sigma)\\w\in V_{2}(\Sigma)}}\sqrt{deg(v)deg(w)}.$$
\end{lemma}
\begin{proof}
The first result follows as $||A(\Sigma)||_{\infty}=\max_{v\in V(\Sigma)}deg(v)$, and it is standard that $||A(\Sigma)||_{\infty}$ is an upper bound for the absolute values of the eigenvalues of $A(\Sigma)$.
The second inequality follows from e.g. \cite[3.7.2]{hornjohnson}, as we now detail. In this case, we have $$A(\Sigma)=\begin{pmatrix}
0&B\\B^{T}&0
\end{pmatrix},$$ for some matrix $B$. By definition, the set of eigenvalues of $A$ are the set of \emph{singular values} of $B$, $\{\sigma_{j}(B)\}_{j}$. Therefore, $\max_{i}\vert \lambda_{i}(A(G))\vert =\max_{j}\vert \sigma_{j}(B)\vert$. By \cite[3.7.2]{hornjohnson}, $$\max_{j}\vert \sigma_{j}(B)\vert\leq \sqrt{\vert \vert B\vert \vert_{\infty} \vert \vert B\vert \vert_{1}}=\max_{\substack{v\in V_{1}(\Sigma)\\w\in V_{2}(\Sigma)}}\sqrt{deg(v)deg(w)}.$$
\end{proof}
We now note some matrix inequalities, which we will use to analyse eigenvalues of graphs.
\begin{lemma*}[Weyl's inequality, \cite{Weyl}]
Let $A$ and $B$ be symmetric $m\times m$ real matrices. For $i=1,\hdots ,m$: $\mu_{i}(A)+\mu_{m}(B)\leq \mu_{i}(A+B)\leq \mu_{i}(A)+\mu_{1}(B).$
\end{lemma*}
In a similar spirit to this, we provide an application of the Horn inequalities.
\begin{lemma}\cite[Problem III.6.14.]{BhatiaRajendra1997Ma/R}
Let $A$ be a $\vert V\vert \times \vert V\vert$ positive real diagonal matrix with minimum entry $A_{min}$ and maximum entry $A_{max}$, and let $B$ be a $\vert V\vert \times \vert V\vert$ symmetric real matrix. Then $$A_{min}\mu_{2}(B)\leq \mu_{2}(AB)\leq A_{max}\mu_{2}(B).$$
\end{lemma}
\begin{definition}
For a vector $\vect{v}=(v_{1},\hdots ,v_{m})$ we define $\norm{v}=\sqrt{\sum v_{i}^{2}}.$ For an $m\times m$ matrix $A$, we define
\begin{enumerate}[label={$\arabic* )$}]
\item $\vert\vert A\vert\vert_{2}=\max \limits_{\vect{v}\neq \vect{0}}\dfrac{\vert\vert A\vect{v}\vert\vert}{\norm{v}},$
\item $\vert\vert A\vert\vert_{1}$ as the maximum absolute column sum of $A$, and
\item $\vert\vert A\vert\vert_{\infty }$ as the maximum absolute row sum of $A$.
\end{enumerate}
\end{definition}
We note that $(\vert\vert A\vert\vert_{2})^{2}\leq \vert\vert A\vert\vert_{1} \vert\vert A\vert\vert_{\infty}.$ Using Weyl's inequality and the fact that $\max_{i}\vert\mu_{i}(A)\vert\leq \vert\vert A\vert \vert_{2}$, we can deduce the following, which will be used implicitly throughout the paper.
\begin{lemma}
Let $A$ and $B$ be symmetric $m\times m$ real matrices. For $i=1,\hdots, m$:
$$\mu_{i}(A)-\vert\vert A-B\vert\vert_{2}\leq \mu_{i}(B)\leq \mu_{i}(A)+\vert\vert A-B\vert\vert_{2}$$
\end{lemma}
Finally, we note the following consequence of the Courant-Fischer theorem: see, e.g. \cite[Corollary III.1.2]{BhatiaRajendra1997Ma/R}.
\begin{theorem*}
Let $A$ be a symmetric $m\times m$ real matrix with first eigenvalue $\mu_{1}(A)$ and corresponding eigenvector $\vect{e}$. Then
$$\mu_{2} (A)= \max\limits_{\substack{\vect{v}\perp \vect{e}\\ \vert \vert \vect{v}\vert \vert =1}}\langle A\vect{v},\vect{v}\rangle=\max\limits_{\vect{v}\perp \vect{e}}\dfrac{\langle A\vect{v},\vect{v}\rangle}{\langle \vect{v},\vect{v}\rangle}.$$
\end{theorem*}
\section{A spectral criterion for Property (T)}\label{sec: a spectral criterion for Property (T)}
In this section we deduce a spectral criterion for Property (T); we first remind the reader of some of the relevant definitions. We focus only on finitely generated groups; for a further exposition the reader should see, for example, \cite{bekkadelaharpe}.
\begin{definition}
Let $\Gamma$ be a finitely generated group with finite generating set $S$, let $\mathcal{H}$ be a Hilbert space, and let $\pi: \Gamma\rightarrow\mathcal{U}(\mathcal{H})$ be a unitary representation of $\Gamma.$ We say that $\pi$ has \emph{almost-invariant vectors} if for every $\epsilon>0$ there is some non-zero $u_{\epsilon}\in\mathcal{H}$ such that for every $s\in S$, $||\pi(s) u_{\epsilon}-u_{\epsilon}||<\epsilon ||u_{\epsilon}||.$
We say that $\Gamma$ has \emph{Property (T)} if for every Hilbert space $\mathcal{H}$ and unitary representation $\pi:\Gamma\rightarrow\mathcal{U}(\mathcal{H})$ with almost-invariant vectors, there exists a non-zero invariant vector for $\pi$.
\end{definition}
\begin{remark}
It is standard that the above definition is not reliant on the choice of generating set $S$. Let $\Gamma$ be a finitely generated group. If $H$ is a finite index subgroup of $\Gamma$, then $\Gamma$ has Property (T) if and only if $H$ has Property (T) (see e.g. \cite[Theorem Theorem 1.7.1]{bekkadelaharpe}). If $\Gamma$ has Property (T) and $\Gamma'$ is a homomorphic image of $\Gamma,$ then $\Gamma'$ has Property (T) (see e.g. \cite[Thm 1.3.4]{bekkadelaharpe}).
\end{remark}
Now, let $G=\langle S\;\vert\;R\rangle$ be a finite presentation of a group. We assume that we have fixed a language $\mathcal{L}^{G}\subseteq (S\sqcup S^{-1})^{*}$ with the following properties:
\begin{enumerate}[label={$\arabic* )$}]
\item $\mathcal{L}^{G}$ is regular, prefix and suffix closed, and consists of geodesics words, and
\item for any $l\geq 1$, the set, $\mathcal{L}^{G}_{l}$, of words of length $l$ in $\mathcal{L}^{G}$ generates a finite index subgroup of $G$.
\end{enumerate}
Recall that a language $\mathcal{L}$ is \emph{prefix closed} if for any word $uv\in \mathcal{L}$, $u\in \mathcal{L}$. It is \emph{suffix closed} if for any word $uv\in \mathcal{L}$, $v\in \mathcal{L}$. Finally, it is \emph{regular} if it can be recognised by a finite state automaton. In our application, the language $\mathcal{L}^{G}$ will appear as a set of specific subwords of the language recognised by the canonical automatic structure on a hyperbolic group.
We define $\mathcal{L}^{G}_{l}$
to be the set of words in $\mathcal{L}^{G}$ of length $l$, and $\mathcal{L}^{G}_{l,\omega}$
to be the set of words in $\mathcal{L}^{G}$ of length between $l-\omega (l)$ and $l$.
Since the evaluation map $\mathcal{E}:\mathcal{L}^{G}\rightarrow G$ is not necessarily injective, we need to make the following definitions.
\begin{definition}
Let $G$ and $\mathcal{L}^{G}$ be as above, let $g\in G$ and $w\in \mathcal{L}^{G}$. We define the following.
\begin{enumerate}[label={$\arabic* )$}]
\item $\overline{w}$ is the image of $w$ under $\mathcal{E}$.
\item We say $g\in \mathcal{L}^{G}$ if there exists some $w\in \mathcal{L}^{G}$ with $\overline{w}=_{G}g$.
\item We define $word(g):=\{w\in \mathcal{L}^{G}\;:\; \overline{w}=_{G}g\}$, and
$$\Phi (g):=\vert word (g)\vert.$$
\end{enumerate}
\end{definition}
We now define the graph that plays a central role in our spectral criterion.
\begin{definition}[{\bf The graph} $\boldsymbol{\Upsilon(G,\mathcal{L}^{G},R)}$]
Let $$R=\{r=(x_{r},y_{r},z_{r})\}_{r}$$ be a finite set of triples of elements in $\mathcal{L}^{G}$, such that for each $r\in R$: $x_{r}y_{r}z_{r}$ is reduced without cancellation and lies in $\mathcal{L}^{G}$; and each $x_{r},y_{r},z_{r}$ is reduced and is not the empty word. Let $\mathcal{R}=\{x_{r}y_{r}z_{r}\;:\;r\in R\}$ and $\mathcal{W}(R):=\cup_{r\in R}\{x_{r},y_{r},z_{r}\}.$ Define the directed graph $\Upsilon(G,\mathcal{L}^{G},R)$ as follows. Let $$V(\Upsilon(G,\mathcal{L}^{G},R))=\mathcal{W}(R)\cup(\mathcal{W}(R))^{-1}:=\mathcal{W}\cup \{w^{-1}\;\vert\; w\in \mathcal{W}(R)\}.$$
In particular, we add in all words in $\mathcal{W}(R)$ and their inverses. For each triple $r=(x_{r},y_{r},z_{r})\in R$, add the directed edges $(x_{r},z_{r}^{-1}),(x_{r}^{-1},z_{r}).$ By construction, $A(\Upsilon(G,\mathcal{W},R))$ is symmetric, so that
$$L(\Upsilon(G,\mathcal{W},R)):=I-D(\Upsilon(G,\mathcal{W},R))^{-1\slash 2}A(\Upsilon(G,\mathcal{W},R))D(\Upsilon(G,\mathcal{W},R))^{-1\slash 2}$$ has real eigenvalues, all of which lie in $[0,2]$.
For a vertex $w\in V$ (so that $w\neq w^{-1}$) define $$rep (w):=\vert\{(u,w,v)\in R\} \vert+\vert\{(u,w^{-1},v)\in R\} \vert.$$ For a fixed choice of $R$ and for an element $g\in G$, we define $$deg (g):=\sum\limits_{\substack{w\in V\\\overline{w}=g}}deg_{\Upsilon(G,\mathcal{W},R)}(w),\mbox{ and }rep (g):=\sum\limits_{\substack{w\in V\\\overline{w}=g}}rep(w).$$
\end{definition}
We may now introduce our spectral theorem. Recall for $g\in G$, we write $g\in V(\Upsilon(G,\mathcal{L}^{G},R))$ when there exists $w\in V(\Upsilon(G,\mathcal{L}^{G},R))$ with $\overline{w}=_{G}g.$
\begin{theorem}\label{thm: directed new spectral criterion}
Let $G=\langle S\;\vert\; T\rangle$ be a finite presentation of a hyperbolic group. Let $R=\{r=(x_{r},y_{r},z_{r})\}_{r}$ be a finite collection of triples in $\mathcal{L}^{G}$ as above, such that $\mathcal{W}(R)$ generates a finite index subgroup of $G$. Let $\mathcal{R}=\{x_{r}y_{r}z_{r}:r\in R\}$.
Suppose that there exist constants $\epsilon,\;\delta>0$ such that for all $g\in G$ with $g\in V(\Upsilon(G,\mathcal{L}^{G},R))$: $ deg (g)\leq (1+\epsilon) deg(g^{-1})$ and $rep (g)\leq (1+\delta)deg(g). $
If $$\lambda_{1}\bigg(\Upsilon(G,\mathcal{L}^{G},R)\bigg)>\dfrac{1+\delta}{2-4\epsilon^{2}},$$ then $G\slash \langle \langle \mathcal{R}\rangle \rangle$ has Property (T).
\end{theorem}
In order to prove this, we will use a criterion due to Ozawa \cite{ozawapropertyt}. For $\Gamma$ a finitely generated group, let $\mathbb{R}[\Gamma]$ be the real group algebra, i.e. the set of all finitely supported functions $\Gamma\rightarrow \mathbb{R}$. We can write an element $\xi \in\RG{\Gamma}$ as $\sum_{g\in \Gamma}\xi (g)g$. The group algebra comes equipped with an involution map, $*$, given by $\xi^{*}(g)=\xi (g^{-1})$. We define the positive cone $$\Sigma^{2}\RG{\Gamma}=\left\{\sum_{i=1}^{n}\xi_{i}\xi^{*}_{i}\;:\;n\in\mathbb{N},\;\xi_{i}\in \RG{\Gamma}\right\}.$$ For $\xi\in \RG{\Gamma}$, we write $\xi\geq_{\RG{\Gamma}}0$ to indicate that $\xi \in \Sigma^{2}\RG{\Gamma}.$ Similarly, we write $\xi\geq_{\RG{\Gamma}} \phi$ if $\xi -\phi\geq_{\RG{\Gamma}} 0$.
Now, let $A$ be a finite symmetric generating set for $\Gamma$, and let $\nu$ be a symmetric probability measure with support $A$. The Laplacian $\Delta_{\nu}$ is defined as \begin{equation}\label{eq: definition of symmetric group laplacian}\Delta_{\nu}=\frac{1}{2}\sum \limits_{a\in A}\nu(a)(2-a-a^{*})=\frac{1}{2}\sum \limits_{a\in A}\nu(a)(1-a)(1-a)^{*}=1-\sum_{a\in A} \nu(a)a
\end{equation} If $\mu$ is a non-symmetric probability measure with finite support $A$, we define \begin{equation}\label{eq: definition of nonsymmetric group laplacian}
\Delta_{\mu}:=1-\sum_{a\in A} \mu (a)a.
\end{equation}
We will use the following result of \cite{ozawapropertyt}.
\begin{theorem*}\cite[Main Theorem]{ozawapropertyt}
Let $\Gamma$ be a finitely generated group and $\nu$ a symmetric probability measure on $\Gamma$ whose support generates $\Gamma$. The group $\Gamma$ has Property (T) if and only if there exists a constant $\kappa>0$ such that $\Delta_{\nu}^{2}-\kappa\Delta_{\nu}\geq_{\RG{\Gamma}}0.$
\end{theorem*}
We may now turn to proving Theorem \ref{thm: directed new spectral criterion}. Given an element $\xi\in\RG{\Gamma}$, its $1$-norm is $\vert\vert \xi\vert\vert_{1}=\sum \vert \xi(g)\vert.$ Define $$\I{\Gamma}:=\left\{\xi\in \RG{\Gamma}\;:\;\sum \xi (g)=0\right\},$$ and $$\I{\Gamma}^{h}=\{\xi\in \I{\Gamma}\;:\;\xi^{*}=\xi\}.$$
\begin{proof}[Proof of Theorem \ref{thm: directed new spectral criterion}]
For notational ease, let $\Upsilon=\Upsilon(G,\mathcal{L}^{G},R)$ and $\Gamma=G\slash \langle \langle \mathcal{R}\rangle \rangle$. Suppose $\Upsilon = (V,E)$. The proof of the theorem proceeds in two parts. Firstly, we find: a finite symmetric generating set, $A$, for $\Gamma$; two probability measures, $\mu$, $\overline{\mu},$ on $A$ that will (often) not be symmetric; and a probability measure $\xi$ on $A$ that is symmetric, such that $\Delta_{\mu}\Delta_{\overline{\mu}}-\Delta_{\mu}-\Delta_{\overline{\mu}}+\lambda_{1}(\Upsilon)^{-1}\Delta_{\xi}\geq_{\RG{\Gamma}} 0.$ We then show how to construct a symmetric probability measure $\nu$ such that $\vert \vert \Delta_{\mu}-\Delta_{\nu}\vert \vert_{1}\leq 2\epsilon$. Finally, we show that we can replace $\Delta_{\mu}$, $\Delta_{\overline{\mu}}$, and $\Delta_{\xi}$ by $\Delta_{\nu}$ in the above equation, at the expense of replacing the constant $\lambda_{1}(\Upsilon)^{-1}$ by the constant $(1+\delta)\lambda_{1}(\Upsilon)^{-1}-4\epsilon^{2}.$ The first part of the proof proceeds extremely similarly to \cite[Example 5]{ozawapropertyt}.
We may assume that $V$ generates $G$. By assumption, $V$ generates a finite index subgroup of $G$ and furthermore, $\mathcal{R}\subseteq \langle V\rangle$. As Property (T) is preserved under finite index extensions and quotients, we may replace $G \slash \langle \langle \mathcal{R}\rangle\rangle $ by the subgroup $\langle V\rangle \slash \langle \langle \mathcal{R}\rangle\rangle $ if $[G:\langle V\rangle]>1$.
Let $\phi:G\twoheadrightarrow \Gamma $ be the quotient map, and $A=\{\phi(\overline{w}): w\in V\}$, so that $A$ is a symmetric generating set for $\Gamma$. Let $\mu'$ be the probability measure on $V:=V(\Upsilon)$ given by $\mu' (w)=deg(w)\slash \vert E\vert .$ Let $\mu$ be the probability measure on $A$ given by $$\mu(a):=\sum\limits_{\substack{w\in V\\\phi(\overline{w})=_{\Gamma}a}}deg(w)\slash \vert E\vert=\sum\limits_{\substack{w\in V\\\phi(\overline{w})=_{\Gamma}a}}\mu' (w).$$
Let $\overline{\mu}$ be the probability measure on $A$ defined by $\overline{\mu}(a):=\mu (a^{-1}).$ Next, we define the probability measure $\xi$ on $A$ by
\begin{align}\label{eq: def of omega}
\xi(a)&:=\sum\limits_{\substack{v\in V\\\phi(\overline{v})=_{\Gamma}a}}\dfrac{rep(v)}{\vert E\vert}\nonumber\\
&=\sum\limits_{\substack{v\in V\\\phi(\overline{v})=_{\Gamma}a}}\dfrac{\vert \{(w^{-1},v)\in E\;:\;(v,u,w)\in R\}\vert+\vert \{(v,w^{-1})\in E\;:\;(v,u^{-1},w)\in R\}\vert}{\vert E\vert }\end{align}
By our assumptions in the statement of the theorem, we have that for all $a\in A$:
\begin{align}\label{eq: omega, mu inequalities}
\begin{split}
\xi (a)&\leq (1+\delta)\mu (a),\;\xi (a)\leq (1+\delta)\overline{\mu} (a),\\\;\overline{\mu}(a)&\leq (1+\epsilon)\mu(a),\;\mu (a)\leq (1+\epsilon)\overline{\mu}(a).
\end{split}
\end{align}
Let $\sigma$ be the uniform probability measure on $E$: we define $d:L^{2}(V,\mu')\rightarrow L^{2}(E,\sigma)$ by $(d\zeta)(x,y)=\zeta (y)-\zeta(x)$. Consider the Laplacian $\Lambda:=d^{*}d\slash 2$. As a matrix, $\Lambda=L(\Upsilon)$. In particular, $\Lambda$ has real eigenvalues, all lying in $[0,2]$. Let $\lambda =\lambda_{1}(\Lambda)=\lambda_{1}(\Upsilon)$. Let $P$ be the orthogonal projection from $L^{2}(V,\sigma )$ to constant functions, and $I$ be the identity operator. It is standard (see e.g. \cite[Example 5]{ozawapropertyt}) that $\lambda^{-1}\Lambda+P-I\geq 0 $, so that there is an operator $T$ on $L^{2}(V,\mu')$ with $\lambda^{-1}\Lambda+P-I=T^{*}T$. Given $v,w\in V$ and $O$ an operator on $L^{2}(V,\mu')$, let $O_{v,w}:=\langle O\delta_{v},\delta_{w}\rangle$.
In $\RG{\Gamma}$, we have that:
\begin{align}\label{eq: left hand side of sum is >=0}
& \sum\limits_{v,w\in V}(\lambda^{-1}\Lambda_{v,w}+P_{v,w}-I_{v,w})\phi(\overline{w})^{-1}\phi(\overline{v})\nonumber\\
&=_{\RG{\Gamma}} \sum\limits_{v,w\in V}\langle T^{*}T\delta_{v},\delta_{w}\rangle\phi(\overline{w})^{-1}\phi(\overline{v}\nonumber)\\
& =_{\RG{\Gamma}}\sum\limits_{v,w\in V}\langle T\delta_{v},T\delta_{w}\rangle\phi(\overline{w})^{-1}\phi(\overline{v})\nonumber\\
(\dagger)\;\;& =_{\RG{\Gamma}}\sum\limits_{v,w\in V}\bigg(\sum_{i\in V} \mu(i)'^{-1}\langle T\delta_{v},\delta_{i}\rangle\langle \delta_{i},T\delta_{w}\rangle\bigg)\phi(\overline{w})^{-1}\phi(\overline{v})\nonumber\\
&=_{\RG{\Gamma}}\sum\limits_{v,w\in V}\sum\limits_{i\in V}\mu(i)^{-1}T_{i,g}T_{i,h}\phi(\overline{w})^{-1}\phi(\overline{v})\nonumber\\
&=_{\RG{\Gamma}}\sum\limits_{i\in V }\zeta_{i}^{*}\zeta_{i}\geq_{\RG{\Gamma}}0,\end{align}
where we have defined $$\zeta_{i}=\mu'(i)^{-1\slash 2}\sum \limits_{v\in V}T_{i,v}\phi(\overline{v})\in \RG{\Gamma}.$$ The inequality $(\dagger)$ follows by definition of the inner product on $L^{2}(V,\mu')$.
We calculate that:
\begin{align*}
\Lambda_{v,v}&=\dfrac{1}{2\vert E\vert}\sum_{(x,y)\in E}(\delta_{v}(x)-\delta_{v}(y))^{2}\\
&=\sum_{y\neq v,(v,y)\in E}\dfrac{1}{2\vert E\vert }+\sum_{y\neq v,(y,v)\in E}\dfrac{1}{2\vert E\vert }\\
&=\dfrac{\left\vert \{(v,y)\in E\} \right\vert-\left\vert \{(v,v)\in E\} \right\vert}{\vert E\vert }\\
&=\mu'(v)-\dfrac{\left\vert \{(v,v)\in E\} \right\vert}{\vert E\vert }
\end{align*}
For $v\neq w$, $\Lambda_{v,w}=-\vert E\vert ^{-1}$ if $(v,w)\in E$, and $\Lambda_{v,w}=0$ otherwise. Hence:
\begin{align}\label{eq: Relating sum Lvw to Delta omega}
&\sum\limits_{v,w\in V}\lambda^{-1}\Lambda_{v,w}\phi(\overline{w})^{-1}\phi(\overline{v}\nonumber)\\
&=_{\RG{\Gamma}}\sum \limits_{v\in V}\mu'(v)\phi(\overline{v})^{-1}\phi(\overline{v})-\sum \limits_{(v,w)\in E}\dfrac{\phi(\overline{w})^{-1}\phi(\overline{v})}{\vert E\vert }\nonumber\\
&=_{\RG{\Gamma}}\sum \limits_{v\in V}\mu'(v)-\sum \limits_{(v,w)\in E}\dfrac{\phi(\overline{w})^{-1}\phi(\overline{v})}{\vert E\vert }\nonumber\\
(\ddag)& =_{\RG{\Gamma}}1-\dfrac{1}{\vert E\vert }\sum_{a\in A}\sum\limits_{\substack{u\in V\\\phi(\overline{u})=_{\Gamma}a}}\vert\{(y^{-1},x)\in E\;:\;(x,u,y)\in R\}\vert a\nonumber\\
&\hspace*{35 pt}-\dfrac{1}{\vert E\vert }\sum_{a\in A}\sum\limits_{\substack{u\in V\\\phi(\overline{u})=_{\Gamma}a}}\vert \{(x,y^{-1})\in E\;:\;(x,u^{-1},y)\in R\}\vert a\nonumber\\
&=_{\RG{\Gamma}}\Delta_{\xi}\;\;(\mbox{by equations }(\ref{eq: definition of nonsymmetric group laplacian})\mbox{ and }(\ref{eq: def of omega})).
\end{align}
The equation $(\ddag)$ follows since $(x,u,y)\in R$ implies that $xuy=_{\Gamma}1$, so that $a=\phi(u)=\phi(x)^{-1}\phi(y)^{-1}$, and hence $a$ arises from the edge $(y^{-1},x).$
Similarly, we have that $P_{v,w}=\mu'(v)\mu'(w),$ and $ I_{v,w}=\delta_{v,w}\mu'(v).$ Therefore:
\begin{align}\label{eq: calculation LHS of P and I}
&\sum\limits_{v,w\in V}(P_{v,w}-I_{v,w})\phi(\overline{w})^{-1}\phi(\overline{v})\nonumber\\
&=_{\RG{\Gamma}}\sum\limits_{v,w\in V}\mu'(v)\mu'(w)\phi(\overline{w})^{-1}\phi(\overline{v})-\sum\limits_{v\in V}\mu'(v)\phi(\overline{v})^{-1}\phi(\overline{v})\nonumber\\
&=_{\RG{\Gamma}}\sum\limits_{v,w\in V}\mu'(v)\mu'(w)\phi(\overline{w})^{-1}\phi(\overline{v})-\sum\limits_{v\in V}\mu'(v)\nonumber\\
&=_{\RG{\Gamma}}\bigg(\sum\limits_{v\in V}\mu'(v)\phi(\overline{v})\bigg)\bigg(\sum\limits_{v\in V}\mu'(v^{-1})\phi(\overline{v})\bigg)-1\nonumber\\
&=_{\RG{\Gamma}}(1-\Delta_{\mu})(1-\Delta_{\overline{\mu}})-1\nonumber\\
&=_{\RG{\Gamma}}\Delta_{\mu}\Delta_{\overline{\mu}}-\Delta_{\mu}-\Delta_{\overline{\mu}}.
\end{align}
Therefore, by equations $(\ref{eq: left hand side of sum is >=0})$, $(\ref{eq: Relating sum Lvw to Delta omega})$, and $(\ref{eq: calculation LHS of P and I})$:
$$\Delta_{\mu}\Delta_{\overline{\mu}}-\Delta_{\mu}-\Delta_{\overline{\mu}}+\lambda^{-1}\Delta_{\xi}\geq_{\RG{\Gamma}}0.$$
Note that $\Delta_{\overline{\mu}}:=_{\RG{\Gamma}}\Delta_{\mu}^{*}$. Since $\xi$ is symmetric, by taking the involution, we see that
$$\Delta_{\overline{\mu}}\Delta_{\mu}-\Delta_{\mu}-\Delta_{\overline{\mu}}+\lambda^{-1}\Delta_{\xi}\geq_{\RG{\Gamma}}0.$$
Therefore:
\begin{equation}\label{eq: first symmetric inequality}
\dfrac{1}{2}\bigg(\Delta_{\overline{\mu}}\Delta_{\mu}+\Delta_{\mu}\Delta_{\overline{\mu}}\bigg)-\Delta_{\mu}-\Delta_{\overline{\mu}}+\lambda^{-1}\Delta_{\xi}\geq_{\RG{\Gamma}}0.
\end{equation}
Let $\nu:=(\mu+\overline{\mu})\slash 2$, so that $\nu$ is a symmetric probability measure on $A$. We see that $\Delta_{\nu}=_{\RG{\Gamma}}(\Delta_{\mu}+\Delta_{\overline{\mu}})\slash 2$. Furthermore:
\begin{align}\label{eq: delta nu squared}
\Delta_{\nu}^{2}&=_{\RG{\Gamma}}\dfrac{1}{4}(\Delta_{\mu}+\Delta_{\overline{\mu}})^{2}=_{\RG{\Gamma}}\dfrac{1}{4}\bigg(\Delta_{\mu}^{2}+\Delta_{\overline{\mu}}^{2}+\Delta_{\mu}\Delta_{\overline{\mu}}+\Delta_{\overline{\mu}}\Delta_{\mu}\bigg)\nonumber\\
&=_{\RG{\Gamma}}\dfrac{1}{2}\bigg(\Delta_{\mu}\Delta_{\overline{\mu}}+\Delta_{\overline{\mu}}\Delta_{\mu}\bigg)+\dfrac{1}{4}\bigg(\Delta_{\mu}^{2}+ \Delta_{\overline{\mu}}^{2}-\Delta_{\mu}\Delta_{\overline{\mu}}-\Delta_{\overline{\mu}}\Delta_{\mu}\bigg)\nonumber\\
&=_{\RG{\Gamma}}\dfrac{1}{2}\bigg(\Delta_{\mu}\Delta_{\overline{\mu}}+\Delta_{\overline{\mu}}\Delta_{\mu}\bigg)+\dfrac{1}{4}\bigg(\Delta_{\mu}-\Delta_{\overline{\mu}}\bigg)\bigg(\Delta_{\mu}-\Delta_{\overline{\mu}}\bigg).
\end{align}
It is easy to see that $(1+\delta)\Delta_{\nu}\geq_{\RG{\Gamma}} \Delta_{\xi},$ since $(1+\delta)\mu(a)\geq \xi(a)$ for all $a\in A$, and similarly for $\overline{\mu}(a)$. Let $\Sigma=(\Delta_{\mu}-\Delta_{\overline{\mu}})(\Delta_{\mu}-\Delta_{\overline{\mu}})\in \I{\Gamma}.$ By equations $(\ref{eq: first symmetric inequality})$ and $(\ref{eq: delta nu squared})$, we have $$\Delta_{\nu}^{2}-2\Delta_{\nu}+\lambda^{-1}\Delta_{\xi}\geq_{\RG{\Gamma}} -\Sigma\slash 4,$$ and hence \begin{equation}\label{eq: second symmetric inequality}
\Delta_{\nu}^{2}-(2-(1+\delta)\lambda^{-1})\Delta_{\nu}\geq_{\RG{\Gamma}} -\Sigma\slash 4.
\end{equation} To apply Ozawa's criterion, it therefore remains to analyze $\Sigma.$ We note that, by e.g. \cite[Lemma 2]{ozawapropertyt}, for any $a,b\in \Gamma$:
\begin{equation}\label{eq: Ozawas lemma}
(2-ab-(ab)^{*})\leq_{\RG{\Gamma}} 2(2-a-a^{*})+2(2-b-b^{*}).
\end{equation} Now, $\Sigma^{*}=\Sigma$ (in particular $\Sigma(g)=\Sigma(g^{-1})$), and $\sum \Sigma (g)=0,$ so that $\Sigma\in\I{\Gamma}^{h}$. We may therefore write $$\Sigma = \dfrac{1}{2}\sum \limits_{a,b\in A}\Sigma (ab) (2-(ab)^*-ab).$$ This is a standard characterization of such elements, as the cone $\I{\Gamma}^{h}$ is spanned by the elements $2-x-x^{-1}$: see e.g. \cite{ozawapropertyt}. By comparing coefficients with the definition of $\Sigma$, we see that $$\Sigma(ab)=(\mu(a)-\mu(a^{-1}))(\mu(b)-\mu(b^{-1})).$$
Therefore, as $\mu(a)\leq (1+\epsilon)\mu(a^{-1})$ for all $a\in A$:
\begin{align}
\Sigma&=_{\RG{\Gamma}}\dfrac{1}{2}\sum\limits_{a,b\in A}(\mu(a)-\mu(a^{-1}))(\mu(b)-\mu(b^{-1}))(2-ab-(ab)^{*})\nonumber\\
(\ddag \dagger)\;&\leq _{\RG{\Gamma}}\dfrac{1}{2}\sum\limits_{a,b\in A}2(\mu(a)-\mu(a^{-1}))(\mu(b)-\mu(b^{-1}))(2-a-a^{*})\nonumber\\
&\;\;\;\;+\dfrac{1}{2}\sum\limits_{a,b\in A}2(\mu(a)-\mu(a^{-1}))(\mu(b)-\mu(b^{-1}))(2-b-b^*)\nonumber\\
&=_{\RG{\Gamma}} 2\sum_{b\in A}(\mu(b)-\mu(b^{-1}))\sum_{a\in A}(\mu(a)-\mu(a^{-1}))(2-a-a^{*})\nonumber\\
&\leq_{\RG{\Gamma}}2\sum_{b\in A}2\epsilon \min\{\mu (b),\mu (b^{-1})\} \sum_{a\in A}(\mu(a)-\mu(a^{-1}))(2-a-a^{*})\nonumber\\
&\leq_{\RG{\Gamma}}4\epsilon \sum_{a\in A}(\mu(a)-\mu(a^{-1}))(2-a-a^{*})\nonumber\\
(\ddag \ddag)&\leq_{\RG{\Gamma}}4\epsilon\sum_{a\in A}2\epsilon \min\{\mu (a),\mu (a^{-1})\}(2-a-a^{*})\nonumber\\
(\ddag \ddag \dagger)&\leq_{\RG{\Gamma}}8\epsilon^{2}\sum_{a\in A}\nu(a)(2-a-a^{*}) \nonumber\\
&=_{\RG{\Gamma}}16\epsilon^{2}\Delta_{\nu}.
\end{align}
We have the (in)equality $(\ddag \dagger)$ by equation $(\ref{eq: Ozawas lemma})$, $(\ddag \ddag)$ by equation $(\ref{eq: omega, mu inequalities})$, and $(\ddag \ddag\dagger )$ by the definition of $\nu$.
Dividing by $-1\slash 4$, this implies that
\begin{equation}\label{eq: Sigma inequality}
-\Sigma\slash 4\geq -4\epsilon^{2}\Delta_{\nu}.
\end{equation}
Hence, by equations $(\ref{eq: second symmetric inequality})$ and $(\ref{eq: Sigma inequality})$, we have that:
\begin{align*}
\Delta_{\nu}^{2}-(2-(1+\delta)\lambda^{-1})\Delta_{\nu} \geq_{\RG{\Gamma}}-\Sigma\slash 4 \geq_{\RG{\Gamma}}-4\epsilon^{2}\Delta_{\nu},
\end{align*}
so that $$\Delta_{\nu}^{2}-(2-4\epsilon^{2}-(1+\delta)\lambda^{-1})\Delta_{\nu}\geq_{\RG{\Gamma}}0.$$ Set $\kappa = 2-4\epsilon^{2}-(1+\delta)\lambda^{-1};$ since $\lambda>(1+\delta)\slash(2-4\epsilon^{2}),$ we have that $\kappa >0$. Therefore, $\nu$ is a symmetric probability measure on $\Gamma$, with finite support that generates $\Gamma$, and there exists $\kappa>0$ such that $$\Delta_{\nu}^{2}-\kappa \Delta_{\nu} \geq_{\RG{\Gamma}}0.$$ Hence, by \cite{ozawapropertyt}, $\Gamma=G\slash \langle \langle R\rangle \rangle$ has Property (T).
\end{proof}
This allows us to deduce the following.
\begin{cor}\label{cor: directed new spectral criterion}
Let $G=\langle S\;\vert\; T\rangle$ be a hyperbolic group. For each $l\geq 1$, choose a set of triples $R_{l}=\{r=(x_{r},y_{r},z_{r})\}_{r}$ in $\mathcal{L}^{G}$ such that: each $x_{r}$, $y_{r}$, $z_{r}$ has length between least $l\slash 3-\omega (l)\slash 3$ and $l\slash 3+2$; and $x_{r}y_{r}z_{r}$ is reduced without cancellation and lies in $\mathcal{L}^{G}_{l,\omega}$. Let $\mathcal{R}_{l}=\{x_{r}y_{r}z_{r}\;:\;r\in R\},$ and $G\slash \langle \langle \mathcal{R}_{l}\rangle \rangle $ be a series of quotients of $G$. Suppose that $\mathcal{L}_{\lfloor l\slash 3\rfloor}^{G}\subseteq V(\Upsilon(G,\mathcal{L}^{G},R_{l}))$, and for all $g \in V(\Upsilon(G,\mathcal{L}^{G},R_{l}))$: $$ deg (g)=(1+o_{l}(1))deg(g^{-1})= (1+o_{l}(1)) rep (g), $$
where the $o_{l}(1)$ term is independent of $g$. If $\lim_{l\rightarrow\infty}\lambda_{1}(\Upsilon(G,\mathcal{L}^{G},R_{l}))>1\slash 2$, then $G\slash \langle \langle \mathcal{R}_{l}\rangle \rangle$ has Property (T) for all sufficiently large $l$.
\end{cor}
\begin{proof}
Since $\lambda=\lim_{l\rightarrow \infty}\lambda_{1}(\Upsilon(G,\mathcal{L}^{G},R_{l}))>1\slash 2,$ we may choose $\epsilon, \delta >0$ such that
$$\lambda>(1+\delta)\slash (2-4\epsilon^{2}).$$ Then $$\lambda_{1}(\Upsilon(G,\mathcal{L}^{G},R_{l}))>(1+\delta)\slash (2-4\epsilon^{2})$$ for all sufficiently large $l$. Similarly, by the assumptions of the corollary, we may assume that for all sufficiently large $l$ and for all $g\in V(\Upsilon(G,\mathcal{L}^{G},R_{l}))$: $ deg (g)\leq(1+\epsilon)deg(g^{-1})$ and $rep (g)\leq(1+\delta)deg (g).$ Since $\mathcal{L}^{G}_{\lfloor l\slash 3\rfloor}\subseteq V(\Upsilon(G,\mathcal{L}^{G},R_{l}))$, we see that $\mathcal{W}(R)$ generates a finite index subgroup of $G$. Applying Theorem \ref{thm: directed new spectral criterion} allows us to conclude the result.
\end{proof}
\section{Counting geodesics in hyperbolic groups}\label{sec: counting geodesics in hyperbolic groups}
Now that we have a spectral criterion for Property (T), we have two remaining tasks. Firstly, we need to find our language $\mathcal{L}^{G}$, and secondly, we have to analyse the resulting graphs $\Upsilon$. We turn to the theory of geodesics in hyperbolic groups. We wish to encode the geodesics in a hyperbolic group as an easily understood dynamical system. The most natural way to do this is to use \emph{Markov chains} and \emph{subshifts}. The reader should see, for example, \cite{calegari2013ergodic} for an in-depth discussion of these concepts. Let $G=\langle S\;\vert\;T\rangle$ be a finite presentation of a non-elementary hyperbolic group (we may assume that $S$ is inverse-closed). Assign an ordering to $S$, and extend this to the \emph{lexicographical ordering} of words in $S^{*}$.
\begin{definition}
A \emph{finite state automaton} (\emph{FSA}) is a finite directed graph with a single distinguished start vertex $v_{0}$, with edges are labelled by elements of $S$ such that for each vertex $v$ and each $s\in S$ there is at most one edge incident to $v$ bearing the label $s$. Some of the vertices are marked as accept vertices. The \emph{language parameterized} by the FSA is the set of words that can be read by starting at $v_{0}$ and concatenating edge labels along a path that ends in an accept state.
\end{definition}
The automatic structure of hyperbolic groups was analysed in \cite{Cannon_combinatorial_structure_hyperbolic_groups}. We will make great use of the following.
\begin{definition}
A subset $\mathcal{L}\subseteq S^{*}$ is a \emph{combing for} $G$ if: $\mathcal{L}$ bijects with $G$ under evaluation; $\mathcal{L}$ is regular (i.e. there exists an FSA parameterizing $\mathcal{L}$); $\mathcal{L}$ is prefix closed, i.e. if $vw\in \mathcal{L}$, then $w\in \mathcal{L}$; and each word in $\mathcal{L}$ is geodesic in $G$.
\end{definition}
Given a language $\mathcal{L}$ and $l\geq 1$, $\mathcal{L}_{l}$ is the subset of $\mathcal{L}$ of words of length $l$. We define $\mathcal{L}_{\infty}$ to be the set of right-infinite words of $\mathcal{L}$. The following is one of the major results in the study of hyperbolic groups.
\begin{theorem*}\cite{Cannon_combinatorial_structure_hyperbolic_groups}
Let $G=\langle S\;\vert\;T\rangle$ be a finite presentation of a hyperbolic group, and let $\mathcal{L}^{geo}$ be the language of lexicographically first geodesics. Then $\mathcal{L}^{geo}$ is a combing for $G$.
\end{theorem*}
In particular, there is a directed graph $\Sigma = (V,E)$ parameterizing $\mathcal{L}^{geo}$ with a unique start vertex $v_{0}$ and all other vertices accept vertices \cite{Cannon_combinatorial_structure_hyperbolic_groups} (see e.g. \cite{calegari2013ergodic} for an overview of its construction). For $n\geq 1$, let $Y_{n}$ be the set of paths of length $n$ in $\Sigma$ starting at $v_{0}$. Define $X_{n}$ as the set of all paths of length $n$ in $\Sigma$. Note that $Y_{n}\subseteq X_{n}$. Let $\mathcal{E}:\sqcup_{n}X_{n}\rightarrow G$ be the evaluation map, i.e. map a path to the corresponding path in $G$ starting at $e$ and take its endpoint in $G$. We can prove the following.
\begin{lemma}\label{lem: evaluation map is bounded to one}
The evaluation map $\mathcal{E}:\sqcup_{n}X_{n}\rightarrow G$ is bounded-to-one, with bound $\vert V(\Sigma)\vert.$
\end{lemma}
\begin{proof}
Suppose that $\sigma, \sigma ' $ are two paths in $\Sigma$ starting from the same vertex, $v$, and evaluating to the same element in $G$. Let $\gamma$ be a path from the start vertex $v_{0}$ to $v$. Then $\gamma \sigma$ and $\gamma \sigma'$ are two paths in $Y_{\vert \gamma\vert +\vert \sigma\vert}$ evaluating to the same element. Since $\mathcal{L}^{geo}$ is a combing, we have that $\gamma\sigma=\gamma \sigma'$ as words, and so $\sigma =\sigma'$ as words. Therefore, for any $g\in G$ and any vertex $v$ of $\Sigma$, there is at most one path in $\Sigma$ starting at $v$ and evaluating to $g$. This implies that the evaluation map has preimage size bounded by $\vert V(\Sigma ) \vert$.
\end{proof}
A \emph{component} $C$ of $\Sigma$ is a maximal subgraph of $\Sigma$ such that there is a directed path from any vertex $v$ in $C$ to any vertex $w$ in $C$. In this case, by an abuse of notation we can view a component $C$ as a FSA, $\Gamma_{C}$, by adding a unique start vertex $v_{0}'$ adjacent to every vertex in $C$ with each of these added edges having label the empty word, and make all other vertices accept states. Therefore, we can write $\mathcal{L}(C)=\mathcal{L}(\Gamma_{C}).$ In fact, $C$ is an \emph{irreducible (topological) Markov chain} (see e.g. \cite[p. 23]{calegari2013ergodic}).
For such a component, let $M(C)$ be its adjacency matrix, and let $M$ be the adjacency matrix of $\Sigma$. Let $\mu$ be the maximal real eigenvalue of $M$ and $\mu(C)$ the maximal real eigenvalue of $M(C)$. A component $C$ is \emph{maximal} if $\mu(C)=\mu$. We remark that in actuality $\mu = \exp\{\mathfrak{h}\}$ and the number of paths of length $n$ in $M(C)$ is $\eta (C)(1+o(1))\mu(C)^{n}$ for some constant $\eta(C)>0$.
Since each component $C$ is connected, by the Perron--Frobenius theorem, $M_{C}$ has a unique real eigenvalue $\mu (C)$ of maximal norm (with algebraic multiplicity $1$).
The following Lemma appears as a component of the proof of Proposition 6.2 in \cite{gekhtman2018counting}.
\begin{lemma}\label{lem: Cmax gens finite index}\cite[Proof of Proposition 6.2]{gekhtman2018counting}
There exists a maximal component, $C^{max}$, of $\Sigma$ such that for any $l\geq 1$, the set $\mathcal{L}_{l}(C^{max})$ generates a finite-index subgroup of $G$.
\end{lemma}
We may now fix $C^{max}$ to be the desired maximal component in the above lemma and let $\mathcal{L}^{G}=\mathcal{L}(C^{max})$ be the language of $C^{max}.$
We wish to encode the geodesics in a hyperbolic group as an easily understood dynamical system. The most natural way to do this is to use \emph{shifts}.
\begin{definition}
Let $V$ be a finite set. The \emph{two-sided shift on $V$} is the topological space $X:=V^{\mathbb{Z}}$. We endow $V$ with the discrete topology and $X$ with the product topology. It comes endowed with the \emph{shift map} $\sigma:X\rightarrow X$ defined by $(\sigma(\vect{x}))_{i}=\vect{x}_{i+1}.$
\end{definition}
A \emph{subshift} is a closed $\sigma$-invariant subspace of $X$. We wish to look at a specific type of type of shift, called a \emph{subshift of finite type}.
\begin{definition}
Let $A$ be a $\vert V\vert\times \vert V\vert$ $0-1$ matrix. The \emph{subshift of finite type associated to $A$} is the subshift $X_{A}\subset X$ defined by
$$X_{A}=\left\{\vect{x}\in X\;:\;A_{\vect{x}_{i},\vect{x}_{i+1}}=1\right\}.$$
\end{definition}
If we take $\Sigma = (V,E)$ to be a finite directed graph, the \emph{edge subshift} associated to $\Sigma$ is the set $X_{\Sigma}\subseteq (E)^{\mathbb{Z}}$ that is defined by the incidence matrix of $\Sigma$ (recall the incidence matrix $B$ is the $\vert E\vert\times \vert E\vert $ $0-1$ matrix, where $B_{i,j}=1$ if the endpoint of $e_{i}$ is equal to the startpoint of $e_{j}$, and is zero otherwise).
Let $\mathcal{Z}$ be the set of bi-infinite paths in $C^{max},$ i.e. it is the edge subshift of $C^{max}$. Since $C^{max}$ is connected, we see that $B$ (the incidence matrice) is irreducible, and furthermore, by the Perron-Frobenius theorem, it has unique largest real eigenvalue; using the growth rate of $\mathcal{Z}$, this eigenvalue is exactly $\mu$.
\begin{definition}[The Parry measure]
Let $\alpha,\beta$ be the left and right $\mu$-eigenvectors for $B$, normalised so that $\alpha\beta =1.$ Let $p_{i}=u_{i}v_{i}$, and $P_{i,j}=B_{i,j}\beta_{j}\slash \mu \beta_{i}$.
For an edge path $\gamma=(e_{i_{0}},e_{i_{1}},\hdots ,e_{i_{n}})$ in $C^{max}$ and $n\geq 0$, the \emph{cylinder set} $\mathcal{Z}[\gamma,m]$ is defined as
$$\mathcal{Z}[\gamma,m]=\left \{\underline{z}\in\mathcal{Z}\;:\;(z_{m},z_{m+1},\hdots ,z_{m+n})=\gamma\right\}.$$
These sets form a clopen basis for the subshift. The \emph{Parry measure} on $\mathcal{Z}$, $\nu$, defined in \cite{parry1964intrinsic}, is given by its definition on cyclinder sets
$$\nu( \mathcal{Z}[(e_{i_{0}},e_{i_{1}},\hdots ,e_{i_{n}}),m])=p_{i_{0}}P_{i_{0},i_{1}}\hdots P_{i_{n-1},i_{n}}=\dfrac{\alpha_{i_{0}}\beta_{i_{n}}}{\mu^{n}}.$$
\end{definition}
We now begin to analyse the number of paths in $C^{max}$, weighted by the Parry measure. First, we prove the following.
\begin{lemma}\label{lem: convergence of approximation}
For $\omega(L)$ a small-growing function, define $$B_{L,\omega}=\dfrac{1}{\omega(L)}\sum_{m=L-\omega}^{L}\dfrac{1}{\mu^{m}}B^{m}.$$ Then $B_{L,\omega}\rightarrow \beta\alpha$.
\end{lemma}
\begin{proof}
We calculate that:
\begin{align*}
\left \vert\left\vert B_{L+1,\omega}-B_{L,\omega}\right\vert\right\vert
&\leq \dfrac{1}{\omega (L+1)\mu^{L+1}}\left\vert \left\vert B^{L+1}\right\vert\right\vert+\dfrac{1}{\omega (L)}\sum_{m=L-\omega (L)}^{L+1-\omega (L+1)}\dfrac{1}{\mu^{m}}\vert\vert B^{m}\vert\vert\\
&+\sum_{m=\max\left\{\substack{L+1-\omega (L+1)\\L-\omega (L)}\right\}}^{L}\left(\dfrac{1}{\omega(L)}-\dfrac{1}{\omega(L+1)}\right)\dfrac{1}{\mu^{m}}\vert\vert B^{m}\vert\vert\\
&\leq \dfrac{1}{\omega(L+1)}+\dfrac{\omega(L+1)+1-\omega(L)}{\omega (L)}+\sum\limits_{m=\min\left\{\substack{L+1-\omega (L+1)\\L-\omega (L)}\right\}}^{L}
\dfrac{\omega (L+1)-\omega (L)}{\omega(L)\omega(L+1)}\\
&\leq o_{L}(1)+ 2\dfrac{\omega(L+1)-\omega (L)}{\omega (L)}\\
&=o_{L}(1).
\end{align*}
Hence $B_{L,\omega}$ converges to a matrix $B_{\omega}$. Next,
\begin{align*}
\left \vert\left\vert \dfrac{1}{\mu}BB_{L,\omega}-B_{L+1,\omega}\right\vert\right\vert
&\leq \sum_{m=L-\omega (L)}^{L+1-\omega (L+1)}\dfrac{1}{\omega(L)}+ \sum_{m=L+1-\omega (L+1)}^{L+1}\left(\dfrac{1}{\omega(L)}-\dfrac{1}{\omega(L+1)}\right)\dfrac{1}{\mu^{m}}\vert\vert B^{m}\vert\vert\\
&\leq o_{L}(1).
\end{align*}
Therefore $$\dfrac{1}{\mu}BB_{\omega}=B_{\omega}=\dfrac{1}{\mu}B_{\omega}B.$$
Since $B$ has a unique real eigenvalue $\mu$ of largest norm, by applying the above we see that $B_{\omega}$ has eigenvalues $1$ (of algebraic multiplicity $1$) and $0$. In particular, $B_{\omega}$ is a rank one projection, and so can be written $$B_{\omega}=\beta\alpha,$$
where $\alpha$ (respectively $\beta$) is the left (respectively right) $\mu$-eigenvector of $B_{\omega}$, and hence of $B$.
\end{proof}
We now observe the following.
\begin{lemma}\label{lem: measure of sets between two paths}
Let $\gamma=(e_{i_{1}},\hdots ,e_{i_{m}}), \gamma'=(e_{j_{1}},\hdots ,e_{j_{n}})$ be paths in $C^{max}$. Then $$\sum \limits_{\substack{L-\omega (L)\leq s\leq L\\\sigma=(e_{k_{1}},\hdots e_{k_{s}})}}\nu (\mathcal{Z}[\gamma,\sigma,\gamma'],0)=(1+o_{L}(1))\omega (L) \alpha_{i_{1}}\beta_{i_{m}}\alpha_{j_{1}}\beta_{j_{n}}\slash \mu^{m+n-1}.$$
\end{lemma}
\begin{proof}
We see that
\begin{align*}
\sum \limits_{\substack{L-\omega (L)\leq s\leq L\\\sigma=(e_{k_{1}},\hdots e_{k_{s}})\\\gamma\sigma\gamma'\mbox{ a path in }C^{max}}}\nu (\mathcal{Z}[\gamma,\sigma,\gamma'],0)&=\sum \limits_{\substack{L-\omega (L)\leq s\leq L\\\sigma=(e_{k_{1}},\hdots e_{k_{s}})\\\gamma\sigma\gamma'\mbox{ a path in }C^{max}}}\alpha_{i_{1}}\beta_{j_{n}}\slash\mu^{m+n+s-1}\\
&= \dfrac{\alpha_{i_{1}}\beta_{j_{n}}}{\mu^{m+n-1}}\sum \limits_{\substack{L-\omega (L)\leq s\leq L\\\sigma=(e_{k_{1}},\hdots e_{k_{s}})\\\gamma\sigma\gamma'\mbox{ a path in }C^{max}}}1\slash\mu^{s}\\
&=\dfrac{\alpha_{i_{1}}\beta_{j_{n}}}{\mu^{m+n-1}}\sum \limits_{L-\omega (L)\leq t\leq L}B^{t}_{i_{0},j_{n}}1\slash\mu^{s}\\
&=(1+o_{L}(1))\omega (L) \alpha_{i_{1}}\beta_{i_{m}}\alpha_{j_{1}}\beta_{j_{n}}\slash \mu^{m+n-1}
\end{align*}
by Lemma \ref{lem: convergence of approximation}.
\end{proof}
We can also immediately deduce the following lemmas.
\begin{lemma}\label{lem: measure of starting path}
Let $\gamma=(e_{i_{1}},\hdots ,e_{i_{m}}),$ be a path in $C^{max}$. Then
\begin{align*}
\sum \limits_{\substack{L-\omega (L)\leq n\leq L\\\sigma=(e_{j_{1}},\hdots ,e_{j_{n}})}}&\sum \limits_{\substack{L-\omega (L)\leq s\leq L\\\sigma'=(e_{k_{1}},\hdots e_{k_{s}})}}\nu (\mathcal{Z}[\gamma,\sigma,\sigma'],0)=\omega (L)^{2} \nu (\mathcal{Z}[\gamma])\\
&=\sum \limits_{\substack{L-\omega (L)\leq n\leq L\\\sigma=(e_{j_{1}},\hdots ,e_{j_{n}})}}\sum \limits_{\substack{L-\omega (L)\leq s\leq L\\\sigma'=(e_{k_{1}},\hdots e_{k_{s}})}}\nu (\mathcal{Z}[\sigma,\sigma'\gamma],0).
\end{align*}
\end{lemma}
\begin{lemma}\label{lem: measure of sets middle path}
Let $\gamma=(e_{i_{1}},\hdots ,e_{i_{m}})$ be a path in $C^{max}$. Then $$\sum \limits_{\substack{L-\omega (L)\leq n\leq L\\\sigma=(e_{j_{1}},\hdots ,e_{j_{n}})}}\sum \limits_{\substack{L-\omega (L)\leq l\leq s\\\sigma'=(e_{k_{1}},\hdots e_{k_{s}})}}\nu (\mathcal{Z}[\sigma,\gamma,\sigma'],0)= \omega(L)^{2}\nu(\mathcal{Z}[\gamma,0]).$$
\end{lemma}
\section{The first eigenvalue of some random graphs}\label{appendix: Spectral theory of restricted graphs}
To prove Theorem \ref{mainthm: property t in random quotients of hyperbolic groups}, we are required to analyse the eigenvalues of a particular model of random graphs. We first introduce the following well known model.
\begin{definition}
Let $m\geq 1$ and let $\underline{w}=(w_{1},\hdots, w_{m})\in\mathbb{Z}_{+}^{m}.$
The random graph $\Gamma(m,\underline{w})$ is the graph with vertex set $V=\{u_{1},\hdots,u_{m}\}$, and each edge between $u_{i}$ and $u_{j}$ is added with probability $w_{i}w_{j}\rho$, where $\rho=1\slash\sum w_{i}.$ We define $w_{min}=\min\{w_{i}\}$ and $\overline{w}=\sum w_{i}\slash m.$
\end{definition}
The eigenvalues of this graph were analysed in \cite{chungrandomgraph}.
\begin{theorem*}\cite[Theorem 5]{chungrandomgraph}
Suppose that $m$ and $\underline{w}$ are defined such that \\$w_{min}=\Omega_{m} (\sqrt{\;\overline{w}}\log^{3}m)$. Let $\Sigma\sim \Gamma(m,\underline{w})$. Then a.a.s.$(m)$, $$\max_{i\neq 0}\vert 1-\lambda_{i}(\Sigma) \vert= 2[1+o_{m}(1)]\slash \sqrt{\;\overline{w}}.$$
\end{theorem*}
The above inequality can be rewritten as $$\max_{i\neq 1}\vert \mu_{i}(D(\Sigma)^{-1}A(\Sigma)) \vert= 2[1+o_{m}(1)]\slash \sqrt{\;\overline{w}}.$$
We now introduce several models of random graphs specific to our needs. Ultimately, we wish to model our graph $\Upsilon$ by a suitable random graph model $\mathkey(m_{1},m_{2},\pi,p,*).$ However, the random graph $\mathkey(m_{1},m_{2},\pi,p,*)$ is somewhat complicated, and so, we build up to understanding the eigenvalues of this graph by first passing through some simpler models of random graphs. In particular, $\mathkey(m_{1},m_{2},\pi,p,*)$ is formed from the union of $\Gamma(m,\pi,p,*)$, $\mathcal{B}(m_{1},m_{2},\pi,p)$ and $\Lambda(m_{1},m_{2},\pi,p,*)$; we study each of these graphs in turn.
In what follows, $\pi$ will be a vector in $(0,\infty)^{\vert V\vert}$, where $V$ will be the vertex set of the graph under consideration. Informally, $\pi_{i}$ corresponds to the `weight' of the $ith$ vertex of the graph. For notational ease we will write $\pi(v)$ to be the entry of $\pi$ corresponding to the vertex $v$. Furthermore, to be technically accurate, the below constructions are actually constructions of sequences of graphs. We therefore require a choice of $\pi_{m}\in\mathbb{Z}^{m}$ for each $m$: we will ignore the subscripts $m$ for notational ease. For the remainder of this text, we \emph{always} assume that $p<1\slash 2$.
\begin{definition}[{\bf The graph }$\boldsymbol{\Gamma(m,\pi,p,*)}$]
Let $m\geq 1$, $\pi_{min}$ be a constant independent of $m$, and let $\pi\in[\pi_{min},\infty)^{m}$. The random graph $\Gamma(m,\pi,p,*)$ has vertex set $V=\{v_{1},\hdots,v_{m}\}$. The map $*:V\rightarrow V$ a fixed-point free involution specified in advance. Let $v_{i}^{-1}:=*(v_{i})$. Each edge $(v_{i},v_{j})$ is added with probability $(\pi (v_{i})\pi(v_{j}^{-1})+\pi(v_{i}^{-1})\pi(v_{j}))p$.
\end{definition}
\begin{definition}[{\bf The graph} $\boldsymbol{\mathcal{B}(m_{1},m_{2},\pi,p)}$]
Let $m_{1},m_{2}\geq 1$, $\pi_{min}>0$ be a constant independent of $m_{1}$ and $m_{2}$, and $\pi\in[\pi_{min},\infty)^{m_{1}+m_{2}}$. The random graph $\mathcal{B}(m_{1},m_{2},\pi,p)$ is the bipartite graph with vertex partition $V_{1}\sqcup V_{2},$ where $V_{1}=\{u_{1},\hdots,u_{m_{1}}\}$ and $ V_{2}=\{v_{1},\hdots ,v_{m_{2}}\}$. Each edge $(u_{i},v_{j})$ is added with probability $\pi (u_{i})\pi(v_{j})p$.
\end{definition}
\begin{definition}[{\bf The graph} $\boldsymbol{\Lambda(m_{1},m_{2},\pi,p,*)}$]
Let $m_{1},m_{2}\geq 1$, $\pi_{min}>0$ be a constant independent of $m_{1}$ and $m_{2}$, and let $\pi\in[\pi_{min},\infty)^{m_{1}+m_{2}}$. The random graph $\Lambda(m_{1},m_{2},\pi,p,*)$ is defined as follows. It has vertex set $V=V_{1}\sqcup V_{2}\sqcup V_{3}$, where $V_{1}=\{u_{1},\hdots,u_{m_{1}}\}$, $V_{2}=\{v_{1},\hdots,v_{m_{2}}\},$ and $V_{3}=\{v_{1}^{-1},\hdots,v_{m_{2}}^{-1}\}.$ The map $*:V_{1}\rightarrow V_{1}$ is a fixed-point free involution specified in advance. Let $u_{i}^{-1}:=*(u_{i})$. Each edge $(u_{i},v_{j})$ is added with probability $\pi(u_{i}^{-1})\pi(v_{j})p$. Each edge $(u_{i},v_{j}^{-1})$ is added with probability $\pi(u_{i})\pi(v_{j})p$.
\end{definition}
\begin{definition}[{\bf The graph} $\boldsymbol{\mathkey(m_{1},m_{2},\pi,p,*)}$]
Let $m_{1},m_{2}\geq 1$, $\pi_{min}>0$ be a constant independent of $m_{1}$ and $m_{2}$, and let $\pi\in[\pi_{min},\infty)^{m_{1}+m_{2}}$. The random graph $\mathkey(m_{1},m_{2},\pi,p,*)$ is the random graph defined as follows. It has vertex set $V=V_{1}\sqcup V_{2}\sqcup V_{3}$, where $V_{1}=\{u_{1},\hdots,u_{m_{1}}\}$, $V_{2}=\{v_{1},\hdots,v_{m_{2}}\},$ and $V_{3}=\{v_{1}^{-1},\hdots,v_{m_{2}}^{-1}\}.$ The map $*:V_{1}\rightarrow V_{1}$ is a fixed-point free involution specified in advance. Let $u_{i}^{-1}:=*(u_{i})$.
\begin{enumerate}[label={$\arabic* )$}]
\item Each edge $(u_{i},u_{j})$ is added with probability $$(\pi (u_{i})\pi(u_{j}^{-1})+\pi(u_{i}^{-1})\pi(u_{j}))p.$$
\item Each edge $(u_{i},v_{j})$ is added with probability $\pi(u_{i}^{-1})\pi(v_{j})p$.
\item Each edge $(u_{i},v_{j}^{-1})$ is added with probability $\pi(u_{i})\pi(v_{j})p$.
\item Finally, each edge $(v_{i},v_{j}^{-1})$ is added with probability $\pi(v_{i})\pi(v_{j})p$.
\end{enumerate}
\end{definition}
\begin{definition}
For all of the above graphs, we define $$\pi_{max}=\max\{\pi(v)\;:\;v\in V\}.$$
\end{definition}
\subsection{Switching between the adjacency matrix and expected adjacency matrix}
Define the matrix $$M(\Gamma(m,\pi,p,*)) = \mathbb{E}(D(\Sigma))^{-1}\mathbb{E}(A(\Sigma))$$ for $\Sigma\sim\Gamma(m,\pi,p,*).$ We define similarly the matrices $M(\Gamma(m,\underline{w}))$,\\ $M(\mathcal{B}(m_{1},m_{2},\pi,p)),\;M(\Lambda(m_{1},m_{2},\pi,p,*)),$ and $M(\mathkey(m_{1},m_{2},\pi,p,*)).$
It will be important for us to switch between the random matrices $D^{-1}A$ associated to a random graph and the fixed matrix $\mathbb{E}(D^{-1})\mathbb{E}(A).$ In order to do this, we will use the matrix Bernstein inequality.
\begin{theorem}\label{thm: Matrix bernstein}(\cite[Theorem 1.4]{tropp2012user}, c.f. \cite{oliveira2009concentration})
Let $X_{1},\hdots ,X_{M}$ be a set of independent random symmetric $m\times m$ matrices. Suppose for $i=1,\hdots M$: $ \vert\vert X_{i}-\mathbb{E}(X_{i})\vert\vert_{2}\leq L$.
Let $X=\sum_{i}X_{i}$, and $$\sigma(X)=\left\vert \left\vert\sum_{i}\mathbb{E}\left[(X_{i}-\mathbb{E}(X_{i}))(X_{i}-\mathbb{E}(X_{i}))^{T}\right]\right\vert\right\vert.$$
Then for all $t>0$:
$$\mathbb{P}(\vert\vert X-\mathbb{E}(X)\vert\vert_{2}\geq t)\leq 2m\exp\left\{\dfrac{-t^{2}\slash 2}{\sigma(X)+Lt\slash 3}\right\}.$$
\end{theorem}
We will also need to use the Chernoff bounds: for $X\sim Bin(m,p)$ and $\sigma\in [0,1],$ $$\mathbb{P}(\vert X - mp\vert \geq \delta mp)\leq 2\exp(-mp \delta^{2}\slash 3).$$
This allows us to prove the following.
\begin{lemma}\phantom{x}
\begin{enumerate}[label=$\roman*)$]
\item Let $\Sigma\sim\Gamma(m,\pi,p,*)$ with $mp=\Omega_{m}(\pi_{max}\log m)$ or $\Sigma\sim\Gamma(m,\underline{w})$ with $w_{min}p=\Omega_{m} (w_{max}\log m)$. Then
$$\vert\vert D(\Sigma)^{-1}A(\Sigma)-\mathbb{E}(D(\Sigma)^{-1})\mathbb{E}(A(\Sigma))\vert \vert_{2}=o_{m}(1)$$
with probability tending to $1$ as $m$ tends to infinity.
\item Let $\Sigma\sim \mathcal{B}(m_{1},m_{2},\pi,p)$ with $\min\{m_{1},m_{2}\}p=\Omega_{m_{1}}(\pi_{max}\log (m_{1}+m_{2}))$. Then
$$\vert\vert D(\Sigma)^{-1}A(\Sigma)-\mathbb{E}(D(\Sigma)^{-1})\mathbb{E}(A(\Sigma))\vert \vert_{2}=o_{m_{1}}(1)$$
with probability tending to $1$ as $m_{1}$ tends to infinity.
\end{enumerate}
\end{lemma}
\begin{proof}
Consider statement $i)$. Let $\Sigma\sim\Gamma(m,\pi,p,*)$. For $1\leq i<j\leq m$ let $E^{i,j}$ be the matrix with all zero entries except for entry $1$ in position $(i,j)$. Let $X_{i,j}$ be the random symmetric matrix $X_{i,j}=\xi_{i,j}(E^{i,j}+E^{j,i})$, where $\xi_{i,j}\sim Bernoulli(p_{i,j})$ for $p_{i,j}:=(\pi(u_{i})\pi(u_{j}^{-1})+\pi(u_{i}^{-1})\pi(u_{j}))p.$
For each $i,j;$ $$\vert\vert X_{i,j}-\mathbb{E}(X_{i,j})\vert\vert_{2}\leq \max \left\{\vert\vert X_{i,j}-\mathbb{E}(X_{i,j})\vert \vert_{1},\vert\vert X_{i,j}-\mathbb{E}(X_{i,j})\vert \vert_{\infty}\right\}\leq1.$$
Furthermore, for each $i,j$; $$ \mathbb{E}[(X_{i,j}-\mathbb{E}(X_{i,j}))(X_{i,j}-\mathbb{E}(X_{i,j}))^{T}]= p_{i,j}(1-p_{i,j})(E^{i,j}+E^{j,i}).$$
Let $X=\sum_{i,j}X_{i,j}$, so that $X=A(\Sigma)$.
Since $p_{i,j}\leq \pi_{max}2p$, we have that
\begin{align*}
\sigma(X)&= \left\vert\left\vert\sum_{i,j}p_{i,j}(1-p_{i,j})(E^{i,j}+E^{j,i}) \right\vert \right\vert_{2}\\
&\leq \max \left\{\left\vert\left\vert\sum_{i,j}p_{i,j}(1-p_{i,j})(E^{i,j}+E^{j,i}) \right\vert \right\vert_{1},\left\vert\left\vert\sum_{i,j}p_{i,j}(1-p_{i,j})(E^{i,j}+E^{j,i}) \right\vert \right\vert_{\infty}\right\}\\&\leq 2\pi_{max}mp.
\end{align*}
Therefore, by Theorem \ref{thm: Matrix bernstein} (the matrix Bernstein inequality), letting $\omega$ be any function such that $
\omega=\Omega_{m}(1)$ and $mp=\Omega_{m}(\pi_{max}\omega \log m) :$
\begin{align*}
\mathbb{P}\left(\vert \vert X-\mathbb{E}(X)\vert\vert_{2}\geq \sqrt{\pi_{max}mp\omega \log m}\right)&\leq \exp\left\{\dfrac{-\pi_{max}mp\omega \log m}{2mp\pi_{max}+\sqrt{mp\omega \log m}\slash 3}\right\}\\
&\leq m^{-\omega\slash 100}\\
&=o_{m}(1).
\end{align*}
Next, $\mathbb{E}(D)_{min}\geq \pi_{min}^{2}mp$. Note that $X=A(\Sigma),$ and so:
\begin{align*}
& \mathbb{P}\left(\vert \vert \mathbb{E}(D)_{min}^{-1}[A(\Sigma)-\mathbb{E}(A(\Sigma))]\vert\vert_{2} \geq \sqrt{\pi_{max}mp\omega \log m}\slash \pi_{min}mp\right)\\ &\leq \mathbb{P}\left(\vert \vert A(\Sigma)-\mathbb{E}(A(\Sigma)\vert\vert_{2}\geq \sqrt{\pi_{max}mp\omega \log m}\right)\\&=o_{m}(1).
\end{align*}
Finally, since $mp=\Omega_{m}(\pi_{max}\log m)$, it is follows by a standard application of the Chernoff bounds (see e.g. \cite[Theorem 3.4]{frieze_karonski}) that with probability tending to $1$,
$$\vert \vert I-D(\Sigma)\mathbb{E}(D(\Sigma))^{-1}\vert\vert_{2}\leq o_{m}(1),$$
i.e. the actual degree of a vertex is close to its expected degree. Hence
\begin{align*}
\vert \vert D^{-1}A(\Sigma)-\mathbb{E}(D)^{-1}A(\Sigma)\vert \vert_{2}&\leq \vert \vert I-\mathbb{E}(D)^{-1}D\vert \vert_{2}\vert \vert D^{-1}A(\Sigma)\vert \vert_{2}\\
&\leq o_{m}(1)
\end{align*}
with probability $1-o_{m}(1)$, and the result follows. The remaining cases follow similarly.
\end{proof}
As $A(\Lambda(m_{1},m_{2},\pi,p,*))$ and $A(\mathkey(m_{1},m_{2},\pi,p,*))$ are formed by summing matrices of the above form, this immediately implies the following.
\begin{lemma}\label{lem: difference between adjacency matrix and expected adjacency matrix}
Let $\Sigma\sim \Lambda(m_{1},m_{2},\pi,p,*)$ or $\Sigma\sim \mathkey(m_{1},m_{2},\pi,p,*)$ where $(m_{1}+2m_{2})p=\Omega_{m}(\pi_{max}\log (m_{1}+2m_{2}))$. Then
$$\vert\vert D(\Sigma)^{-1}A(\Sigma)-\mathbb{E}(D(\Sigma)^{-1})\mathbb{E}(A(\Sigma))\vert \vert_{2}=o(1)$$
with probability tending to $1$ as $m_{1}+m_{2}$ tends to infinity.
\end{lemma}
\subsection{The first eigenvalue of \texorpdfstring{$\boldsymbol{\Gamma(m,\pi,p,*)}$}{Gamma(m,pi,p)}}
We now turn to analysing the first eigenvalue of the graph $\Gamma(m,\pi,p,*)$.
\begin{lemma}\label{lem: eigenvalue of MGpp}
Suppose that $m\geq 1$, and $p$ is such that that $mp=\Omega_{m}\left(\pi_{max}\log^{6}m\right).$ Then $ \mu_{2}(M(\Gamma(m,\pi,p,*)))=o_{m}(1).$
\end{lemma}
\begin{proof}
Let $M=M(\Gamma(m,\pi,p,*))$, and let $W$ be the expected degree matrix of $\Gamma(m,\pi,p,*)$. Define \begin{itemize}
\item $N_:=\sum_{v\in V}\pi(v),$
\item $\underline{v}(x_{i})=\pi(x_{i})Np$,
\item $\underline{v}'(x_{i})=\pi(x_{i}^{-1})Np,$ and
\item $\underline{w}(x_{i})=2(\pi(x_{i})+\pi(x_{i}^{-1}))Np$.
\end{itemize}
Let $W_{1}=\mathbb{E}(D(\Gamma(m,\underline{v})))$ and $W_{2}=\mathbb{E}(D(\Gamma(m,\underline{v}'))$. Define similarly the matrices $A_{1}$ and $A_{2}$ as the expected adjacency matrices of the graphs $\Gamma(m,\underline{v})$ and $\Gamma (m,\underline{v'})$. Let $W'=\mathbb{E}(D(\Gamma(m,\underline{w}))$. Note that $(WW_{1}^{-1})_{max}\leq 1\slash \pi_{min}.$
Let $M'=M+W^{-1}A_{1}+W^{-1}A_{2}=2M(\Gamma(m,\underline{w})).$ By our assumptions, we have that $w_{min}=\Omega (w_{max}\sqrt{\overline{w}}\log^{3}(m))$, and similarly for $\underline{v}$ and $\underline{v}'$. Therefore $\mu_{2}(M')=o_{m}(1)$ by \cite[Theorem 5]{chungrandomgraph} and Lemma \ref{lem: difference between adjacency matrix and expected adjacency matrix}. Similarly $$\mu_{1}(-W_{1}^{-1}A_{1})=-\mu_{m}(W_{1}^{-1}A_{1})=o_{m}(1),$$ and $\mu_{2}(-W_{2}^{-1}A_{2})=o_{m}(1)$.
By Weyl's inequality and the Courant-Fischer theorem, we have that
\begin{align*}
\mu_{2}(M)&=\mu_{2}\bigg(M'-W^{-1}W_{1}(W_{1}^{-1}A_{1})-W^{-1}W_{2}(W_{2}^{-1}A_{2})\bigg)\\
&\leq \mu_{2}(M')+[W^{-1}W_{1}]_{max}\mu_{1}(-W_{1}^{-1}A_{1})+[W^{-1}W_{2}]_{max}\mu_{1}(-W_{2}^{-1}A_{2})\\
&\leq \mu_{2}(M')+\mu_{1}(-W_{1}^{-1}A_{1})+\mu_{1}(-W_{2}^{-1}A_{2})\\
& \leq o_{m}(1).
\end{align*}
\end{proof}
\subsection{The first eigenvalue of \texorpdfstring{$\boldsymbol{\mathcal{B}(m_{1},m_{2},\pi,p)}$}{B(m,pi,p)}}
We now aim to prove a similar theorem for $\mathcal{B}(m_{1},m_{2},\pi,p)$. In particular, we prove the following.
\begin{lemma}\label{lem: eigenvalue of MBpp}
Suppose that $m_{1},m_{2}\geq 1$, and $p$ is such that that $$\min\{m_{1},m_{2}\}p=\Omega_{m_{1}}\left(\pi_{max}\log^{6}(m_{1}+m_{2})\right).$$ Then $ \mu_{2}(M(\mathcal{B}(m_{1},m_{2},\pi,p)) )\leq o_{m_{1}}(1).$
\end{lemma}
\begin{proof}
The proof proceeds as in the proof of Theorem \ref{lem: eigenvalue of MGpp}. Let $A_{1}$ be the matrix with $(A_{1})_{i,j}=\pi(u_{i})\pi (u_{j})p$ for $i,j\leq m$ and $0$ otherwise. Let $A_{2}$ be the matrix with $(A_{2})_{i,j}=\pi(v_{i})\pi(v_{j})p$ for $i,j> m$ and $0$ otherwise.
Then $M'=M+W^{-1}A_{1}+W^{-2}A_{2}=2M(\Gamma(m_{1}+m_{2},\underline{w})),$ where $\underline{w}$ is defined similarly to the previous lemma. We observe $$\mu_{2}(M'),\mu_{1}(-W^{-1}A_{1}),\mu_{1}(-W^{-1}A_{2})=o_{m_{1}+m_{2}}(1).$$ The proof now follows similarly to Lemma \ref{lem: eigenvalue of MGpp}.
\end{proof}
\subsection{The first eigenvalue of \texorpdfstring{$\boldsymbol{\Lambda(m_{1},m_{2},\pi,p,*)}$}{Lambda(m1,m2,pi,p)}}
We now turn to analysing the eigenvalues of $\Lambda(m_{1},m_{2},\pi,p,*)$.
\begin{lemma}\label{lem: eigenvalue of MLpp}
Suppose that $m_{1},m_{2}\geq 1$, and $p$ is such that that $$\min\{m_{1},m_{2}\}p=\Omega_{m_{1}}\left(\pi_{max}\log^{6}(m_{1}+m_{2})\right).$$ Then $$ \mu_{2}(M(\Lambda(m_{1},m_{2},\pi,p,*)) )\leq o_{m_{1}}(1).$$
\end{lemma}
\begin{proof}
We will use the Rayleigh quotient theorem. We note that
\begin{align*}
M:= M(\Lambda(m_{1},m_{2},\pi,p,*))=W_{1}M_{1}+W_{2}M_{2},
\end{align*}
where $M_{1}$ corresponds to the edges in $V_{1}\times V_{2}$ and $M_{2}$ corresponds to the edges in $V_{1}\times V_{3}$ (normalised by the expected degrees). In particular $M_{1}=M(\mathcal{B}(m_{1},m_{2},\pi,p))$, and $M_{2}=M(\mathcal{B}(m_{1},m_{2},\pi\circ *,p))$. Each $M_{i}$ has largest eigenvalue equal to $1$, with corresponding eigenvector $\vect{1}_{V_{1}\sqcup V_{2}}$ and $\vect{1}_{V_{1}\sqcup V_{3}}$ respectively. Similarly, the first eigenvalue of $M$ is $1$, with corresponding eigenvector $\vect{1}_{V_{1}\sqcup V_{2}\sqcup V_{3}}$.
Therefore, to estimate $\mu_{2}(M)$, we analyse
$$\mu_{2}:=\max\limits_{\substack{\vect{x}\perp \vect{1}_{V}\\ \vert \vert \vect{x}\vert \vert =1}}\langle M\vect{x},\vect{x}\rangle.$$ Let $\vect{\phi}$ be a unit vector with $\vect{\phi}\cdot \vect{1}_{V}=0$ achieving the above maximum, whose existence is guaranteed by the Courant-Fischer theorem. We may write
\begin{align*}
\vect{\phi}&=\vect{\xi}+\vect{\psi}\\
&=\begin{pmatrix}
\alpha \vect{1}_{V_1}\\
\beta \vect{1}_{V_{2}}\\
\gamma \vect{1}_{V_{3}}
\end{pmatrix}+\begin{pmatrix}
\vect{\psi}_{1}\\
\vect{\psi}_{{2}}\\
\vect{\psi}_{{3}}
\end{pmatrix},
\end{align*}
where $\vect{\psi}_{i}\cdot \vect{1}_{V_{i}}=0$ for $i=1,2,3.$ Since $\vect{\phi}\cdot\vect{1}_{V}=0,$ we have that $$\alpha \vert V_{1}\vert+\beta\vert V_{2}\vert+\gamma \vert V_{3}\vert=\alpha m_{1}+(\beta +\gamma)m_{2}=0,$$ and so $\alpha(\beta+\gamma)\leq 0.$
Define $$N_{1}=\sum_{u\in V_{1}}\pi (u),\mbox{ and }N_{2}=\sum_{v\in V_{2}}\pi (v).$$ Furthermore, define
\begin{align*}
D_{1}&=diag\bigg(\pi(u_{1}^{-1})N_{2},\hdots,\pi(u_{m_{1}}^{-1})N_{2},\underbrace{0,\hdots ,0}_{m_{2}},\pi(v_{1})N_{1},\hdots, \pi(v_{m_{2}})N_{1}\bigg),
\\D_{2}&=diag\bigg(\pi(u_{1})N_{2},\hdots,\pi(u_{m_{1}})N_{2},\pi(v_{1})N_{1},\hdots,\pi(v_{m_{2}})N_{1},\underbrace{0,\hdots ,0}_{m_{2}}\bigg).\end{align*}
Then $$M_{1}=D_{1}^{-1}\begin{pmatrix}
0&A_{1}&0\\
A_{1}^{T}&0&0\\
0&0&0,
\end{pmatrix}$$
where $(A_{1})_{i,j}=\pi(u_{i}^{-1})\pi(v_{j})$, and similarly $$M_{2}=D_{2}^{-1}\begin{pmatrix}
0&0&A_{2}\\
0&0&0\\
A_{2}^{T}&0&0,
\end{pmatrix},$$
where $(A_{2})_{i,j}=\pi(u_{i})\pi(v_{j})$.
We can see that
\begin{align*}
W_{1}&=diag\bigg(\dfrac{\pi(u_{1}^{-1})}{\pi(u_{1})+\pi(u_{1}^{-1})},\hdots,\dfrac{\pi(u_{m_{1}}^{-1})}{\pi(u_{m_1})+\pi(u_{m_1}^{-1})},\underbrace{0,\hdots ,0}_{m_{2}},\underbrace{1,\hdots, 1}_{m_{2}}\bigg),
\end{align*}
so that $\vert \vert W_{1}\vert\vert_{2}\leq 1$. Similarly, \begin{align*}
W_{2}&=diag\bigg(\dfrac{\pi(u_{1})}{\pi(u_{1})+\pi(u_{1}^{-1})},\hdots,\dfrac{\pi(u_{m_{1}})}{\pi(u_{m_1})+\pi(u_{m_1}^{-1})},\underbrace{1,\hdots, 1}_{m_{2}},\underbrace{0,\hdots ,0}_{m_{2}}\bigg),,
\end{align*} so that $\vert\vert W_{2}\vert\vert_{2}\leq 1$. Now, $\vert \vert \vect{\psi}\vert\vert \leq 1$ and $\vect{\psi}\cdot \vect{1}_{V_{1}\sqcup V_{2}}=0$, and so by the Courant-Fischer theorem:
$$\langle M_{1}\vect{\psi},\vect{\psi}\rangle\leq \mu_{2}(M_{1})\vert\vert \vect{\psi}\vert\vert^{2}\leq \mu_{2}(M_{1}).$$
Therefore:
\begin{align*}
\langle W_{1}M_{1}\vect{\psi},\vect{\psi}\rangle &=\vect{\psi}^{T} W_{1}M_{1}\vect{\psi}\\
&=\vect{\psi}^{T} W_{1}\vect{\psi}\vect{\psi}^{T}M_{1}\vect{\psi}\slash \vert\vert\vect{\psi}\vert\vert^{2}\\ &= \vect{\psi}^{T}M_{1}\vect{\psi}\left(\vect{\psi}^{T}W_{1}\vect{\psi}^{T}\slash \vect{\psi}^{T}\vect{\psi}\right)\\
&\leq \vect{\psi}^{T}M_{1}\vect{\psi}\max_{\vect{x}\neq \vect{0}}\left(\vect{x}^{T}W_{1}\vect{x}^{T}\slash \vect{x}^{T}\vect{x}\right)\\
&\leq\langle M_{1}\vect{\psi},\vect{\psi}\rangle \left(\vert \vert W_{1}\vert \vert_{2}\right)^{2}\\
&\leq \mu_{2}(M_{1}) \vert\vert \vect{\psi}\vert\vert^{2}\left(\vert \vert W_{1}\vert \vert_{2}\right)^{2}\\
&\leq \mu_{2}(M_{1}) \left(\vert \vert W_{1}\vert \vert_{2}\right)^{2}\\
&\leq \mu_{2}(M_{1})=o_{m_{1}}(1).
\end{align*}
We can also see that $\langle W_{2}M_{2}\vect{\psi},\vect{\psi}\rangle\leq o_{m_{1}}(1).$ It therefore remains to analyse $\langle M\vect{\xi},\vect{\xi}\rangle.$ We have that
\begin{align*}
M\vect{\xi}&=\begin{pmatrix}
\begin{pmatrix}
\dfrac{\gamma \pi (u_{1})+\beta \pi(u_{1}^{-1})}{\pi(u_{1})+\pi(u_{1}^{-1})}\\
\vdots\\
\dfrac{\gamma \pi (u_{m_1})+\beta \pi (u_{m_1}^{-1})}{\pi(u_{m_1})+\pi(u_{m_1}^{-1})}\\
\end{pmatrix}\\
\alpha\vect{1}_{V_2}\\
\alpha \vect{1}_{V_{3}}
\end{pmatrix}.
\end{align*}
Therefore, as $\vert V_{1}\vert ,\vert V_2\vert =\vert V_3\vert\geq 0$ and $*$ is fixed-point free:
\begin{align*}
\langle M\vect{\xi},\vect{\xi}\rangle &=\sum_{u\in V_{1}}\dfrac{\alpha\gamma\pi(u)+\alpha\beta\pi(u^{-1})}{\pi(u)+\pi(u^{-1})} +\alpha\beta \vert V_2\vert +\alpha \gamma \vert V_3\vert\\
&=\alpha \gamma\vert V_{1}\vert+\alpha \beta \vert V_{1}\vert +\alpha\beta \vert V_2\vert +\alpha \gamma \vert V_3\vert\\
&=\alpha(\beta +\gamma)(\vert V_{1}\vert +\vert V_2\vert)\leq 0.
\end{align*}
Hence
\begin{align*}
\mu_{2}(M)=\langle M\vect{\phi},\vect{\phi}\rangle
&=\langle M\vect{\xi},\vect{\xi}\rangle+\langle W_{1}M_{1}\vect{\psi},\vect{\psi}\rangle+\langle W_{2}M_{2}\vect{\psi},\vect{\psi}\rangle\\
&\leq \langle W_{1}M_{1}\vect{\psi},\vect{\psi}\rangle+\langle W_{2}M_{2}\vect{\psi},\vect{\psi}\rangle\\
&\leq o_{m_{1}}(1).
\end{align*}
\end{proof}
\subsection{The first eigenvalue of \texorpdfstring{$\boldsymbol{\mathkey(m_{1},m_{2},\pi,p,*)}$}{Delta(m1,m2,pi,p)}}
We finally have all the pieces in place to analyse the first eigenvalue of the random graph $\mathkey(m_{1},m_{2},\pi,p,*)$. We will approach this in a similar manner to the previous subsection. However, we need three different results: we split the subsequent analysis into three smaller sections.
\subsubsection{The case \texorpdfstring{$\boldsymbol{m_{1}=\Omega_{m_{1}}(m_{2})}$}{m1=Omega(m2)}}
\begin{lemma}\label{lem: eigenvalue of Dpp m1=Omega(m2)}
Let $m_{1},m_{2}\geq 1$ with $m_{1}=\Omega_{m_{1}}(\pi_{max}m_{2})$. Suppose that $p$ satisfies $(m_{1}+2m_{2})p=\Omega_{m_{1}}(\pi_{max}\log^{6}(m_{1}+2m_{2}))$. Let $\Sigma\sim\mathkey(m_{1},m_{2},\pi,p,*)$. Then with probability tending to $1$ as $m_{1}$ tends to infinity,
$$\lambda_{1}(\Sigma)\geq 1-o_{m_{1}}(1).$$
\end{lemma}
\begin{proof}
Since $m_{1}=\Omega_{m_{1}}(\pi_{max}m_{2})$, we have that $m_{1}p=\Omega_{m_{1}}(\pi_{max}\log^{6}(m_{1})).$ Let $\Sigma_{1}$ be the subgraph of $\Sigma$ corresponding to the edges in $V_{1}^{2}$, so that $\Sigma_{1}\sim \Gamma(m,\pi,p,*)$. Similarly let $\Sigma_{2}$ be the subgraph of $\Sigma$ corresponding to the edges in $V_{1}\times (V_{2}\sqcup V_{3})$, so that $\Sigma_{2}\sim \Lambda(m_{1},m_{2},\pi,p,*)$. Finally, let $\Sigma_{3}$ be the subgraph of $\Sigma$ corresponding to the edges in $V_{2}\times V_{3}$, so that $\Sigma_{3}\sim \mathcal{B}(m_{1},m_{2},\pi,p)$.
We note that $D(\Sigma)_{min}\geq \pi_{min}^{2}m_{1}p\slash 2$ almost surely. Furthermore, with probability tending to $1$,
$$\max_{v\in V_{1}(\Sigma_{2})}deg_{\Sigma_{2}}(v) =o_{m_{1}}(m_{1}p\slash \pi_{max}),$$
since $m_{1}=\Omega_{m_{1}}(m_{2}\pi_{max}).$ Similarly, for some fixed $C>0$, with probability tending to $1$: $$\max_{v\in V_{2}(\Sigma_{2})}deg_{\Sigma_{2}}(v) \leq C\pi_{max}m_{1}p.$$
By Lemma \ref{lem: max eigenvalue of adjacency matrix}, almost surely $$\max_{v\in V(\Sigma_{3})}deg_{\Sigma_{3}}(v)=o_{m_{1}}(m_{1}p\slash \pi_{max}).$$
We have that $A(\Sigma)=A(\Sigma_{1})+A(\Sigma_{2})+A(\Sigma_{3}),$ and by Lemma \ref{lem: max eigenvalue of adjacency matrix} we see that: $$\vert \vert A(\Sigma_{2})\vert\vert_{2},\vert \vert A(\Sigma_{3})\vert\vert_{2}=o_{m_{1}}(m_{1}p)$$ with probability tending to $1$. Therefore, $$\vert \vert D(\Sigma)^{-1}A(\Sigma_{2})\vert\vert_{2},\vert \vert D(\Sigma)^{-1}A(\Sigma_{3})\vert\vert_{2}=o_{m_{1}}(1)$$ almost surely. Hence $$\vert \vert D(\Sigma)^{-1}A(\Sigma)-D(\Sigma)^{-1}A(\Sigma_{1})\vert\vert_{2}=o_{m_{1}},$$ so that $\mu_{i}(D(\Sigma)^{-1}A(\Sigma))=\mu_{i}(D(\Sigma)^{-1}A(\Sigma_{1}))+o_{m_{1}}(1)$ for all $i$. By applying the Chernoff bounds, we see that almost surely
$$\vert \vert D(\Sigma)\vert_{\Sigma_{1}}-D(\Sigma_{1})\vert \vert =o_{m_{1}}\min\left\{\vert \vert D(\Sigma)\vert_{\Sigma_{1}}\vert \vert ,\vert \vert D(\Sigma_{1})\vert \vert \right\},$$ so that
$$\vert \vert D(\Sigma)^{-1}A(\Sigma_{1})-D(\Sigma_{1})^{-1}A(\Sigma_{1})\vert\vert_{2}=o_{m_{1}}(1)$$
with probability tending to $1$. By assumption, $\Sigma_{1}\sim\Gamma(m,\pi,p,*)$ with $m_{1}p=\Omega(\pi_{max}\log^{6}m_{1}),$ and hence $\mu_{2}(D(\Sigma_{1})^{-1}A(\Sigma_{1}))=o_{m_{1}}(1)$ by Lemmas \ref{lem: difference between adjacency matrix and expected adjacency matrix} and \ref{lem: eigenvalue of MGpp}. Hence $$\mu_{2}(D(\Sigma)^{-1}A(\Sigma))=o_{m_{1}}(1),$$ almost surely and the result follows.
\end{proof}
\subsubsection{The case o\texorpdfstring{$\boldsymbol{m_{2}=\Omega_{m_{2}}(m_{1})}$}{m1=Omega(m2)}}
\begin{lemma}\label{lem: eigenvalue of Dpp m2=Omega(m1)}
Let $m_{1},m_{2}\geq 1$ with $m_{2}=\Omega(\pi_{max}m_{1})$. Suppose that $p$ satisfies $(m_{1}+2m_{2})p=\Omega_{m_{2}}(\pi_{max}\log^{6}(m_{1}+2m_{2}))$. Let $\Sigma\sim\mathkey(m_{1},m_{2},\pi,p,*)$. Then with probability tending to $1$ as $m_{2}$ tends to infinity,
$$\lambda_{1}(\Sigma)\geq 1-o_{m_{2}}(1).$$
\end{lemma}
\begin{proof}
Since $m_{2}=\Omega_{m_{2}}(\pi_{max}m_{1})$, we have that $m_{2}p=\Omega_{m_{2}}(\pi_{max}\log^{6}(m_{2})).$ Let $\Sigma_{1}$ be the subgraph of $\Sigma$ corresponding to the edges in $V_{1}^{2}$, so that $\Sigma_{1}\sim \Gamma(m,\pi,p,*)$. Similarly let $\Sigma_{2}$ be the subgraph of $\Sigma$ corresponding to the edges in $V_{1}\times (V_{2}\sqcup V_{3})$, so that $\Sigma_{2}\sim\Lambda(m_{1},m_{2},\pi,p,*)$. Finally, let $\Sigma_{3}$ be the subgraph of $\Sigma$ corresponding to the edges in $V_{2}\times V_{3}$, so that $\Sigma_{3}\sim \mathcal{B}(m_{1},m_{2},\pi,p)$.
We note that $D(\Sigma)_{min}\geq \pi_{min}^{2}m_{2}\slash 2$ almost surely. Furthermore, we may deduce similarly to the previous lemma that, with probability tending to $1$: $$\vert \vert A(\Sigma_{1})\vert\vert_{2},\vert \vert A(\Sigma_{2})\vert\vert_{2}=o_{m_{2}}(m_{2}p\slash \pi_{max}).$$
We have that $A(\Sigma)=A(\Sigma_{1})+A(\Sigma_{2})+A(\Sigma_{3}).$ Therefore $$\vert \vert D(\Sigma)^{-1}A(\Sigma_{1})\vert\vert_{2},\vert \vert D(\Sigma)^{-1}A(\Sigma_{2})\vert\vert_{2}=o_{m_{2}}(1)$$ almost surely. Hence $$\vert \vert D(\Sigma)^{-1}A(\Sigma)\vert_{\Sigma_{3}}-D(\Sigma)^{-1}A(\Sigma_{3})\vert\vert_{2}=o_{m_{2}}(1)$$ almost surely, so that $\mu_{i}(D(\Sigma)^{-1}A(\Sigma))=\mu_{i}(D(\Sigma)^{-1}A(\Sigma_{3}))+o_{m_{2}}(1)$ for all $i$. Finally,
$$\vert \vert D(\Sigma)^{-1}A(\Sigma_{3})-D(\Sigma_{3})^{-1}A(\Sigma_{3})\vert\vert_{2}\leq o_{m_{2}}(1)$$
with probability tending to $1$. By assumption, $\Sigma_{3}\sim\mathcal{B}(m_{1},m_{2},\pi,p)$ with $m_{2}p=\Omega_{m_{2}}(\pi_{max}\log^{6}m_{2}),$ so that $\mu_{2}(D(\Sigma_{3})^{-1}A(\Sigma_{3}))\leq o_{m_{2}}(1)$ by Lemmas \ref{lem: difference between adjacency matrix and expected adjacency matrix} and \ref{lem: eigenvalue of MBpp}. Therefore $$\mu_{2}(D(\Sigma)^{-1}A(\Sigma))\leq o_{m_{2}}(1),$$
as required.
\end{proof}
\subsubsection{The case \texorpdfstring{$\boldsymbol{m_{1}=\Theta ( m_{2})}$}{m1=Theta(m2)}}
We now turn to the final case.
\begin{lemma}\label{lem: eigenvalue of MDpp m1=theta(m2)}
Let $m_{1},m_{2}\geq 1$ with $$O_{m_{1}+m_{2}}(m_{2}\slash \pi_{max})\leq m_{1}\leq O_{m_{1}+m_{2}}(m_{2}\pi_{max}).$$ Suppose that $p$ satisfies $(m_{1}+2m_{2})p=\Omega_{m_{1}}(\pi_{max}^{2}\log^{6}(m_{1}+2m_{2}))$. Then
$$\mu_{2}\left(M(\mathkey(m_{1},m_{2},\pi,p,*))\right)\leq o_{m_{1}}(1).$$
\end{lemma}
\begin{proof}
We may write
\begin{align*}
M:&=M(\mathkey(m_{1},m_{2},\pi,p,*))=W_{1}M_{1}+W_{2}M_{2}+W_{3}M_{3},
\end{align*}
where $M_{1}=M(\Gamma(m,\pi,p,*))$, $M_{2}=M(\Lambda(m_{1},m_{2},\pi,p,*))$, and\\ $M_{3}=M(\mathcal{B}(m_{1},m_{2},\pi,p))$.
Now, as previously, let $\vect{\phi}$ be a vector with $\norm{\phi}=1,$ $\vect{\phi}\cdot \vect{1}_{V}=0,$ and $$\mu_{2}(M)=\langle M\vect{\phi},\vect{\phi}\rangle.$$ This vector is guaranteed to exist by the Courant-Fischer theorem. We again write \begin{align*}
\vect{\phi}&=\begin{pmatrix}
\alpha\vect{1}_{V_{1}}\\
\beta\vect{1}_{V_{2}\sqcup V_{3}}
\end{pmatrix}+\begin{pmatrix}
\vect{\psi}_{1}\\
\vect{\psi}_{2}
\end{pmatrix}\\
&=\vect{\xi}+\vect{\psi},
\end{align*}
where $\vect{\psi}_{1}\cdot \vect{1}_{V_{1}}=\vect{\psi}_{2}\cdot \vect{1}_{V_{2}\sqcup V_{3}}=0.$
As we previously argued, by our assumptions on $p$ we have that:
$$\langle W_{i}M_{i}\vect{\psi},\vect{\psi}\rangle=o_{m_{1}}(1),$$
for $i=1,2,3.$
It therefore remains to analyse $\langle M\vect{\xi},\vect{\xi}\rangle.$ Recall that we defined $$N_{1}=\sum_{u\in V_{1}}\pi (u)\;N_{2}=\sum_{v\in V_{2}}\pi (v).$$
For $i\leq m_{1}$, we calculate $$M_{i,j}:=
\begin{cases}
\dfrac{\pi (u_{i})\pi(u_{j}^{-1})+\pi (u_{i}^{-1})\pi(u_{j})}{{(\pi(u_{i})+\pi(u_{i}^{-1}))(N_{1}+N_{2})}}: \;j\leq m_{1},\\
\dfrac{ \pi(u_{i}^{-1})\pi(v_{j})}{(\pi(u_{i})+\pi(u_{i}^{-1}))(N_{1}+N_{2})}: \;m_{1}+1\leq j\leq m_{1}+m_{2},\\
\dfrac{\pi (u_{i})\pi(v_{j})}{(\pi(u_{i})+\pi(u_{i}^{-1}))(N_{1}+N_{2})}: \;m_{1}+m_{2}+1\leq j.
\end{cases}$$
For $m_{1}+1\leq i\leq m_{1}+m_{2},$ we see that
$$M_{i,j}:=
\begin{cases}
\dfrac{\pi (v_{i})\pi(u_{j}^{-1})}{\pi(v_{i})(N_{1}+N_{2})}: \;j\leq m_{1},\\
0: \;m_{1}+1\leq j\leq m_{1}+m_{2},\\
\dfrac{\pi (v_{i})\pi(v_{j})}{\pi(v_{i})(N_{1}+N_{2})}: \;m_{1}+m_{2}+1\leq j.
\end{cases}$$
For $m_{1}+m_{2}+1\leq i\leq m_{1}+2m_{2},$ we see that
$$M_{i,j}:=
\begin{cases}
\dfrac{\pi (v_{i})\pi(u_{j})}{\pi(v_{i})(N_{1}+N_{2})}: \;j\leq m_{1},\\
\dfrac{\pi(v_{i})\pi(v_{j})}{\pi(v_{i})(N_{1}+N_{2})}: \;m_{1}+1\leq j\leq m_{1}+m_{2},\\
0: \;m_{1}+m_{2}+1\leq j.\\
\end{cases}$$
We therefore have:
$$M\vect{\xi}=\begin{pmatrix}
\dfrac{N_{1}\alpha +N_{2}\beta}{N_{1}+N_{2}}\vect{1}_{V_{1}}\\
\dfrac{N_{1}\alpha +N_{2}\beta}{N_{1}+N_{2}}\vect{1}_{V_{2}\sqcup V_{3}}
\end{pmatrix}.$$
Recall that $\vert V_{2}\vert=\vert V_{3}\vert.$ We calculate:
\begin{align*}
& \langle M\vect{\xi},\vect{\xi}\rangle=\begin{pmatrix}
\dfrac{N_{1}\alpha +N_{2}\beta}{N_{1}+N_{2}}\vect{1}_{V_{1}}\\
\dfrac{N_{1}\alpha +N_{2}\beta}{N_{1}+N_{2}}\vect{1}_{V_{2}\sqcup V_{3}}
\end{pmatrix}\cdot \begin{pmatrix}
\alpha\vect{1}_{V_{1}}\\
\beta\vect{1}_{V_{2}\sqcup V_{3}}\end{pmatrix}\\
&=\dfrac{N_{1}\alpha^{2}+N_{2}\alpha\beta }{N_{1}+N_{2}}\vert V_{1}\vert +\dfrac{N_{1}\alpha\beta +N_{2}\beta^{2}}{N_{1}+N_{2}}\vert V_{2}\sqcup V_{3}\vert\\
&=\dfrac{1}{N_{1}+N_{2}}\bigg[
N_{1}\alpha^{2} \vert V_{1}\vert+N_{2}\alpha\beta\vert V_{1}\vert+N_{1}\alpha \beta \vert V_{2}\sqcup V_{3}\vert+N_{2}\beta^{2} \vert V_{2}\sqcup V_{3}\vert
\bigg]\\
&=\dfrac{N_{1}\alpha+N_{2}\beta}{N_{1}+N_{2}}\bigg[
\alpha \vert V_{1}\vert+\beta \vert V_{2}\sqcup V_{3}\vert\bigg]
\\&=0,
\end{align*}
since $$\vect{\xi}\cdot\vect{1}_{V}=\alpha \vert V_{1}\vert +\beta \vert V_{2}\sqcup V_{3}\vert=0.$$
Hence,
\begin{align*}
\mu_{2}(M)=\langle M\vect{\phi},\vect{\phi}\rangle&=\langle M\vect{\xi},\vect{\xi}\rangle+\langle W_{1}M_{1}\vect{\psi},\vect{\psi}\rangle+\langle W_{2}M_{2}\vect{\psi},\vect{\psi}\rangle+\langle W_{3}M_{3}\vect{\psi},\vect{\psi}\rangle\\
&\leq 0+o_{m_{1}}(1)+o_{m_{1}}(1)+o_{m_{1}}(1).
\end{align*}
\end{proof}
Applying Lemma \ref{lem: difference between adjacency matrix and expected adjacency matrix} to Lemmas \ref{lem: eigenvalue of Dpp m1=Omega(m2)}, \ref{lem: eigenvalue of Dpp m2=Omega(m1)}, and \ref{lem: eigenvalue of MDpp m1=theta(m2)}, we deduce the following.
\begin{lemma}\label{lem: eigenvalue of Dpp}
Let $m_{1},m_{2}\geq 1$. Suppose that $p$ satisfies $(m_{1}+2m_{2})p=\Omega_{m_{1}+2m_{2}}(\pi_{max}^{2}\log^{6}(m_{1}+2m_{2}))$. Let $\Sigma\sim \mathkey(m_{1},m_{2},\pi,p,*)$. Then
$$\lambda_{1}(\Sigma)\geq 1-o_{m_{1}+2m_{2}}(1)$$ with probability tending to $1$ as $m_{1}+2m_{2}$ tends to infinity.
\end{lemma}
\section{Property (T) in random quotients of hyperbolic groups}\label{sec: Property (T) in quotients of hyperbolic groups}
We can now turn to proving Theorem \ref{mainthm: property t in random quotients of hyperbolic groups}. We will first focus on a result for a slightly different model of random groups, where each relator is added independently with probability close to some $p$. We now fix $\omega$ a slow-growing function.
\begin{definition}
Given a word $w$, let $Path(w)$ be the set of paths in $C^{max}$ with label $w$. Note that for any word $w$, this set is of size at most $\vert V(C^{max})\vert.$
For a number $l-\omega (l)\leq L\leq l$, let $Sol_{l}(L)$ be the set of ordered triples $(a,b,c)$ with $a+b+c=L,l\slash 3-\omega(l)\slash 3 \leq a,b,c\leq l\slash 3+2$ (note that $\vert Sol_{l}(L)\vert \geq 1$ for all $l-\omega (l)\leq L\leq l$).
The \emph{binomial model of random groups at length $\omega$-near $l$} is obtained as follows. Let $\Xi=\Xi(l)\leq \mu^{l-\omega(l)}.$ For each path $\gamma$ in $C^{max}$ of length between $l-\omega(l)$ and $l$, and each $(a,b,c)\in Sol(\vert \gamma\vert )$, add $lab(\gamma)$ to $R_{l;a,b,c}$ with probability $$\nu (\mathcal{Z}[\gamma,0])\Xi\slash (\omega(l)\slash 3)^{2}.$$
Let $\mathcal{R}_{l}=\sqcup_{a,b,c}\{x_{r}y_{r}z_{r}\;:\;r=(x_{r},y_{r},z_{r})\in R_{l;a,b,c}\},$ and consider $$G\slash \langle \langle \mathcal{R}_{l}\rangle\rangle.$$
\end{definition}
Note that $$Pr(w\in \mathcal{R}_{l})=(1+o_{l}(1))\vert Path(w)\vert\Xi \nu (\mathcal{Z}[w,0])\leq \vert V(C^{max})\vert K\slash \mu^{\vert w\vert}$$
for some constant $K>0.$
We can prove the following.
\begin{theorem}\label{thm: (T) in binomial hyperbolic quotients}
Let $G=\langle S\;\vert \; T\rangle $ be a finite presentation of a hyperbolic group with growth rate $\mathfrak{h}>0.$ Let $\mathcal{R}_{l}$ be obtained as above. If $\Xi\exp\{-\mathfrak{h}l\slash 3\}=\Omega_{l}(l^{7}),$ then $G\slash \langle\langle \mathcal{R}_{l}\rangle\rangle$ has Property (T) with probability tending to $1$ as $l$ tends to infinity.
\end{theorem}
This allows us to prove Theorem \ref{mainthm: property t in random quotients of hyperbolic groups}.
\begin{proof}[Proof of Theorem \ref{mainthm: property t in random quotients of hyperbolic groups}]
Consider the following model of random quotients of $G$, which we denote by $G_{1}(l,p)$. Let $p_{l}:\mathcal{L}^{G}\rightarrow [0,1]$. Add each word $w\in\mathcal{L}^{G}_{l,\omega (l)}$ to the relator set $\mathfrak{A}_{l}$ with probability $p_{l}(w)$, and consider $G\slash \langle \langle \mathfrak{A}_{l}\rangle \rangle$.
First note that, in the binomial model, there exists a constant $C$ such that $p(w\in \mathcal{R}_{l})\leq C\nu^{ld-\vert w\vert}$, independent of $w$. Therefore, as $(T)$ is preserved under quotients, Theorem \ref{thm: (T) in binomial hyperbolic quotients} implies that $G_{1}(l,p_{l})$ has (T) with probability $1-o_{l}(1)$ for all $p_{l}$ with $\min_{w\in\mathcal{L}^{G}_{l,\omega (l)}}\{p_{l}(w)\}\exp\{2\mathfrak{h}l\slash 3\}=\Omega_{l}(l^{7}).$
Fix $d>1\slash 3$ and choose $d>d_{1}>d_{2}>1\slash 3.$ Recall that Property (T) is preserved under adding relators. We have that $$\vert \mathcal{L}^{G}_{l}\vert =(1+o_{l}(1))\eta ' \exp\{\mathfrak{h}l\}$$ for some $\eta'>0$. We previously defined $$\Phi(g):=\left\vert\{w\in \mathcal{L}^{G}\;:\;\overline{w}=g\}\right \vert,$$ and proved that $\Phi(g)$ is uniformly bounded above by some constant $\Phi$.
We refer to the model in Theorem \ref{mainthm: property t in random quotients of hyperbolic groups} (i.e. killing a random subset $\mathfrak{B}_{l}\subseteq Ann_{l,\omega}(G)$ of size $\exp\{\mathfrak{h}ld\}$) by $G(l,d)$. Finally, for $\rho_{l}: G\rightarrow [0,1]$ we produce the model $G_{2}(l,\rho)$ as follows. Add each element $g\in Ann_{l,\omega(l)}(G)$ to $X_{l} $ with probability $\rho_{l}(g)$, and consider $G\slash \langle \langle X_{l}\rangle \rangle$.
Now, consider $p_{l}$ defined by $$p_{l}(w):=\dfrac{\exp\{\mathfrak{h}(ld_{2}-\vert w\vert )\}}{\Phi(\overline{w})},$$ and the model $G_{1}(l,p)$. We may assume that $p_{l}(w,d_{2})=o_{l}(1)$ for all words $w$. The probability that an element $g$ appears in $\mathfrak{A}_{l}$ is
\begin{align*}
1-(1-p_{l}(w,d_{2}))^{\Phi (g)}&\approx (1+o_{l}(1))\Phi(g)p_{l}(w,d_{2})\\
&=(1+o_{l}(1))\exp\{\mathfrak{h}(ld_{2}-\vert w\vert)\}.
\end{align*}
Therefore the model $G_{1}(l,p)$ is effectively the same as the model $G_{2}(l,\rho)$, where we add each element $g$ of $Ann_{l,\omega(l)}(G)$ to $Y_{l}$ with probability $\rho_{l}:=\exp\{\mathfrak{h}ld_{2}-\mathfrak{h}\vert g\vert \}$. In particular, if $d_{2}>1\slash 3,$ then by Theorem \ref{thm: (T) in binomial hyperbolic quotients}, we have that $G_{2}(l,\rho_{l})$ has property $(T)$ with overwhelming probability.
Now, it is easily seen that for $G\slash \langle \langle \mathfrak{B}_{l}\rangle\rangle\sim G(l,d),$ we have almost surely that for $0\leq n\leq \omega(l)$: $$\vert \mathfrak{B}_{l}\cap S_{l-n}\vert \geq (1+o_{l}(1))\exp\{\mathfrak{h}ld-\mathfrak{h}n\}\geq \exp\{\mathfrak{h}ld_{1}\}\;(*),$$ since $n\leq \omega (l)\leq \log\log l$.
Therefore, let $\Gamma=G\slash \langle \langle X_{l}\rangle \rangle \sim G_{2}(l,p_{l}(d_{2}))$: by the Chernoff bounds we have that $$\vert X_{l}\cap S_{l-n}\vert =(1+o_{l}(1))\exp\{\mathfrak{h}ld_{2}\}$$ for each $0\leq n\leq \omega (l).$
Choose a random subset $X_{l}\subset \mathfrak{B}_{l}$ of size $\exp\{\mathfrak{h}ld\}$: then $G\slash \langle \langle \mathfrak{B}_{l}\rangle\rangle\sim G(l,d)$, conditioned on $G$ having $(*)$. Since $\Gamma$ has property (T) with probability $1-o_{l}(1)$, and $G\sim G(l,d)$ has $(*)$ with probability $1-o_{l}(1),$ the result follows.
\end{proof}
It thus remains to prove Theorem \ref{thm: (T) in binomial hyperbolic quotients}. There is one small issue, in that we do not wish to study directed graphs. Given $G$ and $R_{l}$ as above, we can define the graph $\Psi(G,R_{l})$ by collapsing each pair of directed edges $(w,v^{-1}), (v^{-1},w)$ in $\Upsilon(G,\mathcal{L}^{G},R_{l})$ to a single undirected edge. In particular, the edge set of $\Psi(G,R_{l})$ is obtained by adding the undirected edge $\{w_{1},w_{3}^{-1}\}$ for each $r=(w_{1},w_{2},w_{3})\in R_{l}$. Since $L(\Upsilon(G,\mathcal{L}^{G},R_{l}))=L(\Psi(G,R_{l}))$, we can instead just analyse $\Psi(G,R_{l}).$
Let $\mathcal{L}^{+}_{l\slash 3,\omega}$ be the set of $w\in\mathcal{L}^{G}_{l\slash 3,\omega(l)\slash 3}$ with $w^{-1}\notin \mathcal{L}^{G}$. Define the set $$\mathcal{L}^{-}_{l\slash 3,\omega }:=\{w\in\mathcal{L}^{G}_{l\slash 3,\omega (l)\slash 3}\;:\;w^{-1}\in \mathcal{L}^{+}_{l\slash 3,\omega(l)\slash 3}\}.$$ Finally, let $\mathcal{L}^{+-}_{l\slash 3,\omega(l)\slash 3}$ be the set $$\mathcal{L}^{+-}_{l\slash 3,\omega}:=\{w\in \mathcal{L}^{G}\;:\;w^{-1}\in\mathcal{L}^{G}_{l\slash 3,\omega\slash 2}\}.$$
Note that, if $w=w^{-1}$, then $w$ is the trivial word. We now apply our previous work on geodesics in hyperbolic groups to count the degrees of vertices in our random graphs. Recall the Chernoff bounds: for $X\sim Bin(m,p)$ and $\delta\in [0,1],$ $$\mathbb{P}(\vert X - mp\vert \geq \delta mp)\leq 2\exp(-mp \delta^{2}\slash 3).$$ We now prove the following.
\begin{lemma}
\label{lem: degrees l=0 mod 3}
Let $G=\langle S\;\vert \; T\rangle $ be a hyperbolic group with growth rate $\mathfrak{h}>0.$ Let $R_{l}$ be obtained as in Theorem \ref{thm: (T) in binomial hyperbolic quotients}, where $p\exp\{-\mathfrak{h}l\slash 3\}=\Omega_{l}(l)$. Then with probability tending to $1$ as $l$ tends to infinity, for any $v\in \mathcal{L}^{+}_{l,\omega (l)\slash 3},w\in \mathcal{L}^{+-}_{l,\omega (l)\slash 3}$.
\begin{align*}
rep(v)
&=(1+o_{l}(1))\Xi\nu(\mathcal{Z}[v]),\\
rep(w)
&=(1+o_{l}(1))\Xi(\nu(\mathcal{Z}[w])+\nu(\mathcal{Z}[w^{-1}])).
\end{align*}
\end{lemma}
\begin{proof}
Let us suppose that $w\in \mathcal{L}^{+-}_{l\slash 3,\omega (l)\slash 3}$. Then we calculate, using Lemma \ref{lem: measure of sets middle path}.
\begin{align*}
\mathbb{E}(rep(w))&=\sum \limits_{\substack{\sigma\gamma\sigma' \mbox{a path }\\lab(\gamma)=w}} Pr(\sigma\gamma\sigma'\mbox{ added to } R_{\vert \sigma\vert,\vert w\vert,\vert\sigma'\vert})\\&\hspace*{30 pt}+\sum \limits_{\substack{\sigma\gamma\sigma' \mbox{a path }\\lab(\gamma)=w^{-1}}} Pr(\sigma\gamma\sigma'\mbox{ added to } R_{\vert \sigma\vert,\vert w^{-1}\vert,\vert\sigma'\vert})\\
&=\Xi\nu (\mathcal{Z}[w])+\nu (\mathcal{Z}[w^{-1}]).
\end{align*}
The result follows by an application of the Chernoff bounds.
\end{proof}
We next need to analyse edge probabilities.
\begin{lemma}
\label{lem: edge probs l=0 mod 3}
Let $G=\langle S\;\vert \; T\rangle $ be a hyperbolic group with growth rate $\mathfrak{h}>0.$ Let $l\geq 1$. Let $R_{l}$ be obtained as in Theorem \ref{thm: (T) in binomial hyperbolic quotients}, where $\Xi \exp\{-\mathfrak{h}l\slash 3\}=\Omega_{l}(l)$. Let $E$ be the edge set of $\Psi(G,\mathfrak{R}_{l})$. There exists a constant $\pi_{min}>0$ and a function $\pi:L^{G}\rightarrow [\pi_{min},\infty)$ such that for any $u\in \mathcal{L}^{+}_{l\slash 3,\omega (l)\slash 3},v\in\mathcal{L}^{-}_{l\slash 3,\omega (l)\slash 3},w,w'\in \mathcal{L}^{+-}_{l\slash 3,\omega (l)\slash 3}$:
\begin{align*}
Pr(\{u,v\}\in E)&=(1+o_{l}(1))\pi(u)\pi(v^{-1})\exp\{-2\mathfrak{h}l\slash3 \}3\Xi\slash \omega (l),\\
Pr(\{u,w\}\in E)&=(1+o_{l}(1))\pi(u)\pi(w^{-1})\exp\{-2\mathfrak{h}\slash3 \}3\Xi\slash \omega(l),\\
Pr(\{v,w\}\in E)&=(1+o_{l}(1))\pi(v^{-1})\pi(w)\exp\{-2\mathfrak{h}l\slash 3\}3\Xi\slash \omega(l),\\
Pr(\{w,w'\}\in E)&=(1+o_{l}(1))[\pi(w)\pi(w'^{-1})+\pi(w^{-1})\pi(w')]\exp\{-2\mathfrak{h}l\slash 3\}3\Xi\slash \omega(l).
\end{align*}
\end{lemma}
Clearly $u$ is never connected to $v^{-1}$, etc.
\begin{proof}
We prove Statement $4$ only. For a path $\gamma=(e_{i_{1}},\hdots ,e_{i_{n}})$, let $$\pi (\gamma) =\alpha_{i_{1}}\beta_{i_{n}} \exp\{\mathfrak{h}(l\slash 3 -n)\}.$$ Define $$\pi (w)=\sum_{lab(\gamma)=w}\pi (\gamma).$$ The edge $\{w,w'\}$ can occur when either $w uw'^{-1}$ is added to $R_{l;\vert w\vert,\vert u\vert ,\vert w'\vert}$, or $w' uw^{-1}$ is added to $R_{l;\vert w'\vert,\vert u\vert ,\vert w\vert}$. By Lemma \ref{lem: measure of sets between two paths}, the probability such a word appears in the correct $R_{l;a,b,c}$ is $$(1+o_{l}(1))\dfrac{\omega (l)}{3}(\pi (w)\pi (w'^{-1})+\pi (w^{-1})\pi(w')) \Xi\exp\{-2\mathfrak{h}l\slash 3\}(\omega(l)\slash 3)^{-1}.$$
The probability the word is then added to the correct $R_{l;a,b,c}$ to obtain the desired edge is $(\omega(l)\slash 3)^{-2}$, and the result follows.
\end{proof}
Finally, we can easily observe the following.
\begin{lemma}
Let $G=\langle S\;\vert \; T\rangle $ be a hyperbolic group with growth rate $\mathfrak{h}>0.$ Let $l\geq 1$, and let $R_{l}$ be obtained as in Theorem \ref{thm: (T) in binomial hyperbolic quotients}. Then for all $g\in G$ with $g\in V$:
$$deg(g)=(1+o_{l}(1))deg(g^{-1})=(1+o_{l}(1))rep(g).$$
\end{lemma}
We now note that, importantly, for small $p$, there are not too many double edges in $\Psi(G,R_{l}).$ A \emph{matching} in a graph is a set, $\mathcal{M}$, of edges in the graph such that no two edges in $\mathcal{M}$ share a common endpoint.
\begin{lemma}\label{lem: double edges form matching} (c.f. \cite{antoniuktriangle})
Let $G=\langle S\;\vert \; T\rangle $ be a hyperbolic group, with growth rate $\mathfrak{h}>0.$ Let $R_{l}$ be constructed as in Theorem \ref{thm: (T) in binomial hyperbolic quotients}. If $\Xi\leq \exp\{\mathfrak{h}ld\}$ for some $d<5\slash 12$, then with probability tending to $1$ as $l$ tends to infinity, the set of double edges in $\Psi(G,R_{l})$ forms a matching, and there are no triple edges in $\Psi(G,R_{l})$ .
\end{lemma}
The proof of the above is effectively that of \cite[Lemma 6.3]{ashcroft2021property}, which closely follows the \cite[p.12]{antoniuktriangle}.
\begin{proof}
Fix $\Xi\leq \exp\{\mathfrak{h}ld\}$ for some $d<5\slash 12$.
The probability, $\mathbb{P}_{3}$, that there exists a pair of vertices $u,v$ with at least three edges between $u$ and $v$ is bounded above by
\begin{align*}
\mathbb{P}_{3}&\leq O_{l}\left((\exp\{\mathfrak{h}l\slash 3\})^{2}(\exp\{\mathfrak{h}l\slash 3\})^{3}\Xi^{3}\exp\{-2\mathfrak{h}l\}\log l\right)\\
&\leq O_{l}\left(\exp\{\mathfrak{h}l\slash 3\}\right)^{(5+9d-9)\mathfrak{h}l}2\log l)\\
&=o_{l}(1),
\end{align*}
since $d<5\slash 12<4\slash 9$. The probability, $\mathbb{P}_{doub}$ that there are vertices $u,v,w$ with double edges between $u$ and $v$ and $u$ and $w$ is bounded by
\begin{align*}
\mathbb{P}_{doub}&= O_{l}(\left (\exp\{\mathfrak{h}l\slash 3\})^{3}(\eta\exp\{\mathfrak{h}l\slash 3\})^{4}\Xi^{4}\exp\{-4\mathfrak{h}l\}\log l\right)\\
&\leq O_{l}(\left(\exp\{\mathfrak{h}l\slash 3\}\right)^{(12d-5)}\log l)\\
&=o_{l}(1),
\end{align*}
since $d<5\slash 12.$
\end{proof}
This allows us to prove Theorem \ref{thm: (T) in binomial hyperbolic quotients}.
\begin{proof}[Proof of Theorem \ref{thm: (T) in binomial hyperbolic quotients}]
As (T) is an increasing property, we may assume that $\Xi$ satisfies the conditions of Theorem \ref{thm: (T) in binomial hyperbolic quotients} and Lemma \ref{lem: double edges form matching}. For example, we may take $\Xi=\exp\{\mathfrak{h}ld\}$ for $1\slash 3 <d<5\slash 12$. Note that the minimum degree of $\Psi(G,\mathfrak{R}_{l})$ tends to infinity with probability tending to $1$. Therefore, as the set of double edges forms a matching in $\Psi(G,\mathfrak{R}_{l})$, by collapsing double edges in $\Psi(G,\mathfrak{R}_{l})$, we find a graph $\Psi'_{l}$ with at most one edge between any two vertices such that $$\lambda_{1}(\Psi'_{l})=\lambda_{1}(\Psi(G,\mathfrak{R}_{l}))+o_{l}(1)$$
almost surely. Applying Lemma \ref{lem: edge probs l=0 mod 3}, we observe that $$\Psi'\sim\mathring{\Delta}\left(\vert \mathcal{L}^{+-}_{l\slash 3,\omega (l)\slash 3}\vert,\vert \mathcal{L}^{+}_{l\slash 3,\omega (l)\slash 3}\vert,\pi ,\Xi\exp\{-2\mathfrak{h}l\slash 3\}\slash \omega(l),*\right),$$
where $*:V_{1}\rightarrow V_{1}$ maps $w\mapsto w^{-1}$ (and hence is fixed point free). Now, $\vert \mathcal{L}^{G}_{l,\omega(l)\slash 3}\vert = \vert \mathcal{L}^{+-}_{l\slash 3,\omega(l)\slash 3}\vert+2\vert \mathcal{L}^{+}_{l\slash 3,\omega(l)\slash 3}\vert$, $\log^{6}(\vert \mathcal{L}^{G}_{l\slash 3,\omega(l)\slash 3}\vert)=O_{l}(l^{6})$. Furthermore, $\pi_{max}\leq \log\log l$. As $\Xi$ satisfies the assumptions of Theorem \ref{thm: (T) in binomial hyperbolic quotients}, we may apply Lemma \ref{lem: eigenvalue of Dpp}, to deduce that with probability $1-o_{l}(1)$; $$\lambda_{1}(\Psi')\geq 1+o_{l}(1).$$ Since $$\lambda_{1}(\Psi(G,\mathfrak{R}_{l}))=\lambda_{1}(\Psi'(G,\mathfrak{R}_{l}))+o_{l}(1)$$ almost surely, we have $$\lambda_{1}(\Psi(G,\mathfrak{R}_{l}))\geq 1+o_{l}(1)$$ almost surely. Furthermore, with probability $1-o_{l}(1)$, $\mathcal{L}^{G}_{l\slash 3}\subseteq \mathcal{W}(R_{l})$, and so $\mathcal{W}(R_{l})$ generates a finite index subgroup of $G$ by Lemma \ref{lem: Cmax gens finite index}. The result now follows by Corollary \ref{cor: directed new spectral criterion} and Lemma \ref{lem: degrees l=0 mod 3}.
\end{proof}
We now very briefly indicate how to alter the preceding proofs to obtain a proof of Theorem \ref{mainthm: property t in random quotients of hyperbolic groups aperiodic case}. Firstly, we can replace Lemma \ref{lem: convergence of approximation} by the following classical result in ergodic Markov chains.
\begin{lemma}\label{lem: convergence of approximation aperiodic}
Let $B$ be the edge incidence matrix of $C^{max}$, with left (right) $\mu$ eigenvector $\alpha$ ($\beta)$. If $C^{max}$ is aperiodic, we have convergence $$\dfrac{B^{m}}{\mu^{m}}\rightarrow \beta\alpha.$$
\end{lemma}
We then switch to a new binomial model as follows.
\begin{definition}
The \emph{binomial model of random groups at length $3l$} is obtained as follows. Let $\Xi=\Xi(l)\leq \mu^{l-\omega(l)}.$ For each path $\gamma$ in $C^{max}$ of length $3l$, add $lab(\gamma)$ to $R_{3l;l,l,l}$ with probability $$\nu (\mathcal{Z}[\gamma,0])\Xi,$$
and consider $$G\slash \langle \langle \mathcal{R}_{l}\rangle\rangle.$$
\end{definition}
The aim is to prove the following.
\begin{theorem}\label{thm: binomial (T) aperiodic case}
Let $G=\langle S\;\vert \; T\rangle $ be a finite presentation of a hyperbolic group with growth rate $\mathfrak{h}>0,$ and $C^{max}$ aperiodic, Let $\mathcal{R}_{3l}$ be obtained as above. If $\Xi\exp\{-\mathfrak{h}l\}=\Omega_{l}(l^{7}),$ then $G\slash \langle\langle \mathcal{R}_{3l}\rangle\rangle$ has Property (T) with probability tending to $1$ as $l$ tends to infinity.
\end{theorem}
The proof of the above follows immediately as the proof of Theorem \ref{thm: (T) in binomial hyperbolic quotients}, using the following lemma, which is obtained by an application of Lemma \ref{lem: convergence of approximation aperiodic}.
\begin{lemma}\label{lem: measure of starting path aperiodic}
Let $\gamma=(e_{i_{1}},\hdots ,e_{i_{l}}),\gamma'=(e_{i'_{1}},\hdots ,e_{i'_{l}}),$ be paths in $C^{max}$. The following hold.
\begin{align*}
&\sum \limits_{\sigma=(e_{k_{1}},\hdots e_{k_{l}})}\nu (\mathcal{Z}[\gamma,\sigma,\gamma'],0)=(1+o_{L}(1)) \alpha_{i_{1}}\beta_{i_{m}}\alpha_{j_{1}}\beta_{j_{n}}\slash \mu^{m+n-1}.\\
&\sum \limits_{\sigma=(e_{j_{1}},\hdots ,e_{j_{l}})}\sum \limits_{\sigma'=(e_{k_{1}},\hdots e_{k_{l}})}\nu (\mathcal{Z}[\gamma,\sigma,\sigma'],0)=\nu (\mathcal{Z}[\gamma,0])\\
& \sum \limits_{\sigma=(e_{j_{1}},\hdots ,e_{j_{l}})}\sum \limits_{\sigma'=(e_{k_{1}},\hdots e_{k_{l}})}\nu (\mathcal{Z}[\sigma,\sigma',\gamma],0)=\nu (\mathcal{Z}[\gamma,0])\\
& \sum \limits_{\sigma=(e_{j_{1}},\hdots ,e_{j_{l}})}\sum \limits_{\sigma'=(e_{k_{1}},\hdots e_{k_{l}})}\nu (\mathcal{Z}[\sigma,\gamma,\sigma'],0)=\nu(\mathcal{Z}[\gamma,0]).
\end{align*}
\end{lemma}
We then prove Theorem \ref{mainthm: property t in random quotients of hyperbolic groups aperiodic case} from Theorem \ref{thm: binomial (T) aperiodic case} in a manner similar to that employed to deduce Theorem \ref{mainthm: property t in random quotients of hyperbolic groups} from Theorem \ref{thm: (T) in binomial hyperbolic quotients}. The main observation is that for a choice $T_{l}\subseteq Ann_{l,2}(G)$ of size at least $\exp\{\mathfrak{h}ld\}$ for any $\epsilon>0$, we have $\vert T_{l}\cap S_{3\lfloor l\slash 3\rfloor} \vert \geq \exp\{\mathfrak{h}l(d-\epsilon)\}$ almost surely. Theorem \ref{thm: binomial (T) aperiodic case} then implies that the group $G\slash \langle \langle T_{l}\cap S_{3\lfloor l\slash 3\rfloor} \rangle \rangle$ has (T) almost surely.
\bibliographystyle{alpha}
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1806.10850
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\section{Introduction}
Recent evidence has highlighted that breast cancer is a biologically, clinically, and molecularly heterogeneous entity [1]. Uncontrolled proliferation is one of the key hallmarks of cancer [2]. Proliferation is often measured by using the Ki67 biomarker in breast tissue images. Ki67 is a nuclear protein expressed in all active phases (G1, S, G2 and M) of the cell cycle [3]. Cell proliferation is controlled by regulatory proteins and the complex tumor environment transcends through the checkpoints of the cell cycle. Ki67 has been validated as a biomarker for evaluating clinical benefits from endocrine treatment. Percentage of Ki67 positivity has been shown to be associated with patient prognosis [4]. However, manual estimation of Ki67 proliferation index in breast carcinoma can be laborious and prone to intra- and inter observer variations [5-7]. \\
\\
Therefore, there is an urgent need to build robust image analysis pipeline and offer standardized diagnostic solution for Ki67 immunohistochemistry (IHC) assay. Previous work in GeparTrio study [8] used Ki67 automated scoring and its validation was selectively based on few regions. Usage of only few regions for validation restricts the complete heterogeneity based challenges underlying in the Ki67 images. A software platform (QuPath) was developed to process TMA images of Ki67, ER, PR, HER2 and p53 expression [9]. This study uses classical texture feature and the machine learning pipeline that could be interactively trained and adapted to the dataset. However, diagnostic slides from clinical trials are highly heterogeneous, posing challenges such as consistent tissue identification across needle biopsy and whole-slide images, intra-class variability of cells and staining variability. Thus, it is critical to develop improved, automated and efficient ways to identify Ki67-positive and negative nuclei on diagnostic Ki67 images.\\
\\
Recent developments in deep learning, in the context of histology image processing for detection of mitosis in hematoxylin and eosin (H\&E) images based on convolution neural network (CNN) was the earliest implementation [10]. In this approach, the CNN network was trained to regress for probability of each pixel extracted from the patch. For cell classification, spatially constrained CNN has been proposed by adding a constrained layer to the network and regressing the final probability map to find the local maxima [11] \\
\\
In the context of nucleus detection in Ki67 images, a major challenge is the uneven chromatin distribution of hematoxylin stain that frequently occurs in clinical samples. The sharp contrast between heterogeneous weak hematoxylin stains and the Ki67-positive cells often results in failed cell detection. To address this, we design a new approach for localization, segmentation and classification of cells based on hypercolumn descriptors. These multi-scale descriptors not only capture the semantics but also aid localization of information across multiple patches [12]. Thus, our main contribution is the development and extraction of hypercolumn descriptors for a given pixel, which integrate activations from multiple convolutional layers of our SDCS network to yield accurate detection of all Ki67-positive and negative nuclei simultaneously in a large image. Specifically, we exploit the semantic and localization features retained in hypercolumns to address cell segmentation and classification challenges and demonstrated its performance in two clinical cohorts.
\section{Methodology}
\subsection{Overview}
Given an input immunohistological image {\textit{ i}}, the problem is to find a set of detections and then classify the detections into Ki67-positive, Ki67 -negative, stroma and lymphocyte. The problem is solved by training the SDCS detection network and propagating it to the constrained network using the same ground truth annotation coordinates used in the training stages of detection and classification.
\subsection{Manual annotation of training and testing samples}
Twenty whole slide images were selected as training images for the study. An expert pathologist annotated the Ki67-positive cancer cells, Ki67-negative cancer cells (henceforth referred to as Ki67-positive and Ki67-negative cells), stromal cells and lymphocytes. We included samples across the spectrum of Ki67 expression positivity in our training set. Figure. 1 shows two exemplary images from a training sample that has low Ki67 expression. Quality control was performed to exclude the out of focus samples from the training dataset. We used two sets of testing data acquired at different time points to test the performance of cell classification. Breakdown of the number of cells in training validation and test set samples is tabulated in Table 1.
\begin{figure}
\includegraphics[width=1.0\textwidth]{fig1new_v2.png}
\hrulefill
\caption{{a) }An exemplary Ki67 biopsy image, with the annotations for the positive Ki67 nuclei (red), negative Ki67 nuclei (green) marked by the expert pathologist for training our SDCS network.{b) }Exemplary sparsely scattered positive nuclei showing the proliferation and dense negative nuclei pattern on the biopsy image. {c) } Tile images (2000x2000) are extracted using openslide library and the annotations are saved in the high resolution 20x tiles. The coordinates are extracted from the base magnification to extract the training patches.}
\end{figure}
\begin{table}[ht]
\centering
\caption{Breakdown of the number of annotated cells in training, validation and two testing sets}
\label{tab:sample-table1}
\begin {tabular} { l | c | c | c | c }
\hline
Cell types &Training &Validation &Testing set1 &Testing set2 \\ \hline
Ki67 Positive &1626 &201 &254 &386 \\ \hline
Ki67 Negative &733 &232 &416 &744 \\ \hline
Stroma &933 &267 &109 &213 \\ \hline
Lymphocyte &928 &109 &256 &362 \\ \hline
Total &4220 &809 &1035 &1705 \\ \hline
\end{tabular}
\end{table}
\subsection{Hypercolumns}
A typical architecture of CNNs consists of convolution layers and max pooling layers followed by a fully connected layer. CNNs use convolution and pooling operations and the pooled image downsamples the input images by the pooling parameter. \\
\\
During the training phase, we feed the input image to a VGG16 network [13] and extract the sparse hypercolumn descriptors from selected convolutional layers. The hypercolumns are formed by concatenating a series of activations of the convolutional layers. In our implementation, we chose the activations from the final convolutional layer of$c_1$ (p), $c_2 $(p) and $c_5$ (p) from the customized VGG16 like architecture (i.e. conv1; conv2; and conv5) to form the hypercolumn. The fully connected layers in the original VGG16 network are implemented as 1x1 convolution layers. As each convolutional block is preceded by a max-pooling operation that downsamples the activations, we perform bilinear upsampling using an appropriate scaling factor such that the resulting resolution for the activations of each layer forming the hypercolumn is 64x64. Then, we sparsely sample random points from the dense hypercolumns to form rich descriptors for a given pixel in the input image. The sparse hypercolumn descriptors are then fed to a non-linear classifier, in our case, a 2-layered multi-layer perceptron network (again, implemented as 1x1 convolutions) with 256 neurons respectively. We use a sparsely-sampled output mask, whose pixels correspond to the location of the sparse hypercolumns to learn pixel-wise class predictions as shown in Figure 2. The network was implemented using Keras package [12].\\
\begin{figure}
\centering
\includegraphics[width=1.0\textwidth]{fig2_v2.png}
\hrulefill
\caption{Schematic of our simultaneous detection and cell segmentation (SDCS) network architecture. The patches extracted from the cell annotation locations are used to extract the hypercolumn descriptors. The hypercolumn descriptors are then fed to the dense multi-layer perceptron network to segment them into Ki67 positive and Ki67 negative cells.}
\end{figure}
Hypercolumn descriptor can be expressed as in (1)
\begin{equation}
h(p)= [c_1(p), c_2(p) , c_5(p)]
\end{equation}
where h(p) is the hyper column feature for the pixel, $c_i$(p) where {\textit{ i}} = 1,2,5 represents the feature vector corresponding to the ith layer. Given an input, our SDCS network generates pixel-level predictions by operating over the hypercolumns. The final prediction of the pixel p is given by equation (2). We estimate the prediction probability $f_{\theta,p}$(X) for every pixel p as a function of activation given a set of hypercolumn descriptors and its parameter set $\theta$:
\begin{equation}
f_{\theta,p}(X)=g(h_p(X))
\end{equation}
\subsection{Spatially Constrained Convolutional Neural Network (SCCNN)}
We used the same training patches of SDCS network to further classify the detected nucleus using SCCNN framework. We augmented the training regions by flip and mirror operations on individual patches. The per pixel prediction centers of individual cells forms the input to the spatially constrained layer. This framework uses the sliding window strategy with overlapping windows. The predicted probability of being center of the nucleus is generated for each patch size of 51x51 from the constrained network. Subsequently, the results are aggregated to form the probability map representing the local maxima at the nuclei centers. An empirical threshold is determined to our dataset to remove any over prediction. All local maxima whose probability is less than the threshold are not considered in our final output detection. The nucleus centers are classified into respective four classes using the integrated SDCS detection and the SCNN framework (Figure. 3).
\begin{figure}
\centering
\includegraphics[width=1.0\textwidth]{fig3_v3.png}
\hrulefill
\caption{a) Schematic of our simultaneous detection and cell segmentation (SDCS) network hypercolumn generation. b) The exemplary hypercolumn feature representation from the convolution ($c_1$(p)) for the original image to extract the hypercolumn descriptors. c) The hypercolumn descriptors is then fed to the dense multi- layer perceptron network to segment them into Ki67 positive and hematoxylin positive cells. d) Output of the integrated SDCS framework classifying the individual nuclei centers into Ki67 positive, Ki67 negative, stromal cell and lymphocyte respectively. e) Regional hypercolumn descriptor feature maps generated from $c_1$(p), $c_2$(p),$c_5$(p).}
\end{figure}
\subsection{SVM classification}
Localization of individual cells involves morphological image processing and it includes noise removal using median filter, morphological gradient calculation, otsu threshold and distance transformation followed by watershed segmentation to identify the local maxima. The segmented cell center is used to estimate the features. 101 hand crafted features were estimated from each nucleus, comprising 26 haralick feature [14] extracted for the training images in the gray channel and hematoxylin channel, 49 translational and rotational invariant zernike moment features [15], 11 nuclei based features (mean distance of the nuclei center from the perimeter, standard deviation of the distance of nuclei center to the perimeter, nuclear area, major axis, minor axis, eccentricity, orientation, first Hu moment, second Hu moment, nuclei shape roundedness), 15 intensity based features from individual color channel of the RGB image (mean, standard deviation, variance, skewness, kurtosis). The features were normalized and then used to normalize the test dataset. A support vector machine (SVM) with a radial basis function (RBF) was trained to classify the cells into 4 classes. Results from the classification were further used to evaluate the performance of the model on two independent test sets (Table 2).
\section{Experimental Results }
\label{gen_inst}
\subsection{Quantitative Comparison}
We compare cell segmentation and classification results between the integrated SDCS network and machine-learning based SVM method. As demonstrated in our experimental results (Table 2), average classification accuracy increased from 82.05\% (SVM) to 99.06\% (deep SDCS) on the test set 1 and from 76.5\% to 89.59 \% on test set 2. Surprisingly, we also noted that a network configuration with the conventional fine tuning of VGG16 with the training regions but without extraction of hypercolumns resulted in a lower accuracy, compared to our integrated SDCS network that uses activation from few convolution layers to construct the hypercolumn based feature maps extracted from $c_1$(p), $c_2$(p),$c_5$(p)] (Table 2).
\begin{table}[ht]
\centering
\caption{Comparison of classification accuracy of machine learning approach based on SVM classification and deep learning approach based on integrated SDCS network classification on the test dataset comprising images from two sets.}
\label{tab:sample-table2}
\begin {tabular} { c | c | c }
\hline
Approach &Average accuracy set1 &Average accuracy set2 \\ \hline
Haralick Texture &82.05\% &76.5\% \\ \hline
SDCS Network without hypercolumns &85.08\% &82.3\% \\ \hline
Integrated SDCS Network &99.06\% &89.59\% \\ \hline
\end{tabular}
\end{table}
Further breakdown of evaluation metrics for the integrated SDCS network on two test sets indicates better F1-score, precision and recall in test set 1 (Table 3). To evaluate the integrated SDCS framework for the classification of Ki67 images, we use following two test sets. POETIC (1035 cells in 2 slide images from test set 1) and pilot POETIC (1705 in 5 slide images from test set 2).
\begin{table}[ht]
\centering
\caption{Breakdown of the evaluation metrics on two test sets based on integrated SDCS network classification.}
\label{tab:sample-table3}
\begin {tabular} { c | c | c }
\hline
SDCS Network &Test set 1 &Test set 2 \\ \hline
Average accuracy &99.06\% &89.59\% \\ \hline
Average precision &99.06\% &82.49\% \\ \hline
Average Recall &99.06\% &75.70\% \\ \hline
Average F1-score &99.5\% &77.17\% \\ \hline
Average specificity &99.7\% &90.20\% \\ \hline
Average sensitivity &99.5\% &75.70 \% \\ \hline
\end{tabular}
\end{table}
\subsection{Visual Inspection}
We show that the classification results (Figure. 4.) based on our integrated SDCS method and the manual method from a tile of individual slide used in our calculation of evaluation metrics on test dataset in Figure.4. For our validation experiment, we used our integrated model to predict class label for known cell locations that had manual annotations. The nuclei in the test dataset set1 were weakly stained and were frequently hollow with non-uniform staining. It can be observed that our integrated SDCS based classification results were closer to the manual annotations than classical SVM based method (more obvious in the zoomed views of the region in Figure. 4). The model has learnt the complex representations of the Ki67 -negative nuclei and positive nuclei.
\begin{figure}
\centering
\includegraphics[width=1.0\textwidth]{fig4n5_v5.png}
\hrulefill
\caption{Visual comparison of classification based on weakly stained region of biopsy indicating the classification based on integrated SDCS network and SVM classification. a) original image tile (2000x2000) b) Manual annotated tile c) SDCS detection and segmentation d) Integrated SDCS classification e) SVM classification. Black arrow indicates the misclassification error between the two approaches. Yellow arrow indicates the under segmentation error due to the morphological processing. Integrated SDCS detection and classification closer to manual annotation.}
\end{figure}
\section{Discussion}
In this work we propose an integrated deep SDCS framework for simultaneous detection and classification of multiple cells in Ki67-stained IHC images. A major challenge in analyzing these images is the heterogeneous staining, which often results in under detection as we demonstrated with classical machine learning methods. To detect weakly stained cells without resulting in over-detection, hypercolumn descriptors were used to integrate activations from multiple convolutional layers, capturing image semantics across granularities. Another advantage for this network is that only annotations for cell centroids are needed. Therefore, compared with existing networks such as U-net [16, 17], our method removes the need for laborious cell area annotations, which can dramatically speed up the annotation collection and training processes. Moreover, our recognition framework integrates problem-specific design such as a tissue region mask to remove coverslip artefact, which is not part of other standard networks. \\
\\
The architecture based on hypercolumns was empirically chosen based on our evaluation of the training data. The integrated SDCS approach localizes features from multi-scale RGB patches, and extract them from the activation maps to construct the hypercolumn descriptors. The importance of such hypercolumn descriptors was reflected in our experimental results comparing SDCS network with and without hypercolumn. Detection was followed by non-linear classification by the SCCNN method. Overall, we found that this pipeline was able to overcome the challenge in detecting weakly stained Ki67-negative nuclei. No empirical threshold was needed in our segmentation and detection framework. \\
\\
In comparison, the texture-based approach uses features extracted from grey and hematoxylin channels. Visual inspection of the results from the svm classification approach indicated that hand-crafted feature-based model output detecting fewer Ki67 –negative cells compared to the manual annotation. Firstly, this is due to weakly stained samples, where the segmentation needs further refinement and classification output cannot overcome this under segmentation. Secondly, we observed misclassifications of Ki67 -positive cell as Ki67 -negative cell in svm classification than the SDCS approach. (Figure. 4). Thus, the hand-crafted feature extracted from single resolution lacks information to discern weakly stained nuclei as opposed to the multi-scale feature extraction in the integrated SDCS method. Additionally, empirical thresholds for otsu thresholding and distance transform were necessary for obtaining the cell segmentation. Bankhead, P. et al. [9] used QuPath, a texture based method based on classical machine learning, to analyse tissue micro arrays. This method is yet to be analytically validated for biopsy and whole slide images. We were unable to directly compare with QuPath, however, due to the difficulty in parsing our training data.\\
\\
Direct output of the proposed framework is the single-cell classification of stroma, lymphocytes and Ki67-positive and negative cancer cells. Limitations include the relatively small training and test sets, which we plan to expand on by taking more images from the POETIC study consisting of a total of 9,000 Ki67 slides. In addition, direct comparison of our method with the other state-of-the-art networks for classification was not possible, because often the entire image patch is classified. Because our training data are limited to cell centroids instead of cell areas, we cannot directly evaluate cell segmentation accuracy. \\
\\
In summary, visual assessment is considered as the gold standard for Ki67 scoring. Inter- and intra- observer bias may, however, affect the results. Automated estimation of Ki67 score using our integrated SDCS method has the potential to provide consistent and reproducible scores, following training by expert pathologists. Our method also classifies lymphocytes and stromal cells, which could be further used in clinical studies to better understand the complex tumor microenviroment.
\section{Conclusion }
We have demonstrated the performance of our new pipeline for simultaneous cell detection and classification in samples from a large breast cancer clinical trial. Specifically, its use of hypercolumn descriptors aids cell segmentation and further classification in heterogeneously stained Ki67 immunohistochemistry images. Compared with a classical machine learning method, this approach detects weakly stained nucleus with higher precision. This integrated framework will allow us to further validate and test automated Ki67 scores in large breast cancer patient cohorts.
\section{Acknowledgments }
This work is supported by Breast Cancer Now (2015NovPR638). Also the authors would like to thank Prof Nasir Rajpoot at Tissue Image Analytics Lab, University of Warwick for their help in implementation of SCCNN [11].
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\end{document}
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1806.11191
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\section{Introduction}
Generating multi-view images from a single-view input is an interesting problem with broad applications in vision, graphics, and robotics. Yet, it is a challenging problem since 1) computers need to ``imagine'' what a given object would look like after a 3D rotation is applied; and 2) the multi-view generations should preserve the same ``identity''.
Generally speaking, previous solutions to this problem include model-driven synthesis \cite{blanz1999morphable}, data-driven generation \cite{zhu2014multi,yan2016perspective}, and a combination of the both \cite{zhuxiangyu2016,rezende2016unsupervised}. Recently, generative adversarial networks (GANs) \cite{goodfellow2014generative} have shown impressive results in multi-view generation \cite{tran2017disentangled,zhao2017multi}.
\begin{figure}[t]
\centering
\includegraphics[width=0.96\linewidth]{fig_intro}
\caption{Top: The limitation of existing GAN-based methods. They can generate good results if the input is mapped into the learned subspace (Row 1). However, ``unseen'' data may be mapped out of the subspace, leading to poor results (Row 2). Bottom: Our results. By learning complete representations, the proposed CR-GAN can generate realistic, identity-preserved images from a single-view input.}
\label{fig:fig_intro}
\vspace{-4mm}
\end{figure}
These GAN-based methods usually have a single-pathway design: an encoder-decoder network is followed by a discriminator network. The encoder ($E$) maps input images into a latent space ($Z$), where the embeddings are first manipulated and then fed into the decoder ($G$) to generate novel views.
However, our experiments indicate that this single-pathway design may have a severe issue: they can only learn ``incomplete'' representations, yielding limited generalization ability on ``unseen'' or unconstrained data. Take Fig. \ref{fig:fig_intro} as an example. During the training, the outputs of $E$ constitute only a subspace of $Z$ since we usually have a limited number of training samples. This would make $G$ only ``see'' part of $Z$. During the testing, it is highly possible that $E$ would map an ``unseen'' input outside the subspace. As a result, $G$ may produce poor results due to the unexpected embedding.
To address this issue, we propose CR-GAN to learn \textit{Complete Representations} for multi-view generation. The main idea is, in addition to the reconstruction path, we introduce another generation path to create view-specific images from embeddings that are randomly sampled from $Z$. Please refer to Fig. \ref{fig:our_model} for an illustration. The two paths share the same $G$. In other words, $G$ learned in the generation path will guide the learning of both $E$ and $D$ in the reconstruction path, and vice versa. $E$ is forced to be an inverse of $G$, yielding complete representations that would span the entire $Z$ space. More importantly, the two-pathway learning can easily utilize both labeled and unlabeled data for self-supervised learning, which can largely enrich the $Z$ space for natural generations.
To sum up, we have following contributions:
\begin{itemize}
\item To the best of our knowledge, we are the first to investigate ``complete representations'' of GAN models;
\item We propose CR-GAN that can learn ``complete'' representations, using a two-pathway learning scheme;
\item CR-GAN can leverage unlabeled data for self-supervised learning, yielding improved generation quality;
\item CR-GAN can generate high-quality multi-view images from even ``unseen'' dataset in wild conditions.
\end{itemize}
\section{Related Work}
\textbf{Generative Adversarial Networks (GANs)}.
Goodfellow~\emph{et al}\onedot~\cite{goodfellow2014generative} introduced GAN to estimate target distribution via an adversarial process.
Gulrajani~\emph{et al}\onedot~\cite{gulrajani2017improved} presented a more stable approach to enforce \textit{Lipschitz Constraint} on Wasserstein GAN~\cite{arjovsky2017wasserstein}.
AC-GAN~\emph{et al}\onedot~\cite{odena2016conditional} extended the discriminator by containing an auxiliary decoder network to estimate class labels for the training data. BiGANs~\cite{donahue2016adversarial,dumoulin2016adversarially} try to learn an inverse mapping to project data back into the latent space. Our method can also find an inverse mapping, make a balanced minimax game when training data is limited.
\textbf{Multi-view Synthesis}.
Hinton~\emph{et al}\onedot~\cite{hinton2011transforming} introduced transforming auto-encoder to generate images with view variance. Yan~\emph{et al}\onedot~\cite{yan2016perspective} proposed Perspective Transformer Nets to find the projection transformation. Zhou~\emph{et al}\onedot~\cite{zhou2016view} propose to synthesize views by appearance flow. Very recently, GAN-based methods usually follow a single-pathway design: an encoder-decoder network~\cite{peng2016recurrent} followed by a discriminator network. For example, to normalize the viewpoint, {\it e.g.} face frontalization, they either combine encoder-decoder with 3DMM~\cite{blanz1999morphable} parameters~\cite{yin2017towards}, or use duplicates to predict global and local details~\cite{huang2017beyond}. DR-GAN~\cite{tran2017disentangled} follows the single-pathway framework to learn identity features that are invariant to viewpoints. However, it may learn ``incomplete'' representations due to the single-pathway framework. In contrast, CR-GAN can learn complete representations using a two-pathway network, which guarantees high-quality generations even for ``unseen'' inputs.
\textbf{Pose-Invariant Representation Learning}.
For representation learning~\cite{li2016multi,fan2016learning}, early works may use \textit{Canonical Correlation Analysis} to analyze the commonality among different pose subspaces~\cite{hardoon2004canonical,peng2015circle}. Recently, deep learning based methods use synthesized images to disentangle pose and identity factors by cross-reconstruction ~\cite{zhu2014multi,peng2017recons}, or transfer information from pose variant inputs to a frontalized appearance~\cite{zhu2013deep}. However, they usually use only labeled data, leading to a limited performance. We proposed a two-pathway network to leverage both labeled and unlabeled data for self-supervised learning, which can generate realistic images in challenging conditions.
\section{Proposed Method} \label{sec:method}
\subsection{A Toy Example of Incomplete Representations} \label{sec:toy}
A single-pathway network, {\it i.e.} an encoder-decoder network followed by a discriminator network, may have the issue of learning ``incomplete'' representations. As illustrated in Fig.~\ref{fig:our_model} left, the encoder $E$ and decoder $G$ can ``touch'' only a subspace of $Z$ since we usually have a limited number of training data. This would lead to a severe issue in testing when using ``unseen'' data as the input. It is highly possible that $E$ may map the novel input out of the subspace, which inevitably leads to poor generations since $G$ has never ``seen'' the embedding.
\begin{figure}[t]
\centering
\includegraphics[width=0.9\linewidth]{our_model}
\caption{Left: Previous methods use a single path to learn the latent representation, but it is incomplete in the whole space. Right: We propose a two-pathway network combined with self-supervised learning, which can learn complete representations. }
\label{fig:our_model}
\vspace{-4mm}
\end{figure}
A toy example is used to explain this point. We use Multi-PIE~\cite{Gross2010MultiPIE} to train a single-pathway network. As shown in the top of Fig.~\ref{fig:fig_intro}, the network can generate realistic results on Multi-PIE (the first row), as long as the input image is mapped into the learned subspace. However, when testing ``unseen'' images from IJB-A~\cite{klare2015pushing}, the network may produce unsatisfactory results (the second row). In this case, the new image is mapped out of the learned subspace.
This fact motivates us to train $E$ and $G$ that can ``cover'' the whole $\mathit{Z}$ space, so we can learn complete representations. We achieve this goal by introducing a separate generation path, where the generator focuses on mapping the entire $\mathit{Z}$ space to high-quality images. Fig.~\ref{fig:our_model} illustrates the comparison between the single-pathway and two-pathway networks. Please refer to Fig.~\ref{fig:fig_framework} (d) for an overview of our approach.
\vspace{-1mm}
\subsection{Generation Path} \label{sec:generation-path}
The generation path trains generator $G$ and discriminator $D$. Here the encoder $E$ is not involved since $G$ tries to generate from random noise. Given a view label $v$ and random noise $\mathbf{z}$, $G$ aims to produce a realistic image $G(v,\mathbf{z})$ under view $v$. $D$ is trying to distinguish real data from $G$'s output, which minimizes:
\begin{equation}
\label{WACGAN-gp_D}
\small
\begin{gathered}
\underset{\mathbf{z}\sim\mathbb{P}_{\mathbf{z}}}{\mathbb{E}}[D_s(G(v, \mathbf{z}))] - \underset{\mathbf{x}\sim\mathbb{P}_{\mathbf{x}}}{\mathbb{E}}[D_s(\mathbf{x})] +
\\
\lambda_1 \underset{\hat{\mathbf{x}}\sim\mathbb{P}_{\hat{{\mathbf{x}}}}}{\mathbb{E}}[(\left\| \bigtriangledown_{\hat{\mathbf{x}}}D(\hat{\mathbf{x}}) \right \|_2 - 1)^2] - \lambda_2\underset{\mathbf{x}\sim\mathbb{P}_{\mathbf{x}}}{\mathbb{E}}[P(D_v(\mathbf{x})=v)],
\end{gathered}
\end{equation}
where $\mathbb{P}_{\mathbf{x}}$ is the data distribution and $\mathbb{P}_{\mathbf{z}}$ is the noise uniform distribution, $\mathbb{P}_{\hat{\mathbf{x}}}$ is an interpolation between pairs of points sampled from data distribution and the generator distribution~\cite{gulrajani2017improved}. $G$ tries to fool $D$, it maximizes:
\begin{equation}
\label{WACGAN-gp_G}
\small
\underset{\mathbf{z}\sim\mathbb{P}_{\mathbf{z}}}{\mathbb{E}}[D_s(G(v, \mathbf{z}))] + \lambda_3\underset{\mathbf{z}\sim\mathbb{P}_{\mathbf{z}}}{\mathbb{E}}[P(D_v({G(v, \mathbf{z})})=v)],
\end{equation}
where $(D_v(\cdot), D_s(\cdot)) = D(\cdot)$ denotes pairwise outputs of the discriminator. $D_v(\cdot)$ estimates the probability of being a specific view, $D_s(\cdot)$ describes the image quality, \emph{i.e}\onedot} \def\Ie{\emph{I.e}\onedot, how real the image is. Note that in Eq.~\ref{WACGAN-gp_D}, $D$ learns how to estimate the correct view of a real image~\cite{odena2016conditional}, while $G$ tries to produce an image with that view in order to get a high score from $D$ in Eq.~\ref{WACGAN-gp_G}.
\begin{algorithm}[t]
\KwIn{Sets of view labeled images $\mathit{X}$, max number of steps $T$, and batch size $m$.}
\KwOut{Trained network $E$, $G$ and $D$.}
\For{$t = 1$ \KwTo $T$}{
\For{$i = 1$ \KwTo $m$}{
1. Sample $\mathbf{z}\sim\mathit{P}_{\mathbf{z}}$ and $\mathbf{x}_i \sim \mathit{P}_{\mathbf{x}}$ with $v_i$\;
2. $\bar{\mathbf{x}} \gets G(v_i, \mathbf{z})$\;
3. Update $D$ by Eq.~\ref{WACGAN-gp_D}, and $G$ by Eq.~\ref{WACGAN-gp_G}\;
4. Sample $\mathbf{x}_j \sim \mathit{P}_{\mathbf{x}}$ with $v_j$ (where $\mathbf{x}_j$ and $\mathbf{x}_i$ share the same identity)\;
5. $(\bar{\mathbf{v}}, \bar{\mathbf{z}}) \gets E(\mathbf{x}_i)$\;
6. $\tilde{\mathbf{x}}_j \gets G(v_j, \bar{\mathbf{z}})$\;
7. Update $D$ by Eq.~\ref{Reconstruction_D}, and $E$ by Eq.~\ref{Reconstruction_E}\;
}
}
\caption{Supervised training with two paths}\label{xxx-GAN}
\end{algorithm}
\vspace{-1mm}
\subsection{Reconstruction Path}\label{sec:reconstruction-path}
The reconstruction path trains $E$ and $D$ but keeping $G$ fixed. $E$ tries to reconstruct training samples, this would guarantee that $E$ will be learned as an inverse of $G$, yielding complete representations in the latent embedding space.
The output of $E$ should be identity-preserved so the multi-view images will present the same identity. We propose a cross reconstruction task to make $E$ disentangle the viewpoint from the identity. More specifically, we sample a real image pair $(\mathbf{x}_i, \mathbf{x}_j)$ that share the same identity but different views $v_i$ and $v_j$. The goal is to reconstruct $x_j$ from $x_i$. To achieve this, $E$ takes $\mathbf{x}_i$ as input and outputs an identity-preserved representation $\bar{\mathbf{z}}$ together with the view estimation $\bar{\mathbf{v}}$: $ (\bar{\mathbf{v}}, \bar{\mathbf{z}}) = (E_v(\mathbf{x}_i), E_z(\mathbf{x}_i)) = E(\mathbf{x}_i)$. Note that $\bar{\mathbf{v}}$ is learned for further self-supervised training as shown in Sec.~\ref{sec:unlabeled-dataset}.
$G$ takes $\bar{\mathbf{z}}$ and view $v_j$ as the input. As $\bar{\mathbf{z}}$ is expected to carry the identity information of this person, with view $v_j$'s help, $G$ should produce $\tilde{\mathbf{x}}_j$, the reconstruction of $\mathbf{x}_j$. $D$ is trained to distinguish the fake image $\tilde{\mathbf{x}}_j$ from the real one $\mathbf{x}_i$. Thus $D$ minimizes:
\begin{equation}
\small
\begin{gathered}
\label{Reconstruction_D}
\underset{\mathbf{x}_i, \mathbf{x}_j\sim\mathbb{P}_{\mathbf{x}}}{\mathbb{E}}[D_s(\tilde{\mathbf{x}}_j) - D_s(\mathbf{x}_i)] +
\\
\lambda_1 \underset{\hat{\mathbf{x}}\sim\mathbb{P}_{\hat{\mathbf{x}}}}{\mathbb{E}}[(\left\| \bigtriangledown_{\hat{\mathbf{x}}}D(\hat{\mathbf{x}}) \right \|_2 - 1)^2] - \lambda_2\underset{\mathbf{x}_i\sim\mathbb{P}_x}{\mathbb{E}}[P(D_v(\mathbf{x}_i)=v_i)],
\end{gathered}
\end{equation}
where $\tilde{\mathbf{x}}_j = G(v_j, E_z(\mathbf{x}_i))$. $E$ helps $G$ to generate high quality image with view $v_j$, so $E$ maximizes:
\begin{equation}
\small
\begin{gathered}
\label{Reconstruction_E}
\underset{\mathbf{x}_i, \mathbf{x}_j\sim\mathbb{P}_{\mathbf{x}}}{\mathbb{E}}[D_s(\tilde{\mathbf{x}}_j) + \lambda_3 P(D_v(\tilde{\mathbf{x}}_j)=v_j) -
\\
\lambda_4 L_1(\tilde{\mathbf{x}}_j, \mathbf{x}_j) - \lambda_5 L_v(E_v(\mathbf{x}_i), v_i)],
\end{gathered}
\end{equation}
where $L_1$ loss is utilized to enforce that $\tilde{x}_j$ is the reconstruction of $x_j$. $L_v$ is the cross-entropy loss of estimated and ground truth views, to let $E$ be a good view estimator.
The two-pathway network learns complete representations: First, in the generation path, $G$ learns how to produce real images from \textit{any} inputs in the latent space. Then, in the reconstruction path, $G$ retains the generative ability since it keeps unchanged. The alternative training details of the two pathways are summarized in Algorithm~\ref{xxx-GAN}.
\begin{algorithm}[t]
\KwIn{Sets of view labeled and unlabeled images $\mathit{X}$, max number of steps $T$, and batch size $m$.}
\KwOut{Trained network $E$, $G$ and $D$.}
Pre-train $E$, $G$ and $D$ according to Algorithm~\ref{xxx-GAN}\;
\For{$t = 1$ \KwTo $T$}{
\For{$i = 1$ \KwTo $m$}{
Sample $\mathbf{z}\sim\mathit{P}_{\mathbf{z}}$ and $\mathbf{x} \sim \mathit{P}_{\mathbf{x}}$\;
\eIf{$\mathbf{x}$ is labeled}{
1. $\mathbf{x}_i \gets \mathbf{x}$\;
2. Get the label $v_i$ of $\mathbf{x}_i$\;
3. Repeat the step 2 to 7 in Algorithm~\ref{xxx-GAN}\;
}{
4. $(\bar{\mathbf{v}}, \bar{\mathbf{z}}) \gets E(\mathbf{x})$\;
5. Compute $\hat{v}$ (the estimation of $\bar{\mathbf{v}}$)\;
6. Update $D$ by Eq.~\ref{self_supervise_recon_D} and $E$ by Eq.~\ref{self_supervise_recon_E}\;
7. Update $D$ by Eq.~\ref{self_supervise_gen_D} and $G$ by Eq.~\ref{self-supervise_gen_G}\;
}
}
}
\caption{Self-supervised training with two paths}\label{xxx-GAN_unlabeled}
\end{algorithm}
\begin{figure*}[t]
\centering
\includegraphics[width=0.96\linewidth]{fig_framework}
\caption{Comparison of BiGAN~\protect\cite{donahue2016adversarial}, DR-GAN~\protect\cite{tran2017disentangled}, TP-GAN~\protect\cite{huang2017beyond} and our method.}
\label{fig:fig_framework}
\vspace{-4mm}
\end{figure*}
\vspace{-1mm}
\subsection{Self-supervised Learning}\label{sec:unlabeled-dataset}
Labeled datasets are usually limited and biased. For example, Multi-PIE~\cite{Gross2010MultiPIE} is collected in a constrained setting, while large-pose images in 300wLP~\cite{zhuxiangyu2016} are distorted. As a result, $G$ would generate low-quality images since it has only ``seen'' poor and limited examples.
To solve this issue, we further improve the proposed CR-GAN with self-supervised learning. The key idea is to use a pre-trained model to estimate viewpoints for unlabeled images. Accordingly, we modify the supervised training algorithm into two phases. In the first stage, we pre-train the network on labeled data to let $E$ be a good view estimator. In the second stage, both labeled and unlabeled data are utilized to boost $G$. When an unlabeled image $\mathbf{x}$ is fed to the network, a view estimation $\bar{\mathbf{v}}$ is obtained by $E_v(\cdot)$. Denote $\hat{v}$ to be the closest one-hot vector of $\bar{\mathbf{v}}$, in the reconstruction path, we let $E$ minimize $L_v(\bar{\mathbf{v}}, \hat{v})$ and then reconstruct $\mathbf{x}$ to itself. Similar to Eq.~\ref{Reconstruction_D}, $D$ minimizes:
\begin{equation}
\small
\begin{gathered}
\label{self_supervise_recon_D}
\underset{\mathbf{x}\sim\mathbb{P}_{\mathbf{x}}}{\mathbb{E}}[D_s(G(\hat{v}, E_z(\mathbf{x}))) - D_s(\mathbf{x})] +
\\
\lambda_1 \underset{\hat{\mathbf{x}}\sim\mathbb{P}_{\hat{\mathbf{x}}}}{\mathbb{E}}[(\left\| \bigtriangledown_{\hat{\mathbf{x}}}D(\hat{\mathbf{x}}) \right \|_2 - 1)^2] - \lambda_2\underset{\mathbf{x}\sim\mathbb{P}_{\mathbf{x}}}{\mathbb{E}}[P(D_v(\mathbf{x})=\hat{v})],
\end{gathered}
\end{equation}
similar to Eq.~\ref{Reconstruction_E}, $E$ maximizes:
\begin{equation}
\small
\begin{gathered}
\label{self_supervise_recon_E}
\underset{\mathbf{x}\sim\mathbb{P}_{\mathbf{x}}}{\mathbb{E}}[D_s(G(\hat{v}, E_z(\mathbf{x}))) + \lambda_3 P(D_v({G(\hat{v}, E_z(\mathbf{x}))})=\hat{v}) -
\\
\lambda_4 L_1(G(\hat{v}, E_z(\mathbf{x})), \mathbf{x}) - \lambda_5 L_v(E_v(\mathbf{x}), \hat{v})].
\end{gathered}
\end{equation}
In the generation path, we let $\hat{v}$ be the ground truth of $\mathbf{x}$, and generate an image in view $\hat{v}$. So similar to Eq.~\ref{WACGAN-gp_D}, $D$ minimizes:
\begin{equation}
\small
\begin{gathered}
\label{self_supervise_gen_D}
\underset{\mathbf{z}\sim\mathbb{P}_{\mathbf{z}}}{\mathbb{E}}[D_s(G(\hat{v}, \mathbf{z}))] - \underset{\mathbf{x}\sim\mathbb{P}_{\mathbf{x}}}{\mathbb{E}}[D_s(\mathbf{x})] +
\\
\lambda_1 \underset{\hat{\mathbf{x}}\sim\mathbb{P}_{\hat{\mathbf{x}}}}{\mathbb{E}}[(\left\| \bigtriangledown_{\hat{\mathbf{x}}}D(\hat{\mathbf{x}}) \right \|_2 - 1)^2] - \lambda_2\underset{\mathbf{x}\sim\mathbb{P}_{\mathbf{x}}}{\mathbb{E}}[P(D_v(\mathbf{x})=\hat{v})],
\end{gathered}
\end{equation}
similar to Eq.~\ref{WACGAN-gp_G}, $G$ maximizes:
\begin{equation}
\small
\label{self-supervise_gen_G}
\underset{\mathbf{z}\sim\mathbb{P}_{\mathbf{z}}}{\mathbb{E}}[D_s(G(\hat{v}, \mathbf{z}))] + \lambda_3\underset{\mathbf{z}\sim\mathbb{P}_{\mathbf{z}}}{\mathbb{E}}[P(D_v({G(\mathbf{z})})=\hat{v})].
\end{equation}
Once we get the pre-trained model, the encoder $E$ predicts the probabilities of the input image belonging to different views. We choose the view with the highest probability as the estimation. Our strategy is similar to RANSAC algorithm~\cite{fischler1987random}, where we treat the estimations with higher confidence as ``inliers'' and use them to make view estimation more accurate. We summarize the self-supervised training in Algorithm~\ref{xxx-GAN_unlabeled}.
Compared with the single-pathway solution, the proposed two-pathway network boosts the self-supervised learning in two aspects: 1) it provides a better pre-trained model for viewpoint estimation as a byproduct; and 2) it guarantees that we can take full advantage of unlabeled data in training since CR-GAN learns complete representations.
\vspace{-1mm}
\subsection{Discussion}\label{sec:comparisons}
To highlight the novelty of our method, we compare CR-GAN with the following three GANs. In Fig.~\ref{fig:fig_framework}, we show their network structures as well as ours for visual comparison.
\textbf{BiGAN}~\cite{donahue2016adversarial,dumoulin2016adversarially} jointly learns a generation network $G$ and an inference network $E$. The authors proved that $E$ is an inverse network of $G$.
However, in practice, BiGAN produces poor reconstructions due to finite data and limited network capacity. Instead, CR-GAN uses explicit reconstruction loss to solve this issue.
\textbf{DR-GAN}~\cite{tran2017disentangled} also tries to learn an identity preserved representation to synthesize multi-view images. But we have two distinct differences. First, the output of its encoder, also acts as the decoder's input, completely depends on the training dataset. Therefore, it can not deal with new data. Instead, we use the generation path to make sure that the learning of our $G$ is ``complete''. Second, we don't let $D$ estimate the identity for training data, because we employ unlabeled dataset in self-supervised learning which has no identity information. The involvement of unlabeled dataset also makes our model more robust for ``unseen'' data.
\textbf{TP-GAN}~\cite{huang2017beyond} uses two pathway GANs for frontal view synthesis. Their framework is different from ours: First, they use two distinct encoder-decoder networks, while CR-GAN shares all modules in the two pathways. Besides, they use two pathways to capture global features and local details, while we focus on learning complete representations in multi-view generation.
\section{Experiments}\label{sec:experiments}
CR-GAN aims to learn complete representations in the embedding space. We achieve this goal by combining the two-pathway architecture with self-supervised learning. We conduct experiments to evaluate these two contributions respectively. Then we compare our CR-GAN with DR-GAN~\cite{tran2017disentangled}, both the visual results and t-SNE visualization in the embedding space are shown. We also compare CR-GAN and BiGAN with an image reconstruction task.
\subsection{Experimental Settings}
\textbf{Datasets}. We evaluate CR-GAN on datasets with and without view labels. Multi-PIE~\cite{Gross2010MultiPIE} is a labeled dataset collected under constrained environment. We use 250 subjects from the first session with 9 poses within $\pm 60^{\circ}$, 20 illuminations, and two expressions. The first 200 subjects are for training and the rest 50 for testing. 300wLP~\cite{zhuxiangyu2016} is augmented from 300W~\cite{sagonas2013300} by the face profiling approach~\cite{zhuxiangyu2016}, which contains view labels as well. We employ images with yaw angles ranging from $-60^{\circ}$ to $+60^{\circ}$, and discretize them into 9 intervals.
For evaluation on unlabeled datasets, we use CelebA~\cite{liu2015deep} and IJB-A~\cite{klare2015pushing}. CelebA contains a large amount of celebrity images with unbalanced viewpoint distributions. Thus, we collect a subset of 72,000 images from it, which uniformly ranging from $-60^{\circ}$ to $+60^{\circ}$. Notice that the view labels of the images in CelebA are only utilized to collect the subset, while no view or identity labels are employed in the training process. We also use IJB-A which contains 5,396 images for evaluation. This dataset is challenging, since there are extensive identity and pose variations.
\textbf{Implementation Details}. Our network implementation is modified from the residual networks in WGAN-GP~\cite{gulrajani2017improved}, where $E$ shares a similar network structure with $D$. During training, we set $v$ to be a one-hot vector with 9 dimensions and $z \in [-1, 1]^{119}$ in the latent space. The batch size is 64. Adam optimizer~\cite{kingma2014adam} is used with the learning rate of $0.0005$ and momentum of $[0, 0.9]$. According to the setting of WGAN-GP, we let $\lambda_1 = 10$, $\lambda_2 \sim \lambda_4 = 1$, $\lambda_5 = 0.01$. Moreover, all the networks are trained after 25 epochs in supervised learning; we train 10 more epochs in self-supervised learning.
\subsection{Single-pathway {\it vs.} Two-pathway}\label{sec-single-two}
We compare two-pathway network with the one using a single reconstruction path. All the networks are trained on Multi-PIE. When test with Muli-PIE, as shown in Fig.~\ref{fig:single-two-rotation}~(a), both models produce desirable results. In each view, facial attributes like glasses are kept well. However, single-pathway model gets unsatisfactory results on IJB-A, which is an ``unseen'' dataset. As shown in Fig.~\ref{fig:single-two-rotation}~(b), two-pathway model consistently produce natural images with more details and fewer artifacts. Instead, the single-pathway model cannot generate images with good quality. This result prove that our two-pathway network handles ``unseen'' data well by learning a complete representation in the embedding space.
\subsection{Supervised {\it vs.} Self-supervised Learning}\label{sec:sup-self-sup}
The two-pathway network is employed in the following evaluations. We use Multi-PIE and 300wLP in supervised learning. For self-supervised learning, in addition to the above datasets, CelebA is employed as well. Note that we don't use view or identity labels in CelebA during training.
\begin{figure*}[t]
\centering
\includegraphics[width=\linewidth]{sup-self-sup-IJBA}
\caption{Multi-view face generation results on IJB-A~\protect\cite{klare2015pushing}. In each case, from top to bottom: the results generated by our supervised model, DR-GAN~\protect\cite{tran2017disentangled} and our self-supervised model. DR-GAN fails to produce favourable images of large poses, while our method can synthesize reasonable profile images.}
\label{fig:IJBA}
\vspace{-4mm}
\end{figure*}
\textbf{Evaluation on CelebA}. Fig.~\ref{fig:celeba} shows the results on CelebA. In Fig.~\ref{fig:celeba} (a), although the supervised model generates favorable results, there are artifacts in all views. As the supervised model is only trained on Multi-PIE and 300wLP, it is difficult to ``approximate'' the data in the wild. Instead, the self-supervised model has learned a latent representation where richer features are embedded, so it generates more realistic results while the identities are well preserved. We can make the similar observation in Fig.~\ref{fig:celeba} (b). The supervised model can only generate images that are similar to Multi-PIE, while the self-supervised model can generate novel identities. In Fig.~\ref{fig:celeba} (c) and (d), the self-supervised model preserve identity and attributes in a better way than others.
\textbf{Evaluation on IJB-A}. Fig.~\ref{fig:IJBA} shows more results on IJB-A. We find that our self-supervised model successfully generalize what it has learned from CelebA to IJB-A. Note that it is our self-supervised learning approach that makes it possible to train the network on unlabeled datasets.
\subsection{Comparison with DR-GAN}\label{sec:unlabeled}
Furthermore, we compare our self-supervised CR-GAN with DR-GAN~\cite{tran2017disentangled}. We replace DC-GAN~\cite{radford2015unsupervised} network architecture used in DR-GAN with WGAN-GP for a fair comparison.
\textbf{Evaluation on IJB-A}. We show the results of DR-GAN and CR-GAN in Fig.~\ref{fig:IJBA} respectively. DR-GAN produces sharp images, but the facial identities are not well-kept. By contrast, in Fig.~\ref{fig:IJBA} (a) and (b), CR-GAN produces face images with similar identities. In all cases, DR-GAN fails to produce high-quality images with large poses. Although not perfect enough, CR-GAN can synthesize reasonable profile images.
\begin{figure}[t]
\centering
\includegraphics[width=\linewidth]{fig_tsne}
\caption{t-SNE visualization for the embedding space of CR-GAN (\textbf{left}) and DR-GAN (\textbf{right}), with 10 subjects from IJB-A~\protect\cite{klare2015pushing}. The same marker shape (color) indicates the same subject. For CR-GAN, multi-view images of the same subject are embedded close to each other, which means the identities are better preserved.}
\label{fig:t-SNE}
\vspace{-1mm}
\end{figure}
\textbf{Identity Similarities on IJB-A}. We generate 9 views for each image in IJB-A both using DR-GAN and CR-GAN. Then we obtain a 128-dim feature for each view by FaceNet~\cite{SchroffCVPR15}. We evaluate the identity similarities between the real and generated images by feeding them to FaceNet. The squared L2 distances of the features directly corresponding to the face similarity: faces of the same subjects have small distances, while faces of different subjects have large distances. Table~\ref{tb:facenet} shows the results of the average L2 distance of CR-GAN and DR-GAN in different datasets. Our method outperforms DR-GAN on all datasets, especially on IJB-A which contains unseen data. Fig.~\ref{fig:t-SNE} shows the t-SNE visualization in the embedding space of DR-GAN and CR-GAN respectively. For clarity, we only visualize 10 randomly selected subjects along with 9 generated views of each. Compared with DR-GAN, CR-GAN produces tighter clusterings: multi-view images of the same subject are embedded close to each other. It means the identities are better preserved.
\begin{table}[t]
\begin{center}
\setlength\tabcolsep{2.0pt}
\begin{tabular}{c|ccc}
\hline
& Multi-PIE & CelebA & IJB-A \\
\hline
DR-GAN & $1.073 \pm 0.013$ & $1.281 \pm 0.007$ & $1.295 \pm 0.008$ \\
CR-GAN & $\textbf{1.018} \pm \textbf{0.019}$ & $\textbf{1.214} \pm \textbf{0.009}$ & $\textbf{1.217} \pm \textbf{0.010}$ \\
\hline
\end{tabular}
\end{center}
\vspace{-3mm}
\caption{Identity similarities between real and generated images.}
\label{tb:facenet}
\vspace{-3mm}
\end{table}
\textbf{Generative Ability}. We utilize DR-GAN and CR-GAN to generate images from random noises. In Fig.~\ref{fig:DR-GAN-ours-random}, CR-GAN can produce images with different styles, while DR-GAN leads to blurry results. This is because the single-pathway generator of DR-GAN learns incomplete representations in the embedding space, which fails to handle random inputs. Instead, CR-GAN produces favorable results with complete embeddings.
\begin{figure}[t]
\centering
\includegraphics[width=\linewidth]{DR-GAN-ours-random}
\caption{Generating multi-view images from the random noise. (a) DR-GAN~\protect\cite{tran2017disentangled} generates blurry results and many artifacts. (b) CR-GAN generates realistic images of different styles.}
\label{fig:DR-GAN-ours-random}
\vspace{-2mm}
\end{figure}
\subsection{Comparison with BiGAN}
To compare our method with BiGAN, we qualitatively show the image reconstruction results of both methods on CelebA in Fig.~\ref{fig:bigan}. We can find that as demonstrated by~\cite{donahue2016adversarial,dumoulin2016adversarially}, BiGAN cannot reconstruct the data correctly, while CR-GAN keeps identities well due to the explicit reconstruction loss we employed.
\begin{figure}[t]
\centering
\includegraphics[width=\linewidth]{bigan}
\caption{Reconstruction results on CelebA. BiGAN (Row 2) cannot keep identity well. Ours (Row 3) produces better results.}
\label{fig:bigan}
\vspace{-4mm}
\end{figure}
\section{Conclusion}
In this paper, we investigate learning ``complete representations" of GAN models. We propose CR-GAN that uses a two-pathway framework to achieve the goal. Our method can leverage both labeled and unlabeled data for self-supervised learning, yielding high-quality multi-view image generations from even ``unseen'' data in wild conditions.
\section*{Acknowledgements}
This work is partly supported by the Air Force Office of Scientific Research (AFOSR) under the Dynamic Data-Driven Application Systems program, NSF CISE program, and NSF grant CCF 1733843.
|
1801.08124
|
\section{Introduction}
Angle-resolved photoelectron spectroscopy (ARPES) using synchrotron radiation has become an essential tool for condensed matter physics and surface science. The high spectral brightness of synchrotron radiation allows photoelectron spectra to be recorded with photocurrents in the nano-Ampere range. These large photocurrents, parsed by sophisticated electron energy analyzers \cite{Medjanik_NatMat2017, Jozwiak_SRN2012, Iwasawa_JSR2017}, enable detailed studies of the electronic structure of solids and surfaces in energy, momentum, and spin. Tuning the photon energy throughout the extreme ultraviolet (10-100 eV) allows experiments to map the energy dispersion relation for momentum perpendicular to the surface ($k_z$), interpret the contributions of final state effects to the measured energy distribution curves (EDC), and choose between increased surface or bulk sensitivity~\cite{Hufner_book2003}.
\hspace{0.2 in} Shortly after the development of high-power femtosecond lasers and discovery of high order harmonic generation (HHG), in which a broad range of laser-harmonics are coherently emitted from a field-ionized medium \cite{Brabec_RMP2000}, HHG was applied to surface photoemission experiments \cite{Haight_RSI1994, Bauer_PRL2001, Haarlammert_CurrOpSSMS2009}. Indeed, the range of photon energies typically emitted from HHG driven by Ti:Sapphire lasers in noble gasses is nicely coincident with the range of photon energies used by ARPES beamlines at synchrotrons. In addition to the dramatically reduced cost and footprint compared to a synchrotron source, the HHG pulses had the advantage that they could be orders of magnitude shorter than the $\sim$100 ps pulse durations of synchrotrons, enabling ultrafast time-domain studies. However, it also became immediately apparent that photoemission experiments using HHG would be drastically limited compared to what is possible at synchrotrons \cite{Haarlammert_CurrOpSSMS2009, Mathias_Collection2010}.
\hspace{0.2 in} The principle limitation comes from the so-called ``vacuum space-charge" effect \cite{Hellmann_PRB2009}. Consider a synchrotron experiment illuminating the sample with $\sim 10^{12}$ photons/second causing $\sim 10^{11}$ electrons/second (16 nA) to be emitted from the surface. The synchrotron photon pulses arrive at $\sim 100$ MHz repetition rate, so each burst of electrons emitted concurrently from a single pulse contains only $\sim 1000$ electrons. In contrast, due to the high peak powers required to drive the HHG process efficiently, HHG is typically restricted to $<$100 kHz repetition rates. In order to maintain the same photocurrent, the electrons must then be concentrated by more than 1000 times, to more than $10^6$ electrons/pulse. The charging of space at these electron densities distorts the photoelectron spectrum on the eV energy scale, whereas synchrotron beamlines now routinely record photoelectron spectra with meV resolution \cite{Reininger_AIP2007}. Practitioners of time-resolved photoelectron spectroscopy using HHG are then forced to compromise on the applied photon flux, focused spot size, resolution, fidelity of the signal, or some combination thereof \cite{Plotzing_RSI2016, Mathias_Collection2010, Al-Obaidi_Thesis2016}. Compounding the problem is the fact that a time-resolved pump-probe experiment inherently requires much more data than a static one, since one must record spectra at many pump-probe delays and only a small fraction of the sample's electrons should be excited by the pump.
\hspace{0.2 in} With the constraint of space charge setting the fundamental limits on the performance of photoemission experiments, this phenomenon has been extensively studied over a wide range of electron kinetic energies and pulse durations, both experimentally and theoretically \cite{Hellmann_PRB2009, Plotzing_RSI2016, Frietsch_RSI2013, Mathias_Collection2010, Graf_JApplPhys2010, Passlack_JApplPhys2006, Zhou_JElecSpec2005}. For sub-ps pulses and electron kinetic energies in the $\sim$ 5-100 eV range produced from conductive samples, both shifts and broadening of the photoelectron spectra features are observed to scale with linear electron density $\rho \equiv N/D$, where $N$ is the number of electrons emitted from the sample per pulse and $D$ is the spot size of the light on the sample. Expressed in terms of the average sample current ($I_{\textss{sample}}$) and repetition rate ($f_{\textss{rep}}$), this gives:
\begin{equation}\label{eqn:sc}
\Delta E_{\textss{s,b}} = m_{\textss{s,b}} \frac{I_{\textss{sample}}}{e f_{\textss{rep}} D}
\end{equation}
where $\Delta E_{\textss{s,b}}$ is the energy shift (s) or energy broadening (b) of the photoelectron spectrum, $e$ is the charge of the electron, and $m_{\textss{s,b}}$ are empirical scaling factors. Reports of the slope parameters $m_{\textss{s,b}}$ in the literature have varied by a factor of 2 \cite{Hellmann_PRB2009, Plotzing_RSI2016, Frietsch_RSI2013, Mathias_Collection2010}. Recently, Pl\"otzing et al. \cite{Plotzing_RSI2016} have studied these effects for multiple spot sizes and determined $m_b = 2.1 \times 10^{-6}$ eV-mm and and $m_s = 3.2 \times 10^{-6}$ eV-mm with an estimated systematic uncertainty of less than 20\%.
\hspace{0.2 in} Figure \ref{fig:sourcecomparison} illustrates the constraints on attainable sample current for a given resolution according to Eq.~(\ref{eqn:sc}). The dashed lines indicate the space charge limits for sub-ps laser-based systems of different repetition rates assuming a 1 mm spot size - large by ARPES standards. Even at the high repetition rate of 100 kHz and the coarse resolution of 100 meV, space charge constraints still limit the sample current to 760 pA. These low data rates then often restrict experiments to strongly excited samples using absorbed fluences on the order $\sim$ 1 mJ/cm$^2$ \cite{Mathias_NatComm2016, Borgwardt_JPhysChemC2015, Saathoff_PRA2008}. At these fluences ultrashort pump pulses also produce many electrons through multiphoton processes which add to the space-charge problem \cite{Ultstrup_JElecSpec2015, Oloff_JApplPhys2016, Borgwardt_Thesis2016}. For non-conducting samples such as insulators or liquids, where positive charges left behind in the sample are immobile, shifts of the photoelectron spectra can be much larger and show a non-trivial delay dependence in pump-probe experiments which can be difficult to separate from the dynamics of interest \cite{Al-Obaidi_NJP2015}.
\begin{figure}[t!]
\begin{center}
\includegraphics[width = 9.2 cm]{SpaceChargeComparison_updatedUBC.pdf}
\caption{Dashed lines are created by evaluating Eq.~\ref{eqn:sc} for different repetition rates assuming 1 mm spot size and symbols represent published results applying HHG to surface photoemission. Yellow circles represent results from tunable HHG systems and purple squares represent setups where the photon energy is not tunable {in-situ}. Symbols with black edges represent space-charge limited spectrometers, and symbols with red edges represent systems that are not yet space-charge limited. For space-charge-limited systems, the $(x,y)$ positions represent the case where space charge broadening and the photon bandwidth add equally in quadrature. See appendix~\ref{ap:fig1} for detailed explanation on how symbol placement was calculated based on published results \cite{Mills_CLEO2017,*Jones_Private,Buss_SPIE2017,Frietsch_RSI2013,Artemis_Private,Chiang_NJP2015,Plotzing_RSI2016,Ojeda_SD2016,EICH_JElecSpec2014}.}
\label{fig:sourcecomparison}
\end{center}
\end{figure}
\hspace{0.2 in} Motivated by the inverse dependence of Eq.~(\ref{eqn:sc}) on $f_{\textss{rep}}$, in this article we demonstrate the application of a widely tunable cavity-enhanced high-harmonic generation (CE-HHG) light source \cite{Mills_JPhysB2012} to the difficult problem of time-resolved surface photoemission. By performing experiments with high flux at 88 repetition rate, nanoamperes of sample current can be generated from a sub-100 micron laser spot with space charge effects less than 10 meV, comparable to synchrotron-based ARPES experiments \cite{Zhou_JElecSpec2005, Hoesch_RSI2017}. As we show, this enables time-resolved photoelectron spectroscopy in a qualitatively different regime of resolution and pump-fluence than space-charge limited systems. In section \ref{sec:lightsource}, we describe critical and unique details of the light source along with its performance. In section \ref{sec:photspec} we demonstrate both static and time-resolved photoelectron spectroscopy measurements with the high dynamic range enabled by nA space-charge free sample currents. In section \ref{sec:conclusions} we discuss the comparison of this work to previous efforts and describe how the system can be further improved.
\section{Light Source and Beamline}\label{sec:lightsource}
\begin{figure*}[t!]
\begin{center}
\includegraphics[width = 0.95\textwidth]{Apparatus8.pdf}
\caption{CE-HHG source and beamline. High-order harmonics of a resonantly enhanced Yb:fiber frequency comb are generated at the focus of a 6 mirror enhancement cavity and coupled into an XUV beamline. A pulse-preserving monochromator selects one harmonic which is focused on a sample under UHV conditions. BP = Brewster plate, VPD = vacuum photodiode, TM = toroidal mirror, PD = XUV photodiode, GJ = gas jet, IC = input coupler.}
\label{fig:setup}
\end{center}
\end{figure*}
The experimental setup is shown in Fig.~\ref{fig:setup}. A home-built 80 W, 155 fs frequency comb laser with a repetition rate of 88 and and a center wavelength of 1.035~$\mu$m ($h \nu = 1.2$ eV) is passively amplified in a 6 mirror enhancement cavity with a 1\% transmission input coupler. We have described the laser in detail previously \cite{Li_RSI2016}. The laser is locked to the cavity using a two-point Pound-Drever-Hall lock as described in \cite{Foltynowicz_APB2013, Corder_SPIE2018}. Harmonics are generated at a 24 $\mu$m FWHM intracavity focus and reflected from a sapphire wafer placed at Brewster's angle for the resonant 1.035 $\mu$m light. Noble gasses are injected to the focus using a fused silica capillary with a 100 $\mu$m inside diameter. We optimize the nozzle position by moving it to maximize the photocurrent observed on a stainless steel vacuum photodiode (VPD1) \cite{Feuerbacher_JApplPhys1972}. Typical photocurrents from VPD1 are in the range of 100 to 300 nA. When generating harmonics, we also dose each intracavity optic with a mix of ozone and O$_2$ from a commercial ozone generator to prevent hydrocarbon contamination allowing continuous operation.
\hspace{0.2 in} Typical intracavity powers for generating harmonics range from 5-11 kW, depending on the generating gas and desired harmonic spectrum, corresponding to intracavity peak intensities in the range of 0.6 to 1.3 $\times 10^{14}$ W/cm$^2$. Intracavity nonlinear effects are observed from both the HHG gas and self-phase modulation in the Brewster plate, dropping the cavity's power enhancement and necessitating careful tuning of servo loop offsets \cite{Allison_PRL2011, Yost_OptExp2011, Corder_SPIE2018}. For example the power enhancement drops from 270 at low power to 200
at 7.5 kW of intracavity power when generating harmonics in krypton.
\hspace{0.2 in} The outcoupled harmonics are collimated by a 350 mm focal length toroidal mirror at 3 degrees grazing angle (TM1) that forms the first part of a single off-plane grating pulse-preserving monochromator similar to the design of Frassetto et al. \cite{Frassetto_OptExp2011}. The harmonics strike a motorized grating at a 4 degree grazing angle and are refocused by a second $f=350$ mm toroidal mirror (TM2) at an adjustable slit. For all data presented here, the monochromator grating has 150 grooves/mm and is blazed for optimum diffraction efficiency for $\lambda = 35$ nm. With this grating the monochromator selects an individual harmonic with tolerable pulse broadening but does not narrow the transmitted harmonic bandwidth. The exit slit plane of the monochromator is 1:1 imaged to the sample using another 350 mm focal length toroid at 3 degrees grazing angle (TM3). Mirror TM3 is electrically floated such that the photocurrent of electrons ejected from the mirror surface can be used as a passive XUV intensity monitor. All beamline optics are gold coated and the XUV light is polarized perpendicular to the plane of incidence (s-polarization).
\hspace{0.2 in} We detect the XUV flux exiting the monochromator and delivered to the sample using four separate detectors: an aluminum coated silicon photodiode (PD, Optodiode AXUV100Al), the photocurrent from TM3, the photocurrent from the sample, and the photocurrent from an Al$_2$O$_3$ vacuum photodiode \cite{CANFIELD_NBST1987} (VPD2) placed at the end of the surface science chamber. Figure \ref{fig:HHGflux}a) shows a typical HHG spectrum from xenon gas measured using each of the four detectors as the monochromator grating is rotated. The observed harmonic linewidths in Fig.~\ref{fig:HHGflux}b) are due to the intentionally small resolving power of the pulse-preserving monochromator, not the intrinsic harmonic linewidth.
\hspace{0.2 in} The photon flux can be calculated using the measured photocurrent from all the detectors and literature values for the quantum efficiencies. All of these separate calculations agree within a factor of 2. Since contamination and surface oxidation only cause the quantum efficiency of XUV detectors to decrease, all calculated photon fluxes represent lower limits. In Fig.~\ref{fig:HHGflux}b), the higher of the two lower limits from the PD or VPD2 are plotted as a function of photon energy and for three different generating gasses: argon, krypton, and xenon. As can be seen in Fig.~\ref{fig:HHGflux}b), even using a single monochromator grating, by changing the generating gas, a flux of more than $10^{11}$ photons per second is delivered to the sample over a broad tuning range. At lower photon energies, the higher efficiency of generation in Kr and Xe compensates the reduced diffraction of efficiency of the grating blazed for 35 eV. Higher fluxes can be obtained at lower photon energies using different gratings. For example, we have observed $7\times 10^{11}$ photons/s in the 21st harmonic from krypton ($h \nu = 25.1$ eV) using a 100 groove/mm grating blazed for 55 nm. These fluxes are within one order of magnitude of what is available from many state-of-the-art synchrotron beamlines dedicated to ARPES \cite{Hoesch_RSI2017, Reininger_AIP2007, Iwasawa_JSR2017}. Critically, since at 88 MHz, $7 \times 10^{11}$ photons/s corresponds to only 8,000 photons/pulse - also comparable to synchrotrons - all of this flux is usable for high-resolution photoemission experiments.
\begin{figure}[t!]
\begin{center}
\includegraphics[width = 8.5 cm]{MonoFigv5.pdf}
\caption{a) An HHG spectrum from xenon gas measured by rotating the monochromator grating while using the four detectors after the exit slit. The photodiode current (black) uses the left y-axis whereas the the photoemission current from three downstream surfaces uses the right y-axis. Note that the harmonic linewidths are not resolved with the pulse-preserving monochromator design. b) The photon flux delivered to the sample for each harmonic generated with the three gases. The flux has been calibrated using literature values for quantum efficiencies and no corrections for mirror losses have been made. c) The 27th harmonic from Ar imaged with a Ce:YAG crystal at the sample position. d) Lineouts through the centroid of (c) fit with Gaussian functions demonstrating 58 $\mu$m x 100 $\mu$m spot size (FWHM).}
\label{fig:HHGflux}
\end{center}
\end{figure}
\hspace{0.2 in} To measure the XUV spot at the sample, we image the fluorescence from a Ce:YAG scintillator plate placed at the sample position. Figure \ref{fig:HHGflux}c) shows the image and Fig.~\ref{fig:HHGflux}d) shows Gaussian fits in the both the horizontal and vertical along lineouts corresponding to the image centroid. The data indicate a clean elliptical beam with a FWHM of 58 $\mu$m in the horizontal 100 $\mu$m in the vertical. Also, we measure that approximately 70\% of the XUV light can be transmitted through a 100 $\mu$m diameter pinhole oriented at 45 degrees to the beam axis. This spot size is again similar to what is used at synchrotron beamlines \cite{Hoesch_RSI2017, Reininger_AIP2007}. When comparing to previous HHG results it is important to note that in our case this small spot size and high flux are actually usable for experiments due to the absence of space-charge effects at 88 repetition rate. A small spot size enables studying spatially inhomogeneous samples (for example produced by exfoliation \cite{Huang_ACSNano2015}), requires less pump-pulse energy in pump/probe experiments, and is necessary for achieving high angular resolution in ARPES.
\begin{figure}[t!]
\begin{center}
\includegraphics[width = 8.5 cm]{stabilityFig5.pdf}
\caption{a) Normalized XUV flux measured with PD after the monochromator for harmonics 15-25 from Kr over 1 hour without human intervention. c) Relative intensity noise (RIN) of the 23rd harmonic (red), intracavity laser light (green), and Yb:fiber laser (blue), along with the detector noise floor (black dashed).}
\label{fig:spotstable}
\end{center}
\end{figure}
\hspace{0.2 in} We evaluated the amplitude noise and long term stability of the system using the PD detector. For the long term stability, we recorded a series of monochromator scans over a one hour period without any human tuning of the laser alignment or servo loop. Figure \ref{fig:spotstable}a) shows the relative power in the harmonics from Kr with a delivered flux greater than $10^{11}$ photons/s measured every 3 minutes. The RMS fluctuations averaged over all the harmonics over this period are 5\%. Similar results are obtained for HHG in Ar or Xe as well. On longer time-scales, slow drifts in the laser alignment into the cavity and servo-loop offsets require occasional tuning to maintain the flux levels at those shown in Fig.~\ref{fig:HHGflux}b). It is also important to note that since more than 100 pA of photocurrent is observed from TM3, drifts in the XUV flux can also be normalized using this in-situ monitor, as is commonly done at synchrotrons. At the time of writing we have run the source on a near-daily basis without venting the vacuum system or performing any alignment of the in-vacuum optics for more than 2 months with stable and reproducible results, enabling the photoelectron spectroscopy experiments discussed in the next section.
\hspace{0.2 in} For pump-probe experiments it is often advantageous to use lock-in detection to extract small signals from large backgrounds. Fig.~\ref{fig:spotstable}b) shows the amplitude noise (RIN) of the 23rd harmonic from Kr measured using the photodiode current amplified with a transimpedance amplifier and recorded with an FFT spectrum analyzer. For frequencies above 400 Hz, the RIN level is below -60 dBc/Hz, which can enable small differences in the photoelectron spectra to be recorded via lock-in detection. At this noise level, EDC or ARPES signals up to $10^6$ counts/second/bin can be photoelectron-shot-noise limited with proper correction for drift using the TM3 photocurrent.
\section{Photoemission}\label{sec:photspec}
\hspace{0.2 in} Photoelectron spectroscopy measurements are preformed under ultra-high vacuum conditions in a surface science endstation equipped with a hemispherical electron energy analyzer (VSW HA100). The analyzer is specified to have an angular acceptance of $\pm$ 4 degrees at the input and has a channeltron detector at the exit.
The endstation is also equipped with a sputter gun, a LEED, a quadrupole mass spectrometer, an Al K$\alpha$ x-ray source, and a sample manipulator that can be cooled and heated between 100 and 1000 K. Also mounted on the sample manipulator are the Ce:YAG scintillator and pinhole mentioned in section \ref{sec:lightsource}. For all data presented here, the sample is oriented normal to the analyzer axis and 45 degrees to the XUV beam. The electric field vector of the XUV light is in the plane of incidence (p-polarized) and the analyzer axis.
\begin{figure}[t!]
\includegraphics[width = 8.0 cm]{allUPS8.pdf}
\caption{Static photoelectron spectra of a Au (111) surface taken with harmonics 7 through 33, vertically offset for clarity. The color indicates the gas used to generate the harmonic; Ar (black), Kr (red), Xe (blue). Each EDC is normalized to the photocurrent measured at TM3, and spectra taken with photon energies above 25.1 eV have been enlarged by $\times5$.}
\label{fig:allUPS}
\end{figure}
\hspace{0.2 in} Figure \ref{fig:allUPS} shows photoelectron spectra from an Au~(111) surface at 100 K temperature obtained using each harmonic between the 7th ($h \nu = 8.4$ eV) and 33rd ($ h \nu $ = 39.5 eV). Each spectrum was acquired with 34 meV steps individually measured with 1 second of integration for a total scan time of $\sim$6 minutes or less.
At the d-band peaks, the electron count rates can exceed 1 MHz. These static spectra are in good agreement with those recorded by Kevan et al. \cite{Mills_PRB1980,Kevan_PRB1987} using tunable synchrotron radiation. The clearly visible dispersion of the d-bands at binding energies between 3 and 7 eV and the large photon energy dependence of the relative amplitudes of the peaks highlight the importance of conducting photoemission experiments with a tunable source. The same final state effects also strongly influence time-resolved photoelectron spectra and tunability should be considered no less important, as has been emphasized by previous authors \cite{Zhu_JElecSpec2015}.
\hspace{0.2 in} The resolution of the setup can be determined by analyzing the sharpness of the Fermi edge and is dominated by the energy analyzer. Fermi edge widths as low as 110~meV are measured depending on alignment. The best resolution we have been able to observe in any photoemission experiment using this analyzer is 89 meV using a He I lamp and a Kr gas target. From this data we can place a conservative upper limit on the single harmonic photon energy bandwidth of $\sqrt{(110 \units{meV})^ 2 - (89 \units{meV})^2}$ = 65 meV. We have several reasons to believe that the single harmonic linewidth is substantially lower than this. First, the observed Fermi edge sharpness is found to be completely independent of the HHG generating conditions or harmonic used under variation of a large range of parameters. For example, reducing the driving laser intensity or using the cavity to narrow the bandwidth of the driving pulse \cite{Foltynowicz_APB2013} should both reduce the harmonic bandwidth, but no change in the the photoelectron spectrum is observed. Furthermore, Mills et al. \cite{Mills_CLEO2017,*Jones_Private} have reported single harmonic linewidths from a cavity-enhanced HHG source similar to ours, but using even shorter driving pulses, with FWHM as low as 32 meV. Starting with longer driving pulses in our setup, we expect harmonic linewidths narrower than this are obtainable. Further investigation of the energy resolution will be the subject
of future work as we upgrade our electron energy analyzer.
\hspace{0.2 in} Even with the current analyzer-limited resolution, we demonstrate here that the absence of space-charge allows for time-resolved photoemission experiments that are both qualitatively and quantitatively different than what is done with space-charge limited systems. Figure \ref{fig:TRUPS} shows two photoelectron spectra near the Fermi edge of the Au (111) on a logarithmic scale, one with and another without a parallel polarized 1.035 $\mu$m wavelength laser excitation overlapped in space and time. The spectra were taken with 3 nA of sample current, or approximately 215 electrons/pulse. Consider first the black curve taken with the pump laser off. For a 100 kHz system with our spot size (or even somewhat larger), this sample current would result in broadening and shifting of the Fermi edge on the eV scale instead of the $<$ 10 meV effects here. Furthermore, on a logarithmic scale, space charge effects can cause long high energy tails in the photoelectron spectrum \cite{Hellmann_PRB2009, Borgwardt_Thesis2016, Saathoff_PRA2008} that make it difficult to observe small signals from weakly excited samples. Here, with excellent harmonic isolation from our pulse-preserving monochromator and the absence of space charge effects, a precipitous drop of four orders of magnitude is observed in the EDC at the Fermi edge.
\begin{figure}[t!]
\begin{center}
\includegraphics[width = 8.5 cm]{TRFigurev6.pdf}
\caption{a) The photoelectron spectrum of the Au (111) Fermi edge taken without (black) and with a 1.035 $\mu$m pump pulse (red) at a peak intensity of 1.3$\times 10^9$ W/cm$^2$. A LAPE sideband of the surface state peak at 24.8 eV is observed at 26 eV. b) The magnitude of the sideband at a kinetic energy of 26 eV as a function of pump probe time delay. The cross-correlation has a FWHM of 181 fs. c) The amplitude of the sideband at 26 eV as a function of pump peak intensity. A fit to the data gives a slope of $1.34\times10^{-12}$ cm$^2$/W.}
\label{fig:TRUPS}
\end{center}
\end{figure}
\hspace{0.2 in} Next consider the red curve taken with the pump laser on. Before discussing the laser-induced features of the spectrum, consider first what is \emph{not} observed. The photoelectron spectrum is not shifted, broadened, or distorted due to space charge produced by the laser excitation as is commonly observed in pump probe experiments \cite{Ultstrup_JElecSpec2015, Oloff_JApplPhys2016, Borgwardt_Thesis2016, Saathoff_PRA2008}. This is because the high data rate enables the experiment to be performed under the low pump intensity of $1.3 \times 10^{9}$~W/cm$^2$. We measured the sample current from the pump excitation alone and found it to depend strongly on the region of sample probed, as in \cite{Saathoff_PRA2008}, but always at least one order of magnitude less than the XUV probe, or less than 22 electrons/pulse produced by the pump.
\hspace{0.2 in} The reflectivity of the gold sample is $>$ 97\% and the the laser-induced features in the EDC are dominated by the now well-known laser-assisted photoelectric effect (LAPE) \cite{Saathoff_PRA2008}. Briefly, LAPE is a dressing of the ionized electrons by the IR laser field producing sidebands at intervals of the photon energy \cite{Saathoff_PRA2008, Madsen_AmJPhys2005, Mahmood_NatPhys2016, Glover_PRL1996}. In Fig.~\ref{fig:TRUPS}a) a sideband of the surface state peak at 24.8 eV is observed 1.2 eV higher at 26 eV. Figure~\ref{fig:TRUPS}b) shows a measurement of the sideband amplitude at 26.0 eV kinetic energy as a function of the time delay between the IR pump and XUV probe. The data were accumulated in 10 minutes. Within statistical error, identical widths and time-zero positions are observed for cross correlations taken at both higher and lower kinetic energies, further confirming the LAPE mechanism, as hot electrons closer to the Fermi energy would show observable lifetimes \cite{FANN:1992uk,Cao:1998tp}.
\hspace{0.2 in} LAPE can be used to determine the time resolution of the instrument. A gaussian fit to the cross-correlation in Fig.~\ref{fig:TRUPS}b) gives a FWHM of 181 fs. The pump laser pulse duration at the sample position was not independently measured for this experiment, but at the output of the laser the pulse duration was measured to be $165 \pm 10$ fs and optimal compression gives 155 fs \cite{Li_RSI2016}. Taking the lowest possible value of the laser pulse duration then gives a conservative upper limit for the XUV pulse duration at the sample of $\sqrt{(181 \units{fs})^2 - (155 \units{fs})^2}$ = 93 fs.
\hspace{0.2 in} Figure \ref{fig:TRUPS}c) shows amplitudes for the sideband at 26~eV kinetic energy for different pump pulse intensities obtained from fits to time-resolved scans as shown in Fig.~\ref{fig:TRUPS}b). The sideband amplitude is observed to be linear in the laser intensity with a slope of $1.34\pm0.13\times10^{-12}$ cm$^2$/W, and in excellent agreement with theory ($1.3\times10^{-12}$ cm$^2$/W) for our laser and experimental geometry, as calculated in appendix~\ref{ap:LAPE}. Even with the modest scan times of 10 minutes used to acquire this data, sideband amplitudes as small as $6\times10^{-4}$ are easily observed. We also note that with a multichannel electron analyzer, the delay-dependence for the full energy window of Fig.~\ref{fig:TRUPS}a) (or larger) could be obtained in parallel with no increase in data acquisition time.
\section{Conclusions}\label{sec:conclusions}
In this article, we have described ultrafast time-resolved photoemission experiments using a high-flux, high repetition rate, tunable XUV light source. The absence of space-charge effects in photoelectron spectra taken with nano-Ampere photocurrents have allowed us to observe LAPE with sideband amplitudes in the $10^{-3}$ to $10^{-4}$ range, orders of magnitude smaller than typical \cite{Saathoff_PRA2008, Tao_Science2016, Mathias_NatComm2016}. To our knowledge, these are the smallest LAPE signals observed, but more importantly they mimic the response of weakly excited samples. Being able to observe changes in the photoelectron spectra at this level enables studying samples excited weakly at low excitation fluences less than 10 $\mu$J/cm$^2$, which is typical for the more sensitive techniques of optical spectroscopy \cite{Huang_NatPhot2013}, two-photon photoemission \cite{Chan_Science2011_SingletFission}, and laser-based ARPES using 6 eV probe light \cite{Ishida_RSI2016}, but extremely difficult using space-charge limited HHG systems.
\hspace{0.2 in} Whereas space-charge sets a fundamental limit on most HHG-based photoemission instruments, the current time and energy resolution of the system do not represent any inherent limitations of the frequency-comb based methods used here, and are instead limited simply by the laser pulse duration and energy analyzer performance. Both of these are straightforward to improve. For example, in our setup sub-100 fs resolution could be obtained leaving the XUV probe arm unchanged and implementing nonlinear pulse compression in the pump arm \cite{Eidam_AppPhysB2008, Seidel_SciRep2017}, which has no demands on the temporal coherence of the pulse train. CE-HHG can also be performed with shorter driving pulses \cite{Pupeza_NatPhot2013, Lilienfein_OptLett2017}, if desired. Single-grating pulse preserving monochromators have been shown to be compatible with temporal resolutions as small as 8 fs \cite{Gierz_PRL2015}.
\hspace{0.2 in} Returning to Fig.~\ref{fig:sourcecomparison}, we use the conservative upper limit of 65 meV to compare the performance of our system against previous time-resolved photoemission work \cite{Mills_CLEO2017,*Jones_Private,Buss_SPIE2017,Frietsch_RSI2013,Artemis_Private,Chiang_NJP2015,Plotzing_RSI2016,Ojeda_SD2016,EICH_JElecSpec2014}. As can be seen, the present work enables a dramatic improvement over space-charge limited systems operating at lower repetition rate. Notable also is the work of Jones and co-workers \cite{Mills_CLEO2017,*Jones_Private} who have demonstrated to our knowledge the best resolution in HHG-based ARPES using a fixed photon-energy CE-HHG platform based on grating output coupling \cite{Yost_OptLett2008}. However, the grating output coupling method makes dynamic harmonic selection difficult, and also introduces larger pulse front tilts than the pulse-preserving monochromator, resulting in larger focused spot sizes and XUV pulse durations.
\hspace{0.2 in} In the current experimental setup, the single-channel energy analyzer, which measures only one angle and energy at a time, currently sets the primary limit on the data rate. Even with this limitation, space-charge free time-resolved photoelectron spectra with high dynamic range can be recorded in minutes. Upgrading the electron analyzer to a multi-channel version will allow frames of time-resolved ARPES measurements to be accumulated at rates comparable to synchrotron beamlines. Notably, the 88 repetition rate is well suited to the recent advance of time-of-flight momentum microscopy developed by Sch\"onhense and coworkers \cite{Schonhense_JElecSpec2015}.
\section{Acknowledgements}
This work was supported by the U.S. Air Force Office of Scientific Research under Award No. FA9550-16-1-0164, the U.S. Dept. of Energy, Office of Science, Office of Basic Energy Sciences, under Award No. DE- SC0016017, and the Stony Brook Foundation Discovery Prize. MDK, ARM and MGW were supported by the U.S Department of Energy, Office of Science, and supported by its Division of Chemical Sciences, Geosciences, and Biosciences within the Office of Basic Energy Sciences under Contract No. DE-SC0012704. We thank Arthur K. Mills and David J. Jones at the University of British Columbia for many generous and fruitful discussions and T. C. Weinacht at Stony Brook for encouragement regarding LAPE. We are also grateful for useful data regarding space-charge effects in photoemission experiments from Robert A. Kaindl at Lawrence Berkeley National Laboratory and Cephise Cacho at the United Kingdom's Central Laser Facility.
|
2211.07425
|
\section{Introduction}
Deep learning segmentation models are already being used for easing the burden of cancer histopathology whole slide image (WSI) annotation \cite{van2021deep}. However, supervised training requires annotated images, often long training times and computational resources such as graphical processing units (GPU). To overcome these bottlenecks, approaches such as active learning \cite{ren2021survey} are gaining popularity, as they reduce the time needed to perform the segmentation while retaining high performance via expert supervision. One strategy is to have pathologists annotate a small part of the image, either by clicking or scribbling around the regions of interest. Then, features are extracted from those sparsely annotated regions and the dense segmentation masks are obtained by retraining segmentation models. This strategy, however, requires both interactive and GPU support at run time.
For this proof-of-concept, we instead propose seeded interactive clustering (SIC) to decouple the problem by (1) precomputing the latent network representations, which does not require a GPU at run time so it can be done remotely in a computing cluster without interaction and, once the embeddings are obtained, (2) perform interactive clustering locally that does not involve any further training and is relatively cheaper computationally.
\subsection{Previous work}
Different authors have approached this problem by creating interactive tools based on extracting classical features \cite{hollandi2020annotatorj, arzt2022labkit} or on deep learning approaches \cite{lindvall2020tissuewand, jaber2021deep, miao2021quick}. Some of these approaches are intended for cell detection on smaller images, and therefore are harder to scale to bigger WSI. The approaches that are meant for digital pathology usually require a GPU while performing the interactive annotations, which may not always be available.
The re-purposing of convolutional neural networks (CNN) as feature extractors was proposed very early after the first models trained on large datasets were made avaialable \cite{zeiler2014visualizing, donahue2014decaf, oquab2014learning, sharif2014cnn}. Since then, using model architectures such as ResNet-50 \cite{he2016deep} trained on ImageNet has become common practice for extracting features and performing transfer learning, even if some authors argue that these models may not capture all the relevant histology features \cite{chen2022self}.
Recent work also proposes using self-supervised learning approaches for obtaining the embeddings \cite{chen2020simple, ciga2022self}, under the assumption that the feature space would be more representative of the tissue differences. However comparing different embeddings is not a trivial task and is influenced by many factors \cite{kornblith2019similarity, neyshabur2020being}.
Finally, image-set clustering is a field where all these concepts, including the choice of architecture and layer for feature extraction, or the clustering algorithms, has been extensively studied \cite{guerin2021combining}.
\begin{figure}[h!]
\centering
\includegraphics[width=1\linewidth]{sic.png}
\caption{Visualization of the seeded interactive clustering process. The \textit{Remote} box includes the ground truth (left) and illustrates how tissue patch embeddings are extracted (right). The \textit{Local} box, from left to right: the top row shows a tissue sample, where seeds (red and green markers) are manually placed on the tissue image. The bottom row shows the embedding space location of the corresponding tissue patches, where black dots are patches not yet assigned to a category. As the SIC iterations continue, the patches are re-clustered according to embedding similarity. Far right: dots representing patches in UMAP space are colored according to their position in RGB color space, and re-mapped to their coordinates in tissue space. Note that the clustering is made on the full embedding space and the reduced space is only intended for visualization purposes. Visualizations created using TissUUmaps \cite{pielawski2022tissuumaps}.}
\label{fig1}
\end{figure}
\section{Method}
\subsection{Seeded iterative clustering}
The proposed pipeline is presented in Figure \ref{fig1} and has two parts: one remote and one local. The first remote part of the process starts by dividing the image into smaller patches. The size of the patches, the stride between them and the resolution of choice defines the granularity of the final tissue clustering. In this case, we decided to extract $256\times256$ patches with $50\%$ overlap at $20\times$ resolution. Next, each of the patches are embedded using the pretrained model of choice by extracting the last layer latent representation. Finally, we store the matrix of $number\:of\:patches \times embedding\:size$ together with the center coordinates of each patch to represent the complete WSI.
The second part of the pipeline can be run locally. The seeded iterative clustering (SIC) approach, presented in Algorithm \ref{alg:cap}, divides the previously generated patch embedding space in a binary fashion, guided by small seeds -or sparse annotations- selected by the user in representative regions. Then seeded clustering \cite{basu2002semi} is performed iteratively until a score relative to the sparse annotations does not improve. We used F1-score, but any classification metric could be used to optimize the algorithm. The seeded KMeans algorithm only differs from the original KMeans in the first assignment of cluster centers, which are not initialized randomly but based on the seeds. An additional step of conventional KMeans is added as it had shown to improve the performance when the tumor region of the WSI was not big enough.
\begin{algorithm}[ht]
\caption{Seeded Iterative Clustering}\label{alg:cap}
\begin{algorithmic}
\State $X \gets embeddings$ \Comment{Embeddings of patches}
\State $y \gets seeds$ \Comment{Sparse annotations}
\State $score_{prev} \gets 0$ \Comment{Initialize score}
\State $labels \gets SeededKMeans(n_{clusters}=2).fit(X, y)$ \Comment{Initialize labels}
\While{$score \leq score_{prev}$} \Comment{Run until defined score goes down}
\State $X \gets embeddings[labels==1]$ \Comment{Select data only in the postive class}
\State $y \gets seeds[labels==1]$
\If{$\{0,1\} \subset y$} \Comment{If there are still annotations of both classes}
\State $labels \gets SeededKMeans(n_{clusters}=2).fit(X, y)$ \Comment{Re-cluster positive cluster}
\Else{}
\If{$\{1\} \subset y$} \Comment{If there are only negative annotations}
\State $labels \gets KMeans(n_{clusters}=2).fit(X)$ \Comment{Perform conventional clustering}
\EndIf{}
\EndIf{}
\State $score \gets metric(y, label)$ \Comment{Compare only sparse annotations}
\EndWhile{}
\end{algorithmic}
\end{algorithm}
\subsection{Feature space visualization}
Inspired by the spatial-omics field, an intuitive way of visualizing the embeddings is by reducing the dimensionality of the space to three dimensions using Uniform Manifold Approximation and Projection for Dimension Reduction (UMAP) \cite{mcinnes2018umap}. This 3D space can be mapped to the RGB space and visualized directly on top of the images as colored dots at the center of each patch in the image, as previously presented by \citeauthor{chelebian2021morphological} \cite{chelebian2021morphological} and shown in Figure \ref{fig1} (bottom-right). The code for doing this can be found at \href{https://github.com/eduardchelebian/histology-umap}{github.com/eduardchelebian/histology-umap}.
In the UMAP space, it is possible to manually select regions that are clustered together and see if they correspond to biologically relevant regions on the image. This approach however, is highly dependent on the dimensionality reduction algorithm and should be used only for qualitative visualization purposes.
\section{Experimental results}
\subsection{Dataset}
For this proof-of-concept we used the DigestPath challenge cancer segmentation dataset \cite{da2022digestpath}. It is an open database including 250 images of tissue from 93 positive WSI (i.e. with pixel-level annotations) slides in JPEG format which were used for the experimental results. The additional 410 images from 231 WSI negative slides were only used together with the positive slides when self-supervised pretraining was performed. All WSI are hematoxylin-eosin (H\&E) stained and scanned at $20\times$ resolution.
\subsection{Experiments}
In order to explore how different embedding spaces would affect the clustering performance, we chose four different feature extraction frameworks: (1) ResNet-18 pretrained on ImageNet (ResNet18), (2) ResNet-50 pretrained on ImageNet (ResNet50), (3) ResNet-18 pretrained on the dataset using SimCLR self-supervision \cite{chen2020simple} (SimCLR) and (4) ResNet-18 pretrained on 57 histopathology datasets using SimCLR self-supervision \cite{ciga2022self} (HistoSSL).
To mimic the complete workflow where the pathologist would select small benign and cancerous regions, we chose to experiment by randomly selecting 1, 3 and 5 patches that were located inside the benign and cancerous masks, respectively. Knowing that this would probably underestimate the results, as the seeds may not be representative of what a pathologist would choose, we report the F1-score for the patch classification task of the best run per slide in a 5-fold experiment, to simulate the selection an expert would make. The ground-truth coordinates are generated by selecting the patches that are at least 50\% in a cancer region.
\subsection{Results}
\begin{figure}[ht]
\centering
\includegraphics[width=1\linewidth]{fig2.png}
\caption{F1-score for the different models with varying number of simulated clicked patches.}
\label{fig2}
\end{figure}
The results are summarized in Figure \ref{fig2}. The highest per-slide F1-score for the ImageNet pretrained models are very similar regardless of the architecure, $0.57 \pm 0.22$ for ResNet18 and $0.54 \pm 0.21$ for ResNet50, both lower than the self-supervised approaches. There seems to be some benefit for pretraining (SimCLR) on the dataset at hand ($0.74 \pm 0.23$), but this may not be always possible, so it is very promising that a publicly available model (HistoSSL) also achieves higher results ($0.68 \pm 0.22$).
The method itself shows high classification variability, although the worse performing slides are usually the ones that have low positive patch ratio, that is, when the lesion is very small compared to the whole tissue, such as Figure \ref{fig3} (upper right). There is no clear difference from having more seeds, except when it comes to the standard deviation. As they are only used for initialization it seems that if the feature space is descriptive enough, adequately selected patches are sufficient for a good clustering result.
\begin{figure}[ht]
\centering
\includegraphics[width=1\linewidth]{fig3.jpg}
\caption{Examples of predictions using seeded iterative clustering. Visualizations created using TissUUmaps \cite{pielawski2022tissuumaps}.}
\label{fig3}
\end{figure}
\section{Discussion}
We presented seeded iterative clustering (SIC) as a way of leveraging latent representations of neural networks to speed-up the time-consuming process of manual annotation in histopathology. From an annotation tool point of view, the local clustering module takes seconds, even on a normal computer. The embedding time depends on the amount of patches or coordinates one wants. However, contrary to previous work, this can be run remotely without the pathologist interaction and does not need to be recalculated at any point. Additionally, performing the clustering per-slide is more robust to domain shift artifacts.
From a technical point of view, it allowed to compare the representations from different networks. When performing transfer learning, there is a separate effect of the feature reuse and learning low-level statistics from the data \cite{neyshabur2020being}. Retraining, then, may not be ideal to explore the quality of the features. This is precisely why we find that ImageNet pretrained ResNet off-the-shelf are not comparable to using self-supervised pretraining with SimCLR.
Work in progress includes validating this approach on other datasets and developing it into a full interactive tool called in which pathologists select a handful of patches and the method proposes clusters for the rest. Additionally, the method needs to be compared against other approaches in order to confirm its feasibility. Finally, extending this approach in a multi-class problem (e.g. cancer, benign and stroma regions, instead of binary), would provide further insights into how the feature space generated by the different models as well as increase the usability of the tool.
\section*{Broader Impact}
The uncoupling of the feature extraction and clustering modules could allow for the remote generation of embeddings that can then be shared with the community. This would allow for groups with less computational resources to work with the data and circumvent the potential privacy issues for sharing medical images, sharing instead the latent representation matrices per coordinate.
\begin{ack}
This research was funded by the European Research Council via ERC Consolidator grant CoG 682810 to C.W. and the SciLifeLab BioImage Informatics Facility.
\end{ack}
\medskip
\small
\printbibliography
\end{document}
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2211.07317
|
\section{Introduction}
It is a common yet challenging task to acquire visually appealing photos with appropriate brightness under a low light environment.
Traditional ways to increase the image brightness include enlarging the aperture, adopting a higher ISO, and reducing the shutter speed (\emph{i.e}\onedot} \def\Ie{\emph{I.e}\onedot, lengthening exposure time).
As for smartphone cameras with fixed aperture, brightness can only be adjusted by setting the sensor gain (\emph{i.e}\onedot} \def\Ie{\emph{I.e}\onedot, ISO) and exposure time.
Nonetheless, they are negatively correlated to maintain the appropriate brightness level of the image, \emph{i.e}\onedot} \def\Ie{\emph{I.e}\onedot, the shorter-exposure image generally adopts a higher ISO, while the longer-exposure image usually has a lower ISO.
Moreover, high ISO configuration introduces inevitable and complex noise due to the limited photon amount and the process of camera image signal processing (ISP) pipeline, while long-exposure is prone to produce blurry images due to the camera shake and scene variations.
Consequently, photographers have to make a compromise between noise and blur.
Recent advances in image restoration make it possible to further improve the visual quality of the acquired low light images by leveraging deep image denoising or deblurring networks.
Taking supervised denoising as an example,
synthetic or real-world noisy-clean image pairs are required to train the deep networks~\cite{DnCNN,FFDNet,MWCNN,CBDNet,DANet,MIRNet,Restormer}.
However, the models trained with synthetic training pairs are hard to generalize to real noisy images, and the real-world clean reference images are usually obtained by averaging hundreds of noisy ones~\cite{SIDD} or with complicated capturing and processing procedure~\cite{DND}, making the collection of large scale real-world training datasets laborious, expensive, and time-consuming.
Such problems greatly limit the deployment of models on more devices with different noise distributions.
Alternatively, a surge of self-supervised image denoising methods~\cite{DIP,Noise2Noise,Nei2Nei,N2V,Laine19,DBSN,Blind2Unblind,AP-BSN,R2R,soltanayev2018training,zhussip2019extending,CVF-SID,IDR} have been developed to avoid the collection of ground-truth (GT) training images, yet are limited in handling complex real-world image noise.
Another possible solution is to perform motion deblurring on long-exposure images.
Early explorations~\cite{fergus2006removing,shan2008high,xu2013unnatural,michaeli2014blind,schuler2015learning,ren2018deep} are mainly given to spatially uniform deblurring caused by camera motion.
Recently, the proposal of non-uniform datasets (\emph{e.g}\onedot} \def\Eg{\emph{E.g}\onedot, GoPro~\cite{DeepDeblur}, REDS~\cite{nah2019ntire}, HIDE~\cite{HIDE}, RealBlur~\cite{RealBlur}, \emph{etc}\onedot} \def\vs{\emph{vs}\onedot) has greatly boosted the research of deblurring in more practical scenes involving both camera shake and object motion~\cite{DeepDeblur,zhang2018dynamic,tao2018scale,MIMO-UNet,MPRNet}.
In this paper, we suggest to improve low light imaging by jointly leveraging the short-exposure noisy and long-exposure blurry images.
First, such setting is practically feasible.
For example, multiple cameras have been equipped in modern smartphones, which can be designed to acquire short-exposure and long-exposure images, respectively.
Moreover, one can also synthesize a pair of blurry and noisy images from a long burst of images captured by a camera.
Second, the noisy and blurry images convey complementary information, which is beneficial to improve restoration performance and makes self-supervised image restoration (SelfIR) possible.
We note that several methods~\cite{SIGGRAPH-2007,LSD2,Low-light-TMM,ks2022content} have been suggested to combine the blurry image with their noisy counterpart for better image restoration, yet it remains
uninvestigated under the self-supervised regime.
We further present a SelfIR model with blurry and noisy pairs.
Even though the blurry and noisy images are both \textit{disturbed}, the short-exposure images taken with high ISO are \textit{hardly blurry}, while the long-exposure images taken with low ISO are generally near \textit{noise-free}.
Thus, the long-exposure and short-exposure images can be used to provide some supervision information for each other.
On the one hand, the noisy image can serve as an alternative of sharp image to supervise deblurring with negligible performance degradation.
On the other hand, the static regions in long-exposure images are noise-free and sharp, which in turn can provide auxiliary supervision information for image denoising.
Taking these two aspects into account, we present a collaborative learning (co-learning) method termed SelfIR for deblurring and denoising, which is effective in leveraging the complementary information of long- and short-exposure images and can be learned in a self-supervised manner.
Extensive experiments on synthetic data are conducted to evaluate our SelfIR.
Both quantitative and qualitative results show that SelfIR outperforms the state-of-the-art self-supervised denoising methods, as well as the supervised denoising and deblurring counterparts.
To further verify the practicality of our SelfIR model, we have also collected a set of 61 real-world blurry and noisy pairs using smartphones.
Since there are no corresponding ground-truth images for calculating full-reference image quality assessment (IQA) metrics, we evaluate the restoration results using no-reference IQA metrics.
The results show that our method also performs favorably against the competing methods.
To sum up, the main contributions of this work include:
\begin{itemize}
\item We take a step forward in leveraging blurry and noisy image pairs for image restoration. Going beyond leveraging their complementarity in improving restoration performance, we show that it can also be utilized for self-supervised learning of the restoration model.
%
\item A self-supervised image restoration model (SelfIR) is proposed, where short-exposure images serve as supervision for the corresponding deblurring task, while the sharp regions in long-exposure images provide auxiliary supervision for self-supervised denoising.
%
\item Extensive experiments on both synthetic and real-world image pairs show that our SelfIR performs favorably against the state-of-the-art self-supervised denoising methods, as well as the baseline supervised deblurring and denoising methods.
\end{itemize}
\clearpage
\section{Related Work}\label{sec:related_work}
In this section, we briefly review burst image denoising and deblurring, as well as self-supervised image denoising and deblurring methods.
In addition, we recommend \cite{delbracio2021mobile} for a comprehensive introduction to the relevant mobile computational photography.
\noindent\textbf{Burst Image Denoising and Deblurring.}
In comparison with a single image, burst images can provide more information that is beneficial for image restoration.
Hasinoff~\emph{et al}\onedot~\cite{HDRplus} utilize an FFT-based alignment algorithm and a hybrid 2D/3D Wiener filter to denoise and merge a burst of underexposed frames for low-light photography.
KPN~\cite{KPN} predicts spatially variant kernels for every burst noisy image to merge them.
BPN~\cite{BPN} extends the KPN method with a basis prediction network and achieves larger denoising kernels under certain computing resource constraints.
Aittala~\emph{et al}\onedot~\cite{aittala2018burst} take both noise and blur into account, and restore sharp and noise-free images from burst images in an order-independent manner.
\noindent\textbf{Self-Supervised Image Denoising and Deblurring.}
Recently, self-supervised learning has drawn upsurging attention in low-level vision.
DIP~\cite{DIP} utilizes the image prior implicitly captured by the network structure to repair corrupted images.
SelfDeblur~\cite{SelfDeblur} respectively models the deep priors of clear image and blur kernel for self-supervised deblurring.
For these methods, the networks are required to re-train from scratch for each test image, which is less efficient, especially for mobile or edge devices.
Noise2Noise~\cite{Noise2Noise} demonstrates that noisy pairs with mutually independent noise can be used to train a denoising network, opening the door to self-supervised denoising.
Neighbor2Neighbor~\cite{Nei2Nei} utilizes a random neighbor sub-sampler to generate the training pairs from noisy images themselves.
In addition, some works~\cite{N2V,Laine19,DBSN,Blind2Unblind,AP-BSN} elaborately design blind-spot networks to avoid learning the identity mapping for self-supervised denoising.
However, the self-supervised denoising methods are limited in handling complex image noise.
In this work, we utilize the complementarity of long-exposure blurry and short-exposure noisy images for better self-supervised image restoration.
\section{Proposed Method}\label{sec:proposed_method}
In this section, we first show the feasibility of taking noisy images as the supervision of deblurring.
Then, we introduce the sharp area detection method in long-exposure images and auxiliary loss for self-supervised denoising.
Finally, we present the proposed co-learning framework SelfIR.
\subsection{Deblurring with Noisy Image}\label{sec:3.1}
When taking long-exposure photos, the shake of the camera and the motion of objects usually lead to a blurry image $\mathbf{I}_\mathcal{B}$, which can be formulated by,
\begin{equation}
\mathbf{I}_\mathcal{B} = \mathcal{K}(\mathbf{I})+\mathbf{N}_\mathcal{B},
\label{eqn:blur}
\end{equation}
where $\mathbf{I}$ is the latent clear image, $\mathcal{K}$ denotes the blur process with non-uniform kernels, $\mathbf{N}_\mathcal{B}$ represents the low-intensity noise.
Existing supervised deblurring methods generally utilize a deep neural network (denoted by $\mathcal{D_B}$) to estimate $\mathbf{I}$ from $\mathbf{I}_\mathcal{B}$.
For training the parameters of $\mathcal{D_B}$, which is denoted by $\Theta_\mathcal{B}$, the optimization objective can be defined by,
\begin{equation}
\Theta_\mathcal{B}^\ast = \arg\min_{\Theta_\mathcal{B}} \mathbb{E}_{\mathbf{I}_\mathcal{B},\mathbf{I}} \left[\mathcal{L}\left(\mathcal{D_B}(\mathbf{I}_\mathcal{B}; \Theta_\mathcal{B}), \mathbf{I} \right)\right],
\label{eqn:deblur}
\end{equation}
where $\mathcal{L}$ denotes the loss functions for supervised learning.
However, collecting clear images is troublesome in real-world scenes.
Inspired by Noise2Noise~\cite{Noise2Noise}, we show that the noisy short-exposure image can be a substitution of the latent clear image to supervise the task of deblurring.
When taking short-exposure photos under low light environment, the limited photon amount and inherent defects of camera ISP make the images noisy (denoted by $\mathbf{I}_\mathcal{N}$), which can be formulated by,
\begin{equation}
\mathbf{I}_\mathcal{N} = \mathbf{I}+\mathbf{N}_\mathcal{N},
\label{eqn:noise}
\end{equation}
where $\mathbf{N}_\mathcal{N}$ represents the noise (with much higher intensity than $\mathbf{N}_\mathcal{B}$).
When using the noisy image $\mathbf{I}_\mathcal{N}$ as the supervision of deblurring, the optimization of $\Theta_\mathcal{B}$ can be expressed as,
\begin{equation}
\small
\Theta_\mathcal{B}^\ast = \arg\min_{\Theta_\mathcal{B}} \mathbb{E}_{\mathbf{I}_\mathcal{B},\mathbf{I}_\mathcal{N}} \left[\mathcal{L}\left(\mathcal{D_B}(\mathbf{I}_\mathcal{B}; \Theta_\mathcal{B}), \mathbf{I}_\mathcal{N} \right)\right] = \arg\min_{\Theta_\mathcal{B}} \mathbb{E}_{\mathbf{I}_\mathcal{B}} \left[\mathbb{E}_{\mathbf{I}_\mathcal{N}|\mathbf{I}_\mathcal{B}} \left[\mathcal{L}\left(\mathcal{D_B}(\mathbf{I}_\mathcal{B}; \Theta_\mathcal{B}), \mathbf{I}_\mathcal{N} \right)\right]\right].
\label{eqn:deblur_un}
\end{equation}
Suppose that the loss function $\mathcal{L}$ in \cref{eqn:deblur_un} is the $\ell_2$ loss, we have,
\begin{equation}
\begin{split}
\mathbb{E}_{\mathbf{I}_\mathcal{N}|\mathbf{I}_\mathcal{B}} \left[\mathcal{L}\left(\mathcal{D_B}(\mathbf{I}_\mathcal{B}; \Theta_\mathcal{B}), \mathbf{I}_\mathcal{N} \right)\right]
& = \mathbb{E}_{\mathbf{I}_\mathcal{N}|\mathbf{I}_\mathcal{B}} \left[\| \mathcal{D_B}(\mathbf{I}_\mathcal{B}; \Theta_\mathcal{B}) - \mathbf{I}_\mathcal{N} \|_2^2 \right] \\
& = \mathbb{E}_{\mathbf{I}, \mathbf{N}_\mathcal{N}|\mathbf{I}_\mathcal{B}} \left[\| \mathcal{D_B}(\mathbf{I}_\mathcal{B}; \Theta_\mathcal{B}) - (\mathbf{I} + \mathbf{N}_\mathcal{N}) \|_2^2 \right] \\
& = \mathbb{E}_{\mathbf{I}|\mathbf{I}_\mathcal{B}} \left[\| \mathcal{D_B}(\mathbf{I}_\mathcal{B}; \Theta_\mathcal{B}) - \mathbf{I} \|_2^2 \right] - \\
& \quad \ \, 2 \mathbb{E}_{\mathbf{I}, \mathbf{N}_\mathcal{N}|\mathbf{I}_\mathcal{B}} \left[(\mathcal{D_B}(\mathbf{I}_\mathcal{B}; \Theta_\mathcal{B}) - \mathbf{I})^\top \mathbf{N}_\mathcal{N}\right] + \\
& \quad \ \, \mathbb{E}_{\mathbf{N}_\mathcal{N}|\mathbf{I}_\mathcal{B}} \left[\| \mathbf{N}_\mathcal{N} \|_2^2 \right],
\end{split}
\label{eqn:deblur_un1}
\end{equation}
where $\mathbb{E}_{\mathbf{N}_\mathcal{N}|\mathbf{I}_\mathcal{B}} \left[\| \mathbf{N}_\mathcal{N} \|_2^2\right]$ can be regarded as a constant and be safely discarded from \cref{eqn:deblur_un1}.
Further, assume that $\mathbf{N}_\mathcal{N}$ is zero-mean, $\mathbf{N}_\mathcal{N}$ and $\mathbf{I}$ are independent, we can get,
\begin{equation}
\mathbb{E}_{\mathbf{I}, \mathbf{N}_\mathcal{N}|\mathbf{I}_\mathcal{B}} \left[(\mathcal{D_B}(\mathbf{I}_\mathcal{B}; \Theta_\mathcal{B}) - \mathbf{I})^\top \mathbf{N}_\mathcal{N}\right]=0 .
\end{equation}
In this case, the optimal solution $\Theta_\mathcal{B}^\ast$ in \cref{eqn:deblur_un} and that in \cref{eqn:deblur} are the same.
Thus, it is feasible to utilize noisy short-exposure images instead of clear ones as the supervision of deblurring.
\subsection{Denoising with Long-Exposure Image}\label{sec:3.2}
{\parfillskip0pt\par}
\begin{wrapfigure}{R}{0.47\textwidth}
\centering
\includegraphics[width=0.97\linewidth]{blur_image.pdf}
\caption{A blurry image example where the blur is non-uniform in (a). It includes some common blurry areas (b), severe blurry areas with ringing artifacts (c), and some approximately sharp areas (d).}
\label{fig:blurry}
\vspace{-2mm}
\end{wrapfigure}
Self-supervised denoising makes it possible to remove the noise without clean image supervision, but may give rise to obvious performance degradation, especially when handling complex real-world noises.
Here we propose to alleviate this problem by introducing some extra supervision from the long-exposure counterpart.
Obviously, taking the whole long-exposure image as supervision will bring adverse effects, making the results to be blurry.
Nonetheless, it is worth noting that, the blur process $\mathcal{K}$ in \cref{eqn:blur} is generally non-uniform, and sometimes not all areas are blurry.
As shown in \cref{fig:blurry}(d), there exist some approximately sharp regions in the long-exposure image, which can provide partial supervision information that benefits self-supervised denoising.
Therefore, it is crucial to pinpoint the sharp areas in the long-exposure image.
Otherwise, we would prefer to go without sharp areas than accept a shoddy option, as misjudgments of sharp areas will lead to worse denoising results.
However, without any discriminative clues, it is very likely to misjudge the sharp areas.
Considering that $\mathbf{I}_\mathcal{N}$ is nearly non-blurry due to the short exposure time, it may be a reference for sharp area detection.
To avoid noise interference, we pre-process $\mathbf{I}_\mathcal{N}$ with a self-supervised denoising model, and take the result $\mathbf{\tilde{I}}_\mathcal{N}$ to help detect sharp regions in the corresponding blurry image $\mathbf{I}_\mathcal{B}$.
Specifically, we first divide $\mathbf{I}_\mathcal{B}$ and $\mathbf{\tilde{I}}_\mathcal{N}$ into $\mathit{N}$ non-overlapping patches.
For each patch pair $\mathbf{I}_\mathcal{B}^\mathit{n}$ and $\mathbf{\tilde{I}}_\mathcal{N}^\mathit{n}$ ($ 1 \leq \mathit{n} \leq \mathit{N} $), our goal is to obtain a mask $\mathit{m}^\mathit{n} \in \{0,1\}$ that indicates whether $\mathbf{I}_\mathcal{B}^\mathit{n}$ is a sharp patch.
Since $\mathbf{\tilde{I}}_\mathcal{N}$ is nearly non-blurry, when some severe motion blurs exist in $\mathbf{I}_\mathcal{B}^\mathit{n}$, the difference between $\mathbf{I}_\mathcal{B}^\mathit{n}$ and $\mathbf{\tilde{I}}_\mathcal{N}^\mathit{n}$ in textures and edges should be evident.
Taking the above into account, we adopt a similarity metric $\mathit{s}$ to detect the areas with severe motion blurs, \emph{i.e}\onedot} \def\Ie{\emph{I.e}\onedot,
\begin{equation}
\mathit{m}^\mathit{n} = \mathtt{sgn}(\mathtt{max}(0, \mathit{s}(\mathbf{I}_\mathcal{B}^\mathit{n},\mathbf{\tilde{I}}_\mathcal{N}^\mathit{n}) - \epsilon_\mathit{s})),
\label{mask_1}
\end{equation}
where structural similarity (SSIM)~\cite{SSIM} is utilized for the similarity metric $\mathit{s}$, $\epsilon_s$ denotes the threshold, while $\mathtt{max}(a,b)$ and $\mathtt{sgn}(\cdot)$ denote the maximum and sign function, respectively.
However, the initial denoising result $\mathbf{\tilde{I}}_\mathcal{N}$ may be over-smooth, in other words, \cref{mask_1} may fail when facing some mildly blurred regions in $\mathbf{I}_\mathcal{B}^\mathit{n}$.
Therefore, we further measure the difference in variance between $\mathbf{I}_\mathcal{B}^\mathit{n}$ and $\mathbf{\tilde{I}}_\mathcal{N}^\mathit{n}$.
When the variance of $\mathbf{I}_\mathcal{B}^\mathit{n}$ is greater than that of $\mathbf{\tilde{I}}_\mathcal{N}^\mathit{n}$, we consider that $\mathbf{I}_\mathcal{B}^\mathit{n}$ is potential to be a sharp patch.
It should be noted that the difference in variance is not suitable for detecting some severely blurry areas with ringing artifacts (see \cref{fig:blurry}(c)).
In such blurry regions, the variance of $\mathbf{I}_\mathcal{B}^\mathit{n}$ may also be greater than that of $\mathbf{\tilde{I}}_\mathcal{N}^\mathit{n}$.
When synthesizing blurry images, some works~\cite{nah2019ntire,wieschollek2017learning,zhou2019davanet} remove the artifacts by interpolating the frames before averaging the sharp images.
However, the artifacts also exist in real-world blurry images, especially in areas with flickering lights.
Therefore, we still take the ringing artifacts into consideration in this work.
As a result, we jointly use the SSIM and variance measure for judging sharp regions, $\mathit{m}^\mathit{n}$ can be formulated as,
\begin{equation}
\mathit{m}^\mathit{n} = \mathtt{sgn}(\mathtt{max}(0, \mathit{s}(\mathbf{I}_\mathcal{B}^\mathit{n},\mathbf{\tilde{I}}_\mathcal{N}^\mathit{n}) - \epsilon_\mathit{s})) * \mathtt{sgn}(\mathtt{max}(0, \mathtt{var}(\mathbf{I}_\mathcal{B}^\mathit{n}) - \mathtt{var}(\mathbf{\tilde{I}}_\mathcal{N}^\mathit{n}) - \epsilon_\mathit{v})),
\label{mask_2}
\end{equation}
where $\mathtt{var}( \cdot ) $ and $\epsilon_\mathit{v}$ denote the variance function and the threshold, respectively.
$\epsilon_s$ and $\epsilon_v$ are set to 0.99 and 1e-5 when color intensity values of $\mathbf{I}_\mathcal{B}^\mathit{n}$ and $\mathbf{\tilde{I}}_\mathcal{N}^\mathit{n}$ are normalized to $[0,1]$.
Defining the final output image as $\mathbf{\hat{I}}$, the auxiliary loss function for denoising can be denoted as,
\begin{equation}
\mathcal{L}_\mathit{aux}(\mathbf{\hat{I}}, \mathbf{I}_\mathcal{B}) = {\sum}_{\mathit{n}=1}^\mathit{N} \mathit{m}^\mathit{n} \| \mathbf{\hat{I}}^\mathit{n} - \mathbf{I}_\mathcal{B}^\mathit{n} \|_2^2 .
\label{loss:aux}
\end{equation}
\subsection{Co-Learning of Deblurring and Denoising}\label{sec:3.3}
\begin{figure}[t]
\centering
\begin{overpic}[width=0.93\linewidth]{overview.pdf}
\put(21.5,57.1){\scriptsize$\mathit{g}_1(\cdot)$}
\put(23.3,46.8){\scriptsize$\mathit{g}_1(\cdot)$}
\put(23.3,36.5){\scriptsize$\mathit{g}_2(\cdot)$}
\put(35.3,20.6){\scriptsize$\mathit{g}_1(\cdot)$}
\put(35.3,9.9){\scriptsize$\mathit{g}_2(\cdot)$}
\put(14.3,47.8){\scriptsize($\mathbf{I}_\mathcal{B}$)}
\put(14,31.8){\scriptsize($\mathbf{I}_\mathcal{N}$)}
\put(30,50.9){\scriptsize$\mathit{g}_1(\mathbf{I}_\mathcal{B})$}
\put(30,40.5){\scriptsize$\mathit{g}_1(\mathbf{I}_\mathcal{N})$}
\put(30,30.2){\scriptsize$\mathit{g}_2(\mathbf{I}_\mathcal{N})$}
\put(44.8,62.8){\scriptsize$\mathcal{L}_\mathit{aux}$}
\put(44.2,60.8){\scriptsize\cref{loss:aux}}
\put(42.4,35.8){\scriptsize$\mathcal{L}_\mathit{rec}\ \&\ \mathcal{L}_\mathit{reg}$}
\put(40.8,33.6){\scriptsize\cref{loss:rec,loss:reg}}
\put(50.6,49.0){\scriptsize$\mathcal{D}$}
\put(86.5,31.6){\scriptsize$\mathcal{D}$}
\end{overpic}
\vspace{-1mm}
\caption{Overview of our proposed SelfIR framework.
(a) Training phase of SelfIR. Sub-sampled blurry image $\mathit{g}_1(\mathbf{I}_\mathcal{B})$ and noisy image $\mathit{g}_1(\mathbf{I}_\mathcal{N})$ are taken as the inputs. $\mathit{g}_2(\mathbf{I}_\mathcal{N})$ is used for calculating the reconstruction loss $\mathcal{L}_\mathit{rec}$ (see \cref{loss:rec}) and regularization loss $\mathcal{L}_\mathit{reg}$ (see \cref{loss:reg}), while $\mathit{g}_1(\mathbf{I}_\mathcal{B})$ is taken for calculating auxiliary loss (see \cref{loss:aux}).
(b) Example of neighbor sub-sampler. In each $2\times2$ cell, two pixels are randomly selected for respectively composing the neighboring sub-images.
(c) Testing phase of SelfIR. The blurry and noisy images can be directly taken for restoration.}
\label{fig:framework}
\vspace{-2mm}
\end{figure}
As illustrated in \cref{sec:3.1,sec:3.2}, the noisy images can provide effective supervision information for deblurring, while the blurry images can also provide auxiliary supervision for self-supervised denoising.
In other words, we can learn to deblur or denoise without additional clear and clean ground-truths.
However, learning them independently will lead to limited performance.
As some works~\cite{SIGGRAPH-2007,LSD2,Low-light-TMM,ks2022content} have suggested, better restoration performance can be obtained by incorporating blurry with noisy images.
Thus, it should be further considered to achieve self-supervised image restoration that takes two disturbed images as input.
In this section, we delicately design the self-supervised model SelfIR that learns from both deblurring and denoising tasks in a collaborative learning manner.
For training SelfIR, it is required to avoid the trivial solution when taking both blurry and noisy images as the input.
For example, when taking noisy images for supervising deblurring, the model may simply output the noisy images instead of learning the deburring results.
Fortunately, Neighbor2Neighbor~\cite{Nei2Nei} shows the noises in two sub-sampled images (\emph{i.e}\onedot} \def\Ie{\emph{I.e}\onedot, $\mathit{g}_1(\mathbf{I}_\mathcal{N})$ and $\mathit{g}_2(\mathbf{I}_\mathcal{N})$, see \cref{fig:framework}(b)) from noisy image $\mathbf{I}_\mathcal{N}$ are almost independent.
Therefore, they can be respectively taken as the input and target for training the denoising subtask in a self-supervised manner.
Simultaneously, combining the derivation in \cref{eqn:deblur_un1}, $\mathit{g}_1(\mathbf{I}_\mathcal{B})$ and $\mathit{g}_2(\mathbf{I}_\mathcal{N})$ can be respectively taken as the input and target to train the deblurring subtask.
Therefore, our SelfIR can take a step forward and deliver both $\mathit{g}_1(\mathbf{I}_\mathcal{B})$ and $\mathit{g}_1(\mathbf{I}_\mathcal{N})$ ($i.e.$, $\{\mathit{g}_1(\mathbf{I}_\mathcal{B}, \mathit{g}_1(\mathbf{I}_\mathcal{N})\}$) into the restoration network $\mathcal{D}$, as shown in \cref{fig:framework}(a).
The sub-sampled image $\mathit{g}_2(\mathbf{I}_\mathcal{N})$ is taken for calculating reconstruction loss $\mathcal{L}_\mathit{rec}$, which can be written as,
\begin{equation}
\mathcal{L}_\mathit{rec} = \| \mathcal{D}(\mathit{g}_1(\mathbf{I}_\mathcal{B}), \mathit{g}_1(\mathbf{I}_\mathcal{N})) - \mathit{g}_2(\mathbf{I}_\mathcal{N}) \|_2^2.
\label{loss:rec}
\end{equation}
We also calculate regularization loss $\mathcal{L}_\mathit{reg}$ similar to \cite{Nei2Nei},
\begin{equation}
\mathcal{L}_\mathit{reg} = \| \mathcal{D}(\mathit{g}_1(\mathbf{I}_\mathcal{B}), \mathit{g}_1(\mathbf{I}_\mathcal{N})) - \mathit{g}_2(\mathbf{I}_\mathcal{N}) - ( \mathit{g}_1(\hat{\mathcal{D}}(\mathbf{I}_\mathcal{B}, \mathbf{I}_\mathcal{N})) - \mathit{g}_2(\hat{\mathcal{D}}(\mathbf{I}_\mathcal{B}, \mathbf{I}_\mathcal{N})) ) \|_2^2,
\label{loss:reg}
\end{equation}
where $\hat{\mathcal{D}}$ has same parameters with $\mathcal{D}$ but has no gradient for back-propagation.
Moreover, the sharp areas in $\mathit{g}_1(\mathbf{I}_\mathcal{B})$ can provide auxiliary supervision for the model learning and the performance can be further improved.
Specifically, we calculate auxiliary loss $\mathcal{L}_\mathit{aux}(\mathcal{D}(\mathit{g}_1(\mathbf{I}_\mathcal{B}), \mathit{g}_1(\mathbf{I}_\mathcal{N}))), \mathit{g}_1(\mathbf{I}_\mathcal{B}))$ in \cref{loss:aux}.
For obtaining the mask $\mathit{m}^\mathit{n}$, we replace $\mathbf{I}_\mathcal{B}$ and $\mathbf{\tilde{I}}_\mathcal{N}$ in~\cref{mask_2} with $\mathit{g}_1(\mathbf{I}_\mathcal{B})$ and $\hat{\mathcal{D}}(\mathit{g}_1(\mathbf{I}_\mathcal{B}), \mathit{g}_1(\mathbf{I}_\mathcal{N}))$, respectively.
Finally, the learning objective of parameters $\Theta_\mathcal{D}$ with the co-learning manner can be formulated as,
\begin{equation}
\Theta_\mathcal{D}^\ast = \argmin_{\Theta_\mathcal{D}} \mathbb{E}_{\mathbf{I}_\mathcal{B},\mathbf{I}_\mathcal{N}} ( \mathcal{L}_\mathit{rec} + \lambda_\mathit{reg} \mathcal{L}_\mathit{reg} + \lambda_\mathit{aux} \mathcal{L}_\mathit{aux}),
\label{eqn:co-learning}
\end{equation}
where $\lambda_\mathit{reg}$ and $\lambda_\mathit{aux}$ are hyper-parameters for balancing the loss terms.
\section{Experiments}\label{sec:experiments}
In this section, we first describe the dataset configuration and the training / evaluation protocols in detail.
Then, both the quantitative and qualitative results on synthetic and real-world images are given for comprehensive evaluation.
\subsection{Implementation Details}\label{sec:training_details}
\noindent\textbf{Synthetic Datasets.}
For synthesizing blurry images, early methods~\cite{SIGGRAPH-2007,LSD2} convolve sharp images with simulated blur kernels, which is quite different from the real-world blur model.
Recently, the GoPro dataset\footnote{\url{https://seungjunnah.github.io/Datasets/gopro.html}}~\cite{DeepDeblur} offers a more realistic way to synthesize blurry images, which has been widely adopted for motion deblurring tasks~\cite{DeepDeblur,zhang2018dynamic,tao2018scale,MIMO-UNet,MPRNet}.
In the dataset, although slight inherent noise exists in the sharp video frames that are captured by a high-speed camera (\emph{i.e}\onedot} \def\Ie{\emph{I.e}\onedot, GoPro Hero4 Black), it has little effect on model training and evaluation.
We directly regard the sharp images as clean images.
The blurry image is generated by averaging consecutive sharp frames, which can better simulate both camera shake and object motion.
Thus, we can get the blurry-clear pairs as $\mathbf{I}_\mathcal{B}$ and $\mathbf{I}$, respectively.
For synthesizing the noisy image $\mathbf{I}_\mathcal{N}$, we consider three noise distributions: 1)~Gaussian noise with $\sigma\!\in\![5/255, 50/255]$, 2)~Poisson noise with $\lambda\!\in\![5, 50]$, and 3)~sensor noise~\cite{UPI}.
Please refer to the supplementary material for the noise formulations.
Experiments are conducted in both sRGB and raw-RGB spaces.
For the sRGB space, we add Gaussian or Poisson noise to the sRGB sharp images.
While for the raw-RGB space, the more complex sensor noise is added to the sharp raw-RGB images, and the blurry images are also converted back to the raw-RGB space through the unprocessing~\cite{UPI} pipeline.
Finally, there are 2,103 image pairs for training, and we use the remaining 1,111 pairs for testing.
\noindent\textbf{Real-world Datasets.}
Furthermore, we have also collected 61 real-world blurry and noisy raw-RGB pairs with Huawei P40 smartphones in the professional camera mode.
In particular, the blurry images are captured with a low ISO and long exposure time, while the corresponding noisy images are captured with a high ISO and short exposure time in the same scene.
Please refer to the supplementary material for detailed collection process.
Due to the limited amount of real-world images, we use 30 pairs to fine-tune the model pre-trained on raw-RGB images with synthetic sensor noise, and take the remaining 31 pairs for evaluation.
\noindent\textbf{Training Details.}
Following~\cite{Noise2Noise,Laine19,Nei2Nei}, we adopt a U-Net~\cite{UNet} architecture as our restoration network, where two encoders are respectively deployed to the blurry and noisy input images for better domain-specific feature extraction, and then blurry and noisy features are fused in the decoder part.
Detailed architecture is given in the supplementary material.
During training, the batch size is set to 16 and the patch size is $128 \times 128$.
Adam optimizer~\cite{Adam} with $\beta_1=0.9$ and $\beta_2=0.999$ is used to train the network for 200 epochs.
The learning rate is initially set to $3\times10^{-4}$ for synthetic experiments and $1\times10^{-4}$ for real-world experiments.
And it reduces by half every 50 epochs.
For the hyper-parameters in~\cref{eqn:co-learning}, $\lambda_\mathit{aux}$ is set to 2, $\lambda_\mathit{reg}$ is set to 2 and 4 for experiments in sRGB space and raw-RGB space, respectively.
All experiments are conducted with PyTorch~\cite{PyTorch} on an Nvidia GeForce RTX 2080Ti GPU.
\noindent\textbf{Evaluation Configurations.}
We convert all raw-RGB results to sRGB space through post-processing pipeline, and the quantitative metrics are computed in the sRGB space.
For the results of synthetic experiments, we take peak signal to noise ratio (PSNR), structural similarity (SSIM)~\cite{SSIM} and learned perceptual image patch similarity (LPIPS)~\cite{LPIPS} as evaluation metrics.
For the results of real-world experiments, due to the lack of ground-truth images, we utilize no-reference IQA metrics (\emph{i.e}\onedot} \def\Ie{\emph{I.e}\onedot, NIQE~\cite{NIQE}, NRQM~\cite{NRQM}, and PI~\cite{PI}) to evaluate the generated images.
\begin{table}[t]
\small
\renewcommand\arraystretch{1
\begin{center}
\caption{Quantitative results on synthetic sRGB images.}
\label{table:sRGB}
\vspace{2mm}
\scalebox{1}{
\begin{tabular}{cccccc}
\toprule
& Method
& \tabincell{c}{Gaussian $\sigma \in [5/255,50/255]$ \\ \footnotesize{PSNR}\scriptsize{$\uparrow$} \small{/} \footnotesize{SSIM}\scriptsize{$\uparrow$} \small{/} \footnotesize{LPIPS}\scriptsize{$\downarrow$}}
& \tabincell{c}{Poisson $\lambda \in [5,50]$ \\ \footnotesize{PSNR}\scriptsize{$\uparrow$} \small{/} \footnotesize{SSIM}\scriptsize{$\uparrow$} \small{/} \footnotesize{LPIPS}\scriptsize{$\downarrow$}} \\
\midrule
\multirow{2}{*}{\begin{tabular}[c]{@{}c@{}}Supervised\\Deblurring \end{tabular}}
& Baseline$_\mathcal{B}$ & \multicolumn{2}{c}{28.24 / 0.8561 / 0.191} \\
& DeepDeblur~\cite{DeepDeblur} & \multicolumn{2}{c}{30.04 / 0.9015 / 0.133} \\
\midrule
\multirow{2}{*}{\begin{tabular}[c]{@{}c@{}}Supervised\\Denoising \end{tabular}}
& Baseline$_\mathcal{N}$ & 34.91 / 0.9360 / 0.098 & 33.15 / 0.9225 / 0.126 \\
& DnCNN~\cite{DnCNN} & 34.63 / 0.9308 / 0.121 & 32.45 / 0.9084 / 0.128 \\
\midrule
Supervised IR & Baseline$_\mathcal{R}$ & 36.15 / 0.9534 / 0.070 & 34.74 / 0.9454 / 0.084 \\
\midrule
\multirow{8}{*}{\begin{tabular}[c]{@{}c@{}}Self-Supervised\\Denoising \end{tabular}}
& N2N~\cite{Noise2Noise} & 34.88 / 0.9354 / 0.100 & 33.09 / 0.9216 / 0.129 \\
& N2V~\cite{N2V} & 33.09 / 0.9180 / 0.115 & 31.81 / 0.8999 / 0.137 \\
& Laine19-mu~\cite{Laine19} & 33.61 / 0.9227 / 0.104 & 32.29 / 0.9091 / 0.131 \\
& Laine19-pme~\cite{Laine19} & 34.76 / 0.9322 / 0.086 & 32.77 / 0.9147 / 0.116 \\
& DBSN~\cite{DBSN} & 33.72 / 0.9224 / 0.111 & 31.46 / 0.8883 / 0.144 \\
& R2R~\cite{R2R} & 33.74 / 0.9223 / 0.100 & 30.05 / 0.7649 / 0.230 \\
& Neighbor2Neighbor~\cite{Nei2Nei} & 34.29 / 0.9271 / 0.085 & 32.68 / 0.9160 / 0.111 \\
& Blind2Unblind~\cite{Blind2Unblind} & 34.69 / 0.9353 / 0.107 & 33.09 / 0.9216 / 0.132 \\
\midrule
\multirow{1}{*}{Ours}
& SelfIR & 35.74 / 0.9499 / 0.076 & 34.27 / 0.9404 / 0.092 \\
\bottomrule
\end{tabular}}
\end{center}
\end{table}
\subsection{Experimental Results in sRGB Space}
To assess our proposed SelfIR, we build several baseline methods for comparison, including 1)~a supervised deblurring baseline (denoted by Baseline$_\mathcal{B}$), 2)~a supervised denoising baseline (denoted by Baseline$_\mathcal{N}$), 3)~a supervised image restoration baseline which takes both blurry and noisy images as input (denoted by Baseline$_\mathcal{R}$), and 4)~Neighbor2Neighbor~\cite{Nei2Nei} since the denoising part of our co-learning framework is based on it.
Note that the network structure is not the focus of this paper, so we deploy a simple U-Net~\cite{UNet} architecture for our restoration network $\mathcal{D}$ and the first three baseline methods.
Besides, we choose two classical supervised methods (\emph{i.e}\onedot} \def\Ie{\emph{I.e}\onedot, DeepDeblur~\cite{DeepDeblur} and DnCNN~\cite{DnCNN}) for comparison.
Self-supervised denoising methods (\emph{e.g}\onedot} \def\Eg{\emph{E.g}\onedot, N2N~\cite{Noise2Noise}, N2V~\cite{N2V}, Laine19~\cite{Laine19}, DBSN~\cite{DBSN}, R2R~\cite{R2R}, and Blind2Unblind~\cite{Blind2Unblind}) are also compared.
For DeepDeblur~\cite{DeepDeblur}, we use the officially released model for testing.
For other methods, the models are retrained with our synthetic images for a fair comparison.
The quantitative results of synthetic experiments on Gaussian and Poisson noise are given in \cref{table:sRGB}.
One can see that our SelfIR outperforms all other methods except the supervised IR model (Baseline$_\mathcal{R}$), which can be seen as the upper limit of the restoration performance with the utilized U-Net architecture and the blurry-noisy pair input.
In comparison to the self-supervised denoising baseline Neighbor2Neighbor~\cite{Nei2Nei}, 1.45 dB and 1.59 dB PSNR gains are obtained by our SelfIR on Gaussian and Poisson noise, respectively.
Even compared to the supervised denoising baseline model (Baseline$_\mathcal{N}$), our method also improves the PSNR index by 0.83 dB and 1.12 dB.
Due to the severely ill-posed nature of deblurring, SelfIR brings a PSNR gain by more than 6 dB comparing to the supervised deblurring baseline model (Baseline$_\mathcal{B}$).
The results clearly show that the blurry images can indeed provide beneficial features for denoising, and so do the noisy images for deblurring.
The qualitative results are shown in \cref{fig:gau1,fig:poi1}.
One can see that our results are sharper and clearer than other denoising or deblurring methods.
In terms of visual quality, the proposed SelfIR is almost comparable to the supervised image restoration model (Baseline$_\mathcal{R}$), and combines the advantages of supervised denoising Baseline$_\mathcal{N}$ and deblurring Baseline$_\mathcal{B}$.
More results can be found in the supplemental material.
\begin{figure}[t]
\vspace{-2mm}
\begin{minipage}[t]{\linewidth}
\centering
\tiny
\subfloat[\scriptsize Noisy Image]
{
\includegraphics[height=.15\linewidth, width=.15\linewidth]{img/gau/384_11_00_000009_noise.png}
}
\hfill
\subfloat[\scriptsize Baseline$_\mathcal{N}$]
{
\includegraphics[height=.15\linewidth, width=.15\linewidth]{img/gau/384_11_00_000009_denoise.png}
}
\hfill
\subfloat[\scriptsize N2V~\cite{N2V}]
{
\includegraphics[height=.15\linewidth, width=.15\linewidth]{img/gau/384_11_00_000009_n2v.png}
}
\hfill
\subfloat[\scriptsize Laine19-pme~\cite{Laine19}]
{
\includegraphics[height=.15\linewidth, width=.15\linewidth]{img/gau/384_11_00_000009_pme.png}
}
\hfill
\subfloat[\scriptsize DBSN~\cite{DBSN}]
{
\includegraphics[height=.15\linewidth, width=.15\linewidth]{img/gau/384_11_00_000009_dbsn.png}
}
\hfill
\subfloat[\scriptsize R2R~\cite{R2R}]
{
\includegraphics[height=.15\linewidth, width=.15\linewidth]{img/gau/384_11_00_000009_r2r.png}
}
\vspace{-2mm}
\subfloat[\scriptsize Blurry Image]
{
\includegraphics[height=.15\linewidth, width=.15\linewidth]{img/gau/384_11_00_000009_blur.png}
}
\hfill
\subfloat[\scriptsize Baseline$_\mathcal{B}$]
{
\includegraphics[height=.15\linewidth, width=.15\linewidth]{img/gau/384_11_00_000009_deblur.png}
}
\hfill
\subfloat[\tiny Neighbor2Neighbor~\cite{Nei2Nei}]
{
\includegraphics[height=.15\linewidth, width=.15\linewidth]{img/gau/384_11_00_000009_nei2nei.png}
}
\hfill
\subfloat[\scriptsize Blind2Unblind~\cite{Blind2Unblind}]
{
\includegraphics[height=.15\linewidth, width=.15\linewidth]{img/gau/384_11_00_000009_b2u.png}
}
\hfill
\subfloat[\scriptsize SelfIR (Ours)]
{
\includegraphics[height=.15\linewidth, width=.15\linewidth]{img/gau/384_11_00_000009_ours.png}
}
\hfill
\subfloat[\scriptsize Baseline$_\mathcal{R}$]
{
\includegraphics[height=.15\linewidth, width=.15\linewidth]{img/gau/384_11_00_000009_full.png}
}
\end{minipage}
\vspace{-1mm}
\caption{Visual comparison on Gaussian noise. The texts in our result are sharper and clearer. The proposed SelfIR is almost comparable to supervised Baseline$_\mathcal{R}$, which can be seen as the upper limit of the restoration performance with the utilized U-Net architecture and the blurry-noisy pair input.}
\label{fig:gau1}
\vspace{-3mm}
\end{figure}
\begin{figure}[t]
\vspace{-2mm}
\begin{minipage}[t]{\linewidth}
\centering
\tiny
\subfloat[\scriptsize Noisy Image]
{
\includegraphics[height=.15\linewidth, width=.15\linewidth]{img/poi/410_11_00_000137_noise.png}
}
\hfill
\subfloat[\scriptsize Baseline$_\mathcal{N}$]
{
\includegraphics[height=.15\linewidth, width=.15\linewidth]{img/poi/410_11_00_000137_denoise.png}
}
\hfill
\subfloat[\scriptsize N2V~\cite{N2V}]
{
\includegraphics[height=.15\linewidth, width=.15\linewidth]{img/poi/410_11_00_000137_n2v.png}
}
\hfill
\subfloat[\scriptsize Laine19-pme~\cite{Laine19}]
{
\includegraphics[height=.15\linewidth, width=.15\linewidth]{img/poi/410_11_00_000137_pme.png}
}
\hfill
\subfloat[\scriptsize DBSN~\cite{DBSN}]
{
\includegraphics[height=.15\linewidth, width=.15\linewidth]{img/poi/410_11_00_000137_dbsn.png}
}
\hfill
\subfloat[\scriptsize R2R~\cite{R2R}]
{
\includegraphics[height=.15\linewidth, width=.15\linewidth]{img/poi/410_11_00_000137_r2r.png}
}
\vspace{-2mm}
\subfloat[\scriptsize Blurry Image]
{
\includegraphics[height=.15\linewidth, width=.15\linewidth]{img/poi/410_11_00_000137_blur.png}
}
\hfill
\subfloat[\scriptsize Baseline$_\mathcal{B}$]
{
\includegraphics[height=.15\linewidth, width=.15\linewidth]{img/poi/410_11_00_000137_deblur.png}
}
\subfloat[\tiny Neighbor2Neighbor~\cite{Nei2Nei}]
{
\includegraphics[height=.15\linewidth, width=.15\linewidth]{img/poi/410_11_00_000137_nei2nei.png}
}
\hfill
\subfloat[\scriptsize Blind2Unblind~\cite{Blind2Unblind}]
{
\includegraphics[height=.15\linewidth, width=.15\linewidth]{img/poi/410_11_00_000137_b2u.png}
}
\hfill
\subfloat[\scriptsize SelfIR (Ours)]
{
\includegraphics[height=.15\linewidth, width=.15\linewidth]{img/poi/410_11_00_000137_ours.png}
}
\hfill
\subfloat[\scriptsize Baseline$_\mathcal{R}$]
{
\includegraphics[height=.15\linewidth, width=.15\linewidth]{img/poi/410_11_00_000137_full.png}
}
\end{minipage}
\vspace{-1mm}
\caption{Visual comparison on Poisson noise. In terms of the visual result, SelfIR combines the advantages of supervised denoising Baseline$_\mathcal{N}$ and deblurring Baseline$_\mathcal{B}$.}
\label{fig:poi1}
\vspace{-3mm}
\end{figure}
\begin{table}[t]
\small
\renewcommand\arraystretch{1
\begin{center}
\caption{Quantitative results on synthetic and real-world raw-RGB images.}
\label{table:raw}
\vspace{2mm}
\scalebox{1}{
\begin{tabular}{cccccc}
\toprule
& Method
& \tabincell{c}{Sensor Noise~\cite{UPI} \\ \footnotesize{PSNR}\scriptsize{$\uparrow$} \small{/} \footnotesize{SSIM}\scriptsize{$\uparrow$} \small{/} \footnotesize{LPIPS}\scriptsize{$\downarrow$}}
& \tabincell{c}{Real-World Images \\ \footnotesize{NIQE}\scriptsize{$\downarrow$} \small{/} \footnotesize{NRQM}\scriptsize{$\uparrow$} \small{/} \footnotesize{PI}\scriptsize{$\downarrow$}} \\
\midrule
\multirow{2}{*}{\begin{tabular}[c]{@{}c@{}}Supervised\\Deblurring \end{tabular}}
& Baseline$_\mathcal{B}$ & 28.14 / 0.8547 / 0.162 & 6.26 / 5.04 / 5.62 \\
& DeepDeblur~\cite{DeepDeblur} & 29.75 / 0.8881 / 0.115 & 6.76 / 4.78 / 6.00 \\
\midrule
\multirow{2}{*}{\begin{tabular}[c]{@{}c@{}}Supervised\\Denoising \end{tabular}}
& Baseline$_\mathcal{N}$ & 34.52 / 0.9461 / 0.053 & 5.69 / 4.85 / 5.43 \\
& DnCNN~\cite{DnCNN} & 33.81 / 0.9325 / 0.076 & 6.05 / 5.10 / 5.48 \\
\midrule
Supervised IR & Baseline$_\mathcal{R}$ & 36.10 / 0.9574 / 0.035 & 5.54 / 5.14 / 5.18 \\
\midrule
\multirow{8}{*}{\begin{tabular}[c]{@{}c@{}}Self-Supervised\\Denoising \end{tabular}}
& N2N~\cite{Noise2Noise} & 34.67 / 0.9472 / 0.053 & 6.10 / 4.93 / 5.59 \\
& N2V~\cite{N2V} & 31.39 / 0.9227 / 0.076 & 5.82 / 5.52 / 5.17 \\
& Laine19-mu~\cite{Laine19} & 32.74 / 0.9304 / 0.073 & 5.87 / 5.67 / 5.10 \\
& Laine19-pme~\cite{Laine19} & 33.28 / 0.9119 / 0.095 & 7.26 / 6.03 / 5.62 \\
& DBSN~\cite{DBSN} & 33.59 / 0.9389 / 0.060 & 6.57 / 5.48 / 5.54 \\
& R2R~\cite{R2R} & 32.21 / 0.8807 / 0.117 & 5.63 / 5.63 / 4.99 \\
& Neighbor2Neighbor~\cite{Nei2Nei} & 32.82 / 0.9275 / 0.087 & 6.47 / 5.86 / 5.33 \\
& Blind2Unblind~\cite{Blind2Unblind} & 33.30 / 0.9380 / 0.061 & 5.28 / 5.22 / 5.04 \\
\midrule
\multirow{1}{*}{Ours}
& SelfIR & 34.51 / 0.9440 / 0.053 & 5.48 / 5.83 / 4.86 \\
\bottomrule
\end{tabular}}
\end{center}
\vspace{-2mm}
\end{table}
\subsection{Experimental Results in Raw-RGB Space}
\cref{table:raw} shows the quantitative results of synthetic and real-world experiments on raw-RGB images.
For synthetic experiments with more complex and realistic sensor noise~\cite{UPI}, all models are retrained with our synthetic images.
Our SelfIR achieves a 1.69 dB PSNR gain in comparison with Neighbor2Neighbor~\cite{Nei2Nei}.
For evaluation on real-world images, we directly use the models of N2N~\cite{Noise2Noise}, Laine19-pme~\cite{Laine19}, R2R~\cite{R2R} and all supervised methods, which are pre-trained for sensor noise.
For SelfIR and other self-supervised methods, we fine-tune the pre-trained model on our real-world training set.
The PI~\cite{PI} metric of SelfIR is improved by 0.55 on real-world testing images through fine-tuning.
And the results on no-reference IQA show that our method is very competitive in comparison with the competing methods on real-world images.
The visual results are given in the supplemental material.
\begin{table}[t]
\small
\vspace{1mm}
\renewcommand\arraystretch{1
\begin{center}
\caption{Results of deblurring with clear images and noisy images as the supervision.}
\label{table:un_deblurring}
\vspace{2mm}
\scalebox{1}{
\begin{tabular}{cccccc}
\toprule
\tabincell{c}{Supervision \\Information}
& \tabincell{c}{Gaussian $\sigma \in [5/255,50/255]$ \\ \footnotesize{PSNR}\scriptsize{$\uparrow$} \small{/} \footnotesize{SSIM}\scriptsize{$\uparrow$} \small{/} \footnotesize{LPIPS}\scriptsize{$\downarrow$}}
& \tabincell{c}{Poisson $\lambda \in [5,50]$ \\ \footnotesize{PSNR}\scriptsize{$\uparrow$} \small{/} \footnotesize{SSIM}\scriptsize{$\uparrow$} \small{/} \footnotesize{LPIPS}\scriptsize{$\downarrow$}}
& \tabincell{c}{Sensor Noise~\cite{UPI} \\ \footnotesize{PSNR}\scriptsize{$\uparrow$} \small{/} \footnotesize{SSIM}\scriptsize{$\uparrow$} \small{/} \footnotesize{LPIPS}\scriptsize{$\downarrow$}} \\
\midrule
Clear Images & 28.24 / 0.8561 / 0.191 & 28.24 / 0.8561 / 0.191 & 28.14 / 0.8547 / 0.162 \\
Noisy Images & 28.29 / 0.8578 / 0.190 & 28.23 / 0.8563 / 0.191 & 28.16 / 0.8545 / 0.164 \\
\bottomrule
\end{tabular}}
\end{center}
\end{table}
\begin{table}[t!]
\small
\renewcommand\arraystretch{1
\begin{center}
\caption{Ablation study of auxiliary loss on Gaussian noise.}
\label{table:aux_loss}
\vspace{2mm}
\scalebox{1}{
\begin{tabular}{cccccc}
\toprule
& \tabincell{c}{Neighbor2Neighbor~\cite{Nei2Nei} \\ \footnotesize{PSNR}\scriptsize{$\uparrow$} \small{/} \footnotesize{SSIM}\scriptsize{$\uparrow$} \small{/} \footnotesize{LPIPS}\scriptsize{$\downarrow$}}
& \tabincell{c}{SelfIR \\ \footnotesize{PSNR}\scriptsize{$\uparrow$} \small{/} \footnotesize{SSIM}\scriptsize{$\uparrow$} \small{/} \footnotesize{LPIPS}\scriptsize{$\downarrow$}} \\
\midrule
w/o $\mathcal{L}_{aux}$ & 34.29 / 0.9271 / 0.085 & 35.65 / 0.9492 / 0.080 \\
w/ $\mathcal{L}_{aux}$ & 34.45 / 0.9307 / 0.093 & 35.74 / 0.9499 / 0.076 \\
\bottomrule
\end{tabular}}
\end{center}
\end{table}
\section{Ablation Study}
\subsection{Feasibility of Deblurring with Noisy Images}
In order to verify the feasibility of taking noisy images as the supervision of deblurring in \cref{sec:3.1}, we replace the clear supervision of Baseline$_\mathcal{B}$ with noisy images of different noise distributions, and the results are shown in \cref{table:un_deblurring}.
It can be seen that taking noisy or clear images as the supervision leads to similar performance on the deblurring task, which confirms that the noisy images can be an alternative of clear images to supervise the deblurring task.
\subsection{Effect of Auxiliary Loss}
In order to evaluate the effect of auxiliary loss $\mathcal{L}_{aux}$ in \cref{loss:aux}, we add $\mathcal{L}_{aux}$ to the loss terms of self-supervised denoising method Neighbor2Neighbor~\cite{Nei2Nei}.
As shown in \cref{table:aux_loss}, we obtain 0.16 dB PSNR gain against the baseline Neighbor2Neighbor~\cite{Nei2Nei}.
The result indicates that blurry images can provide some auxiliary supervision information and improve performance for self-supervised denoising.
When removing $\mathcal{L}_{aux}$ from our SelfIR, the PSNR dropped by 0.09 dB.
Since the long-exposure image is taken directly as input, the loss term may be less effective, but is still beneficial.
\begin{minipage}[t]{\linewidth}
\small
\centering
\begin{minipage}[t]{0.45\linewidth}
\centering
\captionof{table}{\centering Ablation study on different weights ($\lambda_{reg}$ values) of regularization loss.}
\label{table:reg_loss_w}
\begin{tabular}{cc}
\toprule
$\lambda_{reg}$ & \footnotesize{PSNR}\scriptsize{$\uparrow$} \small{/} \footnotesize{SSIM}\scriptsize{$\uparrow$} \small{/} \footnotesize{LPIPS}\scriptsize{$\downarrow$} \\
\midrule
0 & 35.20 / 0.9473 / 0.097 \\
1 & 35.64 / 0.9492 / 0.082 \\
2 & 35.74 / 0.9499 / 0.076 \\
4 & 35.72 / 0.9496 / 0.075 \\
8 & 35.73 / 0.9497 / 0.072 \\
\bottomrule
\end{tabular}
\end{minipage}
\hspace{3mm}
\begin{minipage}[t]{0.45\linewidth}
\centering
\captionof{table}{\centering Ablation study on different weights ($\lambda_{aux}$ values) of auxiliary loss.}
\label{table:aux_loss_w}
\begin{tabular}{cc}
\toprule
$\lambda_{aux}$ & \footnotesize{PSNR}\scriptsize{$\uparrow$} \small{/} \footnotesize{SSIM}\scriptsize{$\uparrow$} \small{/} \footnotesize{LPIPS}\scriptsize{$\downarrow$} \\
\midrule
0 & 35.65 / 0.9492 / 0.080 \\
1 & 35.73 / 0.9498 / 0.078 \\
2 & 35.74 / 0.9499 / 0.076 \\
4 & 35.73 / 0.9499 / 0.076 \\
8 & 35.67 / 0.9496 / 0.077 \\
\bottomrule
\end{tabular}
\end{minipage}
\end{minipage}
\vspace{3mm}
\subsection{Effect of Different Loss Weights}
We conduct ablation studies on different weighting hyper-parameters (\emph{i.e}\onedot} \def\Ie{\emph{I.e}\onedot, $\lambda_{reg}$ and $\lambda_{reg}$) for balancing regularization and auxiliary loss terms.
The experiments are conducted on Gaussian noise in sRGB space.
When varying one hyper-parameter, the other one is set to 2 by default.
As shown in Tabs.~\ref{table:reg_loss_w} and ~\ref{table:aux_loss_w}, it can be seen that the sensitivity to $\lambda_{reg}$ and $\lambda_{reg}$ of SelfIR is acceptable.
\section{Conclusion}
The complementarity between long-exposure blurry and short-exposure noisy images not only improves the performance of image restoration, but also makes it possible to learn a restoration model in a self-supervised manner.
Jointly leveraging the long- and short-exposure images, we present a self-supervised image restoration method named SelfIR.
On the one hand, we take the short-exposure images as the supervision information for deblurring.
On the other hand, we utilize the sharp areas in the long-exposure images as auxiliary supervision information in aid of self-supervised denoising.
SelfIR combines these two aspects by a collaborative learning scheme, and makes the deblurring and denoising tasks benefit from each other.
Experiments on synthetic and real-world images show the effectiveness and practicality of our proposed method.
\section{Limitation, Impact, Etc}\label{sec:limit}
This work is still limited in assessing the results on real-world image pairs.
Although no-reference IQA metrics are adopted, these may be too unstable to report the performance consistent with humans accurately.
We will establish a real-world blurry-noisy pair dataset including high-quality GTs to solve this problem in the future work.
As for societal influence, this work is promising to be applied to terminal devices (\emph{e.g}\onedot} \def\Eg{\emph{E.g}\onedot., smartphones) for obtaining better images under low-light environments.
It has no foreseeable negative impact.
Besides, the images used in this work are from natural or human scenes.
The GoPro dataset used for synthetic experiments is public under CC BY 4.0 license.
There is no personally identifiable information or offensive content in the experimental data.
\section*{Acknowledgement}
This work was supported by the Major Key Project of Peng Cheng Laboratory (PCL2021A12) and the National Natural Science Foundation of China (NSFC) under Grants No.s U19A2073 and 61871381.
\clearpage
|
1810.00919
|
\section{Introduction}
\label{sec:intro} Many econometric data are big data in the form of time series that can be seen as functions (\cite{Tsay16}).
In Functional Data Analysis (FDA) the observations are functional time series or multivariate functions. \cite{Ramsay05} provide an excellent overview of FDA. FDA has been applied in many different fields (\cite{Ramsay02}), and although it is a relatively new area of research in the business and economic sectors, applications are beginning to proliferate in these fields (\cite{ASMB:ASMB388,doi:10.1080/07350015.2015.1092976,doi:10.1080/07350015.2017.1279058}).
Data mining of functional time series (\cite{FU2011164}) is as important as in the classical multivariate version. It is desirable to understand and describe the entire data set and to be able to extract information that is easily interpretable even by non-experts. We deal with an unsupervised statistical learning problem since only input features and not output are present. Data decomposition techniques to find the latent components are usually employed in classical multivariate statistics (see \cite[Chapter 14]{HTF09} for a complete review of unsupervised learning techniques). A data matrix is considered as a linear combination of several factors. The constraints on the factors and how they are combined give rise to different unsupervised techniques (\cite{Morup2012,Thurau12,Vinue15}) with different goals. For instance, Principal Component Analysis (PCA) explains data variability adequately at the expense of the interpretability of the factors, which is not always straightforward since factors are linear combinations of the features. For their part, clustering techniques such as $k$-means or $k$-medoids return factors that are easily interpreted. Note that data are explained through several centroids, which are means of groups of data in the case of $k$-means, or medoids, which are concrete observations in the case of $k$-medoids. Nevertheless, the binary assignment of data to the clusters diminishes their modeling flexibility as compared with PCA.
Archetype analysis (AA) lies somewhere in between these two methods, as the interpretability of its factors is as easy as with clustering techniques but its modeling flexibility is higher than for clustering methodologies. \cite{Vinue15} provide a summary table showing the relationship between several multivariate unsupervised techniques, as do \cite{Morup2012}. \cite{STONE1996110} also showed that AA may be more appropriate than PCA when the data do not have elliptical distributions.
AA was formulated by \cite{Cutler1994}. In AA each observation in the data set is approximated by a mixture (convex combination) of pure or extremal types called archetypes. Archetypes themselves are restricted to be convex combinations of the individuals in the data set. However, AA may not be satisfactory in some fields since, being artificial constructions, nothing guarantees the existence of subjects in our sample with characteristics similar to those of the archetypes (\cite{Seiler2013}).
In order to solve this issue, the new concept of archetypoids was introduced by \cite{Vinue15}. In Archetypoid Analysis (ADA) each observation in the data set is approximated by a convex combination of a set of real extreme observations called archetypoids. AA and ADA were extended to dense functional data by \cite{Epifanio2016} and to sparse functional data by \cite{VinEpi17}. In the functional context, functions from the data set, which can be multivariate functions, are explained by mixtures of archetypal functions.
This process not only allows us to relate the subjects of the sample to extreme patterns but also facilitates comprehension of the data set. Humans understand the data better when the individuals are shown through their extreme constituents (\cite{Davis2010}) or when features of an individual are shown as opposed to those of another (\cite{Thurau12}). In other words, as regards human interpretability, the central points returned by clustering methods do not seem as good as extreme types, which are also more easily understandable than a linear combination of data.
AA and ADA have therefore aroused the interest of researchers working in different fields, such as astrophysics (\cite{Chan2003}), biology (\cite{Esposito2012}), climate (\cite{doi:10.1175/JCLI-D-15-0340.1,w9110873,EpiIbSi18}), developmental
psychology (\cite{SAM16}), e-learning (\cite{Theodosiou}), engineering (\cite{EpiVinAle,EpiIbSi17,MillanEpi}), genetics (\cite{Morup2013}), machine learning (\cite{Morup2012,EugsterPAMI,Eugster14,Rago15}), multi-document summarization (\cite{Canhasi13,Canhasi14}), nanotechnology (\cite{doi:10.1021/acsnano.5b05788}), neuroscience (\cite{griegos,Morup16}) and sports (\cite{Eugster2012}). AA has also been applied in market research (\cite{Li2003,Porzio2008,Midgley2013}) in the multivariate context. Despite the fact that financial time series are commonly analyzed by unsupervised techniques ranging from PCA (\cite{Alexander, Tsay,Ingrassia2005}) to clustering (\cite{DOSE2005145,BASALTO2007635,TSENG201211,DURSO20132114,DIAS2015852,AnnMaharaj2010,CAPPELLI201323,doi:10.1002/cem.2565,DURSO201233,doi:10.1142/S0218488504002849,1516151,Alonso2018}), including robust versions of these (\cite{6737103,Verdonck,DURSO20161,DURSO201812}), to the best of our knowledge functional archetypal analysis has not been used in financial or business applications until now.
As archetypes are situated on the boundary of the convex hull of data (\cite{Cutler1994}), AA and ADA solutions are sensitive to outliers.
The problem of robust AA in the multivariate case was addressed by \cite{Eugster2010}. The idea is to find the archetypes of the large majority of the data set rather than of the totality. \cite{Eugster2010} considered a kind of M-estimators for multivariate real-valued data ($m$-variate), where the domain of their loss function is not $\mathbb{R}^+$, but $\mathbb{R}^m$. Recently, \cite{Sinova} considered functional M-estimators for the first time, where the domain of their loss function is $\mathbb{R}^+$. We base our proposal to robustify archetypal solutions in the multivariate and functional case on this last kind of loss function, which is commonly used in robust analysis (\cite{maronna}).
The main novelties of this work consist of: 1. Proposing a robust methodology for classical multivariate and functional AA and ADA; 2. Introducing a new visualization procedure that makes it easy to summarize the results and multivariate time series; 3. Applying functional archetypal analysis to financial time series for the first time, more specifically to multivariate financial time series.
Section \ref{metodologia} reviews AA and ADA for the multivariate and functional case and Section \ref{metodologiarobusta} introduces their respective robust versions. Our proposal is compared with a previously existing methodology for robust multivariate AA in Section \ref{simulacion}, where a simulation study with functional data is also carried out to validate our procedure. In Section \ref{resultados}, robust ADA is applied to a data set of 496 companies that are characterized
by two financial time series. Furthermore, some visualization tools are also introduced in this section. Finally, conclusions
and future work are discussed in Section \ref{conclusiones}. The code in R (\cite{R}) and data for reproducing the results are available at \url{www3.uji.es/~epifanio/RESEARCH/rofada.rar}.
\section{Archetypal analysis}
\label{metodologia}
\subsection{AA and ADA in the classical multivariate case}
Let $\mathbf{X}$ be an $n \times m$ matrix with $n$ cases and $m$ variables. In AA, three matrices are sought: a) the $k$ archetypes $\mathbf{z}_j$, which are the rows of a $k \times m$ matrix $\mathbf{Z}$; b) an $n \times k$ matrix $\mathbf{\alpha} = (\alpha_{ij})$ that contains the mixture coefficients that approximate each case $\mathbf{x}_i$ by a mixture of the archetypes ($\mathbf{\hat{x}}_i = \displaystyle \sum_{j=1}^k \alpha_{ij} \mathbf{z}_j$); and c) a $k \times n$ matrix $\mathbf{\beta} = (\beta_{jl})$ that contains the mixture coefficients that define each archetype ($\mathbf{z}_j$ = $\sum_{l=1}^n \beta_{jl} \mathbf{x}_l$). To determine these matrices, the following residual sum of squares (RSS) with the respective constraints is minimized ($\| \cdot\|$ denotes the Euclidean norm for vectors):
\begin{equation} \label{RSSar}
RSS = \displaystyle \sum_{i=1}^n \| \mathbf{x}_i - \sum_{j=1}^k \alpha_{ij} \mathbf{z}_j\|^2 = \sum_{i=1}^n \| \mathbf{x}_i - \sum_{j=1}^k \alpha_{ij} \sum_{l=1}^n \beta_{jl} \mathbf{x}_l\|^2{,}
\end{equation}
under the constraints
\begin{enumerate}
\item[1)] $\displaystyle \sum_{j=1}^k \alpha_{ij} = 1$ with $\alpha_{ij} \geq 0$ {for} $i=1,\ldots,n$ {and}
\item[2)] $\displaystyle \sum_{l=1}^n \beta_{jl} = 1$ with $\beta_{jl} \geq 0$ {for} $j=1,\ldots,k${.}
\end{enumerate}
It is important to mention that archetypes do not necessarily match real cases. Specifically, this will only happen when one and only one $\beta_{jl}$ is equal to one for each archetype, i.e. when each archetype is composed of only one case that presents the entire weight. Therefore, in ADA the previous constraint 2) is changed by the following one, and as a consequence the previous continuous optimization problem of AA is transformed into a mixed-integer optimization problem:
\begin{enumerate}
\item[2)] $\displaystyle \sum_{l=1}^n \beta_{jl} = 1$ with $\beta_{jl} \in \{0,1\}$ and $j=1,\ldots,k$.
\end{enumerate}
\cite{Cutler1994} demonstrated that archetypes are located on the boundary of the convex hull of the data if $k$ $>$ 1, although this does not necessarily happen for archetypoids (see \cite{Vinue15}). However, if $k$ = 1, the archetype coincides with the mean and the archetypoid with the medoid (\cite{Kaufman90}).
\cite{Cutler1994} developed an alternating minimizing algorithm to estimate the matrices in the AA problem. It alternates between calculating the best $\mathbf{\alpha}$ for given archetypes $\mathbf{Z}$ and the best archetypes $\mathbf{Z}$ for a given $\mathbf{\alpha}$. In each step a
penalized version of
the non-negative least squares algorithm by \cite{Lawson74} is used to solve the convex least squares problems. That algorithm, with certain modifications (previous data standardization and use of spectral norm in equation \ref{RSSar} instead of Frobenius norm for matrices), was implemented by \cite{Eugster2009} in the R package {\bf archetypes}. In our R implementation those modifications were canceled and the data are not standardized by default and the objective function to optimize coincides with equation \ref{RSSar}.
As regards the estimation of the matrices in the ADA problem, \cite{Vinue15} developed an algorithm based on the idea of the Partitioning Around Medoids (PAM) clustering algorithm (\cite{Kaufman90}). This algorithm consists of two stages: the BUILD phase and the SWAP phase. In the BUILD phase, an initial set of archetypoids is computed, while that set is improved during the SWAP phase by exchanging the chosen observations for unselected cases and by inspecting whether these replacements diminish the RSS. \cite{JSSv077i06} implemented that algorithm in the R package {\bf Anthropometry} with three possible initial sets in the BUILD step. One of them is referred to as the $cand_{ns}$ set and consists of
the nearest neighbors in Euclidean distance to the $k$ archetypes. The second candidates, the {$cand_{\alpha}$} set, consist of the observations with the maximum $\alpha$ value for each archetype $j$, i.e. the observations with the largest relative share for the respective archetype. The third initial candidates, the so-called {$cand_{\beta}$} set, are the cases with the maximum $\beta$ value for each archetype $j$, i.e. the cases that most influence the construction of the archetypes. Each of these three sets goes through the SWAP phase and three sets are obtained. From these three sets, the one with lowest RSS (often the same set is retrieved from the three initializations) is selected as the ADA solution.
Archetypes are not necessarily nested and neither are archetypoids. Therefore, changes in $k$ will yield different conclusions. This is why the selection criterion is particularly important. Thus, if the researcher has prior knowledge of the structure of the data, the value of $k$ can be selected based on that information. Otherwise, the elbow criterion, which has been widely used (\cite{Cutler1994,Eugster2009,Vinue15}), could be considered. With the elbow criterion, the RSS is represented for different $k$ values and the value $k$ is chosen as the point where the elbow is found.
\subsection{AA and ADA in the functional case}
In FDA each datum is a function. In this context, the values of the $m$ variables in the classical multivariate context become function values with a continuous index $t$, and the data set adopts the form $\{x_1(t),...,x_n(t)\}$ with $t \in [a,b]$.
It is assumed that these functions belong to a Hilbert space, they satisfy reasonable smoothness conditions and are square-integrable functions on that interval. In addition, in the definition of the inner product, the sums are replaced by integrals.
Again, the goal of functional archetype analysis (FAA) is to approximate the functional data sample by mixtures of $k$ archetype functions.
The difference with the multivariate case is that now both archetypes and observations are functions. In FAA, two matrices $\alpha$ and $\beta$ are also calculated to minimize the RSS. However, certain aspects should be highlighted. On the one hand, RSS are now calculated with a functional norm (the $L^2$-norm, $\|f\|^2= <f,f> = \int_a^b f(t)^2 dt$, is considered) instead of a vector norm. On the other hand, observational and archetype vectors $\mathbf{x}_i$ and $\mathbf{z}_i$ now correspond to observational and archetype functions $x_i(t)$ and $z_i(t)$. In any case, the interpretation of matrices $\alpha$ and $\beta$ is the same as in the standard multivariate case.
Functional archetypoid analysis (FADA) is also an adaptation of ADA, changing vectors for functions. In this regard, FADA aims to find $k$ functions of the sample (archetypoids) that approximate the functions of the sample through the mixtures of these functional archetypoids. Again, vector norms are replaced by functional norms. Interpretation of the matrices is the same as before.
In practice, the functions are recorded at discrete times. Standard AA and ADA could be applied to the function values of $m$ equally-spaced values from $a$ to $b$ to obtain FAA and FADA. However, this approach is not computationally efficient (\cite{Epifanio2016}). Therefore, we represent data by means of basis functions. This reduces noise, i.e. functions are smoothed. Furthermore, data observations do not have to be equally spaced, the number of observed points can vary across records and they can be measured at different time points. This also makes it possible to perform a more efficient analysis, since the number of coefficients of the basis functions is usually smaller than the number of time points evaluated.
The crux of the matter is to choose an appropriate basis together with the number of basis elements. Nevertheless, this issue appears repeatedly in all FDA problems. Functions of the sample should be expanded by basis functions that share common features (see \cite{Ramsay05} for a detailed explanation about smoothing functional data). For densely observed functions, the case that concerns us, the basis coefficients are computed separately for each function, while information from all functions should be used to calculate the coefficients for each function (\cite{James}) for sparsely observed functions.
Let us see how the RSS is formulated in terms the coefficients $\mathbf{b}_i$, the vector of length $m$ that approximates $x_i(t)$ $\approx$ $\sum_{h=1}^m b_i^h B_h(t)$ with the basis functions $B_h$ ($h$ = 1, ..., $m$) (see \cite{Epifanio2016} for details):
\begin{equation}\label{RSSfar}
\begin{split}
RSS = \displaystyle \sum_{i=1}^n \| {x}_i - \sum_{j=1}^k \alpha_{ij} {z}_j\|^2 = \sum_{i=1}^n \| {x}_i - \sum_{j=1}^k \alpha_{ij} \sum_{l=1}^n \beta_{jl} {x}_l\|^2 =
\sum_{i=1}^n {\mathbf{a}}'_i \mathbf{W} {\mathbf{a}}_i
{,}
\end{split}
\end{equation}
where $\mathbf{a'}_i$ = $\mathbf{b'}_i - \sum_{j=1}^k \alpha_{ij} \sum_{l=1}^n \beta_{jl} \mathbf{b'}_l$ and $\mathbf{W}$ is the order $m$ symmetric matrix with the inner products of the pairs of basis functions $w_{m_1,m_2}$ = $\int B_{m_1}B_{m_2}$. If the basis is orthonormal, for instance the Fourier basis, $\mathbf{W}$ is the order $m$ identity matrix and FAA and FADA can be computed using AA and ADA with the basis coefficients. If not, $\mathbf{W}$ has to be computed previously one single time by numerical integration.
{Let us see a toy example to illustrate what archetypes mean and the differences compared with PCA and clustering. We use a functional version, previously considered by \citet{Ferraty} and \citet{Epifanio08}, of the well-known simulated
data known as waveform data \citep{Breiman}. Functions $x$ are discretized
at 101
points ($t$ = 1, 1.2, 1.4, ..., 21) such that
\begin{itemize}
\item $x(t) = uh_1(t) + (1 - u)h_2(t) + \epsilon(t)$ for class 1,
\item
$x(t) = uh_1(t) + (1
- u)h_3(t) + \epsilon(t)$ for class 2, and
\item $x(t) = uh_2(t) + (1 - u)h_3(t) + \epsilon(t)$ for class 3,
\end{itemize}
where $u$ is uniform on $[0, 1]$, $\epsilon(t)$ are standard
normal variables, and $h_i$ are the shifted triangular
waveforms: $h_1(t)$ = $max(6 - |t - 11|, 0)$, $h_2(t)$ = $h_1(t -
4)$ and $h_3(t)$ = $h_1(t + 4)$. Note that $x$ are mixtures of $h_j$ ($j$ = 1, 2, 3), and therefore $h_j$ are archetype functions by definition. The toy example has 150 waveforms in each of the 3 classes. Figure \ref{waveform} shows the simulated data set, together with the results for AA, PCA and clustering. Note that archetypes are estimations of $h_j$ functions, the purest profiles, unlike cluster centers, which are not so extreme and whose profiles are not as clear as those of archetypes, since they are central points: the mean of each class. PC 1 quantifies a discriminability trade-off between classes 1 and 2, whereas PC 2 quantifies a discriminability trade-off between class 3 and the rest of the classes. In summary, finding
extreme profiles, which are easily interpretable, is not the objective of clustering or PCA, but
it is the intention of AA and ADA, together with the expression of the data as a mixture of those extreme profiles.}
\begin{figure}
\begin{center}
\begin{tabular}{ccc}
\includegraphics[width=.3\linewidth]{fig1sample.pdf} & \includegraphics[width=.3\linewidth]{fig1original.jpg} &
\includegraphics[width=.3\linewidth]{fig1archetypes.jpg} \\
(a) & (b) & (c)\\
\includegraphics[width=.3\linewidth]{fig1kmeans.jpg} & \includegraphics[width=.3\linewidth]{fig1pca1.jpg} &
\includegraphics[width=.3\linewidth]{fig1pca2.jpg} \\
(d) & (e) & (f)
\end{tabular}
\end{center}
\caption{{Waveform. (a) Five functions for each of the three
classes, respectively. (b) $h_j$ functions together with the data set in gray. (c) Archetypes. (d) Centers of $k$-means. (e) Effect of adding and subtracting a multiple of PC 1 to the mean curve. (f) Effect of adding and subtracting a multiple of PC 2 to the mean curve. }
\label{waveform}}
\end{figure}
\subsubsection{Multivariate functional archetypal analysis}
It is common to analyze data with more than one dimension. In our context, this means working with samples in which we analyze more than one function for each individual, so each function describes a characteristic of the subject.
First of all, we need to define an inner product between multivariate functions. The simplest definition is to add up the inner products of the multivariate functions. Therefore, the squared norm of a $P$-variate function is the sum of the squared norms of the $P$ components. Consequently, FAA and FADA for $P$-variate functions is equivalent to $P$ independent FAA and FADA with shared parameters $\alpha$ and $\beta$.
In practical terms, this means to work with a composite function formed by stringing the $P$ functions together.
Without loss of generality, let $f_i(t)=(x_i(t),y_i(t))$ be a bivariate function. So, its squared norm is $\|f_i\|^2= \int_a^b x_i(t)^2 dt + \int_a^b y_i(t)^2 dt$. Let $\mathbf{b}_i^x$ and $\mathbf{b}_i^y$ be the vectors of length $m$ of the coefficients for $x_i$ and $y_i$ for the basis functions $B_h$. Therefore, to compute FAA and FADA, the RSS is reformulated as:
\begin{equation}\label{RSSfarb}
\begin{gathered}
RSS ={} \sum_{i=1}^{n}||f_i- \sum_{j=1}^{k} \alpha_{ij}z_j||^2
= \sum_{i=1}^{n}||f_i- \sum_{j=1}^{k} \alpha_{ij}\sum_{l=1}^{n}\beta_{jl}f_l||^2
= \sum_{i=1}^{n}||x_i- \sum_{j=1}^{k} \alpha_{ij}\sum_{l=1}^{n}\beta_{jl}x_l||^2 \\ + \sum_{i=1}^{n}||y_i- \sum_{j=1}^{k} \alpha_{ij}\sum_{l=1}^{n}\beta_{jl}y_l||^2
= \sum_{i=1}^{n}{\mathbf{a}^x}'_i \mathbf{Wa}_i^x+\sum_{i=1}^{n}{\mathbf{a}^y}'_i \mathbf{Wa}_i^y
\end{gathered}
\end{equation}
where $\mathbf{{a^x}}'_i={\mathbf{b}^x}'_i - \sum_{j=1}^{k} \alpha_{ij}\sum_{l=1}^{n}\beta_{jl} {\mathbf{b}^x}'_l $ and ${\mathbf{a}^y}'_i={\mathbf{b}^y}'_i - \sum_{j=1}^{k} \alpha_{ij}\sum_{l=1}^{n}\beta_{jl} \mathbf{{b^y}}'_l $ with the corresponding AA or ADA constraints for $\mathbf{\alpha}$ and $\mathbf{\beta}$. The union of $\mathbf{b}^x_i$ and $\mathbf{b}^y_i$ composes the observations. If the basis functions are orthonormal, FAA and FADA are reduced to computing standard AA and ADA for the $n \times 2m$ coefficient matrix composed by joining the coefficient matrix for $x$ and $y$ components.
\section{Robust archetypal analysis} \label{metodologiarobusta}
The RSS is formulated as the sum of the squared (vectorial or functional) norm of the residuals, $r_i$ ($i$ = 1, ..., $n$). Here, $r_i$ denote vectors of length $m$ in the multivariate case or (univariate or multivariate) functions in the functional case. The least squares loss function does not provide robust solutions since it favors outliers; large residuals have large effects. M-estimators try to lower the large influence of outliers by changing the square loss
function for a less rapidly increasing loss function. \cite{Eugster2010} defined a loss function from $\mathbb{R}^m$ to $\mathbb{R}$. However, \cite{Sinova} established several conditions of the loss function $\rho$ for functional M-estimators, the first of which is that the loss function is defined from $\mathbb{R}^+$ to $\mathbb{R}$ and the loss is specified as $\rho(||r_i||)$. {Furthermore, $\rho(0)$ should be zero, $\rho(x)/x$ should tend towards zero, when $x$ tends towards zero, and $\rho$ should be differentiable and $\rho'$ and $\phi(x)$ = $\rho'(x)/x$ should be both continuous and
bounded, where we assume that $\phi(0)$ := $lim_{x \rightarrow 0}$ $\rho'(x)/x$ exists and is finite. This last condition is not satisfied by the standard least squares loss function $\rho(x)$ = $x^2$ ($\rho'$ is not bounded). Details about properties of functional M-estimators, such as their consistency and robustness by means of
their breakdown point and their influence function can be found in \cite{Sinova}.}
\cite{Sinova} also analyzed the common families of loss functions. We follow the ideas of \cite{Sinova} and the Tukey biweight or bisquare family of loss function (\cite{10.2307/1267936}) with tuning parameter $c$ is considered, since this loss function copes with extreme outliers well. Therefore, RSS in equations \ref{RSSar}, \ref{RSSfar} and \ref{RSSfarb} are replaced by $\sum_{i=1}^n \rho_c(||r_i||)$, where $\| \cdot\|$ denotes the Euclidean norm for vectors, the $L^2$-norm for univariate functions and corresponding norm for $P$-variate functions, and $\rho_c(||r_i||)$ is given by
\begin{equation} \label{robustrho}
\rho_c(||r_i||) =\left\{ \begin{array}{ll}
c^2/6 \cdot (1 - (1 - ||r_i||^2/c^2)^3) & \mbox{if 0 $\leq$ $||r_i||$ $\leq$ $c$}\\
c^2/6 & \mbox{if $c$ $<$ $||r_i||$}
\end{array}
\right.
\end{equation}
For the tuning parameter $c$, we follow \cite{Cleveland}, as did \cite{Eugster2010}, and $c$ = 6$me$ with $me$ being the median of the residual norms unequal to zero{, although other alternatives are analyzed in the simulation study}. From the computational point of view, we only have to replace RSS with this new objective function in the previous algorithms. It only depends on the norm of the residuals, so in the functional case, it can be expressed in terms of the coefficients in the basis and $\mathbf{W}$, and no integration is needed.
{Note that $\rho_c$ is not scale equivariant, i.e. the results depend heavily on the units of measurement, which is why in both the real case \citep{maronna} and the functional case \citep{Sinova} the tuning parameter should take into account the
distribution of the data (the residuals in our case), in particular certain percentile of this distribution. Theoretically, the tuning parameter should be chosen such that the loss function is well adapted to the data \citep{Sinova}, but in practice there is no one gold standard method for selecting the tuning parameter.}
Another possibility would be to use the Huber family of loss functions (\cite{huber:1964}) that depends on a tuning parameter. For example, this loss function was used by \cite{chen:hal-00995911} {and \cite{SUN2017147}, where} the tuning parameter was manually set and no suggestion was given about its selection. An important difference between Huber and the bisquare family is that residuals larger than $c$ contribute the same to the loss in this last family, which is not the case with the Huber family. For that reason, the bisquare family can better cope with extreme outliers.
\section{Simulation study}
\label{simulacion}
\subsection{Multivariate data} To compare the performance of our proposal, the same procedures and data set, known as ozone, examined by \cite{Eugster2010} are considered. The data, which are also used as a demo in the R library {\bf archetypes}, consist of 330 observations of 9 standardized variables that are related to air pollution. We create a corrupted data set by adding a total of 5 outliers, as \cite{Eugster2010} did. Table \ref{percentiles} shows the percentiles of 3 archetypes (A1, A2 and A3) extracted in four situations: a) original AA with the original data set before adding the outliers (AAO), this solutions plays the gold standard reference role; b) original AA with the corrupted data set (AAC); c) robust AA by \cite{Eugster2010} with the corrupted data set (RAA-EL); d) our proposal of robust AA with the corrupted data set (RAA-ME).
At a glance it can be seen that the archetypes returned by our proposal are the most similar to the original ones. To corroborate it, the Frobenius norm of the difference between the gold standard reference and the different alternatives applied to the corrupted data set has been computed with the following results for each situation: AAC 206.3; RAA-EL 221.3; RAA-ME 64.5. Our proposal provides the solution with the smallest difference with respect to the gold standard reference, while the robust AA solution by \cite{Eugster2010} provides a greater difference than the non-robust AA solution. For both AAC and RAA-EL one of the archetypes (A2) is built entirely of a mixture of the added outliers, which is not the case with our proposal (0.07 is the only $\beta$ weight for the outliers). Therefore, a more robust solution is achieved with our proposal.
\begin{table}
\caption{Percentiles profiles in four situations for the ozone data set (see text for details). \label{percentiles}}
\centering
\begin{tabular}{rrrrrrrrrrrrr}
Situation & \multicolumn{3}{c}{AAO} & \multicolumn{3}{c}{AAC} & \multicolumn{3}{c}{RAA-EL} & \multicolumn{3}{c}{RAA-ME}\\
Variable & A1 & A2 & A3 & A1 & A2 & A3 & A1 & A2 & A3 & A1 & A2 & A3\\ \hline
OZONE&12&89&12&3&100&70& 12&99&81&3&97&12\\
500MH& 3&92&65&6&100&54&7&99&73&3&99&50\\
WDSP& 96&43&5&63&100&27& 43&99&27&78&78&8\\
HMDTY&56&82&11&19&100&45& 20&99&56&45&97&11\\
STMP&4&93&22&5&100&66& 10&99&80&5&98&16\\
INVHT&100&14&63&99&100&4& 70&99&5&99&41&56\\
PRGRT&92&64&2&36&100&36& 34&99&40&79&78&2\\
INVTMP&1&93&49&5&100&75&9&99&84&3&97&40\\
VZBLTY&77&15&77&87&100&9& 76&99&9&76&38&76
\end{tabular}
\end{table}
{We have also analyzed the influence that the tuning parameter has on the results. Instead of $c$ = 6$me$, we consider the following alternatives: $c$ = $P_j$, with $j$ = 25, 50 and 75, representing the 25th, 50th and 75th percentiles of the residual norms unequal to zero, and the same but multiplied by 6, i.e. $c$ = 6 $P_j$. Table \ref{cnorm} shows the Frobenius norm of the difference between the gold standard reference and RAA-ME computed using the different values of $c$. The same results are obtained if we use $c$ = 6$P_{25}$ and our selection, $c$ = 6$P_{50}$, but they are worse if $c$ = 6$P_{75}$ is used; even so the result continues to be better than that for RAA-EL and similar to AAC. A slight improvement is achieved using $c$ = $P_{75}$, but if we use $c$ = $P_{25}$ or $c$ = $P_{50}$, the results are worse, although better than those obtained by using AAC and RAA-EL. So, except for one case, $c$ = 6$P_{75}$, where the results are similar to those without robustifying, we obtain more robust results for all the $c$ considered.}
\begin{table}
\caption{{Frobenius norm of the difference between AAO and RAA-ME for different $c$.} \label{cnorm}}
\centering
\begin{tabular}{lrrr}
Multiplicative & \multicolumn{3}{c}{Percentile}\\
Factor & 25th & 50th & 75th\\ \hline
1 & 155.37 & 155.37 & 63.38\\
6 & 64.5 & 64.5 & 211.79
\end{tabular}
\end{table}
\subsection{Functional data} To check the robustness of our proposal, we now consider a set of $n$ = 100 functions that are generated from the following model, which was analyzed previously by \cite{ENV:ENV878}, \cite{FRAIMAN2013326} and \cite{Arribas} for functional outlier detection procedures. $n - \left\lceil cr \cdot n\right\rceil$ are generated from $X(t)$ = $30t(1-t)^{3/2} + \epsilon(t)$, whereas the remaining $\left\lceil cr \cdot n\right\rceil$ functions are generated from this contamination model: $30t^{3/2} (1-t) + \epsilon(t)$, where $t \in \left[ 0, 1 \right]$ and $\epsilon(t)$ is a Gaussian process with
zero mean and covariance function $\gamma (s,t)$ = $0.3 \exp \{ - \left| s - t \right| /0.3\}$. The functions are measured at 50 equispaced points between 0 and 1. A total of 100 simulations have been run with two contamination rates $cr$ = 0.1 and 0.15. Original and robust ADA have been applied with $k$ = 2 archetypoids. With $cr$ = 0.1, 10\% of the times one outlier belongs to the solution for original ADA, while no outlier is included as an archetypoid for robust ADA. With $cr$ = 0.15, 78\% of the times one outlier belongs to the solution for original ADA, while this percentage was only 32\% for robust ADA. Therefore, our proposal provides robust solutions.
{Let us analyze the influence that $c$ has on the results. Table \ref{cout} shows the percentage of times one outlier belongs to the robust ADA solution for different $c$ values. It seems that, for these data, a multiplicative factor of 1 gives more robust results than if we used a multiplicative factor of 6. Nevertheless, even with a multiplicative factor of 6 and for all the percentiles considered, the results are more robust than those obtained using the original ADA.}
\begin{table}
\caption{{Percentage of times one outlier belongs to the robust ADA solution for different $c$ and $cr$.} \label{cout}}
\centering
\begin{tabular}{lrrrrrr}
& \multicolumn{3}{c}{$cr$ = 0.1} & \multicolumn{3}{c}{$cr$ = 0.15} \\
Multiplicative & \multicolumn{3}{c}{Percentile} & \multicolumn{3}{c}{Percentile}\\
Factor & 25th & 50th & 75th & 25th & 50th & 75th\\ \hline
1 & 0 & 0 & 0 & 0 & 1 & 2\\
6 & 0 & 0 & 2 & 8 & 32 & 52
\end{tabular}
\end{table}
{Let us visually compare the solutions obtained for one of the simulation with $cr$ = 0.1, i.e. with 10 outliers, when an outlier is selected as an archetypoid by the original ADA algorithm. Remember that no outlier is selected as an archetypoid with our robust proposal. In Figure \ref{adapcafig}, we compare those solutions with the solutions obtained by PCA and robust PCA, as developed by \cite{doi:10.1198/004017004000000563} and \cite{Engelen_Hubert_Vanden_Branden_2016} and implemented in the function $robpca$ from the R package {\bf rospca} \citep{rospca}. PC 1 is nearly zero in the first half of the interval, but 0.2 in the second half of the interval. This result is highly influenced by the outliers. The two archetypoids with the original ADA return a similar base as with PCA. However, the robust versions of PCA and ADA return results that are similar to each other, but different from their respective non-robust versions. In both cases, the robust version returns solutions that are like a band \citep{doi:10.1198/jasa.2009.0108} of the non-contaminated data. The robust PC 1 is nearly constant (-0.15) along the entire interval. Depending on the multiple considered the band covering the data is more or less wide.}
\begin{figure}
\begin{center}
\begin{tabular}{cc}
\includegraphics[width=.5\linewidth]{fig2PCA.pdf} & \includegraphics[width=.5\linewidth]{fig2RPCA}
\\
(a) & (b) \\
\includegraphics[width=.5\linewidth]{fig2OADA.pdf} & \includegraphics[width=.5\linewidth]{fig2RADA.pdf}
\\
(c) & (d)
\end{tabular}
\end{center}
\caption{{Simulated model with 10 outliers. Data are shown in gray and outliers in black. (a) Effect of adding and subtracting a multiple of PC 1 to the mean curve. (b) Effect of adding and subtracting two different multiples of robust PC 1 to the center curve (the multiple is 2 for blue functions and 5 for red and purple functions). (c) Archetypoids with the original ADA algorithm. (d) Archetypoids with the proposed robust ADA algorithm.}
\label{adapcafig}}
\end{figure}
{The previous comparison is qualitative. To make a quantitative comparison we can take into account that outliers can be detected by their large deviation from the robust fit, as obtained with the function $robpca$. For the ADA methodology, we propose to compute the robust archetypoids with $c$ = $P_{50}$ because it is conservative and ensures a more robust solution without outliers as archetypoids. Then, we compute $||r_i||$ and a box plot is applied to this distribution to detect the outliers. We call this methodology RADAB. Furthermore, we also consider the method by \cite{HYNDMAN20074942} (ISFE), who use integrated square forecast errors and robust principal component analysis to detect outliers, with the function $foutliers$ and the option $HUoutliers$ from the R package {\bf rainbow} \citep{rainbow}. Table \ref{medidasoutliers} shows the results (True Positive Rate, TPR, False Positive Rate, FPR, and Matthews correlation coefficient, MCC) for different $cr$ values. $robpca$ has the highest TPR, but at the expense of having the highest FPR and the lowest MCC. With no outlier ($cr$ =0), ISFE has the lowest FPR, but FPR is smaller for RADAB with $cr$ = 0.1 and $cr$ = 0.15. On the other hand, RADAB reports excellent results with TPR, with nearly 100\% for TPR, which is not the case with ISFE. In fact, the maximum for MCC is achieved with RADAB in all the cases. }
\begin{table}
\caption{{Mean and standard deviation, in brackets, of TPR (percentage), FPR (percentage) and MCC for different $cr$.} \label{medidasoutliers}}
\tiny
\centering
\begin{tabular}{c@{\hspace{0.3\tabcolsep}}c@{\hspace{0.3\tabcolsep}}c@{\hspace{0.3\tabcolsep}}c@{\hspace{0.3\tabcolsep}}c@{\hspace{0.3\tabcolsep}}c@{\hspace{0.3\tabcolsep}}c@{\hspace{0.3\tabcolsep}}c@{\hspace{0.3\tabcolsep}}c@{\hspace{0.3\tabcolsep}}c@{\hspace{0.3\tabcolsep}}c@{\hspace{0.3\tabcolsep}}}
Method & $cr=0$ & \multicolumn{3}{c}{$cr$ = 0.05} & \multicolumn{3}{c}{$cr$ = 0.1} & \multicolumn{3}{c}{$cr$ = 0.15} \\
& FPR & TPR & FPR & MCC & TPR & FPR & MCC & TPR & FPR & MCC\\ \hline
$robpca$ & 10.75 (2.94) & 100 (0) & 8.89 (2.69) & 0.59 (0.06) & 100 (0) & 7.69 (2.36) & 0.74 (0.06) & 100 (0) & 5.94 (2.08) & 0.84 (0.05)\\
RADAB & 4.82 (2.14) & 99.40 (4.45) & 3.00 (1.98) & 0.80 (0.10) & 99.0 (7.18) & 1.58 (1.66) & 0.93 (0.09) & 97.93 (10.58) & 0.52 (0.89) & 0.98 (0.03)\\
ISFE & 2.99 (2.0) & 86.00 (29.75) & 2.94 (1.84) & 0.70 (0.26)& 84.90 (30.99) & 2.71 (1.77) & 0.78 (0.26) & 77.13 (33.10) & 2.41 (1.85) & 0.75 (0.29)
\end{tabular}
\end{table}
\section{Application}
\label{resultados} When dealing with time series, the theoretical complexity of many of the statistical methods available for analysis
leads to periodic summaries of the data series being commonly used in practice. Furthermore, many of the techniques, such as the classic Box-Jenkins theory (\cite{box1976time}), involve verifying a set of quite restrictive hypotheses, such as stationarity, equally-spaced observations or belonging to a specific kind of well-known processes. What makes our proposal attractive is not only the lack of these restrictive hypotheses (it could also be applied to sparsely measured time series), but also the data speak for themselves and the results can be interpreted easily by non-specialists. It also allows them to be visualized, which is an important task (\cite{JSSv025c01}).
To illustrate these claims, robust bivariate FADA is applied to the company stock prices as detailed below.
\subsection{Data}
We consider a data set from \cite{quantcuote2017}, which is composed of a collection of daily resolution data with the typical open, high, low, close, volume (OHLCV) structure. This collection runs from 01/01/1998 to 07/31/2013 for 500 currently active symbols in the S\&P 500. In addition, the aggregate S\&P 500 index OHLCV daily tick data (\cite{yahoofinance2017}) have been added to the data set. The length of this time series has been selected so that its range coincides with that of the QuantQuote disaggregated series. We have also collected information from
\cite{sectorSPDR2017}, which classifies stocks on the S\&P 500 index within ten major sectors, namely:
Consumer Discretionary (CD),
Consumer Staples (CS),
Energy (E),
Financials (F),
Health Care (HC),
Industrials (I),
Materials (M),
Real Estate (RE),
Technology (T), and
Utilities (U).
We are interested in extracting financially relevant information. With regard to investments, and more specifically to portfolio theory, it is widely accepted that the two key variables are risk and profitability (\cite{markowitz1952portfolio}). On the one hand, we have chosen as a measure of profitability in a time $t$ the aggregate returns over a period of length N, {$r_N$}. So, for each price series, $s_i,\, i=1,...,501$ ${r_N}^{s_i}(t)=\frac{X^{s_i}_{t}-X^{s_i}_{t-N}}{X^{s_i}_{t-N}}$
where $X_t^{s_i}$ is the value of the stock $i$ at the time $t$. In our case, we have chosen $N$ = 250, approximately the number of days that stock markets remain open in a year. On the other hand, we have chosen as a measure of volatility the beta or $\beta$ coefficients, which are widely used in portfolio theory. This is defined for a particular stock $s_i$ in a time $t$ as
$\beta_N^{s_i}(t)= \frac{Cov({r_N}^{s_i}(t),{r_N}^{index}(t))}{Var({r_N}^{index}(t))}$
where ${r_N}^{s_i}(t)$ stands for the aggregated returns of $s_i$ in time $t$ over the last $N$ days and ${r_N}^{index}(t)$ are the returns of the aggregated S\&P 500 index in the same period. To be consistent, we perform the calculations with a time frame of 250 days.
Although FDA can handle missing data well, as explained previously, our sample presents a different problem: some stocks do not exist throughout the entire time series. This is not because we have missing data, but because the companies were founded or were first listed on the stock exchange after 01/01/1998, i.e. the domains of the functions are different. Here we propose an alternative to deal with this problem, trying to maximize the size of our sample and minimize the stocks that must be discarded by taking into account observations since 2000-01-01 and dropping the stocks with more than 20\% missing values. In this way, only four companies from 500 are left out.
The following step is to transform discrete data to functional data. The {cubic B-spline} basis {with equally spaced knots} has been considered because the series are {not periodic}. As regards the number of bases $m$, as suggested by \cite{Ramsay05}, we computed an unbiased estimate of the residual variance using 4 to 22 bases
and selected $m$ = {13}, which is the number of bases that makes decreasing the residual variance substantially. In summary, for each stock $s_i$, both variables, return and beta coefficient in a 250 day time frame, could be expressed as the functions: $r_{250}^{s_i}(t)=\sum_{h=1}^{{13}}\mathbf{a}_h^{s_i}\phi_h(t)$ and $\beta_{250}^{s_i}(t)=\sum_{h=1}^{{13}}\mathbf{b}_h^{s_i}\phi_h(t)$ where $\mathbf{a}^{s_i}$ stands for the vector of coefficients on the basis functions for $r_{250}(t)$ corresponding to the particular stock $s_i$ and, in the same way, $\mathbf{b}^{s_i}$ stands for the vector of coefficients for $\beta_{250}(t)$ corresponding to the particular stock $s_i$, and $\phi_h$ are the {B-splines} basis functions. Therefore, after smoothing, the original data set of dimensions $496 \times 3422 \times 2$ is reduced to $496 \times {13} \times 2$.
As both functions are measured
in non-compatible units, each functional variable is standardized before analysis by standardizing the coefficients in the basis as explained by \cite{Epifanio2016}. This data set was analyzed in a non-robust way by \cite{MolEpi18}{, using the Fourier basis}.
\subsection{Robust bivariate FADA}
Table \ref{tb:ArchetypoidTable} shows the companies obtained as archetypoids for different $k$ values. When $k$ increases, the number of sectors represented in the set of archetypoids increases too. Since archetypoids are not nested, archetypoids may not coincide at all when the $k$ value varies. However, we find a nested structure in the order in which sectors appear, and in fact, some companies remain when $k$ increases, such as {AKAM or XLNX}.
\begin{table}
\centering
\caption{Archetypoids for different $k$ values. We use the same symbols for company names as \cite{sectorSPDR2017}. The abbreviation of the economic sector to which each company belongs appears in parentheses. \label{tb:ArchetypoidTable}}
\begin{tiny}
{
\begin{tabular}{lllllllllllll}
$k$ & A1 & A2 & A3 & A4 & A5 & A6 & A7 & A8 & A9 & A10 \\
3 &ALTR (T) & LNC (F) & WEC (U) & & & & & & & \\
4 & ALTR (T) & LNC (F) & NUE (M) & GIS (CS) & & & & & & \\
5 & ALTR (T) & HIG (F) & NUE (M) & GIS (CS) & KIM (RE) & & & & & \\
6 & BRCM (T) & IPG (CD) & NUE (M) & GIS (CS) & KIM (RE) & DNR (U) & & & & \\
7 & AKAM (T) & C (F) & WMT (CS) & HRL (CS) & BXP (RE) & TER (T) & SWN (E) & & & \\
8 & AKAM (T) & RF (F) & NUE (M) & GIS (CS) & EQR (RE) & XLNX (T) & NBR (E) & CVC (CD) & & \\
9 & AKAM (T) & MS (F) & X (M) & SO (U) & AIV (RE) & XLNX (T) & SWN (E) & IPG (CD) & BRKB (F) & \\
10 & AKAM (T) & MS (F) & ATI (M) & FLIR (T) & KIM (RE) & XLNX (T) & EOG (E) & GCI (CD) & CMCSA (CD) & GIS (CS) \\
\end{tabular}}
\end{tiny}
\end{table}
In the interests of brevity and as an illustrative example we analyze the
results of $k$ = {4}, which is the value selected by the elbow criterion. We include a brief description of the data-driven selected companies based on the information in \cite{sectorSPDR2017}: {Altair Engineering Inc. (ALTR), together with its subsidiaries, provides enterprise-class engineering software worldwide; Lincoln National Corp.(LNC) is a holding company. Through subsidiary companies, the company operates multiple insurance and investment management businesses; Nucor Corporation (NUE) manufactures and sells steel and steel products in the United States and internationally; General Mills, Inc. (GIS) is a leading global food company. Its brands include Cheerios, Annie's, Yoplait, Nature Valley, Fiber One, Haagen-Dazs, Betty Crocker, Pillsbury, Old El Paso, Wanchai Ferry, Yoki and others.}
Figure \ref{fig:archoverlaped} shows the functions of each variable for the {4} archetypoids.
{It can be seen that GIS is a company that, in comparison with the rest of the archetypoids, presents low and constant values for both variables. Looking at ALTR, it presents the highest returns and volatilities at the beginning of the time series probably as a result of the .com bubble. LNC presents the typical profile of a financial company, with moderate profitability and volatility during the first three quarters of the time series. Once the crisis broke out in 2007, volatility shot up to unprecedented levels while profitability plummeted. Finally, NUE is characterized by having bell-shaped functions, that is, with relatively low values at the extremes of the temporal domain and higher values at the center.}
\begin{figure}
\begin{center}
\includegraphics[width=.7\linewidth]{Arch_overlacped.pdf}
\end{center}
\caption{$r_{250}$ and $\beta_{250}$ functions of the {4} archetypoids.
\label{fig:archoverlaped}}
\end{figure}
What makes this technique very interesting is that the companies are expressed as mixtures of those extreme profiles through the $\alpha_{ij}$ values. For a visual overview of the S\&P 500 stocks, we propose the following procedure that provides a data-driven taxonomy of the S\&P 500 stocks. The idea is to group together companies that share similar $\alpha_{ij}$ profiles. A simple method to do this is to establish a threshold $U$, so that if the weight of an archetypoid for a given individual is greater than $U$, we will say that this subject is in the cluster generated by this archetypoid. If we repeat this process for each archetypoid, we will generate {4} ``pure" clusters of subjects, i.e., clusters of subjects that are represented mostly by a single archetypoid. To take this a little further, this process is repeated with combinations of two archetypoids. Thus, we will group in the same cluster the individuals whose sum of {$\alpha$} weights for two concrete archetypoids is greater than $U$, thus generating ${\binom{4}{2}=6}$ additional clusters. In this case, it might happen that for a given subject, there is more than one combination of archetypoids whose sum exceeds $U$. So as not to complicate the graphic representation, we will classify these subjects generically as mixtures, even though these mixtures will be composed of different sets of archetypoids.{Note that we do not carry out a clustering method, but we form clusters according to the alpha values.}
Figure \ref{fig:red80r} condenses all the information extracted by means of a network that is built through an adjacency matrix that gathers the cluster information. The smaller $U$ is, the larger the number of companies that belong to any cluster, and vice versa. We have chosen a value of $U$ = 0.8, which is high but not so much, i.e. $U$ = 0.8 is a breakeven between having a considerable number of companies, but at the same time not too many to be able to read the graphic representation concisely. Archetypoids are highlighted with a gray square, the lines and color codes allow us to differentiate the structure of the clusters, and different sectors are represented through different geometrical shapes.
\begin{sidewaysfigure}
\centering
\includegraphics[width=1\linewidth]{red80_A4.pdf}
\caption{Cluster structure with $U$=0.8. Sector indicates the economic sector and tipo the type of cluster. \label{fig:red80r}}
\end{sidewaysfigure}
{Starting with the pure clusters we see that NUE generates a small heterogeneous group with 3 other companies.
GIS is the archetypoid that generates the largest cluster on its own. Most of the companies in this cluster belong to the C. Staples sector, although we also find some from the Utilities sector.
LNC generates a small cluster with seven other companies. Five of them, Bank of America Corp (BAC) and Hartgord Financial Services Group INC. (HIG), American International Group Inc (AIG), Fifth Third Bancorp (FITB ) and XL Group Ltd (XL) belong to the same sector and have similar profiles. The other two are Gannett Co. Inc. (GCI) related to the publishing industry and Host Hotels and Resorts Inc (HST), a real estate investment trust.
For its part, ALTR generates a larger cluster in which 10 other companies are included. Most of these companies belong to the Technology sector. }
{
Regarding mixed clusters, we will point out some general characteristics, since detailing all the relationships shown in Figure \ref{fig:red80r} would be too extensive.
NUE and GIS form the largest cluster for this level of $U$. Focusing on this, the NUE-GIS cluster has just a few companies from the C. Staples sector, since these companies are mainly classified in the cluster generated by GIS alone. Many of the companies classified in the NUE-GIS cluster belong to the Utilities, Energy and Health sectors.
It could be said that the common feature of the companies in this cluster is that they are not very sensitive to the economic cycle.}
{
The following most important clusters according to their size are LNC-GIS and ALTR-GIS. The LNC-GIS cluster is certainly heterogeneous in terms of the sectors that comprise it.
However, it can be seen that the majority of Financial and Real Estate companies are part of this group.}
{
The third largest cluster is formed by ALTR-GIS and is quite homogeneous. The majority of the companies in this group belong to the technology sector. If we look at the other two clusters in which ALTR intervenes we see that this feature is repeated. Both ALTR-NUE and ALTR-LNC have a large presence of technology companies and, what is more, we do not find technological companies outside these groups.}
{
The last cluster is formed by LNC-NUE. It is formed by 14 companies and the sector that appears the most times is the C. Discretionary sector.
}
Now we analyze the composition of the sectors managed by market analysts to evaluate the performance of the results qualitatively. Figure \ref{fig:archbysector} shows the normalized relative weight of archetypoids in each of the ten sectors. {It can be seen that each archetypoid represents the component with the greatest weight of the sector to which it belongs. Thus, LNC accounts for more than 45\% of the financial sector, GIS represents almost 80\% of the weight of the C. Staples sector and in the technology sector, the weight of ALTR exceeds the weight of the other three archetypoids.
}
\begin{figure}
\centering
\includegraphics[width=0.8\linewidth]{sectors_by_archetypes.pdf}
\caption{Relative weight of archetypoids in each sector.\label{fig:archbysector}}
\end{figure}
{But it is not only interesting to analyze the weights of the most relevant components. The composition of the mixtures also gives clues about the similarities and dependencies between the different sectors. For example, if we compare Consumer Discretionary sector with Consumer Staples sector, we see that the weights keep a proportion that we would expect. For example, the companies that manufacture durable goods, which are included in the Consumer Discretionary sector, have a direct relationship with those that provide the investment to finance these purchases represented by the LNC archetypoid, which is why that archetypoid has higher weight in this sector.
On the other hand, companies that provide basic or non-durable goods, which belong to the Consumer Staples sector, have a minor relationship with the financial sector. It may be obvious, but it is worth emphasizing that, by definition, non-durable goods are those that are purchased without resorting to financing. Instead, this sector has a great similarity with the Utilities sector (distributors of electricity, water, gas, etc.). This makes economic sense, since basic goods and services distributed by companies in the Utilities sector have similar demand curves. In other words, in the expansive cycles of the economy, consumers decide to increase their investments in goods that require financing such as a car, a washing machine or a computer. However, households' spending on electricity, water, gas or telecommunications will remain relatively constant, as will spending on basic products such as bread, milk, oil, soap and toothpaste.
}
{
Regarding the composition of the energy sector (extractors of oil, gas etc), the small weight of the technological archetypoid is noteworthy. This may point to the weak relationship between the technologies used in each sector. On the one hand, the Energy sector carries out activities where heavy machinery and mechanical operations in general (drilling, mining, extraction etc.) are fundamental. This is completely opposite to the dynamics that prevail in the technological universe, where the main elements are computer applications, digital technology and patents.}
{The Industrials and Materials sectors have similar profiles, which shows the strong interrelation between them. Regarding the Real Estate sector, we see how the LNC archetypoid, belonging to the financial companies sector, has the greatest weight outside its own sector. The relationship between these two sectors is also evident. Finally, we can say that the Utilities and C. Staples sectors are in some way the purest, in the sense that the weight of the dominant archetypoid of this sector is greater than the weight of the other three archetypoids all together.
}
\section{Conclusion}
\label{conclusiones}
This paper introduces robust archetypal analysis (AA and ADA) for multivariate data and functional data, both for univariate and multivariate functions. A simulation study has demonstrated the good performance of our proposal.
Furthermore, the application to the time series of stock quotes in the S\&P 500 from 2000 to 2013 has illustrated the potential of these unsupervised techniques as an alternative to the commonly used clustering of time series, which are usually described by complex models.
Understanding the results of these or other statistical learning models is not always an easy task. Additionally, if these results have to be explained to a public with little or no mathematical knowledge, things can be even worse. In this regard, ADA is particularly suitable since its results can be interpreted in a very simple way by any non-expert person. For instance, anyone with minimal knowledge of investments understands what we mean if we say that a certain stock behaves like a mixture of {Nucor Corporation and Lincoln National} stocks. Another advantage of the applied model is that the functional version of archetypoid analysis (FADA) allows us to condense vectors of observations of any length into a few coefficients, which provides an improvement in computational efficiency and makes this method highly recommended when working with long time series.
With regard to the financial conclusions, in the first place, it has been seen that when we increase the number of chosen archetypoids, the Technology sector appears repeatedly while other sectors do not appear. Therefore, there are companies within this sector that exhibit very different behaviors, such as {XLNX}, FLIR and AKAM.
Furthermore, we have proposed a visual representation of the companies through the definition of clusters based on the mixtures obtained.
Finally, we have analyzed the sectors according to the normalized relative weight of archetypoids that compose each sector and we have seen that some sectors present certain similarities. It is worth mentioning that sectors like Consumer Discretionary, Materials or Industrials offer better opportunities for diversifying risks, since their composition is more heterogeneous. On the other hand, sectors such as Utilities or Consumer Staples present more homogeneous structures, where the weight of the dominant archetypoids of the sector can exceed {70\%}.
As regards future work, from the mathematical point of view, {an open question, both in the real and functional case, is the selection of the tuning parameter of the loss functions. The research line of using robust archetypal analysis for outlier detection is another open path. Another} open problem is the extension of archetypal analysis to mixed data (functional and vector parts). An appropriate definition of the inner product is needed {since functional and vector parts will not be measured in directly compatible units}. Nevertheless, applications, even beyond econometrics, are the main direction of future work. Even so, the application of these models to the world of finance is still a relatively unexplored field. The application of models with functional data allows us to take into account variables collected with different frequencies, such as daily quotes, quarterly balances or annual results, which makes these models especially suitable for financial time series.
Taking this into account, a future development may be to extend the implementation of the bivariate model to a $P$-variable model that makes it possible to work with a large amount of data from each company. From a financial point of view, it may be possible to develop investment strategies using the results shown here to improve performance and reduce the risk of investment decisions. {In fact, an interesting open problem to study is whether the estimated archetypoids may be useful for constructing small portfolios of stocks or a variety
of small (i.e., $k$ stock) portfolios.}
\section{Acknowledgements}
{The authors are grateful to the Editor and reviewers for their very
constructive suggestions, which have led to improvements in the manuscript.}
This work is supported by DPI2017-87333-R from the Spanish Ministry of Economy and Competitiveness (AEI/FEDER, EU) and UJI-B2017-13 from Universitat Jaume I.
|
1001.1786
|
\section{Introduction}
\label{sec:intro}
The spontaneous chiral symmetry breaking plays a central role
in the low-energy Quantum Chromodynamics (QCD).
It is understood that this phenomena is the source of the hadron masses
of order $\Lambda_{\mathrm{QCD}}$, the QCD scale.
An important exception is the pion, which is nearly massless, as it is
a pseudo-Nambu-Goldstone boson.
The pion dynamics is well described by an effective theory, known
as chiral perturbation theory (ChPT) \cite{Gasser:1983yg}, which is
constructed based on the pattern of spontaneous symmetry breaking.
Parameters in ChPT are not known a priori.
In phenomenological analysis, they are determined with experimental
data as inputs, but it is more desirable if they can be calculated
starting from the first-principles of QCD.
This sets a challenge for lattice QCD.
At the leading order, there are two parameters: the chiral condensate
$\Sigma$ and pion decay constant $F$ in the chiral limit.
Calculation of these parameters has long been one of the main
issues in lattice QCD.
In particular, the calculation of $\Sigma$ has been notoriously
difficult, as it survives only in the thermodynamical limit,
{\it i.e.} the limit of massless sea quarks after taking infinite
volume limit.
The determination of other parameters in ChPT, such as the low energy
constant at the next-to-leading (NLO) order can be done only after the
leading order parameters are determined precisely.
Lattice QCD has become the most powerful tool for non-perturbative
calculation of strong interaction of hadrons, with the help of the
rapid speed-up of computers.
In fact, lattice QCD has even played a leading role in the development
of high-end computers.
Still, the mechanism of chiral symmetry breaking remained not entirely
clear until recently, since the chiral symmetry itself was violated in
the simulations with the conventional lattice fermion formulations.
It is theoretically known that the use of Neuberger's overlap-Dirac
operator \cite{Neuberger:1997fp} is a solution to this problem
as it realizes exact chiral symmetry at finite lattice spacings
\cite{Ginsparg:1981bj,Luscher:1998pq}.
Because of its numerical cost, however, it was only recently
that the large-scale simulation of dynamical overlap fermions became
feasible.
The numerical cost of the overlap-Dirac operator is high, compared to
other non-chiral or non-flavor-symmetric lattice fermions, as it
involves an approximation of the sign function of the hermitian
Wilson-Dirac operator.
The cost increases even more
when the Atiyah-Singer index of the Dirac operator, which corresponds
to the topological charge of the background gauge field, changes its
value by $\pm$1.
This is because the molecular dynamics steps have to involve an extra
procedure \cite{Fodor:2003bh} in order to catch a sudden jump of the
fermion determinant on the topology boundary.
This additional procedure, known as the reflection/refraction, needs
numerical cost potentially proportional to the lattice volume squared.
Recently, the JLQCD and TWQCD collaborations have performed
large-scale simulations of 2- and 2+1-flavor QCD
employing the overlap fermions for sea quarks \cite{Aoki:2008tq}.
We avoid the extra numerical cost due to the change of topology
by a modification of the lattice action to suppress the topology
tunneling as proposed in
\cite{Izubuchi:2002pq, Vranas:2006zk, Fukaya:2006vs}.
Our lattice simulations are confined in a fixed topological
sector, so that an expectation value of any operator could be
deviated from the value in the true QCD vacuum.
This effect can be understood as a finite volume effect and estimated
in a theoretically clean manner as discussed below.
It is worth noticed that the simulation parameters contain those in
the $\epsilon$-regime on a $L\sim 2$~fm lattice, as well as in the
conventional $p$-regime
\cite{Fukaya:2007fb, Fukaya:2007yv, Fukaya:2007pn}.
This enables us to study the chiral dynamics in an entirely different
set-up and to determine the low-energy constants at the point very
close to the chiral limit.
With exact chiral symmetry, the study of spontaneous chiral symmetry
breaking is theoretically clean, but it still requires a good control
of the systematic effects due to the finite volume \cite{DeGrand:2006nv}.
For such infra-red effects, the lightest particle, which is the pion,
gives a dominant contribution.
It should therefore be possible to use analytic calculations within
ChPT in order to predict the finite volume corrections for a quantity
of interest.
Then, the lattice results can be directly fitted with these finite-volume
formulae of ChPT to determine the relevant low-energy constants.
The effect of fixed topology can also be understood as one of such
infra-red effects since the {\it global} topological charge should not
affect the physics at a local sub-volume when the entire volume $V$ is
large enough \cite{Brower:2003yx,Aoki:2007ka}.
In a calculation of the topological susceptibility
\cite{Aoki:2007pw,Chiu:2008kt, Chiu:2008jq, Hsieh}
through topological charge density correlator,
we can actually see that local topological excitations are active even
when the global topological charge is kept fixed.
Its result is consistent with an expectation of ChPT, which implies
that the ChPT-based analysis is valid for the effects due to the fixed
topological charge\cite{Mao:2009sy, Aoki:2009mx}.
There have been a number of analytical works that aimed at controlling
the infrared effects occurring in the lattice simulations.
A well-known example is the finite volume correction due the pions
wrapping around the lattice \cite{Bernard:2001yj,Colangelo:2005gd}.
Extended works are necessary when the system enters the so-called
$\epsilon$-regime \cite{Gasser:1987ah,Leutwyler:1992yt} by reducing
the sea quark mass to the vicinity of the chiral limit.
In this regime, the vacuum fluctuation of the pion field plays a
special role and a non-perturbative approach is needed in ChPT.
Namely, the zero-momentum pion mode has to be integrated over the
group manifold of the chiral symmetry in contrast to the case of the
conventional $p$-regime where a certain vacuum is (randomly) chosen by
the spontaneous symmetry breaking.
Recently, the partition functions with fully non-degenerate flavors
\cite{Splittorff:2002eb} were calculated, so that even
the (partially) quenched analysis \cite{Bernardoni:2007hi} of the
meson correlators is possible.
To study more realistic set-up, {\it i.e.} including the strange quark
in the $p$-regime, several hybrid method to treat both the $\epsilon$-
and $p$-regimes have been proposed
\cite{Bernardoni:2008ei,Damgaard:2008zs}.
The effect of fixed topology is worked out in
\cite
Aoki:2009mx}.
We also note that the effects of explicit violation of chiral symmetry
due to the Wilson term are also discussed \cite{Bar:2008th,Shindler:2009ri},
which is needed to study the Wilson fermion simulations near the chiral limit
\cite{Hasenfratz:2008ce, Giusti:2008vb}.
In this talk, the dynamical overlap fermion simulation
by the JLQCD and TWQCD collaborations
is reviewed in Section~\ref{sec:ov}.
In Section~\ref{sec:ChPT}, we discuss the finite size scaling
as well as the global topological effects within ChPT.
As an example, our recent result for chiral condensate
\cite{:2009fh, Hashimoto:2009iv, Hashimoto:2009iy}
is presented in Section \ref{sec:cond}.
Summary and conclusion are given in Section \ref{sec:summary}.
\section{Dynamical overlap fermion at fixed topology}
\label{sec:ov}
We employ the overlap-Dirac operator
\cite{Neuberger:1997fp}
\begin{equation}
\label{eq:ov}
D(m) =
\left(m_0+\frac{m}{2}\right)+
\left(m_0-\frac{m}{2}\right)
\gamma_5 \mbox{sgn}[H_W(-m_0)],
\end{equation}
for the quark action. Here $m$ denotes the quark mass and
$H_W\equiv\gamma_5D_W(-m_0)$ is the Hermitian
Wilson-Dirac operator with a large negative mass $-m_0$.
We take $m_0=1.6$ throughout our simulations.
(Here and in the following the parameters are
given in the lattice unit.)
In the chiral limit $m\to 0$,
the overlap-Dirac operator (\ref{eq:ov}) satisfies
the Ginsparg-Wilson relation \cite{Ginsparg:1981bj}
\begin{equation}
\label{eq:GW}
D(0)\gamma_5 + \gamma_5D(0)=\frac{1}{m_0}D(0)\gamma_5D(0).
\end{equation}
With this relation, the fermion action constructed from
(\ref{eq:ov}) has exact chiral symmetry under a
modified chiral transformation \cite{Luscher:1998pq}.
Moreover, it is known that the overlap-Dirac operator has
an index which corresponds to the topological charge
in the continuum limit \cite{Hasenfratz:1998ri}.
In the numerical implementation of the overlap-Dirac operator
(\ref{eq:ov}), the profile of near-zero modes of the kernel
operator $H_W(-m_0)$
largely affects the numerical cost of the overlap fermion
(The presence of such near-zero modes is also a problem for
the locality property of the overlap operator \cite{Hernandez:1998et}.).
For the approximation of the sign function in
(\ref{eq:ov}), the number of operations of the Wilson-Dirac operator
needed to keep a certain precision monotonically increases as the
condition number
$\lambda^{max}_{W}/\lambda^{min}_{W}$ grows,
where $\lambda^{max/min}_W$ denotes
the maximum/minimum eigenvalue of the operator $|H_W(-m_0)|$.
Moreover, since the overlap-Dirac operator is not uniquely determined
when $H_W(-m_0)$ has a zero eigenvalue,
the overlap fermion determinant has a discontinuity.
This discontinuity of the determinant prevents
smooth evolution of the molecular dynamics steps and
requires a special treatment,
known as the reflection/refraction procedure \cite{Fodor:2003bh}.
It needs an extra numerical cost, which is potentially
proportional to the lattice volume squared.
At currently available lattice spacings with
conventional gauge actions,
the spectral density $\rho_W(\lambda_W)$ of the
operator $H_W(-m_0)$ is
non-zero at zero eigenvalue $\lambda_W$ = 0 \cite{Edwards:1998sh}.
Note that the appearance of $\rho_W(\lambda_W=0)$ is, however,
a lattice artifact due to the so-called dislocations:
local lumps of gauge configurations \cite{Berruto:2000fx},
which disappears in the continuum limit.
To avoid the problem of the large extra numerical cost and of the
potentially ill-defined overlap operator,
we introduce additional Wilson fermions and
twisted-mass bosonic spinors to generate a weight
\begin{equation}
\label{eq:detHw}
\frac{\det[H_W(-m_0)^2]}{\det[H_W(-m_0)^2+\mu^2]},
\end{equation}
in the functional integrals
\cite{Izubuchi:2002pq, Vranas:2006zk, Fukaya:2006vs}.
Both of fermions and ghosts are unphysical
as their masses are of order of the lattice
cutoff, and thus do not affect low-energy physics.
The numerator suppresses the appearance of near-zero modes,
while the denominator cancels unwanted effects from higher modes.
The ``twisted-mass'' parameter $\mu$ controls the value below which
the eigenmodes are suppressed.
In our numerical studies, we set $\mu$ = 0.2.
\begin{figure}[tbp]
\centering
\includegraphics[width=10cm,clip=true]{hist_qrg.eps}
\caption{
Histogram of the spectral density of $H_W(-m_0)$.
Data for three values of $\mu$ ($\mu$ = 0.0, 0.2, and
0.4) are shown in the plot.
Note that $\mu=0$ corresponds to the case where the extra
fermion determinant is turned off.
}
\label{fig:eigen_hist}
\end{figure}
As Fig.~\ref{fig:eigen_hist} shows,
the near-zero modes of $H_W(-m_0)$
are actually washed out when $\mu$ is non-zero
in quenched QCD simulations.
This leads to a large reduction of the numerical cost
to approximate the sign function in (\ref{eq:ov}) \cite{Fukaya:2006vs}.
We also find that the molecular dynamics evolution
is smooth in the hybrid Monte Carlo updates
and we can turn off the
reflection/refraction procedure.
\if0
In our simulations, the sign function is
approximated by a rational function of the form
(see, {e.g.}, \cite{vandenEshof:2002ms,Chiu:2002eh})
\begin{equation}
\label{eq:rational}
\frac{H_W}{\sqrt{H_W^2}} = H_w \times \frac{d_0}{\lambda_{th}}
(h_W^2+c_{2n}) \sum_{l=1}^n \frac{b_l}{h_W^2+c_{2l-1}},
\end{equation}
where $\lambda_{th}$ is the lower threshold of the range of
approximation and $h_W\equiv H_W/\lambda_{th}$.
The coefficients $b_l$, $c_l$ and $d_0$ are
determined analytically (the Zolotarev approximation)
so as to optimize the accuracy of the approximation.
Since we have to fix the lower limit $\lambda_{th}$, we
calculate a few lowest-lying eigenvalues and project them
out before applying (\ref{eq:rational})
when their absolute value is smaller than $\lambda_{th}$.
The value of $\lambda_{th}$ is set to 0.144 in our studies.
The accuracy of the approximation improves exponentially
as the number of poles $n$ increases.
With $n=10$, the sign function $\mbox{sgn}[H_W(-m_0)]$ is
approximated to a $10^{-8}$-$10^{-7}$
level.
Since the multi-shift conjugate gradient method can be used
to invert all the $(h_W^2+c_{2l-1})^{-1}$ terms at once, the
numerical cost depends on $n$ only weakly.
\fi
The presence of zero-mode of $H_W(-m_0)$ is related to a topology
change:
the Atiyah-Singer index or the topological charge of gauge fields
changes its value when an eigenvalue $H_W(-m_0)$ crosses zero.
The condition $H_W(-m_0)=0$, thus, forms a topology boundary
on the gauge configuration space.
With the lattice action including (\ref{eq:detHw}), therefore,
the topological charge never changes
during the molecular dynamics steps of the Hybrid Monte Carlo
(HMC) simulations.
In this work, the simulations are mainly performed in the trivial
topological sector $Q=0$.
In order to check the topological charge dependence,
we also carry out independent simulations at $Q=+1$, $-2$ and $-4$
at some parameter choices.
The configuration space of a given fixed topology is simply
connected in the continuum limit, hence it is natural to assume
that the ergodicity of the Monte Carlo simulation is satisfied within
in a given topological sector.
In the Monte Carlo simulations, we choose 5-6
different points of the up and down quark mass $m_{ud}$
in a range $0.002 \leq m_{ud}\leq 0.100$.
For the $N_f=2+1$ runs, two values of the strange quark mass:
$m_s=0.080$ and 0.100 are taken.
For the gauge part, we use the Iwasaki gauge action \cite{Iwasaki:1985we}
at $\beta=2.3$ (except for the case of
$m_{ud}=0.002$ in the $N_f=2$ run where $\beta=2.35$ is chosen).
The lattice volumes are $V=L^3T=16^3\times 32\;(N_f=2)$
and $V=L^3T=16^3\times 48\;(N_f=2+1)$.
For the latter, we also carry out a run on a
$V=L^3T=24^3\times 48$ lattice at $m_{ud}=0.025$ and $m_s=0.080$,
in order to check the finite volume effect.
The lattice scales $a^{-1}=1.667$ GeV ($N_f=2$) and
$a^{-1}=1.833$ GeV ($N_f=2+1$) are determined
from the heavy quark potential, using $r_0=0.49$ fm as an input
\cite{Sommer:1993ce}.
The lattice size is then estimated as $L\sim 1.9$ fm for
$N_f=2$, and $L\sim 1.7$ fm for $N_f=2+1$ runs.
Note that for the lightest quark mass $m_{ud}=0.002$ $\sim3$ MeV,
the system of pions is inside the $\epsilon$-regime.
Since our gauge configurations are generated
in a fixed topological sector,
expectation value of any operator
could be different from those in the QCD vacuum.
Also, our lattice size is $\sim 2$ fm and
considerable finite volume effects, especially in the
$\epsilon$-regime, are expected.
As our lattice size is, however, still kept
larger than the inverse of QCD scale,
{\it i.e.} $\Lambda_{QCD}L \gg 1$,
both effects can be considered as a part of
infra-red physics for which pions
are most responsible.
We, therefore, expect that chiral perturbation theory
(ChPT) can correct these systematic effects.
In the next section, we discuss how to evaluate
physical observables in a fixed topological sector within ChPT
at finite $V$.
Non-perturbative treatment of the zero momentum mode,
as well as the Fourier transform
with respect to the vacuum angle $\theta$, play a key role.
Using their analytic formulae, we can
convert the lattice QCD results on a finite lattice to
the values in the true QCD vacuum
in the infinite space-time volume.
\section{Finite $V$ and fixed $Q$ effects
within ChPT}
\label{sec:ChPT}
In this section, we first discuss how to evaluate
the effect of fixing topology.
A general argument leads to a consequence that
the dependence on the global topological charge
only appears with a suppression factor $1/V$.
Namely, it is a part of the finite volume effects.
Recent studies of the finite volume scaling
within ChPT are then reviewed.
Once we assume that the heavier hadrons, such as
rho mesons, baryons {\it etc}, are all decoupled from
the theory at the scale of $1/V^{1/4}$,
only pions describe the difference of the
finite volume system from the infinite volume one.
We discuss, in particular,
a non-perturbative approach to integrate over the chiral
field's vacuum, which is necessary in the $\epsilon$-regime.
\subsection{Topology as an infra-red physics}
Let us start our discussion with an intuitively noticeable difference
between the trivial topological sector ($Q=0$) and
the first non-trivial one ($Q=1$).
In the weak coupling limit $g\ll 1$, it is well-known that a self-dual
solution, the so-called one-instanton solution, dominates
the configuration space of the $Q=1$ sector
and its relative weight is given by $\sim \exp(-8\pi^2/g^2)$.
For larger value of $Q$, the weight is expected to be
$\sim \exp(-8\pi^2|Q|/g^2)$.
As the coupling constant becomes strong,
$g\sim 1$, more complicated configurations with
many pairs of instantons and anti-instantons
are more favored, since the entropy gives more impact
on the free energy than the action density.
Suppose that the number of such pairs generated
in a typical configuration is $Q_{ave}$.
The trivial sector $Q=0$ then has
$Q_{ave}$ instantons and $Q_{ave}$ anti-instantons
while in the $Q=1$ sector $(Q_{ave}+1)$ instantons
and $Q_{ave}$ instantons are there.
As $Q_{ave}$ grows, the difference between the global topological charge,
$Q=0$ and $Q=1$ would become less important.
If the theory has a mass gap $\Lambda_{gap}$
(it is natural to assume $\Lambda_{gap}=\Lambda_{QCD}$ for
the pure gauge theory while $\Lambda_{gap}$ is the pion mass $m_\pi$ for QCD),
the typical size of an instanton or anti-instanton
should be given by $1/\Lambda_{gap}$ and their
density is estimated as $\sim \Lambda_{gap}^4$.
The value of $Q_{ave}$ discussed above
is then estimated by $\sim \Lambda_{gap}^4 V$
and one can easily see how the
difference between $Q=0$ and $Q=1$ (or higher)
disappears as $\sim 1/V$ when $V$ is sent to infinity
or equivalently $Q_{ave}\to \infty$.
The effect of the global topological charge thus
should be understood as a finite volume effect.
Brower et al. \cite{Brower:2003yx} and Aoki et al. \cite{Aoki:2007ka}
gave a more theoretical and solid formulation for the effect of the
global topological charge.
The partition function of the theory at a fixed topological
charge $Q$ is obtained from those at the $\theta$ vacua
by a Fourier transformation
\begin{eqnarray}
Z_Q &=& \int d\theta\; e^{i\theta Q}Z(\theta) = \int d\theta\;
e^{i\theta Q}\exp(-f(\theta)V),
\end{eqnarray}
where $f(\theta)$ denotes a free-energy density of the $\theta$ vacuum.
When the vacuum angle $\theta$ is small,
$f(\theta)$ can be expanded in $\theta^2$ as \cite{Vafa:1984xg}
\begin{eqnarray}
f(\theta) &=& \frac{\chi_t}{2}\theta^2 + c_4 \theta^4 +c_6 \theta^6+ \cdots,
\end{eqnarray}
where a constant term is omitted.
Here, $\chi_t$ corresponds to the topological susceptibility.
Assuming that all the constants, $\chi_t$, $c_4$, $c_6$ {\it etc}.
are of the order of $\sim (\Lambda_{gap})^4$
and the volume is large enough to satisfy $L \Lambda_{gap}\gg 1$,
the above $\theta$ integral can be
evaluated by a saddle-point expansion as
\begin{eqnarray}
Z_Q &=& \frac{1}{\sqrt{2\pi \chi_t V}}
\exp\left(-\frac{Q^2}{2\chi_t V}\right)
\left[1-\frac{c_4}{8\chi_t^2 V}+\cdots\right],
\end{eqnarray}
which clearly shows that the global topological charge dependence
disappears in the limit $V\to\infty$.
It is also important to notice that the distribution of
the global topological charge converges
to the Gaussian distribution
as the volume increases, which agrees well with
the intuitive picture above that only the entropy given by
the distribution of instantons and anti-instantons becomes
important in the thermodynamical limit.
Under those minimal assumptions on the vacuum free energy, one can
prove that $\chi_t$, $c_4$ {\it etc}. can be extracted from lattice
QCD simulations at a fixed topological charge \cite{Aoki:2007ka}.
For instance, $\chi_t$ appears as a constant mode in the two-point
correlator in the flavor singlet channel
\begin{eqnarray}
\langle \eta^\prime (x)\eta^\prime (y)\rangle_Q
&=& -\frac{\chi_t}{V} + {\cal O}(1/V^2)+{\cal O}(e^{-m_{\eta^\prime}|x-y|}),
\end{eqnarray}
for a large separation $|x-y|$.
The excitation in this channel corresponds to the $\eta'$
meson whose non-zero mass is given by
$m_{\eta^\prime}$.
The constant correlation has a negative sign when the global
topological charge $Q$ is zero, because at long distances there is
more chance to find oppositely charged local topological excitations
when the sum is constrained to zero.
In the numerical simulations \cite{Aoki:2007pw, Chiu:2008kt, Chiu:2008jq, Hsieh}
the presence of this constant mode is confirmed
as Fig.~\ref{fig:etaprime} shows.
Moreover, the extracted values of $\chi_t$ via above formula
are found to agree with the ChPT prediction
\cite{Leutwyler:1992yt}
\begin{equation}
\chi_t = \frac{\Sigma}{\sum_f^{N_f}1/m_f},
\end{equation}
as seen in Fig.~\ref{fig:chit}.
The value of $\Sigma$ extracted from this analysis is consistent with
a nominal value $\Sigma\simeq$ (250~MeV)$^3$.
Chiral fit including the next-to-leading chiral corrections
\cite{Mao:2009sy,Aoki:2009mx} is underway.
\begin{figure}[tbp]
\centering
\includegraphics[width=10cm,clip=true]{etaprime.eps}
\caption{
The eta-prime correlator (circles)
at $m=0.002$ and $Q=0$ obtained in the two-flavor QCD simulation.
A negative constant contribution is seen.
The triangles are its connected and disconnected diagram parts.
}
\label{fig:etaprime}
\end{figure}
\begin{figure}[tbp]
\centering
\includegraphics[width=10cm,clip=true]{chit.eps}
\caption{
$\chi_t$ extracted from the $\eta'$ meson correlators.
A good agreement with ChPT predictions (solid lines)
is seen both in the $N_f=2$ and $N_f=2+1$ lattice data.
}
\label{fig:chit}
\end{figure}
There are two remarkable conclusions that may be drawn from
these lattice data.
First, local fluctuation of topology
exists even when the global topological charge is fixed
in Monte Carlo simulations.
There was some doubt about the ergodicity of the Monte Carlo
simulation with the topology fixing term, but as far as the numerical
data imply there is no evidence of the problem.
Second, the topological charge actually feels the presence of
dynamical fermions and the $\chi_t$ vanishes in the chiral limit as
expected from ChPT.
Topology is a part of the infrared physics that can
be well described by the pion physics.
\subsection{Finite $V$ and fixed $Q$ within ChPT}
The Lagrangian of ChPT is given by \cite{Gasser:1983yg}
\begin{eqnarray}
\mathcal{L}&=& \frac{F^2}{4}
{\rm Tr}[\partial_\mu U(x)^\dagger \partial_\mu U(x)]
-\frac{\Sigma}{2}{\rm Tr}
[\mathcal{M}^\dagger e^{-i\theta/N_f}U(x)
+U(x)^\dagger e^{i\theta/N_f}\mathcal{M}]+\cdots,
\end{eqnarray}
where the chiral field $U(x)$ is an element of $SU(N_f)$ group.
Here the pion decay constant and the chiral condensate are
denoted by $F$ and $\Sigma$, respectively.
The vacuum angle $\theta$ is given as a phase in front of the
mass matrix $\mathcal{M}=\mbox{diag}(m_u, m_d, m_s, \cdots)$.
In the conventional $p$-expansion, we treat
the exponent of $U(x)$ as the Nambu-Goldstone modes (here we denote as
$\xi(x)$), or pions, and expand the chiral field as
\begin{eqnarray}
U(x) &=& \exp\left(i\frac{\sqrt{2}\xi(x)}{F}\right)=
1+i\frac{\sqrt{2}}{F}\xi(x)-\frac{1}{F^2}\xi^2(x)+\cdots.
\end{eqnarray}
With the counting rule
\begin{eqnarray}
\label{eq:pcounting}
\mathcal{M}\sim p^2,\;\;\; \partial_\mu \sim p,\;\;\;
1/L, 1/T \sim p,\;\;\; \xi(x) \sim p,
\end{eqnarray}
physical amplitudes are systematically expanded in terms of $p^2$.
In the $p$-regime,
the finite volume effect appears in the pion propagator since
the momentum space is discretized \cite{Bernard:2001yj}.
Pion correlator reads
\begin{eqnarray}
\label{eq:pcorr}
\langle \xi^a(x)\xi^b(y)\rangle = \delta_{ab}
\sum_p \frac{e^{ip(x-y)}}{p^2+m_\pi^2},
\end{eqnarray}
where $a$($b$) denotes the $a$($b$)-th generator of $SU(N_f)$
and the summation is taken over the 4-momentum
$
p=2\pi(n_t/T, n_x/L, n_y/L, n_z/L),
$
with integer $n_\mu$'s.
As a consequence,
all the correlators become periodic.
Even at a contact point
$x=y$, there exists a finite volume correction
\begin{eqnarray}
\label{eq:g1}
\langle \xi^a(x)\xi^b(x)\rangle &=&\delta_{ab}\left(
\frac{m_\pi^2}{16\pi^2}\ln \frac{m_\pi^2}{\mu_{sub}^2}
+g_1(m_\pi^2)\right),\\
g_1(M^2)&=& \sum_{a\neq 0}\frac{M}{4\pi^2|a|}K_1(M|a|),
\end{eqnarray}
which is understood as an effect of pion wrapping around the lattice.
Here $K_1(x)$ is the modified Bessel function and the summation
is taken over the 4-vector $a_\mu=n_\mu L_\mu$ with
$L_i=L$ for $i=1,2,3$ and $L_4=T$,
except for $a_\mu=(0,0,0,0)$.
Note that the subtraction of the ultraviolet divergence is done
at a scale $\mu_{sub}$, which can be made in exactly the same way
as in the infinite volume.
In a similar perturbative manner, the effect of global topology
is recently calculated to the next-to-leading order \cite{Aoki:2009mx}.
In the $\epsilon$-regime, the above $p$-expansion
(\ref{eq:pcounting})
fails because the zero-momentum mode contribution
induces an unphysical infrared divergence,
which has to be circumvented
by exactly treating the vacuum fluctuation of the chiral field.
Namely, using a parameterization
\begin{eqnarray}
U(x) &=& U_0 \exp\left(i\frac{\sqrt{2}\xi^\prime(x)}{F}\right),
\end{eqnarray}
where $U_0\in SU(N_f)$ and $\xi^\prime$ satisfies
\begin{eqnarray}
\int d^4 x\; \xi^\prime(x) &=&0,
\end{eqnarray}
one can explicitly factorize the zero momentum part as $U_0$.
Since $U_0$ has no dependence on $x$,
the group integral can be non-perturbatively performed
as in the calculation of random matrix models.
The non-zero momentum modes $\xi^\prime$'s
are perturbatively treated as an expansion in $\epsilon^2$ according
to the counting rule
\begin{eqnarray}
\mathcal{M}\sim \epsilon^4,\;\;\; \partial_\mu \sim \epsilon,\;\;\;
1/L, 1/T \sim \epsilon,\;\;\; \xi'(x) \sim \epsilon.
\end{eqnarray}
This $\epsilon$-expansion \cite{Gasser:1987ah, Leutwyler:1992yt}
is useful when the quark mass is
so small that the pion correlation length exceeds
the spatial extent, $m_\pi L \ll 1$.
The zero momentum component $U_0$ can be explicitly integrated out and
written in terms of analytic functions.
This fact opens an interesting theoretical opportunities.
In particular, at the leading order of the $\epsilon$-expansion, the
system is proven to be equivalent to the Random Matrix Theory
\cite{Shuryak:1992pi,Damgaard:1998xy,Basile:2007ki}.
In the context of the QCD study, this provides a new method to
determine the chiral condensate by matching the low-lying eigenvalues
of the Dirac operator.
At an early stage, a simple setup with all degenerate quark masses
were studied.
As lattice QCD is developed to reach the simulations near the
chiral limit, calculations in a more realistic setup has become
relevant, and partially quenched calculations of various quantities
have been carried out \cite{Bernardoni:2007hi,Damgaard:2000di}.
With strange quark mass kept at its physical value, the finite volume
system is not purely in the $\epsilon$-regime even when the up and
down quark masses are sent close to the chiral limit, because the kaon
and $\eta$ are heavy and do not satisfy $m_{K,\eta} L \ll 1$.
For this mixed-regime, a hybrid method is proposed
\cite{Bernardoni:2008ei} and even extended to the case of heavy-light
mesons \cite{Bernardoni}.
More recently, a theoretical framework in which the $\epsilon$- and
$p$-regimes are treated in a unified manner is proposed
\cite{Damgaard:2008zs} of which
details are described in the next section.
As a final remark of this section, we note the role of topological
charge in the $\epsilon$-regime.
Intuitively, the global topological charge become relevant to the
dynamics of the system when the volume is small.
This can be explicitly studied within ChPT.
The $\theta$ integral can be absorbed in the zero-mode integrals
\begin{eqnarray}
\int \frac{d\theta}{2\pi}\int_{SU(N_f)} dU_0 e^{i\theta Q}
=\int_{U(N_f)} d U_0 (\det U_0)^Q,
\end{eqnarray}
where $U_0$ is integrated over U($N_f$) manifold.
The effect of the topological charge
enters through a factor $(\det U_0)^Q$.
For instance, the spectral density of low-lying Dirac eigenmodes is
largely affected by $Q$, of which dependence can be used
to test the validity of ChPT, in addition to the quark mass dependence.
\section{Determination of the chiral condensate}
\label{sec:cond}
\subsection{Analytic results beyond the leading order}
The chiral condensate is related to the Dirac eigenvalue density
$\rho(\lambda)$ at $\lambda=0$ in the thermodynamical limit
\cite{Banks:1979yr} as $\rho(0)=\Sigma/\pi$.
This relation can be easily extended to non-zero eigenvalues
by an analytical continuation of the valence mass $m_v$
to a pure imaginary value $i\lambda$:
\begin{eqnarray}
\rho(\lambda) &=& \frac{1}{\pi}{\rm Re}\langle \bar{q_v}q_v \rangle |_{m_v=i\lambda}.
\end{eqnarray}
Here, $\bar{q}_vq_v$ is the scalar density operator made of the
valence quark field.
This general formula is valid for both $p$- and $\epsilon$-regimes.
In the $p$-regime, using the partial quenching technique for the
imaginary valence quark mass,
Osborn {\it et al.} \cite{Osborn:1998qb} (see also \cite{Smilga:1993in})
found that the Dirac spectrum contains a logarithmic dependence on
$\lambda$.
This calculation is done in the infinite volume limit with degenerate
quark masses.
For small eigenvalues, the effect of finite volume becomes important.
The ChPT calculation is simplified if one consider the
$\epsilon$-expansion and taking its leading order contribution.
The integral over the zero momentum pion mode can be done
analytically, and the spectral function has been obtained as a
function of $N_f$, sea quark masses, and topological charge $Q$
\cite{Damgaard:2000ah,Verbaarschot:1993pm,Akemann:1998ta}.
Except for the exact zero-modes associated with $Q$, there is
a finite gap from zero (of order $1/\Sigma V$, which is called the
microscopic region) in the Dirac operator spectrum.
These analytic ChPT results can be used to extract $\Sigma$ by
comparing with the lattice data in the $\epsilon$-regime
\cite{Fukaya:2007fb,DeGrand:2006nv}.
But, since the formulae are obtained at the leading order, the value
of $\Sigma$ thus obtained is a subject of the NLO corrections of the
$\epsilon$-expansion.
Furthermore, it requires that the system is in the $\epsilon$-regime,
which is numerically demanding.
For common lattice QCD configurations produced in a $p$-regime
set-up, these analytical results cannot be applied.
Here we introduce a new method of the chiral expansion
\cite{Damgaard:2008zs}.
It is based on the $p$-expansion, but includes the pion zero-mode
integral explicitly so that a transition to the $\epsilon$-regime is
smooth.
In this scheme, one may predict the eigenvalue spectrum in the
microscopic region for the system in the $p$-regime.
With the so-called replica trick, the calculation is extended to the
case of non-degenerate quarks of arbitrary number of flavors.
At a fixed topological charge $Q$, we obtain \cite{Damgaard:2008zs}
\begin{eqnarray}
\label{eq:rho}
\rho_Q(\lambda) &=& \Sigma_{\rm eff}
\hat{\rho}^\epsilon_Q(\lambda \Sigma_{\rm eff}V,\{m_{sea}\Sigma_{\rm eff}V\})
+\rho^p(\lambda,\{m_{sea}\})
\end{eqnarray}
where $\lambda$ denotes the Dirac eigenvalue,
$\{m_{sea}\Sigma_{\rm eff}V\}=\{m_1\Sigma_{\rm eff}V,m_2\Sigma_{\rm eff}V,\cdots\}$
is a set of the sea quark masses normalized by
an {\it effective} chiral condensate $\Sigma_{\rm eff}$
(the definition is given below) and $V$.
The first term on the right hand side of (\ref{eq:rho}) contains
the one in the leading-order $\epsilon$-expansion
$\hat{\rho}^\epsilon_Q(\zeta,\{\mu_{sea}\}=\{\mu_1,\mu_2\cdots\})$,
which is rescaled so that the physical scale $\Sigma$ is factored
out.
This is a known function given by determinants of the Bessel functions
\cite{Akemann:1998ta}:
\begin{eqnarray}
\hat{\rho}^\epsilon_Q(\zeta,\{\mu_{sea}\})
\equiv
C_2 \frac{|\zeta|}{2\prod^{N_f}_f(\zeta^2 + \mu^2_f)}
\frac{\det \tilde{\mathcal{B}}}{\det \mathcal{A}},
\end{eqnarray}
where an $N_f\times N_f$ matrix $\mathcal{A}$ and an
$(N_f+2)\times (N_f+2)$ matrix $\tilde{\mathcal{B}}$ are defined by
\begin{eqnarray}
\mathcal{A}_{ij}&=& \mu_i^{j-1}I_{Q+j-1}(\mu_i),\\
\tilde{\mathcal{B}}_{1j} &=& \zeta^{j-2}J_{Q+j-2}(\zeta),\;\;\;
\tilde{\mathcal{B}}_{2j} = \zeta^{j-1}J_{Q+j-1}(\zeta),\nonumber\\
\tilde{\mathcal{B}}_{ij} &=& (-\mu_{i-2})^{j-1}I_{Q+j-1}(\mu_{i-2})
\;\;\;(i\neq 1,2).
\end{eqnarray}
Here, the overall sign is $C_2=+1$ for the $N_f=2$ and 3 cases.
The second term in (\ref{eq:rho}) is a logarithmic
NLO correction as always seen in the conventional $p$-expansion.
Defining $M_{ij}^2\equiv (m_i+m_j)\Sigma/F^2$, the function is given by
\begin{eqnarray}
\label{eq:rhop}
\rho^p(\lambda,\{m_{sea}\}) &\equiv&
-\frac{\Sigma}{\pi F^2}{\rm Re}[
\sum^{N_f}_f (\bar{\Delta}(M^2_{fv})
-\bar{\Delta}(M^2_{ff}/2))
\left.-(\bar{G}(M^2_{vv})-\bar{G}(0))
\right]_{m_v=i\lambda},
\end{eqnarray}
where
\begin{eqnarray}
\bar{G}(M^2) &=& \left\{
\begin{array}{l}
\frac{1}{2}\left[
\bar{\Delta}(M^2) +
(M^2-M_{ud}^2)\partial_{M^2}\bar{\Delta}(0,M^2)\right]
\hspace{1.4in}(N_f=2),\\
\frac{1}{3}\left[
-\frac{2(M^2_{ud}-M^2_{ss})^2}
{9(M^2-M^2_{\eta})^2}\bar{\Delta}(M_\eta^2)
\;\;\;+\left(1 + \frac{2(M_{ud}^2-M_{ss}^2)^2}{9(M^2-M^2_{\eta})^2}\right)
\bar{\Delta}(M^2)
\right.\\\left.
\;\;\;
+\frac{(M^2-M^2_{ud})(M^2-M^2_{ss})}{(M^2-M^2_{\eta})}
\partial_{M^2}\bar{\Delta}(M^2)\right]
\hspace{1.5in}
(N_f=2+1),\\
\end{array}\right.\nonumber\\\\
\bar{\Delta}(M^2)&=&
\frac{M^2}{16\pi^2}\ln \frac{M^2}{\mu_{sub}^2}
+\bar{g}_1(M^2).
\label{eq:Deltabar}
\end{eqnarray}
Here
$M_{ud}^2=2m_u\Sigma/F^2=2m_d\Sigma/F^2$, $M_{ss}^2=2m_s\Sigma/F^2$
and $M_\eta^2=(M_{ud}^2+2M_{ss}^2)/3$.
The function $\bar{g}_1(M^2)=g_1(M^2)-1/M^2V$ denotes
the well-known finite volume correction from non-zero modes
\cite{Bernard:2001yj}
(see also (\ref{eq:g1})).
The scale $\mu_{sub}^2$(=770 MeV in this work) is a subtraction scale.
Note that $\rho^p(\lambda,\{m_{sea}\}$ is insensitive to the
topological charge.
The {\it effective}
condensate in (\ref{eq:rho}) is also expressed in terms of
$\bar{\Delta}(M^2)$ and $\bar{G}(M^2)$ as
\begin{eqnarray}
\label{eq:Sigmaeff}
\Sigma_{\rm eff} &\equiv & \Sigma \left[1-\frac{1}{F^2}\left(
\sum^{N_f}_f \bar{\Delta}(M^2_{ff}/2)-\bar{G}(0)
-16L^r_6\sum^{N_f}_f M^2_{ff}\right)\right].
\end{eqnarray}
This depends on the sea quark masses, volume $V$ and $L_6^r$ of which value is
renormalized (at $\mu_{sub}=770$ MeV in this work).
\begin{figure}[tbp]
\centering
\includegraphics[width=10cm,clip=true]{rho_ChPT.eps}
\caption{
The first term $\Sigma_{\rm eff}\hat{\rho}_Q^\epsilon$ (solid-thin curve),
the second term $\rho^p$ (dashed)
and the total contribution (solid-thick)
of the spectral density (\protect\ref{eq:rho}) are shown.
The curves are multiplied by $\pi$.
We use $\Sigma=[240\mbox{MeV}]^3$, $F=94$ MeV, $L_6^r=-0.0001$, $L=T/3=1.9$ fm,
$m_{ud}=20$ MeV and $m_s=120$ MeV as inputs.
}
\label{fig:rhoChPT}
\end{figure}
For an illustration,
we draw curves given by the formula (\ref{eq:rho})
in Fig.~\ref{fig:rhoChPT}.
The contributions from
the first term $\Sigma_{\rm eff}\hat{\rho}^\epsilon_Q$ (solid-thin curve),
the second term $\rho^p$ (dashed)
and the total contribution $\rho_Q(\lambda)$ (solid-thick)
are shown separately.
We use typical parameters
$\Sigma=[240\mbox{MeV}]^3$,
$F$ = 94~MeV,
$L_6^r=-0.0001$,
$L=T/2$ = 1.9~fm,
$m_{ud}$ = 20~MeV and
$m_s$ = 120~MeV as inputs.
One can see that the second term gives a negative contribution and
shows a significant curvature in the lower end of the spectrum.
This is the effect of the chiral logarithm.
For this quark mass, the formula starts to deviate from
the leading order expression in the $\epsilon$-expansion
already at $\lambda\sim 5$ MeV.
\subsection{A numerical analysis}
Our simulation details and parameters have been
already presented in Section.~\ref{sec:ov}.
For the study of the Dirac spectrum,
80 lowest pairs of eigenvalues of the overlap-Dirac
operator $D(0)$ are calculated at every 5--10 trajectories.
We employ the implicitly restarted Lanczos algorithm for the
chirally projected operator $P_+\,D(0)\,P_+$, where
$P_+\!=\!(1+\gamma_5)/2$.
From its eigenvalue ${\rm Re} \lambda^{ov}$,
the pair of eigenvalues $\lambda^{ov}$ (and its complex conjugate)
of $D(0)$ is extracted through the relation
$|1-\lambda^{ov}/m_0|^2=1$, that forms a circle on a complex plane.
For the comparison with the effective theory,
the lattice eigenvalue
$\lambda^{ov}$ is projected onto the imaginary axis as
$\lambda\equiv\mathrm{Im}\lambda^{ov}/(1-\mathrm{Re}\lambda^{ov}/(2m_0))$.
Note that the real part of $\lambda^{ov}$ is negligible
(within 1\%) for the low-lying modes.
When we match the lattice data for the spectral density with the
analytic calculation (\ref{eq:rho}),
two parameters are to be determined at each set of
the quark masses: $\Sigma_{\rm eff}$ and $F$.
In the second NLO term of (\ref{eq:rho}), the difference between
$\Sigma_{\rm eff}$ and $\Sigma$ is a higher order effect.
We therefore take two reference values of $\lambda$ to give inputs to
determine $\Sigma_{\rm eff}$ and $F$.
The reference points are chosen such that they have maximum
sensitivity to the parameters
in the convergence range of the chiral expansion:
$\lambda =0.004$ ($\sim 7$ MeV) and 0.017 ($\sim 30$ MeV)
except for the case with $m_{ud}=0.002$ and $Q=1$,
for which we choose $\lambda=0.01$ and 0.02
(because of its weaker sensitivity to the NLO effects).
At these two reference points, we compare the mode number
below a given value of $\lambda$ \cite{Giusti:2008vb},
with an integrated formula of ChPT (\ref{eq:rho})
\begin{eqnarray}
\label{eq:cum}
N_Q(\lambda) &\equiv& V\int^{\lambda}_0 d\lambda^\prime \rho_Q (\lambda^\prime),
\end{eqnarray}
and determine $\Sigma_{\rm eff}$ and $F$.
As Giusti and L\"uscher \cite{Giusti:2008vb} studied,
it is also useful to define a quantity
\begin{eqnarray}
\label{eq:Sigmamode}
\Sigma^{mode}_Q(\lambda)
&\equiv& \frac{\pi N_Q(\lambda)}{\lambda V},
\end{eqnarray}
to see the NLO effects, or the chiral
logarithmic effects to $\Sigma$.
We test the both of $N_f=2+1$ and $N_f=2$ ChPT formulae.
For the latter case, the strange quark is assumed
to be decoupled from the theory.
\begin{figure*}[tbp]
\centering
\includegraphics[width=9.5cm]{rho-lattice.eps}
\includegraphics[width=9.5cm]{cum-lattice.eps}
\includegraphics[width=9.5cm]{Sigmamode0.015.eps}
\caption{
The spectral density $\pi\rho_Q(\lambda)$ (top), the mode
number $N_Q(\lambda)$ (center) and $\Sigma^{mode}_Q(\lambda)$ (bottom)
of the Dirac operator at $m_{ud}$ = 0.015, $m_s$ = 0.080 and $Q=0$.
The lattice result (histogram (top) or solid symbols (center and bottom)) is
compared with the ChPT formula drawn by solid curves.
For comparison, the prediction of the leading
$\epsilon$-expansion (dashed curves) is also shown.
}
\label{fig:rho015}
\end{figure*}
\begin{figure*}[tbp]
\centering
\includegraphics[width=9.5cm]{rho-lattice-epsilon.eps}
\includegraphics[width=9.5cm]{cum-lattice-epsilon.eps}
\includegraphics[width=9.5cm]{Sigmamode0.002.eps}
\caption{
Same as Fig.~\protect\ref{fig:rho015}, but at
$m_{ud}=0.002$.
The NLO correction is smaller in the $\epsilon$-regime.
}
\label{fig:rho002}
\end{figure*}
Figures~\ref{fig:rho015} and \ref{fig:rho002} show the lattice data
for the spectral density (upper panel), its integral (middle)
and $\Sigma^{mode}_Q(\lambda)$ defined by (\ref{eq:Sigmamode})
at two different sea quark masses:
one in the $p$-regime ($m$ = 0.015, Fig.~\ref{fig:rho015}) and the
other in the $\epsilon$-regime ($m$ = 0.002, Fig.~\ref{fig:rho002}).
The analytic formula is also plotted with two parameters fixed at two
reference points of the mode number.
The leading-order contribution is given by dotted curves while the
full result is shown by solid curves.
In the $p$-regime result (Fig.~\ref{fig:rho015}),
the effect of the NLO term in the $p$-expansion is clearly seen as a
deviation from the leading-order density
$\Sigma_{\rm eff}\hat{\rho}_Q^\epsilon$
(dotted curve) in the histogram.
The deviation starting already around $\lambda \sim 0.005$
is also clear in the mode number $N_Q(\lambda)$ and $\Sigma_Q^{mode}(\lambda)$.
On the other hand, the NLO formula (solid curve) describes the lattice
data very nicely up to $\lambda\sim m_s/2$.
The convergence of the chiral expansion is better for the
$\epsilon$-regime data (Fig.~\ref{fig:rho002}), but the difference
between LO and NLO still exists.
We also observe that there is a wider gap near $\lambda=0$, which is
expected because the value of the sea quark mass $m=0.002$ is similar
to the lowest eigenvalue, so that the suppression due to the fermionic
determinant $\prod(\lambda^2+m^2)$ works strongly.
\begin{figure*}[tbp]
\centering
\includegraphics[width=10cm]{rho-latticeQ1.eps}
\includegraphics[width=10cm]{Sigmamode0.015Qdep.eps}
\caption{
The spectral density at $m=0.015$ and $Q=1$ (top)
and comparison of $\Sigma_Q^{mode}(\lambda)$ at $Q=0$ and 1 (bottom).
In the ChPT curves, the same values of $\Sigma_{eff}$ and $F$
are used as inputs.
}
\label{fig:rhoQ}
\end{figure*}
\begin{figure*}[tbp]
\centering
\includegraphics[width=10cm]{rho-latticeL24.eps}
\includegraphics[width=10cm]{Sigmamode0.025Vdep.eps}
\caption{
The spectral density at $m=0.025$ and $L=24$ (top)
and comparison of $\Sigma_Q^{mode}(\lambda)$ at $L=24$ and 16 (bottom).
In the ChPT curves, the same values of $\Sigma_{eff}$
(but the volume dependence is corrected within ChPT) and $F$
are used as inputs.
}
\label{fig:rhoV}
\end{figure*}
One of the significant consequences of the ChPT formula (\ref{eq:rho})
is that the spectral function for different topological charge $Q$ and
volume $V$ should be described by the same set of the parameters,
{\it i.e.} $\Sigma_{\mathrm{eff}}$ and $F$.
This provides a highly non-trivial cross-check of the formula.
For this purpose we produced data
at non-zero topological charge $Q=1$.
The results are shown in Fig.~\ref{fig:rhoQ}.
Here the curves of the NLO ChPT is drawn with inputs from the $Q=0$
data and there is no further free parameter to adjust.
The good agreement below $\lambda\simeq$ 0.03 gives further
confidence on the analysis.
A similar check can be done with the lattice data obtained from a
larger volume lattice $24^3\times 48$, for which the data are shown in
Fig.~\ref{fig:rhoV}.
The comparison is a bit more tricky for different volumes, because the
definition of $\Sigma_{\mathrm{eff}}$ (\ref{eq:Sigmaeff}) depends on $V$.
Namely the function $\bar{\Delta}(M^2)$ contains $\bar{g}_1(M^2)$,
which represents the finite volume effect.
It is possible to convert the value of $\Sigma_{\mathrm{eff}}$ for
different volumes.
If we convert the result at $L=24$, $\Sigma_{\rm eff}=0.00306(7)$ to
the one on a $L=16$ lattice, it becomes 0.00341(18), which may be
compared with the independent calculation at $L=16$ at the same sea
quark mass $m=0.025$, which is 0.0333(18).
Therefore, the finite volume scaling is confirmed at least on two
different volumes, whose difference is a factor of 3.
The curves in Figures~\ref{fig:rho015}--\ref{fig:rhoV}
are drawn using the $N_f=2+1$ ChPT results, but we found
the difference from $N_f=2$ ChPT formula is hardly
visible in the scale of this plot, which confirms decoupling
of the strange quark from the low energy theory.
From these analysis the values of $\Sigma_{\rm eff}$ and $F$ are
extracted for each sea quark mass.
Note that $\Sigma_{\rm eff}$ is extracted at the NLO accuracy,
while the value of $F$, which first appears at the NLO term,
might have larger systematic corrections from NNLO contributions.
We find that the results for $\Sigma_{\rm eff}$ are stable under
change of two reference points in a range $\lambda < 0.03$.
As noted above, there is little difference between
$N_f=2$ and $N_f=2+1$ formulae;
$\Sigma_{\rm eff}$ and $F$
are almost equal well within the statistical error.
The difference between $m_s=0.080$ and $m_s=0.100$
is even weaker.
In the following analysis, we concentrate on the data at $m_s=0.080$.
\subsection{Chiral extrapolation of $\Sigma_{\rm eff}$}
We next consider the sea quark mass dependence of $\Sigma_{\rm eff}$.
As shown in (\ref{eq:Sigmaeff}), $\Sigma_{\rm eff}$ is a function of
$\Sigma$, $F$, $L_6$, which can be determined from the lattice data.
The chiral condensate $\Sigma$ thus obtained should have the NLO accuracy.
In the fitting of the lattice data,
we attempt
(A) 3-parameter ($\Sigma$, $F$, $L_6$) fit without any inputs
and (B) 2-parameter ($\Sigma$, $L_6$) fit with
$F=0.0410$ (for $N_f=3$ ChPT) or with $F=0.0406$ (for $N_f=2$ ChPT).
These values of $F$ correspond to the chiral limit of $F$ extracted
from the analysis of the spectral function.
\begin{figure}[tbp]
\centering
\includegraphics[width=10cm]{Nf3latNf3ChPTSigmaeff3prmfitms0.080.eps}
\caption{
Three parameter fit of $\Sigma_{\rm eff}$.
The $N_f=3$ ChPT formula is used.
}
\label{fig:SigmaefffitNf3}
\end{figure}
The fitting is shown in Fig.~\ref{fig:SigmaefffitNf3}
for the case (A) with the $N_f=3$ ChPT formula.
We use the lightest 4, 5, and 6 data points.
All the curves are consistent with the lattice data used in the fit
and in fact the $\chi^2$ per degrees of freedom is reasonable (between
0.6 and 1.5).
A remarkable fact is that the chiral limit (shown by a square) is not
sensitive to the number of data points used.
The chiral limit is very stable because of the presence of the
$\epsilon$-regime data point.
Similar curves are obtained for the case with $N_f=2$ and for the case
(B).
With the 2-parameter fit (the case(B)) the heaviest data point cannot
be well described, {\it i.e.} $\chi^2/\mathrm{d.o.f.}$ is about 2.5.
From these curves, one can extract the low energy constants of ChPT.
Note in the case of $N_f=3$ ChPT, we have two limits of chiral
condensate: $\Sigma^{N_f=3}$, where ``three'' flavor massless limit is
taken, and $\Sigma^{\rm phys}$, which is a two-flavor chiral limit
with strange quark mass fixed at a finite value $m_s=0.08$.
As already mentioned, the strange quark dependence is so small
that the difference from the value at the physical
strange quark mass is negligible.
The extracted values of $\Sigma^{\rm phys}$
are stable against the different choice of fitting function and
fitting range, while $\Sigma^{N_f=3}$ shows strong sensitivity to
them.
It means that the determination of $\Sigma^{N_f=3}$ is not feasible
with our current data set.
This is natural because the strange quark mass dependence is not
well controlled by the lattice data.
On the other hand, the determination of $\Sigma^{\rm phys}$ is very
stable, thanks to the $\epsilon$-regime data point.
Our estimate of systematic effects due to the chiral extrapolation
is $\sim 2$ \%.
From the above analysis, we determine the low-energy constants for
2+1-flavor QCD as
\begin{eqnarray}
\Sigma^{\rm phys} &=& 0.00186(10)(44) \sim [226(4)(18) \mbox{MeV}]^3,\\
F &=& 0.0406(05)(41) \sim 74(1)(8)\mbox{MeV},\\
L^r_6(770~{\rm MeV}) &=& -0.00011(25)(11),
\end{eqnarray}
where the first error is statistical and
the second error is systematic, respectively.
To obtain the final result,
we convert the value of $\Sigma^{\rm phys}$
to the definition in the $\overline{\mathrm{MS}}$ scheme,
by using the non-perturbative renormalization factor
\cite{Noaki:2009xi}calculated
through the RI/MOM scheme \cite{Martinelli:1994ty}.
The result \cite{:2009fh}, $\Sigma^{\rm phys}$
in the limit of $m_{ud}=0$ and $m_s$ fixed at
its physical value, is
\begin{equation}
\label{eq:final}
\Sigma^{\overline{\mathrm{MS}}}(\mathrm{2~GeV}) =
[242(04)(^{+19}_{-18})~\mbox{MeV}]^3.
\end{equation}
Let us here discuss possible systematic errors in (\ref{eq:final}).
Since our lattice studies are done at only one value of $\beta$,
it is difficult to estimate the discretization errors.
But it should be partly reflected
in the mismatch of the observables measured in different ways.
We here estimate it from a mismatch of the lattice spacing;
0.1003(46) fm from the pion decay constant \cite{Noaki} and
0.1087(15) fm from the $\Omega$ baryon mass \cite{Noaki2}.
This 7.4\% deviation is added in the systematic error.
The systematic error due to finite volume is estimated as $\sim 1.4$\%
using the lattice data at two different volumes.
\if0
The pion decay constant $F$ extracted from the
NLO terms in the spectral function analysis has
the accuracy of the leading order.
Its quark mass dependence is therefore not under control
unless we carry out a two-loop analysis.
It is still notable that the value obtained in the 3 parameter
fit is consistent with the one in the chiral limit
of data in Table.\ref{tab:SigmaeffFeff}.
Here we just use the difference between the value from
the $\Sigma_{\rm eff}$ fit, linear chiral limit and
the one at $ma=0.015$ as a systematic error for the final result of $F$,
which is hopefully a good estimate for the higher order correction.
\begin{figure}[tbp]
\centering
\includegraphics[width=10cm]{modenum.eps}
\caption{
Dependence on $m_{ud}$ of $\Sigma_{mode}$ is plotted.
The dotted and dashed curves are ChPT predictions (at $V=\infty$)
with $N_f=2$ and $N_f=2+1$ respectively.
}
\label{fig:mode}
\end{figure}
Let us compare our result with a previous work by
Giusti and L\"uscher \cite{Giusti:2008vb}.
They have studied the low-lying mode number of the
Wilson-Dirac operator $\nu(M=(\lambda^2+m_{ud}^2)^{1/2}, m_{ud})$,
which corresponds to $2A_Q(\lambda)$ in our notation
(except for the summation over topological charge).
Note here that the factor ``2'' comes from
extending the integration range in (\ref{eq:cum})
to negative $\lambda$ using $\rho_Q(-\lambda)=\rho_Q(\lambda)$,
which is guaranteed by the symmetry of the overlap-Dirac operator.
According to their analysis, we compare
\begin{eqnarray}
\Sigma_{mode}&\equiv &\frac{\pi A_Q(\lambda_{ref})}{\lambda_{ref} V},
\end{eqnarray}
with the ChPT prediction in the infinite volume.
Although a good linear $\lambda$ dependence of the mode number
even at $\lambda \sim 95$ MeV was reported in \cite{Giusti:2008vb},
we choose a rather smaller value of the reference scale
$\lambda_{ref}=0.015$ ($\sim 30$ MeV)
since our above analysis shows a deviation
from the ChPT formula already at $\sim 60$ MeV.
In Fig.\ref{fig:mode}, we plot the sea quark mass
dependence of $\Sigma_{mode}$ as well as
the ChPT predictions in the infinite volume limit $V\to \infty$
with dotted ($N_f=2$) and dashed ($N_f=2+1$) curves.
Here $\Sigma=0.00186, F=0.0406, L_6^r =-0.00011$
from our 5pt fit are used as inputs.
The lowest two mass points clearly deviate from
the ChPT formula at $V=\infty$, which indicates
a precise control of finite $V$ effects described by
(\ref{eq:rho}) is essential for our analysis on a $L\sim 2$ fm lattice.\\
\fi
\if0
The data for the larger lattice $L^3T=24^3\times 48$
in Table~\ref{tab:SigmaeffFeff}
provides a good estimate for the finite volume effects.
The extracted value $\Sigma_{\rm eff}=0.00306(7)$
deviates from that with
$L=16$ by $\sim 9$ \%.
But in the definition of $\Sigma_{\rm eff}$ in (\ref{eq:Sigmaeff}),
finite $V$ correction is already included at NLO, from which
one can convert the value into the one with the same volume as
$\Sigma_{\rm eff}(L=24)\to \Sigma_{\rm eff}(L=16)=0.00341(18)$,
which is consistent with the value obtained from the lattice at $L=16$.
The deviation is only $1.4$\%.
\fi
\section{Summary and Conclusion}
\label{sec:summary}
In this talk, a study of the spontaneous chiral symmetry breaking
performed by the JLQCD and TWQCD collaborations has been presented.
We discussed that fixing topology is an essential part
for the dynamical overlap fermion simulations on the lattice.
By reducing the numerical cost with the topology fixing determinant,
we have performed the first large-scale simulations of
dynamical overlap quarks.
The up and down quark masses are reduced close to the physical point.
We have then discussed that the global topology,
as well as the finite volume effect,
can be described well within the chiral perturbation theory.
In fact, we have found a good agreement of our lattice data
for the Dirac operator spectrum with the ChPT predictions even in the
region where its finite size effect is large.
We extract the chiral condensate in 2+1-flavor QCD.
\begin{acknowledgments}
The author thanks P.H.~Damgaard and members of JLQCD and TWQCD collaborations
for useful discussions.
Numerical simulations are performed on IBM System Blue Gene
Solution at High Energy Accelerator Research Organization
(KEK) under a support of its Large Scale Simulation
Program (No. 06-13).
This work was supported by the Global COE program
of Nagoya Univ. ``Quest for Fundamental Principles in the Universe''
from JSPS and MEXT of Japan.
\end{acknowledgments}
|
1001.2168
|
\section{Introduction} \label{sec:intro}
Clusters of galaxies form in correspondence of the peaks of the
primordial matter density field as a result of the
gravitational collapse of both dark matter and baryons. In the
framework of the standard $\Lambda$CDM\ cosmological model and the
hierarchical clustering of the large scale structure
formation they constitute both the most recent and the
largest virialised objects of the Universe.
Nowadays, it is clear that gravitation is not the only process that
influences the physics of the intracluster medium (ICM, hereafter).
The electrons
of the ionized plasma emit via free-free interaction with the protons,
making clusters bright X-ray sources and allowing the gas to cool
efficiently, particularly in the central regions. During the last
decade, the observations of the {\it XMM-Newton}\ and {\it Chandra}\ X-ray satellites
highlighted the presence of several interaction mechanisms between
the galactic component and the ICM, showing that the
evolution of the two is intimately tied. For
instance, the accretion of cold gas onto the brightest cluster
galaxies (BCGs) at the cluster center is strongly suspected to fuel
the super-massive blackholes they host. This process is able to
trigger
star-formation within the BCGs \citep{odea08,pipino09} and power
episodic violent outbursts of their central active galactic nuclei
(AGNs), whose energy injection into the ICM prevents the overcooling
of the gas \citep{fabian06,mcnamara07}. AGNs feedback and other
non-gravitational processes, such as supernovae (SNe) powered
galactic
winds \citep{kapferer06,sijacki06,schindler08}, preheating
of the ICM \citep[see e.g.][]{fang08}, together with gravitational ones (i.e.
galaxy-galaxy interactions, ram-pressure stripping, galaxy mergers)
induce energy and matter exchanges between the galactic medium and the
ICM. To date these complex physical processes and their impact on the
statistical cluster properties, thus on our understanding of
structure formation \citep{voit05} and use of the
cluster population in cosmological studies
\citep{mantz08,vikhlinin09,mantz09}, are under scrutinous
observational and
theoretical investigations \citep[see, e.g.,][and references
therein]{arnaud05,borgani08a}.
The presence of heavy elements in the ICM is the most
direct evidence and consequence of the ejection of galactic material.
Within clusters their abundance has been widely
measured making use of X-ray observations \citep[see the reviews
by][]{sarazin88, arnaud05,werner08} with a typical abundance of
0.3 $ Z_\odot$.
Stars and SNe constitute the most efficient way to produce and
disperse metals. Heavy
elements in the ICM originate from different processes: early enrichment
\citep{aguirre07}, continuous injection from galaxy members and
in situ production by intra-cluster stars \citep[sources of the
intra-cluster light, see][]{arnaboldi04,krick07,murante07,conroy07,dolag09}.
The aforementioned processes do not discriminate between the natures
of the ejected galactic material, therefore these enrichments
are unambiguously
linked also to the ejection of neutral gas and dust in the ICM.
Observations indicate that in galactic environments
dust is a minor component, with dust-to-gas ratio
$M_{\rm dust}/M_{\rm gas} \approx 0.01$
\citep{mathis77} and
it could be as low as $M_{\rm dust}/M_{\rm gas} = 10^{-5}-10^{-4}$
in the ICM \citep{popescu00,aguirre01}. Nonetheless, dust particles
have lifetimes long enough to be heated by collisions with the
hot electrons and re-emit at the infrared (IR) wavelengths.
In this
way they can constitute an additional cooling agent
of the gas \citep{montier04} which might play an important role in ICM
physics, as seen with the implementation of dust cooling
in hydrodynamical simulations \citep{dasilva09}.
However obtaining observational constraints on the possible
IR signal coming from intracluster dust is a very challenging issue, since
the average sky fluctuations caused by background galaxies
and galactic cirrus clouds are comparable or even higher
than the overall flux coming from a single cluster, which
is anyway expected to be dominated by the dust emission from
star-forming galaxies. In fact, nowadays the only claimed (and
still controversial) detection of this
IR emission comes from the studies on the Coma cluster of
\cite{stickel98,stickel02}, who measured a diffuse dust mass of
$M_{\rm dust} \approx 10^7 M_\odot$, while other attempts resulted
in non detections \citep[see, e.g.,][]{bai07}.
If studies on single objects appear very problematic, the
low signal-to-noise problem can be overcome by adopting a
statistical approach, taking advantage of the high number
of clusters detected mainly with optical surveys \citep[see]
[for a review]{biviano08}. This concept was first used at IR
wavelengths by \cite{montier05} who performed a
stacking analysis of the {\it Infra Red Astronomical Satellite}
(IRAS) survey in the direction of
more than 11~000 known galaxy
clusters and groups and obtained a statistical
detection of the overall clusters signal at 12, 25, 60 and 100
$\mu$m. Starting from this result, \cite{giard08} characterised
the IR luminosity evolution of the stacked sample, analysing
also its correlation with their X-ray luminosities. These works
constitute a first important result on the global IR properties
of galaxy clusters. However, since the clusters IR emission
could be dominated by star-forming galaxies, it
becomes crucial to disentangle the galactic signal from a
possible diffuse component in order to quantify its
implications on cluster physics. On the other side,
understanding the IR properties of the cluster galaxies could
provide hints on the environmental effects over their star-formation,
which are nowadays objects of studies at different wavelengths,
from the radio to the ultraviolet \citep[UV, see][for a review]{boselli06a}.
In this work we attempt to reconstruct the observed stacked IR
emission measured in the direction of galaxy clusters in the IRAS
whole sky survey. We model in details the
contribution of cluster galaxies to the total IR emission in order
to understand whether this contribution is sufficient to explain the
fluxes and luminosities derived by \cite{montier05} and
\cite{giard08} or if, on the contrary, there is an indication of
a non-galactic component, possibly associated to intracluster dust.
Since for our modelisation we make use of the known
spectral properties of galaxies as observed mainly in the field, the
comparison between our results and observed data can
be useful also to highlight possible differences between
cluster and field galaxies.
This paper is organised as follows. In the next
Section we briefly present the statistical detection of
the clusters IR emission of \cite{montier05} and \cite{giard08} and we
focus in particular on the subsample of the SDSS-maxBCG catalogue. In
Section~\ref{sec:model} we describe the details of our modelisation
for the IR luminosity of cluster galaxies and discuss the possible
presence of other galactic components in Section~\ref{sec:other}.
In Section~\ref{sec:flux}
we apply our model to estimate the IRAS\ fluxes, then our
results are discussed in Section~\ref{sec:results}. Finally, we
present our conclusions in Section~\ref{sec:concl}.
Throughout this paper we assume a flat $\Lambda$CDM\ cosmological model
with $(\Omega_\Lambda,\Omega_{\rm m},h)=(0.7,0.3,0.7)$.
\section{The statistical IR emission of clusters} \label{sec:stack}
Working on a list of 11~507 groups and clusters
selected from 14 publicly available catalogues,\footnote{The list was
build making use of the SIMBAD database, operated at CDS,
Strasbourg, France (http://simbad.u-strasbg.fr/simbad/) and
NASA/IPAC Extragalactic Database (NED), operated by the
JPL/Caltech (http://nedwww.ipac.caltech.edu), among which the
ABELL catalogue \citep{abell89} and the {\it ``Northern Sky
Optical Cluster Survey''} \citep[NSC,][]{gal03}.}
\citet{montier05} performed a clear statistical detection of
the IR flux in the direction of galaxy clusters by
stacking their corresponding fields (within 10' from the cluster center)
in the IRAS\ all sky survey. After dealing
carefully with the point sources contamination, foreground/background
subtraction and other various systematic effects, they measured
stacked fluxes at 60 and 100~$\mu$m with signal-to-noise ratio of 57
and 43 respectively. Over the four IRAS wavelengths this emission
proves to be consistent with the spectral signature of galactic
IR emission.
More recently, on the basis of \cite{montier05} results,
\cite{giard08} performed a statistical analysis in redshift,
presenting for the first time the evolution of the IR luminosity of
galaxy clusters. In addition to \cite{montier05} original list of
11~507 clusters,
\cite{giard08} backed-up their analysis on two
{\it standalone} catalogues -- i.e.
the {\it ``Northern Sky Optical Cluster Survey''}
\citep[NSC,][]{gal03}, and the SDSS-maxBCG catalogue \citep{koester07}
-- for which a richness information was available.
They extended their IRAS stacking analysis to the {\it Rosat All Sky
Survey} \citep[RASS,][]{voges92} in order to compare the evolution in
redshifts of the IR and X-ray luminosity. They showed that the
stacked IR luminosities are on average 20 times higher than the X-ray
luminosities. They also found that the IR luminosity is evolving
rapidly as $(1+z)^5$ in the $0.1 < z < 1$ interval. \cite{giard08}
also made use
of the richness information (i.e. correlated to the halo occupation
number), contained in the SDSS-maxBCG and NSC catalogues, to constrain
the dependence of the IR luminosities with cluster richness. They
derived a correlation following $L_{\rm IR} \propto (N_{\rm
gal}^{R200})^{0.8\pm0.2} $.
In order to understand the aforementioned results at
60 and 100 $\mu$m, in this work we focus on the modelisation of
the IR luminosity and fluxes of galaxy clusters. The IR spectrum
of galaxies, which is dominated by dust emission
\citep{lagache05,soifer08}, is expected to make a major contribution
to the IR emission detected in the direction of clusters \citep{giard08}.
Therefore, a careful modeling of the IR emission of member galaxies
is needed in order to understand the global properties of clusters in
the IR, and in particular to disentangle between the relative contribution
of galaxies and intracluster dust. Given the
inhomogeneity of the main dataset used by \cite{montier05} and
\cite{giard08} (i.e. the 11~507 groups and clusters) in terms of cluster
size and detection
method, it becomes very difficult to define a global selection
function of such sample, that would be necessary in order to
have a suitable starting base for our
work. In order to overcome these difficulties we decided to base our
modelisation on a restricted and well defined cluster sample by
considering a single catalogue. For the purpose of our calculations
we chose the SDSS-maxBCG catalogue.
\subsection{The SDSS-maxBCG catalogue} \label{ssec:maxbcg}
Since we want to reconstruct the total IR luminosity and flux due to
cluster galaxies, we require an information on the halo occupation number
(i.e. the number of galaxies within the cluster potential well). In this
view the SDSS-maxBCG catalogue \citep{koester07} is the most fitted to
our needs.
Indeed, since the SDSS-maxBCG catalogue is created by
identifying overdensities in the galaxy distribution, it contains
information about the richness of the cluster itself that we can use
directly without having to deal with the uncertainties connected
with the $N-M$ and $L_X-M$ relations, that would be necessary, for
instance, in case of X-ray selected clusters. Moreover, the number
of objects contained in the SDSS-maxBCG catalogue is the highest
among all the others (i.e. 13~807 groups and clusters), thus providing
a good statistics which is needed to assess the robustness of our
results. Finally, \cite{koester07} provide also
the total luminosity in the $r$ and $i$-band of the identified
cluster members which also have a well defined spectral type
(E/S0 galaxies). We will make use of this information
to obtain an estimate of their flux in the IRAS\ bands (see
Sect.~\ref{sec:model}).
The SDSS-maxBCG catalogue has been obtained by analysing the
clustering properties of more than 500~000 SDSS galaxies in
an area of $\sim$7~500 deg$^2$ in the redshift range
$0.1<z<0.3$. For each cluster the catalogue contains a
photometric redshift and an indication of the cluster
richness, $N_{\rm gal}^{R200}$. This quantity corresponds to the
number of early-type galaxy members at a distance lower
than $R_{200}$ from the central BCG. The adopted definition
of $R_{200}$ is the radius at which the deprojected galactic
density is 200 $\Omega_{\rm m}$$^{-1}$ times the average galactic
density on the large scale: in the approximation of
galaxies following the matter distribution, this is equivalent
to the usual cosmological definition of $R_{200}$ as the
radius enclosing a density 200 times larger than the critical
density of the Universe \citep[see the discussion in][for details]
{hansen05}.
It is important to note that $N_{\rm gal}^{R200}$\ does \emph{not} represent the
total cluster richness because it does not include spiral
galaxies: in fact, the cluster members candidates have been
selected for having colors matching the E/S0 ridgeline
\citep{bower92} and $M_r<-16$. We will explain in the detail
our definition of the cluster richness in Section~\ref{ssec:late}.
The SDSS-maxBCG catalogue includes 13~823 groups and clusters with
$N_{\rm gal}^{R200}$$\geq 10$ and it is widely dominated by the presence of
groups and small clusters ($N_{\rm gal}^{R200}$$\approx 10-15$). As a reference,
when computing their corresponding $M_{200}$ with the relation of
\cite{rykoff08}, the median of the mass distribution is $4.5 \times
10^{13} \, h^{-1} M_\odot$.
In this work, we adopted the trimming done by \cite{giard08} on
the SDSS-maxBCG catalogue of objects located in regions of the
sky not covered by the IRAS data, with noisy images, or whose
fields contain strong IR sources. This selection process
ended up in a sample of 7~476 clusters, corresponding to a total
of 121~318 E/S0 galaxy members, that will be the object of our
analysis.
\section{Modeling galaxy luminosities} \label{sec:model}
The main source of the IR radiation of galaxy clusters is expected
to be the thermal emission due to the dust reprocessing the
UV photons emitted by stars inside cluster galaxies. Since we are
basing our work on galaxies observed in the optical band, we need
to construct a valid model that is able to connect their galactic
$r$ band emission
to the corresponding dust emission in the IR. In particular,
we will focus on the 60 $\mu$m\ and 100 $\mu$m\ IRAS\ bands fluxes, in
order to compare our results with the most relevant
measurements of \cite{giard08}.
The characteristics of the galactic IR emission
depend on the amount of dust present in the galaxies and are
strictly connected with their star-formation rate (SFR) and,
therefore, their spectral type.
\begin{figure}
\includegraphics[width=0.50\textwidth]{seds_freq.ps}
\caption{SEDs as a function of wavelength for the 5
\textsc{grasil} \protect \citep{silva98} templates used in this work.
The red line shows our reference
model for E/S0 galaxies, the three blue lines represent normal
spiral galaxies Sa, Sb, Sc (light blue, average, dark blue,
respectively) and the green line represents our starburst galaxy
model, corresponding to M82. All of the templates have been
arbitrarily normalized to the same integrated luminosity
of $10^{30}$ erg/s in the band 8-1000 $\mu$m\
(i.e. IR luminosity, $L_{\rm IR}$). The four shaded
regions identify the optical $r$ and $i$-band and the two
IRAS\ bands of 60 $\mu$m\ and 100 $\mu$m, from left to right. }
\label{fig:seds}
\end{figure}
Spectral energy distribution (SED) templates obtained with the spectral
synthesis code \textsc{grasil} \citep{silva98} were used in order to
represent the typical emission of galaxies in the local universe.
\textsc{grasil} SEDs extend from UV to radio wavelengths, including
dust reprocessing and nebular line emission. We used four templates for normal
galaxies, with Salpeter's initial mass function and age 13Gyr, namely an
elliptical galaxy and
three different spiral galaxies (Sa, Sb and Sc).
We also included the \textsc{grasil} model fit to
multi-wavelength observations of M82 as a semi-empirical template to represent
the typical SED of local starburst galaxies \citep{silva98}.
For spiral galaxies, the SEDs correspond to a weighted average over the
different lines of sight, from face-on to edge-on, in order to statistically
account for the mean spatial orientations of cluster galaxies.
The five templates are displayed in Fig.~\ref{fig:seds}.
The SED models described above were used to derive the expected IR
luminosities from observed fluxes and corresponding luminosities in the $r$
band. For a given template, the luminosity in the $r$ band is given by
\begin{equation}
L_r = \int{L(\lambda)T_r(\lambda)d\lambda} \ ,
\end{equation}
where $L(\lambda)$ is the luminosity per unit wavelength and $T_r(\lambda)$ is
the SDSS $r$ filter transmission. For the purpose of this work, we
define the IR luminosity following \cite{lefloch05}:
\begin{equation}
L_{\rm IR} =
\int_{8\mu{\rm m}}^{1000\mu{\rm m}}{L(\lambda)d\lambda} \ .
\end{equation}
Therefore, for each template SED, the scaling ratio can be defined as follows
\begin{equation}
R_{{\rm IR},r} \equiv \frac{L_{\rm IR}}{L_r} \ .
\label{eq:f_ir}
\end{equation}
Table~\ref{tab:gal_pop} summarizes the $R_{{\rm IR},r}$ values for the
different templates used in this paper. For a given $L_r$, the corresponding
$L_{\rm IR}$\ can be different by almost three orders of magnitude depending
on the spectral type of the galaxy. The modelisation of the luminosity of
early-type and late-type galaxy populations is explained in the details in
the next sections.
\begin{table}
\begin{center}
\caption{
Reference spectral types and galactic population models used in this work.
}
\begin{tabular}{lrrrcr}
\hline
\hline
Spectral type & \multicolumn{3}{c}{Population (\%)} & $u-r$ & $R_{{\rm IR},r}$ \\
& no ev. & ref. & cons. & & \\
\hline
\it{-Early-type} & & & & & \\
BCG & 4.3 & 4.0 & 4.2 & -- & 0.12 \\
E/S0 & 65.4 & 61.5 & 63.3 & -- & 0.12 \\
(Tot. early-type) &(69.7)&(65.5)&(67.5)& & \\
\\
\it{-Late-type} & & & & & \\
Sa & 4.4 & 5.1 & 6.5 & $>$2.2 & 2.36 \\
Sb & 9.6 & 10.7 & 11.7 & 1.8--2.2 & 13.82 \\
Sc & 15.8 & 18.2 & 14.1 & 1.0--1.8 & 33.80 \\
Starburst & 0.5 & 0.5 & 0.2 & $<$1.0 & 73.16 \\
(Tot. late-type) &(30.3)&(34.5)&(32.5)& & \\
\hline
\hline
\label{tab:gal_pop}
\end{tabular}
\end{center}
The fraction (in percentile) of each type of galaxies in the three
population models is shown:
without redshift evolution, in our reference model and for the
conservative scenario
(second to fourth column, respectively). The fifth
column shows the corresponding intrinsic $u-r$ color intervals
used for the modelisation of the spirals (early-type galaxies are
taken directly from the SDSS-maxBCG catalogue). The last
column shows the values of the ratio
$L_{\rm IR}/L_r$ for each template.
\end{table}
\subsection{Early-type galaxies} \label{ssec:early}
Early-type galaxies are characterized by an old stellar population and
a star-formation history which is essentially compatible with
passive evolution:
for this reason they are not expected to dominate the
emission in the IRAS\ bands. In fact, for a fixed $L_{\rm IR}$, the
amount of energy
emitted at wavelengths $\lambda \ga 40$ $\mu$m\ is one order of
magnitude lower with respect to normal spirals (see
Fig.\ref{fig:seds}). However, it is known that in dense environments
elliptical galaxies largely dominate the galactic population, being
about 4 times more frequent than spirals \citep[see,
e.g.,][]{dressler80,dressler97}. Moreover red galaxies are usually
more massive
and more luminous in the $r$ band than blue ones. For these reasons
their total contribution on the clusters IR signal may be non-negligible.
As mentioned in Section~\ref{ssec:maxbcg}, \cite{koester07} identified
the number $N_{\rm gal}^{R200}$\ of E/S0 galaxies ($M_r<-16$)
inside $R_{200}$ of each of the 7~476
clusters of our sample. For each cluster, the authors provide the
luminosity, $k$-corrected at $z$=0.25, in the $r$ band. of the BCG
and of the other E/S0 members as a whole. We corrected these
luminosities into rest-frame luminosities by using the LRG template of
\textsc{kcorrect} \citep{blanton07}: we refer to $L_{r}^{\rm BCG}$ and
$L_{r}^{{\rm memb}}$ for the luminosity of the BCG and of the cluster as a
whole, respectively, after this correction. Given the uniform
properties of early-type galaxies, and the high number of
such objects in our sample, their typical SED should be well
represented by the E/S0 template described above. In fact, although
the IR signal of every single galaxy is lost in the stacking process,
the E/S0 template still represents the average behavior of the
early-type population of galaxies.
For every cluster we assign a $r$-band luminosity
to all of its early-type members contained in the SDSS-maxBCG
catalogue. For the BCG we use the value of
$L_{r}^{\rm BCG}$, while for the other galaxies we use the
average luminosity of the non-BCG E/S0 galaxies, determined
as
\begin{equation}
L_{r}^{\rm avg} =
\frac{L_{r}^{\rm memb}-L_{r}^{\rm BCG}}{N_{\rm gal}^{R200}-1} \ .
\label{eq:lr_others}
\end{equation}
From eq.~\ref{eq:f_ir}, we translated these $r$-band luminosities
into IR luminosities: $L_{\rm IR} =
R_{{\rm IR},r} \times L_r$, subsequently used to normalise the
SED and further on to derive the fluxes in the 60 $\mu$m\ and 100
$\mu$m\ IRAS\ bands (see Sect.~\ref{sec:flux}).
\subsection{Contribution of late-type galaxies} \label{ssec:late}
Although the majority of optically bright galaxies in clusters
environment are
elliptical, and despite spiral galaxies in dense
environments tend to be quickly stripped of their gas and have
their star-formation quenched, they are still expected to provide a
dominant contribution to the IR emission, due to their higher
SFRs. Therefore, the contribution of late-type galaxies to the
IR emission is crucial for our purposes and should be carefully
estimated. Given the fact that
late-type galaxies are not considered in the SDSS-maxBCG catalogue,
their contribution was computed based on available modelings
of galaxy populations, in particular the distribution of
galaxies as a function of spectral type and luminosity. Since
our sample goes from galaxy groups to rich clusters, we have
included the environment dependence of these variables.
We based our model on the known properties of galaxies in the local
universe as given by \cite{balogh04} who analysed the bimodal distribution
of galaxies in a local ($z<0.08$) sample of SDSS galaxies (DR1). These authors
performed a detailed study on the color and luminosity distribution of
galaxies as a function of their environment for both early and late-type
galaxies. For each galaxy they define a density estimator
$\Sigma_5$ which represents the local projected galaxy density ($M_r<-20$) in
Mpc$^{-2}$ and they use it to classify the different environmental
regimes. Beside $N_{\rm gal}^{R200}$\ mentioned before (see Sect.~\ref{ssec:maxbcg}), the
SDSS-maxBCG
dataset contains also information about the number
of galaxies $N_{\rm gal}$ projected within a distance lower than 1
$h^{-1}$ Mpc from the BCG, with the magnitude limit of $M_r<-16$.
Therefore for each cluster we can assume a
unique value of $\Sigma_5$ for all of its members by establishing a
relation with $N_{\rm gal}$ and, finally, with the properties of
the population of late-type galaxies present.
We proceed as follows.
\begin{itemize}
\item[i)]First of all, we integrate the double gaussian
distributions of red and blue galaxies
\citep[Fig. 1 of][]{balogh04} obtaining a ratio of red and blue
galaxies for the 5 values of $\Sigma_5$ considered: for our work, we
only need the three density bins corresponding to dense environments (i.e.
groups and clusters, $\Sigma_5 > 0.5$). In these bins the resulting spiral
fractions are $f_{\rm spi}=$0.597, 0.478 and 0.286, in order of
increasing density.
\item[ii)] In order to use these values for our
calculations we need, at first, to correct the value of $N_{\rm gal}$.
The galaxy samples of
\cite{koester07}, in fact, has a magnitude limit
$M_r < -16$ while the definition of $\Sigma_5$ includes only
galaxies with $M_r < -20$: for this reason we use the red
galaxies luminosity function (LF) given by \cite{baldry04} to
calculate the number of expected galaxy members with
$M_r < -20$. We obtained $<N_{\rm gal}^{-20}>=0.32 \, N_{\rm gal}$.
As for \cite{balogh04}, the sample of \cite{baldry04} is constituted
by local SDSS galaxies, with the difference that it contains no
density distinction (e.g. field galaxies). Therefore, by
using the LF of \cite{baldry04} for this
calculation we neglect the effect of environmental properties;
anyway, since red galaxies are mostly present in galaxy
clusters, we do not expect this to change significantly our
results.
\item[iii)] We are now able to obtain a first estimation
of the density parameter associated to every cluster with
the formula
\begin{equation}
\Sigma_{5,0} = \frac{<N_{\rm gal}^{-20}>}{1-f_{{\rm spi},0}}
\frac{h^2}{\pi (1 {\rm Mpc})^2} \ ,
\label{eq:sigma5}
\end{equation}
where $f_{{\rm spi},0}$ is the fraction of spiral galaxy
(with $M_r < -20$) that for this calculation we fix arbitrarily
to the initial value to $f_{{\rm spi},0}=0.3$.
\item[iv)] Using the value of $\Sigma_{5,0}$ we
calculate a new estimate $f_{{\rm spi},1}$ of the spiral
fraction by interpolating between the values of
$f_{\rm spi}$ obtained by \cite{balogh04}; this allows the
calculation of a new value $\Sigma_{5,1}$ of the galaxy density
with equation~\ref{eq:sigma5}.
\item[v)] We repeat this procedure until $\Sigma_{5,i}$ and
$f_{{\rm spi},i}$ converge to definite values and check that
the result does not depend on the initial choice of
$f_{{\rm spi},0}$, thus obtaining the expected number
of spiral galaxies with $M_r < -20$
\begin{equation}
<N_{\rm spi}^{-20}> = <N_{\rm gal}^{-20}> \times
\frac{f_{{\rm spi}}}{1-f_{{\rm spi}}} \times
\frac{N_{\rm gal}^{R200}}{N_{\rm gal}} \ ,
\end{equation}
where the factor $N_{\rm gal}^{R200}/N_{\rm gal}$ accounts for the
ratio between the galaxies inside $R_{200}$ and $1 h^{-1}$ Mpc.
\item[vi)]Finally, we add the expected number of spiral galaxies in the
magnitude range $-20 < M_r < -18$ by referring again to the galaxy
distributions of \cite{balogh04}, thus obtaining for every cluster the
estimated value of the number of spiral members with
$M_r < -18$, $<N_{\rm spi}>$. This value is then used as the
average of a poissonian distribution, that we assumed
to assign a randomised number of spiral $N_{\rm spi}$ to each
cluster, thus to determine the total cluster galaxies: $N_{200} =
N_{\rm gal}^{R200}+N_{\rm spi}$.
\end{itemize}
\begin{figure}
\includegraphics[width=0.50\textwidth]{spiral_frac.ps}
\caption{Fraction of spiral galaxies within $R_{200}$
(see Sect.~\ref{ssec:maxbcg} for the definition of $R_{200}$) as a
function of $N_{\rm gal}$. The solid line represents the result
obtained when considering only galaxies with $M_r < -20$ \protect
\citep[i.e. the magnitude limit used to calculate $\Sigma_5$ in][]
{balogh04}. The dashed
line is the result obtained by
considering all the galaxies included in our model: late-type
galaxies with $M_r < -18$ and early-type galaxies with $M_r < -16$
from the SDSS-maxBCG catalogue. These
functions have been calculated neglecting the evolution of the
spiral fraction with redshift.}
\label{fig:spiral_frac}
\end{figure}
This formalism led us to derive an average fraction of
about 30\% spiral galaxies in our sample of SDSS-maxBCG clusters
(see Table~\ref{tab:gal_pop}).
The relation between $N_{\rm gal}$ and the
fraction of spiral galaxies inside the whole cluster is shown in Fig.
\ref{fig:spiral_frac}. When considering galaxies with the
same magnitude limit (dashed line), the spiral fraction
can be higher than 0.5 for groups and small clusters, and then
lessens with increasing $N_{\rm gal}$ down to $\sim$0.4 for
$N_{\rm gal}$=70. When considering all the galaxies included in
our model (i.e. $M_r<-16$ for early-type and $M_r < -18$ for late-type,
solid line) the spiral fraction becomes lower by about 0.2.
The morphology-density study of \cite{balogh04} becomes
useful also to assign a magnitude and a color to each
spiral galaxy. For a given value of $\Sigma_5$ it is
possible to construct a probability distribution
$\mathcal{P}(M_r,u-r)$ for a late-type galaxy
to have a given magnitude $M_r$ and a given intrinsic color $u-r$.
The former value is used to determine the luminosity $L_r$ in
the $r$-band as
\begin{equation}
L_r = L_{0,r} 10^{-0.4M_r} \ ,
\label{eq:mag2lum}
\end{equation}
where $L_{0,r}$=2.15$\times$10$^{34}$ erg/s is the luminosity of
an object with $M_r=0$, while the latter is used to assign the
spectral type, according to the intervals shown in Table~\ref{tab:gal_pop}.
To define these intervals we calculated the $u-r$ colors of our
\textsc{grasil} templates (2.40, 2.01 and 1.55 for the Sa, Sb and Sc,
respectively) and defined the limits in order to associate to every
galaxy the template that best approximates its value of $u-r$.
We also checked the consistency of these numbers with the observed
colors of local galaxies as reported by \cite{fukugita95} for the
SDSS photometric system\footnote{Referring to their Table~3, they
obtain $u-r$ colors of 2.26 for Sab and 1.68 for Sbc, thus close to
our assumptions for the Sa/Sb and Sb/Sc limits, respectively.}.
For what concerns starburst galaxies, the limit of
$u-r < 1.0$ has been taken directly from \cite{fukugita95} for irregular
galaxies, since the $u-r$ of our M82 templates ($u-r=2.10$) is not
representative of the starburst galaxies population. However, we will
show that the choice of this limit has a negligible impact on our
final results. Finally, the luminosity $L_{\rm IR}$
is obtained from Eq.~\ref{eq:f_ir},
where $R_{{\rm IR},r}$ is chosen according to the spectral
type.
\subsection{Evolution with redshift of spiral galaxies} \label{ssec:z_evol}
Despite being the SDSS-maxBCG a catalogue of relatively nearby
groups and clusters ($0.1<z<0.3$), it is necessary to take into
account the evolution with redshift of the late-type galaxies
properties with respect to the results of \cite{balogh04}
which are derived from a local (i.e. $z<0.08$) sample
of SDSS galaxies. For the purpose of our modelisation we consider the
sample of \cite{balogh04} as representative of the galaxy population
at $z_{\rm B}=0.04$, the central value of the interval,
and apply two independent evolution effects. \newline
\newline
{\bf Spiral fraction evolution} \newline
It is generally accepted that in a hierarchical structure formation
scenario spiral galaxies tend to evolve towards S0 or ellipticals.
This can happen either passively, with the consumption of the gas
reservoir and its ejection with SNe explosions, or it can also
be triggered by environmental interactions with other galaxies
and with the ICM \citep[see][for a review]{boselli06a}: it is
therefore expected that
the fraction of late-type galaxies increases with redshift
\citep[the so called Butcher-Oemler effect,][]{butcher84}.
In fact, \cite{lagana09} find a change from about 0.1 to 0.3
in the spiral fraction of 20 galaxy clusters in the
interval $0<z<0.25$. Fitting the data on the fraction of
star-forming galaxies of these clusters, they quantify its
redshift evolution as: $\frac{df_{\rm spi}}{dz} = 1.3\pm 0.6$
(private communication). A similar result has been
obtained by \cite{delucia07}, who find $\frac{df_{\rm spi}}{dz}
\sim 1.1$ in the range $0.4<z<0.8$. Given the redshift interval
of the SDSS-maxBCG catalogue, we take as a reference the value of
\cite{lagana09}. Despite this sample contains rather massive
objects ($k_{\rm B}T$ from $\sim3$ to $\sim10$ keV), it should provide
a first-order representation of the spiral fraction evolution
in the redshift range of the SDSS-maxBCG catalogue. We
therefore correct the number of spiral galaxies obtained in
Section~\ref{ssec:late} with the following formula:
\begin{equation}
<N_{\rm spi}(z)> = [1+1.3 \, (z-z_{\rm B})]<N_{\rm spi}>_{z=0} \ .
\end{equation}
Including this effect in our modelisation raises the global spiral
fraction of our sample from 30\% to 35\%, as shown in
Table~\ref{tab:gal_pop}.
Although this number may seem high for clusters \citep[see e.g.]
[for the Coma cluster]{bai06} we must consider that the SDSS-maxBCG
catalogue is dominated by groups and small clusters, as mentioned
in Section~\ref{ssec:maxbcg}. However, when applying our model to
the case of MS 1054-03 as observed by \cite{bai07}
($z=0.83$, $N_{200}=144$), we obtain $f_{\rm spi}=0.16$ which is
consistent within a 1$\sigma$ limit with the value of $0.13
\pm 0.03$ found by the authors.
\newline
\newline
{\bf Luminosity evolution} \newline The current hierarchical structure
formation scenario predicts that the peak of the star-formation is to
be placed at $z \approx 2-3$ and that at later epochs galaxies are
mainly consuming their reservoir of gas. For this reason, when
modeling the IR emission in the redshift range of the SDSS-maxBCG cluster
sample we must consider that higher SFRs are expected with respect to
the local galaxies. This effect has been widely observed and
quantified. We take as a reference the evolution of the IR-luminosity
observed by \cite{lefloch05}: when parameterising it in the form of
$L_{\rm IR} \propto (1+z)^{\alpha_L}$, they obtain $\alpha_L=3.2^
{+0.7}_{-0.2}$ over a sample of 2 635 objects identified at 24 $\mu$m\
in the {\it Chandra}\ Deep Field South in the redshift range $0 < z \la 1$.
This result has been
obtained by analysing field galaxies and this might introduce an
environment bias on our results. Anyway, since it is expected that
local star-forming galaxies inside clusters were the latest accreted,
we can assume that this evolution scenario is valid also in cluster
and group environments. Moreover, as shown in figure~3 of \cite{giard08},
this assumption is backed-up by the {\it Spitzer} observation of the
distant cluster MS1054-03 \citep{bai07}. Another bias may be introduced
by the fact that these objects were identified directly in the mid-IR
and not in the optical as for the galaxies of our sample. Therefore
the galaxies of this sample are brighter than the ones of
the SDSS-maxBCG, both in the IR and in the optical.
However,
the total luminosity of the sample of \cite{lefloch05} is dominated by
luminous ($L_{\rm IR}$$> 10^{11} L_\odot$) massive galaxies, which are the ones
that provide most of the signal also in the IRAS\ bands. Therefore, we
assume that their result
can be applied also to the SDSS-maxBCG cluster sample to describe its
global IR luminosity evolution.
Consequently, we determine
the IR luminosity $L_{\rm IR}$ as a function of redshift:
\begin{equation}
L_{\rm IR}(z) =
L_{{\rm IR},0}(1+z-z_{\rm B})^{3.2} \ ,
\end{equation}
where $L_{{\rm IR},0}$ is the luminosity obtained
by the magnitude $M_r$ as from eq.\ref{eq:mag2lum}.
We do not include any specific effect of spiral population
evolution towards bluer types with increasing redshift. In fact,
the fraction of the different spectral types of spirals in the
redshift range we are interested in is not expected to differ
significantly with respect to the local galaxies inside galaxy
clusters \citep[see e.g.][]{delucia07}. Moreover the consequent
effect of slightly increased global luminosity is anyway
partially included when assuming the luminosity evolution
with redshift aforementioned.
\subsection{Uncertainties on the model parameters}
\label{ssec:unc}
The choice of the values of the different parameters
introduces a degree of uncertainty in our modelisation, the most
important of which are connected with the choice of the redshift
evolution parameters and the $u-r$ color intervals used to define the
spiral types (see Table~\ref{tab:gal_pop}). In order to quantify these
uncertainties, we introduce two other sets of parameters to define a
conservative and an extreme scenario and we will use the values of the
fluxes estimated with these models to define the degree of confidence
of our results. More precisely, for what concerns the redshift
evolution we adopt the $\pm 1 \sigma$ values for $\frac{df_{\rm spi}}
{dz}=(0.7,1.9)$ and $\alpha_L=(3.0,3.9)$, as measured by
\cite{lagana09} and \cite{lefloch05}, respectively. Moreover, we also
artificially add a shift in the $u-r$ color intervals of $\pm 0.1$ in
order to obtain a redder (conservative scenario) and a bluer (extreme)
spiral population. Anyway, these changes introduce only minor
modifications in the galaxy population: the difference in the
global spiral fraction is of $\sim2$\%, while the only significant
change is in
the fraction of starburst galaxies that goes from 0.2\% in the
conservative model to 1.1\% in the extreme one. On the whole, the
difference in the final fluxes is mainly introduced by the modified
luminosity evolution. The results on the galaxy
population for the conservative model are also shown in
Table~\ref{tab:gal_pop}.
\subsection{The IR luminosity function of the SDSS-maxBCG galaxies}
\label{ssec:lumfun}
We show in Fig.~\ref{fig:lum_func} the luminosity function of
the galaxies included in our reference model, for the different
spectral types. The luminosity considered is given in the
rest-frame 60 $\mu$m\ IRAS\ band.
The luminosities go from a minimum of $\sim10^8$$L_\odot$\ for the
fainter E/S0 galaxies up to a maximum of some $10^{11}$$L_\odot$\
for the brightest spirals.
Each type of spiral has a luminosity range of about 2 orders
of magnitudes: this is a consequence of our method (see
Sect.~\ref{ssec:late}), that held the spiral magnitudes within
the interval $-23<M_r<-18$ which corresponds to a factor of
$10^{0.4 \ \Delta M_r}=100$ in luminosity
with the only possible modification introduced by the
redshift evolution. Anyway, we don't expect this limitation
to affect significantly our results. In fact, this limit
range is wider than the one covered by the distribution of the BCGs,
which are the only galaxies for which we have direct
observational constraints. On the contrary the E/S0 curve
appears narrower because we considered the average
luminosity of the E/S0 galaxies inside each cluster, thus
limiting the dispersion of the E/S0 galaxies to the
dispersion of the cluster luminosities.
When considering the luminosity function in the 100 $\mu$m\
IRAS\ band the range and shape is similar for all the
spectral types. Only starbursts are a factor $\sim$3 fainter
in this band due to the drop in luminosity (see
Fig.~\ref{fig:seds}) which is due to an average
higher galactic dust temperature compared to non-starburst.
\begin{figure}
\includegraphics[width=0.50\textwidth]{lum_func2.ps}
\caption{Luminosity functions of each spectral type
implemented in our model in the IRAS\ 60~$\mu$m\ (i.e. top hat
band-pass within [40,80]~$\mu$m). Spectral types are color-coded
as follow: (i) early-type galaxies as (dark-red) for the BCGs and
(light-red) for the other E/S0s (we assigned to every galaxy the
average luminosity of the E/S0 members of each cluster, as shown in
eq.~\ref{eq:lr_others}); (ii) normal late-type galaxies, with
Sa, Sb, Sc going from (light-blue) to (dark-blue), respectively;
(iii) starburst galaxies as (green).}
\label{fig:lum_func}
\end{figure}
\section{Other galactic components}
\label{sec:other}
In this Section we try to put some constraints on the impact
of three components of galactic origin which are not included
in our modelisation of the IR emission of cluster galaxies:
a missed population of faint galaxies, the possible presence of
dust-embedded AGNs and the IR emission coming from
heavily obscured star-forming galaxies. In all of these cases we
conclude that their impact on our final results is probably
very small, if not completely negligible.
\subsection{Faint galaxy population}
As mentioned in Sect.~\ref{sec:model}, our modelisation includes
red galaxies with $M_r < -16$ (the limit of the SDSS-maxBCG
catalogue) and blue galaxies with $M_r < -18$, thus
neglecting the signal of fainter objects, that may not be
identified by the SDSS observations although present inside the
cluster. We try to obtain
a rough estimation of the impact of these objects on the
total emission by referring to the luminosity functions
obtained by \cite{baldry04} (see their Fig. 7).
For what concerns early-type galaxies, the faint-end of the
LF ($M^*=-21.49$, $\alpha=-0.83$ in the Schecter function
parameterisation) has a negative slope, so the number of galaxies
is expected to diminish at higher magnitudes. For this
reason no significant impact can be associated to these
objects.
On the other side, the faint-end of the blue galaxies LF
($M^*=-20.60$, $\alpha=-1.35$) has a positive slope, so an
increasing
number of galaxies is expected with lower luminosities.
For simplicity, we consider the assumption that all
galaxies with $M_r < -18$ have been included in our
modelisation, then we integrate the late-type LF
$\phi_{\rm b}(M_r)$ of \cite{baldry04} splitting it in
two at this magnitude limit. We obtain
\begin{equation}
\frac{L_{\rm faint}}{L_{\rm bright}} =
\frac{\int_{-18}^\infty \phi_{\rm b}(M_r)L(M_r)dM}
{\int_{-23}^{-18}\phi_{\rm b}(M_r)L(M_r)dM}
= 0.13 \ ,
\label{eq:lfaint}
\end{equation}
where $L(M_r)$ is the luminosity as a function of the
magnitude $M_r$, thus indicating that the emission of the
faint galaxy population is marginal with respect to the
bright one.
Moreover, since the LF of \cite{baldry04} is
obtained with the observation of field galaxies, it is
reasonable to expect that in cluster environments the
faint-end of the LF would be shallower, if not even
negative, due to the processes of merging that affect
particularly the smaller objects. Therefore the ratio
of 0.13 obtained in eq.~\ref{eq:lfaint} can be safely
considered an upper limit of the true value.
For these reasons, and considering the assumption of the
connection between the luminosity in the optical and IR
bands used throughout this work, we can conclude that even
if we can not exclude the presence of the signal of
a population of faint ($M_r > -18$) unresolved galaxies,
we expect that the contribution of faint star-forming
galaxies to be marginal.
\subsection{Dusty AGNs}
When estimating the total IR luminosities of the galaxies
of our sample, we do not take into account what could be
the contribution of AGN deeply embedded in dusty cocoons
within cluster galaxies.
In fact, recent results indicate the existence of
a population of heavily absorbed AGNs in the field \citep[see e.g.][]
{fiore09,lanzuisi09}: these objects are detectable at IR wavelengths.
Although optical observations indicate a very small fraction of
cluster galaxies with detected AGNs \citep[$\sim1$\%, see
e.g.][]{dressler99}, some X-ray observations have detected an excess
of point sources associated to AGNs in cluster fields. For example,
\cite{martini06} observed with {\it Chandra} a sample of 8 low redshift
($z \lesssim 0.3$) clusters, finding $\sim$5\% of their galaxy members
hosting an AGN, most of which are not detected with optical
surveys.
These AGNs are also present in E/S0 galaxies, and whilst
they are obscured in the optical (their emission being heavily
absorbed), they could be bright in the IR. Therefore, they could
contribute to the total IR cluster emission.
It is difficult to quantify with precision the impact on the total IR
signal of these kinds of objects. We take as a reference the work done
by \cite{bai07}, who studied the IR properties of the galaxies of
MS1054--03 ($z=0.83$) with the {\it Spitzer}
satellite. They identified eight point sources with X-ray and
radio observations that could be associated with AGNs: anyway, only
three of these objects have an IR counterpart over the 144 IR-detected
cluster members, and only one of these is associated with a bright
star-forming object. For these reasons the authors conclude that the
contamination from AGNs is negligible. Similar conclusions have been
drawn by \cite{bai06} in their study of the Coma cluster ($z=0.023$).
Even if they are based on single cluster studies,
these results, which have been obtained both at
redshift higher and lower than our cluster sample,
indicate that the impact of AGN contamination in
the observations of \cite{giard08} is probably
very small.
\subsection{Heavily obscured star-forming galaxies}
Optical observations have highlighted the fact that some
galaxy clusters host heavily obscured star-forming galaxies.
In the framework of our modelisation, the presence of these
objects would lead to an underestimate of their IR emission
based on the optical one \citep[see][for a review]{metcalfe05}.
In particular, IR observations on A1689 \citep{duc02},
J1888.16CL \citep{duc04},
CL0024+1654 \citep{coia05} and, more recently, on A1758
\citep{haines09} have revealed galaxies with much higher SFRs
with respect to what is expected from optical diagnostics (e.g.
[O{\sc ii}]). However, all of these clusters show clear signs
of recent dynamical activities, like major mergers and
significant accretion of galaxies from the field, thus
suggesting that these phenomena are responsible of the
star-formation triggering. Since our cluster sample is at a
relatively low redshift, we do not expect it to contain a high
fraction of dynamically active haloes and, therefore, to be
significantly affected by the presence of heavily obscured
star-formation.
\section{Reconstructing the stacked IR flux}
\label{sec:flux}
As mentioned before the main objective of this work is
to explain the origin of the IR emission observed in the direction
of galaxy clusters by \cite{montier05} and \cite{giard08}. Therefore,
we want to compare
the flux expected out of our model from the galactic emission in the
60 $\mu$m\ and 100 $\mu$m\ IRAS\ bands with the one measured by
\cite{giard08}. In order to do so, we need to compute for every
single cluster the expected flux taking into account the instrumental
beam and the spectral band pass of the IRAS\ satellite.
We assume that every galaxy is placed at the
redshift $z$ of the cluster to which it belongs and we compute
its flux $F_{\lambda_0}$\footnote{This quantity corresponds to a
flux per unit frequency (e.g. Jy). Anyway we prefer to use the
notation $F_\lambda$, rather than $F_\nu$, to refer directly to the
fluxes $F_{60}$ and $F_{100}$ in the 60 and 100 $\mu$m\ bands,
respectively.}
in a given band
\begin{equation}
F_{\lambda_0} =
\frac{1}{4\pi d_{\rm L}^2 \Delta \nu_0}
\int L \left( \frac{\lambda}{1+z} \right)
f_{\lambda_0}(\lambda) d\lambda \ ,
\end{equation}
where $\Delta \nu_0$ is the frequency interval of the corresponding band,
\begin{equation}
d_{\rm L}(z) = \frac{c(1+z)}{H_0}\int_0^z
\frac{dz'}{\sqrt{\Omega_{\rm m}(1+z')^3+\Omega_\Lambda}}
\end{equation}
is the luminosity distance of the cluster
($H_0 \equiv 100\, h$ km s$^{-1}$ Mpc$^{-1}$),
$f_{\lambda_0}(\lambda)$ is the IRAS\ spectral response function
in a given band (i.e. $\lambda_0 =60, 100$ $\mu$m) which is
convolved\footnote{For these calculations we used the
tools of the \textsc{dustem} code, an updated version of the
emission model of \cite{desert90}, which contains details on
the IRAS\ response functions in the different bands.
}
with the SED $L(\lambda)$ of the
spectral type of each cluster galaxy, normalised according to
the value of $L_{\rm IR}$\ (see Sect.~\ref{sec:model}).
\begin{figure*}
\includegraphics[width=1.00\textwidth]{hist_r200.ps}
\caption{Distribution of the angular sizes of $R_{200}$ for the cluster
sample in 4 redshift intervals chosen in order to contain the same number of
clusters (i.e. 1869). The bin size of each histogram is 0.5 arcmin.
}
\label{fig:hist_r200}
\end{figure*}
The stacked signals by \cite{giard08} are integrated
fluxes over an angular area of 10' of radius centered on the cluster
(see Sect.~\ref{sec:stack} for details). Since the IRAS\ FWHM
beams are 4' and 4.5' at 60 and 100 $\mu$m\, respectively, thus
comparable to the size of the observed field, we need to take into
account the possible loss of signal from galaxies
distant from the cluster center due to the convolution with the
instrumental beam. In order to model this effect, we
have to distribute the cluster galaxies in the
cluster potential well. We thus randomly assign to every galaxy a
distance from the BCG (considered to be at the cluster center, thus
at the center of the 10' field) by assuming that their spatial
distribution follows a NFW profile \citep{navarro97}.
The two parameters needed
to characterise the NFW profile are $R_{200}$ and the concentration
$c$. They were obtained from the richness $N_{\rm gal}^{R200}$\
of each halo (see Sect.~\ref{ssec:maxbcg}) and by adopting the
$N - M$ scaling relation
of \cite{rykoff08} derived from the SDSS-maxBCG, and
the $(M,z)-c$ relation of \cite{dolag04} derived from numerical
simulations. Then for every galaxy we compute the fraction of the
signal that falls
inside the observed region in the two bands. This effect
proves to be completely negligible as globally less than one per
cent of the signal falls outside the field in both bands.
Since galaxies of later spectral types are expected to be more
spread than ellipticals, as a result of their recent accretion into
the cluster
\citep[see, e.g., the discussion in][]{popesso05},
we repeated our estimation on the possible
lost signal by assuming that late-type galaxies are distributed
uniformly in a sphere of radius $R_{200}$: even with this extreme
hypothesis the amount of signal that is expected to fall outside the
field remains negligible (about 3\% of the total).
The low impact of this effect on the global results
can also be seen from the distributions of the
angular sizes of $R_{200}$ obtained with this method shown in
Fig.~\ref{fig:hist_r200} for different
redshift bins. More than 70\% of the clusters have angular sizes
lower than 5' and only in the lowest redshift bin some objects (43)
exceed the 10' aperture.
With the stacking technique
adopted by \cite{giard08}, it is clear that also the signal
coming from foreground/background objects is present in the
observed fields. Anyway, as said in Section~\ref{sec:stack},
the authors adopted a background subtraction based on the signal
in the fields adjacent to every map, relying on the high statistical
robustness of their sample. For this reason we can safely conclude
that the emission of foreground/background objects has been
successfully subtracted, thus we do not need to include it in
our modelisation.
\section{Results and discussion}
\label{sec:results}
\subsection{Predicted and observed IR fluxes}
Our estimations of the 60 and 100~$\mu$m\ fluxes from cluster
galaxies are reported in Table~\ref{tab:results},
associated to the emission calculated with our model
for the different galaxy populations.
\begin{table}
\begin{center}
\caption{Estimated fluxes in the 60 $\mu$m\ and 100
$\mu$m\ IRAS\ bands from the galaxy population of the SDSS-maxBCG
clusters.
}
\begin{tabular}{lcrr}
\hline
\hline
Spectral type & & $F_{60}$ & $F_{100}$ \\
& & (Jy) & (Jy) \\
\hline
\it{-Early-type} & & & \\
BCG & & 8.2 & 16.5 \\
E/S0 & & 31.1 & 62.9 \\
(Total early-type) & & (39.3) & (79.4) \\
\\
\it{-Late-type} & & & \\
Sa & & 36.4 & 58.6 \\
Sb & & 259.5 & 621.3 \\
Sc & & 333.9 & 1133.4 \\
Starburst & & 15.4 & 12.1 \\
(Total late-type) & & (645.2) & (1825.4) \\
\\
\hline
Total & & 684.5$_{-138.0}^{+211.3}$ & 1904.8$_{-429.1}^{+617.1}$ \\
Observed & & 570.1$\pm36.1$ & 1359.9$\pm249.1$ \\
\hline
\hline
\label{tab:results}
\end{tabular}
\end{center}
We show the total fluxes and contribution of each galaxy population
included in our model. The quoted errors on the total values
correspond to the differences between our reference model and the
conservative and extreme scenarios described in Section~\ref{ssec:unc}.
The measurements by \cite{giard08} are reported together with their
1$\sigma$ error bars.
\end{table}
The total flux due to the early-type galaxies estimated with our model
is 39.3~Jy at 60 $\mu$m\ band and 79.4~Jy in the 100 $\mu$m\ band,
accounting for about 7\% of and 6\%, respectively, of
the fluxes measured in the same bands by \cite{giard08}. In both bands
about 20\% of the E/S0 signal comes from the BCGs.
As expected, late-type galaxies constitute by far
the most significant contribution to the IR emission. According to our
reference model they contribute to 645.2~Jy at 60~$\mu$m\ and
1904.8~Jy at 100~$\mu$m\
thus accounting for about 95\% of the total galactic
emission. This contribution comes mostly from the Sb and Sc population,
with the Sc being widely dominant in the 100~$\mu$m\ band.
Despite their high luminosities, we predict that starburst galaxies do
not provide a significant contribution to the IR emission, due to
their low expected number. We obtain a contribution of
15.4~Jy at 60 $\mu$m\ and 12.1~Jy at 100~$\mu$m\,
corresponding respectively to about 2\% and 0.5\% of total predicted
signal. Only in our extreme scenario their contribution becomes
non-negligible in the 60 $\mu$m\ band, reaching the 5\% of
the total predicted flux. This low contribution agrees with the
low rate of starburst galaxies as found in the field by
\citet{lefloch05} at the redshift range of the SDSS-maxBCG
catalogue (i.e. $0.1<z<0.3$).
The total fluxes associated to the galactic emission predicted
by our reference model are $684.5 \ [546.5,895.8]$~Jy at 60~$\mu$m\
and $1904.8 \ [1475.7,2521.9]$~Jy at 100~$\mu$m\ (the bracketed interval
indicate the values derived from our conservative and extreme
models, see Sect.~\ref{ssec:unc}).
It appears that the reconstructed IR emission due to
the galactic dust emission of the cluster members can
explain the entire signal measured by \cite{giard08}, with an
indication that our reference model overestimates the total flux,
particularly at 100~$\mu$m. Indeed, these
authors obtained $570.1\pm36.1$~Jy and $1359.9\pm249.1$~Jy at
60 $\mu$m\ and 100 $\mu$m, respectively. We will propose an explanation
of this discrepancy in the next sections. However, when considering
our conservative scenario, the predicted emission is in good agreement
with the total measured signal in both bands.
Given these results, modulo the uncertainties of our
modelisation, we obtain that the IR emission of the galaxy members
is consistent with the total observed emission of our clusters
sample, leaving little space to the possible presence of
other components like intracluster dust.
\subsection{Redshift evolution}
\label{ssec:z_ev}
In this Section we characterise the redshift evolution of the average
clusters luminosity. Following \cite{desert90}, we define the total
luminosity in the 60 and 100 $\mu$m\ IRAS\ bands as
\begin{equation}
\label{eq:lum_60+100}
L_{60+100} \equiv 4 \pi d_{\rm L}^2 \times
\left[ \lambda \frac{\Delta \nu}{\Delta \lambda}
F_\lambda(60 \mu {\rm m}) +
\lambda \frac{\Delta \nu}{\Delta \lambda}
F_\lambda(100 \mu {\rm m}) \right] \ ,
\end{equation}
where $\Delta \nu$ and $\Delta \lambda$ are the bandwidths in
frequency and wavelength, respectively, of the two IRAS bands.
This formula represents a good approximation of the total luminosity of our
haloes at $\lambda \gtrsim 40$$\mu$m.
We show in Fig.~\ref{fig:lum_z} the average luminosity as obtained
from eq.~\ref{eq:lum_60+100} for the clusters at different
redshifts, both as predicted by our model and as observed by
\cite{giard08}. Our 4 bins are defined in order to contain the same
number of clusters (i.e. 1869).
This analysis shows clearly that the discrepancy between
our reference model with respect to the observed fluxes is due to the
low-redshift clusters only, which are responsible for most
of the predicted flux. In fact, while our reference model is compatible
within $1 \sigma$ with the results of \cite{giard08} in the 3
high-redshift bins, in the first bin ($0.10<z<0.17$) the predicted
luminosity is higher at $6 \sigma$ confidence. Our conservative
scenario is also compatible with observed luminosities at $z>0.17$,
while the discrepancy for the low-redshift clusters persists at
more than $3 \sigma$. On the contrary, our extreme model is excluded
by the results of \cite{giard08} through all the redshift range.
\begin{figure}
\includegraphics[width=0.50\textwidth]{lum_z.ps}
\caption{Average cluster luminosities (see eq.~\ref{eq:lum_60+100})
as a function of redshift.
Diamonds represent the measurements and associated $1 \sigma$ error
bars by \cite{giard08} in 4 redshift bins, defined in
order to contains the same number of clusters (i.e. 1869).
The solid line represents the average luminosity of the clusters
in our reference model. The two dashed lines show the
corresponding luminosity of the conservative and
extreme model (see Section~\ref{ssec:unc} for the details).}
\label{fig:lum_z}
\end{figure}
\subsection{Selection bias}
Our results indicate that although our model
well describes the global IR emission of the SDSS-maxBCG
galaxy clusters at $z \gtrsim 0.17$, the predicted galactic
emission at low redshift clearly exceeds the measured stacked
signal. The strong decrease in the IR luminosity is present also in the
NED and NSC clusters samples analysed by \cite{giard08} and, as
discussed in their work, it can be interpreted as a selection
effect which biases towards rich and massive clusters at high
redshift as
it is evident for the NSC sample (see their Fig.~6). However,
this is not the case for the clusters object of our analysis. In
Fig.~\ref{fig:ngal_z} we show the average cluster richness in
the same redshift bins both for the values of $N_{\rm gal}^{R200}$,
directly obtained from the SDSS-maxBCG catalogue, and $N_{200}$
which includes late-type galaxies, as described in
Section~\ref{ssec:late}. It is clear that no selection bias is
present in our clusters sample: on the contrary low-redshift
clusters show slightly higher values of $N_{\rm gal}^{R200}$. Even when including
late-type galaxies this trend does not change significantly.
\begin{figure}
\includegraphics[width=0.50\textwidth]{ngal_z.ps}
\caption{Average richness as a function of
redshift. Diamonds represent the average values of $N_{\rm gal}^{R200}$\
(i.e. the number of E/S0 members) with associated $1 \sigma$
errorbars in 4 redshift bins defined as in Fig.~\ref{fig:hist_r200}
and \ref{fig:lum_z}.
Triangles represent the values of $N_{200}$ (i.e. total cluster
members, see the definition in Section~\ref{ssec:late}).}
\label{fig:ngal_z}
\end{figure}
\subsection{Effect of the cluster environment}
These considerations indicate that the discrepancy
between our model and the decrease of luminosity towards lower
redshift must be connected with an evolution of the galaxy
IR luminosity driven by the cluster environment, associated to gas
and/or dust removal and consequent star-formation quenching.
This can happen via ram-pressure stripping and tidal interaction.
This picture is in agreement with the low number
of high IR-to-optical galaxies observed in local galaxy clusters
\citep[see, e.g.,][]{bicay87}. A similar phenomenon
is also seen in the observed properties of the galaxies of the Virgo
cluster. In fact, \cite{boselli06b} observed a truncation in the
disk of NGC 4569 both in the UV and in the IR (8--70 $\mu$m) that they
associate to a ram-pressure stripping of the external regions.
More in general, \cite{gavazzi06} observe a consistent
$H_\alpha$ deficiency in Virgo galaxies which indicates a low
SFR and diminished dust heating/emission \citep[see also the
discussion in][]{boselli06a}.
In this scenario, once the star-formation quenching happens,
the SEDs of the galaxy members would be consistently modified
at the IR and UV wavelengths, being not compatible anymore with
field galaxies, thus explaining the excess predicted by our model.
However, we must point out that the environmental effects on the Virgo
cluster are likely to be much stronger than what is expected for
the population of clusters considered in our analysis. In fact,
the Virgo cluster is
about one order of magnitude more massive than the majority of the
SDSS-maxBCG galaxy clusters and it shows several evidences of recent
dynamical activity. On the contrary, being the SDSS-maxBCG cluster
sample mainly constituted by small haloes (see Section~\ref{ssec:maxbcg}),
in the context of the
hierarchical structure formation scenario it is reasonable to expect
that they correspond to relaxed and dynamically old systems
\citep[see, for instance, the results on the concentration-mass
relation of][]{dolag04,buote07}.
\subsection{Constraints on intracluster dust}
If we take as a reference our conservative scenario and compare
its results with the average observed luminosity in the
three high-redshift bins, we can estimate an upper limit on the possible
emission due to extragalactic dust of about 10\% of the total luminosity,
which translates into a dust-to-gas mass abundance of
$Z_{\rm d} \lesssim 5 \times 10^{-5}$ \citep[see the discussion
in][]{giard08}. This figure is in agreement with the expectations from
theoretical models \citep{popescu00,aguirre01} and with
current observational upper limits on the dust abundance.
\cite{chelouche07} analysed the reddening of a sample of quasars
in the direction of the clusters of the SDSS-maxBCG catalogue,
obtaining an estimate of $Z_{\rm d} \approx 10^{-5}$. A similar
result has also been obtained by \cite{bovy08} by measuring
the dust absorption on the spectra of galaxies located behind local
($z \sim 0.05$) galaxy clusters.
If on one side we do not see an evidence of intracluster dust
emission in our sample, these last considerations leave an open
question of how much dust has been lost by cluster galaxies polluting
the ICM and if it can live enough to produce a significant diffuse
IR emission. According to \cite{popescu00}, the dust stripped from
infalling galaxies will probably remain localised close to their parent
galaxies without diffusing efficiently into the ICM.
In any case, the strong decrease in the total cluster luminosity
observed by \cite{giard08} compared to our results, suggests that,
if present, the signal of intracluster dust should be very small.
\section{Summary and conclusions}
\label{sec:concl}
In this work we performed, for the first time, a
thorough modelisation of the overall IR emission of galaxy clusters
due to cluster galaxies IR emission. We tested the results of our
model against the statistical stacking measurements of clusters IR
emission in the 60 and 100~$\mu$m\ IRAS\ bands by \cite{giard08},
making use of the SDSS-maxBCG catalogue of groups and clusters
\citep{koester07}.
We used the available SDSS-maxBCG data on the
luminosity in the $r$-band of the cluster early-type members, and
converted them into IR luminosities by adopting the model templates of
the \textsc{grasil} code \citep{silva98}. Since the
SDSS-maxBCG catalogue does not contain information on the late-type
galaxies component, we used the results
on the morphology-density relation obtained by \cite{balogh04} on a
local SDSS galaxy sample to construct a model to
statistically associate to every cluster its spiral
galaxy population, namely the number of spiral members, their
spectral type and their $M_r$. Again, this
information on the optical properties
has been used to obtain the corresponding IR emission by making
use of 4 other \textsc{grasil} templates to represent
normal late-type (Sa, Sb and Sc) and starburst galaxies.
We also included in our model the expected redshift evolution of the
spiral fraction and the IR luminosity by using results on observed
galaxies in order to account for the possible difference in the spiral
galaxy population of the SDSS-maxBCG catalogue ($0.1<z<0.3$) and the more
local objects of the \cite{balogh04} sample ($z<0.08$).
Finally, we used our predictions to calculate the total expected
galactic flux, by considering also the possible loss of signal
due to the IRAS\ beam smoothing and the IRAS\ response
function, and compared it with the measurements of \cite{giard08}.
Our main results can be summarized as follows.
\begin{itemize}
\item[i)] According to our model, late-type galaxies
represent about $\sim 35$\% of the galaxy population in nearby
groups and clusters (i.e. $z<0.3$). This figure is coherent with the
fact that this sample is dominated by groups and small clusters
($N_{\rm gal}^{R200}$$= 10-20$), thus letting this fraction of
late-types falling between the field ($f_{\rm spi} \approx 0.5$)
and the rich cluster ($f_{\rm spi} \approx 0.15$) regime.
\item[ii)] As expected, normal late-type galaxies constitute the most
important contribution to the total IR luminosity
at 60 and 100 $\mu$m\, accounting for $\sim 95$\% of the
total galactic emission. Oppositely, the impact of
starburst galaxies is marginal.
\item[iii)] Early-type galaxies dominate the faint end of the LF
in the IR and they account for the remaining $\sim 5$\%
of the galactic contribution of the cluster IR emission.
\item[iv)] Our model shows that the total flux estimated from the galaxy
accounts, within uncertainties, for the entire stacked signal
measured by \cite{giard08}. With our reference model we
obtain $684.5$~Jy at 60 $\mu$m\ and $1904.8$~Jy at 100~$\mu$m,
which exceeds the stacked fluxes measured by \cite{giard08}.
However, when considering the uncertainties in our parameters,
we showed that in a conservative scenario this discrepancy
disappears.
\item[v)] We compared the redshift distribution of the
IR emission predicted by our model to the measurements of
\cite{giard08} and found that the excess flux is present
only in the clusters at lower redshift ($z \lesssim 0.17$).
\end{itemize}
The results presented here show that the IR emission
of galaxy clusters can be explained with its galactic component only,
leaving very little room to the possible presence of any diffuse
emission associated to intracluster dust. In this framework, the upper
limit on the dust-to-gas mass abundance obtained by \cite{giard08} can
be reduced by an order of magnitude, down to $Z_{\rm d} \lesssim
5 \times 10^{-5}$.
This result is in
agreement with current estimations on the dust abundance
in the ICM obtained from extinction and reddening measurements
in the direction of SDSS clusters \citep{chelouche07,bovy08}.
The lack of diffuse dust, however, does not mean that the
environment is not influencing the IR properties of its galaxy
members. The fact that the predicted excess emission is concentrated
in the lowest redshift objects indicates that the IR emission
of late-type galaxies in local clusters is not completely
compatible with their field equivalents. This result is in
agreement with the lack of bright IR galaxies observed in several
local clusters \citep[see][and references therein]{boselli06a} and
it is likely connected to the quenching of the star-forming
activity driven by the cluster environment on its galaxy members, due
to gas or dust removal: in this last case, the dust injected
in the ICM is probably quickly depleted via sputtering processes.
However, it is not clear why this could act significantly only
at $z \lesssim 0.2$ when most of the large-scale structures are
already formed. For this reason, and given the limited redshift
range of the SDSS-maxBCG catalogue, it would be interesting to
extend our analysis to higher redshifts, where the SFR is expected
to be higher and more dynamical activity is expected at cluster
scales. \\
\begin{acknowledgements}
We thank an anonymous referee who helped improving the presentation
of our results. We are grateful to L. Silva for publishing some
\textsc{grasil} templates in her webpage. We thank M. Balogh and T.
F. Lagan\'a for
providing some additional data not published in their papers.
We acknowledge useful discussions with S. Bardelli, J.-P. Bernard,
A. Boselli, J. Lanoux, L. Pozzetti, F. Pozzi and C. Tonini. We are
particularly grateful to A. Bongiorno
for the help provided in the treatment of optical data. The authors
L. Montier, E. Pointecouteau and M. Roncarelli acknowledge the support
of grant ANR-06-JCJC-0141.
\end{acknowledgements}
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\title{On the List-Decodability of Random Linear Codes}
\author{{\sc Venkatesan Guruswami}\thanks{Computer Science Department, Carnegie Mellon University. {\tt guruswami@cmu.edu}. Supported in part by NSF CCF 0953155 and a Packard Fellowship.}
\and {\sc Johan H{\aa}stad}\thanks{School of Computer Science and Communication, KTH. {\tt johanh@csc.kth.se}. Research supported by ERC grant 226203.}
\and {\sc Swastik Kopparty}\thanks{CSAIL, MIT. {\tt swastik@mit.edu.} Work was partially done while the author
was an intern at Microsoft Research, New England.}
}
\begin{document}
\maketitle
\thispagestyle{empty}
\begin{abstract}
For every fixed finite field ${\mathbb F}_q$, $p \in (0,1-1/q)$ and $\varepsilon >
0$, we prove that with high probability a random subspace $C$ of
${\mathbb F}_q^n$ of dimension $(1-H_q(p)-\varepsilon)n$ has the
property that every Hamming ball of radius $pn$ has at most
$O(1/\varepsilon)$ codewords.
This answers a basic open question concerning the list-decodability
of linear codes, showing that a list size of $O(1/\varepsilon)$
suffices to have rate within $\varepsilon$ of the ``capacity''
$1-H_q(p)$. This matches up to constant factors the list-size
achieved by general random codes, and gives an exponential improvement
over the best previously known list-size bound of $q^{O(1/\varepsilon)}$.
The main technical ingredient in our proof is a strong upper bound
on the probability that $\ell$ random vectors chosen from a Hamming
ball centered at the origin have too many (more than $\Theta(\ell)$)
vectors from their linear span also belong to the ball.
\end{abstract}
\newpage
\section{Introduction}
One of the central problems in coding theory is to understand the
trade-off between the redundancy built into codewords (aka the rate of
the code) and the fraction of errors the code enables correcting.
Suppose we are interested in codes over the binary alphabet (for
concreteness) that enable recovery of the correct codeword $c \in
\{0,1\}^n$ from any noisy received word $r$ that differs from $c$ in
at most $pn$ locations. For each $c$, there are about
${n \choose
{pn}} \approx 2^{H(p) n}$ such possible received words $r$, where
$H(x) = -x \log_2 x - (1-x) \log_2(1-x)$ stands for the binary entropy
function.
\vnote{I removed the footnote on approximate equality, as in the proofs we use ${n \choose {pn}} \le 2^{H(p) n}$ which is true without fudging.}
Now for each such $r$, the error-recovery procedure must
identify $c$ as a possible choice for the true codeword. In fact, even if
the errors are randomly distributed and not worst-case, the algorithm
must identify $c$ as a candidate codeword for {most} of these $2^{H(p)
n}$ received words, if we seek a low decoding error probability.
\iffalse
Therefore, if we require the error-recovery procedure to pin down the
codeword (or even narrow down the codeword to a few possibilities) for
all (or even most) received words, then the codewords must have
Hamming balls of volume $2^{H(p) n}$ with limited overlap centered
around them.
\fi
This implies that there can be at most $\approx
2^{(1-H(p))n}$ codewords, or equivalently the largest rate $R$ of the
code one can hope for is $1-H(p)$.
If we could pack about $2^{(1-H(p))n}$ pairwise disjoint Hamming balls
of radius $p n$ in $\{0,1\}^n$, then one can achieve a rate
approaching $1-H(p)$ while guaranteeing correct and unambiguous
recovery of the codeword from an arbitrary fraction $p$ of
errors. Unfortunately, it is well known that such an asymptotic
``perfect packing'' of Hamming balls in $\{0,1\}^n$ does not exist,
and the largest size of such a packing is at most $2^{(\alpha(p)+o(1))
n}$ for $\alpha(p) < 1-H(p)$ (in fact $\alpha(p)=0$ for $p \ge 1/4$).
Nevertheless, it turns out that it is possible to pack
$2^{(1-H(p)-\varepsilon)n }$ such Hamming balls such that no $O(1/\varepsilon)$ of
them intersect at a point, for any $\varepsilon > 0$. In fact a random
packing has such a property with high probability.
\noindent {\bf List Decoding.}
This fact implies that it is possible to achieve rate approaching the
optimal $1-H(p)$ bound for correcting a fraction $p$ of {\em
worst-case} errors in a model called {\em list decoding}. List
decoding, which was introduced independently by Elias and Wonzencraft
in the 1950s~\cite{elias,wozencraft}, is an error-recovery model where
the decoder is allowed to output a small list of candidate codewords
that must include all codewords within Hamming distance $pn$ of the
received word. Note that if at most $pn$ errors occur, the list
decoder's output will include the correct codeword. In addition to the
rate $R$ of the code and the error fraction $p$, list decoding has an
important third parameter, the ``list-size,'' which is the largest
number $L$ of codewords the decoder is allowed to output on any
received word. The list-size thus bounds the maximum ambiguity in the
output of the decoder.
For codes over an alphabet of size $q$, all the above statements hold
with $H(p)$ replaced by $H_q(p)$, where $H_q(x) = x \log_q(q-1) - x
\log_q x - (1-x) \log_q(1-x)$ is the $q$-ary entropy function.
\begin{definition}[Combinatorial list decodability property]
Let $\Sigma$ be a finite alphabet of size $q$, $L \ge 1$ an integer, and $p \in (0,1-1/q)$.
A code $C \subseteq \Sigma^n$ is said to be $(p,L)$-list-decodable, if for every $x \in \Sigma^n$, there are at most $L$ codewords of $C$ that are at Hamming distance $pn$ or less from $x$. Formally, $|B_n^q(x,p) \cap C| \le L$ for every $x$, where $B^q_n( x, p) \subseteq \Sigma^n$ is the ball of radius $pn$ centered at $x \in \{0,1\}^n$.
\end{definition}
We restrict $p < 1-1/q$ in the above definition since a random string
differs from each codeword in at most a fraction $1-1/q$ of positions,
and so over alphabet size $q$ decoding from a fraction $1-1/q$ or more
errors is impossible (except for trivial codes).
\noindent {\bf Combinatorics of list decoding.}
A fundamental question in list decoding is to understand the trade-off
between rate, error-fraction, and list-size. For example, what list-size suffices if we want codes of rate within $\varepsilon$ of the optimal $1-H_q(p)$ bound? That is, if we define $L_{q,p}(\varepsilon)$ to be the minimum integer $L$ for which there are $q$-ary $(p,L)$-list-decodable codes of rate at least $1-H_q(p)-\varepsilon$ for infinitely many lengths $n$, how does $L_{q,p}(\varepsilon)$ behave for small $\varepsilon$ (as we keep the alphabet size $q$ and $p \in (0,1-1/q)$ fixed)?
It is known that unbounded list-size is needed as one approaches the
optimal rate of $1-H_q(p)$. In other words, $L_{q,p}(\varepsilon) \to \infty$
as $\varepsilon \to 0$. This was shown for the binary case in
\cite{blinovsky}, and his result implicitly implies $L_{2,p}(\varepsilon)\ge
\Omega(\log (1/\varepsilon))$ (see \cite{atri-cocoon} for an explicit
derivation of this). For the $q$-ary case, $L_{q,p}(\varepsilon) =
\omega_\varepsilon(1)$ was shown in \cite{blin-q-ary,blin-convexity}. In the
language of list-decoding, the above-mentioned result on
``almost-disjoint'' sphere packing states that for large enough block lengths, a random code of rate
$1-H_q(p)-\varepsilon$ is $(p,\frac{1}{\varepsilon})$-list-decodable with high
probability. In other words, $L_{q,p}(\varepsilon) \le 1/\varepsilon$. This
result appears in \cite{elias91} (and is based on a previous random
coding argument for linear codes from \cite{ZP81}). The result is
explicitly stated in \cite{elias91} only for $q=2$, but trivially
extends for arbitrary alphabet size $q$. This result is also tight,
in the sense that with high probability a {\em random} code of rate
$1-H_q(p)-\varepsilon$ is {\em not} $(p,c_{p,q}/\varepsilon)$-list-decodable
w.h.p. for some constant $c_{p,q}>0$~\cite{atri-cocoon}.
An interesting question is to close the exponential gap in the lower and upper bounds on $L_{2,p}(\varepsilon)$, and more generally pin down the asymptotic behavior of $L_{q,p}(\varepsilon)$ for every $q$. The upper bound of $O(1/\varepsilon)$ is perhaps closer to the truth, and it is probably the lower bound that needs strengthening.
\noindent {\bf Context of this work.}
In this work, we address another fundamental combinatorial question
concerning list-decodable codes, namely the behavior of $L_{q,p}(\varepsilon)$
when restricted to {\em linear codes}. For $q$ a prime power, a
$q$-ary linear code is simply a subspace of ${\mathbb F}_q^n$ (${\mathbb F}_q$ being the
field of size $q$).
Most of the well-studied and practically used codes are linear codes.
Linear codes admit a succinct representation in terms of its basis
(called generator matrix). This aids in finding and representing such
codes efficiently, and as a result linear codes are often useful as
``inner'' codes in concatenated code constructions.
In a linear code,
by translation invariance, the neighborhood of every codeword looks
the same, and this is often a very useful symmetry property. For
instance, this property was recently used in \cite{GS-additive} to
give a black-box conversion of linear list-decodable codes to codes
achieving capacity against a worst-case additive channel (the
linearity of the list-decodable code is crucial for this connection).
Lastly, list-decodability of linear codes brings to the fore some
intriguing questions on the interplay between the geometry
of linear spaces and Hamming balls, and is therefore interesting in
its own right. For these and several other reasons, it is desirable
to achieve good trade-offs for list decoding via linear codes.
Since linear codes are a highly structured subclass of all codes,
proving the existence of linear codes with list-decodability
properties similar to general codes can be viewed as a strong
``derandomization'' of the random coding argument used to construct
good list-decodable codes. A derandomized family of codes called
``pseudolinear codes'' were put forth in \cite{GI-focs01} since linear
codes were not known to have strong enough list-decoding properties.
Indeed, prior to this work, the results known for linear codes were
substantially weaker than for general codes (we discuss the details
next). {\em Closing this gap is the main motivation behind this
work.}
\noindent {\bf Status of list-decodability of linear codes.}
Zyablov and Pinsker proved that a random binary linear code of rate
$1-H(p)-\varepsilon$ is $(p,2^{O(1/\varepsilon)})$-list-decodable with high
probability~\cite{ZP81}. The proof extends in a straightforward way to
linear codes over ${\mathbb F}_q$, giving list-size $q^{O(1/\varepsilon)}$ for rate
$1-H_q(p)-\varepsilon$. Let us define $L^{\mathsf{lin}}_{q,p}(\varepsilon)$ to be the
minimum integer $L$ for which there is an infinite family of
$(p,L)$-list-decodable linear codes over ${\mathbb F}_q$ of rate at least
$1-H_q(p)-\varepsilon$. The results of \cite{ZP81} thus imply that
$L^{\mathsf{lin}}_{q,p}(\varepsilon) \le \exp(O_q(1/\varepsilon))$.
Note that this bound is {\em exponentially worse} than the $O(1/\varepsilon)$
bound known for general codes. In \cite{elias91}, Elias mentions the
following as the most obvious problem left open left by the random
coding results: {\em Is the requirement of the much larger list size for
linear codes inherent, or can one achieve list-size closer to the
$O(1/\varepsilon)$ bound for general random codes?}
For the {\em binary} case, the {\em
existence} of $(p,L)$-list-decodable linear codes of rate at least
$1-H(p)-1/L$ is proven in \cite{GHSZ}. This implies that
$L_{2,p}^{\mathsf{lin}} \le 1/\varepsilon$. There are some results which obtain lower
bounds on the rate for the case of small fixed list-size (at most
$3$)~\cite{blinovsky,blin-book,wei-feng}; these bounds are complicated
and not easily stated, and as noted in \cite{blin2}, are weaker for the
linear case for list-size as small as $5$.
The proof in \cite{GHSZ} is based on a carefully designed potential
function that quantifies list-decodability, and uses the
``semi-random'' method to successively pick good basis vectors for the
code. The proof only guarantees that such binary linear codes exist
with positive probability, and does not yield a high probability
guarantee for the claimed list-decodability property.
Further, the proof relies crucially on the binary alphabet and
extending it to work for larger alphabets (or even the ternary case)
has resisted all attempts. Thus, for $q > 2$, $L_{q,p}(\varepsilon) \le
\exp(O_q(1/\varepsilon))$ remained the best known upper bound on list-size.
A high probability result for the binary case, and an upper bound of
$L_{q,p}(\varepsilon) \le O(1/\varepsilon)$ for ${\mathbb F}_q$-linear codes, were
conjectured in \cite[Chap. 5]{G-thesis}.
\noindent {\bf Our contribution.} In this work, we resolve the above
open question concerning list-decodability of linear codes over {\em
all} alphabets. In particular, we prove that $L^{\mathsf{lin}}_{q,p}(\varepsilon)
\le C_{q,p}/\varepsilon$ for a constant $C_{q,p} < \infty$. Up to constant factors,
this matches the best known result for general, non-linear
codes. Further, our result in fact shows that a random ${\mathbb F}_q$-linear code of
rate $1-H_q(p)-\varepsilon$ is $(p,C_{p,q}/\varepsilon))$-list-decodable {\em with
high probability}. This was not known
even for the case $q=2$. The high probability claim implies an
efficient randomized Monte Carlo construction of such list-decodable
codes.
We now briefly explain the difficulty in obtaining good bounds for
list-decoding linear codes and how we circumvent it. This is just a
high level description; see the next section for a more technical
description of our proof method.
Let us recall the straightforward random coding method that shows the
list-decodability of random (binary) codes. We pick a code $C
\subseteq \{0,1\}^n$ by uniformly and independently picking $M=2^{Rn}$
codewords. To prove it is $(p,L)$-list-decodable, we fix a center $y$
and a subset $S$ of $(L+1)$ codewords of $C$. Since these codewords are
independent, the probability that all of them land in the ball of radius
$pn$ around $y$ is at most $\bigl(\frac{2^{H(p)
n}}{2^n}\bigr)^{L+1}$. A union bound over all $2^n$ choices of $y$
and at most $M^{L+1}$ choices of $S$ shows that if $R \le 1-H(p) - 1/L$,
the code fails to be $(p,L)$-list-decodable with probability at most
$2^{-\Omega(n)}$.
Attempting a similar argument in the case of random linear codes,
defined by a random linear map $A : {\mathbb F}_2^{Rn} \rightarrow {\mathbb F}_2^n$,
faces several immediate obstacles. The $2^{Rn}$ codewords of a random
linear code are not independent of one another; in fact the points of
such a code are highly correlated and not even $3$-wise independent
(as $A(x+y) = Ax+Ay$). However, any $(L+1)$ distinct codewords
$Ax_1,Ax_2,\dots,Ax_{L+1}$ must contain a subset of $\ell \ge
\log_2(L+1)$ independent codewords, corresponding to a subset
$\{x_{i_1}, \dots, x_{i_\ell}\}$ of {\em linearly independent}
message vectors. This lets one mimic the argument for the random code case
with $\log_2(L+1)$ playing the role of $L+1$. However, as a result, it
leads to the exponentially worse list-size bounds.
To get a better result, we somehow need to control the ``damage''
caused by subsets of codewords of low rank. This is the crux of our
new proof. Stated loosely and somewhat imprecisely, we prove a strong
upper bound on the fraction of such low rank subsets, by proving that
if we pick $\ell$ random vectors from the Hamming ball $B_n(0,p)$ (for
some constant $\ell$ related to our target list-size $L$), it is
rather unlikely that more than $\Theta(\ell)$ of the $2^\ell$ vectors
in their span will also belong to the ball $B_n(0,p)$. (See
Theorem~\ref{thm:cbound} for the precise statement.) This ``limited
correlation'' between linear subspaces and Hamming balls is the main
technical ingredient in our proof. It seems like a basic and powerful
probabilistic fact that might find other applications. The argument
also extends to linear codes over ${\mathbb F}_q$ after some adaptations.
\section{Results and Methods}
Our main result is that random linear codes in ${\mathbb F}_2^n$ of rate $1 - H(p)- \epsilon$ can be list-decoded
from $p$-fraction errors with list-size only $O(\frac{1}{\epsilon})$. We also show the analogous result
for random $q$-ary linear codes.
\begin{theorem}
\label{thm:main}
Let $p \in (0, 1/2)$. Then there exist constants $C_p, \delta > 0$, such that
for all $\epsilon > 0$ and all large enough integers $n$, letting $R = 1 - H(p) - \epsilon$, if $\mathcal C \subseteq {\mathbb F}_2^n$
is a random linear code of rate $R$, then
$$\Pr[\mathcal C \mbox{ is $(p, \frac{C_p}{\epsilon})$-list-decodable}] > 1 - 2^{-\delta n}.$$
\end{theorem}
The proof begins by simplifying the problem to its combinatorial
core. Specifically, we reduce the problem of studying the {\em
list-decodability} of a random linear code of {\em linear} dimension
to the problem of studying the {\em weight-distribution} of certain
random linear codes of {\em constant} dimension. The next theorem
analyzes the weight distribution of these constant-dimensional random
linear codes. The notation $B_n(x,p)$ refers to the Hamming ball of radius $pn$ centered at $x \in {\mathbb F}_2^n$.
\begin{theorem}[Span of random points in $B_n(0,p)$]
\label{thm:cbound}
For every $p \in ( 0, 1/2)$, there is a constant $C > 0$, such that
for all $n$ large enough and all $\ell = o(\sqrt{n})$, if $X_1,
\ldots, X_\ell$ are picked independently and uniformly at random from
$B_n(0, p)$, then
$$ \Pr [ | \mathrm{span}(\{X_1, \ldots, X_\ell\}) \cap B_n(0, p) | > C \cdot\ell] \leq 2^{-5n}.$$
\end{theorem}
We now give a brief sketch of the proof of Theorem~\ref{thm:cbound}.
Index the elements of $\mathrm{span}(\{X_1, \ldots, X_\ell\})$ as
follows: for $v \in {\mathbb F}_2^{\ell}$, let $X_v$ denote the random vector
$\sum_{i=1}^{\ell} v_i X_i$. Fix an arbitrary $S \subseteq {\mathbb F}_2^\ell$
of cardinality $C \cdot\ell$, and let us study the event $E_S$: that all
the vectors $(X_v)_{v \in S}$ lie in $B_n(0,p)$. If none of the events
$E_S$ occur, we know that $| \mathrm{span}(\{X_1, \ldots, X_\ell\})
\cap B_n(0, p) | \leq C\cdot\ell.$
The key technical step is a Ramsey-theoretic lemma (Lemma~\ref{lem:cinc}, stated below) which says
that large sets $S$ automatically have the property that some
translate of $S$ contains a certain structured subset
(which we call an ``increasing chain'').
This structured subset allows us to give strong upper bounds on the probability that all the vectors
$(X_v)_{v \in S}$ lie in $B_n(0,p)$.
Applying this to each $S \subseteq {\mathbb F}_2^\ell$ of cardinality $C \ell$ and taking a union bound gives
Theorem~\ref{thm:cbound}.
To state the Ramsey-theoretic lemma (Lemma~\ref{lem:cinc}), we first define increasing chains. For a vector $v \in {\mathbb F}_2^{\ell}$, the {\em support} of $v$, denoted ${\mathrm {supp}}(v)$, is defined to be the set of its nonzero coordinates.
\begin{definition}
A sequence of vectors $v_1, \ldots, v_d \in {\mathbb F}_2^{\ell}$ is called an $c$-increasing chain of length $d$,
if for all $j \in [d]$,
$$\left|{\mathrm {supp}}(v_j) \setminus \left(\bigcup_{i = 1}^{j-1}{\mathrm {supp}}(v_i)\right)\right| \geq c.$$
\end{definition}
We now state the Ramsey-theoretic lemma that plays the central role in Theorem~\ref{thm:cbound}.
The proof appears in Section~\ref{sec:cinc}, where it is proved using the Sauer-Shelah lemma.
\begin{lemma}
\label{lem:cinc}
For all positive integers $c,\ell$ and $L \le 2^\ell$, the following holds.
For every $S \subseteq
{\mathbb F}_2^\ell$ with $|S| = L$, there is a $w \in {\mathbb F}_2^\ell$ such that $S+ w$
has an $c$-increasing chain of length at least $\frac{1}{c}(\log
\frac{L}{2}) - (1 - \frac{1}{c})(\log \ell) $.
\end{lemma}
\subsection{Larger alphabet}
Due to their geometric nature, our arguments generalize to the case of $q$-ary alphabet
(for arbitrary constant $q$) quite easily. Below we state our main theorem for the case of $q$-ary alphabet.
\begin{theorem}
\label{thm:qmain}
Let $q$ be a prime power and let $p \in (0, 1-1/q)$. Then there exist constants $C_{p,q}, \delta > 0$, such that
for all $\epsilon > 0$, letting $R = 1 - H_q(p) - \epsilon$, if $\mathcal C \subseteq {\mathbb F}_q^n$
is a random linear code of rate $R$, then
$$\Pr[\mathcal C \mbox{ is $(p, \frac{C_{p,q}}{\epsilon})$-list-decodable}] > 1 - 2^{-\delta n}.$$
\end{theorem}
The proof of Theorem~\ref{thm:qmain} has the same basic outline as the
proof of Theorem~\ref{thm:main}. In particular, it proceeds via a
$q$-ary analog of Theorem~\ref{thm:cbound}. The only notable
deviation occurs in the proof of the $q$-ary analog of
Lemma~\ref{lem:cinc}. The traditional generalization of the
Sauer-Shelah lemma to larger alphabets turns out to be unsuitable for
this purpose. Instead, we formulate and prove a non-standard
generalization of the Sauer-Shelah lemma for the larger alphabet case
which is more appropriate for this situation. Details appear in
Section~\ref{sec:qary}.
\section{Proof of Theorem~\ref{thm:main}}
\label{sec:ctogeneral}
Let us start by restating our main theorem.
\noindent {\bf Theorem~\ref{thm:main} (restated)}\ \
{\it
Let $p \in (0, 1/2)$. Then there exist constants $C_p, \delta > 0$, such that
for all $\epsilon > 0$ and all large enough integers $n$, letting $R = 1 - H(p) - \epsilon$, if $\mathcal C \subseteq {\mathbb F}_2^n$
is a random linear code of rate $R$, then
$$\Pr[\mathcal C \mbox{ is $(p, \frac{C_p}{\epsilon})$-list-decodable}] > 1 - 2^{-\delta n}.$$
}
\begin{proof}
Pick $C_p = 4 C$, where $C$ is the constant from Theorem~\ref{thm:cbound}. Pick $\delta = 1$. Take $L = \frac{C_p}{\epsilon}$.
Let $\mathcal C$ be a random $Rn$ dimensional linear subspace of ${\mathbb F}_2^n$.
We want to show that
\begin{equation}
\label{eq:bound1}
\Pr_{\mathcal C} [\exists x \in {\mathbb F}_2^n \mbox{ s.t. } |B_n(x, p) \cap \mathcal C | > L] < 2^{-\delta n}.
\end{equation}
Let $x \in {\mathbb F}_2^n$ be picked uniformly at random. We will work towards Equation~\eqref{eq:bound1} by
studying the following quantity.
$$ \Delta {\stackrel{\rm def}{=}} \Pr_{\mathcal C, x}[ |B_n(x, p) \cap \mathcal C | > L ].$$
Note that to prove Equation~\eqref{eq:bound1}, it suffices to show
that\footnote{We could even replace the $2^{-n}$ by
$2^{-(1-R)n}$. Indeed, for every $\mathcal C$ for which there is a
``bad'' $x$, we know that there are $2^{Rn}$ ``bad'' $x$'s (the
translates of $x$ by $\mathcal C$).}
$$ \Delta < 2^{-\delta n} \cdot 2^{-n}.$$
\vnote{We can also replace $2^{-n}$ by $2^{(H(p)-1)n}$. This would better than $2^{(R-1)n}$ for low rates or $p \to 1/2$. A commented footnote is in the latex file.}
Now for each $\ell \in [\log(L+1), L+1]$, let
$\mathcal F_\ell$ be the set of all $(v_1, \ldots, v_{\ell}) \in B_n(0,p)^\ell$ such
that $v_1, \ldots, v_\ell$ are linearly independent and
$|\mathrm{span}(v_1, \ldots, v_{\ell}) \cap B_n(0,p)^{\ell}| > L$.
Let $\mathcal F = \bigcup_{\ell = \log (L+1)}^{L+1} \mathcal F_\ell$
For each $\mathbf v = (v_1, \ldots, v_\ell) \in \mathcal F$, let $\{ \mathbf v \}$
denote the set $\{ v_1, \ldots, v_{\ell}\}$.
We now bound $\Delta$. Notice that if $|B_n(x,p) \cap \mathcal C| >
L$, then there must be some $\mathbf v \in \mathcal F$ for which
$B_n(x,p) \cap \mathcal C \supseteq x + \{ \mathbf v \}$. Indeed, we
can simply take $\mathbf v$ to be a maximal linearly independent
subset of $(B_n(x,p) \cap \mathcal C) + x$ if this set has size at
most $L+1$, and any linearly independent subset of $(B_n(x,p) \cap
\mathcal C) + x$ of size $L+1$ otherwise.
Therefore, by the union bound,
\begin{align}
\Delta &\leq \sum_{\mathbf v \in \mathcal F} \Pr_{\mathcal C, x} [ B_n(x,p) \cap \mathcal C \supseteq x + \{ \mathbf v \}]\\
&= \sum_{\mathbf v \in \mathcal F} \Pr_{\mathcal C, x} [B_n(0, p) \cap (\mathcal C + x) \supseteq \{ \mathbf v \}]\\
&\leq \sum_{\mathbf v \in \mathcal F} \Pr_{\mathcal C, x} [B_n(0, p) \cap (\mathcal C + \{0, x\} ) \supseteq \{ \mathbf v \}]\\
&= \sum_{\mathbf v \in \mathcal F} \Pr_{\mathcal C^*} [B_n(0,p) \cap \mathcal C^* \supseteq \{ \mathbf v \}],
\label{eq:b2}
\end{align}
where $\mathcal C^*$ is the code $\mathcal C + \{0, x\}$ which is a
random $Rn + 1$ dimensional subspace.
The last probability can be bounded as follows. By the linear
independence of $v_1, \ldots, v_\ell$, the probability that $v_j \in
\mathcal C^*$ conditioned on $\{v_1, \ldots, v_{j-1}\} \subseteq
\mathcal C^*$ is precisely the probability that a given point in a
$n+1-j$ dimensional space lies in a $Rn+1-j$ dimensional subspace, and
hence this conditional probability is exactly $2^{Rn+1-n}$. We can
hence conclude that
\begin{equation}
\label{eq:b3}
\Pr_{\mathcal C^*}[ \mathcal C^*\supseteq \{ \mathbf v \} ] = \left(\frac{ 2^{Rn + 1} }{2^n} \right)^\ell.
\end{equation}
Putting together Equations~\eqref{eq:b2} and \eqref{eq:b3}, we have
\begin{align*}
\Delta &\leq \sum_{\mathbf v \in \mathcal F} \Pr_{\mathcal C^*} [B_n(0,p) \cap \mathcal C^* \supseteq \{ \mathbf v \}]
\leq \sum_{\ell = \log (L+1)}^{L+1} \sum_{\mathbf v \in \mathcal F_\ell} \Pr_{\mathcal C^*} [\mathcal C^* \supseteq \{ \mathbf v \}]\\
&\leq \sum_{\ell = \log (L+1)}^{L+1} \sum_{\mathbf v \in \mathcal F_\ell} \left(\frac{ 2^{Rn + 1} }{2^n} \right)^\ell
\leq \sum_{\ell = \log (L+1)}^{L+1} |\mathcal F_{\ell}| \cdot \left(\frac{ 2^{Rn + 1} }{2^n} \right)^\ell \\
\end{align*}
We now obtain an upper bound on $|\mathcal F_\ell|$.
We have two cases depending on the size of $\ell$.
\begin{itemize}
\item {\bf Case 1:} $\ell < 4/\epsilon$. In this
case, we notice that $\frac{|\mathcal F_{\ell}|}{|B_n(0,p)|^\ell}$ is a lower bound on the probability that $\ell$ points $X_1, \ldots, X_\ell$ chosen uniformly at random from $B_n(0,p)$
have $|\mathrm{span}(\{X_1, \ldots, X_\ell\}) \cap B_n(0,p)| > L.$
Since $L > C \cdot \ell$, Theorem~\ref{thm:cbound} tells us
that this probability is bounded from above by $2^{-5n}$. Thus, in this case $|\mathcal F_{\ell}| \leq |B_n(0,p)|^{\ell} 2^{-5n} \leq 2^{n \ell H(p)} \cdot 2^{-5n}.$
\item {\bf Case 2:} $\ell \geq 4/\epsilon$. In this case, we have the trivial bound of $|\mathcal F_\ell| \leq |B_n(0,p)|^{\ell} \leq 2^{n\ell H(p)}$.
\end{itemize}
\noindent
Thus, we may bound $\Delta$ by:
\begin{align*}
\Delta & \leq \sum_{\ell=\log L}^{\lfloor 4/\epsilon\rfloor} |\mathcal F_{\ell}| \cdot \left(\frac{ 2^{Rn+1} }{2^n} \right)^\ell + \sum_{\ell=\lceil 4/\epsilon\rceil}^{L} |\mathcal F_{\ell}| \cdot \left(\frac{ 2^{Rn+1} }{2^n} \right)^\ell\\
&\leq \sum_{\ell = \log L}^{\lfloor 4/\epsilon\rfloor} 2^{n\ell H(p)} 2^{-5n} \left(\frac{ 2^{Rn+1} }{2^n} \right)^\ell + \sum_{\ell=\lceil 4/\epsilon\rceil}^{L} 2^{n\ell H(p)} \left(\frac{ 2^{Rn+1} }{2^n} \right)^{\ell}\\
& \leq 2^{-5n} \cdot 4/ \epsilon + L \cdot 2^{-(\epsilon n)\cdot (4/\epsilon) }\\
& \leq 2^{-\delta n} \cdot 2^{-n}
\end{align*}
as desired.
\end{proof}
\section{Proof of Theorem~\ref{thm:cbound}}
\label{sec:cbound}
In this section, we prove Theorem~\ref{thm:cbound} which bounds the probability that the span of $\ell$ random points
in $B_n(0,p)$ intersects $B_n(0,p)$ in more than $C \cdot \ell$ points, for some large constant $C$.
We use the following simple fact.
\begin{lemma}
\label{lem:deltap}
For every $p \in (0, 1/2)$, there is a $\delta_p > 0$ such that for
all large enough integers $n$ and every $x \in {\mathbb F}_2^n$, the
probability that two uniform independent samples $w_1, w_2$ from
$B_n(0,p)$ are such that $w_1 + w_2 \in B_n(x,p)$ is at most
$2^{-\delta_p n}$.
\end{lemma}
\noindent
{\bf Sketch of proof.} The point $w_1+w_2$ is essentially
a random point in $B_n(0,2p-2p^2)$. The probability that it lies
in the smaller ball $B_n(x,p)$ is easily seen to be maximal
when $x=0$ and is then exponentially small.\hfill\rule{7pt}{7pt}
\noindent
{\bf Theorem~\ref{thm:cbound} (restated)}\ \
{\it
For every $p \in ( 0, 1/2)$, there is a constant $C > 0$, such that for all $n$ large enough and all
$\ell = o(\sqrt{n})$, if
$X_1, \ldots, X_\ell$ are picked independently and
uniformly at random from $B_n(0, p)$,
then
$$ \Pr [ | \mathrm{span}(\{X_1, \ldots, X_\ell\}) \cap B_n(0, p) | > C\cdot\ell] \leq 2^{-5n}.$$
}
\begin{proof}
Set $L = C \cdot \ell$ and let $c = 2$. Let $\delta_p > 0$ be the
constant given by Lemma~\ref{lem:deltap}. Let
\[ d = \bigg\lfloor \frac{1}{c}\log \frac{L}{2} - \Bigl(1 - \frac{1}{c}\Bigr)\log\ell \bigg\rfloor \ge \frac{1}{2} \log \frac{L}{2\ell} - 1 = \frac{1}{2}
\log \frac{C}{8} \ . \]
\noindent For a vector $u \in {\mathbb F}_2^{\ell}$, let $X_u$
denote the random variable $\sum_{i} u_i X_i$.
We begin with a claim which bounds the probability of a particular
collection of linear combinations of the $X_i$ all lying within
$B_n(0,p)$. At the heart of this claim lies the Ramsey-theoretic
Lemma~\ref{lem:cinc}.
\begin{claim}
\label{claim:perS}
For each $S \subseteq {\mathbb F}_2^\ell$ with $|S| =L+1$,
\begin{equation}
\label{eq:perS}
\Pr[ \forall v \in S, X_v \in B_n(0,p)] < 2^n \cdot 2^{-\delta_p dn}.
\end{equation}
\end{claim}
\begin{proof}
Let $w$ and $v_1, \ldots, v_d \in S$ be as given by
Lemma~\ref{lem:cinc}. That is, $v_1+w, v_2+w, \cdots, v_d+w$ is an
$c$-increasing sequence.
Then,
\begin{align}
\Pr [ \forall v \in S, X_v \in B_n(0,p)] &\leq \Pr [ \forall j \in [d], X_{v_j} \in B_n(0,p)] \\
&= \Pr [ \forall j \in [d], X_{v_j} + X_w \in B_n(X_w,p)]\\
&= \Pr [ \forall j \in [d], X_{v_j + w} \in B_n(X_w,p)]
\label{eq:bnxw}
\end{align}
We now bound the probability that there exists $y \in {\mathbb F}_2^n$
such that for all $j \in [d]$, $X_{v_j + w} \in B_n(y, p)$.
Fix $y \in {\mathbb F}_2^n$. We have:
\begin{align}
\Pr [ \forall j \in [d], X_{v_j +w} \in B_n(y,p)] &\leq \prod_{j = 1}^d \Pr\biggl[ X_{v_j + w} \in B_n(y,p) \mid ( X_t : t \in \Bigl(\bigcup_{i = 1}^{j-1}{\mathrm {supp}}(v_i + w)\Bigr)\biggr]\\
&\leq \bigl( 2^{-\delta_p n}\bigr)^d.
\label{eq:bnxw1}
\end{align}
The last inequality follows from applying Lemma~\ref{lem:deltap} with
$w_1$ and $w_2$ being vectors $X_{i_1}$ and $X_{i_2}$, where $i_1, i_2$ are
two distinct elements of ${\mathrm {supp}}(v_j + w) \setminus \left(\bigcup_{i = 1}^{j-1}{\mathrm {supp}}(v_i +
w)\right)$, and $x = y + \sum_{k \in [\ell], k\not\in \{i_1, i_2\}}
(v_j+w)_k X_k$.
Taking a union bound of Equation~\eqref{eq:bnxw1} over all $y \in {\mathbb F}_2^n$, we see that
$$ \Pr[ \exists y \in F_2^n \mbox{ s.t. } \forall j \in [d], X_{v_j + w} \in B_n(y,p)] \leq 2^n \cdot 2^{-\delta_p n d}.$$
Combining this with Equation~\eqref{eq:bnxw} completes the proof of the claim.
\end{proof}
Given this claim, we now bound the probability that more than $L$
elements of ${\mathrm {span}}(\{X_1, \ldots, X_\ell\})$ lie inside
$B_n(0,p)$. This event occurs if and only if for some set $S
\subseteq {\mathbb F}_2^\ell$ with $|S| = L+1$, it is the case that $\forall v
\in S$, $X_v \in B_n(0,p)$. Taking a union bound of~\eqref{eq:perS}
over all such $S$, we see that the probability that there exists some
$S \subseteq {\mathbb F}_2^\ell$ with $|S| = L+1$ such that $\forall v \in S,
X_v \in B_n(0,p)$ is at most $2^{\ell (L+1)} \cdot 2^{n} \cdot
2^{-\delta_p d n}$. Taking $C$ to be a large enough constant so that
$d \ge \frac{1}{2} \log\frac{C}{8} > \frac{12}{\delta_p}$, the theorem
follows.
\end{proof}
\section{Proof of Lemma~\ref{lem:cinc}}
\label{sec:cinc}
In this section, we will prove Lemma~\ref{lem:cinc}, which finds a
large $c$-increasing chain in some translate of any large enough set
$S \subseteq {\mathbb F}_2^\ell$.
We will use the Sauer-Shelah Lemma.
\begin{lemma}[Sauer-Shelah~\cite{sauer, shelah}]
For all integers $\ell, c$, and for any set $S \subseteq \{0,1\}^\ell$, if $|S| > 2 \ell^{c-1}$, then
there exists some set of coordinates $U \subseteq [\ell]$ with $|U| = c$ such that
$\{ v|_U \mid v \in S \} = \{0, 1\}^U$.
\end{lemma}
\noindent
{\bf Lemma~\ref{lem:cinc} (restated)}\ \
{\it
For all positive integers $c,\ell$ and $L \le 2^\ell$, the following holds.
For every $S \subseteq
{\mathbb F}_2^\ell$ with $|S| = L$, there is a $w \in {\mathbb F}_2^\ell$ such that $S+ w$
has an $c$-increasing chain of length at least $\frac{1}{c}(\log
\frac{L}{2}) - (1 - \frac{1}{c})(\log \ell) $.
}
\begin{proof}
We prove this by induction on $\ell$. The claim holds trivially for $\ell \le c$, so assume $\ell > c$.
If $L \leq 2 \ell^{c-1}$, then again the lemma holds trivially.
Otherwise, by the Sauer-Shelah lemma, we get a set $U$ of $c$
coordinates such that for each $u \in {\mathbb F}_2^U$, there is some $v \in S$
such that $v|_U = u$. We will represent elements of ${\mathbb F}_2^{\ell}$ in
the form $(u, v')$ where $u \in {\mathbb F}_2^U$ and $v' \in
{\mathbb F}_2^{[\ell]\setminus U}$.
Let $u_0 \in {\mathbb F}_2^U$ be a vector such that $|\{ v \in S \mid v|_{U} =
u_0\}|$ is at least $L / 2^c$ (we know that such a $u$ exists by
averaging). Let $S' \subseteq {\mathbb F}_2^{[\ell]\setminus U}$ be given by
$S' = \{ v|_{[\ell]\setminus U} \mid v|_{U} = u\}$. By choice of $u$,
we have $|S'| \geq L/2^c$.
By the induction hypothesis, there exist $w' \in {\mathbb F}_2^{\ell-c}$ and $v'_1,
\ldots, v'_{d'} \in S'$ such that for each $j \in [d']$,
$$\Bigl|{\mathrm {supp}}(v'_j + w') \setminus \Bigl(\bigcup_{i = 1}^{j-1}{\mathrm {supp}}(v'_i + w')\Bigr)\Bigr| \geq c.$$
for $d' \ge \frac{1}{c} \log (L/2^{c+1}) - ( 1- \frac{1}{c}) \log (\ell - c)$.
Let $d=d'+1$. Note that $d
\ge \frac{1}{c} \log (L/2) - ( 1- \frac{1}{c}) \log (\ell - c) \ge
\frac{1}{c} \log (L/2) - ( 1- \frac{1}{c}) \log \ell$. For $i \in
[d']$, let $v_i = (u_0, v'_i) \in {\mathbb F}_2^{\ell}$. Let $v_d$ be any
vector in $S$ with $(v_d)|_U = \neg u_0$, the bitwise complement of
$u_0$. Let $w = (u_0, w')$. We claim that $w$ and $v_1, \ldots, v_d$
satisfy the desired properties.
\noindent
Indeed, for each $j \in [d']$, we have
\begin{align*}
\biggl|{\mathrm {supp}}(v_j + w) \setminus \Bigl(\bigcup_{i = 1}^{j-1}{\mathrm {supp}}(v_i + w)\Bigr)\biggr| &=
\biggl|{\mathrm {supp}}(v'_j + w') \setminus \Bigl(\bigcup_{i = 1}^{j-1}{\mathrm {supp}}(v'_i + w')\Bigr)\biggr| \geq c.
\end{align*}
\noindent
Also
$$\left|{\mathrm {supp}}(v_d + w) \setminus \left(\bigcup_{i = 1}^{d-1}{\mathrm {supp}}(v_i + w)\right)\right| \geq \left| {\mathrm {supp}}(v_d + w) \setminus ([\ell] \setminus U )\right| = |U| = c.$$
\noindent
Thus for all $j \in [d]$, we have $\Bigl|{\mathrm {supp}}(v_j + w) \setminus \bigl(\bigcup_{i = 1}^{j-1}{\mathrm {supp}}(v_i + w)\bigr)\Bigr| \geq c$,
as desired.
\end{proof}
\section{Larger alphabets}
\label{sec:qary}
As mentioned in the introduction the case of $q$-ary alphabet is nearly identical to the
case of binary alphabet. We only highlight the differences.
As before, the crux turns out to be the problem of studying the weight distribution of certain random constant-dimensional codes.
\begin{theorem}[$q$-ary span of random points in $B^q_n(0,p)$]
\label{thm:qcbound}
For every prime-power $q$ and every $p \in ( 0, 1-1/q)$, there is a constant $C_q > 0$, such that for all $n$ large enough and all
$\ell = o(\sqrt{n})$, if
$X_1, \ldots, X_\ell$ are picked independently and
uniformly at random from $B^q_n(0, p)$,
then
$$ \Pr [ | \mathrm{span}(\{X_1, \ldots, X_\ell\}) \cap B^q_n(0, p) | > C_q \cdot\ell] \leq q^{-5n}.$$
\end{theorem}
The proof of Theorem~\ref{thm:qcbound} proceeds as before, by bounding the probability via a large $c$-increasing
chain. The $c$-increasing chain itself is found in an analog of Lemma~\ref{lem:cinc} for $q$-ary alphabet. We first need a definition.
\begin{definition}
A sequence of vectors $v_1, \ldots, v_d \in [q]^{\ell}$ is called an $c$-increasing chain of length $d$,
if for all $j \in [d]$,
$$\left|{\mathrm {supp}}(v_j) \setminus \left(\bigcup_{i = 1}^{j-1}{\mathrm {supp}}(v_i)\right)\right| \geq c.$$
\end{definition}
Now we have the following lemma.
\begin{lemma}[$q$-ary increasing chains Ramsey]
\label{lem:qcinc}
For every prime power $q$, and all positive integers $c,\ell$ and $L \le q^\ell$, the following holds. For every $S \subseteq {\mathbb F}_q^\ell$ with $|S| = L$, there is a
$w \in {\mathbb F}_q^\ell$ such that
$S+ w$ has an $c$-increasing chain of length at least $\frac{1}{c}\log_q \bigl(\frac{L}{2}\bigr) - (1 - \frac{1}{c})\log_q ((q-1)\ell)$.
\end{lemma}
The proof of Lemma~\ref{lem:qcinc} needs a non-standard generalization of the Sauer-Shelah lemma to larger alphabet described in the next section.
\subsection{A $q$-ary Sauer-Shelah lemma}
The traditional generalization of the Sauer-Shelah lemma to large
alphabets is the Karpovsky-Milman lemma~\cite{KMqary}, which roughly states that
given $S \subseteq [q]^\ell$ of cardinality at least $(q-1)^l
l^{c-1}$, there is a set $U$ of $c$ coordinates such that for every $u
\in [q]^U$, there is some $v \in S$ such that the restriction $v|_U$
equals $u$. Applying this lemma in our context, once $q > 2$,
requires us to have a set $S > 2^\ell$, which turns out to lead to
exponential list size bounds. Fortunately, the actual property needed
for us is slightly different. We want a bound $B$ (ideally polynomial
in $\ell$) such that for any set $S \subseteq [q]^\ell$ of cardinality
at least $B$, there is a set $U$ of $c$ coordinates such that for
every $u \in [q]^U$, there is some $v \in S$ such that the restriction
$v|_U$ {\em differs from $u$ in every coordinate of $U$}. It turns
out that this weakened requirement admits polynomial-sized $B$.
We state and prove this generalization of the Sauer-Shelah lemma below.
\begin{lemma}[$q$-ary Sauer-Shelah]
\label{lem:qsauer}
For all integers $q, \ell, c$, for any set $S \subseteq [q]^\ell$, if $|S| > 2 \cdot ((q-1) \cdot \ell)^{c-1}$, then
there exists some set of coordinates $U \subseteq [\ell]$ with $|U| = c$ such that for every $u \in [q]^U$,
there exists some $v \in S$ such that $u$ and $v|_{U}$ differ in every coordinate.
\end{lemma}
\begin{proof}
We prove this by induction on $\ell$ and $c$.
If $c= 1$, then $|S| > 2$ and the result holds by letting $U$ equal any coordinate on which
not all elements of $S$ agree.
Now assume $c > 1$.
Represent an element $x$ of $[q]^{\ell}$ as a pair $(y, b)$, where $y \in [q]^{\ell-1}$ consists of the first
$\ell-1$ coordinates of $x$ and $b \in [q]$ is the last coordinate of $x$.
Consider the following subsets of $[q]^{\ell-1}$.
$$S_1 = \{ y \in [q]^{\ell-1} \mid \mbox{ for at least 1 value of $b \in [q]$, $(y, b) \in S$}\}.$$
$$S_2 = \{ y \in [q]^{\ell-1} \mid \mbox{ for at least 2 values of $b \in [q]$, $(y, b) \in S$}\}.$$
\noindent
Note that $|S| \leq (|S_1|-|S_2|) + q |S_2| = |S_1| + (q-1) |S_2|$.
By assumption,
$$|S| > 2 \cdot ((q-1) \cdot \ell)^{c-1} \geq 2 \cdot ((q-1) \cdot (\ell-1))^{c-1} + (q-1) \left( 2 \cdot ((q-1) \cdot (\ell-1))^{c-2}\right),$$
(using the elementary inequality $\ell^{c-1} \geq (\ell-1)^{c-1} +
(\ell-1)^{c-2}$). Thus, either $|S_1| > 2\cdot ((q-1) \cdot
(\ell-1))^{c-1}$, or else $|S_2| > 2 \cdot ((q-1) \cdot
(\ell-1))^{c-2}$.
\noindent
We now prove the desired claim in each of these cases.
\smallskip
\noindent {\bf Case 1:} $|S_1| > 2\cdot ((q-1) \cdot (\ell-1))^{c-1}$. In this case,
we can apply the induction hypothesis to $S_1$ with parameters $\ell-1$ and $c$, and get a
subset of $U$ of $[\ell-1]$ of cardinality $c$. Then the set $U$ has the desired property.
\smallskip
\noindent {\bf Case 2:} $|S_2| > 2 \cdot ((q-1) \cdot
(\ell-1))^{c-2}$. In this case, we apply the induction hypothesis to
$S_2$ with parameters $\ell-1$ and $c-1$, and get a subset $U$ of
$[\ell -1]$ of cardinality $c-1$. Then the set $U \cup \{ \ell\}$ has
the desired property. Indeed, take any vector $u \in [q]^{U \cup
\{\ell\}}$. Let $u' = u |_U$. By the induction hypothesis, we know
that there is a $v \in S_2$ such that $v|_U$ differs from $u'$ in
every coordinate of $U$. Now we know that there are at least two $b
\in [q]$ such that $(v, b) \in S$. At least one of these $b$ will be
such that $(v, b)$ differs from $u$ in every coordinate of $U \cup
\{\ell\}$, as desired.
\end{proof}
In the next section, we use the above lemma to prove the
Ramsey-theoretic $q$-ary increasing chain claim
(Lemma~\ref{lem:qcinc}).
\section{Proof of $q$-ary increasing chain lemma}
\label{app:q-ary-proof}
In this section, we prove Lemma~\ref{lem:qcinc}, which we restate
below for convenience.
\smallskip
\noindent {\bf Lemma~\ref{lem:qcinc} (restated)}\ \ {\it For every
prime power $q$, and all positive integers $c,\ell$ and $L \le
q^\ell$, the following holds. For every $S \subseteq {\mathbb F}_q^\ell$ with
$|S| = L$, there is a $w \in {\mathbb F}_q^\ell$ such that $S+ w$ has an
$c$-increasing chain of length at least $\frac{1}{c}\log_q
\bigl(\frac{L}{2}\bigr) - (1 - \frac{1}{c})\log_q ((q-1)\ell)$. }
\begin{proof}
We prove this by induction on $\ell$. The claim holds trivially for $\ell \le c$, so assume $\ell > c$.
If $L \leq 2 ((q-1) \cdot \ell)^{c-1}$, then again the lemma holds
trivially. Otherwise, by Lemma~\ref{lem:qsauer} we get a set $U$ of
$c$ coordinates such that for each $u \in {\mathbb F}_q^U$, there is some $v
\in S$ such that $v|_U$ differs from $u$ in every coordinate. We will
represent elements of ${\mathbb F}_q^{\ell}$ in the form $(u, v')$ where $u \in
{\mathbb F}_q^U$ and $v' \in {\mathbb F}_q^{[\ell]\setminus U}$.
Let $u_0 \in {\mathbb F}_q^U$ be a vector such that $|\{ v \in S \mid v|_{U} = u_0\}|$ is
at least $L / q^c$ (we know that such a $u$ exists by averaging).
Let $S' \subseteq {\mathbb F}_q^{[\ell]\setminus U}$ be given by
$S' = \{ v|_{[\ell]\setminus U} \mid v|_{U} = u\}$.
By choice of $u$, we have $|S'| \geq L/q^c$.
\noindent By the induction hypothesis, for
\[ d' \geq \frac{1}{c} \log \Bigl(\frac{L}{2q^{c}}\Bigr) - \Bigl( 1- \frac{1}{c}\Bigr) \log ((q-1)(\ell - c)) \ , \]
there exist $w' \in {\mathbb F}_q^{\ell-c}$ and $v'_1, \ldots, v'_{d'} \in S'$ such
that for each $j \in [d']$,
$$\biggl|{\mathrm {supp}}(v'_j + w') \setminus \Bigl(\bigcup_{i = 1}^{j-1}{\mathrm {supp}}(v'_i + w')\Bigr)\biggr| \geq c.$$
Let $d=d'+1$. Note that
\[ d \ge \frac{1}{c}\log_q\Bigl(
\frac{L}{2}\Bigr) - \Bigl(1 - \frac{1}{c}\Bigr)\log_q ((q-1)\ell) \ .\]
For $i \in [d']$, let $v_i = (u_0, v'_i) \in {\mathbb F}_q^{\ell}$. Let $v_d$ be any vector in $S$ where $(v_d)|_U$ differs
from $u_0$ in every coordinate of $U$. Let $w = (-u_0, w')$.
We claim that $w$ and $v_1, \ldots, v_d$ satisfy the desired properties.
\noindent Indeed, for each $j \in [d']$, we have
\begin{align*}
\biggl|{\mathrm {supp}}(v_j + w) \setminus \Bigl(\bigcup_{i = 1}^{j-1}{\mathrm {supp}}(v_i + w)\Bigr)\biggr| &=
\biggl|{\mathrm {supp}}(v'_j + w') \setminus \Bigl(\bigcup_{i = 1}^{j-1}{\mathrm {supp}}(v'_i + w')\Bigr)\biggr| \geq c.
\end{align*}
\noindent Also,
$$\biggl|{\mathrm {supp}}(v_d + w) \setminus \Bigl(\bigcup_{i = 1}^{d-1}{\mathrm {supp}}(v_i + w)\Bigr)\biggr| \geq \Big| {\mathrm {supp}}(v_d + w) \setminus ([\ell] \setminus U )\Bigr| = |U| = c.$$
\noindent Thus for all $j \in [d]$, we have
$$\biggl|{\mathrm {supp}}(v_j + w) \setminus \Bigl(\bigcup_{i = 1}^{j-1}{\mathrm {supp}}(v_i + w)\Bigr)\biggr| \geq c,$$
as desired.
\end{proof}
Given Lemma~\ref{lem:qcinc}, the proof of
Theorem~\ref{thm:qcbound} is virtually identical to the proof of its
binary analog Theorem~\ref{thm:cbound}. Theorem~\ref{thm:qmain} can
then be proved (using Theorem~\ref{thm:qcbound}) in the same manner as
Theorem~\ref{thm:main} was proved.
\section*{Acknowledgements}
Some of this work was done when we were all participating in the Dagstuhl seminar 09441
on constraint satisfaction. We thank the
organizers of the seminar for inviting us, and Schloss Dagstuhl for the wonderful hospitality.
\bibliographystyle{alpha}
|
1001.2400
|
\section{Introduction}
\label{intro}
The main difficulty of the coronal heating and solar wind acceleration
is how to lift up nonthermal (e.g. magnetic) energy from the photosphere
to upper regions because the hotter corona cannot stably exist above the
cooler photosphere by upward thermal heat flux. In addition,
it is not well understood how to let the energy dissipate at appropriate
positions.
The origin of energy that heats up the corona and accelerates the solar winds
is in the surface convective layer. The solar atmosphere is filled with
complicated structure of magnetic field which is supposed to be amplified
by dynamo mechanism in the interior \citep[e.g.,][]{bru04}.
The turbulent motions of the surface convection drive various modes of
upward propagating waves. The turbulent motions may also trigger
magnetic reconnections which result in flares and flare-like events,
which drive various mode of waves at locations above the
photosphere \citep{str99}.
In this paper, we focus on the roles of the waves in heating and accelerating
the solar winds.
The compressive waves that are excited at the photosphere cannot travel to
sufficiently upper regions because
they easily steepen the wave fronts to form shocks after the amplification
of the amplitude in the density decreasing atmosphere. For instance,
the acoustic waves mostly dissipate before reaching the corona, and therefore
they cannot contribute to the heating of the corona and the acceleration of
the solar wind \citep{ss72,suz02}.
Fast mode waves in low $\beta$ (magnetically dominated) condition suffer
refraction so that they hardly reach the coronal height if the fast mode
speed varies in horizontal direction due to complicated magnetic structure
\citep{kc06}.
In general, the compressive waves
that are excited from the photosphere cannot give
a significant contribution to the coronal heating and solar wind acceleration.
On the other hand, Alfv\'{e}n~ waves can travel a longer distance owing
to the incompressive characters. A fraction of the Alfv\'{e}n~ waves excited
at the photospheric level is supposed to propagate
to the corona and the solar wind acceleration region so that they
play a more dominant role in the heating and acceleration of the solar wind
than the compressive waves.
Alfv\'{e}nic~ oscillations are also detected in various regions on the Sun by
the recent observations using the HINODE satellite,
{\it e.g.}, in a solar prominence
\citep{oka07}, in spicules \citep{dep07}, and in a chromospheric jet
\citep{nis08}.
To understand the dissipation mechanism of the Alfv\'{e}n~ waves is a key to
understand the heating and acceleration
of the solar wind because this corresponds to the transfer of the energy
flux of the Alfv\'{e}n~ waves to the ambient plasma.
The Alfv\'{e}n~ speed in general largely changes with height because the
density and magnetic field strength decrease in a different manner.
\citet{shi10} have estimated the distribution of Alfv\'{e}n~ speeds in a
polar region from HINODE observation and found that the Alfv\'{e}n~ speeds
actually vary in both vertical and horizontal directions.
Under these circumstances, Alfv\'{e}n~ waves suffer reflection through the
upward propagation as a result of the deformation of the wave shape
\citep{an90,moo91,hi07}.
Interactions between the outgoing Alfv\'{e}n~ waves and the incoming component
possibly lead to the damping through turbulent-like cascade \citep[e.g.,][]
{dmi03,cra07,vv07}.
This process might further generate high-frequency
ioncyclotron waves, which are widely discussed especially on the preferential
heating of the perpendicular components with respect to magnetic field of
minor heavy ions \citep[e.g.,][]{tm97,koh98,cra99}.
The variation of the Alfv\'{e}n~ speeds also anticipate phase mixing of Alfv\'{e}n~
waves along neighboring field lines \citep{hp83,sg84,vg05}, which might
further generate fast mode waves \citep{nak97}.
The density stratification and the variation of the Alfv\'{e}n~ speed also
enhances the parametric decay instability of outgoing Alfv\'{e}n~ waves
(Suzuki \& Inutsuka 2006; SI06 hereafter).
The parametric decay excites compressive waves in addition
to incoming Alfv\'{e}n~ waves.
In contrast to those excited at the photosphere, these compressive waves
are generated in the upper regions and directly contribute to the heating
of the corona and solar winds by the shock dissipation.
Although the final wave dissipation mechanism and heating process
are still not well figured out, understanding the wave reflection in
stratified atmosphere with varying Alfv\'{e}n~ speed is primarily important.
Alfv\'{e}n~ waves are reflected most effectively in the
chromosphere and transition region because of the rapid change of the
density.
While most of the Alfv\'{e}n~ waves from the surface cannot penetrate into
the corona, the remaining waves sufficiently heat and accelerate the solar
wind. Our previous work (Suzuki \& Inutsuka 2005; SI05 hereafter) showed
that the
transmitted fraction into the corona is an order of 10$\%$ and that this is
enough for the coronal heating and solar wind acceleration in open coronal
holes.
Although turbulent-like cascade of Alfv\'{e}n~(ic) waves is not treated in this
work because we adopted one dimensional (1D) magnetohydrodynamical (MHD)
simulations, this is the first attempt that dynamically treats wave reflection
from the photosphere to solar wind regions simultaneously with heating of
the gas by the wave damping. In this review paper, we firstly summarize
our dynamical simulations of solar winds, focusing on the reflection
of Alfv\'{e}n~ waves under various conditions.
Actually, the recent HINODE observation also detected a signature of
reflected Alfv\'{e}n~ waves \citep{ft09}, which will be introduced in more
detail later.
The concept of Alfv\'{e}n~ wave-driven winds is not limited
to the Sun. It is expected that the similar mechanism operates in
winds from other stars \citep[e.g.][]{vel93}, such as protostars \citep{cra08},
low- and intermediate-mass main sequence stars, red giant stars
\citep[][S07 hereafter]{suz07}, and proto-neutron stars with strong magnetic
field (Suzuki \& Nagataki 2005).
Among these objects, we show Alfv\'{e}n~ wave driven
red giant winds in this paper based on our work (S07).
In red giant stars, the stratification properties (e.g. decrease of density)
of the atmosphere are different from those in the Sun because of the smaller
surface gravity. This affects the propagation and reflection of
the Alfv\'{e}n~ waves. Here we discuss the change of the wave reflection with
stellar evolution.
In the last part of this paper, we further extend to winds from accretion
disks. To date, it has been widely discussed that accretion disk winds
can be driven by centrifugal force \citep{bp82,ks98}.
Global magnetic field plays a key role in this process; if there is poloidal
field that is sufficiently tilted with respect to an accretion disk,
the gas can flow out along with the field lines.
On the other hand, it is expected that small-scale MHD turbulence also
potentially drives disk winds, similarly to the solar winds,
although such mechanism has not been well studied so far.
It is now widely accepted that the outward transport of
disk angular momentum and inward mass accretion are realized by the
effective viscosity owing to turbulence. Magnetorotational instability (MRI)
is now known to be a very effective mechanism that drives turbulence if weak
magnetic fields exist initially (Balbus \& Hawley 1991).
The excited MHD turbulence is supposed to accelerate the disk material
to upper regions as well as transport the angular momentum outwardly.
In Suzuki \& Inutsuka (2009), we studied the disk winds driven by MHD
turbulence by local disk simulations. In this paper, we introduce the
results with emphasizing the similarity to and difference from the Alfv\'{e}n~
wave-driven solar and stellar winds.
\section{Solar Winds}
\label{sec:slw}
\subsection{Simulation Set-up}
\begin{figure}
\includegraphics[width=0.8\textwidth]{sprd1.eps}
\caption{Set-up of solar wind simulations. The cartoon shows
a schematic view of a super-radially open flux tube, which we set
in our simulations.}
\label{fig:scm}
\end{figure}
We simulate the heating and acceleration of solar winds in 1D super-radially
expansing flux tubes from the photosphere to sufficiently distant locations
(0.1 -- 0.3 AU). Here we summarize the basic points of the simulation set-up.
For the details, please see SI05 and SI06.
We determine radial magnetic field strength, $B_r$, from the conservation
of magnetic flux :
\begin{equation}
B_r f r^2= {\rm const.},
\end{equation}
where $r$ is heliocentric distance, and $f$ is a super-radial expansion
factor. We give radial variation of $f$ in advance.
The structure of $B_r(r)$ is determined by surface radial field,
$B_{r,0}$, and $f$, and is fixed during the simulations.
In our simulations (SI05 \& SI06), we consider the magnetic field of
$\sim$ few hundred Gauss for $B_{r,0}$ and the total super-radial expansion
factor of 75-450, which give the average magnetic field strength of open
field regions, $B_{r,0}/f$, of an order of 1 G.
We should remark that recent HINODE observations found that open
magnetic flux tubes in polar regions are connected to ubiquitously
distributed unipolar patches with more than kilo-Gauss \citep{tsu08b,shi10}.
These open flux tubes super-radially open with a factor of several hundred to
a thousand.
Although the obtained $B_{r,0}$ and $f$ are slightly larger than those
adopted in our simulations, the values of the average strength, $B_{r,0}/f$,
are similar. As shown in SI06 and \citet{suz06}, $B_{r,0}/f$ is a more
important parameter in determining the properties of the solar winds.
We inject transverse perturbations of magnetic field with various spectra
from the photosphere. Outgoing Alfv\'{e}n~ waves are excited by these
fluctuations. The propagation and dissipation of the waves are dynamically
treated by the following MHD equations :
\begin{equation}
\label{eq:mass}
\frac{d\rho}{dt} + \frac{\rho}{r^2 f}\frac{\partial}{\partial r}
(r^2 f v_r ) = 0 ,
\end{equation}
\begin{equation}
\label{eq:mom}
\rho \frac{d v_r}{dt} = -\frac{\partial p}{\partial r}
- \frac{1}{8\pi r^2 f}\frac{\partial}{\partial r} (r^2 f B_{\perp}^2)
+ \frac{\rho v_{\perp}^2}{2r^2 f}\frac{\partial }{\partial r} (r^2 f)
-\rho \frac{G M_{\odot}}{r^2} ,
\end{equation}
\begin{equation}
\label{eq:moc1}
\rho \frac{d}{dt}(r\sqrt{f} v_{\perp}) = \frac{B_r}{4 \pi} \frac{\partial}
{\partial r} (r \sqrt{f} B_{\perp}).
\end{equation}
$$
\rho \frac{d}{dt}\left(e + \frac{v^2}{2} + \frac{B^2}{8\pi\rho}
- \frac{G M_{\odot}}{r} \right)
+ \frac{1}{r^2 f}
\frac{\partial}{\partial r}\left[r^2 f \left\{ \left(p
+ \frac{B^2}{8\pi}\right) v_r \right. \right.
$$
\begin{equation}
\label{eq:eng}
\left. \left.
- \frac{B_r}{4\pi} (\mbf{B \cdot v})\right\}\right]
+ \frac{1}{r^2 f}\frac{\partial}{\partial r}(r^2 f F_{\rm c})
+ q_{\rm R} = 0,
\end{equation}
\begin{equation}
\label{eq:ct}
\frac{\partial B_{\perp}}{\partial t} = \frac{1}{r \sqrt{f}}
\frac{\partial}{\partial r} [r \sqrt{f} (v_{\perp} B_r - v_r B_{\perp})],
\end{equation}
where $\rho$, $\mbf{v}$, $p$, $\mbf{B}$ are density, velocity, pressure,
and magnetic field strength, respectively, and subscripts
$r$ and $\perp$ denote radial and tangential components;
$\frac{d}{dt}$ and $\frac{\partial}{\partial t}$ denote Lagrangian and
Eulerian derivatives, respectively;
$e=\frac{1}{\gamma -1}\frac{p}{\rho}$ is specific energy and we assume
the equation of state for ideal gas with a ratio of specific heat,
$\gamma=5/3$;
$G$ and $M_{\odot}$ are the
gravitational constant and the solar mass; $F_{\rm c}(=\kappa_0 T^{5/2}
\frac{dT}{dr})$ is thermal conductive flux by Coulomb collisions, where
$\kappa_0=10^{-6}$ in c.g.s unit \citep{brg65};
$q_{\rm R}$ is radiative cooling.
We use optically thin radiative loss \citep{LM90}
in the corona and take into account
optically thick effects in the chromosphere \citep[][SI06]{aa89,mor04}.
We adopt the second-order MHD-Godunov-MOCCT scheme to update the physical
quantities.
Each cell boundary is treated as discontinuity, and for the time evolution
we solve nonlinear Riemann shock tube problems with the magnetic pressure term
by using the Rankin-Hugoniot relations. Therefore, heating is automatically
calculated from the shock jump condition.
A great advantage of our code is that no artificial viscosity is required
even for strong MHD shocks; numerical diffusion is
suppressed to the minimum level for adopted numerical resolution.
We initially set static atmosphere with a temperature $T=10^4$K to see
whether the atmosphere is heated up to coronal temperature and accelerated
to a transonic flow.
At $t=0$ we start the inject of the transverse fluctuations from the
photosphere and continue the simulations until the quasi-steady states
are achieved.
\subsection{Results}
\begin{figure}
\includegraphics[width=1.15\textwidth]{swoutputobs20.ps}
\caption{Comparison of the simulation result with observations of fast solar
wind.
Outflow speed, $v_r$(km s$^{-1}$) (Top-left), temperature, $T$(K) (top-right),
rms transverse amplitude, $\langle dv_{\perp} \rangle$(km s$^{-1}$)
(bottom-left), and density in logarithmic scale,
$\log(\rho({\rm g\;cm^{-3}}))$ (bottom-right), are plotted.
Observational data in the temperature panel are electron density,
$\log(N_e({\rm cm^{-3}}))$ which is to be referred to the right axis.
Dashed lines indicate the initial conditions and solid lines
are the results at $t=2573$ minutes. In the bottom panel, the initial
value ($\langle dv_{\perp} \rangle=0$) dose not appear.
{\it Top-left}: Green vertical error bars are proton outflow speeds in an
interplume region by UVCS/SoHO \citep{tpr03}.
Dark blue vertical error bars are proton outflow speeds by the Doppler
dimming technique using UVCS/SoHO data \citep{zan02}.
A dark blue open square with errors is velocity by IPS measurements
averaged in 0.13 - 0.3AU of high-latitude regions \citep{koj04}.
Light blue data are taken from \citet{gra96}; crossed bars are
IPS measurements by EISCAT,
crossed bars with open circles are by VLBA measurements, and
vertical error bars with open circles are data based on observation
by SPARTAN 201-01 \citep{hab94}. {\it top-right}: Pink circles are
electron temperatures by CDS/SoHO \citep{fdb99}.
{\it bottom-left}: Blue circles are non-thermal broadening inferred from
SUMER/SoHO measurements \citep{ban98}. Cross hatched region
is an empirical constraint of non-thermal broadening based on
UVCS/SoHO observation \citep{ess99}. Green error bars are transverse velocity
fluctuations derived from IPS measurements by EISCAT\citep{can02}.
{\it bottom-right}: Circles and stars are observations by SUMER/SoHO
\citep{wil98} and by CDS/SoHO \citep{tpr03}, respectively.
Triangles \citep{tpr03} and squares \citep{lql97} are observations
by LASCO/SoHO.
}
\label{fig:fstwd}
\end{figure}
Before discussing the reflection of the Alfv\'{e}n~ waves in various cases,
we explain how the coronal heating and the solar wind acceleration
were accomplished in the typical case which we studied in SI05 and SI06
for the fast solar wind.
We adopt $B_{r,0} =161$ G, the total $f=75$, and the root-mean-squared (rms)
surface amplitude, $\langle dv \rangle = 0.7$ km s$^{-1}$.
Figure \ref{fig:fstwd} plots the final structure of the simulated solar wind
after the quasi-steady state is achieved
in comparison with observations of fast solar winds.
In the four panels $v_r$(km s$^{-1}$), $T$(K),
mass density, $\rho$(g cm$^{-3}$), and rms transverse amplitude,
$\langle dv_{\perp} \rangle$(km s$^{-1}$) are plotted.
As for the density, we compare our result with observed electron density,
$N_e$, in the corona.
When deriving $N_e$ from $\rho$ in the corona, we assume H and He are
fully ionized, and $N_e({\rm cm^{-3}}) = 6\times 10^{23}\rho$(g cm$^{-3}$).
Figure \ref{fig:fstwd} shows that the initially cool and static atmosphere
is effectively heated and accelerated by the dissipation of the Alfv\'{e}n~ waves.
The sharp TR which divides the cool chromosphere with $T\sim 10^4$K and
the hot corona with $T\sim 10^6$K is formed owing to a thermally unstable
region around $T\sim 10^5$K in the radiative cooling function \citep{LM90}.
The hot corona streams out as the transonic solar wind.
The simulation naturally explains the observed trend quite well.
(see SI05 and SI06 for more detailed discussions.)
\begin{figure}
\includegraphics[width=1.\textwidth]{swoutput31_lres_blue.ps}
\caption{$r-t$ diagrams for $v_r$ (upper-left), $\rho$ (lower-left),
$v_{\perp}$ (upper-right), and $B_{\perp}/B_r$ (lower-right.)
The horizontal axises cover from $R_{\odot}$ to $15R_{\odot}$, and
the vertical axises cover from $t=2570$ minutes to $2600$ minutes, where
$R_{\odot}$ is the solar radius.
Blue and gray shaded regions indicate positive and negative amplitudes
which exceed certain thresholds. The thresholds are $d v_r
=\pm 96$km/s for $v_r$, $d\rho /\rho=\pm0.25$ for $\rho$,
$v_{\perp}=\pm 180$km/s for $v_{\perp}$, and $B_{\perp}/B_r=\pm
0.16$ for $B_{\perp}/B_r$, where $d \rho$ and $d v_r$ are
differences from the averaged $\rho$ and $v_r$.
Arrows on the top panels indicate characteristics of Alfv\'{e}n~, slow MHD
and entropy waves at the respective locations (see text). }
\label{fig:tdd}
\end{figure}
The reflection of Alfv\'{e}n~ waves are easily seen in $r-t$ diagrams.
Figure \ref{fig:tdd}
presents contours of amplitude of $v_r$, $\rho$, $v_{\perp}$, and
$B_{\perp}/B_r$ in $R_{\odot} \le r \le 15 R_{\odot}$
from $t=2570$ min. to $2600$ min.
Blue (gray) shaded regions denote positive (negative) amplitude. Above
the panels, we indicate the directions of the local 5 characteristics, two
Alfv\'{e}n~, two slow, and one entropy waves at the respective positions.
In our simple 1D geometry, $v_r$ and $\rho$ trace the slow modes
which have longitudinal wave components, while $v_{\perp}$ and $B_{\perp}$
trace the Alfv\'{e}n~ modes which are transverse (note that fast-mode and
Alfv\'{e}n~ mode degenerate in the simple 1D treatment, and then we simply
call them Alfv\'{e}n~ waves).
One can clearly see the Alfv\'{e}n~ waves in $v_{\perp}$ and $B_{\perp}/B_r$
diagrams, which have the same slopes with the Alfv\'{e}n~ characteristics shown
above.
One can also find the incoming modes propagating from lower-right to
upper-left as well as the outgoing modes generated from the surface\footnote{
It is instructive to note that the incoming Alfv\'{e}n~ waves have the positive
correlation between $v_{\perp}$ and $B_{\perp}$ (dark-dark or light-light
in the figures), while the outgoing modes have the negative correlation
(blue-gray or gray-blue).}.
These incoming waves are generated by the reflection at the `density mirrors'
of the slow modes in addition to the reflection owing to the shape
deformation (SI05 \& SI06).
At intersection points of the outgoing and incoming characteristics
the non-linear wave-wave interactions take place, which play a role
in the wave dissipation.
The slow modes are seen in $v_r$ and $\rho$ diagrams. Although it might
be difficult to distinguish, the most of the patterns are due to the outgoing
slow modes\footnote{The phase correlation of the longitudinal slow
waves is opposite to that of the transverse Alfv\'{e}n~ waves.
The outgoing slow modes
have the positive correlation between amplitudes of $v_r$ and $\rho$,
($\delta v_r \delta \rho > 0$), while the incoming modes have the negative
correlation ($\delta v_r \delta \rho < 0$).}
which are generated from the perturbations of the Alfv\'{e}n~ wave
pressure, $B_{\perp}^2/8\pi$ \citep{ks99,tsu02}.
These slow waves steepen eventually and lead to the shock dissipation.
The processes discussed here are the combination of the direct mode conversion
to the compressive waves and the parametric decay instability due to
three-wave (outgoing Alfv\'{e}n~, incoming Alfv\'{e}n~, and outgoing slow waves)
interactions \citep{gol78,ter86} of the Alfv\'{e}n~ waves.
Although they are not generally efficient in the homogeneous background
since they are the nonlinear mechanisms, the density gradient of the
background plasma totally changes the situation.
Owing to the gravity, the density rapidly decreases in the corona as $r$
increases, which results in the amplification of the wave amplitude so that
the waves easily become nonlinear.
Furthermore, the Alfv\'{e}n~ speed varies a lot due to the change of the density
even within one wavelength of Alfv\'{e}n~ waves with periods of minutes or longer.
This leads to both variation of the wave pressure in one wavelength
and partial reflection through the deformation of the wave shape
\citep{moo91}.
The dissipation is greatly enhanced by the density stratification,
in comparison with the case of the homogeneous background.
Thus, the low-frequency Alfv\'{e}n~ waves are effectively dissipated,
which results in the heating and acceleration of the coronal plasma.
\begin{figure}
\includegraphics[width=1.\textwidth]{reflc1.ps}
\caption{The top panels show the wave actions ($S_{\rm c}$) of the out-going
(solid) and incoming (dashed) Alfv\'{e}n~ waves. The bottom panels show
density (solid) and temperature (dashed).
The left panels are the results of the surface
amplitude, $\langle dv\rangle =1.4$km s$^{-1}$, the middle panels are
the results of $\langle dv\rangle =0.7$km s$^{-1}$, and the right panels
are the results of $\langle dv\rangle =0.4$km s$^{-1}$.}
\label{fig:depdv}
\end{figure}
From now, we examine how the reflection properties are affected by
changing input parameters.
In order to quantitatively study the dissipation and reflection of
the Alfv\'{e}n~ waves, we use an adiabatic constant,
$S_{\rm c}$, of the outgoing Alfv\'{e}n~ wave derived from wave action
\citep{jaq77}:
\begin{equation}
S_{c,\pm} = \rho \langle \delta v_{\rm A,+}^2 \rangle
\frac{(v_r + v_{\rm A})^2}{v_{\rm A}} \frac{r^2 f(r)}{r_c^2 f(r_c)},
\end{equation}
where $v_{\rm A}= B_r/\sqrt{4\pi\rho}$ is Alfv\'{e}n~ velocity and
\begin{equation}
\delta v_{\rm A,\pm} = \frac{1}{2}\left(v_{\perp} \mp B_{\perp}/\sqrt{4\pi\rho}
\right)
\end{equation}
is the amplitudes of the outgoing (for $+$ sign) and incoming (for $-$ sign)
Alfv\'{e}n~ waves (Els\"{a}sser variables).
First, we show the results with different values of surface fluctuations,
$\langle dv \rangle = 0.4, 0.7, 1.4$ km s$^{-1}$ (Figure \ref{fig:depdv}).
In the upper panel, we compare $S_{c,+}$ and $S_{c,-}$ of these three cases,
whereas we use the absolute values for $S_{c,-}$ to show in logarithmic scale.
As shown in the bottom panels, the density is very sensitive to the input
$\langle dv \rangle$. The density of the $\langle dv \rangle=1.4$ km s$^{-1}$
case is 1000 times larger than the density of the
$\langle dv \rangle=0.4$ km s$^{-1}$ case. Accordingly the mass fluxes of the
solar winds differs about several hundred times.
The differences of the densities and the mass fluxes between the two
cases are much larger than the ratio of the input energy ($\propto dv^2$)
$\approx 12$.
The reasons why the density sensitively depends on the input
surface perturbation can be explained by the reflection and the nonlinear
damping of Alfv\'{e}n~ waves.
The difference of the wave reflection in different $\langle dv \rangle$
cases are illustrated in the upper panels of Figure \ref{fig:depdv}.
In the smaller $\langle dv \rangle$ cases, the values of the outgoing (solid)
and incoming (dashed) components are very similar especially in the
chromospheric regions ($r-R_{\odot}\; \buildrel < \over \sim \; 0.01 R_{\odot}$).
This indicates that most of the outgoing Alfv\'{e}n~ waves generated from the
surface is reflected back downward.
Because the heating is smaller, the temperature is lower in
the smaller $\langle dv \rangle$ cases. Then, the scale height becomes smaller
and the density decreases rapidly. The Alfv\'{e}n~ speed changes
more rapidly and the wave shape is largely deformed, which enhances
the reflection. When the input wave energy decreases, a positive feedback
operates; a smaller fraction of the energy can reach the coronal region.
As a result, the density and mass flux of the solar wind becomes much
smaller than the decreasing factor of the input energy.
In addition to the effect of the wave reflection, the nonlinear dissipation
of the Alfv\'{e}n~ waves also plays a role in the sensitive behavior of
the density on the input wave energy (SI06). When the input wave energy
becomes smaller, the density becomes smaller as explained above.
Then, the nonlinearity of Alfv\'{e}n~ wave, $\delta v_{\rm A,+}/v_{\rm A}$,
decreases not only because the amplitude, $\delta v_{\rm A,+}$, is small but
also because the Alfv\'{e}n~ speed, $v_{\rm A}(\propto 1/\sqrt{\rho})$, is larger.
Therefore, the Alfv\'{e}n~ waves do not dissipate and the heating is
reduced, which further decreases the density; this is another type
of positive feedbacks, which also results in the sensitive dependence
of the density on the input wave energy.
\begin{figure}
\includegraphics[width=1.\textwidth]{reflc2.ps}
\caption{Same as Figure \ref{fig:depdv} but for the cases with different
spectra of surface perturbation.
The left panels are the results of $P(\nu)\propto \nu^{-1}$ (pink noise),
the middle panels are the results of $P(\nu)\propto \nu^0$ (white noise),
and the right panels are the results of sinusoidal waves with 3 minutes. }
\label{fig:depsp}
\end{figure}
Next, we investigate how the wave reflection is affected by the spectra of the
input fluctuations.
Here we study the three cases : The first case adopts the power spectrum,
$P(\nu)\propto \nu^{-1}$ (pink noise), with respect to frequency, $\nu$;
the second case adopts $P(\nu) \propto
\nu^0$ (white noise); the third case adopts sinusoidal perturbation with
period of 3 minutes. In the first and second cases, the range of
the period is set from 20 seconds to 30 minutes.
Figure \ref{fig:depsp} compares these three cases.
The first and second cases show similar results, but in the chromospheric
regions the first case suffers more reflection. This is because larger
power is in longer wavelengths (lower frequency). If the wavelength is longer
than the characteristic variation scale of Alfv\'{e}n~ speed, Alfv\'{e}n~ waves
suffer more reflection.
The third case (sinusoidal perturbation) shows considerably different
structure from these two cases. The transition region is not so sharp, which
is seen in the temperature structure (dashed line in the bottom panel),
because the heating is localized near the wave crests which are formed as
a result of steepening of nonlinearly excited compressive waves.
Then, the heating becomes more sporadic and responding to this intermittent
heating the transition region moves up and down.
This is a clear contrast to the first and second cases in which the sharp
transition regions work as
walls against the Alfv\'{e}n~ waves because the Alfv\'{e}n~ speeds change drastically.
On the other hand, in the sinusoidal case Alfv\'{e}n~ waves become more
transmittant owing to the fluctuating transition region. As shown in the top
panel, the reflected fraction (the ratio of the incoming component to
the outgoing one in the chromosphere) is smaller in this case.
\subsection{Discussions}
\subsubsection{Recent Observation of Waves}
Wave activities are observed in various portions of the solar atmosphere.
Propagating slow MHD waves are often observed in the corona as the
fluctuations of intensity \citep{ofm99,sak02,ban09a}.
\citet{tom07} detected Alfv\'{e}n~ waves propagating along with magnetic field
lines in the corona. Although the obtained wave amplitude is smaller than
the value required for the coronal heating, this may be due to the
line-of-sight effect \citep{mci08}. Indeed, nonthermal broadening of spectral
lines, which are supposed to be closely linked to Alfv\'{e}n~ waves, gives
much larger amplitudes \citep{sin04,ban09b}.
Alfv\'{e}n~(ic) waves are detected at the photospheric level in quiet regions
\citep{ulr96} by analyzing the phase correlation of magnetic field and
velocity perturbations. Similar attempts have been carried out in sunspot
regions as well \citep{lit98,bel00},
Recently, \citet{ft09} carried out very detailed analysis of the observed
fluctuations
by the spectro-polarimeter (SP) of the Solar Optical Telescope
\citep[SOT;][]{tsu08,sue08,ich08,shi08} aboard the HINODE satellite.
They obtained the fluctuations of magnetic field, $\delta B$, the
fluctuations of velocity field, $\delta v$, and the fluctuations of
intensity, $\delta I$,
which reflects density perturbation, $\delta \rho$ of pores, and magnetic
concentrations in plages. The periods of the oscillations are
distributed from 3 to 6 minutes in the pores and from 4 to 9 minutes in
the plages.
From the phase correlation of $\delta B$, $\delta v$, and $\delta I$
they investigated the modes and directions of the waves to that are attributed
the observed fluctuations.
One of the most important results is that the phase differences between
$\delta v$ and $\delta B$ are nearly -90 degrees \footnote{\citet{ft09}
picked up the regions with positive magnetic polarity (magnetic fields toward
observers), which is the reason why there are no data with the phase shift of
$+90$ degrees. Data with negative polarity actually show the phase shift peaked
around $+90$ degrees (Fujimura \& Tsuneta, private
communication). }. The obtained phase correlation is consistent with
standing kink-mode (transverse) or sausage-mode (longitudinal) waves,
which are, roughly speaking, surface-mode counterparts of Alfv\'{e}n~ and
slow MHD waves.
Namely, waves that propagate to positive and negative directions are almost
equally exist at the photospheric level.
Although it is quite difficult to estimate the fraction of sausage
(compressive) mode and kink (nearly incompressive) mode because the
transformation from $\delta I$ to $\delta \rho$ is not straightforward,
it is expected that regions with smaller $\delta I$ are dominated by
kink-mode oscillations. Based on this consideration, they picked up a
particular pore region with relatively small $\delta I$.
In this region, the phase difference between $\delta B$ and
$\delta v$ is -96 degrees, which indicates that the upward propagating flux is
slightly larger than the downward propagating component. Assuming the observed
fluctuations attribute to kink-mode waves, they obtained the net Poynting flux
that propagates upward as $2.7\times 10^6$ erg cm$^{-2}$s$^{-1}$.
Interestingly enough, our simulations also show very effective reflection
of Alfv\'{e}n~ waves below the transition region.
In the reference case for the fast solar wind (Figure \ref{fig:fstwd}),
85\% of the initial upward Poynting flux of the Alfv\'{e}n~ waves from the
photosphere is reflected back downward before reaching the corona.
This implies that one can observe the comparable amount of downward flux
to the upward component.
The net leakage of the upgoing flux to the corona in this simulation is
15\% of the input at the photosphere ($\sim 5\times 10^5$ erg cm$^{-2}$s$^{-1}$).
These features seem quite consistent,
at least in a qualitative sense, with the observation by \citet{ft09}.
The specific values of the leaking and reflected fluxes
depend on strength and configuration of magnetic flux tubes and amplitudes
of surface fluctuations.
\subsubsection{Data Driven Simulations}
The observation data are also used in numerical simulations. Recently,
various groups are performing so-called data-driven simulations
by adopting magnetic field data on the Sun \citep[e.g.][]{man08,kat09}.
While these works are mainly aiming at global heliospheric phenomena such as
propagation of coronal mass ejections,
observed data by HINODE are also applied for MHD simulations of surface
activities.
\citet{ms09} have carried out numerical simulations in open field regions
by using observed transverse motions on the Sun.
They obtained horizontal motions of granules by local correlation tracking
\footnote{To do so HINODE/SOT is the best telescope, because horizontal
oscillations are largely affected by the atmospheric seeing.}.
These transverse motions are expected to excite Alfv\'{e}nic~ waves that
propagate along the vertical magnetic fields.
They observed 14 different regions and derive the power spectrum of the
transverse oscillations. Using the obtained spectrum as the input from the
photosphere, they performed 1D MHD simulations in open field regions up to
the corona following \citet{ks99}.
They found that, compared to artificial inputs of white noise or pink noise,
more fraction of the Alfv\'{e}n~ wave energy transmits
through the transition region. They interpret that this is because more energy
is resonantly trapped between the photosphere and the transition region and
eventually leaks into the corona.
\subsubsection{Slow Solar Winds}
In this paper, we have introduced our simulation results for
the fast solar winds from polar coronal holes.
In SI06 and \citet{suz06}, we tried to explain observed data of
slow solar winds in the same framework of the nonlinear dissipation of
the Alfv\'{e}n~ waves. We have shown that the slow winds from flux tubes
with smaller $B_{r,0}/f$ and slightly larger $\langle dv_{\perp} \rangle$
well explain the observed properties of the slow streams from mid- to
low-latitude regions; the same physical mechanism operates but the
environments (e.g. geometries of flux tubes) make the differences between
the fast and slow solar winds.
This interpretation is quite consistent with the observed data by
Interplanetary scintillation measurements \citep{kfh05}.
HINODE observations detected mass outflows from open regions in the
vicinities of active regions \citep{sak07,ima07,har08}. These outflows
might become slow solar winds and possibly give the significant
contribution to the total mass loss from the Sun \citep{sak07}.
What is puzzling is that the acceleration seems to take place at a very low
altitude.
\citet{harr08} measured the Doppler velocities of the outflows and
concluded that the outflow speeds reach $\approx 100$ km s$^{-1}$ at
the height of $\sim 0.1 R_{\odot}$ from the surface.
Such rapid acceleration is very different from those inferred in the
classical slow winds.
For instance, the observations by \citet{she97} shows that
the slow winds associated with the streamer belt reach $\simeq 100$ km s$^{-1}$
above 2-3 solar radii.
It is difficult to explain the observed rapid acceleration by
our simulations of the nonlinear Alfv\'{e}n~ waves. If we increase the input
energy ($\sim \langle dv^2\rangle$), density, rather than outflow
speeds, increases owing to the larger heating. Probably other calculations
or simulations will give similar tendency.
To explain the outflow from the near active regions, pure momentum inputs
are necessary.
\section{Evolution to Red Giants}
\label{sec:stw}
When the Sun evolve to a red giant star, the properties of the stellar
wind also change.
When the stellar atmosphere becomes cool enough to form dusts during
the asymptotic giant branch (AGB), the stellar wind is mainly driven by the
radiation pressure on the dust particles \citep{bow88}.
Before the AGB phase, namely from the main sequence to
(early) red giant branch (RGB), the driving mechanism of the stellar winds
is regarded to be similar to the solar wind; the origin of the driver is the
kinetic energy of the surface convection, and MHD waves generated by the
turbulent motion of the surface convection accelerates the stellar winds
\citep[e.g.][]{hm80,jo89,cm95}.
However, the properties of the stellar
winds change with stellar evolution primarily because the surface gravity
decreases.
\begin{figure*}
\includegraphics[width=0.8\textwidth]{stwndfig21_lres.ps}
\caption{Evolution of stellar wind from the main sequence to RGB.
From top to bottom, radial outflow velocity, $v_r$ (km s$^{-1}$),
temperature, $T$ (K), and density, $\rho$(g cm$^{-3}$), are plotted.
The solid lines are the results
of the 1 $M_{\odot}$ stars ; the black, blue, green, and red lines are the
results of the star with the surface gravity, $\log g=4.4$ (Sun), 3.4, 2.4,
and 1.4, respectively.
The dashed lines are the results of the $3\; M_{\odot}$ stars; the black and
light-blue lines are the results of the stars with the surface gravity,
$\log g=2.4$ and 1.4. }
\label{fig:evl}
\end{figure*}
Based on these considerations, we studied the evolution of the stellar
winds with the stellar evolution from the main sequence to the RGB (S07).
We extend the solar wind simulation introduced in
the previous section to simulate the red giant winds.
We change the stellar radius, which controls the surface gravity, and
effective temperature (or sound speed).
We input the surface fluctuations that excite outgoing waves from the
photosphere in the same manner as in the solar wind simulation.
The amplitudes and spectra of the perturbations are estimated
from the scaling relation of surface convective flux and period
\citep[][see also Brun \& Palacios 2009 for recent simulations]{ren77}.
We simulated the stellar winds from a $1M_{\odot}$ star with different
surface gravities, $\log g=4.4$, 3.4, 2.4, and 1.4 (the Sun has $\log g=4.4$)
to investigate the effect of the stellar evolution. In addition, we simulated
the stellar winds from a $3M_{\odot}$ star with $\log=2.4$ and 1.4.
In all the models we use the photospheric magnetic field, $B_{r,0}=240$ G, and
the total super-radial expansion factor, $f=240$. Note that as for
the main sequence star (the Sun in this case) these values are between the
models for fast and slow solar winds (SI06).
The stellar wind structures after the quasi-steady states are presented
in Figure \ref{fig:evl}.
\begin{figure}
\includegraphics[width=0.8\textwidth]{coolingfig1.ps}
\caption{Cooling function of the solar metallicity gas from \citet{LM90}.
The main coolants in the respective temperature ranges are also indicated with
arrows.}
\label{fig:clg}
\end{figure}
The figure shows that hot coronae with $T\; \buildrel > \over \sim \; 10^6$ disappears when
the star evolves to $\log g \le 2.4$, and the density in the atmospheres and
winds increase (when measured in stellar radii).
The increase of the density is owing to the decrease of the surface
gravity as a result of the expansion of the star; more mass can be lifted
up to upper regions because of the smaller gravity.
The decrease of the temperature can also be partly explained by the decrease
of the gravity.
When the star evolves to
the red giant, the escape velocity becomes comparable to the sound speed;
for example, the sound speed of the coronal gas ($\sim 150$ km s$^{-1}$)
exceeds the escape speed of the red giant stars with $\log g \sim 2$ at a few
stellar radii. Then, the hot corona with $T\; \buildrel > \over \sim \; 10^6$ K cannot be confined
by the gravity and inevitably streams out.
While the increase of the density is rather continuous with the stellar
evolution, the temperature rapidly drops from the star with $\log g=3.4$
to the star with 2.4.
Thermal instability plays a role in this rapid decrease of the temperature,
in addition to the gravity effect explained above.
Figure \ref{fig:clg} shows the radiative cooling function adopted from
\citet{LM90} of the solar metallicity gas under the optically thin
approximation, which we use in the simulation.
As shown in the figure, the radiative flux decreases for increasing
temperature in $10^5\;({\rm K}) \; \buildrel < \over \sim \; T \; \buildrel < \over \sim \; 10^6\;({\rm K})$.
In this region, the cooling of gas is suppressed when it is heated up;
it is thermally unstable. Then, the gas is stable in only $T\; \buildrel < \over \sim \; 10^5$ K
or $T\; \buildrel > \over \sim \; 10^6$ K in which thermal conduction also plays a role in
the stabilization.
This is the main reason why the temperature drops
from $\sim 10^6$ K to $\; \buildrel < \over \sim \; 10^5$ with stellar evolution.
For more detail please see S07.
\begin{figure}
\includegraphics[width=1.\textwidth]{reflc3.ps}
\caption{The left panels are the results
of the main sequence star with the surface gravity of $\log g=4.4$ and
the right panels are the results of the red giant star with $\log g =1.4$
of 1 $M_{\odot}$ star.
The top panels show the wave actions ($S_{\rm c}$) of the out-going
(solid) and incoming (dashed) Alfv\'{e}n~ waves. The bottom panels show
density (solid) and temperature (dashed). The wave action ($S_{\rm c}$)
of the red giant case is reduced to 0.1 times of the original value for
comparison. }
\label{fig:depev}
\end{figure}
The properties of the reflection of the Alfv\'{e}n~ waves are also
affected by the stellar evolution.
Figure \ref{fig:depev} compares the reflection of the outgoing Alfv\'{e}n~ waves
between the main sequence star ($\log g=4.4$) and the moderately evolved
red giant star ($\log g=1.4$).
As clearly illustrated in the bottom panels, the reflection of the outgoing
Alfv\'{e}n~ waves is significantly suppressed in the evolved star compared
to the main sequence star.
only 30\% of the input Alfv\'{e}n~ waves from the surface is reflected back in
the evolved $1 M_{\odot}$ star with $\log g=1.4$, 90\% is reflected back
in the main sequence star.
This is mainly because the density slowly decreases
in the red giant star and the variation scale of the Alfv\'{e}n~ speed is larger.
Then, the outgoing Alfv\'{e}n~ waves do not suffer reflection so much and
more fraction of the input energy can reach higher altitudes.
However, because of the larger density the radiative loss is more effective
in the evolved stars (S07).
The mass loss rate, $4\pi \rho v_r r^2$, of the red giant star with
$\log g=1,4$ of 1 $M_{\odot}$ star is more than $10^5$ times larger
than the mass loss rate of the main sequence star. This is much larger
than the increase of the stellar surface area ($=1000$ times); the increase
of the mass flux, $\rho v_r$, itself contributes significantly
\citep[S07; see also][]{sc05}.
\begin{figure}
\includegraphics[width=1.\textwidth]{stwndfig20_lres.ps}
\caption{Plasma $\beta$ values as functions of stellar radii.
The solid lines are the results of the $1 M_{\odot}$ star; the black, blue,
green, and red lines are the
results of the star with the surface gravity, $\log g=4.4$ (Sun), 3.4, 2.4,
and 1.4, respectively.
The dashed lines are the results of the $3\; M_{\odot}$ stars; the black and
light-blue lines are the results of the stars with the surface gravity,
$\log g=2.4$ and 1.4. }
\label{fig:beta}
\end{figure}
The Alfv\'{e}n~ wave-driven stellar wind plays an important role in terms
of the energy conversion from magnetic energy to other types of energy.
To study the energy conversion, a plasma $\beta$ value, which is
defined as the ratio of gas pressure to magnetic pressure,
\begin{equation}
\beta=8\pi p / B^2,
\end{equation}
is a useful parameter.
Figure \ref{fig:beta} shows the $\beta$ values for the different six
models\footnote{
This figure is modified from the bottom panel of Figure 11 of S07.
In S07, the $\beta$ values were estimated from the radial magnetic
field strength, $B_r$. In this paper, we calculate $\beta$ from the total
magnetic field strength, the sum of radial and transverse components.}.
In the region closed to the surface, the $\beta$'s decrease on $r$ at first,
which is more clearly seen in the unevolved stars.
In this region, the atmosphere is mostly static and the density decreases
rapidly according to an exponential manner. The decrease of the density is
faster than the decrease of the magnetic energy $\propto B^2$, and then,
the atmosphere becomes magnetically dominated, $\beta <1$.
If the static atmosphere continued to the upper altitudes without the
dissipation of the Alfv\'{e}n~ waves, the $\beta$ values would keep decreasing.
In reality, however, the density structure is redistributed by the heating
from the wave dissipation. In the main sequence ($\log g=4.4$) and subgiant
($\log g=3.4$) stars,
the hot coronae are formed, which give larger pressure scale heights (
$\propto$ temperature). Therefore, the densities decrease more slowly
than the magnetic energy, and the $\beta$ values increase in the coronal
regions.
In the red giant stars, although hot coronae do not form as in the main
sequence and subgiant stars, the gas is lifted up directly by magnetic
pressure associated with Alfv\'{e}n~ waves in the weak gravity conditions,
which are gradually connected to the stellar wind regions.
Interestingly, the final $\beta$ values
are $0.01-0.1$, being independent from the surface gravities.
The redistribution of the density through the wave dissipation makes the
plasma $\beta$ values stay at the moderate values.
\section{Turbulent-driven Accretion Disk Winds}
\label{sec:dsw}
\begin{figure}
\includegraphics[width=1.\textwidth]{localshb1.eps}
\caption{Schematic picture of the local shearing box simulation of an
accretion disk.}
\label{fig:schdsk}
\end{figure}
We investigated accretion disk winds that are driven by MHD turbulence
in accretion disks \citep{si09}.
The geometrical aspects of disk winds are different from those of the stellar
winds as easily expected.
On the other hand, the driving mechanism shows similarities to the solar
and red giant winds, which we discuss from now.
To study turbulence driven accretion disk winds, we performed 3D MHD
simulations in a local patch of an accretion disk (Hawley et al. 1995;
Matsumoto \& Tajima 1995; Figure \ref{fig:schdsk}).
Here, we neglect the effects of curvature and the simulations are carried
out in a local Cartesian box that co-rotates with a Keplerian rotating
disk. Following the convention (Hawley et al. 1995), $x$, $y$, and
$z$ coordinates corresponds to radial, azimuthal, and vertical directions,
respectively. While in the $x$ direction we do not take into
account the radial gradients of the background density, in the $z$ direction
the density stratification owing to the gravity by a central star is included.
The size of the simulation box is $(x,y,z)=(\pm 0.5 H, \pm 2H, \pm 4H)$, where
a scale height, $H$, is defined from rotation frequency, $\Omega$,
and sound speed as $H=\sqrt{2}c_s/\Omega$.
We use $(32,64,256)$ mesh points for the $(x,y,z)$ coordinates.
The $z$ size of the simulation box covers from the sufficiently lower
region to upper region of an entire disk so that we can simultaneously
treat the amplification of magnetic field by MRI and excitation of disk
winds in a self-consistent manner.
This is an advantage of the simulation of the accretion disk winds in
comparison to the simulations of solar/stellar winds, in which we give the
conditions at the surface based observationally inferred values without
considering the generation of the magnetic field in the solar/stellar
interiors.
Because the co-rotating box is not in an inertial coordinate, inertial forces
need to be taken into account in the momentum equations :
\begin{equation}
\frac{d\mbf{v}}{d t} = -\frac{1}{\rho}\nabla(p+\frac{B^2}{8\pi})
+ \frac{(\mbf{B}\cdot\nabla)\mbf{B}}{4\pi\rho}
- 2\mbf{\Omega_0}\times\mbf{v} + 3\Omega_0^2 \mbf{x}
- \Omega_0^2 \mbf{z},
\label{eq:mom}
\end{equation}
where the third term on the right hand side is Coriolis force, the fourth
term corresponds to tidal expansion which is the sum of centrifugal
force and the radial direction of the gravity by a central star, and the
last term is the vertical component of the gravity which gives the
stratification of the density in the vertical direction.
We assume isothermal gas instead of solving the energy equation in order
to focus of the dynamics of the disk winds.
In the $x$ direction, the shearing boundary condition (Hawley et al.1995)
is adopted to mimic the differential rotation of a Keplerian rotating disk.
The simple periodic boundary condition is applied to the $y$ direction.
The outgoing condition (SI06) is prescribed to the $z$ direction to
properly treat streaming out disk winds.
We would like to note that our simulation is the first attempt that adopts
the real outgoing boundary to the disk wind simulation in the local shearing
box, whereas the simple zero-gradient boundary was used in previous
studies \citep{ms00,hir06}.
As the initial condition, we give Keplerian rotation, $v_x =-3/2\Omega
x$, with small perturbations as seeds for MRI and set up weak vertical
magnetic fields with $\beta=10^6$ at the midplane. In the vertical direction,
we give the hydrostatic density structure, $\rho \propto \exp(-z^2/H^2)$.
\begin{figure}
\includegraphics[width=1.\textwidth]{var_t_av_fig31_wnd5.ps}
\caption{Time-averaged disk structure during $t=200-400$ rotations.
The variables are also averaged on $x-y$ plane at each $z$ grid.
The top left panel shows $v_z/c_s$ (solid), whereas the dotted line is the
initial condition ($v_z/c_s=0$).
The bottom left panel presents the density (solid) in comparison with the
initial condition (dotted). The top right panel
presents magnetic energy, $B^2/4\pi$. The dashed, solid, and dotted lines
correspond to $x$, $y$, and $z$ components, whereas the $y$ component
shows both mean (thick) and fluctuation (thin) components.
The bottom right panel shows the plasma $\beta$ (solid) in comparison with
the initial value (dotted).
}
\label{fig:zav}
\end{figure}
In this review paper we mainly focus on the time-averaged
structure of disk winds. We would like to briefly describe the time evolution
and dynamical properties of the simulated local accretion disk
(see Suzuki \& Inutsuka 2009 for more detail).
After $\sim 3$ rotations, MRI triggers MHD turbulence firstly around
$z\sim \pm (2-3) H$. In the higher regions ($|z|>3H$), MRI does not set in
because the circumstance is magnetically dominate from the beginning and
stable against MRI. On the other hand, the delay of the MRI trigerring
in the near-midplane region is an artifact. Since in this region
the plasma $\beta$ is too high (gas pressure dominated) at first, we cannot
resolve the most unstable wavelength of MRI. However, MRI gradually takes
place later on. As the magnetic field strength increases, the most unstable
wavelength can be eventually resolved at the midplane.
After 100 rotation, the entire region except the surface
regions ($|z|>3H$) becomes turbulent and the disk winds are driven by
MHD turbulent pressure from the upper and lower surfaces. At $200$ rotations,
the magnetic field strength almost saturates, balancing the amplification
by MRI with the cancellation by magnetic reconnections and the escape with
the disk winds\footnote{Since the ideal MHD condition is assumed in the
simulation, the magnetic reconnections take place due to the numerical
resistivity determined by the grid scale.
Then, the saturation level might depend on the resolutions of simulations,
whereas this problem is still under debate \citep{san04,pcp07}.}. At this time,
the magnetic energy is amplified 1000 times of the initial energy of the weak
vertical field, and the field lines are dominated by the toroidal ($y$)
component (see below).
An interesting feature is that the disk winds are blow off intermittently.
The mass fluxes of the disk winds become strong every 5-10 rotations
quasi-periodically. This is a consequence of breakups of large scale channel
flows at $z\approx \pm 2H$ (Suzuki \& Inutsuka 2009).
Figure \ref{fig:zav} presents the disk wind structure averaging over 200 - 400
rotations. The variables are averaged on the $x-y$ plane at each $z$ point.
The top left panel shows that the gas streams out of the upper and lower
surfaces. The average outflow velocity is nearly the sound speed at the upper
and lower boundaries. The disk winds are dominantly accelerated by magnetic
energy (Poyinting flux) of the MHD turbulence, which we inspect from now.
The top right panel shows magnetic energy at the saturated state.
The dashed, solid, and dotted lines are $x$, $y$, and $z$ components.
In the $y$ component we are showing both mean, $\langle B_y\rangle^2$ and
fluctuation, $\delta B_y^2$, components. $\langle B_y\rangle^2$
is the simple average on $x-y$ planes, $\langle B_y(z)\rangle=\int\int
dx dy B_y(x,y,z)/(L_{x}L_{y})$, and the the fluctuations are
determined from $\delta B_y^2(z)=\int\int dx dy(B_y(x,y,z)-\langle
B_y(z)\rangle)^2/(L_{x}L_{y})$, where $L_x(=H)$ and
$L_y(=4H)$ are the $x$ and $y$ lengths of the simulation box.
As for $B_x$ and $B_z$ the fluctuation components greatly dominate the means,
and so we simply present $B_x^2$ and $B_z^2$.
The magnetic energy, which is dominated by the toroidal ($y$) component
as a consequence of winding, is amplified by $\approx$ 1000 times of
the initial value ($B_{z,0}^2/4\pi=10^{-6}$) in most of the region
($|z|<3H$).
While in the region near the mid-plane ($|z|<1.5H$),
the magnetic field is dominated by fluctuating component ($\delta B_y$),
the coherent component ($\langle B_y \rangle$) dominates in the regions
near the surfaces ($|z|>1.5H$).
In the surface regions the magnetic pressure
is comparable to or larger than the gas pressure ($\beta\; \buildrel < \over \sim \; 1$),
and thus, the gas motions cannot control the configuration of
the magnetic fields.
Therefore, the field lines tend to be straightened by magnetic tension
to give $\langle B\rangle^2>\delta B^2$ there,
even if the gas is turbulent.
We also note that $\langle B_z^2\rangle $
is amplified by MRI and Parker (1966) instability,
whereas $\langle B_z\rangle^2$ is strictly conserved.
The comparison of the final density structure (solid) with the initial
hydrostatic structure (dotted) in the bottom left panel shows that the
mass is loaded up to the onset regions of outflows from $z\approx\pm 2H$
by the amplified magnetic energy.
The plasma $\beta$ value in the bottom right panel is a good indicator to
understand the locations of the mass loading.
The panel shows that the wind onset regions correspond to $\beta\approx 1$,
which indicates that the winds start to be accelerated when the magnetic
pressure exceeds the gas pressure.
In the further upper regions, $|z|\; \buildrel > \over \sim \; 3H$, $\beta$ stays almost constant
slightly below unity. Without disk winds, $\beta$ decreases with increasing
height in the hydrostatic structure. However, the mass loading and disk winds
inhibit the decrease of $\beta$.
We can interpret that the density structure is redistributed to give the almost
constant $\beta$ in the wind regions. This is, in a qualitative sense,
similar to the $\beta$ structures in the stellar winds (Figure \ref{fig:beta}).
The detailed features are different mainly because we do not solve the energy
equation with radiative cooling and conduction in the disk wind simulations.
\begin{figure*}
\includegraphics[width=0.8\textwidth]{var_t_av_fig31_efx4.ps}
\caption{Time-averaged Poynting associated magnetic tension, $-B_z v_{\perp}
B_{\perp}$, flux of magnetic tension. }
\label{fig:efx}
\end{figure*}
The disk winds are driven by Poynting flux; both magnetic pressures and
tension almost equally contribute. The magnetic tension term, $-B_z v_{\perp}
B_{\perp}$ ($\perp$ denotes $x$ and $y$) shows an interesting distribution,
which is shown in Figure \ref{fig:efx}.
In the surface regions, the absolute values decrease
with increasing $z$, because the Poynting flux is converted to the kinetic
energy of the disk winds; this is natural behavior. However, at $|z|\approx
\pm 1.5H$, the solid line crosses the 0 line. This indicates that the Poynting
flux of the tension force directs toward both surface and midplane from here.
This is a result of the breakups of large scale channel flows at
$|z|\approx \pm 1.5H$ \citep{si09}.
The Poynting flux associated with Alfv\'{e}n~ waves is expressed as the same
shape, $-B_z v_{\perp} B_{\perp}$, because they are propagating by tension
restoring force. The properties of the Poynting flux of the magnetic
tension shown in Figure \ref{fig:efx} are different from the Alfv\'{e}n~ waves
discussed in the solar and stellar winds, because $\delta B_{\perp}
\gg B_z$ in the disk wind situation. The Poynting flux here is
mainly by large-scale channel flows in mid altitude regions ($z\sim 2H$)
and is associated with a U-shape magnetic field by Parker instability
in the surface regions ($|z|\; \buildrel > \over \sim \; 3H$: Suzuki et al.2010).
\section{Summary}
We have introduced the results of our simulation studies on Alfv\'{e}n~
wave-driven winds. In the section \ref{sec:slw} we summarized the solar
wind simulations presented in SI05 and SI06,
emphasizing the reflection of Alfv\'{e}n~ waves. In the standard run, 15\% of
the initial wave energy flux from the photosphere can transmit into
the corona, which is sufficient for the heating and acceleration of the
solar winds. The Alfv\'{e}n~ waves nonlinearly generate compressive waves
which eventually steepen and heat up the ambient gas by the shock heating.
In another way of the explanation, the outgoing Alfv\'{e}n~ waves dissipate
by parametric decay to incoming Alfv\'{e}n~ waves and outgoing compressive
waves. This process is very efficient in the density stratified
atmosphere.
The structure of the solar winds shows very sensitive dependence on
the input wave amplitude by the positive feedback mechanism: When increasing
the amplitude, the coronal density becomes larger and the pressure scale
height becomes longer because of the larger heating. The larger density
enhances the wave dissipation because the wave nonlinearly is larger
owing to the smaller Alfv\'{e}n~ speed ($\propto 1/\sqrt{\rho}$). The longer
pressure scale height reduces the wave reflection. Therefore, the heating
is enhanced when the surface amplitude slightly increases.
This shows that the structure of solar wind is largely altered on a small
change of the input energy through the reflection and the nonlinear dissipation
of the Alfv\'{e}n~ waves.
We introduced recent HINODE observations of Alfv\'{e}nic~ oscillations.
The obtained data by \citet{ft09} can be explained by the sum of upgoing
Alfv\'{e}n~ waves and the comparable amount of the reflected component.
They can estimate the leakage Poynting flux of the Alfv\'{e}nic~ oscillations to
the upper location from the phase correlation of the perturbing magnetic
field and the velocity.
They analyzed one case and found that the sufficient energy to heat the corona
and solar wind is going upward.
These observed properties are quite consistent with what we get in the
simulations.
We discussed the roles of Alfv\'{e}n~ waves in the context of stellar
evolution. Before reaching AGB phase when dusts are formed in cool atmosphere,
it is expected that, similarly to the solar wind, the Alfv\'{e}n~ waves play a
major role in driving the stellar winds.
By using the surface fluctuation strength estimated from the convective flux,
we performed the MHD simulations.
When a star evolves to RGB phase, the hot corona suddenly disappears and
the atmosphere streams out as the cool chromospheric winds in which magnetized
hot bubbles intermittently float. The main reason of the disappearance of
the hot corona is the gravity effect; the atmosphere flows out before heated
up owing to the weak gravity confinement. In addition, the thermal instability
also plays a role in the sudden drop of the temperature.
In evolved stars, the Alfv\'{e}n~ waves do not suffer reflection because the
density decreases more slowly and the change of Alfv\'{e}n~ speeds become more
gradual. This is a part of the reason why the mass loss rate jumps up
with stellar evolution.
We also introduced the accretion disk winds by MHD turbulence amplified
by MRI. The disk winds are driven by Poynting flux when the magnetic energy
dominates the gas energy. In the disk wind regions the plasma $\beta$ value
is kept slightly below unity by the redistribution of the
density structure, which is qualitatively similar to what we got in the
simulation of the solar and stellar winds.
On the other hands, the disk winds are intermittent mainly because of
the breakups of large scale channel flows.
In the simulations, we only model the onset region of the disk winds.
In further upper region with magnetically dominated condition, Alfv\'{e}n~ waves
might play a role in accerating the winds.
\begin{acknowledgements}
This work was supported in part by Grants-in-Aid for
Scientific Research from the MEXT of Japan
(19015004 and 20740100) and Inamori Foundation.
\end{acknowledgements}
\bibliographystyle{aps-nameyear}
|
1802.01019
|
\section{Introduction}\label{introduction}
In general relativity, the vacuum Einstein field equations of a spacetime $(M,\gamma)$ with cosmological constant $\Lambda$ can be written as
\[\text{Ric}_\gamma=\Lambda \gamma.\]
When $\Lambda<0$ the lowest energy solution is called the anti-de Sitter(AdS) spacetime. The significance of AdS spacetimes has increased especially from the AdS/CFT correspondence. In this context, asymptotically hyperbolic Riemannian 3-manifolds arise naturally as spacelike hypersurfaces of AdS spacetimes. Moreover, these manifolds can be considered spacelike hypersurfaces in asymptotically flat spacetimes which approach null infinity.
The construction of asymptotically flat solutions to the Einstein constraint equations, which provide Cauchy data for Einstein equation with $\Lambda=0$, has been studied extensively. From physical motivation, the dominant energy condition requires the scalar curvature of such metrics to be nonnegative. Due to this condition, R. Bartnik \cite{Bartnik:1993er} introduced a construction of $3$-metrics with prescribed scalar curvature by considering $3$-manifolds foliated by round spheres. There have been other interesting results inspired by this foliation construction. See \cite{Shi:2002dv, Smith:2004wp, Lin:2014bk}. In particular, C. -Y. Lin \cite{Lin:2014bk} used Hamilton's modified Ricci flow on surfaces as foliation to construct an asymptotically flat end. Let $(\Sigma,g)$ be a surface diffeomorphic to $\mathbb{S}^2$ whose area is $4\pi$. Recall that Hamilton's modified Ricci flow in \cite{Hamilton:1988bp} is defined as the family $(\Sigma,g(t))$ satisfying
\begin{equation}\label{RF}
\left\{
\begin{split}
\frac{\partial}{\partial t}g_{ij}&=(r-R)g_{ij}+2D_iD_j f=2M_{ij},\\
g(1)&=g,
\end{split}\right.
\end{equation}
where $R=R(t)$ is the scalar curvature of $g(t)$ on $\Sigma$ and
$$r=\frac{1}{|\Sigma_{t}|}\int_\Sigma R(t)\,d\mu_t=2,$$
where $|\Sigma_{t}|$ is the area of $\Sigma$ with respect to $g(t)$ and $f=f(t,x)$ is the Ricci potential satisfying the equation
$$\Delta f=R-r.$$
Note that this flow converges to a metric of constant curvature exponentially fast in any $C^k$-norm (see \cite[Appendix B]{Chow:vq}). Consider a metric $\overline{g}$ on $N=[1,\infty)\times \Sigma$ of the form
$$\overline{g}=u^2 dt^2 +t^2 g(t).$$
The unknown function $u$ on $N$ with prescribed scalar curvature $\overline{R}$ satisfies a quasilinear second order parabolic equation derived from the Gauss equation for each slice $\{t\}\times\Sigma$. Therefore, Lin obtained asymptotically flat $3$-metrics by solving this equation with some conditions for $\overline{R}$. Moreover, C. Sormani and Lin \cite{Lin:2016ca} studied the class of asymptotically flat three-dimensional Riemannian manifolds foliated by Hamilton's modified Ricci flow, and they used these manifolds to estimate the Bartnik mass. In addition, they showed rigidity and monotonicity of the Hawking mass of level sets given in the foliation (see \cite[Theorem 5]{Lin:2016ca}).
In this paper, we construct an asymptotically hyperbolic $3$-metric using Ricci flow foliation method and investigate properties of the metric. First we recall the most general form of definition of asymptotically hyperbolic manifolds in \cite{Chrusciel:2003fg}. Let $\mathbb{H}^n$ denote the standard hyperbolic space. The metric $g_0$ on $\mathbb{H}^n$ can be written as
\[g_0=dr^2+(\sinh r)^2 h_0,\]
where $r$ is the $g_0$-distance to a fixed point $o$ and $h_0$ is the standard metric on $\mathbb{S}^{n-1}$. Let $\varepsilon_0=\partial_r,\varepsilon_\alpha=\frac{1}{\sinh\rho}f_\alpha,\alpha=1,\ldots,n-1$, where $\{f_\alpha\}_{1\leq\alpha\leq n-1}$ is a local orthonormal frame of $(\mathbb{S}^{n-1},h_0)$ so that $\{\varepsilon_i\}_{0\leq i\leq n-1}$ forms a local orthonormal frame on $\mathbb{H}^n$. Let $S_r$ denote the geodesic sphere in $\mathbb{H}^n$ of radius $r$ centered at $o$.
\begin{definition}\label{defn AH}
A manifold $(M^n,g)$ is called \emph{asymptotically hyperbolic} if, outside a compact set, $M$ is diffeomorphic to the exterior of some geodesic sphere $S_{r_0}$ in $\mathbb{H}^n$ such that the metric components $g_{ij}=g(\varepsilon_i,\varepsilon_j),0\leq i,j\leq n-1$, satisfy
\[|g_{ij}-\delta_{ij}|=O(e^{-\tau r}),|\varepsilon_k(g_{ij})|=O(e^{-\tau r}),|\varepsilon_k(\varepsilon_l(g_{ij}))|=O(e^{-\tau r})\]
for some $\tau>\frac{n}{2}$.
\end{definition}
There are several other versions of the definition with different contexts, see \cite{Chen:2016hm}, \cite{Wang:2001hg}, \cite{Neves:2010hb}.
In section \ref{construction}, we derive the quasilinear parabolic equation for a function $u$ from prescribed scalar curvature $\overline{R}$ on $N$, and prove the existence of a solution. To estimate $C^0$ bounds, we introduce the substitution $w=u^{-2}$ which provides an explicit form of the bounds. (c.f. Remark \ref{substitution remark})
In section \ref{metric section}, we construct an asymptotically hyperbolic $3$-metric by the result in section \ref{construction}.
\begin{theorem}\label{theorem 1}
Let $(\Sigma,g)$ be a $2$-manifold which is diffeomorphic to $\mathbb{S}^2$ with area $4\pi$ and let $N$ be the product manifold $[1,\infty)\times\Sigma$. Then for any $H\in C^{\infty}(\Sigma)$ with $H>0$, there exists an asymptotically hyperbolic $3$-metric on $N$ of the form
\begin{equation}\label{metric}
\overline{g}=\frac{u^2}{1+t^2}dt^2+t^2g(t),
\end{equation}
with the scalar curvature $\overline{R}\equiv-6$ where $u\in C^\infty(N)$ is positive everywhere, and $g(t)$ is the solution to Hamilton's modified Ricci flow (\ref{RF}). Here $H$ is the mean curvature in direction $\partial_{t}$ on $\{1\}\times\Sigma$.
\end{theorem}
As in \cite{Lin:2014bk}, the crucial step here is to verify when the solution of the equation in section \ref{construction} exists. In fact, we can construct an asymptotically hyperbolic $3$-metric with a more general condition on the scalar curvature as
\[\overline{R}=-6+O(t^{-5})\geq -6.\]
The dominant energy condition requires that $\overline{R}\geq -6$, and the decay is needed to control the behavior of $u$ near infinity. See Theorem \ref{main theorem}.
In section \ref{hawking mass section}, we prove the corresponding rigidity and monotonicity result of the hyperbolic analogue of the Hawking mass as in \cite[Theorem 5]{Lin:2016ca}.
The mass of asymptotically hyperbolic Riemannian manifolds is defined as a linear functional by P. T. Chru\'sciel and M. Herzlich \cite{Chrusciel:2003fg}.
\begin{definition}
The mass functional $\mathbf{M}(g)$ is a linear functional on the kernel of $(\mathcal{DS})^*_{g_0}$, where $(\mathcal{DS})^*_{g_0}$ is the formal adjoint of the linearization of the scalar curvature at $g_0$. Let $\theta=(\theta^1,\ldots,\theta^n)\in \mathbb{S}^{n-1}\subset\mathbb{R}^n$ and the functions
\[V^{(0)}=\cosh r,\, V^{(j)}=\theta^j\sinh r\text{ for }1\leq j\leq n.\]
forms a basis of the kernel of $(\mathcal{DS})^*_{g_0}$. Using this basis, $\mathbf{M}(g)$ is defined as
\[\begin{split}
\mathbf{M}(g)(V^{(i)})&=\frac{1}{2(n-1)\omega_{n-1}}\lim_{r\rightarrow\infty}\int_{S_r}\left[V^{(i)}(\text{div}_0 h-d\text{tr}_0 h)\right.\\
&\left.\hspace{2in}-h(\nabla_0 V^{(i)},\cdot)+(\text{tr}_0 h)dV^{(i)}\right](\nu_0)d\sigma_0
\end{split}\]
where $h=g-g_0,\nu_0$ is the $g_0$-unit outward normal to $S_r$ and $d\sigma_0$ is the volume element on $S_r$ of the metric induced from $g_0$. The notation $\text{div}_0,\text{tr}_0,\nabla_0$ denotes the divergence, trace, the covariant derivative with respect to $g_0$, respectively.
\end{definition}
The Hawking mass on asymptotically hyperbolic manifolds is introduced by X. Wang \cite{Wang:2001hg}. (see also \cite{Neves:2010hb}).
\begin{definition}
Let $(M^3,g)$ be a $3$-dimensional asymptotically hyperbolic manifold, and let $\Sigma\subset M^3$ be a closed $2$-surface. Then the Hawking mass $\tilde{\mathfrak{m}}_H(\Sigma)$ of $\Sigma$ is defined as
\begin{equation}\label{hawking mass definition}
\tilde{\mathfrak{m}}_H(\Sigma)=\sqrt{\frac{|\Sigma|}{16\pi}}\left(1-\frac{1}{16\pi}\int_{\Sigma}H^2\, d\sigma+\frac{|\Sigma|}{4\pi}\right)
\end{equation}
where $d\sigma$ is the induced volume form with respect to $g$.
\end{definition}
To study the rigid case, we use the following result about the Hawking mass and the mass functional
\begin{equation}\label{hawking mass relation}
\mathbf{M}(g)(V^{(0)})=\lim_{r\rightarrow\infty}\tilde{\mathfrak{m}}_H(S_r)
\end{equation}
proved by P. Miao, L. -F. Tam, and N. Xie \cite{Miao:2017df}.
\begin{theorem}\label{rigidity theorem}
Let $(\Sigma,g_1)$ be a surface diffeomorphic to $\mathbb{S}^2$ with positive mean curvature (not necessarily constant) and let $N=[1,\infty)\times\Sigma$ be an asymptotically hyperbolic extension obtained in Theorem \ref{main theorem}. Then $\tilde{\mathfrak{m}}_H(\Sigma_{t})$ is nondecreasing, where $\Sigma_{t}=\{t\}\times\Sigma$. Furthermore, if
\begin{equation}
\mathbf{M}(\overline{g})(V^{(0)})=\tilde{\mathfrak{m}}_H(\Sigma)
\end{equation}
then $\overline{R}=-6$ everywhere, $\Sigma$ is isometric to the standard sphere, and $N$ is rotationally symmetric. If $\tilde{\mathfrak{m}}_H(\Sigma)=0$ then $N$ is isometric to a rotationally symmetric region in a hyperbolic space. If $\tilde{\mathfrak{m}}_H(\Sigma)=m>0$ then $N$ is isometric to a rotationally symmetric region in anti-de Sitter Schwarzschild space of mass $m$.
\end{theorem}
\subsection*{Acknowledgments} The author would like to express my gratitude to my advisor, Professor Lan-Hsuan Huang, for all her support, guidance, and motivation. The author is also grateful to Professors Xiadong Yan, Ovidiu Munteanu and Guozhen Lu for their helpful discussions. The author was partially supported by the NSF under grant DMS 1452477.
\section{Parabolic equation with Ricci flow foliation}\label{construction}
In this section, we will derive the equation for prescribed scalar curvature $\overline{R}$ on $N$ from (\ref{metric}) and obtain a priori estimates for a solution $u$. The argument is slightly modified from \cite{Bartnik:1993er} to be suitable for the derived equation.
From the Gauss equation for each slice $\{t\}\times \Sigma$, we have
$$\overline{R}=R_t+2\overline{\text{Ric}}\left(\frac{\sqrt{1+t^2}}{u}\partial_t,\frac{\sqrt{1+t^2}}{u}\partial_t \right)+||h||^2-H^2,$$
where $R_t$ is the scalar curvature on $\{t\}\times\Sigma$ with the induced metric $t^2g(t)$, $h$ is the second fundamental form, and $H$ is the mean curvature in direction $\partial_t$. By direct computation, we have
\[\begin{split}\overline{\Gamma}_{ij}^{0} =\frac{1}{2}\overline{g}^{0l}(\overline{g}_{lj,i}+\overline{g}_{il,j}-\overline{g}_{ij,l})&=\frac{1+t^{2}}{2u^{2}}\left(-\frac{\partial}{\partial t}(t^{2}g(t)_{ij})\right)\\
& =\frac{1+t^{2}}{u^{2}}\left(-2tg_{ij}-2t^{2}M_{ij}\right),\\
h_{ij} =-\overline{g}\left(\overline{\nabla}_{\partial_{i}}\partial_{j},\frac{\sqrt{1+t^{2}}}{u}\partial_{0}\right)&=-\frac{\sqrt{1+t^{2}}}{u}\overline{g}(\overline{\Gamma}_{ij}^{0}\partial_{0},\partial_{0})\\
& =\frac{\sqrt{1+t^{2}}}{2u}(2tg_{ij}+2t^{2}M_{ij}),\\\end{split}
\]\[\begin{split}
H &=\overline{g}^{ij}h_{ij}=t^{-2}g^{ij}\left(\frac{\sqrt{1+t^{2}}}{u}tg_{ij}\right)=\frac{2\sqrt{1+t^{2}}}{tu},\\
||h||^{2} & =\frac{2(1+t^{2})}{t^{2}u^{2}}+\frac{1+t^{2}}{u^{2}}|M|_{g(t)}^{2}.
\end{split}
\]
Then, we obtain
\[\overline{\text{Ric}}\left(\frac{\sqrt{1+t^{2}}}{u}\partial_{t},\frac{\sqrt{1+t^{2}}}{u}\partial_{t}\right)=-\frac{1}{u}\Delta_{\overline{g}|_{\Sigma_{t}}}u+\frac{\sqrt{1+t^{2}}}{u}\frac{\partial H}{\partial t}-||h||^{2},\]
\[\begin{split}
h_{ij} & =\frac{\sqrt{1+t^{2}}}{u}(tg_{ij}+t^{2}M_{ij}),\\
H & =\frac{2\sqrt{1+t^{2}}}{tu},\quad ||h||^{2}=\frac{2(1+t^{2})}{t^{2}u^{2}}+\frac{1+t^{2}}{u^{2}}|M|_{g(t)}^{2}.
\end{split}
\]
Thus we get the following quasilinear second order parabolic equation
\begin{equation}\label{targetPDE}
\begin{split}
t(1+t^{2})\frac{\partial u}{\partial t}=\frac{u^{2}\Delta_{g(t)}u}{2}&-\frac{u^{3}}{4}(R_{g(t)}-t^{2}\overline{R})\\
&+u\left(\frac{1+3t^{2}}{2}+\frac{t^{2}(1+t^{2})|M|_{g(t)}^{2}}{4}\right).
\end{split}
\end{equation}
Here $\Delta_{g(t)}$ and $R_{g(t)}$ are the Laplace operator and the scalar curvature on $(\{t\}\times\Sigma,g(t))$ respectively. For any interval $I\subset\mathbb{R}^{+}$, let $A_{I}=I\times\Sigma$.
For sake of convenience, we will use the following notations: $\Delta=\Delta_{g(t)},$ $\nabla=\nabla^{g(t)}$, for any $f\in C^{0}(A_{I})$, $f^{*},f_{*}:I\rightarrow\mathbb{R}$ are defined by
\[f_{*}(t)=\inf\{f(t,x):x\in\Sigma\},\qquad f^{*}(t)=\sup\{f(t,x):x\in\Sigma\}.\]
Now from the parabolicity of (\ref{targetPDE}), the local existence can be obtained by standard Schauder theory \cite[Theorem 8.2]{Lieberman:1996ue}.
\begin{proposition}\label{proposition short time}
Let $I=[t_{0},t_{1}],1\leq t_{0}<t_{1}<\infty$, and let $\overline{R}\in C^{\alpha,\alpha/2}(A_{I})$. Then for any initial condition
\begin{equation}\label{initial}
u(t_{0},x)=\varphi(x),\quad x\in\Sigma,
\end{equation}
where $\varphi\in C^{2,\alpha}(\Sigma)$ satisfies
\begin{equation}\label{initialP}
0<\delta_{0}\leq\varphi^{-2}(x)\leq\delta_{0}^{-1},\quad x\in\Sigma,
\end{equation}
for some constant $\delta_{0}>0$, the parabolic equation (\ref{targetPDE}) with the initial condition (\ref{initial}) has a unique solution $u\in C^{2+\alpha,1+\alpha/2}(A_{[t_{0},t_{0}+T]})$ for some $T>0$. Here $T$ depends on $\delta_{0},t_{0},||R||_{\alpha,\alpha/2;A_I}, ||M||_{\alpha,\alpha/2;A_I},$ $||\overline{R}||_{\alpha,\alpha/2;A_I}$ and $||\varphi||_{2,\alpha}$.
\end{proposition}
To state the existence of global solution, we need the following a priori $C^0$ estimates for the solution $u$ which control the parabolicity and prevent the finite-time blow up.
\begin{proposition}\label{proposition for delta}
Suppose $u\in C^{2+\alpha,1+\alpha/2}(A_{[t_{0},t_{1}]}),1\leq t_{0}<t_{1}$, is a positive solution to (\ref{initial}). If we further assume that $\overline{R}$ is defined on $A_{[1,\infty)}$ such that the functions
\begin{equation}\label{lower delta}
\delta_{*}(t)=\frac{1}{t(1+t^{2})}\int_{1}^{t}\frac{\left(R_{g(s)}-s^{2}\overline{R}\right)_{*}}{2}\exp\left(-\int_{s}^{t}\frac{\tau(|M|^{*})^{2}}{2}d\tau\right)ds
\end{equation}
and
\begin{equation}\label{upper delta}
\delta^{*}(t)=\frac{1}{t(1+t^{2})}\int_{1}^{t}\frac{\left(R_{g(s)}-s^{2}\overline{R}\right)^{*}}{2}\exp\left(-\int_{s}^{t}\frac{\tau(|M|_{*})^{2}}{2}d\tau\right)ds
\end{equation}
are defined and finite for all $t\in[t_{0},\infty)$, then for $t_{0}\leq t\leq t_{1}$, we have
\begin{equation}\label{lower bound}
u^{-2}(t,x)\geq\delta_{*}(t)+\frac{t_{0}(1+t_{0}^{2})}{t(1+t^{2})}(u^{*}(t_{0})^{-2}-\delta_{*}(t_{0}))\exp\left(-\int_{t_{0}}^{t}\frac{\tau(|M|^{*})^{2}}{2}d\tau\right)
\end{equation}
and
\begin{equation}\label{upper bound}
u^{-2}(t,x)\leq\delta^{*}(t)+\frac{t_{0}(1+t_0^{2})}{t(1+t^{2})}(u_{*}(t_{0})^{-2}-\delta^{*}(t_{0}))\exp\left(-\int_{t_{0}}^{t}\frac{\tau(|M|_{*})^{2}}{2}d\tau\right).
\end{equation}
\end{proposition}
\begin{proof}
Let $w=u^{-2}$ then we have
\[\Delta u
=-\frac{3}{2}w^{-1}\nabla u\cdot\nabla w-\frac{1}{2}w^{-\frac{3}{2}}\Delta w.
\]
Substituting the Laplace term in (\ref{targetPDE}), we obtain
\begin{equation}\label{after substitution}
\begin{split}
\frac{\partial w}{\partial t} =\frac{1}{t(1+t^{2})}\left[\frac{3}{2}u\nabla u\cdot\nabla w+\frac{1}{2w}\right.&\Delta w+\frac{1}{2}(R_{g(t)}-t{}^{2}\overline{R})\\
&\left.-w\left(1+3t^{2}+\frac{t^{2}(1+t^{2})|M|^{2}}{2}\right)\right].
\end{split}
\end{equation}
By applying the maximum principle, we have
\[t\frac{dw_{*}}{dt}\geq\frac{1}{1+t^{2}}\left(-w_{*}\left(1+3t^{2}+\frac{t^{2}(1+t^{2})\left(|M|^{*}\right)^{2}}{2}\right)+\frac{1}{2}(R_{g(t)}-t{}^{2}\overline{R})_{*}\right)\] at the maximum of $u(t,x)$. We solve the following ODE
\begin{equation}\label{ODE maximum principle}
t\frac{dw_{*}}{dt}=-w_{*}\left(1+t\left(\frac{2t}{1+t^{2}}+\frac{t\left(|M|^{*}\right)^{2}}{2}\right)\right)+\frac{(R_{g(t)}-t^{2}\overline{R})_{*}}{2(1+t)^{2}}.
\end{equation}
Using the integrating factor method, we let \[\varphi(t)=\exp\left(\int_{1}^{t}\frac{2s}{1+s^{2}}+\frac{s\left(|M|^{*}\right)^{2}}{2}ds\right).\] Then we have
\[\frac{d(t\varphi(t)w_{*}(t))}{dt}=\frac{(R_{g(t)}-t^{2}\overline{R})_{*}}{2(1+t^{2})}\varphi(t).\] Integrating to solve $t\varphi(t)w_*(t)$ and noting $u^{-2}(t,x)=w(t,x)\geq w_*(t)$, we derive
\[\begin{split}u^{-2} & \geq\frac{1}{t}\int_{t_{0}}^{t}\frac{\left(R_{g(s)}-s^{2}\overline{R}\right)_{*}}{2(1+s^{2})}\,\exp\left(-\int_{s}^{t}\left(\frac{2\tau}{1+\tau^{2}}+\frac{\tau(|M|^{*})^{2}}{2}\right)d\tau\right)ds\\
&\qquad\qquad\qquad\qquad\qquad +\frac{t_{0}}{t}\exp\left(-\int_{t_{0}}^{t}\left(\frac{2\tau}{1+\tau^{2}}+\frac{\tau(|M|^{*})^{2}}{2}\right)d\tau\right)w_{*}(t_{0})\\
&=\delta_{*}(t)+\frac{t_{0}(1+t_0^2)}{t(1+t^2)}(w_{*}(t_{0})-\delta_{*}(t_{0}))\exp\left(-\int_{t_{0}}^{t}\left(\frac{\tau(|M|^{*})^{2}}{2}\right)d\tau\right).
\end{split}\]
Similarly, applying the maximum principle to $w^{*}$, we get the upper bound of $u^{-2}$.
\end{proof}
\begin{remark}\label{substitution remark}
In proof of Proposition \ref{proposition for delta}, the idea of substitution as $w=u^{-2}$ first appeared in Bartnik's work \cite[Proposition 3.3]{Bartnik:1993er}. The advantage of this is that we can simplify the coefficient of the term $R_{g(t)}-t^2\overline{R}$ as in (\ref{after substitution}) so that we can find the explicit solution of the equation (\ref{ODE maximum principle}) when applying the maximum principle. This will be used not only to prove the global existence of solution but also to show that $\overline{g}=\frac{u^2}{1+t^2}dt^2+t^2g(t)$ is asymptotically hyperbolic with a certain initial condition on $\overline{R}$.
\end{remark}
With Propositions \ref{proposition short time} and \ref{proposition for delta}, we can prove the global existence of solution as the following.
\begin{theorem}\label{global existence theorem}
Assume that $\overline{R}\in C^{\alpha,\alpha/2}(N)$ and the constant $K$ is defined by
\begin{equation}\label{constant K}
\begin{split}K =\sup_{1\leq t<\infty}\left\{ -\int_{1}^{t}\frac{\left(R_{g(s)}-s^{2}\overline{R}\right)_{*}}{4}\exp\left(\int_{1}^{s}\frac{\tau(|M|^{*})^{2}}{2}d\tau\right)ds\right\} <\infty.
\end{split}
\end{equation}
Then for every $\varphi\in C^{2,\alpha}(\Sigma)$ such that
\begin{equation}\label{global condition}
0<\varphi(x)<\frac{1}{\sqrt{K}}\text{ for all }x\in\Sigma,
\end{equation}
there is a unique positive solution $u\in C^{2+\alpha,1+\alpha/2}(N)$ with the initial condition
\begin{equation}\label{target PDE initial}
u(1,\cdot)=\varphi(\cdot).
\end{equation}
\end{theorem}
\begin{proof}
By considering (\ref{lower bound}) and (\ref{constant K}) simultaneously, we have
\[\begin{split}
\left(u^{-2}\right)_*(t)&>\delta_{*}(t)+\frac{2K}{t(1+t^2)}\exp\left(-\int_{1}^t\frac{\tau\left(|M|^*\right)^2}{2}d\tau\right)\\
&\geq\frac{2}{t(1+t^{2})}\exp\left(-\int_{1}^t\frac{\tau\left(|M|^*\right)^2}{2}d\tau\right)\\
&\hspace{1in}\times\left(\int_{1}^{t}\frac{\left(R_{g(s)}-s^{2}\overline{R}\right)_{*}}{4}\exp\left(\int_{1}^{s}\frac{\tau(|M|^{*})^{2}}{2}d\tau\right)ds+K\right)\\
&\geq 0
\end{split}
\]
for all $t\geq 1$. Hence it follows from Proposition \ref{proposition for delta} that $u$ doesn't blow up for all $t\geq 1$. Combining this and Proposition \ref{proposition short time} which states the local existence, we get the desired result.
\end{proof}
\section{Asymptotically hyperbolic $3$-metric with Ricci flow foliation}\label{metric section}
Using Theorem \ref{global existence theorem}, we can construct a metric with prescribed scalar curvature $\overline{R}$ along the Ricci flow foliation. By assuming the approximate decay for $\overline{R}$, we prove that the metric is asymptotically hyperbolic.
\begin{theorem}\label{main theorem}
Let $(\Sigma,g)$ be a $2$-manifold which is diffeomorphic to $\mathbb{S}^2$ with area $4\pi$. Let $N$ be the product manifold $[1,\infty)\times\Sigma$. Assume that $\overline{R}\in C^{\infty}(N)$ satisfies
\begin{equation}\label{source condition}
\overline{R}=-6+O(t^{-5})\geq -6.
\end{equation}
Then for any $H\in C^{\infty}(\Sigma)$ with the condition
\begin{equation}\label{extension condition}
H>2\sqrt{2K}
\end{equation}
where $K$ is defined as (\ref{constant K}), there exists an asymptotically hyperbolic $3$-metric on $N$ of the form
\begin{equation}
\overline{g}=\frac{u^2}{1+t^2}dt^2+t^2g(t).
\end{equation}
Here $g(t)$ is the solution to Hamilton's modified Ricci flow (\ref{RF}), such that $\overline{R}$ and $H$ are the scalar curvature on $(N,\overline{g})$ and the mean curvature in direction $\partial_{t}$ on $\{1\}\times\Sigma$, respectively.
\end{theorem}
To prove the theorem we need the following lemma which investigates the decay of $u$ by the assumption (\ref{source condition}). The method is similar to the proof of Y. Shi and L. -F. Tam in \cite{Shi:2002dv}.
\begin{lemma}\label{decay Lemma}
Let $u$ be the solution of (\ref{targetPDE}) with the initial condition $u(1,x)=\varphi(x)$, where $\varphi(x)$ satisfies (\ref{global condition}). Then for sufficiently large $t$, we have the estimate
\begin{equation}
\begin{split}
\left|\left(\frac{\partial}{\partial t}\right)^k\left(\frac{\partial^{|\beta|}}{\partial x^\beta}\right)(u-1)\right|\leq \frac{C}{t^3}
\end{split}
\end{equation}
where $\beta$ is a multi-index.
\end{lemma}
\begin{proof}[Proof of Lemma \ref{decay Lemma}]
First we need to verify the $C^0$ bounds. Since $|R_{g(s)}-2|$ is bounded we have
\[\begin{split}
&\int_{1}^{t}\frac{(R_{g(s)}-2)^*}{2}\exp\left(-\int_{s}^{t}\frac{\tau(|M|_{*})^{2}}{2}d\tau\right)ds\leq C_1,\\
&\int_{1}^{t}\frac{(R_{g(s)}-2)_*}{2}\exp\left(-\int_{s}^{t}\frac{\tau(|M|^*)^2}{2}d\tau\right)ds\geq C_2.
\end{split}\]
From the scalar curvature condition (\ref{source condition}), we obtain
\[\begin{split}\delta^{*}(t) & =\frac{1}{t(1+t^{2})}\int_{1}^{t}\frac{\left(R_{g(s)}-s^{2}\overline{R}\right)^{*}}{2}\exp\left(-\int_{s}^{t}\frac{\tau(|M|_{*})^{2}}{2}d\tau\right)ds\\
& =\frac{1}{t(1+t^{2})}\int_{1}^{t}\exp\left(-\int_{s}^{t}\frac{\tau(|M|_{*})^{2}}{2}d\tau\right)ds\\
& \qquad+\frac{1}{t(1+t^{2})}\int_{1}^{t}\frac{\left(R_{g(s)}-2+6s^{2}+O(s^{-3})\right)^{*}}{2}\exp\left(-\int_{s}^{t}\frac{\tau(|M|_{*})^{2}}{2}d\tau\right)ds.
\end{split}\]
Since $\exp\left(-\int_{s}^{t}\frac{\tau(|M|_{*})^{2}}{2}d\tau\right)\leq 1$, we have
\[
\int_{1}^{t}\exp\left(-\int_{s}^{t}\frac{\tau(|M|_{*})^{2}}{2}d\tau\right)\leq t-1.
\]
Thus we get
\[\begin{split}
\delta^*(t)&\leq \frac{1}{1+t^2}+\frac{C_1-1}{t(1+t^2)}+1-\frac{1}{1+t^2}+O(t^{-3})\\
&\leq 1+\frac{C_3}{t^3}.
\end{split}
\]
To get the lower bound estimate, similarly we have
\[\begin{split}\delta_{*}(t) & =\frac{1}{t(1+t^{2})}\int_{1}^{t}\frac{\left(R_{g(s)}-s^{2}\overline{R}\right)_{*}}{2}\exp\left(-\int_{s}^{t}\frac{\tau(|M|^{*})^{2}}{2}d\tau\right)ds\\
& =\frac{1}{t(1+t^{2})}\int_{1}^{t}\exp\left(-\int_{s}^{t}\frac{\tau(|M|^{*})^{2}}{2}d\tau\right)ds\\
& \qquad+\frac{1}{t(1+t^{2})}\int_{1}^{t}\frac{\left(R_{g(s)}-2+6s^{2}+O(s^{-3})\right)_{*}}{2}\exp\left(-\int_{s}^{t}\frac{\tau(|M|^{*})^{2}}{2}d\tau\right)ds.\\
\end{split}\]
Let $t_{0}\geq 1$ such that \[\int_{t_{0}}^{\infty}\frac{\tau(|M|^{*})^{2}}{2}d\tau\leq 1.\]
Then, from \cite[Lemma 4.1]{Bartnik:1993er}, we have
\[1-\frac{C_4}{t}\leq\frac{1}{t}\int_{1}^{t}\exp\left(-\int_{s}^{t}\frac{\tau(|M|^*)^2}{2}d\tau\right)\leq 1+\frac{C_4}{t}.\]
Then we get
\[\begin{split}\delta_{*}(t) & \geq\frac{1}{1+t^{2}}-\frac{C_4}{t(1+t^{2})}+\frac{C_2}{t(1+t^{2})}\\
&\qquad\qquad+\frac{1}{t(1+t^{2})}\int_{1}^{t}3s^{2}\exp\left(-\int_{s}^{t}\frac{\tau(|M|^{*})^{2}}{2}d\tau\right)ds+O(t^{-3})\\
& \geq\frac{1}{1+t^{2}}-\frac{C_4}{t(1+t^{2})}+\frac{C_2}{t(1+t^{2})}+\frac{1}{t(1+t^{2})}\left[\int_{1}^{t_{0}}3s^{2}\exp\left(-\int_{s}^{t}\frac{\tau(|M|^{*})^{2}}{2}d\tau\right)ds\right.\\
& \qquad\qquad\qquad\qquad\left.+t^{3}-t_{0}^{3}+\int_{t_{0}}^{t}3s^{2}\left(\exp\left(-\int_{s}^{t}\frac{\tau(|M|^{*})^{2}}{2}d\tau\right)-1\right)ds\right]+O(t^{-3})\\
&\geq 1-\frac{C_4+t_0^3-C_2}{t(1+t^{2})}-\frac{1}{t(1+t^{2})}\left[\int_{t_{0}}^{t}3s^{2}\int_{s}^{t}\tau(|M|^{*})^{2}d\tau ds\right]+O(t^{-3}).\\
\end{split}\]
We used the fact that $e^{\eta}-1\geq-2|\eta|\text{ for }|\eta|\leq 1$ to get the third inequality. It follows from the fact that \[|M|^2\leq C_5e^{-ct}\] that we have
\[\begin{split}
\int_{t_0}^{t}3s^2\int_s^t \tau(|M|^*)^2 d\tau ds &\leq\int_{t_0}^{t}3s^2\int_s^t \tau C_5 e^{-c\tau} d\tau ds\\
&\leq C_5\int_{t_0}^{t}3s^2\int_s^t \tau e^{-c\tau} d\tau ds\\
&\leq O(e^{-ct})+C_5\int_{t_0}^{t}3s^2 \left(\frac{1}{c}se^{-cs}+\frac{1}{c^2}e^{-cs}\right) ds=O(e^{-ct}).
\end{split}
\]
Hence we obtain
\[
\begin{split}
\delta_{*}(t)
& \geq1-\frac{C_6}{t^3},
\end{split}
\]
and thus
\[|u(t,x)-1|\leq\frac{C}{t^3}.\]
Now we find the estimate for derivatives. Write the equation (\ref{targetPDE}) as follows
\[\begin{split}
&t(1+t^{2})\frac{\partial u}{\partial t}\\
&\qquad=\frac{\partial}{\partial x^{i}}\left(\frac{1}{2}u^{2}g^{ij}\frac{\partial u}{\partial x^{j}}\right)-\left[\frac{\partial}{\partial x^{i}}\left(\frac{u^{2}}{2\sqrt{|g|}}\right)\right]\left(\sqrt{|g|}g^{ij}\frac{\partial u}{\partial x^{j}}\right)\\
& \qquad\qquad-\frac{u^{3}}{4}(R_{g(t)}-t^{2}\overline{R})+u\left(\frac{1+3t^{2}}{2}+\frac{t^{2}(1+t^{2})|M|^{2}}{4}\right)\\
&\qquad =\frac{\partial}{\partial x^{i}}\left(\frac{1}{2}u^{2}g^{ij}\frac{\partial u}{\partial x^{j}}\right)-\left(ug^{ij}\frac{\partial u}{\partial x^{i}}\frac{\partial u}{\partial x^{j}}\right)\\
&\qquad\qquad-\left[\frac{\partial}{\partial x^{i}}\left(\frac{1}{2\sqrt{|g|}}\right)\right]\left(u^{2}\sqrt{|g|}g^{ij}\frac{\partial u}{\partial x^{j}}\right) -\frac{u^{3}}{4}(R_{g(t)}-t^{2}\overline{R})\\
&\qquad\qquad+u\left(\frac{1+3t^{2}}{2}+\frac{t^{2}(1+t^{2})|M|^{2}}{4}\right).\\
\end{split}\]
Then by letting $s=\log \left(\frac{t}{\sqrt{1+t^2}}\right)+1$, we get the following form
\begin{equation}
\frac{\partial u}{\partial s}=\frac{\partial}{\partial x^{i}}a_{i}(x,s,u,Du)-a(x,s,u,Du)
\end{equation}
where \[a_{i}(x,s,u,p)=\frac{1}{2}u^{2}g^{ij}p_{j},\]
\[\begin{split}
a(x,s,u,p)=ug^{ij}p_{i}&p_{j}+\left[\frac{\partial}{\partial x^{i}}\left(\frac{1}{2\sqrt{|g|}}\right)\right]\left(u^{2}\sqrt{|g|}g^{ij}p_{j}\right)\\ &+\frac{u^{3}}{4}(R_{g(t)}-t^{2}\overline{R})-u\left(\frac{1+3t^{2}}{2}+\frac{t^{2}(1+t^{2})|M|^{2}}{4}\right).
\end{split}\]
It follows from the $C^0$ estimate of $u$ that\[a_{i}p_{i}\geq C|p|^{2},\quad|a_{i}|\leq C|p|,\quad|a|\leq C(1+|p|^{2}),\] where $C$ is independent of $s$. By \cite[Theorem V.1.1]{Ladyzhenskaya:1988wu}, for any $s_0,s_1\in[1-\frac{1}{2}\log 2,1)$ with $s_0<s_1$, there are constants $\beta>0$ and $C_1>0$ independent of $s_0,s_1$, such that
\[||u||_{\beta,\beta/2;A_{[s_0,s_1]}}\leq C_1.\]
Now consider the function $v=u-1$, we have the linear parabolic equation in terms of $v$
\[\begin{split}
\frac{\partial v}{\partial s}&=\frac{u^{2}}{2}g^{ij}\frac{\partial^{2}v}{\partial x^{i}\partial x^{j}}-\frac{u^{2}}{2}\frac{\partial}{\partial x^{i}}\left(\sqrt{|g|}g^{ij}\right)\frac{\partial v}{\partial x^{j}}\\
&\qquad-\frac{u^{3}}{4}(R_{g(t)}-t^{2}\overline{R})+u\left(\frac{1+3t^{2}}{2}+\frac{t^{2}(1+t^{2})|M|^{2}}{4}\right)\\
&:=Lv-\frac{1}{4}(R_{g(t)}-t^2\overline{R})+\frac{1+3t^2}{2}+\frac{t^2(1+t^2)|M|^2}{4}\\
&=Lv+f,
\end{split}\]
where $f(x,t)=-\frac{R_{g(t)}-2}{4}+\frac{t^2(1+t^2)|M|^2}{4}+O(t^{-3})=O(t^{-3})$, since $|R_{g(t)}-2|$ and $|M|$ converge to $0$ exponentially fast. Therefore the usual Schauder interior estimates \cite[Theorem IV.10.1]{Ladyzhenskaya:1988wu} and bootstrap argument give the desired result.
\end{proof}
\begin{proof}[Proof of Theorem \ref{main theorem}]
It suffices to show that the metric $\overline{g}$ obtained from Theorem \ref{global existence theorem} is asymptotically hyperbolic. From Lemma \ref{decay Lemma}, we have the following expression of the metric $\overline{g}$:
\begin{equation*}
\begin{split}
\overline{g}&=\frac{u^2}{1+t^2}dt^2+t^2g(t)\\
&=\frac{dt^2}{1+t^2}+O(t^{-5})dt^2+t^2g(t).
\end{split}
\end{equation*}
This implies that
\begin{equation*}
\begin{split}
\overline{g}_{tt}&=\frac{1}{t^2}-\frac{1}{t^4}+\frac{\overline{g}_{tt}^{(-5)}}{t^5}+\frac{\overline{g}_{tt}^{(-6)}}{t^6}+O(t^{-7}),\\
\overline{g}_{ij}&=t^2\sigma_{ij}+O(e^{-ct})
\end{split}
\end{equation*}
where $\sigma_{ij}$ is the standard metric on the sphere $S^2$ and $\overline{g}_{tt}^{(-5)},\overline{g}_{tt}^{(-6)}\in C^\infty(\Sigma)$. By adopting the definition in \cite{Chen:2016hm}, $\overline{g}$ is asymptotically hyperbolic.
\end{proof}
\begin{corollary}
Let $(\Sigma,\sigma)$ be the $2$-sphere with the standard metric, and fix any $0<m<1$. Then by prescribing the scalar curvature $\overline{R}\equiv -6$ on $N=[1,\infty)\times\Sigma$, the metric $\overline{g}$ obtained from Theorem \ref{main theorem}, with the initial condition for the constant mean curvature $H$ on $\{1\}\times\Sigma$ as
\[H\equiv \sqrt{8(1-m)},\]
is the anti-de Sitter Riemannian Schwarzschild metric with the mass $m$.
\end{corollary}
\begin{proof}
Note that from the initial metric $(\Sigma,\sigma)$ the solution to Hamilton's modified Ricci flow is constant, i.e., $|M|\equiv 0$ and $R_{g(t)}\equiv 2$. Then from (\ref{constant K}), we have
\[K=\sup_{1\leq t<\infty}\left\{-\int_{1}^{t}\frac{2+6s^2}{4}ds\right\}=0.\]
Thus by Theorem \ref{main theorem}, there exists an asymptotically hyperbolic metric $\overline{g}$ with mean curvature $H=\sqrt{8(1-m)}$ on $\{1\}\times\Sigma$. It is easy to see that from Proposition \ref{proposition for delta} we have
\[u^{-2}(t,x)=1-\frac{1}{t(1+t^2)}\left(2-\frac{H^2}{4}\right),\]
and hence the metric on $N$ we obtained is
\[\begin{split}
\overline{g}&=\left(1+t^2-\frac{1}{t}\left(2-\frac{H^2}{4}\right)\right)^{-1}dt^2+t^2\sigma\\
&=\left(1+t^2-\frac{2m}{t}\right)^{-1}dt^2+t^2\sigma
\end{split}
\]
Notice that the boundary at $t=1$ is not totally geodesic. However, once we obtain the explicit form, we can extend this metric on $N=[1,\infty)\times\Sigma$ up to the totally geodesic boundary as $\overline{N}=[t_0,\infty)\times\Sigma$ where $t_0$ is the largest zero of the polynomial $t^3+t-2m$.
\end{proof}
\section{Rigidity and Monotonicity of the Hawking Mass}\label{hawking mass section}
In this section we prove Theorem \ref{rigidity theorem} regarding rigidity and monotonicity of the Hawking mass with the foliation we used in previous sections.
The proof basically follows an argument in \cite[Theorem 5]{Lin:2016ca}.
\begin{proof}[Proof of Theorem \ref{rigidity theorem}]
Consider $N=[1,\infty)\times\Sigma$ equipped with the metric
\begin{equation*}
\overline{g}=\frac{u^2}{1+t^2}dt^2+t^2g(t)
\end{equation*}
where $g(t)$ is the solution of the modified Ricci flow.
From (\ref{hawking mass relation}) we have
\begin{equation*}
\mathbf{M}(\overline{g})(V^{(0)})=\lim_{t\rightarrow\infty}\tilde{\mathfrak{m}}_H(\Sigma_{t}).
\end{equation*}
We compute the Hawking mass of $\Sigma_{t}=\{t\}\times\Sigma$
\begin{equation}\label{hawking mass formula}
\begin{split}
\tilde{\mathfrak{m}}_H(\Sigma_{t})&=\sqrt{\frac{|\Sigma_{t}|}{16\pi}}\left(1-\frac{1}{16\pi}\int_{\Sigma_{t}}H^2\, d\sigma_t+\frac{|\Sigma_{t}|}{4\pi}\right)\\
&=\sqrt{\frac{4\pi t^2}{16\pi}}\left(1-\frac{1}{16\pi}\int_{\Sigma}\frac{4(1+t^2)}{t^2u^2}t^2\, d\sigma+\frac{4\pi t^2}{4\pi}\right)\\
&=\frac{t(1+t^2)}{2}\left(1-\frac{1}{4\pi}\int_{\Sigma}u^{-2}d\sigma\right)\\
&=\frac{1}{4\pi}\int_{\Sigma}\frac{t(1+t^2)}{2}(1-u^{-2})d\sigma.
\end{split}
\end{equation}
Hence we have the following equality
\begin{equation}
\mathbf{M}(\overline{g})(V^{(0)})=\lim_{t\rightarrow\infty}\frac{1}{4\pi}\int_{\Sigma}\frac{t(1+t^2)}{2}(1-u^{-2})d\sigma.
\end{equation}
Now by Gauss-Bonnet theorem, we have
\begin{equation*}
\begin{split}
\frac{d}{dt}\tilde{\mathfrak{m}}_H(\Sigma_{t})&=\frac{1}{4\pi}\int_{\Sigma}\frac{3t^2+1}{2}(1-u^{-2})+\frac{t(1+t^2)}{2}2u^{-3}\frac{\partial u}{\partial t}d\sigma\\
&=\frac{1}{4\pi}\int_{\Sigma}\frac{3t^2+1}{2}+\frac{u^{-1}\Delta u}{2}-\frac{R}{4}+\frac{t^2\overline{R}}{4}+\frac{t^2(1+t^2)}{4u^2}|M|^2 d\sigma\\
&=\frac{1}{4\pi}\int_{\Sigma}\frac{(\overline{R}+6)t^2}{4}+\frac{u^{-1}\Delta u}{2}+\frac{t^2(1+t^2)}{4u^2}|M|^2 d\sigma\\
&=\frac{1}{8\pi}\int_{\Sigma}\frac{(\overline{R}+6)t^2}{2}+\frac{|\nabla u|^2}{u^2}+\frac{t^2(1+t^2)}{2u^2}|M|^2 d\sigma\geq 0\\
\end{split}
\end{equation*}
given $\overline{R}\geq -6$. Thus the condition $\mathbf{M}(\overline{g})(V^{(0)})=\tilde{\mathfrak{m}}_H(\Sigma)$ implies that $\frac{d}{dt}\tilde{\mathfrak{m}}_H(\Sigma_{t})=0$, that is, $\overline{R}=-6,|M|=0,$ and $\nabla u=0$. It follows from $|M|=0$ that $(\Sigma,g_1)$ is isometric to a standard sphere. Since $\nabla u=0$, $N$ is rotationally symmetric. From the result \cite[Theorem 3.3]{Sakovich:2017jo} by Sakovich and Sormani, if $\tilde{\mathfrak{m}}_H(\Sigma)=0$ then $N$ is isometric to a hyperbolic space or if $\tilde{\mathfrak{m}}_H(\Sigma)=m>0$ then $N$ is isometric to a Riemannian anti-de Sitter Schwarzschild manifold of mass $m$.
\end{proof}
|
1905.02428
|
\section{Motivation}
Due to current trends in distributed systems and information integration, there is an increasing need for accessing external information sources from within knowledge representation formalisms such as \emph{Answer Set Programming} (\emph{ASP}) \cite{GelfondL91}.
For instance, it might be necessary to integrate information derived from a (possibly remote) \emph{Description Logic} (\emph{DL}) ontology into the computation of an answer set. If the derivation in the ontology is relative to information in the ASP part, a bidirectional exchange between a DL reasoner and an ASP solver is required.
This kind of interaction is not provided by ordinary ASP, and pre-computing all possible derivations from the ontology and adding them to the answer set program is often not feasible.
Motivated by this, the \textsc{hex}-formalism \cite{eiter2016model} has been developed, where external sources can be referenced in a program, and are evaluated during solving.
The approach is related to \emph{SMT}, but the focus is more on techniques for evaluating general external sources represented by arbitrary computations, i.e.\ it enables an API-like approach such that a user can define plugins without expert knowledge on solver construction.
By employing \textsc{hex}, a user can, e.g., define a library function for concatenating strings, accessed via an external predicate $\&concat$.
It could be used as illustrated by the following rule, where a first name and a last name are provided, and $\&concat$ returns the full name:
$ \textit{fullname}(\textit{Full}) \leftarrow \&\textit{concat}[\textit{F},\textit{L}](\textit{Full}), \textit{firstname}(\textit{F}), \textit{lastname}(\textit{L}).$
\textsc{hex}-programs are very expressive since they enable a bidirectional exchange of information between a logic program and external sources and thus, encompasses the formalization of nonmonotonic and recursive aggregates. Consequently, \textsc{hex} is suited for a wide range of applications, but also requires sophisticated evaluation algorithms to deal with the complexity that goes along with the high expressiveness. For this reason, my thesis work aims at the design and implementation of novel integrated solving techniques for improving the
efficiency of the formalism in general, as well as for specific classes
of programs.
Another focus of my work is on new applications that leverage the provided techniques, and in turn push the advancement of the formalism.
\section{Goals of My PhD Thesis}
Challenges regarding efficient evaluation of \textsc{hex}-programs comprise the lack of a tight integration of the solving process with the evaluation of external sources and with the grounding procedure.
Accordingly, the main goals of my doctoral research are:
\begin{enumerate}
\item to design advanced reasoning techniques that improve the evaluation of \textsc{hex}-pro\-grams by tightly integrating processes which have so far been treated as mostly independent sub-problems.
\item to develop innovative applications of \textsc{hex}-programs
that utilize external atoms for integrating as well as realizing methods from the area of \emph{Machine Learning} (\emph{ML}).
\item to implement newly developed evaluation algorithms in the \textsc{hex}-program solver \textsc{dlvhex} \cite{EiterIST06}, and to investigate their performance using benchmark problems.
\end{enumerate}
\section{Background}
Here, I start by briefly summarizing the work most related to \hex\ and its applications, and I introduce the theoretical background which my thesis work is based on.
\leanparagraph{Related Work}
As there are many scenarios where it is more natural, and often more efficient, to outsource some information or computation in ASP, several approaches exist for this purpose, realizing different degrees of integration.
\textsc{dlv-ex} programs \cite{CalimeriCI07} represent an early approach, which enables bidirectional communication with an external source, and allows the introduction of new terms by \emph{value invention}.
The \textsc{clingo} system also provides a mechanism for importing the extension of user-defined predicates via function calls during grounding \cite{GebserKKS14}.
In both cases, the interaction is more restricted than in \hex{} such that, e.g., nonmonotonic aggregates cannot be expressed.
\textsc{clingo 5} \cite{GebserKKOSW16} provides generic interfaces for combining theory solving with ASP, but its semantics differs from \hex\ and the approach is targeted at system developers.
Besides, there are extensions of ASP towards
the integration of specific sources; e.g.,
the \textsc{clingcon} system \cite{OstrowskiS12} implements \emph{constraint ASP} relying on a tailored integration of a constraint solver.
The setting of \hex{} differs as its goal is to enable a broad range of users to implement custom external sources and to harness efficient solving techniques.
\textsc{hex} has been applied to a wide range of use cases. Among them are a framework for executing scheduled actions in external environments (\emph{Act}\textsc{hex} \cite{FinkGIRS13}); a system for merging belief sets based on nested \textsc{hex} programs (\emph{MELD system}, \cite{RedlEK11}); and an artificial agent able of playing the computer game \emph{Angry Birds} (\emph{Angry}\textsc{hex}, \cite{EiterFS16}).
\leanparagraph{\textsc{hex}-Programs}
\textsc{hex}-programs \cite{eiter2016model} extend ASP by allowing the use of \emph{external atoms} of the form $\&g[p_1,...,p_k](c_1,...,c_l)$ in rule bodies, where $\&g$ is an \emph{external predicate} name, $p_1,...,p_k$ are input predicate names or constants, and $c_1,...,c_l$ are output constants. The ground
semantics of an external atom $\&g[p_1,...,p_k](c_1,...,c_l)$ is given by a \emph{Boolean} $1+k+l$-ary oracle function $f_{\&g}$ s.t.\ an external atom evaluates to \emph{true} for a given assignment $\mathbf{A}$ over ordinary atoms if the oracle function returns \emph{true}, i.e.\ $f_{\&g}(\mathbf{A},p_1,...,p_k,c_1,...,c_l) = \mathbf{t}$, and to \emph{false} otherwise. The notion of satisfaction is extended to \textsc{hex}-rules and programs in the obvious way. Answer sets of a \textsc{hex}-program~$\Pi$ are those assignments $\mathbf{A}$ to ordinary atoms which are minimal models of the program consisting of all rules in $\Pi$ of which the body is satisfied under $\mathbf{A}$ (the so-called \emph{FLP-reduct} \cite{FaberLP04}, an alternative to the well-known \emph{GL-reduct}).
The basic procedure for computing the answer sets of a \textsc{hex}-program $\Pi$ consists in replacing each (ground) external atom $\&g[p_1,...,p_k](c_1,...,c_l)$ by an ordinary atom of the form $e_{\&g[p_1,...,p_k]}(c_1,...,c_l)$ and adding a guess $e_{\&g[p_1,...,p_k]}(c_1,...,c_l) \vee ne_{\&g[p_1,...,p_k]}(c_1,...,c_l) \leftarrow$ for its evaluation \cite{TUW-140622}. The result of this translation is an ordinary answer set program; and ordinary ASP solvers such as \textsc{clasp} can be used for computing \emph{model candidates}.
However, guesses for external atoms must be checked afterwards for compatibility with the external semantics.
By integrating \emph{Conflict-Driven Nogood Learning} (\emph{CDNL}) search into the \textsc{hex}-algorithm, the input-output relations can be learned from these checks in form of \emph{nogoods} to avoid wrong guesses in the future search.
Moreover, even if a model candidate complies with the answers of the corresponding oracle calls, it still needs to be checked for minimality wrt.\ the FLP-reduct.
\section{Research Progress}
In this section, I present the research results obtained since I started my PhD studies.
\subsection{Integration of Solving and External Evaluations}
In the beginning of my doctoral research, I worked on the tighter integration
of the solving process and external calls \cite{DBLP:journals/jair/EiterKRW18}, which required an extension
of the oracle semantics.
Before, oracle functions were only defined for complete inputs to external atoms, such that they could only be evaluated after the whole input was decided.
As a result, many wrong guesses could only be detected late during search and nogoods were large as they usually entailed the complete input assignment.
However, this could not be improved when using two-valued assignments since external sources might be nonmonotonic, and they are \emph{black boxes} such that theory specific techniques like in SMT cannot be applied.
We have overcome the mentioned challenges by extending the two-valued semantics to three-valued assignments that use the classical values \emph{true} and \emph{false}, and the new value \emph{unassigned}.
Based on partial assignments, we have introduced new evaluation techniques to increase the performance of \textsc{hex}-evaluation, which can be utilized in the search for model candidates as well as the search applied during checking minimality wrt.\ the FLP-reduct.
First, we have extended two-valued oracle functions to three-valued ones, which allows evaluation at any point during search under partial input.
Second, nogoods now can also be learned under partial assignments, which are often significantly smaller.
Moreover, given some input-output nogood, we obtain a set of minimal nogoods by applying a minimization procedure similar to the one in \cite{OstrowskiS12}.
As an alternative, we also incorporated the \textsc{QuickXplain} algorithm \cite{Junker04} for conflict minimization, which is more suited for nogoods that contain many irrelevant literals.
The benefit of the new solving techniques has been verified by experiments using \dlvhex.
\subsection{Integration of External Sources and Minimality Checking}
In addition to the usual minimality check of ASP,
a special minimality check wrt.\ the FLP-reduct is required during
the evaluation of \hex-programs
to avoid
cyclic justifications via external sources.
The check is a bottleneck in practice
as it often accounts for most of the time required to evaluate \hex-programs.
For this reason,
syntactic information regarding atom dependencies has been used to detect
situations where
the external minimality
check can be skipped \cite{efkrs2014-jair}.
However, this approach overapproximates the real dependencies
as a result of the black-box nature of external sources.
In our most recent work \cite{DBLP:conf/lpnmr/EKLPNMR2019}, we considered a tighter integration of minimality checking and external sources by showing how the real external dependencies can be approximated more closely
by also taking semantic dependency information into account.
The additional dependency information can be provided by a user or even generated automatically.
This brings us closer to a clear-box view of external sources,
and allows us to skip the external minimality check in more cases.
Furthermore, we
stated conditions
under which the costs for checking and generating
semantic dependency information can be reduced.
Using an experimental evaluation, we could verify that having more fine-grained information about the actual dependencies among atoms is crucial for applications where otherwise the overestimation makes the minimality check infeasible.
\subsection{Integration of Grounding and Solving}
During the second year of my PhD studies, I worked on integrating
the lazy-grounding ASP solver \emph{Alpha} \cite{DBLP:conf/lpnmr/Weinzierl17} into the
\dlvhex{} system, with the goal to achieve a tighter integration of \hex-sovling and grounding.
Lazy grounding avoids an exponential blowup of the grounding by interleaving
grounding and solving, whereby rules are grounded on-the-fly depending on the satisfaction
of their bodies.
The resulting approach exhibits promising results for classes of programs
where the grounding bottleneck of ASP is an issue \cite{DBLP:conf/ijcai/EiterKW17}.
This issue is even more challenging to tackle within the framework of \hex\
due to the need for grounding external atoms; and
nonmonotonic dependencies and value
invention (i.e., import of new constants) from external sources make
the integration nontrivial.
As a result, we needed to introduce a novel external source interface to
incrementally extend a \hex-program grounding, where new output
terms may appear \emph{during solving}. This resulted in a
novel evaluation algorithm for \hex-programs that
can incorporate lazy-grounding solvers as backend solvers.
\subsection{Applications of HEX-Programs in Machine Learning}
As \hex\ allows to integrate different formalisms,
it is well-suited for combining diverse forms of reasoning.
In this branch of my research, my goal was to exploit this strength for two new applications in the area of ML.
The first one integrates an external statistical classifier,
while the second encodes an existing approach for logic-based ML.
\subsubsection{Hybrid Classification of Visual Objects}
A basic task in \emph{Statistical Relational Learning} \cite{getoor2007introduction} is \emph{Collective Classification} \cite{SenNBG10},
which
is simultaneously finding
correct labels for a number of interrelated objects, e.g., predicting the classes of objects in a
complex visual scene.
Even if advanced algorithms for object recognition have
been developed,
they may fail unavoidably and yield ambiguous results due to few
training data, noisy inputs, or inherent ambiguity
of visual appearance.
It is then still possible to
draw on further information from the context in which
an object occurs to disambiguate its label.
We approached the Collective Classification problem in ASP by defining \emph{Hybrid Classifiers} (\emph{HC}) that combine a local classifier, which
predicts the probability of each local label
based on object features, with context constraints (weighted ASP constraints) using object relations \cite{EiterK16}.
At this, external atoms of \hex\ can be used to interface an ontology reasoner, a spatial reasoning calculus as well as the local classifier directly from within the encoding.
This has not been realized in the first version of the approach, where the integration was created ad-hoc using a pre-processing step. However, the usage of external atoms for this purpose will be described in my dissertation.
To obtain a probabilistic semantics, we embedded our encoding into the formalism $LP^{MLN}$ \cite{LeeW16}, such that an HC corresponds to an $LP^{MLN}$ program.
We showed that solutions of the resulting \emph{HC encoding} can be
obtained efficiently via a backtranslation from $LP^{MLN}$ into classical ASP with weak constraints \cite{BuccafurriLR00}, and by leveraging combinatorial optimization capabilities of ASP solvers.
Experiments wrt.\ object classification in indoor and outdoor scenes exhibit significant accuracy improvements compared to using only a local classifier.
\subsubsection{Meta-Interpretive Learning}
In the area of \emph{Inductive Logic Programming} (\emph{ILP}),
\emph{Meta-Interpretive Learning} (\emph{MIL}) is a recent approach, introduced by Muggleton et al.\ \cite{MuggletonLT15}, that learns logic programs from examples and background knowledge
by instantiating meta-rules.
The Metagol system \cite{metagol} efficiently
solves MIL-problems by relying on the query-driven search of
Prolog.
Its focus on positive examples, however, effects that Metagol
can detect the derivability of negative examples only at a later check,
which can severely hit performance.
Viewing MIL-problems as
combinatorial search problems, they can alternatively be solved by
employing ASP, which may result in performance
gains as a result of efficient conflict propagation.
By employing modern ASP solvers, violations of
negative examples can potentially be propagated earlier.
However, a
straightforward ASP-encoding of MIL results in a huge search space due
to a lack of procedural bias and the need for grounding. To address
this challenge, we have encoded MIL in the \hex-formalism \cite{DBLP:journals/tplp/KaminskiEI18}.
Our encoding utilizes external atoms to outsource background knowledge that defines manipulations of complex
terms such as lists and strings, which is easy to realize in Prolog but less supported by ASP.
Moreover, we identified a class of MIL-problems which can be solved efficiently by using our \hex-encodings, and showed empirically that the performance can be increased compared to Metagol by employing \hex.
In addition, by abstracting from term manipulations in the encoding
and by exploiting the \hex-interface mechanism, the import of
constants can be entirely avoided in order to mitigate the grounding
bottleneck.
\section{Future Work}
The overarching theme of my thesis is to tightly integrate different mechanisms employed during
\hex-solving,
and a number of new evaluation techniques has been developed for this purpose.
However, there are several further ways in which these techniques can be
combined, extended and exploited for different parts of solving in the future.
First, while we have only employed simple heuristics for deciding the frequency
of external evaluations during \hex-solving, dynamic heuristics could also be used,
where the frequency is adjusted according to the amount of information gained from previous
calls.
Second, our \hex-algorithm that exploits lazy-grounding could be combined with a pre-grounding
algorithm, where the respective grounding mechanisms are applied for different modules of a \hex-program
based on their properties.
Moreover, additional semantic dependency information, which we used for deciding the necessity of the external minimality check, is also valuable for reducing the number of external evaluations during the model search and grounding, and could be utilized there as well.
\bibliographystyle{plain}
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1309.5553
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\section{Introduction}
\label{sec:intro}
If a group $G$ has a finite $K(G, 1)$
and does not contain any Baumslag-Solitar groups, is $G$ hyperbolic?
(See \cite{Questions}.)
This is one of the most famous questions on hyperbolic groups.
Probably, many people expect that the answer is negative,
and it would be better to restrict our attention to some good class of groups.
In this paper we consider automatic groups.
If an automatic group $G$ does not contain any $\mathbb{Z} + \mathbb{Z}$ subgroups,
is $G$ hyperbolic?
Our problem is listed in \cite{openproblem} and attributed to Gersten.
Note that the class of all automatic groups contains
the class of all hyperbolic groups, all virtually abelian groups
and all geometrically finite hyperbolic groups \cite{MR1161694}.
A geometrically finite hyperbolic group is, in some sense,
similar to hyperbolic groups, but it might contain a $\mathbb{Z}+\mathbb{Z}$ subgroup.
Thus the class of automatic groups is a nice target
to consider the original question mentioned before.
Let us recall some related works very briefly.
If the group is the fundamental group of a closed $3$-manifold,
our question corresponds to the so-called ``weak hyperbolization''
of $3$-manifolds \cite{MR1362788}.
Also, D.~Wise proved that if the group satisfies
the small cancellation condition $B(6)$,
then the above question is answered affirmatively \cite{MR2053602}.
P.~Papasoglu proved that the Cayley graph of a group
which is semihyperbolic but not hyperbolic
contains a subset quasi-isometric to $\mathbb{Z} + \mathbb{Z}$ \cite{MR1459144}.
In this paper, we define the notion of ``$n$-tracks of length $n$'',
which suggests a clue of the existence of $\mathbb{Z} + \mathbb{Z}$ subgroup,
and show its existence in every non-hyperbolic automatic groups
with mild conditions.
As an application, we will show the next theorem:
{\bf Theorem\ \ref{thm:cube}\ }
{\it Let $G$ be a group acting effectively, cellularly,
properly discontinuously and cocompactly
on a CAT(0) cube complex $X$.
If each hyperplane in $G\backslash X$ embeds
and does not self-osculate,
and $G$ is not word hyperbolic,
then $G$ contains $\mathbb{Z} + \mathbb{Z}$ as a subgroup.}
We remark that the assumption ``no self-osculating hyperplanes''
can be made weaker.
See section $4$ for the precise conditions we need.
See also Sageev and Wise \cite{MR2821442}.
They considered a similar problem for groups acting on CAT(0) cube-complex,
and introduced the notion of ``facing triple''.
We do not know the relation between our condition ``no direct self-contact''
in section 4
and theirs.
This paper is organized as follows.
In section $2$, we review definitions and some properties
of hyperbolic groups and automatic groups.
In section $3$, we introduce the notion of ``$n$-tracks''
and show its existence.
In section $4$, we review automatic structure of groups
which act effectively, cellularly, properly discontinuously
and cocompactly on CAT(0) cube complex due to
Niblo--Reeves~\cite{MR1604899},
and prove our theorem mentioned before.
\section{Hyperbolic groups and automatic groups}\label{sec:handa}
In this section, we briefly review definitions
and some properties of hyperbolic groups and automatic groups.
We refer to \cite{MR1161694} for the general theory.
Let $G$ be a finitely generated group with a set of generators $A$.
In this paper, we will always assume that $A^{-1} = A$.
The \textit{Cayley graph} $\Gamma := \Gamma(G, A)$ of $G$
with respect to $A$ is a directed, labeled graph defined as follows:
the set of vertices is $G$ itself.
For $g, h \in G$, there is a directed edge $(g h)$,
source $g$ and target $h$, with label $a \in A$ if and only if $g a = h$.
Let $w$ be a word over $A$.
A \textit{prefix} of a word is any number of leading letters of that word.
We denote by $\ell(w)$ the word length of $w$
and by $w(t)$ the prefix of $w$ with length $t$.
The image of $w$ in $G$ by the natural projection is denoted by $\overline{w}$.
In this paper, we denote by $\overline{w(t_{1}, t_{2})}$
the subpath of the image of $w$ by the natural projection on $\Gamma$
from the vertex $\overline{w(t_{1})}$ to the vertex $\overline{w(t_{2})}$.
The Cayley graph $\Gamma$ is a metric space by its path metric.
We denote this metric by $d(g, h)$ for $g, h \in G$.
\begin{definition}
A geodesic space is said to be \textit{hyperbolic}
(in the sense of Gromov \cite{MR919829})
if there is a number $\delta > 0$ such that,
for any triangle $\triangle x y z$ with geodesic sides,
the distance from a point $u$ on one side to the union of the other two sides
is bounded by $\delta$.
A group $G$ with a set of generators $A$ is called word hyperbolic
if the Cayley graph $\Gamma$ is hyperbolic.
\end{definition}
It should be noted that the definition of a word hyperbolic group
does not depend on the choice of generators.
One of the most important properties of word hyperbolic groups
is the following theorem.
\begin{theorem}\label{thm:z2nothyperbolic}
If $G$ contains a $\mathbb{Z} + \mathbb{Z}$ subgroup, then $G$ can not be word hyperbolic.
(See \cite{MR1170363}.)
\end{theorem}
Next, we recall the concept of automatic structure.
Again, let $G$ be a finitely generated group with a set of generators $A$.
We denote by $\varepsilon$ the identity element of $G$.
A special letter $\$ \not\in A$ is used
to define the automatic structure of the group.
A finite state automaton $M$ over an alphabet $A$ is a machine
that determines ``accept'' or ``reject'' for a given word over $A$.
See \cite{MR1161694} for detail.
The language given by all the accepted words of a finite state automaton $M$
is denoted by $L(M)$.
\begin{definition}
\textit{An automatic structure} on $G$ consists of
a finite state automaton $W$ over $A$
and finite state automata $M_{x}$ over
$\left(A \cup \{\$\}\right) \times \left(A \cup \{\$\} \right)$,
for $x \in A \cup \{\varepsilon\}$, satisfying the following conditions:
\begin{enumerate}
\item The natural projection from $L(W)$ to $G$ is surjective.
\item For $x \in A \cup \{ \varepsilon \}$,
we have $(w, w') \in L(M_x)$ if and only if
$\overline{w \, x\mathstrut} = \overline{w'\mathstrut}$
and both $w$ and $w'$ are elements of $L(W)$.
\end{enumerate}
For $M_{x}$ ($x \in A \cup \{\varepsilon\}$),
we think of the input $(w, w')$,
where $w = x_1 \, x_2 \cdots x_n$ and $w' = x_1' \, x_2' \cdots x_m'$
($x_i, x_j' \in A$, $i = 1, \ldots, n$ and $j= 1, \ldots, m$),
as the string $(x_1, x_1') \, (x_2, x_2') \cdots$ defined over
$\left( A \cup \{ \$ \} \right) \times \left( A \cup \{ \$ \} \right)$.
If the word length of $w$ is not equal to $w'$,
say $\ell(w) < \ell(w')$,
we use the pudding letter $\$ $ and the input for the automaton is
$(x_1, x_1') \, (x_2, x_2') \cdots (x_n, x_n') (\$, x_{n+1}') \cdots (\$, x_m')$.
\end{definition}
$W$ is called the {\em word acceptor},
and each $M_{x}$ is called a \textit{compare automaton}
for the automatic structure.
An automatic group is one that admits an automatic structure.
\begin{lemma}[Lemma 2.3.2 of \cite{MR1161694}] \label{lem:kfellow}
If $G$ has an automatic structure,
there is a constant $k$ with the following property:
If $(w_1, w_2)$ is accepted by one of the automata
$M_x$, for $x \in A \cup \{\varepsilon\}$,
then $d(\overline{w_1(t)} ,\overline{w_2(t)}) < k$
for any integer $t\geq 0$.
\end{lemma}
Such a number $k$ is called a {\em fellow traveler's constant}
for the structure.
We define some properties of automatic structures which will be assumed later.
\begin{definition}
Let $W$ be the word accepter of an automatic structure of a group $G$
with generators $A$.
\begin{enumerate}
\item The automatic structure is \textit{prefix closed} if, for every
$w \in L\left( W \right)$, any prefix $w\left( t \right) \, \left( 0
\leq t \leq \ell\left( w \right) \right)$ is an element of
$L\left( W \right)$.
\item The automatic structure has the \textit{uniqueness property} if
the natural projection from $L\left( W \right)$ to $G$ is injective,
thus bijective.
\item The group is \textit{weakly geodesically automatic} if any word
$w \in L\left( W \right)$ is a geodesic with respect to path metric
of $\Gamma$.
\item The group is \textit{strongly geodesically automatic} if
$L\left( W \right)$ is equal to the set of all geodesic words.
\end{enumerate}
\end{definition}
In this paper, we investigate the relation between the word hyperbolicity
and the automaticity for finitely generated groups.
Here is the basic fact about the relation.
\begin{theorem}[Papasoglu \cite{MR1346209}]
Any finitely presented group is word hyperbolic
if and only if it is strongly geodesically automatic.
\end{theorem}
In the proof of the above theorem, the following lemma was proved.
\begin{lemma}\label{lem:mthickbigon}
If a group is not hyperbolic and weakly geodesically automatic,
then for any large $M > 0$,
there exists a pair of geodesic words $(b_1, b_2)$
such that $\overline{b_1 \mathstrut} = \overline{b_2 \mathstrut}$
and $d\left(\overline{b_1(r)}, \overline{b_2(r)}\right) > M$ for some $r$.
\end{lemma}
See Fig.~\ref{fig:mthick}.
We call $(b_1, b_2)$ in Fig.~\ref{fig:mthick}
\textit{$M$-thick bigon} with side $b_1, b_2$.
\section{Existence of $n$-tracks in non-hyperbolic automatic groups}
\label{sec:track}
Let $G$ be an automatic group with automatic structure
$(A, W, \{ M_x \}_{x \in A \cup \{ \varepsilon \}})$
where $A$ is the set of generators with $A^{-1} = A$,
$W$ the word acceptor
and $M_{x}$ the compare automaton for $x \in A \cup \{ \varepsilon \}$.
The following is the key concept in this paper.
\begin{definition}\label{def:track}
Let $T = \{t_1, t_2, \ldots, t_n\}$
be a set of mutually disjoint $n$ paths of length $n$ in $\Gamma$.
We call $T$ $n$-tracks of length $n$
if there exist $2 \, n$ words
$w_1, w_1', w_2, w_2',$ $\ldots, w_n, w_n'$ of $L(W)$
and a positive integer $r$ such
that $(w_i', w_{i+1})$ is accepted by some compare automaton
for $i = 1, 2,\ldots , n-1$,
and that $t_i = \overline{w_i(r, r+n) \mathstrut}
= \overline{w_i'(r, r+n) \mathstrut}$ for $i=1, 2, \ldots, n$.
See Fig.~\ref{fig:deftrack}.
\end{definition}
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\caption{$4$-track $T=\{t_1, t_2, t_3, t_4\}$ and its related paths}
\label{fig:deftrack}
\end{figure}
The purpose of this section is to prove the following theorem.
\begin{theorem}\label{thm:track}
Let $G$ be a weakly geodesically automatic group
whose automatic structure $(A, W, \{ M_x \}_{x \in A \cup \{\varepsilon\}})$
is prefix closed and has the uniqueness property.
If $G$ is not hyperbolic,
then it contains $n$-tracks of length $n$ for any $n > 0$.
\end{theorem}
\begin{proof}
Let $k$ be a fellow traveler's constant for the automatic structure
and set $M = 2 \, k \, (n+1)^{2}$.
By Lemma~\ref{lem:mthickbigon}, there exists a $M$-thick bigon in $\Gamma$.
We denote by $b_1$ and $b_2$ the two sides of this $M$-thick bigon,
and by $e$ and $g$ the two end points.
Without loss of generality,
we may assume that $e$ is the identity vertex $\varepsilon$.
Since the automatic structure is weakly geodesically automatic,
there exists a word $p_0$ in $L(W)$
whose image $\overline{p_0}$ in $\Gamma$ is a geodesic from $e$ to $g$.
Then, at least one of two bigons $(p_0, b_1)$ and $(p_0, b_2)$ is $(M/2)$-thick.
We denote this bigon by $B = (p_0, b)$.
(See Fig.~\ref{fig:mthick})
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\label{fig:mthick}
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By definition, we can find paths $p_i \in L(W)$ from $e$
to $\overline{b(\ell(b) - i)}$ for $i = 0, 1, 2, \ldots, \ell(b)$.
Write $P = \{ p_i \}_{i = 0}^{\ell(b)}$.
Since the automatic structure is weakly geodesically automatic,
each $p_i \in P$ is geodesic
and $\ell(p_i) = \ell(b(\ell(b)-i)) = \ell(b) - i$.
We claim that the intersection of two distinct paths $p_i$ and $p_j$
($i \neq j$) of $P$ is their common prefix
(possibly the identity vertex $e$) only.
To see this, suppose that $\overline{p_i}$ and $\overline{p_j}$ in $\Gamma$
have an intersection $\overline{p_i(t_i)} = \overline{p_j(t_j)}$ in $G$.
Since the automatic structure is prefix closed,
both prefixes $p_i(t_i)$ of $p_i$ and $p_j(t_j)$ of $p_j$ are in $L(W)$.
Then, uniqueness property implies that $p_i(t_i) = p_j(t_j)$
(thus $t_i = t_j$) and the claim is proved.
Since $B = (p_0, b)$ is $(M/2)$-thick bigon,
there exists a number $r$ such that $d(p_0(r), b(r)) \geq M/2$.
Let $\Lambda_j$ \, ($j = 0,1,2, \ldots$) be a graph
whose vertex set $V_j$ is the subset
$\left\{\overline{p_i(r+jn)} \, \mathstrut
\vert \, \ell(p_i) \geq r+jn \right\}$ of $V(\Gamma)$,
and whose edge set $E_j$ is
$\left\{ \left( \overline{p_i(r+jn)},\, \overline{p_{i+1}(r+jn)} \right) \,
\mathstrut \vert \, i = 0, 1, 2, \ldots , \ell(b)-(r+jn)-1 \right\}$.
Let $\gamma_j$ be a shortest simple path in $\Lambda_j$
from $\overline{p_0(r+jn)}$ to $\overline{p_{\ell(b)-(r+jn)}(r+jn)}$,
and $\gamma_J$ be the longest path
in $\left\{ \gamma_0, \gamma_1, \gamma_2, \ldots \right\}$.
We set $R = r+Jn$.
Let $\gamma_{J+1}^0, \gamma_{J+1}^1, \gamma_{J+1}^2, \ldots ,
\gamma_{J+1}^{\ell(\gamma_{J+1})}$ be the geodesic paths in $L(W)$
from $e$ to the vertices of $\gamma_{J+1}$.
Note that $\gamma_{J+1}^0=p_0(R+n)$
and $\gamma_{J+1}^{\ell(\gamma_{J+1})}=p_{\ell(b)-R+n}(R+n)$.
Let $t^i = \overline{\gamma_{J+1}^i(R, R+n)}$.
By construction, $t^i$ and $t^{i+1}$ are subpaths of $p_m$ and $p_{m+1}$
respectively for some $m$,
and $(p_m, p_{m+1})$ is accepted by $\mathcal M$.
Let $T' = \left\{t^0, t^1, \ldots , t^{\ell(\gamma_{J+1})} \right\}$.
If consecutive $n$ paths in $T'$ are mutually disjoint,
then they give $n$-tracks of length $n$.
Note that if some of $T'$ intersect each other,
this gives branches in the union of $T'$ by the above claim.
Let $y$ be the number of branches in the union of the paths
$t^0, t^1, \ldots , t^{\ell(\gamma_{J+1})}$.
(See thick curves in Fig.~\ref{fig:ntrack2}.)
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\caption{Finding $n$-track of length $n$.}
\label{fig:ntrack2}
\end{figure}
We claim that $y \leq n$.
To see this, let $\widehat{\gamma_{J+1}}$ be the image of $\gamma_{J+1}$
by the natural projection $\pi$ from $\Lambda_{J+1}$ to $\Lambda_{J}$,
that is, $\pi(\overline{\gamma_{J+1}^i(R+n)}) = \overline{\gamma_{J+1}^i(R)}$.
We have $\ell(\widehat{\gamma_{J+1}}) \leq \ell(\gamma_{J+1}) - y$.
Since there are $n$ endpoints $\overline{p_{\ell(b)-R}(R)}$,
$\overline{p_{\ell(b)-(R+1)}(R+1)}$, $\ldots$,
$\overline{p_{\ell(b)-(R+n-1)}(R+n-1)}$ of $p_i$'s
between $\Lambda_J$ and $\Lambda_{J+1}$,
and $\gamma_{J}$ is the shortest in $\Lambda_{J}$,
we have $\ell(\gamma_{J}) \leq \ell(\widehat{\gamma_{J+1}}) + n$.
Since $\gamma_J$ is the longest in
$\left\{ \gamma_0, \gamma_1, \gamma_2, \ldots \right\}$,
it follows that $\ell({\gamma_{J+1}}) \leq \ell(\gamma_J)$.
Therefore
$\ell(\widehat{\gamma_{J+1}}) + y \leq \ell(\gamma_{J+1})
\leq \ell(\gamma_{J}) \leq \ell(\widehat{\gamma_{J+1}}) + n$,
thus we have $y \leq n$ and the claim is proved.
Recall that we set $M = 2k(n+1)^2$ at the beginning of this proof,
where $k$ is a fellow traveler's constant.
Hence we have $\ell(\gamma_J) \geq \ell(\gamma_0) > M/2k = (n+1)^2$.
Since there are at most $n$ branches in
$T'$ and the number of elements in $T'$ is $(n+1)^2$,
there exist consecutive $n$ paths in $T'$ with the desired property,
and the theorem is proved.
\end{proof}
\section{CAT(0) cube complexes}
\label{sec:cube}
Does the existence of $n$-track of length $n$ for any $n$
imply the existence of $\mathbb{Z} + \mathbb{Z}$ subgroup?
We do not have the complete answer.
But, as an application of the theorem in the previous section,
we give a partial answer to this question
for the groups acting on CAT(0) cube complexes.
See \cite{MR2605177}, for CAT(0) and its relation to hyperbolicity.
See also \cite{MR2949207}, \cite{MR2821442}.
\subsection{The automatic structure for groups acting on CAT(0) cube complex}
\label{subsec:NR}
In this subsection, we briefly review
the automatic structures given by
Niblo and Reeves \cite{MR1604899}.
An $n$-cube is a copy of $[-1,1]^n$.
A cube complex is obtained from a collection of cubes
of various dimensions by identifying certain subcubes.
A flag complex is a simplicial complex
with the property that every finite set of pairwise
adjacent vertices spans a simplex.
Let $X$ be a cube complex.
The link of a vertex $v$ in $X$ is a complex built from simplices
corresponding to the corners of cubes adjacent to $v$.
\begin{definition}
A cube complex $X$ is nonpositively curved
if, for each vertex $v$ in $X$, $\text{link}(v)$ is a flag complex.
\end{definition}
Gromov \cite{MR919829} showed that a cube complex is CAT(0)
if and only if it is simply connected and nonpositively curved.
Many groups studied in combinatorial group theory act
properly and cocompactly on CAT(0) cube complexes.
Let us recall the definition of hyperplane for cube complex.
Our reference here is \cite{MR2377497}.
See also \cite{MR2979855}.
A midplane in a cube $[-1,1]^n$ is the subspace
obtained by restricting exactly one coordinate to $0$.
Given an edge in a cube, there is a unique midplane
which cuts the edge transversely.
A hyperplane $H$ of a cube complex $X$
is obtained by developing the midplanes in $X$,
i.e., identifying common subcubes of midplanes which cuts the same edge.
These edges are said to be dual to $H$.
This is a basic fact about hyperplane.
\begin{lemma}[Proposition 2.7 in \cite{MR1604899}]\label{lem:sep}
Every hyperplane in CAT(0) cube complex $X$ separates $X$ into exactly two components.
\end{lemma}
Each component is referred to as the halfspace associated with $H$.
Let $X$ be a CAT(0) cube complex and
consider a sequence of cubes $\{C_i\}_0^n$ in $X$,
each of dimension at least 1,
such that each cube meets its successor in a single vertex
$\tilde v_i = C_{i-1} \cap C_i$.
This sequence is called a cube-path
if $C_i$ is the cube of minimal dimension containing $\tilde v_i$ and $\tilde v_{i+1}$.
Let $\tilde v_0$ to be the vertex of $C_0$, which is diagonally opposite to $\tilde v_l$,
and $\tilde v_{n+1}$ the vertex of $C_n$, diagonally opposite to $\tilde v_n$.
$\tilde v_0$ is called the initial vertex and $\tilde v_{n+1}$ the terminal vertex.
For a cube $C\in X$, $St(C)$ is the union of all cubes
which contain $C$ as a subface (including C itself).
\begin{definition}[Definition 3.1 in \cite{MR1604899}]
A cube-path is called a normal cube-path if $C_i \cap St(C_{i-1}) = \tilde v_i$.
\end{definition}
\begin{lemma}\label{lem:unqp}
Given two vertices $\iota, \tau\in V(X)$,
there is a unique normal cube-path from $\iota$ to $\tau$.
(Proposition 3.3 in \cite{MR1604899}).
A normal cube-path achieves the minimum length among
all cube-paths joining the endpoints
(See remark in section 3 in \cite{MR1604899} and \cite{ReevesPHD}.)
\end{lemma}
\begin{remark}[Remark at the end of section 3 in \cite{MR1604899}]\label{rem:NR3}
Given a vertex $\tilde v$ on a normal cube-path, which terminates at $\tau$,
the cube following $\tilde v$ is spanned by the planes
which meet $St(\tilde v)$ and separate $v$ from $\tau$.
\end{remark}
Let $X$ be a CAT(0) cube complex, and $V(X)$ its vertex set.
Let $G$ be a group acting effectively, cellularly, properly
discontinuously and cocompactly on $X$.
Let $G\backslash X$ denote the quotient
of the complex $X$ by the action of $G$.
The fundamental groupoid $\pi(G\backslash X)$ is the groupoid
whose objects are the points of $G\backslash X$ and morphisms
between points $v, v'$ are homotopy classes of paths in $G\backslash X$
beginning at $v$ and ending at $v'$.
The multiplication in $\pi(G\backslash X)$
is induced by composition of paths.
A directed cube is a cube
with two ordered diagonally opposite vertices specified.
Let $A$ be the set of
homotopy classes of the diagonal of all directed cubes in $G\backslash X$.
The correspondence between $A$ and directed cubes in $G\backslash X$
is one to one.
The directed cubes in $X$ can be labelled equivariantly by
(the lifts of) $A$,
so each cube-path in X defines a word in $A^*$.
Let $\mathcal L$ be the subset of $A^*$
which corresponds to normal cube-paths.
\begin{lemma}\label{lem:as}
Let $A$ and $\mathcal L$ be as above. Then we have:
\begin{enumerate}
\item There exists an isometry between $\pi(G\backslash X)$ with the
word metric given by $A$ and $V(X)$ with the metric given by
normal cube paths.
(Lemma 4.1 in \cite{MR1604899})
\item $\mathcal L$ is regular over $A$.
(Proposition 5.1 in \cite{MR1604899})
\item $\mathcal L$ satisfies 1-fellow travel property.
(Proposition 5.2 in \cite{MR1604899})
\end{enumerate}
In particular, $({A, L})$ induces an automatic structure for
$\pi(G\backslash X)$. (See Theorem 5.3 in \cite{MR1604899})
This structure is prefix closed, weakly geodesically automatic
with uniqueness property. ( (1) and Lemma~\ref{lem:unqp} )
\end{lemma}
The set of states of (non-deterministic) finite-state automaton
for $\mathcal L$ is $A$. (Proposition 5.1 in \cite{MR1604899})
Thus, There is a natural map from the set of states of the word acceptor
of $\pi(G\backslash X)$ to $G\backslash X$
by taking the tail of directed cubes.
In this section, for vertices $\tilde{v}, \tilde{u}$ in $X$,
we denote by $d(\tilde{v},\tilde{u})$ the distance
given by normal-cube paths.
Let $v$ be a vertex in $G\backslash X$.
The group $G$ is realized as a subgroupoid $\pi(G\backslash X, \{v\})$
whose object is $v$ only,
and whose morphisms are all the morphisms of $\pi(G\backslash X)$
between $v$.
It is easy to construct an automatic structure
for the group $G=\pi(G\backslash X, \{v\})$
from the automatic structure for the groupoid $\pi(G\backslash X)$.
\subsection{Standard automata}
Let $G$ be a group or groupoid with automatic structure $M =
\left( A, W, \left\{ M_{x} \right\}_{x \in A \cup \left\{ \varepsilon \right\}} \right)$,
and $k$ a fellow traveler's constant for $M$.
For later purpose, we construct an automaton $\mathcal M$ from $M$.
It is called standard automata in \cite{MR1161694}
when $G$ is a group.
(See Definition 2.3.3 in \cite{MR1161694}.)
Put
\[
S':= \left\{ ( s,t,g ) \; \vert \;
s, t \in S_{W}, \: s, t \neq F_{W}, \: \ell(g) \leq k \right\}
\]
where $S_{W}$ is the state set of $W$ and $F_{W}$ is the failure state of $W$.
The state set $S$ of $\mathcal M$ is $S' \cup \{ \mbox{failure state } F \}$.
The initial state of $\mathcal M$ is $\left( s_{0}, s_0, id \right)$,
where $s_{0}$ is the initial state of $W$.
The transition function $\mu$ of $\mathcal M$ is:
\[
\mu((s,t,g), (x,y)) =
\begin{cases}
\left(\mu_W(s,x),\: \mu_W(t,y),\: x^{-1}\,g\,y\right)\;
& \mbox{ if it is in } S',\\
F & \mbox{ otherwise,}
\end{cases}
\]
where $\mu_{W}$ is the transition function of $W$.
Note that $\mu$ can be extended in a unique way
to a map $A^{\ast} \times A^{\ast} \to S$ and we also denote it by $\mu$.
\subsection{Groups acting on CAT(0) cube complexes}
Let $G$ be a group acting effectively,
cellularly, properly discontinuously and cocompactly
on a CAT(0) cube complex $X$.
Let $\tilde v$ be a vertex and $H$ a hyperplane in $X$.
Let ${\mathcal H}^-$ be the halfspace associated with $H$ that does not contain $\tilde v$.
We define the distance between $\tilde v$ and $H$ as $d(\tilde v, H) = d+\frac12$,
where $d = \min \{d(\tilde v, \tilde v') | \tilde v' \in {\mathcal H}^-\} - 1$.
We denote by $N(H) \cong H\times [-1,1]$ the cubical neighborhood of $H$.
\begin{lemma}\label{lem:separate}
If $d(\tilde v, H) \geq \frac32$, there exists a hyperplane $H'$
such that $d(\tilde v,H')=\frac12$ and $H'$ separates $\tilde v$ and $H$.
\end{lemma}
We remark that $d(\tilde v,H')=\frac12$ if and only if $St(\tilde v)\cap H'\neq\emptyset$.
\begin{proof}
We denote the halfspace associated with $H$ that contains $\tilde v$ by ${\mathcal H}^+$.
We prove the lemma by induction on $d(\tilde v, H)$.
Suppose that $d(\tilde v, H) = \frac32$.
Let $\tilde v'$ be a vertex in ${{\mathcal H}^+} \cap N(H)$
such that $d(\tilde v, \tilde v') = 1$.
Let $C_1$ be the cube spanned by $\tilde v'$ and $\tilde v$.
There exists an edge $e$ in $C_1$ adjacent to $\tilde v'$, not contained in $N(H)$
such that the hyperplane $H'$ defined by $e$ (i.e., dual to $e$) separates $\tilde v'$ and $\tilde v$.
Then, $H'$ does not intersect $H$,
(See, for example, the proof of Lemma 2.14 in \cite{MR1604899}.)
and $H'$ separates $\tilde v$ and $H$.
Next, suppose that $d+\frac12 = d(\tilde v, H) > \frac32$.
Let $\tilde v'$ be a vertex in ${\mathcal H}^+ \cap N(H)$
such that $d(\tilde v, \tilde v') = d$.
Let $C_1, \dots, C_d$ be the normal cube-path from $\tilde v'$ to $\tilde v$.
Denote the vertex $C_1\cap C_2$ by $\tilde v_1$.
As in the base case, there exists a hyperplane $H''$
that separates $\tilde v_1$ and $\tilde v'$ and
$H''$ does not intersect $H$.
Since $C_1, \dots, C_d$ is a normal cube-path, $H''$ separates $\tilde v$ and $H$,
and $d(\tilde v, H'') < d(\tilde v, H)$.
By induction, there exists a hyperplane $H'$ such that $d(\tilde v, H')=\frac12$
and $H'$ separates $\tilde v$ and $H''$.
Clearly, $H'$ separates $\tilde v$ and $H$ and the lemma is proved.
\end{proof}
Next lemma is our key technical lemma.
\begin{lemma}\label{lem:distance}
Let $C_0, \dots, C_n$ be a normal cube-path, and
$\tilde v_0, \dots, \tilde v_{n+1}$ the vertices of this cube path,
that is, $\tilde v_i=C_{i-1}\cap C_i$ for $i=1, \dots, n$
with $\tilde v_0$ the initial vertex and $\tilde v_{n+1}$ the terminal vertex.
Let $H$ be a hyperplane separating $\tilde v_0$ and $\tilde v_n$.
If $d(\tilde v_0, H)=d+\frac12$, then $H$ separates $\tilde v_d$ and $\tilde v_{d+1}$.
\end{lemma}
\begin{proof}
We prove the lemma by induction on $d$.
The base case $d=0$ is trivial
as $d(\tilde v_0,H)=\frac12$ and $H$ meets $St(\tilde v)$.
Then, $H$ separates $\tilde v_0$ and $\tilde v_1$ by Remark~\ref{rem:NR3}.
Now, consider the case $d>0$.
Assume that $d(\tilde v_0,H)=d+\frac12$ and $H$ does not separate $\tilde v_d$ and $\tilde v_{d+1}$.
Since $d(\tilde v_0,H)=d+\frac12$,
$H$ can not separate $\tilde v_0$ and $\tilde v_d$.
Hence, $H$ separates $\tilde v_{d+1}$ and $\tilde v_n$.
By Remark~\ref{rem:NR3}, we see that $H\cap St(\tilde v_d)=\emptyset$.
By Lemma~\ref{lem:separate}, there exists a hyperplane $H'$
such that $H'\cap St(\tilde v_d) \neq \emptyset$ and
$H'$ separates $\tilde v_d$ and $H$.
Then, $H'$ separates $\tilde v_d$ and $\tilde v_n$, and it follows that
$H'$ separates $\tilde v_d$ and $\tilde v_{d+1}$.
(See Remark~\ref{rem:NR3} again.)
By the induction hypothesis, $d(\tilde v_0, H') \geq d+\frac12$,
so we have $d(\tilde v_0, H')=d+\frac12.$
It follows that $d(\tilde v_0, H) > d(\tilde v_0, H') = d+\frac12$ and this is a contradiction.
\end{proof}
Let $\mathcal M$ be the standard automaton for the
automatic structure of the groupoid $\pi(G\backslash X)$
given in~\ref{subsec:NR}.
We use the same symbols as in the previous subsection.
Let $(s,t,g)$ be a state in $\mathcal M$.
Since $\mathcal L$ (the set of words corresponding to normal cube-paths)
satisfies 1-fellow travel property,
$g$ is in $A$ (the set of generators).
Recall that $A$ consists of directed cubes in $G\backslash X$.
We define the dimension the the state $(s,t,g)$,
denoted by $\dim(s,t,g)$, as the dimension of $g$ as a (directed) cube.
We also define $\dim(\text{failure state }F) = +\infty.$
\begin{lemma}\label{lem:mono}
For any transition $(s',t',g') = \mu((s,t,g), (x,y))$ in $\mathcal M$,
with $x\neq e$ and $y\neq e$,
we have $\dim(s',t',g') \geq \dim(s,t,g)$.
\end{lemma}
\begin{proof}
Fix $\iota\in V(X)$ as the base point of $X$.
Let $\tau_0, \tau_1$ be two points in $X$ such that
$d(\tau_0, \tau_1)=1$ and $d(\iota, \tau_0) = d(\iota, \tau_1)$.
Denote the associated vertices by
$\iota=\tilde v_0, \ldots, \tilde v_n=\tau_0$ and
$\iota=\tilde u_0, \ldots, \tilde u_n=\tau_1$.
Let $H$ be a hyperplane and put $d=d(\iota, H)$.
If $H$ separates $\tau_0$ and $\tau_1$,
then, by Remark~\ref{rem:NR3} and Lemma~\ref{lem:distance},
$H$ does not separate $\tilde v_i$ and $\tilde u_i$ for $i=0, \dots, d$,
and separates $\tilde v_i$ and $\tilde u_i$ for $i=d+1, \dots, n$.
If $H$ does not separate $\tau_0$ and $\tau_1$,
but separate $\iota$ from both $\tau_0$, $\tau_1$,
then $H$ does not separate $\tilde v_i$ and $\tilde u_i$ for all $i=0, \dots, n$,
since $H$ separates $\tilde v_d$ and $\tilde v_{d+1}$ as well as $\tilde u_d$ and $\tilde u_{d+1}$.
It follows that the dimension of the cube spanned by $\tilde v_i$ and $\tilde u_i$ is
equal to the number of hyperplanes separating $\tau_0$ and $\tau_1$
having distance to $\iota$ less than $i$.
Since this dimension is equal to the corresponding state in $\mathcal M$,
the dimensions of the states are
monotone increasing under the transitions in $\mathcal M$.
\end{proof}
For vertices $\tilde v, \tilde v'$ in $X$, we denote the set of hyperplanes
separating $\tilde v$ and $\tilde v'$ by $S(\tilde v, \tilde v')$.
When $d(\tilde v, \tilde v')=1$, we denote by $[\tilde v, \tilde v'] (\in A)$
the label of the directed cube from $\tilde v$ to $\tilde v'$.
Since our automatic structure has uniqueness property,
there is a natural map $P: X \to S_W$,
where $S_W$ is the state set of the word acceptor for $\pi_1(G\backslash X)$.
We say that vertices $\tilde v, \tilde u, \tilde v', \tilde u'$ in $X$
corresponds to a transition $(s',t',g') = \mu((s,t,g), (x,y))$ in $\mathcal M$
if $P(\tilde v)=s, P(\tilde u)=t, P(\tilde v')=s', P(\tilde u')=t',
[v,u]=g, [v,v']=x, [u,u']=y$ and $[v',u']=g'$.
\begin{lemma}\label{lem:spancube}
Suppose that the vertices $\tilde v, \tilde u, \tilde v', \tilde u'$ in $X$
corresponds to a transition $(s',t',g') = \mu((s,t,g), (x,y))$ in $\mathcal M$
with $\dim(s',t',g') = \dim(s,t,g)$.
Then, $\tilde v, \tilde u, \tilde v', \tilde u'$ span a cube in $X$
such that
$S(\tilde v, \tilde v') = S(\tilde u, \tilde u')$ and
$S(\tilde v, \tilde u) = S(\tilde v', \tilde u')$
\end{lemma}
\begin{proof}
By the argument in the proof of Lemma~\ref{lem:mono},
it is clear that $S(\tilde v, \tilde u) = S(\tilde v', \tilde u')$, and
$S(\tilde v, \tilde v') = S(\tilde u, \tilde u')$.
Since $S(\tilde v, \tilde u') = S(\tilde v, \tilde u) \cup S(\tilde u, \tilde u')$,
we have $H\cap St(\tilde v)\neq\emptyset$ for each $H\in S(\tilde v, \tilde u')$.
By Lemma 2.14 in \cite{MR1604899}, there exists a cube that this union spans,
and the lemma is proved.
\end{proof}
\begin{lemma}\label{lem:xy}
Consider two transitions
$(s',t',g') = \mu((s,t,g), (x,y))$ and
$(s'',t'',g'') = \mu((s,t,g), (x',y'))$ in $\mathcal M$,
and suppose that
$\dim(s,t,g) = \dim(s',t',g') = \dim(s'', t'', g'')$.
Then, $x=x'$ implies $y=y'$.
Similarly, $y=y'$ implies $x=x'$.
\end{lemma}
Note that the condition $\dim(s,t,g) = \dim(s',t',g') = \dim(s'', t'', g'')$
implies that both $(s',t',g')$ and $(s'',t'',g'')$ are not failure states,
whose dimension was defined as $+\infty$.
\begin{proof}
We will prove that $x=x'$ implies $y=y'$.
Note that we have $s' = s''$ in this case.
Suppose that vertices $\tilde v, \tilde u, \tilde v', \tilde u'$ in $X$
correspond to a transition $(s',t',g') = \mu((s,t,g), (x,y))$,
and vertices $\tilde v, \tilde u, \tilde v', \tilde u''$ in $X$
correspond to a transition $(s',t'',g') = \mu((s,t,g), (x',y'))$ in $\mathcal M$.
Let $C_1, C_2, C_3$, and $C_4$ be the cubes spanned by
$\{\tilde v, \tilde v'\}$, $\{\tilde v, \tilde u\}$,
$\{\tilde v, \tilde v', \tilde u, \tilde u'\}$,
and
$\{\tilde v, \tilde v', \tilde u, \tilde u''\}$,
respectively.
Then, in $link(v)$, the simplices corresponding to $C_3$ and $C_4$
are spanned by the simplices corresponding to $C_1$ and $C_2$.
Recall that $link(v)$ was a flag complex, because $X$ is nonpositively curved.
Thus, we have $C_3 = C_4$.
Hence, $u'=u''$ and we have $y = y'$.
\end{proof}
To prove the main theorem of this section,
we need a stronger conclusion than the previous lemma.
In order to state the next lemma,
let us introduce some notation. (See \cite{MR2377497} for more details.)
Let $\vec{a}, \vec{b}$ be oriented edges
having a common initial (or terminal) vertex $v$.
Oriented edges $\vec{a}$ and $\vec{b}$ are said to directly osculate at $v$
if they are not adjacent in $\text{link}(v)$.
Let $a, b$ be (unoriented) edges
having a common end point $v$.
Edges $a$ and $b$ are said to osculate at $v$
if they are not adjacent in $\text{link}(v)$.
We consider hyperplanes in $G\backslash X$.
From now on, we assume that each hyperplane in $G\backslash X$ is embedding.
A hyperplane $H$ is said to be 2-sided
if its open cubical neighborhood is isomorphic to the product $H\times (-1,1)$.
If a hyperplane is not 2-sided, then it is said to be 1-sided.
If $H$ is 2-sided, one can orient dual edges in a consistent way.
A 2-sided hyperplane is said to directly self-osculate
if it is dual to distinct oriented edges that directly-osculate.
We say that 1-sided hyperplane self-osculates
if it is dual to distinct (unoriented) edges that osculate.
In this paper, we introduce the following notion:
\begin{definition}
We say that a 2-sided hyperplane $H$ self-contacts
if there are two vertices $u, v$ such that
$d(u,v)=1$ and $H$ directly self-osculates at
$u$ and $v$.
(See Figure~\ref{fig:ws}.)
We say that a 1-sided hyperplane $H$ self-contacts
if there are two vertices $u, v$ such that
$d(u,v)=1$ and $H$ self-osculates at $u$ and $v$.
\end{definition}
\begin{figure}[htbp]
\centering
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\caption{directly self-contact}
\label{fig:ws}
\end{figure}
Let $H$ be a hyperplane in $G\backslash X$ and
$N(H)$ the cubical neighborhood of $H$.
Let $C$ be a cube in $N(H)$ and $M$ the unique midplane of $C$
that belongs to $H$.
$M$ is unique because we are assuming that $H$ embeds in $G\backslash X$.
Let $u$ and $v$ be two vertices of $C$ not separated by $M$
such that $d(u,v) = d-1$, where $d$ is the dimension of $C$.
Then, we say that $C$ is spanned by $u$, $v$ and $H$.
If there is no hyperplanes that self-contacts,
then $C$ can be determined uniquely by $u$, $v$ and $H$.
(See Figure~\ref{fig:ws}).
\begin{remark}\label{rem:413}
By definition, if a cube complex is special in the sense of \cite{MR2377497},
then each hyperplane embeds, and it has no hyperplane of self-contact,
\end{remark}
Let $P_{s, t, g}$ be the set of pairs of letters such that
it may appear in a word in $L({\mathcal M}_{s,t,g})$,
where ${\mathcal M}_{s, t, g}$ is the automaton with the same set of states
and transition as $\mathcal M$ but having the initial state $(s, t, g)$,
and accept states
\[
\{(s',t',g') | \dim(g')=\dim(g)\}.
\]
In other words, $(x,y)$ is in $P_{s, t, g}$
if there exist a sequence $(x_0,y_0), \ldots, (x_{n-1}, y_{n-1}),$ $(x, y)$
which is accepted by ${\mathcal M}_{s, t, g}$.
(Note that $P_{s, t, g}$ depends only on the strongly connected component of $\mathcal M$.)
\begin{lemma}\label{lem:cor}
If each hyperplane in $G\backslash X$ is embedding and
does not self-contact,
then, for each $(s, t, g)$ with $g\neq id$,
there exist two subsets $A', A''\subset A$ and
a one to one correspondence $f:A'\to A''$ such that
\[
P_{s, t, g} = \{ (x,y)\in A'\times A'' | f(x)=y \}
\]
\end{lemma}
\begin{proof}
$A$ and $A'$ are determined by taking first and second projections of
$P_{s, t, g}$.
It suffices to show that if there are two sequences
$(x_1, y_1), \dots, (x_n, y_n)$ and
$(x_1', y_1'), \dots, (x_m', y_m')$
both accepted by ${\mathcal M}_{s,t,g}$,
then $x_n=x_m'$ implies $y_n=y_m'$, as well as
$y_n=y_m'$ implies $x_n=x_m'$.
In this case, we can define $f(x_n)=y_n$.
We prove the former case.
Define $(s_i, t_i, g_i)$ and $(s_j', t_j', g_j')$ inductively by
\begin{align*}
(s_i, t_i, g_i) & = \mu((s_{i-1}, t_{i-1}, g_{i-1}), (x_{i}, y_{i})), \\
(s_j', t_j', g_j') & = \mu((s_{i-j}', t_{i-j}', g_{i-j}'), (x_{j}', y_{j}'))
\end{align*}
for $i=1,\dots, n$ and $j=1,\dots,m$.
Let $v, u, v_i, u_i, v_j', u_j'$ be vertices in $G\backslash X$
corresponding to $s, t, s_i, t_i, s_j'$ and $t_j'$, respectively.
(For the correspondence between the set of states of the word acceptor
and $G\backslash X$, see the paragraph after Lemma~\ref{lem:as}.)
Put $d=\dim(s,t,g)$, and
denote the hyperplanes separating $v$ and $u$ by $H_1,\dots,H_d$.
(Recall that $S(v,u)=
S(u_i,u_i)=S(v_j',u_j')$ for all $i$ and $j$.)
Let $C_k$ be the cube
spanned by $v_{n-1}$, $v_{n}$ and $H_k$ for $k=1,\dots,d$.
Recall that, by the assumption
(each hyperplane embeds and does not self-contact),
each $C_k$ is uniquely determined by $v_{n-1}$, $v_{n}$ and $H_k$.
Let $C$ be the cube spanned by $v_{n-1}$, $v_{n}$, $u_{n-1}$, and $u_{n}$.
Then, $C$ is spanned by $C_1,\dots,C_d$ ($C$ contains $C_1,\dots,C_d$),
and it is uniquely determined by
$v_{n-1}, v_n$ and $S(v,u)$.
Define $C'$ in the same way.
It is uniquely determined by
$v_{n-1}', v_n'$ and $S(v,u)$.
If $x_n=x_n'$, then we have $v_{n-1} = v_{n-1}'$, $v_n=v_n'$ and $C=C'$.
Thus, $u_{n-1} = u_{n-1}'$ and $u_n=u_n'$.
Hence we have $y_n=y_m'$.
\end{proof}
This is our main theorem in this section.
\begin{theorem}\label{thm:cube}
Let $G$ be a group acting effectively,
cellularly, properly discontinuously and cocompactly
on a CAT(0) cube complex $X$.
If each hyperplane in $G\backslash X$ is embedding and
does not self-contact
and $G$ is not word hyperbolic,
then, $G$ contains $\mathbb{Z}+\mathbb{Z}$ subgroup.
\end{theorem}
\begin{proof}
First, note that Theorem~\ref{thm:track} works
for the groupoid $\pi(G\backslash X)$.
By Lemma~\ref{lem:as},
Its Niblo--Reeves automatic structure described in the previous subsection
is prefix closed, weakly geodesic and has uniqueness property.
$X$ is not hyperbolic since $X$ and the Cayley graph of $G$ is quasi-isometric.
Hence, there exists an $n$-track of length $n$ in $X$ for any $n>0$.
For two vertices $\tilde v, \tilde v'$ in $X$ with $d(\tilde v, \tilde v')=1$,
we denote by $\dim(\tilde v, \tilde v')$ the dimension of the cube spanned by $\tilde v$ and $\tilde v'$.
(Recall that we use normal cube-paths to define metric.)
Fix a vertex $\iota$ as the base point in $X$.
Let $T = \{ t_{1}, \ldots, t_{n} \}$ be an $n$-tracks of length $n$ in $X$.
We will improve $T$ in two steps.
Step one.
The vertices in $T$ can be identified with $\{1, 2, \dots, n\} \times \{0, 1, \dots, n\}$,
where $(i,j)$ corresponds to $j$-th vertex in $t_i$,
which was denoted by $w_i(r+j)$ in the previous section.
Here, we denote this vertex by $v_{i,j}$.
We want to consider consecutive subtracks (a block) of $T$ of the form
$T' = \{i_0, i_0+1, \dots, i_0+m-1\} \times \{j_0, j_0+1, \dots, j_0+m\}$
such that
\begin{equation}\label{eqn:dim}
\dim(v_{i,j}, v_{i+1,j}) = \dim(v_{i,j+1}, v_{i+1,j+1})
\text{ for any }(i,j)\in T'.
\end{equation}
By Lemma~\ref{lem:mono}, for each $i\in\{1, \dots, n\}$,
the number of indices $j\in\{0, \dots, n\}$ with
$\dim(v_{i,j}, v_{i+1,j}) \neq \dim(v_{i,j+1}, v_{i+1,j+1})$
is smaller than the maximal dimension of cubes in $G\backslash X$.
Thus, it is easy to see that for any $m>0$,
there exists $n$ such that any $n$-tracks of length $n$ contains
an $m$-tracks of length $m$ that satisfies (\ref{eqn:dim}).
By abusing the notation,
we refer to this subtracks by the same symbol $T= \{ t_{1}, \ldots, t_{n} \}$
and denote its size by $n$.
(Note that we do not assume that
$\dim(v_{i,j}, v_{i+1,j}) = \dim(v_{i+1,j}, v_{i+2,j})$ in $T$.)
Step one is finished.
Step two.
Recall that, for two vertices $\tilde v, \tilde v'$ in $X$,
we denote by $S(\tilde v,\tilde v')$ the set of hyperplanes separating them.
By the condition (\ref{eqn:dim}), for each $i\in\{1,\dots,n-1\}$,
there exists a set of hyperplanes
${\mathfrak H}_i$ such that
$S(v_{i,j}, v_{i+1,j}) = {\mathfrak H}_i$ for any $j\in\{0,\dots,n\}$.
By Lemma~\ref{lem:distance}, for each hyperplane $H\in \cup_i {\mathfrak H}_i$,
we have $d(\iota, H) < d(\iota, v_{1,0})$.
Now,
put $m = d(v_{1,n}, v_{n,n}) + 1$ and
let $C_1,\dots,C_{m-1}$ be the normal cube-path from $v_{1,n}$ to $v_{n,n}$.
We denote the vertices of this normal cube-path by
$v_{1}', v_{2}',\dots, v_{m}'$.
For $i\in\{1,\dots,m\}$,
let $t'_i$ be the postfix (tail) of normal cube-path from the base point $\iota$ to $v_i'$
with length $m$.
We claim that $T' = \{t_1',\dots,t_m'\}$ is a $m$-tracks of length $m$.
To see this,
let $\mathfrak H_i'$ be the set of hyperplanes separating
$v_{i}'$ and $v_{i+1}'$.
Then,
\[
\bigcup_{i=1}^{m-1} {\mathfrak H}_i' = S(v_{1,n}, v_{n,n})
\subset \bigcup_{i=1}^{n-1} {\mathfrak H}_i.
\]
Thus, we have
\begin{equation}\label{eqn:d}
d(\iota, H') < d(\iota, v_{1,0})
\end{equation}
for each hyperplane $H' \in \cup_i {\mathfrak H}_i'$.
Thus, by Lemma~\ref{lem:distance},
we have $t_i' \cap t_j' = \emptyset$ if $i\neq j$, and
$d(\iota, v_i') = d(\iota, v_{1,n})$ for $i\in\{1,\dots,m\}$,
and $T'$ is a $m$-tracks of length $m$.
Moreover, (\ref{eqn:d}) implies that $T'$ satisfies the condition (\ref{eqn:dim}).
(And, needles to say, $v_{1}',\dots,v_{m}'$ was the vertices of a normal cube-path.)
The size $m$ can be smaller than $n$.
But, since $X$ is locally finite, if $n$ was large enough,
we may assume that $m=d(v_{1,n},v_{n,n})-1$ is as large as we want.
By abusing notation,
we refer to this new tracks by the same symbol $T$ and
denote its size by $n$, and vertices by $v_{i,j}$.
Step two is finished.
We denote by $t_{i,j} (\in A)$ the label of the directed edge
from $v_{i,j}$ to $v_{i, j+1}$.
Fix $i\in\{1,\dots,n\}$, and
let us consider a pair of consecutive tracks $t_i$ and $t_{i+1}$ in $T$.
By definition,
for each pair of vertices $(v_{i,j}, v_{i+1,j})$ on $t_i$ and $t_{i+1}$,
there is a corresponding state $(s_j,t_j,g_j)\in\mathcal M$.
(It is unique by the uniqueness property of the automatic structure.)
Since $\dim(v_{i,j}, v_{i+1,j})$ is constant for all $j$,
each state $(s_j,t_j,g_j)$ can be regarded as a state
in ${\mathcal M}_{s_0, t_0, g_0}$,
and clearly, $g_0\neq id$.
(For the definition of ${\mathcal M}_{s_0, t_0, g_0}$,
see paragraphs after Remark~\ref{rem:413}.)
By Lemma~\ref{lem:cor},
there exists a one to one map $f_i:A\to A$
such that $t_{i+1, j} = f_i(t_{i, j})$ for any $j$.
(In Lemma~\ref{lem:cor}, the map was from a subset $A'$ of $A$ to
another subset $A''$, but it is easy to extend this map from $A$ to $A$.
The extension is not unique but the complement of $A'$ will not be used anyway.)
By combining these maps,
for each $i, i'\in\{1,\dots,n\}$,
we have a one to one map
$f_{i,i'}:A \to A$ such that $t_{i', j} = f_{i,i'}(t_{i, j})$ for any $j$.
For vertices $\tilde v, \tilde v'\in X$,
we denote the word in $A^*$
given by the normal cube-path from $\tilde v$ to $\tilde v'$ by $[\tilde v,\tilde v']$.
Let $D$ be the maximal dimension of the cubes in $X$.
We claim that,
if $n$ (the size of tracks) was large enough,
$T$ contains a set of indices $I=\{i_0, i_1, \dots, i_D\}$ and
$J=\{j_0, j_1\}$ which satisfies the following conditions:
\begin{itemize}
\item[(C1)] $f_{i,i'}:A \to A$ is the identity map for each $i, i'\in I$.
\item[(C2)] $v_{i_0,j_0}$ and $v_{i_0,j_1}$ correspond
to the same state in the word acceptor,
\item[(C3)] $v_{i,j}$ projects to the same vertex, say $v$,
in $G\backslash X$ for all $i\in I, j\in J$.
\item[(C4)] $[v_{i,j_0}, v_{i',j_0}] = [v_{i,j_1}, v_{i',j_1}]$
for all $i,i'\in I$.
Moreover, there exists a letter $\alpha\in A$ such that
$[v_{i,j_0}, v_{i',j_0}]$ begins with $\alpha$
for any $i<i'$.
\end{itemize}
To see the claim observe that
\begin{enumerate}
\item the number of permutations $A \to A$ is finite
\item the number of states in the word acceptor is finite
\item the number of vertices in $G\backslash X$ is finite
\item the word lengths of $[v_{i,j}, v_{i',j}]$ is less than $|i-i'|$
for any $j$,
\end{enumerate}
and these numbers does not depend on the choice of $T$.
Then, it is easy to calculate the value of $n$ needed so that $T$
contains $I$ and $J$ which satisfy
the above conditions.
(At the moment,
$D$ does not have to be he maximal dimension of the cubes in $X$.)
From now on, we assume that our $T$ is large enough so that
it satisfies these conditions.
\begin{figure}[htbp]
\centering
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\caption{Finding $\mathbb{Z}+\mathbb{Z}$ subgroup: Vertical solid lines are tracks of $T$. Dashed lines are hyperplanes. We omit $b_{i_0, i_2}, b_{i_0, i_3},$ etc.
to simplify the picture.}
\label{fig:t}
\end{figure}
Now, define $a = [v_{i_0,j_0}, v_{i_0,j_1}]$
and $b_{i,i'} = [v_{i,j_0}, v_{i',j_0}]$
($= [v_{i,j_1}, v_{i',j_1}]$) for $i,i'\in I$.
We consider that these elements are in $\pi_1(G\backslash X, v) \simeq G$,
because of (C3).
Note that these elements connect vertices $v_{i,j}$ ($i\in I, j\in J$)
``vertically'' and ``horizontally.''
By (C1) and (C3), we have $ab_{i,j} = b_{i,j}a$ for any $i\in I$, $j\in J$.
By (C2), $a^n$ (with possibly some prefix) is accepted by the
word acceptor for any $n>0$.
Since the automatic structure is weakly geodesic and
has the uniqueness property.
$a$ is torsion free and $a^n$ is a normal cube-path for any $n>0$.
We claim that, at least one element of $\{b_{i,i'}\}_{i,i'\in I}$ is torsion free.
Let $H$ be a hyperplane separating $v_{i_0,j_0}$ and $v_{i_0+1,j_0}$.
(By (C4), $v_{i_0,j_0}$ and $v_{i_0+1,j_0}$ span a (directed) cube labeled $\alpha$.)
We want to consider the action
of elements in $\pi_1(G\backslash X)$ on hyperplanes.
For the sake of simplicity,
(after conjugation,) suppose that the vertex $v_{i_0,j_0}$ is the
base point of $X$ as Cayley graph of $\pi_1(G\backslash X)$.
Define $H_i$ as the image of $H$ by the action of $b_{i_0,i}$ for each $i\in I$.
By the second condition in (C4), $H_i$
intersects a cube with label $\alpha$ and separates $t_i$ and $t_{i+1}$.
It follows that $H_i \neq H_{i'}$ for any $i\neq i'$,
because $v_{1,n}, v_{2,n}, \dots, v_{n,n}$ are vertices of a normal cube-path
(after step two)
and a hyperplane separates a normal cube-path at most once
(Remark~\ref{rem:NR3}).
Since a family of pairwise intersecting hyperplanes
have a common point of intersection,
the cardinality of such a family is bounded by the maximal dimension $D$ of $X$.
(See Theorem 4.14 in \cite{MR1347406}.)
Thus, in $I=\{i_0, i_1, \dots, i_D\}$, there exist indices $i, i'$ with $i<i'$
such that $H_i$ and $H_{i'}$ do not intersect,
We claim that $b_{i,i'}$ is torsion free.
For the sake of simplicity,
(after conjugation,) suppose that the vertex $v_{i,j_0}$ is the
base point of $X$ as Cayley graph of $\pi_1(G\backslash X)$.
Then, $b_{i,i'}(H_i) = H_{i'}$,
and we have $H_i \cap b_{i,i'}(H_i) = \emptyset$.
Let ${\mathcal H}$ be a halfspace of $H_i$ containing $v_{n,n}$.
By the second condition in (C4),
$b_{i,i'}({\mathcal H})$ also contains $v_{n,n}$.
Thus, $b_{i,i'}({\mathcal H_i}) \subset {\mathcal H_i}$,
Then, the orbit of $H_i$ under any positive power of $b_{i,i'}$
is contained in ${\mathcal H_i}$.
Hence $b_{i,i'}$ is torsion free.
Therefore, at least one element of $\{b_{i,i'}\}_{i,i'\in I}$ is torsion free
and we denote this element by $b$.
Next, we claim that $\langle a, b \rangle$ is rank two.
Let $i$ be the index used to define $b=b_{i,i'}$.
Recall that positive powers of $b$ gives
a nested sequence of halfspaces
${\mathcal H}_i \supset b({\mathcal H_i}) \supset b^2 ({\mathcal H_i})
\supset \cdots$.
If $\langle a, b \rangle$ is cyclic,
then there exist $p,q$ such that $a^p = b^q$.
Since $b^q \in b^{q-1} ({\mathcal H_i})$,
we have $a^p \in b^{q-1} ({\mathcal H_i}) \subset {\mathcal H}_i$.
Recall that $a^n$ is a normal cube-path for any $n>0$.
Recall also that it stayed outside of ${\mathcal H_i}$ when it was in $[v_{i,j_0}, v_{i,j_1}]$.
But, $St(v_{i,j_0}) \cap H_i\neq\emptyset$
and this is a contradiction.
(Recall Remark~\ref{rem:NR3}.)
Thus, this subgroup is not cyclic, but is of rank two.
Hence $\langle a, b \rangle$ is $\mathbb{Z}+\mathbb{Z}$ subgroup, and the theorem is proved.
\end{proof}
Finally, we ask questions that we hope interesting.
\begin{question}
Is the condition "no hyperplane of direct self-contact" necessary?
\end{question}
\begin{question}
More systematic method is known to show that
a group acting on a space is automatic.
See \cite{MR2239447}.
Can one generalize the above result for this setting?
\end{question}
|
2110.12672
|
\section{Introduction} \label{sec:intro}
Coronal holes are areas of low plasma density and relatively low temperature in the outer atmosphere of the Sun.
They are associated with magnetic field rapidly expanding with height and the acceleration of the high-speed solar wind \citep{1975SoPh...40..351W, 2019ARA&A..57..157C}.
Coronal holes predominantly reside above unipolar areas.
The field extending from them forms the interplanetary magnetic field, and the outflowing plasma develops into the fast solar wind \citep{1973SoPh...29..505K,2002ESASP.508..361C,2009SSRv..144..383W}.
The acceleration of the fast solar wind occurs in the transition zone and the lower corona \citep{2005Sci...308..519T}.
The energy flux entering the chromosphere and the transition region, required to maintain the temperature of the corona and accelerate the fast solar wind, should be about 5$\times10^5$\,erg cm$^{-2}$\,s$^{-1}$ \citep[see reviews by][]{1978ARA&A..16..393V, 1981ARA&A..19....7K, 1990SSRv...54..377N, 1996SSRv...75..453N, 1991PPCF...33..539B,1993SoPh..148...43Z,2000PPCF...42..415J}.
Magnetohydrodynamic (MHD) waves dissipating in the upper solar atmosphere may be responsible for transporting a part of this energy.
In the matter of the energy transfer, one of the promising agents may be Alfv\'en waves.
Mainly open magnetic-field configuration in coronal holes provides favorable conditions for their propagation to considerable altitudes \citep{1998A&A...339..208B}.
Alfv\'en waves are thought to be caused by reconnections in the network; they contribute to the turbulence of the plasma flow from coronal holes \citep{1992sws..coll....1A,1993SoPh..144..155A,2018EGUGA..20.1790M}.
\citet{2007ApJS..171..520C} showed that Alfv\'en waves can be generated by granular motions \citep{2009SSRv..144..383W}.
Using \emph{Solar Dynamics Observatory}
(SDO) data, transverse oscillations were shown to be observed in coronal holes in the lower corona \citep{2011Natur.475..477M, 2014ApJ...790L...2T, 2018ApJ...852...57W}.
Nonthermal broadening of spectral lines has also been used as an indication of propagating torsional Alfv\'en waves in and under coronal holes \citep{1990ApJ...348L..77H, 2009A&A...501L..15B, 2012ApJ...751..110B, 2013ApJ...776...78H,2014AstL...40..222Z, 2016AstL...42...55K}.
Sources of the solar wind are located at the heights of the chromosphere and the transition zone \citep{2018EGUGA..20.1790M}, and yet most of the research on coronal holes concerns only their manifestations and characteristics in the upper atmosphere \citep[e.g.][]{2009A&A...499L..29B, 2020arXiv201208802B, 2009A&A...501L..15B, 2010ASSP...19..281B,2011SSRv..158..267B,2014ApJ...789..118K}, while the works studying the lower atmosphere under coronal holes have not been as numerous in the recent decade \citep{2003SoPh..217...53K, 2007Sci...318.1574D, 2007SoPh..243..143T, 2007ARep...51..773K, 2010SoPh..262...53T, 2014Sci...346A.315T, 2014AstL...40..222Z, 2016SoPh..291.1977G}.
With this article, we try to contribute to this area by analyzing the oscillations that we observe in the lower atmosphere under two coronal holes.
\section{Instruments and Data}
We carried out spectral observations for this work with the use of the ground-based \emph{Horizontal Solar Telescope} at the \emph{Sayan Solar Observatory} \citep{2004ARep...48..954K,2011SoPh..268..329K}.
The telescope allows for a spatial resolution of around 1$''$ and spectral resolution 4 to 15\,m{\AA}\ per pixel for the lines used in the observations; the temporal resolution of the series is four seconds.
The slit of the spectrograph covers a 25-arcsecond long region on the Sun’s surface.
During recording, a photoelectric guide moves the image to compensate for the Sun’s rotation.
From the spectrograms, we derived intensity, line-of-sight (LOS) velocity, and line-width signals.
The spectral lines that we used in the observations are the Si\,\textsc{i} 10827\,{\AA}\ and H$\alpha$ lines that form in the photosphere and chromosphere, respectively.
To analyze the oscillation spectral composition, we used the fast Fourier transform algorithm and the Morlet wavelet.
We calculated the confidence levels using the technique described by \citet{1998BAMS...79...61T}.
It is based on comparing dynamic spectrum characteristics with a theoretical background noise.
The method implies a white- or red-noise spectrum, although more complicated models are also used \citep{2016ApJ...825..110A}. The white-noise model is implied for our data, as the wavelet power stays roughly constant over the range of periods of interest 2\,--\,10\,minutes.
For additional analysis, we used data from the \emph{Atmospheric Imaging Assembly} \citep[AIA:][]{sdoaia} onboard the \emph{Solar Dynamics Observatory} (SDO). The instrument provides full-disk images of the Sun in several ultraviolet channels with a 0.6$''$ spatial resolution and a temporal cadence as short as 12\,seconds.
To compensate for the rotation, we used the algorithm available in the SunPy core package \citep{sunpy_community2020}.
\section{Results and Discussion}
\subsection{Intensity and LOS Velocity Oscillations}
We carried out spectral observations in the Si\,\textsc{i} 10827\,{\AA}\ and H$\alpha$ lines in the bases of two coronal holes on 12 August 2020 (80-minute series) and 22 September 2020 (105\,minutes). Figure~\ref{fig:CH-193} shows the locations of the coronal holes.
These lines form in the upper photosphere and chromosphere respectively.
Immediately after recording the coronal hole series on 22 September, we recorded a similar series outside of the coronal-hole region close to it to compare the oscillation parameters in the coronal hole and in a quiet-Sun region.
\begin{figure}
\centerline{
\includegraphics[width=7cm]{CH-193.jpg}
}
\caption{The locations of the spectrograph slit in the coronal-hole regions superimposed on the 193\,{\AA} channel images. The second panel also shows the location of the slit in a region outside the coronal hole for the quiet-Sun series.}
\label{fig:CH-193}
\end{figure}
The distribution of the Fourier oscillatory power over the length of the slit in the coronal holes (Figure~\ref{fig:along-the-slit}) shows that the power is evenly distributed along the slit with areas of a slight increase. The maximum power is at 10$''$ for the first coronal hole and at 6$''$ for the second.
\begin{figure}
\centerline{
\includegraphics[width=8cm]{Fig3.jpg}
}
\caption{Oscillation power distribution along the slit in the first (\emph{upper row}) and second (\emph{bottom row}) coronal holes.}
\label{fig:along-the-slit}
\end{figure}
For the analysis, we used wavelet spectra, which show frequencies dominating in the signal over the recording time of the series.
We constructed cross-spectra based on pairs of signals to show similar frequencies at the same temporal intervals in these signals.
They are calculated by multiplying the wavelet spectrum of one series with the complex conjugate of another.
Cross-spectra indicate the intersections of the time--frequency domains that show oscillation power in two series.
We used cross-spectra to determine frequencies that exist simultaneously at two altitude levels -- in the photosphere and chromosphere (Figure~\ref{fig:CH-wvl}), and therefore can propagate upwards (or downwards) in the atmosphere of the coronal-hole region.
\begin{figure}
\centerline{
\includegraphics[width=8cm]{Fig4.jpg}
}
\caption{Wavelet spectra and cross-spectra in the coronal hole on 22 September 2020 for different parameters in the H$\alpha$ and Si\,\textsc{i} lines in the six arcsecond point. The regions enclosed by the light contours ans dashed lines have significance greater than 99\,\% relative to the corresponding white noise.}
\label{fig:CH-wvl}
\end{figure}
For comparison, Figure~\ref{fig:QS-wvl} shows similar spectra typical of the quiet-Sun region series.
\begin{figure}
\centerline{
\includegraphics[width=8cm]{Fig5.jpg}
}
\caption{Wavelet spectra and Si\,\textsc{i}--H$\alpha$ cross-spectra for the quiet-Sun series on 22 September 2020 for different parameters in the H$\alpha$ and Si\,\textsc{i} lines. The regions enclosed by the light contours ans dashed lines have significance greater than 99\,\% relative to the corresponding white noise.}
\label{fig:QS-wvl}
\end{figure}
Oscillations with periods of four to five minutes dominate both in the photosphere and in the chromosphere of the coronal holes in all of the observed parameters.
This could indicate that these are the periods of waves propagating between the studied atmosphere layers.
By comparison, in the region of the quiet Sun, oscillations at different heights are observed at different frequencies, and the cross-spectra do not show mutual high-power domains.
In this case, one can say that direct propagation of oscillations is not observed, in contrast to coronal holes.
We determined the periods of maximum power on the diagrams of mutual-period oscillations in the coronal-hole regions, averaged over the observation area.
The maximum power of the cross-spectra falls in the ranges of periods 1.6\,minutes wide centered at 4.83 and 5.06\,minutes for the first and second coronal holes.
Earlier, \citet{2010SoPh..262...53T} also noted the prevalence of five- over three-minute oscillations in the Ca\,\textsc{ii} K and 8498\,{\AA}\ lines under coronal holes.
To estimate the parameters of wave propagation between the layers of the solar atmosphere, we measured the phase difference between the signals.
Figure~\ref{fig:V-I-phase} shows the values of the phase difference between the Si\,\textsc{i} and H$\alpha$ intensity signals as well as the Si\,\textsc{i} and H$\alpha$ velocity signals throughout the series for different periods.
\begin{figure}
\centerline{
\includegraphics[width=8cm]{Fig6.jpg}
}
\caption{\textit{Upper row}: cross wavelet power between the H$\alpha$ and Si\,\textsc{i} velocity (left) and intensity (right) signals. \textit{Bottom row}: phase differences between the signals. The areas outside of the 99\,\% significance contour are filled with white and/or shaded.}
\label{fig:V-I-phase}
\end{figure}
We estimated the average value of the phase delay for significant oscillations over all spatial points of the slit.
For the velocity signals, the value of the lag between the photosphere and chromosphere in the 5\,$\pm$\,0.5\,minute period range is 22.6\,$\pm$\,12.8\,degrees.
The intensity signals show a more significant phase difference scatter.
This may indicate that the contribution of standing waves to the observed five-minute oscillations varies during the time series.
Another explanation may be absorption by unresolved chromospheric plasma nonuniformities \citep{2009ApJ...705..272R}.
The observed phase difference may help us to assess the propagation speed.
We assume the height difference between the formation levels of the two lines for the Si\,\textsc{i} and H$\alpha$ lines to be approximately 1\,Mm \citep{2008ApJ...682.1376B,2012ApJ...749..136L}. This gives an average speed of 54\,km\,s$^{-1}$. This speed is greater than the sound speed in the chromosphere; it, however, falls within the reasonable error range given the uncertainties in the difference between the formation heights and the measured time lag.
To analyze the type of observed waves, one can compare the phase difference between the velocity and intensity signals of the same layer.
A zero phase shift between them indicates a propagating slow magnetoacoustic wave.
The diagrams in Figure~\ref{fig:V-I-phase-2} show that in the time-period domains of significant oscillations, a shift from 0 to about 100 degrees is observed.
This may confirm the assumption that the type of waves that exist at the observed heights changes during the time series.
\begin{figure}
\centerline{
\includegraphics[width=8cm]{Fig7.jpg}
}
\caption{Examples of the phase differences between the H$\alpha$ velocity and intensity signals in coronal-hole regions. The areas outside of the 99\,\% significance contour are filled with white and/or shaded.}
\label{fig:V-I-phase-2}
\end{figure}
Another type of MHD waves is the sausage mode.
When it is observed, the line-width signals should demonstrate a double frequency compared to the frequency observed in the intensity signal \citep{2013A&A...555A..74A,2017AstL...43..844K}.
However, this is not found in observations.
On the contrary, the frequencies of significant oscillations of these parameters coincide.
The distribution of dominant frequencies in the SDO channels also shows a slight increase in oscillation power in the five-minute (or 3.3\,mHz) range in the region of the coronal hole with respect to the quiet Sun (Figure~\ref{fig:SDO}).
The dominant frequencies were derived from the FFT-spectra.
To clear the images of the noise-dominated areas, we applied an image-morphology method \citep{Serra1988image} based on the assumption that the values vary to a greater extent in the signal-dominated spatial points: a $5\times5$ window is passed across an image, and the standard deviation within the window is calculated.
Then, for the points where the standard deviation is close to zero, the central point is neglected and filled with white. After this procedure, only the points of the highest variation in the signal are left in the maps.
The resulting images were smoothed using morphological dilation with a $5\times5$ disk as the structuring element.
\begin{figure}
\centerline{
\includegraphics[width=8cm]{SDO.jpg}
}
\caption{\textit{Left}: distribution of dominant frequencies in the AIA channels in the coronal-hole region; the blue contour shows borders of the coronal hole as they appear in the 193\,{\AA}\ channel. \textit{Right}: distribution of dominant frequencies in the quiet-Sun region for comparison.}
\label{fig:SDO}
\end{figure}
To a greater extent, this is seen at coronal heights (the 193\,{\AA}\ channel), but even at the height of the transition region, a denser concentration of significant oscillation areas is seen in the central part of the coronal-hole region.
\subsection{Line-Width Signals}
The observational manifestations of torsional Alfv\'en waves are usually associated with the line width oscillations available in our observations.
In cases where a magnetic tube is located at an angle to the line of sight, the rotating of the tube contributes simultaneously to the red and violet shifts of the spectral line, which leads to its broadening, while such oscillations do not affect the intensity and line-of-sight velocity signals.
Therefore, when observing true torsional Alfv\'en waves, synchronous radial-velocity or intensity signals should not accompany the line-width oscillations.
In our observations, the periods of significant oscillations in the profile line-width signals are distributed in the range from four to six minutes (Figure~\ref{fig:CH-wvl}).
In the coronal holes that we observed, however, the power distribution in the time-period diagrams approximately coincides in the intensity and line-width signals.
In addition, the phase difference between oscillations in intensity and line width in the significant oscillations domains of the diagram remains close to zero during the observation time (Figure~\ref{fig:HW-int-phase}).
\begin{figure}
\centerline{
\includegraphics[width=8cm]{Fig10.jpg}
}
\caption{\textit{Left}: cross-spectrum of the H$\alpha$ line-width and intensity signals in the coronal hole on 22 September 2020; \textit{right}: phase difference between the line-width and intensity signals is close to zero during the time series.}
\label{fig:HW-int-phase}
\end{figure}
In the wave trains of signals filtered in the range of five-minute periods, one can see that the line-width oscillations repeat the intensity signals in phase and amplitude with a high degree of accuracy (Figure~\ref{fig:wave-trains}).
\begin{figure}
\centerline{
\includegraphics[width=8cm]{Fig9.jpg}
}
\caption{Intensity and line-width signals of the of the H$\alpha$ line filtered in the 4.5\,--\,5.5 minute range in a coronal-hole region.}
\label{fig:wave-trains}
\end{figure}
From the analysis of the line-width and intensity oscillation characteristics, we can conclude that most likely both of these signals are a manifestation of the same MHD modes, since the periods and phases of the significant oscillations coincide in them.
This means that the observed oscillations in the line-width of the spectral line are not associated with the manifestations of torsional Alfv\'en waves.
The question remains open about the amplitude of the line-width oscillations, which is too large for the temperature oscillations caused by acoustic waves.
\citet{2015ApJ...799L..12D} provide another possible explanation for non-thermal variations of the line-profile widths.
This explanation suggests that non-thermal broadening of spectral lines can be caused by magnetoacoustic impacts propagating from below along vertical magnetic tubes.
In this case, the sawtooth shape of three-minute LOS-velocity oscillations in the chromosphere above a sunspot umbra indicates the presence of shock waves \citep{2006ApJ...640.1153C, 2014ApJ...786..137T, 2015LRSP...12....6K}, which, according to \citet{2015ApJ...799L..12D}, may cause periodic non-thermal broadening of spectral lines.
This suggestion needs a more detailed study of periodic three-minute variations of the spectral line-width variations above sunspot umbrae.
The waves that we observed in the lower atmosphere of coronal holes apparently cannot be classified as torsional Alfv\'en waves. We may assume that they are slow MHD waves.
\citet{2015NatCo...6.7813M}, observing the lower corona in the areas of open field lines in the \textit{Coronal Multi-Channel Polarimeter} (CoMP) data, noted an increased oscillation power in the three to five mHz range.
They attributed these oscillations to Alfv\'enic kink waves (propagating both upwards and downwards).
We observed signs of the wave propagation in a spectral range close to that of the oscillations observed in the lower corona in coronal holes, which are attributed to the manifestations of Alfv\'en waves.
\citet{2008SoPh..251..251C}, \citet{2012ApJ...751...31H}, and \citet{2015NatCo...6.7813M} showed mechanisms that may cause a transformation of the slow MHD waves ($p$-modes) into Alfv\'en waves.
Slow-mode waves are also found at coronal heights in coronal holes \citep{2011SSRv..158..267B,2014ApJ...789..118K}.
We assume that a part of the slow-mode waves from the lower atmosphere can undergo mode transformation in the upper chromosphere and serve as a source of Alfv\'en waves, which does not exclude the possibility of partial leaking without conversion.
\subsection{Estimating a Possible Input of Unresolved Flows in the Studied Signals}
The lower solar atmosphere -- especially, the chromosphere -- is a highly dynamic medium harboring, alongside with oscillations and waves, spontaneous plasma flows, which may demonstrate quasi-periodic behavior.
This makes them difficult to distinguish from purely wave processes in observations \citep{ 2010ApJ...722.1013D,2011ApJ...727...28T, 2012ApJ...759..144T}.
They are specifically challenging to differentiate in observations with a limited resolution. Such flows, however, may be identified in spectral observations by the asymmetry characteristics of the line profiles.
Recurrent variations in the line asymmetry resulting from quasi-periodic flows may add to the oscillations found in signals in LOS velocity and line-width signals.
To assess the impact of the non-wave dynamics on the signals that we use in the analysis, we studied the asymmetry of the line profiles in the observational series.
Fine unresolved flows may influence the shape of the lines, thus making an input in the LOS-velocity signals.
We measured the red--blue asymmetry profiles as in \citet{2011ApJ...738...18T}: we interpolated the line profiles ten times and subtracted the blue-wing intensity integrated over a narrow spectral range from that at the symmetrical position in the red wing roughly at the intensity level where the LOS velocity signals where taken.
Then we derived from these the variations that the changes in the asymmetry may have introduced to the velocity signals.
The measured asymmetry variations for the H$\alpha$ line are 1.5\,--\,2.2\,m{\AA}, which results in changes in the velocity signals of the order of 70\,--\,100\,m\,s$^{-1}$, while the typical amplitudes of the velocity signals in this line are 1100\,m\,s$^{-1}$.
A typical line-width oscillation amplitude in our analysis is 20\,--\,25\,m{\AA}.
This analysis suggests that the periodic changes in the asymmetry of the line profiles are much lower in magnitude than the line-width variations and LOS-velocity signals derived using the lambda-meter technique.
Thus, the unresolved flows in the aperture slit do not significantly influence these signals.
\section{Conclusion}
In this work, we analyzed the parameters of oscillations in the regions under coronal holes in the photosphere and chromosphere (the Si\,\textsc{i} 10827\,{\AA}\ and H$\alpha$ lines).
Compared to the quiet Sun, significant oscillations in a mutual range of periods was found in the coronal-hole regions at both studied levels. The range is a 1.6-minute wide band centered at 5.0 and 5.1 minutes for the first and second coronal holes.
Based on the phase-shift analysis, we observed predominantly upward propagation with an average phase shift of 22.6\,$\pm$\,12.8 degrees between the oscillations observed at the two levels.
This phase shift yields a propagation speed of 54\,km\,s$^{-1}$, which is close to the sound speed in the chromosphere.
The variations of the phase shift between the velocity and intensity signals in the lower atmosphere may indicate the presence of both standing and propagating waves over the time series. It is also possible that this variation is caused by the complications induced by non-wave phenomena such as spicules or jets.
In our data we tried to find manifestations of Alfv\'en waves under coronal holes.
As a proxy indicator of torsional Alfv\'en waves we used oscillations in the line width signals, whose frequencies found in our observations match those observed in the corona in open-field regions.
However, these line width oscillations seem to be associated with other MHD modes (we assume, slow MHD), since they accompany the intensity and LOS-velocity oscillations.
Nevertheless, physical mechanisms exist that allow both direct leakage and the transformation of the slow MHD waves that we observed in the lower atmosphere into Alfv\'en waves observed in the corona.
\acknowledgments
The reported study was funded by RFBR, project number 20-32-70076 and Project No.\,II.16.3.2 of ISTP SB RAS. Spectral data were recorded at the Angara Multiaccess Center facilities at ISTP SB RAS. We acknowledge the NASA/SDO science teams for providing the data.
We thank the anonymous reviewer for helpful remarks.
|
1905.09957
|
\section{Code}
\label{sec:code}
Code for this paper is publicly available at the following repository:\\
\url{https://github.com/jfc43/robust-attribution-regularization}
\section{Proofs}
\label{sec:proofs}
\subsection{Additional definitions}
\label{sec:additional-defs}
Let $P, Q$ be two distributions, a coupling $M = (Z, Z')$ is a joint distribution, where,
if we marginalize $M$ to the first component, $Z$, it is identically distributed as $P$,
and if we marginalize $M$ to the second component, $Z'$, it is identically distributed as $Q$.
Let $\prod(P, Q)$ be the set of all couplings of $P$ and $Q$,
and let $c(\cdot, \cdot)$ be a ``cost'' function that maps $(z, z')$ to a real value.
Wasserstein distance between $P$ and $Q$ w.r.t. $c$ is defined as
$$W_c(P, Q) = \inf_{M \in \prod(P, Q)}\left\{
\Exp_{(z, z') \sim M}\left[c(z, z')\right]\right\}.$$
Intuitively, this is to find the ``best transportation plan'' (a coupling $M$)
to minimize the expected transportation cost
(transporting $z$ to $z'$ where the cost is $c(z, z')$).
\subsection{Integrated Gradients for an Intermediate Layer}
\label{sec:IG-intermediate}
In this section we show how to compute Integrated Gradients for an intermediate layer of
a neural network. Let $h: \Real^d \mapsto \Real^k$ be a function that computes a hidden
layer of a neural network, where we map a $d$-dimensional input vector to a
$k$-dimensional output vector. Given two points $\bfx$ and $\bfx'$ for computing
attribution, again we consider a parameterization
(which is a mapping $r: \Real \mapsto \Real^d$) such that $r(0) = \bfx$,
and $r(1) = \bfx'$.
The key insight is to leverage the fact that Integrated Gradients is a
\emph{curve integration}. Therefore, given some hidden layer, one can then
naturally view the previous layers as inducing a \emph{curve} $h \circ r$
which moves from $h(\bfx)$ to $h(\bfx')$, as we move from $\bfx$ to $\bfx'$
along curve $r$. Viewed this way, we can thus naturally compute $\IG$
for $\bfh$ in a way that leverages all layers of the network.
Specifically, consider another curve $\gamma(t): \Real \mapsto \Real^k$,
defined as $\gamma(t) = h(r(t))$, to compute a curve integral.
By definition we have $f(\bfx) = g(h(\bfx))$
\begin{align*}
f(\bfx') - f(\bfx)
&= g(h(\bfx')) - g(h(\bfx)) \\
&= g(\gamma(1)) - g(\gamma(0)) \\
&= \int_0^1\sum_{i=1}^k\frac{\partial f(\gamma(t))}{\partial h_i}\gamma_i'(t)dt \\
&= \sum_{i=1}^k\int_0^1\frac{\partial f(\gamma(t))}{\partial h_i}\gamma_i'(t)dt
\end{align*}
Therefore we can define the attribution of $h_i$ naturally as
\begin{align*}
\IG^f_{h_i}(\bfx, \bfx')
= \int_0^1\frac{\partial f(\gamma(t))}{\partial h_i}\gamma_i'(t)dt
\end{align*}
Let's unpack this a little more:
\begin{align*}
\int_0^1\frac{\partial f(\gamma(t))}{\partial h_i}\gamma_i'(t)dt
&= \int_0^1\frac{\partial f(h(r(t)))}{\partial h_i}
\sum_{j=1}^d\frac{\partial h_i(r(t))}{\partial \bfx_j}r_j'(t)dt \\
&= \int_0^1\frac{\partial f(h(r(t)))}{\partial h_i}
\sum_{j=1}^d\frac{\partial h_i(r(t))}{\partial \bfx_j}r_j'(t)dt \\
&= \sum_{j=1}^d \left\{
\int_0^1\frac{\partial f(h(r(t)))}{\partial h_i}
\frac{\partial h_i(r(t))}{\partial \bfx_j}r_j'(t)dt\right\}
\end{align*}
This thus gives the lemma
\begin{lemma}
Under curve $r: \Real \mapsto \Real^d$ where $r(0) = \bfx$ and $r(1) = \bfx'$,
the attribution for $h_i$ for a differentiable function $f$ is
\begin{align}
\label{appendix:eq:intermediate-attribution-integral}
\IG^f_{h_i}(\bfx, \bfx', r) = \sum_{j=1}^d \left\{
\int_0^1\frac{\partial f(h(r(t)))}{\partial h_i}
\frac{\partial h_i(r(t))}{\partial \bfx_j}r'_j(t)dt\right\}
\end{align}
\end{lemma}
Note that (6) nicely recovers attributions for input layer, in which case $h$ is the
identity function.
\vskip 5pt
\noindent\textbf{Summation approximation.} Similarly, we can approximate the above
Riemann integral using a summation. Suppose we slice $[0, 1]$ into $m$ equal segments,
then (\ref{eq:intermediate-attribution-integral}) can be approximated as:
\begin{align}
\label{appendix:eq:intermediate-attribution-sum-approx}
\IG^f_{h_i}(\bfx, \bfx') =
\frac{1}{m}\sum_{j=1}^d
\left\{
\sum_{k=0}^{m-1}
\frac{\partial f(h(r(k/m)))}{\partial h_i}
\frac{\partial h_i(r(k/m))}{\partial \bfx_j}
r'_j(k/m)
\right\}
\end{align}
\subsection{Proof of Proposition~\ref{prop:recover-madry}}
\label{sec:proof-recover-madry}
If we put $\lambda=1$ and let $s(\cdot)$ be the $\sf sum$ function
(sum all components of a vector), then for any curve $r$ and
any intermediate layer $\bfh$, (\ref{eq:rar-objective}) becomes:
\begin{align*}
\rho(\bfx, y; \theta)
&= \ell(\bfx, y; \theta) + \max_{\bfx' \in N(\bfx, \varepsilon)}
\{{\sf sum}(\IG^{\ell_y}(\bfx, \bfx'; r))\} \\
&= \ell(\bfx, y; \theta) + \max_{\bfx' \in N(\bfx, \varepsilon)}
\{\ell(\bfx', y; \theta)
- \ell(\bfx, y; \theta)\} \\
&= \max_{\bfx' \in N(\bfx, \varepsilon)}\ell(\bfx',y; \theta)
\end{align*}
where the second equality is due to the Axiom of Completeness of $\IG$.
\subsection{Proof of Proposition~\ref{prop:recover-input-gradient-regularization}}
Input gradient regularization is an old idea proposed by
Drucker and LeCun~\cite{DL92}, and is recently used by
Ross and Doshi-Velez~\cite{RDV18} in adversarial training setting.
Basically, for $q \ge 1$, they propose
$\rho(\bfx, y; \theta)
= \ell(\bfx, y; \theta) + \lambda \|\nabla_{\bfx} \ell(\bfx, y; \theta)\|_q^q,$
where they want small gradient at $\bfx$. To recover this objective
from robust attribution regularization, let us pick $s(\cdot)$ as the
$\|\cdot\|_1^q$ function (1-norm to the $q$-th power),
and consider the simplest curve
$r(t) = \bfx + t(\bfx' - \bfx)$. With the na\"{i}ve summation approximation
of the integral $\IG^{\ell_y}_i$ we have
$\IG^{\ell_y}_i(\bfx, \bfx'; r)
\approx \frac{(\bfx'_i - \bfx_i)}{m}
\sum_{k=1}^m
\frac{
\partial \ell(\bfx + \frac{k-1}{m}(\bfx'-\bfx), y; \theta)}{\partial \bfx_i}$,
where larger $m$ is, more accurate we approximate the integral.
Now, if we put $m=1$, which is the coarsest approximation, this becomes
$(\bfx'_i - \bfx_i)\frac{\partial\ell(\bfx, y; \theta)}{\partial \bfx_i}$,
and we have $\IG^{\ell_y}(\bfx, \bfx'; \theta)
= (\bfx'-\bfx) \odot \nabla_{\bfx}\ell(\bfx, y; \theta).$
Therefore (\ref{eq:rar-objective}) becomes:
\begin{align*}
\rho(\bfx, y; \theta)
=& \ell(\bfx, y; \theta) + \lambda \max_{\bfx' \in N(\bfx, \varepsilon)}
\{\|\IG^{\ell_y}(\bfx, \bfx'; \theta)\|_1^q\} \\
\approx &\ell(\bfx, y; \theta)
+ \lambda \max_{\bfx' \in N(\bfx, \varepsilon)}\{
\|(\bfx'-\bfx) \odot \nabla_{\bfx}\ell(\bfx, y; \theta)\|_1^q\}
\end{align*}
Put the neighborhood as $\|\bfx' - \bfx\|_p \le \varepsilon$
where $p \in [1, \infty]$ and $\frac{1}{p} + \frac{1}{q} = 1$.
By H\"{o}lder's inequality,
$\|(\bfx'-\bfx) \odot \nabla_{\bfx}\ell(\bfx, y; \theta)\|_1^q
\le \|\bfx' - \bfx\|_p^q\|\nabla\ell(\bfx, y;\theta)\|_q^q
\le \varepsilon^q \|\nabla\ell(\bfx, y;\theta)\|_q^q$
which means that
$\max_{\|\bfx'-\bfx\|_p \le \varepsilon}
\{\|(\bfx'-\bfx) \odot \nabla_{\bfx}\ell(\bfx, y; \theta)\|_1^q\}
= \varepsilon^q \|\nabla\ell(\bfx, y;\theta)\|_q^q.$
Thus by putting $\lambda = \lambda'/\varepsilon^q$,
we recover gradient regularization with regularization parameter $\lambda'$.
\subsection{Proof of Proposition~\ref{prop:regularize-by-loss-output}}
Let us put $s(\cdot) = \|\cdot\|_1$, and $\bfh = \ell_y$
(the output layer of loss function!), then we have
\begin{align*}
\rho(\bfx, y; \theta)
= &\ell_y(\bfx) + \max_{\bfx' \in N(\bfx, \varepsilon)}\{
\|\IG^{\ell_y}_{\ell_y}(\bfx, \bfx'; r)\|_1\} \\
=&\ell_y(\bfx) + \max_{\bfx' \in N(\bfx, \varepsilon)}\{
|\ell_y(\bfx') - \ell_y(\bfx)|\}
\end{align*}
where the second equality is because
$\IG^{\ell_y}_{\ell_y}(\bfx, \bfx'; r) = \ell_y(\bfx') - \ell_y(\bfx)$.
\subsection{Proof of
Proposition~\ref{prop:recover-wasserstein-prediction-robustness-objective}}
Specifically, again, let $s(\cdot)$ be the summation function and
$\lambda=1$, then we have
$\Exp_{Z, Z'}[d_{\IG}(Z, Z')]
= \Exp_{Z, Z'}[{\sf sum}(\IG_{\bfh}^\ell(Z, Z'))]
= \Exp_{Z, Z'}[\ell(Z';\theta) - \ell(Z;\theta)].$
Because $P$ and $Z$ are identically distributed, thus the objective
reduces to
\begin{align*}
&\sup_{Q;M\in \prod(P,Q)}\Big\{
\Exp_{Z, Z'}[\ell(Z;\theta) + \ell(Z';\theta) - \ell(Z;\theta)]\\
&\qquad\qquad\qquad \text{ s.t. } \Exp_{Z,Z'}[c(Z, Z')] \le \rho
\Big\} \\
=& \sup_{Q;M\in \prod(P,Q)}\left\{
\Exp_{Z'}[\ell(Z';\theta)]
\text{ s.t. } \Exp_{Z,Z'}[c(Z, Z')] \le \rho \right\} \\
=& \sup_{Q: W_c(P, Q) \le \rho}\left\{\Exp_Q[\ell(Q; \theta)]\right\},
\end{align*}
which is exactly Wasserstein prediction robustness objective.
\subsection{Proof of Theorem~\ref{thm:dist-rar-duality2}}
\label{sec:proof-dist-rar-duality2}
The proof largely follows that for Theorem 5 in~\cite{SND18}, and we provide it here for completeness.
Since we have a joint supremum over $Q$ and $M \in \prod(P, Q)$ we have that
\begin{align*}
\sup_{Q; M \in \prod(P, Q)}\left\{
\Exp_{M=(Z, Z')}\big[d^\gamma_{\IG}(Z, Z')\big]\right\}
& = \sup_{Q; M \in \prod(P, Q)}
\int [d_{\IG}(z, z') - \gamma c(z, z')] d M(z, z') \\
& \le \int \sup_{z'}\{ d_{\IG}(z, z') - \gamma c(z, z') \} d P(z) \\
& = \Exp_{z \sim P}\left[\sup_{z'}\{d^\gamma_{\IG}(z, z')\}\right].
\end{align*}
We would like to show equality in the above.
Let $\mathcal{Q}$ denote the space of regular conditional probabilities from $Z$ to $Z'$. Then
\begin{align*}
\sup_{Q; M \in \prod(P, Q)} \int [d_{\IG}(z, z') - \gamma c(z, z')] d M(z, z')
\ge \sup_{Q \in \mathcal{Q}} \int [d_{\IG}(z, z') - \gamma c(z, z') ] d Q(z'|z) d P(z).
\end{align*}
Let $\mathcal{Z'}$ denote all measurable mappings $z \rightarrow z'(z)$ from $Z$ to $Z'$. Using the measurability result in Theorem 14.60 in~\cite{rockafellar2009variational}, we have
\begin{align*}
\sup_{z' \in \mathcal{Z'}} \int [d_{\IG}(z, z'(z)) - \gamma c(z, z'(z)) ] d P(z) = \int \sup_{z' } [d_{\IG}(z, z') - \gamma c(z, z') ] d P(z)
\end{align*}
since $\gamma c - d_{\IG} $ is a normal integrand.
Let $z'(z)$ be any measurable function that is $\epsilon$-close to attaining the supremum above, and define the conditional distribution $Q(z'|z)$ to be supported on $z'(z)$. Then
\begin{align*}
\sup_{Q; M \in \prod(P, Q)} \int [d_{\IG}(z, z') - \gamma c(z, z')] d M(z, z')
& \ge \int [d_{\IG}(z, z') - \gamma c(z, z') ] d Q(z'|z) d P(z) \\
& = \int [d_{\IG}(z, z'(z)) - \gamma c(z, z'(z)) ] d P(z) \\
& \ge \int \sup_{z'} [d_{\IG}(z, z') - \gamma c(z, z') ] d P(z) - \epsilon \\
& \ge \sup_{Q; M \in \prod(P, Q)} \int [d_{\IG}(z, z') - \gamma c(z, z')] d M(z, z') - \epsilon.
\end{align*}
Since $\epsilon \ge 0$ is arbitrary, this completes the proof. \qed
\subsection{Proof of Theorem~\ref{thm:dist-rob-duality}:
Connections Between the Distributional Robustness Objectives}
\label{sec:proof-dist-rob-duality}
Let $\theta^*$ denote an optimal solution of (\ref{eq:dist-rar-objective}) and let $\theta'$ be any non-optimal solution. Let $\gamma(\theta^*)$ denote the corresponding $\gamma$ by Lemma~\ref{lem:duality-inside}, and $\gamma(\theta')$ denote that for $\theta'$.
Since $\gamma(\theta')$ achieves the infimum, we have
\begin{align}
& \Exp_{z \sim P}\left[\ell(z; \theta')
+ \lambda \sup_{z'}\{ d_{\IG}(z, z') - \gamma(\theta^*) c(z, z')\} \right]
\\
\ge &
\Exp_{z \sim P}\left[\ell(z; \theta')
+ \lambda \sup_{z'}\{ d_{\IG}(z, z') - \gamma(\theta') c(z, z')\} \right]
\\
> & \Exp_{z \sim P}\left[\ell(z; \theta^*)
+ \lambda \sup_{z'}\{ d_{\IG}(z, z') - \gamma(\theta^*) c(z, z')\} \right].
\end{align}
So $\theta'$ is not optimal for (\ref{eq:dist-rar-objective-lagrange-2}).
This then completes the proof. \qed
\begin{lemma}
\label{lem:duality-inside}
Suppose $c(z,z)=0$ and $d_{\IG}(z,z)=0$ for any $z$, and suppose
$\gamma c(z, z') - d_{\IG}(z, z')$ is a normal integrand.
For any $\rho > 0$, there exists $\gamma \ge 0$ such that
\begin{align}
& \sup_{Q; M \in \prod(P, Q)} \left\{
\Exp_{(Z,Z') \sim M}[d_{\IG}(Z, Z')] \text{ s.t. }
\Exp_{(Z,Z') \sim M}[c(Z, Z')] \le \rho \right\}
\\
= &\inf_{\zeta \ge 0}\Exp_{z \sim P} \left[
\sup_{z'}\{ d_{\IG}(z, z') - \zeta c(z, z') + \zeta \rho \} \right].
\end{align}
Furthermore, there exists $\gamma \ge 0$ achieving the infimum.
\end{lemma}
This lemma generalizes Theorem 5 in~\cite{SND18}
to a larger, but natural, class of objectives.
\begin{proof}
For $Q$ and $M \in \Pi(P,Q)$, let
\begin{align}
\Lambda_{\IG}(Q, M) & := \Exp_{(Z,Z') \sim M}[d_{\IG}(Z, Z')]
\\
\Lambda_{c}(Q, M) & := \Exp_{(Z,Z') \sim M}[c(Z, Z')]
\end{align}
First, the pair $(Q, M)$ forms a convex set, and $\Lambda_{\IG}(Q, M)$ and $\Lambda_{c}(Q, M)$ are linear functionals over the convex set.
Set $Q=P$ and set $M$ to the identity coupling (such that $(Z, Z') \sim M$ always has $Z = Z'$). Then $\Lambda_{c}(Q, M) = 0 < \rho$ and thus the Slater's condition holds. Applying standard infinite dimensional duality results (Theorem 8.6.1 in~\cite{luenberger1997optimization}) leads to
\begin{align}
& \sup_{Q; M \in \prod(P, Q); \Lambda_c(Q, M) \le \rho} \Lambda_{\IG}(Q, M)
\\
= &
\sup_{Q; M \in \prod(P, Q)}
\inf_{\zeta \ge 0} \left\{ \Lambda_{\IG}(Q, M) - \zeta \Lambda_c(Q, M) + \zeta \rho \right\}
\\
= &
\inf_{\zeta \ge 0} \sup_{Q; M \in \prod(P, Q)}
\left\{ \Lambda_{\IG}(Q, M) - \zeta \Lambda_c(Q, M) + \zeta \rho \right\}.
\end{align}
Furthermore, there exists $\gamma \ge 0$ achieving the infimum in the last line.
Now, it suffices to show that
\begin{align}
& \sup_{Q; M \in \prod(P, Q) }
\left\{ \Lambda_{\IG}(Q, M) - \gamma \Lambda_c(Q, M) + \gamma \rho \right\}
\\
= &
\Exp_{z \sim P} \left[ \sup_{z'}\{ d_{\IG}(z, z') - \gamma c(z, z') + \gamma \rho \} \right].
\end{align}
This is exactly what Theorem~\ref{thm:dist-rar-duality2} shows.
\end{proof}
\subsection{Proof of Theorem~\ref{thm:one-layer-neural-networks}}
\label{sec:proof-one-layer-neural-networks}
Let us fix any one point $\bfx$, and consider
$g(-y_i\langle \bfw, \bfx \rangle) + \lambda\max_{\bfx' \in N(\bfx, \varepsilon)}\|\IG^{\ell_y}_{\bfx}(
\bfx, \bfx'; \bfw)\|_1$. Due to the special form of $g$, we know that:
\begin{align*}
\IG^{\ell_y}_i(\bfx, \bfx'; \bfw) = \frac{\bfw_i(\bfx' - \bfx)_i}{\langle \bfw, \bfx'-\bfx \rangle}
\cdot \big(g(-y\langle \bfw, \bfx'\rangle) - g(-y\langle \bfw, \bfx\rangle)\big)
\end{align*}
Let $\Delta = \bfx'-\bfx$ (which satisfies that $\|\Delta\|_\infty \le \varepsilon)$, therefore its
absolute value (note that we are taking 1-norm):
\begin{align*}
\frac{\big|g(-y\langle \bfw, \bfx\rangle - y\langle \bfw, \Delta \rangle)
- g(-y\langle \bfw, \bfx\rangle)\big|)}{|\langle \bfw, \Delta \rangle|}
\cdot |\bfw_i\Delta_i|
\end{align*}
Let $z = -y\langle \bfw, \bfx \rangle$ and $\delta = -y\langle \bfw, \Delta\rangle$, this is further
simplified as $\frac{|g(z+\delta)-g(z)|}{|\delta|}|\delta_i|$. Because $g$ is non-decreasing,
so $g' \ge 0$, and so this is indeed $\frac{g(z+\delta) - g(z)}{\delta}$, which is the slope of the
secant from $(z, g(z))$ to $(z+\delta, g(z+\delta))$. Because $g$ is convex so the secant slopes are
non-decreasing, so we can simply pick $\Delta_i=-y\sign(\bfw_i)\varepsilon$,
and so $\delta = \|\bfw\|_1\varepsilon$, and so that $\|\IG\|_1$ becomes
\begin{align*}
|g(z+\varepsilon\|\bfw\|_1) - g(z)| \cdot \frac{\sum_i|\bfw_i\Delta_i|}{|\delta|}
&= |g(z+\varepsilon\|\bfw\|_1) - g(z)| \cdot \frac{\sum_i|\bfw_i|\varepsilon}{\|\bfw\|_1\varepsilon} \\
&= |g(z+\varepsilon\|\bfw\|_1) - g(z)| \\
&= g(z+\varepsilon\|\bfw\|_1) - g(z)
\end{align*}
where the last equality follows because $g$ is nondecreasing. Therefore the objective simplifies to
$\sum_{i=1}^mg(-y_i\langle \bfw, \bfx_i \rangle + \varepsilon\|\bfw\|_1)$,
which is exactly Madry et al.'s objective under $\ell_\infty$ perturbations.\qed
Let us consider two examples:
\noindent\emph{Logistic Regression}.
Let $g(z)=\ln(1+\exp(z))$. Then $g(-y\langle \bfw, \bfx \rangle)$
recovers the Negative Log-Likelihood loss for logistic regression. Clearly $g$ is nondecreasing
and $g'$ is also nondecreasing. As a result, adversarial training for logistic regression is exactly
``robustifying'' attributions/explanations.
\noindent\emph{Softplus hinge loss}. Alternatively, we can let $g(z) = \ln(1+\exp(1+z))$,
and therefore $g(-y\langle \bfw, \bfx \rangle) = \ln(1+\exp(1-y\langle \bfw, \bfx\rangle))$
is the softplus version of the hinge loss function. Clearly this $g$ also satisfy our requirements,
and therefore adversarial training for softplus hinge loss function is also exactly about
``robustifying'' attributions/explanations.
\section{More Details of Experiments}
\label{sec:experiments-details}
\subsection{Experiment Settings}
\label{sec:experiment-settings}
We perform experiments on four datasets: MNIST, Fashion-MNIST, GTSRB and Flower.
Robust attribution regularization training requires extensive computing power.
We conducted experiments in parallel over multiple NVIDIA Tesla V100 and NVDIA GeForce RTX 2080Ti GPUs both on premises and on cloud. Detailed experiment settings for each dataset are described below.
\subsubsection{MNIST}
\noindent\textbf{Data}.
The MNIST dataset~\cite{mnist} is a large dataset of handwritten digits. Each digit has 5,500 training images and 1,000 test images. Each image is a $28 \times 28$ grayscale. We normalize the range of pixel values to $[0, 1]$.
\noindent\textbf{Model}.
We use a network consisting of two convolutional layers with 32 and 64 filters respectively, each followed by $2 \times 2$ max-pooling, and a fully connected layer of size 1024. Note that we use the same MNIST model as \cite{madry2017towards}.
\noindent\textbf{Training hyper-parameters}.
The hyper-parameters to train different models are listed below:
\noindent\emph{NATURAL}.
We set learning rate as $10^{-4}$, batch size as 50, training steps as 25,000, and use Adam Optimizer.
\noindent\emph{Madry et al.}.
We set learning rate as $10^{-4}$, batch size as 50, training steps as 100,000, and use Adam Optimizer. We use PGD attack as adversary with random start, number of steps of 40, step size of 0.01, and adversarial budget $\epsilon$ of 0.3.
\noindent\emph{IG-NORM}.
We set $\lambda=1$, $m=50$ for gradient step, learning rate as $10^{-4}$, batch size as 50, training steps as 100,000, and use Adam Optimizer. We use PGD attack as adversary with random start, number of steps of 40, step size of 0.01, $m=10$ for attack step, and adversarial budget $\epsilon=0.3$.
\noindent\emph{IG-SUM-NORM}.
We set $\beta$ as 0.1, $m$ in the gradient step as 50, learning rate as $10^{-4}$, batch size as 50, training steps as 100,000, and use Adam Optimizer. We use PGD attack as adversary with random start, number of steps of 40, step size of 0.01, $m=10$ in the attack step, and adversarial budget $\epsilon=0.3$.
\noindent\textbf{Evaluation Attacks}.
For attacking inputs to change model predictions, we use PGD attack with random start, number of steps of $100$, adversarial budget $\epsilon$ of 0.3 and step size of $0.01$. For attacking inputs to change interpretations, we use Iterative Feature Importance Attacks (IFIA) proposed by ~\cite{GAZ17}. We use their top-k attack with $k=200$, adversarial budget $\epsilon=0.3$, step size $\alpha=0.01$ and number of iterations $P=100$. We set the feature importance function as Integrated Gradients(IG) and dissimilarity function $D$ as Kendall's rank order correlation. We find that IFIA is not stable if we use GPU parallel computing (non-deterministic is a behavior of GPU), so we run IFIA three times on each test example and use the best result with the lowest Kendall's rank order correlation.
\subsubsection{Fashion-MNIST}
\noindent\textbf{Data}.
The Fashion-MNIST dataset ~\cite{xiao2017fashion} contains images depicting wearables
such as shirts and boots instead of digits, which is more complex than MNIST dataset. The image format,
the number of classes, as well as the number of examples are
all identical to MNIST.
\noindent\textbf{Model}.
We use a network consisting of two convolutional layers with 32 and 64 filters respectively, each followed by $2 \times 2$ max-pooling, and a fully connected layer of size 1024.
\noindent\textbf{Training hyper-parameters}.
The hyper-parameters to train different models are listed below:
\noindent\emph{NATURAL}.
We set learning rate as $10^{-4}$, batch size as 50, training steps as 25,000, and use Adam Optimizer.
\noindent\emph{Madry et al.}.
We set learning rate as $10^{-4}$, batch size as 50, training steps as 100,000, and use Adam Optimizer. We use PGD attack as adversary with random start, number of steps of 20, step size of 0.01, and adversarial budget $\epsilon$ of 0.1.
\noindent\emph{IG-NORM}.
We set $\lambda=1$, $m=50$ for gradient step, learning rate as $10^{-4}$, batch size as 50, training steps as 100,000, and use Adam Optimizer. We use PGD attack as adversary with random start, number of steps of 20, step size of 0.01, $m=10$ for attack step, and adversarial budget $\epsilon=0.1$.
\noindent\emph{IG-SUM-NORM}.
We set $\beta$ as 0.1, $m$ in the gradient step as 50, learning rate as $10^{-4}$, batch size as 50, training steps as 100,000, and use Adam Optimizer. We use PGD attack as adversary with random start, number of steps of 20, step size of 0.01, $m=10$ in the attack step, and adversarial budget $\epsilon=0.1$.
\noindent\textbf{Evaluation Attacks}.
For attacking inputs to change model predictions, we use PGD attack with random start, number of steps of $100$, adversarial budget $\epsilon$ of 0.1 and step size of $0.01$. For attacking inputs to change interpretations, we use Iterative Feature Importance Attacks (IFIA) proposed by ~\cite{GAZ17}. We use their top-k attack with $k=100$, adversarial budget $\epsilon=0.1$, step size $\alpha=0.01$ and number of iterations $P=100$. We set the feature importance function as Integrated Gradients(IG) and dissimilarity function $D$ as Kendall's rank order correlation. We find that IFIA is not stable if we use GPU parallel computing (non-deterministic is a behavior of GPU), so we run IFIA three times on each test example and use the best result with the lowest Kendall's rank order correlation.
\subsubsection{GTSRB}
\noindent\textbf{Data}.
The German Traffic Sign Recognition Benchmark (GTSRB) \cite{stallkamp2012man} is a dataset of color images depicting 43 different traffic signs. The images are not of a fixed dimensions and have rich background and varying light conditions as would be expected of photographed images of traffic signs. There are about 34,799 training images, 4,410 validation images and 12,630 test images. We resize each image to $32 \times 32$. The pixel values are in range of $[0, 255]$. The dataset has a large imbalance in the number of sample occurrences across classes. We use data augmentation techniques to enlarge the training data and make the number of samples in each class balanced. We construct a class preserving data augmentation pipeline consisting of rotation, translation, and projection transforms and apply this pipeline to images in the training set until each class contained 10,000 training examples. We use this new augmented training data set containing 430,000 samples in total to train models. We also preprocess images via image brightness normalization.
\noindent\textbf{Model} .
We use the Resnet model \cite{he2016deep}. We perform per image standardization before feeding images to the neural network. The network has 5 residual units with (16, 16, 32, 64) filters each. The model is adapted from CIFAR-10 model of \cite{madry2017towards}. Refer to our codes for details.
\noindent\textbf{Training hyper-parameters}.
The hyper-parameters to train different models are listed below:
\noindent\emph{NATURAL}.
We use Momentum Optimizer with weight decay. We set momentum rate as 0.9, weight decay rate as 0.0002, batch size as 64, and training steps as 70,000. We use learning rate schedule: the first 500 steps, we use learning rate of $10^{-3}$; after 500 steps and before 60,000 steps, we use learning rate of $10^{-2}$; after 60,000 steps, we use learning rate of $10^{-3}$.
\noindent\emph{Madry et al.}.
We use Momentum Optimizer with weight decay. We set momentum rate as 0.9, weight decay rate as 0.0002, batch size as 64, and training steps as 70,000. We use learning rate schedule: the first 500 steps, we use learning rate of $10^{-3}$; after 500 steps and before 60,000 steps, we use learning rate of $10^{-2}$; after 60,000 steps, we use learning rate of $10^{-3}$. We use PGD attack as adversary with random start, number of steps of 7, step size of 2, and adversarial budget $\epsilon$ of 8.
\noindent\emph{IG-NORM}.
We set $\lambda$ as 1, $m$ in the gradient step as 50. We use Momentum Optimizer with weight decay. We set momentum rate as 0.9, weight decay rate as 0.0002, batch size as 64, and training steps as 70,000. We use learning rate schedule: the first 500 steps, we use learning rate of $10^{-6}$; after 500 steps and before 60,000 steps, we use learning rate of $10^{-4}$; after 60,000 steps, we use learning rate of $10^{-5}$. We use PGD attack as adversary with random start, number of steps of 7, step size of 2, $m$ in the attack step of 5, and adversarial budget $\epsilon$ of 8.
\noindent\emph{IG-SUM-NORM}.
We set $\beta$ as 1, $m$ in the gradient step as 50. We use Momentum Optimizer with weight decay. We set momentum rate as 0.9, weight decay rate as 0.0002, batch size as 64, and training steps as 70,000. We use learning rate schedule: the first 500 steps, we use learning rate of $10^{-5}$; after 500 steps and before 60,000 steps, we use learning rate of $10^{-4}$; after 60,000 steps, we use learning rate of $10^{-5}$. We use PGD attack as adversary with random start, number of steps of 7, step size of 2, $m$ in the attack step of 5, and adversarial budget $\epsilon$ of 8.
\noindent\textbf{Evaluation Attacks}.
For attacking inputs to change model predictions, we use PGD attack with number of steps of $40$, adversarial budget $\epsilon$ of 8 and step size of $2$. For attacking inputs to change interpretations, we use Iterative Feature Importance Attacks (IFIA) proposed by ~\cite{GAZ17}. We use their top-k attack with $k=100$, adversarial budget $\epsilon=8$, step size $\alpha=1$ and number of iterations $P=50$. We set the feature importance function as Integrated Gradients(IG) and dissimilarity function $D$ as Kendall's rank order correlation. We find that IFIA is not stable if we use GPU parallel computing (non-deterministic is a behavior of GPU), so we run IFIA three times on each test example and use the best result with the lowest Kendall's rank order correlation.
\subsubsection{Flower}
\noindent\textbf{Data}.
Flower dataset \cite{nilsback2006visual} is a dataset of 17 category flowers with 80 images for each class (1,360 image in total). The flowers chosen are some common flowers in the UK. The images have large scale, pose and light variations and there are also classes with large variations of images within the class and close similarity to other classes. We randomly split the dataset into training and test sets. The training set has totally 1,224 images with 72 images per class. The test set has totally 136 images with 8 images per class. We resize each image to $128 \times 128$. The pixel values are in range of $[0,255]$. We use data augmentation techniques to enlarge the training data. We construct a class preserving data augmentation pipeline consisting of rotation, translation, and projection transforms and apply this pipeline to images in the training set until each class contained 1,000 training examples. We use this new augmented training data set containing 17,000 samples in total to train models.
\noindent\textbf{Model}.
We use the Resnet model \cite{he2016deep}. We perform per image standardization before feeding images to the neural network. The network has 5 residual units with (16, 16, 32, 64) filters each. The model is adapted from CIFAR-10 model of \cite{madry2017towards}. Refer to our codes for details.
\noindent\textbf{Training hyper-parameters}.
The hyper-parameters to train different models are listed below:
\noindent\emph{NATURAL}.
We use Momentum Optimizer with weight decay. We set momentum rate as 0.9, weight decay rate as 0.0002, batch size as 16, and training steps as 70,000. We use learning rate schedule: the first 500 steps, we use learning rate of $10^{-3}$; after 500 steps and before 60,000 steps, we use learning rate of $10^{-2}$; after 60,000 steps, we use learning rate of $10^{-3}$.
\noindent\emph{Madry et al.}.
We use Momentum Optimizer with weight decay. We set momentum rate as 0.9, weight decay rate as 0.0002, batch size as 16, and training steps as 70,000. We use learning rate schedule: the first 500 steps, we use learning rate of $10^{-3}$; after 500 steps and before 60,000 steps, we use learning rate of $10^{-2}$; after 60,000 steps, we use learning rate of $10^{-3}$. We use PGD attack as adversary with random start, number of steps of 7, step size of 2, and adversarial budget $\epsilon$ of 8.
\noindent\emph{IG-NORM}.
We set $\lambda$ as 0.1, $m$ in the gradient step as 50. We use Momentum Optimizer with weight decay. We set momentum rate as 0.9, weight decay rate as 0.0002, batch size as 16, and training steps as 70,000. We use learning rate schedule: the first 500 steps, we use learning rate of $10^{-4}$; after 500 steps and before 60,000 steps, we use learning rate of $10^{-3}$; after 60,000 steps, we use learning rate of $10^{-4}$. We use PGD attack as adversary with random start, number of steps of 7, step size of 2, $m$ in the attack step of 5, and adversarial budget $\epsilon$ of 8.
\noindent\emph{IG-SUM-NORM}.
We set $\beta$ as 0.1, $m$ in the gradient step as 50. We use Momentum Optimizer with weight decay. We set momentum rate as 0.9, weight decay rate as 0.0002, batch size as 16, and training steps as 70,000. We use learning rate schedule: the first 500 steps, we use learning rate of $10^{-4}$; after 500 steps and before 60,000 steps, we use learning rate of $10^{-3}$; after 60,000 steps, we use learning rate of $10^{-4}$. We use PGD attack as adversary with random start, number of steps of 7, step size of 2, $m$ in the attack step of 5, and adversarial budget $\epsilon$ of 8.
\noindent\textbf{Evaluation Attacks}.
For attacking inputs to change model predictions, we use PGD attack with number of steps of $40$, adversarial budget $\epsilon$ of 8 and step size of $2$. For attacking inputs to change interpretations, we use Iterative Feature Importance Attacks (IFIA) proposed by ~\cite{GAZ17}. We use their top-k attack with $k=1000$, adversarial budget $\epsilon=8$, step size $\alpha=1$ and number of iterations $P=100$. We set the feature importance function as Integrated Gradients(IG) and dissimilarity function $D$ as Kendall's rank order correlation. We find that IFIA is not stable if we use GPU parallel computing (non-deterministic is a behavior of GPU), so we run IFIA three times on each test example and use the best result with the lowest Kendall's rank order correlation.
\subsection{Why a different $m$ in the Attack Step?}
From our experiments, we find that the most time consuming part during training is using adversary $\calA$ to find $\bfx^*$. It is because we need to run several PGD steps to find $\bfx^*$. To speed it up, we set a smaller $m$ (no more than 10) in the attack step.
\subsection{Choosing Hyper-parameters}
Our IG-NORM (or IG-SUM-NORM) objective contains hyper-parameters $m$ in the attack step, $m$ in the gradient step and $\lambda$ (or $\beta$). From our experiments, we find that if $\lambda$ (or $\beta$) is too large, the training cannot converge. And if $\lambda$ (or $\beta$) is too small, we cannot get good attribution robustness. To select best $\lambda$ (or $\beta$), we try three values: 1, 0.1, and 0.01, and use the one with the best attribution robustness. For $m$ in the attack step, due to the limitation of computing power, we usually set a small value, typically 5 or 10.
We study how $m$ in the gradient step affects results on MNIST using IG-NORM objective. We try $m \in \{10, 20, 30, \cdots, 100\}$, and set $\lambda=1$ and $m$ in the attack step as 10. Other training settings are the same. The results are summarized in Table~\ref{table:different-m}.
\begin{table}[htb]
\centering
\begin{tabular}{ c | c | c | c | c }
\hline
$m$ & {\tt NA} & {\tt AA} & {\tt IN} & {\tt CO} \\ \hline
10 & 98.54\% & 78.05\% & 67.14\% & 0.2574 \\ \hline
20 & 98.72\% & 80.29\% & 70.78\% & 0.2699 \\ \hline
30 & 98.70\% & 80.44\% & 71.06\% & 0.2640 \\ \hline
40 & 98.79\% & 73.41\% & 64.76\% & 0.2733 \\ \hline
50 & 98.74\% & 81.43\% & {\bf 71.36\%} & {\bf 0.2841} \\ \hline
60 & 98.78\% & 89.25\% & 63.55\% & 0.2230 \\ \hline
70 & 98.80\% & 74.78\% & 67.37\% & 0.2556 \\ \hline
80 & 98.75\% & 80.26\% & 69.90\% & 0.2633 \\ \hline
90 & 98.61\% & 78.54\% & 70.88\% & 0.2787 \\ \hline
100 & 98.59\% & 89.36\% & 59.70\% & 0.2210 \\ \hline
\end{tabular}
\vspace{1mm}
\caption{Experiment results for different $m$ in gradient step on MNIST.}
\label{table:different-m}
\vspace{-7mm}
\end{table}
From the results, we can see when $m = 50$, we can get the best attribution robustness.
For objective IG-SUM-NORM and other datasets, we do similar search for $m$ in the gradient step.
We find that usually, $m = 50$ can give good attribution robustness.
\subsection{Dimensionality and effectiveness of attribution attack}
Similar to~\cite{GAZ17}, we observe that IFIA is not so successful when
number of dimensions is relatively small. For example, on GTSRB
dataset the number of dimensions is relatively small ($32\times 32 \times 3$),
and if one uses small adversarial budget ($8/255\approx 0.031$),
the attacks become not very effective.
On the other hand, even though MNIST dimension is small ($28\times 28 \times 1$)
, the attack remains effective for large budget ($0.3$).
On Flower dataset the number of dimension is large ($128 \times 128 \times 3$),
and the attack is very effective on this dataset.
\subsection{Use Simple Gradient to Compute Feature Importance Maps}
We also experiment with Simple Gradient (SG)~\cite{simonyan2013deep}
instead of Integrated Gradients (IG) to compute feature importance map.
\change{The experiment settings are the same as previous ones except that
we use SG to compute feature importance map in order to compute
rank correlation and top intersection, and also in the Iterative Feature
Importance Attacks (IFIA) (evaluation attacks). The results
are summarized in Table \ref{table:simple-gradient}.
Our method produces significantly better attribution robustness
than both natural training and adversarial training, except being
slightly worse than adversarial training on Fashion-MNIST.
We note that Fashion-MNIST is also the only data set in our
experiments where IG results are significantly different from
that of SG (where under IG, IG-SUM-NORM is significantly better).
Note that IG is a \emph{princpled sommothed verison} of SG and so
this result highlights differences between these two attribution methods
on a particular data set. More investigation into this phenomenon
seems warranted.
}
\begin{table}[htb]
\centering
\begin{tabular}{ c | c | c | c | c | c }
\hline
Dataset & Approach & {\tt NA} & {\tt AA} & {\tt IN} & {\tt CO} \\ \hline
\multirow{3}{*}{MNIST}
& NATURAL & 99.17\% & 0.00\% & 16.64\% & 0.0107 \\ \cline{2-6}
& Madry et al. & 98.40\% & 92.47\% & 47.95\% & 0.2524 \\ \cline{2-6}
& IG-SUM-NORM & 98.34\% & 88.17\% & {\bf 61.67\%} & {\bf 0.2918} \\ \hline \hline
\multirow{3}{*}{Fashion-MNIST}
& NATURAL & 90.86\% & 0.01\% & 21.55\% & 0.0734 \\ \cline{2-6}
& Madry et al. & 85.73\% & 73.01\% & {\bf 58.37\%} & {\bf 0.3947} \\ \cline{2-6}
& IG-SUM-NORM & 85.44\% & 70.26\% & 54.91\% & 0.3674 \\ \hline \hline
\multirow{3}{*}{GTSRB}
& NATURAL & 98.57\% & 21.05\% & 51.31\% & 0.6000 \\ \cline{2-6}
& Madry et al. & 97.59\% & 83.24\% & 70.27\% & 0.6965 \\ \cline{2-6}
& IG-SUM-NORM & 95.68\% & 77.12\% & {\bf 75.03\%} & {\bf 0.7151} \\ \hline \hline
\multirow{3}{*}{Flower}
& NATURAL & 86.76\% & 0.00\% & 6.72\% & 0.2996 \\ \cline{2-6}
& Madry et al. & 83.82\% & 41.91\% & 54.10\% & 0.7282 \\ \cline{2-6}
& IG-SUM-NORM & 82.35\% & 47.06\% & {\bf 65.59\%} & {\bf 0.7503} \\ \hline
\end{tabular}
\vspace{1mm}
\caption{Experiment results for using Simple Gradient to compute feature importance maps.}
\label{table:simple-gradient}
\vspace{-8mm}
\end{table}
\subsection{Additional Visualization Results}
\label{sec:additional-visualization-results}
Here we provide more visualization results for MNIST in Figure~\ref{fig:ex1}, for Fashion-MNIST in Figure~\ref{fig:ex2}, for GTSRB in Figure~\ref{fig:ex3}, and for Flower in Figure~\ref{fig:ex4}.
\begin{figure}[htb]
\centering
\begin{minipage}{\linewidth}
\centering
NATURAL \hspace{2.5cm} IG-NORM \hspace{2.5cm} IG-SUM-NORM
\end{minipage}
\begin{subfigure}{\textwidth}
\centering
\begin{subfigure}[b]{.32\textwidth}
\centering
\includegraphics[width=0.48\linewidth,bb=0 0 449 464]{figures/MNIST/50/nat-model-original-image.jpg}
\includegraphics[width=0.48\linewidth,bb=0 0 449 464]{figures/MNIST/50/nat-model-original-ig.jpg} \\
\includegraphics[width=0.48\linewidth,bb=0 0 449 464]{figures/MNIST/50/nat-model-perturbed-image.jpg}
\includegraphics[width=0.48\linewidth,bb=0 0 449 464]{figures/MNIST/50/nat-model-perturbed-ig.jpg}
\captionsetup{justification=centering}
\caption*{Top-100 Intersection: 37.0\% \\ Kendall's Correlation: 0.0567}
\end{subfigure}
\begin{subfigure}[b]{.32\textwidth}
\centering
\includegraphics[width=0.48\linewidth,bb=0 0 449 464]{figures/MNIST/50/ar-model-original-image.jpg}
\includegraphics[width=0.48\linewidth,bb=0 0 449 464]{figures/MNIST/50/ar-model-original-ig.jpg} \\
\includegraphics[width=0.48\linewidth,bb=0 0 449 464]{figures/MNIST/50/ar-model-perturbed-image.jpg}
\includegraphics[width=0.48\linewidth,bb=0 0 449 464]{figures/MNIST/50/ar-model-perturbed-ig.jpg}
\captionsetup{justification=centering}
\caption*{Top-100 Intersection: 64.0\% \\ Kendall's Correlation: 0.1823}
\end{subfigure}
\begin{subfigure}[b]{.32\textwidth}
\centering
\includegraphics[width=0.48\linewidth,bb=0 0 449 464]{figures/MNIST/50/adv-ar-model-original-image.jpg}
\includegraphics[width=0.48\linewidth,bb=0 0 449 464]{figures/MNIST/50/adv-ar-model-original-ig.jpg} \\
\includegraphics[width=0.48\linewidth,bb=0 0 449 464]{figures/MNIST/50/adv-ar-model-perturbed-image.jpg}
\includegraphics[width=0.48\linewidth,bb=0 0 449 464]{figures/MNIST/50/adv-ar-model-perturbed-ig.jpg}
\captionsetup{justification=centering}
\caption*{Top-100 Intersection: 67.0\% \\ Kendall's Correlation: 0.2180}
\end{subfigure}
\caption{For all images, the models give \emph{correct} prediction -- 6.}
\end{subfigure}
\begin{subfigure}{\textwidth}
\centering
\begin{subfigure}[b]{.32\textwidth}
\centering
\includegraphics[width=0.48\linewidth,bb=0 0 449 464]{figures/MNIST/90/nat-model-original-image.jpg}
\includegraphics[width=0.48\linewidth,bb=0 0 449 464]{figures/MNIST/90/nat-model-original-ig.jpg} \\
\includegraphics[width=0.48\linewidth,bb=0 0 449 464]{figures/MNIST/90/nat-model-perturbed-image.jpg}
\includegraphics[width=0.48\linewidth,bb=0 0 449 464]{figures/MNIST/90/nat-model-perturbed-ig.jpg}
\captionsetup{justification=centering}
\caption*{Top-100 Intersection: 43.0\% \\ Kendall's Correlation: 0.0563}
\end{subfigure}
\begin{subfigure}[b]{.32\textwidth}
\centering
\includegraphics[width=0.48\linewidth,bb=0 0 449 464]{figures/MNIST/90/ar-model-original-image.jpg}
\includegraphics[width=0.48\linewidth,bb=0 0 449 464]{figures/MNIST/90/ar-model-original-ig.jpg} \\
\includegraphics[width=0.48\linewidth,bb=0 0 449 464]{figures/MNIST/90/ar-model-perturbed-image.jpg}
\includegraphics[width=0.48\linewidth,bb=0 0 449 464]{figures/MNIST/90/ar-model-perturbed-ig.jpg}
\captionsetup{justification=centering}
\caption*{Top-100 Intersection: 74.0\% \\ Kendall's Correlation: 0.1718}
\end{subfigure}
\begin{subfigure}[b]{.32\textwidth}
\centering
\includegraphics[width=0.48\linewidth,bb=0 0 449 464]{figures/MNIST/90/adv-ar-model-original-image.jpg}
\includegraphics[width=0.48\linewidth,bb=0 0 449 464]{figures/MNIST/90/adv-ar-model-original-ig.jpg} \\
\includegraphics[width=0.48\linewidth,bb=0 0 449 464]{figures/MNIST/90/adv-ar-model-perturbed-image.jpg}
\includegraphics[width=0.48\linewidth,bb=0 0 449 464]{figures/MNIST/90/adv-ar-model-perturbed-ig.jpg}
\captionsetup{justification=centering}
\caption*{Top-100 Intersection: 84.0\% \\ Kendall's Correlation: 0.2501}
\end{subfigure}
\caption{For all images, the models give \emph{correct} prediction -- 3.}
\end{subfigure}
\begin{subfigure}{\textwidth}
\centering
\begin{subfigure}[b]{.32\textwidth}
\centering
\includegraphics[width=0.48\linewidth,bb=0 0 449 464]{figures/MNIST/400/nat-model-original-image.jpg}
\includegraphics[width=0.48\linewidth,bb=0 0 449 464]{figures/MNIST/400/nat-model-original-ig.jpg} \\
\includegraphics[width=0.48\linewidth,bb=0 0 449 464]{figures/MNIST/400/nat-model-perturbed-image.jpg}
\includegraphics[width=0.48\linewidth,bb=0 0 449 464]{figures/MNIST/400/nat-model-perturbed-ig.jpg}
\captionsetup{justification=centering}
\caption*{Top-100 Intersection: 41.0\% \\ Kendall's Correlation: 0.1065}
\end{subfigure}
\begin{subfigure}[b]{.32\textwidth}
\centering
\includegraphics[width=0.48\linewidth,bb=0 0 449 464]{figures/MNIST/400/ar-model-original-image.jpg}
\includegraphics[width=0.48\linewidth,bb=0 0 449 464]{figures/MNIST/400/ar-model-original-ig.jpg} \\
\includegraphics[width=0.48\linewidth,bb=0 0 449 464]{figures/MNIST/400/ar-model-perturbed-image.jpg}
\includegraphics[width=0.48\linewidth,bb=0 0 449 464]{figures/MNIST/400/ar-model-perturbed-ig.jpg}
\captionsetup{justification=centering}
\caption*{Top-100 Intersection: 83.0\% \\ Kendall's Correlation: 0.2837}
\end{subfigure}
\begin{subfigure}[b]{.32\textwidth}
\centering
\includegraphics[width=0.48\linewidth,bb=0 0 449 464]{figures/MNIST/400/adv-ar-model-original-image.jpg}
\includegraphics[width=0.48\linewidth,bb=0 0 449 464]{figures/MNIST/400/adv-ar-model-original-ig.jpg} \\
\includegraphics[width=0.48\linewidth,bb=0 0 449 464]{figures/MNIST/400/adv-ar-model-perturbed-image.jpg}
\includegraphics[width=0.48\linewidth,bb=0 0 449 464]{figures/MNIST/400/adv-ar-model-perturbed-ig.jpg}
\captionsetup{justification=centering}
\caption*{Top-100 Intersection: 84.0\% \\ Kendall's Correlation: 0.3151}
\end{subfigure}
\caption{For all images, the models give \emph{correct} prediction -- 2.}
\end{subfigure}
\caption{Top-100 and Kendall's Correlation are rank correlations
between original and perturbed saliency maps.
NATURAL is the naturally trained model, IG-NORM and IG-SUM-NORM are models trained using our
robust attribution method. We use attribution attacks described in~\cite{GAZ17} to perturb
the attributions while keeping predictions intact. For all images, the models give
\emph{correct} predictions. However, the saliency maps
(also called feature importance maps), computed via IG, show that attributions of
the naturally trained model are very fragile, either visually or quantitatively as measured
by correlation analysis, while models trained using our method are much more robust
in their attributions.}
\label{fig:ex1}
\end{figure}
\begin{figure}[htb]
\centering
\begin{minipage}{\linewidth}
\centering
NATURAL \hspace{2.5cm} IG-NORM \hspace{2.5cm} IG-SUM-NORM
\end{minipage}
\begin{subfigure}{\textwidth}
\centering
\begin{subfigure}[b]{.32\textwidth}
\centering
\includegraphics[width=0.48\linewidth,bb=0 0 449 464]{figures/Fashion-MNIST/0/nat-model-original-image.jpg}
\includegraphics[width=0.48\linewidth,bb=0 0 449 464]{figures/Fashion-MNIST/0/nat-model-original-ig.jpg} \\
\includegraphics[width=0.48\linewidth,bb=0 0 449 464]{figures/Fashion-MNIST/0/nat-model-perturbed-image.jpg}
\includegraphics[width=0.48\linewidth,bb=0 0 449 464]{figures/Fashion-MNIST/0/nat-model-perturbed-ig.jpg}
\captionsetup{justification=centering}
\caption*{Top-100 Intersection: 50.0\% \\ Kendall's Correlation: 0.4595}
\end{subfigure}
\begin{subfigure}[b]{.32\textwidth}
\centering
\includegraphics[width=0.48\linewidth,bb=0 0 449 464]{figures/Fashion-MNIST/0/ar-model-original-image.jpg}
\includegraphics[width=0.48\linewidth,bb=0 0 449 464]{figures/Fashion-MNIST/0/ar-model-original-ig.jpg} \\
\includegraphics[width=0.48\linewidth,bb=0 0 449 464]{figures/Fashion-MNIST/0/ar-model-perturbed-image.jpg}
\includegraphics[width=0.48\linewidth,bb=0 0 449 464]{figures/Fashion-MNIST/0/ar-model-perturbed-ig.jpg}
\captionsetup{justification=centering}
\caption*{Top-100 Intersection: 63.0\% \\ Kendall's Correlation: 0.6099}
\end{subfigure}
\begin{subfigure}[b]{.32\textwidth}
\centering
\includegraphics[width=0.48\linewidth,bb=0 0 449 464]{figures/Fashion-MNIST/0/adv-ar-model-original-image.jpg}
\includegraphics[width=0.48\linewidth,bb=0 0 449 464]{figures/Fashion-MNIST/0/adv-ar-model-original-ig.jpg} \\
\includegraphics[width=0.48\linewidth,bb=0 0 449 464]{figures/Fashion-MNIST/0/adv-ar-model-perturbed-image.jpg}
\includegraphics[width=0.48\linewidth,bb=0 0 449 464]{figures/Fashion-MNIST/0/adv-ar-model-perturbed-ig.jpg}
\captionsetup{justification=centering}
\caption*{Top-100 Intersection: 87.0\% \\ Kendall's Correlation: 0.6607}
\end{subfigure}
\caption{For all images, the models give \emph{correct} prediction -- Ankle boot.}
\end{subfigure}
\begin{subfigure}{\textwidth}
\centering
\begin{subfigure}[b]{.32\textwidth}
\centering
\includegraphics[width=0.48\linewidth,bb=0 0 449 464]{figures/Fashion-MNIST/8/nat-model-original-image.jpg}
\includegraphics[width=0.48\linewidth,bb=0 0 449 464]{figures/Fashion-MNIST/8/nat-model-original-ig.jpg} \\
\includegraphics[width=0.48\linewidth,bb=0 0 449 464]{figures/Fashion-MNIST/8/nat-model-perturbed-image.jpg}
\includegraphics[width=0.48\linewidth,bb=0 0 449 464]{figures/Fashion-MNIST/8/nat-model-perturbed-ig.jpg}
\captionsetup{justification=centering}
\caption*{Top-100 Intersection: 47.0\% \\ Kendall's Correlation: 0.1293}
\end{subfigure}
\begin{subfigure}[b]{.32\textwidth}
\centering
\includegraphics[width=0.48\linewidth,bb=0 0 449 464]{figures/Fashion-MNIST/8/ar-model-original-image.jpg}
\includegraphics[width=0.48\linewidth,bb=0 0 449 464]{figures/Fashion-MNIST/8/ar-model-original-ig.jpg} \\
\includegraphics[width=0.48\linewidth,bb=0 0 449 464]{figures/Fashion-MNIST/8/ar-model-perturbed-image.jpg}
\includegraphics[width=0.48\linewidth,bb=0 0 449 464]{figures/Fashion-MNIST/8/ar-model-perturbed-ig.jpg}
\captionsetup{justification=centering}
\caption*{Top-100 Intersection: 54.0\% \\ Kendall's Correlation: 0.2508}
\end{subfigure}
\begin{subfigure}[b]{.32\textwidth}
\centering
\includegraphics[width=0.48\linewidth,bb=0 0 449 464]{figures/Fashion-MNIST/8/adv-ar-model-original-image.jpg}
\includegraphics[width=0.48\linewidth,bb=0 0 449 464]{figures/Fashion-MNIST/8/adv-ar-model-original-ig.jpg} \\
\includegraphics[width=0.48\linewidth,bb=0 0 449 464]{figures/Fashion-MNIST/8/adv-ar-model-perturbed-image.jpg}
\includegraphics[width=0.48\linewidth,bb=0 0 449 464]{figures/Fashion-MNIST/8/adv-ar-model-perturbed-ig.jpg}
\captionsetup{justification=centering}
\caption*{Top-100 Intersection: 65.0\% \\ Kendall's Correlation: 0.3136}
\end{subfigure}
\caption{For all images, the models give \emph{correct} prediction -- Sandal.}
\end{subfigure}
\begin{subfigure}{\textwidth}
\centering
\begin{subfigure}[b]{.32\textwidth}
\centering
\includegraphics[width=0.48\linewidth,bb=0 0 449 464]{figures/Fashion-MNIST/80/nat-model-original-image.jpg}
\includegraphics[width=0.48\linewidth,bb=0 0 449 464]{figures/Fashion-MNIST/80/nat-model-original-ig.jpg} \\
\includegraphics[width=0.48\linewidth,bb=0 0 449 464]{figures/Fashion-MNIST/80/nat-model-perturbed-image.jpg}
\includegraphics[width=0.48\linewidth,bb=0 0 449 464]{figures/Fashion-MNIST/80/nat-model-perturbed-ig.jpg}
\captionsetup{justification=centering}
\caption*{Top-100 Intersection: 39.0\% \\ Kendall's Correlation: 0.4129}
\end{subfigure}
\begin{subfigure}[b]{.32\textwidth}
\centering
\includegraphics[width=0.48\linewidth,bb=0 0 449 464]{figures/Fashion-MNIST/80/ar-model-original-image.jpg}
\includegraphics[width=0.48\linewidth,bb=0 0 449 464]{figures/Fashion-MNIST/80/ar-model-original-ig.jpg} \\
\includegraphics[width=0.48\linewidth,bb=0 0 449 464]{figures/Fashion-MNIST/80/ar-model-perturbed-image.jpg}
\includegraphics[width=0.48\linewidth,bb=0 0 449 464]{figures/Fashion-MNIST/80/ar-model-perturbed-ig.jpg}
\captionsetup{justification=centering}
\caption*{Top-100 Intersection: 61.0\% \\ Kendall's Correlation: 0.5983}
\end{subfigure}
\begin{subfigure}[b]{.32\textwidth}
\centering
\includegraphics[width=0.48\linewidth,bb=0 0 449 464]{figures/Fashion-MNIST/80/adv-ar-model-original-image.jpg}
\includegraphics[width=0.48\linewidth,bb=0 0 449 464]{figures/Fashion-MNIST/80/adv-ar-model-original-ig.jpg} \\
\includegraphics[width=0.48\linewidth,bb=0 0 449 464]{figures/Fashion-MNIST/80/adv-ar-model-perturbed-image.jpg}
\includegraphics[width=0.48\linewidth,bb=0 0 449 464]{figures/Fashion-MNIST/80/adv-ar-model-perturbed-ig.jpg}
\captionsetup{justification=centering}
\caption*{Top-100 Intersection: 71.0\% \\ Kendall's Correlation: 0.6699}
\end{subfigure}
\caption{For all images, the models give \emph{correct} prediction -- Trouser.}
\end{subfigure}
\caption{Top-100 and Kendall's Correlation are rank correlations
between original and perturbed saliency maps.
NATURAL is the naturally trained model, IG-NORM and IG-SUM-NORM are models trained using our
robust attribution method. We use attribution attacks described in~\cite{GAZ17} to perturb
the attributions while keeping predictions intact. For all images, the models give
\emph{correct} predictions. However, the saliency maps
(also called feature importance maps), computed via IG, show that attributions of
the naturally trained model are very fragile, either visually or quantitatively as measured
by correlation analysis, while models trained using our method are much more robust
in their attributions.}
\label{fig:ex2}
\end{figure}
\begin{figure}[htb]
\centering
\begin{minipage}{\linewidth}
\centering
NATURAL \hspace{2.5cm} IG-NORM \hspace{2.5cm} IG-SUM-NORM
\end{minipage}
\begin{subfigure}{\textwidth}
\centering
\begin{subfigure}[b]{.32\textwidth}
\centering
\includegraphics[width=0.48\linewidth,bb=0 0 449 464]{figures/GTSRB/310/nat-model-original-image.jpg}
\includegraphics[width=0.48\linewidth,bb=0 0 449 464]{figures/GTSRB/310/nat-model-original-ig.jpg} \\
\includegraphics[width=0.48\linewidth,bb=0 0 449 464]{figures/GTSRB/310/nat-model-perturbed-image.jpg}
\includegraphics[width=0.48\linewidth,bb=0 0 449 464]{figures/GTSRB/310/nat-model-perturbed-ig.jpg}
\captionsetup{justification=centering}
\caption*{Top-100 Intersection: 45.0\% \\ Kendall's Correlation: 0.5822}
\end{subfigure}
\begin{subfigure}[b]{.32\textwidth}
\centering
\includegraphics[width=0.48\linewidth,bb=0 0 449 464]{figures/GTSRB/310/ar-model-original-image.jpg}
\includegraphics[width=0.48\linewidth,bb=0 0 449 464]{figures/GTSRB/310/ar-model-original-ig.jpg} \\
\includegraphics[width=0.48\linewidth,bb=0 0 449 464]{figures/GTSRB/310/ar-model-perturbed-image.jpg}
\includegraphics[width=0.48\linewidth,bb=0 0 449 464]{figures/GTSRB/310/ar-model-perturbed-ig.jpg}
\captionsetup{justification=centering}
\caption*{Top-100 Intersection: 78.0\% \\ Kendall's Correlation: 0.7471}
\end{subfigure}
\begin{subfigure}[b]{.32\textwidth}
\centering
\includegraphics[width=0.48\linewidth,bb=0 0 449 464]{figures/GTSRB/310/adv-ar-model-original-image.jpg}
\includegraphics[width=0.48\linewidth,bb=0 0 449 464]{figures/GTSRB/310/adv-ar-model-original-ig.jpg} \\
\includegraphics[width=0.48\linewidth,bb=0 0 449 464]{figures/GTSRB/310/adv-ar-model-perturbed-image.jpg}
\includegraphics[width=0.48\linewidth,bb=0 0 449 464]{figures/GTSRB/310/adv-ar-model-perturbed-ig.jpg}
\captionsetup{justification=centering}
\caption*{Top-100 Intersection: 80.0\% \\ Kendall's Correlation: 0.7886}
\end{subfigure}
\caption{For all images, the models give \emph{correct} prediction -- Dangerous Curve to The Left.}
\end{subfigure}
\begin{subfigure}{\textwidth}
\centering
\begin{subfigure}[b]{.32\textwidth}
\centering
\includegraphics[width=0.48\linewidth,bb=0 0 449 464]{figures/GTSRB/270/nat-model-original-image.jpg}
\includegraphics[width=0.48\linewidth,bb=0 0 449 464]{figures/GTSRB/270/nat-model-original-ig.jpg} \\
\includegraphics[width=0.48\linewidth,bb=0 0 449 464]{figures/GTSRB/270/nat-model-perturbed-image.jpg}
\includegraphics[width=0.48\linewidth,bb=0 0 449 464]{figures/GTSRB/270/nat-model-perturbed-ig.jpg}
\captionsetup{justification=centering}
\caption*{Top-100 Intersection: 56.0\% \\ Kendall's Correlation: 0.6679}
\end{subfigure}
\begin{subfigure}[b]{.32\textwidth}
\centering
\includegraphics[width=0.48\linewidth,bb=0 0 449 464]{figures/GTSRB/270/ar-model-original-image.jpg}
\includegraphics[width=0.48\linewidth,bb=0 0 449 464]{figures/GTSRB/270/ar-model-original-ig.jpg} \\
\includegraphics[width=0.48\linewidth,bb=0 0 449 464]{figures/GTSRB/270/ar-model-perturbed-image.jpg}
\includegraphics[width=0.48\linewidth,bb=0 0 449 464]{figures/GTSRB/270/ar-model-perturbed-ig.jpg}
\captionsetup{justification=centering}
\caption*{Top-100 Intersection: 85.0\% \\ Kendall's Correlation: 0.7963}
\end{subfigure}
\begin{subfigure}[b]{.32\textwidth}
\centering
\includegraphics[width=0.48\linewidth,bb=0 0 449 464]{figures/GTSRB/270/adv-ar-model-original-image.jpg}
\includegraphics[width=0.48\linewidth,bb=0 0 449 464]{figures/GTSRB/270/adv-ar-model-original-ig.jpg} \\
\includegraphics[width=0.48\linewidth,bb=0 0 449 464]{figures/GTSRB/270/adv-ar-model-perturbed-image.jpg}
\includegraphics[width=0.48\linewidth,bb=0 0 449 464]{figures/GTSRB/270/adv-ar-model-perturbed-ig.jpg}
\captionsetup{justification=centering}
\caption*{Top-100 Intersection: 83.0\% \\ Kendall's Correlation: 0.8338}
\end{subfigure}
\caption{For all images, the models give \emph{correct} prediction -- General Caution.}
\end{subfigure}
\begin{subfigure}{\textwidth}
\centering
\begin{subfigure}[b]{.32\textwidth}
\centering
\includegraphics[width=0.48\linewidth,bb=0 0 449 464]{figures/GTSRB/1000/nat-model-original-image.jpg}
\includegraphics[width=0.48\linewidth,bb=0 0 449 464]{figures/GTSRB/1000/nat-model-original-ig.jpg} \\
\includegraphics[width=0.48\linewidth,bb=0 0 449 464]{figures/GTSRB/1000/nat-model-perturbed-image.jpg}
\includegraphics[width=0.48\linewidth,bb=0 0 449 464]{figures/GTSRB/1000/nat-model-perturbed-ig.jpg}
\captionsetup{justification=centering}
\caption*{Top-100 Intersection: 43.0\% \\ Kendall's Correlation: 0.6160}
\end{subfigure}
\begin{subfigure}[b]{.32\textwidth}
\centering
\includegraphics[width=0.48\linewidth,bb=0 0 449 464]{figures/GTSRB/1000/ar-model-original-image.jpg}
\includegraphics[width=0.48\linewidth,bb=0 0 449 464]{figures/GTSRB/1000/ar-model-original-ig.jpg} \\
\includegraphics[width=0.48\linewidth,bb=0 0 449 464]{figures/GTSRB/1000/ar-model-perturbed-image.jpg}
\includegraphics[width=0.48\linewidth,bb=0 0 449 464]{figures/GTSRB/1000/ar-model-perturbed-ig.jpg}
\captionsetup{justification=centering}
\caption*{Top-100 Intersection: 67.0\% \\ Kendall's Correlation: 0.7595}
\end{subfigure}
\begin{subfigure}[b]{.32\textwidth}
\centering
\includegraphics[width=0.48\linewidth,bb=0 0 449 464]{figures/GTSRB/1000/adv-ar-model-original-image.jpg}
\includegraphics[width=0.48\linewidth,bb=0 0 449 464]{figures/GTSRB/1000/adv-ar-model-original-ig.jpg} \\
\includegraphics[width=0.48\linewidth,bb=0 0 449 464]{figures/GTSRB/1000/adv-ar-model-perturbed-image.jpg}
\includegraphics[width=0.48\linewidth,bb=0 0 449 464]{figures/GTSRB/1000/adv-ar-model-perturbed-ig.jpg}
\captionsetup{justification=centering}
\caption*{Top-100 Intersection: 81.0\% \\ Kendall's Correlation: 0.8128}
\end{subfigure}
\caption{For all images, the models give \emph{correct} prediction -- No Entry.}
\end{subfigure}
\caption{Top-100 and Kendall's Correlation are rank correlations
between original and perturbed saliency maps.
NATURAL is the naturally trained model, IG-NORM and IG-SUM-NORM are models trained using our
robust attribution method. We use attribution attacks described in~\cite{GAZ17} to perturb
the attributions while keeping predictions intact. For all images, the models give
\emph{correct} predictions. However, the saliency maps
(also called feature importance maps), computed via IG, show that attributions of
the naturally trained model are very fragile, either visually or quantitatively as measured
by correlation analyses, while models trained using our method are much more robust
in their attributions.}
\label{fig:ex3}
\end{figure}
\begin{figure}[htb]
\centering
\begin{minipage}{\linewidth}
\centering
NATURAL \hspace{2.5cm} IG-NORM \hspace{2.5cm} IG-SUM-NORM
\end{minipage}
\begin{subfigure}{\textwidth}
\centering
\begin{subfigure}[b]{.32\textwidth}
\centering
\includegraphics[width=0.48\linewidth,bb=0 0 449 464]{figures/Flower/14/nat-model-original-image.jpg}
\includegraphics[width=0.48\linewidth,bb=0 0 449 464,bb=0 0 449 464]{figures/Flower/14/nat-model-original-ig.jpg} \\
\includegraphics[width=0.48\linewidth,bb=0 0 449 464]{figures/Flower/14/nat-model-perturbed-image.jpg}
\includegraphics[width=0.48\linewidth,bb=0 0 449 464]{figures/Flower/14/nat-model-perturbed-ig.jpg}
\captionsetup{justification=centering}
\caption*{Top-1000 Intersection: 1.0\% \\ Kendall's Correlation: 0.4601}
\end{subfigure}
\begin{subfigure}[b]{.32\textwidth}
\centering
\includegraphics[width=0.48\linewidth,bb=0 0 449 464]{figures/Flower/14/ar-model-original-image.jpg}
\includegraphics[width=0.48\linewidth,bb=0 0 449 464]{figures/Flower/14/ar-model-original-ig.jpg} \\
\includegraphics[width=0.48\linewidth,bb=0 0 449 464]{figures/Flower/14/ar-model-perturbed-image.jpg}
\includegraphics[width=0.48\linewidth,bb=0 0 449 464]{figures/Flower/14/ar-model-perturbed-ig.jpg}
\captionsetup{justification=centering}
\caption*{Top-1000 Intersection: 65.4\% \\ Kendall's Correlation: 0.7248}
\end{subfigure}
\begin{subfigure}[b]{.32\textwidth}
\centering
\includegraphics[width=0.48\linewidth,bb=0 0 449 464]{figures/Flower/14/adv-ar-model-original-image.jpg}
\includegraphics[width=0.48\linewidth,bb=0 0 449 464]{figures/Flower/14/adv-ar-model-original-ig.jpg} \\
\includegraphics[width=0.48\linewidth,bb=0 0 449 464]{figures/Flower/14/adv-ar-model-perturbed-image.jpg}
\includegraphics[width=0.48\linewidth,bb=0 0 449 464]{figures/Flower/14/adv-ar-model-perturbed-ig.jpg}
\captionsetup{justification=centering}
\caption*{Top-1000 Intersection: 63.9\% \\ Kendall's Correlation: 0.8036}
\end{subfigure}
\caption{For all images, the models give \emph{correct} prediction -- Bluebell.}
\end{subfigure}
\begin{subfigure}{\textwidth}
\centering
\begin{subfigure}[b]{.32\textwidth}
\centering
\includegraphics[width=0.48\linewidth,bb=0 0 449 464]{figures/Flower/15/nat-model-original-image.jpg}
\includegraphics[width=0.48\linewidth,bb=0 0 449 464]{figures/Flower/15/nat-model-original-ig.jpg} \\
\includegraphics[width=0.48\linewidth,bb=0 0 449 464]{figures/Flower/15/nat-model-perturbed-image.jpg}
\includegraphics[width=0.48\linewidth,bb=0 0 449 464]{figures/Flower/15/nat-model-perturbed-ig.jpg}
\captionsetup{justification=centering}
\caption*{Top-1000 Intersection: 6.2\% \\ Kendall's Correlation: 0.3863}
\end{subfigure}
\begin{subfigure}[b]{.32\textwidth}
\centering
\includegraphics[width=0.48\linewidth,bb=0 0 449 464]{figures/Flower/15/ar-model-original-image.jpg}
\includegraphics[width=0.48\linewidth,bb=0 0 449 464]{figures/Flower/15/ar-model-original-ig.jpg} \\
\includegraphics[width=0.48\linewidth,bb=0 0 449 464]{figures/Flower/15/ar-model-perturbed-image.jpg}
\includegraphics[width=0.48\linewidth,bb=0 0 449 464]{figures/Flower/15/ar-model-perturbed-ig.jpg}
\captionsetup{justification=centering}
\caption*{Top-1000 Intersection: 58.20\% \\ Kendall's Correlation: 0.6694}
\end{subfigure}
\begin{subfigure}[b]{.32\textwidth}
\centering
\includegraphics[width=0.48\linewidth,bb=0 0 449 464]{figures/Flower/15/adv-ar-model-original-image.jpg}
\includegraphics[width=0.48\linewidth,bb=0 0 449 464]{figures/Flower/15/adv-ar-model-original-ig.jpg} \\
\includegraphics[width=0.48\linewidth,bb=0 0 449 464]{figures/Flower/15/adv-ar-model-perturbed-image.jpg}
\includegraphics[width=0.48\linewidth,bb=0 0 449 464]{figures/Flower/15/adv-ar-model-perturbed-ig.jpg}
\captionsetup{justification=centering}
\caption*{Top-1000 Intersection: 65.9\% \\ Kendall's Correlation: 0.7970}
\end{subfigure}
\caption{For all images, the models give \emph{correct} prediction -- Cowslip.}
\end{subfigure}
\begin{subfigure}{\textwidth}
\centering
\begin{subfigure}[b]{.32\textwidth}
\centering
\includegraphics[width=0.48\linewidth,bb=0 0 449 464]{figures/Flower/56/nat-model-original-image.jpg}
\includegraphics[width=0.48\linewidth,bb=0 0 449 464]{figures/Flower/56/nat-model-original-ig.jpg} \\
\includegraphics[width=0.48\linewidth,bb=0 0 449 464]{figures/Flower/56/nat-model-perturbed-image.jpg}
\includegraphics[width=0.48\linewidth,bb=0 0 449 464]{figures/Flower/56/nat-model-perturbed-ig.jpg}
\captionsetup{justification=centering}
\caption*{Top-1000 Intersection: 6.8\% \\ Kendall's Correlation: 0.4653}
\end{subfigure}
\begin{subfigure}[b]{.32\textwidth}
\centering
\includegraphics[width=0.48\linewidth,bb=0 0 449 464]{figures/Flower/56/ar-model-original-image.jpg}
\includegraphics[width=0.48\linewidth,bb=0 0 449 464]{figures/Flower/56/ar-model-original-ig.jpg} \\
\includegraphics[width=0.48\linewidth,bb=0 0 449 464]{figures/Flower/56/ar-model-perturbed-image.jpg}
\includegraphics[width=0.48\linewidth,bb=0 0 449 464]{figures/Flower/56/ar-model-perturbed-ig.jpg}
\captionsetup{justification=centering}
\caption*{Top-1000 Intersection: 58.0\% \\ Kendall's Correlation: 0.7165}
\end{subfigure}
\begin{subfigure}[b]{.32\textwidth}
\centering
\includegraphics[width=0.48\linewidth,bb=0 0 449 464]{figures/Flower/56/adv-ar-model-original-image.jpg}
\includegraphics[width=0.48\linewidth,bb=0 0 449 464]{figures/Flower/56/adv-ar-model-original-ig.jpg} \\
\includegraphics[width=0.48\linewidth,bb=0 0 449 464]{figures/Flower/56/adv-ar-model-perturbed-image.jpg}
\includegraphics[width=0.48\linewidth,bb=0 0 449 464]{figures/Flower/56/adv-ar-model-perturbed-ig.jpg}
\captionsetup{justification=centering}
\caption*{Top-1000 Intersection: 63.4\% \\ Kendall's Correlation: 0.8201}
\end{subfigure}
\caption{For all images, the models give \emph{correct} prediction -- Tigerlily.}
\end{subfigure}
\caption{Top-1000 and Kendall's Correlation are rank correlations
between original and perturbed saliency maps.
NATURAL is the naturally trained model, IG-NORM and IG-SUM-NORM are models trained using our
robust attribution method. We use attribution attacks described in~\cite{GAZ17} to perturb
the attributions while keeping predictions intact. For all images, the models give
\emph{correct} predictions. However, the saliency maps
(also called feature importance maps), computed via IG, show that attributions of
the naturally trained model are very fragile, either visually or quantitatively as measured
by correlation analyses, while models trained using our method are much more robust
in their attributions.}
\label{fig:ex4}
\end{figure}
\section{Introduction}
\label{sec:introduction}
\input{introduction}
\section{Preliminaries}
\label{sec:preliminaries}
\input{preliminaries}
\section{Robust Attribution Regularization}
\label{sec:robust-attribution-regularization}
\input{robust-attribution-regularization}
\section{Instantiations and Optimizations}
\label{sec:optimization}
\input{optimization}
\section{Experiments}
\label{sec:experiments}
\input{experiments}
\section{Discussion and Conclusion}
\label{sec:conclusion}
\input{conclusion}
\section{Acknowledgments}
\label{sec:acknowledgments}
\input{acknowledgement}
\newpage
\bibliographystyle{alpha}
\subsection{Uncertainty Set Model}
\label{sec:uncertainty-set-model}
In the uncertainty set model, for any sample $(\bfx, y) \sim P$ for
a data generating distribution $P$, we think of it as a ``nominal'' point and
assume that the real sample comes from a neighborhood around $\bfx$.
In this case, given any intermediate layer $\bfh$, we propose the following
objective function:
\begin{align}
\label{eq:rar-objective}
\begin{split}
&\minimize_\theta \Exp_{(\bfx, y) \sim P}[\rho(\bfx, y; \theta)] \\
&\text{where\ \ }\rho(\bfx, y; \theta) = \ell(\bfx, y; \theta) +
\lambda\max_{\bfx'\in N(\bfx, \varepsilon)}
s(\IG^{\ell_y}_{\bfh}(\bfx, \bfx'; r))
\end{split}
\end{align}
where $\lambda \ge 0$ is a regularization parameter, $\ell_y$ is the
loss function with label $y$ fixed:
$\ell_y(\bfx; \theta) = \ell(\bfx, y; \theta)$,
$r: [0, 1] \mapsto \Real^d$ is a curve parameterization from $\bfx$ to $\bfx'$,
and $\IG^{\ell_y}$ is the integrated gradients of $\ell_y$, and therefore gives
attribution of changes of $\ell_y$ as we go from $\bfx$ to $\bfx'$. $s(\cdot)$
is a size function that measures the ``size'' of the attribution.\footnote{
We stress that this regularization term depends on model parameters $\theta$
through loss function $\ell_y$.}
We now study some particular instantiations of the objective
(\ref{eq:rar-objective}). Specifically, we recover existing robust training
objectives under \emph{weak} instantiations (such as choosing $s(\cdot)$ as
summation function, which is not metric, or use crude approximation of $\IG$),
and also derive new instantiations that are natural extensions to existing ones.
\begin{proposition}[\textbf{Madry et al.'s robust prediction objective}]
\label{prop:recover-madry}
If we set $\lambda=1$ , and let $s(\cdot)$ be the $\sf sum$ function
(sum all components of a vector), then for any curve $r$ and any intermediate
layer $\bfh$, (\ref{eq:rar-objective}) is exactly the objective proposed
by Madry et al.~\cite{MMSTV17} where
$\rho(\bfx, y; \theta) = \max_{\bfx' \in N(\bfx, \varepsilon)}\ell(
\bfx', y; \theta)$.
\end{proposition}
We note that:
(1) $\sf sum$ is a weak size function which does not give a metric.
(2) As a result, while this robust prediction objective falls
within our framework, and regularizes robust attributions,
it allows a small regularization term where attributions actually change
significantly but they cancel each other in summation.
Therefore, the control over robust attributions can be weak.
\begin{proposition}[\textbf{Input gradient regularization}]
\label{prop:recover-input-gradient-regularization}
For any $\lambda' > 0$ and $q \ge 1$, if we set
$\lambda = \lambda'/\varepsilon^q$, $s(\cdot)=\|\cdot\|_1^q$,
and use only the first term of summation approximation
(\ref{eq:intermediate-attribution-sum-approx}) to approximate $\IG$,
then (\ref{eq:rar-objective}) becomes exactly the input gradient
regularization of Drucker and LeCun~\cite{DL92}, where we have
$\rho(\bfx, y; \theta)
= \ell(\bfx, y; \theta)
+ \lambda \|\nabla_{\bfx} \ell(\bfx, y; \theta)\|_q^q$.
\end{proposition}
In the above we have considered instantiations of a weak size function
(summation function), which recovers Madry et al.'s objective,
and of a weak approximation of $\IG$ (picking the first term),
which recovers input gradient regularization. In the next example,
we pick a nontrivial size function, the 1-norm $\|\cdot\|_1$,
use the precise $\IG$, but then we use a \emph{trivial
intermediate layer}, the output loss $\ell_y$.
\begin{proposition}[\textbf{Regularizing by attribution of the loss output}]
\label{prop:regularize-by-loss-output}
Let us set $\lambda=1$, $s(\cdot) = \|\cdot\|_1$, and $\bfh = \ell_y$
(the output layer of loss function!), then we have
$\rho(\bfx, y; \theta)
= \ell_y(\bfx) + \max_{\bfx' \in N(\bfx, \varepsilon)}\{
|\ell_y(\bfx') - \ell_y(\bfx)|\}$.
\end{proposition}
We note that this loss function is a ``surrogate'' loss function for
Madry et al.'s loss function because
$\ell_y(\bfx) + \max_{\bfx' \in N(\bfx, \varepsilon)}\{|\ell_y(\bfx')
- \ell_y(\bfx)|\}
\ge \ell_y(\bfx) + \max_{\bfx' \in N(\bfx, \varepsilon)}\{(\ell_y(\bfx')
- \ell_y(\bfx))\}
= \max_{\bfx' \in N(\bfx, \varepsilon)}\ell_y(\bfx')$.
Therefore, even at such a trivial instantiation,
robust attribution regularization provides interesting guarantees.
\vspace{-2mm}
\subsection{Distributional Robustness Model}
\label{sec:distributional-robustness-model}
A different but popular model for robust optimization is the
distributional robustness model. In this case we consider a family of
distributions $\calP$, each of which is supposed to be a ``slight variation''
of a base distribution $P$. The goal of robust optimization is then
that certain objective functions obtain stable values over this entire family.
Here we apply the same underlying idea to the distributional robustness
model: One should get a small loss value over the base distribution $P$,
and for any distribution $Q \in \calP$, the $\IG$-based \emph{attributions}
change only a little if we move from $P$ to $Q$. This is formalized as:
\begin{align*}
\minimize_\theta \Exp_P[\ell(P; \theta)]
+ \lambda\sup_{Q \in \calP} \left\{W_{d_{\IG}}(P, Q) \right\},
\end{align*}
where the $W_{d_{\IG}}(P, Q)$ is the Wasserstein distance between
$P$ and $Q$ under a distance metric
$d_{\IG}$.\footnote{
For supervised learning problem where $P$ is of the form $Z=(X, Y)$,
we use the same treatment as in~\cite{SND18} so that cost function is
defined as $c(z, z') = c_x(x, x') + \infty\cdot{\bf 1}\{y \neq y'\}$.
All our theory carries over to such $c$ which has range
$\Real_{+} \cup \{\infty\}$.
}
We use $\IG$ to highlight that this metric is related to integrated gradients.
We propose again $d_{\IG}(\bfz, \bfz') = s(\IG^{\ell}_{\bfh}(\bfz, \bfz'))$.
We are particularly interested in the case where $\calP$ is a Wasserstein ball
around the base distribution $P$, using ``perturbation'' cost metric $c(\cdot)$.
This gives regularization term
$\lambda \Exp_{W_c(P, Q) \le \rho}\sup\{W_{d_{\IG}}(P, Q)\}$.
An unsatisfying aspect of this objective, as one can observe now, is that
$W_{d_{\IG}}$ and $W_c$ can take two \emph{different} couplings,
while intuitively we want to use only one coupling to transport $P$ to $Q$.
For example, this objective allows us to pick a coupling $M_1$ under which
we achieve $W_{d_{\IG}}$
(recall that Wasserstein distance is an infimum over couplings),
and a different coupling $M_2$ under which we achieve $W_c$,
but under $M_1 = (Z, Z')$, $\Exp_{z, z' \sim M_1}[c(z, z')] > \rho$,
violating the constraint. This motivates the following modification:
\begin{align}
\label{eq:dist-rar-objective}
\begin{split}
&\minimize_\theta \Exp_P[\ell(P; \theta)] +
\lambda\sup_{Q; M \in \prod(P, Q)} \Big\{
\Exp_{Z,Z'}[d_{\IG}(Z, Z')]
\text{ s.t. } \Exp_{Z,Z'}[c(Z, Z')] \le \rho \Big\},
\end{split}
\end{align}
In this formulation, $\prod(P, Q)$ is the set of couplings of $P$ and $Q$,
and $M=(Z, Z')$ is one coupling. $c(\cdot, \cdot)$ is a metric,
such as $\|\cdot\|_2$, to measure the cost of an adversary perturbing $z$ to $z'$.
$\rho$ is an upper bound on the expected perturbation cost,
thus constraining $P$ and $Q$ to be ``close'' with each together.
$d_{\IG}$ is a metric to measure the change of attributions from $Z$ to $Z'$,
where we want a large $d_{\IG}$-change under a small $c$-change.
The supremum is taken over $Q$ and $\prod(P, Q)$.
\begin{proposition}[\textbf{Wasserstein prediction robustness}]
\label{prop:recover-wasserstein-prediction-robustness-objective}
Let $s(\cdot)$ be the summation function and $\lambda=1$, then
for any curve $\gamma$ and any layer $\bfh$, (\ref{eq:dist-rar-objective})
reduces to $\sup_{Q: W_c(P, Q) \le \rho}\left\{
\Exp_Q[\ell(Q; \theta)]\right\}$, which is the objective proposed by
Sinha, Namhoong, and Duchi~\cite{SND18} for \emph{robust predictions}.
\end{proposition}
\noindent\textbf{Lagrange relaxation}.
For any $\gamma \ge 0$, the Lagrange relaxation of
(\ref{eq:dist-rar-objective}) is
\begin{align}
\label{eq:dist-rar-objective-lagrange}
\begin{split}
\minimize_\theta \bigg\{ \Exp_P[\ell(P; \theta)]
+ \lambda \sup_{Q; M \in \prod(P, Q)}\Big\{
\Exp_{M=(Z, Z')}\big[
d_{\IG}(Z, Z') - \gamma c(Z, Z')\big]\Big\} \bigg\}
\end{split}
\end{align}
where the supremum is taken over $Q$ (unconstrained) and all couplings of $P$
and $Q$, and we want to find a coupling under which $\IG$ attributions change a lot,
while the perturbation cost from $P$ to $Q$ with respect to $c$ is small.
Recall that $g: \Real^d \times \Real^d \rightarrow \Real$
is a \emph{normal integrand} if for each $\alpha$, the mapping
$z \rightarrow \{ z' | g(z, z') \le \alpha \}$
is closed-valued and measurable~\cite{rockafellar2009variational}.
Our next two theorems generalize the duality theory in~\cite{SND18} to
a much larger, but natural, class of objectives.
\begin{theorem}
\label{thm:dist-rar-duality2}
Suppose $c(z,z)=0$ and $d_{\IG}(z,z)=0$ for any $z$,
and suppose $\gamma c(z, z') - d_{\IG}(z, z')$ is a normal integrand.
Then, $\sup_{Q; M \in \prod(P, Q)}\{
\Exp_{M=(Z, Z')}[d^\gamma_{\IG}(Z, Z')]\} =
\Exp_{z \sim P}[\sup_{z'}\{d^\gamma_{\IG}(z, z')\}].$
Consequently, we have (\ref{eq:dist-rar-objective-lagrange}) to be equal to the following:
\begin{align}
\label{eq:dist-rar-objective-lagrange-2}
\begin{split}
\minimize_\theta
\Exp_{z \sim P}\Big[\ell(z; \theta) +
\lambda\sup_{z'}\{d_{\IG}(z, z') - \gamma c(z, z')\} \Big]
\end{split}
\end{align}
\end{theorem}
The assumption $d_{\IG}(z,z)=0$ is true for what we propose,
and $c(z, z) = 0$ is true for any typical cost such as $\ell_p$ distances.
The normal integrand assumption is also very weak,
e.g., it is satisfied when $d_{\IG}$ is continuous and $c$ is closed convex.
Note that (\ref{eq:dist-rar-objective-lagrange-2}) and (\ref{eq:rar-objective}) are very similar,
and so we use (\ref{eq:rar-objective}) for the rest the paper.
Finally, given Theorem~\ref{thm:dist-rar-duality2}, we are also able to connect (\ref{eq:dist-rar-objective})
and (\ref{eq:dist-rar-objective-lagrange-2}) with the following duality result:
\begin{theorem}
\label{thm:dist-rob-duality}
Suppose $c(z,z)=0$ and $d_{\IG}(z,z)=0$ for any $z$,
and suppose $\gamma c(z, z') - d_{\IG}(z, z')$ is a normal integrand.
For any $\rho > 0$, there exists $\gamma \ge 0$
such that the optimal solutions of (\ref{eq:dist-rar-objective-lagrange-2})
are optimal for (\ref{eq:dist-rar-objective}).
\end{theorem}
\subsection{One Layer Neural Networks}
\label{sec:one-layer-neural-networks}
\input{one-layer-neural-networks}
|
1905.09884
|
\section{Proof of Theorem \ref{thm:sparse_bnb}}\label{appendixA}
\setcounter{theorem}{0}
\begin{theorem}[Sparse Bernoulli naive Bayes]
Consider the sparse Bernoulli naive Bayes training problem \eqref{eq:bnb0}, with binary data matrix $X \in \{0,1\}^{n \times m}$. The optimal values of the variables are obtained as follows. Set
\begin{align*}\label{eq:v-w-def-sbnb}
v &:= (f^{+}+f^{-}) \circ \log \Big( \dfrac{f^{+}+f^{-}}{n}\Big) + (n\mathbf 1 -f^{+}-f^{-}) \circ \log \Big(\mathbf 1 - \dfrac{f^{+}+f^{-}}{n}\Big) , \\
w &:= w^+ + w^-, \;\; w^\pm := f^{\pm} \circ \log \dfrac{f^{+}}{n_\pm} + (n_\pm \mathbf 1 - f^{\pm}) \circ \log \Big(\mathbf 1 - \dfrac{f^{\pm}}{n_\pm}\Big) .
\end{align*}
Then identify a set ${\cal I}$ of indices with the $k$ largest elements in $w-v$, and set ${\theta^{+}_\ast},{\theta^{-}_\ast}$ according to
\begin{align*}
{\theta^{+}_{\ast_i}} = {\theta^{-}_{\ast_i}} = \frac{1}{n}(f^{+}_i + f^{-}_i), \;
\forall i \not\in {\cal I}, \;\;\;\;
{\theta^{\pm}_{\ast_i}} = \dfrac{f^{\pm}_i}{n_\pm} , \;\forall i \in {\cal I}.
\end{align*}
\end{theorem}
First note that an $\ell_0$-norm constraint on a $m$-vector $q$ can be reformulated as
\[
\|q\|_0 \leq k \Longleftrightarrow \exists \: {\cal I} \subseteq [m] , \;\; |\mathcal{I}| \leq k ~:~ \forall \: i \not\in {\cal I}, \;\;\; q_i = 0.
\]
Hence problem \eqref{eq:bnb0} is equivalent to
\begin{equation}\label{eq:noncvxNB}
\max_{\theta^{+},\theta^{-} \in [0,1]^m, \mathcal{I}} \mathcal{L}_{\text{bnb}}(\theta^{+},\theta^{-}; X) ~:~ \theta^{+}_i = \theta^{-}_i \;\;\forall i \not\in {\cal I} , \;\; \mathcal{I} \subseteq [m], \;\; |\mathcal{I}| \leq k ,
\end{equation}
where the complement of the index set ${\cal I}$ encodes the indices where variables $\theta^{+}, \theta^{-}$ agree.
Then \eqref{eq:noncvxNB} becomes
\begin{align}
p^\ast := \max_{\mathcal{I} \subseteq [m], \: |\mathcal{I}| \leq k} & \;\; \; \sum_{i \not\in \mathcal{I}} \Big( \max_{\theta_i \in [0,1]} \:
(f^{+}_i+f^{-}_i)\log \theta_i + (n-f^{+}_i-f^{-}_i) \log (1-\theta_i) \Big) \nonumber
\\
&+ \sum_{i \in \mathcal{I}} \Big( \max_{\theta^{+}_i\in [0,1]}
f^{+}_i \log \theta^{+}_i + (n_+ -f^{+}_i) \log (1 -\theta^{+}_i) \Big) \label{eq:decomp} \\
&+ \sum_{i \in \mathcal{I}} \Big( \max_{\theta^{-}_i \in [0,1]} f^{-}_i \log \theta^{-}_i+ (n_- - f^{-}_i) \log(1-\theta^{-}_i)
\Big) . \nonumber
\end{align}
where we use the fact that $n_+ + n_- = n$. All the sub-problems in the above can be solved in closed-form, yielding the optimal solutions
\begin{equation}\label{eq:theta-opt-sbnb}
{\theta^{+}_\ast}_i = {\theta^{-}_\ast}_i = \frac{1}{n}(f^{+}_i + f^{-}_i), \;\;\forall
i \not\in {\cal I}, \;\;\; \text{and} \;\;\;
{\theta^{\pm}_{\ast_i}} = \dfrac{f^{\pm}_i}{n_\pm} , \;\;\forall i \in {\cal I}.
\end{equation}
Plugging the above inside the objective of \eqref{eq:noncvxNB} results in a Boolean formulation, with a Boolean vector $u$ of cardinality $\le k$ such that $\mathbf 1-u$ encodes indices for which entries of $\theta^{+},\theta^{-}$ agree:
\begin{align*}
p^\ast :=& \max_{u \in {\cal C}_{k}} \:
(\mathbf 1 - u)^\top v + u^\top w ,
\end{align*}
where, for $k \in [m]$:
\[
\mathcal{C}_k := \{ u ~:~ u \in \{0,1\}^m, \; \textbf{1}^\top u \leq k\},
\]
and vectors $v,w$ are as defined in \eqref{eq:v-w-def-sbnb}:
\begin{align*}
v &:= (f^{+}+f^{-}) \circ \log \Big( \dfrac{f^{+}+f^{-}}{n}\Big) + (n\mathbf 1 -f^{+}-f^{-}) \circ \log \Big(\mathbf 1 - \dfrac{f^{+}+f^{-}}{n}\Big) , \\
w &:= w^+ + w^-, \;\; w^\pm := f^{\pm} \circ \log \dfrac{f^{+}}{n_\pm} + (n_\pm \mathbf 1 - f^{\pm}) \circ \log \Big(\mathbf 1 - \dfrac{f^{\pm}}{n_\pm}\Big).
\end{align*}
We obtain
\[
p^\ast = \mathbf 1^\top v + \max_{u \in {\cal C}_{k}} \: u^\top (w-v)
= \mathbf 1^\top v + s_{k}(w-v),
\]
where $s_{k}(\cdot)$ denotes the sum of the $k$ largest elements in its vector argument. Here we have exploited the fact that the map $z := w-v \ge 0$, which in turn implies that
\[
s_{k}(z) = \max_{u \in \{0,1\}^m \::\: \textbf{1}^\top u = k} \: u^\top z =
\max_{u \in {\cal C}_{k}} \: u^\top z.
\]
In order to recover an optimal pair $({\theta^{+}_\ast},{\theta^{-}_\ast})$, we simply identify the set ${\cal I}$ of indices with the $m-k$ smallest elements in $w-v$, and set ${\theta^{+}_\ast},{\theta^{-}_\ast}$ according to \eqref{eq:theta-opt-sbnb}.
\iffalse
\section*{Shapley-Folkman}
Here, we show a slight generalization of \citep[Th.\,I.3]{Ekel99}. Given the following problem
\begin{equation}\label{eq:ncvx-pb-const}\tag{cP}
\begin{array}{ll}
\mbox{minimize} & \sum_{i=1}^{n} f_i(x_i) \\
\mbox{subject to} & \sum_{i = 1}^n g_i(x_i) \leq b,
\end{array}\end{equation}
in the variables $x_i\in{\mathbb R}^{d_i}$, where the functions $g_i$ take values in ${\mathbb R}^m$.
\subsection*{Biconjugate and Convex Envelope}
Given a function $f$, not identically $+\infty$, minorized by an affine function, we write $f^*(y)\triangleq \inf_{x\in\mathop{\bf dom} f} \{y^{\top}x - f(x)\}$ the conjugate of $f$, and $f^{**}(y)$ its biconjugate. The biconjugate of $f$ (aka the convex envelope of $f$) is the pointwise supremum of all affine functions majorized by $f$ (see e.g. \citep[Th.\,12.1]{Rock70} or \citep[Th.\,X.1.3.5]{Hiri96}), a corollary then shows that $\mathop{\bf epi}(f^{**})=\overline{{\mathop {\bf Co}}(\mathop{\bf epi}(f))}$. For simplicity, we write $S^{**}=\overline{{\mathop {\bf Co}}(S)}$ for any set $S$ in what follows. We will make the following technical assumptions on the functions $f_i$.
\begin{assumption}\label{as:fi}
The functions $f_i: {\mathbb R}^{d_i} \rightarrow {\mathbb R}$ are proper, 1-coercive, lower semicontinuous and there exists an affine function minorizing them.
\end{assumption}
Note that coercivity trivially holds if $\mathop{\bf dom}(f_i)$ is compact (since $f$ is $+\infty$ outside). When Assumption~\ref{as:fi} holds, $\mathop{\bf epi}(f^{**})$, $f_i^{**}$ and hence $\sum_{i=1}^{n} f_i^{**}(x_i)$ are closed \citep[Lem.\,X.1.5.3]{Hiri96}. Finally, as in e.g. \citep{Ekel99}, we define the lack of convexity of a function as follows.
\begin{definition}\label{def:rho}
Let $f: {\mathbb R}^{d} \rightarrow {\mathbb R}$, we let $\rho(f)\triangleq \sup_{x\in \mathop{\bf dom}(f)} \{f(x) - f^{**}(x)\}$.
\end{definition}
Many other quantities measure lack of convexity (see e.g. \citep{Aubi76,Bert14} for further examples). In particular, the nonconvexity measure $\rho(f)$ can also be written
\[
\rho(f)=\sup_{\substack{x_i\in \mathop{\bf dom}(f)\\ \alpha\in{\mathbb R}^{d+1}}}~\left\{ f\left(\sum_{i=1}^{d+1}\alpha_i x_i\right) - \sum_{i=1}^{d+1}\alpha_i f(x_i): \mathbf 1^T\alpha=1,\alpha \geq 0\right\}
\]
when $f$ satisfies Assumption~\ref{as:fi} (see \citep[Th.\,X.1.5.4]{Hiri96}).
\subsection*{Bounds on the Duality Gap}
Writing the epigraph of problem~\eqref{eq:ncvx-pb-const},
\[
\mathcal{G}\triangleq \left\{(x,r_0,r) \in {\mathbb R}^{d+1+m} :~ \sum_{i=1}^{n} f_i(x_i) \leq r_0,\,\sum_{i = 1}^n g_i(x_i) \leq r \right\},
\]
and its projection on the last $m+1$ coordinates,
\begin{equation}\label{eq:epi}
\mathcal{G}_r\triangleq \left\{(r_0,r) \in {\mathbb R}^{m+1} :~ (x,r_0,r)\in \mathcal{G}\right\},
\end{equation}
we can write the Lagrange dual function of~\eqref{eq:ncvx-pb-const} as
\begin{equation}\label{eq:dual-f}
\Psi(\lambda) \triangleq \inf\left\{ r_0 + \lambda^{\top}r:~ (r_0,r) \in \mathcal{G}_r^{**} \right\},
\end{equation}
in the variable $\lambda\in{\mathbb R}^m$, where $\mathcal{G}^{**}=\overline{{\mathop {\bf Co}}(\mathcal{G})}$ is the closed convex hull of the epigraph $\mathcal{G}$ (the projection being linear here, we have $(\mathcal{G}_r)^{**}=(\mathcal{G}^{**})_r=\mathcal{G}_r^{**}$). The dual takes the generic form
\begin{equation}\label{eq:dual-pb-const}\tag{cD}
\sup_{\lambda \geq 0} \Psi(\lambda),
\end{equation}
where $\Psi$ is the dual function associated to problem~\eqref{eq:ncvx-pb-const}. Note that explicitly deriving this dual may be hard. We will also use the perturbed version of problem~\eqref{eq:ncvx-pb-const}, defined as
\begin{equation}\label{eq:p-ncvx-pb-const}\tag{p-cP}
\begin{array}{rll}
\mathrm{h}_{cP}(u) \triangleq &\mbox{min.} & \sum_{i=1}^{n} f_i(x_i) \\
&\mbox{s.t.} & \sum_{i = 1}^n g_i(x_i) - b \leq u
\end{array}\end{equation}
in the variables $x_i\in{\mathbb R}^{d_i}$, with perturbation parameter $u\in{\mathbb R}^m$. We let $\mathrm{h}_{cD} \triangleq \mathrm{h}_{cP}^{**}$ be the dual of problem~\eqref{eq:p-ncvx-pb-const} and in particular, solving for $\mathrm{h}_{cD}(0)$ is equivalent to solving problem~\eqref{eq:dual-pb-const}. Using these new definitions, we can formulate a more general bound for the duality gap (see \cite[Appendix I, Thm. 3]{Ekel99} for more details).
\begin{proposition}
Suppose the functions $f_i$ and $g_i$ in~\eqref{eq:ncvx-pb-const} are such that all $(f_i + \mathbf 1^{\top} g_i)$ satisfy Assumption~\ref{as:fi}. Then, one has
\[
\mathrm{h}_{cD}((m + 1)\bar{\rho}_g) \leq \mathrm{h}_{cP}((m + 1) \bar{\rho}_g) \leq \mathrm{h}_{cD}(0) + (m + 1) \bar{\rho}_f,
\]
where $\bar{\rho}_f = \sup_{i \in [1,n]} \rho(f_i)$ and $\bar{\rho}_g = \sup_{i \in [1, n]} \rho(g_i)$.
\end{proposition}
\begin{proof}
The global reasoning is similar to Proposition~\ref{prop:gap-sf}, using the graph of $\mathrm{h}_{cP}$ instead of the $\mathcal{F}_i$'s.
\end{proof}
\begin{proof}
Let us write
\[
\mathcal{F}_i = \left\{ \left(f_i(x_i), g_i(x_i) \right):~ x_i \in {\mathbb R}^{d_i} \right\},
\]
we have
\[
\mathcal{G}_r^{**} = \sum_{i=1}^n \mathcal{F}_i^{**}.
\]
With $\mathcal{G}_r^{**}$ closed by construction and the sets ${\mathop {\bf Co}}(\mathcal{F}_i)$ closed by Assumption~\ref{as:fi}, the Shapley-Folkman theorem \citep{Star69} shows that we have in fact
\[
\mathcal{G}_r^{**} = \sum_{i\in [1,n] \setminus \mathcal{S}} \mathcal{F}_i + \sum_{i\in \mathcal{S}} {\mathop {\bf Co}}(\mathcal{F}_i)
\]
with $|\mathcal{S}|\leq m$. This means that there are points $x_i\in{\mathbb R}^{d_i}$, for $i\in [1,n] \setminus \mathcal{S}$ and $x_{ij}\in{\mathbb R}^{d_i}$, for $i\in [1,n] \setminus \mathcal{S}$, $j\in [1,m+1]$ together with coefficients $\lambda_{ij}\geq 0$ such that $\mathbf 1^T\lambda_i=1$ and the corresponding minimizer of~\eqref{eq:dual-f} in $\mathcal{G}_r^{**}$ can be written
\begin{equation}\label{eq:zstar}
z^\star = \sum_{i\in [1,n] \setminus \mathcal{S}}
\begin{pmatrix}
f_i(x_i)\\
g_i(x_i)
\end{pmatrix}
+
\sum_{i\in \mathcal{S}}
\sum_{j\in [1,m+1]}
\lambda_{ij} \begin{pmatrix}
f_i(x_{ij})\\
g_i(x_{ij})
\end{pmatrix}
+
\begin{pmatrix}
0\\
w
\end{pmatrix}
\end{equation}
with $w \in {\mathbb R}^{m}_+$,
which we summarize as
\[
z^\star = \sum_{i=1}^n z^{(i)} +
\begin{pmatrix}
0\\
w
\end{pmatrix},
\]
where $z^{(i)} \in \mathcal{F}_i$.
Since $f_i^{**}(x)=f_i(x)$ when $x \in {\mathop {\bf Ext}}(\mathcal{F}_i)$ because $\mathop{\bf epi}(f^{**})=\overline{{\mathop {\bf Co}}(\mathop{\bf epi}(f))}$ when Assumption~\ref{as:fi} holds, we have
\begin{eqnarray*}
\overbrace{\sum_{i=1}^{n} f_i^{**}(x^\star_i)}^{\ref{eq:cvx-pb}} &=& \sum_{i\in [1,n] \setminus \mathcal{S}} f_i^{**}(x^\star_i) + \sum_{i\in \mathcal{S}}f_i(x^\star_i) + \sum_{i\in \mathcal{S}} f_i^{**}(x^\star_i) - f_i(x^\star_i) \\
& \geq & \sum_{i\in [1,n] \setminus \mathcal{S}} f_i(x^\star_i) + \sum_{i\in \mathcal{S}}f_i(x^\star_i) ~-~ \sum_{i\in \mathcal{S}} \rho(f_i)\\
& \geq & \underbrace{\sum_{i=1}^{n} f_i(\hat{x}^\star_i)}_{\ref{eq:ncvx-pb}} ~-~ \sum_{i\in \mathcal{S}} \rho(f_i)~.
\end{eqnarray*}
The last inequality holds because $x^\star$ is feasible for~\eqref{eq:ncvx-pb}
\end{proof}
\fi
\section{Proof of Theorem \ref{thm:sparse_mnb}}\label{appendixB}
\begin{theorem}[Sparse Multinomial Naive Bayes]
Let $\phi(k)$ be the optimal value of \eqref{eq:mnb0}. Then $\phi(k) \leq \psi(k)$, where $\psi(k)$ is the optimal value of the following one-dimensional convex optimization problem
\begin{equation}\label{eqa:ub}\tag{USMNB}
\psi(k) := C + \min_{\alpha \in [0,1]} \: s_k(h(\alpha) ),
\end{equation}
where $C$ is a constant, $s_{k}(\cdot)$ is the sum of the top $k$ entries of its vector argument, and for $\alpha \in (0,1)$
\[
h(\alpha) := f_+ \circ \log f_+ + f_- \circ \log f_- - (f_++f_-) \circ \log (f_++f_-) - f_+ \log \alpha - f_- \log(1-\alpha).
\]
Further, given an optimal dual variable $\alpha_\ast$ that solves \eqref{eq:ub}, we can reconstruct a primal feasible (sub-optimal) point $(\theta^{+},\theta^{-})$ for \eqref{eq:mnb0} as follows. For $\alpha^\ast$ optimal for \eqref{eq:ub}, let $\mathcal{I}$ be complement of the set of indices corresponding to the top $k$ entries of $h(\alpha_\ast)$; then set $B_\pm := \sum_{i \not\in\mathcal{I}} f^{\pm}_i$, and
\begin{equation}\label{eqa:primalsol}
{\theta^{+}_\ast}_i = {\theta^{-}_\ast}_i = \dfrac{f^{+}_i + f^{-}_i}{\mathbf 1^\top (f^{+}+f^{-})}, \;\forall i \in \mathcal{I} , \;\;\;\;
{\theta^{\pm}_{\ast_i}} = \dfrac{B_++B_-}{B_\pm} \dfrac{f^{\pm}_i}{\mathbf 1^\top (f^{+}+f^{-})}, \;\forall i \not\in \mathcal{I} .
\end{equation}
\end{theorem}
\begin{proof}
We begin by deriving the expression for the upper bound $\psi(k)$.
\paragraph{Duality bound.}
We first derive the bound stated in the theorem.
Problem~\eqref{eq:mnb0} is written
\begin{align*}
({\theta^{+}_\ast},{\theta^{-}_\ast}) &=
\arg\max_{\theta^{+}, \theta^{-} \in [0,1]^m} \: f^{+\top} \log \theta^{+} + f^{-\top} \log \theta^{-} ~:~
\begin{array}[t]{l} \mathbf 1^\top \theta^{+} = \mathbf 1^\top\theta^{-} = 1, \\ \|\theta^{+} - \theta^{-}\|_0 \leq k.
\end{array} \tag{SMNB}
\end{align*}
By weak duality we have $\phi(k) \leq \psi(k)$ where
\begin{align*}
\psi(k) := \min_{\substack{\mu^{+},\mu^{-}\\ \lambda \geq 0}} \:
\max_{\theta^{+}, \theta^{-} \in [0,1]^m} & f^{+\top} \log \theta^{+} + f^{-\top} \log \theta^{-} + \mu^{+} (1 - \textbf{1}^\top \theta^{+}) + \mu^{+} (1 - \textbf{1}^\top \theta^{+}) \\
&+ \lambda (k - \|\theta^{+}-\theta^{-}\|_0).
\end{align*}
The inner maximization is separable across the components of $\theta^{+},\theta^{-}$ since $\|\theta^{+}-\theta^{-}\|_0 = \sum_{i=1}^m \textbf{1}_{\{\theta^{+}_i\neq \theta^{-}_i\}}$. To solve it, we thus only need to consider one dimensional problems written
\begin{align}\label{eq:dual-1dmax}
\max_{q,r\in[0,1]} f^{+}_i \log q + f^{-}_i \log r - \mu^{+} q - \mu^{-} r - \lambda \mathbbm{1}_{\{q \not = r\}},
\end{align}
where $f^{+}_i,f^{-}_i>0$ and $\mu^{\pm} >0$ are given. We can split the max into two cases; one case in which $q = r$ and another when $q \not = r$, then compare the objective values of both solutions and take the larger one. Hence \eqref{eq:dual-1dmax} becomes
\[
\max \Big( \max_{u \in [0,1]} \: (f^{+}_i + f^{-}_i) \log u - (\mu^{+} + \mu^{-}) u, \max_{q,r\in[0,1]} \: f^{+}_i \log q + f^{-}_i \log r - \mu^{+} q - \mu^{-} r -\lambda \Big).
\]
Each of the individual maximizations can be solved in closed form, with optimal point
\begin{equation}\label{eq:opt-theta}
u^\ast = \dfrac{(f^{+}_i + f^{-}_i)}{\mu^{+}+\mu^{-}}, \quad q^\ast = \dfrac{f^{+}_i}{\mu^{+}}, \quad r^\ast = \dfrac{f^{-}_i}{\mu^{-}}.
\end{equation}
Note that none of $u^\ast,q^\ast,r^\ast$ can be equal to either 0 or 1, which implies $\mu^{+}, \mu^{-} > 0$. Hence \eqref{eq:dual-1dmax} reduces to
\begin{equation}\label{eq:dual-max}
\max \Big( (f^{+}_i + f^{-}_i) \log \Big( \dfrac{(f^{+}_i+f^{-}_i)}{\mu^{+} + \mu^{-}} \Big) , f^{+}_i \log \Big(\dfrac{f^{+}_i}{\mu^{+}} \Big) + f^{-}_i \log \Big(\dfrac{f^{-}_i}{\mu^{-}} \Big) - \lambda \Big) - (f^{+}_i + f^{-}_i).
\end{equation}
We obtain, with $S:=\mathbf 1^\top (f^{+}+f^{-})$,
\begin{align}\label{eq:ub2}
\psi(k) &= -S + \min_{\substack{\mu^{+},\mu^{-}>0 \\ \lambda \geq 0}} \: \mu^{+}+\mu^{-} + \lambda k + \sum_{i=1}^m \max (v_i(\mu), w_i(\mu) - \lambda) .
\end{align}
where, for given $\mu=(\mu^{+},\mu^{-})>0$,
\[
v(\mu) := (f^{+}+f^{-}) \circ \log \Big( \dfrac{f^{+}+f^{-}}{\mu^{+}+\mu^{-}}\Big) , \;\;
w(\mu) := f^{+} \circ \log \Big( \dfrac{f^{+}}{\mu^{+}} \Big) + f^{-} \circ \log \Big( \dfrac{f^{-}}{\mu^{-}} \Big).
\]
Recall the variational form of $s_{k}(z)$. For a given vector $z \ge 0$, Lemma~\ref{lem:sk} shows
\[
s_{k}(z) = \min_{\lambda \geq 0} \: \lambda k + \sum_{i=1}^m \max (0, z_i - \lambda) .
\]
Problem \eqref{eq:ub2} can thus be written
\begin{eqnarray*}
\psi(k) &=& -S+\min_{\substack{\mu>0 \\ \lambda \geq 0}} \: \mu^{+}+\mu^{-} + \lambda k + \mathbf 1^\top v(\mu) + \sum_{i=1}^m \max (0 , w_i(\mu)-v_i(\mu) - \lambda) \\
&=& -S + \min_{\mu>0} \: \mu^{+}+\mu^{-} + \mathbf 1^\top v(\mu) + s_{k}(w(\mu) - v(\mu)) ,
\end{eqnarray*}
where the last equality follows from $w(\mu)\ \ge v(\mu)$, valid for any $\mu>0$. To prove this, observe that the negative entropy function $x \rightarrow x\log x$ is convex, implying that its perspective $P$ also is. The latter is the function with domain ${\mathbb R}_+ \times {\mathbb R}_{++}$, and values for $x \ge0$, $t>0$ given by $P(x,t) = x \log (x/t)$. Since $P$ is homogeneous and convex (hence subadditive), we have, for any pair $z_+,z_-$ in the domain of $P$: $P(z_++z_-) \le P(z_+)+P(z_-)$. Applying this to $z_\pm := (f^{\pm}_i,\mu^{+}_i)$ for given $i \in [m]$ results in $w_i(\mu) \ge v_i(\mu)$, as claimed.
We further notice that the map $\mu \rightarrow w(\mu) - v(\mu)$ is homogeneous, which motivates the change of variables $\mu_\pm = t\, p_\pm$, where $t=\mu_++\mu_->0$ and $p_\pm >0$, $p_++p_- = 1$. The problem reads
\begin{align*}
\psi(k) &= -S + (f^{+}+f^{-})^\top \log(f^{+}+f^{-}) +\min_{\substack{t>0, \: p>0,\\p_++p_-=1}} \left\{ t - S \log t + s_{k}(H(p))\right\} \\&
= C + \min_{p>0, \: p_++p_-=1} \: s_k(H(p)),
\end{align*}
where $C :=(f^{+}+f^{-})^\top \log(f^{+}+f^{-}) - S \log S$, because $t = S$ at the optimum, and
\[
H(p) := v - f^{+} \circ \log p_+ - f^{-} \circ \log p_-,
\]
with
\[
v= f^{+} \circ \log f^{+} + f^{-} \circ \log f^{-} - (f^{+}+f^{-})\circ \log(f^{+} + f^{-}).
\]
Solving for $\psi(k)$ thus reduces to a 1D bisection
\[
\psi(k) = C + \min_{\alpha \in [0,1]} \: s_k(h(\alpha)),
\]
where
\[
h(\alpha) := H(\alpha,1-\alpha) = v - f^{+} \log \alpha - f^{-} \log(1-\alpha).
\]
This establishes the first part of the theorem. Note that it is straightforward to check that with $k=n$, the bound is exact: $\phi(n) = \psi(n)$.
\paragraph{Primalization.}
Next we focus on recovering a primal feasible (sub-optimal) point $(\theta^{+ \text{sub}},\theta^{- \text{sub}})$ from the dual bound obtained before. Assume that $\alpha_\ast$ is optimal for the dual problem \eqref{eq:ub}. We sort the vector $h(\alpha_\ast)$ and find the indices corresponding to the top $k$ entries. Denote the complement of this set of indices by $\mathcal{I}$. These indices are then the candidates for which $\theta^{+}_i = \theta^{-}_i$ for $i \in \mathcal{I}$ in the primal problem to eliminate the cardinality constraint. Hence we are left with solving
\begin{align}\label{eq:prml-dual-lb}
(\theta^{+ \text{sub}},\theta^{- \text{sub}}) &=
\arg\max_{\theta^{+}, \theta^{-} \in [0,1]^m} \: f^{+\top} \log \theta^{+} + f^{-\top} \log \theta^{-} \\
&\text{s.t.} \mathbf 1^\top \theta^{+} = \mathbf 1^\top\theta^{-} = 1, \nonumber \\
&\theta^{+}_i = \theta^{-}_i , \;\; i \in \mathcal{I} \nonumber
\end{align}
or, equivalently
\begin{align}\label{eq:prml-dual-chnge}
\max_{\theta,\theta^{+},\theta^{-}, s \in [0,1]} & \sum_{i \in \mathcal{I}} (f^{+}_i + f^{-}_i) \log \theta_i + \sum_{i \not\in \mathcal{I}} (f^{+}_i \log \theta^{+}_i + f^{-}_i \log \theta^{-}_i) \\
\text{s.t.} &\; \mathbf 1^\top \theta^{+} = \mathbf 1^\top \theta^{-} = 1-s , \;\; \textbf{1}^\top \theta = s . \nonumber
\end{align}
For given $\kappa \in [0,1]$, and $f \in {\mathbb R}_{++}^m$, we have
\[
\max_{u \::\: \mathbf 1^\top u = \kappa} \; f^\top \log(u) = f^\top \log f - (\mathbf 1^\top f) \log (\mathbf 1^\top f) + (\mathbf 1^\top f) \log \kappa,
\]
with optimal point given by $u^\ast = (\kappa/(\mathbf 1^\top f))f$. Applying this to problem \eqref{eq:prml-dual-chnge}, we obtain that the optimal value of $s$ is given by
\[
s^\ast = \arg\max_{s \in (0,1)} \: \{ A \log s + B \log (1-s) \} = \frac{A}{A+B} ,
\]
where
\[
A := \sum_{i \in \mathcal{I}} (f^{+}_i+f^{-}_i) , \;\; B_\pm := \sum_{i \not\in \mathcal{I}} f^{\pm}_i, \;\; B :=B_++B_- = \mathbf 1^\top (f^{+}+f^{-}) - A.
\]
We obtain
\[
\theta^{+ \text{sub}}_i = \theta^{- \text{sub}}_i = \frac{s^\ast}{A}(f^{+}_i+f^{-}_i), \;\; i \in \mathcal{I}, \;\;
\theta^{\pm \text{sub}}_i = \frac{(1-s^\ast)}{B_\pm (A+B)} f^{\pm}_i, \;\; i \not\in \mathcal{I},
\]
which further reduces to the expression stated in the theorem.
\end{proof}
\section{Proof of Theorem \protect\ref{thm:sparse_mnb_quality}}
\label{appendixC}
The proof follows from results by \citep{Aubi76} (see also \citep{Ekel99,dAsp17} for a more recent discussion) which are briefly summarized below for the sake of completeness. Given functions $f_i$, a vector $b \in {\mathbb R}^m$, and vector-valued functions $g_i$, $i\in[n]$ that take values in ${\mathbb R}^m$, we consider the following problem:
\begin{equation}\label{eq:p-ncvx-pb-const}\tag{P}
\mathrm{h}_{P}(u) := \min_x \: \sum_{i=1}^{n} f_i(x_i) ~:~ \sum_{i = 1}^n g_i(x_i) \leq b + u
\end{equation}
in the variables $x_i\in{\mathbb R}^{d_i}$, with perturbation parameter $u\in{\mathbb R}^m$. We first recall some basic results about conjugate functions and convex envelopes.
\paragraph{Biconjugate and convex envelope.}
Given a function $f$, not identically $+\infty$, minorized by an affine function, we write
\[
f^*(y)\triangleq \inf_{x\in\mathop{\bf dom} f} \{y^{\top}x - f(x)\}
\]
the conjugate of $f$, and $f^{**}(y)$ its biconjugate. The biconjugate of $f$ (aka the convex envelope of $f$) is the pointwise supremum of all affine functions majorized by $f$ (see e.g. \citep[Th.\,12.1]{Rock70} or \citep[Th.\,X.1.3.5]{Hiri96}), a corollary then shows that $\mathop{\bf epi}(f^{**})=\overline{{\mathop {\bf Co}}(\mathop{\bf epi}(f))}$. For simplicity, we write $S^{**}=\overline{{\mathop {\bf Co}}(S)}$ for any set $S$ in what follows. We will make the following technical assumptions on the functions $f_i$ and $g_i$ in our problem.
\begin{assumption}\label{as:fi}
The functions $f_i: {\mathbb R}^{d_i} \rightarrow {\mathbb R}$ are proper, 1-coercive, lower semicontinuous and there exists an affine function minorizing them.
\end{assumption}
Note that coercivity trivially holds if $\mathop{\bf dom}(f_i)$ is compact (since $f$ can be set to $+\infty$ outside w.l.o.g.). When Assumption~\ref{as:fi} holds, $\mathop{\bf epi}(f^{**})$, $f_i^{**}$ and hence $\sum_{i=1}^{n} f_i^{**}(x_i)$ are closed \citep[Lem.\,X.1.5.3]{Hiri96}. Also, as in e.g. \citep{Ekel99}, we define the lack of convexity of a function as follows.
\begin{definition}\label{def:rho}
Let $f: {\mathbb R}^{d} \rightarrow {\mathbb R}$, we let
\begin{equation}\label{eq:rho}
\rho(f)\triangleq \sup_{x\in \mathop{\bf dom}(f)} \{f(x) - f^{**}(x)\}
\end{equation}
\end{definition}
Many other quantities measure lack of convexity (see e.g. \citep{Aubi76,Bert14} for further examples). In particular, the nonconvexity measure $\rho(f)$ can be rewritten as
\begin{equation}\label{def:alt-lack-cvx}
\rho(f)=\sup_{\substack{x_i\in \mathop{\bf dom}(f)\\ \mu\in{\mathbb R}^{d+1}}}~\left\{ f\left(\sum_{i=1}^{d+1}\mu_i x_i\right) - \sum_{i=1}^{d+1}\mu_i f(x_i): \mathbf 1^\top \mu=1,\mu \geq 0\right\}
\end{equation}
when $f$ satisfies Assumption~\ref{as:fi} (see \citep[Th.\,X.1.5.4]{Hiri96}).
\paragraph{Bounds on the duality gap and the Shapley-Folkman Theorem}
Let $\mathrm{h}_{P}(u)^{**}$ be the biconjugate of $\mathrm{h}_{P}(u)$ defined in~\eqref{eq:p-ncvx-pb-const}, then $\mathrm{h}_{P}(0)^{**}$ is the optimal value of the dual to~\eqref{eq:p-ncvx-pb-const} \citep[Lem.\,2.3]{Ekel99}, and \citep[Th.\,I.3]{Ekel99} shows the following result.
\begin{theorem}\label{th:sf}
Suppose the functions $f_i,g_{ji}$ in problem~\eqref{eq:p-ncvx-pb-const} satisfy Assumption~\ref{as:fi} for $i=1,\ldots,n$, $j=1,\ldots,m$. Let
\begin{equation}\label{eq:sf-pbar}
\bar p_j = (m+1) \max_i \rho(g_{ji}), \quad \mbox{for $j=1,\ldots,m$}
\end{equation}
then
\begin{equation}\label{eq:sf-bnd}
\mathrm{h}_{P}(\bar p) \leq \mathrm{h}_{P}(0)^{**} + (m+1)\max_i \rho(f_i).
\end{equation}
where $\rho(\cdot)$ is defined in Def.~\ref{def:rho}.
\end{theorem}
We are now ready to prove Theorem~\ref{thm:sparse_mnb_quality}, whose proof follows from Theorem~\ref{th:sf} above.
\begin{theorem}[Quality of Sparse Multinomial Naive Bayes Relaxation]\label{thma:sparse_mnb_quality}
Let $\phi(k)$ be the optimal value of \eqref{eq:mnb0} and $\psi(k)$ that of the convex relaxation in~\eqref{eq:ub}, we have for $k \ge 4$,
\begin{align*}
\psi(k-4) \leq \phi(k) \leq \psi(k) \leq \phi(k + 4).
\end{align*}
for $k\geq 4$.
\end{theorem}
\begin{proof}
Problem~\eqref{eq:mnb0} is {\em separable} and can be written in perturbation form as in the result by \citep[Th.\,I.3]{Ekel99} recalled in Theorem~\ref{th:sf}, to get
\begin{equation}\label{eq:pert-p}
\begin{array}{rll}
\mathrm{h}_{P}(u) = & \min_{q,r} & -f^{+\top} \log q -f^{-\top} \log r\\
& \text{subject to} & \textbf{1}^\top q = 1 + u_1,\\
& & \textbf{1}^\top r = 1 + u_2,\\
& & \sum_{i=1}^m \mathbf 1_{q_i \neq r_i} \leq k + u_3
\end{array}\end{equation}
in the variables $q,r\in [0,1]^m$, where $u\in{\mathbb R}^3$ is a perturbation vector. By construction, we have $\phi(k)=-\mathrm{h}_{P}(0)$ and $\phi(k+l)=-\mathrm{h}_{P}((0,0,l))$. Note that the functions $\mathbf 1_{q_i \neq r_i}$ are lower semicontinuous and, because the domain of problem~\eqref{eq:mnb0} is compact, the functions
\[
f^{+}_i \log q_i + q_i + f^{-}_i \log r_i + r_i + \mathbf 1_{q_i \neq r_i}
\]
are 1-coercive for $i=1,\ldots,m$ on the domain and satisfy Assumption~\ref{as:fi} above.
Now, because $q,r\geq 0$ with $\mathbf 1^\top q=\mathbf 1^\top r=1$, we have $q-r\in[-1,1]^m$ and the convex envelope of $\mathbf 1_{q_i \neq r_i}$ on $q,r\in [0,1]^m$ is $|q_i-r_i|$, hence the {\em lack of convexity}~\eqref{def:alt-lack-cvx} of $\mathbf 1_{q_i \neq r_i}$ on $[0,1]^2$ is bounded by one, because
\[
\rho(\mathbf 1_{x \neq y}) := \sup_{x,y\in[0,1]} \{\mathbf 1_{y \neq x} - |x-y|\} = 1
\]
which means that $\max_{i=1,\ldots,n} \rho(g_{3i})=1$ in the statement of Theorem~\ref{th:sf}. The fact that the first two constraints in problem~\eqref{eq:pert-p} are convex means that $\max_{i=1,\ldots,n} \rho(g_{ji})=0$ for $j=1,2$, and the perturbation vector in~\eqref{eq:sf-pbar} is given by $\bar p=(0,0,4)$, because there are three constraints in problem~\eqref{eq:pert-p} so $m=3$ in~\eqref{eq:sf-pbar}, hence
\[
\mathrm{h}_{P}(\bar p)= \mathrm{h}_{P}((0,0,4))=-\phi(k+4).
\]
The objective function being convex separable, we have $\max_{i=1,\ldots,n} \rho(f_i)=0$. Theorem~\ref{th:sf} then states that
\[
\mathrm{h}_{P}(\bar p)= \mathrm{h}_{P}((0,0,4))= -\phi(k+4) \leq \mathrm{h}_{P}(0)^{**} + 0 = -\psi(k)
\]
because $-\mathrm{h}_{P}(0)^{**}$ is the optimal value of the dual to $\phi(k)$ which is here $\psi(k)$ defined in Theorem~\ref{thm:sparse_mnb}. The other bound in~\eqref{eq:gap-bnd}, namely $\phi(k) \leq \psi(k)$,
follows directly from weak duality.
\end{proof}
\paragraph{Primalization.}
We first derive the second dual of problem~\eqref{eq:p-ncvx-pb-const}, i.e. the dual of problem~\eqref{eq:ub}, which will be used to extract good primal solutions.
\begin{proposition}\label{prop:bidual}
A dual of problem~\eqref{eq:ub} is written
\begin{equation}\label{eq:p-cvx-pb-const}\tag{D}
\begin{array}{ll}
\mbox{max.} & z^\top (g\circ \log(g))+ x^\top (f^{+}\circ\log(f^{+})+f^{-}\circ\log(f^{-})) + (x^\top g)\log (x^\top g) - (x^\top g)\\
\\
& - (\mathbf 1^\top g) \log(\mathbf 1^\top g) - (x^\top f^{+}) \log(x^\top f^{+}) - (x^\top f^{-}) \log(x^\top f^{-}) \\
&\\
\mbox{s.t.} & x+z=\mathbf 1, \;\; \mathbf 1^\top x\leq k, \;\; x\geq 0, \;\; z\geq 0
\end{array}\end{equation}
in the variables $x,z\in{\mathbb R}^n$. Furthermore, strong duality holds between the dual~\eqref{eq:ub} and its dual~\eqref{eq:p-cvx-pb-const}.
\end{proposition}
\begin{proof}
The dual optimum value $\psi(k)$ in~\eqref{eq:ub} can be written as in~\eqref{eq:ub2},
\[
\psi(k) = -S + \min_{\substack{\mu^{+},\mu^{-}>0 \\ \lambda \geq 0}} \: \mu^{+}+\mu^{-} + \lambda k + \sum_{i=1}^m \max (v_i(\mu), w_i(\mu) - \lambda).
\]
with $S:=\mathbf 1^\top (f^{+}+f^{-})$, and
\[
v(\mu) := (f^{+}+f^{-}) \circ \log \Big( \dfrac{f^{+}+f^{-}}{\mu^{+}+\mu^{-}}\Big) , \;\;
w(\mu) := f^{+} \circ \log \Big( \dfrac{f^{+}}{\mu^{+}} \Big) + f^{-} \circ \log \Big( \dfrac{f^{-}}{\mu^{-}} \Big).
\]
for given $\mu=(\mu^{+},\mu^{-})>0$. This can be rewritten
\[
\min_{\substack{\mu^{+},\mu^{-}>0 \\ \lambda \geq 0}}~\max_{\substack{x+z=1\\x,z\geq 0}}~
\mu^{+} + \mu^{-} -S + \lambda (k-\mathbf 1^\top x) + z^\top v(\mu) + x^\top w(\mu)\]
using additional variables $x,z\in{\mathbb R}^n$, or again
\begin{equation}\label{eq:D-minmax}
\min_{\substack{\mu^{+},\mu^{-}>0 \\ \lambda \geq 0}}~\max_{\substack{x+z=1\\x,z\geq 0}}~
\begin{array}[t]{l}
\lambda (k-\mathbf 1^\top x) - (x+z)^\top g - (z^\top g) \log(\mu^{+}+\mu^{-}) +z^\top (g\circ\log(g))\\
- (x^\top f^{+})\log(\mu^{+}) - (x^\top f^{-}) \log(\mu^{-}) \\
+ x^\top (f^{+}\circ\log(f^{+})+f^{-}\circ\log(f^{-}))+ \mu^{+} + \mu^{-}
\end{array}\end{equation}
calling $g=f^{+}+f^{-}$. Strong duality holds in this min max problem so we can switch the min and the max.
Writing $\mu_\pm = t\, p_\pm$, where $t=\mu_+ +\mu_-$ and $p_\pm >0$, $p^+ +p^- = 1$ the Lagrangian becomes
\begin{eqnarray*}
L(p_+,p_-,t,\lambda,x,z,\alpha) &=&\mathbf 1^\top \nu - z^\top \nu - x^\top \nu + \lambda k - \lambda \mathbf 1^\top x - \mathbf 1^\top g - (z^\top g) \log(t)\\
&& - (x^\top f^{+})\log(t\,p_+) - (x^\top f^{-}) \log(t\,p_-) + t\\
&& +z^\top (g\circ\log(g)) + x^\top (f^{+}\circ\log(f^{+})+f^{-}\circ\log(f^{-}))\\
&& + \alpha (p_+ +p_- -1),
\end{eqnarray*}
where $\alpha$ is the dual variable associated with the constraint $p_++p_-=1$. The dual of problem~\eqref{eq:ub} is then written
\begin{eqnarray*}
\sup_{\{x\geq 0,z\geq 0,\alpha\}} ~\inf_{\substack{p_+\geq 0,p_-\geq 0,\\t\geq 0,\lambda\geq 0}} L(p_+,p_-,t,\mu^{-},\lambda,x,z,\alpha)
\end{eqnarray*}
The inner infimum will be $-\infty$ unless $\mathbf 1^\top x \leq k$, so the dual becomes
\[
\sup_{\substack{x+z=\mathbf 1, \mathbf 1^\top x\leq k,\\x\geq 0,z\geq 0,\alpha}} ~\inf_{\substack{p_+\geq 0,p_-\geq 0,\\t\geq 0}}
\begin{array}[t]{l}
z^\top (g\circ \log(g))+ x^\top (f^{+}\circ\log(f^{+})+f^{-}\circ\log(f^{-})) \\
- (x^\top f^{+}) (\log t + \log (p_+)) - (x^\top f^{-}) (\log t + \log (p_-))\\
+ t - \mathbf 1^\top g -(z^\top g)\log(t) + \alpha(p_+ + p_- -1)
\end{array}
\]
and the first order optimality conditions in $t,p_+, p_-$ yield
\begin{eqnarray}\label{eq:dual-opt}
t & = & \mathbf 1^\top g\\
p_+ & = & (x^\top f^{+})/\alpha\nonumber\\
p_- & = & (x^\top f^{-})/\alpha\nonumber
\end{eqnarray}
which means the above problem reduces to
\[
\sup_{\substack{x+z=\mathbf 1, \mathbf 1^\top x\leq k,\\x\geq 0,z\geq 0,\alpha}}
\begin{array}[t]{l}
z^\top (g\circ \log(g))+ x^\top (f^{+}\circ\log(f^{+})+f^{-}\circ\log(f^{-})) \\
- (\mathbf 1^\top g) \log(\mathbf 1^\top g) - (x^\top f^{+}) \log(x^\top f^{+}) - (x^\top f^{-}) \log(x^\top f^{-}) \\
+ (x^\top g)\log \alpha -\alpha
\end{array}\]
and setting in $\alpha=x^\top g$ leads to the dual in~\eqref{eq:p-cvx-pb-const}.
\end{proof}
We now use this last result to better characterize scenarios where the bound produced by problem~\eqref{eqa:ub} is tight and recovers an optimal solution to problem~\eqref{eq:mnb0}.
\begin{proposition}\label{prop:nonbinary}
Given $k>0$, let $\phi(k)$ be the optimal value of \eqref{eq:mnb0}. Given an optimal solution $(x,z)$ of problem~\eqref{eq:p-cvx-pb-const}, let $J=\{i:x_i \notin \{0,1\}\}$ be the set of indices where $x_i,z_i$ are not binary in $\{0,1\}$. There is a feasible point $\bar \theta,\bar \theta^+,\bar \theta^-$ of problem~\eqref{eq:mnb0} for $\bar k = k + |J|$, with objective value OPT such that
\[
\phi(k) \leq OPT \leq \phi(k + |J|).
\]
\end{proposition}
\begin{proof}
Using the fact that
\[
\max_{x}~ a \log(x) - bx = a \log \left(\frac{a}{b}\right) -a
\]
the max min problem in \eqref{eq:D-minmax} can be rewritten as
\begin{equation}\label{eq:max-min-max}
\max_{\substack{x+z=1\\x,z\geq 0}}~\min_{\substack{\mu^{+},\mu^{-}>0 \\ \lambda \geq 0}}~\max_{\theta,\theta^+,\theta^-}~
\begin{array}[t]{l}
\lambda (k-\mathbf 1^\top x) + z^\top (g\circ \log\theta)\\
+ x^\top( f^{+} \circ \log\theta^+) + x^\top (f^{-} \circ \log\theta^-)\\
+ \mu^{+}(1 - z^\top \theta - x^\top \theta^+) + \mu^{-}(1 - z^\top \theta - x^\top \theta^-)
\end{array}\end{equation}
in the additional variables $\theta,\theta^+,\theta^-\in{\mathbb R}^n$, with~\eqref{eq:opt-theta} showing that
\[
\theta_i = \dfrac{(f^{+}_i + f^{-}_i)}{\mu^{+}+\mu^{-}}, \quad \theta_i^+ = \dfrac{f^{+}_i}{\mu^{+}}, \quad \theta_i^- = \dfrac{f^{-}_i}{\mu^{-}}.
\]
at the optimum. Strong duality holds in the inner min max, which means we can also rewrite problem~\eqref{eq:p-cvx-pb-const} as
\begin{equation}\label{eq:d-epi}
\max_{\substack{x+z=1\\x,z\geq 0}}~\max_{\substack{z^\top \theta + x^\top \theta^+\leq 1\\z^\top \theta + x^\top \theta^-\leq 1\\ x^\top \mathbf 1 \leq k}}~ z^\top (g\circ \log\theta) + x^\top( f^{+} \circ \log\theta^+ + f^{-} \circ \log\theta^-)
\end{equation}
or again, in epigraph form
\begin{equation}\label{eq:dual-epi}
\begin{array}{ll}
\mbox{max.} & r\\
\mbox{s.t.} &
\begin{pmatrix}
r\\
1\\
1\\
k
\end{pmatrix}
\in
\begin{pmatrix}
0\\
{\mathbb R}_+\\
{\mathbb R}_+\\
{\mathbb R}_+
\end{pmatrix}
+
\sum_{i=1}^n \left\{ z_i
\begin{pmatrix}
g_i \log\theta_i\\
\theta_i\\
\theta_i\\
0
\end{pmatrix}
+ x_i
\begin{pmatrix}
f^{+}_i\log\theta_i^+ + f^{-}_i\log\theta_i^-\\
\theta_i^+\\
\theta_i^-\\
1
\end{pmatrix}
\right\}
\end{array}\end{equation}
Suppose the optimal solutions $x^\star,z^\star$ of problem~\eqref{eq:p-cvx-pb-const} are binary in $\{0,1\}^n$ and let $\mathcal{I}=\{i:z_i=0\}$, then problem~\label{eq:d-epi} (hence problem~\eqref{eq:p-cvx-pb-const}) reads
\begin{align}\label{eq:prml-dual-lb}
(\theta^{+ \text{sub}},\theta^{- \text{sub}}) = \arg\max_{\theta^{+}, \theta^{-} \in [0,1]^m}&
\: f^{+\top} \log \theta^{+} + f^{-\top} \log \theta^{-} \\
\text{s.t.} & \mathbf 1^\top \theta^{+} = \mathbf 1^\top\theta^{-} = 1, \nonumber \\
&\theta^{+}_i = \theta^{-}_i , \;\; i \in \mathcal{I}. \nonumber
\end{align}
which is exactly \eqref{eq:prml-dual-lb}. This means that the optimal values of problem~\eqref{eq:prml-dual-lb} and~\eqref{eq:p-cvx-pb-const} are equal, so that the relaxation is tight and $\theta_i^+=\theta_i^-$ for $i\in\mathcal{I}$.
Suppose now that some coefficients $x_i$ are not binary. Let us call $J$ the set $J=\{i:x_i \notin \{0,1\}\}$. As in \citep[Th.~I.3]{Ekel99}, we define new solutions $\bar \theta,\bar \theta^+,\bar \theta^-$ and $\bar x, \bar z$ as follows,
\[
\left\{\begin{array}{ll}
\bar \theta_i=\theta_i,~\bar \theta^+_i=\theta^+_i,~\bar \theta^-_i=\theta^-_i \mbox{ and } \bar z_i = z_i,~\bar x_i = x_i & \mbox{if $i \notin J$}\\
\bar \theta_i=0,~\bar \theta^+_i = z_i\theta + x_i\theta^+_i ,~\bar \theta^-_i = z_i\theta + x_i\theta^-_i \mbox{ and } \bar z_i = 0,~\bar x_i = 1 & \mbox{if $i \in J$}
\end{array}\right.\]
By construction, the points $\bar \theta,\bar \theta^+,\bar \theta^-$ and $\bar z, \bar x$ satisfy the constraints
$\bar z^\top \bar \theta + \bar x^\top \bar \theta^+\leq 1$, $\bar z^\top \bar \theta + \bar x^\top \bar \theta^-\leq 1$ and $\bar x^\top \mathbf 1 \leq k$. We also have $\bar x^\top \leq k + |J|$ and
\begin{eqnarray*}
&& z^\top ((f^{+}+f^{-})\circ \log\theta) + x^\top( f^{+} \circ \log\theta^+ + f^{-} \circ \log\theta^-)\\
&\leq & \bar z^\top ((f^{+}+f^{-})\circ \log\bar\theta) + \bar x^\top( f^{+} \circ \log\bar\theta^+ + f^{-} \circ \log\bar\theta^-)
\end{eqnarray*}
by concavity of the objective, hence the last inequality.
\end{proof}
We will now use the Shapley-Folkman theorem to bound the number of nonbinary coefficients in Proposition~\ref{prop:bidual} and construct a solution to~\eqref{eq:p-cvx-pb-const} satisfying the bound in Theorem~\ref{thm:sparse_mnb_quality}.
\begin{proposition}\label{prop:sf-feas}
There is a solution to problem~\eqref{eq:p-cvx-pb-const} with at most four nonbinary pairs $(x_i,z_i)$.
\end{proposition}
\begin{proof}
Suppose $(x^\star,z^\star,r^\star)$ and $(\theta,\theta^+_i,\theta^-_i)$ solve problem~\eqref{eq:p-cvx-pb-const} written as in~\eqref{eq:d-epi}, we get
\begin{equation}\label{eq:dual-epi}
\begin{pmatrix}
r^\star\\
1 - s_1\\
1 - s_2\\
k - s_3
\end{pmatrix}
=
\sum_{i=1}^n \left\{ z_i
\begin{pmatrix}
g_i \log\theta_i\\
\theta_i\\
\theta_i\\
0
\end{pmatrix}
+ x_i
\begin{pmatrix}
f^{+}_i\log\theta_i^+ + f^{-}_i\log\theta_i^-\\
\theta_i^+\\
\theta_i^-\\
1
\end{pmatrix}
\right\}
\end{equation}
where $s_1,s_2,s_3 \geq 0$. This means that the point $(r^\star, 1 - s_1,1 - s_1,k - s_3)$ belongs to a Minkowski sum of segments, with
\begin{equation}\label{eq:mink}
\begin{pmatrix}
r^\star\\
1 - s_1\\
1 - s_2\\
k - s_3
\end{pmatrix}
\in
\sum_{i=1}^n {\mathop {\bf Co}} \left( \left\{
\begin{pmatrix}
g_i \log\theta_i\\
\theta_i\\
\theta_i\\
0
\end{pmatrix}
,
\begin{pmatrix}
f^{+}_i\log\theta_i^+ + f^{-}_i\log\theta_i^-\\
\theta_i^+\\
\theta_i^-\\
1
\end{pmatrix}
\right\}\right)
\end{equation}
The Shapley-Folkman theorem~\citep{Star69} then shows that
\begin{eqnarray*}
\begin{pmatrix}
r^\star\\
1 - s_1\\
1 - s_2\\
k - s_3
\end{pmatrix}
& \in &
\sum_{[1,n] \setminus \mathcal {S}} \left\{
\begin{pmatrix}
g_i \log\theta_i\\
\theta_i\\
\theta_i\\
0
\end{pmatrix}
,
\begin{pmatrix}
f^{+}_i\log\theta_i^+ + f^{-}_i\log\theta_i^-\\
\theta_i^+\\
\theta_i^-\\
1
\end{pmatrix}
\right\}\\
&& +
\sum_{\mathcal {S}} {\mathop {\bf Co}} \left( \left\{
\begin{pmatrix}
g_i \log\theta_i\\
\theta_i\\
\theta_i\\
0
\end{pmatrix}
,
\begin{pmatrix}
f^{+}_i\log\theta_i^+ + f^{-}_i\log\theta_i^-\\
\theta_i^+\\
\theta_i^-\\
1
\end{pmatrix}
\right\}\right)
\end{eqnarray*}
where $|\mathcal{S}|\leq 4$, which means that there exists a solution to~\eqref{eq:p-cvx-pb-const} with at most four nonbinary pairs $(x_i,z_i)$ with indices $i\in\mathcal{S}$.
\end{proof}
In our case, since the Minkowski sum in~\eqref{eq:mink} is a polytope (as a Minkowski sum of segments), the Shapley-Folkman result reduces to a direct application of the fundamental theorem of linear programming, which allows us to reconstruct the solution of Proposition~\ref{prop:sf-feas} by solving a linear program.
\begin{proposition}\label{prop:post}
Given $(x^\star,z^\star,r^\star)$ and $(\theta,\theta^+_i,\theta^-_i)$ solving problem~\eqref{eq:p-cvx-pb-const}, we can reconstruct a solution $(x,z)$ solving problem~\eqref{prop:bidual}, such that at most four pairs $(x_i,z_i)$ are nonbinary, by solving
\begin{equation}\label{eq:lp-postprocess}
\begin{array}{ll}
\mbox{min.} & c^\top x\\
\mbox{s.t.} & \sum_{i=1}^n (1-x_i) g_i \log\theta_i + x_i (f^{+}_i\log\theta_i^+ + f^{-}_i\log\theta_i^-) = r^\star\\
& \sum_{i=1}^n (1-x_i) \theta_i + x_i \theta^+_i \leq 1\\
& \sum_{i=1}^n (1-x_i) \theta_i + x_i \theta^-_i \leq 1\\
& \sum_{i=1}^n x_i \leq k\\
& 0 \leq x \leq 1
\end{array}\end{equation}
which is a linear program in the variable $x\in{\mathbb R}^n$ where $c\in{\mathbb R}^n$ is e.g. a i.i.d. Gaussian vector.
\end{proposition}
\begin{proof}
Given $(x^\star,z^\star,r^\star)$ and $(\theta,\theta^+_i,\theta^-_i)$ solving problem~\eqref{eq:p-cvx-pb-const}, we can reconstruct a solution $(x,z)$ solving problem~\eqref{prop:bidual}, by solving~\eqref{eq:lp-postprocess} which is a linear program in the variable $x\in{\mathbb R}^n$ where $c\in{\mathbb R}^n$ is e.g. a i.i.d. Gaussian vector. This program has $2n+4$ constraints, at least $n$ of which will be saturated at the optimum. In particular, at least $n-4$ constraints in $0 \leq x \leq 1$ will be saturated so at least $n-4$ coefficients $x_i$ will be binary at the optimum, idem for the corresponding coefficients $z_i=1-x_i$.
\end{proof}
Proposition~\ref{prop:post} shows that solving the linear program in~\eqref{eq:lp-postprocess} as a postprocessing step will produce a solution to problem~\eqref{eq:p-cvx-pb-const} with at most $n-4$ nonbinary coefficient pairs $(x_i,z_i)$. Proposition~\ref{prop:nonbinary} then shows that this solution satisfies
\[
\phi(k) \leq OPT \leq \phi(k + 4).
\]
which is the bound in Theorem~\eqref{thm:sparse_mnb_quality}.
Finally, we show a technical lemma linking the dual solution $(x,z)$ in~\eqref{eq:p-cvx-pb-const} above and the support of the $k$ largest coefficients in the computation of $s_k(h(\alpha))$ in theorem~\ref{thm:sparse_mnb}.
\begin{lemma}\label{lem:sk}
Given $c\in{\mathbb R}^n_+$, we have
\begin{equation}\label{eq:sk-min}
s_{k}(c) = \min_{\lambda \geq 0} \: \lambda k + \sum_{i=1}^n \max (0, c_i - \lambda)
\end{equation}
and given $k$, $\lambda \in [c_{[k+1]},c_{[k]}]$ at the optimum, where $c_{[1]} \geq \ldots \geq c_{[n]}$. Its dual is written
\begin{equation}\label{eq:dual-mink}
\begin{array}{ll}
\mbox{max.} & x^\top c\\
\mbox{s.t.} & \mathbf 1^\top x \leq k\\
& x + z = 1 \\
& 0 \leq z,x
\end{array}\end{equation}
When all coefficients $c_i$ are distinct, the optimum solutions $x,z$ of the dual have at most one nonbinary coefficient each, i.e. $x_i,z_i \in (0,1)$ for a single $i \in [1,n]$. If in addition $c_{[k]}>0$, the solution to~\eqref{eq:dual-mink} is binary.
\end{lemma}
\begin{proof}
Problem~\eqref{eq:sk-min} can be written
\[\begin{array}{ll}
\mbox{min.} & \lambda k + \mathbf 1^\top t\\
\mbox{s.t.} & c - \lambda \mathbf 1 \leq t\\
& 0 \leq t\\
\end{array}\]
and its Lagrangian is then
\[
L(\lambda,t,z,x) = \lambda k + \mathbf 1^\top t + x^\top(c - \lambda \mathbf 1 - t) + z^\top t.
\]
The dual to the minimization problem~\eqref{eq:sk-min} reads
\[\begin{array}{ll}
\mbox{max.} & x^\top c\\
\mbox{s.t.} & \mathbf 1^\top x \leq k\\
& x + z = 1 \\
& 0 \leq z,x
\end{array}\]
in the variable $w\in{\mathbb R}^n$, its optimum value is $s_{k}(z)$. By construction, given $\lambda \in [c_{[k+1]},c_{[k]}]$, only the $k$ largest terms in $\sum_{i=1}^m \max (0, c_i - \lambda)$ are nonzero, and they sum to $s_{k}(c)-k \lambda$. The KKT optimality conditions impose
\[
x_i(c_i - \lambda - t_i)=0
\quad \mbox{and} \quad
z_it_i=0, \quad i=1,\ldots,n
\]
at the optimum. This, together with $x + z = 1$ and $t,x,z\geq 0$, means in particular that
\begin{equation}\label{eq:slack}
\left\{\begin{array}{ll}
x_i=0, z_i=1, &\mbox{if } c_i-\lambda < 0\\
x_i=0, z_i=1, \mbox{ or } x_i=1, z_i=0 & \mbox{if } c_i-\lambda > 0
\end{array}\right.
\end{equation}
the result of the second line comes from the fact that if $c_i-\lambda > 0$ and $t_i=c_i - \lambda$ then $z_i=0$ hence $x_i=1$, if on the other hand $t_i\neq c_i - \lambda$, then $x_i=0$ hence $z_i=1$. When the coefficients $c_i$ are all distinct, $c_i-\lambda=0$ for at most a single index $i$ and~\eqref{eq:slack} yields the desired result. When $c_{[k]}>0$ and the $c_i$ are all distinct, then the only way to enforce zero gap, i.e.
\[
x^\top c = s_k(c)
\]
is to set the corresponding coefficients of $x_i$ to one.
\end{proof}
\section{Robustness} \label{subsec:rob}
In this section we consider a variant of the original naive Bayes models where the input data (matrix $X$) is imperfectly known, and subject to adversarial noise. We adopt the robust maximum likelihood (maximin) principle developed in~\citet{ben2009robust,lanckriet2002robust,bertsimas2019robust}.
\subsection{Robust maximum likelihood}
Consider a generic maximum likelihood problem
\[
\max_\theta \: \mathcal{L}(\theta; X),
\]
with $X \in {\mathbb R}^{n \times m}$ the data matrix, $\mathcal{L}$ the loss function and $\theta \in {\mathbb R}^m$ containing the parameters of the model.
We now assume that $X$ is subject to additive noise. Modeling the noisy matrix as $X+U$, where matrix $U$ is only known to belong to a given ``uncertainty set'' ${\cal U} \subseteq {\mathbb R}^{n \times m}$, the robust version of the above problem is defined as
\begin{equation}\label{eq:rob-maximin}
\max_\theta \: \min_{U \in {\cal U}} \: \mathcal{L}(\theta; X+U),
\end{equation}
where $\mathcal{L}$ is the loss function and $\theta \in {\mathbb R}^m$ contains the parameters of the model. Note that the approach generalizes well-known models, for example with the loss function $\mathcal{L}(\theta; X) = -\|X\theta - y\|_2$, with $y \in {\mathbb R}^n$ a given response vector, and the uncertainty set ${\cal U}$ chosen to be the set of matrices with largest singular value norm less than a given positive number $\rho$, the resulting robust counterpart is the so-called ``square-root'' version of ridge regression:
\[
\max_\theta \: -\|X\theta-y\|_2 - \rho \|\theta\|_2.
\]
This illustrates the connections between robust counterparts and penalization, with a penalty that depends specifically on the uncertainty set, as further elaborated in \citet{ben2009robust}. We will observe a similar connection in the context of naive Bayes.
We now consider such robust formulations of \eqref{eq:bnb_train} and \eqref{eq:mnb_train}. First, we define noise models that make sense from a practical point of view.
\subsection{Adversarial noise models}
We assume that the uncertainty affects each data point independently. This leads to uncertainty sets of the form
\[
{\cal U}(X) = \left\{ [u_1, \ldots,u_n] \in {\mathbb R}^{n \times m} ~:~ u_i \in {\cal V}(x_i) , \;\; i \in [n] \right\},
\]
where a set ${\cal V}(x) \subseteq {\mathbb R}^m$ models the uncertainty on a generic data point $x \in {\mathbb R}^n$. We focus on the following uncertainty sets, depending on the nature of the data point, either binary-, integer-, or real-valued.
In the Bernoulli model, the input is binary, so it makes sense to consider flips as a primary source of uncertainty. For a given binary vector $x \in \{0,1\}^n$ and integer $p \in [m]$,
\begin{align}\label{eq:unc-bnb}
{\cal V}_\text{flip}(x) &= \{ \delta ~:~ \delta + x \in \{0,1\}^m, \;\; \|\delta\|_1 \leq p\} . \tag{$p$-flips}
\end{align}
Thus, ${\cal V}_\text{flip}(x)$ represents the set of vectors that can be obtained by flipping at most $p\leq m$ components in a binary vector $x$.
For the multinomial model, we often deal with count vectors $x \in \mathbb{N}^n$. The set
\begin{align}\label{eq:unc-mnb}
{\cal V}_\text{typo}(x) &= \{ \delta ~:~ \delta + x \in \mathbb{N}^m, \|\delta\|_1 \leq t\} \tag{$t$-typos}
\end{align}
models the fact that count vectors can be altered by adding or substracting at most $t$ counts to $x$. This can represent the presence of typos in text data.
Finally, for a real-valued vector $x \in {\mathbb R}^n$:
\begin{align}\label{eq:unc-lmbda}
{\cal V}_\text{shift}(x) &= \{ \delta ~:~ \delta + x \in \mathbb{N}^m, \delta \in [-\gamma, \gamma]^m\cap \mathbb{Z}^m \} , \tag{$\gamma$-shift}
\end{align}
with $\gamma>0$ given. Here, ${\cal V}_\text{shift}(x)$ represents an interval uncertainty affecting each feature independently.
\subsection{Robust counterparts}
Applying the modified formulation~\eqref{eq:rob-maximin} in lieu of the classical naive Bayes ones \eqref{eq:bnb_train} and \eqref{eq:mnb_train} leads to, respectively:
\begin{align} \label{eq:r-bnb-flip}
&\max_{\theta^{+}, \theta^{-} \in [0,1]^m} \:
\min_{U \in {\cal U}_\text{flip}(X)} \: \mathcal{L}_{\text{bnb}}(\theta^{+},\theta^{-}; X+U) , \tag{R-BNB-flip}
\end{align}
and
\begin{align}\label{eq:r-mnb}
&\max_{\theta^{+}, \theta^{-} \in [0,1]^m} \:
\min_{U \in {\cal U}(X)} \:
\mathcal{L}_{\text{mnb}}(\theta^{+},\theta^{-}; X+U) ~:~ \mathbf 1^\top \theta^{+} = \mathbf 1^\top\theta^{-} = 1,\tag{R-MNB}
\end{align}
where ${\cal U}(X)$ is one of the uncertainty sets introduced above, with the exception of the first \eqref{eq:unc-bnb}.
The inner minimization can be solved exactly for all the proposed uncertainty sets. Below we derive the subsequent robust, convex optimization problems corresponding to each of the uncertainty sets.
\paragraph{Bernoulli model with flip uncertainty.}
Under $p$-flip uncertainty, \eqref{eq:r-bnb-flip} becomes
\begin{align*}
\max_{\theta^{\pm}\in[0,1]^m} \: \min_{\delta_\pm \in R_\pm} &(f^{+}+\delta_+)^\top \log \theta^{+} + (\mathbf 1 - (f^{+} + \delta_-))^\top \log (\mathbf 1 - \theta^{+}) \\
&+ (f^{-}+\delta_-)^\top \log \theta^{-} + (\mathbf 1 - (f^{-}+\delta_-))^\top \log (\mathbf 1 - \theta^{-}) ,
\end{align*}
where for given $f \in \{0,1\}^m$
\[
R(f) := \left\{\delta ~:~ \delta + f \in \{0,1\}^m, \;\; \|\delta\|_1 \leq p \right\}.
\]
First analyze the following sub-problem, for given $\theta \in (0,1)^m$ and $f \in \{0,1\}^m$:
\begin{align}\label{eq:r-decomp}
&\min_{\delta \in R(f)} \: (f+\delta)^\top \log \theta + (\mathbf 1 - (f + \delta))^\top \log (\mathbf 1 - \theta) \\&= f^\top \log \theta +
(\mathbf 1-f)^\top \log(\mathbf 1 - \theta) + \min_{\delta \in R(f)} \: \delta^\top q(\theta) , \nonumber
\end{align}
where $q(\theta):= \log \theta - \log (\mathbf 1 -\theta)$.
Since $f \in \{0,1\}^m$, the constraint $\delta + f \in \{0,1\}^m$ implies that $\delta \in \{-1,0,1\}^m$. For given $i \in [m]$, if $f_i = 0$ then $\delta_i \in \{0,1\}$ and if $f_i = 1$ then $\delta_i \in \{-1,0\}$. Then for $q \in {\mathbb R}^m$ given, define
\begin{align}
h(q;f) &:= \min_{\delta \in R(f)}\: q^\top \delta \\
& = \min_\delta \: ((\mathbf 1 - 2f) \circ q )^\top \delta ~:~ \delta \in \{0,1\}^m , \;\; \mathbf 1^\top \delta \le p \\
&=-s_p (((2f-\mathbf 1)\circ q )_+),
\end{align}
with $s_p$ the sum of the $p$ largest entries in its vector argument. To see why this is true, note that $\delta \in \{(\mathbf 1- 2f) \cup \textbf{0} \; | \; \|\delta\|_1 \leq p\}$. Hence we can either set $\delta_i =0$ or $\delta_i = (\mathbf 1 - 2f)_i$; the optimal choice will clearly depend on the sign of $q_i$ and the result follows.
Returning to the original max-min problem, we have that \eqref{eq:r-bnb-flip} becomes
\begin{align*}
\max_{\theta^{+},\theta^{-} \in [0,1]^m} &\;\;\;\;f^{+\top} \log \theta^{+} + (\mathbf 1 - f^{+})^\top \log (\mathbf 1 - \theta^{+}) - s_p (((2f^{+} - \mathbf 1)\circ \log (\theta^{+}/ (1-\theta^{+})) )_+) \\
&+f^{-\top} \log \theta^{-} + (\mathbf 1 - f^{-})^\top \log (\mathbf 1 - \theta^{-}) - s_p (((2f^{-} - \mathbf 1)\circ \log (\theta^{-}/ (1-\theta^{-})) )_+).
\end{align*}
This problem is convex, since the inner minimization is the pointwise minimum of jointly concave functions in $(\theta^{+},\theta^{-})$.
\paragraph{Multinomial model with typo uncertainty.}
Under the typo uncertainty, \eqref{eq:r-mnb} becomes
\begin{align*}
\max_{\theta^{+},\theta^{-} \in [0,1]^m} \: \min_{\delta_\pm \in R(f^{\pm})} &(f^{+} + \delta_+)^\top \log \theta^{+} + (f^{-} + \delta_-)^\top \log \theta^{-} ~:~ \mathbf 1^\top \theta^{+} = 1, \;\; \mathbf 1^\top \theta^{-} = 1 ,
\end{align*}
where
\[
R(f_i) = \left\{ \delta ~:~ \delta + f_i \in \mathbb{N}^m, \; \|\delta\|_1 \leq t \right\} .
\]
Since the uncertainty acts independently on $f^{+}$ and $f^{-}$ we can consider the sub-problem
\begin{align*}
\min_{\delta \in r(f)} (f+\delta)^\top \log \theta^{+} = f^\top \log \theta^{+} + \min_{\delta \in r(f)} \delta^\top \log \theta^{+}
\end{align*}
Note since the objective function above is linear, we can perform the minimization over the convex hull of the feasible region. Since $f \geq 0$, it follows that $$\textbf{Co}(r(f)) = \{ \delta \:|\:\delta + f\geq 0, \|\delta\|_1 \leq t\}$$
In this reformulation, the solution to the minimization is trivial. Since $\theta^{+} \in [0,1]^m$, we have that $\log \theta^{+} \leq 0$ and to minimize the inner product $\delta^\top \log \theta^{+}$ we set $\delta = t \cdot e_j$ where $e_j$ is the $j$th unit basis vector and $j$ is the index corresponding to the largest absolute value entry of $\theta^{+}$. In summary, we have that
\begin{align*}
\min_{\delta \in r(f)} \: \delta^\top \log \theta^{+} = \min_{\delta \in \textbf{Co}(r(f))} \delta^\top \log \theta^{+} = -t \|\log \theta^{+}\|_\infty
\end{align*}
Returning to the original max-min problem, we have that \eqref{eq:r-mnb} becomes
\begin{align*}
\max_{\theta^{+},\theta^{-} \in [0,1]^m} & f^{+\top} \log \theta^{+} + f^{-\top} \log \theta^{-} - t \Big( \| \log \theta^{+} \|_\infty + \| \log \theta^{-}\|_\infty \Big)\\
& \text{s.t} \;\; \mathbf 1^\top \theta^{+} = 1, \;\; \mathbf 1^\top \theta^{-} = 1
\end{align*}
Note by monotonicity of the logarithm and positivity of $\theta^{+},\theta^{-}$, the problem is equivalent to
\begin{align*}
\max_{\theta^{+},\theta^{-} \in [0,1]^m} & f^{+\top} \log \theta^{+} + f^{-\top} \log \theta^{-} - t \Big( |\log (\displaystyle\max_{1 \le i \le m} \: \theta^{+}_i)| + |\log (\displaystyle\max_{1 \le i \le m} \: \theta^{-}_i)| \Big)\\
& \text{s.t} \;\; \mathbf 1^\top \theta^{+} = 1, \;\; \mathbf 1^\top \theta^{-} = 1
\end{align*}
By construction, the problem is convex.
Note that the above problem is separable across $\theta^{+},\theta^{-}$. We now show how to reduce the problem with typo uncertainty into a one dimensional problem that can be solved using bisection. Let
\begin{align*}
p^\ast = \max_{\theta \in \Delta^m} \; f^\top \log \theta - t |\log (\displaystyle\max_{1 \le i \le m} \: \theta_i)|
\end{align*}
where $\Delta^m$ denote the simplex in $\mathbb{R}^m$. Then we have that
\begin{align*}
p^\ast &= \max_{\substack{\theta \in \Delta^m\\ \log \theta \geq z\mathbf 1}} \; f^\top \log \theta + tz \\
&= \min_\nu \max_{\substack{\theta \geq 0\\ \log \theta \geq z \mathbf 1}} \; f^\top \log \theta + tz + \nu (1 - \mathbf 1 ^\top \theta) \\
&= \min_\nu \max_{\substack{\theta \geq 0\\ \theta \geq u\mathbf 1}} \; f^\top \log \theta + t\log u + \nu (1 - \mathbf 1 ^\top \theta)
\end{align*}
where the second equality follows by forming the lagrangian and using strong duality, the third line we make the substitution $u = e^z$. We first solve the inner maximization over $\theta$. The subproblem writes
\begin{align*}
\max_{\theta \geq u\mathbf 1} \; f^\top \log \theta - \nu \mathbf 1^\top \theta
\end{align*}
Note if $\nu < 0$, then the problem is unbounded; hence $\nu \geq 0$. We then have that
\begin{align*}
\theta^\ast = \max\Big( \dfrac{f}{\nu}, u\mathbf 1 \Big)
\end{align*}
As a result, we have that $\max_i f_i := F \leq \nu$ amd $u \leq 1$ to ensure that $\theta^\ast_i \leq 1$ for all $i \in [m]$. Plugging back into the objective and rearranging we have
\begin{align*}
p^\ast &= \min_{F \leq \nu} \max_{u \in [0,1]} \; f^\top \log \Big(\max(f/\nu,u\mathbf 1)\Big) + t\log u + \nu(1 - \mathbf 1^\top \max(f/\nu,u\mathbf 1)) \\
&= \min_{F \leq \nu} \max_{u \in [0,1]} \; f^\top \log \Big(\max(f,\nu u\mathbf 1)\Big) - f^\top \mathbf 1 \log \nu + t\log u + \nu - \mathbf 1^\top \max(f, \nu u\mathbf 1) \\
&= \min_{F \leq \nu} \max_{\eta \in [0,\nu]} \; f^\top \log \Big(\max(f,\eta\mathbf 1)\Big) - f^\top \mathbf 1 \log \nu + t\log \eta/\nu + \nu - \mathbf 1^\top \max(f, \eta \mathbf 1) \\
\end{align*}
\paragraph{Multinomial model with shift uncertainty.}
Under the shift uncertainty, \eqref{eq:r-mnb} becomes
\begin{align*}
\max_{\theta^{+},\theta^{-} \in [0,1]^m} \min_{\delta_i \in r(f_i)} &(f^{+} + \delta_+)^\top \log \theta^{+} + (f^{-} + \delta_-)^\top \log \theta^{-} \\
& \text{s.t} \;\; \mathbf 1^\top \theta^{+} = 1, \;\; \mathbf 1^\top \theta^{-} = 1
\end{align*}
where $$r(f_i) = \{ \delta \; | \; \delta + f_i \in \mathbb{N}^m, \; \delta \in [-\gamma,\gamma]^m \cap \mathbb{Z}^m\}$$
Since the uncertainty acts independently on $f^{+}$ and $f^{-}$ we can consider the sub-problem
\begin{align*}
\min_{\delta \in r(f)} (f+\delta)^\top \log \theta^{+} = f^\top \log \theta^{+} + \min_{\delta \in R(f)} \delta^\top \log \theta^{+}
\end{align*}
Since the objective function above is linear, we can perform the minimization over the convex hull of the feasible set. Since $f \geq 0$, it follows that $$\textbf{Co}(r(f)) = \{ \delta \:|\: \delta + f \geq 0, \; \|\delta\|_\infty \leq \gamma\}$$
In this reformulation, the solution to the minimization is trivial. Since $\theta^{+} \in [0,1]^m$, we have that $\log \theta^{+} \leq 0$ and in order to minimize the inner product, we simply set $\delta = \gamma \mathbf 1$ (note $\gamma \mathbf 1 + f \geq 0$ since $f \geq 0$). In summary, we have that
\begin{align*}
\min_{\delta \in r(f)} \delta^\top \log \theta^{+} = \min_{\delta \in \textbf{Co}(r(f))} \delta^\top \log \theta^{+} = -\gamma \|\log \theta^{+}\|_1 = \gamma \sum_{i=1}^m \log \theta^{+}_i
\end{align*}
returning to the original max-min problem, we have that \eqref{eq:r-mnb} becomes
\begin{align*}
\max_{\theta^{+},\theta^{-} \in [0,1]^m} & (f^{+} + |C_1| \gamma \mathbf 1)^\top \log \theta^{+} + (f^{-} + |C_2| \gamma \mathbf 1)^\top \log \theta^{-} \\
& \text{s.t} \;\; \mathbf 1^\top \theta^{+} = 1, \;\; \mathbf 1^\top \theta^{-} = 1
\end{align*}
which is a convex problem (recall $|C_1|$ and $|C_2|$ are the number of data points in class 1 and class 2 in the training set).
Note the parallel between $\mathcal{V}_\text{shift}(x)$ and traditional Laplace smoothing for the Multinomial Naive Bayes model. If we assume $|C_1| = |C_2|$ and scale $\gamma$ by $1/|C_1|$, then the solution to the above problem is simply the MLE estimate and is
\begin{align*}
\theta^{\pm}^\ast = \dfrac{f^{\pm} + \gamma \mathbf 1}{\sum_{i=1}^n (f^{\pm}_i + \gamma)}
\end{align*}
which is the same performing Laplace smoothing with parameter $\gamma$. Similarly if $|C_1| \not = |C_2|$ we can set two different hyper parameters $\gamma_1, \gamma_2$ for each class and scale them by $1/|C_1|$ and $1/|C_2|$ to get the same formula as for traditional Laplacian smoothing for Multinomial Naive Bayes.
\section{Details on Datasets}\label{appendixF}
This section details the data sets used in our experiments.
\paragraph{Downloading data sets.}
\begin{enumerate}
\item \underline{AMZN} The complete Amazon reviews data set was collected from \href{https://drive.google.com/drive/folders/0Bz8a_Dbh9Qhbfll6bVpmNUtUcFdjYmF2SEpmZUZUcVNiMUw1TWN6RDV3a0JHT3kxLVhVR2M}{here}; only a subset of this data was used which can be found \href{https://gist.github.com/kunalj101/ad1d9c58d338e20d09ff26bcc06c4235}{here}. This data set was randomly split into 80/20 train/test.
\item \underline{IMDB} The large movie review (or IMDB) data set was collected from \href{http://ai.stanford.edu/~amaas//data/sentiment/}{here} and was already split 50/50 into train/test.
\item \underline{TWTR} The Twitter Sentiment140 data set was downloaded from \href{http://cs.stanford.edu/people/alecmgo/trainingandtestdata.zip}{here} and was pre-processed according to the method highlighted \href{https://towardsdatascience.com/another-twitter-sentiment-analysis-bb5b01ebad90}{here}.
\item \underline{MPQA} The MPQA opinion corpus can be found \href{http://mpqa.cs.pitt.edu/}{here} and was pre-processed using the code found \href{https://github.com/AcademiaSinicaNLPLab/sentiment_dataset}{here}.
\item \underline{SST2} The Stanford Sentiment Treebank data set was downloaded from \href{https://nlp.stanford.edu/sentiment/}{here} and the pre-processing code can be found \href{https://github.com/AcademiaSinicaNLPLab/sentiment_dataset}{here}.
\end{enumerate}
\paragraph{Creating feature vectors.} After all data sets were downloaded and pre-processed, the diffeent types of feature vectors were constructed using \texttt{CounterVectorizer} and \texttt{TfidfVectorizer} from Sklearn \citep{scikit}. Counter vector, tf-idf, and tf-idf word bigrams use the \texttt{analyzer = `word'} specification while the tf-idf char bigrams use \texttt{analyzer = `char'}.
\paragraph{Two-stage procedures.} For experiments 2 and 3, all standard models were trained in Sklearn \citep{scikit}. In particular, the following settings were used in stage 2 for each model
\begin{enumerate}
\item \texttt{LogisticRegression(penalty=`l2', solver=`lbfgs', C =1e4, max\_iter=1e2)}
\item \texttt{LinearSVC(C = 1e4)}
\item \texttt{MultinomialNB(alpha=a)}
\end{enumerate}
In the first stage of the two stage procedures, the following settings were used for each of the different feature selection methods
\begin{enumerate}
\item \texttt{LogisticRegression(random\_state=0, C = $\lambda_1$,penalty=`l1',solver=`saga',
max\_iter=1e2)}
\item \texttt{clf = LogisticRegression(C = 1e4, penalty=`l2', \
solver = `lbfgs', max\_iter = 1e2).fit(train\_x,train\_y)} \\
\texttt{selector\_log = RFE(clf, $k$), step=0.3)}
\item \texttt{Lasso(alpha = $\lambda_2$,
selection=`cyclic', tol = 1e-5)}
\item \texttt{LinearSVC(C =$\lambda_3$, \
penalty=`l1',dual=False)}
\item \texttt{clf = LinearSVC(C = 1e4, penalty=`l2',dual=False).fit(train\_x,train\_y)} \\
\texttt{selector\_svm = RFE(clf,$k$, step=0.3)}
\item \texttt{MultinomialNB(alpha=a)}
\end{enumerate}
where $\lambda_i$ are hyper-parameters used by the $\ell_1$ methods to achieve a desired sparsity level $k$. $a$ is a hyper-parameter for the different MNB models which we compute using cross validation (explained below).
\paragraph{Hyper-parameters.}
For each of the $\ell_1$ methods we manually do a grid search over all hyper-parameters to achieve an approximate desired sparsity pattern. For determining the hyper-parameter for the MNB models, we employ 10-fold cross validation on each data set for each type of feature vector and determine the best value of $a$. In total, this is $16 + 20 = 36$ values of $a$ -- $16$ for experiment 2 and $20$ for experiment 3. In experiment 2, we do not use the twitter data set since computing the $\lambda_i$'s to achieve a desired sparsity pattern for the $\ell_1$ based feature selection methods was computationally intractable.
\paragraph{Experiment 2 and 3: full results.}
Here we show the results of experiments 2 and 3 for all the data sets.
\begin{figure}[h!]
\centering
\makebox[\textwidth]{\includegraphics[height = 1\textheight]{figures/amzn_log_featselect_1d.pdf}}
\caption{Experiment 2: AMZN - Stage 2 Logistic}
\end{figure}
\begin{figure}[h!]
\centering
\makebox[\textwidth]{\includegraphics[height = 1\textheight]{figures/amzn_svm_featselect_1d.pdf}}
\caption{Experiment 2: AMZN - Stage 2 SVM}
\end{figure}
\begin{figure}[h!]
\centering
\makebox[\textwidth]{\includegraphics[height = 1\textheight]{figures/amzn_mnb_featselect_1d.pdf}}
\caption{Experiment 2: AMZN - Stage 2 MNB}
\end{figure}
\begin{figure}[h!]
\centering
\makebox[\textwidth]{\includegraphics[height = 1\textheight]{figures/imdb_log_featselect_1d.pdf}}
\caption{Experiment 2: IMDB - Stage 2 Logistic}
\end{figure}
\begin{figure}[h!]
\centering
\makebox[\textwidth]{\includegraphics[height = 1\textheight]{figures/imdb_svm_featselect_1d.pdf}}
\caption{Experiment 2: IMDB - Stage 2 SVM}
\end{figure}
\begin{figure}[h!]
\centering
\makebox[\textwidth]{\includegraphics[height = 1\textheight]{figures/imdb_mnb_featselect_1d.pdf}}
\caption{Experiment 2: IMDB - Stage 2 MNB}
\end{figure}
\begin{figure}[h!]
\centering
\makebox[\textwidth]{\includegraphics[height = 1\textheight]{figures/mpqa_log_featselect_1d.pdf}}
\caption{Experiment 2: MPQA - Stage 2 Logistic}
\end{figure}
\begin{figure}[h!]
\centering
\makebox[\textwidth]{\includegraphics[height = 1\textheight]{figures/mpqa_svm_featselect_1d.pdf}}
\caption{Experiment 2: MPQA - Stage 2 SVM}
\end{figure}
\begin{figure}[h!]
\centering
\makebox[\textwidth]{\includegraphics[height = 1\textheight]{figures/mpqa_mnb_featselect_1d.pdf}}
\caption{Experiment 2: MPQA - Stage 2 MNB}
\end{figure}
\begin{figure}[h!]
\centering
\makebox[\textwidth]{\includegraphics[height = 1\textheight]{figures/sst2_log_featselect_1d.pdf}}
\caption{Experiment 2: SST2 - Stage 2 Logistic}
\end{figure}
\begin{figure}[h!]
\centering
\makebox[\textwidth]{\includegraphics[height = 1\textheight]{figures/sst2_svm_featselect_1d.pdf}}
\caption{Experiment 2: SST2 - Stage 2 SVM}
\end{figure}
\begin{figure}[h!]
\centering
\makebox[\textwidth]{\includegraphics[height = 1\textheight]{figures/sst2_mnb_featselect_1d.pdf}}
\caption{Experiment 2: SST2 - Stage 2 MNB}
\end{figure}
\begin{figure}[h!]
\centering
\makebox[\textwidth]{\includegraphics[height = 1\textheight]{figures/amzn_nomax_mnb_featselect_1d.pdf}}
\caption{Experiment 3: AMZN - Stage 2 MNB}
\end{figure}
\begin{figure}[h!]
\centering
\makebox[\textwidth]{\includegraphics[height = 1\textheight]{figures/imdb_nomax_mnb_featselect_1d.pdf}}
\caption{Experiment 3: IMDB - Stage 2 MNB}
\end{figure}
\begin{figure}[h!]
\centering
\makebox[\textwidth]{\includegraphics[height = 1\textheight]{figures/twtr_nomax_mnb_featselect_1d.pdf}}
\caption{Experiment 3: TWTR - Stage 2 MNB}
\end{figure}
\begin{figure}[h!]
\centering
\makebox[\textwidth]{\includegraphics[height = 1\textheight]{figures/mpqa_nomax_mnb_featselect_1d.pdf}}
\caption{Experiment 3: MPQA - Stage 2 MNB}
\end{figure}
\begin{figure}[h!]
\centering
\makebox[\textwidth]{\includegraphics[height = 1\textheight]{figures/sst2_nomax_mnb_featselect_1d.pdf}}
\caption{Experiment 3: SST2 - Stage 2 MNB}
\end{figure}
\section*{Amazon Data set}
\begin{figure}[H]
\vspace*{-0.5cm}
\makebox[\linewidth]{
\includegraphics[width=1.25\linewidth]{figures/amzn_log_featselect.png}
}
\caption{Amazon Dataset. Stage 2: Logistic Regression.}
\end{figure}
\begin{figure}[H]
\vspace*{-0.5cm}
\makebox[\linewidth]{
\includegraphics[width=1.25\linewidth]{figures/amzn_mnb_featselect.png}
}
\caption{Amazon Dataset. Stage 2: MNB.}
\end{figure}
\begin{figure}[H]
\vspace*{-0.5cm}
\makebox[\linewidth]{
\includegraphics[width=1.25\linewidth]{figures/amzn_svm_featselect.png}
}
\caption{Amazon Dataset. Stage 2: SVM.}
\end{figure}
\section*{IMDB Data set}
\begin{figure}[H]
\vspace*{-0.5cm}
\makebox[\linewidth]{
\includegraphics[width=1.25\linewidth]{figures/imdb_log_featselect.png}
}
\caption{IMDB Dataset. Stage 2: Logistic Regression.}
\end{figure}
\begin{figure}[H]
\vspace*{-0.5cm}
\makebox[\linewidth]{
\includegraphics[width=1.25\linewidth]{figures/imdb_mnb_featselect.png}
}
\caption{IMDB Dataset. Stage 2: MNB.}
\end{figure}
\begin{figure}[H]
\vspace*{-0.5cm}
\makebox[\linewidth]{
\includegraphics[width=1.25\linewidth]{figures/imdb_svm_featselect.png}
}
\caption{IMDB Dataset. Stage 2: SVM.}
\end{figure}
\section*{MPQA Data set}
\begin{figure}[H]
\vspace*{-0.5cm}
\makebox[\linewidth]{
\includegraphics[width=1.25\linewidth]{figures/mpqa_log_featselect.png}
}
\caption{MPQA Dataset. Stage 2: Logistic Regression.}
\end{figure}
\begin{figure}[H]
\vspace*{-0.5cm}
\makebox[\linewidth]{
\includegraphics[width=1.25\linewidth]{figures/mpqa_mnb_featselect.png}
}
\caption{MPQA Dataset. Stage 2: MNB.}
\end{figure}
\begin{figure}[H]
\vspace*{-0.5cm}
\makebox[\linewidth]{
\includegraphics[width=1.25\linewidth]{figures/mpqa_svm_featselect.png}
}
\caption{MPQA Dataset. Stage 2: SVM.}
\end{figure}
\section*{SST2 Data set}
\begin{figure}[H]
\vspace*{-0.5cm}
\makebox[\linewidth]{
\includegraphics[width=1.25\linewidth]{figures/sst2_log_featselect.png}
}
\caption{SST2 Dataset. Stage 2: Logistic Regression.}
\end{figure}
\begin{figure}[H]
\vspace*{-0.5cm}
\makebox[\linewidth]{
\includegraphics[width=1.25\linewidth]{figures/sst2_mnb_featselect.png}
}
\caption{SST2 Dataset. Stage 2: MNB.}
\end{figure}
\begin{figure}[H]
\vspace*{-0.5cm}
\makebox[\linewidth]{
\includegraphics[width=1.25\linewidth]{figures/sst2_svm_featselect.png}
}
\caption{SST2 Dataset. Stage 2: SVM.}
\end{figure}
\section{Introduction}
Machine learning models are being trained on data sets where the inputs are often very high dimensional (particularly in text data sets) and feature selection is a key component of model training pipelines. We expect sparse models to be easier to interpret and to generalize better.
Feature selection means introducing $\ell_0$ constraints or $\ell_1$ regularization terms. While the former is non-convex and in general NP hard to solve, the latter is the method of choice for almost all models; it is convex and can be added to most existing training problems. However, this comes at a cost and makes optimization problems non-smooth and potentially more difficult to train.
One machine learning model that has persisted despite its simplicity is Na\"ive Bayes. It is still one of the models of choice for text applications ({\color{blue} XXX insert citation XXX}) because of both classification performance and computational efficiency. Although there have been many improvements to the baseline Na\"ive Bayes model ({\color{blue} XXX insert citations XXX}), there were few attempts so far to make it sparse.
\subsection{Paper Contributions}
In this paper, we construct a sparse Na\"ive Bayes model and test it on five different text data sets. Our model is able to maintain predictive power while only using a fraction of the features compared to the traditional Na\"ive Bayes model. Furthermore, we compare our method with other popular feature selection methods and show that our method is competitive while only taking a fraction of computational time. Furthermore, we show how in addition to sparsity we are able to make the model robust to different types of uncertainty.
\subsection{Paper Contributions}
In this paper, we construct a sparse naive Bayes model and test it on five different text data sets. Our model is able to maintain predictive power while only using a fraction of the features compared to the traditional naive Bayes model. Furthermore, we compare our method with other popular feature selection methods and show that our method is competitive while only taking a fraction of computational time. Furthermore, we show how in addition to sparsity we are able to make the model robust to different types of uncertainty.
\subsection{Related Work}
\paragraph{naive Bayes Improvements.} A large body of literature builds on the traditional naive Bayes classifier. A non-extensive list includes the seminal work by \citet{frank2002locally} introducing Weighted naive Bayes, Lazy Bayesian Learning by \citet{zheng2000lazy}, and Tree Augmented naive Bayes by \citet{friedman1997bayesian}. \citet{webb2005not} improves the computational complexity of the aforementioned methods while maintaining the same accuracy. For a more complete discussion of different modifications to naive Bayes, we refer the reader to \citet{jiang2007survey} and the references therein.
\paragraph{naive Bayes \& Feature Selection} Of particular interest to this work are methods that employ feature selection. These methods would improve classification accuracy by efficiently selecting a relevant subset of features from the training set. \citet{kim2006some} use information theoretic quantities for feature selection in text classification while \citet{mladenic1999feature} compare a myriad of different methods and show the efficacy of the Odds Ratio metric. These feature selection methods often use adhoc scoring functions to rank the importance of the different features. The first work that employs an $\ell_1$ penalized optimization problem at training time is \citet{zheng2018sparse} which introduces an $\ell_1$ penalty at training time. However, they work in the weighted naive bayes setting and their classifier is sparse in the logarithm of the conditional probability of the features given the classes whereas for interpretability reasons, we seek sparsity in the features themselves. Additionally, they are need to solve a relaxation of their training problem using a possibly ill-conditioned design matrix of posterior probabilities whereas we are able to solve the $\ell_0$ and $\ell_1$ variant of our problem either in closed form, or with provably small gap.
\paragraph{naive Bayes \& Robustness}
To the best of our knowledge, there is no work that attempts to make the classical naive Bayes classifier robust to various types of uncertainty. In this work we show the parallels between robust optimization and traditional Laplace smoothing used in naive Bayes.
\section{Background and Notation}
\paragraph{Notation} We are given a data matrix $X \in \mathbb{R}^{m \times n} = [x^{(1)},x^{(2)},\hdots,x^{(n)}]^\top$ consisting of $n$ data points, each with $m$ features. $Y \in [k]^n$ are the $k$ different class labels for the $n$ data points; with $[k] = \{1,\hdots,k\}$. $C_i$ with $i \in [k]$ refers to the $i$th class and $C(x)$ denotes the class label of $x$. We write $x_{j}$ the $j$th component of a vector and let $\textbf{1}$ be a vector of ones. $|\mathcal{I}|$ refers to the cardinality of a set $\mathcal{I}$. Finally, we let $f_i = \sum_{\{x | x \in C_i\}} x$ be the sum of the feature vectors belonging to class $C_i$. Unless otherwise specified, functional operations on vectors are performed element-wise. We say that a vector $w \in \mathbb{R}^m$ is $s$-sparse or has sparsity level $\alpha \%$ if at most $s$ of $\alpha \%$ of its coefficients are nonzero respectively. $p(\cdot)$ denotes the probability of an event.
\paragraph{naive Bayes} We are interested in predicting the class label of a test point $x^{(t)}$. To do so, naive Bayes selects $C(x^{(t)}) = \arg \max_{i \in [k]} p(C_i | x^{(t)})$. To calculate this posterior probability, we employ Bayes Rule and then use the ``naive" assumption that $p(x^{(t)}|C_i) = \prod_{j=1}^m p(x_{j}^{(t)} | C_i)$ leading to
\begin{align}\label{eq:nb_test}
C(x^{(t)}) &= \arg \max_{i \in [k]}\; \log p(C_i) + \sum_{j=1}^m \log p(x_{j}^{(t)} | C_i)
\end{align}
where we drop the proportionality constant from Bayes' theorem and use monotonicty of the logarithm. In \eqref{eq:nb_test}, we need to have an explicit model for $p(x_j|C_i)$ whose parameters we train using $X$. In the case of binary or integer-valued features, we use bernoulli/categorical distributions and in the case of real valued features we use a Gaussian distribution. For $p(C_i)$ we use a categorical distribution. We then use maximum likelihood estimates (MLE) to determine the parameters of the distributions. For $p(C_i)$, this simply becomes the number of data points in $X$ belonging to class $i$ divided by $n$.
For simplicity, we only deal with binary classification ($k = 2$) in what follows.
\paragraph{Bernoulli naive Bayes}
With binary features, i.e. $x \in \{0,1\}^m$, we have the following conditional probability distribution
\[
p(x_j|C_i) = \theta_{ij}^{x_j}(1-\theta_{ij})^{1-x_j}, \quad \mbox{for all $j=1,\ldots,m,~ i=1,2$},
\]
hence
\[
\log p(x|C_i) = x^\top \log \theta_i - (\textbf{1} - x)^\top \log(\textbf{1} - \theta_i).
\]
Learning the model means learning $\theta_1 = [\theta_{11},\theta_{12},\hdots,\theta_{1m}]$ and $\theta_2 = [\theta_{21},\theta_{22},\hdots,\theta_{2m}]$, and the classical Bernoulli naive Bayes learning problem becomes
\begin{align} \label{eq:bnb_train}
(\theta_1^\ast,\theta_2^\ast) &= \mathop{\mathrm{argmax}}_{\theta_1,\theta_2 \in [0,1]^m} \mathcal{L}_{\text{bnb}}(\theta_1,\theta_2)
\end{align}
where
\begin{align*}
\mathcal{L}_{\text{bnb}}(\theta_1,\theta_2; X) &= \sum_{\{x| x\in C_1\}} \log p(x|C_1) + \sum_{\{x | x\in C_2\}} \log p(x|C_2) \\
&= f_1^\top \log \theta_1 + (\textbf{1} - f_1)^\top \log (\textbf{1} - \theta_1) + f_2^\top \log \theta_2 + (\textbf{1} - f_2)^\top \log (\textbf{1} - \theta_2) \nonumber
\end{align*}
Note that \eqref{eq:bnb_train} jointly concave and is decomposable across features and the optimal solution is simply the MLE estimate. From \eqref{eq:nb_test}, we get a linear classification rule, with $C(x^{(t)}) = C_1$ if and only if
\begin{align}\label{eq:b-rule}
w_{0b} + w_b^\top x^{(t)} \geq 0
\end{align}
with
\begin{align*}
w_{0b} &= \log p(C_1) - \log p(C_2) + \textbf{1}^\top \Big(\log (\textbf{1} - \theta_1^\ast) - \log(\textbf{1} - \theta_2^\ast)\Big) \\
w_b &= \log \dfrac{\theta_1^\ast \circ (\textbf{1} - \theta_2^\ast)}{\theta_2^\ast\circ (\textbf{1} - \theta_1^\ast)}
\end{align*}
where $\circ$ denotes the Hadamard (elementwise) product.
\paragraph{Multinomial naive Bayes} With integer-valued features, i.e. $x \in \mathbb{N}^m$, we have the following conditional probability distribution
\begin{align*}
p(x | C_i) = \dfrac{(\sum_{j=1}^m x_j)!}{\prod_{j=1}^m x_j!} \prod_{j=1}^m \theta_{ij}^{x_j}
\end{align*}
and thus
\begin{align*}
\log p(x|C_i) = x^\top \log \theta_i + \log \Big(\dfrac{(\sum_{j=1}^m x_j)!}{\prod_{j=1}^m x_j!} \Big)
\end{align*}
The traditional Multinomial naive Bayes model is often used when $x$ is not integral; however in the Bernoulli naive Bayes model, it is essential $x$ be binary. Note we have the constraint $\textbf{1}^\top \theta_i = 1$. This amounts again to learning $\theta_1$ and $\theta_2$ just as in the Bernoulli naive Bayes case but with additional constraints. Hence the classical Multinomial naive Bayes learning problem becomes
\begin{align} \label{eq:mnb_train}
(\theta_1^\ast,\theta_2^\ast) &= \mathop{\mathrm{argmax}}_{\substack{\textbf{1}^\top \theta_1 = 1,~ \textbf{1}^\top \theta_2 = 1\\\theta_1,\theta_2 \in [0,1]^m}} \mathcal{L}_{\text{mnb}}(\theta_1,\theta_2)
\end{align}
where
\begin{align*}
\mathcal{L}_{\text{mnb}}(\theta_1,\theta_2; X) &= \sum_{\{x| x\in C_1\}} \log p(x|C_1) + \sum_{\{x | x\in C_2\}} \log p(x|C_2) \\
&= f_1^\top \log \theta_1 + f_2^\top \log \theta_2
\end{align*}
Again, \eqref{eq:mnb_train} is jointly concave and the objective is decomposable across features. However, we have the added complexity of equality constraints on $\theta_1, \theta_2$. As before, for a test point $x^{(t)}$, we get a linear classification rule, with $C(x^{(t)}) = C_1$ if and only if
\begin{align}\label{eq:m-rule}
w_{0m} + w_m^\top x^{(t)} \geq 0
\end{align}
where
\begin{align*}
w_{0m} = \log p(C_1) - \log p(C_2), \quad\mbox{and}\quad w_m = \log \theta_1 - \log \theta_2.
\end{align*}
\section{naive Feature Selection}
In this section, we incorporate sparsity constraints into the aforementioned models.
\subsection{Sparsity} \label{subsec:sparse} In order to encourage sparsity in our classifier and select features, we want $w_b$ and $w_m$ to be sparse; that is
\[
\log \dfrac{\theta_1 \circ (\textbf{1} - \theta_2)}{\theta_2\circ (\textbf{1} - \theta_1)},\quad\mbox{and}\quad
\log \theta_1 - \log \theta_2
\]
are sparse. For both $w_b$ and $w_m$, respectively defined in~\eqref{eq:b-rule} and~\eqref{eq:m-rule}, this happens if and only if the difference vector $\theta_1 - \theta_2$ is sparse. By enforcing sparsity in $\theta_1 - \theta_2$, the classifier only uses a subset of the features of $x$ for classification, making the model more interpretable.
\paragraph{Sparse Bernoulli naive Bayes} We can enforce sparsity using both the $\ell_0$ and $\ell_1$ norm, to form two new sparse Bernoulli naive Bayes training problems,
\begin{align}\label{eq:bnb0}
(\theta_1^\ast,\theta_2^\ast) &= \mathop{\mathrm{argmax}}_{\substack{\|\theta_1 - \theta_2\|_0 \leq s\\\theta_1,\theta_2 \in [0,1]^m}} \mathcal{L}_{bnb}(\theta_1,\theta_2; X)\tag{SBNB-$\ell_0$}
\end{align}
and
\begin{align}\label{eq:bnb1}
(\theta_1^\ast,\theta_2^\ast) = \mathop{\mathrm{argmax}}_{\theta_1,\theta_2 \in [0,1]^m} \mathcal{L}_{bnb}(\theta_1,\theta_2; X) - \lambda \|\theta_1 - \theta_2\|_1 \tag{SBNB-$\ell_1$}
\end{align}
for \eqref{eq:bnb_train} where $\lambda$ and $s$ are hyperparameters influencing the sparsity pattern (note because of strong duality, \eqref{eq:bnb1} is equivalent to removing the $\ell_1$ norm from the objective and adding it as a constraint). For \eqref{eq:bnb0}, a constrained formulation will guarantee a sparsity level of $s$. In the case of \eqref{eq:bnb1}, since the $\ell_1$ norm is a convex-proxy for sparsity, we need to tune a hyperparameter in either the constrained or unconstrained formulation.
\paragraph{Sparse Multinomial naive Bayes} To enforce sparsity, as in the Bernoulli model, we form two new sparse Multinomial naive Bayes training problems,
\begin{align}\label{eq:mnb0}
(\theta_1^\ast,\theta_2^\ast) &= \mathop{\mathrm{argmax}}_{\substack{\textbf{1}^\top \theta_1 = 1, \textbf{1}^\top \theta_2 = 1 \\\|\theta_1 - \theta_2\|_0 \leq s\\\theta_1,\theta_2 \in [0,1]^m}} \mathcal{L}_{mnb}(\theta_1,\theta_2; X)\tag{SMNB-$\ell_0$}
\end{align}
in the variables $\theta_1,\theta_2 \in [0,1]^m$, and
\begin{align}\label{eq:mnb1}
(\theta_1^\ast,\theta_2^\ast) &= \mathop{\mathrm{argmax}}_{\substack{\textbf{1}^\top \theta_1 = 1,~ \textbf{1}^\top \theta_2 = 1\\\theta_1,\theta_2 \in [0,1]^m}} \mathcal{L}_{mnb}(\theta_1,\theta_2; X) - \lambda \|\theta_1 - \theta_2\|_1 \tag{SMNB-$\ell_1$}
\end{align}
in light of \eqref{eq:mnb_train}.
\paragraph{Solving naive Feature Selection Problems.}
At first, the four sparse training problems look much more challenging to solve compared to \eqref{eq:bnb_train} and \eqref{eq:mnb_train}. $\theta_1$ and $\theta_2$ are now coupled together by the norms and by equality constraints in the multinomial case whereas before they could be optimized independently using the MLE estimates. Furthermore, \eqref{eq:bnb0} and \eqref{eq:mnb0} are non-convex and potentially hard. However, we will see that \eqref{eq:bnb0} and \eqref{eq:bnb1} can both be solved efficiently in \textit{closed form}; in fact the complexity of solving them is the same as the complexity of solving \eqref{eq:bnb_train}. Additionally, \eqref{eq:mnb1} and a relaxation on \eqref{eq:mnb0} can be solved efficiently. This is addressed in Theorems \ref{thm:sparse_bnb} and \ref{thm:sparse_mnb}. {\color{blue} talk about solving \eqref{eq:mnb1}}\\
\begin{theorem}\label{thm:sparse_bnb}
Consider the binary classification training problem with $n$ binary feature vectors; that is $x^{(i)} \in \{0,1\}^m$ for $i \in [n]$. Then \eqref{eq:bnb0} and \eqref{eq:bnb1} can both be solved exactly with time complexity linear in the number of features.
\end{theorem}
\begin{proof}
See Appendix A.
\end{proof}
\paragraph{Solving \eqref{eq:mnb0} via Relaxation.} While \eqref{eq:bnb0} can be solved in closed form despite the $\ell_0$ constraint, \eqref{eq:mnb0} does not admit a similar solution due to equality constraints. However, we can solve a convex upperbound.
\begin{theorem}\label{thm:cvx-rlx}
Let $\phi(s)$ be the optimal value of \eqref{eq:mnb0}. Then $\phi(s) \leq \psi(s)$ where $\psi(s)$ is the optimal value of the following convex optimization problem
\begin{align*}\label{eq:ub}
\psi = \min_{\mu_1,\mu_2} \mu_1 + \mu_2 + \textbf{1}^\top w + s_{m-s}(v-w) \tag{UB-$\ell_0$}
\end{align*}
where
\begin{align*}
v &= (f_1+f_2) \circ \log \Big( \dfrac{f_1+f_2}{\mu_1 + \mu_2}\Big) - (f_1 + f_2) \\
w &= f_1 \circ \log \Big( \dfrac{f_1}{\mu_1} \Big) + f_2 \circ \log \Big( \dfrac{f_2}{\mu_2} \Big) - (f_1 + f_2)
\end{align*}
and where $s_{m-s}(z)$ denotes the sum of the top $m-s$ entries of a vector. Furthermore, we have
\begin{align*}
\psi(s-4) \leq \phi(s) \leq \psi(s) \leq \phi(s + 4), \;\; \text{for} \;\; s \geq 4
\end{align*}
\end{theorem}
\begin{proof}
See Appendix A.
\end{proof}
We can form another convex relaxation of the Sparse Multinomial naive Bayes problem in~\eqref{eq:mnb0} using duality. We have the following result.
The key point here is that, while problem~\eqref{eq:mnb0} is nonconvex and potentially hard, the dual problem is a convex optimization problem in dimension three, which can be solved very efficiently. The bound in~\eqref{eq:gap-bnd} implies
\[
\xi(k) \leq \phi(k) \leq \xi(k) + \Delta_k, \quad \mbox{for $k\geq4$,}
\]
where $\Delta_k=\xi(k)-\xi(k-4)$. This means that where $\xi(k)$ does not vary too fast, so $\Delta_k$ is small, then the duality gap in problem~\eqref{eq:mnb0} is itself small, bounded by $\Delta_k$, and solving the convex problem~\eqref{eq:dual-mnb0} will yield a good approximate solution to~\eqref{eq:mnb0}. Furthermore, if $(y_1^*,y_2^*,\lambda^*)$ solves~\eqref{eq:dual-mnb0}, feature $i$ will be selected if and only if
\[
f_{1i} \log\left(\frac{-f_{1i}}{y_1^*}\right) + f_{2i} \left(\frac{-f_{2i}}{y_2^*}\right) - \lambda^* > (f_{1i}+f_{2i}) \log\left(\frac{-(f_{1i}+f_{2i})}{(y_1^*+y_2^*)}\right)
\]
in the definition of $h_{f_i,\lambda}^{*}(y_1,y_2)$. We now show the following result on the complexity of solving \eqref{eq:ub} and \eqref{eq:mnb1} to arbitrary precision $\epsilon$.
\section{Robustness} \label{subsec:rob} As we did with their sparse counterparts, we would like to consider robust formulations of \eqref{eq:bnb_train} and \eqref{eq:mnb_train}. For \eqref{eq:bnb_train} and \eqref{eq:mnb_train}, we consider robustness to the following uncertainty sets:
\begin{align}\label{eq:unc-bnb}
\Delta_\text{flip}(x) &= \{ \delta \;|\; \delta + x \in \{0,1\}^m, \|\delta\|_1 \leq p\} \tag{$p$-flips}
\end{align}
\begin{align}\label{eq:unc-mnb}
\Delta_\text{typo}(x) &= \{ \delta \;|\; \delta + x \in \mathbb{N}^m, \|\delta\|_1 \leq t\} \tag{$t$-typos}
\end{align}
\begin{align}\label{eq:unc-lmbda}
\Delta_\text{shift}(x) &= \{ \delta \;|\; \delta + x \in \mathbb{N}^m, \delta \in [-\gamma, \gamma]^m\cap \mathbb{Z}^m \} \tag{$\gamma$-shift}
\end{align}
Here, $\Delta_\text{flip}(x)$ represents the set of vectors that can be constructed by flipping at most $p\leq m$ components of $x$ in the Bernoulli model. $\Delta_\text{typo}(x)$ represents the set of vectors that can be constructed by adding at most $t$ counts in $x$. $\Delta_\text{shift}(x)$ represents an uncertainty set where each entry of $x$ is uncertain by $\pm \gamma$ (this is in fact equivalent to Laplace smoothing -- see Appendix A for further details).
Under the assumption that each data point is perturbed independently, the robust training problems become
\begin{align} \label{eq:r-bnb}
\theta_1^\ast,\theta_2^\ast &= \arg\max_{\theta_1,\theta_2 \in [0,1]^m} \min_{\Delta \in \Delta_\text{flip}(X)} \mathcal{L}_{bnb}(\theta_1,\theta_2; X + \Delta) \tag{R-BNB}
\end{align}
and
\begin{align}\label{eq:r-mnb}
\theta_1^\ast,\theta_2^\ast &= \arg\max_{\theta_1,\theta_2 \in [0,1]^m} \min_{\Delta \in \Delta(X)} \mathcal{L}_{mnb}(\theta_1,\theta_2; X + \Delta) \tag{R-MNB} \\
&\hspace{10mm} \text{s.t.} \;\; \textbf{1}^\top \theta_1 = 1,\; \textbf{1}^\top \theta_2 = 1 \notag
\end{align}
where in \eqref{eq:r-mnb}, $\Delta(X)$ is any of the uncertainty sets but \eqref{eq:unc-bnb}.
Again, this problem seems daunting due to non convexity of the uncertainty sets but \eqref{eq:r-bnb} and \eqref{eq:r-mnb} reduce to a \textit{convex} problems and the inner minimization can be solved exactly for all the proposed uncertaitny sets. This is addressed in Theorem \ref{thm:rob}.
\begin{theorem}\label{thm:rob}
Consider the binary classification training problem with $n$ binary feature vectors; that is $x^{(i)} \in \{0,1\}^m$ for $i \in [n]$. Furthermore, consider the uncertainty set given by \eqref{eq:unc-bnb}. Then \eqref{eq:r-bnb} reduces to a convex optimization problem and the inner minimization can be solved exactly {\color{blue} talk about \eqref{eq:r-mnb}}.
\end{theorem}
\begin{proof}
See Appendix A.
\end{proof}
\subsection{Sparsity \& Robustness} Any one of the sparse models in Section \eqref{subsec:sparse} can be combined with the appropriate robustness model in \eqref{subsec:rob} to come up with a sparse, robust variant of naive Bayes. By Theorem \ref{thm:rob}, the jointly sparse and robust model is a convex optimization problem. For example, \eqref{eq:mnb0} and \eqref{eq:mnb1} with \eqref{eq:unc-lmbda}, the training problems becomes
\begin{align}\label{eq:srnb0}
(\theta_1^\ast,\theta_2^\ast) &= \mathop{\mathrm{argmax}}_{\substack{\textbf{1}^\top \theta_1 = 1, \textbf{1}^\top \theta_2 = 1 \\\|\theta_1 - \theta_2\|_0 \leq s\\\theta_1,\theta_2 \in [0,1]^m}} (f_1 + |C_1| \gamma \textbf{1})^\top \log \theta_1 + (f_2 + |C_2| \gamma \textbf{1})^\top \log \theta_2 \tag{SRNB-$\ell_0$}
\end{align}
and
\begin{align}\label{eq:srnb1}
(\theta_1^\ast,\theta_2^\ast) &= \mathop{\mathrm{argmax}}_{\substack{\textbf{1}^\top \theta_1 = 1, \textbf{1}^\top \theta_2 = 1 \\\theta_1,\theta_2 \in [0,1]^m}} (f_1 + |C_1| \gamma \textbf{1})^\top \log \theta_1 + (f_2 + |C_2| \gamma \textbf{1})^\top \log \theta_2 - \lambda \|\theta_1 - \theta_2\|_1 \tag{SRNB-$\ell_1$}
\end{align}
respectively, where $|C_1|, |C_2|$ are the number of training points in class 1 and class 2 respectively. For more combinations of sparsity and uncertainty sets, see the proof of Theorem \eqref{thm:rob} in Appendix A.
\section{Experiments}\label{sec:expt}
In this section we compare \eqref{eq:mnb0} with other feature selection methods for sentiment classification on 5 different text data sets. We run three different sets of experiments.
\subsection{Experiment 1: Feature selection}
For each data set and each type of feature vector, we perform the following two-stage procedure: 1) employ a feature selection method to attain a desired sparsity level (0.1\%, 1\%, 5\%, 10\%) and 2) train a classifier based on the sub-selected features from part 1. Specifically, we use $\ell_1$-regularized logistic regression, logistic regression with recursive feature elimination (RFE), $\ell_1$-regularized support vector machine (SVM), SVM with RFE, LASSO, thresholded Multinomial naive Bayes (TMNB), the Odds Ratio metric described by \citep{mladenic1999feature} and \eqref{eq:mnb0} in Step 1. Then using the sub-selected features, in Step 2 we train a logistic model, a Multinomial naive Bayes model, and a SVM. For TMNB, we train a MNB model and then sub-select the features corresponding to indicides of the largest absolute value entries of $w_m$ defined in \eqref{eq:m-rule}. We use \eqref{eq:srnb0} as opposed to \eqref{eq:srnb1} since we can immediately achieve a desired sparsity pattern without having to tune any hyperparameters. For each desired sparsity level and each data set in stage 1, we do a grid search over the optimal Laplace smoothing parameter for MNB for each type of feature vector. We use this same hyper parameter in \eqref{eq:srnb0}. Table 1 summarizes the data used in Experiment 1.
\newlength{\oldintextsep}
\setlength{\oldintextsep}{\intextsep}
\begin{table}[h!]
\centering
\begin{tabular}{lrrrrr}
\toprule
\textsc{Feature Vectors} & \textsc{Amazon} & \textsc{IMDB} & \textsc{Twitter} & \textsc{MPQA} & \textsc{SST2}\\
\midrule
\textsc{Count Vector} & \textsc{31,666} & \textsc{103,124} & \textsc{273,779} & \textsc{6,208} & \textsc{16,599} \\
\textsc{tf-idf} & \textsc{5000} & \textsc{5000} & \textsc{5000} & \textsc{5000} & \textsc{5000} \\
\textsc{tf-idf wrd bigram} & \textsc{5000} & \textsc{5000} & \textsc{5000} & \textsc{5000} & \textsc{5000} \\
\textsc{tf-idf char bigram} & \textsc{5000} & \textsc{5000} & \textsc{5000} & \textsc{4838} & \textsc{5000} \\ \midrule
\textsc{$n_\text{train}$} & \textsc{8000} & \textsc{25,000} & \textsc{1,600,000} & \textsc{8484} & \textsc{76,961}\\
\textsc{$n_\text{test}$} & \textsc{2000} & \textsc{25,000} & \textsc{498} & \textsc{2122} & \textsc{1821}\\
\bottomrule\\
\end{tabular}
\caption{\textbf{Experiment 1 data}: Number of features for each type of feature vector for each data set. For tf-idf feature vectors, we fix the number of features to 5000 for all data sets. For more on how data sets were preprocessed, see Appendix B.}
\label{tab:dsinfo}
\end{table}
\setlength\intextsep{7pt}
\subsection{Experiment 2: Large Scale Feature selection}
For this experiment, we consider the same data sets but do not put any limit on the number of features for the tf-idf vectors. Because of the large scale nature of the data sets, most of the feature selection methods in Experiment 1 are not scalable. We use the same two-stage procedure as before: 1) do feature selection using TMNB, the Odds Ratio and \eqref{eq:srnb0} and 2) train a MNB model using the features subselected in stage 1. We tune the hyperparameters for MNB and \eqref{eq:srnb0} the same way as in Experiment 1. Table 2 summarizes the data used in Experiment 2.
\begin{table}[h!]
\centering
\begin{tabular}{lrrrrr}
\toprule
\textsc{Feature Vectors} & \textsc{Amazon} & \textsc{IMDB} & \textsc{Twitter} & \textsc{MPQA} & \textsc{SST2}\\
\midrule
\textsc{Count Vector} & \textsc{31,666} & \textsc{103,124} & \textsc{273,779} & \textsc{6,208} & \textsc{16,599} \\
\textsc{tf-idf} & \textsc{31,666} & \textsc{103,124} & \textsc{273,779} & \textsc{6,208} & \textsc{16,599} \\
\textsc{tf-idf wrd bigram} & \textsc{870,536} & \textsc{8,950,169} & \textsc{12,082,555} & \textsc{27,603} & \textsc{227,012} \\
\textsc{tf-idf char bigram} & \textsc{25,019} & \textsc{48,420} & \textsc{17,812} & \textsc{4838} & \textsc{7762} \\
\bottomrule\\
\end{tabular}
\caption{\textbf{Experiment 2 data}: Number of features for each type of feature vector for each data set with no limit on the number of features for the tf-idf vectors. The train/test split is the same as in Table \ref{tab:dsinfo}.}
\label{tab:dsinfo2}
\end{table}
\setlength\intextsep{7pt}
\subsection{Experiment 3: XXX $k/n$}
Using the IMDB dataset in Table \ref{tab:dsinfo}, we perform the following experiment: we fix a sparsity pattern $s/m = 0.05$ and then increase $s$ and $m$. Since we artifically set the number of tf-idf features to 5000 in Experiment 1, here we let the number of tf-idf features vary from 10000 to 80000. We then plot the the time it takes to train an $\ell_1$-regularized logistic model and \eqref{eq:mnb0} at a 0.05 sparsity level.
\begin{figure}[h!]
\centering
\includegraphics[width=\linewidth]{figures/imdb_kovern_time.png}
\caption{Experiment 3:}
\label{fig:expt3}
\end{figure}
\section{Conclusion}
XXX
\subsubsection*{Acknowledgments}
The authors would like to thank Yeshwanth Cherapanamjeri for helpful discussions.
\newpage
\bibliographystyle{apalike}
\section{Introduction}
Modern, large-scale data sets call for classification methods that scale mildly (e.g. linearly) with problem size. In this context, the classical naive Bayes model remains a very competitive baseline, due to its linear complexity in the number of training points and features. In fact, it is sometimes the only feasible approach in very large-scale settings, particularly in text applications, where the number of features can easily be in the millions or higher.
Feature selection, on the other hand, is a key component of machine learning pipelines, for two main reasons: i) to reduce effects of overfitting by eliminating noisy, non-informative features and ii) to provide interpretability. In essence, feature selection is a combinatorial problem, involving the selection of a few features in a potentially large population. State-of-the-art methods for feature selection employ some heuristic to address the combinatorial aspect, and the most effective ones are usually computationally costly. For example, LASSO \citep{tibshirani1996regression} or $l_1$-SVM models \citep{fan2008liblinear} are based on solving a convex problem involving an $l_1$-norm penalty on the vector of regression coefficients, a heuristic to constrain its cardinality which requires tuning a hyper parameter to achieve a desired sparsity level.
Since naive Bayes corresponds to a linear classification rule, feature selection in this setting is directly related to the sparsity of the vector of classification coefficients, just as in LASSO or $l_1$-SVM. This work is devoted to a sparse variant of naive Bayes. Our main contributions are as follows.
\begin{itemize}
\item We formulate a sparse naive Bayes problem that involves a direct constraint on the cardinality of the vector of classification coefficients, leading to an interpretable naive Bayes model. No hyper-parameter tuning is required in order to achieve the target cardinality.
\item We derive an exact solution of sparse naive Bayes in the case of binary data, and an approximate upper bound for general data, and show that it becomes increasingly tight as the marginal contribution of features decreases. Both models can be trained very efficiently, with an algorithm that scales almost linearly with the number of features and data points, just like classical naive Bayes.
\item We show in numerical experiments that our model significantly outperforms simple baselines ({\it e.g.}, thresholded naive Bayes, odds ratio), and achieves similar performance as more sophisticated feature selection methods, at a fraction of the computing cost.
\end{itemize}
\paragraph{Related Work on Naive Bayes Improvements.} A large body of literature builds on the traditional naive Bayes classifier. A non-extensive list includes the seminal work by \citet{frank2002locally} introducing Weighted naive Bayes; Lazy Bayesian Learning by \citet{zheng2000lazy}; and the Tree-Augmented naive Bayes method by \citet{friedman1997bayesian}. The paper \citep{webb2005not} improves the computational complexity of the aforementioned methods, while maintaining the same accuracy. For a more complete discussion of different modifications to naive Bayes, we refer the reader to \citep{jiang2007survey} and the references therein.
\paragraph{Related Work on Naive Bayes and Feature Selection.} Of particular interest to this work are methods that employ feature selection. \citet{kim2006some} use information-theoretic quantities for feature selection in text classification, while \citet{mladenic1999feature} compare a host of different methods and shows the comparative efficacy of the Odds Ratio method. These methods often use ad hoc scoring functions to rank the importance of the different features.
To our knowledge, the first work to directly address sparsity in the context of naive Bayes, with binary data only, is \citet{zheng2018sparse}. Their model does not directly address the requirement that the weight vector of the classification rule should be sparse, but does identify key features in the process. The method requires solving an approximation to the combinatorial feature selection problem via $l_1$-penalized logistic regression problem with non-negativity constraints, that has the same number of features and data points as the original one. Therefore the complexity of the method is the same as ordinary $l_1$-penalized logistic regression, which is relatively high. In contrast, our binary (Bernoulli) naive Bayes bound is exact, and has complexity almost linear in training problem size.
\section{Background on Naive Bayes}
In this paper, for simplicity only, we consider a two-class classification problem; the extension to the general multi-class case is straightforward.
\paragraph{Notation.} For an integer $m$, $[m]$ is the set $\{1,\ldots,m\}$. The notation $\mathbf 1$ denotes a vector of ones, with size inferred from context. The cardinality (number of non-zero elements) in a $m$-vector $x$ is denoted $\|x\|_0$, whereas that of a finite set $\mathcal{I}$ is denoted $|\mathcal{I}|$. Unless otherwise specified, functional operations (such as $\max(0,\cdot)$) on vectors are performed element-wise. For $k \in [n]$, we say that a vector $w \in \mathbb{R}^m$ is $k$-sparse or has sparsity level $\alpha \%$ if at most $k$ or $\alpha \%$ of its coefficients are nonzero respectively. For two vectors $f,g \in {\mathbb R}^m$, $f \circ g \in {\mathbb R}^m$ denotes the Hadamard (elementwise) product. For a vector $z$, the notation $s_{k}(z)$ is the sum of the top $k$ entries. Finally, $\mathop{\bf Prob}(A)$ denotes the probability of an event $A$.
\paragraph{Data Setup.}
We are given a non-negative data matrix $X \in \mathbb{R}_+^{n \times m} = [x^{(1)},x^{(2)},\hdots,x^{(n)}]^\top$ consisting of $n$ data points, each with $m$ dimensions (features), and a vector $y \in \{-1,1\}^n$ that encodes the class information for the $n$ data points, with $C_+$ and $C_-$ referring to the positive and negative classes respectively. We define index sets corresponding to each class $C_+,C_-$, and their respective cardinality, and data averages:
\[
{\cal I}_\pm := \left\{ i \in [n] ~:~ y_i = \pm 1 \right\}, \;\; n_\pm = |{\cal I}_\pm|, \;\;
f_\pm := \sum_{i \in {\cal I}_\pm} x^{(i)} = \pm (1/2)X^\top (y \pm \mathbf 1)
\]
\paragraph{Naive Bayes.} We are interested in predicting the class label of a test point $x \in {\mathbb R}^n$ via the rule $\hat{y}(x) = \arg \max_{\epsilon \in \{-1,1\}} \mathop{\bf Prob}(C_\epsilon \: | \: x )$. To calculate the latter posterior probability, we employ Bayes' rule and then use the ``naive'' assumption that features are independent of each other: $\mathop{\bf Prob}(x \: | \: C_\epsilon) = \prod_{j=1}^m \mathop{\bf Prob}(x_{j} \: | \: C_\epsilon)$, leading to
\begin{align}\label{eq:nb_test}
\hat{y}(x) &= \arg \max_{\epsilon \in \{-1,1\}}\; \log \mathop{\bf Prob}(C_\epsilon) + \sum_{j=1}^m \log \mathop{\bf Prob}(x_{j} | C_\epsilon) .
\end{align}
In \eqref{eq:nb_test}, we need to have an explicit model for $\mathop{\bf Prob}(x_j|C_i)$; in the case of binary or integer-valued features, we use Bernoulli or categorical distributions, while in the case of real-valued features we can use a Gaussian distribution. We then use the maximum likelihood principle (MLE) to determine the parameters of those distributions. Using a categorical distribution, $\mathop{\bf Prob}(C_\pm)$ simply becomes the number of data points in $X$ belonging to class $\pm 1$ divided by $n$.
\paragraph{Bernoulli Naive Bayes.}
With binary features, that is, $X \in \{0,1\}^{n \times m}$, we choose the following conditional probability distributions parametrized by two positive vectors $\theta^{+}, \theta^{-} \in (0,1)^m$. For a given vector $x \in \{0,1\}^m$,
\[
\mathop{\bf Prob}(x_j ~|~ C_\pm) = (\theta^{\pm}_j)^{x_j}(1-\theta^{\pm}_j)^{1-x_j}, \;\; j \in [m],
\]
hence
\[
\sum_{j=1}^n \log \mathop{\bf Prob}(x_j ~|~ C_\pm) = x^\top \log \theta^{\pm} + (\mathbf 1 - x)^\top \log(\mathbf 1 - \theta^{\pm}).
\]
Training a classical Bernoulli naive Bayes model reduces to the problem
\begin{align} \label{eq:bnb_train}
({\theta^{+}_\ast},{\theta^{-}_\ast}) &= \arg\max_{\theta^{+}, \theta^{-} \in [0,1]^m} \mathcal{L}_{\text{bnb}}(\theta^{+},\theta^{-}; X)
\end{align}
where the loss is a concave function
\begin{align}
\mathcal{L}_{\text{bnb}}(\theta^{+}, \theta^{-}) =& \sum_{i \in {\cal I}_+} \log \mathop{\bf Prob}(x^{(i)} ~|~ C_+) + \sum_{i \in {\cal I}_-} \log \mathop{\bf Prob}(x^{(i)} ~|~ C_-) \label{eq:loss-def-bnb}\\
=& f^{+\top} \log \theta^{+} + (n_+ \mathbf 1 - f^{+})^\top \log (\mathbf 1 - \theta^{+})
\\
&+ f^{-\top} \log \theta^{-} + (n_-\mathbf 1 - f^{-})^\top \log (\mathbf 1 - \theta^{-}) . \nonumber
\end{align}
Note that problem \eqref{eq:bnb_train} is decomposable across features and the optimal solution is simply the MLE estimate, that is, $\theta^{\pm}_* = f^{\pm}/ n_\pm$. From \eqref{eq:nb_test}, we get a linear classification rule: for a given test point $x \in {\mathbb R}^m$, we set $\hat{y}(x) = \mbox{\bf sign}(v + w_b^\top x)$, where
\begin{align}\label{eq:b-rule}
v := \log \dfrac{\mathop{\bf Prob}(C_+)}{\mathop{\bf Prob}(C_-)} + \mathbf 1^\top \Big(\log (\mathbf 1 - {\theta^{+}_\ast}) - \log(\mathbf 1 - {\theta^{-}_\ast})\Big), \;\;
w_b := \log \dfrac{{\theta^{+}_\ast} \circ (\mathbf 1 - {\theta^{-}_\ast})}{\theta^{-}_\ast\circ (\mathbf 1 - {\theta^{+}_\ast})} .
\end{align}
\paragraph{Multinomial naive Bayes.} With integer-valued features, that is, $X \in \mathbb{N}^{n \times m}$, we choose the following conditional probability distribution, again parameterized by two positive $m$-vectors $\theta^{\pm} \in [0,1]^m$, but now with the constraints $\mathbf 1^\top \theta^{\pm} = 1$: for given $x \in \mathbb{N}^m$,
\begin{align*}
\mathop{\bf Prob}(x ~|~ C_\pm) = \dfrac{(\sum_{j=1}^m x_j)!}{\prod_{j=1}^m x_j!} \prod_{j=1}^m (\theta^{\pm}_j)^{x_j} ,
\end{align*}
and thus
\begin{align*}
\log \mathop{\bf Prob}(x ~|~ C_\pm) = x^\top \log \theta^{\pm} + \log \left(\dfrac{(\sum_{j=1}^m x_j)!}{\prod_{j=1}^m x_j!} \right) .
\end{align*}
While it is essential that the data be binary in the Bernoulli model seen above, the multinomial one can still be used if $x$ is non-negative real-valued, and not integer-valued. Training the classical multinomial model reduces to the problem
\begin{align} \label{eq:mnb_train}
(\theta^{+}_\ast,\theta^{-}_\ast) &= \arg\max_{\theta^{+}, \theta^{-} \in [0,1]^m} \mathcal{L}_{\text{mnb}}(\theta^{+},\theta^{-}) ~:~ \mathbf 1^\top \theta^{+} = \mathbf 1^\top\theta^{-} = 1,
\end{align}
where the loss is a again a concave function
\begin{align}
\mathcal{L}_{\text{mnb}}(\theta^{+}, \theta^{-}) =& \sum_{i \in {\cal I}_+} \log \mathop{\bf Prob}(x^{(i)} ~|~ C_+) + \sum_{i \in {\cal I}_-} \log \mathop{\bf Prob}(x^{(i)} ~|~ C_-) \nonumber\\
=& f^{+\top} \log \theta^{+} + f^{-\top} \log \theta^{-} . \label{eq:loss-def-mnb}
\end{align}
Again, problem \eqref{eq:mnb_train} is decomposable across features, with the added complexity of equality constraints on $\theta^{\pm}$. The optimal solution is the MLE estimate $\theta^{\pm}_* = {f^\pm}/({\mathbf 1^\top f^\pm})$. As before, we get a linear classification rule: for a given test point $x \in {\mathbb R}^m$, we set $\hat{y}(x) = \mbox{\bf sign}(v + w_m^\top x)$, where
\begin{align}\label{eq:m-rule}
v := \log \mathop{\bf Prob}(C_+) - \log \mathop{\bf Prob}(C_-), \;\; w_m := \log {\theta^{+}_\ast} - \log {\theta^{-}_\ast} .
\end{align}
\section{Naive Feature Selection}
In this section, we incorporate sparsity constraints into the aforementioned models.
\subsection{Naive Bayes with Sparsity Constraints}
\label{subsec:sparse}
For a given integer $k \in [m]$, with $k <m$, we seek to obtain a naive Bayes classifier that uses at most $k$ features in its decision rule.
For this to happen, we need the corresponding coefficient vector, denoted $w_b$ and $w_m$ for the Bernoulli and multinomial cases, and defined in~\eqref{eq:b-rule} and~\eqref{eq:m-rule} respectively, to be $k$-sparse. For both Bernoulli and multinomial models, this happens if and only if the difference vector ${\theta^{+}_\ast} - {\theta^{-}_\ast}$ is sparse. By enforcing $k$-sparsity on the difference vector, the classifier uses less than $m$ features for classification, making the model more interpretable.
\paragraph{Sparse Bernoulli Naive Bayes.} In the Bernoulli case, the sparsity-constrained problem becomes
\begin{align}\label{eq:bnb0}
({\theta^{+}_\ast},{\theta^{-}_\ast}) &=
\arg\max_{\theta^{+}, \theta^{-} \in [0,1]^m} \: \mathcal{L}_{\text{bnb}}(\theta^{+},\theta^{-}; X) ~:~ \|\theta^{+} - \theta^{-}\|_0 \leq k, \tag{SBNB}
\end{align}
where ${\cal L}_{\text{bnb}}$ is defined in \eqref{eq:loss-def-bnb}. Here, $\|\cdot\|_0$ denotes the $l_0$-norm, or cardinality (number of non-zero entries) of its vector argument, and $k <m$ is the user-defined upper bound on the desired cardinality.
\paragraph{Sparse Multinomial Naive Bayes.} In the multinomial case, in light of \eqref{eq:mnb_train}, our model is written
\begin{align}\label{eq:mnb0}
({\theta^{+}_\ast},{\theta^{-}_\ast}) &=
\arg\max_{\theta^{+}, \theta^{-} \in [0,1]^m} \: \mathcal{L}_{\text{mnb}}(\theta^{+},\theta^{-}; X) ~:~
\begin{array}[t]{l} \mathbf 1^\top \theta^{+} = \mathbf 1^\top\theta^{-} = 1, \\ \|\theta^{+} - \theta^{-}\|_0 \leq k.
\end{array} \tag{SMNB}
\end{align}
where ${\cal L}_{\text{mnb}}$ is defined in \eqref{eq:loss-def-mnb}.
\subsection{Main Results}
Due to the inherent combinatorial and non-convex nature of the cardinality constraint, and the fact that it couples the variables $\theta^{\pm}$, the above sparse training problems look much more challenging to solve when compared to their classical counterparts, \eqref{eq:bnb_train} and \eqref{eq:mnb_train}. We will see in what follows that this is not the case.
\paragraph{Sparse Bernoulli Case.} The sparse counterpart to the Bernoulli model, \eqref{eq:bnb0}, can be solved efficiently in \textit{closed form}, with complexity comparable to that of the classical Bernoulli problem \eqref{eq:bnb_train}.
\begin{theorem}[Sparse Bernoulli naive Bayes]\label{thm:sparse_bnb}
Consider the sparse Bernoulli naive Bayes training problem \eqref{eq:bnb0}, with binary data matrix $X \in \{0,1\}^{n \times m}$. The optimal values of the variables are obtained as follows. Set
\begin{align}\label{eq:v-w-def-sbnb}
v &:= (f^{+}+f^{-}) \circ \log \Big( \dfrac{f^{+}+f^{-}}{n}\Big) + (n\mathbf 1 -f^{+}-f^{-}) \circ \log \Big(\mathbf 1 - \dfrac{f^{+}+f^{-}}{n}\Big) , \\
w &:= w^+ + w^-, \;\; w^\pm := f^{\pm} \circ \log \dfrac{f^{+}}{n_\pm} + (n_\pm \mathbf 1 - f^{\pm}) \circ \log \Big(\mathbf 1 - \dfrac{f^{\pm}}{n_\pm}\Big) . \nonumber
\end{align}
Then identify a set ${\cal I}$ of indices with the $m-k$ smallest elements in $w-v$, and set ${\theta^{+}_\ast},{\theta^{-}_\ast}$ according to
\begin{align}
{\theta^{+}_{\ast_i}} = {\theta^{-}_{\ast_i}} = \frac{1}{n}(f^{+}_i + f^{-}_i), \;\forall
i \in {\cal I}, \;\;\;\;
{\theta^{\pm}_{\ast_i}} = \dfrac{f^{\pm}_i}{n_\pm} , \; \forall i \not\in {\cal I}.
\end{align}
\end{theorem}
\begin{proof}
See Appendix~\ref{appendixA}.
\end{proof}
Note that the complexity of the computation (including forming the vectors $f^{\pm}$, and finding the $k$ largest elements in the appropriate $m$-vector) grows as $O(mn \log(k))$. This represents a very moderate extra cost compared to the cost of the classical naive Bayes problem, which is $O(mn)$.
\paragraph{Multinomial Case.}
In the multinomial case, the sparse problem \eqref{eq:mnb0} does not admit a closed-form solution. However, we can obtain an easily computable upper bound.
\begin{theorem}[Sparse multinomial naive Bayes]\label{thm:sparse_mnb} \label{thm:cvx-rlx}
Let $\phi(k)$ be the optimal value of \eqref{eq:mnb0}. Then $\phi(k) \leq \psi(k)$, where $\psi(k)$ is the optimal value of the following one-dimensional convex optimization problem
\begin{equation}\label{eq:ub}\tag{USMNB}
\psi(k) := C + \min_{\alpha \in [0,1]} \: s_k(h(\alpha) ),
\end{equation}
where $C$ is a constant, $s_{k}(\cdot)$ is the sum of the top $k$ entries of its vector argument, and for $\alpha \in (0,1)$,
\[
h(\alpha) := f_+ \circ \log f_+ + f_- \circ \log f_- - (f_++f_-) \circ \log (f_++f_-) - f_+ \log \alpha - f_- \log(1-\alpha).
\]
Furthermore, given an optimal dual variable $\alpha_\ast$ that solves \eqref{eq:ub}, we can reconstruct a primal feasible (sub-optimal) point $(\theta^{+},\theta^{-})$ for \eqref{eq:mnb0} as follows. For $\alpha^\ast$ optimal for \eqref{eq:ub}, let $\mathcal{I}$ be complement of the set of indices corresponding to the top $k$ entries of $h(\alpha_\ast)$; then set $B_\pm := \sum_{i \not\in\mathcal{I}} f^{\pm}_i$, and
\begin{equation}\label{eq:primalsol}
{\theta^{+}_\ast}_i = {\theta^{-}_\ast}_i = \dfrac{f^{+}_i + f^{-}_i}{\mathbf 1^\top (f^{+}+f^{-})}, \;\forall i \in \mathcal{I} , \;\;\;\;
{\theta^{\pm}_{\ast_i}} = \dfrac{B_+ + B_-}{B_\pm} \dfrac{f^{\pm}_i}{\mathbf 1^\top (f^{+}+f^{-})}, \;\forall i \not\in \mathcal{I} .
\end{equation}
\end{theorem}
\begin{proof}
See Appendix~\ref{appendixB}.
\end{proof}
The key point here is that, while problem~\eqref{eq:mnb0} is nonconvex and potentially hard, the dual problem is a one-dimensional convex optimization problem which can be solved very efficiently, using bisection. The number of iterations to localize an optimal $\alpha^*$ with absolute accuracy $\epsilon$ grows slowly, as $O(\log(1/\epsilon))$; each step involves the evaluation of a sub-gradient of the objective function, which requires finding the $k$ largest elements in a $m$-vector, and costs $O(m\log k)$. As before in the Bernoulli case, the complexity of the sparse variant in the multinomial case is $O(mn \log k)$, versus $O(mn)$ for the classical naive Bayes.
\paragraph{Quality estimate.} The quality of the bound in the multinomial case can be analysed using bounds on the duality gap based on the Shapley-Folkman theorem, as follows.
\begin{theorem}[Quality of Sparse Multinomial Naive Bayes Relaxation]\label{thm:sparse_mnb_quality} \label{thm:cvx-rlx-quality}
Let $\phi(k)$ be the optimal value of \eqref{eq:mnb0} and $\psi(k)$ that of the convex relaxation in~\eqref{eq:ub}, we have, for $k \ge 4$,
\begin{equation}\label{eq:gap-bnd}
\psi(k-4) \leq \phi(k) \leq \psi(k) \leq \phi(k + 4) .
\end{equation}
\end{theorem}
\begin{proof}
See Appendix~\ref{appendixC}.
\end{proof}
The bound in Theorem \ref{thm:cvx-rlx-quality} implies in particular
\[
\psi(k-4) \leq \phi(k) \leq \psi(k-4) + \Delta(k), \mbox{ for $k\geq4$,}
\]
where $\Delta(k):=\psi(k)-\psi(k-4)$. This means that if $\psi(k)$ does not vary too fast with $k$, so that $\Delta(k)$ is small, then the duality gap in problem~\eqref{eq:mnb0} is itself small, bounded by $\Delta(k)$; then solving the convex problem \eqref{eq:ub} will yield a good approximate solution to~\eqref{eq:mnb0}. This means that when the marginal contribution of additional features, i.e. $\Delta(k)/\psi(k)$ becomes small, our bound becomes increasingly tight. The ``elbow heuristic'' is often used to infer the number of relevant features $k^*$, with $\psi(k)$ increasing fast when $k<k^*$ and much more slowly when $k\geq k^*$. In this scenario, our bound becomes tight for $k\geq k^*$ .
\section{Experiments}\label{sec:expt}
\subsection{Experiment 1: Duality Gap}
\begin{figure}[h!]
\centering
\begin{minipage}[b]{0.495\textwidth}
\includegraphics[width=\textwidth]{figures/v2_dualbound_30.pdf}
\end{minipage}
\hfill
\begin{minipage}[b]{0.495\textwidth}
\includegraphics[width=\textwidth]{figures/v2_dualbound_3000.pdf}
\end{minipage}
\caption{\textbf{Experiment 1:} Duality gap bound versus sparsity level for $m = 30$ (left panel) and $m= 3000$ (right panel), showing that the duality gap quickly closes as $m$ or $k$ increase.}
\label{fig:expt1}
\end{figure}
In this experiment, we generate random synthetic data with uniform independent entries: $f^{\pm} \sim U[0,1]^m$, where $m \in \{30,3000\}$. We then normalize $f^{\pm}$ and compute $\psi(k)$ and $\psi(k-4)$ for $4 \leq k \leq m$ and plot how this gap evolves as $k$ increases. For each value of $k$, we also plot the value of the reconstructed primal feasible point, as detailed in Theorem~\ref{eq:ub}. The latter serves as a lower bound on the true value $\phi(k)$ as well, which can be used to test \textit{a posteriori} if our bound is accurate.
Figure \ref{fig:expt1} shows that, as the number of features $m$ or the sparsity parameter $k$ increases, the duality gap bound decreases. Figure \ref{fig:expt1} also shows that the \textit{a posteriori} gap is almost always zero, implying strong duality. In particular, as shown in Figure \ref{fig:expt1}(b), as the number of features increases, the gap between the bounds and the primal feasible point's value becomes negligible for all values of $k$.
\subsection{Experiment 2: Feature Selection}
In the next three experiments, we compare our sparse multinomial model \eqref{eq:mnb0} with other feature selection methods for sentiment classification on five different text data sets. Some details on the data sets sizes are given in Table~\ref{tab:dsinfo}. More information on these data sets and how they were pre-processed are given in Appendix \ref{appendixF}.
\begin{table}[h!]
\centering
{\small
\begin{tabular}{lrrrrr}
\toprule
\textsc{Feature Vectors} & \textsc{Amazon} & \textsc{IMDB} & \textsc{Twitter} & \textsc{MPQA} & \textsc{SST2}\\
\midrule
\textsc{Count Vector} & \textsc{31,666} & \textsc{103,124} & \textsc{273,779} & \textsc{6,208} & \textsc{16,599} \\
\textsc{tf-idf} & \textsc{5000} & \textsc{5000} & \textsc{5000} & \textsc{5000} & \textsc{5000} \\
\textsc{tf-idf wrd bigram} & \textsc{5000} & \textsc{5000} & \textsc{5000} & \textsc{5000} & \textsc{5000} \\
\textsc{tf-idf char bigram} & \textsc{5000} & \textsc{5000} & \textsc{5000} & \textsc{4838} & \textsc{5000} \\ \midrule
\textsc{$n_\text{train}$} & \textsc{8000} & \textsc{25,000} & \textsc{1,600,000} & \textsc{8484} & \textsc{76,961}\\
\textsc{$n_\text{test}$} & \textsc{2000} & \textsc{25,000} & \textsc{498} & \textsc{2122} & \textsc{1821}\\
\bottomrule\\
\end{tabular}}
\caption{\textbf{Experiment 2 data}: Number of features for each type of feature vector for each data set. For tf-idf feature vectors, we fix the maximum number of features to 5000 for all data sets. The last two rows show the number of training and test samples.}
\label{tab:dsinfo}
\end{table}
For each data set and each type of feature vector, we perform the following two-stage procedure. In the first step, we employ a feature selection method to attain a desired sparsity level of (0.1\%, 1\%, 5\%, 10\%); in the second step, we train a classifier based on the selected features. Specifically, we use $\ell_1$-regularized logistic regression, logistic regression with recursive feature elimination (RFE), $\ell_1$-regularized support vector machine (SVM), SVM with RFE, LASSO, thresholded Multinomial naive Bayes (TMNB), the Odds Ratio metric described by \cite{mladenic1999feature} and \eqref{eq:mnb0} in the first step. Then using the selected features, in the second step we train a logistic model, a Multinomial naive Bayes model, and a SVM. Thresholded multinomial naive Bayes (TMNB) means we train a multinomial naive Bayes model and then select the features corresponding to indices of the largest absolute value entries of the vector of classification coefficients $w_m$, as defined in \eqref{eq:m-rule}. For each desired sparsity level and each data set in the first step, we do a grid search over the optimal Laplace smoothing parameter for MNB for each type of feature vector. We use this same parameter in \eqref{eq:mnb0}. All models were implemented using Scikit-learn \citep{scikit}.
\begin{figure}[h!]
\centering
\includegraphics[height=.45\textheight]{figures/expt2_imdb.pdf}
\caption{\textbf{Experiment 2:} Accuracy (top panel) and computing speedup for our method (bottom panel), with the IMDB dataset/Count Vector with MNB in stage 2, showing performance on par with the best feature selection methods, at fraction of computing cost. Times \textit{do not} include the cost of grid search to reach the target cardinality for $\ell_1$-based methods.}
\label{fig:expt2}
\end{figure}
Figure \ref{fig:expt2} shows that \eqref{eq:mnb0} is competitive with other feature selection methods, consistently maintaining a high test set accuracy, while only taking a fraction of the time to train; for a sparsity level of $5\%$, a logistic regression model with $\ell_1$ penalty takes more than $1000$ times longer to train.
\subsection{Experiment 3: large-scale feature selection}
For this experiment, we consider the same data sets as before, but do not put any limit on the number of features for the tf-idf vectors. Due to the large size of the data sets, most of the feature selection methods in Experiment 1 are not feasible. We use the same two-stage procedure as before: 1) do feature selection using TMNB, the Odds Ratio method and our method \eqref{eq:ub}, and 2) train a MNB model using the features selected in stage 1. We tune the hyperparameters for MNB and \eqref{eq:ub} the same way as in Experiment 2. In this experiment, we focus on sparsity levels of $0.01\%, 0.05\%, 0.1\%, 1\%$. Table \ref{tab:dsinfo2} summarizes the data used in Experiment 2 and in Table \ref{tab:time} we display the average training time for \eqref{eq:ub}.
\begin{table}[h!]
\centering
{\small \begin{tabular}{lrrrrr}
\toprule
\textsc{Feature Vectors} & \textsc{Amazon} & \textsc{IMDB} & \textsc{Twitter} & \textsc{MPQA} & \textsc{SST2}\\
\midrule
\textsc{Count Vector} & \textsc{31,666} & \textsc{103,124} & \textsc{273,779} & \textsc{6,208} & \textsc{16,599} \\
\textsc{tf-idf} & \textsc{31,666} & \textsc{103,124} & \textsc{273,779} & \textsc{6,208} & \textsc{16,599} \\
\textsc{tf-idf wrd bigram} & \textsc{870,536} & \textsc{8,950,169} & \textsc{12,082,555} & \textsc{27,603} & \textsc{227,012} \\
\textsc{tf-idf char bigram} & \textsc{25,019} & \textsc{48,420} & \textsc{17,812} & \textsc{4838} & \textsc{7762} \\
\bottomrule\\
\end{tabular}}
\caption{\textbf{Experiment 3 data}: Number of features for each type of feature vector for each data set with no limit on the number of features for the tf-idf vectors. The train/test split is the same as in Table \ref{tab:dsinfo}.}
\label{tab:dsinfo2}
\end{table}
\begin{table}[h!]
\centering
{\small \begin{tabular}{lrrrrr}
\toprule
\textsc{} & \textsc{Amazon} & \textsc{IMDB} & \textsc{Twitter} & \textsc{MPQA} & \textsc{SST2}\\
\midrule
\textsc{Count Vector} & \textsc{0.043} & \textsc{0.22} & \textsc{1.15} & \textsc{0.0082} & \textsc{0.037} \\
\textsc{tf-idf} & \textsc{0.033} & \textsc{0.16} & \textsc{0.89} & \textsc{0.0080} & \textsc{0.027} \\
\textsc{tf-idf wrd bigram} & \textsc{0.68} & \textsc{9.38} & \textsc{13.25} & \textsc{0.024} & \textsc{0.21} \\
\textsc{tf-idf char bigram} & \textsc{0.076} & \textsc{0.47} & \textsc{4.07} & \textsc{0.0084} & \textsc{0.082} \\
\bottomrule\\
\end{tabular}}
\caption{\textbf{Experiment 3 run times}: Average run time (in seconds, with a standard CPU and a non-optimized implementation) over $4 \times 30 = 120$ values for different sparsity levels and 30 randomized train/test splits per sparsity level for each data set and each type of feature vector. On the largest data set (\textsc{Twitter}, $\sim 12$M features, $\sim 1.6$M data points), the computation takes less than 15 seconds. For the full distribution of run times, see Appendix \ref{appendixF}.}
\label{tab:time}
\end{table}
\begin{figure}[h!]
\centering
\begin{minipage}[b]{0.495\textwidth}
\includegraphics[width=\textwidth]{figures/expt3_gains.pdf}
\end{minipage}
\hfill
\begin{minipage}[b]{0.495\textwidth}
\includegraphics[width=\textwidth]{figures/expt4_imdb.pdf}
\end{minipage}
\caption{\textbf{Experiment 3 (Left):} Accuracy gain for our method (top panel) and factor slower (bottom panel) over all data sets listed in Table \ref{tab:dsinfo2} with MNB in stage 2, showing substantial performance increase with a constant increase in computational cost. \textbf{Experiment 4 (Right):} Run time with IMDB dataset/tf-idf vector data set, with increasing $m,k$ with fixed ratio $k/m$, empirically showing (sub-) linear complexity.}
\label{fig:expt34}
\end{figure}
Figure \ref{fig:expt34} shows that, even for large datasets with millions of features and data points, our method, implemented on a standard CPU with a non-optimized solver, takes at most a few seconds, while providing a significant improvement in performance. See Appendix \ref{appendixF} for the accuracy versus sparsity plot for each data set and each type of feature vector.
\subsection{Experiment 4: complexity}
Using the IMDB dataset in Table \ref{tab:dsinfo}, we perform the following experiment: we fix a sparsity pattern $k/m = 0.05$ and then increase $k$ and $m$. Where we artificially set the number of tf-idf features to 5000 in Experiment 1, here we let the number of tf-idf features vary from $10,000$ to $80,000$. We then plot the the time it takes to train \eqref{eq:mnb0} at a the fixed $5 \%$ sparsity level. Figure \ref{fig:expt34} shows that for a fixed sparsity level, the complexity of our method appears to be sub-linear.
\subsubsection*{Acknowledgments}
AA is at CNRS \& d\'epartement d'informatique, \'Ecole normale sup\'erieure, UMR CNRS 8548, 45 rue d'Ulm 75005 Paris, France, INRIA and PSL Research University. AA would like to acknowledge support from the {\em ML \& Optimisation} joint research initiative with the {\em fonds AXA pour la recherche} and Kamet Ventures, as well as a Google focused award.
\newpage
\bibliographystyle{apalike}
\section{Robustness} \label{subsec:rob}
In this section we consider a variant of the original naive Bayes models where the input data (matrix $X$) is imperfectly known, and subject to adversarial noise. We adopt the robust maximum likelihood (maximin) principle developed in~\citet{ben2009robust,lanckriet2002robust,bertsimas2019robust}.
\subsection{Robust maximum likelihood}
Consider a generic maximum likelihood problem
\[
\max_\theta \: \mathcal{L}(\theta; X),
\]
with $X \in {\mathbb R}^{n \times m}$ the data matrix, $\mathcal{L}$ the loss function and $\theta \in {\mathbb R}^m$ containing the parameters of the model.
We now assume that $X$ is subject to additive noise. Modeling the noisy matrix as $X+U$, where matrix $U$ is only known to belong to a given ``uncertainty set'' ${\cal U} \subseteq {\mathbb R}^{n \times m}$, the robust version of the above problem is defined as
\begin{equation}\label{eq:rob-maximin}
\max_\theta \: \min_{U \in {\cal U}} \: \mathcal{L}(\theta; X+U),
\end{equation}
where $\mathcal{L}$ is the loss function and $\theta \in {\mathbb R}^m$ contains the parameters of the model. Note that the approach generalizes well-known models, for example with the loss function $\mathcal{L}(\theta; X) = -\|X\theta - y\|_2$, with $y \in {\mathbb R}^n$ a given response vector, and the uncertainty set ${\cal U}$ chosen to be the set of matrices with largest singular value norm less than a given positive number $\rho$, the resulting robust counterpart is the so-called ``square-root'' version of ridge regression:
\[
\min_\theta \: \|X\theta-y\|_2 + \rho \|\theta\|_2.
\]
This illustrates the connections between robust counterparts and penalization, with a penalty that depends specifically on the uncertainty set, as further elaborated in \citet{ben2009robust}. We will observe a similar connection in the context of naive Bayes.
We now consider such robust formulations of \eqref{eq:bnb_train} and \eqref{eq:mnb_train}. First, we define noise models that make sense from a practical point of view.
\subsection{Adversarial noise models}
We assume that the uncertainty affects each data point independently. This leads to uncertainty sets of the form
\[
{\cal U}(X) = \left\{ [u_1, \ldots,u_n] \in {\mathbb R}^{n \times m} ~:~ u_i \in {\cal V}(x_i) , \;\; i \in [n] \right\},
\]
where a set ${\cal V}(x) \subseteq {\mathbb R}^m$ models the uncertainty on a generic data point $x \in {\mathbb R}^n$. We focus on the following uncertainty sets, depending on the nature of the data point, either binary-, integer-, or real-valued.
In the Bernoulli model, the input is binary, so it makes sense to consider flips as a primary source of uncertainty. For a given binary vector $x \in \{0,1\}^n$ and integer $p \in [m]$,
\begin{align}\label{eq:unc-bnb}
{\cal V}_\text{flip}(x) &= \{ \delta ~:~ \delta + x \in \{0,1\}^m, \;\; \|\delta\|_1 \leq p\} . \tag{$p$-flips}
\end{align}
Thus, ${\cal V}_\text{flip}(x)$ represents the set of vectors that can be obtained by flipping at most $p\leq m$ components in a binary vector $x$.
For the multinomial model, we often deal with count vectors $x \in \mathbb{N}^n$. The set
\begin{align}\label{eq:unc-mnb}
{\cal V}_\text{typo}(x) &= \{ \delta ~:~ \delta + x \in \mathbb{N}^m, \|\delta\|_1 \leq t\} \tag{$t$-typos}
\end{align}
models the fact that count vectors can be altered by adding or substracting at most $t$ counts to $x$. This can represent the presence of typos in text data.
Finally, for a real-valued vector $x \in {\mathbb R}^n$:
\begin{align}\label{eq:unc-lmbda}
{\cal V}_\text{shift}(x) &= \{ \delta ~:~ \delta + x \in \mathbb{N}^m, \delta \in [-\gamma, \gamma]^m\cap \mathbb{Z}^m \} , \tag{$\gamma$-shift}
\end{align}
with $\gamma>0$ given. Here, ${\cal V}_\text{shift}(x)$ represents an interval uncertainty affecting each feature independently.
\subsection{Robust counterparts}
Applying the modified formulation~\eqref{eq:rob-maximin} in lieu of the classical naive Bayes ones \eqref{eq:bnb_train} and \eqref{eq:mnb_train} leads to, respectively:
\begin{align} \label{eq:r-bnb-flip}
&\max_{\theta^{+}, \theta^{-} \in [0,1]^m} \:
\min_{U \in {\cal U}_\text{flip}(X)} \: \mathcal{L}_{\text{bnb}}(\theta^{+},\theta^{-}; X+U) , \tag{R-BNB-flip}
\end{align}
and
\begin{align}\label{eq:r-mnb}
&\max_{\theta^{+}, \theta^{-} \in [0,1]^m} \:
\min_{U \in {\cal U}(X)} \:
\mathcal{L}_{\text{bnb}}(\theta^{+},\theta^{-}; X+U) ~:~ \mathbf 1^\top \theta^{+} = \mathbf 1^\top\theta^{-} = 1,\tag{R-MNB}
\end{align}
where ${\cal U}(X)$ is one of the uncertainty sets introduced above, with the exception of the first \eqref{eq:unc-bnb}.
The inner minimization can be solved exactly for all the proposed uncertainty sets. The following result illustrates this for one of the uncertainty models.
\begin{theorem}\label{thm:rob}
In the ``typo'' uncertainty set given by \eqref{eq:unc-bnb}, the robust variant of naive Bayes \eqref{eq:r-mnb} reduces to a convex optimization problem:
\[
\begin{array}{ll}
\displaystyle\max_{\theta^{+}, \theta^{-} \in [0,1]^m} & f^{+\top} \log \theta^{+} + (\mathbf 1 - f^{+})^\top \log (\mathbf 1 - \theta^{+}) + f^{-\top} \log \theta^{-} + (\mathbf 1 - f^{-})^\top \log (\mathbf 1 - \theta^{-}) \\
& - \log (\displaystyle\max_{1 \le i \le m} \: \theta^{+}_i) - \log (\displaystyle\max_{1 \le i \le m} \: \theta^{-}_i) ~:~ \mathbf 1^\top \theta^{+} = \mathbf 1^\top \theta^{-} = 1
\end{array}
\]
which can be solved in $O(m \log m)$.
\end{theorem}
\begin{proof}
See Appendix~\ref{appendixD}.
\end{proof}
\subsection{Combining sparsity and robustness}
Any one of the sparse models in Section \eqref{subsec:sparse} can be combined with the appropriate robustness model in \eqref{subsec:rob} to results in a robust counterpart to sparse naive Bayes. By Theorem \ref{thm:rob}, the jointly sparse and robust model is a convex optimization problem. For example, \eqref{eq:mnb0} and \eqref{eq:mnb1} with \eqref{eq:unc-lmbda}, the training problems becomes
\begin{align}\label{eq:srnb0}
(\theta_1^\ast,\theta_2^\ast) &= \arg\max_{\substack{\mathbf 1^\top \theta_1 = 1, \mathbf 1^\top \theta_2 = 1 \\\|\theta_1 - \theta_2\|_0 \leq s\\\theta_1,\theta_2 \in [0,1]^m}} (f^{+} + |C_1| \gamma \mathbf 1)^\top \log \theta_1 + (f^{-} + |C_2| \gamma \mathbf 1)^\top \log \theta_2 \tag{SRNB-$\ell_0$}
\end{align}
and
\begin{align}\label{eq:srnb1}
(\theta_1^\ast,\theta_2^\ast) &= \arg\max_{\substack{\mathbf 1^\top \theta_1 = 1, \mathbf 1^\top \theta_2 = 1 \\\theta_1,\theta_2 \in [0,1]^m}} (f^{+} + |C_1| \gamma \mathbf 1)^\top \log \theta_1 + (f^{-} + |C_2| \gamma \mathbf 1)^\top \log \theta_2 - \lambda \|\theta_1 - \theta_2\|_1 \tag{SRNB-$\ell_1$}
\end{align}
respectively, where $|C_1|, |C_2|$ are the number of training points in class 1 and class 2 respectively. For more combinations of sparsity and uncertainty sets, see the proof of Theorem \eqref{thm:rob} in Appendix A.
\end{comment}
|
1511.05145
|
\section{Introduction}
A fundamental probe in modern astronomy to understand the Universe, its history and evolution, is the measurement of distances. Stellar parallax and the spectroscopic parallax allow us to reach $\sim$ 100-1000 pc, respectively but farther afield other methods are needed. A traditional technique for measuring distances consists in applying the inverse square law for astrophysical sources with known absolute magnitudes, aka, as standard candles. One of the first such objects used in astronomy were Cepheid stars. A Cepheid star's period is directly related to its intrinsic luminosity \citep{Leavitt08,benedict07} and allows one to probe the Universe to 15 Mpc. To attain larger distances brighter objects are required. Type Ia supernovae (SNe~Ia), have an absolute $B$-band magnitude about -19.5 -- -19.2 mag (depending on the assumptions of H$_{0}$ \citealt{richardson02,riess11}) which can be precisely calibrated using photometric and/or spectroscopic information from the SN itself, and be used as excellent distance indicators. Indeed, there are two parameters correlated to the luminosity. The first one is the decline rate: SNe~Ia with fast decline rates are fainter and have narrower light-curve peaks \citep{phillips93} and the second one, the colour \citep{riess96,tripp98}: redder SNe~Ia are fainter. The standardisation of SNe~Ia to a level $\sim$ 0.15--0.2 mag \citep{phillips93,hamuy96,riess96}, led to the measurement of the expansion history of the Universe and showed that, contrary to expectations, the Universe is undergoing an accelerated expansion \citep{riess98,perlmutter99,schmidt98}. Within this new paradigm, one of the greatest challenges is the search for the mechanism that causes the acceleration, an endeavour that will require exquisitely precise measurements of the cosmological parameters that characterise the current cosmological concordance model, i.e., $\Lambda$$CDM$ model. Several techniques that offer the promise to provide such constraints have been put forward over recents years: refined versions of the SNe~Ia method (\citealt{betoule14}), cosmic microwave background radiation measurements (Cosmic Microwave Background Explorer, \citealt{fixsen96,jaffe01}; Wilkinson Microwave Anisotropy Probe, \citealt{spergel07,bennett03}; and more recently the Planck mission, \citealt{planck13}), and baryon acoustic oscillation measurements (\citealt{blake03,seo03}). All of the above techniques have their own merits, but also their own systematic uncertainties that could become dominant with the increasingly higher level of precision required. Thus, it is important to develop as many methods as possible, since the truth will likely emerge from the combination of different independent approaches.\\
\indent
While SNe~Ia have been used as the primary diagnostic in constraining cosmological parameters, type IIP supernovae (SNe~IIP) have also been established to be useful independent distance indicators. SNe~IIP are 1--2 mag less luminous than the SNe~Ia however, their intrinsic rate is higher than SNe~Ia rate \citep{li2011}, and additionally the rate peaks at higher redshifts than SNe~Ia \citep{taylor14}, which motivates their use in the cosmic distance scale (see \citealt{hamuy02}). Also the fact that in principle they are the result of the same physical mechanism, and their progenitors are better understood than those of SNe~Ia, further encourages investigations in this direction. SNe~IIP are thought to be core-collapse supernovae (CCSNe), i.e., the final explosion of stars with zero-age main-sequence mass $\geq$ 8 ${\rm M}_{\odot}$ \citep{smart09b}. CCSNe have diverse classes, with a large range of observed luminosities, light-curve shapes, and spectroscopic features. CCSNe are classified in two groups according to the absence (SNe~Ib/c :\citealt{filippenko93,dessart11,bersten14,Kuncarayakti15}) or presence (SNe~II) of \ion{H}{1} lines (\citealt{min41,filippenko97} and references therein). Additional of the SNe~IIP and SNe~IIL which are discussed later, SNe~II are composed by SNe~IIb which evolve spectroscopically from SNe~IIP at early time to \ion{H}{1} deficient few weeks to a month past maximum \citep{woosley87} and SNe~IIn which have narrow \ion{H}{1} emission lines (\citealt{che81,fra82,sch90,chu94,van00,kan12,dejaeger15a}).
\indent
Historically, SNe~II were separated in two groups: SNe~IIP (70\% of CCSNe; \citealt{li2011}), which are characterised by long duration plateau phases ($\leq$~100~days) of constant luminosity, and SNe~IIL which have linearly declining light-curve morphologies \citep{barbon79}. However, as discussed in detail in \citet{anderson14a}, it is not clear how well this terminology describes the diversity of SNe~II. There are few SNe~II which show flat light-curves, and in addition there are very few (if any) SNe which decline linearly before falling onto the radioactive tail. Therefore, henceforth we simply refer to all SNe with distinct decline rates collectively as SNe~II, and later further discuss SNe in terms of their ``$s_{2}$'' plateau decline rates \citep{anderson14a}. \citet{sanders15} also suggested that the SNe~II family forms a continuous class, while \citet{arcavi13} and \citet{faran14b,faran14a} have argued for two separate populations.\\
\indent
The most noticeable difference between SNe~II occurs during the plateau phase. The optically thick phase is physically well-understood and is due to a change in opacity and density in the outermost layers of the SN. At the beginning the hydrogen present in the outermost layers of the progenitor star is ionised by the shock wave, which implies an increase of the opacity and the density which prevent the radiation from the inner parts to escape. After a few weeks, the star has cooled to the temperature allowing the recombination of ionised hydrogen (higher than 5000 K due to the large optical depth). The ejecta expand and the photosphere recedes in mass space, releasing the energy stored in the corresponding layers. The plateau morphology requires a recession of the photosphere in mass that corresponds to a fixed radius in space so that luminosity appears constant. As \citet{anderson14a} show, this delicate balance is rarely observed and there is significant diversity observed in the $V$-band light-curve. To reproduce the plateau morphology, hydrodynamical models have used red supergiant progenitors with extensive H envelopes \citep{grassberg71,falk77,chevalier76}. Direct detections of the progenitor of SNe~IIP have confirmed these models (\citealt{vandyk03,smartt09a}). It has also been suggested that SN~IIL progenitors may be more massive in the zero age main sequence than SNe~IIP \citep{EliasRosa10,EliasRosa11} and with smaller hydrogen envelopes \citep{popov93}.\\
\indent
To date several methods have been developed to standardise SNe~II. The first method called the ``Expanding Photosphere Method'' (EPM) was developed by \citet{kirshner74} and allows one to obtain the intrinsic luminosity assuming that SNe~II radiate as dilute blackbodies, and that the SN freely expands with spherical symmetry. The EPM was implemented for the first time on a large number of objects by \citet{schmidt94} and followed by many studies \citep{hamuy01,leonard03,dessart05,dessart06,jones09,enriquez11}. One of the biggest issues with this method is the EPM only works if one corrects for the blackbody assumptions which requires corrections factors computed from model atmospheres (\citealt{eastman96,dessart05} and see \citealt{dessart06} for the resolution of the EPM-based distance problem to SN 1999em). Also to avoid the problem in the estimation of the dilution factor, \citet{baron04} proposed a distance correcting factor that takes into account the departure of the SN atmosphere from a perfect blackbody, the ``Spectral-fitting Expanding Atmosphere Method'' (SEAM, updated in \citealt{dessart08}). This method consists of fitting the observed spectrum using an accurate synthetic spectrum of SNe~II, and then since the spectral energy distribution is completely known from the calculated synthetic spectra, one may calculate the absolute magnitude in any band.\\
\indent
A simpler method, also based on photometric and spectroscopic parameters, the ``Standardised Candle Method'' (SCM) was first introduced by \citet{hamuy02}. They found that the luminosity and the expansion velocity are correlated when the SN is in its plateau phase (50 days post explosion). This relation is physically well understood: for a more luminous SN, the hydrogen recombination front will be at a larger radius thereby the velocity of the photosphere will be greater \citep{kasen09} for a given post-explosion time. Thanks to this method the scatter in the Hubble diagram (hereafter Hubble diagram) drops from 0.8 mag to 0.29 mag in the $I$-band. \citet{nugent06} improved this method by adding an extinction correction based on the $(V-I)$ colour at day 50 after maximum. This new method is very powerful and many other studies \citep{nugent06,poznanski09,olivares10,andrea10} have confirmed the possibility to use SNe~II as standard candles finding a scatter between 10 and 18\% in distance. Recently \citet{maguire10} suggested that using near-infrared (NIR) filters, the SCM, the dispersion can drop to a level of 0.1--0.15 mag (using 12 SNe~IIP). Indeed, in the NIR the host-galaxy extinction is less important, thus there may be less scatter in magnitude. Note also the work done by \citet{rodriguez14} where the authors used the Photospheric Magnitude Method (PMM) which correspond to a generalisation of the SCM for various epochs throughout the photospheric phase and found a dispersion of 0.12 mag using 13 SNe. This is an intrinsic dispersion and is not the RMS.\\
\indent
The main purpose of this work is to derive a method to obtain purely photometric distances, i.e, standardise SNe~II only using light-curves and colour-curve parameters, unlike other methods cited above which require spectroscopic parameters. This is a big issue, and purely photometric methods will be an asset for the next generation of surveys such as the large synoptic survey telescope (LSST; \citealt{ivezic09,lien14LSST}). These surveys will discover such a large number of SNe that spectroscopic follow-up will be impossible for all but only for small number of events. This will prevent the use of current methods to standardise SNe~II and calculate distances. Therefore deriving distances with photometric data alone is important and useful for the near future but also allows us to reach higher distance due to the fact that getting even one spectrum for a SN~II at $z\geq 1$ is very challenging.\\
\indent
The paper is organised as follows. In section 2 a description of the data set is given and in section 3 we explain how the data are corrected for Milky Way (MW) extinction and how the K-correction is applied. In section 4 we describe the photometric colour method (PCM) using optical and NIR filters and we derive a photometric Hubble diagram. In section 5 we present a comparative Hubble diagram using the SCM. In section 6 we compare our method with the SCM and we conclude with a summary in Section 7.
\section{Data Sample}
\subsection{Carnegie Supernova Project}
The \textit{Carnegie Supernova Project}\footnote{\url{http://csp.obs.carnegiescience.edu/}} (CSP, \citealt{ham06}) provided all the photometric and spectroscopic data for this project. The goal of the CSP was to establish a high-cadence data set of optical and NIR light-curves in a well-defined and well-understood photometric system and obtain optical spectra for these same SNe. Between 2004 and 2009, the CSP observed many low redshift SNe~II ($N_{SNe}\sim 100$ with $z \leq 0.04$), 56 had both optical and NIR light-curves with good temporal coverage; one of the largest NIR data samples. Two SN 1987A-like events were removed (SN 2006V and SN 2006au see \citealt{taddia12}) living the sample listed in Table~\ref{parameters} with photometric parameters measured by \citet{anderson14a}. Note that we do not include SNe~IIb or SNe~IIn.\\
\begin{table*}
\begin{center}
\caption{SN~II parameters}
\begin{tabular}{cccccccc}
\hline
SN & AvG & v$_{helio}$ & v$_{CMB}$& Explosion date & $s_{1}$ & $s_{2}$
&OPTd\\
&(mag)&(km s$^{-1}$)&(km s$^{-1}$)&(MJD)&(mag 100d$^{-1}$)&(mag 100d$^{-1}$)&(days)\\
\hline
\hline
2004ej &0.189 &2723(6) &3045(23) &53224.90(5) &$\cdots$ &1.07(0.04) &96.14\\
2004er &0.070 &4411(33) &4186(37) &53271.80(4) &1.28(0.03) &0.40(0.03) &120.15\\
2004fc &0.069 &1831(5) &1560(20) &53293.50(10) &$\cdots$ &0.82(0.02) &106.06\\
2004fx &0.282 &2673(3) &2679(3) &53303.50(4) &$\cdots$ &0.09(0.03) &68.40\\
2005J &0.075 &4183(1) &4530(24) &53382.78(7) &2.11(0.07) &0.96(0.02) &94.03\\
2005Z &0.076 &5766(10) &6088(25) &53396.74(8) &$\cdots$ &1.83(0.01) &78.84\\
2005an &0.262 &3206(31) &3541(39) &53428.76(4) &3.34(0.06) &1.89(0.05) &77.71\\
2005dk &0.134 &4708(25) &4618(26) &53599.52(6) &2.26(0.09) &1.18(0.07) &84.22\\
2005dn &0.140 &2829(17) &2693(20) &53601.56(6) &$\cdots$ &1.53(0.02) &79.76\\
2005dw &0.062 &5269(10) &4974(23) &53603.64(9) &$\cdots$ &1.27(0.04) &92.59\\
2005dx &0.066 &8012(31) &7924(31) &53615.89(7) &$\cdots$ &1.30(0.05) &85.59\\
2005dz &0.223 &5696(8) &5327(27) &53619.50(4) &1.31(0.08) &0.43(0.04) &81.86\\
2005es &0.228 &11287(49) &10917(55) &53638.70(10) &$\cdots$ &1.31(0.05) &$\cdots$\\
2005gk &0.154 &8773(10) &8588(30) &$\cdots$ &$\cdots$ &1.25(0.07) &$\cdots$\\
2005hd &0.173 &8323(10) &8246(30) &$\cdots$ &$\cdots$ &1.83(0.13) &$\cdots$\\
2005lw &0.135 &7710(29) &8079(39) &53716.80(10) &$\cdots$ &2.05(0.04) &107.23\\
2006Y &0.354 &10074(10) &10220(30) &53766.50(4) &8.15(0.76) &1.99(0.12) &47.49\\
2006ai &0.347 &4571(10) &4637(30) &53781.80(5) &4.97(0.17) &2.07(0.04) &63.26\\
2006bc &0.562 &1363(10) &1476(13) &53815.50(4) &1.47(0.18) &-0.58(0.04) &$\cdots$\\
2006be &0.080 &2145(9) &2243(11) &53805.81(6) &1.26(0.08) &0.67(0.02) &72.89\\
2006bl &0.144 &9708(49) &9837(50) &53823.81(6) &$\cdots$ &2.61(0.02) &$\cdots$\\
2006ee &0.167 &4620(19) &4343(27) &53961.88(4) &$\cdots$ &0.27(0.02) &85.17\\
2006it &0.273 &4650(9) &4353(23) &54006.52(3) &$\cdots$ &1.19(0.13) &$\cdots$\\
2006ms &0.095 &4543(18) &4401(21) &54034.00(13) &2.07(0.30) &0.11(0.48) &$\cdots$\\
2006qr &0.126 &4350(5) &4642(21) &54062.80(7) &$\cdots$ &1.46(0.02) &96.85\\
2007P &0.111 &12224(25) &12570(35) &54118.71(3) &$\cdots$ &2.36(0.04) &84.33\\
2007U &0.145 &7791(9) &7795(9) &54134.61(6) &2.94(0.02) &1.18(0.01) &$\cdots$\\
2007W &0.141 &2902(2) &3215(22) &54136.80(7) &$\cdots$ &0.12(0.04) &77.29\\
2007X &0.186 &2837(6) &3055(16) &54143.85(5) &2.43(0.06) &1.37(0.03) &97.71\\
2007aa &0.072 &1465(4) &1826(26) &54135.79(5) &$\cdots$ &-0.05(0.02) &67.26\\
2007ab &0.730 &7056(13) &7091(13) &54123.86(6) &$\cdots$ &3.30(0.08) &71.30\\
2007av &0.099 &1394(3) &1742(24) &54175.76(5) &$\cdots$ &0.97(0.02) &$\cdots$\\
2007hm &0.172 &7540(15) &7241(26) &54335.64(6) &$\cdots$ &1.45(0.04) &$\cdots$\\
2007il &0.129 &6454(10) &6146(24) &54349.77(4) &$\cdots$ &0.31(0.02) &103.43\\
2007oc &0.061 &1450(5) &1184(19) &54382.51(3) &$\cdots$ &1.83(0.01) &77.61\\
2007od &0.100 &1734(3) &1377(25) &54402.59(5) &2.37(0.05) &1.55(0.01) &$\cdots$\\
2007sq &0.567 &4579(4) &4874(21) &54421.82(3) &$\cdots$ &1.51(0.05) &88.34\\
2008F &0.135 &5506(21) &5305(25) &54470.58(6) &$\cdots$ &0.45(0.10) &$\cdots$\\
2008K &0.107 &7997(10) &8351(27) &54477.71(4) &$\cdots$ &2.72(0.02) &87.1\\
2008M &0.124 &2267(4) &2361(8) &54471.71(9) &$\cdots$ &1.14(0.02) &75.34\\
2008W &0.267 &5757(45) &6041(49) &54485.78(6) &$\cdots$ &1.11(0.04) &83.86\\
2008ag &0.229 &4439(6) &4428(6) &54479.85(6) &$\cdots$ &0.16(0.01) &102.95\\
2008aw &0.111 &3110(4) &3438(23) &54517.79(10) &3.27(0.06) &2.25(0.03) &75.83\\
2008bh &0.060 &4345(8) &4639(22) &54543.54(5) &3.00(0.27) &1.20(0.04) &$\cdots$\\
2008bk &0.054 &230(4) &-50(20) &54542.89(6) &$\cdots$ &0.11(0.02) &104.83\\
2008bu &1.149 &6630(9) &6683(10) &54566.78(5) &$\cdots$ &2.77(0.14) &44.75\\
2008ga &1.865 &4639(3) &4584(5) &54711.85(4) &$\cdots$ &1.17(0.08) &72.79\\
2008gi &0.181 &7328(34) &7103(37) &54742.72(9) &$\cdots$ &3.13(0.08) &$\cdots$\\
2008gr &0.039 &6831(41) &6549(46) &54766.55(4) &$\cdots$ &2.01(0.01) &$\cdots$\\
2008hg &0.050 &5684(10) &5449(19) &54779.75(5) &$\cdots$ &-0.44(0.01) &$\cdots$\\
2009N &0.057 &1036(2) &1386(25) &54846.79(5) &$\cdots$ &0.34(0.01) &89.50\\
2009ao &0.106 &3339(5) &3665(23) &54890.67(4) &$\cdots$ &-0.01(0.12) &41.71\\
2009bu &0.070 &3494(9) &3372(13) &54907.91(6) &0.98(0.16) &0.18(0.04) &$\cdots$\\
2009bz &0.110 &3231(7) &3393(13) &54915.83(4) &$\cdots$ &0.50(0.02) &$\cdots$\\
\hline
\hline
\setcounter{table}{2}
\end{tabular}
\tablecomments{SN and light curve parameters. In the first column the SN name, followed by its reddening due to dust in our Galaxy \citep{schlafly11} are listed. In column 3 we list the host-galaxy heliocentric recession velocity. These are taken from the NASA Extragalactic Database (NED: \url{http://ned.ipac.caltech.edu/}). In column 4 we list the host-galaxy velocity in the CMB frame using the CMB dipole model presented by \citet{fixsen96}. In column 5 the explosion epochs is presented. In columns 6 and 7 we list the decline rate $s_{1}$ and $s_{2}$ in the $V$-band, where $s_{1}$ is the initial, steeper slope of the light-curve and $s_{2}$ is the decline rate of the plateau as defined by \citet{anderson14a} . Finally column 8 presents the optically thick phase duration (OPTd) values, i.e., the duration of the optically thick phase from explosion to the end of the plateau (see \citealt{anderson14a})}
\label{parameters}
\end{center}
\end{table*}
\subsection{Data reduction}
\subsubsection{Photometry}
All the photometric observations were taken at the Las Campanas Observatory (LCO) with the Henrietta Swope 1-m and the Ir\'en\'ee du Pont 2.5-m telescopes using optical ($u$, $g$, $r$, $i$, $B$, and $V$), and NIR filters ($Y$, $J$, and $H$, see \citealt{stritzinger11}).
\indent
All optical images were reduced in a standard way including bias subtractions, flat-field corrections, application of a linearity correction and an exposure time correction for a shutter time delay. The NIR images were reduced through the following steps: dark subtraction, flat-field division, sky subtraction, geometric alignment and combination of the dithered frames. Due to the fact that SN measurements can be affected by the underlying light of their host galaxies, we took care in correctly removing the underlying host-galaxy light. The templates used for final subtractions were always taken months/years after each SN faded and under seeing conditions better than those of the science frames. Because the templates for some SNe were not taken with the same telescope, they were geometrically transformed to each individual science frame. These were then convolved to match the point-spread functions, and finally scaled in flux. The template images were then subtracted from a circular region around the SN position on each science frame (see \citealt{contreras10}).\\
\indent
Observed magnitudes for each SN was derived relative to local sequence stars and calibrated from observations of standard stars in the \citet{lan92} ($BV$), \citet{smithja2002} ($u'g'r'i'$), and \citet{persson04} ($YJHKs$) systems. The photometry of the local sequence stars are on average based on at least three photometric nights. Magnitudes are expressed in the natural photometric system of the Swope+CSP bands. Final errors for each SN are the result of the instrumental magnitude uncertainty and the error on the zero point. The full photometric catalog will be published in an upcoming paper (note that the $V$-band photometry has been already published in \citealt{anderson14a}).\\
\subsubsection{Spectroscopy}
The majority of our spectra were obtained with the 2.5m Ir\'en\'ee du Pont telescope using the WFCCD- and Boller and Chiven spectrographs (the last is now decommissioned) at LCO. Additional spectra were obtained with the 6.5m Magellan Clay and Baade telescopes with LDSS-2, LDSS-3, MagE (see \citealt{massey12} for details) and IMACS together with the CTIO 1.5m telescope and the Ritchey-Chr\'etien Cassegrain Spectrograph, and the New Technology Telescope (NTT) at La Silla observatory using the EMMI and EFOSC instruments. The majority of the spectra are the combination of three exposures to facilitate cosmics-ray rejection. Information about the grism used, the exposure time, the observation strategy can be found in \citet{ham06,folatelli10}. All spectra were reduced in a standard way as described in \citet{ham06} and \citet{folatelli13}. Briefly, the reduction was done with IRAF\footnote{IRAF is distributed by the National Optical Astronomy Observatory, which is operated by the Association of Universities for Research in Astronomy (AURA) under cooperative agreement with the National Science Foundation.} using the standard routines (bias subtraction, flat-field correction, 1-D extraction, wavelength and flux calibration). The full spectroscopic sample will be published in an upcoming paper and the reader can refer to \citet{anderson14b} and \citet{gutierrez14} for a thorough analysis of this sample.\\
\section{First photometric corrections}
In order to proceed with our aim of creating a Hubble diagram based on photometric measurements using the PCM, in this section we show how to correct apparent magnitudes for MW extinction (AvG) and how to apply the K-correction, without the use of observed SN spectra but only with model spectra.
\subsection{MW correction}
In the $V$ band the determination of AvG can be applied using the extinction maps of \citealt{schlafly11}. To convert AvG to extinction values in other bands we need to adopt: an extinction law and the effective wavelength for each filter.\\
\indent
SN~II spectra evolve with time from a blue continuum at early times to a redder continuum with many absorption/emission features at later epochs. This implies that the effective wavelength of a broad-band filter also changes with time (see the formula given \citealt{Bessell12} A.21). To calculate effective wavelengths at different epochs we adopt a sequence of theoretical spectral models from \citet{dessart13} consisting of a SN progenitor with a main-sequence mass of 15 ${\rm M}_{\odot}$, solar metallicity Z=0.02, zero rotation and a mixing-length parameter of 3\footnote{More information about this model (named m15mlt3) can be found in \citet{dessart13}}. The choice of this model is based on the fact that it provided a good match to a prototypical SN~II such as SN~1999em. For each photometric epoch, we choose the closest theoretical spectrum in epoch since the explosion, the extinction law from \citet{car89} and in time $R_{V}=3.1$ to obtain the MW extinction in the other filters.\\
\subsection{K-correction}
Having corrected the observed magnitudes for Galactic extinction, we need to apply also a correction attributable to the expansion of the Universe called the K-correction (KC). A photon received in one broad photometric band--pass in the observed referential has not necessarily been emitted (rest--frame referential) in the same filter, that is why this correction is needed. For each epoch of each filter we use the same procedure to estimate the KC. Here we describe our method step by step for one epoch and a given filter X.
\begin{enumerate}
\item{We choose in our model spectral library (\citealt{dessart13}, model m15mlt3) the theoretical spectrum (rest
frame) closest to the photometric epoch since explosion time (corrected for time dilatation), with a rest-frame spectral energy distribution (SED), $f^{rest}({\lambda}_{rest})$. Because our library covers a limited range of epochs from 12.2 to 133 days relative to explosion, observations outside these limits are ignored.}
\item{We bring the rest-frame theoretical spectrum to the observer's frame using $(1+z_{hel})$ the correction, where $z_{hel}$ is the heliocentric redshift of the SN, $f^{obs}({\lambda})={f^{rest}({\lambda}_{rest}(1+z_{hel}))} \times 1/(1+z_{hel})$ where $\lambda$ is the wavelength in the observer's frame.}
\item{We match the theoretical spectrum to the observed photometric magnitudes of the SN \citep{hsiao07}. For this we calculate synthetic magnitudes (from the model in the observer's frame,$f^{obs}({\lambda})$) and compare them to the observed magnitudes corrected for MW extinction. We use all the filters available at this epoch. Then we obtain a warping function $W(\lambda)$ (quadratic, cubic, depending on the number of filters used) and do a constant extrapolation for the wavelengths outside of the range of filters used. With our warping function we correct our model spectrum and obtain $f^{obs}_{warp}({\lambda})=W(\lambda) \times f^{obs}({\lambda})$. We compute the magnitude in the observer's frame :
\[m^{X}_{z}=-2.5log_{10}\left(\frac{1}{hc} \int f^{obs}_{warp}(\lambda)S^{X}_{\lambda}\lambda d\lambda\right)+ZP_{X} \]
with c the light velocity in $\rm{\AA}$ s$^{-1}$, $h$ the Planck constant in
ergs s, $\lambda$ is wavelength, $S^{X}_{\lambda}$ the transmission function of filter $X$
and $ZP_{X}$ is the zero point of filter $X$ (see \citealt{contreras10,stritzinger11}).}
\item{We bring back the warping spectrum to the rest frame $f^{rest}_{warp}(\lambda)=(1+z_{hel})f^{obs}_{warp}(\lambda \times 1/(1+z_{hel}))$ and we obtain and calculate the magnitude :
\[m^{X}_{0}=-2.5log_{10}\left(\frac{1}{hc} \int f^{rest}_{warp}(\lambda)S^{X}_{\lambda}\lambda d\lambda\right)+ZP_{X} \]}
\item{Finally we obtain the KC for this epoch as the difference
between the observed and the rest frame magnitude,
$KC_{X}=m^{X}_{z}-m^{X}_{0}$.}
\item{To estimate the associated errors, we follow the same procedure
but instead of using the observed magnitudes for the warping,
we use the upper limit, i.e., observed magnitudes plus associated
uncertainties.}
\end{enumerate}
As a complementary work on the KC and to validate our method we compare the KC values found using the \citet{dessart13} model to those computed from our database of observed spectra. In both cases we use exactly the same procedure. First the observed spectrum is corrected in flux using the observed photometry (corrected for AvG) in order to match the observed magnitudes. The photometry is interpolated to the spectral epoch. In Figure~\ref{model_obs} we show a comparison between the KC obtained with the theoretical models and using our library of observed spectra at different redshifts. As we can see that the KC values calculated with both methods are very consistent. This exercise validates the choice of using the \citet{dessart13} models to calculate the KC. There are two advantages to use the theoretical models. First we can obtain the KC for NIR filters ($Y$, $J$, $H$) for which we do not have observed spectra, and secondly this method does not require observed spectra which are expensive to obtain in terms of telescope time, and virtually impossible to get at higher redshifts.
\begin{figure}
\includegraphics[width=10.0cm]{fig_1.eps}
\caption{Comparison between the KC calculated using the theoretical
models and observed spectra at different redshifts in $V$ band. The black dotted line
represents x=y. Each square represents one observed spectrum of our database. The colour bar on the right side represents the different redshifts.}
\label{model_obs}
\end{figure}
\section{The photometric colour Method: PCM}
In this section we present our PCM with which we derive the corrected magnitudes necessary for constructing the Hubble diagram solely with photometric data. Since we want to examine Hubble diagrams from photometry obtained at different epochs, we start by linearly interpolating colours on a daily basis from colours observed at epochs around the epoch of interest. The same procedure is used to interpolate magnitudes.\\
\subsection{Methodology}
To correct and standardise the apparent magnitude we use two photometric parameters: $s_{2}$ which is the slope of the plateau measured in the $V$-band \citep{anderson14a}, and a colour term at a specific epoch. The colour term is mainly used to take into account the dispersion caused by the host-galaxy extinction. The magnitude is standardised using a weighted least-squares routine by minimising the equation below :
\begin{equation}
m_{\lambda 1}+\alpha s_{2} - \beta_{\lambda 1} (m_{\lambda 2}-m_{\lambda 3}) = 5 log (cz)+ZP,
\end{equation}
where $c$ is the speed of light, $z$ the redshift, $m_{\lambda_{1,2,3}}$
the observed magnitudes with different filters, and corrected for AvG and KC, while $\alpha$, $\beta_{\lambda_{1}}$ and $ZP$ are free fitting parameters. The errors on these parameters are derived assuming a reduced chi square equal to one. In order to obtain the errors on the standardised magnitudes, an error propagation is performed in an iterative manner.
Note that $\beta_{\lambda_{1}}$ is related to host-galaxy $R_{V}$ if we assume that the colour-magnitude relation is due to extrinsic factors (the intrinsic colour is degenerate with the $ZP$). We obtain :
\begin{equation}
\beta_{\lambda 1}=\frac{A_{\lambda 1}}{E(m_{\lambda 2}-m_{\lambda 3})} ,
\end{equation}
where $A_{\lambda 1}$ is the host-galaxy extinction in the $\lambda 1$ filter and $E$ the colour excess.
Assuming a \citet{car89} law, there is one to one relationship between $R_{V}$ and $\beta_{\lambda_{1}}$. First we obtain the theoretical $\beta$ for different $R_{V}$ values using the \citet{car89} coefficients (a and b):
\begin{equation}
\beta(R_V)=\frac{a_{\lambda 1}+\frac{b_{\lambda 1}}{R_V}}{(a_{\lambda 2}+\frac{b_{\lambda 2}}{R_V})-(a_{\lambda 3}+\frac{b_{\lambda 3}}{R_V})} .
\end{equation}
Then we derive $R_{V}$ from the value of $\beta_{\lambda 1}$ determined from the least-squares fit (Eq 1). We will discuss the resulting $R_{V}$ values in section 6.5.\\
\subsection{Hubble flow sample}
We select only SNe located in the Hubble flow, i.e., with $cz_{CMB}$ $\geq$ 3000 km s$^{-1}$ in order to minimize the effect of peculiar galaxy motions. Our available sample is composed of the entire sample in the Hubble flow but 3 SNe. We eliminate two SNe due to the fact that the warping function cannot be computed, thus the K-correction (SN 2004ej and SN 2008K). We also take out the outlier SN 2007X and found for this object particular characteristics like clear signs of interaction with the circumstellar medium (flat H alpha P-Cygni profile, see Guti\'errez et al. in prep.).\\
\indent
SNe~II are supposedly characterised by similar physical conditions (e.g. temperature) when they arrive towards the end of the plateau \citep{hamuy02} that is why we use the end of the optically thick phase measured in the $V$ band (as defined by \citealt{anderson14a}) as the time origin in order to bring all SNe to the same time scale. When the end of the plateau is not available we choose 80 days post explosion, which is the average for our sample.\\
\indent
Given that SNe~II show a significant dispersion in the plateau duration driven by different evolution speeds, we decide to take a fraction of the plateau duration and not an absolute time, to ensure that we compare SNe~II at the same evolutionary phase. Thus, in the following analysis, we adopt OPTd*X\% as the time variable where OPTd is the optically thick phase duration and X is percentage ranging between 1-100\%.\\
\indent
In Figure~\ref{RMS_time} we present the variation with evolutionary phase of the dispersion in the Hubble diagram using the filters available and the $(V-i)$ colour. The lowest root mean square (RMS) values in the optical is found for the $r$ band, and at NIR wavelength using the $Y/J$ band. Note that the coverage in the $Y$ band is better than in the $J$ band hence, hereafter we use the $Y$ band. For these two bands we can obtain the median RMS over all the epochs (from 0.2*OPTd to 1.0*OPTd) and the standard deviation. We find for $r$ band 0.47 $\pm$ 0.04 mag and for $Y$ band 0.48 $\pm$ 0.04 mag. In Figure~\ref{RMS_time_couleur} we do as above but this time we change colours. Fixing the $r$ band and using different colours we show the variation of the RMS. This figure shows that the colour that minimises the RMS is $(V-i)$ ($(r-J)$ yields a lower dispersion but the time coverage is significantly less). We find a median RMS over all the epochs of 0.47 $\pm$ 0.05. For this reason we decide to combine the $r$ band and the $(V-i)$ colour for the Hubble diagram. Note also that the best epoch for the $r$ band is close to the middle of the plateau, 55\% of the time from the explosion to the end of the plateau, whereas in the $Y$ band is later in phase post explosion, around 65\%. In general the best epoch to standardise the magnitude is between 60--70\% of the OPTd for NIR filters and for optical filters between 50--60\% of OPTd. Physically these epochs correspond in both cases more or less to the middle of the plateau. Note that we tried other time origin such as the epoch of maximum magnitude instead of the end of the plateau but changing the reference does not lower the RMS.\\
\indent
In Figure~\ref{Hubble diagram_hubble_flow} we present a Hubble diagram based entirely on photometric data using $s_{2}$ and colour term for two filters, $r$ band and $Y$ band. In the $r$ band the RMS is 0.44 mag (with 38 SNe) which allows us to measure distances with an accuracy of $\sim$ 20\%. We find the same precision using the $Y'$-band with a RMS of 0.43 mag (30 SNe). Note that the colour term is more important for the optical filter than for the NIR filter. Indeed, for the $r$ band the RMS decreases from 0.50 to 0.44 mag when the colour term is added whereas for the NIR filter the improvement is only of 0.004 mag. Using all available epochs we find a mean improvement of 0.025 $\pm$ 0.011 in $r$ band and 0.014 $\pm$ 0.013 in $Y$ band. This shows that the improvement is significant in optical but less in NIR. The drop using the optical filter is not surprising because this term is probably at least partly related to host-galaxy extinction which is more prevalent in optical wavelengths than in the NIR, so adding a colour term for NIR filters does not significantly influence the dispersion. Note that if we use the weighted root mean square (WRMS) as defined by \citet{blondin11} we find 0.40 mag and 0.36 mag for the $r$ band and $Y$ band, respectively, after $s_{2}$ and colour corrections.\\
\indent
In the literature the majority of the studies used SNe~IIP for their sample. To check if we can include all the SNe~II (fast- and slow-decliners) we did some analysis of the SNe and investigate if any of the higher residuals arise from intrinsic SN properties. The overall conclusion is that at least to first order, we did not find any correlation between SNe~II intrinsic differences ($s_{2}$, OPTd,...) and the Hubble residuals. This suggests that SNe within the full range of $s_{2}$ values (i.e., all SNe~II) should be include in Hubble diagram.\\
\indent
Following the work of \citet{folatelli10} for SNe~Ia, we investigated the combined Hubble diagram using all the filters available (by averaging the distance moduli derived in each filter) but the dispersion obtained is not much better. We found the same correlation between the distance-modulus residuals in one band versus those in another band as found by \citet{folatelli10}, suggesting that the inclusion of multiple bands does not improve the distance estimate.\\
\indent
If we include SNe in the Hubble flow ($cz \geq$~3000 km s$^{-1}$) and very nearby SNe ($cz \leq$~3000 km s$^{-1}$) for the $r$ band the dispersion increases from 0.44 mag to 0.48 mag (46 SNe) whereas in the $Y$ band the RMS increase from 0.43 mag to 0.45 (41 SNe).\\
\indent
We also try to use two different epochs, one for the magnitude and the other for the colour but again, this does not improve the RMS. Finally, we try also to use the total decline rate (between maximum to the end of the plateau) instead of the plateau slope. Using the total decline rate does not lower the RMS (dispersion around 0.47 mag for 45 SNe in the $r$ band) but could be useful for high redshift SNe.
\begin{figure}
\includegraphics[width=9.5cm]{fig_2a.eps}
\caption{Variation in phase of the dispersion in the Hubble diagram for different filters and using a colour term $(V-i)$. In the x-axis we present the time as (explosion time+OPTd*X\%). The black squares present the $B$ band, dark blue circle the $g$ band, blue cross the $V$ band, dark green diamonds the $r$ band, green hexagons for the $i$ band, yellow pentagons for the $Y$ band, red plus symbol for the $J$ band. The $H$ band is not presented because the sampling is as good as it is in the other bands.}
\label{RMS_time}
\end{figure}
\begin{figure}
\includegraphics[width=9.5cm]{fig_2b.eps}
\caption{Variation in phase of the dispersion in the Hubble diagram using the $r$ band and different colours. In the x-axis we present the time as the OPTd*X\%. The black stars present $(r-i)$ colour, dark blue squares are for $(V-r)$, blue circle $(B-V)$, cyan cross $(g-r)$, green diamonds $(V-Y)$, yellow pentagons for $(r-J)$, and red hexagons for $(V-i)$.}
\label{RMS_time_couleur}
\end{figure}
\begin{figure*}
\center
\includegraphics[width=6.5cm]{r_PCM_a.eps}
\includegraphics[width=6.5cm]{r_PCM_b.eps}\\
\includegraphics[width=9.5cm]{r_PCM_c.eps}\\
\includegraphics[width=6.5cm]{Y_PCM_a.eps}
\includegraphics[width=6.5cm]{Y_PCM_b.eps}\\
\includegraphics[width=9.5cm]{Y_PCM_c.eps}
\caption{In the figures, we present the dispersion (RMS) using the PCM, the
number of SNe ($N_{SNe}$) and the epoch chosen with respect to OPTd (OPTd*X\%) for our Hubble flow sample. On the bottom of each plot, the residuals are shown. In all the residual plots, the dashed line correspond to the RMS. \textit{Top Left:} Apparent magnitude corrected for MW extinction and KC in the $r$ band plotted against $cz_{CMB}$; \textit{Top Right:} Apparent magnitude corrected for MW extinction, KC and $s_{2}$ term in the $r$ band plotted against $cz_{CMB}$. \textit{Top Center: } Apparent magnitude corrected for MW
extinction, KC, $s_{2}$ term in the $r$ band, and by colour term, $(V-i)$ plotted against $cz_{CMB}$. \textit{Bottom Left:} Apparent magnitude corrected for MW extinction and KC in the $Y$ band plotted against $cz_{CMB}$; \textit{Bottom Right:}
Apparent magnitude corrected for MW extinction, KC and $s_{2}$ term in the $Y$ band plotted against $cz_{CMB}$. \textit{Bottom Center:} Apparent magnitude corrected for MW extinction, KC, $s_{2}$ term in the $Y$ band, and by colour term $(V-i)$ plotted against $cz_{CMB}$.}
\label{Hubble diagram_hubble_flow}
\end{figure*}
\section{The standard Candle Method (SCM)}
The SCM as employed by various authors gives a Hubble diagram dispersion of 0.25--0.30 mag \citep{hamuy02,nugent06,poznanski09,olivares10,andrea10}. Here we present the Hubble diagram using the SCM for our sample.
\subsection{\ion{Fe}{2} velocity measurements}
To apply the SCM, we need to measure the velocity of the SN ejecta. One of the best features is \ion{Fe}{2} $\lambda 5018$ because other iron lines such as \ion{Fe}{2} $\lambda 5169$ can be blended by other elements. Expansion velocities are measured through the minimum flux of the absorption component of P-Cygni line profile after correcting the spectra for the heliocentric redshifts of the host-galaxies. Errors were obtained by measuring many times the minimum of the absorption changing the trace of the continuum. The range of velocities is 1800--8000 km s$^{-1}$ for all the SNe. Because we need the velocities for different epochs in order to find the best epoch (as done for the PCM), i.e., with less dispersion, we do an interpolation/extrapolation using a power law \citep{hamuyphd} of the form:
\begin{equation}
V(t) = A \times t^{\gamma},
\end{equation}
where $A$ and $\gamma$ are two free parameters obtained by least-squares minimisation for each individual SN and $t$ the epoch since explosion. In order to obtain the velocity error, we perform a Monte Carlo simulation, varying randomly each velocity measurement according to the observed velocity uncertainties over more than 2000 simulations. From this, for each epoch (from 1 to 120 days after explosion) we choose the velocity as the average value and the incertainty to the standard deviation of the simulations. The median value of $\gamma$ is $-$0.55$\pm$ 0.25. This value is comparable with the value found by other authors ($-$0.5 for \citealt{olivares10} and $-$0.464 by \citealt{nugent06} and $-$0.546 by \citet{takats12}). Note that, as found by \citet{faran14b}, the iron velocity for the fast-decliners (SNe~IIL) also follow a power law but with more scatter. Indeed for the slow-decliners ($s_{2} \leq 1.5$) we find a median value, $\gamma = -0.55 \pm 0.18 $ whereas for the fast-decliners ($s_{2} \geq 1.5$) we obtain $\gamma = -0.56 \pm 0.35 $. More details will be published in an upcoming paper (Guti\'errez et al.).
\subsection{Methodology}
To standardise the apparent magnitude, we perform a least-squares minimisation on :
\small
\begin{equation}
m_{\lambda 1}+\alpha log(\frac{v_{Fe II}}{5000~km~s^{-1}})-\beta_{\lambda_{1}} (m_{\lambda 2}-m_{\lambda 3} ) = 5 log (cz)+ZP ,
\end{equation}
\normalsize
\\
where $c$, $z$, $m_{\lambda 1,2,3}$ are defined in section 4.1 and $\alpha$, $\beta_{\lambda_{1}}$, and $ZP$ are free fitting parameters. The errors on the magnitude are obtained in the same way as for the PCM but the epoch is different. For the SCM, the photospheric expansion velocity is very dependent on the explosion date that is why after trying different epochs and references, we found that the best reference is the explosion time as used in \citet{nugent06}, \citet{poznanski09}, \citet{rodriguez14}. The same epoch for the magnitude, the colour and the iron velocity is employed. Just like for the PCM, we use the same colour, $(V-i)$, and the same filters ($r$, $Y$ band). For some SNe we are not able to measure an iron velocity due to the lack of spectra (only one epoch) and our sample is thus composed of 26 SNe.
\subsection{Results}
In Figure~\ref{Hubble diagram_SCM} we present the Hubble diagram and the residual for two different filters. The dispersion is 0.29 mag (or 0.30--0.28 mag in WRMS for the $Y$ band and $r$ band respectively) for 24 SNe (some SNe do not have colour at this epoch). These values are some what better than previous studies \citep{hamuy02,nugent06,poznanski09,olivares10,andrea10} where the authors found dispersions around 0.30 mag with 30 SNe (more details in section 6.3). Note the major differences between our study and theirs is that they included very nearby SNe ($cz \leq$~3000 km s$^{-1}$), only slow-declining SNe~II (SNe~II with low $s_{2}$, historically referred to as SNe~IIP), did not calculate a power-law for each SN as we do, and used a different epoch. Note also the work done by \citet{maguire10} where they applyed the SCM to NIR filters ($J$-band and $(V-J)$ colour) using nearby SNe (92\% of their sample with $cz \leq$~3000 km s$^{-1}$), finding a dispersion of 0.39 mag with 12 SNe (see section 6.3). To finish we tried a combination of the PCM and SCM, i.e., adding a $s_{2}$ term to the SCM but this does not improve the dispersion.
\begin{figure}
\begin{center}
\includegraphics[width=9cm]{r_SCM.eps}\\
\includegraphics[width=9cm]{Y_SCM.eps}
\caption{In all the figures, we present the dispersion (RMS) using the SCM. The number of SNe ($N_{SNe}$) and the epoch chosen with respect to explosion date in days. Both plots present the Hubble diagram using the SNe in the Hubble flow. On the top we present the Hubble diagram using the $r$ band and the colour $(V-i)$. On the bottom is the same but we use a NIR filter, $Y$ band. On the bottom of each plot we present the residual. In the residuals plot, the dashed line correspond to the RMS.}
\label{Hubble diagram_SCM}
\end{center}
\end{figure}
\section{Discussion}
Above we demonstrate that using two terms, $s_{2}$ and a colour, we are able to obtain a dispersion of 0.43 mag (optical bands). In this section we try to reduce the RMS by using well-observed SNe and we compare the PCM to the SCM. We also discuss comparisons between the SCM using the CSP sample with other studies. Because the value of the RMS is the crucial parameter to estimate the robustness of the method, we also discuss statistical errors. Finally, we briefly present the values of $R_{V}$ derived from the colour term both from PCM and SCM.
\subsection{Golden sample}
A significant fraction of values from \citet{anderson14a} do not correspond to the slope of the plateau but sometimes to a combination of $s_{1}$ (initial decline) and $s_{2}$. Indeed, for some SNe, it was impossible to distinguish two slopes and the best fit was only one slope. For this reason we decide to define a new sample composed only by 12 SNe with values of $s_{1}$ and $s_{2}$ and with $cz_{CMB}$ $\geq$ 3000 km s$^{-1}$. From this sample and using the $r$/$(V-i)$ combination we obtain a dispersion of 0.39 mag with 12 SNe, which compares to 0.48 mag from the entire sample. From the $Y$ band the dispersion drops considerably from 0.44 to 0.18 mag with only 8 SNe. However this low value should be taken with caution due to possible statistical effects which are discussed later (see section 6.4).
\subsection{Method comparisons}
In Figure~\ref{Hubble diagram_compa_r} and in Figure~\ref{Hubble diagram_compa_Y} we compare the Hubble diagram obtained using the SCM and the PCM. For both methods we use the same SNe (Hubble flow sample), and the same set of magnitude-colour. The dispersion using the $r$ band and the $Y$ band is 0.43 mag for the PCM whereas for the SCM is 0.29.\\
\indent
In general the SCM is more precise than the PCM but the dispersion found with the PCM is consistent with the results found by the theoretical studies done by \citet{kasen09} (distances accurate to $\sim$ 20\%) but the authors used other photometric correlations (plateau duration). Unfortunately, as suggested by \citet{anderson14a} using this parameter the prediction is not seen in the observations. We tried to use the OPTd values as an input instead of the $s_{2}$ and we did not see any improvement on the dispersion. Note also the recent work of \citet{faran14a}, in which the authors found a correlation between the iron velocity and the $I$-band total decline rate. Although in this paper we do not use the total decline rate but another quantity related to the plateau slope, our work confirms the possibility of using photometric parameters instead of spectroscopic.\\
\begin{figure*}
\includegraphics[width=9cm]{r_PCM_SCM_b.eps}
\includegraphics[width=9cm]{r_PCM_SCM_a.eps}
\caption{In all the figures, we present the dispersion (RMS), the
number of SNe ($N_{SNe}$) and the epoch chosen with respect to the
end of the plateau (OPTd*X\%) for the SCM and with respect to the explosion date for the SCM. On the bottom of each plot, the residuals are shown. In all the residual plots, the dashed line correspond to the RMS. For both methods we use the Hubble flow sample, $cz_{CMB}$ $\geq$ 3000 km s$^{-1}$, the $r$ band and the colour $(V-i)$. Plotted on the left is the SCM whereas in the right is for the PCM}
\label{Hubble diagram_compa_r}
\end{figure*}
\begin{figure*}
\includegraphics[width=9cm]{Y_PCM_SCM_b.eps}
\includegraphics[width=9cm]{Y_PCM_SCM_a.eps}
\caption{In all the figures, we present the dispersion (RMS), the
number of SNe ($N_{SNe}$) and the epoch chosen with respect to the
end of the plateau (OPTd*X\%) for the SCM and with respect to the explosion date for the SCM. On the bottom of each plot, the residuals are shown. In all the residuals plot, the dashed line correspond to the RMS. For both methods we use the Hubble flow sample, $cz_{CMB}$ $\geq$ 3000 km s$^{-1}$, the $Y$ band and the colour $(V-i)$. Plotted on the left is the SCM whereas in the right is for the PCM}
\label{Hubble diagram_compa_Y}
\end{figure*}
\subsection{SCM comparisons}
In this section we compare our SCM with other studies. First we use only optical filters to compare with \citet{poznanski09} and \citet{olivares10}. Both studies used the $(V-I)$ colour and also the $I$ band. Note that \citet{olivares10} also used the $B$ and $V$ band but here we consider only the $I$ band for consistency. \citet{poznanski09} found a dispersion of 0.38 mag using 40 slow-decliners. In our sample instead of using the $I$ band we used the sloan filter, $i$ band and $(V-i)$ colour. Using our entire sample, i.e., SNe (37 SNe in total for all the redshift range) we derive a dispersion similar to \citet{poznanski09} of 0.32 mag (epoch: 35 days after explosion). We can also compare the parameter $\alpha$ derived from the fit. Again we obtain a consistent value, $\alpha = 4.40 \pm 0.52 $ whereas \citet{poznanski09} found $\alpha = 4.6 \pm 0.70$. The other parameters are not directly comparable due to the fact that the authors assumed an intrinsic colour which is not the case in the current work. Using a Hubble constant ($H_{0}$) equal to 70 km s$^{-1}$ Mpc$^{-1}$ we can translate our ZP to an absolute magnitude ($ZP = M_{corr}-5log(H_{0})+25$) M$_{i}$ = $-$17.12 $\pm$ 0.10 mag that it is lower than the results obtained by \citet{poznanski09} (M$_{I}$ = $-$17.43 $\pm$ 0.10 mag). This difference is probably due to the fact that the corrected magnitude has not been corrected for the intrinsic colour in our work.\\
\indent
Using 30 slow-declining SNe in the Hubble flow and very nearby SNe (z between 0.00016 and 0.05140), $(V-I)$ colour and the $I$ band, \citet{olivares10} derived a dispersion of 0.32 mag which is the same that we obtained. However the parameters derived by \citet{olivares10} are different. Indeed using the same equation (5), and the entire sample they obtained $\alpha = 2.62 \pm 0.21 $, $\beta = 0.60 \pm 0.09 $ and $ZP =-2.23 \pm 0.07 $ instead of $\alpha = 4.40 \pm 0.52 $, $\beta = 0.98 \pm 0.31 $ and $ZP =-1.34 \pm 0.10 $ for us. From their ZP ($H_{0}$=70 km s$^{-1}$ Mpc$^{-1}$) we derive M$_{I}$ = -18.00 $\pm$ 0.07 mag (M$_{i}$ = -17.12 $\pm$ 0.15 mag for us). When the authors restrict the sample to objects in the Hubble flow, they end up with 20 SNe and a dispersion of 0.30 mag. If we do the same cut, we find a dispersion of 0.29 for 24 SNe. We obtain consistent dispersion for both samples using similar filters. Note that reducing our sample to slow-decliners alone ($s_{2} \leq 1.5$, the classical SNe~IIP in other studies) in the Hubble flow does not improve the dispersion. As mentioned in section 5.3, the difference in dispersion between \citet{olivares10} and our study can be due, among other things, to the difference in epoch used, or that we calculate a power-law for each SN for the velocity.\\
\indent
With respect to the NIR filters \citet{maguire10} suggested that it may be possible to reduce the scatter in the Hubble diagram to 0.1--0.15 mag and this should then be confirmed with a larger sample and more SNe in the Hubble flow. The authors used 12 slow-decliners but only one SN in the Hubble flow. Using the $J$ band and the colour $(V-J)$ they found a dispersion of 0.39 mag against 0.50 mag using the $I$ band. From this drop in the NIR, the authors suggested that using this filter and more SNe in the Hubble flow could reduce the scatter from 0.25-0.3 mag (optical studies) to 0.1--0.15 mag. With the same filters used by \citet{maguire10}, and using the Hubble flow sample, we find a dispersion of 0.28 mag with 24 SNe. This dispersion is 0.1 mag higher than that predicted by \citet{maguire10} (0.1--0.15 mag). To derive the fit parameters, the authors assumed an intrinsic colour $(V-J)_{0}$ $=$ 1 mag. They obtained $\alpha = 6.33 \pm 1.20 $ and an absolute magnitude M$_{J}$=$-$18.06 $\pm$ 0.25 mag ($H_{0}$=70 km s$^{-1}$ Mpc$^{-1}$). If we use only the SNe with $cz_{CMB}$ $\geq$ 3000 km s$^{-1}$ (24 SNe), we find $\alpha = 4.64 \pm 0.64 $ and ZP = $-$2.44 $\pm$ 0.18 which corresponds to M$_{J}$=$-$18.21 $\pm$ 0.18 mag assuming H$_{0}$ = 70 km s$^{-1}$ Mpc$^{-1}$. If we include all SNe at any redshift, the sample goes up to 34 SNe and the dispersion is 0.31 mag. From all SNe we derive $\alpha = 4.87 \pm 0.52 $ and ZP =$-$2.44 $\pm$ 0.20 which corresponds to M$_{J}$=-18.21 $\pm$ 0.20. To conclude, the Hubble diagram derived from the CSP sample using the SCM is consistent and some what better with those found in the literature.\\
\indent
More recently, \citet{rodriguez14} proposed another method to derive a Hubble diagram from SNe~II. The PMM corresponds to the generalisation of the SCM, i.e., the distances are obtained using the SCM at different epochs and then averaged. Using the $(V-I)$ colour, and the filter $V$, the authors found an intrinsic scatter of 0.19 mag. Given that the intrinsic dispersion used by \citet{rodriguez14} is a different metric than that used by us (the RMS dispersion) we computed the latter from their data, obtaining 0.24 mag for 24 SNe in the Hubble flow. Using the $V$ band and the $(V-i)$ colour and doing an average over several epochs we found a dispersion of 0.28 mag which is similar to the value found from the SCM and comparable with the value derived by \citet{rodriguez14}. From the $Y$ band and $(V-i)$ colour we find an identical dispersion of 0.29 mag.
\subsection{Low number effects}
In analysing the Hubble diagram, the figure of merit is the RMS and the holy grail is to obtain very low dispersion in the Hubble diagram (i.e. low distance errors). In our work we show that in the $Y$ band we can achieve a RMS around 0.43--0.48 mag using the Hubble flow sample (30 SNe) and the entire sample (41 SNe), whereas using the golden sample (8 SNe) we obtain a dispersion of 0.18 mag. It is important to know if this decrease in RMS is due to the fact that we used well-studied SNe within the golden sample or if it is due to the low number of SNe. For this purpose we do a test using the Monte Carlo bootstrapping method.\\
\indent
From our Hubble flow sample, we remove randomly one SN and compute the dispersion. We do that for 30000 simulations and the final RMS corresponds to the median, and the errors to the standard deviation. Then after removing one SN, we remove randomly two SNe and again estimate the RMS and the dispersion over 30000 simulations. We repeat this process until we have only 4 SNe, i.e., we remove from one SN to (size available sample - 4 SNe). For each simulation we compute a new model, i.e., new fit parameters ($\alpha$, $\beta$, and $ZP$).\\
\indent
From this test we conclude that when the number of SNe is lower than 10--12 SNe the RMS is very uncertain because the parameters (i.e., $\alpha$, $\beta$, and $ZP$) start diverging (see Appendix). This implies that the RMS is driven by the reduced number of objects so it is difficult to conclude if the model for the golden sample is better because the RMS is smaller or because it is due to a statistical effect.
\subsection{Low $R_{V}$}
As stated in section 4.1, the $\beta_{\lambda_{1}}$ colour term is related to the total-to-selective extinction ratio if the colour-magnitude relation is due to extrinsic factors (dust). In the literature, for the MW, $R_{V}$ is known to vary from one line of sight to another, from values as low as 2.1 \citep{welty92} to values as large as 5.6-5.8 \citep{car89,fitzpatrick99,draine03}. In general for the MW, a value of 3.1 is used which corresponds to an average of the Galactic extinction curve for diffuse interstellar medium (ISM). Using the minimisation of the Hubble diagram with a colour term, in the past decade the SNe~Ia community has derived lower $R_{V}$ for host-galaxy dust than for the MW. Indeed they found $R_{V}$ between 1.5--2.5 \citep{krisciunas07,eliasrosa08,goobar08,folatelli10,phillips13,burns14}. This trend was also seen more recently using SNe~II \citep{poznanski09,olivares10,rodriguez14}. This could be due to unmodeled effects such as a dispersion in the intrinsic colours (e.g. \citealt{scolnic14}).\\
\indent
We follow previous work in using the minimisation of the Hubble diagram to obtain constraints on $R_{V}$ for host-galaxy dust. Using the PCM, the Hubble flow sample and the $r$ band, we find $\beta_{r}$ close to 0.98. Using a \citet{car89} law we can transform this value in the total-to-selective extinction ratio, and we obtain, $R_{V}=1.01_{-0.41}^{+0.53}$. Following the same procedure but using the SCM, we also derive low $R_{V}$ values, but consistent with those derived using the PCM.\\
\indent
At first sight, our analysis would suggest a significantly different nature of dust in our Galaxy and other spiral galaxies, as previously seen in the analysis of SNe~Ia and SNe~II. However, we caution the reader that the low $R_{V}$ values could reflect instead intrinsic magnitude-colour for SNe~II not properly modelled. To derive the $R_{V}$ (or pseudo $R_{V}$) values we assume that all the SNe~II have the same intrinsic colours and same intrinsic colour-luminosity relation, however theoretical models with different masses, metallicity, show different intrinsic colours \citep{dessart13}. Disentangling both effects would require to know the intrinsic colours of our SN sample. Indeed, with intrinsic colour-luminosity corrections the $\beta_{\lambda_{1}}$ colour term could change and thus we will be able to derive an accurate $R_{V}$. In a forthcoming paper we will address this issue through different dereddening techniques (de Jaeger, in prep.) that we are currently investigating.
\section{Conclusions}
Using 38 SNe~II in the Hubble flow we develop a technique based solely on photometric data (PCM) to build a Hubble diagram based on SNe~II. In summary :
\begin{enumerate}
\item{Using PCM we find a dispersion of 0.44 mag using the $r$ band and 0.43 mag with the $Y$ band,thus using NIR filters the improvement is not so significant for the PCM.}
\item{The $s_{2}$ plays a useful role, allowing us to reduce the dispersion from 0.58 mag to 0.50 mag for $r$ band.}
\item{The colour term does not have so much influence on the NIR filters because
it is related to the host-galaxy extinction.}
\item{We find very low ($\beta$) values (the colour-magnitude coefficient). If $\beta$ is purely extrinsic, it implies very low $R_{V}$ values.}
\item{The Hubble diagram derived from the CSP sample using the SCM yields to a dispersion of 0.29 mag, some what better than those found in the literature and emphasising the potential of SCM in cosmology.}
\end{enumerate}
It is interesting also to obtain more data and SNe for which the initial decline rate and the plateau are clearly visible to try to reduce this dispersion. The PCM is very promising, and more efforts must be done in this direction, i.e., trying to use only photometric parameters. In the coming era of large photometric wide--field surveys like LSST, having spectroscopy for every SNe will be impossible hence the PCM which is the first purely photometric method could be very useful.
\acknowledgments
The referee is thanked for their through reading of the manuscript, which helped clarify and improve the paper. Support for T. D., S. G., L. G., M. H. , C. G., F. O., H. K., is provided by the Ministry of Economy, Development, and Tourism's Millennium Science Initiative through grant IC120009, awarded to The Millennium Institute of Astrophysics, MAS. S. G., L. G., H. K. and F.O. also acknowledge support by CONICYT through FONDECYT grants 3130680, 3140566, 3140563 and 3140326, respectively. The work of the CSP has been supported by the National Science Foundation under grants AST0306969, AST0607438, and AST1008343. M. D. S., C. C. and E. H. gratefully acknowledge generous support provided by the Danish Agency for Science and Technology and Innovation realized through a Sapere Aude Level 2 grant. The authors thank F. Salgado for his work done with the CSP. This research has made use of the NASA/IPAC Extragalactic Database (NED) which is operated by the Jet Propulsion Laboratory, California Institute of Technology, under contract with the National Aeronautics and Space Administration and of data provided by the Central Bureau for Astronomical Telegrams.
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1911.04922
|
\section{Introduction}
\IEEEPARstart{M}ACHINE intelligence is revolutionizing every branch of science and technology \cite{ml1,dl1}.
If a machine wants to learn, it requires at least two ingredients: information and computation, which are usually separated from each other in machine-type communication (MTC) systems \cite{iot1}.
Nonetheless, sending vast volumes of data from MTC devices to the cloud not only leads to a heavy communication burden but also increases the transmission latency.
To address this challenge brought by MTC, a promising solution is the \emph{edge machine learning} technique \cite{edge1,edge2,edge3,edge4,fed1,fed2,fed3,fed4} that trains a machine learning model or \emph{fine-tunes a pre-trained model} at the edge, i.e., at a nearby radio access point with computation resources.
In general, there are two ways to implement edge machine learning: data sharing and model sharing.
Data sharing uses the edge to collect data generated from MTC devices for machine learning \cite{edge1,edge2,edge3,edge4}, while model sharing uses federated learning \cite{fed1,fed2,fed3,fed4} to exchange model parameters (instead of data) between the edge and users.
Both approaches are recognized as key paradigms in the sixth generation (6G) wireless communications \cite{6g1,6g2,6g3}.
However, since the MTC devices often cannot process the data due to limited computational power, this paper focuses on data sharing.
\subsection{Motivation and Related Work}
In contrast to conventional communication systems, edge machine learning systems aim to maximize the learning performance instead of the communication throughput.
Therefore, edge resource allocation becomes very different from traditional resource allocation schemes that merely consider the wireless channel state information \cite{waterfilling,fair,sumrate,gradient}.
For instance, the celebrated water-filling scheme allocates more resources to better channels for throughput maximization \cite{waterfilling}, and the max-min fairness scheme allocates more resources to cell-edge users to maintain certain quality of service \cite{fair}.
While these two schemes have proven to be very efficient in traditional wireless communication systems, they could lead to poor learning performance in edge learning systems because they do not account for the machine learning factors such as model and dataset complexities.
Imagine training a deep neural network (DNN) and a support vector machine (SVM) at the edge.
Due to much larger number of parameters in DNN, the edge should allocate more resources to MTC devices that upload data for the DNN than those for the SVM.
Nonetheless, in order to maximize the learning performance, we need a mathematical expression of the learning performance with respect to the number of samples, which does not exist to the best of the authors' knowledge.
While the sample complexity of a learning task can be related to the Vapnik-Chervonenkis (VC) dimension \cite{ml1}, this theory only provides a vague estimate that is independent of the specific learning algorithm or data distribution.
To better understand the learning performance, it has been proved in \cite{model3,model4} that the generalization error can be upper bounded by the summation of the bias between the model's prediction and the optimal prediction, the variance due to training datasets, and the noise of the target example.
With the bound being tight for certain loss functions (e.g., squared loss and zero-one loss), the bias-variance decomposition theory gives rise to an empirical nonlinear classification error model \cite{model1,model2,model5} that is also theoretically supported by the inverse power law derived via statistical mechanics \cite{model6}.
\subsection{Summary of Results}
In this paper, we adopt the above nonlinear model to approximate the learning performance, and a \emph{learning centric power allocation (LCPA)} problem is formulated with the aim of minimizing classification error subject to the total power budget constraint.
When the data is non-independent-and-identically-distributed (non-IID) among users, the LCPA problem can also be flexibly integrated with uncertainty sampling.
Since the formulated machine learning resource allocation problem is nonconvex and nonsmooth, it is nontrivial to solve.
By leveraging the majorization minimization (MM) framework from optimization, an MM-based LCPA algorithm that obtains a Karush-Kuhn-Tucker (KKT) solution is proposed.
To get deeper insights into LCPA, asymptotic analysis with the number of antennas at the edge going to infinity is provided.
The asymptotic optimal solution discloses that the transmit powers are inversely proportional to the channel gain, and scale exponentially with the classification error model parameters.
This result reveals that machine learning has a stronger impact than wireless channels in LCPA.
To enable affordable computational complexity when the number of MTC devices is large, a variant of LCPA, called mirror-prox LCPA, is proposed.
The algorithm is a first-order method (FOM), implying that its complexity is linear with respect to the number of users.
Extensive experimental results based on public datasets show that the proposed LCPA scheme is able to achieve a higher classification accuracy than that of the sum-rate maximization and max-min fairness power allocation schemes.
For the first time, the benefit brought by joint communication and learning design is quantitatively demonstrated in edge machine learning systems.
Our results also show that the mirror-prox LCPA reduces the computation time by orders of magnitude compared to the MM-based LCPA but still achieves satisfactory performance.
To sum up, the contributions of this paper are listed as follows.
\begin{itemize}
\item The LCPA scheme is developed for the edge machine learning problem, which maximizes the learning accuracy instead of the communication throughput.
\item To understand how LCPA works, an asymptotic optimal solution to the edge machine learning problem is derived, which, for the first time, discloses that the transmit power obtained from LCPA grows linearly with the path-loss and grows exponentially with the learning parameters.
\item To reduce the computation time of LCPA in the massive multiple-input multiple-output (MIMO) setting, a variant of LCPA based on FOM is proposed, which enables the edge machine learning system to scale up the number of MTC users.
\item Extensive experimental results based on public datasets (e.g., MNIST, CIFAR-10, ModelNet40) show that the proposed LCPA is able to achieve a higher accuracy than that of the sum-rate maximization and max-min fairness schemes.
\end{itemize}
\subsection{Outline}
The rest of this paper is organized as follows.
System model and problem formulation are described in Section II.
Classification error modeling is presented in Section III.
The MM-based LCPA algorithm, the asymptotic analysis, and the large-scale optimization method are derived in Sections IV, V, and VI, respectively.
Finally, experimental results are presented in Section VII, and conclusions are drawn in Section~VIII.
\emph{Notation}:
Italic letters, lowercase and uppercase bold letters represent scalars, vectors, and matrices, respectively.
Curlicue letters stand for sets and $|\cdot|$ is the cardinality of a set.
The operators $(\cdot)^{T}$, $(\cdot)^{H}$ and $(\cdot)^{-1}$ take the transpose, Hermitian and inverse of a matrix, respectively.
We use $(a_1,a_2,\cdots)$ to represent a sequence, $[a_1,a_2,\cdots]^{T}$ to represent a column vector, and $\left\Vert\cdot\right\Vert_p$ to represent the $\ell_p$-norm of a vector.
The symbol $\mathbf{I}_{N}$ indicates the $N\times N$ identity matrix,
$\mathbf{1}_{N}$ indicates the $N\times 1$ vector with all entries being unity,
and $\mathcal{CN}(0,1)$ stands for complex Gaussian distribution with zero mean and unit variance.
The function $[x]^+=\mathrm{max}(x,0)$, while $\mathrm{exp}(\cdot)$ and $\mathrm{ln}(\cdot)$ denote the exponential function and the natural logarithm function, respectively.
The function $\lfloor x\rfloor=\mathrm{max}\{n\in\mathbb{Z}:n\leq x \}$.
Finally, $\mathbb{E}(\cdot)$ means the expectation of a random variable, $\mathbb{I}_{\mathcal{A}}(x)=1$ if $x\in\mathcal{A}$ and zero otherwise, and $\mathcal{O}(\cdot)$ means the order of arithmetic operations.
For ease of retrieval, important variables and parameters to be used in this paper are listed in Table~I.
\begin{table*}[!t]
\caption{Summary of Important Variables and Parameters}
\centering
\begin{tabular}{|l|l|l|}
\hline
\textbf{Symbol} & \textbf{Type} & \textbf{Description} \\
\hline
$p_{k}\in\mathbb{R}_+$ & Variable & Transmit power (in $\mathrm{Watt}$) at user $k$. \\
$v_{m}\in\mathbb{Z}_+$ & Variable & Number of training samples for task $m$. \\
\hline
$P$ & Parameter & Total transmit power budget (in $\mathrm{Watt}$). \\
$B$ & Parameter & Communication bandwidth (in $\mathrm{Hz}$). \\
$T$ & Parameter & Transmission time (in $\mathrm{s}$). \\
$\xi$ & Parameter & The factor accounting for packet loss and network overhead. \\
$\sigma^2$ & Parameter & Noise power (in $\mathrm{Watt}$). \\
$D_m$ & Parameter & Data size (in $\mathrm{bit}$) per sample for task $m$ \\
$A_m$ & Parameter & Number of historical samples at the edge for task $m$. \\
$G_{k,l}$ & Parameter & The composite channel gain from user $l$ to the edge when decoding data of user $k$. \\
\hline
$\mathcal{D}_k,\mathcal{H}_k,\mathcal{T}_k,\mathcal{V}_k$ & Dataset & The full dataset, historical dataset, training dataset, validation dataset of user $k$. \\
\hline
$\Psi_m(v_m)$ & Function & Classification error of the learning model $m$ when the sample size is $v_m$. \\
$\Theta_m(v_m|a_m,b_m)$ & Function & Empirical classification error model for task $m$ with parameters $(a_m,b_m)$. \\
$U(\mathbf{d})$ & Function & Prediction confidence of sample $\mathbf{d}$. \\
\hline
\end{tabular}
\end{table*}
\section{System Model and Problem Formulation}
\setcounter{secnumdepth}{4}
We consider an edge machine learning system shown in Fig.~1, which consists of an intelligent edge with $N$ antennas and $K$ users with datasets $\{\mathcal{D}_1,\cdots,\mathcal{D}_K\}$.
The goal of the edge is to train $M$ classification models by collecting data from $M$ user groups (e.g., UAVs with camera sensors) $\{\mathcal{Y}_1,\mathcal{Y}_2,\cdots,\mathcal{Y}_M\}$, with the group $\mathcal{Y}_m$ containing all users having data for training the model $m$.
In case where some data from a particular user is used to train both model $m$ and model $j$, we can allow $\mathcal{Y}_m$ and $\mathcal{Y}_j$ to include a common user, i.e., $\mathcal{Y}_m\bigcap\mathcal{Y}_j\neq \emptyset$ for $m\neq j$.
For the classification models, without loss of generality, Fig.~1 depicts a convolutional neural network (CNN) and a support vector machine (SVM) (i.e., $M=2$), but more user groups and other classification models are equally valid.
It is assumed that the data are labeled at the edge.
This can be supplemented by the recent self-labeled techniques \cite{self,self2}, where a classifier is trained with an initial small number of labeled examples, and then the model is retrained with its own most confident predictions, thus enlarging its labeled training set.
After training the classifiers, the edge can feedback the trained models to users for subsequent use (e.g., object detection).
Notice that if the classifiers are pre-trained at the cloud and deployed at the edge, the task of edge machine learning is to fine-tune the pre-trained models at the edge, using local data generated from MTC users.
\begin{figure*}[!t]
\centering
\includegraphics[width=170mm]{./img/WANG1.eps}
\caption{System model of machine intelligence at the edge.
}
\label{fig1}
\end{figure*}
More specifically, the user $k\in\{1,\cdots,K\}$ transmits a signal $s_{k}$ with power $\mathbb{E}[|s_{k}|^2]=p_{k}$.
Accordingly, the received signal $\mathbf{r}=[r_1,\cdots,r_N]^T\in\mathbb{C}^{N\times 1}$ at the edge is
$\mathbf{r}=\sum_{k=1}^K\mathbf{h}_{k}\, s_{k}+\mathbf{n}$,
where $\mathbf{h}_{k}\in \mathbb{C}^{N\times 1}$ is the channel vector from the $k^{\mathrm{th}}$ user to the edge,
and $\mathbf{n}\sim \mathcal{CN}(\mathbf{0},\sigma^2\mathbf{I}_N)$.
By applying the maximal ratio combining (MRC) receiver $\mathbf{h}_{k}/\left\Vert\mathbf{h}_{k}\right\Vert_2$ to $\mathbf{r}$, the data-rate of user $k$ is
\begin{align}
&R_{k}=\mathrm{log}_2\left(1+\frac{G_{k,k}p_{k}}{\sum_{l=1,l\neq k}^KG_{k,l}p_{l}+
\sigma^2} \right), \label{Rk}
\end{align}
where $G_{k,l}$ represents the composite channel gain (including channel fading and MIMO processing) from user $l$ to the edge when decoding data of user $k$:
\begin{align}
&G_{k,l}=
\left\{
\begin{aligned}
&\left\Vert\mathbf{h}_{k}\right\Vert_2^2
,&\mathrm{if}~k=l
\\
&\frac{|\mathbf{h}_k^H\mathbf{h}_{l}|^2}{\left\Vert\mathbf{h}_{k}\right\Vert_2^2}
,&\mathrm{if}~k\neq l
\end{aligned}
\right.
.
\end{align}
With the expression of $R_k$ in \eqref{Rk}, the amount of data in $\mathrm{bit}$ received from user $k$ is $BTR_{k}$, where constant $B$ is the bandwidth in $\mathrm{Hz}$ that is assigned to the system (e.g., a standard MTC system would have $180~\mathrm{kHz}$ bandwidth \cite{iot3}), and $T$ is the total number of transmission time in second.
As a result, the total number of training samples that are collected at the edge for training the model $m$ is
\begin{align}\label{sample size}
&v_m= \xi\sum_{k\in\mathcal{Y}_m} \left\lfloor \frac{BTR_{k}}{D_m} \right\rfloor +A_m \approx \sum_{k\in\mathcal{Y}_m}\frac{\xi BTR_{k}}{D_m} +A_m,
\end{align}
where $A_m$ is the number of historical samples for task $m$ residing at the edge, $D_m$ is the number of bits for each data sample, and $\xi\leq1$ is a factor accounting for the reduced number of samples due to packet loss and network overhead.
The approximation is due to $\lfloor x\rfloor\to x$ when $x\gg 1$.
For example, in real-world edge/cloud machine learning applications, the data needs to be uploaded multiple times \cite{history1,history2} (e.g., twice a day).
If the historical dataset of user $k$ at the edge is $\mathcal{H}_k$, then $A_m=\sum_{k\in\mathcal{Y}_m}|\mathcal{H}_k|$.
For $D_m$, if we consider the MNIST dataset \cite{MNIST}, since the handwritten digit images are gray scale with $28\times 28$ pixels (each pixel has $8$ bits), in this case $D_m=8\times28\times28+4=6276~\mathrm{bits}$ ($4$ bits are reserved for the labels of $10$ classes \cite{MNIST} in case the users also transmit labels).
Lastly, if the system reserves $30\%$ of the resource blocks for network overhead \cite{PDCCH}, and the probability of packet error rate is $0.1$ \cite{xi2}, then $\xi$ can be estimated as $\xi=0.7\times 0.9=0.63$.
In the considered system, the design variables that can be controlled are the transmit powers of different users $\mathbf{p}=[p_1,\cdots,p_K]^T$ and the sample sizes of different models $\mathbf{v}=[v_1,\cdots,v_M]^T$.
Since the power costs at users should not exceed the total budget $P$, the variable $\mathbf{p}$ needs to satisfy
$\sum_{k=1}^Kp_k\leq P$.
Since a larger $\sum_{k=1}^Kp_k$ always improves the learning performance, we can rewrite $\sum_{k=1}^Kp_k\leq P$ as $\sum_{k=1}^Kp_k=P$.
Having the transmit power satisfied, it is then crucial to minimize the classification errors (i.e., the number of incorrect predictions divided by the number of total predictions),
which leads to the following learning centric power allocation (LCPA) problem:
\begin{subequations}
\begin{align}
\mathrm{P}:\mathop{\mathrm{min}}_{\substack{\mathbf{p},\,\mathbf{v}}}
\quad&\mathop{\mathrm{max}}_{m=1,\cdots,M}~\Psi_m(v_m), \nonumber\\
\mathrm{s. t.}\quad &\sum_{k=1}^Kp_k=P,\quad p_k\geq 0,\quad k=1,\cdots,K, \\
&\sum_{k\in\mathcal{Y}_m}\frac{\xi BT}{D_m}\mathrm{log}_2\left(1+\frac{G_{k,k}p_{k}}{\sum_{l=1,l\neq k}^KG_{k,l}p_{l}+
\sigma^2} \right)
\nonumber\\
&
+A_m
=v_m,\quad m=1,\cdots,M,
\label{P0b}
\end{align}
\end{subequations}
where $\Psi_m(v_m)$ is the classification error of the learning model $m$ when the sample size is $v_m$, and the min-max operation at the objective function is to guarantee the worst-case learning performance.
The key challenge to solve $\rm{P}$ is that functions $(\Psi_1,\cdots,\Psi_M)$ represent generalization errors, and to the best of the authors' knowledge, currently there is no exact expression of $\Psi_m(v_m)$.
To address this issue, Section III will adopt an empirical classification error model to approximate $\Psi_m$.
The problem formulation $\mathrm{P}$ has assumed that the data is IID among different users (i.e., different users have identical distributions).
However, practical applications may involve non-IID data distributions.
For example, user $1$ has $1000$ samples but they are exactly the same (e.g., repeating the first sample in the MNIST dataset); and user $2$ has $50$ different samples (e.g., randomly drawn from the MNIST dataset).
In this case, upper bound and lower bound data-rate constraints should be imposed depending on the quality of users' data (i.e., whether their data improves the learning performance and how much the improvement could be):
\begin{itemize}
\item If the data transmitted from user $k$ does not help to improve the learning performance, a maximum amount of transmitted data $Z_k^{\rm{max}}$ should be imposed.
This is the case of user $1$, since the well-understood data should not be transmitted to the edge over and over again.
\item On the other hand, if the data transmitted from user $k$ helps to improve the learning performance, a minimum amount of data $Z_k^{\rm{min}}$ should be imposed on this user.
This is the case of user $2$, since more data from this user would reduce the learning error.
\end{itemize}
Notice that both the lower and upper bounds can be imposed on the same user simultaneously.
Therefore, we can add the following constraint to problem $\mathrm{P}$:
\begin{align}\label{ratebounds}
Z_k^{\rm{min}}&\leq\frac{\xi BT}{D_m}\mathrm{log}_2\left(1+\frac{G_{k,k}p_{k}}{\sum_{l=1,l\neq k}^KG_{k,l}p_{l}+
\sigma^2} \right)
\nonumber\\
&\leq Z_k^{\rm{max}},\quad k=1,\cdots,K.
\end{align}
Of course, the remaining question is how to determine whether the data from a particular user is useful or not.
To this end, the uncertainty sampling method \cite{uncertainty} can be adopted.
In particular, let the function $\mathbb{P}(\mathbf{c}|\mathbf{d}, \mathbf{w})$ denote the probability of the label being $\mathbf{c}$ given sample $\mathbf{d}\in\mathbb{R}^{S\times 1}$ and the model parameter vector $\mathbf{w}$ (e.g., $\mathbf{c}$ is a one-hot vector containing $10$ elements if the learning problem is a $10$-classes classification; for the MNIST dataset, we have $S=784$; and in the considered CNN model, $\mathbf{w}$ contains the convolution matrices and the bias vectors).
The confidence of the predicted label $\mathbf{c}^*=\mathop{\mathrm{argmax}}_{\mathbf{c}}\mathbb{P}(\mathbf{c}|\mathbf{d},\mathbf{w})$ is
\begin{align}\label{uncertain}
&U(\mathbf{d})=\mathbb{P}(\mathbf{c}^*|\mathbf{d},\mathbf{w}).
\end{align}
Therefore, we can compute the $U(\mathbf{d})$ for all the historical samples from user $k$.
If the average (or median, minimum) value of $U(\mathbf{d})$ is large (e.g., $>0.9$) for the historical data from user $k$, then user $k$ should have its data-rate upper bounded.
On the other hand, if the average (or median, minimum) value of $U(\mathbf{d})$ is small (e.g., $<0.5$), then user $k$ should have its data-rate lower bounded.
The entire LCPA scheme with uncertainty sampling is summarized in Fig.~2.
\begin{figure}[!t]
\centering
\includegraphics[width=75mm]{./img/uncertainty.eps}
\caption{Illustration of the LCPA scheme with uncertainty sampling.
}
\label{fig2}
\end{figure}
\emph{Remark 1 (Practical Power Control Procedure):
The LCPA problem will be solved at the edge, which can then inform the users about their transmit powers through downlink control channels.
For example, the 3GPP standard reserves some resource blocks for Physical Downlink Control Channel (PDCCH) \cite{PDCCH}, which are used for sending control signals such as users' transmit powers and modulation orders.
}
\section{Modeling of Classification Error}
\subsection{Classification Error Rate Function}
In general, the classification error $\Psi_m(v_m)$ is a nonlinear function of $v_m$ \cite{model1,model2,model3,model4,model5,model6}.
Particularly, this nonlinear function should satisfy the following properties:
\begin{itemize}
\item[(i)] As $\Psi_m$ is a percentage, it satisfies $0\leq \Psi_m(v_m) \leq 1$;
\item[(ii)] Since more data would provide more information, $\Psi_m(v_m)$ is a monotonically decreasing function of $v_m$ \cite{model1};
\item[(iii)] As $v_m$ increases, the magnitude of the partial derivative $|\partial \Psi_m/\partial v_m|$ would gradually decrease and become zero when $v_m$ is sufficiently large \cite{model2}, meaning that increasing sample size no longer helps machine learning.
\end{itemize}
Based on the properties (i)--(iii), the following nonlinear model $\Theta_m(v_m|a_m,b_m)$ \cite{model1,model2,model5} can be used to capture the shape of $\Psi_m(v_m)$:
\begin{align}
\Psi_m(v_m) \approx
\Theta_m(v_m|a_m,b_m)
=a_m\, v_m^{-b_m}, \label{model1}
\end{align}
where $a_m,b_m\geq 0$ are tuning parameters.
It can be seen that $\Theta_m$ satisfies all the above properties.
Moreover, $\Theta_m(v_m|a_m,b_m)\to 0$ if $v_m\to +\infty$, meaning that the error is $0$ with infinite data.\footnote{We assume the model is powerful enough such that given infinite amount of data, the error rate can be driven to zero.}
\textbf{Interpretation from Learning Theory.}
Apart from (i)--(iii), the model \eqref{model1} corroborates the \emph{inverse power relationship} between learning performance $\Psi_m$ and the amount of training data $v_m$ from the perspective of statistical mechanics \cite{model6}.
In particular, according to \cite{model6}, the training procedure can be modeled as a Gibbs distribution of networks characterized by a temperature parameter $T_g$.
The asymptotic generalization error as the number of samples $v_m$ goes to infinity is expressed as \cite[Eq. (3.12)]{model6}
\begin{align}\label{RQ2_A2}
\Psi_m(v_m)&\rightarrow\epsilon_{\mathrm{min}}+\left(\frac{T_g}{2}+\frac{\mathrm{Tr}(\mathbf{U}_m\mathbf{V}_m^{-1})}{2W_m}\right)W_mv_m^{-1},
\nonumber\\
\mathrm{if}~v_m&\rightarrow +\infty,
\end{align}
where $\epsilon_{\mathrm{min}}\geq 0$ is the minimum error for all possible learning models and $W_m$ is the number of parameters.
The matrices $(\mathbf{U}_m,\mathbf{V}_m)$ contain the second-order and first-order derivatives of the generalization error function with respect to the parameters of model $m$.
By comparing \eqref{RQ2_A2} with \eqref{model1}, we can see that $a_m=\left(\frac{T_g}{2}+\frac{\mathrm{Tr}(\mathbf{U}_m\mathbf{V}_m^{-1})}{2W_m}\right)W_m$ and $b_m=-1$ in the asymptotic case.
Therefore, as the number of samples goes to infinity, the weighting factor $a_m$ in \eqref{model1} accounts for the \emph{model complexity} of the classifier $m$.
Moreover, in the finite sample size regime, the slopes of learning curves may be different for different datasets even for the same machine learning model.
This means that $v_m^{-1}$ is not always suitable in practice.
Therefore, $b_m$ is introduced as a tuning parameter accounting for the correlation in a dataset.
\subsection{Parameter Fitting of CNN and SVM Classifiers}
We use the public MNIST dataset \cite{MNIST} as the input images, and train the $6$-layer CCN (shown in Fig.~1) with training sample size $v_m^{(i)}$ ranging from $100$ to $10000$.
In particular, the input image is sequentially fed into a $5\times 5$ convolution layer (with ReLu activation, 32 channels, and SAME padding), a $2\times 2$ max pooling layer, then another $5\times 5$ convolution layer (with ReLu activation, 64 channels, and SAME padding), a $2\times 2$ max pooling layer, a fully connected layer with $128$ units (with ReLu activation), and a final softmax output layer (with $10$ ouputs).
The training procedure is implemented via Adam optimizer with a learning rate of $10^{-4}$ and a mini-batch size of $100$.
After training for $5000$ iterations, we test the trained model on a dataset with $1000$ unseen samples, and compute the corresponding classification error.
By varying the sample size $v_m$ as $(v_m^{(1)},v_m^{(2)},\cdots)=(100,150,200,300,500,1000,5000,10000)$, we can obtain the classification error $\Psi_m(v_m^{(i)})$ for each sample size $v_m^{(i)}$, where $i=1,\cdots,Q$, and $Q=8$ is the number of points to be fitted.
With $\{v_m^{(i)},\Psi_m(v_m^{(i)})\}_{i=1}^Q$, the parameters $(a_m,b_m)$ in $\Theta_m$ can be found via the following nonlinear least squares fitting:
\begin{align}
\mathop{\mathrm{min}}_{a_m,\,b_m}\quad&\frac{1}{Q}\mathop{\sum}_{i=1}^Q\Big|\Psi_m\left(v_m^{(i)}\right)-\Theta_m\left(v_m^{(i)}|a_m,b_m\right)\Big|^2,
\nonumber\\
\mathrm{s.t.}\quad
&a_m\geq 0,\quad b_m\geq 0.
\label{fitting}
\end{align}
The above problem can be solved by two-dimensional brute-force search, or gradient descent method.
Since the parameters $(a_m,b_m)$ for different tasks are obtained independently, the total complexity is linear in terms of the number of tasks.
The fitted classification error versus the sample size is shown in Fig.~3a.
It is observed from Fig.~3a that with the parameters $(a_m,b_m)=(9.27,0.74)$, the nonlinear classification error model in \eqref{model1} matches the experimental data of CNN very well.
To demonstrate the versatility of the model, we also fit the nonlinear model to the classification error of a support vector machine (SVM) classifier.
The SVM uses penalty coefficient $C=1$ and Gaussian kernel function $K(\mathbf{x}_i,\mathbf{x}_j)=\mathrm{exp}\left(-\widetilde{\gamma}\, \left\Vert\mathbf{x}_i-\mathbf{x}_j\right\Vert_2^2\right)$ with $\widetilde{\gamma}=0.001$ \cite{sklearn}.
Moreover, the SVM classifier is trained on the digits dataset in the Scikit-learn Python machine learning tookbox, and the dataset contains $1797$ images of size $8\times 8$ from $10$ classes, with $5$ bits (corresponding to integers $0$ to $16$) for each pixel \cite{sklearn}.
Therefore, each image needs $D_m=8\times 8\times 5+4=324~\mathrm{bits}$.
Out of all images, we train the SVM using the first $1000$ samples with sample size $(v_m^{(1)},v_m^{(2)},\cdots)=(30,50,100,200,300,400,500,1000)$,
and use the latter $797$ samples for testing.
The parameters $(a_m,b_m)$ for the SVM are obtained following a similar procedure in \eqref{fitting}.
It is observed from Fig.~3a that with $(a_m,b_m)=(6.94,0.8)$, the model in \eqref{model1} fits the experimental data of SVM.
\begin{figure*}
\centering
\subfigure[]{
\includegraphics[width=58mm]{./img/model.eps}}
\subfigure[]{
\includegraphics[width=58mm]{./img/implement1.eps}}
\subfigure[]{
\includegraphics[width=58mm]{./img/implement2.eps}}
\caption{
a) Comparison between the experimental data and the nonlinear classification error model. The parameters in the models are given by
$(a_m,b_m)=(9.27,0.74)$ for CNN and $(a_m,b_m)=(6.94,0.8)$ for SVM;
b) Fitting the error function to historical datasets. The parameters in the models are given by $(a_m,b_m)=(7.3,0.69)$ for CNN and $(a_m,b_m)=(5.2,0.72)$ for SVM;
c) Comparison between different classification tasks.}
\end{figure*}
\subsection{Practical Implementation}
One may wonder how could one obtain the fitted classification error model before the actual machine learning model is being trained.
There are two ways to address this issue.
\textbf{1) Extrapolation.} More specifically, the error function can be obtained by training the machine learning model on the historical dataset at the edge, and the performance on a future larger dataset can be predicted.
This is called extrapolation \cite{model1}.
For example, by fitting the error function to the first half experimental data of CNN in Fig.~3b (i.e., $v_m=(100,150,200,300)$), we can obtain $(a_m,b_m)=(7.3,0.69)$, and the resultant curve predicts the errors at $v_m=(500,1000,5000,10000)$ very well as shown in Fig.~3b.
Similarly, with $(a_m,b_m)=(5.2,0.72)$ obtained from the experimental data of $v_m=(30,50,100,200)$, the proposed model for SVM matches the classification errors at $v_m=(300,400,500,1000)$.
It can be seen that the fitting performance in Fig.~3b is slightly worse than that in Fig. 3a, as we use smaller number of pilot data.
But since our goal is to distinguish different tasks rather than accurate prediction of the classification errors, the extrapolation method can guide the resource allocation at the edge.
\textbf{2) Approximation.} This means that we can pre-train a large number of commonly-used models offline (not at the edge) and store their corresponding parameters of $(a_m,b_m)$ in a look-up table at the edge.
Then by choosing a set of parameters from the table, the unknown error model at the edge can be approximated.
This is because the error functions can share the same trend for two similar tasks, e.g., classifying digit `$8$' and `$9$' with SVM as shown in Fig.~3c.
Notice that there may be a mismatch between the pre-training task and the real task at the edge.
This is the case between classifying digit `$8$' and `$5$' in Fig.~3c.
As a result, it is necessary to carefully measure the similarity between two tasks when choosing the parameters.
\subsection{Parameter Fitting of ResNet and PointNet}
To verify the nonlinear model in \eqref{model1} under deeper learning models and larger datasets,
we train the $110$-layer deep residual network (ResNet-$110$ with $1.7~$M parameters) \cite{resnet} using the CIFAR-10 dataset as the input images, with training sample size ranging from $5000$ to $50000$.
The image in the CIFAR-10 dataset has $32\times32$ pixels (each pixel has $3~$Bytes representing RGB), and each image sample has a size of $(32\times32\times3+1)\times8=24584~\mathrm{bits}$.
The training procedure is implemented with a diminishing learning rate and a mini-batch size of $100$.
After training for $50000$ iterations ($\sim2.5$ hours), we test the trained model on a dataset with $10000$ unseen samples, and obtain the corresponding classification error.
It can be seen from Fig.~4a that the proposed model with $(a_m,b_m)=(8.15,0.44)$ matches the experimental data of ResNet-$110$ very well.
Moreover, we also consider the PointNet ($3.5~$M parameters), which applies feature transformations and aggregates point features by max pooling \cite[Fig.~2]{pointnet} to classify $3$D point clouds dataset ModelNet40 (see examples in Fig.~4b).
In ModelNet40, there are $12311$ CAD models from $40$ object categories, split into $9843$ for training and $2468$ for testing.
Each sample has $2000$ points with three single-precision floating-point coordinates ($4~$Bytes), and the data size per sample is $(2000\times3\times4+1)\times8=192008~\mathrm{bits}$.
After training for $250$ epochs ($\sim5.5$ hours) with a mini-batch of $32$, the classification error versus the number of samples is obtained in Fig.~4a, and the proposed classification error model with $(a_m,b_m)=(0.96,0.24)$ matches the experimental data of PointNet very well.
\begin{figure*}[!t]
\centering
\hspace{0.1in}
\subfigure[]{
\includegraphics[width=58mm]{./img/model2.eps}}
\subfigure[]{
\includegraphics[width=50mm]{./img/example.eps}}
\subfigure[]{
\includegraphics[width=58mm]{./img/flowchart.eps}}
\caption{a) Comparison between the experimental data and the nonlinear classification error model. The parameters in the models are given by
$(a_m,b_m)=(8.15,0.44)$ for ResNet-$110$ and $(a_m,b_m)=(0.96,0.24)$ for PointNet; b) Examples of the 3D point clouds in ModelNet40; c) The flowchart for LCPA Algorithms.}
\end{figure*}
\section{MM-Based LCPA Algorithm}
Based on the results in Section III, we can directly approximate the true error function $\Psi_m$ by $\Theta_m$.
However, to account for the approximation error between $\Psi_m$ and $\Theta_m$ (e.g., due to noise in samples or slight mismatch between data used for fitting and data observed in MTC devices), a weighting factor $\beta_m\geq 1$ can be applied to $\Theta_m$,
where a higher value of $\beta_m$ accounts for a larger approximation error.\footnote{Since the real classification error is scattered around the fitted one, introducing $\rho_m\geq1$ means that we need to elevate the fitted curves to account for the possibilities that worse classification results may happen compared to our prediction.
In other words, we should be more conservative (pessimistic) about our prediction.}
Then by replacing $\Psi_m$ with $\beta_m\Theta_m$ and putting \eqref{P0b} into $\Theta_m(v_m|a_m,b_m)$ to eliminate $\mathbf{v}$, problem $\mathrm{P}$ becomes:
\begin{subequations}
\begin{align}
\mathrm{P}1:\mathop{\mathrm{min}}_{\substack{\mathbf{p}}}
\quad&\mathop{\mathrm{max}}_{m=1,\cdots,M}~\beta_m\,\Phi_m(\mathbf{p}), \nonumber\\
\mathrm{s. t.}\quad&\sum_{k=1}^Kp_k=P,\quad p_k\geq 0,\quad \forall k,
\label{P1a} \\
&G_{k,k}p_{k}\geq\left(2^{D_mZ_k^{\rm{min}}/(\xi BT)}-1\right)
\nonumber\\
&
\times\left(\sum_{l\neq k}G_{k,l}p_{l}+\sigma^2\right),\quad \forall k,
\label{P1b}
\\
&G_{k,k}p_{k}\leq\left(2^{D_mZ_k^{\rm{max}}/(\xi BT)}-1\right)
\nonumber\\
&\times\left(\sum_{l\neq k}G_{k,l}p_{l}+\sigma^2\right),\quad \forall k,
\label{P1c}
\end{align}
\end{subequations}
where constraints \eqref{P1b}--\eqref{P1c} come from equation \eqref{ratebounds} in the non-IID case and
\begin{align}
&\Phi_m(\mathbf{p}):=
\nonumber\\
&a_m\left[
\sum_{k\in\mathcal{Y}_m} \frac{\xi BT}{D_m}\mathrm{log}_2\left(1+\frac{G_{k,k}p_{k}}{\sum_{l\neq k}G_{k,l}p_{l}+
\sigma^2} \right)
+A_m
\right]^{-b_m}.
\end{align}
It can be seen that $\mathrm{P}1$ is a nonlinear optimization problem due to the nonlinear classification error model \eqref{model1}.
Moreover, the $\mathrm{max}$ operator introduces non-smoothness to the problem, and the objective function is not differentiable.
Thus the existing method based on gradient descent \cite{gradient} is not applicable.
To solve $\mathrm{P}1$, we propose to use the framework of MM \cite{mm1,mm2,mm3,wang}, which constructs a sequence of upper bounds $\{\widetilde{\Phi}_m\}$ on $\{\Phi_m\}$ and replaces $\{\Phi_m\}$ in $\rm{P}1$ with $\{\widetilde{\Phi}_m\}$ to obtain the surrogate problems.
More specifically, given any feasible solution $\mathbf{p}^\star$ to $\mathrm{P}1$, we define surrogate functions
\begin{align}
&\widetilde{\Phi}_{m}(\mathbf{p}|\mathbf{p}^\star)
\nonumber\\
&=a_m\Bigg\{
\sum_{k\in\mathcal{Y}_m} \frac{\xi BT}{D_m\mathrm{ln}2\,}
\Bigg[
\mathrm{ln}\left(\sum_{l=1}^K\frac{G_{k,l}p_{l}}{\sigma^2}+1\right)
\nonumber\\
&\quad
{}
-
\mathrm{ln}\left(\sum_{l=1,l\neq k}^K\frac{G_{k,l}p^\star_{l}}{\sigma^2}+1\right)-\left(\sum_{l=1,l\neq k}^K\frac{G_{k,l}p^\star_{l}}{\sigma^2}+1\right)^{-1}
\nonumber
\\&\quad
{}
\times\left(\sum_{l=1,l\neq k}^K\frac{G_{k,l}p_{l}}{\sigma^2}+1\right)
+1
\Bigg]+A_m
\Bigg\}^{-b_m}, \label{Phi}
\end{align}
and the following proposition can be established.
\begin{proposition}
The functions $\{\widetilde{\Phi}_m\}$ satisfy the following conditions:
\noindent(i) Upper bound condition: $\widetilde{\Phi}_{m}(\mathbf{p}|\mathbf{p}^\star)\geq \Phi_{m}(\mathbf{p})$.
\noindent(ii) Convexity: $\widetilde{\Phi}_m(\mathbf{p}|\mathbf{p}^{\star})$ is convex in $\bm{\mathbf{p}}$.
\noindent(iii) Local equality condition: $\widetilde{\Phi}_{m}(\mathbf{p}^\star|\mathbf{p}^\star)=\Phi_{m}(\mathbf{p}^\star)$ and $\nabla_{\mathbf{p}}\widetilde{\Phi}_{m}(\mathbf{p}^\star|\mathbf{p}^\star)=\nabla_{\mathbf{p}}\Phi_{m}(\mathbf{p}^\star)$.
\end{proposition}
\begin{proof}
See Appendix A.
\end{proof}
With part (i) of \textbf{Proposition 1}, an upper bound can be directly obtained if we replace the functions $\{\Phi_m\}$ by $\{\widetilde{\Phi}_m\}$ around a feasible point.
However, a tighter upper bound can be achieved if we treat the obtained solution as another feasible point and continue to construct the next-round surrogate function.
In particular, assuming that the solution at the $n^{\mathrm{th}}$ iteration is given by $\mathbf{p}^{[n]}$, the following problem is considered at the $(n+1)^{\mathrm{th}}$ iteration:
\begin{align}
\mathrm{P}1[n+1]:\mathop{\mathrm{min}}_{\substack{\mathbf{p}}}
\quad&\mathop{\mathrm{max}}_{m=1,\cdots,M}~\beta_m\widetilde{\Phi}_m(\mathbf{p}|\mathbf{p}^{[n]}),
\nonumber\\
\mathrm{s. t.}\quad&\rm{constraints}~\eqref{P1a}-\eqref{P1c}. \label{P1[n+1]}
\end{align}
Based on part (ii) of \textbf{Proposition 1}, the problem $\mathrm{P}1[n+1]$ is convex and can be solved by off-the-shelf software packages (e.g., CVX Mosek \cite{opt1}) for convex programming.
Denoting its optimal solution as
$\mathbf{p}^*$, we can set
$\mathbf{p}^{[n+1]}=\mathbf{p}^*$, and the process repeats with solving the problem $\mathrm{P}1[n+2]$.
According to part (iii) of \textbf{Proposition 1} and \cite[Theorem 1]{mm1}, every limit point of the sequence
$(\mathbf{p}^{[0]},\mathbf{p}^{[1]},\cdots)$ is a KKT solution to $\mathrm{P}1$ as long as the starting point $\mathbf{p}^{[0]}$ is feasible to $\mathrm{P}1$ (e.g., $\mathbf{p}^{[0]}=P/K\,\bm{1}_K$).
The entire procedure of the MM-based LCPA is summarized in the left hand branch of Fig.~4c.
In terms of computational complexity, $\mathrm{P}1[n+1]$ involves $K$ primal variables and $M+3K+1$ dual variables.
The dual variables correspond to $M+3K+1$ constraints in $\mathrm{P}1[n+1]$, where $M$ constraints come from the $\mathrm{max}$ operator, $K$ constraints come from nonnegative power constraints, $2K$ constraints come from the data-rate bounds, and one constraint comes from the power budget.
Therefore, the worst-case complexity for solving $\mathrm{P}1[n+1]$ is
$\mathcal{O}\Big((M+4K+1)^{3.5}\Big)$ \cite{opt2}.
Consequently, the total complexity for solving $\mathrm{P}1$ is $\mathcal{O}\Big(\mathcal{I}\,(M+4K+1)^{3.5}\Big)$, where $\mathcal{I}$ is the number of iterations needed for the algorithm to converge.
\section{Asymptotic Analysis and Insights to LCPA}
To understand how LCPA works, this section investigates the asymptotic case when the number of antennas at the edge approaches infinity (i.e., $N \to +\infty$).
Moreover, we consider the special case of $|\mathcal{Y}_m|=1$ and $\mathcal{Y}_m\bigcap\mathcal{Y}_j=\emptyset$ (i.e., each user group has only one unique user).
For notational simplicity, we denote the unique user in group $\mathcal{Y}_m$ as user $m=k$.
As each task only involves one user, non-IID data distribution among users in a single task does not exist, and therefore constraints \eqref{P1b}--\eqref{P1c} can be removed.
On the other hand, as $N\to+\infty$, the channels from different users to the edge would be asymptotically orthogonal \cite{massive1,massive2,massive3} and we have
\begin{align}
&G_{k,l}=\frac{|\mathbf{h}_k^H\mathbf{h}_l|^2}{\left\Vert\mathbf{h}_k\right\Vert_2^2}\to 0,\quad\forall k\neq l.
\end{align}
Based on such orthogonality feature, and putting $G_{k,l}=0$ for $k\neq l$ into $\Phi_m$ in $\rm{P}1$, problem $\mathrm{P}1$ under $N\to+\infty$ and $|\mathcal{Y}_m|=1$ is rewritten as
\begin{align}
&\mathrm{P}2:\mathop{\mathrm{min}}_{\substack{\mathbf{p},\,\mu}}
\quad\mu,
\nonumber\\
&\mathrm{s. t.}\quad
\beta_ka_k\left(
\frac{\xi BT}{D_k}\mathrm{log}_2\left(1+\frac{G_{k,k}p_{k}}{\sigma^2} \right)+A_k
\right)^{-b_k}
\leq\mu,\quad \forall k,
\nonumber\\
&\quad\quad
\sum_{k=1}^Kp_k= P,\quad p_k\geq 0,\quad\forall k,
\label{P2b}
\end{align}
where $\mu\in[0,1]$ is a slack variable and has the interpretation of \emph{classification error level}.
The following proposition gives the optimal solution to $\mathrm{P}2$.
\begin{proposition}
The optimal $\mathbf{p}^*$ to $\mathrm{P}2$ is
\begin{align}\label{pk*}
p_k^*(\mu)&=\Bigg[\frac{\sigma^2}{G_{k,k}}\,\mathrm{exp}\left(\frac{D_k\mathrm{ln}2\,}{\xi BT}\left[\left(\frac{\mu}{\beta_ka_k}\right)^{-1/b_k}-
A_k\right]\right)
\nonumber\\
&\quad{}
-\frac{\sigma^2}{G_{k,k}}\Bigg]^+,\quad k=1,\cdots,K,
\end{align}
where $\mu$ satisfies $\sum_{k=1}^Kp_k^*(\mu)=P$.
\end{proposition}
\begin{proof}
See Appendix B.
\end{proof}
To efficiently compute the classification error level $\mu$, it is observed that the function $p_k^*(\mu)$ is a decreasing function of $\mu$.
Therefore, the classification error level $\mu$ can be obtained from solving $\sum_{k=1}^Kp_k^*(\mu)=P$ using bisection method within interval $[0,1]$.
More specifically, given $\mu_{\mathrm{max}}$ and $\mu_{\mathrm{min}}$ (initially $\mu_{\mathrm{max}}=1$ and $\mu_{\mathrm{min}}=0$), we set $\mu=(\mu_{\mathrm{max}}+\mu_{\mathrm{min}})/2$.
If $\sum_kp_k^*(\mu)\geq P$, we update $\mu_{\mathrm{min}}=\mu$; otherwise, we update $\mu_{\mathrm{max}}=\mu$. This procedure is repeated until $|\mu_{\mathrm{max}}-\mu_{\mathrm{min}}|<\epsilon$ with $\epsilon=10^{-8}$.
Since bisection method has a linear convergence rate \cite{bisection}, and in each iteration we need to compute $K$ scalar functions $p_1^*(\mu),\cdots,p_K^*(\mu)$, the bisection method has a complexity of $\mathcal{O}(\mathrm{log}\left(1/\epsilon\right)K)$.
\textbf{Scaling Law of Learning Centric Communication.}
According to \textbf{Proposition 2}, the user transmit power $p_k$ is inversely proportional to the wireless channel gain $G_{k,k}=\left\Vert\mathbf{h}_k\right\Vert_2^2$.
On the other hand, it is exponentially dependent on the classification error level $\mu$ and the learning parameters $(a_k,b_k,D_k,A_k)$.
Moreover, among all parameters, $b_k$ is the most important factor, since $b_k$ is involved in both the power and exponential functions.
The above observations disclose that in edge machine learning systems, the learning parameters will have more significant impacts on the radio resource allocation than those of the wireless channels.
\textbf{Learning Centric versus Communication Centric Power Allocation.} Notice that the result in \eqref{pk*} is fundamentally different from the most well-known resource allocation schemes (e.g., iterative water-filling \cite{waterfilling} and max-min fairness \cite{fair}).
For example, the water-filling solution for maximizing the system throughput under $N\to+\infty$ is given by
\begin{align}\label{water}
&p_k^{\rm{WF}}=
\left(\frac{1}{\lambda\mathrm{ln}2\,}-\frac{\sigma^2}{G_{k,k}}\right)^+,
\end{align}
where $\lambda$ is a constant chosen such that $\sum_{k=1}^Kp_k^{\rm{WF}}=P$.
On the other hand, the max-min fairness solution under $N\to+\infty$ is given by
\begin{align}\label{fair}
&p_k^{\rm{FAIR}}=
P\left(\sum_{k=1}^K\frac{\sigma^2}{G_{k,k}} \right)^{-1}\,\frac{\sigma^2}{G_{k,k}}.
\end{align}
It can be seen from \eqref{water} and \eqref{fair} that the water-filling scheme would allocate more power resources to better channels, and the max-min fairness scheme would allocate more power resources to worse channels.
But no matter which scheme we adopt, the only impact factor is the channel condition $\sigma^2/G_{k,k}$.
\section{Large-scale Optimization under IID Datasets}
Although a KKT solution to $\mathrm{P}1$ has been derived in Section IV, it can be seen that MM-based LCPA requires a cubic complexity with respect to $K$.
This leads to time-consuming computations if $K$ is in the range of hundreds or more.
As a result, low-complexity large-scale optimization algorithms are indispensable.
To this end, in this section we consider the case of $N \to +\infty$ under IID datasets, and develop an algorithm based on the FOM.
As $N\to+\infty$, we put $G_{k,l}=0$ for $k\neq l$ into $\Phi_m$ in $\rm{P}1$, and the function $\Phi_m$ is asymptotically equal to
\begin{align}
&\Xi_m(\mathbf{p})
=a_m\left[
\sum_{k\in\mathcal{Y}_m} \frac{\xi BT}{D_m}\mathrm{log}_2\left(1+\frac{G_{k,k}p_k}{\sigma^2} \right)+A_m
\right]^{-b_m}. \label{Xi}
\end{align}
Therefore, the problem $\mathrm{P}1$ in the case of IID datasets and $N \to +\infty$ is equivalent to
\begin{align}
\mathrm{P}3:\mathop{\mathrm{min}}_{\substack{\mathbf{p}}}
\quad&\mathop{\mathrm{max}}_{m=1,\cdots,M}~\beta_m\Xi_m(\mathbf{p}),
\nonumber\\
\mathrm{s. t.}\quad&\sum_{k=1}^Kp_k=P,\quad p_k\geq 0,\quad k=1,\cdots,K. \label{P3}
\end{align}
The major challenge for solving $\mathrm{P}3$ comes from the nonsmooth operator $\rm{max}$ in the objective function, which hinders us from computing the gradients.
To deal with the non-smoothness, we reformulate $\mathrm{P}3$ into a smooth bilevel optimization problem with $\ell_1$-norm (simplex) constraints.
Observing that the projection onto a simplex in Euclidean space requires high computational complexities, a mirror-prox LCPA method working on non-Euclidean manifold is proposed.
In this way, the distance is measured by Kullback-Leibler (KL) divergence, and the non-Euclidean projection would have analytical expressions.
Lastly, with an extragradient step, the proposed mirror-prox LCPA converges to the global optimal solution to $\mathrm{P}3$ with an iteration complexity of $\mathcal{O}(1/\epsilon)$ \cite{BFOM1,BFOM2,BFOM3}, where $\epsilon$ is the target solution accuracy.
More specifically, we first equivalently transform $\mathrm{P}3$ into a smooth bilevel optimization problem.
By defining set $\mathcal{P}=\left\{\mathbf{p}\in\mathbb{R}^{K\times 1}_+:\left\Vert\mathbf{p}\right\Vert_1=P\right\}$ and introducing variables $\bm{\alpha}\in\mathbb{R}^{M\times 1}$ such that $\bm{\alpha}\in\mathcal{A}=\{\bm{\alpha}\in\mathbb{R}^{M\times 1}_+:\left\Vert\bm{\alpha}\right\Vert_1=1\}$, $\mathrm{P}3$ is rewritten as
\begin{align}\label{saddle}
&\mathrm{P}4:\mathop{\mathrm{min}}_{\substack{\mathbf{p}\in\mathcal{P}}}
~\mathop{\mathrm{max}}_{\bm{\alpha}\in\mathcal{A}}~\underbrace{\mathop{\sum}_{m=1}^{M}\alpha_m\times\beta_m\Xi_m(\mathbf{p})}_{\Upsilon(\bm{\alpha},\mathbf{p})}.
\end{align}
It can be seen from $\mathrm{P}4$ that $\Upsilon(\bm{\alpha},\mathbf{p})$ is differentiable with respect to either $\mathbf{p}$ or $\bm{\alpha}$, and the corresponding gradients are
\begin{subequations}
\begin{align}
\nabla_{\mathbf{p}}\Upsilon(\bm{\alpha},\mathbf{p})&=
\mathop{\sum}_{m=1}^{M}\alpha_m\beta_m\nabla_\mathbf{p}\Xi_m(\mathbf{p}), \label{gradient1}
\\
\nabla_{\bm{\alpha}}\Upsilon(\bm{\alpha},\mathbf{p})
&=
\left[\beta_1\Xi_1(\mathbf{p}),\cdots,\beta_M\Xi_M(\mathbf{p}) \right]^T, \label{gradient2}
\end{align}
\end{subequations}
where
\begin{align}
\nabla_\mathbf{p}\Xi_m(\mathbf{p})
&=
\left[\frac{\partial \Xi_m}{\partial p_1},\cdots,\frac{\partial \Xi_m}{\partial p_K} \right]^T, \label{gradient}
\end{align}
with its $j^{\mathrm{th}}$ element being
\begin{align}
\frac{\partial \Xi_m}{\partial p_j}&=
-\frac{a_mb_m\xi BT\,\mathbb{I}_{\mathcal{Y}_m}(j)}{D_m\mathrm{ln}2\,(\sigma^2G_{j,j}^{-1}+p_j)}
\nonumber\\
&\quad{}
\times
\left[
\sum_{k\in\mathcal{Y}_m} \frac{\xi BT}{D_m}\mathrm{log}_2\left(1+\frac{G_{k,k}p_{k}}{\sigma^2} \right)+A_m
\right]^{-b_m-1}.
\label{ximpartial}
\end{align}
However, $\mathrm{P}4$ is a bilevel problem, with both the upper layer variable $\mathbf{p}$ and the lower layer variable $\bm{\alpha}$ involved in the simplex constraints.
In order to facilitate the projection onto simplex constraints, below we consider a non-Euclidean (Banach) space induced by $\ell_1$-norm.
In such a space, the Bregman distance between two vectors $\mathbf{x}$ and $\mathbf{y}$ is the KL divergence
\begin{align}\label{W}
&\mathbb{KL}\left(\mathbf{x},\mathbf{y}\right)=\mathop{\sum}_{l=1,2,\cdots}x_l\,\mathrm{ln}\left(\frac{x_l}{y_l}\right),
\end{align}
and the following proposition can be established.
\begin{proposition}
If the classification error $\mu=\mathop{\mathrm{max}}_{m}\beta_m\Xi_m(\mathbf{p})$ is upper bounded by $\mu\leq\mu_0$, then $\Upsilon(\bm{\alpha},\mathbf{p})$ is $(L_{1},L_{2},L_{2},0)$--smooth in Banach space induced by $\ell_1$-norm, where
\begin{subequations}
\begin{align}
L_{1}&=\mathop{\mathrm{max}}_{\substack{m=1,\cdots,M\\k=1,\cdots,K}}~
\frac{\beta_ma_mb_m\xi BTG_{k,k}}{D_m\mathrm{ln}2\,\sigma^4}
\left(\frac{\mu_0}{\beta_ma_m}\right)^{1+1/b_m}
\nonumber\\
&\quad\times
\left[G_{k,k}+
\frac{(b_m+1)\xi BTH_m}{D_m\mathrm{ln}2}
\left(\frac{\mu_0}{\beta_ma_m}\right)^{1/b_m}\right], \label{L11}
\\
L_{2}&=\mathop{\mathrm{max}}_{m=1,\cdots,M}~
\frac{\beta_ma_mb_m\xi BTH_m}{D_m\mathrm{ln}2\,\sigma^2}
\left(\frac{\mu_0}{\beta_ma_m}\right)^{1+1/b_m}, \label{L12}
\end{align}
\end{subequations}
with $H_m:=\left\Vert[\mathbb{I}_{\mathcal{Y}_m}(1)\,G_{1,1},\cdots,\mathbb{I}_{\mathcal{Y}_m}(K)\,G_{K,K}]^T\right\Vert_2$.
\end{proposition}
\begin{proof}
See Appendix C.
\end{proof}
The smoothness result in \textbf{Proposition 3} enables us to apply mirror descent (i.e., generalized gradient descent in non-Euclidean space) to $\mathbf{p}$ and mirror ascent to $\bm{\alpha}$ in the $\ell_1$-space \cite{BFOM2}.
This leads to the proposed mirror-prox LCPA, which is an iterative algorithm that involves i) a proximal step and ii) an extragradient step.
In particular, the mirror-prox LCPA initially chooses a feasible $\mathbf{p}=\mathbf{p}^{[0]}\in\mathcal{P}$ and $\bm{\alpha}=\bm{\alpha}^{[0]}\in\mathcal{A}$ (e.g., $\mathbf{p}^{[0]}=P/K\,\bm{1}_K$ and $\bm{\alpha}^{[0]}=1/M\,\bm{1}_M$).
Denoting the solution at the $n^{\mathrm{th}}$ iteration as $(\mathbf{p}^{[n]},\bm{\alpha}^{[n]})$, the following equations are used to update the next-round solution \cite{BFOM2}:
\begin{subequations}
\begin{align}
\mathbf{p}^{\diamond}&=
\mathop{\mathrm{argmin}}_{\mathbf{p}\in\mathcal{P}}~
\mathbb{KL}\left(\mathbf{p},\mathbf{p}^{[n]}\right)
\nonumber\\
&\quad{}
+\eta \, \mathbf{p}^T
\left[\mathop{\sum}_{m=1}^{M}\alpha_m^{[n]}\beta_m\nabla_\mathbf{p}\Xi_m(\mathbf{p}^{[n]})\right],
\label{BFOM1}
\\
\bm{\alpha}^{\diamond}&=
\mathop{\mathrm{argmin}}_{\bm{\alpha}\in\mathcal{A}}~
\mathbb{KL}\left(\bm{\alpha},\bm{\alpha}^{[n]}\right)
\nonumber\\
&\quad{}
-\eta
\left[\beta_1\Xi_1(\mathbf{p}^{[n]}),\cdots,\beta_M\Xi_M(\mathbf{p}^{[n]}) \right]\bm{\alpha}, \label{BFOM2}
\\
\mathbf{p}^{[n+1]}&=
\mathop{\mathrm{argmin}}_{\mathbf{p}\in\mathcal{P}}~
\mathbb{KL}\left(\mathbf{p},\mathbf{p}^{[n]}\right)
\nonumber\\
&\quad{}
+\eta \, \mathbf{p}^T
\left[\mathop{\sum}_{m=1}^{M}\alpha_m^{\diamond}\beta_m\nabla_\mathbf{p}\Xi_m(\mathbf{p}^{\diamond})\right],
\label{BFOM3}
\\
\bm{\alpha}^{[n+1]}&=
\mathop{\mathrm{argmin}}_{\bm{\alpha}\in\mathcal{A}}~\mathbb{KL}\left(\bm{\alpha},\bm{\alpha}^{[n]}\right)
\nonumber\\
&\quad{}
-\eta
\left[\beta_1\Xi_1(\mathbf{p}^{\diamond}),\cdots,\beta_M\Xi_M(\mathbf{p}^{\diamond}) \right]\bm{\alpha}, \label{BFOM4}
\end{align}
\end{subequations}
where $\eta$ is the step-size, and the terms inside $[\cdots]$ in \eqref{BFOM1}--\eqref{BFOM4} are obtained from \eqref{gradient1}--\eqref{gradient2}.
Notice that a small $\eta$ would lead to slow convergence of the algorithm while a large $\eta$ would cause the algorithm to diverge.
According to \cite{BFOM2}, $\eta$ should be chosen inversely proportional to Lipschitz constant $L_1$ or $L_2$ derived in \textbf{Proposition 3}.
In this paper, we set $\eta=10^3/L_2$ with $\mu_0=0.1$, which empirically provides fast convergence of the algorithm.
\textbf{How Mirror-prox LCPA Works.}
The formulas \eqref{BFOM1}--\eqref{BFOM2} update the variables along their gradient direction, while keeping the updated point $\{\mathbf{p}^{\diamond},\bm{\alpha}^{\diamond}\}$ close to the current point $\{\mathbf{p}^{[n]},\bm{\alpha}^{[n]}\}$.
This is achieved via the \emph{proximal operator} that minimizes the distance $\mathbb{KL}\left(\mathbf{p},\mathbf{p}^{[n]}\right)$ (or $\mathbb{KL}\left(\bm{\alpha},\bm{\alpha}^{[n]}\right)$) plus a first-order linear function.
Since the KL divergence is the Bregman distance, the update \eqref{BFOM1}--\eqref{BFOM2} is a Bregman proximal step.
On the other hand, the gradients in \eqref{BFOM3} and \eqref{BFOM4} are computed using $\mathbf{p}^{\diamond}$ and $\bm{\alpha}^{\diamond}$, respectively.
By doing so, we can obtain the look-ahead gradient at the intermediate point $\mathbf{p}^{\diamond}$ and $\bm{\alpha}^{\diamond}$ for updating $\mathbf{p}^{[n+1]}$ and $\bm{\alpha}^{[n+1]}$.
This ``look-ahead'' feature is called \emph{extragradient}.
Lastly, we put the Bregman distance $\mathbb{KL}$ in \eqref{W}, the function $\Xi_m$ in \eqref{Xi}, the gradient in \eqref{gradient}, and a proper $\eta$ into \eqref{BFOM1}--\eqref{BFOM4}.
Based on the KKT conditions, the equations \eqref{BFOM1}--\eqref{BFOM2} are shown to be equivalent to
\begin{subequations}
\begin{align}
p^{\diamond}_k&=\left\{
\sum_{i=1}^K
p^{[n]}_i\mathrm{exp}\left[
-\eta\left(\mathop{\sum}_{m=1}^{M}\alpha_m^{[n]}\beta_m\nabla_{p_i}\Xi_m(\mathbf{p}^{[n]})\right)
\right]\right\}^{-1}
\nonumber
\\
&\quad{}
\times
P\, p^{[n]}_k\mathrm{exp}\left[
-\eta\left(\mathop{\sum}_{m=1}^{M}\alpha_m^{[n]}\beta_m\nabla_{p_k}\Xi_m(\mathbf{p}^{[n]})\right)
\right]
,
\nonumber\\
&\quad{}
k=1,\cdots,K,
\label{BFOM21}
\\
\alpha^{\diamond}_m&=
\left(
\sum_{i=m}^M
\alpha^{[n]}_i\mathrm{exp}\left[
\eta\,\beta_i
\Xi_i(\mathbf{p}^{[n]})
\right]\right)^{-1}
\nonumber\\
&\quad{}
\times
\alpha^{[n]}_m\mathrm{exp}\left[
\eta\,\beta_m
\Xi_m(\mathbf{p}^{[n]})
\right]
,\quad
m=1,\cdots,M.
\label{BFOM22}
\end{align}
\end{subequations}
The equations \eqref{BFOM3}--\eqref{BFOM4} can be similarly reduced to closed-form expressions.
According to \textbf{Proposition 3} and \cite{BFOM1}, the mirror-prox LCPA algorithm is guaranteed to converge to the optimal solution to $\mathrm{P}4$.
But in practice, we can terminate the iterative procedure when the norm $\left\Vert\mathbf{p}^{[n]}-\mathbf{p}^{[n-1]}\right\Vert_{\infty}$ is small enough, e.g., $\left\Vert\mathbf{p}^{[n]}-\mathbf{p}^{[n-1]}\right\Vert_{\infty}<10^{-8}$.
The entire procedure for computing the solution to $\mathrm{P}4$ using the mirror-prox LCPA is summarized in the right hand branch of Fig.~4c.
In terms of computational complexity, computing \eqref{BFOM21} requires a complexity of $\mathcal{O}(MK)$.
Since the number of iterations for the mirror-prox LCPA to converge is $\mathcal{O}(1/\epsilon)$ with $\epsilon$ being the target solution accuracy, the total complexity of mirror-prox LCPA would be $\mathcal{O}(MK/\epsilon)$.
\section{Simulation Results and Discussions}
This section provides simulation results to evaluate the performance of the proposed algorithms.
It is assumed that the noise power $\sigma^2=-87~\mathrm{dBm}$ (corresponding to power spectral density $-140~\mathrm{dBm/Hz}$ with $180~\mathrm{kHz}$ bandwidth \cite{iot3}), which includes thermal noise and receiver noise.
The total transmit power at users is set to $P=13~\mathrm{dBm}$ (i.e., $20~\rm{mW}$), with the communication bandwidth $B=180~\mathrm{kHz}$.
The path loss of the $k^{\mathrm{th}}$ user $\varrho_{k}=-100~\rm{dB}$ is adopted \cite{wang}, and $\mathbf{h}_{k}$ is generated according to $\mathcal{CN}(\mathbf{0},\varrho_{k}\mathbf{I}_N)$ \cite{massive2}.
Without otherwise specified, it is assumed that $\xi=1$.
We set $Z_k^{\mathrm{min}} =0$ and $Z_k^{\mathrm{max}}=+\infty$ for all $k$ in Sections VII-A to VII-C.
Simulations with upper and lower bounds on data amount will be presented in Section VII-D.
Each point in the figures is obtained by averaging over $10$ simulation runs, with independent channels in each run.
All optimization problems are solved by Matlab R2015b on a desktop with Intel Core i5-4570 CPU at 3.2~GHz and 8~GB RAM.
All the classifiers are trained by Python 3.6 on a GPU server with Intel Core i7-6800 CPU at 3.4 GHz and GeForce GTX 1080 GPU.
\subsection{CNN and SVM}
We consider the aforementioned CNN and SVM classifiers with the number of learning models $M=2$ at the edge: i) Classification of MNIST dataset \cite{MNIST} via CNN; ii) Classification of digits dataset in Scikit-learn \cite{sklearn} via SVM.
The data size of each sample is $D_1=6276~\mathrm{bits}$ for the MNIST dataset and $D_2=324~\mathrm{bits}$ for the digits dataset in Scikit-learn.
It is assumed that there are $A_1=300$ CNN samples and $A_2=200$ SVM samples in the historical dataset.
The parameters in the two classification error models are assumed to be perfectly known and they are given by $(a_1,b_1)=(7.3,0.69)$ for CNN and $(a_2,b_2)=(5.2,0.72)$ for SVM as in Fig.~3b.
Finally, it is assumed that $(\beta_1,\beta_2)=(1,1.2)$ since the approximation error of SVM in Fig.~3b is larger than that of CNN.
To begin with, the case of $N=20$ and $K=4$ with $\mathcal{Y}_1=\{1\}$ and $\mathcal{Y}_2=\{2,3,4\}$ is simulated.
Under the above settings, we compute the collected sample sizes by executing the proposed MM-based LCPA, and the maximum error of classifiers (i.e., the worse classification result of CNN and SVM, where the classification error for each task is obtained from the machine learning experiment using the sample sizes from the power allocation algorithms) versus the total transmission time $T$ is shown in Fig. 5a.
Besides the proposed MM-based LCPA, we also simulate two benchmark schemes: 1) Max-min fairness scheme \cite[Sec. II-C]{fair}; 2) Sum-rate maximization scheme \cite[Sec. IV]{sumrate}.
It can be seen from Fig.~5a that the proposed MM-based LCPA algorithm with $\mathcal{I}=10$ has a significantly smaller classification error compared to other schemes, and the gap concisely quantifies the benefit brought by more training images for CNN under joint communication and learning design.
For example, at $T=20$ in Fig.~5a, the proposed MM-based LCPA collects $2817$ MNIST images on average, while the sum-rate maximization and the max-min fairness schemes collect $1686$ images and $1781$ images, respectively.
Furthermore, if we target at the same learning error, the proposed algorithm saves the transmission time by at least $30\%$ compared to benchmark schemes.
This can be seen from Fig.~5a at the target error $4.5\%$, where the proposed algorithm takes $10$ seconds for transmission, but other methods require about $20$ seconds.
The saved time enables the edge to collect data for other edge computing tasks \cite{mec}.
\begin{figure*}[!t]
\centering
\subfigure[]{
\includegraphics[width=58mm]{./img/T.eps}}
\subfigure[]{
\includegraphics[width=58mm]{./img/N.eps}}
\subfigure[]{
\includegraphics[width=58mm]{./img/large.eps}}
\caption{a) Maximum error of classifiers versus total transmission time $T$ when $N=20$ and $K=4$; b) Maximum error of classifiers versus the number of antennas $N$ when $M=K=2$; c) Maximum error of classifiers versus the number of users $K$ when $N=100$;
}
\end{figure*}
To get more insight into the edge learning system, the case of $K=2$ with $\mathcal{Y}_1=\{1\}$ and $\mathcal{Y}_2=\{2\}$ at $T=5~\mathrm{s}$ is simulated, and the classification error versus the number of antennas $N=\{10,20,40,100\}$ is shown in Fig.~5b.
It can be seen from Fig.~5b that the classification error decreases as the number of antennas increases, which demonstrates the advantage of employing massive MIMO in edge machine learning.
More importantly, the proposed analytical solution in \textbf{Proposition 2} outperforms the water-filling\footnote{In the case of large $N$, sum-rate maximization scheme \cite[Sec. IV]{sumrate} would reduce to the iterative water-filling scheme \cite{waterfilling}, which allocates power according to \eqref{water}.} and max-min fairness schemes even at a relatively small number of antennas $N=10$.
This is achieved by allocating much more power resources to the first user (i.e., user uploading datasets for CNN), because training CNN is more difficult than training SVM.
In particular, the transmit powers in $\rm{mW}$ are given by: 1) $(p_1,p_2)=(19.8473, 0.1524)$ for the analytical LCPA scheme; 2) $(p_1,p_2)=(9.9862, 10.0138)$ for the water-filling scheme; and 3) $(p_1,p_2)=(10.0869, 9.9131)$ for the max-min fairness scheme.
Notice that the performance gain brought by LCPA in Fig.~5b is slightly smaller than that in Fig.~5a, since the ratio $|\mathcal{Y}_1|/|\mathcal{Y}_2|$ is increased. But no matter what value $|\mathcal{Y}_1|$ and $|\mathcal{Y}_2|$ take, the proposed LCPA always outperforms existing algorithms due to its learning centric feature.
To verify the performance and the low complexity nature of the mirror-prox LCPA in Section VI when the number of antennas is large, the case of $N=100$
and $K\in\{50,100\}$ at $T=5~\mathrm{s}$ is simulated, with $\mathcal{Y}_1$ containing the first $1/5$ users and $\mathcal{Y}_2$ containing the rest $4/5$ users.
The maximum error of classifiers versus the number of users $K$ is shown in Fig.~5c.
It can be seen that the proposed mirror-prox LCPA algorithm significantly reduces the classification error compared to the water-filling and max-min fairness schemes, and it achieves performance close to that of the MM-based LCPA.
On the other hand, the average execution time at $K=100$ is given by: 1) $18.2496~\rm{s}$ for the MM-based LCPA; 2) $0.5673~\rm{s}$ for the mirror-prox LCPA; 3) $0.0044~\rm{s}$ for the water-filling scheme; and 4) $0.0054~\rm{s}$ for the max-min fairness scheme.
Compared with MM-based LCPA, the mirror-prox LCPA saves at least $95\%$ of the computation time, which corroborates the linear complexity derived in Section VI.
\subsection{Deep Neural Networks}
Next, we consider the ResNet-$110$ as task $1$ and the CNN in Section III as task $2$ at the edge, with $D_1=24584~\mathrm{bits}$ and $D_2=6276~\mathrm{bits}$.
The error rate parameters are given by $(a_1,b_1)=(8.15,0.44)$ and $(a_2,b_2)=(7.3,0.69)$.
In addition, it is assumed that $(\beta_1,\beta_2)=(1,1)$, $T=200~\mathrm{s}$, and there is no historical sample at the edge (i.e., $A_1=A_2=0$).
We simulate the case of $N=20$ and $K=4$ with $\mathcal{Y}_1=\{1,2\}$ and $\mathcal{Y}_2=\{3,4\}$.
For ResNet-$110$, we assume that the datasets $\{\mathcal{D}_1,\mathcal{D}_2\}$ are formed by dividing the CIFAR-10 dataset into two parts, each with $30000$ different samples.
For CNN, we assume that the datasets $\{\mathcal{D}_3,\mathcal{D}_4\}$ are formed by dividing the MNIST dataset into two parts, each also with $30000$ different samples.
The worse classification error between the two tasks (obtained from the machine learning experiment using the sample sizes from the power allocation algorithms) is: 1) $14.13\%$ for MM-based LCPA; 2) $16.79\%$ for the sum-rate maximization scheme; and 3) $16.42\%$ for the max-min fairness scheme.
It can be seen that the proposed LCPA achieves the smallest classification error, which demonstrates the effectiveness of learning centric resource allocation under deep neural networks and large datasets.
\begin{figure*}[!t]
\centering
\subfigure[]{
\includegraphics[width=58mm]{./img/practical.eps}}
\subfigure[]{
\includegraphics[width=58mm]{./img/xi.eps}}
\subfigure[]{
\includegraphics[width=58mm]{./img/nonIID.eps}}
\caption{a) Comparison of classification error when $N=100$ and $K=5$ at $T=10~\rm{s}$ with $(a_m,b_m)$ estimated from historical data; b) Maximum error of classifiers versus $\xi$ when $N=100$ and $K=5$; c) Comparison of classification error when $N=100$ and $K=6$ at $T=10~\rm{s}$ in the non-IID case.
}
\end{figure*}
\subsection{Practical Considerations}
In Sections VII-A and VII-B, we have assumed that the parameters $(a_m,b_m)$ are perfectly known.
In practice, we need to estimate them from the historical data.
More specifically, denote the historical data as $\mathcal{H}_k\subseteq \mathcal{D}_k$.
We take $2/3$ of the historical data in $\mathcal{H}_k$ as the training dataset $\mathcal{T}_k$, and the remaining $1/3$ as the validation dataset $\mathcal{V}_k$ for all $k$.
Therefore, the number of training samples for estimating $(a_m,b_m)$ is $\sum_{k\in\mathcal{Y}_m}|\mathcal{T}_k|$.
Based on the above dataset partitioning, the parameters $\{a_m, b_m\}$ are estimated in two stages:
\begin{itemize}
\item In the training stage, we can vary the sample size $v_m$ as $(v_m^{(1)},v_m^{(2)},\cdots)$ within $[0,\sum_{k\in\mathcal{Y}_m}|\mathcal{T}_k|]$.
For each $v_m^{(i)}$, we train the learning model for $E$ epochs and test the trained model on the validation dataset $\{\mathcal{V}_k\}_{k\in\mathcal{Y}_m}$.
The classification errors corresponding to the different sample sizes are given by $(\Psi_m(v_m^{(1)}),\Psi_m(v_m^{(2)}),\cdots)$.
\item In the fitting stage, with the classification error versus sample size, the parameters $(a_m,b_m)$ in $\Theta_m$ are found via the nonlinear least squares fitting in \eqref{fitting}.
\end{itemize}
Assuming that the complexity of processing each image (e.g., for CNN, each processing stage includes a backward pass and a forward pass) is $\mathcal{O}(W_m)$ where $W_m$ is the number of parameters in the learning model, the complexity in the training stage is $\mathcal{O}(EW_m\sum_{i}v_m^{(i)})$.
If $\sum_{i}v_m^{(i)}$ is smaller than the total number of samples after data collection, then $\mathcal{O}(EW_m\sum_{i}v_m^{(i)})$ would be smaller than the actual model training complexity.
On the other hand, the complexity in the fitting stage is negligible, since the problem \eqref{fitting} only has two variables.
Even using a naive brute-force search with a step size of $0.001$ for each dimension, the running time for solving \eqref{fitting} is only $5$ seconds on a desktop with Intel Core i5-4570 CPU at 3.2~GHz and 8~GB RAM.
To illustrate the above procedure, we consider the case of $N=100$ and $K=5$ with $M=2$, where the first task is to train the CNN with user $\mathcal{Y}_1=\{1\}$ and the second task is to train the SVM with users $\mathcal{Y}_1=\{2,3,4,5\}$.
For CNN, we assume that the dataset $\mathcal{D}_1$ contains $10000$ different samples from the MINST dataset.
For SVM, we assume that the datasets $\{\mathcal{D}_2,\mathcal{D}_3,\mathcal{D}_4,\mathcal{D}_5\}$ are formed by dividing the scikit-learn digit dataset into four parts, each with $250$ different samples.
To conform with the size of $\mathcal{D}_1$, the datasets $\{\mathcal{D}_2,\mathcal{D}_3,\mathcal{D}_4,\mathcal{D}_5\}$ are augmented by appending $9750$ samples (generated via random rotations of the original samples) to the $250$ original samples.
The test dataset for CNN contains another $1000$ samples from the MNIST dataset and the test dataset for SVM contains another $797$ samples from the scikit-learn digits dataset.
It is assumed that $|\mathcal{H}_1|=450$ and $|\mathcal{H}_2|=\cdots=|\mathcal{H}_5|=75$.
According to the ``$2/3$-training and $1/3$-validation'' partitioning of $\mathcal{H}_k$, we have $|\mathcal{T}_1|=300$ and $|\mathcal{T}_2|=\cdots=|\mathcal{T}_5|=50$, meaning that there are $300$ and $200$ historical samples for training CNN and SVM, respectively.
For CNN, we vary the sample size $v_1$ as $(v_1^{(1)},v_1^{(2)},\cdots)=(100,150,200,300)$ and perform training for $E=200$ epochs.
The classification errors on the validation dataset $\mathcal{V}_1$ are given by $(0.300, 0.200, 0.140, 0.120)$, and after model fitting we have $(a_1,b_1)=(9.74,0.77)$.
On the other hand, for SVM, we vary the sample size $v_2$ as $(v_2^{(1)},v_2^{(2)},\cdots)=(30,50,100,200)$.
The classification errors on the validation dataset $\{\mathcal{V}_2,\mathcal{V}_3,\mathcal{V}_4,\mathcal{V}_5\}$ are given by $(0.510, 0.280, 0.220, 0.050)$, and after model fitting we have $(a_2,b_2)=(14.27,0.85)$.
Based on the above estimation results for $(a_m,b_m)$, the classification errors at $T=10~\mathrm{s}$ for different schemes are compared in Fig.~6a.
It can be seen from Fig.~6a that the proposed LCPA reduces the minimum classification error by at least $20\%$ compared with other simulated schemes.
To demonstrate the performance of the proposed LCPA under various values of $\xi$, the classification error versus $\xi=\{1/4,1/3,1/2,1\}$ at $T=10~\mathrm{s}$ is shown in Fig.~6b.
It can be seen from Fig.~6b that the classification error increases as $\xi$ decreases, which is due to the loss of samples during wireless transmission.
However, no matter what value $\xi$ takes, the proposed LCPA achieves the minimum classification error among all the simulated schemes.
\subsection{Non-IID Dataset}
In the non-IID case, we add one more user (denoted as user 6) to the system considered in Section VII-C, with the first task training the CNN with users $\mathcal{Y}_1=\{1,6\}$ and the second task training the SVM with users $\mathcal{Y}_1=\{2,3,4,5\}$.
The dataset $\mathcal{D}_6$ repeats the first sample in the MINST dataset for $10000$ times and $|\mathcal{H}_6|=450$.
Out of this $\mathcal{H}_6$, we use $2/3$ as the training dataset $\mathcal{T}_6$ and $1/3$ as the validation dataset $\mathcal{V}_6$.
We train the CNN in Fig.~1 using the training dataset $\{\mathcal{T}_1,\mathcal{T}_6\}$ for $200$ epochs, and obtain the parameter $\mathbf{w}$ in the CNN.
We then compute the prediction confidence for all the data $\mathbf{d}\in\{\mathcal{V}_1,\mathcal{V}_6\}$ based on equation \eqref{uncertain} with the probability function being the softmax output of CNN.
It turns out that the least confident samples in $\mathcal{V}_1$ and $\mathcal{V}_6$ have the following scores: $\mathop{\mathrm{min}}_{\mathbf{d}\in\mathcal{V}_1}U(\mathbf{d})=0.274$ and $\mathop{\mathrm{min}}_{\mathbf{d}\in\mathcal{V}_6}U(\mathbf{d})=0.999$.
For user $1$, since the CNN model is not sure about its prediction, we set $Z_1^{\mathrm{min}}=100$ and $Z_1^{\mathrm{max}}=10000$.
For user $6$, since the CNN model can predict its validation data with very high confidence, we set $Z_6^{\mathrm{min}}=0$ and $Z_6^{\mathrm{max}}=10$.
Based on the ratio between $Z_1^{\mathrm{max}}$ and $Z_6^{\mathrm{max}}$, we use $450$ samples from $\mathcal{H}_1$ and $10/10000\times450$ samples from $\mathcal{H}_6$ for estimating $(a_m,b_m)$.
Then we have $(a_1,b_1)=(9.74,0.77)$.
For the SVM model, since the data from users $\{2,3,4,5\}$ are assumed to be IID, $(a_2,b_2)$ is the same as that in Section VII-C, i.e., $(a_2,b_2)=(14.27,0.85)$.
The comparison of classification error in the non-IID case is shown in Fig.~6c.
It can be seen from Fig.~6c that the performance of MM-based LCPA (without uncertainty sampling) becomes worse than that in Fig.~6a, which is due to the additional resources allocated to user $6$.
However, the proposed MM-based LCPA still outperforms the sum-rate maximization and max-min fairness schemes.
More importantly, by adopting the uncertainty sampling method to distinguish the users' data quality, the classification error of LCPA can be further reduced (e.g., from the black bar to the yellow bar in Fig.~6c).
This demonstrates the benefit brought by \emph{joint estimation of sample size and prediction confidence} in edge machine learning systems.
\section{Conclusions}
This paper has introduced the LCPA concept to edge machine learning.
By adopting an empirical classification error model, learning efficient edge resource allocation has been obtained via the MM-based LCPA algorithm.
In the large-scale settings, a fast FOM has been derived to tackle the curse of high-dimensionality.
Simulation results have shown that the proposed LCPA algorithms achieve lower classification errors than existing power allocation schemes.
Furthermore, the proposed fast algorithm significantly reduces the execution time compared with the MM-based LCPA while still achieving satisfactory performance.
\appendices
\section{Proof of Proposition 1}
To prove part (i), consider the following inequality:
\begin{align}
&-\mathrm{ln}\left(\sum_{l=1,l\neq k}^K\frac{G_{k,l}p^\star_{l}}{\sigma^2}+1\right)
-\frac{\sum_{l=1,l\neq k}^KG_{k,l}p_{l}/\sigma^2+1}{\sum_{l=1,l\neq k}^KG_{k,l}p^\star_{l}/\sigma^2+1}
+1
\nonumber\\
&
\leq
-
\mathrm{ln}\left(\sum_{l=1,l\neq k}^K\frac{G_{k,l}p_{l}}{\sigma^2}+1
\right), \label{A1}
\end{align}
which is obtained from $-\mathrm{ln}(x')-\frac{1}{x'}(x-x')\leq-\mathrm{ln}(x)$ for any $(x,x')$ due to the convexity of $-\mathrm{ln}(x)$.
Adding $\mathrm{ln}\left(\sum_{l=1}^KG_{k,l}p_{l}/\sigma^2+1 \right)$ on both sides of \eqref{A1}, we obtain
\begin{align}
&\mathrm{ln}\left(\sum_{l=1}^K\frac{G_{k,l}p_{l}}{\sigma^2 }+1
\right)
-
\mathrm{ln}\left(\sum_{l=1,l\neq k}^K\frac{G_{k,l}p^\star_{l}}{\sigma^2}+1\right)
\nonumber\\
&
-\frac{\sum_{l=1,l\neq k}^KG_{k,l}p_{l}/\sigma^2+1}{\sum_{l=1,l\neq k}^KG_{k,l}p^\star_{l}/\sigma^2+1}
+1
\nonumber
\\
&
\leq
\mathrm{ln}\left(\sum_{l=1}^K\frac{G_{k,l}p_{l}}{\sigma^2}+1\right)
-
\mathrm{ln}\left(\sum_{l=1,l\neq k}^K\frac{G_{k,l}p_{l}}{\sigma^2}+1\right)
\nonumber
\\
&
=\mathrm{ln}\left(1+\frac{G_{k,k}p_{k}}{\sum_{l=1,l\neq k}^KG_{k,l}p_{l}+
\sigma^2}\right). \label{A2}
\end{align}
Putting the result of \eqref{A2} into $\widetilde{\Phi}_{m}(\mathbf{p}|\mathbf{p}^\star)$ in \eqref{Phi} and since $a_mx^{-b_m}$ is a decreasing function of $x$, we immediately prove
\begin{align}
&\widetilde{\Phi}_{m}(\mathbf{p}|\mathbf{p}^\star)
\nonumber\\
&\geq a_m\Bigg[
\sum_{k\in\mathcal{Y}_m} \frac{\xi BT}{D_m\mathrm{ln}2\,}
\mathrm{ln}\left(1+\frac{G_{k,k}p_{k}}{\sum_{l\neq k}G_{k,l}p_{l}+
\sigma^2}\right)+A_m
\Bigg]^{-b_m}
\nonumber\\
&
=
\Phi_{m}(\mathbf{p}).
\end{align}
To prove part (ii), we first notice that $\widetilde{\Phi}_m(\mathbf{p}|\mathbf{p}^{\star})=h_m\left( g_m(\mathbf{p}|\mathbf{p}^\star)\right)$ is a composition function of $h_m\circ g_m$, where
$h_m(x)=a_mx^{-b_m}$ and
\begin{align}
g_m(\mathbf{p}|\mathbf{p}^\star)
&=
\sum_{k\in\mathcal{Y}_m} \frac{\xi BT}{D_m\mathrm{ln}2\,}
\Bigg[
\mathrm{ln}\left(\sum_{l=1}^K\frac{G_{k,l}p_{l}}{\sigma^2}+1\right)
\nonumber\\
&\quad{}
-
\mathrm{ln}\left(\sum_{l=1,l\neq k}^K\frac{G_{k,l}p^\star_{l}}{\sigma^2 }+1\right)
\nonumber
\\&
\quad{}
-\frac{\sum_{l=1,l\neq k}^KG_{k,l}p_{l}/\sigma^2+1}{\sum_{l=1,l\neq k}^KG_{k,l}p^\star_{l}/\sigma^2+1}
+1
\Bigg]
+A_m. \label{B1}
\end{align}
Since $\nabla h_m(x)=-a_mb_mx^{-b_m-1}\leq 0$ and $\nabla^2 h_m(x)=a_mb_m(b_m+1)x^{-b_m-2}\geq0$, the function $h_m(x)$ is convex and nonincreasing.
Adding to the fact that $g_m(\mathbf{p}|\mathbf{p}^\star)$ is a concave function of $\mathbf{p}$, we immediately prove the convexity of $\widetilde{\Phi}_m$ using the composition rule \cite[Ch. 3, pp. 84]{opt1}.
Finally, to prove $\widetilde{\Phi}_{m}(\mathbf{p}^\star|\mathbf{p}^\star)=\Phi_{m}(\mathbf{p}^\star)$ and $\nabla_{\mathbf{p}}\widetilde{\Phi}_{m}(\mathbf{p}^\star|\mathbf{p}^\star)=\nabla_{\mathbf{p}}\Phi_{m}(\mathbf{p}^\star)$, we put $\mathbf{p}=\mathbf{p}^\star$ into the functions $\{\widetilde{\Phi}_{m},\Phi_m,\nabla_{\mathbf{p}}\widetilde{\Phi}_{m},\nabla_{\mathbf{p}}\Phi_{m}\}$, and we immediately obtain part (iii).
\section{Proof of Proposition 2}
To prove this proposition, the Lagrangian of $\mathrm{P}2$ is
\begin{align}
L&=\mu+\sum_{k=1}^K\nu_k\Bigg[\beta_ka_k\left(
\frac{\xi BT}{D_k}\mathrm{log}_2\left(1+\frac{G_{k,k}p_{k}}{\sigma^2} \right)+A_k
\right)^{-b_k}
\nonumber\\
&\quad{}
-\mu\Bigg]
+\chi\left(\mathop{\sum}_{k=1}^Kp_k-P\right)-\sum_{k=1}^K\theta_kp_k, \label{C0}
\end{align}
where $\{\nu_k,\chi,\theta_k\}$ are non-negative Lagrange multipliers.
According to the KKT conditions $\partial L/\partial \mu^*=0$ and $\partial L/\partial p_k^*=0$ \cite{opt1}, the optimal $\{\mu^*,\,p_k^*,\,\nu_k^*,\,\chi^*,\theta_k^*\}$ must together satisfy
\begin{align}
&1-\sum_{k=1}^K\nu_k^*=0,\quad
\chi^*+\nu_k^*\times F_k(p_k^*)
=\theta_k^*,\quad \forall k, \label{C1}
\end{align}
where
\begin{align}
F_k(x)&=-\frac{\beta_ka_kb_k\xi BT}{D_k\mathrm{ln}2\,(\sigma^2G_{k,k}^{-1}+x)}
\nonumber\\
&\quad{}
\times
\left[\frac{\xi BT}{D_k}\mathrm{log}_2\left(1+\frac{G_{k,k}x}{\sigma^2} \right)+A_k
\right]^{-b_k-1}
,
\end{align}
with $x\geq 0$ ($x\neq0$ if $A_k=0$).
Notice that $F_k(x)<0$ holds for any $x\geq 0$.
Based on the result of \eqref{C1}, it is clear that
$\sum_{k=1}^K\frac{\theta_k^*-\chi^*}{F_k(p_k^*)}=1$.
Adding to the fact that $\theta_k^*\geq 0$ and $F_k(p_k^*)<0$, we must have $\chi^*\neq 0$.
Now we will consider two cases.
\begin{itemize}
\item $p_k^*=0$. In this case, $\beta_ka_ku_k^{-b_k}\leq \mu^*$ must hold.
\item $p_k^*>0$.
In such a case, based on the complementary slackness condition, we must have $\theta_k^*=0$.
Putting $\theta_k^*=0$ into \eqref{C1} and using $\chi^*\neq 0$, $\nu_k^*\neq0$ holds.
Using $\nu_k^*\neq0$ and the complementary slackness condition, $\beta_ka_k\left(
\frac{\xi BT}{D_k}\mathrm{log}_2\left(1+\frac{G_{k,k}p_{k}^*}{\sigma^2} \right)+A_k
\right)^{-b_k}=\mu^*$ for all $k$.
\end{itemize}
Combining the above two cases gives \eqref{pk*} and the proposition is proved.
\section{Proof of Proposition 3}
To prove this proposition, we need the following lemma for $\nabla_{\mathbf{p}}\Xi_m(\mathbf{p})$.
\begin{lemma}
If $\mu\leq\mu_0$, the gradient function $\nabla_{\mathbf{p}}\Xi_m(\mathbf{p})$ satisfies the following:
\noindent(i) $\left\Vert\nabla_{\mathbf{p}}\Xi_m(\mathbf{p})\right\Vert_{2}\leq
L_{2}/\beta_m$;
\noindent(ii) $\left\Vert\nabla_{\mathbf{p}}\Xi_m(\mathbf{p})-\nabla_{\mathbf{p}}\Xi_m(\mathbf{p}')\right\Vert_{\infty}\leq
L_{1}/\beta_m\left\Vert\mathbf{p}-\mathbf{p}'\right\Vert_2$.
\end{lemma}
\begin{proof}
To begin with, the assumption $\mu=\mathrm{max}_m\beta_m\Xi_m(\mathbf{p})\leq \mu_0$ gives
\begin{align}
&\sum_{k\in\mathcal{Y}_m} \frac{\xi BT}{D_m}\mathrm{log}_2\left(1+\frac{G_{k,k}p_{k}}{\sigma^2} \right)+A_m
\geq\left(\frac{\beta_ma_m}{\mu_0}\right)^{\frac{1}{b_m}}. \label{lemma1}
\end{align}
Based on \eqref{lemma1} and the expression of $\partial \Xi_m/\partial p_j$ in \eqref{ximpartial}, we have
\begin{align}
\Big|\frac{\partial \Xi_m}{\partial p_j}\Big|
&\leq a_mb_m\left(\frac{\beta_ma_m}{\mu_0}\right)^{-1-1/b_m}
\frac{\xi BT\mathbb{I}_{\mathcal{Y}_m}(j)}{D_m\mathrm{ln}2\,(\sigma^2G_{j,j}^{-1}+p_j)}
\nonumber \\
&\leq
\frac{a_mb_m\xi BTG_{j,j}\mathbb{I}_{\mathcal{Y}_m}(j)}{D_m\mathrm{ln}2\,\sigma^2}
\left(\frac{\mu_0}{\beta_ma_m}\right)^{1+1/b_m},
\end{align}
where the second inequality is due to $p_j\geq 0$.
Putting the above result into \eqref{gradient}, and based on the definition of $L_{2}$ in \eqref{L12}, part (i) is immediately proved.
Next, to prove part (ii),
we notice that the derivative in \eqref{ximpartial} can be rewritten as $\nabla_{p_j}\Xi_m(\mathbf{p})=h_m(\mathbf{p})\times g_{m,j}(\mathbf{p})$, with the auxiliary functions
\begin{subequations}
\begin{align}
h_m(\mathbf{p})&=
-a_mb_m\Bigg[
\sum_{k\in\mathcal{Y}_m} \frac{\xi BT}{D_m}\mathrm{log}_2\left(1+\frac{G_{k,k}p_{k}}{\sigma^2} \right)
\nonumber\\
&\quad{}
+A_m
\Bigg]^{-b_m-1},
\\
g_{m,j}(\mathbf{p})&=\frac{\xi BT\mathbb{I}_{\mathcal{Y}_m}(j)}{D_m\mathrm{ln}2\,\,(\sigma^2G_{j,j}^{-1}+p_j)}.
\end{align}
\end{subequations}
Using the result in \eqref{lemma1} and due to $p_j\geq 0$, we have
\begin{align}
|h_m(\mathbf{p})|&\leq
a_mb_m\left(\frac{\mu_0}{\beta_ma_m}\right)^{1+1/b_m},
\nonumber\\
|g_{m,j}(\mathbf{p})|&\leq \frac{\xi BTG_{j,j}\mathbb{I}_{\mathcal{Y}_m}(j)}{D_m\mathrm{ln}2\,\sigma^2}.
\label{D3-1}
\end{align}
Furthermore, according to Lipschitz conditions \cite{BFOM2} of $h_m$ and $g_{m,j}$, they satisfy
\begin{subequations}
\begin{align}
|h_m(\mathbf{p})-h_m(\mathbf{p}')|
&\leq
\mathop{\mathrm{sup}}_{\mathbf{p}\in\mathcal{P}}\,\left\Vert\nabla_{\mathbf{p}}h_m(\mathbf{p})||_{2}\times||\mathbf{p}-\mathbf{p}'\right\Vert_2
\nonumber\\
&\leq
\frac{a_mb_m(b_m+1)\xi BTH_m}{D_m\mathrm{ln}2\,\sigma^2}
\nonumber\\
&\quad{}
\times
\left(\frac{\mu_0}{\beta_ma_m}\right)^{1+2/b_m}
\left\Vert\mathbf{p}-\mathbf{p}'\right\Vert_2, \label{D3-2a}
\\
|g_{m,j}(\mathbf{p})-g_{m,j}(\mathbf{p}')|&\leq
\mathop{\mathrm{sup}}_{\mathbf{p}\in\mathcal{P}}\,\left\Vert\nabla_{\mathbf{p}}g_{m,j}(\mathbf{p})\right\Vert_{2}\times\left\Vert\mathbf{p}-\mathbf{p}'\right\Vert_2
\nonumber\\
&
\leq
\frac{\xi BTG_{j,j}^2\mathbb{I}_{\mathcal{Y}_m}(j)}{D_m\mathrm{ln}2\,\sigma^4}\left\Vert\mathbf{p}-\mathbf{p}'\right\Vert_2. \label{D3-2b}
\end{align}
\end{subequations}
As a result, the following inequality is obtained:
\begin{align}
&|\nabla_{p_j}\Xi_m(\mathbf{p})-\nabla_{p_j}\Xi_m(\mathbf{p}')|
\nonumber\\
&
\leq
|h_m(\mathbf{p})|\,|g_{m,j}(\mathbf{p})-g_{m,j}(\mathbf{p}')|
\nonumber\\
&\quad{}
+|h_m(\mathbf{p})-h_m(\mathbf{p}')|\,|g_{m,j}(\mathbf{p}')| \nonumber
\\
&
\leq
\Bigg[
a_mb_m\left(\frac{\mu_0}{\beta_ma_m}\right)^{1+1/b_m}\frac{\xi BTG_{j,j}^2}{D_m\mathrm{ln}2\,\sigma^4}
+
\left(\frac{\mu_0}{\beta_ma_m}\right)^{1+2/b_m}
\nonumber\\
&\quad{}
\times
\frac{a_mb_m(b_m+1)\xi^2B^2T^2G_{j,j}H_m}{D_m^2\mathrm{ln}^22\sigma^4}\Bigg]
\left\Vert\mathbf{p}-\mathbf{p}'\right\Vert_2
\nonumber\\
&
\leq L_{1}/\beta_m\left\Vert\mathbf{p}-\mathbf{p}'\right\Vert_2
, \label{D4}
\end{align}
where the first inequality is due to $|ab+cd|\leq|a||b|+|c||d|$, and the second inequality is obtained from \eqref{D3-1} and \eqref{D3-2a}--\eqref{D3-2b}.
By taking the maximum of \eqref{D4} for all $j$, part (ii) is proved.
\end{proof}
Based on \textbf{Lemma 1}, we are now ready to prove the proposition.
In particular, according to \cite{BFOM1,BFOM2,BFOM3}, the function $\Upsilon(\bm{\alpha},\mathbf{p})$ is $(L_{1},L_{2},L_{2},0)$--smooth if and only if
\begin{subequations}
\begin{align}
&\left\Vert
\left[\mathop{\sum}_{m=1}^{M}\alpha_m\beta_m\nabla_\mathbf{p}\Xi_m(\mathbf{p})\right]
-
\left[\mathop{\sum}_{m=1}^{M}\alpha_m\beta_m\nabla_\mathbf{p}\Xi_m(\mathbf{p}')\right]
\right\Vert_{\infty}
\nonumber\\
&
\leq
L_{1}
\left\Vert\mathbf{p}-\mathbf{p}'\right\Vert_1,
\label{Lips1}
\\
&\left\Vert
\left[\mathop{\sum}_{m=1}^{M}\alpha_m\beta_m\nabla_\mathbf{p}\Xi_m(\mathbf{p})\right]
-
\left[\mathop{\sum}_{m=1}^{M}\alpha_m'\beta_m\nabla_\mathbf{p}\Xi_m(\mathbf{p})\right]
\right\Vert_{\infty}
\nonumber\\
&
\leq
L_{2}
\left\Vert\bm{\alpha}-\bm{\alpha}'\right\Vert_1, \label{Lips2}
\\
&\Big\Vert
\left[\beta_1\Xi_1(\mathbf{p}),\cdots,\beta_M\Xi_M(\mathbf{p}) \right]
\nonumber\\
&-\left[\beta_1\Xi_1(\mathbf{p}'),\cdots,\beta_M\Xi_M(\mathbf{p}') \right]
\Big\Vert_{\infty}
\leq
L_{2}
\left\Vert\mathbf{p}-\mathbf{p}'\right\Vert_1,
\label{Lips3}
\\
&\Big\Vert
\left[\beta_1\Xi_1(\mathbf{p}),\cdots,\beta_M\Xi_M(\mathbf{p}) \right]
\nonumber\\
&
-\left[\beta_1\Xi_1(\mathbf{p}),\cdots,\beta_M\Xi_M(\mathbf{p}) \right]
\Big\Vert_{\infty}
\leq
0\,
\left\Vert\bm{\alpha}-\bm{\alpha}'\right\Vert_1,
\label{Lips4}
\end{align}
\end{subequations}
for any $\mathbf{p}, \mathbf{p}'\in\mathcal{P}$ and $\bm{\alpha}, \bm{\alpha}'\in\mathcal{A}$.
To prove the above inequalities, we need to upper bound the left hand sides of \eqref{Lips1}--\eqref{Lips4} and compare the bounds with the right hand sides of \eqref{Lips1}--\eqref{Lips4}.
To this end, the left hand side of \eqref{Lips1} is bounded as
\begin{align}
&\left\Vert
\mathop{\sum}_{m=1}^{M}\alpha_m\beta_m\left[\nabla_\mathbf{p}\Xi_m(\mathbf{p})
-\nabla_\mathbf{p}\Xi_m(\mathbf{p}')\right]
\right\Vert_{\infty}
\nonumber\\
&\leq\mathop{\sum}_{m=1}^{M}\alpha_m\beta_m \left\Vert
\nabla_\mathbf{p}\Xi_m(\mathbf{p})
-\nabla_\mathbf{p}\Xi_m(\mathbf{p}')
\right\Vert_{\infty} \nonumber
\\
&
\leq
L_{1}\left\Vert\mathbf{p}-\mathbf{p}'\right\Vert_2,
\end{align}
where the last inequality is from \textbf{Lemma 1}.
Further due to $\left\Vert\mathbf{p}-\mathbf{p}'\right\Vert_2\leq\left\Vert\mathbf{p}-\mathbf{p}'\right\Vert_1$, the equation \eqref{Lips1} is proved.
On the other hand, the left hand side of \eqref{Lips2} is upper bounded as
\begin{align}
&\left\Vert
\mathop{\sum}_{m=1}^{M}(\alpha_m-\alpha_m')\beta_m\nabla_\mathbf{p}\Xi_m(\mathbf{p})
\right\Vert_{\infty}
\nonumber\\
&\leq\mathop{\sum}_{m=1}^{M}|\alpha_m-\alpha_m'| \left\Vert\beta_m
\nabla_\mathbf{p}\Xi_m(\mathbf{p})
\right\Vert_{\infty}
\nonumber\\
&
\leq
L_{2}
\left\Vert\bm{\alpha}-\bm{\alpha}'\right\Vert_1,
\end{align}
where the last inequality is from \textbf{Lemma 1}.
In addition, the left hand side of \eqref{Lips3} can be upper bounded as
\begin{align}
&\left\Vert
\left[\beta_1\Xi_1(\mathbf{p})-\beta_1\Xi_1(\mathbf{p}'),\cdots,\beta_M\Xi_M(\mathbf{p})-\beta_M\Xi_M(\mathbf{p}') \right]
\right\Vert_{\infty}
\nonumber\\
&\leq
\mathop{\mathrm{max}}_{m=1,\cdots,M}~\beta_m\times\mathop{\mathrm{sup}}_{\mathbf{p}\in\mathcal{P}}\,\left\Vert\nabla_{\mathbf{p}}\Xi_m(\mathbf{p})\right\Vert_{2}\times
\left\Vert\mathbf{p}-\mathbf{p}'\right\Vert_2
\nonumber\\
&\leq
L_{2}
\left\Vert\mathbf{p}-\mathbf{p}'\right\Vert_1.
\end{align}
Finally, since the left hand side of \eqref{Lips4} is zero, we immediately have that \eqref{Lips4} holds.
|
2210.14059
|
\section{Introduction} \label{sect_intro}
We present new results in the study of self-similar measures on
path space for infinite graph models,here called generalized
Bratteli diagrams. By this, we mean particular
graph systems $B$ with the property that the corresponding sets $V$
of vertices, and $E$ of edges, admit discrete level-structures,
defined below. This means that $V$ is a disjoint union of the
sets $V_n$
and $E$ is a disjoint union of the sets $E_n$. We emphasize that
the class of generalized Bratteli diagrams
include two cases: (1) all the sets $V_n, E_n$ are countable (if all
$V_n$'s are finite, then we have a standard Bratteli diagram);
(2) all the set $V_n, E_n$ are standard Borel spaces. In case (2),
we will say that $B$ is a measurable (or continuous) Bratteli
diagram.
Our present framework is motivated by, but more versatile than,
the more familiar setting
in earlier studies of Bratteli diagrams. We use the discrete levels in order to identify a
class of self-similar path space measures, called iterated function system (IFS) measures.
The latter in turn are inspired by earlier studies
of IFS systems arising the analysis of fractals, such as various
Sierpinski constructs, or conformal attractors. However, by
contrast our
analysis centers around a new path space analysis for
discrete-time
random walk models in generalized Bratteli diagrams. We also
discuss IFS measures for the case when the system of levels making
up $B$ are instead standard measure spaces, so non-discrete.
We recall that a measure $\mu$ on a standard Borel space $X$ is
called \textit{self-similar} with respect to an iterated function
system $f_i : X \to X, i \in I,$ if
$$
\mu = \sum_{i \in I} p_i \mu\circ f_i^{-1},
$$
where $p = (p_i)_{i\in I}$ is a probability vector and
$I$ is finite or countable.
A main aim of our paper is an identification of, and an analysis of,
iterated function system (IFS) measures on the path space $X_B$
of generalized
Bratteli diagrams. This entails two tasks, (a) an analysis of IFS
structure of the path spaces, and (b) a study of the particular
IFS measures on them.
Our main results consist of finding an explicit construction of
IFS measures for both discrete and measurable generalized
Bratteli diagrams.
In order to motivate, and to place this in context, we add the
following three comments here: (i) Previously, IFS measures have
been considered in special geometries of self-similar fractals (see
e.g., \cite{Jorgensen2018(book)} and the papers cited there). These
standard self-similar fractals may be realized in finite-dimensional
Euclidean space. By contrast, there are no previous studies of
IFS measures in path space, i.e., IFS measures realized in the class
of path space structures considered here, standard and
generalized Bratteli
diagrams. (ii) While there are earlier results on other, different
but related, classes of measures on path space of generalized
Bratteli diagrams, e.g., tail invariant measures, and Markov
measures, our present identification of IFS measures on path space
is new. (iii) In order to prepare the reader for the IFS path space
measures, it will be necessary for us to begin with an account of
tail invariant measures, and Markov measures. The tools involved
there for generalized Bratteli diagrams are also needed in our
introduction of the new IFS measures. But this means our main
results for IFS measures will be postponed to sections
\ref{s:discrete GBD} and \ref{sect MBD} below, after
the necessary preliminaries have been presented.
Generalized Bratteli diagrams considered here arise in various
areas and have many applications. We mention Cantor and Borel
dynamics where they are used to construct models of
transformations, see \cite{HermanPutnamSkau1992},
\cite{GiordanoPutnamSkau1995}, \cite{Durand2010},
\cite{BezuglyiKarpel2016}, \cite{BezuglyiDooleyKwiatkowski2006}.
Measurable Bratteli diagrams can be met in the theory of Markov
chains, we refer to \cite{Nummelin_1984} as an example of such
literature. Several other
applications come to mind,
(1) models from financial mathematics, and (2) neural networks. In
each case, the role of the IFS measures must be specified.
In some applications, there are important limit theorems,
for example for financial derivative models both discrete and
continuous pricing formulas are important. And it may be stated in
the setting of generalized Bratteli diagrams, measurable setting.
We mean first of all
financial derivative models, e.g., binomial models; vs
continuous/measurable (cm), e.g., pricing of options via Ito
calculus, see e.g., \cite{BrunickShreve2013},
\cite{KaratzasShreve1998}. In these applications, we also have
theorems to the effect that the continuous models are limits of
discrete counterparts. Typically, the limit arguments involve the
Central Limit Theorem from probability.
Another case of discrete models take the form of deep neural network
models, deep means a “large number of levels”, so many steps through
the levels in our diagrams.
The organization of the paper is the following. Section
\ref{s:Basics} contains the basic definitions related to the
concept of generalized Bratteli diagram (discrete case),
and the description of various classes of measures on the path
space of a generalized Bratteli diagrams. We consider
tail invariant, shift-invariant, and Markov measures and their
relations to a semibranching function system generated by a
stationary generalized Bratteli diagram.
In section \ref{s:discrete GBD}, we consider a semibranching
function system $\{\tau_e\}$ defined on the path space of a
stationary generalized Bratteli diagram and indexed by the
edge set $E$. We prove there one of the main results by giving
necessary and sufficient conditions on the existence of an
IFS measure for $\{\tau_e\}$, see Theorem \ref{thm IFS stat BD}.
Section \ref{sect MBD} focuses on
measurable Bratteli diagrams. Since this notion is relatively new,
we give detailed definitions and discuss the properties of such
diagrams. Then we prove a measurable version of the main theorem
about the existence of an IFS measure, Theorem \ref{thm IFS MBD}.
At the end of this introduction, we mention the literature
that may be interesting for the reader. The literature on standard
Bratteli diagrams their applications in
dynamics is very extensive. We mention here the following
pioneering papers
\cite{HermanPutnamSkau1992}, \cite{GiordanoPutnamSkau1995},
\cite{DurandHostSkau1999}, a recent book \cite{Putnam2018}, and
surveys \cite{Durand2010}, \cite{BezuglyiKarpel2016},
\cite{BezuglyiKarpel2020}.
Generalized Bratteli diagrams are less studied. The stationary
case uses the Perron-Frobenius theory for infinite matrices.
We refer to the book \cite{Kitchens1998} and the literature there.
These diagrams are discussed in \cite{BezuglyiJorgensen2022} and
\cite{Bezuglyi_Jorgensen_Sanadhya_2022}.
More references and numerous connections with other areas can be
found therein. In particular, the following papers on IFS measures
and fractals are related to the current paper
\cite{BratteliJorgensen1999}, \cite{BratteliJorgensen2002},
\cite{DutkayJorgensen2009}, \cite{DutkayJorgensen2010},
\cite{DutkayJorgensen2014a}, \cite{Jorgensen2006},
\cite{Jorgensen2018(book)}, \cite{RavierStrichartz_2016},
\cite{Alonso-Strichartz_2020}, \cite{CaoHassleretal_2021}.
\section{Basics on generalized Bratteli diagrams} \label{s:Basics}
We consider the fundamentals of path space for generalized Bratteli
diagrams in this section.
This notion was first introduced in
\cite{BezuglyiDooleyKwiatkowski2006} under
the name of Borel-Bratteli diagram. More detailed exposition of
this concept can be found in \cite{BezuglyiJorgensen2022}. For the
reader's convenience we give a concise version here.
\subsection{Main definitions}\label{ss_main def}
In the introduction, we described the notion of a Bratteli diagram
as an infinite graded graph. A natural extension
of this concept consists of consideration of diagrams with
countably infinite levels.
\begin{definition}\label{def GBD} (\textit{Generalized Bratteli
diagrams, vertices, edges, incidence matrices})
Let $V_0$ be a countable set (which can be identified with either
$\mathbb N$ or $\mathbb Z$ if necessary).
Set $V_i = V_0$ for all $i \geq 1$, and $V = \bigsqcup_{i=0}^\infty
V_i$.
A countable graded graph $B = (\mathcal{V, E})$ is called a
\textit{generalized Bratteli diagram}
if it has the following properties.
(i) The set of edges $\mathcal E$ of $B$ is represented as
$\bigsqcup_{i=0}^\infty
E_i$ where $E_i$ is the set of edges between the vertices of levels
$V_i$
and $V_{i+1}, i \geq 0$.
(ii) The set $E(w, v)$ of edges $e$ between the vertices $w \in V_i$
and $v \in V_{i+1}$ is finite (or empty).
Let $f^{(i)}_{v,w} = |E(w, v)|$ where $|\cdot |$ denotes the
cardinality of a set. It defines a sequence of infinite
(countable-by-countable)
\textit{incidence matrices} $(F_i : i \in \mathbb N_{0})$ with
entries $F_i = (f^{(i)}_{v,w} : v \in V_{i+1}, w\in V_i),\ \
f^{(i)}_{v,w} \in \mathbb N_0.$
(iii) It is required that the matrices
$F_i$ have at most \textit{finitely many non-zero entries} in each
row. In general, we do not impose any restrictions on the columns
of $F_i$.
(iv) The maps $r,s : E \to V$ are defined on the diagram $B$:
for every $e \in \mathcal E$, there are $w, v$ such that $e \in E(w, v)$;
then $s(e) =w$ and
$r(e) = v$. They are called the \textit{range} ($r$) and
\textit{source}
($s$) maps.
(v) For every $w \in V_i, \; i \geq 0$,
there exist an edge $e \in E_i$ such that $s(e) = w$; for every
$v\in V_i,\; i >1$, there exists an edge $e' \in
E_{i-1}$ such that $r(e') = w$. In other
words, every row and every column of the incidence matrix $F_i$ has
non-zero entries.
(vi) If all entries of incidence matrices $F_n$ are zero or ones,
the corresponding generalized Bratteli diagram is called a 0-1
diagram.
\end{definition}
\begin{remark}
(1) It follows from Definition \ref{def GBD} that the structure of
every generalized
Bratteli diagram is completely determined by a sequence of matrices
$(F_n)$ such that every matrix $F_n$ satisfies (iii) and (iv).
Indeed, the entry $f_{v,w}^{(n)}$ indicates the number of edges
between the vertex $w \in V_n$ and vertex
$v\in V_{n+1}$. It defines the set $E(w, v)$; then one takes
$$
E_n = \bigcup_{w\in V_n, v \in V_{n+1}} E(w, v).
$$
In this case, we write $B= B(F_n)$. If all $F_n = F$, the
corresponding generalized Bratteli diagram $B(F)$ is called
\textit{stationary.}
(2) If $V_0$ is a singleton, and each $V_n$ is a finite set,
then we obtain the standard definition of a Bratteli diagram
originated in
\cite{Bratteli1972}. Later it was used in the theory
of $C^*$-algebras and dynamical systems for solving important
classification problems of Cantor dynamics and construction of
various models of homeomorphisms of a Cantor set (for references,
see Introduction).
\end{remark}
\begin{definition}\label{def path space}
(\textit{Path space and cylinder sets})
A finite or infinite \textit{path} in a generalized
Bratteli diagram $B = (V,E)$ is a
sequence of edges $(e_i : i \geq 0)$ such that $r(e_i) = s(e_{i+1})$
for all $i\geq 0$.
Denote by $X_B$ the set of all infinite paths. Every finite path
$\overline e =
(e_0, \ldots , e_n)$ determines a \textit{cylinder subset}
$[\overline e]$ of $X_B$:
$$
[\overline e] := \{x = (x_i) \in X_B : x_0 = e_0, \ldots, x_n = e_n\}.
$$
The collection of all cylinder subsets forms a base of neighborhoods
for a
topology on $X_B$. In this topology, $X_B$ is a Polish
zero-dimensional space, and every cylinder set is clopen.
In general, $X_B$ is not locally compact.
But if the set $s^{-1}(v)$ is
finite for every vertex $v \in \mathcal V$, then the path space
$X_B$ is locally compact.
\end{definition}
In the following remark, we formulate several
statements about properties of generalized Bratteli diagrams and
their path spaces.
\begin{remark} \label{rem prop BD}
(1) If $x = (x_i)$ is a point in $X_B$, then it is obviously
represented as the intersection of clopen sets:
\begin{equation} \label{eq_ x as cylinders}
\{x\} = \bigcap_{n\geq 0} [\overline e]_n
\end{equation}
where $[\overline e]_n = [x_0, \ldots ,x_n]$.
(2) Define a metric on $X_B$ compatible with the clopen topology:
for $x = (x_i)$ and $y = (y_i)$ from $X_B$, we set
$$
\mathrm{dist}(x, y) = \frac{1}{2^N},\ \ \ N = \min\{i \in
\mathbb N_0 : x_i \neq y_i\}.
$$
(3) We will \textit{assume}
that the diagram $B$ is chosen so that the space $X_B$
\textit{has no isolated points}. This means that every column of the
incidence matrix $F_n, n \in \mathbb N_0,$ has more than one non-zero
entry.
(4) Let $\mathbbm 1 = (..., 1, 1,....)$ be the vector indexed by
$v\in V_0$ such that every entry equals 1. Define $H^{(n)} := F_{n-1}
\cdots F_0 \mathbbm 1$. Let $E(V_0, v), v \in V_n,$ denote the set
of all
finite paths between $V_0$ and a fixed vertex $v\in V_n$. Then
$H^{(n)}_v = |E(V_0, v)|$.
(5) Let $X_v^{(n)}$ be a subset of $X_B$ such that
\begin{equation}\label{eq_X_v}
X_v^{(n)} = \bigcup_{v_0 \in V_0}\bigcup_{\overline e \in E(v_0, v)}
[\overline e].
\end{equation}
For every level $V_n$, the collection $\{X_v^{(n)} : v \in V_n \}$
forms a partition $\zeta_n$ of $X_B$ into disjoint clopen sets.
Every set $X_v^{(n)}$ is a \textit{finite union} of cylinder sets.
The
number of the cylinder sets here is exactly $H^{(n)}_v$.
The sequence of partitions $(\zeta_n)$ is refining.
According to \eqref{eq_X_v}, the cylinder sets from all $X_v^{(n)}$
generate the topology (and Borel $\sigma$-algebra) on $X_B$.
\end{remark}
For a generalized Bratteli diagram $B$, define the
\textit{tail equivalence relation} $\mathcal R$ on the path space
$X_B$.
\begin{definition}\label{def tail}
(\textit{Tail equivalence relation})
It is said that two infinite paths $x = (x_i)$ and $y = (y_i)$ are
\textit{tail equivalent} if there
exists $m \in \mathbb N$ such that $x_i = y_i$ for all $i \geq m$. Let
$[x]_{\mathcal R} := \{ y \in X_B : (x,y) \in \mathcal R\}$
be the set of points
tail equivalent to $x$. We say that a point $x$ is
\textit{periodic}
if $| [x]_{\mathcal R} | < \infty$. If there is no periodic
points, then the tail equivalence relation is called
\textit{aperiodic}.
\end{definition}
Without loss of generality, we will consider generalized Bratteli
diagrams with aperiodic $\mathcal R$.
Clearly, $\mathcal R$ is a hyperfinite countable Borel equivalence
relation, see \cite{DoughertyJacksonKechris1994} for definitions.
\begin{definition}\label{def irreducible}
It is said that a generalized Bratteli diagram $B$ is
\textit{irreducible} if, for any two vertices $v$ and
$w$ and any $n \in \mathbb N_0$, there exists a level $V_m (m >n)$
such that $w \in V_n$ and $v\in V_m$
are connected by a finite path. This is equivalent to the property
that, for any fixed $v,w$, there exists $m \in \mathbb N$ such that the
product of matrices $F_{m-1} \cdots F_n$ has a non-zero
$(v,w)$-entry.
\end{definition}
\subsection{Measures on the path space of a Bratteli diagram}
\label{ss_measures}
In this subsection, we will consider two classes of Borel measures
on the path space $X_B$ of a generalized Bratteli diagram. They are
tail invariant measures and Markov measures.
\begin{definition}\label{def tail inv m}
(\textit{Tail equivalent measures})
Let $B = (\mathcal{\mathcal{V, E}})$ be a
generalized Bratteli diagram, and $X_B$ the path space of $B$.
A Borel measure $\mu$ on $X_B$ (finite or $\sigma$-finite) is
called \textit{tail invariant} if, for any two finite paths $\overline e$
and $\overline e'$ such that $r(\overline e) = r(\overline e')$, one has
\begin{equation}\label{eq_inv measures}
\mu([\overline e]) = \mu([\overline e']),
\end{equation}
where $[e]$ and $[e']$ denote the corresponding cylinder sets.
\end{definition}
If $\mu(X_B) = 1$, then the property of tail invariance means that
the probability to arrive
at a vertex $v \in V_n$ does not depend on a starting point $w \in
V_0$ and does not depend on the path connecting $w$ and $v$.
Let $\mu$ be a Borel tail invariant measure on $X_B$. Relation
\eqref{eq_inv measures} defines a sequence of
non-negative vectors $(\mu^{(n)})$ where $\mu^{(n)} =
( \mu^{(n)}_v : v \in V_n)$:
\begin{equation}\label{eq def og mu(n)}
\mu^{(n)}_v = \mu([\overline e]), \ \ \ \overline e\in E(V_0, v), \ v \in V_n.
\end{equation}
Because $\mu$ is tail invariant the value $\mu^{(n)}_v $ does
not depend on the choice of $\overline e\in E(V_0, v)$.
The following theorem is a key tool in the study of tail invariant
measures, see \cite{BezuglyiKwiatkowskiMedynetsSolomyak2010},
\cite{Durand2010}, \cite{BezuglyiKarpel2016},
\cite{BezuglyiJorgensen2022}.
\begin{theorem}\label{thm inv measures}
Let $B =(\mathcal V,\mathcal E)$ be a generalized Bratteli diagram defined
by a sequence $(F_n)$ of incidence matrices.
Let $\mu$ be a Borel probability measure on the path space $X_B$
of
$B$ which is tail invariant. Then the corresponding sequence of
vectors
$\mu^{(n)}$ (defined as in \eqref{eq def og mu(n)}) satisfies the
property
\begin{equation}\label{eq_inv meas via A_n}
A_n \mu^{(n+1)} = \mu^{(n)},
\end{equation}
where $A_n = F_n^T$ is the transpose of $F_n$.
Conversely, if a sequence of vectors $\mu^{(n)}$ satisfies
\eqref{eq_inv meas via A_n}, then it defines a unique tail
invariant measure $\mu$.
The theorem remains true for $\sigma$-finite measures $\nu$
satisfying the property
$\nu([\overline e]) < \infty$ for every cylinder set $[\overline e]$.
\end{theorem}
The other interesting class of measures on Bratteli diagrams is
Markov
measures. In the context of Bratteli diagrams, these measures were
considered in \cite{DooleyHamachi2003}, \cite{Renault2018},
\cite{BezuglyiJorgensen2022} and some other papers.
\begin{definition}\label{def Mark meas} (\textit{Markov measures})
Let $B = (\mathcal{V, E})$ be a generalized Bratteli diagram
constructed by
a sequence of incidence matrices $(F_n)$. Let $q = (q_{v})$ be a
strictly positive vector, $q_v >0, v\in V_0$, and let $(P_n)$
be a sequence of non-negative infinite matrices with entries
$(p^{(n)}_{v,e})$ where $v \in V_n, e \in E_{n}, n= 0, 1, 2,
\ldots $. To define a
\textit{Markov measure} $m$, we require that the sequence $(P_n)$
satisfies the following properties:
\begin{equation}\label{defn of P_n}
(a)\ \ p^{(n)}_{v,e} > 0 \ \Longleftrightarrow \ (s(e) = v); \ \ \ \
(b)\ \ \sum_{e : s(e) = v} p^{(n)}_{v,e} =1.
\end{equation}
Condition \eqref{defn of P_n}(a) shows that $p^{(n)}_{v,e}$ is
positive
only on the edges outgoing from the vertex $v$, and therefore the
matrices $P_n$ and $A_n =F_n^T$ share the same set of zero entries.
For any cylinder set $[\overline e] = [(e_0, e_1, \ldots , e_n)]$
generated by the path $\overline e$ with $v =s(e_0) \in V_0$, we set
\begin{equation}\label{eq m([e])}
m([\overline e]) = q_{s(e_0)}p^{(0)}_{s(e_0), e_0} \cdots
p^{(n)}_{s(e_n), e_n}.
\end{equation}
Relation \eqref{eq m([e])} defines the value of the measure $m$
of the set $[\overline e]$.
By \eqref{defn of P_n}(b), this measure satisfies the
\textit{Kolmogorov consistency condition} and can be extended to the
$\sigma$-algebra of Borel sets.
To emphasize that $m$ is generated by a sequence of
\textit{stochastic matrices}, we will also write $m = m(P_n)$.
If all stochastic matrices $P_n$ are equal to a matrix $P$, then
the corresponding measure $m(P)$ is \textit{called stationary Markov
measure.}
\end{definition}
We refer to \cite{BezuglyiJorgensen2022} for a detailed study of
Markov measures. We mention here only the following result.
\begin{theorem}\label{thm inv meas is Markov}
Let $\nu$ be a tail invariant probability
measure on the path space $X_B$ of a generalized Bratteli diagram
$B = (\mathcal{V, E})$. Then there
exists a sequence of Markov matrices $(P_n)$ such that $\nu =
m(P_n)$.
\end{theorem}
For stationary generalized Bratteli diagrams, we can find explicit
formulas for tail invariant measures. In the following statement,
we use the terminology from the Perron-Frobenius theory, see
\cite{Kitchens1998} for details.
\begin{theorem} \label{thm inv meas stat BD}
\cite{BezuglyiJorgensen2022}
(1) Let $B = B(F)$ be a stationary Bratteli diagram such
that the incidence matrix $F$ (and therefore $A = F^T$) is
irreducible, aperiodic, and recurrent. Let $t = (t_v : v \in V_0)$
be a right eigenvector corresponding to the Perron eigenvalue
$\lambda$ for $A$, $At = \lambda t$.
Then there exists a tail
invariant measure $\mu$ on the path space $X_B$ whose values on
cylinder sets are determined by the following rule: for every finite
path $\overline e(w, v)$ that begins at $w \in V_0$ and
terminates at $v \in V_n$, $n \in \mathbb N_0$, we set
\begin{equation}\label{eq inv meas stat BD}
\mu^{(n)}_v = \mu([\overline e(w, v)]) = \frac{t_v}{\lambda^{n}}.
\end{equation}
(2) The measure $\mu$ is finite if and only if the right eigenvector
$t = (t_v)$ has the property $\sum_v t_v <\infty$.
\end{theorem}
In particular, $\mu(X_w^{(0)}) = t_w, w \in V_0$, and
\begin{equation}
\mu(X_v^{(n)}) = H_v^{(n)} \frac{t_v}{\lambda^{n}}.
\end{equation}
\subsection{Semibranching function systems on Bratteli diagrams}
\label{subs s.b.s.}
Here we give the definition of a semibranching function system
following
\cite{MarcolliPaolucci2011} and \cite{BezuglyiJorgensen2015}.
This notion was used in the literature, in particular, for the
construction of representations
of Cuntz-Krieger algebras, see \cite{BezuglyiJorgensen2015},
\cite{FarsiJKP2018a}, \cite{FarsiJKP2018b}.
\begin{definition}\label{s.f.s.}
(\textit{Semibranching function systems and coding maps})
(1) Let $(X,\mu)$ be a probability
measure space with non-atomic measure $\mu$. We consider a family
$\{\sigma_i : i\in \Lambda\}$ of one-to-one $\mu$-measurable maps
indexed by a finite (or countable) set $\Lambda$.
The family $\{\sigma_i\}$ is called a \textit{semibranching
function
system (s.f.s.)} if the following conditions hold:
(i) $\sigma_i$ is defined on a subset $D_i$ of $X$ and takes values
in $R_i = \sigma_i(D_i)$ such that $\mu(R_i \cap R_j) = 0$ for
$i\neq j$ and $\mu(X\setminus \bigcup_{i\in \Lambda} R_i) = 0$;
(ii) the measure $\mu \circ \sigma_i$ is equivalent to $\mu$ and,
i.e.,
$$
\rho{_\mu}(x, \sigma_i) := \frac{d\mu\circ \sigma_i}{d\mu}(x) > 0
\ \ \mbox{for $\mu$-a.e. $x\in D_i$};
$$
(iii) there exists an endomorphism $\sigma : X \to X$ (called a
\textit{coding map}) such that $\sigma\circ \sigma_i(x) = x$ for
$\mu$-a.e. $x\in D_i, \ i \in \Lambda$.
If, additionally to properties (i) - (iii), we have $\bigcup_{i\in
\Lambda}D_i = X$ ($\mu$-a.e.), then the s.f.s. $\{\sigma_i : i\in
\Lambda\}$ is called \textit{saturated.}
(2) It is said that a saturated s.f.s. satisfies
\textit{condition C-K}
\footnote{C-K stands for Cuntz-Krieger.}
if for any $i\in \Lambda$ there exists a subset $\Lambda_i \subset
\Lambda$ such that up to a set of measure zero
$$
D_i = \bigcup_{j\in \Lambda_i} R_j.
$$
In this case, condition C-K defines a 0-1 matrix $\widetilde A$ by the
rule:
\begin{equation}\label{C-K defines A}
\widetilde a_{i,j} =1 \ \ \Longleftrightarrow \ \ j\in \Lambda_i, \ \ i \in
\Lambda.
\end{equation}
Then the matrix $\widetilde A$ is of the size $|\Lambda|\times |\Lambda|$.
\end{definition}
\textbf{Semibranching function system associated with a generalized
stationary Bratteli diagram}.
Let $B$ be a generalized stationary 0-1 Bratteli diagram. We
construct an s.f.s.
$\Sigma$ which is defined on the path space $X_B$ endowed with a
Markov measure $m$. As we will see below, this measure must
have some
additional properties to satisfy Definition \ref{s.f.s.}.
The role of the index set $\Lambda$ for this s.f.s.
is played by the edge set $E$ which is the set of edges between
any two consecutive levels of $B$. For
any $e \in E$, we denote
\begin{equation}\label{defn of D_e}
D_e = \{y = (y_i) \in X_B : s(y) = s(y_0) = r(e)\},
\end{equation}
\begin{equation}\label{defn of R_e}
R_e = \{y = (y_i) \in X_B : y_0 = e\}.
\end{equation}
We see that $D_e$ depends on $r(e)$ only so that $D_e = D_{e'}$
if and only if $r(e) = r(e')$.
The collection of maps $\{\tau_e : e \in E\}, \tau_e : D_e \to
R_e$, is defined by the formula
\begin{equation}\label{defn of sigma_e}
\tau_e (y) := (e, y_0, y_1, .... ), \quad y = (y_i).
\end{equation}
Since $s(y_0) = r(e)$, the map $\tau_e$ is well defined on $D_e$.
\begin{remark}\label{rem contractive}
We defined the metric $\mbox{dist}$ in Remark \ref{rem prop BD}.
It follows from the definition of $\tau_e$ that
$$
\mbox{dist}(\tau_e(x), \tau_e(y) ) = \frac{1}{2} \mbox{dist}
(x, y), \ e \in E,
$$
that is $\tau_e$ is a contractive map for every $e$.
\end{remark}
\begin{proposition}
The system $\{D_e, R_e, \tau_e : e \in E\}$ is a saturated
s.f.s. on the
path space $X_B$ of a generalized stationary Bratteli diagram
satisfying conditions (i), (iii) of Definition
\ref{s.f.s.} and the C-K condition.
\end{proposition}
\begin{proof}
Let $\sigma : X_B \to X_B$ be defined as follows:
for any $x = (x_i)_{i \geq 0} \in X_B$,
\begin{equation}\label{defn of sigma}
\sigma(x) := (x_1, x_2, ...)
\end{equation}
It follows from (\ref{defn of sigma_e}) and (\ref{defn of sigma})
that the map $\sigma$ is onto and
$$
\sigma\circ\tau_e(x) = x,\ \ \ x \in D_e;
$$
Hence $\sigma$ is a coding map.
We deduce from (\ref{defn of R_e}) that $\{R_e :
e\in E\}$ constitutes a partition of $X_B$ into clopen sets.
Relation
(\ref{defn of D_e}) implies that $\{\tau_e : e \in E\}$
is a saturated s.f.s. Moreover, we claim that it satisfies
condition C-K. Indeed,
\begin{equation}\label{checking C-K}
D_e = \bigcup_{f : s(f) = r(e)} R_f,\ \ \ e \in E,
\end{equation}
because $y = (y_i) \in D_e \ \Longleftrightarrow \ s(y_0) = r(e) \
\Longleftrightarrow \ \exists f = y_0\ \mbox{such\ that}\ y =
(f, y_1, ...) \ \Longleftrightarrow \ y \in
\bigcup_{f : s(f) = r(e)} R_f$. Thus, $\Lambda_e = \{f : s(f) =
r(e)\}$, see \eqref{C-K defines A}.
Relation (\ref{checking C-K}) shows that the non-zero entries of
the 0-1 matrix $\widetilde A$ from Definition \ref{s.f.s.} are defined by
the rule:
\begin{equation}\label{eq-matrix tilde A}
(\widetilde a_{e,f} =1) \ \Longleftrightarrow \ (s(f) = r(e)).
\end{equation}
We observe that the matrix $\widetilde A$ has the following property:
there are finitely many nonzero entries in every column of $\widetilde A$,
but the rows of $\widetilde A$ may contain infinitely many nonzero entries.
Next, we observe that $\sigma: X_B \to X_B$ is a finite-to-one
continuous map. Indeed,
$$
|\sigma^{-1}(x)| = |r^{-1}(r(x_1))| = \sum_{u\in V_0} f_{v,u}.
$$
The latter is finite by Definition \ref{def GBD}.
\end{proof}
Thus, it remains to find out under what conditions property (ii) of
Definition \ref{s.f.s.} holds. We consider here two classes: tail
invariant measures and Markov measures.
Let $m$ be a Borel probability measure
on $X_B$. Since $X_B$ is naturally partitioned
into a refining sequence of clopen partitions $\mathcal Q_n$
formed by cylinder sets of length $n$, we can apply de Possel's
theorem (see, for instance, \cite{ShilovGurevich1977}).
We have that for $m$-a.a $x$,
\begin{equation}\label{eq:Possel's}
\rho_m(x, \tau_e) = \lim_{n\to \infty} \frac{m(\tau_e([\overline
e(n)])}{m[\overline e(n)]}
\end{equation}
where $\{x\} = \bigcap_n [\overline e(n)]$.
\textit{Tail invariant measure}.
We first consider the case when $m$ is the tail invariant measure
$\mu$ determined in Theorem \ref{thm inv meas stat BD}. Let
$At = \lambda t$ where $t = (t_v)$.
If $\overline f = (f_0, f_1, ... , f_n) \in D_e$, then $r(\overline f)
= r(f_n)$ and
$\tau_e(\overline f) = (e, f_0, ... , f_n)$. By
(\ref{eq inv meas stat BD}), we have
$$
\mu([\overline f ]) = \frac{t_{r(\overline f)}}{\lambda^{n}},\ \ \
\mu(\tau_e([\overline f ])) =
\frac{t_{r(\overline f)}}{\lambda^{n+1}},
$$
and therefore
\begin{equation}\label{RN for mu}
\rho_{\mu}(x, \tau_e) = \lambda^{-1}.
\end{equation}
\textit{Stationary Markov measure}. Let $m$ be a stationary Markov
measure determined by a stochastic matrix $P$, $m = m(P)$ as in
\eqref{eq m([e])}. If $\{x\} =\bigcap_n [f(n)] \in D_e$, then
$$
m([\overline f(n)]) = q_{s(f_0)}p_{s(f_0), f_0} \cdots
p_{s(f_n), f_n},
$$
$$
m(\tau_e([\overline f(n)])) = q_{s(e)} p_{s(e), e}p_{s(f_0), f_0}
\cdots p_{s(f_n), f_n},
$$
and the Radon-Nikodym derivative can be found by
\begin{equation}\label{RN for nu}
\rho_{m}(x, \tau_e) = \frac{q_{s(e)} p_{s(e),e}} {q_{s(f_0)}}.
\end{equation}
It follows from (\ref{RN for mu}) and (\ref{RN for nu}) that
$\rho_{m}$ is positive on $D_e$ if and only if all entries of the
vector $q = (q_v)$ are positive. The latter means that the support
of $m$ is the entire space $X_B$.
\textit{Non-stationary Markov measure}. In this case, we need
some additional conditions to guarantee that the Radon-Nikodym
derivative $\rho_m(x, \tau_e)$ is positive on $D_e$. As above, we
represent $x = (x_i)\in D_e$ by means of the sequence
$[\overline f(n)] = [(f_0, f_1, ... , f_n)]$ such that $x_i = f_i,
i= 0,1, ... ,n$, and find
$$
m([f(n)]) = q_{s(f_0)} p^{(0)}_{s(f_0), f_0} \cdots
p^{(n)}_{s(f_n), f_n}
$$
and
$$
m(\tau_e([f(n)])) = q_{s(e)} p^{(0)}_{s(e), e}p^{(1)}_{s(f_0),
f_0} \cdots p^{(n+1)}_{s(f_n), f_n}.
$$
A direct computation gives the following result.
\begin{lemma}\label{RN for m}
Let $m$ be a Markov measure on the path space of a
generalized stationary 0-1
Bratteli diagram. Then $\rho_m(x, \tau_e) >0$ on $D_e$
if and only if
\begin{equation}\label{product convergence}
0< \prod_{i=1}^\infty \frac{p^{(i+1)}_{s(f_i),
f_i}}{p^{(i)}_{s(f_i), f_i}} <\infty
\end{equation}
for any $x = \bigcap_n[\overline f(n)] \in D_e$
\end{lemma}
A Markov measure satisfying (\ref{product convergence}) is called a
\textit{quasi-stationary measure}. Condition
(\ref{product convergence}) appeared first in
\cite{DutkayJorgensen2014}, \cite{DutkayJorgensen2015} in a
different context.
We summarize the above results in the following theorem.
\begin{theorem}\label{example summary}
Given a generalized stationary 0-1 Bratteli diagram $B$ with the
edge set $E$, the collection of maps $\{\tau_e : D_e \to R_e\},\
e \in E$, defined in \eqref{defn of sigma_e} on the space $(X_B,
m)$, forms a saturated s.f.s. $\Sigma$ satisfying C-K condition
where the Markov measure $m$ is either the tail invariant measure
$\mu$, or a stationary Markov measure $m(P)$ of
full support, or a quasi-stationary measure Markov measure of full
support.
\end{theorem}
\begin{remark}
Let $B$ be a generalized stationary Bratteli diagram of
\textit{bounded size}, see \cite{BJKS_2022} for the definition.
In particular, if the incidence matrix $A$ is banded, then $B$ is
of bounded size. In this case, the definition of the
Cuntz-Krieger algebra
$\mathcal O_A$ can be given similar to the case of finite matrices.
Then we can use the methods developed in
\cite[Theorem 4.12]{BezuglyiJorgensen2015}
to construct a representation of the Cuntz-Krieger algebra
$\mathcal O_A$ generated by the s.f.s. defined in
Theorem \ref{example summary}. We omit the details.
\end{remark}
\subsection{Shift invariant measures on stationary Bratteli
diagrams}\label{ss_shift inv}
Let $B =(\mathcal{V,E})$ be a generalized stationary Bratteli
diagram defined
by the incidence infinite matrix $F$ and $A = F^T$.
In this subsection, we discuss $\sigma$-invariant measures on
$X_B$. We will consider two cases, tail invariant measures and
Markov measures.
Recall that, for every $v \in V_1$, the row sum $H^{(1)}_v =
\sum_{w \in V_0} f_{v,w}$ is
finite, see Remark \ref{rem prop BD} for notation.
In \eqref{defn of sigma}, we defined a finite-to-one endomorphism
$\sigma$ acting on the path space $X_B$. For $x = (e_0. e_1, ...)$,
we have
\begin{equation} \label{eq preimage for sigma}
\sigma^{-1}(x) = \{y = (y_i) \in X_B : r(y_0) = s(e_0), y_i =
e_{i-1}, i \geq 1\}
\end{equation}
and $|\sigma^{-1}(x)| = H^{(1)}_{s(e_0)}$.
\textit{Tail invariant measures}. There is a special case of
a generalized stationary Bratteli diagram such that the
tail invariant measure is also shift-invariant.
\begin{proposition}
Let the matrix $A$ has a Perron-Frobenius eigenpair $(\xi,\lambda)$.
Let $\mu$ be the tail invariant measure on $X_B$ defined as in
Theorem \ref{thm inv meas stat BD}. Then $\mu$ is $\sigma$-invariant
if and only if $H^{(1)}_v = \lambda$ for all $v\in V_0$.
\end{proposition}
\begin{proof} For a cylinder set $[\overline e] = [e_0, e_1, ...,
e_n]$, calculate $m([\overline e])$ and $m(\sigma^{-1}([\overline e]))$.
it follows from \eqref{eq preimage for sigma} that
\begin{equation}
\sigma^{-1}([\overline e]) = \bigcup_{f : r(f) =s(e_0)}
[f, e_0, ..., e_n] = \bigcup_{f : r(f) =s(e_0)} [f, \overline e]
\end{equation}
and this union is disjoint. Let $\lambda$ and $\xi$ be a
Peron-Frobenius eigenpair, $A\xi = \lambda\xi$.
By Theorem \ref{thm inv meas stat BD}, we have
$$
\mu([\overline e]) = \frac{\xi_v}{\lambda^n}, \qquad
\mu([f, \overline e]) = \frac{\xi_v}{\lambda^{n+1}},\ v = r(\overline e)
$$
Then
$$
\mu(\sigma^{-1}[\overline e]) = \sum_{f : r(f) = s(e_0)}
\mu ([f, \overline e]) = \sum_{f : r(f) = s(e_0)}
\frac{\xi_v}{\lambda^{n+1}} = |\sigma^{-1} ([\overline e])|
\frac{\xi_v}{\lambda^{n+1}} = H^{(1)}_{w}
\frac{\xi_v}{\lambda^{n+1}}
$$
where $w = s(e_0) \in V_1$. By de Possel's theorem
\eqref{eq:Possel's},
$$
\frac{d\mu\circ\sigma^{-1}}{d\mu}(x) = \lim_{n \to\infty}
\frac{\mu\circ\sigma^{-1}(\overline e]_n)}{\mu([\overline e]_n)} =
H^{(1)}_{w} \lambda^{-1}
$$
where $x = \bigcap_n [\overline e]_n$. Therefore,
$$
\mu\circ\sigma^{-1} = \mu \ \Longleftrightarrow \
H^{(1)}_{w} = \sum_{u \in V_0} f_{w, u} = \lambda, \ \forall
w \in V_0.
$$
\end{proof}
\begin{remark}
If $\max\{ H^{(1)}_v : v \in V_1\} = M <\infty$, then
$$
\lambda^{-1} \leq \frac{d\mu\circ\sigma^{-1}}{d\mu}(x) \leq
M\lambda^{-1}.
$$
This means that $\mu$ is equivalent to a $\sigma$-invariant measure
$\mu'$. In particular, this is true for bounded size diagrams.
\end{remark}
\vskip 2mm
\textit{Markov measures}. Consider first a stationary Markov measure
on the path space of a generalized stationary Bratteli diagram,
see Definition \ref{def Mark meas}. For simplicity we will work
with a $0-1$ diagram; the general case is considered similarly.
Let $q = (q_v : v \in V_0)$ be a positive probability vector
(called initial distribution). Let $P$ be a Markov matrix. For
a cylinder set $[\overline e] = [e_0, e_1,..., e_n]$, we use
\eqref{eq m([e])} to determine the value of the corresponding
Markov measure $m(P)$.
\begin{theorem}
For $B, q, P, m(P)$ as above, the measure $m = m(P)$ is
$\sigma$-invariant if and only if $q P = P$.
\end{theorem}
\begin{proof} Suppose that $q P = q$.
We know that $m([\overline e]) = q_{s(e_0)}p_{s(e_0), r(e_0)} \cdots
p_{s(e_n), r(e_n)}$ and
\begin{equation} \label{eq: qP = q}
\begin{aligned}
m(\sigma^{-1}[\overline e]) = & \sum_{f : r(f) = s(e_0)}
m([f, \overline e]) \\
= & \sum_{f : r(f) = s(e_0)} q_{s(f)} p_{s(f), r(f)}
p_{s(e_0), r(e_0)} \cdots p_{s(e_n), r(e_n)} \\
=& q_{r(f)} p_{s(e_0), r(e_0)} \cdots p_{s(e_n), r(e_n)} \\
=& m([\overline e])
\end{aligned}
\end{equation}
because $q_{r(f)} = q_{s(e_0)}$ and $q$ is $P$-invariant. Since
$\overline e$ is arbitrary, we see that $m$ is $\sigma$-invariant.
The converse statement follows from \eqref{eq: qP = q}: the equality
$m([\overline e]) = m(\sigma^{-1}[\overline e])$ implies $q P =q$.
\end{proof}
It remains to consider a non-stationary Markov measure $m$ defined
on the path space of a generalized stationary Bratteli diagram $B$.
In this case, the measure $m$ is defined by a sequence of Markov
matrices $(P_n)$ and an initial distribution $q = (q_v)$:
$$
m([e_0, ..., e_n]) = q_{s(e_0)}p^{(0)}_{s(e_0), r(e_0)} \cdots
p^{(n)}_{s(e_n), r(e_n)}.
$$
\begin{proposition}
Suppose that $q P_0 =q$. Then the measures $m = m(P_n)$ is
$\sigma$-invariant if and only if
$$
\prod_{i=1}^\infty \frac{p^{(i+1)}_{s(e_i), r(e_i)}}
{p^{(i)}_{s(i_i), r(e_i)}} = 1.
$$
\end{proposition}
\begin{proof}
Using the same technique, we can show that, under the condition
$q P = q$,
$$
\frac{d\mu\circ\sigma^{-1}}{d\mu}(x) = \prod_{i=1}^\infty
\frac{p^{(i+1)}_{s(e_i), r(e_i)}} {p^{(i)}_{s(i_i), r(e_i)}}
$$
where $x = \bigcap_n [e_0, ..., e_n$. We omit the details.
\end{proof}
\section{IFS measures on discrete generalized Bratteli diagrams}
\label{s:discrete GBD}
In this section, we consider the notion of \textit{iterated
function system (IFS)}. This concept have been discussed in many
books and papers. We mention here only several related references
such as \cite{Hutchinson1995},
\cite{HutchinsonRueschendorf1998}, \cite{Jorgensen2006},
\cite{Barnsley2006}, \cite{Jorgensen2018(book)},
\cite{DutkayJorgensen2007}, \cite{DutkayJorgensen2009},
\cite{JKS_2011}, \cite{MorrisSert2021}.
Our goal is to describe IFS measures defined on the path space of
a generalized Bratteli diagram.
\subsection{Iterated function systems and measures}
By an \textit{endomorphism} $\sigma$ we mean a finite-to-one
(or countable-to-one) Borel map of a standard Borel space $(X, \mathcal B)$
onto itself. In this case, there exists a
family of one-to-one maps $\{\tau_i\}_{i\in \Lambda}$ such that
$\tau_i : X \to X$ and $\sigma \circ \tau_i = \mbox{id}_{X}$ where
$\Lambda$ is at most countable. The maps
$\tau_i$ are called the \emph{inverse branches} for $\sigma$. The
collection of maps $(\tau_i : 1 \in \Lambda)$ gives an
example of \textit{iterated function system (IFS)}.
In general, an IFS
is defined by a collection of maps $\{\tau_i : i\in \Lambda\}$
of a complete metric space (or a compact space) such that
$\tau_i$'s are continuous (or even contractions). In this work, we
focus on the following features of the family $\{\tau_i : i \in
\Lambda\}$: (i) each $\tau_i$ is a one-to-one map defined on a
subset $D_i$ of $X$, and (ii) there exists an endomorphism $\sigma$
of $X$ such that $\sigma\circ\tau_i = \mathrm{id}_X$ for each $i$.
The general theory of \textit{infinite iterated function systems} is
more detailed and requires additional assumptions
(see, for example, the expository article \cite{Mauldin1998}).
As an example, one can consider the Gauss map
$x \mapsto \{1/x\}$ of the unit interval
with the piecewise monotone inverse branches $\tau_i : 1/(i+x)$
defined on $(1/(i+1), 1/i)$.
\begin{definition}\label{def_IFS measure} (\textit{IFS measures})
Suppose that
$(\tau_i : i \in \Lambda)$ is a given IFS on a Borel space
$(X, \mathcal B)$. Let $p = (p_i : i \in \Lambda)$ be a strictly positive
vector indexed by a countable set $\Lambda$.
A measure $\mu_p$ on $(X, \mathcal B)$ is called an \emph{IFS measure}
for $(\tau_i)$ if
\begin{equation}\label{eq IFS measure def}
\mu_p = \sum_{i\in \Lambda} p_i\; \mu_p\circ \tau_i^{-1},
\end{equation}
or, equivalently,
$$
\int_X f(x)\; d\mu_p(x) = \sum_{i\in \Lambda} p_i
\int_X f(\tau_i(x))\; d\mu_p(x), \ \qquad f\in L^1(\mu_p).
$$
\end{definition}
The main tool in the study of an IFS $(\tau_i : i \in \Lambda)$ on a
space $X$ is a realization of the IFS as the full one-sided shift
$S$ on a symbolic product space $\Omega$. Then any $S$-invariant and
ergodic product-measure $\mathbb P$ on $\Omega$ can be pulled back
to $X$. This construction gives ergodic invariant measures for IFSs.
In the next subsection, we will find an explicit method for
construction of an IFS measure on the path space of a generalized
stationary Bratteli diagram.
We discuss here a method that leads to construction of IFS measures.
This approach works perfectly for many specific applications under
some additional conditions on $X$ and maps $\tau_i$.
Suppose that $(X; \tau_i, i \in \lambda)$ is a given IFS.
Let $\Omega$ be the infinite direct product
$$
\Omega = \prod_{i\in \mathbb N_0} \Lambda_i,\qquad \Lambda_i = \Lambda.
$$
For $\omega = (\omega_1, \omega_2, ...) \in
\Omega$, let $\omega|_n $ denote the finite word $(\omega_0, ... ,
\omega_n)$. Then, $\omega|_n$ defines a map $\tau_{\omega|_n}$
acting on $X$ by the formula:
$$
\tau_{\omega|_n} (x) := \tau_{\omega_0} \cdots \tau_{\omega_n}
(x), \qquad x \in X, \ \ \omega \in \Omega, \ n \in \mathbb N_0.
$$
It is said that $\Omega$ is an \emph{encoding space} if, for every
$\omega\in \Omega$,
\begin{equation}\label{eq singleton}
F(\omega) = \bigcap_{n \geq 1}\tau_{\omega|_n}(X)
\end{equation}
is a singleton. Relation \eqref{eq singleton} defines a Borel map
$F : \Omega \to F(\Omega)$.
If each $\tau_i$ is a contraction and $X$ is a complete metric,
then a coding map $F :\Omega\to X$ always exists.
Let $S$ be the left shift on $\Omega$:
$$
S(\omega_0, \omega_1, ... ) = (\omega_1, \omega_2, ... ).
$$
The inverse branches $s_i, i \in \Lambda,$ of $S$ are
$$
s_i (\omega_0, \omega_1, ...) = (i, \omega_0, \omega_1, ...).
$$
Clearly,
$$
s_i (\Omega) = C(i) = \{\omega \in \Omega : \omega_0 = i\},
$$
and the space $\Omega $ is partitioned into the sets $C(i), \in
\Lambda$.
Let $p = (p_i : i \in \Lambda)$ be a positive probability vector.
It defines the product measure $\mathbb P = p \times p \times \cdots$
on $\Omega$.
We observe that the maps $(s_i : i\in \Lambda)$ constitute
an IFS on $\Omega$ such that $\mathbb P$ is an IFS measure:
\begin{equation}\label{eq P is IFS}
\mathbb P = \sum_{i\in \Lambda} p_i \; \mathbb P\circ s_i^{-1}.
\end{equation}
The following result shows how IFS measures arise.
\begin{proposition} Suppose that $(X; \tau_i, i\in \Lambda)$
is an IFS that admits a coding map $F : \Omega \to X$. Let
$p = (p_i)$ be a probability vector generating the product measure
$\mathbb P = p \times p \times \cdots$. Then the measure
$\mu := \mathbb P\circ F^{-1}$ is an IFS measure satisfying
$$
\mu = \sum_{i=1}^N p_i \mu\circ \tau_i^{-1}.
$$
Moreover, if $F$ is continuous, then $\mu$ has full support.
\end{proposition}
The proof can be found in \cite{BezuglyiJorgensen2018(book)}.
In Section \ref{sect MBD}, we will consider a analogue of the above
construction for measurable Bratteli diagrams.
\subsection{IFS measures on generalized stationary Bratteli
diagrams}
In this subsection we discuss an explicit formula for an IFS measure
on the path space of a generalized stationary 0-1 Bratteli diagram
$B = (\mathcal V,\mathcal E)$.
In the following remark, we show that the requirement to be
a 0-1 Bratteli diagram is not restrictive.
\begin{remark}
Every generalized stationary Bratteli diagram $B= (\mathcal V, \mathcal E)$
can be represented as a 0-1
Bratteli diagram. Indeed, we can use the matrix $\widetilde A$
defined in \eqref{eq-matrix tilde A} to built a new generalized
Bratteli diagram $\widetilde B = (\widetilde V, \widetilde E)$.
Note that $\widetilde A$ is a 0-1 matrix such
that every row of $\widetilde F = \widetilde A^T$ has finitely many non-zero
entries so that $\widetilde F$ can be viewed as an incidence matrix
of $\widetilde B$. In this case, the set of vertices $\widetilde V$ for each
level is coincides with $E$ and two vertices $\widetilde v =f \in E$ and
$\widetilde w = f \in E$ are connected by a singe edge if and only if
$r(e) = s(f)$. Moreover, the path spaces $X_{\widetilde B}$ coincides
with $X_B$.
The above argument can be applied to any non-stationary
generalized Bratteli diagram.
\end{remark}
We can now apply the results of Subsection \ref{subs s.b.s.}
and construct an s.f.s. $\Psi = (X_B; \tau_e, e \in E)$ where
$\tau_e : D_e \to R_e$ is defined in \eqref{defn of sigma_e}.
\begin{theorem}\label{thm IFS stat BD}
Let $B = (\mathcal V,\mathcal E)$ be a generalized stationary
0-1 Bratteli diagram, $p = (p_e : e \in E)$ a probability vector,
and $\Psi$ an s.f.s. defined by $\tau_e : D_e \to R_e$, $e \in E$.
We identify every edge
$e \in E$ with the pair of vertices $(s(e), r(e))$.
Let $\nu$ be a Borel probability full measure on $X_B$; we define a
probability positive vector $q = (q_v : v \in V_0)$ by setting
$q_v = \nu([w])$ where $[w] = \{ x in X_B : s(x) = w\}$.
Then
\begin{equation}\label{eq_meas nu}
\nu = \sum_{e\in E} p_e \; \nu\circ \tau_e^{-1}
\end{equation}
if and only if
\begin{equation}\label{eq: q =qP}
q_w = \sum_{w \in V_0} p_{w,v} q_v = \sum_{e : s(e) = w}
p_{s(e), r(e)} q_{r(e)},
\end{equation}
that is $Pq = q$ where the matrix $P$ has the entries
$(p_{s(e), r(e)} : e \in E) $.
\end{theorem}
\begin{proof} We use in the proof notation from \eqref{defn of D_e}
- \eqref{checking C-K}.
We note that relation \eqref{eq_meas nu} will be proved if we
show that it holds for any cylinder set $C \subset X_B$. Recall
that
it follows from the definition of $\tau_e$ that $\tau_e^{-1}$ is
uniquely determined on $R_e$ and $\tau_e^{-1}(e, y_1, y_2, ...) =
( y_1, y_2, ...)$.
Let $q = (q_w) >0$ be a solution to $q = P q$. Define
$\nu([w]) = q_w$, $w\in V_0$.
For any edge $f \in E$ and the corresponding cylinder set $[f]$,
we set
\begin{equation}\label{eq:nu([f])}
\nu([f]) = q_{r(f)}p_f.
\end{equation}
Check that this definition of $\nu$ satisfies \eqref{eq_meas nu}
for the cylinder sets $[f]$. Indeed, since $\tau_e^{-1}$ is defined
only on $R_e = [e]$ and $\tau_f^{-1}([f]) = [r(f)]$, we have
$$
\sum_{e \in E} p_e \nu\circ\tau_e^{-1}([f]) = p_f
\nu\circ\tau_f^{-1}([f]) = p_f \nu([r(f)]) = q_{r(f)}p_f
= \nu([f]).
$$
Then, by induction, we define the values of $\nu$ on all cylinder
sets of length $n$:
$$
\nu([f_0, ..., f_{n-1}]) := p_{f_0}\cdots p_{f_{n-1}}
q_{r(f_{n-1})}.
$$
Verify that this definition satisfies the Kolmogorov extension
theorem. Because
$$
[f_0, ..., f_{n-1}] = \bigcup_{e : s(e) = r(f_{n-1})}
[f_0, ..., f_{n-1}, e],
$$
we find that
$$
\begin{aligned}
\sum_{e : s(e) = r(f_{n-1})} \nu([f_0, ..., f_{n-1}, e]) = &
\sum_{e : s(e) = r(f_{n-1})} p_{f_0}\cdots p_{f_{n-1}}p_e q_{r(e)}\\
= & p_{f_0}\cdots p_{f_{n-1}} \sum_{e : s(e) = r(f_{n-1})}
p_e q_{r(e)}\\
= & p_{f_0}\cdots p_{f_{n-1}} q_{r(f_{n-1})}\\
= & \nu([f_0, ..., f_{n-1}])
\end{aligned}
$$
(we used here \eqref{eq: q =qP}).
It remains to show that \eqref{eq_meas nu} holds for
any cylinder $[f_0, ..., f_{n}]$. Indeed,
$$
\begin{aligned}
\sum_{e \in E} p_e \nu\circ\tau_e^{-1}([f_0, ..., f_{n}]) =&
p_{f_0} \nu([f_1, ..., f_{n}]) \\
=& p_{f_0} p_{f_1}\cdots p_{f_{n}} q_{r(f_{n})} \\
=& \nu([f_0, ..., f_{n}]).
\end{aligned}
$$
Remark that the condition $P q = q$ is used to check that the
measure $\nu$ defined inductively on cylinder sets can be extended
to all Borel sets.
\end{proof}
\begin{theorem}\label{thm-IFS-inv}
Let $B$ be an irreducible generalized stationary Bratteli diagram,
and let $\nu$ be the IFS
measure defined in Theorem \ref{thm IFS stat BD}. Then:
(i) $\nu$ is not tail invariant;
(ii) $\nu$ is shift-invariant.
\end{theorem}
\begin{proof}
(i) Suppose that $\nu$ is tail invariant. Let $f,e \in E$ be two
edges with $r(f) = r(e)$, i.e., $f, e$ are tail-equivalent.
Then it follows from \eqref{eq:nu([f])} that $\nu([f]) = \nu([e])$
or
$$
q_{r(f)} p_f = q_{r(e)} p_e.
$$
This means that $p_f = p_e$. In other words, the matrix
$$
P = (p_{w,v}) : e = (w, v)\in E)
$$
has constant columns.
Consider two cylinder sets
$[f_0, f_1]$ and $[e_0, e_1]$ such that $r(f_1) = r(e_1)$ and
there exist edges $g, h$ satisfying the properties:
$s(g) = s(h)$, $r(g) = r_{f_0}$, and $r(h) = r_{e_0}$.
From the tail invariance of $\nu$ and the case considered above,
we obtain that $p_{f_1} = p_{e_1}$ and
$$
q_{r(f_1)} p_{f_0}p_{f_1} = q_{r(e_1)} p_{e_0}p_{e_1}
$$
Hence, $p_{f_0} = p_{e_0}$ and therefore $p_g = p_h$. This proves
that the rows of the matrix $P$ are constant. But the vector
$ (p_e : e \in E)$ is probability, contradiction.
\\
(ii) Let $\varphi(x)$ be a bounded positive Borel function on the
path space $X_B$. For the left shift $\sigma$ on $X_B$, compute
\begin{equation}
\begin{aligned}
\int_{X_B} \varphi(x) \; d\nu\circ\sigma^{-1}(x) = &
\int_{X_B} \varphi(\sigma x) \; d\nu(x) \\
=& \sum_{e\in E} \int_{X_B} p_e\varphi(\sigma x) \; d\nu\circ
\tau_e^{-1}(x) \\
= & \sum_{e\in E} \int_{X_B} p_e\varphi(\sigma (\tau_e x)) \;
d\nu(x) \\
= & \sum_{e\in E} \int_{X_B} p_e\varphi(x) \; d\nu(x) \\
= & \int_{X_B} \varphi(x) \; d\nu(x).
\end{aligned}
\end{equation}
This proves that $\nu\circ \sigma^{-1} = \nu$.
\end{proof}
\section{Measurable Bratteli diagrams and IFS measures}
\label{sect MBD}
In this section, we discuss a measurable analogue of generalized
Bratteli diagrams. The principal differences between
discrete and measurable generalized Bratteli diagrams are:
(a) the levels
$V_n$ of a measurable Bratteli diagram are formed by standard Borel
spaces $(X_n, \mathcal A_n)$, and (b) the sets of edges $E_n$ are Borel
subsets of $X_n \times X_{n+1}$. Because these objects are
non-discrete, we need to use new methods and techniques.
\subsection{Measurable Bratteli diagrams and path space measures}
\label{ss MBD-IFS}
We give the definitions of main objects in this subsection.
\begin{definition}\label{def meas BD}
(\textit{Measurable Bratteli diagrams})
Let $\{(X_n, \mathcal A_n) : n \in \mathbb N_0\}$ be a sequence of standard Borel
spaces. Let $\{E_n : n \in N_0\}$ be a
sequence of Borel subsets such that $ E_n \subset
X_n \times X_{n+1}$. Denote by $\mathcal E = \bigsqcup_{i\geq 0}
E_i$ and $\mathcal X = \bigsqcup_{i\geq 0} X_i$ the sets of
``edges'' and ``vertices'', respectively.
For $e = (x,y) \in E_i$, the maps $s_i(x,y) = x$ and $r_i(x,y) =y$
are
onto projections of $E_i$ to $X_i$ and $X_{i+1}$. Define
$s, r$ on $\mathcal E$ by setting $s = s_i, r = r_i$ on $E_i$.
Then we call $\mathcal B = (\mathcal X, \mathcal E)$ a \textit{measurable Bratteli
diagram}. The pair $(X_n, E_n)$ is called the $n$-th level of the
measurable Bratteli diagram $\mathcal B$.
If all $E_n = E$, then the measurable Bratteli diagram $\mathcal B$ is
called \textit{stationary}.
\end{definition}
\begin{remark}\label{rem MBD}
(1) We will identify the standard Borel spaces $\{(X_n, \mathcal A_n) :
n \in \mathbb N_0\}$ with an uncountable standard Borel space $(X, \mathcal A)$.
This means that a paint $x$ can be seen in all levels
$(X_n, \mathcal A_n)$. This fact explains why we do not require that all
levels $X_n = X_0$ in the definition of a stationary Bratteli
diagram. Nevertheless, we will keep using subindeces to indicate
the level of a measurable Bratteli diagram.
(2) The set of edges $E_i, i \geq 0,$ can be represented as
follows:
$$
E_i = \bigcup_{x\in X_i} s^{-1}(x) = \bigcup_{x\in X_{i+1}}
r^{-1}(x).
$$
This means that $E_i$ can be seen as the union of ``vertical'' and
``horizontal'' sections.
The fact that $r$ is an onto map says that $\forall y \in X_{i+1}
\ \exists x \in X_i \ \mbox{such\ that} \ (x, y) \in E_i$.
A similar property holds for the map $s$.
(3) To define the \textit{path space} $\mathcal X_{\mathcal B}$ of a
measurable Bratteli diagram $\mathcal B = (\mathcal X, \mathcal E)$,
we take a sequence $\overline x = \{ e_i = (x_i, x_{i+1})\}, e_i
\in E_i$, such that $r(e_i) = s(e_{i+1})$ for all $i$.
Equivalently, $\overline x = (x_0, x_1, x_2, ...)$ where every pair
$(x_i, x_{i+1})$ is in $E_i$, $i\in \mathbb N_0$. Then $s(\overline x) = x_0$.
(4) We denote
\begin{equation}\label{eq_paths from w}
\mathcal X_\mathcal B(w) = \{ \overline x \in \mathcal X_\mathcal B : s(\overline x) = w\}.
\end{equation}
Clearly, the sets $\mathcal X_\mathcal B(w)$ form a partition $\eta$ of
$\mathcal X_\mathcal B$, and $X_0$ is the quotient space with respect to
this partition.
\end{remark}
We will consider below \textit{ measurable stationary Bratteli
diagrams}, i.e., $E_i =E$. For every $e = (w,v)\in E$, we set
$$
D_e = \{\overline x \in \mathcal X_{\mathcal B} : s(\overline x) = r(e) = v\},
$$
$$
R_e = \{ \overline x \in \mathcal X_{\mathcal B} : (x_0, x_1) = e\}.
$$
For $\overline x = (x_0, x_1, ...) \in D_e$, define the map $\tau_e : D_e
\to R_e$ by setting
$$
\tau_e(\overline x) = (w, v, x_1, x_2, ...).
$$
We note that $\overline x \in D_e$ means that $x_0 = v$ so that
$\tau_e(\overline x)$ is well defined and belongs to $R_e$. Moreover, the
map $\tau_e$ is one-to-one on its domain and
the map $\tau_e^{-1} : R_e \to D_e$ is defined by
$$
\tau_e^{-1}(e, \overline y) = \overline y.
$$
We can consider the metric $\mbox{dist}$ on the path space
$\mathcal X_\mathcal B$
similarly to the case of generalized Bratteli diagrams. As shown
in Remark \ref{rem contractive}, the maps $\tau_e$ are contractive
for all $e\in E$.
Let $\mathcal B$ be a measurable Bratteli diagram with the path space
$\mathcal X_{\mathcal B}$. By a measure $\mu$ on $X_\mathcal B$, we mean a Borel
positive (finite or sigma-finite) measure.
Consider a sequence of Borel sets $C_i \subset X_i : i\in \mathbb N_0$.
Then
$(C_0 \times C_1 \times\cdots \times C_N) \cap \mathcal X_\mathcal B$ is
called a \textit{cylinder subset} of $X_\mathcal B$ of length $N$ and
denoted by $[C_0, ... , C_N]$. In particular, we can consider the
set $R_e$ as a cylinder set $[e]$ defined by the edge
$e$, $e\in E$. It defines the partition
\begin{equation}\label{eq xi}
\xi = \{[e] : e \in E\}
\end{equation}
of $\mathcal X_\mathcal B$. We remark that the quotient
$\mathcal X_\mathcal B/\xi$ is isomorphic to the set $E$.
Since $D_e = \mathcal X_\mathcal B(r(e))$ for every $e \in E$,
the partition $\eta$
(see Remark \ref{rem MBD} (4)) is an enlargement of $\xi$ because
$$
D_e = \bigcup_{f : s(f) = r(e)} [f].
$$
\vskip 3mm
\textbf{Observation}. The collection of cylinder sets
$[C_0, ... , C_N]$ of any finite
length generates the Borel sigma-algebra of $X_\mathcal B$. A measure
$\mu$ on $X_\mathcal B$ is completely determined by its values on cylinder
sets. These facts are obvious.
\vskip 3mm
\subsection{IFS measures on measurable Bratteli diagrams}
Let $\mathcal B = (\mathcal X, \mathcal E)$ be a stationary measurable Bratteli
diagram. Recall that $E$ denotes a Borel subset of $X_0 \times
X_1$ (in fact, $E$ can be viewed as a subset of $X_0\times X_0$).
Suppose that $p$ is a Borel measure on $E$, then $(E,p)$ becomes
a standard measure space. We consider
both cases, finite and sigma-finite measures $p$.
The key tool of this subsection is the disintegration theorem
for a finite or sigma-finite measure with respect to a measurable
partition of a measure space. The literature on this subject is
very extensive. We refer to the original paper by
Rokhlin \cite{Rohlin1949} and more recent paper
\cite{Simmons2012} (see also \cite{BezuglyiJorgensen2018(book)}).
\begin{definition} \label{def cond meas}
Suppose that $(Z, \mathcal C, \mu)$ and $(Y, \mathcal D, \nu)$ are standard
$\sigma$-finite measure spaces. Let $\pi : Z \to Y$ be a
measurable function. A system of \textit{conditional measures}
for $\mu$ with respect to $\pi$ is a collection of measures
$\{\mu_y : y \in Y\}$ such that \\
(i) $\mu_y$ is a Borel measure on $\pi^{-1}(y)$,\\
(ii) for every $B \in \mathcal C$, the function $y\mapsto \mu_y(B)$ is
measurable and $\mu(B) = \int_Y \mu_y(B) \; d\nu(y)$.
\end{definition}
\begin{theorem}[\cite{Simmons2012}] \label{thm Simmons}
Let $(Z, \mathcal C, \mu)$ and $(Y, \mathcal D, \nu)$ be as above.
For any measurable function $\pi : Z \to Y$
such that $\mu\circ \pi^{-1} \ll \nu$ there exists a uniquely
determined system of conditional measures $(\mu_y)_{y \in Y}$ which
disintegrates the measure $\mu$, i.e.
$$
\mu = \int_{Y} \mu_y d\nu(y).
$$
\end{theorem}
\begin{remark} \label{rem meas p}
For the edge set $E$ of a stationary measurable Bratteli diagram
$\mathcal B$ and a Borel measure $p$ on $E$, consider the partition of
$E$ into the sets $r(s^{-1}(x)), x\in X_0$. This partition
is measurable in the sense of \cite{Rohlin1949}. We can apply
the disintegration theorem cited above. More precisely, the
sets $r^{-1}(x) = \{(x, y) \in E : y \in X_1\}$, where $x\in X_0$,
are vertical sections of the set $E$. Setting $\widehat p (\cdot) =
p(s^{-1}(\cdot))$, we have the projection of the measure $p$
onto $X_0$. Denoting by $x \mapsto p(x, \cdot)$ the corresponding
system of conditional measures, we have
$$
p= \int_{X_0} p(x, dy)\; d\widehat p(x)
$$
where the measure $p(x, dy)$ is supported by the set $r(s^{-1}(x))$.
\end{remark}
We will consider several versions of disintegration theorem
related to measures on the path space $\mathcal
X_\mathcal B$ of a measurable Bratteli diagram $\mathcal B$. We assume that the
measures will satisfy conditions (i) and (ii) formulated below.
\vskip 0.3cm
(i) For a given measure $m$ on $\mathcal X_\mathcal B$ and the set
$\mathcal X_\mathcal B(y)$ defined in \eqref{eq_paths from w},
the measurable function
$$
q_m(y) = m(\mathcal X_\mathcal B(y)), \ y \in X_0,
$$
takes finite values.
(ii) The partition $\xi$ defined in \eqref{eq xi} is obviously
measurable, and the measure $m$ on $\mathcal X_\mathcal B$ can
be disintegrated with respect to $\xi$. Denote by $\widehat m$ the
projection of $m$ onto $E = X/\xi$. Let $p$ be a measure on $E$.
Assuming that $\widehat m \ll p$ and applying Theorem \ref{thm Simmons},
we have
\begin{equation}\label{eq:dis}
m = \int_{E} m_{e} \; d p(e),
\end{equation}
where $e \mapsto m_e$ is the system of conditional measures of
$m$ with respect to $(E, p)$.
We note that the conditional measure $m_e$ is supported by the set
$[e]$, $e \in E$.
\vskip 0.3cm
The class of IFS measures is determined by a special form of
measures $m_e$ defined in \eqref{eq:dis}. We will assume that
the measures considered on the path space $\mathcal X_\mathcal B$ satisfy
conditions (i) and (ii).
\begin{definition} (\textit{IFS measures and disintegration})
Let $\mathcal B$ be a stationary measurable Bratteli diagram, and
let $\{\tau_e, e \in E\}$ be the system of contractive maps
defined in
subsection \ref{ss MBD-IFS}. Suppose $p$ is a fixed probability
measure on
the set $E$. A Borel measure $\mu$ is called an \textit{IFS measure}
with respect to $p$ if
\begin{equation}\label{eq-IFS for MBD}
\mu = \int_E \mu\circ\tau_e^{-1}\; dp(e).
\end{equation}
\end{definition}
Our goal is to find conditions under which a
measure $m$ defined on the path space of a stationary measurable
Bratteli diagram is an IFS measure.
\begin{theorem}\label{thm IFS MBD} Let $p$ be a Borel
probability measure on
$E$ where $E$ is the edge set of a stationary measurable Bratteli
diagram $\mathcal B = (\mathcal X, \mathcal E)$.
Let $\mu$ be a measure on $\mathcal X_\mathcal B$ and $q(x) =
\mu(\mathcal X_\mathcal B(x))$.
Then $\mu$ is and IFS measure if and only if the following
condition holds:
\begin{equation}\label{eq P-MBD}
\int_{X_1} p(x, dy) q(y) = q(x).
\end{equation}
\end{theorem}
\begin{proof}
In the proof of the theorem, we apply the idea used in Theorem
\ref{thm IFS stat BD}.
We will construct an IFS measure $\mu$ by defining its values on
cylinder sets of consequently increasing length. This construction
will be based on the application of \eqref{eq-IFS for MBD} so that
the measure $\mu$ will be an IFS measure automatically.
Relation \eqref{eq P-MBD} is used to satisfy the Kolmogorov
extension theorem.
As mentioned in Remark \ref{rem meas p},
the projection of $p$ onto $X_0$ defines the measure $\widehat p$.
This allows us to define $\mu$ on cylinder subsets of
$\mathcal X_\mathcal B$ generated by Borel sets on $X_0$:
$$
\mu([C_0]) = \int_{C_0} q(x_0) \; d\widehat p(x_0),
$$
where $C_0$ is a Borel subset of $X_0$.
For a cylinder set $[C_0, C_1] =\{ x= (x_i) \in \mathcal X_\mathcal B :
x_0 \in C_0, x_1 \in C_1\}$ of length two, we define
\begin{equation}\label{eq_cyl set 2}
\mu([C_0, C_1]) = \int_{C_0} \left(\int_{C_1 \cap r(s^{-1}(x_0))}
p(x_0, dy) q(y) \right) \; d\widehat p(x_0).
\end{equation}
Show that this definition of the measure $\mu$ satisfies
\eqref{eq-IFS for MBD}:
$$
\begin{aligned}
\int_E \mu\circ \tau_e^{-1} ([C_0, C_1]) \;
d p(e)
= & \int_E \mu\circ \tau_e^{-1} ([C_0, C_1]\cap [e]) \; d p(e)\\
=& \int_{C_0} \int_{C_1 \cap r(s^{-1}(x_0))} \mu(\mathcal X_\mathcal B(y)\;
p(x_0, dy)\; d\widehat p(x_0)\\
= & \int_{C_0} \int_{C_1 \cap r(s^{-1}(x_0))} q(y) \;
p(x_0, dy)\; d\widehat p(x_0) \\
= & \mu([C_0, C_1])
\end{aligned}
$$
In general, if we defined the measure $\mu$ on cylinder sets
of the form $[C_0, ..., C_k], k = 1,..., n-1,$, then we set
$$
\mu ([C_0, ..., C_n]) = \int_E \mu\circ \tau_e^{-1}
([C_0, ..., C_{n-1}]) \; dp(e).
$$
This means that this definition of $\mu$ shows that relation
\eqref{eq-IFS for MBD} holds automatically.
It remains to show that the measure $\mu$ is well defined, that is
it satisfies the Kolmogorov extension theorem. We check this
property for the cylinder sets of length two. Indeed, in this case
we take $C_1 = X_1$ and compute
$$
\begin{aligned}
\mu([C_0, X_1]) = & \int_{C_0} \left( \int_{X_1 \cap r(s^{-1}(x_0))}
q(y) p(x_0, dy) \right) d\widehat p(y)\\
= & \int_{C_0} \left( \int_{r(s^{-1}(x_0))}
q(y) p(x_0, dy) \right) d\widehat p(y)\\
=& \int_{C_0} q(x_0) \; d\widehat p(x_0)\\
=& \mu([C_0]).
\end{aligned}
$$
The general case is proved analogously.
\end{proof}
\begin{remark}
We note that the existence of an IFS measure on $\mathcal X_\mathcal B$
can be proved by using a fixed point theorem following
\cite{Hutchinson1981}. We consider the metric $\mathrm{dist}$ on
$\mathcal X_\mathcal B$ defined in Remark \ref{rem prop BD}.
Let
$$
\mbox{Lip}_1 = \{ f : \mathcal X_\mathcal B \to \mathbb R :
|f(x) - f(y) \leq \ \mbox{dist}(x, y)\}.
$$
For a measure $\nu$ on $\mathcal X_\mathcal B$, set
$$
L(\nu) = \int_E \nu\circ\tau_e^{-1} \; dp(e).
$$
Since $\tau_e$ is a contractive map for every $e$, one can show that
$\rho(L(\nu), L(\mu)) \leq \rho (\nu, \mu)$ where the metric
$\rho$ is defined by
$$
\rho (\nu, \mu) = \sup \Bigl\{ \big| \int f \; d\nu - \int f\;
d\mu\; \big| : f \in \mathrm{Lip}_1\Bigr\}.
$$
Then we conclude that there exists a measure $\mu_0$ such that
$L(\mu_0) = \mu_0$.
\end{remark}
\begin{remark}
The IFS measure $\mu$ defined in Theorem \ref{thm IFS MBD} is
shift invariant. The proof of this fact is similar to that
giving in Theorem \ref{thm-IFS-inv}.
\end{remark}
\textbf{Acknowledgements}. The authors are pleased to thank our
colleagues and collaborators, especially, R. Curto, H. Karpel,
P. Muhly, W. Polyzou, S. Sanadhya. We are
thankful to the members of the seminars in Mathematical Physics and
Operator Theory at the University of
Iowa for many helpful conversations.
\textbf{Declaration}.
The authors declare that they have no conflict of interest.
\bibliographystyle{alpha}
|
1704.05900
|
\section{Introduction}
\label{s1}
Supernovae (SNe) mark the deaths of stars. They can be divided into
two broad categories: core-collapse SNe from the gravitationally
powered explosion of massive stars \citep[e.g.][]{smartt_progenitors_2009},
and Type Ia SNe from the thermonuclear destruction of white dwarfs
\citep[e.g.][]{hillebrandt_type_2000}. In both cases, there has been
considerable interest in recent years in understanding their
progenitor systems for reasons as diverse as testing stellar
evolutionary models, improving their use as cosmological probes, and
understanding their role in driving galactic evolution.
The most direct method for studying the progenitors of supernovae is
to detect them in pre-explosion imaging. This is limited however to
the handful of cases where SNe have exploded in a nearby galaxy with
deep, high-resolution images. Furthermore, it is most suited for
studying core-collapse supernova progenitors
\citep{smartt_progenitors_2009} which are luminous supergiants
(although see \citealp{li_exclusion_2011} and \citealp{mccully_luminous_2014} for applications to Type Ia
SNe). Alternative techniques to infer core-collapse SN progenitor
properties using nucleosynthetic yields from late-time spectroscopy of
SNe \citep[e.g.][]{jerkstrand_nebular_2014}, or hydrodynamic estimates
of ejecta mass \citep{bersten_iptf13bvn:_2014} are always model
dependent. For Type Ia SNe, the spectroscopic and photometric
signatures of interaction between SN ejecta and a companion may be
used to constrain the progenitor system, but are relatively weak
effects \citep{maeda_signatures_2014}.
Another means to study SN progenitors is to search for their former
binary companions that have survived the explosion. At least 70\% of
massive stars are seen to be in binary systems
\citep[e.g.][]{sana_binary_2012}. Furthermore, in a handful of cases, a
surviving binary companion has been detected in deep imaging of
extragalactic core-collapse SNe
\citep{maund_disappearance_2009,folatelli_blue_2014}. Type Ia SNe
require a binary companion to explode \citep{hillebrandt_type_2000},
and may leave behind a detectable non-degenerate companion
\citep[e.g.][and references
therin]{han_companion_2008,pan_search_2014,noda_brightness_2016}. For
both Type Ia and core-collapse SNe, the stellar parameters of a
surviving binary companion can constrain the evolutionary
status of the SN progenitor at the point of explosion
\citep{maund_disappearance_2009,bersten_type_2012}. A SN progenitor
companion may also be polluted with metals from the explosion
(\citealp{israelian_evidence_1999} and more recently \citealp{liu_interaction_2015}).
Several searches have already been made for runaway stars in Galactic
Type Ia SN remnants, most notably in Tycho's SN where a possible
candidate (designated Tycho G) has been claimed to be the former
binary companion
\citep{ruiz-lapuente_binary_2004,gonzalez_hernandez_chemical_2009,bedin_improved_2014}. This
association has since been disputed
\citep{kerzendorf_subaru_2009,kerzendorf_high-resolution_2013,xue_newly_2015}. Searches
within other Galactic remnants such as that of SN 1006
\citep{gonzalez_hernandez_no_2012} and Kepler's SN
\citep{kerzendorf_reconnaissance_2014} have failed to yield a companion,
while a non-degenerate companion has been almost completely ruled out
for SNR 0509$-67.5$ in the Large Magellanic Cloud
\citep{schaefer_absence_2012}.
Searches for companions to core-collapse SNe have mostly focussed on
runaway OB stars near SN remnants
\citep{blaauw_origin_1961,guseinov_searching_2005}. HD 37424, a main
sequence B star, has been proposed to be associated with the SN
remnant S147 \citep{dincel_discovery_2015}. The pulsar PSR
J0826+2637 has been suggested to share a common origin with the
runaway supergiant G0 star HIP 13962 \citep{tetzlaff_origin_2014}, although there is no identified SNR. In
the Large Magellanic Cloud, the fastest rotating O-star (VFTS102) has
been suggested to be a spun-up SN companion associated with the young
pulsar PSR J0537$-6910$ \citep{dufton_vlt-flames_2011}.
Recently \citet{kochanek_cas_2017} used Pan-STARRS1 photometry \citep{chambers_pan-starrs1_2016}, the \citet{green_three-dimensional_2015} dust-map and the NOMAD \citep{zacharias_vizier_2005} and HSOY \citep{altmann_hot_2017} proper motion catalogues to search for runaway former companions of the progenitors of the three most recent, local core-collapse SNe: the Crab, Cas A and SN 1987A. Based on a null detection of any reasonable candidates \citet{kochanek_cas_2017} put limits on the initial mass ratio $q=M_2/M_1\lesssim0.1$ for the nominal progenitor binary of these SNRs. \citet{kochanek_cas_2017} note that this limit implies a 90\% confidence upper limit on the $q\gtrsim0.1$ binary fraction at death of $f_{\mathrm{b}}<44\%$ in tension with observations of massive stars.
The reason to search for runaway companions of core-collapse
supernovae is that their presence or absence can be used to constrain
aspects of binary star evolution. These include mass transfer rates,
common-envelope evolution and the period and binary fraction
distributions. Runaway stars are interesting in their own right for
their dynamical properties with the fastest runaway stars being
unbound from the Milky Way.
In this work, we present a systematic search for SN-ejected binary
companions within the {\it Gaia} Data Release 1
\citep[DR1;][]{gaia_collaboration_gaia_2016}. Unfortunately, the Kepler SNR is too distant at around $6\;\mathrm{kpc}$ to
search for a companion using DR1, however ten other remnants including
S147 lie within our distance cut-off of $2\;\mathrm{kpc}$. The data sources for both the
runaways and the SNR are discussed in Section~\ref{sec:data}. We then
outline two approaches to this problem -- the first using purely
kinematic methods in Section~\ref{sec:simple}, the second exploiting
colour, magnitude and reddening together with the peculiar velocity of the
progenitor binary with a Bayesian framework in Section~\ref{sec:bayesgrid}. For
four SNRs (the Cygnus Loop, HB 21, S147 and the Monoceros Loop) we identify
likely runaway companions which are discussed in detail in
Section~\ref{sec:results}.
\section{Sources of data}
\label{sec:data}
The list of candidate stellar companions for each SNR is taken from a cross-match of
TGAS and APASS (Sec. \ref{sec:tgas}). There is no analogously uniform
catalogue for SNRs and so we conduct a literature review for each SNR
to establish plausible estimates for the central position, distance
and diameter, which we discuss in Section \ref{sec:snr} and in
Appendix \ref{sec:boutique}.
\subsection{Summary of stellar data}
\label{sec:tgas}
On the 14th September 2016 the {\it Gaia} Data Release 1 (GDR1) was made
publicly available \citep{gaia_collaboration_gaia_2016-1,gaia_collaboration_gaia_2016}. The primary astrometric component of the release
was the realisation of the Tycho-{\it Gaia} astrometric solution
(TGAS), theoretically developed by \citet{michalik_tycho-gaia_2015},
which provides positions, parallaxes and proper motions for the stars
in common between the GDR1 and \emph{Tycho-2} catalogues. At
$1\;\mathrm{kpc}$ the errors in the parallax from TGAS typically
exceed $30\%$. To constrain the distance of the candidates we
nearest-neighbour cross-match with the AAVSO Photometric All-Sky
Survey (APASS) DR9 to obtain the $B{-}V$ colour. {\it Gaia} Data Release 2
is anticipated in early 2018 and will contain the additional blue
$G_{\mathrm{BP}}$ and red $G_{\mathrm{RP}}$ magnitudes. Substituting
for $B{-}V$ with the $G_{\mathrm{BP}}{-}G_{\mathrm{RP}}$ colour will allow
our method to be applied using only data from {\it Gaia} DR2.
\subsection{Summary of individual SNRs}
\label{sec:snr}
We select SNRs that are closer than $2\;\mathrm{kpc}$, having stars
in our TGAS-APASS cross-match within the central 25\% of the SNR by radius,
and lying within the footprint of Pan-STARRS so we can use the 3D
dustmap of \citet{green_three-dimensional_2015}. The SNRs in our sample are typically older than $10\;\mathrm{kyr}$ and so will have swept up more mass from the ISM than was ejected, which makes it difficult to type them from observations of their ejecta. We can say that these SNRs are likely the remnants of core-collapse SNe since around $80\%$ of Galactic SNe are expected to be core-collapse SNe \citep[e.g.][]{mannucci_supernova_2005,li_nearby_2011}. Moreover, several are identified with regions of recent star formation (i.e. G205.5$+00.5$ with Mon OB2) or molecular clouds in OB associations (i.e. G089.0$+04.7$ with molecular clouds in Cyg OB7). The properties of this sample of ten SNRs are given in Table \ref{tab:snr}, where $N_{\mathrm{TGAS}}$ is the number of candidates found in TGAS and $N_{\mathrm{TGAS+APASS}}$ is the number remaining after the cross-match with APASS.
\begin{table*}[]
\centering
\caption{Assumed properties of the sample of supernova remnants where the errors on the distance are $1\sigma$ and are described in Appendix \ref{sec:boutique}.}
\label{tab:snr}
\begin{tabular}{llllllll}
\hline \hline SNR & Known as & RA & Dec & Diameter (arcmin) & Distance (kpc) & $N_{\mathrm{TGAS}}$ & $N_{\mathrm{TGAS+APASS}}$ \\ \hline
G065.3$+05.7$ & --- & 19:33:00 & +31:10 & 310$\times$240 & $0.77\pm0.2$ & 294 & 7 \\
G069.0$+02.7$ & CTB 80 & 19:53:20 & +32:55 & 80 & $1.5\pm0.5$ & 14 & 11 \\
G074.0$-08.5$ & Cygnus Loop & 20:51:00 & +30:40 & 230$\times$160 & $0.54_{-0.08}^{+0.10}$ & 115 & 76 \\
G089.0$+04.7$ & HB 21 & 20:45:00 & +50:35 & 120$\times$90 & $1.7\pm0.5$ & 25 & 3 \\
G093.7$-00.2$ & CTB 104A, DA 551 & 21:29:20 & +50:50 & 80 & $1.5\pm0.2$ & 10 & 10 \\
G114.3$+00.3$ & --- & 23:37:00 & +61:55 & 90$\times$55 & $0.7\pm0.35$ & 19 & 17 \\
G119.5$+10.2$ & CTA 1 & 00:06:40 & +72:45 & 90 & $1.4\pm0.3$ & 8 & 7 \\
G160.9$+02.6$ & HB 9 & 05:01:00 & +46:40 & 140$\times$120 & $0.8\pm0.4$ & 19 & 18 \\
G180.0$-01.7$ & S147 & 05:39:00 & +27:50 & 180 & $1.30_{-0.16}^{+0.22}$ & 36 & 31 \\
G205.5$+00.5$ & Monoceros Loop & 06:39:00 & +06:30 & 220 & $1.2\pm0.4$ & 53 & 47 \\ \hline \hline
\end{tabular}
\end{table*}
Establishing a potential association between a star and a SNR requires
us to demonstrate a spatial coincidence at around the time of the
supernova explosion. The relevant properties of each SNR are then the location
of the centre $(\alpha,\delta)_{\mathrm{SNR}}$, distance
$d_{\mathrm{SNR}}$, age $t_{\mathrm{SNR}}$, angular diameter
$\theta_{\mathrm{SNR}}$ and either the proper motion
$(\mu_{\alpha\ast},\mu_{\delta})_{\mathrm{SNR}}$ or peculiar velocity
$(v_{\mathrm{R}},v_{\mathrm{z}},v_{\phi})_{\mathrm{SNR}}$.
We take the \citet[known as
\emph{SNRcat}]{ferrand_census_2012} and \citet{green_catalogue_2014} catalogues as the primary
sources of SNR properties. We use the positions and angular
diameters from the detailed version of the Green catalogue that is
available online\footnote{Green D. A., 2014, `A Catalogue of Galactic
Supernova Remnants (2014 May version)', Cavendish Laboratory,
Cambridge, United Kingdom (available at
\url{http://www.mrao.cam.ac.uk/surveys/snrs/}).}.
The distance to a SNR is usually uncertain and so we describe the origin of each distance in Appendix \ref{sec:boutique}.
We do not use estimates of the ages of SNRs because distance estimates to SNRs are degenerate with
the age, so these two measurements are not
independent. We thus conservatively assume that the
supernova must be older than $1\;\mathrm{kyr}$ and younger than
$150\;\mathrm{kyr}$. A younger supernova at $1\;\mathrm{kpc}$ would very likely
be in the historical record \citep{stephenson_historical_2002,green_historical_2003} and the shell of an older SNR would no longer be
detectable.
Determining the location of the centre of a SNR is usually not straightforward. The
standard method to obtain the centre is to calculate the centroid of
the projected structure of the SNR shell on the sky, but this position
can be obfuscated by various effects such as the interaction between the ejecta and the local ISM,
overlap between SNRs, and background objects misclassified as
belonging to the SNR. G074.0$-08.5$ (Cygnus Loop) is notable for its peculiarity with a substantial blowout region to the south of the primary spherical shell \citep[e.g.][]{fang_numerically_2017}. A naive calculation of the centroid for this SNR would result in a centre which is around $10\;\mathrm{arcmin}$ away from the centroid of the shell. We have verified that our results for G074.0$-08.5$ are robust to this level of systematic error. Some of our SNR central positions have
associated statistical errors, but because these estimates do not in
general account for systematics we instead use a more conservative
constraint. We adopt a prior for the true position of the SNR centre
which is a two-dimensional Gaussian with a FWHM given by
\begin{equation}
\theta'=\mathrm{max}(5',0.05\theta_{\mathrm{SNR}}).
\end{equation}
These values were chosen to attempt to balance the statistical and systematic errors which are present.
We assume that the progenitor system was a typical binary in the Milky
Way thin disk and so is moving with the rotational velocity of the
disk together with an additional peculiar motion. We sample a peculiar velocity from the
velocity dispersions of the thin disk and propagate it into a
heliocentric proper motion. We take the Sun to be at $R_{\sun}=8.5\;\mathrm{kpc}$
and the Milky Way's disk rotation speed to be $v_{\mathrm{disk}} = 240\;
\mathrm{km}\;\mathrm{s}^{-1}$ with a solar peculiar velocity of
$(U_{\sun},V_{\sun},W_{\sun})=(11.1,12.24,7.25)\;\mathrm{km}\;\mathrm{s}^{-1}$
\mbox{\citep{schonrich_local_2010}}. We neglect uncertainties in
these values since they are subdominant.
\begin{figure}[t]
\includegraphics[scale=0.45,trim = 0mm 0mm 0mm 2mm, clip]{{allsimple}.pdf}
\caption{The fraction $F$ of realisations of each star in each SNR which are consistent with being spatially coincident with the centre of the SNR at one point in the past $150\;\mathrm{kyr}$. The criteria for determining whether a realisation is consistent are described in Section \ref{sec:simple}.}
\label{fig:simple}
\end{figure}
\section{Searches with only kinematic constraints}
\label{sec:simple}
The typical expansion velocities of supernova remnant shells are more
than $1000\;\mathrm{km}\;\mathrm{s}^{-1}$ for the first few $10^4$
years of their evolution \citep{reynolds_supernova_2008}. Thus, since recent estimates for the maximum
velocity of runaways are $540\;\mathrm{km}\;\mathrm{s}^{-1}$ for late
B-types and $1050\;\mathrm{km}\;\mathrm{s}^{-1}$ for G/K-dwarfs
\citep{tauris_maximum_2015}, it is reasonable to assume that the former
companion to the SNR progenitor still resides in the SNR. For each
SNR we select all stars in TGAS that are within 25\% of the radius of the SNR
giving us somewhere in the range of $10\text{--}300$ stars per
SNR. Less than 1\% of runaways are ejected with velocities in excess of $200\;\mathrm{km}\;\mathrm{s}^{-1}$ \citep[e.g.][]{eldridge_runaway_2011} thus considering every star in the SNR would increase the number of potential candidates by an order of magnitude while negligibly increasing the completeness of our search. Our choice to search the inner 25\% by radius is more conservative than the one sixth by radius searched by previous studies \citep[e.g.][]{guseinov_searching_2005,dincel_discovery_2015}. For each of these stars, we have positions $(\alpha,\delta)$,
parallax $\omega$ and proper motions $(\mu_{\alpha*},\mu_{\delta})$ as well
as the mean magnitude G, with a full covariance matrix $\mathrm{Cov}$ for the
astrometric parameters.
Given the geometric centre
$(\alpha_{\mathrm{SNR}},\delta_{\mathrm{SNR}})$ and proper motion
$(\mu_{\alpha,\mathrm{SNR}},\mu_{\mathrm{\delta,SNR}})$ of the
remnant and their errors, we can estimate the past location at
time $-t$ of each star by the equations of motion,
\begin{align}
\alpha_{*}(t)&=\alpha_{*}-t\mu_{\alpha\ast} \\
\delta(t)&=\delta-t\mu_{\delta},
\end{align}
and we can write similar expressions for the remnant centre. Note we use $*$ to denote quantities we have transformed to a flat space, for instance $\alpha_{*}=\alpha\cos\delta$. The angular separation $\Delta \theta$ is then approximated by,
\begin{equation}
\label{eq:theta}
\Delta\theta(t) = \sqrt{\left[\alpha_{*}(t)-\alpha_{*,\mathrm{SNR}}(t)\right]^2+\left[\delta(t)-\delta_{\mathrm{SNR}}(t)\right]^2}.
\end{equation}
Since the typical angular separations involved are less than a few degrees at all times this approximation is valid to first order. This expression has a clearly defined global minimum given by,
\begin{equation}
T_{\mathrm{min}}=\frac{(\alpha_{*}\!-\!\alpha_{*,\mathrm{SNR}})(\mu_{\alpha*}\!-\!\mu_{\alpha*,\mathrm{SNR}})+(\delta\!-\!\delta_{\mathrm{SNR}})(\mu_\delta\!-\!\mu_{\delta,\mathrm{SNR}})}{(\mu_{\alpha*}\!-\!\mu_{\alpha*,\mathrm{SNR}})^2+(\mu_\delta\!-\!\mu_{\delta,\mathrm{SNR}})^2},
\end{equation}
which can be substituted back into Equation \ref{eq:theta} to obtain the minimum separation $\Delta\theta_{\mathrm{min}}$.
We construct the covariance matrix
$\mathrm{Cov}=\mathrm{D}^{1/2}\mathrm{Corr}\mathrm{D}^{1/2}$ using the
correlation matrix $\mathrm{Corr}$ and the diagonal matrix of
errors
$\mathrm{D}=\mathrm{Diag}(\sigma_{\alpha}^2,\sigma_{\delta}^2,\sigma_{\omega}^2+(0.3\;\mathrm{mas})^2,\sigma_{\mu_{\alpha}}^2,\sigma_{\mu_{\delta}}^2)$ which are given in TGAS. We
have added on the $0.3\;\mathrm{mas}$ systematic error in parallax
recommended by \citet{gaia_collaboration_gaia_2016-1}. We draw samples
from the multivariate Gaussian distribution defined by the mean position
$(\alpha,\delta,\omega,\mu_{\alpha},\mu_{\delta})$ and the
covariance matrix $\mathrm{Cov}$ and from the distributions of the SNR
centre, distance and peculiar velocity. These latter distributions are
described in Section \ref{sec:snr}. We calculate $T_{\mathrm{min}}$
and $\theta_{\mathrm{min}}$ for each of the samples which can be combined into distributions for the predicted minimum separation and time
at which it occurs.
Once we have these distributions we classify stars by the
plausibility of them being the former companion. We do this
in a qualitative way by finding the fraction $F$ of realizations of each
star which satisfy $1<(T_{\mathrm{min}}/\mathrm{kyr})<150$, the
line-of-sight distance between the star and the SNR is less than
$153\;\mathrm{pc}$ and has $\theta_{\mathrm{min}}$ corresponding to a
physical separation less than $1\;\mathrm{pc}$. The latter two of
these constraints use the distance to the location of the progenitor
binary when the supernova exploded which can be calculated using the
sampled parameters and the time of the minimum separation. The $153\;\mathrm{pc}$
limit of the second constraint is simply the distance travelled by a star
at $1000\;\mathrm{km}\;\mathrm{s}^{-1}$ over $150\;\mathrm{kyr}$ and
is the maximum likely distance travelled by a runaway associated with
a SNR. The $1\;\mathrm{pc}$ limit of the third constraint is approximately the maximum likely separation of two stars in a binary and is smaller than the $153\;\mathrm{pc}$ limit in the radial direction because we have a measurement of the proper motion of each candidate. The value of $F$ is shown for every star in each SNR in Figure \ref{fig:simple}.
We rank the candidates in each SNR by this quasi-statistical measure and
consider the star with the highest $F$ to be the most likely
candidate. For some of these stars, we can obtain APASS $B{-}V$
photometry from a TGAS/APASS cross-match and, were this method
effective, most of the best candidates would be blue. Of
the ten best candidates two have no associated $B{-}V$ in the
cross-match and five have $B{-}V>1.3$ hence are unlikely to be OB
stars. One of the two best candidates without a measured $B{-}V$ was HD 37424 in G180.0$-01.7$ (S147), which is one of the only five stars in G180.0$-01.7$ which did not have APASS magnitudes. HD 37424 has been previously suggested to be the runaway companion of G180.0$-01.7$ \citep{dincel_discovery_2015} and taking $B{-}V=0.073\pm0.025$ from that paper we see that our kinematic method would have proposed this star as a candidate if it had magnitudes in APASS. The remaining three stars with magnitudes are TYC 2688-1556-1 in G074.0$-08.5$ (Cygnus Loop) with
$B{-}V=0.43$, BD+50 3188 in G089.0$+04.7$ (HB 21) with $B{-}V=0.39$ and TYC
4280-562-1 in G114.3$+00.3$ with $B{-}V=0.39$. Of these stars only BD+50
3188 is specifically mentioned in the literature with
\citet{chojnowski_high-resolution_2015} concluding that it is a B star. That one of the stars is B type suggests that the other two stars with similar colour are also B type by association, although it is possible that these two stars are less reddened by interstellar dust. G089.0$+04.7$ is at a distance of $1.7\pm0.5\;\mathrm{kpc}$ while the other
SNRs are much closer at $0.54_{-0.08}^{+0.10}\;\mathrm{kpc}$ and
$0.70\pm0.35\;\mathrm{kpc}$. The consequence of the $B{-}V$ measurement being less reddened by dust is that the star is intrinsically redder and so the two untyped candidates may be A type or later.
This conclusion has some obvious problems. First, we have not
established what fraction of runaways from core-collapse SNe we would
expect to be later-type than OB. The distribution of mass ratios in massive binary systems is observed to be flat \citep[e.g.][]{sana_binary_2012,duchene_stellar_2013,kobulnicky_toward_2014} which suggests that we would expect most runaways from
core-collapse SNe to be bright, blue, OB-type stars. Thus, while it is
possible to have low-mass companions of primaries with masses
$M>8\;M_{\sun}$, around 80\% of companions to massive stars
will have masses in excess of $3\;M_{\sun}$ with a median
of $7\;M_{\sun}$. This expectation is in conflict with the result above where five out of the ten best candidates are likely to be low-mass stars. One explanation for this seemingly
large fraction of contaminants is that the method efficiently rules
out those stars which are travelling in entirely the wrong direction
to have originated in the centre of the SNR, but leaves in background
stars which are co-incident on the sky with the centre of the SNR and
whose proper motion is not constrained. This can explain the large
fraction of our best candidates being stars whose photometry indicates
that if they are at the distance of the SNR then they must be faint,
red, late-type stars, because more distant stars will be more dust-obscured and so appear redder.
A second problem is that, because we have not accounted for the
reddening in $E(B{-}V)$ in a quantitative way, the estimated spectral type of
our candidates depended on one of them having already been typed. This
estimated type is very uncertain, and two of the stars could be A type
or even later. The method we used also did not make use of the {\it
Gaia} $G$ magnitude, despite it being the most accurate
magnitude contained in the {\it Gaia}-APASS cross-match. Third, while we have generated a list of candidates, the
ranking in the list is not on a firm statistical basis. There are four
stars in G074.0$-08.5$ which have $0.25<F<0.28$, only one of which is
our best candidate. It is difficult to defend a candidate when a
different statistical measure could prefer a different star. The
fourth problem is that using the $B{-}V$ photometry to further constrain
our list of candidates relies on an expectation that most runaways from
core-collapse supernova should be OB stars. Ideally our statistical
measure should incorporate this prior but include the possibility that
some runaways will be late-type.
These problems point towards the need for an algorithm that
incorporates kinematics with photometry, dust maps and binary star
simulations in a Bayesian framework.
\section{Bayesian search with binaries, light and dust}
\label{sec:bayesgrid}
\subsection{Binary star evolution grid}
\label{sec:bingrid}
The three most important parameters which determine the evolution of a binary star are the initial primary mass $M_1$, initial secondary mass $M_2$ and initial orbital period $P_{\mathrm{orb}}$. Empirical probability distributions have been determined for these parameters and combining these with a model for binary evolution allows us to calculate a probability distribution for the properties of runaway stars. The properties of runaway stars which we are interested in are the ejection
velocity $v_{\mathrm{ej}}$, the intrinsic colour $(B{-}V)_0$ and the
intrinsic {\it Gaia} G magnitude $G_0$ at the time of the supernova.
There are two standard formalisms used when evolving a large number of binary stars to evaluate the probability distribution for an outcome. The first is Monte Carlo-based and involves sampling initial properties from the distributions and evolving each sampled binary. In this approach the initial properties of the evolved binaries are clustered in the high-probability regions of the initial parameter space and low-probability regions which may have interesting outcomes might not be sampled at all. The other method is grid-based and selects binaries to evolve on a regularly-spaced grid across the parameter space. This grid divides the parameter space into discrete elements (voxels) and the probability of a binary having initial properties which lie in that voxel can be found by integrating the probability distributions over the voxel. This probability is assigned to the outcome of the evolution of the binary that was picked in that voxel. The probability distribution for the runaway properties can be determined by either method. In the Monte Carlo approach the resulting runaways from the binary evolution are samples from the probability distribution for runaway properties while in the grid approach the distribution can be obtained by summing the probabilities which were attached to the runaway from each evolved binary.
Our choice of a grid over a Monte Carlo approach was motivated by our
need to probe unusual areas of the parameter space. A Monte Carlo
approach would require a large number of samples to fully explore
these areas, while a grid approach gives us the location and
associated probability of each voxel that produces a runaway star as
well as the properties of the corresponding runaway star. These
probabilities and other properties are thus functions of this initial
grid.
We model the properties of stars ejected from binary systems in which
one component goes supernova using the {\sc binary\_c}
population-nucleosynthesis framework
\citep{izzard_new_2004,izzard_population_2006,izzard_population_2009}.
This code is based on the binary-star evolution ({\sc bse}) algorithm
of \citet{hurley_evolution_2002} expanded to incorporate nucleosynthesis,
wind-Roche-lobe-overflow \citep{abate_wind_2013,abate_modelling_2015},
stellar rotation \citep{de_mink_rotation_2013}, accurate stellar
lifetimes of massive stars \citep{schneider_ages_2014}, dynamical
effects from asymmetric supernovae \citep{tauris_runaway_1998}, an
improved algorithm describing the rate of Roche-lobe overflow
\citep{claeys_theoretical_2014}, and core-collapse supernovae
\citep{zapartas_delay-time_2017}. In particular, we take our black
hole remnant masses from \citet{spera_mass_2015} and use a fit to the
simulations of \citet{liu_interaction_2015} to determine the impulse
imparted by the supernova ejecta on the companion, both of which were
options previously implemented in {\sc binary\_c}. We use version
2.0pre22, SVN 4585. Grids of stars are modelled using the {\sc
binary\_grid2} module to explore the single-star parameter space as
a function of stellar mass $M$, and the binary-star parameter space in
primary mass $M_{1}$, secondary mass $M_{2}$ and orbital period $P_{\mathrm{orb}}$.
We pre-compute this binary grid of 8,000,000 binaries with
primary mass $M_1$, mass ratio $q=M_2/M_1$ and orbital period $P_{\mathrm{orb}}$ having the ranges,
\begin{align}
8.0 \leq M_1 / M_{\sun} &\leq 80.0,\nonumber\\
0.1 \; M_{\sun}/M_1 \leq q &\leq 1,\\
-1.0 \leq \log_{10} (P_{\mathrm{orb}}/\mathrm{days}) &\leq 10.0.\nonumber
\end{align}
We assume the primary mass has the
\mbox{\cite{kroupa_variation_2001}} IMF,
\begin{equation}
N(M_1)\propto
\begin{cases}
M_1^{-0.3}, & \mathrm{if}\ 0.01<M_1/M_{\sun}<0.08, \\
M_1^{-1.3}, & \mathrm{if}\ 0.08<M_1/M_{\sun}<0.5, \\
M_1^{-2.3}, & \mathrm{if}\ 0.5<M_1/M_{\sun}<80.0, \\
0, & \mathrm{otherwise.}
\end{cases}
\end{equation}
We assume a flat mass-ratio distribution for each system over the
range $0.1\;M_{\sun}/M_1<q<1$. We use a hybrid period
distribution \citep{izzard_temp_2017} which gives the
period distribution as a function of primary mass and bridges the
log-normal distribution for low-mass stars
\citep{duquennoy_multiplicity_1991} and a power law
\citep{sana_binary_2012} distribution for
OB-type stars. The grid was set at solar metallicity to model recent
runaway stars from nearby SNRs.
It is useful to distinguish between the runaway parameter space
$(B{-}V)_0$--$G_0$--$v_{\mathrm{ej}}$, which is best for highlighting
the different runaway production channels, and the progenitor space
$M_1$--$q$--$P_{\mathrm{orb}}$, which is best for investigating
the connection of those channels to other binary phenomena. For
instance, our plot of runaway space in Figure \ref{fig:bvvej} has
several gaps towards the top right, which, when viewed instead in the
progenitor space, turn out to be regions where the binary has merged
prior to the primary going supernova. There are several prominent
trends in Figure \ref{fig:bvvej} which will be discussed in detail in
a forthcoming paper. We note that most of the probability is
concentrated on the left edge of the plot in slow runaways of
all colours. These correspond to the scenario of binary ejection where
the two stars do not interact and the ejection velocity is purely the
orbital velocity of the companion at the time of the supernova. The
rest of the structure corresponds to cases when at some point in the
evolution the primary overflows onto the secondary and forms a common
envelope \citep{izzard_common_2012,ivanova_common_2013}. The drag force of the gas on the two stellar cores causes an
in-spiral, while the lost orbital energy heats and ejects the common
envelope. These runaways are faster due to the larger orbital velocity
from the closer orbit, but there is a small additional kick from the
impact of SN ejecta on their surface.
\begin{figure}[t]
\includegraphics[scale=0.53,trim = 2mm 7mm 30mm 0mm, clip]{{./custombvvej}.pdf}
\caption{Probability distribution in velocity-colour space of the
runaways produced by our binary evolution grid. The top and right plots show 1D projections of the joint probability distribution.}
\label{fig:bvvej}
\end{figure}
\subsection{Algorithm}
\label{sec:algo}
We want to assess the hypothesis that a given observed star with
observables $\boldsymbol{x}$ is a runaway from a SNR. Our null hypothesis
$H_0$ is that a particular star is not the runaway companion
and we wish to test this against the hypothesis $H_1$ that it is. In
Bayesian inference each hypothesis $H$ has a set of model parameters
${\boldsymbol \theta}$ which can take values in the region $\mathit{\Omega}$. $H$ is
defined by a prior $\mathcal{P}({\boldsymbol \theta}|H)$ and a likelihood
$\mathcal{L}(\boldsymbol{x}|\mathbf{\boldsymbol \theta},H)$. The Bayesian evidence for
the hypothesis is then $\mathcal{Z}$, which is given by the integral
\begin{equation}
\mathcal{Z}=\int_{\mathit{\Omega}}\mathcal{P}({\boldsymbol \theta}|H)\; \mathcal{L}(\boldsymbol{x}|{\boldsymbol \theta},H) \;\mathrm{d}{\boldsymbol \theta}.
\label{eq:evidence}
\end{equation}
The evidence is equivalent to $\mathrm{Pr}(\boldsymbol{x}|H)$, i.e. the
probability of the data given the hypothesis.
To compare the background ($H_0$) and runaway ($H_1$) hypotheses we calculate the
Bayes factor $K=\mathcal{Z}_1/\mathcal{Z}_0$, where $\mathcal{Z}_0$
and $\mathcal{Z}_1$ are the evidences for $H_0$ and $H_1$
respectively. The interpretation of Bayes factors is subjective but a Bayes factor greater than one indicates that $H_1$ is
more strongly supported by the data than $H_0$ and vice versa. A review on the use of Bayes factors is given by \citet{kass_bayes_1995} who provide a table of approximate descriptions for the weight of evidence in favour of $H_1$ indicated by a Bayes factor $K$. To aid the interpretation of our results we replicate this table in Table \ref{tab:bayes}.
\begin{table}[]
\centering
\caption{Subjective interpretation of Bayes factors $K$ (taken from \citealp{kass_bayes_1995}).}
\label{tab:bayes}
\begin{tabular}{lll}
\hline \hline $2\ln K$ & $K$ & Evidence against $H_0$ \\ \hline
0 to 2 & 1 to 3 & Not worth more than a bare mention \\
2 to 6 & 3 to 20 & Positive \\
6 to 10 & 20 to 150 & Strong \\
$>10$ & $>150$ & Very strong \\ \hline \hline
\end{tabular}
\end{table}
To obtain the evidence for $H_0$ we define a probability distribution
using the stars in the TGAS/APASS cross-match that lie in an annulus
of width $10\degr$ outside the circle from which we draw our
candidates. Assuming that the locations of the stars in the space
$(\omega,\mu_{\alpha *},\mu_{\delta},G,B{-}V)$ can be described by a probability distribution we can approximate that distribution in a non-parametric way by placing Gaussians at the location of each star and summing up their contributions over the entire space. This method is called kernel density estimation (KDE). Note that we normalise the value in each
dimension by the standard deviation in that dimension for the entire
sample. This normalisation is necessary because the different
dimensions have different units. The prior for each candidate is a
Gaussian in each dimension centred on the measured value with a
standard deviation given by the measurement error. The likelihood for
a point sampled from the prior is the KDE evaluated at that
point. Strictly speaking this is the wrong way round. The KDE should
define the prior and the likelihood should be a series of Gaussians
centred on the data, but, since the definition of the evidence is
symmetric in the prior and likelihood (Eq. \ref{eq:evidence}), we are
free to switch them.
The evidence for $H_1$ is more complicated to calculate because the
model parameters ${\boldsymbol \theta}$ are properties of the SNR and
progenitor binary and thus need to be transformed into predicted
observables $\boldsymbol{\tilde{x}}$ of the runaway. The likelihood is
\begin{equation}
\mathcal{L}(\boldsymbol{x}|{\boldsymbol \theta})=\mathcal{N}(\boldsymbol{x}|\boldsymbol{\tilde{x}}({\boldsymbol \theta}),\mathrm{Cov}(\boldsymbol{x})),
\end{equation}
where $\boldsymbol{\tilde{x}}$ is a function of ${\boldsymbol \theta}$ and
$\mathcal{N}(\boldsymbol{a}|\boldsymbol{b},\boldsymbol{C})$ denotes the PDF of a
multivariate Gaussian distribution evaluated at $\boldsymbol{a}$ with mean
$\boldsymbol{b}$ and covariance matrix $\boldsymbol{C}$. For this preliminary
work we neglect the off-diagonal terms of the covariance matrix.
\begin{table}
\caption{Model parameters for the runaway hypothesis.}
\begin{tabular}{ l l }
\hline \hline
Parameter & Description \\ \hline
$\alpha_{\mathrm{SNR}}$ & RA of the true centre of the SNR \\
$\delta_{\mathrm{SNR}}$ & DEC of the true centre of the SNR \\
$d_{\mathrm{SNR}}$ & Distance to the true centre of the SNR \\
$t_{\mathrm{SNR}}$ & Age of the SNR \\
$M_1$ & Primary mass of the progenitor binary \\
$q=M_2/M_1$ & Mass ratio of the progenitor binary \\
$P_{\mathrm{orb}}$ & Orbital period of the progenitor binary \\
$E(B{-}V)$ & Reddening along the LoS to the candidate \\
$v_{\mathrm{R,pec}}$ & Peculiar velocity in Galactic $R$ \\
$v_{\mathrm{z,pec}}$ & Peculiar velocity in Galactic $z$ \\
$v_{\phi\mathrm{,pec}}$ & Peculiar velocity in Galactic $\phi$ \\\hline \hline
\end{tabular}
\label{tab:param}
\end{table}
The prior combines the primary mass $M_1$, mass ratio $q$ and
period $P_{\mathrm{orb}}$ of the progenitor binary with the location
($\alpha_{\mathrm{SNR}}$, $\delta_{\mathrm{SNR}}$), age
$t_{\mathrm{SNR}}$, distance $d_{\mathrm{SNR}}$ and peculiar velocity
$\boldsymbol{v}_{\mathrm{pec}}$ of the SNR and the reddening
$E(B{-}V)$ along the line of sight. These model parameters are
given in Table \ref{tab:param} for reference. The prior is
\begin{align}
\mathcal{P}({\boldsymbol \theta})=&\mathcal{N}(\alpha_{\mathrm{SNR}},\delta_{\mathrm{SNR}})\mathcal{N}(d_{\mathrm{SNR}})\mathcal{U}(t_{\mathrm{SNR}})P(M_1,q,P_{\mathrm{orb}}) \nonumber \\
&P(E(B{-}V)|d_{\mathrm{run}})\mathcal{N}(v_{\mathrm{R,pec}})\mathcal{N}(v_{\mathrm{z,pec}})\mathcal{N}(v_{\phi\mathrm{,pec}}),
\end{align}
where $\mathcal{N}(a)$ denotes a univariate Gaussian
distribution in $a$, $\mathcal{N}(a,b)$
denotes a multivariate Gaussian distribution in $a$ and
$b$, $\mathcal{U}(a)$ denotes a uniform distribution
in $a$ and the other components are non-analytic. The
additional variable $d_{\mathrm{run}}$ is the predicted distance
between the observer and runaway and is a function of the other model
parameters. The ranges, means and standard deviations for the first
three and last three distributions are given in Section \ref{sec:snr}
and were used for the simple method in Section \ref{sec:simple}.
The function $P(M_1,q,P_{\mathrm{orb}})$ is the
probability that, if there is a runaway star, it originates in a
progenitor binary with those properties. This probability can be
obtained directly from the PDFs of the binary properties (Section
\ref{sec:bingrid}) after renormalising to remove the binaries which do
not produce runaway stars.
The other non-analytic function
$P(E(B{-}V)|d_{\mathrm{run}})$ expresses the probability of the reddening along the line of sight to
the observed star if it is at a distance $d_{\mathrm{run}}$. \citet{green_three-dimensional_2015} used Pan-STARRS 1 and 2MASS
photometry to produce a 3D dustmap covering three quarters of the sky
and extending out to several
kiloparsec. \citet{green_three-dimensional_2015} provide samples from
their posterior for $E(B{-}V)$ in each distance modulus bin for each
HEALPix ($\mathtt{nside}=512$, corresponding to a resolution of approximately $7\;\mathrm{arcmin}$) on the sky. We use a Gaussian KDE to obtain a smooth probability distribution for
$E(B{-}V)$ in each distance modulus bin. We then interpolate between
those distributions to obtain a smooth estimate of $\mathrm{P}(E(B{-}V)|\mu)$
which we illustrate for one sight-line towards the centre of S147 in
Figure \ref{fig:ebv}. Note that $\log_{10}d_{\mathrm{run}}=1+\mu/5$,
where $d_{\mathrm{run}}$ is a function of our other model
parameters. \citet{green_three-dimensional_2015} used the same
definition of $E(B{-}V)$ as \citet{schlegel_maps_1998} so we have
converted their $E(B{-}V)$ to the Landolt filter system using
coefficients from \citet{schlafly_measuring_2011}.
\begin{figure}[t]
\includegraphics[scale=0.50,trim = 8mm 0mm 0mm 12mm, clip]{{ebvinterp}.pdf}
\caption{The conditional PDF for the dust extinction along the
line of sight to G180.0$-1.7$ calculated by interpolating samples from the \citet{green_three-dimensional_2015} dust map.}
\label{fig:ebv}
\end{figure}
The rest of this section is devoted to describing the transform
$\boldsymbol{\tilde{x}}({\boldsymbol \theta})$ between the model parameters and
the predicted observables. The outcome of the binary evolution is a
function solely of the progenitor binary model parameters. The
pre-calculated grid of binary stars thus provides the ejection
velocity $v_{\mathrm{ej}}$, intrinsic colour $(B{-}V)_0$ and intrinsic
magnitude $G_0$, which are essential to mapping the model parameters to
predicted observables. In addition, we obtain other parameters of
interest such as the present day mass of the runaway star
$M_{\mathrm{run}}$ and the age $T_{\mathrm{run}}$.
The kinematics of the SNR centre are fully determined by the position,
distance and peculiar velocity, under the assumption that the velocity
is composed of a peculiar velocity on top of the rotation of the
Galactic disk at the location of the SNR centre. The location of
the runaway on the sky is known because the errors on the observed
position of a star with {\it Gaia} are small enough to be
negligible. The remaining kinematics that need to be predicted are the
distance $d_{\mathrm{run}}$ and proper motion. The velocity vector of
the runaway $\boldsymbol{v}_{\mathrm{run}}$ is the sum of
the velocity of the SNR and the ejection velocity vector
$\boldsymbol{v}_{\mathrm{ej}}$. The location of the explosion, now the centre of the SNR, continues along the orbit of the progenitor binary within the Galaxy. We advance the centre of the SNR and the runaway along their
orbits for the current age of the SNR $t_{\mathrm{SNR}}$, noting that
this time is so short that any acceleration is negligible and thus the
orbits are essentially straight lines. The separation of the centre of the SNR and
the runaway at this point is then simply the difference of their
velocity vectors multiplied by $t_{\mathrm{SNR}}$,
i.e. $\boldsymbol{v}_{\mathrm{ej}}t_{\mathrm{SNR}}$. We then fix the
kinematics of the model by denoting the present-day centre of the SNR
to be at $(\alpha_{\mathrm{SNR}}, \delta_{\mathrm{SNR}})$ and the
present-day distance to the centre of the SNR to be
$d_{\mathrm{SNR}}$.
To obtain predictions for the proper motions and parallax we consider
the intersection of the half-line defined by the observed position of
the candidate on the sky and a sphere centred at the distance and
position of the SNR. This sphere has a radius given by
$v_{\mathrm{ej}}t_{\mathrm{SNR}}$, which is the distance travelled by
the runaway since the supernova. A diagram of this geometry is shown
in Figure \ref{fig:geometry}. If the distance travelled by the runaway
is not large enough then the sphere fails to intersect the line and
thus the likelihood of this set of parameters is zero. In almost every case there are two intersections which correspond to the runaway
moving either away from or towards us. If the SNR is close and old and the
runaway is travelling rapidly, there is a pathological case in which
there is only one solution because the solution which corresponds to a
runaway moving towards us is already behind us. The geometry of the
intersection point gives us the distance to the star which we can use
to predict the parallax. The predicted proper motion of the runaway
depends on the velocity of progenitor binary. We sample in the
velocity dispersion of the Milky Way thin disk and add on the rotation
of the disk and ejection velocity of the runaway. This velocity is
converted to proper-motions and line-of-sight radial velocities using
the transforms of \citet{johnson_calculating_1987}.
\begin{figure}[h]
\begin{tikzpicture}
\node[inner sep=0pt] (whitehead) at (0,0)
{\includegraphics[scale=0.75,trim = 0mm 200mm 390mm 4mm, clip]{{./geometry}.pdf}};
\node[] at (-2.2,1.3) {\large$(\alpha_{\mathrm{SNR}},\delta_{\mathrm{SNR}})$};
\node[] at (3.6,1.3) {\large$(\alpha,\delta)$};
\node[] at (-0.2,2.4) {\large$v_{\mathrm{ej}}t_{\mathrm{SNR}}$};
\node[] at (-2.3,-1.1) {\large$d_{\mathrm{SNR}}$};
\end{tikzpicture}
\caption{A diagram of the geometry described in Section
\ref{sec:algo}. The observer is at {\large$\odot$} and the
centre of the supernova remnant is at {\Large$+$}. The two
possible locations of the candidate if it is the runaway are
marked by {\large$\times$} and correspond to the
intersection of the sphere of radius
$v_{\mathrm{ej}}t_{\mathrm{SNR}}$ centred on the SNR and the
half-line defined by the coordinates of the candidate.}
\label{fig:geometry}
\end{figure}
We then obtain a prediction for $B{-}V$ by simply using $B{-}V=(B{-}V)_0 +
E(B{-}V)$. The {\it Gaia} $G$ band is broader than the $V$ band and so is more sensitive to the slope of the spectrum (Sanders et al., in preparation). One consequence of this is that the relative reddening in the $G$ band $A(G)/A(V)$ is a function of the intrinsic colour of the star.
Assuming that $A(V)=R_{V}E(B{-}V)$, where the constant $R_V=3.1$ is related to the average size of the dust grains and has been empirically determined in the Milky Way \citep{schultz_interstellar_1975}, we recast this dependency as
$A(G)/E(B{-}V)$ as a function of $(B{-}V)_0$. This relation has been calculated empirically by Sanders et al. (in preparation) and thus we have an expression for the apparent
magnitude $G=G_0+A(G)+\mu$.
We elected to use nested sampling \citep{skilling_nested_2006} to
explore the parameter space since it is optimised with estimating the evidence as the primary goal while more standard
Markov Chain Monte Carlo (MCMC) methods are targeted at obtaining samples from the posterior which afterwards can be used to estimate the evidence. We use the {\sc MultiNest}
implementation of nested sampling
\citep{feroz_multimodal_2008,feroz_multinest:_2009,feroz_importance_2013}
which we access through the {\sc PyMultiNest} {\sc Python} module
\citep{buchner_x-ray_2014}. {\sc MultiNest} requires that we express
our prior as a transform from a unit hypercube to the space covered by
our prior. For independent parameters, this is a trivial application
of inverting the cumulative distribution function. However, we have
two prior probability distributions $P(E(B{-}V)|\mu)$ and
$P(M_1,q,P_{\mathrm{orb}})$ for which there are no suitable
transforms. Note that $\mu$ is the distance modulus to the runaway
which is a complicated function of the position,
distance and age of the SNR and the ejection velocity of the
runaway. For these parameters, we use the standard method of moving
the probability distribution into the likelihood, which is implemented
in {\sc MultiNest} by assuming a uniform distribution in the prior and
including a factor in the likelihood to remove this extra
normalisation. Some technical details of the implementation of {\sc
MultiNest} are discussed in Appendix~\ref{sec:multinest}.
Using nested sampling, we explore the parameter space and obtain a
value for the log of the evidence for each candidate. We then obtain
the Bayes factor by dividing the evidence for $H_1$ by the evidence
for $H_0$. A Bayes factor less than one indicates that the null hypothesis
is more strongly favoured, i.e. this star is likely a background star. A
Bayes factor greater than one suggests that the runaway model is
preferred.
\subsection{Fraction of supernovae with runaways}
\label{sec:runfrac}
Only a fraction of supernovae will result in a runaway companion. Some
massive stars are born single and companions are not always
gravitationally unbound from the compact remnant after the
supernova. Companions of massive stars tend to also be massive and so
some will themselves explode as a core-collapse supernova, either in a
bound system with the compact remnant of the primary or after being
ejected as a runaway star \citep[e.g.][]{zapartas_delay-time_2017}. A further contaminant is that binary
evolution can cause stars to merge before the primary supernova
occurs, through dynamical mass transfer leading to a spiral-in during
common envelope evolution. Our model assumes that there was a runaway
companion to the SNR and thus the calculated evidence needs to be
multiplied by the fraction of SNRs with a runaway.
Evolving a population of binary stars as described above we find that
the average number of core-collapse supernovae per binary system with
a primary more massive than $8\;M_{\sun}$ is 1.22. All single
stars in the mass range $8<M/M_{\odot}<40$ are expected to go supernova, with most stars more massive than $40\;M_{\odot}$ probably collapsing directly to black holes \citep{heger_how_2003}. Note that in the version of {\sc binary\_c} used for this work a
core-collapse supernova is signalled whenever the core of a star
collapses to a neutron star or black hole, including the case where the primary collapses directly to a black hole. Such collapses are
sufficiently rare that we do not correct for this effect. An
assumption on the binary fraction is required to combine statistics
for single and binary populations. \cite{arenou_simulated_2010}
provides an analytic empirical fit to the observed binary fraction of
various stellar masses,
\begin{equation}
F_{\mathrm{bin}}(M_1)=0.8388\tanh(0.079+0.688M_1).
\end{equation}
Based on this binary fraction and grids of single and binary stars evolved with {\sc binary\_c} we estimate that 32.5\% of core-collapse supernovae have a runaway
companion. This fraction is best described as `about a third' given
the approximate nature of the prescriptions used to model the binary evolution and the uncertainties in the empirical distributions of binary properties.
\subsection{Verification}
\label{sec:verification}
We verify our calculation of the evidence above by sampling runaways
from the model and using their
$(\omega,\mu_{\alpha,\ast},\mu_{\delta},G, B{-}V)$ to generate a kernel
density estimate of their PDF. The evidence for a candidate to be a
runaway can then be computed identically to the background
evidence. In contrast to the method described in Section
\ref{sec:algo}, where the prior and likelihood are functions of the model parameters which are described in Table \ref{tab:param}, this
method casts the prior and likelihood as a function of the model observables. In the limit where we draw infinite samples from our model this method will give the same result as the method in Section \ref{sec:algo}. Drawing samples from the model and constructing a KDE is advantageous for its simplicity. The likelihood function is
an evaluation of a KDE and thus is guaranteed to be smooth and
non-zero everywhere, meaning that the considerations discussed in
Appendix~\ref{sec:multinest} are not relevant. The first disadvantage of
calculating the evidence by this method is that it only gives accurate
values of the evidence for regions of the parameter space which are
well sampled. The second disadvantage is that by not being explicit
about the model parameters we cannot directly constrain them,
and so this method does not output the maximum-likelihood
distance to the SNR or the mass of the progenitor primary. The implicit method is used in this work solely as a
cross-check of our results.
\subsection{Validation}
\label{sec:validation}
We validate our method by considering approximations to the false
positive and the false negative rate. For each SNR we assume there is
a nominal SNR at Galactic coordinates
$(l,b)=(l_{\mathrm{SNR}}-1\degr,b_{\mathrm{SNR}})$ with the same
distance and diameter estimates as the true SNR. We acquire candidates
from our TGAS and APASS cross-match and inject an equal number of
model runaway stars sampled from our binary grid, calculating
equatorial coordinates, parallaxes and proper motions which would
correspond to a runaway from that location ejected in a random
direction. These artificial measurements are convolved with a typical
covariance matrix of errors, here using the mean covariance matrix in
our list of candidates for this nominal SNR. For the dust correction,
we randomly select one of the twenty samples provided by the
\citet{green_three-dimensional_2015} dustmap in each distance modulus
bin along each line of sight, corresponding to the sight-line and
distance modulus that we have sampled for the runaway. The injected
runaways and real candidates are shuffled together so that the
algorithm described in Section \ref{sec:algo} is applied in the same manner to both the real stars and fake runaways. Since there is not a real SNR at this location all the real stars selected from the cross-match should be preferred to be background stars, while by construction the fake injected runaways should prefer the runaway hypothesis. An injected runaway which returns a Bayes factor $K<1$
is a false negative and a real star with $K>1$ is a false positive.
At the bottom of Figure \ref{fig:bayes}, we show the calculated Bayes
factor $K$ for all the real stars and injected stars. There are 217
stars in each series. Only three of the real stars are returned as
false positives giving a false positive rate of 1.3$\%$. All three of
these false positives are from the fake version of G065.3$+05.7$ which
we find is because of the large photometric errors of APASS in
this field. These errors are around $\pm0.142$ in $B{-}V$ which compare
to $\pm0.055\;\mathrm{mag}$ for G180.0$-01.7$. This suggests that the
millimag precision of the $G_{\mathrm{BP}}$ and $G_{\mathrm{RP}}$
bands in {\it Gaia} DR2 will further reduce the false positive rate.
There are 22 false negatives which corresponds to a false negative
rate of about $10\%$. Given that we only expect a third of SNRs to
have an associated runaway companion (see Sec. \ref{sec:runfrac}) and
that we only consider ten SNRs, we should have at most one false negative in our observed sample.
We note that there are more stars closer to the $2\ln{K}=0$ boundary in our science runs (above the line in Fig. \ref{fig:bayes})
than were found in the false positive test. This is because runaways
are more likely to be OB stars and that OB stars are typically found
in star-forming regions. If a SN has occurred then a star-forming
region is nearby and so there are OB stars in or close to the SNR
which act as contaminants.
\begin{figure*}[t]
\includegraphics[scale=0.48,trim = 4mm 6mm 3mm 0mm, clip]{{allfalse}.pdf}
\caption{Bayes factors for the hypothesis that each star in each SNR is a
runaway star versus the hypothesis that it is a
contaminant. The false positive and false negative series are described in Section \ref{sec:validation}.}
\label{fig:bayes}
\end{figure*}
\section{Results}
\label{sec:results}
We report the seven stars for which the Bayes factor is
greater than one by at least the error on the evidence estimated by {\sc MultiNest}. In Figure
\ref{fig:bayes}, we show the calculated Bayes factor $K$ for the
candidates in each SNR over the range $(-20,20)$. In Section
\ref{sec:considerations} we discuss the three contaminant stars which
we are able to rule out and in Section \ref{sec:indivcand} we analyze
each of the four real candidates individually. Three of our candidates are new while HD 37424 in S147 has previously been suggested by \citet{dincel_discovery_2015}.
The only SNR in common with the search for OB runaways by
\citet{guseinov_searching_2005} is G089.0$+04.7$ and they proposed a
different candidate, GSC 03582$-00029$. This star appears to be
significantly brighter in the infrared ($J=10.2, H=9.7, K=9.6$)
than in the optical ($B=11.9$) while an OB star should have $B{-}K=-1$ \citep{castelli_new_2004}, so the classification of this star as
OB seems unlikely.
\begin{table}[]
\centering
\caption{Table of median posterior values for our new candidates.}
\label{tab:candidates}
\resizebox{0.48\textwidth}{!}{%
\begin{tabular}{llll}
\hline \hline
SNR & G074.0$-08.5$ & G089.0$+04.7$ & G205.5$+00.5$ \\
Candidate & TYC 2688-1556-1 & BD+50 3188 & HD 261393 \\ \hline
Sp. Type & --- & OB- & B5V \\
$2\ln K$ & $0.81\pm0.15$ & $3.10\pm0.12$ & $1.78\pm0.14$ \\
$\Delta\theta\;(\mathrm{arcmin})$ & 11.45 & 2.43 & 8.54 \\
$d_{\mathrm{SNR}}\;(\mathrm{kpc})$ & $0.57\pm0.07$ & $1.69\pm0.26$ & $1.32\pm0.24$ \\
$t_{\mathrm{SNR}}\;(\mathrm{kyr})$ & $100.28\pm30.02$ & $107.17\pm27.77$ & $115.64\pm23.28$ \\
$\log_{10}M_1\;(M_{\odot})$ & $1.11\pm0.17$ & $1.23\pm0.18$ & $1.07\pm0.12$ \\
$q$ & $0.16\pm0.05$ & $0.52\pm0.18$ & $0.44\pm0.11$ \\
$\log_{10}P_{\mathrm{orb}}\;(\mathrm{days})$ & $3.72\pm0.65$ & $2.41\pm1.36$ & $2.86\pm1.34$ \\
$E(B{-}V)$ & $0.06\pm0.01$ & $0.55\pm0.08$ & $0.16\pm0.04$ \\
$v_{\mathrm{pec}}\;(\mathrm{km}\;\mathrm{s}^{-1})$ & $29.20\pm10.34$ & $13.70\pm10.66$ & $16.14\pm20.41$ \\
$(B{-}V)_0$ & $0.20\pm0.06$ & $-0.25\pm0.02$ & $-0.19\pm0.03$ \\
$G_0$ & $2.39\pm0.27$ & $-2.52\pm0.47$ & $-0.92\pm0.49$ \\
$v_{\mathrm{ej}}\;(\mathrm{km}\;\mathrm{s}^{-1})$ & $161.60\pm193.32$ & $32.25\pm17.73$ & $38.82\pm26.09$ \\
$v_{\mathrm{r}}\;(\mathrm{km}\;\mathrm{s}^{-1})$ & $45.74\pm246.32$ & $-16.99\pm41.04$ & $23.82\pm51.94$ \\
$M_{\mathrm{run}}\;(M_{\odot})$ & $1.73\pm0.13$ & $10.85\pm2.93$ & $5.78\pm1.15$ \\ \hline \hline
\end{tabular}}
\end{table}
\subsection{Eliminating the contaminants}
\label{sec:considerations}
In Figure \ref{fig:bayes}, there are six stars which have Bayes
factors greater than one. The presence of two of these stars for
G074.0$-08.5$ makes it clear that there is at least some level of
contamination. We found empirically that there are two ways to produce
false positives in our model. First, if the star is a high
proper-motion star in the foreground then the evidence for it in the
background model can be spuriously low. This can occur because the
background is constructed by taking a kernel density estimate of stars
around the SNR and it may not contain enough foreground stars to
reproduce this population. A low evidence in favour of the background
model boosts the Bayes factor so that the runaway model is preferred,
even if the star would be a very low-likelihood runaway. Second, if
the errors on the photometry from APASS are greater than around
$0.1\;\mathrm{mag}$ in each of $B$ and $V$ then it is possible for the
algorithm to ascribe a high probability to a far-away blue star when
the candidate is actually a nearby red star. This increases the
likelihood in favour of the runaway hypothesis.
If a contaminant is caused by the first of these possibilities, then
this is clear from an unusually jagged posterior of the runaway model. Foreground high proper-motion stars tend to not be OB
stars and so to explain the star under the runaway hypothesis {\sc
MultiNest} is forced to sample in regions of the progenitor binary
parameter space that produce fast, red runaways. These are rare and lie in the region to the top right of Figure \ref{fig:bvvej} that is not well sampled in the binary grid because there are very few of them. This under-sampling results in
a jagged posterior dominated by spikes of high probability, with
reported modes that are poorly converged with large errors on
$\ln\mathcal{Z}$. The stars with the highest Bayes factor in both
G074.0$-08.5$ and G160.9$+02.6$ are contaminants of this first kind,
which can clearly be seen in Figure \ref{fig:bayes} as both these
stars have much broader error bars than the typical candidate.
The second type of contaminant is only a problem in this work because
we have chosen to take the photometry from APASS for all the SNRs,
while for some fields Tycho2 has much smaller errors. This is mainly
caused by a known problem in measurements taken for APASS DR8 in
Northern fields where the blue magnitudes have larger
errors than expected\footnote{\url{https://www.aavso.org/apass}}. If
the best measurement of $B{-}V$ has a large error then the problem
discussed above is a feature, because the Bayesian evidence is the
likelihood integrated against the probability of every possible
combination of model parameters. The star with the highest Bayes
factor in G065.3$+05.7$ is one such contaminant. BD+30 3621 was the
only star in G065.3$+05.7$ with a Bayes factor greater than one. APASS
reports a measurement $(B{-}V)=1.10\pm0.88$ for this star, but {\sc
MultiNest} picked out a most likely value of
$(B{-}V)_0=-0.22\pm0.02$. The Tycho 2 catalogue reports
$B{-}V=1.37\pm0.02$ confirming BD+30 3621 as a late-type star. The large
measurement error reported in APASS allowed the model to explore
parameter space where this star is much bluer than in reality. This is a preliminary study in preparation for {\it Gaia} DR2,
which will provide $G_{\mathrm{BP}}$ and $G_{\mathrm{RP}}$ with
millimag precision across the entire sky. This second type of
contaminant will not be a problem in {\it Gaia} DR2, because there
will not be more accurate photometry that we could use to
`double check' the measurements.
\begin{figure}[h]
\includegraphics{{./rosette}.pdf}
\caption{Digitized Sky Survey image of the vicinity of the
Rosette Nebula. The runaway star candidate HD 261393 is marked by a white
cross, the white cross hairs indicate the geometric centre
of the Monoceros Loop and the white circle approximately
shows the inner edge of the Monoceros Loop shell. The
Rosette Nebula is in the bottom right and the Mon OB2
association extends $3\degr$ to the east and north-east
towards the centre of the Monoceros Loop.}
\label{fig:rosette}
\end{figure}
\begin{figure*}[h]
\includegraphics[scale=0.28,trim = 55mm 45mm 40mm 40mm, clip]{{G074.0-8.5_55_marg}.pdf}
\caption{Corner plots of the posterior samples from the model
of TYC 2688-1556-1 in G074.0$-08.5$ with $1\sigma$, $2\sigma$
and $3\sigma$ contours. The 1D histograms include the CDF of
that parameter and the error bars indicate the median and
$1\sigma$ errorbars of each mode. \textbf{Bottom left:} A
corner plot showing the model parameters, excluding the five
parameters related to the position and peculiar velocity of
the SNR which did not have covariances with the other
parameters. \textbf{Top right:} A corner plot showing a
selection of the derived parameters which are functions of
the model parameters.}
\label{fig:marginalG074.0}
\end{figure*}
\begin{figure*}[h]
\includegraphics[scale=0.28,trim = 55mm 45mm 40mm 40mm, clip]{{G089.0+4.7_8_marg}.pdf}
\caption{As Fig. \ref{fig:marginalG074.0} but for the
candidate BD+50 3188 in G089.0$+04.7$.}
\label{fig:marginalG089.0}
\end{figure*}
\begin{figure*}[h]
\includegraphics[scale=0.28,trim = 55mm 45mm 40mm 40mm, clip]{{G180.0-1.7_16_marg}.pdf}
\caption{As Fig. \ref{fig:marginalG074.0} but for the
candidate HD 37424 in G180.0$-01.7$.}
\label{fig:marginalG180.0}
\end{figure*}
\begin{figure*}[h]
\includegraphics[scale=0.28,trim = 55mm 45mm 40mm 40mm, clip]{{G205.5+0.5_43_marg}.pdf}
\caption{As Fig. \ref{fig:marginalG074.0} but for the
candidate HD261393 in G205.5$+00.5$.}
\label{fig:marginalG205.5}
\end{figure*}
\subsection{Individual candidates}
\label{sec:indivcand}
\textbf{TYC 2688-1556-1} The star TYC 2688-1556-1 in G074.0$-08.5$ (Cygnus Loop) has
no known references in the literature. It is a relatively high proper-motion
star with $(\mu_{\alpha
*},\mu_{\delta})=(3.92\pm0.83,-21.03\pm1.25)\;\mathrm{mas}\;\mathrm{yr}^{-1}$
reported in TGAS. The colour and magnitude of this star in the
TGAS/APASS cross-match suggest this star is likely A type,
which agrees with the posterior for the current mass of the
runaway of $1.73\pm0.13\;M_{\odot}$. The posterior for the
ejection velocity includes a second mode which corresponds to the clump of stars at $v_{\mathrm{ej}}=700\;\mathrm{km}\;\mathrm{s}^{-1}$ in Figure \ref{fig:bvvej}. Runaways in this
region of $(B{-}V)_0\textbf{--}v_{\mathrm{ej}}$ space have undergone
significant mass exchange with the primary and will have had a common
envelope phase. This mass exchange shrinks the orbit of the binary which increases the orbital velocity and is the origin of the high velocity of these stars. If this mode is the true origin of TYC 2688-1556-1,
then the star is predicted to have lost several solar masses of
material, having started off at around $6\;M_{\odot}$ and ended with around $2\;M_{\odot}$. In
this case the star may be chemically peculiar. A more prominent
observable of this channel is that it would predict a heliocentric
radial velocity around $+ 600\;\mathrm{km}\;\mathrm{s}^{-1}$ or $- 600\;\mathrm{km}\;\mathrm{s}^{-1}$, where
the uncertainty is due to the degeneracy in whether the star is moving towards or
away from us. Looking at Figure \ref{fig:marginalG074.0}, this
degeneracy appears as a `v'-shaped contour in the
$v_{\mathrm{r}}\text{--}v_{\mathrm{ej}}$ plot. If the star is from
this mode, it is likely unbound from the Milky Way. The
covariance in the most probable mode between $M_1$ and $q$ is
simply the relationship $M_2=qM_1=\mathrm{const}$. This covariance is
interpreted as there being minimal mass transfer in the binary system
so that the mass of the runaway now is approximately the mass it was
born with. The secondary mode is clearly visible as lying off this
relationship.
\textbf{BD+50 3188} The B-type star BD+50 3188 in G089.0$+04.7$
exhibits emission lines in its spectra and so is classed as a Be star
\citep[most recently studied
by][]{chojnowski_high-resolution_2015}. The emission lines in Be
stars are thought to originate from a low-latitude disk or ring-like
envelope \citep{kogure_astrophysics_2007}, which in the case of BD+50
3188 is measured to be rotating at $138\;\mathrm{km}\;\mathrm{s}^{-1}$
\citep{chojnowski_high-resolution_2015}. Be stars are also
characterised by rapid rotation, which can be close to their break-up
speed, and this is thought to be related to their formation mechanism
\citep{kogure_astrophysics_2007}. Be stars are observed in both single
and binary systems. There are plausible formation mechanisms in the
literature which can produce Be stars that are single or binary
\citep{kogure_astrophysics_2007}. A Be star can be formed if it is the
mass gaining component in a binary in the Roche-lobe Overflow (RLOF)
phase \citep[see][]{harmanec_emission-line_1987} where the emission-lines originate in the accretion disk formed of material lost by the Roche-lobe filling companion. \citet{pols_formation_1991} argues that the duration of the RLOF phase, and hence the lifetime of the accretion disk,
is not sufficiently long to explain the high fraction of B type stars
which are Be stars. \citet{pols_formation_1991} instead
propose the post-mass-transfer model where most Be stars are in systems after the end of RLOF. During RLOF the mass-gaining component is spun up by the angular momentum of the accreted mass. If the mass gainer is rotating at close to break-up by
the end of the RLOF phase we may see emission-lines from a decretion disk around the equator of the star. Shortly after the RLOF phase, the mass loser in such a system
detonates as a supernova which may unbind the system and produce
a runaway Be star
\citep{kogure_astrophysics_2007}. \citet{rinehart_single_2000} used
proper-motions from Hipparcos to compare the velocity distributions of
B and Be type stars and, finding that they were consistent to within
$1\sigma$, argued that they were not consistent with most Be type
stars being runaways. \citet{berger_search_2001} performed a similar
analysis but with the inclusion of radial velocities and found instead
that around $7\%$ of Be stars have large peculiar
velocities. \citet{berger_search_2001} points out that the runaway
fraction among all B stars is about $2\%$ and thus this result alone
supports a runaway origin for at least some Be
stars. \citet{berger_search_2001} goes on to argue that the fraction
of Be stars which have been spun up by binary interaction is
unknown, because some binaries will remain bound post-supernova (Be +
neutron star binary) and others will either be low-mass or experience
extreme mass-loss and bypass the supernova altogether (Be + Helium
star or Be + white dwarf binaries). In more recent work, \citet{de_mink_rotation_2013} modelled massive binary stars and found that it is possible for all early-type Be stars to originate in binaries through mass transfer and mergers. \citet{rivinius_classical_2013} review the origin and physics of Be stars and conclude that the emission-lines in a majority of the systems are due to a decretion disk around a rapidly rotating star, however they conclude that binarity is not a widespread mechanism because the statistical properties of B and Be binaries appear to be identical \citep{abt_binaries_1978,oudmaijer_binary_2010}. BD+50 3188 is the only Be star
within a $2\degr$ radius of G089.0$+04.7$ and is only
$2.4\;\mathrm{arcmin}$ from the centre. That it is a Be star with no
known binary companion which is spatially co-located with the SNR
lends circumstantial evidence to it being the runaway companion of
G089.0$+04.7$.
\textbf{HD 37424} This star is our most likely candidate with a Bayes factor $K$ of $2\ln K=17.72\pm0.13$. A connection between this star and the SNR G180.0$-01.7$ was previously drawn by \citet{dincel_discovery_2015} who used the kinematics of the star and the associated central compact object PSR J0538+2817 to show both were in the same location $30\pm4\;\mathrm{kyr}$ ago. \citet{dincel_discovery_2015} estimated that this star has spectral type $\mathrm{B}0.5\mathrm{V}\pm0.5$ and a mass around $13\;M_{\odot}$, while our method found $M_{\mathrm{run}}=10.38\pm1.04\;M_{\odot}$. \citet{dincel_discovery_2015} used this mass and the lack of nearby O-type stars to argue that the progenitor primary must have a mass that is at most $20\text{--}25\;M_{\odot}$, with the possibility that the system may have been a twin binary. The most likely mode in our posterior (Fig. \ref{fig:marginalG180.0}) corresponds to a scenario where the initial masses in the binary were $M_1=20\pm5\;M_{\odot}$ and $M_2=7\pm2\;M_{\odot}$. Our favoured initial primary mass is consistent with the lack of O-type stars while we find that the secondary has actually increased in mass because of mass transfer from the primary to the companion. The possibility of a twin progenitor binary is strongly excluded under our model.
Similarly to Section \ref{sec:simple} we took $B{-}V=0.073\pm0.025$ from \citet{dincel_discovery_2015} because HD 37424 is one of the five stars in G180.0$-01.7$ without APASS photometry. We were motivated to investigate this star despite it not having APASS photometry because it had been previously suggested to be the runaway companion.
\textbf{HD 261393} The star HD 261393 in G205.5$+00.5$ is given a
spectral type of B5V by \citet{voroshilov_catalog_1985} who also
assigned it membership of NGC 2244, an open cluster at the centre of
the Rosette Nebula. However, HD 261393 is $2\fdg5$ from the
centre of the Rosette Nebula (Fig. \ref{fig:rosette}), so it is
more likely to be a member of the adjoining Monoceros OB2 association
which extends to the east and north-east by several
degrees. \citet{odegard_decameter_1986} established that the Monoceros
Loop is within the Mon OB2 association and is interacting with, and
lies behind, the Rosette Nebula. This conclusion was supported by
later work (see \citealp{xiao_radio_2012} for a review).
\citet{martins_quantitative_2012} modelled the stellar properties of
ten O type stars in NGC 2244 and the surrounding Monoceros OB2
association and found that the age of the stars is in the range
$1\text{--}5\;\mathrm{Myr}$. In order for HD 261393 to be a runaway
with an age less than $5\;\mathrm{Myr}$, our model would require the
primary of the progenitor binary to be at least
$40\;M_{\odot}$. In the posterior shown in Figure
\ref{fig:marginalG205.5} a primary of this mass would lie between the
$2$ and $3\sigma$ contours. This extra constraint would decrease the
Bayesian evidence for a runaway origin and may be enough to result in
the background being more favourable. A similar line of reasoning for
the mass of the primary was put forward by \citet{gebel_nature_1972}
who argued that the minimum possible mass of the progenitor must exceed the $25\;M_{\odot}$ mass of the most massive O
star in the SNR. The models used by \citet{martins_quantitative_2012} to estimate the age of Mon OB2 did not include the possibility of rejuvenation by mass transfer or merger in binaries which can result in an underestimated age of OB associations (e.g. \citealp{schneider_ages_2014} used binary evolution simulations to predict that the $9\pm3$ and $8\pm3$ most massive stars in the Arches and Quintuplet star clusters respectively are likely merger products). Including the possibility of binary evolution would increase the estimated age of the Mon OB2 association and thus decrease the tension with our model. \citet{gebel_nature_1972} further speculated that the
B type star HD 258982 might be the associated runaway star because it
is the only B type star observed at that time in the SNR which
displays the \ion{CaK}{} absorption line at the
$16\;\mathrm{km}\;\mathrm{s}^{-1}$ of the expanding SNR shell. HD
258982 is around $1\fdg5$ away from the geometric centre of
the Monoceros Loop and the proper motion of this star had not been
measured at the time of \citet{gebel_nature_1972}. In TGAS, this star
has a measured proper motion of around
$3\;\mathrm{mas}\;\mathrm{yr}^{-1}$ meaning that the star can have
travelled at most $0\fdg1$ in the
$150\;\mathrm{kyr}$ age of the SNR and is effectively ruled
out as a possible candidate.
\section{Conclusions}
\label{sec:conclusions}
We have used two methods to search for and quantify the significance
of runaway former companions of the progenitors of nearby SNRs. The first method used
kinematics from the Tycho-{\it Gaia} astrometric solution (TGAS) to
find the star most likely to have been spatially coincident with the
SNR centre in the past $150\;\mathrm{kyr}$ and further filtered those
candidates based on their $B{-}V$ colour. This filtering was done to
select likely OB stars. The second method is more elaborate and was
designed to make full use of the available photometry, to incorporate
3D dustmaps, to be explicit about our expectation that most but not
all runaways are OB type, and to be statistically rigorous. This
Bayesian method has the advantage that it constrains the properties of
both the progenitor binary and the present day runaway.
Both methods returned four candidates and reassuringly three of those were in common. These are TYC 2688-1556-1 in G074.0$-08.5$, BD+50 3188
in G089.0$+04.7$ and HD 37424 in G180.0$-01.7$. The remaining candidate
from the kinematic method is TYC 4280-562-1 in G114.3$+00.3$ which has
$2\ln K=-4.69\pm0.14$ in the Bayesian method and thus is the seventh
most likely runaway in this SNR. The remaining candidate from the
Bayesian method is HD 261393 in G205.5$+00.5$, which was ranked fourth
in this SNR by the kinematic method.
Three of the candidates proposed by our Bayesian method are new, while
HD 37424 was previously suggested by \citet{dincel_discovery_2015}. It
is reassuring that this star was picked out by both methods and was
already in the literature. It has a Bayes factor $K$ of $2\ln
K=17.72\pm0.13$, which makes it a very strong candidate. The posterior
suggests that this star may have gained several solar masses from the
primary prior to the supernova. The best of our new candidates is
BD+50 3188. This is a Be star which can be explained by the star
being spun up by mass transfer from the primary prior to the
supernova. It is also the only Be star within several degrees of this
SNR and is only $2.4\;\mathrm{arcmin}$ from the geometric centre. If
TYC 2688-1556-1 is the runaway companion of G074.0$-08.5$ then it is
likely to be an A type. There is a second mode in the posterior for
TYC 2688-1556-1 which would correspond to this star having mass
transferred onto its primary. It predicts that this star may be
chemically peculiar and have a velocity greater than
$600\;\mathrm{km}\;\mathrm{s}^{-1}$, making it a hypervelocity
star. The final candidate from the Bayesian method is HD 261393. It is
possible that the progenitor of the Monoceros Loop is part of the
recent burst of star formation that has occurred in the Mon OB2
association over the last $1\text{--}5\;\mathrm{Myr}$. If this is
true, then this extra constraint may mean HD 261393 is more likely to
be a background star.
The method that \citet{dincel_discovery_2015} used to propose HD 37424
as a candidate was based on a coincident spatial location with the
pulsar in the past and thus is independent from our method which
relates the star to the properties of the SNR. One advantage of our
method is that it does not require there to be a known associated
pulsar. Our Bayesian method could be altered to include stellar radial
velocities and pulsar properties. The radial velocities would be an
additional constraint on the model, the pulsar parallax could provide
a more accurate distance to the SNR, and the pulsar proper motion
combined with a time since the SN would set the location of the
progenitor binary at the time of the SN. {\sc Gaia} is aiming to provide radial velocities for a bright subset of the main photometric and astrometric sample. It is estimated that for a B1V star with apparent magnitude $V=11.3$ the end-of-survey error on the radial velocity\footnote{\url{https://www.cosmos.esa.int/web/gaia/science-performance}} will be $15\;\mathrm{km}\;\mathrm{s}^{-1}$ which is sufficiently precise for tight constraints to be placed on runaway candidates.
A requirement of our Bayesian framework is the probability of a SNR to
have a runaway companion. Accounting for single stars, merging stars,
binaries that remain bound post-supernova and runaways that themselves
go supernova, we find that one third of core-collapse SNRs should have
a runaway companion. In agreement with this result, we find three
runaway candidates from the ten SNRs considered.
As mentioned previously, \citet{kochanek_cas_2017} ruled out runaway companions of the Crab, Cas A and SN 1987A SNRs with initial mass ratios $q\gtrsim0.1$. Including this null result for these three SNRs does not change our conclusion that the number of runaway candidates is consistent with the expected number of runaways, but if our two weaker candidates (TYC 2688-1556-1 and HD 261393) are subsequently ruled out a significant tension could arise. The SNRs considered by \citet{kochanek_cas_2017} are all younger ($t_{\mathrm{SNR}}<1\;\mathrm{kyr}$) and more distant ($d_{\mathrm{SNR}}>2\;\mathrm{kpc}$) than our SNR sample, making the two works complementary. The advantage of considering young SNRs is that a runaway companion is constrained to be much nearer to the centre of the SNR which limits the region that must be searched. The main disadvantage is the lack of parallaxes for distant stars which makes it harder to exclude candidates because of the degeneracy between distance, reddening and photometry. In terms of method \citet{kochanek_cas_2017} used PARSEC isochrones to carry out a pseudo-Bayesian fit to the photometry of each star while accounting for the distance and extinction to the SNR, which we would categorize as a middle ground between our simple and fully Bayesian approaches. \citet{kochanek_cas_2017} noted that a full simulation of binary evolution was beyond the scope of their work. It is the integration of binary evolution with a fully Bayesian method which is the main advance of this work. Future \emph{Gaia} data releases will allow our fully Bayesian method to be applied to both the Crab and Cas A SNRs.
{\it Gaia} Data Release 2 (DR2) will contain positions, parallaxes,
proper-motions and $G$, $G_{\mathrm{BP}}$ and $G_{\mathrm{RP}}$ for
over a billion stars. This dataset is the reason for constructing our
Bayesian framework. The millimagnitude precision of the photometry
will remove poorly measured stars as contaminants, while the
milliarcsecond precision of the parallaxes will remove high
proper-motion foreground stars. The final {\it Gaia} data release aims
to be complete down to $G\approx20.5$ and at that completeness we will
be able to test the existence of a runaway companion for all the nearby SNRs.
Future spectroscopic observations of BD+50 3188, TYC 2688-1556-1 and
HD 261393 will test whether they truly are SN companions, allowing
them to be used to test binary star evolution. With {\it Gaia} DR2 in
early 2018, our Bayesian framework provides a sharp set of tools that
will allow us to find any runaways there are to find.
\begin{acknowledgements}
We thank Sergey Koposov, Vasily Belokurov, Jason Sanders and other members of the
Streams group at the Institute of Astronomy in Cambridge for comments
while this work was in progress. We also thank Gerry Gilmore for early discussions on this work. We are especially appreciative of Mathieu Renzo, Simon
Stevenson, Manos Zapartas and the many other authors cited above who
have contributed to the development of {\sc binary\_c}. DPB is grateful to the Science and
Technology Facilities Council (STFC) for providing PhD funding. MF is supported by a Royal Society - Science Foundation Ireland University Research Fellowship. This work was partly supported by the European Union FP7 programme through ERC grant number 320360. RGI thanks the STFC for funding his Rutherford fellowship under grant ST/L003910/1 and Churchill College, Cambridge for his fellowship. This
work has made use of data from the European Space Agency (ESA) mission
{\it Gaia} (\url{https://www.cosmos.esa.int/gaia}), processed by the
{\it Gaia} Data Processing and Analysis Consortium (DPAC,
\url{https://www.cosmos.esa.int/web/gaia/dpac/consortium}). Funding
for the DPAC has been provided by national institutions, in particular
the institutions participating in the {\it Gaia} Multilateral
Agreement. This research has made use of the APASS database, located
at the AAVSO web site. Funding for APASS has been provided by the
Robert Martin Ayers Sciences Fund. Figures \ref{fig:bvvej} and \ref{fig:ebv} made use of the {\sc cubehelix} colour scheme \citep{green_colour_2011}.
\end{acknowledgements}
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\section{Introduction}
Recommender systems have become quite essential for our modern information society. Applied within a variety of engines, they predict human behaviour (e.g. ratings a user might give to a specific item) and thus models a user's preferences. In doing so, those algorithms use specific machine learning techniques to learn about one's personal interests and to develop empathy for multiple as well as variable human aspects.
Unfortunately, human beings can not be deemed as constant functions. It has recently been shown, that users provide inconsistent ratings when requested to rate same films at different times \cite{Hill}. This \textbf{Human Uncertainty}, as we understand it in this contribution, appears to be a characteristic feature of the cognitive process of decision making which influences its outcome, making it circumstantial and temporally unstable; the outcome appears to be more or less fluctuating randomly when repeating a decision making. Consequently, we may assume that observed decisions are drawn from individual distributions \cite{delia}.
Accordingly, this complicates the evaluation of recommender systems, since it is not clear whether the difference between a given rating and the prediction is induced by the system or just a matter of Human Uncertainty. If we are able to improve the system-induced prediction quality to such an extent that only the factor of human uncertainty is left, then all visible differences within a quality metric would only exist due to this uncertainty and may vary with each repeated rating trial. This implies that rankings of different (well improved) recommender systems would shuffle with each repetition as well, i.e sound rankings do no longer exist for excellent systems but there is an equivalence class of indistinguishable optimal systems. This leads to the assumption of some Magic Barrier where natural variability may prevent us from getting much more accurate \cite{Herlocker}.
\paragraph{Motivating Example}
As a motivating example, we consider the task of rating prediction, along with the Root Mean Square Error (RMSE) as a widely used metric for prediction quality. In a systematic experiment with real users (described in more detail in forthcoming sections), individuals rated theatrical trailers multiple times. Figure \ref{fig:IntroA} shows that only 35\% of all users show constant rating behaviour, whereas about 50\% use two different answer categories and 15\% of all users make use of three or more categories. Based on these observations, we compute the RMSE for three recommender systems (designed by definition of their predictors $\pi$) for each rating trial. Figure \ref{fig:IntroB} depicts the RMSE outcomes and their frequency. It becomes apparent at once that the RMSE itself yields a particular degree of uncertainty, emerged from uncertain user feedback. When ranking these recommender systems, Figure \ref{fig:IntroB} allows for three possible results
\begin{equation}
(R1\prec R2\prec R3) \;\lor\; (R2\prec R1\prec R3) \;\lor\; (R1\prec R3\prec R2),
\end{equation}
where the relation $\prec$ denotes ``better than''.
\begin{figure}[b]
\centering
\begin{subfigure}{0.3\textwidth}
\includegraphics[width=\textwidth]{CatFreq}
\caption{Frequency of used answer categories}
\label{fig:IntroA}
\end{subfigure}
\begin{subfigure}{0.3\textwidth}
\includegraphics[width=\textwidth]{DiscreteRMSE}
\caption{Distribution of RMSE outcomes}
\label{fig:IntroB}
\end{subfigure}
\caption{Uncertain user ratings and impact on the RMSE}
\end{figure}
The ranking problem is most obvious for recommender $R1$ as it could be both, the best or the worst recommender, although it operates for the same users rating the same items. In addition, it may be possible that further repetitions of ratings would lead to even more ranking possibilities. This naturally implies to deem those RMSE scores as single draws from distributions that are strongly overlapping. As will be revealed later, recommender $R1$ is the Magic Barrier itself. Therefore, our considerations above - the indistinguishability of excellent systems close to the Magic Barrier - hold even for straightforward investigations.
\paragraph{The Problem}
The problem of Human Uncertainty - if not explicitly considered - is that any improvement to an existing system or even the assessment of different systems might not be statistically sound.
This, in particular, has financial implications when money is invested in the further development of a system but as a result, there is merely an overfitting instead of real improvements. Therefore, the crux is to recognise whether the prediction quality has really improved or is just some random artefact. So there is a need for a decision criterion whether a system still has room for improvements.
For the RMSE in particular, a criterion has recently been developed which allows for a dichotomous consideration (yes or no)\cite{MagicBarrier1}. But while the uncertainty of users is considered, its influence on the precise localisation of the Magic Barrier is negated. However, in our example (Fig \ref{fig:IntroB}) we have seen that the RMSE (esp. the Magic Barrier) itself follows a distribution due to Human Uncertainty. As a consequence, systems with an RMSE near the ``old'' Magic Barrier might already be interfered by this Human Uncertainty and respectively, achieving an RMSE less than the ``old'' Magic Barrier does not always mean that this system is already interfered. So the question changes from ``Is the prediction quality interfered by Human Uncertainty?'' to ``How likely is it that the prediction quality is interfered by Human Uncertainty?'', which allows for more differentiated evaluation of recommender systems.
\paragraph{Our Objective}
In this contribution, we present a method by which the Magic Barrier can be estimated for any quality assessment metric.
For this purpose, we will embed the Magic Barrier into a complete probabilistic framework and deduce a pragmatic theory through complexity reduction. We aim to generate concrete and action-oriented quantities that can easily be embedded in existing approaches to recommender assessment. We also provide our data records for modelling Human Uncertainty and demonstrate its transferability using the example of Netflix Prize.
\section{Related Work}
\paragraph{Recommender Systems and Assessment}
The central role of recommender systems led to a lot of research and produced a variety of techniques and approaches. A good introduction and overview is given by \cite{Jannach, Handbook}. For the comparative assessment, different metrics are used to determine the prediction quality, such as the root mean squared error (RMSE), the mean absolute error (MAE), the mean average precision (MAP) along with many others \cite{Herlocker, Bobadilla, workshop12}. Although we exemplify our methodology in accordance with the RMSE, the main results of this contribution can be easily adopted for alternative assessment metrics without substantial loss of generality, insofar they require for (uncertain) human input.
\paragraph{Dealing with Uncertainties}
The relevance of our contribution arises from the fact that the unavoidable human uncertainty sometimes has a vast influence on the evaluation of different prediction algorithms \cite{LikeLikeNot, noise2}. The idea of uncertainty is not only related to predictive data mining but also to measuring sciences such as metrology. Recently, a paradigm shift was initiated on the basis of a so far incomplete theory of error \cite{Grabe, Buffler}. In consequence, measured properties are currently modelled by probability density functions and quantities calculated therefrom are then assigned a distribution by means of a convolution of their argument densities. This model is described in \cite{GUM}. A feasible framework for computing these convolutions via Monte-Carlo-Simulation is given by \cite{GUMsupp1}. We take this as a basis for our own modelling of uncertainty for addressing similar issues in the field of computer science. To derive a pragmatic and easy to handle theory, we will refer to the Gaussian Error Propagation which is commonly used in physics as well \cite{Ku,Bevington,Taylor}.
\paragraph{The Magic Barrier}
One of the first works addressing Human Uncertainty and its impact on recommender systems was presented in \cite{Hill}, where users have been proven to give inconsistent ratings on movies. The authors claim that it will never be possible to perfectly predict ratings and that there must exist an upper bound on rating prediction accuracy. Later, this upper bound was mentioned once again in \cite{Herlocker} and received the name Magic Barrier, which is still in use nowadays. A first calculation of the Magic Barrier can be found in \cite{MagicBarrier1}. Derived by risk function minimisation, the authors defined the Magic Barrier as the square root of the averaged user variances (gathered from repeated ratings). Even though this approach accounts for Human Uncertainty, its influence - namely the uncertainty of the Magic Barrier itself - remains unconsidered. In our contribution, we complete this theory and therefore allow a more differentiated analysis of recommender assessment.
\paragraph{Experimental Designs}
The complexity of human perception and cognition can be addressed by means of latent distributions \cite{delia}. This idea is widely used in cognitive science and in statistical modelling of ordinal data \cite{cub}.
We adopt the idea of modelling user uncertainty by means of individual Gaussians following the argumentation in \cite{GaussModel} for constructing our individual response models. The methodology applied in our experiments is adopted from experimental psychology \cite{psycho} and works on repeating rating scenarios for same users-items-pairs as done before in \cite{RateAgain}.
\section{Modelling a Magic Barrier}
In this section, we embed human uncertainty into a mathematical construct and introduce an approach for estimating a Magic Barrier for a given evaluation metric. Although the term ``Magic Barrier'' is related to the RMSE in particular, such a barrier does basically exist for any metric comparing (uncertain) user inputs with predicted scores. Therefore, we first develop a general framework which will then be illustrated for the RMSE as a prominent example.
\subsection{Changing Paradigms}
As mentioned above, various experiments \cite{RateAgain,Hill} along with our own have shown that users are scattering around their true value of preference. Consequently, we may assume that observed decisions are drawn from individual distributions, as a result of complex cognition processes, and influenced by multiple factors (e.g. mood, media literacy, etc.) \cite{delia}. Therefrom, a paradigm shift has to be carried out, which is similar to the recent change of perspectives on measurement errors in metrology \cite{Buffler}: Every measurable quantity that is somehow related to human cognition is no longer considered as a single point (point-paradigm) but rather as a whole interval of possible values (set-paradigm) that is somehow distributed (distribution-paradigm). In the context of this paper, we will, therefore, consider user ratings as random variables. On this basis, we develop statistical methodologies that are to be explored hereinafter.
\subsection{Composed Quantities}
Composed quantities, in this contribution, are quantities $Z$ that compute from a continuous function $Z=g(X_1,\ldots,X_n)$ of large amounts of uncertain arguments $X_i$ (random variables). Hence, $Z$ becomes a random variable itself. This reasoning can be understood heuristically: For each draw, there is a variety of possibilities for a single outcome $x_i$ of a random variable $X_i$. The outcomes $x_1,\ldots,x_n$ of all random variables altogether result into a single outcome for the composed quantity $Z$ by means of $z = g(x_1,\ldots,x_n)$. Accounting for all the possibilities for $x_1,\ldots,x_n$ (e.g. when repeating draws infinitely) will then result in a variety of possible outcomes $z$. Thus, the distribution of $Z$ emerges as a convolution of $n$ density functions with respect to the mapping $g$ \cite{GUM, GUMsupp1}.
\subsection{Magic Barrier Estimation}
The Magic Barrier is defined as the minimum of an evaluation metric when explicitly accounting for Human Uncertainty.
Therefore, we must first specify an optimal recommender by defining its predictors. Then we have to compute the probability density function of the evaluation metric which arises for this optimal recommender.
\paragraph{What is an optimal recommender?}
The choice of predictors depends on the evaluation metric and the underlying data model. We will demonstrate this by using an example. In the case of the Root Mean Square Error (RMSE)
\begin{equation} \label{eq:RMSE}
\text{RMSE} = \sqrt{\tfrac{1}{N}\textstyle{\sum_{\nu}} (X_\nu - \pi_\nu)^2},
\end{equation}
the comparison of a rating $X_\nu$ and a prediction $\pi_\nu\in\mathbb{R}$ is done via
$c(X_\nu)=(X_\nu - \pi_\nu)^2$, whose expectation reaches its minimum when
\begin{equation}
\tfrac{d}{d\pi} \textstyle{\sum_{i=0}^N} (x_i - \pi)^2 = 2\cdot \textstyle{\sum_{i=0}^N} (\pi-x_i) = 0
\;\Leftrightarrow\; \pi = \frac{1}{N}\textstyle{\sum_{i=0}^N} x_i \quad
\end{equation}
where $x_i$ denote the realisations of the random variable $X_\nu$.
Hence, the optimal recommender system with respect to the RMSE is defined by $\pi_\nu := \mathbb{E}[X_\nu]$ for each user-item-pair $\nu$.
This might be totally different when considering the Mean Absolute Error (MAE), whose primary comparison is based on the function $c(X_\nu)=\vert X_\nu - \pi_\nu \vert$, reaching a minimum for its expectation when $\pi_\nu$ is the median of $X_\nu$. The median corresponds to the expected value, only if a symmetrical distribution is chosen as the underlying data model. Consequently, when assuming all $X_\nu\sim\mathcal{N}(\mu_\nu,\sigma_\nu)$ to be normally distributed (symmetric density function), the optimal recommender system does not differ for the RMSE and the MAE respectively. Having $X_\nu\sim\Gamma(\alpha_\nu,\beta_\nu)$ being gamma-distributed instead, the optimal recommender may be different for both metrics, depending on the extent of asymmetry.
\paragraph{Monte-Carlo-Simulation}
Now having the definition of an optimal recommender system, we need to deduce the probability density function of the evaluation metric for this optimum. In theory, this is done by a convolution of all density functions $f_i$ of $X_i$, but what sounds simple at first, turns out to be quite laborious and inapplicable as demonstrated in \cite{Chan}. For this reason, metrologists typically apply statistical simulations.
In this paper we use \textbf{Monte-Carlo-Simulations} as described in \cite{GUMsupp1}: For each of our ratings $X_\nu$, we compute a sample $\mathcal{S}(X_\nu):= \{ x^1_\nu,\ldots, x^\tau_\nu\}$ of
$\tau$ pseudo-random numbers (trials) that are drawn from a distribution (underlying data model).
Then, we yield a sample for the evaluation metric $Z=g(X_1,\ldots,X_N)$ via
\begin{equation}\label{eq:RMSE_MCM}
\mathcal{S}(Z) =\left\lbrace z_k = g(x^k_1,\ldots, x^k_N) \colon k=1,\ldots,\tau \right\rbrace.
\end{equation}
Post hoc illustration of this sample by a normed histogram with $b$ bins leads to an approximation for the density of $Z$.
Although the statistical simulation of convolutions produces excellent results while also being easy to realise, we are facing a blatant run-time problem as soon as we are entering the realm of big data. For example, for $N = 80\,000$ ratings, the simulation already takes up to an hour of runtime\footnote{Mac mini, i5 processor, 8GB DDR3-RAM}. To compute the Magic Barrier on the Netflix test record ($N = 2.8\cdot 10^6$), we need about 35 hours.
In the following sections, we will derive a pragmatic estimate for the desired density function of the Magic Barrier for arbitrary metrics. With this, we get same results but need only a mere fraction of the simulation runtime. For example, the probability density for the Magic Barrier on the Netflix test record can be computed in less than 80 milliseconds.
\paragraph{Estimation Analytics}
Even before the technical possibilities of statistical simulations existed, metrologists had estimated the expected value and the variance of quantities $Z=g(X)$. The core these estimations is to expand $g\in C^\infty(\mathbb{R})$ into its Taylor series
\begin{equation}
g(X) = \sum_{k=0}^\infty \frac {g^{(k)}(\mu)}{k!} \, (X-\mu)^k
\end{equation}
where $g^{(k)}(\mu)$ denotes the $k^\text{th}$ derivative of $g$ evaluated at the expectation of $X$. Due to the linearity of the expectation\footnote{$\mathbb{E}[aX+b] = a\mathbb{E}[X]+b$ holds for $a,b\in\mathbb{R}$ and arbitrary random variable $X$}, we yield
\begin{eqnarray} \label{eq:TaylorExpect}
\mathbb{E}[g(X)]
&=& \mathbb{E} \left[ \sum_{k=0}^\infty \frac {g^{(k)}(\mu)}{k!} \, (X-\mu)^k \right] \nonumber
= \sum_{k=0}^\infty \frac {g^{(k)}(\mu)}{k!} \mathbb{E}\left[(X-\mu)^k\right] \\
&=&\sum_{k=0}^\infty \frac {g^{(k)}(\mu)}{k!} m_k
\end{eqnarray}
where $m_k$ is the $k$-th central moment. For the variance and its quasi-linearity\footnote{$\mathbb{V}[aX+b] = a^2 \mathbb{V}[X]$ holds for $a,b\in\mathbb{R}$ and arbitrary random variable $X$}, we yield
\begin{small}
\begin{eqnarray}\label{eq:TaylorVar}
\mathbb{V}[g(X)]
&=& \mathbb{V} \left[ \sum_{k=0}^\infty \frac {f^{(k)}(\mu)}{k!} \, (X-\mu)^k \right] \nonumber
= \sum_{k=0}^\infty \left(\frac{f^{(k)}(\mu)}{k!}\right)^2 \mathbb{V}\left[(X-\mu)^k\right] \\
&=& \sum_{k=0}^\infty \left(\frac{f^{(k)}(\mu)}{k!}\right)^2 (m_{2k}-m_k^2)
\end{eqnarray}
\end{small}
\hspace{-1ex} where the last line has been simplified by using the common identity
$\mathbb{V}[(X-\mu)^k] = \mathbb{E}[(X-\mu)^{2k}] - \mathbb{E}[(X-\mu)^k]^2 = m_{2k}-{m_k}^2$. The usual approximation is to omit terms of higher orders, like
\begin{eqnarray*}
\mathbb{E}[g(X)] &=& g(\mu) + g'(\mu) \cdot m_1 + \ldots \approx g(\mu) \\
\mathbb{V}[g(X)] &=& g'(\mu)^2 m_1 + g''(\mu)^2 (m_4-m_2^2)/4+ \ldots \approx g'(\mu)^2 m_1.
\end{eqnarray*}
We have so far only considered a smooth function with just one argument in order to guarantee an easy understanding of the methodology. When considering $n$ arguments, we use a Taylor series in more dimensions and yield equivalent results which, together with the assumption of normality, form the Gaussian Error Propagation \cite{Ku,Bevington,Taylor}.
\section{Magic Barrier for the RMSE}
\subsection{Application of Gaussian Error Propagation}
In this section we will derive closed form approximations for the RMSE and therefore define
\begin{equation}
\mathcal{MB} = g(X_1,\ldots,X_N) := \sqrt{\tfrac{1}{N}\textstyle{\sum_{\nu}} (X_\nu - \mathbb{E}[X_\nu])^2}.
\end{equation}
Since we have to face multiple arguments, we would usually need a Taylor series in several variables, which is quite ugly for demonstration purposes. Therefore, we first condense all ratings $X_1,\ldots,X_N$ into a single random variable and then use the one-dimensional Taylor approximation.
In doing so, we choose Gaussians as the underlying data model for our ratings. By this means, every rating $X_\nu \sim\mathcal{N}(\mu_\nu,\sigma_\nu)$ can be written as
$X_\nu = \sigma_\nu\mathbb{I}+\mu_\nu$ where $\mathbb{I}\sim\mathcal{N}(0,1)$.
Hence, $Y_\nu:=(X_\nu - \mathbb{E}[X_\nu])^2$ receives the expectation
\begin{eqnarray}
\mathbb{E}[Y_\nu] &=& \mathbb{E}[(\sigma_\nu\mathbb{I}+\mu_\nu -\mu_\nu)^2]
= \mathbb{E}[(\sigma_\nu\mathbb{I})^2] \nonumber\\
&=& \mathbb{E}[\sigma_\nu^2\mathbb{I}^2]
= \sigma_\nu^2 \mathbb{E}[\mathbb{I}^2]
= \sigma_\nu^2 \mathbb{V}[\mathbb{I}]
= \sigma_\nu^2
\end{eqnarray}
as well as the variance
\begin{eqnarray}
\mathbb{V}[Y_\nu] &=& \mathbb{V}[(\sigma_\nu\mathbb{I}+\mu_\nu -\mu_\nu)^2]
= \mathbb{V}[(\sigma_\nu\mathbb{I})^2] \nonumber\\
&=& \mathbb{V}[\sigma_\nu^2\mathbb{I}^2]
= \sigma_\nu^4 \mathbb{V}[\mathbb{I}^2]
= \sigma_\nu^4 \left(\mathbb{E}[\mathbb{I}^4]-\mathbb{E}[\mathbb{I}^2]^2 \right)\nonumber\\
&=& \sigma_\nu^4 \left(3\mathbb{V}[\mathbb{I}]^2-\mathbb{V}[\mathbb{I}]^2 \right)
= 2\sigma_\nu^4
\end{eqnarray}
We thus obtain a $\chi^2$-distribution for $Z:=\tfrac{1}{N}\sum_{\nu}Y_\nu$ which converges into a Gaussian for a large number $N$ of ratings by means of the central limit theorem. The parameters of this Gaussian are
\begin{eqnarray}
\mathbb{E}[Z] &=& \mathbb{E}\left[\frac{1}{N}\sum_{\nu}Y_\nu \right]
= \frac{1}{N}\sum_{\nu}\mathbb{E}[Y_\nu]
= \frac{1}{N}\sum_{\nu} \sigma_\nu^2 \\
\mathbb{V}[Z] &=& \mathbb{V}\left[\frac{1}{N}\sum_{\nu}Y_\nu \right]
= \frac{1}{N^2}\sum_{\nu}\mathbb{V}[Y_\nu]
= \frac{2}{N^2}\sum_{\nu} \sigma_\nu^4.
\end{eqnarray}
Now we can consider the Magic Barrier to be the image of the root function of a single random variable, i.e. $\mathcal{MB} = g(X_1,\ldots,X_N) \equiv h(Z):=\sqrt{Z}$ where
$Z\sim \mathcal{N}(\frac{1}{N}\sum_{\nu} \sigma_\nu^2 \, , \, \frac{2}{N^2}\sum_{\nu} \sigma_\nu^4)$. Applying the one-dimensional Taylor approximation from equations \ref{eq:TaylorExpect} and \ref{eq:TaylorVar} leads to
\begin{eqnarray}
\mathbb{E}[\mathcal{MB}] &=& \sqrt{\mathbb{E}[Z]} - \frac{\mathbb{V}[Z] }{8\mathbb{E}[Z]^{3/2}} - \ldots
\approx \sqrt{\mathbb{E}[Z]} \\
\mathbb{V}[\mathcal{MB}] &=& \frac{\mathbb{V}[Z] }{4\mathbb{E}[Z]} + \frac{\mathbb{V}[Z]^2}{32\mathbb{E}[Z]^3} +\ldots \approx \frac{\mathbb{V}[Z]}{4\mathbb{E}[Z]}.
\end{eqnarray}
With additional assumption of normality (which is indeed a suitable model, as we will confirm soon), the approximated distribution of the Magic Barrier for the RMSE is
\begin{equation} \label{eq:MBapp}
\mathcal{MB}\sim\mathcal{N} \left(\sqrt{\frac{1}{N} \textstyle{\sum_{\nu}} \sigma_\nu^2} \; ,\,
\frac{1}{2N} \frac{\textstyle{\sum_{\nu}} \sigma_\nu^4}{\textstyle{\sum_{\nu}} \sigma_\nu^2} \right)
\end{equation}
where $\mathbb{E}[\mathcal{MB}] \approx (\sum_{\nu}\sigma_\nu^2/N)^{1/2}$ exactly meets the traditional Magic Barrier as defined in \cite{MagicBarrier1} and
$\mathbb{V}[\mathcal{MB}] \approx (\sum_{\nu}\sigma_\nu^4)/(2N\sum_{\nu} \sigma_\nu^2)$ represents the traditionally neglected uncertainty of this Magic Barrier, emerged from uncertain user ratings.
\subsection{Goodness of Approximation}
As mentioned above, the method presented here is merely an approximation, since we omit terms of higher orders. At this point, one may wonder how well this estimate actually matches the true state.
To answer this question, we first compare the simulated expectations and variances with the calculated ones in a regression analysis. Concerning the distribution model, we investigate the degree similarity using the Jensen–Shannon-Divergence.
\paragraph{Regression analysis}
We keep the following simulations as general as possible.
To this end we gradually fix a particular number $N$ of ratings from the set $\{50, 100, 150, 200, 500, 1000\}$ and sample $N$ expectations $\mu_\nu$ uniformly from the interval $[1,5]$ as well as $N$ variances $\sigma^2_\nu$ uniformly from $[\sigma^2_{min},\sigma^2_{max}]$. These intervals result from the assumption of five repeated ratings (as happened in our experiments) with the commonly used 5-star scale. Under these conditions, the positive variance yields limitations\footnote{Samples are only examples producing the minimum/maximum variance}
\begin{eqnarray}
\sigma^2_{min} &=& \operatorname{var}(\{1,1,1,1,2\})= 0.16 \\
\sigma^2_{max} &=& \operatorname{var}(\{1,1,1,5,5\})= 3.86
\end{eqnarray}
For each pair $(\mu_\nu,\sigma^2_\nu)$ we then compute a sample $\mathcal{S}(X_\nu)$ with $\tau=10^7$ random numbers drawn from the specified Gaussian to perform the convolution via equation \ref{eq:RMSE_MCM}. For many repetitions, we receive a lot of simulated expectations/variances to be plotted against the approximated ones by means of linear regression. A perfect match between simulation and approximation would lead to the regression $y=1\cdot x+0$ with correlation coefficient $R^2=1$. The results
\begin{eqnarray}
\operatorname{Sim}(\mathbb{E}) &=& 0.999 \cdot \operatorname{Apr}(\mathbb{E}) - 0.003 \qquad (R^2=0.99)\\
\operatorname{Sim}(\mathbb{V}) &=& 0.981 \cdot \operatorname{Apr}(\mathbb{V}) + 0.000 \qquad (R^2=1.00)
\end{eqnarray}
show that this condition is almost fully achieved and hence we may consider these approximations as appropriate.
\paragraph{Jensen–Shannon-Divergence}
\begin{figure}[t]
\centering
\includegraphics[width=\linewidth]{JSD}
\caption{Jensen–Shannon-Divergence for comparing the simulated distribution with a predetermined Gaussian}
\label{fig:JSD}
\end{figure}
When modelling the Magic Barrier, not only the expectation and the variance are of great importance, but rather the entire probability density. While the simulated distribution arises naturally from convolution, it is predetermined for the approximation. Therefore, it is necessary to evaluate the degree of deviation of both distributions. In doing so, we proceed as done in the regression analysis above, but instead of computing means and variances, we transform our samples into discrete probability distributions $P_{sim}$ and $P_{apr}$ and analyse the Jensen–Shannon-Divergence (JSD)
\begin{equation}
\operatorname{JSD}(P_{sim}| P_{apr})
= \frac{1}{2} D_\mathrm{KL}(P_{sim}| M) + \frac{1}{2} D_\mathrm{KL}(P_{apr}| M)
\end{equation}
where $D_\mathrm{KL}(P_1|P_2)=\sum _{i}P_1(i)\,\log_2 (P_1(i)/P_2(i))$ denotes the Kullback-Leibler-Divergence and $M=\frac {1}{2}(P_{sim}+P_{apr})$. Since we use the base $2$ logarithm, the JSD yields the boundaries
\begin{equation}
0 \leq \operatorname{JSD} \leq 2\log(2) \quad\text{or}\quad 0 \leq \tfrac{\operatorname{JSD}}{2\log(2)} \leq 1
\end{equation}
The outcomes for the normed JSD is shown in Figure \ref{fig:JSD}. We observe that the mid-range of all outcomes is located between $0.01$ and $0.08$ confirming high similarity of the simulated distribution and the assumed Gaussian. There are, however, some outliers which only occur for $N=50$ ratings. This can be explained by the fact that the RMSE contains the sum of squared normal distributions, which is $\chi^2$-distributed, but quickly converges to the normal distribution for $N>100$. Thus, the more ratings we have, the more adequate is a Gaussian as the assumed density. For a visual comparison of both distributions, Figure \ref{fig:ExpBarrierComp} depicts the simulated density as well as the approximation for our experiment with $N=213$.
\subsection{Understanding the Magic Barrier}
\begin{figure*}
\centering
\begin{subfigure}{0.24\textwidth}
\includegraphics[width=\textwidth]{SensMEAN}
\caption{$\mathbb{E}[\mathcal{MB}]$ with respect to $N$}
\label{fig:SensA}
\end{subfigure}
\hfill
\begin{subfigure}{0.24\textwidth}
\includegraphics[width=\textwidth]{SensVar}
\caption{$\mathbb{V}[\mathcal{MB}]$ with respect to $N$}
\label{fig:SensB}
\end{subfigure}
\hfill
\begin{subfigure}{0.24\textwidth}
\includegraphics[width=\textwidth]{SensMeanSigma}
\caption{$\mathbb{V}[\mathcal{MB}]$ with respect to $\sigma^2$}
\label{fig:SensC}
\end{subfigure}
\hfill
\begin{subfigure}{0.24\textwidth}
\includegraphics[width=\textwidth]{SensVarSigma}
\caption{$\mathbb{V}[\mathcal{MB}]$ with respect to $\sigma^2$}
\label{fig:SensD}
\end{subfigure}
\caption{Sensitivity analysis of the Magic Barrier varying the number of ratings and the extent of rating variances}
\end{figure*}
In this section, we will take a closer look at the properties of the Magic Barrier. For this purpose, the individual dependencies of the Magic Barrier and their effects are analysed in a sensitivity analysis.
In addition, we will generalise the dichotomous decision criterion from \cite{MagicBarrier1} and develop a pragmatic rule of thumb to ascertain whether a deeper consideration of the Magic Barrier seems worthwhile.
\paragraph{Sensitivity analysis}
A sensitivity analysis is used to determine how a quantity responds to the variation of its arguments. Therefore, we vary one argument within reasonable boundaries while fixing all the other arguments at the same time.
In Figure \ref{fig:SensA} and \ref{fig:SensB}, one can observe the Magic Barrier's reaction to an increasing number $N$ of ratings. It is seen that the expectation remains unaffected by the number of uncertain ratings. Only the extent of the uncertainty raises or lowers the mean value. On a 5-star scale together with five re-ratings, the expected value yields limitations (green and red) due to the minimum and maximum variance possible. The growth behaviour of the expectation under rating uncertainty is asymptotic.
However, Figure \ref{fig:SensB} reveals that the Magic Barrier's variance is heavily impacted by the number of ratings, i.e. the precision of the Magic Barrier even gains when more uncertain ratings are added. The extent of rating uncertainty also leads to boundaries, but this influence gradually disappears for increasing $N$. A comparison of Figure \ref{fig:SensB} and Figure \ref{fig:SensD} reveals that the number of ratings significantly affects the first two decimal places of the variance, whereas the influence of the rating uncertainty affects only the third and fourth decimal places at most. In summary, it can be said that the extent of Human Uncertainty alone is responsible for the location of the Magic Barrier, whilst its spread can be reduced by adding ratings. However, the degree of this improvement decreases very rapidly.
What we have omitted here is the influence of the underlying data model and the applied rating scale. The rating scale limits the variance of a user and thus has a great impact of the possible location of the Magic Barrier. The underlying data model has also a great impact on the Magic Barrier but will be the discussed separately in further research.
\paragraph{Do we need a Magic Distribution?}
Now having in mind that the variance of the Magic Barrier decreases for large $N$, one may ask if we really need the Magic Barrier to be a distribution rather that a single score. The answer depends on many factors. First of all, the world of recommender assessment does not entirely consist of large-scale experiments, so that the variance can not be deemed to equal zero. In the case of large-scale experiments, the predefined accuracy of computed scores does matter quite a lot. For example, all RMSE scores were given to the fourth decimal place in Netflix Prize \cite{netflixrules}. As shown in the following sections, the standard deviation of the Magic Barrier for the Netflix data set can be assumed to be $\sigma=0.0007$, which is still seven times larger than the specified rounding accuracy of four decimal places.
In this example, we see that even for large data records the effort of considering the Magic Barrier as a distribution is quite meaningful.
\begin{figure}[b]
\includegraphics[width=\linewidth]{ErrorIllustration}
\caption{Interference of RMSE with the Magic Barrier}
\label{fig:ErrExa}
\end{figure}
Furthermore, we need a non-vanishing variance for a statistically sound decision whether a system can still be improved.
Following \cite{MagicBarrier1}, any improvement of a recommender system is pointless, if the RMSE score is below the Magic Barrier, i.e. $\mathbb{E}[\text{RMSE}]<\mathbb{E}[\mathcal{MB}]$. But since both quantities are distributed, their density functions may nevertheless overlap. Figure \ref{fig:ErrExa} illustrates the interference of the Magic Barrier with a recommender system used in our experiments. Although the decision criterion from \cite{MagicBarrier1} holds, there is a significant probability that the RMSE outcome is already affected by the Magic Barrier. This probability is given by
\begin{equation}\label{eq:BarrierInterference}
P(\mathcal{MB}>\text{RMSE})
= \int_{-\infty}^\infty f_{\text{RMSE}}(x)\cdot\big( 1-F_{\mathcal{MB}}(x) \big) \,\mathrm{d}x
\end{equation}
where $F_{\mathcal{MB}}(x)$ denotes the cumulative distribution function. In our example from Figure \ref{fig:ErrExa}, this probability is around $0.33$, i.e. the RMSE is interfering with the Magic Barrier in one of three outcomes. For this reason, an analysis of possible improvements can not be answered by a dichotomous decision criterion (yes or no), but has to be answered by means of probabilities (How likely is it that my system can still be improved and what risk am I willing to accept?).
\paragraph{When is a differentiated consideration needed?}
However, such a differentiated approach is not always worth it.
Therefore, it would be useful to have a rule of thumb to find out whether a differentiated consideration is fruitful or not. For example, a possible criterion might be the intersection of the 99\%-confidence intervals of the RMSE and the Magic Barrier. Due to normality, further analysis should be taken into consideration, when
\begin{equation} \label{eq:criterion}
\mathbb{E}[\mathcal{MB}] + 3\sqrt{\mathbb{V}[\mathcal{MB}] }> \mathbb{E}[\text{RMSE}] - 3\sqrt{\mathbb{V}[\text{RMSE}]}.
\end{equation}
By assuming $\mathbb{V}[\mathcal{MB}] \approx \mathbb{V}[\text{RMSE}]$, which usually holds when both quantities are computed on the same data record, this criterion can be simplified to
$\mathbb{E}[\text{RMSE}] - \mathbb{E}[\mathcal{MB}] < 6\mathbb{V}[\mathcal{MB}]^{1/2}$.
\section{Experiments}
In this section, we examine our theoretical considerations in reality.
To this end, we conducted a controlled experiment with real users and measured their uncertainty. We are thus able to support the chosen data model and verify our approximation on a real data set. On this basis, possible applications can be illustrated (e.g. transferring our variances to other situations where no Human Uncertainty was explicitly measured).
\subsection{The Experiment}
Our experiment is set up with Unipark's\footnote{http://www.unipark.com/de/} survey engine while our participants were committed from the crowdsourcing platform Clickworker\footnote{https://www.clickworker.de/}.
To derive a user's rating distributions, we use the method of re-rating, which was successfully used in \cite{RateAgain, Hill} before. For this purpose, participants watched theatrical trailers of popular movies and television shows and provided ratings in five repetition trials\footnote{A full description can be found on https://jasbergk.wixsite.com/research}. User ratings have been recorded for five out of ten fixed trailers so that remaining trailers act as distractors triggering the misinformation effect, i.e. memory is becoming less accurate due to interference from post-event information.
We received a rating tensor $R_{u,i,t}$ with $\dim(R)=(67,5,5)$, having $N=1\,675$ ratings in total, where the coordinates $(u,i,t)$ encode the rating that has been given to item $i$ by user $u$ in the $t$-th trial. From this record we derive a unique rating distribution for each user-item-pair by considering tensor-slices in trial-dimension
$R_{u,i}:= \{ R_{u,i,t} \vert t=1,\ldots ,5 \}$ for which we compute Maximum-Likelihood-Parameters given a predetermined data model (e.g. Gaussians, CUB-Models, etc.). Altogether $67$ people from Germany, Austria and Switzerland participated in this experiment. This group can be parted into $57\%$ females and $43\%$ males whose ages range from $20$ to $60$ years while over $60\%$ of our participants where aged between $20$ and $40$. This group also includes a good average of lower, medium and higher educational levels. The rating frequency habits range from ``rarely'' to ``often'' in uniform distribution.
\subsection{Data Model and Uncertainties}
\paragraph{Proving the data model}
In this contribution we opt for Gaussians since they are strongly associated to human characteristics \cite{PerceptionCognition} and have also been proven to be appropriate user models in \cite{GaussModel}. Additionally, Gaussians exhibit maximum entropy along all distributions with finite mean/variance and support on $\mathbb{R}$.
For each recorded item, all tensor slices having a non-vanishing variance are checked for normality by means of a one-sample KS-test \cite{KS} with confidence level $\alpha=0.05$. The null hypothesis was never rejected, allowing to keep the Gaussian distribution as a possible model.
\paragraph{Proving Human Uncertainty}
For each of the user-item-pairs $R_{u,i}$, we compute the Gaussian ML-Parameters and consider the variances $\mathbb{V}(R_{u,i})$ as representations of the Human Uncertainty. In our experiment, only few tensor slices contain constant ratings and hence lead to a vanishing variance. Performing an item-wise analysis, the fraction of tensor slices with non-zero variance ranges from 50 to 90\% that is, only every second participant is able to reproduce its own decisions for the best case. For the worst case, only one out of ten participants is able to precisely reproduce a rating.
Figure \ref{fig:VarDist} depicts the distribution of variances emerged from repeated ratings within our experiment.
We observe that the overall variance follows an exponential distribution $\mathbb{V}\sim\operatorname{Exp}(\lambda)$ with parameter $\lambda=2.11$. This power-law distribution literally means, that many users have a low degree of uncertainty while only a few users have a very high degree of uncertainty.
\begin{figure}[b]
\includegraphics[width=.9\linewidth]{TotalVarDist}
\caption{Distribution of variances emerged from repeated ratings within our experiment}
\label{fig:VarDist}
\end{figure}
\subsection{The Magic Barrier}
Figure \ref{fig:SensA} shows that the expected value of the Magic Barrier depends solely on the Human Uncertainty.
For our five-star scale as well as its minimum and maximum variances, the expectation should - when equation \ref{eq:MBapp} holds - be located in the interval $[0.40 \, ;\, 1.55]$. In the case of our experiment, we have $N=213$ rating distributions with non-vanishing variance. It is clear from Figure \ref{fig:SensB} that for this sample size, the distribution of the Human Uncertainty has a large impact on the variance of the Magic Barrier. According to equation\ref{eq:MBapp}, the variance of the Magic Barriers should be found in the interval $[0.0008 \, ;\, 0.0113]$.
On the basis of our data record, the simulation and approximation lead to well matching expectations (ca. $ 0.733$) and variances (ca. $0.003$) for the Magic Barrier. It is apparent, that the true values are located near the lower bound of the previously estimated intervals. This can be explained by the power-law distribution, i.e. a lot of variances are near the minimum and only a few have got higher extents.
The difference of expectations is about $0.2\%$ while the difference of variances is about $1.2\%$. The matching between the simulated and the assumed data model of a Gaussian can be clearly confirmed in Figure \ref{fig:ExpBarrierComp}. The corresponding normed JSD is $0.05$.
\begin{figure}[b]
\includegraphics[width=.9\linewidth]{ExperimentalBarrierComparison.pdf}
\caption{Visual comparison of simulated and approximation Magic Barrier based on experimental data records}
\label{fig:ExpBarrierComp}
\end{figure}
\subsection{Application}
\paragraph{Implicit Impact on Recommender Assessment}
So far we have only discussed the explicit impact on the assessment of recommender systems, that is: How likely is it that a system can still be improved, just before the RMSE solely depends on Human Uncertainty itself. Now we want to investigate the implicit influence, which affects any recommender comparison, even if the corresponding RMSE distributions are not directly overlapping with the Magic Barrier.
In doing so, we generate two copies of the Magic Barrier (as the optimal recommender). Each of these copies is gradually distorted by adding artificial noise to their predictors in such a way that the relative noise difference of both copies remain constant. By increasing the noise for both copies whilst keeping their relative difference constant, we generate an offset (distance from the Magic Barrier). This offset is plotted against the probabilities of error when using the traditional point-paradigm ranking, which is given by the generalisation of equation \ref{eq:BarrierInterference}.
\begin{figure}[t]
\includegraphics[width=\linewidth]{NoiseDistancesSplines.pdf}
\caption{Error Probabilities for a point-paradigm ranking of systems with constant RMSE difference according to their overall distance to the Magic Barrier.}
\label{fig:BarrierDistance}
\end{figure}
Figure \ref{fig:BarrierDistance} depicts the family of curves, mapping the distance from the Magic Barrier to the corresponding error probabilities. This distance (x-axis) represents the overall quality of a system, i.e. the larger this quantity, the worse the prediction quality. The colours encode the relative difference $\Delta$ of two recommender systems among each other. For the green curve (representing 10\% noise of difference), an $x$-value of $0.15$ means that system 1 has a noise of 15\% whereas system 2 has a noise of 25\%. The corresponding $y$-value indicates the error probability for ranking both systems using the traditional point-paradigm. It is apparent, that two recommender systems can not be brought into a ranking order without considerable error probability if their relative difference is less than 15\%, regardless of their basic prediction quality.
As a result, we recognise the following: The distance from the Magic Barrier has a great influence on the overlaps in two constantly different recommender systems, i.e. for a fixed difference in prediction quality, they can be distinguished much better if they are bad systems, rather than good ones. On the contrary, the better a system becomes, the more improvement does a revision need, in order to be detected with statistical evidence. This basically means that a recommender system within a repeated process of improvement will certainly reach a prediction quality so that there is probably no sufficiently large amount of optimisation left, in order to distinguish further improvements from the old system with statistical evidence. This convergence is actually the true nature of the Magic Barrier, which could not have been shown without switching perspectives to the distribution-paradigm.
\paragraph{Transferability: The Netflix Prize}
Unfortunately, existing records have not gathered Human Uncertainty.
Therefore, we examine the possibility of applying the findings of our experiment to such data records. To this end, we assume the distribution of Human Uncertainty, emerged from our experiment, to be valid for a larger number of ratings. Under this condition, we will examine possible consequences on Netflix Prize as an example.
The Netflix test record consists of $N = 2.8\cdot 10^6$ ratings in total. For each of these ratings, we randomly assign a variance drawn from the Pareto distribution in Figure \ref{fig:VarDist}. According to this data, the Magic Barrier can be estimated to $\mathcal{MB}\sim\mathcal{N}(0.6687, 0.0007)$. Even though the standard deviation seems small, it is still in the range of Netflix's rounding accuracy of four decimal places. To estimate whether the contest winner \cite{netflixleaderboard} might interfere with the Magic Barrier, we use the simplification of Equation \ref{eq:criterion}. Since $\mathbb{E}[\text{RMSE}_{best}] - \mathbb{E}[\mathcal{MB}] = 0.8567 - 0.6687 = 0.1880$ is greater than
$6\mathbb{V}[\text{MB}]^{1/2} = 0.1587$, it can be assumed that the Magic Barrier has not yet been reached. In fact, there is still the potential for about 20\% of improvement when taking the winner as reference.
\section{Discussion and Conclusions}
\paragraph*{Discussion}
In our experiment, the existence of Human Uncertainty is proven and it has been shown that it corresponds to a power-law distribution, i.e. there are many users having a small variance and there are only a few users having a large variance.
This implies the existence of an offset within every prediction quality metric that emerges from Human Uncertainty, the so-called Magic Barrier. Having several recommender systems whose RMSEs, for example, are lower than this Magic Barrier, every repetition of the rating proceeding would very likely result into rearrangements of the ranking order, i.e. a reliable ranking can not be built.
In this article, we have lifted an existing theory of this Magic Barrier into a completely probabilistic methodology, providing a generalisation for any quality related metric. Our estimation provides processing of big data in little time while additionally being very precise. With our probabilistic approach, the true nature of the Magic Barrier can be demonstrated:
When approaching the Magic Barrier, the distinguishability of many recommender systems automatically decreases, supporting the idea of one equivalence class of optimal systems.
Likewise, essential properties of the Magic Barrier have been revealed, for example, the expectation does not change for a higher number of ratings. In contrast, the variance even decreases for an additional number of uncertain ratings and allows to locate the Magic Barrier more precisely. Finally, we have demonstrated the possibility to transfer our results onto other data records in order to make careful predictions of possible interference.
\paragraph*{Conclusion}
What are the consequences for the assessment of recommender systems in general? The essence of our contribution is the revelation of the following problems:
\begin{enumerate}
\item People are not able to tell us what they really mean.
\item Human Uncertainty creates a barrier from which below any assessment results are just random.
\item This barrier also implicitly influences recommender assessments; the better our systems become, the more indistinguishable they become.
\end{enumerate}
At this point it must be said that these problems are not grounded in this new perspective presented here, but have always been present in data analysis. The approach used in this contribution is just able to make these problems visible. Furthermore, these problems do not only occur within our experiments, but have also been proven by other authors in different situations of user feedback. This may have far-reaching consequences, especially in the area of the recommender systems, when the selection of a supposedly better system is a monetary decision. For example, financial resources may be invested in improving a system but the improvements achieved are purely random, which remains unnoticed.
For this reason, it becomes crucial to further examine the extent of impact of Human Uncertainty within this field of research. It is also necessary to find proper solutions for these problems, e.g. designing sophisticated mechanisms to identify uncertainty and developing novel strategies to efficiently deal with it. This naturally involves research that connects the fields of behavioural decision making, cognitive psychology and recommender systems to create interdisciplinary synergy effects. We will continue to address these issues in further research.
\newpage
\vfill\eject
\bibliographystyle{acm}
|
1611.05466
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\section{Introduction}
\label{sec:intro}
The recent discovery of the Higgs boson \cite{Aad:2012tfa, Chatrchyan:2012xdj} and the ongoing measurements of its properties \cite{Khachatryan:2016vau} are in good agreement with the hypothesis that this particle is a remnant of the Brout-Englert-Higgs mechanism, i.e. the spontaneous breaking of $\mathrm{SU}(2)_L \times \mathrm{U}(1)_Y \to \mathrm{U}(1)_\mathrm{QED}$.
While the precise determination of the Higgs and gauge boson masses, as well as the interactions of the Higgs boson with elementary particles, including itself, will continue to improve our understanding of the scalar potential's local structure in the vicinity of the vacuum, its global structure, which can possibly explain the nature of electroweak symmetry breaking, is very difficult to probe experimentally.
For example, the nature of the Higgs, whether elementary or composite, is still an open question. Even if the Higgs is assumed to be elementary, the shape of its potential remains unknown. It could be of mexican-hat shape as in
the Standard Model (SM), or it could be deformed by strong quantum corrections due to virtual effects of additional fields. Were the Higgs boson to be a composite pseudo-Nambu-Goldstone boson of a strongly-coupled sector, one would expect
a periodic potential involving trigonometric functions. In all cases, the Higgs mass is fixed by the curvature of the potential at its minimum, and so in the vicinity of the latter the shape of the potential will be similar in all possible models. Nevertheless, deviations are allowed away from the minimum. For example, one could have a barrier at zero temperature between the vacuum and the origin of field-space. Moreover, in
composite Higgs models the relation between the Higgs field's vacuum expectation value (VEV) and the gauge boson masses differs from its SM counterpart, and thus the location of the minimum in field-space may vary.
Discriminating between the different possibilities is of fundamental importance for our understanding of nature and, hence, the embedding of the effective Standard Model in an underlying UV theory. This motivates to consider possible observables which could be sensitive to the Higgs potential beyond its minimum. A possible candidate is the energy scale of baryon-number-violating processes. If baryon number is only violated by the anomaly under the weak interactions, then it follows that processes that violate baryon-number are associated with transitions between vacua classified by their weak topological charge. The minimum energy barrier between these vacua thus sets the expected scale of baryon-violating processes, which is an observable that could potentially be probed by experiments, either at colliders \cite{Aoyama:1986ej,Ringwald:1989ee,Espinosa:1989qn,Farrar:1990vb,Ringwald:1990qz,Gibbs:1994cw} or cosmic ray and neutrino detectors \cite{Morris:1991bb,Morris:1993wg,Han:2003ru,Ahlers:2005zy,Anchordoqui:2005ey,Fodor:2003bn}. Getting accurate predictions for the rates of baryon-number-violating interactions is a difficult problem, due to a possible breakdown of
the semiclassical expansion used to compute vacuum transitions. There have been extensive discussions in the literature (see for example \cite{McLerran:1989ab,Cornwall:1990hh,Arnold:1990va,Khlebnikov:1990ue,Porrati:1990rk,Khoze:1990bm,Khoze:1991mx,Rubakov:1992ec,Bezrukov:2003er,Ringwald:2003ns,Tye:2015tva}), which has not led to a definite consensus. Recent estimates point towards rates that could be probed by future experiments \cite{Ringwald:2003ns,Tye:2015tva}. However, these estimates use different methods than previous calculations giving more negative results, and a detailed understanding of the reasons for the discrepancies is still lacking. For recent analyses of measurement prospects at colliders, cosmic ray and neutrino detectors, see for example \cite{Ellis:2016ast,Brooijmans:2016lfv,Ellis:2016dgb}.
Aside from determining the rate of observable baryon-violation effects, it should be noted that the energy barrier between topological vacua can also play a crucial role in potential explanations of the baryon asymmetry of the Universe. In the early Universe, finite temperature effects become important and affect the height of the barrier. At temperatures at which the electroweak symmetry is restored, the barrier effectively disappears and vacuum transitions are unsuppressed \cite{Arnold:1987mh,Khlebnikov:1988sr,Dine:1989kt}, while below the electroweak phase transition the tunneling rate becomes Boltzmann suppressed. In scenarios of
electroweak baryogenesis \cite{Kuzmin:1985mm} (for reviews, see \cite{Trodden:1998ym,Morrissey:2012db}), the baryon asymmetry is created during the nucleation of bubbles of the broken electroweak phase in a first order transition, in such a way that unsuppressed vacuum transitions in the unbroken phase convert a chiral asymmetry into net baryon number. The latter can then survive in the broken phase only if the corresponding vacuum transitions are strongly suppressed, which enforces a bound on the relative size of the energy barrier with respect to the temperature at the onset of bubble nucleation. On the other hand, in mechanisms of leptogenesis \cite{Fukugita:1986hr} (see \cite{Buchmuller:2004nz,Davidson:2008bu} for reviews), out-of equilibrium decays or oscillations of heavy neutrinos generate a net lepton asymmetry, which is then partly reprocessed into baryon number by vacuum transitions. A viable mechanism then requires the lepton asymmetry to be generated while vacuum transitions are still active.
The existence of a minimum energy barrier between vacua can be inferred from topological arguments \cite{Manton:1983nd}, and indeed one can calculate the field configurations at the top of the barrier. These are the so called sphalerons,
which correspond to saddle-points of a bosonic energy functional. This functional depends on the spatial derivatives of the gauge and scalar fields, as well as the scalar potential. The resulting sphaleron configurations
involve a nontrivial profile for the scalar fields, which probe field values beyond the minimum of the scalar potential. Thus the resulting sphaleron energy is potentially sensitive to the details of the potential away from the Higgs vacuum. On the other hand, non-standard derivative interactions can also affect the energy functional and the sphaleron barrier.
The previous considerations motivate us to calculate the sphaleron barrier in nonstandard realizations of the Higgs vacuum, in order to look for possible deviations with respect to the SM value coming from
a modified potential and/or derivative terms. Sphaleron configurations have been calculated not only for the Standard Model \cite{Dashen:1974ck,Klinkhamer:1984di} (with a resulting energy barrier of the order of 9 TeV for the observed value of the Higgs mass), but also in a number of extensions of the Standard Model
involving an elementary Higgs and other scalars \cite{Kastening:1991nw,Enqvist:1992kd,Bachas:1996ap,Moreno:1996zm,Kleihaus:1998bh,Grant:1998ci,Grant:2001at,Funakubo:2005bu,Ahriche:2007jp,Ahriche:2009yy,Ahriche:2014jna}. In many of these models, the deviations from the SM behaviour arise mainly due to the existence of additional scalars with electroweak charges, all of them acquiring nontrivial profiles in the sphaleron configuration. Still, the sphaleron barrier was never found to deviate substantially from its SM value. In this work, we restrict to models with a single electroweak scalar, and focus on possible
large deformations of the SM case, either through sizable interactions that change the shape of the potential for an elementary Higgs, or by considering composite Higgs models, in which not only the potential is modified, but there are also new derivative interactions.
In the first case, a good example of a potential which is very different from that of the SM is one
in which the Higgs vacuum is separated from the origin by a potential energy barrier at zero temperature. Such type of scenarios was introduced in reference \cite{Grojean:2004xa}, using higher-dimensional operators, and motivated by electroweak baryogenesis. A UV completion involving extra scalars with strong couplings to the Higgs was found in \cite{Espinosa:2007qk,Espinosa:2008kw}, and the large couplings were shown not to spoil perturbation theory in \cite{Tamarit:2014dua}.
Hence, we will here adopt a general parametrization of the potential, capturing its features without worrying about the concrete realization in terms of additional scalars. We assume additional scalars to be stabilized at the origin, without inducing tadpoles in a given Higgs background, and thus playing no role in the calculation of sphaleron configurations.
Composite Higgs scenarios, well motivated by naturalness considerations, realise the Higgs boson as a pseudo-Goldstone boson with a potential that remains protected from large quantum corrections due to
an approximate global symmetry. We will center our attention on the minimal composite scenarios of reference \cite{Agashe:2004rs}, in which the pattern of global symmetry breaking is SO(5)$\rightarrow$SO(4).
The organization of the paper is as follows. In section \ref{sec:csn} we summarize the link between B+L violating processes and the sphaleron barrier. The calculation of the sphaleron configuration in
the SM is reviewed in section \ref{sec:sphalsm}, while \ref{sec:sphaldef} focuses on the case of a deformed potential. Section \ref{sec:sphalcomp} focuses on the sphaleron energy in minimal composite Higgs models and in Section~\ref{sec:summary} we offer a summary.
\section{Overview of sphalerons and B+L violation}
\label{sec:csn}
In a nonabelian gauge theory, vacua are associated with pure gauge configurations: since the Hamiltonian is gauge invariant, such configurations have the same energy as the one with zero gauge fields. Fore more general field configurations, the requirement of finite action demands them to tend to such vacuum configurations at infinity. ``Infinity'' can be understood as a 3-sphere $S_3$ of infinite radius within $\mathbb{R}^4$, and thus finite action configurations are associated with mappings from $S_3$ to the gauge group. If the group is compact, such as the electroweak SU(2)$_L$, which itself
has the topology of a sphere, the mappings are classified by an integer
winding number or topological charge, counting the number of times that the compact group can be wrapped around $S_3$.
This topological charge $q$ can be written in terms of the nonabelian field-strength as
\begin{equation}
q=\frac{1}{16\pi^2}\int d^4x\, {\rm tr} \,\tilde F_{\mu\nu}F^{\mu\nu},
\end{equation}
where $\tilde F^{\mu\nu}\equiv\frac{1}{2} \epsilon^{\mu\nu\rho\sigma}F_{\rho\sigma}$.
The integrand above is a total derivative, and thus only picks a contribution from the boundary at infinity, as expected from the fact that $q$ is associated with mappings of the sphere at infinity into the gauge group. One can always choose a so-called topological gauge in which $A_0=0$ and all the gauge field components go to zero at spatial infinity. Then the only nonzero contributions to $q$ at the boundary of $\mathbb{R}^4$ are localized at the two space slices at $t=\pm\infty$. It can then be seen that one may write
\begin{align}
\label{eq:qNS}
q=N_{CS}(t=\infty)-N_{CS}(t=-\infty),
\end{align}
where $N_{CS}(t)$, known as the Chern-Simons number, is given in the topological gauge by the following integral over a spatial slice with fixed $t$:
\begin{align}
\label{eq:NCS}
N_{CS}(t)=\frac{1}{16\pi^2}\int_{t}\!d^3x\,\epsilon_{ijk}\left(A^a_i \partial_j A^a_k+\frac{1}{3}\epsilon^{abc} A^a_i A^b_j A^c_k\right).
\end{align}
Although the topological charge $q$ is an integer, $N_{CS}$ is not necessarily so. The Chern-Simons number becomes an integer only when evaluated over pure gauge configurations. Note that, since arbitrary gauge configurations of finite action tend to a pure gauge transformation at infinity, the topological charge given by \eqref{eq:qNS} is indeed an integer.
We conclude that vacua can be characterized by integer values of the Chern-Simons number. This implies that there can be an
energy barrier between configurations with integer $N_{CS}$. One can then consider paths in field space between vacuum configurations along which the height of the barrier is minimized. The field configurations at the top of this minimal
barriers are known as sphalerons, and the height of the barrier is the sphaleron energy.
In order to be more precise about the aforementioned energy of the gauge field configurations, it can be defined, in analogy with a zero-dimensional quantum mechanics problem, from the contributions to the Hamiltonian that do not involve time derivatives. This gives a functional $V_{\rm bos}$ which in the topological gauge adopts the form
\begin{equation}
\label{eq:Vbos}
V_{\rm bos}[A_\mu^a,\phi_i]\equiv\int\!d^3x\,\left\{\frac{1}{4g^2}F^a_{ij}F^{a}_{ij}+{\cal L}_{\rm kin,sp}^{\rm matter}[A_\mu^a,\phi_i]+V^{\rm matter}[\phi_i]\right\},
\end{equation}
where $\phi_i$ represents generic scalar fields, ${\cal L}_{\rm kin,sp}^{\rm matter}$ stands for the contributions of spatial derivatives to their kinetic terms, while $V^{\rm matter}$ denotes their potential energy density. Sphalerons correspond
to saddle points of $V_{\rm bos}$, as is intuitively clear from their role as configurations with maximal energy along minimal-barrier paths between vacua. Being extremal points of $V_{\rm bos}$, sphalerons are static solutions of the Euclidean equations of motion of the theory, i.e. satisfying
\begin{align}
\partial_\nu\frac{\delta V_{\rm bos}}{\delta \partial_\nu A^a_\mu}-\frac{\delta V_{\rm bos}}{\delta A^a_\mu}=0,\quad \partial_\nu\frac{\delta V_{\rm bos}}{\delta\partial_\nu \phi_i}-\frac{\delta V_{\rm bos}}{\delta\phi_i}=0.
\end{align}
As emphasized in Sec.~\ref{sec:intro}, because $V_{\rm bos}$ is sensitive to the potential energy density of the scalars and contributions involving their spatial derivatives, the sphaleron energy can vary if either of them is modified.
Aside from the sphaleron configurations, which are extrema of $V_{\rm bos}$, one can also define constrained extrema of $V_{\rm bos}$ by demanding a fixed value of $N_{CS}$. This gives a function $V^{\rm saddle}_{\rm bos}[N_{CS}]$. Sphalerons correspond to local maxima of $V^{\rm saddle}_{\rm bos}[N_{CS}]$. As $V_{\rm bos}$ is invariant under gauge transformations, and because gauge transformations
with nontrivial topological charge change $N_{CS}$ by integer quantities, $V^{\rm saddle}_{\rm bos}[N_{CS}]$ is a periodic function of $N_{CS}$. Further, $V_{\rm bos}$ is also invariant under parity transformations of the fields, under which $N_{CS}$ changes sign. It then follows that the graph of the function $V^{\rm saddle}_{\rm bos}[N_{CS}]$ is invariant under reflections around lines with constant half-integer and integer values of $N_{CS}$, as illustrated in Fig.~\ref{fig:VbosNCS}. It is known that $V^{\rm saddle}_{\rm bos}[N_{CS}]$ can be multivalued away from integer values of $N_{CS}$, implying the existence of multiple families of extrema. In the SM, for $m_h< 12~m_W$, there is a single branch, with $V^{\rm saddle}_{\rm bos}[N_{CS}]$ having negative second derivatives in between the vacua, as in the left plot of Fig.~\ref{fig:VbosNCS}. The reflection symmetry implies then that the maximum of the curve $V^{\rm saddle}_{\rm bos}[N_{CS}]$ in between integer values of $N_{CS}$ lies at half-integer values of $N_{CS}$. Or, in other words, sphaleron configurations are invariant under parity transformations for $m_h = 125$ GeV. For $m_h\geq 12~m_W$, a new branch of sphalerons appears \cite{Kunz:1988sx,Yaffe:1989ms}, which come in pairs related by parity transformations; these are known as bisphalerons. In this case the extremal path in field-space defining $V^{\rm saddle}_{\rm bos}[N_{CS}]$ becomes multivalued, and $V^{\rm saddle}_{\rm bos}[N_{CS}]$ develops cusps.
Nevertheless one can define deformed paths for which sphaleron configurations do indeed sit atop an energy barrier \cite{Kunz:1994ah}. The situation is schematically depicted in the right plot in Fig.~\ref{fig:VbosNCS}.
\begin{figure}[h]
\begin{center}
\includegraphics[width=0.49\textwidth]{pot1.pdf}
\includegraphics[width=0.49\textwidth]{pot2.pdf}
\caption{\label{fig:VbosNCS} Left: Schematic representation of $V^{\rm saddle}_{\rm bos}[N_{CS}]$ in the presence of a single-valued branch of extremal solutions. Note the translation and reflection symmetries of the graph. Right: Illustration of $V_{\rm bos}[N_{CS}]$ evaluated at non-extremal paths between vacua when bisphalerons are present (right).}
\end{center}
\end{figure}
The former definition of bosonic potential energy, inspired by quantum mechanics, might seem ad-hoc, so that the physical meaning of $E_{\rm sph}$ needs some further clarification. In fact, it is not obvious to see how $E_{\rm sph}$ may play a role in tunneling processes between the topological vacua. The reason is that tunneling rates are computed from solutions to the full Euclidean equations of motion, known as instantons \cite{Belavin:1975fg,'tHooft:1976fv}. These differ from sphalerons
because the latter are static solutions, while instantons depend as well on time. Despite this, the sphaleron energy can play a role when considering not just spontaneous vacuum transitions, but scattering processes at a fixed energy. The existence of multiple topological vacua affects the wave function of the true vacuum, and this effect can be incorporated in a path integral formalism by including sums over field configurations
around instanton backgrounds. This gives rise to new effective instanton vertices that can be incorporated in diagrammatic expansions, which encode the nontrivial effects of the vacuum transitions. In principle, these vertices are suppressed by exponential factors involving the Euclidean action of the instantons, $\exp(-S^E_{\rm inst})$, which as said before also determine the tunneling rates. Actual calculations show that when the external
particles have energies of the order of $E_{\rm sph}$, the exponential suppression of the instanton effects can be lifted \cite{ McLerran:1989ab,Cornwall:1990hh,Arnold:1990va,Khlebnikov:1990ue,Porrati:1990rk,Khoze:1990bm,Khoze:1991mx,Rubakov:1992ec,Bezrukov:2003er,Ringwald:2003ns,Tye:2015tva}. Thus, $E_{\rm sph}$ can indeed be interpreted as a physical energy barrier between topological vacua, because the effect of vacuum transitions becomes unsuppressed when one prepares states with $E>E_{\rm sph}$.
A more direct connection between sphalerons and energy barriers can be established at finite temperature. Thermal fluctuations allow states with energies above the barrier, which can then induce classical
vacuum transitions. The thermal transition rate is determined from static solutions to the Euclidean equations of motion -- i.e. sphalerons -- and the rate scales as $\exp(-S^{E,3D}_{\rm sph}/T)=\exp(-E_{\rm sph}/T)$, where $S^{E,3D}_{\rm sph}$ is the thermal Euclidean action, defined as the spatial integral of the Euclidean Lagrangian evaluated on time-independent configurations. Thermal fluctuations induce
excitations with average energy of the order of $T$, and when $T\gtrsim E_{\rm sph}$ the rate becomes unsuppressed. Again, $E_{\rm sph}$ can be interpreted as an energy barrier between the topological vacua.
We can conclude this section by reviewing the link between sphalerons and B+L violation. In the SM, B-L is conserved while B+L is an anomalous symmetry. Denoting the SU(2)$_L$ field strength as $W_{\mu\nu}$, the B+L
current satisfies the following anomalous conservation equation,
\begin{align}
\label{eq:anomeq}
\partial_\mu J^\mu_{B+L}=\frac{3}{8\pi^2}\,{\rm tr}\, \tilde W_{\mu\nu}W^{\mu\nu}.
\end{align}
This means that a given gauge field background with topological charge $q$ induces the following change of B+L between $t=-\infty$ and $t=\infty$:
\begin{align}
\label{eq:anomaly}
\Delta(B+L)=\int d^3x \left[J^0_{B+L}(t=\infty)-J^0_{B+L}(t=-\infty)\right]=\int d^4 x \,\partial_0 J^0_{B+L}= \frac{3}{8\pi^2}\int d^4x\,{\rm tr}\, \tilde W_{\mu\nu}W^{\mu\nu}=6 q,
\end{align}
where we used Eq.~\eqref{eq:anomeq} with the assumption that the current vanishes at spatial infinity. Tunneling between topological vacua is associated with instanton configurations
which tend towards pure gauge configurations with different integer values of $N_{CS}$ at $t=\pm \infty$. Thus the instanton configurations have a nonzero topological charge $q=N_{CS}(\infty)-N_{CS}(-\infty)$, which implies that vacuum transitions are immediately associated with violations of B+L. In this way, the sphaleron energy sets the scale of baryon-number-violating processes. Equation \eqref{eq:anomaly} implies that in a vacuum transition with $\Delta N_{CS}=1$, there is a change of B+L by six units. Thus, sphaleron-related processes involve the production of large numbers of particles. The allowed processes can be identified by using the
effective instanton vertices mentioned earlier. For an instanton background with topological charge $q$, the vertices involve a number of fermion fields related to the number of fermionic zero modes of the background; the resulting interaction violates B+L by $6q$ units. For example, a one-instanton vertex inducing a transition with $\Delta N_{CS}=1$, generates an interaction with twelve fermion fields, of the form $\Pi_i (u_L d_L d_L \nu_L)_i,$ with $i=1,\dots 3$ labelling the generations \cite{Harvey:1990qw}. This can for example give rise to the creation of three baryons and three neutrinos from the vacuum, or can induce $2\rightarrow 10$ processes with quarks and leptons. As mentioned before, the production cross sections are up for debate.
In the following, we will calculate the sphaleron energy in elementary Higgs boson scenarios with a modified potential, and in composite Higgs boson scenarios. In the first case, the modified Higgs potential can be understood
as arising from the virtual effects of heavier fields. In the second case, the sphaleron energy can be calculated in an effective
theory arising after integrating out modes of the strongly coupled sector. We have argued before that sphaleron effects become relevant at processes with energies of the order of the sphaleron energy. Then if $E_{\rm sph}$ is larger than the mass of the heavy fields or the compositeness scale, the question might arise of whether at those energies one can still trust the original calculation of the minimum energy barrier. This is the case because the effective theory in which the heavy fields are integrated out describes the dynamics when those fields lie at their energy minima, and so minimal energy configurations of the full theory can be reliably calculated in the effective description. In the composite case, it should be noted that the Higgs, being a pseudo-Goldstone boson, is protected by the global symmetry of the composite sector. Interactions inside the latter cannot generate contributions to the Higgs potential, which arises from interactions that break the global symmetry and are already taken into account in the effective theory. The situation is then similar to the case of an elementary Higgs with an effective potential induced by heavy fields, and the previous conclusion applies.
\section{Sphaleron energy in the Standard Model}
\label{sec:sphalsm}
In this section we review the calculation of the SM sphaleron configuration, mostly following the treatment in \cite{Schaldach}. As we are considering the minimum barrier between vacua with different weak topological charge, we can simply restrict to field
trajectories connecting the vacua without exciting degrees of freedom that do not couple to the weak bosons -- doing otherwise would just give higher energy configurations. This allows to ignore gluons, and forces
to consider the Higgs field. As in a nonzero Higgs background the weak bosons mix with the hypercharge boson, in principle one should take it into account it as well, but because the mixing is small, the effect is subleading (less than $1\%$, \cite{Kleihaus:1991ks,Ahriche:2014jna}) and will
be ignored. Thus one has to consider the functional
\begin{equation}
\label{eq:VbosSM}
V^{\rm SM}_{\rm bos}[A_\mu^a,H]=\int\!d^3x\,\left\{\frac{1}{4g^2}W^a_{ij}W^{a}_{ij}+D_i H^\dagger D_i H+V(H)\right\},
\end{equation}
where $D_i H=\partial_i H-i\sigma^a A^a_i H$, with $\sigma^a$ being the usual Pauli matrices. $V(H)$ is the Higgs potential normalized to be zero at the Higgs vacuum, so that $V^{\rm SM}_{\rm bos}$ evaluated at the sphaleron configuration can be directly interpreted as the energy barrier between topological vacua.\footnote{This is because with this choice of normalization, the bosonic energy of the vacuum configuration with zero gauge fields and the Higgs at its VEV becomes zero.} At tree level $V(H)$ is given in terms of the Higgs mass squared $m^2_h$ and the Higgs VEV $v$ by
\begin{align}
V(H)=-\frac{m^2_h}{2v^2}\left(H^\dagger H-\frac{v^2}{2}\right)^2.
\end{align}
It is useful to work in dimensional units, and to do so we rescale the fields and coordinates in units of the $W$ mass, which in the limit of zero Weinberg angle is $m^2_W=g^2 v^2/4$:
\begin{align}
\label{eq:rescaling}
x^\mu&\rightarrow\frac{1}{m_W} y^\mu,& A^a_\mu&\rightarrow m_W \tilde A^a_\mu, & H&\rightarrow\frac{m_W}{\sqrt{2}g} \tilde H.
\end{align}
Then one can find the sphaleron configuration by extremising the dimensionless functional
\begin{align}
\label{eq:rescaledVbosSM}
\tilde V^{\rm SM}_{\rm bos}=\frac{1}{g^2}\int\!d^3y\,\left\{\frac{1}{4g^2}\tilde W^a_{ij}\tilde W^{a}_{ij}+\frac{1}{2}D_i \tilde H^\dagger D_i \tilde H+\tilde V(\tilde H)\right\},
\end{align}
where $ \tilde V(\tilde H)\equiv\frac{\kappa^2}{32}(\tilde H^\dagger \tilde H-4)^2$ and $\kappa^2\equiv\frac{m^2_h}{m^2_W}$. The equations of motion of the sphaleron configuration are
\begin{eqnarray}\begin{aligned}
\label{eq:eomSM}
&({\cal D}_j \tilde W_{ij})^a+\frac{i}{4}\left(\tilde H^\dagger \sigma^a D_i \tilde H-D_i \tilde H^\dagger \sigma^a D_i\tilde H\right)=0,\\
&\left[D^2_i-2\frac{\partial}{\partial ( \tilde H^\dagger \tilde H)}\tilde V(\tilde H)\right]\tilde H=0.
\end{aligned}\end{eqnarray}
To solve the former equations, we impose a rotationally symmetric ansatz\footnote{This ansatz is often called hedgehog solution.} \cite{Dashen:1974ck,Manton:1983nd,Klinkhamer:1984di,Akiba:1988ay}. Defining $r\equiv \sqrt{\sum y^2_i}$ and $n_i\equiv y_i/r$, the ansatz is given by
\begin{equation}
\label{eq:ansatz}
\begin{aligned}
&\tilde A_i^a=\epsilon_{aij}n_j\frac{1-A(r)}{r}+(\delta_{ai}-n_an_i)\frac{B(r)}{r}+n_an_i\frac{C(r)}{r},\\
&\tilde H(r)=2\left(F(r)\,\mathbb{I}+i G(r)\,\vec{n}\cdot\vec{\sigma}\right)\left[
\begin{array}{c}
0\\
1
\end{array}\right].
\end{aligned}
\end{equation}
One can consider SU(2)$_L$ transformations preserving the $A_0=0$ gauge condition. Taking a group element of the form
\begin{equation}
U(r)=\exp[{\vec{n}\cdot\vec{\sigma}} P(r)]=\cos P(r)+i{\vec{n}\cdot\vec{\sigma}}\sin P(r),
\end{equation}
the functions in the ansatz of \eqref{eq:ansatz} transform as
\begin{equation}
\label{eq:gaugetr}
\begin{aligned}
A\rightarrow &\, A\cos 2P-B \sin 2P,& B\rightarrow& B\cos 2 P+ A\sin 2P, &C\rightarrow& C+ 2r P',\\
F\rightarrow &\, F \cos P - G\sin P, & G\rightarrow& G\cos P+F \sin P.
\end{aligned}
\end{equation}
We can use this freedom to set $C(r)=0$, although the price to pay is that one will lose the topological gauge condition $A_i\rightarrow0$ for $r\rightarrow\infty$.\footnote{In the topological gauge, given the ansatz \eqref{eq:ansatz}, $A_i\rightarrow0$ implies for example $A(r)\rightarrow1$, which is not respected by the gauges transformations of equation \eqref{eq:gaugetr}.} Inserting the ansatz \eqref{eq:ansatz} with $C(r)=0$ into the first of the equations in \eqref{eq:eomSM}, one gets an equation of the form
\begin{equation}
\label{eq:eqgauge}
\begin{aligned}
E_1 \frac{2n_an_i}{r^2}+E_2\frac{n_an_i-\delta_{ai}}{r}+E_3\,\epsilon_{aij}\frac{n_j}{r}=0.
\end{aligned}
\end{equation}
The orthogonality of the 2-index objects with indices $a,i$ of Eq.~(\ref{eq:eqgauge}) means that its
solutions must satisfy $E_1=E_2=E_3=0$, which yields
\begin{equation}
\label{eq:Es}
\begin{aligned}
&\,B A'-AB'+r^2 (GF'-FG')=0,\\
&\,B''-\frac{B}{r^2}\left(A^2+B^2-1\right)+2GF-B(G^2+F^2)=0,\\
&\,A''-\frac{A}{r^2}\left(A^2+B^2-1\right)-A(G^2+F^2)-G^2+F^2=0.
\end{aligned}
\end{equation}
The second equation in \eqref{eq:eomSM}, after substitution of the ansatz, adopts the form
\begin{align}
E_4\,\mathbb{I}+E_5\,{\vec{n}\,\vec{\sigma}}=0.
\end{align}
This implies $E_4=E_5=0$, which gives
\begin{eqnarray}\label{eq:E4E5SM}\begin{aligned}
&\frac{2}{r^2}(r^2G')'-\frac{G}{r^2}\left((A+1)^2+B^2\right)+\frac{2BF}{r^2}-\kappa^2G(F^2+G^2-1)=0,\\
&\frac{2}{r^2}(r^2F')'-\frac{F}{r^2}\left((A-1)^2+B^2\right)+\frac{2BG}{r^2}-\kappa^2F(F^2+G^2-1)=0.
\end{aligned}
\end{eqnarray}
By calculating the derivative with respect to $r$, one can show that the first equation of Eqs.~\eqref{eq:Es} is not independent of the others, leaving four equations with four unknown functions.
Solving them requires to impose boundary conditions for the unknown functions and their derivatives.
At large $r$, finiteness of $V_{\rm bos}$ evaluated with the sphaleron solution implies that gauge fields must approach a pure gauge configuration, while the scalar fields must tend to a minimum of their potential. The choice of boundary conditions can be simplified by obtaining asymptotic solutions with the desired properties,
which will depend on fewer parameters. For the SM, the asymptotic solutions for large and small $r$ at the chosen accuracy level depend each on 3 parameters, and are given in appendix \ref{app:asympt}.
A regular sphaleron solution can be found by applying an iterative numerical procedure such that, at each step, one obtains two solutions to the sphaleron equations by imposing boundary conditions at large and small r, respectively, while the steps are repeated with varying boundary conditions until the two solutions match smoothly at an intermediate value of $r$.
Before illustrating the solutions, it should be noted that one can reduce the equations further by redefining the unknown functions. Given
the gauge transformation properties \eqref{eq:gaugetr}, one may define gauge-invariant quantities $R^2\equiv A^2+B^2$, $S^2\equiv H^2+G^2$. Then one has
\begin{equation}
\label{eq:RS}
\begin{aligned}
A=&R \cos\theta, & B=& R \sin\theta,\\
F= &\,S \cos \phi, & G=& S\sin\phi.
\end{aligned}
\end{equation}
The above mapping does not uniquely define the variables $R,S,\theta,\phi$, since $A,B,F,G$ are invariant under two discrete transformations, i.e.
\begin{eqnarray}\begin{aligned}
\label{eq:jumps1}
R\rightarrow R,\quad \theta=\theta+2m\pi, \\
S\rightarrow S,\quad \phi\rightarrow\phi+2n\pi,
\end{aligned}\end{eqnarray}
and
\begin{eqnarray}\begin{aligned}
\label{eq:jumps2}
R\rightarrow-R,\quad \theta=\theta+(2m+1)\pi, \\
S\rightarrow-S,\quad \phi\rightarrow\phi+(2n+1)\pi,
\end{aligned}\end{eqnarray}
with $m,n,\in\mathbb{Z}$.
When looking for smooth sphaleron profiles, it should be noted that the former discrete changes in $R,S,\theta,\phi$ can still be admitted, since they don't affect
the functions $A,B,F,G$. In terms of the new variables the four independent equations become
\begin{eqnarray}
\label{eq:RSSM}
\begin{aligned}
&r^2R''+r^2S^2 \cos [2 \phi -\theta ]+R-R \left(R^2+r^2(\theta '^2+S^2\right))=0,\\
&2 r^2 S''-2r^2 S \phi'^2+4r S'-S \left(\kappa^2 r^2 \left(S^2-1\right)-2 R \cos [2 \phi -\theta ]+R^2+1\right)=0,\\
&R \theta ''+2 \theta ' R'+S^2 \sin [2 \phi -\theta ]=0,\\
&r^2 S \phi''+2 r\phi' \left(r S'+S\right)-R S \sin [2 \phi -\theta ]=0.
\end{aligned}\end{eqnarray}
The last two equations can be solved by
\begin{align}
\label{eq:thetas}
\theta'=\phi'=0~~\mathrm{and}~~\phi=\frac{\theta}{2}+\omega\frac{\pi}{2},\,~~\mathrm{with}~~\omega\in \mathbb{Z},
\end{align}
which finally yields
\begin{equation}
\label{eq:eqsRsimpleSM}
\begin{aligned}
&r^2 R''-{R^3}+ R \left(1-r^2 S^2\right)\pm r^2 S^2=0,\\
&2r^2 S''+4 r S'-S \left((R\mp1)^2+\kappa^2 r^2(S^2-1)\right)=0.
\end{aligned}
\end{equation}
The upper and lower signs are associated with even and odd $\omega$ in \eqref{eq:thetas}, and the corresponding equations can be related by the transformation $R\rightarrow-R$. However, if the sign of $R$ is fixed at large values of $r$ with a suitable boundary condition, both types of equations could give rise to different branches of sphalerons. For $m_h=125$ GeV and $R>0$ at large values of $r$, only the upper-sign branch has solutions. Equations \eqref{eq:thetas}, \eqref{eq:eqsRsimpleSM} do not allow to fix the constant values of $\theta,\phi$, which, given the identities in Eq.~\eqref{eq:RS}, prevents to reconstruct the values of the four unknown functions $A,B,H,G$ in the ansatz \eqref{eq:ansatz} in the gauge $C=0$. Nevertheless, $\theta$ can be determined from the generic properties of the functional $V^{\rm saddle}_{\rm bos}[N_{CS}]$ introduced in section \ref{sec:csn}, up to the ambiguity of Eqs.~\eqref{eq:jumps1} and \eqref{eq:jumps2}. As mentioned before, for the observed value of the Higgs mass there is a single branch of parity-invariant sphaleron solutions, and the symmetries
of $V^{\rm saddle}_{\rm bos}[N_{CS}]$ then imply that sphalerons have $N_{CS}=1/2+n,\,\, n\in \mathbb{Z}$. In order to get the expression of $N_{CS}$ in the $R,S,\theta,\phi$
field coordinates, one has to be careful because the relation of Eq.~\eqref{eq:NCS} for $N_{CS}$ is only valid in a topological gauge with $A_i\rightarrow0$ for $r\rightarrow\infty$. However, in order to eliminate
the function $C(r)$ from the ansatz \eqref{eq:ansatz} we performed a further gauge transformation which can violate the previous gauge condition. Nevertheless, one can use the properties of gauge transformations in Eq.~\eqref{eq:gaugetr} to map the fields in the $C=0$ gauge into fields in the topological gauge, where Eq.~\eqref{eq:NCS} holds. Expressing the result in terms of functions in the $C=0$ gauge one finally obtains:
\begin{equation}
\label{eq:NCSC0}
N_{CS}=\frac{1}{2\pi}\int dr (A'B-B'A)+\frac{1}{2\pi}\arctan\frac{B_\infty}{A_\infty}=\frac{\theta_\infty+ n\pi}{2\pi}-\frac{1}{2\pi}\int dr R^2\theta', \,\,n\in\mathbb{Z}.
\end{equation}
$B_\infty, A_\infty,\theta_\infty$ denote the values of the corresponding functions at infinity. In the $C=0$ gauge it no longer holds that the gauge fields vanish at $r\rightarrow\infty$. The ambiguity in $\theta_\infty$ up to multiples of $\pi$ is due to the discrete redundancy of Eqs.~\eqref{eq:jumps1} and \eqref{eq:jumps2}. From Eq.~\eqref{eq:NCSC0}, when imposing Eq.~\eqref{eq:thetas} one can see that the sphaleron solutions with $N_{CS}=1/2$ have constant $\arctan B_\infty/A_\infty=\theta_\infty+ n\pi=\pi$. This, together with Eq.~\eqref{eq:thetas}, allows to fix the ansatz \eqref{eq:ansatz} in the $C=0$ gauge by simply solving the two differential equations in \eqref{eq:eqsRsimpleSM}.
The boundary conditions for the two functions $R$ and $S$ can be obtained from the asymptotic solutions for the functions $A,B,G,F$ in appendix \ref{app:asympt}, imposing Eq.~$\eqref{eq:thetas}$ and $\theta=\pi+n\pi$.
At the chosen level of accuracy, this reduces the free parameters of the asymptotic solutions from six to four.
For $N_{CS}=1/2$ and $R>0$ at large values of $r$, only the upper sign choice in Eq.~\eqref{eq:eqsRsimpleSM} gives a solution, and one can choose $\theta=\pi$. The upper sign choice corresponds to even $\omega$ in Eq.~\eqref{eq:thetas}, i.e. $\phi=\theta/2+n \pi,$ with $n\in\mathbb{Z}$. As is clear from Eq.~\eqref{eq:RS}, this implies that the sphaleron has $F=B=0$ for all $r$. As mentioned earlier, for $r\rightarrow\infty$ the scalar field must lie in a minimum of its potential energy in order for the sphaleron to have finite energy. This is satisfied for $F^2+G^2=S^2=1$, as can be seen from the ansatz \eqref{eq:ansatz} and the rescaled potential term in $\tilde V_{\rm bos}^{\rm SM}$ in Eq.~\eqref{eq:rescaledVbosSM}. On the other hand, regularity at $r=0$ forces $G(0) = 0$, which, together with the condition
$F(r)=0\,\forall\, r$, means that the scalar field must be zero at $r=0$. Thus, the sphaleron probes the Higgs potential between the origin ($F=G=0$) and the vacuum configuration ($F^2+G^2=1$).
The sphaleron energy can be obtained from $m_W \tilde V_{\rm bos}$ evaluated in the sphaleron configuration; in terms of the $R,S,\theta,\phi$ variables, $\tilde V_{\rm bos}$ is equal to
\begin{eqnarray}\begin{aligned}
\label{eq:tildeVbosSM}
\tilde V_{\rm bos}=\frac{2\pi }{g^2}\int \frac{dr}{r^2}\,&\left\{2 r^2 \left[{R'}^2+R^2 {\theta '}^2+2 r^2 \left(S^2 {\phi '}^2+{S'}^2\right)\right]+\kappa ^2 r^4 \left(S^2-1\right)^2+2 R^2 \left(r^2 S^2-1\right)\right.\\
&\left.-4 r^2 R S^2 \cos [\theta -2 \phi ]+2 r^2 S^2+R^4+1\right\}.
\end{aligned}\end{eqnarray}
Solving the different systems of equations -- either \eqref{eq:Es} and \eqref{eq:E4E5SM}, or the system \eqref{eq:RSSM}, or the reduced system \eqref{eq:eqsRsimpleSM} -- with the iterative procedure described above, fixing $m_h=125.09$ GeV and $m_W=80.398$ GeV \cite{Olive:2016xmw} we recover in all cases the known value of the SM sphaleron barrier,
\begin{align}
E^{SM}_{\rm sph}=9.11\, {\rm TeV}.
\end{align}
Fig. \ref{fig:profiles_SM} illustrates the profiles for $R$ and $S$ in the sphaleron solution, as well as the contributions to the dimensionless bosonic energy density -- defined as the integrand in equation \eqref{eq:tildeVbosSM} -- from the derivatives of the gauge fields, those of the scalars, and the scalar potential. The contribution from the potential is substantially lower than that of the derivatives. This hints towards a limited sensitivity of $E_{\rm sphal}$ to the details of the scalar potential, and greater sensitivity to modified derivative interactions. This will be confirmed in the following section dedicated to nonstandard Higgs scenarios.
\begin{figure}[h!]
\begin{center}
\begin{minipage}{0.5\textwidth}
\includegraphics[width=.95\textwidth]{SM_2.pdf}
\end{minipage}%
\begin{minipage}{0.5\textwidth}
\hfil\hskip0.8cm\includegraphics[width=.95\textwidth]{VbosSM.pdf}
\end{minipage}
\caption{Left: Profiles for $R,S$, in units of $m_W$, in the SM sphaleron configuration obtained by solving the reduced system of 2 differential equations. The vertical line marks the scale at which the low $r$ solution (red) was matched with the high $r$ solution (blue). Right: Contributions to the dimensionless integrand in $\tilde V_{\rm bos}^{SM}$, evaluated on the sphaleron solution, due to the gauge fields (solid blue), derivatives of the scalar field (dashed orange) and the potential
energy density of the Higgs (dotted green).}
\label{fig:profiles_SM}
\end{center}
\end{figure}
\section{Sphaleron energy for an elementary Higgs in a deformed potential}
\label{sec:sphaldef}
As an illustration of the effect of a modified potential away from the Higgs vacuum, in this section we consider a theory with an elementary Higgs, yet with a nonstandard potential. The experience with the SM shows that the sphaleron configuration for an elementary Higgs is sensitive to field values between the origin and the vacuum configuration, as follows from the
boundary conditions at $r\rightarrow\infty$ and $r\rightarrow0$. Thus we may consider potentials which deviate from the SM in this
region, while having a minimum whose VEV and curvature reproduce the correct Higgs and gauge boson masses. A potential which is very different from the SM can be achieved for example if the Higgs vacuum at zero
temperature is
separated from the origin of field space by a potential energy barrier. Such type of scenarios was introduced in reference \cite{Grojean:2004xa}, using higher-dimensional operators. A UV completion involving
extra scalars with strong couplings to the Higgs boson was found in \cite{Espinosa:2007qk,Espinosa:2008kw}, and the large couplings were shown not to spoil perturbation theory in \cite{Tamarit:2014dua}. Here we will adopt a practical
approach and simply model the Higgs potential with strong logarithmic corrections, i.e.
\begin{align}
\label{eq:VHdef}
V(H)=V_0+m_H^2H^\dagger H+(H^\dagger H)^2\left(-\lambda+\beta\log\left[\gamma+\frac{2H^\dagger H}{\phi^2_0}\right]\right).
\end{align}
In the equation above, $\beta$ represents an effective quartic coupling arising from loop corrections, and $\gamma$ -- which would be associated with the field-independent contributions to the masses of the
particles running in the loop corrections -- guarantees that the potential is analytic at $H=0$. $\phi_0$ can be chosen at will to be the Higgs VEV $v$ (the difference can be compensated by a redefinition
of the other couplings), and $V_0$ is fixed by requiring as before that the potential is zero at the minimum. Imposing that the correct Higgs and $W$ masses are generated at tree-level, one can eliminate the couplings $m^2_H$ and $\lambda$, and end up with a potential
\begin{align}
\label{eq:VHdef2}
V^{\rm log}(H)=H^\dagger H\left(-\frac{m^2_h}{2}+\frac{(2+3\gamma)\beta v^2}{2(1+\gamma)^2}\right)+(H^\dagger H)^2\left(\frac{m^2_h}{2v^2}-\frac{\beta(3+4\gamma)}{2(1+\gamma)^2}+
\beta\log\left[\frac{\gamma v^2+2H^\dagger H}{v^2(1+\gamma)}\right]\right),
\end{align}
with only $\beta$, $\gamma$ as free parameters.
The parameter $\beta$ controls the size of the barrier with respect to the origin and the energy of the Higgs vacuum. A barrier appears for $\beta>0$, yet increasing $\beta$ too much ($\gtrsim 0.26$ for the measured values of $m_h$ and $m_W$) raises the Higgs vacuum above the origin, so that the symmetric phase becomes preferred. A negative value of $\beta$ causes an instability at values of the field beyond the Higgs vacuum,
which captures the situation in the SM for the measured value of the Higgs and top masses. The allowed window of values of $\beta$ can be obtained by requiring that the electroweak vacuum is sufficiently long-lived with
respect to tunneling towards large values of the fields (for $\beta<0$) or towards the origin ($\beta>0$). The tunneling rate can be calculated from the exponential of the Euclidean action of the scalar field evaluated at a bounce solution \cite{Coleman:1977py}. We have computed the latter numerically for $\beta>0$, while for $\beta<0$, in the presence of a runaway as in the SM, we used the analytic approximation of \cite{Isidori:2001bm}. Doing so we obtain the following window of allowed parameters:
\begin{align}
\label{eq:stabwind}
-0.005\lesssim\beta\lesssim0.5.
\end{align}
The shape
of the potential is illustrated in figure \ref{fig:potentialdef} for different values of $\beta$, including the extrema of the above interval.
\begin{figure}[h!]
\begin{center}
\begin{minipage}{0.5\textwidth}
\includegraphics[width=0.95\textwidth]{Vdef.pdf}
\end{minipage}%
\begin{minipage}{0.5\textwidth}
\includegraphics[width=0.95\textwidth]{Vdef2.pdf}
\end{minipage}
\caption{Deformed Higgs potential for $\gamma=0.1$ and varying values of $\beta$. The dots represent the Higgs minimum, with VEV and curvature fixed by the $W$ and Higgs masses. On the left hand, the values of $\beta$ ensure absolute stability of the Higgs vacuum. The right
hand plot shows the extremal values of $\beta$ for which the metastable Higgs vacuum is still sufficiently long lived at zero temperature. The shaded areas reflect the region of the potential probed by sphaleron configurations.}
\label{fig:potentialdef}
\end{center}
\end{figure}
If the extra scalar fields that originate the logarithmic corrections are stabilized at the origin and do not receive induced tadpoles in a given Higgs background (as can be ensured
with appropriate discrete or global symmetries), they will not play a role in the calculation
of the sphaleron barrier and can be set to zero. Thus the sphaleron configuration can be obtained, as in the SM case, by extremising the bosonic energy functional involving the spatial
derivatives of the gauge fields and the Higgs, and the modified Higgs potential of equation \eqref{eq:VHdef2}. Performing the same rescalings as in Eq.~\eqref{eq:rescaling}, the rescaled bosonic energy looks as Eq.~\eqref{eq:rescaledVbosSM}, but with $\tilde V(\tilde H)$ substituted by
\begin{align}
\tilde V(\tilde H)=\frac
{\kappa^2}{32}(\tilde H^\dagger \tilde H-4)^2-\frac{2\hat\beta(1+2\gamma)}{(1+\gamma)^2}+\frac{\hat\beta(2+3\gamma)}{(1+\gamma)^2}\tilde H^\dagger \tilde H+\frac{\hat\beta(\tilde H^\dagger\tilde H)^2}{4}\left\{\log\left[\frac{4\gamma+\tilde H^\dagger \tilde H}{4(1+\gamma)}\right]-\frac{(3+4\gamma)}{2(1+\gamma)^2}\right\},
\end{align}
where we defined $\hat\beta\equiv \beta/g^2$. The equations of the sphaleron are formally the same as in Eq.~\eqref{eq:eomSM}, but with the above potential. Introducing again the ansatz
\eqref{eq:ansatz} and choosing the gauge $C(r)=0$, one gets identical results as before for the first family of equations in \eqref{eq:eomSM}, i.e. Eqs \eqref{eq:Es}, as they are not sensitive to the potential, while the second family of equations is modified. Once more, there are only four independent equations, which are:
\begin{equation}
\label{eq:Esdef}
\begin{aligned}
&\,B''-\frac{B}{r^2}\left(A^2+B^2-1\right)+2GF-B(G^2+F^2)=0,\\
&\,A''-\frac{A}{r^2}\left(A^2+B^2-1\right)-A(G^2+F^2)-G^2+F^2=0,\\
&\frac{2}{r^2}(r^2G')'-\frac{G}{r^2}\left((A+1)^2+B^2\right)+\frac{2BF}{r^2}-\kappa^2G(F^2+G^2-1)+G\Delta=0,\\
&\frac{2}{r^2}(r^2F')'-\frac{F}{r^2}\left((A-1)^2+B^2\right)+\frac{2BG}{r^2}-\kappa^2F(F^2+G^2-1)+F\Delta=0,
\end{aligned}
\end{equation}
where we defined
\begin{eqnarray}
\label{eq:gammadeltasdef}
\begin{aligned}
\Delta\equiv&\,\frac{4 \beta }{\gamma _0^2 \gamma _1}\Big\{2 \gamma _0^2 \gamma _1 \left(F^2+G^2\right)\log \frac{\gamma _0}{\gamma _1} -\left(F^2+G^2-1\right) \left(\gamma ^2 \left(F^2+G^2
-3\right)-2 \gamma \left(F^2+G^2+1\right)-2 \left(F^2+G^2\right)\right)\Big\},\\
\gamma_0\equiv&\,(1+\gamma),\\
\gamma_1\equiv&\,(F^2+G^2+\gamma).
\end{aligned}
\end{eqnarray}
The presence of $F, G$ in the gauge-invariant combination suggests that the equations will be simpler using the variables $R,S,\theta,\phi$ as in Eq.~\eqref{eq:RS}. We find
\begin{eqnarray}
\label{eq:eqsRdef}
\begin{aligned}
&r^2R''+r^2S^2 \cos [2 \phi -\theta ]+R-R \left(R^2+r^2(\theta '^2+S^2\right))=0,\\
&2 r^2 S''-2r^2 S \phi'^2+4r S'-S \left(\kappa^2 r^2 \left(S^2-1\right)-2 R \cos [2 \phi -\theta ]+R^2+1\right)+r^2 S\Delta_S=0,\\
&R \theta ''+2 \theta ' R'+S^2 \sin [2 \phi -\theta ]=0,\\
&r^2 S \phi''+2 r\phi' \left(r S'+S\right)-R S \sin [2 \phi -\theta ]=0,
\end{aligned}
\end{eqnarray}
where now, using $\gamma_0=1+\gamma$ and $\gamma_1=S^2+\gamma$ as in Eqs.~\eqref{eq:gammadeltasdef},
\begin{eqnarray}
\begin{aligned}
\Delta_S\equiv-\frac{4\hat\beta }{\gamma_0^2\gamma_1}\left\{\gamma _1 \left(3 \gamma -(4 \gamma +3) S^2+2\right)+\gamma _0^2 S^2 \left(2 \gamma _1 \log\frac{\gamma _1}{\gamma_0}+S^2\right)\right\}.
\end{aligned}
\end{eqnarray}
As in the SM case, the last two equations are solved by Eq.~\eqref{eq:thetas}, and one gets a simplified set of only two differential equations:
\begin{equation}
\begin{aligned}
\label{eq:eqsRsimpledef}
&r^2 R''-{R^3}+ R \left(1-r^2 S^2\right)\pm r^2 S^2=0,\\
&2r^2 S''+4 r S'-S \left((R\mp1)^2+\kappa^2 r^2(S^2-1)\right)+r^2 S\Delta_S=0.
\end{aligned}
\end{equation}
Once more, the upper and lower sign correspond to branches of solutions with either even or odd $\omega$ in Eq.~\eqref{eq:thetas}.
The asymptotic solutions and the boundary conditions for the equations are similar to those in the SM case, and discussed in appendix \ref{app:asympt}.
Using the same iterative method, we have solved the systems
\eqref{eq:Esdef}, \eqref{eq:eqsRdef} and \eqref{eq:eqsRsimpledef}, obtaining compatible results in all cases. As in the SM, for the measured Higgs and $W$ boson masses we only found a single branch
of sphaleron solutions, which for $R>0$ at large $r$ corresponds to the upper sign choice in \eqref{eq:eqsRsimpledef}, with $N_{CS}=1/2$, and with $\theta=\pi,\phi=\pi/2$, i.e. $F=B=0$.
The resulting energy barrier differs at the level of $\lesssim9\%$ from the SM one, even for the limiting cases in the stability window of equation \eqref{eq:stabwind}. For absolutely stable Higgs vacua, the deviations are below 3\%; these results are illustrated in Fig.~\ref{fig:Esph_def}. Figure \ref{fig:profiles_def} shows the profiles and the contributions to the bosonic energy density coming from
a sphaleron configuration with $\beta$ near the upper stability limit of Eq.~\eqref{eq:stabwind}. This gives the largest deviation from the SM, with the Higgs minimum above the origin (see Fig.~\ref{fig:potentialdef}). Note that with the potential normalized to zero at the former minimum, the energy density at the origin becomes negative, and the sphaleron configuration probes negative energies, as shown on the right plot in Fig.~\ref{fig:profiles_def}. This plays a role in lowering the sphaleron energy barrier, which becomes $E_{\rm sph}[\beta=0.495]=8.29$ TeV.
\begin{figure}[h!]
\begin{center}
\includegraphics[width=.5\textwidth]{Esph_def.pdf}
\caption{In red, sphaleron energy as a function of $\beta$ in models with a deformed Higgs potential, for $\beta$ in the allowed stability window. The dash-dotted gray line represents the SM result. $\gamma$ was fixed to $0.1$, and hardly influences the results. The shaded band corresponds to absolutely stable Higgs vacua, as in the left plot in figure \ref{fig:potentialdef}.}
\label{fig:Esph_def}
\end{center}
\end{figure}
\begin{figure}[h!]
\begin{center}
\begin{minipage}{0.5\textwidth}
\includegraphics[width=.95\textwidth]{def_2.pdf}
\end{minipage}%
\begin{minipage}{0.5\textwidth}
\hfil\hskip0.8cm\includegraphics[width=.95\textwidth]{Vbosdef.pdf}
\end{minipage}
\caption{Properties of sphaleron configurations with $\gamma=0.1$ and $\beta=0.495$. The left plot shows the profiles for $R,S$, in units of $m_W$, with the vertical lines marking the scale at which the low $r$ solution (red) was matched with the high $r$ solution (blue). The plot on the right shows the contributions to the dimensionless integrand in $\tilde V_{\rm bos}$, evaluated on the sphaleron solution, due to the gauge fields (solid blue), derivatives of the scalar field (dashed orange) and the potential
energy density of the Higgs (dotted green). In all plots, the gray dash-dotted lines correspond to the SM results.}
\label{fig:profiles_def}
\end{center}
\end{figure}
\section{Sphaleron energy in composite Higgs scenarios}
\label{sec:sphalcomp}
In this section we will study sphaleron configurations in minimal composite Higgs scenarios, in which the Higgs arises as a pseudo-Goldstone boson of a global SO(5) symmetry broken down to SO(4) \cite{Agashe:2004rs}.
The pattern of symmetry breaking enforces non-standard derivative interactions for the Higgs, as well as a modified relation between the Higgs VEV and the weak boson masses. Interactions that break the global symmetry generate a nonstandard Higgs potential, which still exhibits a discrete translational symmetry. Both the modified derivative interactions and potential can affect the sphaleron energy, and we expect larger deviations
than before. This is not only due to the modified derivative interactions, but also to the modified relation between the Higgs VEV and the $W$ boson mass. In models with elementary Higgses, we saw that the sphaleron probes
the potential between the origin and the minimum. With the location and curvature of the minimum fixed by the Higgs and gauge boson masses, the potential of an elementary Higgs can only be modified by changing the depth of
the minimum, or the shape of the potential in between the latter and the origin. In composite Higgs models there is in principle a further degree of freedom associated with the location of the minimum, as a result of the modified relation between the $W$ mass and the Higgs VEV.
Following \cite{Agashe:2004rs}, one can work in an effective theory involving the gauge and pseudo-Goldstone fields. The breaking of SO(5) into SO(4) leaves four Goldstones $h^m,m=1,\dots,4$, which can be included
in a multiplet $\Sigma$ carrying a nonlinear representation of the broken SO(5). The breaking is assumed to originate from a field $\Sigma_0$
in the fundamental of SO(5), which acquires a VEV involving a scale $f_\pi$: $\langle\Sigma_0\rangle^\top=[0,0,0,0,f_\pi]$. Then the field multiplet $\Sigma$ is given by
\begin{equation}
\Sigma=\exp\frac{\Pi}{f_\pi}\times\Sigma_0,
\end{equation}
with $\Pi$ given by a sum over broken global SO(5) generators $\tilde G^a$ multiplied by its corresponding Goldstone fields $\Pi=i\sqrt{2} \tilde G^a h^a$. The result is
\begin{equation}
\label{eq:Sigma}
\Sigma^\top=\frac{\sin\frac{h}{f_\pi}}{h}[h^1,h^2,h^3,h^4,h\cot\frac{h}{f_\pi}],
\end{equation}
where we defined $h\equiv(\sum_m(h^m)^2)^{1/2}$. With these conventions, and gauging an $SU(2)$ subgroup of the surviving SO(4) symmetry of the composite sector,
the effective Lagrangian of the gauge and pseudo-Goldstone fields becomes \cite{Agashe:2004rs}
\begin{equation}
\label{eq:L}
{\cal L}= \,\frac{f_\pi^2}{2}(D_\mu\Sigma)^\top D^\mu\Sigma-\frac{1}{4g^2}W^a_{\mu\nu}W^{a,\mu\nu}-\alpha\cos\frac{h}{f_\pi}+\beta\sin^2\frac{h}{f_\pi}.
\end{equation}
The last two terms represent the scalar potential for the pseudo-Goldstones, arising from explicit sources of SO(5) breaking, such as Yukawas and the gauging of $\mathrm{SU}(2)_L$. We may identify the Goldstones $h^m$ with the usual elementary Higgs multiplet as
\begin{equation}
\label{eq:Hh}
H=\left[
\begin{array}{c}
H^+\\
H^0
\end{array}
\right]=\frac{1}{\sqrt{2}}\left[
\begin{array}{c}
h^1+ih^2\\
h^3+ih^4
\end{array}
\right].
\end{equation}
In the Higgs vacuum we have respectively $\langle h\rangle=\langle h^3\rangle\neq0$, and
\begin{align}
\label{eq:vacuum}
\langle\Sigma^\top\rangle=[0,0,\epsilon,0,\sqrt{1-\epsilon^2}],\quad\epsilon=\sin\frac{\langle h\rangle}{f_\pi}.
\end{align}
The covariant derivatives in the Lagrangian of Eq.~\eqref{eq:L} include SU(2)$_L$ generators. From the identification of Eq.~\eqref{eq:Hh} we may construct the representation of SU(2)$_L$ on the Goldstone
multiplet $\Sigma$ as follows,
\begin{equation}
\label{eq:SU2gen}
\begin{aligned}
D_\mu\Sigma=&\partial_\mu\Sigma-i A_\mu^a T^a\Sigma,\\
T^1=&\frac{i}{2}\left[
\begin{array}{ccccc}
0 & 0 & 0 & 1 & 0\\
0 & 0 & -1 & 0 & 0\\
0 & 1 & 0 & 0 & 0\\
-1 & 0 & 0 & 0 & 0\\
0 & 0 & 0 & 0 & 0
\end{array} \right],
\,T^2=&\frac{i}{2}\left[
\begin{array}{ccccc}
0 & 0 & -1 & 0 & 0\\
0 & 0 & 0 & -1 & 0\\
1 & 0 & 0 & 0 & 0\\
0 & 1 & 0 & 0 & 0\\
0 & 0 & 0 & 0 & 0
\end{array} \right],
\,T^3=&\frac{i}{2}\left[
\begin{array}{ccccc}
0 & 1 & 0 & 0 & 0\\
-1 & 0 & 0 & 0 & 0\\
0 & 0 & 0 & -1 & 0\\
0 & 0 & 1 & 0 & 0\\
0 & 0 & 0 & 0 & 0
\end{array} \right].
\end{aligned}
\end{equation}
Note how the generators are a subset of the unbroken SO(4) symmetry acting on the first four components of $\Sigma$.
The mass of the gauge bosons in the vacuum is defined by
\begin{equation}
m^2_W=\frac{g^2\epsilon^2 f^2_\pi}{4}\equiv\frac{g^2v^2}{4},
\label{eq:mw}
\end{equation}
where at tree-level $v= \epsilon f_\pi = 246$ GeV. Note that $v$ here does not represent the pseudo-Goldstone VEV, but rather parameterizes the $W$ mass. For $|\alpha/(2\beta)|\leq1$ the scalar potential has a minimum at
\begin{equation}
\label{eq:vev2}
\cos\frac{\langle h\rangle}{f_\pi}=-\frac{\alpha}{2\beta},
\end{equation}
and the fluctuations of the field $h^3$ -- the pseudo-Goldstone version of the Higgs -- acquire a mass
\begin{align}
m^2_h=\frac{2\beta\epsilon^2}{f_\pi^2}.
\end{align}
Note that a positive Higgs mass requires $\beta>0$, while Eq.~\eqref{eq:vev2} will have a solution for either positive or negative $\alpha$. It appears that there are two families of solutions, but as we will
argue later they are physically equivalent. Beyond the known value of the weak gauge coupling, the bosonic low-energy Lagrangian of Eq.~\eqref{eq:L} has three parameters $f_\pi,\alpha,\beta$, and there is freedom in choosing the sign of $\alpha$. Requiring that the masses of the gauge bosons and the Higgs
reproduce their measured values leaves one free parameter, which we take as $f_\pi$. Thus we may write $\alpha,\beta$ in terms of the physical masses $m_W,m_h$ and $f_\pi$:
\begin{align}
\label{eq:abeta}
\alpha=\pm\frac{g f_\pi^4}{4}\frac{m^2_h}{m_W}\sqrt{\frac{g^2}{m^2_W}-\frac{4}{f_\pi^2}},\quad \beta=\frac{g^2 f^4_\pi}{8}\frac{m^2_h}{m^2_W}.
\end{align}
Note that consistency demands
\begin{align}
\label{eq:fmin}
f_\pi > \frac{2m_W}{g}.
\end{align}
Picking the lowest possible value of $\langle h\rangle$ yielding the correct gauge boson mass, (i.e. $\sin h/f_\pi>0$) one has
\begin{equation}
\label{eq:vev}
\begin{aligned}
&\langle h\rangle_-=v+\frac{v^3}{6f_\pi^2}+\frac{3v^5}{40f_\pi^4}+{\cal O}\left(\frac{1}{f_\pi^6}\right)~~\mathrm{for}~~\alpha<0,\\
&\langle h\rangle_+=\pi f_\pi-\langle h\rangle_-=\pi f_\pi-v-\frac{v^3}{6f_\pi^2}-\frac{3v^5}{40f_\pi^4}+{\cal O}\left(\frac{1}{f_\pi^6}\right)~~\mathrm{for}~~\alpha>0,
\end{aligned}
\end{equation}
which shows explicitly the modified relation between the Higgs VEV and the $W$ masses alluded to before. At this point, one would be tempted to argue that models in the large $f_\pi$, $\alpha>0$ branch,
with $\langle h\rangle \gg v$, should develop a larger sphaleron barrier. In previous sections we saw that the sphaleron configurations probe
the potential between the origin and the Higgs VEV, and so for large $\langle h\rangle$ the sphaleron profile would have to cover a larger amount of field-space,
implying larger kinetic contributions to the bosonic energy. Alas, this intuition is misleading. An important difference with respect to the elementary Higgs case is that the theory has a discrete selection rule.
It can be easily seen that in a unitary gauge with $h^i=0,i\neq3$, and thus $h^3=h$, the Lagrangian of Eq.~\eqref{eq:L} is invariant under the discrete transformation
\begin{align}
\label{eq:alphalaw}
\alpha\rightarrow-\alpha,\quad h\rightarrow \pi f_\pi-h.
\end{align}
This means that sphaleron solutions for one choice of sign of $\alpha$ can be related to sphaleron solutions for the other choice, with matching energies. For this reason the two choices of sign of $\alpha$ are physically equivalent. This equivalence can also be seen by expanding the Lagrangian in the unitary gauge around the vacuum configurations, $ h=\langle h\rangle+\delta h$, with $\langle h\rangle$ given in Eq.~\eqref{eq:vev}, and with $\alpha$ either positive or negative as in Eq.~\eqref{eq:abeta}. Doing a $1/f_\pi$ expansion, the resulting terms are related by an unphysical $Z_2$ transformation $\delta h\rightarrow-\delta h$. The same conclusion applies to fermionic couplings, which we did not discuss here but can be modelled again with $\sin h/f$ interactions \cite{Agashe:2004rs}. Therefore both scenarios with $\alpha>0$ and $\alpha<0$ are indistinguishable, and we will focus on the $\alpha<0$ case. We show in figure \ref{fig:Vcomp} the potential energy density of $h$ for $f=260$ GeV, showing the two equivalent realizations with different signs of $\alpha$. Note the different position of the VEVs and how the potentials are related by the transformation \eqref{eq:alphalaw}.
\begin{figure}[h!]
\begin{center}
\begin{minipage}{0.5\textwidth}
\includegraphics[width=0.95\textwidth]{Vcompneg.pdf}
\end{minipage}%
\begin{minipage}{0.5\textwidth}
\includegraphics[width=0.95\textwidth]{Vcomppos.pdf}
\end{minipage}
\caption{Composite Higgs potential for $f_\pi=260$ GeV, for the physically equivalent realizations with $\alpha<0$ (left) and $\alpha>0$ (right). Both cases yield the correct Higgs and $W$ masses at tree-level. The darker shade shows the region of the potential probed by the
sphaleron branch in common with the SM, while the lighter shade corresponds to the region probed by the new, higher-energy sphaleron branch present in composite models.}
\label{fig:Vcomp}
\end{center}
\end{figure}
Given the row of zeros in the SU(2)$_L$ generators in Eq.~\eqref{eq:SU2gen}, it is clear that $\Sigma$ involves two irreducible representations. We may write $\Sigma=\tilde\Sigma\oplus\Lambda$, with
\begin{align}
\label{eq:tildeSigma}
\tilde\Sigma^\top=\frac{\sin\frac{h}{f_\pi}}{h}[h^1,h^2,h^3,h^4],\quad
\Lambda=\cos\frac{h}{f_\pi}.
\end{align}
Then $D_\mu\tilde\Sigma=\partial_\mu\tilde\Sigma-iA^a_\mu\tilde T^a\tilde\Sigma$, with the $\tilde T^a$ given by the upper-left $4\times4$ blocks of the generators in Eq.~\eqref{eq:SU2gen}. In this way
we can rewrite Eq.~\eqref{eq:L} as
\begin{equation}
\label{eq:L2}
{\cal L}= \,\frac{f_\pi^2}{2}\partial_\mu\Lambda\partial^\mu\Lambda+\frac{f_\pi^2}{2}(D_\mu\tilde\Sigma)^\top D^\mu\tilde\Sigma-\frac{1}{4g^2}W^a_{\mu\nu}W^{a,\mu\nu}-\alpha\cos\frac{h}{f_\pi}+\beta\sin^2\frac{h}{f_\pi}.
\end{equation}
We are now ready to define the bosonic potential energy relevant for sphaleron configurations. As in the previous cases, this is just given by the Hamiltonian in the temporal $A_0=0$ gauge, with time derivatives
omitted (or equivalently, the Euclidean Lagrangian evaluated in static configurations). Performing rescalings as in Eq.~\eqref{eq:rescaling}, with $h_i\rightarrow m_W/g h_i$, the dimensionless bosonic energy functional becomes
\begin{equation}
\tilde V_{\rm bos}=\frac{1}{g^2}\int\!d^3y\,\left\{\frac{1}{4}\tilde W^a_{ij}\tilde W^{a}_{ij}+\frac{\hat f_\pi^2}{2}\partial_i\Lambda \partial_i\Lambda+\frac{\hat f_\pi^2}{2}(D_i\tilde\Sigma)^\top D_i\tilde\Sigma+\hat\alpha\cos\frac{\tilde h}{\hat f_\pi}-\hat\beta\sin^2\frac{\tilde h}{\hat f_\pi}\right\},
\end{equation}
where we defined the following modified couplings,
\begin{align}
\hat f_\pi\equiv\frac{g f_\pi}{m_W},\quad \hat \alpha\equiv\frac{\alpha g^2}{m^4_W},\quad \hat \beta\equiv\frac{\beta g^2}{m^4_W}.
\end{align}
The equations for the sphaleron configurations that extremise $\tilde V_{\rm bos}$ are:
\begin{equation}
\label{eq:eom}
\begin{aligned}
({\cal D}_j W_{ij})^a+\frac{i\hat f_\pi^2}{2}\left(\tilde\Sigma^\top T^a D_i\tilde\Sigma-(D_i\tilde\Sigma)^\top T^a \Sigma\right)=0,\quad i,a=1,\dots,3,\\
\hat f^2_\pi\left(\partial^2\Lambda\frac{\partial\Lambda}{\partial h^m}+(D^2\tilde\Sigma)^n\frac{\partial\tilde\Sigma^n}{\partial h^m}\right)+\frac{h^m}{h\hat f_\pi}\sin\frac{h}{\hat f_\pi}
\left(\hat\alpha+2\hat\beta\cos\frac{h}{\hat f_\pi}\right)=0,\quad m=1,\dots,4,
\end{aligned}
\end{equation}
with
\begin{align}
\frac{\partial\Lambda}{\partial h^m} &=-\frac{h^m}{h\hat f_\pi}\sin\frac{h}{\hat f_\pi},& \frac{\partial\tilde\Sigma^n}{\partial h^m} &=\frac{1}{h^3\hat f_\pi}\left(h \cos\frac{h}{\hat{f}_\pi}
-\hat{f}_\pi\sin\frac{h}{\hat{f}_\pi}\right)h^m h^n+\frac{\delta^{mn}}{h}\sin\frac{h}{\hat{f}_\pi}.
\end{align}
We shall proceed as before and introduce the same rotationally symmetric ansatz as in Eq.~\eqref{eq:ansatz}, with $\tilde H$ interpreted in terms of the dimensionless Goldstone fields $\tilde h^i$ as
$\tilde H=[\tilde h_1+i\tilde h_2, \tilde h_3+i\tilde h_4]^\top$.
After introducing the ansatz in the equations of motion, and going as before into the $C=0$ gauge, one gets four independent equations of motion, which can be written as:
\begin{equation}
\label{eq:Escomp}
\begin{aligned}
&\,B''-\frac{B}{r^2}\left(A^2+B^2-1\right)+{\cal F}^2_1\left[2GF-B(G^2+F^2)\right]=0,\\
&\,A''-\frac{A}{r^2}\left(A^2+B^2-1\right)-{\cal F}^2_1\left[A(G^2+F^2)+G^2-F^2\right]=0,\\
\end{aligned}
\end{equation}
\begin{equation}
\begin{aligned}
&\,{{{\cal F}_1}}\left\{\hat{f}_\pi^2 \left[{{\cal F}_2} G \left(4 F \left(2 B G+r \left(r F''+2 F'\right)\right)-G^2 \left(2 A^2+4 A+2 B^2+2\right)-F^2 \left(2 A^2-4 A+2 B^2+2\right)\right.\right.\right.\\
&\left.\left.+4 r G\left(r G''+2 G'\right)\right)+{{\cal F}_1} F \left(F \left(4 r^2 G''+8 r G'-8 A G\right)-4 G^2 B+4F^2 B-4 G r \left(r F''+2 F'\right)\right)\right]\\
&\left.+4 \hat\alpha G^3 r^2+4 \hat\alpha G F^2 r^2+8 \hat\beta {{\cal F}_2} G r^2 \left(G^2+F^2\right)\right\}\\
&+{2\hat{f}_\pi^2 \,r}\left\{{{\cal F}_1}' \left[4 {{\cal F}_2} G\left(G^2+F^2\right) +4 r G' \left({{\cal F}_1} F^2+{{\cal F}_2} G^2\right)+4 r G F ({{\cal F}_2}-{{\cal F}_1}) F'\right]+2G \left(G^2+F^2\right)( r{{\cal F}_2} {{\cal F}_1}'' \right.\\
&\left.-r {{\cal F}_1} {{\cal F}_2}'' -2 {{\cal F}_1} {{\cal F}_2}' )\right\}=0
,\\
\ \\
&\,{{{\cal F}_1}}\left\{\hat{f}_\pi^2 \left[{{\cal F}_2} F \left(4 F \left(2 B G+r(r F''+2 F')\right)-G^2 \left(2 A^2+4 A+2 B^2+2\right)-F^2 \left(2 A^2-4 A+2 B^2+2\right)\right.\right.\right.\\
&\left.\left.+4 r G \left(r G''+2 G'\right)\right)+ {{\cal F}_1} G \left(G(4r^2 F''+8 r F'+8 A F)+4 B G^2-4 B F^2-4 r F(r G''+2G')\right)\right]\\
&\left.+4 \hat\alpha G^2 F r^2+4 \hat\alpha F^3 r^2+8 \hat\beta {{\cal F}_2} F r^2 \left(G^2+F^2\right)\right\}\\
&+{2 \hat{f}_\pi^2\, r}\left[ {{\cal F}_1}' \left(4{{\cal F}_2} F\left(G^2+F^2\right) +4 r F' \left({{\cal F}_1} G^2+{{\cal F}_2} F^2\right)+4 G F r ({{\cal F}_2}-{{\cal F}_1}) G'\right)+2 F \left(G^2+F^2\right) \left(r{{\cal F}_2} {{\cal F}_1}''\right.\right.\\
&\left.\left.-r{{\cal F}_1} {{\cal F}_2}''-2 {{\cal F}_1} {{\cal F}_2}'\right)\right]=0.
\end{aligned}
\end{equation}
In the previous equations, we defined the ``form-factors''
\begin{eqnarray}\begin{aligned}
{\cal F}_1\equiv&\,\frac{\hat{f}_\pi\sin\frac{h}{\hat{f}_\pi}}{h}=\frac{\hat{f}_\pi}{2\sqrt{G^2+F^2}}\sin\left[\frac{2\sqrt{G^2+F^2}}{\hat{f}_\pi}\right],\\
{\cal F}_2\equiv&\,\cos\frac{h}{\hat{f}_\pi}=\cos\left[\frac{2\sqrt{G^2+F^2}}{\hat{f}_\pi}\right].
\end{aligned}
\end{eqnarray}
These form factors ${\cal F}_1,{\cal F}_2$ encode the nontrivial interactions of the composite Higgs. They tend to $1$ for large $\hat{f}_\pi$, for which one recovers the limiting case of the SM Higgs. This can be explicitly
checked from the above sphaleron equations, or by realizing that in this limit the bosonic Lagrangian of Eq.~\eqref{eq:L} coincides with the SM one.
The variables introduced in Eq.~\eqref{eq:RS} allow for a substantial simplification for the equations, which become
\begin{equation}
\label{eq:eqsR}
\begin{aligned}
& R''+\frac{\hat{f}_\pi^2}{4} \sin ^2\left[\frac{2 S}{\hat{f}_\pi}\right] \cos [2 \phi-\theta]+R \left(\frac{\hat{f}_\pi^2}{8} \cos \left[\frac{4 S}{\hat{f}_\pi}\right]-\frac{\hat{f}_\pi^2}{8}+\frac{1}{r^2}-\theta'^2\right)-\frac{R^3}{r^2}=0,\\
&2 r \left(r S''\!+2 S'\right)+\frac{\hat{f}_\pi}{4} \sin \left[\frac{4 S}{\hat{f}_\pi}\right] \left(2 R \cos [2 \phi-\theta]\!-\!2 r^2 \phi'^2\!-\!R^2\!-\!1\right)+\frac{r^2}{\hat{f}_\pi} \sin \left[\frac{2 S}{\hat{f}_\pi}\right] \left({\hat\alpha}+2 {\hat\beta} \cos \left[\frac{2 S}{\hat{f}_\pi}\right]\right)=0,\\
& 4 R \theta''+8 R' \theta'+\hat{f}_\pi^2 \sin ^2\left[\frac{2 S}{\hat{f}_\pi}\right] \sin [2 \phi-\theta]=0,\\
& r \left(\hat{f}_\pi r \phi''+2 \phi' \left(2 r S' \cot \left[\frac{2 S}{\hat{f}_\pi}\right]+\hat{f}_\pi\right)\right)-\hat{f}_\pi R \sin [2 \phi-\theta]=0.
\end{aligned}
\end{equation}
In these variables we may write $\tilde V_{\rm bos}$ as
\begin{equation}
\label{eq:Vbospol}
\begin{aligned}
\tilde V_{\rm bos}=&\frac{4\pi}{g^2}\int dr\left\{\frac{1}{4} \left[\hat{f}_\pi^2 \sin ^2\left[\frac{2 S}{\hat{f}_\pi}\right] \left((R-1)^2+2 R(1- \cos [2 \phi-\theta])\right)+4 R^2{ \theta'}^2+4 {R'}^2\right]\right.\\
&\left.+\frac{1}{2} \left(R^2-1\right)^2+\frac
{r^2}{4}\left[\frac{1}{\beta}\left(\hat\alpha+2 \hat\beta \cos \left[\frac{2 S}{\hat{f}_\pi}\right]\right)^2+2 \hat{f}_\pi^2 {\phi'}^2 \sin ^2\left[\frac{2 S}{\hat{f}_\pi}\right]+8 {S'}^2\right]
\right\}.
\end{aligned}
\end{equation}
As in the previous cases, the last two equations in \eqref{eq:eqsR} can be solved as in \eqref{eq:thetas}. This gives the simplified system
\begin{equation}
\label{eq:eqsRsimple}
\begin{aligned}
&R''+ R \left(\frac{1}{r^2}-\frac{1}{4} \hat{f}_\pi^2 \sin ^2\left[\frac{2 S}{\hat{f}_\pi}\right]\right)\pm\frac{1}{4} \hat{f}_\pi^2 \sin ^2\left[\frac{2 S}{\hat{f}_\pi}\right]-\frac{R^3}{r^2}=0,\\
&r^2 S''+2 r S'+\frac{1}{8\hat{f}_\pi}\left(4 \hat\alpha r^2 \sin \left[\frac{2 S}{\hat{f}_\pi}\right]-\sin \left[\frac{4 S}{\hat{f}_\pi}\right] \left(\hat{f}_\pi^2 (R\mp1)^2-4 \hat\beta r^2\right)\right)=0.
\end{aligned}
\end{equation}
The aforementioned physical equivalence between models with $\alpha>0$ and $\alpha<0$ can be understood from the bosonic energy \eqref{eq:Vbospol} and equations \eqref{eq:eqsR}, \eqref{eq:eqsRsimple} by noting that they are invariant under the discrete symmetry
\begin{align}
\hat\alpha\rightarrow-\hat\alpha,\quad S\rightarrow S+\frac{\pi}{2} \hat{f}_\pi.
\end{align}
Thus, solutions with one sign of $\hat\alpha$ can always be mapped onto solutions with the other sign. With the sign of $\hat\alpha$ fixed, another discrete symmetry of the bosonic energy and the sphaleron equations is
\begin{align}
\label{eq:discretesym}
S\rightarrow \pi \hat{f}_\pi\pm S.
\end{align}
The asymptotic solutions for Eq.~\eqref{eq:Escomp} in the limit of
large and small $r$ are given in appendix \ref{app:asympt}, and for fixed $f_\pi$ depend on the same number of parameters as in cases with an elementary Higgs. The corresponding solutions for \eqref{eq:eqsR} and \eqref{eq:eqsRsimple} can be obtained by using the definitions in \eqref{eq:RS}. Once again, reconstructing the full profile of the sphaleron from the solutions to the simplified system \eqref{eq:eqsRsimple} requires to fix the ambiguity in the solution \eqref{eq:thetas} for $\theta,\phi$. As in the SM, parity-invariant sphalerons are expected to have $N_{CS}=1/2$, which fixes $\theta=\pi+n\pi$. Fixing $R>0$ at large $r$, we have found solutions with $\theta=\pi$ in the upper branch of Eq.~\eqref{eq:eqsRsimple}, corresponding to $\phi=\pi/2$ (see \eqref{eq:thetas}).
A distinguishing feature of composite Higgs scenarios is that there are new types of asymptotic solutions at $r\rightarrow0$ that can support novel sphaleron solutions with $N_{CS}=1/2$. In the SM and in the case of a deformed potential, sphalerons ended up having $S(r=0)=0$, as can be seen in figures \ref{fig:profiles_SM} and \ref{fig:profiles_def}. The existence of new solutions with
$S(0)\neq0$ can be understood as follows. By continuity with the SM case, one expects solutions with $S(0)=0$ for both $\hat\alpha>0$ and $\hat\alpha<0$. However, as was just argued, solutions with $\hat\alpha>0$ and $S(0)=0$ can be mapped to solutions with the opposite sign of $\hat\alpha$ by doing $S\rightarrow S+\pi/2 \hat{f}_\pi$, so that one ends up with $S(0)=\pi/2\hat{f}_\pi$. For $\hat\alpha<0$ this corresponds to a local maximum of the potential.
We see that sphalerons interpolating between the minimum of the scalar potential and the origin, for a given choice of $\hat\alpha$, are equivalent to sphalerons that interpolate between the minimum and a local maximum for the opposite choice of $\hat\alpha$ (see figure \ref{fig:Vcomp}). The existence of new solutions with different behaviour near $r=0$ implies that there must be a new family of asymptotic solutions for small $r$,
which is given in appendix \ref{app:asympt}. Amusingly, as further discussed in the appendix, although the choices of opposite values $\hat\alpha$ are equivalent, regular solutions for a given sign of $\hat\alpha$ correspond to singular solutions with the opposite $\hat\alpha$, although the singularity is unphysical, as it can be removed with a gauge transformation. The existence of a new family of sphalerons with $N_{CS}=1/2$ and for the observed value of the Higgs mass is a novel effect which is not present in models with elementary Higgses. In that case, as in the SM, new branches of sphalerons typically have $N_{CS}\neq 1/2$ and only appear if the Higgs is much more massive than observed.
We have computed numerically the sphaleron energy in both families of sphalerons, using the iterative method described in previous sections. The solutions using the four differential equations \eqref{eq:eqsR} confirm the constant values of the angles, $\theta=\pi=2\phi$, derived from \eqref{eq:thetas} and the requirement for $N_{CS}=1/2$. Restricting the analysis to $\hat\alpha<0$, the family of solutions with the usual $S(0)=0$ behaviour gives a sphaleron barrier which, as expected, recovers the SM result in the limit of large $f_\pi$. The new family of $N_{CS}=1/2$ solutions
has greater energies.
\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.48\textwidth]{E_vs_f_smallh.pdf}
\includegraphics[width=0.48\textwidth]{E_vs_f_largeh_2.pdf}
\caption{Energy barrier between topological vacua as a function of $f_\pi$, in the $S(0)=0$ (left) and $S(0)\neq0$ branches. The horizontal line marks the SM limit.}
\label{fig:barrier_smallh}
\end{center}
\end{figure}
As anticipated earlier, the fact that in composite models not only the potential of the Higgs but also its derivative interactions are modified allows for larger deviations from the value of $E_{\rm sph}$, even for the usual family with $S(0)=0$. In this case, $E_{\rm sph}$ reaches nearly 12 TeV for the theoretical minimum
$f_\pi=v$ (see equation \eqref{eq:fmin}), while it decreases rapidly with growing $f_\pi$. With current collider bounds demanding $f_\pi\gtrsim 0.5$ TeV \cite{Aad:2015pla, Azatov:2013hya}, the sphaleron barrier differs from the SM by less than three percent. In the $S(0)\neq0$ branch, the sphaleron barrier starts similarly at 12 TeV and grows linearly with $f_\pi$. The dependence of $E_{\rm sph}$ with the compositeness scale in the two branches is illustrated in Fig.~\ref{fig:barrier_smallh}. The energies in the $S(0)\neq0$ branch are subject to more numerical uncertainties due to the function $S$ becoming very steep at the origin, which prevents convergence of the iterative approach for large enough values of $f_\pi$. Still, our calculations show a linear growth which, when extrapolated, predicts a barrier of around 28 TeV for $f_\pi=500$ GeV and 70 TeV for $f_\pi=1.2$ TeV, respectively.
Example profiles for $R,S$ of the resulting solutions are shown in Fig.~\ref{fig:profiles}. Note the steepness of $S$ near the origin in the lower graphs corresponding to the $S(0)\neq0$ branch, which affects numerical convergence. For the same examples, Fig.~\ref{fig:Ecomp} shows the contributions to the integrand of the dimensionless bosonic energy functional $\tilde V_{\rm bos}$.
For the $S(0)\neq0$ branch and for large enough $f_\pi$, $E_{\rm sph}$ becomes dominated
by the scalar derivatives, in contrast to the cases with elementary fields (see figures \ref{fig:profiles_SM} and \ref{fig:profiles_def}). This is a consequence of the fact that, in this branch, the sphaleron profile interpolates between the electroweak vacuum and the maximum of the scalar potential at $\pi f_\pi$. The distance in field space travelled by the sphaleron increases linearly with $f_\pi$, and we observe the same for the integral yielding $E_{\rm sph}$.
\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.49\textwidth]{RS_250_2_small.pdf}
\includegraphics[width=0.49\textwidth]{RS_1000_2_small.pdf}
\includegraphics[width=0.49\textwidth]{RS_250_2.pdf}
\includegraphics[width=0.49\textwidth]{RS_260_2.pdf}
\caption{Profile functions for $R,S$ in the sphaleron configurations obtained by solving the system of 2 differential equations. The vertical line marks the scale at which the low $r$ solution (red) was matched with the high $r$ solution (blue). Top: solutions in the $S(0)=0$ branch, with $f_\pi=250$ GeV (left) and $f_\pi=1$ TeV (right). Bottom: solutions in the $S(0)\neq0$ branch, with $f_\pi=250$ GeV (left) and $f_\pi=1$ TeV (right).}
\label{fig:profiles}
\end{center}
\end{figure}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.49\textwidth]{Vbosdef_comp_small.pdf}
\includegraphics[width=0.49\textwidth]{Vbosdef_comp_large.pdf}
\caption{Contributions to the integrand of the dimensionless bosonic energy functional for $\hat{f}_\pi=260$ GeV in the $S(0)=0$ branch (left) and for $S(0)\neq0$ GeV (right). The contributions in solid blue are due to the gauge fields; those coming from derivatives of the scalar fields are shown with dashed orange lines, while those coming from the scalar potential
energy density are shown with dotted green curves. }
\label{fig:Ecomp}
\end{center}
\end{figure}
\section{Summary}
\label{sec:summary}
In this paper we have investigated the size of the sphaleron energy barrier in non-standard realizations of the Higgs vacuum. The sphaleron energy, which in the Standard Model lies near 9 TeV, sets the scale of baryon number violating processes mediated by the B+L anomaly, and is sensitive to properties of the Higgs potential away from the electroweak minimum. Thus, sphaleron-induced processes -- which would manifest themselves through the production of a large number
of quarks and leptons -- could offer new perspectives on the nature of the Higgs particle, offering global information about its potential. This is in contrast to the perturbative processes usually considered at colliders, which only probe the Higgs interactions locally, i.e. at the electroweak vacuum.
The sphaleron energy can also be affected by modifications of the derivative interactions of the Higgs. In order to quantify the possible variations of $E_{\rm sph}$ in non-standard scenarios, we have calculated
$E_{\rm sph}$ in models which exhibit either a modified potential for an elementary Higgs, or both a modified potential and modified derivative terms for a composite Higgs. Such examples capture quite generic possibilities for the Higgs interactions, and can be considered as benchmarks for the possible variations of the scale of baryon number violation processes in theories beyond the Standard Model.
For an elementary Higgs with a modified potential, we considered a generic parametrization of the former involving a logarithmic term, which can introduce a barrier with respect to the origin and
modify the depth of the electroweak vacuum. For long-lived electroweak vacua we find deviations of the sphaleron barrier which are at most of the order of 10\%. Such small deviations become less surprising after realizing that already in the SM the sphaleron energy is dominated by derivative contributions, which mostly depend on the distance on field space covered by the scalar profile of the sphaleron. In models with an elementary Higgs, the sphaleron interpolates between the origin and the electroweak vacuum, whose position is fixed by the masses of the weak gauge bosons.
In composite Higgs models the situation could be in principle different, since both the derivative interactions and the relation between the weak boson masses and the Higgs VEV are modified. We centered our study in minimal composite Higgs models, in which the Higgs is a pseudo-Goldstone of a global SO(5) symmetry broken down to SO(4). We have found that,
in contrast with models with elementary Higgses, for which there are no multiple sphaleron branches for the observed values of the Higgs and gauge boson masses, composite Higgs models exhibit at least two branches of sphaleron solutions. The existence of a new branch can be understood from the discrete translation symmetries of the effective action for the pseudo-Goldstone fields. In contrast to known non-standard sphaleron branches for heavy elementary Higgses, sphalerons in this new branch still have half-integer Chern-Simons number, and an energy higher than the sphalerons in the usual branch. In the latter, although large deviations of $E_{\rm sph}$ are possible at low values of the compositeness scale, they are ruled out by collider bounds, so that the minimum sphaleron energy can only differ from the SM one by less than 3\%. On the other hand, sphalerons in the new branch have an energy that grows linearly with the compositeness scale, and would reach around 28 TeV if extrapolated to $\hat{f}_\pi=500$ GeV, and 70 TeV for $\hat{f}_\pi=1.2$ TeV. The new branch of sphaleron configurations is suggestive of a new high-energy threshold for baryon-violating processes in addition to the SM-like threshold at 9 TeV.
Concerning the theoretical precision precision of our calculations, it should be noted that we set the weak mixing angle $\theta_W$ to zero. In models with elementary Higgses, a nonzero $\theta_W$ is known to induce changes in the sphaleron energy of less than a percent \cite{Kleihaus:1991ks,Ahriche:2014jna}. Such modifications are smaller than the largest deviations of $E_{\rm sph}$ with respect to its SM value that were calculated in the models analyzed in this work. Hence, we expect our estimates to be robust with respect to the inclusion of mixing-angle effects.
\acknowledgements
We want to thank Valya Khoze and Kazuki Sakurai for very helpful discussions and comments. MS is supported in part by the European Commission through the ``HiggsTools'' Inital Training Network PITN-GA-2012-316704.
|
2211.03537
|
\section{Introduction}
In a seminal work~\cite{VonNeumannBook}, von Neumann discovered a continuous analogue of finite-dimensional projective geometry. \emph{Continuous geometries}, i.e., complete, complemented, modular lattices whose algebraic operations possess certain natural continuity properties, are the central objects of this theory. A cornerstone in von Neumann's study is his \emph{coordinatization theorem}~\cite{VonNeumannBook}, which states that, firstly, the set $\lat(R)$ of all principal left ideals of every \emph{regular} ring $R$, ordered by set-theoretic inclusion, constitutes a complemented, modular lattice, and secondly, every complemented, modular lattice of an order at least four arises in this way from an up to isomorphism unique regular ring. A \emph{continuous} ring is a regular ring $R$ whose corresponding lattice $\lat(R)$ is a continuous geometry. Building on a dimension theory for (directly) irreducible continuous geometries, another profound achievement of~\cite{VonNeumannBook}, von Neumann proved that an irreducible, regular ring $R$ is continuous if and only if there exists a (necessarily unique) rank function $\rk \colon R \to [0,1]$ such that $R$ is complete with respect to the induced \emph{rank metric} $R \times R \to [0,1], \, (a,b) \mapsto \rk(a-b)$. Thus, any irreducible, continuous ring $R$ admits a natural topology---the \emph{rank topology} generated by its rank metric---which turns $R$ into a topological ring.
While the discrete irreducible, continuous rings are precisely the ones isomorphic to a matrix ring $\Mat_{n}(D)$ for some division ring $D$ and some positive integer $n$, the class of \emph{non-discrete} irreducible, continuous rings appears intriguingly vast. The initial example of an irreducible continuous geometry is the projection lattice of an arbitrary von Neumann factor $M$ of type $\mathrm{II}_{1}$, in which case the corresponding irreducible, continuous ring is non-discrete and can be described as the algebra $\aff(M)$ of densely defined, closed, linear operators \emph{affiliated with $M$}~\cite{MurrayVonNeumann}. For another example, given a field $F$, one may consider the inductive limit $\varinjlim\limits \Mat_{2^{n}}(F)$ of matrix rings \begin{displaymath}
F \, \cong \, \Mat_{2^{0}}(F) \, \stackrel{\iota_{0}}{\longrightarrow} \, \ldots \, \stackrel{\iota_{n-1}}{\longrightarrow} \, \Mat_{2^{n}}(F) \, \stackrel{\iota_{n}}{\longrightarrow} \, \Mat_{2^{n+1}}(F) \, \stackrel{\iota_{n+1}}{\longrightarrow} \, \ldots
\end{displaymath} along the embeddings \begin{displaymath}
\iota_{n} \colon \, \Mat_{2^{n}}(F) \, \longrightarrow \, \Mat_{2^{n+1}}(F), \quad A \, \longmapsto \, \begin{pmatrix} A & 0 \\ 0 & A \end{pmatrix} \qquad (n \in \N) .
\end{displaymath} Since the maps $(\iota_{n})_{n \in \N}$ are isometric with respect to the normalized rank metrics \begin{displaymath}
d_{n} \colon \, \Mat_{2^{n}}(F) \times \Mat_{2^{n}}(F) \, \longrightarrow \, [0,1], \quad (A,B) \, \longmapsto \, \tfrac{\rank(A-B)}{2^{n}} \qquad (n \in \N) ,
\end{displaymath} those metrics admit a joint extension to $\varinjlim\limits \Mat_{2^{n}}(F)$. The completion $\Mat_{\infty}(F)$ of $\varinjlim\limits \Mat_{2^{n}}(F)$ with respect to the resulting metric constitutes a non-discrete irreducible, continuous ring~\cite{NeumannExamples}. An abstract characterization of continuous rings arising in this manner can be found in~\cite{AraClaramunt18}.
There has been recent interest in concrete occurrences of continuous rings, for instance, in the context of with Kaplansky's direct finiteness conjecture~\cite{ElekSzabo,linnell} and the Atiyah conjecture~\cite{LinnellSchick,elek}. The present note is concerned with topological dynamics of the unit group $\GL(R)$ of an irreducible, continuous ring $R$, equipped with the relative rank topology. In~\cite{CarderiThom}, Carderi and Thom showed that, if $F$ is a finite field, then the topological group $\GL(\Mat_{\infty}(F))$ is \emph{extremely amenable}, i.e., every continuous action of $\GL(\Mat_{\infty}(F))$ on a non-void compact Hausdorff space has a fixed point. By work of the present author~\cite[Prop.~8.5 + Cor.~11.9]{FMS22}, for every non-discrete irreducible, continuous ring $R$, the union of extremely amenable topological subgroups of $\GL(R)$ is dense in $\GL(R)$. This illustrates that the phenomenon of extreme amenability is---to some extent---inherent to topological unit groups of non-discrete irreducible, continuous rings. On the other hand, by a well-known consequence of the ping-pong lemma, for every division ring $D$ of characteristic zero and every natural number $n \geq 2$, the unit group of the discrete irreducible, continuous ring $\Mat_{n}(D)$ is non-amenable, which raises the question as to whether there exist non-discrete irreducible, continuous rings with topologically non-amenable unit groups, too. This question is answered affirmatively by our main result, which concerns the ring of densely defined, closed, linear operators affiliated with the group von Neumann algebra $\vN(G)$ of a discrete group $G$.
\begin{mthm}[Corollary~\ref{corollary:main}] Let $G$ be a group that is not inner amenable.\footnote{See Proposition~\ref{proposition:robin} and Theorem~\ref{theorem:haagerup.olesen} for examples of such groups.} Then $\aff(\vN(G))$ is a non-discrete irreducible, continuous ring whose unit group is non-amenable with respect to the rank topology. \end{mthm}
The argument proving our main result proceeds via inspecting several isometric group actions for Eymard--Greenleaf amenability. More precisely, if $G$ is a non-inner amenable group, then the natural action of $G$ on the space of $1/2$-trace projections of the $\mathrm{II}_{1}$ factor $\vN(G)$, equipped with the trace metric, is not Eymard--Greenleaf amenable (Theorem~\ref{theorem:first}), which witnesses non-amenability of the topological group $\GL(\aff(\vN(G)))$, by virtue of a general mechanism comparing certain actions of $\GL(\aff(M))$ and the unitary group $\U(M) \leq \GL(\aff(M))$ for an arbitrary $\mathrm{II}_{1}$ factor $M$ (Lemma~\ref{lemma:amenable}).
This article is organized as follows. After recollecting some general background material on topological dynamics in Section~\ref{section:amenability}, we turn to continuous geometries and unit groups of their coordinate rings in Section~\ref{section:continuous.rings}. The subsequent Section~\ref{section:projections} contains a discussion of Eymard--Greenleaf amenability for actions of unitary groups of $\mathrm{II}_{1}$ factors on the associated projection spaces. Finally, in Section~\ref{section:groups}, we specify to group von Neumann algebras and connect our previous considerations with inner amenability of discrete groups, finishing the proof of our main result.
\section{Eymard--Greenleaf amenability}\label{section:amenability}
An action of a group $G$ by isomorphisms on a uniform space $(X,\mathscr{E})$ is said to be \emph{Eymard--Greenleaf amenable}\footnote{This term was coined by Pestov~\cite[Def.~3.5.9, p.~64]{PestovBook}, referencing works of Eymard~\cite{eymard} and Greenleaf~\cite{greenleaf}.} if the algebra \begin{displaymath}
\UCB(X,\mathscr{E}) \, \defeq \, \{ f \in \ell^{\infty}(X,\R) \mid \forall \epsilon \in \R_{>0} \, \exists E \in \mathscr{E} \, \forall (x,y) \in E \colon \, \vert f(x) - f(y) \vert \leq \epsilon \}
\end{displaymath} of all uniformly continuous bounded real-valued functions on $(X,\mathscr{E})$ admits a $G$-invariant mean, i.e., a positive unital linear map \begin{displaymath}
\mu \colon \, \UCB(X,\mathscr{E}) \, \longrightarrow \, \R
\end{displaymath} such that \begin{displaymath}
\forall g \in G \ \forall f \in \UCB(X,\mathscr{E}) \colon \quad \mu(f \circ \tilde{g}) \, = \, \mu(f) ,
\end{displaymath} where we let $\tilde{g} \colon X \to X, \, x \mapsto gx$ for each $g \in G$. In particular, this yields a concept of amenability for isometric group actions on metric spaces, where a metric space $(X,d)$ is being viewed as a uniform space carrying the induced uniformity \begin{displaymath}
\{ E \subseteq X \times X \mid \exists r \in \R_{>0} \, \forall x,y \in X \colon \, d(x,y) < r \Longrightarrow (x,y) \in E \} .
\end{displaymath} Furthermore, Eymard--Greenleaf amenability naturally gives rise to a notion of amenability for topological groups. To be more precise, let $G$ be a topological group. Considering the neighborhood filter $\mathscr{U}(G)$ of the neutral element in $G$, one may endow $G$ with its \emph{right uniformity} \begin{displaymath}
\mathscr{E}_{\Rsh}(G) \, \defeq \, \left\{ E \subseteq G \times G \left\vert \, \exists U \in \mathscr{U}(G) \, \forall x,y \in G \colon \, xy^{-1}\! \in U \Longrightarrow \, (x,y) \in E \right\} \right.\! .
\end{displaymath} The topological group $G$ is called \emph{amenable} the action of the group $G$ by left translations on the uniform space $(G,\mathscr{E}_{\Rsh}(G))$ is Eymard--Greenleaf amenable. By classical work of Rickert~\cite[Thm.~4.2]{rickert}, the topological group $G$ is amenable if and only if every continuous\footnote{Continuity of an action means \emph{joint} continuity.} action of $G$ on a non-void compact Hausdorff space admits an invariant regular Borel probability measure, or equivalently, if every continuous action of $G$ by affine homeomorphisms on a non-void compact convex subset of a locally convex topological vector space has a fixed point.
A topological group is said to have \emph{small invariant neighborhoods} if its neutral element admits a neighborhood basis consisting of conjugation-invariant subsets. The following is well known.
\begin{lem}\label{lemma:pestov} Let $G$ be an amenable topological group having small invariant neighborhoods. Then every continuous isometric action of $G$ on a non-empty metric space is Eymard--Greenleaf amenable. \end{lem}
\begin{proof} Consider a continuous isometric action of $G$ on a non-empty metric space $X$. Pick any $x \in X$. Since $G$ has small invariant neighborhoods, \begin{displaymath}
\UCB(X) \, \longrightarrow \, \UCB(G,\mathscr{E}_{\Rsh}(G)), \quad f \, \longmapsto \, (g \mapsto f(gx))
\end{displaymath} constitutes a well-defined operator, which is moreover unital, positive, and $G$-equivariant with respect to the left-translation action on $G$ (for details, see~\cite[Lem.~3.6.5, p.~71]{PestovBook} or~\cite[Prop.~3.9]{JuschenkoSchneider}). Thus, via composition with this operator, any $G$-left-invariant mean on $\UCB(G,\mathscr{E}_{\Rsh}(G))$ gives rise to a $G$-invariant mean on $\UCB(X)$. \end{proof}
\section{Continuous rings and their unit groups}\label{section:continuous.rings}
We recollect some elements of von Neumann's continuous geometry~\cite{VonNeumannBook}. By a \emph{lattice}
we mean a partially ordered set $L$ in which every pair of elements $x,y \in L$ admits both a (necessarily unique) supremum $x\vee y \in L$ and a (necessarily unique) infimum $x\wedge y \in L$. A \emph{complete lattice} is a partially ordered set $L$ such that every subset $S \subseteq L$ has a (necessarily unique) supremum $\bigvee S \in L$. If $L$ is a complete lattice, then every $S \subseteq L$ admits a (necessarily unique) infimum $\bigwedge S \in L$, too. A lattice $L$ is called \emph{bounded} if it has both a (necessarily unique) greatest element $1 = 1_{L} \in L$ and a (necessarily unique) least element $0 = 0_{L} \in L$. Clearly, any complete lattice is bounded. A lattice $L$ is said to be \emph{(directly) irreducible} if $\vert L \vert \geq 2$ and $L$ is not isomorphic to a direct product of two lattices of cardinality at least two. A \emph{continuous geometry} is a complete lattice $L$ such that \begin{itemize}
\item[---] $L$ is \emph{complemented}, i.e., \begin{displaymath}
\qquad \forall x \in L \ \exists y \in L \colon \quad x \vee y = 1, \ \, x\wedge y = 0 ,
\end{displaymath}
\item[---] $L$ is \emph{modular}, i.e., \begin{displaymath}
\qquad \forall x,y,z \in L \colon \quad x \leq y \ \, \Longrightarrow \ \, x \vee (y \wedge z) = y \wedge (x \vee z) ,
\end{displaymath}
\item[---] and, for every chain $C \subseteq L$ and every element $x \in L$, \begin{displaymath}
\qquad x \wedge \bigvee C \, = \, \bigvee \{ x \wedge y \mid y \in C\}, \quad \, x \vee \bigwedge C \, = \, \bigwedge \{ x \vee y \mid y \in C\} .
\end{displaymath}
\end{itemize} A \emph{dimension function} on a bounded lattice $L$ is a map $\Delta \colon L \to [0,1]$ such that \begin{itemize}
\item[---] $\Delta (0_{L}) = 0$ and $\Delta (1_{L}) = 1$,
\item[---] $\Delta (x\vee y) + \Delta (x \wedge y) \, = \, \Delta (x) + \Delta (y)$ for all $x,y \in L$,
\item[---] for all $x,y \in L$, \begin{displaymath}
\qquad x < y \ \, \Longrightarrow \ \, \Delta (x) < \Delta (y).
\end{displaymath} \end{itemize} If $\Delta \colon L \to [0,1]$ is a dimension function on a bounded lattice $L$, then \begin{displaymath}
\delta_{\Delta} \colon \, L \times L \, \longrightarrow \, [0,1] , \quad (x,y) \, \longmapsto \, \Delta(x \vee y)-\Delta(x\wedge y)
\end{displaymath} is a metric on $L$ (see~\cite[V.7, Lem.~on p.~76]{BirkhoffBook} or~\cite[I, Satz~6.2, p.~46]{MaedaBook}). By work of von Neumann~\cite{VonNeumannBook}\footnote{Existence is due to~\cite[I.VI, Thm.~6.9, p.~52]{VonNeumannBook} (see also~\cite[V, Satz~2.1, p.~118]{MaedaBook}), uniqueness is due to~\cite[I.VII, Cor.~1 on p.~60]{VonNeumannBook} (see also~\cite[V, Satz~2.3, p.~120]{MaedaBook}).}, every irreducible continuous geometry $L$ admits a unique dimension function, which will be denoted by $\Delta_{L} \colon L \to [0,1]$. If $L$ is an irreducible continuous geometry, then we let $\delta_{L} \defeq \delta_{\Delta_{L}}$.
We proceed to some basic remarks concerning von Neumann's continuous rings~\cite{VonNeumannBook} (see also~\cite{MaedaBook,GoodearlBook}). A ring will be called \emph{(directly) irreducible} if it is non-zero and not isomorphic to a direct product of two non-zero rings. Given a unital ring $R$, we consider the set \begin{displaymath}
\lat(R) \, \defeq \, \{ Ra \mid a \in R \} ,
\end{displaymath} partially ordered by set-theoretic inclusion. A unital ring $R$ is called \emph{(von Neumann) regular} if \begin{displaymath}
\forall a \in R \ \exists b \in R \colon \qquad aba \, = \, a .
\end{displaymath} Due to~\cite[II.II, Thm.~2.4, p.~72]{VonNeumannBook}, if $R$ is a regular ring, then the partially ordered set $\lat(R)$ is a complemented, modular lattice, in which \begin{displaymath}
I\vee J \, = \, I+J, \quad I\wedge J \, = \, I\cap J \qquad (I,J \in \lat(R)) .
\end{displaymath} A \emph{continuous} ring is a regular ring $R$ such that $\lat(R)$ is a continuous geometry. A \emph{rank function} on a regular $R$ is a map $\rk \colon R \to [0,1]$ such that \begin{itemize}
\item[---] $\rk(1)=1$,
\item[---] $\rk(ab) \leq \min \{ \rk(a),\rk(b)\}$ for all $a,b \in R$,
\item[---] for all $e,f \in R$, \begin{displaymath}
\qquad e^{2}=e, \ f^{2}=f, \ ef=fe=0 \ \, \Longrightarrow \ \, \rk(e+f) = \rk(e) + \rk(f),
\end{displaymath}
\item[---] $\rk (a) > 0$ for every $a \in R\setminus \{ 0 \}$.\footnote{The third condition readily entails that $\rk(0) = 0$.}
\end{itemize} For any rank function $\rk \colon R \to [0,1]$ on a regular ring $R$, \begin{displaymath}
d_{\rk} \colon \, R \times R \, \longrightarrow \, [0,1], \quad (a,b) \, \longmapsto \, \rk(a-b)
\end{displaymath} is a metric on $R$ (see~\cite[II.XVIII, Lem.~18.1]{VonNeumannBook} or~\cite[VI, Satz~5.1]{MaedaBook}), and the rank function $\rk \colon R \to [0,1]$ is called \emph{complete} if the metric space $(R,d_{\rk})$ is complete. By fundamental work of von Neumann~\cite[II.XVIII, Thm.~18.1]{VonNeumannBook} (see also~\cite[VII, Satz~2.2]{MaedaBook}), every irreducible, continuous ring $R$ admits a unique complete rank function, which will be denoted by $\rk_{R} \colon R \to [0,1]$.
\begin{remark}\label{remark:rank.vs.dimension} According to~\cite[II.II, Thm.~2.9, p.~76]{VonNeumannBook}, a regular ring $R$ is irreducible if and only if the lattice $\lat(R)$ is irreducible. Furthermore, due to~\cite[II.XVIII, Thm.~18.1, p.~237]{VonNeumannBook}, if $R$ is an irreducible, continuous ring, then $\rk_{R}(a) = \Delta_{\lat(R)}(Ra)$ for every $a \in R$. \end{remark}
Let $R$ be an irreducible, continuous ring. The topology on $R$ generated by the \emph{rank metric} $d_{\rk, R} \defeq d_{\rk_{R}}$ will be referred to as the \emph{rank topology} of $R$. We call $R$ \emph{discrete} if its rank topology is discrete. The unit group \begin{displaymath}
\GL(R) \, \defeq \, \{ a \in R \mid \exists b \in R \colon \, ab=ba=1 \}
\end{displaymath} endowed with the relative rank topology constitutes a topological group (cf.~\cite[Rem.~7.8]{FMS22}), which will be denoted by $\GL(R)_{\rk}$. Since its topology is generated by a bi-invariant metric (namely, the restriction of $d_{\rk,R}$), the topological group $\GL(R)_{\rk}$ has small invariant neighborhoods.
\begin{lem}\label{lemma:isometric.action} Let $R$ be an irreducible, continuous ring. Then \begin{displaymath}
\GL(R) \times \lat(R) \, \longrightarrow \, \lat(R), \quad (g,I) \, \longmapsto \, Ig^{-1}
\end{displaymath} is an continuous isometric action of $\GL(R)_{\rk}$ on $(\lat(R),\delta_{\lat(R)})$. Furthermore, for each $t \in [0,1]$, \begin{displaymath}
\lat_{t}(R) \, \defeq \, \{ I \in \lat(R) \mid \Delta_{\lat(R)}(I) = t \} \, = \, \{ Ra \mid a \in R, \, \rk_{R}(a) = t \}
\end{displaymath} is $\GL(R)$-invariant.\end{lem}
\begin{proof} Clearly, $\GL(R) \times \lat(R) \to \lat(R), \, (g,I) \mapsto Ig^{-1}$ is a well-defined action. If $g \in \GL(R)$, then $\rk_{R}(a) = \rk_{R}\left(ag^{-1}g\right) \leq \rk_{R}\left(ag^{-1}\right) \leq \rk_{R}(a)$ and hence \begin{equation}\tag{1}\label{unit}
\Delta_{\lat(R)}\left(Rag^{-1}\right) \, \stackrel{\ref{remark:rank.vs.dimension}}{=} \, \rk_{R}\left( ag^{-1} \right) \, = \, \rk_{R}(a) \, \stackrel{\ref{remark:rank.vs.dimension}}{=} \, \Delta_{\lat(R)}(Ra)
\end{equation} for all $a \in R$, thus \begin{align*}
\delta_{\lat(R)}\left(Ig^{-1},Jg^{-1}\right) \, &= \, \Delta_{\lat(R)}\left(Ig^{-1} + Jg^{-1}\right)-\Delta_{\lat(R)}\left(Ig^{-1} \cap Jg^{-1}\right) \\
& = \, \Delta_{\lat(R)}\left((I + J)g^{-1}\right)-\Delta_{\lat(R)}\left((I \cap J)g^{-1}\right) \\
& \stackrel{\eqref{unit}}{=} \, \Delta_{\lat(R)}(I + J)-\Delta_{\lat(R)}(I \cap J) \, = \, \delta_{\lat(R)}(I,J)
\end{align*} for all $I,J \in \lat(R)$, which shows that the considered action is isometric. Furthermore, as proved in~\cite[Lem.~7.9(3)]{FMS22}, \begin{equation}\tag{2}\label{orbital}
\forall I \in \lat(R) \ \forall a,b \in R \colon \qquad \delta_{\lat(R)}(Ia,Ib) \, \leq \, 2\min \{ \Delta_{\lat(R)}(I),\rk_{R}(a-b)\} .
\end{equation} In turn, \begin{displaymath}
\delta_{\lat(R)}\left(Ig^{-1},Ih^{-1}\right) \, \stackrel{\eqref{orbital}}{\leq} \, 2\rk_{R}\left(g^{-1}-h^{-1}\right) \, \stackrel{\eqref{unit}}{=} \, 2\rk_{R}(g-h) \, = \, d_{\rk,R}(g,h)
\end{displaymath} for all $I \in \lat(R)$ and $g,h \in \GL(R)$. This means that, for each $I \in \lat(R)$, the map \begin{displaymath}
(\GL(R),d_{\rk,R}) \, \longrightarrow \, (\lat(R),\delta_{\lat(R)}), \quad g \, \longmapsto \, Ig^{-1}
\end{displaymath} is $2$-Lipschitz, in particular continuous. Since the action is also isometric, thus the map $\GL(R) \times \lat(R) \to \lat(R), \, (g,I) \mapsto Ig^{-1}$ is continuous. The final assertion is an immediate consequence of Remark~\ref{remark:rank.vs.dimension} and~\eqref{unit}. \end{proof}
\begin{lem}\label{lemma:skew.amenable} Let $R$ be a non-discrete, irreducible, continuous ring, let $t \in [0,1]$. If $\GL(R)_{\rk}$ is amenable, then the action of $\GL(R)$ on $(\lat_{t}(R),\delta_{\lat(R)})$ is Eymard--Greenleaf amenable. \end{lem}
\begin{proof} Note that $\lat_{t}(R) \ne \emptyset$ by non-discreteness of $R$ (see, e.g.,~\cite[Rem.~7.2]{FMS22}). Since the topological group $\GL(R)_{\rk}$ has small invariant neighborhoods, thus the claim is a direct consequence of Lemma~\ref{lemma:isometric.action} and Lemma~\ref{lemma:pestov}. \end{proof}
\section{Geometry of projections and affiliated operators}\label{section:projections}
In this section we prove that, if the unit group of the ring of operators affiliated with a $\mathrm{II}_{1}$ factor $M$ is amenable with respect to the corresponding rank topology, then the action of the unitary group of $M$ on the space of its $1/2$-trace projections is Eymard--Greenleaf amenable (Lemma~\ref{lemma:amenable}).
We start off with some very general remarks on von Neumann algebras. For background, the reader is referred to~\cite{KadisonRingrose,ConwayBook}. Given a von Neumann algebra $M$, we consider its \emph{unitary group} \begin{displaymath}
\U(M) \, \defeq \, \{ u \in M \mid uu^{\ast} = u^{\ast}u = 1 \} ,
\end{displaymath} as well as the set \begin{displaymath}
\Pro(M) \, \defeq \, \left\{ p \in M \left\vert \, p^{2} = p = p^{\ast} \right\} \right.
\end{displaymath} of all \emph{projections} of $M$. If $M$ is a von Neumann factor of type $\mathrm{II}_{1}$, then we let $\tr_{M} \colon M \to \C$ denote its (necessarily faithful, normal) unique tracial state (cf.~\cite[Thm.~8.2.8, p.~517]{KadisonRingrose}), which in turn gives rise to the \emph{trace metric} \begin{displaymath}
d_{\tr,M} \colon \, M \times M \, \longrightarrow \, \R_{\geq 0} , \quad (x,y) \, \longmapsto \, \sqrt{\tr_{M}((x-y)^{\ast}(x-y))} .
\end{displaymath}
\begin{remark}\label{remark:isometric.action} Let $M$ be a von Neumann factor of type $\mathrm{II}_{1}$. Then \begin{displaymath}
\U(M) \times \Pro(M) \, \longrightarrow \, \Pro(M) , \quad (u,p) \, \longmapsto \, upu^{\ast}
\end{displaymath} is an isometric action of $\U(M)$ on $(\Pro(M),d_{\tr,M})$. For each $t \in [0,1]$, \begin{displaymath}
\Pro_{t}(M) \, \defeq \, \{ p \in \Pro(M) \mid \tr_{M}(p) = t \}
\end{displaymath} is a $\U(M)$-invariant subset of $\Pro(M)$. \end{remark}
The following remark summarizes several facts about the geometry of projections in $\mathrm{II}_{1}$ factors.
\begin{remark}\label{remark:projection.lattice} Let $M$ be a von Neumann algebra. We equip $\Pro(M)$ with the partial order defined by \begin{displaymath}
p \leq q \quad :\Longleftrightarrow \quad pq = p \qquad (p,q \in \Pro(M)) .
\end{displaymath} Observe that, for any two $p,q \in \Pro(M)$, \begin{equation}\tag{$\dagger$}\label{swap}
p \leq q \ \ \Longleftrightarrow \ \ pq=p \ \ \Longleftrightarrow \ \ (pq)^{\ast} = p^{\ast} \ \ \Longleftrightarrow \ \ q^{\ast}p^{\ast} = p^{\ast} \ \ \Longleftrightarrow \ \ qp = p .
\end{equation} Then $\Pro(M)$ is a complete lattice on which the map \begin{displaymath}
\Pro(M) \, \longrightarrow \, \Pro(M), \quad p \, \longmapsto \, 1-p
\end{displaymath} constitutes an orthocomplementation (see~\cite[Prop.~6.3, p.~82]{redei}). Suppose now that $M$ is finite. Then $\Pro(M)$ is also modular (see~\cite[Prop.~6.14, p.~99]{redei}), thus a continuous geometry by~\cite{kaplansky}. Moreover, $M$ is a non-zero factor if and only if $\Pro(M)$ is irreducible (cf.~\cite[1.1, \S6, Ex.~11C, p.~39]{BerberianBook}), in which case \begin{displaymath}
\Delta_{\Pro(M)} \, = \, {\tr_{M}}\vert_{\Pro(M)}
\end{displaymath} (see~\cite[8.4, p.~530]{KadisonRingrose}).
In particular, if $M$ is a factor of type $\mathrm{II}_{1}$, then \begin{displaymath}
\Delta_{\Pro(M)}(\Pro(M)) \, = \, \tr_{M}(\Pro(M)) \, = \, [0,1]
\end{displaymath} (see~\cite[Thm.~8.4.4(ii), p.~533]{KadisonRingrose}).
\end{remark}
By work of Murray and von Neumann~\cite[Thm.~XV, p.~229]{MurrayVonNeumann}, if $M$ is a finite von Neumann algebra acting on a Hilbert space~$H$, then the set $\aff(M)$ of all densely defined, closed, linear operators on $H$ \emph{affiliated with $M$}, i.e., those\footnote{That is, a densely defined, closed, linear operator $a$ on $H$ belongs to $\aff(M)$ if and only if $ua=au$ for every $u \in \U(M')$ (which entails that the domain of $a$ is $\U(M')$-invariant).} commuting with every unitary in the commutant of $M$, equipped with the addition \begin{displaymath}
\aff(M) \times \aff(M) \, \longrightarrow \, \aff(M), \quad (a,b) \, \longmapsto \, \overline{a+b}
\end{displaymath} and the multiplication \begin{displaymath}
\aff(M) \times \aff(M) \, \longrightarrow \, \aff(M), \quad (a,b) \, \longmapsto \, \overline{ab} ,
\end{displaymath} is a unital ring, of which $M$ constitutes a unital subring. The reader is referred to~\cite{KadisonLiu} for a comprehensive account on and to~\cite{Berberian57,BerberianBook,Berberian82} for alternative algebraic descriptions of this construction. We confine ourselves to the following remark, summarizing the information relevant for our purposes.
\begin{remark}\label{remark:affiliated} Let $M$ be a von Neumann factor of type $\mathrm{II}_{1}$. Then \begin{itemize}
\item[$(1)$] $\aff(M)$ is a regular ring by~\cite[II.II, Appx.~2, (VI), p.~89--90]{VonNeumannBook},
\item[$(2)$] due to~\cite[Cor.~7.1 + Thm.~5.2]{Berberian57}, $\kappa_{M} \colon \Pro(M) \to \lat(\aff(M)), \, p \mapsto \aff(M)p$ is a bijection, thus an order isomorphism, since for any $p,q \in \Pro(M)$, \begin{displaymath}
\qquad p \leq q \quad \Longleftrightarrow \quad pq=q \quad \Longleftrightarrow \quad \aff(M)p \subseteq \aff(M)q ,
\end{displaymath}
\item[$(3)$] ${\tr_{M}}\vert_{\Pro(M)} \stackrel{\ref{remark:projection.lattice}}{=} \Delta_{\Pro(M)} \stackrel{(2)}{=} {\Delta_{\lat(\aff(M))}} \circ {\kappa_{M}}$,
\item[$(4)$] $\aff(M)$ is continuous and irreducible by (2) and Remarks~\ref{remark:projection.lattice},~\ref{remark:rank.vs.dimension},
\item[$(5)$] $\aff(M)$ is non-discrete, since \begin{displaymath}
\qquad \rk_{\aff(M)}(\aff(M)) \, \stackrel{\ref{remark:rank.vs.dimension}}{=} \, \Delta_{\lat(\aff(M))}(\lat(\aff (M))) \, \stackrel{(3)}{=} \, \tr_{M}(\Pro (M)) \, \stackrel{\ref{remark:projection.lattice}}{=} \, [0,1] .
\end{displaymath}
\end{itemize} \end{remark}
The map from Remark~\ref{remark:affiliated}(2) has the following additional properties.
\begin{lem}\label{lemma:isometric} Let $M$ be a $\mathrm{II}_{1}$ factor and let $R \defeq \aff(M)$. Then \begin{itemize}
\item[$(1)$] $\kappa_{M} \colon \Pro(M) \to \lat(R)$ is $\U(M)$-equivariant,
\item[$(2)$] $\kappa_{M}(\Pro_{t}(M)) = \lat_{t}(R)$ for each $t \in [0,1]$, and
\item[$(3)$] $\kappa_{M}^{-1} \colon (\lat(R),\delta_{\lat(R)}) \to (\Pro(M),d_{\rk,R})$ is $1$-Lipschitz.
\end{itemize} \end{lem}
\begin{proof} (1) For all $p \in \Pro(M)$ and $u \in \U(M)$, \begin{displaymath}
\kappa_{M}(upu^{\ast}) \, = \, Rupu^{\ast} \, = \, Rpu^{\ast} \, = \, Rpu^{-1} \, = \, \kappa_{M}(p)u^{-1} .
\end{displaymath}
(2) This is an immediate consequence of Remark~\ref{remark:affiliated}(2)$+$(3).
(3) Let $I,J \in \lat(R)$. Consider $p \defeq \kappa_{M}^{-1}(I),\, q \defeq \kappa_{M}^{-1}(J) \in \Pro(M)$. Straightforward calculations using Remark~\ref{remark:projection.lattice}\eqref{swap} and the fact that $p\wedge q \leq p\vee q$ show that $e \defeq (p\vee q) - (p \wedge q) \in \Pro(M)$ and $(p\wedge q)e = e(p \wedge q) = 0$.
Thus, \begin{equation}\tag{$\ast$}\label{difference}
\rk_{R}(p \vee q) \, = \, \rk_{R}(e + (p\wedge q)) \, = \, \rk_{R}(e) + \rk_{R}(p\wedge q) .
\end{equation} Moreover, \begin{equation}\tag{$\ast\ast$}\label{null}
(p-q)(p\wedge q) \, = \, p(p\wedge q) - q(p\wedge q) \, = \, (p\wedge q) - (p\wedge q) \, = \, 0 .
\end{equation} We conclude that \begin{align*}
d_{\rk,R}\left(\kappa_{M}^{-1}(I),\kappa_{M}^{-1}(J)\right) \, &= \, d_{\rk,R}(p,q) \, = \, \rk_{R}(p-q) \, = \, \rk_{R}(p(p\vee q) - q(p\vee q)) \\
& = \, \rk_{R}((p-q)(p\vee q)) \, = \, \rk_{R}((p-q)(e + (p\wedge q))) \\
& = \, \rk_{R}((p-q)e + (p-q)(p\wedge q)) \, \stackrel{\eqref{null}}{=} \, \rk_{R}((p-q)e) \\
& \leq \, \rk_{R}(e) \, \stackrel{\eqref{difference}}{=} \, \rk_{R}(p \vee q) - \rk_{R}(p\wedge q) \\
& \stackrel{\ref{remark:affiliated}(4)+\ref{remark:rank.vs.dimension}}{=} \, \Delta_{\lat(R)}(\kappa_{M}(p \vee q)) - \Delta_{\lat(R)}(\kappa_{M}(p \wedge q)) \\
& \stackrel{\ref{remark:affiliated}(2)}{=} \, \Delta_{\lat(R)}(\kappa_{M}(p) + \kappa_{M}(q)) - \Delta_{\lat(R)}(\kappa_{M}(p)\cap \kappa_{M}(q)) \\
& = \, \Delta_{\lat(R)}(I + J) - \Delta_{\lat(R)}(I\cap J) \, = \, \delta_{\lat(R)}(I,J) . \qedhere
\end{align*} \end{proof}
Our proof of Lemma~\ref{lemma:amenable} also uses the following well-known inequality.
\begin{lem}\label{lemma:trace.topology} Let $M$ be a von Neumann factor of type $\mathrm{II}_{1}$. For every $a \in M$, \begin{displaymath}
\tr_{M}(a^{\ast}a) \, \leq \, \Vert a \Vert^{2}\rk_{\aff(M)}(a) .
\end{displaymath} \end{lem}
\begin{proof} First, being a positive linear functional on a $C^{\ast}$-algebra, $\tr_{M}$ satisfies \begin{equation}\tag{$1$}\label{murphy}
\forall x,y \in M \colon \qquad \tr_{M}(y^{\ast} x^{\ast}xy) \, \leq \, \Vert x^{\ast}x \Vert \tr_{M}(y^{\ast}y)
\end{equation} (see, e.g.,~\cite[Thm.~3.3.7, p.~90]{MurphyBook}). Now, consider $R \defeq \aff(M)$ and let $a \in M$. By Remark~\ref{remark:affiliated}(2), there exists $p \in \Pro(M)$ with $Rp = Ra$. It follows that \begin{equation}\tag{$2$}\label{ideal}
a \, = \, ap
\end{equation} and \begin{equation}\tag{$3$}\label{dimension.vs.trace}
\rk_{R}(a) \, \stackrel{\ref{remark:rank.vs.dimension}}{=} \, \Delta_{\lat(R)} (Ra) \, = \, \Delta_{\lat(R)} (Rp) \, = \Delta_{\lat(R)}(\kappa_{M}(p)) \, \stackrel{\ref{remark:affiliated}(3)}{=} \, \tr_{M}(p) .
\end{equation} We conclude that \begin{align*}
\tr_{M}(a^{\ast}a) \, &\stackrel{\eqref{ideal}}{=} \, \tr_{M}(p^{\ast}a^{\ast}ap) \, \stackrel{\eqref{murphy}}{\leq} \, \Vert a^{\ast}a \Vert \tr_{M}(p^{\ast}p) \, = \, \Vert a \Vert^{2}\tr_{M}(p) \, \stackrel{\eqref{dimension.vs.trace}}{=} \, \Vert a \Vert^{2}\rk_{R}(a) . \qedhere
\end{align*} \end{proof}
\begin{lem}\label{lemma:amenable} Let $M$ be a von Neumann factor of type $\mathrm{II}_{1}$, let $R \defeq \aff(M)$, and let $t \in [0,1]$. Suppose that the topological group $\GL(R)_{\rk}$ is amenable. Then the action of $\U(M)$ on $(\Pro_{t}(M),d_{\rk,R})$ is Eymard--Greenleaf amenable. In particular, the action of $\U(M)$ on $(\Pro_{t}(M),d_{\tr,M})$ is Eymard--Greenleaf amenable. \end{lem}
\begin{proof} Since $R$ is non-discrete by Remark~\ref{remark:affiliated}(5) and $\GL(R)_{\rk}$ is amenable, the action of $\GL(R)$ on $(\lat_{t}(R),\delta_{\lat(R)})$ is Eymard--Greenleaf amenable due to Lemma~\ref{lemma:skew.amenable}, i.e., there is a $\GL(R)$-invariant mean $\mu \colon \UCB(\lat_{t}(R),\delta_{\lat(R)}) \to \R$. In particular, $\mu$ is $\U(M)$-invariant. By Lemma~\ref{lemma:isometric}, \begin{displaymath}
\UCB(\Pro_{t}(M),d_{\rk,R}) \, \longrightarrow \, \UCB(\lat_{t}(R),\delta_{\lat(R)}), \quad f \, \longmapsto \, f \circ \kappa_{M}^{-1}
\end{displaymath} is a well-defined, $\U(M)$-equivariant, positive, unital, linear operator, thus \begin{displaymath}
\UCB(\Pro_{t}(M),d_{\rk,R}) \, \longrightarrow \, \R, \quad f \, \longmapsto \, \mu\left( f \circ \kappa_{M}^{-1} \right)
\end{displaymath} constitutes a well-defined $\U(M)$-invariant mean. Hence, the action of $\U(M)$ on $(\Pro_{t}(M),d_{\rk,R})$ is Eymard--Greenleaf amenable. Furthermore, since \begin{align*}
d_{\tr, M}(u,v) \, &= \, \sqrt{\tr_{M}((u-v)^{\ast}(u-v))} \, \stackrel{\ref{lemma:trace.topology}}{\leq} \, \Vert u-v \Vert \sqrt{\rk_{R}(u-v)} \\
& \leq \, 2 \sqrt{\rk_{R}(u-v)} \, = \, 2 \sqrt{d_{\rk, R}(u,v)}
\end{align*} for all $p,q \in P(M)$, we see that $\UCB(\Pro_{t}(M),d_{\tr,M}) \subseteq \UCB(\Pro_{t}(M),d_{\rk,R})$. Thus, via restriction, any $\U(M)$-invariant mean on $\UCB(\Pro_{t}(M),d_{\rk,R})$ gives rise to a $\U(M)$-invariant mean on $\UCB(\Pro_{t}(M),d_{\tr,M})$.
\end{proof}
\section{Group von Neumann algebras and inner amenability}\label{section:groups}
In this section we prove that a non-trivial ICC group acting in an amenable fashion (in the sense of Eymard--Greenleaf) on the space of $1/2$-trace projections of its group von Neumann algebra must be inner amenable (Theorem~\ref{theorem:first}). Combining this with Lemma~\ref{lemma:amenable}, we deduce that, if the unit group of the ring affiliated with the group von Neumann algebra of a non-trivial ICC group $G$ is amenable with respect to the rank topology, then $G$ must be inner amenable (Theorem~\ref{theorem:second}). This way, we produce examples of non-discrete irreducible, continuous rings whose unit groups are non-amenable with respect to the rank topology (Corollary~\ref{corollary:main}).
Let us briefly recall the definition of a group von Neumann algebra. For background on this construction, the reader is referred to~\cite[7, \S43 $+$ \S53]{ConwayBook}. Let $G$ be a group and consider the complex Hilbert space $\ell^{2}(G) = \ell^{2}(G,\C)$ spanned by the orthonormal standard basis $(b_{g})_{g \in G}$. The left and right regular representations $\lambda_{G},\rho_{G} \colon G \to \U(\B(\ell^{2}(G)))$ are given by \begin{displaymath}
\lambda_{G}(g)(f) \, \defeq \, f(g^{-1}h), \qquad \rho_{G}(g)(f)(h) \, \defeq \, f(hg) \qquad \left(g,h \in G, \, f \in \ell^{2}(G)\right) ,
\end{displaymath} the \emph{adjoint representation}~\cite{effros} is defined as \begin{displaymath}
\alpha_{G} \colon \, G \, \longrightarrow \, \U(\B(\ell^{2}(G))), \quad g \, \longmapsto \, \lambda_{G}(g)\rho_{G}(g) = \rho(g)_{G}\lambda_{G}(g) .
\end{displaymath} The \emph{group von Neumann algebra} of $G$ is defined as the bicommutant \begin{displaymath}
\vN(G) \, \defeq \, \lambda_{G}(G)'' \, \subseteq \, \B\left(\ell^{2}(G)\right)
\end{displaymath} and comes along equipped with the faithful, normal, tracial state \begin{displaymath}
\tr_{\vN(G)} \colon \, \vN(G) \, \longrightarrow \, \C, \quad a \, \longmapsto \, \langle a(b_{e}), b_{e} \rangle .
\end{displaymath} Furthermore, let us consider the $\alpha_{G}$-invariant closed linear subspace \begin{displaymath}
\left. \ell^{2}_{0}(G) \, \defeq \, \{ b_{e} \}^{\perp} \, = \, \left\{ f \in \ell^{2}(G) \, \right\vert \langle f,b_{e} \rangle = 0 \right\} \, = \, \left. \left\{ f \in \ell^{2}(G) \, \right\vert f(e) = 0 \right\} .
\end{displaymath}
\begin{remark}\label{remark:icc} (1) A group $G$ is said to have the \emph{infinite conjugacy class property}, or to be an \emph{ICC group}, if the conjugacy class of every element of $G\setminus \{ e \}$ is infinite. It is well known (see, e.g.,~\cite[Thm.~43.13, p.~249]{ConwayBook}) that a group $G$ has the infinite conjugacy class property if and only if $\vN(G)$ is a factor. Moreover, if $G$ is a non-trivial ICC group, then the factor $\vN(G)$ is of type $\mathrm{II}_{1}$ (cf.~\cite[Thm.~53.1, p.~301]{ConwayBook}).
(2) Let $G$ be a non-trivial ICC group. Since $G \to \U(\vN(G)), \, g \mapsto \lambda_{G}(g)$ is a homomorphism, Remark~\ref{remark:isometric.action} entails that \begin{displaymath}
G \times \Pro(\vN(G)) \, \longrightarrow \, \Pro(\vN(G)) , \quad (g,p) \, \longmapsto \, \lambda_{G}(g)p\lambda_{G}(g)^{\ast}
\end{displaymath} is an isometric action of $G$ on $(\Pro(\vN(G)),d_{\tr,\vN(G)})$, which leaves each of the sets $\Pro_{t}(\vN(G))$ ($t \in [0,1]$) invariant. Henceforth, this action will be referred to as the \emph{natural} action of $G$ on $\Pro(\vN(G))$ (or $\Pro_{t}(\vN(G))$, for $t \in [0,1]$, resp.). \end{remark}
For a Hilbert space $H$, we consider its unit sphere $\Sph_{H} \defeq \{ x \in H \mid \Vert x \Vert = 1 \}$ equipped with the induced metric $\Sph_{H} \times \Sph_{H} \to \R, \, (x,y) \mapsto \Vert x-y \Vert$.
\begin{lem}\label{lemma:map} Let $G$ be a non-trivial ICC group. The map given by \begin{displaymath}
\phi_{G} \colon \, \Pro_{1/2}(\vN(G)) \, \longrightarrow \, \Sph_{\ell^{2}_{0}(G)}, \quad p \, \longmapsto \, p(b_{e}) - (\id-p)(b_{e}) \, = \, (2p-\id)(b_{e})
\end{displaymath} is $2$-Lipschitz with respect to the trace metric on $\Pro_{1/2}(\vN(G))$. Furthermore, \begin{displaymath}
\phi_{G}(\lambda_{G}(g)p\lambda_{G}(g)^{\ast}) \, = \, \alpha_{G}(g)(\phi_{G}(p))
\end{displaymath} for all $p \in \Pro_{1/2}(\vN(G))$ and $g \in G$. \end{lem}
\begin{proof} First of all, we need to show that $\phi_{G}$ is well defined. To this end, let $p \in \Pro_{1/2}(\vN(G))$. Evidently, \begin{displaymath}
\langle \phi_{G}(p),b_{e} \rangle \, = \, 2\langle p(b_{e}),b_{e} \rangle - \langle b_{e},b_{e} \rangle \, = \, 2\tr_{\vN(G)}(p) - 1 \, = \, 1-1 \, = \, 0 ,
\end{displaymath} hence $\phi_{G}(p) \in \ell^{2}_{0}(G)$. Note that $p^{\ast}(\id-p) = p(\id-p) = p-p^{2} = p-p =0$, thus
\begin{displaymath}
\langle p(b_{e}),(\id-p)(b_{e}) \rangle \, = \, \langle b_{e},p^{\ast}(\id-p)(b_{e}) \rangle \, = \, \langle b_{e},0 \rangle \, = \, 0 ,
\end{displaymath} that is, $p(b_{e}) \perp (\id-p)(b_{e})$. From this and the fact that $\{ p, \id-p \} \subseteq \Pro_{1/2}(M)$, we infer that \begin{align*}
\langle \phi_{G}(p),\phi_{G}(p) \rangle \, & = \, \langle p(b_{e}),p(b_{e}) \rangle + \langle (\id-p)(b_{e}),(\id-p)(b_{e}) \rangle \\
& = \, \langle p^{\ast}p(b_{e}),b_{e} \rangle + \langle (\id -p)^{\ast}(\id -p)(b_{e}),b_{e} \rangle \\
& = \, \langle p(b_{e}),b_{e} \rangle + \langle (\id -p)(b_{e}),b_{e} \rangle \\
& = \, \tr_{\vN(G)}(p) + \tr_{\vN(G)}(\id -p) \, = \, 1/2 + 1/2 \, = \, 1 ,
\end{align*} therefore $\Vert \phi_{G}(p) \Vert_{2} = 1$, i.e., $\phi_{G}(p) \in \Sph_{\ell^{2}_{0}(G)}$. This shows that $\phi_{G}$ is well defined. Concerning Lipschitz continuity, we observe that \begin{align*}
\Vert \phi_{G}(p) - \phi_{G}(q) \Vert_{2} \, & = \, 2 \Vert (p-q)(b_{e}) \Vert_{2} \, = \, 2 \sqrt{\langle (p-q)(b_{e}),(p-q)(b_{e}) \rangle} \\
& = \, 2 \sqrt{\langle (p-q)^{\ast}(p-q)(b_{e}),b_{e} \rangle} \, = \, 2 \sqrt{\tr_{M}((p-q)^{\ast}(p-q))} \\
& = \, 2d_{\tr,\vN(G)}(p,q)
\end{align*} for all $p,q \in \Pro_{1/2}(\vN(G))$. Finally, since $\rho_{G}(G) \subseteq \lambda_{G}(G)' = \lambda_{G}(G)''' = \vN(G)'$, we see that, for all $p \in \Pro_{1/2}(\vN(G))$ and $g \in G$, \begin{align*}
\phi_{G}(\lambda_{G}(g)p\lambda_{G}(g)^{\ast}) \, &= \, (2\lambda_{G}(g)p\lambda_{G}(g)^{\ast}-\id)(b_{e}) \\
& = \, \lambda_{G}(g)(2p-\id)\lambda_{G}(g)^{\ast}(b_{e}) \, = \, \lambda_{G}(g)(2p-\id)(b_{g^{-1}}) \\
& = \, \lambda_{G}(g)(2p-\id)\rho_{G}(g)(b_{e}) \\
& \stackrel{\rho_{G}(g) \in \vN(G)'}{=} \, \lambda_{G}(g)\rho_{G}(g)(2p-\id)(b_{e}) \\
& = \, \alpha_{G}(g)(2p-\id)(b_{e}) \, = \, \alpha_{G}(g)(\phi_{G}(p)) . \qedhere
\end{align*} \end{proof}
Before elaborating on consequences of Lemma~\ref{lemma:map}, let us recall another basic fact (Lemma~\ref{lemma:map.2}). To clarify some relevant notation, let $X$ be a set. Then $\Sym(X)$ denotes the full symmetric group over $X$, which consists of all bijections from $X$ to itself. Furthermore, let us equip the set \begin{displaymath}
\left. \Prob(X) \, \defeq \, \left\{ f \in \ell^{1}(X,\R) \, \right\vert \Vert f \Vert_{1} = 1, \, f \geq 0 \right\}
\end{displaymath} with the metric \begin{displaymath}
\Prob(X) \times \Prob(X) \, \longrightarrow \, \R, \quad (f,g) \, \longmapsto \, \Vert f-g \Vert_{1} .
\end{displaymath}
\begin{remark}\label{remark:isometry} Let $G$ be a group and let $X \defeq G\setminus \{ e \}$. Consider the homomorphism $\gamma_{G} \colon G \to \Sym(X)$ given by \begin{displaymath}
\gamma_{G}(g)(x) \, \defeq \, gxg^{-1} \qquad (g \in G, \, x \in X) .
\end{displaymath} We note that $\ell^{2}_{0}(G) \to \ell^{2}(X), \, f \mapsto f\vert_{X}$ is a isometric linear isomorphism, and \begin{displaymath}
\forall g \in G \ \forall f \in \ell^{2}_{0}(G) \colon \quad \alpha_{G}(g)(f)\vert_{X} \, = \, (f\vert_{X}) \circ \gamma_{G}\left(g^{-1}\right) .
\end{displaymath} \end{remark}
\begin{lem}\label{lemma:map.2} Let $X$ be a set. Then \begin{displaymath}
\psi_{X} \colon \, \Sph_{\ell^{2}(X)} \, \longrightarrow \, \Prob (X), \quad f \, \longmapsto \, \vert f \vert^{2}
\end{displaymath} is $2$-Lipschitz. Also, $\psi_{X}(f \circ \sigma) = \psi_{X}(f) \circ \sigma$ for all $f \in \Sph_{\ell^{2}(X)}$ and $\sigma \in \Sym(X)$. \end{lem}
\begin{proof} First of all, let us note that $\psi_{X}$ is well defined: indeed, if $f \in \Sph_{\ell^{2}(X)}$, then $\vert f \vert^{2}(x) = \vert f(x) \vert^{2} \in \R_{\geq 0}$ for every $x \in X$ and also $\sum_{x \in X} \vert f\vert^{2}(x) = \Vert f \Vert_{2}^{2} = 1$, wherefore $\vert f \vert^{2} \in \Prob(X)$. Furthermore, by the Cauchy--Schwarz inequality, \begin{align*}
\Vert \psi_{X}(f) - \psi_{X}(g) \Vert_{1} \, & = \, \left\lVert \vert f \vert^{2} - \vert g \vert^{2} \right\rVert_{1} \, = \, \sum\nolimits_{x \in X} \left\lvert \vert f(x)\vert^{2} - \vert g(x) \vert^{2} \right\rvert \\
& = \, \sum\nolimits_{x \in X} \vert \, \vert f(x) \vert + \vert g(x)\vert \, \vert \cdot \vert \, \vert f(x)\vert - \vert g(x)\vert \, \vert \\
& \leq \, \sum\nolimits_{x \in X} ( \vert f(x) \vert + \vert g(x)\vert) \vert f(x) - g(x) \vert \\
& = \, \lvert \langle \vert f\vert + \vert g \vert ,\vert f - g \vert \rangle \rvert \, \leq \, \Vert \, \vert f \vert + \vert g \vert \, \Vert_{2}\cdot \Vert f - g \Vert_{2} \, \leq \, 2 \Vert f-g \Vert_{2}
\end{align*} for all $f,g \in \Sph_{\ell^{2}(X)}$. Finally, if $f \in \Sph_{\ell^{2}(X)}$ and $\sigma \in \Sym(X)$, then \begin{displaymath}
\psi_{X}(f \circ \sigma) \, = \, \vert f \circ \sigma \vert^{2} \, = \, \vert f \vert^{2} \circ \sigma \, = \, \psi_{X}(f) \circ \sigma . \qedhere
\end{displaymath} \end{proof}
Now we turn back to the map devised in Lemma~\ref{lemma:map}.
\begin{lem}\label{lemma:map.3} Let $G$ be a non-trivial ICC group and let $X \defeq G\setminus \{ e\}$. Then \begin{displaymath}
\xi_{G} \colon \, \Pro_{1/2}(\vN(G)) \, \longrightarrow \, \Prob (X), \quad p \, \longmapsto \, \psi_{X}(\phi_{G}(p)\vert_{X})
\end{displaymath} is $4$-Lipschitz with respect to the trace metric on $\Pro_{1/2}(\vN(G))$. Furthermore, \begin{displaymath}
\xi_{G}(\lambda_{G}(g)p\lambda_{G}(g)^{\ast}) \, = \, \xi_{G}(p) \circ \gamma_{G}\left(g^{-1}\right)
\end{displaymath} for all $p \in \Pro_{1/2}(\vN(G))$ and $g \in G$. \end{lem}
\begin{proof} Thanks to Lemma~\ref{lemma:map}, Remark~\ref{remark:isometry} and Lemma~\ref{lemma:map.2}, the map $\xi_{G}$ is well defined. We also see that \begin{align*}
\Vert \xi_{G}(p) - \xi_{G}(q) \Vert_{1} \, &= \, \Vert \psi_{X}(\phi_{G}(p)\vert_{X}) - \psi_{X}(\phi_{G}(q)\vert_{X}) \Vert_{1} \, \stackrel{\ref{lemma:map.2}}{\leq} \, 2\Vert \phi_{G}(p)\vert_{X} - \phi_{G}(q)\vert_{X} \Vert_{2} \\
& \stackrel{\ref{remark:isometry}}{=} \, 2\Vert \phi_{G}(p) - \phi_{G}(q) \Vert_{2} \, \stackrel{\ref{lemma:map}}{\leq} \, 4d_{\tr,\vN(G)}(p,q)
\end{align*} for all $p,q \in \Pro_{1/2}(\vN(G))$, that is, $\xi_{G}$ is $4$-Lipschitz with respect to $d_{\tr,\vN(G)}$. Moreover, for all $g \in G$ and $p \in \Pro_{1/2}(\vN(G))$, \begin{align*}
\xi_{G}(\lambda_{G}(g)p&\lambda_{G}(g)^{\ast}) \, = \, \psi_{X}(\phi_{G}(\lambda_{G}(g)p\lambda_{G}(g)^{\ast})\vert_{X}) \, \stackrel{\ref{lemma:map}}{=} \, \psi_{X}((\alpha_{G}(g)\phi_{G}(p))\vert_{X}) \\
& \stackrel{\ref{remark:isometry}}{=} \, \psi_{X}\left((\phi_{G}(p)\vert_{X}) \circ \gamma_{G}\left(g^{-1}\right)\right) \, \stackrel{\ref{lemma:map.2}}{=} \, \psi_{X}(\phi_{G}(p)\vert_{X}) \circ \gamma_{G}\left(g^{-1}\right) \\
&= \, \xi_{G}(p) \circ \gamma_{G}\left(g^{-1}\right) .\qedhere
\end{align*} \end{proof}
\begin{lem}\label{lemma:map.4} Let $G$ be a non-trivial ICC group and let $X \defeq G\setminus \{ e\}$. Then \begin{displaymath}
\Xi_{G} \colon \, \ell^{\infty}(X,\R) \, \longrightarrow \, \UCB(\Pro_{1/2}(\vN(G)),d_{\tr,\vN(G)})
\end{displaymath} given by \begin{displaymath}
\Xi_{G}(f)(p) \, \defeq \, \sum\nolimits_{x \in X}f(x)\xi_{G}(p)(x) \qquad \left(f \in \ell^{\infty}(X,\R), \, p \in \Pro_{1/2}(\vN(G))\right)
\end{displaymath} is a positive, unital, linear operator. Furthermore, \begin{displaymath}
\Xi_{G}(f \circ {\gamma_{G}(g)})(p) \, = \, \Xi_{G}(f)(\lambda_{G}(g)p\lambda_{G}(g)^{\ast})
\end{displaymath} for all $f \in \ell^{\infty}(X,\R)$, $g \in G$ and $p \in \Pro_{1/2}(\vN(G))$. \end{lem}
\begin{proof} In order to prove that $\Xi_{G}$ is well defined, consider any $f \in \ell^{\infty}(X,\R)$. Since $\xi_{G}( \Pro_{1/2}(\vN(G))) \subseteq \Prob(X)$ by Lemma~\ref{lemma:map.3}, it follows that \begin{displaymath}
\sup\nolimits_{p \in \Pro_{1/2}(\vN(G))} \vert \Xi_{G}(f)(p) \vert \, \leq \, \sup\nolimits_{p \in \Pro_{1/2}(\vN(G))} \sum\nolimits_{x \in X} \vert f(x)\vert \xi_{G}(p)(x) \, \leq \, \Vert f \Vert_{\infty},
\end{displaymath} thus $\Xi_{G}(f) \in \ell^{\infty}(\Pro_{1/2}(\vN(G)),\R)$. For all $p,q \in \Pro_{1/2}(\vN(G))$, we see that \begin{align*}
\vert \Xi_{G}(f)(p) - \Xi_{G}(f)(q) \vert \, &\leq \, \sum\nolimits_{x \in X} \vert f(x) \vert \cdot \vert \xi_{G}(p)(x) - \xi_{G}(q)(x) \vert \\
& \leq \, \Vert f \Vert_{\infty} \Vert \xi_{G}(p) - \xi_{G}(q) \Vert_{1} \, \stackrel{\ref{lemma:map.3}}{\leq} \, \, 4\Vert f \Vert_{\infty} d_{\tr,\vN(G)}(p,q) .
\end{align*} That is, $\Xi_{G}(f) \colon \Pro_{1/2}(\vN(G)) \to \R$ is $4\Vert f \Vert_{\infty}$-Lipschitz with respect to~$d_{\tr,\vN(G)}$. In particular, $\Xi_{G}(f) \in \UCB(\Pro_{1/2}(\vN(G)),d_{\tr,\vN(G)})$. Hence, $\Xi_{G}$ is well defined. It is straightforward to check that $\Xi_{G}$ is linear. As $\xi_{G}( \Pro_{1/2}(\vN(G))) \subseteq \Prob(X)$ again by Lemma~\ref{lemma:map.3}, the operator $\Xi_{G}$ is moreover unital and positive. Finally, for all $f \in \ell^{\infty}(X,\R)$, $g \in G$ and $p \in \Pro_{1/2}(\vN(G))$, \begin{align*}
\Xi_{G}(f \circ {\gamma_{G}(g)})(p) \, &= \, \sum\nolimits_{x \in X} f(\gamma_{G}(g)(x))\xi_{G}(p)(x) \\
& = \, \sum\nolimits_{x \in X} f(x)\xi_{G}(p)\left(\gamma_{G}\left(g^{-1}\right)(x)\right) \\
& \stackrel{\ref{lemma:map.3}}{=} \, \sum\nolimits_{x \in X} f(x)\xi_{G}(\lambda_{G}(g)p\lambda_{G}(g)^{\ast})(x) \\
& = \, \Xi_{G}(f)(\lambda_{G}(g)p\lambda_{G}(g)^{\ast}) . \qedhere
\end{align*} \end{proof}
Recall that a group $G$ is said to be \emph{inner amenable}~\cite{effros} if the action of $G$ on the (discrete) set $G\setminus \{ e\}$ given by conjugation is amenable, i.e., if there exists a $\gamma_{G}(G)$-invariant mean on $\ell^{\infty}(G\setminus \{ e\},\R)$. Note that every non-inner amenable group is a non-trivial ICC group.
\begin{thm}\label{theorem:first} Let $G$ be a non-trivial ICC group. If the natural action of $G$ on $(\Pro_{1/2}(\vN(G)),d_{\tr,\vN(G)})$ is Eymard--Greenleaf amenable, then $G$ is inner amenable. \end{thm}
\begin{proof} In the light of Remark~\ref{remark:icc}(2), for each $g \in G$, we consider \begin{displaymath}
\pi_{G}(g) \colon \, \Pro_{1/2}(\vN(G)) \, \longrightarrow \, \Pro_{1/2}(\vN(G)), \quad p \, \longmapsto \, \lambda_{G}(g)p\lambda_{G}(g)^{\ast} .
\end{displaymath} Suppose that the action of $G$ on $(\Pro_{1/2}(\vN(G)),d_{\tr,\vN(G)})$ is Eymard--Greenleaf amenable, i.e., there is a mean $\mu \colon \UCB(\Pro_{1/2}(\vN(G)),d_{\tr,\vN(G)}) \to \R$ such that \begin{equation}\tag{$\ast$}\label{invariant.mean}
\forall g \in G \ \forall f \in \UCB(\Pro_{1/2}(\vN(G)),d_{\tr,\vN(G)}) \colon \quad \mu(f \circ {\pi_{G}(g)}) \, = \, \mu(f) .
\end{equation} Since $\Xi_{G}$ is a a positive, unital, linear operator by Lemma~\ref{lemma:map.4}, \begin{displaymath}
\nu \defeq \mu \circ {\Xi_{G}} \colon \, \ell^{\infty}(G\setminus \{ e\},\R) \, \longrightarrow \, \R
\end{displaymath} constitutes a mean. Furthermore, for all $g \in G$ and $f \in \ell^{\infty}(G\setminus \{ e \},\R)$, \begin{displaymath}
\nu(f \circ {\gamma_{G}(g)}) \, = \, \mu(\Xi_{G}(f \circ {\gamma_{G}(g)})) \, \stackrel{\ref{lemma:map.4}}{=} \, \mu(\Xi_{G}(f) \circ {\pi_{G}(g)}) \, \stackrel{\eqref{invariant.mean}}{=} \, \mu(\Xi_{G}(f)) \, = \, \nu(f) .
\end{displaymath} Thus, $G$ is inner amenable. \end{proof}
\begin{thm}\label{theorem:second} Let $G$ be a non-trivial ICC group and let $R \defeq \aff(\vN(G))$. If the topological group $\GL(R)_{\rk}$ is amenable, then $G$ is inner amenable. \end{thm}
\begin{proof} Suppose that $\GL(R)_{\rk}$ is amenable. Then, by Lemma~\ref{lemma:amenable}, the action of $\U(\vN(G))$ on $(\Pro_{1/2}(\vN(G)),d_{\tr,\vN(G)})$ is Eymard--Greenleaf amenable, whence the natural action of $G$ on $(\Pro_{1/2}(\vN(G)),d_{\tr,\vN(G)})$ is Eymard--Greenleaf amenable, too. Hence, $G$ is inner amenable by Theorem~\ref{theorem:first}. \end{proof}
\begin{cor}\label{corollary:main} Let $G$ be a group that is not inner amenable. Then $\aff(\vN(G))$ is a non-discrete irreducible, continuous ring whose unit group is non-amenable with respect to the rank topology. \end{cor}
\begin{proof} Not being inner amenable, $G$ must be a non-trivial ICC group. Thus, Remark~\ref{remark:icc}(1) and Remark~\ref{remark:affiliated}(1)$+$(4)$+$(5) assert that $R \defeq \aff(\vN(G))$ is a non-discrete irreducible, continuous ring. By Theorem~\ref{theorem:second}, the topological group $\GL(R)_{\rk}$ is non-amenable. \end{proof}
For the sake of completeness, we mention two prominent results negating inner amenability for certain concrete groups, thus providing specific examples of continuous rings such as in Corollary~\ref{corollary:main}.
\begin{prop}[Effros~\cite{effros}]\label{proposition:robin} Let $X$ be a set with $\vert X \vert > 1$. Then the free group $\free (X)$ is not inner amenable. \end{prop}
\begin{proof}\!\!\!\footnote{This argument, which is simpler than the original one from~\cite{effros} (based on~\cite{MurrayVonNeumannIV}), was kindly pointed out to the author by Robin Tucker-Drob.} For each $g \in \free (X)\setminus \{ e \}$, the centralizer $\Cent_{\free (X)}(g) = \{ h \in \free (X) \mid gh=hg\}$ is cyclic\footnote{By the Nielsen--Schreier theorem, the subgroup $\Cent_{\free (X)}(g) \leq \free (X)$ is free, i.e., $\Cent_{\free (X)}(g) \cong \free (Y)$ for some set $Y$. Since the center of $\Cent_{\free (X)}(g)$ contains the non-trivial element $g$, we conclude that $\vert Y \vert = 1$. Hence, $\Cent_{\free (X)}(g) \cong \free (Y) \cong \Z$ is cyclic.}, thus amenable. Since the (discrete) group $\free (X)$ is non-amenable, this implies by~\cite[Cor.~4.3]{HaagerupOlesen} (which is a consequence of a result due to Rosenblatt~\cite[Prop.~3.5]{Rosenblatt81}) that $\free (X)$ is not inner amenable. \end{proof}
The proof of the following result in~\cite{HaagerupOlesen} has the same global structure as the one above, but requires a much more delicate analysis of centralizers.
\begin{thm}[Haagerup \& Olesen~\cite{HaagerupOlesen}]\label{theorem:haagerup.olesen} The Thompson groups $T$ and $V$ are not inner amenable. \end{thm}
\section*{Acknowledgments}
The author is grateful to Andreas Thom, Maxime Gheysens, and Robin Tucker-Drob for their comments on earlier versions of this note. Moreover, the author would like to thank Robin Tucker-Drob for having pointed out the simple proof of Proposition~\ref{proposition:robin}.
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\section{Introduction}
The design of efficient scheduling algorithms is a fundamental problem in wireless networks. In each time slot, a scheduling algorithm aims to determine a subset of non-interfering links such that the system of queues in the network is stabilized. Depending on the interference model and the network topology, it is known that there exists a \textit{`rate region'} - a maximal set of arrival rates - for which the network can be stabilized. A scheduling algorithm that can support any arrival rate in the rate region is said to be throughput optimal. A well-known algorithm called the Max-Weight scheduling algorithm \cite{182479} is said to be throughput optimal. However, the Max-Weight scheduler is not practical for distributed implementation due to the following reasons: (i) global network state information is required, and (ii) requires the computation of maximum-weighted independent set problem in each time slot, which is an NP-hard problem.
There have been several efforts in the literature to design low-complex, distributed approximations to the Max-Weight algorithm \cite{11222334,max_w_2}. Greedy approximation algorithms such as the \textit{maximal} scheduling policies, which can support a fraction of the maximum throughput, are one such class of approximations \cite{Wan2013}. On the other hand, we have algorithms like carrier sense multiple access (CSMA) algorithms \cite{csma_1,csma_2}, which are known to be near-optimal in terms of the throughput performance but known to suffer from poor delay performance.
Inspired by the success of deep-learning-based algorithms in various fields like image processing and natural language processing, recently, there has been a growing interest in their application in wireless scheduling as well \cite{ml_1,ml_2,ml_3}. Initial research in this direction focused on the adaption of widely used neural architectures like multi-layer perceptrons or convolutional neural networks (CNNs) \cite{cnn_1} to solve wireless scheduling problems. However, these architectures are not well-suited for the scheduling problem because they do not explicitly consider the network graph topology. Hence, some of the recent works in wireless networks study the application of the Graph Neural Network (GNN) architectures for solving the scheduling problem \cite{gnn_1}. For instance, a recent work \cite{9414098} has proposed a GNN based algorithm, where it has been observed that the help of Graph Neural networks can improve the performance of simple greedy scheduling algorithms like Longest-Queue-First (LQF) scheduling.
However, this result is observed on a simple interference model called the conflict graph model, which captures only binary relationships between links. Nevertheless, in real wireless networks, the interference among the links is additive, and the cumulative effect of all the interfering links decides the feasibility of any transmission. Hence, it is essential to study whether the GNN based approach will improve the performance of greedy LQF scheduling under a realistic interference model like the (Signal-to-interference-plus-noise ratio) SINR model, which captures the cumulative nature of interference.
One of the challenges in conducting such a study is that the concept of graph neural networks is not readily applicable for the SINR interference model since a graph cannot represent it. Hence, we introduce a new interference model which retains the cumulative interference nature yet is amenable to a graph-based representation and conduct our study on the proposed interference model. This approach will provide insights into whether the GNN-based improvement for LQF will work for practical interference models.
To that end, in this paper, we study whether GNN based algorithms can be used for designing efficient scheduling under this general interference model. Specifically, we consider a $k$-tolerant conflict graph model, where a node can successfully transmit during a time slot if not more than $k$ of its neighbors are transmitting in that time slot. Moreover, when $k$ is set to zero, the $k$-tolerance model can be reduced to the standard conflict graph model, in which a node cannot transmit if any of its neighbors is transmitting. We finally tabulate our results and compare them with other GNN-based distributed scheduling algorithms under a standard conflict-graph-based interference model.
In sum, our contributions are as follows:
\begin{itemize}
\item[(i)] We propose a GCN-based distributed scheduling algorithm for a generalized interference model called the $k$-tolerant conflict graph model.
\item[(ii)] The training of the proposed GCN does
not require a labeled data set (involves solving an NP-hard
problem). Instead, we design a loss function that utilizes an existing greedy approach and trains a GCN that improves the performance of the greedy approach by $4$ to $20$ percent.
\end{itemize}
The remainder of the paper is organized as follows. In Sec.~\ref{section2}, we briefly present our network model. In Sec.~\ref{section3}, an optimal scheduling policy for $k$-tolerance conflict graph interference model, a GCN-based $k$-tolerant independent set solver, is presented. In Sec.~\ref{section4}, we conduct experiments on different data sets and show the numerical results of the GCN-based scheduling approach. Finally, the paper is concluded in Sec.~\ref{section5}.\\
\noindent
\textbf{\textit{Motivation:}} In the SINR interference model, a link can successfully transmit if the cumulative interference from all nodes within a radius is less than some fixed threshold value. The conflict graph model insists that all the neighbours should not transmit when a link is transmitting. However, in a real-world situation, a link can successfully transmit as long as the cumulative interference from all its neighbours (the links which can potentially interfere with a given link) is less than a threshold value. As a special case, in this paper, we consider a conservative SINR model called k-tolerance model in which, if $i_{max}$ is the estimated strongest interference that a link can cause to another and let $i_{th}$ be the cumulative threshold interference that a link can tolerate, then a conservative estimate of how many neighbouring links can be allowed to transmit without violating the threshold interference is given by $k= i_{th}/i_{max}$. In other words, $k$-neighbours can transmit while a given link is transmitting.
It can be seen that this conservative model retains the cumulative nature of the SINR interference model. Hence a study on this model should give us insights into the applicability of GNN based solutions for realistic interference models.
\section{Network Model}\label{section2}
We model the wireless network as an undirected graph
$\mathcal{G} = (V,E)$ with $N$ nodes.
Here, the set of nodes $V = \{v_i\}_{i=1}^N$ of the graph
represents links in the wireless network i.e.,
a transmitter-receiver pair.
We assume an edge between two nodes, if
the corresponding links could potentially interfere
with each other.
Let $E$ and $\textbf{A}$ denote the set of edges and the adjacency matrix of
graph $\mathcal{G}$ respectively. We denote the set
of neighbors of node $v$ by $\mathcal{N}(v)$ i.e.,
a node $v^{\prime} \in \mathcal{N}(v)$,
if the nodes $v$ and $v^{\prime}$ share an edge between them.
We say a node is $k$-tolerant,
if it can tolerate at most $k$ of its transmitting neighbors. In other words, a
$k$-tolerant node can successfully transmit, if the number of neighbors
transmitting at the same time is at most $k$.
We define a \textit{$k$}-\textit{tolerant conflict graph} as a graph in which each node
is $k$-tolerant, and model the wireless network as a
\textit{$k-$tolerant conflict graph}. Note that this is a generalization of the
popular conflict graph model, where a node can tolerate none of
its transmitting neighbors. The conflict graph model corresponds to $0$-tolerant conflict
graph ($k=0$).
We assume that the time is slotted. In each time slot,
the scheduler has to decide on the set of links to transmit in that time slot.
A feasible schedule is a set of links that can successfully transmit at the same time.
At any given time $t$, a set of links can successfully transmit,
if the corresponding nodes form a \textit{$k$-independent set} (defined below) in graph
$\mathcal{G}$.
Thus, a feasible schedule corresponds to a $k$-independent set in $\mathcal{G}$.
\begin{definition}
($k$-independent set) A subset of vertices of a
graph~$\mathcal{G}$ is $k$\hyp{}independent, if it induces
in $\mathcal{G}$, a sub-graph of maximum degree at most $k$.
\end{definition}
\par A scheduler has to choose a feasible schedule at any given time.
Let $\mathcal{S_G}$ denotes the collection of all possible $k-$independent sets i.e., the
feasible schedules. We denote the schedule at time $t$ by an $N$
length vector $\sigma(t) =\left(\sigma_v(t), \; v \in V\right)$.
We say $\sigma_v(t) = 1$ if at time $t$, node $v$ is scheduled to transmit and
$\sigma_v(t)=0$, otherwise.
Depending on the scheduling decision $\sigma(t) \in \mathcal{S_G}$
taken at time $t$, node $v \in V$ (a link in the original wireless network)
gets a rate of $\mu_v(t,\sigma)$.
We assume that packets arriving at node $v$ can be stored in an infinite buffer.
At time $t$, let $\lambda_v(t)$ be the number of packets that arrive at node
$v \in V$. We then have the following queuing dynamics at node $v$:
\begin{align}
q_v(t+1) = \left[q_v(t) + \lambda_v(t) - \mu_v(t,\sigma) \right]^+ .
\end{align}
The set of arrival rates for which there exist a scheduler that
can keep the queues stable is known as the rate region
of the wireless network.
\subsection{Max-Weight Scheduler}
A well known scheduler that stabilises the network is the Max-Weight algorithm~\cite{182479}.
The Max-Weight algorithm chooses a schedule $\sigma^*(t) \in \mathcal{S_G}$ that
maximizes the sum of queue length times the service rate, i.e.,
\begin{align}
\label{eqn: Max-Weight algorithm}
\sigma^*(t) = \arg \max_{\sigma \in \mathcal{S_G}} \sum_v q_v(t)
\mu_v(t,\sigma).
\end{align}
We state below one of the celebrated results in radio resource allocation.
\begin{theorem} \cite{182479}
Let the arrival process $\lambda_v(t)$ be an
ergodic process with mean $\lambda_v$.
If the mean arrival rates ($\lambda_v$) are within the rate
region, then the Max-Weight
scheduling algorithm is throughput optimal.
\end{theorem}
In spite of such an attractive result, the Max-Weight algorithm is seldom implemented in practice. This is because, the scheduling decision in \eqref{eqn: Max-Weight algorithm} has complexity that is exponential in the number of nodes.
Even with the simplistic assumption of a conflict graph model,
\eqref{eqn: Max-Weight algorithm} reduces to the NP-hard problem of finding the maximum weighted independent set. At the timescale of these scheduling decisions, finding the exact solution to \eqref{eqn: Max-Weight algorithm} is practically infeasible. Hence, we resort to solving \eqref{eqn: Max-Weight algorithm} using a Graph Neural Network (GNN) model. Before we explain our GNN based algorithm, we shall rephrase the problem in \eqref{eqn: Max-Weight algorithm} for the k-tolerant conflict graph model below.
\subsection{Maximum weighted k-independent set}
In the $k$-tolerant conflict graph model $\mathcal{G}$, the Max-Weight problem
is equivalent to the following integer program:
\begin{align}
\label{eqn:k-independent set problem}
\begin{aligned}
\mbox{Maximize: } & \sum_v \sigma_v w_v \\
\mbox{Such that: } & \sigma_v \left(\sum_{v^{\prime} \in \mathcal{N}(v)}
\sigma_{v^\prime}\right) \leq k \\
& \sigma_{v} \in \{0,1\}, \text{ for all } v \in \mathcal{V}
\end{aligned}
\end{align}
Here $\bm{w} = (w_v:\; v \in V)$ is the weight vector. The constraint in
\eqref{eqn:k-independent set problem} ensures that whenever a node is
transmitting, at most $k$ of its neighbors can
transmit.
It can be observed that the maximum weight
problem in \eqref{eqn: Max-Weight algorithm}
corresponds to using the weights
$w_v = q_v(t) \mu_v(t,\sigma)$ in the above formulation.
Henceforth, the rest of this paper is devoted to solving the maximum weighted
$k$-independent set problem using a graph neural network.\\
\section{Graph Neural Network based Scheduler}\label{section3}
\begin{center}
\begin{figure}[h!]
\hspace{-1.1in}
\includegraphics[width = 1.5\textwidth]{figure2.pdf}
\caption{The architecture of the Graph Convolutional Neural Network based maximum weighted $k-$independent set problem solver.}
\vspace{-2em}
\label{fig:gcn_schematic}
\end{figure}
\end{center}
In this section, we present a graph neural network
based solution to solve the maximum weighted
$k$-independent set problem. We use the Graph Convolution
Neural network (GCN) architecture from \cite{kipf2017semi,graph_conv}.
The GCN architecture is as follows:
We use a GCN with $L$ layers. The input of each layer is a
feature matrix $\textbf{Z}^l \in \mathbb{R}^{N \times C^l} $
and its output is fed as the input to the next layer.
Precisely, at the $(l+1)$th layer, the feature matrix $\textbf{Z}^{l+1}$ is
computed using the following graph convolution operation:
\begin{align}
\textbf{Z}^{l+1} = \Phi(\textbf{Z}^l \bm{\Theta}_0^l + \bm{\mathcal{L}} \textbf{Z}^l \bm{\Theta}_1^l),
\end{align}
where $\bm{\Theta}_0^l, \bm{\Theta}_1^l \in \mathbb{R}^{C^l \times C^{l+1}}$ are the trainable weights of the neural network, $C^l$~denotes the number of feature channels
in $l$-th layer, $\Phi(.)$ is a nonlinear activation function and $\bm{\mathcal{L}}$ is the normalized Laplacian of the input graph $\mathcal{G}$ computed as follows:
$\bm{\mathcal{L}} = \textbf{I}_N -
\textbf{D}^{-\frac{1}{2}} \textbf{A} \textbf{D}^{-\frac{1}{2}}.$
Here, $\textbf{I}_N$ denotes the $N \times N$ identity matrix and $\textbf{D}$ is the diagonal matrix with entries $\textbf{D}_{ii} = \sum_j \textbf{A}_{ij}$.
\par We take the input feature matrix
$\textbf{Z}^0 \in \mathbb{R}^{N \times 1}$
as the weights $\bm{w}$ of the nodes (hence $C^0 = 1$)
and $\Phi(.)$ as a ReLU activation function
for all layers except for the last layer. For the last layer,
we apply sigmoid activation
function to get the likelihood of the nodes to be included in the $k$-independent set.
We represent this likelihood map from the GCN network using an
N length vector $\bm{\pi} = (\pi_v,\; v \in V) \in [0,1]^N$.
In summary, the GCN takes a graph $\mathcal{G}$ and the node
weights $\bm{w}$ as input and returns a $N$ length
likelihood vector $\bm{\pi}$ (see Figure \ref{fig:gcn_schematic}).
However, we need a $k$-independent
set. In usual classification problems,
such a requirement is satisfied by projecting the likelihood
maps to a binary vector. Projecting the likelihood map onto the
collection of $k$-independent sets is not straightforward, since
the collection of $k$-independent sets are $N$ length binary vectors
that satisfy the constraints in \eqref{eqn:k-independent set problem}.
Such a projection operation by itself might be costly in terms of
computation. Instead, by taking inspiration from
\cite{NEURIPS2018_8d3bba74}, we pass the likelihood map through
a greedy algorithm\footnote{In practice, the greedy algorithm can be replaced with a distributed greedy algorithm \cite{7084695} and train the GCN model w.r.t the distributed greedy algorithm.} to get a $k$-independent set.
The greedy algorithm requires each node to keep track of the
number of its neighbours already added in $k$-independent set.
We sort the nodes in the descending order of the product of the
likelihood and the weight i.e., $\pi_v w_v$. We add the
node with highest likelihood-weight product to the
$k$-independent set, if at most $k$ of its neighbors
are already added in the $k$-independent set.
We remove the nodes that are neighbours to a node
which has already added to the set and also reached a tolerance
of $k$. We then repeat the procedure until
no further nodes are left to be added.
We use a set of node-weighted graphs to train the GCN. Since
the problem at hand is NP-hard, we refrain from finding
the true labels (maximum weighted $k$-independent set) to train the
GCN. Instead, we construct penalty
and reward functions using the desirable properties of the output $\bm{\pi}$. We then
learn the parameters by optimizing over a weighted sum of the constructed
penalties and rewards. We desire the output
$\bm{\pi}$
to predict the maximum weighted
$k$-independent set.
With this in mind we construct the
following rewards and penalties:
\par
\begin{enumerate}
\item [a)] The prediction $\bm{\pi}$ needs to maximize the sum of the weights. So, our prediction needs to maximize
$ R_1 = \sum_v \pi_v w_v $.
\item [b)] The prediction $
\bm{\pi}$ needs
to satisfy the
$k$-independent set constraints.
Therefore, we add a penalty,
if $\bm{\pi}$ violates the independent set
constraints in \eqref{eqn:k-independent set problem}, i.e.,
$P_1 = \sum_{v \in V} \left(\sigma_v \Big( \sum_{v^{\prime} \in \mathcal{N}(v)}
\sigma_{v^\prime} - k \Big) \right)^2$.
\item [c)] Recall that we use the greedy algorithm to predict the $k$-independent set from $\bm{\pi}$. The
greedy algorithm takes $(\pi_v w_v, \; v \in V)$ as the
input and returns a $k$-independent set.
We desire the total weight of the output
$\bm{\pi}$, i.e., $\sum_v \pi_v w_v$ to be close to the
total weight of the $k$-independent returned by the
greedy algorithm. Let $W_{gcn}$ be the total weight of
the independent set predicted by the
greedy algorithm. Then, we penalise the
output $\bm{\pi}$ if it deviates from $W_{gcn}$, i.e.,
$P_2 = \left|\sum_v \pi_v w_v - W_{gcn} \right|^2$.
\end{enumerate}
We finally construct our cost function as a weighted sum of the
above i.e., we want the GCN to minimize the cost function:
\begin{align}\label{eq_cost_function}
C = \beta_1 P_1 + \beta_2 P_2 - \beta_3 R_1
\end{align}
where $\beta_1$, $\beta_2$ and $\beta_3$ denotes the optimization weights of the cost function defined in equation~\eqref{eq_cost_function}.
\input{Results}
\section{Conclusion}\label{section5}
In this paper, we investigated the well-studied
problem of link scheduling in
wireless adhoc networks using the recent
developments in graph neural networks.
We modelled the wireless network as a $k$-tolerant conflict graph and demonstrated that using a GCN, we can improve the
performance of existing greedy algorithms.
We have shown experimentally that this GCN
model improves the performance of the greedy algorithm
by at least $4$-$6$ percent for the ER model and
$11$-$22$ percent for the BA model
(depending on the value of $k$).
In future, we would like to extend the model to a node
dependent tolerance value $k_v$ and pass the tolerance value
as the node features of the GNN in addition to the weights.
\bibliographystyle{ieeetr}
\section{Experiments}\label{section4}
We perform our experiments on a single GPU GeForce GTX 1080 Ti
\footnote{Training the models took around two hours.}.
The data used for training, validation and testing
are described in the subsection below.
\subsection{Dataset}
\par We train our GCN using randomly generated graphs.
We consider two graph distributions, namely Erdos-Reyni (ER) and
Barbasi-Albert (BA) models. These distributions were also used in \cite{9414098}.
Our choice of these graph models is to ensure fair comparison with prior work
on conflict graph model \cite{9414098} ($k=0$).
In ER model with $N$ nodes, an edge is introduced between two nodes with
a fixed probability $p$, independent of the generation of other edges.
The BA model generates a graph with $N$ nodes (one node at a time),
preferentially attaching the node to $M$ existing nodes with probability
proportional to the degree of the existing nodes.
\par For training purpose, we generate $5000$ graphs of each of
these models. For the ER model, we choose
$p \in \{0.02, 0.05, 0.075, 0.10, 0.15\}$ and for the
BA model we choose $M = Np$. The weights of the nodes are chosen
uniformly at random from the interval $[0,1]$. We use an additional
$50$ graphs for validation and $500$ graphs for testing.
\subsection{Choice of hyper-parameters}
We train a GCN with $3$ layers consisting i) an input layer
with the weights of the nodes as input features
ii) a single hidden layer with $32$ features and
iii) an output layer with $N$ features (one for each node)
indicating the likelihood of choosing the corresponding node in
the $k$-independent set.
This choice of using a smaller number of layers ensures
that the GCN operates with a minimal number of communications
with its neighbors. We fix $k = 0$, and experiment training the GCN
with different choices of the optimization weights $\beta_1$, $\beta_2$
and $\beta_3$. The results obtained are tabulated in Figure \ref{fig:table1}. Let $W_{gr}$ denote the total weight of
the plain greedy algorithm i.e., without any GCN and $W_{gcn}$
denote the total weight of the independent set predicted by the
GCN-greedy combination. We have tabulated the average ratio
between the total weight of the nodes in the independent set
obtained from the GCN-greedy and the total weight
of the nodes in the independent set obtained from the plain greedy algorithm, i.e., $W_{gcn}/W_{gr}$. The average is taken over the
test data set.
\begin{figure}
\begin{center}
\begin{tabular}{ |c|c|c|c|c|c|c|c| }
\hline
& & & & \multicolumn{2}{c|}{Test Data = ER} &
\multicolumn{2}{c|}{Test Data = BA} \\ \cline{5-8}
Training Data & $\beta_1$ & $\beta_2$ & $\beta_3$ & Average & Variance & Average & Variance \\
& & & &$W_{gcn}/W_{gr}$ & $\times \; 10^{-3}$ & $W_{gcn}/W_{gr}$& $\times \; 10^{-3}$ \\
\hline
\multirow{8}{*}{BA} & 5 & 5 & 10 & 1.038 & 3.047
& 1.11 & 10.16 \\
& 10 & 10 & 1 & 1.035 & 3.297 & 1.11 & 10.37 \\
& 5 & 5 & 1 & 1.035 & 3.290 & 1.11 & 10.14 \\
& 1 & 1 & 1 & 1.034 & 3.253 & 1.10 & 10.23 \\
& 5 & 5 & 30 & 1.041 & 3.230 & 1.10 & 10.39 \\
& 5 & 5 & 50 & 1.041 & 3.214 & 1.10 & 10.28 \\
& 5 & 5 & 100 & 1.035 & 2.838 & 1.09 & 10.02 \\
& 30 & 1 & 1 & 1.031 & 2.401 & 1.07 & 8.25 \\
\hline
\multirow{8}{*}{ER} & 5 & 5 & 30 & 1.040 & 2.929
& 1.10 & 10.12 \\
& 5 & 5 & 10 & 1.039 & 3.145 &
1.11 & 10.71 \\
& 5 & 5 & 50 & 1.039 & 2.957 &
1.09 & 9.92 \\
& 1 & 1 & 1 & 1.038 & 3.135 & 1.11 & 10.74 \\
& 1 & 20 & 1 & 1.036 & 3.070 & 1.11 & 10.55 \\
& 10 & 10 & 1 & 1.034 & 3.428 & 1.11 & 10.34 \\
& 5 & 5 & 1 & 1.034 & 3.331 & 1.11 & 10.34\\
& 5 & 5 & 100 & 1.031 & 2.420 & 1.08 & 8.42\\
\hline
\multicolumn{4}{|c|}{Distributed scheduling using GNN
\cite{9414098}} & 1.039 & 3.5 & 1.11 & 11.0 \\
\hline
\end{tabular}
\caption{Table showing the average and variance of the ratio of the total
weight of the nodes in the independent set ($K=0$) obtained using GCN to
that of the independent set obtained using greedy algorithm.
We observe a $3$ percent increase in the total weight for the ER model and
$11$ percent increase in the total weight for the BA model.
Our performance matches with the performance of the GCN used in
\cite{9414098}.}
\vspace{-1.5em}
\label{fig:table1}
\end{center}
\end{figure}
The training was done with BA and ER models separately. We test
the trained models also with test data from both models to understand if the trained models are transferable.
We see that GCN trained with
parameters $\beta_1 = 5$, $\beta_2 = 5$ and $\beta_3 = 10$
performs well
for both ER and BA graph models.
The GCN improves the total weight of the
greedy algorithm by $4$ percent for the ER model and
by $11$ percent for the BA model.
Also, we see that the GCN trained with ER
model performs well with BA data and vice versa.
\subsection{Performance for different $k$}
We also evaluate the performance for different tolerance values
$k \in \{1,2,3,4\}$. We use the parameters
$\beta_1 = 5$, $\beta_2 = 5$ and $\beta_3 = 10$
in the cost function. Recall that we have come up with this
choice using extensive simulations for $k=0$.
In Figure~\ref{fig:table2}, we tabulate the average ratio
between the total weight of the $k$-independent set obtained
using the GCN-greedy combo and that of the plain greedy algorithm
i.e., $W_{gcn}/W_{gr}$.
We have also included the variance from this performance.
We observe that the performance for a general $k$ is even better
as compared to $k=0$. For example, we see that for $k = 2,3,4$,
we see $6$ percent improvement for the ER model and close to $20$
percent improvement for the BA model.
\begin{figure}
\begin{center}
\begin{tabular}{ |c|c|c|c|c|c| }
\hline
& & \multicolumn{2}{c|}{Test Data = ER} &
\multicolumn{2}{c|}{Test Data = BA} \\ \cline{3-6}
Training Data & $k$ & Average & Variance & Average & Variance \\
& &$W_{gcn}/W_{gr}$ & $\times \; 10^{-3}$ & $W_{gcn}/W_{gr}$& $\times \; 10^{-3}$ \\
\hline
\multirow{4}{*}{BA} & 1 & 1.056 & 4.07 & 1.143 & 10.22 \\
& 2 & 1.062 & 5.26 & 1.193 & 10.92 \\
& 3 & 1.067 & 5.55 & 1.209 & 20.14 \\
& 4 & 1.063 & 4.53 & 1.241 & 20.57 \\
\hline
\multirow{4}{*}{ER} & 1 & 1.056 & 3.99 & 1.143 & 10.18 \\
& 2 & 1.064 & 5.12 & 1.187 & 10.81 \\
& 3 & 1.066 & 4.82 & 1.205 & 20.13 \\
& 4 & 1.062 & 4.18 & 1.225 & 20.29 \\
\hline
\end{tabular}
\caption{The table shows the average and variance of the
ratio between the total weight of the $k$-independent set obtained using GCN-greedy combo to that of the plain greedy algorithm for $k \in \{1,2,3,4\}$. We observe that the improvement is consistently
above $5$ percent for the ER model and above $14$ percent
for the BA model.}
\vspace{-1.5em}
\label{fig:table2}
\end{center}
\end{figure}
Interestingly, the GCN trained with ER graphs performs well on
the BA data set as well. This indicates that the trained GCN
is transferable to other models.
\begin{comment}
\begin{figure}[t!]
\centering
\begin{subfigure}[t]{0.5\textwidth}
\centering
\pgfplotstabletypeset[col sep=comma,
columns={Model,b1,b2,b3,avg,var},
columns/Model/.style={string type, column name = Training Data},
every head row/.style={before row=\hline,after row=\hline},
every last row/.style={after row=\hline},
every column/.style={column type/.add={|}{}},
every last column/.style={column type/.add={}{|}},
columns/b1/.style={column name = $\beta_1$},
columns/b2/.style={column name = $\beta_2$},
columns/b3/.style={column name = $\beta_3$},
columns/avg/.style={column name = Average, precision=3},
columns/var/.style={column name = Variance, precision=3}
] {./Results/K0_ER.csv}
\caption{Caption}
\label{fig:my_label}
\end{subfigure}%
\begin{subfigure}[t]{0.5\textwidth}
\centering
\pgfplotstabletypeset[col sep=comma,
columns={b1,b2,b3,avg,var},
every head row/.style={before row=\hline,after row=\hline},
every last row/.style={after row=\hline},
every column/.style={column type/.add={|}{}},
every last column/.style={column type/.add={}{|}},
columns/b1/.style={column name = $\beta_1$},
columns/b2/.style={column name = $\beta_2$},
columns/b3/.style={column name = $\beta_3$},
columns/avg/.style={column name = Average},
columns/var/.style={column name = Variance}] {./Results/K0_BA.csv}
\caption{Caption}
\label{fig:my_label}
\end{subfigure}
\end{figure}
\end{comment}
\section{Experiments}\label{section4}
We perform our experiments on a single GPU GeForce GTX 1080 Ti
\footnote{Training the models took around two hours.}.
The data used for training, validation and testing
are described in the subsection below.
\subsection{Dataset}
\par We train our GCN using randomly generated graphs.
We consider two graph distributions, namely Erdos-Reyni (ER) and
Barbasi-Albert (BA) models. These distributions were also used in \cite{9414098}.
Our choice of these graph models is to ensure fair comparison with prior work
on conflict graph model \cite{9414098} ($k=0$).
In ER model with $N$ nodes, an edge is introduced between two nodes with
a fixed probability $p$, independent of the generation of other edges.
The BA model generates a graph with $N$ nodes (one node at a time),
preferentially attaching the node to $M$ existing nodes with probability
proportional to the degree of the existing nodes.
\par For training purpose, we generate $5000$ graphs of each of
these models. For the ER model, we choose
$p \in \{0.02, 0.05, 0.075, 0.10, 0.15\}$ and for the
BA model we choose $M = Np$. The weights of the nodes are chosen
uniformly at random from the interval $[0,1]$. We use an additional
$50$ graphs for validation and $500$ graphs for testing.
\subsection{Choice of hyper-parameters}
We train a GCN with $3$ layers consisting i) an input layer
with the weights of the nodes as input features
ii) a single hidden layer with $32$ features and
iii) an output layer with $N$ features (one for each node)
indicating the likelihood of choosing the corresponding node in
the $k$-independent set.
This choice of using a smaller number of layers ensures
that the GCN operates with a minimal number of communications
with its neighbors. We fix $k = 0$, and experiment training the GCN
with different choices of the optimization weights $\beta_1$, $\beta_2$
and $\beta_3$. The results obtained are tabulated in Figure \ref{fig:table1}. Let $W_{gr}$ denote the total weight of
the plain greedy algorithm i.e., without any GCN and $W_{gcn}$
denote the total weight of the independent set predicted by the
GCN-greedy combination. We have tabulated the average ratio
between the total weight of the nodes in the independent set
obtained from the GCN-greedy and the total weight
of the nodes in the independent set obtained from the plain greedy algorithm, i.e., $W_{gcn}/W_{gr}$. The average is taken over the
test data set.
\begin{figure}
\begin{center}
\begin{tabular}{ |c|c|c|c|c|c|c|c| }
\hline
& & & & \multicolumn{2}{c|}{Test Data = ER} &
\multicolumn{2}{c|}{Test Data = BA} \\ \cline{5-8}
Training Data & $\beta_1$ & $\beta_2$ & $\beta_3$ & Average & Variance & Average & Variance \\
& & & &$W_{gcn}/W_{gr}$ & $\times \; 10^{-3}$ & $W_{gcn}/W_{gr}$& $\times \; 10^{-3}$ \\
\hline
\multirow{8}{*}{BA} & 5 & 5 & 10 & 1.038 & 3.047
& 1.11 & 10.16 \\
& 10 & 10 & 1 & 1.035 & 3.297 & 1.11 & 10.37 \\
& 5 & 5 & 1 & 1.035 & 3.290 & 1.11 & 10.14 \\
& 1 & 1 & 1 & 1.034 & 3.253 & 1.10 & 10.23 \\
& 5 & 5 & 30 & 1.041 & 3.230 & 1.10 & 10.39 \\
& 5 & 5 & 50 & 1.041 & 3.214 & 1.10 & 10.28 \\
& 5 & 5 & 100 & 1.035 & 2.838 & 1.09 & 10.02 \\
& 30 & 1 & 1 & 1.031 & 2.401 & 1.07 & 8.25 \\
\hline
\multirow{8}{*}{ER} & 5 & 5 & 30 & 1.040 & 2.929
& 1.10 & 10.12 \\
& 5 & 5 & 10 & 1.039 & 3.145 &
1.11 & 10.71 \\
& 5 & 5 & 50 & 1.039 & 2.957 &
1.09 & 9.92 \\
& 1 & 1 & 1 & 1.038 & 3.135 & 1.11 & 10.74 \\
& 1 & 20 & 1 & 1.036 & 3.070 & 1.11 & 10.55 \\
& 10 & 10 & 1 & 1.034 & 3.428 & 1.11 & 10.34 \\
& 5 & 5 & 1 & 1.034 & 3.331 & 1.11 & 10.34\\
& 5 & 5 & 100 & 1.031 & 2.420 & 1.08 & 8.42\\
\hline
\multicolumn{4}{|c|}{Distributed scheduling using GNN
\cite{9414098}} & 1.039 & 3.5 & 1.11 & 11.0 \\
\hline
\end{tabular}
\caption{Table showing the average and variance of the ratio of the total
weight of the nodes in the independent set ($K=0$) obtained using GCN to
that of the independent set obtained using greedy algorithm.
We observe a $3$ percent increase in the total weight for the ER model and
$11$ percent increase in the total weight for the BA model.
Our performance matches with the performance of the GCN used in
\cite{9414098}.}
\vspace{-1.5em}
\label{fig:table1}
\end{center}
\end{figure}
The training was done with BA and ER models separately. We test
the trained models also with test data from both models to understand if the trained models are transferable.
We see that GCN trained with
parameters $\beta_1 = 5$, $\beta_2 = 5$ and $\beta_3 = 10$
performs well
for both ER and BA graph models.
The GCN improves the total weight of the
greedy algorithm by $4$ percent for the ER model and
by $11$ percent for the BA model.
Also, we see that the GCN trained with ER
model performs well with BA data and vice versa.
\subsection{Performance for different $k$}
We also evaluate the performance for different tolerance values
$k \in \{1,2,3,4\}$. We use the parameters
$\beta_1 = 5$, $\beta_2 = 5$ and $\beta_3 = 10$
in the cost function. Recall that we have come up with this
choice using extensive simulations for $k=0$.
In Figure~\ref{fig:table2}, we tabulate the average ratio
between the total weight of the $k$-independent set obtained
using the GCN-greedy combo and that of the plain greedy algorithm
i.e., $W_{gcn}/W_{gr}$.
We have also included the variance from this performance.
We observe that the performance for a general $k$ is even better
as compared to $k=0$. For example, we see that for $k = 2,3,4$,
we see $6$ percent improvement for the ER model and close to $20$
percent improvement for the BA model.
\begin{figure}
\begin{center}
\begin{tabular}{ |c|c|c|c|c|c| }
\hline
& & \multicolumn{2}{c|}{Test Data = ER} &
\multicolumn{2}{c|}{Test Data = BA} \\ \cline{3-6}
Training Data & $k$ & Average & Variance & Average & Variance \\
& &$W_{gcn}/W_{gr}$ & $\times \; 10^{-3}$ & $W_{gcn}/W_{gr}$& $\times \; 10^{-3}$ \\
\hline
\multirow{4}{*}{BA} & 1 & 1.056 & 4.07 & 1.143 & 10.22 \\
& 2 & 1.062 & 5.26 & 1.193 & 10.92 \\
& 3 & 1.067 & 5.55 & 1.209 & 20.14 \\
& 4 & 1.063 & 4.53 & 1.241 & 20.57 \\
\hline
\multirow{4}{*}{ER} & 1 & 1.056 & 3.99 & 1.143 & 10.18 \\
& 2 & 1.064 & 5.12 & 1.187 & 10.81 \\
& 3 & 1.066 & 4.82 & 1.205 & 20.13 \\
& 4 & 1.062 & 4.18 & 1.225 & 20.29 \\
\hline
\end{tabular}
\caption{The table shows the average and variance of the
ratio between the total weight of the $k$-independent set obtained using GCN-greedy combo to that of the plain greedy algorithm for $k \in \{1,2,3,4\}$. We observe that the improvement is consistently
above $5$ percent for the ER model and above $14$ percent
for the BA model.}
\vspace{-1.5em}
\label{fig:table2}
\end{center}
\end{figure}
Interestingly, the GCN trained with ER graphs performs well on
the BA data set as well. This indicates that the trained GCN
is transferable to other models.
\begin{comment}
\begin{figure}[t!]
\centering
\begin{subfigure}[t]{0.5\textwidth}
\centering
\pgfplotstabletypeset[col sep=comma,
columns={Model,b1,b2,b3,avg,var},
columns/Model/.style={string type, column name = Training Data},
every head row/.style={before row=\hline,after row=\hline},
every last row/.style={after row=\hline},
every column/.style={column type/.add={|}{}},
every last column/.style={column type/.add={}{|}},
columns/b1/.style={column name = $\beta_1$},
columns/b2/.style={column name = $\beta_2$},
columns/b3/.style={column name = $\beta_3$},
columns/avg/.style={column name = Average, precision=3},
columns/var/.style={column name = Variance, precision=3}
] {./Results/K0_ER.csv}
\caption{Caption}
\label{fig:my_label}
\end{subfigure}%
\begin{subfigure}[t]{0.5\textwidth}
\centering
\pgfplotstabletypeset[col sep=comma,
columns={b1,b2,b3,avg,var},
every head row/.style={before row=\hline,after row=\hline},
every last row/.style={after row=\hline},
every column/.style={column type/.add={|}{}},
every last column/.style={column type/.add={}{|}},
columns/b1/.style={column name = $\beta_1$},
columns/b2/.style={column name = $\beta_2$},
columns/b3/.style={column name = $\beta_3$},
columns/avg/.style={column name = Average},
columns/var/.style={column name = Variance}] {./Results/K0_BA.csv}
\caption{Caption}
\label{fig:my_label}
\end{subfigure}
\end{figure}
\end{comment}
\section{Introduction}
The design of efficient scheduling algorithms is a fundamental problem in wireless networks. In each time slot, a scheduling algorithm aims to determine a subset of non-interfering links such that the system of queues in the network is stabilized. Depending on the interference model and the network topology, it is known that there exists a \textit{`rate region'} - a maximal set of arrival rates - for which the network can be stabilized. A scheduling algorithm that can support any arrival rate in the rate region is said to be throughput optimal. A well-known algorithm called the Max-Weight scheduling algorithm \cite{182479} is said to be throughput optimal. However, the Max-Weight scheduler is not practical for distributed implementation due to the following reasons: (i) global network state information is required, and (ii) requires the computation of maximum-weighted independent set problem in each time slot, which is an NP-hard problem.
There have been several efforts in the literature to design low-complex, distributed approximations to the Max-Weight algorithm \cite{11222334,max_w_2}. Greedy approximation algorithms such as the \textit{maximal} scheduling policies, which can support a fraction of the maximum throughput, are one such class of approximations \cite{Wan2013}. On the other hand, we have algorithms like carrier sense multiple access (CSMA) algorithms \cite{csma_1,csma_2}, which are known to be near-optimal in terms of the throughput performance but known to suffer from poor delay performance.
Inspired by the success of deep-learning-based algorithms in various fields like image processing and natural language processing, recently, there has been a growing interest in their application in wireless scheduling as well \cite{ml_1,ml_2,ml_3}. Initial research in this direction focused on the adaption of widely used neural architectures like multi-layer perceptrons or convolutional neural networks (CNNs) \cite{cnn_1} to solve wireless scheduling problems. However, these architectures are not well-suited for the scheduling problem because they do not explicitly consider the network graph topology. Hence, some of the recent works in wireless networks study the application of the Graph Neural Network (GNN) architectures for solving the scheduling problem \cite{gnn_1}. For instance, a recent work \cite{9414098} has proposed a GNN based algorithm, where it has been observed that the help of Graph Neural networks can improve the performance of simple greedy scheduling algorithms like Longest-Queue-First (LQF) scheduling.
However, this result is observed on a simple interference model called the conflict graph model, which captures only binary relationships between links. Nevertheless, in real wireless networks, the interference among the links is additive, and the cumulative effect of all the interfering links decides the feasibility of any transmission. Hence, it is essential to study whether the GNN based approach will improve the performance of greedy LQF scheduling under a realistic interference model like the (Signal-to-interference-plus-noise ratio) SINR model, which captures the cumulative nature of interference.
One of the challenges in conducting such a study is that the concept of graph neural networks is not readily applicable for the SINR interference model since a graph cannot represent it. Hence, we introduce a new interference model which retains the cumulative interference nature yet is amenable to a graph-based representation and conduct our study on the proposed interference model. This approach will provide insights into whether the GNN-based improvement for LQF will work for practical interference models.
To that end, in this paper, we study whether GNN based algorithms can be used for designing efficient scheduling under this general interference model. Specifically, we consider a $k$-tolerant conflict graph model, where a node can successfully transmit during a time slot if not more than $k$ of its neighbors are transmitting in that time slot. Moreover, when $k$ is set to zero, the $k$-tolerance model can be reduced to the standard conflict graph model, in which a node cannot transmit if any of its neighbors is transmitting. We finally tabulate our results and compare them with other GNN-based distributed scheduling algorithms under a standard conflict-graph-based interference model.
In sum, our contributions are as follows:
\begin{itemize}
\item[(i)] We propose a GCN-based distributed scheduling algorithm for a generalized interference model called the $k$-tolerant conflict graph model.
\item[(ii)] The training of the proposed GCN does
not require a labeled data set (involves solving an NP-hard
problem). Instead, we design a loss function that utilizes an existing greedy approach and trains a GCN that improves the performance of the greedy approach by $4$ to $20$ percent.
\end{itemize}
The remainder of the paper is organized as follows. In Sec.~\ref{section2}, we briefly present our network model. In Sec.~\ref{section3}, an optimal scheduling policy for $k$-tolerance conflict graph interference model, a GCN-based $k$-tolerant independent set solver, is presented. In Sec.~\ref{section4}, we conduct experiments on different data sets and show the numerical results of the GCN-based scheduling approach. Finally, the paper is concluded in Sec.~\ref{section5}.\\
\noindent
\textbf{\textit{Motivation:}} In the SINR interference model, a link can successfully transmit if the cumulative interference from all nodes within a radius is less than some fixed threshold value. The conflict graph model insists that all the neighbours should not transmit when a link is transmitting. However, in a real-world situation, a link can successfully transmit as long as the cumulative interference from all its neighbours (the links which can potentially interfere with a given link) is less than a threshold value. As a special case, in this paper, we consider a conservative SINR model called k-tolerance model in which, if $i_{max}$ is the estimated strongest interference that a link can cause to another and let $i_{th}$ be the cumulative threshold interference that a link can tolerate, then a conservative estimate of how many neighbouring links can be allowed to transmit without violating the threshold interference is given by $k= i_{th}/i_{max}$. In other words, $k$-neighbours can transmit while a given link is transmitting.
It can be seen that this conservative model retains the cumulative nature of the SINR interference model. Hence a study on this model should give us insights into the applicability of GNN based solutions for realistic interference models.
\section{Network Model}\label{section2}
We model the wireless network as an undirected graph
$\mathcal{G} = (V,E)$ with $N$ nodes.
Here, the set of nodes $V = \{v_i\}_{i=1}^N$ of the graph
represents links in the wireless network i.e.,
a transmitter-receiver pair.
We assume an edge between two nodes, if
the corresponding links could potentially interfere
with each other.
Let $E$ and $\textbf{A}$ denote the set of edges and the adjacency matrix of
graph $\mathcal{G}$ respectively. We denote the set
of neighbors of node $v$ by $\mathcal{N}(v)$ i.e.,
a node $v^{\prime} \in \mathcal{N}(v)$,
if the nodes $v$ and $v^{\prime}$ share an edge between them.
We say a node is $k$-tolerant,
if it can tolerate at most $k$ of its transmitting neighbors. In other words, a
$k$-tolerant node can successfully transmit, if the number of neighbors
transmitting at the same time is at most $k$.
We define a \textit{$k$}-\textit{tolerant conflict graph} as a graph in which each node
is $k$-tolerant, and model the wireless network as a
\textit{$k-$tolerant conflict graph}. Note that this is a generalization of the
popular conflict graph model, where a node can tolerate none of
its transmitting neighbors. The conflict graph model corresponds to $0$-tolerant conflict
graph ($k=0$).
We assume that the time is slotted. In each time slot,
the scheduler has to decide on the set of links to transmit in that time slot.
A feasible schedule is a set of links that can successfully transmit at the same time.
At any given time $t$, a set of links can successfully transmit,
if the corresponding nodes form a \textit{$k$-independent set} (defined below) in graph
$\mathcal{G}$.
Thus, a feasible schedule corresponds to a $k$-independent set in $\mathcal{G}$.
\begin{definition}
($k$-independent set) A subset of vertices of a
graph~$\mathcal{G}$ is $k$\hyp{}independent, if it induces
in $\mathcal{G}$, a sub-graph of maximum degree at most $k$.
\end{definition}
\par A scheduler has to choose a feasible schedule at any given time.
Let $\mathcal{S_G}$ denotes the collection of all possible $k-$independent sets i.e., the
feasible schedules. We denote the schedule at time $t$ by an $N$
length vector $\sigma(t) =\left(\sigma_v(t), \; v \in V\right)$.
We say $\sigma_v(t) = 1$ if at time $t$, node $v$ is scheduled to transmit and
$\sigma_v(t)=0$, otherwise.
Depending on the scheduling decision $\sigma(t) \in \mathcal{S_G}$
taken at time $t$, node $v \in V$ (a link in the original wireless network)
gets a rate of $\mu_v(t,\sigma)$.
We assume that packets arriving at node $v$ can be stored in an infinite buffer.
At time $t$, let $\lambda_v(t)$ be the number of packets that arrive at node
$v \in V$. We then have the following queuing dynamics at node $v$:
\begin{align}
q_v(t+1) = \left[q_v(t) + \lambda_v(t) - \mu_v(t,\sigma) \right]^+ .
\end{align}
The set of arrival rates for which there exist a scheduler that
can keep the queues stable is known as the rate region
of the wireless network.
\subsection{Max-Weight Scheduler}
A well known scheduler that stabilises the network is the Max-Weight algorithm~\cite{182479}.
The Max-Weight algorithm chooses a schedule $\sigma^*(t) \in \mathcal{S_G}$ that
maximizes the sum of queue length times the service rate, i.e.,
\begin{align}
\label{eqn: Max-Weight algorithm}
\sigma^*(t) = \arg \max_{\sigma \in \mathcal{S_G}} \sum_v q_v(t)
\mu_v(t,\sigma).
\end{align}
We state below one of the celebrated results in radio resource allocation.
\begin{theorem} \cite{182479}
Let the arrival process $\lambda_v(t)$ be an
ergodic process with mean $\lambda_v$.
If the mean arrival rates ($\lambda_v$) are within the rate
region, then the Max-Weight
scheduling algorithm is throughput optimal.
\end{theorem}
In spite of such an attractive result, the Max-Weight algorithm is seldom implemented in practice. This is because, the scheduling decision in \eqref{eqn: Max-Weight algorithm} has complexity that is exponential in the number of nodes.
Even with the simplistic assumption of a conflict graph model,
\eqref{eqn: Max-Weight algorithm} reduces to the NP-hard problem of finding the maximum weighted independent set. At the timescale of these scheduling decisions, finding the exact solution to \eqref{eqn: Max-Weight algorithm} is practically infeasible. Hence, we resort to solving \eqref{eqn: Max-Weight algorithm} using a Graph Neural Network (GNN) model. Before we explain our GNN based algorithm, we shall rephrase the problem in \eqref{eqn: Max-Weight algorithm} for the k-tolerant conflict graph model below.
\subsection{Maximum weighted k-independent set}
In the $k$-tolerant conflict graph model $\mathcal{G}$, the Max-Weight problem
is equivalent to the following integer program:
\begin{align}
\label{eqn:k-independent set problem}
\begin{aligned}
\mbox{Maximize: } & \sum_v \sigma_v w_v \\
\mbox{Such that: } & \sigma_v \left(\sum_{v^{\prime} \in \mathcal{N}(v)}
\sigma_{v^\prime}\right) \leq k \\
& \sigma_{v} \in \{0,1\}, \text{ for all } v \in \mathcal{V}
\end{aligned}
\end{align}
Here $\bm{w} = (w_v:\; v \in V)$ is the weight vector. The constraint in
\eqref{eqn:k-independent set problem} ensures that whenever a node is
transmitting, at most $k$ of its neighbors can
transmit.
It can be observed that the maximum weight
problem in \eqref{eqn: Max-Weight algorithm}
corresponds to using the weights
$w_v = q_v(t) \mu_v(t,\sigma)$ in the above formulation.
Henceforth, the rest of this paper is devoted to solving the maximum weighted
$k$-independent set problem using a graph neural network.\\
\section{Graph Neural Network based Scheduler}\label{section3}
\begin{center}
\begin{figure}[h!]
\hspace{-1.1in}
\includegraphics[width = 1.5\textwidth]{figure2.pdf}
\caption{The architecture of the Graph Convolutional Neural Network based maximum weighted $k-$independent set problem solver.}
\vspace{-2em}
\label{fig:gcn_schematic}
\end{figure}
\end{center}
In this section, we present a graph neural network
based solution to solve the maximum weighted
$k$-independent set problem. We use the Graph Convolution
Neural network (GCN) architecture from \cite{kipf2017semi,graph_conv}.
The GCN architecture is as follows:
We use a GCN with $L$ layers. The input of each layer is a
feature matrix $\textbf{Z}^l \in \mathbb{R}^{N \times C^l} $
and its output is fed as the input to the next layer.
Precisely, at the $(l+1)$th layer, the feature matrix $\textbf{Z}^{l+1}$ is
computed using the following graph convolution operation:
\begin{align}
\textbf{Z}^{l+1} = \Phi(\textbf{Z}^l \bm{\Theta}_0^l + \bm{\mathcal{L}} \textbf{Z}^l \bm{\Theta}_1^l),
\end{align}
where $\bm{\Theta}_0^l, \bm{\Theta}_1^l \in \mathbb{R}^{C^l \times C^{l+1}}$ are the trainable weights of the neural network, $C^l$~denotes the number of feature channels
in $l$-th layer, $\Phi(.)$ is a nonlinear activation function and $\bm{\mathcal{L}}$ is the normalized Laplacian of the input graph $\mathcal{G}$ computed as follows:
$\bm{\mathcal{L}} = \textbf{I}_N -
\textbf{D}^{-\frac{1}{2}} \textbf{A} \textbf{D}^{-\frac{1}{2}}.$
Here, $\textbf{I}_N$ denotes the $N \times N$ identity matrix and $\textbf{D}$ is the diagonal matrix with entries $\textbf{D}_{ii} = \sum_j \textbf{A}_{ij}$.
\par We take the input feature matrix
$\textbf{Z}^0 \in \mathbb{R}^{N \times 1}$
as the weights $\bm{w}$ of the nodes (hence $C^0 = 1$)
and $\Phi(.)$ as a ReLU activation function
for all layers except for the last layer. For the last layer,
we apply sigmoid activation
function to get the likelihood of the nodes to be included in the $k$-independent set.
We represent this likelihood map from the GCN network using an
N length vector $\bm{\pi} = (\pi_v,\; v \in V) \in [0,1]^N$.
In summary, the GCN takes a graph $\mathcal{G}$ and the node
weights $\bm{w}$ as input and returns a $N$ length
likelihood vector $\bm{\pi}$ (see Figure \ref{fig:gcn_schematic}).
However, we need a $k$-independent
set. In usual classification problems,
such a requirement is satisfied by projecting the likelihood
maps to a binary vector. Projecting the likelihood map onto the
collection of $k$-independent sets is not straightforward, since
the collection of $k$-independent sets are $N$ length binary vectors
that satisfy the constraints in \eqref{eqn:k-independent set problem}.
Such a projection operation by itself might be costly in terms of
computation. Instead, by taking inspiration from
\cite{NEURIPS2018_8d3bba74}, we pass the likelihood map through
a greedy algorithm\footnote{In practice, the greedy algorithm can be replaced with a distributed greedy algorithm \cite{7084695} and train the GCN model w.r.t the distributed greedy algorithm.} to get a $k$-independent set.
The greedy algorithm requires each node to keep track of the
number of its neighbours already added in $k$-independent set.
We sort the nodes in the descending order of the product of the
likelihood and the weight i.e., $\pi_v w_v$. We add the
node with highest likelihood-weight product to the
$k$-independent set, if at most $k$ of its neighbors
are already added in the $k$-independent set.
We remove the nodes that are neighbours to a node
which has already added to the set and also reached a tolerance
of $k$. We then repeat the procedure until
no further nodes are left to be added.
We use a set of node-weighted graphs to train the GCN. Since
the problem at hand is NP-hard, we refrain from finding
the true labels (maximum weighted $k$-independent set) to train the
GCN. Instead, we construct penalty
and reward functions using the desirable properties of the output $\bm{\pi}$. We then
learn the parameters by optimizing over a weighted sum of the constructed
penalties and rewards. We desire the output
$\bm{\pi}$
to predict the maximum weighted
$k$-independent set.
With this in mind we construct the
following rewards and penalties:
\par
\begin{enumerate}
\item [a)] The prediction $\bm{\pi}$ needs to maximize the sum of the weights. So, our prediction needs to maximize
$ R_1 = \sum_v \pi_v w_v $.
\item [b)] The prediction $
\bm{\pi}$ needs
to satisfy the
$k$-independent set constraints.
Therefore, we add a penalty,
if $\bm{\pi}$ violates the independent set
constraints in \eqref{eqn:k-independent set problem}, i.e.,
$P_1 = \sum_{v \in V} \left(\sigma_v \Big( \sum_{v^{\prime} \in \mathcal{N}(v)}
\sigma_{v^\prime} - k \Big) \right)^2$.
\item [c)] Recall that we use the greedy algorithm to predict the $k$-independent set from $\bm{\pi}$. The
greedy algorithm takes $(\pi_v w_v, \; v \in V)$ as the
input and returns a $k$-independent set.
We desire the total weight of the output
$\bm{\pi}$, i.e., $\sum_v \pi_v w_v$ to be close to the
total weight of the $k$-independent returned by the
greedy algorithm. Let $W_{gcn}$ be the total weight of
the independent set predicted by the
greedy algorithm. Then, we penalise the
output $\bm{\pi}$ if it deviates from $W_{gcn}$, i.e.,
$P_2 = \left|\sum_v \pi_v w_v - W_{gcn} \right|^2$.
\end{enumerate}
We finally construct our cost function as a weighted sum of the
above i.e., we want the GCN to minimize the cost function:
\begin{align}\label{eq_cost_function}
C = \beta_1 P_1 + \beta_2 P_2 - \beta_3 R_1
\end{align}
where $\beta_1$, $\beta_2$ and $\beta_3$ denotes the optimization weights of the cost function defined in equation~\eqref{eq_cost_function}.
\input{Results}
\section{Conclusion}\label{section5}
In this paper, we investigated the well-studied
problem of link scheduling in
wireless adhoc networks using the recent
developments in graph neural networks.
We modelled the wireless network as a $k$-tolerant conflict graph and demonstrated that using a GCN, we can improve the
performance of existing greedy algorithms.
We have shown experimentally that this GCN
model improves the performance of the greedy algorithm
by at least $4$-$6$ percent for the ER model and
$11$-$22$ percent for the BA model
(depending on the value of $k$).
In future, we would like to extend the model to a node
dependent tolerance value $k_v$ and pass the tolerance value
as the node features of the GNN in addition to the weights.
\bibliographystyle{ieeetr}
|
2206.06192
|
\section{Introduction}
This work is about the very well-known ``Switchboard'' subset of an evaluation of US English conversational telephone speech recognition, originally conducted by NIST in 2000. \cite{fiscus20002000}
The current best published result is 4.3\% WER \cite{tuske2021limit}, which also acknowledged that
``most of the speakers appear in the training data, hyperparameters are optimized on [the test set], and the human error rate might also have been overestimated''.
Other recent results demonstrate 5.0\% with low-latency streaming \cite{nguyen2021super},
and many works reference \cite{XiongEtAl:msr-tr2016} and \cite{saon2017interspeech} as the first systems to achieve the milestone of parity with human performance, which is described as 5.1\% to 5.9\% WER.
A careful analysis in \cite{mansfield2021revisiting} notes that
``humans are more likely to miss words than to misrecognize them'',
and is notable in several regards:
code was provided to specify a non-standard data cleaning and text normalization process,
while output from a research system was re-scored in an (unsuccessful) attempt to replicate a published result.
Our work continues in this effort to fully describe and improve upon the standard scoring methodology,
sharing data and software to enable reproducible results.
This work benchmarks commercial ASR systems, inspired by \cite{del2021earnings},
which archived outputs from the dates of collection.
For this Switchboard benchmark, a particular advantage of benchmarking commercial systems is that the evaluation simulates a more realistic scenario of presenting each conversation side as an entire 5-minute audio file.
By contrast, the NIST evaluation allowed research systems to use the reference segmentation as input,
which can result in artificially low WER scores.
This work is similar to \cite{kim2021semantic} by presenting transcript precision and recall as possibly more insightful alternatives to WER, particularly for highlighting characteristics of human performance.
The use of an ``oracle'' word error rate that is optimistically calculated from ASR alternatives is similar to \cite{moriya2021asr} which reranks N-best alternatives for spoken content retrieval, as well as our prior work \cite{wegmann2013tao} in evaluating systems for spoken term detection.
While evaluating traditional N-best lists, we also introduce a novel representation for phrase-level alternatives. This captures the full expressiveness of an ASR lattice \cite{povey2012generating}, but in a more compact and linear data structure that can be conveniently manipulated as input to ASR scoring software, or indexed by a text-based search engine infrastructure.
The aim of this work is to show how this representation enables nearly perfect oracle accuracy (0.18\% WER) on a well-established ASR task.
This theoretical result motivates the further use of phrase alternatives toward a highly practical goal of enabling spoken term detection (i.e. audio search) applications that exhibit perfect recall.
\section{Scoring the Switchboard Benchmark}
\begin{table*}[t]
\caption{
Switchboard WER scored with corrected references, optional deletions \& exclusions, using differing segmentations.
\\ Italicized results in all tables used the reference segmentation,
which can be considered a bound on expected real-world performance.
}
\label{tab:wer}
\centering
\begin{tabular}{ l r r r r r | r}
\toprule
& ASR1 & ASR2 & ASR3 & ASR4 & ASR5 & \textit{ASR6} \\
\midrule
\midrule
LDC STM \& GLM & 10.18 & 12.37 & 11.10 & 8.25 & 8.62 & \textit{4.63} \\
+ RT-03 GLM & 9.94 & 12.20 & 10.88 & 8.07 & 7.96 & \textit{4.40} \\
+ RT-04F GLM & 9.92 & 12.20 & 10.86 & 8.05 & 7.95 & \textit{4.39} \\
\midrule
\midrule
STM with corrections & 8.15 & 10.42 & 8.65 & 5.65 & 6.08 & \textit{2.72} \\
+ GLM with alternations & 8.07 & 10.29 & 8.41 & 5.51 & 5.79 & \textit{2.63} \\
\midrule
+ Exclude hesitations & 7.86 & 9.99 & 8.41 & 5.28 & 5.30 & \textit{2.45} \\
+ Optional backchannels & 7.77 & 9.77 & 7.55 & 5.17 & 4.54 & \textit{2.43} \\
+ Exclude backchannels & 6.48 & 9.72 & 7.55 & 5.08 & 4.54 & \textit{2.37} \\
\midrule
+ Single-segment STM & \textbf{6.43} & \textbf{9.67} & \textbf{6.42} & \textbf{5.01} &\textbf{4.50} & \textit{2.30} \\
+ \textit{Reference segmentation} & \textit{5.94} & \textit{9.66} & \textit{5.09} & \textit{4.29} &\textit{4.03} & \textbf{\textit{2.30}} \\
\bottomrule
\end{tabular}
\end{table*}
\subsection{Corrected reference files}
Reference files from the original NIST evaluation are now distributed by the Linguistic Data Consortium (LDC)\footnote{https://catalog.ldc.upenn.edu/LDC2002T43},
but differ from what was later used in DARPA-funded evaluations known as ``RT-03'' and ``RT-04F''.
For example, the newer GLM files include backchannel mappings that generally improve scores.
Human transcribers disagree on this very difficult task
\cite{XiongEtAl:msr-tr2016,saon2017interspeech,mansfield2021revisiting,greenberg1996insights,stolcke2017comparing},
so it should not be surprising that there are inevitably some errors in these reference transcripts and mappings.
For this work, a professional linguist was commissioned to very carefully audit and correct these references.
In addition to the original transcripts, they could refer to four independent results from human speech recognition (HSR) services,
but not any of the ASR systems.
This paper's authors further corrected the GLM file with ad-hoc normalization of number formatting.
However, the vast majority of changes were related to an artifact of the \verb|make_reference| script
that is distributed with the test set;
it was used to create the reference STM by converting transcripts from an original TXT file format.
Unfortunately, every contraction in the original transcript is always expanded into multiple words
(see lines 126-131 in \verb|make_reference|).
This does not seem to be sensible,
especially considering that the GLM filtering would also redundantly expand all contractions.
We thus decided to reverse this automatic expansion of contractions,
and directed the highly skilled linguist to transcribe each instance correctly as either its contracted or expanded form
by carefully listening to each acoustic realization,
favoring the contracted form in cases of true ambiguity.
These corrected reference files are shared publicly,\footnote{https://mod9.io/switchboard-benchmark.\{glm,stm\}}
and should lead to substantial improvements across all systems.
\subsection{Expansions vs. alternations}
The NIST SCTK software\footnote{https://github.com/usnistgov/SCTK}
can use a GLM mapping file to filter reference STM and hypothesis CTM files by applying a set of transformation rules.
For example, contracted or compound words can be expanded with a rule such as
\verb|I'M => I AM|.
However, by always expanding contractions in both the reference STM and hypothesis CTM,
this rule often double-counts correct matches as well as errors.
A better approach is to denote \textit{alternations} to be applied in the GLM file, e.g.
\verb|I'M => { I'M / I AM }|, which will be scored as one or two matches or errors as appropriate.
The original form should be listed first in the alternation, since SCTK will favor it when multiple alignments have the same number of errors; otherwise, favoring the expanded form results in overly optimistic scoring.
\subsection{Optional deletions and excluded words}
Another effect of the filtering is to treat some words as \textit{optional deletions},
marked by parentheses in the STM file, in particular \texttt{(\%HESITATION)}.
An ASR system should exclude such difficult words from CTM hypotheses,
due to the asymmetric risk:
an error can have a larger effect on the numerator of WER, compared to a correct match incrementing the denominator (Eq. \ref{eq:wer}).
One major commercial system (ASR3) never hypothesizes hesitations --
nor any backchannels such as ``uh-huh'', which are not optional deletions under the NIST scoring rules.
So that their system is not disadvantaged by a design choice, we can consider backchannels to be optional deletions as well.
So that other ASR systems are not then disadvantaged by hypothesizing backchannels, we also exclude those from their CTM files.
\subsection{Segmentation}
The NIST tools can misalign hypotheses with word-level timestamps that differ slightly from the reference utterance-level segmentation of an STM file.
A solution is to convert the multi-segment STM into one long segment.
This can improve WER for ASR systems with consistent timestamp drift, and is needed to score any HSR (human speech recognition) result.
This problem is not observed in academic research experiments,
because the reference segmentation is assumed to be a valid input to the ASR system.
This practice may be unrealistic in real-world settings, however,
as seen in the bottom rows of \tablename~\ref{tab:wer}:
it can have a rather significant effect on WER results.
\subsection{Measuring accuracy with precision and recall}
\begin{equation}
\label{eq:wer}
\textrm{WER} =
100\% \times
\frac{\textrm{\#Inserted} + \textrm{\#Deleted} + \textrm{\#Substituted}}{\textrm{\#Correct} + \textrm{\#Deleted} + \textrm{\#Substituted}}
\end{equation}
\begin{equation}
\label{eq:precision}
\textrm{Precision} =
\frac{\textrm{\#Correct}}{\textrm{\#Correct} + \textrm{\#Inserted} + \textrm{\#Substituted}}
\end{equation}
\begin{equation}
\label{eq:recall}
\textrm{Recall} =
\frac{\textrm{\#Correct}}{\textrm{\#Correct} + \textrm{\#Deleted} + \textrm{\#Substituted}}
\end{equation}
Whereas the WER metric can be computed as in Eq. \ref{eq:wer},
a pair of non-standard metrics can also be useful to consider when evaluating ASR accuracy.
Transcript precision is the proportion of hypothesized words that are correct.
It does not penalize deletions and scores consistently well for human transcripts,
since it forgives the common tendency to omit words or phrases that do not convey much meaning
(e.g. stuttering ``i i i i ...'').
The recall metric can be rather variable for HSR results;
it could be useful for evaluating against non-verbatim reference transcriptions.
\begin{table}[h]
\caption{
Automatic (ASR) vs. human (HSR) speech recognition.
\\ Human speech recognition marked $^*$ was not speaker-labeled; it was scored against a force-aligned speaker-merged STM file.}
\label{tab:precision}
\centering
\begin{tabular}{ l r c c r }
\toprule
& WER & Precision & Recall & Cost/min. \\
\midrule
\midrule
ASR1 & 6.43 & .950 & .945 & --- \\
ASR2 & 9.67 & .930 & .916 & 4.0\textcent{} \\
ASR3 & 6.42 & .953 & .943 & 7.2\textcent{} \\
ASR4 & 5.01 & .961 & .962 & 4.8\textcent{} \\
ASR5 & 4.50 & \textbf{.964} & .960 & 3.3\textcent{} \\
\midrule
\textit{ASR1} & \textit{5.94} & \textit{.953} & \textit{.947} & --- \\
\textit{ASR2} & \textit{9.66} & \textit{.929} & \textit{.913} & 2.5\textcent{} \\
\textit{ASR3} & \textit{5.09} & \textit{.965} & \textit{.953} & 16.5\textcent{} \\
\textit{ASR4} & \textit{4.29} & \textit{.969} & \textit{.963} & 11.0\textcent{} \\
\textit{ASR5} & \textit{4.01} & \textit{.968} & \textit{.964} & 3.0\textcent{} \\
\textit{ASR6} & \textit{2.30} & \textbf{\textit{.981}} &\textit{.981} & --- \\
\midrule
\midrule
HSR1 & 4.84 & \textbf{.973} & .957 & \$1.25 \\
HSR2 & 4.33 & \textbf{.975} & .962 & \$2.75 \\
\midrule
HSR3$^*$ & 12.95 & \textbf{.973} & .877 & \$0.79 \\
HSR4$^*$ & 11.72 & \textbf{.972} & .891 & \$2.00 \\
\bottomrule
\end{tabular}
\end{table}
\section{Representations of ASR Alternatives}
Lattices can be generated by some ASR decoders, particularly in a WFST system such as Kaldi \cite{povey2012generating},
to represent the inherent ambiguity and uncertainty of hypotheses.
However, the lattices are large and difficult to use in applications that
require properties such as time-synchronous word sub-sequences.
Let $L_u$ be the formal language representing the set of all word sequences encoded in the lattice for a given utterance $u$.
\subsection{Utterance-level alternatives (i.e. N-best lists)}
\begin{table}[t]
\caption{
Oracle WER for utterance-level ($N$-best) alternatives.
}
\label{tab:alternatives-utterance}
\centering
\begin{tabular}{ l c r r r r r }
\toprule
& WER & $N$ & $N_{\max}$ & $N_{.9}$ & $N_{.5}$ & MB \\
\midrule
\midrule
\textit{ASR1} & \textit{4.61} & 2 & 2 & 2 & 2 & 0.2 \\
\textit{ASR1} & \textit{2.70} & 10 & 10 & 10 & 10 & 0.5 \\
\textit{ASR1} & \textit{1.58} & 100 & 100 & 100 & 100 & 1.9 \\
\textit{ASR1} & \textit{1.09} & 1000 & 1000 & 1000 & 1000 & 15.2 \\
\midrule
\textit{ASR2} & \textit{7.39} & 2 & 2 & 2 & 2 & 0.2 \\
\textit{ASR2} & \textit{5.41} & 10 & 10 & 10 & 10 & 0.5 \\
\textit{ASR2} & \textit{4.35} & 100 & 100 & 100 & 29 & 1.5 \\
\textit{ASR2} & \textit{4.05} & 1000 & 1000 & 1000 & 29 & 7.6 \\
\midrule
\textit{ASR3} & \textit{3.95} & 2 & 2 & 2 & 2 & 0.2 \\
\textit{ASR3} & \textit{2.38} & 10 & 10 & 10 & 7 & 0.4 \\
\textit{ASR3} & \textit{2.06} &$\infty$& 20 & 20 & 7 & 0.5 \\
\midrule
\textit{ASR4} & \textit{3.12} & 2 & 2 & 2 & 2 & 0.2 \\
\textit{ASR4} & \textit{2.01} &$\infty$& 10 & 10 & 10 & 0.5 \\
\midrule
\textit{ASR5} & \textit{2.98} & 2 & 2 & 2 & 2 & 0.2 \\
\textit{ASR5} & \textit{2.29} &$\infty$& 5 & 5 & 5 & 0.4 \\
\bottomrule
\end{tabular}
\end{table}
Utterance-level alternatives, better known as N-best lists, can be used to enumerate a formal language $L_u(N)$,
a set comprising up to $N$ most likely word sequences in the lattice.
The lattice's language is a superset, with equality in the theoretical limit:
\begin{equation}
L_u \supseteq \lim_{N \to \infty} L_u(N)
\end{equation}
\subsection{Word-level alternatives}
Word-level alternatives, sometimes known as \textit{sausages}, can be derived by aligning paths in a lattice \cite{mangu2000finding}
or from statistics used in Minimum Bayes' Risk decoding \cite{xu2011minimum}.
These represent a smaller formal language of up to $N$ single-word sequences $L_w(N)$ at each word position $w$.
Due to 1-to-1 word alignments, the lattice's language cannot be decomposed as a
cross-product and concatenation (indicated by $\prod$) of component sets:
\begin{equation}
L_u \neq \prod_{w \in u} L_w(N)
\end{equation}
There may be sequences in $L_u$ that cannot be represented as a concatenation of elements in $L_w(N)$, even for large $N$.
\begin{table}[t]
\caption{
Oracle WER for word-level alternatives.
}
\label{tab:alternatives-word}
\centering
\begin{tabular}{ l c r r r r r }
\toprule
& WER & $N$ & $N_{\max}$ & $N_{.9}$ & $N_{.5}$ & MB \\
\midrule
\midrule
\textit{ASR1} & \textit{2.69} & 2 & 2 & 2 & 2 & 0.2 \\
\textit{ASR1} & \textit{1.35} & 10 & 10 & 10 & 2 & 0.4 \\
\textit{ASR1} & \textit{1.19} & 100 & 100 & 12 & 2 & 0.5 \\
\textit{ASR1} & \textit{1.19} &$\infty$& 323 & 12 & 2 & 0.5 \\
\midrule
\textit{ASR2} & \textit{6.98} & 2 & 2 & 2 & 1 & 0.2 \\
\textit{ASR2} & \textit{5.75} & 10 & 10 & 3 & 1 & 0.2 \\
\textit{ASR2} & \textit{5.74} &$\infty$& 25 & 3 & 1 & 0.2 \\
\bottomrule
\end{tabular}
\end{table}
\subsection{Phrase-level alternatives}
\begin{table}[t]
\caption{Oracle WER for phrase-level alternatives.}
\label{tab:alternatives-phrase}
\centering
\begin{tabular}{ l c r r r r r }
\toprule
& WER & $N$ & $N_{\max}$ & $N_{.9}$ & $N_{.5}$ & MB \\
\midrule
\midrule
\textit{ASR1} & \textit{2.92} & 2 & 2 & 2 & 2 & 0.3 \\
\textit{ASR1} & \textit{1.08} & 10 & 10 & 10 & 3 & 0.6 \\
\textit{ASR1} & \textit{0.65} & 100 & 100 & 22 & 3 & 1.0 \\
\textit{ASR1} & \textit{0.57} & 1000 & 1000 & 22 & 3 & 1.3 \\
\bottomrule
\end{tabular}
\end{table}
By contrast, all paths in the lattice can be represented as a subset of the crossed and concatenated phrase-level alternatives \cite{faria2021phrase}:
\begin{equation}
L_u \subseteq \lim_{N \to \infty} \prod_{p \in u} L_p(N)
\end{equation}
In this formulation $L_p(N)$ is a set of up to $N$ word sequences, which may be of varying lengths, at phrase position $p$.
\subsection{Converting lattices to phrase alternatives}
Phrase alternatives can be derived from a lattice as follows:
\begin{enumerate}
\item Word-align the lattice, which may need determinization.
\item Establish phrase boundaries as those times not crossed by non-silence arcs (above some arc posterior threshold).
\item For each phrase, mask the lattice arcs outside the phrase boundaries by setting their output symbols as epsilon.
\item Determinize each phrase-masked lattice, which removes most epsilon arcs, and find $N$ best paths (i.e. phrases).
\end{enumerate}
The phrase alternatives representation is motivated by its compactness compared to utterance-level alternatives,
since it decomposes the utterance as a concatenation of word sequences that are assumed to be independent of each other.
It is also more expressive since this cross product generates additional word sequences that may not have been present in the lattice.
\subsection{Representing alternative hypotheses in NIST SCTK}
A lesser-known feature of the CTM file format is that it can be used to represent \textit{alternatives} in ASR hypotheses, for example:
\begin{verbatim}
sw_4390 A * * <ALT_BEGIN>
sw_4390 A 4.49 0.66 UM
sw_4390 A * * <ALT>
sw_4390 A 4.49 0.66 I'M
sw_4390 A * * <ALT_END>
\end{verbatim}
While this is typically used to represent \textit{alternations} created by filtering with the GLM file,
it can be further leveraged to enable oracle scoring of ASR alternatives at various levels.
However, this functionality requires a minor modification\footnote{https://github.com/usnistgov/SCTK/pull/34} to the \verb|sclite| source code,
as well as auxiliary software\footnote{https://pypi.org/project/mod9-asr} that can create the CTM files while fixing a couple of related bugs in SCTK
(such as expanding doubly-nested alternatives after GLM filtering).
\section{Speech Recognition Systems}
Automatic (ASR) and human (HSR) systems were evaluated:
\
\textbf{ASR1} is a Kaldi baseline. An OPGRU acoustic model and a trigram language model were trained only on Switchboard plus Fisher. These models were loaded by the Mod9 ASR Engine to produce utterance-, word-, and phrase-level alternatives.
\textbf{ASR1$^*$} customized the decoding graph by adding the 28 words that were out-of-vocabulary (OOV)
with respect to the system's relatively small lexicon (about 40,000 words that appeared in the training data).
Pronunciations were automatically generated with a grapheme-to-phoneme model \cite{novak2016phonetisaurus}
by requesting the Mod9 ASR Engine's \verb|add-words| command.
\textbf{ASR1$^\dagger$} used non-default pruning beam sizes to produce denser lattices, by requesting a \verb|speed:3| option of the Mod9 ASR Engine, a trade-off with more compute and memory usage.
\textbf{ASR1$^{*\dagger}$} combined both of the above settings.
\
\textbf{ASR2} is IBM Watson with an older ``Narrowband'' model,
instead of using a more accurate ``next-generation'' model,
because this system is uniquely capable of demonstrating utterance- and word-level alternatives at extreme depths.
\textbf{ASR3} is Google Cloud Platform's STT service, using an ``enhanced'' variant of their ``phone\_call'' model.
\textbf{ASR4} is Amazon Transcribe, configured for US English.
\textbf{ASR5} is Microsoft Azure's Speech-to-Text service, which generates utterance-level alternatives of very limited depth.
\
\textbf{ASR6} is the system in \cite{tuske2021limit}, from which IBM Research shared CTM-formatted system outputs for evaluation purposes.
\
\textbf{HSR1} is the Rev.com service, which has speaker labeling.
\textbf{HSR2} is the TranscribeMe service, requesting ``verbatim'' quality transcripts that include speaker labeling.
\textbf{HSR3} is the TranscribeMe service, requesting ``first draft'' quality transcripts that do not include speaker labeling.
\textbf{HSR4} is the cielo24 service, with no speaker labeling.
\section{Results}
All results can be reproduced from system outputs\footnote{https://mod9.io/switchboard-benchmark-results.tar.gz}
that were archived in early 2022,
using open-source scoring scripts.\footnote{https://mod9.io/switchboard-benchmark-scripts.tar.gz}
The bottom row and right column of \tablename~\ref{tab:wer}, middle section of \tablename~\ref{tab:precision}, and left columns of other tables have italicized font.
This convention is used to clarify which results might be considered unrealistic,
due to use of a reference segmentation or also because of the oracle nature of selecting a best alternative.
\tablename~\ref{tab:wer} presents the WER results from scoring each of the ASR systems with successively improved configurations of the scoring tools,
as described in Sections 2.1 through 2.4.
\tablename~\ref{tab:precision} compares the ASR and HSR systems, including precision and recall metrics in addition to WER.
The results for HSR3 and HSR4 are exceptional because they required conversion of reference STM files into a single-channel format,
using forced-alignment with an HTK-based ASR system; regions of overlapped speech may be incorrectly merged in some cases.
Dual-channel audio files were submitted to the HSR services, so transcribers could understand conversations sides in context.
\tablename~\ref{tab:precision} also reports the cost of processing the Switchboard test set, based on its duration of 100 minutes.
For ASR without reference segmentation, audio was presented as channel-separated files, thus totaling 200 minutes, much of which was silence.
For ASR that exploited reference segmentation, audio was presented as a collection of 1,834 short audio files, totaling 123 minutes.
Note: ASR3 and ASR4 costs increase even as less data is processed, since their respective policies are to bill requests by rounding up to 15s granularity or at minimum 15s.
Tables 3, 4, 5, and 6 report the oracle WER when the NIST SCTK scoring software is presented with CTM files that represent utterance-, word-, and phrase-level alternatives.
These results all use the reference segmentation, since the software cannot score alternatives that cross STM segment boundaries.
Each table reports the parameter $N$ that was requested, which may be greater than the actual $N_{\max}$ returned.
The $N_{.9}$ and $N_{.5}$ columns indicate the depths of alternatives at the top decile and median results; these convey the distribution more clearly than the mean statistic.
The rightmost columns report the storage size of the \verb|gzip|-compressed CTM files in megabytes.
The last row of Table \ref{tab:alternatives-phrase-2} relates a hypothetical oracle selecting the best transcript from a phrase-level representation of alternatives, derived from very dense lattices, decoded with added knowledge of all OOV words, using a reference segmentation.
\begin{table}[t]
\caption{
Oracle WER for phrase-level alternatives: adding all OOV words (ASR1$^*$);
denser lattices (ASR$^\dagger$); and both (ASR$^{*\dagger}$).
}
\label{tab:alternatives-phrase-2}
\centering
\begin{tabular}{ l c r r r r r }
\toprule
& WER & $N$ & $N_{\max}$ & $N_{.9}$ & $N_{.5}$ & MB \\
\midrule
\midrule
\textit{ASR1}$^*$ & \textit{5.79} & 1 & 1 & 1 & 1 & 0.1 \\
\textit{ASR1}$^*$ & \textit{0.49} & 100 & 100 & 22 & 3 & 1.0 \\
\textit{ASR1}$^*$ & \textit{0.42} & $\infty$ & 5250 & 22 & 3 & 1.4 \\
\midrule
\textit{ASR1}$^\dagger$ & \textit{0.36} & 1000 & 1000 & 125 & 14 & 5.4 \\
\textit{ASR1}$^\dagger$ & \textit{0.33} & 10000 & 10000 & 125 & 14 & 7.6 \\
\midrule
\textit{ASR1}$^{*\dagger}$ & \textit{0.21} & 1000 & 1000 & 124 & 14 & 5.4 \\
\textit{ASR1}$^{*\dagger}$ & \textbf{\textit{0.18}} & 10000 & 10000 & 124 & 14 & 7.4 \\
\bottomrule
\end{tabular}
\end{table}
\section{Conclusion}
This work highlighted subtle issues with evaluating the famous Switchboard benchmark.
It presented a reproducible Kaldi ASR baseline, comparing major cloud platforms to
human transcription services, and clarified that IBM's research system achieves a \textbf{super-human record of 2.3\% instead of 4.3\% WER}.
Some experiments are unrealistic to varying degrees, ranging from the assumption of an oracle to the accepted use of a reference segmentation.
Nonetheless, such results demonstrate the potential for \textbf{lattice-based ASR approaching 0.18\% WER}.
These results motivate future work to improve lattice generation \cite{rybach2017lattice,lv2021let},
particularly in E2E ASR systems.
Our current research also explores open-vocabulary decoding in a WFST framework,
in which novel words may be included in a lattice and derived phrase alternatives.
These advances enable new applications, e.g. audio search or machine-assisted transcription,
that can be designed to mitigate inevitable errors in 1-best ASR.
\section{Acknowledgments}
Thanks to our many friends from ICSI: \\
$\bigstar$ Michael Ellsworth, who carefully audited the references.\\
$\bigstar$ Andreas Stolcke, who clarified many evaluation practices.\\
$\bigstar$ Brian Kingsbury, who shared results from IBM Research.\\
$\bigstar$ Deanna Gelbart, who wrote code for phrase alternatives.
\vfill
\bibliographystyle{IEEEtran}
|
1507.04297
|
\section{Introduction and the main result}
In this paper, we prove that given a non-contractible hyperbolic periodic orbit of a symplectomorphism isotopic to the identity there are infinitely many periodic points provided the symplectomorphism satisfies some constraints on its flux. The result holds for certain symplectic manifolds such as products of some surfaces with complex projective spaces (see Section~\ref{section:mainthm} for more details).
The main theorem of this paper fits into the conjecture originally (as far as we know) stated by Hofer and Zehnder claiming (see \cite[p.263]{HZ11})
"(...) that every Hamiltonian map on a compact symplectic manifold $(M, \omega)$ possessing more fixed points than necessary required by the V. Arnold conjecture possesses always infinitely many periodic points."
For instance, the expected bound for $\CP^n$ is $n+1$. This conjecture was motivated by the results of Gambaudo and Le Calvez in \cite{Gam_LeCa} and Franks in \cite{Fr88} (see also \cite{Fr92,Fr96}) where they prove that an area preserving diffeomorphism of $S^2$ with more than two fixed points has infinitely many periodic points; see also \cite{BH11,CKRTZ12,Ke12} for symplectic topological proofs.
More accurately, our result fits in a variant of this conjecture described by G{\"u}rel in \cite{Gu_noncon,Gu_linear}; see also \cite{GG:hyp12}. This version suggests the presence of infinitely many periodic points provided the existence of a fixed point which is \emph{unnecessary} from a homological or geometrical perspective.
In fact, Theorem~\ref{maintheo} asserts that, for a certain class of symplectic manifolds, a symplectomorphism isotopic to the identity with a (non-contractible) hyperbolic periodic orbit possesses infinitely many periodic points, as long as it satisfies some condition on its flux. The theorem is a symplectic analogue of a result proved in \cite{GG:hyp12} for Hamiltonian diffeomorphisms and also generalizes the theorem in \cite{Ba_hyp} for contractible periodic orbits of symplectomorphisms. Moreover, in dimension greater than two, the conjecture for Hamiltonian diffeomorphisms is proved in \cite{Gu_noncon} and \cite{Gu_linear} where the fixed points are assumed to be \emph{unnecessary} from a homological perspective.
Recall that, in general, a symplectomorphism need not have periodic points, as is the case of $\psi\colon \T^{2n}\rightarrow \T^{2n}$ defined by
\[
\psi(x,y)=(x+\theta,y)\quad\quad\text{with}\quad \theta\not=1,
\]
whereas, for many symplectic manifolds, a Hamiltonian diffeomorphism has an infinite number of periodic points. In fact, Conley conjectured (\cite{Co}) the existence of infinitely many periodic orbits of a Hamiltonian diffeomorphism on the $2$-torus and the statement has now also been proved, e.g. for negative monotone manifolds or symplectic manifolds with ${c_1}|_{\pi_2(M)}=0$; see \cite{CGG11,GG:action09,He12} and also \cite{FrHa03,Gi:conley,GG:conley12,Hi,LeC,SZ92}. Observe that in the conjecture mentioned above described by G\"urel there is an assumption, for instance, on the existence of a periodic orbit of a certain geometrical type. This is in contrast with the Conley conjecture which is unconditional. Hence, owing to the above example we expect some condition in order for a symplectomorphism, which is not necessarily a Hamiltonian diffeomorphism, to have infinitely many periodic points.
\subsection{Existence of infinitely many periodic orbits}\label{section:mainthm}
Consider a closed monotone symplectic manifold $(M^{2n}, \omega)$ with minimal Chern number $c_1^{\min}$ and toroidal minimal Chern number $c_{1,T}^{\min}$ (for the exact definitions see \eqref{eqn:minimalChernnumb} and \eqref{eqn:toroidalc1min}, respectively). We denote by $q$ the generator of the Novikov ring $\Lambda=\mathbb{Z}_2[q^{-1},q]$ normalized with degree $-2c_1^{\min}$ (see Section~\ref{section:loopspaces}).
Given a homotopy class (with fixed end points) of a symplectic isotopy, $[\phi_t]$, connecting the identity to some symplectomorphism $\phi$, we denote by $\varrho$ the cohomology class on $\mathcal{L}M$ (i.e. the space of loops in $M$)
\[
\varrho\colon=-\overline{[\omega]} + \text{ev}^*(\text{Flux}[\phi_t])\in H^1(\mathcal{L}M;\reals),
\]
where $\overline{[\omega]}$ is the cohomology class in $H^1(\mathcal{L}M;\reals)$ associated with the symplectic form $\omega$ (see Section~\ref{section:loopspaces}), $\text{ev}\colon \mathcal{L}M\rightarrow M$ is the evaluation map $x\mapsto x(0)$ and $\text{Flux}([\phi_t])$ is the image of the class $[\phi_t]$ by the homomorphism (see Section~\ref{section:po_af})
\[\text{Flux}\colon \widetilde{\text{Symp}_0}(M,\omega) \rightarrow H^1(M,\reals); \quad
[\phi_t] \mapsto \left[ \int_0^1 \omega (X_t, \cdot) dt\right].
\]
The following theorem on periodic orbits of symplectomorphisms is the main result of this paper.
\begin{theorem}\label{maintheo}
\
Let $(M^{2n}, \omega)$ be a closed, connected, strictly spherically monotone symplectic manifold. Assume that \[c^{\min}_{1,T}\geq \frac{n}{2}+1,\]
\begin{eqnarray}\label{l:1}\beta*\alpha =q[M] \quad \quad \text{in}\; HQ_*(M)=H_*(M)\otimes \Lambda\end{eqnarray}
for some ordinary homology classes $\alpha, \beta\in H_*(M)$ with $\deg(\alpha), \deg(\beta)<2n$
and
\begin{eqnarray}\label{item:qs} \text{either} \quad c^{\min}_{1} =c^{\min}_{1,T} \quad \text{or} \quad \deg(\alpha)\geq 3n+1-2c^{\min}_{1,T}.
\end{eqnarray}
Then any symplectomorphism $\phi$ in $\text{Symp}_0(M,\omega)$ with
\begin{enumerate}
\item a hyperbolic (non-contractible) periodic orbit $\gamma$, with free homotopy class $\zeta$,
\item $\varrho$ and $\overline{c_1}$ strictly toroidally proportional and strictly $\zeta$-toroidally proportional
\end{enumerate}
has infinitely many periodic points.
The corresponding periodic orbits lie in the free homotopy classes formed by iterations of the hyperbolic periodic orbit $\gamma$.
\end{theorem}
The conditions in the hypotheses of this theorem involve both the manifold and the symplectomorphism.
Among the manifolds meeting the requirements of Theorem~\ref{maintheo} are products of surfaces with genus greater than or equal to two, $\Sigma_{g\geq 2}$, and complex projective spaces (see Examples~\ref{example:firstchern}~and~\ref{example:quantumprod} and Remark~\ref{rmk:exampleManifolds}). These manifolds satisfy the first requirement in \eqref{item:qs}. More generally, instead of $\Sigma_{g\geq 2}$, we may consider closed K\"ahler manifolds with negative sectional curvature (e.g., complex hyperbolic
spaces); see, e.g., \cite{Gu_noncon} for a discussion of these manifolds.
The hyperbolicity assumption (i) on the orbit $\gamma$ is crucial (see Remark~\ref{rmk:hyp}). Ginzburg and G{\"u}rel (\cite[Section~3]{GG:hyp12}) proved that the energy required for a (\emph{Floer--Novikov}) trajectory to approach a $k$-th iteration of $\gamma$ and crossing its fixed neighborhood is bounded from below by a strictly positive constant independent of $k$. The requirement that $\gamma$ is hyperbolic is so that the orbit has the feature just mentioned and which is described in Section~\ref{section:BCEthm}. Hence, more \emph{generally}, there are infinitely many periodic points if in the hypotheses of Theorem~\ref{maintheo} the hyperbolicity condition is replaced by the property described in Theorem~\ref{thm:ballcrossing}.
Condition (ii) involves both the manifold and the symplectomorphism $\phi$. It gives constraints on the flux of an isotopy $\{\phi_t\}$ (connecting $id$ and $\phi$). Here, there is a relation between the cohomology classes $\varrho$ and $\overline{c_1}$ on the loop space $\mathcal{L}M$. As explained in the beginning of this section, the class $\varrho$ depends on the cohomology class of the symplectic form $\omega$ and on the flux of the considered isotopy $\{\phi_t\}$ and the class $\overline{c_1}$ depends on the first Chen class of the manifold. There are maps meeting these requirements. Namely, consider on $M=\Sigma_{g\geq 2}\times \CP^n$ a symplectic isotopy $\phi_t=\psi_t\times id \colon M\rightarrow M$ where $\psi_t$ is a symplectic isotopy of $\Sigma_{g\geq 2}$ and $id\colon \CP^n \rightarrow \CP^n$ is the identity map on $\CP^n$. Let $\gamma$ be a (non-contractible) periodic orbit of $\phi_t$ in $M$ with free homotopy class $\zeta$. Assume also that $\text{Flux}([\phi_t])\in H^1(M;\reals)$ and $\overline{[\omega]}\in H^1(\mathcal{L}M;\reals)$ are positively proportional on $\pi_1(\mathcal{L}M)$ and on $\pi_1(\mathcal{L}_{\zeta}M)$, i.e.
\[
\text{ev}^*\text{Flux}([\phi_t])|_{\pi_1(\mathcal{L}M)}=\lambda_1 \overline{[\omega]}|_{\pi_1(\mathcal{L}M)} \quad \text{and} \quad
\text{ev}^*\text{Flux}([\phi_t])|_{\pi_1(\mathcal{L}_{\zeta}M)}=\lambda_2 \overline{[\omega]}|_{\pi_1(\mathcal{L}_{\zeta}M)}
\]
for some positive constants $\lambda_1,\lambda_2>0$. Since $M$ is strictly spherically and toroidally monotone, such an isotopy $\phi_t$ satisfies condition (ii). (Cf. \cite[Proposition~1.2 and Example~1]{Ba_hyp} for the case where $\gamma$ is contractible.)
Moreover, if the periodic orbit $\gamma$ is contractible, then, when the manifold $M$ is toroidally monotone and $c^{\min}_{1} =c^{\min}_{1,T}$, Theorem~\ref{maintheo} generalizes the main results in \cite{Ba_hyp} and \cite{GG:hyp12}.
\subsection{Acknowledgments}
The author is grateful to Viktor Ginzburg for valuable discussions. Part of this work was carried out at ICMAT. The author would like to thank the institute for their warm hospitality and support.
\section{Preliminaries}\label{section:prelim}
In this section, we introduce the notation used throughout the paper and review some facts needed to prove the results.
\subsection{Symplectic manifolds}
\label{section:sympmnfd}
Let $(M^{2n},\omega)$ be a \emph{closed} (i.e. compact and with no boundary) symplectic manifold and consider an almost complex
structure $J$ on $TM$ compatible with $\omega,$ i.e. such that $g(X, Y) := \omega(X, JY)$ is a Riemannian metric on $M.$
Since the space of almost complex structures compatible with $\omega$ is connected, the first Chern class $c_1\in H^2(M;\Z)$ is uniquely determined by $\omega$.
The \emph{minimal Chern number} of a symplectic manifold $(M,
\omega)$ is the positive integer $c_1^{\min}$ which generates the discrete group formed by the integrals
of $c_1$ over the spheres in $M$, i.e.,
\begin{eqnarray}\label{eqn:minimalChernnumb}
\left<c_1, \pi_2(M)\right> = c_1^{\min} \Z
\end{eqnarray}
where $c_1^{\min}\in\N$ with the convention $c_{1}^{\text{min}}=\infty$ if the image of $c_1\colon \pi_2(M)\xrightarrow{} \Z$ is $0$.
The symplectic manifold $(M,\omega)$ is called \emph{spherically monotone} if the integrals of the cohomology classes $c_1$ and
$[\omega]$ over spheres satisfy the proportionality condition
\[
[\omega]|_{\pi_2(M)}=\lambda_{S} {c_1}|_{\pi_2(M)},
\]
for some non-negative constant $\lambda_{S}\in\reals.$ If $[\omega]|_{\pi_2(M)}$ and ${c_1}|_{\pi_2(M)}$ are non-zero, then we say that $(M,\omega)$ is \emph{strictly spherically monotone}. Observe that, in this case, $c_1^{\min}<\infty$.
\subsection{Loop spaces}\label{section:loopspaces} In this section, we follow \cite{BH01_non-con}, \cite{O_flux06} and references there in.
Let $\mathcal{L}M:=\mathcal{C}^{\infty}(S^1,M)$ be the space of smooth free loops in $M$ where $S^1=\reals / \Z$. The first Chern class $c_1\in H^2(M;\Z)$ of $(M,\omega)$ defines a cohomology class
\[
\overline{c_1}\in H^1(\mathcal{L}M;\Z)=\text{Hom}(H_1(\mathcal{L}M;\Z),\Z),
\]
by interpreting a class in $H_1(\mathcal{L}M;\Z)$ as a linear combination of tori, i.e. maps $S^1\times S^1 \rightarrow M$, and $\overline{c_1}$ as integrating $c_1$ over it. Similarly, $[\omega]\in H^2(M;\reals)$ gives rise to a cohomology class
\[
\overline{[\omega]}\in H^1(\mathcal{L}M;\reals)=\text{Hom}(H_1(\mathcal{L}M;\Z),\reals).
\]
The manifold $(M,\omega)$ is called \emph{atoroidal} if the cohomology classes $\overline{c_1}$ and $\overline{[\omega]}$ satisfy the conditions
\[
\overline{[\omega]}|_{\pi_1(\mathcal{L}M)}=0= \overline{c_1}|_{\pi_1(\mathcal{L}M)}.
\]
i.e.
\[
\displaystyle\int_{S^1\times S^1} v^*\omega=0=\displaystyle\int_{S^1\times S^1} v^*\eta \quad\quad \text{for all }v:S^1\times S^1 \rightarrow M,
\]
where $\eta$ is a $2$-form in $M$ that represents the first Chern class $c_1\in H^2(M;\Z)$.
Moreover, $(M,\omega)$ is called \emph{toroidally monotone} if
\[
\overline{[\omega]}|_{\pi_1(\mathcal{L}M)}=\lambda_T \overline{c_1}|_{\pi_1(\mathcal{L}M)}
\]
for some non-negative constant $\lambda_{T}\in\reals$ and, if $\overline{[\omega]}|_{\pi_1(\mathcal{L}M)}$ and $\overline{c_1}|_{\pi_1(\mathcal{L}M)}$ are non-zero, the manifold is called \emph{strictly toroidally monotone}.\\
Given a cohomology class $[\theta]\in H^1(M;\reals)$, we consider the following cohomology class on the loop space $\mathcal{L}M$
\begin{eqnarray*}
\varrho:= -\overline{[\omega]} + \text{ev}^*[\theta]\in H^1(\mathcal{L}M;\reals)
\end{eqnarray*}
where $\text{ev}\colon \mathcal{L}M \rightarrow M $ denotes the evaluation map $x \mapsto x(0)$.
We say that $\varrho$ and $\overline{c_1}$ are \emph{toroidally proportional} if
\[
\varrho|_{\pi_1(\mathcal{L}M)}=\lambda_{\varrho}\overline{c_1}|_{\pi_1(\mathcal{L}M)}
\]
i.e.
\begin{eqnarray*}\label{eqn:toroprop}
-\displaystyle\int_{S^1\times S^1} v^*\omega + \displaystyle\int_{S^1} v_0^*\theta=\lambda_{\varrho}\displaystyle\int_{S^1\times S^1} v^*\eta \quad\quad \text{for all }v:S^1\times S^1 \rightarrow M,
\end{eqnarray*}
for some non-negative $\lambda_{\varrho}\in \reals$, where $v_0\colon S^1 \rightarrow M$ is given by $v_0=v(\cdot,0)$ and $\eta$ is a $2$-form that represents the class $c_1$.
If $\varrho|_{\pi_1(\mathcal{L}M)}$ and $\overline{c_1}|_{\pi_1(\mathcal{L}M)}$ are non-zero, we say $\varrho$ and $\overline{c_1}$ are \emph{strictly toroidally proportional}.
Moreover, we say that $\varrho$ is \emph{rational} if the group of integrals of $\varrho$ over tori is discrete, i.e.,
\begin{eqnarray}\label{eqn:varphi_rational}
\left< \varrho, \pi_1(\mathcal{L}M)\right>=h_{\varrho}\Z
\end{eqnarray}
for some non-negative $h_{\varrho}\in \reals$. Notice that if $\varrho$ and $\overline{c_1}$ are strictly toroidally monotone then $\varrho$ is rational and if $[\theta]=0$ then $\varrho$ and $\overline{c_1}$ are toroidally proportional if and only if $(M,\omega)$ is toroidally monotone.
Define the \emph{toroidal minimal Chern number} $c^{\min}_{1,T}\in\N$ by
\begin{eqnarray}\label{eqn:toroidalc1min}c^{\min}_{1,T}\Z=\left< \overline{c_1}, \pi_1(\mathcal{L}M)\right>,\end{eqnarray}
with convention $c^{\min}_{1,T}=\infty$ if the image of $\overline{c_1}\colon \pi_1(\mathcal{L}M) \rightarrow \Z$ is 0,
and hence, when $\varrho$ and $\overline{c_1}$ are toroidally proportional, we have
\begin{eqnarray}\label{eqn:c1minTprop}
c^{\min}_{1,T}=\frac{h_{\varrho}}{\lambda_{\varrho}}.
\end{eqnarray} Moreover define
\begin{eqnarray}\label{eqn:nut}
\nu_{T}:= \frac{c^{\min}_1}{c^{\min}_{1,T}} > 0.
\end{eqnarray}
Notice that $c^{\min}_{1,T}\not= 0$ and, when $M$ is strictly spherically monotone, $c^{\min}_1 <\infty$.
\begin{example}\label{example:firstchern}
Consider $M=\CP^n\times \Sigma_{g\geq 2}$, where $\Sigma_{g\geq 2}$ is a surface with genus greater than or equal to 2. The surface $\Sigma_{g\geq 2}$ is aspherical and atoroidal and $c_{1,T}^{\min}(\CP^n)=c_{1}^{\min}(\CP^n)=n+1$. Hence, the toroidal minimal Chern number of
$M$ is $c_{1,T}^{\min}(M)=c_{1}^{\min}(M)=n+1$ (see, e.g., \cite{MS12} for more details). In Theorem~\ref{maintheo}, the assumption on the minimal Chern number
$$
c_{1,T}^{\min}(M)\geq \dim(M)/4 +1,
$$
in this case, is
$$
n+1\geq \frac{n+1}{2} +1,
$$
which is equivalent to $n\geq 1$.
\end{example}
Let $\zeta$ be a free homotopy class of maps $S^1\rightarrow M$, (i.e. let $\zeta\in \pi_0(\mathcal{L}M)$), fix a reference loop $z\in \zeta$ and a symplectic trivialization of $TM$ along $z$. Denote by $\mathcal{L}_{\zeta}M$ the component of $\mathcal{L}M$ with loops in the free homotopy class $\zeta$ (i.e. $\mathcal{L}_{\zeta}M=p_{\pi_0}^{-1}(\zeta)$ is the preimage of $\zeta$ under the natural projection $p_{\pi_0}\colon \mathcal{L}M \rightarrow \pi_0(\mathcal{L}M)$).
Consider the connected abelian principal covering $p \colon \widetilde{\mathcal{L}}_{\zeta}M \rightarrow \mathcal{L}_{\zeta}M$ with structure group
\begin{eqnarray}\label{eqn:Gamma_z}
\Gamma_{\zeta}:= \frac{\pi_1(\mathcal{L}_{\zeta}M)}{\ker \overline{c_1} \cap \ker \varrho}
\end{eqnarray}
where $\overline{c_1}$ and $\varrho$ are considered as homomorphisms $\pi_1(\mathcal{L}_{\zeta}M)\rightarrow \reals$ and we have $p^*\varrho=0=p^*\overline{c_1}\in H^1(\widetilde{\mathcal{L}}_{\zeta}M;\reals)$. These homomorphisms induce homomorphisms on $\Gamma_{\zeta}$:
\begin{eqnarray*}
\overline{c_1}\colon \Gamma_{\zeta} \rightarrow \Z \quad \text{and} \quad \varrho\colon \Gamma_{\zeta} \rightarrow \reals.
\end{eqnarray*}
Define the \emph{$\zeta$-minimal Chern number} $c_{1,\zeta}^{\text{min}}\in \N$ by
\begin{eqnarray}\label{eqn:c1minzeta}
c_{1,\zeta}^{\min}\Z= \left<\overline{c_1},H_1(\mathcal{L}_{\zeta}M;\Z)\right> \subseteq \Z
\end{eqnarray}
with the convention $c_{1,\zeta}^{\text{min}}=\infty$ if the image of $\overline{c_1}\colon H_1(\mathcal{L}_{\zeta}M;\Z)\xrightarrow{} \Z$ is $0$. The number $c_{1,\zeta}^{\text{min}}$ depends only on $(M,\omega, \zeta)$.
\begin{remark}
If $\zeta$ is trivial, the minimal Chern number $c_{1,\zeta=0}^{\min}$ is the usual minimal Chern number $c_{1}^{\min}$ as defined in \eqref{eqn:minimalChernnumb} but in general it is smaller. Indeed the diagram
\begin{equation}\label{diagram:c1}
\xymatrix{
{H_1(\mathcal{L}_{\zeta}M;\Z)} \ar[r]^{\quad\quad\overline{c_1}}& {\Z} \\
{\pi_1(\mathcal{L}_{\zeta}M)} \ar[u]& {\pi_2(M)} \ar[l] \ar[u]^{c_1} .
}
\end{equation}
commutes, where, if we denote by $\Omega(M,z(0))$ the space of based pointed loops with base point $z(0)$, the bottom mapping $\pi_2(M)=\pi_1(\Omega(M,z(0))\rightarrow \pi_1(\mathcal{L}_{\zeta}M)$ is induced from the mapping $\Omega(M,z(0))\rightarrow \mathcal{L}M$ given by concatenating a loop based at $z(0)$ with z. Then $c_{1,\zeta}^{\min}<\infty$ if and only if $c_{1}^{\min}<\infty$ and, in this case, $c_{1,\zeta}^{\min}$ divides $c_{1}^{\min}$ (see also Remark~\ref{rmk:nu}).
\end{remark}
We interpret a class in $H^1(\mathcal{L}_{\zeta}M;\Z)$ as a linear combination of tori, i.e., maps $v\colon S^1\times S^1 \rightarrow M$ such that the homotopy class of $v(s,\cdot)\colon S^1 \rightarrow M$ is $\zeta,$ for all $s\in S^1$. The cohomology classes $\varrho$ and $\overline{c_1}$ are said strictly $\zeta$-toroidally proportional if
\[\varrho|_{\pi_1(\mathcal{L}_{\zeta}M)}=\lambda_{\zeta}\overline{c_1}|_{\pi_1(\mathcal{L}_{\zeta}M)},\]
for some $\lambda_{\zeta}>0$, and $\varrho|_{\pi_1(\mathcal{L}_{\zeta}M)}\not=0\not= \overline{c_1}|_{\pi_1(\mathcal{L}_{\zeta}M)}$ (see also~\cite[Section~8]{PolShe15}).
In fact, if $\varrho$ and $\overline{c_1}$ are also toroidally proportional, we have $\lambda_{\zeta}=\lambda_{\varrho}$ and moreover
\[
\left<\varrho, \pi_1(\mathcal{L}_{\zeta}M) \right>=h_{\zeta}\Z
\]
for some non-negative $h_{\zeta}\in \reals$. Observe that, in this case, $c^{\min}_{1,T}\not=0$ divides $c^{\min}_{1,\zeta}<\infty$ and, hence, we have
\begin{eqnarray}\label{eqn:h_zeta}
h_{\zeta}=\frac{c_{1,\zeta}^{\min}}{c_{1,T}^{\min}}h_{\varrho},
\end{eqnarray}
with $h_{\zeta}={c_{1,\zeta}^{\min}}/{c_{1,T}^{\min}}h_{\varrho}\in \N$.
\begin{remark}\label{rmk:nut=1}
If $\varrho$ and $\overline{c_1}$ are strictly toroidally proportional and $\zeta$-toroidally proportional, then, when $\nu_T=1$ (recall \eqref{eqn:nut}), $c_{1}^{\min}=c_{1,T}^{\min}=c_{1,\zeta}^{\min}$.
\end{remark}
\subsubsection{The covering space $\widetilde{\mathcal{L}}_{\zeta}M$.}\label{section:decrp_cov_sp}
Consider the space of pairs $(x,v)$ where $x\colon S^1 \rightarrow M$ is a loop in $M$ with homotopy class $\zeta$ and $v$ a cylinder (i.e. $v\colon [0,1]\times S^1 \rightarrow M$ is a homotopy) connecting $x$ to $z$. The cylinder $v$ is called a capping of $x$.
\begin{remark}
Observe that if $\zeta$ is the trivial homotopy class and $z$ is a constant loop then $v$ can be viewed as a disc in $M$.
\end{remark}
We say that $(x,v)$ is in relation with $(x',v')$ if the following conditions hold:
\[x=x{'},\quad\quad
\displaystyle\int_{S^1\times S^1} u^*\eta =0
\]
and
\[
\int_{S^1\times S^1} u^*\omega = \int_{S^1} {u_0}^*\theta
\]
where $u$ is the two-torus obtained by attaching $v$ and $-v'$ (that denotes $v'$ with reversed orientation) to each other, $\eta$ is a $2$-form representing the first Chern class $c_1$ and $u_0=u|_{S^1 \times \{0\}}$ (see Figure~\ref{figure:u0}). The equivalence class $\overline{x}=[x,v]$ is called a lift of the loop $x\in\mathcal{L}_{\zeta}M$ and the space of such equivalence classes is denoted by $\widetilde{\mathcal{L}}_{\zeta}M$.
\begin{figure}[htb!]
\centering
\def\svgwidth{150pt}
\input{u0.pdf_tex}
\caption{The circle $u_0$.}\label{figure:u0}
\end{figure}
\subsection{Periodic orbits and the action functional}\label{section:po_af} In this section, we follow \cite{BH01_non-con}, \cite{LO_fixedpts95} and references there in.
Let $\phi\in \text{Symp}_0(M)$, $\phi_t$ be a symplectic isotopy connecting $\phi_0=id$ to $\phi_1=\phi$ with $\text{Flux}([\phi_t])=:[\theta]$ and $X_t$ be the vector field associated with $\phi_t$, i.e.
\[
\frac{d}{dt}\phi_t = X_t \circ \phi_t.
\]
Recall that $\text{Flux}\colon \widetilde{\text{Symp}_0}(M,\omega) \rightarrow H^1(M,\reals)$ is the homomorphism defined by
\begin{eqnarray}\label{eqn:flux}
[\phi_t] \mapsto \left[ \int_0^1 \omega (X_t, \cdot) dt\right]
\end{eqnarray}
and its kernel is given by $\widetilde{\text{Ham}}(M,\omega)$, i.e. the universal covering of the group of Hamiltonian diffeomorphisms.
L\^e and Ono proved in \cite[Lemma~2.1]{LO_fixedpts95} that we can deform $\{\phi_t\}$ through symplectic isotopies (keeping the end points fixed) so that the cohomology classes $[\omega(X^{\prime}_t,\cdot)]$, for all $t\in[0,1]$, and $\text{Flux}([\phi'_t])=[\theta]$ are the same (where $X^{\prime}_t$ is the vector field associated with the deformed symplectic isotopies $\phi'_t$). Namely, each element in $\widetilde{\text{Symp}_0}(M,\omega)$ admits a representative symplectic isotopy generated by a smooth path of closed $1$-forms $\theta_t$ on $M$ whose cohomology class is identically equal to the flux, i.e.
\begin{eqnarray}\label{eqn:Ham}
-\omega(X^{\prime}_t,\cdot)= \theta + dH_t=:\theta_t
\end{eqnarray}
for some one-periodic in time Hamiltonian $H_t\colon M \rightarrow M, \;t\in S^1$.
The fixed points of $\phi=\phi_1$ are in one-to-one correspondence with one-periodic solutions of the differential equation
\begin{eqnarray}\label{eqn:de_theta}
\dot{x}(t)=X_{\theta_t}(t,x(t))
\end{eqnarray}
where $X_{\theta_t}$ is defined by $\omega(X_{\theta_t},\cdot)=-\theta_t$. From now on we denote the vector field $X_{\theta_t}$ by $Z_t$.
The set of one-periodic solutions of \eqref{eqn:de_theta} is denoted by $\mathcal{P}(\theta_t)$ and coincides with the zero set of the closed $1$-form $\alpha:=\alpha_{\{\phi_t\}}$ defined on the loop space of $M$, $\mathcal{L}M$, by
\begin{eqnarray*}\label{eqn:mv-af}
\alpha_{\{\phi_t\}}(x,\xi)&=& \displaystyle\int_0^1 \omega(\dot{x}(t)-Z_t(x(t)), \xi) dt\\
&=&\displaystyle\int_0^1 \omega(\dot{x}(t), \xi) + \theta_t(x(t))(\xi) dt\\
&=& \displaystyle\int_0^1 \omega(\dot{x}(t), \xi) dt + \displaystyle\int_0^1 (\theta+ dH_t)(x(t))(\xi) dt
\end{eqnarray*}
where $x\in \mathcal{L}M$ and $\xi\in T_x\mathcal{L}M$ (i.e. $\xi$ is a tangent vector field along the loop $x$).
From \eqref{eqn:Gamma_z} we then see that $p^*\alpha=d\CA$ for some $\CA\in C^{\infty}(\widetilde{\mathcal{L}}_{\zeta}M,\reals)$.
Then an action functional $\CA=\mathcal{A}_{\{\phi_t\}}$ is given by
\begin{eqnarray}\label{eqn:action_flnov}
\mathcal{A}_{\{\phi_t\}}([x,v])=-\int_{[0,1]\times S^1} v^*\omega +\displaystyle\int_0^1 \left[\left(\int_0^1 (v_t)^*\theta\right)+ H_t(x(t)) \right]dt
\end{eqnarray}
where $[x,v]\in \widetilde{\mathcal{L}}_{\zeta}M$ is as in Section~\ref{section:decrp_cov_sp} and $v_t\colon [0,1]\rightarrow M$ is defined by $s \mapsto v(s,t)$ (see Figure~\ref{figure:vt}).
\begin{figure}[htb!]
\centering
\def\svgwidth{100pt}
\input{x_prime.pdf_tex}
\caption{The segment $v_t$.}\label{figure:vt}
\end{figure}
The action functional $\mathcal{A}_{\{\phi_t\}}$ is homogeneous with respect to iterations in the following sense
\[
\mathcal{A}_{\{\phi_t^{ k}\}}([x,v]^{k})=k\mathcal{A}_{\{\phi_t\}}([x,v]),
\]
where $[x,v]^{k}$ is the $k$-th iteration of $[x,v]$, and satisfies
\[ \CA_{\{\phi_t\}}([x,v]\#A)=\CA_{\{\phi_t\}}([x,v]) + \varrho (A),
\]
where $A\in \pi_1(\mathcal{L}_{\zeta}M, z)$.
\begin{remark}\label{rmk:action}
\begin{enumerate}
\item\label{rmk:action1} The action of a (lifted) periodic orbit depends on the initial choice of $z$. If $z'\in \zeta$, there is a homotopy $F\colon [0,1]\times S^1 \rightarrow M$ connecting $z$ to $z'$, i.e., $F(0, \cdot)=z$ and $F(1, \cdot)=z'$. Then $\CA_{\{\phi_t\}}([x,v]\#F)=\CA_{\{\phi_t\}}([x,v])+\CA(F)$ where $[x,v]\#F$ denotes the class $[x,w]$ with $w$ the cylinder obtained by concatenating $v$ with $F$ and
\[\CA(F)=-\displaystyle\int_{[0,1]\times S^1} F^*\omega + \displaystyle\int_{0}^1 (\displaystyle\int_{0}^1 F_t^*\theta) dt\] with $F_t\colon [0,1] \rightarrow M$, $F_t(s):=F(t,s), t\in S^1.$
\item Whenever we consider the $k$-th iteration $\phi_t^k$ of $\phi_t$ we simultaneously iterate the class $\zeta$ and the reference curve $z$ (i.e. we pass to $k\zeta$ and $z^k$, respectively).
\end{enumerate}
\end{remark}
\subsection{The mean index and augmented action} Let $x$ be a periodic orbit of $Z_t$. If the eigenvalues of the map
\[
d\phi_{x(0)}:T_{x(0)}M\rightarrow T_{x(0)}M
\]
are not equal to one, then the orbit $x$ is called \emph{non-degenerate}. If, in addition, none of the eigenvalues of the linearized return map $d\phi_{x(0)}$ has absolute value equal to one, then we say $x$ is \emph{hyperbolic}. The symplectomorphism $\phi$ is said to be non-degenerate if all its one-periodic orbits are non-degenerate. Moreover, if all periodic orbits of $Z_t$ with homotopy class $\zeta$ are non-degenerate, then the set of periodic orbits of $Z_t$ in $\mathcal{L}_{\zeta}M$, denoted by $\mathcal{P}_{\zeta}$, is finite.
The homotopy class $\zeta$ gives rise to a well defined, up to homotopy, $\C$-vector bundle trivialization of $x^*TM$ for every $x\in \mathcal{L}_{\zeta}M$ and, for a one-periodic orbit of $\phi$, the linearized flow along $x$
\[
d\phi_t\colon T_{x(0)}M \rightarrow T_{x(t)}M
\]
can be viewed as a symplectic path $\Phi\colon[0,1] \rightarrow \Sp(2n).$ Set $\overline{\mathcal{P}_{\zeta}}:= p^{-1}(\mathcal{P}_{\zeta})$. Then one gets a well defined mean index
$\D_{\phi_t}\colon \overline{\mathcal{P}_{\zeta}} \rightarrow \reals$ and, for non-degenerate orbits, the Conley--Zehnder index $\MUCZ([x,v])\in \Z$ satisfying
\[
\D_{{\{\phi_t^{k}\}}}([x,v]^{k})=k\D_{\{\phi_t\}}([\widetilde{x},v]),
\]
\[
\D_{\{\phi_t\}}([x,v]\#A)=\D_{\{\phi_t\}}([x,v])- 2\overline{c_1}(A)
\]
and
\[
\MUCZ([x,v]\#A)=\MUCZ([x,v])-2\overline{c_1}(A) \quad\quad\text{(when } x\text{ is non-degenerate),}
\]
for every $[x,v]\in \overline{\mathcal{P}_{\zeta}}$ and $A\in \Gamma_{\zeta}$ (cf. \cite{BH01_non-con} and \cite{SZ92}). Moreover
\begin{eqnarray}\label{eqn:mi_czi}
0\not =|\D_{\{\phi_t\}}([x,v])-\MUCZ([x,v])|< n,
\end{eqnarray}
when $x$ is non-degenerate.
The mean index $\D_{\{\phi_t\}} ([x,v])\in \reals$ measures, roughly speaking, the total angle swept by certain eigenvalues with absolute value one of the symplectic path $\Phi$ induced by the linearized flow and a trivialization as above.
\begin{remark}\label{rmk:index}
The Conley--Zehnder index depends on the initial choice of the trivialization of $TM$ along $z$; see \cite[Section~3]{BH01_non-con}.
\end{remark}
The \emph{augmented action} is defined by
\begin{eqnarray}\label{eqn:augmentedaction}
\underline{\mathcal{A}}_{\{\phi_t\}}([x,v]):= \mathcal{A}_{\{\phi_t\}}([x,v]) - \frac{\lambda_{\varrho}}{2} \D_{\{\phi_t\}}([x,v])
\end{eqnarray}
and it is homogeneous with respect to iterations in the following sense
\[
\underline{\mathcal{A}}_{\{\phi_t^{k}\}}([x,v]^{k})=k\underline{\mathcal{A}}_{\{\phi_t\}}([x,v]).
\]
\begin{remark}\label{rmk:augmented_independent}
Note that the augmented action is independent of the lift of a loop, i.e.
\[
\underline{\mathcal{A}}_{\{\phi_t\}}([x,v])=\underline{\mathcal{A}}_{\{\phi_t\}}([x,v{'}])
\]
where $v$ and $v{'}$ are cylinders connecting $x$ to $z$.
\end{remark}
\subsection{The Floer--Novikov homology}\label{section:fnhomol}
In this section, we revisit the definition of the Floer--Novikov homology for non-contractible periodic orbits following \cite{BH01_non-con}. Consider a smooth almost complex structure $J$ on $M$ compatible with $\omega$, i.e. such that
\[
g(X,Y):=\omega(X,JY)
\]
defines a Riemannian metric on $M$. We will denote by $\mathcal{J}$ the set of almost complex structures compatible with $\omega$.
Choose $J\in \mathcal{J}$ and let $\widetilde{g}$ denote the induced weak Riemannian metric on $\mathcal{L}M$ given by
\[
\widetilde{g}(X_x,Y_x)=\displaystyle\int_{S^1} g(X_x(t), Y_x(t)) dt,
\]
where $X_x$ and $Y_x$ are vector fields along $x$. A gradient flow line is a mapping $u\colon \reals\times S^1\rightarrow M$ satisfying
\begin{eqnarray}\label{eqn:gradientflowline}
\partial_s u + J(\partial_t u - Z_t(u(s,t)))=0.
\end{eqnarray}
The maps $u\colon \reals \rightarrow \mathcal{L}M$ which satisfy \eqref{eqn:gradientflowline} with boundary conditions
\begin{eqnarray}\label{eqn:boundconds_lim_tilde}
\displaystyle\lim_{s\rightarrow\pm\infty} u(s,t) = x_{\pm}(t)\in \mathcal{P}_{\zeta}
\end{eqnarray}
can be seen as connecting orbits between $\overline{x}_{-}:= [x_{-}, v_{-}]$ and $\overline{x}_{+}:= [{x}_{+}, v_{+}]$ in $\widetilde{\mathcal{L}}_{\zeta}M$ where
\begin{eqnarray}\label{eqn:transform_cap}
v^{-}\#u=v^{+}.
\end{eqnarray}
We denote by $\CM(\overline{x}_{-},\; \overline{x}_{+})$ the space of finite energy solutions of (\ref{eqn:gradientflowline}--\ref{eqn:boundconds_lim_tilde}) that transform \emph{cappings} as in \eqref{eqn:transform_cap}. The energy of a connecting orbit in this space is given by
\[
E(u):= \displaystyle\int_{\reals\times S^1} |\p_s u|^2_{g} dsdt=\CA_{\{\phi_t\}}(\overline{x}_{+})-\CA_{\{\phi_t\}}(\overline{x}_{-}).
\]
For any $\overline{x}_{-},\;\overline{x}_{+}\in \overline{\mathcal{P}_{\zeta}},$ the space $\CM(\overline{x}_{-},\; \overline{x}_{+})$ is a smooth manifold of dimension $\MUCZ(\overline{x}_{+})-\MUCZ(\overline{x}_{-})$. It admits a natural $\reals$-action given by reparametrization. For $\overline{x}_{-},\; \overline{x}_{+} \in \overline{\mathcal{P}_{\zeta}}$ such that $\MUCZ(\overline{x}_{+})-\MUCZ(\overline{x}_{-})=1$, we have that $\CM(\overline{x}_{-},\; \overline{x}_{+})/ \reals$ is finite and set
\[
n_2(\overline{x},\; \overline{y}):= \# \CM(\overline{x},\; \overline{y}) \quad \text{modulo two}.
\]
Denote by $\overline{\mathcal{P}_{\zeta,k}}$ the subset of $\overline{\mathcal{P}_{\zeta}}$ whose elements $\overline{x}$ satisfy
${\MUCZ(\overline{x})=k}$. Consider the chain complex where the $k$-th chain group $C_k$ consists of all formal sums
\begin{eqnarray*}
\sum \xi_{\overline{x}}\cdot \overline{x}
\end{eqnarray*}
with $\overline{x}\in \overline{\mathcal{P}_{\zeta,k}},\;\xi_{\overline{x}}\in \mathbb{Z}_2$ and such that, for all $c\in \reals$, the set
\[
\big{\{}\overline{x}\;|\;\xi_{\overline{x}}\not=0,
\;\mathcal{A}_{\{\phi_t\}}(\overline{x})>c\big{\}}
\]
is finite. For a generator $\overline{x}$ in $C_k$, the boundary operator $\partial_k$ is defined as follows
\[
\partial_k(\overline{x})=\displaystyle\sum_{\MUCZ(\overline{y})=k-1} n_2(\overline{x},\overline{y})\overline{y}.
\]
The boundary operator $\partial$ satisfies $\partial^2=0$ and we have the Floer--Novikov homology groups
\begin{eqnarray*}
HFN_k([\theta], \zeta)=\frac{\ker\partial_k}{\im \partial_{k+1}}.
\end{eqnarray*}
\begin{remark}[\cite{GG:hyp12}]
The pair-of-pants product is not defined on the "non-contractible" Floer--Novikov homology for an individual class $\zeta$. Such a product between $HF_*([\theta], \zeta_1)$ and $HF_*([\theta], \zeta_2)$ takes values in $HF_*([\theta], \zeta_1+\zeta_2)$.
\end{remark}
\subsubsection{The Filtered Floer--Novikov homology}
The total chain Floer complex $C_*=\colon$ $C_*^{(-\infty, \infty)}$ admits a filtration by $\reals$. Let $\CS$ be the set of critical values of the functional $\mathcal{A}_{\{\phi_t\}}$ (defined in \eqref{eqn:action_flnov}) which is called the \emph{action spectrum} of $\{\phi_t\}$. For each
$b\in(-\infty,\infty]$ outside $\CS$, the chain complex $C_*^{(-\infty, b)}$ is generated by lifted loops $\overline{x}$ with action $\A_{\{\phi_t\}}(\overline{x})$ less than~$b.$ For $-\infty\leq a < b \leq \infty$ outside $\CS,$ set
\[
C_*^{(a,b)} :=C_*^{(-\infty,b)} / C_*^{(-\infty,a)}.
\]
The boundary operator $\p\colon C_*\rightarrow C_{*-1}$ descends to $C_*^{(a,b)}$ and hence the \emph{filtered Floer--Novikov homology} $HFN_*^{(a,b)}(\theta_t, \zeta)$ is well defined.
This construction also extends to all symplectomorphisms in $\text{Symp}_0(M,\omega)$ (see \cite[Sections~3~and~4]{BH01_non-con}). For a path $\{\phi_t\}$ connecting an arbitrary $\phi\in\text{Symp}_0(M,\omega)$ to the identity, set
\begin{eqnarray}\label{eqn:homol_deg}
HFN_*^{(a,b)}(\theta_t, \zeta)\colon=
HFN_*^{(a,b)}(\theta_t^{\prime}, \zeta)
\end{eqnarray}
where $\theta_t$ is the $1$-form associated with the path $\{\phi_t\}$ and $\theta_t^{\prime}$ is the $1$-form associated with a non-degenerate perturbation $\{\phi'_t\}$ of $\{\phi_t\}$. Here $-\infty\leq a<b\leq \infty$ are outside the action spectrum of $\{\phi_t\}$.
\begin{remark}
Considering Remark~\ref{rmk:action}~\ref{rmk:action1} and Remark~\ref{rmk:index}, observe that the filtered Floer--Novikov homology when $\zeta=0$ is a shift of the standard grading and filtration in the (contractible) Floer--Novikov homology (considered in \cite{O_flux06}).
\end{remark}
\subsection{Novikov rings and quantum homology}
In this section we introduce the Novikov rings used in this paper in order to define the quantum homology and the quantum product action. Here we follow \cite{GG:hyp12} and references therein.
\subsubsection{The Novikov ring $\Lambda$ and quantum homology with coefficients in $\Lambda$.}\label{section:novikov_L} Consider the Novikov ring $\Lambda:=\Lambda(\Gamma, [\omega], \F)$ associated with $\Gamma$ and the weighting homomorphism $[\omega]\colon \Gamma \rightarrow \F$ with values in the field $\F=\Z_2$ (see \cite{HS95}). Here the group $\Gamma$ is the quotient
\[
\Gamma:= \frac{\pi_2(M)}{\ker c_1 \cap \ker [\omega]}
\]
where $[\omega]\colon \pi_2(M)\rightarrow \reals$ and $c_1\colon \pi_2(M)\rightarrow \Z$ are the maps given by the integration of the corresponding $2$-forms. These maps descend to $\Gamma$. When $M$ is strictly monotone, the Novikov
ring $\Lambda$ may simply be taken to be the group algebra $\F[\Gamma]$ over $\F$.
Moreover, in this case, the group $\Gamma$ is isomorphic to $\Z$ and denote by $A_0$ the generator of $\Gamma$ with $c_1(A_0)=-2c_1^{\text{min}}$.
Furthermore, an element in $\Lambda=\F[\Gamma]$ is a formal finite linear combination of $\alpha_A e^{A}$, with $\alpha_A\in \F$ and $A\in \Gamma$, and $\Lambda$ is graded by setting $\deg(e^{A})=-2c_1(A)$ with $A\in \Gamma$. Hence, setting $q=e^{A_0}$, we have $\deg(q)=-2c_1^{\text{min}}$ and $\Lambda=\F[q,q^{-1}]$. \\
The \emph{quantum homology} of $M$ is defined
as
\[
HQ_*(M)= H_*(M)\otimes\Lambda
\]
where the degree of a generator $\alpha \otimes e^{A}$, with $\alpha\in H_*(M)$ and $A\in \Gamma$, is $\deg(\alpha)+2c_1(A)$. The product structure is given by the quantum product:
\[
\alpha*\beta =\displaystyle\sum_{A\in \Gamma} (\alpha*\beta)_A e^A
\]
where $(\alpha*\beta)_{A}\in H_*(M)$ is defined via some Gromov--Witten invariants of $M$ and has degree ${\deg(\alpha)+\deg(\beta)-2n+2c_1(A)}$. Thus \[{\deg(\alpha*\beta)=\deg(\alpha)+\deg(\beta)-2n}.\] When $A=0$, $(\alpha*\beta)_0=\alpha\cap\beta$, where $\cap$ stands for the intersection product of ordinary homology classes.
It suffices to restrict the summation to the negative cone $[\omega](A)\leq 0$ and we can write
\[
\alpha*\beta=\alpha\cap\beta+\displaystyle\sum_{k>0}(\alpha*\beta)_k q^k,
\]
where $\deg((\alpha*\beta)_k)=\deg(\alpha)+\deg(\beta)-2n+2c^{\min}_{1}k$ and the sum is finite.
The unit in the algebra $HQ_*(M)$ is the fundamental class $[M]$ and, for $a\in \Lambda$ and $\alpha\in H_*(M)$,
\[a\alpha=(a[M])*\alpha\] where degree of $a\alpha$ is $\deg(a\alpha)=\deg(a)+\deg(\alpha)$. Then the ordinary homology $H_*(M)$ is canonically embedded in $HQ_*(M)$.
The map $[\omega]\colon \Gamma \rightarrow \reals$ can be extended to $HQ_*(M)$ by
\[
[\omega](\alpha)=\max\{[\omega](A)\;|\; \alpha_{A} \not= 0 \}=\max\{-h_0k\;|\;\alpha_k\not=0\}
\]
where $\alpha=\sum\alpha_{A} e^A=\sum\alpha_k q^k$ and it satisfies
\begin{eqnarray*}
[\omega](\alpha+\beta)\leq\max\{[\omega](\alpha),[\omega](\beta)\}
\end{eqnarray*}
and
\begin{eqnarray*}\label{eqn:I_omega_inequality_2}
[\omega](\beta*\alpha)\leq [\omega](\alpha)+[\omega](\beta).
\end{eqnarray*}
\begin{example}\label{example:quantumprod}
Consider $M=\CP^n,\;\alpha=[\CP^{n-1}]\in H_{2n-2}(M)$ and $\beta=[\text{pt}]\in H_{0}(M)$ (where $[\text{pt}]$ is the class of a point in $\CP^n$). The fact that there is a unique line through any two points is reflected in the identity $(\beta*\alpha)_A=[\CP^n]$ where $c_1(A)=c_1^{\min}$. Hence, in $HQ_*(\CP^n)$, $\beta*\alpha=q[\CP^n].$
Similarly, for
$M= \CP^n\times \Sigma_{g\geq 2}$,
\[
\alpha=[\CP^{n-1}\times \Sigma_{g\geq 2}]\in H_{2n}(M)
\]
and
\[
\beta=[\text{pt} \times \Sigma_{g\geq 2}]\in H_{2}(M),
\]
the quantum product $\beta*\alpha$ satisfies the homological condition \eqref{l:1}. (See \cite{MS12} for these and further computations.)
\end{example}
\begin{remark}\label{rmk:exampleManifolds}
The manifold $\CP^n\times \Sigma_{g\geq 2}$ admits symplectomorphisms which need not be Hamiltonian diffeomorphisms and, as seen in Examples~\ref{example:firstchern} and ~\ref{example:quantumprod}, satisfies the conditions on the manifold of Theorem~\ref{maintheo}.
Observe that $\CP^n$ also satisfies these conditions. However, $\CP^n$ is simply connected and hence (see~\eqref{eqn:flux}) has no symplectomorphism which is not a Hamiltonian diffeomorphism.
Let us also point out that there are symplectic flows on $\Sigma_{g\geq 2}$ with only (finitely many) hyperbolic fixed points and no other periodic orbits (see e.g.~\cite[Exercise~14.6.1, Chapter~14]{KH95}).
\end{remark}
\subsubsection{\emph{The Novikov ring $\Lambda_{\zeta}$.}}\label{section:novikov_Lz} For a given homotopy class $\zeta$, consider the Novikov ring $\Lambda_{\zeta}:=\Lambda(\Gamma_{\zeta}, \varrho, \F)$ associated with $\Gamma_{\zeta}$ (defined in \eqref{eqn:Gamma_z}) and weighting homomorphism $\varrho\colon \Gamma_{\zeta} \rightarrow \reals$ (defined in \eqref{eqn:toroprop}) with values in the field $\F=\Z_2$. When the homomorphisms $\varrho$ and $\overline{c_1}\colon H_1(M;\Z)\rightarrow \reals$ are strictly toroidally proportional, the Novikov ring $\Lambda_{\zeta}$ may be taken to be the group algebra $\F[\Gamma_{\zeta}]$ over $\F$. For all $\zeta$,
\[
\Lambda_{\zeta}=\F[q_{\zeta}, q_{\zeta}^{-1}]
\]
where $q_{\zeta}=e^{A_{\zeta}}$ and $A_{\zeta}$ is the element which generates $\Gamma_{\zeta}\cong \Z$ with degree $-2c^{\min}_{1,\zeta}$. This isomorphism holds under the hypotheses of the main theorem, namely, that $\varrho$ and $\overline{c_1}$ are $\zeta$-strictly toroidally proportional.
\begin{remark}\label{rmk:nu}
When $M$ is strictly spherically monotone, we have $c_{1}^{\text{min}}<\infty$. In this case, by diagram~\eqref{diagram:c1}, we have ${\text Im}(\pi_2(M)\xrightarrow{c_1}\Z) \subset {\text Im}(H_1(\mathcal{L}_{\zeta}M;\Z)\xrightarrow{\overline{c_1}}\Z) $. Hence $c_{1,\zeta}^{\text{min}}$ divides $c_{1}^{\text{min}}$ and $q=q^{\nu_{\zeta}}_{\zeta}$ with $\nu_{\zeta}=c_{1}^{\text{min}} / c_{1,\zeta}^{\text{min}}\in \N$. Moreover, under the conditions of Remark~\ref{rmk:nut=1} we have $q=q_{\zeta}$.
\end{remark}
\subsubsection{\emph{The quantum product action.}}\label{section:capproduct}
We describe an action of the quantum homology $HQ_*(M)$ on the filtered Floer--Novikov homology. We follow \cite[Section~2.3]{GG:hyp12} in the Floer--Novikov setting; see \cite[Section~3]{LO_cup96} for more details.
Let $\phi$ be a non-degenerate symplectomorphism, $J$ be a generic almost complex structure and $[\sigma]$ be a class in $H_*(M)$. Denote by $\mathcal{M}(\overline{x},\overline{y};\sigma)$ the moduli space of solutions $u$ of \eqref{eqn:gradientflowline}--\eqref{eqn:boundconds_lim_tilde} that transform cappings as in \eqref{eqn:transform_cap} with $\overline{x}, \overline{y}\in \overline{\mathcal{P}}_{\zeta}$ and such that $u(0,0)\in \sigma$ where $\sigma$ is a generic cycle representing $[\sigma]$.
Then the dimension of this moduli space is
\[
\dim \;\mathcal{M}(\overline{x},\overline{y};\sigma)=\MUCZ(\overline{x})-
\MUCZ(\overline{y})
-\text{codim}(\sigma).
\]
Let $m(\overline{x},\overline{y};\sigma)\in\Z_2$ be $\# \mathcal{M}(\overline{x},\overline{y};\sigma)$ modulo 2 when this moduli space is zero-dimensional and zero otherwise. For any $c,\;c'\not\in \mathcal{S}$, there is a map
\[
\Phi_{\sigma}:C_*^{(c,c')}\rightarrow C_{*-\text{codim}(\sigma)}^{(c,c')}
\]
induced by
\[
\Phi_{\sigma}(\overline{x})=\displaystyle\sum_{\overline{y}} m(\overline{x},\overline{y};\sigma)\overline{y}.
\]
This map commutes with the Floer--Novikov differential $\p$ and descends (independently of the choice of the cycle representing the class $[\sigma]$) to a map
\[
\Phi_{[\sigma]}:HFN_*^{(c,c')}(\theta_t, \zeta)\rightarrow HFN_{*-\text{codim}(\sigma)}^{(c,c')}(\theta_t, \zeta).
\]
The action of a class $\alpha=q^{k}[\sigma]$ is induced by the map
\begin{eqnarray}\label{eqn:action_induced}
\Phi_{q^{k}\sigma}(\overline{x}):=\displaystyle\sum_{\overline{y}}m(q^{k}\overline{x}, \overline{y}; \sigma)\overline{y}
\end{eqnarray}
Recall that, by Remark~\ref{rmk:nu}, $q=q_{\zeta}^{\nu_{\zeta}}$ with $\nu_{\zeta}\in \N.$ Then, in \eqref{eqn:action_induced}, the element $q^{k}\overline{x}$, with $\overline{x}=[x,v]$, is the lift $[x,w]$ of $x$ where $w$ is obtained by attaching $k\nu_{\zeta} A_{\zeta}$ to $v$.
The map $\Phi_{\alpha}$ can be extended to all $\alpha\in H_*(M)\otimes\Lambda$ by linearity over $\Lambda$ and then we have
\begin{eqnarray*}\label{eqn:action_quantum}
\Phi_{\alpha}: HFN_*^{(c,c')}(\theta_t, \zeta)\rightarrow HFN_{*-2n+\deg(\alpha)}^{(c,c')+\varrho(\alpha)}(\theta_t,\zeta).
\end{eqnarray*}
\begin{remark}\label{rmk:varphi_zeta} Note that the map $\varrho\colon \Gamma_{\zeta} \rightarrow \reals$ extends to $ H_{*}(M)\otimes \Lambda $ as
\[
\varrho(\alpha):=\max\{-k\nu_{\zeta}h_{\zeta}\;|\;\alpha_k\not=0\}
=\max\{-k\nu_{T}h_{\varrho}\;|\;\alpha_k\not=0\}
\]
where $\alpha=\sum\alpha_k q^k=\sum\alpha_k q_{\zeta}^{k\nu_{\zeta}}$. The above equality follows from
\begin{eqnarray}\label{eqn:nus}
\nu_{\zeta}h_{\zeta} = \nu_{\zeta} \frac{c_{1,\zeta}^{\min}}{c_{1,T}^{\min}}h_{\varrho}=\frac{c_{1}^{\min}}{c_{1,T}^{\min}}h_{\varrho}=\nu_{T}h_{\varrho}
\end{eqnarray}
which holds by \eqref{eqn:h_zeta}, Remark~\ref{rmk:nu} and \eqref{eqn:nut}.
We have
\begin{eqnarray*
\varrho(\alpha+\beta)\leq\max\{\varrho(\alpha), \varrho(\beta)\}
\end{eqnarray*}
and
\begin{eqnarray}\label{eqn:varphi_inequality_2}
\varrho(\alpha*\beta)\leq \varrho(\alpha)+\varrho(\beta).
\end{eqnarray}
\end{remark}
The maps $\Phi_{\alpha}$, for all $\alpha\in HQ_*(M)$, give an action of the quantum homology on the filtered Floer--Novikov homology.\\
This action has the following properties:
\[
\Phi_{[M]}=id
\]
and
\begin{eqnarray}\label{eqn:multip_quantum}
\Phi_{\beta}\Phi_{\alpha}=\Phi_{\beta*\alpha}.
\end{eqnarray}
\begin{remark}
Observe that in the multiplicativity property \eqref{eqn:multip_quantum}, in general, the maps on both side of the equality have different target spaces. We should understand the identity in \eqref{eqn:multip_quantum} as that the following diagram commutes for any interval $(d,d^{'})$ with $d\geq c+\varrho(\alpha)+\varrho(\beta)$ and $d^{'}\geq c'+\varrho(\alpha)+\varrho(\beta)$:
\begin{equation}\label{diagram:cap}
\xymatrix{
{HFN_*^{(c,c^{\prime})}(\theta_t,\zeta)} \ar[r]^{\Phi_{\alpha}}\ar[dr]_{\Phi_{\beta*\alpha}}
& {HFN_{*-2n+\deg(\alpha)}^{(c,c^{\prime})+\varrho(\alpha)}(\theta_t,\zeta)} \ar[r]^{\Phi_{\beta}} & {HFN_{*-4n+\deg(\alpha)+\deg(\beta)}^{(c,c^{\prime})+\varrho(\alpha)+\varrho(\beta)}}(\theta_t,\zeta) \ar[d] \\
{} & {HFN_{*-2n+\deg(\beta*\alpha)}^{(c,c^{\prime})+\varrho(\beta*\alpha)}(\theta_t,\zeta)} \ar[r] & {HFN_{*-2n+\deg(\beta*\alpha)}^{(d,d^{'})}(\theta_t,\zeta)}
}
\end{equation} where the vertical and the bottom horizontal arrows are the natural quotient-inclusion maps whose existence is guaranteed by the choice of $d$ and $d^{'}$ and \eqref{eqn:varphi_inequality_2}.
\end{remark}
\section{Proof of the main theorem}\label{section:proof}
\subsection{The Ball-Crossing Energy Theorem}\label{section:BCEthm} In this section, we state the theorem which corroborates the importance of the assumption on the hyperbolicity of the orbit $\gamma$. See \cite{GG:hyp12} for the original statement and proof of the theorem and a discussion on the result.
Let $\phi$ be a symplectomorphism (isotopic to the identity) on a symplectic manifold $(M,\omega)$ and fix a one-periodic in time almost complex structure $J$ compatible with $\omega$. For a closed domain $\Sigma \subset \reals \times S^1_k$ (i.e. a closed subset with non-empty interior), where $S^1_k=\reals/k\Z$, the \emph{energy} of a solution ${u:\Sigma \rightarrow M}$ of the equation \eqref{eqn:gradientflowline} is, by definition,
\[
E(u):= \displaystyle\int_{\Sigma} |\p_s u|^2_{g} dsdt.
\]
Let $\gamma$ be a hyperbolic one-periodic solution of $Z_t$ in $M$ and $\overline{\gamma}\in \widetilde{\mathcal{L}}_{\zeta}M$ be a lift of $\gamma$.
A solution ${u:\Sigma \rightarrow M}$ of the equation \eqref{eqn:gradientflowline} is said to be \emph{asymptotic to} $\gamma^k$ as $s\rightarrow \infty$ if $\Sigma$ contains a cylinder $[s_0,\infty)\times S^1_k$ and $u(s,t)\rightarrow \gamma^k(t)$ $C^{\infty}$-uniformly in $t$ as $s\rightarrow \infty$.
Consider a small closed neighborhood U of $\gamma$ with smooth boundary.
\begin{theorem}[({\cite[Ball-Crossing Energy Theorem]{GG:hyp12}})]
\label{thm:ballcrossing}
\
There exists a constant $c_{\infty}>0$ (independent of $k$ and $\Sigma$) such that for any solution $u$ of the equation \eqref{eqn:gradientflowline}, with $u(\partial\Sigma)\subset\partial U$ and $\partial\Sigma\not=\emptyset$, which is asymptotic to $\gamma^k$ as $s\rightarrow\infty$, we have
\begin{eqnarray}\label{eqn:energy_bound}
E(u)>c_{\infty}.
\end{eqnarray}
Moreover, the constant $c_{\infty}$ can be chosen to make \eqref{eqn:energy_bound} hold for all $k$-periodic almost complex structures (with varying $k$) $C^{\infty}$-close to $J$ uniformly on $\reals\times U$.
\end{theorem}
\begin{remark}\label{rmk:hyp}
The assumption that the orbit is hyperbolic is essential in Theorem~\ref{thm:ballcrossing}. For instance, there are "non hyperbolic" examples where the ball-crossing energy can get arbitrary small for arbitrarily large iterations $k$; see~\cite[Remark~3.4]{GG:hyp12}.
\end{remark}
\subsection{Proof of Theorem~\ref{maintheo}}
Let $\zeta$ be the free homotopy class of the hyperbolic periodic orbit $\gamma$. By passing, if necessary, to the second iteration assume
\[
d\phi\colon T_{\gamma(0)}M \rightarrow T_{\gamma(0)}M
\]
has an even number of real eigenvalues in $(-1,0)$. Then there exists a trivialization of $TM$ along $\gamma$ such that the mean index of $\gamma$ (and, since $\gamma$ is hyperbolic, the Conley--Zehnder index) is equal to zero. Moreover, add a constant to the associated hamiltonian $H_t$ (see \eqref{eqn:Ham}) so that the action of some lift of $\gamma$ is
\begin{eqnarray}\label{eqn:action_gamma}
\CA_{\{\phi_t\}}(\overline{\gamma})=0.
\end{eqnarray}
We reason by contradiction and suppose $\phi$ has finitely many periodic points $x_1,\ldots,x_m$.
Consider an almost complex structure $J^{'}$ on $M$. Let $U$ be a small closed neighborhood of $\gamma$ such that $\phi$ has no periodic orbit intersecting $U$ except $\gamma$. By the Ball-Crossing Energy Theorem, there exists a constant $c_{\infty}>0$ such that, for all $k$, the energy of any solution of \eqref{eqn:gradientflowline} of period $k$ asymptotic to $\gamma^k$ as $s\rightarrow\infty$ is greater than $c_{\infty}$.\\
For each $i=1,\ldots,m$, fix a reference loop $z_i$ in $\zeta_i$ (e.g. take $z_i=x_i$) and consider a lift $\overline{x_i}:=[x_i, v_i]\in \widetilde{\mathcal{L}}_{\zeta_i}M$ of $x_i\in \mathcal{L}_{\zeta_i}M$, where $\zeta_i$ is the free homotopy class of $x_i$ and $v_i$ is a cylinder connecting $x_i$ to $z_i$. We have that
\[
a_i:=\mathcal{A}_{\{\phi_t\}}(\overline{x_i}) \; (\text{mod } h_{\varrho})\; \text{in}\; S^1_{h_{\varrho}}\quad\quad\text{and}\quad\quad
\underline{a_i}:=\underline{\mathcal{A}}_{\{\phi_t\}}(\overline{x_i})\] are independent of the considered lift of $x_i$.
\begin{remark}
Observe that, for all $i=1,\ldots,m$, the difference between the actions of two lifts of $x_i$ is a multiple of $h_{\zeta_i}$ when $\varrho$ is rational in the sense of \eqref{eqn:varphi_rational}. Since, for all $i=1,\ldots,m,$ the constant $h_{\varrho}$ divides $h_{\zeta_i}$ (recall \eqref{eqn:h_zeta}) it follows that the difference between these action values is also a multiple of $h_{\varrho}$ and hence $\CA_{\phi_t}(\overline{x_i})$ (mod $h_{\varrho}$) is independent of the considered lift of $x_i$. The fact that the second expression is independent of the lift of $x_i$ follows from Remark~\ref{rmk:augmented_independent}.
\end{remark}
Take $\epsilon,\;\delta>0$ sufficiently small so that
\begin{eqnarray}\label{eqn:epsdelt}
2(\epsilon+\delta)<\lambda_{\varrho} \text{ and } \epsilon<c_{\infty}
\end{eqnarray}
and $C$ a sufficiently large constant so that
\begin{eqnarray}\label{eqn:constant_C}
C> \nu_{T}h_{\varrho}+\frac{\lambda_{\varrho}}{2}(n+1).
\end{eqnarray}
Then, by Kronecker's Theorem (see e.g. \cite[Theorem~7.10]{Apostol90}), there exists $k$ such that for all $i=1,\ldots,m$
\begin{eqnarray}\label{eqn:actioneps}
||k a_i||_{h_{\varrho}}<\epsilon\quad
\end{eqnarray} and \begin{eqnarray}\label{eqn:actionC} \text{either } |\underline{a_i}|=0 \text{ or } |k\underline{a_i}|>C.
\end{eqnarray}
where we denote by $||a||_{h}\in [0,h / 2]$ the distance from $a\in S^1_{h}=\reals/{h\Z}$ to $0$. Observe that $k$ depends on $\epsilon$ (and $\delta$) and $C$, hence it also depends on $c_{\infty}$ and also on the fixed neighborhood $U$.
Consider a non-degenerate perturbation $\phi{'}$ of $\phi^k$ such that (\ref{eqn:homol_deg}) holds and such that the Hamiltonian $K$ associated with $\phi{'}$ (in the sense of \eqref{eqn:Ham}) satisfies the following properties:
\begin{enumerate}\label{enumerate:K}
\item \label{enumerate:K2}$K$ coincides with $H^{\natural k}$ on the neighborhood $U$,
\item \label{enumerate:K3}$K$ is $k$-periodic and non-degenerate and
\item \label{enumerate:K1}$K$ is sufficiently $C^2$-close to $H^{\natural k}$.
\end{enumerate}
Here, we consider $K$ sufficiently $C^2$-close to $H^{\natural k}$ in order to have the existence of $k$ such that for all $x$ $k$-periodic solution of $K$
\begin{eqnarray}\label{eqn:action_epsbounds}
\big\|\mathcal{A}^{h_{\varrho}}_{\{\phi_t^{'}\}}(\overline{x})\big\|_{h_{\varrho}}<\epsilon
\end{eqnarray}
and
\begin{eqnarray}\label{eqn:augmented_bounds}
\text{either } \big|\underline{\mathcal{A}}_{\{\phi_t^{'}\}}(\overline{x})\big|<\delta
\text{ or } \big|\underline{\mathcal{A}}_{\{\phi_t^{'}\}}(\overline{x})\big|>C
\end{eqnarray}
where $\mathcal{A}^{h_{\varrho}}_{\{\phi_t^{'}\}}(\overline{x})$ stands for $\mathcal{A}_{\{\phi_t^{'}\}}(\overline{x})$ mod $h_{\varrho}$. Note that, as long as $\delta<h_{\varrho},$ conditions \eqref{eqn:action_epsbounds} and \eqref{eqn:augmented_bounds} follow from \eqref{eqn:actioneps} and \eqref{eqn:actionC}, respectively. Observe that if $\phi^k$ is non-degenerate, then we can take $\phi'=\phi^k$.
For any $k$-periodic almost complex structure $J$ sufficiently close to (the $k$-periodic extension of) $J'$, all non-trivial $k$-periodic solutions of the equation \eqref{eqn:gradientflowline} for the pair ($\phi'$, $J$) asymptotic to $\gamma^k$ as $s\rightarrow\infty$ have energy greater than $c_{\infty}$.
\begin{lemma}\emph{{\cite[Lemma~4.1]{GG:hyp12}}}\label{lemma:tau}
Let $\tau := C - \frac{\lambda_{\varrho}}{2}(n+1)$. The orbit ${\overline{\gamma}}^k$ is not connected by a solution of \eqref{eqn:gradientflowline} to any $\overline{x}\in \overline{\mathcal{P}_{k\zeta}}$ with Conley--Zehnder index $\pm 1$ and action in $(-\tau,\tau)$, where $\zeta$ is the free homotopy class of $\gamma$.
In particular, ${\overline{\gamma}}^k$ is closed in $C^{(-\tau,\tau)}_*$ and $0\not=[{\overline{\gamma}}^k]\in HFN^{(-\tau,\tau)}_*(\theta^{\prime}_t, k\zeta)$. Moreover, ${\overline{\gamma}}^k$ must enter every cycle representing its homology class $[{\overline{\gamma}}^k]$ in $HFN^{(-\tau,\tau)}_*(\theta^{\prime}_t, k\zeta)$.
\end{lemma}
\begin{proof}
Assume the orbit ${\overline{\gamma}}^k$ is connected, by a solution $u$ of \eqref{eqn:gradientflowline}, to some $\overline{x}\in \overline{P_{k\zeta}}$ with index $\MUCZ(\overline{x})=\pm 1$ with action in $(-\tau,\tau)$.
Consider the first case in \eqref{eqn:augmented_bounds}, i.e. $\big|\underline{\mathcal{A}}_{\{\phi_t^{'}\}}(\overline{x})\big|<\delta$: since
\begin{itemize}
\item[i)] $\big\|\mathcal{A}^{h_{\varrho}}_{\{\phi_t^{'}\}}(\overline{x})\big\|_{h_{\varrho}}<\epsilon$ (by \eqref{eqn:action_epsbounds}),
\item[ii)]$E(u)>c_{\infty}>\epsilon$ (by Theorem~\ref{thm:ballcrossing} and \eqref{eqn:epsdelt}) and
\item[iii)]$\mathcal{A}_{\{\phi_t^{'}\}}({\overline{\gamma}}^k)=0$ (by \eqref{eqn:action_gamma}),
\end{itemize}
we have
\[
\big|\mathcal{A}_{\{\phi_t^{'}\}}(\overline{x})\big|>h_{\varrho} - \epsilon.
\]
Then, by the definition of augmented action \eqref{eqn:augmentedaction} and since
\begin{itemize}
\item[i)] $\big|\underline{\mathcal{A}}_{\{\phi_t^{'}\}}(\overline{x})\big|<\delta$ and
\item[ii)] $2(\epsilon +\delta)<\lambda_{\varrho}$ (by \eqref{eqn:epsdelt}),
\end{itemize}
we have
\[
\big|\D_{\{\phi_t^{'}\}}(\overline{x})\big|> \frac{2}{\lambda_{\varrho}}(h_{\zeta}-\epsilon-\delta)\geq2c_{1,T}^{\min}-\frac{2(\epsilon+\delta)}{\lambda_{\varrho}} >2c_{1,T}^{\min}-1.
\]
The second inequality follows from $h_{\zeta}/\lambda_{\varrho}=c_{1,\zeta}^{\min}$ and $c_{1,\zeta}^{\min}\geq c_{1,T}^{\min}.$
Thus, by \eqref{eqn:mi_czi},
\[
\big|\MUCZ(\overline{x})\big|>2c_{1,T}^{\min}-1-n\geq n+2-1-n=1
\]
where the second inequality follows from the requirement that $c_{1,T}^{\min}\geq n/2 +1$. We obtained a contradiction since $\MUCZ(\overline{x})=\pm 1$.
Consider now the second case in \eqref{eqn:augmented_bounds}, i.e. $\big|\underline{\mathcal{A}}_{\{\phi_t^{'}\}}(\overline{x})\big|>C$: by the definition of augmented action \eqref{eqn:augmentedaction}, we obtain
\[
\big|\mathcal{A}_{\{\phi_t^{'}\}}(\overline{x})\big|> C - \frac{\lambda_{\varrho}}{2} \big|\D_{\{\phi_t^{'}\}}(\overline{x})\big| >
C-\frac{\lambda_{\varrho}}{2}(n+1)=:\tau
\]
where the second inequality follows from the fact that $\big|\D_{\{\phi_t\}}(\overline{x})\big|<n+1$ (which holds since $\MUCZ(\overline{x})=\pm 1$ and by \eqref{eqn:mi_czi}). Hence the action of $\overline{x}$ is outside the interval $(-\tau,\tau)$ and we obtained a contradiction.
\end{proof}
The previous lemma also holds for $q{\overline{\gamma}}^k$ (where $q\overline{x}$ is as defined in Section~\ref{section:capproduct}) with the shifted range of actions $(-\tau,\tau)-\nu_{T}h_{\varrho}.$
For an interval $(a,b)$ containing the interval $[-\nu_{T}h_{\varrho},0]$ and contained in the intersection of the action intervals $(-\tau,\tau)$ and $(-\tau,\tau)-\nu_{T}h_{\varrho}$,
Lemma~\ref{lemma:tau} holds for both ${\overline{\gamma}}^k$ and $q{\overline{\gamma}}^k$ and the interval $(a,b)$.
\begin{remark}
Observe that the existence of such an interval $(a,b)$ is guaranteed by $-\tau<-\nu_{T}h_{\varrho}<0<\tau-\nu_{T}h_{\varrho}$ that follows from \eqref{eqn:constant_C}.
\end{remark}
For the sake of completeness, we state this result in the following lemma.
\begin{lemma}\label{lemma:qgamma}
The \emph{orbits} ${\overline{\gamma}}^k$ and $q{\overline{\gamma}}^k$ are not connected by a solution of \eqref{eqn:gradientflowline} to any $\overline{x}\in \overline{\mathcal{P}_{k\zeta}}$ with Conley--Zehnder index $\pm 1$ and action in $(a,b)$ where
\[ [-\nu_{T}h_{\varrho},0]\subset (a,b)\subseteq (-\tau,\tau)\cap (-\tau-\nu_{T}h_{\varrho},\tau-\nu_{T}h_{\varrho}).\]
In particular, ${\overline{\gamma}}^k$ and $q{\overline{\gamma}}^k$ are closed in $C^{(a,b)}_*$ and $[{\overline{\gamma}}^k]\not=0\not=[q{\overline{\gamma}}^k]\in HFN^{(a,b)}_*(\theta^{\prime}_t, k\zeta)$.
Moreover, the orbits ${\overline{\gamma}}^k$ and $q{\overline{\gamma}}^k$ must enter every cycle representing their homology classes, respectively $[{\overline{\gamma}}^k]$ and $[q{\overline{\gamma}}^k]$, in $HFN^{(a,b)}_*(\theta^{\prime}_t, k\zeta)$.
\end{lemma}
Recall that, by \eqref{eqn:action_epsbounds}, all periodic orbits of $\phi'$ have action values in the $\epsilon$-neighborhood of $h_{\varrho}\Z$. The next lemma yields a contradiction and the main theorem follows.
\begin{lemma}\emph{{\cite[Lemma~4.2]{GG:hyp12}}}\label{lemma:orbity}
The symplectomorphism $\phi'$ has a periodic orbit with action outside the $\epsilon$-neighborhood of $h_{\varrho}\Z$.
\end{lemma}
\begin{proof} For ordinary homology classes $\alpha,\;\beta\in H_*(M)$ with $\deg(\alpha),\;\deg(\beta)<2n$ as in the statement of Theorem~\ref{maintheo}, consider $\Phi_{\beta*\alpha}([\overline{\gamma}^k])$ as an element of the group $HFN_*^{(a,b)}(\theta^{\prime}_t, \zeta)$ with $(a,b)$ as in Lemma~\ref{lemma:qgamma}. Since $\beta*\alpha=q[M]$, it follows, by \eqref{eqn:multip_quantum} and \eqref{diagram:cap}, that
\[
\Phi_{\beta}\Phi_{\alpha}([{\overline{\gamma}}^k])=\Phi_{\beta*\alpha}([{\overline{\gamma}}^k])=
\Phi_{q[M]}([{\overline{\gamma}}^k])=q\Phi_{[M]}([{\overline{\gamma}}^k])= q[{\overline{\gamma}}^k].
\]
Take $\sigma$ and $\eta$ generic cycles representing the ordinary homology classes $\alpha$ and $\beta$, respectively. The chain $\Phi_{\eta}\Phi_{\sigma}({\overline{\gamma}}^k)$ represents the homology class $q[\overline{\gamma}^k]$ and hence the \emph{orbit} $q{\overline{\gamma}}^k$ enters the chain $\Phi_{\eta}\Phi_{\sigma}({\overline{\gamma}}^k)$ (by Lemma~\ref{lemma:qgamma}). Recall that $q[x,v]$ is the class $[x,w]$ where $w$ is the cylinder obtained by attaching $\nu_{\zeta} A_{\zeta}$ to $v$. Hence (see Figure~\ref{fig:existence_y}), there exists an \emph{orbit} $\overline{y}$ in $\Phi_{\sigma}({\overline{\gamma}}^k)$ connected to ${\overline{\gamma}}^k$ and $q{\overline{\gamma}}^k$ by trajectories which are solutions of \eqref{eqn:gradientflowline}. By the Ball-Crossing Energy Theorem, \eqref{eqn:epsdelt} and
\begin{enumerate}
\item $\mathcal{A}_{\{\phi_t^{'}\}}({\overline{\gamma}}^k)=0$
\item\label{item:action}$\mathcal{A}_{\{\phi_t^{'}\}}(q{\overline{\gamma}}^k)=-\nu_{T} h_{\varrho}$,
\end{enumerate}
we obtain
\begin{eqnarray}\label{eqn:proof_actionbounds}
-\epsilon > \mathcal{A}_{\{\phi_t\}}(\overline{y})> -\nu_{T} h_{\varrho} +\epsilon.
\end{eqnarray}
The fact in point \ref{item:action} follows from $\varrho(\nu_{k\zeta}A_{k\zeta})=-\nu_{k\zeta}h_{k\zeta}= -\nu_T h_{\varrho}$ where the second equality holds by \eqref{eqn:nus}.
\begin{figure}[htb!]
\centering
\def\svgwidth{300pt}
\input{enbhd_nc_rho.pdf_tex}
\caption{The orbit $y$.}\label{fig:existence_y}
\end{figure}
Let us consider the two cases in assumption \eqref{item:qs} of the main theorem. Namely, first assume $c^{\min}_{1}=c^{\min}_{1,T}$ (i.e.,
$\nu_T=1$). In this case, we obtain, by \eqref{eqn:proof_actionbounds}, that the action value of $\overline{y}$ is outside the $\varepsilon$-neighborhood of $h_{\varrho}\Z$:
\[
-\epsilon > \mathcal{A}_{\{\phi'_t\}}(\overline{y}) > -h_{\varrho} +\varepsilon.
\]
Now, assume $\deg(\alpha)\geq 3n-2c^{\min}_{1,T} +1$. As above, we distinguish the two cases in \eqref{eqn:augmented_bounds}: $\underline{\mathcal{A}}_{\{\phi'_t\}}(\overline{y})<\delta$ and $\underline{\mathcal{A}}_{\{\phi'_t\}}(\overline{y})>C$. Firstly, when $\underline{\mathcal{A}}_{\{\phi'_t\}}(\overline{y})<\delta$, assume $\mathcal{A}_{\{\phi'_t\}}(\overline{y})$ is in the $\eps$-neighborhood of $h_{\varrho}\Z$. Then \eqref{eqn:proof_actionbounds} implies that
\[\mathcal{A}_{\{\phi'_t\}}(\overline{y})<-h_{\varrho}+\varepsilon.\]
This condition and the fact that the augmented action is less than
$\delta$ give the following upper bound for the mean index of $\overline{y}$
\[\Delta_{\{\phi'_t\}}(\overline{y})<\frac{2}{\lambda_{\varrho}}(-h_{\varrho}+\epsilon+\delta).
\]
The right hand side of the inequality is equal to $-2c^{\text{min}}_{1,T}+ \frac{2}{\lambda_{\varrho}}(\varepsilon+\delta)$
(by \eqref{eqn:c1minTprop}) and, by the choices in \eqref{eqn:epsdelt}, it is less than $-2c^{\text{min}}_{1,T}+1$.
Moreover, since $\gamma$ is a hyperbolic periodic orbit and $\MUCZ(\overline{y})=\MUCZ(\overline{\gamma}^k)-2n+\deg(\alpha)$, we have
\begin{eqnarray} \label{eqn:CZ_alpha}
\MUCZ(\overline{y}) =-2n+\deg(\alpha)
\end{eqnarray}
which implies, by \eqref{eqn:mi_czi}, that $\deg(\alpha)<3n - 2c^{\text{min}}_{1,T}+1$ which contradicts our hypothesis. Before we consider the second case in \eqref{eqn:augmented_bounds}, observe that the mean index of $\overline{y}$ is bounded from above by $n+1$:
\begin{eqnarray}\label{eqn:Delta}
\Delta(\overline{y})<n+1.
\end{eqnarray}
This follows from \eqref{eqn:mi_czi}, \eqref{eqn:CZ_alpha}
and our assumptions on the degree of the homology class $\alpha$. Now, when $\underline{\mathcal{A}}_{\{\phi'_t\}}(\overline{y})>C$, we have $|\mathcal{A}_{\{\phi'_t\}}(\overline{y})|>C-\frac{\lambda_{\varrho}}{2}|\Delta(\overline{y})|$. Then, by \eqref{eqn:constant_C},
\[
|\mathcal{A}_{\{\phi'_t\}}(\overline{y})|>C-\frac{\lambda_{\varrho}}{2}|\Delta(\overline{y})|>\nu_{T}h_{\varrho}\]
which is impossible due to \eqref{eqn:proof_actionbounds}.
\end{proof}
\bibliographystyle{amsalpha}
|
1507.03920
|
\section{Introduction}\label{sec:intro}
Answer set programming (ASP) \cite{DBLP:journals/ngc/GelfondL91,DBLP:journals/amai/Niemela99,mare-trus-99} is a declarative language for knowledge representation, particularly suitable to model common non-monotonic tasks such as reasoning by default, abductive reasoning, and belief revision \cite{bara-2002,DBLP:conf/nmr/MarekR04,DBLP:journals/ai/LinY02,DBLP:conf/kr/DelgrandeSTW08}.
If on the one hand ASP makes logic closer to the real world allowing for reasoning on incomplete knowledge, on the other hand it is unable to model imprecise information that may arise from the intrinsic limits of sensors, or the vagueness of natural language.
Fuzzy answer set programming (FASP) \cite{DBLP:journals/amai/NieuwenborghCV07} overcomes this limitation by interpreting propositions with a truth degree in the real interval $[0,1]$.
Intuitively, the higher the degree assigned to a proposition, the \emph{more true} it is, with $0$ and $1$ denoting \emph{totally false} and \emph{totally true}, respectively.
The notion of fuzzy answer set, or fuzzy stable model, was recently extended to arbitrary propositional formulas \cite{DBLP:conf/jelia/LeeW14}.
\citeANP{DBLP:conf/jelia/LeeW14} also propose an example on modeling dynamic \emph{trust} in social networks, which inspired the following simplified scenario that clarifies how truth degrees increase the knowledge representation capability of ASP.
\begin{example}\label{ex:social}
A user of a social network may trust or distrust another user, and these are vague concepts that can be naturally modeled by truth degrees.
These degrees may change over time.
For example, if at some point $A$ has a conflict with $B$, it is likely that her distrust on $B$ will increase and her trust on $B$ will decrease.
These are non-monotonic concepts that can be naturally handled in FASP.
\end{example}
In practice, however, ASP offers many efficient solvers such as \textsc{dlv} \cite{DBLP:conf/datalog/AlvianoFLPPT10}, \textsc{cmodels} \cite{DBLP:conf/lpnmr/LierlerM04}, \textsc{clasp} \cite{DBLP:journals/ai/GebserKS12}, and \textsc{wasp} \cite{DBLP:conf/lpnmr/AlvianoDFLR13}, which is not the case for FASP.
A preliminary FASP solver for programs with atomic heads and \L ukasiewicz\xspace conjunction, called \textsc{fasp}, was presented at
ICLP'13 by \cite{DBLP:journals/tplp/AlvianoP13}.
It implements approximation operators and a translation into bilevel programming \cite{DBLP:journals/ijar/BlondeelSVC14}.
A more general solver, called \textsc{ffasp}
\cite{DBLP:conf/ecai/MushthofaSC14},
is based on a translation into ASP for computing stable models whose truth degrees are in the set
$\ensuremath{\mathbb{Q}}\xspace_k := \{i/k \mid i \in [0..k]\}$, for a fixed $k$.
In general, exponentially many $k$ must be tested for checking the existene of a stable model, which is infeasible in practice. Therefore, \textsc{ffasp} tests by default a limited set of values.
Neither \textsc{fasp} nor \textsc{ffasp} accept nesting of negation, which would allow to encode \emph{choice rules}, a convenient way for guessing truth degrees without using auxiliary atoms \cite{DBLP:conf/jelia/LeeW14}.
Indeed, choice rules allow to check satisfiability of fuzzy propositional formulas without adding new atomic propositions.
Our aim is to provide a more flexible FASP solver supporting useful patterns like choice rules.
Satisfiability modulo theories (SMT) \cite{DBLP:series/faia/BarrettSST09} extends propositional logic with external background theories---e.g. real arithmetic \cite{DBLP:journals/tocl/Ratschan06,DBLP:journals/jar/AkbarpourP10}---for which specialized methods provide efficient decision procedures.
SMT is thus a good candidate as a target framework for computing fuzzy answer sets efficiently.
This is non-trivial because the minimality condition that fuzzy stable models must satisfy makes the problem hard for the second level of the polynomial hierarchy; indeed, the translation provided in Section~\ref{sec:translation} produces quantified theories in general.
However, structural properties of the program that decrease the complexity to NP can be taken into account in order to obtain more tailored translations.
For example, disabling head connectives and recursive definitions yields a compact translation into fuzzy propositional logic known as \emph{completion} \cite{DBLP:journals/tplp/JanssenVSC12}, which in turn can be expressed in SMT (see Section~\ref{sec:completion}).
Since completion is unsound for programs with recursive definitions, the notion of \emph{ordered completion} has arisen in the ASP literature \cite{DBLP:journals/amai/Ben-EliyahuD94,DBLP:conf/ecai/Janhunen04,DBLP:journals/amai/Niemela08,DBLP:journals/ai/AsuncionLZZ12}.
In a nutshell, stable models of ASP programs with atomic heads can be recasted in terms of program reducts and fixpoint of the immediate consequence operator, where the computation of the fixpoint defines a ranking of the derived atoms.
Fuzzy stable models of programs with atomic heads can also be defined in terms of reducts and fixpoint of the immediate consequence operator \cite{DBLP:journals/tplp/JanssenVSC12}, although the notion of ranking can be extended to FASP only when recursive \L ukasiewicz\xspace disjunction is disabled.
Using these notions, ordered completion is defined for FASP programs in Section~\ref{sec:ordered}.
In ASP, completion and ordered completion are also applicable to disjunctive programs having at most one recursive atom in each rule head.
Such programs, referred to as \emph{head cycle free} (HCF) \cite{DBLP:journals/amai/Ben-EliyahuD94}, are usually translated into equivalent programs with atomic heads by a so-called \emph{shift} \cite{DBLP:journals/tocl/EiterFW07}.
The same translation also works for HCF FASP programs using \L ukasiewicz\xspace disjunction in rule heads.
On the other hand, \L ukasiewicz\xspace conjunction and G\"odel\xspace disjunction require more advanced constructions (Section~\ref{sec:shift}) which introduce recursive \L ukasiewicz\xspace disjunction in rule bodies to restrict auxiliary atoms to
be Boolean.
Such rules are handled by integrality constraints in the theory produced by the completion, while they inhibit the application of the ordered completion.
As in ASP, the shift is unsound in general for FASP programs with head cycles, and complexity arguments given in Section~\ref{sec:hard} prove that it is unlikely that head connectives other than G\"odel\xspace conjunction can be eliminated in general.
The general translation into SMT, completion, and ordered completion are implemented in a new FASP solver called \textsc{fasp2smt} (\url{http://alviano.net/software/fasp2smt/}; see Section~\ref{sec:experiment}).
\textsc{fasp2smt} uses \textsc{gringo} \cite{DBLP:conf/lpnmr/GebserKKS11} to obtain a ground representation of the input program, and \textsc{z3} \cite{DBLP:conf/tacas/MouraB08} to solve SMT instances encoding ground programs.
Efficiency of \textsc{fasp2smt} is compared with the previously implemented solver \textsc{ffasp} \cite{DBLP:conf/ecai/MushthofaSC14}, showing strengths and weaknesses of the proposed approach.
\section{Background}\label{sec:background}
We briefly recall the syntax and semantics of FASP \cite{DBLP:journals/amai/NieuwenborghCV07,DBLP:conf/jelia/LeeW14} and SMT \cite{DBLP:series/faia/BarrettSST09}.
Only the notions needed for the paper are introduced; for example, we only consider real arithmetic for SMT.
\subsection{Fuzzy Answer Set Programming}
Let $\ensuremath{\mathcal{B}}\xspace$ be a fixed set of propositional atoms.
A \emph{fuzzy atom} (\emph{atom} for short) is either a propositional atom from $\ensuremath{\mathcal{B}}\xspace$, or a numeric constant in $[0,1]$.
\emph{Fuzzy expressions} are defined inductively as follows:
every atom is a fuzzy expression;
if $\alpha$ is a fuzzy expression then $\ensuremath{\raise.17ex\hbox{\ensuremath{\scriptstyle\mathtt{\sim}}}}\xspace} \DeclarePairedDelimiter\norm{\lVert}{\rVert \alpha$ is a fuzzy expression, where \ensuremath{\raise.17ex\hbox{\ensuremath{\scriptstyle\mathtt{\sim}}}}\xspace} \DeclarePairedDelimiter\norm{\lVert}{\rVert denotes \emph{negation as failure};
if $\alpha$ and $\beta$ are fuzzy expressions, and $\odot \in \{\otimes,\oplus,\veebar,\barwedge\}$ is a connective,
$\alpha \odot \beta$ is a fuzzy expression.
Connectives $\otimes,\oplus$ are known as the \L ukasiewicz\xspace connectives, and $\veebar,\barwedge$ are the G\"odel\xspace
connectives.
A \emph{head expression} is a fuzzy expression of the form $p_1 \odot \cdots \odot p_n$, where $n \geq 1$, $p_1,\ldots,p_n$
are atoms, and $\odot \in \{\otimes,\oplus,\veebar,\barwedge\}$.
A \emph{rule} is of the form $\alpha \leftarrow \beta$, where $\alpha$ is a head expression, and $\beta$ is a fuzzy expression.
A \emph{FASP program} $\Pi$ is a finite set of rules.
Let $\ensuremath{\mathit{At}}\xspace(\Pi)$ denote the set of atoms used by $\Pi$.
A \emph{fuzzy interpretation} $I$ for a FASP program $\Pi$ is a function $I : \ensuremath{\mathcal{B}}\xspace \rightarrow [0,1]$ mapping
each propositional atom of $\ensuremath{\mathcal{B}}\xspace$ into a truth degree in $[0,1]$.
$I$ is extended to fuzzy expressions as follows:
$I(c) = c$ for $c \in [0,1]$;
$I(\ensuremath{\raise.17ex\hbox{\ensuremath{\scriptstyle\mathtt{\sim}}}}\xspace} \DeclarePairedDelimiter\norm{\lVert}{\rVert \alpha) = 1 - I(\alpha)$;
$I(\alpha \otimes \beta) = \max\{I(\alpha) + I(\beta) - 1, 0\}$;
$I(\alpha \oplus \beta) = \min\{I(\alpha) + I(\beta), 1\}$;
$I(\alpha \veebar \beta) = \max\{I(\alpha), I(\beta)\}$; and
$I(\alpha \barwedge \beta) = \min\{I(\alpha), I(\beta)\}$.
\begin{comment}
Let $\I$ be the set of all interpretations and $I,J \in \I$.
$I$ is a \emph{subset} of $J$, denoted $I \subseteq J$, if $I(a) \le J(a)$ for each $a \in \ensuremath{\mathcal{B}}\xspace$.
$I$ is a strict subset of $J$, denoted $I \subset J$, if $I \subseteq J$ and $I \neq J$.
Fuzzy set intersection ($I \cap J$), union ($I \cup J$), and difference ($I \setminus J$) are defined as follows:
for every $a \in \ensuremath{\mathcal{B}}\xspace$,
$[I \cap J](a) := \min\{I(a),J(a)\}$,
$[I \cup J](a) := \max\{I(a),J(a)\}$, and
$[I \setminus J](a) := \max\{I(a) - J(a), 0\}$.
\end{comment}
$I$ satisfies a rule $\alpha \leftarrow \beta$ ($I \models \alpha \leftarrow \beta$) if
$I(\alpha) \geq I(\beta)$; $I$ is a model of a FASP program $\Pi$, denoted $I \models \Pi$, if $I \models r$ for each $r \in \Pi$.
$I$ is a \emph{stable model} of the FASP program $\Pi$ if $I \models \Pi$ and there is no interpretation $J$ such that
$J \subset I$ and $J \models \Pi^I$, where the \emph{reduct} $\Pi^I$ is obtained from $\Pi$ by replacing each occurrence of a
fuzzy expression $\ensuremath{\raise.17ex\hbox{\ensuremath{\scriptstyle\mathtt{\sim}}}}\xspace} \DeclarePairedDelimiter\norm{\lVert}{\rVert \alpha$ by the constant $1-I(\alpha)$.
Let $\ensuremath{\mathit{SM}}\xspace(\Pi)$ denote the set of stable models of $\Pi$.
A program $\Pi$ is \emph{coherent} if $\ensuremath{\mathit{SM}}\xspace(\Pi) \neq \emptyset$; otherwise, $\Pi$ is \emph{incoherent}.
Two programs $\Pi,\Pi'$ are equivalent w.r.t.\ a crisp set $S \subseteq \ensuremath{\mathcal{B}}\xspace$, denoted $\Pi \equiv_S \Pi'$, if $|\ensuremath{\mathit{SM}}\xspace(\Pi)| = |\ensuremath{\mathit{SM}}\xspace(\Pi')|$ and $\{I \cap S \mid I \in \ensuremath{\mathit{SM}}\xspace(\Pi)\} = \{I \cap S \mid I \in \ensuremath{\mathit{SM}}\xspace(\Pi')\}$, where $I \cap S$ is the interpretation assigning $I(p)$ to all $p \in S$, and 0 to all $p \notin S$.
\begin{example}\label{ex:social:encoding}
Consider the scenario described in Example~\ref{ex:social}.
Let $U$ be a set of users, and $[0..T]$ the timepoints of interest, for some $T \geq 1$.
Let $\mathit{trust}(x,y,t)$ be a propositional atom expressing that $x \in U$ trusts $y \in U$ at time $t \in [0..T]$.
Similarly, $\mathit{distrust}(x,y,t)$ represents that $x$ distrusts $y$ at time $t$, and $\mathit{conflict}(x,y,t)$ encodes that $x$ has a conflict with $y$ at time $t$.
The social network example can be encoded by the FASP program $\Pi_1$ containing the following rules, for all $x \in U$, $y \in U$, and $t \in [0..T-1]$:
\[
\begin{array}{rcl}
\mathit{distrust}(x,y,t+1) & \!\!\!\leftarrow\!\!\! & \mathit{distrust}(x,y,t) \oplus \mathit{conflict}(x,y,t) \\
\mathit{trust}(x,y,t+1) & \!\!\!\leftarrow\!\!\! & \mathit{trust}(x,y,t) \otimes \ensuremath{\raise.17ex\hbox{\ensuremath{\scriptstyle\mathtt{\sim}}}}\xspace} \DeclarePairedDelimiter\norm{\lVert}{\rVert(\mathit{distrust}(x,y,t+1) \otimes \ensuremath{\raise.17ex\hbox{\ensuremath{\scriptstyle\mathtt{\sim}}}}\xspace} \DeclarePairedDelimiter\norm{\lVert}{\rVert\mathit{distrust}(x,y,t))
\end{array}
\]
The second rule above states that the trust degree of $x$ on $y$ decreases when her distrust degree on $y$ increases.
A stable model $I$ of $\Pi_1 \cup \{\mathit{trust}(\mathit{Alice},\mathit{Bob},0) \leftarrow 0.8$, $\mathit{conflict}(\mathit{Alice},\mathit{Bob},1) \leftarrow 0.2\}$ is such that $I(\mathit{distrust}(\mathit{Alice},\mathit{Bob},2)) = 0.2$, and $I(\mathit{trust}(\mathit{Alice},\mathit{Bob},2)) = 0.6$.
\end{example}
ASP programs are FASP programs such that all head connectives are $\veebar$, all body connectives are $\barwedge$, and all numeric constants are $0$ or $1$.
Moreover, an ASP program $\Pi$ implicitly contains \emph{crispifying} rules of the form $p \leftarrow p \oplus p$, for all $p \in \ensuremath{\mathit{At}}\xspace(\Pi)$.
In ASP programs, $\veebar$ and $\barwedge$ are usually denoted $\vee$ and $\wedge$, respectively.
\subsection{Satisfiability Modulo Theories}
Let $\Sigma = \Sigma^V \cup \Sigma^C \cup \Sigma^F \cup \Sigma^P$ be a \emph{signature} where $\Sigma^V$ is a set of
\emph{variables}, $\Sigma^C$ is a set of \emph{constant} symbols, $\Sigma^F$ is the set of binary \emph{function} symbols $\{+,-\}$, and $\Sigma^P$ is the set of binary \emph{predicate} symbols $\{<,\leq,\geq,>,=,\neq\}$.
\emph{Terms} and \emph{formulas} over $\Sigma$ are defined inductively, where we use infix notation for all binary
symbols.
Constants and variables are terms.
If $t_1,t_2$ are terms and $\odot \in \Sigma^F$ then $t_1 \odot t_2$ is a term.
If $t_1,t_2$ are terms and $\odot \in \Sigma^P$ then $t_1 \odot t_2$ is a formula.
If $\varphi$ is a formula and $t_1,t_2$ are terms then $\ensuremath{\mathit{ite}}\xspace(\varphi,t_1,t_2)$ is a term (\ensuremath{\mathit{ite}}\xspace stands for \emph{if-then-else}).
If $\varphi_1,\varphi_2$ are formulas and $\odot \in \{\vee, \wedge, \rightarrow, \leftrightarrow\}$ then $\varphi_1 \odot \varphi_2$ is a formula.
If $x$ is a variable and $\varphi$ is a formula then $\forall x.\varphi$ is a formula.
We consider only closed formulas, i.e., formulas in which all free variables are universally quantified.
For a term $t$ and integers $a,b$ with $a < b$, we use $t \in [a..b]$ in formulas to represent the subformula $\bigvee_{i = a}^b t = i$.
Similarly, for terms $t,t_1,t_2$, $t \in [t_1,t_2]$ represents $t_1 \leq t \wedge t \leq t_2$.
A $\Sigma$-theory $\Gamma$ is a set of $\Sigma$-formulas.
A $\Sigma$-structure \ensuremath{\mathcal{A}}\xspace is a pair $(\ensuremath{\mathbb{R}}\xspace,\cdot^\ensuremath{\mathcal{A}}\xspace)$, where $\cdot^\ensuremath{\mathcal{A}}\xspace$ is a mapping such that
$p^\ensuremath{\mathcal{A}}\xspace \in \ensuremath{\mathbb{R}}\xspace$ for each constant symbol $p$,
$(c)^\ensuremath{\mathcal{A}}\xspace = c$ for each number $c$,
$\odot^\ensuremath{\mathcal{A}}\xspace$ is the binary function $\odot$ over reals if $\odot \in \Sigma^F$, and the binary relation $\odot$ over reals if $\odot \in \Sigma^P$.
Composed terms and formulas are interpreted as follows:
for $\odot \in \Sigma^F$, $(t_1 \odot t_2)^\ensuremath{\mathcal{A}}\xspace = t_1^\ensuremath{\mathcal{A}}\xspace \odot t_2^\ensuremath{\mathcal{A}}\xspace$;
$\ensuremath{\mathit{ite}}\xspace(\varphi,t_1,t_2)^\ensuremath{\mathcal{A}}\xspace$ equals $t_1^\ensuremath{\mathcal{A}}\xspace$ if $\varphi^\ensuremath{\mathcal{A}}\xspace$ is true, and $t_2^\ensuremath{\mathcal{A}}\xspace$ otherwise;
for $\odot \in \Sigma^P$, $(t_1 \odot t_2)^\ensuremath{\mathcal{A}}\xspace$ is true if and only if $t_1^\ensuremath{\mathcal{A}}\xspace \odot t_2^\ensuremath{\mathcal{A}}\xspace$;
for $\odot \in \{\vee,\wedge,\rightarrow,\leftrightarrow\}$, $(\varphi_1 \odot \varphi_2)^\ensuremath{\mathcal{A}}\xspace$ equals $\varphi_1^\ensuremath{\mathcal{A}}\xspace \odot \varphi_2^\ensuremath{\mathcal{A}}\xspace$ (in propositional logic);
$(\forall x.\varphi)^\ensuremath{\mathcal{A}}\xspace$ is true if and only if $\varphi[x/n]$ is true for all $n \in \ensuremath{\mathbb{R}}\xspace$, where $\varphi[x/n]$ is the formula obtained by substituting $x$ with $n$ in $\varphi$.
\ensuremath{\mathcal{A}}\xspace is a $\Sigma$-model of a theory $\Gamma$, denoted $\ensuremath{\mathcal{A}}\xspace \models \Gamma$, if $\varphi^\ensuremath{\mathcal{A}}\xspace$ is true for all $\varphi \in \Gamma$.
\begin{example}
Let $\Sigma^C$ be $\{p,q,s,z\}$, $x$ be a variable, and
$\Gamma_1=\{z \in [0,1], \forall x.(x \geq z)\}$ be a $\Sigma$-theory.
Any $\Sigma$-model of $\Gamma_1$ maps $z$ to 0.
If $\ensuremath{\mathit{ite}}\xspace(p + q \leq 1, p + q, 1) \geq s \leftrightarrow (p \geq \ensuremath{\mathit{ite}}\xspace(s - q \geq 0, s - q, 0) \wedge q \geq \ensuremath{\mathit{ite}}\xspace(s - p \geq 0, s - p, 0))$ is added to $\Gamma_1$,
then any $\Sigma$-model of $\Gamma_1$ maps $z$ to 0, and $p,q,s$ to real numbers in the interval $[0,1]$.
\end{example}
\section{Structure Simplification}\label{sec:simplification}
The structure of FASP programs can be simplified through rewritings that leave at most one connective in each rule body \cite{DBLP:conf/ecai/MushthofaSC14}.
Essentially, a rule of the form $\alpha \leftarrow \beta \odot \gamma$, with $\odot \in \{\otimes,\oplus,\veebar,\barwedge\}$, is replaced by the rules
$\alpha \leftarrow p \odot q$, $p \leftarrow \beta$, and $q \leftarrow \gamma$, with $p$ and $q$ fresh atoms.
A further simplification, implicit in the translation into crisp ASP by \cite{DBLP:conf/ecai/MushthofaSC14}, eliminates $\barwedge$ in rule heads and $\veebar$ in rule bodies:
a rule of the form $p_1 \barwedge \cdots \barwedge p_n \leftarrow \beta$, $n \geq 2$, is equivalently replaced by
$n$ rules $p_i \leftarrow \beta$, for $i \in [1..n]$;
and a rule of the form $\alpha \leftarrow \beta \veebar \gamma$ is replaced by $\alpha \leftarrow \beta$, $\alpha \leftarrow \gamma$.
Moreover, a rule of the form $\alpha \leftarrow \ensuremath{\raise.17ex\hbox{\ensuremath{\scriptstyle\mathtt{\sim}}}}\xspace} \DeclarePairedDelimiter\norm{\lVert}{\rVert \beta$ can be equivalently replaced by the rules $\alpha \leftarrow \ensuremath{\raise.17ex\hbox{\ensuremath{\scriptstyle\mathtt{\sim}}}}\xspace} \DeclarePairedDelimiter\norm{\lVert}{\rVert p$ and $p \leftarrow \beta$, where $p$ is a fresh atom.
Let $\ensuremath{\mathit{simp}}\xspace(\Pi)$ be the program obtained from $\Pi$ by applying these substitutions.
\begin{restatable}{proposition}{PropSimp}\label{prop:simp}
For every FASP program $\Pi$,
it holds that $\Pi \equiv_{\ensuremath{\mathit{At}}\xspace(\Pi)} \ensuremath{\mathit{simp}}\xspace(\Pi)$,
i.e., $|\ensuremath{\mathit{SM}}\xspace(\Pi)| = |\ensuremath{\mathit{SM}}\xspace(\ensuremath{\mathit{simp}}\xspace(\Pi))|$ and $\{I \cap \ensuremath{\mathit{At}}\xspace(\Pi) \mid I \in \ensuremath{\mathit{SM}}\xspace(\Pi)\} = \{I \cap \ensuremath{\mathit{At}}\xspace(\Pi) \mid I \in \ensuremath{\mathit{SM}}\xspace(\ensuremath{\mathit{simp}}\xspace(\Pi))\}$.
\end{restatable}
\citeANP{DBLP:conf/ecai/MushthofaSC14} also simplify rule heads:
$\alpha \odot \beta \leftarrow \gamma$ is replaced by
$p \odot q \leftarrow \gamma$, $p \leftarrow \alpha$, $\alpha \leftarrow p$, $q \leftarrow \beta$, and $\beta \leftarrow q$, where $p$ and $q$ are fresh atoms.
We do not apply these rewritings as they may inhibit other simplifications introduced in Section~\ref{sec:shift}.
\subsection{Hardness results}\label{sec:hard}
A relevant question is whether more rule connectives can be eliminated in order to further simplify the structure of FASP programs.
We show that this is not possible, unless the polynomial hierarchy collapses, by adapting the usual reduction of
2-QBF$_\exists$ satisfiability to ASP coherence testing \cite{DBLP:journals/amai/EiterG95}:
for $n > m \geq 1$, $k \geq 1$ and formula
$\phi := \exists x_1,\ldots,x_m \forall x_{m+1},\ldots,x_n\ \bigvee_{i=1}^k L_{i,1} \wedge L_{i,2} \wedge L_{i,3}$,
test the coherence of $\Pi_\phi$ below
\begin{eqnarray}
x_i^T \vee x_i^F \leftarrow 1 && \forall i \in [1..n] \label{eq:hard:1} \\
x_i^T \leftarrow \mathit{sat} \quad x_i^F \leftarrow \mathit{sat} \quad 0 \leftarrow \ensuremath{\raise.17ex\hbox{\ensuremath{\scriptstyle\mathtt{\sim}}}}\xspace} \DeclarePairedDelimiter\norm{\lVert}{\rVert \mathit{sat} && \forall i \in [m+1..n] \label{eq:hard:3} \\
\mathit{sat} \leftarrow \sigma(L_{i,1}) \wedge \sigma(L_{i,2}) \wedge \sigma(L_{i,3}) && \forall i \in [1..k] \label{eq:hard:4}
\end{eqnarray}
where $\sigma(x_i) := x_i^T$, and $\sigma(\neg x_i) := x_i^F$, for all $i \in [1..n]$.
$\Sigma^P_2$-hardness for FASP programs with $\veebar$ in rule heads is proved by defining a FASP program $\Pi_\phi^\veebar$ comprising (\ref{eq:hard:1})--(\ref{eq:hard:4}) (recall that $\vee$ is $\veebar$, and $\wedge$ is $\barwedge$).
This also holds if we replace $\wedge$ with $\otimes$ in (\ref{eq:hard:4}).
Another possibility is to replace $\vee$ with $\oplus$ in (\ref{eq:hard:1}), and add $p \leftarrow p \oplus p$ for all atoms in $\ensuremath{\mathit{At}}\xspace(\Pi_\phi)$, showing $\Sigma^P_2$-hardness for FASP programs with $\oplus$ in rule heads, a result already proved by \citeN{DBLP:journals/ijar/BlondeelSVC14} with a different construction.
The same result also applies to $\otimes$, but we need a more involved argument.
Let $\Pi_\phi^\otimes$ be the program obtained from $\Pi_\phi$ by replacing $\wedge$ with $\otimes$, substituting the rule (\ref{eq:hard:1}) with the following three rules for each $i \in [1..n]$:
\begin{eqnarray*}
x_i^T \otimes x_i^F \leftarrow 0.5 \quad &
x_i^T \otimes x_i^T \otimes x_i^T \leftarrow x_i^T \otimes x_i^T \quad &
x_i^F \otimes x_i^F \otimes x_i^F \leftarrow x_i^F \otimes x_i^F
\end{eqnarray*}
For all interpretations $I$, the first rule enforces $I(x_i^T) + I(x_i^F) \geq 1.5$.
The second rule enforces $3 \cdot I(x_i^T) - 2 \geq 2 \cdot I(x_i^T) - 1$ whenever $2 \cdot I(x_i^T) - 1 > 0$, i.e., $I(x_i^T) \geq 1$ whenever $I(x_i^T) > 0.5$.
Similarly, the third rule enforces $I(x_i^F) \geq 1$ whenever $I(x_i^F) > 0.5$.
Hence, one of $x_i^T,x_i^F$ is assigned 1, and the other 0.5.
Since conjunctions are modeled by $\otimes$, and each conjunction contains three literals whose interpretation is either 0.5 or 1, it follows that the interpretation of the conjunction is 1 if all literals are 1, and at most 0.5 otherwise.
Hence, $\phi$ is satisfiable if and only if $\Pi_\phi^\otimes$ is coherent.
\begin{comment}
In fact, let $\ensuremath{\mathit{crisp}}\xspace_\otimes(S)$ comprise the following rule for each atom $p \in S$:
\begin{eqnarray}
p \otimes p & \leftarrow & p
\end{eqnarray}
for modeling that if $I(p) > 0$ then $2 \cdot I(p) - 1 \geq I(p)$, i.e., $I(p) \geq 1$.
It can be checked that $\phi$ is satisfiable if and only if $\ensuremath{\mathit{fuzzy}}\xspace(\Pi_\phi) \cup \ensuremath{\mathit{crisp}}\xspace_\otimes(\ensuremath{\mathit{At}}\xspace(\Pi_\phi))$ is coherent.
\end{comment}
\begin{restatable}{theorem}{ThmHard}\label{thm:hard}
Checking coherence of FASP programs is $\Sigma^P_2$-hard already in the following cases:
(i) all connectives are $\otimes$;
(ii) head connectives are $\veebar$, and body connectives are $\barwedge$ (or $\otimes$); and
(iii) head connectives are $\oplus$, and body connectives are $\barwedge$ (or $\otimes$) and $\oplus$.
\end{restatable}
\subsection{Shifting heads}\label{sec:shift}
Theorem~\ref{thm:hard} shows that $\oplus$, $\otimes$, and $\veebar$ cannot be eliminated from rule heads in general by a polytime translation, unless the polynomial hierarchy collapses.
This situation is similar to the case of disjunctions in ASP programs, which cannot be eliminated
either.
However, \emph{head cycle free} (HCF) programs admit
a translation known as \emph{shift} that eliminates $\vee$ preserving stable models \cite{DBLP:journals/tocl/EiterFW07}.
We extend this idea to FASP connectives.
The definition of HCF programs relies on the notion of \emph{dependency graph}.
Let $\ensuremath{\mathit{pos}}\xspace(\alpha)$ denote the set of propositional atoms occurring in $\alpha$ but not under the scope of any $\ensuremath{\raise.17ex\hbox{\ensuremath{\scriptstyle\mathtt{\sim}}}}\xspace} \DeclarePairedDelimiter\norm{\lVert}{\rVert$ symbol.
The dependency graph $\ensuremath{\mathcal{G}}\xspace_\Pi$ of a FASP program $\Pi$ has vertices $\ensuremath{\mathit{At}}\xspace(\Pi)$, and an arc $(p,q)$ if there is a rule $\alpha \leftarrow \beta \in \Pi$ such that $p \in \ensuremath{\mathit{pos}}\xspace(\alpha)$, and $q \in \ensuremath{\mathit{pos}}\xspace(\beta)$.
A \emph{(strongly connected) component} of $\Pi$ is a maximal set containing pairwise reachable vertices of $\ensuremath{\mathcal{G}}\xspace_\Pi$.
A program $\Pi$ is \emph{acyclic} if $\ensuremath{\mathcal{G}}\xspace_\Pi$ is acyclic;
$\Pi$ is HCF if there is no rule $\alpha \leftarrow \beta$ where $\alpha$ contains two atoms from the same component of $\Pi$;
$\Pi$ has non-recursive $\odot \in \{\otimes,\oplus,\veebar,\barwedge\}$ in rule bodies if whenever $\odot$ occurs in the body of a rule $r$ of $\ensuremath{\mathit{simp}}\xspace(\Pi)$ but not under the scope of a \ensuremath{\raise.17ex\hbox{\ensuremath{\scriptstyle\mathtt{\sim}}}}\xspace} \DeclarePairedDelimiter\norm{\lVert}{\rVert symbol then for all $p \in H(r)$ and for all $q \in \ensuremath{\mathit{pos}}\xspace(B(r))$ atoms $p$ and $q$ belong to different components of $\ensuremath{\mathit{simp}}\xspace(\Pi)$.
\begin{example}
The program $\{p \leftarrow q \oplus \ensuremath{\raise.17ex\hbox{\ensuremath{\scriptstyle\mathtt{\sim}}}}\xspace} \DeclarePairedDelimiter\norm{\lVert}{\rVert\naf p\}$ is acyclic.
Note that $\ensuremath{\raise.17ex\hbox{\ensuremath{\scriptstyle\mathtt{\sim}}}}\xspace} \DeclarePairedDelimiter\norm{\lVert}{\rVert\naf p$ does not provide an arc to the dependency graph.
Adding the rule $q \otimes s \leftarrow p$ makes the program cyclic but still HCF because $q$ and $s$ belong
to two different components.
If also $q \leftarrow s$ is added, then the program is no more HCF.
Finally, note that $\Pi_1$ in Example~\ref{ex:social:encoding} is acyclic.
\end{example}
It should now be clear why we decided not to reduce the number of head connectives in the
translation \ensuremath{\mathit{simp}}\xspace defined at the beginning of this section. By removing a connective in the head of a rule
of an HCF program, we might produce a program that is not HCF.
Consider for example the HCF program $\{p \otimes q \otimes s \leftarrow 1\}$.
To reduce one of the occurrences of $\otimes$, we can introduce a fresh atom $\mathit{aux}$ that
stands for $q\otimes s$.
However, $q$ and $s$ would belong to the same component of the resulting program
$\{p \otimes \mathit{aux} \leftarrow 1,$ $q \otimes s \leftarrow \mathit{aux},$ $\mathit{aux} \leftarrow q \otimes s\}$.
We now define the \emph{shift} of a rule for all types of head connectives.
The essential idea is to move all head atoms but one to the body (hence the name
shift).
To preserve stable models, this has to be repeated for all head atoms, and some additional conditions might be required.
For a rule of the form $p_1 \oplus \cdots \oplus p_n \leftarrow \beta$, the shift essentially mimics the original
notion for ASP programs, and produces
\begin{equation}\label{eq:shift:oplus}
p_i \leftarrow \beta \otimes \ensuremath{\raise.17ex\hbox{\ensuremath{\scriptstyle\mathtt{\sim}}}}\xspace} \DeclarePairedDelimiter\norm{\lVert}{\rVert p_1 \otimes \cdots \otimes \ensuremath{\raise.17ex\hbox{\ensuremath{\scriptstyle\mathtt{\sim}}}}\xspace} \DeclarePairedDelimiter\norm{\lVert}{\rVert p_{i-1} \otimes \ensuremath{\raise.17ex\hbox{\ensuremath{\scriptstyle\mathtt{\sim}}}}\xspace} \DeclarePairedDelimiter\norm{\lVert}{\rVert p_{i+1} \otimes \cdots \otimes \ensuremath{\raise.17ex\hbox{\ensuremath{\scriptstyle\mathtt{\sim}}}}\xspace} \DeclarePairedDelimiter\norm{\lVert}{\rVert p_n
\end{equation}
for all $i \in [1..n]$.
Intuitively, the original rule requires any model $I$ to satisfy the condition
$I(p_1) + \cdots + I(p_n) \geq I(\beta)$. This is the case if and only if
$$I(p_i) \geq I(\beta) + \sum_{j \in [1..n], j \neq i} (1 - I(p_j)) - (n-1) = I(\beta) - \sum_{j \in [1..n], j \neq i} I(p_j);$$
i.e., if and only if \eqref{eq:shift:oplus} is satisfied, for all $i \in [1..n]$.
The shift of rules with other connectives in the head is more elaborate.
For $p_1 \otimes \cdots \otimes p_n \leftarrow \beta$, it produces
\begin{eqnarray}\label{eq:shift:otimes}
p_i \leftarrow q \otimes (\beta \oplus \ensuremath{\raise.17ex\hbox{\ensuremath{\scriptstyle\mathtt{\sim}}}}\xspace} \DeclarePairedDelimiter\norm{\lVert}{\rVert p_1 \oplus \cdots \oplus \ensuremath{\raise.17ex\hbox{\ensuremath{\scriptstyle\mathtt{\sim}}}}\xspace} \DeclarePairedDelimiter\norm{\lVert}{\rVert p_{i-1} \oplus \ensuremath{\raise.17ex\hbox{\ensuremath{\scriptstyle\mathtt{\sim}}}}\xspace} \DeclarePairedDelimiter\norm{\lVert}{\rVert p_{i+1} \oplus \cdots \oplus \ensuremath{\raise.17ex\hbox{\ensuremath{\scriptstyle\mathtt{\sim}}}}\xspace} \DeclarePairedDelimiter\norm{\lVert}{\rVert p_n) & q \leftarrow \beta & q \leftarrow q \oplus q \quad
\end{eqnarray}
for all $i \in [1..n]$, where $q$ is a fresh atom.
The last two rules enforce $I(q) = 1$ whenever $I(\beta) > 0$, and $I(q) = 0$ otherwise.
For all $i \in [1..n]$, $I(q) = 0$ implies that the body of the first rule is interpreted as 0, and $I(q) = 1$ implies $I(q \otimes \gamma) = I(\gamma)$, where $\gamma$ is $\beta \oplus \ensuremath{\raise.17ex\hbox{\ensuremath{\scriptstyle\mathtt{\sim}}}}\xspace} \DeclarePairedDelimiter\norm{\lVert}{\rVert p_1 \oplus \cdots \oplus \ensuremath{\raise.17ex\hbox{\ensuremath{\scriptstyle\mathtt{\sim}}}}\xspace} \DeclarePairedDelimiter\norm{\lVert}{\rVert p_{i-1} \oplus \ensuremath{\raise.17ex\hbox{\ensuremath{\scriptstyle\mathtt{\sim}}}}\xspace} \DeclarePairedDelimiter\norm{\lVert}{\rVert p_{i+1} \oplus \cdots \oplus \ensuremath{\raise.17ex\hbox{\ensuremath{\scriptstyle\mathtt{\sim}}}}\xspace} \DeclarePairedDelimiter\norm{\lVert}{\rVert p_n$.
Since the original rule is associated with the satisfaction of $\sum_{i \in [1..n]} I(p_i) - (n-1) \geq I(\beta)$, which is the case if and only if $I(p_i) \geq I(\beta) + \sum_{j \in [1..n], j \neq i} (1 - I(p_j))$, for all $i \in [1..n]$, this
translation preserves stable models for HCF programs.
The shift of $p_1 \veebar \cdots \veebar p_n \leftarrow \beta$ requires an even more advanced construction.
Notice first that since the program is HCF, we can order head atoms such that for every
$1\le i<j\le n$, $p_i$ does not reach $p_j$ in $\ensuremath{\mathcal{G}}\xspace_\Pi$. Assume w.l.o.g. that one such ordering is given.
Then, the shift of this rule is the program containing the rules
\begin{eqnarray}
p_i \leftarrow \beta \barwedge \ensuremath{\raise.17ex\hbox{\ensuremath{\scriptstyle\mathtt{\sim}}}}\xspace} \DeclarePairedDelimiter\norm{\lVert}{\rVert q_1 \barwedge \cdots \barwedge \ensuremath{\raise.17ex\hbox{\ensuremath{\scriptstyle\mathtt{\sim}}}}\xspace} \DeclarePairedDelimiter\norm{\lVert}{\rVert q_{i-1} \barwedge q_i \label{eq:veebar:1}\\
q_i \leftarrow (p_i \veebar \cdots \veebar p_n) \otimes \ensuremath{\raise.17ex\hbox{\ensuremath{\scriptstyle\mathtt{\sim}}}}\xspace} \DeclarePairedDelimiter\norm{\lVert}{\rVert(p_{i+1} \veebar \cdots \veebar p_n) & q_i \leftarrow q_i \oplus q_i & q_n \leftarrow 1 \label{eq:veebar:2}
\end{eqnarray}
for all $i \in [1..n]$, where each $q_i$ is a fresh atom.
Intuitively, (\ref{eq:veebar:2}) enforces $I(q_i) = 1$ whenever $I(p_i) > \max\{I(p_{i+1}), \ldots, I(p_n)\}$, and $I(q_i) = 0$ otherwise, with the exception of $I(q_n)$ which is always 1.
The rule (\ref{eq:veebar:1}) enforces that $I(p_i) \geq I(\beta)$ whenever $I(p_i) \geq \max\{I(p_1), \ldots, I(p_{i-1})\}$, and either $I(p_i) > \max\{I(p_{i+1}), \ldots, I(p_n)\}$ or $i = n$.
In the following, let $\ensuremath{\mathit{shift}}\xspace(\Pi)$ denote the program obtained by shifting all rules of $\Pi$.
\begin{restatable}{theorem}{ThmShift}\label{thm:shift}
Let $\Pi$ be FASP program.
If $\Pi$ is HCF then $\Pi \equiv_{\ensuremath{\mathit{At}}\xspace(\Pi)} \ensuremath{\mathit{shift}}\xspace(\Pi)$.
\end{restatable}
\begin{comment}
\section{Tractability result}
Define a class of normal programs that guarantee the absence of loops involving different rules with the same head atom.
Introduce the linear program we discuess in Cosenza and prove that its optimal solution is the fixpoint of $T_\Pi$.
Do we have any counterexample for this?
\end{comment}
\section{Translation into SMT}\label{sec:translation}
We now define a translation $\ensuremath{\mathit{smt}}\xspace$ mapping $\Pi$ into a $\Sigma$-theory, where
$\Sigma^C = \ensuremath{\mathit{At}}\xspace(\Pi)$, and $\Sigma^V = \{x_p \mid p \in \ensuremath{\mathit{At}}\xspace(\Pi)\}$.
The theory has two parts, \ensuremath{\mathit{out}}\xspace and \ensuremath{\mathit{inn}}\xspace,
for producing a model and checking its minimality, respectively.
In more detail, $f \in \{\ensuremath{\mathit{out}}\xspace, \ensuremath{\mathit{inn}}\xspace\}$ is the following:
for $c \in [0,1]$, $f(c) = c$;
for $p \in \ensuremath{\mathit{At}}\xspace(\Pi)$, $f(p)$ is $p$ if $f = \ensuremath{\mathit{out}}\xspace$, and $x_p$ otherwise;
$f(\ensuremath{\raise.17ex\hbox{\ensuremath{\scriptstyle\mathtt{\sim}}}}\xspace} \DeclarePairedDelimiter\norm{\lVert}{\rVert \alpha) = 1 - \ensuremath{\mathit{out}}\xspace(\alpha)$;
$f(\alpha \oplus \beta) = \ensuremath{\mathit{ite}}\xspace(t \leq 1, t, 1)$, where $t$ is $f(\alpha) + f(\beta)$;
$f(\alpha \otimes \beta) = \ensuremath{\mathit{ite}}\xspace(t \geq 0, t, 0)$, where $t$ stands for $f(\alpha) + f(\beta) - 1$;
$f(\alpha \veebar \beta) = \ensuremath{\mathit{ite}}\xspace(f(\alpha) \geq f(\beta), f(\alpha), f(\beta))$;
$f(\alpha \barwedge \beta) = \ensuremath{\mathit{ite}}\xspace(f(\alpha) \leq f(\beta), f(\alpha), f(\beta))$;
$f(\alpha \leftarrow \beta) = f(\alpha) \geq f(\beta)$.
Note that propositional atoms are mapped to constants by \ensuremath{\mathit{out}}\xspace, and to variables by \ensuremath{\mathit{inn}}\xspace.
Moreover, negated expressions are always mapped by \ensuremath{\mathit{out}}\xspace.
Define $\ensuremath{\mathit{smt}}\xspace(\Pi):=\{p \in [0,1] \mid p \in \ensuremath{\mathit{At}}\xspace(\Pi)\} \cup \{\ensuremath{\mathit{out}}\xspace(r) \mid r \in \Pi\} \cup \{\varphi_{\ensuremath{\mathit{inn}}\xspace}\}$, where
\begin{equation}
\varphi_\ensuremath{\mathit{inn}}\xspace := \forall\{x_p \mid p \in \ensuremath{\mathit{At}}\xspace(\Pi)\}.\bigwedge_{p \in \ensuremath{\mathit{At}}\xspace(\Pi)} x_p \in [0,p] \wedge \bigwedge_{r \in \Pi} \ensuremath{\mathit{inn}}\xspace(r) \rightarrow \bigwedge_{p \in \ensuremath{\mathit{At}}\xspace(\Pi)} x_p = p.
\end{equation}
\begin{example}\label{ex:naive}
Consider the program $\Pi_2=\{p \leftarrow q \veebar \ensuremath{\raise.17ex\hbox{\ensuremath{\scriptstyle\mathtt{\sim}}}}\xspace} \DeclarePairedDelimiter\norm{\lVert}{\rVert s,$ $q \oplus s \leftarrow \ensuremath{\raise.17ex\hbox{\ensuremath{\scriptstyle\mathtt{\sim}}}}\xspace} \DeclarePairedDelimiter\norm{\lVert}{\rVert\naf p\}$.
The theory $\ensuremath{\mathit{smt}}\xspace(\Pi_2)$ is
$\{p \in [0,1],$ $q \in [0,1],$ $s \in [0,1]\} \cup \{p \geq \ensuremath{\mathit{ite}}\xspace(q \geq 1-s, q, 1-s),$
$\ensuremath{\mathit{ite}}\xspace(q + s \leq 1, q+s, 1) \geq 1 - (1 - p)\} \cup \{\forall x_p.\forall x_q.\forall x_s.x_p \in [0,p] \wedge x_q \in [0,q] \wedge x_s \in [0,s] \wedge x_p \geq \ensuremath{\mathit{ite}}\xspace(x_q \geq 1-s, x_q, 1-s) \wedge \ensuremath{\mathit{ite}}\xspace(x_q + x_s \leq 1, x_q+x_s, 1) \geq 1 - (1 - p) \rightarrow x_p = p \wedge x_q = q \wedge x_s = s\}$.
\begin{comment}
\begin{align*}
\{ & p \in [0,1],\ q \in [0,1],\ s \in [0,1],\ p \geq \ensuremath{\mathit{ite}}\xspace(q \geq 1-s, q, 1-s), \\
& \ensuremath{\mathit{ite}}\xspace(q + s \leq 1, q+s, 1) \geq 1 - (1 - p), \\
&\forall x_p.\forall x_q.\forall x_s.\left( x_p \in [0,p] \wedge x_q \in [0,q] \wedge x_s \in [0,s] \wedge x_p \geq \ensuremath{\mathit{ite}}\xspace(x_q \geq 1-s, x_q, 1-s) \right. \\
& {}\wedge \left. \ensuremath{\mathit{ite}}\xspace(x_q + x_s \leq 1, x_q+x_s, 1) \geq 1 - (1 - p) \right) \rightarrow x_p = p \wedge x_q = q \wedge x_s = s \}.
\end{align*}
\end{comment}
Let \ensuremath{\mathcal{A}}\xspace be a $\Sigma$-structure such that $p^\ensuremath{\mathcal{A}}\xspace = q^\ensuremath{\mathcal{A}}\xspace = 1$ and $s^\ensuremath{\mathcal{A}}\xspace = 0$.
It can be checked that $\ensuremath{\mathcal{A}}\xspace \models \ensuremath{\mathit{smt}}\xspace(\Pi_2)$.
Also note that $I(p) = I(q) = 1$ and $I(s) = 0$ implies $I \in \ensuremath{\mathit{SM}}\xspace(\Pi_2)$.
\end{example}
For an interpretation $I$ of $\Pi$, let $\ensuremath{\mathcal{A}}\xspace_I$ be the one-to-one $\Sigma$-structure for $\ensuremath{\mathit{smt}}\xspace(\Pi)$ such that $p^{\ensuremath{\mathcal{A}}\xspace_I} = I(p)$, for all $p \in \ensuremath{\mathit{At}}\xspace(\Pi)$.
\begin{restatable}{theorem}{ThmSmt}\label{thm:smt}
Let $\Pi$ be a FASP program.
$I \in \ensuremath{\mathit{SM}}\xspace(\Pi)$ if and only if $\ensuremath{\mathcal{A}}\xspace_I \models \ensuremath{\mathit{smt}}\xspace(\Pi)$.
\end{restatable}
\subsection{Completion}\label{sec:completion}
A drawback of \ensuremath{\mathit{smt}}\xspace is that it produces quantified theories, which are usually handled by incomplete
heuristics in SMT solvers \cite{DBLP:conf/cav/GeM09}.
Structural properties of FASP programs may be exploited to obtain a more tailored translation that extends \emph{completion} \cite{DBLP:conf/adbt/Clark77} to the fuzzy case.
Completion is a translation into propositional theories used to compute stable models of acyclic ASP programs with atomic heads.
Intuitively, the models of the completion of a program $\Pi$ coincide with the \emph{supported models} of $\Pi$,
i.e., those models $I$ with $I(p) = \max\{I(\beta) \mid p \leftarrow \beta \in \Pi\}$, for each $p \in \ensuremath{\mathit{At}}\xspace(\Pi)$.
This notion was extended to FASP programs by \citeN{DBLP:journals/tplp/JanssenVSC12}, with fuzzy propositional theories as target framework.
We adapt it to produce
$\Sigma$-theories, for the $\Sigma$ defined before.
Let $\Pi$ be a program with atomic heads, and $p \in \ensuremath{\mathit{At}}\xspace(\Pi)$.
We denote by $\ensuremath{\mathit{heads}}\xspace(p,\Pi)$ the set of rules in $\Pi$ whose head is $p$, and by $\ensuremath{\mathit{constraints}}\xspace(\Pi)$ the set of rules in $\Pi$ whose head
is a numeric constant.
The completion of $\Pi$ is the $\Sigma$-theory:
\begin{equation}
\begin{split}
\ensuremath{\mathit{comp}}\xspace(\Pi) :={} & \{p \in [0,1] \wedge p = \ensuremath{\mathit{supp}}\xspace(p,\ensuremath{\mathit{heads}}\xspace(p,\Pi)) \mid p \in \ensuremath{\mathit{At}}\xspace(\Pi)\} \cup {} \\
& \{\ensuremath{\mathit{out}}\xspace(r) \mid r \in \ensuremath{\mathit{constraints}}\xspace(\Pi)\},
\end{split}
\end{equation}
where $\ensuremath{\mathit{supp}}\xspace(p,\emptyset):=0$, and for $n \geq 1$,
$\ensuremath{\mathit{supp}}\xspace(p,\{p \leftarrow \beta_i \mid i \in [1..n]\}):=\ensuremath{\mathit{ite}}\xspace(\ensuremath{\mathit{out}}\xspace(\beta_1) \geq t, \ensuremath{\mathit{out}}\xspace(\beta_1), t)$,
where $t$ is $\ensuremath{\mathit{supp}}\xspace(p,\{p \leftarrow \beta_i \mid i \in [2..n]\})$.
Basically, $\ensuremath{\mathit{supp}}\xspace(p,\ensuremath{\mathit{heads}}\xspace(p,\Pi))$ yields a term interpreted as $\max\{\ensuremath{\mathit{out}}\xspace(\beta)^{\ensuremath{\mathcal{A}}\xspace_I} \mid p \leftarrow \beta \in \Pi\}$ by all $\Sigma$-structures $\ensuremath{\mathcal{A}}\xspace$.
\begin{example}
Since $\Pi_2$ in Example~\ref{ex:naive} is acyclic, $\Pi_2 \equiv_{\ensuremath{\mathit{At}}\xspace(\Pi_2)} \ensuremath{\mathit{shift}}\xspace(\Pi_2)$.
The theory $\ensuremath{\mathit{comp}}\xspace(\ensuremath{\mathit{shift}}\xspace(\Pi_2))$ is
$\{p \in [0,1] \wedge p = \ensuremath{\mathit{ite}}\xspace(q \geq 1-s, q, 1-s),$
$q \in [0,1] \wedge q = \ensuremath{\mathit{ite}}\xspace(t_1 \geq 0, t_1, 0),$
$s \in [0,1] \wedge s = \ensuremath{\mathit{ite}}\xspace(p-q \geq 0, p-q, 0)\}$, where $t_1$ is $(1 - (1-p)) + (1-s) -1$, and $t_2$ is $(1 - (1-p)) + (1-q) -1$.
\end{example}
Since $\ensuremath{\mathit{smt}}\xspace(\Pi)$ and $\ensuremath{\mathit{comp}}\xspace(\Pi)$ have the same constant symbols, $\ensuremath{\mathcal{A}}\xspace_I$ defines a one-to-one mapping between interpretations of $\Pi$ and $\Sigma$-structures of $\ensuremath{\mathit{comp}}\xspace(\Pi)$.
An interesting question is whether correctness can be extended to HCF programs, for example by first shifting heads.
Notice that (\ref{eq:shift:otimes}) and (\ref{eq:veebar:2}) introduce rules of the form $q \leftarrow q \oplus q$ through the shift of $\otimes$ or $\veebar$, breaking acyclicity.
However, $q \leftarrow q \oplus q$ is a common pattern to force a Boolean interpretation of $q$, which can be encoded by integrality constraints in the theory.
The same observation applies to rules of the form $q \otimes q \leftarrow q$.
Define $\ensuremath{\mathit{bool}}\xspace(\Pi):=\{p \leftarrow p \oplus p \in \Pi\} \cup \{p \otimes p \leftarrow p \in \Pi\}$,
and let $\ensuremath{\mathit{bool}}\xspace^-(\Pi)$ be the program obtained from $\Pi \setminus \ensuremath{\mathit{bool}}\xspace(\Pi)$ by performing the following operations for each $p \in \ensuremath{\mathit{At}}\xspace(\ensuremath{\mathit{bool}}\xspace(\Pi))$:
first, occurrences of $p$ in rule bodies are replaced by $b_p$, where $b_p$ is a fresh atom;
then, a choice rule $b_p \leftarrow \ensuremath{\raise.17ex\hbox{\ensuremath{\scriptstyle\mathtt{\sim}}}}\xspace} \DeclarePairedDelimiter\norm{\lVert}{\rVert\naf b_p$ is added.
The refined completion is the following:
\begin{equation}
\ensuremath{\mathit{rcomp}}\xspace(\Pi) := \ensuremath{\mathit{comp}}\xspace(\ensuremath{\mathit{bool}}\xspace^-(\Pi)) \cup \{b_p = \ensuremath{\mathit{ite}}\xspace(p > 0, 1, 0) \mid p \in \ensuremath{\mathit{At}}\xspace(\ensuremath{\mathit{bool}}\xspace(\Pi))\},
\end{equation}
and the associated $\Sigma$-structure $\ensuremath{\mathcal{A}}\xspace_I^r$ is such that $p^{\ensuremath{\mathcal{A}}\xspace_I^r} = I(p)$ for $p \in \ensuremath{\mathit{At}}\xspace(\Pi)$, and $b_p^{\ensuremath{\mathcal{A}}\xspace_I^r}$ equals 1 if $I(p) > 0$, and 0 otherwise, for $p \in \ensuremath{\mathit{At}}\xspace(\ensuremath{\mathit{bool}}\xspace(\Pi))$.
\begin{restatable}{theorem}{ThmComp}\label{thm:comp}
Let $\Pi$ be a program such that $\Pi \setminus \ensuremath{\mathit{bool}}\xspace(\Pi)$ is acyclic.
Then, $I \in \ensuremath{\mathit{SM}}\xspace(\Pi)$ if and only if $\ensuremath{\mathcal{A}}\xspace_I^r \models \ensuremath{\mathit{rcomp}}\xspace(\ensuremath{\mathit{shift}}\xspace(\ensuremath{\mathit{simp}}\xspace(\Pi)))$.
\end{restatable}
Note that in the above theorem $\ensuremath{\mathit{simp}}\xspace$ and $\ensuremath{\mathit{shift}}\xspace$ are only required because $\ensuremath{\mathit{comp}}\xspace$ and $\ensuremath{\mathit{rcomp}}\xspace$ are defined for normal programs.
\subsection{Ordered Completion}\label{sec:ordered}
Stable models of recursive programs do not coincide with supported models, making completion unsound.
To regain soundness, \emph{ordered completion} \cite{DBLP:journals/amai/Ben-EliyahuD94,DBLP:conf/ecai/Janhunen04,DBLP:journals/amai/Niemela08,DBLP:journals/ai/AsuncionLZZ12}
uses a notion of \emph{acyclic support}.
Let $\Pi$ be an ASP program with atomic heads.
$I$ is a stable model of $\Pi$ if and only if there exists a \emph{ranking} $r$ such that, for each $p \in I$, $I(p) = \max\{I(\beta) \mid p \leftarrow \beta \in \Pi,$ $r(p) = 1+\max(\{0\} \cup \{r(q) \mid q \in \ensuremath{\mathit{pos}}\xspace(\beta)\})\}$ \cite{DBLP:conf/ecai/Janhunen04}.
This holds because the reduct $\Pi^I$ is also \ensuremath{\raise.17ex\hbox{\ensuremath{\scriptstyle\mathtt{\sim}}}}\xspace} \DeclarePairedDelimiter\norm{\lVert}{\rVert-free, and thus its unique minimal model is the least fixpoint of the immediate consequence operator $\ensuremath{\mathcal{T}}\xspace_{\Pi^I}$, mapping interpretations $J$ to $\ensuremath{\mathcal{T}}\xspace_{\Pi^I}(J)$ where
$\ensuremath{\mathcal{T}}\xspace_{\Pi^I}(J)(p) := \max\{J(\beta) \mid p \leftarrow \beta \in \Pi^I\}$.
Since $J(\alpha \wedge \beta) \leq J(\alpha)$ and $J(\alpha \wedge \beta) \leq J(\beta)$, for all interpretations $J$, the limit is reached in $|\ensuremath{\mathit{At}}\xspace(\Pi)|$ steps.
For FASP programs, however, the least fixpoint of $\ensuremath{\mathcal{T}}\xspace_{\Pi^I}$ is not reached within a linear number of applications \cite{DBLP:books/daglib/0035275}.
For example, $2^n$ applications are required for the program
$\{p \leftarrow p \oplus c\}$, for $c = 1/2^n$ and $n \geq 0$ \cite{DBLP:journals/ijar/BlondeelSVC14}.
On the other hand, for $\odot \in \{\barwedge,\otimes\}$ and all interpretations $J$, we have
$J(\alpha \odot \beta) \leq J(\alpha)$ and $J(\alpha \odot \beta) \leq J(\beta)$.
The claim can thus be extended to the fuzzy case if recursion over $\oplus$ and $\veebar$ is disabled.
\begin{comment}
Moreover, the immediate consequence operator can be refined to handle crispifying rules of the form $p \leftarrow p \oplus p$.
Let $C_{\Pi}$ be an operator such that $C_{\Pi}(J)(p)$ equals $\lceil J(p)\rceil$ if $p \in \ensuremath{\mathit{At}}\xspace(\ensuremath{\mathit{bool}}\xspace(\Pi))$, and $J(p)$ otherwise.
The refined immediate consequence operator $\ensuremath{\mathcal{T}}\xspace_{\Pi^I}^*$ is the composition $C_\Pi \circ \ensuremath{\mathcal{T}}\xspace_{\Pi^I}$.
Note that the fixpoints of $\ensuremath{\mathcal{T}}\xspace_{\Pi^I}$ and $\ensuremath{\mathcal{T}}\xspace_{\Pi^I}^*$ coincides.
\end{comment}
\begin{restatable}{lemma}{LemRank}\label{lem:rank}
Let $\Pi$ be such that $\Pi$ has atomic heads and non-recursive $\oplus,\veebar$ in rule bodies.
Let $I$ be an interpretation for $\Pi$.
The least fixpoint of $\ensuremath{\mathcal{T}}\xspace_{\Pi^I}$ is reached in $|\ensuremath{\mathit{At}}\xspace(\Pi)|$ steps.
\end{restatable}
Ordered completion can be defined for this class of FASP programs.
Let $J$ be the least fixpoint of $\ensuremath{\mathcal{T}}\xspace_{\Pi^I}$.
The \emph{rank} of $p \in \ensuremath{\mathit{At}}\xspace(\Pi)$ in $J$ is the step at which $J(p)$ is derived.
Let $r_p$ be a constant symbol expressing the rank of $p$.
Define $\ensuremath{\mathit{rank}}\xspace(\emptyset):=1$, and $\ensuremath{\mathit{rank}}\xspace(\{q_i \mid i \in [1..n]\}):=\ensuremath{\mathit{ite}}\xspace(r_{q_1} \geq t, r_{q_1}, t)$ for $n \geq 1$, where $t=\ensuremath{\mathit{rank}}\xspace(\{q_i \mid i \in [2..n]\})$.
Also define $\ensuremath{\mathit{osupp}}\xspace(p,\emptyset):=0$, and for $n \geq 1$,
\begin{equation*}
\ensuremath{\mathit{osupp}}\xspace(p,\{p \leftarrow \beta_i \mid i \in [1..n]\}):=\bigvee_{i \in [1..n]}(p = \ensuremath{\mathit{out}}\xspace(\beta_i) \wedge r_p = 1 + \ensuremath{\mathit{rank}}\xspace(\ensuremath{\mathit{pos}}\xspace(\beta_i))).
\end{equation*}
The ordered completion of $\Pi$, denoted $\ensuremath{\mathit{ocomp}}\xspace(\Pi)$, is the following theory:
\begin{comment}
\begin{equation}
\begin{split}
\ensuremath{\mathit{ocomp}}\xspace(\Pi) :={} & \{p \in [0,1] \wedge r_p\in [1..n] \wedge p = 0 \rightarrow r_p = 1 \mid p \in \ensuremath{\mathit{At}}\xspace(\Pi)\} \\
{}\cup{} & \{p = \ensuremath{\mathit{ocomp}}\xspace(p,\ensuremath{\mathit{heads}}\xspace(p,\Pi)) \mid p \in \ensuremath{\mathit{At}}\xspace(\Pi) \setminus \ensuremath{\mathit{At}}\xspace(\ensuremath{\mathit{bool}}\xspace(\Pi))\} \\
{}\cup{} & \{p = \ensuremath{\mathit{ite}}\xspace(\ensuremath{\mathit{ocomp}}\xspace(p,\ensuremath{\mathit{heads}}\xspace(p,\Pi)) > 0, 1, 0) \mid p \in \ensuremath{\mathit{At}}\xspace(\ensuremath{\mathit{bool}}\xspace(\Pi))\} \\
{}\cup{} & \{\ensuremath{\mathit{out}}\xspace(r) \mid r \in \ensuremath{\mathit{constraints}}\xspace(\Pi)\}.
\end{split}
\end{equation}
\end{comment}
\begin{equation}
\ensuremath{\mathit{comp}}\xspace(\Pi) \cup \{r_p\in [1..|\ensuremath{\mathit{At}}\xspace(\Pi)|] \wedge p > 0 \rightarrow \ensuremath{\mathit{osupp}}\xspace(p,\ensuremath{\mathit{heads}}\xspace(p,\Pi)) \mid p \in \ensuremath{\mathit{At}}\xspace(\Pi)\}.
\end{equation}
\begin{example}
The $\Sigma$-theory $\ensuremath{\mathit{ocomp}}\xspace(\{p \leftarrow 0.1,$ $p \leftarrow q$, $q \leftarrow p\})$ is the following:
\begin{align*}
& \{p \in [0,1] \wedge p = \ensuremath{\mathit{ite}}\xspace(0.1 \geq q, 0.1, q)\} \cup \{q \in [0,1] \wedge q = p\} \\
{}\cup{} & \{r_p \in [1..2] \wedge p > 0 \rightarrow (p = 0.1 \wedge r_p = 1 + 0) \vee (p = q \wedge r_p = 1 + r_q)\} \\
{}\cup{} & \{r_q \in [1..2] \wedge q > 0 \rightarrow q = p \wedge r_q = 1 + r_p)\}.
\end{align*}
The theory is satisfied by \ensuremath{\mathcal{A}}\xspace if $p^\ensuremath{\mathcal{A}}\xspace = q^\ensuremath{\mathcal{A}}\xspace = 0.1$, $r_p^\ensuremath{\mathcal{A}}\xspace = 1$, and $r_q^\ensuremath{\mathcal{A}}\xspace = 2$.
\end{example}
\begin{comment}
Lemma~\ref{lem:rank} and ordered completion can be extended to programs with non-recursive use of $\oplus$ in the body.
Such programs are partitioned in strata according to the dependency graph, and the fixpoint of the operator $\ensuremath{\mathcal{T}}\xspace_{\Pi^I}$ can be computed following the order of these strata.
When processing a stratum, $\oplus$ is applied to atoms belonging to a preceding stratum, whose interpretation is already fixed.
\end{comment}
The correctness of \ensuremath{\mathit{ocomp}}\xspace, provided that $\Pi$ satisfies the conditions of Lemma~\ref{lem:rank}, is proved by the following mappings:
for $I \in \ensuremath{\mathit{SM}}\xspace(\Pi)$, let $\ensuremath{\mathcal{A}}\xspace^o_I$ be the $\Sigma$-model for $\ensuremath{\mathit{ocomp}}\xspace(\Pi)$ such that $p^{\ensuremath{\mathcal{A}}\xspace^o_I} = I(p)$ and $r_p^{\ensuremath{\mathcal{A}}\xspace^o_I}$ is the rank of $p$ in $I$, for all $p \in \ensuremath{\mathit{At}}\xspace(\Pi)$;
for $\ensuremath{\mathcal{A}}\xspace$ such that $\ensuremath{\mathcal{A}}\xspace \models \ensuremath{\mathit{ocomp}}\xspace(\Pi)$, let $I_\ensuremath{\mathcal{A}}\xspace$ be the interpretation for $\Pi$ such that $I_\ensuremath{\mathcal{A}}\xspace(p) = p^\ensuremath{\mathcal{A}}\xspace$, for all $p \in \ensuremath{\mathit{At}}\xspace(\Pi)$.
\begin{restatable}{theorem}{ThmOcomp}\label{thm:ocomp}
Let $\Pi$ be an HCF program with non-recursive $\oplus$ in rule bodies, and whose head connectives are $\barwedge,\oplus$.
If $I \in \ensuremath{\mathit{SM}}\xspace(\Pi)$ then $\ensuremath{\mathcal{A}}\xspace^o_I \models \ensuremath{\mathit{ocomp}}\xspace(\ensuremath{\mathit{shift}}\xspace(\ensuremath{\mathit{simp}}\xspace(\Pi)))$.
Dually, if $\ensuremath{\mathcal{A}}\xspace \models \ensuremath{\mathit{ocomp}}\xspace(\ensuremath{\mathit{shift}}\xspace(\ensuremath{\mathit{simp}}\xspace(\Pi)))$ then $I_\ensuremath{\mathcal{A}}\xspace \in \ensuremath{\mathit{SM}}\xspace(\Pi)$.
\end{restatable}
The above theorem does not apply in case of recursive $\oplus$ in rule bodies.
For example, $\{p \leftarrow p \oplus 0.1\}$ has a unique stable model assigning $1$ to $p$, while its ordered completion is
the following $\Sigma$-theory with no $\Sigma$-model:
$\{p \in [0,1] \wedge p = \ensuremath{\mathit{ite}}\xspace(p + 0.1 \leq 1, p + 0.1, 1)\} \cup \{r_p \in [1..1] \wedge p > 0 \rightarrow p = \ensuremath{\mathit{ite}}\xspace(p + 0.1 \leq 1, p + 0.1, 1) \wedge r_p = 1 + r_p\}$.
\section{Implementation and Experiment}\label{sec:experiment}
We implemented the translations from Section~\ref{sec:simplification} in the new FASP solver \textsc{fasp2smt}.
\textsc{fasp2smt} is written in \textsc{python}, and uses \textsc{gringo} \cite{DBLP:conf/lpnmr/GebserKKS11} to obtain a ground representation of the input program, and \textsc{z3} \mbox{\cite{DBLP:conf/tacas/MouraB08}} to solve SMT instances encoding ground programs.
The output of \textsc{gringo} encodes a propositional program, say $\Pi$, that is conformant with the syntax in Section~\ref{sec:background}.
The components of $\Pi$ are computed, and the structure of the program is analyzed.
If $\Pi \setminus \ensuremath{\mathit{bool}}\xspace(\Pi)$ is acyclic, $\ensuremath{\mathit{rcomp}}\xspace(\ensuremath{\mathit{shift}}\xspace(\ensuremath{\mathit{simp}}\xspace(\Pi)))$ is built.
If $\Pi$ is HCF with non-recursive $\oplus$ in rule bodies, and only $\barwedge$ and $\oplus$ in rule heads, then $\ensuremath{\mathit{ocomp}}\xspace(\ensuremath{\mathit{shift}}\xspace(\ensuremath{\mathit{simp}}\xspace(\Pi)))$ is built.
In all other cases, $\ensuremath{\mathit{smt}}\xspace(\ensuremath{\mathit{simp}}\xspace(\Pi))$ is built.
The built theory is fed into \textsc{z3}, and either a stable model or the string \textsc{incoherent} is reported.
The performance of \textsc{fasp2smt} was assessed on instances of a benchmark used to evaluate the FASP solver \textsc{ffasp} \cite{DBLP:conf/ecai/MushthofaSC14}.
The benchmark comprises two (synthetic) problems, the fuzzy versions of \emph{Graph Coloring} and \emph{Hamiltonian Path}, originally considered by \citeN{DBLP:journals/tplp/AlvianoP13}.
In Graph Coloring edges of an input graph are associated with truth degrees, and each vertex $x$ is non-deterministically colored with a shadow of gray, i.e., truth degree 1 is distributed among the atoms $\mathit{black}_x$ and $\mathit{white}_x$.
The truth degree of each edge $xy$, say $d$, enforces $d \otimes \mathit{black}_x \otimes \mathit{black}_y = 0$ and $d \otimes \mathit{white}_x \otimes \mathit{white}_y = 0$, i.e., adjacent vertices must be colored with sufficiently different shadows of gray.
Similarly, in Hamiltonian Path vertices and edges of an input graph are associated with truth degrees, and Boolean connectives are replaced by \L ukasiewicz\xspace connectives in the usual ASP encoding.
The truth degree of each edge $xy$, say $d$, is non-deterministically distributed among the atoms $\mathit{in}_{xy}$ and $\mathit{out}_{xy}$.
Reaching a vertex $y$ from the initial vertex $x$ via an edge $xy$ guarantees that $y$ is reached with truth degree $\mathit{in}_{xy}$.
Reaching a third vertex $z$ via an edge $yz$, instead, guarantees that $z$ is reached with truth degree $\mathit{in}_{xy} \otimes \mathit{in}_{yz}$.
In other words, the more uncertain is the selection of an edge $xy$, the more uncertain is the membership of $y$ in the selected path, which in turn implies an even more uncertain membership of any $z$ reached by an edge $yz$.
In the original encodings, \L ukasiewicz\xspace disjunction was used to guess (fuzzy) membership of elements in one of two sets.
For example, Hamiltonian Path used a rule of the form
$\mathit{in}(X,Y) \oplus \mathit{out}(X,Y) \leftarrow \mathit{edge}(X,Y)$, which was shifted and replaced by
$\mathit{in}(X,Y) \leftarrow \mathit{edge}(X,Y) \otimes \ensuremath{\raise.17ex\hbox{\ensuremath{\scriptstyle\mathtt{\sim}}}}\xspace} \DeclarePairedDelimiter\norm{\lVert}{\rVert\mathit{out}(X,Y)$ and
$\mathit{out}(X,Y) \leftarrow \mathit{edge}(X,Y) \otimes \ensuremath{\raise.17ex\hbox{\ensuremath{\scriptstyle\mathtt{\sim}}}}\xspace} \DeclarePairedDelimiter\norm{\lVert}{\rVert\mathit{in}(X,Y)$ by \citeANP{DBLP:journals/tplp/AlvianoP13}.
In fact, in 2013 the focus was on FASP programs with atomic heads and only $\otimes$ in rule bodies, and the shift of $\oplus$ for these programs was implicit in the work of \citeN{DBLP:journals/ijar/BlondeelSVC14}.
Since our focus is now on a more general setting, the original encodings were restored, even if it is clear that \textsc{fasp2smt} shifts such programs by itself.
In fact, Graph Coloring is recognized as acyclic, and Hamiltonian Path as HCF with no $\oplus$ in rule bodies.
It turns out that \textsc{fasp2smt} uses completion for Graph Coloring, and ordered completion for Hamiltonian Path.
The experiment was run on an Intel Xeon CPU 2.4 GHz with 16 GB of RAM.
CPU and memory usage were limited to 600 seconds and 15 GB, respectively.
\textsc{fasp2smt} and \textsc{ffasp} were tested with their default settings, and the performance was measured by \textsc{pyrunlim} (\url{http://alviano.net/software/pyrunlim/}), the tool used in the last ASP Competitions \cite{DBLP:conf/lpnmr/AlvianoCCDDIKKOPPRRSSSWX13,DBLP:journals/corr/CalimeriGMR14}.
\begin{table}[b]
\caption{Performance of \textsc{fasp2smt} and \textsc{ffasp} (average execution time in seconds; average memory consumption in MB).}\label{tab:experiment}
\begin{tabular}{rrrrrrrrrrrrrrrrrr}
\toprule
&&& \multicolumn{3}{c}{\textsc{fasp2smt}} & \multicolumn{3}{c}{\textsc{ffasp}} & \multicolumn{3}{c}{\textsc{ffasp} (shifted enc.)} \\
\cmidrule{4-6}\cmidrule{7-9}\cmidrule{10-12}
& \bf den & \bf inst & \bf sol & \bf time & \bf mem & \bf sol & \bf time & \bf mem & \bf sol & \bf time & \bf mem \\
\cmidrule{1-12}
\parbox[t]{2mm}{\multirow{5}{*}{\rotatebox[origin=c]{90}{\bf graph--col}}}
& 20 & 6 & 6 & 94.0 & 174 & 6 & 5.3 & 302 & 6 & 1.5 & 69 \\
& 40 & 6 & 6 & 102.4 & 178 & 6 & 19.8 & 1112 & 6 & 5.3 & 181 \\
& 60 & 6 & 6 & 107.6 & 180 & 6 & 46.7 & 2472 & 6 & 11.8 & 342 \\
& 80 & 6 & 6 & 111.1 & 181 & 6 & 90.1 & 4420 & 6 & 21.0 & 550 \\
& 100 & 6 & 6 & 111.7 & 181 & 6 & 151.9 & 7025 & 6 & 33.6 & 812 \\
\rule{0pt}{2.5ex}
\parbox[t]{2mm}{\multirow{10}{*}{\rotatebox[origin=c]{90}{\bf ham--path}}}
& 20 & 10 & 10 & 1.7 & 25 & 10 & 17.3 & 410 & 10 & 3.5 & 101 \\
& 40 & 10 & 10 & 1.8 & 25 & 10 & 20.3 & 462 & 10 & 2.3 & 105 \\
& 60 & 10 & 10 & 2.1 & 25 & 10 & 13.2 & 481 & 10 & 2.0 & 107 \\
& 80 & 10 & 10 & 2.4 & 25 & 10 & 32.9 & 868 & 10 & 3.9 & 188 \\
& 100 & 10 & 10 & 2.1 & 25 & 10 & 69.0 & 1385 & 10 & 6.5 & 323 \\
& 120 & 10 & 10 & 2.0 & 25 & 10 & 125.5 & 2042 & 10 & 10.5 & 475 \\
& 140 & 10 & 10 & 1.9 & 25 & 10 & 176.8 & 2821 & 10 & 14.7 & 669 \\
& 160 & 10 & 10 & 2.2 & 25 & 9 & 139.6 & 3769 & 10 & 20.8 & 960 \\
& 180 & 10 & 10 & 2.4 & 26 & 8 & 203.1 & 4914 & 10 & 28.9 & 1270 \\
\bottomrule
\end{tabular}
\end{table}
The results are reported in Table~\ref{tab:experiment}.
Instances are grouped according to the granularity of numeric constants, where instances with $\textbf{den} = d$ are characterized by numeric constants of the form $n/d$.
There are 6 instances of Graph Coloring and 10 of Hamiltonian Path in each group.
All instances of Graph Coloring are coherent, while there is an average of 4 incoherent instances in each group of Hamiltonian Path.
All instances are solved by \textsc{fasp2smt} (column \textbf{sol}), and the granularity of numeric constants does not really impact on execution time and memory consumption.
The performance is particularly good for Hamiltonian Path, while \textsc{ffasp} is faster than \textsc{fasp2smt} in Graph Coloring for numeric constants of limited granularity.
The performance of \textsc{ffasp} deteriorates when the granularity of numeric constants increases, and 6 timeouts are reported for the largest instances of Hamiltonian Path.
Another strength of \textsc{fasp2smt} is the limited memory consumption compared to \textsc{ffasp}.
If we decrease the memory limit to 3 GB, \textsc{ffasp} runs out of memory on 12 instances of Graph Coloring and 34 instances of Hamiltonian Path, while \textsc{fasp2smt} still succeeds in all instances.
For the sake of completeness, manually shifted encodings were also tested.
The performance of \textsc{fasp2smt} did not change, while \textsc{ffasp} improves considerably, especially regarding memory consumption.
We also tested 180 instances (not reported in Table~\ref{tab:experiment}) of two simple problems called \emph{Stratified} and \emph{Odd Cycle} \cite{DBLP:journals/tplp/AlvianoP13,DBLP:conf/ecai/MushthofaSC14}, which both \textsc{fasp2smt} and \textsc{ffasp} solve in less than 1 second.
The main picture resulting from the experimental analysis is that \textsc{fasp2smt} is slower than \textsc{ffasp} in Graph Coloring, but it is faster in Hamiltonian Path.
The reason for these different behaviors can be explained by the fact that all tested instances of Graph Coloring are coherent, while incoherent instances are also present among those tested for Hamiltonian Path.
To confirm such an intuition, we tested the simple program $\{p \oplus q \leftarrow 1, 0 \leftarrow p \oplus q\}$.
Its incoherence is proved instantaneously by \textsc{fasp2smt}, while \textsc{ffasp} requires 71.8 seconds and 446 MB of memory (8.3 seconds and 96 MB of memory if the program is manually shifted).
\section{Conclusions}
SMT proved to be a reasonable target language to compute fuzzy answer sets efficiently.
In fact, when structural properties of the evaluated programs are taken into account, efficiently evaluable theories are produced by \textsc{fasp2smt}.
This is the case for acyclic programs, for which completion can be used, as well as for HCF programs with only $\oplus$ in rule heads and no recursive $\oplus$ in rule bodies, for which ordered completion is proposed.
Moreover, common patterns to \emph{crispify} atoms, which would introduce recursive $\oplus$ in rule bodies, are possibly replaced by integrality constraints.
The performance of \textsc{fasp2smt} was compared with \textsc{ffasp}, which performs multiple calls to an ASP solver.
An advantage of \textsc{fasp2smt} is that, contrary to \textsc{ffasp}, its performance is not affected by the approximation used to represent truth degrees in the input program.
On the other hand, \textsc{ffasp} is currently faster than \textsc{fasp2smt} for instances having a stable model with truth degrees in $\ensuremath{\mathbb{Q}}\xspace_k$, for some small $k$, which however cannot be determined a priori.
Such a $k$ does not exist for incoherent instances, and indeed in this case \textsc{fasp2smt} significantly overcomes \textsc{ffasp}.
It is also important to note that in general the amount of memory required by \textsc{fasp2smt} is negligible compared to \textsc{ffasp}.
Future work will evaluate the possibility to extend the approximation operators by \citeN{DBLP:journals/tplp/AlvianoP13} to the broader language considered in this paper, with the aim of identifing classes of programs for which the fixpoints are reached within a linear number of applications.
\section*{Acknowledgement}
Mario Alviano was partially supported by MIUR within project ``SI-LAB BA2\-KNOW -- Business Analitycs to Know'', by Regione Calabria, POR Calabria FESR 2007-2013, within projects ``ITravel PLUS'' and ``KnowRex'', by the National Group for Scientific Computation (GNCS-INDAM), and by Finanziamento Giovani Ricercatori UNICAL.
Rafael Pe\~naloza was partially supported by the DFG within the Cluster of Excellence `cfAED;' this work was
developed while still being affiliated with TU Dresden and the Center for Advancing Electronics Dresden, Germany.
\bibliographystyle{acmtrans}
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1908.00578
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\subsection{Notes}
\section{Introduction
In this article we consider the problem of finding the visibility set from a given viewpoint given a set of known obstacles using a Partial Differential Equation (PDE).
In principle, the visibility set is simply given by ray tracing and there are numerous algorithms for solving the visibility problem using explicit representations of the obstacles \cite{Coorg1997,Durand2000,AgarwalRayShooting2D,AgarwalRayShooting3D}.
Finding the visibility set plays a crucial role in numerous applications including rendering, visualization \cite{VisualizationReference}, etching \cite{EtchingReference}, surveillance, exploration \cite{TsaiInformationVisibility}, navigation \cite{LandaTsaiVisibilityPointClouds}, and inverse problems, to only name a few.
Specifically, in \cite{TsaiVisibilityExtensions} the level set framework \cite{OSnum,SethianBook} developed in \cite{TsaiVisibilityPDE} was extended to deal with the optimal placing of a single viewer or a group of viewers and $A$-to-$B$ optimal path planning, where optimality is measured in terms of the volume of the visible region.
More recently, in \cite{TsaiDeepLearning} a convolutional neural network is proposed to determine the vantage points that maximize visibility in the context of surveillance and exploration, with the visibility sets of the training data being computed efficiently using the PDE formulation introduced in \cite{TsaiVisibilityPDE}.
For applications which involve optimization of the viewpoint, the discontinuity of the visibility can make optimization more difficult.
The advantage of using level set/PDE methods is the improved regularity of the solution.
It is clear then that the ultimate goal of the work is inverse problems involving visibility.
As is the case with inverse problems, a better understanding of the forward problem is essential for better results of the more challenging inverse problem.
In this work, we focus our attention in the forward problem and do not go further and study the inverse problem.
We propose a simple formulation of the visibility problem - the visibility set is the subzero level set of the solution of a nonlinear obstacle problem.
In \cite{TsaiVisibilityPDE} the visibility problem is presented as a boundary value problem for a first order differential equation: the visibility set to a given viewpoint $x^*$ is given by $\{\psi(x) \geq 0\}$ where the function $\psi(x)$ is the solution of
\begin{equation}\label{PDE:Tsai}
\grad \psi \cdot \frac{x-x^*}{\abs{x-x^*}} = \min\left\{H(\psi-g)\grad g \cdot \frac{x-x^*}{\abs{x-x^*}},0\right\}
\end{equation}
with $\psi(x^*) = g(x^*)$.
Here $H(z) = \chi_{[0,\infty)}(z)$ is the characteristic function of $[0,\infty)$ and $g$ is a signed distance function to the obstacles, positive outside the obstacles and negative inside.
Despite the complex nature of the operator in \eqref{PDE:Tsai}, in \cite{TsaiPDEProof} the visibility function $\psi$ is shown to be the viscosity solution of an equivalent Hamilton-Jacobi type equation involving jump discontinuities in the Hamiltonian.
A numerical scheme to solve this equation is presented and its convergence is established.
For our formulation, $g$ is still a signed distance function to the obstacles, but is instead negative outside and positive inside. Then the visibility set is given as $\{u(x) \leq 0\}$ where the function $u(x)$ solves the following nonlinear first order local PDE
\[
\min\{u(x)-g(x), (x-x^*)\cdot \grad u(x)\} = 0
\]
with $u(x^*) = g(x^*)$. This is not only considerably simpler than \eqref{PDE:Tsai}, but can also be generalized to allow multiple viewpoints as we will show. Moreover, each sublevel set of $u$ is in fact the visibility set of the corresponding superlevel set of $g$. Efficiencies then arise when the obstacles are given by the graph of a function (for example, heights of buildings). In this case, we can reduce the dimension of the problem, and compute the horizontal visibility set from a given height, using the level set representation. Similarly, if the $t$ superlevel set of $g$ represents the position of the obstacles at a certain time $t$ ($g$ can for instance be the solution of an Eikonal equation), then the PDE needs to be solved only once and the visibility set at any given time can be extracted from the corresponding sublevel set.
The paper is organized as follows.
In \autoref{sec:SS}, we characterize visibility sets as star-shaped envelopes.
In \autoref{sec:PDE} we derive the new visibility PDE and its generalization to multiple viewpoints.
In \autoref{sec:scheme} we present the numerical scheme, while in \autoref{sec:convergence} we establish its convergence.
Finally, in \autoref{sec:numerics} we present both two-dimensional and three-dimensional examples of visibility sets computed using the new proposed PDE.
\section{Star shaped sets and functions}\label{sec:SS
In this section we give an interpretation of the visibility set from a given point $x^*$ as the star-shaped envelope with respect to the point $x^*$. The definitions of star-shaped sets and envelopes are then extended to functions. Finally, we provide explicit formulas for the star-shaped envelopes of a function.
We star by recalling the definition of a star-shaped set.
\begin{definition}
We say the set $S \subset \R^n$ is \emph{star-shaped} with respect to $x^*$ if
\[
x \in S \implies tx^* + (1-t)x \in S, \quad \text{ for all } t \in [0,1].
\]
\end{definition}
A simple example of a star-shaped set is a convex set. Indeed, convex sets are star-shaped with respect to every point inside. Moreover, intersections of convex sets are convex, but unions are not. As a consequence there is a natural (outer) convex envelope, but not an inner one. On the other hand, star-shaped sets are closed under both intersections and unions, which means one can define two star-shaped envelopes (with respect to $x^*$) for sets, the inner and outer envelopes.
\begin{definition}
Given $S\subset \mathbb{R}^n$ and $x^* \in S$, the outer star-shaped envelope of $S$ with respect to $x^*$ is the intersection of all star-shaped sets with respect to $x^*$ which contain $S$. The inner star-shaped envelope of $S$ is the union of all star-shaped sets contained in it.
\end{definition}
\begin{figure}[h]
\centering
\subfigure{\includegraphics[width=0.31\textwidth,trim={4cm 2.3cm 5cm 2.3cm},clip]{original_set}}
\subfigure{\includegraphics[width=0.31\textwidth,trim={4cm 2.3cm 5cm 2.3cm},clip]{inner_envelope}}
\subfigure{\includegraphics[width=0.31\textwidth,trim={4cm 2.3cm 5cm 2.3cm},clip]{outer_envelope}}
\caption{The inner and outer star-shaped envelopes of a set: the original set (left, dashed); inner star-shaped envelope / visibility set (center); outer star-shaped envelope (right).}
\label{fig:SSenvelopes}
\end{figure}
Looking at Figure \ref{fig:SSenvelopes}, one immediately sees how the inner star-shaped envelope corresponds to the \emph{visibility} subset of $S$ if there is an illumination source at the point $x^*$.
The remainder of $S$ is the invisible part.
One the other hand, the outer star-shaped envelope minus $S$ corresponds to the least amount of obstacles which would need to be removed so that all of $S$ is visible from $x^*$.
We now discuss star-shaped functions which are the main building block to characterize the visibility set as the solution of a nonlinear obstacle PDE. We write $S_\alpha(u)\equiv\{x \in \R^n \mid u(x)\leq\alpha\}$ for the $\alpha$-sublevel set of a function $u$ and let $\Omega$ be a star-shaped domain with respect to $x^*$.
\begin{definition}
We say that the function $u: \Omega \to \mathbb{R}$ is star-shaped with respect to $x^*$ if
\[
\text{$S_\alpha(u)$ is star-shaped with respect to $x^*$}
\]
for all $\alpha\in\mathbb{R}$.
\end{definition}
\begin{remark}
Notice the similarity to quasiconvex functions: $u$ is said to be quasiconvex if $S_\alpha(u)$ is convex for all $\alpha\in\mathbb{R}$.
\end{remark}
We now characterize star-shaped functions with a zero-order condition.
\begin{lemma}\label{lemma:SSequivalence}
A function $u:\Omega \to \mathbb{R}$ is star-shaped with respect to $x^*$ if and only if
\begin{equation}
\label{eq:SScond0}
u(t x^* + (1-t)y) \le u(y),
\quad \text{ for all } y \in \Omega, 0 \le t \le 1.
\end{equation}
\end{lemma}
\begin{proof}
By definition, $u$ is star-shaped with respect to $x^*$ if and only if for all $\alpha\in\mathbb{R}$, $S_\alpha(u)$ is star-shaped with respect to $x^*$. This is equivalent to the condition
\[
u(y) \le \alpha \implies u(t x^* + (1-t)y) \le \alpha,
\quad \text{ for all } y \in \Omega, 0 \le t \le 1
\]
for all all $\alpha \in \mathbb{R}$, which in turn is equivalent to \eqref{eq:SScond0}.
\end{proof}
We use this result to describe the monotonicity of a star-shaped function.
\begin{proposition}\label{prop:SSequivalence}
Let $u:\Omega\to\mathbb{R}$ be a function. Then $u$ is star-shaped with respect to $x^*$ if and only if $u$ is increasing along rays from $x^*$ to $x$. Moreover, if $u$ is star-shaped with respect to $x^*$, then $x^*$ is a global minimum of $u$.
\end{proposition}
\begin{proof}
This follows immediately from Lemma \ref{lemma:SSequivalence}.
\end{proof}
Next, in a similar way to star-shaped envelopes of a set, we define upper and lower star-shaped envelopes of a function $g$ with respect to a point $x^*$.
\begin{definition}
Let $g\in C(\Omega)$ be bounded by below. The lower star-shaped envelope of $g$ with respect to $x^*$ is defined as
\begin{equation}
SS^-(g)(x) = \sup \{ v(x) \mid \text{ $v$ is star-shaped with respect to $x^*$ and $v \leq g$}\},
\end{equation}
while the upper star-shaped envelope is given by
\begin{equation}
SS^+(g)(x) = \inf \{ v(x) \mid \text{ $v$ is star-shaped with respect to $x^*$ and $v \geq g$}\}.
\end{equation}
\end{definition}
\begin{remark}
We require that $g$ is bounded by below in order for the lower star-shaped envelope to be well defined since otherwise there would no star-shaped function with respect to $x^*$ bounded from above by $g$.
\end{remark}
We finish this section by proving the following simple explicit solution formulas for the star-shaped envelope of a function.
\begin{proposition}\label{prop:LowerEnvExplicit}
Let $g \in C(\Omega)$ be bounded by below and define $w:\Omega\to\mathbb{R}$ to be given by
\begin{equation}\label{eq:lowerss_formula}
w(x) = \min \left\{g(y) \mid y = x + t(x-x^*) \in \Omega, t\geq 0\right\}
\end{equation}
with $w(x^*) = \min_{x\in\Omega} g(x)$. Then $w$ is star-shaped with respect to $x^*$ and $w = SS^-(g)$.
\end{proposition}
\begin{proof}
By the assumptions on $g$, $w$ is well-defined. By construction, $w$ is increasing along rays from $x^*$ to $x$, and so, by Proposition \ref{prop:SSequivalence}, $w$ is star-shaped with respect to $x^*$. Moreover, it is clear that $w \leq g$.
We want to show that $w = SS^-(g)$. Suppose by contradiction that it is not. This means that there exists a star-shaped with respect to $x^*$ function $v$ with $v\leq g$ such that $v(x) > w(x)$ for some $x\in\Omega$. Without loss of generality, assume that $x\neqx^*$. We have $w(x) < g(x)$ and that there exists $y \in \Omega$ such that
\[
y \in \argmin \left\{g(y) \mid y = x+t(x-x^*)\in\Omega,t > 0\right\}.
\]
Hence $v(x) > w(x) = w(y) = g(y) \geq v(y)$ and therefore $v$ is not increasing along the ray from $x^*$ to $x$. Finally, we invoke Proposition \ref{prop:SSequivalence} to conclude that $v$ is not star-shaped with respect to $x^*$, which leads to the desired contradiction.
\end{proof}
\begin{remark}\label{rmk:prop_lower_ss}
Intuitively, we can find $w$ by tracing the values from the boundary along rays to $x^*$ and taking the minimum of $g$ along the way.
\end{remark}
\begin{proposition}\label{prop:UpperEnvExplicit}
Let $g \in C(\Omega)$ and define $u:\Omega\to\mathbb{R}$ to be given by
\begin{equation}\label{eq:explicit_visibility}
u(x) = \max \left\{g(y) \mid y = x^* + t(x-x^*) \in \Omega, t\in[0,1]\right\}.
\end{equation}
Then $u$ is star-shaped with respect to $x^*$ and $u = SS^+(g)$.
\end{proposition}
\begin{proof}
From the definition of $u$, it is clear that $u$ is increasing along rays from $x^*$ to $x$, and so, by Proposition \ref{prop:SSequivalence}, $u$ is star-shaped with respect to $x^*$. Moreover, $u \geq g$, again by definition of $u$.
We want to show that $u = SS^+(g)$. Suppose by contradiction that it is not. This means that there exists a star-shaped with respect to $x^*$ function $v$ with $v\geq g$ such that $v(x) < u(x)$ for some $x\in\Omega$. Without loss of generality assume $x\neqx^*$. We have $u(x) > g(x)$ and that there exists $y\in\Omega$ such that
\[
y \in \argmax \left\{g(y) \mid y = x^*+t(x-x^*)\in\Omega,t\in[0,1)\right\}.
\]
Hence $v(x) < u(x) = u(y) = g(y) \leq v(y)$, which means, just like in the proof of Proposition \ref{prop:LowerEnvExplicit}, that $v$ is not increasing along the ray from $x^*$ to $x$. Hence $v$ is not star-shaped with respect to $x^*$ according to Proposition \ref{prop:SSequivalence} and we have obtained our contradiction.
\end{proof}
\begin{remark}\label{rmk:prop_upper_ss}
This formula corresponds to the classic ray tracing algorithm to find the visibility set. We trace the values towards the boundary along rays from $x^*$ taking the maximum of $g$ along the way.
\end{remark}
\section{PDEs and Visibility}\label{sec:PDE
In this section, we present the new PDE formulation of visibility sets from a single viewpoint and its extension to multiple viewpoints.
We start with a level set PDE interpretation for star-shaped envelopes.
Given that the inner star-shaped envelope of a set corresponds to its visibility set, the PDE obtained computes the visibility set for each sublevel set. We then generalize it to multiple viewpoints.
\subsection{Viscosity Solutions
Viscosity solutions \cite{CIL} provide the correct notion of weak solution to a class of degenerate elliptic PDEs which includes the PDEs considered here.
We review it briefly here.
Let $S^n$ be the set of real symmetric $n\times n$ matrices, and take $N \leq M$ to denote the usual partial ordering on $S^n$, namely that $N-M$ is negative semi-definite.
\begin{definition}
The operator $F(x,r,p,M):\Omega\times \mathbb{R}\times \mathbb{R} \times S^n \to \mathbb{R}$ is degenerate elliptic if
\[
F(x, r, p, M) \leq F(x, s, p, N) \quad \text{whenever } r \leq s \text{ and } N \leq M.
\]
\begin{remark}
For brevity we use the notation $F[u](x) \equiv F(x, u(x), \grad u(x), D^2u(x))$.
\end{remark}
\end{definition}
\begin{definition}[Upper and lower semi-continuous envelopes]
The upper and lower semicontinuous envelopes of a function $u(x)$ are defined, respectively, by
\begin{align*}
u^*(x) & = \limsup_{y\to x} u(y),\\
u_*(x) & = \liminf_{y\to x} u(y).
\end{align*}
\end{definition}
\begin{definition}[Viscosity solutions]
Let $F:\Omega \times \mathbb{R} \times \R^n$. We say the upper semi-continuous (lower semi-continuous) function $u:\Omega\to\mathbb{R}$ is a viscosity subsolution (supersolution) of $F[u] = 0$ in $\Omega$ if for every $\phi \in C^1(\Omega)$, whenever $u-\phi$ has a local maximum (minimum) at $x\in\Omega$,
\[
F(x, u(x), \grad \phi(x)) \leq 0 \ (\geq 0).
\]
Moreover, we say u is a viscosity solution of $F[u] = 0$ if $u$ is both a viscosity sub- and supersolution.
\end{definition}
\begin{remark}
For brevity we use the notation $F[u](x) \equiv F(x,u(x),\grad u(x))$. In addition, when checking the definition of a viscosity solution we can limit ourselves to considering unique, strict, global maxima (minima) of $u-\phi$ with a value of zero at the extremum. See, for example, \cite[Prop 2.2]{KoikeViscosity}.
\end{remark}
\subsection{Regularity
We briefly discuss the regularity of the star-shaped envelopes. We start by observing that the star-shaped functions need not be continuous.
\begin{example}
In one dimension, the function $u(x)=1$ for $x\neq 0$ and $u(0)=0$ is star-shaped with respect to $0$.
In two dimensions, take $A = \{(x,y)\in\mathbb{R}^2: xy = 0\}$ and define $u$ as the characteristic function of the complement of $A$, i.e., $u(x) = 0$ if $x \in A$ and $u(x) = 1$ otherwise.
Once again $u$ is star-shaped with respect to the origin, but it is not continuous.
In fact, $u$ is lower semicontinuous.
\end{example}
As for the star-shaped envelopes of functions, the upper star-shaped envelope is continuous while the lower star-shaped envelope is only lower semicontinuous as it may be discontinuous at $x^*$.
\begin{proposition}\label{prop:continuity}
Let $g \in C(\Omega)$ be bounded by below. Then $w = SS^-(g)$ is lower semicontinuous in $\Omega$ and continuous in $\Omega$ except at $x^*$, while $u = SS^+(g)$ is continuous in $\Omega$. If $x^*$ is a global minimum of $g$ then $w = SS^-$ is also continuous in $\Omega$.
\end{proposition}
\begin{proof}
The proof follows from the solutions formulas \eqref{eq:lowerss_formula} and \eqref{eq:explicit_visibility} since we take the minimum and maximum of a continuous function $g$ along rays to and from $x^*$, respectively, as pointed out in Remarks \ref{rmk:prop_lower_ss} and \ref{rmk:prop_upper_ss}.
\end{proof}
\begin{example}\label{ex:regularity}
Let $g(x) = \abs{x+1}$ and let $x^* = 0$. Then the lower and upper star-shaped envelopes are given by
\[
w(x) = \begin{cases}
-x-1 & \text{if } x < -1,\\
0 & \text{if } -1 \leq x \leq 0,\\
x+1 & \text{if } x>0,
\end{cases}
\quad \text{and} \quad u(x) = \begin{cases}
-x-1 & \text{if } x < -2,\\
1 & \text{if } -2 \leq x \leq 0,\\
x+1 & \text{if } x>0.
\end{cases}
\]
A two-dimensional example is given in Figure~\ref{fig:Ex1Envelope}: $g$ is given by the distance to two points and $x^*$ is chosen as a point on the $.3$ level set of $g$.
The upper and lower envelopes are pictured.
The lower one is discontinuous at $x^*$.
Replacing $g$ with $\max(g,g(x^*)$ leads to a function whose global minimum is attained at $x^*$ and therefore both star-shaped envelopes are continuous in this case. This is depicted in Figure~\ref{fig:Ex1Envelope2}.
\end{example}
\begin{figure}[h]
\subfigure{\includegraphics[width=0.31\textwidth]{contour_g}}
\subfigure{\includegraphics[width=0.31\textwidth]{contour_lower_envelope}}
\subfigure{\includegraphics[width=0.31\textwidth]{contour_upper_envelope}}
\caption{Contour plot of $g$ given in Example \ref{ex:regularity} (left), its lower star-shaped envelope $w$ (center) and its upper (visibility) envelope $u$ (right). The point $x^*$ is marked by $*$. The lower star-shaped envelope is discontinuous.}
\label{fig:Ex1Envelope}
\end{figure}
\begin{figure}[h]
\subfigure{\includegraphics[width=0.31\textwidth]{contour_g_regular}}
\subfigure{\includegraphics[width=0.31\textwidth]{contour_upper_envelope_regular}}
\subfigure{\includegraphics[width=0.31\textwidth]{contour_lower_envelope_regular}}
\caption{Contour plot of $\max(g,g(x^*)$ where $g$ given in Example \ref{ex:regularity} (left), its lower star-shaped envelope $w$ (center) and its upper (visibility) envelope $u$ (right). The point $x^*$ is marked by $*$. Since $x^*$ is a minimizer of $\max(g,g(x^*))$ both star-shaped envelopes are continuous.}
\label{fig:Ex1Envelope2}
\end{figure}
\subsection{A local PDE for star-shaped envelopes
We start by establishing a first order condition for star-shaped functions.
\begin{proposition}\label{prop:SSfirstorder}
Suppose $u:\R^n\to\mathbb{R}$ is differentiable. Then $u$ is star-shaped with respect to $x^*$ if and only if $\grad u(x) \cdot (x-x^*) \geq 0$ for all $x\in\mathbb{R}$.
\end{proposition}
\begin{proof}
By the mean value theorem, given any $t\in[0,1]$ there exists $s\in (0,t)$ such that
\[
u(tx^* + (1-t)x)-u(x) = t \grad u(x+s(x^*-x)) \cdot (x^*-x).
\]
Thus if $u$ is star-shaped with respect to to $x^*$ we obtain using Lemma \ref{lemma:SSequivalence}
\[
\grad u(x+s(x^*-x)) \cdot (x^*-x) \leq 0.
\]
Taking the limit as $t\to 0$ leads to the desired inequality $\grad u(x) \cdot (x-x^*) \geq 0$.
Now, suppose that $\grad u(x) \cdot (x-x^*) \geq 0$ for all $x \in \R^n$. We argue by contradiction. Assume that $u$ is not star-shaped with respect to $x^*$. Hence, by Lemma \ref{lemma:SSequivalence}, there are $y \in \Omega$ and $t\in[0,1]$ such that
\[
u(t x^* + (1-t)y) > u(y).
\]
Then, again by the mean value theorem,
\[
\grad u(y+s(x^*-y)) \cdot (x^*-y) > 0
\]
for some $s\in (0,t)$. However, taking $x = y+s(x^*-y)$ in $\grad u(x) \cdot (x-x^*) \geq 0$ leads to
\begin{align*}
& \grad u(y+s(x^*-y)) \cdot (y+s(x^*-y)-x^*) \geq 0\\
\Longleftrightarrow & \grad u(y+s(x^*-y)) \cdot (y-x^*)(1-s) \geq 0\\
\Longleftrightarrow & \grad u(y+s(x^*-y)) \cdot (y-x^*) \geq 0\\
\Longleftrightarrow & \grad u(y+s(x^*-y)) \cdot (x^*-y) \leq 0.
\end{align*}
We have the desired contradiction and so the proof is complete.
\end{proof}
We are interested in star-shaped functions that may not be differentiable since typically the visibility set has corners.
Thus we need to characterize star-shaped functions in the weak sense. We will do so using viscosity solutions. Throughout the rest of this section let $\Omega$ be a bounded star-shaped domain with respect to $x^* \in \R^n$ and denote its boundary by $\partial \Omega$.
\begin{proposition}\label{prop:SSviscosityequivalence}
Suppose $u\in USC(\Omega)$. Then $u$ is star-shaped with respect to $x^*$ if and only if $u$ is a viscosity subsolution of $\grad u(x) \cdot (x^*-x) = 0$.
\end{proposition}
\begin{proof}
Suppose $u$ is star-shaped with respect to $x^*$. Let $\phi \in C^1(\Omega)$ be such that $u-\phi$ has a local maximum at $x \in \Omega$. Without loss of generality assume that $\phi(x)=u(x)$. Then we have $u(y) \leq \phi(y)$ in a neighborhood of $x$. Since $u$ is star-shaped with respect to $x^*$, $u$ is increasing along arrays from $x^*$ to $x$ by Proposition \ref{prop:SSequivalence} and therefore
\[
u(x) \leq u(x+h(x-x^*))
\]
for $h > 0$. Hence, given the choice of $\phi$,
\[
\phi(x) \leq \phi(x+h(x-x^*))
\]
for $h > 0$ sufficiently small. Since
\[
\frac{\phi(x+h(x-x^*))-\phi(x)}{h} = \grad \phi(x+s(x^*-x)) \cdot (x-x^*)
\]
for $s \in (0,h)$, we obtain $\grad\phi(x) \cdot (x^*-x) \leq 0$ as $h \to 0$. This shows that $u$ is a viscosity subsolution of $\grad u(x) \cdot (x^*-x) = 0$.
Suppose now that $u$ is a viscosity subsolution of $\grad u(x) \cdot (x^*-x) = 0$ and that $u$ is not star-shaped with respect to $x^*$.
Then there exists $y,z$ such that $u(y) \geq u(x^*)$ and $u(y) > u(z)$ with $y$ lying on the line segment from $x^*$ to $z$.
Without loss of generality assume that $y = \argmax_x u(x)$.
We can then construct a linear $\phi \in C^1(\Omega)$ such that $\grad \phi = (u(z)-u(y))(z-y)$ and $u-\phi$ has a local maximum at $x$ with $(x-x^*) \cdot (z-y) > 0$.
But then
\[
\grad\phi(x) \cdot (x-x^*) = (u(z)-u(y)) (z-y)\cdot (x-x^*) < 0
\]
which contradicts our assumption.
\end{proof}
We can now finally write the PDEs for the upper and lower star-shaped envelopes of $g$ with respect to $x^*$.
\begin{proposition}\label{prop:LowerEnv}
Let $g \in C(\Omega)$ be bounded by below and let $w = SS^-(g)$ be the lower star-shaped envelope $w$ of $g$ with respect to $x^*$. Assume $w$ is continuous. Then $w$ is the viscosity solution of the obstacle problem
\[
\max\{w(x)-g(x), (x^*-x)\cdot \grad w(x)\} = 0
\]
along with boundary conditions $w = g$ on $\partial \Omega$.
\end{proposition}
\begin{proof}
By Proposition \ref{prop:LowerEnvExplicit}, $w$ is star-shaped with respect to $x^*$ and therefore we can write
\[
w(x) = \sup \{ v(x) \mid \text{ $v\in USC(\Omega)$ is star-shaped with respect to $x^*$ and $v \leq g$}\}.
\]
Now, according to Proposition \ref{prop:SSviscosityequivalence}, $w$ is precisely the supremum of all subsolutions of the PDE and so the proof follows directly from Perron's method.
\end{proof}
\begin{proposition}\label{prop:UpperEnv}
Let $g\in C(\Omega)$ and let $u = SS^+(g)$ be the upper star-shaped envelope (visibility) of $g$ with respect to $x^*$. Then $u$ is the viscosity solution of the obstacle problem
\begin{equation}\label{PDE:visibility}
\min\{u(x)-g(x), (x-x^*)\cdot \grad u(x)\} = 0,
\end{equation}
along with $u(x^*) = g(x^*)$.
\end{proposition}
\begin{proof}
By definition the upper star-shaped envelope is given by
\[
u = \inf \{ v(x) \mid v(y) \geq g(y) \text{ for all }y, \text{ $v$ is star-shaped with respect to $x^*$} \}.
\]
We observe that this is equivalent to
\[
-u = \sup \{ v(x) \mid v(y) \leq -g(y) \text{ for all }y, \text{ $-v$ is star-shaped with respect to $x^*$} \}\\
\]
and so $-u$ is the solution of
\[
\max\{U-(-g),-(x^*-x)\cdot \grad U(x)\} = 0,
\]
by a similar reasoning to the one in Proposition \ref{prop:LowerEnv}. Now, since the equation can be rewritten as
\[
\min\{-U-g,(x^*-x)\cdot \grad U(x)\} = 0,
\]
we conclude that $u$ is the solution of
\[
\min\{u-g,(x-x^*)\cdot \grad u(x)\} = 0
\]
as desired.
\end{proof}
\begin{remark}
Contrary to Proposition \ref{prop:LowerEnv} we do not need to assume the continuity of the start-shaped envelope as this follows directly from the assumptions on $g$ (see Proposition \ref{prop:continuity}).
\end{remark}
\subsection{Comparison Principle
An important property of elliptic equations, from which uniqueness follows, is the comparison principle that states that subsolutions lie below supersolutions. In addition, it also plays a crucial role when establishing the convergence of approximation schemes using the the theory of Barles and Souganidis \cite{BSnum}, which we intend to do later on. However, in such setting, the comparison principle required is a strong comparison principle: The boundary conditions are satisfied in the viscosity sense. In general, such a comparison is only available when the solutions are continuous up to the boundary (see \cite{CIL} for more details).
We focus our attention in PDE \eqref{PDE:visibility} as it is the one we are most interested in: Its solution allows us to determine the visibility set. A strong comparison principle is however not satisfied: Notice that any function that is nonincreasing along any direction away from $x^*$ is a subsolution and thus given any supersolution we can always construct a subsolution that lies above it by adding a large enough constant. We can however circumvent this by requiring that the subsolution $u$ satisfies $u(x^*) \leq g(x^*)$. This will also prove to be enough to establish convergence of our numerical scheme. Intuitively, imposing that the subsolution $u$ satisfies $u(x^*) \leq g(x^*)$ guarantees that $u$ lies below $g$. This, together with the fact that the solution of \eqref{PDE:visibility} is the infimum of all supersolutions according to Perron's method, is enough to reach the desired conclusion.
\begin{proposition}\label{prop:comparison}
Let $g \in C(\Omega)$. Let $u$ be a viscosity subsolution of \eqref{PDE:visibility} such that $u(x^*) \leq g(x^*)$ and let $v$ be a supersolution of \eqref{PDE:visibility}. Then $u \leq v$ in $\Omega$.
\end{proposition}
\begin{proof}
Suppose there exists $x \in \Omega$ such that $u(x) > v(x)$ by contradiction. Since $v$ is a supersolution \eqref{PDE:visibility},
\[
v(x) \geq SS^+(g)(x) = \max \{ g(y) \mid y = x^* + t(x-x^*) \in \Omega, t\in[0,1]\}
\]
where we used Perron's characterization of $SS^+(g)$ and Proposition \ref{prop:UpperEnvExplicit}. In particular, we have $v(x) \geq g(x)$ and $v(x) \geq g(x^*)$. Hence $u(x) > g(x)$ and $u(x) >~u(x^*)$ by assumption on $u$. Therefore there exists a linear function $\phi$ such that $u-\phi$ has a local maximum at $y$ with $u(y) > g(y)$ and $(y-x^*)\cdot \grad \phi(y) = u(x)-u(x^*)> 0$. We have derived a contradiction with the assumption that $u$ is a subsolution of \eqref{PDE:visibility} and the proof is complete.
\end{proof}
\subsection{Visibility from multiple viewpoints
We are now interested in the visibility set from multiple viewpoints where a point is consider visible if is seen by at least one viewpoint. We follow the same ideas as before, starting by generalizing the definition of star-shaped set.
\begin{definition}\label{def:SSone}
We say that a set $S \subset \R^n$ is star-shaped with respect to $\{x^*_1,\ldots,x^*_r\}$ if
\[
y \in S \implies \exists_{x^*\in\{x^*_1,\ldots,x^*_r\}} \forall_{t\in[0,1]} \: ty + (1-t)x^* \in S.
\]
\end{definition}
As in \autoref{sec:SS} we can define star-shaped functions with respect to $\{x^*_1,\ldots,x^*_n\}$ according to Definition \ref{def:SSone}. More importantly, Proposition \ref{prop:SSfirstorder} can be generalized.
\begin{proposition}
Suppose $u:\R^n\to\mathbb{R}$ is differentiable and bounded by below. Then $u$ is star-shaped with respect to $\{x^*_1,\ldots,x^*_r\}$ according to Definition \ref{def:SSone} if and only if
\[
\max_{i=1,\ldots,r} \min_{t\in[0,1]} \grad u(x^*_i+t(x-x^*_i)) \cdot (x-x^*_i) \geq 0.
\]
\end{proposition}
\begin{remark}
The presence of the minimum in $t$ may not appear obvious at first, but it is crucial here. In order for a point $x$ to be visible from $x^*_i$ then $u$ must be increasing along the ray from $x^*_i$ to $x$ which guarantees that all the points along the ray will be in the visible set. Without the minimum in $t$ a point $x$ could be consider visible by first moving along a ray towards $x^*_i$ and then follow a different viewpoint.
\end{remark}
In this case, $u = SS^+_{\{x^*_1,\ldots,x^*_r\}}(g)$ is the solution of the following PDE
\begin{equation}\label{PDE:visibility_alo}
\min\{u(x)-g(x), \max_{i = 1,\ldots,r}\min_{t\in[0,1]} \grad u(x^*_i+t(x-x^*_i)) \cdot (x-x^*_i)\} = 0.
\end{equation}
Despite being a non-local PDE, unlike the previous ones with a single viewpoint $x^*$, we still have a fast solver available: The solution $u$ is given by
\[
u = \min_{i = 1,\ldots,r} u_i,
\]
where $u_i$ is the solution of
\[
\min\{u(x)-g(x), (x-x^*_i)\cdot \grad u(x)\} = 0.
\]
This follows from noticing that
\[
SS^+_{\{x^*_1,\ldots,x^*_r\}}(g) = \min_{i = 1,\ldots,r} SS^+_{x^*_i}(g).
\]
In addition, the following solution formula is also available
\[
u(x) = \min_{i = 1,\ldots,r}\max \{ g(y) \mid y \in x^*_i + t(x-x^*_i) \in \Omega, t\in[0,1]\}.
\]
At this point, a natural question to ask is what other visibility definitions will lead to PDEs following the approach taken here. For instance, given $3$ viewpoints $\{x^*_1,x^*_2,x^*_3\}$ a point can be considered visible if (i) it is seen by at least two viewpoints; (ii) it is seen by $x^*_1$ or both $x^*_2$ and $x^*_3$ (iii) it is seen by all viewpoints. All these can be computed efficiently computed by first determining the visibility of each viewpoint and then taking the appropriate combination of maximums and minimums. However, determining a corresponding PDE following the approach considered here will in general fail. Consider for instance case (iii) where a point is visible if it is seen by all viewpoints. In this setting, one can no longer define star-shaped like sets as the visibility set may be disconnected (see Figure \ref{fig:ex_multiple_viewpoints} for a simple example with two viewpoints). The fundamental difference is that it is no longer true that if a point $x$ is visible then all points along the ray from $x$ to the viewpoint are visible.
\begin{figure}[htp]
\centering
\subfigure{\includegraphics[width=0.48\textwidth]{ex_visibility_alo}}
\subfigure{\includegraphics[width=0.48\textwidth]{ex_visibility_all}}
\caption{Visibility set from multiple viewpoints when a point is visible if it seen by at least one viewpoint (left) and by all the viewpoints (right).}
\label{fig:ex_multiple_viewpoints}
\end{figure}
\section{Convergent finite difference schemes}\label{sec:scheme
In this section, we discuss the numerical schemes used to solve the PDEs introduced in the previous section, focusing our attention on the visibility PDE \eqref{PDE:visibility}.
As we will see, the scheme proposed here is degenerate elliptic finite difference schemes for which there exists a well established convergence framework.
Before we begin we introduce some notation. For simplicity, we will assume we are working on the hypercube $[-1, 1]^n \subseteq \R^n$. We write $x = (x_1, \ldots, x_n) \in [-1,1]^n$. The domain is discretized with a uniform grid, resulting in the following spatial resolution:
\[
h \equiv \frac{2}{N-1},
\]
where $N$ is the number of grid points used to discretize $[-1, 1]$. We denote by $\Omega^h$ the computational domain which in our case reduces to $[-1, 1]^n \cap h\mathbb{Z}^n$.
Our schemes are written as the operators $F^h[u] : C(\Omega^h) \to C(\Omega^h)$, where $C(\Omega^h)$ is the set of grid functions $u : \Omega^h \to \mathbb{R}$. We assume they have the following form:
\[
F^h[u](x) = F^h(u(x), u(x) - u(\cdot)) \quad \text{for } x \in \Omega^h
\]
where $u(\cdot)$ corresponds to the value of u at points in $\Omega^h$.
\subsection{Finite difference for the visibility PDE
For simplicity, we present the scheme in the two-dimensional setting. The generalization to higher dimensions is straightforward.
Let $x \in \Omega^h$. Define the vector $v^* = x-x^*$ for $x\in\Omega^h$. If $x$ is one of the four grid points enclosing $x^*$, let $\tilde{x} = x^*$. Otherwise, let $\tilde{x}$ denote the intersection between the line that passes through $x$ and $x^*$ and a line segment formed by 2 of the 8 neighbors of $x$.
Then $u_{v^*}(x) = (x-x^*) \cdot \grad u(x)$ and its upwind approximation is given by
\[
u^h_v(x) \equiv \frac{u(x)-\mathcal{I}_h u(\tilde{x})}{\abs{x-\tilde{x}}},
\]
where $\mathcal{I}_h$ is the piecewise linear Lagrange interpolant. The numerical scheme for the visibility PDE \eqref{PDE:visibility} is then given by
\begin{equation}\label{scheme:visibility}
F^h[u](x) = \min\{u(x)-g(x), u^h_v(x)\}, \quad x \in \Omega^h.
\end{equation}
\subsection{Fast sweeping solver
We implement a fast sweeping solver to compute the solutions of \eqref{scheme:visibility}. Solving the equation $F^h[u](x) = 0$ for the reference variable, $u(x)$, leads to the update formula
\[
u(x) = \max\{g(x),\mathcal{I}_h u(\tilde{x})\},
\]
where $\tilde{x}$ was defined in the previous section. Since all characteristic are straight lines and flow away from $x^*$, the domain $\Omega^h$ only needs to be swept once (but in a very specific ordering). For simplicity, we will assume that $x^* \in \Omega^h$. Let $u_{i,j}$ denote the solution at $x_{i,j}$ where
\[
x_{i,j} = (-1+(i-1)h,-1+(j-1)h).
\]
Set $(i^*,j^*)$ such that $x_{i^*,j^*} \leq x^* < x_{i^*+1,j^*+1}$ (here the inequalities are interpreted component-wise). Set as well $u_{i^*,j^*} = g(x^*)$. The domain is then divided into four quadrants and each is swept in the following way:
\begin{itemize}
\item $i = i^*,\ldots,N$, $j = j^*,\ldots, 1$ (sweeping bottom right square)
\item $i = i^*,\ldots,N$, $j = j^*,\ldots, N$ (sweeping top right square)
\item $i = i^*+1,\ldots,1$, $j = j^*,\ldots, 1$ (sweeping bottom left square)
\item $i = i^*+1,\ldots,1$, $j = j^*,\ldots, N$ (sweeping top left square)
\end{itemize}
\begin{remark}
Alternatively, the solver can be seen as a direct consequence of the solution formula \eqref{eq:explicit_visibility}.
\end{remark}
\section{Convergence of Numerical Solutions}\label{sec:convergence
In this section we recall the notion of degenerate elliptic schemes and show that the solutions of the proposed numerical scheme converges to the solutions of \eqref{PDE:visibility} as the discretization parameter tends to zero. The standard framework used to establish convergence is that of Barles and Souganidis \cite{BSnum}, which we state below. In particular, it guarantees that the solutions of any monotone, consistent, and stable scheme converge to the unique viscosity solution of the PDE.
\subsection{Degenerate elliptic schemes
Consider the Dirichlet problem for the degenerate elliptic PDE, $F[u] = 0$, and recall its corresponding finite difference formulation:
\[
\begin{cases}
F_h(x,u(x), u(x)-u(\cdot)) = 0, & x \in \Omega^h,\\
u(x)-g(x) = 0, & x \in \partial \Omega^h,
\end{cases}
\]
where $h$ is the discretization parameter.
\begin{definition}
$F_h[u]$ is a degenerate elliptic scheme if it is non-decreasing in each of its arguments.
\end{definition}
\begin{remark}
Although the convergence theory in \cite{BSnum} is originally stated in terms of monotone approximation schemes (schemes with non-negative coefficients), ellipticity is an equivalent formulation for finite difference operators \cite{ObermanSINUM}.
\end{remark}
\begin{definition}
The finite difference operator $F_h[u]$ is consistent with $F[u]=0$ if for any smooth function $\phi$ and $x\in\Omega$,
\begin{align*}
\lim_{h \to 0,y \to x, \xi \to 0} F_h(y,\phi(y)+\xi,\phi(y)-\phi(\cdot)) = F(x,\phi(x),\nabla \phi(x)).
\end{align*}
\end{definition}
\begin{definition}
The finite difference operator $F_h[u]$ is stable if there exists $M > 0$ independent of $h$ such that if $F_h[u] = 0$ then $\norm{u}_\infty \leq M$.
\end{definition}
\begin{remark}[Interpolating to the entire domain]
The convergence theory assumes that the approximation scheme and the grid function are defined on all of $\Omega$. Although the finite difference operator acts only on functions defined on $\Omega^h$, we can extend such functions to $\Omega^h$ via piecewise linear interpolation. In particular, performing piecewise linear interpolation maintains the ellipticity of the scheme, as well as all other relevant properties. Therefore, we can safely interchange $\Omega$ and $\Omega^h$ in the discussion of convergence without any loss of generality
\end{remark}
\subsection{Convergence of numerical approximations
Next we will state the theorem for convergence of approximation schemes, tailored to elliptic finite difference schemes, and demonstrate that the proposed scheme fits in the desired framework. In particular, we will show that the schemes are elliptic, consistent, and have stable solutions.
\begin{proposition}[Convergence of approximation schemes \cite{BSnum}]\label{prop:convergence}
Let $u$ denote the unique viscosity of the degenerate elliptic PDE $F[u]=0$ with Dirichlet boundary conditions for which there exists a strong comparison principle. For each $h$, let $u^h$ denote the solutions of $F^h[u]=0$, where the finite difference scheme $F_h[u]$ is a consistent, stable and elliptic scheme. Then $u^h \to u$ locally uniformly on $\Omega$ as $h \to 0$.
\end{proposition}
Now, we check that our scheme is consistent, stable and elliptic.
\begin{lemma}[Consistency]\label{lemma:consistency}
The scheme is consistent.
\end{lemma}
\begin{proof}
It is sufficient to show that
\[
(x-x^*) \cdot \grad u(x) = u^h_v(x) + \mathcal{O}(h)
\]
which follows immediately from a Taylor expansion argument and the definition of $u^h_v$.
\end{proof}
\begin{lemma}[Stability]\label{lemma:stability}
Suppose $g$ is bounded. Then the scheme is stable.
\end{lemma}
\begin{proof}
Since the solution $u^h$ satisfies
\[
u^h(x) = \max\{g(x),\mathcal{I}_h u^h(\tilde{x})\}, \quad x \in \Omega^h,
\]
it follows that $g(x^*) \leq u^h(x) \leq \norm{g}$.
\end{proof}
\begin{lemma}[Ellipticity]\label{lemma:ellipticity}
The scheme is elliptic.
\end{lemma}
\begin{proof}
The term $u(x)-g(x)$ is trivially elliptic. Since $\tilde{x}$ belongs to the line segment of two grid points, $\mathcal{I}_h u(\tilde{x})$ is a convex combination of neighboring grid values and therefore $u^h_v$ is elliptic. Hence $F^h$, being the minimum of two elliptic schemes, is also elliptic.
\end{proof}
\begin{theorem}
Suppose $g \in C(\Omega)$. Then the solutions of \eqref{scheme:visibility} converge locally uniformly on $\Omega$ as $h\to 0$ to the unique viscosity solution of \eqref{PDE:visibility} along with $u(x^*) = g(x^*)$.
\end{theorem}
\begin{proof}
By Lemmas \ref{lemma:consistency}, \ref{lemma:stability} and \ref{lemma:ellipticity}, we see that $F^h$ \eqref{scheme:visibility} is consistent, stable and elliptic.
In order to apply Proposition \ref{prop:convergence}, our PDE must satisfy a strong comparison principle which is not the case here. However, inspecting the proof of Proposition \ref{prop:convergence} in \cite{BSnum}, one notices that it is enough to show that $\overline{u} \leq \underline{u}$ in $\Omega$, where
\[
\overline{u}(x) = \limsup_{h\to 0,y\to x} u^h(x) \text{ and } \underline{u}(x) = \liminf_{h\to 0,y\to x} u^h(x).
\]
This will follow from Proposition \ref{prop:comparison} by proving that $\overline{u}(x^*) \leq g(x^*)$. Indeed, due to the continuity of $g$,
\begin{align*}
\overline{u}(x^*) & = \limsup_{h\to 0,y\to x^*} u^h(x^*)\\
& = \lim_{\epsilon\to 0} \sup\{u^h(y): y \in B(x^*,\epsilon)\setminus\{x^*\}, 0 < h < \epsilon\}\\
& \leq \lim_{\epsilon\to 0} \sup_{B(x^*,\epsilon)} g(x)\\
& = g(x^*).
\end{align*}
\end{proof}
\section{Numerical results}\label{sec:numerics
In this section we present the numerical results. We focus on solving equation \eqref{PDE:visibility} as it is the one we are mainly interested since the sublevel sets of its solution provide us with the visibility set. We present results both in two and three dimensions.
\begin{example}\label{ex_convergence}
We start with a simple example to show a numerical convergence test. We take $x^*=(-1,-1)$ as the viewpoint and consider as the obstacle function the cone $g(x) = -\sqrt{x_1^2+x_2^2}$ and therefore each level-set of $g$ corresponds to a circle with center at the origin and a different radius. The exact solution was obtained using \eqref{eq:explicit_visibility}. The difference between the numerical solution and the exact solution in the $l_\infty$ norm is presented in Table \ref{table:ex_convergence}, where we also confirm the expected first order convergence. Moreover, in Figure \ref{fig:ex_convergence}, we plot the level sets of both $g$ and the numerical solution, as well as their respective surface plots. As we can see, the visibility set is computed for each level set of $g$.
\end{example}
\begin{figure}[htp]
\centering
\subfigure{\includegraphics[width=0.31\textwidth]{ex_convergence_levelsets}}
\subfigure{\includegraphics[width=0.31\textwidth]{ex_convergence_surfplot_g}}
\subfigure{\includegraphics[width=0.31\textwidth ]{ex_convergence_surfplot_u}}
\caption{Level sets of the obstacle $g$ and the solution $u$ of \eqref{PDE:visibility} (left) and the respective surface plots (center and right).}
\label{fig:ex_convergence}
\end{figure}
\begin{table}[h]
\centering
\begin{tabular}{cccc}
N & $h$ & Error & Order\\
\hline
32 & \num{1.29e-01} & \num{9.12e-02} & - \\
64 & \num{6.35e-02} & \num{4.49e-02} & 1.02 \\
128 & \num{3.15e-02} & \num{2.23e-02} & 1.01 \\
256 & \num{1.57e-02} & \num{1.11e-02} & 1.01 \\
512 & \num{7.83e-03} & \num{5.54e-03} & 1.00 \\
1024 & \num{3.91e-03} & \num{2.76e-03} & 1.00 \\
2048 & \num{1.95e-03} & \num{1.38e-03} & 1.00 \\
4096 & \num{9.77e-04} & \num{6.91e-04} & 1.00 \\
\hline
\end{tabular}
\caption{Errors and order of convergence for Example \ref{ex_convergence}.}
\label{table:ex_convergence}
\end{table}
\begin{example}\label{ex2}
In this example we are interested in computing the visibility set where we have four different obstacles: two squares with centers $(-1.5,-0.2)$, $(0,0.3)$ and side lengths $0.5$, $1$ respectively and two circles with origins $(-0.3,1.5)$, $(-0.3,-1.4)$ and radius $0.5$. We achieve this by considering the obstacle function
\[
g(x) = -\min\left\{g_1(x),g_2(x),g_3(x),g_4(x)\right\}
\]
where
\begin{align*}
g_1(x) & = 2\max\{\abs{x_1+1.5},\abs{x_2+0.2}\},\\
g_2(x) & = \max\{\abs{x_1},\abs{x_2-0.3}\},\\
g_3(x) & = \sqrt{(x_1+0.3)^2+(x_2-1.5)^2},\\
g_4(x) & = \sqrt{(x_1+0.3)^2+(x_2+1.4)^2},
\end{align*}
and looking into the $0.5$ level set. We first solve \eqref{PDE:visibility} with $x^* = (-1.5,-1.4)$ and $x^* = (1.5,-0.3)$. We compute as well the visibility set with respect to $\{x^*_1,x^*_2\} = \{(-1.5,-1.4),(1.5,-0.3)\}$ when a point is consider visible if it is seen by any of the viewpoints and by both viewpoints simultaneous. The former corresponds to the solution of \eqref{PDE:visibility_alo}. All results are displayed in Figure \ref{fig:ex2}.
\end{example}
\begin{figure}[htp]
\centering
\subfigure{\includegraphics[width=0.48\textwidth]{ex2_visibility_1}}
\subfigure{\includegraphics[width=0.48\textwidth]{ex2_visibility_2}}
\subfigure{\includegraphics[width=0.48\textwidth]{ex2_visibility_alo}}
\subfigure{\includegraphics[width=0.48\textwidth]{ex2_visibility_all}}
\caption{Results for Example \ref{ex2}: solution to \eqref{PDE:visibility} with $x^* = (-1.5,-1.4)$ (top-right); solution to \eqref{PDE:visibility} with $x^* = (1.5,-0.3)$ (top-left); solution to \eqref{PDE:visibility_alo} with $\{x^*_1,x^*_2\} = \{(-1.5,-1.4),(-1.5,-1.4)\}$ (bottom-left), i.e., set of points visible by any of the viewpoints; set of points visible by both viewpoins (bottom-right). All visibility sets are displayed in red, while the obstacles are displayed in black.}
\label{fig:ex2}
\end{figure}
\begin{example}\label{ex3}
We consider a simple three-dimensional example where the obstacle function is given by
\[
g(x) = -\min\{g_1(x),g_2(x)\}
\]
where
\[
g_1(x) = \max(\abs{x_1+2},\abs{x_2},x_3)-1 \quad\text{and}\quad g_2(x) = \max(\abs{x_1-3}),\abs{x_2-4},x_3)-2.
\]
It can be interpreted as the visibility set of a $360^\circ$ camera in the middle of two buildings.
The results are displayed in Figure \ref{fig:ex3}.
\end{example}
\begin{figure}[htp]
\centering
\subfigure{\includegraphics[width=0.48\textwidth]{ex3_obstacles}}
\subfigure{\includegraphics[width=0.48\textwidth]{ex3_visibility}}
\caption{Results for Example \ref{ex3}. The obstacles are displayed in blue, while the contour of the visibility set is displayed in red.}
\label{fig:ex3}
\end{figure}
\begin{example}\label{ex4}
The Stanford 3D Scanning Repository provides a dataset of 35947 distinct points that form what is known as the ``Stanford Bunny''. In this example we considered it as the obstacle by taking the function $g$ to be the signed distance function to the dataset points. The results are displayed in Figure \ref{fig:ex4}.
\end{example}
\begin{figure}[htp]
\centering
\subfigure{\includegraphics[width=0.48\textwidth,trim={2cm 3cm 5cm 1cm},clip]{ex4_obstacles}}
\subfigure{\includegraphics[width=0.48\textwidth,trim={2cm 3cm 5cm 1cm},clip]{ex4_visibility}}
\caption{Results for Example \ref{ex4}. The obstacles are displayed in blue, while the contour of the visibility set is displayed in red.}
\label{fig:ex4}
\end{figure}
\section{Conclusions
In this article, we described a new simpler PDE to compute the visibility set from a given viewpoint given a set of known obstacles. We proposed a finite difference numerical scheme to compute its solution and showed its convergence. We discuss the generalization of the result to multiple viewpoints and present a PDE for the visibility set where a point is visible if it is seen by at least one of many viewpoints. Numerical examples of different visibility sets computed as the solution of the new proposed PDE in both two and three dimensions are presented.
\section*{Acknowledgments}
This material is based upon work supported by the Air Force Office of Scientific Research under award number FA9550-18-1-0167.
The second author thanks the hospitality of the Mathematics and Statistics department of McGill University during its visit where the work for this paper was carried out.
\bibliographystyle{amsalpha}
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1702.00844
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\section{Introduction}
A recent development in radio astronomy is to build instruments where
traditional dishes are replaced with many small and simple
omni-directional antennas. The signals of the antennas are combined to
form one large virtual telescope. Examples include current and future
instruments such as LOFAR (LOw Frequency Array)~\cite{ppopp2010},
MeerKAT (Karoo Array Telescope)~\cite{meerkat},
ASKAP (Australian Square Kilometre Array Pathfinder)~\cite{askap},
and SKA (Square Kilometre Array)~\cite{ska}. These new generation telescopes produce
enormous data streams. The data streams from the different antennas
must be cross-correlated to filter out noise. The correlation process
also performs a data reduction by integrating samples over time. The
correlation step is especially challenging, since the computational
demands grow \emph{quadratically} with the number of data streams. The
correlator is extremely demanding, since it is not only
computationally intensive, but also very data intensive. In the
current field of radio astronomy, the number of operations that has to
be performed per byte of I/O is exceptionally small.
For astronomy, high-performance computing is of key
importance. Instruments like LOFAR are essentially software
telescopes, requiring massive amounts of compute power and data
transport capabilities. Future instruments, like the SKA~\cite{ska},
need in the order of exaflops of computation, and petabits/s of I/O.
Traditionally, the online processing for radio-astronomy instruments
is done on special-purpose hardware. A relatively recent
development is the use of supercomputers~\cite{spaa-06,ppopp2010}. Both
approaches have several important disadvantages. Special-purpose
hardware is expensive to design and manufacture and, equally
important, it is inflexible. Furthermore, the process from creating a
hardware design and translating that into a working implementation
takes a long time. Solutions that use a supercomputer (e.g., a Blue
Gene/P in the LOFAR case) are more flexible~\cite{ppopp2010}, but are
expensive to purchase, and have high maintenance and electrical power
cost. Moreover, supercomputers are not always well-balanced for our
needs. For instance, most supercomputers feature highly efficient
double-precision operations, while single precision is sufficient for
our applications.
\begin{figure*}[t]
\begin{center}
\includegraphics[width=12cm]{processing-overview.eps}
\end{center}
\caption{An extremely simplified view of LOFAR processing.}
\label{fig-processing-overview}
\end{figure*}
In this paper, we investigate the correlation algorithm on many-core
hardware, such as graphics processors (GPUs)~\cite{gpgpu-hardware} and
the \mbox{Cell/B.E.}~\cite{cell}. In contrast to many others, we do not
only use NVIDIA GPUs, but also include ATI hardware. In addition, we
compare with the LOFAR production implementation on a Blue Gene/P
supercomputer~\cite{ppopp2010}. As a reference, we also include
multi-core general-purpose processors. There are
many advantages to the use of many-core systems: it is a flexible
software solution, has lower costs in terms of purchase and maintenance,
and the power usage is significantly lower than that of a traditional supercomputer.
The correlator differs from applications that were
investigated on many-core hardware in the past, because of the correlator's low flop/byte
ratio. In addition, it is a streaming real-time application, so host-to-device
data transfers are on the critical path. In many other studies, these
transfers are not considered.
The production correlator on the Blue Gene/P achieves 96\% of the
theoretical peak performance. We demonstrate that the processing
power and memory bandwidth of current GPUs are highly imbalanced for
correlation purposes. This leads to suboptimal performance. Still, GPUs are
considerably more power efficient than the BG/P (4x for NVIDIA, and 2.4x for ATI).
\shortversion{
The \mbox{Cell/B.E.} processor, in contrast, achieves an excellent 92\% efficiency. The \mbox{Cell/B.E.}
runs the correlator 4-6 times (depending on the manufacturing process of the \mbox{Cell/B.E.}) more energy
efficiently than the Blue Gene/P.
The research presented in this paper is an
important pathfinder for next-generation telescopes.
}
\longversion{
The
correlator achieves only only 8\% of the peak performance on ATI GPUs,
and 18\% on NVIDIA GPUs.
The \mbox{Cell/B.E.} processor, in contrast,
achieves an excellent 92\% efficiency. The \mbox{Cell/B.E.} is also the most
energy-efficient solution. It runs the correlator 5 times more energy
efficiently than the Blue Gene/P.
The research presented in this paper is an
important pathfinder for next-generation telescopes.
}
The rest of this paper is structured as follows.
Section~\ref{correlation-alg} explains how the correlation algorithm
works, and why it is important. In Section~\ref{hardware}, we describe
the many-core architectures that we evaluate in detail, finishing
with a comparison and discussion. Next, in Section~\ref{sec:perf}, we
explain how we implemented the correlator algorithm on each of these
architectures, and describe the performance we achieve. In
Section~\ref{sec:perf-compare}, we evaluate, compare, and discuss the
results, while identifying the weak and strong points of the
architectures. Section~\ref{related} discusses related work. In
Section~\ref{sec:discussion}, we investigate if our results and
insights can be applied to other applications. Additionally, we discuss scalability
issues. Finally, we conclude in Section~\ref{conclusions}.
\section{Correlating Radio Astronomy Signals}
\label{correlation-alg}
We call a set of receivers that are grouped closely
together a \emph{station}. The data streams from the different
stations must be filtered, delays in the signal path must be
compensated for, and the data streams from different stations must be
cross-correlated. The correlation process performs a data reduction by
integrating samples over time. In this paper, we use the LOFAR
telescope as an example, but the results apply equally well
to other instruments. An overview of the processing needed for the
standard imaging pipeline of LOFAR is shown in
Figure~\ref{fig-processing-overview}. The pipeline runs from left to right.
The thickness of the lines indicates the size of the data streams.
In this paper, we focus on the
correlator step (the gray box in
Figure~\ref{fig-processing-overview}), because its costs grow
quadratically with the number of stations. All other steps have a
lower time complexity. We choose 64 as the number of stations, since
that is a realistic number for LOFAR. Future instruments will likely
have even more stations. We call the combination of two stations a
\emph{baseline}. The total number of baselines is $(nrStations \times
(nrStations + 1)) / 2$, since we need each pair of correlations only
once. This includes the autocorrelations (the correlation of a station with itself),
since we need this later in the pipeline for calibration purposes.
Although the autocorrelations can be computed with fewer instructions, we
ignore this here, since the number of autocorrelations is small, and grows
linearly with the number of stations, while the number of normal correlations
grows quadratically.
The correlator algorithm itself is straightforward, and can be
written in a single formula:
$C_{s_1,s_2\geq s_1,p_1\in\{X,Y\},p_2\in\{X,Y\}} = \displaystyle\sum_{t} Z_{s_1,t,p_1} * Z_{s_2,t,p_2}^\ast$
\begin{figure*}[t]
\lstset{language=C,basicstyle={\fontencoding{T1}\fontfamily{pcr}\fontseries{m}\fontshape{n}\fontsize{7}{10pt}\selectfont}}
\begin{lstlisting}{}
for (ch=0; ch<nrChannels; ch++)
for (station2=0; station2<nrStations; station2++)
for (station1=0; station1<=station2; station1++)
for (pol1 = 0; pol1 < nrPolarizations; pol1++)
for (pol2 = 0; pol2 < nrPolarizations; pol2++) {
complex float sum = 0 + i*0;
for (time=0; time < integrationTime; time++) {
sum += samples[ch][station1][time][pol1]
* ~samples[ch][station2][time][pol2];
}
baseline = computeBaseline(station1, station2);
correlation[baseline][ch][pol1][pol2] = sum;
}
\end{lstlisting}
\caption{Pseudo code for the correlation algorithm.}
\label{correlator-code}
\end{figure*}
Pseudo code for the algorithm is shown in Figure~\ref{correlator-code}.
A sample is a ($2 \times 32-bit$) complex number that represents the
amplitude and phase of a signal at a particular time. The receivers
are polarized; they take separate samples from orthogonal (X and Y)
directions. The received signals from sky sources are so weak, that the antennas
mainly receive noise. To see if there is statistical coherence
in the noise, simultaneous samples of each pair of stations are correlated,
by multiplying the sample of one station with the complex
conjugate (i.e., the imaginary part is negated) of the sample of the other station.
To reduce the output size, the products are integrated, by accumulating all products.
For the LOFAR telescope, we accumulate 768 correlations at 763 Hz,
so that the integration time is approximately one second.
This is much shorter than for current telescopes. The short integration time
leads to more output data.
Since the correlation of station A and B is
the complex conjugate of the correlation of station B and A, only
one pair is computed.
Stations are also autocorrelated, i.e., with
themselves. Both polarizations of a station A are correlated with both polarizations
of a station B, yielding correlations in XX, XY, YX, and YY
directions.
The correlator is mostly multiplying and adding complex numbers.
\longversion{
A complex multiplication of two complex numbers $a$ and $b$ can be written as follows, where $a_r$ is the real part of
$a$, while $a_i$ is the imaginary part: \\
\noindent $(a_r + a_ii)(b_r + b_ii)$ \\
\noindent For a correlation, we need to multiply a sample from station A with
the complex conjugate (i.e., the imaginary part is negated) from
station B, and add it to an accumulator to integrate over time.
}
\longversion{.
\noindent $acc = acc + (a_r + a_ii)(b_r - b_ii)$ \\
\noindent This the same as: \\
\noindent $acc = acc + a_r b_r - a_r b_ii + a_ii b_r - a_ii b_ii$ \\
\noindent The last term ($- a_ii b_ii$) is the same as $+ a_i b_i$, since $i^2 = -1$.
Thus, the total term becomes: \\
\noindent $acc = acc + (a_r b_r + a_i b_i) + (-a_r b_i + a_i b_r )i$ \\
\noindent If we split the real and imaginary parts, and add them to the corresponding accumulators, we get: \\
\noindent $acc_r = acc_r + a_rb_r + a_ib_i$ \\
\noindent $acc_i = acc_i - a_rb_i + a_ib_r$ \\
\noindent Thus, we use both the real and imaginary parts of each sample \emph{two} times.
With the use of fused-multiply add instructions (fma), this becomes: \\
}
We can implement the correlation operation very efficiently, with only
four fma instructions, doing eight floating-point operations in
total. For each pair of stations, we have to do this four times, once
for each combination of polarizations. Thus, in total we need 32
operations and load 8 floats (32 bytes) from memory, resulting in \emph{exactly
one FLOP/byte}. The number of operations that is performed per byte
that has to be loaded from main memory is called the \emph{arithmetic
intensity}~\cite{system-performance}. For the correlation algorithm,
the arithmetic intensity is extremely low.
\begin{figure}[t]
\begin{center}
\includegraphics[width=4.2cm]{correlation-triangle.eps}
\end{center}
\caption{An example correlation triangle.}
\label{fig-correlation}
\end{figure}
An important optimization that we implemented is the reduction
of memory loads by the correlator.
This is achieved
by keeping correlations that are being accumulated in registers, and
by reusing samples that are loaded from memory as many times as
possible.
A sample can be used multiple times by correlating it
with the samples from multiple other stations in the same loop iteration.
For example, a sample from station A in the X polarization
that is loaded into a register pair can be correlated with the X and
Y polarizations of stations B, C and D, using it 6 times.
\longversion{
In fact,
it is used 12 times, since a correlation requires 2 complex, fused
multiply-add instructions.
}
Figure~\ref{fig-correlation} shows how we correlate multiple
stations at the same time. Each square represents the XX, XY,
YX, and YY correlations of the stations as indicated by row and
column number. The figure is triangular, because we compute
the correlation of each pair of stations only once. The squares labeled \emph{A} are
autocorrelations, which could be treated specially since they require less
computations. The triangle is divided into larger tiles, in this case
2x3 tiles (the dark gray boxes), but arbitrary sizes are possible.
A tile is correlated as a unit. For example, the lower
right-hand-side rectangle correlates stations 9 and 10 with stations
0, 1, and 2.
It is important to tune the tile size to the architecture. We want to
make the tile size as large as possible, while still fitting in the
register file. This offers the highest level of data reuse.
If we have a $w \times h$ tile size, the number of operations is given by $flops = 32wh$.
The number of bytes that has to loaded from memory is $16(w+h)$.
The minimum number of registers that is required is $4 (1 + min(w,h)) + 8 w h$.
This is the total number of registers, including accumulators, while reusing
registers if a value is no longer needed (hence the $min$ operation). However,
this formula does not count additional registers that could be needed for data prefetching,
address calculations and loop counters.
The number of registers is expressed in single-precision float registers. If an architecture has vector
registers, the result can be divided by the vector length.
Table~\ref{tile-size-table} shows the properties of different tile sizes.
Despite the division of the correlation triangle in tiles, there
still is opportunity for additional data reuse \emph{between} tiles.
The tiles
within a row or column in the triangle still need the same samples.
In
addition to registers, caches can thus also be used to increase data
reuse. Since we know exactly what data can be reused at what moment, we
found it is important to have direct influence on the caches and the thread scheduler. This
way, we can make sure that tiles in same row or column are calculated
at the same time by different threads.
Because the algorithm is
extremely data intensive, the resulting optimized implementation on
many-cores is typically limited by the architecture's memory
bandwidth. The memory aspects of the algorithm are twofold.
There is an algorithmic part, the tile size, which is limited
by the number of registers. The second aspect is architectural in nature: the cache
sizes, cache hierarchy and hit ratio. Together, these two aspects dictate the
memory bandwidth that is needed to keep the ALUs busy.
\begin{table}
\caption{Properties of different tile sizes.}
\label{tile-size-table}
{\small
\begin{tabular}{l|r|r|r|r}
tile & floating point & memory loads & arithmetic & minimum nr. \\
size & operations & (bytes) & intensity & registers (floats) \\
\hline
1x1 & 32 & 32 & 1.00 & 16 \\
1x2 & 64 & 48 & 1.33 & 24 \\
1x4 & 128 & 80 & 1.60 & 40 \\
2x2 & 128 & 64 & 2.00 & 44 \\
3x2 & 192 & 80 & 2.40 & 60 \\
3x3 & 288 & 96 & 3.00 & 88 \\
4x3 & 384 & 112 & 3.43 & 112 \\
4x4 & 512 & 128 & 4.00 & 148 \\
\end{tabular}
}
\end{table}
In this paper, we focus on the maximal performance that can be achieved
with a \emph{single many-core chip}. It is important to realize that the
correlator itself is \emph{trivially parallel}, since tens of thousands of
frequency channels can be processed independently. This allows us to
efficiently exploit many-core hardware. We use floating point instead
of integer operations, since all architectures support this
well. Single precision floating point is accurate enough for our
purposes.
Since the code is relatively straightforward, we implemented the
performance-critical kernel in assembly on all
architectures. Therefore, this paper really compares the hardware
architectures; \emph{compilers do not influence the
performance}. Although we wrote the critical parts in assembly, the
additional code was written in the natural programming model for each
architecture. Therefore, we also take programmability into
account.
We do this both for the assembly parts, and the high-level code.
The first is a measure of the programmability of the hardware architecture itself.
The second gives an indication of the quality of the software stack.
\section{Many-Core Hardware}
\label{hardware}
Recent many-core architectures present enormous computational
performance at very low costs. For example, the recently introduced
ATI 4870x2 GPU has 1600 compute cores, and achieves a theoretical
performance of 2.4 teraflops, all on a single PCI card. This GPU costs
less than 500 dollars, resulting in an unprecedented price/performance
ratio. The number of cores in the accelerators also increases rapidly. In only a
few years, the number of cores has increased from about 16, e.g., in
the NVIDIA GeForce~6 series in 2004, to the 1600 cores in the ATI
4870x2, that became available in 2008. In only 4 years, a 100-fold
increase in the number of cores has been realized. During this time,
the theoretical peak performance increased from 12~gigaflops to
2.4~teraflops, a growth of a factor 200.
Reductions in power costs become increasingly
important. For LOFAR, for example, a large part of the operational
costs is the electrical power consumption. Current supercomputers
already focus on power efficiency. The IBM Blue Gene/P, for
instance, can perform 384 gflops/kW~\cite{bgp}, which is significantly better
than current general-purpose systems. Nevertheless, modern many-core
accelerators are, in theory, even more efficient. For example, an NVIDIA GTX280
GPU has a theoretical peak performance of 3952 gflops/kW, more than ten times
better than a Blue Gene/P. The IBM Roadrunner system~\cite{roadrunner}, which is based
on the \mbox{Cell/B.E.} processor, became the first super computer to achieve one
petaflops. The three most energy efficient supercomputers on the
Green500 list\footnote{See http://www.green500.org.}, are similarly based on the \mbox{Cell/B.E.}
Although the number of cores and the theoretical peak performance have
increased dramatically, the memory bandwidth did not increase equally
fast. In fact, the available memory bandwidth \emph{per core} is
decreasing for many-core architectures at the moment. This introduces
significant performance bottlenecks for data intensive
applications~\cite{gpgpu-hardware,larrabee}. The important question
is: How much of the theoretical performance can be reached in
practice? In this paper we answer this question for correlation
process, we extrapolate our results to data-intensive applications in
general. The increase of both the number of cores and the peak
performance is much larger for the accelerators than for normal
CPUs. This makes many-core accelerators interesting for
high-performance computing. In the remainder of this section, we
discuss several many-core architectures in detail, and conclude with a
summary and discussion of the differences that are essential for the
correlator, and for data-intensive applications in general.
\subsection{General Purpose multi-core CPU (Intel Core i7 920)}
As a reference, we implemented the correlator on a multi-core general
purpose architecture. We use a quad core Intel Core~i7 920 CPU
(code name Nehalem) at 2.67~GHz.
\longversion{
The chip is manufactured with a 45~nm
process. The on-chip L1 cache has 32 KB for
instructions, and 32~KB for data per core.
There is 256 KB L2 cache (combined instruction and data) per core, and
8 MB L3, shared by all cores.
The system uses a 4.8 GT/s QuickPath interface, and has a thermal design power of 80~Watts.
}
\shortversion{
There is 32~KB of on-chip L1 data cache per core, 256~KB L2 cache per core, and 8~MB
of shared L3 cache.
The thermal design power (TDP) is 130~Watts.
}
The theoretical
peak performance of the system is 85~gflops, in single precision.
The parallelism comes from four cores with two-way hyperthreading, and a vector length of four floats,
provided by the SSE4 instruction set.
The architecture has several
important drawbacks for our application. First, there is no fused
multiply-add instruction. Since the correlator performs mostly
multiplies and adds, this can cause a performance penalty. The
processor does have multiple pipelines, and the multiply and add
instructions are executed in different pipelines, allowing eight
flops per cycle per core.
Another problem is that SSE's shuffle instructions
to move data around in vector registers are more limited
than for instance on the \mbox{Cell/B.E.} processor. This complicates an
efficient implementation.
For the future Intel Larrabee GPU, and for the next
generation of Intel processors, both a fused multiply-add instruction
and improved shuffle support has been announced.
The number
of SSE registers is small (sixteen 128-bit registers), allowing only little
data reuse.
\longversion{At most 64 single-precision floats can be kept in registers.}
This is a problem for the correlator, since
the tile size is limited by the number of registers. A smaller tile
size means less opportunity for data reuse, increasing the memory
bandwidth that is required.
\subsection{IBM Blue Gene/P}
The IBM Blue Gene/P~(BG/P)~\cite{bgp} is the architecture that is
currently used for the LOFAR correlator~\cite{ppopp2010}. Four 850~MHz
PowerPC~450 processors are integrated on each Blue Gene/P chip. We
found that the BG/P is extremely suitable for our application, since
it is highly optimized for processing of complex numbers. The BG/P
performs \emph{all} floating point operations in double precision, which is
overkill for our application.
The L2 prefetch unit prefetches the sample data efficiently from
memory. In contrast to all other architectures we evaluate, the
problem is compute bound instead of I/O bound, thanks to the BG/P's high
memory bandwidth per operation. It is 3.5--10 times higher than
for the other architectures.
The ratio between flops and bytes/sec of
memory bandwidth is exactly 1.0 for the BG/P.
The BG/P has a register file with 32 vector registers of width 2.
Therefore, 64 floating point numbers (with double precision) can be
kept in the register file simultaneously. This is the same amount as
on the general purpose Intel chip, but an important difference is that the
BG/P has 32 registers of width 2, compared to Intel's 16 of width 4.
The smaller vector size reduces the amount of shuffle instructions
needed.
\longversion{
Also, the available memory bandwidth per operation is much
higher on the Blue Gene/P, so this is not a problem.
}
The BG/P is an energy efficient supercomputer. This is
accomplished by using many small, low-power chips, at a low clock
frequency. The supercomputer also has excellent I/O capabilities,
there are five specialized networks for communication.
\subsection{ATI 4870 GPU (RV 770)}
\shortversion{
The most high-end GPU provided by ATI (recently acquired by AMD) is the 4870~\cite{amd-manual}.
The RV770 processor in the 4870 runs at 750 MHz, and has a thermal design
power of 160 Watts.
The RV770 chip has ten SIMD cores, each containing 16
superscalar streaming processors. Each streaming
processor has five independent scalar ALUs. Therefore, the GPU
contains 800 ($10 \times 16 \times 5$) scalar 32-bit streaming processors. The
Ultra-Threaded Dispatch Processor controls how the execution units
process streams.
The theoretical peak performance is 1.2~teraflops.
The 4870 has 1~GB of GDDR5 memory with a theoretical bandwidth of 115.2~GB/s.
The board uses a PCI-express~2.0 interface for communication with
the host system.
Each of the ten SIMD cores contains 16 KB of local memory and separate L1
texture cache.
The L2 cache is shared. The
maximum L1 bandwidth is 480 GB/sec. The bandwidth between the L1 and
L2 Caches is 384 GB/sec. The application can specify if a read should be cached or not.
The SIMD cores can exchange data using 16 KB of global
memory.
}
\longversion{
ATI (recently acquired by AMD) GPUs can also be used for high-performance
computing~\cite{amd-manual}. We use the ATI 4870 GPU, which currently is the most
high-end product. The RV770 processor in the 4870 has 956 million
transistors with 55 nm process technology, and has a thermal design
power of 160 Watts. the RV770 has 800 unified shader processor
cores, running at 750 MHz, with a theoretical peak performance of 1.2~teraflops
in total.
The 4870 has 1~GB of GDDR5 memory clocked at 900~MHz,
with a 256-bit interface. The theoretical memory bandwidth is
115.2~GB/s.
The board uses a PCI-express~2.0 interface for communication with
the host system.
The RV770 chip has ten SIMD cores, each containing 16 blocks of
superscalar streaming processors (i.e., 160 in total). Each streaming
processor has five independent scalar ALUs. Therefore, the GPU
contains 800 ($160 \times 5$) scalar 32-bit streaming processors. The
Ultra-Threaded Dispatch Processor controls how the execution units
process streams.
Together, the streaming processors can execute five FMA (Fused Multiply Add)
instructions per cycle. One of five ALUs can execute a double
precision floating point instruction, or a more complex instruction
such as SIN, COS, LOG, EXP, etc. The processor also contains a branch
execution unit that offloads the ALUs and reduces performance
penalties caused by jump instructions.
Each of the ten SIMD cores contains 16 KB of local memory, separate L1
texture cache and four texture units. Each texture unit contains an
address processor and four 32-bit texture fetch units. Thus, there
is one fetch unit for five ALUs.
The maximal resolution of textures is
$8192\times8192$. This limitation is reflected in the programming models
that AMD provides: arrays are also maximally $8192\times8192$ large.
The L2 cache is shared, and tied to four 64-bit memory channels. The
maximum L1 bandwidth is 480 GB/sec. The bandwidth between the L1 and
L2 Caches is 384 GB/sec. The application can specify if a read should be cached or not.
The SIMD cores can exchange data using 16 KB of global
memory.
}
The ATI 4870 GPU has the largest number of cores of all architectures
we evaluate (800). However, the architecture has several important
drawbacks for data-intensive applications. First, the
host-to-device bandwidth is too low. In practice, the achieved
PCI-express bandwidth is far from the theoretical limit. We will
explain this in more detail in Section~\ref{perf-ati}. The achieved
bandwidth is not enough to keep all cores busy. Second, we found that
overlapping communication with computation by performing asynchronous
data transfers between the host and the device has a large impact on
kernel performance. We observed kernel slowdowns of \emph{a factor of
three} due to transfers in the background. Third, the architecture
does not provide random write access to device memory, but only to
\emph{host} memory. However, for our application which is mostly
read-performance bound, this does not have a large impact (see
Section~\ref{perf-ati}).
\subsection{NVIDIA GPU (Tesla C1060)}
\shortversion{
NVIDIA's Tesla C1060 contains a GTX~280 GPU (code-named GT200), is
manufactured using a 65 nm process, and has 1.4 billion
transistors. The device has 30 cores (called multiprocessors) running
at 1296 MHz, with 8 single precision ALUs, and one double precision
ALU per core. Current NVIDIA GPUs thus have fewer cores than ATI
GPUs, but the individual cores are faster. The memory architecture is
also quite different. NVIDIA GPUs still use GDDR3 memory, while ATI
already uses GDDR5 with the 4870~GPU. The GTX~280 in the Tesla
configuration has 4~GB of device memory, and has a thermal design
power of 236 Watts. The theoretical peak performance is 933 gflops.
The number of registers is large: there are 16384 32-bit floating
point registers per multiprocessor. There also is 16~KB of shared memory per multiprocessor.
This memory is shared between all threads on a multiprocessor, but not globally.
There is a total amount of 64 KB of constant memory on the chip.
Finally, texture caching hardware is available.
NVIDIA only specifies that ``the cache working set for texture memory
is between 6 and 8 KB per multiprocessor''~\cite{cuda-manual}.
The application has some control over the caching
hardware. It is possible to specify which area of device
memory must be cached, while the shared memory is completely
managed by the application.
On GPUs, it is possible to synchronize the threads within a multiprocessor.
With our application, we exploit this to increase the cache hit
ratio. This improves performance considerably on NVIDIA hardware, but not on ATI hardware.
When accessing device memory, it is important to make sure that simultaneous
memory accesses by different threads are \emph{coalesced} into a
single memory transaction.
In contrast to ATI hardware, NVIDIA GPUs support random write access
to device memory. This allows a programming model that is much closer
to traditional models, greatly simplifying software development.
The NVIDIA GPUs suffer from a similar
problem as the ATI GPUs: the host-to-device bandwidth is equally
low.
}
\longversion{
The GTX~280 GPU (code-named GT200) is manufactured with a 65 nm
process, and has 1.4 billion transistors and 30 cores (called multiprocessors)
at 1296 MHz, with 8 single precision ALUs, and one double precision ALU per core.
Current NVIDIA GPUs thus have fewer cores than ATI GPUs, but the
individual cores are faster. The memory architecture is also quite
different. NVIDIA GPUs still use GDDR3 memory, while ATI already uses
GDDR5 with the 4870~GPU. The GTX~280 has 1~GB of device memory,
and has a thermal design power of 236 Watts.
The theoretical peak performance is 933 gflops.
The number of registers is large: there are 16384 32-bit floating
point registers per multiprocessor.
These are shared between all threads in the multiprocessor.
Also, there is a total amount of
64 KB of constant memory on the chip. Additionally, there is 16 KB of
shared memory (SRAM) available per multiprocessor. Finally,
texture caching hardware is available.
The exact specifications are not disclosed,
but it is believed that each multiprocessor has a dedicated L1 cache
which is likely 16~KB and a shared L2 cache of 256~KB~\cite{beyond3d}.
NVIDIA only specifies that ``the cache working set for texture memory
is between 6 and 8 KB per multiprocessor''~\cite{cuda-manual}.
Threads are grouped in blocks, where threads inside a block run on the
same multiprocessor. The application has some control over the caching
hardware. First, it is possible to specify which area of device
memory must be cached by the texture cache, and whether a 1D or 2D
layout should be used. Second, the shared memory is completely
managed by the application. The memory is shared between threads in
the same thread block (at most 512 threads), but not globally between
blocks. It is possible to synchronize the threads within a
block. With our application, we exploit this to improve the cache hit
ratio. This improves performance considerably.
The shared memory is
organized into 16 banks, which can be accessed simultaneously. If
threads access data in the same bank, this leads to a bank conflict,
and access is serialized. If all threads access the same address, the
data is broadcast.
NVIDIA GPUs are capable of reading 32-bit, 64-bit, or 128-bit words from global
memory into registers in a single instruction. For our application, this means that
we can load a complex sample with two polarizations with one instruction.
When accessing device memory, it is important to make sure that simultaneous
memory accesses by different threads are \emph{coalesced} into a
single memory transaction. With more recent
hardware (with compute capability 1.2 or higher), the restrictions on
the access patterns that can be coalesced are loosened considerably.
If all threads access memory that lies within the same 128-byte segment,
the access will be coalesced.
In contrast to ATI hardware, NVIDIA GPUs support random write access
to device memory. This allows a programming model that is much closer
to traditional models, greatly simplifying software development.
The NVIDIA GPUs suffer from similar architectural
problems as the ATI GPUs. The host-to-device bandwidth is equally
low, and we found that memory transfers also have a large influence on
kernel performance.
}
\begin{table}
\caption{Differences between many-core memory architectures.}
\label{memory-properties}
{\small
\begin{tabular}{l|l|l}
feature & Cell/B.E. & GPUs \\
\hline
\hline
access times & uniform & non-uniform \\
\hline
cache sharing level & single thread (SPE) & all threads in a \\
& & multiprocessor \\
\hline
access to off-chip memory & not possible, only through DMA & supported \\
\hline
memory access & asynchronous DMA & hardware-managed \\
overlapping & & thread preemption \\
\hline
communication & communication between & independent thread blocks \\
& SPEs through EIB & + shared memory within a block \\
\end{tabular}
}
\end{table}
\subsection{The Cell Broadband Engine (QS21 blade server)}
\shortversion{
The Cell Broadband Engine (\mbox{Cell/B.E.})~\cite{cell} is a heterogeneous many-core
processor, designed by Sony, Toshiba and IBM (STI).
The \mbox{Cell/B.E.} has nine cores: the Power Processing Element
(PPE), acting as a main processor, and eight Synergistic Processing
Elements (SPEs) that provide the real processing power. All cores run at 3.2 GHz.
The cores, the main memory, and the external I/O are connected by a
high-bandwidth (205 GB/s) Element Interconnection Bus (EIB).
The main memory has a high-bandwidth (25 GB/s), and uses XDR (Rambus).
The PPE's main role is
to run the operating system and to coordinate the SPEs.
An SPE contains a RISC-core (the Synergistic Processing Unit (SPU)),
a 256KB Local Store (LS), and a memory flow controller.
The LS is an extremely fast local
memory (SRAM) for both code and data and is managed entirely by the
application with explicit DMA transfers. The LS can be considered
the SPU's L1 cache.
The LS bandwidth is 47.7 GB/s per SPU.
The \mbox{Cell/B.E.} has a large number of registers: each SPU has 128, which are
128-bit (4 floats) wide. The theoretical peak performance of one SPU
is 25.6 single-precision gflops.
The SPU can dispatch two instructions in each clock cycle using
the two pipelines designated \emph{even} and
\emph{odd}. Most of the arithmetic instructions execute on the even
pipe, while most of the memory instructions execute on the odd pipe.
We use a QS21 Cell blade with two \mbox{Cell/B.E.} processors and 2 GB
main memory (XDR). This is divided into 1 GB per processor.
A single \mbox{Cell/B.E.} in our system has a TDP of 70~W.
Recently, an equally fast version with a 50~W TDP has been announced.
The 8 SPEs of a single chip in the system have a total theoretical single-precision peak performance of 205 gflops.
}
\longversion{
The Cell Broadband Engine (\mbox{Cell/B.E.}) is a heterogeneous multi-core
processor, initially designed by Sony, Toshiba and IBM (STI) for the
PlayStation 3 (PS3) game console. Given the \mbox{Cell/B.E.}’s peak
performance of 204 single precision gflops~\cite{cell-network}, it was
quickly considered a good target platform for HPC
applications. \mbox{Cell/B.E.} has nine cores: the Power Processing Element
(PPE), acting as a main processor, and eight Synergistic Processing
Elements (SPEs) that provide the real processing power. All cores run at 3.2 GHz.
The cores, the main memory, and the external I/O are connected by a
high-bandwidth Element Interconnection Bus (EIB).
The EIB has an
extremely high performance, the aggregate bandwidth is 204.8 GB/s.
The main memory has a high-bandwidth (25 GB/s), and uses XDR (Rambus).
The PPE contains the Power Processing Unit (PPU), a 64-bit PowerPC
core with a VMX/AltiVec unit, separated L1 caches (32KB for data and
32KB for instructions), and 512KB of L2 Cache.
The PPE’s main role is
to run the operating system and to coordinate the SPEs.
An SPE contains a RISC-core (the Synergistic Processing Unit (SPU)),
a 256KB Local Storage (LS), and
a memory flow controller. The LS is an extremely fast local
memory (SRAM) for both code and data and is managed entirely by the
application with explicit DMA transfers. The LS can be considered as
the SPU's L1 cache. With the DMA transfers, random write access to
memory is available. Sixteen bytes can be transferred between the LS
and the SPU registers per cycle. The bandwidth thus is 47.7 GB/s per SPU.
All SPU instructions operate on 128-bit quantities. The \mbox{Cell/B.E.} has
a large number of registers: each SPU has 128 registers, which are
128-bit (4 floats) wide. The theoretical peak performance of one SPU
is 25.6 single-precision gflops.
The SPU can dispatch two instructions in each clock cycle using
the two pipelines designated \emph{even} and
\emph{odd}. Most of the arithmetic instructions execute on the even
pipe, while most of the memory instructions execute on the odd pipe.
The Cell/B.E was originally manufactured with a 90 nm process. This
version has a TDP of approximately 100 W. The version we use in this
paper is a more recent version that uses a 65 nm process, which
reduced the TDP to about 70 W. Recently, a 45 nm version has also
been introduced, reducing the power requirements even further, to a
TDP of 50 W.
Although the smaller manufacturing size would allow for
higher clock frequencies (more than 6 GHz has been reported), IBM
chose to keep the frequencies the same, leading to a much better flop
per Watt ratio.
In May 2008, an Opteron- and \mbox{Cell/B.E.}-based supercomputer, the IBM
Roadrunner system~\cite{roadrunner}, became the world's first system
to achieve one petaflops. The roadrunner features a newer version of
the \mbox{Cell/B.E.} processor, the PowerXCell~8i. This version is
manufactured with a 65 nm process, and adds support for up to 32GB of
DDR2 memory, as well as dramatically improving double-precision
floating-point performance on the SPEs. The world's three most energy
efficient supercomputers, as represented by the Green500 list, are
similarly based on the PowerXCell~8i.
In this paper, we use a QS21 Cell blade, a second-generation blade
system. The QS21 features two 3.2 GHz \mbox{Cell/B.E.} processors and 2 GB
main memory (XDR). This is divided into 1 GB per processor. The SPEs
in the system have a total theoretical peak performance of 410 gflops.
}
\subsection{Hardware Comparison and Discussion}
\label{hardware-comparision}
The memory architectures of the many-core systems are of particular
interest, since our application is mostly memory-throughput
bound (as will be discussed in Section~\ref{sec:perf}).
Table~\ref{memory-properties} shows some key differences of the memory
architectures of the many-core systems.
\longversion{
The caching
architectures of the many-core systems show several remarkable
differences.
}
Both ATI and NVIDIA GPUs have a hardware L1 and L2
cache, where the application can control which memory area is cached,
and which is not. The GPUs also have shared memory, which is
completely managed by the application.
\longversion{
This memory is shared between
threads in the same block, but not globally between blocks.
}
Also,
coalescing and bank conflicts have to be taken into account, at the
cost of significant performance penalties~\cite{cuda-manual}. Therefore, the
memory access times are \emph{non-uniform}. The access
times of the local store of the \mbox{Cell/B.E.}, in contrast, are completely
uniform (6 cycles). Also, each \mbox{Cell/B.E.} SPE has its own \emph{private} local store, there
is no cache that is shared between threads. While the GPUs can
directly access device memory, the \mbox{Cell/B.E.} does not provide access to
main memory. All data has to be loaded and stored into the local store
first. Also, the way that is used to overlap memory accesses
with computations is different. The \mbox{Cell/B.E.} uses
asynchronous DMA transfers, while the GPUs use hardware-managed thread
preemption to hide load delays. Finally, the SPEs of the \mbox{Cell/B.E.} can
communicate using the Element Interconnection Bus, while the multiprocessors
of a GPU execute completely independently.
\begin{table}[t]
\caption{Properties of the different many-core hardware platforms. For the Cell/B.E., we consider the local store to be L1 cache.}
\label{architecture-properties}
{\small
\begin{tabular}{l|l|l|l|l|l}
& Intel & IBM & ATI & NVIDIA & STI \\
Architecture & Core i7 & BG/P & 4870 & Tesla C1060 & Cell/B.E.\\
\hline
cores x FPUs per core & 4x4 & 4x2 & 160x5 & 30x8 & 8x4 \\
operations per cycle per FPU & 2 & 2 & 2 & 2 & 2 \\
Clock frequency (GHz) & 2.67 & 0.850 & 0.75 & 1.296 & 3.2 \\
\textbf{gflops per chip} & \textbf{85} & \textbf{13.6} & \textbf{1200} & \textbf{936} & \textbf{204.8}\\
\hline
registers per core x register width & 16x4 & 64x2 & 1024x4 & 2048x1 & 128x4 \\
\hline
total L1 data cache size per chip (KB) & 32 & 128 & ??? & ??? & 2048 \\
total L1 cache bandwidth (GB/s) & ??? & 54.4 & 480 & ??? & 409.6 \\
total device RAM bandwidth (GB/s) & n.a. & n.a. & 115.2 & 102 & n.a. \\
\textbf{total host RAM bandwidth (GB/s)} & \textbf{25.6} & \textbf{13.6} & \textbf{8.0} & \textbf{8.0} & \textbf{25.8}\\
\hline
Process Technology (nm) & 45 & 90 & 55 & 65 & 65 \\
TDP (W) & 130 & 24 & 160 & 236 & 70 \\
\textbf{gflops / Watt (based on TDP)} & \textbf{0.65} & \textbf{0.57} & \textbf{7.50} & \textbf{3.97} & \textbf{2.93}\\
\hline
\textbf{gflops/device bandwidth (gflops / GB/s)}& n.a. & n.a. & \textbf{10.4} & \textbf{9.2} & n.a. \\
\textbf{gflops/host bandwidth (gflops / GB/s)} & \textbf{3.3}& \textbf{1.0} & \textbf{150} & \textbf{117} & \textbf{7.9} \\
\end{tabular}
}
\end{table}
Table~\ref{architecture-properties} shows the key properties of the
different architectures we discuss here. Note that the performance
numbers indicate the \emph{theoretical} peak. The memory bandwidths of the
different architectures show large differences. Due to the PCI-e bus,
the host-to-device bandwidth of the GPUs is low.
The number of gflops per byte of memory bandwidth gives an indication of the performance of the
memory system. A lower number means a better balance between memory
and compute performance. For the GPUs, we can split this number into
a device-to-host component and an internal component. It is clear that
the \emph{relative} performance of the memory system in the Blue
Gene/P system is significant higher than that of all the other
architectures.
The number of gflops that can be achieved per Watt is
an indication of the theoretical power efficiency of the hardware.
In theory, the many-core architectures are more power efficient than
general-purpose systems and the BG/P.
The Bound and Bottleneck analysis~\cite{system-performance,roofline}
is a method to gain insight into
the performance that can be achieved in practice on a particular platform.
Performance is bound \emph{both}
by theoretical peak performance in flops, and the product of the
memory bandwidth and the arithmetic intensity $AI$ (the flop/byte
ratio): \\
$\mathit{perf_{max} = min(perf_{peak}, AI \times memoryBandwidth)}$.
Several important assumptions are made with this method. First, it
assumes that the memory bandwidth is independent of the access
pattern. Second, it assumes a complete overlap of communication and
communication, i.e., all memory latencies are completely
hidden. Finally, the method does not take caches into
account. Therefore, if the correlator can make effective use of the
caching mechanisms, performance can actually be better than
$\mathit{perf_{max}}$. Nevertheless, the $\mathit{perf_{max}}$ gives a rough idea of the
performance than can be achieved.
With the GPUs, there are several communication steps that influence
the performance. First, the data has to be transferred from the host to
the device memory. Next, the data is read from the device memory into
registers.
\shortversion{
Although the GPUs offer high internal memory bandwidths,
the host-to-device bandwidth is limited by the low
PCI-express throughput (8~GB/s for PCI-e~2.0~16X).
In practice, we measured even lower throughputs.
With the NVIDIA GPU, we achieved 5.58~GB/s, and with the ATI GPU 4.62 GB/s.
}
\longversion{
Although the GPUs offer high internal memory bandwidths,
(115.2~GB/s for the ATI~4870, and 141.7~GB/s for the NVIDIA~GTX280)
the bandwidth from the host to the device is limited by the low
PCI-express bandwidth (8~GB/s for PCI-e~2.0~16X).
Therefore, we
investigate the performance that can be achieved both with and without
communication from the host to the device.
In practice, we measured even lower throughputs from the host to the GPUs.
With the NVIDIA GPU, we achieved 5.58~GB/s, and with the ATI GPU 4.62 GB/s.
The throughputs of the different GPUs thus is almost identical, and much lower than
the theoretical limit.
}
For both communication steps, we can compute the arithmetic intensity and the
$\mathit{perf_{max}}$. The sample data must be loaded into the device memory, but
is then reused several times, by the different tiles. We call the
arithmetic intensity from the point of view of the entire computation $AI_{global}$.
The number of flops in the computation is the number of baselines times 32 operations,
while the number of bytes that have to be loaded in total is 16 bytes times the number of stations.
As explained in Section~\ref{correlation-alg}, the number of baselines is $(nrStations \times (nrStations + 1)) / 2$.
If we substitute this, we find that $AI_{global} = nrStations + 1$.
Since we use 64 stations, the $AI_{global}$ is 65 in our case.
The $AI_{local}$ is the arithmetic intensity on the device itself. The value depends on the tile size,
and was described in Section~\ref{correlation-alg}.
For both ATI and NVIDIA hardware, the $\mathit{perf_{max,global}}$ is $65 \times 8.0 =
520$ gflops, if we use the theoretical PCI-e~2.0~16X bandwidth of 8~GB/s.
If
we look at the PCI-e bandwidth that is achieved in practice (4.62 and 5.58 GB/s respectively),
the GPUs have a $\mathit{perf_{max,global}}$ of \emph{only 300 gflops for ATI, and 363 gflops for NVIDIA}.
Since there is no data reuse
between the computations of different frequency channels, this is a
realistic upper bound for the performance that can be achieved, assuming there is no performance penalty for
overlapping device computation with the host-to-device transfers. We
conclude that due to the low PCI-e bandwidth, only a small fraction of
the theoretical peak performance can be reached, even if the kernel
itself has ideal performance.
In the following sections we will
evaluate the performance we achieve with the correlator in detail,
while comparing to $\mathit{perf_{max}}$.
\section{Correlator Implementation and Performance}
\label{sec:perf}
This section describes the implementation of the correlator on
the different architectures.
In many cases, we experimented with
different versions, because it is often unclear beforehand what the
best implementation strategy is.
We evaluate the performance in detail. For comparison reasons, we use the performance
\emph{per chip} for each architecture.
We also calculate the achieved memory bandwidths for all
architectures in the same way. We know the number of bytes that has
to be loaded by the kernel, depending on the tile size that is used.
We divide this by the
execution time of the kernel to calculate the bandwidth.
Thanks to data reuse with caches and local stores, the achieved bandwidth can be \emph{higher}
than the memory bandwidth.
\subsection{General Purpose multi-core CPU (Intel Core i7 920)}
\longversion{
\begin{table}[t]
\begin{center}
\begin{tabular}{l||r|r||r|r}
& \multicolumn{2}{c||}{1 core } &\multicolumn{2}{c}{4 cores} \\
tile & & throughput & & throughput \\
size & gflops & (GB/s) & gflops & (GB/s) \\
\hline
1x1 & 9.2 (51.2\%) & 8.6 (33.6\%) & 35.7 (49.6\%) & 33.3 (130.1\%) \\
2x2 & 11.1 (61.5\%) & 5.2 (20.3\%) & 44.3 (61.5\%) & 20.6 ( 80.5\%) \\
3x2 & 10.7 (59.7\%) & 4.1 (16.0\%) & 43.0 (59.7\%) & 16.7 ( 65.2\%) \\
\end{tabular}
\end{center}
\caption{Performance of the correlator on a core i7 920.}
\label{results-cell}
\end{table}
}
We use the SSE3 instruction set to exploit vector parallelism.
Due to the limited shuffle instructions, computing the correlations of the four
polarizations within a vector is inefficient. We achieve only a speedup of a factor
of 2.8 compared to a version without SSE3.
We found that, unlike on all
other platforms, vectorizing the integration time loop works significantly better.
This way, we compute four samples with subsequent time stamps in a vector.
The use of SSE3 improves the performance by a factor of 3.6 in this case.
In addition, we use multiple threads to utilize all four cores.
To benefit from hyperthreading, we need twice as many threads as cores (i.e., 8 in our case).
Using more threads does not help. Hyperthreading increases performance by 6\%.
The most efficient version uses a tile size of
$2 \times 2$\longversion{and explicitly spills registers to the stack}. Larger tile sizes are
inefficient due to the small SSE3 register file. We achieve a
performance of 48.0 gflops, 67\% of the peak, while using 73\% of the peak bandwidth.
\subsection{IBM Blue Gene/P}
The LOFAR production correlator is implemented on the Blue Gene/P platform.
We use it as the reference for performance comparisons.
\longversion{
For optimal performance, most time-intensive code is written in
assembly, since we could not get satisfactory performance from
compiled C++ code.
}
The (assembly) code hides load and
instruction latencies, issues concurrent floating point, integer, and
load/store instructions, and uses the L2 prefetch buffers in the most
optimal way.
\longversion{
Most instructions are parallel fused multiply-adds,
that sustain four operations per cycle.
We optimally exploited the
large, $32 \times 2$ FPU register file.
}
We use a cell size of $2 \times 2$, since this offers the highest
level of reuse, while still fitting in the register file.
\longversion{
Each tile
requires 16 complex registers to accumulate the correlations. With 32
complex register available, there are 16 left to load the X and Y
samples from the stations. The correlation of multiple stations in the
same loop iteration also helps to hide the 5-cycle instruction latencies of
the fused multiply-add instructions, since the correlations are
independently computed.
}
The performance we achieve with this version is 13.1 gflops per chip,
96\% of the theoretical peak performance.
\longversion{
The Blue Gene/P has enough
memory bandwidth to keep all cores busy.
}
The problem is compute bound, and not I/O bound, thanks to the
high memory bandwidth per flop, as is shown in Table~\ref{architecture-properties}.
For more information, we refer to~\cite{ppopp2010}.
\subsection{ATI 4870 GPU (RV 770)}
\label{perf-ati}
ATI offers two separate programming models, at
different abstraction levels. The low-level programming model is
called the ``Compute Abstraction Layer'' (CAL).
\longversion{
CAL provides
direct access to GPU without needing to learn graphics-specific
programming languages.
}
CAL provides communication primitives and an
intermediate assembly language, allowing fine-tuning of
device performance. For
high-level programming, ATI adopted \emph{Brook}, which was originally developed at
Stanford~\cite{brook}. ATI's extended version is called
\emph{Brook+}~\cite{amd-manual}.
We implemented the correlator both with Brook+ and with CAL.
\longversion{
The maximal resolution of textures is $8192\times8192$. This
hardware limitation is reflected in the programming models that AMD provides:
arrays are also maximally $8192\times8192$ large.
}
With both Brook+ and CAL, the programmer has to do the vectorization, unlike with NVIDIA GPUs.
CAL provides a feature called \emph{swizzling}, which is used to
select parts of vector registers in arithmetic operations.
For
example, it is possible to perform operations like \texttt{mul r0,
r1.xxyy, r2.zwzw}, which selects the $x$ and $y$ components of $r1$
twice, and performs a vector multiplication with the $z$ and $w$
components of $r2$, resulting in the four combinations.
We extensively exploit this feature in the correlator, and
found that this
improves readability of the code significantly.
Unlike the other architectures, the ATI GPUs are not well documented.
Essential information, such as the number of registers, cache sizes,
and memory architecture is missing, making it hard to write
optimized code. Although the situation improved recently, the documentation is still inadequate.
Moreover, the programming tools are insufficient. The
high-level Brook+ model does not achieve acceptable performance for our application. The
low-level CAL model does, but it is difficult to use.
\longversion{
\begin{table}[t]
\begin{center}
{\small
\begin{tabular}{l|r||r|r||r|r||r}
& & \multicolumn{2}{c||}{no communication} &\multicolumn{2}{c||}{with communication} & cache \\
& tile & & throughput & & throughput & hit \\
implementation & size & gflops & (GB/s) & gflops & (GB/s) & ratio \\
\hline
Brook+ regs & 1x1 & ( \%) & ( \%) & ( \%) & ( \%) & unknown \\
Brook+ regs & 2x2 & ( \%) & ( \%) & ( \%) & ( \%) & unknown \\
\hline
CAL regs & 1x1 & 97 ( 8\%) & 91 (79\%) & 75 ( 6\%) & 70 (61\%) & 23\% \\
CAL regs & 2x2 & 163 (14\%) & 76 (58\%) & 88 ( 7\%) & 50 (33\%) & 13\% \\
\hline
CAL mem & 1x1 & 73 ( 6\%) & 68 (59\%) & 59 ( 5\%) & 55 (48\%) & 16\% \\
CAL mem & 2x2 & 171 (14\%) & 80 (69\%) & 110 ( 9\%) & 52 (45\%) & 15\% \\
CAL mem & 3x2 & 182 (15\%) & 71 (61\%) & 113 ( 9\%) & 44 (38\%) & 12\% \\
CAL mem & 3x3 & 296 (25\%) & 92 (80\%) & 147 (12\%) & 46 (40\%) & 27\% \\
CAL mem & 4x3 & 404 (34\%) &110 (95\%) & 171 (14\%) & 47 (41\%) & 42\% \\
\end{tabular}
}
\end{center}
\caption{Performance of the correlator on an ATI GPU.}
\label{results-ati}
\end{table}
}
Synchronizning the threads within a multiprocessor can increase the cache hit ratio, by
ensuring that threads that access the same samples are scheduled at roughly
the same time. With NVIDIA hardware, this leads to a considerable
performance improvement (see Section~\ref{nvidia-perf}).
However, although the ATI hardware can synchronize the threads within a multiprocessor,
we could not achieve performance increases this way.
\longversion{
With ATI
hardware, there is no way to influence the thread scheduler. Moreover,
the way the scheduler works is not documented.
}
\shortversion{
The architecture also does not provide random write access to device
memory. The kernel output can be written to at most 8 output
registers (each 4 floats wide). The hardware stores these to
predetermined locations in device memory. When using the output
registers, at most 32 floating point values can be stored. This
effectively limits the tile size to $2\times2$.
Random write access to \emph{host} memory is provided.
The correlator reduces the data by a large amount, and the results are
never reused by the kernel. Therefore, they can be directly streamed
to host memory.
The theoretical operations/byte ratio of the ATI 4870 architecture is
10.4 for device memory (see Table~\ref{architecture-properties}). In
order to achieve this ratio with our application, a minimal tile size
of $10 \times 10$ would be needed. This would require at least 822
registers per thread. This is unfeasible, so we cannot achieve the
peak performance. Data sharing between tiles using the hardware
caches could improve this situation.
The best performing implementation streams the result data directly to
host memory, and uses a tile size of 4x3, thanks to the large number
of registers. The kernel itself achieves 420 gflops, which is 35\% of
the theoretical peak performance. The achieved device memory bandwidth
is 114~GB/s, which is 99\% of the theoretical maximum. Thanks to the
large tile size, the cache hit ratio is 65\%. As is shown in
Table~\ref{tile-size-table}, the arithmetic intensity with this tile
size is 3.43. Therefore, \\
$\mathit{perf_{max} = min(1200, 3.43 \times 115.2) = 395}$.
We achieve significantly more than this, thanks to the texture cache.
If we also take the host-to-device transfers into account, performance
becomes much worse.
We found that the host-to-device throughput is only 4.62 GB/s in practice,
although the theoretical PCI-e bus bandwidth is 8 GB/s.
The transfer can be done asynchronously, overlapping the
computation with host-to-device communication. However, we discovered that the
performance of the compute kernel decreases significantly
if transfers are performed concurrently. For the $4\times3$
case, the compute kernel becomes 2.2 times slower, which can be fully
attributed to the decrease of device memory throughput. Due to the
low I/O performance, we achieve only 190 gflops, 16\% of the theoretical
peak. This is 63\% of the $\mathit{perf_{max,global}}$ of 300 gflops
that we calculated in Section~\ref{hardware-comparision}.
}
\longversion{
The architecture
does not provide random write access to device memory. The kernel output
must be written to at most 8 output registers (each 4 floats wide). The hardware
stores these to predetermined locations in device memory.
Random write access to \emph{host} memory is provided.
The correlator reduces the data by a large amount, and the results are
never reused by the kernel. Therefore, they can be directly streamed
to host memory. For our application which is mostly read-performance
bound, this does not have a large impact.
The theoretical operations/byte ratio of the ATI 4870 architecture is
10.4 for device memory (See Table~\ref{architecture-properties}). In order to achieve this
ratio with our application, a minimal tile size of $10 \times 10$ would be
needed. This would require at least 822 registers per thread. This is unfeasible,
so we cannot achieve the peak performance.
Data sharing between tiles using the hardware caches could
improve this situation.
As is shown in Table~\ref{results-ati}, the best performing
implementation uses a tile size of 4x3. Since the number of registers
is not disclosed, we did not know beforehand what the best tile size
would be.
CAL automatically performs spilling if insufficient registers are
available.
The kernel itself achieves 297 gflops, which is 25\% of
the theoretical peak performance. The achieved device memory bandwidth
is 81~GB/s, which is 70\% of the theoretical maximum. Thanks to
the large tile size, the cache hit ratio is 47\%.
When using the eight output registers, at most 32 floating
point values can be stored. This effectively limits the tile size to $2\times2$.
In our case, there is a way around this problem. The ATI hardware does not
support random write access to device memory. However, since we write all
correlation data only once, we can stream it directly to \emph{host}
memory. We found that this does not affect performance.
In Table~\ref{results-ati}, the implementations labeled with ``regs'' are using the
output registers, while the implementations with ``mem'' are streaming directly
to host memory.
With a tile size of 4x3, the arithmetic intensity is 3.43 (see Table~\ref{tile-size-table}).
Therefore, \\
$\mathit{perf_{max} = min(1200, 3.43 \times 115.2) = 395}$. \\
We do not achieve this
performance, because the memory bandwidth that is achieved in practice
is significantly lower than the theoretical bandwidth of 115.2 GB/s.
The cache
allows for the sharing of sample data between tiles, and
increases the arithmetic intensity from 3.4 to 3.7 (297/81).
If we also take the host-to-device transfers into account, performance
becomes much worse.
We found that only 2.27 GB/s is reached in practice by the ATI
hardware.
The transfer can be done asynchronously, overlapping the
computation with host-to-device communication. However, we discovered that the
performance of the compute kernel decreases significantly
if transfers are being performed in the background. For the $4\times3$
case, the compute kernel becomes 3.0 times slower, which can be fully
attributed to the decrease of device memory throughput. Due to the
low I/O performance, we achieve only 98 gflops, 8\% of the theoretical
peak. However, this is 66\% of the $\mathit{perf_{max,global}}$ of 147 gflops
that we calculated in Section~\ref{hardware-comparision}. Without host-to-device I/O,
we get 25\% of the peak performance: 297 gflops. If we compare this
with the $\mathit{perf_{max}}$ of 395 gflops, this is 75\%.
}
\subsection{NVIDIA GPU (Tesla C1060)}
\label{nvidia-perf}
NVIDIA's programming model is called Cuda~\cite{cuda-manual}.
Cuda is relatively high-level, and achieves good performance.
However, the programmer still has to think about many details such as
memory coalescing, the texture cache, etc.
An advantage of NVIDIA hardware and Cuda is that the application does not have to do
vectorization. This is thanks to the fact that all cores have their own address generation units.
All data parallelism is expressed by using threads.
The correlator uses 128-bit reads to load a complex sample with two
polarizations with one instruction. Since random write access to
device memory is supported (unlike with the ATI hardware), we can
simply store the output correlations to device memory. We use the
texture cache to speed-up access to the sample data. We do not use it for the
output data, since that is written only once, and never read back by the kernel.
With Cuda, threads
within a thread block can be synchronized. We exploit this feature to let
the threads that access the same samples run in lock step. This way,
we pay a small synchronization overhead, but we can increase the cache hit
ratio significantly. We found that this optimization improved performance by a factor of 2.0.
We also investigated the use of the per-multiprocessor shared memory as an
application-managed cache. Others report good results with this
approach~\cite{gpu-cache}. However, we found that, for our
application, the use of shared memory only led to performance
degradation.
\longversion{
\begin{table}[t]
\caption{Performance of the correlator on an NVIDIA Tesla GPU.}
{\small
\begin{tabular}{r||r|r||r|r}
& \multicolumn{2}{c||}{no communication} &\multicolumn{2}{c}{with communication} \\
tile & & throughput & & throughput \\
size & gflops & (GB/s) & gflops & (GB/s) \\
\hline
1x1 & 158 (17\%) & 147 (144\%) & 145 (16\%) & 135 (132\%) \\
2x2 & 228 (24\%) & 106 (104\%) & 202 (22\%) & 94 ( 92\%) \\
3x2 & 285 (31\%) & 110 (108\%) & 243 (26\%) & 94 ( 93\%) \\
3x3 & 280 (30\%) & 87 ( 85\%) & 239 (26\%) & 74 ( 73\%) \\
4x3 & 101 (11\%) & 28 ( 27\%) & 95 (10\%) & 26 ( 25\%) \\
\end{tabular}
}
\label{results-nvidia}
\end{table}
}
\longversion{
As is shown in Table~\ref{results-nvidia}, the best performing
implementation uses a tile size of 3x2.
}
\shortversion{
The best performing
implementation uses a tile size of 3x2.
}
The optimal tile size is influenced by the way the available registers are used.
The register file is a shared resource. A smaller tile size means less register usage,
which allows the use of more concurrent threads, hiding load delays.
On NVIDIA hardware, we found that the using a relatively small tile size and many threads increases performance.
The kernel itself, without host-to-device communication achieves 314
gflops, which is 34\% of the theoretical peak performance. The
achieved device memory bandwidth is 122~GB/s, which is 120\% of the
theoretical maximum. We can reach more than 100\% because we include
data reuse. The performance we get with the correlator is
significantly improved thanks to this data reuse, which we achieve by
exploiting the texture cache. The advantage is large, because
separate bandwidth tests show that the theoretical bandwidth cannot be
reached in practice. Even in the most optimal case, only 71\% (72
GB/s) of the theoretical maximum can be obtained. The arithmetic
intensity with this tile size is 2.4. We can use this to calculate the
maximal performance without communication.
$\mathit{perf_{max} = min(966, 2.4 \times 102) = 245}$ gflops. In practice, the
performance is better than that: we achieve 128\% of this, thanks to
the efficient texture cache.
If we include communication, the performance drops by 15\%, and we
only get 274 gflops. Just like with the ATI hardware, this is caused
by the low PCI-e bandwidth. With NVIDIA hardware and our
data-intensive kernel, we do see significant performance gains by
using asynchronous I/O. With synchronous I/O, we achieve only 162
gflops (compared to the 274 we get with asynchronous I/O). Therefore,
the use of asynchronous I/O is essential.
\begin{figure*}[t]
\begin{center}
\includegraphics[width=0.4\columnwidth]{memory-layout.eps}
\end{center}
\caption{Transposing the memory layout for better memory coalescing on the GPU.}
\label{memory-layout}
\end{figure*}
Since memory coalescing is so important on GPUs, we also investigated
an alternative implementation, that first performs a memory transpose.
This way, we can make sure that the GPU threads always read the
samples in a coalesced way as much as possible. Normally, the memory
layout is such that a sequence of samples over time are subsequently
stored in memory per station. For the earlier steps in the correlator
pipeline, this is the natural order. However, if we change this
around, and store the data from all stations for each time step
together, we can achieve better performance on the GPU. The two different
memory layouts are shown in Figure~\ref{memory-layout}.
Of course,
this data transpose is costly. Nevertheless, we found that the total
performance in the system can be increased, if we perform the
transpose on the host, while the GPU is correlating. Thus, we overlap
the computation on the GPU with both the transpose on the host and
the data transfer from the host to the device. Performing the
transpose on the GPU puts it on the critical path, and decreases
performance considerably. Our transpose on the host is implemented
using multiple threads to exploit the multi-core architecture, and
using SSE instructions for better memory throughput. With this
approach, we were able to fully overlap the transpose with useful work
on the GPU. Therefore, the transpose is moved completely off the
critical path.
A drawback of this approach is that it puts additional stress on the
host CPU and memory bus. In a production setup, the host is also used
to receive the sample data from the network.
Moreover, the CPU could
be used to perform additional tasks, such as pre- or
post-processing of the data. Possible steps could be filtering,
bandpass and phase corrections, etc. It is unclear what would be more
efficient: using the host for the data transpose as described above, or
using it for additional processing steps. Further research is needed
to clarify thus further.
With the alternative memory layout, the performance of the kernel itself
increases from 314 to 357 gflops (38\% of the peak).
For the version that includes the PCI-e I/O, the
performance increases from 274 to 300 gflops (32\% of the peak).
The internal memory bandwidth we achieve is 139~GB/s. This thus is
significantly higher than the 122~GB/s we got without the data transpose.
In Section~\ref{hardware-comparision}, we calculated that the
$\mathit{perf_{max,global}}$ for our hardware is 363 gflops. In
practice, we achieve 83\% of this limit due to the external I/O
problems.
\subsection{The Cell Broadband Engine (QS21 blade server)}
The basic \mbox{Cell/B.E.} programming is based on multi-threading:
the PPE spawns threads that execute asynchronously on SPEs.
The SPEs can
communicate with other SPEs and the PPE, using mechanisms like signals and mailboxes
for synchronization and small amounts of data, or DMA transfers for
larger data. With the \mbox{Cell/B.E.} it is important to exploit all levels of parallelism.
Applications deal with task and data parallelism across multiple SPEs, vector parallelism
inside the SPEs, and double or triple-buffering for DMA
transfers~\cite{cell}. The \mbox{Cell/B.E.} can be
programmed in C or C++, while using intrinsics to exploit vector
parallelism.
The large number of registers (128 times 4 floats) allows a big tile size of
$4\times4$, leading to a lot of data reuse.
We exploit the vector parallelism of the \mbox{Cell/B.E.} by computing the four
polarization combinations in parallel. We found that this performs
better than vectorizing over the integration time. This is thanks to the \mbox{Cell/B.E.}'s
excellent support for shuffling data around in the vector registers.
\longversion{
This is useful for our application, since the different polarizations
and real and imaginary parts of the samples have to be multiplied and
added in several different ways.
}
The shuffle instruction is executed
in the odd pipeline, while the arithmetic is executed in the even
pipeline, allowing them to overlap.
We identified a minor performance problem with the pipelines of the
\mbox{Cell/B.E.} Regrettably, there is no (auto)increment instruction in the odd
pipeline. Therefore, loop counters and address calculations have to
be performed on the critical path, in the even pipeline. In the time
it takes to increment a simple loop counter, four multiply-adds, or 8
flops could have been performed. To circumvent this, we performed loop
unrolling in our kernels. This solves the performance problem, but has
the unwanted side effect that it uses local store memory, which is
better used as data cache.
\longversion{
\begin{table}[t]
\caption{Performance of the correlator on a QS21 Cell Blade, 8 SPEs.}
{\small
\begin{tabular}{l||r|r||r|r}
& \multicolumn{2}{c||}{no communication} &\multicolumn{2}{c}{with communication} \\
tile & & throughput & & throughput \\
size & flops & (GB/s) & flops & (GB/s) \\
\hline
1x1 & 51 (25\%) & 47 (182\%) & 50 (25\%) & 47 (181\%) \\
2x2 & 169 (83\%) & 79 (305\%) & 164 (80\%) & 77 (296\%) \\
3x2 & 182 (89\%) & 71 (273\%) & 168 (82\%) & 65 (253\%) \\
3x3 & 181 (89\%) & 56 (218\%) & 174 (85\%) & 54 (209\%) \\
\textbf{4x3} & \textbf{197 (97\%)} & \textbf{53 (207\%)} & \textbf{187 (92\%)} & \textbf{50 (192\%)} \\
\end{tabular}
}
\label{results-cell}
\end{table}
}
A distinctive property of the architecture is that cache transfers are
explicitly managed by the application, using DMA. This is unlike other
architectures, where caches work transparently.
Although issuing explicit DMA commands complicates programming,
for our application this is not problematic.
By dividing the
integration time into smaller intervals, we can keep the sample data
for \emph{all stations} in the local store.
We overlap communication with computation, by using multiple buffers.
For the sample data we use double buffering.
Thanks to the explicit cache,
the correlator implementation fetches each sample from main memory
\emph{only exactly once}.
For the correlation output data, several approaches are possible.
In~\cite{ics}, we describe an implementation that loads and stores a strip of
tiles into the local store with one DMA operation. Consider the
example of Figure~\ref{fig-correlation}, for instance, with a tile
size of $2\times3$. In this case, we load three rows of correlations
(the height of a tile) at once, for instance the rows 0, 1, and 2.
Because of this, we have to load and store the correlations to main
memory several times, since the sub-results have to be accumulated.
Since the correlations are both read and written, we use triple
buffering in this case.
We also developed a version that keeps the correlation results in the
local store, and stores the correlation output data to host memory
only once. Double buffering is enough in this case. We can fit the
result in the local store if the number of receivers is not too large
(e.g., 64). For the LOFAR instrument, this is the case. The achieved
performance of the two versions is identical. However, both have their
own advantages. The first version scales to larger numbers of
receivers, but uses more bandwidth between the host memory and the
SPEs. This bandwidth is available on the \mbox{Cell/B.E.}, so this
does not hurt performance. The second version uses less bandwidth,
but is limited in the number of receivers. However, the amount of
host memory bandwidth used is important, since we may want to run
additional operations on the host (the PPE), such as receiving data
from the network, and preprocessing it.
If an even larger number of receivers is used (e.g., 256),
good performance can still be achieved on the \mbox{Cell/B.E.}.
This can be done by splitting the correlation triangle of
Figure~\ref{fig-correlation} in blocks (e.g., of size $32\times32$), which are
in turn divided into the tiles we already used in the previous
implementations. On the SPEs, we now only load the samples inside the
block into the local store, and not the entire triangle. A complication is that the X and
Y directions of the blocks no longer deal with the same samples in all
cases. For example, in the X direction, the samples could run from
0--31, while the Y-axis runs from 64--96. Therefore, the number of
samples that has to be loaded into the local store increases with a
factor of two. Nevertheless, the \mbox{Cell/B.E.} has the bandwidth
that is needed for this.
\longversion{
The performance we achieve on a single
\mbox{Cell/B.E.} chip is shown in Table~\ref{results-cell}.
}
Due to the high
memory bandwidth and the ability to reuse data, we achieve 188
gflops, including all memory I/O. This is 92\% of the peak
performance on one chip. If we use both chips in the cell blade, the
performance drops only with a small amount, and we still achieve
91\% (374 gflops) of the peak performance. Even though the memory
bandwidth per operation of the \mbox{Cell/B.E.} is eight times lower than
that of the BG/P, we still achieve excellent performance, thanks to
the high data reuse factor.
\section{Comparison and Evaluation}
\label{sec:perf-compare}
In this section, we compare the performance, power efficiency, and
programmablity of the different architectures. We also discuss a relatively
new development: OpenCL. We describe how we implemented the correlator using this language.
\subsection{Performance}
\begin{figure*}[t]
\begin{center}
\includegraphics[width=\columnwidth]{performance-graph.eps}
\end{center}
\caption{Achieved performance on the different platforms.}
\label{performance-graph}
\end{figure*}
Figure~\ref{performance-graph} shows the performance on all
architectures we evaluated. The NVIDIA GPU achieves the highest
\emph{absolute} performance. Nevertheless, the GPU \emph{efficiencies}
are much lower than on the other platforms. The \mbox{Cell/B.E.}
achieves the highest efficiency of all many-core architectures, close
to that of the BG/P. Although the theoretical peak performance of the
\mbox{Cell/B.E.} is 4.6 times lower than the NVIDIA chip, the absolute
performance is only 1.6 times lower. If both chips in the QS21 blade
are used, the \mbox{Cell/B.E.} also has the highest absolute
performance. For the GPUs, it is possible to use more than one chip as
well. This can be done in the form of multiple PCI-e cards, or with
two chips on a single card, as is done with the ATI 4870x2
device. However, we found that this does not help, since the
performance is already limited by the low PCI-e throughput, and the
chips have to share this resource. The graph indeed shows that the
host-to-device I/O has a large impact on the GPU performance, even
when using one chip. With the \mbox{Cell/B.E.}, the I/O (from main
memory to the Local Store) only has a very small impact.
\subsection{Power Efficiency}
\begin{table}[t]
\caption{Measured performance of the different many-core hardware platforms.}
\label{architecture-measurements}
{\small
\begin{tabular}{l|l|l|l|l|l}
& Intel & IBM & ATI & NVIDIA & STI Cell \\
Architecture & Core i7 & BG/P & 4870 & Tesla C1060 & (full blade) \\
\hline
measured gflops (including I/O) & 48.0 & 13.1 & 190 & 300 & 374 \\
achieved efficiency & 67\% & 96\% & 16\% & 32\% & 91\% \\
\hline
measured bandwidth (GB/s) & 18.6 & 6.6 & 81 & 139 & 49.5 \\
bandwidth efficiency & 73\% & 48\% & 70\% & 136\% & 192\% \\
\hline
TDP chip (W) & 130 & 24 & 160 & 236 & 140 \\
theoretical gflops / Watt & 0.65 & 0.57 & 7.50 & 3.97 & 2.93 \\
achieved gflops/Watt (TDP) & 0.37 & 0.54 & 1.19 & 1.27 & 2.67 \\
power efficiency compared to BG/P (TDP) & 0.69 x & 1.0 x & 2.20 x & 2.35 x & 4.9 x \\
\hline
\textbf{measured power, full system} (W) & 208 & 44 & 259 & 250 & 315 \\
\textbf{achieved gflops/Watt full system} & 0.23 & 0.30 & 0.73 & 1.20 & 1.18 \\
\textbf{power efficiency compared to BG/P}& 0.77 x & 1.0 x & 2.4 x & 4.0 x & 3.9 x \\
\end{tabular}
\end{table}
In Table~\ref{architecture-measurements}, we present the power efficiency for
the different architectures.
The results show that the \mbox{Cell/B.E.} chip is
about five times more energy efficient than the BG/P. This is not a
fair comparison, since the BG/P includes a lot of network hardware on
chip, while the other architectures do not offer this. Nevertheless,
it is clear that the \mbox{Cell/B.E.} is significantly more efficient.
A 45 nm version of the \mbox{Cell/B.E.} has been announced for early 2009.
With this version, which has identical performance, but reduces the TDP to about 50W,
the \mbox{Cell/B.E.} chip even is seven times more efficient than the BG/P.
The 65 nm version of the \mbox{Cell/B.E.} chip already is
about 2.3 times more energy efficient than the GPUs (based on TDP).
The fact that the three most energy efficient supercomputers on the
Green500 list\footnote{See http://www.green500.org.} are based on the
\mbox{Cell/B.E.} supports our findings.
The Green500 list also specifies the achieved power efficiency for
entire supercomputers, i.e. including memory, chipsets, networking hardware, etc.
PowerXCell-based systems achieve 0.54 glops/W, while the Blue Gene/P
is less power efficient, and achieves 0.37 gflops/W. Systems based on
general-purpose CPUs only achieve 0.27 gflops/W.
The situation is different if we do not just look at the chips
themselves, but also take the full host system into account. The
lower three lines in Table~\ref{architecture-measurements} show the
\emph{measured} power and efficiencies of the \emph{full systems},
while utilized. For the GPUs, this means that we integrated them in
the Core~i7 host system that we also use for the general purpose CPU
measurements. We measured the dissipated power with a VoltCraft-3000
power meter. The results show that the full Cell blade uses
relatively much power in addition to the TDP of the two
\mbox{Cell/B.E.} chips. This is partially caused by the XDR memory in
the system. As a result of this, the achieved performance per Watt is
lower than expected, but still 3.9 times higher than that of the
BG/P. For the NVIDIA GPU, however, the opposite is true. In practice,
the device consumes much less power than specified by the TDP.
Therefore, the achieved efficiency of a full system with the GPU is
almost the same as that of the \mbox{Cell/B.E.}. The ATI GPU is much
less efficient, due to the small fraction of the theoretical peak
performance that is reached in practice.
\subsection{Programmablity}
The performance gap between assembly and a high-level programming language
is quite different for the different platforms. It also
depends on how much the compiler is helped by manually unrolling
loops, eliminating common sub-expressions, the use of register variables,
etc., up to a level that the C code becomes almost as low-level as assembly
code. The difference varies between only a few percent to a factor of 10.
For the BG/P, the performance from compiled C++ code was by far not
sufficient. The assembly version hides load and instruction
latencies, issues concurrent floating point, integer, and load/store
instructions, and uses the L2 prefetch buffers in the most optimal
way. The resulting code is approximately 10 times faster than C++
code. For both the Cell/B.E. and the Intel core~i7, we found that
high-level code in C or C++ in combination with the use of intrinsics
to manually describe the SIMD parallelism yields acceptable
performance compared to optimized assembly code. Thus, the programmer
specifies which instructions have to be used, but can typically leave
the instruction scheduling and register allocation to the compiler.
On NVIDIA hardware, the high-level Cuda model delivers excellent
performance, as long as the programmer helps by using SIMD data types
for loads and stores, and separate local variables for values that
should be kept in registers. With ATI hardware, this is different. We
found that the high-level Brook+ model does not achieve acceptable
performance compared to hand-written CAL code. Manually written assembly
is more than three times faster. Also, the Brook+ documentation is insufficient.
\begin{table}[t]
\caption{Strengths and weaknesses of the different platforms for data-intensive applications.}
\label{architecture-results-table}
{\small
\begin{tabular}{l|l|l|l|l}
Intel & IBM & ATI & NVIDIA & STI \\
Core i7 920 & Blue Gene/P & 4870 & Tesla C1060 & Cell/B.E. \\
\hline
+ well-known & + L2 prefetch unit & + largest number & + random & + random \\
& ~~~works well & ~~~of cores & ~~~write access & ~~~write access \\
& + high memory & + swizzling & + Cuda is & + shuffle \\
& ~~~bandwidth & & ~~~high-level & ~~~capabilities \\
& & & & + explicit cache \\
& & & & + power efficiency \\
& & & & \\
& & & & \\
\hline
- few registers & - everything & - low PCI-e & - low PCI-e & - multiple \\
- no fma & ~~~double precision & ~~~bandwidth & ~~~bandwidth & ~~~parallelism levels \\
- limited & - expensive & - transfer slows & & - no increment \\
~~~shuffling & & ~~~down kernel & & ~~~in odd pipe \\
& & - no random & & \\
& & ~~~write access & & \\
& & - CAL is low-level & & \\
& & - bad Brook+ & & \\
& & ~~~performance & & \\
& & - not well & & \\
& & ~~~documented & & \\
\end{tabular}
}
\end{table}
In Table~\ref{architecture-results-table} we summarize the
architectural strengths and weaknesses that we identified. Although
we focus on the correlator application in this paper, the
results are applicable to applications with low flop/byte ratios in
general.
\subsection{Implementing the Correlator with OpenCL}
An interesting recent development is the OpenCL programming
model~\cite{opencl}. OpenCL is an open standard for parallel
programming of heterogeneous systems, developed by the Khronos
Group. Many important industry partners participate in the effort.
Many-core vendors (e.g., AMD, IBM, Intel, ATI, NVIDIA) have
pledged to support OpenCL to increase both portability and
programmability of their hardware. The main idea of OpenCL is that
a single language is used to program the many-core hardware of all
different vendors.
We have implemented the correlator
application in OpenCL, and tested it with the latest available OpenCL
implementations. Currently, AMD provides OpenCL support for multi-core \emph{CPUs} (not
GPUs) in the second beta release of their stream SDK version
2.0. NVIDIA released OpenCL support for their \emph{GPUs} in a
closed beta program. Since both implementations are still in the beta
stages, we do not discuss performance, but only programmability and portability.
OpenCL consists of two parts: a host API
(to interface with the C program that runs on the CPU), and a special
C-based language to develop kernels that run on the accelerators. The
host interface is very low-level, and is similar to ATI's CAL, or
NVIDIA's driver API. The Cuda model provided by NVIDIA has a much
higher abstraction level than OpenCL, and is significantly easier to
use. AMD provides an additional set of C++ wrappers around OpenCL's C
interface, but this set is not standardized. An important aspect of
OpenCL is that it uses \emph{run-time compilation} for the kernels.
To demonstrate the level of abstraction of the host part of the OpenCL programming model,
we will describe the many steps that are necessary to run a kernel. An OpenCL program has to
create a \emph{context} and a \emph{command queue}. Next, the kernel
program has to be loaded from disk (or constructed in memory), and a
program object has to be created. Then, the program has to be
compiled, and a kernel object has to be created. Subsequently, the
arguments to the kernel invocation have to be set, using a call per
parameter. Similar to CAL and Cuda, all buffers have to be explicitly
allocated and transferred from the host to the device. Finally,
a domain has to be specified, indicating the number of global and local threads to use.
Only after all
these steps, can the kernel be launched. Also, if texture-cached
memory is used, as we do with the correlator on GPUs, the programmer
has to explicitly create 2D or 3D image objects, which also use a
different set of calls for the data transfers. Here, the graphics
origins of OpenCL clearly shine through.
The language that is used to write the kernels is based on C, with
additional extensions. The level of abstraction is similar to Cuda,
and of higher level than CAL and the assembly that is needed to program
the Cell or CPUs efficiently. A large step forward is that the
language has a \emph{standard interface to perform vectorization}, and also
supports swizzling to shuffle data around inside vectors. Special
annotations are used to declare data structures in the different
memory regions (global, constant, local, and private). Finally,
special operations have to be used to load data from texture-cached
memory.
\begin{table}[t]
\caption{Strengths and weaknesses of OpenCL.}
\label{opencl-table}
{\small
\begin{tabular}{l|l}
Strengths & Weaknesses \\
\hline
+ high-level kernel language & - low-level host API \\
+ vectorization and swizzling & - only C binding standardized (no C++, Java, or Python) \\
+ portability & - performance portability not solved \\
+ runtime compilation & \\
\end{tabular}
}
\end{table}
We wrote several versions of the correlator in OpenCL, using different
tile sizes, and using normal and texture-cached memory. A large
benefit of OpenCL is that the code indeed does compile and run without
any changes on both CPUs and GPUs. The only exception was a correlator version
that uses the texture cache: AMD's beta version does not support this yet.
We assume this problem will be solved for the final release.
However, the problem of \emph{performance} portability is not fully solved
by OpenCL yet. First, we had to vectorize the correlator kernel
code. With NVIDIA hardware (and the Cuda model), this is not
necessary. Therefore, NVIDIA's runtime compiler removes the
vectorization. On the CPU, the vectorization did increase
performance. Second, the different platforms have very different
memory models. The programmer still has to deal with these differences
to reach the best performance. The texture cache of the GPUs and the local store of
the Cell/B.E. are examples of this. Moreover, we found that our
algorithmic changes are still necessary: the different platforms need
different tile sizes to perform optimally, due to the different numbers of registers they
have. Finally, the different architectures require very different
numbers of global and local threads: only a few global threads for
CPUs, and many thousands, divided over global and local threads, for
the GPU. Table~\ref{opencl-table} summarizes the strengths and weaknesses of
OpenCL that we identified.
\section{Related Work}
\label{related}
Intel's 80-core Terascale Processor~\cite{terascale} was the first
generally programmable microprocessor to break the teraflop barrier. It
has a good flop/Watt ratio, making it an interesting candidate for
future correlators.
Intel's Larrabee~\cite{larrabee} (to be released) is another promising
architecture. Larrabee will be a hybrid between a GPU and a
multi-core CPU. It will be compatible with the x86 architecture, but
will have 4-way simultaneous multi-threading, 512-bit wide vector
units, shuffle and multiply-add instructions, and special texturing
hardware. Larrabee will use in-order execution, and will have coherent
caches. Unlike current GPUs, but similar to the \mbox{Cell/B.E.},
Larrabee will have a ring bus for communication between cores and for
memory transactions.
Another interesting architecture to implement correlators are
FPGAs~\cite{fpga-correlator}. LOFAR's on-station correlators are also
implemented with FPGAs. Solutions with FPGAs combine good performance with
flexibility. A disadvantage is that FPGAs are relatively
difficult to program efficiently.
Also, we want to run more than just the correlator on our hardware.
LOFAR is the first of a new generation of software telescopes, and how the
processing is done best is still the topic of research, both in
astronomy and computer science. We perform the initial processing steps on
FPGAs already, but find that this solution is not flexible enough for
the rest of the pipeline. For LOFAR, currently twelve different
processing pipelines are planned. For example, we would like to do the
calibration of the instrument and pulsar detection online on the same
hardware, before storing the data to disk. We even need to support
multiple different observations simultaneously. All these issues
together require enormous flexibility from the processing solution.
Therefore, we restrict us to many-cores, and leave application-specific
instructions and FPGAs as future work.
Once the processing pipelines are fully understood, future
instruments, such as the SKA, will likely use ASICs.
Williams et al.~\cite{peri} describe an auto-tuning framework for
multi-cores. The framework can automatically perform
different low-level optimizations to increase performance. However,
GPUs are not considered in this framework. We performed all
optimizations manually, which is possible in our case, since the
algorithm is relatively straightforward. More important, we found that
in our case, algorithmic changes are required to achieve good performance.
Examples include the use of different tile sizes, and vectorizing
over the different polarizations instead of the inner time loop.
A software-managed cache is used on the \mbox{Cell/B.E.} processor. GPUs
typically have a small amount of shared memory that can be used in a
similar way~\cite{gpu-cache}. An important difference is that in the
\mbox{Cell/B.E.} the memory is private for a thread, while with GPUs all
threads on a multiprocessor share the memory. The available
memory per thread is also much smaller. We applied
the technique described in~\cite{gpu-cache}, but found it did not
increase performance for our application.
Wayth et al.\ describe a GPU correlator for
the Murchison Widefield Array (MWA)~\cite{mwa-gpu-correlator}.
They optimize their code by tiling the correlator triangle in one dimension
(a technique described by Harris et.al.~\cite{gpu-correlator-harris}),
while tiling in two dimensions, as we described in this paper, is much more efficient.
For instance, a 2x2 tile requires the same amount of operations as a 1x4 tile,
but performs fewer memory operations~(see table~\ref{tile-size-table}).
For larger tiles, the arithmetic intensity of two-dimensional tiles is even
better.
Also, the MWA GPU version does not use the texture cache, but shared memory.
We found that this was significantly slower.
Their claim that their GPU implementation is 68~times faster than their CPU
implementation is highly biased, since their CPU implementation is not
optimized, single threaded, and does not use SSE.
As a result, our CPU version is 48 times faster than their CPU version,
while our GPU version is 4.2 times faster than their GPU version (even
though our data rates are four times as high due to our larger sample sizes).
Hence, their GPU implementation is only 1.4 times faster than an optimized
CPU implementation, not~68 times.
\section{Discussion}
\label{sec:discussion}
A key application characteristic of the correlator is that it is extremely
regular. This means that we know exactly which memory is referenced at
what time. In this paper, we explained that this property makes many
optimizations possible. We also implemented several other signal-processing algorithms we did not
discuss here, albeit not on all many-core architectures. Most
of our conclusions hold for all
(data-intensive) applications. However, this paper does not compare the ability of the
architectures to cope with \emph{unpredictable} memory access
patterns. We know, for example, that a particular radio-astronomy
imaging algorithm (W-projection) exhibits random memory access, and as a result
performs poorly on at least some of these architectures, and probably
all~\cite{gridding-08}. Also, the software-manged cache of the \mbox{Cell/B.E.}
is less effective here, since the programmer cannot predict the
accesses in advance. Fortunately, not all applications behave so
unpredictably. In general, we advocate that the focus for
optimizations for many-core architectures should be on memory
bandwidth, access patterns, and efficient use of the caches, even at
the cost of increased synchronization and extra computation.
In this paper, we focus on the maximal performance that can be achieved
with a single many-core chip. An exiting result we present here is
that even extremely data-intensive applications, such as the correlator,
can perform well on many-core architectures, in particular on the
\mbox{Cell/B.E.}. These results allows us to move forward, and bring up the
question of scalability: can we scale the results to a full system
that processes all telescope data? In this context, it is important
to emphasize that the correlator itself is trivially parallel, since
tens of thousands of frequency channels can be processed
independently. However, in case of an FX correlator, a major data
exchange is necessary \emph{prior} to correlation: each input contains all
frequency channels of a single receiver, but the correlator requires a
single frequency channel of all receivers. We implemented this for the
LOFAR correlator on the 3-D torus of the Blue Gene/P, where we
exchange all data asynchronously. Although an efficient implementation
is complex, the time required for this exchange
is small compared to the time to correlate the data. Moreover, the
data rates grow linearly with the number of receivers, while the
compute time of the correlator algorithm grows quadratically.
We also experimented on a PC cluster with a Myrinet switch, which was
able to handle the all-to-all exchange at the required data rates. On the Blue Gene/P, we can
scale the application to more than 10.000 cores. For more information, we
refer to~\cite{spaa-06,ppopp2010}.
\section{Conclusions}
\label{conclusions}
Current and future telescopes have high computational and I/O demands. Therefore,
we evaluated the performance of the extremely data-intensive
correlator algorithm on today's many-core architectures. This research
is an important pathfinder for future radio-astronomy instruments.
The algorithm is simple, we can therefore afford to optimize and
analyze the performance by hand, even if this requires assembly,
application-managed caches, etc. The performance of compiler-generated
code is thus not an issue: \emph{we truly compared the architectural
performance}.
Compared to the BG/P, many-core architectures have a significantly
lower memory bandwidth \emph{per operation}. Minimizing the number of
memory loads per operation is of key importance. We do this by
extensively optimizing the algorithm for each architecture. This
includes making optimal use of caches and registers. A high memory
bandwidth per flop is not strictly necessary, as long as the
architecture allows efficient data reuse.
This can be achieved through caches, local stores and registers.
Only two architectures perform well with our application. The BG/P
supercomputer achieves high efficiencies thanks to the high memory
bandwidth per FLOP. The \mbox{Cell/B.E.} also performs excellently, even
though its memory bandwidth per operation is eight times lower. We
achieve this by exploiting the
application-managed cache and the large number of registers,
optimally reusing all sample data.
The \mbox{Cell/B.E.} is about five to seven times more energy efficient than the
BG/P, if we do not take the network hardware into account.
It is clear that application-level control of cache behavior (either
through explicit DMA or thread synchronization) has a substantial
performance benefit, and is of key importance for data intensive
high-performance computing. The results also demonstrated that, for
data-intensive applications, the recent trend of increasing the number
of cores does not work if I/O is not scaled accordingly.
\begin{acknowledgements}
This work was performed in the context of the NWO STARE
AstroStream project. We gratefully acknowledge NVIDIA, and in
particular Dr. David Luebke, for providing freely some of the GPU
cards used in this work. Finally, we thank Chris Broekema, Jan David
Mol, and Alexander van Amesfoort for their comments on an earlier
version of this paper.
\end{acknowledgements}
\bibliographystyle{abbrv}
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2103.09165
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\section{Introduction}
Superposition is one of the most striking phenomena which distinguishes quantum from classical physics. The degree to which a system is superposed between different orthogonal states is known as \emph{coherence} \cite{RevModPhys.89.041003,aberg2006quantifying,PhysRevLett.119.230401}. Much like entanglement \cite{RevModPhys.81.865}, coherence is considered to be a valuable resource in quantum information processes. In Quantum computing \cite{nielsen2002quantum,Preskill2018quantumcomputingin}, where information is encoded in the states of two-level systems, algorithms designed to operate in superposition, are exponentially faster than their classical counterparts \cite{10.1137/S0097539795293172,PhysRevLett.79.325,Arute2019}. Coherence is so central to the development of a universal quantum computer that it is used as a metric for the quality of a quantum processor. The time that it takes for a qubit to effectively decohere due to noise is known as the \emph{dephasing time} with current processors achieving times of a few hundred microseconds. Coherent phenomena are important in other fields of research, such as quantum metrology \cite{PhysRevA.94.052324} and thermodynamics \cite{Lostaglio2015,PhysRevLett.115.210403,PhysRevX.5.021001,Narasimhachar2015,Korzekwa_2016} for example. Surprisingly it has been suggested that these phenomena might also be present in biological processes and more specifically in the efficiency of energy transport during photosynthesis \cite{Lloyd_2011}.
A simple method of obtaining coherence is by extracting it from another system. When this process involves a quantum field as the source then it is known as a \emph{coherence harvesting protocol}. Despite an extensive amount of research on entanglement harvesting protocols (see, e.g., \cite{VALENTINI1991321,Reznik2003,PhysRevA.71.042104,Salton_2015,PhysRevD.92.064042,PhysRevD.96.025020,PhysRevD.96.065008, PhysRevD.98.085007,Cong:2020nec,Tjoa:2020eqh}) and the deep connection that exists between entanglement and coherence \cite{PhysRevLett.115.020403,PhysRevLett.117.020402,PhysRevA.96.032316}, coherent harvesting has not received any attention. By employing the Unruh-DeWitt (UDW) particle detector model \cite{Unruh,DeWitt,birrell}, it was shown recently that a two-level pointlike detector, initially in its ground energy state, interacting with a coherent massless scalar field in $1+1$ flat spacetime, can harvest a small amount of coherence \cite{KBM}. As it turns out, this amount depends on the initial energy of the field, the mean interaction duration and the detector's state of motion. For a detector moving at relativistic speeds and initial field energies lower than the gap between its energy levels, it is possible to extract a larger amount of coherence than when it is static, a phenomenon dubbed by the authors as ``swelling''.
In this article, we provide a thorough study of the conditions under which coherence harvesting is possible for any initial state of the field in $n+1$ dimensional Minkowski spacetime. In order to achieve this and to avoid the problem of IR divergences that are present in the $1+1$ dimensional case of a linear coupling between detector and field \cite{BJA}, we instead consider an interaction in which the former is coupled to the proper time derivative of the latter. Both models contain all the essential features of matter interacting with radiation \cite{Wavepacket:det,CHLI2}, so they provide a useful benchmark for studying possible applications of relativistic effects in quantum information processing. Acknowledging the fact that a pointlike detector is not a physical system--an atom or an elementary particle, for example, has finite size-- and to make our results as relevant as possible we will take into consideration the spatial extension of the detector.
We show that when the interaction is instantaneous harvesting is catalytic \cite{PhysRevLett.113.150402}. At the cost of some energy, which assists in the extraction process, it is possible to repeatedly extract the same amount of coherence each time. For an inertial detector moving at a constant velocity and under suitable conditions, it is proven that this is also the maximum amount that can be obtained. As an example we consider the case of harvesting coherence from a coherent scalar field and find that the process depends on the phase of its coherent amplitude distribution, its initial energy, the mean radius of the detector and the mean interaction duration between the two. For a mean radius comparable to the inverse of its transition frequency, it is shown that although the amount of coherence extracted is of the same order as the coupling constant the process can be repeated to obtain a single unit of coherence in a very short time. We conclude that even in the case of a spatially extended detector swelling effects are still present but these are weaker in a $3+1$ compared to a $1+1$ dimensional spacetime.
\section{Quantum coherence}\label{Qcoh}
From a physical point of view coherence reflects the degree of superposition that a quantum system exhibits when it simultaneously occupies different orthogonal eigenstates of an observable of interest \cite{PhysRevLett.119.230401}. Coherent systems are considered to be valuable resources in quantum information processes, because with their help it is possible, at the cost of consuming some of the coherence that they contain, to simulate transformations that violate conservation laws associated with the corresponding observable.
Mathematically, let $\{\ket{i}\}$ denote a set of basis states spanning a finite discrete Hilbert space $\mathcal{H}$, which correspond to the eigenstates of an observable $\hat{O}$. Any state $\rho$ which is diagonal in this basis
\begin{equation}\label{definition}
\rho=\sum_ip_i\ketbra{i}
\end{equation}
is called \emph{incoherent} and commutes with the observable. If $\rho$ contains non-diagonal elements then it is called \emph{coherent} \cite{RevModPhys.89.041003}. In this case $[\rho,\hat{O}]\neq 0$ \footnote{assuming that the spectrum of $\hat{O}$ is non-degenerate}, and the state changes under the action of the one parameter group of symmetry transformations $U(s)=\text{exp}(-is\hat{O})$ generated by the observable. This makes coherent systems useful as reference frames and reservoirs for the implementation of non-symmetric transformations \cite{RevModPhys.79.555,Gour_2008,Marvian2014,PhysRevA.90.062110}. For example, for a fixed Hamiltonian $\hat{H}$, any system that possesses coherence with respect to the energy basis can be used as a clock since in this case its rate of change is non-zero, $\dot\rho(t)\neq0$, so it necessarily changes with the passage of time. The same system could alternatively be utilised as a coherent energy reservoir with the help of which it is possible to perform incoherent transformations on other systems \cite{PhysRevLett.113.150402}.
The amount of coherence present in a system can be quantified with the help of a \emph{coherence measure}. This is a real valued function $C(\cdot)$ on the set of density matrices $\mathcal{D}$ such that
\begin{equation}
C(\rho)\geq 0,\quad \forall\rho\in\mathcal{D}
\end{equation}
with equality if and only if $\rho$ is incoherent. A simple example of such a function is given by the $\ell_1$-norm of coherence \cite{RevModPhys.89.041003}, which is equal to the sum of the modulus of the system's non-diagonal elements
\begin{equation}\label{coh_measure}
C(\rho)=\sum_{i\neq j}\abs{\rho_{ij}}
\end{equation}
with values ranging between $0$ for an incoherent state and $d-1$ for the maximally coherent $d$-dimensional pure state
\begin{equation}
\ket{\psi}=\frac{1}{\sqrt{d}}\sum_{i=0}^{d-1}\ket{i}.
\end{equation}
In order to extract coherence from a coherent system $\sigma$ to an incoherent system $\rho$ it is necessary to bring the two in contact and make them interact through a completely positive and trace preserving quantum operation. When the latter obeys the conservation law associated with the observable and is strictly incoherent (in the sense that it maps incoherent states to incoherent states) the process is called \emph{faithful} \cite{PhysRevA.101.042325}. When this is no longer the case the operation generates extra coherence, which increases the amount stored in the combined system and can assist in the extraction process \cite{PhysRevA.92.032331,BU20171670}, in much the same way that a quantum operation which is non-local can create entanglement between two spacelike separated systems.
We shall now demonstrate how to construct such an assisted protocol for harvesting coherence onto an UDW detector from a scalar field. In what follows we shall assume a flat $n+1$ dimensional spacetime with metric signature $(-+\cdots+)$. We will denote spacetime vectors by sans-serif characters, and the scalar product of vectors $\mathsf{x}$ and $\mathsf{y}$ as $\mathsf{x}\cdot\mathsf{y}$. Boldface letters represent spatial n-vectors. Throughout, we make use of natural units in which $\hbar=c=1$ and employ the interaction picture for operators and states.
\section{Unruh-DeWitt detector model}\label{UDW}
To study the amount of coherence harvested from a massless scalar field we will employ an UDW detector coupled to the proper time derivative of the field \cite{Hinton,Takagi,DM}. In the simplest case considered here, the latter is modeled as a qubit with two energy levels, ground $\ket{g}$ and excited $\ket{e}$ and energy gap equal to $\Omega$, with Hamiltonian
\begin{equation}
\hat{H}_\text{\tiny\ensuremath D}=\frac\Omega2(\ketbra{e}-\ketbra{g})
\end{equation}
which is moving along a worldline $\mathsf{x}(\tau)$ parametrized by its proper time $\tau$. The detector is interacting with a massless scalar field in $n+1$ dimensions
\begin{equation}\label{field}
\hat\phi(\mathsf{x})=\int\frac{d^n\mathbf{k}}{\sqrt{(2\pi)^n2\abs{\mathbf{k}}}}\left[\hat a_{\mathbf{k}}e^{i\mathsf{k}\cdot\mathsf{x}}+\text{H.c.}\right],
\end{equation}
with a normal-ordered Hamiltonian of the form
\begin{equation}
\hat{H}_\phi=\int \abs{\mathbf{k}}\hat a^\dagger_{\mathbf{k}}\hat a^{}_{\mathbf{k}}d^n\mathbf{k},
\end{equation}
where $\hat a_{\mathbf{k}}$, and $\hat a^\dagger_{\mathbf{k}}$ are the creation and annihilation operators of the mode with momentum $\mathbf{k}$ that satisfy the canonical commutation relations
\begin{equation}\label{commutation}
[\hat a_{\mathbf{k}},\hat a_{\mathbf{k'}}]=[\hat a^\dagger_{\mathbf{k}},\hat a^\dagger_{\mathbf{k'}}]=0,\quad [\hat a_{\mathbf{k}},\hat a^\dagger_{\mathbf{k'}}]=\delta(\mathbf{k}-\mathbf{k'}).
\end{equation}
The interaction between detector and field is constructed by coupling the former's monopole moment operator
\begin{equation}\label{dipole}
\hat{\mu}(\tau)=e^{i\Omega\tau}\ketbra{e}{g}+ e^{-i\Omega\tau}\ketbra{g}{e},
\end{equation}
to the momentum degrees of freedom of the latter through the following interaction Hamiltonian
\begin{equation}\label{interaction}
\hat{H}_{\text{int}}(\tau)=\lambda\chi(\tau)\hat{\mu}(\tau)\otimes\partial_\tau\hat{\phi}_f(\mathsf{x}(\tau)).
\end{equation}
Here $\lambda$ is a coupling constant with dimensions $(\mbox{length})^{\frac{n+1}{2}}$, $\chi(\tau)$ is a real valued \emph{switching function} that describes the way the interaction is switched on and off; and $\hat{\phi}_f(\mathsf{x}(\tau))$ is a smeared field on the detector's center of mass worldline $\mathsf{x}(\tau)=(t(\tau),\mathbf{x}(\tau))$,
\begin{equation}\label{smeared}
\hat{\phi}_f(\mathsf{x}(\tau))=\int_{\mathcal{S}(\tau)} f(\boldsymbol{\xi})\hat{\phi}(\mathsf{x}(\tau,\boldsymbol{\xi}))d^n\boldsymbol\xi,
\end{equation}
where
\begin{equation}\label{Fermi-Walker}
\mathsf{x}(\tau,\boldsymbol\xi)=\mathsf{x}(\tau)+\boldsymbol\xi
\end{equation}
are the Fermi-Walker coordinates \cite{MTW} on the simultaneity hyperplane $S(\tau)$, which is defined by all those space-like vectors $\boldsymbol\xi$ normal to the detector's four-velocity, $\mathcal{S}(\tau)=\left\{\boldsymbol\xi|\mathsf u\cdot\boldsymbol\xi=0\right\}$ (see Fig. \ref{fig:Fermi-Walker}).
\begin{figure}
\centering
\includegraphics[width=\columnwidth]{Plots/spacetime_diagram.jpg}
\caption{Any point in the neighbourhood of the detector's worldline can be described by its Fermi-Walker coordinates $(\tau,\boldsymbol\xi)$, where the proper time $\tau$ indicates its position along the trajectory and $\boldsymbol\xi$ is the displacement vector from this point lying on the simultaneity hyperplane consisting of all those space-like vectors normal to its four-velocity $\mathsf{u}$.}\label{fig:Fermi-Walker}
\end{figure} The real valued function $f(\boldsymbol\xi)$ in Eq. (\ref{smeared}) is known as the \emph{smearing function} and is a physical reflection of the finite size and shape of the detector \cite{Schlicht_2004,Louko_2006,Wavepacket:det,CHLI2}.
Compared to the usual UDW interaction in which the detector is linearly coupled to the field, the derivative coupling is free of the issue of IR divergences in the $1+1$ dimensional case which arise due to the massless nature of the field \cite{BJA}. The Hamiltonian in Eq. \eqref{interaction} resembles closely the dipole interaction between an atom with dipole moment $\mathbf{d}$ and an external electromagnetic field, since in this case the electric field operator is defined, in the Coulomb gauge, by means of the vector potential $\hat{\mathbf{A}}(t,\boldsymbol{x})$ as $\hat{\mathbf{E}}(t,\boldsymbol{x})=-\partial_t \hat{\mathbf{A}}(t,\boldsymbol{x})$ \cite{Scully}.
Combining Εq. (\ref{field}) with Εqs. (\ref{smeared})-(\ref{Fermi-Walker}) the smeared field operator reads
\begin{equation}\label{mod_smeared}
\hat{\phi}_f(\mathsf{x}(\tau))=\int\frac{d^n\mathbf{k}}{\sqrt{(2\pi)^n2\abs{\mathbf{k}}}}\left[F(\mathsf k,\tau)\hat a_{\mathbf{k}}e^{i\mathsf{k}\cdot\mathsf{x}(\tau)}+\text{H.c.}\right],
\end{equation}
where
\begin{equation}\label{Fourier}
F(\mathsf k,\tau)=\int_{\mathcal{S}(\tau)} f(\boldsymbol\xi)e^{i\mathsf k\cdot\boldsymbol\xi}d^n\boldsymbol\xi
\end{equation}
is the Fourier transform of the smearing function. Now $\mathsf k$ can always be decomposed as
\begin{equation}
\mathsf k=(\mathsf k\cdot\mathsf u)\mathsf u+(\mathsf k\cdot{\mathsf {\boldsymbol\zeta}}){\mathsf {\boldsymbol\zeta}}
\end{equation}
for some unit vector ${\mathsf{\boldsymbol\zeta}}\in \mathcal{S}(\tau)$. Since for a massless scalar field $\mathsf k$ is light-like, it follows that $(\mathsf k\cdot \mathsf u)^2=(\mathsf k\cdot{\mathsf {\boldsymbol\zeta}})^2$. This means that for a spherically symmetric smearing function the Fourier transform in Eq. (\ref{Fourier}) is real and depends only on $\abs{\mathsf{k}\cdot\mathsf{u}}$,
\begin{equation}\label{const-smearing}
F(\mathsf{k},\tau)=F(\abs{\mathsf{k}\cdot\mathsf{u}}).
\end{equation}
\section{Assisted harvesting and catalysis of quantum coherence}\label{sec:harvesting}
Suppose now that before the interaction is switched on at a time $\tau_\text{on}$, the combined system of detector and field starts out in a separable state of the form
\begin{equation}\label{separable}
\ketbra{g}\otimes\sigma_\phi,
\end{equation}
where the detector occupies its lowest energy level and the field is in a state $\sigma_\phi$. The final state of the system after a time $\tau_\text{off}$ at which the interaction is switched off, can be obtained by evolving Eq. (\ref{separable}) with the unitary operator
\begin{equation}\label{evolution}
\hat{U}=\mathcal{T}\text{exp}\left(-i\int\limits_{\tau_\text{on}}^{\tau_\text{off}}\hat{H}_{\text{int}}(\tau)d\tau\right),
\end{equation}
where $\mathcal{T}$ denotes time ordering. Assuming that the switching function has a compact support we can extend the limits over $\pm\infty$. Setting
\begin{equation}\label{Phi}
\hat\Phi=\int\limits_{-\infty}^{+\infty}\chi(\tau)e^{-i\Omega\tau}\partial_\tau\hat\phi_f(\mathsf{x}(\tau))d\tau,
\end{equation}
Eq. (\ref{evolution}) can then be rewritten as
\begin{equation}\label{eq-evolution}
\hat{U}=\text{exp}\left[-i\lambda(\ketbra{e}{g}\otimes\hat\Phi^\dagger+\ketbra{g}{e}\otimes\hat\Phi)\right].
\end{equation}
Tracing out the field degrees of freedom, one can obtain the state of the detector after the interaction which in this case is equal to
\begin{equation}\label{det-state}
\rho_\text{\tiny\ensuremath D}=\left(\begin{array}{cc}
1-\lambda^2\tr^{}(\hat\Phi^\dagger\sigma_\phi\hat\Phi)& i\lambda \tr^{}(\hat\Phi\sigma_\phi) \\
-i\lambda \tr^{}(\hat\Phi^\dagger\sigma_\phi)& \lambda^2\tr^{}(\hat\Phi^\dagger\sigma_\phi\hat\Phi)
\end{array}\right)+\mathcal{O}(\lambda^3).
\end{equation}
In a similar fashion, by taking the partial trace over the detector's Hilbert space, we can obtain the state of the field after harvesting,
\begin{equation}\label{field-state}
\sigma_\phi'=\sigma_\phi+\lambda^2\left(\hat\Phi^\dagger\sigma_\phi\hat\Phi-\frac12\left\{\hat\Phi\hat\Phi^\dagger,\sigma_\phi\right\}\right)+\mathcal{O}(\lambda^4).
\end{equation}
With the help of Eqs. (\ref{coh_measure}) and (\ref{det-state}) the amount of coherence harvested to the detector to lowest order in the coupling constant is equal to
\begin{equation}\label{coher}
C=2\lambda\abs{\tr^{}(\hat\Phi\sigma_\phi)}.
\end{equation}
Defining
\begin{equation}
\mathcal{F_\pm}(\mathbf{k})=\int\limits_{-\infty}^{+\infty}\chi(\tau)e^{\pm i\Omega\tau}\partial_\tau \left(F(\mathsf k,\tau)e^{i\mathsf k\cdot\mathsf x(\tau)}\right)d\tau,
\end{equation}
Eq. (\ref{coher}) can be written as
\begin{equation}\label{explicit-coher}
C=2\lambda\abs{\int\frac{d^n\mathbf{k}}{\sqrt{(2\pi)^n2\abs{\mathbf{k}}}}\Big(\mathcal{F}_-(\mathbf{k})a(\mathbf{k})+\mathcal{F}^*_+(\mathbf{k})a^*(\mathbf{k})\Big)},
\end{equation}
where
\begin{equation}\label{ampl-distr}
a(\mathbf{k})=\tr^{}(\hat a_{\mathbf{k}}\sigma_\phi)
\end{equation} is the \emph{coherent amplitude distribution} of the field.
Suppose that we wish to repeat the process and extract coherence onto a fresh detector copy. It is straightforward to see that for the $m$-th harvest one can extract an amount of
\begin{equation}\label{m-coher}
C^{(m)}=2\lambda\abs{\tr^{}(\hat\Phi\sigma_\phi^{(m)})}
\end{equation}
units of coherence from a perturbed field in the state
\begin{equation}\label{m-field-state}
\sigma_\phi^{(m)}=\sigma_\phi^{(m-1)}+\lambda^2\left(\hat\Phi^\dagger\sigma_\phi^{(m-1)}\hat\Phi-\frac12\left\{\hat\Phi\hat\Phi^\dagger,\sigma_\phi^{(m-1)}\right\}\right).
\end{equation}
By combining Eqs. (\ref{m-coher}) and (\ref{m-field-state}) and exploiting the cyclic property of the trace as well as the fact that $[\hat\Phi,\hat\Phi^\dagger]$ is a $c$-number (for proof see Appendix \ref{appendix-useful}) it follows that
\begin{equation}\label{repeat-coh}
C^{(m+1)}=C^{(m)}\abs{1+\frac{\lambda^2}{2}\left[\hat\Phi,\hat\Phi^\dagger\right]},
\end{equation}
so to lowest order in the coupling constant the amount of coherence harvested each time remains the same.
\begin{figure}
\centering
\includegraphics[width=\columnwidth]{Plots/harvesting.jpg}
\caption{\textbf{Assisted harvesting of quantum coherence.} A moving two-level system, initially in its ground state at some time $t<t_\text{on}$, interacts with a massless scalar field through a derivative coupling. The process requires an external flow of energy which assists harvesting by increasing the combined system's coherence. After the interaction is switched off at a time $t_\text{off}$ the detector will find itself in a superposition between its energy levels.}
\label{fig:harvesting}
\end{figure}
Let's focus our attention on normalised smearing and switching functions such that
\begin{equation}\label{fixing}
\int\limits_{-\infty}^{+\infty}\chi(\tau)d\tau=\int_{\mathcal S(\tau)}f(\boldsymbol\xi)d^n\boldsymbol\xi=1,
\end{equation}
and define
\begin{equation}
R=\int_{\mathcal S(\tau)}\abs{\xi}f(\boldsymbol\xi)d^n\boldsymbol\xi
\end{equation}
as the mean radius of the detector and
\begin{equation}
T=\int\limits_{-\infty}^{+\infty}\abs{\tau}\chi(\tau)d\tau
\end{equation}
as the mean interaction duration respectively. This will make it easier to compare different setups and will allow the study, in a unified way, of the effects that different sizes and finite interaction durations have on harvesting as well as the limiting case of an instantaneous interaction in which $\chi(\tau)=\delta(\tau)$. In this limit, $[\hat\Phi,\hat\Phi^\dagger]=0$ and the amount harvested each time is exactly the same to any order (for more details see Appendix \ref{second-appendix}). It seems that when the detector interacts with the field through a delta coupling, coherence harvesting is catalytic \cite{PhysRevLett.113.150402}. Even though in principle this is allowed for infinite dimensional systems that act as coherence reservoirs \cite{PhysRevLett.123.020403,PhysRevLett.123.020404}, it is not certain if this is the case here. Since the interaction Hamiltonian does not commute with the unperturbed part, $\hat H_\text{\tiny\ensuremath D}+\hat H_\phi$, of the total Hamiltonian, the process requires an outside supply of positive energy $\Delta E$ each time \cite{Hackl2019minimalenergycostof,Bény2018}. Energy non-conserving unitaries like the one in Eq. (\ref{evolution}) can increase the coherence of the combined system assisting in the extraction process \cite{PhysRevA.92.032331,BU20171670} (see Fig. \ref{fig:harvesting}). Nonetheless a necessary condition for extracting a non trivial amount of coherence is for the field to be in a state with a non-zero coherent amplitude distribution.
\section{Inertial detectors}\label{Sec:Inert}
We will now consider an inertial detector which is moving along a worldline with a constant velocity $\boldsymbol\upsilon$, and whose center of mass coordinates is given by
\begin{equation}\label{coordinates}
\mathsf{x}(\tau)=\mathsf{u}\tau,
\end{equation}
where $\mathsf{u}=\gamma(1,\boldsymbol{\upsilon})$ is its four-velocity, with $\gamma=1/\sqrt{1-\upsilon^2}$ the Lorentz factor. For a spherically symmetric smearing function with a positive Fourier transform, it can be proven that
\begin{thrm}
For a suitable choice of the coherent amplitude distribution's phase the maximum amount of harvested coherence to lowest order, is obtained by a detector interacting instantaneously with the field.
\end{thrm}
\begin{proof}
Taking the absolute value inside the integral in Eq. (\ref{explicit-coher}) we find that
\begin{equation}
C\leq2\lambda{\int\frac{d^n\mathbf{k}}{\sqrt{(2\pi)^n2\abs{\mathbf{k}}}}\abs{a(\mathbf{k})}\left(\abs{\mathcal{F}_-(\mathbf{k})}+\abs{\mathcal{F}_+(\mathbf{k})}\right)}.
\end{equation}
For a detector moving with a constant velocity the Fourier transform of the smearing function no longer depends on its proper time, in this case
\begin{equation}
\mathcal{F}_-(\mathbf{k})=i(\mathsf k\cdot\mathsf u)F(\abs{\mathsf k\cdot\mathsf u})X^*(\Omega-\mathsf k\cdot\mathsf u)
\end{equation}
and
\begin{equation}
\mathcal{F}_+(\mathbf{k})=i(\mathsf k\cdot\mathsf u)F(\abs{\mathsf k\cdot\mathsf u})X(\Omega+\mathsf k\cdot\mathsf u)
\end{equation}
where
\begin{equation}\label{Xi}
X(\Omega\pm\mathsf k\cdot\mathsf u)=\int\limits_{-\infty}^{+\infty}\chi(\tau)e^{i(\Omega\pm\mathsf k\cdot\mathsf u)\tau}d\tau.
\end{equation}
Because of the normalization property in Eq. (\ref{fixing}), $\abs{X(\Omega\pm\mathsf k\cdot\mathsf u)}\leq 1$ so finally
\begin{equation}\label{inequality}
C\leq4\lambda\int\frac{(-\mathsf k\cdot\mathsf u)}{\sqrt{(2\pi)^n2\abs{\mathbf{k}}}}F(\abs{\mathsf k\cdot\mathsf u})\abs{a(\mathbf{k})}d^n\mathbf{k},
\end{equation}
where equality holds for $\chi(\tau)=\delta(\tau)$ and a coherent amplitude distribution with phase, $\text{arg}(a(\mathbf{k}))=\frac{\pi}{2}$ \footnote{In the Unruh-DeWitt interaction where the factor $(-\mathsf k\cdot\mathsf u)$ in the numerator is absent, the Theorem holds for an arbitrary motion of the detector as long as $\mathsf{x}(0)=0$.}.
Note that if the Fourier transform of the smearing function is not positive then Eq. (\ref{inequality}) is only an upper bound on the amount of harvested coherence.
\end{proof}
If the amplitude distribution is also spherically symmetric then
\begin{multline}\label{moving-coher}
C=2\lambda\left|\int\frac{(-\mathsf k\cdot\mathsf u)F(\abs{\mathsf k\cdot\mathsf u})}{\sqrt{(2\pi)^n2\abs{\mathbf{k}}}}\left[a(\abs{\mathbf{k}})X^*(\Omega-\mathsf k\cdot\mathsf u)\right.\right.\\\left.-{a}^*(\abs{\mathbf{k}})X(\Omega+\mathsf k\cdot\mathsf u)\right]d^n\mathbf{k}\Bigg|,
\end{multline}
which for a static detector reduces to
\begin{multline}\label{static-coher}
C=\frac{2\lambda s_n}{\sqrt{2(2\pi)^n}}\left|\int_0^{\infty}k^{n-\frac12}F(k)\left[a(k)X^*(\Omega+k)\right.\right.\\\left.-a^*(k)X(\Omega-k)\right]dk\Bigg|, \end{multline}
where $s_n=\frac{2\pi^{n/2}}{\Gamma(n/2)}$ is the surface area of the unit $n$-sphere.
By boosting the four-momentum $\mathsf k$ to the detector's frame of reference it can be shown that Eq. (\ref{moving-coher}) is equivalent to Eq. (\ref{static-coher}) with a symmetric coherent amplitude distribution of the form
\begin{equation}\label{moving-distribution}
a_{\upsilon}(k)=\frac1{s_n}\int a\left(\frac{k}{\gamma(1-\mathbf{\boldsymbol{\upsilon}}\cdot\hat\mathbf{k})}\right)\frac{d\hat\mathbf{k}}{{[\gamma(1-\mathbf{\boldsymbol{\upsilon}}\cdot\hat\mathbf{k})]^{n-\frac12}}}.
\end{equation}
From the detector's point of view, the field's coherent amplitude is equivalent to a mixture of Doppler shifted distributions with weight equal to $[s_n\gamma(1-\boldsymbol{\upsilon}\cdot\hat\mathbf{k})^{n-\frac12}]^{-1}$. For a similar result regarding the interaction of an inertial detector with a heat bath see \cite{PhysRevD.102.085005}.
\section{Assisted harvesting and catalysis from a coherent field}\label{sec:harvest}
For a coherent state $\ket{a}$ of the field, the coherent amplitude distribution in Eq. (\ref{ampl-distr}) is equal to the eigenvalue of the annihilation operator with mode $\mathbf{k}$
\begin{equation}
\hat{a}_{\mathbf{k}}\ket{a}=a(\mathbf{k})\ket{a},
\end{equation}
in this case the amount of harvested coherence to lowest order is given by the expectation value of the field operator $\hat\Phi$
\begin{equation}
C=2\lambda|{\bra{a}\hat{\Phi}\ket{a}}|.
\end{equation}
The energetic cost associated with harvesting is equal to the energy difference between the final and initial states of the combined system of detector and field
\begin{equation}
\Delta E=\tr(\hat H_\text{\tiny\ensuremath D}(\rho_\text{\tiny\ensuremath D}-\ketbra{g}))+\tr(\hat H_\phi(\sigma_\phi'-\ketbra{a})).
\end{equation}
To lowest order this splits into two contributions
\begin{equation}
\Delta E=\Delta E_\text{coh}+\Delta E_\text{vac},
\end{equation}
where
\begin{equation}
\Delta E_\text{coh}=\frac{C^2}{4}\left(\Omega+4\Re\left[\frac{\bra{a}[\hat\Phi,\hat{H}_\phi]\ket{a}}{\bra{a}\hat\Phi\ket{a}}\right]\right)
\end{equation}
is the cost associated with harvesting and
\begin{equation}
\Delta E_\text{vac}=\frac{\lambda^2}{2(2\pi)^n}\int\left(1+\frac{\Omega}{\abs{\mathbf{k}}}\right)\abs{\mathcal{F}_-(\mathbf{k})}^2d^n\mathbf{k}.
\end{equation}
is the cost of interacting with the vacuum \cite{PhysRevD.96.025020}.
Let us consider an inertial detector and a harvesting process in which the switching and smearing functions are respectively given by the following Gaussians
\begin{equation}\label{gaussian-switching}
\chi(\tau)=\frac{\text{exp}\left(-\frac{\tau^2}{\pi T^2}\right)}{\pi T}
\end{equation}
\begin{equation}\label{gaussian-smearing}
f(\boldsymbol\xi)=\frac{\text{exp}\left(-\frac{\boldsymbol\xi^2}{\pi R_n^2}\right)}{(\pi R_n)^n},
\end{equation}
while the state of the field is described by a coherent amplitude distribution with a unit average number of excited quanta of the form
\begin{equation}\label{amplitude}
a(\mathbf{k})=\frac{\text{exp}(-\frac{k^2}{2\pi {E}_n^2}+i\frac{\pi r}{2})}{(\pi {E}_n)^{n/2}},\quad r=0,1
\end{equation}
where
\begin{equation}
E_n=\frac{s_{n+1}}{\pi s_n}E\quad\text{and}\quad R_n=\frac{s_{n+1}}{\pi s_n}R,
\end{equation}
with $E=\bra{a}\hat H_\phi\ket{a}$ the mean initial energy of the field. Note that even though the support of Eq. (\ref{gaussian-switching}) is no longer compact, as was originally required, the analysis is expected to present a good approximation to a compact switching function of the form
\begin{equation}
\chi(\tau)=\begin{cases}
{\text{exp}(-\frac{\tau^2}{\pi T^2})}/(\pi T),&|\tau|\leq\mathcal{T}\\
0,&\text{otherwise}
\end{cases}
\end{equation}
provided that $\mathcal{T}\geq 4\sqrt{\pi}T$. We will now treat the static and moving cases separately.
\subsection{Static detector}
For $\upsilon=0$ the Fourier transforms of the switching and smearing functions are equal to
\begin{equation}\label{tint}
X(\Omega\pm k)=\text{exp}\left[{-\frac{\pi(\Omega\pm k)^2T^2}{4}}\right]
\end{equation}
and
\begin{equation}\label{calculated-Fourier}
F(\mathsf{k})=\text{exp}\left[-\frac{\pi k^2R_n^2}{4}\right]
\end{equation}
respectively. Inserting these into Eq. (\ref{static-coher}) we obtain that the amount of harvested coherence, which now depends on the initial energy of the field, the mean interaction duration and the mean radius of the detector is
\begin{multline}\label{C}
C(E,T,R)=\frac{4\lambda s_n}{\sqrt{2(2\pi^2{E}_n)^n}}e^{-\frac{\pi\Omega^2T^2}{4}}\\
\times\int_0^\infty k^{n-\frac12}e^{-\mathrm{a}{k^2}}\sinh^{1-r}(bk)\cosh^r(bk)dk,
\end{multline}
with
\begin{equation}
\mathrm{a}=\frac{1}{2\pi {E}_n^2}\left[1+\frac{\pi^2E_n^2(R^2_n+T^2)}{2}\right], \quad b=\frac{\pi \Omega T^2}{2}.
\end{equation}
The integral on the right hand side is equal to
\begin{figure*}
\begin{minipage}{\textwidth}
\subfloat[$r=1$]{\includegraphics[width=0.45\textwidth]{Plots/1dphu0.png}}\hspace{0.5cm}
\subfloat[$r=0$]{\includegraphics[width=0.45\textwidth]{Plots/1du0.png}}
\end{minipage}
\caption{Amount of harvested coherence $C/\bar\lambda$ from a coherent scalar field in $1+1$ dimensions and a Gaussian amplitude distribution with phase a) $\phi=\frac{\pi}{2}$ and b) $\phi=0$, as a function of the mean initial energy of the field (in units $\Omega$) and the mean interaction duration (in units $1/\Omega$), for a detector with mean radius $R=1/\Omega$.}
\label{fig:static1}
\begin{minipage}{\textwidth}
\subfloat[$r=1$]{\includegraphics[width=0.45\textwidth]{Plots/3dphu0.png}}\hspace{0.5cm}
\subfloat[$r=0$]{\includegraphics[width=0.45\textwidth]{Plots/3du0.png}}
\end{minipage}
\caption{Amount of harvested coherence $C/\bar\lambda$ from a coherent scalar field in $3+1$ dimensions and a Gaussian amplitude distribution with phase a) $\phi=\frac{\pi}{2}$ and b) $\phi=0$, as a function of the mean initial energy of the field (in units $\Omega$) and the mean interaction duration (in units $1/\Omega$), for a detector with mean radius $R=1/\Omega$}\label{fig:static3}
\end{figure*}
\begin{widetext}
\begin{equation}\label{cylindrical}
\int\limits_{0}^\infty k^{n-\frac{1}{2}}e^{-\mathrm{a}k^2}\sinh^{1-r}(bk)\cosh^r(bk)dk=
\frac{\Gamma(n+\frac12)}{2(2\mathrm{a})^{\frac{n}2+\frac14}}e^\frac{b^2}{8\mathrm{a}}\left[D_{-n-\frac12}\left(-\frac{b}{\sqrt{2\mathrm{a}}}\right)-(-1)^{r}D_{-n-\frac12}\left(\frac{b}{\sqrt{2\mathrm{a}}}\right)\right],\quad b>0
\end{equation}
where $D_p(z)$ denotes the \emph{parabolic cylinder function} \cite{gradshteyn2014table}. In a similar way it can be shown that
\begin{equation}
\Delta E_\text{coh}=\frac{C^2}{4}\left[\Omega-\frac{4(n+\frac12)}{\sqrt{2\mathrm{a}}}\frac{D_{-n-\frac32}\left(-\frac{b}{\sqrt{2\mathrm{a}}}\right)+(-1)^{r}D_{-n-\frac32}\left(\frac{b}{\sqrt{2\mathrm{a}}}\right)}{D_{-n-\frac12}\left(-\frac{b}{\sqrt{2\mathrm{a}}}\right)-(-1)^{r}D_{-n-\frac12}\left(\frac{b}{\sqrt{2\mathrm{a}}}\right)}\right]
\end{equation}
and
\begin{equation}
\Delta E_\text{vacuum}=\frac{\lambda^2\pi s_n\Gamma(n+1)}{(8\pi^2\mathrm{a}')^\frac{n+1}{2}}e^{-\frac{\pi\Omega^2T^2}{2}+\frac{b^2}{8\mathrm{a}'}}\left[\frac{n+1}{\sqrt{2\mathrm{a}'}}D_{-n-2}\left(\frac{2b}{\sqrt{2\mathrm{a}'}}\right)+\Omega D_{-n-1}\left(\frac{2b}{\sqrt{2\mathrm{a}'}}\right)\right],
\end{equation}
where
\begin{equation}
\mathrm{a}'=\frac{\pi(R_n^2+T^2)}{2}.
\end{equation}
\end{widetext}
In Figs. \ref{fig:static1} and \ref{fig:static3} we present the amount of coherence harvested, scaled by the dimensionless coupling constant $\bar\lambda=\lambda\Omega^\frac{n+1}{2}$, as a function of the initial mean energy $E$ of the field (in units $\Omega$) and the interaction duration $T$ (in units $1/\Omega$) for a $1+1$ and a $3+1$ dimensional Mikowski spacetime respectively. In order to simplify the situation we will tacitly assume from now on that the mean radius of the qubit is equal to its transition wavelength $R=1/\Omega$. It is clear from both figures that the harvesting profile depends strongly on the phase of the coherent amplitude distribution. For $r=1$ and for a fixed initial field energy, the maximum amount that can be harvested is obtained through the use of an instantaneous interaction ($T=0$), in agreement with the Theorem of Sec. \ref{Sec:Inert}. When $r=0$ it is impossible to harvest coherence to a qubit interacting instantaneously with the field, in this case the maximum is obtained for interaction durations comparable to the mean radius. In both settings, if the initial energy of the field is zero the amount of coherence harvested vanishes. This is also true for very large energy values. Qualitatively, harvesting is more efficient for field energies comparable to the energy gap. For a resonant energy of the field, $E=\Omega$, it is possible to extend the process to greater interaction times compared to other energies and still extract a small amount of coherence.
Now with the help of Eq. (\ref{commutator}) of Appendix \ref{appendix-useful}, Eqs. (\ref{tint})-(\ref{calculated-Fourier}) and Eq. (\ref{cylindrical}) it can be shown that
\begin{multline}
\lambda^2[\hat\Phi,\hat\Phi^\dagger]=-\frac{2ns_n\bar\lambda^2}{s_{2n}[4\pi\Omega^2(R_n^2+T^2)]^\frac{n+1}{2}}e^{-\frac{\pi\Omega^2T^2(2R^2_n+T^2)}{4(R^2_n+T^2)}}\\\times\left[D_{-n-1}\left(\!-\sqrt\frac{\pi\Omega^2 T^4}{R^2_n+T^2}\right)\!-D_{-n-1}\left(\sqrt\frac{\pi\Omega^2T^4}{R^2_n+T^2}\right)\right].
\end{multline}
From Fig. \ref{fig:com} it can be seen that for $\bar\lambda<<1$ and $R=1/\Omega$ this term is negligible. Since the maximum amount of harvested coherence is of the same order as $\bar\lambda$ then, according to Eq. (\ref{repeat-coh}), we can repeat the process $m$ times for a total of $C_{\text{tot}}=\mathcal{O}(m\bar\lambda)$ units of coherence. Assuming that for a phase-less coherent amplitude distribution obtaining the maximum in each harvest requires a time of approximately $T=1/\Omega$ it follows the total duration is of the order $\mathcal{O}(m/\Omega)$. To extract a single unit of coherence requires therefore approximately $\mathcal{O}(1/\bar\lambda\Omega)$ seconds. For a transition frequency in the optical spectrum and $\bar\lambda=10^{-3}$ this time is of the order of $10^{-12}$ seconds.
\begin{figure}
\includegraphics[width=\columnwidth]{Plots/comm.png}
\caption{$\lambda^2[\hat\Phi,\hat\Phi^\dagger]/\bar\lambda^2$ as a function of the mean interaction duration (in units $1/\Omega$), for a detector with mean radius $R=1/\Omega$.}
\label{fig:com}
\end{figure}
\subsection{Detector moving at a constant velocity}
\begin{figure*}\begin{minipage}{\textwidth}
\subfloat[$C_{0.8}/\bar\lambda$ ($r=1$)]{\includegraphics[scale=0.36]{Plots/1dphu08.png}}\hspace{0.45cm}
\subfloat[$C_0/C_{0.8}$ ($r=1$)]{\includegraphics[scale=0.36]{Plots/1dphswu08.png}}\hspace{0.45cm}
\subfloat[$E=0.1\Omega$ ($r=1$)]{\includegraphics[scale=0.38]{Plots/1Cph08.png}}\\
\subfloat[$C_{0.8}/\bar\lambda$ ($r=0$)]{\includegraphics[scale=0.36]{Plots/1du08.png}}\hspace{0.42cm}
\subfloat[$C_0/C_{0.8}$ ($r=0$)]{\includegraphics[scale=0.36]{Plots/1dswu08.png}}\hspace{0.45cm}
\subfloat[$E=0.1\Omega$ ($r=0$)]{\includegraphics[scale=0.38]{Plots/1C08.png}}
\end{minipage}
\caption{Left: Amount of harvested coherence, $C_{0.8}/\bar\lambda$, in $1+1$ dimensions. Center: Amount of swelling $C_0/C_{0.8}$. Right: Comparison between a static and a moving detector for an initial energy of the field $E=0.1\Omega$.}
\label{fig:6}
\begin{minipage}{\textwidth}
\subfloat[$C_{0.8}/\bar\lambda$ ($r=1$)]{\includegraphics[scale=0.36]{Plots/3dphu08.png}\hspace{0.5cm}}
\subfloat[$C_0/C_{0.8}$ ($r=1$)]{\includegraphics[scale=0.36]{Plots/3dphswu08.png}}\hspace{0.45cm}
\subfloat[$E=0.2\Omega$ ($r=1$)]{\includegraphics[scale=0.38]{Plots/3Cph08.png}}\\
\subfloat[$C_{0.8}/\bar\lambda$ ($r=0$)]{\includegraphics[scale=0.36]{Plots/3du08.png}}
\hspace{0.4cm}
\subfloat[$C_0/C_{0.8}$ ($r=0$)]{\includegraphics[scale=0.36]{Plots/3dswu08.png}}\hspace{0.45cm}
\subfloat[$E=0.2\Omega$ ($r=0$)]{\includegraphics[scale=0.38]{Plots/3C08.png}}
\end{minipage}
\caption{Left: Amount of harvested coherence, $C_{0.8}/\bar\lambda$, in $3+1$ dimensions. Center: Amount of swelling $C_0/C_{0.8}$. Right: Comparison between a static and a moving detector for an initial energy of the field $E=0.2\Omega$.}
\label{fig:7}
\end{figure*}
According to Eq. (\ref{moving-distribution}), a detector moving at a constant velocity still perceives the field as a coherent state but in a mixture of static coherent amplitude distributions of the form (\ref{amplitude}) with Doppler shifted energies equal to
\begin{equation}
E(\boldsymbol\upsilon)=E\gamma(1-\boldsymbol\upsilon\cdot\hat\mathbf{k}).
\end{equation}
The amount of harvested coherence in this case is given by
\begin{equation}
C_\upsilon(E,T,R)=\frac{1}{s_n}\int \frac{C(E(\boldsymbol\upsilon),T,R)}{\gamma(1-\boldsymbol\upsilon\cdot\hat\mathbf{k})^\frac{n-1}{2 }}d\hat{\mathbf{k}}.
\end{equation}
In Figs. \ref{fig:6} and \ref{fig:7} we numerically evaluate this amount for a detector moving at a constant relativistic speed of $\upsilon=0.8$, in $1+1$ and $3+1$ dimensions respectively. We observe that close to resonance the amount of coherence harvested decreases with an increasing value of the detector's speed. As in \cite{KBM}, for lower and higher initial energies of the field there exist ``swelling" regions, where it is possible to extract more coherence to a moving than to a static detector. However, this effect becomes less intense for a higher spacetime dimension.
\subsection{Assisted catalysis}
For an instantaneous interaction coherence harvesting is catalytic. Despite the fact that after each harvest the state of the field has changed, it is possible to extract the same amount of coherence to a sequence of detectors. Ignoring the trivial case of $r=0$, for a coherent amplitude distribution with phase $\phi=\frac{\pi}{2}$ each detector will harvest
\begin{equation}
C_\upsilon(E)=\frac{2\lambda\Gamma(3/4) }{(2\pi)^\frac14}\left[\frac{E_+}{\left(1+\frac{\pi^2E_+^2}{\Omega^2}\right)^\frac34}+\frac{E_-}{\left(1+\frac{\pi^2E_-^2}{\Omega^2}\right)^\frac34}\right]
\end{equation}
units of coherence in $1+1$ and
\begin{multline}
C_\upsilon(E)=\\\frac{16\bar\lambda \Gamma(3/4)}{(2\pi^9)^\frac14\gamma\upsilon}
\left[\left(1+\frac{\pi^2E_-^2}{32\Omega^2}\right)^{-\frac{3}{4}}-\left(1+\frac{\pi^2E_+^2}{32\Omega^2}\right)^{-\frac{3}{4}}\right]
\end{multline}
in $3+1$ dimensions, where $E_\pm=E\gamma(1\pm\upsilon)$ denote the field's relativistic Doppler shifted energies. As has already been mentioned in Sec. \ref{sec:harvesting}, catalysis is an energy consuming process. The cost of each extraction to lowest order in this case is equal to
\begin{equation}\label{Enikost}
\Delta E=\begin{cases}
\frac{C_\upsilon^2(E)\Omega}{4}+\frac{\bar\lambda^2\Omega}{\pi^2}(1+\frac{\gamma}{\sqrt{2}}),&n=1\\
&\\
\frac{C_\upsilon^2(E)\Omega}{4}+\frac{8\bar\lambda^2\Omega}{\pi^4}\left(1+\frac{3\gamma}{\sqrt{2}}\right),&n=3.
\end{cases}
\end{equation}
In Fig. \ref{fig:Ecost} we plot the
amount of coherence harvested through catalysis along with its energy cost (in units $\Omega$) as a function of the initial energy of the field.
For field energies close to resonance the amount obtained is maximized. Once again it can be seen that this amount decreases for an increasing value of the detector's speed. This is also true for the energy cost associated with harvesting. On the other hand, the cost associated with the vacuum remains relatively constant.
\section{Conclusions}\label{conclus}
\begin{figure*}
\subfloat[$\upsilon=0$]{\includegraphics[width=0.32\textwidth]{Plots/Enc1u0.png}}\hspace{0.2cm}
\subfloat[$\upsilon=0.6$]{\includegraphics[width=0.32\textwidth]{Plots/Enc1u06.png}}\hspace{0.2cm}
\subfloat[$\upsilon=0.8$]{\includegraphics[width=0.32\textwidth]{Plots/Enc1u08.png}}\\
\subfloat[$\upsilon=0$]{\includegraphics[width=0.32\textwidth]{Plots/Enc3d.png}}\hspace{0.2cm}
\subfloat[$\upsilon=0.6$]{\includegraphics[width=0.32\textwidth]{Plots/Enc3u06.png}}\hspace{0.2cm}
\subfloat[$\upsilon=0.8$]{\includegraphics[width=0.32\textwidth]{Plots/Enc3u08.png}}
\caption{Amount of harvested coherence $C_\upsilon/\bar\lambda$ and cost in energy $\Delta E/\Omega\bar\lambda^2$ as a function of the initial energy of the field (in units $\Omega$) for various detector speeds. Upper: 1+1 dimensions. Lower: 3+1 dimensions.}
\label{fig:Ecost}
\end{figure*}
We have thoroughly investigated the conditions under which an UDW detector, coupled to a massless scalar field through a derivative coupling, succeeds in harvesting quantum coherence. It was proven that for an instantaneous interaction between detector and field, harvesting is catalytic, i.e., the same amount can be repeatedly extracted. For a suitable choice of the field's coherent amplitude distribution and an inertial detector, when the Fourier transform of the smearing function is positive this is also the maximum amount that can be obtained. By considering as an example a harvesting protocol in which the switching, smearing and coherent amplitude functions are Gaussian, it was demonstrated that for a coherent state of the field the process depends on the phase of the amplitude, the mean initial field energy, the mean interaction duration and the mean radius of the detector. We observed that, for a resonant energy of the field, it is possible to extend the process to longer interaction durations. It was also shown that the total time required to harvest, through repeated applications of the protocol, a single unit of coherence to a sequence of detectors is very short.
For a detector moving at a constant velocity and for a mean radius equal to the inverse of its transition frequency we verify the presence of swelling affects as was reported in \cite{KBM}. Nonetheless, since energy non-conserving interactions such as the one considered here are coherence generating \cite{PhysRevA.92.032331,BU20171670}, it is possible that this increase is due to the interaction. To avoid this possibility and in order to be able to determine how different parameters which are intrinsic to the combined system of qubit and field affect harvesting, we will study, in future work , protocols under energy conserving interactions such as the one given by the Glauber photodetection model \cite{PhysRev.130.2529,PhysRevD.46.5267} for example.
\acknowledgments{
The authors wish to thank Lena S. Peurin for fruitful discussions during preparation of this manuscript. D. M.'s research is co-financed by Greece and the European Union (European Social Fund-ESF) through the Operational Programme ``Human Resources Development, Education and Lifelong Learning" in the context of the project ``Reinforcement of Postdoctoral Researchers - 2nd Cycle" (MIS-5033021), implemented by the State Scholarships Foundation (IKY).}
|
1612.00467
|
\section{Introduction}
The steadily growing amount of digitized clinical data such as health records, scholarly medical literature, systematic reviews of substances and procedures, or descriptions of clinical trials holds significant potential for exploitation by automatic inference and data mining techniques. Besides the wide range of clinical research questions such as drug-to-drug interactions~\cite{wienkers2005predicting} or quantitative population studies of disease properties~\cite{wren2005data}, there is a rich potential for applying data-driven methods in daily clinical practice for key tasks such as decision support~\cite{kawamoto2005improving} or patient mortality prediction~\cite{moreno2005saps}. The latter task is especially important in clinical practice when prioritizing allocation of scarce resources or determining the frequency and intensity of post-discharge care.
There has been an active line of work towards establishing probabilistic estimators of patient mortality both in the clinical institution as well as after discharge~\cite{pirracchio2015mortality,che2016recurrent,johnson2016machine}. The authors report solid performance on both publicly available and proprietary clinical datasets.
In spite of these encouraging findings, we note that most competitive approaches rely on time series and demographic information while algorithmic processing of the unstructured textual portion of clinical notes remains an important, yet, to date, insufficiently studied problem. The few existing advances towards tapping into this rich source of information rely on term-wise representations such as tf-idf embeddings~\cite{ghassemi2014unfolding} or distributions across latent topic spaces~\cite{lehman2012risk}.
This intuitively appears sub-optimal since several studies have independently highlighted the importance of accounting for phrase compositionality manifested, \textit{e.g.}, in the form of negations~\cite{kuhn2016implicit}, or long-range dependencies in clinical resources. Models that solely rely on point estimates of term semantics cannot be assumed to adequately capture such interactions.
In this paper, we aim to address these shortcomings by presenting a convolutional neural network architecture that explicitly represents not just individual terms but also entire phrases or documents in a way that preserves such subtleties of natural language.
The remainder of this paper is structured as follows: Section~\ref{sec:model} introduces our model and our objective function. Subsequently, in Section~\ref{sec:experiments}, we empirically evaluate the model against two competitive baselines on the task of intensive care unit (ICU) mortality prediction on the popular MIMIC-III database~\cite{johnson2016mimic}. Finally, Section~\ref{sec:conclusion} concludes with a brief discussion of our findings.
\section{Model}
\label{sec:model}
While simple feed-forward architectures, such as the doc2vec scheme~\cite{le2014distributed}, have been established as versatile plug-in modules in many machine learning applications~\cite{lee2016sentiment,lau2016empirical}, they are inherently incapable of directly recognizing complex multi-word or multi-sentence patterns. However, constructions such as \textit{no sign of pneumothorax} are frequently encountered in clinical notes and encode crucial information for the task of mortality prediction.
Following recent work in document classification~\cite{yang2016} and dialogue systems~\cite{serbanSBCP15}, we adopt a two-layer architecture. Let $d=\langle s_1,\dots,s_n\rangle$ denote a patient's record comprising $n$ sentences. Our first layer independently maps sentences $s_i$ to sentence vectors $\mathbf x_i\in\mathbb R^{D_S}$. The second layer combines $\langle \mathbf x_1,\dots,\mathbf x_n\rangle$ into a single patient representation $\mathbf x\in\mathbb R^{D_P}$. For both levels we use convolutional neural networks (CNNs) with max-pooling which have shown excellent results on binary text classification tasks \cite{kim2014}, \cite{severyn2015}. Following work by Severyn \textit{et al}~\cite{severyn2015}, we use word-embeddings to provide vector-input for the first CNN layer. Finally, the output of our model is $p(y), y\in[0,1]$, the estimated mortality probability, and our objective is the cross entropy $l(y, y^\star)$ where $y^\star$ is the ground-truth label. The graph rendered in black in Figure \ref{figArchitecture} depicts this basic architecture.
\begin{figure}[ht]
\begin{tikzpicture}[scale=0.5]
\matrix (m) [matrix of math nodes,row sep=1.4em,column sep=0.4em,minimum width=0.1em]
{
s_1 & s_2 & \cdots & s_n & & &\\
\mathbf x_1 & \mathbf x_2 & & \mathbf x_n & \mathbf x_{\phantom{n}}& p( y|\mathbf x) & l(y,y^\star|\mathbf x)\\
\colA{p( y|\mathbf x_1)} & \colA{p( y|\mathbf x_2)} &\colA{\dots} & \colA{p( y|\mathbf x_n)} & & & \mathcal L = l( y, y^\star|\mathbf x) \colA{+ \lambda\mathcal R} \\
\colA{\frac 1 n l_1(y,y^\star|\mathbf x_1)} & \colA{\frac 1 n l_2(y,y^\star|\mathbf x_2)} & & \colA{\frac 1 n l_n(y,y^\star|\mathbf x_n)} & \colA{=\mathcal R}& {}& {\phantom{\cdot}}\\
};
\path[->] (m-1-1) edge [double] node [left, align=center] {\textit{word-level}\\ \textit{CNN}} (m-2-1);
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\draw[->] (m-2-6) -- (m-2-7);
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\draw[->,color=rosso] (m-2-2) -- (m-3-2);
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\node [above = 0 em of m-2-5, align=center] {\textit{patient} \\ \textit{vector}};
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\end{tikzpicture}
\caption{Model architecture: In black, our basic architecture. In red, target replication. In violet, optional note information $z_i$ introduced in the next section. The CNN layers are depicted by double arrows. For clarity, we omit the word-vectors that serve as input to the initial CNN.}
\label{figArchitecture}
\end{figure}
\paragraph{Target replication} The performance of the basic model presented above is promising but not yet satisfying. For similar long-sequence prediction problems,~\cite{lipton15replication} and~\cite{dai2015} have noted that it is beneficial to replicate the loss at intermediate steps. Following their approach, we compute an individual softmax mortality probability $p_i(y|\mathbf x_i)$ for every sentence $i=1\dots n$ and incorporate $n$ additional cross entropy terms into our final objective. For a corpus $\mathcal D$ containing patients $d_1,\dots d_N$ and corresponding labels $y^\star_1,\dots d^\star_N$ we seek to minimize:
\begin{align}
\mathcal L &=\sum_{(d^{(j)},{y^\star}^{(j)})\in\mathcal D}\mathcal L(d^{(j)},{y^\star}^{(j)})\\
\mathcal L (d=\langle s_1,\dots,s_n\rangle,y^\star)&= l(y, {y^\star}|\mathbf x) + \lambda\mathcal R = l(y, {y^\star}|\mathbf x) + \frac{\lambda}{n}\sum_{i=1}^nl_i(y, y^\star|\mathbf x_i)
\end{align}
$\mathcal R$ can be interpreted as the average prediction error at the sentences level, effectively bringing the classification loss closer to the word-level and regularizing the first CNN to learn sentence representations tailored to the mortality prediction problem. The hyper-parameter $\lambda$ determines the strength of the regularizer.
\paragraph{Incorporating note information} End-to-end neural network architectures such as ours allow for easy incorporation of additional information that can increase predictive power. Every note in our collection has a \emph{category} associated such as \textit{nursing}, \textit{physician} or \textit{social work}. Providing this information to our classifier can help to reliably assess the importance of individual sentences for the classification task. To exploit this information, we embed all 14 categories into a vector space $\mathbb R^{D_C}$ and concatenate every sentence vector $\mathbf x_i$ with its associated category vector $\mathbf z_i$.
\section{Experiments}
\label{sec:experiments}
We evaluate the proposed method on three standardized ICU mortality prediction tasks. On the basis of a patient's electronic health record, we predict whether the patient will die (1) during the hospital stay, (2) within 30 days after discharge, or, (3) within 1 year after discharge, and report AUC as an evaluation measure.
\subsection{Data}
\label{sub:data}
MIMIC-III~\cite{johnson2016mimic} is an openly-accessible critical care database, comprising 46,520 patients with 58,976 hospital stays. It contains measurements of patient state (through vital sign, lab tests and other variables) as well as procedures and treatments. Crucially, it also contains over 2 million unstructured textual notes written by healthcare providers.
Following the data filtering and pre-processing steps in~\cite{ghassemi2014unfolding}, we restrict to adults ($\geq$18 years old) with only one hospital admission. Most importantly, we exclude notes from the \emph{discharge summary} category and any notes recorded after the patient was discharged. This results in 31,244 patients with 812,158 notes. 13.82\% of patients died in the hospital, 3.70\% were discharged and died within thirty days, and 12.06\% were discharged and died within a year. We randomly sample 10\% of the patients for the test set, and 10\% for the validation set. The remaining 80\% of the patients are used during training. We construct the vocabulary by keeping the 300K most frequent words across all notes and replace all the words which are not part of the vocabulary with an out-of-vocabulary token.
\subsection{Baselines}
\label{sub:baseline}
\paragraph{LDA based model}
We recreate the LDA-based Retrospective Topic Model from ~\cite{ghassemi2014unfolding}. This model is the state-of-the-art method for mortality prediction on unstructured data from MIMIC II. We recreate the model on MIMIC III, and closely follow their preprocessing and hyperparameter settings. We tokenize each note and remove all stopwords using the Onix stopword list \footnote{\url{www.lextek.com/manuals/onix}}. The vocabulary is constructed as the union of the 500 most informative words in each patient's note based on a tf-idf metric. All words which are not part of the vocabulary are removed. We keep the number of topics to be 50 and set the LDA priors for the topic distributions and the topic-word distributions to $\alpha = \frac{50}{numberTopics}$ and $\beta = \frac{200}{vocabularySize}$, respectively. We train a separate linear kernel SVM on the per-note topic distributions to predict the mortality for each task.
Since SVM classifiers are sensitive to significant class-imbalances, we follow~\cite{ghassemi2014unfolding} in randomly sub-sampling the patients who did not die in the training sets to reach a ratio of 70\%/30\% between the negative and positive class. We do not modify the distribution of classes within the test and validation set. The LDA vectors are trained on the entire training data, but the SVM classifiers are trained using the vectors from the down-sampled training sets only.
\paragraph{Feed-forward Neural Network}
\label{sub:d2v}
As our second baseline we use the popular distributed bag of words (DBOW) scheme proposed by Le and Mikolov~\cite{le2014distributed}. In a range of initial experiments, we determined the DBOW architecture (rather than the distributed memory alternative) and an embedding space dimensionality of $400$ to be optimal in terms of accuracy and generality. Using the same pre-processing as for the LDA baseline, we train separate linear SVMs for each task.
\subsection{Parameters and Pretraining}
\label{sub:parameters}
We pre-train 50-dimensional word vectors on the training data using the word2vec implementation of the gensim~\cite{gensim} toolbox. Our word-level CNN uses 50 filters of sizes 3,\ 4 and 5 resulting in a sentence representation of size $D_S=150$. We embed categories in $D_C=10$ dimensional space and use 50 filters of size 3 for the sentence-level CNN resulting in a patient representation of size $D_P=50$. Furthermore, we regularize the fully connected layer before our final softmax by l2-regularization on the weights and dropout with keep probability 0.8.
\subsection{Results}
\label{sub:results}
Table~\ref{tab:results} summarizes the results of the three models on all tasks. Across all methods there seems to be a general tendency that labels further in the future are harder to predict. We observe that both neural models are superior to the LDA baseline, in particular on the two harder tasks. Furthermore, our two-level CNN model outperforms doc2vec by a significant margin on all tasks.
\begin{table}[ht]
\caption{MIMIC-III Mortality Prediction AUC}
\label{tab:results}
\centering
\begin{tabular}{llll}
\toprule
Task & LDA & doc2vec & CNN \\
\midrule
Hospital & 0.930 & 0.930 & \textbf{0.963}\\
30-day & 0.800 & 0.831 & \textbf{0.858}\\
1-year & 0.790 & 0.824 & \textbf{0.853}\\
\bottomrule
\end{tabular}
\end{table}
To highlight the effectiveness of the target replication, Table \ref{tab:CNNresults} shows the results of our model with and without target replication. We report on 30-days post-discharge, but performance on the other tasks is comparable.
\begin{table}[ht]
\caption{Performance analysis for target replication}
\label{tab:CNNresults}
\centering
\begin{tabular}{lllll}
\toprule
Model & without target replication & with target replication \\
\midrule
AUC & 0.682 & 0.858\\
\bottomrule
\end{tabular}
\end{table}
The results of our CNN show that modeling sentence and document structure explicitly results in noticeable performance gains. In addition, learning sentence representations and training them in our regularizer on the classification task, enables us to retrieve a patient's most informative sentences. This allows an inspection of the model's features, similar to LDA's topic distributions but on the sentence level. This stands in stark contrast to doc2vec's generic document representations. To showcase these features, Table~\ref{tab:sentences} shows a patient's top five sentences indicating likelihoods of survival and death respectively.
\begin{table}[ht]
\caption{The three highest and three lowest scoring sentences of one patient in the 1-year task.}
\begin{tabular}{ll}
\toprule
P(survival) high& the remaining support lines are unchanged .\\
& no effusion .\\
&the cardiomediastinal contours are normal .\\
\midrule
P(survival) low&now found to have metastatic lesions in her brain .\\
&impression UNK multiple large enhancing masses within the brain with \\
&\phantom{bla}\qquad\ $\hookrightarrow$ surrounding vasogenic edema most consistent with .\\
&enhancing lesions in the right temporal lobe and right mid brain consistent\\
&\phantom{bla}\qquad $\hookrightarrow$ with metastatic disease .\\
\bottomrule
\end{tabular}
\label{tab:sentences}
\end{table}
While most patients' top-scoring sentences look promising, a careful study of the predictions reveals that some neutral sentences can be ranked too highly in either direction. This is due to the model's inability to appropriately handle sentences that do not help to distinguish the two classes. We plan to address this in the future by a more advanced attention mechanism.
\section{Conclusion}
\label{sec:conclusion}
In this paper we developed a two-layer convolutional neural network for the problem of ICU mortality prediction. On the MIMIC-III critical care database our model outperforms both existing BOW approaches and the popular doc2vec neural document embedding technique on all three tasks. We conclude that accounting for word and phrase compositionality is crucial for identifying important text patterns. Such findings have impact beyond the immediate context of automatic prediction tasks and suggest promising directions for clinical machine learning research to reduce patient mortality.
\small
\bibliographystyle{plain}
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2110.06742
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\section{INTRODUCTION}
\label{sec:intro}
\subsection{\acl{MORL}}
\ac{MORL} is the subfield of \ac{RL} that attempts to find optimal policies for problems with at least two objectives. Within this field, various techniques have been proposed, such as \acp{MOEA}\cite{Deb2002}, Deep Q-Learning with Lexicographical Thresholding~\cite{Hayes2020} and Outer-Loop Approaches like Deep OLS~\cite{Mossalam2016}. An essential property of \ac{MORL} or even \ac{MOO} in general is that each of the objectives must be in conflict with the others. When two objectives can be optimized concurrently, the problem becomes equivalent to the problem of optimizing the sum of both objectives, effectively reducing the number of objectives by one. The problem discussed in this paper consists of 2 to 3 conflicting objectives, and can thus be classified as a \ac{MORL} problem.
\subsection{\acl{MaOO}}
While there is no exact definition of when a \acl{MOO} problem becomes a \ac{MaOO} problem, the line is typically drawn around 4 objectives, with anything with more than 1, but less than 4 objectives considered a \acl{MOO} problem. \cite{Saxena2009} define a \ac{MaOO} problem to have `significantly more than five' objectives, while \cite{Cheng2016}, \cite{Mane2017}, \cite{Wang2015} and \cite{Yang2013} say \ac{MaOO} problems are any problem with more than three objectives. Despite these differences, it is generally accepted that \ac{MaOO} problems are significantly more challenging and less intuitive than \ac{MOO} problems. While the authors thought it important to point out the difference between \ac{MOO} and \ac{MaOO}, this paper will restrict itself to \ac{MOO} problems.
\subsection{\aclp{MDP}}
\acp{MDP} are the most commonly used way of describing problem statements in the field of \acl{RL}. An \ac{MDP} is usually described as a 4-tuple \cite{Sutton1998}:
$$MDP = \left(\mathcal{S}, \mathcal{A}, p\left(s' | s, a\right), \mathcal{R}\left(s, a\right)\right)$$
The elements of the \ac{MDP} are the state-space ($S$), the action-space ($A$), the dynamics ($ p\left(s' | s, a\right)$) and the reward function ($\mathcal{R}\left(s, a\right)$).
The state-space describes the environment's state. Some \ac{RL} problems are fully-observable, meaning that the agent gets to observe the complete state of the environment, while others are partially observable, meaning that the agent only gets to observe part of the state of the environment.
The action-space determines the actions which an agent can take while operating in an environment. It is possible for this action space to be dependent on the current state of the environment, in which case it is usually denoted as $\mathcal{A}\left(s\right)$. In this paper however, this is not the case, so we will simply denote the action space as $\mathcal{A}$.
The dynamics of an environment ($p\left(s' | s, a\right)$) determine how the environment evolves and reacts to an agent's actions. Essentially, it is a function that gives the probability distribution of new states, conditional on the current state, and the action which the agent took.
Finally, there is the reward function $\mathcal{R}\left(s, a\right)$. In most \ac{RL} literature, the reward is defined as
$$\mathcal{R}: \left(S, A\right) \in \mathcal{S} \times \mathcal{A} \mapsto \mathbb{R}$$
Unfortunately, when dealing with \ac{MORL} this formulation requires some changes, since our environments return a reward vector consisting of $N$ objectives, resulting in a definition of
$\mathcal{R}$ as $$\mathcal{R}: \left(S, A\right) \in \mathcal{S} \times \mathcal{A} \mapsto \mathbb{R}^{N}$$ with $N \in \left\{2, 3, \ldots\right\}$.
In section~\ref{sec:sota}, the authors discuss existing \ac{MOO} and \ac{MORL} benchmarks, and open-source \ac{DST} implementations. This is followed by a detailed description of the original \ac{DST} problem proposed by Vamplew et al.\cite{Vamplew2011} in section~\ref{sec:dst}. After discussing the existing problem, the authors showcase their improved version of the problem in section~\ref{sec:newdst}, and followed by a discussion of their implementation~\ref{sec:impl}. Finally, the authors provide some potential directions for future research in section~\ref{sec:discussion-future_work}.
\section{STATE OF THE ART}
\label{sec:sota}
\subsection{Optimality Criteria}
\label{ssec:sota:criteria}
When considering scalarizing \ac{MORL} approaches, there are two different ways of performing this scalarization, and each of these techniques corresponds to a different notion of a good solution. In a recent survey paper on utility-based \ac{MORL}, \cite{Radulescu2019} described these two optimality criteria: \ac{ESR} and \ac{SER}. In their paper, they formulate \ac{ESR} as shown in equation~\ref{eq:sota:esr}, where $u()$ is the scalarization function:
\begin{equation}
V_{u}^{\pi} = \mathbb{E}\left[u\left(\sum_{t=0}^{\infty}{\gamma^{t}r_{t}}\right)| \pi, \mu_{0}\right]
\label{eq:sota:esr}
\end{equation}
The mathematical formulation of \ac{SER}, on the other hand, is shown in equation~\ref{eq:sota:ser}
\begin{equation}
V_{u}^{\pi} = u\left(\mathbb{E}\left[\sum_{t = 0}^{\infty}{\gamma^{t}r_{t} | \pi, \mu_{0}}\right]\right)
\label{eq:sota:ser}
\end{equation}
Under \ac{ESR}, a user derives utility from a single roll-out of its policy, while under \ac{SER} utility is derived from the expected outcome (mean over multiple roll-outs). In the context of \ac{MOO}, the notions of \ac{ESR} and \ac{SER} are very valuable, since they describe two fundamentally different types of solutions. A practical example of \ac{ESR} is the LEMONADE algorithm, from Elsken et al.~\cite{Elsken2018}. Using LEMONADE, Elsken et al. find the optimal neural network architecture for image classification on CIFAR-10 and ImageNet64x64. In this paper, the architecture search problem is formulated as an \ac{ESR} problem, since the author's policy (an evolutionary algorithm) is only executed once to obtain results. An example of \ac{SER} is demonstrated by Khamis et al. in their paper on \ac{MORL} for traffic signal control~\cite{Khamis2014}. The policy found by Khamis et al. is executed many times, and is only optimal if it can consistently route traffic in an optimal way (\ac{SER}), rather than generating a single optimal traffic light configuration (\ac{ESR}).
\subsection{Existing Benchmark Suites}
\label{ssec:sota:benchmarks}
In \ac{MOO} Literature, there exist numerous benchmarks, such as the \ac{WFG} Toolkit~\cite{Huband2005}, ZTD Benchmark~\cite{Zitzler2000}, DTLZ Benchmark~\cite{Deb2001}. Vamplew et al.~\cite{Vamplew2011} also included a link to a list of benchmarks in their original publication. Unfortunately, at the time of writing, the given link\footnote{\href{http://hdl.handle.net/102.100.100/4461}{http://hdl.handle.net/102.100.100/4461}} no longer works. When analyzing benchmarks like ZTD, DTLZ and WFG, as well as the problems proposed in~\cite{Vamplew2011}, we notice that almost all of them are tailored towards \ac{ESR}-style optimization. While this provides a nice playground for \acp{MOEA}, it doesn't suit many \ac{RL} problems well, since they are usually formulated under the \ac{SER} criterion, as opposed to \acp{MOEA}, which are usually built around the concept of \ac{ESR} (An example of \ac{RL} being used with an \ac{ESR} criterion is the architecture search performed by Pham et al.~\cite{Pham2018}).
\subsection{\acl{DST}}
\label{ssec:sota:dst}
One commonly occurring \ac{MORL} benchmark problem is the \acf{DST} problem. In this problem, a submarine is tasked with collecting treasures. For each timestep the submarine spends collecting treasures, it is penalized with a time-score of -1. Every step that the submarine spends not finding a treasure, it is given a treasure score of 0, and when the submarine finds a treasure, it is given a treasure score equal to the value of the treasure. This problem was originally proposed by Vamplew et al.~\cite{Vamplew2011} as part of an effort to provide a benchmark suite for \ac{MORL} algorithms. Since its introduction, the \ac{DST} problem has been used numerous times as a benchmark problem in \ac{MORL} literature \cite{VanMoffaert2014a}, \cite{VanMoffaert2013}, \cite{Mossalam2016}
\subsection{Open Source Implementations}
\label{ssec:sota:impls}
When benchmarking \ac{RL} algorithms, a commonly used \ac{API} is that of OpenAI's gym~\cite{Brockman2016}. Gym proposes a simple software \ac{API} that encapsulates the underlying \ac{MDP}. Through this \ac{API}, gym allows for algorithms to be easily tested on multiple different problems to show an algorithms capability to generalize to different problems.\\
A number of open-source implementations of the \ac{DST} problem exist. Van Moffaert and Nowé provide an implementation of the \ac{DST} problem as part of their paper on Pareto-Q Learning~\cite{VanMoffaert2014a}. Their implementation\footnote{\href{https://gitlab.ai.vub.ac.be/mreymond/deep-sea-treasure}{https://gitlab.ai.vub.ac.be/mreymond/deep-sea-treasure}} is written in Python, and compliant with the gym interface. This implementation does not allow for arbitrary Pareto-fronts to be used, although they do provide an implementation of the Bountyful Sea Treasure variant~\cite{VanMoffaert2014b}.
Another implementation is that of Nguyen et al.~\cite{Nguyen2018}, as part of their \ac{MODRL} framework. Their framework contains both problem statements and solution strategies. The framework\footnote{\href{https://personal-sites.deakin.edu.au/~thanhthi/drl.htm}{https://personal-sites.deakin.edu.au/~thanhthi/drl.htm}} is written in Python. The environment isn't fully compatible with the OpenAI gym interface (The implementation is missing certain attributes such as \texttt{action\_space} and \texttt{observation\_space}, and likely doesn't implement all functionality present in a regular gym environment, since it doesn't inherit from \texttt{gym.Env} class or any of its subclasses). While the \ac{API} doesn't match that of gym, it does present an interface that is similar enough to allow research to rapidly adapt it to a gym-compatible environment. This implementation is much more customizable, and while it doesn't allow for the specification of arbitrary Pareto-fronts, it does have the built-in capability to present a convex, concave, linear and mixed (convex and concave) Pareto-front.
\section{BI-OBJECTIVE DEEP SEA TREASURE}
\label{sec:dst}
First, we will analyze the original \ac{DST} Problem, as proposed by \cite{Vamplew2011}. We will start this discussion by defining the exact \ac{MDP} that the \ac{DST} Problem proposes. In the spirit of keeping this paper compact, theoretical proofs a basic set of properties of the \ac{DST} problem and its solutions are omitted from this paper, and presented in the supplementary material.
The \ac{MDP} that defines the \ac{DST} Problem is a finite-horizon \ac{MDP} (Vamplew et al. originally limited the environment to 1000 time steps~\cite{Vamplew2011}) with the state and action spaces defined in equation~\ref{eq:dst:mdp}. The dynamics and reward function will be elaborated further in section~\ref{ssec:dst:dynamics} and section~\ref{ssec:dst:rewards} (We follow Sutton and Barto's convention of using $\doteq$ for definitions~\cite{Sutton1998}).
\begin{equation}
MDP_{2DST} \doteq \left(\left\{0 \ldots 9\right\} \times \left\{0 \ldots 10\right\}, \left\{0, 1, 2, 3\right\}, p\left(s' | s, a\right), \mathcal{R}\left(s, a\right)\right)
\label{eq:dst:mdp}
\end{equation}
Through this section, and section~\ref{sec:newdst}, we will use the letter $t$ to denote time.
\subsection{State Space}
\label{ssec:dst:state-space}
Looking at the state-space from equation~\ref{eq:dst:mdp}, we can see the agent observes 2 positive integers, representing the agent's current position as an $(x, y)$ pair. We note that the agent starts at the point with coordinate $(0, 0)$ in the top-left corner of the environment, and as the agent moves left-to-right, and top-to-bottom, its respective x and y coordinates increase. The original DST problem consisted of a $10 \times 11$ grid, resulting in a maximum state space size of 110 (The actual size of the state space is 60, since not all squares of the grid are accessible to the agent).
\subsection{Action Space}
\label{ssec:dst:action-space}
The agent has 4 potential actions, corresponding to each of the cardinal directions in the world, enumerated in clockwise fashion (UP, RIGHT, DOWN, and LEFT). Given the state space of size 60, and an action space of size 4, we can determine that a solution using tabular Q-learning would only require a table with around 240 entries. This is a first indicator of the simplicity inherent in the original \ac{DST} problem, and presents a strong argument in favour of more complicated problems when using Deep Learning-based techniques.
\subsection{Dynamics}
\label{ssec:dst:dynamics}
The environment dynamics for the \ac{DST} problem are completely deterministic:
\begin{equation}
s \doteq \left(x_{t}, y_{t}\right)
\end{equation}
\begin{align}
\Delta x\left(a\right) \doteq
\begin{cases}
1 & \mbox{if } a = 1\\
-1 & \mbox{if } a = 3\\
0 & \mbox{otherwise}
\end{cases}
\end{align}
\begin{equation}
x_{t + 1}\left(s, a\right) \doteq
\begin{cases}
x_{t} + \Delta x\left(a\right) & \mbox{if } \mbox{ collides}\left(x_{t} + \Delta x\left(a\right)\right) = 0\\
x_{t} & \mbox{otherwise}
\end{cases}
\label{eq:dst:collides}
\end{equation}
\begin{align}
p\left(s' | s, a\right) =
\begin{cases}
1 & \mbox{if } s' = \left(x_{t + 1}\left(s, a\right), y_{t + 1}\left(s, a\right)\right)\\
0 & \mbox{otherwise}
\end{cases}
\end{align}
The $collides\left(\right)$ function in equation~\ref{eq:dst:collides} returns an integer in $\left\{0, 1\right\}$ indicating if the new position would cause a collision or not. We only describe the procedure for obtaining the next x-value from the current x value and the action, but the procedure for obtaining the next y-value is analogous.
\subsection{Rewards}
\label{ssec:dst:rewards}
The reward function in the original \ac{DST} problem returns a 2-dimensional reward, consisting of a time component, and a treasure component. We denote the time component as $r_{b}\left(s, a\right)$ and the treasure component as $r_{p}\left(s, a\right)$ (P for plunder), resulting in the following reward function:
$$\mathcal{R}\left(s, a\right) = \left(r_{b}\left(s, a\right), r_{p}\left(s, a\right)\right)$$
with
\begin{equation}
r_{b}\left(s, a\right) \doteq -1
\end{equation}
and
\begin{equation}
r_{p}\left(s, a\right) \doteq
\begin{cases}
p \in \mathbb{R}^{+}_{0} & \mbox{in terminal state}\\
0 & \mbox{if state is not terminal}
\end{cases}
\end{equation}
The exact value of the treasures in the \ac{DST} environment can vary between publications. Vamplew et al. proposed an original set of treasures in their paper~\cite{Vamplew2011}, Mossalam et al. proposed an alternative set of treasures to make the problem convex in their publication~\cite{Mossalam2016}. Another set of treasure values can be found in \cite{VanMoffaert2014b}, where the problem was renamed to Bountiful Sea Treasure. For the remainder of this publication, we will not consider any particular set of treasure values, rather, we will simply assume that all treasure values are $>0$.
In the bi-objective \ac{DST} environment, it can be proven that actions causing a collision always result in a solution which is Pareto-dominated by the same solution where the colliding action is removed. It can also be shown that any solution that results in a time-out, rather than the agent finding a treasure will be Pareto-dominated by a solution that finds a treasure. For proof of these properties, we refer to the supplementary material included with this paper. Using these properties we can exclude certain solutions from the search space, making it substantially smaller.
\section{\newrlenv}
\label{sec:newdst}
In this section, we will introduce an improved alternative to the original \ac{DST} problem. This version of the \ac{DST} problem involves the optimization of three conflicting objectives: Time, Treasure and Fuel, and has a larger observation and action space than the original \ac{DST} problem. Through the changes in observation space, the new \ac{DST} problem more closely resembles an \ac{SER} problem than the original \ac{DST} problem, bringing it more in-line with traditional \ac{RL} problems. Similar to the \ac{DST} problem, we define this problem to be a finite-horizon \ac{MDP}, limited to 1000 time steps.
\begin{equation}
MDP_{3DST} \doteq (\mathbb{Z}^{2 \times 11}, \left\{-3, -2, \ldots, 2, 3\right\}^{2}, p\left(s' | s, a\right), \mathcal{R}\left(s, a\right))
\end{equation}
\subsection{State Space}
\label{ssec:newdst:state-space}
The state space of this new \ac{DST} problem is significantly larger than the old one. The new state space consists of eleven two-element column vectors. The first column vector represents the agent's current velocity, expressed as a separate x- and y-component. The following ten column vectors each represent the agent's relative coordinates to each of the treasures. This observation differs from the original \ac{DST} problem in that it directly tells the agent where each of the potential solutions are. By changing the formulation of the observation space, the agent's task changes from memorizing the treasure locations and a path to them, to learning how to traverse the ocean towards whatever treasure the agent finds more interesting. Through this reformulation of the problem, the optimality criterion for the \ac{DST} problem can also shift to \ac{SER}, rather than \ac{ESR} if, for example, treasure locations were randomized. When analyzing this environment's state space, we note that the final ten columns of the observation all capture the same state (the agent's position). From section~\ref{ssec:dst:state-space}, we know that there are 60 positions the agent can visit. In the tri-objective environment, the agent's velocity is normally capped at 5 in any direction, meaning that there are a total of 121 different velocity vectors the agent can achieve ($\left(5 + 1 + 5\right)^2$). While not every velocity vector is achievable in every position, this still allows us to place an upper bound on the size of our state-space of 7260, which is already significantly larger than the original \ac{DST} problem, which had a state-space of 60 elements.
\subsection{Action Space}
\label{ssec:newdst:action-space}
The action-space consists of a two-element vector. The elements of this vector represent the respective x- and y-components of the acceleration the agent would like to make. In the new \ac{DST} environment, the agent no longer takes discrete steps in one of the cardinal directions, but rather, the agent can accelerate in one or both dimensions, allowing for more efficient ways of reaching each treasure at the cost of a higher fuel consumption. The total size of our action space is 49 ($7 \times 7$). When combining this with our state-space, we get a combined state-action space of $355740$ elements. While this is an upper bound, and the actual number will likely be lower, it is still a significant increase over the original \ac{DST} problem, which had a state-action space of 240 elements.
\subsection{Dynamics}
\label{ssec:newdst:dynamics}
At a high-level, the dynamics of the new \ac{MDP} are similar to that of the old \ac{MDP}: If the agent attempts to make a move that would result in a collision, the state if left unchanged, otherwise, the agent's desired action is executed. Similar to section~\ref{ssec:dst:dynamics}, we will only show the dynamics function for the x-component for brevity, noting that a set of identical operations are executed for the y-component.
We start off by defining our action $a$:
\begin{equation}
a \doteq \left(a_{x}, a_{y}\right)
\end{equation}
Next, we determine a preliminary next velocity and position, which will be used for collision checking.
\begin{align}
v'_{t + 1, x} &= v_{t, x} + a_{x}\\
x'_{t + 1} &= x_{t} + v'_{t + 1, x}
\end{align}
Knowing the position of the submarine if the action was executed, we can check for collisions, and update our actual velocity and position accordingly.
\begin{equation}
v_{t + 1, x} =
\begin{cases}
v_{t, x} + a_{x} &\mbox{ if } collides\left(x'_{t + 1}\right) = 0\\
0 &\mbox{ otherwise }
\end{cases}
\end{equation}
\begin{equation}
x_{t + 1} = x_{t} + v_{t + 1, x}
\end{equation}
We note that in this case, the $collides()$ function should not only check for collisions in the cardinal directions, but also in diagonal directions. Another important note is that the agent's velocity is reduced to zero if an action would cause a collision, which is a behaviour an agent could attempt to exploit to arrest movement without consuming fuel.
\subsection{Rewards}
\label{ssec:newdst:rewards}
In terms of rewards, the new \ac{DST} problem inherits the two objectives from the old \ac{DST} problem (time and treasure), and adds a third objective, fuel. The fuel objective was designed in such a way that it conflicts with the two already existing objectives. When the agent attempts to find far-away treasures, it will need to spend more fuel to cover the distance, and when the agent wants to cover a given distance quicker, it can spend more fuel accelerating and decelerating.
Formally, we define the fuel objective $r_{f}$ (F for fuel) as the sum of the squares of the acceleration in both dimensions, if the action does not cause a collision, or 0 if the action would cause a collision:
\begin{equation}
r_{f} =
\begin{cases}
-\left(a_{x}^{2} + a_{y}^{2}\right) &\mbox{ if } collides\left(x'_{t + 1}\right) = 0\\
0 &\mbox{ otherwise }
\end{cases}
\end{equation}
We note that this function creates the exploit mentioned earlier in section~\ref{ssec:newdst:dynamics}. The formulation of this reward function also does not necessarily make collisions sub-optimal, since coasting (Moving without accelerating using built-up inertia) uses the same amount of fuel as colliding.
This leads us to $\mathcal{R}\left(s, a\right)$ as:
\begin{equation}
\mathcal{R}\left(s, a\right) = \left(r_{b}\left(s, a\right), r_{p}\left(s, a\right), r_{f}\left(s, a\right)\right)
\end{equation}
For the tri-objective \ac{DST} problem, we show that there exists at least 1 solution ending in a time-out that is part of the Pareto-front. The proof for this property can be found in the supplementary material. Besides this, the authors also attempt to prove whether or not a collision impacts the optimality of a solution. Unfortunately, the new acceleration-based action space makes this proof significantly more complicated than for the bi-objective environment. Following a line of reasoning similar to that used for this proof in the bi-objective case, it quickly becomes clear that this can not be proven in the same way. In the tri-objective case, a collision necessarily must be considered as consisting of multiple timesteps, since it is possible for an agent to accumulate more velocity than it can arrest in a single step. Proving or disproving this property would thus require a fundamentally different line of reasoning, which the authors feel that this speaks to the added complexity inherent in the tri-objective environment.
\section{IMPLEMENTATION}
\label{sec:impl}
The authors also provide a reference implementation of both the new and old \ac{DST} problem, written in Python\footnote{\environmentURL}. The authors also published their code as a PyPI package\footnote{\environmentPackage}. The implementation provides numerous options. The proofs in the supplementary materials only consider the default options, since changes to these options would change the properties of the environment, but we believe they would still be valuable for researchers regardless. When building the implementation, care was taken to ensure the implementation conforms to the gym \ac{API}, to ease adoption among \ac{RL} practitioners.\\
\subsection{Configurability}
\label{ssec:impl:config}
The original \ac{DST} problem proposed by Vamplew et al.~\cite{Vamplew2011} can easily be recreated using a separate wrapper in combination with the regular environment. The fuel objective is not an inherent part of the environment, but rather can be added by utilizing a different wrapper. Through the use of gym's wrapper system, the authors hope to provide a sufficiently composable environment that can be used for future research.\\
The implementation allows its users to re-define the action space by specifying the positive acceleration levels in 1 dimension that the agent can use, along with this, the fuel costs for each acceleration level can also be set, allowing the user to completely re-define their fuel objective based on their preferences.\\
The default treasure location and value are set to match those described by Vamplew et al.~\cite{Vamplew2011}, but can be reconfigured. This serves two functions, first, it allows users to redefine their Pareto-fronts for time and treasure, by moving treasures closer or further away and changing their exact treasure values. Secondly, it also allows users to build \ac{RL} agents that do not converge to a single solution, but rather that learn to search for and find solutions, bringing the optimization criterion closer to an \ac{SER}-like formulation, rather than an \ac{ESR}-like formulation.
An example of a custom set of treasures is shown in Fig.~\ref{fig:impl:debug-info}a.\\
In some cases, it may be interesting to intrinsically disregard collisions. While they can easily be proven to be sub-optimal in the bi-objective case, this is not true for the tri-objective case. The authors' implementation allows users to make collisions guaranteed to be sub-optimal, however. By setting the \texttt{implicit\_collision\_constraint} option, the environment will return a vector of immediate rewards that are worse than any other set of possible reward values.\\
The implementation also has some additional functionality to help with debugging, such as the capability to display debug information, shown in Fig. ~\ref{fig:impl:debug-info}b. The environment provides many more options. For a detailed overview, we refer to the implementations documentation.
\begin{figure}[h]
\centering
\subfloat[Custom Treasures and No Debug Information]{{\includegraphics[width = 0.21 \textwidth]{figure1a.png} }}%
\qquad%
\subfloat[Default Treasures and Debug Information]{{\includegraphics[width = 0.21 \textwidth]{figure1b.png} }}%
\caption{Submarine moving through \ac{DST} environment}
\label{fig:impl:debug-info}
\end{figure}
\subsection{Pareto-Front}
\label{ssec:impl:paretofront}
Finally, we provide the Pareto-front for the tri-objective environment. While we do not discuss it here, the Pareto-front data for the bi-objective environment is also available in our repository. While we were unable to prove certain properties of the tri-objective environment, we were able to empirically determine the Pareto-front of the tri-objective environment through exhaustive graph search. Solving this environment took 33.22 hours using 28 Intel E5-2680v4 Broadwell CPUs. The numerical data for the Pareto-front can be found in the same repository that contains the code for the environment. Figure~\ref{fig:pf} shows the obtained Pareto-front for the tri-objective environment. It is important to note that the solution resulting in a time-out reported in section~\ref{sec:newdst} was omitted for clarity, it is however present in the numerical dataset in the repository. Each point on the Pareto-front has been projected onto the time-treasure plane for clarity, and horizontal lines have been drawn for all solutions that lead to the same treasure.
\begin{figure}
\centering
\includegraphics[width = 0.52 \textwidth]{figure2.png}
\caption{3-Objective Pareto-front. Points in blue lie on the convex hull, while points in red are contained inside the convex hull.}
\label{fig:pf}
\end{figure}
\begin{figure}[h]
\centering
\includegraphics[width = 0.52 \textwidth]{figure3.png}
\caption{Detailed view of the low-treasure region of the 3-Objective Pareto-front. Points in blue lie on the convex hull, while points in red are contained inside the convex hull. A convex hull is the smallest, convex set of points that contains a given set of points.}
\label{fig:pfdetail}
\end{figure}
Looking at Figure~\ref{fig:pf}, a first notable property of this Pareto front is that the number of points on the Pareto-front (25 + 1) no longer corresponds to the number of treasures (10). In our analysis, we will ignore the 26th point generated by the submarine performing the ``idle" action 1000 times. While this point is theoretically optimal, we argue that it does not produce a useful solution, and hence will disregard it in further analysis. We can see that for each treasure, there exist different trade-offs between fuel and time. We also note, that not all treasures are part of the Pareto-front, the treasures that make up the Pareto-front are those with values $\left\{1, 2, 3, 8, 16, 50, 74, 124\right\}$, while the treasures with values $\left\{5, 24\right\}$ are not part of the Pareto-front, as can be seen in the detailed view of the Pareto-front in~\ref{fig:pfdetail}.\\
By looking at Figure~\ref{fig:pf} it can also be easily ascertained that the Pareto-front for this environment is not convex. In this Pareto-front, 9 out of 25 solutions lie inside the convex hull, rather than on it. In the original \ac{DST} problem, 3 out of 10 points formed local concavities. Similar to the original \ac{DST} problem, this presents an interesting challenge for \ac{MORL} algorithms, testing their ability to handle non-convex Pareto-fronts.
\subsection{Guidelines}
\label{ssec:impl:guidelines}
While this implementation is highly configurable, the authors recommend anyone using their implementation to always report performance on both default configurations (\texttt{VamplewWrapper + DeepSeaTreasureV0} and \texttt{FuelWrapper + DeepSeaTreasureV0}) While it may be interesting to alter the default configuration in the context of a specific paper, using a non-standard configuration makes comparing the used algorithm to that of other researchers very difficult, or even impossible. Some options (like \texttt{render\_grid} or \texttt{render\_treasure\_values}) do not impact the found solutions, but other options can have a profound impact on the solutions that are found, and how quickly/easily they can be found (options such as \texttt{implicit\_collision\_constraint} and \texttt{treasure\_values}). We encourage users to make maximal use of the configurability offered by the environment, we also advocate for always reporting performance on custom configurations in addition to the default ones, to allow for fair comparisons between algorithms.\\
When utilizing custom configurations, we also believe it to be critical to report the complete configuration of the environment. The authors have made sure to provide a simple way to report an environments' configuration through the use of the \texttt{config()} method. This method returns the complete configuration of the environment, and can be used to create a new environment with an identical configuration. The authors encourage anyone using this environment to always report the complete configuration of their environment by including this data in a \texttt{.json} file in the supplementary materials or appendix of the publication. The use of machine-readable format like JSON over a human-readable format like a table in a PDF file makes it easy for researchers aiming to reproduce each others work to easily copy and paste a configuration from a publication into their own code.
The implementation and Pareto-front dataset provided by the authors is independently citable through Zenodo under DOI \environmentDOI.
\section{DISCUSSION \& FUTURE WORK}
\label{sec:discussion-future_work}
One major omission in this paper is the concept of constraints. Constraints in the context of \ac{MOO} are usually formulated as a set of (in)equalities that check certain properties of a solution. In the case of the \ac{DST} problem, a constraint could be ``The agent is not allowed to visit square (5, 3)". While this may seem arbitrary in the context of the \ac{DST} problem, constraints are often found in real-life engineering applications, such as in the placement of distributed tasks in fog environments~\cite{Eyckerman2020} or the automatic design of neural networks targeting embedded devices~\cite{Cassimon2020}. Constraints also pose an interesting challenge to the \ac{API} proposed by gym~\cite{Brockman2016}, since the current \ac{API} provides no natural way of indicating that certain solutions are ``unacceptable" or ``invalid". While it is usually possible to work around this, by ending episodes early, and modifying reward functions to make constraint violations guaranteed to be sub-optimal, this is not always easy to achieve, and prompts the question of whether or not gym provides the most well-suited \ac{API} for solving constrained optimization problems using \ac{RL} agents.\\
While much research has been done on solving \ac{MORL} problems under \ac{ESR} optimality criteria, with both evolutionary and \ac{RL} techniques, literature on algorithms targeting \ac{SER} optimality criteria is relatively sparse, even though this is the dominant problem formulation in \ac{RL}. With this in mind, the authors believe this to be a valuable research direction, since many \ac{RL} problems whose formulation is simplified to single-objective problems currently, could be revisited, while examining their complete solution space.\\
The authors would also like to point out that, even though the complexity of the original \ac{DST} problem was significantly increased in the new, tri-objective \ac{DST} problem, the \ac{DST} problem should still be considered a toy problem. Problems like Multi-Objective traffic signal control, like the one tackled by Khamis et al. ~\cite{Khamis2014} could provide valuable use-cases to show the suitability of \ac{MORL} algorithms in more practical use-cases.\\
In this paper, the authors proposed a new environment, and tackled it from both a theoretical and practical perspective, providing a working, powerful and easy-to-use implementation. In order to reduce the scope of this paper, the authors decided against the execution of various \ac{MOO} and \ac{MORL} algorithms on the environment. While the comparison between algorithms and environments would undoubtedly provide valuable data and insights, the authors feel like this would be better suited for a separate publication.
\section*{ACKNOWLEDGEMENTS}
\acknowledgement
\bibliographystyle{apalike}
|
1702.03903
|
\section{Introduction}
Active matter, i.e., the emergent behavior of a large number of agents
that can produce mechanical forces via energy dissipation
\cite{Ramaswamy_10}, is recently proving itself as an extremely rich
context for non-equilibrium phenomena. Instances range from schools of
fish or bird flocks, to vibrated granular rods or propelled
nanoscale or colloidal particles, for all of which fluctuations play a
conspicuous role in the collective dynamics \cite{Marchetti_13}.
Bacterial systems \cite{Ben-Jacob_00} provide further instances of
active matter, from microswimmer suspensions in which single cell
motility plays a crucial role \cite{Sokolov_07,Zhang_10} to bacterial
colonies, in which motility can be hampered
\cite{Ben-Jacob_98,Matsushita_04,Bonachela_11}. Actually, the fronts
of bacterial colonies have long been held as textbook examples
\cite{Vicsek_92,Barabasi_95,Meakin_98} on how interactions among
individuals lead to collective, highly-correlated behavior. For
experiments frequently done using {\em Bacillus subtilis} or {\em
Escherichia coli}, this ranges from the formation of characteristic
patterns ---like diffusion-limited aggregation (DLA) fractals,
concentric rings, or dense-branched morphologies--- to formation of
disks or of compact, but rough, morphologies
\cite{Fujikawa_89,Vicsek_90,Wakita_97,Matsushita_98,Matsushita_04},
all of which are also found in other, non-living, systems.
The simplest situation in which individual bacterial motility is fully
suppressed by a high agar concentration on the supporting Petri dish
has received particular attention, as it paradigmatically demonstrates
a change from DLA branches to compact, Eden-like, clusters, for an
increasing nutrient concentration \cite{Fujikawa_89,Bonachela_11},
akin to that found for many other diffusion-limited (DL) growth
systems \cite{Nicoli_09}. This morphological transition has been
recently shown to bear direct importance on the biological performance
of the colony \cite{Nadell_10,Mitri_11,Nadell_13}: branches
enable the space segregation of cell lines which respond differently
with respect to the production of enzymes needed for biofilm
formation, enhancing the prevalence of cooperative cells. Biofilms are
surface-attached communities hosting most living bacteria in nature,
of paramount importance to medicine and technology, from infections to
energy harvesting \cite{Costerton_95,Wilking_11}.
Furthermore, front fluctuations of Eden clusters \cite{Eden_61} are in
the celebrated Kardar-Parisi-Zhang (KPZ) \cite{Kardar_86} universality
class of kinetic roughening
\cite{Barabasi_95,Alves_11,Takeuchi_12}. Sparked by breakthroughs on
exact solutions of the KPZ equation and related growth models, that
have been experimentally validated (see \cite{Halpin-Healy_15} for a review),
this class is recently being demonstrated as a paradigm for strong fluctuations in one dimension (1D), as found e.g.\ in non-linear oscillators \cite{VanBeijeren_12}, stochastic hydrodynamics \cite{Mendl_13}, quantum liquids \cite{Kulkarni_13},
or reaction-diffusion systems \cite{Nesic_14}. Remarkably,
in the low motility case, most experimental values
found for the scaling exponents of compact Eden-like
bacterial colonies {\em do not} coincide with the KPZ values
\cite{Vicsek_90,Wakita_97,Bonachela_11}. This fact has been reconciled with a
putative Eden behavior via e.g.\ effective quenched
disorder \cite{Bonachela_11}, unexpectedly for a system which is
succesfully described by continuum
\cite{Lacasta_99,Mimura_00,Dockery_02,Kobayashi_04,Giverso_15,Giverso_16}
or discrete \cite{Nadell_10,Farrell_13,Farrell_17} models with no
source of quenched disorder.
In this article, we report colony growth experiments for {\em
B.\ subtilis} and {\em E.\ coli} under suppressed-motility conditions \cite{Matsushita_98,Rafols_98} in the alleged Eden regime. We explain the
non-KPZ kinetic roughening that we indeed observe as non-universal
scaling behavior induced by the diffusive instabilities that
occur. This is achieved by comparing our data with simulations of a
continuum model that we formulate, indicating that these experimental
conditions keep the system within a DL transient for all accessible
times. Moreover, the increase of front branching with time for the experimental colonies prevents asymptotics from being in the KPZ universality class under our suppressed-motility conditions. Analogous non-universal behavior has been identified in other DL systems, like thin film growth by electrodeposition (ECD), by chemical vapor deposition (CVD)
\cite{Castro_00,Nicoli_09}, or in coffee ring formation by evaporating
colloidal suspensions \cite{Yunker_13,Nicoli_13,Yunker_13_2,Oliveira_14}.
The paper is organized as follows. Our experimental setup and methods are described in Sec.\
\ref{sec:experiments}, while a continuum model which we employ to rationalize our observations is described in Sec.\ \ref{sec:model}. This is followed by our experimental results, which are reported in Sec.\ \ref{sec:analysis}. Further discussion is provided in Sec.\ \ref{sec:conclusions}, which also contains our conclusions and an outlook on future work. Further technical details on error estimates are left to an appendix.
\section{Experimental Setup}
\label{sec:experiments}
We have grown colonies of {\em B.\ subtilis} 168 (BS) and {\em
E.\ coli} ATCC 25922 (EC) on Petri dishes as in
\cite{Matsushita_98,Rafols_98}, in the high agar concentration (i.e.,
low motility) regime for different concentrations of
nutrients. Specifically, we have kept a constant agar concentration
$C_a=10$ g/l while considering different values of the nutrient
concentration, $C_n = 10, 15$, or $20$ g/l, within the Eden-like
region in the morphological space of
\cite{Matsushita_98,Rafols_98}. These conditions correspond to a value
for the non-dimensional thickness $\delta$ of the active layer within
the bacterial colony, where the nutrient concentration has
non-negligible gradients
\cite{Dockery_02,Nadell_10,Farrell_13,Giverso_16,Farrell_17}, which is
large enough for the colony to look compact on the accesible
space-time scales.
For inoculating Petri dishes, bacteria were grown overnight in
nutritive liquid medium [5 g/l NaCl (Merck, Germany), 3 g/l meat
extract (Merck, Germany), 10 g/l bacto-peptone (Lab.\ Conda, Spain)]
and the OD600 was measured. Cells were pelleted at 12 krpm in a
microcentrifuge, and resuspended to 0.5 OD600 in minimal medium
without bacto-peptone. Two replica Petri dishes were prepared
following \cite{Rafols_98}: a 3 mm thick agar plate in nutritive
medium [5 g/l NaCl (Merck, Germany), 5 g/l K${}_2$HPO${}_4$ (Carlo
Erba, Italy) and bacto-peptone (Lab.\ Conda, Spain)] inoculated at
the center with 1 $\mu$l of the cell suspension was incubated at 35
${}^\circ$C in a sealed humid chamber for up to 33 days, leading to
growth of quasi-2D colonies. No swarming of bacteria has been
detected.
Pictures were taken at different incubation times using a digital
camera (Olympus SC30, Japan; 3.3 Mp) coupled to a stereo microscope
(Olympus SZX10), or a digital camera (Nikon D5000, Japan; 12.3 Mp) for
large enough colonies. These photographs were treated to extract the
position of the colony front at each growth time, see
Fig.\ \ref{fig:profiles}.
\begin{figure}
\centering
\mbox{
\includegraphics[width=0.25\textwidth]{019f_2.pdf}
\includegraphics[width=0.25\textwidth]{019p_2.pdf}}
\\
\mbox{
\includegraphics[width=0.25\textwidth]{168f_2.pdf}
\includegraphics[width=0.25\textwidth]{168p_2.pdf}}
\\
\mbox{
\includegraphics[width=0.25\textwidth]{792f_3.pdf}
\includegraphics[width=0.25\textwidth]{792p_2.pdf}}
\caption{Experimental photographs of the bacterial
colonies (left column, (a, c, e)) and profiles extracted using the procedure
described in the text (right column, (b, d, f)). All these examples corresponds
to {\em B.\ subtilis} with $C_n=20$ g/l. The growth times are: (a, b) 19 h, (c, d) 168 h, and (e, f) 792 h, top to bottom.}
\label{fig:profiles}
\end{figure}
\subsection{Extraction of front profiles}
We next consider the protocol that we have
followed in order to extract the position of the fronts of the
bacterial colonies from the photographs. The analysis was
semi-automatic. An algorithm was developed, which works in the
majority of the cases without supervision. The images were digitized
and subject to a constrast filter in order to highlight the
interface. The resulting image can be regarded as a matrix with entries equal to $1$ inside
the colony and equal to $0$ outside the colony. Then, a discretized
continuous curve was obtained as follows. First, the geometric center
of the colony bulk was estimated. Then we obtained the intensity curve
along rays emanating from that point for different angles,
$I_\theta(r)$. For each angle $\theta$, we obtained the distance
$r(\theta)$ from the center, such that a certain threshold value of
the total intensity was found below it. Mathematically,
\begin{equation}
\int_0^{r(\theta)} dr \, I_\theta(r) = \mu \int_0^\infty dr \, I_\theta(r) ,
\label{eq:intensity_threshold}
\end{equation}
where $\mu$ is the threshold parameter. In our present case, $\mu=0.99$ was
employed, i.e., the radius $r(\theta)$ is defined as the first
percentile of the intensity distribution. As an illustration,
Fig.\ \ref{fig:profiles} shows a set of experimental photographs and
their corresponding profiles. Note the compact form of the bacterial
colony, delimited by a well-defined front that fluctuates around an
average circular shape.
\section{Effective model}
\label{sec:model}
The evolution of the colony front can be rationalized through a
kinetic continuum model for the dynamics of the front
position. In this model the detailed dynamics of relevant
physical fields (e.g., bacterial and nutrient densities) other than
the position, $\vec{r}(t)$, of the front itself, is neglected. The model
is tailored so as to capture purely the form and the dynamics of the front, in a similar
way to many other instances of diffusion-limited growth, like thin solid films
\cite{Bales_90,Ojeda_00,Castro_12} or combustion fronts
\cite{Frankel_95,Blinnikov_96}, in which this type of approach has proven useful.
Specifically, we consider
\begin{equation}
\partial_t \vec r =
\( A_0 + A_1 K(\vec r) + A_a \Theta_a(\vec r) + A_n \eta \) \vec n ,
\label{eq:bacterial_growth}
\end{equation}
where $\vec r$ is an interface point, $\vec n$ is the local exterior
normal, $K(\vec r)$ denotes the curvature of the interface at that
point, $\Theta_a(\vec r)$ is the local aperture angle and $\eta$ is a
zero-average Gaussian uncorrelated space-time noise. Furthermore,
$A_0$, $A_1$, $A_a$ and $A_n$ are parameters which quantify,
respectively, the relative strengths of the average growth velocity of
a planar front, surface tension, the dependence on the aperture angle,
and fluctuations. Equation \eqref{eq:bacterial_growth} is similar to
continuum models earlier put forward in the context of growth of thin
solid films limited by diffusive transport, see
e.g.\ \cite{Meakin_98}. Note that, in contrast with many works in that
field, Eq.\ \eqref{eq:bacterial_growth} applies to interfaces with an
arbitrary geometry, in particular with an average circular shape, and
is not affected by small-slope, nor no-overhang approximations. In
this sense, the model can be considered a stochastic generalization of
a previous equation put forward in the context of combustion fronts
\cite{Frankel_95,Blinnikov_96}, for which transport also takes place
by diffusion.
In our model, we assume that growth resources increase locally with the angle under which a given point $\vec{r}$ at the interface sees the exterior world, which we describe as the {\em
aperture angle}, $\Theta_a(\vec{r})$, wich is illustrated by the sketch on Fig.\ \ref{fig:shadowing} and further in Fig.\ \ref{fig:foto}. Intuitively, points inside cavities get less nutrient than those at local protuberances. As frequently done in the context of diffusion-limited growth, one may make an analogy \cite{Meakin_98} to an ensemble of grass leaves which are striving to collect sunlight: taller leaves cast shadows on shorter ones, hindering growth of the latter. With this metaphor in mind, we consider this term to implement a {\em shadowing effect}, as frequently done in the context of DL growth \cite{Meakin_98}.
\begin{figure}
\begin{center}
\includegraphics[width=5cm]{figillust_3.pdf}
\end{center}
\caption{Given an interface (shown by the curved red
solid line), and a point $\vec{r}$ on it, let us consider all the rays
emanating from this point (dashed straight lines), and find out the fraction of rays that do not intersect the interface again (those delimited on the right side of the figure by the straight black solid lines). Such a fraction provides the local aperture angle, $\Theta_a(\vec{r})$.}
\label{fig:shadowing}
\end{figure}
Mathematically, the computation of the aperture angle is performed as
follows. Let $\Gamma$ be the interface, with $\vec r_0$ and $\vec r$ being
points on it. Let $A(\vec r,\vec r_0)$ be the angle under which
$\vec r$ is seen from $\vec r_0$. Then, the aperture angle from point
$\vec r_0$ is given by
\begin{equation}
\Theta_a(\vec r_0)=2\pi -
\left|\mbox{Range}_{\vec r\in \Gamma}\left( A(\vec r,\vec r_0) \right)\right|.
\label{eq:material_angle}
\end{equation}
i.e. the measure of the range of function $A(\vec r,\vec r_0)$ when
$\vec r$ takes values on $\Gamma$.
Equation \eqref{eq:bacterial_growth} implements the basic mechanisms influencing
growth dynamics: on average, the front tends to minimize its length and
grows along the local normal direction, faster at those locations $\vec{r}$ which are more exposed [larger aperture angle $\Theta_a(\vec r)$] to the external diffusive fluxes; moreover, the front position experiences stochastic fluctuations related with microscopic events (nutrient transport and consumption, as well as cell division and relocation). The choice of these mechanisms is supported by more detailed continuum models of bacterial colonies \cite{Dockery_02,Giverso_15,Giverso_16} which find the front to be unconditionally unstable to perturbations. In particular, no quenched disorder is assumed.
In order to simulate Eq.\ \eqref{eq:bacterial_growth}, we have proceeded along the lines of \cite{Rodriguez-Laguna_11,Santalla_14}: the interface is discretized
in an adaptive way, adding and removing points dynamically in order to
keep a constant spatial resolution. The normal vector and the local
curvature are computed using concepts from discrete geometry. An
important element of the simulation is that the interface is always a
simple curve, although it can have {\em overhangs}: self-intersections
are removed.
The evaluation of the aperture angle is the most costly part of the
calculation to simulate Eq.\ \eqref{eq:bacterial_growth}, since it is
a global measurement. We have devised the following algorithm in order
to compute it. Given a point $P$ and a segment
$P_1P_2$, we define the minimal angle-interval as the counterclockwise ordered
pair $\alpha(P,P_1P_2)\equiv (\alpha_0,\alpha_1)$ of angles, with
respect to the horizontal, under which the segment is viewed from the
point. If a segment is extended to a chain $P_1\cdots P_n$, we just
compute the union of all angle-intervals. The aperture angle is the complementary of the measure of the final angle-interval.
In order to assess the type of morphological instability implied by
the aperture term in Eq.\ \eqref{eq:bacterial_growth}, we have
simulated it numerically by setting to zero all other terms in the
equation. We have performed a linear stability analysis of the ensuing
model by studying the rate of growth or decay in time for
sinusoid-like perturbations of an overall circular shape (not
shown). We have verified the expected unstable behavior: the amplitude
of a small perturbation grows with a velocity proportional to the
wave-number $k$ of the perturbation. In the case of a band geometry
with periodic boundary conditions, this means that, according to
Eq.\ \eqref{eq:bacterial_growth},
\begin{equation}\label{eq:h_k}
\partial_t h_k(t) \simeq |k| \, h_k(t) + \cdots ,
\end{equation}
where $h_k(t)$ is the amplitude of a small sinusoidal perturbation of
a flat profile with wave-vector $k$. This is indeed the well-known
behavior of the aperture-angle term, as elucidated in other
diffusion-limited systems
\cite{Meakin_98,Frankel_95,Blinnikov_96}. The growth law
Eq.\ \eqref{eq:h_k} corresponds specifically to the destabilizing component
of the classic Mullins-Sekerka instability, paradigmatic of diffusion-limited growth
\cite{Vicsek_92,Meakin_98}.
\begin{figure}[t!]
\centering
\includegraphics[width=0.45\textwidth]{fig_profiles_2.pdf}
\caption{Interfaces from
numerical simulations of Eq.\ \eqref{eq:bacterial_growth} for
$A_0=0$, $A_1=0.1$, $A_a=1$, $A_n=0.1$ and a circular initial
condition. Sketches further illustrate the meaning of the local aperture
angle $\Theta_a(\vec r)$. The growth time for each profile can be read
from the color bar on the right. Space and time units are
arbitrary.}
\label{fig:foto}
\end{figure}
Figure \ref{fig:foto} shows the time evolution of an initially
circular interface described by Eq.\ \eqref{eq:bacterial_growth}, as obtained from numerical
simulations for a representative choice of parameters. Once the
interface perimeter grows large enough, the shadowing instability indeed sets
in, reflecting the preferential growth at front protrusions, as
compared with front troughs. In strong similarity with the
experimental profile on the Fig.\ \ref{fig:profiles}, the colony
remains a compact aggregate for all $t$, with a front that fluctuates
around an average circular shape.
\section{Experimental Results}
\label{sec:analysis}
In this section we report our experimental results for BS and EC conlonies. Along with the various experimental properties studied, we additionally consider numerical simulations of Eq.\ \eqref{eq:bacterial_growth} as aids to interpret the experimental results.
\subsection{Time evolution: Radius and global roughness}
We first consider quantitatively the time evolution of our experimental BS and EC colonies through the average radius and global roughness of the colony fronts: After front extraction as described in Sec.\ \ref{sec:experiments}, each profile is a set of $N$ points on the plane, $\{x_i,y_i\}_{i=1}^N$. This set is employed to obtain the radius, $R$, of the best fitting circle, using a minimization procedure to find the corresponding center $(x_{\rm C},y_{\rm C})$. The deviations from the fitting circle provide the {\em global roughness} or surface width,
\begin{equation}
W \equiv
{ \left< \frac{1}{N}
\sum_{i=1}^{N}
\left(\sqrt{(x_i-x_{\rm C})^2+(y_i-y_{\rm C})^2} - R\right)^2
\right> }^{1/2},
\label{eq:def_W}
\end{equation}
where brackets denote averages over experimental realizations.
\begin{figure}[t!]
\includegraphics[width=.5\textwidth]{fig_roughness_3.pdf}
\caption{\label{fig:wrt} (a) Experimental radius $R$ (open
symbols) and roughness $W$ (solid symbols) vs growth time. Purple
and blue (red) symbols are for BS (EC), with $C_n$ as in the
legend.
Lines are fits to power laws, $R \sim t^n$ and $W \sim t^{\beta}$,
with $n\approx 0.4$ and $\beta\approx 0.5$. (b) $+$
($\times$) symbols are data for $R$ ($W$) from numerical
simulations of Eq.\ \eqref{eq:bacterial_growth} for parameters as
in Fig.\ \ref{fig:foto}, averaged over 500 noise
realizations. The lines represent power-laws $R\sim
t^{n}$ and $W\sim t^{\beta}$ with different values of $n$ and $\beta$ for short and long times, as indicated. Units are arbitrary.}
\end{figure}
Both the radius and the global roughness of the experimental colony fronts depend on growth time. Results for $R(t)$ and $W(t)$ are provided in the top panel of Fig.\ \ref{fig:wrt}. Data can be fit by power laws in both cases, $R(t) \sim t^n$ and $W(t) \sim t^{\beta}$, with $n \simeq 0.38$--$0.43$ and $\beta \simeq 0.47$--$0.52$ values which are similar
for different nutrient concentration values and bacterial
species. Usually, for experimental circular interfaces that display
Eden/KPZ fluctuations \cite{Takeuchi_10,Yunker_13} ---conspicuously
including (Vero) cell aggregates \cite{Huergo_11}---, the average
front velocity is constant, hence the average front position increases
linearly with time. At variance with this, the radial growth rate we
measure is sublinear, i.e., $n < 1$. On the other hand, $W$ follows
power-law behavior with time as in standard kinetic roughening
systems. Taking into account that uncorrelated surface growth (so-called random deposition, RD) is characterized by $\beta_{\rm RD}=0.5$ \cite{Barabasi_95}, our relatively large experimental $\beta$ values are suggestive of uncorrelated, or possibly unstable growth
wherein front fluctuations are amplified and grow even faster than in mere RD
\cite{Vicsek_92,Barabasi_95,Meakin_98}. As noted in
\cite{Bonachela_11}, to date no other experimental work on bacterial
colony growth provides information on the time evolution of $R(t)$ or
$W(t)$ under our working conditions, in spite of the fact that
universality classes are defined by two independent exponents
\cite{Vicsek_92,Barabasi_95,Meakin_98}, one of them related with
time-dependent behavior.
For the sake of comparison, the bottom panel of Fig.\ \ref{fig:wrt} shows the average radius and global roughness obtained from numerical simulations of our model, Eq.\ \eqref{eq:bacterial_growth}. Apparently in contrast with the experiment, for each magnitude two different regimes can be distinguished, one for short times and a different one for long times, within which the power laws are characterized by different exponent values. Note that the numerical values of the exponents which are closest to those of the experiments correspond to the model short-time regime. Actually, taking e.g.\ BS colonies with $C_n = 20$ g/l as a representative case, we can make a more detailed comparison between the experimental behavior of $W(t)$ and $R(t)$ with that predicted by Eq.\ \eqref{eq:bacterial_growth}.
\subsubsection{Simulations in physical units}
\label{sec:phys_units}
The experimental data for the evolution of the global roughness agree closely with the early time behavior of the simulations; these were performed for several sets of parameter values, with very similar results. The specific choice given in Fig.\ \ref{fig:foto} (namely, $A_0=0$, $A_1=0.1$, $A_a=1$ and $A_n=0.1$) turned out to be the most relevant one to our experimental system. Of course, the units for these constants are arbitrary in principle. However, we can convert them into physical units through detailed comparison with the experimental data, as follows.
\begin{figure}[t!]
\includegraphics[width=.5\textwidth]{fig_sup.pdf}
\caption{\label{fig:numbers} (a) Magnified view of the global roughness of a BS bacterial colony studied in Fig.\ \ref{fig:wrt}. The time marked as $T_0$ indicates a change in the power-law behavior. (b) Magnified view of the roughness from numerical simulations of Eq.\ \eqref{eq:bacterial_growth} using parameters as in Fig.\ \ref{fig:wrt}. Time $t_0$ corresponds to the initial change in scaling behavior. Time $t_1$, signalling the beginning of asymptotic, long-time behavior, is also indicated.}
\end{figure}
In Fig.\ \ref{fig:numbers}(a) we show the roughness of the interface, $W(t)$, for the same {\em B.\ subtilis} experiments with $C_n=20$ g/l considered in Fig.\ \ref{fig:wrt}, but in a magnified view. A certain time $T_0=297$ hour (h) can be identified which marks a change in the power-law behavior of the data, at which the global roughness is $W_0=0.25$ mm. The experiment ends at time $T_e=801$ h, when $W_e=0.47$ mm. Thus, we have $W_e/W_0=1.9$ and $T_e/T_0=2.7$. The physical occurrence of $T_0$ can be confirmed by in other measurable quantitites, such as the average front velocity, see Fig. \ref{fig:speed}.
\begin{figure}[t!]
\includegraphics[width=.48\textwidth]{fig_sup_2_2.pdf}
\caption{\label{fig:speed} Average front speed as a function of time for BS experiments using $C_n=20$ g/l in linear (a) and doubly-logarithmic (b) displays. The data group themselves into two scaling regimes, approximately separated at $T_0=297$ h.}
\end{figure}
The front speed is estimated by comparing consecutive measurements of the radius and
using a finite-differences approach. The two panels show the same data, the only difference between them being that the bottom one is shown in logarithmic scale. We can see how the data divide into two sequences of points with slightly different scaling behavior, with the
division approximately corresponding to $T_0=297$ h.
Coming back to the simulations of Eq.\ \eqref{eq:bacterial_growth}, Fig.\ \ref{fig:numbers}(b) indicates a change in the scaling behavior of the global roughness at time $t_0=0.14$ [T], with a roughness of $w_0=0.044$ [L], where [L] and [T] are length and time units, respectively. Thus, the end of the experiment should correspond to a roughness
$w_e=0.044$ [L] $\times 1.9=0.084$ [L], which takes place at $t_e\sim
0.44$ [T]. We make this time correspond to $T_e=801$ h. Thus, the
numerical conversion from arbitrary time units to hours is $801$ h$/0.44$ [T] $\approx 1800$ h/[T]. The same reasoning can be performed with the length unit and we obtain a conversion factor of $0.47$ mm$/0.044$ [L] $\approx 11$ mm/[L]. Alternative procedures can be designed to obtain the conversion factors, and they all provide similar results. At any rate, using the indicated conversion factors we can estimate the physical values of the equation parameters in physical units, namely, $A_0=0$ mm/h, $A_1=0.067$ mm${}^2$/h, $A_a=6.1\cdot 10^{-3}$ mm/h, $A_n=0.086$ mm${}^{3/2}$/h${}^{1/2}$. Experimental data are compared with simulations for this parameter choice in Fig.\ \ref{fig:wrt2}.
\begin{figure}[t!]
\includegraphics[width=.45\textwidth]{fig_roughness_6.pdf}
\caption{\label{fig:wrt2} Evolution of the radius and global roughness predicted by Eq.\ \eqref{eq:bacterial_growth} with physical parameters $A_0=0$ mm/h, $A_1=0.067$ mm${}^2$/h, $A_a=6.1\cdot 10^{-3}$ mm/h, $A_n=0.086$ mm${}^{3/2}$/h${}^{1/2}$. Squares (circles) are experimental radius (roughness) for BS with $C_n=20$ g/l; error bars are of the same size as symbols or smaller, see appendix A.}
\end{figure}
With respect to $W(t)$, agreement is reached for essentially the full duration of the experiments. For times longer than approximately $800$ hours (which remain beyond our experimental setup), Eq.\ \eqref{eq:bacterial_growth} predicts almost linear increase
with time for $W(t)$ and $R(t)$. The agreement between experimental and
simulation data is slightly worse in the case of $R(t)$, for which the
initial condition plays a stronger role in the continuum model.
Nevertheless, agreement also becomes quantitative for $t > 100$
hours. Note, the time $t_1$ required for the onset of long-time, asymptotic behavior in the experiments can be assessed from the numerical simulation of Eq.\ \eqref{eq:bacterial_growth}, see the bottom panel of Fig.\ \ref{fig:numbers}. We estimate $t_1=0.8$, which approximately corresponds to $1440$ hours. Overall, Fig.\ \ref{fig:wrt2} suggests that the scaling behavior reached in the experiments is preasymptotic, clear-cut asymptotics occurring for $t > t_1$, approximately twice our longest experimental growth time.
\subsection{Geometrical properties: Local roughness and radial correlations}
Further non-trivial properties of the experimental colonies involve the spaial dependence of front fluctuations. We can characterize them quantitatively by considering the so-called local roughness, $w(\ell)$, which evaluates interface deviations from an average position, within observation windows of size $\ell$ \cite{Vicsek_92,Barabasi_95,Meakin_98}. We proceed as is customary for systems with an overall circular symmetry \cite{Meakin_98,Santalla_14}: Namely,
each point on the front is converted to polar coordinates emanating
from the geometric center, $(x_i,y_i)\to (\theta_i,r_i)$, whereby
$\theta_i$ ($r_i$) is considered a new independent (dependent)
variable. Given an initial point $\vec r_0$ and a length scale $\ell$,
we consider the set of points within a circle centered at $\vec r_0$
with radius $\ell$. Then, we make a fit to the straight line which
minimizes the deviations. The mean-square distance of the front
positions to that fitting line provides the {\em local} roughness
$w(\ell)$.
Results for our experimental BS and EC colonies are displayed in Fig.\ \ref{fig:morphology}.
\begin{figure*}[t!]
\includegraphics[width=\textwidth]{fig_morph_old6.pdf}
\caption{\label{fig:morphology} Experimental length-scale dependence of
the local roughness of BS (a) and EC (b) colony fronts for different times and $C_n$ as indicated in the legends. (c) Same observable for numerical simulations of Eq.\ \eqref{eq:bacterial_growth} as in Fig.\ \ref{fig:foto}. All straight lines represent $w(\ell) \sim
\ell^{0.75}$.}
\end{figure*}
An approximate power-law dependence, $w(\ell)\sim \ell^{\alpha}$, holds at intermediate scales above 100 $\mu$m, and up to $3$ mm for the most favorable cases, with $\alpha\simeq 0.75$.
For the sake of comparison, we recall that a one-dimensional interface provided by the world-line of an uncorrelated random walk features $\alpha_{\rm RW}=1/2$ \cite{Vicsek_92,Barabasi_95,Meakin_98}. Our experimental value for $\alpha$ is in the same range as those found earlier for
similar bacterial colony experiments \cite{Vicsek_90,Wakita_97,Bonachela_11} and is also similar to values measured in other DL systems, like 1D ECD \cite{Pastor_96,Huo_01} or 2D thin films grown by CVD under low sticking conditions \cite{Ojeda_00,Zhao_00}.
In these contexts, such large $\alpha$ are known not to correspond to any well-defined universality class of kinetic roughening \cite{Castro_00,Nicoli_09,Castro_12}, but to merely reflect the large surface slopes that ensue, due to diffusive instabilities. Such instabilities are actually well-known to correlate with front branching \cite{Vicsek_92,Barabasi_95,Meakin_98}, which in our experiments can be assessed through the behavior of the autocorrelation of the radial interface fluctuations as a function of the angular distance,
\begin{equation}
C(\Delta\theta,t) = \langle [r(\theta,t)-R(t)][r(\theta+\Delta\theta,t)-R(t)]\rangle .
\label{eq:C}
\end{equation}
As seen in Fig.\ \ref{fig:corr}, and in spite of the compactness of the colonies, $C(\Delta\theta,t)$ vanishes approximately at the same angular distance for different times, indicating fronts that develop well-defined branches. Moreover, the importance of such branching increases monotonically along the experimental time evolution. Such a behavior is analogous to the result of detailed continuum models of bacterial colony growth put forward
in \cite{Giverso_15,Giverso_16}, which predict unconditional instability of the colony front to perturbations for a variety of relaxation mechanisms that include both, chemotactic and volumetric expansions. In application of the analysis in \cite{Giverso_15,Giverso_16} to our data, Fig.\ \ref{fig:arearadio} shows the time evolution of the area/perimeter ratio for the experimental colonies, compared to the $R(t)/2$ value that would correspond to a perfectly circular front in each case; clearly the actual perimeter grows too fast with time relative to the enclosed area, as compared with expectations for an ideally circular front. Such a behavior is inconsistent in particular with the occurrence of Eden behavior at long times \cite{Vicsek_92,Barabasi_95,Meakin_98}.
\begin{figure*}[t!]
\includegraphics[width=\textwidth]{fig_corr_8.pdf}
\caption{\label{fig:corr} Left to right: Autocorrelation function of experimental radial fluctuations vs angular distance rescaled by $\theta_0$, for BS (a), EC (b), and simulations (c) of Eq.\ \eqref{eq:bacterial_growth} with parameters as in Fig.\ \ref{fig:foto}, where $\theta_0=30^\circ, 60^\circ$, and $15^\circ$, respectively. Times and $C_n$ are as given in the legends.}
\end{figure*}
\begin{figure
\includegraphics[width=.45\textwidth]{fig_arearadio_2.pdf}
\caption{\label{fig:arearadio} Experimental area/perimeter ratio vs time. Symbols are direct measurements for conditions described in the legend; lines show $R(t)/2$ as obtained from Fig.\ \ref{fig:wrt} for the corresponding sets of data.}
\end{figure}
The geometrical properties of the front observed in the experiments are very similarly found also in the simulations of Eq.\ \eqref{eq:bacterial_growth}. Figure \ref{fig:morphology} shows the dependence of the simulated local roughness with length scale for different times, readily compared with the experimental data in the same figure. Indeed, for small scales the model yields $w(\ell)\sim \ell^{\alpha}$, with $\alpha$ increasing with time up to 0.75, very close to the experimental value. Comparison between the model and the experiments improves with increasing times, as typical length-scales also increase. Note, these simulation data include the long-time, asymptotic regime identified for the model in Fig.\ \ref{fig:wrt}. Finally, the behavior in simulations of the radial autocorrelation function $C(\Delta\theta,t)$ also supports our interpretation on branching at the interface: Figure \ref{fig:corr} indeed shows the selection of a precise correlation angle value $\theta_0$, analogous to the experimental morphologies.
\section{Discussion and Conclusions}
\label{sec:conclusions}
Summarizing our experimental observations for both BS and EC colonies in the suppressed-motility conditions \cite{Wakita_97,Matsushita_98,Rafols_98} which are in the classically alleged Eden regime, we obtain branched interfaces with scaling exponents $\beta\simeq0.5$ and $\alpha\simeq0.75$, which unambiguously differ from Eden/KPZ behavior, characterized by non-branched interfaces, $\beta_{\rm KPZ}=1/3$, and $\alpha_{\rm KPZ}=1/2$ \cite{Vicsek_92,Barabasi_95,Meakin_98}. Our experimental data are also inconsistent with quenched noise effects which, e.g., allegedly induce $\beta=0.61, \alpha=0.68$ in agent-based simulations of bacterial colonies \cite{Bonachela_11}, or with the so-called quenched KPZ (qKPZ) equation \cite{Barabasi_95}. Unconditioned by any comparison to models, the fact that our experimental colony profiles become increasingly branched during all accessible times (Figs.\ \ref{fig:corr} and \ref{fig:arearadio}) moreover suggests that the observed scaling is preasymptotic behavior for a system whose asymptotics is not Eden, and we speculate that this could also be the case for other, classical, experiments \cite{Vicsek_90,Wakita_97} performed under conditions which are similar to ours.
Given the semi-quantitative agreement between our experiments and simulations of the effective
model, Eq.\ \eqref{eq:bacterial_growth}, we can consider the latter in order to predict what would be the actual asymptotic behavior of the former. Indeed, Eq.\ \eqref{eq:bacterial_growth} predicts a long-time behavior with $\beta=0.93$ (Fig.\ \ref{fig:wrt}) and $\alpha=0.75$ (Fig.\ \ref{fig:morphology}). Actually, a small-slope approximation of Eq.\ \eqref{eq:bacterial_growth} yields dimension-independent exponents $\alpha=\beta=1$ \cite{Nicoli_09,Nicoli_09b} ---recently measured in CVD under DL conditions \cite{Castro_12}---, which are definitely non-KPZ and are expected to characterize Eq.\ \eqref{eq:bacterial_growth} at long times. Note, for interfaces with $\alpha\gtrsim 1$, local measurements using $w(\ell)$ are known to underestimate the correct value of the roughness exponent \cite{Castro_00}, explaining our $\alpha=0.75$ value. Parameter conditions in our experiments would make such a long-time regime hardly accessible, requiring growth times at least twice the longest time that we have been able to reach, as estimated in Sec.\ \ref{sec:phys_units}. On the other hand, the preasymptotic ($t< 800$ h) behavior in Eq.\ \eqref{eq:bacterial_growth} ---during which $W(t)$ evolves as in our experiments--- is dominated by the diffusive (shadowing) instabilities that induce branching of the front and large exponent values. In such a case, and as shown for other DL systems \cite{Nicoli_09,Castro_00,Castro_12}, the exponent values are non-universal and may depend on parameter values and even on the specific space/time ranges in which power-law fits are attempted.
In conclusion, bacterial colonies where individual motility is
suppressed form compact aggregates whose front morphology can still be
dominated by diffusive instabilities. For our experimental conditions, similar to those in
\cite{Wakita_97,Matsushita_98,Rafols_98}, preasymptotic
scaling seems to occur at the accessed times, which in any case is not in the Eden/KPZ universality class. There is no need to invoke quenched disorder to account for this
discrepancy. Rather, the shadowing instability induces large front
fluctuations with non-universal scaling. This behavior is strongly
reminiscent of many other experimental systems
\cite{Pastor_96,Ojeda_00,Zhao_00,Huo_01,Yunker_13} in which transport-induced instabilities induce effective scaling. In some of these cases
\cite{Yunker_13} the observed kinetic roughening properties have also been associated with the qKPZ universality class, due to accidental similarities in the values of the scaling
exponents \cite{Nicoli_13,Yunker_13_2,Oliveira_14}. Note, attributing a set of scaling exponents to a well-defined, asymptotic, universality class like qKPZ, or to non-universal preasymptotic behavior as we are presently advocating for, are conceptually very different interpretations.
Non-KPZ exponents due to diffusive instabilities are also predicted by
agent-based simulations \cite{Farrell_13,Farrell_17} for small values
of the active layer thickness $\delta$. However, for sufficiently
large $\delta$ very compact colonies with extremely flat fronts are
found \cite{Farrell_13,Farrell_17}. While this seemingly questions the
prevalence of diffusive instabilities, continuum models
\cite{Giverso_15,Giverso_16} analytically predict such flat front
conditions to be a finite-size effect. Thus, parameter conditions
select a typical length-scale $\ell_0$ for the instabilities,
which is well defined for any value of $\delta$. As
standard in pattern formation \cite{Cross_09}, the correlation length
along the front (initially a few cell sizes across) needs to increase
up to $\ell_0$ for the instability to set in. If $\ell_0$ is very
large (in band geometry, for systems smaller than $\ell_0$), the front
may effectively be flat. In circular geometries, for sufficiently
(perhaps, exceedingly) long times, the instability will still occur.
We should still note that additional systems exist, which are closely
related to the ones we study, and for which Eden/KPZ scaling does
occur. For instance, bacterial colonies for which individual motility
is non-negligible \cite{Wakita_97} yield a roughness exponent
compatible with the 1D KPZ value. Also, aggregates of non-cancerous
(Vero) or cancerous (HeLa) primate cells display unambiguous KPZ
\cite{Huergo_11}, and even qKPZ \cite{Huergo_14,Muzzio_16}, scaling,
as is the case with fungal growth \cite{Lopez_02}. Experimentally, KPZ
scaling also applies to fluctuating frontiers between different genetic
strains in range expansion of {\em E.\ coli} \cite{Hallatschek_07},
although deviations from Eden behavior can also occur
\cite{Kuhr_11,Reiter_14}. In general, individual cell motility seems
to play a relevant role, to the extent that instabilities associated
with nutrient transport can eventually be superseded. Indeed, the Eden
model \cite{Eden_61} will at any rate stand as the prime example for
reaction-limited growth \cite{Vicsek_92,Barabasi_95,Meakin_98}, where
nutrient transport is, effectively, infinitely fast and irrelevant to
front fluctuations.
\begin{acknowledgments}
We acknowledge fruitful conversations with G.\ Melaugh and
K.\ A.\ Takeuchi. This work has been supported by Ministerio de Economía y
Competitividad, Agencia Estatal de Investigación, and Fondo Europeo de
Desarrollo Regional (Spain and European Union) through grants FIS2015-66020-C2-1-P, FIS2015-69167-C2-1-P, FIS2015-73337-JIN, and BIO2016-79618-R, and by Comunidad Aut\'onoma de Madrid (Spain) Grant NANOAVANSENS S2013/MIT-3029.
\end{acknowledgments}
|
1702.03707
|
\section{Introduction}
The aim of this article is twofold. In the spirit of Graham-Yao \cite{GrY90}, we give a ``whirlwind tour'' of two areas of Geometric Ramsey Theory, and make some modest contributions to them.
\smallskip
The {\em diameter} of a finite point set $P$, denoted by ${\rm diam}(P)$, is the largest distance that occurs between two points of $P$. Borsuk's famous conjecture~\cite{Bor33}, restricted to finite point sets, states that any such set of unit diameter in $\mathbb{R}^d$ can be colored by $d+1$ colors so that no two points of the same color are at distance {\em one}. This conjecture was disproved in a celebrated paper of Kahn and Kalai~\cite{KaK93}. We extend the theorem of Kahn and Kalai as follows.
\medskip
\noindent{\bf Theorem 1.} {\em For any integer $r\ge 2$, there exist $\varepsilon=\varepsilon(r)>0$ and $d_0=d_0(r)$ with the following property. For every $d\ge d_0$, there is a finite point set $P\subset\mathbb{R}^d$ of diameter $1$ such that no matter how we color the elements of $P$ with fewer than $(1+\varepsilon)^{\sqrt{d}}$ colors, we can always find $r$ points of the same color, any two of which are at distance $1$.}
\medskip
In a seminal paper of Erd\H{o}s, Graham, Montgomery, Rothschild, Spencer, and Strauss~\cite{ErGM73}, the following notion was introduced. A finite set $P$ of points in a Euclidean space is a {\em Ramsey configuration} or, briefly, is {\em Ramsey}
if for every $r\ge 2$, there exists an integer $d=d(P,r)$ such that no matter how we color all points of~$\mathbb{R}^d$ with $r$ colors, we can always find a monochromatic subset of $\mathbb{R}^d$ that is congruent to $P$. In two follow-up articles~\cite{ErGM75a},~\cite{ErGM75b}, Erd\H os, Graham, and their coauthors established many important properties of these sets.
In the present paper, we introduce a related notion.
\medskip
\noindent{\bf Definition 2.} {\em A finite set $P$ of points in a Euclidean space is {\em diameter-Ramsey} if for every integer $r\ge 2$, there exist an integer $d=d(P,r)$ and a finite subset $R\subset \mathbb{R}^d$ with ${\rm diam}(R)={\rm diam}(P)$ such that no matter how we color all points of $R$ with $r$ colors, we can always find a monochromatic subset of $R$ that is congruent to $P$.}
\medskip
Obviously, every diameter-Ramsey set is Ramsey, but the converse is not true. For example, we know that all triangles are Ramsey, but not all of them are diameter-Ramsey.
\medskip
\noindent{\bf Theorem 3.} {\em All acute and all right-angled triangles are diameter-Ramsey.}
\medskip
\noindent{\bf Theorem 4.} {\em No triangle that has an angle larger than $150^\circ$ is diameter-Ramsey.}
\medskip
There is another big difference between the two notions: By definition, every subset of a Ramsey configuration is Ramsey. This is not the case for diameter-Ramsey sets.
\medskip
\noindent{\bf Theorem 5.} {\em The $7$-element set consisting of a vertex of a $6$-dimensional cube and its $6$ adjacent vertices is not diameter-Ramsey.}
\medskip
We will see that the vertex set of a cube (in fact, the vertex set of any brick) is diameter-Ramsey; see Lemma~\ref{brick}. Therefore, the property that a set is diameter-Ramsey is not hereditary.
\smallskip
It appears to be a formidable task to characterize all diameter-Ramsey simplices. It easily follows from the definition that all regular simplices are diameter-Ramsey; see Proposition~\ref{simplex}. We will show that the same is true for ``almost regular'' simplices.
\medskip
\noindent{\bf Theorem 6.}
{\em For every integer $n\ge 2$, there exists a positive real number $\varepsilon=\varepsilon(n)$ such that every $n$-vertex simplex whose side lengths belong to the interval
$[1-\varepsilon, 1+\varepsilon]$ is diameter-Ramsey.}
\medskip
This article is organized as follows: In Section 2, we give a short survey of problems and results on the structure of diameters and related coloring questions. In Section 3, we prove Theorem 1. In Section 4, we establish some simple properties of diameter-Ramsey sets and prove Theorems 3, 4, and 6, in a slightly stronger form. The proof of Theorem 5 is presented in Section 5. The last section contains a few open problems and concluding remarks.
\section{A short history}
{\bf I. The number of edges of diameter graphs and hypergraphs.}
Hopf and Pannwitz~\cite{HoP34} noticed that in any set $P$ of $n$ points in
the plane, the diameter occurs at most~$n$ times. In other words, among the
${n\choose 2}$ distances between pairs of points from~$P$ at most $n$ are
equal to $\mathrm{diam}(P)$. This bound can be attained for every $n\ge 3$. For odd $n$
this is shown by the vertex set of a regular $n$-gon, and for even $n$ it is
not hard to observe that one may add a further point to the vertex set of a
regular $(n-1)$-gon so as to obtain such an example. In fact all extremal
configurations were characterized by Woodall~\cite{Wo71}.
\smallskip
The same question in $\mathbb{R}^3$ was raised by V\'azsonyi, who conjectured
that the maximum number of times the diameter can occur among $n\ge 4$ points
in $3$-space is $2n-2$. V\'azsonyi's conjecture was proved independently
by Gr\"unbaum~\cite{Gr56}, by Heppes~\cite{He56}, and by Straszewicz~\cite{St57};
see also~\cite{Sw08} for a simple proof.
The extremal configurations were characterized in terms of ball polytopes
by Kupitz, Martini, and Perles~\cite{KuMP10}.
\smallskip
In dimensions larger than $3$, the nature of the problem is radically different.
\begin{thm}\label{erdos}\rm (Erd\H os~\cite{Er60}) For any integer $d>3$, the maximum number of occurrences
of the diameter (and, in fact, of any fixed distance) in a set of $n$ points in $\mathbb{R}^d$
is $\frac12\left(1-\frac{1}{\lfloor d/2\rfloor}+o(1)\right)n^2.$
\end{thm}
More recently, Swanepoel~\cite{Sw09} determined the exact maximum number of
appearances of the diameters for all $d>3$ and all $n$ that are sufficiently
large depending on $d$.
\smallskip
The {\em diameter graph} associated with a set of points $P$ is a graph with
vertex set $P$, in which two points are connected by an edge if and only if
their distance is $\mathrm{diam}(P)$. Erd\H{o}s noticed that there is an intimate
relationship between the above estimates for the number of edges of diameter
graphs and the following attractive conjecture of Borsuk~\cite{Bor33}:
Every (finite) $d$-dimensional point set can be decomposed into at most $d+1$
sets of smaller diameter. If it were true, this bound would be best possible, as
demonstrated by the vertex set of a regular simplex in $\mathbb{R}^d$.
\smallskip
One can generalize the notion of diameter graph as follows. Given a point set
$P\subset \mathbb{R}^d$ and an integer $r\ge 2$, let $H_r(P)$ denote the hypergraph
with vertex set $P$ whose hyperedges are all $r$-element subsets
$\{p_1,\ldots,p_r\}\subseteq P$ with $|p_i-p_j|=\mathrm{diam}(P)$
whenever $1\le i\neq j\le r$. Obviously, $H_2(P)$ is the diameter graph of $P$,
and $H_r(P)$ consists of the vertex sets of all {\em $r$-cliques}
(complete subgraphs with $r$ vertices) in the diameter graph.
Note that every $r$-clique corresponds to a regular $(r-1)$-dimensional simplex
with side length ${\rm diam}(P)$. We call $H_r(P)$ the {\em $r$-uniform diameter hypergraph} of $P$.
\smallskip
It was conjectured by Schur that the Hopf-Pannwitz theorem mentioned at the
beginning of this subsection can be extended to higher dimensions in the following
way: For any $d\ge 2$ and any $d$-dimensional $n$-element point set $P$,
the hypergraph $H_d(P)$ has at most $n$ hyperedges.
This was proved for $d=3$ by Schur, Perles, Martini, and Kupitz~\cite{ScPMK03}.
Building on work of Mori\'c and Pach~\cite{MoP15}, the case $d=4$ was resolved
by Kupavskii~\cite{Ku14}, and the general case of Schur's conjecture was subsequently settled by Kupavskii and Polyanskii~\cite{KuP14}.
\smallskip
However, for $2<r<d$ we know very little about the number of edges of the
diameter hypergraphs $H_r(P)$ and it would be interesting to investigate
this matter further.
\medskip
\noindent{\bf II. The chromatic number of diameter graphs and hypergraphs.} Erd\H os~\cite{Er46} pointed out that if we could prove that the number of edges
of the diameter graph of every $n$-element point set $P\subset\mathbb{R}^d$ is
smaller than $\frac{d+1}{2}n$, then this would imply that there is a vertex of
degree at most $d$. Hence, the chromatic number of the diameter graph would be at
most $d+1$, and the color classes of any proper coloring with $d+1$ colors would
define a decomposition of $P$ into at most $d+1$ pieces of smaller diameter, as
required by Borsuk's conjecture. For $d=2$ and $3$, this is the case. However, as is shown by Theorem~\ref{erdos}, in higher dimensions the number of edges of an $n$-vertex diameter graph can grow quadratically in $n$. Based on this, Erd\H os later suspected that Borsuk's conjecture may be false (personal communication).
This was verified only in 1993 by Kahn and Kalai~\cite{KaK93}.
\smallskip
Using a theorem of Frankl and Wilson~\cite{FrW81}, Kahn and Kalai established
the following much stronger statement.
\begin{thm}
{\rm (Kahn-Kalai)} For any sufficiently large $d$,
there is a finite point set $P$ in the \mbox{$d$-dimensional} Euclidean space such that
no matter how we partition it into fewer than $(1.2)^{\sqrt{d}}$ parts, at least
one of the parts contains two points whose distance is $\mathrm{diam}(P)$.
\end{thm}
In other words, the chromatic number of the diameter graph of $P$ is at least
$(1.2)^{\sqrt{d}}$.
Today Borsuk's conjecture is known to be false for all dimensions
$d\ge 64$; cf.~\cite{JeB14}.
\smallskip
\begin{dfn} The {\em chromatic number} of a hypergraph $H$ is the smallest number
$\chi=\chi(H)$ with the property that the vertex set of $H$ can be colored
with $\chi$ colors such that no hyperedge of $H$ is monochromatic.
\end{dfn}
Clearly, we have
\[
\chi(H_r(P))\le \chi(H_{r-1}(P))\le\ldots \le\chi(H_2(P))\,,
\]
for every $P$ and $r\ge 2$. Moreover,
\[
\chi(H_r(P))\le \biggl\lceil\frac{\chi(H_2(P))}{r-1}\biggr\rceil\,.
\]
To see this, take a proper coloring of the diameter graph $H_2(P)$ with the
minimum number, ${\chi=\chi(H_2(P))}$, of colors and let $P_1,\ldots, P_{\chi}$
be the corresponding color classes. Coloring all elements of
\[
P_{(i-1)(r-1)+1}\cup P_{(i-1)(r-1)+2}\cup\ldots\cup P_{i(r-1)}
\]
with color $i$ for $1\le i\le\frac{\chi}{r-1}$, we obtain a proper coloring of the hypergraph $H_r(P)$. (Here we set $P_s=\emptyset$ for all $s>\chi$.)
\smallskip
Using the above notation, the Kahn-Kalai theorem states that for any
sufficiently large integer~$d$, there exists a set $P\subset\mathbb{R}^d$
with $\chi(H_2(P))\ge (1.2)^{\sqrt{d}}$. According to a result of
Schramm~\cite{Sch88}, we have
$\chi(H_2(P))\le \bigl(\sqrt{3/2}+\varepsilon\bigr)^d$
for every $\varepsilon>0$, provided that $d$ is sufficiently large.
\smallskip
In the next section, we prove Theorem 1 stated in the Introduction. It extends the Kahn-Kalai theorem to
$r$-uniform diameter hypergraphs with $r\ge 2$. Using the above notation, we will prove the following.
\begin{thm}\label{thm:KK}
For any integer $r\ge 2$, there exist $\varepsilon=\varepsilon(r)>0$ and
$d_0=d_0(r)$ with the following property. For every $d\ge d_0$, there is a
finite point set $P\subset\mathbb{R}^d$ of diameter $1$ such that
\[
\chi(H_r(P))\ge (1+\varepsilon)^{\sqrt{d}}\,.
\]
That is, for any partition of $P$ into fewer than $(1+\varepsilon)^{\sqrt{d}}$ parts at least one of the parts contains $r$ points any two of which are at distance $1$.
\end{thm}
\medskip
\noindent{\bf III. Geometric Ramsey theory.} Recall from the Introduction that, according to the definition of Erd\H os, Graham {\it et al.}~\cite{ErGM73}, a finite set of points in some Euclidean space is said to be {\em Ramsey} if for every $r\ge 2$, there exists an integer $d=d(P,r)$ such that no matter how we color all points of $\mathbb{R}^d$ with $r$ colors, we can always find a monochromatic subset of $\mathbb{R}^d$ that is congruent to~$P$. Erd\H os, Graham {\it et al.} proved, among many other results, that every Ramsey set is {\em spherical}, i.e., embeddable into the surface of a sphere. Later Graham~\cite{Gr94} conjectured that the converse is also true: every spherical configuration is {\em Ramsey}. An important special case of this conjecture was settled by Frankl and R\"odl.
\begin{thm}\label{franklrodl} {\rm ~\cite{FrR90}} Every simplex is Ramsey.
\end{thm}
It was shown in \cite{ErGM73} that the class of all Ramsey sets is closed both under taking subsets and taking Cartesian products. This implies
\begin{cor} {\rm \cite{ErGM73}}\label{bricks}
All {\em bricks}, i.e., Cartesian products of finitely many $2$-element sets, are Ramsey.
\end{cor}
Further progress in this area has been rather slow. The first example of a planar Ramsey configuration with at least {\em five} elements was exhibited by K\v{r}\'i\v{z}, who showed that every regular polygon is Ramsey. He also proved that the same is true for every Platonic solid. Actually, he deduced both of these statements from the following more general theorem.
\begin{thm} {\rm ~\cite{Kr91}}
If there is a soluble group of isometries acting on a finite set of points $P$ in $\mathbb{R}^d$, which has at most $2$ orbits, then $P$ is Ramsey.
\end{thm}
Graham's conjecture is still widely open. In fact, it is not even known whether all quadrilaterals inscribed in a circle are Ramsey.
\smallskip
An alternative conjecture has been put forward by Leader, Russell, and Walters~\cite{LRW12}.
They call a point set {\it transitive} if its symmetry group is transitive. A subset of a transitive set is said to be {\it subtransitive}. Leader {\it et al.} conjecture that a set is Ramsey if and only if it is subtransitive. It is not obvious {\em a priori} that this conjecture is different from Graham's, that is, if there exists any spherical set which is not subtransitive. However, this was shown to be the case in~\cite{LRW12}. In~\cite{LRW11} the same authors showed further that not all quadrilaterals inscribed in a circle are subtransitive.
\smallskip
The ``compactness'' property of the chromatic number, established by Erd\H{o}s and de Bruijn~\cite{BrE51}, implies that for every Ramsey set $P$ and every positive integer $r$, there exists a {\em finite} configuration $R=R(P,r)$ with the property that {\em no matter how we color the points of $R$ with $r$ colors, we can find a congruent copy of $P$ which is monochromatic}. Following the (now standard) notation introduced by Erd\H os and Rado, we abbreviate this property by writing
$$R\longrightarrow (P)_r.$$
In Section 4, we address the problem how small the diameter of such a set $R$ can be. In particular, we investigate the question whether there exists a set $R$ with ${\rm diam}(R)={\rm diam}(P)$ such that $R\longrightarrow (P)_r.$ If such a set exists for every $r$, then according to Definition 2 (in the Introduction), $P$ is called {\rm diameter-Ramsey.}
\section{Proof of Theorem 1}
\label{sec:KK}
The proof of Theorem 1, reformulated as Theorem~\ref{thm:KK}, is based on the construction used by Kahn and Kalai in~\cite{KaK93}.
\smallskip
Suppose for simplicity that $d={2n\choose 2}$ holds for some {\em even}
integer $n$ and set $[2n]=\{1,2,\ldots,2n\}$. The construction takes place in
$\mathbb{R}^d$ and in the following we will index the coordinates of this space by the
$2$-element subsets of $[2n]$.
\smallskip
To each partition $[2n]=X\cup Y$ of $[2n]$ into two $n$-element subsets
$X$ and $Y$, we assign the point $p(X,Y)=p(Y,X)\in\mathbb{R}^d$ whose
coordinate $p_T(X, Y)$ corresponding to some unordered pair $T\subseteq [2n]$
is given by
\[
p_T(X,Y)=\left\{
\begin{array}{ll}
1 & {\mbox {\rm if}}\;\; |T\cap X|=|T\cap Y|=1, \\
0 & {\mbox {\rm otherwise.}}
\end{array}
\right.
\]
Let $P\subseteq\mathbb{R}^d$ be the set of all such points $p(X,Y)$.
We have $|P|=\frac12{2n\choose n}$.
\smallskip
Each point $p(X,Y)\in P$ has precisely $|X|\,|Y|=n^2$ nonzero coordinates. The squared Euclidean distance between $p(X,Y)$ and $p(X',Y')$, for two different partitions of $[2n]$, is equal to the number of coordinates in which $p(X,Y)$ and $p(X',Y')$ differ. The number of coordinates in which both $p(X,Y)$ and $p(X',Y')$ have a $1$ is equal to
\[
|X\cap X'||Y\cap Y'|+|X\cap Y'||X'\cap Y|\,.
\]
Denoting $|X\cap X'|=|Y\cap Y'|$ by $t$, the last expression is equal to $t^2+(n-t)^2$. Thus, we have
\[
\|p(X,Y)-p(X',Y')\|^2=2n^2-2(t^2+(n-t)^2)\,,
\]
which attains its maximum for $t=\frac{n}{2}$. The maximum is $n^2$, so that
$\mathrm{diam}(P)=n.$
\begin{fact}\label{fact:n2}
An $r$-element subset $\{p(X_1,Y_1),\ldots,p(X_r,Y_r)\}\subseteq P$ is a hyperedge of $H_r(P)$, the $r$-uniform diameter hypergraph of $P$, if and only if
\[
|X_i\cap X_j|=\frac{n}{2}\;\;\; \mbox{for all }\, 1\le i\neq j\le r. \;\;\;\;\;\;\; \Box
\]
\end{fact}
We need the following important special case of a result of Frankl and
R\"odl~\cite{FrR87} from extremal set theory.
The set of all $n$-element subsets of $[2n]$ is denoted by ${[2n]\choose n}$.
\begin{thm} {\rm \cite{FrR87}}\label{thm:FR}
For every integer $r\ge 2$, there exists $\gamma=\gamma(r)>0$ with the following
property. Every family of subsets ${\mathcal F}\subseteq {[2n]\choose n}$ with
$|{\mathcal F}|\ge (2-\gamma)^{2n}$ has $r$ members,
$F_1,\ldots,F_r\in{\mathcal F}$, such that
\[
|F_i\cap F_j|=\left\lfloor\frac{n}{2}\right\rfloor\;\;\; \mbox{for all }\, 1\le i\neq j\le r\,.
\]
\end{thm}
\smallskip
To establish Theorem~\ref{thm:KK}, fix a subset $Q$ of the set $P$ defined above.
The elements of $Q$ are points $p(X,Y)\in\mathbb{R}^d$ for certain partitions
$[2n]=X\cup Y$. Let ${\mathcal F}(Q)\subseteq{[2n]\choose n}$ denote the family
of all sets $X$ and $Y$ defining the points in $Q$. Notice that
$|{\mathcal F}(Q)|=2\,|Q|$.
\smallskip
By definition, $\chi=\chi(H_r(P))$ is the smallest number for which there is
a partition
\[
P=Q_1\cup\ldots\cup Q_{\chi}
\]
such that no $Q_k$ contains any hyperedge belonging to $H_r(P)$.
According to Fact~\ref{fact:n2}, this is equivalent to the condition that
${\mathcal F}(Q_k)$ does not contain $r$ members such that any two have precisely
$\frac{n}{2}$ elements in common. Now Theorem~\ref{thm:FR} implies that
\[
|{\mathcal F}(Q_k)|=2\,|Q_k|<\bigl(2-\gamma(r)\bigr)^{2n}
\quad \text{whenever} \quad 1\le k\le\chi\,.
\]
Thus, we have
\[
|P|=\sum_{k=1}^{\chi}|Q_k|<\frac{\chi}{2}\bigl(2-\gamma(r)\bigr)^{2n}\,.
\]
Comparing the last inequality with the equation $|P|=\frac12{2n\choose n}$,
we obtain
\[
\chi=\chi(H_r(P))>\frac{{2n\choose n}}{\bigl(2-\gamma(r)\bigr)^{2n}}
>\left(1+\frac{\gamma(r)}{3}\right)^{\sqrt{2d}}\,.
\]
This completes the proof of Theorem~\ref{thm:KK}.
\medskip
The proof of Theorem~\ref{thm:KK} gives the following result. The {\em regular} simplex $S_r$ with $r$ vertices and {\em unit side length} is not only a Ramsey configuration, but for every $k$ there exists set $P(k)\subseteq \mathbb{R}^d$ of {\em unit diameter} with $d\le c(r)\log^2 k$ such that no matter how we color $P(k)$ with $k$ colors, it contains a monochromatic congruent copy of $S_r$. (Here $c(r)>0$ is a suitable constant that depends only on~$r$.)
\section{Diameter-Ramsey sets -- Proofs of Theorems 3, 4, and 6}
According to Definition 2 (in the Introduction), a finite point set $P$ is diameter-Ramsey if for every $r\ge 2$, there exists a finite set $R$ in some Euclidean space with ${\rm diam}(R)={\rm diam}(P)$ such that no matter how we color all points of $R$ with $r$ colors, we can always find a monochromatic subset of $R$ that is congruent to $P$. Before proving Theorems 3, 4, and 6, we make some general observations about diameter-Ramsey sets.
\medskip
\begin{prop}\label{simplex}
Every regular simplex is diameter-Ramsey.
\end{prop}
\noindent{\bf Proof.} Let $P$ be (the vertex set of) a $d$-dimensional regular simplex. For a fixed integer $r\ge 2$, let $R$ be an $rd$-dimensional regular simplex of the same side length. By the pigeonhole principle, no matter how we color the vertices of $R$ with $r$ colors, at least $d+1$ of them will be of the same color, and they induce a congruent copy of $P$. \hfill $\Box$
\medskip
Recall that a {\em brick} is the vertex set of the Cartesian product of finitely many $2$-element sets.
\begin{lemma}\label{brick}
If $P$ and $Q$ are diameter-Ramsey sets, then so is their Cartesian product $P\times Q$. Consequently, any brick is diameter-Ramsey.
\end{lemma}
\noindent{\bf Proof.} It was shown in \cite{ErGM73} that for any Ramsey sets $P$ and $Q$, their Cartesian product,
\[
P\times Q = \{ p\times q\;|\;p\in P, q\in Q\}\,,
\]
is also a Ramsey set. Their argument, combined with the equation
\[
\mathrm{diam}^2(P\times Q)=\mathrm{diam}^2(P)+\mathrm{diam}^2(Q)\,,
\]
proves the lemma. \hfill $\Box$
\medskip
\noindent{\bf Proof of Theorem 3.}
Consider a right-angled triangle $T$ whose legs are of length $l_1$ and $l_2$. Let $P$ (resp., $Q$) be a set consisting of two points at distance $l_1$ (resp., $l_2$) from each other, so that we have
$T\subseteq P\times Q$. By Lemma~\ref{brick}, $P\times Q$ is diameter-Ramsey. Since $\mathrm{diam}(T)=\mathrm{diam}(P\times Q)$, we also have that $T$ is diameter-Ramsey.
\smallskip
Now let $T$ be an acute triangle with sides $a$, $b$, and $c$, where $a\le b\le c$.
Set
\[
l_1=\sqrt{c^2-a^2}, \quad l_2=\sqrt{c^2-b^2}, \quad \text{ and } \quad
x=\sqrt{a^2+b^2-c^2}\,.
\]
Since $T$ is acute, we have $a^2+b^2-c^2>0$. Therefore, $x$ is well defined. We have $l_1\ge l_2\ge 0$. Suppose first that $l_1\ge l_2>0$. Let $T_0$ be a right angled triangle with legs $l_1$ and $l_2$, and let $S$ be an equilateral triangle of side length $x$. We have $a^2=l_2^2+x^2$, $b^2=l_1^2+x^2$, and $c^2=l_1^2+l_2^2+x^2$. Thus,
\[
T\subseteq T_0\times S \quad \text{ and } \quad \mathrm{diam}(T)=\mathrm{diam}(T_0\times S)=c\,.
\]
By Proposition~\ref{simplex} and Lemma~\ref{brick}, we conclude that $T$ is diameter-Ramsey. In the remaining case, we have $l_2=0$. Now $T_0$ degenerates into a line segment or a point. It is easy to see that the above proof still applies. \hfill$\Box$
\medskip
We will prove Theorem 4 in a more general form. For this, we need a definition.
\begin{dfn}\label{degenerate}
Let $t$ be a positive integer. A finite set of points $P$ in some Euclidean space is said to be {\em $t$-degenerate} if it has a point $p\in P$ such that for the vertex set $S$ of any regular $t$-dimensional simplex with $p\in S$ and ${\rm diam}(S)={\rm diam}(P)$, we have
$${\rm diam}(P\cup S)>{\rm diam}(P).$$
\end{dfn}
\begin{thm}\label{degthm}
Let $t\ge 1$ and let $P$ be a finite $t$-degenerate set of points in some Euclidean space, which contains the vertex set of a regular $t$-dimensional simplex of side length ${\rm diam}(P)$. Then $P$ is not diameter-Ramsey.
\end{thm}
\noindent{\bf Proof.} Suppose for contradiction that $P$ is diameter-Ramsey. This implies that there exists a set~$R$ with ${\rm diam}(R)={\rm diam}(P)$ such that no matter how we color it by two colors, it always contains a monochromatic congruent copy of $P$.
Color the points of $R$ with red and blue, as follows. A point is colored {\em red} if it belongs to a subset $S\subset R$ that spans a $t$-dimensional simplex of side length ${\rm diam}(R)
$. Otherwise, we color it {\rm blue}. Let $P'$ be a monochromatic copy of $P$. By the assumptions, $P'$ contains the vertices of a regular $t$-dimensional simplex of side length ${\rm diam}(P)$, and all of these vertices are red. Since $P$ is $t$-degenerate, the point of $P'$ corresponding to $p$ is blue, which is a contradiction. \hfill $\Box$
\medskip
Theorem 4 is an immediate corollary of Theorem~\ref{degthm} and the following statement.
\begin{lemma}\label{150degrees1}
Every triangle that has an angle larger than $150^\circ$ is $1$-degenerate.
\end{lemma}
With no danger of confusion, for any two points $p$ and $p'$, we write $pp'$ to denote both the segment connecting them and its length.
\smallskip
To establish Lemma~\ref{150degrees1}, it is sufficient to verify the following.
\begin{lemma}\label{150degrees2}
Let $T=\{p_1,p_2,p_3\}$ be the vertex set of a triangle and $q$ another point in some Euclidean space such that
$$\max(p_2q, p_3q)\le p_1q\le p_2p_3.$$
Then the angle of $T$ at $p_1$ is at most $150^\circ$.
\end{lemma}
First, we show why Lemma~\ref{150degrees2} implies Lemma~\ref{150degrees1}. Let $T=\{p_1,p_2,p_3\}$ be a triangle whose angle at $p_1$ is larger than $150^\circ$, so that $\mathrm{diam}(T)=p_2p_3$. Suppose without loss of generality that $\mathrm{diam}(T)=1$. To prove that $T$ is $1$-degenerate, it is enough to show that for any unit segment $S=p_1q$, we have $\mathrm{diam}(T\cup S)>1$. Suppose not. Then we have
$\max(p_2q,p_3q)\le p_1q=p_2p_3=1.$ Hence, by Lemma~\ref{150degrees2}, the angle of $T$ at $p_1$ is at most $150^\circ$, which is a contradiction.
\medskip
\noindent{\bf Proof of Lemma~\ref{150degrees2}.} Proceeding indirectly, we assume that
\begin{equation}\label{eq:more150}
\sphericalangle{p_2p_1p_3}>150^\circ\,.
\end{equation}
Let $\Pi$ denote a ($2$-dimensional) plane containing $T$, and let $q'$ denote the orthogonal projection of $q$ to $\Pi$. In the plane $\Pi$, let $g$ and $h$ denote the perpendicular bisectors of the segments $p_1p_2$ and $p_2p_3$, respectively.
\smallskip
\parpic[l]{\includegraphics[width=6cm]{150.pdf}} Since $p_1q\ge p_2q$, we have $p_1q'\ge p_2q'$. Thus, $q'$ belongs to the closed half-plane of $\Pi$ bounded by $g$ where $p_2$ lies. By symmetry, $q'$ belongs to the half-plane bounded by $h$ that contains~$p_3$. This implies that the intersection of these two half-planes is nonempty. In particular, $p_1$ cannot be an interior point of $p_2p_3$ and, by \eqref{eq:more150}, it follows that the triangle $T$ must be non-degenerate. Hence, $g$ and $h$ must meet at a point $o$, the circumcenter of $T$.
\smallskip
Due to the inscribed angle theorem, we have
\[
\sphericalangle{p_2p_1p_3}+\tfrac12 \sphericalangle{p_2op_3}=180^\circ
\]
and hence $\sphericalangle{p_2op_3}< 60^\circ$ by~\eqref{eq:more150}.
This, in turn, implies that $p_2o, p_3o> p_2p_3$.
Thus, we have $$p_2q'\le p_2q\le p_2p_3<p_2o$$ and, in particular, $q'\ne o$. If one side of a triangle is smaller than another, then the same is true for the opposite angles. Applying this to the triangle $p_2q'o$, we obtain that $\sphericalangle{q'op_2}<90^\circ$. Analogously, we have
$\sphericalangle{q'op_3}<90^\circ$, which contradicts the position of $q'$ described in the previous paragraph. \hfill$\Box$
\medskip
We have been unable to answer
\begin{quest}\label{qu:obtuse}
Does there exist any obtuse triangle that is diameter-Ramsey?
\end{quest}
We would like to remark, however, that the answer would be affirmative if we would
just consider colourings with two colours. This is shown by the following example.
\begin{exmp}
Let $R$ be the vertex set of a regular heptagon $p_1p_2\dots p_7$ and let
$P=\{p_1, p_2, p_4\}$. Clearly, $P$ is the vertex set of an obtuse triangle
having an angle of size $\tfrac 47\cdot 180^\circ>90^\circ$ and ${\mathrm{diam}(R)=\mathrm{diam}(P)}$.
Moreover, we
have $R\longrightarrow (P)_2$, because the triple system with vertex set~$R$
whose edges are all sets of the form $\{p_i, p_{i+1}, p_{i+3}\}$ (the addition being
performed modulo $7$) is known to be isomorphic to the Fano plane, which
in turn is known to have chromatic number $3$.
\end{exmp}
It seems to be quite difficult to characterize all diameter-Ramsey simplices. According to Proposition~\ref{simplex}, every regular simplex is diameter-Ramsey. Theorem 6 states that this remains true for ``almost regular'' simplices. It is a direct corollary of the following statement.
\begin{lemma}
Every simplex $S$ with vertices $p_1, p_2, \ldots, p_n$ satisfying
\[
\sum_{1\le i<j\le n}(p_ip_j)^2\ge \left(\binom{n}{2}-1\right)\mathrm{diam}^2(S)
\]
is diameter-Ramsey.
\end{lemma}
\noindent{\bf Proof.}
Suppose without loss of generality that $\mathrm{diam}(S)=p_1p_2=1$. Our strategy is to embed~$S$ into the Cartesian product $R$ of $1+\binom{n}{2}$ regular simplices, some of which might degenerate to a point. We will be able to achieve this, while making sure that $\mathrm{diam}(R)=1$. Thus, in view of Proposition~\ref{simplex} and Lemma~\ref{brick}, we will be done.
\smallskip
Set
\[
a=\sqrt{\sum_{i<j}(p_ip_j)^2-\binom{n}{2}+1}\,\,\,\, {\rm and}\,\,\,\, x_{ij}=\sqrt{1-(p_ip_j)^2}
\]
for every $i<j$. Let $T_0$ be a regular simplex of side length $a$ with $n$ vertices. Let $S_{ij}$ be a regular simplex of side length $x_{ij}$ with $n-1$ vertices, $1\le i<j\le n$. For the Cartesian product of these simplices,
\[
R=T_0\times\prod_{i<j}S_{ij}\,,
\]
we have
\[
\mathrm{diam}^2(R)=a^2+\sum_{i<j}x_{ij}^2=1\,,
\]
as required.
\smallskip
Let $\pi_0\colon R\longrightarrow T_0$ and $\pi_{ij}\colon R\longrightarrow S_{ij}$
denote the canonical projections. Choose $n$ points, $q_1, \ldots, q_n\in R$
such that
\[
T_0=\{\pi_0(q_1), \ldots, \pi_0(q_n)\}\,,\,\,\,
S_{ij}=\{\pi_{ij}(q_1), \ldots, \pi_{ij}(q_n)\}\,\,\,
\text{ and } \,\,\, \pi_{ij}(q_i)=\pi_{ij}(q_j)\,,
\]
for $1\le i<j\le n$. It remains to check that the
simplex $\{q_1, \ldots, q_n\}$ is congruent to $S$. However, this is obvious, because
\[
(q_kq_\ell)^2=a^2+\sum_{i<j}x_{ij}^2-x_{k\ell}^2=1-x_{k\ell}^2=(p_kp_\ell)^2\,,
\]
for every $1\le k<\ell\le n$.
\hfill $\Box$
\section{\bf Proof of Theorem 5}
Throughout this section, let $d\ge 6$, let $p_0$ denote the origin of $\mathbb{R}^d$, and let $S=\{p_0,p_1,p_2,p_3\}\subset\mathbb{R}^d$ be the vertex set of a regular tetrahedron of side length $\sqrt2$. Further, let $P\subset\mathbb{R}^d$ denote the $7$-element set consisting of the origin $p_0\in \mathbb{R}^d$ and the (endpoints of the) first $6$ unit coordinate vectors $q_1=(1,0,0,0,0,0,\ldots),$ $q_2=(0,1,0,0,0,0,\ldots),$ $\ldots,$ $q_6=(0,0,0,0,0,1,\ldots).$
Obviously, we have $\mathrm{diam}(S)=\mathrm{diam}(P)=\sqrt2$.
\smallskip
In view of Theorem~\ref{degthm}, in order to establish Theorem 5, it is sufficient to prove that $P$ is $3$-degenerate. That is, we have to show that $\mathrm{diam}(P\cup S)> \sqrt2$. In other words, we have to establish
\begin{clm}\label{claim}
There exist integers $i$ and $j$\, $(1\le i\le 3,\, 1\le j\le 6)$ with $p_iq_j>\sqrt2$.
\end{clm}
The rest of this section is devoted to the proof of this claim.
\smallskip
For $i=1,2,3$, decompose $p_i$ into two components: the orthogonal projection of $p_i$ to the subspace induced by the first 6 coordinate axes and its orthogonal projection to the subspace induced by the remaining coordinate axes. That is, if $p_i=(x_i(1),\ldots,x_i(d))$, let $p_i=p'_i+p''_i$, where
\[
p_i'=(x_i(1),\ldots,x_i(6),0,\ldots,0)\;\;\;{\rm and}\;\;\;
p_i''=(0,\ldots,0,x_i(7),\ldots,x_i(d))\,.
\]
Obviously, we have
\begin{equation}\label{egyenlet1}
|p_i|^2=|p_i'|^2+|p_i''|^2=2\,.
\end{equation}
The proof of Claim~\ref{claim} is indirect. Suppose, for the sake of contradiction, that
\[
\mathrm{diam}\{p_0,p_1,p_2,p_3,q_1,\ldots,q_6\}=\sqrt2\,.
\]
Since $q_j$ and $p_0$ differ only in their $j$th coordinate and $p_iq_j\le p_ip_0$,
the points $p_i$ and $q_j$ lie on the same side of the hyperplane perpendicularly
bisecting the segment $p_0q_j$. That is,
\begin{equation}\label{egyenlet2}
x_i(j)\ge \frac12\;\;\;\text{for every}\;i,j\;\; (1\le i\le 3,\, 1\le j\le 6).
\end{equation}
Hence, we have $|p_i'|^2=\sum_{j=1}^6x_i^2(j)\ge\frac32$ and, by (\ref{egyenlet1}),
\begin{equation}\label{egyenlet3}
|p_i''|^2=|p_i|^2-|p_i'|^2\le\frac12\;\;\;\text{for every}\;i\;\; (1\le i\le 3).
\end{equation}
Moreover, if $i, i'\in\{1,2,3\}$ are distinct, then
\[
\langle p_i, p_{i'}\rangle=\tfrac12\bigl(|p_i|^2+|p_{i'}^2|-|p_i-p_{i'}|^2\bigr)
=\tfrac12(2+2-2)=1\,,
\]
whence~\eqref{egyenlet2} implies
\[
\langle p''_i, p''_{i'}\rangle=1-\sum_{j=1}^6x_i(j)x_{i'}(j)\le-\tfrac12\,.
\]
In view of~\eqref{egyenlet3} it follows that
\[
|p''_1+p''_2+p''_3|^2
=|p''_1|^2+|p''_2|^2+|p''_3|^2
+2\bigl(\langle p''_1, p''_{2}\rangle+\langle p''_1, p''_{3}\rangle
+\langle p''_2, p''_{3}\rangle\bigr)\le -\tfrac32\,,
\]
which is a contradiction. This concludes the proof of Claim~\ref{claim}
and, hence, also the proof of Theorem~5.
\section{Concluding remarks}
\noindent{\bf I. Kneser graphs and hypergraphs.} Let $d=rn+(k-1)(r-1)$,
where $r,k\ge 2$ are integers. Assign to each $n$-element subset $X\subseteq[d]$
the characteristic vector of $X$. That is, assign to $X$ the point
$p(X)\in\mathbb{R}^d$, whose $i$-th coordinate is
\[
p_i(X)=\left\{
\begin{array}{ll}
1 & {\mbox {\rm if}}\;\; i\in X , \\
0 & {\mbox {\rm if}}\;\; i\not\in X.
\end{array}
\right.
\]
Let $P\subseteq\mathbb{R}^d$ be the set of all points $p(X)$.
We have $|P|=\binom{d}{n}$ and $\mathrm{diam}(P)=\sqrt{2n}$.
\smallskip
For $r=2$, we have $P\subset\mathbb{R}^{2n+k-1}$, and the diameter graph
$H_2(P)$ is called a {\em Kneser graph}. It was conjectured by Kneser~\cite{Kn55}
and proved by Lov\'asz~\cite{Lo78} that $\chi(H_2(P))> k.$ On the other hand, if
$k\le n$, we have $H_3(P)=\emptyset$.
This was generalized to any value of $r$ by Alon, Frankl, and
Lov\'asz~\cite{AlFL86}, who showed that $\chi(H_r(P))>k$, while
$H_{r+1}(P)=\emptyset$, provided that $(k-1)(r-1)<n$. In other words,
the fact that the chromatic number of the $r$-uniform diameter hypergraph of a
point set is high does not imply that the same must hold for its $(r+1)$-uniform
counterpart.
\smallskip
For any integers $r, d\ge 2,$ let $\chi_r(d)$ denote the maximum chromatic number which
an $r$-uniform diameter hypergraph of a point set $P\subseteq\mathbb{R}^d$ can have.
\begin{quest}
Is it true that for every $r\ge 2$, we have $\chi_{r+1}(d)=o(\chi_r(d))$, as $d$ tends to infinity?
\end{quest}
\bigskip
\noindent{\bf II. Relaxations of the diameter-Ramsey property.} Diameter-Ramsey configurations seem to constitute a somewhat peculiar
subclass of the class of all Ramsey configurations. We suggest to classify
all Ramsey configurations $P$ according to the growth rate of the minimum diameter of a point set $R$ with $R\longrightarrow (P)_r$, as $r\rightarrow\infty$.
\begin{dfn}
Given a Ramsey configuration $P$ and an integer $r$, we define
\[
d_P(r)=\inf\,\{\mathrm{diam}(R)\,|\,R\longrightarrow (P)_r\}\,.
\]
\end{dfn}
We have $d_P(r)\ge \mathrm{diam}(P)$, for any Ramsey set $P$ and any integer
$r$, and this holds with equality if and only if for every $\varepsilon>0$ there
exists a configuration $R$ with $R\longrightarrow (P)_r$ and
$\mathrm{diam}(R)\le (1+\varepsilon)\mathrm{diam}(P)$. Certainly, all diameter-Ramsey sets $P$ satisfy
$d_P(r)=\mathrm{diam}(P)$ for all~$r$, but perhaps the configurations with the latter
property form a broader class.
\begin{dfn}
We call a Ramsey set $P$, lying in some Euclidean space,
\begin{enumerate}
\item[(a)] {\it almost diameter-Ramsey} if $d_P(r)=\mathrm{diam}(P)$ holds for all
positive integers $r$;
\item[(b)] {\it diameter-bounded} if there is $C_P>0$ such that
$d_P(r)<C_P$ holds for every positive integer~$r$;
\item[(c)] {\it diameter-unbounded} if $d_P(r)$ tends to infinity, as $r\rightarrow \infty$.
\end{enumerate}
\end{dfn}
We do not know whether there exists any almost diameter-Ramsey configuration
that fails to be diameter-Ramsey. Thus, we would like to ask the following
\begin{quest}
Is it true that every almost diameter-Ramsey set is diameter-Ramsey?
\end{quest}
To establish the diameter-boundedness of certain sets, we may utilize a result of
Matou\v sek and R\"odl~\cite{MaR95}. They showed that, given a
simplex $S$ with circumradius $\varrho$, any number of colors $r$, and
any $\varepsilon>0$, there exists an integer $d$ such that the $d$-dimensional sphere
of radius $\varrho+\varepsilon$ contains a configuration $R$ with
$R\longrightarrow (S)_r$. In particular, this implies the
following
\begin{cor}
Every simplex is diameter-bounded Ramsey.
\end{cor}
Consequently, every diameter-unbounded Ramsey set must be affinely dependent.
We cannot decide whether there exists any diameter-unbounded Ramsey set,
but the regular pentagon may serve as a good candidate. K\v{r}\'i\v{z}'s proof
establishing that the regular pentagon is Ramsey~\cite{Kr91} does not seem to
imply that it is also diameter-bounded.
\begin{quest}
Is the regular pentagon diameter-unbounded?
\end{quest}
Finally we mention that one can also define these notions for families
of configurations and ask, e.g., whether they be uniformly diameter-bounded
Ramsey. As an example, we remark that a slight modification of a colouring appearing
in~\cite{ErGM73} shows that no bounded subset of any Euclidean space
can simultaneously arrow all triangles whose diameter is $2$ with $8$ colours.
To see this, one may colour each point $x$ with the residue class of $\lfloor 2\|x\|^2\rfloor$
modulo $8$. Given any $K>1$ we set $\xi=\frac{1}{17K^2}$ and consider the isosceles
triangle with legs of length $1+\xi$ and base of length $2$. Assume for the sake
of contradiction that there is a monochromatic copy $abc$ of this triangle with
apex vertex $b$ and with $\|a\|, \|b\|, \|c\|\le K$. Let $m$ denote the mid-point of the
segment $ac$ and observe that $bm=\sqrt{\xi}$. The triangle inequality yields
\[
\sqrt{\xi}=\|b-m\|\ge \big|\|b\|-\|m\|\big|
\]
and, hence, we have
\[
\sqrt{\xi}\cdot \bigl(\|b\|+\|m\|\bigr)\ge \big|\|b\|^2-\|m\|^2\big|\,.
\]
Multiplying by $4$, and applying triangle inequality to the left-hand side and the
parallelogram law to the right-hand side we infer
\begin{align*}
2\sqrt{\xi}\cdot \bigl(\|a\|+2\|b\|+\|c\|\bigr) & \ge \big| 4\|b\|^2-\|a+c\|^2\big| \\
& = \big| 4\|b\|^2-2\|a\|^2-2\|c\|^2+\|a-c\|^2\big| \\
& = \big|4+\bigl(2\|b\|^2-2\|a\|^2\bigr) +\bigl(2\|b\|^2-2\|c\|^2\bigr)\big|\,,
\end{align*}
which due to
$\lfloor 2\|a\|^2\rfloor\equiv \lfloor 2\|b\|^2\rfloor\equiv\lfloor 2\|c\|^2\rfloor\pmod{8}$
leads to $8K\sqrt{\xi}\ge 2$, contrary to our choice of $\xi$.
\begin{rem}
While revising this article, we learned from Nora Frankl about some
progress regarding Question~\ref{qu:obtuse} obtained jointly with Jan Corsten~\cite{CF17}.
They proved that the bound of $150^\circ$ appearing in Theorem~4 above can be lowered
to $135^\circ$. Their elegant proof involves the spherical colouring and Jung's inequality.
\end{rem}
|
1702.03874
|
\section{Introduction}
Many signal processing and resource allocation problems can be posed as an optimization problem which aims at minimizing a system cost consisting of local costs of subsystems and a shared cost of the whole system. For instance, consider a power system divided into multiple subsystems depending on either the geographical locations or the power line connections \cite{gan2013optimal}. On one hand, if a subsystem receives some amount of power supplies, the consumption or storage of these supplies enables the subsystem to gain some utility. On the other hand, the generation of the total power supplies of all the subsystems incurs some cost for the whole power system due to factors such as the consumption of natural resources, the pollution and the human efforts. The goal of the designer or controller of the power system is to maximize the social welfare or minimize the overall system cost including the negative of the total utilities of all the subsystems and the power generation cost of the whole system. This structure of local costs plus shared common cost arises in many applications such as smart grids, communication networks, or more generally, signal processing and control in multi-agent systems. Optimization problems with such structure are called the sharing problems as there is a term of the shared cost of the whole system in the objective function \cite{boyd2011distributed}.
One implicit assumption of the conventional sharing problem is that both the local cost functions and the shared cost function are static, i.e., they do not vary with time. However, in practice, the cost or utility functions of many applications are intrinsically time-varying. For example, in power grids, the utility functions of the subsystems vary across time as the power users' demands evolve, e.g., the demands climax during evening and decline between midnight and early morning. The generation cost of the power system also varies with time owing to the changing and somewhat unpredictable renewable energy sources (e.g., wind and solar energy) as well as the fluctuation of the market prices of the traditional energy. As another example, in real-time signal processing or online learning of multi-agent systems, the data stream arrive sequentially as opposed to in a single batch. This also makes the cost functions involved in the estimators or learners vary with time. Therefore, we are motivated to study a dynamic version of the sharing problem in this paper.
In the literature, dynamic optimization problems arise in various research fields and have been studied from different perspectives. In adaptive signal processing such as the recursive least squares (RLS), the input/output data arrive sequentially, resulting in a time-varying objective function (the discounted total squared errors) to be minimized \cite{Haykin:1996:AFT:230061}. The RLS algorithm is able to track the unknown time-variant weight vectors relating the input and output data in real time. More recently, the concept of adaptive signal processing has been extended to adaptive networks, leading to dynamic distributed optimization problems over networked systems \cite{sayed2014adaptive,jiang2013distributed}. Another line of research for dynamic optimization is online convex optimization (OCO) \cite{hazan2016introduction,hall2015online,mahdavi2012trading,koppel2015saddle,flaxman2005online,zinkevich2003online}. In OCO, the time-varying cost functions are unknown a-priori and the goal is to design online algorithms with low (e.g., sublinear) regrets, i.e., the solution from the algorithms are not too worse than the optimal offline benchmarks. More broadly speaking, online learning (e.g., the weighted majority algorithm and the multiplicative weight update method) \cite{littlestone1989weighted,arora2012multiplicative,kivinen1997exponentiated,tekin2016adaptive,tekin2015distributed,tekin2014distributed} and (stochastic) dynamic control/programming (e.g., Markov decision processes) \cite{puterman2014markov,bertsekas1976dynamic,altman1999constrained} also lie in the category of dynamic optimizations, though their problem formulations are very distinct from that of this paper.
To solve the dynamic sharing problem in an online manner, in this paper, we present a dynamic ADMM algorithm. As a dual domain method, the ADMM is superior to its primal domain counterparts such as the gradient descent method in terms of convergence speed. Due to its broad applicability, the ADMM has been exploited in various signal processing and control problems including distributed estimation \cite{ling2010decentralized}, decentralized optimization \cite{shi2014linear}, wireless communications \cite{shen2012distributed}, power systems \cite{zhang2016admm} and multi-agent coordination \cite{chang2014proximal}. While most existing works only consider static ADMM in which the objectives and constraints are time-invariant, a few recent studies have investigated the ADMM in a dynamic scenario. When the time-varying objective functions are unknown a-priori, an online ADMM algorithm is proposed in \cite{hosseini2014online} to generate solutions with low regrets compared to the optimal static offline solution. This online ADMM is not directly applicable to many dynamic sharing problems in which the goal is to track the time-varying optimal points and a static offline benchmark is not very meaningful. Additionally, several stochastic ADMM algorithms \cite{ouyang2013stochastic,zhong2014fast,suzuki2013dual} have been proposed to solve stochastic programs iteratively using sequential samples. Though the stochastic ADMM operates in a time-varying manner as the new samples arrive sequentially, the underlying stochastic program itself does not change over time, which makes the problem setup very distinct from the dynamic sharing problem considered here. A more closely related work is \cite{ling2014decentralized}, in which a dynamic ADMM algorithm is applied to the consensus optimization problems. However, the convergence analysis of the dynamic ADMM in \cite{ling2014decentralized} significantly relies on the special structure of the consensus optimization problems, in which all agents share the same decision variable. This leaves the performance of the dynamic ADMM in other optimization scenarios largely unknown.
Our goal in this work is to investigate the convergence behaviors of the dynamic ADMM for the dynamic sharing problem both theoretically and empirically. The main contributions of this paper can be summarized as follows.
\begin{itemize}
\item We motivate and formally formulate the dynamic sharing problem (Problem \eqref{dynamic_share}). A dynamic ADMM algorithm (Algorithm 1) is proposed for a more general dynamic optimization problem (Problem \eqref{dynamic_admm}), which encompasses the dynamic sharing problem as a special case. The dynamic ADMM can adapt to the time-varying cost functions and track the optimal points in an online manner.
\item We analyze the convergence properties of the proposed dynamic ADMM algorithm. We show that, under standard technical assumptions, the dynamic ADMM converges linearly to some neighborhoods of the time-varying optimal points. The sizes of the neighborhoods are related to the drifts of the dynamic optimization problem: the more drastically the dynamic problem evolves with time, the larger the sizes of the neighborhoods. We also study the impact of the drifts on the steady state convergence behaviors of the dynamic ADMM.
\item Two numerical examples are presented to validate the effectiveness of the dynamic ADMM algorithm. The first example is a dynamic sharing problem while the second one is the dynamic least absolute shrinkage and selection operator (LASSO). We observe that the dynamic ADMM can track the time-varying optimal points quickly and accurately. For the dynamic LASSO, the dynamic ADMM has competitive performance compared to the benchmark offline optimizor while the former is computationally superior to the latter dramatically.
\end{itemize}
The remaining part of this paper is organized as follows. In Section \Rmnum{2}, the dynamic sharing problem is formally defined and a dynamic ADMM algorithm is proposed. In Section \Rmnum{3}, theoretical analysis of the convergence properties of the dynamic ADMM is presented. Two numerical examples are shown in Section \Rmnum{4}, following which we conclude this work in Section \Rmnum{5}.
\section{Problem Statement and Algorithm Development}
In this section, we first formally state the dynamic sharing problem and give some examples and motivations for it. Then, we present some rudimentary knowledge of the standard static ADMM. Finally, we develop a dynamic ADMM algorithm for a more general dynamic optimization problem, which encompasses the dynamic sharing problem as a special case.
\subsection{The Statement of the Problem}
Consider the sharing problem \cite{boyd2011distributed}:
\begin{align}\label{share}
\text{Minimize}~~\sum_{i=1}^nf^{(i)}\left(\mathbf{x}^{(i)}\right)+g\left(\sum_{i=1}^n\mathbf{x}^{(i)}\right),
\end{align}
with variables $\mathbf{x}^{(i)}\in\mathbb{R}^p$, $i=1,...,n$, where $f^{(i)}:\mathbb{R}^p\mapsto\mathbb{R}$ is the local cost function of subsystem $i$ and $g:\mathbb{R}^p\mapsto\mathbb{R}$ is the global cost function of some commonly shared objective of all subsystems. The global cost function $g$ takes the sum of all $\mathbf{x}^{(i)}$ as its input argument. The sharing problem \eqref{share} is a canonical problem with broad applications in resource allocation and signal processing \cite{boyd2011distributed}. One limitation of the problem formulation in \eqref{share} and its solution methods is that all the cost functions are static, i.e., they do not vary over time. This can be a major obstacle when the application is inherently time-variant and real-time, in which the cost functions change with time and online processing/optimization is imperative. In such circumstances, dynamic algorithms adaptive for the variation of the cost functions are more favorable. This motivates us to study a dynamic version of the sharing problem:
\begin{align}\label{dynamic_share}
\text{Minimize}~~\sum_{i=1}^nf_k^{(i)}\left(\mathbf{x}^{(i)}\right)+g_k\left(\sum_{i=1}^n\mathbf{x}^{(i)}\right),
\end{align}
where $k$ is the time index. $f_k^{(i)}:\mathbb{R}^p\mapsto\mathbb{R}$ is the local cost function of subsystem $i$ at time $k$ and $g_k:\mathbb{R}^p\mapsto\mathbb{R}$ is the global cost function of the shared objective at time $k$. The dynamic sharing problem in \eqref{dynamic_share} can be applied to many dynamic resource allocation problems, among which we name two in the following.
\begin{itemize}
\item Consider a power grid which is divided into $n$ power subsystems according to either geographical locations or power line connections. If subsystem $i$ receives $\mathbf{x}^{(i)}$ amount of power supplies at time $k$, then it gains a utility of $-f_k^{(i)}\left(\mathbf{x}^{(i)}\right)$ by either consuming or storing the supplies. In other words, $f_k^{(i)}$ is the negative of the utility function of power subsystem $i$ at time $k$. The utility function is time-variant because users often have different power demands at different time, e.g., 6-11pm may be the peak demand period while 2-6am may be a low demand period. On the other hand, the generation of the total power supplies of $\sum_{i=1}^n\mathbf{x}^{(i)}$ can incur a cost of $g_k\left(\sum_{i=1}^n\mathbf{x}^{(i)}\right)$ for the power generator due to resource consumptions, human efforts and pollution. The generation cost function $g_k$ also varies across time owing to factors such as the changing and somewhat unpredictable renewable energy sources and the variant prices of the traditional energy sources. Thus, the overall social welfare maximization problem can be posed as a dynamic sharing problem as in \eqref{dynamic_share}.
\item Consider a cognitive radio network composed of $n$ secondary users. If secondary user $i$ obtains spectrum resources $\mathbf{x}^{(i)}$ at time $k$, then her utility is $-f_k^{(i)}\left(\mathbf{x}^{(i)}\right)$. The negative utility function $f_k^{(i)}$ is time-variant because users have different spectrum demands as they change their wireless applications. For example, a user has a high spectrum demand if she is watching online videos. But if she is just making a phone call, her spectrum demand is small. Moreover, for the network operator to provide the total spectrum resources of $\sum_{i=1}^n\mathbf{x}^{(i)}$, he needs to pay a cost of $g_k\left(\sum_{i=1}^n\mathbf{x}^{(i)}\right)$, in which the cost function $g_k$ is also time-varying due to factors including the uncertainty of the sensed temporarily unused spectrum by primary users and the changing market prices of the spectrum. As such, the overall system cost minimization problem can be casted into the form of dynamic sharing problem in \eqref{dynamic_share}.
\end{itemize}
A well-known method to decouple the local cost functions $f^{(i)}$ and the global cost function $g$ in the sharing problem \eqref{share} is the ADMM \cite{boyd2011distributed}. As a dual domain optimization method, the ADMM has faster convergence than the primal domain methods such as the gradient descent algorithm. This inspires us to develop and analyze a dynamic ADMM algorithm to solve the dynamic sharing problem in \eqref{dynamic_share} in this work. Before formal development of the algorithm, we first present a brief review of the basics of ADMM in the next subsection.
\subsection{Preliminaries of ADMM}
ADMM is an optimization framework widely applied to various signal processing applications, including wireless communications \cite{shen2012distributed}, power systems \cite{zhang2016admm} and multi-agent coordination \cite{chang2014proximal}. It enjoys fast convergence speed under mild technical conditions \cite{deng2016global} and is especially suitable for the development of distributed algorithms \cite{boyd2011distributed,bertsekas1989parallel}. ADMM solves problems of the following form:
\begin{eqnarray}\label{admm_prime}
\text{Minimize}_{\mathbf{x},\mathbf{z}} f(\mathbf{x})+g(\mathbf{z})~~\text{s.t.}~~\mathbf{Ax+Bz=c},
\end{eqnarray}
where $\mathbf{A}\in\mathbb{R}^{p\times n},B\in\mathbb{R}^{p\times m},c\in\mathbb{R}^p$ are constants and $\mathbf{x}\in\mathbb{R}^n,\mathbf{z}\in\mathbb{R}^m$ are optimization variables. $f:\mathbb{R}^n\mapsto\mathbb{R}$ and $g:\mathbb{R}^m\mapsto\mathbb{R}$ are two convex functions. The augmented Lagrangian can be formed as:
\begin{equation}
\mathfrak{L}_\rho(\mathbf{x,z,y})=f(\mathbf{x})+g(\mathbf{z})+\mathbf{y}^\mathsf{T}(\mathbf{Ax+Bz-c})+\frac{\rho}{2}\|\mathbf{Ax+Bz-c}\|_2^2,
\end{equation}
where $\mathbf{y}\in\mathbb{R}^p$ is the Lagrange multiplier and $\rho>0$ is some constant. The ADMM then iterates over the following three steps for $k\geq0$ (the iteration index):
\begin{eqnarray}
&&\mathbf{x}^{k+1}=\arg\min_\mathbf{x} \mathfrak{L}_\rho\left(\mathbf{x},\mathbf{z}^k,\mathbf{y}^k\right),\label{x_prime}\\
&&\mathbf{z}^{k+1}=\arg\min_\mathbf{z} \mathfrak{L}_\rho\left(\mathbf{x}^{k+1},\mathbf{z},\mathbf{y}^k\right),\label{z_prime}\\
&&\mathbf{y}^{k+1}=\mathbf{y}^{k}+\rho\left(\mathbf{Ax}^{k+1}+\mathbf{Bz}^{k+1}-\mathbf{c}\right).\label{multiplier_prime}
\end{eqnarray}
The ADMM is guaranteed to converge to the optimal point of \eqref{admm_prime} as long as $f$ and $g$ are convex \cite{boyd2011distributed,bertsekas1989parallel}. It is recently shown that global linear convergence can be ensured provided additional assumptions on problem \eqref{admm_prime} holds \cite{deng2016global}.
\subsection{Development of the Dynamic ADMM}
Define $\mathbf{x}=\left[\mathbf{x}^{(1)\mathsf{T}},...,\mathbf{x}^{(n)\mathsf{T}}\right]^\mathsf{T}\in\mathbb{R}^{np}$, $\mathbf{A}=[\mathbf{I}_p,...,\mathbf{I}_p]\in\mathbb{R}^{p\times np}$ and
\begin{align}\label{f_decompose}
f_k(\mathbf{x})=\sum_{i=1}^nf_k^{(i)}\left(\mathbf{x}^{(i)}\right).
\end{align}
Then, the dynamic sharing problem can be reformulated as:
\begin{align}
\label{dynamic_share_admm}\text{Minimize}_{\mathbf{x}\in\mathbb{R}^{np},\mathbf{z}\in\mathbb{R}^p}~~&f_k(\mathbf{x})+g_k(\mathbf{z})\\
\text{s.t.}~~&\mathbf{Ax-z=0}.
\end{align}
In the remaining part of this paper, we study the following more general dynamic optimization problem:
\begin{align}
\label{dynamic_admm}\text{Minimize}_{\mathbf{x}\in\mathbb{R}^N,\mathbf{z}\in\mathbb{R}^M}~~&f_k(\mathbf{x})+g_k(\mathbf{z})\\
\text{s.t.}~~&\mathbf{Ax+Bz=c},
\end{align}
where $f_k:\mathbb{R}^N\mapsto\mathbb{R}$ and $g_k:\mathbb{R}^{M}\mapsto\mathbb{R}$ are two functions and $\mathbf{A}\in\mathbb{R}^{M\times N},\mathbf{B}\in\mathbb{R}^{M\times M}$ are two matrices. The problem \eqref{dynamic_share_admm} is clearly a special case of the problem \eqref{dynamic_admm} by taking $N=np,M=p,\mathbf{B=-I},\mathbf{c=0}$ and $f_k$ decomposable as in \eqref{f_decompose}. To apply the ADMM, we form the augmented Lagrangian of the problem \eqref{dynamic_admm}:
\begin{align}
\mathfrak{L}_{\rho,k}(\mathbf{x,z},\boldsymbol{\lambda})=f_k(\mathbf{x})+g_k(\mathbf{z})+\boldsymbol{\lambda}^\mathsf{T}(\mathbf{Ax+Bz-c})+\frac{\rho}{2}\|\mathbf{Ax+Bz-c}\|_2^2,
\end{align}
where $\boldsymbol{\lambda}\in\mathbb{R}^M$ is the Lagrange multiplier and $\rho>0$ is some positive constant. Thus, applying the traditional static ADMM \eqref{x_prime}, \eqref{z_prime} and \eqref{multiplier_prime} to the dynamic augmented Lagrangian $\mathfrak{L}_{\rho,k}$, we propose a dynamic ADMM algorithm, as specified in Algorithm 1. The main difference between the dynamic ADMM in Algorithm 1 and the traditional static ADMM described in subsection \Rmnum{2}-B is that the functions $f_k$ and $g_k$ varies across iterations of the ADMM. The aim of this paper is to study the impact of these varying functions on the ADMM algorithm. Lastly, we introduce the following two linear convergence concepts which shall be used later.
\begin{defi}
A sequence $\mathbf{s}_k$ is said to converge \underline{Q-linearly} to $\mathbf{s}^*$ if there exists some constant $\theta\in(0,1)$ such that $\|\mathbf{s}_{k+1}-\mathbf{s}^*\|_2\leq\theta\|\mathbf{s}_k-\mathbf{s}^*\|_2$ for any positive integer $k$.
\end{defi}
\begin{defi}
A sequence $\mathbf{v}_k$ is said to converge \underline{R-linearly} to $\mathbf{v}^*$ if there exists a positive constant $\tau>0$ and some sequence $\mathbf{s}_k$ Q-linearly converging to some point $\mathbf{s}^*$ such that $\|\mathbf{v}_k-\mathbf{v}^*\|_2\leq\tau\|\mathbf{s}_k-\mathbf{s}^*\|_2$ for every positive integer $k$.
\end{defi}
\begin{algorithm}[!htbp]
\renewcommand{\algorithmicrequire}{\textbf{Inputs:} }
\renewcommand\algorithmicensure {\textbf{Outputs:} }
\caption{The dynamic ADMM algorithm for the dynamic problem \eqref{dynamic_admm}}
\begin{algorithmic}[1]
\STATE \texttt{Initialize $\mathbf{x}_0=\mathbf{0},\mathbf{z}_0=\boldsymbol{\lambda}_0=\mathbf{0},k=0$
\STATE \underline{Repeat}:
\STATE $k\leftarrow k+1$
\STATE Update $\mathbf{x}$ according to:
\begin{align}\label{x_update}
\mathbf{x}_k=\arg\min_\mathbf{x}f_k(\mathbf{x})+\boldsymbol{\lambda}_{k-1}^\mathsf{T}\mathbf{Ax}+\frac{\rho}{2}\|\mathbf{Ax+Bz}_{k-1}-\mathbf{c}\|_2^2.
\end{align}
\STATE Update $\mathbf{z}$ according to:
\begin{align}\label{z_update}
\mathbf{z}_k=\arg\min_\mathbf{z}g_k(\mathbf{z})+\boldsymbol{\lambda}_{k-1}^\mathsf{T}\mathbf{Bz}+\frac{\rho}{2}\|\mathbf{Bz+Ax}_k-\mathbf{c}\|_2^2.
\end{align}
\STATE Update $\boldsymbol{\lambda}$ according to:
\begin{align}\label{lambda_update}
\boldsymbol{\lambda}_k=\boldsymbol{\lambda}_{k-1}+\rho(\mathbf{Ax}_k+\mathbf{Bz}_k-\mathbf{c}).
\end{align}
}
\end{algorithmic}
\end{algorithm}
\section{Convergence Analysis}
In this section, convergence analysis for the dynamic ADMM algorithm, i.e., Algorithm 1, is conducted. We first make several standard assumptions for algorithm analysis. Then, we show that the iterations of the dynamic ADMM converge linearly (either Q-linearly or R-linearly) to some neighborhoods of their respective optimal points (Theorem 1 and 2). The sizes of these neighborhoods depend on the \emph{drift} (to be formally defined later) of the dynamic optimization problem \eqref{dynamic_admm}. Finally, we demonstrate the impact of the drift of the dynamic optimization problem \eqref{dynamic_admm} on the steady state convergence properties of the dynamic ADMM.
\subsection{Assumptions}
Throughout the convergence analysis, we make the following assumptions on the functions $f_k$ and $g_k$, all of which are standard in the analysis of optimization algorithms \cite{boyd2004convex,shi2014linear,deng2016global}.
\begin{assump}
For any positive integer $k$, $g_k$ is strongly convex with constant $m>0$ ($m$ is independent of $k$), i.e., for any positive integer $k$:
\begin{align}\label{g_strong}
\left(\nabla g_k(\mathbf{z})-\nabla g_k(\mathbf{z}')\right)^\mathsf{T}(\mathbf{z-z'})\geq m\|\mathbf{z}-\mathbf{z}'\|_2^2,~~\forall\mathbf{z,z'}\in\mathbb{R}^M.
\end{align}
\end{assump}
\begin{assump}
For any positive integer $k$, $f_k$ is strongly convex with constant $\widetilde{m}>0$ ($\widetilde{m}$ is independent of $k$), i.e., for any positive integer $k$:
\begin{align}\label{f_strong}
\left(\nabla f_k(\mathbf{x})-\nabla f_k(\mathbf{x}')\right)^\mathsf{T}(\mathbf{x-x'})\geq \widetilde{m}\|\mathbf{x}-\mathbf{x}'\|_2^2,~~\forall\mathbf{x,x'}\in\mathbb{R}^N.
\end{align}
\end{assump}
\begin{assump}
For any positive integer $k$, $\nabla g_k$ is Lipschitz continuous with constant $L>0$ ($L$ is independent of $k$), i.e., for any positive integer $k$ and any $\mathbf{z,z'}\in\mathbb{R}^M$:
\begin{align}
\|\nabla g_k(\mathbf{z})-\nabla g_k(\mathbf{z}')\|_2\leq L\|\mathbf{z-z}'\|_2.
\end{align}
\end{assump}
We note that, if $g_k$ is twice differentiable, the condition \eqref{g_strong} is equivalent to $\nabla^2g_k(\mathbf{z})\succeq m\mathbf{I},\forall\mathbf{z}$. Similarly, if $f_k$ is twice differentiable, the condition \eqref{f_strong} is equivalent to $\nabla^2f_k(\mathbf{x})\succeq \widetilde{m}\mathbf{I},\forall\mathbf{x}$. This second order definition of strong convexity is more intuitively acceptable and has been used in the analysis of convex optimization algorithms in the literature \cite{boyd2004convex}. But it requires twice differentiability and is not directly useful in the analysis in this work. We further note that when $f_k$ is decomposable as in \eqref{f_decompose} of the dynamic sharing problem, if for any $i=1,...,n$ and positive integer $k$, $f_k^{(i)}$ is strongly convex with constant $\widetilde{m}_i>0$, then Assumption 2 holds with $\widetilde{m}=\min_{i=1,...,n}\widetilde{m}_i>0$. We present the following fact from convex analysis \cite{vandenberghe2016gradient}, which will be used in the later analysis.
\begin{lem}
For any differentiable convex function $h:\mathbb{R}^l\mapsto\mathbb{R}$ and positive constant $L>0$, the following two statements are equivalent:
\begin{enumerate}
\item $\nabla h$ is Lipschitz continuous with constant $L$, i.e., $\left\|\nabla h(\mathbf{x})-\nabla h\left(\mathbf{x}'\right)\right\|_2\leq L\left\|\mathbf{x}-\mathbf{x}'\right\|_2,\forall \mathbf{x},\mathbf{x}'\in\mathbb{R}^l$.
\item $\left\|\nabla h(\mathbf{x})-\nabla h\left(\mathbf{x}'\right)\right\|_2^2\leq L\left(\mathbf{x}-\mathbf{x}'\right)^\mathsf{T}\left(\nabla h(\mathbf{x})-\nabla h\left(\mathbf{x}'\right)\right),\forall\mathbf{x},\mathbf{x}'\in\mathbb{R}^l$.
\end{enumerate}
\end{lem}
We further assume that the matrix $\mathbf{B}\in\mathbb{R}^{M\times M}$ is nonsingular.
\begin{assump}
$\mathbf{B}$ is nonsingular.
\end{assump}
\subsection{Convergence Analysis}
In this subsection, we study the convergence behavior of the proposed dynamic ADMM algorithm under the Assumptions 1-4. Due to the strong convexity assumption in Assumptions 1 and 2, there is a unique primal/dual optimal point pair $(\mathbf{x}_k^*,\mathbf{z}_k^*,\boldsymbol{\lambda}_k^*)$ for the dynamic optimization problem \eqref{dynamic_admm} at time $k$. Denote $\mathbf{u}_k=\left[\mathbf{z}_k^\mathsf{T},\boldsymbol{\lambda}_k^\mathsf{T}\right]^\mathsf{T}$ and $\mathbf{u}_k^*=\left[\mathbf{z}_k^{*\mathsf{T}},\boldsymbol{\lambda}_k^{*\mathsf{T}}\right]^\mathsf{T}$. Since $\mathbf{B}$ is a square matrix, the eigenvalues of $\mathbf{BB}^\mathsf{T}$ are the same as those of $\mathbf{B}^\mathsf{T}\mathbf{B}$. Denote the smallest eigenvalue of $\mathbf{BB}^\mathsf{T}$, which is also the smallest eigenvalue of $\mathbf{B}^\mathsf{T}\mathbf{B}$, as $\alpha$. According to Assumption 4, $\mathbf{B}$ is nonsingular, so $\mathbf{BB}^\mathsf{T}$ and $\mathbf{B}^\mathsf{T}\mathbf{B}$ are positive definite and $\alpha>0$. Define matrix $\mathbf{C}\in\mathbb{R}^{2M\times 2M}$ to be:
\begin{align}
\mathbf{C}=\left[
\begin{array}{cc}
\frac{\rho}{2}\mathbf{B}^\mathsf{T}\mathbf{B}&\\
&\frac{1}{2\rho}\mathbf{I}_M
\end{array}
\right]
\end{align}
Since $\mathbf{B}$ is nonsingular (Assumption 4), we know that $\mathbf{C}$ is positive definite. Therefore, we can define a norm on $\mathbb{R}^{2M}$ as $\|\mathbf{u}\|_\mathbf{C}=\sqrt{\mathbf{u}^\mathsf{T}\mathbf{Cu}}$. Define $t$ to be any arbitrary number within the interval $(0,1)$. A positive constant $\delta>0$ is defined as:
\begin{align}\label{delta_def}
\delta=\min\left\{\frac{2mt}{\rho\|\mathbf{B}\|_2^2},\frac{2\alpha\rho(1-t)}{L}\right\},
\end{align}
where $\|\mathbf{B}\|_2$ is the spectral norm, i.e., the maximum singular value, of $\mathbf{B}$. We further note the following fact, which shall be invoked in later analysis.
\begin{lem}
For any symmetric matrix $\mathbf{A}\in\mathbb{R}^{l\times l}$ and any vectors $\mathbf{x,y,z}\in\mathbb{R}^l$, we have:
\begin{align}
2(\mathbf{x-y})^\mathsf{T}\mathbf{A(z-x)}=\mathbf{(z-y)}^\mathsf{T}\mathbf{A(z-y)}-\mathbf{(x-y)}^\mathsf{T}\mathbf{A(x-y)}-\mathbf{(z-x)}^\mathsf{T}\mathbf{A(z-x)}.
\end{align}
\end{lem}
Now, we are ready to show the first intermediate result.
\begin{prop}
For any positive integer $k$, we have:
\begin{align}\label{intermediate1}
\|\mathbf{u}_k-\mathbf{u}_k^*\|_\mathbf{C}\leq\frac{1}{\sqrt{1+\delta}}\|\mathbf{u}_{k-1}-\mathbf{u}_k^*\|_\mathbf{C}.
\end{align}
\end{prop}
\begin{proof}
Due to the strong convexity assumption in Assumptions 1 and 2, the subproblems in \eqref{x_update} and \eqref{z_update} are unconstrained convex optimization problems. Thus, vanishing gradient is necessary and sufficient for optimality of the subproblems \eqref{x_update} and \eqref{z_update}. The updates of $\mathbf{x}$ and $\mathbf{z}$ can be rewritten as:
\begin{align}
&\label{x_upd}\nabla f_k(\mathbf{x}_k)+\mathbf{A}^\mathsf{T}\boldsymbol{\lambda}_{k-1}+\rho\mathbf{A}^\mathsf{T}(\mathbf{Ax}_k+\mathbf{Bz}_{k-1}-\mathbf{c})=\mathbf{0},\\
&\label{z_upd}\nabla g_k(\mathbf{z}_k)+\mathbf{B}^\mathsf{T}\boldsymbol{\lambda}_{k-1}+\rho\mathbf{B}^\mathsf{T}(\mathbf{Ax}_k+\mathbf{Bz}_k-\mathbf{c})=\mathbf{0}.
\end{align}
Combining \eqref{z_upd} and \eqref{lambda_update} yields:
\begin{align}\label{g_relation}
\nabla g_k(\mathbf{z}_k)+\mathbf{B}^\mathsf{T}\boldsymbol{\lambda}_k=\mathbf{0}.
\end{align}
Combining \eqref{x_upd} and \eqref{lambda_update} gives:
\begin{align}\label{f_relation}
\nabla f_k(\mathbf{x}_k)+\mathbf{A}^\mathsf{T}(\boldsymbol{\lambda}_k+\rho\mathbf{B}(\mathbf{z}_{k-1}-\mathbf{z}_k))=\mathbf{0}.
\end{align}
According to Assumptions 1 and 2, the problem \eqref{dynamic_admm} is a convex optimization problem. Thus, Karush-Kuhn-Tucker (KKT) conditions are necessary and sufficient for optimality. Hence,
\begin{align}
\label{f_kkt}&\nabla f_k(\mathbf{x}_k^*)+\mathbf{A}^\mathsf{T}\boldsymbol{\lambda}_k^*=\mathbf{0},\\
\label{g_kkt}&\nabla g_k(\mathbf{z}_k^*)+\mathbf{B}^\mathsf{T}\boldsymbol{\lambda}_k^*=\mathbf{0},\\
\label{feasible_kkt}&\mathbf{Ax}_k^*+\mathbf{Bz}_k^*=\mathbf{c}.
\end{align}
Because of the convexity of $g_k$ (Assumption 1) and Lipschitz continuity of its gradient (Assumption 3), by invoking Lemma 1, we get:
\begin{align}
\|\nabla g_k(\mathbf{z}_k)-\nabla g_k(\mathbf{z}_k^*)\|_2^2\leq L(\mathbf{z}_k-\mathbf{z}_k^*)^\mathsf{T}(\nabla g_k(\mathbf{z}_k)-\nabla g_k(\mathbf{z}_k^*)).
\end{align}
Further using \eqref{g_kkt} and \eqref{g_relation}, we obtain:
\begin{align}\label{1}
(\mathbf{z}_k-\mathbf{z}_k^*)^\mathsf{T}\mathbf{B}^\mathsf{T}(\boldsymbol{\lambda}_k^*-\boldsymbol{\lambda}_k)\geq\frac{1}{L}\left\|\mathbf{B}^\mathsf{T}(\boldsymbol{\lambda}_k-\boldsymbol{\lambda}_k^*)\right\|_2^2.
\end{align}
According to the strong convexity of $g_k$ (Assumption 1), we have:
\begin{align}\label{2}
m\|\mathbf{z}_k-\mathbf{z}_k^*\|_2^2\leq(\nabla g_k(\mathbf{z}_k)-\nabla g_k(\mathbf{z}_k^*))^\mathsf{T}(\mathbf{z}_k-\mathbf{z}_k^*)=\left(-\mathbf{B}^\mathsf{T}\boldsymbol{\lambda}_k+\mathbf{B}^\mathsf{T}\boldsymbol{\lambda}_k^*\right)^\mathsf{T}(\mathbf{z}_k-\mathbf{z}_k^*).
\end{align}
Combining \eqref{1} and \eqref{2}, we know that for any $t\in(0,1)$:
\begin{align}\label{3}
(\mathbf{z}_k-\mathbf{z}_k^*)^\mathsf{T}\mathbf{B}^\mathsf{T}(\boldsymbol{\lambda}_k^*-\boldsymbol{\lambda}_k)\geq tm\|\mathbf{z}_k-\mathbf{z}_k^*\|_2^2+\frac{1-t}{L}\left\|\mathbf{B}^\mathsf{T}(\boldsymbol{\lambda}_k-\boldsymbol{\lambda}_k^*)\right\|_2^2.
\end{align}
According to the convexity of $f_k$ (Assumption 2), we have:
\begin{align}
0\leq (\mathbf{x}_k-\mathbf{x}_k^*)^\mathsf{T}(\nabla f_k(\mathbf{x}_k)-\nabla f_k(\mathbf{x}_k^*)).
\end{align}
Further making use of \eqref{f_relation} and \eqref{f_kkt}, we get:
\begin{align}\label{4}
0\leq(\mathbf{x}_k-\mathbf{x}_k^*)^\mathsf{T}\mathbf{A}^\mathsf{T}(\boldsymbol{\lambda}_k^*-\boldsymbol{\lambda}_k+\rho\mathbf{B}(\mathbf{z}_k-\mathbf{z}_{k-1})).
\end{align}
Adding \eqref{3} and \eqref{4} leads to:
\begin{align}
\begin{split}\label{6}
&(\mathbf{z}_k-\mathbf{z}_k^*)^\mathsf{T}\mathbf{B}^\mathsf{T}(\boldsymbol{\lambda}_k^*-\boldsymbol{\lambda}_k)+(\mathbf{x}_k-\mathbf{x}_k^*)^\mathsf{T}\mathbf{A}^\mathsf{T}(\boldsymbol{\lambda}_k^*-\boldsymbol{\lambda}_k+\rho\mathbf{B}(\mathbf{z}_k-\mathbf{z}_{k-1}))\\
&\geq tm\|\mathbf{z}_k-\mathbf{z}_k^*\|_2^2+\frac{1-t}{L}\left\|\mathbf{B}^\mathsf{T}(\boldsymbol{\lambda}_k-\boldsymbol{\lambda}_k^*)\right\|_2^2.
\end{split}
\end{align}
From \eqref{lambda_update} and \eqref{feasible_kkt}, we get:
\begin{align}\label{5}
\mathbf{A}(\mathbf{x}_k-\mathbf{x}_k^*)+\mathbf{B}(\mathbf{z}_k-\mathbf{z}_k^*)=\frac{1}{\rho}(\boldsymbol{\lambda}_k-\boldsymbol{\lambda}_{k-1}).
\end{align}
Making use of \eqref{5}, we derive:
\begin{align}
&(\mathbf{z}_k-\mathbf{z}_k^*)^\mathsf{T}\mathbf{B}^\mathsf{T}(\boldsymbol{\lambda}_k^*-\boldsymbol{\lambda}_k)+(\mathbf{x}_k-\mathbf{x}_k^*)^\mathsf{T}\mathbf{A}^\mathsf{T}(\boldsymbol{\lambda}_k^*-\boldsymbol{\lambda}_k+\rho\mathbf{B}(\mathbf{z}_k-\mathbf{z}_{k-1}))\\
&=(\boldsymbol{\lambda}_k^*-\boldsymbol{\lambda}_k)^\mathsf{T}[\mathbf{B}(\mathbf{z}_k-\mathbf{z}_k^*)+\mathbf{A}(\mathbf{x}_k-\mathbf{x}_k^*)]+\rho(\mathbf{x}_k-\mathbf{x}_k^*)^\mathsf{T}\mathbf{A}^\mathsf{T}\mathbf{B}(\mathbf{z}_k-\mathbf{z}_{k-1})\\
&=\frac{1}{\rho}(\boldsymbol{\lambda}_k^*-\boldsymbol{\lambda}_k)^\mathsf{T}(\boldsymbol{\lambda}_k-\boldsymbol{\lambda}_{k-1})+\rho(\mathbf{A}(\mathbf{x}_k-\mathbf{x}_k^*))^\mathsf{T}\mathbf{B}(\mathbf{z}_k-\mathbf{z}_{k-1})\\
&=\frac{1}{\rho}(\boldsymbol{\lambda}_{k-1}-\boldsymbol{\lambda}_k)^\mathsf{T}(\boldsymbol{\lambda}_k-\boldsymbol{\lambda}_k^*)+(\boldsymbol{\lambda}_k-\boldsymbol{\lambda}_{k-1}-\rho\mathbf{B}(\mathbf{z}_k-\mathbf{z}_k^*))^\mathsf{T}\mathbf{B}(\mathbf{z}_k-\mathbf{z}_{k-1})
\end{align}
Together with \eqref{6}, we get:
\begin{align}
&\frac{1}{\rho}(\boldsymbol{\lambda}_{k-1}-\boldsymbol{\lambda}_k)^\mathsf{T}(\boldsymbol{\lambda}_k-\boldsymbol{\lambda}_k^*)+\rho(\mathbf{z}_k-\mathbf{z}_k^*)^\mathsf{T}\mathbf{B}^\mathsf{T}\mathbf{B}(\mathbf{z}_{k-1}-\mathbf{z}_k)\\
&\geq (\boldsymbol{\lambda}_k-\boldsymbol{\lambda}_{k-1})^\mathsf{T}\mathbf{B}(\mathbf{z}_{k-1}-\mathbf{z}_k)+tm\|\mathbf{z}_k-\mathbf{z}_k^*\|_2^2+\frac{1-t}{L}\left\|\mathbf{B}^\mathsf{T}(\boldsymbol{\lambda}_k-\boldsymbol{\lambda}_k^*)\right\|_2^2,
\end{align}
which is equivalent to:
\begin{align}
&\frac{1}{\rho}(\boldsymbol{\lambda}_{k-1}-\boldsymbol{\lambda}_k)^\mathsf{T}(\boldsymbol{\lambda}_k-\boldsymbol{\lambda}_{k-1}+\boldsymbol{\lambda}_{k-1}-\boldsymbol{\lambda}_k^*)+\rho(\mathbf{z}_{k-1}-\mathbf{z}_k)^\mathsf{T}\mathbf{B}^\mathsf{T}\mathbf{B}(\mathbf{z}_k-\mathbf{z}_{k-1}+\mathbf{z}_{k-1}-\mathbf{z}_k^*)\\
&\geq (\boldsymbol{\lambda}_k-\boldsymbol{\lambda}_{k-1})^\mathsf{T}\mathbf{B}(\mathbf{z}_{k-1}-\mathbf{z}_k)+tm\|\mathbf{z}_k-\mathbf{z}_k^*\|_2^2+\frac{1-t}{L}\left\|\mathbf{B}^\mathsf{T}(\boldsymbol{\lambda}_k-\boldsymbol{\lambda}_k^*)\right\|_2^2.
\end{align}
This can be further rewritten as:
\begin{align}
\begin{split}\label{9}
&\frac{1}{\rho}(\boldsymbol{\lambda}_{k-1}-\boldsymbol{\lambda}_k)^\mathsf{T}(\boldsymbol{\lambda}_{k-1}-\boldsymbol{\lambda}_k^*)+\rho(\mathbf{z}_{k-1}-\mathbf{z}_k)^\mathsf{T}\mathbf{B}^\mathsf{T}\mathbf{B}(\mathbf{z}_{k-1}-\mathbf{z}_k^*)\\
&\geq \frac{1}{\rho}\|\boldsymbol{\lambda}_{k-1}-\boldsymbol{\lambda}_k\|_2^2+\rho\|\mathbf{Bz}_{k-1}-\mathbf{Bz}_k\|_2^2+(\boldsymbol{\lambda}_k-\boldsymbol{\lambda}_{k-1})^\mathsf{T}\mathbf{B}(\mathbf{z}_{k-1}-\mathbf{z}_k)\\
&~~~+mt\|\mathbf{z}_k-\mathbf{z}_k^*\|_2^2+\frac{1-t}{L}\left\|\mathbf{B}^\mathsf{T}(\boldsymbol{\lambda}_k-\boldsymbol{\lambda}_k^*)\right\|_2^2.
\end{split}
\end{align}
Making use of Lemma 2, we obtain:
\begin{align}
\begin{split}\label{7}
&\frac{1}{\rho}(\boldsymbol{\lambda}_{k-1}-\boldsymbol{\lambda}_k)^\mathsf{T}(\boldsymbol{\lambda}_{k-1}-\boldsymbol{\lambda}_k^*)\\
&=-\frac{1}{2\rho}\|\boldsymbol{\lambda}_k^*-\boldsymbol{\lambda}_k\|_2^2+\frac{1}{2\rho}\|\boldsymbol{\lambda}_{k-1}-\boldsymbol{\lambda}_k\|_2^2+\frac{1}{2\rho}\|\boldsymbol{\lambda}_k^*-\boldsymbol{\lambda}_{k-1}\|_2^2,
\end{split}
\end{align}
and
\begin{align}
\begin{split}\label{8}
&\rho(\mathbf{z}_{k-1}-\mathbf{z}_k)^\mathsf{T}\mathbf{B}^\mathsf{T}\mathbf{B}(\mathbf{z}_{k-1}-\mathbf{z}_k^*)\\
&=-\frac{\rho}{2}\|\mathbf{Bz}_k^*-\mathbf{Bz}_k\|_2^2+\frac{\rho}{2}\|\mathbf{Bz}_{k-1}-\mathbf{Bz}_k\|_2^2+\frac{\rho}{2}\|\mathbf{Bz}_k^*-\mathbf{Bz}_{k-1}\|_2^2.
\end{split}
\end{align}
Combining \eqref{7} and \eqref{8} and further utilizing \eqref{9} gives:
\begin{align}
&\frac{1}{2\rho}\|\boldsymbol{\lambda}_{k-1}-\boldsymbol{\lambda}_k^*\|_2^2+\frac{\rho}{2}\|\mathbf{Bz}_{k-1}-\mathbf{Bz}_k^*\|_2^2-\frac{1}{2\rho}\|\boldsymbol{\lambda}_k-\boldsymbol{\lambda}_k^*\|_2^2-\frac{\rho}{2}\|\mathbf{Bz}_k-\mathbf{Bz}_k^*\|_2^2\\
&=\frac{1}{\rho}(\boldsymbol{\lambda}_{k-1}-\boldsymbol{\lambda}_k)^\mathsf{T}(\boldsymbol{\lambda}_{k-1}-\boldsymbol{\lambda}_k^*)+\rho(\mathbf{z}_{k-1}-\mathbf{z}_k)^\mathsf{T}\mathbf{B}^\mathsf{T}\mathbf{B}(\mathbf{z}_{k-1}-\mathbf{z}_k^*)\nonumber\\
&~~~-\frac{1}{2\rho}\|\boldsymbol{\lambda}_{k-1}-\boldsymbol{\lambda}_k\|_2^2-\frac{\rho}{2}\|\mathbf{Bz}_{k-1}-\mathbf{Bz}_k\|_2^2\\
&\geq \frac{1}{2\rho}\|\boldsymbol{\lambda}_{k-1}-\boldsymbol{\lambda}_k\|_2^2+\frac{\rho}{2}\|\mathbf{Bz}_{k-1}-\mathbf{Bz}_k\|_2^2+(\boldsymbol{\lambda}_k-\boldsymbol{\lambda}_{k-1})^\mathsf{T}\mathbf{B}(\mathbf{z}_{k-1}-\mathbf{z}_k)\nonumber\\
&~~~+mt\|\mathbf{z}_k-\mathbf{z}_k^*\|_2^2+\frac{1-t}{L}\left\|\mathbf{B}^\mathsf{T}(\boldsymbol{\lambda}_k-\boldsymbol{\lambda}_k^*)\right\|_2^2\\
&=\frac{1}{2\rho}\left\|\boldsymbol{\lambda}_k-\boldsymbol{\lambda}_{k-1}+\rho(\mathbf{Bz}_{k-1}-\mathbf{Bz}_k)\right\|_2^2+mt\|\mathbf{z}_k-\mathbf{z}_k^*\|_2^2+\frac{1-t}{L}\left\|\mathbf{B}^\mathsf{T}(\boldsymbol{\lambda}_k-\boldsymbol{\lambda}_k^*)\right\|_2^2\\
&\geq mt\|\mathbf{z}_k-\mathbf{z}_k^*\|_2^2+\frac{1-t}{L}\left\|\mathbf{B}^\mathsf{T}(\boldsymbol{\lambda}_k-\boldsymbol{\lambda}_k^*)\right\|_2^2\\
&\geq\frac{mt}{\|\mathbf{B}\|_2^2}\|\mathbf{Bz}_k-\mathbf{Bz}_k^*\|_2^2+\frac{\alpha(1-t)}{L}\|\boldsymbol{\lambda}_k-\boldsymbol{\lambda}_k^*\|_2^2\\
&\geq\delta\left(\frac{1}{2\rho}\|\boldsymbol{\lambda}_k-\boldsymbol{\lambda}_k^*\|_2^2+\frac{\rho}{2}\|\mathbf{Bz}_k-\mathbf{Bz}_k^*\|_2^2\right),
\end{align}
where the last step is due to the definition of $\delta$ in \eqref{delta_def}. Noticing the definition of $\|\cdot\|_\mathbf{C}$, we get:
\begin{align}
\|\mathbf{u}_{k-1}-\mathbf{u}_k^*\|_\mathbf{C}^2\geq(1+\delta)\|\mathbf{u}_k-\mathbf{u}_k^*\|_\mathbf{C}^2,
\end{align}
which is tantamount to \eqref{intermediate1}.
\end{proof}
\begin{rem}
Proposition 1 states that $\mathbf{u}_k$ is closer to $\mathbf{u}_k^*$ than $\mathbf{u}_{k-1}$ with a shrinkage factor of $\delta$. The bigger the $\delta$, the stronger the shrinkage. Note that there is an arbitrary factor $t\in(0,1)$ in the definition of $\delta$ in \eqref{delta_def}. By choosing $t=\frac{\alpha\rho^2\|\mathbf{B}\|_2^2}{mL+\alpha\rho^2\|\mathbf{B}\|_2^2}$, we get the maximum $\delta$ as $\delta_{\max}=\frac{2m\alpha\rho}{mL+\alpha\rho^2\|\mathbf{B}\|_2^2}$. In the expression of $\delta_{\max}$, only $\rho$ is a tunable algorithm parameter while all other parameters are given by the optimization problem. The fact that $\delta_{\max}$ increases with $\rho$ may partially justify the need of a relatively large $\rho$ for good convergence behaviors of the dynamic ADMM. We will investigate the impact of $\rho$ on algorithm performance empirically in Section \Rmnum{4}.
\end{rem}
Proposition 1 establishes a relation between $\|\mathbf{u}_k-\mathbf{u}_k^*\|_\mathbf{C}$ and $\|\mathbf{u}_{k-1}-\mathbf{u}_k^*\|_\mathbf{C}$. However, to describe the convergence behavior of the dynamic ADMM algorithm, what we really want is the relation between $\|\mathbf{u}_k-\mathbf{u}_k^*\|_\mathbf{C}$ and $\|\mathbf{u}_{k-1}-\mathbf{u}_{k-1}^*\|_\mathbf{C}$. This is accomplished by the following theorem.
\begin{thm}
Define the drift $d_k$ of the dynamic problem \eqref{dynamic_admm} to be:
\begin{align}\label{drift_def}
d_k=\sqrt{\frac{\rho}{2}}\|\mathbf{B}\|_2\|\mathbf{z}_{k-1}^*-\mathbf{z}_k^*\|_2+\frac{1}{\sqrt{2\rho\alpha}}\|\nabla g_{k-1}(\mathbf{z}_{k-1}^*)-\nabla g_k(\mathbf{z}_k^*)\|_2.
\end{align}
Then, for any integer $k\geq2$, we have:
\begin{align}
\|\mathbf{u}_k-\mathbf{u}_k^*\|_\mathbf{C}\leq\frac{1}{\sqrt{1+\delta}}(\|\mathbf{u}_{k-1}-\mathbf{u}_{k-1}^*\|_\mathbf{C}+d_k).
\end{align}
\end{thm}
\begin{proof}
According to \eqref{g_kkt}, we have:
\begin{align}
\label{a1}&\nabla g_k(\mathbf{z}_k^*)+\mathbf{B}^\mathsf{T}\boldsymbol{\lambda}_k^*=\mathbf{0},\\
\label{a2}&\nabla g_{k-1}(\mathbf{z}_{k-1}^*)+\mathbf{B}^\mathsf{T}\boldsymbol{\lambda}_{k-1}^*=\mathbf{0}.
\end{align}
Substraction of \eqref{a2} from \eqref{a1} yields:
\begin{align}
\mathbf{B}^\mathsf{T}(\boldsymbol{\lambda}_k^*-\boldsymbol{\lambda}_{k-1}^*)=-\nabla g_k(\mathbf{z}_k^*)+\nabla g_{k-1}(\mathbf{z}_{k-1}^*).
\end{align}
Hence,
\begin{align}
&\|\nabla g_{k-1}(\mathbf{z}_{k-1}^*)-\nabla g_k(\mathbf{z}_k^*)\|_2^2\\
&=\|\mathbf{B}^\mathsf{T}(\boldsymbol{\lambda}_{k-1}^*-\boldsymbol{\lambda}_k^*)\|_2^2\\
&=(\boldsymbol{\lambda}_{k-1}^*-\boldsymbol{\lambda}_k^*)^\mathsf{T}\mathbf{B}\mathbf{B}^\mathsf{T}(\boldsymbol{\lambda}_{k-1}^*-\boldsymbol{\lambda}_k^*)\\
\label{a3}&\geq\alpha\|\boldsymbol{\lambda}_{k-1}^*-\boldsymbol{\lambda}_k^*\|_2^2.
\end{align}
On the other hand,
\begin{align}\label{a4}
(\mathbf{z}_{k-1}^*-\mathbf{z}_k^*)^\mathsf{T}\mathbf{B}^\mathsf{T}\mathbf{B}(\mathbf{z}_{k-1}^*-\mathbf{z}_k^*)\leq\|\mathbf{B}\|_2^2\|\mathbf{z}_{k-1}^*-\mathbf{z}_k^*\|_2^2.
\end{align}
Combining \eqref{a3} and \eqref{a4}, we get:
\begin{align}
&\|\mathbf{u}_{k-1}^*-\mathbf{u}_k^*\|_\mathbf{C}^2\\
&=\frac{\rho}{2}(\mathbf{z}_{k-1}^*-\mathbf{z}_k^*)^\mathsf{T}\mathbf{B}^\mathsf{T}\mathbf{B}(\mathbf{z}_{k-1}^*-\mathbf{z}_k^*)+\frac{1}{2\rho}\|\boldsymbol{\lambda}_{k-1}^*-\boldsymbol{\lambda}_k^*\|_2^2\\
&\leq\frac{\rho}{2}\|\mathbf{B}\|_2^2\|\mathbf{z}_{k-1}^*-\mathbf{z}_k^*\|_2^2+\frac{1}{2\rho\alpha}\|\nabla g_{k-1}(\mathbf{z}_{k-1}^*)-\nabla g_k(\mathbf{z}_k^*)\|_2^2\\
&\leq\left(\sqrt{\frac{\rho}{2}}\|\mathbf{B}\|_2\|\mathbf{z}_{k-1}^*-\mathbf{z}_k^*\|_2+\frac{1}{\sqrt{2\rho\alpha}}\|\nabla g_{k-1}(\mathbf{z}_{k-1}^*)-\nabla g_k(\mathbf{z}_k^*)\|_2\right)^2\\
&=d_k^2.
\end{align}
Thus, $\|\mathbf{u}_{k-1}^*-\mathbf{u}_k^*\|_\mathbf{C}\leq d_k$ and:
\begin{align}\label{a5}
\|\mathbf{u}_{k-1}-\mathbf{u}_k^*\|_\mathbf{C}\leq\|\mathbf{u}_{k-1}-\mathbf{u}_{k-1}^*\|_\mathbf{C}+\|\mathbf{u}_{k-1}^*-\mathbf{u}_k^*\|_\mathbf{C}\leq\|\mathbf{u}_{k-1}-\mathbf{u}_{k-1}^*\|_\mathbf{C}+d_k.
\end{align}
Combining \eqref{a5} and \eqref{intermediate1} in Proposition 1 gives:
\begin{align}
\|\mathbf{u}_k-\mathbf{u}_k^*\|_\mathbf{C}\leq\frac{1}{\sqrt{1+\delta}}(\|\mathbf{u}_{k-1}-\mathbf{u}_{k-1}^*\|_\mathbf{C}+d_k).
\end{align}
\end{proof}
\begin{rem}
Theorem 1 means that $\mathbf{u}_k$ converges Q-linearly (with contraction factor $\sqrt{1+\delta}$) to some neighborhood of the optimal point $\mathbf{u}_k^*$. The size of the neighborhood is characterized by $d_k$, the drift of the dynamic problem \eqref{dynamic_admm}, which is determined by the problem formulation instead of the algorithm. The more drastically the dynamic problem \eqref{dynamic_admm} varies across time, the bigger the drift $d_k$, and the larger the size of that neighborhood. When the dynamic problem \eqref{dynamic_admm} degenerates to its static counterpart, i.e., $f_k$ and $g_k$ does not vary with $k$, $d_k$ becomes zero. In such a case, Theorem 1 degenerates to the linear convergence result of static ADMM in \cite{deng2016global}.
\end{rem}
Q-linear convergence of $\mathbf{u}_k$ to some neighborhood of the optimal point $\mathbf{u}_k^*$ is established in Theorem 1. A more meaningful result will be about the convergence properties of $\mathbf{x}_k,\mathbf{z}_k,\boldsymbol{\lambda}_k$. To this end, we want to link the quantities $\|\mathbf{x}_k-\mathbf{x}_k^*\|_2$, $\|\mathbf{z}_k-\mathbf{z}_k^*\|_2$, $\|\boldsymbol{\lambda}_k-\boldsymbol{\lambda}_k^*\|_2$ with $\|\mathbf{u}_k-\mathbf{u}_k^*\|_\mathbf{C}$. This is accomplished by the following theorem.
\begin{thm}
For any integer $k\geq2$, we have:
\begin{align}
\begin{split}\label{a8}
&\|\mathbf{x}_k-\mathbf{x}_k^*\|_2\\
&\leq\frac{1}{\widetilde{m}}\|\mathbf{A}\|_2\left[\left(\sqrt{2\rho}+\|\mathbf{B}\|_2\sqrt{\frac{2\rho}{\alpha}}\right)\|\mathbf{u}_k-\mathbf{u}_k^*\|_\mathbf{C}+\|\mathbf{B}\|_2\sqrt{\frac{2\rho}{\alpha}}\|\mathbf{u}_{k-1}-\mathbf{u}_{k-1}^*\|_\mathbf{C}+\sqrt{2\rho}d_k\right],
\end{split}
\end{align}
where $\|\mathbf{A}\|_2$ is the spectral norm, i.e., the largest singular value, of $\mathbf{A}$.
Furthermore, for any positive integer $k$, we have:
\begin{align}
\label{a6}&\|\mathbf{z}_k-\mathbf{z}_k^*\|_2\leq\sqrt{\frac{2}{\alpha\rho}}\|\mathbf{u}_k-\mathbf{u}_k^*\|_\mathbf{C},\\
\label{a7}&\|\boldsymbol{\lambda}_k-\boldsymbol{\lambda}_k^*\|_2\leq\sqrt{2\rho}\|\mathbf{u}_k-\mathbf{u}_k^*\|_\mathbf{C}.
\end{align}
\end{thm}
\begin{proof}
\eqref{a6} and \eqref{a7} are straightforward. According to the definition of $\|\cdot\|_\mathbf{C}$, we have $\|\mathbf{u}_k-\mathbf{u}_k^*\|_\mathbf{C}^2\geq\frac{1}{2\rho}\|\boldsymbol{\lambda}_k-\boldsymbol{\lambda}_k^*\|_2^2$, which results in \eqref{a7}. Moreover, we note:
\begin{align}
\|\mathbf{Bz}_k-\mathbf{Bz}_k^*\|_2^2=(\mathbf{z}_k-\mathbf{z}_k^*)^\mathsf{T}\mathbf{B}^\mathsf{T}\mathbf{B}(\mathbf{z}_k-\mathbf{z}_k^*)\geq\alpha\|\mathbf{z}_k-\mathbf{z}_k^*\|_2^2,
\end{align}
and
\begin{align}
\|\mathbf{u}_k-\mathbf{u}_k^*\|_\mathbf{C}^2\geq\frac{\rho}{2}\|\mathbf{Bz}_k-\mathbf{Bz}_k^*\|_2^2,
\end{align}
which together lead to \eqref{a6}. Now, we proceed to prove \eqref{a8}. From the strong convexity of $f_k$ (Assumption 2) and equations \eqref{f_relation}, \eqref{f_kkt}, we derive:
\begin{align}
&\widetilde{m}\|\mathbf{x}_k-\mathbf{x}_k^*\|_2^2\\
&\leq(\mathbf{x}_k-\mathbf{x}_k^*)^\mathsf{T}(\nabla f_k(\mathbf{x}_k)-\nabla f_k(\mathbf{x}_k^*))\\
&=(\mathbf{x}_k-\mathbf{x}_k^*)^\mathsf{T}\left[\mathbf{A}^\mathsf{T}(-\boldsymbol{\lambda}_k+\rho\mathbf{B}(\mathbf{z}_k-\mathbf{z}_{k-1}))+\mathbf{A}^\mathsf{T}\boldsymbol{\lambda}_k^*\right]\\
&=(\mathbf{x}_k-\mathbf{x}_k^*)^\mathsf{T}\mathbf{A}^\mathsf{T}[\boldsymbol{\lambda}_k^*-\boldsymbol{\lambda}_k+\rho\mathbf{B}(\mathbf{z}_k-\mathbf{z}_{k-1})]\\
&\leq\|\mathbf{x}_k-\mathbf{x}_k^*\|_2\|\mathbf{A}\|_2(\|\boldsymbol{\lambda}_k-\boldsymbol{\lambda}_k^*\|_2+\rho\|\mathbf{B}\|_2\|\mathbf{z}_k-\mathbf{z}_{k-1}\|_2).
\end{align}
Therefore,
\begin{align}
&\|\mathbf{x}_k-\mathbf{x}_k^*\|_2\\
&\leq\frac{1}{\widetilde{m}}\|\mathbf{A}\|_2(\|\boldsymbol{\lambda}_k-\boldsymbol{\lambda}_k^*\|_2+\rho\|\mathbf{B}\|_2\|\mathbf{z}_k-\mathbf{z}_{k-1}\|_2)\\
&\leq\frac{1}{\widetilde{m}}\|\mathbf{A}\|_2[\|\boldsymbol{\lambda}_k-\boldsymbol{\lambda}_k^*\|_2+\rho\|\mathbf{B}\|_2(\|\mathbf{z}_k-\mathbf{z}_k^*\|_2+\|\mathbf{z}_k^*-\mathbf{z}_{k-1}^*\|_2+\|\mathbf{z}_{k-1}-\mathbf{z}_{k-1}^*\|_2)]
\end{align}
Further exploiting \eqref{a6} (for $k$ and $k-1$) and \eqref{a7}, we obtain:
\begin{align}
\begin{split}\label{a9}
&\|\mathbf{x}_k-\mathbf{x}_k^*\|_2\\
&\leq\frac{1}{\widetilde{m}}\|\mathbf{A}\|_2\Bigg[\left(\sqrt{2\rho}+\|\mathbf{B}\|_2\sqrt{\frac{2\rho}{\alpha}}\right)\|\mathbf{u}_k-\mathbf{u}_k^*\|_\mathbf{C}+\|\mathbf{B}\|_2\sqrt{\frac{2\rho}{\alpha}}\|\mathbf{u}_{k-1}-\mathbf{u}_{k-1}^*\|_\mathbf{C}\\
&~~~~+\rho\|\mathbf{B}\|_2\|\mathbf{z}_k^*-\mathbf{z}_{k-1}^*\|_2\Bigg]
\end{split}
\end{align}
According to the definition of the drift $d_k$ in \eqref{drift_def}, we know that $\|\mathbf{z}_k^*-\mathbf{z}_{k-1}^*\|_2\leq\frac{1}{\|\mathbf{B}\|_2}\sqrt{\frac{2}{\rho}}d_k$. Substituting this relation into \eqref{a9} leads to \eqref{a8}.
\end{proof}
\begin{rem}
Sine $\mathbf{u}_k$ converges Q-linearly to some neighborhood of $\mathbf{u}_k^*$ (Theorem 1), Theorem 2 indicates that $\mathbf{x}_k,\mathbf{z}_k,\boldsymbol{\lambda}_k$ converge R-linearly to some neighborhoods of $\mathbf{x}_k^*,\mathbf{z}_k^*,\boldsymbol{\lambda}_k^*$, respectively. When the dynamic optimization problem \eqref{dynamic_admm} degenerates to its static version, i.e., $f_k$ and $g_k$ does not vary with $k$, Theorem 2 also degenerates to its static counterpart in \cite{deng2016global,shi2014linear}.
\end{rem}
To see the impact of the drift $d_k$ (and thus the difference between the dynamic ADMM and the static ADMM) on the steady state convergence behaviors, we present the following result.
\begin{thm}
Suppose the drift defined in \eqref{drift_def} satisfies $d_k\leq d,\forall k$, for some $d\in\mathbb{R}$. Then, we have:
\begin{align}
\label{b1}&\limsup_{k\rightarrow\infty}\|\mathbf{u}_k-\mathbf{u}_k^*\|_\mathbf{C}\leq\frac{d}{\sqrt{1+\delta}-1},\\
\label{b2}&\limsup_{k\rightarrow\infty}\|\mathbf{x}_k-\mathbf{x}_k^*\|_2\leq\frac{\|\mathbf{A}\|_2}{\widetilde{m}}\left[\frac{\sqrt{2\rho}+\|\mathbf{B}\|_2\sqrt{\frac{8\rho}{\alpha}}}{\sqrt{1+\delta}-1}+\sqrt{2\rho}\right]d,\\
\label{b3}&\limsup_{k\rightarrow\infty}\|\mathbf{z}_k-\mathbf{z}_k^*\|_2\leq\sqrt{\frac{2}{\alpha\rho}}\frac{d}{\sqrt{1+\delta}-1},\\
\label{b4}&\limsup_{k\rightarrow\infty}\|\boldsymbol{\lambda}_k-\boldsymbol{\lambda}_k^*\|_2\leq\sqrt{2\rho}\frac{d}{\sqrt{1+\delta}-1}.
\end{align}
\end{thm}
\begin{proof}
According to Theorem 1, we have:
\begin{align}
&\left(\sqrt{1+\delta}\right)^k\|\mathbf{u}_k-\mathbf{u}_k^*\|_\mathbf{C}\leq\left(\sqrt{1+\delta}\right)^{k-1}\|\mathbf{u}_{k-1}-\mathbf{u}_{k-1}^*\|_\mathbf{C}+\left(\sqrt{1+\delta}\right)^{k-1}d_k,\\
&\left(\sqrt{1+\delta}\right)^{k-1}\|\mathbf{u}_{k-1}-\mathbf{u}_{k-1}^*\|_\mathbf{C}\leq\left(\sqrt{1+\delta}\right)^{k-2}\|\mathbf{u}_{k-2}-\mathbf{u}_{k-2}^*\|_\mathbf{C}+\left(\sqrt{1+\delta}\right)^{k-2}d_{k-1},\\
&\cdots\cdots\nonumber\\
&\left(\sqrt{1+\delta}\right)^2\|\mathbf{u}_2-\mathbf{u}_2^*\|_\mathbf{C}\leq\sqrt{1+\delta}\|\mathbf{u}_1-\mathbf{u}_1^*\|_\mathbf{C}+\sqrt{1+\delta}d_2.
\end{align}
Summing them together gives:
\begin{align}
&\left(\sqrt{1+\delta}\right)^k\|\mathbf{u}_k-\mathbf{u}_k^*\|_\mathbf{C}\\
&\leq\sum_{i=1}^{k-1}\left(\sqrt{1+\delta}\right)^id_{i+1}+\sqrt{1+\delta}\|\mathbf{u}_1-\mathbf{u}_1^*\|_\mathbf{C}\\
&\leq d\sqrt{1+\delta}\sum_{i=0}^{k-2}\left(\sqrt{1+\delta}\right)^i+\sqrt{1+\delta}\|\mathbf{u}_1-\mathbf{u}_1^*\|_\mathbf{C}.
\end{align}
Hence,
\begin{align}
\|\mathbf{u}_k-\mathbf{u}_k^*\|_\mathbf{C}\leq\frac{d}{\sqrt{1+\delta}-1}+\frac{\|\mathbf{u}_1-\mathbf{u}_1^*\|_\mathbf{C}}{\left(\sqrt{1+\delta}\right)^{k-1}},
\end{align}
which results in \eqref{b1}. Combining \eqref{b1} with \eqref{a6} and \eqref{a7} immediately leads to \eqref{b3} and \eqref{b4}. In addition, according to \eqref{a8}, we have:
\begin{align}
\begin{split}
&\|\mathbf{x}_k-\mathbf{x}_k^*\|_2\\
&\leq\frac{1}{\widetilde{m}}\|\mathbf{A}\|_2\left[\left(\sqrt{2\rho}+\|\mathbf{B}\|_2\sqrt{\frac{2\rho}{\alpha}}\right)\|\mathbf{u}_k-\mathbf{u}_k^*\|_\mathbf{C}+\|\mathbf{B}\|_2\sqrt{\frac{2\rho}{\alpha}}\|\mathbf{u}_{k-1}-\mathbf{u}_{k-1}^*\|_\mathbf{C}+\sqrt{2\rho}d\right].
\end{split}
\end{align}
Therefore,
\begin{align}
&\limsup_{k\rightarrow\infty}\|\mathbf{x}_k-\mathbf{x}_k^*\|_2\\
&\leq\frac{\|\mathbf{A}\|_2}{\widetilde{m}}\left[\left(\sqrt{2\rho}+\|\mathbf{B}\|_2\sqrt{\frac{8\rho}{\alpha}}\right)\limsup_{k\rightarrow\infty}\|\mathbf{u}_k-\mathbf{u}_k^*\|_\mathbf{C}+\sqrt{2\rho}d\right]\\
&\leq\frac{\|\mathbf{A}\|_2}{\widetilde{m}}\left[\frac{\sqrt{2\rho}+\|\mathbf{B}\|_2\sqrt{\frac{8\rho}{\alpha}}}{\sqrt{1+\delta}-1}+\sqrt{2\rho}\right]d.
\end{align}
\end{proof}
\section{Numerical Examples}
In this section, two numerical examples are presented to validate the effectiveness of the proposed dynamic ADMM algorithm, Algorithm 1. The first example is a dynamic sharing problem and the second one is the dynamic least absolute shrinkage and selection operator (LASSO). Through these two examples, we confirm that the proposed dynamic ADMM algorithm is suitable for not only the dynamic sharing problem \eqref{dynamic_share} (e.g., the first numerical example) but also the general form of dynamic optimization problem \eqref{dynamic_admm} (e.g., the second example, dynamic LASSO, which is not a sharing problem).
\subsection{The Dynamic Sharing Problem}
\subsubsection{Problem Formulation and Algorithm Development}
We first consider the following dynamic sharing problem:
\begin{align}\label{numerical_sharing}
\text{Minimize}_{\mathbf{x}^{(1)},...,\mathbf{x}^{(n)}\in\mathbb{R}^p}~~\sum_{i=1}^n\left(\mathbf{x}^{(i)}-\boldsymbol{\theta}_k^{(i)}\right)^\mathsf{T}\mathbf{\Phi}_k^{(i)}\left(\mathbf{x}^{(i)}-\boldsymbol{\theta}_k^{(i)}\right)+\gamma\left\|\sum_{i=1}^n\mathbf{x}^{(i)}\right\|_1,
\end{align}
where $\boldsymbol{\theta}_k^{(i)}\in\mathbb{R}^p$, $\mathbf{\Phi}_k^{(i)}\in\mathbb{R}^{p\times p}$ positive definite, $\gamma>0$ are given problem data. A motivating application instance of the problem \eqref{numerical_sharing} can be as follows. Suppose there are $n$ subsystems and $p$ quantities (such as data flow in communication networks or currents in power grids) distributed over these subsystems. The amount of the $p$ quantities at subsystem $i$ is described by the vector $\mathbf{x}_i\in\mathbb{R}^p$. Our goal is to estimate the vectors $\mathbf{x}_i,i=1,2,...,n$. The problem data at time $k$ are $\mathbf{\Phi}_k^{(i)},\boldsymbol{\theta}_k^{(i)}$, which vary across time as we keep obtaining new measurements and updating the problem data. We assume that the statistical model of the vectors $\mathbf{x}_i$ is Gaussian so that the first term in \eqref{numerical_sharing} corresponds to the negative log likelihood. Suppose the sum of most quantities across all subsystems cancel out (such as the generation/consumption of power due to the energy conservation rule and the incoming/outgoing current or data flow due to the Kirchhoff's laws) while a few do not cancel out because of abnormality such as leakage. This implies that the sum $\sum_{i=1}^n\mathbf{x}^{(i)}$ should be sparse, i.e., most entries are zero. To incorporate this prior knowledge of sparsity into the estimator, we introduce the $l_1$ regularization term, i.e., the second term in \eqref{numerical_sharing}. Therefore, the estimator is tantamount to the dynamic sharing problem in \eqref{numerical_sharing}.
The problem \eqref{numerical_sharing} is clearly in the form of \eqref{dynamic_share} with:
\begin{align}
&f_k^{(i)}\left(\mathbf{x}^{(i)}\right)=\left(\mathbf{x}^{(i)}-\boldsymbol{\theta}_k^{(i)}\right)^\mathsf{T}\mathbf{\Phi}_k^{(i)}\left(\mathbf{x}^{(i)}-\boldsymbol{\theta}_k^{(i)}\right),\\
&g_k(\mathbf{z})=\gamma\|\mathbf{z}\|_1.
\end{align}
Define:
\begin{align}
\mathbf{x}=\left[
\begin{array}{c}
\mathbf{x}^{(1)}\\
\mathbf{x}^{(2)}\\
\vdots\\
\mathbf{x}^{(n)}
\end{array}
\right],~~\boldsymbol{\theta}_k=\left[
\begin{array}{c}
\boldsymbol{\theta}_k^{(1)}\\
\boldsymbol{\theta}_k^{(2)}\\
\vdots\\
\boldsymbol{\theta}_k^{(n)}
\end{array}
\right],~~\mathbf{\Phi}_k=\left[
\begin{array}{cccc}
\mathbf{\Phi}_k^{(1)}&&&\\
&\mathbf{\Phi}_k^{(2)}&&\\
&&\ddots&\\
&&&\mathbf{\Phi}_k^{(n)}
\end{array}
\right].
\end{align}
Thus, in terms of problem \eqref{dynamic_share_admm}, we have:
\begin{align}
f_k(\mathbf{x})=\left(\mathbf{x}-\boldsymbol{\theta}_k\right)^\mathsf{T}\mathbf{\Phi}_k\left(\mathbf{x}-\boldsymbol{\theta}_k\right).
\end{align}
Applying the dynamic ADMM algorithm, i.e., Algorithm 1, to this dynamic sharing problem, we obtain Algorithm \ref{num_share}. The soft-threshold function $\mathcal{S}$ is defined for $a\in\mathbb{R},\kappa>0$ as follows:
\begin{align}
\mathcal{S}_\kappa(a)=
\begin{cases}
a-\kappa,\text{~~if~~}a>\kappa,\\
0,\text{~~if~~}|a|\leq\kappa,\\
a+\kappa,\text{~~if~~}a<\kappa.
\end{cases}
\end{align}
In \eqref{soft}, an entrywise extension of the soft-threshold function to vector input is used.
\begin{algorithm}[!htbp]
\caption{The dynamic ADMM algorithm for the dynamic sharing problem \eqref{numerical_sharing}}
\begin{algorithmic}[1]\label{num_share}
\STATE \texttt{Initialize $\mathbf{x}_0=\mathbf{0},\mathbf{z}_0=\boldsymbol{\lambda}_0=\mathbf{0},k=0$
\STATE \underline{Repeat}:
\STATE $k\leftarrow k+1$
\STATE Update $\mathbf{x}$ according to:
\begin{align}
\mathbf{x}_k=\left(2\mathbf{\Phi}_k+\rho\mathbf{A}^\mathsf{T}\mathbf{A}\right)^{-1}\left(2\mathbf{\Phi}_k\boldsymbol{\theta}_k-\mathbf{A}^\mathsf{T}\boldsymbol{\lambda}_{k-1}+\rho\mathbf{A}^\mathsf{T}\mathbf{z}_{k-1}\right).
\end{align}
\STATE Update $\mathbf{z}$ according to:
\begin{align}\label{soft}
\mathbf{z}_k=\mathcal{S}_\frac{\gamma}{\rho}\left(\mathbf{Ax}_k+\frac{\boldsymbol{\lambda}_{k-1}}{\rho}\right).
\end{align}
\STATE Update $\boldsymbol{\lambda}$ according to:
\begin{align}
\boldsymbol{\lambda}_k=\boldsymbol{\lambda}_{k-1}+\rho(\mathbf{Ax}_k-\mathbf{z}_k).
\end{align}
}
\end{algorithmic}
\end{algorithm}
\subsubsection{Generation of $\mathbf{\Phi}_k^{(i)}$ and $\boldsymbol{\theta}_k^{(i)}$}
We generate the problem data $\mathbf{\Phi}_k^{(i)}$ and $\boldsymbol{\theta}_k^{(i)}$ recursively as follows. Given $\mathbf{\Phi}_{k-1}^{(i)}~(k\geq 1)$, we first generate $\widetilde{\mathbf{\Phi}}_k^{(i)}$ according to $\widetilde{\mathbf{\Phi}}_k^{(i)}=\mathbf{\Phi}_{k-1}^{(i)}+\eta_k^{(i)}\mathbf{E}_k^{(i)}$, where $\eta_k^{(i)}$ is some small positive number and $\mathbf{E}_k^{(i)}$ is a random symmetric matrix with entries uniformly distributed on $[-1,1]$. Then, we construct the matrix $\mathbf{\Phi}_k^{(i)}$ as:
\begin{align}
\mathbf{\Phi}_k^{(i)}=
\begin{cases}
\widetilde{\Phi}_k^{(i)},~~\text{if}~~\lambda_{\min}\left(\widetilde{\mathbf{\Phi}}_k^{(i)}\right)\geq\epsilon,~\text{i.e.,}~\widetilde{\mathbf{\Phi}}_k^{(i)}\succeq\epsilon\mathbf{I},\\
\widetilde{\Phi}_k^{(i)}+\left[\epsilon-\lambda_{\min}\left(\widetilde{\mathbf{\Phi}}_k^{(i)}\right)\right]\mathbf{I},~~\text{otherwise},
\end{cases}
\end{align}
where $\lambda_{\min}(\cdot)$ denotes the smallest eigenvalue and $\epsilon>0$ is some positive constant. Through this construction, we ensure that $\mathbf{\Phi}_k^{(i)}\succeq\epsilon\mathbf{I},k=1,2,...$. In addition, $\mathbf{\Phi}_0$ is a random symmetric matrix whose entries are uniformly distributed on $[-1,1]$.
Given $\boldsymbol{\theta}_{k-1}^{(i)}~(k\geq1)$, we generate $\boldsymbol{\theta}_k^{(i)}$ according to:
\begin{align}
\boldsymbol{\theta}_k^{(i)}=\boldsymbol{\theta}_{k-1}^{(i)}+\eta_k^{(i)}\mathbf{h}_k^{(i)},
\end{align}
where $\mathbf{h}_k^{(i)}$ is a random $p$-dimensional vector whose entries are uniformly distributed on $[-1,1]$. $\boldsymbol{\theta}_0^{(i)}$ is also a random $p$-dimensional vector with entries uniformly distributed on $[-1,1]$.
\subsubsection{Simulation Results}
\begin{figure}
\centering
\includegraphics[scale=.3]{figs/resource.eps}\\
\caption{The convergence curve of $\|\mathbf{x}_k-\mathbf{x}_k^*\|_2$. $\mathbf{x}_k^*$ is the optimal point of the dynamic sharing problem \eqref{numerical_sharing} at time $k$ computed by an offline optimizor. $\mathbf{x}_k$ is the online solution given by the proposed dynamic ADMM algorithm, i.e., Algorithm \ref{num_share}.}\label{resource}
\end{figure}
In the first simulation, we set the parameters as $\eta=0.2,\epsilon=1,\gamma=1,\rho=1,p=5,n=20$. We use the CVX package \cite{cvx,gb08} to compute the optimal point $\mathbf{x}_k^*$ of the instance of the dynamic sharing problem \eqref{numerical_sharing} at time $k$ in an offline manner. The convergence curve of $\|\mathbf{x}_k-\mathbf{x}_k^*\|_2$ ($\mathbf{x}_k$ is the online solution given by the proposed dynamic ADMM algorithm, i.e., Algorithm \ref{num_share}) is shown in Fig. \ref{resource}. The result is the average of 100 independent trials. We observe that $\mathbf{x}_k$ can converge to some neighborhood of $\mathbf{x}_k^*$ after about 30 iterations. This corroborates the theoretical results (Theorem 2 and Theorem 3) and the effectiveness of the proposed dynamic ADMM algorithm.
\begin{figure}
\centering
\includegraphics[scale=.3]{figs/resource_rho.eps}\\
\caption{The impact of the algorithm parameter $\rho$ on the convergence behaviors ($\|\mathbf{x}_k-\mathbf{x}_k^*\|_2$) of the dynamic ADMM.}\label{resource_rho}
\end{figure}
In the second simulation, we investigate the impact of the algorithm parameter $\rho$ on the convergence performance of the dynamic ADMM. We consider three different values for $\rho$: $0.01,0.1,1$. The corresponding convergence curves ($\|\mathbf{x}_k-\mathbf{x}_k^*\|_2$) are shown in Fig. \ref{resource_rho}. We find that $\rho=0.1$ yields the best convergence performance among the three circumstances. This indicates that the importance of an appropriate value of $\rho$, which should be neither too large nor too small. We note that similar observations have been made in the traditional static ADMM \cite{boyd2011distributed}.
\subsection{Dynamic LASSO}
\subsubsection{Problem Formulation}
Least absolute shrinkage and selection operator (LASSO) is an important and renowned problem in statistics and signal processing. It embodies sparsity-aware linear regression. Here, we consider a dynamic version of the LASSO since the problem data often vary with time in many real-time applications as new measurements arrive sequentially:
\begin{align}\label{lasso_num}
\text{Minimize}_{\mathbf{x}\in\mathbb{R}^p}~~\frac{1}{2}\|\mathbf{F}_k\mathbf{x}-\mathbf{h}_k\|_2^2+\gamma\|\mathbf{x}\|_1,
\end{align}
where $\mathbf{F}_k\in\mathbb{R}^{m\times p}$, $\mathbf{h}_k\in\mathbb{R}^m$ are time-variant problem data and $\gamma>0$ is some positive constant controlling the sparsity of the solution. The problem \eqref{lasso_num} is clearly in the form of \eqref{dynamic_admm} with $f_k(\mathbf{x})=\frac{1}{2}\|\mathbf{F}_k\mathbf{x}-\mathbf{h}_k\|_2^2$, $g_k(\mathbf{z})=\gamma\|\mathbf{z}\|_1$, $\mathbf{A}=\mathbf{I}$, $\mathbf{B}=-\mathbf{I}$, $\mathbf{c=0}$. Thus, we can apply Algorithm 1 to the problem \eqref{lasso_num}, where both \eqref{x_update} and \eqref{z_update} admit closed-form solutions. Note that the problem \eqref{lasso_num} does not fall into the category of dynamic sharing problem \eqref{dynamic_share} as $f_k(\mathbf{x})$ cannot be decomposed across several parts of $\mathbf{x}$. Our goal in this numerical example is to show that the proposed dynamics algorithm works well for the general dynamic optimization problem \eqref{dynamic_admm}, not just the dynamic sharing problem.
\subsubsection{Generation of $\mathbf{F}_k$ and $\mathbf{h}_k$}
The problem data $\mathbf{F}_k$ and $\mathbf{h}_k$ are generated as follows. Given $\mathbf{F}_{k-1}~(k\geq1)$, we generate $\mathbf{F}_k$ according to:
\begin{align}
\mathbf{F}_k=\mathbf{F}_{k-1}+\eta_k\mathbf{W}_k,
\end{align}
where $\eta_k$ is some small positive constant and $\mathbf{W}_k\in\mathbb{R}^{m\times p}$ is a random matrix with entries uniformly distributed on $[-1,1]$. $\mathbf{F}_0$ is also a random matrix with entries uniformly distributed on $[-1,1]$.
To generate the sequence $\mathbf{h}_k$, we construct an auxiliary ground-truth sequence $\widetilde{\mathbf{x}}_k$ as follows. We randomly select $q$ different numbers $\{j_1,...j_q\}$ from the set $\{1,...,p\}$, where $q\ll p$. Given $\widetilde{\mathbf{x}}_{k-1}~(k\geq1)$, we generate $\widetilde{\mathbf{x}}_k$ based on:
\begin{align}
\widetilde{\mathbf{x}}_k=\widetilde{\mathbf{x}}_{k-1}+\eta_k\mathbf{u}_k,
\end{align}
where $\mathbf{u}_k\in\mathbb{R}^p$ is a random vector with $j_l$-th entry uniformly distributed on $[-1,1]$, $l=1,...,q$ and other entries equal to zero. $\widetilde{\mathbf{x}}_0$ is a random vector whose $j_l$-th entry is uniformly distributed on $[0,1]$, $l=1,...,q$ and other entries are zero. This enforces sparsity of the ground-truth $\widetilde{\mathbf{x}}_k$ to be estimated, which is the underlying hypothesis of the LASSO. With $\widetilde{\mathbf{x}}_k$ and $\mathbf{F}_k$ in hands, we generate $\mathbf{h}_k$ according to:
\begin{align}
\mathbf{h}_k=\mathbf{F}_k\widetilde{\mathbf{x}}_k+\mathbf{v}_k,
\end{align}
where $\mathbf{v}_k\sim\mathcal{N}(\mathbf{0},\sigma^2\mathbf{I})$ is a $m$-dimensional Gaussian random vector.
\subsubsection{Simulation Results}
\begin{figure}
\renewcommand\figurename{\small Fig.}
\centering \vspace*{8pt} \setlength{\baselineskip}{10pt}
\subfigure[Slowly time-variant case, i.e., $\eta=0.01$.]{
\includegraphics[scale = 0.3]{figs/lasso_error.eps}}
\subfigure[Fast time-variant case, i.e., $\eta=0.1$.]{
\includegraphics[scale = 0.3]{figs/lasso_high_vary_error.eps}}
\caption{The gaps between the online estimate generated by applying the dynamic ADMM to the dynamic LASSO \eqref{lasso_num}, the estimate given by the offline optimizor through the CVX package (i.e., the optimal point of \eqref{lasso_num}) and the ground-truth: $\|\mathbf{x}_k-\mathbf{x}_k^*\|_2,\|\mathbf{x}_k-\widetilde{\mathbf{x}}_k\|_2$ and $\|\mathbf{x}_k^*-\widetilde{\mathbf{x}}_k\|_2$.}
\label{lasso_error}
\end{figure}
In the simulations, we set the parameters as: $m=10,p=30,q=2,\rho=1,\gamma=0.2,\sigma=0.1$. All results except Fig. \ref{lasso_track} are the average of 100 independent trials. We consider two values, 0.01 and 0.1, for $\eta$, the parameter controlling the variation of the problem data across time. We call $\eta=0.01$ and $\eta=0.1$ the slowly time-variant case and the fast time-variant case, respectively. Denote the online estimate generated by applying the dynamic ADMM to the dynamic LASSO \eqref{lasso_num}, the estimate given by the offline optimizor through the CVX package (i.e., the optimal point of \eqref{lasso_num}) and the ground-truth as $\mathbf{x}_k,\mathbf{x}^*_k$ and $\widetilde{\mathbf{x}}_k$, respectively. The gaps between these three quantities, i.e., $\|\mathbf{x}_k-\mathbf{x}_k^*\|_2,\|\mathbf{x}_k-\widetilde{\mathbf{x}}_k\|_2$ and $\|\mathbf{x}_k^*-\widetilde{\mathbf{x}}_k\|_2$, in the slowly time-variant case and the fast time-variant case are reported in Fig. \ref{lasso_error}-(a) and Fig. \ref{lasso_error}-(b), respectively. A few remarks are in order. First, the solution of the optimizor $\mathbf{x}_k^*$ should be regarded as the benchmark for the dynamic ADMM as the former is the optimal point of \eqref{lasso_num}, or in other words, the best that the dynamic LASSO can achieve. For both slowly and fast time-variant cases, the gaps between the dynamic ADMM and the offline optimizor, i.e., the blue line with square marker, converge to some small values after about 40 iterations. This indicates that the dynamic ADMM can track the optimal point of \eqref{lasso_num} well. Second, the gaps between the dynamic ADMM and the truth (red line with cross markers) as well as the gaps between the offline optimizor and the truth (black line with triangle markers) are similar after some 50 iterations in both slowly and fast time-variant cases. This suggests that in terms of tracking the ground-truth, the dynamic ADMM and the offline optimizor have similar performances while the former has much less computational complexity than the latter. Third, unsurprisingly, comparing \ref{lasso_error}-(a) with \ref{lasso_error}-(b), we observe that the tracking performances of both the dynamic ADMM and the offline optimizor are related to the value of $\eta$: the larger the $\eta$, the more drastically the change of the problem data across time, the poorer the tracking performance.
\begin{figure}
\renewcommand\figurename{\small Fig.}
\centering \vspace*{8pt} \setlength{\baselineskip}{10pt}
\subfigure[Slowly time-variant case, i.e., $\eta=0.01$.]{
\includegraphics[scale = 0.3]{figs/lasso_zero.eps}}
\subfigure[Fast time-variant case, i.e., $\eta=0.1$.]{
\includegraphics[scale = 0.3]{figs/lasso_high_vary_zero.eps}}
\caption{$\|\check{\mathbf{x}}_k\|_2$ and $\|\check{\mathbf{x}}_k^*\|_2$, the deviations of the dynamic ADMM and the offline optimizor from the true sparsity pattern.}
\label{lasso_zero}
\end{figure}
Since the ground-truth $\widetilde{\mathbf{x}}_k$ is sparse and the aim of LASSO is to promote sparsity, a critical performance metric of a solution is how well can it recover the true sparsity pattern. To this end, we compute $\|\check{\mathbf{x}}_k\|_2$ and $\|\check{\mathbf{x}}_k^*\|_2$. Here $\check{\mathbf{x}}_k$ denotes the subvector of $\mathbf{x}_k$ composed of those entries at positions $\{1,...,p\}\backslash\{j_1,...,j_q\}$, i.e., the positions at which the ground-truth is identically zero. Similar definition holds for $\check{\mathbf{x}}_k^*$. $\|\check{\mathbf{x}}_k\|_2$ and $\|\check{\mathbf{x}}_k^*\|_2$ characterize the deviations of the dynamic ADMM ($\mathbf{x}_k$) and the offline optimizor ($\mathbf{x}_k^*$) from the true sparsity pattern. The convergence curves of $\|\check{\mathbf{x}}_k\|_2$ and $\|\check{\mathbf{x}}_k^*\|_2$ are illustrated in Fig. \ref{lasso_zero}-(a) and Fig. \ref{lasso_zero}-(b) for the slowly time-variant case and the fast time-variant case, respectively. We remark that in both cases, after some 50 iterations, the sparsity pattern deviation of the dynamic ADMM is close to that of the offline optimizor. This demonstrates that the dynamic ADMM has the same capability of identifying the true sparsity pattern as the offline optimizor does while the former enjoys significant computational advantage over the latter.
\begin{figure}
\centering
\includegraphics[scale=.3]{figs/lasso_high_vary_track.eps}\\
\caption{The trajectories of the two nonzero dimensions (i.e., $i_1,i_2$, corresponding to the horizontal axis and the vertical axis, respectively) of the dynamic ADMM, the offline optimizor and the ground-truth in one trial of the fast time-variant case}\label{lasso_track}
\end{figure}
Lastly, a more palpable result of the tracking performance is shown in Fig. \ref{lasso_track}, in which the trajectories of the two nonzero dimensions (i.e., $i_1,i_2$, corresponding to the horizontal axis and the vertical axis, respectively) of the dynamic ADMM, the offline optimizor and the ground-truth in one trial of the fast time-variant case are shown. The starting point corresponds to $k=10$ and the time gap between two adjacent points is 10. We observe that the dynamic ADMM can track the truth well. The tracking performance of the offline optimizor is somewhat better, but at the expense of its heavy or even intractable computational burden in many real-time applications.
\section{Conclusion}
In this paper, motivated by the dynamic sharing problem, we propose and study a dynamic ADMM algorithm, which can adapt to the time-varying optimization problems in an online manner. Theoretical analysis is presented to show that the dynamic ADMM converges linearly to some neighborhood of the optimal point. The size of the neighborhood depends on the inherent evolution speed, i.e., the drift, of the dynamic optimization problem across time: the more drastically the problem evolves, the bigger the size of the neighborhood. The impact of the drift on the steady state convergence behaviors of the dynamic ADMM is also investigated. Two numerical examples, namely a dynamic sharing problem and the dynamic LASSO, are presented to corroborate the effectiveness of the dynamic ADMM. We remark that the dynamic ADMM can track the time-varying optimal points quickly and accurately. For the dynamic LASSO, the dynamic ADMM has competitive performance compared to the benchmark offline optimizor while the former possesses significant computational advantage over the latter.
|
2201.13437
|
\section{Conclusion}
We have reported quantum simulations on realistic models of multi-layered rGO which reveal the complex interplay between disorder and interlayer interactions in dictating the dominant transport mechanism.
Depending on the concentration of defects, multilayer interaction can enhance or suppress the system conductance, which results from the competition between the mean free path $\ell_\mathrm{mfp}$ and the interlayer diffusion length $\ell_\mathrm{inter}$. When about 5\% of the carbon atoms are involved in defected regions,
$\ell_\mathrm{inter}$ becomes much longer than $\ell_\mathrm{mfp}$.
In that case, intralayer scattering largely dominates over interlayer diffusion, leading to a weak dependence of $\ell_\mathrm{mfp}$ on $N_\mathrm{layer}$.
On the other hand, $\ell_\mathrm{loc}\sim \ell_\mathrm{inter}$, so that a localized state in one of the layers has enough extension for tunneling to an adjacent one.
If $\ell_\mathrm{inter}$ was much larger than $\ell_\mathrm{loc}$, the tunneling rates would be much smaller.
Once $\ell_\mathrm{inter}\sim\ell_\mathrm{loc}$, tunneling rates are appreciable and charge delocalization is promoted.
While $\ell_\mathrm{mfp}$ is weakly dependent on $N_\mathrm{layer}$, $\ell_\mathrm{loc}$ increases with $N_\mathrm{layer}$ as expected from a generalization of the Thouless relationish for one-dimensional conductors.
This unprecedented interplay between transport length scales is a specific result of 2D layered nature of the multilayer rGO systems. Such mechanism enables hopping transport to overcome the diffusion limit, which is usually the upper bound in bulk systems.
Our theoretical analysis enables us to derive a novel scaling rule, which is in perfect agreement with experimental data at various temperature, and consistent with the Thouless relationship.
The fundamental findings of this study are not limited to multilayered reduced graphene oxide but could find applications in other two-dimensional stacks as well.
\medski
{\noindent{\textbf{Acknowledgements.}}}
The authors acknowledge support from the Flag-Era JTC 2017 project `ModElling
Charge and Heat trANsport in 2D-materIals based Composites$-$ MECHANIC'.
MN\c{C} and HS acknowledge support from T\"UB\.ITAK (117F480). AA and SR are supported by - MECHANIC reference number: PCI2018-093120 funded by Ministerio de Ciencia, Innovacion y Universidades and the European Union Horizon
2020 research and innovation programme under Grant Agreement No.
881603 (Graphene Flagship). ICN2 is funded by the CERCA Programme/
Generalitat de Catalunya, and is supported by the Severo Ochoa program
from Spanish MINECO (Grant No. SEV-2017-0706).
V.-H.N. and J.-C.C. acknowledge financial support from the F\'ed\'eration Wallonie-Bruxelles through the ARC on 3D nano-architecturing of 2D crystals (N$^\circ$16/21-077), from the European Union's Horizon 2020 Research Project and Innovation Program --- Graphene Flagship Core3 (N$^\circ$881603), and from the Belgium FNRS through the research project (N$^\circ$T.0051.18).
Computational resources have been provided by the CISM supercomputing facilities of UCLouvain and the CECI consortium funded by F.R.S. -FNRS of Belgium (N$^\circ$2.5020.11).
\haldun{Authors are particularly grateful to Prof.~Paolo Samorì and Marco Gobbi for the scientific assistance to prepare the devices, Valentina Mussi for some supporting measurements and for enlightening discussions.}
\clearpage
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\newpage
\clearpage
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\setcounter{figure}{0}
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\newcommand{Mustafa~Ne\c{s}et~\c{C}{\i}nar, Aleandro~Antidormi, Viet-Hung~Nguyen, Alessandro~Kovtun, Samuel~Lara~Avila, Andrea~Liscio, Jean-Christophe~Charlier, Stephan~Roche, H\^{a}ldun~Sevin\c{c}li}{Mustafa~Ne\c{s}et~\c{C}{\i}nar, Aleandro~Antidormi, Viet-Hung~Nguyen, Alessandro~Kovtun, Samuel~Lara~Avila, Andrea~Liscio, Jean-Christophe~Charlier, Stephan~Roche, H\^{a}ldun~Sevin\c{c}li}
\begin{center}
\fontsize{18}{30} \selectfont\sffamily{Supporting Information}\\
\fontsize{20}{30} \selectfont\sffamily\textbf{Towards Optimized Charge Transport in Multilayer Reduced Graphene Oxides}\\\vspace{10mm}
\fontsize{14}{14} \selectfont\sffamily{Mustafa~Ne\c{s}et~\c{C}{\i}nar, Aleandro~Antidormi, Viet-Hung~Nguyen, Alessandro~Kovtun, Samuel~Lara~Avila, Andrea~Liscio, Jean-Christophe~Charlier, Stephan~Roche, H\^{a}ldun~Sevin\c{c}li}\\\vspace{5mm}
\fontsize{12}{12} \selectfont\sffamily{E-mail: stephan.roche@icn2.cat; haldunsevincli@iyte.edu.tr}
\end{center}
\section{Kubo-Greenwood and Landauer-B\"uttiker methods}
An efficient linear scaling approach~\cite{fan_linear_2020} is used in Kubo transport to estimate the energy- and time-dependent mean squared displacement of the wave-packet that spreads into the investigated atomic structure,
\begin{equation}
\label{eqn:mean-square-displacement}
\Delta X^2(E,t) = \frac{ \mathrm{Tr}{ \left[ \delta(E-\hat{H}) |\hat{X}(t) - \hat{X}(0)|^2 \right] } }{ g(E)},
\end{equation}
where $g(E)= \mathrm{Tr} [\delta(E - \hat{H})] $ is the density at energy $E$. The time-dependent semiclassical diffusion coefficient $D(E,t) = \frac{\partial }{\partial t} \Delta X^2(E,t)$ and its asymptotic limit $\tilde{D}(E)$ can then be calculated, allowing for the computation of both the electron conductivity and the mean free path as $\sigma(E) = e^2 g(E) \tilde{D}(E)$ and $\ell_\mathrm{mfp}(E) = 2\tilde{D}(E)/v_F(E)$,
respectively (with $v_F(E)$ being the carrier velocity).
The conductivity values in rGO systems have been averaged over 10 different randomly chosen initial wave packets, and the calculation of the mean-square displacement was carried out through an efficient decomposition in terms of Chebyshev polynomials, with 5000 moments. Note that periodic boundary conditions are employed in both longitudinal and transverse directions.
Concerning the transport simulations performed using the Landauer-B\"uttiker technique, the rGO system is partitioned into three regions, namely the left and right electrodes (leads) and the central region. The {leads} are modeled as scattering free regions made up of the same {ideal} material. The Green function for the central region is calculated as $\mathcal{G}(E)=\left[(E+i0^+)I-H_C-\Sigma \right]^{-1}$, where $I$ is identity matrix, $H_C$ is the Hamiltonian matrix for the central region, and the self-energy term includes effects of the left and right reservoirs as $\Sigma=\Sigma_L+\Sigma_R$.
In this work, systems containing as many as $10^{6}$ atoms have been simulated, for which efficient decimation algorithms are implemented.~\cite{ryndyk_book_2015,sevincli_srep_2013}
The transmission amplitude is obtained from $\mathcal{T}(E)=\mathrm{Tr}\left[\Gamma_L\mathcal{G}\Gamma_R\mathcal{G}^\dagger\right]$, where $\Gamma_{L(R)}=i [\Sigma_{L(R)}-\Sigma_{L(R)}^\dagger]$ are the left (right) broadening matrices.
Conductance values are calculated using the Landauer formula,
\begin{equation}
G=\frac{2e^2}{h}\int \left(- \frac{\partial f_{FD}(E,\mu,T)}{\partial E} \right) \mathcal{T}(E) dE,
\end{equation}
where $e$ is electron charge, $f_{FD}$ is Fermi-Dirac distribution function, $T$ is temperature (herewith 100~K) and $\mathcal{T}(E)$ is the transmission probability for a given energy $E$.
Since low-energy properties are only of interest, transmission coefficients are integrated over 40 $k$-points in the transverse direction in order to reach sufficiently accurate energy resolution. At last, a geometric average on the transmission function over an ensemble of 20 samples is applied to overcome the sample size effects.
\clearpage
\section{Supplementary details on simulated structures}
\haldun{
The types and amounts of defects are deduced from MD simulations replicating the thermal annealing of GO on a computer. In particular, following the protocol in Ref. 6, several atomistic samples of GO have been generated with a total number of atoms as large as 10000 and an initial oxygen concentration of 35\%. The thermal reduction of the systems has been simulated for different annealing temperatures and a statistical analysis of the chemical and morphological properties of the resulting rGO samples has been performed. In the table below, a summary of the values of the most relevant chemical species observed in the the samples after reduction is given. At the annealing temperature of 900oC, a final concentration of oxidizing agents of 5\% has been found, in extremely good agreement with the experimental samples. From a detailed exploration of the atomistic structures, the concentration and type of defects has been derived and used to model rGO samples for transport calculations.
}
\begin{table}[]
\haldun{
\begin{tabular}{lccccccc}
\hline
& C sp² & C sp³ & C-OH & C-O-C & C=O & O-C=O & O/CFIT \\
& (\%) & (\%) & (\%) & (\%) & (\%) & (\%) & \\
\hline
GO & 32.9 & 11.5 & 3.0 & 43 & 6.5 & 3.0 & 0.35 (imposed) \\
rGO 300 °C & 75.4 & 10.5 & 2.9 & 3.9 & 4.8 & 2.5 & 0.154 \\
rGO 600 °C & 83.5 & 8.8 & 1.1 & 3.7 & 1.1 & 1.8 & 0.11 \\
rGO 900 °C & 94.1 & 3.6 & 0.3 & 0.8 & 0.6 & 0.6 & 0.05 \\
\hline
\end{tabular}
}
\caption{
\haldun{Concentration of different chemical species observed in the atomistic samples of rGO after the reduction process at different annealing temperatures.
}
}
\end{table}
\clearpage
\section{Supplementary details regarding tight-binding parameters}
As explained in the main text, while the change in C-C bond length is modeled by the distance dependence of hopping energies, the local doping due to the localized states induced by defects and impurities are included by adding on-site energies to C-atoms surrounding their position. These on-site energies have the following common form
\begin{itemize}
\item $\varepsilon_D$ applied to C atoms directly connected to impurities/defects
\item $\varepsilon_n$ applied to other surrounding C atoms
\end{itemize}
\begin{eqnarray}
\varepsilon_n =\frac{\varepsilon_0}{1+(d_n/\lambda_D)^\kappa}
\end{eqnarray}
where $d_n$ is the distance from the $n^{th}$ C atom to the considered impurity, $\varepsilon_0$ is the maximum value of $\varepsilon_n$, $\lambda_D$ is the decay length and the number $\kappa$ is determined depending on the defect/impurity types.
\vspace{10mm}
\noindent
\begin{tabular}{lllllc}
\hline
Oxygen impurity & $\lambda_D=1.0$~$\textup\AA$ & $\kappa=3$ &$\varepsilon_{0}=1.6$~eV &$\varepsilon_{D}=-28.0$~eV & \\
OH group & $\lambda_D=1.0$~$\textup\AA$& $\kappa=3$ &$\varepsilon_{0}=1.8$~eV &$\varepsilon_{D}=-28.0$~eV & \\
585 defect & $\lambda_D=5.0$~$\textup\AA$ & $\kappa=5$ &$\varepsilon_{0}=1.0$~eV &$\varepsilon_{D}=-1.3$~eV & \\
Stone-Waled defect & $\lambda_D=1.6$~$\textup\AA$ & $\kappa=5$ &$\varepsilon_{0}=-1.5$~eV &$\varepsilon_{D}=-1.2$~eV & \\
555-777 defect & $\lambda_D=8.0$~$\textup\AA$ & $\kappa=5$& $\varepsilon_{0}=0.4$~eV& $\varepsilon_{D1}=-1.8$~eV & $\varepsilon_{D2}=-0.5$~eV \\
\hline
\end{tabular}
\vspace{5mm}
The electronic band structures obtained using the proposed tight binding Hamiltonians are presented in Figs.~\ref{fig_supp2} and \ref{fig_supp3}. Indeed, our proposed tight-binding models reproduce well the low energy bands, compared to the DFT results.
\begin{figure}
\includegraphics[width=1\textwidth]{fig_supp_viet1.png}
\caption{Defects and impurities investigated in this work.}
\label{fig_supp1}
\end{figure}
\begin{figure}
\includegraphics[width=1.\textwidth]{fig_supp_viet2.png}
\caption{The electronic band structure of graphene with oxygen impurity (left) and OH group (right). TB calculations fit to the DFT results.}
\label{fig_supp2}
\end{figure}
\begin{figure}
\includegraphics[width=1.\textwidth]{fig_supp_viet3.png}
\caption{The electronic bandstructure of graphene with structural defects.
TB calculations fit to the DFT results.}
\label{fig_supp3}
\end{figure}
\clearpage
\haldun{
Other sophisticated models (i.e., larger distance neighbor as well as Slater-Koster like models) generally present a disadvantage that a large number of adjusted parameters are required to model accurately the considered defective systems. This disadvantage also gives rise to some difficulties for the implementation of transport calculations in the large scale devices while the accuracy is not significantly improved.
In Fig.~\ref{fig:slater-koster} a comparison of the computed electronic band strucures of bilayer graphene obtained using our used model and Slater-Koster like models in Ref.~\citenum{laissardiere:nanolett_2010} that has been shown to compute well the electronic structure of both Bernal stacking and twisted bilayer graphene systems.}
\begin{figure}
\includegraphics[width=.7\textwidth]{fig_slater-koster.png}
\caption{
\haldun{
Comparison of the electronic bands of bilayer graphene obtained using our TB Hamiltonian and Slater-Koster like models in Nano Lett. 10, 804–808 (2010).}
}
\label{fig:slater-koster}
\end{figure}
\clearpage
\section{Note on parametrization and system setups}
The tight-binding parametrization for structural defects involves modifying onsite energies of $sp^2$ orbitals by a value which decays exponentially with the distance, and onsite energies due to defects surrounding a particular atom are taken additive. For Landuer-B\"uttiker calculations, we also introduced defect-free buffers of 2.57~nm in length between the scattering region and the leads to saturate the effects due to the onsite energies (see Fig.\ref{fig1} right panel), and cutoff radius of 2.57~nm was used as the range of the modification. For the scaling analysis, the central transport channel is lengthened by adding 1.284~nm-length blocks (which corresponds to a mesh resolution of the same length).
\clearpage
\section{Supplementary mean-free-path plots}
Mean-free-paths as obtained from Landauer-B\"uttiker and Kubo-Greenwood simulations with changing layer thickness are shown for comparison in Figure~\ref{fig_supp_mfp}.
The agreement between two methodologies is remarkable.
\begin{figure}[h]
\includegraphics[width=1\textwidth]{fig_supp_mfp_comparison.pdf}
\caption{
Mean-free-paths for different numbers of layers as obtained from NEGF and KG simulations are shown. Blue, red and green curves represent mono/bi/tri-layer rGO systems.
}
\label{fig_supp_mfp}
\end{figure}
\clearpage
\section{Supplementary results with lower defect concentrations}
We have shown in the main text that interlayer coupling affects charge transport of defect-free and defective systems in opposite ways. Namely, in defect-free systems conductance is reduced with the number of layers, whereas it is enhanced in rGO, which contains 95\% sp$^2$ carbon. Reducing the amount of impurities, it is possible to observe the transition. In Fig.~\ref{fig_supp_lowdefect}, defect concentration is 10 times lower than those in the main text (99.5\% sp$^2$ carbon), where monolayer is observed to have the highest transmission values around the charge neutrality point.
\begin{figure}[h]
\includegraphics[width=.75\textwidth]{fig_supp_lowdefect.pdf}
\caption{Transmission spectrum through rGO with low concentration of imperfections. The device sizes are the same with those in the main text, impurity concentration is 10 times lower with 99.5\% sp$^2$ ratio. Blue, red and green curves correspond to mono/bi/trilayer rGO, respectively.}
\label{fig_supp_lowdefect}
\end{figure}
\clearpage
\section{Supplementary LDOS plots}
\begin{figure}[h]
\includegraphics[width=.5\textwidth]{fig_supp_leftInjectedLDOS2.png}
\caption{
Left injected LDOS plots for mono/bi/tri-layer rGO. High and low LODS values are distinguihed on the left/right electrodes. Localization is more pronounced in monolayer, whereas the carrier density more dispersed in trilayer sample.
}
\label{fig_supp_ldos}
\end{figure}
Using the Green's function method, we could compute the left (right, respectively) injected LDOS \cite{datta_nanoscale_2000}, reflecting the propagation of electrons from left to right (right to left, respectively) electrodes. In particular, the left-injected LDOS is given by
\begin{eqnarray}
LDOS_L = \frac{G\Gamma_LG^\dagger}{2\pi}.
\end{eqnarray}
LDOS in the mulitlayer zones are averaged over the layers to show the contributions from all layers. The decay of the presented left-injected LDOS along the Ox axis in Fig.S6 is essentially due to scatterings with defects/impurities, manifesting as the electronic localization in the device region. In the monolayer case, charge localization is more pronounced than bilayer and trilayer systems, in agreement with Fig.~\ref{fig2}f. Importantly, it is shown that the improved propagation of electrons from left to right electrodes is obtained when increasing number of graphene layers, thus illustrating the transport properties discussed in the main text.
\clearpage
\section{Temperature-dependence of electrical resistivity $\rho(T)$, Efros-Shklovskii variable range hopping model}
At low temperature, charge transport in graphene-based materials is typically occurring via charge hopping in a disorder-broadened density of states near the Fermi level $g(E_F)$.~\cite{cheah_jpcm_2013} In the Ohmic regime, the resistivity is tipycally modelled by a stretched exponential behavior:
\begin{eqnarray}
\label{eqn:vrh}
\rho(T)=\rho_{0,VRH} \exp
\left(
\frac{T_0}{T}
\right)^\beta,
\end{eqnarray}
where $\rho_{0,VRH}$ is a prefactor and $\beta$ is a characteristic exponent. $T_0$ represents a characteristic temperature correlated to the localization length $(\ell_\mathrm{loc})$, the higher the first one, the lower the latter. The $\ell_\mathrm{loc}$ is defined as the average spatial extension of the charge carrier wave function: the lower the $T_0$, the larger the $\ell_\mathrm{loc}$.
\begin{figure}
\includegraphics[width=0.99\textwidth]{fig_OpticalImages.png}
\caption{
\ciz{
Sample thickness was measured by atomic force microscopy (AFM) and the graphitic structure is confirmed by Raman measurements.}
\haldun{
Optical images of devices fabricated using (a) single rGO nanosheet and (b) partially oxidized graphite. Samples thickness were measured by atomic force microscopy (AFM) and the graphitic structure of (b) was confirmed by Raman measurements.}
}
\label{fig:supp_oxidized-graphite}
\end{figure}
\begin{figure}[t]
\centering
\includegraphics[width=1.0\textwidth]{figS8.png}
\caption{Correlation plots. Experimental data of multilayer RGO (circles for devices reported in Ref.~\citenum{kovtun_acsnano_2021} and acquired at different temperatures. Samples with both sp$^2$ contents (86\% and 77\%) show linear dependence, in good agreement with the theoretical prediction for scaling (cf. Eqn.~\ref{eqn:multilayerscaling} in the main text). All the linear fitting curves calculated at different temperatures are included between the two curves acquired at 5~K and 300~K (dashed area).}
\label{fig:supp_correlation}
\end{figure}
The analytic expression reported in Eqn.~\ref{eqn:vrh} is quite general depending on the model commonly referred as variable range hopping (VRH). The stretching exponent $\beta$ is strongly dependent on the shape of $g(E_F)$, e.g. when the density of states is constant (Mott-VRH model),~\cite{mott_cjp_1956} the $\beta$ value directly depends on the system’s dimension ($D$) with the form $\beta= 1/(D+1)$. Reduced graphene oxide thin films show the presence of a gap at the Fermi level due to the Coulomb interaction between the occupied, excited state above $E_F$ and the hole left by the same electron below $E_F$. This case is described by the so called Efros Shklovskii model (ES-VRH)~\cite{efros_jpcm_1975}
with characteristic exponent of Eqn.~\ref{eqn:vrh} $\beta=1/2$, which does not depend on the system dimensionality. The characteristic temperature $T_0$ for 2D materials is given by
\begin{eqnarray}
\label{eqn:T0}
T_0=\frac{2.8e^2}{4\pi\epsilon_0\epsilon_rk_B\ell_\mathrm{loc}}
=\frac{1}{A\epsilon_r\ell_\mathrm{loc}},
\end{eqnarray}
where $e$ is the elementary charge, $\epsilon_0$ and $\epsilon_r$ represent the vacuum permittivity and the relative permittivity of the material and $k_B$ is the Boltzmann constant. For the sake of simplicity, all the universal constants are collected by the parameter $A = 0.021 \mu\mathrm{m}^{-1}\mathrm{K}^{-1}$.
Combining Eqns.~\ref{eqn:vrh} and \ref{eqn:T0}, we obtain the mathematical expression reported in the main text,
\begin{eqnarray}
\rho(T)=\rho_0\exp\left(\frac{1}{A\epsilon_r\ell_\mathrm{loc} T}\right)^{1/2}.
\end{eqnarray}
\clearpage
\section{Electrical resistivity measurements $\rho(T)$}
\haldun{
Single rGO nanosheet and partially oxidized graphite were prepared by thermal annealing ($T_\mathrm{ann} = 900^o$C) of GO and oxidized nanographite, respectively, deposited on clean SiO$_2$/Si substrates (2,000 rpm for 60s).
}
\haldun{
The micrometric electrodes were lithographically patterned to characterize the electrical transport across a limited number of overlapping flakes. Lithography was carried out by exposing a standard photoresist (AZ1505, Microchemicals) with the 405~nm laser of a Microtech laser writer. A 30-nm-thick Au film (without adhesion layer) was thermally evaporated onto the patterned photoresist and lift-off was carried out in warm acetone (40$^o$C).
}
\haldun{
The resistance vs temperature measurements were carried out with a Quantum Design Physical Properties Measurements System (PPMS), using an external Keithley 2636 Source-Meter. The resistance was measured in the temperature range between 300~K to 5~K with a slow ramp (1~K/min). The Ohmic behavior of the device was checked by the linearity of the I-V curves.
}
\haldun{
Typically, each acquired $\rho(T)$ curve corresponds to an array of $>$50,000 resistivity values at different temperatures. For sake of simplicity, such $\rho(T)$ curves were sampled at 43~temperature values with logarithmic steps, as reported in Fig.~\ref{fig:supp_rho-logT}. Three values acquired at 5~K, 100~K and 300~K are reported in Figure~5 in the main text.
}
\begin{figure}[t]
\includegraphics[width=14cm]{fig_supp__rho_logT.png}
\caption{
\haldun{Example of measured $\rho(T)$ curve and sampled temperature values. Red circles correspond to the resistivity values depicted in the correlation plots.}
}
\label{fig:supp_rho-logT}
\end{figure}
\begin{landscape}
\begin{table}[]
\haldun{
\caption{Summary of experimental parameters of all the studied devices}
\begin{tabular}{|c|l|l|l|c|c|c|}
\hline sample & & $N_\textrm{layer}$ & $\xi$ (nm) & $\rho_\textrm{5K}$ & $\rho_\textrm{100K}$ & $\rho_\textrm{300K}$ \\
\hline \multirow{3}{8em} {single RGO nanosheet}
& \multirow{3}{5em}{Device 28 [ref.5 main text]} & 1 & 4.0$\pm$0.3 & & & \\
& & & & & & \\
& & & & & & \\
& & 1 & 3.7$\pm$0.4 & & & \\
\hline \multirow{2}{8em}{Bi-layer RGO}
& Device 26 & 2 & 7.5$\pm$0.8 & & & \\
& Device 27 & 2 & 7.8$\pm$1.4 & & & \\
\hline \multirow{2}{8em}{RGO thin film}
& Device 6 & 5$\pm$1 & 18$\pm$2 & -- & $(3.6\pm 12)\times 10^{-4}$ & $(2.8\pm0.9)\times 10^{-4}$ \\
& Device 1 & $6\pm1$ & $26\pm4$ & $(13\pm3)\times 10^{-4}$ & $(6.5\pm1.7)\times10^{-5}$ & $(4.7\pm1.3)\times10^{-4}$ \\
\hline \multirow{2}{8em}{partially oxidized graphite}
& & \multirow{2}{*}{$6\pm1$} & \multirow{2}{*}{$243\pm56$} & \multirow{2}{*}{$(5.9\pm1.3)\times10^{-4}$} & \multirow{2}{*}{$1.1\pm0.3)\times10^{-4}$} & \multirow{2}{*}{$(2.9\pm0.7)\times10^{-5}$} \\
& & & & & & \\
\hline \multirow{9}{8em}{RGO thin film}
& Device 7 & $7\pm1$ & & $(3.1\pm0.7)\times10^{-4}$ & $(3.4\pm0.8)\times10^{-5}$ & $(2.4\pm0.6)\times10^{-5}$ \\
& Device 17 & $7\pm1$ & & $(6.4\pm1.4)\times10^{-4}$ & $(1.1\pm0.2)\times10^{-4}$ & $(4.8\pm1.1)\times10^{-4}$ \\
& Device 2 & $8\pm1$ & & $(2.4\pm0.5)\times10^{-4}$ & $(3.7\pm0.7)\times10^{-5}$ & $(1.9\pm0.4)\times10^{-5}$ \\
& Device 8 & $10\pm1$ & & $(9.7\pm1.5)\times10^{-5}$ & $(1.6\pm0.2)\times10^{-5}$ & $(1.2\pm0.2)\times10^{-5}$ \\
& Device 3 & $11\pm1$ & & $(7.9\pm1.2)\times10^{-5}$ & $(2.9\pm0.4)\times10^{-5}$ & $(1.5\pm0.2)\times10^{-5}$ \\
& Device 9 & $17\pm1$ & & $(3.7\pm0.4)\times10^{-5}$ & $(1.3\pm0.1)\times10^{-5}$ & $(1.3\pm0.1)\times10^{-5}$ \\
& Device 4 & $18\pm1$ & & $(4.3\pm0.4)\times10^{-5}$ & $(1.6\pm0.2)\times10^{-5}$ & $(1.4\pm0.1)\times10^{-5}$ \\
& Device 10 & $33\pm2$ & & $(2.8\pm0.3)\times10^{-5}$ & $(1.2\pm0.1)\times10^{-5}$ & $(1.3\pm0.1)\times10^{-5}$ \\
& Device 5 & $35\pm2$ & & $(2.4\pm0.2)\times10^{-5}$ & $(2.0\pm0.2)\times10^{-5}$ & $(1.4\pm0.2)\times10^{-5}$ \\
\hline
\end{tabular}
}
\end{table}
\end{landscape}
\end{document}
|
2102.03397
|
\section*{Acknowledgments}
This work was supported through the Air Force Office of Scientific Research Grant No. FA9550-15-1-0474, and the National Science Foundation [Platform for the Accelerated Realization, Analysis, and Discovery of Interface Materials (PARADIM)] under Cooperative Agreement No. DMR-1539918, NSF DMR-1709255. This research is funded in part by the Gordon and Betty Moore Foundation's EPiQS Initiative through Grant No. GBMF3850 to Cornell University. B.D.F. and J.N.N. acknowledge support from the NSF Graduate Research Fellowship under Grant No. DGE-1650441. P.M. acknowledges support from the Indo US Science and Technology Forum (IUSSTF). This work made use of the Cornell Center for Materials Research (CCMR) Shared Facilities, which are supported through the NSF MRSEC Program (No. DMR-1719875). Substrate preparation was performed in part at the Cornell NanoScale Facility, a member of the National Nanotechnology Coordinated Infrastructure (NNCI), which is supported by the NSF (Grant No. ECCS-1542081).
|
1304.2976
|
\section{Introduction}
According to galaxy formation theories dwarf spheroidal galaxies are
believed to inhabit massive dark matter halos. Because of their large
mass to light ratio these galaxies are ideal to test fundamental
predictions of the $\Lambda$CDM cosmological paradigm, since it is generally
considered relatively safe to neglect baryons in the construction of
dynamical models.
One of the strongest predictions from $\Lambda$CDM concerns the
dark matter density profile. Early simulations of dark matter halos
assembled in a cosmological context showed that such a profile is
accurately described by a two-sloped form, now known as NFW profile
\citep{nfw1996, nfw1997}. More recently Einasto profiles have been
shown to provide a better fit \citep[e.g.][]{Springel2008,Navarro2010MNRAS402}, in
particular for satellite galaxies \citep{VeraCiro2013MNRAS}. These
predictions are made using dark matter only simulations and therefore
neglect (by construction) the baryonic component. And although baryons
are sub-dominant in the total potential of the system
\citep{Walker2012}, it has been suggested that they could play a role in the
evolution of dwarf spheroidal galaxies, for instance, in modifying the
internal orbital structure \citep{Bryan2012} and the overall density
profile \citep{Governato2012}. The complex evolution of baryons and
its non-trivial interplay with the host halo are difficult to model
and not yet completely understood \citep[see][]{Ponzen2012MNRAS}.
Another effect driving the internal dynamics of satellite halos is the
tidal interaction with the main host. It can change the density
profile \citep{Hayashi2003}, the geometrical shape of the mass
distribution \citep{Kuhlen2007}, and also influence the kinematics of
the embedded stars \citep{Lokas2010ApJ}. Unfortunately these uncertainties imply that even
when the observations of the local dwarf spheroidal galaxies are not
consistent with being embedded in the halos predicted from pure dark
matter N-body simulations, this does not necessarily reflect a
fundamental problem of $\Lambda$CDM.
Thanks to their relative proximity, information for individual stars
in the dwarf galaxies satellites of the Milky Way are relatively easy
to get. Sky positions are easily determined from photometry, and
radial velocity measurements are possible to estimate within an error
of $\sim 2$ km/s. Some of the
datasets compiled to the date include thousands of individual members
with line-of-sight velocities
\citep{Helmi2006ApJ,Battaglia2006AA,Battaglia2008ApJ...681L..13B,Walker2009AJ....137.3100W,Battaglia2011MNRAS}
Proper motions of individual stars are currently
still too difficult to measure. Despite the fact that only three of
the total of six phase space coordinates are available from
measurements, it is possible to create dynamical models of these
systems that can be compared to these observables.
Following the method thoroughly described in \citet{Breddels2012arXiv}
we set out to model Fornax, Sculptor, Carina and Sextans with
orbit-based dynamical methods (Schwarzschild modeling) assuming they
are embedded in spherical halos. As extensively shown in the
literature
\citep[e.g.][]{Richstone1984ApJ...286...27R,Rix1997ApJ...488..702R,vanderMarel1998ApJ...493..613V,Cretton1999ApJS..124..383C,Valluri2004ApJ...602...66V,vdBosch2008MNRAS.385..647V,Jardel2012}
this method allows to construct a non-parametric estimator of the
distribution function. Among many, this method has one advantage over
Jeans modeling, by not having to assume a particular velocity
anisotropy profile, therefore being more general and thus less prone
to biases associated to the assumptions. But even in this case there
are other limitations in the modeling such as the mass-anisotropy
degeneracy. In this work we use higher moments (fourth moment) of the
line of sight velocity distribution to get a better handle on this
degeneracy.
To compare how different shapes for the dark matter profiles fit the
data, we first need to establish a statistical framework. In this
paper we do this in a Bayesian way using the evidence
\citep{Mackay_2003_information}. This method provides a natural way of
comparing models in Bayesian inference and also makes it possible to
combine the data of all the dwarf spheroidals to test for example, if
all dwarf spheroidals could be embedded in a universal halo
\citep{Mateo1993,Walker2009}. Furthermore, the shape may give us hints
to how the dwarf galaxy formed and the anisotropy profile may be used
to distinguish between evolutionary scenarios \citep[see
e.g.][]{Mayer2010AdAst2010E..25M,Kazantzidis2011ApJ,Helmi2012ApJ}.
This paper is organized as follows. We begin in \S \ref{sec:data} by
presenting the data and all the ingredients needed to do the model
comparison. In \S \ref{sec:method} we present our dynamical and
statistical methods. We present the results of our Schwarzschild
models for the four dSph in our sample in \S \ref{sec:results_schw},
while the Bayesian model comparison is done in \S \ref{sec:bayes}. We
discuss the implications of our results in \S \ref{sec:slope} and conclude in \S
\ref{sec:end}.
\section{Data}
\label{sec:data}
\begin{figure*}
\centerline{\includegraphics[scale=0.65]{fig/vlos_vs_r_fnx}\includegraphics[scale=0.65]{fig/vlos_vs_r_scl}
\includegraphics[scale=0.65]{fig/vlos_vs_r_car}\includegraphics[scale=0.65]{fig/vlos_vs_r_sxt}}
\caption{Radius versus line of sight velocity for Fornax, Sculptor Carina and Sextans. The horizontal lines show the borders of the bins, the vertical lines denote the mean systemic velocity of the galaxy together with the $\pm 3\sigma$ region. \label{fig:vlos}}
\end{figure*}
\begin{figure*}
\centerline{\includegraphics[scale=0.45]{fig/moments_fnx}\includegraphics[scale=0.45]{fig/moments_scl}}
\centerline{\includegraphics[scale=0.45]{fig/moments_car}\includegraphics[scale=0.45]{fig/moments_sxt}}
\caption{Line of sight velocity moments for Fornax, Sculptor Carina and Sextans. For each galaxy we show the velocity dispersion and the kurtosis. The black dots show the mean, and the error bars the 1$\sigma$ error. The blue regions show the confidence interval for the NFW fit, similar to \cite{Breddels2012arXiv}. \label{fig:moments}}
\end{figure*}
\begin{table}
\centering
\begin{tabular}{l|r|r|r}
Name & $N_\text{Batt}$ & $N_\text{Walker}$ & $N_\text{member}$ \\
\hline
Fornax & 945$^{(1)}$ & 2633$^{(5)}$& 2936 \\
Sculptor & 1073$^{(2)}$ & 1541$^{(5)}$& 1685 \\
Carina & 811$^{(3)}$ & 1982$^{(5)}$ & 885 \\
Sextans & 792$^{(4)}$ & 947$^{(5)}$ & 541 \\
\end{tabular}
\caption{Number of stars in the kinematic samples used in this paper. Sources: $^{(1)}$\citet{Battaglia2006AA}, $^{(2)}$\citet{Battaglia2008ApJ...681L..13B}, $^{(3)}$\citet{Helmi2006ApJ,Koch2006AJ,Starkenburg2010AA}, $^{(4)}$\citet{Battaglia2011MNRAS}, $^{(5)}$\citet{Walker2009AJ....137.3100W}}
\label{tab:data_kin}
\end{table}
\begin{table*}
\centering
\begin{tabular}{l|c|c|c|c|c|c}
Name & $R_{e,\text{max,Batt}}$ & $R_{e,\text{max,walker}}$ & $\mu_{\rm MW}$ & $\sigma_{\rm MW}$ & $\mu_{\rm dwarf}$ & $\sigma_{\rm dwarf}$ \\
& (kpc) & (kpc) & (km/s) & (km/s) & (km/s) & (km/s) \\
\hline
Fornax & 1.82 & 2.21 & $41.1$ & $ 38.9$ & $55.1 $ & $12.1$ \\
Sculptor & 1.37 & 1.65 & $17.9$ &$47.4$ & $110.6$ & $ 10.1$ \\
Carina & 0.86 & 0.96 & $70.9$ & $62.5$ & $222.9$ & $6.6$ \\
Sextans & 1.86 & 1.65 & $67.5$ & $74.5$ & $224.3$ & $7.9$\\
\end{tabular}
\caption{Parameters of the foreground plus dwarf galaxy model used for determining membership, as well as
for deriving the radial profiles for the second and fourth velocity moments for each dSph.}
\label{tab:data_params}
\end{table*}
\begin{table*}
\centering
\begin{tabular}{l|r|c|c|c}
Name & distance & profile & scale radius & $L_V $ \\
& (kpc) & & (kpc) & $\times 10^5\ensuremath{\text{L}_{\odot}}$ \\
\hline
Fornax & 138$^{(1)}$ & Plummer$^2$ & 0.79 & $100^{(2)}$ \\
Sculptor & 79$^{(3)}$ & Plummer$^3$ & 0.30 & $10^{(3)}$ \\
Carina & 101$^{(1)}$ & Exponential$^4$ & 0.16 & $2.4^{(4)}$\\
Sextans & 86$^{(1)}$ & Exponential$^4$ & 0.39 & $4.37^{(4)}$ \\
\end{tabular}
\caption{Distances, type of photometric profile used, scale radius and stellar luminosity used for the dynamic models. Sources: $(1)$ \citet{Mateo1998ARA}, $(2)$ \citet{Battaglia2006AA}, $(3)$ \citet{Battaglia2008ApJ...681L..13B}, $(4)$ \citet{Irwin1995MNRAS.277.1354I}}
\label{tab:gal_params}
\end{table*}
In this section we present the data that is used for fitting our
dynamical models. The radial velocity measurements of the dwarf
spheroidal galaxies come from
\citet{Helmi2006ApJ,Battaglia2006AA,Battaglia2008ApJ...681L..13B,Walker2009AJ....137.3100W}
and \citet{Battaglia2011MNRAS}. We plot radius versus heliocentric
velocity in Fig.~\ref{fig:vlos} for each galaxy separately.
Figure \ref{fig:vlos} shows that each dSph suffers from foreground
(Milky Way) contamination. To remove this contamination and reliably
identify member stars we have developed a simple analytic model for
the positional and kinematic distribution of both foreground and the
galaxy in question \citep[along the lines of][Breddels et al.\ in
prep]{Battaglia2008ApJ...681L..13B}. For each particular
dataset\footnote{For a given dSph there may be multiple datasets, and
we treat each independently because their sampling might be
different.}, we assume that the foreground has a constant surface
density, and that the dSph follows a specific stellar density
profile. We also assume that the velocity distribution at each radius
may be modeled as sum of two Gaussians. The Gaussian describing the
foreground has the same shape at all radii, while that of the stars
associated with the dwarf can have a varying dispersion with
radius. Their relative amplitude also changes as function of
distance from the dwarf's centre. This model results in a
determination of the relative contribution of member-to-non-member
stars as a function of velocity and radial distance $R$.
Based on this model we calculate the elliptical radius at which the
ratio of dSph:foreground is 3:1 (without using any velocity
information). We remove all stars outside of this radius from the
dataset. A particular star included in more than one dataset is
removed only when it it satisfies the condition for all sets, for
instance a star outside $R_{e,\text{cut, Batt}}$, but inside
$R_{e,\text{cut, Walker}}$ will not be discarded. This simple clipping
in elliptical radius cleans up part of the foreground
contamination. For completeness, the radii for all datasets cleaned up
in this way are presented in Table \ref{tab:data_params}, as well as
the fit to the foreground model. The number of stars and the sources
are listed in Table \ref{tab:data_kin}.
For the resulting dataset, we compute the second and fourth moment of
the radial velocity as a function of circular radius as
follows\footnote{ Elliptical radii are only used for the clipping, for
the rest of the analysis we use the circular radius}. We first
define radial bins such that each bin has at least 250 stars in the
velocity range $v_\text{sys}-3\sigma_v,v_\text{sys}+3\sigma_v$. If the
last bin has less that 150 objects, the last two bins are
merged. After this, we fit our parametric model for the galaxy plus
foreground for each radial bin, to derive new velocity
dispersions. Then for each bin we do a $3\sigma$ clipping on the
velocity using the new velocity dispersion, and from this selection we
calculate the second and fourth moments. The errors on the moments are
computed using Eqs.~(17) and (19) in
\citet{Breddels2012arXiv}. The second moment and the kurtosis\footnote{The kurtosis is defined as $\gamma_2 = \mu_4/\mu_2^2$, where $\mu_4$ is the fourth and $\mu_2$ is the second moment of the line of sight velocity distribution.} are
shown in Fig. \ref{fig:moments} for each galaxy, where the black dot
corresponds to the mean, and the error bars indicate the 1$\sigma$
error bar. The blue region shows the confidence interval for the NFW
model found in \S \ref{sec:results_schw}.
For the photometry we use analytic fits given by various literature
sources as listed in Table \ref{tab:gal_params}. Although the stellar
mass is sub-dominant in the potential, we do include its contribution
in the dynamic models and fix $M/L_V=1$, as in
\citet{Breddels2012arXiv}.
\section{Methods}
\label{sec:method}
\subsection{Dynamical models}
Our aim is to compare different models to establish what type of dark
matter profile best matches the kinematical data of local dSph
galaxies. Here we consider the following profiles to describe the
dark matter halos of the dwarfs in our sample:
\begin{align}
\rho(r) &= \frac{\rho_0}{ x \left(1+x \right)^2}, && \text{NFW}\\
\rho(r) &= \frac{\rho_0}{\left(1+ x^\gamma \right)^{\beta/\gamma}}, && \text{(cored) $\beta\gamma$-profile} \\
\rho(r) &= \rho_0 \exp \left( -\frac{2}{\alpha'} \left( x^{\alpha'} -1 \right) \right), && \text{Einasto}
\end{align}
where $x=r/r_s$ and $r_s$ is the scale radius. Each model has at least
two unknown parameters $r_s$ and $\rho_0$. As we did in
\citet{Breddels2012arXiv}, we transform these two parameters to $r_s$
and \ensuremath{\text{M}_{1\text{kpc}}}\xspace (the mass within 1 kpc). As discussed in the Introduction,
the NFW and Einasto models are known to fit the halos dark matter distributions extracted from
cosmological N-body simulations. On the other hand, we explore the $\beta\gamma$ models
to test the possibility of a core in the dark halo. Note that, in comparison to the NFW profile, the Einasto
model has one extra parameter ($\alpha'$), but here we consider only
two values for $\alpha'=0.2, 0.4$ to cover the range suggested by
\citet{VeraCiro2013MNRAS}. On the other hand, the $\beta\gamma$
profiles have two extra parameters, but we limit ourselves here to two
different outer slopes ($\beta=3,4$) and two different transition
speeds between the inner and the outer slopes ($\gamma=1,2$). Note that the $\beta\gamma$
models have a true core only for $\gamma > 1$, however in all cases the central logarithmic slope vanishes,
$d \log \rho/d\log r = 0$. However, we loosely refer to these models as cored in what follows.
Note that, with these choices, all of our profiles
ultimately have just two free parameters. The list of models explored and their
parameters are summarized in Table
\ref{tab:model_names}.
\begin{table}
\centering
\begin{tabular}{l|c|l}
Name & Fixed parameters & Free parameters \\
\hline
NFW & - & \ensuremath{\text{M}_{1\text{kpc}}}\xspace, $r_s$ \\
core13 & $\beta=3,\gamma=1$ & \ensuremath{\text{M}_{1\text{kpc}}}\xspace, $r_s$ \\
core14 & $\beta=4,\gamma=1$ & \ensuremath{\text{M}_{1\text{kpc}}}\xspace, $r_s$ \\
core23 & $\beta=3,\gamma=2$ & \ensuremath{\text{M}_{1\text{kpc}}}\xspace, $r_s$ \\
core24 & $\beta=4,\gamma=2$ & \ensuremath{\text{M}_{1\text{kpc}}}\xspace, $r_s$ \\
einasto.2 & $\alpha'=0.2$ & \ensuremath{\text{M}_{1\text{kpc}}}\xspace, $r_s$ \\
einasto.4 & $\alpha'=0.4$ & \ensuremath{\text{M}_{1\text{kpc}}}\xspace, $r_s$ \\
\end{tabular}
\caption{Model names and their characteristic parameters of the various dark matter density profiles explored.}
\label{tab:model_names}
\end{table}
The orbit-based dynamical (Schwarzschild) models of each dwarf galaxy
are obtained as follows \citep[see][for a more detailed
description]{Breddels2012arXiv}. For each of the dark halo profiles,
with its own set of parameters, we integrate a large number of orbits
in the respective gravitational potential (including also the
contribution of the stars). We then find a linear combination of
these orbits that fits both the light and the kinematics. The orbital
weights found in this way have a physical meaning and can be used to
obtain the distribution function of the system. As data we have the
line of sight velocity moments (second and fourth depicted in
Fig. \ref{fig:moments}), and the light profile (Table
\ref{tab:data_params}). The best fit models (which give us the values of the
parameters for a specific dark matter halo profile) are those that
minimize the $\chi^2 = \chi^2_{\rm kin} + \chi^2_{\rm reg}$, under the
condition that the orbital weights are positive, and that the observed
light distribution is fit to better than 1\% at each radius. Here
$\chi^2_{\rm kin} = \sum_k (\mu_{2,k} -
\mu^{model}_{2,k})^2/{\rm var}(\mu_{2,k}) + \sum_k (\mu_{4,k} -
\mu^{model}_{4,k})^2/{\rm var}(\mu_{4,k})$. The $\chi^2_{\rm reg}$ is a
regularization term to make sure that the solution for the orbit
weights leads to a relatively smooth distribution function.
\citet{Breddels2012arXiv} calibrated the amplitude of this term for
Sculptor. To have the regularization term for the other dwarfs of the
same relative strength, we note that $\chi^2_\text{reg} \propto 1/N$,
where $N$ is the number of members with radial velocities, since the
$\chi_\text{kin}^2$ term also scales as $1/N$. Therefore, normalizing
its amplitude to that of Sculptor we get $\chi^2_\text{reg, dwarf} =
\chi^2_\text{reg, \text{Scl}} \times N_\text{Scl}/N_{\rm dwarf}$.
\begin{figure*}
\begin{tabular}{c @{}c@{} @{}c@{} @{}c@{} @{}c@{}}
\textbf{Fornax} & \textbf{Sculptor} & \textbf{Carina} & \textbf{Sextans}\\
\includegraphics[scale=0.6]{fig/pdf_2d_cdm_fnx} & \includegraphics[scale=0.6]{fig/pdf_2d_cdm_scl} & \includegraphics[scale=0.6]{fig/pdf_2d_cdm_car} & \includegraphics[scale=0.6]{fig/pdf_2d_cdm_sxt} & \rotatebox{90}{\hspace{1cm}\textbf{NFW/Einasto}}\\
\includegraphics[scale=0.6]{fig/pdf_2d_cored_fnx} & \includegraphics[scale=0.6]{fig/pdf_2d_cored_scl} & \includegraphics[scale=0.6]{fig/pdf_2d_cored_car} & \includegraphics[scale=0.6]{fig/pdf_2d_cored_sxt} & \rotatebox{90}{\hspace{2cm}\textbf{cored}}\\
\end{tabular}
\caption{Pdf for the two free parameters characterizing the dark halo
profiles for each dSph galaxy obtained using Schwarzschild
modeling. The top row shows the pdfs with NFW/Einasto models, the
bottom panel those for all cored models explored. The contours show
the 1 and 2$\sigma$ confidence levels (the 3$\sigma$ contour is not
shown to avoid crowding the image). \label{fig:pdfs}}
\end{figure*}
\subsection{Bayesian model comparison}
Background on Bayesian model comparison may be found in
\citet{Mackay_2003_information}. For completeness we discuss it here
briefly, but we assume the reader is familiar with the basics of Bayesian
inference.
Given the data $\mathcal{D}$ and assuming a model $M_i$, the posterior for the parameters $\theta_i$ of this model is:
\begin{equation}
p(\theta_i|\mathcal{D}, M_i) = \frac{p(\mathcal{D}|\theta_i, M_i) p(\theta_i|M_i)}{p(\mathcal{D}|M_i)}.
\end{equation}
The normalization constant $p(\mathcal{D}|M_i)$, also called the evidence, is of little interest in parameter inference, but is useful in Bayesian model comparison. To assess the probability of a particular model given the data, we find
\begin{equation}
p(M_i|\mathcal{D}) = \frac{p(\mathcal{D}|M_i)p(M_i)}{p(\mathcal{D})},
\end{equation}
where we see the evidence is needed. In this case $p(\mathcal{D})$ is the uninteresting normalization constant, as it cancels out if we compare two models:
\begin{equation}
\frac{p(M_i|\mathcal{D})}{p(M_j|\mathcal{D})} = \frac{p(\mathcal{D}|M_i)}{p(\mathcal{D}|M_j)}\frac{p(M_i)}{p(M_j)} = B_{i,j}\frac{p(M_i)}{p(M_j)}, \label{eq:bayes}
\end{equation}
where $B_{i,j}$ is called the Bayes factor. If we take the priors on the different models to be equal (i.e. $p(M_i)=p(M_j)$), the ratio of the evidence (the Bayes factor $B_{i,j}$) gives the odds ratio of the two models given the data $\mathcal{D}$.
Using these results we can perform model comparison between dark
matter density profiles, i.e. $\mathcal{M}=\{M_\text{nfw},
M_\text{Einasto}, ...\}$, and calculate for instance the odds that a
given galaxy is embedded in an NFW profile compared to an Einasto
model, $B_{\text{NFW},\text{Einasto}}$.
Not only can we do model comparison on a single object, but we may
also test if our objects share a particular model (e.g. they are all
embedded in NFW halos). If our dataset $\mathcal{D}$ consists of the
observations of two galaxies, i.e. $\mathcal{D} = \mathcal{D}_1 \cup \mathcal{D}_2$ and
assuming the datasets are uncorrelated and independent, we obtain:
\begin{equation}
\frac{p(M_i|\mathcal{D})}{p(M_j|\mathcal{D})} = \frac{p(\mathcal{D}_1|M_i)p(\mathcal{D}_2|M_i)}{p(\mathcal{D}_1|M_j)p(\mathcal{D}_2|M_j)}\frac{p(M_i)}{p(M_j)} = B_{i,j,1}B_{i,j,2}\frac{p(M_i)}{p(M_j)} \label{eq:bayes_mul}
\end{equation}
where each factor $p(\mathcal{D}_k|M_i)$ should be marginalized over its (own) characteristic parameters. From Eq.~(\ref{eq:bayes_mul}) we can see that the odds ratio of the models and Bayes factor from different measurements can be multiplied to give combined evidence for a particular model.
Behind each $p(M_i|\mathcal{D})$ is a set of orbit based dynamical
(Schwarzschild) models, obtained as described above. For each of the
models we calculate the evidence. Later on we compare each model's evidence to that
of an NFW profile, i.e.\ we compute the Bayes factor $B_{i,\text{NFW}}$,
where $i$ can be e.g.\ Einasto. By definition
$B_{\text{NFW},\text{NFW}}=1$, and again assuming equal priors on the
different models, the Bayes factor equals the odds ratio of the
models, such that for $B_{i,\text{NFW}} > 0$, model $i$ is favored
over an $\text{NFW}$ profile.
\section{Results}
\label{sec:apply}
\subsection{Schwarzschild models}
\label{sec:results_schw}
\begin{figure*}
\centerline{\includegraphics[scale=0.45]{fig/moments_fnx_bestfit}\includegraphics[scale=0.45]{fig/moments_scl_bestfit}}
\centerline{\includegraphics[scale=0.45]{fig/moments_car_bestfit}\includegraphics[scale=0.45]{fig/moments_sxt_bestfit}}
\caption{Similar to Fig. 2, except now we show the different best fit models for the various density profiles explored, which are indicated with different colors (the color scheme is the same as in Fig.~\ref{fig:pdfs}). \label{fig:moments_bestfit}}
\end{figure*}
\begin{figure*}
\centerline{\includegraphics[scale=0.6]{fig/evidence}}
\caption{Evidences for all models listed in Table \ref{tab:model_names}, relative to the NFW case. The last column shows the combined evidence for all galaxies together, and shows that the core23 and core24 are strongly disfavored.\label{fig:evidence}}
\end{figure*}
As a result of our Schwarzschild modeling technique, we obtain a two
dimensional probability density function (pdf) of the two parameters,
\ensuremath{\text{M}_{1\text{kpc}}}\xspace and $r_s$, for each galaxy and for each dark matter halo profile. In Fig. \ref{fig:pdfs} we plot the
pdf for the cored models and the NFW and Einasto models separately for
each galaxy. The colored dots correspond to the maximum likelihood for
each of the corresponding models as indicated by the legend. The
contours show the 1 and 2$\sigma$ equivalent confidence intervals
(the 3$\sigma$ contour is not shown for clarity). For both Fornax and
Sculptor the parameters for all profiles are relatively well
determined, while for Carina and Sextans this is less so. This can be
attributed to the difference in sample size (and hence to the smaller
number of members) in these systems, which has translated into fewer
bins where the moments can be computed (see
Fig. \ref{fig:moments}). In general for all four galaxies the scale
radius for the cored profiles is found to be smaller than that for the
NFW/Einasto profiles. We come back to this point in section \ref{sec:slope}.
Our model's masses at $r_{1/2}$, the 3d radius enclosing half of the stellar mass,
are compatible with those of \citet{Wolf2010MNRAS}. However, our results for Fornax do not agree
with those of \citet{Jardel2012}. These authors prefer a cored profile with a much
larger scale radius, and their enclosed mass is smaller in comparison to \citet{Wolf2010MNRAS}. We note that
this might be partly related to the fact that the amplitude of their line of sight
velocity dispersion profile (see their Fig. 2) is slightly lower than what we have determined here.
In Fig.~\ref{fig:moments_bestfit} we overlay on the kinematic observables the predictions
from the best fit Schwarzschild models. We note that all models provide very similar
and virtually indistinguishable fits, especially for the 2nd moment. Some slight differences
are apparent in the kurtosis, but in all cases, the differences are smaller than the error bars
on the moments.
In general we find all anisotropy profiles to be roughly constant with
radius and slightly tangentially biased on average. We do not find
significant differences between the profiles for cored and NFW models (the
reason for this will become clear in \S \ref{sec:slope}).
Fornax' s anisotropy $\beta \sim -0.2\pm0.2$, while Sculptor and Carina
have on average $\beta \sim -0.5\pm0.3$. For Sextans the anisotropy
cannot be determined reliably, $\beta \sim -0.3\pm0.5$. These values are
compatible with those of \citet{Walker2007ApJ}, which were derived using the spherical Jeans
equation assuming a constant velocity anisotropy profile.
\subsection{Bayesian comparison of the models}
\label{sec:bayes}
We compute the evidence relative to the NFW using Eq.~(\ref{eq:bayes})
by integrating over the parameters (in our case the scale radius and
the mass) the pdfs shown in Fig.~\ref{fig:pdfs}. We do this for each
dwarf galaxy and for all the models listed in Table
\ref{tab:model_names}. The different Bayes factors are shown in
Fig.~\ref{fig:evidence}. Each set of bars shows the Bayes factors for
the given dSph galaxy ($B_{i,NFW,k}$), while the last set shows the
combined result ($B_{i,NFW,\text{comb}} = \prod_k B_{i,NFW,k}$). We
note that an odds ratio between 1:2 till 1:3 is considered ``Barely
worth mentioning'' \citep{jeffreys1998theory}, and only odds ratios
above 1:10 are considered ``strong'' evidence.
For each galaxy there is hardly any evidence for or against an Einasto
profile (with $\alpha'=0.2,0.4$) compared to NFW. This is not
unexpected since these profiles are quite similar over a large region
\citep{VeraCiro2013MNRAS}. Also in the case of the combined evidence
the NFW and Einasto are hard to distinguish. Comparing the NFW or
Einasto profiles for individual galaxies to the cored models, one
cannot strongly rule out a particular model. For Fornax, Sculptor and
Carina, the $\gamma = 2$ models (where the transition speed is fast)
appear to be less likely, but this is not the case for Sextans. However, when we look
at the combined evidence, i.e. we explore whether all dwarfs are
embedded in the same halos, such $\gamma = 2$ models are clearly disfavored.
The results for Sculptor may be compared to those of
\citet{Breddels2012arXiv}. In that paper, the authors found that the maximum
likelihood value for the central slope of the density profile corresponded to
a cored model. Since the evidence is the integral of the pdf, and not
directly related to the maximum likelihood (except for a Gaussian
distribution), we should not be surprised to find a slightly stronger
evidence for the NFW case here. In any case, the differences between
the models are minor as shown graphically in
Fig. \ref{fig:moments_bestfit}, and the evidence and the maximum
likelihood (marginally) favoring different models can be
attributed simply to not being able to distinguish amongst these.
\newcommand{0.55}{0.55}
\begin{figure*}
\begin{tabular}{c @{}c@{} @{}c@{} @{}c@{} @{}c@{}}
& \textbf{Fornax} & \textbf{Sculptor} & \textbf{Carina} & \textbf{Sextans}\\
& \includegraphics[scale=0.55]{fig/mass_profiles_fnx} & \includegraphics[scale=0.55]{fig/mass_profiles_scl} & \includegraphics[scale=0.55]{fig/mass_profiles_car} & \includegraphics[scale=0.55]{fig/mass_profiles_sxt}
\end{tabular}
\caption{{\bf Top row}: Enclosed mass as a function of radius for the different dark matter density profiles, with the stellar component in black. {\bf Middle row}: Logarithmic density slope as a function or radius, where the black curve corresponds again to the stellar component. The red dashed line indicates $r_{-3}$, the radius at which the light profile has a logarithmic slope of $-3$, while the black line indicates $r_{1/2}$, the radius at which half of the stellar mass in enclosed (in 3d). {\bf Bottom row}: Cumulative density distribution of the (2d) radial distribution of the data (black), and the light (red) showing the kinematic data is sampled more concentrated towards the center. \label{fig:mass_profiles}}
\end{figure*}
\subsection{A robust slope measurement}
\label{sec:slope}
We now inspect in more detail the shape of the mass distributions found for the various
best fitting models. We are interested in exploring why the differences between the various
models as small as apparent in Fig.~\ref{fig:moments_bestfit}.
The top row of Fig.~\ref{fig:mass_profiles} shows the enclosed
dark matter mass for the best fit models (indicated by the solid dots
in Fig.~\ref{fig:pdfs}) for each galaxy separately. We use the same
color coding as in Fig.~\ref{fig:pdfs}, and also include the stellar
mass in black. The red-dashed vertical lines denote $r_{-3}$, the
radius at which the light density profile has a logarithmic slope of $-3$,
while the black line indicates $r_{1/2}$. This
remarkable figure shows that for each galaxy there is a region where
the mass distributions are truly almost indistinguishable from one
another. The different profiles, each characterized by its own
functional form, scale radius $r_s$ and mass $M_{\rm 1kpc}$, conspire to
produce a unique mass distribution. This region extends from slightly
below $r_{-3}$ to approximately the location of the outermost data
point (see bottom panel). Here $M(r) \propto r^x$, where $x$ ranges from $1.65$ for Fnx, to $1.9$ for Sextans.
In the middle row of Fig.~\ref{fig:mass_profiles} we plot the
logarithmic slope of the dark halo density distribution, where the
black line denotes the stellar density. Near the position where the
logslope of the stellar density is $-3$, all the best fit dark matter
density profiles seem to reach a similar logslope, although the value
of the slope varies from galaxy to galaxy.
The radius where the
logslopes coincide lies, as expected, inside the region where the mass distribution
is well determined, since both quantities are related through
derivatives.
To illustrate the distribution of the kinematic sample with respect to
the light, we plot in the bottom row of Fig. \ref{fig:mass_profiles} the cumulative 2d radial
distribution of the kinematic data in black. The cumulative 2d radial
distribution for the light is plotted as the red histogram. All
kinematic datasets are more concentrated than the light, but no clear
trend is visible between the distribution of the kinematic sample with
respect to the light, and the exact location where the logslope of
most accurately determined.
The existence of a finite region where the mass is more accurately
determined has also been observed in the literature in works using MCMC in combination with Jeans
modeling. For example, it is visible in e.g. the
right panel of Fig 1. in \citet{Wolf2010MNRAS}, Fig. 18 in
\citet{Walker2012arXiv}, and Fig. 10 in \cite{Jardel2013ApJ} for
Draco, in the case of a non-parametric density distribution with
Schwarzschild models.
\newcommand{0.6}{0.6}
\begin{figure*}
\begin{tabular}{c @{}c@{} @{}c@{} @{}c@{} @{}c@{}}
& \textbf{Fornax} & \textbf{Sculptor} & \textbf{Carina} & \textbf{Sextans}\\
& \includegraphics[scale=0.6]{fig/pdf-r3_fnx} & \includegraphics[scale=0.6]{fig/pdf-r3_scl} & \includegraphics[scale=0.6]{fig/pdf-r3_car} & \includegraphics[scale=0.6]{fig/pdf-r3_sxt}
\end{tabular}
\caption{Similar to Fig. \ref{fig:pdfs}, except now using $M_{-3}$ and $r_{-3}$ as parameters. Note that the contours for the NFW cannot go beyond $\kappa(r_{-3}) \ge -1$. \label{fig:pdfrThree}}
\end{figure*}
The analysis of \citet[][]{Wolf2010MNRAS} used the light weighted
average of the velocity dispersion to relate the radius at which the
logslope of the light is $-3$, or the half light radius, to the point
where the mass is accurately (being independent on the anisotropy) and
precisely (showing the least uncertainty) determined. Our findings go beyond this result. They
suggest that whatever dynamical model or
method is explored, there is a better set of parameters to describe
the mass distribution of dSph galaxies. Let $r_{-3}$ be the radius at
which the logslope of the (3d) light distribution is $-3$. Since the
mass is accurately determined in this region, a natural parameter
would be $M_{-3}=M(r_{-3})$. And since also the logslope at this
radius is accurately determined, the next parameter should be
$\kappa_{-3} = \frac{\text{d}\log \rho}{\text{d}\log r} |_{r=r_{-3}}$.
For any general model, if the values of $\beta$ and $\gamma$ are fixed, this effectively makes $r_s$ a function of
$\kappa_{-3}$.
Fig.~\ref{fig:pdfrThree} shows the pdf for the $M_{-3}$ and
$\kappa_{-3}$ parameters for both the NFW and core13 models for each
galaxy, assuming a flat prior on these parameters in the domain shown
in this figure, except for the NFW profile which we limit to
$\kappa_{-3}=-1.05$, since for $\kappa_{-3} \ge -1$ the scale radius
is unphysical. As can be seen from the pdf while there is still
uncertainty associated to the logslope at this radius, the value is
nearly model independent and therefore we believe this value to be
accurate, especially for Fornax and Sculptor ($\kappa_{-3}=-1.4\pm0.15$ and $\kappa_{-3}=-1.3\pm0.12$ respectively for the NFW model).
Note also that some uncertainties might arise because the
kinematics are not sampled exactly according to the light.
These results also help us understand why we found that the scale radii of the best fitting
NFW profiles always to be larger than those of the cored models (see Fig.~\ref{fig:pdfs}).
For the NFW, we have
\begin{equation}
\kappa(r) = \frac{{\rm d} \log \rho(r)}{{\rm d} \log r} = \frac{-2 r}{r + r_s} - 1,
\end{equation}
which can be easily solved for $r_s$:
\begin{equation}
r_{s,\text{nfw}} = -r \frac{\kappa + 3}{\kappa + 1}.
\end{equation}
A similar solution can be found for the other parametric models, for instance the $\gamma\beta$ model gives:
\begin{equation}
r_{s,\gamma\beta} = r \left( \frac{-\kappa}{\beta+\kappa} \right)^{-1/\gamma}.
\end{equation}
If we now require that the slopes are the same at $r_{-3}$ for the NFW and core13 models, we find
\begin{equation}
\frac{r_{s,\text{nfw}}}{r_{s,\text{core13}}} = \frac{\kappa}{\kappa+1},
\end{equation}
which is $> 1$ for $\kappa < -1$, explaining why the cored profiles
have smaller scale radii than the NFW profile, i.e. to get the same
logslope at the same location for the cored models, their scale radius
needs to be smaller than that of the NFW profile. A similar result holds for
the other cored models.
\section{Conclusions}
\label{sec:end}
In this paper we have presented a comparison of dynamical models using
different dark matter profiles for four dwarf spheroidal galaxies in
the Local Group, namely Fornax, Sculptor, Carina and Sextans. The
model comparison was done using Bayesian evidence. We have found that
no particular model is significantly preferred, and that all four
dwarf spheroidals are compatible with either NFW/Einasto or any of the
explored cored profiles. Only Sextans shows a slight preference for
cored models, but not with high odds. Nonetheless, we find that it is
very unlikely that all four dwarf spheroidals are each embedded in a
cored dark matter halo of the form $\rho_{DM} \propto 1/(1 +
r^2)^{\beta/2}$, with $\beta = 3, 4$.
Our best fit models however, conspire to produce the same mass
distribution over a relatively large range in radii, from $r_{-3}$
up to the last measured data point (which is often close to the nominal
tidal radius obtained from fitting the light profile). This $M(r) \sim
r^{x}$, with $x = 1.65-1.9$, is similar to that suggested by \citet{Walker2009} albeit with a slightly
steeper exponent. Another (related) quantity that is robustly
determined and independent of the assumed dark matter density profile,
is the logslope of the density distribution at $r_{-3}$. We find for
the dwarfs in our sample, that this slope ranges from $\kappa_{-3} \sim
-1.4$ at $r_{-3} = 0.96$ kpc for Fornax, to $\kappa_{-3} \sim -1.1$ at $r_{-3} =
0.98$ kpc for Sextans.
These findings can be seen as an extension of the results of
\citet{Wolf2010MNRAS}, who showed that the mass at $r_{-3}$ can be
determined very accurately in a model independent fashion. These
authors demonstrated that this result might be understood from the
Jeans equation. Although we do not have yet a solid mathematical
explanation for our new findings, we suspect that this might be
obtained using the virial theorem, which is effectively another, yet
independent moment of the collisionless Boltzmann equation.
In the near future, we will apply Schwarzschild modeling to the same
data but instead of the moments, we will use the discrete
individual measurements directly. This approach should allows us to
get the most out of the data, since no information is lost. When
binning, one loses spatial resolution, but also the higher moments of
the line-of-sight velocity distribution are not included in the
fitting procedure because of their large and asymmetric errors.
Furthermore, the use of the full line-of-sight velocity distribution
should improve the precision of the anisotropy profile, which may be
an interesting quantity to discriminate formation scenarios.
This moments-to-discrete modeling step must be carried out before
deciding if and how much more data is needed to discriminate among
various dark matter density profiles. Nonetheless, we have learned
here that the functional form of the mass distribution may be
determined over a large distance range, even when only a few hundred
velocity measurements are available (as in the case of
Sextans). However, the uncertainty on the value of the exact slope of
the density profile at e.g. $r_{-3}$ is driven by the sample size.
An obvious next step is to establish if the subhalos extracted from
cosmological simulations have the right characteristics to host the
dSph of the Milky Way, now that not only the mass, but also its
functional form (1st and 2nd derivatives), of their dark halos have
been determined reliably.
\section*{Acknowledgments}
We are grateful to Giuseppina Battaglia, Glenn van de Ven and Remco van den Bosch for
discussions that led to the work presented here. We acknowledge financial support from
NOVA (the Netherlands Research School for Astronomy), and
European Research Council under ERC-StG grant GALACTICA-24027.
|
1304.3340
|
\section{Introduction}
Several criteria to detect non-classicality of quantum states of a
harmonic oscillator have been introduced, mostly
based on phase-space distributions
\cite{Wig32,Gla63,Sud63,CTL91,CTL92,Arv97,Arv98,Mar01,Par01,Dod02,Ken04},
ordered moments \cite{Shc05,Kie08,Vog08}, or on information-theoretic arguments
\cite{Sim00,Dua00,Mar02,Gio10,Fer12,Geh12,Por12}. At the same time, an
ongoing research line addresses the characterization of quantum states
according to their Gaussian or non-Gaussian character
\cite{nonGHS,nonGRE,nonGL,barbieri10,Rad11,Jez11,Jez12,predojevic12,Rad13,Dau10},
and a question arises on whether those two
different hierarchies are somehow linked each other. \par
As a matter of
fact, if we restrict our attention to pure states, Hudson's theorem
\cite{hudson74,soto83} establishes that the border between Gaussian and
non-Gaussian states coincides exactly with the one between states with
positive and negative Wigner functions. However, if we move to mixed
states, the situation gets more involved. Attempts to extend
Hudson's theorem have been made, by looking at upper bounds on
non-Gaussianity measures for mixed states having positive Wigner
function \cite{mandilara09}. In this framework, by focusing on states
with positive Wigner function, one can define an additional border
between states in the {\em Gaussian convex hull} and those in the
complementary set of {\em quantum non-Gaussian states}, that is, states
that can not be expressed as mixtures of Gaussian states. The situation
is summarized in Fig. \ref{f:PosWigner}: the definition of the
Gaussian convex hull generalizes the notion of Glauber's
non-classicality \cite{titulaer65}, with coherent states replaced by
generic pure Gaussian states, i.e. squeezed coherent states.
\par
Quantum non-Gaussian states with positive Wigner function are not useful
for quantum computation \cite{mari12,veitch13}, and are not necessary for
entanglement distillation, {\em e.g.} the non-Gaussian entangled resources
used in \cite{heersink06} are mixtures of Gaussian states. On the other
hand, they are of fundamental interest for quantum information
and quantum optics. In particular, since no negativity of the
Wigner function can be detected for optical losses higher than $50\%$
\cite{cahill69} (or equivalently, for detector efficiencies below $50\%$)
criteria able to detect quantum non-Gaussianity are needed in order to
certify that a {\em highly non linear} process (such as Fock state generation,
Kerr interaction, photon addition/subtraction operations or conditional
photon number detections) has been implemented in a noisy environment, even
if no negativity can be observed in the Wigner function.
\par
Different measures of non-Gaussianity for quantum states have been
proposed \cite{nonGHS,nonGRE,nonGL}, but these cannot
discriminate between quantum non-Gaussian states and mixtures of
Gaussian states. An experimentally friendly criterion for quantum
non-Gaussianity, based on photon number probabilities, has been
introduced \cite{Rad11}, and then employed
in different experimental settings to prove the generation of quantum
non-Gaussian states, such as heralded single-photon states
\cite{Jez11}, squeezed single-photon states \cite{Jez12} and Fock
states from a semiconductor quantum dot \cite{predojevic12}.
\begin{figure}[t!]
\includegraphics[width=0.95\columnwidth]{f1_wigs.pdf}
\caption{Venn diagram description for continuous-variable quantum states
with positive Wigner function. The quantum states can be divided in two
sets: {\em quantum non-Gaussian} states and states belonging to the
Gaussian convex hull. The latter trivally includes (Glauber) classical
states and Gaussian states. \label{f:PosWigner}} \end{figure}
\par
In this paper we introduce a family of criteria which are able to detect
quantum non-Gaussianity for single-mode quantum states of a harmonic
oscillator based on the Wigner function.
As we already pointed out, according to Hudson's theorem, the only pure states having a
positive Wigner function are Gaussian states. One can then wonder if any
bound exists on the values that the Wigner function of convex mixtures
of Gaussian states can take. By following this intuition we
present several bounds on the values of the Wigner function for
convex mixtures of Gaussian states, consequently defining a class of sufficient
criteria for quantum non-Gaussianity.
\par
In the next section we will introduce some notation and the preliminary
notions needed for the rest of the paper. In Sec. \ref{s:criteria} we
will prove and discuss our Wigner function based criteria for quantum
non-Gaussianity and in Sec. \ref{s:examples} we will prove their
effectiveness by considering different families of non-Gaussian states
evolving in a lossy (Gaussian) channel. We will conclude the paper in
Sec. \ref{s:conclusions} with some remarks.
\section{Preliminary notions} \label{s:preliminary}
Throughout the paper we
will use the quantum optical terminology, where excitations of a quantum
harmonic oscillator are called photons. All the results can be naturally
applied to any bosonic continuous-variable (CV) system. We will
consider a single mode described by a mode operator $a$, satisfying the
commutation relation $[a,a^\dag] = \mathbbm{1}$. A quantum state
$\varrho$ is fully described by its
characteristic function \cite{cahill69}
\begin{align}
\chi[\varrho](\gamma) = {\rm Tr}[\varrho D(\gamma)]\:,
\end{align}
where $D(\gamma) = \exp\{\gamma a^\dag - \gamma^* a \}$ represents the displacement
operator. In addition, the quantum state $\varrho$ can be fully described by the
Fourier transform of the characteristic function, {\em i. e.} the Wigner
function \cite{cahill69}
\begin{align}
W[\varrho](\alpha) = \int \frac{d^2 \gamma}{\pi^2} e^{\gamma^* \alpha - \gamma \alpha^*}
\chi[\varrho](\gamma)\:.
\end{align}
A state is defined to be {\em Gaussian} if and only if its Wigner function
(or equivalently its characteristic function) is Gaussian. All
single-mode Gaussian states can be expressed as $$\varrho= D(\alpha)
S(\xi) \nu_\beta S^\dag(\xi) D^\dag(\alpha)\,,$$ where
$
S(\xi) = \exp\left\{ \frac12 \xi (a^\dag)^2 - \frac12 \xi^* a^2 \right\}
$,
is the squeezing operator, and
$\nu_\beta = e^{-\beta a^\dag a}/{\rm Tr}[ e^{-\beta a^\dag a}]$ is a
thermal state ($\alpha, \xi \in \mathbbm{C}$ and $\beta>0$).
Pure Gaussian states can be written as $|\psi_{\sf G}\rangle =
D(\alpha) S(\xi)|0\rangle$, and, according to Hudson's theorem \cite{hudson74,soto83},
they are the only pure states having a positive Wigner function.
Together with that of a Gaussian state, one can define the concept of {\em Gaussian
map}: a quantum (completely positive)
map is defined Gaussian iff it transforms Gaussian states
into Gaussian states. All unitary Gaussian maps can be expressed as
$U_{\sf G} = \exp\{- i H_{\sf bil} t\}$, and they correspond to
Hamiltonian operators $H_{\sf bil}$ at most bilinear in the mode
operators. Similarly, a generic Gaussian map can be decomposed as a
Gaussian unitary acting on the system plus an ancilla (the latter
prepared in a Gaussian state), followed by partial tracing over the
ancillary mode \cite{eisert_wolf}.
\par
Another complete description of a CV quantum state $\varrho$ may be
given in terms of the so-called
$P$-function $P[\varrho](\alpha)$ \cite{cahill69}, defined
implicitly via the formula
\begin{align}
\varrho = \int d^2\alpha\: P[\varrho](\alpha)
|\alpha\rangle\langle \alpha|\:, \label{eq:pfunction}
\end{align}
where $|\alpha\rangle = D(\alpha) |0\rangle$ represents a
coherent state. According to Glauber a state
is {\em non-classical} iff its $P$-function is not a proper probability
distribution, {\em e.g.} the $P$-function is more singular than a
Dirac-delta function. Note that the negativity of the Wigner function
is a more restrictive definition of non-classicality: there exists
non-classical states having a positive Wigner function ({\em e.g.} squeezed states),
while all the states having a non-positive Wigner function are
non-classical according to Glauber.
\par
In a similar spirit as in Glauber's approach to
non-classicality, in this paper we study the concept of
{\em quantum non-Gaussian} states. These are defined as
follows. The Gaussian convex hull is the set of states
\begin{align}\label{eq:Ghull}
\mathcal{G} = \left\{ \varrho \in \mathcal{H} \: \lvert \: \varrho = \int d{\boldsymbol \lambda} \: p({\boldsymbol \lambda}) \: |\psi_{\sf G}({\boldsymbol \lambda})\rangle\langle\psi_{\sf G}({\boldsymbol \lambda})| \right\} \:,
\end{align}
where $\mathcal{H}$ denotes the Hilbert space of continuous-variable quantum states,
$p({\boldsymbol \lambda})$ is a proper probability
distribution and $|\psi_{\sf G}({\boldsymbol \lambda})\rangle$ are pure
Gaussian states, i.e. , in the single mode case,
squeezed coherent states identified by the set of
parameters ${\boldsymbol \lambda}\equiv \{\alpha,\xi\}$.
Since Gaussian states do not form a convex set, the set in Eq.~\eqref{eq:Ghull}
includes states which are not Gaussian. Moreover any mixed Gaussian
state can be written as a weighted sum of pure Gaussian states, and hence
the set above also includes convex mixtures of mixed Gaussian states.
\par
The definition of {\em quantum non-Gaussianity} naturally follows:
\par\noindent
{\bf Definition}. {\em A quantum state $\varrho$ is quantum non-Gaussian
iff it is not possible to express it as a convex mixture of Gaussian
states, that is iff $\varrho \notin \mathcal{G}$}.
\par
As illustrated in Fig. \ref{f:PosWigner}, the border here defined dividing
quantum non-Gaussian states and mixtures of Gaussian states falls in
between the border dividing classical and non-classical states, and the
one which divides states with positive and non-positive Wigner functions. The
importance of such a further distinction is evident if we note that all
states in ${\cal G}$ can be prepared through a combination of Gaussian
operations and classical randomization.
On the contrary, if $\varrho\notin{\cal G}$ then a highly {\em non-linear} process
(due to a non-Gaussian operation or measurement) had necessarily taken place
in the generation the quantum state $\varrho$. While the negativity of
the Wigner function is always sufficient to certify it, more elaborated
criteria, as those elaborated in this paper, are needed in order to
detect such a characteristic when
quantum states exhibit a positive Wigner function.
\section{Criteria to detect quantum non-Gaussianity}\label{s:criteria}
In order to find criteria for the detection of
quantum non-Gaussian states, we follow the
intuition given by Hudson's theorem for pure Gaussian states. We will
focus on lower bounds on the values taken by the Wigner function of
states which belong to the Gaussian convex hull $\mathcal{G}$. In this
section we present our main findings as one lemma leading to two final
propositions and two additional corollaries. The `quantum
non-Gaussianity criteria', derived directly from these results, are
presented at the end of the section.
\begin{lemma}[Lower bound on the Wigner
function at the origin of phase space for a pure Gaussian state]\label{l:boundpure}
For any given pure single-mode Gaussian state $|\psi_{\sf G}\rangle$,
the value of the Wigner function at the origin of the phase space is bounded from below as
\begin{align}
W[|\psi_{\sf G}\rangle\langle \psi_{\sf G}|](0) \geq \frac2\pi \exp\{ -2 n (1+n)\} \:, \label{eq:boundpure}
\end{align}
where $n=\langle \psi_{\sf G}| a^\dag a |\psi_{\sf G}\rangle$.
\end{lemma}
\proof{A generic pure single-mode Gaussian state can be always written as $|\psi_{\sf G}\rangle =
D(\alpha) S(\xi) |0\rangle$, where $\alpha=|\alpha|e^{i\theta}$ and $\xi=r e^{i\phi}$ ($r>0$)
are two complex numbers. We can thus write the Wigner function evaluated in zero as
\begin{align}
W\left[\ket{\psi_g}\bra{\psi_g}\right]&(0)=\frac{2}{\pi}\exp\left\{-2|\alpha|^2 \left[\cosh{2r} \right.
\right.\nonumber \\
&\left.\left. -\cos{(2\theta+\phi)}\sinh{2r} \right] \right\} .
\label{eq:wigGausszero}
\end{align}
Our goal is to minimize the value of the Wigner function or equivalently to maximize
the function
\begin{align}
f(\alpha,\xi)&=2|\alpha|^2 \left(\cosh{2r}-\cos{(2\theta+\phi)}\sinh{2r} \right).
\end{align}
A first maximization is obtained by considering
\begin{align}
2\theta+\phi=\pi + 2k\pi\:\:\:{\rm with}\:\: k\in \mathbbm{N}\:, \label{eq:phases}
\end{align}
which yields
\begin{align}
f(\alpha,\xi) \leq 2|\alpha|^2 e^{2 r} = 2n_d\left( 2n_s + 1 +2\sqrt{n_s(1+n_s)}\right).
\label{eq:max1}
\end{align}
In the last equation we introduced the displacement and squeezing photon numbers, $n_d = |\alpha|^2$ and $n_s =\sinh^2 r$, and we used the formula ${\rm arcsinh}(x)=\log(x+\sqrt{1+x^2})$. Note that these two parameters obey
$$
n=\langle \psi_{\sf G} | a^\dag a | \psi_{\sf G}\rangle =n_d + n_s,
$$
where $n$ is the average photon number of the state $| \psi_{\sf G}\rangle$. We can thus express the right hand side (rhs) of Eq. (\ref{eq:max1}) in terms of
$n$ and $n_s$, obtaining
\begin{align}
f(\alpha,\xi) \leq 2(n-n_s)\left(2n_s+1+2\sqrt{n_s(1+n_s)}\right).
\end{align}
For a given average photon number $n$, the above function is maximized with regard to the parameter $n_s$ by choosing
\begin{align}
n_s = \frac{n^2}{1+2n}, \label{eq:optsq}
\end{align}
and obtaining
\begin{align}
f(\alpha,\xi) \leq 2n(1+n).
\end{align}
This leads to
\begin{align}
W[|\psi_{\sf G}\rangle\langle\psi_{\sf G}|](0) \geq \frac 2\pi \exp\left\{-2n(1+n)\right\} .
\end{align}
}
\par\noindent
By looking at the proof, we remark that the bound obtained is tight: given a fixed energy $n$, by choosing the phases according to condition (\ref{eq:phases}) and
the squeezing energy according to (\ref{eq:optsq}), it is always possible to find a family of pure Gaussian states saturating the inequality. In particular the maximization obtained via condition (\ref{eq:phases}) simply corresponds, at fixed $n_d$ and $n_s$, to displace the state along the direction of the squeezed quadrature. The condition (\ref{eq:optsq}) shows that for small values of $n$, the minimum of the Wigner function is obtained by using the energy in displacement, while for larger values of $n$, the optimal squeezing fraction $n_s$ tends to an asymptotic value $n_s^{\sf (as)} =n/2$.\\
Let us now generalize the bound obtained to a generic convex mixture of Gaussian states.
\begin{proposition}[Lower bound on the Wigner function at the origin for a convex mixture of Gaussian states]\label{p:boundmix}
For any single-mode quantum state $\varrho$ which belongs to the Gaussian convex hull
$\mathcal{G}$, the value of the Wigner function at the origin is bounded by
\begin{align}
W[\varrho](0) \geq \frac2\pi \exp\{ -2 \bar{n} (1+\bar{n})\}
\label{eq:boundmix}
\end{align}
where $\bar{n}= {\rm Tr}[\varrho a^\dag a]$.
\end{proposition}
\proof{The multi-index ${\boldsymbol \lambda}$, which labels every Gaussian state in the convex mixture $|\psi_{\sf G}({\boldsymbol \lambda})\rangle = D(\alpha) S(\xi) |0\rangle$, contains the information about the squeezing $\xi$ and displacement $\alpha$. We can then equivalently consider as variables ${\boldsymbol \lambda}=\{n,n_s, \theta,\phi\}$. By exploiting the linearity property of the Wigner function we obtain
\begin{align}
W[\varrho](0) &= \int d{\boldsymbol \lambda} \: p({\boldsymbol \lambda}) W[|\psi_{\sf G}({\boldsymbol \lambda})\rangle\langle\psi_{\sf G}({\boldsymbol \lambda})|](0) \nonumber \\
&\geq \frac 2\pi \int d{\boldsymbol \lambda} \: p({\boldsymbol \lambda}) \exp \{ -2n(1+n) \} \:, \label{eq:integral}
\end{align}
where inequality (\ref{eq:boundpure}) has been used. By defining
\begin{align}
\widetilde{p}(n) = \int_0^{n} dn_s \int_0^{2\pi} d\phi \int_0^{2\pi} d\theta \: p({\boldsymbol \lambda}),
\end{align}
which is a valid probability distribution with respect to the variable $n$, Eq. (\ref{eq:integral}) becomes\begin{align}
W[\varrho](0) \geq \frac 2\pi \int_0^{\infty} dn \: \widetilde{p}(n) \: \exp \{ -2n(1+n)\}.
\end{align}
Studying the second derivative of
\begin{align}
B_{\sf min}(n) = \frac 2 \pi \exp\{ -2n(1+n)\} \:,
\end{align}
we conclude that the function is {\it convex} in the whole {\em physical} region ({\em i.e.} $n\geq0$).
As a consequence,
\begin{align}
\int_0^{\infty} dn \: \widetilde{p}(n) \: B_{\sf min} (n) \geq B_{\sf min} \left(\int_0^{\infty}
dn\: \widetilde{p}(n)\: n \right) = B_{\sf min}(\bar{n})
\end{align}
where $\bar{n}=\int_0^{\infty} dn\: \widetilde{p}(n)\: n = {\rm Tr}[\varrho a^\dag a]$. From
the last inequality we obtain straightforwardly the thesis
\begin{align}
W[\varrho](0) \geq \frac 2\pi \exp\{ -2 \bar{n} ( 1+\bar{n}) \} \:.
\end{align}
}
\par\noindent
The following proposition generalizes the bound obtained above.
\begin{proposition}\label{p:bound2mix}
For any single-mode quantum state $\varrho \in \mathcal{G}$, and for any given Gaussian map $\mathcal{E}_{\sf G}$ (or alternatively a convex mixture thereof), the following inequality holds
\begin{align}
W[\mathcal{E}_{\sf G}(\varrho)](0) \geq \frac2\pi \exp\{ -2 \bar{n}_\mathcal{E} (1+\bar{n}_\mathcal{E})\}\:, \label{eq:bound2mix}
\end{align}
where $\bar{n}_\mathcal{E} = {\rm Tr}[\mathcal{E}_{\sf G}(\varrho) a^{\dag}a]$.
\end{proposition}
\proof{Given a quantum state $\varrho$ which can be expressed as a mixture of Gaussian state, and a Gaussian map $\mathcal{E}_{\sf G}$ (or a convex mixture thereof), the output state
\begin{align}
\varrho^\prime = \mathcal{E}_{\sf G} (\varrho)
\end{align}
can still be expressed as a mixture of Gaussian states. As a consequence we can apply to the state $\varrho^\prime$ the result in Proposition \ref{p:boundmix}, obtaining the thesis.}
\\
\par\noindent
Proposition \ref{p:bound2mix} leads to two corollaries that will be used in the rest of the paper.
\begin{corollary}
For any single-mode quantum state $\varrho \in \mathcal{G}$,
the following inequality holds
\begin{align}
W[\varrho](\beta) \geq \frac2\pi \exp\{ -2 \bar{n}_\beta (1+\bar{n}_\beta)\}\:, \:\: \forall \beta \in \mathbbm{C} \:, \label{eq:boundDISP}
\end{align}
where $\bar{n}_\beta = {\rm Tr}[\varrho D(\beta) a^\dag a D^\dagger(\beta)]$.
\end{corollary}
\proof{
The proof is straightforward from Proposition \ref{p:bound2mix} with the Gaussian map
$\mathcal{E}_{\sf G}(\varrho) = D(-\beta)\varrho D^\dagger(-\beta)$. We also use the
property of the Wigner function
$$
W[\varrho](\beta) = W[D^\dag(\beta)\varrho D(\beta)](0) \:,
$$
and $D^\dagger(\beta)=D(-\beta)$.}
\begin{corollary}
For any single-mode quantum state $\varrho$ belonging to the Gaussian convex hull
$\mathcal{G}$,
the following inequality holds
\begin{align}
W[\varrho](0) \geq \max_{\xi \in \mathbbm C} \left( \frac2\pi \exp\{ -2 \bar{n}_\xi (1+\bar{n}_\xi)\} \right)\:, \label{eq:boundSQ}
\end{align}
where $\bar{n}_\xi = {\rm Tr}[ \varrho S^\dagger(\xi) a^\dag a S(\xi)]$.
\end{corollary}
\proof{The proof follows from Proposition \ref{p:bound2mix}, by
considering the Gaussian map $\mathcal{E}_{\sf G}(\varrho) = S(\xi) \varrho S^\dag(\xi)$.
Moreover, since the value of the Wigner function at the origin is invariant under any
squeezing operation, {\em i.e.}
\begin{align}
W[S(\xi)\varrho S^\dag(\xi)](0) = W[\varrho](0) \:,
\end{align}
one can maximize the rhs of
inequality (\ref{eq:bound2mix}) with regard to the squeezing parameter $\xi$.}\\
\par\noindent
The violation of any of the inequalities presented in the last two propositions and two corollaries provides a {\it sufficient} condition to conclude that a state is quantum non-Gaussian. We formalize this by re-expressing the previous results in the form of two criteria for the detection of quantum non-Gaussianity.
\begin{criterion}\label{c:uno}
Let us consider a quantum state $\varrho$ and define the quantity
\begin{align}
\Delta_1[\varrho] = W[\varrho](0) - \frac2\pi \exp\{-2 \bar{n}(\bar{n}+1)\} \:. \label{eq:Delta1}
\end{align}
Then,
\begin{align}
\Delta_1[\varrho] < 0 \:\: \Rightarrow\:\: \varrho \notin \mathcal{G}, \nonumber
\end{align}
that is, $\varrho$ is quantum non-Gaussian.
\end{criterion}
\begin{criterion}\label{c:due}
Let us consider a quantum state $\varrho$, a Gaussian map $\mathcal{E}_{\sf G}$ (or a convex mixture thereof), and define the quantity
\begin{align}
\Delta_2[\varrho,\mathcal{E}_{\sf G}] = W[\mathcal{E}_{\sf G}(\varrho)](0) - \frac2\pi \exp\{-2 \bar{n}_\mathcal{E}(\bar{n}_\mathcal{E}+1)\}\:. \label{eq:Delta2}
\end{align}
Then,
\begin{align}
\exists\, \mathcal{E}_{\sf G}\:\: {\rm s.t.} \:\: \Delta_2[\varrho,\mathcal{E}_{\sf G}] < 0
\Rightarrow \: \varrho \notin \mathcal{G}. \nonumber
\end{align}
\end{criterion}
\par
Typically, Criterion \ref{c:uno} can be useful to detect quantum non-Gaussianity of phase-invariant states having the minimum of the Wigner function at the origin of phase space. On the other hand, Criterion \ref{c:due} is of broader applicability. To give two paradigmatic examples, the latter criterion can be useful if: (i) the minimum of the Wigner function is far from the origin, so that one may be able to violate inequality (\ref{eq:boundDISP}) by considering displacement operations; (ii) the state is not phase-invariant and presents some squeezing, and thus one may be able to violate inequality (\ref{eq:boundSQ}) by using single-mode squeezing operations.
\section{Violation of the criteria for non-Gaussian states evolving in a lossy Gaussian channel}\label{s:examples}
In this section we test the effectiveness of our criteria, by applying them to typical quantum states that are of relevance to the quantum optics community. We shall consider pure, non-Gaussian states evolving in a lossy channel, and test their quantum non-Gaussianity after such evolution. Specifically, we focus on the family of quantum channels associated to the Markovian master equation
\begin{align}\label{eq:Markov}
\frac{d\varrho}{dt} = \gamma a \varrho a^\dag - \frac\gamma 2\left(a^\dag a \varrho + \varrho a^\dag a \right).
\end{align}
The resulting time evolution, characterized by the parameter $\epsilon = 1-e^{-\gamma t}$,
models both the incoherent loss of photons in a dissipative zero temperature environment, and
inefficient detectors with an efficiency parameter $\eta = 1-\epsilon$. The evolved
state $\mathcal{E}_\epsilon(\varrho_0)$ can be equivalently derived by
considering the action of a beam splitter with reflectivity $\epsilon$, which
couples the system to an ancillary mode prepared in a vacuum state.
The corresponding average photon number reads
\begin{align}
\bar{n}_\epsilon = {\rm Tr}[\mathcal{E}_\epsilon
(\varrho_0) a^\dag a] = (1-\epsilon)\: \bar{n}_0 \:,
\end{align}
where $\bar{n}_0 = {\rm Tr}[\varrho_0 a^\dag a]$ is the initial average photon number.
\\
It is well known that, for $\epsilon > 0.5$ ({\em i.e.} for detector efficiencies
$\eta<0.5$), no negativity of the Wigner function can be observed. We will focus
then on the violation of our criteria for larger values of $\epsilon$, which ensures that the evolved states have a positive Wigner function. \\
Notice that the quantum map $\mathcal{E}_\epsilon$
is a Gaussian map. As a consequence, by combining the divisibility property of the map (inherited from the Markovian structure of Eq.~\eqref{eq:Markov}) and Criterion \ref{c:due}, if a violation is observed for a given loss parameter $\bar{\epsilon}$, then the state is quantum non-Gaussian for any lower value $\epsilon \leq \bar{\epsilon}$ \cite{divisibility}. For this reason we will focus on the maximum values of the loss parameter $\epsilon$ for which
a violation of the bounds is observed, {\em i.e.}
\begin{align}
\epsilon_{\sf max}^{(1)}[\varrho] &= \max\{ \epsilon \: : \Delta_1[\mathcal{E}_\epsilon(\varrho)] \leq 0 \} \:, \\
\epsilon_{\sf max}^{(2)}[\varrho] &= \max\{ \epsilon \: :\exists\mathcal{E}_{\sf G}\text{ s.t.} \ \Delta_2[\mathcal{E}_\epsilon(\varrho),\mathcal{E}_{\sf G}] \leq 0 \} \:.
\end{align}
\\
In what follows, we start by focusing on Criterion \ref{c:uno}, and thus we will look for negative values of the non-Gaussianity indicator $\Delta_1[\varrho]$ defined in Eq. (\ref{eq:Delta1}).
We will consider different families of states, namely Fock states,
photon-added coherent states and photon-subtracted squeezed states.
In section \ref{s:violation2}, we will study how to improve the results obtained,
by considering the second criterion and thus by studying the
non-Gaussianity indicator $\Delta_2 [\varrho, \mathcal{E}_{\sf G}]$.
\subsection{Violation of the first criterion} \label{s:violation1}
\subsubsection{Fock states}
Let us start by considering Fock states $|m\rangle$, that is
the eigenstates of the number operator: $a^\dag a |m\rangle = m |m\rangle$.
A fock state evolved in a lossy channel can be written as a mixture
of Fock states as
\begin{equation}
\mathcal{E}_\epsilon (|m\rangle\langle m|) =\sum_{l=0}^{m}\alpha_{l,m}(\epsilon)\ket{l}\bra{l}\;,
\label{eq:FockEv}
\end{equation}
with
\begin{equation}
\alpha_{l,m}(\epsilon)=
\binom{m}{l}
(1-\epsilon)^{l}\epsilon^{m-l}\;.
\end{equation}
We recall here that the Wigner function at the origin is proportional
to the expectation value of the parity operator $\Pi = (-)^{a^\dag a}$,
that is
\begin{align}
W[\varrho] ( 0) =\frac2\pi {\rm Tr}[\varrho \Pi] = \frac2\pi \left(P_{\sf even} - P_{\sf odd} \right)\:,
\label{eq:evenodd}
\end{align}
where $P_{\sf even}$ ($P_{\sf odd}$) represents the probability of detecting
an even (odd) number of photons. By using Eq. (\ref{eq:FockEv}) one obtains
\begin{align}
W[\mathcal{E}_\epsilon(|m\rangle\langle m|](0) = \frac2\pi (2\epsilon -1)^m \:,
\end{align}
and thus the non-Gaussianity indicator reads
\begin{align}
\Delta_1[\mathcal{E}_\epsilon(|m\rangle\langle m|] = \frac2\pi\left\{ (2\epsilon -1)^m - e^{-2(1-\epsilon) m [ (1-\epsilon) m +1]} \right\} \:.
\end{align}
\begin{figure}[t!]
\includegraphics[width=0.495\columnwidth]{Fock1a.pdf}
\includegraphics[width=0.495\columnwidth]{Fock1b.pdf}
\caption{(Left) Non-Gaussianity indicator $\Delta_1[\mathcal{E}_\epsilon(|m\rangle\langle m|)]$ for the first
three Fock states: red-dotted line: $m=1$; green-dashed line: $m=2$; blue-solid line: $m=3$. \\
(Right) Maximum value of the noise parameter $\epsilon_{\sf max}^{(1)}$ such that the
bound (\ref{eq:boundmix}) is violated for the state $\mathcal{E}_\epsilon(|m\rangle\langle m|)$, as a function
of the Fock number $m$.
\label{f:Fock1}}
\end{figure}
The behavior of $\Delta_1[\mathcal{E}_\epsilon(|m\rangle\langle m|]$ as a function of $\epsilon$ for the first three Fock states is plotted in Fig. \ref{f:Fock1} (left panel).
One can observe that the criterion works really well for the
Fock state $|1\rangle$, which is proven to be quantum non-Gaussian for all values of $\epsilon <1$. For the Fock states $|2\rangle$ and $|3\rangle$, a non monotonous behavior of $\Delta_1$ is observed as a function of the loss parameter. Still, negative values of the non-Gaussian indicator are observed in the region of interest $\epsilon>0.5$. However, the maximum value of the noise parameter $\epsilon_{\sf max}^{(1)}$ decreases monotonically as a function of $m$, as shown in Fig. \ref{f:Fock1} (right panel). By increasing the Fock number $m$, it settles to the asymptotic value $\epsilon_{\sf max}^{(1)}\rightarrow 0.5$. As one would expect by looking at the bound in Eq. (\ref{eq:boundmix}), for high values of the average photon
number, the criterion becomes practically equivalent to the detection of negativity of the
Wigner function, and thus the maximum noise corresponds to $\epsilon=0.5$.
\subsubsection{Photon-added coherent states}
A \emph{photon-added coherent} (PAC) state is defined as
\begin{equation}
\ket{\psi_{\sf pac}}=\frac{1}{\sqrt{1+|\alpha|^2}}a^{\dagger}\ket{\alpha}.
\end{equation}
The operation of photon-addition has been implemented in different contexts
\cite{zavatta04,zavatta07,parigi07,zavatta09}, and in particular non-Gaussianity
and non-classicality of PAC states have been investigated in \cite{barbieri10}. \\
Being the non-Gaussianity indicator $\Delta_1[\varrho]$ phase insensitive, we can consider $\alpha \in \mathbbm{R}$
without loss of generality. The average photon number can be easily calculated
obtaining
\begin{align}
\bar{n}_0^{\sf (pac)} = \langle \psi_{\sf pac} | a^\dag a | \psi_{\sf pac} \rangle = \frac{\alpha^4 + 3 \alpha^2 + 1}{1+\alpha^2} \:,
\end{align}
while its Wigner function reads
\begin{align}
W[|\psi_{\sf pac}\rangle] (\lambda)&=\frac{2}{\pi}\frac{e^{-2(\alpha-\lambda)(\alpha-\lambda^*)}}{1+\alpha^2}\times\\
&\left(-1+\alpha^2+4|\lambda|^2-2\alpha(\lambda+\lambda^*)\right)\;.
\end{align}
The Wigner function of the state after the loss channel $\mathcal{E}_\epsilon(|\psi_{\sf pac}\rangle\langle \psi_{\sf pac}|)$ can be evaluated by means of the formula
\begin{align}
W[\mathcal{E}_\epsilon(\varrho)](\lambda)=\int \!\!d^2 \lambda' K_\epsilon(\lambda, \lambda')W[\varrho](\lambda^\prime)\;, \label{eq:WigEv}
\end{align}
where
\begin{align}
K_ \epsilon(\lambda, \lambda')=\frac{2}{\pi\epsilon}\exp\left\{-\frac{2\left| \lambda-\lambda' \sqrt{1-\epsilon}\right|^2}{\epsilon} \right\} \;.
\end{align}
\begin{figure}[t!]
\includegraphics[width=0.495\columnwidth]{PACS1a.pdf}
\includegraphics[width=0.495\columnwidth]{PACS1b.pdf}
\caption{(Left) Non-Gaussianity indicator $\Delta_1[\mathcal{E}_\epsilon(|\psi_{\sf pac}\rangle\langle \psi_{\sf pac}|)]$ for PAC states as a function of $\epsilon$ and for different values of $\alpha$: red-dotted line: $\alpha=0.2$; green-dashed line: $\alpha=0.4$; blue-solid line: $\alpha=0.6$. \\
(Right) Maximum value of the noise parameter $\epsilon_{\sf max}^{(1)}$ such that the
bound (\ref{eq:boundmix}) is violated for the state $\mathcal{E}_\epsilon(|\psi_{\sf pac}\rangle\langle \psi_{\sf pac}|)$, as a function of the parameter $\alpha$.
\label{f:PACS1}}
\end{figure}
The non-Gaussianity indicator $\Delta_1[\mathcal{E}_\epsilon(|\psi_{\sf pac}\rangle\langle\psi_{\sf pac}|)$ can then
be straightforwardly evaluated and is plotted in Fig. \ref{f:PACS1} (left) as a function of
$\epsilon$ for different values of $\alpha$.
We note that negative values of the indicator can be observed in an interval for the noise parameter $\epsilon$, which decreases with the increase of $\alpha$. We can explain this feature by noting that, as $\alpha$ decreases, the PAC state approaches the Fock state $\ket{1}$: as a consequence its quantum non-Gaussianity can be more easily detected via Criterion \ref{c:uno}, in particular due to the minimum value of the Wigner function approaching the origin of the phase space. We plotted in Fig. \ref{f:PACS1} (right) the maximum value $\epsilon_{\sf max}^{(1)}$ at which the violation of the bound is observed as a function of $\alpha$. Similarly to Fock states, we observe that by increasing the energy this value tends to the asymptotic value $\epsilon_{\sf max}^{(1)} \rightarrow 0.5$.
\subsubsection{Photon-subtracted squeezed states}
Let us consider now another important class of non-Gaussian states that can be engineered
with current technology. The {\em photon-subtracted squeezed} (PSS) states are defined as
\begin{align}
\ket{\psi_{{\sf pss}}}=\frac{1}{\sinh r}aS(r)\ket{0}.
\end{align}
For low values of squeezing, these states approximate the {\em Schr\"{o}dinger kitten} states,
that is, superpositions of coherent states $|\pm \alpha\rangle$ with opposite phase and
small amplitude ($|\alpha| \lesssim 1$) \cite{dakna97}.
The generation of this kind of states has been demonstrated experimentally \cite{ourjoumtsev06,wakui07,jonas06,gerrits10}, and it relies on performing conditional photon number measurements.\\
Without loss of generality we shall consider a real squeezing parameter $r\in\mathbbm{R}$;
the corresponding average photon number of a PSS state reads
\begin{align}
\bar{n}_0^{\sf (pss)} = 3 \sinh^2 r +1\:,
\end{align}
while its Wigner function is
\begin{align}
W[|\psi_{\sf pss}\rangle](\lambda) &= - \frac2\pi e^{-2|\lambda|^2 \cosh{2 r} + (\lambda^2 + \lambda^{*2}) \sinh{2 r}} \times \nonumber \\
&\left[1-4|\lambda|^2 \cosh{2 r} + 2 (\lambda^2 + \lambda^{*2}) \sinh{2 r} \right] \:.
\end{align}
As for the PAC states, the Wigner function of the evolved state can be evaluated by means
of Eq.~(\ref{eq:WigEv}) and the non-Gaussianity indicator $\Delta_1[\mathcal{E}_\epsilon(|\psi_{\sf pss}\rangle
\langle\psi_{\sf pss}|]$ can be evaluated accordingly. Its behavior as a function of $\epsilon$ and for different values of the squeezing factor $r$ is plotted in Fig. \ref{f:PSSS1} (left).
\begin{figure}[t!]
\includegraphics[width=0.495\columnwidth]{PSSS1a.pdf}
\includegraphics[width=0.495\columnwidth]{PSSS1b.pdf}
\caption{(Left) Non-Gaussianity indicator $\Delta_1[\mathcal{E}_\epsilon(|\psi_{\sf pss}\rangle\langle \psi_{\sf pss}|)]$ for PSS states as a function of $\epsilon$ and for different values of $r$: red-dotted line: $r=0.1$; green-dashed line: $r=0.3$; blue-solid line: $r=0.5$. \\
(Right) Maximum value of the noise parameter $\epsilon_{\sf max}^{(1)}$ such that the
bound (\ref{eq:boundmix}) is violated for the state $\mathcal{E}_\epsilon(|\psi_{\sf pss}\rangle\langle \psi_{\sf pss}|)$, as a function of the initial squeezing parameter $r$.
\label{f:PSSS1}}
\end{figure}
In the right panel of Fig. \ref{f:PSSS1} we plot the maximum noise parameter $\epsilon_{\sf max}^{(1)}$ as a function of the squeezing parameter $r$, observing the same behavior obtained for Fock and PAC states: the value of $\epsilon_{\sf max}^{(1)}$ decreases monotonically with the energy of the state, approaching the asymptotic value $\epsilon_{\sf max}^{(1)}
\rightarrow 0.5$.
\subsection{Violation of the second criterion} \label{s:violation2}
We will now show how the second criterion, which is based on the violation of the inequality
(\ref{eq:bound2mix}), can be exploited in order to improve the results
shown in the previous section. Since in this case one can optimize the procedure
over an additional Gaussian channel, in general one has $\epsilon_{\sf max}^{(2)} \geq \epsilon_{\sf max}^{(1)}$. The simplest Gaussian maps that one can consider are displacement and squeezing operations; correspondingly we are going to seek violation of the bounds described by Eqs. (\ref{eq:boundDISP}) and (\ref{eq:boundSQ}). As anticipated in Sec. \ref{s:criteria}, these new criteria are useful for states which are not phase invariant: the paradigmatic examples are
states {\em displaced} in the phase-space, that is, having the minimum of the Wigner function
outside the origin, or states that exhibit squeezing in a certain quadrature. Due
to this fact, the bounds based on Eqs. (\ref{eq:boundDISP}) and (\ref{eq:boundSQ})
cannot help in optimizing the results we obtained for Fock states. We will focus then
on the other classes of states we introduced, that is PAC and PSS states.
\begin{figure}[t!]
\includegraphics[width=0.9\columnwidth]{WigPac0.pdf}
\caption{ Contour plot of the Wigner function of the photon-added coherent state $|\psi_{\sf pac}\rangle$ for $\alpha=1$. The minimum of the Wigner function is not at the origin of the phase-space, and the state has non-zero first moments.
\label{f:WigPac0}}
\end{figure}
\subsubsection{Photon-added coherent states}
By looking at the PAC state Wigner function in Fig. \ref{f:WigPac0}, one observes that its minimum is not at the origin of the phase space. Moreover, these states have non-zero first moments, implying that one can
decrease their average photon number by applying an appropriate displacement. Both
observations suggest that it is possible to decrease the value the quantum non-Gaussianity indicator defined in Eq. (\ref{eq:Delta2}),
\begin{align}
\Delta_{\sf pac}(\beta) = \Delta_2[\mathcal{E}_\epsilon(|\psi_{\sf pac}\rangle\langle\psi_{\sf pac}|), \mathcal{D}_\beta] \:,
\label{eq:DeltaPAC}
\end{align}
by means of a displacement operation $\mathcal{D}_\beta(\varrho) = D(\beta) \varrho D(\beta)^\dag$. To evaluate $\Delta_{\sf pac}(\beta)$ according to Eq. (\ref{eq:DeltaPAC}) one has simply to evaluate the Wigner function of the state $\varrho=\mathcal{E}_\epsilon(|\psi_{\sf pac}\rangle\langle\psi_{\sf pac}|)$ in a displaced point in the phase space, {\em i.e.} $W[\varrho](-\beta)$, and its average photon
number
\begin{align}
\bar{n}^{\sf (pac)}(\beta) &= (1-\epsilon)n_0^{\sf (pac)} + |\beta|^2 \: + \nonumber \\
& \:\: + \sqrt{1-\epsilon}( \beta^* \langle a \rangle_0 +
\beta \langle a^\dag \rangle_0 ) \:\:,
\end{align}
where $\langle A \rangle_0 = \langle \psi_0 | A |\psi_0\rangle$, and for
$|\psi_0\rangle=| \psi_{\sf pac}\rangle$,
\begin{align}
\langle a \rangle_0 = \langle a^\dag \rangle_0 = \frac{\alpha(2+\alpha^2)}{1+\alpha^2}\:.
\end{align}
Our goal is then to minimize $\Delta_{\sf pac}(\beta)$
over the possible displacement parameters $\beta$. \\
\begin{figure}[t]
\includegraphics[width=0.495\columnwidth]{PACS2a.pdf}
\includegraphics[width=0.495\columnwidth]{PACS2b.pdf}
\caption{(Left) Non-Gaussianity indicator $\Delta_{\sf pac}(\beta)$ as a function
of the additional displacement parameter $\beta$, for $\epsilon=0.8$ and for
different values of the initial parameter $\alpha$: :
red-dotted line: $\alpha=0.2$; green-dashed line: $\alpha=0.4$; blue-solid line: $\alpha=0.6$. \\
(Right) Optimized non-Gaussianity indicator $\Delta_{\sf pac}(\beta_{\sf opt})$ as a function
of $\epsilon$ and for different values of $\alpha$, where the displacement parameter $\beta_{\sf opt}$ has been chosen as in Eq. (\ref{eq:betaopt}): $\alpha=0.2$; green-dashed line: $\alpha=0.4$; blue-solid line: $\alpha=0.6$.
\label{f:PACS2}}
\end{figure}
In Fig. \ref{f:PACS2} (left) we
plot $\Delta_{\sf pac}(\beta)$ as a function
of $\beta$ for different values of the coherent state parameter $\alpha$ and for
$\epsilon=0.8$. We observe that, while for $\beta=0$ the bound is not always
violated, it is possible to find values such that $\Delta_{\sf pac}(\beta) <0$ and thus
prove that the state is quantum non-Gaussian. Unfortunately the optimal value $\beta_{\sf opt}$, which minimizes $\Delta_{\sf pac}(\beta)$, can not be obtained analytically. However we observed that for large values of $\epsilon$
and for $\alpha \gtrsim 1.5$ one can approximate it as
\begin{align}
\beta_{\sf opt}\simeq - \alpha\sqrt{1-\epsilon} = -\alpha e^{-\gamma t/2} . \label{eq:betaopt}
\end{align}
The behavior of $\Delta_{\sf pac}(\beta_{\sf opt})$ as a function of $\epsilon$ shown in Fig. \ref{f:PACS2} (right), for different values of $\alpha$ and fixing $\beta_{\sf opt}$ as in Eq.~\eqref{eq:betaopt}. If we compare this with Fig. \ref{f:PACS1}, not only we observe an improvement in our capacity to witness quantum non-Gaussianity for these states, but we also see that $\Delta_{\sf pac}(\beta_{\sf opt})$ remains negative for all values of $\epsilon$. Indeed, numerical investigations seem to suggest that $\epsilon_{\sf max}^{(2)}\simeq 1$ for all the possible values of $\alpha$: we indeed conjecture that any initial PAC state remains quantum non-Gaussian during the lossy evolution induced by Eq.~\eqref{eq:Markov}, and that this feature can be captured by our second criterion. However, as one can observe from Fig. \ref{f:PACS2} (right), the non-Gaussianity indicator approaches zero quite fast with both $\alpha$ and $\epsilon$, and thus it may be more challenging to detect its negativity in an actual experiment for states with a high average photon number and for large losses.
\subsubsection{Photon-subtracted squeezed states}
\begin{figure}[t!]
\includegraphics[width=0.9\columnwidth]{WigPSS0.pdf}
\caption{ Contour plot of the Wigner function of the photon-subtracted squeezed state $|\psi_{\sf pss}\rangle$ for $r=0.3$. The minimum of the Wigner function is at the origin of the phase-space, and the state exhibits squeezing in one of the quadratures.
\label{f:WigPss0}}
\end{figure}
Like PAC states inherit a displacement in phase space from the initial coherent states,
PSS states inherit squeezing, as we can observe by looking at
the Wigner function in Fig. \ref{f:WigPss0}.
This motivates us to make use of Corollary \ref{c:due}, and thus optimize the non-Gaussianity indicator in Eq. (\ref{eq:Delta2}) as
\begin{align}
\Delta_{\sf pss}(s) = \Delta_2[\mathcal{E}_\epsilon(|\psi_{\sf pss}\rangle\langle \psi_{\sf pss}|, \mathcal{S}_s] \:, \label{eq:DeltaPSS}
\end{align}
that is by considering an additional squeezing operation $\mathcal{S}_s(\varrho) =S(s) \varrho S^\dag(s)$ on the evolved state $\varrho=E(|\psi_{\sf pss}\rangle\langle \psi_{\sf pss}|)$. As pointed out in the proof of inequality (\ref{eq:boundSQ}),
the Wigner function at the origin is invariant under squeezing operations. Hence, the optimal
value $s_{\sf opt}$ that minimizes $\Delta_{\sf pas}(s)$ coincides with the value
which minimizes the average photon number of $\mathcal{S}_s =S(s) \varrho S^\dag(s)$,
\begin{align}
\bar{n}^{\sf (pss)}(s) &= (1-\epsilon)\left[ n_0^{\sf (pss)} \left( \mu_s^2 + \nu_s^2 \right) \right.\nonumber \\
& \:\: \left. + \mu_s \nu_s\left( \langle a^2 \rangle_0 + \langle a^{\dag 2} \rangle_0 \right)
\right] + \nu_s^2\,,
\end{align}
where $\mu_t = \cosh t$, $\nu_t =\sinh t$ and for an initial PSS state (with a real squeezing parameter $r$),
\begin{align}
\langle a^2 \rangle_0 = \langle a^{\dag 2} \rangle_0 = 3\mu_r\nu_r \:.
\end{align}
\begin{figure}[t!]
\includegraphics[width=0.495\columnwidth]{PSSS2a.pdf}
\includegraphics[width=0.495\columnwidth]{PSSS2b.pdf}
\caption{(Left) Non-Gaussianity indicator $\Delta_{\sf pss}(s)$ as a function
of the additional squeezing parameter $s$, for $\epsilon=0.7$ and for
different values of the initial parameter $r$:
red-dotted line: $r=0.1$; green-dashed line: $r=0.3$; blue-solid line: $r=0.5$. \\
(Right) Optimized non-Gaussianity indicator $\Delta_{\sf pss}(s_{\sf opt})$ as a function
of $\epsilon$ and for different values of $r$, where the squeezing parameter is given by:
red-dotted line: $r=0.1$; green-dashed line: $r=0.3$; blue-solid line: $r=0.5$.
\label{f:PSSS2}}
\end{figure}
The behavior of $\Delta_{\sf pas}(s)$ as a function of the additional squeezing
$s$ is plotted in Fig. \ref{f:PSSS2}. As we observed in the previous case, the
optimised criterion works in cases where
the bound (\ref{eq:boundmix}) (corresponding to $s=0$) was not violated. \\
Moreover the optimal squeezing value can be evaluated analytically,
yielding
\begin{align}
s_{\sf opt} &= - \textrm{arccosh}( \mu_{\sf opt} )\:, \\
\mu_{\sf opt}&= \frac1{\sqrt{2}} \left(1 + \frac{6 (1-\epsilon) \mu_r^2+4\epsilon-3}{\sqrt{
(4\epsilon-3)^2+12(1-\epsilon)\epsilon \mu_r^2}}\right)^{1/2} .
\end{align}
\begin{figure}[h!]
\includegraphics[width=0.8\columnwidth]{PSSS3.pdf}
\caption{
Maximum values of the noise parameter $\epsilon_{\sf max}^{(2)}$ and $\epsilon_{\sf max}^{(1)}$, obtained respectively by means of the optimized and not-optimized criteria,
for the state $\mathcal{E}_\epsilon(|\psi_{\sf pss}\rangle\langle \psi_{\sf pss}|)$, as a function of the initial squeezing parameter $r$.
Red-dotted line: $\epsilon_{\sf max}^{(2)}$; blue-solid line: $\epsilon_{\sf max}^{(1)}$.
\label{f:PSSS3}}
\end{figure}
The optimized quantum non-Gaussianity indicator $\Delta_{\sf pss}(s_{\sf opt})$ is
plotted in Fig. \ref{f:PSSS2} (right), where we observe that negative values are obtained for
large values of losses. However, while for PAC states we had evidence that the
maximum value of losses is $\epsilon_{\sf max}^{(2)}\simeq 1$ for all the possible initial
states, this is no longer true for PSS states. The behavior of $\epsilon_{\sf max}^{(2)}$ as a function
of $r$ is plotted in Fig. \ref{f:PSSS3}, together with the previously obtained
$\epsilon_{\sf max}^{(1)}$. We can notice the big improvement in our detection capability, obtained by exploiting Corollary~\ref{c:due}; however for large values of $r$ we still observe that $\epsilon_{\sf max}^{(2)}$ decreases towards the same limiting value $\epsilon_{\sf max}^{(2)}\rightarrow 0.5$. Moreover, as it can be observed
in Fig. \ref{f:PSSS2} (right), the indicator $\Delta_{\sf pas}(s_{\sf opt})$ approaches zero
by increasing $r$ and $\epsilon$, and thus also in this case it can become challenging to witness quantum non-Gaussianity with our methods, in experiments with large values of the initial squeezing $r$ and large losses.
\section{Conclusions} \label{s:conclusions}
We have presented a set of criteria to detect quantum non-Gaussian
states, that is, states that can not be expressed as mixtures of
Gaussian states. The first criterion is based on seeking the violation of a
lower bound for the values that the Wigner function can take at the
origin, depending only on the average photon number of the state. To
verify the effectiveness of the criterion, we considered the evolution
of non-Gaussian pure states in a lossy Gaussian channel, looking for the
maximum value of the noise where such bound is violated. We observed
that the criterion works well, detecting quantum non-Gaussianity in the
non-trivial region of the noise parameters where no negativity of the
Wigner function can be observed.
\par
We have also shown how the criterion can be generalized and improved,
by optimising over additional Gaussian operations applied to
the states of interest. Notice that in a possible experimental
implementation one does not need to perform such additional Gaussian
operations, such as displacement or squeezing, in the actual
experiment. Indeed, it suffices to use the data obtained on the state
itself, and then apply suitable post-processing to evaluate the
optimized non-Gaussianity indicator.
\par
Our criterion, which expresses a
sufficient condition for quantum non-Gaussianity, shares some
similarities with Hudson's theorem for pure Gaussian states, in the
sense that it establishes a relationship between the concept of
Gaussianity (combined with classical mixing), and the possible values
that a Wigner function can take.
The successful implementation of our criteria corresponds to the
measurement of the Wigner function at the origin of the phase space
which, in turn, corresponds to the (photon) parity of the state under
investigation. This may be obtained with current technology
by direct parity measurement \cite{haroche07}, or by reconstruction
of the photon distribution either by tomographic reconstruction
or by the on/off method
\cite{Wal96,Ban96,Opa97,Mun95,Ray04,Mog98,ros04,cvp05,All09,ms07,mt03,zam04,NIST,yam99}.
When the criterion is satisfied, one can confirm that
the quantum state at disposal has been generated by means of a
highly non-linear process, even in the cases where, perhaps due to
inefficient detectors or other types of noise, negativity of the Wigner
function can not be detected.
\section{Acknowledgments} MGG, TT and MSK thank Radim Filip for discussions.
MGAP thanks Vittorio Giovannetti for discussions. MGG acknowledges support from UK EPSRC (EP/I026436/1). T.T. and M.S.K. acknowledge support from the NPRP 4- 426 554-1-084 from Qatar National Research Fund. SO and MGAP acknowledge support
from MIUR (FIRB ``LiCHIS'' No. RBFR10YQ3H).
|
1304.3328
|
\section{Introduction}
In \cite{AS}, Aganagic and Shakirov defined refined invariants of the $(m,n)$ torus knot by constructing two matrices $S$ and $T$ that act on an appropriate quotient of the Fock space, and satisfy the relations in the group $SL_2(\BZ)$ (as in the work of Etingof and Kirillov). They conjectured that these invariants match the Poincar\'e polynomials of Khovanov-Rozansky HOMFLY homology (\cite{KhR2,KhSoergel}) of torus knots. The computation of Khovanov-Rozansky homology for torus knots is a hard open problem in knot theory, and the Aganagic-Shakirov conjecture has been verified in a few cases when this homology can be explicitly computed from the definition.
In \cite{Ch}, Cherednik reinterpreted the construction of \cite{AS} in terms of the spherical double affine Hecke algebra $\mathbf{SH}$ of type $A$, by replacing the $SL_2(\BZ)$ action on the Fock space representation by an $SL_2(\BZ)$ action on the DAHA. As such, he obtained a conjectural definition for the three variable torus knot invariant known as the \textbf{superpolynomial} $\mathcal{P}_{n,m}^\lambda(u,q,t)$, defined for all partitions $\lambda$ and all pairs of coprime integers $m,n$. In Cherednik's viewpoint, these superpolynomials arise as evaluations of certain elements $P_{n,m}^\lambda$ in the DAHA. These elements are polynomials in:
$$
P_{kn,km} \in \mathbf{SH}, \qquad \forall k\in \mathbb{Z},
$$
by the same formula as the well-known Macdonald polynomials $P_\lambda$ are polynomials in the power sum functions $p_k$, for all partitions $\lambda$. The DAHA $\mathbf{SH}$ is bigraded in such a way that $P_{kn,km}$ lies in bidegrees $(kn,km)$, and the action of $SL_2(\BZ)$ on $\mathbf{SH}$ is by automorphisms which permute the bidegrees of the elements $P_{kn,km}$. Cherednik constructs this action by a sequence of elementary transformations, which are however rather difficult to describe by a closed formula.
Schiffmann and Vasserot (\cite{svhilb,svmacd}) give an alternate description of the DAHA by showing that it is isomorphic to the elliptic Hall algebra. Under this automorphism, $P_{kn,km}$ correspond to the standard generators of the elliptic Hall algebra described by Burban and Schiffmann in \cite{BS}. Moreover, Schiffmann and Vasserot show that $\mathbf{SH}$ acts on the $K-$theory of the Hilbert scheme of points on the plane, and we will show that Cherednik's superpolynomials can be computed in this representation. We use this viewpoint to prove two conjectures announced by Cherednik in \cite{Ch} (let us remark that while the present paper was being written, Cherednik also announced independent proofs in \cite[Section 2.4.1]{Ch}). The occurrence of the Hilbert scheme is not so surprising: Nakajima (\cite{Nak}) already used it to compute the matrices $S$ and $T$ of Aganagic and Shakirov, although by using a different construction from ours.
An explicit description of the action of $P_{kn,km}$ on the $K-$theory of the Hilbert scheme was obtained in \cite{Negut}, where these operators were shown to be described by a certain geometric correspondence called the flag Hilbert scheme. We believe that a certain line bundle on this moduli space is related to the unique finite-dimensional irreducible module $L_{\frac mn}$ of the rational Cherednik algebra (see Conjecture \ref{conj:big} for a precise statement). A computational consequence of our approach is the following formula for uncolored superpolynomials as a sum over standard Young tableaux.
\begin{theorem}
\label{thm:main}
The superpolynomial $\mathcal{P}_{n,m}(u,q,t)$, defined as in \cite{Ch}, is given by:
\begin{equation}
\label{eqn:for}
\mathcal{P}_{n,m}(u,q,t) =\sum_{\mu \vdash n} \frac {\widetilde{\gamma}^n}{\widetilde{g}_\mu} \sum^{\textrm{SYT}}_{\text{of shape }\mu} \frac {\prod_{i=1}^{n} \chi_i^{S_{m/n}(i)} (1-u\chi_i)(q \chi_i - t)}{\left(1-\frac {q\chi_{2}}{t\chi_{1}}\right) \ldots \left(1-\frac {q\chi_{n}}{t\chi_{n-1}}\right)}\prod_{1\leq i < j\leq n} \frac {(\chi_j-q\chi_i)(t\chi_j-\chi_i)}{(\chi_j-\chi_i)(t\chi_j-q\chi_i)}
\end{equation}
where the sum is over all standard Young tableaux of size $n$, and $\chi_i$ denotes the $q,t^{-1}$-weight of the box labeled by $i$ in the tableau. The constants in the above relation are given by:
\begin{equation}
\label{def:smn}
S_{m/n}(i)=\left\lfloor \frac {im}n\right\rfloor - \left\lfloor \frac {(i-1)m}n \right\rfloor, \qquad \qquad \widetilde{\gamma}=\frac {(t-1)(q-1)}{(q-t)}
\end{equation}
and:
$$
\widetilde{g}_\mu =\prod_{\square \in \lambda}(1-q^{a(\square)}t^{l(\square)+1}) \prod_{\square \in \lambda}(1-q^{-a(\square)-1}t^{-l(\square)})
$$
The notions of arm-length $a(\square)$ and leg-length $l(\square)$ of a box in a Young diagram will be recalled in Figure \ref{fig}.
\end{theorem}
We also give a prescription to compute general colored superpolynomials $\mathcal{P}_{n,m}^\lambda$, for example on a computer, although we do not yet have any ``nice'' formula. We use formulas such as \eqref{eqn:for} to explore many combinatorial consequences, such as to prove or formulate conjectures about $q,t-$Catalan numbers, parking functions and Tesler matrices in Section \ref{sec:comb}. The highlight is an $\frac mn$ version of the shuffle conjecture \cite{HHRLU}, where a certain combinatorial sum over parking functions in an $m \times n$ rectangle obtained by Hikita \cite{Hikita} is connected with the operators $P_{n,m}$ (see Conjecture \ref{conj:shuffle} for all details). A sample of these results is a corollary of Conjecture \ref{conj:shuffle}, which appears to be new and interesting by itself:
\begin{conjecture}
\label{conj:super}
The ``superpolynomial'' $\mathcal{P}_{n,m}(u,q,t)$ can be written as a following sum:
$$
\mathcal{P}_{n,m}(u,q,t) = \sum_{D} q^{\delta_{m,n}-|D|}t^{-h_{+}(D)}\prod_{P\in v(D)}(1-ut^{\beta(P)})
$$
Here the summation is over all lattice paths $D$ contained below the main diagonal of an $m\times n$l, $P$ goes over all vertices of $D$, $\delta_{m,n}=\frac{(m-1)(n-1)}{2}$ and
$h_{+}(D)$ and $\beta(P)$ are certain combinatorial statistics (see Section \ref{sec:catalan} for details).
\end{conjecture}
For $m=n+1$, the above identity was conjectured in \cite{EHKK} and proved in \cite{HSchroeder}. In the present paper, we prove this conjecture in the limit $t=1$.
For many values of $m$ and $n$, the above conjecture has been verified on a computer.
We also note that Conjecture \ref{conj:super} can be regarded as a refinement of \cite[Conjectures 23,24]{ORS}. Indeed, in \cite{ORS} the authors conjectured
a relation between the Khovanov-Rozansky homology of an algebraic knot and the homology of the Hilbert schemes of points on the corresponding plane curve singularity.
For torus knots, these Hilbert schemes admit pavings by affine cells, and the homology can be computed combinatorially by counting these cells weighted with their dimensions.
It has been remarked in \cite[Appendix A.3]{ORS} that the corresponding combinatorial sum can be rewritten as a sum over lattice paths matching the combinatorial side
of Conjecture \ref{conj:super}. Furthermore, it has been conjectured in \cite[Conjectures 23,24]{ORS}
that the Poincar\'e polynomial for the Khovanov-Rozansky homology of torus knots can be rewritten as an equivariant character of the space of sections of a certain sheaf
on the Hilbert scheme of points on $\mathbb{C}^2$, and this sheaf was written explicitly for $m=kn\pm 1$. For general $m$, a construction of such a sheaf or its class in the equivariant $K$-theory was not accessible by the methods of \cite{ORS}
(see \cite[p. 24]{ORS} for the extensive computations).
In the present paper, we present a (conjectural) candidate for such a sheaf for all $m$ coprime with $n$, using the geometric realization of $\mathcal{P}_{n,m}(u,q,t)$.
To summarize, one can say that Conjecture \ref{conj:super} would imply the agreement between the Aganagic-Shakirov and Oblomkov-Rasmussen-Shende conjectural descriptions of HOMFLY homology of torus knots,
though, indeed, the relation between both of these descriptions to the actual definition of \cite{KhR2,KhSoergel} remains unknown.
Another conjecture relates the operator $P_{m,n}$ to the finite-dimensional representation $L_{\frac mn}$ of the rational Cherednik algebra with parameter $c=\frac{m}{n}$
equipped with a certain filtration defined in \cite{GORS}. Such a representation is naturally graded and carries an action of symmetric group $S_n$ preserving both the grading and the filtration.
\begin{conjecture}
The bigraded Frobenius character of $L_{\frac mn}$, equipped with the natural grading and extra filtration (defined in \cite{GORS}) equals $P_{m,n}\cdot 1$.
\end{conjecture}
This conjecture has bee mainly motivated by \cite{GORS} where $L_{\frac mn}$ has been related to the Hilbert schemes on the
singular curve $\{x^m=y^n\}$ and to the knot homology. For $m=kn\pm 1$, the conjecture follows from the results of \cite{GS,GS2}.
The structure of this paper is the following: in Section \ref{sec:sym}, we recall the basics on symmetric functions, Macdonald polynomials, the double affine Hecke algebra, we state Cherednik's conjectures \ref{conj1} and \ref{conj2}, and recall how they relate to the original construction of Aganagic and Shakirov in Chern-Simons theory. In Section \ref{sec:stab}, we use the stabilization procedure of Schiffmann-Vasserot to prove Cherednik's conjectures by recasting his superpolynomials as matrix coefficients. In Section \ref{sec:hilb}, we discuss the Hilbert scheme and the flag Hilbert scheme, and show how the machinery of \cite{Negut} gives new formulas for torus knot invariants. In Section \ref{sec:rat}, we discuss the connection between the Hilbert scheme and the rational Cherednik algebra, and conjecture that a certain line bundle on the flag Hilbert scheme corresponds to the representation $L_{\frac mn}$ under the Gordon-Stafford functor. Finally, in Section \ref{sec:comb}, we present certain aspects from the combinatorics of symmetric functions which arise in connection to our work, state several conjectures about $q,t-$Catalan numbers, parking functions and Tesler matrices, which we prove in several special cases.
\section*{Acknowledgments}
We are deeply grateful to Andrei Okounkov, who spurred this paper by explaining to us the relation between the earlier results of the second author to the work of the first author on knot invariants. He has helped us greatly with a lot of advice, and many interesting and educating discussions. We are grateful to M. Aganagic, F. Bergeron, R. Bezrukavnikov, I. Cherednik, P. Etingof, A. Garsia, I. Gordon, J. Haglund, M. Haiman, A. Kirillov Jr., I. Losev, M. Mazin, H. Nakajima, N. Nekrasov, A. Oblomkov, J. Rasmussen, S. Shadrin, S. Shakirov, A. Smirnov, A. Sleptsov and V. Shende for their interest and many useful discussions. Special thanks to Maxim Kazaryan for helping us with the {\tt Mathematica} code. The work of E. G. was partially supported by the NSF grant DMS-1403560, RFBR grants RFBR-10-01-678, RFBR-13-01-00755 and the Simons foundation.
\section{DAHA and Macdonald polynomials}
\label{sec:sym}
\subsection{Symmetric functions}
\label{sub:symfunc}
Consider formal parameters $q$ and $t$, and let us define the following ring of constants and its field of fractions:
\begin{equation}
\label{eqn:rings}
\mathbb{K}_0 = \mathbb{C}[q^{\pm 1},t^{\pm 1}], \qquad \qquad \mathbb{K} = \mathbb{C}(q,t)
\end{equation}
Among our basic objects of study will be the algebras of symmetric polynomials:
$$
V = \mathbb{K}[x_1,x_2,\ldots]^\textrm{Sym}, \qquad \qquad V_N = \mathbb{K}[x_1,\ldots,x_N]^\textrm{Sym}
$$
In fact, the algebras $V_N$ form a projective system $V_N \longrightarrow V_{N-1}$, with the maps given by setting $x_N=0$, and the inverse limit of the system is $V$. An important system of generators for these vector spaces consists of power-sum functions $p_k = \sum_i x_i^k$, for which we have:
$$
V = \mathbb{K}[p_1,p_2,\ldots], \qquad \qquad V_N = \mathbb{K}[p_1,\ldots,p_N]
$$
A linear basis of $V$ is given by:
$$
p_\lambda = p_{\lambda_1} p_{\lambda_2}\ldots, \qquad \text{as} \quad \lambda = (\lambda_1 \geq \lambda_2\geq\ldots)
$$
go over all integer partitions. We have the scalar product $\langle \cdot, \cdot \rangle$ on $V$ given by:
\begin{equation}
\label{eqn:innerp}
\langle p_{\lambda},p_{\mu}\rangle = \delta_{\lambda}^{\mu} z_{\lambda} \qquad \forall \ \lambda,\mu
\end{equation}
where $z_{1^{n_1}2^{n_2}\ldots} = \prod_{i\geq 1} i^{n_i} n_i!$. Then another very important basis of $V$ is given by the Schur functions $s_\lambda$, which are orthogonal under the above scalar product and satisfy:
$$
s_\lambda = m_\lambda + \sum_{\mu < \lambda} c_\lambda^\mu m_\mu \qquad \textrm{for some } c^\mu_\lambda \in \mathbb{Z}
$$
where $m_\lambda = \textrm{Sym} \left(z_1^{\lambda_1}z_2^{\lambda_2}\ldots \right)$ are the monomial symmetric functions, and $<$ denotes the dominance partial ordering on partitions: $\mu \leq \lambda$ if $\mu_1+\ldots+\mu_i \leq \lambda_1+\ldots+\lambda_i$ for all $i\geq 1$.
\subsection{Macdonald polynomials}
Another remarkable inner product on $V$ was introduced by Macdonald \cite{Macdonald}:
\begin{equation}
\label{eqn:innerpqt}
\langle p_{\lambda},p_{\mu}\rangle_{q,t}= \delta_{\lambda}^{\mu} z_{\lambda}\prod_{i}\frac{1-q^{\lambda_{i}}}{1-t^{\lambda_{i}}} \qquad \forall \ \lambda,\mu
\end{equation}
The Macdonald polynomials $P_\lambda$ are defined by the property of being orthogonal with respect to $\langle \cdot, \cdot \rangle_{q,t}$ and upper triangular in the basis of monomial symmetric functions:
$$
P_\lambda = m_\lambda + \sum_{\mu < \lambda} d^\mu_\lambda m_\mu \qquad \textrm{for some } d^\mu_\lambda \in \mathbb{K}
$$
The square norm of $P_{\lambda}$ is given by:
$$
\langle P_{\lambda},P_{\lambda}\rangle_{q,t} = \frac{h'_{\lambda}}{h_{\lambda}},
$$
where:
$$
h_{\lambda} = \prod_{\square \in \lambda}(1-q^{a(\square)}t^{l(\square)+1}), \qquad h'_{\lambda}=\prod_{\square \in \lambda}(1-q^{a(\square)+1}t^{l(\square)})
$$
Here $\square$ goes over all the boxes in the Young diagram associated to the partition $\lambda$. The arm-length $a(\square)$ (respectively, the leg-length $l(\square)$) is defined as the number of boxes above (respectively, to the right) of the box $\square$. For illustration, see Figure \ref{fig}.
\begin{figure}[ht]
\begin{tikzpicture}
\draw (0,0)--(0,5)--(1,5)--(1,4)--(3,4)--(3,2)--(5,2)--(5,0)--(0,0);
\draw (2,1)--(2,1.5)--(2.5,1.5)--(2.5,1)--(2,1);
\draw (2.25,1.25) node {};
\draw [<->,>=stealth] (2.5,1.25)--(5,1.25);
\draw [<->,>=stealth] (0,1.25)--(2,1.25);
\draw [<->,>=stealth] (2.25,1.5)--(2.25,4);
\draw [<->,>=stealth] (2.25,0)--(2.25,1);
\draw (4,1) node {$a(\square)$};
\draw (1,1) node {$a'(\square)$};
\draw (1.9,2.8) node {$l(\square)$};
\draw (2.8,0.5) node {$l'(\square)$};
\end{tikzpicture}
\caption{Arm, leg, co-arm and co-leg}
\label{fig}
\end{figure}
There are various normalizations of Macdonald polynomials, and we will encounter their integral form $J_{\lambda}=h_{\lambda}P_{\lambda}$. We have:
\begin{equation}
\label{eqn:normj}
\langle J_{\lambda},J_{\lambda}\rangle_{q,t} = h_{\lambda}^2\langle P_{\lambda},P_{\lambda}\rangle_{q,t} =h_{\lambda} h'_{\lambda}.
\end{equation}
All these constructs make sense both in $V$ (infinitely many variables) and in $V_N$ (finitely many variables), and they are compatible under the maps $V \longrightarrow V_N$. Let us now focus on the case of finitely many variables. Fix a positive integer $N$ and let:
$$
\rho_N=(N-1,N-2,\ldots,1,0)
$$
Consider the evaluation homomorphism:
\begin{equation}
\label{eqn:eval}
\varepsilon_N: V_N \longrightarrow \mathbb{K}, \qquad \varepsilon_N(f) = f(t^{\rho}) = f(t^{N-1},t^{N-2},\ldots,1)
\end{equation}
A simple computation reveals that:
\begin{equation}
\label{power sum eval}
\varepsilon_N(p_{k}) = \frac{1-t^{kN}}{1-t^k}=\frac{1-u^{k}}{1-t^{k}},
\end{equation}
where we capture the $N$-dependence in the new variable $u=t^{N}$. Therefore, one can compute $\varepsilon_N(f)$ for a general symmetric function $f$ by expanding it in terms of the $p_{k}$ and then using \eqref{power sum eval}.
\begin{theorem}(\cite[eq. VI.6.17 and VI.8.8]{Macdonald})
\label{thm:jeval}
The following equations hold:
\begin{equation}
\label{J eval}
\varepsilon_N(P_{\lambda}) = \frac 1{h_\lambda} \prod_{\square\in \lambda} (t^{l'(\square)}-uq^{a'(\square)})
\qquad \Longrightarrow \qquad \varepsilon_N(J_{\lambda}) = \prod_{\square\in \lambda}\left(t^{l'(\square)}-uq^{a'(\square)}\right)
\end{equation}
where $a'(\square)$ (respectively, $l'(\square)$) denote the co-arm and co-leg of $\square$ in $\lambda$ (see Figure \ref{fig}).
\end{theorem}
\subsection{Double affine Hecke algebras}
Following \cite{ChBook}, we will define the double affine Hecke algebra (DAHA) of type $A_N$.
\begin{definition}
The algebra $\mathbf{H}_{N}$ is defined over $\mathbb{K}$ by generators $T_i^{\pm 1}$ for $i \in \{1,\ldots,N-1\}$, and ${X_j}^{\pm 1}, {Y_j}^{\pm 1}$ for $j\in \{1,\ldots,N\}$, under the following relations:
\begin{multline}
\label{DAHA relations}
(T_i+t^{-1/2})(T_i-t^{1/2})=0,\qquad T_{i}T_{i+1}T_{i}=T_{i+1}T_{i}T_{i+1},\qquad [T_i,T_k]=0\ \mbox{for}\ |i-k|>1\\
T_iX_iT_i=X_{i+1},\qquad \qquad {T_i}^{-1}Y_i{T_{i}}^{-1}=Y_{i+1}\\
[T_i,X_k]=0,\qquad [T_i,Y_k]=0\ \mbox{for}\ |i-k|>1\\
[X_j,X_k]=0,\qquad [Y_j,Y_k]=0, \qquad Y_1X_1\ldots X_N=qX_1\ldots X_NY_1,$$
$${X_1}^{-1}Y_2=Y_2{X_1}^{-1}{T_1}^{-2} \\
\end{multline}
\end{definition}
The algebra $\mathbf{H}_N$ contains the Hecke algebra generated by $T_i$, and two copies of the affine Hecke algebra generated by $(T_i,X_j)$ and $(T_i,Y_j)$, respectively. The basic module of $\mathbf{H}_N$ is the polynomial representation:
$$
\mathbf{H}_N \longrightarrow \textrm{End}(\mathbb{C}[X_1,\ldots, X_N]),
$$
The element $X_i$ acts as multiplication by $x_i$, and $T_i$ acts by the Demazure-Lusztig operator:
$$
T_i=t^{1/2}s_{i}+(t^{1/2}-t^{-1/2}) \frac {s_i-1}{x_i/x_{i+1}-1}
$$
where $s_i=(i , i+1)$ are the simple reflections. Let us define the operators $\partial_i$ on $\mathbb{C}[X_1,\ldots, X_N])$ by:
$$
\partial_{i}(f)=f(x_1,\ldots,x_{i-1},qx_i,x_{i+1},\ldots x_N).
$$
and introduce the operator $\gamma=s_{N-1}\cdots s_{1}\partial_{1}$
Set:
$$
Y_i=t^{\frac {N-1}2} T_{i}\cdots T_{N-1}\gamma T_{1}^{-1}\cdots T_{i-1}^{-1}.
$$
Then the operators $T_i, X_j, Y_j$ on $V_N$ satisfy the relations of the DAHA. A priori, these operators do not necessarily send the subspace of symmetric polynomials $V_N$ to itself. However, it is well-known that for any symmetric polynomial $f$, we have:
$$
f(X_1,\ldots,X_n):V_N \rightarrow V_N, \qquad f(Y_1,\ldots,Y_n):V_N \rightarrow V_N.
$$
We will need the following result of Macdonald:
\begin{proposition}(\cite[eq. VI.3.4]{Macdonald})
On the subspace $V_N$ of symmetric polynomials, the operator $\delta_1:=Y_1+\ldots+Y_N$ can be rewritten as:
\begin{equation}
\label{delta1}
\delta_1=\sum_{i}A_i\partial_i, \qquad \text{where} \quad A_i=\prod_{j\neq i}\frac{tx_i-x_j}{x_i-x_j}.
\end{equation}
\end{proposition}
In fact, the operator $\delta_1$ is diagonal in the basis of Macdonald polynomials of $V_N$, with eigenvalues given by:
$$
\delta_1 \cdot P_{\lambda}(x)= \left(\sum_{i}t^{N - i}q^{\lambda_i} \right)P_{\lambda}(x).
$$
This is part of an alternate definition of Macdonald polynomials as common eigenfunctions of a collection of commuting differential operators, the first of which is $\delta_1$. Generalizing this, the following result is the main theorem of \cite{Ch2}:
\begin{theorem}[\cite{Ch2}]
Let $f$ be a symmetric polynomial in $N$ variables. The operator:
$$L_{f}:=f(Y_1,\ldots,Y_N)$$
is diagonal in the basis of Macdonald polynomials in $V_N$, with eigenvalues:
\begin{equation}
\label{eigenvalues}
L_{f} \cdot P_{\lambda}(x)=f(t^{\rho_N}q^{\lambda})P_{\lambda}(x).
\end{equation}
\end{theorem}
\subsection{The $SL_2(\BZ)$ action}
\label{sub:slzact}
Letting $e \in \mathbf{H}_N$ denote the complete idempotent, the spherical DAHA is defined as the subalgebra:
$$
\mathbf{SH}_N = e\cdot \mathbf{H}_N \cdot e
$$
There is an action of $SL_2(\BZ)$ on $\mathbf{H}_N$ that preserves the subalgebra $\mathbf{SH}_N$. To define it, let:
$$
\tau_{+}=\left(\begin{matrix}1 & 1\\ 0& 1 \end{matrix}\right),\quad \tau_{-}=\left(\begin{matrix}1 & 0\\ 1& 1 \end{matrix}\right)
$$
be the generators of $SL_2(\BZ)$. Then Cherednik (\cite{ChBook}, see also \cite{svmacd}) shows that:
\begin{multline}
\label{SL2 action}
\tau_{+}(X_i)=X_i,\quad \tau_{+}(T_i)=T_i,\quad \tau_{+}(Y_i)=Y_iX_i(T_{i-1}^{-1}\cdots T_i^{-1})(T_{i}^{-1}\cdots T_{i-1}^{-1}) \\
\tau_{-}(Y_i)=Y_i,\quad \tau_{-}(T_i)=T_i,\quad \tau_{-}(X_i)=X_iY_i(T_{i-1}\cdots T_i)(T_{i}\cdots T_{i-1})
\end{multline}
extend to automorphisms of $\mathbf{H}_N$, and they respect the relations in $SL_2(\BZ)$. This action allows one to construct certain interesting elements in $\mathbf{H}_N$. Start by defining:
$$
P^{\lambda,N} := P_\lambda(Y_1,\ldots,Y_N) \in \mathbf{SH}_N
$$
For example, $P^{(1),N} = \delta_1$ is the sum of the $Y_i$. For any pair of integers $(n,m)$ with $\gcd(n,m)=1$, let us choose any matrix of the form:
$$
\gamma_{n,m} = \left(\begin{matrix} x & n \\ y & m \\ \end{matrix}\right) \in SL_2(\BZ)
$$
\begin{definition} (and \textbf{Proposition}, see \cite{ChBook}, \cite{svmacd}) The elements:
$$
P_{n,m}^{\lambda,N} := \gamma_{n,m}(P^{\lambda,N}) \in \mathbf{SH}_N
$$
do not depend on the choice of $\gamma_{n,m}$.
\end{definition}
The same construction can be done starting from the power-sum symmetric functions:
$$
p_k^N:=Y_1^k+\ldots+Y_N^k \in \mathbf{SH}_N.
$$
Acting on them with $\gamma_{n,m}$ gives rise to operators:
$$
P^N_{kn,km} := \gamma_{n,m}(p_k^N) \in \mathbf{SH}_N
$$
As in the above Proposition, these do not depend on the particular choice of $\gamma_{n,m}$. Since the Macdonald polynomials $P^{\lambda,N}$ are polynomials in the power-sum functions $p_k^N$, then the $P_{n,m}^{\lambda,N}$ are polynomials in $P^N_{kn,km}$. These can be easily calculated on a computer.
\subsection{Cherednik's conjectures}
Following \cite{Ch}, we define the \emph{DAHA-superpolynomials}:
$$
\mathcal{P}_{n,m}^{\lambda,N}(q,t) =\varepsilon_N(P_{n,m}^{\lambda,N}\cdot 1) \in \mathbb{K}, \qquad \qquad \mathcal{P}_{kn,km}^{N}(q,t) =\varepsilon_N(P_{kn,km}^{N}\cdot 1) \in \mathbb{K}
$$
where $\varepsilon_N$ is the evaluation map of \eqref{eqn:eval}. Since the operators $P_{n,m}^{\lambda,N}$ can be expressed as sums of products of $P_{kn,km}^{N}$, we will see that the superpolynomials $\mathcal{P}_{n,m}^{\lambda,N}(q,t)$ can be expressed in terms of the matrix elements of $P_{kn,km}^{N}$. Therefore, we will mostly focus on the latter, and in Section \ref{sec:hilb} we will show how to compute them. The following conjectures were stated in \cite{Ch}:
\begin{conjecture}[Stabilization]
\label{conj1}
There exists a polynomial $\mathcal{P}_{n,m}^{\lambda}(u,q,t)$ such that:
$$
\mathcal{P}_{n,m}^{\lambda,N}(q,t) = \mathcal{P}_{n,m}^{\lambda}(u=t^N,q,t)
$$
\end{conjecture}
\begin{conjecture}[Duality]
\label{conj2}
Define the reduced superpolynomial by the equation:
$$\mathcal{P}_{n,m}^{\lambda,red}(u,q,t)=\frac{\mathcal{P}_{n,m}^{\lambda}(u,q,t)}{\mathcal{P}_{1,0}^{\lambda}(u,q,t)}$$
Then the polynomials $\mathcal{P}$ for transposed diagrams are related by the equation:
$$
q^{(1-n)|\lambda|} \mathcal{P}_{n,m}^{\lambda^t,red}(u,q,t) = t^{(n-1)|\lambda|} \mathcal{P}_{n,m}^{\lambda,red}(u,t^{-1},q^{-1}).
$$
\end{conjecture}
One of our main results is a proof of the above conjectures, to be given in Subsections \ref{sub:proofs} and \ref{sub:refined}.
\subsection{Refined Chern-Simons theory}
In this section we describe the approach of Aganagic and Shakirov, which has its roots in Chern-Simons theory. A 3-dimensional topological quantum field theory associates a number $Z(M)$ to every 3-manifold, a Hilbert space $Z(N)$ to a closed 2-manifold $N$, and a vector $Z(M)\in Z(\partial{M})$ to a 3-manifold $M$ with boundary. All our 3-manifolds may come with closed knots embedded in them.
We will be interested in invariants of torus knots inside the sphere $S^3$. The sphere can be split into two solid tori glued along the boundary, and the Hilbert space $Z(T^2)$ associated to their common boundary can be identified with a suitable quotient of $V$. This quotient has a basis consisting of Macdonald polynomials labeled by Young diagrams inscribed in a $k\times N$ rectangle. One can think of Macdonald polynomials as invariants of the meridian of the solid torus colored by these diagrams. The space $Z(T^2)$ is acted on by the mapping class group of the torus, namely $SL_2(\BZ)$, as constructed in \cite{AS}.
Consider the $(m,n)$ torus knot colored by a partition $\lambda$ inside a solid torus, linked with a meridian colored by $\mu$. For such a link, a TQFT should produce a vector $v_{n,m,\mu}^{\lambda}$ in $Z(T^2)$. One defines a {\em knot operator} $W_{n,m}^{\lambda}$ by the formula:
$$
W_{n,m}^{\lambda}|P_{\mu}\rangle=v_{n,m,\mu}^{\lambda}.
$$
In particular, $W_{n,m}^{\lambda}| 1 \rangle$ is the vector in $Z(T^2)$ associated to a solid torus with a $\lambda$-colored $(m,n)$ torus knot inside. The construction of \cite{AS} uses the equation:
\begin{equation}
\label{conj}
W_{n,m}^{\lambda}=K^{-1}W_{1,0}^{\lambda}K,
\end{equation}
where $W_{1,0}^{\lambda}$ is the operator of multiplication by $P_{\lambda}$ and $K$ is any element of $SL_2(\BZ)$ taking $(1,0)$ to $(n,m)$, seen as an endomorphism of $Z(T^2)$. The action of $SL_2(\BZ)$ was introduced by A. Kirillov Jr. in \cite{K} and is given by the two matrices (\cite{AS,EK,K}) written in the Macdonald polynomial basis \footnote{The operators in \cite{AS} differ from these by overall scalar factors, which are not important for us}:
\begin{equation}
\label{S and T}
S_{\lambda}^{\mu}=P_{\lambda}(t^{\rho}q^{\mu})P_{\mu}(t^{\rho}),\qquad T_{\lambda}^{\mu}=\delta_{\lambda}^{\mu}q^{\frac{1}{2}\sum_{i}\lambda_i(\lambda_i-1)}t^{\sum_{i}\lambda_i(i-1)}.
\end{equation}
which correspond to the matrices:
$$
\sigma = \left( \begin{array}{cc}
0 & 1 \\
-1 & 0 \end{array} \right), \qquad \tau = \left( \begin{array}{cc}
1 & 1 \\
0 & 1 \end{array} \right) \in SL_2(\BZ).
$$
The refined Chern-Simons knot invariant is then defined as
\begin{equation}
\label{CS evaluation}
\mathcal{P}^{CS}_{n,m,\lambda}(q,t):=\langle 1|SW_{n,m}^{\lambda}|1 \rangle_{q,t}.
\end{equation}
The extra $S$-matrix in \eqref{CS evaluation} is responsible for the gluing of two solid tori into $S^3$.
The equations (\ref{conj})-(\ref{CS evaluation}) give a rigorous definition of the polynomials $\mathcal{P}^{CS}_{n,m\lambda}(q,t)$, which a priori depend on the choice of $k$ and $N$. It is conjectured in \cite{AS} that for large enough $k$ and $N$ the answer does not depend on $k$ and its $N$-dependence can be captured in a variable $u=t^N$.
The approach of \cite{Ch} is to identify $Z(T^2)$ with the finite-dimensional representation of $\mathbf{SH}_N$. Such representations were classified in \cite{VV}, and in type $A_N$ they occur when $q$ is a root of unity of degree $k+\beta N$, and $t=q^{\beta}$. It is known (\cite{Ch3}) that the bilinear form $\langle\cdot, \cdot \rangle_{q,t}$ is nondegenerate on $Z(T^2)$, which is a quotient of the polynomial representation by the kernel of this form.
The following lemma shows the equivalence of the approaches of \cite{AS} and \cite{Ch}.
\begin{lemma}
\label{lem:conj}
In the representation $Z(T^2)$, any $W\in \mathbf{SH}_N$ satisfies:
\begin{equation}
\label{eqn:lemma}
SWS^{-1}=\sigma(W).
\end{equation}
where $\sigma$ acts on the DAHA as in Subsection \ref{sub:slzact}.
\end{lemma}
\begin{proof} By \eqref{eigenvalues}, we have $S_{\lambda}^{\mu}=\varepsilon_N(P_{\lambda}(Y)\cdot P_{\mu}(x))$. Consider the vector:
$$
v=S(1)=\sum_{\mu}\varepsilon_N(P_{\mu})P_{\mu}
$$
Then:
$$
S(P_{\lambda})=\sum_{\mu}S_{\lambda}^{\mu}P_{\mu}=P_{\lambda}(Y)\cdot v
$$
hence we conclude that for any function $f \in V_N$, one has $S(f)=\sigma(f)\cdot v$. Therefore:
$$
S(Wf)=\sigma(Wf)\cdot v=\sigma(W)\sigma(f)\cdot v=\sigma(W)S(f)
$$
holds for any $W\in \mathbf{H}_N$ and any $f\in V_N$.
\end{proof}
\begin{corollary}
The definitions of the refined knot invariants in \cite{AS} and \cite{Ch}
are equivalent to each other:
$$\mathcal{P}^{CS}_{n,m,\lambda}(q,t)=\mathcal{P}^{\lambda,N}_{n,m}(q,t).$$
\end{corollary}
\begin{proof}
Similarly to Lemma \ref{lem:conj}, one can show that the equation $TWT^{-1}=\tau(W)$ holds for any $W\in \mathbf{SH}_N$. Together with (\ref{eqn:lemma}), this implies the equation $KWK^{-1}=\kappa(W)$, where $K$ is an arbitrary operator from $SL_2(\BZ)$ and $\kappa$ is the corresponding automorphism of the spherical DAHA. In other words, the two actions of $SL_2(\BZ)$ on the image of $\mathbf{SH}_N$ in the automorphisms of $Z(T^2)$ agree with each other. Since both $P_{1,0}^{\lambda,N}$ and $W_{1,0}^{\lambda,N}$ are defined as multiplication operators by the Macdonald polynomial $P_{\lambda}$, one has $P_{1,0}^{\lambda,N}=W_{1,0}^{\lambda,N}$ and
$$P_{n,m}^{\lambda,N}=W_{n,m}^{\lambda,N}$$
for all $m$, $n$ and $\lambda$. It remains to notice that the covector
$\langle 1|S|\cdot\rangle_{q,t}$ coincides with the evaluation map $\varepsilon_N$, hence
$$\mathcal{P}^{CS}_{n,m\lambda}(q,t)=\langle 1|SW_{n,m}^{\lambda}|1 \rangle_{q,t}=
\varepsilon_{N}(W_{n,m}^{\lambda}\cdot 1)=\varepsilon_{N}(P_{n,m}^{\lambda}\cdot 1)=\mathcal{P}^{\lambda,N}_{m,n}(q,t).
$$
\end{proof}
\section{Stabilization and $N$-dependence}
\label{sec:stab}
Conjecture \ref{conj1} involves the $N$-dependence of the expression $\mathcal{P}_{n,m}^{\lambda,N} = \varepsilon_N (P^{\lambda,N}_{n,m} \cdot 1)$, and we will understand this in three steps. First, we will recast the evaluation $\varepsilon_N$ as a certain matrix coefficient of the operator $P^{\lambda,N}_{n,m}$. Secondly, we will describe the behaviour of these operators as $N \rightarrow \infty$ and show that they stabilize to an operator $P^\lambda_{n,m}$. Thirdly, we will discuss the behaviour of the matrix coefficients as $N \rightarrow \infty$.
\subsection{Matrix Coefficients}
Let us start from the polynomial representation $\mathbf{SH}_N \longrightarrow \text{End}(V_N)$. We will consider the \emph{evaluation vector}:
\begin{equation}
\label{eqn:evalv}
\mathbf{v}(u) := \sum_{\lambda} \frac {J_\lambda }{h_\lambda h'_\lambda} \prod_{\square\in \lambda}\left(t^{l'(\square)}-u q^{a'(\square)}\right)\in V_N
\end{equation}
In the above, the first sum goes over all partitions, and so $\mathbf{v}(u)$ rightfully takes values in a completion of $V_N$. Then for any $f\in V_N$, we have:
$$
\varepsilon_N(f) = \langle f, \mathbf{v}(u) \rangle_{q,t} \Big |_{u=t^N}
$$
Indeed, since the above relation is linear, it is enough to check it for $f=J_\lambda$, where it follows from \eqref{eqn:normj} and \eqref{J eval}. Then our knot invariants are given by the formula:
\begin{equation}
\label{eqn:pose}
\mathcal{P}^{\lambda,N}_{n,m} = \varepsilon_N(P^{\lambda,N}_{n,m}\cdot 1) = \langle \mathbf{v}(u)|P^{\lambda,N}_{n,m}|1\rangle_{q,t} \Big |_{u=t^N}
\end{equation}
where $\langle \cdot | * | \cdot \rangle_{q,t}$ denotes matrix coefficients with respect to the Macdonald scalar product $\langle \cdot , \cdot \rangle_{q,t}$.
\subsection{Stabilization of operators}
We will now let $N$ vary. Recall that the spaces of symmetric polynomials $V_N$ form a projective system under the maps $\eta_{N}:V_N \longrightarrow V_{N-1}$ that set $x_N=0$, and $V$ is the inverse limit if this system. Therefore, we have projection maps $\eta_{\infty,N}: V \longrightarrow V_N$. We will rescale our operators:
\begin{equation}
\label{eqn:rescale}
\overline{P}^{N}_{0,k}=t^{-k(N-1)}\left(P^{N}_{0,k} - \frac {t^{kN} - 1}{t^k-1} \right), \qquad \qquad \overline{P}^{N}_{kn,km} = P^N_{kn,km} \quad \text{for }n \neq 0
\end{equation}
\begin{proposition}(cf. \cite[Prop. 1.4]{svhilb}) The following relation holds:
$$
\overline{P}^{N-1}_{kn,km}\circ \eta_{N}=\eta_{N}\circ \overline{P}^{N}_{kn,km}
$$
\end{proposition}
Therefore, there exist limiting operators $P_{kn,km}:=\lim_{N\to\infty} \overline{P}^{N}_{kn,km}$ on $V$, such that:
$$
\overline{P}^{N}_{kn,km}\circ \eta_{\infty,N}=\eta_{\infty,N}\circ P_{kn,km}
$$
For a general partition, the operators $P_{n,m}^{\lambda,N}$ on $V_N$ are sums of products of $P_{kn,km}^N$, and therefore there exist operators $P_{n,m}^\lambda$ on $V$ which stabilize the operators $P_{n,m}^{\lambda,N}$:
$$
\overline{P}^{\lambda,N}_{n,m} \circ \eta_{\infty,N} = \eta_{\infty,N} \circ P_{n,m}^\lambda
$$
All these new operators $P^\lambda_{n,m}$ and $P_{kn,km}$ lie in the algebra $\mathbf{SH}$, defined in \cite{svhilb} as the stabilization of the spherical DAHA's $\mathbf{SH}_N$ as $N \rightarrow \infty$.
\subsection{Commutation relations}
The isomorphism between $\mathbf{SH}$ and the elliptic Hall algebra, established in \cite{svmacd} (see also \cite{Negut2}), allows one to present some explicit commutation relations between the operators $P_{n,m}$. These commutation relations were discovered by Burban and Schiffmann in \cite{BS}. It is sometimes convenient to represent the operator $P_{n,m}$ by the vector $(n,m)$ in the integer lattice $\mathbb{Z}^2$ (we assume $n>0$). The action of $SL_2(\BZ)$ on the algebra $\mathbf{SH}$ is then just given by the linear action on this lattice.
\begin{definition}(\cite{Negut2})
A triangle with vertices $X=(0,0)$, $Y=(n_2,m_2)$ and $Z=(n_1+n_2,m_1+m_2)$ is called quasi-empty if $m_1n_2-m_2n_1>0$ and there are no lattice points neither inside the triagle, nor on at least one of the edges $XY, YZ$.
\end{definition}
Let us define the constants:
\begin{equation}
\label{eq:alpha}
\alpha_n=\frac{(q^n-1)(t^{-n}-1)(q^{-n}t^{n}-1)}{n},
\end{equation}
and the operators $\theta_{kn,km}$ (for coprime $m,n$) by the equation:
$$
\sum_{n=0}^{\infty}z^{n}\theta_{kn,km}=\exp\left(\sum_{n=1}^{\infty}\alpha_nz^{n}P_{kn,km}\right).
$$
The elliptic Hall algebra is defined by the following commutation relations (\cite{BS}):
$$
[P_{n_1,m_1},P_{n_2,m_2}]=0,
$$
if the vectors $(n_1,m_1)$ and $(n_2,m_2)$ are collinear, and:
\begin{equation}
\label{commutation}
[P_{n_1,m_1},P_{n_2,m_2}]=\frac{\theta_{n_1+n_2,m_1+m_2}}{\alpha_1},
\end{equation}
if the points $(0,0), (n_2,m_2)$ and $(n_1+n_2,m_1+m_2)$ form a quasi-empty triangle.
\begin{figure}[ht]
\begin{tikzpicture}
\draw [->,>=stealth] (0,0)--(4,0);
\draw [->,>=stealth] (0,0)--(0,4);
\draw [->,>=stealth] (0,0)--(1,1);
\draw [->,>=stealth] (1,1)--(2,3);
\draw [->,>=stealth] (0,0)--(2,3);
\draw (0,0) node {\tiny $\bullet$};
\draw (1,0) node {\tiny $\bullet$};
\draw (2,0) node {\tiny $\bullet$};
\draw (3,0) node {\tiny $\bullet$};
\draw (0,1) node {\tiny $\bullet$};
\draw (1,1) node {\tiny $\bullet$};
\draw (2,1) node {\tiny $\bullet$};
\draw (3,1) node {\tiny $\bullet$};
\draw (0,2) node {\tiny $\bullet$};
\draw (1,2) node {\tiny $\bullet$};
\draw (2,2) node {\tiny $\bullet$};
\draw (3,2) node {\tiny $\bullet$};
\draw (0,3) node {\tiny $\bullet$};
\draw (1,3) node {\tiny $\bullet$};
\draw (2,3) node {\tiny $\bullet$};
\draw (3,3) node {\tiny $\bullet$};
\end{tikzpicture}
\caption{Example of a commutation relation: $[P_{1,2},P_{1,1}]=P_{2,3}$.}
\end{figure}
It would be interesting to find a topological interpretation of the equation \eqref{commutation}. Note that the complete statement of this equation requires the RHS of \eqref{commutation} to be multiplied by a central element, which must be included in the definition of the elliptic Hall algebra. For the sake of brevity, we will neglect this technical point, whose discussion can be found in \cite{BS}.
\subsection{Proof of the stabilization conjecture \ref{conj1}}
\label{sub:proofs}
In this Subsection, we will use the elliptic Hall algebra viewpoint to prove Conjecture \ref{conj1}. The inner product $\langle \cdot, \cdot \rangle_{q,t}$, the vacuum vector $|1\rangle$ and the evaluation vector $\mathbf{v}(u)$ are all preserved under the maps $\eta_N:V_N \longrightarrow V_{N-1}$. Therefore, all of these notions are compatible with the above limiting procedure, so we conclude that:
\begin{equation}
\label{eqn:superpoly}
\mathcal{P}^\lambda_{n,m} := \langle \mathbf{v}(u)|P^{\lambda}_{n,m}|1\rangle_{q,t}
\end{equation}
is the same whether we compute it in $V$ or $V_N$. Comparing this with \eqref{eqn:pose}, we conclude that:
$$
\mathcal{P}^\lambda_{n,m} = \mathcal{P}^{\lambda,N}_{n,m} \Big |_{u=t^N}
$$
This proves Conjecture \ref{conj1}. We will prove Conjecture \ref{conj2} in the next section, when we will realize the elliptic Hall algebra and the polynomial representation geometrically.
\section{The Hilbert scheme of points}
\label{sec:hilb}
\subsection{Basic Definitions} Let $\Hilb_d = \Hilb_d(\mathbb{C}^2)$ denote the moduli space of colength $d$ ideal sheaves $\mathcal{I} \subset \mathcal{O}_{\mathbb{C}^2}$. It is a smooth and quasi-projective variety of dimension $2d$. Pushing forward the universal quotient sheaf on $\Hilb_d \times \mathbb{C}^2$ gives rise to the rank $d$ tautological vector bundle on $\Hilb_d$:
$$
\mathcal{T}|_{\mathcal{I}} = \Gamma(\mathbb{C}^2, \mathcal{O}/\mathcal{I})
$$
The torus $T = \mathbb{C}^* \times \mathbb{C}^*$ acts on $\mathbb{C}^2$ by dilations, and therefore acts on sheaves on $\mathbb{C}^2$ by direct image. This gives an action of $T$ on the moduli spaces $\Hilb_d$. We can then consider the equivariant $K-$theory groups $K_T^*(\Hilb_d)$, all of which will be modules over:
$$
K_T^*(\textrm{pt}) = \mathbb{K}_0 = \mathbb{C}[q^{\pm 1}, t^{\pm 1}]
$$
where $q$ and $t$ are a fixed basis of the characters of $T = \mathbb{C}^* \times \mathbb{C}^*$. As early as the work of Nakajima, it became apparent that one needs to study the direct sum of these $K-$theory groups over all degrees $d$. In other words, we will consider the vector space:
$$
K = \bigoplus_{d\geq 0} K_T^*(\Hilb_d) \otimes_{\mathbb{K}_0} \mathbb{K}
$$
This vector space comes with the geometric pairing:
\begin{equation}
\label{eqn:geom}
(\cdot,\cdot): K \otimes K \longrightarrow \mathbb{K}, \qquad \qquad (\alpha, \beta) = \pi_*(\alpha \otimes \beta)
\end{equation}
where $\pi:\Hilb_d \longrightarrow \text{pt}$ is the projection map. Feigin-Tsymbaliuk and Schiffmann-Vasserot independently proved the following result:
\begin{theorem} \label{thm:sv} (\cite{FT}, \cite{svhilb}) There exists a geometric action of the algebra $\mathbf{SH}$ on $K$, which becomes isomorphic to the polynomial representation $V$. \end{theorem}
\subsection{Torus fixed points}
We have the following localization theorem in equivariant $K-$theory:
\begin{equation}
\label{eqn:localization}
K \cong \bigoplus_{d\geq 0} K_T^*(\Hilb^T_d) \otimes_{\mathbb{K}_0} \mathbb{K}
\end{equation}
There are finitely many torus fixed points in $\Hilb_d$, and they are all indexed by partitions $\lambda \vdash d$:
$$
\mathcal{I}_\lambda = (x^{\lambda_1} y^0, x^{\lambda_2}y^1,\ldots) \subset \mathbb{C}[x,y]
$$
The skyscraper sheaves at these fixed points give a basis $I_\lambda = [\mathcal{I}_\lambda]$ of the right hand side of \eqref{eqn:localization}, and therefore also of the vector space $K$. This basis is orthogonal with respect to the pairing \eqref{eqn:geom}:
$$
(I_\lambda, I_\mu) = \delta_{\lambda,\mu} \cdot g_\lambda
$$
where:
\begin{equation}
\label{eqn:defg}
g_\lambda := \Lambda^\bullet(T^\vee_\lambda \Hilb_d) = \prod_{\square \in \lambda}(1-q^{a(\square)}t^{-l(\square)-1}) \prod_{\square \in \lambda}(1-q^{-a(\square)-1}t^{l(\square)})
\end{equation}
\begin{theorem} (e.g. \cite{haimlectures2,OP})
\label{thm:fixed}
Under the isomorphism $K \cong V$ of Theorem \ref{thm:sv}, the classes $I_\lambda \in K$ correspond to the modified Macdonald polynomials:
$$
\widetilde{H}_{\lambda}(q;t) = t^{n(\lambda)} \varphi_{\frac 1{1-1/t}} \left[J_{\lambda}(q;t^{-1}) \right] \in V
$$
where $\varphi_{\frac 1{1-1/t}}:V \longrightarrow V$ is the plethystic homomorphism defined by $p_k \longrightarrow \frac {p_k}{1-1/t^{k}}$, and:
\begin{equation}
\label{eqn:defn}
n(\lambda)=\sum_{\square\in \lambda} l(\square)=\sum_{\square\in \lambda} l'(\square)
\end{equation}
\end{theorem}
The polynomials $\widetilde{H}_{\lambda}$ were introduced by Garsia and Haiman \cite{GH93}, and they are called {\em modified Macdonald polynomials}. They behave nicely under transpose:
\begin{equation}
\label{hmu symmetry}
\widetilde{H}_{\lambda^{t}}(q;t)=\widetilde{H}_{\lambda}(t;q).
\end{equation}
Following \cite{BG}, let us define the operator $\nabla$ on $V$ by the formula:
\begin{equation}
\label{def:nabla}
\nabla \widetilde{H}_{\lambda}=q^{n(\lambda^t)}t^{n(\lambda)}\widetilde{H}_{\lambda}.
\end{equation}
Under the isomorphism $V\cong K$, $\nabla$ corresponds to the operator of multiplication by the line bundle $\mathcal{O}(1)$, while the geometric inner product \eqref{eqn:geom} corresponds to the following twist of the Macdonald inner product:
\begin{equation}
\label{eqn:twist}
(\cdot, \cdot) = (-q)^{-d} \left\langle \varphi^{-1}_{\frac 1{1-1/t}} \left(\nabla^{-1}(\cdot)\right) , \varphi^{-1}_{\frac 1{1-1/t}}(\cdot) \right\rangle_{q,t^{-1}} \qquad \text{on} \quad K_d \cong V_d
\end{equation}
\subsection{Geometric operators}
\label{sub:fhilb}
The most interesting elements of $\mathbf{SH}$ to us are $P_{kn,km}$, for all $k\neq 0$ and $\gcd(n,m)=1$. In the current geometric setting, we will study their conjugates:
\begin{equation}
\label{eq:TP0}
\widetilde{P}_{kn,km} = \varphi_{\frac 1{1-1/t}} \circ P_{kn,km}(q;t^{-1}) \circ \varphi^{-1}_{\frac 1{1-1/t}}
\end{equation}
Let us now describe how these operators act on $K$, and we will start with the simplest case, namely $n=0$. We define the polynomial:
\begin{equation}
\label{eqn:tautclass}
\Lambda(z) = \sum_{i=0}^d (-z)^i [\Lambda^i \mathcal{T}] \in K[z]
\end{equation}
where $\mathcal{T}$ is the tautological rank $d$ vector bundle on $\Hilb_d$. As was shown in \cite{Negut}, we have:
\begin{equation}
\label{eqn:0}
\exp\left( \sum_{k \geq 0} \alpha_k \widetilde{P}_{0,k} z^{k} \right) \cdot c = \frac {\Lambda\left(\frac qz\right)\Lambda\left(\frac tz\right)\Lambda\left(\frac 1{zqt}\right)}{\Lambda\left(\frac 1{zq}\right)\Lambda\left(\frac 1{zt}\right)\Lambda\left(\frac {qt}{z}\right)} \cdot c \qquad \forall c\in K
\end{equation}
where the constants $\alpha_k$ are given by \eqref{eq:alpha}. In order to compute how each $\widetilde{P}_{0,\pm k}$ acts, we need to expand the right hand side in powers of $z$ and take the appropriate coefficient. We will give a more computationally useful description of $\widetilde{P}_{0,k}$ in \eqref{eqn:cartan} below.
For $n > 0$, we consider the flag Hilbert scheme:
$$
\Hilb_{d,d+kn} = \{\mathcal{I}_0 \supset \mathcal{I}_{1} \supset \ldots \supset \mathcal{I}_{kn}\} \subset \Hilb_d \times \Hilb_{d+1} \times \ldots \times \Hilb_{d+kn}
$$
where the inclusions are all required to be supported at the same point of $\mathbb{C}^2$. This variety comes with projection maps:
$$
p^-: \Hilb_{d,d+kn} \longrightarrow \Hilb_d, \qquad p^+: \Hilb_{d,d+kn} \longrightarrow \Hilb_{d+kn}
$$
that forget all but the first/last ideal in the flag, and with tautological line bundles $\mathcal{L}_1,\ldots,\mathcal{L}_{kn}$ given by:
$$
\mathcal{L}_i|_{\mathcal{I}_0 \supset \ldots \supset \mathcal{I}_{kn}} = \Gamma(\mathbb{C}^2, \mathcal{I}_{i-1}/\mathcal{I}_{i})
$$
As explained in \cite{Negut}, the flag Hilbert scheme is not simply the iteration of $kn$ individual Nakajima correspondences. The reason for this is that the convolution product of Nakajima correspondences is not a complete intersection, and thus the intersection-theoretic composition is a codimension $kn-1$ class on the scheme $\Hilb_{d,d+kn}$. For this reason, the composition differs significantly from the fundamental class of $\Hilb_{d,d+kn}$. In {\em loc. cit.}, we have the following description of the operators $\widetilde{P}_{kn,km}$ acting on $K$:
\begin{equation}
\label{eqn:cre}
\widetilde{P}_{kn,km}\cdot c = \frac 1{[k]} \cdot p^{+}_* \left[\prod_{i=1}^{kn} [\mathcal{L}_{kn+1-i}]^{\otimes S_{m/n}(i)} \otimes \left( \sum_{j=0}^{k-1} \frac {(qt)^j \cdot [\mathcal{L}_n][\mathcal{L}_{2n}]\cdots [\mathcal{L}_{jn}]}{[\mathcal{L}_{n+1}][\mathcal{L}_{2n+1}]\cdots [\mathcal{L}_{jn+1}]} \right) \cdot p^{- *}(c)\right]
\end{equation}
where we denote $[k] = \frac {(q^k-1)(t^k-1)}{(q-1)(t-1)}$ and $S_{m/n}$ are the integral parts of \eqref{def:smn}. Note that the above differs by an overall normalizing constant from the operators of \cite{Negut}. The above formula has geometric meaning, and in the next section we will make it more suitable for computations.
\subsection{Matrix Coefficients} Let us compute the matrix coefficients $(I_\lambda|\widetilde{P}_{kn,km}|I_\mu )$. Define:
\begin{equation}
\label{eqn:defomega}
\omega(x) = \frac {(x-1)(x-qt)}{(x-q)(x-t)}
\end{equation}
The easiest case for us is $n=0$, since $\widetilde{P}_{0,k}$ is multiplication with a given $K-$theory class, and thus is diagonal in the basis $I_\lambda$. More concretely, \eqref{eqn:0} yields:
$$
\exp\left( \sum_{k \geq 0} \alpha_k ( I_\lambda|\widetilde{P}_{0,k}|I_\mu ) z^{k} \right) = \delta_{\lambda}^{\mu} \prod_{\square \in \lambda} \frac {\omega\left( \frac z{\chi(\square)} \right)}{\omega \left( \frac {\chi(\square)}z\right)}
$$
where given a box $\square = (i,j)$ in a Young diagram, its \emph{weight} is $\chi(\square) = q^{i-1}t^{j-1}$. Taking the logarithm of the above gives us:
\begin{equation}
\label{eqn:cartan}
( I_\lambda|\widetilde{P}_{0,k}|I_\mu ) = \delta_{\lambda}^{\mu} \left( \frac {(qt)^k}{(q^k-1)(t^k-1)} + \sum_{\square \in \lambda} \chi(\square)^{k} \right)
\end{equation}
The matrix coefficients of $\widetilde{P}_{kn,km}$ for $n>0$ are written in terms of \textit{standard Young tableaux} (abbreviated $\textrm{SYT}$), so let us recall this notion. Given two Young diagrams $\rho_1 \supset \rho_2$, a $\textrm{SYT}$ between them is a way to index the boxes of $\rho_1 \backslash \rho_2$ with different numbers $1,\ldots,l$ such that any two numbers on the same row or column decrease as we go to the right or up. We will often write:
$$
\rho_1 = \rho_2 + \square_1 + \ldots + \square_l
$$
if we want to point out that the box indexed by $i$ is $\square_i$. Then \cite{Negut} gives us the formula:
$$
( I_\mu|\widetilde{P}_{kn,km}|I_\lambda ) = \frac {\gamma^{kn}}{[k]} \cdot \frac {g_\lambda}{g_\mu} \sum^{\textrm{SYT}}_{\mu = \lambda + \square_1 + \ldots + \square_{kn}} \quad \left[\sum_{j=0}^{k-1} (qt)^{j} \frac {\chi_{n(k-1)+1}\chi_{n(k-2)+1}\cdots \chi_{n(k-j)+1}}{\chi_{n(k-1)}\chi_{n(k-2)}\cdots \chi_{n(k-j)}} \right] \cdot
$$
\begin{equation}
\label{eqn:fix3}
\cdot \frac {\prod_{i=1}^{kn} \chi_i^{S_{m/n}(i)} (qt \chi_i - 1 ) }{\left(1 - qt\frac {\chi_2}{\chi_1}\right)\cdots \left(1 - qt\frac {\chi_{kn}}{\chi_{kn-1}} \right)} \prod_{1\leq i<j \leq kn} \omega^{-1} \left(\frac {\chi_j}{\chi_i} \right) \prod_{1\leq i \leq kn}^{\square \in \lambda} \omega^{-1} \left(\frac {\chi(\square)}{\chi_i} \right)
\end{equation}
where $\chi_i=\chi(\square_i)$, $\gamma = \frac {(q-1)(t-1)}{qt(qt-1)}$ and $g_\lambda$ are the equivariant constants of \eqref{eqn:defg}.
\begin{remark}
Since the multiplication operators by power sums $p_n$ coincide with the operators $P_{n,0}$, equation \eqref{eqn:fix3} can be used to compute their matrix elements in the modified Macdonald basis. Indeed, this computation will agree with the Pieri rules for Macdonald polynomials \cite{Macdonald}, so \eqref{eqn:fix3} can be considered as a generalization of Pieri rules.
\end{remark}
\subsection{Refined invariants via Hilbert schemes}
\label{sub:refined}
Similarly to \eqref{eq:TP0}, we can define operators $\widetilde{P}_{n,m}^{\lambda}$ as conjugates to $P_{n,m}^{\lambda}$ under $\varphi_{\frac 1{1-1/t}}$.
In particular, the operator $\widetilde{P}_{1,0}^{\lambda}$ is conjugate by $\varphi_{\frac 1{1-1/t}}$ to the multiplication operator by $P_{\lambda}(q,t^{-1})$.
Since $\varphi_{\frac 1{1-1/t}}$ is a ring homomorphism, $\widetilde{P}_{1,0}^{\lambda}$ is a multiplication operator by
$$
\varphi_{\frac 1{1-1/t}}(P_{\lambda}(q,t^{-1}))=\frac{t^{-n(\lambda)}\widetilde{H}_{\lambda}(q,t)}{h_{\lambda}(q,t^{-1})}.
$$
and the action of $SL_2(\BZ)$ implies:
\begin{equation}
\label{eq:TP}
\widetilde{P}_{n,m}^{\lambda}:= \frac{t^{-n(\lambda)}}{h_{\lambda}(q,t^{-1})}\widetilde{H}_{\lambda}\left[p_k\to \widetilde{P}_{kn,km}\right].
\end{equation}
As was shown in \eqref{eqn:superpoly}, the super-polynomials are given by:
$$
\mathcal{P}^\lambda_{n,m}(u,q,t) = \left\langle \mathbf{v}(u)|P^{\lambda}_{n,m}|1\right \rangle_{q,t},\ \text{where}\ \mathbf{v}(u) = \sum_{\mu \vdash n} \frac {J_\mu }{h_\mu(q;t) h'_\mu(q;t)} \prod_{\square\in \mu}\left(t^{l'(\square)}-uq^{a'(\square)}\right)
$$
Under the isomorphism of Theorem \ref{thm:fixed}, we can write the above as a matrix coefficient in $K \cong V$:
\begin{equation}
\label{eqn:super2}
\widetilde{\CP}^\lambda_{n,m}(u,q,t) =\left ( \Lambda(u) |\widetilde{P}^{\lambda}_{n,m}|1 \right) ,\ \text{where}\ \Lambda(u) = \sum_{\mu \vdash n} \frac {I_\mu}{g_\mu} \prod_{\square\in \mu}\left(1 - u q^{a'(\square)}t^{l'(\square)}\right)
\end{equation}
and the change of variables is:
\begin{equation}
\label{eqn:change}
\widetilde{\CP}^\lambda_{n,m}(u,q,t) = (-q)^{-n|\lambda|}\mathcal{P}^\lambda_{n,m}(u,q,t^{-1})
\end{equation}
By the equivariant localization formula, we see that the $K-$theory class $\Lambda(u)$ defined above as a sum of fixed points coincides with the exterior class of \eqref{eqn:tautclass}. We have thus expressed our super-polynomials in terms of the geometric operators \eqref{eqn:cre} on the $K-$theory of the Hilbert scheme.
\begin{proof}[Proof of Theorem \ref{thm:main}] By \eqref{eqn:super2}, the uncolored DAHA -- superpolynomial is given by:
$$
\widetilde{\CP}_{n,m}(u,q,t) = ( \Lambda(u) | \widetilde{P}_{n,m} | 1 ) = \sum_{\mu \vdash n} ( I_\mu | \widetilde{P}_{n,m} | 1 ) \cdot \frac {\prod_{\square \in \mu} (1-u\chi(\square))}{g_\mu}
$$
We can use \eqref{eqn:fix3} to compute the above matrix coefficients, and we obtain:
\begin{equation}
\label{eqn:sumsyt}
\widetilde{\CP}_{n,m}(u,q,t) = \sum^{\textrm{SYT}}_{\mu = \square_1+\ldots+\square_n} \frac {\gamma^n}{g_\mu} \cdot \frac {\prod_{i=1}^{n} \chi_i^{S_{m/n}(i)} (1-u\chi_i)(qt\chi_i - 1)}{\left(1 - qt \frac {\chi_{2}}{\chi_{1}}\right) \cdots \left(1 - qt \frac {\chi_{n}}{\chi_{n-1}}\right)}\prod_{1\leq i < j\leq n} \omega^{-1} \left( \frac {\chi_j}{\chi_i} \right)
\end{equation}
Changing $t\rightarrow t^{-1}$ gives us formula \eqref{eqn:for}.
\end{proof}
As for the colored knot invariant $\widetilde{\CP}_{n,m}^\lambda$
, it is also a particular matrix coefficient of the operator $\widetilde{P}_{n,m}^\lambda$:
$$
\widetilde{\CP}^\lambda_{n,m}(u,q,t) = ( \Lambda(u) | \widetilde{P}^\lambda_{n,m} | 1 ) = \sum_{\mu \vdash n|\lambda|} ( I_\mu | \widetilde{P}^\lambda_{n,m} | 1 ) \cdot \frac {\prod_{\square \in \mu} (1 - u\chi(\square))}{g_\mu}
$$
By \eqref{eq:TP}, the operator $\widetilde{P}_{n,m}^\lambda$ expands in the operators $\widetilde{P}_{nk,mk}$ given by the same formula as modified Macdonald polynomials expand in the power-sum functions $p_k$. For computational purposes, the task then becomes to compute expressions of the form:
$$
\sum_{\mu \vdash n(k_1+\ldots+k_t)} ( I_\mu | \widetilde{P}_{nk_1,mk_1}\cdots\widetilde{P}_{nk_t,mk_t} | 1 ) \cdot \frac {\prod_{\square \in \mu} (1 - u\chi(\square))}{g_\mu}
$$
for any $k_1,\ldots,k_t$. These can be computed by iterating \eqref{eqn:fix3} and the result will be a sum over standard Young tableaux. The summand will be in general more complicated than \eqref{eqn:sumsyt}, but it can be taken care of by a computer. While we do not yet have a ``nice" formula suitable for writing down in a theoretical paper, we may use the geometric viewpoint to prove Cherednik's second conjecture:
\begin{proof}[Proof of Conjecture \ref{conj2}] In terms of the modified superpolynomials $\widetilde{\CP}$, the desired identity becomes:
$$
\widetilde{\CP}_{n,m}^{\lambda^t,red}(u,q,t) = \widetilde{\CP}_{n,m}^{\lambda,red}(u,t,q),\ \text{where}\ \widetilde{\CP}_{n,m}^{\lambda,red}(u,q,t):=(-q)^{-(n-1)|\lambda|}\mathcal{P}_{n,m}^{\lambda,red}(u,q,t^{-1}).
$$
Recall that
$$
\mathcal{P}_{1,0}^{\lambda}(u,q,t)=\frac{1}{h_{\lambda}(q,t)}\prod_{\square\in \lambda}(t^{l'(\square)}-uq^{a'(\square)}),
$$
so
$$
\widetilde{\CP}_{1,0}^{\lambda}(u,q,t)=\frac{t^{-n(\lambda)}}{h_{\lambda}(q,t^{-1})}\prod_{\square\in \lambda}(1-uq^{a'(\square)}t^{l'(\square)}),
$$
hence
$$
\widetilde{\CP}_{n,m}^{\lambda,red}(u,q,t)=\frac{\widetilde{\CP}_{n,m}^{\lambda}(u,q,t)}{\widetilde{\CP}_{1,0}^{\lambda}(u,q,t)}=
\frac{h_{\lambda}(q,t^{-1})(\Lambda(u) |\widetilde{P}^{\lambda}_{n,m}|1 )}{t^{-n(\lambda)}\prod_{\square\in \lambda}(1-uq^{a'(\square)}t^{l'(\square)})}.
$$
Therefore by \eqref{eq:TP} we get
$$
\widetilde{\CP}_{n,m}^{\lambda,red}(u,q,t)=\frac{\left(\Lambda(u) |\widetilde{H}_{\lambda}(q,t)\left[p_k\to \widetilde{P}_{km,kn}\right]|1\right)}{\prod_{\square\in \lambda}(1-uq^{a'(\square)}t^{l'(\square)})}.
$$
By \eqref{hmu symmetry} $\widetilde{H}_{\lambda}(q,t)=\widetilde{H}_{\lambda^t}(t,q)$, and by \eqref{eqn:fix3} the matrix coefficients of $\widetilde{P}_{km,kn}$ are invariant under the
switching $q\leftrightarrow t$ and the transposition of the axis, as well as $\Lambda(u)$.
\end{proof}
\subsection{Constant term formulas} Formula \eqref{eqn:sumsyt} can be repackaged as a contour integral. We may write:
$$
\widetilde{\CP}_{n,m}(u,q,t) = ( \Lambda(u) |\widetilde{P}_{n,m}|1 ) = (1|\widetilde{P}_{-n,m}|\Lambda(u) )
$$
where $\widetilde{P}_{-n,m}$ is the adjoint of $\widetilde{P}_{n,m}$. Formula (4.12) of \cite{Negut} gives us the following integral formula for this expression:
\begin{equation}
\label{eqn:integral}
\widetilde{\CP}_{n,m}(u,q,t) = \int
\frac {\prod_{i=1}^n z_i^{S_{m/n}(i)} \cdot \frac {1 - u z_i}{z_i-1} }{\left(1-qt\frac {z_{2}}{z_{1}}\right) \cdots \left(1-qt\frac {z_{n}}{z_{n-1}}\right)} \prod_{1\leq i < j\leq n} \omega \left( \frac {z_i}{z_j} \right) \frac {dz_1}{2\pi i z_1}\cdots \frac {dz_n}{2\pi i z_n}
\end{equation}
where the contours of the variables $z_i$ surround 1, with $z_1$ being the outermost and $z_{n}$ being the innermost (we take $q,t$ very close to 1). We can move the contours so that they surround $0$ and $\infty$, and then the integral comes down to the following residue computation:
$$
\widetilde{\CP}_{n,m}(u,q,t) = \left(\textrm{Res}_{z_{n}=0} - \textrm{Res}_{z_{n}=\infty} \right) \cdots \left( \textrm{Res}_{z_1=0} - \textrm{Res}_{z_1=\infty}\right)
$$
\begin{equation}
\label{eqn:resknot}
\frac 1{z_1\cdots z_n} \cdot \frac {\prod_{i=1}^n z_i^{S_{m/n}(i)} \cdot \frac {1-uz_i}{z_i-1} }{\left(1 - qt\frac {z_{2}}{z_{1}}\right) \cdots \left(1 - qt\frac {z_{n}}{z_{n-1}}\right)} \prod_{1\leq i < j\leq n} \omega \left( \frac {z_i}{z_j} \right)
\end{equation}
We will compute the above residue in section \ref{sub:tesler}, which will give another combinatorial way to compute $\widetilde{\CP}_{n,m}$.
\subsection{} Let us say a few words about the viewpoint of Nakajima in \cite{Nak}, which relates knot invariants to the following map:
$$
\Psi_d:V \longrightarrow K_d, \qquad \qquad \Psi_d(f) = f(\mathcal{W}^\vee)
$$
where $\mathcal{W}$ is the universal bundle on $\Hilb_d$. As a $K-$theory class (and this will be sufficient for the purposes of the present paper), it is given by $[\mathcal{W}] = 1 - (1-q)(1-t)[\mathcal{T}]$. We may take the direct product of the above maps over all $d$ and define:
\begin{equation}
\label{eqn:s}
\Psi:V \longrightarrow K, \qquad \qquad \Psi = \prod_{d=0}^\infty \Psi_d
\end{equation}
The map $\Psi$ defined above takes values in a certain completion of $K$, since we consider the direct product. Via equivariant localization, we see that:
\begin{equation}
\label{eqn:sloc}
\Psi(f) = \sum_{\lambda} \frac {I_\lambda}{g_\lambda} \cdot f\left( 1 - (1-q^{-1})(1-t^{-1})\sum_{\square \in \lambda} \chi(\square)^{-1} \right)
\end{equation}
The right hand side of \eqref{eqn:sloc} uses plethystic notation of symmetric functions, which is described in \cite{Nak}. In \emph{loc. cit.}, Nakajima uses the map \eqref{eqn:s} to study knot invariants, essentially by using the viewpoint given by the left hand side of relation \eqref{eqn:lemma} where the $S$-matrix is realized as an operator on $V_N$. Our viewpoint, outlined in the previous sections, is to compute the same knot invariants by using the right hand side of \eqref{eqn:lemma} and interpret $S$ as an automorphism of the algebra $\mathbf{SH}$. The two perspectives produce significantly different formulas.
\section{Representations of the rational Cherednik algebra}
\label{sec:rat}
\subsection{The rational Cherednik algebra}
Rational Cherednik algebras were introduced in \cite{EG} as degenerations of the DAHA.
\begin{definition}
The rational Cherednik algebra of type $A_{n-1}$ with parameter $c$ is:
$$
\mathbf{H}_{c}=\mathbb{C}[\mathfrak{h}]\otimes \mathbb{C}[\mathfrak{h}^{*}]\rtimes \mathbb{C}[S_n],
$$
where $\mathfrak{h}$ is the Cartan subalgebra of $\mathfrak{sl}_n$, and the commutation relations between the various generators are:
$$[x,x']=0,\quad [y,y']=0,\quad gxg^{-1}=g(x), \quad gyg^{-1}=g(y),$$
$$[x,y]=(x,y)-c\sum_{s\in \mathcal{S}}(\alpha_s,x)(\alpha_s^{*},y)s,$$
for any $x\in \mathfrak{h}^{*}$, $y\in \mathfrak{h}$, $g\in S_n$. Here $\mathcal{S}$ denotes the set of all reflections in $S_n$ and $\alpha_s$ is the equation of the reflecting hyperplane of $s\in \mathcal{S}$.
\end{definition}
The polynomial representation also makes sense for rational Cherednik algebras, and its representation space is $M_{c}(n)=\mathbb{C}[\mathfrak{h}]$. The symmetric group acts naturally, $\mathfrak{h}$ acts by multiplication operators and elements of $\mathfrak{h}^{*}$ act by Dunkl operators:
$$
D_y=\partial_y-c\sum_{s\in \mathcal{S}}\frac{(\alpha_s,y)}{\alpha_s}(1-s)
$$
As a generalization of this construction, one can consider the standard module:
$$
M_{c}(\lambda)=\tau_{\lambda}\otimes \mathbb{C}[\mathfrak{h}],
$$
where $\lambda$ is any partition and $\tau_{\lambda}$ is the corresponding irreducible representation of $S_n$. It is well-known that $M_{c}(\lambda)$ has a unique simple quotient $L_{c}(\lambda)$.
\subsection{Finite-dimensional representations}
It turns out that the representation theory of the rational Cherednik algebra depends crucially on the parameter $c$. For example, we have the following classification of finite-dimensional representations.
\begin{theorem}[\cite{BEG}]
\label{def Lmn}
The algebra $\mathbf{H}_c$ only has finite-dimensional representations if $c = \frac mn$ for some $\gcd(m,n)=1$, in which case it has a unique irreducible representation $$L_{\frac mn}=L_{\frac mn}(n).$$
Furthermore (if $m>0$), one has $$\dim L_{\frac mn}=m^{n-1}, \qquad \dim (L_{\frac mn})^{S_n}=\frac{(m+n-1)!}{m!n!}.$$
\end{theorem}
The representation $L_{\frac{m}{n}}$ is canonically graded and carries a grading-preserving action of $S_n$. In particular, it is a representation of $S_n$, so we can define its Frobenius character:
$$
\ch L_{\frac mn} = \frac{1}{n!}\sum_{\sigma\in S_n}\Tr_{L_{\frac mn}}(\sigma)p_1^{k_1}\ldots p_r^{k_r}
$$
where $p_i$ are power sums, and $k_i$ is the number of cycles of length $i$ in the permutation $\sigma$. The Frobenius character makes sense for any representation of $S_n$, and in particular the Frobenius character of the irreducible $\tau_\lambda$ equals the Schur polynomial $s_{\lambda}$.
\begin{theorem}[\cite{BEG}]
The graded Frobenius character of $L_{\frac mn}$ equals
$$
\ch_{q} L_{\frac mn}=\frac{q^{-\frac{(m-1)(n-1)}{2}}}{[m]_{q}}\phi_{[m]}(h_{n}),
$$
where $[m]_{q} = \frac {1-q^{m}}{1-q}$ and $\phi_{[m]}:\Lambda\to \Lambda$ is the homomorphism defined by $\phi_{[m]}(p_k)=p_k\frac{1-q^{km}}{1-q^{k}}$.
\end{theorem}
For $m=n+1$, Gordon observed a close relation between the representation $L_{\frac{n+1}{n}}$ and Haiman's work. Gordon constructs a certain filtration on $L_{\frac{n+1}{n}}$ and proves the following result.
\begin{theorem}[\cite{Gordon}]
\label{thm:gordon}
The bigraded Frobenius character of $\gr L_{\frac{n+1}{n}}$ is given by the formula
$$\ch_{q,t} \gr L_{\frac{n+1}{n}}=\nabla e_n.$$
\end{theorem}
In \cite{GORS}, Gordon's filtration was generalized to all finite-dimensional representations $L_{\frac{m}{n}}$ and it was conjectured that the bigraded character $\gr L_{\frac{m}{n}}$ is tightly related to the Khovanov-Rozansky homology of the $(m,n)$ torus knot. In light of the conjectures of \cite{AS}, we formulate the following:
\begin{conjecture}
\label{conj:frob}
The bigraded Frobenius character of $\gr L_{\frac{m}{n}}$ is given by:
$$
\ch_{q,t} \gr L_{\frac{m}{n}} = \widetilde{P}_{n,m}\cdot 1
$$
where $\widetilde{P}_{n,m}$ are the transformed DAHA elements of Subsection \ref{sub:fhilb}.
\end{conjecture}
When $m=n+1$, the conjecture follows from Theorem \ref{thm:gordon} and Corollary \ref{pn1} below. Conjecture \ref{conj:frob} is also supported by numerical computations, and it is compatible with some structural properties. For example, the symmetry between the $q$ and $t$ gradings of the character has been proven in \cite{GORS}, and this symmetry is manifest in the operators $\widetilde{P}_{n,m}$. Moreover, Conjecture \ref{conj:frob} was proved in \emph{loc. cit.} at $t=q^{-1}$, by showing that the knot invariant equals the singly graded Frobenius character, see also Section \ref{sec:CY limit} for details. It was also observed by Gordon and Stafford that:
$$\ch_{q,t}\gr L_{\frac{m+n}{n}}=\nabla \ch_{q,t} \gr L_{\frac{m}{n}},$$
where $\nabla$ is the operator of \eqref{def:nabla}. This matches with the equality:
\begin{equation}
\label{eqn:nabla}
\widetilde{P}_{n,n+m} = \nabla \widetilde{P}_{n,m} \nabla^{-1}
\end{equation}
which follows easily from the definition of $\widetilde{P}_{n,m}$ in Subsection \ref{sub:fhilb}.
\begin{remark}
The above only deals with the uncolored case, since the representation-theoretic interpretation of colored refined knot invariants has yet to be developed. It is proved in \cite{EGL} that at $t=q^{-1}$ the unrefined $\lambda$-colored invariant of the $(m,n)$ torus knot is given by the character of the infinite-dimensional irreducible representation $L_{\frac{m}{n}}(n\lambda)$. It would be interesting to define a filtration on $L_{\frac{m}{n}}(n\lambda)$ that matches their character with refined invariants.
\end{remark}
\subsection{The Gordon-Stafford construction}
Conjecture \ref{conj:frob} is part of a correspondence between representations of the rational Cherednik algebra and coherent sheaves on Hilbert schemes, which we will now discuss. Kashiwara and Rouquier (\cite{KR}) have constructed a quantization of the Hilbert scheme depending on the parameter $c$, such that the category of coherent sheaves over this quantization is equivalent to the category of representations of $\mathbf{H}_{c}$. In characteristic $p$, the analogous construction has been carried out by Bezrukavnikov-Finkelberg-Ginzburg (\cite{BFG}). We will only be concerned with characteristic 0, in which case the initial result of Gordon and Stafford (\cite{GS}) claims the existence of a map:
\begin{equation}
\label{eqn:gs}
D^{b}\text{Rep}(\mathbf{H}_{c}) \longrightarrow D^{b}\text{Coh}(\Hilb_n)
\end{equation}
for all $c$. The category on the left consists of filtered representations (see \cite{GS} for the exact definition) of the rational Cherednik algebra. One may ask about the image of the unique irreducible finite-dimensional representation $L_{\frac mn}$ under the above assignemnt. During our discussions with Andrei Okounkov, the following conjecture was proposed:
\begin{conjecture}
\label{conj:big}
Under the Gordon-Stafford map \eqref{eqn:gs}, $L_{\frac mn}$ is sent to:
\begin{equation}
\label{eqn:gs2}
\mathcal{F}_{\frac mn} := p_* \left(\mathcal{L}_n^{S_{m/n}(1)} \otimes \ldots \otimes \mathcal{L}_1^{S_{m/n}(n)} \right)
\end{equation}
where $p:\Hilb_{0,n} \longrightarrow \Hilb_n$ is the projection map from the flag Hilbert scheme to the Hilbert scheme (see Subsection \ref{sub:fhilb} for the notations), and $\mathcal{L}_1,\ldots,\mathcal{L}_n$ are the tautological line bundles.
\end{conjecture}
The flag Hilbert scheme together with the projection $p$ should be understood in the DG sense, see \cite{Negut} for details. In fact, the above conjecture is a particular case of a far-reaching conjectural framework of Bezrukavnikov--Okounkov, concerning filtrations on the derived category of the Hilbert scheme.
To support Conjecture \ref{conj:big}, note that the functor \eqref{eqn:gs} matches the bigraded character of representations with the biequivariant $K-$theory classes of coherent sheaves. Therefore, Conjecture \ref{conj:big} implies that:
$$
\ch_{q,t} \gr L_{\frac{m}{n}} = \left[ p_*\left(\mathcal{L}_n^{S_{m/n}(1)} \otimes \cdots \otimes \mathcal{L}_1^{S_{m/n}(n)}\right) \right]
$$
Comparing with \eqref{eqn:cre}, we see that the object in the right hand side is simply $\widetilde{P}_{n,m} \cdot 1 \in K$. Therefore, Conjecture \ref{conj:big} implies Conjecture \ref{conj:frob}.
\subsection{Affine Springer fibres}
\label{sec:affine flag}
Yet another geometric realization of $L_{\frac mn}$ is provided by the affine Springer fibres in the affine flag variety. Let us recall that the affine Grassmannian $\Gr_n$ of type $A_{n-1}$ can be defined as the moduli space of subspaces
$V\subset \mathbb{C}((t))$ satisfying $t^nV\subset V$ and a certain normalization condition. Similarly, the affine flag variety $\Fl_n$ can be defined as the moduli space of flags of subspaces in $\mathbb{C}((t))$ of the form
$V_1\supset V_2\supset\ldots\supset V_{n}\supset V_{n+1}=t^{n}V_{1}$ such that $\dim V_i/V_{i+1}=1$.
Recall that an affine permutation (of type $A_{n-1}$) is a bijection $\omega:\mathbb{Z}\to \mathbb{Z}$ such that $\omega(x+n)=\omega(x)+n$ for all $x$ and $\sum_{i=1}^{n}\omega(i)=\frac{n(n+1)}{2}.$
It is well known that $\Fl_n$ is stratified by the affine Schubert cells $\Sigma_{\omega}$ labelled by the affine permutations. The {\em homogeneous affine Springer fiber}:
$$
\Sigma_{\frac mn} \subset \Fl_n \qquad (\text{resp. } \Sigma^{\Gr}_{\frac mn} \subset \Gr_n)
$$
is defined as as set of flags (resp. subspaces) invariant under multiplication by $t^m$, where, as above, we assume that $\gcd(m,n)=1$. It is known to be a finite-dimensional projective variety \cite{GKM,KL,LS} and the total dimension of the homology equals (\cite{Hikita,LS}):
$$
\dim H^{*}(\Sigma_{\frac mn})=m^{n-1}, \qquad \dim H^{*}(\Sigma^{\Gr}_{\frac mn})=\frac{(m+n-1)!}{m!n!}.
$$
The similarity between this equation and Theorem \ref{def Lmn} suggests a relation between $L_{\frac mn}$ and the homology of $\Sigma_{\frac mn}$. Indeed, in \cite{OY,VV} the authors constructed geometric actions of the DAHA and trigonometric / rational Cherednik algebras on the space $H^{*}(\Sigma_{\frac mn})$ equipped with certain filtrations. In all these constructions, the spherical parts of the corresponding representations can be naturally identified with $H^{*}(\Sigma^{\Gr}_{\frac mn})$, also equipped with certain filtrations. It is important to mention that the homological grading on $H^{*}(\Sigma_{\frac mn})$ {\em does not} match the representation-theoretic grading on $L_{\frac mn}$. On the other hand, the {\em bigraded} character of $\gr H^{*}(\Sigma_{\frac mn})$
is expected to match the bigraded character of $\gr L_{\frac mn}$ after some regrading, when one takes into account both the geometric filtration on the homology and the generalized Gordon filtration on $L_{\frac mn}$ (see \cite{GORS} for the precise conjecture).
In the next section we give an explicit combinatorial counterpart of this conjecture (Conjecture \ref{conj:shuffle}), which can be explicitly verified on a computer. By Conjecture \ref{conj:frob}, the bigraded character of $\gr L_{\frac mn}$ is given by $\widetilde{P}_{n,m}\cdot 1$ and hence can be computed combinatorially using \eqref{eqn:fix3}. On the other hand, one can try to compute the bigraded character of $\gr H^{*}(\Sigma_{\frac mn})$ using some natural basis in the homology, which is expected to be compatible with the geometric filtration.
\begin{definition}(\cite{GMV})
We call an affine permutation $\omega$ $m$--stable, if $\omega(x+m)>\omega(x)$ for all $x$.
\end{definition}
\begin{theorem}(\cite{GMV})
The intersection of an affine Schubert cell with the affine Springer fiber $\Sigma_{\frac mn}$ is either empty or isomorphic to an affine space. The nonempty intersections correspond to the $m$-stable affine permutations $\omega$, and the dimension of the corresponding cell in $\Sigma_{\frac mn}$ equals:
$$
\dim \Sigma_{\omega}\cap \Sigma_{\frac mn}=|\left\{(i,j)|\omega(i)<\omega(j), 0<i-j<m, 1\le j\le n\right\}|.
$$
\end{theorem}
In the homology of $\Sigma_{\frac mn}$ one then have a combinatorial basis corresponding to these cells, with the homological gradings given by the above equation.
In \cite{GMV} the $m$-stable affine permutations has been identified by an explicit bijection with another combinatorial object, the so-called $m/n$-parking functions, see Section \ref{sec:pf}. The dimension of a cell is translated to a certain combinatorial statistics $\textrm{dinv}$ on parking functions, which has been obtained earlier by Hikita in \cite{Hikita}.
We also conjecture that the geometric filtration on the homology is compatible with the basis of cells, and admits an easy combinatorial description as the ``area" of the corresponding parking function. Modulo this conjecture, one can show that the bigraded character of $\gr H^{*}(\Sigma_{\frac mn})$ coincides with the combinatorial expression \eqref{eq:def Fr}.
One can similarly describe the cell decomposition of $\Sigma^{\Gr}_{\frac mn}$, which turns out to coincide with the compactified Jacobian of the plane curve singularity $\{x^m=y^n\}$. The affine Schubert cells in $\Gr_n$ cut out affine cells in $\Sigma^{\Gr}_{\frac mn}$, which can be labeled either by the $m$--stable permutations with additional restrictions
(\cite{GMV}) or by the Dyck paths in the $m\times n$ rectangle (\cite{GM1,GM2}). The dimension of such a cell can be rewritten as $\frac{(m-1)(n-1)}{2}-h_{+}(D)$, where $h_{+}(D)$ is an explicit combinatorial statistic on the corresponding Dyck path $D$
(see Section \ref{sec:catalan}).
\section{Combinatorial consequences}
\label{sec:comb}
\subsection{} In this section we focus on the combinatorial structure of uncolored refined knot polynomials. By (\ref{eqn:sumsyt}), we have for $\gcd(n,m)=1$:
\begin{equation}
\label{eqn:adriano}
\widetilde{P}_{n,m}\cdot 1 = \sum_{\lambda \vdash n}c_{n,m}(\lambda) \frac {\widetilde{H}_{\lambda}}{g_\lambda},
\end{equation}
where $c_{n,m}(\lambda)$ is the sum of terms $c_{n,m}(T)$ over all standard Young tableaux $T$ of shape $\lambda$:
\begin{equation}
\label{cnd}
c_{n,m}(T)= \gamma^n\frac {\prod_{i=1}^{n} \chi_i^{S_{m/n}(i)} (qt\chi_i-1)}{\prod_{i=1}^{n-1}\left(1 - qt \frac {\chi_{i+1}}{\chi_i} \right)} \prod_{1\leq i < j\leq n} \omega^{-1} \left( \frac {\chi_j}{\chi_i} \right)
\end{equation}
Recall that $\chi_i$ denotes the weight of box $i$ in the standard Young tableau $T$ and the constants $S_{m/n}(i)$ are defined by (\ref{def:smn}). Some of the coefficients $c_{n,m}(\lambda)$ have appeared in various sources: for $m=1$ they are remarkably simple and were computed first in \cite{GH} and later rediscovered in \cite{ORS,Sha}. For general $m$ and small $n$ some of these coefficients were computed in \cite[Section 5.3]{ORS}, \cite{DMMSS} and \cite{Sha}. Although the individual terms $c_{n,m}(T)$ have a nice factorized form, their sums $c_{n,m}(\lambda)$ look less attractive, for example:
$$
c_{7,2}(4,3) = (1-q)^2 (1-t)^2 (1-t^2)(1-t^3)(qt-1)
$$
$$
(q^3 t^3+q^3 t^2-q^3+q^2 t^5+2 q^2 t^4+q^2 t^3-q^2 t+q t^6+q t^5-q t^4-2 q t^3-q t^2+t^7-t^5-t^4-t^3)
$$
For hook shapes of size $n$, the coefficients $c_{n,m}(k,1,\ldots,1)$ are equal to a product of linear factors times a sum of $n$ terms. Explicitly, the following formula was computed in \cite{Pieri} using shuffle algebra machinery:
$$
c_{n,m}(k,1,\ldots,1) =\frac {(1 - q)(1 - t)}{q^nt^n} \prod_{i=1}^{k-1} (1-q^i) \prod_{i=1}^{n-k} (1-t^i) \left(\sum_{i=0}^{n-1} q^{\sum_{j=0}^{k-1} \left \lfloor \frac {mj+i}n \right\rfloor}t^{\sum_{j=1}^{n-k} \left \lceil \frac {mj-i}n \right\rceil} \right)
$$
for all coprime $m$ and $n$, and all $1\leq k \leq n$. For small $k$, this agrees with the computations in \cite[Section 5.3]{ORS}.
Further, we give a combinatorial interpretation of uncolored refined knot invariants, generalizing the so-called {\em ``Shuffle Conjecture''} of \cite{HHRLU}. In \cite{GH} A. Garsia and M. Haiman introduced a bivariate deformation of Catalan numbers, and in \cite{GHagl} (see also \cite{HCatalan}) it was proved that it can be obtained as a weighted sum over Dyck paths. In \cite{GM1} (see also \cite{GM2}) this weighted sum was reinterpreted as a sum over cells in a certain affine Springer fiber and generalized to the rational case. We conjecture that the rational extension of $q,t-$Catalan numbers is given by the $u=0$ specialization of the refined invariant (Conjecture \ref{conj:catalan}) and thus can be computed as a certain sum over tableaux.
The coefficients of the full $u-$expansion of the refined invariant are given by the generalized Schr\"oder numbers. The combinatorial statistics for these numbers was conjectured in \cite{EHKK} and proved in \cite{HSchroeder}, and the rational extension of these statistics was conjectured in \cite{ORS}. We give a conjectural formula for them in terms of tableaux in Conjecture \ref{conj:schroeder}.
It was conjectured in \cite{HHRLU} that the vector $\nabla e_n = \widetilde{P}_{n,n+1} \cdot 1$ can be written as a certain sum over parking functions on $n$ cars, and it was shown that the combinatorial formulas for $q,t-$Catalan and $q,t-$Schr\"oder numbers follow from this conjecture. This combinatorial sum was reinterpreted in \cite{Hikita} as a weigthed sum over the cells in a certain parabolic affine Springer fiber, and a rational extension of the combinatorial statistics of \cite{HHRLU} has been proposed. We conjecture that the symmetric polynomials constructed in \cite{Hikita} coincide with $\widetilde{P}_{n,m} \cdot 1$. This conjecture is supported by vast experimental data provided to us by Adriano Garsia.
It has been conjectured in \cite{GORS} that the weighted sums of \cite{Hikita} (also \cite{GM1,ORS})
compute the bigraded Frobenius characters of the finite-dimensional representations $L_{\frac{m}{n}}$ (and their specializations), and the Poincar\'e polynomials of Khovanov-Rozansky homology of torus knots. On the other hand, it has been conjectured in \cite{AS,Ch} that refined knot invariants compute the Poincar\'e polynomials of Khovanov-Rozansky homology. Although all of these conjectures remain open, the ``rational Shuffle Conjecture" (Conjecture \ref{conj:shuffle}) provides a consistency check for them, since its left and right hand side are explicit combinatorial expressions independent of knot homology or filtration on $L_{\frac{m}{n}}$ .
Finally, we use the notion of Tesler matrices introduced in \cite{HTesler} (see also \cite{AGHRS,HTesler2}) to compute the residue \eqref{eqn:resknot}, and thus give an explicit formula for refined knot invariants. We will use this to prove the specialization of the rational Shuffle Conjecture at $t=1$.
\subsection{Generalized $q,t$-Catalan numbers}
\label{sec:catalan}
We define a $m/n$ Dyck path to be a lattice path in a $m \times n$ rectangle from the top left to the bottom right corner, which always stays below the diagonal connecting these two corners. Alternatively, a Dyck path is a Young diagram inscribed in the right triangle with vertices $(0,0),(m,0)$ and $(0,n)$. We denote the set of all $m/n$ Dyck paths by $Y_{m/n}$, and it is well known that:
$$
|Y_{m/n}|=\frac{(m+n-1)!}{m!n!}.
$$
Given a Dyck path $D$, we define, following \cite{GM1} and \cite{GM2}, the statistic:
$$
h_{+}(D)=\left\{x\in D\ \vline\ \frac{a(x)}{l(x)+1}<\frac{m}{n}<\frac{a(x)+1}{l(x)}\right\}.
$$
We define the $m/n$ rational Catalan number as the following weighted sum over Dyck paths:
$$
C_{n,m}(q,t)=\sum_{D\in Y_{m/n}} q^{\delta_{m,n}-|D|}t^{h_{+}(D)},
$$
where $\delta_{m,n}=\frac{(m-1)(n-1)}{2}.$ The polynomial $C_{n,m}(q,t)$ is symmetric in $m$ and $n$ by construction, and it has been conjectured in \cite{GM2} that it is symmetric in $q$ and $t$ as well. Here we propose strengthening the $q,t-$symmetry conjecture by the following:
\begin{conjecture}
\label{conj:catalan}
The following relation holds:
$$
C_{n,m}(q,t) = (h_n|\widetilde{P}_{n,m}|1) = \sum_{\lambda \vdash n} \frac {c_{n,m}(\lambda)}{g_\lambda}
$$
where $h_n \in V$ is the complete symmetric function.
\end{conjecture}
Indeed, the matrix coefficient in the right hand side of the above relation is symmetric in $q$ and $t$, since \eqref{cnd} implies:
$$
c_{n,m}(\lambda;q,t) = c_{n,m}(\lambda^t;t,q)
$$
Therefore, Conjecture \ref{conj:catalan} implies that:
$$
C_{n,m}(q,t) = C_{n,m}(t,q)
$$
Note that for $m=n+1$, Conjecture \ref{conj:catalan} follows from the results of \cite{GHagl} and Corollary \ref{pn1}.
In \cite{ORS}, the polynomials $C_{n,m}(q,t)$ have been extended to accommodate the extra variable $u$. Given a Dyck path $D$ and an internal vertex $P$, we define $\beta(P)$ to be the number of horizontal segments of $D$ intersected by the line passing through $P$ and parallel to the diagonal (see Figure \ref{fig:beta}). Let $v(D)$ denote the set of internal vertices of $D$.
\begin{figure}[ht]
\begin{tikzpicture}
\draw (0,0)--(0,5)--(7,5)--(7,0)--(0,0);
\draw [dashed] (0,5)--(7,0);
\draw [ultra thick] (0,5)--(0,3.5)--(0.3,3.5)--(0.3,3)--(2,3)--(2,2)--(4,2)--(4,1)--(5,1)--(5,0)--(7,0);
\draw (3.81,1.01) node {$P \bullet$};
\draw [dashed] (-1.6,5)--(5.4,0);
\end{tikzpicture}
\caption{Computation of the statistic $\beta(P)$}
\label{fig:beta}
\end{figure}
\begin{conjecture}
\label{conj:schroeder}
The following equation holds:
$$
\sum_{D\in Y_{m/n}} q^{\delta_{m,n}-|D|}t^{h_{+}(D)}\prod_{P\in v(D)}(1-ut^{-\beta(P)}) =
\widetilde{\CP}_{n,m}(u,q,t)
$$
\end{conjecture}
For $m=n+1$ a similar identity was conjectured in \cite{EHKK} and proved in \cite{HSchroeder} (see \cite{HBook} and \cite[Section A.3]{ORS} for more details).
\subsection{The Rational Shuffle Conjecture}
\label{sec:pf}
The symmetric polynomial $\widetilde{P}_{n,m}\cdot 1 \in V$ has a combinatorial interpretation.
Let us define a $m/n$ parking function as a function:
$$
f:\{1,\ldots,m\}\to \{1,\ldots,n\},\ \text{such that }\ |f^{-1}([1,i])|\ge \frac{mi}{n} \quad \forall \ i
$$
Alternatively, a parking function can be presented as a standard Young tableau $F$ of skew shape $(D+1^{m})\setminus D$, where $D$ is a $m/n$ Dyck path. Given such a tableau, the function $f$ can be reconstructed by sending each $i$ to the $x$-coordinate of the box labeled by $i$ in the tableau. It is clear that this correspondence is bijective.
\begin{figure}[ht]
\begin{tikzpicture}
\draw (0,0)--(0,5)--(3,5)--(3,0)--(0,0);
\draw (0,1)--(3,1);
\draw (0,2)--(3,2);
\draw (0,3)--(3,3);
\draw (0,4)--(3,4);
\draw (1,0)--(1,5);
\draw (2,0)--(2,5);
\draw [dashed] (3,0)--(0,5);
\draw [ultra thick] (0,5)--(0,2)--(1,2)--(1,1)--(2,1)--(2,0)--(3,0);
\draw (0.5,4.5) node {$5$};
\draw (0.5,3.5) node {$2$};
\draw (0.5,2.5) node {$1$};
\draw (1.5,1.5) node {$4$};
\draw (2.5,0.5) node {$3$};
\end{tikzpicture}
\caption{A $3/5$ Dyck path and a parking function}
\label{expf}
\end{figure}
Given a box $x=(i,j)$, let us define $r(x)=mn-m-n-mi-nj$. Given a $m/n$ parking function $F$,
define:
$$
s(F)=|\left\{(x,y)\ :\ x>y \ \text{ such that } \ r(F(y))<r(F(x))<r(F(x))+m\right\}|
$$
and define $s_{\max}(D)$ to be the maximum of $s(F)$ over all parking functions $F$ constructed on the Dyck path $D$.
Let
$$
\textrm{dinv}(F)=s(F)+h_{+}(D)-s_{\max}(D).
$$
Finally, define the descent set of $F$ by:
$$
\textrm{Des}(F)=\{x\ :\ r(F(x))>r(F(x+1))\}
$$
Let $\text{PF}_{m/n}$ denote the set of all $m/n$ parking functions. The following symmetric function has been independently constructed in \cite{armstrong} and \cite{Hikita}:
\begin{equation}
\label{eq:def Fr}
\Fr_{n,m}=\sum_{F\in \text{PF}_{m/n}}q^{\delta_{m,n}-|D|}t^{\textrm{dinv}(F)}Q_{\textrm{Des}(F)},
\end{equation}
where $Q_{\textrm{Des}(F)}$ is the Gessel quasisymmetric function \cite{Gessel,HHRLU} associated with the set $\textrm{Des}(F)$. In \cite{Hikita}, it was proved that $\Fr_{n,m}$ specializes to the symmetric function from \cite{HHRLU} when $m=n+1$, and that $\Fr_{n,m}$ computes the Frobenius character of the $S_n$ action in the homology of a certain Springer fiber in the affine flag variety equipped with extra filtration, as in Section \ref{sec:affine flag}. The following conjecture generalizes this fact for all $m$
and it arose during private communication between the first author and Adriano Garsia:
\begin{conjecture}
\label{conj:shuffle}
The following identity holds:
$$
\Fr_{n,m}=\widetilde{P}_{n,m}\cdot 1
$$
\end{conjecture}
When $m=n+1$, it follows from Corollary \ref{pn1} that this conjecture specializes to the main conjecture of \cite{HHRLU}. At the suggestion of Adriano Garsia, we give a constant term formula for $\widetilde{P}_{n,m}\cdot 1$, which is related to the SYT formula of \eqref{eqn:adriano} in the same way as formula \eqref{eqn:integral} is related to formula \eqref{eqn:sumsyt}. The following formula follows from \cite{Negut}:
\begin{equation}
\label{eqn:constant}
\widetilde{P}_{n,m}\cdot 1 = \int
\frac {\Psi_n(e(qtz_1)\cdots e(qtz_n)) \prod_{i=1}^n z_i^{S_{m/n}(i)} }{\left(1 - qt\frac {z_{2}}{z_{1}}\right) \cdots \left(1 - qt\frac {z_{n}}{z_{n-1}}\right)} \prod_{1\leq i < j\leq n} \omega \left( \frac {z_i}{z_j} \right) \frac {dz_1}{2\pi i z_1}\cdots \frac {dz_n}{2\pi i z_n}
\end{equation}
where $\Psi_n$ is the map of \eqref{eqn:sloc} and $e(z) = \sum_i (-z)^i e_i$. The above integral goes over contours that surround $0$ and $\infty$, with $z_1$ being the innermost and $z_{n}$ being the outermost contour. One can compute the above residues in $z_i$ and produce a sum of symmetric functions indexed by certain matrices of natural numbers. We will show how to do this in the slightly simpler case of the super-polynomials $\widetilde{\CP}_{n,m}$.
\subsection{The $m=n+1$ case}
For $m=n+1$, one can prove that the rational $q,t$--Catalan numbers agree with the $q,t$--Catalan numbers defined in \cite{GH}, and Conjecture \ref{conj:shuffle} agrees with the ``Shuffle conjecture'' of \cite{HHRLU}.
The following proposition follows from the results of \cite{GH}, but we present its proof here for completeness.
\begin{proposition}
One has the identity: $\widetilde{P}_{0,1}(p_n)=e_n$.
\end{proposition}
\begin{proof}
It is enough to prove this above in any $V_N$, since $V$ is the inverse limit of these vector spaces.
Let us recall that:
$$
\overline{P}^{N}_{0,k}=t^{-k(N-1)} \left(P^{N}_{0,k}-\frac {t^{kN}-1}{t^k-1}\right)
$$
If we keep $N$ finite but make the change of variables $\varphi_{\frac 1{1-1/t}}$ of Theorem \ref{thm:fixed}, we obtain the operator:
\begin{equation}
\label{p01}
\widetilde{P}^N_{0,1}=\varphi_{\frac 1{1-1/t}} \circ \overline{P}^{N}_{0,1} \circ \varphi^{-1}_{\frac 1{1-1/t}} = t^{-N+1} \varphi_{\frac 1{1-1/t}} \circ \left(\delta_1-\frac {t^{N}-1}{t-1}\right) \circ \varphi^{-1}_{\frac 1{1-1/t}}
\end{equation}
where $\delta_1$ is the operator of \eqref{delta1}. The operators $\widetilde{P}^N_{0,1}$ stabilize to $\widetilde{P}_{0,1}$.
Using (\ref{p01}), we can rewrite the desired identity as:
\begin{equation}
\label{delta 1 of p}
\delta_1(p_n)=\frac{1-t^N}{1-t}p_n+(-1)^{n}\frac{t^N(1-q^n)}{t^n(1-t)}\varphi^{-1}_{\frac 1{1-1/t}}(e_{n}).
\end{equation}
Indeed, $\partial_{q}^{(i)}p_n=p_n+(q^n-1)x_i^{n},$ so by (\ref{delta1})
$$
\delta_1(p_n)=p_n\sum_{i}A_{i}(x)+(q^n-1)\sum_{i}A_{i}(x)x_i^{n}.
$$
Consider the function $F(z) = \prod_{i=1}^{N}\frac{1-zx_i}{1-ztx_i}=\sum_{z=0}^{\infty} z^{n} F_{n}.$
It has the following partial fraction decomposition:
$$F(z)=\frac{1}{t^N}+\frac{t-1}{t^N}\sum_{i=1}^{N}\frac{A_i(x)}{1-tzx_i},$$
hence
\begin{equation}
\sum_{i}A_{i}(x)x_i^{n}=\begin{cases}\frac{1-t^N}{1-t}\qquad n=0 \\ \frac{F_n t^N}{t^{n}(t-1)} \ \quad n>0 ,\\\end{cases}
\end{equation}
Therefore we have
$$
\delta_1(p_n)=\frac{1-t^N}{1-t}p_n+\frac{t^N(1-q^n)}{t^n(1-t)}F_n.
$$
On the other hand,
$$\ln F(z)=\sum_{i=1}^{N}(\ln(1-zx_i)-\ln(1-ztx_i))=-\sum_{k=1}^{\infty}(1-t^{k})\frac{z^{k}p_{k}}{k},$$
hence
$$F(z)=\varphi^{-1}_{\frac{1}{1-t}}\left[\exp (-\sum_{k=1}^{\infty}\frac{z^{k}p_{k}}{k})\right]=\varphi^{-1}_{\frac{1}{1-t}}\left[\prod_{i}(1-zx_i)\right],$$
and $F_{n}=(-1)^{n}\varphi^{-1}_{\frac{1}{1-t}}(e_{n}).$
\end{proof}
\begin{corollary}
\label{pn1}
The following identities hold:
$$\widetilde{P}_{n,1}\cdot 1 =e_n,\ \widetilde{P}_{n,n+1}\cdot 1=\nabla e_n$$
\end{corollary}
\begin{proof}
It follows from \eqref{commutation} that $\widetilde{P}_{n,1}=[\widetilde{P}_{0,1},\widetilde{P}_{n,0}]$, hence:
$$\widetilde{P}_{n,1}\cdot 1=[\widetilde{P}_{0,1},\widetilde{P}_{n,0}]\cdot 1=\widetilde{P}_{0,1}\widetilde{P}_{n,0}\cdot 1=\widetilde{P}_{0,1}\cdot p_n=e_n.$$
The second identity follows from the equation $\widetilde{P}_{n,n+1}=\nabla \widetilde{P}_{n,1} \nabla^{-1}$.
\end{proof}
As a corollary, we get the following decomposition of the Garsia-Haiman coefficients:
\begin{proposition}
Given a Young diagram $\lambda$, the following identity holds:
\begin{equation}
\label{n1decomposition}
\Pi_{\lambda}B_\lambda = \frac 1{M} \sum^{\text{SYT }}_{T\text{ of shape }\lambda}c_{n,1}(T),
\end{equation}
where:
$$\Pi_{\lambda}=\prod_{\square \in \lambda}^{\chi(\square)\neq 1} (1-\chi(\square)), \qquad B_{\lambda}= \sum_{\square \in \lambda} \chi(\square)$$
The coefficients $c_{n,1}(T)$ are defined by \eqref{cnd}.
\end{proposition}
\begin{proof}
It has been shown in \cite[Theorem 2.4]{GH} that the left hand side of (\ref{n1decomposition}) coincides with the coefficient:
$$
g_\lambda \cdot \langle e_n, \widetilde{H}_{\lambda} \rangle_{q,t^{-1}}
$$
By Corollary \ref{pn1}, this is equal to $g_\lambda \cdot \langle\widetilde{H}_{\lambda}|\widetilde{P}_{n,1}| 1 \rangle_{q,t^{-1}}$, which equals the right hand side of \eqref{n1decomposition} by \eqref{cnd}.
\end{proof}
\subsection{Tesler matrices}
\label{sub:tesler}
By \eqref{eqn:resknot}, the super-polynomials $\mathcal{P}_{n,m}$ ultimately come down to computing the residue:
$$
\widetilde{\CP}_{n,m}(u,q,t) = \left(\textrm{Res}_{z_{n}=0} - \textrm{Res}_{z_{n}=\infty} \right) \cdots \left( \textrm{Res}_{z_1=0} - \textrm{Res}_{z_1=\infty}\right)
$$
\begin{equation}
\label{eqn:res2}
\frac 1{z_1\cdots z_n} \cdot \frac {\prod_{i=1}^n z_i^{S_{m/n}(i)} \cdot \frac {1-uz_i}{z_i-1} }{\left(1 - qt\frac {z_{2}}{z_{1}}\right) \cdots \left(1 - qt\frac {z_{n}}{z_{n-1}}\right)} \prod_{1\leq i < j\leq n} \omega \left( \frac {z_i}{z_j} \right)
\end{equation}
For $m>0$, the above only has residues at $z_i = \infty$. Therefore, let us consider the expansions:
$$
\frac {1 - ux}{x-1} = -1 + (u - 1) \sum_{k\geq 1} x^{-k}, \qquad \omega(x) = 1 + \sum_{k=1} A(k) x^{-k}, \qquad \frac {\omega(x)}{1 - \frac {qt}x} = \sum_{k=1} B(k) x^{-k}
$$
where:
$$
A(k) = -(q-1)(t-1)\frac {q^{k}-t^{k}}{q-t}, \qquad B(k) = \frac {(q^{k+1}-q^{k})-(t^{k+1}-t^{k})}{q-t}
$$
Using these, we can compute the \eqref{eqn:res2} inductively. Take first the residue in the variable $z_1$:
$$
\widetilde{\CP}_{n,m}(u,q,t) = \sum^{x_n^i\geq 0}_{x_{n}^1+\ldots+x_n^n = S_{m/n}(n)} (1-u+u\delta_{x_n^n}^0) \cdot A(x_n^{n-1}) \prod^{x_n^i>0}_{i<n-1} B(x^i_n)
$$
$$
\textrm{Res}_{z_{n}=\infty} \cdots \textrm{Res}_{z_{2}=\infty} \frac 1{z_1 \cdots z_{n-1}} \frac {\prod_{i=2}^{n} z_i^{S_{m/n}(i) + x_i^1} \cdot \frac {1-uz_i}{z_i-1}}{\left(1 - qt\frac {z_{2}}{z_{1}} \right) \cdots \left(1 - qt\frac {z_{n}}{z_{n-1}} \right)}\prod_{1\leq i < j\leq n-1} \omega \left( \frac {z_i}{z_j} \right)
$$
Following \cite{HTesler} (see also \cite{HTesler2} and \cite{AGHRS}), we introduce the notion of {\em Tesler matrix}. An upper triangular matrix $X=\{x_j^i \geq 0\}_{1\leq i\leq j\leq n}$ is called a $m/n$ Tesler matrix if it satisfies the following system of equations:
\begin{equation}
\label{eq:tesler}
x_{i}^{i}+\sum_{j>i}x_{j}^{i}-\sum_{j<i}x_{i}^{j} = S_{m/n}(i) \qquad \forall \ i
\end{equation}
We will denote the set of all $m/n$ Tesler matrices by $\textrm{Tes}_{m/n}$. Taking next the residues in the variables $z_{2},\ldots,z_n$ gives us:
\begin{equation}
\label{eqn:tes1}
\widetilde{\CP}_{n,m}(u,q,t) = \sum_{X\in \textrm{Tes}_{m/n}} \prod_{1\leq i \leq n}^{x_i^i >0} (1-u) \prod_{1\leq i \leq n-1} B(x_{i+1}^i)\prod^{x_j^i>0}_{i<j-1} A(x_j^i)
\end{equation}
Therefore, the above computes the uncolored knot invariant as a sum of certain simple terms over all Tesler matrices. Note that the whole sum depends very strongly on $n$, while the $m$ dependence is captured only in the equation \eqref{eq:tesler}. However, the superpolynomial $\mathcal{P}_{n,m}$ is conjecturally symmetric in $m$ and $n$, and this is not manifest from the above formula.
\subsection{Degeneration at $t=1$}
In fact, $\widetilde{\CP}_{n,m}|_{u=0}$ is a polynomial in $q$ and $t$ with positive coefficients, which is not manifest from \eqref{eqn:tes1} above. This follows from the fact that it is the Euler characteristic of a certain line bundle on the flag Hilbert scheme $\Hilb_{0,n}$, as in Subsection \ref{sub:fhilb}. The higher cohomology groups of this line bundle vanish, and $H^0$ only produces positive coefficients (this vanishing result is outside the scope of this paper and will be presented in a future work). However, we can completely describe this polynomial when $t=1$ (or when $q=1$, since the right hand side of \eqref{eqn:tes1} is clearly symmetric in $q$ and $t$).
\begin{theorem}
Conjectures \ref{conj:catalan} and \ref{conj:schroeder} hold for $t=1$ and any coprime $n,m$.
\end{theorem}
\begin{proof}
Let us first remark that any Tesler matrix from $\textrm{Tes}_{m,n}$ gives rise to a Dyck path in the $n\times m$ rectangle, with horizontal steps $x_{i}^{i}$. Indeed, for any $k\le n$ we have the equation:
$$
S_{m/n}(1)+\ldots+S_{m/n}(k) = \sum_{i=1}^{k} \left(x_{i}^{i}+\sum_{j>i}x_{j}^{i}-\sum_{j<i}x_{i}^{j}\right)=
\sum_{i=1}^{k} \left(x_{i}^{i} +\sum_{j>k}x_{j}^{i}\right)
$$
Therefore:
$$
\sum_{i=1}^{k} x_{i}^{i} \leq S_{m/n}(1)+\ldots+S_{m/n}(k) = \left\lfloor \frac{k m}{n}\right\rfloor.
$$
Any given Dyck path may correspond to many Tesler matrices. However, note that:
\begin{equation}
\label{eqn:eval0}
A(x)|_{t=1}= \delta_x^0, \qquad \qquad B(x)|_{t=1} = q^{x}
\end{equation}
so any summand of \eqref{eqn:tes1} that has some $x_j^i>0$ for $j>i+1$ will vanish. Therefore, the only summands of \eqref{eqn:tes1} that survive are those such that $x_j^i=0$ for all $j>i+1$. Such Tesler matrices will be called \emph{quasi-diagonal}, and the set of quasi-diagonal $m/n$ Tesler matrices will be denoted by $\textrm{qTes}_{m/n}$. Therefore, \eqref{eqn:tes1} becomes:
\begin{equation}
\label{eqn:dyck1}
\widetilde{\CP}_{n,m}(0,q,1) = \sum_{X\in \textrm{qTes}_{m/n}} q^{x_n^{n-1}+\ldots+x_2^1} \prod_{1\leq i \leq n}^{x_i^i>0}(1-u)
\end{equation}
The condition $x_{i}^{i}>0$ specifies the corners of a Dyck path, while the sum
$\sum_{i=1}^{n-1}x_{i+1}^{i}=\sum_{i=1}^{n}i(S_{m/n}(i)-x_{i}^{i})$ computes the area between the Dyck path and the diagonal. Therefore:
$$
\widetilde{\CP}_{n,m}(0,q,1) = \sum_{D\in Y_{m/n}} q^{\frac{(m-1)(n-1)}{2}-|D|} (1-u)^{\# \text{ of corners of } D}
$$
This implies Conjecture \ref{conj:schroeder} at $t=1$. When we set $u=0$, we obtain Conjecture \ref{conj:catalan} at $t=1$.
\end{proof}
\subsection{Degeneration at $t=q^{-1}$}
\label{sec:CY limit}
We will compute the knot invariant at $t=q^{-1}$, and show that it is a $q-$analogue of the $m,n-$Catalan number. This proves Conjecture \ref{conj:catalan} at $t=q^{-1}$.
\begin{proposition}
For any $m$ and $n$ with $\gcd(m,n)=1$, we have:
\begin{equation}
\label{eq:qfactorial}
\widetilde{\CP}_{n,m}(0,q,q^{-1}) = \frac {[m+n-1]!}{[m]! [n]!},
\end{equation}
where $[k] = \frac {q^{k/2}-q^{-k/2}}{q^{1/2}-q^{-1/2}}$ are the $q-$integers and $[k]!=[1]\cdots [k]$ are the $q-$factorials.
\end{proposition}
\begin{proof}
Let us describe the degeneration of all constructions that we used to the case $t=q^{-1}$, where $q$ and $t$ are the equivariant parameters on $\mathbb{C}^2$. Macdonald polynomials $P_{\lambda}(q,t^{-1})$ will degenerate to Schur polynomials $s_{\lambda}$, hence modified Macdonald polynomials $\widetilde{H}_{\lambda}$ will degenerate to modified Schur polynomials $\varphi_{\frac 1{1-1/t}}(s_{\lambda})$.
As it was explained in \cite{AS}, the case $t=q^{-1}$ corresponds to the classical Chern-Simons theory.
The corresponding knot invariants and operators were widely discussed in the mathematical and physical literature,
see e.g. \cite{K,stevan} for more details. In particular, it is shown in \cite[section 3.4]{stevan} that:
$$\widetilde{P}_{n,m}(q,q^{-1})=D\widetilde{P}_{n,0}(q,q^{-1})D^{-1},\ \text{where}\ D=\nabla(q,q^{-1})^{\frac{m}{n}}.$$
Remark that $p_{n}=\sum_{k=0}^{n-1}(-1)^{k}s_{(n-k,1^{k})},$ and:
$$\nabla(s_{(n-k,1^{k})})=q^{\frac{(n-k)(n-k-1)}{2}-\frac{k(k+1)}{2}}s_{(n-k,1^{k})}=q^{\frac {n(n-2k-1)}{2}}s_{(n-k,1^{k})},$$
hence:
$$D(s_{(n-k,1^{k})})=q^{\frac {m(n-2k-1)}{2}}s_{(n-k,1^{k})}=q^{\frac{(m-1)(n-1)}{2}+\frac{n-1}{2}-km}s_{(n-k,1^{k})}.$$
Therefore:
$$\widetilde{P}_{n,m}(1)=D(p_n)=q^{\frac{(m-1)(n-1)}{2}}\sum_{k=0}^{n-1}(-1)^{k}q^{\frac {n-1}{2}-km}\varphi_{\frac 1{1-1/t}}(s_{(n-k,1^{k})}).$$
By \cite[Theorem 1.6]{BEG}, this vector coincides with the graded Frobenius character
of the finite-dimensional representation $L_{m/n}$. One can also check (see e.g \cite{g,GORS} for details) that
its evaluation is given by the equation \eqref{eq:qfactorial}.
\end{proof}
|
1706.01017
|
\section{Introduction}
\label{sec:intro}
Among galaxy scalings, the correlation of baryonic mass with rotation velocity (baryonic Tully--Fisher relation; BTFR) stands out. In addition to possessing a very small intrinsic scatter, the BTFR is an almost perfect power-law over six decades of mass, describes galaxies with a wide range of morphologies, and has residuals systematically uncorrelated with other galaxy variables. This makes it at once a strong test of galaxy formation theories and an important source of information on their degrees of freedom.
Recently,~\citet[hereafter L16]{Lelli} have presented the BTFR of the \textsc{sparc} sample~\citep{SPARC}, a compilation of $175$ galaxies with high-quality H\textsc{i} rotation curves (RCs) and \textit{Spitzer} imaging at $3.6 \, \mu m$. The authors claim two features of the \textsc{sparc} BTFR to be very difficult for $\Lambda$CDM-based models to account for: its small intrinsic scatter $s_\text{BTFR}$ ($\sim0.11$ dex in baryonic mass) and the negligible correlation $\rho$ of its residuals with galaxy size. In standard galaxy formation, $s_\text{BTFR}$ should receive contributions from the halo mass--concentration and halo mass--galaxy mass relations, both of which themselves have scatter $0.1-0.2$ dex, and a simple application of Kepler's laws may be expected to yield an anticorrelation of velocity and size residuals.
Although valid, these arguments lack the statistical evidence required to claim a significant discrepancy. The aim of this work is to supply that evidence. In particular, we will calculate the expectation for $s_\text{BTFR}$ and $\rho$ in a vanilla $\Lambda$CDM model described by abundance matching (AM) by constructing mock data sets identical to \textsc{sparc} in all baryonic variables and analysed in precisely the same way. We will find two additional effects to be important, neither of which have previously been considered in detail: 1) sample variance in BTFR statistics between \textsc{sparc}-like realisations of the full galaxy--halo population, and 2) the falloff with galactocentric radius of the sensitivity of the baryonic component of the RC to galaxy size. We will show that when these effects are accounted for in a complete and fully self-consistent comparison with the \textsc{sparc} data, the discrepancies in $s_\text{BTFR}$ and $\rho$ are $\sim3.6\:\sigma$, and $2.2\:\sigma$ respectively. In addition, we investigate two further statistics that are constraining for galaxy formation models: the BTFR curvature and the fraction of RCs that are flat. The significance levels at which the \textsc{sparc} values for these quantities differ from those of the model are $\sim3.0\: \sigma$ and $2.2\: \sigma$. We conclude that the BTFR statistics of the \textsc{sparc} data pose a challenge to AM models that is moderately statistically significant.
\textcolor{black}{Our work builds on a number of studies aimed at assessing the consistency between the observed BTFR and the $\Lambda$CDM prediction, which have been carried out within both AM (e.g.~\citealt{TG, D12, DC}) and hydrodynamical (e.g.~\citealt{Santos-Santos, Dutton17}) frameworks. Despite a broad consensus that the general shape of the relation is compatible with $\Lambda$CDM, the precise extent of this agreement -- as well as the significance of more detailed features such as intrinsic scatter, curvature and residual correlations -- remains unclear. We intend our focused work on \textsc{sparc} to pave the way for more general statistical analyses in the future.}
\section{Method}
\label{sec:method}
Before detailing our procedure, we describe the twofold novelty of our approach. First, by using mock galaxies with baryonic properties identical to those of \textsc{sparc} and sampled at the same radii in the same ways, we eliminate systematic error in the comparison of BTFR statistics and ensure that any differences with the observed dynamics are due solely to the distribution of dark matter. Second, by thoroughly sampling the set of halo properties that may be associated with a given galaxy by AM, we robustly calculate the sample variance of each BTFR statistic in the model. Simple frequentist methods will then allow us to determine the significance of differences with the corresponding statistics in the data. Our method is similar to that of~\citet{D17}, on which it is based. The steps are the following.
\begin{enumerate}
\item{} From the full \textsc{sparc} data set, remove starburst dwarfs and galaxies with $i<30^{\circ}$ or quality flag 3. These are all the selection criteria of L16, except for a cut on RC flatness which we will come to shortly. We denote the resulting sample, containing $150$ galaxies, as ``\textsc{sparc}'' hereafter.
\item{} Estimate the true stellar and gas masses of each \textsc{sparc} galaxy by scattering the measured values (assuming $M_*/L=0.5$ for the disc, $M_*/L=0.7$ for the bulge, and $M_\text{gas}=1.33\:M_\text{H\textsc{i}}$;~\citealt{SPARC}) by the measurement uncertainties calculated using L16, eq. 5. Use the stellar mass to assign each galaxy a halo by the technique of abundance matching (AM;~\citealt{Kravtsov,Conroy}). In particular, we will use the AM model that~\citet{Lehmann} find to reproduce best the correlation function of SDSS, and match to halos in the \textsc{darksky-400} simulation~\citep{DarkSky}, a $(400 \: \mathrm{Mpc~h^{-1}})^3$ box with $4096^3$ particles run with the \textsc{2hot} code~\citep{Warren}. We identify halos using \textsc{rockstar}~\citep*{Rockstar}.
\item{} Use an NFW profile with the concentration and mass \textcolor{black}{(subtracting the baryon fraction)} of the assigned N-body halo to calculate the velocity due to the dark matter at each of the radii at which the RC of each \textsc{sparc} galaxy was probed~\citep{SPARC}. Add in quadrature the fixed baryonic contribution (imported directly from the \textsc{sparc} data) to calculate the total velocity, then scatter by the corresponding uncertainty (L16, eq. 3) to model observational error.
\item{} Use the algorithm of L16 (eqs. 1-2) to determine whether a given model galaxy has a flat RC, and if so to calculate the corresponding velocity $V_\text{f}$. If the RC is not flat, discard the galaxy. Denote by $N_\text{f}$ the total number of galaxies in the mock data set removed in this way.
\item{} Use $V_\text{f}$ and the baryonic masses ($M_\text{b}$) of the remaining galaxies to calculate the BTFR statistics. Begin by fitting to the BTFR and $M_\text{b}-R_\text{eff}$ relation quadratic curves in log-log space, with Gaussian scatter in $\log(V_\text{f})$ and $\log(R_\text{eff})$ respectively, by maximising the corresponding likelihood model. Subtract the $V_\text{f}$ and $M_\text{b}$ measurement uncertainties in quadrature from the total scatter to estimate the intrinsic scatter $s_\text{BTFR}$. Take the best-fitting coefficient of the quadratic term, $q$, as a measure of the BTFR curvature.
\item{} Calculate the velocity and radius residuals as
\begin{equation}
\Delta V_\text{f} \equiv V_\text{f} - \langle V_\text{f}|M_\text{b} \rangle
\end{equation}
\noindent and
\begin{equation}
\Delta R_\text{eff} \equiv R_\text{eff} - \langle R_\text{eff}|M_\text{b} \rangle,
\end{equation}
\noindent where $\langle Y|M_\text{b} \rangle$ denotes the expectation for $Y$ at fixed $M_\text{b}$ given the fit to the full relation, and calculate $\rho$ as the Spearman's rank coefficient of their correlation. This completes the treatment of a single mock data set.
\item{} Repeat steps (ii)-(vi) for 2000 mock data sets, in each case randomly drawing for each \textsc{sparc} galaxy a different \textsc{darksky-400} halo consistent with the AM model. This generates distributions of $s_\text{BTFR}$, $\rho$, $N_\text{f}$ and $q$ that fully capture the sample variance of the model predictions.
\item{} Calculate the significance of the difference between model and data for each of $X \equiv \{s_\text{BTFR}, \: \rho, \: N_f, \: q\}$ as
\begin{equation}
\sigma_X \equiv (\langle X \rangle - X_d)/s_X,
\end{equation}
\noindent where $\langle X \rangle$ is the mean of $X$ over all mock data sets, $s_X$ is the standard deviation of the distribution, and $X_d$ is the corresponding value in the \textsc{sparc} data.
\end{enumerate}
\section{Results}
\label{sec:results}
We present the predicted vs observed \textsc{sparc} BTFR in Fig.~\ref{fig:btfr}, the correlation of $V_\text{f}$ and $R_\text{eff}$ residuals in Fig.~\ref{fig:res}, and the distribution of each statistic $X$ in Fig.~\ref{fig:hists}. Table~\ref{tab:table} lists the mean and standard deviations of these distributions, along with the significances of offsets from the data. Here we describe these results: Section~\ref{sec:residuals} focuses on $\rho$, Section~\ref{sec:scatter} on $s_\text{BTFR}$, Section~\ref{sec:N_f} on $N_\text{f}$ and Section~\ref{sec:curvature} on curvature $q$.
\subsection{The $\Delta{R_\textrm{eff}}-\Delta{V_\text{f}}$ correlation}
\label{sec:residuals}
We begin with the correlation of $R_\text{eff}$ and $V_\text{f}$ residuals. In Figure~\ref{fig:res} we stack $\Delta R_\text{eff}$ and $\Delta V_\text{f}$ of all mock data sets to form a contour plot, on which we overlay the \textsc{sparc} data. In Figure~\ref{fig:spear} we compare the distribution of $\rho$ in the mock data to the corresponding value in the real data, and in the first row of Table~\ref{tab:table} we report $\langle \rho \rangle$, $s_\rho$, $\rho_d$ and $\sigma_\rho$.
It is clear that the model prediction is not particularly discrepant with the data: neither show a strong $\Delta R_\text{eff}-\Delta V_\text{f}$ correlation. This may be understood as follows. $V_\text{f}$ is calculated from the flat part of the RC defined by the final measured points. As this is typically several times beyond $R_\text{eff}$ ($\sim2-10$ for the \textsc{sparc} sample), the velocity contribution due to the galaxy is effectively that of a point mass at its centre. Variations in galaxy size at fixed $M_\text{b}$ provide only a small perturbation to this leading order term, rendering $\langle \rho \rangle$ negligible. (Similar results obtain replacing $R_\text{eff}$ by the scale length of the stellar or gas disc.) Note that this is very different to the results of~\citet{DW15} (their figs. 6-7), which L16 cite as evidence for the expectation $\rho \ll 0$. This is because~\citet{DW15} use the velocity at the radius enclosing 80\% of the $i$-band light, where the baryonic contribution to the RC is not only larger but depends much more sensitively on $R_\text{eff}$.
In fact, $\rho_d$ is \emph{more} negative than $\langle \rho \rangle$, indicating a \emph{stronger} residual anticorrelation in the data than predicted by the model. Although only $2.2\:\sigma$ significant, this provides evidence within our framework for a second component of the galaxy--halo connection: an anticorrelation of $R_\text{eff}$ with halo mass $M_\text{vir}$ or concentration $c$ at fixed $M_\text{b}$. This would give smaller galaxies on average more dark matter within $R_\text{f}$, and hence larger $V_\text{f}$. Such a correlation has already been suggested by~\citet{D17} on the basis of the correlation of the residuals of the mass discrepancy--acceleration relation (MDAR) with galaxy size. The red histogram in Figure~\ref{fig:spear} shows the result for the best-fitting correlation found there, $\Delta R_\text{eff} \sim -0.4 \: \Delta c$; that this model gives a good fit to $\rho_d$ suggests the BTFR and MDAR to contain similar information in this regard. This correlation may however be in disagreement with the observed $\Delta R-\Delta V$ correlation when $V$ is measured further in~\citep{DW15}.
\begin{table}
\begin{center}
\begin{tabular}{l|r|r|r|}
\hline
Statistic & \textsc{sparc} & Model mean & Discrepancy/$\sigma$\\
\hline
\rule{0pt}{3ex}
$\rho$ & $-0.20$ & $0.00$ & $2.2$\\
\rule{0pt}{3ex}
$s_\text{BTFR}$ (dex) & $0.029$ & $0.064$ & $3.6$\\
\rule{0pt}{3ex}
\textquotedbl \: ($M_\text{b}>10^{9.5} M_\odot$) & $0.027$ & $0.053$ & $2.1$\\
\rule{0pt}{3ex}
$N_\text{f}$ & $27$ & $33.4$ & $2.2$\\
\rule{0pt}{3ex}
$q$ & $0.003$ & $0.039$ & $3.0$\\
\hline
\end{tabular}
\caption{\textcolor{black}{Comparison of \textsc{sparc} and model BTFR statistics. $\rho$ is the Spearman's rank coefficient of the $\Delta R_\text{eff}-\Delta V_\text{f}$ correlation, $s_\text{BTFR}$ is the intrinsic BTFR scatter, $N_\text{f}$ is the number of galaxies with non-flat RCs, and $q$ is the quadratic BTFR curvature. The 3$^\text{rd}$ row shows $s_\text{BTFR}$ for $M_\text{b} > 10^{9.5} M_\odot$ galaxies only.}}
\label{tab:table}
\end{center}
\end{table}
\begin{figure*}
\subfigure[]
{
\includegraphics[width=0.45\textwidth]{fig1a}
\label{fig:btfr}
}
\subfigure[]
{
\includegraphics[width=0.45\textwidth]{fig1b}
\label{fig:res}
}
\caption{The prediction of abundance matching applied to the \textsc{sparc} sample for the BTFR (Fig.~\ref{fig:btfr}) and $\Delta R_\text{eff}-\Delta V_\text{f}$ correlation (Fig.~\ref{fig:res}), compared to the data itself. Blue stars show the modal $V_\text{f}$ over all mock data sets for each \textsc{sparc} galaxy, and error bars show the $1\: \sigma$ variation. While the model BTFR is curved and has higher scatter than is observed, its residuals are correctly uncorrelated with galaxy size.}
\label{fig:fig1}
\end{figure*}
\begin{figure*}
\subfigure[]
{
\includegraphics[width=0.45\textwidth]{fig2a}
\label{fig:spear}
}
\subfigure[]
{
\includegraphics[width=0.45\textwidth]{fig2b}
\label{fig:sig}
}
\subfigure[]
{
\includegraphics[width=0.45\textwidth]{fig2c}
\label{fig:nfail}
}
\subfigure[]
{
\includegraphics[width=0.45\textwidth]{fig2d}
\label{fig:quad}
}
\caption{The distributions of four key BTFR statistics predicted by AM -- the strength of the $\Delta R_\text{eff}-\Delta V_\text{f}$ correlation $\rho$, the intrinsic scatter $s_\text{BTFR}$, the number $N_\text{f}$ of galaxies with non-flat RCs, and the curvature $q$ -- compared to the values in the \textsc{sparc} data. The results are quantified in Table~\ref{tab:table}. `$\Delta R$-$\Delta c$ corr' in Fig.~\ref{fig:spear} denotes an anticorrelation of $R_\text{eff}$ and $c$ residuals as described in Section~\ref{sec:residuals}.}
\label{fig:hists}
\end{figure*}
\subsection{Scatter}
\label{sec:scatter}
We now proceed to $s_\text{BTFR}$. The blue histogram in Figure~\ref{fig:sig} shows the distribution of this statistic in the model, and the second row of Table~\ref{tab:table} lists $\langle s_\text{BTFR} \rangle$, $s_s$, $s_d$ and $\sigma_s$. As anticipated by L16\textcolor{black}{, and in agreement with~\citet{D12} and~\cite{DC}}, we find the predicted BTFR scatter to be upwards of 0.15 dex in $M_\text{b}$, with typical mock data sets having $\sim0.25$ dex.\footnote{The scatters in $M_\text{b}$ may be approximately obtained from the quoted scatters in $V_\text{f}$ by dividing by the BTFR slope, $\sim0.25$.} Given the spread among mock data sets, this is $3.6\: \sigma$ discrepant with the \textsc{sparc} value of $\sim0.11$ dex. This is significant -- none of our 2000 mock data sets have $s_\text{BTFR} < s_d$ -- but not phenomenally so. $s_\text{BTFR}$ can be reduced to a small degree by tightening the galaxy--halo connection: adopting an AM scatter of $0$ reduces $\langle s_\text{BTFR} \rangle$ to 0.061, with a corresponding discrepancy of $3.2\: \sigma$.
It is evident from Fig.~\ref{fig:btfr} that the predicted BTFR scatter rises towards lower $M_\text{b}$. To quantify this effect, we show in green in Figure~\ref{fig:sig} the $s_\text{BTFR}$ distribution with all $M_\text{b}<10^{9.5} M_\odot$ galaxies removed; this reduces the discrepancy to $2.1\: \sigma$ (Table~\ref{tab:table}, row 3). \textcolor{black}{The model prediction may be unreliable for $M_\text{b} \lesssim 10^{9.5} M_\odot$, as the stellar mass function requires extrapolation, AM cannot be directly tested with clustering, and low-mass halos may not be fully resolved}. We note also that gas mass fractions rise rapidly below $M_\text{b} \sim 10^{9.5} M_\odot$, amplifying any potential error incurred by performing AM with stellar as opposed to total baryonic mass. An $M_\text{b}$-based AM would correlate $M_\text{b}$ more strongly with $M_\text{vir}$ and $c$, likely reducing $s_\text{BTFR}$.
\subsection{Rotation curve flatness}
\label{sec:N_f}
Since we eliminate galaxies in each mock data set with non-flat RCs, Figs.~\ref{fig:fig1},~\ref{fig:spear},~\ref{fig:sig} and~\ref{fig:quad} pertain only to a subset of the full sample. An orthogonal statistic with which to compare model and data, therefore, is the number of galaxies out of the original $150$ that fail the flatness cut, which we denote $N_\text{f}$. Fig.~\ref{fig:nfail} shows the distribution of $N_\text{f}$ over all the mock data sets compared to the value in \textsc{sparc} (27), and the corresponding statistics are shown in the 4$^\text{th}$ row of Table~\ref{tab:table}. We find a larger fraction of our model galaxies to have rising RCs at the last measured point than in the data (a $2.2\: \sigma$ discrepancy), reflecting the fact the NFW density profile falls as $\sim r^{-1}$ out to large radius. This quantifies the longstanding ``disc--halo conspiracy''~\citep{conspiracy}, and deserves attention in future studies.
\subsection{Curvature}
\label{sec:curvature}
A final significant feature of the model BTFR is its curvature, which is $\sigma_q = 3.0\: \sigma$ discrepant with the data (Fig.~\ref{fig:quad} and Table~\ref{tab:table}, final row). We caution however that this prediction is sensitive to the low-$M_\text{b}$ model uncertainties described in Section~\ref{sec:scatter}, and removing the $M_\text{b} < 10^{9.5} M_\odot$ region reduces $\sigma_q$ below $1 \: \sigma$ (Fig.~\ref{fig:quad}, green histogram).
\textcolor{black}{In the context of AM, BTFR curvature follows from the well-known mismatch between the slopes of the stellar and halo mass functions, and is therefore present to some extent in all AM-based studies (e.g.~\citealt{TG, Desmond}). The precise magnitude of $q$, however, depends on the details of the AM: lower curvature follows from a shallower bright end to the SMF (e.g. the photometry of~\citealt{Bernardi_SMF}, used here), a halo proxy with less concentration dependence, and a lower AM scatter. Clustering studies have begun to set strong constraints on these variables, and we find that $q$ varies by only $\sim5$ per cent as the halo proxy and AM scatter span the ranges allowed by~\citet{Lehmann}. This suggests that spatial statistics provide sufficient information on AM for the shape of the predicted BTFR to follow almost uniquely. Older AM schemes which match to halo mass directly -- as used for example in~\citet{DC} -- produce a straighter BTFR, although that change alone reduces $q$ by $15$ per cent at most and cannot lower $\sigma_q$ below $2.5 \: \sigma$. A further reduction requires a preferential decrease in halo $M_\text{vir}$ or $c$ at the faint and/or bright ends (e.g. by baryonic feedback), biases from selection effects, or systematic errors in stellar mass measurements. Some hydrodynamical simulations incorporating these effects have achieved a straighter BTFR (e.g.~\citealt{Santos-Santos, Dutton17}). $q$ likely depends in addition on the velocity measure: when sampled much beyond $R_\text{max}$, $V$ will be lowered for high-mass galaxies with falling RCs, reducing the upturn in the BTFR at the bright end. This may also contribute to the lower curvature of~\citet{DC}, who measure $V$ at $8 R_\text{d}$.}
\section{Discussion and Conclusions}
\label{sec:discussion}
\textcolor{black}{A range of approaches have been developed in the past two decades to elucidate the nature and origin of the baryonic Tully--Fisher relation (BTFR), but only recently have data and models become sufficiently sophisticated for statistically rigorous analysis to be possible. To advance this programme, we calculate the significance levels at which four key statistics of a state-of-the-art observational BTFR dataset, \textsc{sparc}, differ from those expected in a modern $\Lambda$CDM abundance matching model. We create mock datasets with precisely the baryonic properties of \textsc{sparc}, and analyse them in an identical fashion to the real data~\citep{Lelli}. Any differences between model and measured galaxies must therefore derive solely from differences in the distribution of dark matter, and hence be attributable to the galaxy--halo connection.} Our findings are the following:
\begin{itemize}
\item{} When defined using the flat part of the RC, the BTFR's residuals would \emph{not} be expected to anticorrelate with galaxy size in $\Lambda$CDM; a significant test of galaxy formation requires that velocity be measured at a \emph{smaller} radius, where the contribution of the baryonic mass depends more strongly on its concentration. On the other hand, the fact that the baryonic part of $V_\text{f}$ depends little on $R_\text{eff}$ makes their correlation more sensitive to a second global galaxy--halo correlation (after $M_*$--$f(M_\text{vir},c)$ imposed by AM), viz the relation between galaxy size and halo properties. We find $\sim 2 \: \sigma$ evidence for an anticorrelation of $R_\text{eff}$ with $M_\text{vir}$ or $c$ at fixed $M_\text{b}$.
\item{} The predicted BTFR scatter is $3.6\:\sigma$ larger than observed, and cannot be appreciably lowered by tightening the galaxy--halo connection. However, simulation and model uncertainties may impact the prediction at $M_\text{b} \lesssim 10^{9.5} M_\odot$, and excising this region reduces the discrepancy to $2.1 \: \sigma$.
\item{} \textcolor{black}{A further BTFR statistic with significant constraining power for models of the galaxy--halo connection is the quadratic curvature, which we find to be overpredicted by AM at the $3.0 \: \sigma$ level. This may indicate mass-dependent baryonic effects on the dark matter halos (e.g.~\citealt{DC1, Sawala}) or a correlation of TFR selection criteria with dynamical halo properties.}
\item{} For BTFR studies that remove galaxies with non-flat RCs (e.g. \textsc{sparc}), the fraction of such galaxies in a given dataset is orthogonal to other BTFR statistics and provides an additional handle on RC shape. We find AM to overpredict the fraction of galaxies with rising RCs at $2.2\:\sigma$, suggesting that its application to N-body halo density profiles does not fully satisfy the observed ``disc--halo conspiracy''.
\item{} The significance of discrepancies between theoretical and observed BTFR statistics is set by sample variance among model realisations, which scales inversely with the size of the data set. As statistical tests are rarely performed in the literature, the importance of sample size is often overlooked (but see~\citealt{Sorce}). Assuming BTFR statistics to obey the central limit theorem, the widths of mock data distributions -- and hence significance levels -- will vary with $\sim \sqrt{N}$, suggesting that increasing sample size may be preferable for increasing the power of statistical tests than applying stringent cuts on data quality. Because our model explicitly includes the size of the \textsc{sparc} sample (in addition to baryonic galaxy properties imported directly from the observations), our results should strictly be considered to apply only to the \textsc{sparc} BTFR, not to the BTFR per se.
\end{itemize}
We propose three directions in which this work could be taken: 1) seek the features of the BTFR -- or galaxy dynamics more generally -- with most constraining power for galaxy formation, and calculate the significance of their deviations from AM predictions; 2) modify the model to alleviate the aforementioned discrepancies, for example by complexifying the galaxy--halo connection or introducing new degrees of freedom for baryonic effects; 3) incorporate improvements in simulation resolution and survey depth to strengthen the AM prediction for $M_\text{b} < 10^{9.5} M_\odot$. This regime is not only critical for the statistical power of BTFR tests, but also connects to topical galaxy formation issues at the dwarf scale.
\section*{Acknowledgements}
I thank Federico Lelli for guidance with the \textsc{sparc} data, and Federico Lelli, Stacy McGaugh, Risa Wechsler and an anonymous referee for comments on the manuscript.
This work used the DarkSky simulations, made using an INCITE 2014 allocation on the Oak Ridge Leadership Computing Facility at Oak Ridge National Laboratory. I thank the DarkSky collaboration for creating and providing access to these simulations, and Sam Skillman and Yao-Yuan Mao for running \textsc{rockstar} and \textsc{consistent trees} on them.
This work received support from the U.S.\ Department of Energy under contract number DE-AC02-76SF00515.
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1504.01938
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\section{Introduction}\label{SecIntro}
Consider the following class of definable ccc posets (see examples in Section \ref{SecBorelccc}).
\begin{definition}\label{DefcccBorel}
A poset $\Sor$ is \emph{ccc Borel} if it is ccc, the relations $\leq_{\Sor}$ and $\perp_{\Sor}$ are Borel, it adds a (generic) real $\dot{\eta}$ and there is a Borel relation $E\subseteq\omega^\omega\times\omega^\omega$ such that,
\begin{enumerate}[(i)]
\item $E(z,\mathds{1}_{\Sor})$ is true for any real $z$ and
\item in any $\Sor$-extension, $p\in\Sor$ is in the generic filter iff $E(\dot{\eta},p)$.
\end{enumerate}
A subposet $\Qor$ of $\Sor$ is \emph{nice} if $\Qor=\Sor^M$ for some transitive model $M$ of (a large fragment of) $\thzfc$ that contains $\omega_1$, $\dot{\eta}$ and the parameters of $\Sor$ and $E$.
\end{definition}
It is very common to use finite support iterations of nice subposets of Borel ccc posets (and also of quite small ccc posets) to obtain models where many cardinal invariants assume different values (see, for example, \cite{Br-Cichon}, \cite{JuSh-KunenMillerchart}, \cite{Me-MatIt} and \cite{Me-Matit02}). In \cite{Left Cichon}, the same technique is used to prove the consistency of $\bfrak<\non(\Mwf)<\cov(\Mwf)$ but, as it is hard to preserve unbounded families while using nice subposets of $\Eor$ (used to increase $\non(\Mwf)$, see Example \ref{ExpBorellinked}), new ideas like a construction of chains of ultrafilters had to be introduced to guarantee that $\leq^*$-increasing unbounded families in the ground model are preserved through the iteration. Here, it is necessary to code countable delta systems of conditions in the iteration without complete knowledge of what the iteration would be, that is, the code of these delta system can be interpreted once the iteration is constructed. This coding is possible because names of reals can be coded by Borel functions, as illustrated in the following fact.
\begin{theorem}[{\cite{Left Cichon}}]\label{fsiBorelComp}
Let $\Por=\langle\Por_\alpha,\Qnm_\alpha\rangle_{\alpha<\delta}$ be a finite support iteration, $\delta=B\cup C$ disjoint union such that, for $\alpha\in B$, $\Qnm_\alpha$ is a $\Por_\alpha$-name of a nice subposet of a Borel ccc poset coded in the ground model and, for $\alpha\in C$, $\Qnm_\alpha$ is a $\Por_\alpha$-name of a ccc poset which domain, without loss of generality, is assumed to be an ordinal\footnote{In practice, these posets are small with respect to some fixed cardinal, this in order to have nice preservation properties for the iteration.}. If $\dot{x}$ is a $\Por$-name for a real, then there is a Borel function $F$ in the ground model such that $\Vdash\dot{x}=F(\langle\dot{\eta}_\alpha\rangle_{\alpha\in N})$ for some countable subset $N$ of $\delta$, where
\begin{enumerate}[(i)]
\item if $\alpha\in B\cap N$, $\dot{\eta}_\alpha$ is the name of the generic real added by $\Qnm_\alpha$ and
\item if $\alpha\in C\cap N$, $\dot{\eta}_\alpha=\dot{\chi}_\alpha\frestr W_\alpha$ where $\dot{\chi}_\alpha$ is the characteristic function of the generic set added by $\Qnm_\alpha$ and $W_\alpha$ is a countable set, where $\langle W_\alpha\rangle_{\alpha\in C\cap N}$ belongs to the ground model.
\end{enumerate}
\end{theorem}
The main objective of this text is to extend this coding of names by Borel functions to the context of iterations along a template. This is possible by considering template iterations that alternates between nice subposets of Borel $\sigma$-linked posets (some of them correctness-preserving, see Definition \ref{DefCorrPres}), coded in the ground model, and arbitrary $\sigma$-linked posets (which in practice, are quite small). We are going to call these \emph{simple template iterations} (see Definition \ref{DefSimpleiteration} for details). The main result is stated in detail in Theorem \ref{MainThm}.
The theory of template iterations was originally introduced Shelah \cite{Sh-TempIt} to construct a model of $\aleph_1<\dfrak<\afrak$. Further applications and generalizations of the template iteration theory are presented, for example, in \cite{Br-TempIt,Br-CtbleCof, Br-Luminy}, \cite{Me-TempIt}, \cite{MaxCofGr} and \cite{Fischer-Mejia}. Our notation about template iterations corresponds to \cite{Me-TempIt}.
\begin{acknowledgements}
This paper was motivated from the talk ``$\sfrak\bfrak\afrak$" (joint work with V. Fischer \cite{Fischer-Mejia}) that the author contributed to the RIMS 2014 Workshop on Infinitary Combinatorics in Set Theory and Its Applications. The author is deeply thankful with T. Usuba for organizing such a wonderful conference.
\end{acknowledgements}
\section{Simple template iterations}\label{SecBorelccc}
In this section, we want to define the type of iterations we are interested in for the main result, which we call simple (template) iterations.
\begin{notation}\label{NotationNames}
Given a ccc poset $\Por$, without loss of generality, we assume that any $\Por$-name $\dot{x}$ for a real is of the form $\langle h_n^{\dot{x}},A_n^{\dot{x}}\rangle_{n<\omega}$ where, for each $n<\omega$, $A_n=A_n^{\dot{x}}$ is a maximal antichain in $\Por$, $h_n=h_n^{\dot{x}}:A_n\to\omega$ and each $p\in A_n$ decides $\dot{x}(n)$ to be $h_n(p)$.
\end{notation}
\begin{lemma}\label{forceSimga1-2}
Let $\Sor$ be a Suslin ccc poset. If $\varphi(z)$ is a $\boldsymbol{\Sigma}^1_1$-statement of reals and $\dot{x}$ is a $\Sor$-name for a real, then the statement ``$p\Vdash\varphi(\dot{x})$" is $\boldsymbol{\Sigma}^1_2$. On the other hand, if $\varphi(z)$ is a $\boldsymbol{\Pi}^1_1$-statement of reals, then ``$p\Vdash\varphi(\dot{x})$" is $\boldsymbol{\Pi}^1_2$.
\end{lemma}
\begin{proof}
We first prove that, if $T\subseteq\omega^{<\omega}$ is a tree, then the statement ``$p\Vdash\dot{x}\in[T]$" is $\boldsymbol{\Sigma}^1_1\boldsymbol{\cup}\boldsymbol{\Pi}^1_1$ (the smallest $\sigma$-algebra containing both $\boldsymbol{\Sigma}^1_1$ and $\boldsymbol{\Pi}^1_1$). As in Notation \ref{NotationNames}, $\dot{x}=\langle h,A_n\rangle_{n<\omega}$ were $A_n=\{q_{n,i}\ /\ i<|A_n|\}$ is countable and $h_n:|A_n|\to\omega$ (in the sense that $q_{n,i}$ decides $\dot{x}(n)=h_n(i)$), so $\dot{x}$ can be seen as a real itself. Therefore, ``$\dot{x}$ is a $\Sor$-name for a real" is a $\boldsymbol{\Sigma}^1_1\boldsymbol{\cup}\boldsymbol{\Pi}^1_1$-statement (it is just $\boldsymbol{\Pi}^1_1$ if $\Sor$ is Borel ccc). Now, notice that $p\Vdash\dot{x}\frestr k\in T$ iff $p\in\Sor$, $\dot{x}$ is a $\Sor$-name for a real and, for every $s\in\omega^k$, if $\{q_{i,s(i)}\ /\ i<k\}\cup\{p\}$ has a common stronger condition in $\Sor$, then $\langle h_i(s(i))\rangle_{i<k}\in T$, which is a $\boldsymbol{\Sigma}^1_1\boldsymbol{\cup}\boldsymbol{\Pi}^1_1$-statement (or just $\boldsymbol{\Pi}^1_1$ if $\Sor$ is Borel).
Recall that an analytic statement is the projection of $[T]$ for some tree $T\subseteq(\omega\times\omega)^\omega$. Note that $p\Vdash\exists_y((\dot{x},y)\in[T])$ iff $p\in\Sor$, $\dot{x}$ is a $\Sor$-name for a real and there is a $\Sor$-name for a real $\dot{y}$ such that $p\Vdash(\dot{x},\dot{y})\in[T]$, which is clearly a $\boldsymbol{\Sigma}^1_2$-statement.
The other affirmation is proven similarly (because $p\Vdash\dot{x}\notin[T]$ is $\boldsymbol{\Pi}^1_2$).
\end{proof}
As a consequence of this Lemma we have that the generic filter of any nice subposet of a Borel ccc poset is also well described by the Borel relation of the Borel poset, as shown in the following result.
\begin{corollary}\label{Nicesubposet}
Let $\Sor$ be a Borel ccc poset as in Definition \ref{DefcccBorel} and $\Qor$ a nice subposet of $\Sor$. If $G$ is $\Qor$-generic over $V$ and $p\in\Qor$, then $p\in G$ iff $E(\dot{\eta},p)$.
\end{corollary}
\begin{proof}
$\Vdash p\in\dot{G}\sii E(\dot{\eta},p)$ is equivalent to say that $p\Vdash E(\dot{\eta},p)$ and, for every $q\in\Sor$, if $q\Vdash E(\dot{\eta},p)$ then $q\parallel p$, which is a $\boldsymbol{\Pi}^1_2$-statement by Lemma \ref{forceSimga1-2}. So $\forall_{p\in\Sor}(\Vdash p\in\dot{G}\sii E(\dot{\eta},p))$ is also $\boldsymbol{\Pi}^1_2$.
Now, let $M$ a transitive model of (a large fragment of) $\thzfc$ that contains $\omega_1$, $\dot{\eta}$ and the parameters of $\Sor$ and $E$, such that $\Qor=\Sor^M$. By the absoluteness of $\boldsymbol{\Pi}^1_2$-statements, $M\models\forall_{p\in\Sor}(\Vdash p\in\dot{G}\sii E(\dot{\eta},p))$. If $G$ is $\Qor$-generic over $V$, then it is $\Qor$-generic over $M$, so $M[G]\models``p\in G\sii E(\eta[G],p)"$ for any $p\in\Qor$. Therefore, as $E$ is a Borel relation, the equivalence $``p\in G\sii E(\eta[G],p)"$ is also true in $V[G]$.
\end{proof}
Definable posets that are involved in simple iterations should satisfy the following two notions.
\begin{definition}[{\cite{Br-Luminy}}]\label{DefBorelsigmalinked}
A poset $\Sor$ is \emph{Borel $\sigma$-linked} if it is Borel ccc (see Definition \ref{DefcccBorel}) and there is a sequence $\{S_n\}_{n<\omega}$ of linked sets such that the statement ``$x\in S_n$" is Borel. In addition, if all those $S_n$ are centered, we say that $\Sor$ is \emph{Borel $\sigma$-centered}.
\end{definition}
\begin{definition}[{\cite{Br-Luminy}}]\label{DefCorrPres}
\begin{enumerate}[(1)]
\item A system of posets $\langle\Por_0,\Por_1,\Qor_0,\Qor_1\rangle$ is \emph{correct} if $\Por_i$ is a complete subposet of $\Qor_i$ for $i=0,1$, $\Por_0$ is a complete subposet of $\Por_1$, $\Qor_0$ is a complete subposet of $\Qor_1$ and, whenever $p\in\Por_0$ is a reduction of $q\in\Qor_0$, then $p$ is a reduction of $q$ with respect to $\Por_1,\Qor_1$.
\item A Suslin ccc poset $\Sor$ is \emph{correctness-preserving} if, for any $\langle\Por_0,\Por_1,\Qor_0,\Qor_1\rangle$ as in (1), the system $\langle\Por_0\ast\Snm^{V^{\Por_0}},\Por_1\ast\Snm^{V^{\Por_1}},
\Qor_0\ast\Snm^{V^{\Qor_0}},\Qor_1\ast\Snm^{V^{\Qor_1}}\rangle$ is correct.
\end{enumerate}
\end{definition}
\begin{example}\label{ExpBorellinked}
\begin{enumerate}[(1)]
\item Consider $\Eor$ the canonical forcing that adds an eventually different real, that is, conditions are of the form $(s,F)\in\omega^{<\omega}\times[\omega^\omega]^{<\omega}$ and the order is given by $(s',F')\leq(s,F)$ iff $s\subseteq s'$, $F\subseteq F'$ and $s(i)\neq x(i)$ for all $x\in F$ and $i\in|s'|\menos|s|$. It is clear that this poset has a Borel definition. $\dot{e}=\bigcup\{s\ /\ \exists_F((s,F)\in\dot{G})\}$ is the name of the generic real and, with the closed-relation $E(z,(s,F))$ defined as ``$s\subseteq z$ and $\forall_{i\geq|s|}\forall_{x\in F}(z(i)\neq x(i))$", it is clear that $\Vdash``(s,F)\in\dot{G}\sii E(\dot{e},(s,F))"$, so $\Eor$ is Borel ccc. It is also clear that $\Eor$ is Borel $\sigma$-centered.
\item Classical forcing notions like Cohen forcing and Hechler forcing are Borel $\sigma$-centered, while localization forcing and random forcing are Borel $\sigma$-linked.
\item All the previous posets are correctness-preserving, due to Brendle \cite{Br-Luminy,Br-Shat} (see also \cite[Sect. 2]{Me-TempIt}).
\end{enumerate}
\end{example}
\begin{definition}\label{DefSimpleiteration}
Let $\langle L,\bar{\Iwf}\rangle$ be an indexed template. A \emph{simple (template) iteration $\Por\frestr\langle L,\bar{\Iwf}\rangle$} consists of the following components:
\begin{enumerate}[(i)]
\item $L=B\cup R\cup C$ as a disjoint union.
\item For $x\in B\cup R$ let $\Sor_x$ be a Borel $\sigma$-linked correctness-preserving poset, where $E_x$ is its corresponding Borel relation and $\dot{\eta}_x$ the name of its generic real.
\item For $x\in R$ fix $C_x\in\hat{\Iwf}_x$.
\item For $x\in C$ fix an ordinal $\gamma_x$ and $C_x\in\hat{\Iwf}_x$.
\end{enumerate}
For $x\in L$ and $A\in\hat{\Iwf}_x$, $\Qnm^A_x$ (the $\Por\frestr A$-name of the poset used at coordinate $x$ of the iteration) is defined as follows.
\begin{enumerate}[(i)]
\setcounter{enumi}{4}
\item If $x\in B$ then $\Qnm^A_x=\Sor_x^{V^{\Por\upharpoonright A}}$.
\item If $x\in R$, fix $\Qnm_x$ a $\Por\frestr C_x$-name of a nice subposet of $\Sor_x^{V^{\Por\upharpoonright C_x}}$. $\Qnm^A_x=\Qnm_x$ if $C_x\subseteq A$, or it is the trivial poset otherwise.
\item If $x\in C$, fix $\Qnm_x$ a $\Por\frestr C_x$-name of a $\sigma$-linked poset with domain $\gamma_x$. $\Qnm^A_x=\Qnm_x$ if $C_x\subseteq A$, or it is the trivial poset otherwise.
\end{enumerate}
For $x\in C$, denote by $\dot{\eta}_x$ the $\Por\frestr(C_x\cup\{x\})$-name of the characteristic function of the generic subset of $\Qnm_x$. Besides, for $B\in\hat{\Iwf}_x$, $\dot{\eta}^B_x$ denotes the $\Por\frestr(B\cup\{x\})$-name of the generic subset of $\Qnm^B_x$, so $\dot{\eta}^B_x=\dot{\eta}_x$ if $C_x\subseteq B$ or $\dot{\eta}^B_x=\{(0,1)\}$ otherwise.
If $A\subseteq L$, define $\Por^*\frestr A$ as the set of conditions $p\in\Por\frestr A$ such that, for $x\in C\cap\dom p$, $p(x)$ is an ordinal in $\gamma_x$ (not just a name).
\end{definition}
\begin{remark}\label{RemonSmallposets}
\begin{enumerate}[(1)]
\item We could just ignore the ordinal in (iv) and state in (vii) that $\Qnm$ is a $\Por\frestr C_x$-name for a $\sigma$-linked poset. This is because, by ccc-ness, we can find an ordinal $\gamma_x$ such that $\Por\frestr C_x$ forces that $\Qnm$ is densely embedded into a poset with domain $\gamma_x$. In practice, these ordinals are meant to be small (in \cite{Left Cichon}, assuming $\kappa=\bfrak=\cfrak$ in the ground model, ``small" means ``of size $<\kappa$").
\item It is easy to see that, in a simple iteration as in Definition \ref{DefSimpleiteration}, $\Por^*\frestr A$ is dense in $\Por\frestr A$ for all $A\subseteq L$. By induction on $\mathrm{Dp}(A)$: let $p\in\Por\frestr A$, $x=\max(\dom p)$, so there exists an $A'\in\Iwf_x\frestr A$ such that $p\frestr L_x\in\Por\frestr A'$ and $p(x)$ is a $\Por\frestr A'$-name for a condition in $\Qnm_x^{A'}$. Assume $x\in C$. If $C_x\subseteq A'$, get $p'\leq p\frestr L_x$ in $\Por\frestr A'$ and some $\xi<\gamma_x$ such that $p'\Vdash p(x)=\xi$. Now, find $q'\leq p'$ in $\Por^*\frestr A'$ (by induction hypothesis), so $q=q'\cup\{(x,\xi)\}$ is in $\Por^*\frestr A$ and it is stronger that $p$. On the other hand, if $C_x\nsubseteq A'$, then $p(x)$ is the trivial condition, which can be assumed to be 0, so this case is handled like before. The case $x\in B\cup R$ is also similar (and simpler).
\item In Definition \ref{DefSimpleiteration}, we can add more conditions to $\Por^*\frestr A$ depending on the posets used at coordinates $x\in B\cup R$. For example, for such an $x$ where $\Sor_x=\Eor$, we could further assume that, if $x\in\dom p$, then $p(x)=(s,\dot{F})$ where $s$ and $|\dot{F}|$ are already decided. Again, we obtain that $\Por^*\frestr A$ is dense in $\Por\frestr A$.
\end{enumerate}
\end{remark}
\section{Borel computation}\label{SecBorelComp}
Throughout this section, fix an indexed template $\langle L,\bar{\Iwf}\rangle$ and a simple iteration $\Por\frestr\langle L,\bar{\Iwf}\rangle$ as in Definition \ref{DefSimpleiteration}.
\begin{definition}\label{DefHistory}
By recursion on $\mathrm{Dp}(A)$, define, for any $p\in\Por^*\frestr A$ and $\dot{x}=\langle h_n,A_n\rangle_{n<\omega}$ a $\Por^*\frestr A$-name for a real:
\begin{enumerate}[(1)]
\item $H^A(p)\subseteq A$ and $H^A(\dot{x})\subseteq A$ as follows:
\begin{enumerate}[(i)]
\item If $x=\max(\dom p)$, choose $A'\in\Iwf_x\frestr A$ such that $p\frestr L_x\in\Por^*\frestr A'$ and $p(x)$ is a $\Por^*\frestr A'$-name for a condition in $\Qnm^{A'}_x$. Put $H^A(p)=H^{A'}(p\frestr L_x)\cup H^{A'}(p(x))\cup\{x\}$, but ignore $H^{A'}(p(x))$ when $x\in C$. On the other hand, if $p=\langle\ \rangle$, put $H^A(p)=\varnothing$.
\item $H^A(\dot{x})=\bigcup\{H^A(p)\ /\ p\in A_n,\ n<\omega\}$.
\end{enumerate}
\item Sequences $\overline{W}^A(p)\in\prod_{z\in H^A(p)\cap C}\Pwf(\gamma_z)$ and $\overline{W}^A(\dot{x})\in\prod_{z\in H^A(\dot{x})\cap C}\Pwf(\gamma_z)$ as follows:
\begin{enumerate}[(i)]
\item If $x=\max(\dom p)$, choose $A'\in\Iwf_x\frestr A$ as in (1)(i) and put, for $z\in (H^{A'}(p\frestr L_x)\cup H^{A'}(p(x)))\cap C$, $W^A(p)_z=W^{A'}(p\frestr L_x)_z\cup W^{A'}(p(x))_z$ (ignore undefined terms in this union) and, if $x\in C$, put $W^A(p)_x=\{p(x)\}$ (recall that the trivial condition in $\Qnm_x^{A'}$ is 0 for $x\in C$).
\item $W^A(\dot{x})_z=\bigcup\{W^A(p)_z\ /\ z\in H^A(p)\cap C,\ p\in A_n,\ n<\omega\}$ for $z\in H^A(\dot{x})\cap C$.
\end{enumerate}
\end{enumerate}
\end{definition}
It is necessary to see that both functions $H^A(\cdot)$ and $\overline{W}^A(\cdot)$ are well defined. Moreover, they do not depend on $A$, as follows from the following result.
\begin{lemma}\label{HistoryWellDef}
Let $A'\subseteq A\subseteq L$.
\begin{enumerate}[(a)]
\item If $p\in\Por^*\frestr A'$ then $H^{A'}(p)=H^{A}(p)$ and $\overline{W}^{A'}(p)=\overline{W}^A(p)$.
\item If $\dot{x}$ is a $\Por^*\frestr A'$-name for a real, then $H^{A'}(\dot{x})=H^{A}(\dot{x})$ and $\overline{W}^{A'}(\dot{x})=\overline{W}^A(\dot{x})$.
\end{enumerate}
\end{lemma}
\begin{proof}
We prove both (a) and (b) simultaneously by induction on $\mathrm{Dp}(A)$. Let $A'\subseteq A$ and $p\in\Por^*\frestr A'$. If $p=\langle\ \rangle$, clearly $H^{A'}(p)=H^{A}(p)$ and $\overline{W}^{A'}(p)=\overline{W}^A(p)$, so assume that $p\neq\langle\ \rangle$ and let $x=\max(\dom p)$. Then, there exists $K'\in\Iwf_x\frestr A'$ such that $p\frestr L_x\in\Por^*\frestr K'$ and $p(x)$ is a $\Por^*\frestr K'$-name for a condition in $\Qnm_x^{K'}$. Clearly, there is a $K\in\Iwf_x\frestr A$ containing $K'$ so, by induction hypothesis, $H^{K'}(p\frestr L_x)\cup H^{K'}(p(x))=H^K(p\frestr L_x)\cup H^K(p(x))$ and, for $z$ in this set and in $C$, $W^{K'}(p\frestr L_x)_z\cup W^{K'}(p(x))_z=W^K(p\frestr L_x)_z\cup W^K(p(x))_z$ (ignore undefined objects). Therefore, $H^{A'}(p)=H^A(p)$ and $\overline{W}^{A'}(p)=\overline{W}^A(p)$.
If $\dot{x}$ is a $\Por^*\frestr A'$-name for a real, $H^{A'}(\dot{x})=H^A(\dot{x})$ and $\overline{W}^{A'}(\dot{x})=\overline{W}^A(\dot{x})$ follow straightforward.
\end{proof}
Lemma \ref{HistoryWellDef} allows us to denote $H(\cdot)=H^L(\cdot)$ and $\overline{W}(\cdot)=\overline{W}^L(\cdot)$. The intension of these two functions, which is materialized in Theorem \ref{MainThm} is that any $p\in\Por^*\frestr L$ can be reconstructed from the generic objects added at stages $x\in H(p)$ in the iteration and, for $z\in H(p)\cap C$, $p$ only depends on the information given by the set $W(p)_z$. Therefore, the same applies for $\Por^*\frestr L$-names for reals, which allows to define a Borel function in the ground model that determines $\dot{x}$ when it is evaluated at the generic reals from $H(\dot{x})$ where, for $z\in H(\dot{x})\cap C$, it is only needed to look at $W(\dot{x})_z$ intersected the generic set added at $z$. All these information from where conditions and names depend are countable, which is easily proved by induction on $\mathrm{Dp}(A)$ for $A\subseteq L$.
\begin{lemma}\label{CtbleHistory}
For each $p\in\Por^*\frestr L$, $H(p)$ is a countable subset of $L$ and, for each $z\in H(p)\cap C$, $W(p)_z$ is a countable subset of $\gamma_z$. The same applies to $\Por^*\frestr L$-names for reals.
\end{lemma}
\begin{notation}\label{NotationPolish}
Given a triple $\mathbf{t}=(H_S,H_C,\bar{W})$ where $H_S$ and $H_C$ are countable disjoint sets and $\bar{W}=\langle W_a\rangle_{a\in H_C}$ is a sequence of countable sets, define $\R(\mathbf{t})=(\omega^\omega)^{H_S}\times\prod_{a\in H_C}2^{W_a}$, which is clearly a Polish space. Additionally, for an arbitrary sequence $\bar{z}$ of functions, if $H_S\cup H_C\subseteq\dom(\bar{z})$, denote $\bar{z}\frestr\mathbf{t}=\langle z_a\rangle_{a\in H_S}\widehat{\ \ }\langle z_a\frestr W_a\rangle_{a\in H_C}$.
Fix $A\subseteq L$, $p\in\Por^*\frestr A$ and $\dot{x}$ a $\Por^*\frestr A$-name for a real. For $p\in\Por^*\frestr A$, let $\mathbf{t}_p=(H(p)\menos C,H(p)\cap C,\bar{W}_p)$ and $\R(p):=\R(\mathbf{t}_p)$. Likewise, define $\mathbf{t}_{\dot{x}}$ and $\R(\dot{x})$.
In particular, considering $\tilde{\eta}=\langle\dot{\eta}_z\rangle_{z\in L}$, $\tilde{\eta}\frestr\mathbf{t}_p$ and $\tilde{\eta}\frestr\mathbf{t}_{\dot{x}}$ are $\Por^*\frestr A$-names for reals in $\R(p)$ and $\R(\dot{x})$, respectively.
\end{notation}
We are now ready to state and prove the main result of this text.
\begin{theorem}\label{MainThm}
Let $\Por\frestr\langle L,\bar{\Iwf}\rangle$ be a simple iteration as in Definition \ref{DefSimpleiteration}.
\begin{enumerate}[(a)]
\item There is a relation $\Ewf\subseteq\{(\bar{z},p)\ /\ p\in\Por^*\frestr L\textrm{\ and }\bar{z}\in\R(p)\}$ such that, for any $p\in\Por^*\frestr L$,
\begin{enumerate}[(i)]
\item $\Ewf(\cdot,p)$ is Borel in $\R(p)$ and
\item $\Vdash_{\Por^*\upharpoonright L}p\in\dot{G}\sii\Ewf(\tilde{\eta}\frestr{\mathbf{t}_p},p)$.
\end{enumerate}
\item If $\dot{x}$ is a $\Por^*\frestr L$-name for a real, there exists a Borel function $F_{\dot{x}}:\R(\dot{x})\to\omega^\omega$ such that $\Vdash_{\Por^*\upharpoonright L}\dot{x}=F_{\dot{x}}(\tilde{\eta}\frestr\mathbf{t}_{\dot{x}})$.
\end{enumerate}
\end{theorem}
\begin{proof}
By recursion on $\mathrm{Dp}(A)$ for $A\subseteq L$, we define a relation $\Ewf^A\subseteq\{(\bar{z},p)\ /\ p\in\Por^*\frestr A\textrm{\ and }\bar{z}\in\R(p)\}$ such that, for any $p\in\Por^*\frestr A$,
\begin{enumerate}[(i)]
\item $\Ewf^A(\cdot,p)$ is Borel in $\R(p)$,
\item $\Vdash_{\Por^*\upharpoonright A}p\in\dot{G}\sii\Ewf^A(\tilde{\eta}\frestr{\mathbf{t}_p},p)$ and
\item for all $K\subseteq A$, $q\in\Por^*\frestr K$ and $\bar{z}\in\R(q)$, $\Ewf^K(\bar{z},q)$ iff $\Ewf^A(\bar{z},q)$.
\end{enumerate}
Within this recursion, for any $\Por^*\frestr A$-name for a real $\dot{x}$, we construct a Borel function $F^A_{\dot{x}}:\R(\dot{x})\to\omega^\omega$ such that
\begin{enumerate}[(i)]
\setcounter{enumi}{3}
\item $\Vdash_{\Por^*\upharpoonright A}\dot{x}=F^A_{\dot{x}}(\tilde{\eta}\frestr\mathbf{t}_{\dot{x}})$ and
\item for all $K\subseteq A$ and $\dot{y}$ $\Por^*\frestr K$-name for a real, $F^K_{\dot{y}}=F^A_{\dot{y}}$.
\end{enumerate}
This implies that $\Ewf^L$ and $F_{\dot{x}}=F^L_{\dot{x}}$ is as we want for (a) and (b).
We proceed with the construction by the following cases.
\begin{enumerate}[(1)]
\item \emph{$A$ has a maximum $x$ and $A_x=A\cap L_x\in\hat{\Iwf}_x$.} We consider cases on $x$.
\begin{itemize}
\item[(1.1)] If $x\in B\cup R$, $\Ewf^A(\bar{z},p)$ iff $p\in\Por^*\frestr A$, $\bar{z}\in\R(p)$ and, either $x\notin\dom p$ and $\Ewf^{A_x}(\bar{z},p)$, or $x\in\dom p$, $\Ewf^{A_x}(\bar{z}\frestr\mathbf{t}_{p\upharpoonright L_x},p\frestr L_x)$ and $E_x(z_x,F^{A_x}_{p(x)}(\bar{z}\frestr\mathbf{t}_{p(x)}))$. Note that, when $x\in R$ and $p(x)$ is the trivial condition, $H(p(x))=\varnothing$ and $F^{A_x}_{p(x)}(\bar{z}\frestr\mathbf{t}_{p(x)})=\mathds{1}_{\Sor_x}$, so $E_x(z_x,F^{A_x}_{p(x)}(\bar{z}\frestr\mathbf{t}_{p(x)}))$ is true.
\item[(1.2)] If $x\in C$, $\Ewf^A(\bar{z},p)$ iff $p\in\Por^*\frestr A$, $\bar{z}\in\R(p)$ and, either $x\notin\dom p$ and $\Ewf^{A_x}(\bar{z},p)$, or $x\in\dom p$, $\Ewf^{A_x}(\bar{z}\frestr\mathbf{t}_{p\upharpoonright L_x},p\frestr L_x)$ and $z_x(p(x))=1$.
\end{itemize}
\item \emph{$A$ has a maximum $x$ but $A_x\notin\hat{\Iwf}_x$.} $\Ewf^A(\bar{z},p)$ iff there is an $A'\subseteq A$ such that $A'\cap L_x\in\Iwf_x\frestr A$, $p\in\Por^*\frestr A'$, $\bar{z}\in\R(p)$ and $\Ewf^{A'}(\bar{z},p)$. By (iii), this is equivalent to say that, for any $A'\subseteq A$, if $A'\cap L_x\in\Iwf_x\frestr A$, $p\in\Por^*\frestr A'$ and $\bar{z}\in\R(p)$, then $\Ewf^{A'}(\bar{z},p)$.
\item \emph{$A$ does not have a maximum.} Two cases
\begin{itemize}
\item[(3.1)] \emph{$A=\varnothing$.} $\Ewf^A(\bar{z},p)$ iff $p=\langle\ \rangle$ and $\bar{z}\in\R(p)=\{\langle\ \rangle\}$.
\item[(3.2)] \emph{$A\neq\varnothing$.} $\Ewf^A(\bar{z},p)$ iff there are $x\in A$ and $A'\in\Iwf_x\frestr A$ such that $p\in\Por^*\frestr A'$, $\bar{z}\in\R(p)$ and $\Ewf^{A'}(\bar{z},p)$. By (iii), this is equivalent to say that, for any $x\in A$ and $A'\in\Iwf_x\frestr A$, if $p\in\Por^*\frestr A'$ and $\bar{z}\in\R(p)$ then $\Ewf^{A'}(\bar{z},p)$.
\end{itemize}
\end{enumerate}
(i), (ii) and (iii) can be checked by simple calculations. We only show one case of (iii) to give an idea of how to proceed. Assume that $A$ is as in case (2) and $K\subseteq A$, also as in case (2), where $y=\max(K)\leq x$. Let $p\in\Por^*\frestr K$. We can find $A'\subseteq A$ and $K'\subseteq K\cap A'$ such that $K'\cap L_y\in\Iwf_y\frestr K$, $A'\cap L_x\in\Iwf_x\frestr A$ and $p\in\Por^*\frestr K'$ so, for $\bar{z}\in\R(p)$, $\Ewf^K(\bar{z},p)$ iff $\Ewf^{K'}(\bar{z},p)$ (by (2)) iff $\Ewf^{A'}(\bar{z},p)$ (by induction hypothesis) iff $\Ewf^A(\bar{z},p)$ (by (2)).
Let $\dot{x}=\langle h_n,A_n\rangle_{n<\omega}$ as in Notation \ref{NotationNames}, where $A_n=\{p_{n,k}\ /\ k<\beta_n\}$ is a maximal antichain with $\beta_n=|A_n|$. Define $D\subseteq\R(\dot{x})$ such that $\bar{z}\in D$ iff, for all $n<\omega$, there is a unique $k<\beta_n$ such that $\Ewf^A(\bar{z}\frestr\mathbf{t}_{p_{n,k}},p_{n,k})$. Clearly, by (i), $D$ is Borel. Let $F:D\to\omega^\omega$ defined as $F(\bar{z})(n)=h_n(p_{n,k})$ where $k<\beta_n$ is the unique such that $\Ewf^A(\bar{z}\frestr\mathbf{t}_{p_{n,k}},p_{n,k})$. It is easy to see that $\Vdash_{\Por^*\upharpoonright A}``\tilde{\eta}\frestr\mathbf{t}_{\dot{x}}\in D\textrm{\ and }\dot{x}=F(\tilde{\eta}\frestr\mathbf{t}_{\dot{x}})"$ by (ii). In a trivial way, we can extend $F$ to a Borel function $F^A_{\dot{x}}$ with domain $\R(\dot{x})$. (v) is an immediate consequence of (iii).
\end{proof}
{\small
\bibliographystyle{spmpsci}
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1504.01704
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\section{preliminaries}
We consider the case that $m=2k+1$ for $k\in\mathbb{N}$, whence the digits set $\Omega=\{0,1,\cdots, 2k+1\}$,
the generalized golden ration $\beta=\mathcal{G}(m)=\frac{k+1+\sqrt{k^{2}+6k+5}}{2}$, and $I_{\beta, m}=[0,\frac{m}{\beta-1}]=[0, \beta-k]$.
Recall that $\beta$ satisfies the algebraic equation $$\frac{k+1}{\beta}+\frac{k+1}{\beta^{2}}=1.$$
There are several digit sets need to be considered. We define the small-digit set to be $S=\{0, 1, \ldots, k\}$, and the big-digit set
$B=\{k+1, k+2, \ldots, 2k+1\}.$ Also we put $S^{-}=\{0, 1, \ldots, k-1\} $ and $S_{-}=\{1,2,\ldots, k\}$ by removing the smallest element and the biggest one from $S$ respectively. In the same way,
$B_-=\{k+2,\ldots,2k+1\}$ and $B^-=\{k+1,\ldots, 2k\}$.
\subsection{Sequence $F_{n}$}
In this subsection, we define a sequence $F_{n}$ which has some properties relating with $\beta$.
\begin{lemma}\label{n=fn}
Define $\{F_{n}\}_{n\geq1}$ to be the integer sequence satisfying $$F_{n+1}=(k+1)(F_{n}+F_{n-1})\quad (n\geq2)$$ with $F_{1}=1, F_{2}=k+1$. Then for any integer $n\ge0$, there exists a finite sequence $\{n_{i}\}_{i=1}^{l}\in\{0, 1, \ldots, (k+1)\}^{\ast}$ such that
\begin{equation}\label{n}
n=\sum_{i=1}^{l}n_{i}F_{i},
\end{equation}
where $l=l(n)$ is dependent of $n$.
\end{lemma}
We remark that the expansion (\ref{n}) of $n$ is by no means unique. While we can require that $l(n)<l$ if $n<F_{l}$.
\begin{proof}
The case $n=0$ or $n=1$ is trivial. Now by induction, we assume that the conclusion holds for $0\leq n<F_{m}$.
Noticing the fact that $F_{m+1}<(k+2)F_{m}$ and $$[F_{m},(k+2)F_{m})=\bigcup_{j=0}^{k+1}[jF_{m}, (j+1)F_{m}). $$
When $F_{m}\leq n <F_{m+1}$, we have that
$n\in[jF_{m}, (j+1)F_{m})$ for some $j\in\{0, 1, \ldots, k+1\}$, and $0\leq n-jF_{m}<F_{m}$. By the hypothesis of induction, $$n-jF_{m}=\sum_{i=1}^{m-1}\widetilde{n_{i}}F_{i}, $$ hence, we take $n_{m}=j$ together with $n_{i}=\widetilde{n_{i}}$ for $i<m$ to obtain an expansion of $n$.
\end{proof}
\begin{lemma}\label{fn}
For any $n\in\mathbb{N}$, $F_{n}\beta=F_{n+1}-(-\frac{k+1}{\beta})^{n}$.
\end{lemma}
\begin{proof}
Since $$F_{n+1}=(k+1)(F_{n}+F_{n-1})$$ and $$(-\frac{k+1}{\beta})^{n}=(k+1)(-\frac{k+1}{\beta})^{n-1}+(k+1)(-\frac{k+1}{\beta})^{n-2}, $$
the conclusion follows by induction.
\end{proof}
\subsection{Properties on the $\beta$-expansion}
\begin{lemma}\label{1}
The number $1$ has countably many expansions under the base $\beta$.
\end{lemma}
\begin{proof}
Recalling the fact that $1.00=0.(k+1)(k+1), $ we have
$0. (k+2)=1. 00(k+1)$ and $0. (2k+1)(2k+1)\ldots=1. (k+1). $
Let $0. \delta_{1}\delta_{2}\ldots\in\{0, 1, \ldots, 2k+1\}^{\infty}$ be an expansion of $1$.
We consider four cases according as the value of the the first digit $\delta_{1}$.
\begin{case} $\delta_{1} \in S^{-}. $\end{case}
Since $0. \delta_{2}\delta_{3}\ldots\leq0. (2k+1)(2k+1)(2k+1)\ldots=0.1(k+1)$,
\begin{eqnarray*}
0. \delta_{1}\delta_{2}\ldots &\leq& 0. \delta_{1}(2k+1)(2k+1)(2k+1)\ldots \\
&=& 0. (\delta_{1}+1)(k+1) \\
&<& 0.(k+1)(k+1)=1.00.
\end{eqnarray*}
This case is impossible.
\begin{case} $\delta_{1}=k.$\end{case} We know that $0. (2k+1)(2k+1)\ldots=1. (k+1)$, then $0. \delta_{1}\delta_{2}\ldots=0. (k+1)(k+1)$. Thus in this case the only possibility is that $0. \delta_{1}\delta_{2}\ldots=0. k(2k+1)(2k+1)\ldots. $
\begin{case} $\delta_{1}=k+1.$\end{case}
\begin{itemize}
\item $\delta_{1}=k+1, \delta_{2}\in S^{-}. $
Since \begin{eqnarray*}
0. (k+1)\delta_{2}(2k+1)(2k+1)\ldots &=& 0. (k+1)(\delta_{2}+1)(k+1) \\
&<& 0.(k+1)(k+1).
\end{eqnarray*}
Thus this subcase is impossible.
\item $\delta_{1}=k+1, \delta_{2}=k. $
In this subcase, we readily check that $0. \delta_{3}\delta_{4}\ldots$ is again an expansion of 1.
\item $\delta_{1}=k+1, \delta_{2}=k+1. $
Clearly, in this subcase the only possibility is that $1=0.(k+1)(k+1)000\ldots.$
\item $\delta_{1}=k+1, \delta_{2}\in B_{-}.$
Since $0. (k+1)\delta_{2}>0. (k+1)(k+1)=1, $ it is impossible.
\end{itemize}
\begin{case}
$\delta_{1}\in B_{-}. $
\end{case}
Since $0.(k+2)=1.00(k+1)>1, $ it is impossible.
In conclusion, the expansion of 1 takes one of the forms:
\begin{enumerate}
\item $1=0. k(2k+1)(2k+1)(2k+1)\ldots$
\item $1=0. (k+1)(k+1)$
\item $1=0. (k+1)k \delta_{3}\delta_{4}\ldots$ with $0.\delta_{3}\delta_{4}\ldots $again an expansion of 1.
\end{enumerate}
By an easy induction(on the number of the block $((k+1)k)'$s occurring in the beginning of the expansion), we then obtain all of the expansion of 1 as follows
\begin{itemize}
\item $0.((k+1)k)^{n}k(2k+1)(2k+1)(2k+1)\ldots$
\item $0.((k+1)k)^{n}(k+1)(k+1)$
\item $0.((k+1)k)^{\infty}, $ i.e. $0. (k+1)k (k+1)k (k+1)k \ldots$.
\end{itemize}
\end{proof}
In the following, we study the ``carry" and ``borrow" of the expansion in the light of the formula $1.00=0.(k+1)(k+1)$.
First we introduce a notation ``index" Ind$^{+}$($x$) for a sequence $x=0.x_{1}x_{2}\ldots x_{n}\ldots\in\{0,1,\ldots, 2k+1\}^{\infty}$ as follows: for $i\geq1$,
$$ \text{Ind}^{+}(x)=\left\{
\begin{array}{ll}
2i-1, & \hbox{if $x_{2j-1}\in S, x_{2j}\in B$ for $j=1,2,\ldots, i-1$ and $x_{2i-1}\in B$;} \\
2i, & \hbox{if $x_{2j-1}\in S, x_{2j}\in B$ for $j=1,2,\ldots, i-1$ and $x_{2i-1}, x_{2i}\in S$;}\\
\infty, & \hbox{if $x_{2j-1}\in S, x_{2j}\in B$ for $j=1,2,3,\ldots$.}\\
\end{array}
\right.$$
Then for any sequence $0.bx_{1}x_{2}x_{3}\ldots$ with $b\in B_-=\{k+2,\ldots,2k+1\}$, we can define the
``carry" map $T^{+}$
according as the value of the Ind$^{+}$($x$) as follows:
\begin{itemize}
\item If Ind$^{+}$($0. x_{1}x_{2}x_{3}\ldots$)=1, then
$$T^{+}(0.bx_{1}x_{2}x_{3}\ldots) = 1.(b-k-1)(x_{1}-k-1)x_{2}x_{3}\ldots. $$
\item If Ind$^{+}$($0. x_{1}x_{2}x_{3}\ldots$)=$2i-1$ for any $i\geq2$, then
\begin{eqnarray*}
&& T^{+}(0.bx_{1}x_{2}x_{3}\ldots)\\
&=& 1.(b-k-2)(x_{1}+1)(x_{2}-1)\ldots(x_{2i-1}+1) x_{2i-2}(x_{2i-1}-k-1)x_{2i}x_{2i+1}\ldots.
\end{eqnarray*}
\item If Ind$^{+}$($0. x_{1}x_{2}x_{3}\ldots$)=$2i$ for any $i\geq1$, then
\begin{eqnarray*}
&&T^{+}(0.bx_{1}x_{2}x_{3}\ldots) \\
&=& 1.(b-k-2)(x_{1}+1)(x_{2}-1)\ldots x_{2i-1}(x_{2i}+k+1)x_{2i+1}x_{2i+2}\ldots.
\end{eqnarray*}
\item If Ind$^{+}$($0. x_{1}x_{2}x_{3}\ldots$)=$\infty$, then
\begin{eqnarray*}
&&T^{+}(0.bx_{1}x_{2}x_{3}\ldots) \\
&=& 1.(b-k-2)(x_{1}+1)(x_{2}-1)(x_{3}+1)(x_{4}-1)\ldots.
\end{eqnarray*}
\end{itemize}
Obviously, for any $k\in\{1,2,\cdots\}\cup\{\infty\}$ the restriction of the map $T^{+}$ to the sequences $0.bx_{1}x_{2}x_{3}\ldots$ with $b\in B_-$ and Ind$^{+} ( 0. x_{1}x_{2}x_{3}\ldots)=k $ is injective.
In a dual way, we define the Ind$^{-}$($x$): for $i\geq1$
$$ \text{Ind}^{-}(x)=\left\{
\begin{array}{ll}
2i-1, & \hbox{if $x_{2j-1}\in B, x_{2j}\in S$ for $j=1,2,\ldots, i-1$ and $x_{2i-1}\in S$;} \\
2i, & \hbox{if $x_{2j-1}\in B, x_{2j}\in S$ for $j=1,2,\ldots, i-1$ and $x_{2i-1}, x_{2i}\in B$;}\\
\infty, & \hbox{if $x_{2j-1}\in B, x_{2j}\in S$ for $j=1,2,\ldots$ .}
\end{array}
\right.$$
Then for any sequence $1.a x_{1}x_{2}x_{3}\ldots$ with $a\in S^{-}=\{0,\ldots, k-1\}$, we define the
``borrow" map $T^{+}$ as follows:
\begin{itemize}
\item If Ind$^{-}$($0. x_{1}x_{2}x_{3}\ldots$)=1, then
$$T^{-}(1.a x_{1}x_{2}x_{3}\ldots) = 0.(a+k+1)(x_{1}+k+1)x_{2}x_{3}\ldots. $$
\item If Ind$^{-}$($0. x_{1}x_{2}x_{3}\ldots$)=$2i-1$ for any $i\geq2$, then
\begin{eqnarray*}
&& T^{-}(1. a x_{1}x_{2}x_{3}\ldots)\\
&=& 0.(a+k+2)(x_{1}-1)(x_{2}+1)\ldots (x_{2i-3}-1)x_{2i-2}(x_{2i-1}+k+1)x_{2i}\ldots.
\end{eqnarray*}
\item If Ind$^{-}$($0. x_{1}x_{2}x_{3}\ldots$)=$2i$ for any $i\geq1$, then
\begin{eqnarray*}
&&T^{-}(1. a(k+2)x_{1}x_{2}x_{3}\ldots) \\
&=& 0. (a+k+2)(x_{1}-1)(x_{2}+1)\ldots (x_{2i-2}+1)x_{2i-1}(x_{2i}-k-1)x_{2i+1}\ldots.
\end{eqnarray*}
\item If Ind$^{-}$($0. x_{1}x_{2}x_{3}\ldots$)=$\infty$, then
\begin{eqnarray*}
&&T^{-}(1. a(k+2)x_{1}x_{2}x_{3}\ldots) \\
&=& 0. (a+k+2)(x_{1}-1)(x_{2}+1)(x_{3}-1)(x_{4}+1)\ldots.
\end{eqnarray*}
\end{itemize}
\medskip
\begin{lemma}
The set $\{0, 1, \ldots, 2k+1\}^{\ast}$ is closed under multiplication by $\beta$, more precisely, for any $x\in(0, \frac{\beta-k}{\beta})$ which has a finite expansion $x=0. \varepsilon_{1}\ldots\varepsilon_{n}$, there exist $\eta_{1}\ldots\eta_{m}\in\{0, 1, \ldots, 2k+1\}^{\ast}$ such that $\beta x =0. \eta_{1}\ldots\eta_{m}.$
\end{lemma}
\begin{proof}
Let $x\in(0, \frac{\beta-k}{\beta})$ be a number with a finite expansion, and $0. \varepsilon_{1}\ldots\varepsilon_{n}$ be an expansion of $x$. Then $0. \varepsilon_{1}\ldots\varepsilon_{n}<0. 1(k+1)$ since $\frac{\beta-k}{\beta}=0. 1(k+1). $
We need to find a finite expansion for $\beta x$. We consider three cases according as the value of the first two digits $\varepsilon_{1}\varepsilon_{2}.$
\addtocounter{case}{-4}
\begin{case} $\varepsilon_{1}=0$.
\end{case}
In this case, it is clearly that $\beta x=0.\varepsilon_{2}\ldots\varepsilon_{n}$.
\begin{case}\label{10}
$\varepsilon_{1}=1, \varepsilon_{2}\in S^{-}.$
\end{case}
Recall that $0.(2k+1)(2k+1)(2k+1)\ldots=1.(k+1)$. In this case, the digits can take any value in $\{0, 1,2, \ldots, 2k+1\}$, since $$0.1\varepsilon_{2}(2k+1)(2k+1)\ldots(2k+1)<0.1(k+1).$$
Putting Ind$^{-}$($\varepsilon_{3}\ldots\varepsilon_{n}$)=$s$, we have that
\begin{itemize}
\item If $s=1$, then $$\beta x=0.(\varepsilon_{2}+k+1)(\varepsilon_{3}+k+1)\varepsilon_{4}\varepsilon_{5}\ldots \varepsilon_{n};$$
\item If $s=2i-1$ for some $i\geq2$, then
\begin{eqnarray*}
&& \beta x \\
&=& 0.(\varepsilon_{2}+k+2)(\varepsilon_{3}-1)(\varepsilon_{4}+1)\ldots (\varepsilon_{2i-1}-1)\varepsilon_{2i}(\varepsilon_{2i+1}+k+1)\varepsilon_{2i+2}\ldots \varepsilon_{n};
\end{eqnarray*}
\item If $s=2i$ for some $i\geq 1$, then
\begin{eqnarray*}
&& \beta x \\
&=& 0. (\varepsilon_{2}+k+2)(\varepsilon_{3}-1)(\varepsilon_{4}+1)\ldots (\varepsilon_{2i}+1)\varepsilon_{2i+1}(\varepsilon_{2i+2}-k-1)\varepsilon_{2i+3}\ldots \varepsilon_{n}.
\end{eqnarray*}
\end{itemize}
\begin{case}
$\varepsilon_{1}=1, \varepsilon_{2}=k. $
\end{case}
In this case, since $0.1k\varepsilon_{3}\varepsilon_{4}\ldots<0.1(k+1)$, we have the constraint that $$0.\varepsilon_{3}\varepsilon_{4}\ldots<1. $$
Since $1=0.(k+1)k(k+1)k(k+1)k\ldots, $ the expansion of $x$ is of one of the following forms:
\begin{enumerate}
\item $x=0.1k((k+1)k)^{p}a\varepsilon_{2p+4}\ldots\varepsilon_{n}\text{ for }a\in S, $
\item $x=0.1k((k+1)k)^{p}(k+1)b\varepsilon_{2p+5}\ldots\varepsilon_{n}$
$\text{ for }b\in S^{-}, $
\end{enumerate}
where $p\in\mathbb{N}\cup\{0\}$. \
In the first subcase, we have $$\beta x=1. k((k+1)k)^{p}a\varepsilon_{2p+4}\ldots\varepsilon_{n}. $$
By $1.00=0.(k+1)(k+1)$, it follows that
$$\beta x=0.(2k+1)\ldots(2k+1)(a+k+1)\varepsilon_{2p+4}\ldots\varepsilon_{n},$$
which is in $\{0,1,\ldots,2k+1\}^{\ast}$.
In the second subcase, we have
\begin{eqnarray*}
&& \beta x \\
&=& 1. k((k+1)k)^{p}(k+1)b \varepsilon_{2p+5}\ldots\varepsilon_{n} \\
&=& 0.(2k+1)\ldots(2k+1)(2k+2)b \varepsilon_{2p+5}\ldots\varepsilon_{n}.
\end{eqnarray*}
Writing $\varepsilon_{2p+5}\varepsilon_{2p+6}\ldots\varepsilon_{n}=x_{1}x_{2}\ldots x_{q}$ and Ind$^{-}$($x_{1}x_{2}\ldots x_{q}$)=$s$, we have that
\begin{itemize}
\item If $s=1$, then $$\beta x=0.(2k+1)\ldots(2k+1)(2k+1)(b+k+1)(x_{1}+k+1)x_{2}\ldots x_{q};$$
\item If $s=2i-1$ for some $i\geq2$, then
\begin{eqnarray*}
&& \beta x \\
&=& 0.(2k+1)\ldots(2k+1)(b+k+2)(x_{1}-1)(x_{2}+1)\ldots (x_{2i-3}-1)x_{2i-2}\\&&(x_{2i-1}+k+1)x_{2i}\ldots x_{q};
\end{eqnarray*}
\item If $s=2i$ for some $i\geq 1$, then
\begin{eqnarray*}
&& \beta x \\
&=& 0.(2k+1)\ldots(2k+1)(b+k+2)(x_{1}-1)(x_{2}+1)\ldots (x_{2i-2}+1)x_{2i-1}\\&&(x_{2i}-k-1)x_{2i+1}\ldots x_{q}.
\end{eqnarray*}
\end{itemize}
\end{proof}
Let $x=0. \varepsilon_{1}\ldots\varepsilon_{n}\in\{0, 1, \ldots, 2k+1\}^{\ast}$. If the condition that $\varepsilon_{i}\in S$ for $1\leq i\leq n-1 $ implies that $\varepsilon_{i+1}\in B$, which means the digit $k+1, k+2, \ldots, $ or $2k+1$ is always separated by the digit 0, 1,$\ldots,$ or $k$, we say that this sequence $0. \varepsilon_{1}\ldots\varepsilon_{n}$ is $B$-separated.
For $x=0.\varepsilon_{1}\varepsilon_{2}\ldots\varepsilon_{n}$, we define
$$l=\min\left\{1 \leq i\leq n-1: \varepsilon_{i}\varepsilon_{i+1}\in B^{2}\right\}$$ with the convention that $\min\emptyset=\infty$.
It is easy to see that $l=\infty$ when and only when the sequence $0.\varepsilon_{1}\varepsilon_{2}\ldots\varepsilon_{n}$ is $B$-separated.
We then define an operator $C_{r}$ as follows:
\begin{eqnarray*}
&& C_{r}(0.\varepsilon_{1}\varepsilon_{2}\ldots\varepsilon_{n}) \\
&=& \left\{
\begin{array}{ll}
0.\varepsilon_{1}\ldots\varepsilon_{l-2}(\varepsilon_{l-1}+1)(\varepsilon_{l}-(k+1))(\varepsilon_{l+1}-(k+1))\varepsilon_{l+2}\ldots\varepsilon_{n}, & \hbox{\text{ if } $l < \infty$;} \\
0.\varepsilon_{1}\varepsilon_{2}\ldots\varepsilon_{n}, & \hbox{\text{ if } $l=\infty$.}
\end{array}
\right.
\end{eqnarray*}
It is easy to see that the operator $C_{r}$ preserves the value. On the other hand when $l<\infty$, the operator $C_{r}$ reduces by $2k+1$ the summation of all the digits in the expansion. Thus there exists $k\in\mathbb{N}$ such that $$C_{r}^{k+1}(0.\varepsilon_{1}\varepsilon_{2}\ldots\varepsilon_{n})=C_{r}^{k}(0.\varepsilon_{1}\varepsilon_{2}\ldots\varepsilon_{n}).$$
Whence the sequence $C_{r}^{k}(0.\varepsilon_{1}\varepsilon_{2}\ldots\varepsilon_{n})$ is $B$-separated, and then we define the map $T$ as
$$T(0.\varepsilon_{1}\varepsilon_{2}\ldots\varepsilon_{n})=C_{r}^{k}(0.\varepsilon_{1}\varepsilon_{2}\ldots\varepsilon_{n}), $$ for such $k$.
If $\varepsilon_{1}\varepsilon_{2}\in B^{2}$, then $0.\varepsilon_{1}\varepsilon_{2}=1.(\varepsilon_{1}-k-1)(\varepsilon_{2}-k-1)$. Denote $T(0.\varepsilon_{1}\varepsilon_{2}\ldots\varepsilon_{n})=\eta_{0}.\eta_{1}\ldots\eta_{n}$, where $\eta_{0}\in\{0, 1\}$ and $\eta_{i}\in\{0,\ldots,2k+1\}$ for $1\leq i\leq n$. Clearly, the map $T$ satisfies the following properties:
\begin{itemize}
\item If $\eta_{i}\in B_{-}$ for some $i$, then $\varepsilon_{i}=\eta_{i}$. In other words, there is no new occurrence of the digit $\varepsilon\in B_{-}$ in the process, and thus we have $$|\eta_{0}.\eta_{1}\ldots\eta_{n}|_{\varepsilon}\leq|0.\varepsilon_{1}\varepsilon_{2}\ldots\varepsilon_{n}|_{\varepsilon}, $$ where $|\cdot|_{\varepsilon}$ denotes the total number of occurrences of the digit $\varepsilon$ in the sequence.
\item The sequence $\eta_{0}.\eta_{1}\ldots\eta_{n}$, or equivalently, the sequence $0.\eta_{1}\ldots\eta_{n}$ is $B$-separated.
\end{itemize}
\begin{lemma}\label{finite}
For any $0. \varepsilon_{1}\ldots\varepsilon_{n}\in\{0, 1, \ldots, 2k+1\}^{\ast}$, there exists $\widetilde{\varepsilon_{0}}. \widetilde{\varepsilon_{1}}\ldots\widetilde{\varepsilon_{m}}$ such that $\widetilde{\varepsilon_{0}}. \widetilde{\varepsilon_{1}}\ldots\widetilde{\varepsilon_{m}}=0. \varepsilon_{1}\ldots\varepsilon_{n}$, where $\widetilde{\varepsilon_{0}}\in\{0, 1\}$ and $\widetilde{\varepsilon_{i}}\in\{0, 1, \ldots, k+1\}$ for $i=1, \ldots, m$.
\end{lemma}
\begin{proof}
Applying the map $T$ on the expansion $0. \varepsilon_{1}\ldots\varepsilon_{n}$ if necessary, we can suppose without loss of generality that $0. \varepsilon_{1}\ldots\varepsilon_{n}$ is $B$-separated. Our aim is to eliminate all the digits $k+2, \ldots, 2k+1$ from the expansion.
Now we want to eliminate the digit $2k+1$ in the first step. We regroup the expansion $0. \varepsilon_{1}\ldots\varepsilon_{n}$ according as the position of the last $2k+1$ as follows:
\begin{equation}\label{eta}
0. \varepsilon_{1}\ldots\varepsilon_{n}=0. \varepsilon_{1}\ldots\varepsilon_{m}(2k+1) \delta_{1}\ldots\delta_{p},
\end{equation}
where $m+k+1=n$ and $\delta=\delta_{1}\ldots\delta_{p}\in\{0, 1, \ldots, 2k\}^{\ast}$. Moreover, we have that $\varepsilon_{m}, \delta_{1}\in S$ from the $B$-separation property.
We claim that we can eliminate the last occurrence of the digit $2k+1$ from the expansion without change of the value. Meanwhile, in the process there are no new occurrences of $2k+1$ in the resulted expansion. We will show the claim by induction on the length of $\delta$.
When $ \delta=\emptyset$, that is, $m+1=n$, recalling the fact that $0.(2k+1)=1.(k-1)0(k+1)$, we have
$$0. \varepsilon_{1}\ldots\varepsilon_{n}=0. \varepsilon_{1}\ldots\varepsilon_{n-2}(\varepsilon_{n-1}+1)(k-1)0(k+1). $$
When the length of $\delta$ is 1, we have
$$0. \varepsilon_{1}\ldots\varepsilon_{n}=0. \varepsilon_{1}\ldots\varepsilon_{m-1}(\varepsilon_{m}+1)(k-1)\delta_{1}(k+2). $$
Now by induction, we assume that the conclusion holds for the expansion with the length of $\delta$ less than $p$.
When the length of $\delta$ is $p$,
according to the value of the digit $\delta_{2}$, we consider the following three cases:
\addtocounter{case}{-3}
\begin{case}$\delta_{2}\in S^{-}$. \end{case}
In this case, by $0.(k+2)=1.00(k+1)$, we have
$$0. \varepsilon_{1}\ldots\varepsilon_{n}=0. \varepsilon_{1}\ldots\varepsilon_{m-1}(\varepsilon_{m}+1)(k-1)\delta_{1}(\delta_{2}+k+1)\delta_{3}\ldots\delta_{p}. $$
Thus the conclusion follows.
\begin{case}$\delta_{2}=k. $\end{case}
By $0.(k+2)=1.00(k+1)$, we have
$$0. \varepsilon_{1}\ldots\varepsilon_{n}=0. \varepsilon_{1}\ldots\varepsilon_{m-1}(\varepsilon_{m}+1)(k-1)\delta_{1}(2k+1)\delta_{3}\ldots\delta_{p}. $$
Applying the map $T$ on the above expansion, we have the conclusion by the hypothesis of induction.
\begin{case}$\delta_{2}\in B^{-}. $\end{case}
Recalling the fact that the sequence $0. \varepsilon_{1}\ldots\varepsilon_{n}$ is $B$-separated, the block $\delta$ can be regrouped as
\begin{equation}
\delta_{1}a_{1}\delta_{3}a_{2}\delta_{5}a_{2}\delta_{7}\ldots a_{t}\delta_{2t+1}\delta_{2t+2} \ldots \delta_{p},
\end{equation}
where $a_{j}\in B^{-}$ for $j\in\{1, 2, \ldots, t\}$, and when $p \geq 2t+2$, $\delta_{2t+2}\in S$.
By $0.(k+2)=1.00(k+1)$, we obtain that
\begin{eqnarray*}
&& 0.(2k+1)\delta \\
&=& 1.(k-1)(\delta_{1}+1)(a_{1}-1)\ldots (a_{t}-1)\delta_{2t+1}(\delta_{2t+2}+k+1)\delta_{2t+3} \ldots \delta_{p}.
\end{eqnarray*}
All the digits in the latter expansion are in $\{0, 1, \ldots, 2k\}$ expect at most that $\delta_{2t+2}+k+1\in\{0,1,\ldots,2k+1\}$. If $\delta_{2t+2}+k+1<2k+1$, we are done; otherwise we apply the map $T$, and then use the induction hypothesis.
The claim then follows.
Up to now, we have already eliminate the last digit $2k+1$ in the expansion without new occurrence of $2k+1$ , and then we continue this process to eliminate all the digits $2k+1$ from the expansion.
Using the same argument, we eliminate the other digit in $\{k+2, \ldots, 2k\}$.
\end{proof}
From the compactness of the symbol space $\{0, 1, \ldots, k+1\}^{\infty}$ ,we remark that any infinite sequence in $\{0, 1, \ldots, 2k+1\}^{\infty}$ has the property as in Lemma \ref{finite}, too.
\begin{remark}\label{infinite}
For any $0. \varepsilon_{1}\varepsilon_{2}\ldots\in\{0, 1, \ldots, 2k+1\}^{\infty}$, there exists $\widetilde{\varepsilon_{0}}. \widetilde{\varepsilon_{1}}\widetilde{\varepsilon_{2}}\ldots$ such that
$$0. \varepsilon_{1}\varepsilon_{2}\ldots=\widetilde{\varepsilon_{0}}. \widetilde{\varepsilon_{1}}\widetilde{\varepsilon_{2}}\ldots, $$
where $\widetilde{\varepsilon_{i}}\in\{0, 1, \ldots, k+1\}$ for $i\geq1$ and $\widetilde{\varepsilon_{0}}\in\{0,1\}$.
\end{remark}
\begin{lemma}\label{addition}
The set $\{0, 1, \ldots, 2k+1\}^{\ast}$ is closed under addition, more precisely, for any two finite expansions $\xi, \eta$ in $\{0, 1, \ldots, 2k+1\}^{\ast},$
there exists a sequence $\delta_{0}. \delta_{1}\delta_{2}\ldots\delta_{s}$ such that $$\xi+\eta=\delta_{0}. \delta_{1}\delta_{2}\ldots\delta_{s},$$ where $0. \delta_{1}\delta_{2}\ldots\delta_{s}$ is in $\{0, 1, \ldots, 2k+1\}^{\ast}. $
\end{lemma}
\begin{proof}
Adding 0's at the end of the expansion if necessary, we cam assume without loss of generality that the length of $\xi$ and $\eta$ are equal.
By Lemma \ref{finite}, we can suppose without loss of generality that
$x=\xi_{0}. \xi_{1}\ldots\xi_{n}$, $y=\eta_{0}. \eta_{1}\ldots\eta_{n}$, where $0.\xi_{1}\ldots\xi_{n}$ and $0. \eta_{1}\ldots\eta_{n}$ are in $\{0, 1, \ldots, k+1\}^{\ast}$.
Then
$$\xi+\eta=z_{0}. z_{1}z_{2}\ldots z_{n}, $$
where
$z_{j}=\xi_{j}+\eta_{j}\in\{0, 1, 2, \ldots, 2k+2\}$ for $j=0,1,2,3,\ldots, n$.
Now we define two new sequences as follows.
For any $j\in\{1,2, \ldots, n\}$, define
$$x_{j}=\left\{
\begin{array}{ll}
2k+1, & \hbox{ if $z_{j}=2k+2$;} \\
z_{j}, & \hbox{ \text{others} .}
\end{array}
\right. \text{and }
y_{j}=\left\{
\begin{array}{ll}
1, & \hbox{ if $z_{j}=2k+2$;} \\
0, & \hbox{ \text{others} .}
\end{array}
\right.$$
Obviously, we have $z_{j}=x_{j}+y_{j}$ for $j=1,2,3,\ldots, n$
and $0.y_{1}\ldots y_{n}\in\{0, 1\}^{\ast}$, $0.x_{1}\ldots x_{n}\in\{0, 1, \ldots, 2k+1\}^{\ast}.$
By the Lemma $\ref{finite}$ again, there is a sequence $\widetilde{x_{0}}. \widetilde{x_{1}}\ldots\widetilde{x_{q}}$ such that
$$0.x_{1}\ldots x_{n}=\widetilde{x_{0}}. \widetilde{x_{1}}\ldots\widetilde{x_{q}},$$
where $0. \widetilde{x_{1}}\ldots\widetilde{x_{q}}\in\{0, 1, \ldots, k+1\}^{\ast}$. Hence, it follows $$\xi+\eta=(\xi_{0}+\eta_{0}+\widetilde{x_{0}}).(\widetilde{x_{1}}+y_{1})\ldots(\widetilde{x_{q}}+y_{q})\quad (q>n),$$
where $0.(\widetilde{x_{1}}+y_{1})\ldots(\widetilde{x_{q}}+y_{q})\in\{0, 1, \ldots, k+2\}^{\ast}.$
\end{proof}
\begin{lemma}$\label{dividingk}$
The set $\{0, 1, \ldots, 2k+1\}^{\ast}$ is closed under dividing by $k+1$.
\end{lemma}
\begin{proof}
For each $\varepsilon\in\{0, 1, \ldots, 2k+1\}$, we define
$$i(\varepsilon)=\left\{
\begin{array}{ll}
0, & \hbox{if $\varepsilon\in S$;} \\
k+1, & \hbox{if $\varepsilon\in B$.}
\end{array}
\right.
\text{ and }
t(\varepsilon)=\left\{
\begin{array}{ll}
(k+1)\varepsilon, & \hbox{if $\varepsilon\in S$;} \\
(k+1)(\varepsilon-k-1), & \hbox{if $\varepsilon\in B$.}
\end{array}
\right.
$$
Let $x=0. \varepsilon_{1}\ldots\varepsilon_{n}\in\{0, 1, \ldots, 2k+1\}^{\ast}$. Define $0.\eta_{1}\ldots\eta_{n+2}$ as
$$\eta_{j}=i(\varepsilon_{j})+t(\varepsilon_{j-1})+t(\varepsilon_{j-2})\quad(j=1, 2,\ldots, n+2),$$
with $\varepsilon_{-1}=\varepsilon_{0}=0. $ Clearly, the sequence $0.\eta_{1}\ldots\eta_{n+2}$ has the following two properties :
\begin{itemize}
\item $\eta_{j}\in\{0, k+1, 2(k+1), \ldots, (2k+1)(k+1)\}$,
\item $0.\eta_{1}\ldots\eta_{n+2}= 0. \varepsilon_{1}\ldots\varepsilon_{n}$.
\end{itemize}
Thus, we obtain
$$\frac{x}{k+1}=0.\frac{\eta_{1}}{k+1}\ldots\frac{\eta_{n+2}}{k+1}\in\{0, 1, \ldots, 2k+1\}^{\ast}.$$
\end{proof}
\begin{corollary}\label{2n}
For any $n\in\mathbb{N}\cup\{0\}$, $\frac{1}{(k+1)^{n}}$ has at least a finite expansion.
\end{corollary}
\begin{proof}
This is a direct result of Lemma \ref{1} and $\ref{dividingk}$.
\end{proof}
\section {proof of theorem \ref{theorem}}
We are now in a position to proof Theorem \ref{theorem}. Recalling that
$$\mathcal{S}=\{\frac{p\beta+q}{(k+1)^{n}}\in(0, \beta-k): n, p, q\in\mathbb{Z}\}.$$
\begin{lemma}\label{xs}
Let $x \in (0, \beta-k)$. Then $x$ has a finite expansion if and only if $x\in\mathcal{S}$.
\end{lemma}
\begin{proof}
Let $x\in(0, \beta-k)$ be a number with a finite expansion $0. \varepsilon_{1}\ldots\varepsilon_{2n}$, i.e.,
$$x=\sum_{i=1}^{2n}\varepsilon_{i}\beta^{-i}, $$ where $\varepsilon_{i}\in\{0, 1, \ldots, 2k+1\}$ for $i=1, 2, \ldots, 2n $. By Lemma $\ref{fn}$, we have
\begin{eqnarray*}
x &=& \sum_{i=1}^{n}\frac{\varepsilon_{2i-1}(F_{2i-1}\beta-F_{2i})}{(k+1)^{2i-1}}+\sum_{i=1}^{n}\frac{\varepsilon_{2i}(F_{2i+1}-F_{2i}\beta)}{(k+1)^{2i}} \\
&=& \frac{1}{(k+1)^{2n}}[(\sum_{i=1}^{n}\varepsilon_{2i-1}F_{2i-1}(k+1)^{2n-(2i-1)}-\sum_{i=1}^{n}\varepsilon_{2i}F_{2i}(k+1)^{2n-2i})\beta \\ &+&(\sum_{i=1}^{n}\varepsilon_{2i}F_{2i+1}(k+1)^{2n-2i}-\sum_{i=1}^{n}\varepsilon_{2i-1}F_{2i}(k+1)^{2n-(2i-1)})]
\end{eqnarray*}
Putting $$p=\sum_{i=1}^{n}\varepsilon_{2i-1}F_{2i-1}(k+1)^{2n-(2i-1)}-\sum_{i=1}^{n}\varepsilon_{2i}F_{2i}(k+1)^{2n-2i}$$ and $$q=\sum_{i=1}^{n}\varepsilon_{2i}F_{2i+1}(k+1)^{2n-2i}-\sum_{i=1}^{n}\varepsilon_{2i-1}F_{2i}(k+1)^{2n-(2i-1)}, $$ we show that, $x\in\mathcal{S}$.
On the other hand, if $x\in\mathcal{S}$, then there exist $p, q, n\in\mathbb{Z}$ such that $$x=\frac{p\beta+q}{(k+1)^{n}} $$
with $x\in(0, \beta-1)$. Applying Lemma \ref{n=fn} with $p$, we have
\begin{equation}\label{p}
p=\sum_{i=1}^{2p_{0}}n_{i}F_{i},
\end{equation}
where $n_{i}\in\{0, 1, \ldots, k+1\}$ for any $i\in\{1, \ldots, 2p_{0}\}$, or $n_{i}\in\{0, -1, \ldots, -(k+1)\}$ for any $i\in\{1, \ldots, 2p_{0}\}$.
From Lemma $\ref{fn}$ and (\ref{p}), we have
\begin{eqnarray*}
x &=&\frac{1}{(k+1)^{n}}\left[\sum_{i=1}^{p_{0}}n_{2i-1}(F_{2i}+(\frac{k+1}{\beta})^{2i-1})+\sum_{i=1}^{p_{0}}n_{2i}(F_{2i+1}-(\frac{k+1}{\beta})^{2i})+q\right] \\
&=& \sum_{i=1}^{p_{0}}\frac{n_{2i-1}}{\beta^{2i-1}(k+1)^{n-(2i-1)}}- \sum_{i=1}^{p_{0}}\frac{n_{2i}}{\beta^{2i}(k+1)^{n-2i}}+\frac{M}{(k+1)^{n}},
\end{eqnarray*}
where $M=\sum_{i=1}^{2p_{0}}n_{i}F_{i+1}+q$.
Hence, from Lemma \ref{addition} and Corollary \ref{2n}, we obtain that any $x\in\mathcal{S}$ has a finite expansion.
\end{proof}
\begin{remark}\label{finite-coun}
By Lemma \ref{1}, the number 1 has countably many expansions, and thus for any $x\in S$, we can deduce at least countably many expansions for $x$ from its finite expansion.
\end{remark}
\begin{lemma}\label{coun-finite}
If $x\in(0, \beta-k)$ has countably many expansions then $x$ at least has a finite expansion.
\end{lemma}
\begin{proof}
Let
\begin{equation}\label{x}
0. \varepsilon_{1}\varepsilon_{2}\varepsilon_{3}\ldots
\end{equation}
be an infinite expansion of $x$. We consider the 2-blocks appearing in the expansion.
\addtocounter{case}{-3}
\begin{case}\label{bb}
There are infinite many blocks in $B^2$ or $S^2$ appearing in (\ref{x}).
\end{case}
\begin{enumerate}
\item There are infinite many blocks of form $xb_{1}b_{2}$ with $x\in\{0,1, \ldots, 2k\}$, $b_{1}, b_{2}\in B$, or $ya_{1}a_{2}$ with $y\in\{1,2, \ldots, 2k+1\}$, $a_{1},a_2\in S. $
Notice that $$0.xb_{1}b_{2}=0.(x+1)(b_{1}-k-1)(b_{2}-k-1), $$ and $$0.ya_{1}a_{2}=0.(y-1)(a_{1}+k+1)(a_{2}+k+1). $$ The number $x$ has uncountably many expansions.
\item The expansion ends with $(2k+1)^{\infty}$ or $0^{\infty}$.
Since $1=0.k(2k+1)^{\infty}$, the number $x$ has a finite expansion.
\end{enumerate}
\begin{case}
The blocks in $B^2\cup S^2$ appears for finite times.
\end{case}
In this case, the expansion is of the type
\begin{equation}\label{x1}
x=0.\varepsilon_{1}\ldots\varepsilon_{n_{1}}a^{(1)}_{1}b^{(1)}_{1}a^{(1)}_{2}b^{(1)}_{2}a^{(1)}_{3}b^{(1)}_{3}\ldots,
\end{equation}
with $a^{(1)}_{i}\in S$ and $b^{(1)}_{i}\in B$ for $i\geq 1$.
Put $l=\min\{i\geq1: b_{i}\geq k+2 \}$ with the convention that $\min\emptyset=\infty. $ Thus $l=\infty$ when and only when $b_{i}=k+1$ for all $i\geq1$.
When $l=\infty$, we have
\begin{eqnarray*}
x &=& 0.\varepsilon_{1}\ldots\varepsilon_{n_{1}}a^{(1)}_{1}(k+1)a^{(1)}_{2}(k+1)a^{(1)}_{3}(k+1)\ldots\\
&=& 0.\varepsilon_{1}\ldots\varepsilon_{n_{1}}a^{(1)}_{1}k (a^{(1)}_{2}+k+1)(2k+1)(a^{(1)}_{3}+k+1)(2k+1)\ldots
\end{eqnarray*}
There are infinitely many blocks in $B^2$, whence just as in Case \ref {bb}, either $x$ has a finite expansion, or
$x$ has uncountably many expansions.
When $l<\infty$, we have, by $0.(k+2)=1.00(k+1)$, that
\begin{eqnarray*}
\mathcal{E}_{1} &=& 0.\varepsilon_{1}\ldots\varepsilon_{n_{1}}a^{(1)}_{1}b^{(1)}_{1}\ldots a^{(1)}_{l-1}b^{(1)}_{l-1}(a^{(1)}_{l}+1)0(a^{(1)}_{l+1}+1)(b^{(1)}_{l+1}-1)\ldots.
\end{eqnarray*}
We rewrite this expansion as
\begin{equation}\label{x2}
\mathcal{E}_{2}= 0.\varepsilon_{1}\ldots\varepsilon_{n_{2}}a^{(2)}_{1}0a^{(2)}_{2}b^{(2)}_{2}a^{(2)}_{3}b^{(2)}_{3}\ldots,
\end{equation}
with $$a^{(2)}_{i}\in S+1=\{x+1: x\in S\}=\{1, 2, \ldots, k+1\}$$ and $$b^{(2)}_{i}\in B-1=\{y-1: y\in B\}=\{k, k+1, \ldots, 2k
\}$$.
Meanwhile, using the same argument as the Case\ref{bb}, we may assume that the blocks in $B^2\cup S^2$ appears for only finite times, and thus in the expansion (\ref{x2}), the subword $a^{(2)}_{i}b^{(2)}_{i}$ for $i$ large enough
\begin{itemize}
\item can not in $\{1k, 2k, \ldots, kk, (k+1)(k+1), (k+1)(k+2), \ldots, (k+1)(2k+1)\}$,
\item may not be equal to $(k+1)k$, expect the case when $a^{(2)}_{i}b^{(2)}_{i}=(k+1)k$ eventually. Whence by $1.00=0.((k+1)k)^{\infty}$, the expansion can be transformed into a finite one.
\end{itemize}
In the light of these, we may suppose that the expansion of $x$ is of the form:
\begin{equation}\label{x3}
x= 0.\varepsilon_{1}\ldots\varepsilon_{n_{3}}a^{(3)}_{1}b^{(3)}_{1}a^{(3)}_{2}b^{(3)}_{2}a^{(3)}_{3}b^{(3)}_{3}\ldots,
\end{equation}
with $$a^{(3)}_{i}\in (S+1)^{-}=\{1, 2, \ldots, k\}$$ and $$b^{(3)}_{i}\in (B-1)_{-}=\{k+1, \ldots, 2k
\}$$ for $i\geq 1$.
Up to now, we only need to consider the expansion (\ref{x3}). Compared with expansion (\ref{x1}), all $a_{i}b_{i}$ may take the values in $(S+1)^-\times(B-1)_{-}$ rather than $SB$. We continue this process to reach an expansion with all $a_{i}b_{i}\in (S+2)^-\times(B-2)_{-}$, and so on. Finally, we obtain an expansion ending by $((k+1)k)^{\infty}, $ and it can be transformed into a finite one as before.
\end{proof}
\begin{lemma}\label{finite-coun}
Any number in $\mathcal{S}$ has countably many expansions.
\end{lemma}
\begin{proof} By Remark \ref{finite-coun}, we only need show that any number in $\mathcal{S}$ can not have uncountably many expansions.
To this end, we suppose that
$x\in\mathcal{S} $ has have uncountably many expansions. Considering for $\beta^{-n}x$ instead, we may assume $x<1.$ Thus
$$y=1-x\in\mathcal{S}.$$
By Lemmas \ref{xs} and \ref{finite}, the number $y$ has a finite expansion
$$y=0. \varepsilon_{1}\ldots\varepsilon_{n}\in\{0,1,\ldots,k+1\}^{\ast}.$$
Since $x$ has uncountable many expansions, there exist an uncountable index set $\Lambda$ and a block $\omega=\omega_{1}\omega_{2}\ldots\omega_{n}\in\{0,1, \ldots, 2k+1\}^{\ast}$ such that for any $\lambda\in\Lambda$,
$$\omega_{\lambda}=0. \omega_{1}\omega_{2}\ldots\omega_{n}\omega_{n+1}^{\lambda}\omega_{n+2}^{\lambda}\ldots$$
is an expansion of $x$, and thus
$$1=x+y=0.z_{1}z_{2}\ldots z_{n}\omega_{n+1}^{\lambda}\omega_{n+2}^{\lambda}\ldots, $$
where $z_{j}=\omega_{j}+\varepsilon_{j}\in\{0,1,\ldots,3k+2\}$ for $1\leq j\leq n$.
\smallskip
Applying the map $T$ to $0.z_{1}z_{2}\ldots z_{n}$, we obtain that $$T(0.z_{1}z_{2}\ldots z_{n})=\widetilde{z_{0}}. \widetilde{z_{1}}\widetilde{z_{2}}\ldots\widetilde{z_{n}}$$
and if $\widetilde{z_{i}}>k+1$, then we have
$$\left\{
\begin{array}{ll}
\widetilde{z_{i+1}}\leq k, & \hbox{for $i=1,2,\ldots, n$;} \\
\widetilde{z_{i-1}}\leq k, & \hbox{for $i=2,3,\ldots, n$.}
\end{array}
\right.\eqno{(*)}$$
Since $0.z_{1}z_{2}\ldots z_{n}<x+y=1$ and $T$ preserves the value, $\widetilde{z_{0}}=0$. So we may
suppose that the expansion $0.z_{1}z_{2}\ldots z_{n}$ satisfies the above property $(*)$.
\smallskip
We claim that the number $f_{n-1}=1-0.z_{1}z_{2}\ldots z_{n-1}$ has uncountably many expansions.
Since Ind$^{+}(\cdot)$ takes values amongst a countable set, there exists
an uncountable subset $\Lambda_{1}$ of $\Lambda$ such that
Ind$^{+}(\omega_{n+1}^{\lambda}\omega_{n+2}^{\lambda}\ldots)=s$ for some $s\in\{1,2,\ldots\}\cup\{\infty\}$.
If $z_{n}\leq2k+1$, then $$0.0\ldots0z_{n}\omega_{n+1}^{\lambda}\omega_{n+2}^{\lambda}\ldots\in\{0,1,\ldots, 2k+1\}^{\infty}$$ for $\lambda\in\Lambda_{1}$. Hence, $f_{n-1}$ has uncountably many expansions.
If $z_{n}>2k+1$, then for $\lambda\in\Lambda_{1}$,
$$T^+(0.0\ldots0z_{n}\omega_{n+1}^{\lambda}\omega_{n+2}^{\lambda}\ldots)=0.0\ldots01\xi_{n}\xi_{n+1}\ldots,$$
with $\xi_{i}\in\{0,1,\ldots,2k+1\}$. Since the restriction of the map $T^{+}$ to sequences with Ind$^{+}(\cdot)=s$ is an injection, these provide uncountably many expansions of $f_n$.
The claim then follows. Now we show each $f_{l}=1-0.z_{1}z_{2}\ldots z_{l}$ for $l\in\{ n-1,\ldots,1,0\}$ has uncountable many expansions. For this, we consider the following property $(P_q)$:
there exists an uncountable set
$\Lambda_{q}$ such that
\begin{itemize}
\item if $z_{q+1}\leq 2k+1$, then $f_q$ has the expansions of the form $$ 0.0\ldots00\omega_{q+1}^{\lambda}\omega_{q+2}^{\lambda}\ldots ~(\lambda\in\Lambda_{q});$$
\item $z_{q+1}>2k+1$, then $f_q$ has the expansions of the form $$ 0.0\ldots01\omega_{q+1}^{\lambda}\omega_{q+2}^{\lambda}\ldots ~( \lambda\in\Lambda_{q}).$$
\end{itemize}
\smallskip
Suppose $(P_q)$ holds, then
\begin{eqnarray*}
f_{q-1} &=& 1-0.z_{1}z_{2}\ldots z_{q-1} \\
&=& 0.0\ldots0(z_{q}+a)\omega_{q+1}^{\lambda}\omega_{q+2}^{\lambda}\ldots
\end{eqnarray*}
where $a=0$ if $z_{q+1}\leq 2k+1$, and $a=1$ if $z_{q+1}> 2k+1$.
If $z_{q+1}\leq 2k+1$, we reach the property $(P_{q-1})$ using the same argument as above;
if $z_{q+1}> 2k+1$, then due to Property $(*)$, we have $z_{q}\leq k$. Whence $0.0\ldots0(z_{q}+a)\omega_{q+1}^{\lambda}\omega_{q+2}^{\lambda}\ldots $ is the desired expansion, and we obtain $(P_{q-1})$ also.
Therefore, by induction, we know that any $f_{l}=1-0.z_{1}z_{2}\ldots z_{l}$ has uncountable many expansions. This is a contradiction since $f_0=1$ has only countably many expansions.
\end{proof}
Theorem \ref{theorem} then follows from the above lemmas.
\medskip
Following the similar idea with the proof of Theorem \ref{theorem} and with even less effort, we prove Theorem \ref{theorem2}.
\medskip
\begin{flushleft}
{\bf Acknowledgements}~ This work was supported by NSFC Nos. 11171123 and 11222111.
\end{flushleft}
|
1704.05294
|
\section{Introduction} \label{intro}
Perfect teleportation of an arbitrary quantum state using a classical channel would require an infinite amount of classical resources. However, it is possible to teleport an arbitrary quantum state with unit fidelity using a quantum channel and a few bits of classical communication. As perfect teleportation does not have a classical analogue, schemes for quantum teleportation (QT) drew considerable attention of the quantum communication community since its introduction in 1993 by Bennett et al., \cite{bennett1993teleporting}. As a consequence, a large number of modified QT schemes have been proposed. In the present scenario, these schemes span a wide spectrum of quantum communication science. To elucidate this point, we may mention a few schemes of quantum communication tasks, which may be viewed as modified schemes for QT. This set of quantum communication tasks includes--remote state preparation (RSP) \cite{pati2000minimum}, controlled teleportation (CT) \cite{karlsson1998quantum,pathak2011efficient}, bidirectional state teleportation (BST) \cite{huelga2001quantum}, bidirectional controlled state teleportation (BCST) \cite{zha2013bidirectional,shukla2013bidirectional,thapliyal2015general,thapliyal2015applications}, quantum information splitting (QIS) \cite{hillery1999quantum,nie2011quantum}, quantum secret sharing (QSS) \cite{hillery1999quantum}, hierarchical versions of these schemes \cite{shukla2013hierarchical,mishra2015integrated,shukla2016hierarchical}, and many more. Many of these schemes have very interesting applications (for details see \cite{P13}). For example, we may mention that every scheme of BST can be used to design quantum remote control \cite{huelga2001quantum}. It is also worth noting here that although teleportation (and most of its variants) in its original form is not a secure communication scheme, it can be used as a primitive for secure quantum communication. On top of that, there are proposals to employ teleportation in quantum repeaters to enhance the feasibility of quantum communication, and in ensuring security against an eavesdropper's attempt to encroach the private spaces of legitimate users devising trojan-horse attack \cite{lo1999unconditional}. This wide applicability of QT and its variants and the fact that quantum resources are costly (for example, preparation and maintenance of an $n$-qubit entangled state is extremely difficult for large $n$) have motivated us to investigate whether the recently proposed schemes \cite{li2016quantum,hassanpour2016bidirectional,da2007teleportation,li2016asymmetric,song2008controlled,cao2005teleportation,muralidharan2008quantum,tsai2010teleportation,nie2011quantum,tan2016deterministic,wei2016comment,li2016quantum1,yu2013teleportation,nandi2014quantum} are using an optimal amount of quantum resources? If not, how to reduce the amount of quantum resources to be used? An effort to answer these questions has led to the present work, where we aim to propose a scheme for teleportation of an $n$-qubit quantum state using an optimal amount of quantum resources and to experimentally realize a particular case of the proposed scheme using 5-qubit IBM quantum computer. Before we proceed to describe the findings of the present work, we would like to elaborate a bit on what makes it fascinating to work on teleportation even after almost quarter century of its introduction.
As a natural generalization of QT schemes, protocols for teleportion of multi-qubit states have also been proposed. For example, Chen et al. \cite{chen2006general} have presented a general form of a genuine multipartite entangled channel. Their work encapsulates the essence that the bipartite nature of the channel is sufficient for teleportation of multi-qubit quantum states. Furthermore, they concluded that QT of an arbitrary $n$-qubit state can be achieved by performing $n$ rounds of Bennett et al.'s protocol \cite{bennett1993teleporting} for QT (one for each qubit). Later, this scheme was extended to CT of an arbitrary $n$-qubit quantum state \cite{ man2007genuine}. Along the same line, a BST and a BCST schemes may be designed for teleporting two arbitrary multi-qubit states, one each by Alice and Bob. Specifically, a quantum state suitable as a quantum channel for a CT (BCST) scheme should essentially reduce to a useful quantum channel for QT (BST) after the controller's measurement (see \cite{thapliyal2015general} for detail discussion). Most of these results related to teleporation of multi-qubit states are theoretical in nature. In these schemes, multi-qubit entangled states are used without giving much attention to the possibility of experimental generation and maintenance of such entangled states. On the other hand, first experimental realization of Bennett et al.'s protocol was reported by Bouwmeester et al., \cite{bouwmeester1997experimental} in 1997 using photons. Later on, a number of realizations using other architectures have also been reported \cite{bouwmeester1997experimental,nielsen1998complete,furusawa1998unconditional,zhao2004experimental,riebe2004deterministic,barrett2004deterministic}.
However, hardly any proposal for teleportation of multi-qubit quantum states have been tested experimentally becuase experimental realization of those schemes would require quantum resources that are costly. This observation has further motivated us to design a general teleportation scheme that would require lesser amount of quantum resource and/or such resources that can be generated and maintained easily using available technology.
It may be noted that an optimized scheme for multipartite QT must involve optimization of both procedure and resources. Optimization of the procedure demands use of those states as quantum channel, which can be prepared easily and are least affected by decoherence; while the optimization of resources that the scheme should exploit/utilize all the channel qubits that are available for performing QT. The results of Ref. \cite{chen2006general} can be viewed as an optimization of procedure (as Bell states can be easily prepared and are less prone to decoherence in comparison with the multipartite entangled states). The importance of such strategies becomes evident in experimental scenarios, say when we wish to realize QT in a quantum network designed for quantum internet \cite{sun2016quantum}. Naturally, an optimized QT scheme would improve the performance of such a quantum internet. Another kind of optimization has been attempted in some recent works. Specifically, efforts have been made to form teleportation channel (having lesser number of entangled qubits) suitable for the teleportation of specific types of unknown quantum states \cite{li2016quantum,hassanpour2016bidirectional,da2007teleportation,li2016asymmetric,song2008controlled,cao2005teleportation,muralidharan2008quantum,tsai2010teleportation,nie2011quantum,tan2016deterministic,wei2016comment,li2016quantum1,yu2013teleportation,nandi2014quantum}. For example, in \cite{li2016quantum}, the 3-qubit and the 4-qubit unknown quantum states have been teleported using 4 and 5-qubit cluster states, respectively. An extended list of these complex states and the corresponding channels are given in Table \ref{tab:lit}. In fact, some of this set of schemes has exploited the fact that some of the probability amplitudes in the state to be teleported are zero and a QT scheme essentially transfers the unknown probability amplitudes to distant qubits. We noticed that the quantum resources used in these protocols are not optimal and most of the cases involve redundant qubits. Keeping these in mind, here, we set ourselves the task to minimize the number of these qubits exploiting the available information regarding the mathematical structure of the quantum state to be teleported. Specifically, in what follows, we would propose an efficient and optimal (uses minimum number of Bell states as quantum channel) scheme to teleport a multi-qubit state of specific form. Further, it would be established that the proposed scheme is simple in nature and can be extended to corresponding CT and BCST schemes.
Actual relevance of an optimized scheme lies in the experimental realization of the scheme only. An interesting window for experimental realization of the schemes of quantum computation and communication in general and optimized schemes in particular has been opened up recently, when IBM provided free access to a 5-qubit superconducting quantum computer by placing it in cloud \cite{IBMQE,devitt2016performing}. This has provided a platform for experimental testing of various proposals for quantum communication and computation. In the present work, we have used IBM quantum computer to experimentally realize the optimal scheme designed here. Specifically, we have successfully implemented the optimal quantum circuit designed for teleportation of a 2-qubit state. The experiment performed is a proof of principle experiment as the receiver and the sender are not located at two distant places, but it shows successful teleportation with very high fidelity and paves away the path for future realizations of the proposed scheme using optical elements.
Rest of the paper is organized as follows. In Section \ref{mun}, we propose a scheme for the teleporation of a multi-qubit quantum state having $m$ unknown coefficients using optimal quantum resources. We also discuss a specific case of this scheme which corresponds to QT of a two-unknown multi-qubit quantum state using a Bell state as quantum channel. In Section \ref{cbt}, we describe optimal schemes for controlled unidirectional and bidirectional state teleportation the quantum states.
Further, in Section \ref{ex}, an experimental realization of the proposed scheme is reported using the 5-qubit IBM quantum computer available on cloud. Finally, we conclude the paper in section \ref{con}.
\section{Teleportation of an $n$-qubit state with $m$ unknown coefficients} \label{mun}
Consider an $n$-qubit quantum state to be teleported as
\begin{equation}
|\psi\rangle=\sum_{i=1}^{m}\alpha_{i}\ket{x_{i}}, \label{eq:psi}
\end{equation}
where $\alpha_i$s are the probability amplitudes ensuring normalization $\sum_{i=1}^{m}\vert\alpha_{i}\vert^{2}=1$. Additionally, $x_i$s are mutually orthogonal to each other, i.e., $\langle x_i|x_j\rangle=\delta_{ij}\,\forall\,\left(1<i,j<m\right)$. Therefore, one may note that $x_i$s are the elements of an $n$-qubit orthogonal basis iff $m\leq2^n$. For example, we may consider 3-qubit quantum states $\ket{\xi_1}=\alpha_1|000\rangle+\alpha_2|111\rangle$ and $\ket{\xi_2}=\alpha_1|000\rangle+\alpha_2|011\rangle+\alpha_3|100\rangle+\alpha_4|111\rangle$, teleportation schemes for which were proposed in Refs.
in \cite{yu2013teleportation} and \cite{wei2016comment}, respectively. For both the states $n=3$, but we can easily observe that $m=2$ for $\ket{\xi_1}$ and $m=4$ for $\ket{\xi_2}$. In what follows, we will establish that because of this difference in the values of $m$, teleportation of $\ket{\xi_1}$ would require only one Bell state, whereas that of $\ket{\xi_2}$ would require 2 Bell states.
An arbitrary $n$-qubit quantum state possesses $2^n$ superposition, i.e., $m=2^n.$ Here, we set ourselves a task to teleport state $\ket{\psi}:\, m<2^n$ using optimal quantum resources (i.e., using minimum number of qubits in the multiqubit entangled state used as quantum channel). To do so, we will transform the state to be teleported (say, $|\psi\rangle$) to a quantum state of preferred form (say, $|\psi^{\prime}\rangle$). Specifically, the central idea of our scheme lies in finding out a unitary $U$, which transforms state $\ket{\psi}$ into $\ket{\psi^\prime}$, i.e., $U\ket{\psi}= \ket{\psi^\prime}$, such that
\begin{equation}
|\psi^\prime\rangle=\sum_{i=1}^{m}{\alpha_{i}^{\prime}|y_{i}\rangle}. \label{eq:psiD}
\end{equation}
Here, $|\psi^\prime\rangle $ is a unique $n$-qubit quantum state which is written using only $m$ out of total $n$ elements of the computational basis $\{y_i\}$. Specifically, the unitary is expected to possess a one-to-one map for each element of $\{x_i\} \rightarrow \{y_i\}$.
As shown in Figure \ref{fig:mun}, here we prefer $\ket{y_i} = \ket{0}^{n-m^{\prime}} \otimes \ket{\widetilde{m}_i}$, where $m^{\prime}=\left\lceil \log_2{m}\right\rceil $ and $\widetilde{m}_i$ is the binary equivalent of decimal value $m_i$ in an $m^{\prime}$-bit binary string. This step transforms $m$ elements of $\ket{\psi}$ with non-zero projections in Eq. \ref{eq:psi} to that of $m^\prime$ elements of $\ket{\psi^\prime}$ in Eq. \ref{eq:psiD}.
However, to design the unitary able to accomplish such a task the map $\{x_i\} \rightarrow \{y_i\}$ should exist between all the elements (both possessing either zero or non-zero projection in $\ket{\psi}$ and $\ket{\psi^\prime})$ in both the basis. In other words, the unitary $U$ mapping the basis elements from $\{x_i\}$ to $\{y_i\}$ required for the desired transformation would be
\begin{equation}
U=\sum_{i=1}^{2^n}|y_{i}\rangle\langle x_{i}|
\end{equation}
Being a $2^n$ dimensional computational basis $\left\{y_{i}\right\}$ is already known
while state $\ket{\psi}$ reveals only $m$ orthogonal vectors of basis $\{x_i\}$. Therefore, the remaining $\left(2^n-m\right)$ elements of basis $\{x_i\}$ can be obtained by Gram-Schmidt procedure, such that the elements of $\{x_i\}$ follow the completeness relation, i.e., $\sum_{i=1}^{2^n}{\sum}|x_{i}\rangle\langle x_{i}|=\mathbb{I}$.
The obtained quantum state $\ket{\psi^\prime}$ possesses the first ${n-m^{\prime}}$ qubits in $\ket{0}$, while the remaining $m^{\prime}$ qubits hold the complete information of $\ket{\psi}$. Therefore, our task reduces to a teleportation of an $m^{\prime}$-qubit quantum states using an optimal amount of quantum resources. An $m^{\prime}$-qubit quantum state can be teleported either by using at least $2m^{\prime}$-qubit entangled state or $m^{\prime}$ Bell states, one for teleporting each qubit \cite{chen2006general}.
Preparing a multi-qubit entangled state is relatively expensive and such a state is more prone to decoherence than a two qubit entangled state. For the reason, we prefer the second strategy and select Bell states as a quantum channel (see Figure \ref{fig:mun}). Once the quantum state $\ket{\psi^\prime}$ is reconstructed at Bob's port, he would require to perform a unitary operation $U^{\dagger}$.
For the sake of completeness of the paper we may summarize teleportation of an arbitrary $m^{\prime}$-qubit quantum state in the following steps:
\begin{enumerate}
\item The $m^{\prime}$-qubit unknown state to be teleported (whose qubits are indexed by $1,2,\cdots, m^{\prime}$) and the first qubits of each $m^{\prime}$ Bell states (indexed by $A_{1}, A_{2}, \cdots, A_{m^{\prime}}$) are with Alice and all the second qubits (indexed by $B_{1}, B_{2}, \cdots, B_{m^{\prime}}$) are with Bob.
\item Alice performs pairwise Bell measurements on her qubits ($i, A_{i}$) and finally announces $m^{\prime}$ measurement outcomes.
\item Bob applies Pauli operations on each qubit in his possession depending upon the measurement outcome of Alice (see Table 1 of Ref. \cite{thapliyal2015applications} for detail). At the end of this step Bob obtains the $m^{\prime}$-qubit unknown quantum state that Alice has teleported.
\end{enumerate}
Once Bob has access to the $m^{\prime}$-qubit unknown state and knowledge of the unitary $U$ Alice has applied, he prepares $n-m^{\prime}$ qubits in $\ket{0}$. Finally, he applies $U^{\dagger}$ to reconstruct the unknown
quantum state $\begin{array}{lcl}
U^{\dagger}|\psi^{\prime}\rangle & = & |\psi\rangle\end{array}$
\begin{figure}
\begin{center} \includegraphics[scale=0.52]{fig1.pdf} \end{center}\protect\caption{\label{fig:mun} (Color online) The circuit for teleportation of an $n$-qubit quantum states having $m$ unknown coefficients. Here, $m^\prime=\left\lceil \log_2{m}\right\rceil $ is the number of Bell states used which depends on the number of unknowns $m$. Also, $U_i$ is the unitary operation Bob has to apply to reconstruct the teleported quantum state.}
\end{figure}
In Table \ref{tab:lit}, we give unitary operations involved in teleportation of various multiqubit states with different number of unknowns using our scheme. Teleportation of these states using relatively fragile and expensive quantum resources have been reported in the recent past.
To be specific, our technique can be used to teleport any quantum state having two unknown coefficients \cite{da2007teleportation,cao2005teleportation,tsai2010teleportation,yu2013teleportation,li2016quantum} using a single Bell state, irrespective of the number of physical qubits present in the state. In contrast, in the existing literature (cf. Columns 2 and 3 of Table \ref{tab:lit}) it is observed that the number of qubits used in the quantum channel increases with the increase in number of qubits to be teleported. Similarly, a quantum state having four-unknown coefficients can be teleported only using two Bell states, unlike the higher dimensional entangled states used in \cite{wei2016comment,li2016quantum1,tan2016deterministic}. This clearly establishes that our scheme allows one to circumvent the use of redundant qubits and complex entangled states that are used until now, and thus the present proposal increases the possibility of experimental realization.
\begin{center}
\begin{table}
\begin{tabular}{|>{\centering}p{2.6cm}|>{\centering}p{1.6cm}|>{\centering}p{2.9cm}|>{\centering}p{1.8cm}|>{\centering}p{7.4cm}|}
\hline
Quantum state to be teleported & Number of qubits in the state to be teleported & Quantum state used as quantum channel and corresponding reference & Number of Bell states required in our scheme to teleport the state & Unitary to be applied by Alice \tabularnewline
\hline
$\alpha|a_{0}\rangle+\beta|a_{3}\rangle$ & 2-qubit & 4-qubit CS \cite{da2007teleportation}, 3-qubit W class state \cite{cao2005teleportation} & 1 & $|a_{0}\rangle\langle a_{0}|+|a_{1}\rangle\langle a_{3}|+|a_{2}\rangle\langle a_{1}|+|a_{3}\rangle\langle a_{2}|$\tabularnewline
\hline
$\alpha|a_{1}\rangle+\beta|a_{2}\rangle$ & 2-qubit & 3-qubit GHZ like state \cite{tsai2010teleportation} & 1 & $|a_{0}\rangle\langle a_{1}|+|a_{1}\rangle\langle a_{2}|+|a_{2}\rangle\langle a_{0}|+|a_{3}\rangle\langle a_{3}|$\tabularnewline
\hline
$\alpha|a_{0}\rangle+\beta|a_{7}\rangle$ & 3-qubit & 4-qubit CS \cite{yu2013teleportation} & 1 & $\begin{array}{l}
|a_{0}\rangle\langle a_{0}|+|a_{1}\rangle\langle a_{7}|
+|a_{2}\rangle\langle a_{1}|+|a_{3}\rangle\langle a_{2}|\\
+|a_{4}\rangle\langle a_{3}|+|a_{5}\rangle\langle a_{4}|
+|a_{6}\rangle\langle a_{5}|+|a_{7}\rangle\langle a_{6}|
\end{array}$\tabularnewline
\hline
$\begin{array}{l}
\alpha(|a_{0}\rangle+|a_{3}\rangle)\\
+\beta(|a_{4}\rangle+|a_{7}\rangle)
\end{array}$ & 3-qubit & 4-qubit CS \cite{li2016quantum} & 1 & $\begin{array}{l}
\frac{1}{\sqrt{2}}(|a_{0}\rangle\langle(|a_{0}+a_{3})|+|a_{1}\rangle\langle a_{4}+a_{7})|\\
+|a_{2}\rangle\langle(a_{1}+a_{2})|+|a_{3}\rangle\langle(a_{5}+a_{6})|\\
+|a_{4}\rangle\langle(a_{0}-a_{3})|+|a_{5}\rangle\langle(a_{4}-a_{7})|\\
+|a_{6}\rangle\langle(a_{1}-a_{2})|+|a_{7}\rangle\langle(a_{5}-a_{6})|)
\end{array}$\tabularnewline
\hline
$\begin{array}{l}
\alpha(|a_{0}\rangle+|a_{7}\rangle)\\
+\beta(|a_{13}\rangle+|a_{10}\rangle)
\end{array}$ & 4-qubit & 5-qubit CS \cite{li2016quantum} & 1 & $\begin{array}{l}
\frac{1}{\sqrt{2}}(|a_{0}\rangle\langle|a_{0}+a_{7}|+|a_{1}\rangle\langle(a_{13}+a_{10})|\\
+|a_{2}\rangle\langle(a_{1}+a_{2})|+|a_{3}\rangle\langle(a_{3}+a_{4})|\\
+|a_{4}\rangle\langle(a_{5}+a_{6})|+|a_{5}\rangle\langle(a_{8}+a_{9})|\\
+|a_{6}\rangle\langle(a_{11}+a_{12}|+|a_{7}\rangle\langle(a_{14}+a_{15})|\\
+|a_{8}\rangle\langle(a_{0}-a_{7})|+|a_{9}\rangle\langle(a_{13}-a_{10})|\\
+|a_{10}\rangle\langle(a_{1}-a_{2})|+|a_{11}\rangle\langle(a_{3}-a_{4})|\\
+|a_{12}\rangle\langle(a_{5}-a_{6})|+|a_{13}\rangle\langle(a_{8}-a_{9})|\\
+|a_{14}\rangle\langle(a_{11}-a_{12})|+|a_{15}\rangle\langle(a_{14}-a_{15})|)
\end{array}$\tabularnewline
\hline
$\begin{array}{l}
\alpha|a_{0}\rangle+\beta|a_{3}\rangle\\
+\gamma|a_{4}\rangle+\delta|a_{7}\rangle
\end{array}$ & 3-qubit & 5-qubit state \cite{wei2016comment} & 2 & $\begin{array}{l}
|a_{0}\rangle\langle a_{0}|+|a_{1}\rangle\langle a_{3}|
+|a_{2}\rangle\langle a_{4}|+|a_{3}\rangle\langle a_{7}|\\
+|a_{4}\rangle\langle a_{1}|+|a_{5}\rangle\langle a_{2}|
+|a_{6}\rangle\langle a_{5}|+|a_{7}\rangle\langle a_{6}|
\end{array}$\tabularnewline
\hline
$\begin{array}{l}
\alpha|a_{0}\rangle+\beta|a_{3}\rangle\\
+\gamma|a_{12}\rangle+\delta|a_{15}\rangle
\end{array}$ & 4-qubit & 6-qubit CS \cite{li2016quantum1} & 2
& $\begin{array}{l}
|a_{0}\rangle\langle a_{0}|+|a_{1}\rangle\langle a_{3}|
+|a_{2}\rangle\langle a_{12}|+|a_{3}\rangle\langle a_{15}|\\
+|a_{4}\rangle\langle a_{1}|+|a_{5}\rangle\langle a_{2}|
+|a_{6}\rangle\langle a_{4}|+|a_{7}\rangle\langle a_{5}|\\
+|a_{8}\rangle\langle a_{6}|+|a_{9}\rangle\langle a_{7}|
+|a_{10}\rangle\langle a_{8}|+|a_{11}\rangle\langle a_{9}|\\
+|a_{12}\rangle\langle a_{10}|+|a_{13}\rangle\langle a_{11}|
+|a_{14}\rangle\langle a_{13}|+|a_{15}\rangle\langle a_{14}|
\end{array}$\tabularnewline
\hline
$\begin{array}{l}
\alpha|a_{0}\rangle+\beta|a_{15}\rangle\\
+\gamma|a_{63}\rangle+\delta|a_{48}\rangle
\end{array}$ & 6-qubit & 6-qubit CS \cite{tan2016deterministic} & 2 & $\begin{array}{l} |a_{0}\rangle\langle a_{0}|+|a_{1}\rangle\langle a_{15}|+|a_{2}\rangle\langle a_{63}|+|a_{3}\rangle\langle a_{48}|\\
+|a_{4}\rangle\langle a_{1}|+|a_{5}\rangle\langle a_{2}|+|a_{6}\rangle\langle a_{3}|+|a_{7}\rangle\langle a_{4}|\\
+|a_{8}\rangle\langle a_{5}|+|a_{9}\rangle\langle a_{6}|+|a_{10}\rangle\langle a_{7}|+|a_{11}\rangle\langle a_{8}|\\
+|a_{12}\rangle\langle a_{9}|+|a_{13}\rangle\langle a_{10}|+|a_{14}\rangle\langle a_{11}|+|a_{15}\rangle\langle a_{12}|\\
+|a_{16}\rangle\langle a_{13}|+|a_{17}\rangle\langle a_{14}|+|a_{18}\rangle\langle a_{16}|+|a_{19}\rangle\langle a_{17}|\\
+|a_{20}\rangle\langle a_{18}|+|a_{21}\rangle\langle a_{19}|+|a_{22}\rangle\langle a_{20}|+|a_{23}\rangle\langle a_{21}|\\
+|a_{24}\rangle\langle a_{22}|+|a_{25}\rangle\langle a_{23}|+|a_{26}\rangle\langle a_{24}|+|a_{27}\rangle\langle a_{25}|\\
+|a_{28}\rangle\langle a_{26}|+|a_{29}\rangle\langle a_{27}|+|a_{30}\rangle\langle a_{28}|+|a_{31}\rangle\langle a_{29}|\\
+|a_{32}\rangle\langle a_{30}|+|a_{33}\rangle\langle a_{31}|+|a_{34}\rangle\langle a_{32}|+|a_{35}\rangle\langle a_{33}|\\
+|a_{36}\rangle\langle a_{34}|+|a_{37}\rangle\langle a_{35}|+|a_{38}\rangle\langle a_{36}|+|a_{39}\rangle\langle a_{37}|\\
+|a_{40}\rangle\langle a_{38}|+|a_{41}\rangle\langle a_{39}|+|a_{42}\rangle\langle a_{40}|+|a_{43}\rangle\langle a_{41}|\\
+|a_{44}\rangle\langle a_{42}|+|a_{45}\rangle\langle a_{43}|+|a_{46}\rangle\langle a_{44}|+|a_{47}\rangle\langle a_{45}|\\
+|a_{48}\rangle\langle a_{46}|+|a_{49}\rangle\langle a_{47}|+|a_{50}\rangle\langle a_{49}|+|a_{51}\rangle\langle a_{50}|\\
+|a_{52}\rangle\langle a_{51}|+|a_{53}\rangle\langle a_{52}|+|a_{54}\rangle\langle a_{53}|+|a_{55}\rangle\langle a_{54}|\\
+|a_{56}\rangle\langle a_{55}|+|a_{57}\rangle\langle a_{56}|+|a_{58}\rangle\langle a_{57}|+|a_{59}\rangle\langle a_{58}|\\
+|a_{60}\rangle\langle a_{59}|+|a_{61}\rangle\langle a_{60}|+|a_{62}\rangle\langle a_{61}|+|a_{63}\rangle\langle a_{62}|
\end{array}$\tabularnewline
\hline
\end{tabular}
\caption{\label{tab:lit} A list of quantum states and the resources used to teleport them in the recent past. The unitary required to decrease the number of entangled qubits to be used as quantum channel is also mentioned. Here, CS stands for cluster state, and $a_{i}$ corresponds to the binary value of decimal number $i$ expanded upto $k$ digits if the entry in the second column of the same row is $k$ qubit. For example, entry in first row second column is 2 qubit, so the state to be teleported (see first row first column) would be $\alpha|a_{0}\rangle+\beta|a_{3}\rangle=\alpha|00\rangle+\beta|11\rangle$. Thus, $a_{0}$ is expanded here up to two digits, but in the third row $a_{0}$ will be expanded as $|000\rangle$, i.e., up to three digits as the entry in third row second column is 3 qubit. Naturally, in this particular case, state to be teleported is $\alpha|a_{0}\rangle+\beta|a_{7}\rangle=\alpha|000\rangle+\beta|111\rangle$. In our scheme, both $\alpha|00\rangle+\beta|11\rangle$ and $\alpha|000\rangle+\beta|111\rangle$ can be teleported using single Bell state, as the resource required depends only on the number of unknown coefficients.}
\end{table}
\end{center}
\hspace{0.5cm}
\subsection{Teleportation of state of type $\ket{\psi} = \alpha \ket{x_i} + \beta \ket{x_j}$} \label{2un}
As an explicit example of the proposed scheme, consider an $n$-qubit state with only two unknowns, i.e.,
\begin{equation}
\ket{\psi} = \alpha \ket{x_i} + \beta \ket{x_j},
\end{equation}
such that, $\inpr{x_i}{x_j} = \delta_{ij}$ and ${\abs{\alpha}}^{2} + {\abs{\beta}}^{2}= 1$. Here, $x_i$ and $x_j$ are the elements of some $2^{n}$ dimensional basis set. Our task is to teleport state $\ket{\psi}$ using optimal quantum resources (i.e., using minimum number of entangled qubits in quantum channel). As mentioned previously the minimum number of Bell states required in the quantum channel for this kind of state would be $ \left\lceil \log_2{2}\right\rceil =1$.
Therefore, we will transform state $\ket{\psi}$ into $\ket{\psi^\prime}$, such that $U\ket{\psi}= \ket{\psi^\prime}$, such that
\begin{equation}
\ket{\psi^\prime} = \alpha^\prime \ket{y_i} + \beta^\prime \ket{y_j}
\end{equation}
with, $\inpr{y_i}{y_j}= \delta_{ij}$ as $y_{i}$ and $y_{j}$ are the the elements of computational basis. For the simplest choice of $\ket{\psi^\prime}$, we choose $y_{i} = {\ket{0}}^{n-1} \otimes \ket{0}$ and $y_{j} = {\ket{0}}^{n-1}\otimes\ket{1}$.
Now, we will show that it is possible to teleport state $\ket{\psi^\prime}$ using one $e$ bit (Bell state) and classical resource.
The quantum circuit for teleporting state $\ket{\psi}$ is given in Figure \ref{fig:2un}. The first part of the quantum circuit shows transformation of the state $\ket{\psi}$ to the state $\ket{\psi^\prime}$ while the second part of the circuit contains standard scheme for teleporting an arbitrary single qubit state. The third part of the circuit involves application of the unitary ${\mathrm U}^{\dagger}$ to transform the reconstructed state $\ket{\psi^\prime}$ into the unknown state $\ket{\psi}$ to be teleported.
\begin{figure}
\begin{center} \includegraphics[scale=0.52]{fig2.pdf}\end{center}\protect\caption{\label{fig:2un}(Color online) As an explicit example, a quantum circuit is shown that performs teleportation of an $n$-qubit quantum state having 2 unknown coefficients using only a single Bell state.}
\end{figure}
\section{Controlled and Bidirectional teleportation with optimal resource}\label{cbt}
Controlled teleportation of a single qubit involves a third party (Charlie) as supervisor and hence instead of a Bell state we require tripartite entangled state as quantum channel. As mentioned in Section \ref{intro}, many of the papers which involve multiqubit complex states for teleportation also perform controlled teleportation using those multiqubit complex states. Here, we extend our scheme, for optimal QT to optimal CT. The scheme of CT using optimal resources can be explained along the same line of the QT scheme as follows.
To construct an optimal scheme, it is assumed that Charlie also knows the unitary Alice is using to reduce the size of the quantum state to be teleported. In other words, he is aware of the number of entangled qubits Alice and Bob require to perform teleportation. Suppose the reduced quantum state has $m^\prime$ qubits, Charlie prepares $m^\prime$ GHZ states and share the three qubits among Alice, Bob and himself. Charlie measures his qubit in $\{\ket{+},\ket{-}\}$ basis and withholds the measurement outcome. Independently, Alice and Bob perform the QT scheme with the only difference that Bob requires Charlie's measurement disclosure to reconstruct the state. Charlie announces the required classical information when he wishes Bob to reconstruct the state.
Similarly, when both Alice and Bob wish to teleport a quantum
state each to Bob and Alice, respectively, under the supervision of Charlie, they perform QT schemes independently, while Charlie prepares the quantum channel in such a way that after his measurement the reduced state is the product of $2m^\prime$ Bell states (half of which will be used for Alice to Bob, while the remaining half for Bob to Alice communication). Charlie's disclosure of his measurement outcomes end both Alice's and Bob's ignorance regarding the quantum channel they were sharing, and they can subsequently reconstruct the unknown quantum states teleported to them (see \cite{thapliyal2015general} for detail).
In Ref. \cite{thapliyal2015applications}, some of the present authors have shown that BCST can also be accomplished solely using Bell states. Therefore, CT and BCST can also be performed using only bipartite entanglement. In absence of the controller, a BCST scheme can be reduced to a BST scheme.
In the light of our present results, some of the recent schemes of CT using 4-qubit cluster state \cite{song2008controlled} and QIS using 4 and 5-qubit cluster state \cite{muralidharan2008quantum,nie2011quantum}; BST using 3-qubit GHZ state \cite{hassanpour2016bidirectional}; and 6-qubit cluster state \cite{li2016asymmetric}, can also be performed with reduced amount of quantum resources (entangled states involving lesser number of qubits).
\section {Experimental implementation of the proposed efficient QT scheme using IBM's real quantum processor} \label{ex}
Recently, IBM corporation has placed a 5-qubit superconductivity-based quantum computer on cloud \cite{IBMQE}, and has provided its access to everyone. This initiative has enabled the interested researchers to experimentally realize various proposals for quantum information processing tasks. Interestingly, a set of such implementations have already been reported \cite{fedortchenko2016quantum,rundle2016quantum,devitt2016performing}. Currently, this 5-qubit superconductivity-based quantum computer has many limitations, like available gate library is only approximately universal, measurement of individual qubits at different time points is not allowed, limited applicability of C-NOT gate, and short decoherence time \cite{IBMDT}. Further, the real quantum computer (IBM quantum experience) available at cloud allows a user to perform an experiment using at most 5-qubits. Keeping these limitations in mind, we have chosen a 2-qubit two-unknown quantum state $\ket{\psi}=\alpha(\ket{00}+\ket{11})+\beta(\ket{01}-\ket{10})$ : $2(|\alpha|^2+|\beta|^2)=1$ as the state to be teleported. Here, we would like to mention that implementation of a single qubit teleportation protocol (which requires only three qubits) has already been demonstrated using IBM's quantum computer \cite{fedortchenko2016quantum}. The experimental implementation of the present QT scheme is relatively complex and can be divided into four parts as shown in Figure \ref{fig:sim}. Part A involves preparation of state $\ket{\psi}$ (using qubit q[0] and q[1]) and a Bell state (using qubit q[2] and q[3]). The complex circuit comprised of the quantum gates from Clifford group is shown in Figure \ref{fig:sim}. Here, it may be noted that the IBM quantum computer accepts quantum gates from Clifford group only. The state $\ket{\psi}$ in this particular case is prepared with $|\alpha|^2=0.375$ and $|\beta|^2=0.125$. Preparation of the desired 2-qubit state by application of specific quantum gates is detailed in the following.
\begin{figure}
\begin{center} \includegraphics[scale=.95]{fig3.pdf}\end{center}
\protect\caption{\label{fig:sim}(Color online) Teleportation circuit used on the IBM QE. Details of the various parts A-D are as follows. (A) Preparation of an unknown quantum state $\ket{\psi}$ with $\abs{\alpha}^2=0.375$ and $\abs{\beta}^2=0.125$ and the Bell state (EPR channel). (B) The unitary operation $U$ to transform 2-qubit state $\left(\alpha \left(\ket{00}+\ket{11}\right)+ \beta \left(\ket{01}-\ket{10}\right)\right)$ to state $\left(\alpha \ket{00}+ \beta \ket{10}\right)$ in the computational basis. (C) Teleportation of a single qubit state. (D) Reconstructing the state $\ket{\psi}$ from the teleported single qubit state by unitary operation $U^{\dagger}$ followed by projective measurement.}
\end{figure}
$\ket{00}
\xrightarrow{\hspace{0.2cm}{\mathrm{H}^{1},\,\mathrm{T}^{1}}\hspace{0.2cm}} \left(\frac{\ket{0}+ e^{i\frac{\pi}{4}} \ket{1}}{\sqrt{2}}\right) \ket{0}\xrightarrow{\hspace{0.2cm}{\mathrm{H}^{1},\,\mathrm{S}^{1}}\hspace{0.2cm}} e^{i\frac{\pi}{8}}(\cos\left({\frac{\pi}{8}}\right)\ket{0}+\sin\left({\frac{\pi}{8}}\right)\ket{1}) \ket{0}
\xrightarrow{\hspace{0.2cm}{{\mathrm{T}^{\dagger}}^{1},\,\mathrm{X}^{1},\,\mathrm{H}^{1}}\hspace{0.2cm}} \sqrt{2}e^{i\frac{\pi}{8}}({\alpha \ket{0}+\beta \ket{1}}) \ket{0} \xrightarrow{\hspace{0.2cm}{\mathrm{C^{1}-NOT^{2}}}\hspace{0.2cm}}
\sqrt{2}e^{i\frac{\pi}{8}}(\alpha \ket{00}+\beta \ket{11}) \xrightarrow{\hspace{0.2cm} \mathrm{H}^{1}\hspace{0.2cm}} e^{i\frac{\pi}{8}}\left(\alpha \left(\ket{00}+\ket{10}\right)+\beta \left(\ket{01}-\ket{11}\right)\right)
\xrightarrow{\hspace{0.2cm}\mathrm{C^{1}-NOT^{2}} \hspace{0.2cm}}e^{i\frac{\pi}{8}}\left(\alpha\left(\ket{00}+\ket{11}\right)+\beta\left(\ket{01}-\ket{10}\right)\right)$.
Here, $e^{i\frac{\pi}{8}}$ is the global phase in the simulated quantum state with $\alpha= \frac{1}{\sqrt{2}}\left(\cos{\frac{\pi}{8}}+e^{-i\frac{\pi}{4}}\sin{\frac{\pi}{8}}\right) $, and
$\beta=\frac{1}{\sqrt{2}}\left(-\cos{\frac{\pi}{8}}+e^{-i\frac{\pi}{4}}\sin{\frac{\pi}{8}}\right)$. We have explicitly mentioned the qubit-number on which a particular operation is to be performed by mentioning the qubit-number on the the superscript of the corresponding unitary operator.
As described in Sec. \ref{2un}, Part B involves application of a unitary $U= \left(\mathrm{C^{2}-NOT^{1}}\right) \cdot \left(\mathbb{I}\otimes H\right) \cdot \left(\mathrm{C^{2}-NOT^{1}}\right)$ to transform the state $\ket{\psi}$ from the entangled basis to the computational basis and is the bottleneck of the protocol. Such a transformation allows us to render the information encoded into a smaller number of qubits (in our example it is a single qubit) and thus it reduces the amount of resources required. Part C is dedicated to the teleportation of a single qubit state. Here, we have used computational counterpart of teleportation \cite{adami1999quantum}, which can be performed when both Alice's and Bob's qubits are locally available for a 2-qubit operation. Teleportation of a single qubit state in analogy of Ref. \cite{fedortchenko2016quantum} can also be performed. This part of the circuit can be divided into two sub-parts. The first one (left aligned), which includes an EPR circuit, entangles qubit q[1] to the Bell state while the second part (right aligned) disentangles Bob's qubit (q[3]) from Alice's qubits. The need of disentangling Bob's qubit from Alice's qubit is explained above. In the standard protocol for QT \cite{bennett1993teleporting}, Alice measures her qubits and announces measurement outcomes. Depending on the measurement outcome of Alice, Bob applies a unitary operation and reconstructs the unknown state. In IBM's quantum computer, simultaneous measurement of all the qubits is mandatory, which will project Bob's qubit into a mixed state. Therefore, we preferred to disentangle Bob's qubit from Alice's qubits before measurement. At last, in Part D, Bob applies the unitary $U^{\dagger}$ followed by the projective measurement on all qubits, which reveals the state $\ket{\psi}$ teleported to Bob's qubits.
\begin{figure}
\begin{center} \includegraphics[scale=0.65]{fig4.pdf}\end{center}\protect\caption{\label{fig:tom}(Color online) Graphical representation of real (Re) and imaginary (Im) parts of the density matrices of (a) the theoretical state $\alpha(\ket{00}+\ket{11})+\beta(\ket{01}-\ket{10})$, (b) the experimentally prepared state, and (c) the reconstructed state after teleportation.}
\end{figure}
To perform a quantitative analysis of the performance of the QT scheme under consideration, we would require the density matrices of the state to be teleported and that of the teleported state. In a recent implementation of QT on IBM computer only probabilities of various outcomes were obtained \cite{fedortchenko2016quantum}. However, to obtain the full picture, we need to reconstruct the density matrix of the teleported state using quantum state tomography \cite{chuang1998bulk}. Till date, a large number of advanced protocols have been proposed for quantum state tomography \cite{schmied2016quantum}. An advanced protocol suppress the requirement of repeated preparation of the state to be tomographed and hence allows state characterization in dynamical environment using only one experiment \cite{shukla2013ancilla}. Here, we use the method proposed by Chuang et al. \cite{chuang1998bulk}. According to which we would require to obtain fifteen unknown parameters in a 2-qubit density matrix, and the same can be obtained using nine measurements. Using quantum state tomography (see \cite{james2001measurement,hebenstreit2017compressed,alsina2016experimental,rundle2016quantum,filipp2009two} for detail), we have reconstructed the teleported state (using nine rounds of experiments with 8192 runs of each experiment) as
\begin{equation}
\rho^{\prime\prime}=\left[{\begin{array}{cccc}
0.41 & 0.0125 + 0.0775 i & 0.085 - 0.19 i & 0.2035 - 0.05425 i\\
0.0125 - 0.0775 i & 0.134 & -0.0645 - 0.01625i & -0.021 - 0.051 i\\
0.085+ 0.19 i & -0.0645 + 0.01625 i & 0.261 & 0.101 + 0.0355 i\\
0.2035 + 0.05425 i & -0.021 + 0.051 i & 0.101 - 0.0355 i & 0.195\\
\end{array}}\right],
\end{equation}
whereas theoretically the state prepared for the teleportation is $\rho=\ket{\Psi}\bra{\Psi}$ with $\ket{\Psi}=
\left\{\alpha \left(\ket{00}+\ket{11}\right)+ \beta \left(\ket{01}-\ket{10}\right)\right\}.$
During experimental implementation the state prepared may also have some errors. Keeping this in mind, we have reconstructed the density matrix of the quantum state (which is to be teleported) generated in the experiment as
\begin{equation}
\rho^\prime=\left[{\begin{array}{cccc}
0.352 & -0.0805 + 0.1045 i & 0.18- 0.133 i & 0.31325- 0.005 i\\
-0.0805 - 0.1045 i & 0.135 & -0.09275 - 0.017 i & -0.099 - 0.12 i\\
0.18+ 0.133 i & -0.09275 + 0.017 i & 0.175 & 0.1505 + 0.1005 i\\
0.31325 + 0.005 i & -0.099 + 0.12 i & 0.1505 - 0.1005 i & 0.338\\
\end{array}}\right].
\end{equation}
Various elements of all the density matrices are shown pictorially in Figure \ref{fig:tom}.
Finally, we would like to quantize the performance of the QT scheme using a distance based measure, fidelity, defined as $F=Tr[\sqrt{\sqrt{\rho_1}.\rho_2.\sqrt{\rho_1}}].$ Using this we calculated the fidelity between the theoretical state with experimentally generated state (i.e., $\rho_1=\rho$ and $\rho_2=\rho^\prime$) as 0.9221. The same calculation performed between experimentally generated and teleported state (i.e., $\rho_1=\rho$ and $\rho_2=\rho^{\prime\prime}$) yields a higher value for fidelity (0.9378). Thus, the state preparation is relatively more erroneous, due to errors in gate implementation and decoherence. However, the constructed state is found to be teleported with high fidelity.
\section {Conclusion} \label{con}
Teleportation of multi-qubit states with the optimal amount of quantum resources in terms of the number of entangled qubits required in the quantum channel has been performed. Specifically, it has been shown that the amount of quantum resources required to teleport an unknown quantum state depends only on the number of non-zero probability amplitudes in the quantum state and is independent of the number of qubits in the state to be teleported.
The present study establishes a foundationally important fact that the more the information regarding the quantum state to be teleported is available the lesser the quantum resources to teleport that state should be required. Thus, there exists a trade-off between our knowledge about the state to be teleported and the amount of quantum resources required for the teleportation. Every scheme of RSP that performs teleportation of a known state using lesser amount of quantum resources than that required for teleportation of an unknown quantum state \cite{pati2000minimum,sharma2015controlled}, essentially exploits this trade-off.
Further, the relevance of the present work is not restricted to QT. It is also useful in CT, BST and BCST schemes. The relevance of the present work also lies in the fact that the limiting cases of our scheme can perform the same task with reduced amount of quantum resources in comparison with the previously achieved counterparts. In fact, for almost all the existing works reported on teleportation of multi-qubit states with some non-zero unknowns, we have shown a clear prescription to optimize the required quantum resources.
Finally, a proof of principle experimental implementation of the proposed scheme on the IBM quantum computer is shown for teleporation of a 2-qubit quantum state. It is important to note that our scheme enables this experimental realization to be accomplished on a 5-qubit quantum computer. From this implementation, we have computed the fidelity between the theoretical and experimentally generated state, and experimentally generated and teleported state. This quantitative analysis infer that the teleportion circuit implemented here is more efficient when compared with the state preparation part. This fact establishes the relevance of the proposed scheme in context of reduction of the decoherence effects on teleportation, too.
We hope our attempt to optimize the resource requirement for teleportation of multi-qubit quantum states should increase the feasibility of multi-qubit quantum state teleportation performed in various quantum systems. This is also expected to impact the teleportation-based direct secure quantum communication scheme, where resources can be optimized exploiting the form of the quantum state teleported (e.g., \cite{joy2017efficient} and references therein). Along the same line, optimization of quantum resources in CT, without affecting the controller's power, will be performed and reported elsewhere.
\textbf{Acknowledgment:}
AP, AS and KT thank Defense Research \& Development Organization (DRDO), India for the support provided through the project number ERIP/ER/1403163/M/01/1603.
\bibliographystyle{Final}
|
math/0701295
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\section{Introduction}
This paper is a companion to our paper \cite{DenSet}, where we introduced
the category of dendroidal sets, and explained some of its applications.
The main goal of the present paper is to introduce a notion of ''inner
Kan complex'' for dendroidal sets, and to prove some of the fundamental
properties of these inner Kan complexes.
Dendroidal sets provide a generalization of simplicial sets. There
is an embedding $i_{!}:sSet\rightarrow dSet$ along which many properties
of simplicial sets can be extended to dendroidal sets. For example,
the category of dendroidal sets carries a closed symmetric monoidal
structure, which extends the Cartesian structure of simplicial sets
(but is not itself Cartesian). This monoidal structure is closely
related to the Boardman-Vogt tensor product of operads (\cite{BoardmanVogt,Fiedor,Dunn}).
It plays a central role in various uses of dendroidal sets, for example
in the construction of homotopy coherent nerves of operads, and in
the definition of weak higher categories given in \cite{DenSet}.
We recall that a simplicial set is said to satisfy the Kan condition,
or, to be a Kan complex, if every horn $\Lambda^{i}[n]\rightarrow X$
for $0\le i\le n$ has a filler. Boardman and Vogt \cite{BoardmanVogt}
study this filler condition for $0<i<n$, and refer to it as the restricted
Kan condition. Recently, Joyal \cite{JoyalBook} has been studying simplicial sets satisfying this condition, under the name of quasi-categories. The horns $\Lambda^{i}[n]$ for $i$ different from
$0$ and $n$ are called inner horns, and we shall call a simplicial
set satisfying this restricted Kan condition an inner Kan complex.
Note that the nerve of any category is an inner Kan complex.
In this paper, we will define inner horns and inner Kan complexes
for dendroidal sets, in such a way that a simplicial set $X$ is an
inner Kan complex iff $i_{!}(X)$ is a dendroidal inner Kan complex.
The dendroidal nerve of any operad provides an example of such a dendroidal
inner Kan complex. We will prove several fundamental properties of
dendroidal inner Kan complexes. Our main result is that the closed
monoidal structure on dendroidal sets has the property that for any
two dendroidal sets $X$ and $Y$, the internal Hom $\underline{Hom}(X,Y)$
is an inner Kan complex whenever $X$ is normal and $Y$ is inner Kan (Theorem 9.1 below). We will
also show that the homotopy coherent nerve of a topological operad
is an inner Kan complex. (A more general statement for operads in
monoidal model categories is given in Theorem 7.1 below.) These results
specialise to known results for simplicial sets, and provide new proofs
of these. Indeed, Cordier and Porter \cite{CordierPorter} prove that
the homotopy coherent nerve of a locally fibrant simplicial category
is an inner Kan complex. And Joyal \cite{JoyalBook} proves for any
two simplicial sets $X$ and $Y$ that $\underline{Hom}(X,Y)$ is
an inner Kan complex (quasi-category) whenever $Y$ is (see also \cite{Joshua}).
The latter result plays a fundamental role in Joyal's proof of the
existence of a model structure on simplicial sets in which the inner
Kan complexes are the fibrant objects. We expect our result for dendroidal
sets to play a similar role in establishing an analogous model structure
on the category of dendroidal sets.
In \cite{DenSet}, we introduced a Grothendieck construction (homotopy
colimit) for diagrams of dendroidal sets, necessary for our definition
of weak higher categories. In Section 8 of this paper, we will prove
that this dendroidal Grothendieck construction yields an inner Kan
complex when applied to a diagram of inner Kan complexes. In addition,
in Section 6, we will give an explicit description of the operad generated
by an inner Kan complex, modelled on the one given by Boardman and
Vogt \cite{BoardmanVogt} in the simplicial case. We then use this
description to prove that a dendroidal set satisfies the unique filler
condition for inner horns iff it is the nerve of an operad.
\section{The category of dendroidal sets}
The notion of dendroidal set was introduced in \cite{DenSet}. We
briefly recall the relevant definitions here.
To begin with, we introduce a category $\Omega$ whose objects are
finite rooted trees. If we think of a tree as a graph, and call a
vertex unary if it has only one edge attached to it, then our trees
$T$ are equipped with a distinguished unary vertex $o$ called the
output, and a set of unary vertices $I$ (not containing $o$) called
the set of inputs. When drawing such a tree, it is common to orient
the tree ''towards the output'' drawn at the bottom, and delete
the designated output and input vertices from the picture. Thus, in
the tree $T$,
\[
\xymatrix{*{\,}\ar@{-}[dr]_{e} & & *{\,}\ar@{-}[dl]^{f}\\
\,\ar@{}[r]|{\,\,\,\,\,\,\,\,\,\,\,\,\,\, v} & *{\bullet}\ar@{-}[dr]_{b} & & *{\,}\ar@{-}[dl]_{c}\ar@{}[r]|{\,\,\,\,\,\,\,\,\,\,\,\, w} & *{\bullet}\ar@{-}[dll]^{d}\\
& & *{\bullet}\ar@{-}[d]_{a} & \,\ar@{}[l]^{r\,\,\,\,\,\,\,\,\,\,\,}\\
& & *{\,}}
\]
the output vertex at the edge $a$ has been deleted, as have the input
vertices at $e,f,c$. This tree $T$ now has three (remaining) vertices,
three \emph{input edges} attached to the three deleted input vertices,
and one \emph{output edge} (attached to the deleted output vertex).
These input and output edges are called \emph{outer edges} (the output
edge is also called the \emph{root}, while the input edges are also
called \emph{leaves}), the others ($b$ and $d$ in the picture) are
called \emph{inner} edges. From now on, we will not mention the input
and output vertices anymore, and ''vertex'' will always refer to
a remaining vertex.
Attached to each such vertex in the tree, there will be one designated
edge pointing towards the root; the other edges attached to this vertex
are called the input edges of that vertex, and their number is called
the \emph{valence} of the vertex. So in the tree $T$ pictured above,
the vertex $r$ has valence three and the vertex $w$ has valence
zero. The tree with just one edge is now drawn as \[
\xymatrix{*{}\ar@{-}[d]\\
*{}}
\]
and referred to as $\eta$, or sometimes as $\eta_{e}$ if we want
to name its edge $e$. The linear tree, with one input edge and one
output edge and $n$ vertices, is denoted $i[n]$. It has $n+1$ edges
which we usually number from input to output as $0,1,\cdots,n$. Here
is a picture of $i[2]$:\[
\xymatrix{ & *{}\ar@{-}[d]^{0}\\
\ar@{}[d]^{i[2]:} & *{\bullet}\ar@{-}[d]^{1}\\
*{} & *{\bullet}\ar@{-}[d]^{2}\\
& *{}}
\]
Each tree $T$ defines a coloured operad (see \cite{ColWCons} for
a definition of coloured operads) which we denoted $\Omega(T)$ in
\cite{DenSet}. The colours of this operad are the edges of the tree,
and the operations are \emph{generated} by the vertices of the tree.
A planar representation of the tree gives a specific set of generators.
For example, for the tree $T$ pictured above, $\Omega(T)$ has six
colours, $a,b,\cdots,f$. A choice of generating operations is $r\in\Omega(T)(b,c,d;a)$,
$w\in\Omega(T)(-;d)$ and $v\in\Omega(T)(e,f;b)$. The other operations
are units such as $1_{b}\in\Omega(T)(b;b)$, compositions such as
$r\circ_{1}v\in\Omega(T)(e,f,c,d;a)$, and permutations such as $r\cdot\tau\in\Omega(T)(c,b,d;a)$.
Note that the \emph{same} tree $T$ can be given a different planar
structure, e.g. \[
\xymatrix{ & *{}\ar@{-}[dr]_{e} & & *{}\ar@{-}[dl]^{f}\\
& *{}\ar@{-}[dr]_{c} & *{\bullet}\ar@{-}[d]_{b} & *{\bullet}\ar@{-}[dl]^{d}\\
& & *{\bullet}\ar@{-}[d]_{a}\\
& & *{}}
\]
which defines the \emph{same} operad $\Omega(T)$ but suggests a
different choice of generators ($r\cdot\tau$ rather than $r$).
The category $\Omega$ is now defined as the category having these
trees $T$ (with designated output and inputs) as objects, and as
arrows $T\rightarrow T'$ the maps of coloured operads $\Omega(T)\rightarrow\Omega(T')$.
(Note that every such map sends colours to colours, i.e., edges of
$T$ to edges of $T'$, and is in fact completely determined by this).
The category of \emph{dendroidal sets} is the category of functors
$X:\Omega^{op}\rightarrow Set$ and natural transformations between
them. We will denote this category by $dSet$.
For a dendroidal set $X$ and a tree $T$, we will usually write $X_{T}$
for $X(T)$, and call an element of the set $X_{T}$ a \emph{dendrex}
of $X$ of ''shape'' $T$.
The linear trees $i[n]$ for $n\ge0$ define a functor (a full embedding)\[
i:\Delta\rightarrow\Omega\]
from the standard simplicial category, and hence by composition a
functor \[
i^{*}:dSet\rightarrow sSet,\]
from dendroidal sets to the category $sSet$ of simplicial sets.
By Kan extension, this functor has both a left and a right adjoint,
denoted $i_{!}$ and $i_{*}:sSet\rightarrow dSet$, respectively.
The functor $i_{!}$ is ''extension by zero''; for a simplicial
set $X$,\[
i_{!}(X)_{T}=\left\{ \begin{array}{cc}
X_{n}, & \textrm{if $T\cong i[n]$ for some $n\ge0$}\\
\phi, & \textrm{otherwise}\end{array}\right.\]
This defines a full embedding $i_{!}:sSet\rightarrow dSet$, from
simplicial sets into dendroidal sets.
Each coloured operad $\mathcal{P}$ defines a dendroidal set $N_{d}(\mathcal{P})$,
its \emph{dendroidal nerve}, by\[
N_{d}(\mathcal{P})_{T}=Hom(\Omega(T),\mathcal{P}),\]
\emph{Hom} denoting the set of arrows in the category of operads.
If $\mathcal{P}$ is itself an operad of the form $\Omega(S)$, given
by an object $S$ of $\Omega$, then $N_{d}(\mathcal{P})$ is the
representable dendroidal set given by $S$, which we will denote by
$\Omega[S]$; i.e.,\[
N_{d}\Omega(S)=\Omega[S]\]
by definition. The functor $N_{d}$ from operads to dendroidal sets
is fully faithful, and has a left adjoint which we will denote by
\[
\tau_{d}:dSet\rightarrow Operad.\]
For a dendroidal set $X$, we refer to $\tau_{d}(X)$ as \emph{the
operad generated} by $X$.
We also recall from \cite{DenSet} that the Cartesian structure on
$sSet$ extends to a (non-Cartesian) closed symmetric monoidal structure
$\otimes$ on $dSet$. This structure is completely determined by
the identity\[
\Omega[S]\otimes\Omega[T]=N_{d}(\Omega(S)\otimes_{BV}\Omega(T))\]
where $\otimes_{BV}$ denotes the Boardman-Vogt tensor product of
(coloured) operads; see \cite{BoardmanVogt}. The corresponding internal
Hom is then determined by the Yoneda lemma, as\[
\underline{Hom}(X,Y)_{T}=Hom_{dSet}(\Omega[T]\otimes X,Y).\]
We will come back to the monoidal structure in more detail in Section
9.
\section{Faces and degeneracies}
Exactly as for $\Delta$, the maps in $\Omega$ are generated by special
kinds of maps.
(i) Given a tree $T$ and a vertex $v\in T$ of valence 1, there is
a tree $T'$, obtained from $T$ by deleting the vertex $v$ and merging
the two edges $e_{1}$ and $e_{2}$ on either side of $v$ into one
new edge $e$. There is an operad map, i.e. an arrow $\sigma_{v}:T\rightarrow T'$
in $\Omega$, which sends $v$ to the unit $1_{e}$. For example:\[
\begin{array}{ccc}
\xymatrix{*{\,}\ar@{-}[dr] & & *{\,}\ar@{-}[dl]\\
& *{\bullet}\ar@{-}[dr]_{e_{1}} & & *{\,}\ar@{-}[dr] & & *{\,}\ar@{-}[dl]\\
& & *{\bullet}\ar@{-}[dr]_{e_{2}}\ar@{}|{\,\,\,\,\,\,\,\,\,\, v} & & *{\bullet}\ar@{-}[dl]\\
& & & *{\bullet}\ar@{-}[d]\\
& & & *{\,}}
& \xymatrix{\\\\\ar[r]^{\sigma_{v}} & *{}}
& \xymatrix{*{\,}\ar@{-}[dr] & & *{\,}\ar@{-}[dl]\\
& *{\bullet}\ar@{-}[ddrr] & & *{\,}\ar@{-}[dr] & & *{\,}\ar@{-}[dl]\\
& \,\ar@{}[r]|{\,\,\,\, e} & & & *{\bullet}\ar@{-}[dl]\\
& & & *{\bullet}\ar@{-}[d]\\
& & & *{\,}}
\end{array}\]
An arrow in $\Omega$ of this kind will be called a \emph{degeneracy
(map).}
(ii) Given a tree $T$, and a vertex $v$ in $T$ with exactly one
inner edge attached to it, one can obtain a new tree $T/v$ by deleting
$v$ and all the external edges attached to it. The operad $\Omega(T/v)$
associated to $T/v$ is simply a suboperad of the one associated to
$T$, and this inclusion of operads defines an arrow in $\Omega$
denoted \[
\partial_{v}:T/v\rightarrow T.\]
An arrow in $\Omega$ of this kind is called an \emph{outer face (map).}
For example\emph{\[
\begin{array}{ccc}
\xymatrix{\\*{\,}\ar@{-}[dr]_{b} & *{\,}\ar@{-}[d]^{c}\ar@{}[r]|{\,\,\,\,\,\,\,\,\,\,\,\, w} & *{\bullet}\ar@{-}[dl]^{d}\\
\,\ar@{}[r]|{\,\,\,\,\,\,\,\,\,\,\,\, r} & *{\bullet}\ar@{-}[d]_{a}\\
& *{\,}}
& \xymatrix{\\\\\ar[r]^{\partial_{v}} & *{}}
& \xymatrix{*{\,}\ar@{-}[dr]_{e} & & *{\,}\ar@{-}[dl]^{f}\\
\,\ar@{}[r]|{\,\,\,\,\,\,\,\,\,\,\, v} & *{\bullet}\ar@{-}[dr]_{b} & & *{\,}\ar@{-}[dl]_{c}\ar@{}[r]|{\,\,\,\,\,\,\,\,\,\,\,\, w} & *{\bullet}\ar@{-}[dll]^{d}\\
& & *{\bullet}\ar@{-}[d]_{a} & \,\ar@{}[l]|{r\,\,\,\,\,\,\,\,\,\,\,}\\
& & *{\,}}
\end{array}\]
}
Moreover, for any tree $T$ with exactly one vertex $v$, each edge
$e$ of $T$ (necessarily outer), there is an \emph{outer face map}\[
e:\eta\rightarrow T\]
sending the unique edge of $\eta$ to $e$.
(iii) Given a tree $T$ and an inner edge $e$ in $T$, one can obtain
a new tree $T/e$ by contracting the edge $e$. There is a canonical
map of operads $\partial_{e}:\Omega(T/e)\rightarrow\Omega(T)$ which
sends the new vertex in $T/e$ (obtained by merging the two vertices
attached to $e$) into the appropriate composition of these two vertices
in $\Omega(T)$. An arrow $\partial_{e}:T/e\rightarrow T$ in $\Omega$
of this kind is called an \emph{inner face (map).} For example
\[
\begin{array}{ccc}
\xymatrix{*{\,}\ar@{-}[rrd]_{e} & *{\,}\ar@{-}[rd]^{f} & & *{\,}\ar@{-}[dl]_{c}\ar@{}[r]|{\,\,\,\,\,\,\,\,\, w} & *{\bullet}\ar@{-}[lld]^{d}\\
& \,\ar@{}[r]_{\,\,\,\,\,\,\,\,\,\,\, u} & *{\bullet}\ar@{-}[d]^{a}\\
& & *{\,}}
& \xymatrix{\\\ar[r]^{\partial_{b}} & *{}}
& \xymatrix{*{\,}\ar@{-}[dr]_{e} & & *{\,}\ar@{-}[dl]^{f}\\
\,\ar@{}[r]|{\,\,\,\,\,\,\,\,\,\,\,\,\,\, v} & *{\bullet}\ar@{-}[dr]_{b} & & *{\,}\ar@{-}[dl]_{c}\ar@{}[r]|{\,\,\,\,\,\,\,\,\,\,\,\, w} & *{\bullet}\ar@{-}[dll]^{d}\\
& & *{\bullet}\ar@{-}[d]_{a} & \,\ar@{}[l]^{r\,\,\,\,\,\,\,\,\,\,\,}\\
& & *{\,}}
\end{array}\]
(iv) Given two trees $T$ and $T'$, any isomorphism $T\rightarrow T'$
of trees, sending inputs to inputs and output to output, of course
defines an isomorphism of operads $\Omega(T)\rightarrow\Omega(T')$,
and hence is an isomorphism $\xymatrix{T\ar[r]^{\cong} & T'}
$ in $\Omega$. For example, if $C_{n}$ denotes the corolla with just
one vertex, $n$ inputs, and one output, then we might name its input
edges $e_{1},\cdots,e_{n}$ \[
\xymatrix{*{}\ar@{-}[ddrr]_{e_{1}} & *{}\ar@{-}[ddr]_{e_{2}} & & & *{}\ar@{-}[ddll]^{e_{n}}\\
& & \ar@{}[r]^{\cdots} & *{}\\
& & *{\bullet}\ar@{-}[dd]\\
& & *{}\\
& & *{}}
\]
Any permutation $\varphi\in\Sigma_{n}$ defines an automorphism of
$C_{n}$ in $\Omega$.
For a tree $T$, let its \emph{degree} $|T|$ be the number of vertices
in $T$. Then degeneracy maps decrease degree by 1, face maps (outer
or inner) increase degree by 1, and isomorphisms preserve degree.
Any map $\xymatrix{T\ar[r]^{f} & T'}
$ in $\Omega$ can be written as $f=\delta\varphi\sigma$, where $\delta$
is a composition of (inner or outer) faces, $\varphi$ is an isomorphism,
and $\sigma$ is a composition of degeneracies. (This composition
is unique up to isomorphism).
\section{Skeletal filtration}
As for any presheaf category, any dendroidal set $X$ is a colimit
of representables, of the form\[
X=\varinjlim\Omega[T]\]
(see \cite{CWM}). We wish to refine this a little, in a way similar
to the skeletal filtration for simplicial sets. To this end, call
a dendrex $x\in X_{T}$ of shape $T$ \emph{degenerate} if there is
a surjective map $\xymatrix{T\ar[r]^{\alpha} & T'}
$ in $\Omega$ (a composition of degeneracies) such that $x=\alpha^{*}(x')$
for some $x'\in X_{T'}$. Here $\alpha$ should not be an isomorphism
of course, so that $T'$ has strictly fewer vertices then $T$ and
$\alpha$ is a non-empty composition of degeneracies.
Given a dendroidal set $X$ we denote by $Sk_{n}(X)$ the sub dendroidal set of $X$ generated by all non-degenerate dendrices $x\in X_{T}$ where $|T|\le n$. An arbitrary
dendroidal set $X$ is clearly the colimit (union) of the sequence \[
\begin{array}{cccc}
\quad\quad\quad & Sk_{0}(X)\subseteq Sk_{1}(X)\subseteq Sk_{2}(X)\subseteq\cdots & \quad\quad\quad & (1)\end{array}\]
We call this the skeletal filtration of $X$. This filtration extends the skeletal filtration for simplicial sets
in the precise sense that for any dendroidal set $X$ and any simplicial
set $S$, there are canonical isomorphisms \[
i^{*}Sk_{n}(X)=Sk_{n}(i^{*}X)\]
and\[
i_{!}Sk_{n}(S)=Sk_{n}(i_{!}S).\]
Consider now the following diagram: \[
\xymatrix{*++{\coprod_{x,T}\partial\Omega[T]}\ar[r]\ar@{>->}[d] & *++{Sk_{n}(X)}\ar@{>->}[d]\\
\coprod_{x,T}\Omega[T]\ar[r] & Sk_{n+1}(X)}
\]
where the sum is taken over isomorphism classes of pairs $(x,T)$ in the category of elements of $X$ where $T$ is a tree with
$n$ vertices and $x\in X_{T}$ is non-degenerate, and $\partial\Omega[T]$
is the boundary of $\Omega[T]$, i.e., the union of its faces. We call the skeletal filtration of $X$ \emph{normal} if this square is a pushout for each $n>0$.
Following Cisinski \cite{Cisinski} we call a dendroidal set \emph{normal} if for each non-degenerate dendrex $x\in X_{T}$, the only isomorphism fixing $x$ is the identity. Cisinski (loc. cit.) proves that the normal dendroidal sets are precisely those whose skeletal filtrations are normal.
\begin{example}
If $X$ is a simplicial set then $i_{!}(X)$ admits a normal skeletal filtration and in fact that skeletal filtration is isomorphic to the usual skeletal filtration of $X$. If $\mathcal{P}$ is a $\Sigma$-free operad then $N_d(\mathcal{P})$ is normal. In particular if $\mathcal{P}$ is the symmetrization of a planar operad then $N_d(\mathcal{P})$ is normal.
\end{example}
\section{Inner Kan complexes}
We begin by introducing inner horns. For a tree $T$, each face map
$\partial:T'\rightarrow T$ defines a monomorphism $\Omega[T']\rightarrow\Omega[T]$
between (representable) dendroidal sets. The union (pushout) of these
subobjects is the boundary of $\Omega[T]$, denoted \[
\xymatrix{*++{\partial\Omega[T]}\ar@{>->}[r] & \Omega[T],}
\]
as above. If $e$ is an inner edge of $T$, then the union of all
the faces \emph{except} \[
\xymatrix{*+++{\partial_{e}:T/e}\ar@{>->}[r] & T}
\]
defines a subobject of the boundary, denoted \[
\xymatrix{*++{\Lambda^{e}[T]}\ar@{>->}[r] & \Omega[T]}
,\]
and called the \emph{inner horn} associated to $e$ (and to $T$).
This terminology and notation extends the one \[
\xymatrix{*++{\Lambda^{k}[n]}\ar@{>->}[r] & *++{\partial\Delta[n]}\ar@{>->}[r] & \Delta[n]}
\]
for simplicial sets, in the sense that \[
i_{!}(\Lambda^{k}[n])=\Lambda^{k}[i[n]]\]
\[
i_{!}(\partial\Delta[n])=\partial\Omega[i[n]]\]
as subobjects of $i_{!}(\Delta[n])=\Omega[i[n]].$
A dendroidal set $K$ is said to be a (dendroidal) \emph{inner Kan
complex} if, for any tree $T$ and any inner edge $e$ in $T$, the
map\[
K_{T}=Hom(\Omega[T],K)\rightarrow Hom(\Lambda^{e}[T],K)\]
is a surjection of sets. It is called a \emph{strict} inner Kan complex
if this map is a bijection (for any $T$ and $e$ as above). For example,
we will see (Proposition 5.3 below) that the dendroidal nerve of an
operad is always a strict inner Kan complex. This terminology is analogous
to the one introduced by Boardman and Vogt, who say a simplicial set
$X$ satisfies the restricted Kan condition if, for any $0<k<n$,
the map $Hom(\Delta[n],X)\rightarrow Hom(\Lambda^{k}[n],X)$ is a
surjection (\cite{BoardmanVogt} Definition 4.8, page 102). In more
recent work (\cite{JoyalPaper,JoyalBook}) Joyal develops the general
theory of simplicial sets satisfying the restricted Kan condition.
Joyal uses the terminology quasi-categories for such simplicial sets
so as to stress the analogy with category theory. In fact a simplicial
set $X$ is a quasi-category iff $i_{!}(X)$ is a dendroidal inner
Kan complex, and for any dendroidal inner Kan complex $K$, the restriction
$i^{*}(K)$ is a quasi-category in the sense of Joyal.
Let us call a map $u:U\rightarrow V$ of dendroidal sets an \emph{anodyne
extension} if it can be obtained from the set of inner horn inclusions
by coproducts, pushouts, compositions, and retracts (cf, \cite{GZ},
p. 60). Then obviously, the surjectivity property for inner Kan complexes
extends to anodyne extensions, in the sense that the map of sets \[
u^{*}:Hom(V,K)\rightarrow Hom(U,K),\]
given by composition with $u$, is again surjective. Similarly, the
map $u^{*}$ is a bijection for any strict inner Kan complex.
For a tree $T$ let $I(T)$ be the set of inner edges of $T$. For
a non-empty subset $A\subseteq I(T)$ let $\Lambda^{A}[T]$ be the
union of all faces of $\Omega[T]$ except those obtained by contracting
an edge from $A$. Note that if $A=\{ e\}$ then $\Lambda^{A}[T]=\Lambda^{e}[T]$.
\begin{lem}
For any non-empty $A\subseteq I(T)$ the inclusion $\Lambda^{A}[T]\rightarrow\Omega[T]$
is anodyne.
\end{lem}
\begin{proof}
By induction on $n=|A|$. If $n=1$ then $\Lambda^{A}[T]\rightarrow\Omega[T]$
is an inner horn inclusion, thus anodyne. Assume the proposition holds
for $n<k$ and suppose $|A|=k$. Choose an arbitrary $e\in A$ and
put $B=A\backslash\{ e\}$. The map $\Lambda^{A}[T]\rightarrow\Omega[T]$
factors as\[
\xymatrix{\Lambda^{A}[T]\ar[r]\ar[rd] & \Lambda^{B}[T]\ar[d]\\
& \Omega[T]}
\]
The vertical map is anodyne by the induction hypothesis and it therefore
suffices to prove that $\Lambda^{A}[T]\rightarrow\Lambda^{B}[T]$
is anodyne. The following diagram expresses that map as a pushout\[
\xymatrix{\Lambda^{B}[T/e]\ar[r]\ar[d] & \Lambda^{A}[T]\ar[d]\\
\Omega[T/e]\ar[r] & \Lambda^{B}[T]}
\]
and since the map $\Lambda^{B}[T/e]\rightarrow\Omega[T/e]$ is anodyne
(by the induction hypothesis), the proof is complete.
\end{proof}
We denote by $\Lambda^{I}[T]$ the dendroidal set $\Lambda^{A}[T]$
where $A=I(T)$, that is $\Lambda^{I}[T]$ is the union of all outer
faces of $\Omega[T]$. By the above proposition the inclusion $\Lambda^{I}[T]\rightarrow\Omega[T]$
is anodyne.
We now consider grafting of trees. For two trees $T$ and $S$, and
a leaf $l$ of $T$, let $T\circ_{l}S$ be the tree obtained by grafting
$S$ onto $T$ by identifying $l$ with the root (output edge) of
$S$. Then there are obvious inclusions $\Omega[S]\rightarrow\Omega[T\circ_{l}S]$
and $\Omega[T]\rightarrow\Omega[T\circ_{l}S]$, the pushout (union)
of which we denote by $\Omega[T]\cup_{l}\Omega[S]\rightarrow\Omega[T\circ_{l}S]$.
\begin{lem}
(Grafting) For any two trees $T$ and $S$ and any leaf $l$ of $T$,
the inclusion $\Omega[T]\cup_{l}\Omega[S]\rightarrow\Omega[T\circ_{l}S]$
is anodyne.
\end{lem}
\begin{proof}
Let us write $R=T\circ_{l}S$. The case where $T=\eta$ or $S=\eta$
is trivial, we therefore assume that this is not the case. We proceed
by induction on $n=|T|+|S|$, the sum of the degrees of $T$ and $S$.
The cases $n=0$ or $n=1$ are taken care of by our assumption that
$T\ne\eta\ne S$. For the case $n=2$ the same assumption implies
that the inclusion $\Omega[T]\cup_{l}\Omega[S]\rightarrow\Omega[R]$
is an inner horn inclusion. In any case it is anodyne. Assume then
that the result holds for $2\le n<k$ and suppose $|T|+|S|=k$.
Recall that $\Lambda^{I}[R]$ is the union of all the outer faces
of $\Omega[R]$. First notice that $\Omega[T]\cup_{l}\Omega[S]\rightarrow\Omega[R]$
factors as\[
\xymatrix{\Omega[T]\cup_{l}\Omega[S]\ar[r]\ar[dr] & \Lambda^{I}[R]\ar[d]\\
& \Omega[R]}
\]
and the vertical arrow is anodyne by a previous result. We now show
that \[
\Omega[T]\cup_{l}\Omega[S]\rightarrow\Lambda^{I}[R]\]
is anodyne by exhibiting it as a pushout of an anodyne extension.
Recall (\cite{DenSet}) that an external cluster is a vertex $v$
with the property that one of the edges adjacent to it is inner while
all the other edges adjacent to it are outer. Let $Cl(T)$ (resp.
$Cl(S)$) be the set of all external clusters in $T$ (resp. $S$)
which do not contain $l$ (resp. the root of $S$). For each $C\in Cl(T)$
the face of $\Omega[R]$ corresponding to $C$ is isomorphic to $\Omega[(T/C)\circ_{l}S]$
and the map $\Omega[T/C]\cup_{l}\Omega[S]\rightarrow\Omega[(T/C)\circ_{l}S${]}
is anodyne by the induction hypothesis. Similarly for every $C\in Cl(S)$
the face of $\Omega[R]$ that corresponds to $C$ is isomorphic to
$\Omega[T\circ_{l}(S/C)]$ and the map $\Omega[T]\cup_{l}\Omega[S/C]\rightarrow\Omega[T\circ_{l}(S/C)]$
is anodyne by the induction hypothesis. The following diagram is a
pushout\[
\xymatrix{\coprod_{C\in Cl(T)}(\Omega[T/C]\cup_{l}\Omega[S])\amalg\coprod_{C\in Cl(S)}(\Omega[T]\cup_{l}\Omega[S/C])\ar[r]\ar[d] & \Omega[T]\cup_{l}\Omega[S]\ar[d]\\
\coprod_{C\in Cl(T)}(\Omega[(T/C)\circ_{l}S])\amalg\coprod_{C\in Cl(S)}(\Omega[T\circ_{l}(S/C)])\ar[r] & \Lambda^{I}[R]}
\]
where the map on the left is the coproduct of all of the anodyne
extensions just mentioned. Since anodyne extensions are closed under
coproducts, it follows that the map on the left of the pushout is
anodyne and thus also the one on the right, which is what we set out
to prove. This concludes the proof.
\end{proof}
We end this section with two remarks on strict inner Kan complexes.
\begin{prop}
The dendroidal nerve of any operad is a strict inner Kan complex.
\end{prop}
\begin{proof}
Let $\mathcal{P}$ be an operad. A dendrex $x\in N_{d}(\mathcal{P})_{T}$
is a map $x:\Omega[T]\rightarrow N_{d}(\mathcal{P})$ which is a map
of operads $\Omega(T)\rightarrow P$. If we choose a planar representative
for $T$ then $\Omega(T)$ is specifically given in terms of generators
and is a free operad. It follows that $x$ is equivalent to a labeling
of the (planar representative) $T$ as follows. The edges are labeled
by colours of $\mathcal{P}$ and the vertices are coloured by operations
in $\mathcal{P}$ where the input of such an operation is the tuple
of labels of the incoming edges to the vertex and the output is the
label of the outgoing edge from the vertex. Any inner horn $\Lambda^{e}[T]\rightarrow N_{d}(\mathcal{P})$
is easily seen to be equivalent to such a labeling of the tree $T$
and thus determines a unique filler.
\end{proof}
\begin{prop}
Any strict inner Kan complex is $2$-coskeletal.
\end{prop}
\begin{proof}
Let $X$ be a strict inner Kan complex. Let $Y$ be any dendroidal
set and assume $f:Sk_{2}Y\rightarrow Sk_{2}X$ is given. We first
show that $f$ can be extended to a dendroidal map $\hat{f}:Y\rightarrow X$.
Suppose $f$ was extended to a map $f_{k}:Sk_{k}Y\rightarrow Sk_{k}X$
for $k\ge2$. Let $y\in Sk_{k+1}(Y)$ be a non-degenerate dendrex
and assume $y\notin Sk_{k}(Y)$. So $y\in Y_{T}$ and $T$ has exactly
$k+1$ vertices. Choose an inner horn $\Lambda^{\alpha}[T]$ (such
an inner horn exist since $k\ge2$). The set $\{\beta^{*}y\}_{\beta\ne\alpha}$
where $\beta$ runs over all faces of $T$, defines a horn $\Lambda^{\alpha}[T]\rightarrow Y$.
Since this horn factors through the $k$-skeleton of $Y$ we obtain,
by applying $f_{k}$, a horn $\Lambda^{\alpha}[T]\rightarrow X$ in
$X$ given by $\{ f(\beta^{*}y)\}_{\beta\ne\alpha}$. Let $f_{k+1}(y)\in X_{T}$
be the unique filler of that horn. By construction we have for each
$\beta\ne\alpha$ that \[
\beta^{*}f_{k+1}(y)=f(\beta^{*}y)\]
it thus remains to show the same for $\alpha$. The dendrices $f(\alpha^{*}y)$
and $\alpha^{*}f_{k+1}(y)$ both have the same boundary and they are
both of shape $S$ where $S$ has $k$ vertices. Since $k\ge2$, $S$
has an inner face, but then it follows that both $f(\alpha^{*}y)$
and $\alpha^{*}f_{k+1}(y)$ are fillers for the same inner horn in
$X$ which proves that they are equal. By repeating the process for
all dendrices in $Sk_{k+1}(Y)$ it follows that $f_{k}$ can be extended
to $f_{k+1}:Sk_{k+1}(Y)\rightarrow Sk_{k+1}(X)$. This holds for all
$k\ge2$ which implies that $f$ can be extended to $\hat{f}:Y\rightarrow X$.
To show uniqueness of $\hat{f}$ assume that $g$ is another extension
of $f$. Suppose it has been shown that $\hat{f}$ and $g$ agree
on all dendrices of shape $T$ where $T$ has at most $k$ vertices,
and let $y\in X_{S}$ be a dendrex of shape $S$ where $S$ has $k+1$
vertices. But then the dendrices $\hat{f}(y)$ and $g(y)$ are dendrices
in $X$ that have the same boundary. Since $k\ge2$ it follows that
these dendrices are both fillers for the same inner horn and so are
the same. This proves that $\hat{f}=g$.
\end{proof}
\section{The operad generated by an inner Kan complex}
We recall that $\tau_{d}:dSet\rightarrow Operad$ denotes the left
adjoint to the dendroidal nerve functor $N_{d}$. In this section,
we will give a more explicit description of the operad $\tau_{d}(X)$
in the case where $X$ is an inner Kan complex. This description extends
the one in \cite{BoardmanVogt} of the category generated by a simplicial
set satisfying the restricted Kan condition. It will lead to a proof
of the following converse of Proposition 5.3.
\begin{thm}
For any strict inner Kan complex $X$, the canonical map $X\rightarrow N_{d}(\tau_{d}(X))$
is an isomorphism.
\end{thm}
Proposition 5.3 and Theorem 6.1 together state that a dendroidal set
is a strict inner Kan complex iff it is the nerve of an operad.
Consider an inner Kan complex $X$. For the description of $\tau_{d}(X)$,
we first fix some notation. For each $n\ge0$ let $C_{n}$ be the
$n$-corolla:\[
\xymatrix{*{}\ar@{-}[dr]_{1} & & *{}\ar@{-}[dl]^{n}\\
& *{\bullet}\ar@{-}[d]^{0}\\
& *{}}
\]
and for each $0\le i\le n$ recall that $i:\eta\rightarrow C_{n}$
denotes the obvious (outer face) map in $\Omega$ that sends the unique
edge of $\eta$ to the edge $i$ in $C_{n}$. An element $f\in X_{C_{n}}$
will be denoted by \[
\xymatrix{*{}\ar@{-}[dr]_{1} & & *{}\ar@{-}[dl]^{n}\\
\ar@{}[r]|{\quad f} & *{\bullet}\ar@{-}[d]^{0}\\
& *{}}
\]
If $C_{n}^{'}$ is another $n$-corolla together with an isomorphism
$\alpha:C_{n}^{'}\rightarrow C_{n}$ then we will usually write $f$
again instead of $\alpha^{*}(f)$. We will use this convention quite
often in the coming definitions and constructions, and in each case
there will be an obvious choice for the isomorphism $\alpha$ given
by the planar representation of the trees at question, which will
usually not be mentioned.
\begin{defn}
Let $X$ be an inner Kan complex and let $f,g\in X_{C_{n}}$, $n\ge0$.
For $1\le i\le n$ we say that \emph{$f$ is homotopic to $g$ along
the edge $i$,} and write $f\sim_{i}g$, if there is a dendrex $H$
of shape \[
\xymatrix{ & *{}\ar@{-}[d]^{i^{'}}\\
*{}\ar@{-}[dr]_{1} & *{\bullet}\ar@{-}[d]^{i} & *{}\ar@{-}[dl]^{n}\\
& *{\bullet}\ar@{-}[d]^{0}\\
& *{}}
\]
whose three faces are:\[
\begin{array}{ccc}
\xymatrix{*{}\ar@{-}[dr]_{1} & *{}\ar@{-}[d]^{i} & *{}\ar@{-}[dl]^{n}\\
\ar@{}[r]|{\quad\,\,\, f} & *{\bullet}\ar@{-}[d]^{0}\\
& *{}}
& \quad\xymatrix{*{}\ar@{-}[dr]_{1} & *{}\ar@{-}[d]\ar@{}[d(0.6)]^{i^{'}} & *{}\ar@{-}[dl]^{n}\\
\ar@{}[r]|{\quad\,\,\, g} & *{\bullet}\ar@{-}[d]^{0}\\
& *{}}
& \quad\xymatrix{ & *{}\ar@{-}[d]^{i^{'}}\\
\ar@{}[r]|{\quad\,\,\, id} & *{\bullet}\ar@{-}[d]^{i}\\
& *{}}
\end{array}\]
where the third one denotes a degeneracy. Similarly we will say that
$f$ is homotopic to $g$ along the edge $0$ and write $f\sim_{0}g$
if there is a dendrex of shape\[
\xymatrix{*{}\ar@{-}[dr]_{1} & & *{}\ar@{-}[dl]^{n}\\
& *{\bullet}\ar@{-}[d]^{0}\\
& *{\bullet}\ar@{-}[d]^{0^{'}}\\
& *{}}
\]
whose three faces are:\[
\begin{array}{ccc}
\xymatrix{ & *{}\ar@{-}[d]^{0}\\
\ar@{}[r]|{\quad\,\,\, id} & *{\bullet}\ar@{-}[d]^{0^{'}}\\
& *{}}
\quad & \xymatrix{*{}\ar@{-}[dr]_{1} & & *{}\ar@{-}[dl]^{n}\\
\ar@{}[r]|{\quad\,\,\, g} & *{\bullet}\ar@{-}[d]^{0^{'}}\\
& *{}}
\quad & \xymatrix{*{}\ar@{-}[dr]_{1} & & *{}\ar@{-}[dl]^{n}\\
\ar@{}[r]|{\quad\,\,\, f} & *{\bullet}\ar@{-}[d]^{0}\\
& *{}}
\end{array}\]
When $f\sim_{i}g$ for some $0\le i\le n$ we will refer to the corresponding
$H$ as a \emph{homotopy} from $f$ to $g$ along $i$ and will sometimes
write $H:f\sim_{i}g$.
\end{defn}
\begin{prop}
Let $X$ be an inner Kan complex. For each $0\le i\le n$ the relation
$\sim_{i}$ on the set $X_{C_{n}}$ is an equivalence relation.
\end{prop}
\begin{proof}
First we prove reflexivity. For $1\le i\le n$ let \[
\begin{array}{ccc}
\xymatrix{ & *{}\ar@{-}[d]^{i^{'}}\\
*{}\ar@{-}[dr]_{1} & *{\bullet}\ar@{-}[d]^{i} & *{}\ar@{-}[dl]^{n}\\
& *{\bullet}\ar@{-}[d]^{0}\\
& *{}}
& \xymatrix{\\\ar[r]^{\sigma_{i}} & *{}\\
}
& \xymatrix{*{}\ar@{-}[dr]_{1} & *{}\ar@{-}[d]^{i} & *{}\ar@{-}[dl]^{n}\\
& *{\bullet}\ar@{-}[d]_{0}\\
& *{}}
\end{array}\]
and for $i=0$ let\[
\begin{array}{ccc}
\xymatrix{*{}\ar@{-}[dr]_{1} & & *{}\ar@{-}[dl]^{n}\\
& *{\bullet}\ar@{-}[d]^{0}\\
& *{\bullet}\ar@{-}[d]^{0^{'}}\\
& *{}}
& \xymatrix{\\\\\ar[r]^{\sigma_{0}} & *{}\\
}
& \xymatrix{\\*{}\ar@{-}[dr]_{1} & & *{}\ar@{-}[dl]^{n}\\
& *{\bullet}\ar@{-}[d]^{0}\\
& *{}}
\end{array}\]
be the obvious degeneracies. It then follows that for any $f\in X_{C_{n}}$
the dendrex $\sigma_{i}^{*}(f)$ is a homotopy from $f$ to $f$,
thus $f\sim_{i}f$.
To prove symmetry assume $f\sim_{i}g$ for some $1\le i\le n$ and
let $H_{fg}$ be a homotopy from $f$ to $g$ along $i$. Consider
the tree $T$:\[
\xymatrix{\\ & *{}\ar@{-}[d]^{i^{''}}\\
& *{\bullet}\ar@{-}[d]^{i^{'}}\\
*{}\ar@{-}[dr]_{1} & *{\bullet}\ar@{-}[d]^{i} & *{}\ar@{-}[dl]^{n}\\
& *{\bullet}\ar@{-}[d]^{0}\\
& *{}}
\]
For the inner horn $\Lambda^{i}[T]$, corresponding to the edge $i$
in the tree above, we now describe a map $\Lambda^{i}[T]\rightarrow X$.
Such a map is given by specifying three dendrices in $X$ of certain
shapes such that their faces match in a suitable way. We describe
this map by explicitly writing the mentioned dendrices and their faces:\\
\\
\[
\begin{array}{ccc}
\xymatrix{ & H_{i}\\
& *{}\ar@{-}[d]^{i^{''}}\\
\ar@{}[r]|{\quad\quad id}\ar@{}[d] & *{\bullet}\ar@{-}[d]^{i^{'}}\\
*{}\ar@{}[r]|{\quad\quad id} & *{\bullet}\ar@{-}[d]^{i}\\
& *{}\\
}
& \quad\xymatrix{ & H_{f}\\
& *{}\ar@{-}[d]^{i^{''}}\\
*{}\ar@{-}[dr]_{1}\ar@{}[r]|{\quad\quad id} & *{\bullet}\ar@{-}[d]^{i} & *{}\ar@{-}[dl]^{n}\\
*{}\ar@{}[r]_{\quad\,\,\quad f} & *{\bullet}\ar@{-}[d]^{0}\\
& *{}\\
}
\quad & \xymatrix{ & H_{fg}\\
& *{}\ar@{-}[d]^{i^{'}}\\
*{}\ar@{-}[dr]_{1}\ar@{}[r]|{\quad\quad id} & *{\bullet}\ar@{-}[d]^{i} & *{}\ar@{-}[dl]^{n}\\
*{}\ar@{}[r]_{\,\,\quad\quad f} & *{\bullet}\ar@{-}[d]^{0}\\
& *{}\\
}
\end{array}\]
with inner faces of these dendrices:\[
\begin{array}{ccc}
\xymatrix{ & *{}\ar@{-}[d]^{i^{''}}\\
\ar@{}[r]|{\quad\,\, id} & *{\bullet}\ar@{-}[d]^{i}\\
& *{}}
\quad & \xymatrix{*{}\ar@{-}[dr]_{1} & *{}\ar@{-}[d]^{i^{''}} & *{}\ar@{-}[dl]^{n}\\
\ar@{}[r]|{\quad\,\,\, f} & *{\bullet}\ar@{-}[d]^{0}\\
& *{}}
\quad & \xymatrix{*{}\ar@{-}[dr]_{1} & *{}\ar@{-}[d]^{i^{'}} & *{}\ar@{-}[dl]^{n}\\
\ar@{}[r]|{\quad\,\,\, g} & *{\bullet}\ar@{-}[d]^{0}\\
& *{}}
\end{array}\]
where $H_{i}$ is a double degeneracy of $i$, $H_{f}$ is a homotopy
from $f$ to $f$ (along the branch $i$) and $H_{fg}$ is the given
homotopy from $f$ to $g$. It is easily checked that the faces indeed
match so that we have a horn in $X$. Let $x$ be a filler for that
horn and consider $H_{gf}=\partial_{i}^{*}(x)$. This dendrex can
be pictured as\[
\xymatrix{ & *{}\ar@{-}[d]^{i^{''}}\\
*{}\ar@{-}[dr]_{1}\ar@{}[r]|{\quad\,\,\, id} & *{\bullet}\ar@{-}[d]^{i'} & *{}\ar@{-}[dl]^{n}\\
\ar@{}[r]|{\quad\,\,\, g} & *{\bullet}\ar@{-}[d]^{0}\\
& *{}}
\]
with inner face:\[
\xymatrix{*{}\ar@{-}[dr]_{1} & *{}\ar@{-}[d]^{i^{''}} & *{}\ar@{-}[dl]^{n}\\
\ar@{}[r]|{\quad\,\,\, f} & *{\bullet}\ar@{-}[d]^{0}\\
& *{}}
\]
and is thus a homotopy from $g$ to $f$ along $i$, so that $g\sim_{i}f$.
For $i=0$ a similar proof works.
To prove transitivity let $f\sim_{i}g$ and $g\sim_{i}h$ for $1\le i\le n$.
Let $H_{fg}$ be a homotopy from $f$ to $g$ and let $H_{gh}$ be
a homotopy from $g$ to $h$. We again consider the tree $T$:\[
\xymatrix{\\ & *{}\ar@{-}[d]^{i^{''}}\\
& *{\bullet}\ar@{-}[d]^{i^{'}}\\
*{}\ar@{-}[dr]_{1} & *{\bullet}\ar@{-}[d]^{i} & *{}\ar@{-}[dl]^{n}\\
& *{\bullet}\ar@{-}[d]^{0}\\
& *{}}
\]
This time we look at $\Lambda^{i^{'}}[T]$. The following describes
a map $\Lambda^{i^{'}}[T]\rightarrow X$ in $X$:\[
\begin{array}{ccc}
\xymatrix{ & H_{i}\\
& *{}\ar@{-}[d]^{i^{''}}\\
\ar@{}[r]|{\quad\quad id} & *{\bullet}\ar@{-}[d]^{i^{'}}\\
\ar@{}[r]|{\quad\quad id} & *{\bullet}\ar@{-}[d]^{i}\\
& *{}}
\quad & \xymatrix{ & H_{gh}\\
& *{}\ar@{-}[d]^{i^{''}}\\
*{}\ar@{-}[dr]_{1}\ar@{}[r]|{\quad\quad id} & *{\bullet}\ar@{-}[d]^{i^{'}} & *{}\ar@{-}[dl]^{n}\\
\ar@{}[r]|{\quad\quad g} & *{\bullet}\ar@{-}[d]^{0}\\
& *{}}
\quad & \xymatrix{ & H_{fg}\\
& *{}\ar@{-}[d]^{i'}\\
*{}\ar@{-}[dr]_{1}\ar@{}[r]|{\quad\quad id} & *{\bullet}\ar@{-}[d]^{i} & *{}\ar@{-}[dl]^{n}\\
\ar@{}[r]|{\quad\quad f} & *{\bullet}\ar@{-}[d]^{0}\\
& *{}}
\end{array}\]
with inner faces being:\[
\xyC{34pt}\begin{array}{ccc}
\xymatrix{ & *{}\ar@{-}[d]^{i^{''}}\\
\ar@{}[r]|{\,\,\quad\quad id} & *{\bullet}\ar@{-}[d]^{i}\\
& *{}}
\quad & \xymatrix{*{}\ar@{-}[dr]_{1} & *{}\ar@{-}[d]^{i^{''}} & *{}\ar@{-}[dl]^{n}\\
\ar@{}[r]|{\quad\quad h} & *{\bullet}\ar@{-}[d]^{0}\\
& *{}}
\quad & \xymatrix{*{}\ar@{-}[dr]_{1} & *{}\ar@{-}[d]^{i^{'}} & *{}\ar@{-}[dl]^{n}\\
\ar@{}[r]|{\quad\quad g} & *{\bullet}\ar@{-}[d]^{0}\\
& *{}}
\end{array}\]
Let $x$ be a filler for that horn and let $H_{fh}=\partial_{i^{'}}^{*}(x)$.
This dendrex can be pictured as follows:\[
\xymatrix{ & *{}\ar@{-}[d]^{i^{''}}\\
*{}\ar@{-}[dr]_{1}\ar@{}[r]|{\quad\quad id} & *{\bullet}\ar@{-}[d]^{i} & *{}\ar@{-}[dl]^{n}\\
\ar@{}[r]|{\quad\quad f} & *{\bullet}\ar@{-}[d]^{0}\\
& *{}}
\]
with inner face:\[
\xymatrix{*{}\ar@{-}[dr]_{1} & *{}\ar@{-}[d]^{i^{''}} & *{}\ar@{-}[dl]^{n}\\
\ar@{}[r]|{\quad\quad h} & *{\bullet}\ar@{-}[d]^{0}\\
& *{}}
\]
and is thus a homotopy from $f$ to $h$ so that $f\sim_{i}h$. The
proof for $i=0$ is similar.
\end{proof}
\begin{lem}
Let $X$ be an inner Kan complex. The relations $\sim_{0},\cdots,\sim_{n}$
on $X_{C_{n}}$ are all equal.
\end{lem}
\begin{rem}
On the basis of this lemma, we will later just write $f\sim g$ instead
of $f\sim_{i}g$.
\end{rem}
\begin{proof}
Suppose $H:f\sim_{i}g$ for $1\le i\le n$ and let $1\le i<j\le n$.
We consider the tree T:\[
\xyC{15pt}\xymatrix{ & *{}\ar@{-}[d]^{i^{'}} & & *{}\ar@{-}[d]^{j^{'}}\\
*{}\ar@{-}[drr]_{1} & *{\bullet}\ar@{-}[dr]^{i} & & *{\bullet}\ar@{-}[dl]_{j} & *{}\ar@{-}[dll]^{n}\\
& & *{\bullet}\ar@{-}[d]^{0}\\
& & *{}}
\]
and the inner horn $\Lambda^{i}[T]$. The following then describes
a map $\Lambda^{i}[T]\rightarrow X$:\\
\\
\[
\begin{array}{ccc}
\xyC{15pt}\xymatrix{ & & H_{f}^{j}\\
& & & *{}\ar@{-}[d]_{j^{'}}\\
*{}\ar@{-}[drr]_{1} & *{}\ar@{-}[dr]^{i} & & *{\bullet}\ar@{-}[dl]_{j} & *{}\ar@{-}[dll]^{n}\ar@{}[l]|{id\,\,\,}\\
& \ar@{}[r]_{\quad\,\,\, f} & *{\bullet}\ar@{-}[d]^{0}\\
& & *{}}
\quad & \xymatrix{ & & H\\
& *{}\ar@{-}[d]^{i^{'}}\\
*{}\ar@{-}[drr]_{1}\ar@{}[r]|{\,\,\, id} & *{\bullet}\ar@{-}[dr]^{i} & & *{}\ar@{-}[dl]_{j} & *{}\ar@{-}[dll]^{n}\\
& \ar@{}[r]_{\,\,\,\quad f} & *{\bullet}\ar@{-}[d]^{0}\\
& & *{}}
\quad & \xymatrix{ & & H_{f}^{i}\\
& *{}\ar@{-}[d]^{i^{'}}\\
*{}\ar@{-}[drr]_{1}\ar@{}[r]|{\,\,\, id} & *{\bullet}\ar@{-}[dr]^{i} & & *{}\ar@{-}[dl]_{j^{'}} & *{}\ar@{-}[dll]^{n}\\
& \ar@{}[r]_{\quad\,\,\, f} & *{\bullet}\ar@{-}[d]^{0}\\
& & *{}}
\end{array}\]
where $H_{f}^{j}:f\sim_{j}f$ and $H_{f}^{i}:f\sim_{i}f$. The inner
faces of the three dendrices are\[
\begin{array}{ccc}
\xyC{15pt}\xymatrix{*{}\ar@{-}[drr]_{1} & *{}\ar@{-}[dr]^{i} & \ar@{}[d(0.7)]^{j^{'}} & *{}\ar@{-}[dl] & *{}\ar@{-}[dll]^{n}\\
& \ar@{}[r]_{\quad f} & *{\bullet}\ar@{-}[d]^{0}\\
& & *{}}
& \quad\xymatrix{*{}\ar@{-}[drr]_{1} & *{}\ar@{-}[dr]^{i^{'}} & & *{}\ar@{-}[dl]_{j} & *{}\ar@{-}[dll]^{n}\\
& \ar@{}[r]_{\quad g} & *{\bullet}\ar@{-}[d]^{0}\\
& & *{}}
\quad & \xymatrix{*{}\ar@{-}[drr]_{1} & *{}\ar@{-}[dr]^{i^{'}} & \ar@{}[d(0.7)]^{j^{'}} & *{}\ar@{-}[dl] & *{}\ar@{-}[dll]^{n}\\
& \ar@{}[r]_{\quad f} & *{\bullet}\ar@{-}[d]^{0}\\
& & *{}}
\end{array}\]
Let $x$ be a filler for this horn, then $\partial_{i}^{*}(x)$ is
the following dendrex\[
\xymatrix{ & & & *{}\ar@{-}[d]_{j^{'}}\\
*{}\ar@{-}[drr]_{1} & *{}\ar@{-}[dr]^{i^{'}} & & *{\bullet}\ar@{-}[dl]_{j} & *{}\ar@{-}[dll]^{n}\ar@{}[l]|{id\,\quad\,}\\
& \ar@{}[r]_{\quad\,\, g} & *{\bullet}\ar@{-}[d]^{0}\\
& & *{}}
\]
with inner face:\[
\xymatrix{*{}\ar@{-}[drr]_{1} & *{}\ar@{-}[dr]^{i^{'}} & \ar@{}[d(0.7)]^{j^{'}} & *{}\ar@{-}[dl] & *{}\ar@{-}[dll]^{n}\\
& \ar@{}[r]_{\quad\,\, f} & *{\bullet}\ar@{-}[d]^{0}\\
& & *{}}
\]
and is thus a homotopy from $g$ to $f$ along the $j$-th branch.
Thus $g\sim_{j}f$ and so $f\sim_{j}g$ as well. The other cases to
be considered follow in a similar way.
\end{proof}
Given an inner Kan complex $X$ and vertices $x_{1},\cdots,x_{n},x\in X_{\eta}$,
let us write \[
X(x_{1},\cdots,x_{n};x)\subseteq X(C_{n})\]
for the set of dendrices $x$ of shape $C_{n}$ with $0^{*}(x)=x$
and $i^{*}(x)=x_{i}$ for $i=0,\cdots,n$. Here $i:\eta\rightarrow C_{n}$
denotes the map in $\Omega$ sending the unique edge of $\eta$ to
the one of $C_{n}$ with name $i$. The equivalence relation $\sim$
on $X(C_{n})$ given by the preceding lemma defines a quotient of
$X(C_{n})$ which we will denote by \[
Ho(X)(x_{1},\cdots,x_{n};x)=X(x_{1},\cdots,x_{n};x)/\sim.\]
This defines a coloured collection $Ho(X)$, and a canonical quotient
map of collections $Sk_{1}(X)\rightarrow Ho(X)$. We will now proceed
to prove the following.
\begin{prop}
There is a unique structure of a (symmetric, coloured) operad on $Ho(X)$
for which the map of collections $Sk_{1}(X)\rightarrow Ho(X)$ extends
to a map of dendroidal sets $X\rightarrow N_{d}(Ho(X))$. The latter
map is an isomorphism whenever $X$ is a \emph{strict} inner Kan \emph{}complex\emph{. }
\end{prop}
To prepare for the proof of this proposition, we begin by defining
the composition operations $\circ_{i}$ of the operad $Ho(X)$. Let
$X$ be an inner Kan complex and let $f\in X_{C_{n}}$ and $g\in X_{C_{m}}$
be two dendrices in $X$. We will say that a dendrex $h\in X_{C_{n+m-1}}$
is a $\circ_{i}$-composition of $f$ and $g$ if there is a dendrex
$\gamma$ in $X$ as follows:\[
\xyC{25pt}\xyR{25pt}\xymatrix{*{}\ar@{-}[dr]_{1^{'}} & & *{}\ar@{-}[dl]^{m^{'}}\\
*{}\ar@{-}[dr]_{1}\ar@{}[r]|{\,\,\quad g} & *{\bullet}\ar@{-}[d]^{i} & *{}\ar@{-}[dl]^{n}\\
\ar@{}[r]|{\,\,\quad f} & *{\bullet}\ar@{-}[d]^{0}\\
& *{}}
\]
with inner face\[
\xymatrix{*{}\ar@{-}[drr]_{1} & *{}\ar@{-}[dr]^{i^{'}} & \ar@{}[d(0.7)]^{m^{'}\,} & *{}\ar@{-}[dl] & *{}\ar@{-}[dll]^{n}\\
& \ar@{}[r]|{\quad\,\, h} & *{\bullet}\ar@{-}[d]^{0}\\
& & *{}}
\]
We will denote this situation by $h\sim f\circ_{i}g$ and call $\gamma$
a \emph{witness} for the composition.
\begin{rem}
Notice that for $1\le i\le n$ we have by definition that $H:f\sim_{i}g$
iff $H$ is a witness for the composition $g\sim f\circ_{i}id$. Similarly
for $i=0$ we have that $H:f\sim_{0}g$ iff $H$ is a witness for
the composition $g\sim id\circ f$.
\end{rem}
\begin{lem}
In an inner Kan complex $X$, if $h\sim f\circ_{i}g$ and $h'\sim f\circ_{i}g$
then $h\sim h'$.
\end{lem}
\begin{proof}
Let $\gamma$ be a witness for the composition $h\sim f\circ_{i}g$
and $\gamma'$ one for the composition $h'\sim f\circ_{i}g$. We consider
the tree $T$:\[
\xymatrix{*{}\ar@{-}[d]^{1^{''}}\\
*{\bullet}\ar@{-}[dr]_{1^{'}} & & *{}\ar@{-}[dl]^{n^{'}}\\
*{}\ar@{-}[dr]_{1} & *{\bullet}\ar@{-}[d]^{i} & *{}\ar@{-}[dl]^{n}\\
& *{\bullet}\ar@{-}[d]^{0}\\
& *{}}
\]
and the inner horn $\Lambda^{i}[T]$. Let $H_{g}:g\sim_{i}g$ and
consider the following map $\Lambda^{i}[T]\rightarrow X$\[
\begin{array}{ccc}
\xymatrix{ & \gamma\\
*{}\ar@{-}[dr]_{1^{'}} & & *{}\ar@{-}[dl]^{m^{'}}\\
*{}\ar@{-}[dr]_{1}\ar@{}[r]|{\quad\,\, g} & *{\bullet}\ar@{-}[d]^{i} & *{}\ar@{-}[dl]^{n}\\
\ar@{}[r]|{\quad\,\, f} & *{\bullet}\ar@{-}[d]^{0}\\
& *{}}
\quad & \xymatrix{ & \gamma'\\
*{}\ar@{-}[dr]_{1^{''}} & & *{}\ar@{-}[dl]^{m^{'}}\\
*{}\ar@{-}[dr]^{1}\ar@{}[r]|{\quad\,\,\, g} & *{\bullet}\ar@{-}[d]^{i} & *{}\ar@{-}[dl]^{n}\\
\ar@{}[r]|{\quad\,\,\, f} & *{\bullet}\ar@{-}[d]^{0}\\
& *{}}
\quad & \xymatrix{ & H_{g}\\
*{}\ar@{-}[d]^{1^{''}}\\
*{\bullet}\ar@{-}[dr]_{1^{'}} & \ar@{}[l]|{id\quad\quad} & *{}\ar@{-}[dl]^{m^{'}}\\
\ar@{}[r]|{\quad\,\,\, g} & *{\bullet}\ar@{-}[d]^{1}\\
& *{}}
\end{array}\]
with inner faces\[
\begin{array}{ccc}
\xymatrix{*{}\ar@{-}[drr]_{1} & *{}\ar@{-}[dr]^{1^{'}} & & *{}\ar@{-}[dl]_{m^{'}} & *{}\ar@{-}[dll]^{n}\\
& \ar@{}[r]|{\quad h} & *{\bullet}\ar@{-}[d]^{0}\\
& & *{}}
\quad & \xymatrix{*{}\ar@{-}[drr]_{1} & *{}\ar@{-}[dr]^{1^{''}} & & *{}\ar@{-}[dl]_{m^{'}} & *{}\ar@{-}[dll]^{n}\\
& \ar@{}[r]|{\quad h'} & *{\bullet}\ar@{-}[d]^{0}\\
& & *{}}
\quad & \xymatrix{*{}\ar@{-}[dr]_{1^{''}} & & *{}\ar@{-}[dl]^{m^{'}}\\
\ar@{}[r]|{\quad g} & *{\bullet}\ar@{-}[d]^{1}\\
& *{}}
\end{array}\]
Let $x$ be a filler for this horn. The face $\partial_{i}^{*}x$
is then the dendrex\[
\xymatrix{ & *{}\ar@{-}[d]_{i^{''}}\\
*{}\ar@{-}[drr]_{1}\ar@{}[r]|{\,\,\quad id} & *{\bullet}\ar@{-}[dr]^{i^{'}} & & *{}\ar@{-}[dl]_{m^{'}} & *{}\ar@{-}[dll]^{n}\\
& \ar@{}[r]|{\,\,\quad h} & *{\bullet}\ar@{-}[d]^{0}\\
& & *{}}
\]
whose inner face is \[
\xymatrix{*{}\ar@{-}[drr]_{1} & *{}\ar@{-}[dr]^{1^{''}} & & *{}\ar@{-}[dl]_{m^{'}} & *{}\ar@{-}[dll]^{n}\\
& \ar@{}[r]_{\quad\,\, h^{'}} & *{\bullet}\ar@{-}[d]^{0}\\
& & *{}}
\]
which proves that $h\sim h'$.
\end{proof}
\begin{lem}
In an inner Kan complex $X$, let $f\sim f'$ and $g\sim g'$. If
$h\sim f\circ_{i}g$ and $h'\sim f'\circ_{i}g'$ then $h\sim h'$.
\end{lem}
\begin{proof}
Let $H$ be a homotopy from $f$ to $f'$ along the edge $i$, $H'$
a homotopy from $g'$ to $g$ along the root, and $\gamma$ a witness
for the composition $h\sim f\circ_{i}g$. We now consider the tree
T:\[
\xymatrix{*{}\ar@{-}[dr]_{1^{''}} & & *{}\ar@{-}[dl]^{m^{''}}\\
& *{\bullet}\ar@{-}[d]^{i^{'}}\\
*{}\ar@{-}[dr]_{1} & *{\bullet}\ar@{-}[d]^{i} & *{}\ar@{-}[dl]^{n}\\
& *{\bullet}\ar@{-}[d]^{0}\\
& *{}}
\]
and the inner horn $\Lambda^{i}[T]$. The following is then a map
$\Lambda^{i}[T]\rightarrow X$ in $X$:\\
\\
\[
\xyC{25pt}\xyR{25pt}\begin{array}{ccc}
\xymatrix{ & H\\
& *{}\ar@{-}[d]^{i^{'}}\\
*{}\ar@{-}[dr]_{1}\ar@{}[r]|{\,\,\quad id} & *{\bullet}\ar@{-}[d]^{i} & *{}\ar@{-}[dl]^{n}\\
\ar@{}[r]|{\,\,\quad f} & *{\bullet}\ar@{-}[d]^{0}\\
& *{}}
\quad & \xymatrix{ & \gamma\\
*{}\ar@{-}[dr]_{i^{''}} & & *{}\ar@{-}[dl]^{m^{''}}\\
*{}\ar@{-}[dr]_{1}\ar@{}[r]|{\,\,\quad g} & *{\bullet}\ar@{-}[d]^{i} & *{}\ar@{-}[dl]^{n}\\
\ar@{}[r]|{\,\,\quad f} & *{\bullet}\ar@{-}[d]^{0}\\
& *{}}
\quad & \xymatrix{ & H'\\
*{}\ar@{-}[dr]_{1^{''}} & & *{}\ar@{-}[dl]^{m^{''}}\\
\ar@{}[r]|{\,\,\quad g^{'}} & *{\bullet}\ar@{-}[d]^{i^{'}}\\
\ar@{}[r]|{\,\,\quad id} & *{\bullet}\ar@{-}[d]^{i}\\
& *{}}
\end{array}\]
with inner faces:\[
\xyC{25pt}\xyR{25pt}\begin{array}{ccc}
\xymatrix{*{}\ar@{-}[dr]_{1} & *{}\ar@{-}[d]^{i^{'}} & *{}\ar@{-}[dl]^{n}\\
\ar@{}[r]|{\quad\,\, f^{'}} & *{\bullet}\ar@{-}[d]^{0}\\
& *{}}
\quad & \xymatrix{*{}\ar@{-}[drr]_{1} & *{}\ar@{-}[dr]^{i^{''}} & \ar@{}[d(0.7)]^{m^{''}} & *{}\ar@{-}[dl] & *{}\ar@{-}[dll]^{n^{''}}\\
& \ar@{}[r]|{\quad\, h} & *{\bullet}\ar@{-}[d]^{0}\\
& & *{}}
\quad & \xymatrix{*{}\ar@{-}[dr]_{1^{''}} & & *{}\ar@{-}[dl]^{m^{''}}\\
\ar@{}[r]|{\quad\, g} & *{\bullet}\ar@{-}[d]^{i}\\
& *{}}
\end{array}\]
The missing face of a filler for this horn is then:\[
\xyC{25pt}\xyR{25pt}\xymatrix{*{}\ar@{-}[dr]_{1^{''}} & & *{}\ar@{-}[dl]^{m^{''}}\\
*{}\ar@{-}[dr]_{1}\ar@{}[r]_{\quad\quad g^{'}} & *{\bullet}\ar@{-}[d]^{i^{'}} & *{}\ar@{-}[dl]^{n}\\
\ar@{}[r]_{\quad\quad f^{'}} & *{\bullet}\ar@{-}[d]^{0}\\
& *{}}
\]
with inner face\[
\xymatrix{*{}\ar@{-}[drr]_{1} & *{}\ar@{-}[dr]^{1^{''}} & & *{}\ar@{-}[dl]_{m^{''}} & *{}\ar@{-}[dll]^{n}\\
& \ar@{}[r]|{\quad\,\,\, h} & *{\bullet}\ar@{-}[d]^{0}\\
& & *{}}
\]
which proves that $h\sim f'\circ_{i}g'$, and thus by the previous
result also that $h\sim h'$.
\end{proof}
We now proceed to prove Proposition 5.6:
\begin{proof}
(of Proposition 5.6) Lemma 5.8 implies that for \[
[f]\in Ho(X)(x_{1},\cdots,x_{n};x)\]
and \[
[g]\in Ho(X)(y_{1},\cdots,y_{m};x_{i})\]
the assignment \[
[f]\circ_{i}[g]=[f\circ_{i}g]\]
is well-defined. This provides the $\circ_{i}$ operations in the
operad $Ho(X)$. The $\Sigma_{n}$ actions are defines as follows.
Given a permutation $\sigma\in\Sigma_{n}$ let $\sigma:C_{n}\rightarrow C_{n}$
be the obvious induced map in $\Omega$. The map $\sigma^{*}:X_{C_{n}}\rightarrow X_{C_{n}}$
restricts to a function \[
\sigma^{*}:X(x_{1},\cdots,x_{n};x)\rightarrow X(x_{\sigma(1)},\cdots,x_{\sigma(n)};x)\]
and it is trivial to verify that this map respects the homotopy relation.
We thus obtain a map \[
\sigma^{*}Ho(X)(x_{1},\cdots,x_{n};x)\rightarrow Ho(X)(x_{\sigma(1)},\cdots,x_{\sigma(n)};x).\]
We now need to show that these structure maps make the coloured collection
$Ho(X)$ into an operad. The verification is simple and we exemplify
it by proving associativity. Let $[f]\in Ho(X)(x_{1},\cdots,x_{n};x)$,
$[g]\in Ho(X)(y_{1},\cdots,y_{m};x_{i})$ and $[h]\in Ho(X)(z_{1},\cdots,z_{k};y_{m})$.
We need to prove that $[f]\circ([g]\circ[h])=([f]\circ[g])\circ[h]$
(with the obvious choice for subscripts on the $\circ$) which is
the same as showing that $f\circ(g\circ h)\sim(f\circ g)\circ h$
for any choice for compositions $\psi\sim g\circ h$ and $\varphi\sim f\circ g$.
Consider the tree $T$ given by\[
\xymatrix{*{}\ar@{-}[dr]_{1^{''}} & & *{}\ar@{-}[dl]^{k^{''}}\\
*{}\ar@{-}[dr]_{1^{'}} & *{\bullet}\ar@{-}[d]^{j} & *{}\ar@{-}[dl]^{m^{'}}\\
*{}\ar@{-}[dr]_{1} & *{\bullet}\ar@{-}[d]^{i} & *{}\ar@{-}[dl]^{n}\\
& *{\bullet}\ar@{-}[d]^{0}\\
& *{}}
\]
and consider the anodyne extension $\Lambda^{I}[T]\rightarrow\Omega[T]$,
cf. Lemma 5.1. The two given compositions $\psi\sim g\circ h$ and
$\varphi\sim f\circ g$ define a map $\Lambda^{I}[T]\rightarrow X$
depicted by\[
\begin{array}{ccc}
\xymatrix{*{}\ar@{-}[dr]_{1^{'}} & & *{}\ar@{-}[dl]^{m^{'}}\\
*{}\ar@{-}[dr]_{1}\ar@{}[r]|{\quad g} & *{\bullet}\ar@{-}[d]^{i} & *{}\ar@{-}[dl]^{n}\\
\ar@{}[r]|{\quad f} & *{\bullet}\ar@{-}[d]^{0}\\
& *{}}
& \xymatrix{\\and\\
}
& \xymatrix{*{}\ar@{-}[dr]_{1^{''}} & & *{}\ar@{-}[dl]^{k^{''}}\\
*{}\ar@{-}[dr]_{1^{'}}\ar@{}[r]|{\quad h} & *{\bullet}\ar@{-}[d]^{j} & *{}\ar@{-}[dl]^{m^{'}}\\
\ar@{}[r]|{\quad g} & *{\bullet}\ar@{-}[d]^{0}\\
& *{}}
\end{array}\]
whose inner faces are respectively $\psi$ and $\varphi$. Let $x\in X_{T}$
be a dendrex extending this map. Let $c:C_{m}\rightarrow T$ be the
map obtained by contracting both $i$ and $j$ and $\rho=c^{*}x$.
It now follows that $\partial_{i}^{*}x$ is a witness for the the
composition $\rho\sim\psi\circ h$ and $\partial_{j}^{*}x$ is a witness
for the composition $\rho\sim f\circ\varphi$. That proves associativity.
The other axioms for an operad follow in a similar manner.
Next, let us show that the quotient map $q:Sk_{1}(X)\rightarrow Ho(X)$
extends to a map $q:X\rightarrow N_{d}(Ho(X))$ of dendroidal sets.
Since we already know that $N_{d}(X)$ is $2$-coskeletal, it suffices
to give its values for dendrices $x\in X_{T}$ where $T$ is a tree
with two vertices. Let $e$ be the inner edge of this tree. Then $\xymatrix{*++{\Lambda^{e}[T]}\ar@{>->}[r] & \Omega[T]\ar[r]^{x} & X}
$ factors through $Sk_{1}(X)$, so its composition $\Lambda^{e}[T]\rightarrow N_{d}(Ho(X))$
with $q$ has a unique extension (Proposition 5.3), which we take
to be $q(x):\Omega[T]\rightarrow N_{d}(Ho(X))$. This defines $q:Sk_{2}(X)\rightarrow Sk_{2}(N_{d}(Ho(X))),$
and hence all of $q:X\rightarrow N_{d}(Ho(X))$ by $2$-coskeletality,
as said.
Finally, when $X$ is itself a strict inner Kan complex, then the
homotopy relation is the identity relation, so $Sk_{1}(X)\rightarrow Ho(X)$
is the identity map. Since $X$ and $N_{d}(Ho(X))$ are now both strict
inner Kan complexes, the extension $q:X\rightarrow N_{d}(Ho(X))$
is an isomorphism.
\end{proof}
The following Proposition, together with Proposition 6.6, now provide
the proof of Theorem 6.1.
\begin{prop}
For any inner Kan complex $X$, the natural map $\tau_{d}(X)\rightarrow Ho(X)$
is an isomorphism of operads.
\end{prop}
\begin{proof}
It suffices to prove that the map $q:X\rightarrow N_{d}(Ho(X))$ of
Proposition 6.6 has the universal property of the unit of the adjunction.
This means that for any operad $\mathcal{P}$ and any map $\varphi:X\rightarrow N_{d}(\mathcal{P})$,
there is a unique map of operads $\psi:Ho(X)\rightarrow\mathcal{P}$
for which $N_{d}(\psi)q=\varphi$. But $\varphi$ induces a map $Ho(X)\rightarrow Ho(N_{d}(\mathcal{P}))$
for which\[
\xymatrix{Sk_{1}(X)\ar[r]^{\varphi}\ar[d]^{q_{X}} & Sk_{1}N_{d}(\mathcal{P})\ar[d]^{q_{\mathcal{P}}}\\
Ho(X)\ar[r]^{Ho(\varphi)} & Ho(N_{d}(\mathcal{P}))}
\]
commutes, and $Ho(N_{d}(\mathcal{P}))=\mathcal{P}$ while $q_{\mathcal{P}}$
is the identity as we have seen in (the proof of) Proposition 6.6.
So $Ho(\varphi)$ in fact defines a map $\psi:Ho(X)\rightarrow\mathcal{P}$
of collections. It is easily seen that $\psi$ is a map of operads.
It is unique because $q_{X}$ is surjective.
\end{proof}
\section{Homotopy coherent nerves of operads}
In this section, we assume $\mathcal{E}$ is a monoidal model category
with a cofibrant unit $I$. We also assume that $\mathcal{E}$ is
equipped with an \emph{interval} in the sense of \cite{GenWCon}.
Such an interval is given by maps \[
\xymatrix{I\ar@<2pt>[r]^{0}\ar@<-2pt>[r]_{1} & H\ar[r]^{\epsilon} & I}
\]
and \[
\xymatrix{H\otimes H\ar[r]^{\,\,\,\,\,\,\vee} & H}
\]
satisfying certain conditions. In particular, $H$ is an interval
in Quillen's sense (\cite{homotopicalAlg}), so $0$ and $1$ together
define a cofibration $I\coprod I\rightarrow H$, and $\epsilon$ is
a weak equivalence. Such an interval $H$ allows one to construct
for each (coloured) operad $\mathcal{P}$ in $\mathcal{E}$ a ''Boardman-Vogt''
resolution $W_{H}(\mathcal{P})\rightarrow\mathcal{P}$. Each operad
in $Set$ can be viewed as an operad in $\mathcal{E}$ (via the functor
$Set\rightarrow\mathcal{E}$ which preserves coproducts and sends
the one-point set to $I$), and hence has such a Boardman-Vogt resolution.
When we apply this to the operads $\Omega(T)$, we obtain the \emph{homotopy
coherent dendroidal nerve $hcN_{d}(\mathcal{P})$} of any operad $\mathcal{P}$
in $\mathcal{E}$, as the dendroidal set given by \[
hcN_{d}(\mathcal{P})_{T}=Hom(W_{H}(\Omega(T)),\mathcal{P})\]
where the Hom is that of operads in $\mathcal{E}$. See \cite{DenSet}
for a more detailed description and examples. Our goal here is to
prove the following result.
\begin{thm}
Let $\mathcal{P}$ be an operad in $\mathcal{E}$, with the property
that for each sequence $c_{1},\cdots,c_{n};c$ of colours of $\mathcal{P}$,
the object $\mathcal{P}(c_{1},\cdots,c_{n};c)$ is fibrant. Then $hcN_{d}(\mathcal{P})$
is an inner Kan complex.
\end{thm}
\begin{rem}
As explained in \cite{DenSet}, our construction of the dendroidal
homotopy coherent nerve specializes to that of the homotopy coherent
nerve of an $\mathcal{E}$-enriched category, and for the case where
$\mathcal{E}$ is the category of topological spaces or simplicial
sets, one recovers the classical definition (\cite{CordierPorter}). In particular,
as a special case of Theorem 7.1, one obtains that for an $\mathcal{E}$-enriched
category with fibrant Hom objects (in other words, for a locally fibrant
$\mathcal{E}$-enriched category), its homotopy coherent nerve is
a quasi-category in the sense of Joyal. This result was proved, for
the case where $\mathcal{E}$ is simplicial sets, by Cordier and Porter
in \cite{CordierPorter}.
\end{rem}
Before embarking on the proof of Theorem 7.1, we need to be a bit
more explicit about the operads of the form $W_{H}\Omega(T)$ involved
in the definition of the homotopy coherent nerve. Recall first of
all the functor\[
Symm:Operad(\mathcal{E})_{\pi}\rightarrow Operad(\mathcal{E})\]
which is left adjoint to the forgetful functor from symmetric operads
to non-symmetric (i.e., planar) ones. If $T$ is an object in $\Omega$
and $\bar{T}$ is a chosen planar representative of $T$, then $\bar{T}$
naturally describes a planar operad $\Omega(\bar{T})$ for which $\Omega(T)=Symm(\Omega(\bar{T}))$.
Since the $W$-construction commutes with symmetrization (as one readily
verifies), it follows that \[
W_{H}(T)=Symm(W_{H}\Omega(\bar{T})).\]
This latter operad $W_{H}\Omega(\bar{T})$ is easily described explicitly.
The colours of \[
W_{H}(\Omega(\bar{T}))\]
are the colours of $\Omega(\bar{T})$, i.e., the edges of $T$. By
a \emph{signature,} we mean a sequence $e_{1},\cdots,e_{n};e_{0}$
of edges. Given a signature $\sigma=(e_{1},\cdots,e_{n};e_{0})$,
we have that $W_{H}(\Omega(\bar{T}))(\sigma)=0$ whenever $\Omega(\bar{T})(\sigma)=\phi$.
And if $\Omega(\bar{T})(\sigma)\ne\phi$, there is a subtree $T_{\sigma}$
of $T$ (and a corresponding planar subtree $\bar{T}_{\sigma}$ of
$\bar{T}$) whose leaves are $e_{1},\cdots,e_{n},$ and whose root
is $e_{0}$. Then \[
W_{H}\Omega(\bar{T})(e_{1},\cdots,e_{n};e_{0})=\bigotimes_{f\in i(\sigma)}H,\]
where $i(\sigma)$ is the set of \emph{inner} edges of $T_{\sigma}$
(or of $\bar{T}_{\sigma}$). (This last tensor product is to be thought
of as the ''space'' of assignments of lengths to inner edges in
$\bar{T}_{\sigma}$; it is the unit if $i(\sigma)$ is empty.)
\begin{rem}
The composition operations in the operad $W_{H}\Omega(\bar{T})$ are
given in terms of the $\circ_{i}$-operations as follows. For signatures
$\sigma=(e_{1},\cdots,e_{n};e_{0})$ and $\rho=(f_{1},\cdots,f_{m};e_{i})$,
the composition map \[
\begin{array}{cccc}
\quad & \xymatrix{\Omega(\bar{T})(e_{1},\cdots,e_{n};e_{0})\otimes\Omega(\bar{T})(f_{1},\cdots,f_{m};e_{i})\ar[d]^{\circ_{i}} & \ar@{}[d]|{(1)}\\
\Omega(\bar{T})(e_{1},\cdots,e_{i-1},f_{1},\cdots,f_{m},e_{i+1},\cdots,e_{n};e_{0}) & *{}}
& \quad\end{array}\]
is the following one. The trees $\bar{T}_{\sigma}$ and $\bar{T}_{\rho}$
can be grafted along $e_{i}$ to form $\bar{T}_{\sigma}\circ_{e_{i}}\bar{T}_{\rho}$,
again a planar subtree of $\bar{T}$. In fact\[
\bar{T}_{\sigma}\circ_{e_{i}}\bar{T}_{\rho}=\bar{T}_{\sigma\circ_{i}\rho}\]
where $\sigma\circ_{i}\rho$ is the signature $(e_{1},\cdots,e_{i-1},f_{1},\cdots,f_{m},e_{i+1},\cdots,e_{n};e_{0})$,
and for the sets of inner edges we have \[
i(\sigma\circ_{i}\rho)=i(\sigma)\cup i(\rho)\cup\{ e_{i}\}.\]
The composition map in $(1)$ now is the map\[
\xymatrix{H^{\otimes i(\sigma)}\otimes H^{\otimes i(\rho)}\ar@{..>}[r]\ar[d]^{\cong} & H^{\otimes i(\sigma\circ_{i}\rho)}\ar[d]^{\cong}\\
H^{\otimes i(\sigma)\cup i(\rho)}\otimes I\ar[r]^{id\otimes1} & H^{\otimes i(\sigma)\cup i(\rho)}\otimes H}
\]
where $1:I\rightarrow H$ is one of the ''endpoints'' of the interval
$H$, as above.
This description of the operad $W_{H}\Omega(\bar{T})$ is functorial
in the planar tree $T$. In particular, we note that for an inner
edge $e$ of $T$, the tree $T/e$ inherits a planar structure $\overline{T/e}$
from $\bar{T}$, and $W_{H}\Omega(\overline{T/e})\rightarrow W_{H}\Omega(\bar{T})$
is the natural map assigning length $0$ to the edge $e$ whenever
it occurs (in a subtree given by a signature).
\end{rem}
\begin{proof}
(Of Theorem 7.1) Consider a tree $T$ and an inner edge $e$ in $T$.
We want to solve the extension problem\[
\xymatrix{*+++{\Lambda^{e}[T]}\ar[r]^{\varphi\,\,\,\,\,\,}\ar@{>->}[d] & hcN_{d}(\mathcal{P})\\
\Omega[T]\ar@{..>}[ur]}
\]
Fix a planar representative $\bar{T}$ of $T$. Then the desired map
$\psi:\Omega[T]\rightarrow hcN_{d}(\mathcal{P})$ corresponds to a
map of planar operads \[
\hat{\psi}:W_{H}\Omega(\bar{T})\rightarrow\mathcal{P}.\]
Each face $S$ of $T$ inherits a planar structure $\bar{S}$ from
$\bar{T}$, and the given map $\varphi:\Lambda^{e}[T]\rightarrow hcN_{d}(\mathcal{P})$
corresponds to a map of operads in $\mathcal{E}$,\[
\hat{\varphi}:W_{H}(\Lambda^{e}[T])\rightarrow\mathcal{P},\]
where $W_{H}(\Lambda^{e}[T])$ denotes the colimit of operads in $\mathcal{E}$,
\[
\begin{array}{ccc}
\quad\quad\quad\quad\quad\quad & W_{H}(\Lambda^{e}[T])=colim\, W(\Omega(\bar{S})) & \quad\quad\quad\quad\quad\quad(2)\end{array}\]
over all but one of the faces of $T$. In other words, $\varphi$
corresponds to a compatible family of maps \[
\hat{\varphi}_{S}:W_{H}(\Omega(\bar{S}))\rightarrow\mathcal{P}.\]
Let us now show the existence of an operad map $\hat{\psi}$ extending
the $\hat{\varphi}_{S}$ for all faces $S\ne T/e$. First, the colours
of $\Omega(\bar{T})$ are the same as those of the colimit in $(2)$,
so we already have a map $\psi_{0}=\varphi_{0}$ on colours:\[
\psi_{0}:(\textrm{Edges of }T)\rightarrow(\textrm{Colours of $\mathcal{P}$}).\]
Next, if $\sigma=(e_{1},\cdots,e_{n};e_{0})$ is a signature of $T$
for which $W_{H}(\Omega(\bar{T}))\ne\phi$, and if $T_{\sigma}\subseteq T$
is not all of $T$, then $T_{\sigma}$ is contained in an outer face
$S$ of $T$. So $W_{H}(\Omega(\bar{T}))(\sigma)=W_{H}(\Omega(\bar{T}_{\sigma}))(\sigma)=W_{H}(\Omega(\bar{S}))(\sigma)$,
and we already have a map \[
\hat{\varphi}_{S}(\sigma):W_{H}(\Omega(\bar{T}))(\sigma)\rightarrow\mathcal{P}(\sigma),\]
given by $\hat{\varphi}_{S}:W_{H}(\Omega(\bar{S}))\rightarrow\mathcal{P}$.
Thus, the only part of the operad map $\hat{\psi}:W_{H}(\Omega(\bar{T}))\rightarrow\mathcal{P}$
not determined by $\varphi$ is the one for the signature $\tau$
where $T_{\tau}=T$; i.e., $\tau=(e_{1},\cdots,e_{n};e_{0})$ where
$e_{1},\cdots,e_{n}$ are all the input edges of $\bar{T}$ (in the
planar order) and $e_{0}$ is the output edge. For this signature,
$\hat{\psi}(\tau)$ is to be a map\[
\hat{\psi}:W_{H}(\Omega(\bar{T})(\tau)=H^{\otimes i(\tau)}\rightarrow\mathcal{P}(\tau)\]
which (i) is compatible with the $\hat{\psi}(\sigma)=\hat{\varphi}_{S}(\sigma)$
for other signatures $\sigma;$ and (ii) together with these $\hat{\psi}(\sigma)$
respects operad composition. The first condition determines $\hat{\psi}(\tau)$
on the subobject of $H^{\otimes i(\tau)}$ which is given by a value
$0$ on one of the tensor-factors marked by an edge $e_{i}$ \emph{other}
than the given $e$. The second condition determines $\hat{\psi}(\tau)$
on the subobject of $H^{\otimes i(\tau)}$ which is given by a value
$1$ on one of the factors. Thus, if we write $1$ for the map $\xymatrix{*++{I}\ar@{>->}[r]^{1} & H}
$ and $\xymatrix{*++{\partial H}\ar@{>->}[r] & H}
$ for the map $I\coprod I\rightarrow H$, and define $\xymatrix{*++{\partial H^{\otimes k}}\ar@{>->}[r] & H^{\otimes k}}
$ by the Leibniz rule (i.e., $\partial(A\otimes B)=\partial(A)\otimes B\cup A\otimes\partial(b)$),
then the problem of finding $\hat{\psi}(\tau)$ comes down to an extension
problem of the form\[
\xymatrix{*+++{\partial(H^{\otimes i(\sigma)-\{ e\}}\otimes H)\cup H^{\otimes i(\sigma)-\{ e\}}\otimes I}\ar[r]\ar@{>->}[d] & \mathcal{P}(\tau)\\
H^{\otimes i(\sigma)-\{ e\}}\otimes H\ar[r]^{\cong} & H^{\otimes i(\sigma)}\ar@{..>}[u]^{\hat{\psi}(\sigma)}}
\]
This extension problem has a solution, because $\mathcal{P}(\tau)$
is fibrant by assumption, and because the left hand map is a trivial
cofibration (by repeated use of the push-out product axiom for monoidal
model categories). This concludes the proof of the theorem.
\end{proof}
\section{Grothendieck construction for dendroidal sets}
Let $\mathbb{S}$ be a Cartesian category. A functor $X:\mathbb{S}^{op}\rightarrow dSet$
is called a \emph{diagram} of dendroidal sets. In \cite{DenSet} a
construction was given of the dendroidal set $\int_{\mathbb{S}}X$.
This construction was then applied to the specific diagram of dendroidal
sets $X:Set^{op}\rightarrow dSet$, where for a set $A$, $X(A)$
was the dendroidal set of weak $n$-categories having $A$ as set
of objects. The dendroidal set $\int_{\mathbb{S}}X$ was defined to
be the dendroidal set of weak $n$-categories. Our aim in this section
is to prove that for a given diagram of dendroidal sets $X:\mathbb{S}^{op}\rightarrow dSet$,
if each $X(S)$ is an inner Kan complex then $\int_{\mathbb{S}}X$
is also an inner Kan complex. For the convenience of the reader we
repeat here the definition of $\int_{\mathbb{S}}X$.
It will be convenient to consider dendroidal collections. A dendroidal
collection is a collection of sets $X=\{ X_{T}\}_{T\in\Omega}$. Each
dendroidal set has an obvious underlying dendroidal collection. A
map of dendroidal collections $X\rightarrow Y$ is a collection of
functions $\{ X_{T}\rightarrow Y_{T}\}_{T\in\Omega}$. Given a Cartesian
category $\mathbb{S}$, consider the dendroidal nerve $N_{d}(\mathbb{S})$
where $\mathbb{S}$ is regarded as an operad via the Cartesian structure.
There is a natural way of associating an object of $\mathbb{S}$ with
each dendrex of $N_{d}(\mathbb{S})$. For a tree $T$ in $\Omega$,
let $leaves(T)$ be the set of leaves of $T$, and for a leaf $l$,
write $l:\eta\rightarrow T$ also for the map sending the unique edge
in $\eta$ to $l$ in $T$. Then, since $\mathbb{S}$ is assumed to
have finite products, each dendrex $t\in N_{d}(\mathbb{S})_{T}$ defines
an object \[
in(t)=\prod_{l\in leaves(T)}l^{*}(t)\]
in $\mathbb{S}$. Notice that if $\alpha:S\rightarrow T$ is a composition
of face maps, then by using the canonical symmetries and the projections
in $\mathbb{S}$ there is a canonical arrow $in(\alpha):in(t)\rightarrow in(\alpha^{*}t)$
for any $t\in X_{T}$.
\begin{defn}
Let $X:\mathbb{S}^{op}\rightarrow dSet$ be a diagram of dendroidal
sets. The dendroidal set $\int_{\mathbb{S}}X$ is defined as follows.
A dendrex $\Omega[T]\rightarrow\int_{\mathbb{S}}X$ is a pair $(t,x)$
such that $t\in N_{d}(\mathbb{S})_{T}$ and $x$ is a map of dendroidal
collections \[
x:\Omega[T]\rightarrow\coprod_{S\in ob(\mathbb{S})}X(S)\]
satisfying the following conditions. For each $r\in\Omega[T]_{R}$
(that is an arrow $r:R\rightarrow T$), we demand that $x(r)\in X(in(r^{*}t))$.
Furthermore we demand the following compatibility conditions to hold.
For any $r\in\Omega[T]_{R}$ and any map $\alpha:\xymatrix{*++{U}\ar@{>->}[r] & R}
$ in $\Omega$ \[
\alpha^{*}(x(r))=X(in(\alpha))x(\alpha^{*}(r)).\]
\end{defn}
\begin{thm}
Let $X:\mathbb{S}^{op}\rightarrow dSet$ be a diagram of dendroidal
sets. If for any $S\in ob(\mathbb{S})$ the dendroidal set $X(S)$
is a (strict) inner Kan complex then so is $\int_{\mathbb{S}}X$.
\end{thm}
\begin{proof}
Let $T$ be a tree and $e$ an inner edge. We consider the extension
problem\[
\xymatrix{*++{\Lambda^{e}[T]}\ar[r]\ar@{>->}[d] & \int_{\mathbb{S}}X\\
\Omega[T]\ar@{..>}[ur]}
\]
The horn $\Lambda^{e}[T]\rightarrow\int_{\mathbb{S}}X$ is given by
a compatible collection $\{(r,x_{R}):\Omega[R]\rightarrow\int_{\mathbb{S}}X\}_{R\ne T/e}$.
We wish to construct a dendrex $(t,x_{T}):\Omega[T]\rightarrow\int_{\mathbb{S}}X$
extending this family. First notice that the collection $\{ r\}_{R\ne T/e}$
is an inner horn $\Lambda^{e}[T]\rightarrow N_{d}(\mathbb{S})$ (actually
this horn is obtained by composing with the obvious projection $\int_{\mathbb{S}}X\rightarrow N_{d}(\mathbb{S})$
sending a dendrex $(t,x)$ to $t$). We already know $N_{d}(\mathbb{S})$
to be an inner Kan complex (actually a strict inner Kan complex) and
thus there is a (unique) filler $t\in N_{d}(\mathbb{S})_{T}$ for
the horn $\{ r\}_{R\ne T/e}$. We now wish to define a map of dendroidal
collections $x_{T}:\Omega[T]\rightarrow\coprod_{S\in ob(\mathbb{S})}X(S)$
that will extend the given maps $x_{R}$ for $R\ne T/e$. This condition
already determines the value of $x_{T}$ for any dendrex $r:U\rightarrow T$
other then $id:T\rightarrow T$ and $\alpha:T/e\rightarrow T$, since
for each such $r$, the tree $U$ factors through one of the faces
$R\ne T/e$. To determine $x_{T}(id_{T})$ and $x_{T}(\alpha)$ consider
the family $\{ y_{R}=x_{R}(id:R\rightarrow R)\}_{R\ne T/e}$. By definition
we have that $y_{R}\in X(in(r))_{R}$. For each such $R$ let $\alpha_{R}:R\rightarrow T$
be the corresponding face map in $\Omega$. Since $\alpha^{*}t=r$
we obtain the map $in(\alpha_{R}):in(r)\rightarrow in(t)$. We can
now pull back the collection $\{ y_{R}\}_{R\ne T/e}$ using $X(in(\alpha_{R}))$
to obtain a collection $\{ z_{R}=X(in(\alpha_{R}))(y_{R})\}_{R\ne T/e}$.
This collection is now a horn $\Lambda^{e}[T]\rightarrow X(in(T))$
(this follows from the compatibility conditions in the definition
of $\int_{\mathbb{S}}X$). Since $X(in(t))$ is inner Kan there is
a filler $u\in X(in(t))_{T}$ for that horn. We now define $x_{T}(id:T\rightarrow T)=u$
and $x_{T}(\alpha:T/e\rightarrow T)=\alpha^{*}(u)$. Notice that since
$e$ is inner we have that $in(t)=in(\alpha)$ and thus the image
of these dendrices are in the correct dendroidal set, namely $X(in(t))$.
It follows from our construction that this makes $(t,x_{T})$ a dendrex
$\Omega[T]\rightarrow\int_{\mathbb{S}}X$ which extends the given
horn. This concludes the proof.
\end{proof}
\section{The exponential property}
Our aim in this section is to prove the following theorem concerning
the closed monoidal structure of dendroidal sets.
\begin{thm}
Let $K$ and $X$ be dendroidal sets, and assume $X$ is normal. If $K$ is a (strict) inner
Kan complex, then so is $\underline{Hom}_{dSet}(X,K)$.
\end{thm}
The internal \emph{Hom} here is defined by the universal property,
giving a bijective correspondence between maps $Y\otimes X\rightarrow K$
and $Y\rightarrow\underline{Hom}(X,K)$ for any dendroidal set $Y$,
and natural in $Y$. We recall from Section 2 that $\otimes$ is defined
in terms of the Boardman-Vogt tensor product of operads. We remind
the reader that for two (coloured) operads $\mathcal{P}$ and $\mathcal{Q}$
with respective sets of colours $C$ and $D$, this tensor product
operad $\mathcal{P}\otimes_{BV}\mathcal{Q}$ has the product $C\times D$
as its set of colours, and is described in terms of generators and
relations as follows. The operations in $\mathcal{P}\otimes_{BV}\mathcal{Q}$
are generated by the operations\[
p\otimes d\in\mathcal{P}\otimes_{BV}\mathcal{Q}((c_{1},d),\cdots,(c_{n},d);(c,d))\]
for any $p\in\mathcal{P}(c_{1},\cdots,c_{n};c)$ and any $d\in D$,
and \[
c\otimes q\in\mathcal{P}\otimes_{BV}\mathcal{Q}((c,d_{1}),\cdots,(c,d_{m});(c,d))\]
for any $q\in\mathcal{Q}(d_{1},\cdots,d_{m};d)$ and any $c\in C$.
The \emph{relations} between these state, first of all, that for fixed
$c\in C$ and $d\in D$, the maps $p\mapsto p\otimes d$ and $c\mapsto c\otimes q$
are maps of operads. Secondly, there is an \emph{interchange law}
stating that, for $p$ and $q$ as above, the composition $p\otimes d(c\otimes q,\cdots,c\otimes q)$
in\[
\mathcal{P}\otimes_{BV}\mathcal{Q}((c_{1},d_{1}),\cdots,(c_{1},d_{m}),\cdots,(c_{n},d_{1}),\cdots,(c_{n},d_{m});(c,d))\]
and $c\otimes q(p\otimes d,\cdots,p\otimes d)$ in\[
\mathcal{P}\otimes_{BV}\mathcal{Q}((c_{1},d_{1}),\cdots,(c_{n},d_{1}),\cdots,(c_{1},d_{m}),\cdots,(c_{n},d_{m});(c,d))\]
are mapped to each other by the obvious permutation $\tau\in\Sigma_{n\times m}$
which puts the two sequences of input colours in the same order. The
tensor product of dendroidal sets is then uniquely determined (up
to isomorphism) by the fact that it preserves colimits in each variable
separately, together with the identity\[
\Omega[S]\otimes\Omega[T]=N_{d}(\Omega(S)\otimes_{BV}\Omega(T))\]
stated in Section 2, which gives the tensor product of two representable
dendroidal sets.
First of all, let us prove that Theorem 9.1 follows by a standard
argument from the following proposition.
\begin{prop}
For any two objects $S$ and $T$ of $\Omega$, and any inner edge
$e$ in $S$, the map\[
\xymatrix{*+++{\Lambda^{e}[S]\otimes\Omega[T]\cup\Omega[S]\otimes\partial\Omega[T]}\ar@{>->}[r] & \Omega[S]\otimes\Omega[T]}
\]
is an anodyne extension.
\end{prop}
In the proposition above, the union is that of subobjects of the codomain,
which is the same as the pushout over the intersection $\Lambda^{e}[S]\otimes\partial\Omega[T]$.
\begin{proof}
(of Theorem 9.1 from Proposition 9.2) The theorem states that for
any tree $S$ and any inner edge $e\in S$, any map of dendroidal
sets\[
\varphi:\Lambda^{e}[S]\otimes X\rightarrow K\]
extends to some map (uniquely in the strict case)\[
\psi:\Omega[S]\otimes X\rightarrow K.\]
By writing $X$ as the union of its skeleta, \[
X=\varinjlim Sk_{n}(X)\]
as in Section 4, and using the fact that the skeletal filtration is normal, we can build this extension $\psi$ by induction
on $n$. For $n=0$, $Sk_{0}(X)$ is a sum of copies of $\Omega[\eta]$,
the unit for the tensor product, so obviously the restriction $\varphi_{0}:\Lambda^{e}[S]\otimes Sk_{0}(X)\rightarrow K$
extends to a map \[
\psi_{0}:\Omega[S]\otimes Sk_{0}(X)\rightarrow K.\]
Suppose now that we have found an extension $\psi_{n}:\Omega[S]\otimes Sk_{n}(X)\rightarrow K$
of the restriction $\varphi_{n}:\Lambda^{e}[S]\otimes Sk_{n}(X)\rightarrow K$.
Consider the following diagram:\[
\xyC{5pt}\xymatrix{*++{\coprod\Lambda^{e}[S]\otimes\partial\Omega[T]}\ar[rr]\ar@{>->}[dd]\ar[rd] & & *++{\coprod\Lambda^{e}[S]\otimes\Omega[T]}\ar@{>->}'[d][dd]\ar[rd]\\
& *++{\Lambda^{e}[S]\otimes Sk_{n}(X)}\ar[rr]\ar@{>->}[dd] & & *++{\Lambda^{e}[S]\otimes Sk_{n+1}(X)}\ar@{>->}[dd]\\
\coprod\Omega[S]\otimes\partial\Omega[T]\ar[rd]\ar'[r][rr] & & \coprod\Omega[S]\otimes\Omega[T]\ar[rd]\\
& \Omega[S]\otimes Sk_{n}(X)\ar[rr] & & \Omega[S]\otimes Sk_{n+1}(X)}
\]
In this diagram, the top and bottom faces are pushouts given by the normal skeletal filtration of $X$. Now inscribe
the pushouts $U$ and $V$ in the back and front face, fitting into
a square\[
\xyC{15pt}\xymatrix{*+++{U}\ar@{>->}[d]\ar[r] & *+++{\coprod\Omega[S]\otimes\Omega[T]}\ar@{>->}[d]\\
V\ar[r] & \Omega[S]\otimes Sk_{n+1}(X)}
\]
The maps $\psi_{n}:\Omega[S]\otimes Sk_{n}(X)\rightarrow K$ and $\varphi_{n+1}:\Lambda^{e}[S]\otimes Sk_{n+1}(X)\rightarrow K$
together define a map $V\rightarrow K$. So, to find $\psi_{n+1},$
it suffices to prove that \[
\xymatrix{*++{V}\ar@{>->}[r] & \Omega[S]\otimes Sk_{n+1}(X)}
\]
is anodyne. But, by a diagram chase argument, the square above is
a pushout, so in fact, it suffices to prove that $\xymatrix{*++{U}\ar@{>->}[r] & \coprod\Omega[S]\otimes\Omega[T]}
$ is anodyne. The latter map is a sum of copies of anodyne extensions
as in the statement of the proposition.
\end{proof}
\begin{cor}
The monoidal structure on the category of coloured operads given by
the Boardman-Vogt tensor product is closed \emph{(}see \cite{DenSet}\emph{)}. It
is related to the closed monoidal structure on dendroidal sets by
two natural isomorphisms\[
\tau_{d}(N_{d}\mathcal{P}\otimes N_{d}\mathcal{Q})=\mathcal{P}\otimes_{BV}\mathcal{Q}\]
and \[
N_{d}(\underline{Hom}(\mathcal{Q},\mathcal{R}))=\underline{Hom}(N_{d}\mathcal{Q},N_{d}\mathcal{R})\]
for any operads $\mathcal{P},\mathcal{Q}$ and $\mathcal{R}$.
\end{cor}
\begin{proof}
The first isomorphism was proved in \cite{DenSet}. The second isomorphism
follows from the first one together with (the strict version of) Theorem
9.1, Theorem 6.1, and the fact that $N_{d}$ is fully faithful.
\end{proof}
In the rest of this section, we will be concerned with the proof of
Proposition 9.2, and we fix $S,T$ and $e$ as in the statement of
the proposition from now on. Our strategy will be as follows. First,
let us write \[
A_{0}\subseteq\Omega[S]\otimes\Omega[T]\]
for the dendroidal set given by the image of $\Lambda^{e}[S]\otimes\Omega[T]\cup\Omega[S]\otimes\partial\Omega[T]$.
We are going to construct a sequence of dendroidal subsets \[
A_{0}\subseteq A_{1}\subseteq A_{2}\subseteq\cdots\subseteq A_{N}=\Omega[S]\otimes\Omega[T]\]
such that each inclusion is an anodyne extension. This will be done
by writing $\Omega[S]\otimes\Omega[T]$ as a union of representables,
as follows. We will explicitly describe a sequence of trees\[
T_{1},T_{2},\cdots,T_{N}\]
together with canonical monomorphisms (all called)\[
\xymatrix{*++{m:\Omega[T_{i}]}\ar@{>->}[r] & \Omega[S]\otimes\Omega[T]}
,\]
and we will write $m(T_{i})\subseteq\Omega[S]\otimes\Omega[T]$ for
the dendroidal subset given by the image of this monomorphism. We
will then define\[
\begin{array}{cccc}
& A_{i+1}=A_{i}\cup m(T_{i+1}) & \quad\quad\quad\quad & (i=0,\cdots,N-1)\end{array}\]
and prove that each $\xymatrix{*++{A_{i}}\ar@{>->}[r] & A_{i+1}}
$ thus constructed is anodyne. For the rest of this section, we will
fix planar structures on the trees $S$ and $T$. These will then
induce a natural planar structure on each of the trees $T_{i}$, and
avoid unnecessary discussion involving automorphisms in the category
$\Omega$.
To define the $T_{i}$, let us think of the vertices of $S$ as \emph{white}
(drawn $\circ$) and those of $T$ as \emph{black} (drawn $\bullet$).
The edges of $T_{i}$ are (labelled by) pairs $(a,x)$ where $a$
is an edge of $S$ and $x$ one of $T$. We refer to $a$ as the $S$-\emph{colour}
of this edge $(a,x)$, and to $x$ as its $T$-\emph{colour}. There
are two kinds of vertices in $T_{i}$ (corresponding to the generators
for $\Omega[S]\otimes\Omega[T]$ coming from vertices of $S$ or of
$T$). There are \emph{white} vertices in $T_{i}$ labelled \[
\xymatrix{*{}\ar@{-}[dr]_{(a_{1},x)} & & *{}\ar@{-}[dl]^{(a_{n},x)}\\
\ar@{}[r]|{\quad v} & *{\circ}\ar@{}[u]|{\cdots}\ar@{-}[d]^{(b,x)}\\
& *{}}
\]
where $v$ is a vertex in $S$ with input edges $a_{1},\cdots,a_{n}$
and output edge $b$, while $x$ is an edge of $T$; and there are
\emph{black} vertices in $T_{i}$ labelled\[
\xymatrix{*{}\ar@{-}[dr]_{(a,x_{1})} & & *{}\ar@{-}[dl]^{(a,x_{m})}\\
\ar@{}[r]|{\quad w} & *{\bullet}\ar@{}[u]|{\cdots}\ar@{-}[d]^{(a,y)}\\
& *{}}
\]
where $w$ is a vertex in $T$ with input edges $x_{1},\cdots,x_{m}$
and output edge $y$, while $a$ is an edge in $S$. Moreover, each
such tree $T_{i}$ is \emph{maximal,} in the sense that its output
(root) edge is labelled $(r_{S},r_{T})$ where $r_{S}$ and $r_{T}$
are the roots of $S$ and $T$, and its input edges are labelled by
all pairs $(a,x)$ where $a$ is an input edge of $S$ and $x$ one
of $T$.
All the possible such trees $T_{i}$ come in a natural (partial) order.
The minimal tree $T_{1}$ in the poset is the one obtained by stacking
a copy of the black tree $T$ on top of each of the input edges of
the white tree $S$. Or, more precisely, on the bottom of $T_{1}$
there is a copy $S\otimes r_{T}$ of the tree $S$ all whose edges
are renamed $(a,r_{T})$ where $r_{T}$ is the output edge at the
root of $T$. For each input edge $b$ of $S$, a copy of $T$ is
grafted on the edge $(b,r)$ of $S\otimes r$, with edges $x$ in
$T$ renamed $(b,x)$. The maximal tree $T_{N}$ in the poset is the
similar tree with copies of the white tree $S$ grafted on each of
the input edges of the black tree. Pictorially $T_{1}$ looks like
\[
\xyR{10pt}\xyC{10pt}\xymatrix{*{}\ar@{-}[rr]_{T}\ar@{-}[rd] & & *{}\ar@{-}[dl] & *{}\ar@{-}[rr]_{T}\ar@{-}[rd] & & *{}\ar@{-}[dl] & & *{}\ar@{-}[rr]_{T}\ar@{-}[rd] & & *{}\ar@{-}[dl] & *{}\ar@{-}[rr]_{T}\ar@{-}[rd] & & *{}\ar@{-}[dl]\\
& *{}\ar@{-}[ddrrr] & & & *{}\ar@{-}[ddr] & & & & *{}\ar@{-}[ddl] & & & *{}\ar@{-}[ddlll]\\
\\ & & & & *{}\ar@{-}[rrrr]\ar@{-}[ddrr] & *{} & & *{} & *{}\ar@{-}[ddll]\\
& & & & & & S\\
& & & & & & *{}\ar@{-}[dd]\\
\\ & & & & & & *{}}
\]
and $T_{N}$ looks like
\[
\xyR{10pt}\xyC{10pt}\xymatrix{*{}\ar@{-}[dr]\ar@{-}[rr]_{S} & & *{}\ar@{-}[dl] & *{}\ar@{-}[dr]\ar@{-}[rr]_{S} & & *{}\ar@{-}[dl] & *{}\ar@{-}[dr]\ar@{-}[rr]_{S} & & *{}\ar@{-}[dl]\\
& *{}\ar@{-}[ddr] & & & *{}\ar@{-}[dd] & & & *{}\ar@{-}[ddl]\\
\\ & & *{}\ar@{-}[rrrr]\ar@{-}[rrdd] & & *{} & & *{}\ar@{-}[ddll]\\
& & & & T\\
& & & & *{}\ar@{-}[dd]\\
\\ & & & & *{}}
\]
The intermediate trees $T_{k}$ ($1<k<N)$ are obtained by letting
the black vertices in $T_{1}$ slowly percolate in all possible ways
towards the root of the tree. Each $T_{k}$ is obtained from an earlier
$T_{l}$ by replacing a configuration
\[
\xyR{10pt}\xyC{10pt}\xymatrix{ & *{}\ar@{-}[ddrr] & *{}\ar@{-}[ddr] & *{}\ar@{-}[dd] & *{}\ar@{-}[ddl] & *{}\ar@{-}[ddll] & *{}\ar@{-}[ddrr] & *{}\ar@{-}[ddr] & *{}\ar@{-}[dd] & *{}\ar@{-}[ddl] & *{}\ar@{-}[ddll] & *{}\ar@{-}[ddrr] & *{}\ar@{-}[ddr] & *{}\ar@{-}[dd] & *{}\ar@{-}[ddl] & *{}\ar@{-}[ddll] & *{} & \,\\
\\ & & *{}\ar@{}[r]|{w} & *{\bullet}\ar@{-}[rrrrrddd] & & & \cdots & & *{\bullet}\ar@{-}[ddd] & & \cdots & & *{}\ar@{}[r]|{w\,\,\,} & *{\bullet}\ar@{-}[dddlllll]\\
\\\quad & & & & & & & & & & & & & & & & & \quad(A)\\
& & & & & & & \ar@{}[r]|{v\,\,} & *{\circ}\ar@{-}[dd]\\
\\ & & & & & & & & *{}}
\]
in $T_{l}$ by \[
\xyR{10pt}\xyC{10pt}\xymatrix{ & *{}\ar@{-}[ddr] & *{}\ar@{-}[dd] & *{}\ar@{-}[ddl] & *{}\ar@{-}[ddr] & *{}\ar@{-}[dd] & *{}\ar@{-}[ddl] & *{}\ar@{-}[ddr] & *{}\ar@{-}[dd] & *{}\ar@{-}[ddl] & *{}\ar@{-}[ddr] & *{}\ar@{-}[dd] & *{}\ar@{-}[ddl] & *{}\ar@{-}[ddr] & *{}\ar@{-}[dd] & *{}\ar@{-}[ddl] & \,\\
\\ & *{}\ar@{}[r]|{v} & *{\circ}\ar@{-}[dddrrrrrr] & & & *{\circ}\ar@{-}[dddrrr] & & & *{\circ}\ar@{-}[ddd] & & & *{\circ}\ar@{-}[dddlll] & \cdots & *{}\ar@{}[r]|{v\,\,\,} & *{\circ}\ar@{-}[dddllllll]\\
\quad & & & & & & & & & & & & & & & & \quad(B)\\
\\ & & & & & & & *{}\ar@{}[r]|{w\,} & *{\bullet}\ar@{-}[dd]\\
\\ & & & & & & & & *{}}
\]
in $T_{k}$. More explicitly, if $v$ and $w$ are vertices in $S$
and $T$, \[
\begin{array}{ccc}
\xyC{10pt}\xyR{10pt}\xymatrix{*{}\ar@{-}[dr]_{a_{1}} & & *{}\ar@{-}[dl]^{a_{n}}\\
\ar@{}[r]|{\,\, v} & *{\circ}\ar@{}[u]|{\cdots}\ar@{-}[d]^{b}\\
& *{}}
& \quad\quad & \xymatrix{*{}\ar@{-}[dr]_{x_{1}} & & *{}\ar@{-}[dl]^{x_{m}}\\
\ar@{}[r]|{\,\, w} & *{\bullet}\ar@{}[u]|{\cdots}\ar@{-}[d]^{y}\\
& *{}}
\end{array}\]
then the edges in $(A)$ are named \[
\xymatrix{ & *{}\ar@{-}[dr] & & *{}\ar@{-}[dl]^{(a_{i},x_{j})}\\
*{}\ar@{-}[dr] & & *{\bullet}\ar@{-}[dl]^{(a_{i},y)}\ar@{}[u]|{\cdots}\\
& *{\circ}\ar@{-}[d]^{(b,y)}\ar@{}[u]|{\cdots}\\
& *{}}
\]
and those in $(B)$ are named\[
\xymatrix{ & *{}\ar@{-}[dr] & *{}\ar@{-}[d] & *{}\ar@{-}[dl]^{(a_{i},x_{j})}\\
*{}\ar@{-}[dr] & & *{\circ}\ar@{-}[dl]^{(b,x_{j})}\\
& *{\bullet}\ar@{-}[d]^{(b,y)}\\
& *{}}
\]
We will refer to these trees $T_{i}$ as the \emph{percolation schemes}
for $S$ and $T$, and if $T_{k}$ is obtained from $T_{l}$ by replacing
$(A)$ by $(B)$, then we will say that $T_{l}$ is obtained by a
\emph{single percolation step}.
\begin{example}
Many of the typical phenomena that we will encounter already occur
for the following two trees $S$ and $T$; here, we have singled out
one particular edge $e$ in $S$, we've numbered the edges of $T$
as $1,\cdots,5$, and denoted the colour $(e,i)$ in $T_{i}$ by $e_{i}$.
\[
\begin{array}{ccc}
\xymatrix{ & *{\,}\ar@{-}[dr] & & *{\,}\ar@{-}[dl]\\
& & *{\circ}\ar@{-}[d]_{e}\\
S= & & *{\circ}\ar@{-}[d]\\
& & *{\,}}
& \quad\quad & \xymatrix{ & *{\,}\ar@{-}[d]_{3} & & *{\,}\ar@{-}[d]_{5}\\
& *{\bullet}\ar@{-}[dr]_{2} & & *{\bullet}\ar@{-}[dl]_{4}\\
T= & & *{\bullet}\ar@{-}[d]_{1}\\
& & *{\,}}
\end{array}\]
There are 14 percolation schemes $T_{1},\cdots,T_{14}$ in this case.
Here is the complete list of them:
$\xyR{5pt}\xyC{5pt}\xymatrix{*{}\ar@{-}[d] & & *{}\ar@{-}[d] & & *{}\ar@{-}[d] & & *{}\ar@{-}[d]\\
*{\bullet}\ar@{-}[dr] & & *{\bullet}\ar@{-}[dl] & & *{\bullet}\ar@{-}[dr] & & *{\bullet}\ar@{-}[dl]\\
& *{\bullet}\ar@{-}[drr] & & & & *{\bullet}\ar@{-}[dll]\\
& & & *{\circ}\ar@{-}[d]_{e_{1}}\\
& & & *{\circ}\ar@{-}[d]\\
& & & *{}\\
& & & T_{1}}
$~~~~~$\xyR{5pt}\xyC{5pt}\xymatrix{*{}\ar@{-}[d] & & *{}\ar@{-}[d] & & *{}\ar@{-}[d] & & *{}\ar@{-}[d]\\
*{\bullet}\ar@{-}[dr] & & *{\bullet}\ar@{-}[dl] & & *{\bullet}\ar@{-}[dr] & & *{\bullet}\ar@{-}[dl]\\
& *{\circ}\ar@{-}[drr]_{e_{2}} & & & & *{\circ}\ar@{-}[dll]^{e_{4}}\\
& & & *{\bullet}\ar@{-}[d]_{e_{1}}\\
& & & *{\circ}\ar@{-}[d]\\
& & & *{}\ar@{-}[]\\
& & & T_{2}}
$~~~~~$\xyR{5pt}\xyC{5pt}\xymatrix{*{}\ar@{-}[d] & & *{}\ar@{-}[d] & & *{}\ar@{-}[dr] & & *{}\ar@{-}[dl]\\
*{\bullet}\ar@{-}[dr] & & *{\bullet}\ar@{-}[dl] & & & *{\circ}\ar@{-}[d]_{e_{5}}\\
& *{\circ}\ar@{-}[drr]_{e_{2}} & & & & *{\bullet}\ar@{-}[dll]^{e_{4}}\\
& & & *{\bullet}\ar@{-}[d]_{e_{1}}\\
& & & *{\circ}\ar@{-}[d]\\
& & & *{}\\
& & & T_{3}}
$
$\xyR{5pt}\xyC{5pt}\xymatrix{*{}\ar@{-}[dr] & & *{}\ar@{-}[dl] & & *{}\ar@{-}[d] & & *{}\ar@{-}[d]\\
& *{\circ}\ar@{-}[d]_{e_{3}} & & & *{\bullet}\ar@{-}[dr] & & *{\bullet}\ar@{-}[dl]\\
& *{\bullet}\ar@{-}[drr]_{e_{2}} & & & & *{\circ}\ar@{-}[dll]^{e_{4}}\\
& & & *{\bullet}\ar@{-}[d]_{e_{1}}\\
& & & *{\circ}\ar@{-}[d]\\
& & & *{}\\
& & & T_{4}}
$~~~~~$\xyR{5pt}\xyC{5pt}\xymatrix{*{}\ar@{-}[dr] & & *{}\ar@{-}[dl] & & *{}\ar@{-}[dr] & & *{}\ar@{-}[dl]\\
& *{\circ}\ar@{-}[d]_{e_{3}} & & & & *{\circ}\ar@{-}[d]^{e_{5}}\\
& *{\bullet}\ar@{-}[drr]_{e_{2}} & & & & *{\bullet}\ar@{-}[dll]^{e_{4}}\\
& & & *{\bullet}\ar@{-}[d]_{e_{1}}\\
& & & *{\circ}\ar@{-}[d]\\
& & & *{}\\
& & & T_{5}}
$~~~~~$\xyR{5pt}\xyC{5pt}\xymatrix{*{}\ar@{-}[d] & & *{}\ar@{-}[d] & & *{}\ar@{-}[d] & & *{}\ar@{-}[d]\\
*{\bullet}\ar@{-}[dr] & & *{\bullet}\ar@{-}[dl] & & *{\bullet}\ar@{-}[dr] & & *{\bullet}\ar@{-}[dl]\\
& *{\circ}\ar@{-}[d]_{e_{2}} & & & & *{\circ}\ar@{-}[d]_{e_{4}}\\
& *{\circ}\ar@{-}[drr] & & & & *{\circ}\ar@{-}[dll]\\
& & & *{\bullet}\ar@{-}[d]\\
& & & *{}\\
& & & T_{6}}
$
$\xyR{5pt}\xyC{5pt}\xymatrix{*{}\ar@{-}[d] & & *{}\ar@{-}[d] & & *{}\ar@{-}[dr] & & *{}\ar@{-}[dl]\\
*{\bullet}\ar@{-}[dr] & & *{\bullet}\ar@{-}[dl] & & & *{\circ}\ar@{-}[d]_{e_{5}}\\
& *{\circ}\ar@{-}[d]_{e_{2}} & & & & *{\bullet}\ar@{-}[d]\\
& *{\circ}\ar@{-}[drr] & & & & *{\circ}\ar@{-}[dll]\\
& & & *{\bullet}\ar@{-}[d]\\
& & & *{}\\
& & & T_{7}}
$~~~~~$\xyR{5pt}\xyC{5pt}\xymatrix{*{}\ar@{-}[d] & & *{}\ar@{-}[d] & & *{}\ar@{-}[dr] & & *{}\ar@{-}[dl]\\
*{\bullet}\ar@{-}[dr] & & *{\bullet}\ar@{-}[dl] & & & *{\circ}\ar@{-}[d]_{e_{5}}\\
& *{\circ}\ar@{-}[d]_{e_{2}} & & & & *{\circ}\ar@{-}[d]\\
& *{\circ}\ar@{-}[drr] & & & & *{\bullet}\ar@{-}[dll]\\
& & & *{\bullet}\ar@{-}[d]\\
& & & *{}\\
& & & T_{8}}
$~~~~~$\xyC{5pt}\xyR{5pt}\xymatrix{*{}\ar@{-}[dr] & & *{}\ar@{-}[dl] & & *{}\ar@{-}[d] & & *{}\ar@{-}[d]\\
& *{\circ}\ar@{-}[d] & & & *{\bullet}\ar@{-}[dr] & & *{\bullet}\ar@{-}[dl]\\
& *{\bullet}\ar@{-}[d] & & & & *{\circ}\ar@{-}[d]\\
& *{\circ}\ar@{-}[drr] & & & & *{\circ}\ar@{-}[dll]\\
& & & *{\bullet}\ar@{-}[d]\\
& & & *{}\\
& & & T_{9}}
$
$\xyC{5pt}\xyR{5pt}\xymatrix{*{}\ar@{-}[dr] & & *{}\ar@{-}[dl] & & *{}\ar@{-}[dr] & & *{}\ar@{-}[dl]\\
& *{\circ}\ar@{-}[d] & & & & *{\circ}\ar@{-}[d]\\
& *{\bullet}\ar@{-}[d] & & & & *{\bullet}\ar@{-}[d]\\
& *{\circ}\ar@{-}[drr] & & & & *{\circ}\ar@{-}[dll]\\
& & & *{\bullet}\ar@{-}[d]\\
& & & *{}\\
& & & T_{10}}
$~~~~~$\xyR{5pt}\xyC{5pt}\xymatrix{*{}\ar@{-}[dr] & & *{}\ar@{-}[dl] & & *{}\ar@{-}[dr] & & *{}\ar@{-}[dl]\\
& *{\circ}\ar@{-}[d]_{e_{3}} & & & & *{\circ}\ar@{-}[d]_{e_{5}}\\
& *{\bullet}\ar@{-}[d]_{e_{2}} & & & & *{\circ}\ar@{-}[d]\\
& *{\circ}\ar@{-}[drr] & & & & *{\bullet}\ar@{-}[dll]\\
& & & *{\bullet}\ar@{-}[d]\\
& & & *{}\\
& & & T_{11}}
$~~~~~$\xyC{5pt}\xyR{5pt}\xymatrix{*{}\ar@{-}[dr] & & *{}\ar@{-}[dl] & & *{}\ar@{-}[d] & & *{}\ar@{-}[d]\\
& *{\circ}\ar@{-}[d]_{e_{3}} & & & *{\bullet}\ar@{-}[dr] & & *{\bullet}\ar@{-}[dl]\\
& *{\circ}\ar@{-}[d] & & & & *{\circ}\ar@{-}[d]_{e_{4}}\\
& *{\bullet}\ar@{-}[drr] & & & & *{\circ}\ar@{-}[dll]\\
& & & *{\bullet}\ar@{-}[d]\\
& & & *{}\\
& & & T_{12}}
$
$\xyC{5pt}\xyR{5pt}\xymatrix{*{}\ar@{-}[dr] & & *{}\ar@{-}[dl] & & *{}\ar@{-}[dr] & & *{}\ar@{-}[dl]\\
& *{\circ}\ar@{-}[d]_{e_{3}} & & & & *{\circ}\ar@{-}[d]_{e_{5}}\\
& *{\circ}\ar@{-}[d] & & & & *{\bullet}\ar@{-}[d]_{e_{4}}\\
& *{\bullet}\ar@{-}[drr] & & & & *{\circ}\ar@{-}[dll]\\
& & & *{\bullet}\ar@{-}[d]\\
& & & *{}\\
& & & T_{13}}
$~~~~~$\xyC{5pt}\xyR{5pt}\xymatrix{*{}\ar@{-}[dr] & & *{}\ar@{-}[dl] & & *{}\ar@{-}[dr] & & *{}\ar@{-}[dl]\\
& *{\circ}\ar@{-}[d] & & & & *{\circ}\ar@{-}[d]\\
& *{\circ}\ar@{-}[d] & & & & *{\circ}\ar@{-}[d]\\
& *{\bullet}\ar@{-}[drr] & & & & *{\bullet}\ar@{-}[dll]\\
& & & *{\bullet}\ar@{-}[d]\\
& & & *{}\\
& & & T_{14}}
$
\end{example}
As claimed, there is a partial order on the percolation schemes $T_{1},\cdots,T_{N}$
for $S\otimes T$, in which $T_{1}$ (copies of $T$ on top of $S$)
is the minimal element and $T_{N}$ (copies of $S$ on top of $T$)
the maximal one. The partial order is given by defining $T\le T'$
whenever the percolation scheme $T'$ can be obtained from the percolation
scheme $T$ by a sequence of percolations. For example, the poset
structure on the percolation trees above is:\[
\xyC{10pt}\xyR{10pt}\xymatrix{ & & T_{1}\ar@{-}[dd]\\
\\ & & T_{2}\ar@{-}[ddrr]\ar@{-}[dd]\ar@{-}[ddll]\\
\\T_{3}\ar@{-}[dd]\ar@{-}[ddrr] & & T_{6}\ar@{-}'[dl][ddll]\ar@{-}'[dr][ddrr] & & T_{4}\ar@{-}[dd]\ar@{-}[ddll]\\
& \, & & \,\\
T_{7}\ar@{-}[ddrr]\ar@{-}[dd] & & T_{5}\ar@{-}[dd] & & T_{9}\ar@{-}[ddll]\ar@{-}[dd]\\
& \,\\
T_{8}\ar@{-}[ddr] & & T_{10}\ar@{-}[ddr]\ar@{-}[ddl] & & T_{12}\ar@{-}[ddl]\\
\\ & T_{11}\ar@{-}[ddr] & & T_{13}\ar@{-}[ddl]\\
\\ & & T_{14}}
\]
The planar structures of $S$ and $T$ provide a way to refine this
partial order by a linear order. It is not important exactly how this
is done, but we shall from now on assume that the percolation schemes
for $S$ and $T$ are ordered $T_{1},\cdots,T_{N}$ where $T_{i}$
comes before $T_{j}$ only if $T_{i}\le T_{j}$ in the partial order.
\begin{lem}
(and notation) Each percolation scheme $T_{i}$ is equipped with a
canonical monomorphism \[
\xymatrix{*++{m:\Omega[T_{i}]}\ar@{>->}[r] & \Omega[S]\otimes\Omega[T]}
.\]
The dendroidal subset given by the image of this monomorphism will
be denoted \[
m(T_{i})\subseteq\Omega[S]\otimes\Omega[T].\]
\end{lem}
\begin{proof}
The vertices of the dendroidal set $\Omega[T_{i}]$ are the edges
of the tree $T_{i}$. The map $m$ is completely determined by asking
it to map an edge named $(a,x)$ in $T_{i}$ to the vertex with the
same name in $\Omega[S]\otimes\Omega[T]$. This map is a monomorphism.
In fact, any map \[
\Omega[R]\rightarrow X,\]
from a representable dendroidal set to an arbitrary one, is a monomorphism
as soon as the map $\Omega[R]_{\eta}\rightarrow X_{\eta}$ on vertices
is.
\end{proof}
Before we continue, we need to introduce a bit of terminology for
trees, i.e., for objects of $\Omega$. Let $R$ be such a tree. A
map $R'\rightarrow R$ which is a composition of basic face maps (maps
of type $(ii)$ or $(iii)$ in Section 3) will also be referred to
as a \emph{face} of $R$, just like for simplicial sets. If it is
a composition of \emph{inner} faces (resp. \emph{outer} faces), the
map $\xymatrix{*++{R'}\ar@{>->}[r] & R}
$ will be called an \emph{inner face} (resp. \emph{outer face)} of
$R$. A \emph{top face} of $R$ is an outer face map $\partial_{v}:R'\rightarrow R$
where $R'$ is obtained by deleting a top vertex from $R$. An \emph{initial
segment} $\xymatrix{*++{R'}\ar@{>->}[r] & R}
$ is a composition of top faces (it is a special kind of outer face
of $R$). If $v$ is the vertex above the root of $R$ and $e$ is
an input edge of $v$, then $R$ contains a subtree $R'$ whose root
is $e$. We'll refer to an inclusion of this kind as a \emph{bottom} face
of $R$ (it is again a special kind of outer face). In all these cases,
we'll often leave the monomorphism $\xymatrix{*++{R'}\ar@{>->}[r] & R}
$ implicit, and apply the same terminology not only to the map $\xymatrix{*++{R'}\ar@{>->}[r] & R}
$ but also to the tree $R'$.
For example, for the tree $T$ constructed above \[
\xymatrix{ & *{\,}\ar@{-}[d]_{3} & & *{\,}\ar@{-}[d]^{5}\\
& *{\bullet}\ar@{-}[dr]_{2} & & *{\bullet}\ar@{-}[dl]^{4}\\
T= & & *{\bullet}\ar@{-}[d]^{1}\\
& & *{\,}}
\]
The following sub-trees are examples of, respectively, a top face,
an initial segment, a bottom face, and an inner face:\[
\begin{array}{ccccccc}
\xyC{10pt}\xyR{10pt}\xymatrix{ & & *{\,}\ar@{-}[d]^{5}\\
*{\,}\ar@{-}[dr]_{2} & & *{\bullet}\ar@{-}[dl]^{4}\\
& *{\bullet}\ar@{-}[d]^{1}\\
& *{\,}}
& \quad & \xymatrix{*{\,}\ar@{-}[dr]_{2} & & *{\,}\ar@{-}[dl]^{4}\\
& *{\bullet}\ar@{-}[d]^{1}\\
& *{\,}}
& \quad & \xymatrix{*{\,}\ar@{-}[d]^{3}\\
*{\bullet}\ar@{-}[d]^{2}\\
*{\,}}
& \quad & \xymatrix{*{\,}\ar@{-}[dr]_{3} & & *{\,}\ar@{-}[dl]^{5}\\
& *{\bullet}\ar@{-}[d]^{1}\\
& *{\,}}
\end{array}\]
\begin{rem}
We observe the following simple properties, which we will repeatedly
use in the proofs of the lemmas below. In stating these properties
and below, we denote by $m(R)$ the image of the composition of the
inclusion $\xymatrix{*++{\Omega[R]}\ar@{>->}[r] & \Omega[T_{i}]}
$ given by a subtree (a face) $R$ of $T_{i}$ and the canonical monomorphism
\[
\xyC{15pt}\xyR{15pt}\xymatrix{*++{m:\Omega[T_{i}]}\ar@{>->}[r] & \Omega[S]\otimes\Omega[T]}
.\]
(i) Let $R$ be a subtree of $T_{i}$. If $m(R)\subseteq A_{0}$ then
$R$ misses a $T$-colour, or an $S$-colour other than $e$, or a
stump of either $S$ or $T$. Here, a stump is a top vertex of valence
zero (i.e., without input edges). We say that $R$ ''misses'' such
a stump $v\in S$, for example, if $m(R)\subseteq\partial_{v}[S]\otimes\Omega[T]$.
The tree $R$ is a sub-tree of $T_{i}$, where edges
are coloured by pairs $(a,x)$, where $a$ is an $S$-colour and $x$
a $T$-colour. By saying that $R$ ''misses'' a $T$-colour $y$,
we mean that none of the colours $(a,x)$ occurring in $R$ has $x=y$
as second coordinate. ''Missing an $S$-colour'' is interpreted
similarly.
(ii) This implies in particular that for any bottom face $\xymatrix{R\ar[r] & T_{i}}
$ of any percolation scheme $T_{i}$ the dendroidal set $m(R)$ is
contained in $A_{0}$, because it must miss either the root colour
$r_{S}$ (in case the root of $T_{i}$ is white), or the root colour
$r_{T}$ (in case the root of $T_{i}$ is black), and $r_{S}\ne e$
because $e$ is assumed inner.
(iii) If $F,G$ are faces of $T_{i}$ then $F$ is a face of $G$
iff $m(F)\subseteq m(G)$. (This is clear from the fact that the map
from $\Omega[R]$ onto its image $m(R)$ is an isomorphism of dendroidal
sets.)
(iv) Let $Q$ and $R$ be initial segments of $T_{i}$, and let $F$
be an inner face of $Q$. If $m(F)\subseteq m(R)$ then also $m(Q)\subseteq m(R)$
(and hence $Q$ is a face of $R$, by (iii)). In fact, let $Inn(Q)$
denote the set of all inner edges of $Q$ and $\xymatrix{*++{Q/Inn(Q)}\ar@{>->}[r] & Q}
$ the inner face of $Q$ given by contracting all these. Then if $m(Q/Inn(Q))\subseteq m(R/Inn(R))$,
it follows by comparing labels of input edges of $Q$ and $R$ that
$Q$ is a face of $R$.
\end{rem}
These remarks prepare the ground for the following lemma. Recall that
$A_{k}=A_{0}\cup m(T_{1})\cup\cdots\cup m(T_{k})$, where $m(T_{i})$
is the image in $\Omega[S]\otimes\Omega[T]$ of the dendroidal set
$\Omega[T_{i}]$.
\begin{lem}
Let $R,Q_{1},\cdots,Q_{p}$ be a family of initial segments in $T_{k+1}$
and write $B=m(Q_{1})\cup\cdots\cup m(Q_{p})\subseteq\Omega[S]\otimes\Omega[T]$.
Suppose
(i) For every top face $F$ of $R$, $m(F)\subseteq A_{k}\cup B$.
(ii) There exists an edge $\xi$ in $R$ such that for every inner
face $\xymatrix{*++{F}\ar@{>->}[r] & R}
$, if $m(F)$ is not contained in $A_{k}\cup B$ then neither is $m(F/(\xi))$.
\\
Then the inclusion $\xymatrix{*++{A_{k}\cup B}\ar@{>->}[r] & A_{k}\cup B\cup m(R)}
$ is anodyne.
\end{lem}
We call $\xi$ a characteristic edge of $R$ with respect to $Q_{1},\cdots,Q_{p}.$
\begin{proof}
If $m(R)\subseteq A_{k}\cup B$ there is nothing to prove. If not,
then by (ii), $m(R/(\xi))$ is not contained in $A_{k}\cup B$. Let
\[
\xi=\xi_{0},\xi_{1},\cdots,\xi_{n}\]
be all the inner edges in $R$ such that the dendroidal set $m(R/(\xi_{i}))$
is not contained in $A_{k}\cup B$. For a sub-sequence $\xi_{i_{1}},\cdots,\xi_{i_{p}}$
of these $\xi_{0},\cdots,\xi_{n}$, we have the dendroidal subset
of $\Omega[S]\otimes\Omega[T]$,\[
\begin{array}{ccc}
\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad & m(R/(\xi_{i_{1}},\cdots,\xi_{i_{p}})), & \quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad(1)\end{array}\]
obtained by contracting each of $\xi_{i_{1}},\cdots,\xi_{i_{p}}$
and composing with $m:\Omega[T_{k+1}]\rightarrow\Omega[S]\otimes\Omega[T]$.
We are going to consider a sequence of anodyne extensions\[
\xymatrix{A_{k}\cup B=B_{0}\,\,\ar@{>->}[r] & B_{1}\,\,\ar@{>->}[r] & \cdots\,\,\ar@{>->}[r] & B_{2^{n}}=A_{k}\cup B\cup m(R)}
\]
by considering images of faces of $\Omega[R]$ of this type (1).
Consider first \[
R_{(0)}=m(R/(\xi_{1},\cdots,\xi_{n})).\]
If $m(R_{(0)})$ is contained in $A_{k}\cup B$, let $B_{1}=B_{0}=A_{k}\cup B$.
Otherwise, let $B_{1}$ be the pushout\[
\xyC{20pt}\xyR{20pt}\xymatrix{m(\Lambda^{\xi_{0}}R_{(0)})\ar[r]\ar[d] & B_{0}\ar[d]\\
m(R_{(0)})\ar[r] & B_{1}}
\]
Notice that $m(\Lambda^{\xi_{0}}R_{(0)})$ is indeed contained in $B_{0}=A_{k}\cup B$.
For, any outer face $F$ of $R_{(0)}$ is a face of an outer face
$G$ of $R$\[
\xymatrix{F\ar[r]\ar[d] & R/(\xi_{1},\cdots,\xi_{n})=R_{(0)}\ar[d]\\
G\ar[r] & R}
\]
if $G$ is a top face, then $m(G)\subseteq A_{k}\cup B$ by assumption
(i); and if $G$ is a bottom face, it already factors through $A_{0}\subseteq A_{k}$
(cf Remark 9.6 (ii) before the lemma). On the other hand, if $F\subseteq R_{(0)}$
is an inner face of $R_{(0)}$ given by contracting an edge $\zeta$
in $R/(\xi_{1},\cdots,\xi_{n})$, then $F$ is a face of $R/(\zeta)$.
So if $m(F)\nsubseteq B_{0}$ then $m(R/(\zeta))$ wouldn't be contained
in $B_{0}$ either, and hence $\zeta$ must be one of $\xi_{0},\cdots,\xi_{n}$.
But $\xi_{1},\cdots,\xi_{n}$ are no longer edges in $R/(\xi_{1},\cdots,\xi_{n})$,
so $\zeta$ must be $\xi_{0}$. This shows that for any inner face
$F$ of $R_{(0)}$ other then $R_{(0)}/(\xi_{0})$, the dendroidal
set $m(F)$ is contained in $B_{0}$, as claimed.
Next, consider all sub-sequences $(\xi_{1},\cdots,\hat{\xi}_{i},\cdots,\xi_{n})$,
and the faces \[
R_{(i)}=R/(\xi_{1},\cdots,\hat{\xi}_{i},\cdots\xi_{n})\,\,\,\,\,\, i=1,\cdots,n\]
We will define \[
B_{2},\cdots,B_{n+1}\]
by considering these $R_{(1)},\cdots,R_{(n)}$. Suppose $B_{1},\cdots,B_{i}$
have been defined. Consider $R_{(i)}$ to form $B_{i+1}$. If its
image $m(R_{(i)})$ is contained in $B_{i}$, let $B_{i+1}=B_{i}$.
Otherwise, $m(R_{(i)})\rightarrow\Omega[S]\otimes\Omega[T]$ does
not factor through $B_{i}$, and a fortiori doesn't factor through
$A_{k}\cup B=B_{0}$ either. So by assumption (ii), we have that $m(R_{(i)}/(\xi_{0}))\nsubseteq A_{k}\cup B$.
But then $m(R_{i}/(\xi_{0}))$ is not contained in $B_{i}$ either,
because by Remark 9.6(iv), if $m(R_{i}/(\xi_{0}))$ would be contained
in one of $m(R_{(0)}),\cdots,m(R_{(i-1)})$, then $R_{i}/(\xi_{0})$
would be a face of one of $R_{(0)},\cdots,R_{(i-1)}$, which is obviously
not the case. On the other hand, $\xi_{0}$ is the \emph{only} edge
of $R_{(i)}$ for which $m(R_{(i)}/(\xi_{0}))$ is not contained in
$B_{i}$ (indeed, the only other candidate would be $\xi_{i}$, but
$R_{(i)}/\xi_{i}=R_{(0)}$ and $m(R_{(0)})\subseteq B_{1}$). So,
we can form the pushout\[
\xymatrix{*++{m(\Lambda^{\xi_{0}}R_{(i)})}\ar@{>->}[d]\ar[r] & *++{B_{i}}\ar@{>->}[d]\\
m(R_{(i)})\ar[r] & B_{i+1}}
\]
Next, consider for each $i<j$ the tree \[
R_{(ij)}=R/(\xi_{1},\cdots,\hat{\xi}_{i},\cdots,\hat{\xi}_{j},\cdots,\xi_{n}),\]
and order these lexicographically, say as\[
R_{1}^{2},\cdots,R_{u}^{2},\,\,\,\,\,\,\,\,(u={{n \choose 2}}).\]
We are going to form anodyne extensions of $B_{n+1}$ by using these
trees, \[
\xymatrix{B_{n+1}\,\,\ar@{>->}[r] & B_{n+2}\,\,\ar@{>->}[r] & \cdots\,\,\ar@{>->}[r] & B_{n+1+u}}
\textrm{, }\]
treating $R_{p}^{2}$ in the step to form $B_{n+p}\rightarrowtail B_{n+p+1}$
(for each $p=1,\cdots,u$). Suppose $B_{n+p}$ has been formed, and
consider $R_{p}^{2}=R_{(ij)}$ say. If $m(R_{p}^{2})\subseteq B_{n+p}$
then let $B_{n+p+1}=B_{n+p}$. If not, then surely $m(R_{p}^{2})\nsubseteq A_{k}\cup B$,
so by assumption (ii) $m(R_{p}^{2}/(\xi_{0}))=m(R/(\xi_{0},\xi_{1},\cdots,\hat{\xi}_{i},\cdots,\hat{\xi}_{j},\cdots,\xi_{n}))$
is not contained in $A_{k}\cup B$. On the other hand, Remark 9.6(iv)
implies that $m(R_{p}^{2}/(\xi_{0}))$ cannot be contained in any
of $m(R_{1}),\cdots,m(R_{n}),m(R_{1}^{2}),\cdots,m(R_{p-1}^{2})$
either. So $m(R_{p}^{2}/(\xi_{0}))$ is not contained in $B_{n+p}$.
As before, $\xi_{0}$ is the \emph{only} inner edge $\zeta$ for which
$m(R_{p}^{2}/(\zeta))$ is not contained in $B_{n+p}$. So we can
form the pushout\[
\xymatrix{*++{m(\Lambda^{\xi_{0}}(R_{(p)}^{2}))}\ar[r]\ar@{>->}[d] & *++{B_{n+p}}\ar@{>->}[d]\\
m(R_{(p)}^{2})\ar[r] & B_{n+p+1}}
\]
Next consider for each $i_{1}<i_{2}<i_{3}$ the tree \[
R_{(i_{1}i_{2}i_{3})}=R/(\xi_{1},\cdots,\hat{\xi}_{i_{1}},\cdots,\hat{\xi}_{i_{2}},\cdots,\hat{\xi}_{i_{3}},\cdots,\xi_{n})\]
and adjoin the pushout along\[
m(\Lambda^{\xi_{0}}R_{(i_{1}i_{2}i_{3})})\rightarrowtail m(R_{(i_{1}i_{2}i_{3})})\]
if necessary. Continuing in this way for all $l=0,1,\cdots,n-1$
and all sub-sequences $i_{1}<\cdots<i_{l}$ and corresponding trees
\[
R/(\xi_{1},\cdots,\hat{\xi}_{i_{1}},\cdots,\hat{\xi}_{i_{l}},\cdots,\xi_{n})\textrm{, }\]
we end up with a sequence of anodyne extensions\[
B_{1}\rightarrowtail\cdots\rightarrowtail B_{q}\]
where $q=\Sigma_{l=0}^{n-1}{{n \choose l}}=2^{n}-1$, and where $m(R/(\xi_{i}))$
is contained in $B_{q}$ for each $i=1,\cdots,n$. In the final step,
and exactly as before, we let $B_{2^{n}}=B_{2^{n}-1}$ if $m(R)\subseteq B_{2^{n}-1}$;
and if not, we form the pushout\[
\xymatrix{*++{m(\Lambda^{\xi_{0}}(R))}\ar[r]\ar@{>->}[d] & *++{B_{2^{n}-1}}\ar@{>->}[d]\\
m(R)\ar[r] & B_{2^{n}}}
\]
Then $B_{2^{n}}$ is the pushout of $A_{0}\cup B$ and $m(R)$ over $(A_{0}\cup B)\cap m(R)$
(because every face $F$ of $R$ for which $m(F)$ is contained in $A_{0}\cup B$
occurs in some corner of the pushouts taken in the construction of
the $B_{i}$). This proves the lemma.
\end{proof}
Consider the tree $T_{k+1}$, and look at all lowest occurrences of
the $S$-colour $e$ (Recall $e$ is the fixed edge in $S$, occurring
in the statement of Proposition 9.2). More precisely, let $e_{i}=(e,x_{i})$
for $i=1,\cdots,t$ be all the edges in $T_{k+1}$ whose $S$-colour
is $e$, while the $S$-colour of the edge immediately below it isn't.
This means that $(e,x_{i})$ is an edge having a white vertex at its
bottom. Let $\beta_{i}$ be the branch in $T_{k+1}$ from the root
to and including this edge $e_{i}$. Each such $\beta_{i}$ is an
initial segment in $T_{k+1}$, to which we will refer as the \emph{spine}
through $e_{i}$. For example, this is a picture of a spine in $T_{k+1}$,\[
\begin{array}{ccc}
& \quad & \xyC{10pt}\xyR{10pt}\xymatrix{ & & *{}\ar@{-}[dr] & *{}\ar@{-}[d] & *{}\ar@{-}[dl]\\
& & *{}\ar@{}[r]|{w_{i}} & *{\cdot}\ar@{-}[d]_{e_{i}} & *{}\ar@{-}[dl]\\
& & *{}\ar@{-}[dr]\ar@{}[r]|{v} & *{\circ}\ar@{-}[d] & *{}\ar@{-}[dl]\\
& *{}\ar@{-}[dr] & *{}\ar@{-}[d] & *{\cdot}\ar@{-}[dl]\\
& *{}\ar@{-}[dr] & *{\cdot}\ar@{-}[d] & *{}\ar@{-}[l]\\
& *{}\ar@{-}[dr] & *{\cdot}\ar@{-}[d] & *{}\ar@{-}[dl]\\
& *{}\ar@{-}[dr] & *{\cdot}\ar@{-}[d] & *{}\ar@{-}[dl]\\
& & *{}\\
}
\end{array}\]
corresponding to the edge $e$ in $S$\[
\xymatrix{*{}\ar@{-}[dr] & & *{}\ar@{-}[dl]\\
\ar@{}[r]|{ v'} & *{\circ}\ar@{-}[d]_{e} & *{}\ar@{-}[dl]\\
\ar@{}[r]|{ v} & *{\circ}\ar@{-}[d]\\
& *{}}
\]
\begin{lem}
Let $R,Q_{1},\cdots,Q_{p}$ be initial segments in $T_{k+1}$, as
in the preceding lemma, and suppose condition (i) of that lemma is
satisfied. Then for any spine $\beta_{i}$ contained in $R$, the
edge $e_{i}\in\beta_{i}$ is characteristic for $R$ with respect
to $Q_{1},\cdots,Q_{p}$.
\end{lem}
\begin{proof}
We have to check condition (ii) of Lemma 9.7. So, suppose $F$ is
an inner face of $R$, and suppose $m(F/(e_{i}))$ is contained in
$A_{k}\cup B=A_{0}\cup m(T_{1})\cup\cdots\cup m(T_{k})\cup m(Q_{1})\cup\cdots\cup m(Q_{p})\subseteq m(T_{k+1})$.
Since $m(F/(e_{i}))$ is isomorphic to the representable dendroidal
set $\Omega[F/(e_{i})]$, it must be contained in one of the dendroidal
sets constituting this union. But, if $m(F/(e_{i}))$ is contained
in $A_{0}$, then by Remark 9.6 (ii) $m(F)$ is also contained in
$A_{0}$. (the only colour occurring in $F$ but possibly not in $F/(e_{i})$
is the $S$-colour $e$). And, if $m(F/(e_{i}))$ is contained in
$m(T_{j})$ for some $j\le k$, then there must be a tensor product
relation applying to the image of $F/(e_{i})$, which allows a black
vertex to move up so as to get into an earlier $T_{j}$, as in:\[
\begin{array}{ccccc}
\xyC{10pt}\xyR{10pt}\xymatrix{*{}\ar@{-}[dr] & *{}\ar@{-}[d] & *{}\ar@{-}[dl] & & *{}\ar@{-}[dr] & *{}\ar@{-}[d] & *{}\ar@{-}[dl]\\
& *{\circ}\ar@{-}[drr] & & & & *{\circ}\ar@{-}[dll]\\
& & & *{\bullet}\ar@{-}[d]\\
& & & *{}}
& \xymatrix{\\*{}\ar[rr] & & *{}\\
}
& \xymatrix{*{}\ar@{-}[drr] & *{}\ar@{-}[dr] & *{}\ar@{-}[d] & *{}\ar@{-}[dl] & *{}\ar@{-}[dll]\\
& & *{}\ar@{-}[d]\\
& & *{}}
& \xymatrix{\\*{} & & *{}\ar[ll]\\
}
& \xyC{4pt}\xyR{4pt}\xymatrix{*{}\ar@{-}[dr] & & *{}\ar@{-}[dl] & & *{}\ar@{-}[dr] & & {}\ar@{-}[dl] & & *{}\ar@{-}[dr] & & *{}\ar@{-}[dl]\\
& *{\bullet}\ar@{-}[ddrrrr] & & & & *{\bullet}\ar@{-}[dd] & & & & *{\bullet}\ar@{-}[ddllll]\\
\\ & & & & & *{\circ}\ar@{-}[d]\\
& & & & & *{}}
\end{array}\]
where the left tree is in $T_{k+1}$, the right tree is in $T_{j}$,
and the middle one in $F/(e_{i})$.
But then the same relation must apply to the image of $F$, because
the edge $e_{i}$, having a white vertex at its root, cannot contribute
to this relation. Finally, if $m(F/(e_{i}))$ is contained in $m(Q_{l})$
for some $l\le p$, then by Remark 9.6 (iv), we have $m(R)\subseteq m(Q_{l})$.
So a fortiori, $m(F)$ is contained in $m(Q_{i})$. This proves the
lemma.
\end{proof}
Recall that our aim is to prove for $A_{k}=A_{0}\cup m(T_{1})\cup\cdots\cup m(T_{k})$
that each inclusion\[
A_{k}\rightarrowtail A_{k+1}\]
is anodyne. Consider the tree $T_{k+1}$, and let $\beta_{1},\cdots,\beta_{t}$
be all the spines contained in it. We shall prove by induction that
$A_{k}\rightarrowtail A_{k}\cup m(R_{1})\cup\cdots\cup m(R_{q})$
is anodyne, for any family $R_{1},\cdots,R_{q}$ of initial segments
each of which contains at least one such spine. The induction will
be on the number of such initial segments as well as on their size.
When applied to the maximal initial segment $T_{k+1}$ itself, this
will show that $A_{k}\rightarrowtail A_{k}\cup m(T_{k+1})=A_{k+1}$
is anodyne, as claimed. The precise form of induction is given by
the following lemma.
\begin{lem}
Fix $l$ with $0\le l\le t$. Let $Q_{1},\cdots,Q_{p}$ be a family
of initial segments in $T_{k+1}$, each containing at least one and
at most $l$ spines. Let $R_{1},\cdots,R_{q}$ be initial segments,
each of which contains $l+1$ spines. Then the inclusion \[
A_{k}\rightarrowtail A_{k}\cup B\cup C\]
for $B=m(Q_{1})\cup\cdots\cup m(Q_{p})$ and $C=m(R_{1})\cup\cdots\cup m(R_{q})$,
is anodyne.
\end{lem}
\begin{proof}
We can measure the size of each of the initial segments $R_{j}$ by
counting the number of vertices in $R_{j}$ which are not on one of
the $l+1$ spines. If this number is not bigger than $u$, we say
that $R_{j}$ has size at most $u$, and write $size(R_{j})\le u$.
Let $\Lambda(l,u)$ be the assertion that the lemma holds for $l$,
for any families $\{ Q_{i}\}$ and $\{ R_{j}\}$ where the $R_{j}$
all have $size(R_{j})\le u$. We will prove $\Lambda(l,u)$ by induction,
first on $l$ and then on $u$.\emph{}\\
\emph{Case $l=0$:} This is the case where there are no $Q$'s, i.e.,
$p=0$. For $l=0$, first consider the case where $u=0$ also. Then
each $R_{i}$ is itself a spine, say $\beta_{i}$, with top inner
edge $e_{i}$ running from a copy of $v$ to a copy $w_{i}$ of $w$.
We will prove that each of the inclusions\[
A_{k}\cup m(R_{1})\cup\cdots\cup m(R_{i-1})\rightarrowtail A_{k}\cup m(R_{1})\cup\cdots\cup m(R_{i})\]
for $i=0,\cdots,q$, is anodyne. If $R_{i}=\beta_{i}$ coincides
with one of the earlier spines $R_{j},\, j<i$, then there is nothing
to prove. If $R_{i}$ is a different spine, then its outer top face
is contained in $A_{0}$ because it misses the vertex $v'$ which
is above $e$ in $S$. So condition (i) of Lemma 9.7 is satisfied,
where $R_{i},R_{1},\cdots,R_{i-1}$ take the role of $R,Q_{1},\cdots,Q_{p}$
in that lemma. By Lemma 9.8, the edge $e_{i}\in R_{i}$ is characteristic
so Lemma 9.7 gives that $A_{k}\cup m(R_{1})\cup\cdots\cup m(R_{i-1})\rightarrowtail A_{k}\cup m(R_{1})\cup\cdots\cup m(R_{i})$
is anodyne, as claimed. The composition of these inclusions will then
be anodyne also, which proves the statement $\Lambda(0,0)$.\\
Suppose now that $\Lambda(0,u)$ has been proved, and consider families
$R_{1},\cdots,R_{q}$ of initial segments which are each of size not
bigger than $u+1$. Suppose that among these, $R_{1},\cdots,R_{q'}$
actually have size not bigger than $u$, while $R_{q'+1},\cdots R_{q}$
have size $u+1$. We shall prove that \[
A_{k}\rightarrowtail A_{k}\cup m(R_{1})\cup\cdots\cup m(R_{q})\]
is anodyne, by induction on the number $r=q-q'$ of initial segments
that have size $u+1$. If $r=0$, this holds by $\Lambda(0,u)$. Suppose
we have proved this for \emph{any} family with not more than $r$
initial segments of size $u+1$, and consider such a family $R_{1},\cdots,R_{q}$
where $q-q'=r+1$. Write $\beta_{q}$ for the spine contained in $R_{q}$
(there is only one such because we are still in the case $l=0$).
For a top outer face $\partial_{x}(R_{q})$ of $R_{q}$, \emph{either}
$x\ne w_{q}$ so that $\partial_{x}(R_{q})$ still contains $\beta_{q}$
but has size at most $u$, \emph{or} $x=w_{q}$ so that $m(\partial_{x}(R_{i}))$
is contained in $A_{0}$ because it misses the vertex $v'$ immediately
above $e$ in $S$. Thus, if we let \[
P=m(R_{1})\cup\cdots\cup m(R_{q-1})\cup\bigcup_{x}m(\partial_{x}(R_{q}))\]
where $x$ ranges over all the top vertices in $R_{q}$, then by
the fact that $\Lambda(0,u+1)$ is assumed to hold for $r=(q-1)-q'$,
\[
\begin{array}{ccc}
\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad & A_{k}\rightarrowtail A_{k}\cup P & \quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad(1)\end{array}\]
is anodyne. To prove that $A_{k}\cup P\rightarrowtail A_{k}\cup P\cup m(R_{q})=A_{k}\cup m(R_{1})\cup\cdots\cup m(R_{q})$
is anodyne as well, we can now apply Lemma 9.7. Indeed, the family
of initial segments containing $P$ is made to contain the images
of all the top faces of $R_{q}$, and $e_{q}\in R_{q}$ is characteristic
by Lemma 9.8. This proves that $A_{k}\cup P\rightarrowtail A_{k}\cup P\cup m(R_{q})$
is anodyne, as claimed. When composed with (1), we find that $A_{k}\rightarrowtail A_{k}\cup m(R_{1})\cup\cdots\cup m(R_{q})$
is anodyne. This proves $\Lambda(0,u+1)$ and completes the inductive
proof of $\Lambda(l,u)$ for $l=0$ and all $u$. \\
Suppose now that we have proved $\Lambda(l',u)$ for all $l'\le l$
and all $u$. We will now prove $\Lambda(l+1,u)$ by induction on
$u$.
\emph{Case} $u=0$: This is the assertion that for any given initial
segments \[
Q_{1},\cdots,Q_{p},R_{1},\cdots,R_{q}\]
of $T_{k+1}$, where the $Q_{j}$ contain at most $l$ spines while
each $R_{i}$ is made up out of exactly $l+1$ spines (and no other
vertices), the inclusion \[
\begin{array}{ccc}
\quad\quad\quad & A_{k}\rightarrowtail A_{k}\cup m(Q_{1})\cup\cdots\cup m(Q_{p})\cup m(R_{1})\cup\cdots\cup m(R_{q}) & \quad\quad\quad(2)\end{array}\]
is anodyne. We shall prove by induction on $q$ that this holds for
all $p$. For $q=0$, the conclusion follows by the inductive assumption
that $\Lambda(l,u)$ holds. Suppose the assertion holds for $q-1$,
and consider $R_{q}$. Each top vertex of $R_{q}$ lies at the end
of a spine, so $\partial^{x}(R_{q})$ contains at most $l$ spines.
Let \[
D=m(Q_{1})\cup\cdots\cup m(Q_{p})\cup\bigcup_{x}\partial_{x}(R_{q})\]
where $x$ ranges over the top vertices of $R_{q}$. Then, by the
assumption for $q-1$, \[
\begin{array}{ccc}
\quad\quad\quad\quad\quad\quad\quad & A_{k}\rightarrowtail A_{k}\cup D\cup m(R_{1})\cup\cdots\cup m(R_{q-1}) & \quad\quad\quad\quad\quad\quad\quad(3)\end{array}\]
is anodyne. To prove that (2) is anodyne, it then suffices to apply
Lemma 9.7, and show that $R_{q}$ has a characteristic edge with respect
to the family of initial segments containing the union $D\cup m(R_{1})\cup\cdots\cup m(R_{q-1})$
in (3). But by Lemma 9.8, any top edge $e_{q}$ of $R_{q}$ is characteristic.
This proves $\Lambda(l+1,u)$, for $u=0$.
\emph{Case} $u+1$: Suppose now $\Lambda(l+1,u)$ holds. To prove
$\Lambda(l+1,u+1)$, consider families\[
\begin{array}{ccc}
\quad\quad\quad\quad\quad\quad\quad & Q_{1},\cdots,Q_{p},R_{1},\cdots,R_{q'},R_{q'+1},\cdots,R_{q} & \quad\quad\quad\quad\quad\quad\quad(4)\end{array}\]
of initial segments in $T_{k+1}$, where the $Q_{i}$ contain at most
$l$ spines, the $R_{i}$ contain exactly $l+1$ spines, the $R_{1},\cdots,R_{q'}$
are of size not more than $u$, and $R_{q'+1},\cdots,R_{q}$ are of
size exactly $u+1$. We will show by induction on the last number
$r=q-q'$ that for any such family, the inclusion\[
\begin{array}{ccc}
\quad\quad\quad & A_{k}\rightarrowtail A_{k}\cup m(Q_{1})\cup\cdots\cup m(Q_{p})\cup m(R_{1})\cup\cdots\cup m(R_{q}) & \quad\quad\quad(5)\end{array}\]
is anodyne. For $r=q-q'=0$ there is nothing to prove, because this
is the case covered by $\Lambda(l+1,u)$. Suppose we have proved that
(5) is anodyne for \emph{any} family (4) with $q-q'\le r$, and consider
such a family with $q-q'=r+1$. The initial segment $R_{q}$ has two
kinds of top outer faces, namely the $\partial_{x}(R_{q})$ which
remove the top of a spine, and the $\partial_{x}(R_{q})$ where $x$
does not lie on a spine. Outer faces of the first kind contain $l$
spines only, and outer faces of the second kind are of size not more
than $u$. Let \[
D=m(Q_{1})\cup\cdots\cup m(Q_{p})\cup\bigcup_{x}m(\partial_{x}R_{q})\]
where $x$ ranges over the top vertices of $R_{q}$ which are on a
spine. Let\[
E=m(R_{1})\cup\cdots\cup m(R_{q'})\cup\bigcup_{x}m(\partial_{x}(R_{q}))\]
where $x$ ranges over the top vertices of $R_{q}$ which are not on a
spine. Then, by the assumption that $\Lambda(l+1,u+1)$ has been established
for families (4) where $q-q'\le r$, we see that \[
\begin{array}{ccc}
\quad\quad\quad\quad\quad\quad\quad & A_{k}\rightarrowtail A_{k}\cup D\cup E\cup R_{q'+1}\cup\cdots\cup R_{q-1} & \quad\quad\quad\quad\quad\quad\quad(6)\end{array}\]
is anodyne. The union $D\cup E$ is made to contain all the images
$\partial_{x}(R_{q})$ of top faces $\partial_{x}(R_{q})$ of $R_{q}$,
and by Lemma 9.8, any edge $e_{q}$ on the top of a spine $\beta_{q}$
in $R_{q}$ is characteristic with respect to the family of initial
segments making up the union on the right-hand-side of (6). So by
Lemma 9.7, the map \[
A_{k}\cup D\cup E\cup R_{q'+1}\cup\cdots\cup R_{q-1}\rightarrowtail A_{k}\cup D\cup E\cup R_{q'+1}\cup\cdots\cup R_{q}\]
is anodyne. When composed with (6), this gives (5), and proves the
case $u+1$.
This established $\Lambda(l+1,u+1)$ and completes, for $l+1$, the
induction on $u$, thus completing the proof.
\end{proof}
|
cond-mat/0701738
|
\section{Theory}
We propose
the general Hamiltonian of a tightly packed toroid, with any type of short/long-range attractive interactions between segments:
${\cal H}_{cl}(a,l,L,W) = {\textstyle \frac{Ll}{2} a^2} - W \Bigl[ \frac{2\pi}{a} {\cal V}(N{}) + \left( L - \frac{2\pi N{}}{a} \right) $\\
${\textstyle Gap(N{}) \Bigr]\!}.$
$l$ denotes the persistence length of the semiflexible polymer chain of
total contour length $L$. $l$ is assumed to be large enough relative to the bond length $l_b$ to realise its stiffness ($l \gg l_b$). $a$ is an inverse toroidal radius and $W$ is a positive coupling constant of the two-body attractive interaction between polymer segments. $N{}$ is the winding number: $N{}\equiv [aL/2\pi]$.\footnote{Gauss' symbol $[x]$ gives the greatest integer among $m<x$.}
${\cal V}(N{})$ is the attractive potential in the toroid cross section in the unit of $-W$, which falls into the number of interacting segmental pairs for the van der Waals nearest neighbour interaction \cite{IK06a}:
${\cal V}(N)=3N-2\sqrt{3}\sqrt{N\!-\!1/4}$.
$Gap(N{})\equiv{\cal V}(N{} \!\,+\!1) - {\cal V}(N{})$ is introduced to compensate the continuity of the Hamiltonian.
Temperature $\beta=1/(k_BT)$ is implicitly included in $l$ and $W$.
An important assumption is that the attractive potential ({\it i.e.} the second term in ${\cal H}_{cl}$)
is a linear function of the toroid radius, or equivalently of the mean perimeter of the toroid \cite{ALG04}.
With the conformation parameter $c\,\equiv \frac{W}{2l}\!\left(
\frac{L}{2\pi} \right)^{\!2}>0$, and a variable $x = \frac{aL}{2\pi}\geq 0$ ({\it i.e.} $[x] = N{}$), the Hamiltonian is simplified by ${\cal H}_{cl}(a,l,L,W) = WL \cdot {\cal H}(c,x)$ and:
\begin{eqnarray}
{\cal H}(c,x)
= \frac{x^2}{4c} + \frac{f([x])}{x} - Gap([x]),
\label{toroidHcx}
\label{Hcx}
\end{eqnarray}
where
$$
f(N) \equiv N {\cal V}(N\!+\!1) - (N\!+\!1) {\cal V}(N).
$$
For consistency, ${\cal V}(0) =
{\cal V}(1) = 0$, so that $\frac{f([x])}{x} \to 0$ at $x=0$. By varying the
value of $W$, ${\cal V}(2)$ can be normalised to unity. For a chain to form a toroid, $WL\gg 1$ and $c>4$ are assumed.
Most physical observables for the tightly packed toroids can be quite accurately estimated by the dominant toroid winding number $N_c$.
For instance, the mean toroid radius is given by
\begin{eqnarray}
r_c\equiv \frac{L}{2\pi N_c}\sim L^{\nu\left(N_c\right)}
\end{eqnarray}
with the exponent $\nu\!\left(N_c\right)$. $N_c$ is given by the global minimum of the Hamiltonian (\ref{Hcx}).
Since ${\cal H}(c,x)$ is governed by the value of $c$, our aim is now to derive the relation between $c$ and $N_c$.
Let us focus on the form of ${\cal H}(c,x)$ in the $m$-th segment ($m < x <m+1$).
The only extremum is given at $x=x_c(m)\equiv\left( 2c f(m)\right)^{\frac13}$.
The condition for this existence in the segment is $f(m)>0$ and $m < x_c(m) < m+1$, from which it follows
$c_L(m) < c < c_U(m),$
where $c_L(m)=\frac{m^3}{2f(m)}$, $c_U(m)=\frac{(m+1)^3}{2f(m)}$.
Provided $f(m)>0$,
it means $c_U(m) \simeq c_L(m)$ for large $m$, and therefore, $c \simeq \frac{m^3}{2f(m)}$. Hence, for large $N_c$, the $c$-$N_c$ relation is almost uniquely determined by
\begin{eqnarray}
c \simeq \frac{N_c^3}{2f(N_c)}.
\end{eqnarray}
Note that the second $x$-derivative of ${\cal H}(c,x)$ is positive except at $x\in\mathbf Z} \def\R{\mathbf R} \def\C{\mathbf C$ so that all existing extrema are stationary.
It should be emphasised that the $c$-$N_c$ relation for a smaller value of $N_c$ is different from its asymptotic one. This becomes crucial when we compare the theory to experiment in the end: real chains such as DNA have finite length. Instead, we replace the above relation by $c \simeq \frac{c_L(N_c)+c_U(N_c)}{2}$, which works well for the cases examined below. We now provide specific attractive potentials and calculate analytical expressions for the mean toroidal and cross sectional radii.
\vspace{5pt}
{\bf The delta function potential:}
One of the simplest interaction is the delta functional attraction,
$
{\cal H}_{AT}(s) = -W\!\!\int_0^L\!\!\!ds\!\int_0^{s}\!\!ds^\prime \; \!\!\delta\!\left( \left| \vec{r}(s) - \vec{r}(s^\prime) \right| \right).
$
In ref.\cite{IK06a},
we have shown that the toroid of winding number $N\geq4$ is the stable ground state for $c>4$.
In this ``ideal toroid'' with zero thickness of the chain, every chain segment interacts equally with all the other segments accumulated on the same arc of the toroid.
Hence, ${\cal V}(N{})$ becomes the number of interacting segmental pairs:
$$
{\cal V}(N)={}_NC_2=\sum_{k=1}^{N-1} k = \frac{1}{2} N(N-1).
$$
$Gap(N)$ is now given by $Gap(N)=N$. Consequently, $f(N)$ is given by $f(N)=\frac12 N(N+1)>0$. Substituting these into the expressions of $c_{L,U}(N)$, we obtain $c_L(N)=\frac{N^2}{N+1}$ and $c_U(N)=\frac{(N+1)^2}{N}$. In the asymptotic limit, we have
\begin{eqnarray}
N_c \simeq c,
\end{eqnarray}
as $c_L^{(N_c)}\simeq N_c$ and $c_U^{(N_c)} \simeq N_c$. Thus, the radius of the dominant ideal toroid is
\begin{eqnarray}
r_c = \frac{L}{2\pi N_c} = \frac{4\pi l}{W L}.
\end{eqnarray}
\vspace{5pt}
{\bf Van der Waals potential:} In real systems, the toroid has finite thickness hence physics could be different from that of the ideal toroid. Therefore, we must incorporate finite size effect into ${\cal V}(N)$ of the Hamiltonian, and then calculate the mean toroidal and cross sectional radii.
First, we consider the van der Waals type interaction, or equivalently effective short-range dominant attraction. Toroidal cross section can be approximated by the hexagonally arranged chains (Fig.\ref{figHexcross})\cite{IK06a}, interacting via effective nearest neighbour van der Waals attraction.
By definition, the excluded volume effect of the segments is incorporated in the hexagonally packed cross section. This hexagonal arrangement could be a good approximation as it has been experimentally observed for the condensed DNA toroid \cite{HD01}.
\begin{figure}[t]
\onefigure[width=4.7cm]{hexcross.eps}
\caption{A complete hexagon with a side of $5$ segments in the toroidal cross section.}
\label{figHexcross}
\end{figure}
\begin{figure}[t]
\onefigure[width=7.9cm]{hexapprox.eps}
\caption{An exactly counted discrete function ${\cal V}_{discrete}(N)$ and its continuum approximation ${\cal V}(N)$ for the number of interacting pairs in the toroidal cross section.}
\label{figHexapprox}
\end{figure}
If the chains are packed in a complete hexagonal cross section (Fig.\ref{figHexcross}),
the winding numbers are $N = 7, 19, 37$, and so on.
In such cases, the number of nearest neighbour interacting pairs between
segments can be counted by the links between neighbouring pairs in the
hexagonal cross section. We then obtain the number as a discrete
function ${\cal V}_{discrete}(N)$.
In the case of a complete hexagon with a side of $(n+1)$ segments, its winding number is given by
$$
N=1+\sum_{i=1}^{n} \!6\, i=3n(n+1)+1 ,
$$
{\it i.e.}, $n = -\frac12 +\frac{1}{\sqrt{3}}\sqrt{N-\frac14}$.
When we connect the centres of nearest neighbour segments in Fig.\ref{figHexcross}, it results in a complete hexagon consisting of regular triangular cells with a side $1$. Counting the number of the regular triangles' sides gives the number of the nearest neighbour interaction links:
\begin{eqnarray}&&
{\cal V}_{discrete}(N) = \bigl[ 3 \times
\left({\rm No.~ of~ the~ regular~ triangles\!:}\, 6n^2 \right)
\nonumber\\&&\quad
+ \left( {\rm Perimeter~ of~ the~ hexagon\!:}\, 6n \right)
\bigr]/2
\nonumber\\&&
=\left( 3\cdot6n^2+6n\right)/2 =3n(3n+1)
\nonumber\\&&
= 3 N - 2\sqrt{3} \sqrt{N-1/4},
\end{eqnarray}
for $N = 1, 7, 19, 37, \cdots$.
Therefore,
for a general value of $N$, we can approximate ${\cal V}_{discrete}(N)$ or analytically continue it to the analytic
function:
\begin{eqnarray}
{\cal V}(N) = 3 N - 2\sqrt{3} \sqrt{N-1/4}.
\end{eqnarray}
This approximates well an exactly counted discrete function ${\cal V}_{discrete}(N)$ (see Fig.\ref{figHexapprox}).
Note that, up to $N=3$ we need not introduce the finite size effect, since there is no difference in the number of interacting pairs between the ideal and the van der Waals nearest neighbour toroids. Thus, $N{}\geq 4$ for this effect.
Note also that the entanglement (knotting) effect of the chain
arrangement is neglected.
The same analysis presented in the previous section leads to the following ``asymptotic'' $c$-$N_c$ relation of the dominant toroid for large $c$:
\begin{eqnarray}
N_c \simeq \left( 2\sqrt{3} \, c \right)^{\frac{2}{5}}.
\end{eqnarray}
Substituting this into $r_c\equiv \frac{L}{2\pi N_c}$, we obtain the mean toroid radius:
\begin{eqnarray}
r_c \simeq {\left(6\pi\right)}^{-\frac15}{\left(\frac{l}{W}\right)}^{\!\!\frac25}L^{\frac15}.
\end{eqnarray}
Note the mean toroid radius of T$4$ DNA in low ionic conditions and Sperm DNA packaged by protamines are quantitatively fitted by this expression \cite{IK06a}. Also, the scaling property $r_{c} \sim L^{\frac15}$ matches the one in \cite{SIGPB03,L15_lit,MKPW05}.
Similarly, the mean radius of the toroidal cross section can be calculated for the complete hexagonal cross section with a side of $(n+1)$ monomers \cite{IK06a}:
\begin{eqnarray}
r_{cross}
&\!\!=&\!\! \frac{2+\sqrt{3}}{4}\left(n+\frac12\right)l_d
\nonumber\\
&\!\!=&\!\! \frac{2\sqrt{3}+3}{12}N^{\frac12}\left[1-\frac{1}{8N} + O\left(\frac{1}{N^2}\right)\right]l_d,
\end{eqnarray}
where $l_d$ is the diameter of the segment.
As $N_c \simeq \left( 2\sqrt{3} \, c \right)^{\frac{2}{5}}$ for large $c$, we have
$$
r_{cross} \simeq \frac{3\sqrt{3}+6}{12}{\left(6\pi\right)}^{\!-\frac25}\!L^{\!\frac25}\! {\left(\frac{W}{l}\right)}^{\!\!\frac15}\!l_d.
$$
Note that the scaling property $r_{cross} \sim L^{\frac25}$ is in
agreement with the one in \cite{SIGPB03} obtained in the asymptotic limit. Also, we can formally consider the case of the ideal toroid ({\it i.e.} $N_c \simeq c$), although it has zero thickness: $r_{cross} \simeq \frac{2\sqrt{3}+3}{24\pi}L{\left(\frac{W}{2\,l}\right)}^{\!\!\frac12} l_d$.
\section{Yukawa potential and general theory}
In experiment, when we put condensing agents such as multivalent cations into DNA
solution, it can cause DNA to undergo the condensation from a
worm-like chain (whip or coil) to toroidal states \cite{HDetal,HD01,YYK99,SIGPB03,IK06a,Bloomfield,MKPW05}. Due to surrounding
ion clouds, effective interaction between the DNA segments could be described by the screened Coulomb (or Yukawa) potential:
$$
V_Y(r) = -W\frac{\exp(-\kappa{l_d}
\left(r\!/\!{l_d}\!-1\!\right))}
{\left(r\!/\!{l_d}\right)},
$$
where \(r\) is the distance between a pair of segments.
The interaction range is characterised by the screening
parameter $\kappa$ (inverse screening length), which depends on salt concentrations. In the low screening limit $\kappa \to 0$, the potential corresponds to the Coulomb interaction.
Not only for the Yukawa potential with various $\kappa$ values, it is in general very difficult to analytically estimate the function ${\cal V}(N{})$ just by counting the number of interacting segmental pairs. This is due to the long-range nature of the potential. Thus, we numerically compute the exact value of the attractive potential in the toroidal cross section and do a fit by the following function. We assume that the function ${\cal V}(N)$ can be in general expanded as a polynomial in the radius of the cross section ($\propto \left(n+\frac12\right)$):
\begin{eqnarray}
&{\cal V}(N) \!\!&\!\!=\!
b_0 {\left(\!n\!+\!\frac12\!\right)}^{\!\alpha}\!\!+\! b_1 {\left(\!n\!+\!\frac12\!\right)}^{\!\alpha-1}\!\!+\! b_2 {\left(\!n\!+\!\frac12\!\right)}^{\!\alpha-2}\!\!\nonumber\\
&&\, +b_3{\left(\!n\!+\!\frac12\!\right)}^{\!\alpha-3}\!\!+\! b_4{\left(\!n\!+\!\frac12\!\right)}^{\alpha-4}\!\nonumber\\
&\!\!&\!\!=\! A_0{\left(\!N\!-\!\frac14\!\right)}^{\!\!\frac{\alpha}{2}}\!\!\!+\!A_1{\left(\!N\!-\!\frac14\!\right)}^{\!\!\frac{\alpha-1}{2}}\!\!\!\!+\!A_2{\left(\!N\!-\!\frac14\!\right)}^{\!\!\frac{\alpha-2}{2}}\!\!\!\!\nonumber\\
&&\, + A_3{\left(\!N\!-\!\frac14\!\right)}^{\!\!\frac{\alpha-3}{2}}\!\!\!\!+\!A_4{\left(\!N\!-\!\frac14\!\right)}^{\!\!\frac{\alpha-4}{2}}\!\!\!,\label{Vn_polynomial}
\end{eqnarray}
where
$A_0=b_0{\left(\frac{1}{\sqrt3}\right)}^{\!\!\alpha}$,
$A_1=b_1{\left(\frac{1}{\sqrt3}\right)}^{\!\!\alpha-1}\!\!\!\!\!\!$,
$A_2=b_2{\left(\frac{1}{\sqrt3}\right)}^{\!\!\alpha-2}\!\!\!\!\!\!$,
$A_3=b_3{\left(\frac{1}{\sqrt3}\right)}^{\!\!\alpha-3}\!\!\!\!\!\!$,
$A_4=b_4{\left(\frac{1}{\sqrt3}\right)}^{\!\!\alpha-4}\!\!\!\!\!\!$.
This fitting is based on the fact that ${\cal V}(N)$ of ideal and van der Waals types are given by
${\cal V}_{ideal}(N) = \frac92{\left(n+\frac12\right)}^{\!4}\!\!-\frac34{\left(n+\frac12\right)}^{2}\!\!-\frac{3}{32}=\frac{N(N-1)}{2}$
and
${\cal V}_{VDW}(N) = 9{\left(n+\frac12\right)}^{2}\!\!-6{\left(n+\frac12\right)}+\frac34=3 N - 2\sqrt{3} \sqrt{N-\frac14}$.
All segments of the ideal toroid interact with each other with interaction energy $-W$, and ${\cal V}(2)$ is normalised to unity. This means the upper bound of ${\cal V}(N)$ should be ${\cal V}_{ideal}(N)$. On the other hand, the lower bound of ${\cal V}(N)$ should be given by the van der Waals nearest neighbour toroid, interacting with energy $-W$. Hence, we have the inequality
\begin{eqnarray}
{\cal V}_{ideal}(N)\geq{\cal V}(N)\geq{\cal V}_{VDW}(N).
\end{eqnarray}
Therefore, we speculate the Yukawa potential shall be in this region and might have a cubic term in $\left(n+\frac12\right)$ in eq.(\ref{Vn_polynomial}). For the analysis, the cross section has to be small relative to the mean toroid radius. Hence, a toroid-toroidal globule transition point ($r_c \simeq r_{cross} $) may get modified if we consider this effect more seriously.
The corresponding function $f(N)$ is given by:
\begin{eqnarray}
f(N) &\!\!\!=&\!\!\! N {\cal V}(N+1) - (N+1) {\cal V}(N)
\nonumber\\
&\hspace{-50pt}=&\hspace{-25pt} \sum_iC_i \! \Bigl[ \left(\frac{i}{2}-1\!\right)\!N^{\!\frac{i}{2}}\!+\!\frac{i^2}{16}N^{\!\frac{i}{2}\!-\!1}
\nonumber\\&&\hspace{-30pt}
+ \left({\frac{i}{384}\!\left(\frac{i}{2}\!-\!1\right)\left(7i\!-\!34\right)}\!\right)\!N^{\!\frac{i}{2}\!-\!2}
+\!O\!\left(N^{\!\frac{i}{2}\!-\!3}\right)\Bigr]
\end{eqnarray}
where we sum over $i=\alpha, \alpha\!-1\!, \alpha\!-\!2, \alpha\!-\!3, \alpha\!-\!4$ and
$C_{\alpha}\!=\!A_0$, $C_{\alpha-1}\!=\!A_1$, $C_{\alpha-2}\!=\!A_2$, $C_{\alpha-3}\!=\!A_3$, $C_{\alpha-4}\!=\!A_4$.
In the asymptotic limit but below the toroid-toroidal globule transition point, $f(N)$ becomes $A_0\!\left(\frac{\alpha}{2}-1\!\right)\!N^{\!\frac{\alpha}{2}}\!$ for $2< \alpha \leq 4$,
and $A_1\!\left(\frac{\alpha-1}{2}\!-\!1\!\right)\!N^{\!\frac{\alpha\!-\!1}{2}}\!$ for $\alpha=2$ (VDW type). Note that to satisfy the condition $f(N)>0$ ({\it i.e.} $r_c>0$), we require $C_{\alpha}=A_0\,({\sim}\,b_0)>0$ for $2<\alpha\leq 4$ and $C_{\alpha\!-\!1}=A_1\,({\sim}\,b_1)<0$ for $\alpha=2$.
The mean toroidal ($r_c$) and cross sectional ($r_{cross}$) radii are calculated for large $c$:
\begin{itemize}
\setlength{\itemsep}{-3pt}
\item[I)] Ideal type ($\alpha=4$): $r_c = \frac{9}{2b_0}\frac{4\pi l}{W\!L}$, $r_{cross}=\frac{2+\sqrt3}{24\pi}\!{\left(\frac{b_0}{3}\right)}^{\!\frac12}\!{\left(\!\frac{W}{l}\!\right)}^{\!\frac12}l_d L$ ($b_0>0$),
\item[II)] Coulomb type ($\alpha=3$): $r_c = 6\pi^{\!\frac13}b_0^{\!-\!\frac23}\!{\left(\!\frac{l}{W}\!\right)}^{\!\frac23}L^{\!-\!\frac13}$, $r_{cross}=\frac{3+2\sqrt3}{12}{\left(\frac{\sqrt3b_0}{72\pi^2}\right)}^{\!\frac13}\!{\left(\!\frac{W}{l}\!\right)}^{\!\frac13}l_d L^{\!\frac23}$ ($b_0>0$),
\item[III)] VDW type ($\alpha=2$): $r_c = {\left(\frac{\pi}{6}b_1^2\right)}^{\!-\!\frac15}\!{\left(\!\frac{l}{W}\!\right)}^{\!\frac25}L^{\!\frac15}$, $r_{cross}=\frac{2+\sqrt3}{4}{\left(\frac{-\!b_1}{2^33^3\pi^2}\right)}^{\!\frac15}\!{\left(\!\frac{W}{l}\!\right)}^{\!\frac15}l_dL^{\!\frac25}$ ($b_1<0$).
\end{itemize}
Note that the radii of the ideal and the van der Waals nearest neighbour toroids correspond to case I with $b_0=\frac92$ and case III with $b_1=-6$, respectively. We will show below that case II is in fact the Coulomb type.
\def\hspace{-3pt}{\hspace{-3pt}}
\begin{table}
\begin{tabular}{|c|c|c|c|}\hline
$\kappa$ & $\alpha$ & $b_0$ & $b_1$
\\\hline\hline
0 & 3.00007$\pm$0.00003 & 8.38$\pm$0.00 & -6.25$\pm$0.02
\\\hline
\hspace{-3pt} 0.01 \hspace{-3pt}\mh$\!$ & 2.65391$\pm$0.00529& 28.92$\pm$0.74 & -214.27$\pm$13.29
\\\hline
0.1 & 2.00845$\pm$0.00386 & 110.45$\pm$2.02 & -954.14$\pm$37.64
\\\hline
0.2 & 2.00048$\pm$0.00207 & 59.20$\pm$0.59 & -259.93$\pm$9.06
\\\hline
0.3 & 2.00008$\pm$0.00113 & 41.06$\pm$0.22 & -121.61$\pm$3.17
\\\hline
0.4 & 2.00002$\pm$0.00063 & 32.03$\pm$0.10 & -73.32$\pm$1.34
\\\hline
0.5 & 1.99957$\pm$0.00020 & 26.69$\pm$0.02 & -51.12$\pm$0.25
\\\hline
0.6 & 1.99958$\pm$0.00013 & 23.12$\pm$0.01 & -38.56$\pm$0.13
\\\hline
0.7 & 1.99963$\pm$0.00008 & 20.58$\pm$0.01 & -30.68$\pm$0.08
\\\hline
0.8 & 1.99969$\pm$0.00006 & 18.69$\pm$0.01 & -25.38$\pm$0.05
\\\hline
0.9 & 1.99974$\pm$0.00005 & 17.23$\pm$0.00 & -21.63$\pm$0.04
\\\hline
1.0 & 1.99978$\pm$0.00004 & 16.07$\pm$0.00 & -18.87$\pm$0.03
\\\hline
\hspace{-3pt} $\beta\!=\!6$ \hspace{-3pt}\mh$\!$ & 1.99999$\pm0.00001$ & 9.56$\pm$0.00 & -6.85$\pm$0.00
\\\hline
\end{tabular}
\caption{Least Squares Fitting of exactly computed ${\cal V}(N)$ for the Yukawa potential with various screening parameter $\kappa$ and for $-W{\left(\frac{l_d}r\right)}^{\!\beta}$ with $\beta=6$ using eq.(\ref{Vn_polynomial}). For convenience, $W$ and $l_d$ are taken to be unity. Data is presented for $\alpha$, $b_0$ and $b_1$ only.}
\label{Yukawadata}
\end{table}
Table \ref{Yukawadata} shows the least square fit of exactly computed ${\cal V}(N)$ for the potential $-W{\left(\frac{l_d}r\right)}^{\!\beta}$ and the Yukawa potential with inverse screening length $\kappa$ using eq.(\ref{Vn_polynomial}).
In numerics, we considered the complete hexagon with a side of up to $50$ segments ({\it i.e.} winding number $N=3n(n+1)+1=7351$). The diameter of the chain $l_d$ is taken to be unity for brevity.
We can confirm that the $\kappa=0$ (Coulomb interaction) result is grouped by case II (Coulomb type).
The Yukawa potential with $\kappa=0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1.0$ and $\beta=6$ results are classified to case III (VDW type).
Of great interest is the Yukawa potential with $\kappa=0.01, 0.1, 0.2$, where the exponent $\alpha$ takes the values $2\sim3$. The transition from the Coulomb type (II) to the van der Waals type (III) occurs in this region. We note that for small $n<3$, fit results deviate from ${\cal V}(N)$ of the Yukawa with $\kappa\leq 0.3$. However, it does not affect the large $n$ behaviours, thus results reported here sustain. It should be also mentioned that the potential $-W{\left(\frac{l_d}r\right)}^{\!\beta}$ with $\beta=3\sim24$ are categorised by case III (VDW type). But for $\beta=1\sim2$, cubic term becomes important (case II).
The inequality ${\cal V}_{ideal}(N)\geq{\cal V}(N)\geq{\cal V}_{VDW}(N)$ means the exponents $\nu\!\left(N_c\right)$ for the mean toroid radius $r_c\sim L^{\nu\left(N_c\right)}$ are bounded by those of ideal and van der Waals toroids in the asymptotic limit:
\begin{eqnarray}
-1 \,{\leq}\, \nu\!\left(N_c\right) \,{\leq}\, \frac15.
\end{eqnarray}
Also, we derive a significant fact that, as far as ${\cal V}(N)$ is given as a polynomial in $\left(n+\frac12\right)$,
we never have $f(N_c){\sim}N_c$ ({\it i.e.} $N_c^2{\sim}c{\sim}L^2$), thus $r_c{\sim}L^0$ for large $c$. This is contradicting to the experimentally well known observation $r_c \sim L^{0}$. To resolve this problem, we plot the exponents $\nu \!\left(N_c\right)$ of the mean toroid radius $r_c\sim L^{\nu \left(N_c\right)}$ for the ``finite'' dominant winding number $N_c$ (Fig.\ref{figFinitenu}).
\begin{figure}[t]
\hspace{-5pt}
\onefigure[width=8.4cm]{finite_nu.eps}
\vspace{-10pt}
\caption{The exponents $\nu\!\left(N_c\right)$ of the mean toroid radius $r_c\sim L^{\nu\left(N_c\right)}$ for the finite dominant winding number $N_c$ with screening parameter $\kappa=0.5\sim1.0$, Coulomb, full and nearest neighbour Van der Waals, and delta function (ideal) attractions.}
\label{figFinitenu}
\end{figure}
The exponents are now defined by
$\nu\!\left(N_c\right)\!\equiv\!{1-{2}/\!{{\nu_f}(N_c)}}$:
\begin{eqnarray*}&&
r_c =L/(2\pi{N_c})
= L/\!\bigl({2\pi{c^{\frac{1}{{\nu_f}(N_c)}}}}\bigr)
\nonumber\\&&
={{{\left(2\pi\right)}^{\frac{2}{{\nu_f}(N_c)}-1}}}{\left({W}/{2l}\right)}^{\!-\frac{1}{{\nu_f}(N_c)}}\!L^{1-\frac{2}{{\nu_f}(N_c)}},
\end{eqnarray*}
where the finite function $\nu_f(N_c)$ is defined as $c= N_c^{\nu_f(N_c)}$.
We find that for
$N_c=100\sim400$ ({\it i.e.} realistic winding number of DNA toroids such as T$4$ DNA \cite{YYK99} or Sperm DNA \cite{Bloomfield}), we have $\nu\,\,{\simeq}\,\,0$ for the Yukawa interaction with
$\kappa=0.5\sim1.0$, and $\nu=0.1\,\,{\sim}\,\,0.13$ for the van der Waals interaction. These could explain the experimental observation: $\nu\,\,{\simeq}\,\,0$.
It is also possible to describe conformational transitions. When the radius of cross section becomes comparable to the segmental diameter $r_{cross} \simeq l_{d}$, the whip-toroid ($N_c=1$) transition occurs. We have $r_c \sim L$ and the transition line $l/W \sim L^2$, which are independent of the interaction shape. Moreover, for the toroid to toroidal globule transition point $r_c \simeq r_{cross} $, we have $r_c \sim l_d^{\frac23}L^{\frac13}$.
The scaling property $r_c \sim L^{\frac13}$ is similar to the expected scaling of globule like objects. Although the outer radius $r_{outer}$ scales differently for large $c$: $r_{outer}=r_c + r_{cross} \sim r_{cross}$, it scales the same as $r_c$ at the transition point.
\section{Conclusions}
To summarise, we have shown how different microscopic interactions between chain segments can alter the physical properties of the condensed toroid. In the asymptotic limit, exponents of the mean toroidal and cross sectional radii are categorised into three distinct species: van der Waals type, Coulomb type, and ideal type. For the intermediate winding number of $N_c=100\sim400$, we find $\nu\,\,{\simeq}\,\,0$ for the Yukawa interaction with inverse screening length $\kappa=0.5\sim1.0$, and $\nu=0.1\,\,{\sim}\,\,0.13$ for van der Waals interaction. These finding are consistent to the experimentally well known observation $\nu\,\,{\simeq}\,\,0$. It would be of great interest to check experimentally $r_c \sim L^{\nu}$ for a fixed salt concentration, {\it i.e.}, for a fixed screening parameter $\kappa$.
If we naively apply the asymptotic values of $r_c$ and $r_{cross}$, transition lines are found to be interaction dependent and to agree with ref. \cite{SIGPB03} in VDW case. However, as has been shown, the asymptotic relations are not so accurate for finite and realistic winding number $N_c$. Therefore, the transition lines are to be studied with a special care.
Finally, It should be stressed that our generic theory can be applied straightforwardly to any toroidal condensation of semiflexible polymer chains with any type and number of microscopic interactions.
\acknowledgments
We are grateful to W. Paul and S. Stepanow for stimulating discussions. We would also like to thank S. Trimper and P. Bruno for stimulating discussions and suggesting investigation of
$-W{\left(\frac1r\right)}^{\!\beta}$.
N.K. acknowledges the Deutsche Forschungsgemeinschaft for financial support.
|
cs/0701009
|
\section{Introduction}
The problem that we consider in this paper can be formulated as a
clustering problem. These types of problems have been studied for
quite long time and have many theoretical and practical applications
in computer science \cite{datta}. A branch of clustering problems
includes problems in which given a set of points the goal is to find
a ``cluster'' (or clusters) with minimum size or maximum number of
points. Typical examples of clusters include spheres, boxes, or any
other shape of fixed complexity. Of course, the difficulty of the
problem greatly depends on the definition of cluster. The clusters
that we consider here are all the shapes of constant diameter.
The
\emph{diameter} of a set $S$ is defined as
$${\rm diam}(S)= \sup_{x,y \in S} |x-y|,$$
where $|x-y|$ is the Euclidean distance between the two points $x$
and $y$. Specifically, we consider the following problem:
\begin{itemize}
\item[] {\bf Problem 1:} Let $P$ be a set of $n$ points in
$\mathbb{R}^d$ and $r>0$ be a real number. Find a subset $S
\subseteq P$ of maximum size which satisfies ${\rm diam}(S) \le r$.
\end{itemize}
A \emph{clique} is a graph in which every two vertices are adjacent.
For a graph $G$, let $\omega(G)$ denote the size of the maximum
clique in $G$, i.e. $\omega(G)$ is the maximum number of vertices of
$G$ such that every two of them are adjacent. Determining
$\omega(G)$ is called the \emph{maximum clique problem}. A closely
related topic is the notion of an independent set. An
\emph{independent set} in $G$ is a subset of vertices such that no
two of them are adjacent. Let $\alpha(G)$ denote the size of the
maximum independent set in $G$. Determining $\alpha(G)$ is called
the \emph{maximum independent set problem}. Denote by $G^c$ the
complement of $G$, and note that $\omega(G)=\alpha(G^c)$. Thus, the
maximum clique problem for $G$ is equivalent to the maximum
independent set problem for $G^c$. For more on these topics we refer
the reader to~\cite{west}.
\comment{ Problem~1 is equivalent to the maximum clique problem in
\emph{disc graphs}: Build a graph $G$ with the vertex set $V(G)=P$
and connect two points $x,y$ if and only if $|x-y| \le r$. There is
a one to one correspondence between cliques in $G$ and sets of
diameter at most $r$ in $P$.
}
\peyman{Problem~1 is equivalent to the maximum clique problem in
\emph{disc graphs} which are defined as follows. Given a point set
$P \subset \mathbb{R}^d$ and a parameter $r$, a disc graph $G$ is
defined by $V(G)=P$ and $xy \in E(G)$ if and only if $|x-y| \le r$.
There is a one to one correspondence between cliques in $G$ and sets
of diameter at most $r$ in $P$. }
In Problem~1 the diameter is fixed and our objective is to maximize the
number of points. On the other hand, we can fix the number of points
and ask for the minimum diameter:
\begin{itemize}
\item[] {\bf Problem 2:} Let $P$ be a set of $n$ points in
$\mathbb{R}^d$ and $k>0$ be an integer. Find a subset $S \subseteq P$
of size $k$ with minimum diameter.
\end{itemize}
We show that both these problems are NP-hard when the dimension is
sufficiently large, i.e., for some $d = \Theta(\log n)$. In fact, we
prove a stronger result which shows that even certain approximations
of these problems are impossible unless P=NP. These approximations
are defined in the following way:
\begin{definition}
For $t,s \ge 1$, a $(t,s)$-approximation algorithm for Problem~1 is an algorithm
that returns a set $S$ of size at least $\mathit{Opt}/t$ so
that ${\rm diam}(S) \le sr$, where $\mathit{Opt}$ is the size of the
optimal answer to Problem~1.
\end{definition}
\begin{definition}
For $t,s \ge 1$, a $(t,s)$-approximation algorithm for Problem~2 is an algorithm
that returns a set $S$ of size at least $k/t$ so that ${\rm diam}(S) \le
s \times \mathit{Opt}$, where $\mathit{Opt}$ is the optimal answer
to Problem~2.
\end{definition}
These two problems are obtained by relaxing the size and diameter
constraints of the output set. A simple observation shows that
these two problems are equivalent.
\begin{lemma}
For $t,s \ge 1$, there exists a polynomial time $(t,s)$-approximation algorithm for
Problem 1 if and only if there exists a polynomial time $(t,s)$-approximation
algorithm for Problem 2.
\end{lemma}
\begin{proof}
Let ${\cal A}$ be a $(t,s)$-approximation algorithm for Problem 1.
Consider an instance $(P,k)$ of Problem 2. For every pair of
points $x,y \in P$, run ${\cal A}$ with parameter
$r_{x,y}:=|x-y|$. We output the minimum $r_{x,y}$ for which the
answer of ${\cal A}$ is of size at least $k/t$. Let $S_o$ be the
optimal solution to the $(P,k)$ instance of Problem 2. At some
point ${\cal A}$ is called with parameter $r:={\rm diam}(S_o)$. Now the
output of ${\cal A}$ is a set of size at least $|S_o| / t \ge
k/t$ and with diameter at most $rs= s \times {\rm diam}(S_o)$.
To prove the other direction let ${\cal B}$ be a
$(t,s)$-approximation algorithm for Problem 2. Consider an
instance $(P,r)$ for Problem 1. A $(t,s)$-approximation algorithm
for Problem 1 can be obtained in a similar way by running ${\cal
B}$ for every $k=1,\ldots,n$.
\end{proof}
Since these two problems are equivalent we refer to both of them
as the Diameter Approximation Problem.
Both Problems 1 and 2 are solvable in the 2-dimensional plane in
polynomial time \cite{aggarwal,datta,polytopes}. For Problem~2 the
fastest known algorithm achieves the running time $O(n\log n +
k^{2.5}n\log k)$~\cite{polytopes}. It is shown in
\cite{cccg} that in the 3-dimensional space there is a $({\pi \over
\cos^{-1}1/3 },1)$-approximation algorithm. Finally, when the
dimension $d$ is a fixed constant, one can design a polynomial time
approximation scheme achieving a $(1,1+\epsilon)$-approximation, for
every $\epsilon>0$~\cite{cccg}. It is also easy to see that there
exists a trivial $(1,2)$-approximation algorithm for this problem in
any dimension: a ball with radius $r$ about a point $x \in P$
containing the maximum number of points is a $(1,2)$-approximation
for Problem 1. Thus, it is interesting to study at which point the
problem turns from polynomially solvable to NP-hard. We have the
following result in this direction:
\begin{theorem}
\label{thm:hardness} For every $\epsilon>0$ there exists
$d=\Theta(\log n)$, so that there is no polynomial time
$(\frac{95}{94}-\epsilon,\sqrt{4/3}-\epsilon)$-approximation
algorithm for the Diameter Approximation Problem in dimension $d$
unless P=NP.
\end{theorem}
We also improve upon the trivial $(1,2)$-approximation algorithm
and obtain the following theorem:
\begin{theorem}
\label{thm:approx} For every $\epsilon>0$ there is a polynomial time
$(1,\sqrt{2}+\epsilon)$-approximation algorithm for the Diameter
Approximation Problem in any dimension.
\end{theorem}
In Section~\ref{sec:hardness} we prove Theorem~\ref{thm:hardness}.
We use spectral properties to move from combinatorics of graphs to
geometry of Euclidean space. This technique in combination with a
hardness result regarding the maximum independent set problem in
$3$-regular graphs proves Theorem~\ref{thm:hardness}. In
Section~\ref{sec:approx} we prove Theorem~\ref{thm:approx} using
simple geometric techniques. Finally, in the ``Corollaries''
subsection we observe that having Theorem~\ref{thm:approx} in hand,
it is possible to move in the other direction.
Corollaries~\ref{cor:SDP} and~\ref{cor:eigenvalue} show that one can
apply Theorem~\ref{thm:approx} to the geometric representation of
the graph to approximate the maximum independent set problem for
certain graphs.
\section{From Graphs to Euclidean Space}
In this section we prove Theorem~\ref{thm:hardness}.
We show that unless P=NP, there is no
$(\frac{95}{94}-\epsilon,\sqrt{4/3}-\epsilon)$-approximation algorithm for the Diameter
Approximation Problem for every $\epsilon>0$.
We use spectral techniques to show that a certain metric on
the graph embeds isometrically into the Euclidean space.
This type of reduction from geometry to graph theory via metric
embedding has been successfully applied to various problems dealing
with the so called $(1-2)$-metrics~\cite{indyk,trevisan}.
Our proof of Theorem~\ref{thm:hardness} relies on the
following result:
\begin{theorem}{ \cite{3MIS}}
\label{thm:3reg} For every $\epsilon>0$, unless P=NP, there is no polynomial time algorithm
that approximates the maximum
independent set problem in $3$-regular graphs within a factor of $\frac{95}{94}-\epsilon$.
\end{theorem}
\subsection{Proof of Theorem~\ref{thm:hardness}}\label{sec:hardness}
To prove the theorem we
reduce the Diameter Approximation Problem to the maximum independent set problem in $3$-regular
graphs.
Consider a $3$-regular graph $G$. Denote by $G^c$ the complement of
$G$, and notice that cliques in $G^c$ correspond to independent sets
in $G$. We begin by finding a lower bound for the minimum eigenvalue
of the adjacency matrix $A_{G^c}$ of $G^c$. Denote by $\lambda_1 \le
\ldots \le \lambda_n$ the eigenvalues of $A_{G^c}$. Since $G^c$ is
an $(n-4)$-regular graph, its Laplacian can be written as
$L_{G^c}=(n-4)I - A_{G^c}$. Thus for a vector $v$, we have
$A_{G^c}v=\lambda v$ if and only if $L_{G^c}v=(n-4-\lambda)v$. This
shows that the eigenvalues of $L_{G^c}$ are $n-4-\lambda_n \le
\ldots \le n-4-\lambda_1$. It is well-known \cite{godsil} that the
maximum eigenvalue of the Laplacian is bounded by $n$. So the
minimum eigenvalue of $A_G$ is at least $-4$. Define
$Q:=A_{G^c}+(4+\gamma)I$ for $\gamma > 0$ chosen sufficiently small.
By the same argument that we used above to find the eigenvalues of
$L_{G^c}$, one can show that the smallest eigenvalue of $Q$ is
$\lambda_1 + 4+\gamma>0$ which means that $Q$ is a symmetric
positive definite matrix. This implies that there exists a
nonsingular matrix $U$, which can be found using elementary
techniques, such that $Q=U^tU$ (see~\cite{lalgebra} page 285 for
instance). Define a function $f:[n] \rightarrow \mathbb{R}^n$ by
setting the value of $f(i)$ to be the $i$-th row of the matrix $U$.
Note that
$$|f(i)|^2=f(i)\cdot f(i)=Q_{i,i}=4+\gamma,$$ and
$$|f(i)-f(j)|^2=|f(i)|^2+|f(j)|^2-2f(i)\cdot f(j)=8+2\gamma-2Q_{i,j}.$$
Thus
$$|f(i)-f(j)|=\left\{\begin{array}{ll}
\sqrt{6+2\gamma} & A^c_{(ij)} = 1 \\
\sqrt{8+2\gamma} & A^c_{(ij)} = 0
\end{array}\right.$$
Consider the vertex $v_i$ of $V(G^c)$ which corresponds to the
$i$-th row and column of $A^c$.
Map $v_i$ to the point $f(i)$ in Euclidean space $\mathbb{R}^n$.
Let $P$ be the resulting point set in $\mathbb{R}^n$.
The above properties imply that every vertex of $V(G^c)$ is mapped to
a vector of magnitude $2$ and the distance between two vertices is $\sqrt{6+2\gamma}$, if
there is an edge between them, and $\sqrt{8+2\gamma}$ if not.
Using the Johnson-Lindenstrauss dimension reduction
lemma~\cite{dimreduction2,dimreduction} there exists a dimension
$d=O(\lambda^{-2}\log n)$, and a polynomial algorithm which maps $P$
into $\mathbb{R}^d$ such that the distance between any two points of $P$
changes by a factor of at most $1+\lambda/2$.
Let $g: V(G^c) \rightarrow \mathbb{R}^d$ be the corresponding map. Thus if
we choose $\lambda$ and $\gamma$ small enough, for every two
vertices $v_i, v_j \in V(G^c)$ the distance between $g(i)$ and
$g(j)$ is at most $(1+\lambda)\sqrt{6}$ if they are connected in
$G^c$ and at least $(1+\lambda)^{-1}\sqrt{8}$ if they are not
connected in $G^c$. So for a set $S \subset V(G^c)$, its geometric
representation will have diameter at most $(1+\lambda)\sqrt{6}$ if
it is a clique but it will have diameter at least
$(1+\lambda)^{-1}\sqrt{8}$ otherwise. By picking $\epsilon$ to be
small enough and applying a
$(\frac{95}{94}-\epsilon,\sqrt{4/3}-\epsilon)$-approximation
algorithm to Problem 1 with $g(G^c)$ and $r=\sqrt{6}$ we can find a
clique of size at least $1/(\frac{95}{94}-\epsilon)$ times the size
of the maximum clique in $G^c$. Theorem~\ref{thm:3reg} shows that
this is impossible unless $P=NP$.
\subsection{Proof of Theorem~\ref{thm:approx}}\label{sec:approx}
In this section we prove Theorem~\ref{thm:approx} by applying
simple geometric techniques. We follow the general ideas and
techniques of \cite{cccg,aggarwal}, borrowing and generalizing the
main tool from \cite{aggarwal}. The idea is to extend and
generalize the trivial $(1,2)$-approximation. One way to interpret
the $(1,2)$-approximation is to say that any set $A$ of diameter $r$
can be placed inside a ball of diameter $2r$ centered at a point in $A$. To obtain a
$(1,\sqrt{2}+\epsilon)$-approximation, we first show that any set of
diameter $r$ can actually be placed inside a ball of diameter
$(\sqrt{2}+\epsilon)r$, and then we produce a polynomial time algorithm to
compute such a ball.
Let $A$ be the optimal answer to Problem~1. We start by proving
that $A$ is inside a ball of diameter $(\sqrt{2}+\epsilon)r$. Let
$B(P,t)$ denote a ball of radius $t$ centered at point $P$.
At the beginning, $P_1$ is an arbitrary point of $A$ and thus
we have $A \subset B(P_1,r)$. At the $i$-th step we assume $A
\subset B(V_i,r_i)$ for a $V_i \in \mathbb{R}^d$ and some value
$r_i \le r$ to be determined later.
Let $P_{i+1}$ be the point of $A$ with maximum distance to $V_i$. This
implies,
$$A \subset B(V_i,r_i) \cap B(P_{i+1},r) \cap B(V_i,|V_i-P_{i+1}|)$$
and since $|V_i-P_{i+1}| \le r_i$, we have:
$$A \subset B(P_{i+1},r) \cap B(V_i,|V_i-P_{i+1}|).$$
If $x=|V_i-P_{i+1}| \le r\sqrt{2}/2$ then the set of points is
inside a ball of diameter $\sqrt{2}r$. So we assume $x >
r\sqrt{2}/2$.
Consider a point $L$ on the intersection of boundaries of the two
balls $B(P_{i+1},r)$ and $B(V_i,x)$ (Figure \ref{fig:step}).
Consider the plane passing through $L$, $P_{i+1}$ and $V_i$ and
draw the line $LV_{i+1}$ perpendicular to the segment
$P_{i+1}V_i$. A simple calculation proves that:
$$|L-V_{i+1}| = r \sqrt{1-{r^2\over 4x^2}}\le r \sqrt{1-{r^2\over 4r_i^2}}$$
Define $r_{i+1} = r \sqrt{1-{r^2\over 4r_i^2}}$. It can be also
verified that if $x > r \sqrt{2}/2$ then $|P_{i+1}-V_{i+1}| <
|V_{i+1}-L|$ and the ball $B(V_{i+1},|L-V_{i+1}|)$ will contain the
intersection $B(P_{i+1},r) \cap B(V_i,x)$. This implies $A \subset
B(V_{i+1},|L-V_{i+1}|) \subset B(V_{i+1},r_{i+1})$. It is easy to
check that the sequence $r_1, r_2, \cdots$ converges to
$r\sqrt{2}/2$. Thus given any $\epsilon>0$ it is possible to fix a
constant $k$ (depending only on $\epsilon$) such that $r_k < r\sqrt{2}/2 + \epsilon$.
To obtain an algorithm from the discussion above, we only need
to consider all different possible choices for $P_1,\ldots,P_k$.
Discarding the invalid choices or choices that result in an
invalid state, each choice for $P_1,\ldots,P_k$ leads to a ball
with radius at most $r\sqrt{2}/2+\epsilon$.
Now the algorithm outputs the one
which contains the maximum number of points. Since $k$ is a
constant, the algorithm is polynomial.
\begin{figure}
\begin{center}
\ \input{step.pstex_t}
\end{center}
\caption{The position of $V_{i+1}$ with respect to the positions of $V_i$, $P_{i+1}$, and $L$. }
\label{fig:step}
\end{figure}
\subsection{Corollaries}
The following corollaries can be obtained using techniques employed
in the previous section.
\begin{corollary}
\label{cor:SDP} Fix $\delta>0$ and let $G$ be a graph such that
there exists a mapping $f:V(G)\rightarrow \mathbb{R}^n$ satisfying
$|f(u)-f(v)|
>\sqrt{2}+\delta $ if $uv \in E(G)$, and $|f(u)-f(v)| \le 1$ otherwise. Then
it is possible to find the size of the maximum independent set of
$G$ in polynomial time.
\end{corollary}
\begin{proof}
Let $V(G)=\{v_1,\ldots,v_n\}$.
Suppose that the $i$-th row of a matrix $M$ be the vector $f(v_i)$. Let further $A=MM^t$. Clearly,
$A$ is positive semi-definite, and $|f(v_i)-f(v_j)|^2=A_{ii}+A_{jj}-2A_{ij}$. Thus finding the map $f$
reduces to finding a positive semi-definite matrix $A$ with
$$A_{ii}+A_{jj}-2A_{ij} > (\sqrt{2}+\delta)^2 \qquad \qquad \forall v_iv_j \in E(G),$$
and
$$A_{ii}+A_{jj}-2A_{ij} \le 1 \qquad \qquad \forall i\neq j, v_iv_j \not\in E(G).$$
As in~(\cite{LLR95}, Theorem 3.2) the ellipsoid algorithm can be invoked to find such a matrix $A$.
Then we apply the algorithm of
Theorem~\ref{thm:approx} to $f(V(G))$ with the setting $r=1$ and $\epsilon=\delta/2$. Let $I$ be an independent
set of maximum size in $G$. Then the diameter of $f(I)$ is at most $1$ because $|f(u)-f(v)| \le 1$,
if $uv \not\in E(G)$. The algorithm given in the proof of Theorem~\ref{thm:approx}
finds a set $P$ of diameter $\sqrt{2}+\delta/2$ whose size is at least $|I|$. Since $|f(u)-f(v)|
>\sqrt{2}+\delta $ when $uv \in E(G)$, we conclude that $f^{-1}(P)$ is an independent set.
This completes the proof.
\end{proof}
\begin{corollary}
\label{cor:eigenvalue} Fix $\epsilon>0$ and let $G$ be a graph
whose minimum eigenvalue is at least $-2+\epsilon$. It is possible
to find the size of the maximum independent set of $G$ in
polynomial time.
\end{corollary}
\begin{proof}
By the proof of Theorem~\ref{thm:hardness} every such graph satisfies the condition of
Corollary~\ref{cor:SDP}.
\end{proof}
\section{Concluding Remarks}
\begin{itemize}
\item We believe that Theorem~\ref{thm:approx} is ``almost''
sharp in the sense that the constant $\sqrt{4/3}-\epsilon$ in
Theorem~\ref{thm:hardness} can be improved to $\sqrt{2}-\epsilon$ becoming arbitrarily close to the $\sqrt{2}+\epsilon$
upper bound of Theorem~\ref{thm:approx}.
\item The main idea behind the proof of Theorem~\ref{thm:approx} was to introduce
a polynomial time algorithm that given $n$ points
computes a ball of diameter
$(\sqrt{2}+\epsilon)r$ which contains the largest subset of the points that has diameter at most $r$.
The fact that such a ball exists was already known, and in fact
stronger results have been obtained using Helly-type arguments (we refer the reader to~\cite{DGK63,MatBook}
for the proofs and the description of the Helly-type theorems). However, the novel part of the proof was
the algorithmic aspect, and showing that there exists a polynomial time algorithm which finds
such a ball.
\item There is already a body of work dedicated to characterization of
all graphs with the smallest eigenvalue of at least
$-2$ (see~\cite{bussemaker,lambda-2}). These graphs have been
characterized as ``generalized line graphs'' plus some finite set
of exceptions. This characterization gives an alternative
proof for Corollary~\ref{cor:eigenvalue} which uses a different polynomial time algorithm.
\end{itemize}
\section*{Acknowledgements}
The authors would like to thank the anonymous referee for his/her
helpful comments and suggestions that greatly helped us to remove
some inaccuracies from the earlier version of this article.
\bibliographystyle{abbrv}
|
physics/0701194
|
\section{}
\section{Introduction}
The importance of cold collisions with regard to the dynamics of
dilute neutral atom clouds was realized soon after the advent of
laser-cooled and trapped atomic samples \cite{weinj03}. Since
then, numerous investigations, both experimental and theoretical, have
been made into the collisional properties of many different
homonuclear
\cite{pritd88jun,wiemc89aug,goulp92aug,aspea92dec,julip93jun,shimf93aug,shimf94nov,rolss95jan,julip95feb,walkt95aug,idot95dec,vassw06mar}
and (later) heteronuclear
\cite{ertmw94oct,ingum99jan,saloc99dec,vieid00jun,mosst01apr,vassw04jul2}
systems. Collisional studies are in themselves interesting, leading to
an in-depth understanding of the various scattering mechanisms present
at such low kinetic energies and methods by which we can have some
measure of control over elastic and inelastic collisions.
As opposed to collisions between thermal atoms, collisions between
cold atoms are sensitive to the long-range part of the interatomic
interaction potential. The de~Broglie wavelength may become comparable
to the characteristic range of the interatomic potential, and (in the
presence of a light field) the possibility of exciting a
quasi-molecular state, and the subsequent decay of that state during
a collision, becomes important. These effects lead to phenomena such as
scattering resonances, interaction retardation, optically assisted
collisions, photoassociation, optical shielding and the formation of
ground-state molecules.
Optically assisted collisions lead to large losses in magneto-optical
traps (MOTs) and are clearly dependant upon experimental parameters
such as the intensity and detuning of the light frequencies present. A
full theoretical treatment of these collisions is often hampered by the
complexity of the molecular hyperfine structure \cite{weinj03},
thus MOTs are usually empirically optimized for their intended
application, and comparisons with theory or other experiments are
difficult. Collisions in the absence of a resonant light field are, on
the other hand, a theoretically more tractable problem and their
associated loss rate coefficients are fundamental properties of a
given system, allowing direct comparisons with theory and between
experiments. Homonuclear and heteronuclear collisions, "in the dark",
between isotopes of a single element are both mediated at long-range
by the van-der-Waals interaction ($\propto 1/\text{R}^6$) and any
differences are due, in the main, to differing atomic structures and
quantum statistical symmetries.
Due to the high internal energy (19.8\,eV) of metastable ($2\,\mbox{$^3 \! S _1$}$)
helium atoms (He*), the spherical symmetry of this atomic state, and
the inverted (hyper)fine structure of the atoms; ionizing collisions
(Penning (PI) and associative (AI)):
\begin{equation}
\begin{array}{lr}
\textup{He*}+\textup{He*}\rightarrow \textup{He}+\textup{He}^++e^- &
\mbox{(PI)},\\
\\
\nonumber \textup{He*}+\textup{He*}\rightarrow \textup{He}^{+}_2+e^- &
\mbox{(AI)},
\end{array}
\label{PenningIonisation}
\end{equation}
dominate losses in trapped samples of laser-cooled He*. This has
provided a unique setting for the study of cold ionizing collisions in
which the highly efficient, direct detection of collisional loss
products using charged-particle detectors is possible. In the past,
several experiments have made use of microchannel plate (MCP)
detectors to measure ion production rates and investigate collisional
losses in $^3\text{He*}$ \cite{morin99apr} and $^4\text{He*}$
\cite{morin99apr,nieha98jun,vassw99aug,vassw00apr}. Having realized
the ability to trap large numbers ($>\!\!10^8$) of both isotopes
(either individually or simultaneously) \cite{vassw04jul2}, we have
previously reported on the isotopic differences between binary
homonuclear collisions of $^3\text{He}$ and $^4\text{He}$
\cite{vassw06mar} in the absence of resonant light, resolving
inconsistencies in prior experimental and theoretical results.
In this article we describe what we believe to be the first study of
heteronuclear binary collisions between metastable atoms. Adapting our
transparent theoretical model \cite{vassw06mar} slightly, we first derive
a value for $K_{34}$ (Sec. \ref{sec:model}), the heteronuclear
ionization rate coefficient in the absence of a resonant light field.
This is complemented by trap loss measurements performed on a
two-isotope magneto-optical trap (TIMOT) of $^3\textup{He*}$ and
$^4\textup{He*}$ \cite{vassw04jul2}, from which we also extract a
value for $K_{34}$ (Sec. \ref{sec:exp}). In Section \ref{sec:conc} we
compare both results and briefly comment upon loss rates in the
presence of the trapping light fields.
\section{Single channel model of the heteronuclear ionizing loss rate}
\label{sec:model}
The following model has been described in depth elsewhere in its
successful application to the description of homonuclear ionizing
collisions \cite{vassw06mar,stasthesis}, where it compares well with
the more comprehensive close coupling theories developed by Venturi
\emph{et al.} \cite{peacg99dec,whiti00may} and Leo \emph{et al.}
\cite{babbj01sep}. Here we extend its applicability to include the
description of heteronuclear He* collisions for which no calculations
have previously been made. In the interests of brevity we only
describe the models salient points and its adaption to the
heteronuclear case.
\subsection{Ionization rate coefficients}
At mK temperatures, collisional processes are dominated by only a few partial waves, $\ell$, and the ionization cross section may be written as a sum over the partial wave contributions:
\begin{equation}
\sigma^{\text{(ion)}}=\sum_{\ell}\sigma_{\ell}^{\text{(ion)}}.
\label{PartialWaves}
\end{equation}
From a semi-classical viewpoint we may further treat the inelastic
scattering as a two-stage process in which (at low energies) elastic
scattering from the interaction potential $V(R)$ occurs at relatively
large internuclear distance ($R\ge 100\, a_0$), whilst ionization
occurs only at small internuclear distance ($R\approx 5\, a_0$)
\cite{yenc84}. The assumption that the processes of elastic scattering
and ionization are uncoupled allows us to factorize the probability of
ionization occurring in a collision, and write the ionization cross
section for collisions with total electronic spin $S$ as
\begin{equation}
\lsup{(2S+1)}{\sigma^{\text{(ion)}}}=\frac{\pi}{k^2}\sum_{\ell}(2\ell+1) \lsup{(2S+1)}{P_\ell^{\text{(tun)}}} \lsup{(2S+1)}{P^{\text{(ion)}}},
\label{CrossSection}
\end{equation}
where $k$ is the wave vector of the relative motion of the two atoms,
$\lsup{(2S+1)}{P_{\ell}^{\text{(tun)}}}$ is the probability of the
atoms reaching small internuclear distance (i.e., not elastically
scattering) and \lsup{(2S+1)}{P^{\text{(ion)}}} is the probability
of ionization occurring at short internuclear distance. In the He*
system the quantum number $S$ is well conserved during ionization and
the application of Wigner's spin conservation rule \cite{massh71} tells
us that \lsup{5}{P^{\text{(ion)}}} is very small \cite{shlyg96mar},
while M\"{u}ller \emph{et al.} \cite{movrm91jun} report ionization
probabilities of 0.975 for other spin states; we thus set
$\lsup{5}{P^{\text{(ion)}}}\!=\!0$ and
$\lsup{1}{P^{\text{(ion)}}}\!=\!\lsup{3}{P^{\text{(ion)}}}\!=\!1$ in
Eq.~(\ref{CrossSection}). Having constructed the $\lsup{1}\Sigma^+_g$,
$\lsup{3}\Sigma^+_u$ and $\lsup{5}\Sigma^+_g$ molecular potentials as
described in \cite{babbj01sep} (to which we may add rotational
barriers as required) we modify them to simulate the losses due to
ionization (Fig.~\ref{pots}) \cite{rolss99mar}.
\begin{figure}
\includegraphics[width=0.9\columnwidth]{Figure1}
\caption{He* potentials (labeled in Hund's case (a) notation)
constructed as described in \cite{babbj01sep} (solid), together with
the modifications made (dashed). Atoms reaching the region of small
$R$ ionize; the corresponding relative particle propagates freely to
$R=-\infty$ in our model, and ionization is accounted for by the loss
of probability flux from the region of the potential well.}
\label{pots}
\end{figure}
By numerically solving the radial wave equation and finding the
stationary states using these potentials, we may calculate the
incident and transmitted probability currents, $J_{\text{in}}$ and
$J_{\text{tr}}$ respectively, the ratio of which
($J_{\text{tr}}/J_{\text{in}}$) gives us
$\lsup{(2S+1)}{P_{\ell}^{\text{(tun)}}}$. The calculated partial wave
ionization cross sections are shown in
Fig.~\ref{partialcrosssections}.
\begin{figure}
\includegraphics[width=0.9\columnwidth]{Figure2}
\caption{Partial wave ionization cross sections $(S\!=\!0)$ for
\mbox{$\lsup{3}{\text{He*}}$-$\lsup{4}{\text{He*}}$} (solid lines),
\mbox{$\lsup{3}{\text{He*}}$-$\lsup{3}{\text{He*}}$} (dotted lines)
and \mbox{$\lsup{4}{\text{He*}}$-$\lsup{4}{\text{He*}}$} (dashed
lines). The familiar quantum threshold behavior
($\sigma^{\text{(ion)}}_{\ell}\!\propto\!
k^{2\ell-1}$~$(k\rightarrow 0)$) is displayed.}
\label{partialcrosssections}
\end{figure}
The ionization rate coefficient $K$
$(\textup{particle}^{-1}\,\textup{cm}^3/\textup{s})$ (determined in
experiments) is temperature dependant, and may be written in terms of
the ionization cross section $\sigma^{\text{(ion)}}(E)$
\cite{miesf89nov,rolss99mar,weinj03}:
\begin{equation}
K(T)=\int^{\infty}_{0}\sigma^{\text{(ion)}}(E)P^{\text{(MB)}}_T(v_r)\,v_r\,dv_r,
\label{IonisationRateCoefficient}
\end{equation}
where $P^{\text{(MB)}}_T(v_r)$ is the Maxwell-Boltzmann
distribution, for a given temperature, of the relative velocities in
the atomic sample. Similarly, we may calculate the partial wave
ionization rate coefficients $\lsup{(2S+1)}{{\cal K}_{\ell}(T)}$
associated with a given \emph{molecular state} (ignoring the radial
contributions: $|s_{1}i_{1},s_{2},S,FM_{F},\ell m_{\ell}\rangle$)~\cite{rolss99mar}.
Although the calculated values of
\lsup{(2S+1)}{\sigma^{\text{(ion)}}_{\ell}}
(Fig.~\ref{partialcrosssections}), and \lsup{(2S+1)}{{\cal
K}_{\ell}(T)}, are very similar for all isotopic combinations, the
description of the collisional process in terms of partial waves for
each combination is very different due to the differing quantum
statistics involved. In the case of homonuclear collisions the
symmetrization postulate limits the number of physical scattering
states describing a colliding pair. However, in the heteronuclear case
there is no symmetry requirement and all partial wave contributions
must be taken into account.
Because the energy gained during a collision is so large, the
evolution of an atomic state
($|s_{1}i_{1}f_{1},s_{2}j_{2},FM_{F},\ell m_{\ell}\rangle$) in the
region where the atomic hyperfine interaction is of the same order of
magnitude as the molecular interaction may be described to a good
approximation as being \emph{diabatic}. Thus, to determine the
ionization rate associated with a given \emph{scattering state}
$K(F)$, we simply expand the atomic states onto the eigenstates of the
short-range molecular Hamiltonian,
\begin{multline}
|s_{1}i_{1}f_{1},s_{2}j_{2},FM_{F},\ell
m_{\ell}\rangle=\sum_{S,I}a_{SI}(F)\times\\
|s_{1}i_{1},s_{2},S,FM_{F},\ell m_{\ell}\rangle,
\end{multline}
determine the fraction of ionizing states $(S\!=\!0,1)$ in this
expansion (see Table~\ref{pairstates}),
\begin{table}
\caption{Expansion coefficients $a_{SI}(F)\!=\!\langle s_1 i_1, s_2 ,S,
F M_F, \ell m_\ell | s_1 i_1 f_1,s_2 j_2, F M_F, \ell m_\ell \rangle$.
The scattering states $|s_1 i_1 f_1,s_2 j_2, F M_F, \ell m_\ell
\rangle$ are indicated by their values of $F$, while the molecular
states $|s_1 i_1, s_2,S, F M_F, \ell m_\ell \rangle$ are given in
Hund's case (a) notation, $^{2S+1}\Sigma_{\text{g/u}}^+$.}
\begin{ruledtabular}
\begin{tabular}{lccc}
F&$^{1}\Sigma_{\text{g}}^+$ &$^{3}\Sigma_{\text{u}}^+$
&$^{5}\Sigma_{\text{g}}^+$ \rule[-2mm]{0mm}{6mm}\\ \hline
1/2&$\sqrt{2/3}$&$-\sqrt{1/3}$&\rule[0mm]{0mm}{5mm}\\
3/2&&$\sqrt{5/6}$&$-\sqrt{1/6}$\rule[0mm]{0mm}{5mm}\\
5/2&&&$1$\rule[0mm]{0mm}{5mm}\\
\end{tabular}
\end{ruledtabular}
\label{pairstates}
\end{table}
and sum over the contributing partial waves:
\begin{equation}
K(F)=\sum_{\ell}\sum_{S,I}|a_{SI}(F)|^2\times \lsup{(2S+1)}{{\cal
K}_{\ell}}.
\end{equation}
Using the partial wave ionization rates obtained we may then derive
the total ionization rate coefficient for an unpolarized sample, i.e.,
the rate coefficient for which the magnetic substates of the atoms in
our sample are evenly populated:
\begin{equation}
K^{\text{(unpol)}}=\frac{1}{(2f_1+1)}\frac{1}{(2f_2+1)}\sum_{F}\sum_{M_F}K(F),
\label{ThRateCoeff}
\end{equation}
which in the heteronuclear case gives:
\begin{equation}
K^{\text{(unpol)}}_{34}\approx\frac{1}{12}\left[\frac{4}{3}(\lsup{1}{{\cal
K}}_0+\lsup{1}{{\cal K}}_1)+{4}(\lsup{3}{{\cal K}}_0+\lsup{3}{{\cal K}}_1)\right].
\label{ApproxThRateCoeff}
\end{equation}
\begin{figure}
\includegraphics[width=0.9\columnwidth]{Figure3}
\caption{Theoretical unpolarized loss rate coefficient curves:
\mbox{$\lsup{3}{\text{He}}\,\text{\textendash}\,\lsup{3}{\text{He}}$}
(dashed),
\mbox{$\lsup{4}{\text{He}}\,\text{\textendash}\,\lsup{4}{\text{He}}$}
(dotted) and,
\mbox{$\lsup{3}{\text{He}}\,\text{\textendash}\,\lsup{4}{\text{He}}$}
(solid), together with our experimental data points and their error
bars:
\mbox{$\lsup{3}{\text{He}}\,\text{\textendash}\,\lsup{3}{\text{He}}$}
(diamond), \mbox{$\lsup{4}{\text{He}}\,\text{\textendash}\,\lsup{4}{\text{He}}$}
(triangle) and
\mbox{$\lsup{3}{\text{He}}\,\text{\textendash}\,\lsup{4}{\text{He}}$}
(square). For the purposes of this figure, the experimental points have
been corrected for the inhomogeneous distribution over the magnetic
substates found in our MOT's (see Sec.~\ref{trappedsamples}).}
\label{ratecoefficients}
\end{figure}
The energy dependant unpolarized ionization rate coefficients are
shown in Fig.~\ref{ratecoefficients} (including those rates previously
calculated \cite{vassw06mar} for $^3\textup{He*}$ and $^4\textup{He*}$)
and for a temperature of $T\!=\!1\,\text{mK}$ we obtain
\begin{equation}
K^{\text{(unpol)}}_{34}=2.9\times 10^{-10}~\text{cm}^3/\text{s}.
\end{equation}
\subsection{Ionization rate coefficient of trapped samples}
\label{trappedsamples}
Optical pumping processes in a MOT cause the distribution over
magnetic substates $P_{m}$ (where $m$ is the azimuthal quantum number)
of the trapped atoms to differ from the uniform (unpolarized)
distribution assumed above. This is important as the contribution of
each collision channel to the ionizing losses depends upon $P_{m}$,
and can be accounted for in our theoretical model by using the density
operator \cite{cohec77}
\begin{equation}
\rho(\bm{r}) = \sum_m \sum_{n\leq m} P_m(\bm{r}) \, P_n(\bm{r}) \, |m, n \rangle \langle m,n |
\label{eq:densityoperator}
\end{equation}
to describe a statistical mixture of magnetic substate pairs
$|m,n\rangle$, where $m$ and $n$ are the azimuthal quantum numbers of
the colliding atoms. The ionization rate coefficient of the mixture
can then be written as
\begin{equation}
K_{34} = \frac{1}{N_{3}N_{4}} \iiint \Bigl( \sum_{\ell} \bigl(\lsup{1}{b\mbox{$\mspace{1.0mu}$}}
\: \lsup{1}{\cal K}_\ell + \lsup{3}{b\mbox{$\mspace{1.0mu}$}}\: \lsup{3}{\cal K}_\ell
\bigr)\Bigr)n_{3}(\bm{r})n_{4}(\bm{r}) \, \text{d}^3r,
\label{eq:Karb}
\end{equation}
where $N_{3}$ and $N_{4}$ are the number of trapped atoms in each
component of the mixture, the coefficients $\lsup{(2S+1)}{{b\mbox{$\mspace{1.0mu}$}}}$
are the sums of the expectation values of the density operator for all
ionizing molecular states with total spin $S$, and, $n_{3}(\bm{r})$ and
$n_{4}(\bm{r})$ are the density distributions of each component in the
sample. Explicit expressions for the coefficients
$\lsup{(2S+1)}{{b\mbox{$\mspace{1.0mu}$}}}$ are given in Table~\ref{coefficients}.
\begin{table*}
\caption{$^{3}\text{He*}$-$^{4}\text{He*}$
$\lsup{(2S+1)}{{b\mbox{$\mspace{1.0mu}$}}}$ coefficients from
Eq.~(\ref{eq:Karb}). The coefficients are the expectation values of
the density operator, Eq.~(\ref{eq:densityoperator}), for all ionizing
molecular states of given $S$ and parity.}
\begin{ruledtabular}
\begin{tabular}{rc}
$^{(2S+1)}b$&$^{3}\text{He*}$-$^{4}\text{He*}$
\rule[-2mm]{0mm}{6mm}\\ \hline
$^{1}b$&$(1/3)(P_{-3/2}P_{1}+P_{3/2}P_{-1})+(2/9)(P_{-1/2}P_{0}+P_{1/2}P_{0})+(1/9)(P_{-1/2}P_{1}+P_{1/2}P_{-1})$\rule[0mm]{0mm}{5mm}\\
$^{3}b$&$(1/2)(P_{-3/2}P_{1}+P_{-3/2}P_{0}+P_{-1/2}P_{1}+P_{1/2}P_{-1}+P_{3/2}P_{0}+P_{3/2}P_{-1})$\rule[0mm]{0mm}{5mm}\\
&$+(1/3)(P_{-1/2}P_{-1}+P_{1/2}P_{1})+(1/6)(P_{-1/2}P_{0}+P_{1/2}P_{0})$\\
\end{tabular}
\end{ruledtabular}
\label{coefficients}
\end{table*}
One may easily check that we recover Eq.~(\ref{ApproxThRateCoeff})
from Eq.~(\ref{eq:Karb}) by substituting the values
$P_{-3/2}\!=\!P_{-1/2}\!=\!P_{1/2}\!=\!P_{3/2}\!=\!\frac{1}{4}$ and
$P_{-1}\!=\!P_{0}\!=\!P_{1}\!=\!\frac{1}{3}$ into the expressions for
the coefficients $\lsup{(2S+1)}{b}$ (Table~\ref{coefficients}) and
evaluating Eq.~(\ref{eq:Karb}).
In order to later make a comparison between theory and experiment, we
determine the distribution $P_{m}(\bm{r})$ by obtaining the
steady-state solution of a rate equation model describing the optical
pumping in our MOT \cite{vassw00apr}. At a temperature of
$T\!=\!1.2\,\text{mK}$ the resulting value of the theoretical
ionization rate coefficient is
\begin{equation}
K^{\text{(th)}}_{34}=2.4\times 10^{-10}~\text{cm}^3/\text{s}
\end{equation}
\section{Experimental Setup}
We investigate cold ionizing collisions in a setup (see
Fig.~\ref{fig:setup}) capable of trapping large numbers ($\gsim10^8$)
of both $^3$He* and $^4$He* atoms simultaneously in a TIMOT
\cite{vassw06mar}. A collimated and Zeeman slowed He* beam is used to
load our TIMOT which is housed inside a stainless steel, ultra-high
vacuum chamber. The beam is produced by a liquid nitrogen (LN$_2$)
cooled dc discharge source supplied with an isotopically enriched
($\approx\! 50/50$) gaseous mixture of $^3\textup{He}$ and
$^4\textup{He}$ held in a helium tight reservoir. During operation the
reservoir is connected such that all helium not entering the
collimation section (the vast majority of it) is pumped back into the
reservoir and recycled, conserving our supply of the relatively
expensive $^3$He gas. Two LN$_2$ cooled molecular sieves, ensuring a
pure supply of helium to the source, are also contained within this
reservoir; the first of these is a type 13X molecular sieve, whilst
the second is type 4A (both are sodium zeolites having pore sizes of
$10\,\text{\AA}$ and $4\,\text{\AA}$ respectively). We then make use
of the curved wavefront technique to collimate our atomic beam in two
dimensions \cite{vassw96jan} before it enters the Zeeman slower. Due
to its lighter mass $^3$He atoms emerge from the source with a greater
mean velocity than $^4$He atoms and in order to achieve a large flux
of both $^3$He* and $^4$He* atoms we have increased the capture
velocity of our Zeeman slower \cite{stasthesis,vassw04jul2}. Our
ultra-high vacuum chamber maintains an operational pressure of $7
\times 10^{-10}\,\text{mbar}$ (with a partial presure of $6.5\times
10^{-10}\,\text{mbar}$, ground state He atoms from the atomic beam are
the major contribution to this) and is based upon the design of our
next-generation BEC chamber \cite{vassw06mar2}. Two coils in an
anti-Helmholtz configuration produce the quadrupole magnetic field
($dB/dz\!=\!0.35\,\text{T/m}$) for the TIMOT; these are placed in
water cooled buckets outside the vacuum and are brought close to the
trapping region by placing them inside reentrant glass windows
situated on either side of the chamber.
\begin{figure}
\includegraphics[width=0.9\columnwidth]{Figure4}
\caption{A schematic overview of the TIMOT experimental apparatus used
in these experiments. Component labels are: AOM, acousto-optic
modulator; QW, quater-wave plate; S, shutter; PBSC, polarizing beam
splitter cube; HW, half-wave plate; CL, cylindrical lens; and NPBSC,
non-polarizing beamsplitter cube. For clarity, all spherical lenses
and excess mirrors have been omitted. }
\label{fig:setup}
\end{figure}
Both helium isotopes are collimated, slowed, and confined in the TIMOT
using 1083~nm light nearly resonant with their $2\,\mbox{$^3 \! S _1$} \rightarrow
2\,\trP{2}$ optical transitions (see Fig.~\ref{transitionspic})
(natural linewidth $\Gamma/2\pi\!=\!1.62\,\text{MHz}$ and saturation
intensity $I_{\text{sat}}\!=\!0.166\,\text{mW\,cm}^{-2}$, for the cycling
transition).
\begin{figure}
\begin{center}
\includegraphics[width=0.75\columnwidth]{Figure5}
\caption{Level scheme for the ground and first excited states in
$^4\textup{He}^*$ and $^3\textup{He}^*$.}
\label{transitionspic}
\end{center}
\end{figure}
As the isotope shift for this transition is $\approx\!34\,\text{GHz}$
the bichromatic beams are produced by overlapping the output from two
ytterbium-doped fiber lasers (IPG Photonics) on a non-polarizing 50/50
beam splitter. One beam is sent to the collimation section, whilst the
second is coupled into a single mode polarization maintaining fiber to
ensure a perfect overlap of the two frequency components, before being
split into the Zeeman slowing beam and the trapping beams. Each fiber
laser is locked to the respective cooling transition using saturated
absorption spectroscopy in an rf-discharge cell; acousto-optic
modulators are then used to generate the slowing and trapping
frequencies which are detuned by -500\,MHz and -40\,MHz respectively
(see Fig.~\ref{transitionspic}). The slowing beam is focused on the
source and has a 1/$e^2$ intensity width of 2.2\,cm at the position of
the trapped cloud, with each frequency component having a peak
intensity of $I_{\text{peak}}\!=\!9\,\text{mW\,cm}^{-2}$
($I_{\text{peak}}/I_{\text{sat}}\!\approx\!54$), while the trapping
beam is split into six independent Gaussian beams with 1/$e^2$
intensity widths of 1.8\,cm and total peak intensity
$I_{\text{peak}}\!=\!57\,\text{mW\,cm}^{-2}$
($I_{\text{peak}}/I_{\text{sat}}\!\approx\!335$). The outputs of two
diode lasers (linewidths $<500\,\text{kHz}$), each locked using
saturated absorption spectroscopy to the helium $2\,\mbox{$^3 \! S _1$} \rightarrow
2\,\trP{2}$ resonance of one of the isotopes, are overlapped on a
second non-polarizing beam-splitter cube and coupled into a single
mode polarization maintaining fiber. This system delivers two, weak,
linearly polarized probe beams ($\Delta\!=\!0$,
$I\!=\!0.05\,I_{\text{sat}}$, with
$I_{\text{sat}}\!=\!0.27\,\text{mW\,cm}^{-2}$ assuming equal
population of all magnetic sublevels of the $2\,\mbox{$^3 \! S _1$}$ state in the
trap) to the chamber for the absorption imaging of each component of
the trapped cloud. Absorption images are recorded using an
IR-sensitive charge-coupled device (CCD) camera \cite{tolthesis}.
Unfortunately the imaging of both trapped components cannot be carried
out in a single experimental cycle; sequential runs must be made,
imaging first one component and then the other.
Trapped clouds are further monitored by two microchannel plate (MCP)
detectors mounted inside the chamber. Operated at a voltage of
-1.5\,kV and positioned 11\,cm from the center of the trap, the MCP's
are used to independently monitor the ions and $\text{He}^*$ atoms
escaping or released from the trap. With an exposed front plate held
at negative high voltage, one MCP mounted above the trap center
attracts all positive ions produced during ionizing collisions (Eq.
(1)) in the trap. The second MCP is shielded by a grounded grid,
mounted below the trap center, and detects only $\text{He}^*$ atoms.
The shielded MCP is used to perform time-of-flight (TOF) measurements
from which we determine the temperature of the trapped atoms. An
absorpton image, in combination with the measured temperature, then
allows us to determine the density distribution, size, and absolute
atom number of the sample. We then use the unshielded MCP to measure
the instantaneous ionization rate in the trapped sample, which, in
combination with the information obtained from the absorption images,
allows us to determine trap loss and ionization rates in the sample.
Our typical TIMOT parameters and measurements have been reported
previously \cite{vassw04jul2,vassw06mar} and are given in
Table~\ref{oldrestable}. The lower temperatures realized in the
present experiments are due to the resolution of a power imbalance in
two of the trapping laser beams.
\begin{table}
\caption{Typical experimental values of the \lsup{3}{\text{He*}} and
\lsup{4}{\text{He*}} components in our TIMOT (error bars correspond to
1 standard deviation).\\}
\label{oldrestable}
\begin{ruledtabular}
\begin{tabular}{lcc}
&$^3\textup{He*}$ &$^4\textup{He*}$\rule[-2mm]{0mm}{6mm}\\ \hline
Temperature \textit{T} (mK) &1.2(1) &1.2(1)\rule[0mm]{0mm}{5mm}\\
Number of atoms \textit{N} &$1.0(3) \times 10^8$ &$1.3(3) \times
10^8$\rule[0mm]{0mm}{5mm}\\
Central density $n_0$ (cm$^{-3}$) &$0.5(1) \times 10^9$ &$1.0(2) \times 10^9$\rule[0mm]{0mm}{5mm}\\
Axial radius $\sigma_p$ (cm) &0.28(4) &0.25(3)\rule[0mm]{0mm}{5mm}\\
Radial radius $\sigma_z$ (cm) &0.16(2) &0.14(1)\rule[0mm]{0mm}{5mm}\\
\end{tabular}
\end{ruledtabular}
\end{table}
\section{Determination of the Heteronuclear Loss Rate}
\label{sec:exp}
The time evolution of the total number of atoms trapped in our TIMOT,
$N\!=\!N_3\!+\!N_4$, may be described by the following phenomenological
equation \cite{pritd88jun}:
\begin{widetext}
\begin{equation}
\frac{dN}{dt}=L_3-\alpha_3 N_3(t)-\beta_{33}\int\!\!\!\int\!\!\!\int
n_3^2(\textbf{r},t)\,d^3\textbf{r}+L_4-\alpha_4N_4(t)-\beta_{44}\int\!\!\!\int\!\!\!\int
n_4^2(\textbf{r},t)\,d^3\textbf{r}-\beta_{34}\int\!\!\!\int\!\!\!\int n_3(\textbf{r},t)n_4(\textbf{r},t)\,d^3\textbf{r},
\label{NumAtoms}
\end{equation}
\end{widetext}
where $t$ denotes time, $L$ is the rate at which atoms are loaded
into the TIMOT, $\alpha$ is the linear loss rate coefficient describing
collisions between trapped He* atoms and background gases, $\beta$ is
the \emph{loss rate} coefficient resulting from binary collisions
between trapped He* atoms, and $n$ is the cloud density. Subscripts
denote whether a given parameter pertains to either of the $^3$He* or
$^4$He* components of the mixture.
It should be noted at this point that the "density limited" regime,
often mentioned with regard to the alkali systems, is not a feature of
the metastable noble gas systems; in the latter the density is not
limited by radiation trapping within the cloud, but by ionizing
collisional losses. This is born out in Eq.~(\ref{NumAtoms}) by the
time dependance of the atom density distributions and hence our
inability to make the simplifying constant density approximation in
the following experiments.
Both linear and quadratic losses in Eq.~(\ref{NumAtoms}) are due to a
number of different mechanisms, and may be subdivided into either
ionizing or non-ionizing categories. The ion production rate may then
be expressed in a manner analogous to Eq.~(\ref{NumAtoms}):
\begin{widetext}
\begin{equation}
\frac{dN_{\text{ion}}}{dt}=\epsilon_a\alpha_3
N_3(t)+\epsilon_bK_{33}\int\!\!\!\int\!\!\!\int
n_3^2(\textbf{r},t)\,d^3\textbf{r}+\epsilon_c\alpha_4N_4(t)+\epsilon_dK_{44}\int\!\!\!\int\!\!\!\int
n_4^2(\textbf{r},t)\,d^3\textbf{r}+\epsilon_eK_{34}\int\!\!\!\int\!\!\!\int n_3(\textbf{r},t)n_4(\textbf{r},t)\,d^3\textbf{r},
\label{NumIons}
\end{equation}
\end{widetext}
where $\epsilon_a, \epsilon_b, \epsilon_c, \epsilon_d$ and
$\epsilon_e$ are the weights of the various ionization mechanisms (and
may include a factor to account for a less than unity detection
efficiency), and the \emph{collision} rate coefficient $K$ has been
introduced. The \emph{loss} and \emph{collision} rate coefficients
are related by the equation $\beta = 2K$, and the appearance of $K$,
instead of $\beta$ in Eq.~\ref{NumIons} expresses the fact that during
each ionizing collision, one ion is produced, but two atoms are lost
from the trap. From an analysis of the trap loss mechanisms
\cite{vassw06mar,stasthesis} it can be seen that to a good
approximation $\epsilon_a\!=\!\epsilon_c\!=\!0$, whilst
$\epsilon_b\!=\!\epsilon_d\!=\!\epsilon_e$. The current signal
measured by the MCP is proportional to the ionization rate,
Eq.~(\ref{NumIons}); and for gaussian spatial density distributions
(centered with respect to each other), the voltage measured by the
oscilloscope may then be written as
\begin{widetext}
\begin{equation}
\phi_{TIMOT}
(t)=eR_{\text{eff}}\left[K_{33}\,n^2_{03}(t)(\pi\sigma_3^2)^{\frac{3}{2}}+K_{44}\,n^2_{04}(t)(\pi\sigma_4^2)^{\frac{3}{2}}+K_{34}\,n_{03}(t)n_{04}(t)\left[\frac{2\pi\sigma^2_3\sigma^2_4}{\sigma_3^2+\sigma_4^2}\right]^\frac{3}{2}\right]+\phi_{\text{bgr}},
\label{MCPVolts}
\end{equation}
\end{widetext}
where $n_0$ is the central density, $\sigma$ is the mean rms radius
(of a given cloud component), $e$ is the electron charge, and
$R_{\text{eff}}$ is an effective resistance.
The experiment is based around the ability of an MCP detector to
measure the ions produced in our metastable isotopic mixture with very
high efficiency, and employs a method first used by Bardou \emph{et
al.}~\cite{aspea92dec} to determine the ionization rate in the absence
of light. To measure the rate in the dark we perform an experiment in
which we load the TIMOT, switch off the Zeeman slower beam and all MOT
beams using the frequency detuning AOM's (the quadrupole field remains
on) for $100\,\mu\text{s}$, before switching the slowing and trapping
light back on again. This on/off cycle is easily repeated many times
while we monitor the ion signal and average it (see
Fig.~\ref{dutycyclepic}). The switch-off time is short compared to the
dynamics of the expanding cloud (we see no variation in the ion signal
during the switch off period) and we switch the light on long enough
(200\,ms) to recapture the cloud and allow it to equilibrate.
\begin{figure}
\includegraphics[width=0.9\columnwidth]{Figure6}
\caption{Averaged ion signals measured during the experiments. The
full line is the signal recorded whilst performing the experiment
with a TIMOT, the dashed line is the corresponding curve for a
\lsup{3}{\text{He*}} MOT (the analogous signal for a
\lsup{4}{\text{He*}} MOT can also be obtained), and the dotted line is
the background signal. The ion rate measured is then averaged over the
$60\,\mu s$ interval indicated by the shaded region in order to obtain
($\phi_{\text{TIMOT}}\!-\!\phi_{\text{bgr}}$) and
($\phi_{^3\text{He*/}^4\text{He*}}\!-\!\phi_{\text{bgr}}$). }
\label{dutycyclepic}
\end{figure}
Equation~(\ref{MCPVolts}) describes the ion production rate in the
dark of our TIMOT; a similar equation describes the ion production
rate of a single-isotope MOT, $\phi_{MOT}$. By combining the equations
for $\phi_{TIMOT}$ and $\phi_{MOT}$ with a measurement of the ratio
$r=(\phi_{TIMOT}-\phi_{bgr})/(\phi_{MOT}-\phi_{bgr})$ (where the time
dependance of the measured signals has been omitted because of the
short duration of the switch-off period), we can derive an expression
for $K_{34}$. It has been verified that under our experimental
conditions the MCP signal varies linearly with the ion production
rate, and all cloud densities and radii may be derived from
absorption images, while we have previously measured $K_{33}$ and
$K_{44}$~\cite{vassw06mar}. As the trap parameters, and therefore the
distribution over the magnetic substates of the atoms, have changed
since the experiments described in Ref.~\cite{vassw06mar} were
performed, we have used the theory described in
Sec.~\ref{trappedsamples} to correct the measured values of $K_{33}$
and $K_{44}$ for these effects, yielding: $K_{33}\!=\!1.6(3)\times
10^{-10}\,\text{cm}^3/\text{s}$ and $K_{44}\!=\!6.5(2)\times
10^{-11}\,\text{cm}^3/\text{s}$.
\section{Results and discussion}
\label{sec:conc}
Performing the experiments described in the previous section, we obtain
the result: $K_{34}^{(exp)}\!=\!2.5(8)\times
10^{-10}\,\text{cm}^3/\text{s}$, which compares well with the
theoretical prediction,
$K^{\text{(th)}}_{34}\!=\!2.4\times 10^{-10}\,\text{cm}^3/\text{s}$,
obtained after undertaking the calculation described in
Sec.~\ref{sec:model}. For future comparison with other theoretical
models, the equivalent theoretical loss-rate coefficient for an
unpolarized gas mixture at a temperature of $T\!=\!1\,\rm{mK}$ is
$K_{34}^{(unpol)}\!=\!2.9\times 10^{-10}\,\text{cm}^3/\text{s}$ (see
Fig.~\ref{ratecoefficients}).These numbers describe the \emph{total}
heteronuclear loss rate in the absence of light; the only other
potential loss process would be hyperfine state changing collisions,
however, due to the inverted nature of the hyperfine structure in
$^3\textup{He*}$ the atoms occupy the lowest hyperfine level of the
$2\,\mbox{$^3 \! S _1$}$ multiplet and so cannot relax to a lower level, while the
endothermic collision necessary to reach the $\text{F}\!=\!1/2$ state
would require $\approx\! 200\,\text{mK}$ to be provided by the trap,
200 times more than the typical collision energy in the TIMOT. The
error in the experimental value is mainly determined by errors in the
absorption imaging, which we find to be $\approx\! 30\%$, and by the
error bars reported on our previous measurements of the homonuclear
loss rates~\cite{vassw06mar}.
With regard to optically assisted collisions, we note that the excited
state potentials in the case of heteronuclear collisions are governed
at long-range by the van-der-Waals interaction
($\propto\!1/\text{R}^6$), having a much shorter range than that of
the resonant dipole interaction ($\propto\!1/\text{R}^3$) dominant in
the homonuclear case. In our TIMOT the laser beams are, in contrast to
most experiments performed on heteronuclear collisions, far-detuned
(25 natural linewidths) from resonance and the atomic excited state
population in the trap is therefore negligible. We can only excite a
molecular state if the correct light frequency is present and the
atoms have reached the Condon point for the transition. As all light
frequencies in our TIMOT are far detuned from any heteronuclear
transition we expect no contribution to the trap loss rate from
optically assisted heteronuclear collisions.
To summarize, we have measured the heteronuclear loss rate
coefficient $K_{34}^{(exp)}$ in the absence of light for a trapped
mixture of $^3\textup{He*}$ and $^4\textup{He*}$ atoms at
$T\!=\!1\,\text{mK}$. The measured value of $K_{34}^{(exp)}$ compares
very well with the value predicted by our single channel model of
ionizing collisions in the He* system. Recently, both helium isotopes
were magnetically trapped in the multi-partial wave regime using
buffer-gas cooling, and a deep magnetic trap~\cite{doylj05dec}. The
probable observation of Penning ionization under these conditions has
been reported~\cite{nguyenthesis}, and it would be interesting to
extend our theory into the multi-partial wave regime and to high
magnetic field values. With the production of quantum degenerate
mixtures of \lsup{3}{\text{He*}} and
\lsup{4}{\text{He*}}~\cite{mcnaj06aug}, we also have the possibility
of investigating both homonuclear and heteronuclear He* collisions at
ultracold temperatures in greater detail. An experiment of interest in
this area would be the implementation of an optical dipole trap in
which it would be possible to prepare ultracold samples in well
defined magnetic substates. In particular, it would be possible to
prepare trapped ultracold samples of \lsup{3}{\text{He*}}
(\lsup{4}{\text{He*}}) in the $m_F\!=\!-3/2$ ($m_J\!=\!-1$) states,
for which (as in the magnetically trapped $m_F\!=\!+3/2$
($m_J\!=\!+1$) states used in the production of ultracold He* gases)
Penning ionization should be suppressed.
\begin{acknowledgments}
We thank Jacques Bouma for technical support. This work was supported
by the ``Cold Atoms'' program of the Dutch Foundation for Fundamental
Research on Matter (FOM), the Space Research Organization Netherlands
(SRON), Grant No. MG-051, and the European Union, Grant No.
HPRN-CT-2000-00125.
\end{acknowledgments}
|
1612.08772
|
\section{Introduction}
\label{sec:Introduction}In this section, we first motivate and describe
our approach for the construction of structured Banach frame decompositions
for decomposition spaces and compare our results to the known literature.
Then, we introduce a few standard and non-standard conventions and
standing assumptions which are used in the remainder of the paper.
Finally, we give a brief overview over the structure of the paper.
\subsection{Motivation and comparison to known results}
Given a Banach space $X$, a family $\Psi=\left(\psi_{i}\right)_{i\in I}$
in $X'$ is called a \textbf{Banach frame\cite{GroechenigDescribingFunctions}}
for $X$ if there is a \textbf{solid sequence space} $Y\leq\mathbb{C}^{I}$
such that
\begin{itemize}
\item the \textbf{analysis operator} $A_{\Psi}:X\to Y,x\mapsto\left(\left\langle x,\psi_{i}\right\rangle _{X,X'}\right)_{i\in I}$
is well-defined and bounded,
\item there is a bounded linear \textbf{reconstruction operator} $R:Y\to X$
satisfying $R\circ A_{\Psi}=\operatorname{id}_{X}$.
\end{itemize}
In particular, this implies $\left\Vert x\right\Vert _{X}\asymp\left\Vert A_{\Psi}x\right\Vert _{Y}$
uniformly over $x\in X$. Here, a Banach space $Y\leq\mathbb{C}^{I}$
is called \textbf{solid} if for all sequences $x=\left(x_{i}\right)_{i\in I}$
and $y=\left(y_{i}\right)_{i\in I}$ with $y\in Y$ and $\left|x_{i}\right|\leq\left|y_{i}\right|$
for all $i\in I$, it follows that $x\in Y$ with $\left\Vert x\right\Vert _{Y}\leq\left\Vert y\right\Vert _{Y}$.
Dual to the notion of a Banach frame, a family $\Phi=\left(\varphi_{i}\right)_{i\in I}$
in $X$ is called an \textbf{atomic decomposition\cite{GroechenigDescribingFunctions}}
for $X$ if there is a solid sequence space $Z\leq\mathbb{C}^{I}$ such
that
\begin{itemize}
\item the \textbf{synthesis operator} $S_{\Phi}:Z\to X,\left(x_{i}\right)_{i\in I}\mapsto\sum_{i\in I}x_{i}\varphi_{i}$
is well-defined and bounded, where convergence of the series occurs
in a suitable (weak) sense,
\item there is a bounded linear \textbf{coefficient operator} $C:X\to Z$
satisfying $S_{\Phi}\circ C=\operatorname{id}_{X}$.
\end{itemize}
In particular, this implies that every $x\in X$ can be written as
$x=\sum_{i\in I}c_{i}\varphi_{i}$ for a suitable sequence $c=\left(c_{i}\right)_{i\in I}=Cx$.
The existence of nice Banach frames and atomic decompositions for
a given (family of) Banach space(s) is extremely convenient, since
the study of many properties like existence of embeddings, boundedness
of operators and description of interpolation spaces, etc., of the
Banach spaces under consideration can be reduced to studying these
properties for the associated sequence spaces, which are often much
easier to understand.
For this reason, much effort has been spent to derive existence of
Banach frames and atomic decompositions for many well-known spaces
like Besov- and Sobolev spaces. The most well-known types of (Banach)
frames are probably the \textbf{wavelet characterization} of Besov
spaces (see e.g.\@ \cite[Theorem 1.64]{TriebelTheoryOfFunctionSpaces3}),
the closely related characterization of these spaces using the $\varphi$-transform
\cite{FrazierJawerthDecompositionOfBesovSpaces,FrazierJawerthDiscreteTransform},
as well as the existence of \textbf{Gabor frames} for modulation spaces\cite{GroechenigTimeFrequencyAnalysis}.
\subsubsection{Classical group-based coorbit theory}
By generalizing the similarities between the theories of wavelet-
and Gabor frames, Feichtinger and Gröchenig initiated the study of
so-called \textbf{coorbit spaces}\cite{FeichtingerCoorbit0,FeichtingerCoorbit1,FeichtingerCoorbit2,GroechenigDescribingFunctions},
which provide a systematic way of obtaining Banach frames and atomic
decompositions for certain Banach spaces. Precisely, one starts with
an irreducible, (square)-integrable representation $\pi:G\to\mathcal{U}\left(\mathcal{H}\right)$
of some locally compact Hausdorff (LCH) topological group $G$. This
representation induces for each $g\in\mathcal{H}$ an associated \textbf{voice
transform}
\[
V_{g}:\mathcal{H}\to C\left(G\right),f\mapsto V_{g}f\qquad\text{ where }\qquad\left(V_{g}f\right)\left(x\right)=\left\langle f,\,\pi\left(x\right)g\right\rangle _{\mathcal{H}}.
\]
For an \textbf{admissible vector} $\psi\in\mathcal{H}\setminus\left\{ 0\right\} $
(which means $V_{\psi}\psi\in L^{2}\left(G\right)$), it follows\cite{DufloMoore}
that $V_{\psi}:\mathcal{H}\to L^{2}\left(G\right)$ is (a scalar multiple
of) an isometry, so that $\left(\pi\left(x\right)\psi\right)_{x\in G}$
is a \textbf{tight continuous frame} for $\mathcal{H}$, since
\[
\left\Vert f\right\Vert _{\mathcal{H}}^{2}=C_{\psi}\cdot\int_{G}\left|\left(V_{\psi}f\right)\left(x\right)\right|^{2}\operatorname{d}\mu\left(x\right)\qquad\forall f\in\mathcal{H}.
\]
In particular, this identity implies $\mathcal{H}=\left\{ f\,\middle|\, V_{\psi}f\in L^{2}\left(G\right)\right\} $.
In generalization of this identity, coorbit theory shows that for
``good enough'' \emph{analyzing windows} $\psi$ and each suitable,
\emph{solid} function space $Y\leq L_{{\rm loc}}^{1}\left(G\right)$,
one can define the associated \textbf{coorbit space} as
\[
{\rm Co}\left(Y\right):=\left\{ f\in\mathcal{R}\,\middle|\, V_{\psi}f\in Y\right\} \qquad\text{ with norm }\qquad\left\Vert f\right\Vert _{{\rm Co}\left(Y\right)}=\left\Vert V_{\psi}f\right\Vert _{Y}.
\]
Here, $\mathcal{R}=\mathcal{R}_{Y}$ is a suitable \emph{reservoir}. Informally,
$\mathcal{R}$ corresponds to the set of (tempered) distributions; but due
to the generality in which coorbit spaces are defined, one has to
use a slightly different definition, intrinsic to the group $G$,
cf.\@ \cite[Section 4]{FeichtingerCoorbit1}.
The main statement of coorbit theory is that one can \emph{discretize}
the (continuous, tight) frame $\left(\pi\left(x\right)\psi\right)_{x\in G}$,
to obtain discrete Banach frames and atomic decompositions, simultaneously
for all spaces ${\rm Co}\left(Y\right)$, where $Y$ ranges over a
suitable set of solid function spaces on $Y$. More precisely, the
following are true:
\begin{itemize}
\item For each translation invariant, solid function space $Y\leq L_{{\rm loc}}^{1}\left(G\right)$,
there is a so-called \textbf{control weight} $w=w_{Y}:G\to\left(0,\infty\right)$,
cf.\@ \cite[equation (4.10)]{FeichtingerCoorbit1}. For the following
statements, we always assume that $w$ is a control weight for $Y$.
\item Associated to each control weight $w$, there is a class $\mathcal{B}_{w}\subset\mathcal{H}$
of \textbf{good (analyzing) vectors} such that for each two $\psi_{1},\psi_{2}\in\mathcal{B}_{w}\setminus\left\{ 0\right\} $,
the identity
\[
\left\{ f\in\mathcal{R}\,\middle|\, V_{\psi_{1}}f\in Y\right\} ={\rm Co}\left(Y\right)=\left\{ f\in\mathcal{R}\,\middle|\, V_{\psi_{2}}f\in Y\right\}
\]
holds, i.e., one has a \emph{consistency statement}.
\item For each control weight $w$ and each $\psi\in\mathcal{B}_{w}\setminus\left\{ 0\right\} $,
there is a unit neighborhood $U=U\left(\psi,w\right)\subset G$, such
that for every $U$-dense and \textbf{relatively separated} family
$X=\left(x_{i}\right)_{i\in I}$ in $G$, the family $\left(\pi\left(x_{i}\right)\psi\right)_{i\in I}$
forms an \textbf{atomic decomposition} of ${\rm Co}\left(Y\right)$,
i.e., there is a solid, discrete sequence space $Y_{d}\left(X\right)\leq\mathbb{C}^{I}$
associated to $Y$ such that the following hold (see \cite[Theorem 6.1 and the associated remark]{FeichtingerCoorbit1}):
\begin{itemize}
\item the synthesis operator
\[
S:Y_{d}\left(X\right)\to{\rm Co}\left(Y\right),\left(\lambda_{i}\right)_{i\in I}\mapsto\sum_{i\in I}\lambda_{i}\cdot\pi\left(x_{i}\right)\psi
\]
is well-defined and bounded (with convergence in the weak-$\ast$-topology
of the reservoir $\mathcal{R}$),
\item there is a bounded linear operator $C:{\rm Co}\left(Y\right)\to Y_{d}\left(X\right)$
satisfying $S\circ C=\operatorname{id}_{{\rm Co}\left(Y\right)}$.
\end{itemize}
\item For each control weight $w$ and each $\psi\in\mathcal{B}_{w}\setminus\left\{ 0\right\} $,
there is a unit neighborhood $V=V\left(\psi,w\right)\subset G$ such
that for every $V$-dense and relatively separated family $X=\left(x_{i}\right)_{i\in I}$
in $G$, the family $\left(\pi\left(x_{i}\right)\psi\right)_{i\in I}$
forms a \textbf{Banach frame} for ${\rm Co}\left(Y\right)$, i.e.,
with the same solid sequence space $Y_{d}\left(X\right)$ as above,
the following hold (see \cite[Theorem 5.3]{GroechenigDescribingFunctions}):
\begin{itemize}
\item the analysis operator
\[
A:{\rm Co}\left(Y\right)\to Y_{d}\left(X\right),f\mapsto\left(\left\langle f,\pi\left(x_{i}\right)\psi\right\rangle \right)_{i\in I}
\]
is well-defined and bounded,
\item there is a bounded linear operator $R:Y_{d}\left(X\right)\to{\rm Co}\left(Y\right)$
satisfying $R\circ A=\operatorname{id}_{{\rm Co}\left(Y\right)}$.
\end{itemize}
\end{itemize}
Here, a family $X=\left(x_{i}\right)_{i\in I}$ is called \textbf{$V$-dense}
if $G=\bigcup_{i\in I}x_{i}V$ and \textbf{relatively separated} if
it is a finite union of separated sets, where a family $Z=\left(z_{j}\right)_{j\in J}$
is called \textbf{separated} if there is a unit neighborhood $W\subset G$
satisfying $z_{j}W\cap z_{\ell}W=\emptyset$ for $j\neq\ell$.
Among other examples, this \emph{group-based} coorbit theory can be
used to obtain Banach frames and atomic decompositions for modulation
spaces as well as for \emph{homogeneous} Besov spaces. There are also
several extensions, for example to the setting of Quasi-Banach spaces\cite{RauhutCoorbitQuasiBanach},
and to the setting of possibly reducible or non-integrable group representations\cite{CoorbitWithVoiceInFrechetSpace}.
The main limitation of this theory, however, is that many relevant
spaces like \emph{inhomogeneous} Besov spaces are \emph{not} covered
by it.
\subsubsection{Generalized coorbit theory}
To overcome this limitation, Fornasier, Rauhut and Ullrich\cite{GeneralizedCoorbit1,GeneralizedCoorbit2}
developed what is called \textbf{generalized coorbit theory}; see
also \cite{NickiErrataForRauhut} for some corrections and extensions
and \cite{QuasiBanachGeneralCoorbit} for a generalization to Quasi-Banach
spaces. For generalized coorbit theory, one starts from a Hilbert
space $\mathcal{H}$, for which one is given a \textbf{continuous
frame} $\Psi=\left(\psi_{x}\right)_{x\in X}$ which is indexed by
some locally compact measure space $X$, equipped with a Radon measure
$\mu$. Formally, this means that for each $f\in\mathcal{H}$, the
function $X\to\mathbb{C},x\mapsto\left\langle f,\,\psi_{x}\right\rangle _{\mathcal{H}}$
is measurable and there are constants $0<A\leq B$ satisfying
\[
A\cdot\left\Vert f\right\Vert _{\mathcal{H}}^{2}\leq\int_{X}\left|\left\langle f,\,\psi_{x}\right\rangle _{\mathcal{H}}\right|^{2}\operatorname{d}\mu\left(x\right)\leq B\cdot\left\Vert f\right\Vert _{\mathcal{H}}^{2}\qquad\forall f\in\mathcal{H}.
\]
In this case, the \textbf{frame operator} $S:\mathcal{H}\to\mathcal{H},f\mapsto\int_{X}\left\langle f,\psi_{x}\right\rangle _{\mathcal{H}}\cdot\psi_{x}\operatorname{d}\mu\left(x\right)$
(with the integral understood in the weak sense) is well-defined,
self-adjoint and positive and thus invertible. Hence, one can form
the \textbf{canonical dual frame} $\tilde{\Psi}=\left(\smash{\tilde{\psi_{x}}}\right)_{x\in X}=\left(S^{-1}\psi_{x}\right)_{x\in X}$.
If the frame $\Psi$ is \textbf{tight}, one can choose $A=B$ and
the dual frame $\tilde{\Psi}$ is simply a scalar multiple of $\Psi$,
but in general, $\Psi$ and $\tilde{\Psi}$ might be very different.
For generalized coorbit theory to be applicable at all, one requires
the \textbf{cross-gramian kernel}
\[
R:X\times X\to\mathbb{C},\left(x,y\right)\mapsto\left\langle \psi_{y},\,S^{-1}\psi_{x}\right\rangle =\left\langle \psi_{y},\,\tilde{\psi_{x}}\right\rangle
\]
to have certain decay/mapping properties; precisely, one requires
$R\in\mathcal{A}_{m}$, where $m=m_{Y}$ is a given control weight
associated to the solid function space $Y\leq L_{{\rm loc}}^{1}\left(X\right)$
in which one is interested. Here, $\mathcal{A}_{m}$ is a suitable
algebra of kernels, cf.\@ \cite[Section 3]{GeneralizedCoorbit1}.
With these two frames $\Psi,\tilde{\Psi}$, there are \emph{two} associated
voice transforms, given by
\[
V_{\Psi}f\left(x\right):=\left\langle f,\,\psi_{x}\right\rangle _{\mathcal{H}}\qquad\text{ and }\qquad W_{\Psi}f\left(x\right):=\left\langle f,\,\smash{\tilde{\psi_{x}}}\right\rangle _{\mathcal{H}}=\left(V_{\Psi}\left[S^{-1}f\right]\right)\left(x\right),
\]
and then (cf.\@ \cite[equation (3.8) and Definition 3.1]{GeneralizedCoorbit1})
also \emph{two} reservoirs $\mathcal{R}_{1}:=\left(\mathcal{K}_{v}^{1}\right)^{\neg}$
and $\mathcal{R}_{2}:=\left(\mathcal{H}_{v}^{1}\right)^{\neg}$ and
\emph{two} coorbit spaces
\[
{\rm Co}\left(Y\right):=\left\{ f\in\mathcal{R}_{1}\,\middle|\, V_{\Psi}f\in Y\right\} \qquad\text{ and }\qquad\widetilde{{\rm Co}}\left(Y\right):=\left\{ f\in\mathcal{R}_{2}\,\middle|\, W_{\Psi}f\in Y\right\} .
\]
Then, if the frame $\Psi$ is ``good enough'' (the precise meaning
of which depends on the space $Y$), one can again discretize the
continuous frame $\Psi$ to obtain atomic decompositions and Banach
frames. However, one has to be a bit careful; assuming that the family
$\left(x_{i}\right)_{i\in I}$ is ``dense enough in $X$'' (cf.\@
\cite[Theorem 5.7]{GeneralizedCoorbit1} for the details), we have
the following:
\begin{itemize}
\item the family $\left(\psi_{x_{i}}\right)_{i\in I}$ is an atomic decomposition
of $\widetilde{{\rm Co}}\left(Y\right)$ with corresponding sequence
space $Y^{\natural}$,
\item the family $\left(\psi_{x_{i}}\right)_{i\in I}$ is a Banach frame
for ${\rm Co}\left(Y\right)$ with corresponding sequence space $Y^{\flat}$.
\end{itemize}
Thus, although generalized coorbit theory is immensely powerful and
general, its main limitation is that one essentially has to start
from a \emph{tight} continuous frame for a Hilbert space $\mathcal{H}$,
since in the non-tight case one faces several limitations:
\begin{itemize}
\item In most cases of continuous \emph{non-tight} frames, one knows very
little about the properties of the (canonical) dual frame $\tilde{\Psi}$,
which makes it hard to verify that the kernel $R\left(x,y\right)=\left\langle \psi_{y},\,\tilde{\psi_{x}}\right\rangle $
satisfies $R\in\mathcal{A}_{m}$.
\item As seen above, one is faced with two \emph{distinct} coorbit spaces
${\rm Co}\left(Y\right)$ and $\widetilde{{\rm Co}}\left(Y\right)$
and obtains a Banach frame for ${\rm Co}\left(Y\right)$ and an atomic
decomposition for $\widetilde{{\rm Co}}\left(Y\right)$. In many cases,
however, it is desired to \emph{simultaneously} have a Banach frame
and an atomic decomposition for \emph{one} common space.
We mention that \cite[Section 4]{GeneralizedCoorbit1} provides criteria
which ensure ${\rm Co}\left(Y\right)=\widetilde{{\rm Co}}\left(Y\right)$,
namely if $\Psi$ and $\tilde{\Psi}$ are $\mathcal{A}_{m}$-self-localized.
To show that this is true, however, one again needs to know a lot
about the dual frame $\tilde{\Psi}$, which in general one does not.
The most convenient way out (outlined in \cite[Theorem 4.7 and the comments afterward]{GeneralizedCoorbit1})
is to find a suitable \textbf{spectral} subalgebra $\mathcal{A}$
of $\mathcal{A}_{m}$ and then to show that the kernel $K\left(x,y\right)=\left\langle \psi_{y},\,\psi_{x}\right\rangle $
satisfies $K\in\mathcal{A}$. Once this is shown, \cite[Theorem 4.7]{GeneralizedCoorbit1}
yields ${\rm Co}\left(Y\right)=\widetilde{{\rm Co}}\left(Y\right)$
as well as $R\in\mathcal{A}\subset\mathcal{A}_{m}$, so that coorbit
theory is applicable. The main limitation of this approach is that
not too many spectral algebras of kernels are known.
\end{itemize}
In total, there are two desirable use cases of (generalized) coorbit
theory in which an actual application is \emph{difficult, or even
impossible:}
\begin{enumerate}
\item In the first case, one is given a (family of) Banach space(s) $B$
and wants to find Banach frames and atomic decompositions for $B$.
To achieve this via (generalized) coorbit theory, one has to find
a (preferably tight) \emph{continuous} frame $\Psi=\left(\psi_{x}\right)_{x\in X}$
and a (family of) solid Banach function space(s) $Y\leq L_{{\rm loc}}^{1}\left(X\right)$
such that $B={\rm Co}\left(Y\right)=\left\{ f\,\middle|\, V_{\Psi}f\in Y\right\} $.
Furthermore, one has to verify that $\Psi$ indeed satisfies all prerequisites
for the application of generalized coorbit theory. Finally, if $\Psi$
is non-tight, one has to verify ${\rm Co}\left(Y\right)=\widetilde{{\rm Co}}\left(Y\right)$,
for example by using the approach using spectral algebras which we
outlined above.
\item In the second case, which occurs e.g.\@ if one wants to study the
approximation theoretic properties of discrete, cone-adapted shearlet
frames\cite{CompactlySupportedShearletFrames}, one starts with a
\emph{discrete} frame $\Psi_{d}=\left(\psi_{i}\right)_{i\in I}$ (or
with a family of such discrete frames, e.g., parametrized by the sampling
density) for a Hilbert space $\mathcal{H}$ and one wants to understand
the space of those functions which are \emph{analysis-sparse} with
respect to this frame, e.g., the space
\[
B_{q}:=\left\{ f\in\mathcal{H}\,\middle|\,\left(\left\langle f,\,\psi_{i}\right\rangle \right)_{i\in I}\in\ell^{q}\left(I\right)\right\} \qquad\text{ for }\qquad q<2.
\]
An important property one might be interested in is \emph{whether
analysis sparsity is equivalent to synthesis sparsity}, i.e., whether
every $f\in B_{q}$ admits an expansion $f=\sum_{i\in I}c_{i}\psi_{i}$
for a sequence $c=\left(c_{i}\right)_{i\in I}\in\ell^{q}\left(I\right)$.
To derive such a statement using coorbit theory, one needs to find
a continuous (preferably tight) frame $\Psi=\left(\psi_{x}\right)_{x\in X}$
for $\mathcal{H}$ such that the discretization $\left(\psi_{x_{i}}\right)_{i\in I}$
of this frame (in the sense of generalized coorbit theory) is equal
to $\Psi_{d}$. Then, provided that ${\rm Co}\left(Y\right)=\widetilde{{\rm Co}}\left(Y\right)=B_{q}$,
coorbit theory will yield the desired statement.
The main problem here—as witnessed by the example of discrete cone-adapted
shearlets—is that it can often be very hard or even impossible to
find such a continuous frame $\Psi$, much less a tight one. There
are tight continuous shearlet frames, e.g.\@ those related to shearlet
coorbit spaces\cite{Dahlke_etal_sh_coorbit1,Dahlke_etal_sh_coorbit2,DahlkeShearletArbitraryDimension,DahlkeShearletCoorbitEmbeddingsInHigherDimensions},
but a discretization of these frames does \emph{not} yield discrete
\emph{cone-adapted} shearlet systems.
\end{enumerate}
As we will see now, our approach does \emph{not} require to have a
continuous frame which can then be discretized. Thus, in this aspect,
our approach improves upon (generalized) coorbit theory. As we will
see in the companion paper \cite{StructuredBanachFrames2}, we are
in particular able to handle discrete cone-adapted shearlet frames;
and for this case, our theory indeed shows that \emph{analysis sparsity
is equivalent to synthesis sparsity}.
\subsubsection{Our approach using decomposition spaces}
For our approach, we start with a \textbf{structured covering} $\mathcal{Q}$
of (an open subset $\mathcal{O}$ of) the frequency space $\mathbb{R}^{d}$.
More precisely (see Subsection \ref{subsec:DecompSpaceDefinitionStandingAssumptions}
for the completely formal assumptions), we assume that
\begin{equation}
\mathcal{Q}=\left(Q_{i}\right)_{i\in I}=\left(T_{i}Q+b_{i}\right)_{i\in I}\label{eq:IntroductionStructuredCoveringAssumption}
\end{equation}
for a fixed open, precompact set $Q\subset\mathbb{R}^{d}$ and certain
linear maps $T_{i}\in\mathrm{GL}\left(\mathbb{R}^{d}\right)$ and translations
$b_{i}\in\mathbb{R}^{d}$. Then, given a suitable partition of unity
$\Phi=\left(\varphi_{i}\right)_{i\in I}$ subordinate to $\mathcal{Q}$
and a suitable weight $w=\left(w_{i}\right)_{i\in I}$ on $I$, as
well as $p,q\in\left(0,\infty\right]$, one defines the decomposition
space (quasi)-norm of a distribution $g$ as
\[
\left\Vert g\right\Vert _{\DecompSp{\mathcal{Q}}p{\ell_{w}^{q}}{}}:=\left\Vert \left(\left\Vert \mathcal{F}^{-1}\left(\varphi_{i}\cdot\widehat{g}\right)\right\Vert _{L^{p}}\right)_{i\in I}\right\Vert _{\ell_{w}^{q}}=\left\Vert \left(\left\Vert \left(\mathcal{F}^{-1}\varphi_{i}\right)\ast g\right\Vert _{L^{p}}\right)_{i\in I}\right\Vert _{\ell_{w}^{q}}\,,
\]
while the \textbf{decomposition space} $\DecompSp{\mathcal{Q}}p{\ell_{w}^{q}}{}$
consists of all distributions for which this (quasi)-norm is finite.
For the exact interpretation of ``distribution'' in this context,
we refer to Subsection \ref{subsec:DecompSpaceDefinitionStandingAssumptions}.
In words, the decomposition space norm is computed by first decomposing
$g$ \emph{in frequency} according to the covering $\mathcal{Q}$ to obtain
the pieces $g_{i}=\mathcal{F}^{-1}\left(\varphi_{i}\cdot\widehat{g}\right)$.
Each of these pieces is then measured in $L^{p}$ and the overall
norm is a certain $\ell_{w}^{q}$-norm over all of these contributions.
In most of the paper, we will even consider the weighted $L^{p}$-spaces
$L_{v}^{p}$ instead of $L^{p}$. But in this introduction, we will
mostly stick to the setting just described, for the sake of simplicity.
Our general aim is to show that one can obtain compactly supported
Banach frames and atomic decompositions $\Psi$ of a very special,
structured form for the decomposition space $\DecompSp{\mathcal{Q}}p{\ell_{w}^{q}}{}$.
In fact, it will turn out that the system $\Psi$ can be taken to
be a generalized shift invariant system generated by a \emph{single
prototype function} $\gamma$, similar to the way in which a prototype
function can generate Gabor, wavelet and shearlet systems.
To see exactly how such a system $\Psi$ might look like, let us write
$S_{i}\xi:=T_{i}\xi+b_{i}$ for $i\in I$. Note that if $\operatorname{supp}\widehat{\gamma}\subset Q$,
then $\operatorname{supp}\left[\widehat{\gamma}\circ S_{i}^{-1}\right]\subset Q_{i}$.
The same remains true in a weak sense if the strict inclusion $\operatorname{supp}\widehat{\gamma}\subset Q$
is replaced by requiring that $\widehat{\gamma}$ be \emph{essentially}
supported in $Q$, which can even hold if $\gamma$ is not band-limited.
Now, note $\mathcal{F}\gamma^{\left(i\right)}=\widehat{\gamma}\circ S_{i}^{-1}$
for $\gamma^{\left(i\right)}:=\left|\det T_{i}\right|\cdot M_{b_{i}}\left[\gamma\circ T_{i}^{T}\right]$.
For consistency with the $L^{2}$-setting, we also consider
\begin{equation}
\gamma^{\left[i\right]}:=\left|\det T_{i}\right|^{1/2}\cdot M_{b_{i}}\left[\gamma\circ T_{i}^{T}\right].\label{eq:IntroductionL2NormalizedFrameElement}
\end{equation}
In fact, we will even allow the generator $\gamma$ to vary with $i\in I$,
i.e., $\gamma^{\left[i\right]}=\left|\det T_{i}\right|^{1/2}\cdot M_{b_{i}}\left[\gamma_{i}\circ T_{i}^{T}\right]$.
An example where this is useful is an inhomogeneous wavelet system:
If the generator $\gamma$ is required to be \emph{independent} of
$i\in I$, the ``low-pass part'' of the wavelet system needs to
be obtained by a frequency shift (i.e., by a modulation) from the
mother wavelet $\gamma$. Indeed, since we consider only affine dilations
of $\widehat{\gamma}$ and since any \emph{linear} dilation of $\widehat{\gamma}$
will vanish at the origin, this is the only way in which one can cover
the origin of the frequency domain. In most cases, the exact shape
of the low-pass part is not important, so that taking a modulation
of the mother wavelet is acceptable. But in other cases, one might
desire more specific properties of the low-pass part; for example,
one could want it to be real-valued. In this case, the added flexibility
of allowing $\gamma$ to depend on $i\in I$ might be valuable. In
this introduction, however, we will only consider the case in which
$\gamma_{i}=\gamma$ is independent of $i\in I$, for the sake of
simplicity.
Now, since the family $\left(\widehat{\gamma^{\left(i\right)}}\right)_{i\in I}$
behaves similarly to the family $\left(\varphi_{i}\right)_{i\in I}$
(at least with respect to the (essential) frequency support), one
could be tempted to conjecture that
\begin{equation}
\left\Vert g\right\Vert _{\DecompSp{\mathcal{Q}}p{\ell_{w}^{q}}{}}\asymp\left\Vert \left(\left\Vert \smash{\gamma^{\left(i\right)}}\ast g\right\Vert _{L^{p}}\right)_{i\in I}\right\Vert _{\ell_{w}^{q}}.\label{eq:IntroductionSemiDiscreteBanachFrame}
\end{equation}
For the special case of $\alpha$-modulation spaces, this statement
was established (for (almost) arbitrary $\gamma\in\mathcal{S}\left(\mathbb{R}^{d}\right)$)
in \cite{EmbeddingsOfAlphaModulationIntoSobolev}. Our first result
(cf.\@ Section \ref{sec:SemiDiscreteBanachFrames}) will be to show
that for $p\in\left[1,\infty\right]$, equation (\ref{eq:IntroductionSemiDiscreteBanachFrame})
is indeed valid under suitable assumptions on $\gamma$. Furthermore,
for $p\in\left(0,1\right)$, we have the slightly modified statement
\begin{equation}
\left\Vert g\right\Vert _{\DecompSp{\mathcal{Q}}p{\ell_{w}^{q}}{}}\asymp\left\Vert \left(\left\Vert \smash{\gamma^{\left(i\right)}}\ast g\right\Vert _{W_{T_{i}^{-T}\left[-1,1\right]^{d}}\left(L^{p}\right)}\right)_{i\in I}\right\Vert _{\ell_{w}^{q}},\label{eq:IntroductionSemiDiscreteQuasiBanachFrame}
\end{equation}
where $W_{T_{i}^{-T}\left[-1,1\right]^{d}}\left(L^{p}\right)$
is a so-called \textbf{Wiener amalgam space} (originally introduced
by Feichtinger\cite{FeichtingerWienerSpaces}). We refer to the results
(\ref{eq:IntroductionSemiDiscreteBanachFrame})-(\ref{eq:IntroductionSemiDiscreteQuasiBanachFrame})
as stating that the family $\left(\gamma^{\left(i\right)}\right)_{i\in I}$
forms a \textbf{\emph{semi-discrete}}\textbf{ Banach frame} for $\DecompSp{\mathcal{Q}}p{\ell_{w}^{q}}{}$.
The reason for this nomenclature is that the index set of the family
$\left(\left(\gamma^{\left(i\right)}\ast g\right)\left(x\right)\right)_{i\in I,x\in\mathbb{R}^{d}}$
has the discrete part $I$, but also the continuous part $\mathbb{R}^{d}$.
Our next results are concerned with a further discretization of this
semi-discrete Banach frame. Indeed, under more stringent assumptions
on $\gamma$, we show in Section \ref{sec:FullyDiscreteBanachFrames}
for $\delta>0$ sufficiently small that the \textbf{structured generalized
shift-invariant system}
\begin{equation}
\Psi_{\delta}:=\left(L_{\delta\cdot T_{i}^{-T}k}\:\widetilde{\gamma^{\left[i\right]}}\right)_{i\in I,k\in\mathbb{Z}^{d}}\qquad\text{ with }\qquad\tilde{f}\left(x\right):=f\left(-x\right)\label{eq:IntroductionBanachFrameFamily}
\end{equation}
generates a Banach frame for $\DecompSp{\mathcal{Q}}p{\ell_{w}^{q}}{}$,
with the associated discrete sequence space
\begin{equation}
Y\!:=\ell_{\left(\left|\det T_{i}\right|^{\frac{1}{2}-\frac{1}{p}}\cdot w_{i}\!\right)_{\!i\in I}}^{q}\!\!\!\!\!\!\!\!\left(\!\left[\ell^{p}\!\left(\mathbb{Z}^{d}\right)\right]_{i\in I}\right)\quad\text{ where }\quad\left\Vert \left(\smash{c_{k}^{\left(i\right)}}\right)_{i\in I,k\in\mathbb{Z}^{d}}\right\Vert _{Y}=\left\Vert \!\left(\left|\det T_{i}\right|^{\frac{1}{2}-\frac{1}{p}}\!\cdot\!w_{i}\!\cdot\!\left\Vert \left(\smash{c_{k}^{\left(i\right)}}\right)_{k\in\mathbb{Z}^{d}}\right\Vert _{\ell^{p}}\right)_{\!i\in I}\right\Vert _{\ell^{q}}.\label{eq:IntroductionSequenceSpaceDefinition}
\end{equation}
Since the system $\Psi_{\delta}$ is generated in a very structured
way—similar to the usual definition of Gabor, wavelet or shearlet
frames—from a \emph{single} prototype function, we call $\Psi_{\delta}$
a \textbf{structured Banach frame} for $\DecompSp{\mathcal{Q}}p{\ell_{w}^{q}}{}$.
Finally, we show in Section \ref{sec:AtomicDecompositions}—again
under slightly different assumptions on $\gamma$—that the family
$\left(L_{\delta\cdot T_{i}^{-T}k}\:\gamma^{\left[i\right]}\right)_{\!i\in I,k\in\mathbb{Z}^{d}}$
forms an \textbf{atomic decomposition} for $\DecompSp{\mathcal{Q}}p{\ell_{w}^{q}}{}$,
with the same associated sequence space $Y$ as above. As above, we
call this family a \textbf{structured atomic decomposition}. Hence,
at least if $\gamma$ is symmetric and fulfills certain technical
conditions, the family $\Psi_{\delta}$ will \emph{simultaneously}
form a Banach frame, as well as an atomic decomposition for $\DecompSp{\mathcal{Q}}p{\ell_{w}^{q}}{}$;
in particular, this implies that \emph{analysis sparsity is equivalent
to synthesis sparsity} for $\Psi_{\delta}$.
We remark that the assumptions placed on the prototype function $\gamma$
are quite technical, even though we will achieve a significant simplification
of these conditions in Section \ref{sec:SimplifiedCriteria}. Indeed,
a slightly simplified version of our main theorem concerning Banach
frames reads as follows:
\begin{thm*}
(cf.\@ Corollary \ref{cor:BanachFrameSimplifiedCriteria} for the
precise statement)
Recall from equation (\ref{eq:IntroductionStructuredCoveringAssumption})
that $\mathcal{Q}=\left(T_{i}Q+b_{i}\right)_{i\in I}$. Assume that there
is an open set $P\subset\mathbb{R}^{d}$ with $\overline{P}\subset Q$
and $\mathcal{O}=\bigcup_{i\in I}T_{i}P+b_{i}$. Let $p,q\in\left(0,\infty\right]$.
Then there are explicitly given $N\in\mathbb{N}$ and $\sigma,\tau>0$, depending
on $d,p,q$, with the following property:
If $w=\left(w_{i}\right)_{i\in I}$ is a $\mathcal{Q}$-moderate\footnote{cf.\@ Section \ref{subsec:DecompSpaceDefinitionStandingAssumptions},
equation (\ref{eq:IntroductionModerateWeightDefinition}) for the
precise definition.} weight and if $\gamma\in L^{1}\left(\mathbb{R}^{d}\right)$ satisfies
the following:
\begin{enumerate}
\item We have $\widehat{\gamma}\in C^{\infty}\left(\mathbb{R}^{d}\right)$,
where all partial derivatives of $\widehat{\gamma}$ are polynomially
bounded.
\item We have $\gamma\in C^{1}\left(\mathbb{R}^{d}\right)$ and $\partial_{\ell}\gamma\in L^{1}\left(\mathbb{R}^{d}\right)\cap L^{\infty}\left(\mathbb{R}^{d}\right)$
for all $\ell\in\left\{ 1,\dots,d\right\} $.
\item We have $\widehat{\gamma}\left(\xi\right)\neq0$ for all $\xi\in\overline{Q}$.
\item We have
\begin{equation}
C_{1}:=\sup_{i\in I}\,\sum_{j\in I}M_{j,i}<\infty\qquad\text{ and }\qquad C_{2}:=\sup_{j\in I}\sum_{i\in I}M_{j,i}<\infty\label{eq:IntroductionSimplifiedCondition}
\end{equation}
with
\[
\qquad M_{j,i}:=\left(\frac{w_{j}}{w_{i}}\right)^{\tau}\cdot\left(1+\left\Vert T_{j}^{-1}T_{i}\right\Vert \right)^{\sigma}\cdot\max_{\left|\beta\right|\leq1}\left(\left|\det T_{i}\right|^{-1}\cdot\int_{Q_{i}}\max_{\left|\alpha\right|\leq N}\left|\left(\partial^{\alpha}\widehat{\partial^{\beta}\gamma}\right)\!\!\left(T_{j}^{-1}\left(\xi-b_{j}\right)\right)\right|\operatorname{d}\xi\right)^{\tau}.
\]
\end{enumerate}
Then, for $\delta>0$ sufficiently small, the family $\Psi_{\delta}$
from equation (\ref{eq:IntroductionBanachFrameFamily}) is a Banach
frame for $\DecompSp{\mathcal{Q}}p{\ell_{w}^{q}}{}$, with associated sequence
space $Y$ as given in equation (\ref{eq:IntroductionSequenceSpaceDefinition}).
\end{thm*}
The conditions which ensure that $\gamma$ generates an atomic decomposition
are similar, but slightly more complicated, cf.\@ Corollary \ref{cor:AtomicDecompositionSimplifiedCriteria}.
For the sake of brevity, we omit them in this introduction.
Of course, condition (\ref{eq:IntroductionSimplifiedCondition}) is
quite technical. The main reason for this is that hugely different
coverings $\mathcal{Q}$ are treated using the same theory. Thus, given
a specific covering (e.g.\@ the ones used to define Besov spaces
or $\alpha$-modulation spaces), the difficulty consists in reducing
the general, abstract criteria provided by the theory to readily verifiable
criteria involving only the smoothness, decay and Fourier decay of
$\gamma$. As we will see in Sections \ref{sec:CompactlySupportedAlphaModulationFrames}
and \ref{sec:BesovFrames}, this is indeed possible for Besov spaces
and $\alpha$-modulation spaces. In addition, in the companion paper
\cite{StructuredBanachFrames2} we will show that the theory also
applies to shearlet smoothness spaces. Furthermore, it turns out that
in each of these cases one can find \emph{compactly supported} prototype
functions $\gamma$ which fulfill the relevant criteria. Thus, although
the decomposition spaces $\DecompSp{\mathcal{Q}}p{\ell_{w}^{q}}{}$ are
defined using the \emph{bandlimited} partition of unity $\left(\varphi_{i}\right)_{i\in I}$,
it is usually possible to give alternative characterizations in terms
of \emph{compactly supported} functions.
\medskip{}
At a first glance, the difficulty pertaining to the technical conditions
described above seems be a major drawback of the theory presented
here in comparison to coorbit theory. But in fact, coorbit theory
faces the same problem: In essentially every example where coorbit
theory is applicable, one has a systematic way of assigning to each
``prototype'' $\psi$ a whole family $\Psi=\left(\psi_{x}\right)_{x\in X}$.
Then, one has to obtain a profound understanding of the mapping $\psi\mapsto\left(\psi_{x}\right)_{x\in X}$
in order to derive readily verifiable conditions on $\psi$ which
ensure that the family $\Psi$ is suitable for the application of
coorbit theory, in particular to ensure that $\Psi$ is a (Hilbert
space) frame and that the kernel $R\left(x,y\right)=\left\langle \psi_{y},\,S^{-1}\psi_{x}\right\rangle $
belongs to $\mathcal{A}_{m}$. As examples for the effort one still
has to put in to apply coorbit theory in specific situations, we mention
\cite{FuehrCoorbit1,FuehrCoorbit2,FuehrGeneralizedCalderonConditions,FuehrSimplifiedVanishingMomentCriteria,GeneralizedCoorbit2,Dahlke_etal_sh_coorbit1,Dahlke_etal_sh_coorbit2,DahlkeShearletArbitraryDimension,DahlkeToeplitzShearletTransform,FuehrVoigtlaenderCoorbitSpacesAsDecompositionSpaces,FuehrWaveletFramesAndAdmissibility,UllrichContinuousCharacterizationsOfBesovTriebelLizorkin}.
We emphasize that this effort should not be seen as a shortcoming
of the mentioned papers or of (generalized) coorbit theory, but rather
as showing that despite the tremendous simplifications coorbit theory
has to offer, one still has to put in work to apply it in concrete
situations. The same is true of the results in this paper.
In fact, there is an intimate connection between the decomposition
space setting considered here and the coorbit setting considered in
\cite{Dahlke_etal_sh_coorbit1,DahlkeShearletArbitraryDimension,FuehrCoorbit2,FuehrSimplifiedVanishingMomentCriteria,UllrichContinuousCharacterizationsOfBesovTriebelLizorkin}:
In all of these papers, the authors consider coorbit spaces of a semi-direct
product $\mathbb{R}^{d}\rtimes H$ for suitable \textbf{dilation
groups} $H\leq\mathrm{GL}\left(\mathbb{R}^{d}\right)$, where the associated
unitary representation $\pi:\mathbb{R}^{d}\rtimes H\to\mathcal{U}\left(L^{2}\left(\mathbb{R}^{d}\right)\right),\left(x,h\right)\mapsto L_{x}D_{h}$
is the \textbf{quasi-regular representation}, i.e., the natural action
of $\mathbb{R}^{d}\rtimes H$ on $L^{2}\left(\mathbb{R}^{d}\right)$
in terms of the translations $L_{x}$ and the dilations $D_{h}$ with
$D_{h}f=\left|\det h\right|^{-1/2}\cdot\left(f\circ h^{-1}\right)$.
The mentioned papers contain—typically somewhat technical and lengthy—sufficient
criteria which ensure that a given \emph{mother wavelet} can serve
as an atom in the coorbit scheme. These conditions heavily depend
on the considered dilation group $H$ and also on the weight $w:\mathbb{R}^{d}\rtimes H\to\left(0,\infty\right)$
which is used for the weighted mixed Lebesgue space $L_{w}^{p,q}\left(\mathbb{R}^{d}\rtimes H\right)$.
For a given mother wavelet $g$ satisfying these criteria, the theory
of coorbit spaces implies that each sufficiently densely sampled family
$\left(\pi\left(x_{j},h_{j}\right)g\right)_{j\in J}$ yields an atomic
decomposition, as well as a Banach frame for the coorbit space ${\rm Co}\left(L_{w}^{p,q}\left(\mathbb{R}^{d}\rtimes H\right)\right)$.
But as shown in \cite{FuehrVoigtlaenderCoorbitSpacesAsDecompositionSpaces}
and in \cite[Section 4]{VoigtlaenderPhDThesis}, we have ${\rm Co}\left(L_{w}^{p,q}\left(\mathbb{R}^{d}\rtimes H\right)\right)=\DecompSp{\mathcal{Q}_{H}}p{\ell_{\tilde{w}}^{q}}{}$
up to canonical identifications, at least if the weight $w=w\left(x,h\right)$
only depends on the second factor, i.e., if $w=w\left(h\right)$.
Here, the so-called \textbf{induced covering} $\mathcal{Q}_{H}=\left(h_{i}^{-T}Q\right)_{i\in I}$
of the \textbf{dual orbit} $\mathcal{O}=H^{T}\xi_{0}\subset\mathbb{R}^{d}$
is determined by an arbitrary well-spread family $\left(h_{i}\right)_{i\in I}$
in $H$. Given this identification, one can then apply the theory
developed in this paper to derive conditions on the prototype $\gamma$
which ensure that the family
\[
\left(L_{\delta\cdot h_{i}k}\:\gamma^{\left[i\right]}\right)_{i\in I,k\in\mathbb{Z}^{d}}=\left(\left|\det h_{i}\right|^{-1/2}\cdot L_{\delta\cdot h_{i}k}\left[\gamma\circ h_{i}^{-1}\right]\right)_{i\in I,k\in\mathbb{Z}^{d}}=\left(\pi\left(\delta h_{i}k,\,h_{i}\right)\gamma\right)_{i\in I,k\in\mathbb{Z}^{d}}
\]
forms a Banach frame, or an atomic decomposition for the decomposition
space $\DecompSp{\mathcal{Q}_{H}}p{\ell_{\tilde{w}}^{q}}{}$ and thus also
for the coorbit space ${\rm Co}\left(L_{w}^{p,q}\left(\mathbb{R}^{d}\rtimes H\right)\right)$.
Note that the family $\left[\left(\delta h_{i}k,\,h_{i}\right)\right]_{i\in I,k\in\mathbb{Z}^{d}}$
is well-spread in $\mathbb{R}^{d}\rtimes H$ since $\left(h_{i}\right)_{i\in I}$
is well-spread in $H$. Hence, the theory developed in this paper
yields Banach frames and atomic decompositions which are \emph{of
the same form} as those obtained via coorbit theory. As future work,
we plan a systematic comparison of the conditions imposed on the prototype
$\gamma$ by coorbit theory (as in \cite{Dahlke_etal_sh_coorbit1,DahlkeShearletArbitraryDimension,FuehrCoorbit2,FuehrSimplifiedVanishingMomentCriteria,UllrichContinuousCharacterizationsOfBesovTriebelLizorkin})
on the one hand and by the theory developed in this paper on the other
hand.
\medskip{}
In spite of the strong connection between coorbit theory and the theory
developed in this paper, they differ in some important aspects:
As a first difference, we observe that coorbit theory requires to
pass from the given continuous frame $\left(\psi_{x}\right)_{x\in X}$
to a sufficiently densely sampled version $\left(\psi_{x_{i}}\right)_{i\in I}$.
This will usually not only require a sufficiently dense sampling \emph{in
the space domain} (which corresponds to $\delta$ in equation (\ref{eq:IntroductionBanachFrameFamily})),
but also to a rather dense sampling \emph{in the frequency domain}.
In contrast, for our approach only the sampling density \emph{in space}
needs to be sufficiently high. The ``frequency sampling density''
is fixed a priori by choosing the covering $\mathcal{Q}=\left(T_{i}Q+b_{i}\right)_{i\in I}$.
Next, the main advantage of our approach in comparison to coorbit
theory is that one does \emph{not} need to start from a given \emph{continuous}
frame $\left(\psi_{x}\right)_{x\in X}$ which is then discretized.
In fact, one can even start from a given discrete frame which is of
the form (\ref{eq:IntroductionBanachFrameFamily}). As long as the
family $\mathcal{Q}=\left(T_{i}Q+b_{i}\right)_{i\in I}$ forms a suitable
covering, one can then consider the associated decomposition spaces
$\DecompSp{\mathcal{Q}}p{\ell_{w}^{q}}{}$ and use the theory presented
here to justify that the discrete frame one started with forms a Banach
frame and an atomic decomposition for $\DecompSp{\mathcal{Q}}p{\ell_{w}^{q}}{}$,
possibly after adjusting the sampling density.
Probably, this intuition is what originally lead Labate et al.\@
to the introduction of the \textbf{shearlet smoothness spaces}\cite{Labate_et_al_Shearlet},
although they did not have the machinery to rigorously prove that
the usual discrete, cone-adapted shearlet systems indeed yield Banach
frames and atomic decompositions for the shearlet smoothness spaces.
Using the theory developed here, we will see in the companion paper
\cite{StructuredBanachFrames2} that this is indeed the case. Furthermore,
we will employ our results to show that suitable discrete, cone-adapted
shearlet systems achieve an almost optimal approximation rate for
the class of \textbf{cartoon-like functions}. At a first glance, this
might appear to be a well-known statement, but a closer inspection
of the classical results about approximation of cartoon-like functions
by shearlets (see e.g.\@ \cite{ShearletsAndOptimallySparseApproximation,CompactlySupportedShearlets,CompactlySupportedShearletsAreOptimallySparse,OptimallySparseMultidimensionalRepresentationUsingShearlets})
reveals that these papers in fact only show that the $N$-term approximation
$f_{N}$ \emph{with respect to the dual frame} of the shearlet frame
satisfies the (almost optimal) rate $\left\Vert f-f_{N}\right\Vert _{L^{2}}\lesssim N^{-1}\cdot\left(\log N\right)^{\theta}$
for suitable $\theta>0$.
\subsubsection{Comparison to other constructions of Banach frame decompositions
of decomposition spaces}
One of the first general constructions of atomic decompositions for
decomposition spaces was given by Borup and Nielsen in \cite{BorupNielsenDecomposition}.
The main difference between their approach and ours is that our frame
elements $\gamma^{\left[i\right]}$ can be chosen to be compactly
supported, while Borup and Nielsen purely focus on bandlimited frame
elements.
There is also a more recent paper by Nielsen and Rasmussen \cite{CompactlySupportedFramesForDecompositionSpaces}
in which they construct compactly supported frames for certain decomposition
spaces. In comparison to that paper, our assumptions concerning the
covering $\mathcal{Q}$ are more general, while our conclusions are more
specific:
\begin{itemize}[leftmargin=0.5cm]
\item In \cite{CompactlySupportedFramesForDecompositionSpaces}, the authors
only consider coverings $\mathcal{Q}$ which are induced by considering
$\mathbb{R}^{d}$ in a certain way as a space of homogeneous type:
More precisely, $\mathcal{Q}$ is assumed to satisfy $\mathcal{Q}=\left(Q_{k}\right)_{k\in\mathbb{Z}^{d}}=\left(\mathcal{B}_{A}\left(\xi_{k},\varrho\cdot h\left(\xi_{k}\right)\right)\right)_{k\in\mathbb{Z}^{d}}$,
where the balls $\mathcal{B}_{A}\left(\xi,r\right)=\left\{ \zeta\in\mathbb{R}^{d}\,\middle|\,\left|\zeta-\xi\right|_{A}<r\right\} $
are defined using the quasi-metric $\left|\cdot\right|_{A}$, which
is induced in a certain way (cf.\@ \cite[Definition 2.1]{CompactlySupportedFramesForDecompositionSpaces})
by the one-parameter group of dilations $\left(\delta_{t}\right)_{t>0}$
where $\delta_{t}=\exp\left(A\cdot\ln t\right)$ for a fixed matrix
$A\in\mathbb{R}^{d\timesd}$ with positive eigenvalues.
As shown between \cite[Lemma 2.6]{CompactlySupportedFramesForDecompositionSpaces}
and \cite[Definition 2.7]{CompactlySupportedFramesForDecompositionSpaces},
we have
\[
Q_{k}=\delta_{h\left(\xi_{k}\right)}\left[\mathcal{B}_{A}\left(0,\varrho\right)\right]+\xi_{k}\qquad\forall k\in\mathbb{Z}^{d},
\]
so that all sets $Q_{k}$ of the covering $\mathcal{Q}$ are affine images
of a fixed set, where the linear parts of the affine maps are all
elements of the one-parameter family $\left(\delta_{t}\right)_{t>0}$.
Note with $\nu:={\rm trace}\,A>0$ that $\det\delta_{t}=t^{\nu}$
for all $t>0$, so that $\delta_{t}$ is uniquely determined by its
determinant. Since the covering used to define shearlet smoothness
spaces uses affine transformations for which many \emph{different}
linear parts have the \emph{same} determinant, this shows—or at least
very strongly indicates—that the covering used to define the shearlet
smoothness spaces does \emph{not} satisfy the assumptions imposed
in \cite{CompactlySupportedFramesForDecompositionSpaces}, while our
theory is able to handle these spaces.
Below, we will give another more rigorous argument which shows that
the theory developed in \cite{CompactlySupportedFramesForDecompositionSpaces}
does in fact \emph{neither} apply to the usual dyadic covering which
is used to define (inhomogeneous) Besov spaces, \emph{nor} to the
covering used to define shearlet smoothness spaces.
\item While each of the compactly supported Banach frames constructed in
\cite{CompactlySupportedFramesForDecompositionSpaces} is a union
of generalized shift invariant systems, it is \emph{not} true that
the frames are generated from a single prototype function in the same
structured way as in our paper. In contrast, the Banach frames constructed
in \cite{CompactlySupportedFramesForDecompositionSpaces} are of the
form
\[
\qquad\left(\psi_{k,n}\right)_{k,n\in\mathbb{Z}^{d}}=\left(\left[h\left(\xi_{k}\right)\right]^{\nu/2}\cdot\tau_{k}\left(\delta_{h\left(\xi_{k}\right)}^{T}\bullet-\frac{\pi}{a}n\right)e^{i\left\langle \cdot,\xi_{k}\right\rangle }\right)_{k,n\in\mathbb{Z}^{d}}\quad\text{ where }\quad\tau_{k}=\sum_{i=1}^{K}a_{i}^{\left(k\right)}g_{m}\left(\bullet+\smash{b_{i}^{\left(k\right)}}\right),
\]
for suitable $K,m\in\mathbb{N}$ and with $g_{m}=C_{g}m^{\nu}\cdot g\circ\delta_{m}^{T}$.
Hence, using notation as in eq.\@ (\ref{eq:IntroductionL2NormalizedFrameElement})
with the covering $\mathcal{Q}$ as defined above and with $b_{k}:=\xi_{k}$,
as well as $T_{k}:=\delta_{h\left(\xi_{k}\right)}$, we have $\left(\psi_{k,n}\right)_{k,n\in\mathbb{Z}^{d}}=\left(L_{\frac{\pi}{a}T_{k}^{-T}n}\:\tau_{k}^{\left[k\right]}\right)_{k,n\in\mathbb{Z}^{d}}$,
while the structured family $\Psi_{\delta}$ defined in equation (\ref{eq:IntroductionBanachFrameFamily})
satisfies $\Psi_{\delta}=\left(\!L_{\delta\cdot T_{k}^{-T}n}\:\widetilde{\psi^{\left[k\right]}}\right)_{\!k,n\in\mathbb{Z}^{d}}$.
In other words, while the structured Banach frames constructed in
this paper arise from a \emph{single} prototype function by translations,
modulations and dilations, the frames constructed in \cite{CompactlySupportedFramesForDecompositionSpaces}
do \emph{not} satisfy this property.
In particular, if the covering $\mathcal{Q}$ is the usual dyadic covering
of $\mathbb{R}^{d}$ used to define (inhomogeneous) Besov spaces,
then $\Psi_{\delta}$ will be an (inhomogeneous) wavelet frame, while
this is \emph{not} in general true of the frame constructed in \cite{CompactlySupportedFramesForDecompositionSpaces}.
Additionally, the results in \cite{CompactlySupportedFramesForDecompositionSpaces}
are not applicable in this setting, as we will see below.
\end{itemize}
This last defect—that the resulting Banach frame is not generated
from a \emph{single} prototype—is addressed in the follow-up paper
\cite{NielsenSinglyGeneratedFrames}. There, Morten Nielsen considers
the same general setting as described above. He then constructs a
\emph{bandlimited} Banach frame for the associated decomposition spaces
which is generated by a \emph{single} prototype function in the same
structured way as proposed in the present paper. Furthermore, Nielsen
then uses a distortion argument to show that one can also obtain a
structured Banach frame with a \emph{single, compactly supported}
generator. Hence, at a first glance, it might seem that all results
of the present paper are already contained in \cite{NielsenSinglyGeneratedFrames}.
This, however, is \emph{not} true for the following reasons:
\begin{itemize}[leftmargin=0.5cm]
\item As already observed above, the coverings considered in \cite{NielsenSinglyGeneratedFrames}
and \cite{CompactlySupportedFramesForDecompositionSpaces} are quite
restricted. They have to be of the form $\mathcal{Q}=\left(Q_{k}\right)_{k\in\mathbb{Z}^{d}}=\left(\mathcal{B}_{A}\left(\xi_{k},\varrho\cdot h\left(\xi_{k}\right)\right)\right)_{k\in\mathbb{Z}^{d}}$,
where the balls $\mathcal{B}_{A}\left(\xi,r\right)=\left\{ \zeta\in\mathbb{R}^{d}\,\middle|\,\left|\zeta-\xi\right|_{A}<r\right\} $
are defined using the quasi-metric $\left|\cdot\right|_{A}$, which
is determined by a suitable matrix $A$.
In particular, the setting considered in \cite{NielsenSinglyGeneratedFrames}
does \emph{neither} include the case of homogeneous or inhomogeneous
Besov spaces, nor the case of shearlet smoothness spaces. To see this,
note that \cite[Proposition 3.6]{NielsenSinglyGeneratedFrames} does
not impose any vanishing moment conditions on the prototype $\gamma$
(which is called $g$ in the notation of \cite{NielsenSinglyGeneratedFrames}).
In fact, it is even required that $\widehat{\gamma}\left(0\right)\neq0$.
But it is folklore that the generator of an (inhomogeneous or homogeneous)
wavelet frame for $L^{2}\left(\mathbb{R}\right)$ has to satisfy certain vanishing
moment conditions; the proof for homogeneous wavelet frames is given
in \cite[Theorem 3.3.1]{DaubechiesTenLecturesOnWavelets}. A proof
of the corresponding statement for discrete cone-adapted shearlet
frames is given in Appendix \ref{sec:ShearletFrameVanishingMomentNecessity}.
In stark contrast, the theory developed in the present paper \emph{is}
able to handle Besov spaces (cf.\@ Section \ref{sec:BesovFrames}),
as well as shearlet smoothness spaces (cf.\@ the companion paper
\cite{StructuredBanachFrames2}).
\item Since a distortion argument is used to obtain a compactly supported
Banach frame from a bandlimited frame, the choice of the generator
$\gamma$ in \cite{NielsenSinglyGeneratedFrames} is quite restricted;
$\gamma$ has to be close enough to the generator of the bandlimited
frame.
In contrast, the assumptions imposed on $\gamma$ in the present paper
are quite mild. In most concrete cases (in particular for $\alpha$-modulation
spaces, Besov spaces and shearlet smoothness spaces), the conditions
reduce to suitable smoothness, decay and vanishing moment criteria,
in conjunction with a certain nonvanishing condition for the Fourier
transform $\widehat{\gamma}$.
\item In the present paper, we also consider the decomposition spaces $\DecompSp{\mathcal{Q}}p{\ell_{w}^{q}}v$
where a \emph{weighted} Lebesgue space $L_{v}^{p}\left(\mathbb{R}^{d}\right)$
is used. In contrast, \cite{NielsenSinglyGeneratedFrames} only considers
the \emph{unweighted} case.
\end{itemize}
We remark however that \cite{NielsenSinglyGeneratedFrames} jointly
considers Triebel-Lizorkin type, as well as Besov type decomposition
spaces. In contrast, at least in its present state, the approach developed
in this paper only applies to the Besov type decomposition spaces.
Finally, we mention the recent paper \cite{CharacterizationOfSparseNonstationaryGaborExpansions}
in which Ottosen and Nielsen take the ``reverse'' of the usual approach:
Instead of starting with a given function space $X$ and then constructing
Banach frames or atomic decompositions for this space, the authors
start with a given \textbf{painless nonstationary Gabor frame} $\left(h_{i,k}\right)_{i,k\in\mathbb{Z}^{d}}$
satisfying
\[
h_{i,k}=L_{a_{i}\cdot k}\,h_{i}\quad\text{ and }\quad\operatorname{supp}\widehat{h_{i}}\subset\left[0,\,a_{i}^{-1}\right]^{d}+b_{i}\quad\text{ for certain }\quad a_{i}>0\text{ and }b_{i}\in\mathbb{R}^{d}.
\]
Under suitable assumptions on the (slightly enlarged) covering
\[
\mathcal{Q}=\left(Q_{i}\right)_{i\in\mathbb{Z}^{d}}\qquad\text{ with }\qquad Q_{i}=a_{i}^{-1}\cdot\left(-\delta,\,1+\delta\right)^{d}+b_{i},
\]
Ottosen and Nielsen then show that the renormalized family $\left(h_{i,k}^{\left(p\right)}\right)_{i,k\in\mathbb{Z}^{d}}$
defined by $h_{i,k}^{\left(p\right)}=a_{i}^{\frac{1}{p}-\frac{1}{2}}\cdot h_{i,k}$
forms a Banach frame for the decomposition space $\DecompSp{\mathcal{Q}}p{\ell_{\omega^{s}}^{q}}{}$,
where $\omega_{i}=1+\left\Vert \xi_{i}\right\Vert ^{2}$ for suitable
$\xi_{i}\in Q_{i}$. In addition, it is shown in \cite[Theorem 6.1]{CharacterizationOfSparseNonstationaryGaborExpansions}
for $p,q\in\left(0,\infty\right)$ that every $f\in\DecompSp{\mathcal{Q}}p{\ell_{\omega^{s}}^{q}}{}$
admits an expansion of the form
\begin{equation}
f=\sum_{i,k\in\mathbb{Z}^{d}}\left\langle h,\,h_{i,k}\right\rangle \cdot\tilde{h}_{i,k},\label{eq:OttosenNielsenBanachFrameExpansion}
\end{equation}
where $\left(\smash{\tilde{h}_{i,k}}\right)_{i,k\in\mathbb{Z}^{d}}$
is the canonical dual frame of the nonstationary Gabor frame $\left(h_{i,k}\right)_{i,k\in\mathbb{Z}^{d}}$.
In summary, the paper \cite{CharacterizationOfSparseNonstationaryGaborExpansions}
starts with a given painless nonstationary Gabor frame and then shows
that the space of analysis-sparse signals w.r.t.\@ the frame coincides
with a suitably defined decomposition space. Note that the painless
nonstationary Gabor frames are always bandlimited. Using the theory
developed in this paper, it should be possible (perhaps with the cost
of changing the sampling density in comparison to the original frame)
to show similar results for nonstationary Gabor frames with \emph{compactly
supported} generators. Furthermore, while the results in \cite{CharacterizationOfSparseNonstationaryGaborExpansions}
only show that each $f\in\DecompSp{\mathcal{Q}}p{\ell_{w}^{q}}{}$ admits
a sparse expansion in terms of the \emph{dual frame} $\left(\smash{\tilde{h}_{i,k}}\right)_{i,k\in\mathbb{Z}^{d}}$,
our results would yield a sparse expansion in terms of the frame itself,
so that analysis sparsity is equivalent to synthesis sparsity.
\subsection{Notation and conventions}
\label{subsec:Notation}We write $\mathbb{N}=\mathbb{Z}_{\geq1}$ for the set of \textbf{natural
numbers} and $\mathbb{N}_{0}=\mathbb{Z}_{\geq0}$ for the set of natural numbers including
$0$. For a matrix $A\in\mathbb{C}^{d\timesd}$, we
denote by $A^{T}$ the transpose of $A$. The norm $\left\Vert A\right\Vert $
of $A$ is the usual \textbf{operator norm} of $A$, acting on $\mathbb{R}^{d}$
equipped with the usual euclidean norm $\left|\cdot\right|=\left\Vert \cdot\right\Vert _{2}$.
The \textbf{open euclidean ball} of radius $r>0$ around $x\in\mathbb{R}^{d}$
is denoted by $B_{r}\left(x\right)$. For a linear (bounded) operator
$T:X\to Y$ between (quasi)-normed spaces $X,Y$, we denote the \textbf{operator
norm} of $T$ by
\[
\vertiii T:=\vertiii T_{X\to Y}:=\sup_{\left\Vert x\right\Vert _{X}\leq1}\left\Vert Tx\right\Vert _{Y}.
\]
For an arbitrary set $M$, we let $\left|M\right|\in\mathbb{N}_{0}\cup\left\{ \infty\right\} $
denote the number of elements of the set. For $n\in\mathbb{N}_{0}=\mathbb{Z}_{\geq0}$,
we write $\underline{n}:=\left\{ 1,\dots,n\right\} $; in particular,
$\underline{0}=\emptyset$. For the \textbf{closure} of a subset $M$
of some topological space, we write $\overline{M}$.
The $d$-dimensional \textbf{Lebesgue measure} of a (measurable)
set $M\subset\mathbb{R}^{d}$ is denoted by $\lambda\left(M\right)$
or by $\lambda_{d}\left(M\right)$. Furthermore, for $M\subset\mathbb{R}^{d}$,
we define the \textbf{indicator function} (or \textbf{characteristic
function}) ${\mathds{1}}_{M}$ of the set $M$ by
\[
{\mathds{1}}_{M}:\mathbb{R}^{d}\to\left\{ 0,1\right\} ,x\mapsto\begin{cases}
1, & \text{if }x\in M,\\
0, & \text{otherwise}.
\end{cases}
\]
For two subsets $A,B\subset\mathbb{R}^{d}$, we define the \textbf{Minkowski
sum} and the \textbf{Minkowski difference} of $A,B$ by
\[
A+B:=\left\{ a+b\,\middle|\, a\in A,\,b\in B\right\} \qquad\text{ and }\qquad A-B:=\left\{ a-b\,\middle|\, a\in A,\,b\in B\right\} .
\]
The Minkowski difference $A-B$ should be distinguished from the \textbf{set-theoretic
difference} $A\setminus B=\left\{ a\in A\,\middle|\, a\notin B\right\} $.
The \textbf{translation} and \textbf{modulation} of a function $f:\mathbb{R}^{d}\to\mathbb{C}^{k}$
by $x\in\mathbb{R}^{d}$ or $\xi\in\mathbb{R}^{d}$ are, respectively,
denoted by
\[
L_{x}f:\mathbb{R}^{d}\to\mathbb{C}^{k},y\mapsto f\left(y-x\right),\qquad\text{ and }\qquad M_{\xi}f:\mathbb{R}^{d}\to\mathbb{C}^{k},y\mapsto e^{2\pi i\left\langle \xi,y\right\rangle }f\left(y\right).
\]
For the \textbf{Fourier transform}, we use the convention $\widehat{f}\left(\xi\right):=\left(\mathcal{F} f\right)\left(\xi\right):=\int_{\mathbb{R}^{d}}f\left(x\right)\cdot e^{-2\pi i\left\langle x,\xi\right\rangle }\operatorname{d} x$
for $f\in L^{1}\left(\mathbb{R}^{d}\right)$. It is well-known that
the Fourier transform extends to a unitary automorphism $\mathcal{F}:L^{2}\left(\mathbb{R}^{d}\right)\to L^{2}\left(\mathbb{R}^{d}\right)$.
The inverse of this map is the continuous extension of the inverse
Fourier transform, given by $\left(\mathcal{F}^{-1}f\right)\left(x\right)=\int_{\mathbb{R}^{d}}f\left(\xi\right)e^{2\pi i\left\langle x,\xi\right\rangle }\operatorname{d}\xi$
for $f\in L^{1}\left(\mathbb{R}^{d}\right)$. We will make frequent
use of the space $\mathcal{S}\left(\mathbb{R}^{d}\right)$ of \textbf{Schwartz
functions} and its dual space $\mathcal{S}'\left(\mathbb{R}^{d}\right)$,
the space of \textbf{tempered distributions}. For more details on
these spaces, we refer to \cite[Section 9]{FollandRA}; in particular,
we note that the Fourier transform restricts to a linear homeomorphism
$\mathcal{F}:\mathcal{S}\left(\mathbb{R}^{d}\right)\to\mathcal{S}\left(\mathbb{R}^{d}\right)$;
by duality, we can thus define $\mathcal{F}:\mathcal{S}'\left(\mathbb{R}^{d}\right)\to\mathcal{S}'\left(\mathbb{R}^{d}\right)$
by $\mathcal{F}\varphi=\varphi\circ\mathcal{F}$ for $\varphi\in\mathcal{S}'\left(\mathbb{R}^{d}\right)$.
Given an open subset $U\subset\mathbb{R}^{d}$, we let $\DistributionSpace U$
denote the space of \textbf{distributions} on $U$, i.e., the topological
dual space of $\TestFunctionSpace U$. For the precise definition
of the topology on $\TestFunctionSpace U$, we refer to \cite[Chapter 6]{RudinFA}.
We remark that the dual pairings $\left\langle \cdot,\cdot\right\rangle _{\mathcal{D}',\mathcal{D}}$
and $\left\langle \cdot,\cdot\right\rangle _{\mathcal{S}',\mathcal{S}}$
are always taken to be \emph{bilinear} instead of sesquilinear.
We write $v_{d}:=\lambda_{d}\left(B_{1}\left(0\right)\right)$
for the $d$-dimensional Lebesgue measure of the euclidean
unit ball. An easy, but sometimes useful estimate is that $v_{d}\leq2^{d}$,
since $B_{1}\left(0\right)\subset\left[-1,1\right]^{d}$.
Furthermore, we let $s_{d}:=\mathcal{H}_{d-1}\left(S^{d-1}\right)$
denote the surface measure of the unit sphere. It is well-known that
$s_{d}=d\cdot v_{d}\leqd\cdot2^{d}\leq2^{2d}$,
since $d\leq2^{d}$. Finally, we have $B_{1}^{\left\Vert \cdot\right\Vert _{\infty}}\left(0\right)\subset B_{\sqrt{d}}\left(0\right)$
and thus $2^{d}=\lambda\left(B_{1}^{\left\Vert \cdot\right\Vert _{\infty}}\left(0\right)\right)\leq\lambda\left(B_{\sqrt{d}}\left(0\right)\right)=v_{d}\cdotd^{d/2}$,
which implies $v_{d}\geq\left(2/\sqrt{d}\right)^{d}$.
The constant $s_{d}$ be important for us due to the following:
For $p\in\left(0,\infty\right)$ and $N>d/p$, we get using
polar coordinates that
\begin{align*}
\left\Vert \left(1+\left|\bullet\right|\right)^{-N}\right\Vert _{L^{p}}^{p}=\int_{\mathbb{R}^{d}}\left(1+\left|x\right|\right)^{-Np}\operatorname{d} x & =\int_{0}^{\infty}r^{d-1}\int_{S^{d-1}}\left(1+\left|r\xi\right|\right)^{-Np}\operatorname{d}\mathcal{H}_{d-1}\left(\xi\right)\,\operatorname{d} r\\
& =\mathcal{H}_{d-1}\left(S^{d-1}\right)\cdot\int_{0}^{\infty}r^{d-1}\cdot\left(1+r\right)^{-Np}\operatorname{d} r\\
& \leq s_{d}\cdot\int_{0}^{\infty}\left(1+r\right)^{d-Np-1}\operatorname{d} r\\
\left({\scriptstyle \text{since }d-Np<0}\right) & =s_{d}\cdot\frac{\left(1+r\right)^{d-Np}}{d-Np}\bigg|_{0}^{\infty}=\frac{s_{d}}{Np-d},
\end{align*}
and hence
\begin{equation}
\left\Vert \left(1-\left|\bullet\right|\right)^{-N}\right\Vert _{L^{p}}\leq\left(\frac{1}{p}\cdot\frac{s_{d}}{N-\frac{d}{p}}\right)^{1/p}\qquad\forall N>d/p,\label{eq:StandardDecayLpEstimate}
\end{equation}
which also remains valid (with the interpretation $x^{0}=1$ for arbitrary
$x\geq0$) for $p=\infty$.
\subsection{Definition of decomposition spaces and standing assumptions}
\label{subsec:DecompSpaceDefinitionStandingAssumptions}For the whole
paper, we fix a \textbf{semi-structured admissible covering} $\mathcal{Q}=\left(Q_{i}\right)_{i\in I}$
of an open subset $\mathcal{O}\subset\mathbb{R}^{d}$. Precisely this means
that for each $i\in I$ there is a measurable subset $Q_{i}'\subset\mathbb{R}^{d}$,
an invertible linear map $T_{i}\in\mathrm{GL}\left(\mathbb{R}^{d}\right)$
and a translation $b_{i}\in\mathbb{R}^{d}$ such that $Q_{i}=S_{i}Q_{i}'=T_{i}Q_{i}'+b_{i}$
for the affine transformation $S_{i}:\mathbb{R}^{d}\to\mathbb{R}^{d},\xi\mapsto T_{i}\xi+b_{i}$
and such that the following properties are fulfilled:
\begin{enumerate}
\item $\mathcal{Q}$ covers $\mathcal{O}$, i.e., $\mathcal{O}=\bigcup_{i\in I}Q_{i}$.
\item $\mathcal{Q}$ is \textbf{admissible}, i.e., we have $\left|i^{\ast}\right|\leq N_{\mathcal{Q}}<\infty$
for all $i\in I$, where
\begin{equation}
i^{\ast}:=\left\{ \ell\in I\,\middle|\, Q_{\ell}\cap Q_{i}\neq\emptyset\right\} .\label{eq:IndexClusterDefinition}
\end{equation}
\item There is some $R_{\mathcal{Q}}>0$ satisfying $Q_{i}'\subset\overline{B_{R_{\mathcal{Q}}}}\left(0\right)$
for all $i\in I$.
\item There is some $C_{\mathcal{Q}}>0$ satisfying $\left\Vert T_{i}^{-1}T_{\ell}\right\Vert \leq C_{\mathcal{Q}}$
for all $i\in I$ and all $\ell\in i^{\ast}$.
\end{enumerate}
The most common form of decomposition spaces uses a (quasi)-norm of
the form $\left\Vert \left(\left\Vert \mathcal{F}^{-1}\left(\varphi_{i}\cdot\widehat{g}\right)\right\Vert _{L^{p}}\right)_{i\in I}\right\Vert _{\ell_{w}^{q}}$,
i.e., the frequency-localized pieces $g_{i}=\mathcal{F}^{-1}\left(\varphi_{i}\cdot\widehat{g}\right)$
of $g$ are measured in $L^{p}\left(\mathbb{R}^{d}\right)$. To achieve
even greater flexibility, we will allow weighted Lebesgue spaces of
the form $L_{v}^{p}\left(\mathbb{R}^{d}\right)$ instead of $L^{p}\left(\mathbb{R}^{d}\right)$.
Here, we write
\[
L_{v}^{p}\left(\smash{\mathbb{R}^{d}}\right):=\left\{ f:\mathbb{R}^{d}\to\mathbb{C}\,\middle|\, f\text{ measurable and }v\cdot f\in L^{p}\left(\smash{\mathbb{R}^{d}}\right)\right\} ,
\]
equipped with the natural (quasi)-norm $\left\Vert f\right\Vert _{L_{v}^{p}}:=\left\Vert v\cdot f\right\Vert _{L^{p}}$.
In order to still obtain reasonable spaces and results, we assume
the following:
\begin{enumerate}[resume]
\item The weights $v,v_{0}:\mathbb{R}^{d}\to\left(0,\infty\right)$ are
measurable and satisfy the following:
\begin{enumerate}
\item $v_{0}\geq1$ and\footnote{One can always assume $v_{0}\geq1$ without loss of generality, since
all properties of $v_{0}$ (including submultiplicativity) are also
fulfilled for $\widetilde{v_{0}}:=1+v_{0}\geq1$, where possibly $\Omega_{1}$
has to be enlarged, since $\widetilde{v_{0}}\left(x\right)\leq1+\Omega_{1}\cdot\left(1+\left|x\right|\right)^{K}\leq\left(1+\Omega_{1}\right)\cdot\left(1+\left|x\right|\right)^{K}$.
Likewise, by switching to $\tilde{v_{0}}\left(x\right):=v_{0}\left(x\right)+v_{0}\left(-x\right)$,
one can always assume $v_{0}$ to be symmetric.} $v_{0}$ is symmetric, i.e., $v_{0}\left(-x\right)=v_{0}\left(x\right)$
for all $x\in\mathbb{R}^{d}$.
\item $v_{0}$ is submultiplicative, i.e., $v_{0}\left(x+y\right)\leq v_{0}\left(x\right)\cdot v_{0}\left(y\right)$
for all $x,y\in\mathbb{R}^{d}$.
\item $v$ is $v_{0}$-moderate, i.e., $v\left(x+y\right)\leq v\left(x\right)\cdot v_{0}\left(y\right)$
for all $x,y\in\mathbb{R}^{d}$.
\item There is some $K\geq0$ and some $\Omega_{1}\geq1$ satisfying $v_{0}\left(x\right)\leq\Omega_{1}\cdot\left(1+\left|x\right|\right)^{K}$
for all $x\in\mathbb{R}^{d}$.
\item The constant $K$ from the previous step satisfies $K=0$ or there
is a constant $\Omega_{0}\geq1$ satisfying $\left\Vert T_{i}^{-1}\right\Vert \leq\Omega_{0}$
for all $i\in I$.
\end{enumerate}
\item There is a \textbf{$\mathcal{Q}$-$v_{0}$-BAPU} (bounded admissible partition
of unity) $\Phi=\left(\varphi_{i}\right)_{i\in I}$ for $\mathcal{Q}$,
which means that:
\begin{enumerate}
\item $\varphi_{i}\in\TestFunctionSpace{\mathcal{O}}$ for all $i\in I$ and furthermore
$\varphi_{i}\equiv0$ on $\mathcal{O}\setminus Q_{i}$.
\item $\sum_{i\in I}\varphi_{i}\equiv1$ on $\mathcal{O}$.
\item For each $p\in\left(0,\infty\right]$, the following expression (then
a constant) is finite:
\[
C_{\mathcal{Q},\Phi,v_{0},p}:=\sup_{i\in I}\left[\left|\det T_{i}\right|^{\max\left\{ \frac{1}{p},1\right\} -1}\cdot\left\Vert \mathcal{F}^{-1}\varphi_{i}\right\Vert _{L_{v_{0}}^{\min\left\{ 1,p\right\} }}\right].
\]
\end{enumerate}
\end{enumerate}
Clearly, if one chooses $K=0$ and $v=v_{0}\equiv1$, then one obtains
the usual decomposition spaces, as considered e.g.\@ in \cite{BorupNielsenDecomposition,VoigtlaenderPhDThesis,DecompositionEmbedding,DecompositionIntoSobolev}.
This will be the most common case. Note that in this case, we do \emph{not}
need to assume $\left\Vert T_{i}^{-1}\right\Vert \leq\Omega_{0}$
for all $i\in I$, i.e., the covering $\mathcal{Q}$ can be very general.
We observe for later use that the preceding assumptions imply
\begin{equation}
\left(1+\left|x\right|\right)^{K}\leq\Omega_{0}^{K}\cdot\left(1+\left|T_{i}^{T}x\right|\right)^{K}\qquad\forall x\in\mathbb{R}^{d}.\label{eq:WeightLinearTransformationsConnection}
\end{equation}
Indeed, in case of $K=0$, this is trivial. In case of $K>0$, our
assumptions imply
\[
\left|x\right|=\left|T_{i}^{-T}T_{i}^{T}x\right|\leq\left\Vert T_{i}^{-T}\right\Vert \cdot\left|T_{i}^{T}x\right|=\left\Vert T_{i}^{-1}\right\Vert \cdot\left|T_{i}^{T}x\right|\leq\Omega_{0}\cdot\left|T_{i}^{T}x\right|
\]
and hence $1+\left|x\right|\leq1+\Omega_{0}\cdot\left|T_{i}^{T}x\right|\leq\Omega_{0}\cdot\left(1+\left|T_{i}^{T}x\right|\right)$,
where the last step used that $\Omega_{0}\geq1$. This easily shows
that equation (\ref{eq:WeightLinearTransformationsConnection}) remains
valid also for $K>0$.
Finally, we observe for later use the convolution relation $L_{v_{0}}^{1}\left(\mathbb{R}^{d}\right)\ast L_{v}^{p}\left(\mathbb{R}^{d}\right)\hookrightarrow L_{v}^{p}\left(\mathbb{R}^{d}\right)$
for $p\in\left[1,\infty\right]$. Indeed, we have
\begin{align*}
v\left(x\right)\cdot\left|\left(f\ast g\right)\left(x\right)\right| & \leq v\left(x\right)\cdot\int_{\mathbb{R}^{d}}\left|f\left(y\right)\right|\cdot\left|g\left(x-y\right)\right|\operatorname{d} y\\
\left({\scriptstyle \text{since }v\left(x\right)=v\left(x-y+y\right)\leq v\left(x-y\right)\cdot v_{0}\left(y\right)}\right) & \leq\int_{\mathbb{R}^{d}}\left|\left(v_{0}\cdot f\right)\left(y\right)\right|\cdot\left|\left(v\cdot g\right)\left(x-y\right)\right|\operatorname{d} y,
\end{align*}
so that Minkowski's inequality for integrals (cf.\@ \cite[Theorem (6.19)]{FollandRA}),
together with the isometric translation invariance of $L^{p}\left(\mathbb{R}^{d}\right)$,
yields
\begin{align}
\left\Vert f\ast g\right\Vert _{L_{v}^{p}}=\left\Vert v\cdot\left(f\ast g\right)\right\Vert _{L^{p}} & \leq\left\Vert x\mapsto\int_{\mathbb{R}^{d}}\left|\left(v_{0}\cdot f\right)\left(y\right)\right|\cdot\left|\left(v\cdot g\right)\left(x-y\right)\right|\operatorname{d} y\right\Vert _{L^{p}}\nonumber \\
& \leq\int_{\mathbb{R}^{d}}\left\Vert x\mapsto\left|\left(v_{0}\cdot f\right)\left(y\right)\right|\cdot\left|\left(v\cdot g\right)\left(x-y\right)\right|\right\Vert _{L^{p}}\operatorname{d} y\nonumber \\
& =\int_{\mathbb{R}^{d}}\left|\left(v_{0}\cdot f\right)\left(y\right)\right|\operatorname{d} y\cdot\left\Vert v\cdot g\right\Vert _{L^{p}}=\left\Vert f\right\Vert _{L_{v_{0}}^{1}}\cdot\left\Vert g\right\Vert _{L_{v}^{p}}<\infty.\label{eq:WeightedYoungInequality}
\end{align}
We will call this estimate the \textbf{weighted Young inequality}.
In particular, it shows that $\left(\left|f\right|\ast\left|g\right|\right)\left(x\right)<\infty$
for almost all $x\in\mathbb{R}^{d}$.
\medskip{}
Given a $\mathcal{Q}$-$v_{0}$-BAPU $\Phi=\left(\varphi_{i}\right)_{i\in I}$,
we define the \textbf{clustered version} of $\Phi$ as $\Phi^{\ast}=\left(\varphi_{i}^{\ast}\right)_{i\in I}$,
where $\varphi_{i}^{\ast}:=\sum_{\ell\in i^{\ast}}\varphi_{\ell}$.
Because of $\sum_{i\in I}\varphi_{i}\equiv1$ on $\mathcal{O}\supset Q_{i}$
and since $\varphi_{\ell}\equiv0$ on $Q_{i}$ for all $\ell\in I\setminus i^{\ast}$,
it is not hard to see $\varphi_{i}^{\ast}\equiv1$ on $\overline{Q_{i}}$,
a property which we will use frequently. In particular, since $\varphi_{i}^{\ast}\in\TestFunctionSpace{\mathcal{O}}$
as a finite sum of elements of $\TestFunctionSpace{\mathcal{O}}$, we see
that $\overline{Q_{i}}\subset\mathcal{O}$ is compact.
\medskip{}
Next, we fix a \textbf{$\mathcal{Q}$-moderate weight} $w=\left(w_{i}\right)_{i\in I}$,
which means that $w_{i}\in\left(0,\infty\right)$ for each $i\in I$
and that there is a constant $C_{\mathcal{Q},w}>0$ such that
\begin{equation}
w_{i}\leq C_{\mathcal{Q},w}\cdot w_{\ell}\qquad\forall\:i\in I\text{ and }\ell\in i^{\ast}.\label{eq:IntroductionModerateWeightDefinition}
\end{equation}
Under these assumptions, it follows from \cite[Lemma 4.13]{DecompositionEmbedding}
that the \textbf{$\mathcal{Q}$-clustering map}
\begin{equation}
\Gamma_{\mathcal{Q}}:\ell_{w}^{q}\left(I\right)\to\ell_{w}^{q}\left(I\right),\left(c_{i}\right)_{i\in I}\mapsto\left(c_{i}^{\ast}\right)_{i\in I}\qquad\text{ with }\qquad c_{i}^{\ast}:=\sum_{\ell\in i^{\ast}}c_{\ell}\label{eq:QClusteringMapDefinition}
\end{equation}
is well-defined and bounded with
\begin{equation}
\vertiii{\Gamma_{\mathcal{Q}}}\leq C_{\mathcal{Q},w}\cdot N_{\mathcal{Q}}^{1+\frac{1}{q}}.\label{eq:WeightedSequenceSpaceClusteringMapNormEstimate}
\end{equation}
Here, the \textbf{weighted sequence space $\ell_{w}^{q}\left(I\right)$}
is given by
\[
\ell_{w}^{q}\left(I\right):=\left\{ c=\left(c_{i}\right)_{i\in I}\in\mathbb{C}^{I}\,\middle|\,\left\Vert c\right\Vert _{\ell_{w}^{q}}:=\left\Vert \left(w_{i}\cdot c_{i}\right)_{i\in I}\right\Vert _{\ell^{q}}<\infty\right\} ,
\]
for arbitrary $q\in\left(0,\infty\right]$.
\medskip{}
Given all of these assumptions, we define for $p,q\in\left(0,\infty\right]$
the \textbf{Fourier-side decomposition space} associated to $\mathcal{Q}$
and the parameters $p,q,v,w$ as
\[
\FourierDecompSp{\mathcal{Q}}p{\ell_{w}^{q}}v:=\left\{ f\in\mathcal{D}'\left(\mathcal{O}\right)\,\middle|\,\left\Vert f\right\Vert _{\FourierDecompSp{\mathcal{Q}}p{\ell_{w}^{q}}v}:=\left\Vert \left(\left\Vert \mathcal{F}^{-1}\left(\varphi_{i}f\right)\right\Vert _{L_{v}^{p}}\right)_{i\in I}\right\Vert _{\ell_{w}^{q}}<\infty\right\} .
\]
Finally, we set $Z\left(\mathcal{O}\right):=\mathcal{F}\left(\TestFunctionSpace{\mathcal{O}}\right)$,
equipped with the unique topology which makes the Fourier transform
$\mathcal{F}:\TestFunctionSpace{\mathcal{O}}\to Z\left(\mathcal{O}\right)$ a topological
isomorphism. Then, with $Z'\left(\mathcal{O}\right)$ denoting the topological
dual space of $Z\left(\mathcal{O}\right)$, we define the (space-side) \textbf{decomposition
space} associated to $\mathcal{Q}$ and the parameters $p,q,v,w$ as
\[
\DecompSp{\mathcal{Q}}p{\ell_{w}^{q}}v:=\left\{ g\in Z'\left(\mathcal{O}\right)\,\middle|\,\left\Vert g\right\Vert _{\DecompSp{\mathcal{Q}}p{\ell_{w}^{q}}v}:=\left\Vert \widehat{g}\right\Vert _{\FourierDecompSp{\mathcal{Q}}p{\ell_{w}^{q}}v}<\infty\right\} ,
\]
where $\mathcal{F} g:=\widehat{g}:=g\circ\mathcal{F}\in\DistributionSpace{\mathcal{O}}$
for $g\in Z'\left(\mathcal{O}\right)$. It is not hard to see that the Fourier
transform $\mathcal{F}:Z'\left(\mathcal{O}\right)\to\DistributionSpace{\mathcal{O}}$
restricts to an isometric isomorphism $\mathcal{F}:\DecompSp{\mathcal{Q}}p{\ell_{w}^{q}}v\to\FourierDecompSp{\mathcal{Q}}p{\ell_{w}^{q}}v$
and that we have $Z'\left(\mathcal{O}\right)=\mathcal{F}^{-1}\left(\mathcal{D}'\left(\mathcal{O}\right)\right)$.
For an explanation for the choice of the reservoirs $\mathcal{D}'\left(\mathcal{O}\right)$
and $Z'\left(\mathcal{O}\right)$, we refer to \cite[Remark 3.13]{DecompositionEmbedding}.
Finally, we mention that \cite[Section 8]{DecompositionEmbedding}
provides a convenient criterion which ensures that each $f\in\DecompSp{\mathcal{Q}}p{\ell_{w}^{q}}{}$
extends to a tempered distribution. In particular, if $v\gtrsim1$,
then clearly $\DecompSp{\mathcal{Q}}p{\ell_{w}^{q}}v\hookrightarrow\DecompSp{\mathcal{Q}}p{\ell_{w}^{q}}{}$.
Hence, if the previously mentioned criterion is fulfilled and if $\mathcal{O}=\mathbb{R}^{d}$,
we have (up to trivial identifications) that
\[
\DecompSp{\mathcal{Q}}p{\ell_{w}^{q}}v=\left\{ g\in\mathcal{S}'\left(\smash{\mathbb{R}^{d}}\right)\,\middle|\,\left\Vert g\right\Vert _{\DecompSp{\mathcal{Q}}p{\ell_{w}^{q}}v}=\left\Vert \left(\left\Vert \mathcal{F}^{-1}\left(\varphi_{i}\widehat{g}\right)\right\Vert _{L_{v}^{p}}\right)_{i\in I}\right\Vert _{\ell_{w}^{q}}<\infty\right\} .
\]
We remark that the usual papers treating general decomposition spaces
(for general $p,q\in\left(0,\infty\right]$) do usually only consider
the case $v\equiv1$. Hence, it is not entirely clear that the spaces
defined here are indeed well-defined (Quasi)-Banach spaces for $v\not\equiv1$.
We will see below (cf.\@ Proposition \ref{prop:WeightedDecompositionSpaceWellDefined}
and Lemma \ref{lem:WeightedDecompositionSpaceComplete}) that this
is indeed the case.
\subsection{Structure of the paper}
The theory of decomposition spaces is highly dependent on convolutions,
since the very definition of the norm involves quantities of the form
\[
\left\Vert \mathcal{F}^{-1}\left(\varphi_{i}\cdot\widehat{g}\right)\right\Vert _{L_{v}^{p}}=\left\Vert \left(\mathcal{F}^{-1}\varphi_{i}\right)\ast g\right\Vert _{L_{v}^{p}}.
\]
For the range $p\in\left[1,\infty\right]$, Young's inequality $L^{1}\ast L^{p}\hookrightarrow L^{p}$
is usually sufficient to handle such convolutions. But in the range
$p\in\left(0,1\right)$, Young's inequality breaks down completely.
For the usual theory of decomposition spaces, one instead invokes
the convolution relation
\[
\left\Vert f\ast g\right\Vert _{L^{p}}\leq C_{p,d}\cdot R^{d\left(\frac{1}{p}-1\right)}\cdot\left\Vert f\right\Vert _{L^{p}}\cdot\left\Vert g\right\Vert _{L^{p}}\qquad\text{ assuming }\qquad\operatorname{supp}\widehat{f}\subset B_{R}\left(\xi_{1}\right)\text{ and }\operatorname{supp}\widehat{g}\subset B_{R}\left(\xi_{2}\right)
\]
for certain $\xi_{1},\xi_{2}\in\mathbb{R}^{d}$. Note though that
this convolution relation only applies to \emph{band-limited} functions.
But since we are interested in characterizations of decomposition
spaces using (possibly) \emph{compactly supported} functions, this
is not of much use to us.
To overcome this problem, we will invoke the theory of the \textbf{Wiener
amalgam spaces} $W_{Q}\left(L^{\infty},L_{v}^{p}\right)$ which were
originally introduced by Feichtinger\cite{FeichtingerWienerSpaces}.
The main idea is to associate to a (measurable) function $f:\mathbb{R}^{d}\to\mathbb{C}$
the local maximal function
\[
M_{Q}f:\mathbb{R}^{d}\to\left[0,\infty\right],x\mapsto\left\Vert {\mathds{1}}_{x+Q}\cdot f\right\Vert _{L^{\infty}}
\]
and to define the Wiener amalgam (quasi)-norm of $f$ as $\left\Vert f\right\Vert _{W_{Q}\left(L^{\infty},L_{v}^{p}\right)}=\left\Vert M_{Q}f\right\Vert _{L_{v}^{p}}$.
Broadly speaking, functions in $W_{Q}\left(L^{\infty},L_{v}^{p}\right)$
are locally in $L^{\infty}$ and globally in $L_{v}^{p}$. For brevity,
we will simply write $W_{Q}\left(L_{v}^{p}\right):=W_{Q}\left(L^{\infty},L_{v}^{p}\right)$.
For these spaces, convolution relations are known, cf.\@ \cite{RauhutWienerAmalgam}
and \cite[Section 2.3]{VoigtlaenderPhDThesis}. For our purposes,
however, these results are not sufficient: They establish estimates
of the form
\[
\left\Vert f\ast g\right\Vert _{W_{Q}\left(L_{v}^{p}\right)}\leq C_{p,Q,v}\cdot\left\Vert f\right\Vert _{W_{Q}\left(L_{v_{0}}^{p}\right)}\cdot\left\Vert g\right\Vert _{W_{Q}\left(L_{v}^{p}\right)},
\]
where the constant $C_{p,Q,v}$ depends heavily—and \emph{in an unspecified
way}—on $Q$. But for our purposes, we will consider the spaces $W_{T_{i}^{-T}\left[-1,1\right]^{d}}\left(L^{\infty},L_{v}^{p}\right)$
where $i\in I$ varies; see for example equation (\ref{eq:IntroductionSemiDiscreteQuasiBanachFrame}).
Then, we will need estimates of the form
\[
\left\Vert f\ast g\right\Vert _{W_{T_{i}^{-T}\left[-1,1\right]^{d}}\left(L_{v}^{p}\right)}\leq C_{i,j,\ell,p,v}\cdot\left\Vert f\right\Vert _{W_{T_{j}^{-T}\left[-1,1\right]^{d}}\left(L_{v_{0}}^{p}\right)}\cdot\left\Vert g\right\Vert _{W_{T_{\ell}^{-T}\left[-1,1\right]^{d}}\left(L_{v}^{p}\right)},
\]
with precise control on the constant $C_{i,j,\ell,p,v}$. Hence, in
Section \ref{sec:QuasiBanachConvolutionWienerAmalgam}, we redevelop
parts of the theory of Wiener amalgam spaces, paying close attention
to the dependence of certain constants on the base-set $Q$.
Next, in Section \ref{sec:SemiDiscreteBanachFrames}, we derive assumptions
on the prototype function $\gamma$ which ensure that the norm equivalences
given in equations (\ref{eq:IntroductionSemiDiscreteBanachFrame})
and (\ref{eq:IntroductionSemiDiscreteQuasiBanachFrame}) are true.
More precisely, we will show that the map
\[
\DecompSp{\mathcal{Q}}p{\ell_{w}^{q}}v\to\ell_{w}^{q}\left(\left[V_{i}\right]_{i\in I}\right),g\mapsto\left(\gamma^{\left(i\right)}\ast g\right)_{i\in I}
\]
forms a Banach frame, where $V_{i}:=L_{v}^{p}\left(\mathbb{R}^{d}\right)$
in case of $p\in\left[1,\infty\right]$ and $V_{i}:=W_{T_{i}^{-T}\left[-1,1\right]^{d}}\left(L_{v}^{p}\right)$
in case of $p\in\left(0,1\right)$ and where finally
\[
\ell_{w}^{q}\left(\left[V_{i}\right]_{i\in I}\right)=\left\{ \left(g_{i}\right)_{i\in I}\,\middle|\,\left(\forall i\in I:g_{i}\in V_{i}\right)\text{ and }\left(\left\Vert g_{i}\right\Vert _{V_{i}}\right)_{i\in I}\in\ell_{w}^{q}\left(I\right)\right\} .
\]
Part of the problem is to explain how the convolution $\gamma^{\left(i\right)}\ast g$
can be interpreted, especially in case of $\mathcal{O}\subsetneq\mathbb{R}^{d}$,
since then each element $g$ of the decomposition space $\DecompSp{\mathcal{Q}}p{\ell_{w}^{q}}v$
is the inverse Fourier transform of the distribution $\widehat{g}\in\DistributionSpace{\mathcal{O}}$,
so that it is not obvious how $\gamma^{\left(i\right)}\ast g$ can
be understood.
In Section \ref{sec:FullyDiscreteBanachFrames}, we further discretize
the Banach frame $\left(\gamma^{\left(i\right)}\right)_{i\in I}$
from above: Under slightly more strict assumptions on $\gamma$ than
before, we will be able to show that the family $\Psi_{\delta}=\left(L_{\delta\cdot T_{i}^{-T}k}\:\widetilde{\gamma^{\left[i\right]}}\right)_{i\in I,k\in\mathbb{Z}^{d}}$
forms a \textbf{Banach frame} for $\DecompSp{\mathcal{Q}}p{\ell_{w}^{q}}v$,
once $\delta>0$ is chosen small enough. Our proof technique is similar
to that of coorbit theory: We use the partition of unity $\left(\varphi_{i}\right)_{i\in I}$
associated to the covering $\mathcal{Q}$ to obtain a kind of reproduction
formula, which we then discretize. The details, however, are quite
technical.
Next, in Section \ref{sec:AtomicDecompositions} we establish the
dual statement that the family $\left(L_{\delta\cdot T_{i}^{-T}k}\:\gamma^{\left[i\right]}\right)_{i\in I,k\in\mathbb{Z}^{d}}$
forms an \textbf{atomic decomposition} for $\DecompSp{\mathcal{Q}}p{\ell_{w}^{q}}v$.
As above, this is based on a suitable discretization of a certain
reproduction formula.
Finally, since the varying assumptions placed on the prototype $\gamma$
are quite technical and hard to verify, Section \ref{sec:SimplifiedCriteria}
is devoted to a considerable simplification of these conditions. While
not exactly straightforward to verify, these conditions can be verified
in practice, where the degree of difficulty mainly depends on the
given covering $\mathcal{Q}=\left(T_{i}Q_{i}'+b_{i}\right)_{i\in I}$.
As a litmus test of our theory, we show in Section \ref{sec:CompactlySupportedAlphaModulationFrames}
that it can be used to obtain compactly supported structured Banach
frames and atomic decompositions for the $\alpha$-modulation spaces
$\AlphaModSpace pq{\alpha}s\left(\mathbb{R}^{d}\right)$, even for
$p,q<1$, thereby extending the state of the art. Furthermore, in
Section \ref{sec:BesovFrames}, we show that our theory can be used
to establish that certain compactly supported wavelet systems generate
Banach frames and atomic decompositions for inhomogeneous Besov spaces.
We emphasize that we consider these two specific examples since they
can be handled with reasonably low effort, but still indicate that—\emph{and
how}—the general theory can be filled with life for concrete special
cases. The theory presented here certainly has more interesting and
more novel applications, in particular to the theory of shearlets.
But in order to keep the size of this paper somewhat manageable, we
postpone these applications to the companion paper \cite{StructuredBanachFrames2}.
\subsection*{Credit where credit is due}
\epigraph{``[...] virtually all of our techniques already exist in some antecedent form. Nevertheless their particular combination here leads to new conclusions and to sharpened versions of known results. Moreover, our presentation reveals a[...] structure underlying a diverse range of topics in harmonic analysis.''}{M.\@ Frazier and B.\@ Jawerth, \cite[Page 36]{FrazierJawerthDiscreteTransform}}
The results and proof techniques employed in this paper were heavily
inspired by a number of earlier results:
The first impulse for writing this paper was caused by my reading
of the paper \cite{EmbeddingsOfAlphaModulationIntoSobolev}. In that
paper, the author characterizes the existence of embeddings between
$\alpha$-modulation spaces and Sobolev spaces. As an intermediate
result, he also proves
\begin{equation}
\left\Vert g\right\Vert _{\AlphaModSpace pq{\alpha}s}\asymp\left\Vert \left(\left\Vert \smash{\gamma^{\left(i\right)}}\ast g\right\Vert _{L^{p}}\right)_{i\in\mathbb{Z}^{d}}\right\Vert _{\ell_{\left(\left\langle k\right\rangle ^{s}\right)_{k\in\mathbb{Z}^{d}}}^{q}}\label{eq:AlphaModulationNormCharacterization}
\end{equation}
for arbitrary $p,q\in\left(0,\infty\right]$, $\alpha\in\left[0,1\right)$
and $s\in\mathbb{R}$, as well as $g\in\AlphaModSpace pq{\alpha}s\left(\mathbb{R}^{d}\right)$,
if the prototype function $\gamma\in\mathcal{S}\left(\mathbb{R}^{d}\right)$
is chosen suitably. Here, the functions $\gamma^{\left(i\right)}$
for $i\in\mathbb{Z}^{d}$ are formed from $\gamma$ as described
before equation (\ref{eq:IntroductionL2NormalizedFrameElement}),
where $\mathcal{Q}=\mathcal{Q}^{\left(\alpha\right)}=\left(\left\langle k\right\rangle ^{\frac{\alpha}{1-\alpha}}\cdot B_{R}\left(0\right)+\left\langle k\right\rangle ^{\frac{\alpha}{1-\alpha}}k\right)_{k\in\mathbb{Z}^{d}}$
is the usual covering used to define $\alpha$-modulation spaces;
see also Section \ref{sec:CompactlySupportedAlphaModulationFrames}.
Note that—at least for $p\in\left[1,\infty\right]$—this result is
a special case of the results about semi-discrete Banach frames from
Section \ref{sec:SemiDiscreteBanachFrames}. Specifically, the paper
\cite{EmbeddingsOfAlphaModulationIntoSobolev} caused me to investigate
whether a norm characterization as in equation (\ref{eq:AlphaModulationNormCharacterization})
was also possible in the more general setting of (essentially) arbitrary
decomposition spaces and not only for $\alpha$-modulation spaces.
In particular, it caused me to consider the structured families of
the form $\left(\gamma^{\left(i\right)}\right)_{i\in I}$ with $\gamma^{\left(i\right)}=\left|\det T_{i}\right|\cdot M_{b_{i}}\left[\gamma\circ T_{i}^{T}\right]$,
where $\mathcal{Q}=\left(T_{i}Q_{i}'+b_{i}\right)_{i\in I}$.
Furthermore, an investigation of the proofs in \cite{EmbeddingsOfAlphaModulationIntoSobolev}
lead me to consider assumptions similar to those stated in Assumption
\ref{assu:MainAssumptions} below. Specifically, it caused me to impose
boundedness of the operator associated to the infinite matrix $\left(\left\Vert \mathcal{F}^{-1}\left(\varphi_{i}\cdot\widehat{\gamma^{\left(j\right)}}\right)\right\Vert _{L^{1}}\right)_{j,i\in I}$.
In summary, at least for the case $p\in\left[1,\infty\right]$, the
results about semi-discrete Banach frames for decomposition spaces
in this paper (cf.\@ Section \ref{sec:SemiDiscreteBanachFrames})
can be seen as a slight generalization of the results in \cite{EmbeddingsOfAlphaModulationIntoSobolev}.
\medskip{}
For the case $p\in\left(0,1\right)$, however, I was not able to adapt
the techniques used in \cite{EmbeddingsOfAlphaModulationIntoSobolev}
to the general setting of decomposition spaces. In fact, for $p\in\left(0,1\right)$,
the results derived in \cite{EmbeddingsOfAlphaModulationIntoSobolev}
differ from those in Section \ref{sec:SemiDiscreteBanachFrames}:
While the characterization from \cite{EmbeddingsOfAlphaModulationIntoSobolev}
(cf.\@ equation (\ref{eq:AlphaModulationNormCharacterization}))
considers the usual $L^{p}$ norm of the convolutions $\gamma^{\left(i\right)}\ast f$,
in Section \ref{sec:SemiDiscreteBanachFrames} we show for $p\in\left(0,1\right)$
that
\[
\left\Vert g\right\Vert _{\DecompSp{\mathcal{Q}}p{\ell_{w}^{q}}{}}\asymp\left\Vert \left(\left\Vert \smash{\gamma^{\left(i\right)}}\ast g\right\Vert _{W_{T_{i}^{-T}\left[-1,1\right]^{d}}\left(L^{p}\right)}\right)_{i\in I}\right\Vert _{\ell_{w}^{q}},
\]
i.e., we use \textbf{Wiener amalgam spaces} instead of the spaces
$L^{p}$ themselves.
Here, again, I was inspired heavily by earlier results: The main limitation
of the spaces $L^{p}\left(\mathbb{R}^{d}\right)$ for $p\in\left(0,1\right)$
in the present setting is that there are no meaningful convolution
relations for them, partly since we do not even have $L^{p}\left(\mathbb{R}^{d}\right)\hookrightarrow L_{{\rm loc}}^{1}\left(\mathbb{R}^{d}\right)$.
Luckily, Holger Rauhut\cite{RauhutCoorbitQuasiBanach} had already
observed—while generalizing coorbit theory \cite{FeichtingerCoorbit0,FeichtingerCoorbit1,FeichtingerCoorbit2,GroechenigDescribingFunctions}
to the setting of Quasi-Banach spaces—that these limitations can be
avoided by considering the Wiener amalgam spaces $W_{Q}\left(L^{p}\right)$
instead of $L^{p}$ itself. Rauhut had also already developed associated
convolution relations\cite{RauhutWienerAmalgam} for these spaces.
Of course, all of this was based on the original invention of Wiener
Amalgam spaces which is due to Hans Feichtinger\cite{FeichtingerWienerInterpolation,FeichtingerWienerSpaces}.
All in all, given these earlier papers, it was natural to consider
Wiener amalgam spaces. The (as far as I know) novel idea was to consider
these Wiener amalgam spaces $W_{Q}\left(L^{p}\right)$ with a definite
choice of the base set $Q$, which was allowed to heavily vary with
$i\in I$. Further, it seems to be a new (or at least not completely
well-known) fact that suitably bandlimited $L^{p}$ functions automatically
belong to $W_{Q}\left(L^{p}\right)$, where this statement comes with
a precise estimate for the Wiener amalgam norm in terms of $Q$ and
the Fourier support of the function.
\medskip{}
At this point, I had managed to generalize the results about semi-discrete
Banach frames developed in \cite{EmbeddingsOfAlphaModulationIntoSobolev}
to the setting of general decomposition spaces. One of my main goals,
however, was a better understanding of the approximation theoretic
properties of discrete, cone-adapted shearlet systems. To achieve
this, a further discretization of these \emph{semi}-discrete Banach
frames was necessary. The inspiration for treating this additional
discretization step came from the theory of coorbit spaces as developed
by Feichtinger and Gröchenig\cite{FeichtingerCoorbit0,FeichtingerCoorbit1,FeichtingerCoorbit2,GroechenigDescribingFunctions}
and also (in more generalized form) by Rauhut, Fornasier and Ullrich\cite{RauhutCoorbitQuasiBanach,GeneralizedCoorbit1,GeneralizedCoorbit2}.
The underlying important idea of coorbit theory is to transfer the
study of certain function spaces via a suitable transform to the study
of certain Banach spaces which have a \textbf{reproducing property}.
Formally, one employs the so-called voice transform $V$ to establish
an isomorphism between the coorbit space ${\rm Co}\left(Y\right)$
and its image $Z:=V\left[{\rm Co}\left(Y\right)\right]$ under the
voice transform. The crucial property of the space $Z$ is that we
have the \textbf{reproducing formula}
\[
F=F\ast G\qquad\forall F\in Z
\]
for a suitable kernel $G$. In fact, in the setting of generalized
coorbit theory, the convolution with $G$ needs to be replaced by
a more general integral operator.
If the kernel $G$ is regular enough, the reproducing formula allows
to show that a sufficiently dense sampling of $F\in Z$ suffices to
reconstruct $F$ uniquely. Proving this is based on a (suitable) notion
of the \textbf{oscillation} of a function. This sampling result can
then be transferred to the coorbit space ${\rm Co}\left(Y\right)$
to obtain Banach frames and atomic decompositions. Similar techniques
are also used in \cite{GroechenigNonuniformSampling2}.
The new contribution was thus to derive a suitable reproducing formula
in the general setting of decomposition spaces, cf.\@ Lemma \ref{lem:SpecialProjection}.
Once this was established, existing ideas and techniques could be
used to obtain the desired discrete Banach frames and atomic decompositions.
We remark, however, that the established reproducing formula for decomposition
spaces is highly nontrivial.
\medskip{}
In total, the present paper would not have been possible without inspiration
from existing results, concepts and techniques (Wiener amalgam spaces
and their convolution relations, oscillation of a function, semi-discrete
Banach frames for $\alpha$-modulation spaces, etc.). The contribution
of the paper is that these results and techniques are combined and
refined to achieve novel and nontrivial results which—due to their
generality—apply in a wide variety of settings.
\subsection*{A comment on constants}
Instead of using only implied constants of the form $C=C\left(d,p,\mathcal{Q},...\right)$,
in this paper we try to provide explicit constants whenever possible.
In principle, this allows one e.g.\@ to determine an \emph{explicit}
$\delta_{0}>0$ such that the family $\Psi_{\delta}$ defined in equation
(\ref{eq:IntroductionBanachFrameFamily}) yields a Banach frame for
the decomposition space under consideration for $0<\delta\leq\delta_{0}$.
We make no effort, however, to produce the optimal (or even good)
constants. Occasionally, we even enlarge appearing constants just
to make the expressions for the constants in question more optically
pleasing (i.e., shorter). Due to these reasons, the resulting sampling
density $\delta_{0}$ will probably be of size $\delta_{0}\approx2^{-1000}$
or even smaller.
Thus, our leading philosophy is that \emph{an arbitrarily bad explicit
constant is still (much) better than an implicit constant which one
does not know at all}.
\section{Convolution in $L^{p},p\in\left(0,1\right)$ and Wiener Amalgam spaces}
\label{sec:QuasiBanachConvolutionWienerAmalgam}The well-known Young
inequality $\left\Vert f\ast g\right\Vert _{L^{p}}\leq\left\Vert f\right\Vert _{L^{1}}\cdot\left\Vert g\right\Vert _{L^{p}}$
fails for $p\in\left(0,1\right)$, cf.\@ \cite[Example 3.1]{DecompositionEmbedding}.
One can solve this in two ways: The first way is given in \cite[Proposition 1.5.1]{TriebelTheoryOfFunctionSpaces},
where it is shown that
\[
\left\Vert f\ast g\right\Vert _{L^{p}}\lesssim\left\Vert f\right\Vert _{L^{p}}\cdot\left\Vert g\right\Vert _{L^{p}}
\]
if $f$ and $g$ are both bandlimited. This theorem, however, has
two disadvantages:
\begin{itemize}
\item The restriction to bandlimited $f,g$ is rather severe; in particular
in our present setting, since we are interested in compactly supported
functions, which can never be bandlimited.
\item The implicit constant in the estimate above depends in a nontrivial
way on the frequency supports of $f,g$.
\end{itemize}
To overcome these limitations, we will develop an improved theory
of convolution for $L^{p},p\in\left(0,1\right)$ using the theory
of \textbf{Wiener amalgam spaces}. As a special case, we will recover
the estimate from above.
Before developing the theory, we remark that essentially everything
mentioned in this section is already known in one form or another.
In particular, Wiener amalgam spaces were originally invented by Feichtinger\cite{FeichtingerWienerInterpolation,FeichtingerWienerSpaces}
and later generalized to Quasi-Banach spaces by Rauhut\cite{RauhutCoorbitQuasiBanach}.
The use of these spaces—and of the oscillation of a function—for obtaining
Banach frames and atomic decompositions for certain spaces goes back
to the theory of coorbit spaces\cite{FeichtingerCoorbit0,FeichtingerCoorbit1,FeichtingerCoorbit2,GroechenigDescribingFunctions,GeneralizedCoorbit1,GeneralizedCoorbit2}
and was also exploited in \cite{GroechenigNonuniformSampling2}. Therefore,
no originality is claimed.
The usual treatments, however, mostly ignore or suppress the dependence
of the Wiener amalgam spaces on the chosen unit neighborhood (see
below for details), whereas this dependence is crucial for us. Hence,
we provide full proofs.
\subsection{Definition of Wiener amalgam spaces}
All of the theory of Wiener amalgam spaces is centered around the
notion of a certain maximal function:
\begin{defn}
\label{def:MaximalFunctionDefinition}(cf.\@ \cite{FeichtingerWienerSpaces},
\cite[Definition 2.2.2]{HeilPhDThesisWienerAmalgam}, \cite{RauhutWienerAmalgam}
and \cite[Definition 2.3.1]{VoigtlaenderPhDThesis}) Let $Q\subset\mathbb{R}^{d}$
be a Borel measurable unit neighborhood and let $f:\mathbb{R}^{d}\to\mathbb{C}^{k}$
be Borel measurable. We then define the \textbf{$Q$-maximal function}
of $f$ as
\[
M_{Q}f:\mathbb{R}^{d}\to\left[0,\infty\right],x\mapsto\esssup_{y\in x+Q}\left|f\left(y\right)\right|=\esssup_{a\in Q}\left|f\left(x+a\right)\right|=\left\Vert L_{-x}f\right\Vert _{L^{\infty}\left(Q\right)}.
\]
For a given $p\in\left(0,\infty\right]$ and a (measurable) weight
$u:\mathbb{R}^{d}\to\left(0,\infty\right)$, we define the \textbf{Wiener
amalgam space} with window $Q$, local component $L^{\infty}$ and
global component $L_{u}^{p}$ as
\[
W_{Q}^{k}\left(L_{u}^{p}\right):=W_{Q}^{k}\left(L^{\infty},L_{u}^{p}\right):=\left\{ f:\mathbb{R}^{d}\to\mathbb{C}^{k}\,\middle|\, f\text{ measurable and }M_{Q}f\in L_{u}^{p}\left(\smash{\mathbb{R}^{d}}\right)\right\} ,
\]
with the natural (quasi)-norm $\left\Vert f\right\Vert _{W_{Q}^{k}\left(L_{u}^{p}\right)}:=\left\Vert M_{Q}f\right\Vert _{L_{u}^{p}}$.
In the most common case $k=1$, we omit the exponent and write $W_{Q}\left(L_{u}^{p}\right)$
instead of $W_{Q}^{1}\left(L_{u}^{p}\right)$.
\end{defn}
\begin{rem*}
\begin{itemize}[leftmargin=0.4cm]
\item One can show for suitable weights $u$ (and we will do so in Lemma
\ref{lem:WienerAmalgamNormEquivalence}) that the space $W_{Q}\left(L_{u}^{p}\right)$
is independent of the choice of the \emph{bounded} measurable unit
neighborhood $Q\subset\mathbb{R}^{d}$, with equivalent quasi-norms
for different choices. Hence, $Q$ is often suppressed in the literature
dealing with Wiener amalgam spaces. For us, however, the precise choice
of $Q$ will be crucial, since we will choose $Q_{i}=T_{i}^{-T}\left[-1,1\right]^{d}$,
so that the sets $Q_{i}$ vary wildly with $i\in I$. Since the constants
appearing in the norm equivalences for different choices of $Q$ depend
heavily on the actual choices of $Q$, we will almost never use the
equivalence for different choices of $Q$, or only in very carefully
chosen ways.
\item Note that $M_{Q}f$ is always a Borel measurable function. Indeed,
since $L^{1}\left(\mathbb{R}^{d}\right)$ is separable, there is
a countable dense family $\left(g_{n}\right)_{n\in\mathbb{N}}$ in $\Gamma:=\left\{ g\in L^{1}\left(\mathbb{R}^{d}\right)\,\middle|\, g\geq0\text{ and }\left\Vert g\right\Vert _{L^{1}}\leq1\right\} $.
Then, we have for an arbitrary measurable function $f$ that
\begin{equation}
\left\Vert f\right\Vert _{L^{\infty}}=\sup_{n\in\mathbb{N}}\int_{\mathbb{R}^{d}}g_{n}\left(x\right)\cdot\left|f\left(x\right)\right|\operatorname{d} x.\label{eq:LInftyNormUsingCountableFamily}
\end{equation}
For $f\in L^{\infty}\left(\mathbb{R}^{d}\right)$, this follows from
the usual characterization of the $L^{\infty}$-norm by duality (cf.\@
\cite[Theorem 6.14]{FollandRA}). In case of $\left\Vert f\right\Vert _{L^{\infty}}=\infty$,
the same theorem shows that for each $M>0$, there is some $g\in\Gamma$
satisfying $\int_{\mathbb{R}^{d}}g\left(x\right)\cdot\left|f\left(x\right)\right|\operatorname{d} x\geq M$.
But by density of the family $\left(g_{n}\right)_{n\in\mathbb{N}}$, there
is then a sequence $\left(n_{k}\right)_{k\in\mathbb{N}}$ such that $g_{n_{k}}\to g$
in $L^{1}\left(\mathbb{R}^{d}\right)$. By switching to a subsequence,
we can also assume $g_{n_{k}}\to g$ almost everywhere. Now, Fatou's
Lemma yields
\[
M\leq\int_{\mathbb{R}^{d}}g\left(x\right)\cdot\left|f\left(x\right)\right|\operatorname{d} x=\int_{\mathbb{R}^{d}}\liminf_{k\to\infty}g_{n_{k}}\left(x\right)\cdot\left|f\left(x\right)\right|\operatorname{d} x\leq\liminf_{k\to\infty}\int_{\mathbb{R}^{d}}g_{n_{k}}\left(x\right)\cdot\left|f\left(x\right)\right|\operatorname{d} x.
\]
Since $M>0$ was arbitrary, this easily yields $\sup_{n\in\mathbb{N}}\int_{\mathbb{R}^{d}}g_{n}\left(x\right)\cdot\left|f\left(x\right)\right|\operatorname{d} x=\infty=\left\Vert f\right\Vert _{L^{\infty}}$.
Now, as a consequence of equation (\ref{eq:LInftyNormUsingCountableFamily}),
we get
\begin{align*}
\left(M_{Q}f\right)\left(x\right) & =\left\Vert {\mathds{1}}_{Q}\cdot L_{-x}f\right\Vert _{L^{\infty}}\\
& =\sup_{n\in\mathbb{N}}\int_{\mathbb{R}^{d}}g_{n}\left(y\right)\cdot{\mathds{1}}_{Q}\left(y\right)\cdot\left|\left(L_{-x}f\right)\left(y\right)\right|\operatorname{d} y\\
& =\sup_{n\in\mathbb{N}}\int_{\mathbb{R}^{d}}g_{n}\left(y\right)\cdot{\mathds{1}}_{Q}\left(y\right)\cdot\left|f\left(x+y\right)\right|\operatorname{d} y.
\end{align*}
But the function $\left(x,y\right)\mapsto g_{n}\left(y\right)\cdot{\mathds{1}}_{Q}\left(y\right)\cdot\left|f\left(x+y\right)\right|$
is Borel measurable, so that measurability of the integrated function
$x\mapsto\int_{\mathbb{R}^{d}}g_{n}\left(y\right)\cdot{\mathds{1}}_{Q}\left(y\right)\cdot\left|f\left(x+y\right)\right|\operatorname{d} y$
follows from the Fubini-Tonelli theorem. Hence, $M_{Q}f$ is Borel
measurable.\qedhere
\end{itemize}
\end{rem*}
It is easy to see that $\left\Vert \cdot\right\Vert _{W_{Q}\left(L_{u}^{p}\right)}$
satisfies the (quasi)-triangle inequality, since $M_{Q}\left(f+g\right)\leq M_{Q}f+M_{Q}g$.
The remaining properties of a (quasi)-norm are also easy to check,
possibly with the exception of definiteness. But this is a consequence
of the following lemma:
\begin{lem}
\label{lem:MaximalFunctionDominatesF}For each Borel measurable unit
neighborhood $Q$ and each measurable $f:\mathbb{R}^{d}\to\mathbb{C}^{k}$,
we have
\[
\left|f\left(x\right)\right|\leq\left(M_{Q}f\right)\left(x\right)\qquad\text{ for almost all }x\in\mathbb{R}^{d}.
\]
In particular, $\left\Vert f\right\Vert _{L_{u}^{p}}\leq\left\Vert f\right\Vert _{W_{Q}\left(L_{u}^{p}\right)}$
and hence $f=0$ almost everywhere if $\left\Vert f\right\Vert _{W_{Q}\left(L_{u}^{p}\right)}=0$.
\end{lem}
\begin{proof}
Since $Q$ is a unit neighbhorhood, there is $\varepsilon>0$ with
$B_{2\varepsilon}\left(0\right)\subset Q$. Since $\mathbb{R}^{d}$
is second countable and since $\left(x+B_{\varepsilon}\left(0\right)\right)_{x\in\mathbb{R}^{d}}$
is an open cover of $\mathbb{R}^{d}$, there is a countable family
$\left(x_{n}\right)_{n\in\mathbb{N}}$ satisfying $\mathbb{R}^{d}=\bigcup_{n\in\mathbb{N}}\left(x_{n}+B_{\varepsilon}\left(0\right)\right)$.
Now, for arbitrary $x\in\mathbb{R}^{d}$ and $y\in x+B_{\varepsilon}\left(0\right)$,
we have $y+B_{\varepsilon}\left(0\right)\subset x+B_{2\varepsilon}\left(0\right)\subset x+Q$
and hence
\[
\left(M_{B_{\varepsilon}\left(0\right)}f\right)\left(y\right)=\left\Vert f\cdot{\mathds{1}}_{y+B_{\varepsilon}\left(0\right)}\right\Vert _{L^{\infty}}\leq\left\Vert f\cdot{\mathds{1}}_{x+Q}\right\Vert _{L^{\infty}}=M_{Q}f\left(x\right)\qquad\forall x\in\mathbb{R}^{d}\,\forall y\in x+B_{\varepsilon}\left(0\right).
\]
Next, for each $n\in\mathbb{N}$, there is a null-set $N_{n}\subset x_{n}+B_{\varepsilon}\left(0\right)$
such that
\[
\left|f\left(x\right)\right|\leq\left\Vert f\cdot{\mathds{1}}_{x_{n}+B_{\varepsilon}\left(0\right)}\right\Vert _{L^{\infty}}=\left(M_{B_{\varepsilon}\left(0\right)}f\right)\left(x_{n}\right)\qquad\text{ for all }\qquad x\in\left[x_{n}+B_{\varepsilon}\left(0\right)\right]\setminus N_{n}.
\]
But for each such $x$, there is some $\gamma\in B_{\varepsilon}\left(0\right)$
such that $x=x_{n}+\gamma$ and thus $x_{n}=x-\gamma\in x+B_{\varepsilon}\left(0\right)$,
so that the equation from above yields $\left|f\left(x\right)\right|\leq\left(M_{B_{\varepsilon}\left(0\right)}f\right)\left(x_{n}\right)\leq M_{Q}f\left(x\right)$.
Recall that this estimate holds for all $x\in\left[x_{n}+B_{\varepsilon}\left(0\right)\right]\setminus N_{n}$.
Since $N:=\bigcup_{n\in\mathbb{N}}N_{n}$ is a null-set and since $\mathbb{R}^{d}=\bigcup_{n\in\mathbb{N}}\left(x_{n}+B_{\varepsilon}\left(0\right)\right)$,
this completes the proof.
\end{proof}
Although easy to prove, the following lemma is frequently helpful,
since it shows that Schwartz functions are contained in arbitrary
Wiener amalgam spaces.
\begin{lem}
\label{lem:SchwartzFunctionsAreWiener}For arbitrary $N\geq0$, we
have
\[
\left[M_{\left[-1,1\right]^{d}}\left(1+\left|\bullet\right|\right)^{-N}\right]\left(x\right)\leq\left(1+2\sqrt{d}\right)^{N}\cdot\left(1+\left|x\right|\right)^{-N}\qquad\forall x\in\mathbb{R}^{d}.
\]
In particular, if $p\in\left(0,\infty\right]$ is arbitrary and if
we set $\left\Vert f\right\Vert _{N}:=\sup_{x\in\mathbb{R}^{d}}\left(1+\left|x\right|\right)^{N}\cdot\left|f\left(x\right)\right|$
for measurable $f:\mathbb{R}^{d}\to\mathbb{C}$, then
\[
\left\Vert f\right\Vert _{W_{\left[-1,1\right]^{d}}\left(L_{\left(1+\left|\cdot\right|\right)^{K}}^{p}\right)}\leq\left(1+2\sqrt{d}\right)^{N}\cdot\left(\frac{1}{p}\frac{s_{d}}{N-K-\frac{d}{p}}\right)^{1/p}\cdot\left\Vert f\right\Vert _{N}\qquad\text{ as soon as }\qquad N>K+\frac{d}{p}.
\]
Hence, $\mathcal{S}\left(\mathbb{R}^{d}\right)\hookrightarrow W_{\left[-1,1\right]^{d}}\left(L_{u}^{p}\right)$
for all $p\in\left(0,\infty\right]$ and $u\in\left\{ v,v_{0},\left(1+\left|\bullet\right|\right)^{K}\right\} $.
\end{lem}
\begin{proof}
To prove the first claim, we distinguish two cases. For $\left|x\right|\leq2\sqrt{d}$,
note that $\left(1+\left|x+a\right|\right)^{-N}\leq1$ for all $a\in\left[-1,1\right]^{d}$,
so that we get
\[
\left[M_{\left[-1,1\right]^{d}}\left(1+\left|\bullet\right|\right)^{-N}\right]\left(x\right)\leq1\leq\left(1+2\sqrt{d}\right)^{N}\cdot\left(1+\left|x\right|\right)^{-N}.
\]
Otherwise, if $\left|x\right|\geq2\sqrt{d}$, we have for
$a\in\left[-1,1\right]^{d}$ that
\[
\left|x-a\right|\geq\left|x\right|-\left|a\right|\geq\left|x\right|-\sqrt{d}=\frac{\left|x\right|}{2}+\frac{\left|x\right|}{2}-\sqrt{d}\geq\frac{\left|x\right|}{2}
\]
and hence $\left(1+\left|x+a\right|\right)^{-N}\leq\left(1+\frac{\left|x\right|}{2}\right)^{-N}\leq\left(\frac{1}{2}\left(1+\left|x\right|\right)\right)^{-N}=2^{N}\left(1+\left|x\right|\right)^{-N}$.
Since $2^{N}\leq\left(1+2\sqrt{d}\right)^{N}=:C$, we get
all in all that $\left(M_{\left[-1,1\right]^{d}}\left(1+\left|\bullet\right|\right)^{-N}\right)\left(x\right)\leq C\cdot\left(1+\left|x\right|\right)^{-N}$
for all $x\in\mathbb{R}^{d}$, as claimed.
For the next claim, we can clearly assume $\left\Vert f\right\Vert _{N}<\infty$.
In this case, we have $\left|f\left(x\right)\right|\leq\left\Vert f\right\Vert _{N}\cdot\left(1+\left|x\right|\right)^{-N}$
for all $x\in\mathbb{R}^{d}$ and hence
\begin{align*}
\left(1+\left|x\right|\right)^{K}\cdot\left(M_{\left[-1,1\right]^{d}}f\right)\left(x\right) & \leq\left\Vert f\right\Vert _{N}\cdot\left(1+\left|x\right|\right)^{K}\cdot\left(M_{\left[-1,1\right]^{d}}\left(1+\left|\bullet\right|\right)^{-N}\right)\left(x\right)\\
& \leq C\cdot\left\Vert f\right\Vert _{N}\cdot\left(1+\left|x\right|\right)^{-\left(N-K\right)}\qquad\forall x\in\mathbb{R}^{d}.
\end{align*}
This yields the claim, since equation (\ref{eq:StandardDecayLpEstimate})
shows $\left\Vert \left(1+\left|\bullet\right|\right)^{-\left(N-K\right)}\right\Vert _{L^{p}}\leq\left(\frac{1}{p}\frac{s_{d}}{N-K-\frac{d}{p}}\right)^{1/p}<\infty$,
as long $N>K+\frac{d}{p}$. The embedding $\mathcal{S}\left(\mathbb{R}^{d}\right)\hookrightarrow W_{\left[-1,1\right]^{d}}\left(L_{u}^{p}\right)$
for all $p\in\left(0,\infty\right]$ and $u\in\left\{ v,v_{0},\left(1+\left|\bullet\right|\right)^{K}\right\} $
is now trivial, since the norm $\left\Vert \bullet\right\Vert _{N}$
is continuous with respect to the topology on $\mathcal{S}\left(\mathbb{R}^{d}\right)$
and since we have
\[
v\left(x\right)=v\left(0+x\right)\leq v\left(0\right)\cdot v_{0}\left(x\right)\leq\Omega_{1}v\left(0\right)\cdot\left(1+\left|x\right|\right)^{K}\qquad\forall x\in\mathbb{R}^{d}.\qedhere
\]
\end{proof}
\begin{lem}
\label{lem:WienerTransformationFormula}For $k\in\mathbb{N}$, a measurable
$f:\mathbb{R}^{d}\to\mathbb{C}^{k}$, $T\in\mathrm{GL}\left(\mathbb{R}^{d}\right)$
and a measurable $Q\subset\mathbb{R}^{d}$, we have
\[
M_{Q}\left(f\circ T\right)=\left(M_{TQ}f\right)\circ T
\]
and hence
\[
\left\Vert f\circ T\right\Vert _{W_{Q}\left(L^{p}\right)}=\left|\det T\right|^{-1/p}\cdot\left\Vert f\right\Vert _{W_{TQ}\left(L^{p}\right)}.\qedhere
\]
\end{lem}
\begin{proof}
Since $T$ and $T^{-1}$ map null-sets to null-sets, we have
\begin{align*}
\left[M_{Q}\left(f\circ T\right)\right]\left(x\right) & =\esssup_{a\in Q}\left|\left(f\circ T\right)\left(x+a\right)\right|\\
& =\esssup_{b\in TQ}\left|f\left(b+Tx\right)\right|\\
& =\left(M_{TQ}f\right)\left(Tx\right)
\end{align*}
for all $x\in\mathbb{R}^{d}$. The final identity is a consequence
of the definitions and of $\left\Vert f\circ T\right\Vert _{L^{p}}=\left|\det T\right|^{-1/p}\cdot\left\Vert f\right\Vert _{L^{p}}$,
which follows easily from the change-of-variables formula.
\end{proof}
Next, we show that iterated applications of $M_{Q}$ can be estimated
using a single $M_{Q'}$.
\begin{lem}
\label{lem:IteratedMaximalFunction}Let $k\in\mathbb{N}$ and assume that
$f:\mathbb{R}^{d}\to\mathbb{C}^{k}$ and $Q_{1},Q_{2}\subset\mathbb{R}^{d}$
are measurable and that $Q_{1}+Q_{2}$ is also measurable. Then
\[
M_{Q_{1}}\left[M_{Q_{2}}f\right]\leq M_{Q_{1}+Q_{2}}f.
\]
In particular, for any measurable $u:\mathbb{R}^{d}\to\left(0,\infty\right)$,
we have
\[
\left\Vert M_{Q_{2}}f\right\Vert _{W_{Q_{1}}\left(L_{u}^{p}\right)}\leq\left\Vert f\right\Vert _{W_{Q_{1}+Q_{2}}^{k}\left(L_{u}^{p}\right)}.\qedhere
\]
\end{lem}
\begin{proof}
For $a\in Q_{1}$, we have $a+Q_{2}\subset Q_{1}+Q_{2}$ and hence
\[
\left(M_{Q_{2}}f\right)\left(x+a\right)=\left\Vert f\cdot{\mathds{1}}_{x+a+Q_{2}}\right\Vert _{L^{\infty}}\leq\left\Vert f\cdot{\mathds{1}}_{x+Q_{1}+Q_{2}}\right\Vert _{L^{\infty}}=\left(M_{Q_{1}+Q_{2}}f\right)\left(x\right).
\]
Since this holds for all $a\in Q_{1}$, we get $\left(M_{Q_{1}}\left[M_{Q_{2}}f\right]\right)\left(x\right)=\esssup_{a\in Q_{1}}\left(M_{Q_{2}}f\right)\left(x+a\right)\leq\left(M_{Q_{1}+Q_{2}}f\right)\left(x\right)$,
as claimed.
\end{proof}
The next three lemmas are important for us, since they imply $\left\Vert f\right\Vert _{W_{T_{j}^{-T}\left[-1,1\right]^{d}}\left(L_{v}^{p}\right)}\!\leq\!C_{i,j,p,v}\cdot\left\Vert f\right\Vert _{W_{T_{i}^{-T}\left[-1,1\right]^{d}}\left(L_{v}^{p}\right)}$,
where the constant $C_{i,j,p,v}$ is explicitly known, cf.\@ Corollary
\ref{cor:WienerLinearCubeNormEstimate}. We begin with an estimate
for the norm of the translation operators on $L_{v}^{p}\left(\mathbb{R}^{d}\right)$.
\begin{lem}
\label{lem:WeightedLpTranslationNorm}For each $y\in\mathbb{R}^{d}$,
the left-translation operator $L_{y}:L_{v}^{p}\left(\mathbb{R}^{d}\right)\to L_{v}^{p}\left(\mathbb{R}^{d}\right)$
is well-defined and bounded with
\[
\vertiii{L_{y}}\leq v_{0}\left(y\right)\leq\Omega_{1}\cdot\left(1+\left|y\right|\right)^{K}.\qedhere
\]
\end{lem}
\begin{rem*}
The only property of $v$ which is used in the proof is that $v\left(x+y\right)\leq v\left(x\right)v_{0}\left(y\right)$.
By submultiplicativity of $v_{0}$, the same estimate holds for $v_{0}$
instead of $v$, so that the claim of the lemma also holds for $v_{0}$
instead of $v$.
\end{rem*}
\begin{proof}
Let $f\in L_{v}^{p}\left(\mathbb{R}^{d}\right)$ be arbitrary and
note
\begin{align*}
v\left(x\right)\cdot\left|\left(L_{y}f\right)\left(x\right)\right| & =v\left(\left(x-y\right)+y\right)\cdot\left|f\left(x-y\right)\right|\\
& \leq v_{0}\left(y\right)\cdot\left|\left(v\cdot f\right)\left(x-y\right)\right|\\
& =v_{0}\left(y\right)\cdot\left[L_{y}\left(v\cdot f\right)\right]\left(x\right).
\end{align*}
By solidity and translation invariance of $L^{p}\left(\mathbb{R}^{d}\right)$,
this implies
\[
\left\Vert L_{y}f\right\Vert _{L_{v}^{p}}=\left\Vert v\cdot L_{y}f\right\Vert _{L^{p}}\leq v_{0}\left(y\right)\cdot\left\Vert L_{y}\left(v\cdot f\right)\right\Vert _{L^{p}}=v_{0}\left(y\right)\cdot\left\Vert v\cdot f\right\Vert _{L^{p}}=v_{0}\left(y\right)\cdot\left\Vert f\right\Vert _{L_{v}^{p}}<\infty.\qedhere
\]
\end{proof}
Now we can derive a first estimate which will allow us to switch from
one ``base set'' $Q$ to another one.
\begin{lem}
\label{lem:WienerAmalgamNormEquivalence}Let $Q_{1},Q_{2}\subset\mathbb{R}^{d}$
and assume that there are $x_{1},\dots,x_{N}\in\mathbb{R}^{d}$ such
that $Q_{1}\subset\bigcup_{i=1}^{N}\left(x_{i}+Q_{2}\right)$. Let
$p\in\left(0,\infty\right]$ and set $s:=\min\left\{ 1,p\right\} $.
Then we have
\[
\left\Vert f\right\Vert _{W_{Q_{1}}^{k}\left(L_{v}^{p}\right)}\leq\left(\sum_{i=1}^{N}\left[v_{0}\left(-x_{i}\right)\right]^{s}\right)^{1/s}\cdot\left\Vert f\right\Vert _{W_{Q_{2}}^{k}\left(L_{v}^{p}\right)}\leq\Omega_{1}\cdot\left(\sum_{i=1}^{N}\left(1+\left|x_{i}\right|\right)^{sK}\right)^{1/s}\cdot\left\Vert f\right\Vert _{W_{Q_{2}}^{k}\left(L_{v}^{p}\right)}.
\]
for all measurable $f:\mathbb{R}^{d}\to\mathbb{C}^{k}$.
\end{lem}
\begin{rem*}
\begin{itemize}[leftmargin=0.4cm]
\item As for the previous lemma, the statement of the lemma remains true
for $v_{0}$ instead of $v$.
\item Note that if $Q_{1},Q_{2}\subset\mathbb{R}^{d}$ are two (Borel measurable)
bounded unit-neighborhoods, compactness of $\overline{Q_{1}}$ yields
finitely many $x_{1},\dots,x_{N}\in\mathbb{R}^{d}$ satisfying $Q_{1}\subset\overline{Q_{1}}\subset\bigcup_{i=1}^{N}\left(x_{i}+Q_{2}^{\circ}\right)\subset\bigcup_{i=1}^{N}\left(x_{i}+Q_{2}\right)$,
so that the preceding lemma yields $\left\Vert f\right\Vert _{W_{Q_{1}}^{k}\left(L_{v}^{p}\right)}\lesssim\left\Vert f\right\Vert _{W_{Q_{2}}^{k}\left(L_{v}^{p}\right)}$,
where the implied constant is independent of $f$. By symmetry, this
argument shows $W_{Q_{1}}^{k}\left(L_{v}^{p}\right)=W_{Q_{2}}^{k}\left(L_{v}^{p}\right)$,
with equivalent (quasi)-norms. But since the constants of the (quasi)-norm
equivalence depend heavily on $Q_{1},Q_{2}$, this statement is not
of too much value for us.\qedhere
\end{itemize}
\end{rem*}
\begin{proof}
We have for any measurable $f:\mathbb{R}^{d}\to\mathbb{C}^{k}$ that
\begin{align*}
\left(M_{Q_{1}}f\right)\left(x\right)=\left\Vert f\cdot{\mathds{1}}_{x+Q_{1}}\right\Vert _{L^{\infty}} & \leq\left\Vert f\cdot{\mathds{1}}_{x+\bigcup_{i=1}^{N}\left(x_{i}+Q_{2}\right)}\right\Vert _{L^{\infty}}\\
& \leq\sum_{i=1}^{N}\left\Vert f\cdot{\mathds{1}}_{x+x_{i}+Q_{2}}\right\Vert _{L^{\infty}}\\
& =\sum_{i=1}^{N}\left(M_{Q_{2}}f\right)\left(x+x_{i}\right)\\
& =\sum_{i=1}^{N}\left(L_{-x_{i}}\left[M_{Q_{2}}f\right]\right)\left(x\right).
\end{align*}
For $p\geq1$, we can thus use the triangle inequality for $L^{p}$
and the estimate for $\vertiii{L_{y}}$ from Lemma \ref{lem:WeightedLpTranslationNorm},
as well as solidity of $L^{p}$ to derive
\[
\left\Vert f\right\Vert _{W_{Q_{1}}^{k}\left(L_{v}^{p}\right)}=\left\Vert M_{Q_{1}}f\right\Vert _{L_{v}^{p}}\leq\sum_{i=1}^{N}\left\Vert L_{-x_{i}}\left(M_{Q_{2}}f\right)\right\Vert _{L_{v}^{p}}\leq\left[\sum_{i=1}^{N}v_{0}\left(-x_{i}\right)\right]\cdot\left\Vert M_{Q_{2}}f\right\Vert _{L_{v}^{p}}=\left[\sum_{i=1}^{N}v_{0}\left(-x_{i}\right)\right]\cdot\left\Vert f\right\Vert _{W_{Q_{2}}^{k}\left(L_{v}^{p}\right)}.
\]
Similarly, for $p\in\left(0,1\right)$, we use the $p$-triangle inequality
(i.e., $\left\Vert \sum_{i=1}^{n}f_{i}\right\Vert _{L^{p}}^{p}\leq\sum_{i=1}^{n}\left\Vert f_{i}\right\Vert _{L^{p}}^{p}$)
to derive
\[
\left\Vert f\right\Vert _{W_{Q_{1}}^{k}\left(L_{v}^{p}\right)}^{p}=\left\Vert M_{Q_{1}}f\right\Vert _{L_{v}^{p}}^{p}\leq\sum_{i=1}^{N}\left\Vert L_{-x_{i}}\left(M_{Q_{2}}f\right)\right\Vert _{L_{v}^{p}}^{p}\leq\left[\sum_{i=1}^{N}\left(v_{0}\left(-x_{i}\right)\right)^{p}\right]\cdot\left\Vert M_{Q_{2}}f\right\Vert _{L_{v}^{p}}^{p}=\left[\sum_{i=1}^{N}\left(v_{0}\left(-x_{i}\right)\right)^{p}\right]\cdot\left\Vert f\right\Vert _{W_{Q_{2}}^{k}\left(L_{v}^{p}\right)}^{p}.\qedhere
\]
\end{proof}
In view of the preceding lemma, our next result becomes relevant:
\begin{lem}
\label{lem:BallCoveringNumber}Let $\left\Vert \cdot\right\Vert $
be any norm on $\mathbb{R}^{d}$ and let $R>0$. For any $r>0$ and
$N:=\left\lfloor \left(3+2r\right)^{d}\right\rfloor $, there
are $x_{1},\dots,x_{N}\in B_{\left(1+r\right)R}^{\left\Vert \cdot\right\Vert }\left(0\right)$
satisfying
\[
B_{\left(1+r\right)R}^{\left\Vert \cdot\right\Vert }\left(0\right)\subset\bigcup_{i=1}^{N}\left[x_{i}+B_{R}^{\left\Vert \cdot\right\Vert }\left(0\right)\right],
\]
where $B_{s}^{\left\Vert \cdot\right\Vert }\left(0\right)=\left\{ x\in\mathbb{R}^{d}\,\middle|\,\left\Vert x\right\Vert <s\right\} $.
\end{lem}
\begin{proof}
First of all, assume we are given $x_{1},\dots,x_{M}\in B_{\left(1+r\right)R}^{\left\Vert \cdot\right\Vert }\left(0\right)$
such that $\left(x_{i}+B_{R/2}^{\left\Vert \cdot\right\Vert }\left(0\right)\right)_{i\in\underline{M}}$
is pairwise disjoint. Because of $x_{i}\in B_{\left(1+r\right)R}^{\left\Vert \cdot\right\Vert }\left(0\right)$,
we have $x_{i}+B_{R/2}^{\left\Vert \cdot\right\Vert }\left(0\right)\subset B_{\left(\frac{3}{2}+r\right)R}^{\left\Vert \cdot\right\Vert }\left(0\right)$,
so that additivity and translation invariance of the Lebesgue-measure
yields
\begin{align*}
M\cdot\left(R/2\right)^{d}\cdot\lambda\left(B_{1}^{\left\Vert \cdot\right\Vert }\left(0\right)\right) & =\sum_{i=1}^{M}\lambda\left(x_{i}+B_{R/2}^{\left\Vert \cdot\right\Vert }\left(0\right)\right)\\
& =\lambda\left(\biguplus_{i=1}^{M}x_{i}+B_{R/2}^{\left\Vert \cdot\right\Vert }\left(0\right)\right)\\
& \leq\lambda\left(B_{\left(\frac{3}{2}+r\right)R}^{\left\Vert \cdot\right\Vert }\left(0\right)\right)\\
& =\left[\left(\frac{3}{2}+r\right)R\right]^{d}\cdot\lambda\left(B_{1}^{\left\Vert \cdot\right\Vert }\left(0\right)\right)
\end{align*}
and thus $M\leq\left(3+2r\right)^{d}$. Since $M\in\mathbb{N}$,
we even get $M\leq N$. In particular, there can be at most a finite
number of such $x_{i}$.
Now (e.g.\@ using Zorn's Lemma), we can find a \emph{maximal} family
$\left(x_{i}\right)_{i\in\underline{M}}$ in $B_{\left(1+r\right)R}^{\left\Vert \cdot\right\Vert }\left(0\right)$
such that the family of sets $\left(x_{i}+B_{R/2}^{\left\Vert \cdot\right\Vert }\left(0\right)\right)_{i\in\underline{M}}$
is pairwise disjoint. As seen above, $M\leq N$.
It remains to show $B_{\left(1+r\right)R}^{\left\Vert \cdot\right\Vert }\left(0\right)\subset\bigcup_{i=1}^{M}\left[x_{i}+B_{R}^{\left\Vert \cdot\right\Vert }\left(0\right)\right]=:\Gamma$.
Thus, let $x\in B_{\left(1+r\right)R}^{\left\Vert \cdot\right\Vert }\left(0\right)$
be arbitrary. In case of $x\in\left\{ x_{1},\dots,x_{M}\right\} $,
we clearly have $x\in\Gamma$. But for $x\notin\left\{ x_{1},\dots,x_{M}\right\} $,
we see by maximality of the family $\left(x_{i}\right)_{i\in\underline{M}}$
that there is some $i\in\underline{M}$ satisfying $\left(x+B_{R/2}^{\left\Vert \cdot\right\Vert }\left(0\right)\right)\cap\left(x_{i}+B_{R/2}^{\left\Vert \cdot\right\Vert }\left(0\right)\right)\neq\emptyset$.
But this easily yields $x\in x_{i}+B_{R}^{\left\Vert \cdot\right\Vert }\left(0\right)\subset\Gamma$,
as desired.
\end{proof}
As announced above, we can now derive a completely quantitative version
of the (quasi)-norm equivalence between $W_{T_{j}^{-T}\left[-R,R\right]^{d}}\left(L_{v}^{p}\right)$
and $W_{T_{i}^{-T}\left[-L,L\right]^{d}}\left(L_{v}^{p}\right)$.
\begin{cor}
\label{cor:WienerLinearCubeNormEstimate}Let $i,j\in I$ and $p\in\left(0,\infty\right]$,
let $R,L\in\left[1,\infty\right)$ and let $f:\mathbb{R}^{d}\to\mathbb{C}^{k}$
be measurable. Then we have
\[
\left\Vert f\right\Vert _{W_{T_{j}^{-T}\left[-R,R\right]^{d}}^{k}\left(L_{v}^{p}\right)}\leq\Omega_{0}^{K}\Omega_{1}\cdot\left(3d\left(L+R\right)\right)^{K+d\cdot\max\left\{ 1,\frac{1}{p}\right\} }\cdot\left(1+\left\Vert T_{j}^{-1}T_{i}\right\Vert \right)^{K+d\cdot\max\left\{ 1,\frac{1}{p}\right\} }\cdot\left\Vert f\right\Vert _{W_{T_{i}^{-T}\left[-L,L\right]^{d}}^{k}\left(L_{v}^{p}\right)}.\qedhere
\]
\end{cor}
\begin{rem*}
As for the preceding results, the statement of the corollary remains
valid if $v$ is replaced by $v_{0}$.
Finally, we explicitly state the two most important special cases
of the preceding corollary:
\begin{itemize}
\item We have $i=j$ and $R=2$, as well as $L=1$. In this case, the corollary
yields
\begin{equation}
\left\Vert f\right\Vert _{W_{T_{i}^{-T}\left[-2,2\right]^{d}}^{k}\left(L_{v}^{p}\right)}\leq\Omega_{0}^{K}\Omega_{1}\cdot\left(18d\right)^{K+d\cdot\max\left\{ 1,\frac{1}{p}\right\} }\cdot\left\Vert f\right\Vert _{W_{T_{i}^{-T}\left[-1,1\right]^{d}}^{k}\left(L_{v}^{p}\right)}.\label{eq:WienerLinearCubeEnlargement}
\end{equation}
\item We have $R=L=1$. In this case, the corollary yields
\begin{equation}
\left\Vert f\right\Vert _{W_{T_{j}^{-T}\left[-1,1\right]^{d}}^{k}\left(L_{v}^{p}\right)}\leq\Omega_{0}^{K}\Omega_{1}\cdot\left(6d\right)^{K+d\cdot\max\left\{ 1,\frac{1}{p}\right\} }\cdot\left(1+\left\Vert T_{j}^{-1}T_{i}\right\Vert \right)^{K+d\cdot\max\left\{ 1,\frac{1}{p}\right\} }\cdot\left\Vert f\right\Vert _{W_{T_{i}^{-T}\left[-1,1\right]^{d}}^{k}\left(L_{v}^{p}\right)}.\qedhere\label{eq:WienerLinearCubeTransformationChange}
\end{equation}
\end{itemize}
\end{rem*}
\begin{proof}
For brevity, set $R':=\left\Vert \left(T_{i}^{-T}\right)^{-1}T_{j}^{-T}\right\Vert _{\ell^{\infty}\to\ell^{\infty}}\cdot R=\left\Vert T_{j}^{-1}T_{i}\right\Vert _{\ell^{1}\to\ell^{1}}\cdot R$
and note
\[
\left(T_{i}^{-T}\right)^{-1}T_{j}^{-T}\left[-R,R\right]^{d}\subset\left[-R',R'\right]^{d}=\overline{B_{R'}^{\left\Vert \cdot\right\Vert _{\infty}}}\left(0\right)\subset\overline{B_{\left(1+\frac{R'}{L}\right)L}^{\left\Vert \cdot\right\Vert _{\infty}}}\left(0\right).
\]
But Lemma \ref{lem:BallCoveringNumber} yields certain $x_{1},\dots,x_{N}\in\overline{B_{\left(1+\frac{R'}{L}\right)L}^{\left\Vert \cdot\right\Vert _{\infty}}}\left(0\right)$
satisfying
\[
\overline{B_{\left(1+\frac{R'}{L}\right)L}^{\left\Vert \cdot\right\Vert _{\infty}}}\left(0\right)\subset\bigcup_{i=1}^{N}\left(x_{i}+\overline{B_{L}^{\left\Vert \cdot\right\Vert _{\infty}}}\left(0\right)\right)=\bigcup_{i=1}^{N}\left(x_{i}+\left[-L,L\right]^{d}\right),
\]
where
\[
N\leq\left(3+2\frac{R'}{L}\right)^{d}=\left(3+2\frac{R}{L}\left\Vert T_{j}^{-1}T_{i}\right\Vert _{\ell^{1}\to\ell^{1}}\right)^{d}\leq3^{d}\left(1+\frac{R}{L}\right)^{d}\left(1+\left\Vert T_{j}^{-1}T_{i}\right\Vert _{\ell^{1}\to\ell^{1}}\right)^{d}.
\]
Next, note $\left\Vert T_{j}^{-1}T_{i}\right\Vert _{\ell^{1}\to\ell^{1}}\leq\sqrt{d}\cdot\left\Vert T_{j}^{-1}T_{i}\right\Vert $
and hence $N\leq\left(3\sqrt{d}\right)^{d}\left(1+\frac{R}{L}\right)^{d}\left(1+\left\Vert T_{j}^{-1}T_{i}\right\Vert \right)^{d}$.
By putting together what we derived above, we get
\[
T_{j}^{-T}\left[-R,R\right]^{d}\subset\bigcup_{i=1}^{N}\left(T_{i}^{-T}x_{i}+T_{i}^{-T}\left[-L,L\right]^{d}\right).
\]
Next, we set $s:=\min\left\{ 1,p\right\} $, note
\begin{align*}
\left\Vert x_{i}\right\Vert _{\infty} & \leq\left(1+\frac{R'}{L}\right)L=\left(L+R'\right)\\
& =\left(L+R\cdot\left\Vert T_{j}^{-1}T_{i}\right\Vert _{\ell^{1}\to\ell^{1}}\right)\\
& \leq\left(L+R\right)\left(1+\left\Vert T_{j}^{-1}T_{i}\right\Vert _{\ell^{1}\to\ell^{1}}\right)\\
& \leq\sqrt{d}\cdot\left(L+R\right)\left(1+\left\Vert T_{j}^{-1}T_{i}\right\Vert \right)
\end{align*}
and thus $1+\left|x_{i}\right|\leq1+d\cdot\left(L+R\right)\left(1+\left\Vert T_{j}^{-1}T_{i}\right\Vert \right)\leqd\cdot\left(1+L+R\right)\left(1+\left\Vert T_{j}^{-1}T_{i}\right\Vert \right)$
and recall equation (\ref{eq:WeightLinearTransformationsConnection})
to derive
\begin{align*}
\left[\sum_{i=1}^{N}\left(1+\left|T_{i}^{-T}x_{i}\right|\right)^{sK}\right]^{1/s} & \leq\Omega_{0}^{K}\cdot\left[\sum_{i=1}^{N}\left(1+\left|x_{i}\right|\right)^{sK}\right]^{1/s}\\
& \leq\left[d\Omega_{0}\left(1+L+R\right)\left(1+\left\Vert T_{j}^{-1}T_{i}\right\Vert \right)\right]^{K}\cdot N^{1/s}\\
\left({\scriptstyle \text{since }R,L\geq1}\right) & \leq3^{K+\frac{d}{s}}d^{K+\frac{d}{2s}}\left(L+R\right)^{K+\frac{d}{s}}\cdot\Omega_{0}^{K}\cdot\left(1+\left\Vert T_{j}^{-1}T_{i}\right\Vert \right)^{K+\frac{d}{s}}\\
& \leq\Omega_{0}^{K}\cdot\left(3d\left(L+R\right)\right)^{K+\frac{d}{s}}\cdot\left(1+\left\Vert T_{j}^{-1}T_{i}\right\Vert \right)^{K+\frac{d}{s}}.
\end{align*}
All in all, Lemma \ref{lem:WienerAmalgamNormEquivalence} implies
\[
\left\Vert f\right\Vert _{W_{T_{j}^{-T}\left[-R,R\right]^{d}}^{k}\left(L_{v}^{p}\right)}\leq\Omega_{0}^{K}\Omega_{1}\cdot\left(3d\left(L+R\right)\right)^{K+\frac{d}{s}}\cdot\left(1+\left\Vert T_{j}^{-1}T_{i}\right\Vert \right)^{K+\frac{d}{s}}\cdot\left\Vert f\right\Vert _{W_{T_{i}^{-T}\left[-L,L\right]^{d}}^{k}\left(L_{v}^{p}\right)},
\]
as desired.
\end{proof}
\subsection{The oscillation of a function}
Later in the paper, we will need to discretize certain reproducing
formulas involving convolutions. As observed in \cite{FeichtingerCoorbit0,FeichtingerCoorbit1,FeichtingerCoorbit2,GroechenigDescribingFunctions,GroechenigNonuniformSampling2},
a central tool for these discretizations is the oscillation of a function
and certain properties of and estimates for it. The goal of this subsection
is to collect these properties and estimates.
\begin{defn}
\label{def:Oscillation}Let $f:\mathbb{R}^{d}\to\mathbb{C}^{k}$ and
let $\emptyset\neq Q\subset\mathbb{R}^{d}$. We define the \textbf{$Q$-oscillation}
of $f$ by
\[
\osc Qf:\mathbb{R}^{d}\to\left[0,\infty\right],x\mapsto\sup_{y,z\in x+Q}\left|f\left(y\right)-f\left(z\right)\right|=\sup_{a,b\in Q}\left|f\left(x+a\right)-f\left(x+b\right)\right|.\qedhere
\]
\end{defn}
\begin{rem*}
Note that if $f$ is continuous, then so is $x\mapsto\left|f\left(x+a\right)-f\left(x+b\right)\right|$,
so that $\osc Qf$ is lower semicontinuous and hence measurable.
\end{rem*}
As our first step, we investigate some elementary properties of the
oscillation, in particular the behaviour of the oscillation under
a linear change of variables and under convolution.
\begin{lem}
\label{lem:OscillationLinearChange}Let $f:\mathbb{R}^{d}\to\mathbb{C}^{k}$,
$T\in\mathrm{GL}\left(\mathbb{R}^{d}\right)$ and let $\emptyset\neq Q\subset\mathbb{R}^{d}$.
Then
\[
\osc Q\left(f\circ T\right)=\left(\osc{TQ}f\right)\circ T.\qedhere
\]
\end{lem}
\begin{proof}
We have
\begin{align*}
\left[\osc Q\left(f\circ T\right)\right]\left(x\right) & =\sup_{a,b\in Q}\left|\left(f\circ T\right)\left(x+a\right)-\left(f\circ T\right)\left(x+b\right)\right|\\
& =\sup_{\alpha,\beta\in TQ}\left|f\left(\alpha+Tx\right)-f\left(\beta+Tx\right)\right|\\
& =\left(\osc{TQ}f\right)\left(Tx\right).\qedhere
\end{align*}
\end{proof}
\begin{lem}
\label{lem:OscillationConvolution}Let $f,g:\mathbb{R}^{d}\to\mathbb{C}$
be measurable, let $\emptyset\neq Q\subset\mathbb{R}^{d}$ and assume
that $\osc Qf$ is measurable and that $\left(\left|f\right|\ast\left|g\right|\right)\left(x\right)<\infty$
for every $x\in\mathbb{R}^{d}$. Then
\[
\left[\osc Q\left(f\ast g\right)\right]\left(x\right)\leq\left[\left(\osc Qf\right)\ast\left|g\right|\right]\left(x\right)\qquad\forall x\in\mathbb{R}^{d}.\qedhere
\]
\end{lem}
\begin{proof}
Let $x\in\mathbb{R}^{d}$ and fix $a,b\in Q$. Then
\begin{align*}
\left|\left(f\ast g\right)\left(x+a\right)-\left(f\ast g\right)\left(x+b\right)\right| & =\left|\int_{\mathbb{R}^{d}}f\left(\left(x+a\right)-y\right)g\left(y\right)\operatorname{d} y-\int_{\mathbb{R}^{d}}f\left(\left(x+b\right)-y\right)g\left(y\right)\operatorname{d} y\right|\\
& \leq\int_{\mathbb{R}^{d}}\left|f\left(\left(x-y\right)+a\right)-f\left(\left(x-y\right)+b\right)\right|\cdot\left|g\left(y\right)\right|\operatorname{d} y\\
& \leq\int_{\mathbb{R}^{d}}\left(\osc Qf\right)\left(x-y\right)\cdot\left|g\left(y\right)\right|\operatorname{d} y\\
& =\left[\left(\osc Qf\right)\ast\left|g\right|\right]\left(x\right),
\end{align*}
as claimed.
\end{proof}
Intuitively, it should be true that smooth functions have a small
oscillation if their derivative is small. The next two lemmas make
this precise:
\begin{lem}
\label{lem:OscillationEstimatedByWienerDerivative}Let $f\in C^{1}\left(\mathbb{R}^{d};\mathbb{C}\right)$.
Then we have for every bounded, convex set $Q\subset\mathbb{R}^{d}$
with nonempty interior that
\[
\osc Qf\leq{\rm diam}\left(Q\right)\cdot M_{Q}\left(\nabla f\right).\qedhere
\]
\end{lem}
\begin{proof}
For $x\in\mathbb{R}^{d}$ and $a,b\in Q$, the fundamental theorem
of calculus yields
\begin{align*}
\left|f\left(x+b\right)-f\left(x+a\right)\right| & =\left|\int_{0}^{1}\frac{\operatorname{d}}{\operatorname{d} t}\bigg|_{t=s}f\left(x+a+t\left(b-a\right)\right)\operatorname{d} s\right|\\
& =\left|\int_{0}^{1}\left\langle \nabla f\left(x+a+s\left(b-a\right)\right),\,b-a\right\rangle \operatorname{d} s\right|\\
& \leq{\rm diam}\left(Q\right)\cdot\sup_{s\in\left[0,1\right]}\left|\nabla f\left(x+sb+\left(1-s\right)a\right)\right|\\
\left({\scriptstyle Q\text{ convex}}\right) & \leq{\rm diam}\left(Q\right)\cdot\sup_{c\in Q}\left|\nabla f\left(x+c\right)\right|\\
& \overset{\left(\dagger\right)}{\leq}{\rm diam}\left(Q\right)\cdot\left[M_{Q}\left(\nabla f\right)\right]\left(x\right).
\end{align*}
Here, it only remains to justify the last step, i.e.\@ that $\left[M_{Q}\left(\nabla f\right)\right]\left(x\right)=\esssup_{c\in Q}\left|\nabla f\left(x+c\right)\right|\overset{!}{=}\sup_{c\in Q}\left|\nabla f\left(x+c\right)\right|$.
Here, only ``$\geq$'' is nontrivial. But by continuity of $\nabla f$,
and since nonempty open sets have positive measure, it is not hard
to see
\[
\esssup_{c\in Q}\left|\nabla f\left(x+c\right)\right|\geq\sup_{c\in Q^{\circ}}\left|\nabla f\left(x+c\right)\right|,
\]
so that it suffices (by continuity) to show that $\overline{Q^{\circ}}\supset Q$.
But for arbitrary $a\in Q^{\circ}$ and $b\in Q$, we have $B_{\varepsilon}\left(a\right)\subset Q$
for some $\varepsilon>0$. For $t\in\left(0,1\right)$, this implies
\[
ta+\left(1-t\right)b\in t\cdot B_{\varepsilon}\left(a\right)+\left(1-t\right)b\subset Q
\]
and hence $ta+\left(1-t\right)b\in Q^{\circ}$. Because of $ta+\left(1-t\right)b\xrightarrow[t\to0]{}b$,
we conclude $b\in\overline{Q^{\circ}}$, as desired.
\end{proof}
\begin{lem}
\label{lem:OscillationSchwartzFunction}Let $f\in C^{1}\left(\mathbb{R}^{d};\mathbb{C}\right)$
and $N\geq0$ and set $C:=\left(3\sqrt{d}\right)^{N+1}$.
Then
\begin{equation}
\left(\osc{\delta\cdot\left[-1,1\right]^{d}}f\right)\left(x\right)\leq C\cdot\left\Vert \nabla f\right\Vert _{N}\cdot\delta\cdot\left(1+\left|x\right|\right)^{-N}\qquad\forall x\in\mathbb{R}^{d}\:\forall\delta\in\left(0,1\right],\label{eq:OscillationPointwiseEstimate}
\end{equation}
where $\left\Vert \nabla f\right\Vert _{N}:=\sup_{x\in\mathbb{R}^{d}}\left(1+\left|x\right|\right)^{N}\left|\nabla f\left(x\right)\right|$.
\end{lem}
\begin{proof}
By Cauchy-Schwarz, we have ${\rm diam}\left(\delta\cdot\left[-1,1\right]^{d}\right)\leq2\sqrt{d}\cdot\delta$.
We can clearly assume $\left\Vert \nabla f\right\Vert _{N}<\infty$.
Since we also have $\delta\left[-1,1\right]^{d}\subset\left[-1,1\right]^{d}$,
we get from Lemmas \ref{lem:OscillationEstimatedByWienerDerivative}
and \ref{lem:SchwartzFunctionsAreWiener} that
\begin{align*}
\left(\osc{\delta\left[-1,1\right]^{d}}f\right)\left(x\right) & \leq2\sqrt{d}\cdot\delta\cdot\left[M_{\delta\left[-1,1\right]^{d}}\left(\nabla f\right)\right]\left(x\right)\\
& \leq2\sqrt{d}\cdot\delta\cdot\left[M_{\left[-1,1\right]^{d}}\left(\nabla f\right)\right]\left(x\right)\\
& \leq2\sqrt{d}\cdot\delta\cdot\left\Vert \nabla f\right\Vert _{N}\cdot\left[M_{\left[-1,1\right]^{d}}\left(1+\left|\bullet\right|\right)^{-N}\right]\left(x\right)\\
\left({\scriptstyle \text{Lemma }\ref{lem:SchwartzFunctionsAreWiener}}\right) & \leq2\sqrt{d}\cdot\left(1+2\sqrt{d}\right)^{N}\cdot\delta\cdot\left\Vert \nabla f\right\Vert _{N}\cdot\left(1+\left|x\right|\right)^{-N}\\
\left({\scriptstyle \text{since }1\leq\sqrt{d}}\right) & \leq\left(3\sqrt{d}\right)^{N+1}\cdot\delta\cdot\left\Vert \nabla f\right\Vert _{N}\cdot\left(1+\left|x\right|\right)^{-N}
\end{align*}
for all $x\in\mathbb{R}^{d}$.
\end{proof}
\subsection{Self-improving properties for bandlimited functions}
Our next aim is to show that bandlimited $L^{p}$-functions automatically
belong to $W_{Q}\left(L^{p}\right)$. More precisely, we will show
for $\operatorname{supp}\widehat{f}\subset T_{i}\left[-R,R\right]^{d}+\xi_{0}$
that
\[
\left\Vert f\right\Vert _{W_{T_{i}^{-T}\left[-1,1\right]^{d}}\left(L^{p}\right)}\leq C_{d,p,R}\cdot\left\Vert f\right\Vert _{L^{p}},
\]
which we call a ``self-improving property'', since we can improve
a simple $L^{p}$ estimate to a Wiener-amalgam estimate, at least
for suitably bandlimited functions. In fact, we will develop a \emph{weighted
version} of the preceding estimate.
All of our results in this section are based on the following convolution
relation for bandlimited $L^{p}$-functions, which we take from \cite[Theorem 3.4]{DecompositionEmbedding}.
We remark that this \emph{pointwise} estimate already appears \emph{in
the proof} of \cite[Proposition 1.5.1]{TriebelTheoryOfFunctionSpaces},
but is not stated explicitly as a theorem.
\begin{thm}
\label{thm:PointwiseQuasiBanachBandlimitedConvolution}Let $Q,\Omega\subset\mathbb{R}^{d}$
be compact and let $p\in\left(0,1\right]$. Furthermore, let $\psi\in L^{1}\left(\mathbb{R}^{d}\right)$
with $\operatorname{supp}\psi\subset Q$ and such that $\mathcal{F}^{-1}\psi\in L^{p}\left(\mathbb{R}^{d}\right)$.
For each $f\in L^{p}\left(\mathbb{R}^{d}\right)\cap\mathcal{S}'\left(\mathbb{R}^{d}\right)$
with $\operatorname{supp}\widehat{f}\subset\Omega$, we have $\mathcal{F}^{-1}\left(\psi\cdot\widehat{f}\right)=\left(\mathcal{F}^{-1}\psi\right)\ast f\in L^{p}\left(\mathbb{R}^{d}\right)$
with
\[
\left(\left|\mathcal{F}^{-1}\psi\right|\ast\left|f\right|\right)\left(x\right)\leq\left[\lambda_{d}\left(Q-\Omega\right)\right]^{\frac{1}{p}-1}\cdot\left[\int_{\mathbb{R}^{d}}\left|\left(\mathcal{F}^{-1}\psi\right)\left(y\right)\right|^{p}\cdot\left|f\left(x-y\right)\right|^{p}\operatorname{d} y\right]^{1/p}
\]
for all $x\in\mathbb{R}^{d}$.
\end{thm}
In order to ``circumvent'' the assumption $f\in L^{p}\left(\mathbb{R}^{d}\right)$,
we also need the following approximation result, a proof of which
can be found in \cite[Lemma 3.2]{DecompositionEmbedding}, or in \cite[Theorem 1.4.1]{TriebelTheoryOfFunctionSpaces}.
In fact, the proof given in \cite{DecompositionEmbedding} is based
on that in \cite{TriebelTheoryOfFunctionSpaces}.
\begin{lem}
\label{lem:BandlimitedPointwiseApproximation}Let $\Omega\subset\mathbb{R}^{d}$
be compact and assume $f\in\mathcal{S}'\left(\mathbb{R}^{d}\right)$
with $\operatorname{supp}\widehat{f}\subset\Omega$. Then $f$ is given by (integration
against) a smooth function $g\in C^{\infty}\left(\mathbb{R}^{d}\right)$
with polynomially bounded derivatives of all orders.
Furthermore, there is a sequence of Schwartz functions $\left(g_{n}\right)_{n\in\mathbb{N}}$
with the following properties:
\begin{enumerate}
\item $\left|g_{n}\left(x\right)\right|\leq\left|g\left(x\right)\right|$
for all $x\in\mathbb{R}^{d}$,
\item $g_{n}\left(x\right)\xrightarrow[n\to\infty]{}g\left(x\right)$ for
all $x\in\mathbb{R}^{d}$,
\item $\operatorname{supp}\widehat{g_{n}}\subset B_{1/n}\left(\Omega\right)$, where $B_{1/n}\left(\Omega\right)$
is the $\frac{1}{n}$-neighborhood of $\Omega$, given by
\[
B_{1/n}\left(\Omega\right)=\left\{ \xi\in\mathbb{R}^{d}\,\middle|\,\operatorname{dist}\left(\xi,\Omega\right)<\frac{1}{n}\right\} .
\]
\end{enumerate}
In the following, we will identify the bandlimited distribution $f$
with its ``smooth version'' $g$.
\end{lem}
Using the two preceding results, we can now establish our first ``self-improving
property''.
\begin{thm}
\label{thm:BandlimitedWienerAmalgamSelfImproving}For $p\in\left(0,\infty\right]$,
$\xi_{0}\in\mathbb{R}^{d}$ and $f\in\mathcal{S}'\left(\mathbb{R}^{d}\right)$
with $\operatorname{supp}\widehat{f}\subset T_{i}\left[-R,R\right]^{d}+\xi_{0}$
for some $i\in I$, we have
\[
\left\Vert f\right\Vert _{W_{T_{i}^{-T}\left[-1,1\right]^{d}}\left(L_{v}^{p}\right)}\leq C\cdot\left\Vert f\right\Vert _{L_{v}^{p}}
\]
with $s:=\min\left\{ 1,p\right\} $ and $N:=\left\lceil K+\frac{d+1}{s}\right\rceil $,
as well as
\[
C=2^{4\left(1+\frac{d}{s}\right)}s_{d}^{\frac{1}{s}}\left(192\cdotd^{3/2}\cdot N\right)^{N+1}\cdot\Omega_{0}^{K}\Omega_{1}\cdot\left(1+R\right)^{\frac{d}{s}}.\qedhere
\]
\end{thm}
\begin{rem*}
The only specific property of $v$ which is used in the proof is that
$v\left(x+y\right)\leq v\left(x\right)v_{0}\left(y\right)$. By submultiplicativity
of $v_{0}$, the same remains true when $v$ is replaced by $v_{0}$.
Hence, we also have $\left\Vert f\right\Vert _{W_{T_{i}^{-T}\left[-1,1\right]^{d}}\left(L_{v_{0}}^{p}\right)}\leq C\cdot\left\Vert f\right\Vert _{L_{v_{0}}^{p}}$
for $f\in\mathcal{S}'\left(\mathbb{R}^{d}\right)$ with $\operatorname{supp}\widehat{f}\subset T_{i}\left[-R,R\right]^{d}+\xi_{0}$.
\end{rem*}
\begin{proof}
We can clearly assume $\left\Vert f\right\Vert _{L_{v}^{p}}<\infty$,
since otherwise the claim is trivial. Using Lemma \ref{lem:BandlimitedPointwiseApproximation},
choose a sequence of Schwartz functions $\left(f_{n}\right)_{n\in\mathbb{N}}$
satisfying $\left|f_{n}\left(x\right)\right|\leq\left|f\left(x\right)\right|$,
as well as $f_{n}\left(x\right)\xrightarrow[n\to\infty]{}f\left(x\right)$
and furthermore $\operatorname{supp}\widehat{f_{n}}\subset B_{1/n}\left(T_{i}\left[-R,R\right]^{d}+\xi_{0}\right)$
for all $n\in\mathbb{N}$. Note that $T_{i}\left(-\left(R+1\right),R+1\right)^{d}+\xi_{0}$
is a neighborhood of the compact(!) set $T_{i}\left[-R,R\right]^{d}+\xi_{0}$,
so that there is some $n_{0}\in\mathbb{N}$ satisfying $\operatorname{supp}\widehat{f_{n}}\subset T_{i}\left[-\left(R+1\right),R+1\right]^{d}+\xi_{0}$
for all $n\geq n_{0}$. By dropping (or modifying) the first $n_{0}$
terms of the sequence $\left(f_{n}\right)_{n\in\mathbb{N}}$, we can assume
that this holds for all $n\in\mathbb{N}$.
Let $s=\min\left\{ 1,p\right\} $ and $N=\left\lceil K+\frac{d+1}{s}\right\rceil \geq2$
as in the statement of the theorem. Using Lemma \ref{lem:SmoothCutOffFunctionConstants}
(with $R+1$ instead of $R$ and with $s=3$) and Corollary \ref{cor:CutoffInverseFourierEstimate},
we get a function $\psi\in\TestFunctionSpace{\mathbb{R}^{d}}$ satisfying
$0\leq\psi\leq1$, as well as $\operatorname{supp}\psi\subset\left[-\left(R+4\right),R+4\right]^{d}$
and $\psi\equiv1$ on $\left[-\left(R+1\right),R+1\right]^{d}$
which also satisfies
\begin{align*}
\left|\left(\mathcal{F}^{-1}\psi\right)\left(x\right)\right| & \leq2\pi\cdot2^{d}\cdot\left(48d\right)^{N+1}\left(N+2\right)!\cdot\left(R+4\right)^{d}\cdot\left(1+\left|x\right|\right)^{-N}\\
& =:C_{1}\cdot\left(1+\left|x\right|\right)^{-N}
\end{align*}
for all $x\in\mathbb{R}^{d}$. Note because of $N\geq2$ that $\left(N+2\right)!=\prod_{\ell=1}^{N+2}\ell\leq\prod_{\ell=2}^{N+2}\left(N+2\right)=\left(N+2\right)^{N+1}\leq\left(2N\right)^{N+1}$
and hence
\[
C_{1}\leq2^{3\left(1+d\right)}\left(96d\cdot N\right)^{N+1}\cdot\left(1+R\right)^{d}.
\]
\medskip{}
Next, define
\[
\varrho:=\mathcal{F}^{-1}\left(L_{\xi_{0}}\left[\psi\circ T_{i}^{-1}\right]\right)=\left|\det T_{i}\right|\cdot M_{\xi_{0}}\left[\left(\mathcal{F}^{-1}\psi\right)\circ T_{i}^{T}\right]
\]
and note $\widehat{\varrho}\equiv1$ on $T_{i}\left[-\left(R+1\right),R+1\right]^{d}+\xi_{0}$,
so that $\widehat{f_{n}}=\widehat{\varrho}\cdot\widehat{f_{n}}$,
which implies $f_{n}=f_{n}\ast\varrho$. Next, we note for arbitrary
$x\in\mathbb{R}^{d}$ and $y\in T_{i}^{-T}\left[-1,1\right]^{d}$
because of
\begin{align*}
1+\left|T_{i}^{T}x\right| & \leq1+\left|T_{i}^{T}\left(x+y\right)\right|+\left|-T_{i}^{T}y\right|\\
& \leq1+\sqrt{d}+\left|T_{i}^{T}\left(x+y\right)\right|\\
& \leq\left(1+\sqrt{d}\right)\left(1+\left|T_{i}^{T}\left(x+y\right)\right|\right)
\end{align*}
that
\begin{align}
\left|\varrho\left(x+y\right)\right| & =\left|\det T_{i}\right|\cdot\left|\left(\mathcal{F}^{-1}\psi\right)\left(T_{i}^{T}\left(x+y\right)\right)\right|\nonumber \\
& \leq C_{1}\cdot\left|\det T_{i}\right|\cdot\left(1+\left|T_{i}^{T}\left(x+y\right)\right|\right)^{-N}\nonumber \\
& \leq\left(1+\sqrt{d}\right)^{N}C_{1}\cdot\left|\det T_{i}\right|\cdot\left(1+\left|T_{i}^{T}x\right|\right)^{-N}.\label{eq:BandlimitedWienerSelfImprovementKernelEstimate}
\end{align}
Now, we distinguish the two cases $p\in\left[1,\infty\right]$ and
$p\in\left(0,1\right)$.
\textbf{Case 1}: We have $p\in\left[1,\infty\right]$ and hence $s=1$.
In this case, note because of
\begin{align}
v\left(x\right)=v\left(z+x-z\right) & \leq v\left(z\right)v_{0}\left(x-z\right)\nonumber \\
& \leq\Omega_{1}\cdot v\left(z\right)\left(1+\left|x-z\right|\right)^{K}\nonumber \\
\left({\scriptstyle \text{eq. }\eqref{eq:WeightLinearTransformationsConnection}}\right) & \leq\Omega_{0}^{K}\Omega_{1}\cdot v\left(z\right)\cdot\left(1+\left|T_{i}^{T}\left(x-z\right)\right|\right)^{K}\label{eq:BandlimitedWienerSelfImprovementWeightProduct}
\end{align}
that
\begin{align*}
v\left(x\right)\cdot\left(M_{T_{i}^{-T}\left[-1,1\right]^{d}}f_{n}\right)\left(x\right) & \leq v\left(x\right)\cdot\sup_{y\in T_{i}^{-T}\left[-1,1\right]^{d}}\left|f_{n}\left(x+y\right)\right|\\
\left({\scriptstyle \text{since }f_{n}=f_{n}\ast\varrho}\right) & \leq v\left(x\right)\cdot\sup_{y\in T_{i}^{-T}\left[-1,1\right]^{d}}\int_{\mathbb{R}^{d}}\left|\varrho\left(\left(x-z\right)+y\right)\right|\cdot\left|f_{n}\left(z\right)\right|\operatorname{d} z\\
\left({\scriptstyle \text{eq. }\eqref{eq:BandlimitedWienerSelfImprovementKernelEstimate}\text{ with }x-z\text{ instead of }x}\right) & \leq\left(1+\sqrt{d}\right)^{N}C_{1}\cdot\left|\det T_{i}\right|\cdot\int_{\mathbb{R}^{d}}v\left(x\right)\cdot\left(1+\left|T_{i}^{T}\left(x-z\right)\right|\right)^{-N}\cdot\left|f_{n}\left(z\right)\right|\operatorname{d} z\\
\left({\scriptstyle \text{eq. }\eqref{eq:BandlimitedWienerSelfImprovementWeightProduct}}\right) & \leq\left(1+\sqrt{d}\right)^{N}\Omega_{0}^{K}\Omega_{1}C_{1}\cdot\left|\det T_{i}\right|\cdot\int_{\mathbb{R}^{d}}\left(1+\left|T_{i}^{T}\left(x-z\right)\right|\right)^{K-N}\cdot\left|\left(v\cdot f_{n}\right)\left(z\right)\right|\operatorname{d} z.
\end{align*}
Now, we simply use Young's inequality $\left\Vert f\ast g\right\Vert _{L^{p}}\leq\left\Vert f\right\Vert _{L^{1}}\cdot\left\Vert g\right\Vert _{L^{p}}$
to conclude
\begin{align*}
\left\Vert f_{n}\right\Vert _{W_{T_{i}^{-T}\left[-1,1\right]^{d}}\left(L_{v}^{p}\right)} & =\left\Vert v\cdot M_{T_{i}^{-T}\left[-1,1\right]^{d}}f_{n}\right\Vert _{L^{p}}\\
& \leq\left(1+\sqrt{d}\right)^{N}\Omega_{0}^{K}\Omega_{1}C_{1}\cdot\left|\det T_{i}\right|\cdot\left\Vert \left(1+\left|T_{i}^{T}\bullet\right|\right)^{-\left(N-K\right)}\right\Vert _{L^{1}}\cdot\left\Vert v\cdot f_{n}\right\Vert _{L^{p}}\\
\left({\scriptstyle N-K\geqd+1,\text{ eq. }\eqref{eq:StandardDecayLpEstimate}\text{ and }\left|f_{n}\right|\leq\left|f\right|}\right) & \leq\left(1+\sqrt{d}\right)^{N}s_{d}\Omega_{0}^{K}\Omega_{1}C_{1}\cdot\left\Vert f\right\Vert _{L_{v}^{p}}.
\end{align*}
But because of $f_{n}\to f$ pointwise, we get
\[
\left(M_{T_{i}^{-T}\left[-1,1\right]^{d}}f\right)\left(x\right)=\left\Vert {\mathds{1}}_{x+T_{i}^{-T}\left[-1,1\right]^{d}}\cdot f\right\Vert _{L^{\infty}}\leq\liminf_{n\to\infty}\left\Vert {\mathds{1}}_{x+T_{i}^{-T}\left[-1,1\right]^{d}}\cdot f_{n}\right\Vert _{L^{\infty}}=\liminf_{n\to\infty}\left(M_{T_{i}^{-T}\left[-1,1\right]^{d}}f_{n}\right)\left(x\right),
\]
so that an application of Fatou's Lemma yields
\[
\left\Vert f\right\Vert _{W_{T_{i}^{-T}\left[-1,1\right]^{d}}\left(L_{v}^{p}\right)}\leq\liminf_{n\to\infty}\left\Vert f_{n}\right\Vert _{W_{T_{i}^{-T}\left[-1,1\right]^{d}}\left(L_{v}^{p}\right)}\leq\left(1+\sqrt{d}\right)^{N}s_{d}\Omega_{0}^{K}\Omega_{1}C_{1}\cdot\left\Vert f\right\Vert _{L_{v}^{p}},
\]
as desired.
\medskip{}
\textbf{Case 2}: We have $p\in\left(0,1\right)$ and hence $s=p$.
In this case, we first note
\begin{align*}
\lambda_{d}\left(\operatorname{supp}\widehat{f_{n}}-\operatorname{supp}\widehat{\varrho}\right) & \leq\lambda_{d}\left(\left[T_{i}\left[-\left(R+1\right),R+1\right]^{d}+\xi_{0}\right]-\left[T_{i}\left[-\left(R+4\right),R+4\right]^{d}+\xi_{0}\right]\right)\\
& \leq\lambda_{d}\left(T_{i}\left[-\left(2R+5\right),2R+5\right]^{d}\right)\\
& =\left|\det T_{i}\right|\cdot\left(4R+10\right)^{d}.
\end{align*}
For brevity, set $C_{2}:=\left(1+\sqrt{d}\right)^{N}\left(4R+10\right)^{d\left(\frac{1}{p}-1\right)}C_{1}$
and apply Theorem \ref{thm:PointwiseQuasiBanachBandlimitedConvolution}
to get
\begin{align*}
v\left(x\right)\cdot\left(M_{T_{i}^{-T}\left[-1,1\right]^{d}}f_{n}\right)\left(x\right) & \leq v\left(x\right)\cdot\sup_{y\in T_{i}^{-T}\left[-1,1\right]^{d}}\left|f_{n}\left(x+y\right)\right|\\
\left({\scriptstyle \text{since }f_{n}=f_{n}\ast\varrho}\right) & \leq v\left(x\right)\cdot\sup_{y\in T_{i}^{-T}\left[-1,1\right]^{d}}\left(\left|f_{n}\right|\ast\left|\varrho\right|\right)\left(x+y\right)\\
\left({\scriptstyle \text{Theorem }\ref{thm:PointwiseQuasiBanachBandlimitedConvolution}}\right) & \leq\left[\left(4R\!+\!10\right)^{d}\cdot\left|\det T_{i}\right|\right]^{\frac{1}{p}-1}\cdot v\left(x\right)\cdot\!\!\!\sup_{y\in T_{i}^{-T}\left[-1,1\right]^{d}}\left(\int_{\mathbb{R}^{d}}\left|\varrho\left(\left(x-z\right)+y\right)\right|^{p}\cdot\left|f_{n}\left(z\right)\right|^{p}\operatorname{d} z\right)^{1/p}\\
\left({\scriptstyle \text{eq. }\eqref{eq:BandlimitedWienerSelfImprovementKernelEstimate}}\right) & \leq C_{2}\cdot\left|\det T_{i}\right|^{1/p}\cdot v\left(x\right)\cdot\left(\int_{\mathbb{R}^{d}}\left(1+\left|T_{i}^{T}\left(x-z\right)\right|\right)^{-Np}\cdot\left|f_{n}\left(z\right)\right|^{p}\operatorname{d} z\right)^{1/p}\\
\left({\scriptstyle \text{eq. }\eqref{eq:BandlimitedWienerSelfImprovementWeightProduct}}\right) & \leq C_{2}\Omega_{0}^{K}\Omega_{1}\cdot\left|\det T_{i}\right|^{1/p}\cdot\left(\int_{\mathbb{R}^{d}}\left(1+\left|T_{i}^{T}\left(x-z\right)\right|\right)^{-p\left(N-K\right)}\cdot\left|\left(v\cdot f_{n}\right)\left(z\right)\right|^{p}\operatorname{d} z\right)^{1/p}.
\end{align*}
Finally, take the $L^{p}$-norm of the preceding estimate to conclude
\begin{align*}
\left\Vert f_{n}\right\Vert _{W_{T_{i}^{-T}\left[-1,1\right]^{d}}\left(L_{v}^{p}\right)}^{p} & \leq\left(C_{2}\Omega_{0}^{K}\Omega_{1}\right)^{p}\cdot\left|\det T_{i}\right|\cdot\int_{\mathbb{R}^{d}}\int_{\mathbb{R}^{d}}\left(1+\left|T_{i}^{T}\left(x-z\right)\right|\right)^{-p\left(N-K\right)}\cdot\left|\left(v\cdot f_{n}\right)\left(z\right)\right|^{p}\operatorname{d} z\operatorname{d} x\\
\left({\scriptstyle \text{Fubini and }y=x-z}\right) & =\left(C_{2}\Omega_{0}^{K}\Omega_{1}\right)^{p}\cdot\left\Vert f\right\Vert _{L_{v}^{p}}^{p}\cdot\left|\det T_{i}\right|\cdot\int_{\mathbb{R}^{d}}\left(1+\left|T_{i}^{T}y\right|\right)^{-p\left(N-K\right)}\operatorname{d} y\\
& =\left(C_{2}\Omega_{0}^{K}\Omega_{1}\right)^{p}\cdot\left\Vert \left(1+\left|\bullet\right|\right)^{-\left(N-K\right)}\right\Vert _{L^{p}}^{p}\cdot\left\Vert f\right\Vert _{L_{v}^{p}}^{p}\\
\left({\scriptstyle \text{eq. }\eqref{eq:StandardDecayLpEstimate}\text{ and }N-K-\frac{d}{p}\geq\frac{1}{p}}\right) & \leq\left(C_{2}\Omega_{0}^{K}\Omega_{1}\right)^{p}\cdot s_{d}\cdot\left\Vert f\right\Vert _{L_{v}^{p}}^{p}.
\end{align*}
The remainder of the proof is now as for $p\in\left[1,\infty\right]$,
but with a slightly different constant.
\end{proof}
Now, we establish our second ``self-improving property'', which
yields an estimate for the $L_{v}^{p}$-norm of the oscillation of
a band-limited function, only in terms of the $L_{v}^{p}$-norm of
the function.
\begin{thm}
\label{thm:BandlimitedOscillationSelfImproving}For each $p\in\left[0,\infty\right)$,
$i\in I$, $\xi_{0}\in\mathbb{R}^{d}$, $R>0$ and $f\in\mathcal{S}'\left(\mathbb{R}^{d}\right)$
with $\operatorname{supp}\widehat{f}\subset T_{i}\left[-R,R\right]^{d}+\xi_{0}$,
we have
\[
\left\Vert \osc{\delta\cdot T_{i}^{-T}\left[-1,1\right]^{d}}\left[M_{-\xi_{0}}f\right]\right\Vert _{W_{T_{i}^{-T}\left[-1,1\right]^{d}}\left(L_{v}^{p}\right)}\leq C\cdot\delta\cdot\left\Vert f\right\Vert _{L_{v}^{p}}
\]
with
\[
C:=\Omega_{0}^{2K}\Omega_{1}^{2}\cdot\left(23040\cdotd^{3/2}\cdot\left(K+1+\frac{d+1}{\min\left\{ 1,p\right\} }\right)\right)^{K+2+\frac{d+1}{\min\left\{ 1,p\right\} }}\cdot\left(1+R\right)^{1+\frac{d}{\min\left\{ 1,p\right\} }}.\qedhere
\]
\end{thm}
\begin{rem*}
As usual, the claim remains valid when $v$ is replaced by $v_{0}$
throughout.
\end{rem*}
\begin{proof}
As usual, since $f$ is a \emph{bandlimited} tempered distribution,
it is actually given by integration against a smooth function with
polynomially bounded derivatives. Furthermore, $\operatorname{supp}\mathcal{F}\left[M_{-\xi_{0}}f\right]=\operatorname{supp} L_{-\xi_{0}}\widehat{f}\subset T_{i}\left[-R,R\right]^{d}$,
so that we can assume $\xi_{0}=0$ for the remainder of the proof.
The first part of the proof is now very similar to that of Theorem
\ref{thm:BandlimitedWienerAmalgamSelfImproving}: We can clearly assume
$\left\Vert f\right\Vert _{L_{v}^{p}}<\infty$, since otherwise the
claim is trivial. Using Lemma \ref{lem:BandlimitedPointwiseApproximation},
choose a sequence of Schwartz functions $\left(f_{n}\right)_{n\in\mathbb{N}}$
satisfying $\left|f_{n}\left(x\right)\right|\leq\left|f\left(x\right)\right|$,
as well as $f_{n}\left(x\right)\xrightarrow[n\to\infty]{}f\left(x\right)$
and furthermore $\operatorname{supp}\widehat{f_{n}}\subset B_{1/n}\left(T_{i}\left[-R,R\right]^{d}\right)$
for all $n\in\mathbb{N}$. Note that $T_{i}\left(-\left(R+1\right),R+1\right)^{d}$
is a neighborhood of the compact(!) set $T_{i}\left[-R,R\right]^{d}$,
so that there is some $n_{0}\in\mathbb{N}$ satisfying $\operatorname{supp}\widehat{f_{n}}\subset T_{i}\left[-\left(R+1\right),R+1\right]^{d}$
for all $n\geq n_{0}$. By dropping (or modifying) the first $n_{0}$
terms of the sequence $\left(f_{n}\right)_{n\in\mathbb{N}}$, we can assume
that this holds for all $n\in\mathbb{N}$.
Let $s:=\min\left\{ 1,p\right\} $ and $N:=\left\lceil K+\frac{d+1}{s}\right\rceil \geq2$.
Using Lemma \ref{lem:SmoothCutOffFunctionConstants} (with $R+1$
instead of $R$ and with $s=3$) and Corollary \ref{cor:CutoffInverseFourierEstimate},
we get a function $\psi\in\TestFunctionSpace{\mathbb{R}^{d}}$ satisfying
$0\leq\psi\leq1$, as well as $\operatorname{supp}\psi\subset\left[-\left(R+4\right),R+4\right]^{d}$
and $\psi\equiv1$ on $\left[-\left(R+1\right),R+1\right]^{d}$
which also satisfies
\begin{align}
\left|\left(\partial^{\alpha}\left[\mathcal{F}^{-1}\psi\right]\right)\left(x\right)\right| & \leq2\pi\cdot2^{d}\cdot\left(48d\right)^{N+1}\left(N+2\right)!\cdot\left(R+5\right)^{\left|\alpha\right|}\left(R+4\right)^{d}\cdot\left(1+\left|x\right|\right)^{-N}\nonumber \\
& \leq2\pi\cdot2^{d}\cdot\left(48d\right)^{N+1}\left(N+2\right)!\cdot\left(R+5\right)^{d+1}\cdot\left(1+\left|x\right|\right)^{-N}\nonumber \\
& =:C_{1}\cdot\left(1+\left|x\right|\right)^{-N}\label{eq:SelfImprovingOscillationKernelDerivativeEstimate}
\end{align}
for all $x\in\mathbb{R}^{d}$ and $\alpha\in\mathbb{N}_{0}^{d}$
with $\left|\alpha\right|\leq1$. Using $\left(N+2\right)!=\prod_{\ell=1}^{N+2}\ell\leq\prod_{\ell=2}^{N+2}\left(N+2\right)=\left(N+2\right)^{N+1}\leq\left(2N\right)^{N+1}$,
we get
\begin{equation}
C_{1}\leq40\cdot10^{d}\cdot\left(96d\cdot N\right)^{N+1}\cdot\left(1+R\right)^{d+1}.\label{eq:SelfImprovingOscillationC1Estimate}
\end{equation}
\medskip{}
Now, let $g_{n}:=f_{n}\circ T_{i}^{-T}$ for $n\in\mathbb{N}$. Using Lemmas
\ref{lem:OscillationLinearChange} and \ref{lem:OscillationEstimatedByWienerDerivative},
we see
\begin{align*}
\osc{\delta\cdot T_{i}^{-T}\left[-1,1\right]^{d}}f_{n} & =\osc{\delta\cdot T_{i}^{-T}\left[-1,1\right]^{d}}\left(g_{n}\circ T_{i}^{T}\right)\\
\left({\scriptstyle \text{Lemma }\ref{lem:OscillationLinearChange}}\right) & =\left(\osc{\delta\cdot\left[-1,1\right]^{d}}g_{n}\right)\circ T_{i}^{T}\\
\left({\scriptstyle \text{Lemma }\ref{lem:OscillationEstimatedByWienerDerivative}}\right) & \leq2\sqrt{d}\cdot\delta\cdot\left(M_{\delta\left[-1,1\right]^{d}}\left[\nabla g_{n}\right]\right)\circ T_{i}^{T}.
\end{align*}
Based on this estimate, Lemmas \ref{lem:WienerTransformationFormula}
and \ref{lem:IteratedMaximalFunction} show
\begin{align}
M_{T_{i}^{-T}\left[-1,1\right]^{d}}\left[\osc{\delta\cdot T_{i}^{-T}\left[-1,1\right]^{d}}f_{n}\right] & \leq2\sqrt{d}\cdot\delta\cdot M_{T_{i}^{-T}\left[-1,1\right]^{d}}\left[\left(M_{\delta\left[-1,1\right]^{d}}\left[\nabla g_{n}\right]\right)\circ T_{i}^{T}\right]\nonumber \\
\left({\scriptstyle \text{Lemma }\ref{lem:WienerTransformationFormula}}\right) & =2\sqrt{d}\cdot\delta\cdot\left[M_{\left[-1,1\right]^{d}}\left(M_{\delta\left[-1,1\right]^{d}}\left[\nabla g_{n}\right]\right)\right]\circ T_{i}^{T}\nonumber \\
\left({\scriptstyle \text{Lemma }\ref{lem:IteratedMaximalFunction}\text{ and }\delta\leq1}\right) & \leq2\sqrt{d}\cdot\delta\cdot\left(M_{\left[-2,2\right]^{d}}\left[\nabla g_{n}\right]\right)\circ T_{i}^{T}\nonumber \\
\left({\scriptstyle \text{Lemma }\ref{lem:WienerTransformationFormula}}\right) & =2\sqrt{d}\cdot\delta\cdot M_{T_{i}^{-T}\left[-2,2\right]^{d}}\left[\left(\nabla g_{n}\right)\circ T_{i}^{T}\right].\label{eq:SelfImprovingOscillationPointwise}
\end{align}
Next, observe $\operatorname{supp}\widehat{g_{n}}=\operatorname{supp}\left[\left|\det T_{i}\right|\cdot\widehat{f_{n}}\circ T_{i}\right]=T_{i}^{-1}\operatorname{supp}\widehat{f_{n}}\subset\left[-\left(R+1\right),R+1\right]^{d}$
for all $n\in\mathbb{N}$, so that we see $\widehat{g_{n}}=\widehat{g_{n}}\cdot\psi$.
Hence, $g_{n}=g_{n}\ast\mathcal{F}^{-1}\psi$. Because of $g_{n},\mathcal{F}^{-1}\psi\in\mathcal{S}\left(\mathbb{R}^{d}\right)$,
this easily implies $\nabla g_{n}=g_{n}\ast\nabla\left(\mathcal{F}^{-1}\psi\right)$.
But for arbitrary Schwartz functions $f,g$ and $T\in\mathrm{GL}\left(\mathbb{R}^{d}\right)$,
we have
\begin{align*}
\left(f\ast g\right)\left(Tx\right) & =\int_{\mathbb{R}^{d}}f\left(Tx-y\right)g\left(y\right)\operatorname{d} y\\
\left({\scriptstyle y=Tz}\right) & =\left|\det T\right|\cdot\int_{\mathbb{R}^{d}}f\left(Tx-Tz\right)g\left(Tz\right)\operatorname{d} z\\
& =\left|\det T\right|\cdot\left[\left(f\circ T\right)\ast\left(g\circ T\right)\right]\left(x\right),
\end{align*}
so that, if we understand the following equation componentwise,
\begin{align}
\left(\nabla g_{n}\right)\left(T_{i}^{T}x\right) & =\left|\det T_{i}\right|\cdot\left[\left(g_{n}\circ T_{i}^{T}\right)\ast\left(\left[\nabla\left(\mathcal{F}^{-1}\psi\right)\right]\circ T_{i}^{T}\right)\right]\left(x\right)\nonumber \\
\left({\scriptstyle \text{with }\eta_{j}:=\partial_{j}\left(\mathcal{F}^{-1}\psi\right)}\right) & =\left|\det T_{i}\right|\cdot\left[\left(f_{n}\ast\left[\eta_{j}\circ T_{i}^{T}\right]\right)\left(x\right)\right]_{j\in\underline{d}}.\label{eq:SelfImprovingOscillationGradientAsConvolution}
\end{align}
\medskip{}
Now, we divide the proof into the two cases $p\in\left[1,\infty\right]$
and $p\in\left(0,1\right)$. In the (easier) case $p\in\left[1,\infty\right]$,
we get
\begin{align*}
v\left(x\right)\cdot\left|\left(\partial_{j}g_{n}\right)\left(T_{i}^{T}x\right)\right| & =\left|\det T_{i}\right|\cdot v\left(x\right)\cdot\left|\left(f_{n}\ast\left[\eta_{j}\circ T_{i}^{T}\right]\right)\left(x\right)\right|\\
& \leq\left|\det T_{i}\right|\cdot\int_{\mathbb{R}^{d}}v\left(x\right)\cdot\left|f_{n}\left(y\right)\right|\cdot\left|\eta_{j}\left(T_{i}^{T}\left(x-y\right)\right)\right|\operatorname{d} y\\
\left({\scriptstyle \text{eq. }\eqref{eq:SelfImprovingOscillationKernelDerivativeEstimate}\text{ and }\eta_{j}=\partial_{j}\left[\mathcal{F}^{-1}\psi\right]}\right) & \leq C_{1}\cdot\left|\det T_{i}\right|\cdot\int_{\mathbb{R}^{d}}v\left(x\right)\cdot\left|f_{n}\left(y\right)\right|\cdot\left(1+\left|T_{i}^{T}\left(x-y\right)\right|\right)^{-N}\operatorname{d} y\\
\left({\scriptstyle \text{since }v\left(x\right)=v\left(x-y+y\right)\leq v\left(y\right)v_{0}\left(x-y\right)}\right) & \leq C_{1}\cdot\left|\det T_{i}\right|\cdot\int_{\mathbb{R}^{d}}\left|\left(v\cdot f_{n}\right)\left(y\right)\right|\cdot v_{0}\left(x-y\right)\cdot\left(1+\left|T_{i}^{T}\left(x-y\right)\right|\right)^{-N}\operatorname{d} y\\
\left({\scriptstyle \text{assumption on }v_{0}\text{ and eq. }\eqref{eq:WeightLinearTransformationsConnection}}\right) & \leq\Omega_{0}^{K}\Omega_{1}C_{1}\cdot\left|\det T_{i}\right|\cdot\int_{\mathbb{R}^{d}}\left|\left(v\cdot f_{n}\right)\left(y\right)\right|\cdot\left(1+\left|T_{i}^{T}\left(x-y\right)\right|\right)^{K-N}\operatorname{d} y
\end{align*}
for arbitrary $x\in\mathbb{R}^{d}$.
But for $z\in T_{i}^{-T}\left[-2,2\right]^{d}$, we have
\begin{align}
1+\left|T_{i}^{T}x\right| & \leq1+\left|T_{i}^{T}\left(x+z\right)\right|+\left|-T_{i}^{T}z\right|\nonumber \\
\left({\scriptstyle \left|T_{i}^{T}z\right|\leq2\sqrt{d}\text{ since }T_{i}^{T}z\in\left[-2,2\right]^{d}}\right) & \leq1+2\sqrt{d}+\left|T_{i}^{T}\left(x+z\right)\right|\nonumber \\
& \leq\left(1+2\sqrt{d}\right)\left(1+\left|T_{i}^{T}\left(x+z\right)\right|\right)\label{eq:SelfImprovingOscillationChineseBracketShift}
\end{align}
and
\begin{align}
v\left(x\right) & =v\left(x+z-z\right)\leq v\left(x+z\right)\cdot v_{0}\left(-z\right)\nonumber \\
& \leq\Omega_{1}\cdot v\left(x+z\right)\cdot\left(1+\left|-z\right|\right)^{K}\nonumber \\
\left({\scriptstyle \text{eq. }\eqref{eq:WeightLinearTransformationsConnection}}\right) & \leq\Omega_{0}^{K}\Omega_{1}\cdot v\left(x+z\right)\cdot\left(1+\left|T_{i}^{T}z\right|\right)^{K}\nonumber \\
& \leq\Omega_{0}^{K}\Omega_{1}\left(1+2\sqrt{d}\right)^{K}\cdot v\left(x+z\right).\label{eq:SelfImprovingOscillationSlightWeightShift}
\end{align}
By applying these two estimates and noting $K-N<0$, we get
\begin{align*}
v\left(x\right)\cdot\left|\left[\left(\partial_{j}g_{n}\right)\circ T_{i}^{T}\right]\left(x+z\right)\right| & \leq\Omega_{0}^{K}\Omega_{1}\cdot\left(1+2\sqrt{d}\right)^{K}\cdot v\left(x+z\right)\cdot\left|\left[\left(\partial_{j}g_{n}\right)\circ T_{i}^{T}\right]\left(x+z\right)\right|\\
& \leq\Omega_{0}^{2K}\Omega_{1}^{2}C_{1}\cdot\left(1+2\sqrt{d}\right)^{K}\cdot\left|\det T_{i}\right|\cdot\int_{\mathbb{R}^{d}}\left|\left(v\cdot f_{n}\right)\left(y\right)\right|\cdot\left(1+\left|T_{i}^{T}\left(x-y+z\right)\right|\right)^{K-N}\operatorname{d} y\\
\left({\scriptstyle \text{eq. }\eqref{eq:SelfImprovingOscillationChineseBracketShift}\text{ with }x-y\text{ instead of }x}\right) & \leq\Omega_{0}^{2K}\Omega_{1}^{2}C_{1}\cdot\left(1+2\sqrt{d}\right)^{N}\cdot\left|\det T_{i}\right|\cdot\int_{\mathbb{R}^{d}}\left|\left(v\cdot f_{n}\right)\left(y\right)\right|\cdot\left(1+\left|T_{i}^{T}\left(x-y\right)\right|\right)^{K-N}\operatorname{d} y.
\end{align*}
Noting that this holds for arbitrary $z\in T_{i}^{-T}\left[-2,2\right]^{d}$
and by taking the $\ell^{2}$-norm over $j\in\underline{d}$,
we conclude
\begin{align*}
v\left(x\right)\cdot\left[M_{T_{i}^{-T}\left[-2,2\right]^{d}}\left(\left[\nabla g_{n}\right]\circ T_{i}^{T}\right)\right]\left(x\right) & \leq C_{2}\cdot\left|\det T_{i}\right|\cdot\int_{\mathbb{R}^{d}}\left|\left(v\cdot f_{n}\right)\left(y\right)\right|\cdot\left(1+\left|T_{i}^{T}\left(x-y\right)\right|\right)^{K-N}\operatorname{d} y\\
& =C_{2}\cdot\left|\det T_{i}\right|\cdot\left(\left|v\cdot f_{n}\right|\ast\left[\left(1+\left|\bullet\right|\right)^{K-N}\circ T_{i}^{T}\right]\right)\left(x\right),
\end{align*}
where $C_{2}:=\Omega_{0}^{2K}\Omega_{1}^{2}\cdot C_{1}\sqrt{d}\left(1+2\sqrt{d}\right)^{N}$.
By taking the $L^{p}$-norm and using Young's theorem for convolutions,
we conclude
\begin{align*}
\left\Vert M_{T_{i}^{-T}\left[-2,2\right]^{d}}\left(\left[\nabla g_{n}\right]\circ T_{i}^{T}\right)\right\Vert _{L_{v}^{p}} & \leq C_{2}\cdot\left|\det T_{i}\right|\cdot\left\Vert v\cdot f_{n}\right\Vert _{L^{p}}\cdot\left\Vert \left(1+\left|\bullet\right|\right)^{K-N}\circ T_{i}^{T}\right\Vert _{L^{1}}\\
& =C_{2}\cdot\left\Vert f_{n}\right\Vert _{L_{v}^{p}}\cdot\left\Vert \left(1+\left|\bullet\right|\right)^{K-N}\right\Vert _{L^{1}}\\
\left({\scriptstyle \text{eq. }\eqref{eq:StandardDecayLpEstimate}\text{ and }K-N=K-\left\lceil K+d+1\right\rceil \leq-\left(d+1\right)}\right) & \leq s_{d}C_{2}\cdot\left\Vert f_{n}\right\Vert _{L_{v}^{p}}\\
\left({\scriptstyle \text{since }\left|f_{n}\right|\leq\left|f\right|}\right) & \leq s_{d}C_{2}\cdot\left\Vert f\right\Vert _{L_{v}^{p}}<\infty.
\end{align*}
In view of equation (\ref{eq:SelfImprovingOscillationPointwise}),
we have thus shown
\begin{align*}
\left\Vert \osc{\delta\cdot T_{i}^{-T}\left[-1,1\right]^{d}}f_{n}\right\Vert _{W_{T_{i}^{-T}\left[-1,1\right]^{d}}\left(L_{v}^{p}\right)} & \leq2\sqrt{d}\cdot\delta\cdot\left\Vert M_{T_{i}^{-T}\left[-2,2\right]^{d}}\left[\left(\nabla g_{n}\right)\circ T_{i}^{T}\right]\right\Vert _{L_{v}^{p}}\\
& \leq2s_{d}\sqrt{d}C_{2}\cdot\delta\cdot\left\Vert f\right\Vert _{L_{v}^{p}}<\infty.
\end{align*}
Now, we note
\begin{align*}
2s_{d}\sqrt{d}C_{2} & =\Omega_{0}^{2K}\Omega_{1}^{2}\cdot C_{1}\cdot2s_{d}\cdotd\cdot\left(1+2\sqrt{d}\right)^{N}\\
\left({\scriptstyle \text{eq. }\eqref{eq:SelfImprovingOscillationC1Estimate}}\right) & \leq\Omega_{0}^{2K}\Omega_{1}^{2}\cdot2s_{d}\cdotd\cdot\left(1+2\sqrt{d}\right)^{N}\cdot40\cdot10^{d}\cdot\left(96d\cdot N\right)^{N+1}\cdot\left(1+R\right)^{d+1}\\
\left({\scriptstyle \text{since }s_{d}\leq4^{d}\text{ and }d\leq2^{d}}\right) & \leq\Omega_{0}^{2K}\Omega_{1}^{2}\cdot2\cdot4^{d}2^{d}\left(1+2\sqrt{d}\right)^{N}\cdot40\cdot10^{d}\cdot\left(96d\cdot N\right)^{N+1}\cdot\left(1+R\right)^{d+1}\\
& \leq\Omega_{0}^{2K}\Omega_{1}^{2}\cdot\left(1+2\sqrt{d}\right)^{N}\cdot80^{d+1}\cdot\left(96d\cdot N\right)^{N+1}\cdot\left(1+R\right)^{d+1}\\
& \leq\Omega_{0}^{2K}\Omega_{1}^{2}\cdot\left(3\sqrt{d}\right)^{N}\cdot80^{d+1}\cdot\left(96d\cdot N\right)^{N+1}\cdot\left(1+R\right)^{d+1}\\
& \leq\Omega_{0}^{2K}\Omega_{1}^{2}\cdot80^{d+1}\cdot\left(288\cdotd^{3/2}\cdot N\right)^{N+1}\cdot\left(1+R\right)^{d+1}\\
\left({\scriptstyle \text{since }N\geqd+1}\right) & \leq\Omega_{0}^{2K}\Omega_{1}^{2}\cdot\left(23040\cdotd^{3/2}\cdot N\right)^{N+1}\cdot\left(1+R\right)^{d+1}\\
\left({\scriptstyle \text{def. of }N}\right) & \leq\Omega_{0}^{2K}\Omega_{1}^{2}\cdot\left(23040\cdotd^{3/2}\cdot\left(K+d+2\right)\right)^{K+d+3}\cdot\left(1+R\right)^{d+1}.
\end{align*}
All that remains is to extend this estimate to $f$ instead of $f_{n}$.
But for $x\in\mathbb{R}^{d}$ and arbitrary $y,z\in\delta\cdot T_{i}^{-T}\left[-1,1\right]^{d}$,
we have
\[
\left|f\left(x+y\right)-f\left(x+z\right)\right|=\liminf_{n\to\infty}\left|f_{n}\left(x+y\right)-f_{n}\left(x+z\right)\right|\leq\liminf_{n\to\infty}\left(\osc{\delta\cdot T_{i}^{-T}\left[-1,1\right]^{d}}f_{n}\right)\left(x\right)
\]
and thus $\left(\osc{\delta\cdot T_{i}^{-T}\left[-1,1\right]^{d}}f\right)\left(x\right)\leq\liminf_{n\to\infty}\left(\osc{\delta\cdot T_{i}^{-T}\left[-1,1\right]^{d}}f_{n}\right)\left(x\right)$
for all $x\in\mathbb{R}^{d}$. A similar estimate holds for the maximal
function $M_{T_{i}^{-T}\left[-1,1\right]^{d}}$ instead of
the oscillation. All in all, an application of Fatou's Lemma yields
\[
\left\Vert \osc{\delta\cdot T_{i}^{-T}\left[-1,1\right]^{d}}f\right\Vert _{W_{T_{i}^{-T}\left[-1,1\right]^{d}}\left(L_{v}^{p}\right)}\leq\liminf_{n\to\infty}\left\Vert \osc{\delta\cdot T_{i}^{-T}\left[-1,1\right]^{d}}f_{n}\right\Vert _{W_{T_{i}^{-T}\left[-1,1\right]^{d}}\left(L_{v}^{p}\right)}\leq2s_{d}\sqrt{d}C_{2}\cdot\delta\cdot\left\Vert f\right\Vert _{L_{v}^{p}}<\infty,
\]
as desired.
\medskip{}
Now, we consider the case $p\in\left(0,1\right)$. Here, we recall
$\eta_{j}=\partial_{j}\left(\mathcal{F}^{-1}\psi\right)$ and observe
\begin{align*}
\operatorname{supp}\mathcal{F}\left[\eta_{j}\circ T_{i}^{T}\right] & =\operatorname{supp}\left[\left|\det T_{i}\right|^{-1}\cdot\widehat{\eta_{j}}\circ T_{i}^{-1}\right]\\
& =T_{i}\operatorname{supp}\widehat{\eta_{j}}\\
& =T_{i}\operatorname{supp}\left(\xi\mapsto2\pi i\xi_{j}\cdot\mathcal{F}\left[\mathcal{F}^{-1}\psi\right]\left(\xi\right)\right)\\
& \subset T_{i}\operatorname{supp}\psi\subset T_{i}\left[-\left(R+4\right),R+4\right]^{d}
\end{align*}
and $\operatorname{supp}\widehat{f_{n}}\subset T_{i}\left[-\left(R+1\right),R+1\right]^{d}$,
so that equation (\ref{eq:SelfImprovingOscillationGradientAsConvolution})
and Theorem \ref{thm:PointwiseQuasiBanachBandlimitedConvolution}
yield
\begin{align*}
& \left|\left(\partial_{j}g_{n}\right)\left(T_{i}^{T}x\right)\right|^{p}\\
& \leq\left|\det T_{i}\right|^{p}\cdot\left[\lambda_{d}\left(T_{i}\left(\left[-\left(R\!+\!4\right),R\!+\!4\right]^{d}-\left[-\left(R\!+\!1\right),R\!+\!1\right]^{d}\right)\right)\right]^{1-p}\cdot\int_{\mathbb{R}^{d}}\left|f_{n}\left(y\right)\right|^{p}\cdot\left|\left(\eta_{j}\circ T_{i}^{T}\right)\left(x-y\right)\right|^{p}\operatorname{d} y\\
\left({\scriptstyle \text{eq. }\eqref{eq:SelfImprovingOscillationKernelDerivativeEstimate}}\right) & \leq\left|\det T_{i}\right|\cdot C_{1}^{p}\left[2\cdot\left(2R+5\right)\right]^{d\left(1-p\right)}\cdot\int_{\mathbb{R}^{d}}\left|f_{n}\left(y\right)\right|^{p}\cdot\left(1+\left|T_{i}^{T}\left(x-y\right)\right|\right)^{-Np}\operatorname{d} y.
\end{align*}
Similar to the case $p\in\left[1,\infty\right]$, this implies
\begin{align*}
\left[v\left(x\right)\!\cdot\!\left|\left(\partial_{j}g_{n}\right)\left(T_{i}^{T}x\right)\right|\right]^{p} & \leq\left|\det T_{i}\right|\!\cdot C_{1}^{p}\left[2\cdot\left(2R+5\right)\right]^{d\left(1-p\right)}\!\cdot\!\int_{\mathbb{R}^{d}}\left|\left(v\!\cdot\!f_{n}\right)\left(y\right)\right|^{p}\!\cdot\!\left[v_{0}\left(x-y\right)\!\cdot\!\left(1\!+\!\left|T_{i}^{T}\!\left(x-y\right)\right|\right)^{-N}\right]^{p}\operatorname{d} y\\
\left({\scriptstyle \text{assump. on }v_{0}\text{ and eq. }\eqref{eq:WeightLinearTransformationsConnection}}\right) & \leq\left[\Omega_{0}^{K}\Omega_{1}\right]^{p}\cdot\left|\det T_{i}\right|\cdot C_{1}^{p}\left[4R+10\right]^{d\left(1-p\right)}\!\cdot\!\int_{\mathbb{R}^{d}}\left|\left(v\!\cdot\!f_{n}\right)\left(y\right)\right|^{p}\cdot\left[\left(1\!+\!\left|T_{i}^{T}\!\left(x-y\right)\right|\right)^{K-N}\right]^{p}\operatorname{d} y.
\end{align*}
As above, equations (\ref{eq:SelfImprovingOscillationChineseBracketShift})
and (\ref{eq:SelfImprovingOscillationSlightWeightShift}) show for
arbitrary $z\in T_{i}^{-T}\left[-2,2\right]^{d}$ that
\begin{align*}
& \left[v\left(x\right)\cdot\left|\left[\left(\partial_{j}g_{n}\right)\circ T_{i}^{T}\right]\left(x+z\right)\right|\right]^{p}\\
\left({\scriptstyle \text{eq. }\eqref{eq:SelfImprovingOscillationSlightWeightShift}}\right) & \leq\left[\Omega_{0}^{K}\Omega_{1}\left(1+2\sqrt{d}\right)^{K}\right]^{p}\cdot\left|\left(v\cdot\left[\left(\partial_{j}g_{n}\right)\circ T_{i}^{T}\right]\right)\left(x+z\right)\right|^{p}\\
& \leq\left[\Omega_{0}^{2K}\Omega_{1}^{2}\left(1+2\sqrt{d}\right)^{K}\right]^{p}\!\!\cdot\!\left|\det T_{i}\right|\cdot C_{1}^{p}\left(10+4R\right)^{d\left(1-p\right)}\!\cdot\!\int_{\mathbb{R}^{d}}\left|\left(v\!\cdot\!f_{n}\right)\left(y\right)\right|^{p}\!\cdot\!\left[\left(1\!+\!\left|T_{i}^{T}\!\left(x+z-y\right)\right|\right)^{K-N}\right]^{p}\operatorname{d} y\\
\left({\scriptstyle \text{eq. }\eqref{eq:SelfImprovingOscillationChineseBracketShift}}\right) & \leq\left[\Omega_{0}^{2K}\Omega_{1}^{2}\left(1+2\sqrt{d}\right)^{N}\right]^{p}\!\!\cdot\!\left|\det T_{i}\right|\cdot C_{1}^{p}\left(10+4R\right)^{d\left(1-p\right)}\!\cdot\!\int_{\mathbb{R}^{d}}\left|\left(v\!\cdot\!f_{n}\right)\left(y\right)\right|^{p}\!\cdot\!\left(1\!+\!\left|T_{i}^{T}\!\left(x-y\right)\right|\right)^{p\left(K-N\right)}\operatorname{d} y.
\end{align*}
Since this holds for arbitrary $z\in T_{i}^{-T}\left[-2,2\right]^{d}$
and $j\in\underline{d}$ and since
\[
\left|\left[\left(\nabla g_{n}\right)\circ T_{i}^{T}\right]\left(x+z\right)\right|\leq\sqrt{d}\cdot\max_{j\in\underline{d}}\left|\left[\left(\partial_{j}g_{n}\right)\circ T_{i}^{T}\right]\left(x+z\right)\right|,
\]
we conclude
\[
\left[v\!\cdot\!M_{T_{i}^{-T}\left[-2,2\right]^{d}}\left(\left[\nabla g_{n}\right]\circ T_{i}^{T}\right)\right]^{p}\!\leq\!\left[\Omega_{0}^{2K}\Omega_{1}^{2}\sqrt{d}\left(1\!+\!2\sqrt{d}\right)^{\!N}\right]^{p}\!\cdot\left|\det T_{i}\right|\cdot C_{1}^{p}\left(10\!+\!4R\right)^{d\left(1-p\right)}\cdot\left|v\!\cdot\!f_{n}\right|^{p}\ast\left[\left(1+\left|\bullet\right|\right)^{p\left(K-N\right)}\circ T_{i}^{T}\right],
\]
so that Young's inequality $L^{1}\ast L^{1}\hookrightarrow L^{1}$
yields
\begin{align*}
\left\Vert \left[\nabla g_{n}\right]\!\circ\!T_{i}^{T}\right\Vert _{W_{T_{i}^{-T}\left[-2,2\right]^{d}}\left(L_{v}^{p}\right)}^{p} & \leq\!\left[C_{1}\Omega_{0}^{2K}\Omega_{1}^{2}\sqrt{d}\left(1\!+\!2\sqrt{d}\right)^{\!\!N}\right]^{p}\!\!\left(10\!+\!4R\right)^{d\left(1-p\right)}\!\cdot\!\left|\det T_{i}\right|\cdot\!\left\Vert \left|v\!\cdot\!f_{n}\right|^{p}\right\Vert _{L^{1}}\!\cdot\left\Vert \left(1\!+\!\left|\bullet\right|\right)^{p\left(K-N\right)}\!\circ\!T_{i}^{T}\right\Vert _{L^{1}}\\
& =\!\left[\Omega_{0}^{2K}\Omega_{1}^{2}\sqrt{d}\left(1\!+\!2\sqrt{d}\right)^{\!\!N}\right]^{p}\!\!\cdot C_{1}^{p}\left(10\!+\!4R\right)^{d\left(1-p\right)}\cdot\left\Vert f_{n}\right\Vert _{L_{v}^{p}}^{p}\cdot\left\Vert \left(1+\left|\bullet\right|\right)^{p\left(K-N\right)}\right\Vert _{L^{1}}\\
\left({\scriptstyle \text{eq. }\eqref{eq:StandardDecayLpEstimate}\text{ and }p\left(K-N\right)\leq-\left(d+1\right)}\right) & \leq\!\left[\Omega_{0}^{2K}\Omega_{1}^{2}\sqrt{d}\left(1\!+\!2\sqrt{d}\right)^{\!\!N}\right]^{p}\!\!\cdot C_{1}^{p}\left(10\!+\!4R\right)^{d\left(1-p\right)}\cdot\left\Vert f_{n}\right\Vert _{L_{v}^{p}}^{p}\cdot s_{d}\\
\left({\scriptstyle \text{since }\left|f_{n}\right|\leq\left|f\right|\text{ and }s_{d}\leq4^{d}}\right) & \leq\!\left[\Omega_{0}^{2K}\Omega_{1}^{2}\sqrt{d}\left(1\!+\!2\sqrt{d}\right)^{\!\!N}4^{d/p}C_{1}\left(10\!+\!4R\right)^{d\left(\frac{1}{p}-1\right)}\right]^{p}\cdot\left\Vert f\right\Vert _{L_{v}^{p}}^{p}.
\end{align*}
In view of equation (\ref{eq:SelfImprovingOscillationPointwise}),
we arrive at
\begin{align*}
\left\Vert \osc{\delta\cdot T_{i}^{-T}\left[-1,1\right]^{d}}f_{n}\right\Vert _{W_{T_{i}^{-T}\left[-1,1\right]^{d}}\left(L_{v}^{p}\right)} & \leq2\sqrt{d}\cdot\delta\cdot\left\Vert \left(\nabla g_{n}\right)\circ T_{i}^{T}\right\Vert _{W_{T_{i}^{-T}\left[-2,2\right]^{d}}\left(L_{v}^{p}\right)}\\
& \leq\delta\cdot2d\cdot\Omega_{0}^{2K}\Omega_{1}^{2}\left(1+2\sqrt{d}\right)^{N}4^{d/p}C_{1}\left(10+4R\right)^{d\left(\frac{1}{p}-1\right)}\cdot\left\Vert f\right\Vert _{L_{v}^{p}}.
\end{align*}
The remainder of the proof is now as for $p\in\left[1,\infty\right]$,
noting that
\begin{align*}
& 2d\cdot\Omega_{0}^{2K}\Omega_{1}^{2}\left(1+2\sqrt{d}\right)^{N}4^{d/p}\cdot C_{1}\left(10+4R\right)^{d\left(\frac{1}{p}-1\right)}\\
\left({\scriptstyle \text{eq. }\eqref{eq:SelfImprovingOscillationC1Estimate}}\right) & \leq2d\cdot\Omega_{0}^{2K}\Omega_{1}^{2}\left(1+2\sqrt{d}\right)^{N}4^{d/p}\cdot40\cdot10^{d}\cdot\left(96d\cdot N\right)^{N+1}\cdot10^{d\left(\frac{1}{p}-1\right)}\left(1+R\right)^{1+\frac{d}{p}}\\
\left({\scriptstyle \text{since }d\leq2^{d}\leq2^{d/p}}\right) & \leq\Omega_{0}^{2K}\Omega_{1}^{2}\cdot\left(3\sqrt{d}\right)^{N}\cdot\left(96d\cdot N\right)^{N+1}\cdot80^{1+\frac{d}{p}}\left(1+R\right)^{1+\frac{d}{p}}\\
\left({\scriptstyle \text{since }N+1\geq\frac{d+1}{p}+1\geq\frac{d}{p}+1}\right) & \leq\Omega_{0}^{2K}\Omega_{1}^{2}\cdot\left(23040\cdotd^{3/2}\cdot N\right)^{N+1}\cdot\left(1+R\right)^{1+\frac{d}{p}}\\
\left({\scriptstyle \text{since }N\leq K+\frac{d+1}{p}+1}\right) & \leq\Omega_{0}^{2K}\Omega_{1}^{2}\cdot\left(23040\cdotd^{3/2}\cdot\left(K+1+\frac{d+1}{p}\right)\right)^{K+2+\frac{d+1}{p}}\cdot\left(1+R\right)^{1+\frac{d}{p}}.\qedhere
\end{align*}
\end{proof}
\subsection{Convolution relation for Wiener amalgam spaces}
Finally, we come to the convolution relation for the Wiener Amalgam
spaces. The theorem stated here is a slight variation (and specialization)
of \cite[Theorem 2.3.24]{VoigtlaenderPhDThesis}, which originally
appeared in \cite{RauhutWienerAmalgam}.
\begin{thm}
\label{thm:WienerAmalgamConvolution}Let $Q_{1},Q_{2}\subset\mathbb{R}^{d}$
be bounded, Borel measurable unit neighborhoods and assume that $Q_{1}-Q_{1}$
and $Q_{2}-Q_{1}$ are measurable. Let $p\in\left(0,\infty\right]$
and set $r:=\min\left\{ 1,p\right\} $.
Assume that there is a countable family $\left(x_{i}\right)_{i\in I}$
in $\mathbb{R}^{d}$ satisfying $\mathbb{R}^{d}=\bigcup_{i\in I}\left(x_{i}+Q_{1}\right)$
and
\[
N:=\sup_{x\in\mathbb{R}^{d}}\left|\left\{ i\in I\,\middle|\, x\in x_{i}+Q_{1}\right\} \right|<\infty.
\]
Then we have for every $f\in W_{Q_{1}-Q_{1}}\left(L_{v_{0}}^{r}\right)$
and every $g\in W_{Q_{2}-Q_{1}}\left(L_{v}^{p}\right)$ that
\begin{itemize}
\item $f\in L_{v_{0}}^{1}\left(\mathbb{R}^{d}\right)$ with $\left\Vert f\right\Vert _{L_{v_{0}}^{1}}\lesssim\left\Vert f\right\Vert _{W_{Q_{1}-Q_{1}}\left(L_{v_{0}}^{r}\right)}$,
where the implied constant only depends on $N,Q_{1},r,v_{0}$.
\item $g\in L_{v}^{\infty}\left(\mathbb{R}^{d}\right)$ with $\left\Vert g\right\Vert _{L_{v}^{\infty}}\lesssim\left\Vert g\right\Vert _{W_{Q_{2}-Q_{1}}\left(L_{v}^{p}\right)}$,
where the implied constant only depends on $N,Q_{1},Q_{2},p,v_{0}$.
In particular,
\[
W_{Q_{2}-Q_{1}}\left(L_{v}^{p}\right)\hookrightarrow L_{v}^{\infty}\left(\smash{\mathbb{R}^{d}}\right)\hookrightarrow L_{\left(1+\left|\bullet\right|\right)^{-K}}^{\infty}\left(\smash{\mathbb{R}^{d}}\right)\hookrightarrow\mathcal{S}'\left(\smash{\mathbb{R}^{d}}\right).
\]
\item The convolution $f\ast g:\mathbb{R}^{d}\to\mathbb{C}$ is a well-defined
continuous function.
\item We have
\[
\left\Vert f\ast g\right\Vert _{W_{Q_{2}}\left(L_{v}^{p}\right)}\leq N^{\frac{1}{r}}\cdot\left[\sup_{x\in Q_{1}}v_{0}\left(x\right)\right]\cdot\left[\lambda_{d}\left(Q_{1}\right)\right]^{1-\frac{1}{r}}\cdot\left\Vert f\right\Vert _{W_{Q_{1}-Q_{1}}\left(L_{v_{0}}^{r}\right)}\cdot\left\Vert g\right\Vert _{W_{Q_{2}-Q_{1}}\left(L_{v}^{p}\right)}.\qedhere
\]
\end{itemize}
\end{thm}
\begin{rem}
\label{rem:WeightedWienerAmalgamTemperedDistribution}Since one can
always choose a compact unit neighborhood $Q_{1}$ for which the assumptions
of the theorem are satisfied (choose e.g.\@ $Q_{1}=\left[-\frac{1}{2},\frac{1}{2}\right]^{d}$
and $x_{i}:=i$ for $i\in I:=\mathbb{Z}^{d}$), we see in view of
Lemma \ref{lem:WienerAmalgamNormEquivalence} that
\begin{equation}
W_{Q}\left(L_{v}^{p}\right)\hookrightarrow L_{v}^{\infty}\left(\smash{\mathbb{R}^{d}}\right)\hookrightarrow L_{\left(1+\left|\bullet\right|\right)^{-K}}^{\infty}\left(\smash{\mathbb{R}^{d}}\right)\hookrightarrow\mathcal{S}'\left(\smash{\mathbb{R}^{d}}\right)\label{eq:WeightedWienerAmalgamTemperedDistribution}
\end{equation}
holds for every bounded unit-neighborhood $Q\subset\mathbb{R}^{d}$.
The same also holds with $v$ instead of $v_{0}$, since $v_{0}$
satisfies all properties that $v$ has.
\end{rem}
\begin{proof}
In the following, we will frequently use the discrete weights $v_{i}^{{\rm disc}}:=v\left(x_{i}\right)$
and $\left(v_{0}^{{\rm disc}}\right)_{i}:=v_{0}\left(x_{i}\right)$,
as well as the constant $C_{1}:=\sup_{x\in Q_{1}}v_{0}\left(x\right)\leq\Omega_{1}\cdot\sup_{x\in Q_{1}}\left(1+\left|x\right|\right)^{K}<\infty$,
which is finite since $Q_{1}$ is bounded. These quantities are important,
since we have for $y=x_{i}+q\in x_{i}+Q_{1}$ the estimates
\[
v\left(y\right)=v\left(x_{i}+q\right)\leq v\left(x_{i}\right)v_{0}\left(q\right)\leq C_{1}\cdot v_{i}^{{\rm disc}}\quad\text{ and similarly }\quad v_{0}\left(y\right)\leq C_{1}\cdot\left(v_{0}^{{\rm disc}}\right)_{i}.
\]
Likewise, by symmetry and submultiplicativity of $v_{0}$, we also
have
\[
\left(v_{0}^{{\rm disc}}\right)_{i}=v_{0}\left(x_{i}\right)=v_{0}\left(y-q\right)\leq v_{0}\left(y\right)\cdot v_{0}\left(-q\right)=v_{0}\left(y\right)\cdot v_{0}\left(q\right)\leq C_{1}\cdot v_{0}\left(y\right).
\]
Completely similar, we also get $v_{i}^{{\rm disc}}\leq C_{1}\cdot v\left(y\right)$
for all $y\in x_{i}+Q_{1}$.
\medskip{}
Now, we first show that we can write each $h\in W_{Q_{1}-Q_{1}}\left(L_{v_{0}}^{r}\right)$
as $h=\sum_{i\in I}h_{i}$, where\footnote{In this proof and the next, but not elsewhere in the paper, we write
$\operatorname{supp} f:=\left\{ x\in\mathbb{R}^{d}\,\middle|\, f\left(x\right)\neq0\right\} $,
which is different from the usual meaning $\operatorname{supp} f:=\overline{\left\{ x\in\mathbb{R}^{d}\,\middle|\, f\left(x\right)\neq0\right\} }$.} $\operatorname{supp} h_{i}\subset x_{i}+Q_{1}$ and where
\begin{equation}
\left\Vert \left(\left\Vert h_{i}\right\Vert _{L^{\infty}}\right)_{i\in I}\right\Vert _{\ell_{v_{0}^{{\rm disc}}}^{r}}\leq C_{1}\cdot\frac{N^{1/r}}{\left[\lambda_{d}\left(Q_{1}\right)\right]^{1/r}}\cdot\left\Vert h\right\Vert _{W_{Q_{1}-Q_{1}}\left(L_{v_{0}}^{r}\right)}.\label{eq:WienerAmalgamDecomposition}
\end{equation}
Indeed, since $I$ is countable (and necessarily infinite, since $\mathbb{R}^{d}=\bigcup_{i\in I}\left(x_{i}+Q_{1}\right)$,
with $Q_{1}$ bounded), we can assume $I=\mathbb{N}$. Then, define $h_{i}:=h\cdot{\mathds{1}}_{\left(x_{i}+Q_{1}\right)\setminus\bigcup_{j=1}^{i-1}\left(x_{j}+Q_{1}\right)}$.
Because of $\mathbb{R}^{d}=\bigcup_{i\in I}\left(x_{i}+Q_{1}\right)$,
this easily yields $h=\sum_{i\in I}h_{i}$ and $\operatorname{supp} h_{i}\subset x_{i}+Q_{1}$
is trivial, so that we only need to verify estimate (\ref{eq:WienerAmalgamDecomposition}).
To this end, first note for $x\in x_{i}+Q_{1}$ that $x_{i}\in x-Q_{1}$
and hence $x_{i}+Q_{1}\subset x+Q_{1}-Q_{1}$, which yields
\[
\left\Vert h_{i}\right\Vert _{L^{\infty}}\leq\left\Vert h\cdot{\mathds{1}}_{x_{i}+Q_{1}}\right\Vert _{L^{\infty}}\leq\left\Vert h\cdot{\mathds{1}}_{x+Q_{1}-Q_{1}}\right\Vert _{L^{\infty}}=\left(M_{Q_{1}-Q_{1}}h\right)\left(x\right).
\]
Now, take the $r$-th power of this estimate, multiply both sides
with $v_{0}^{r}\left(x\right)\cdot{\mathds{1}}_{x_{i}+Q_{1}}\left(x\right)$
and sum over $i\in I$ to arrive at
\[
\sum_{i\in I}\left[\left\Vert h_{i}\right\Vert _{L^{\infty}}^{r}\cdot v_{0}^{r}\left(x\right)\cdot{\mathds{1}}_{x_{i}+Q_{1}}\left(x\right)\right]\leq\left[v_{0}\left(x\right)\cdot\left(M_{Q_{1}-Q_{1}}h\right)\left(x\right)\right]^{r}\cdot\sum_{i\in I}{\mathds{1}}_{x_{i}+Q_{1}}\left(x\right)\leq N\cdot\left[v_{0}\left(x\right)\cdot\left(M_{Q_{1}-Q_{1}}h\right)\left(x\right)\right]^{r}.
\]
As observed at the beginning of the proof, we have ${\mathds{1}}_{x_{i}+Q_{1}}\cdot\left(v_{0}^{{\rm disc}}\right)_{i}\leq C_{1}\cdot v_{0}\cdot{\mathds{1}}_{x_{i}+Q_{1}}$.
By combining this with the preceding estimate and integrating, we
get
\begin{align*}
\lambda_{d}\left(Q_{1}\right)\cdot\left\Vert \left(\left\Vert h_{i}\right\Vert _{L^{\infty}}\right)_{i\in I}\right\Vert _{\ell_{v_{0}^{{\rm disc}}}^{r}}^{r} & =\sum_{i\in I}\left\Vert h_{i}\right\Vert _{L^{\infty}}^{r}\cdot\left(v_{0}^{{\rm disc}}\right)_{i}^{r}\cdot\lambda_{d}\left(x_{i}+Q_{1}\right)\\
& =\int_{\mathbb{R}^{d}}\sum_{i\in I}\left(\left\Vert h_{i}\right\Vert _{L^{\infty}}\cdot\left(v_{0}^{{\rm disc}}\right)_{i}\cdot{\mathds{1}}_{x_{i}+Q_{1}}\left(x\right)\right)^{r}\operatorname{d} x\\
& \leq C_{1}^{r}N\cdot\int_{\mathbb{R}^{d}}\left[v_{0}\left(x\right)\cdot\left(M_{Q_{1}-Q_{1}}h\right)\left(x\right)\right]^{r}\operatorname{d} x\\
& =C_{1}^{r}N\cdot\left\Vert h\right\Vert _{W_{Q_{1}-Q_{1}}\left(L_{v_{0}}^{r}\right)}^{r}<\infty.
\end{align*}
Rearranging shows that equation (\ref{eq:WienerAmalgamDecomposition})
is indeed satisfied.
\medskip{}
Now, since $\ell^{r}\left(I\right)\hookrightarrow\ell^{1}\left(I\right)$
and because of $\operatorname{supp} h_{i}\subset x_{i}+Q_{1}$, so that
\[
v_{0}\cdot\left|h_{i}\right|\leq C_{1}\cdot\left(v_{0}^{{\rm disc}}\right)_{i}\cdot\left|h_{i}\right|\leq C_{1}\cdot\left(v_{0}^{{\rm disc}}\right)_{i}\cdot\left\Vert h_{i}\right\Vert _{L^{\infty}}\cdot{\mathds{1}}_{x_{i}+Q_{1}}\quad\text{ almost everywhere},
\]
we get
\begin{align*}
\left\Vert h\right\Vert _{L_{v_{0}}^{1}}\leq\sum_{i\in I}\left\Vert h_{i}\right\Vert _{L_{v_{0}}^{1}} & \leq\left[\sum_{i\in I}\left\Vert h_{i}\right\Vert _{L_{v_{0}}^{1}}^{r}\right]^{1/r}\\
& \leq C_{1}\cdot\left[\sum_{i\in I}\left(v_{0}^{{\rm disc}}\right)_{i}^{r}\cdot\left\Vert h_{i}\right\Vert _{L^{\infty}}^{r}\cdot\left[\lambda\left(x_{i}+Q_{1}\right)\right]^{r}\right]^{1/r}\\
& \leq C_{1}\cdot\lambda_{d}\left(Q_{1}\right)\cdot\left\Vert \left(\left\Vert h_{i}\right\Vert _{L^{\infty}}\right)_{i\in I}\right\Vert _{\ell_{v_{0}^{{\rm disc}}}^{r}}\\
\left({\scriptstyle \text{eq. }\eqref{eq:WienerAmalgamDecomposition}}\right) & \leq C_{1}^{2}\cdot\left[\lambda_{d}\left(Q_{1}\right)\right]^{1-\frac{1}{r}}\cdot N^{1/r}\cdot\left\Vert h\right\Vert _{W_{Q_{1}-Q_{1}}\left(L_{v_{0}}^{r}\right)}<\infty,
\end{align*}
which proves the first part of the theorem.
\medskip{}
Now, we want to prove the second part of the theorem. For $p=\infty$,
we have $\left\Vert g\right\Vert _{L_{v}^{\infty}}=\left\Vert g\right\Vert _{L_{v}^{p}}\leq\left\Vert g\right\Vert _{W_{Q_{2}-Q_{1}}\left(L_{v}^{p}\right)}$
by Lemma \ref{lem:MaximalFunctionDominatesF}, so that we can assume
$p\in\left(0,\infty\right)$.
Next, we define $g_{i}:=g\cdot{\mathds{1}}_{x_{i}+Q_{1}}$ for $i\in I$
and note for $x\in x_{i}+Q_{1}$ as above that $x_{i}+Q_{1}\subset x+Q_{1}-Q_{1}$,
so that
\[
\left\Vert g_{i}\right\Vert _{L^{\infty}}\leq\left\Vert g\cdot{\mathds{1}}_{x+Q_{1}-Q_{1}}\right\Vert _{L^{\infty}}=\left(M_{Q_{1}-Q_{1}}g\right)\left(x\right).
\]
Hence,
\begin{align*}
\frac{1}{C_{1}}\cdot v_{i}^{{\rm disc}}\cdot\left\Vert g_{i}\right\Vert _{L^{\infty}}\cdot\left[\lambda_{d}\left(Q_{1}\right)\right]^{1/p} & =\frac{1}{C_{1}}\cdot\left[\int_{x_{i}+Q_{1}}\left(v_{i}^{{\rm disc}}\cdot\left\Vert g_{i}\right\Vert _{L^{\infty}}\right)^{p}\operatorname{d} x\right]^{1/p}\\
& \leq\left[\int_{\mathbb{R}^{d}}\left(v\left(x\right)\cdot\left\Vert g_{i}\right\Vert _{L^{\infty}}\cdot{\mathds{1}}_{x_{i}+Q_{1}}\left(x\right)\right)^{p}\operatorname{d} x\right]^{1/p}\\
& \leq\left(\int_{\mathbb{R}^{d}}\left[\left(v\cdot M_{Q_{1}-Q_{1}}g\right)\left(x\right)\right]^{p}\operatorname{d} x\right)^{1/p}\\
& =\left\Vert g\right\Vert _{W_{Q_{1}-Q_{1}}\left(L_{v}^{p}\right)}\\
\left({\scriptstyle \text{Lemma }\ref{lem:WienerAmalgamNormEquivalence}}\right) & \leq C_{2}\cdot\left\Vert g\right\Vert _{W_{Q_{2}-Q_{1}}\left(L_{v}^{p}\right)}.
\end{align*}
Here, the last step used that $Q_{1}-Q_{1}$ and $Q_{2}-Q_{1}$ are
both measurable, bounded unit-neighborhoods, so that Lemma \ref{lem:WienerAmalgamNormEquivalence}
yields a constant $C_{2}=C_{2}\left(Q_{1},Q_{2},v_{0},p\right)>0$
satisfying $\left\Vert g\right\Vert _{W_{Q_{1}-Q_{1}}\left(L_{v}^{p}\right)}\leq C_{2}\cdot\left\Vert g\right\Vert _{W_{Q_{2}-Q_{1}}\left(L_{v}^{p}\right)}$.
But there is a null-set $N_{i}\subset x_{i}+Q_{1}$ satisfying $\left|g\left(x\right)\right|=\left|g_{i}\left(x\right)\right|\leq\left\Vert g_{i}\right\Vert _{L^{\infty}}$
for all $x\in\left(x_{i}+Q_{1}\right)\setminus N_{i}$. Hence,
\[
v\left(x\right)\cdot\left|g\left(x\right)\right|\leq C_{1}\cdot v_{i}^{{\rm disc}}\cdot\left\Vert g_{i}\right\Vert _{L^{\infty}}\leq\frac{C_{1}^{2}C_{2}}{\left[\lambda_{d}\left(Q_{1}\right)\right]^{1/p}}\cdot\left\Vert g\right\Vert _{W_{Q_{2}-Q_{1}}\left(L_{v}^{p}\right)}
\]
for all $x\in\left(x_{i}+Q_{1}\right)\setminus N_{i}$. But since
$N:=\bigcup_{i\in I}N_{i}\subset\mathbb{R}^{d}$ is a null-set and
since $\mathbb{R}^{d}=\bigcup_{i\in I}\left(x_{i}+Q_{1}\right)$,
we get $\left\Vert g\right\Vert _{L_{v}^{\infty}}\leq\frac{C_{1}^{2}C_{2}}{\left[\lambda_{d}\left(Q_{1}\right)\right]^{1/p}}\cdot\left\Vert g\right\Vert _{W_{Q_{2}-Q_{1}}\left(L_{v}^{p}\right)}$,
which proves the main part of the second part of the theorem for $p\in\left(0,\infty\right)$.
To establish the embedding $W_{Q_{2}-Q_{1}}\left(L_{v}^{p}\right)\hookrightarrow L_{v}^{\infty}\left(\mathbb{R}^{d}\right)\hookrightarrow L_{\left(1+\left|\bullet\right|\right)^{-K}}^{\infty}\left(\mathbb{R}^{d}\right)\hookrightarrow\mathcal{S}'\left(\mathbb{R}^{d}\right)$,
we first observe that $L_{\left(1+\left|\bullet\right|\right)^{-K}}^{\infty}\left(\mathbb{R}^{d}\right)\hookrightarrow\mathcal{S}'\left(\mathbb{R}^{d}\right)$
is trivial. Furthermore,
\begin{equation}
v\left(0\right)=v\left(x+\left(-x\right)\right)\leq v\left(x\right)\cdot v_{0}\left(-x\right)\leq\Omega_{1}\left(1+\left|-x\right|\right)^{K}\cdot v\left(x\right)\qquad\forall x\in\mathbb{R}^{d},\label{eq:WeightBoundedBelow}
\end{equation}
so that $v\left(x\right)\geq\frac{v\left(0\right)}{\Omega_{1}}\cdot\left(1+\left|x\right|\right)^{-K}$
and hence $W_{Q_{2}-Q_{1}}\left(L_{v}^{p}\right)\hookrightarrow L_{v}^{\infty}\left(\mathbb{R}^{d}\right)\hookrightarrow L_{\left(1+\left|\bullet\right|\right)^{-K}}^{\infty}\left(\mathbb{R}^{d}\right)$,
as desired.
\medskip{}
Now, note for $f\in L_{v_{0}}^{1}\left(\mathbb{R}^{d}\right)$ and
$g\in L_{v}^{\infty}\left(\mathbb{R}^{d}\right)$ because of $v\left(x\right)=v\left(y+\left(x-y\right)\right)\leq v\left(y\right)\cdot v_{0}\left(x-y\right)$
that
\begin{align}
v\left(x\right)\cdot\int_{\mathbb{R}^{d}}\left|f\left(x-y\right)\right|\cdot\left|g\left(y\right)\right|\operatorname{d} y & \leq\int_{\mathbb{R}^{d}}\left|\left(v_{0}\cdot f\right)\left(x-y\right)\right|\cdot\left|\left(v\cdot g\right)\left(y\right)\right|\operatorname{d} y\nonumber \\
& \leq\left\Vert g\right\Vert _{L_{v}^{\infty}}\cdot\left\Vert f\right\Vert _{L_{v_{0}}^{1}}<\infty\qquad\forall x\in\mathbb{R}^{d}.\label{eq:WeightedLInftyConvolution}
\end{align}
Hence, $\left(f\ast g\right)\left(x\right)$ is well-defined for all
$x\in\mathbb{R}^{d}$ and $\left\Vert f\ast g\right\Vert _{L_{\left(1+\left|\bullet\right|\right)^{-K}}^{\infty}}\lesssim\left\Vert f\ast g\right\Vert _{L_{v}^{\infty}}\leq\left\Vert f\right\Vert _{L_{v_{0}}^{1}}\cdot\left\Vert g\right\Vert _{L_{v}^{\infty}}$.
Now, note that the subspace $C\left(\mathbb{R}^{d}\right)\cap L_{\left(1+\left|\bullet\right|\right)^{-K}}^{\infty}\left(\mathbb{R}^{d}\right)$
of continuous functions in $L_{\left(1+\left|\bullet\right|\right)^{-K}}^{\infty}\left(\mathbb{R}^{d}\right)$
is a closed subspace of $L_{\left(1+\left|\bullet\right|\right)^{-K}}^{\infty}\left(\mathbb{R}^{d}\right)$.
Furthermore, $C_{c}\left(\mathbb{R}^{d}\right)\subset L_{v_{0}}^{1}\left(\mathbb{R}^{d}\right)$
is dense and $L_{v}^{\infty}\left(\mathbb{R}^{d}\right)\hookrightarrow L_{\left(1+\left|\bullet\right|\right)^{-K}}^{\infty}\left(\mathbb{R}^{d}\right)\hookrightarrow L_{{\rm loc}}^{\infty}\left(\mathbb{R}^{d}\right)$.
But for $f\in C_{c}\left(\mathbb{R}^{d}\right)$ and $g\in L_{{\rm loc}}^{\infty}\left(\mathbb{R}^{d}\right)$,
it is not hard to see that $f\ast g$ is continuous.
Altogether, the preceding properties show that $f\ast g\in C\left(\mathbb{R}^{d}\right)\cap L_{\left(1+\left|\bullet\right|\right)^{-K}}^{\infty}\left(\mathbb{R}^{d}\right)$
is well-defined and continuous for all $f\in L_{v_{0}}^{1}\left(\mathbb{R}^{d}\right)$
and $g\in L_{v}^{\infty}\left(\mathbb{R}^{d}\right)$. But in the
setting of the theorem, we have $f\in W_{Q_{1}-Q_{1}}\left(L_{v_{0}}^{r}\right)\hookrightarrow L_{v_{0}}^{1}\left(\mathbb{R}^{d}\right)$
and $g\in W_{Q_{2}-Q_{1}}\left(L_{v}^{p}\right)\hookrightarrow L_{v}^{\infty}\left(\mathbb{R}^{d}\right)$,
so that the third part of the theorem is established.
\medskip{}
It remains to prove the last part of the theorem. To this end, recall
from equation (\ref{eq:WienerAmalgamDecomposition}) that we can write
$f=\sum_{i\in I}f_{i}$, where $\operatorname{supp} f_{i}\subset x_{i}+Q_{1}$ and
such that equation (\ref{eq:WienerAmalgamDecomposition}) is fulfilled,
with $f_{i}$ instead of $h_{i}$ and $f$ instead of $h$.
Next, we estimate $M_{Q_{2}}\left(f_{i}\ast g\right)$ for each $i\in I$
as follows: For $x\in\mathbb{R}^{d}$ and $q\in Q_{2}$, we have
\begin{align*}
\left|\left(f_{i}\ast g\right)\left(x+q\right)\right|\leq\left(\left|f_{i}\right|\ast\left|g\right|\right)\left(x+q\right) & =\int_{\mathbb{R}^{d}}\left|f_{i}\left(y\right)\right|\cdot\left|g\left(x+q-y\right)\right|\operatorname{d} y\\
& \leq\left\Vert f_{i}\right\Vert _{L^{\infty}}\cdot\int_{\mathbb{R}^{d}}{\mathds{1}}_{x_{i}+Q_{1}}\left(y\right)\cdot\left|g\left(x+q-y\right)\right|\operatorname{d} y\\
\left({\scriptstyle z=x+q-y}\right) & =\left\Vert f_{i}\right\Vert _{L^{\infty}}\cdot\int_{\mathbb{R}^{d}}{\mathds{1}}_{x_{i}+Q_{1}}\left(x+q-z\right)\cdot\left|g\left(z\right)\right|\operatorname{d} z\\
\left({\scriptstyle x+q-z\in x_{i}+Q_{1}\text{ implies }z\in x-x_{i}+q-Q_{1}\subset x-x_{i}+Q_{2}-Q_{1}}\right) & \leq\left\Vert f_{i}\right\Vert _{L^{\infty}}\cdot\int_{\mathbb{R}^{d}}{\mathds{1}}_{x_{i}+Q_{1}}\left(x+q-z\right)\operatorname{d} z\cdot\left\Vert g\cdot{\mathds{1}}_{x-x_{i}+Q_{2}-Q_{1}}\right\Vert _{L^{\infty}}\\
& =\left\Vert f_{i}\right\Vert _{L^{\infty}}\cdot\lambda_{d}\left(x+q-x_{i}-Q_{1}\right)\cdot\left(M_{Q_{2}-Q_{1}}g\right)\left(x-x_{i}\right)\\
& =\lambda_{d}\left(Q_{1}\right)\cdot\left\Vert f_{i}\right\Vert _{L^{\infty}}\cdot\left(L_{x_{i}}\left[M_{Q_{2}-Q_{1}}g\right]\right)\left(x\right).
\end{align*}
Since this holds for all $q\in Q_{2}$, we get
\[
\left[M_{Q_{2}}\left(\left|f_{i}\right|\ast\left|g\right|\right)\right]\left(x\right)\leq\lambda_{d}\left(Q_{1}\right)\cdot\left\Vert f_{i}\right\Vert _{L^{\infty}}\cdot\left(L_{x_{i}}\left[M_{Q_{2}-Q_{1}}g\right]\right)\left(x\right)\qquad\forall x\in\mathbb{R}^{d}.
\]
In view of Lemma \ref{lem:WeightedLpTranslationNorm} and by solidity
of $L_{v}^{p}\left(\mathbb{R}^{d}\right)$, this implies
\begin{align*}
\left\Vert M_{Q_{2}}\left[\left|f_{i}\right|\ast\left|g\right|\right]\right\Vert _{L_{v}^{p}} & \leq\left\Vert f_{i}\right\Vert _{L^{\infty}}\cdot\lambda_{d}\left(Q_{1}\right)\cdot\left\Vert L_{x_{i}}\left[M_{Q_{2}-Q_{1}}g\right]\right\Vert _{L_{v}^{p}}\\
& \leq\left(v_{0}^{{\rm disc}}\right)_{i}\cdot\left\Vert f_{i}\right\Vert _{L^{\infty}}\cdot\lambda_{d}\left(Q_{1}\right)\cdot\left\Vert g\right\Vert _{W_{Q_{2}-Q_{1}}\left(L_{v}^{p}\right)}.
\end{align*}
Next, it is not hard to see $M_{Q_{2}}\left(\sum_{i\in I}h_{i}\right)\leq\sum_{i\in I}M_{Q_{2}}h_{i}$,
so that we get because of
\[
\left|\left(f\ast g\right)\left(x\right)\right|\leq\left(\left|f\right|\ast\left|g\right|\right)\left(x\right)\leq\sum_{i\in I}\left(\left|f_{i}\right|\ast\left|g\right|\right)\left(x\right)
\]
that
\begin{align*}
\left\Vert M_{Q_{2}}\left(f\ast g\right)\right\Vert _{L_{v}^{p}}^{r}\leq\left\Vert M_{Q_{2}}\left[\left|f\right|\ast\left|g\right|\right]\right\Vert _{L_{v}^{p}}^{r} & \leq\left\Vert \sum_{i\in I}M_{Q_{2}}\left[\left|f_{i}\right|\ast\left|g\right|\right]\right\Vert _{L_{v}^{p}}^{r}\\
\left({\scriptstyle L_{v}^{p}\text{ satisfies the }r-\text{triangle inequality}}\right) & \leq\sum_{i\in I}\left\Vert M_{Q_{2}}\left[\left|f_{i}\right|\ast\left|g\right|\right]\right\Vert _{L_{v}^{p}}^{r}\\
& \leq\left[\lambda_{d}\left(Q_{1}\right)\cdot\left\Vert g\right\Vert _{W_{Q_{2}-Q_{1}}\left(L_{v}^{p}\right)}\right]^{r}\cdot\sum_{i\in I}\left(v_{0}^{{\rm disc}}\right)_{i}^{r}\cdot\left\Vert f_{i}\right\Vert _{L^{\infty}}^{r}\\
\left({\scriptstyle \text{eq. }\eqref{eq:WienerAmalgamDecomposition}}\right) & \leq\left[\lambda_{d}\left(Q_{1}\right)\cdot\left\Vert g\right\Vert _{W_{Q_{2}-Q_{1}}\left(L_{v}^{p}\right)}\right]^{r}\cdot\left(C_{1}\cdot\frac{N^{1/r}}{\left[\lambda_{d}\left(Q_{1}\right)\right]^{1/r}}\cdot\left\Vert f\right\Vert _{W_{Q_{1}-Q_{1}}\left(L_{v_{0}}^{r}\right)}\right)^{r},
\end{align*}
which finally yields
\[
\left\Vert f\ast g\right\Vert _{W_{Q_{2}}\left(L_{v}^{p}\right)}\leq N^{\frac{1}{r}}C_{1}\cdot\left[\lambda_{d}\left(Q_{1}\right)\right]^{1-\frac{1}{r}}\cdot\left\Vert f\right\Vert _{W_{Q_{1}-Q_{1}}\left(L_{v_{0}}^{r}\right)}\cdot\left\Vert g\right\Vert _{W_{Q_{2}-Q_{1}}\left(L_{v}^{p}\right)}<\infty,
\]
as desired.
\end{proof}
With a very slight variant of the above proof, one can also show the
following modification of the theorem. For completeness, we provide
the proof, but with slightly less details than above.
\begin{prop}
\label{prop:AlternativeWienerAmalgamConvolution}Under the assumptions
of Theorem \ref{thm:WienerAmalgamConvolution}, if $p\in\left(0,1\right]$,
then
\[
\left\Vert f\ast g\right\Vert _{W_{Q_{2}}\left(L_{v}^{p}\right)}\leq N^{\frac{1}{p}}\cdot\left[\sup_{x\in Q_{1}}v_{0}\left(x\right)\right]\cdot\left[\lambda_{d}\left(Q_{1}\right)\right]^{1-\frac{1}{p}}\cdot\left\Vert f\right\Vert _{W_{Q_{2}-Q_{1}}\left(L_{v_{0}}^{p}\right)}\cdot\left\Vert g\right\Vert _{W_{Q_{1}-Q_{1}}\left(L_{v}^{p}\right)}.\qedhere
\]
\end{prop}
\begin{proof}
As in the proof of Theorem \ref{thm:WienerAmalgamConvolution}, let
$C_{1}:=\sup_{x\in Q_{1}}v_{0}\left(x\right)$. Also as in that proof,
we can assume $I=\mathbb{N}$, so that we have $g=\sum_{i\in I}g_{i}$ with
$\operatorname{supp} g_{i}\subset x_{i}+Q_{1}$ for $g_{i}:=g\cdot{\mathds{1}}_{\left(x_{i}+Q_{1}\right)\setminus\bigcup_{j=1}^{i-1}\left(x_{j}+Q_{1}\right)}$.
Furthermore, for arbitrary $x\in x_{i}+Q_{1}$, we have $x_{i}+Q_{1}\subset x+Q_{1}-Q_{1}$
and thus
\[
\left\Vert g_{i}\right\Vert _{L^{\infty}}\leq\left\Vert g\cdot{\mathds{1}}_{x_{i}+Q_{1}}\right\Vert _{L^{\infty}}\leq\left\Vert g\cdot{\mathds{1}}_{x+Q_{1}-Q_{1}}\right\Vert _{L^{\infty}}=\left(M_{Q_{1}-Q_{1}}g\right)\left(x\right).
\]
Now, multiply both sides with $v\left(x\right)$, take the $p$th
power, multiply with ${\mathds{1}}_{x_{i}+Q_{1}}\left(x\right)$ and
sum over $i\in I$ to obtain
\begin{align*}
\sum_{i\in I}\left[v\left(x\right)\cdot\left\Vert g_{i}\right\Vert _{L^{\infty}}\right]^{p}{\mathds{1}}_{x_{i}+Q_{1}}\left(x\right) & \leq\sum_{i\in I}\left[v\left(x\right)\cdot\left(M_{Q_{1}-Q_{1}}g\right)\left(x\right)\right]^{p}{\mathds{1}}_{x_{i}+Q_{1}}\left(x\right)\\
& \leq N\cdot\left[v\left(x\right)\cdot\left(M_{Q_{1}-Q_{1}}g\right)\left(x\right)\right]^{p}.
\end{align*}
But for $x\in x_{i}+Q_{1}$, i.e., $x=x_{i}+q$ with $q\in Q_{1}$,
we have
\[
v\left(x_{i}\right)=v\left(x-q\right)\leq v\left(x\right)\cdot v_{0}\left(-q\right)=v\left(x\right)\cdot v_{0}\left(q\right)\leq C_{1}\cdot v\left(x\right),
\]
so that we arrive at
\[
\sum_{i\in I}\left[v\left(x_{i}\right)\cdot\left\Vert g_{i}\right\Vert _{L^{\infty}}\right]^{p}{\mathds{1}}_{x_{i}+Q_{1}}\left(x\right)\leq C_{1}^{p}\cdot N\cdot\left[v\left(x\right)\cdot\left(M_{Q_{1}-Q_{1}}g\right)\left(x\right)\right]^{p}.
\]
Integrating this estimate over $x\in\mathbb{R}^{d}$ finally yields
\begin{equation}
\lambda_{d}\left(Q_{1}\right)\cdot\sum_{i\in I}\left[v\left(x_{i}\right)\cdot\left\Vert g_{i}\right\Vert _{L^{\infty}}\right]^{p}\leq C_{1}^{p}\cdot N\cdot\left\Vert g\right\Vert _{W_{Q_{1}-Q_{1}}\left(L_{v}^{p}\right)}^{p}<\infty.\label{eq:AlternativeWienerAmalgamConvolutionDecompositionStep}
\end{equation}
Now, let $x\in\mathbb{R}^{d}$ and $q\in Q_{2}$ be arbitrary. Since
$\operatorname{supp} g_{i}\subset x_{i}+Q_{1}$, we have
\begin{align*}
\left(\left|f\right|\ast\left|g_{i}\right|\right)\left(x+q\right) & \leq\left\Vert g_{i}\right\Vert _{L^{\infty}}\cdot\int_{\mathbb{R}^{d}}{\mathds{1}}_{x_{i}+Q_{1}}\left(y\right)\cdot\left|f\left(x+q-y\right)\right|\operatorname{d} y\\
\left({\scriptstyle z=x+q-y}\right) & =\left\Vert g_{i}\right\Vert _{L^{\infty}}\cdot\int_{\mathbb{R}^{d}}{\mathds{1}}_{x_{i}+Q_{1}}\left(x+q-z\right)\cdot\left|f\left(z\right)\right|\operatorname{d} z\\
\left({\scriptstyle x+q-z\in x_{i}+Q_{1}\text{ implies }z\in x+q-x_{i}-Q_{1}\subset x-x_{i}+Q_{2}-Q_{1}}\right) & \leq\left\Vert g_{i}\right\Vert _{L^{\infty}}\cdot\int_{\mathbb{R}^{d}}{\mathds{1}}_{x_{i}+Q_{1}}\left(x+q-z\right)\operatorname{d} z\cdot\left\Vert f\cdot{\mathds{1}}_{x-x_{i}+Q_{2}-Q_{1}}\right\Vert _{L^{\infty}}\\
& \leq\left\Vert g_{i}\right\Vert _{L^{\infty}}\cdot\lambda_{d}\left(x+q-x_{i}-Q_{1}\right)\cdot\left\Vert f\cdot{\mathds{1}}_{x-x_{i}+Q_{2}-Q_{1}}\right\Vert _{L^{\infty}}\\
& =\lambda_{d}\left(Q_{1}\right)\cdot\left\Vert g_{i}\right\Vert _{L^{\infty}}\cdot\left(M_{Q_{2}-Q_{1}}f\right)\left(x-x_{i}\right).
\end{align*}
Since this holds for arbitrary $q\in Q_{2}$, we have shown
\[
\left[M_{Q_{2}}\left(\left|f\right|\ast\left|g_{i}\right|\right)\right]\left(x\right)\leq\lambda_{d}\left(Q_{1}\right)\cdot\left\Vert g_{i}\right\Vert _{L^{\infty}}\cdot\left(M_{Q_{2}-Q_{1}}f\right)\left(x-x_{i}\right).
\]
Hence,
\begin{align*}
v\left(x\right)\cdot\left[M_{Q_{2}}\left(\left|f\right|\ast\left|g_{i}\right|\right)\right]\left(x\right) & \leq\lambda_{d}\left(Q_{1}\right)\cdot\left\Vert g_{i}\right\Vert _{L^{\infty}}\cdot v\left(x\right)\cdot\left(M_{Q_{2}-Q_{1}}f\right)\left(x-x_{i}\right)\\
\left({\scriptstyle \text{since }v\left(x\right)=v\left(x-x_{i}+x_{i}\right)\leq v_{0}\left(x-x_{i}\right)v\left(x_{i}\right)}\right) & \leq\lambda_{d}\left(Q_{1}\right)\cdot v\left(x_{i}\right)\left\Vert g_{i}\right\Vert _{L^{\infty}}\cdot\left[v_{0}\cdot M_{Q_{2}-Q_{1}}f\right]\left(x-x_{i}\right).
\end{align*}
Taking the $L^{p}$ norm on both sides, and using the isometric translation
invariance of $L^{p}$, we conclude
\[
\left\Vert \left|f\right|\ast\left|g_{i}\right|\right\Vert _{W_{Q_{2}}\left(L_{v}^{p}\right)}\leq\lambda_{d}\left(Q_{1}\right)\cdot v\left(x_{i}\right)\left\Vert g_{i}\right\Vert _{L^{\infty}}\cdot\left\Vert f\right\Vert _{W_{Q_{2}-Q_{1}}\left(L_{v_{0}}^{p}\right)}.
\]
Now, we finally combine the estimate $\left|\left(f\ast g\right)\left(x\right)\right|\leq\left(\left|f\right|\ast\left|g\right|\right)\left(x\right)\leq\sum_{i\in I}\left(\left|f\right|\ast\left|g_{i}\right|\right)\left(x\right)$
with solidity of $W_{Q_{2}}\left(L_{v}^{p}\right)$ and with the $p$-triangle
inequality for $W_{Q_{2}}\left(L_{v}^{p}\right)$ (which holds sine
$p\in\left(0,1\right]$) to deduce
\begin{align*}
\left\Vert f\ast g\right\Vert _{W_{Q_{2}}\left(L_{v}^{p}\right)}^{p} & \leq\left[\lambda_{d}\left(Q_{1}\right)\right]^{p}\cdot\left\Vert f\right\Vert _{W_{Q_{2}-Q_{1}}\left(L_{v_{0}}^{p}\right)}^{p}\cdot\sum_{i\in I}\left[v\left(x_{i}\right)\left\Vert g_{i}\right\Vert _{L^{\infty}}\right]^{p}\\
\left({\scriptstyle \text{eq. }\eqref{eq:AlternativeWienerAmalgamConvolutionDecompositionStep}}\right) & \leq\left[\lambda_{d}\left(Q_{1}\right)\right]^{p-1}\cdot C_{1}^{p}\cdot N\cdot\left\Vert f\right\Vert _{W_{Q_{2}-Q_{1}}\left(L_{v_{0}}^{p}\right)}^{p}\cdot\left\Vert g\right\Vert _{W_{Q_{1}-Q_{1}}\left(L_{v}^{p}\right)}^{p},
\end{align*}
which easily yields the claim.
\end{proof}
We now formulate an important special case of Theorem \ref{thm:WienerAmalgamConvolution}
as a corollary:
\begin{cor}
\label{cor:WienerAmalgamConvolutionSimplified}Let $i,j\in I$, $p\in\left(0,\infty\right]$,
$f\in W_{T_{j}^{-T}\left[-1,1\right]^{d}}\left(L_{v_{0}}^{r}\right)$
for $r:=\min\left\{ 1,p\right\} $ and $g\in W_{T_{j}^{-T}\left[-1,1\right]^{d}}\left(L_{v}^{p}\right)$.
Then the convolution $f\ast g$ is pointwise defined and continuous
and we have
\[
\left\Vert f\ast g\right\Vert _{W_{T_{j}^{-T}\left[-1,1\right]^{d}}\left(L_{v}^{p}\right)}\leq\Omega_{0}^{3K}\Omega_{1}^{3}C\cdot\left|\det T_{j}\right|^{\frac{1}{r}-1}\cdot\left\Vert f\right\Vert _{W_{T_{j}^{-T}\left[-1,1\right]^{d}}\left(L_{v_{0}}^{r}\right)}\cdot\left\Vert g\right\Vert _{W_{T_{j}^{-T}\left[-1,1\right]^{d}}\left(L_{v}^{p}\right)}
\]
for $C:=d^{-\frac{d}{2r}}\cdot\left(972\cdotd^{5/2}\right)^{K+\frac{d}{r}}$.
\end{cor}
\begin{proof}
We apply Theorem \ref{thm:WienerAmalgamConvolution} with $Q_{1}=Q_{2}=T_{j}^{-T}\left[-1,1\right]^{d}$.
Note that we have
\[
\mathbb{R}=\bigcup_{k\in\mathbb{Z}}\left(2k+\left[-1,1\right]\right)\qquad\text{ and hence }\qquad\mathbb{R}^{d}=\bigcup_{k\in\mathbb{Z}^{d}}\left(2k+\left[-1,1\right]^{d}\right).
\]
Furthermore, if $x\in\left(2k+\left[-1,1\right]^{d}\right)\cap\left(2\ell+\left[-1,1\right]^{d}\right)$,
we get $2k+\mu=x=2\ell+\nu$ for certain $\mu,\nu\in\left[-1,1\right]^{d}$
and thus $\left\Vert k-\ell\right\Vert _{\infty}=\left\Vert \frac{\nu-\mu}{2}\right\Vert _{\infty}\leq1$.
Thus, we see (by fixing $k\in\mathbb{Z}^{d}$ with $x\in2k+\left[-1,1\right]^{d}$)
that $x\in2\ell+\left[-1,1\right]^{d}$ can hold for at most
$3^{d}$ values of $\ell$, namely for $\ell\in\prod_{j=1}^{d}\left\{ k_{j}-1,k_{j},k_{j}+1\right\} $.
Since $T_{j}^{-T}:\mathbb{R}^{d}\to\mathbb{R}^{d}$ is bijective,
we see
\begin{equation}
\mathbb{R}^{d}=\bigcup_{k\in\mathbb{Z}^{d}}\left(2T_{j}^{-T}k+T_{j}^{-T}\left[-1,1\right]^{d}\right)\quad\text{ and }\quad N:=\sup_{x\in\mathbb{R}^{d}}\left|\left\{ \ell\in\mathbb{Z}^{d}\,\middle|\, x\in2T_{j}^{-T}\ell+T_{j}^{-T}\left[-1,1\right]^{d}\right\} \right|\leq3^{d}.\label{eq:LinearImageOfCubePartition}
\end{equation}
Furthermore, equation (\ref{eq:WeightLinearTransformationsConnection})
yields
\begin{align*}
\sup_{x\in Q_{1}}v_{0}\left(x\right)=\sup_{y\in\left[-1,1\right]^{d}}v_{0}\left(T_{j}^{-T}y\right) & \leq\Omega_{1}\cdot\sup_{y\in\left[-1,1\right]^{d}}\left(1+\left|T_{j}^{-T}y\right|\right)^{K}\\
\left({\scriptstyle \text{eq. }\eqref{eq:WeightLinearTransformationsConnection}}\right) & \leq\Omega_{0}^{K}\Omega_{1}\cdot\sup_{y\in\left[-1,1\right]^{d}}\left(1+\left|y\right|\right)^{K}\\
& \leq\left(2\sqrt{d}\right)^{K}\Omega_{0}^{K}\Omega_{1}.
\end{align*}
All in all, Theorem \ref{thm:WienerAmalgamConvolution} shows
\begin{align*}
\left\Vert f\ast g\right\Vert _{W_{T_{j}^{-T}\left[-1,1\right]^{d}}\left(L_{v}^{p}\right)} & =\left\Vert f\ast g\right\Vert _{W_{Q_{2}}\left(L_{v}^{p}\right)}\\
& \leq3^{\frac{d}{r}}\cdot\left(2\sqrt{d}\right)^{K}\Omega_{0}^{K}\Omega_{1}\cdot\left[\lambda_{d}\left(Q_{1}\right)\right]^{1-\frac{1}{r}}\cdot\left\Vert f\right\Vert _{W_{Q_{1}-Q_{1}}\left(L_{v_{0}}^{r}\right)}\cdot\left\Vert g\right\Vert _{W_{Q_{2}-Q_{1}}\left(L_{v}^{p}\right)}\\
\left({\scriptstyle Q_{2}-Q_{1}=Q_{1}-Q_{1}=T_{j}^{-T}\left[-2,2\right]^{d}}\right) & \leq2^{K}3^{\frac{d}{r}}\cdotd^{\frac{K}{2}}\cdot2^{d\left(1-\frac{1}{r}\right)}\cdot\Omega_{0}^{K}\Omega_{1}\!\cdot\!\left|\det T_{j}^{-T}\right|^{1-\frac{1}{r}}\!\cdot\left\Vert f\right\Vert _{W_{T_{j}^{-T}\left[-2,2\right]^{d}}\left(L_{v_{0}}^{r}\right)}\cdot\left\Vert g\right\Vert _{W_{T_{j}^{-T}\left[-2,2\right]^{d}}\left(L_{v}^{p}\right)}\\
\left({\scriptstyle \text{eq. }\eqref{eq:WienerLinearCubeEnlargement}}\right) & \leq\Omega_{0}^{3K}\Omega_{1}^{3}\cdotd^{-\frac{d}{2r}}\cdot\left(972\!\cdot\!d^{\frac{5}{2}}\right)^{\!K+\frac{d}{r}}\cdot\left|\det T_{j}\right|^{\frac{1}{r}-1}\!\cdot\left\Vert f\right\Vert _{W_{T_{j}^{-T}\left[-1,1\right]^{d}}\left(L_{v_{0}}^{r}\right)}\cdot\left\Vert g\right\Vert _{W_{T_{j}^{-T}\left[-1,1\right]^{d}}\left(L_{v}^{p}\right)}.\qedhere
\end{align*}
\end{proof}
Next, we establish a more quantitative—and weighted—version of the
convolution relation for (suitably) bandlimited functions given in
\cite[Corollary 3.14]{DecompositionEmbedding}, which is in turn a
specialized version of \cite[Proposition in §1.5.1]{TriebelTheoryOfFunctionSpaces}.
The following proposition uses the notation $Q_{i}^{n\ast}:=\bigcup_{\ell\in i^{n\ast}}Q_{\ell}$,
where $i^{1\ast}:=i^{\ast}$ and $i^{\left(n+1\right)\ast}:=\bigcup_{\ell\in i^{n\ast}}\ell^{\ast}$.
For the definition of $i^{\ast}$, cf.\@ equation (\ref{eq:IndexClusterDefinition}).
\begin{prop}
\label{prop:BandlimitedConvolution}Let $p\in\left(0,1\right]$ and
$n\in\mathbb{N}_{0}$. If $i\in I$ and
\begin{itemize}
\item if $\psi\in\TestFunctionSpace{\mathbb{R}^{d}}$ with $\operatorname{supp}\psi\subset\overline{Q_{i}^{n\ast}}$
and
\item if $f\in\mathcal{D}'\left(\mathcal{O}\right)$ with $\operatorname{supp} f\subset\overline{Q_{i}^{n\ast}}$
and $\mathcal{F}^{-1}f\in L_{v}^{p}\left(\mathbb{R}^{d}\right)$,
\end{itemize}
then $\mathcal{F}^{-1}\left(\psi f\right)=\left(\mathcal{F}^{-1}\psi\right)\ast\left(\mathcal{F}^{-1}f\right)\in L_{v}^{p}\left(\mathbb{R}^{d}\right)$
with
\[
\left\Vert \mathcal{F}^{-1}\left(\psi f\right)\right\Vert _{L_{v}^{p}}\leq\left[4R_{\mathcal{Q}}\cdot\left(3C_{\mathcal{Q}}\right)^{n}\right]^{d\left(\frac{1}{p}-1\right)}\cdot\left|\det T_{i}\right|^{\frac{1}{p}-1}\cdot\left\Vert \mathcal{F}^{-1}\psi\right\Vert _{L_{v_{0}}^{p}}\cdot\left\Vert \mathcal{F}^{-1}f\right\Vert _{L_{v}^{p}}
\]
and
\[
\left\Vert \mathcal{F}^{-1}\left(\psi f\right)\right\Vert _{W_{T_{i}^{-T}\left[-1,1\right]^{d}}\left(L_{v}^{p}\right)}\leq C\cdot\left|\det T_{i}\right|^{\frac{1}{p}-1}\cdot\left\Vert \mathcal{F}^{-1}\psi\right\Vert _{L_{v_{0}}^{p}}\cdot\left\Vert \mathcal{F}^{-1}f\right\Vert _{L_{v}^{p}},
\]
where $C:=\Omega_{0}^{K}\Omega_{1}\cdot\left(2^{14}\!\cdot\!d^{\frac{3}{2}}\!\cdot\!\left\lceil K\!+\!\frac{d+1}{p}\right\rceil \right)^{\!\!K+\frac{d+1}{p}+2}\left[1\!+\!4R_{\mathcal{Q}}\left(3C_{\mathcal{Q}}\right)^{n}\right]^{d\left(\frac{2}{p}-1\right)}$.
\end{prop}
\begin{rem*}
\begin{itemize}[leftmargin=0.4cm]
\item Again, the only property of $v$ which we use is that $v$ is measurable
and $v\left(x+y\right)\leq v\left(x\right)v_{0}\left(y\right)$ for
all $x,y\in\mathbb{R}^{d}$. Since this also holds for $v_{0}$ instead
of $v$, the claim also holds with $v$ replaced by $v_{0}$ everywhere.
\item Since $\overline{Q_{j}}\subset\mathcal{O}$ is compact for each $j\in I$,
the same is true of $\overline{Q_{i}^{n\ast}}\subset\mathcal{O}$. Hence,
the distribution $f\in\mathcal{D}'\left(\mathcal{O}\right)$ extends to a tempered
distribution $f\in\mathcal{S}'\left(\mathbb{R}^{d}\right)$, so that
$\mathcal{F}^{-1}f$ is well-defined and such that $\psi f\in\mathcal{S}'\left(\mathbb{R}^{d}\right)$
is a tempered distribution with compact support, since $\psi\in\TestFunctionSpace{\mathbb{R}^{d}}$.
Finally, it follows from \cite[Proposition 2.3.22(11)]{GrafakosClassicalFourierAnalysis}
that $\mathcal{F}^{-1}\left(\psi f\right)=\mathcal{F}^{-1}\psi\ast\mathcal{F}^{-1}f$.\qedhere
\end{itemize}
\end{rem*}
\begin{proof}
First, we note that \cite[Lemma 2.7]{DecompositionEmbedding} yields
\[
Q_{j}\subset T_{i}\left[\overline{B_{\left(1+2C_{\mathcal{Q}}\right)^{n}R_{\mathcal{Q}}}}\left(0\right)\right]+b_{i}\qquad\forall j\in i^{n\ast}.
\]
Hence, setting $R:=\left(1+2C_{\mathcal{Q}}\right)^{n}R_{\mathcal{Q}}$, we have
\begin{equation}
\overline{Q_{i}^{n\ast}}\subset T_{i}\overline{B_{R}}\left(0\right)+b_{i}\subset T_{i}\left[-R,R\right]^{d}+b_{i}=:\Omega.\label{eq:BandlimitedConvolutionStarredSetInclusion}
\end{equation}
Note that, once we have proved the first claimed estimate, the second
one is a consequence of Theorem \ref{thm:BandlimitedWienerAmalgamSelfImproving}
(and some simple estimates of the resulting constant, using $C_{\mathcal{Q}}\geq\left\Vert T_{i}^{-1}T_{i}\right\Vert =1$
and $s_{d}\leq2^{2d}$), since we have $\operatorname{supp}\mathcal{F}\left[\mathcal{F}^{-1}\left(\psi f\right)\right]\subset\operatorname{supp}\psi\subset\overline{Q_{i}^{n\ast}}\subset\Omega$.
As seen in the remark following the proposition, we have $\mathcal{F}^{-1}f\in\mathcal{S}'\left(\mathbb{R}^{d}\right)$
with $\operatorname{supp}\mathcal{F}\left[\mathcal{F}^{-1}f\right]\subset\overline{Q_{i}^{n\ast}}\subset\Omega$
and likewise $\mathcal{F}^{-1}\psi\in\mathcal{S}\left(\mathbb{R}^{d}\right)\subset\mathcal{S}'\left(\mathbb{R}^{d}\right)$
with $\operatorname{supp}\mathcal{F}\left[\mathcal{F}^{-1}\psi\right]\subset\overline{Q_{i}^{n\ast}}\subset\Omega$.
In view of Theorems \ref{thm:BandlimitedWienerAmalgamSelfImproving}
and \ref{thm:WienerAmalgamConvolution}, we thus get $\mathcal{F}^{-1}\psi\in W_{T_{i}^{-T}\left[-1,1\right]^{d}}\left(L_{v_{0}}^{p}\right)\hookrightarrow L_{v_{0}}^{1}\left(\mathbb{R}^{d}\right)$
(cf.\@ Lemmas \ref{lem:SchwartzFunctionsAreWiener} and \ref{lem:WienerAmalgamNormEquivalence})
and $\mathcal{F}^{-1}f\in W_{T_{i}^{-T}\left[-1,1\right]^{d}}\left(L_{v}^{p}\right)\hookrightarrow L_{v}^{\infty}\left(\mathbb{R}^{d}\right)$,
so that $\mathcal{F}^{-1}\psi\ast\mathcal{F}^{-1}f$ is pointwise well-defined
by Corollary \ref{cor:WienerAmalgamConvolutionSimplified}.
Now, Lemma \ref{lem:BandlimitedPointwiseApproximation} ensures existence
of a sequence $\left(h_{n}\right)_{n\in\mathbb{N}}$ of Schwartz functions
satisfying $\left|h_{n}\left(x\right)\right|\leq\left|\left(\mathcal{F}^{-1}f\right)\left(x\right)\right|$,
as well as $h_{n}\left(x\right)\xrightarrow[n\to\infty]{}\left(\mathcal{F}^{-1}f\right)\left(x\right)$
for all $x\in\mathbb{R}^{d}$ and finally $\operatorname{supp}\widehat{h_{n}}\subset B_{1/n}\left(\Omega\right)$
for all $n\in\mathbb{N}$. It is not hard to see $\Omega-B_{1/n}\left(\Omega\right)\subset B_{1/n}\left(\Omega-\Omega\right)$.
Furthermore, by compactness of $\Omega-\Omega$—and using continuity
of the Lebesgue measure from above, cf.\@ \cite[Theorem 1.8(d)]{FollandRA}—we
get
\begin{align*}
\lambda_{d}\left(B_{1/n}\left(\Omega-\Omega\right)\right)\xrightarrow[n\to\infty]{}\lambda_{d}\left(\Omega-\Omega\right) & =\lambda_{d}\left(\left[T_{i}\left[-R,R\right]^{d}+b_{i}\right]-\left[T_{i}\left[-R,R\right]^{d}+b_{i}\right]\right)\\
& \leq\lambda_{d}\left(T_{i}\left[-2R,2R\right]^{d}\right)=\left(4R\right)^{d}\cdot\left|\det T_{i}\right|.
\end{align*}
Now, since $h_{n},\mathcal{F}^{-1}\psi\in\mathcal{S}\left(\mathbb{R}^{d}\right)\subset L^{p}\left(\mathbb{R}^{d}\right)$,
Theorem \ref{thm:PointwiseQuasiBanachBandlimitedConvolution} yields
\begin{align*}
v\left(x\right)\cdot\left(\left|\mathcal{F}^{-1}\psi\right|\ast\left|h_{n}\right|\right)\left(x\right) & \leq\left[\lambda_{d}\left(\operatorname{supp}\mathcal{F}\left[\mathcal{F}^{-1}\psi\right]-\operatorname{supp}\widehat{h_{n}}\right)\right]^{\frac{1}{p}-1}\cdot\left[\int_{\mathbb{R}^{d}}\left[v\left(x\right)\right]^{p}\cdot\left|\mathcal{F}^{-1}\psi\left(x-y\right)\right|^{p}\cdot\left|h_{n}\left(y\right)\right|^{p}\operatorname{d} y\right]^{1/p}\\
\left({\scriptstyle \text{since }v\left(x\right)\leq v_{0}\left(x-y\right)v\left(y\right)}\right) & \leq\left[\lambda_{d}\left(B_{1/n}\left(\Omega-\Omega\right)\right)\right]^{\frac{1}{p}-1}\cdot\left[\int_{\mathbb{R}^{d}}\left|\left(v_{0}\cdot\mathcal{F}^{-1}\psi\right)\left(x-y\right)\right|^{p}\cdot\left|\left(v\cdot h_{n}\right)\left(y\right)\right|^{p}\operatorname{d} y\right]^{1/p}\\
\left({\scriptstyle \text{since }\left|h_{n}\right|\leq\left|\mathcal{F}^{-1}f\right|}\right) & \leq\left[\lambda_{d}\left(B_{1/n}\left(\Omega-\Omega\right)\right)\right]^{\frac{1}{p}-1}\cdot\left[\left(\left|v_{0}\cdot\mathcal{F}^{-1}\psi\right|^{p}\ast\left|v\cdot\mathcal{F}^{-1}f\right|^{p}\right)\left(x\right)\right]^{1/p}.
\end{align*}
Taking the limes inferior on both sides, we get
\[
\liminf_{n\to\infty}\left[v\left(x\right)\cdot\left(\left|\mathcal{F}^{-1}\psi\right|\ast\left|h_{n}\right|\right)\left(x\right)\right]\leq\left[\left(4R\right)^{d}\cdot\left|\det T_{i}\right|\right]^{\frac{1}{p}-1}\cdot\left[\left(\left|v_{0}\cdot\mathcal{F}^{-1}\psi\right|^{p}\ast\left|v\cdot\mathcal{F}^{-1}f\right|^{p}\right)\left(x\right)\right]^{1/p}.
\]
Next, since $h_{n}\to\mathcal{F}^{-1}f$ pointwise, and since we saw
above that $\mathcal{F}^{-1}\psi\ast\mathcal{F}^{-1}f$ is pointwise well-defined,
we get by Fatou's Lemma that
\begin{align*}
\left|\left(\mathcal{F}^{-1}\psi\ast\mathcal{F}^{-1}f\right)\left(x\right)\right| & \leq\int_{\mathbb{R}^{d}}\left|\left(\mathcal{F}^{-1}\psi\right)\left(y\right)\right|\cdot\left|\left(\mathcal{F}^{-1}f\right)\left(x-y\right)\right|\operatorname{d} y\\
& =\int_{\mathbb{R}^{d}}\liminf_{n\to\infty}\left[\left|\left(\mathcal{F}^{-1}\psi\right)\left(y\right)\right|\cdot\left|h_{n}\left(x-y\right)\right|\right]\operatorname{d} y\\
& \leq\liminf_{n\to\infty}\int_{\mathbb{R}^{d}}\left|\left(\mathcal{F}^{-1}\psi\right)\left(y\right)\right|\cdot\left|h_{n}\left(x-y\right)\right|\operatorname{d} y=\liminf_{n\to\infty}\left(\left|\mathcal{F}^{-1}\psi\right|\ast\left|h_{n}\right|\right)\left(x\right)
\end{align*}
for all $x\in\mathbb{R}^{d}$. Hence, we finally see
\begin{align*}
\left\Vert \mathcal{F}^{-1}\left(\psi f\right)\right\Vert _{L_{v}^{p}} & =\left\Vert \mathcal{F}^{-1}\psi\ast\mathcal{F}^{-1}f\right\Vert _{L_{v}^{p}}\leq\left\Vert x\mapsto\liminf_{n\to\infty}v\left(x\right)\cdot\left(\left|\mathcal{F}^{-1}\psi\right|\ast\left|h_{n}\right|\right)\left(x\right)\right\Vert _{L^{p}}\\
& \leq\left[\left(4R\right)^{d}\cdot\left|\det T_{i}\right|\right]^{\frac{1}{p}-1}\cdot\left\Vert x\mapsto\left[\left(\left|v_{0}\cdot\mathcal{F}^{-1}\psi\right|^{p}\ast\left|v\cdot\mathcal{F}^{-1}f\right|^{p}\right)\left(x\right)\right]^{1/p}\right\Vert _{L^{p}}\\
& =\left[\left(4R\right)^{d}\cdot\left|\det T_{i}\right|\right]^{\frac{1}{p}-1}\cdot\left\Vert \left|v_{0}\cdot\mathcal{F}^{-1}\psi\right|^{p}\ast\left|v\cdot\mathcal{F}^{-1}f\right|^{p}\right\Vert _{L^{1}}^{1/p}\\
\left({\scriptstyle \text{Young's inequality}}\right) & \leq\left[\left(4R\right)^{d}\cdot\left|\det T_{i}\right|\right]^{\frac{1}{p}-1}\cdot\left\Vert \left|v_{0}\cdot\mathcal{F}^{-1}\psi\right|^{p}\right\Vert _{L^{1}}^{1/p}\cdot\left\Vert \left|v\cdot\mathcal{F}^{-1}f\right|^{p}\right\Vert _{L^{1}}^{1/p}\\
& =\left[\left(4R\right)^{d}\cdot\left|\det T_{i}\right|\right]^{\frac{1}{p}-1}\cdot\left\Vert \mathcal{F}^{-1}\psi\right\Vert _{L_{v_{0}}^{p}}\cdot\left\Vert \mathcal{F}^{-1}f\right\Vert _{L_{v}^{p}}<\infty.
\end{align*}
Since we have $C_{\mathcal{Q}}\geq\left\Vert T_{i}^{-1}T_{i}\right\Vert =1$,
we get $R=\left(1+2C_{\mathcal{Q}}\right)^{n}R_{\mathcal{Q}}\leq\left(3C_{\mathcal{Q}}\right)^{n}R_{\mathcal{Q}}$,
which easily yields the claim.
\end{proof}
As our last result in this section, we show—as a consequence of our
developed convolution relations—that the decomposition space $\DecompSp{\mathcal{Q}}p{\ell_{w}^{q}}v$
is well-defined, even if $v\not\equiv1$.
\begin{prop}
\label{prop:WeightedDecompositionSpaceWellDefined}Let $\Phi=\left(\varphi_{i}\right)_{i\in I}$
and $\Psi=\left(\psi_{i}\right)_{i\in I}$ be two $\mathcal{Q}$-$v_{0}$-BAPUs.
Then we have
\[
\left\Vert \left(\left\Vert \mathcal{F}^{-1}\left(\varphi_{i}f\right)\right\Vert _{L_{v}^{p}}\right)_{i\in I}\right\Vert _{\ell_{w}^{q}}\asymp\left\Vert \left(\left\Vert \mathcal{F}^{-1}\left(\psi_{i}f\right)\right\Vert _{L_{v}^{p}}\right)_{i\in I}\right\Vert _{\ell_{w}^{q}}
\]
uniformly over $f\in\DistributionSpace{\mathcal{O}}$. In particular, the
decomposition space $\DecompSp{\mathcal{Q}}p{\ell_{w}^{q}}v$ is independent
of the choice of the $\mathcal{Q}$-$v_{0}$-BAPU.
\end{prop}
\begin{proof}
By symmetry, it suffices to establish the estimate ``$\lesssim$''.
We can clearly assume $\left\Vert \left(\left\Vert \mathcal{F}^{-1}\left(\psi_{i}f\right)\right\Vert _{L_{v}^{p}}\right)_{i\in I}\right\Vert _{\ell_{w}^{q}}<\infty$.
Since $L_{v}^{p}\left(\mathbb{R}^{d}\right)$ is a quasi-normed space
and since we have the uniform estimate $\left|i^{\ast}\right|\leq N_{\mathcal{Q}}$
for all $i\in I$, we have
\[
d_{i}:=\left\Vert \mathcal{F}^{-1}\left(\psi_{i}^{\ast}f\right)\right\Vert _{L_{v}^{p}}\leq C\cdot\sum_{\ell\in i^{\ast}}\left\Vert \mathcal{F}^{-1}\left(\psi_{\ell}f\right)\right\Vert _{L_{v}^{p}}=C\cdot\left(\Gamma_{\mathcal{Q}}e\right)_{i},
\]
for a suitable constant $C=C\left(p,N_{\mathcal{Q}}\right)$, where $e=\left(e_{i}\right)_{i\in I}$
is defined by $e_{i}:=\left\Vert \mathcal{F}^{-1}\left(\psi_{i}f\right)\right\Vert _{L_{v}^{p}}$
and where $\Gamma_{\mathcal{Q}}$ is the $\mathcal{Q}$-clustering map, as defined
in Section \ref{subsec:DecompSpaceDefinitionStandingAssumptions},
equation (\ref{eq:QClusteringMapDefinition}).
Now, as seen in Section \ref{subsec:DecompSpaceDefinitionStandingAssumptions},
we have $\psi_{i}^{\ast}\equiv1$ on $Q_{i}$ and thus $\varphi_{i}=\psi_{i}^{\ast}\varphi_{i}$
for all $i\in I$. Hence,
\[
c_{i}:=\left\Vert \mathcal{F}^{-1}\left(\varphi_{i}f\right)\right\Vert _{L_{v}^{p}}=\left\Vert \mathcal{F}^{-1}\left(\varphi_{i}\psi_{i}^{\ast}f\right)\right\Vert _{L_{v}^{p}}=\left\Vert \left[\mathcal{F}^{-1}\varphi_{i}\right]\ast\mathcal{F}^{-1}\left(\psi_{i}^{\ast}f\right)\right\Vert _{L_{v}^{p}}.
\]
In case of $p\in\left[1,\infty\right]$, we can now use the weighted
Young inequality (equation (\ref{eq:WeightedYoungInequality})) to
derive
\[
c_{i}\leq\left\Vert \mathcal{F}^{-1}\varphi_{i}\right\Vert _{L_{v_{0}}^{1}}\cdot\left\Vert \mathcal{F}^{-1}\left(\psi_{i}^{\ast}f\right)\right\Vert _{L_{v}^{p}}\leq C\cdot C_{\mathcal{Q},\Phi,v_{0},p}\cdot\left(\Gamma_{\mathcal{Q}}e\right)_{i}.
\]
Otherwise, if $p\in\left(0,1\right)$, we use Proposition \ref{prop:BandlimitedConvolution}
(with $n=1$, since $\operatorname{supp}\varphi_{i}\subset\overline{Q_{i}^{\ast}}$
and $\operatorname{supp}\psi_{i}^{\ast}\subset\overline{Q_{i}^{\ast}}$) to derive
\begin{align*}
c_{i}=\left\Vert \mathcal{F}^{-1}\left(\varphi_{i}\psi_{i}^{\ast}f\right)\right\Vert _{L_{v}^{p}} & \leq\left[12R_{\mathcal{Q}}C_{\mathcal{Q}}\right]^{d\left(\frac{1}{p}-1\right)}\cdot\left|\det T_{i}\right|^{\frac{1}{p}-1}\cdot\left\Vert \mathcal{F}^{-1}\varphi_{i}\right\Vert _{L_{v_{0}}^{p}}\cdot\left\Vert \mathcal{F}^{-1}\left[\psi_{i}^{\ast}f\right]\right\Vert _{L_{v}^{p}}\\
& \leq C\cdot\left[12R_{\mathcal{Q}}C_{\mathcal{Q}}\right]^{d\left(\frac{1}{p}-1\right)}C_{\mathcal{Q},\Phi,v_{0},p}\cdot\left(\Gamma_{\mathcal{Q}}e\right)_{i}.
\end{align*}
In summary, there is for arbitrary $p\in\left(0,\infty\right]$ a
constant $C'=C'\left(\mathcal{Q},p,\Phi,v_{0}\right)>0$ satisfying $c_{i}\leq C'\cdot\left(\Gamma_{\mathcal{Q}}e\right)_{i}<\infty$
for all $i\in I$. By solidity of $\ell_{w}^{q}\left(I\right)$ and
by boundedness of $\Gamma_{\mathcal{Q}}$, this implies
\[
\left\Vert \left(\left\Vert \mathcal{F}^{-1}\left(\varphi_{i}f\right)\right\Vert _{L_{v}^{p}}\right)_{i\in I}\right\Vert _{\ell_{w}^{q}}\leq C'\cdot\left\Vert \Gamma_{\mathcal{Q}}e\right\Vert _{\ell_{w}^{q}}\leq C'\cdot\vertiii{\smash{\Gamma_{\mathcal{Q}}}}\cdot\left\Vert e\right\Vert _{\ell_{w}^{q}}=C'\cdot\vertiii{\smash{\Gamma_{\mathcal{Q}}}}\cdot\left\Vert \left(\left\Vert \mathcal{F}^{-1}\left(\psi_{i}f\right)\right\Vert _{L_{v}^{p}}\right)_{i\in I}\right\Vert _{\ell_{w}^{q}}.\qedhere
\]
\end{proof}
\section{Semi-discrete Banach Frames}
\label{sec:SemiDiscreteBanachFrames}
\begin{assumption}
\label{assu:MainAssumptions}In the remainder of the paper, we will
use the following assumptions and notations:
\begin{enumerate}
\item We are given a family $\Gamma=\left(\gamma_{i}\right)_{i\in I}$ of
functions $\gamma_{i}:\mathbb{R}^{d}\to\mathbb{C}$ with the following
additional properties:
\begin{enumerate}
\item We have $\gamma_{i}\in L_{\left(1+\left|\bullet\right|\right)^{K}}^{1}\left(\mathbb{R}^{d}\right)\hookrightarrow L_{v_{0}}^{1}\left(\mathbb{R}^{d}\right)\hookrightarrow L^{1}\left(\mathbb{R}^{d}\right)$
for all $i\in I$.
\item We have $\widehat{\gamma_{i}}\in C^{\infty}\left(\mathbb{R}^{d}\right)$ for
all $i\in I$, where all partial derivatives of $\widehat{\gamma_{i}}$
are polynomially bounded, i.e.,
\[
\qquad\qquad\qquad\left|\left(\partial^{\alpha}\widehat{\gamma_{i}}\right)\left(\xi\right)\right|\leq C_{\alpha,i}\cdot\left(1+\left|\xi\right|\right)^{N_{\alpha,i}}\qquad\forall\,\xi\in\mathbb{R}^{d}\,\forall\,\alpha\in\mathbb{N}_{0}^{d}\,\forall\,i\in I,\text{ for suitable }C_{\alpha,i}>0\text{ and }N_{\alpha,i}\in\mathbb{N}_{0}.
\]
\end{enumerate}
\item For $i\in I$, we define
\begin{equation}
\begin{split}\gamma^{\left(i\right)} & :=\mathcal{F}^{-1}\left(\widehat{\gamma_{i}}\circ S_{i}^{-1}\right)\\
& =\mathcal{F}^{-1}\left(L_{b_{i}}\left(\widehat{\gamma_{i}}\circ T_{i}^{-1}\right)\right)\\
& =M_{b_{i}}\left[\mathcal{F}^{-1}\left(\widehat{\gamma_{i}}\circ T_{i}^{-1}\right)\right]\\
& =\left|\det T_{i}\right|\cdot M_{b_{i}}\left[\gamma_{i}\circ T_{i}^{T}\right],
\end{split}
\label{eq:NonCompactFilterDefinition}
\end{equation}
as well as the $L^{2}$-normalized version
\begin{equation}
\gamma^{\left[i\right]}:=\left|\det T_{i}\right|^{1/2}\cdot M_{b_{i}}\left[\gamma_{i}\circ T_{i}^{T}\right]=\left|\det T_{i}\right|^{-1/2}\cdot\gamma^{\left(i\right)}.\label{eq:L2NormalizedFilterDefinition}
\end{equation}
\item For $i\in I$, we set
\[
V_{i}:=\begin{cases}
L_{v}^{p}\left(\mathbb{R}^{d}\right), & \text{if }p\in\left[1,\infty\right],\\
W_{T_{i}^{-T}\left[-1,1\right]^{d}}\left(L_{v}^{p}\right), & \text{if }p\in\left(0,1\right).
\end{cases}
\]
Furthermore, we will occasionally make use of the space
\[
V:=\ell_{w}^{q}\left(\left[V_{i}\right]_{i\in I}\right):=\left\{ \left(f_{i}\right)_{i\in I}\,\middle|\,\left(\forall i\in I:\,f_{i}\in V_{i}\right)\text{ and }\left(\left\Vert f_{i}\right\Vert _{V_{i}}\right)_{i\in I}\in\ell_{w}^{q}\left(I\right)\right\} ,
\]
equipped with the quasi-norm $\left\Vert \left(f_{i}\right)_{i\in I}\right\Vert _{\ell_{w}^{q}\left(\left[V_{i}\right]_{i\in I}\right)}:=\left\Vert \left(\left\Vert f_{i}\right\Vert _{V_{i}}\right)_{i\in I}\right\Vert _{\ell_{w}^{q}}$.
\item Finally, we set
\[
r:=\max\left\{ q,\frac{q}{p}\right\} =\begin{cases}
q, & \text{if }p\in\left[1,\infty\right],\\
\frac{q}{p}, & \text{if }p\in\left(0,1\right)
\end{cases}
\]
and
\[
A_{j,i}:=\begin{cases}
\left\Vert \mathcal{F}^{-1}\left(\varphi_{i}\cdot\widehat{\gamma^{\left(j\right)}}\right)\right\Vert _{L_{v_{0}}^{1}}, & \text{if }p\in\left[1,\infty\right],\\
\left(1+\left\Vert T_{j}^{-1}T_{i}\right\Vert \right)^{d}\cdot\left|\det T_{i}\right|^{1-p}\cdot\left\Vert \mathcal{F}^{-1}\left(\varphi_{i}\cdot\widehat{\gamma^{\left(j\right)}}\right)\right\Vert _{L_{v_{0}}^{p}}^{p}, & \text{if }p\in\left(0,1\right)
\end{cases}
\]
for $i,j\in I$ and we assume that $\overrightarrow{A}$ is a bounded
operator $\overrightarrow{A}:\ell_{w^{\min\left\{ 1,p\right\} }}^{r}\left(I\right)\to\ell_{w^{\min\left\{ 1,p\right\} }}^{r}\left(I\right)$,
where
\[
\overrightarrow{A}\left(c_{i}\right)_{i\in I}:=\left(\sum_{i\in I}A_{j,i}\,c_{i}\right)_{j\in I}.\qedhere
\]
\end{enumerate}
\end{assumption}
\begin{rem}
\label{rem:MainAssumptionsRemark}
\begin{enumerate}
\item The most common case will be to have $\gamma_{i}=\gamma$ for all
$i\in I$, for a fixed prototype $\gamma$. The added flexibility
of allowing $\gamma_{i}$ to vary with $i\in I$ is only rarely needed.
In the cases where it is, we usually have $\gamma_{i}=\gamma_{n_{i}}$
with a given (finite) list of prototypes $\gamma_{1},\dots,\gamma_{N}$.
\item The assumption that $\partial^{\alpha}\widehat{\gamma_{i}}$ is polynomially
bounded for all $\alpha\in\mathbb{N}_{0}^{d}$ is satisfied if $\gamma_{i}\in L^{1}\left(\mathbb{R}^{d}\right)$
has compact support, say $\operatorname{supp}\gamma_{i}\subset\left[-R,R\right]^{d}$
with $R\geq1$, since then differentiation under the integral yields
\begin{align*}
\left|\partial^{\alpha}\widehat{\gamma_{i}}\left(\xi\right)\right| & =\left|\int_{\mathbb{R}^{d}}\gamma_{i}\left(x\right)\cdot\partial_{\xi}^{\alpha}e^{-2\pi i\left\langle x,\xi\right\rangle }\operatorname{d} x\right|\\
& \leq\int_{\left[-R,R\right]^{d}}\left|\gamma_{i}\left(x\right)\right|\cdot\left(2\pi\left|x\right|\right)^{\left|\alpha\right|}\operatorname{d} x\leq\left(2\pi R\right)^{\left|\alpha\right|}\cdot\left\Vert \gamma_{i}\right\Vert _{L^{1}}<\infty
\end{align*}
for all $\xi\in\mathbb{R}^{d}$ and arbitrary $\alpha\in\mathbb{N}_{0}^{d}$.
\item \label{enu:StructuredFamilyFourierTransformPolynomiallyBounded}Under
the above assumptions, the chain rule implies
\begin{align*}
\left|\left(\partial^{\alpha}\widehat{\gamma^{\left(i\right)}}\right)\left(x\right)\right| & =\left|\left(\partial^{\alpha}\left[\widehat{\gamma_{i}}\circ T_{i}^{-1}\right]\right)\left(x-b_{i}\right)\right|\\
& \leq C^{\left(\alpha\right)}\cdot\left\Vert T_{i}^{-1}\right\Vert ^{\left|\alpha\right|}\cdot\max_{\left|\beta\right|\leq\left|\alpha\right|}\left|\left(\partial^{\beta}\widehat{\gamma_{i}}\right)\left(T_{i}^{-1}\left(x-b_{i}\right)\right)\right|\\
& \leq\left(\max_{\left|\beta\right|\leq\left|\alpha\right|}C_{\beta,i}\right)\cdot C^{\left(\alpha\right)}\cdot\left\Vert T_{i}^{-1}\right\Vert ^{\left|\alpha\right|}\cdot\max_{\left|\beta\right|\leq\left|\alpha\right|}\left(1+\left|T_{i}^{-1}\left(x-b_{i}\right)\right|\right)^{N_{\beta,i}}\\
\left({\scriptstyle \text{with }N_{\alpha,i}':=\max_{\left|\beta\right|\leq\left|\alpha\right|}N_{\beta,i}}\right) & \leq C_{\alpha,i}'\cdot\left(1+\left|T_{i}^{-1}\left(x-b_{i}\right)\right|\right)^{N_{\alpha,i}'}\\
\left({\scriptstyle \text{for suitable }C_{\alpha,i}''>0}\right) & \leq C_{\alpha,i}''\cdot\left(1+\left|x\right|\right)^{N_{\alpha,i}'},
\end{align*}
where the last step used
\begin{align*}
1+\left|T_{i}^{-1}\left(x-b_{i}\right)\right| & \leq1+\left\Vert T_{i}^{-1}\right\Vert \left|x-b_{i}\right|\\
& \leq1+\left\Vert T_{i}^{-1}\right\Vert \left|b_{i}\right|+\left\Vert T_{i}^{-1}\right\Vert \left|x\right|\\
& \leq\left(1+\left\Vert T_{i}^{-1}\right\Vert \left|b_{i}\right|+\left\Vert T_{i}^{-1}\right\Vert \right)\cdot\left(1+\left|x\right|\right).
\end{align*}
Hence, all partial derivatives of each $\widehat{\gamma^{\left(i\right)}}$
are polynomially bounded.
\item Using $\gamma_{i}\in L_{\left(1+\left|\bullet\right|\right)^{K}}^{1}\left(\mathbb{R}^{d}\right)$,
we also get
\begin{align*}
\left\Vert \gamma^{\left(i\right)}\right\Vert _{L_{\left(1+\left|\bullet\right|\right)^{K}}^{1}} & =\left|\det T_{i}\right|\cdot\left\Vert \left(1+\left|\bullet\right|\right)^{K}\cdot\left(\gamma_{i}\circ T_{i}^{T}\right)\right\Vert _{L^{1}}\\
& =\left\Vert \left(1+\left|T_{i}^{-T}\bullet\right|\right)^{K}\cdot\gamma_{i}\right\Vert _{L^{1}}\\
\left({\scriptstyle \text{eq. }\eqref{eq:WeightLinearTransformationsConnection}}\right) & \leq\Omega_{0}^{K}\cdot\left\Vert \left(1+\left|\bullet\right|\right)^{K}\cdot\gamma_{i}\right\Vert _{L^{1}}=\Omega_{0}^{K}\cdot\left\Vert \gamma_{i}\right\Vert _{L_{\left(1+\left|\bullet\right|\right)^{K}}^{1}}<\infty
\end{align*}
and thus $\gamma^{\left(i\right)}\in L_{\left(1+\left|\bullet\right|\right)^{K}}^{1}\left(\mathbb{R}^{d}\right)\hookrightarrow L_{v_{0}}^{1}\left(\mathbb{R}^{d}\right)\hookrightarrow L^{1}\left(\mathbb{R}^{d}\right)$,
where the last embedding uses $v_{0}\geq1$.
\item Point (\ref{enu:StructuredFamilyFourierTransformPolynomiallyBounded})
from above shows $\widehat{\gamma^{\left(i\right)}}\cdot\widehat{f}\in\mathcal{S}'\left(\mathbb{R}^{d}\right)$
for arbitrary $f\in\mathcal{S}'\left(\mathbb{R}^{d}\right)$, so that $\gamma^{\left(i\right)}\ast f:=\mathcal{F}^{-1}\left(\widehat{\gamma^{\left(i\right)}}\cdot\widehat{f}\right)$
is a well-defined tempered distribution. Of course, the same also
holds for $\gamma^{\left[i\right]}\ast f:=\mathcal{F}^{-1}\left(\widehat{\gamma^{\left[i\right]}}\cdot\widehat{f}\right)$.
\item Since $\mathbb{R}^{d}$ is $\sigma$-compact, it follows from \cite[Lemma 2.3.7]{VoigtlaenderPhDThesis}
(see also \cite[Theorem 2.6]{RauhutWienerAmalgam}) that for $p\in\left(0,1\right)$,
each of the spaces $V_{i}=W_{T_{i}^{-T}\left[-1,1\right]^{d}}\left(L_{v}^{p}\right)$
is complete (and thus a Quasi-Banach space) for each $i\in I$. Furthermore,
\cite[Lemma 2.3.4]{VoigtlaenderPhDThesis} and \cite[Exercise 1.1.5(c)]{GrafakosClassicalFourierAnalysis}
show $\left\Vert f+g\right\Vert _{V_{i}}\leq2^{\frac{1}{p}-1}\cdot\left[\left\Vert f\right\Vert _{V_{i}}+\left\Vert g\right\Vert _{V_{i}}\right]$
for all $f,g\in V_{i}$. In case of $p\in\left[1,\infty\right]$,
it is clear that $V_{i}=L_{v}^{p}\left(\mathbb{R}^{d}\right)$ is
a Banach space.\qedhere
\end{enumerate}
\end{rem}
Note that in the preceding remark, we \emph{defined} $\gamma^{\left(i\right)}\ast f:=\mathcal{F}^{-1}\left(\widehat{\gamma^{\left(i\right)}}\cdot\widehat{f}\right)$.
We needed to do so, since the usual results about convolution in $\mathcal{S}'\left(\mathbb{R}^{d}\right)$
only define $f\ast\varphi$ for $\varphi\in\mathcal{S}'\left(\mathbb{R}^{d}\right)$
if $f\in\mathcal{S}\left(\mathbb{R}^{d}\right)$ (cf.\@ \cite[Proposition (8.44)]{FollandRA})
or if $f$ is a distribution with compact support. (cf.\@ \cite[Chapter 8, Exercise 35]{FollandRA}).
But note that if we not only know $\varphi\in\mathcal{S}'\left(\mathbb{R}^{d}\right)$,
but the stronger property $\varphi\in\left(L^{1}+L^{\infty}\right)\left(\mathbb{R}^{d}\right)$
and if $f\in L^{1}\left(\mathbb{R}^{d}\right)$, then $f\ast\varphi\in\left(L^{1}+L^{\infty}\right)\left(\mathbb{R}^{d}\right)$
is already defined. Our next result shows that in this (and in a slightly
more general) case, the new definition is consistent.
\begin{lem}
\label{lem:SpecialConvolutionConsistent}Assume $\varphi\in L_{v}^{1}\left(\mathbb{R}^{d}\right)+L_{v}^{\infty}\left(\mathbb{R}^{d}\right)$
and assume that $f\in L_{v_{0}}^{1}\left(\mathbb{R}^{d}\right)$
is such that $\widehat{f}\in C^{\infty}\left(\mathbb{R}^{d}\right)$
and such that all partial derivatives of $\widehat{f}$ have at most
polynomial growth. Then $f\ast\varphi\in L_{v}^{1}\left(\mathbb{R}^{d}\right)+L_{v}^{\infty}\left(\mathbb{R}^{d}\right)\hookrightarrow\mathcal{S}'\left(\mathbb{R}^{d}\right)$
and
\[
f\ast\varphi=\mathcal{F}^{-1}\left[\widehat{f}\cdot\widehat{\varphi}\right].
\]
The assumption on $\varphi$ is in particular fulfilled if $\varphi\in V_{i}$
for some $i\in I$. More precisely,
\begin{equation}
V_{i}\hookrightarrow L_{v}^{1}\left(\smash{\mathbb{R}^{d}}\right)+L_{v}^{\infty}\left(\smash{\mathbb{R}^{d}}\right)\hookrightarrow\mathcal{S}'\left(\smash{\mathbb{R}^{d}}\right).\qedhere\label{eq:ViAreTemperedDistributions}
\end{equation}
\end{lem}
\begin{rem}
\label{rem:WeightedSpacesYieldTemperedDistributions}We saw in the
proof of Theorem \ref{thm:WienerAmalgamConvolution} (cf.\@ equation
(\ref{eq:WeightBoundedBelow})) that $v\left(x\right)\gtrsim\left(1+\left|x\right|\right)^{-K}$.
Furthermore, $v_{0}\geq1$, and thus $L_{v}^{p}\left(\mathbb{R}^{d}\right)\hookrightarrow\mathcal{S}'\left(\mathbb{R}^{d}\right)$
and $L_{v_{0}}^{p}\left(\mathbb{R}^{d}\right)\hookrightarrow L^{p}\left(\mathbb{R}^{d}\right)\hookrightarrow\mathcal{S}'\left(\mathbb{R}^{d}\right)$
for all $p\in\left[1,\infty\right]$. Hence, the expressions $\widehat{f}$
and $\widehat{\varphi}$ above are well-defined tempered distributions.
Since $\widehat{f}\in C^{\infty}\left(\mathbb{R}^{d}\right)$ with
all derivatives of $\widehat{f}$ of at most polynomial growth, we
also see $\widehat{f}\cdot\widehat{\varphi}\in\mathcal{S}'\left(\mathbb{R}^{d}\right)$
and thus also $\mathcal{F}^{-1}\left[\widehat{f}\cdot\widehat{\varphi}\right]\in\mathcal{S}'\left(\mathbb{R}^{d}\right)$.
\end{rem}
\begin{proof}
From the weighted Young inequality (equation (\ref{eq:WeightedYoungInequality})),
we know $L_{v_{0}}^{1}\left(\mathbb{R}^{d}\right)\ast L_{v}^{\infty}\left(\mathbb{R}^{d}\right)\hookrightarrow L_{v}^{\infty}\left(\mathbb{R}^{d}\right)$.
Likewise, the same inequality also yields $\left\Vert f\ast g\right\Vert _{L_{v}^{1}}\leq\left\Vert f\right\Vert _{L_{v_{0}}^{1}}\cdot\left\Vert g\right\Vert _{L_{v}^{1}}<\infty$
and thus in particular $\left(\left|f\right|\ast\left|g\right|\right)\left(x\right)<\infty$
for almost all $x\in\mathbb{R}^{d}$ for $f\in L_{v_{0}}^{1}\left(\mathbb{R}^{d}\right)$
and $g\in L_{v}^{1}\left(\mathbb{R}^{d}\right)$. Hence, together
with Remark \ref{rem:WeightedSpacesYieldTemperedDistributions}, we
see indeed that $f\ast\varphi\in L_{v}^{1}\left(\mathbb{R}^{d}\right)+L_{v}^{\infty}\left(\mathbb{R}^{d}\right)\hookrightarrow\mathcal{S}'\left(\mathbb{R}^{d}\right)$.
\medskip{}
Now, let $\psi\in\mathcal{S}\left(\mathbb{R}^{d}\right)$ be arbitrary.
We have
\begin{align*}
\left\langle \mathcal{F}^{-1}\left[\widehat{f}\cdot\widehat{\varphi}\right],\psi\right\rangle _{\mathcal{S}',\mathcal{S}} & =\left\langle \widehat{f}\cdot\widehat{\varphi},\,\mathcal{F}^{-1}\psi\right\rangle _{\mathcal{S}',\mathcal{S}}\\
& =\left\langle \widehat{\varphi},\,\widehat{f}\cdot\mathcal{F}^{-1}\psi\right\rangle _{\mathcal{S}',\mathcal{S}}\\
& =\left\langle \varphi,\,\mathcal{F}\left[\widehat{f}\cdot\mathcal{F}^{-1}\psi\right]\right\rangle _{\mathcal{S}',\mathcal{S}}\\
& =\left\langle \varphi,\,\left(\mathcal{F}^{-1}\left[\widehat{f}\cdot\mathcal{F}^{-1}\psi\right]\right)\left(-\bullet\right)\right\rangle _{\mathcal{S}',\mathcal{S}}.
\end{align*}
Recall that $v_{0}\geq1$, so that $f\in L_{v_{0}}^{1}\left(\mathbb{R}^{d}\right)\hookrightarrow L^{1}\left(\mathbb{R}^{d}\right)$.
Hence, $h:=f\ast\tilde{\psi}\in L^{1}\left(\mathbb{R}^{d}\right)\ast L^{1}\left(\mathbb{R}^{d}\right)\subset L^{1}\left(\mathbb{R}^{d}\right)$,
where $\tilde{\psi}\left(x\right):=\psi\left(-x\right)$. Thus, the
convolution theorem yields $\widehat{h}=\widehat{f}\cdot\widehat{\tilde{\psi}}=\widehat{f}\cdot\tilde{\widehat{\psi}}=\widehat{f}\cdot\mathcal{F}^{-1}\psi\in\mathcal{S}\left(\mathbb{R}^{d}\right)\subset L^{1}\left(\mathbb{R}^{d}\right)$,
since $\mathcal{F}^{-1}\psi\in\mathcal{S}\left(\mathbb{R}^{d}\right)$
and since all partial derivatives of $\widehat{f}$ are polynomially
bounded. By the Fourier inversion theorem, this implies $f\ast\tilde{\psi}=h=\mathcal{F}^{-1}\widehat{h}=\mathcal{F}^{-1}\left[\widehat{f}\cdot\mathcal{F}^{-1}\psi\right]$,
so that we can continue the calculation from above as follows:
\begin{align*}
\left\langle \mathcal{F}^{-1}\left[\widehat{f}\cdot\widehat{\varphi}\right],\psi\right\rangle _{\mathcal{S}',\mathcal{S}} & =\left\langle \varphi,\,\left(f\ast\tilde{\psi}\right)\left(-\bullet\right)\right\rangle _{\mathcal{S}',\mathcal{S}}\\
& =\int_{\mathbb{R}^{d}}\varphi\left(x\right)\cdot\int_{\mathbb{R}^{d}}f\left(-x-y\right)\cdot\tilde{\psi}\left(y\right)\operatorname{d} y\operatorname{d} x\\
\left({\scriptstyle z=-y}\right) & =\int_{\mathbb{R}^{d}}\varphi\left(x\right)\cdot\int_{\mathbb{R}^{d}}f\left(z-x\right)\cdot\psi\left(z\right)\operatorname{d} z\operatorname{d} x\\
\left({\scriptstyle \text{Fubini}}\right) & =\int_{\mathbb{R}^{d}}\int_{\mathbb{R}^{d}}\varphi\left(x\right)\cdot f\left(z-x\right)\operatorname{d} x\cdot\psi\left(z\right)\operatorname{d} z\\
& =\left\langle f\ast\varphi,\,\psi\right\rangle _{\mathcal{S}',\mathcal{S}},
\end{align*}
which proves the claim.
All that remains is to justify the application of Fubini's theorem.
To this end, we can assume $\varphi\in L_{v}^{1}\left(\mathbb{R}^{d}\right)$
or $\varphi\in L_{v}^{\infty}\left(\mathbb{R}^{d}\right)$, since
then the general case follows by linearity. But for $\varphi\in L_{v}^{\infty}\left(\mathbb{R}^{d}\right)$,
we have because of
\begin{align*}
v\left(0\right) & =v\left(x+\left(-x\right)\right)\\
& \leq v\left(x\right)\cdot v_{0}\left(-x\right)\\
& =v\left(x\right)\cdot v_{0}\left(z-x+\left(-z\right)\right)\\
& \leq v\left(x\right)\cdot v_{0}\left(z-x\right)\cdot v_{0}\left(-z\right)\\
& \leq\Omega_{1}\cdot v\left(x\right)\cdot v_{0}\left(z-x\right)\cdot\left(1+\left|z\right|\right)^{K}
\end{align*}
that
\begin{align*}
\int_{\mathbb{R}^{d}}\int_{\mathbb{R}^{d}}\left|\varphi\left(x\right)\cdot f\left(z-x\right)\cdot\psi\left(z\right)\right|\operatorname{d} x\operatorname{d} z & \leq\frac{\Omega_{1}}{v\left(0\right)}\cdot\int_{\mathbb{R}^{d}}\int_{\mathbb{R}^{d}}\left|\left(v\cdot\varphi\right)\left(x\right)\right|\cdot\left|\left(v_{0}\cdot f\right)\left(z-x\right)\right|\cdot\left(1+\left|z\right|\right)^{K}\left|\psi\left(z\right)\right|\operatorname{d} x\operatorname{d} z\\
& \leq\frac{\Omega_{1}}{v\left(0\right)}\cdot\left\Vert \varphi\right\Vert _{L_{v}^{\infty}}\cdot\int_{\mathbb{R}^{d}}\left(1+\left|z\right|\right)^{K}\left|\psi\left(z\right)\right|\int_{\mathbb{R}^{d}}\left|\left(v_{0}\cdot f\right)\left(z-x\right)\right|\operatorname{d} x\operatorname{d} z\\
\left({\scriptstyle y=z-x}\right) & =\frac{\Omega_{1}}{v\left(0\right)}\cdot\left\Vert \varphi\right\Vert _{L_{v}^{\infty}}\left\Vert f\right\Vert _{L_{v_{0}}^{1}}\int_{\mathbb{R}^{d}}\left(1+\left|z\right|\right)^{K}\left|\psi\left(z\right)\right|\operatorname{d} z<\infty.
\end{align*}
Furthermore, in case of $\varphi\in L_{v}^{1}\left(\mathbb{R}^{d}\right)$,
we get with a similar estimate that
\begin{align*}
\int_{\mathbb{R}^{d}}\int_{\mathbb{R}^{d}}\left|\varphi\left(x\right)\cdot f\left(z-x\right)\cdot\psi\left(z\right)\right|\operatorname{d} x\operatorname{d} z & \leq\frac{\Omega_{1}}{v\left(0\right)}\cdot\int_{\mathbb{R}^{d}}\int_{\mathbb{R}^{d}}\left|\left(v\cdot\varphi\right)\left(x\right)\right|\cdot\left|\left(v_{0}\cdot f\right)\left(z-x\right)\right|\cdot\left(1+\left|z\right|\right)^{K}\left|\psi\left(z\right)\right|\operatorname{d} x\operatorname{d} z\\
& \leq\frac{\Omega_{1}}{v\left(0\right)}\cdot\left[\sup_{z\in\mathbb{R}^{d}}\left(1+\left|z\right|\right)^{K}\left|\psi\left(z\right)\right|\right]\cdot\int_{\mathbb{R}^{d}}\left|\left(v\cdot\varphi\right)\left(x\right)\right|\int_{\mathbb{R}^{d}}\left|\left(v_{0}\cdot f\right)\left(z-x\right)\right|\operatorname{d} z\operatorname{d} x\\
& =\frac{\Omega_{1}}{v\left(0\right)}\cdot\left[\sup_{z\in\mathbb{R}^{d}}\left(1+\left|z\right|\right)^{K}\left|\psi\left(z\right)\right|\right]\cdot\left\Vert \varphi\right\Vert _{L_{v}^{1}}\cdot\left\Vert f\right\Vert _{L_{v_{0}}^{1}}<\infty.
\end{align*}
For the proof of equation (\ref{eq:ViAreTemperedDistributions}),
note for $p\in\left[1,\infty\right]$ that $V_{i}=L_{v}^{p}\left(\mathbb{R}^{d}\right)\hookrightarrow L_{v}^{1}\left(\mathbb{R}^{d}\right)+L_{v}^{\infty}\left(\mathbb{R}^{d}\right)$,
because of the well-known (cf.\@ \cite[Proposition (6.9)]{FollandRA})
embedding $L^{p}\left(\mathbb{R}^{d}\right)\hookrightarrow L^{1}\left(\mathbb{R}^{d}\right)+L^{\infty}\left(\mathbb{R}^{d}\right)$.
But in case of $p\in\left(0,1\right)$, Theorem \ref{thm:WienerAmalgamConvolution}
and the ensuing remark yield $V_{i}=W_{T_{i}^{-T}\left[-1,1\right]^{d}}\left(L_{v}^{p}\right)\hookrightarrow L_{v}^{\infty}\left(\mathbb{R}^{d}\right)\hookrightarrow L_{v}^{1}\left(\mathbb{R}^{d}\right)+L_{v}^{\infty}\left(\mathbb{R}^{d}\right)$,
as desired.
\end{proof}
One of our aims in this section is to show under the conditions of
Assumption \ref{assu:MainAssumptions} (and certain additional assumptions,
cf.\@ Assumption \ref{assu:GammaCoversOrbit}) on $\mathcal{Q},\Gamma=\left(\gamma_{i}\right)_{i\in I}$
and $p,q,v,w$ that we have
\begin{equation}
\left\Vert f\right\Vert _{\DecompSp{\mathcal{Q}}p{\ell_{w}^{q}}v}\asymp\left\Vert \left(\left\Vert \gamma^{\left(i\right)}\ast f\right\Vert _{V_{i}}\right)_{i\in I}\right\Vert _{\ell_{w}^{q}}=\left\Vert \left(\gamma^{\left(i\right)}\ast f\right)_{i\in I}\right\Vert _{V}\label{eq:DesiredCharacterization}
\end{equation}
for all $f\in\DecompSp{\mathcal{Q}}p{\ell_{w}^{q}}v$. Note though, that
it is not a priori clear how the convolution $\gamma^{\left(i\right)}\ast f$
can be interpreted, since we have $f\in\DecompSp{\mathcal{Q}}p{\ell_{w}^{q}}v\leq\mathcal{F}^{-1}\left(\DistributionSpace{\mathcal{O}}\right)\nsubseteq\mathcal{S}'\left(\mathbb{R}^{d}\right)$.
The purpose of the following result is to clarify how $\gamma^{\left(i\right)}\ast f$
can be interpreted and to establish the estimate ``$\gtrsim$''
in equation (\ref{eq:DesiredCharacterization}). We remark that the
theorem uses the notion of \textbf{normal convergence} of a series.
In our context, we say that a series $\sum_{i\in I}g_{i}$ converges
normally in $V_{j}$ if
\[
\begin{cases}
\sum_{i\in I}\left\Vert g_{i}\right\Vert _{V_{j}}<\infty, & \text{if }p\in\left[1,\infty\right],\\
\sum_{i\in I}\left\Vert g_{i}\right\Vert _{V_{j}}^{p}<\infty, & \text{if }p\in\left(0,1\right).
\end{cases}
\]
\begin{thm}
\label{thm:ConvolvingDecompositionSpaceWithGammaJ}If Assumption \ref{assu:MainAssumptions}
is fulfilled, the following hold:
For every $f\in\DecompSp{\mathcal{Q}}p{\ell_{w}^{q}}v$ and $j\in I$, the
distribution $\widehat{\gamma^{\left(j\right)}}\cdot\widehat{f}\in\DistributionSpace{\mathcal{O}}$
extends to a tempered distribution $f_{j}\in\mathcal{F}\left(V_{j}\right)\subset\mathcal{S}'\left(\mathbb{R}^{d}\right)$,
given by
\[
f_{j}:\mathcal{S}\left(\mathbb{R}^{d}\right)\to\mathbb{C},\phi\mapsto\sum_{i\in I}\left\langle \widehat{\gamma^{\left(j\right)}}\cdot\widehat{f},\,\varphi_{i}\phi\right\rangle _{\DistributionSpace{\mathcal{O}},\TestFunctionSpace{\mathcal{O}}}.
\]
Furthermore, the inverse Fourier transform $\mathcal{F}^{-1}f_{j}\in V_{j}$
is given by
\begin{equation}
\left(\mathcal{F}^{-1}f_{j}\right)\left(x\right)=\sum_{i\in I}\left[\mathcal{F}^{-1}\left(\varphi_{i}\widehat{\gamma^{\left(j\right)}}\widehat{f}\right)\right]\left(x\right),\label{eq:SpecialConvolutionInterpretation}
\end{equation}
where the series converges normally in $V_{j}$ and absolutely almost
everywhere.
Finally, the linear map
\[
{\rm Ana}_{\Gamma}:\DecompSp{\mathcal{Q}}p{\ell_{w}^{q}}v\to\ell_{w}^{q}\left(\left[V_{i}\right]_{i\in I}\right),f\mapsto\left(\mathcal{F}^{-1}f_{j}\right)_{j\in I}
\]
is well-defined and bounded, with
\[
\vertiii{{\rm Ana}_{\Gamma}}\leq C\cdot\vertiii{\smash{\Gamma_{\mathcal{Q}}}}\cdot\vertiii{\smash{\overrightarrow{A}}}_{\ell_{w^{\min\left\{ 1,p\right\} }}^{r}\left(I\right)\to\ell_{w^{\min\left\{ 1,p\right\} }}^{r}\left(I\right)}^{\max\left\{ 1,\frac{1}{p}\right\} },
\]
where
\[
C:=\begin{cases}
1, & \text{if }p\in\left[1,\infty\right],\\
N_{\mathcal{Q}}^{\frac{1}{p}-1}\cdot\left(12288\cdotd^{3/2}\cdot\left\lceil K+\frac{d+1}{p}\right\rceil \right)^{\left\lceil K+\frac{d+1}{p}\right\rceil +1}\cdot\left(1+R_{\mathcal{Q}}\right)^{d/p}\left(12R_{\mathcal{Q}}C_{\mathcal{Q}}\right)^{d\left(\frac{1}{p}-1\right)}\cdot\Omega_{0}^{K}\Omega_{1}, & \text{if }p\in\left(0,1\right)
\end{cases}
\]
and where $\Gamma_{\mathcal{Q}}:\ell_{w}^{q}\left(I\right)\to\ell_{w}^{q}\left(I\right)$
is the \textbf{$\mathcal{Q}$-clustering map}, i.e., $\Gamma_{\mathcal{Q}}\left(c_{i}\right)_{i\in I}=\left(c_{i}^{\ast}\right)_{i\in I}$,
with $c_{i}^{\ast}:=\sum_{\ell\in i^{\ast}}c_{\ell}$.
\end{thm}
\begin{rem*}
In the following, we will use the notation $\gamma^{\left(j\right)}\ast f$
instead of $\mathcal{F}^{-1}f_{j}$, so that we have
\[
{\rm Ana}_{\Gamma}\,f=\left(\gamma^{\left(j\right)}\ast f\right)_{j\in I}.
\]
Likewise, because of $\gamma^{\left[j\right]}=\left|\det T_{j}\right|^{-1/2}\cdot\gamma^{\left(j\right)}$,
it is natural to define
\[
\gamma^{\left[j\right]}\ast f:=\left|\det T_{j}\right|^{-1/2}\cdot\gamma^{\left(j\right)}\ast f.
\]
This new notation $\gamma^{\left(j\right)}\ast f$ (and thus also
$\gamma^{\left[j\right]}\ast f$) is consistent in the following sense:
If we have $\mathcal{O}=\mathbb{R}^{d}$ and $\DecompSp{\mathcal{Q}}p{\ell_{w}^{q}}v\hookrightarrow\mathcal{S}'\left(\mathbb{R}^{d}\right)$
(i.e., if every $f\in\DecompSp{\mathcal{Q}}p{\ell_{w}^{q}}v\subset Z'\left(\mathbb{R}^{d}\right)=\left[\mathcal{F}\left(\TestFunctionSpace{\mathbb{R}^{d}}\right)\right]'$
extends to a tempered distribution $f_{\mathcal{S}}$), then our new
definition of the convolution $\gamma^{\left(j\right)}\ast f:=\mathcal{F}^{-1}f_{j}$
agrees with the usual interpretation of $\gamma^{\left(j\right)}\ast f:=\mathcal{F}^{-1}\left(\widehat{\gamma^{\left(j\right)}}\cdot\widehat{f}\right)$
for $f\in\mathcal{S}'\left(\mathbb{R}^{d}\right)\supset\DecompSp{\mathcal{Q}}p{\ell_{w}^{q}}v$,
as we will see now.
First note $\widehat{f_{\mathcal{S}}}|_{\TestFunctionSpace{\mathbb{R}^{d}}}=\widehat{f}$,
where $\widehat{f}=f\circ\mathcal{F}\in\DistributionSpace{\mathbb{R}^{d}}$.
Thus, we have for arbitrary $\phi\in\mathcal{F}\left(\TestFunctionSpace{\mathbb{R}^{d}}\right)$
that
\begin{align*}
\left\langle \mathcal{F}^{-1}f_{j},\,\phi\right\rangle _{\mathcal{S}',\mathcal{S}} & =\left\langle f_{j},\,\mathcal{F}^{-1}\phi\right\rangle _{\mathcal{S}',\mathcal{S}}\\
& =\sum_{i\in I}\left\langle \widehat{\gamma^{\left(j\right)}}\widehat{f},\,\varphi_{i}\cdot\mathcal{F}^{-1}\phi\right\rangle _{\DistributionSpace{\mathbb{R}^{d}},\TestFunctionSpace{\mathbb{R}^{d}}}\\
\left({\scriptstyle \text{since }\mathcal{F}^{-1}\phi\in\TestFunctionSpace{\smash{\mathbb{R}^{d}}}\text{ and }\sum_{i\in I}\varphi_{i}\equiv1\text{ with a locally finite sum}}\right) & =\left\langle \widehat{\gamma^{\left(j\right)}}\cdot\widehat{f},\,\mathcal{F}^{-1}\phi\right\rangle _{\DistributionSpace{\mathbb{R}^{d}},\TestFunctionSpace{\mathbb{R}^{d}}}\\
& =\left\langle \widehat{f},\,\widehat{\gamma^{\left(j\right)}}\cdot\mathcal{F}^{-1}\phi\right\rangle _{\DistributionSpace{\mathbb{R}^{d}},\TestFunctionSpace{\mathbb{R}^{d}}}\\
\left({\scriptstyle \widehat{\gamma^{\left(j\right)}}\cdot\mathcal{F}^{-1}\phi\in\TestFunctionSpace{\smash{\mathbb{R}^{d}}}\subset\mathcal{S}\left(\smash{\mathbb{R}^{d}}\right),\text{ since }\widehat{\gamma^{\left(j\right)}}\in C^{\infty}\left(\smash{\mathbb{R}^{d}}\right)\text{ and }\mathcal{F}^{-1}\phi\in\TestFunctionSpace{\smash{\mathbb{R}^{d}}}}\right) & =\left\langle \widehat{f_{\mathcal{S}}},\,\widehat{\gamma^{\left(j\right)}}\cdot\mathcal{F}^{-1}\phi\right\rangle _{\mathcal{S}',\mathcal{S}}\\
\left({\scriptstyle \widehat{\gamma^{\left(j\right)}}\cdot\widehat{f_{\mathcal{S}}}\in\mathcal{S}'\left(\smash{\mathbb{R}^{d}}\right),\text{ since }\widehat{f_{\mathcal{S}}}\in\mathcal{S}'\left(\smash{\mathbb{R}^{d}}\right)\text{ and all derivatives of }\widehat{\gamma^{\left(j\right)}}\text{ pol. bounded}}\right) & =\left\langle \widehat{\gamma^{\left(j\right)}}\cdot\widehat{f_{\mathcal{S}}},\,\mathcal{F}^{-1}\phi\right\rangle _{\mathcal{S}',\mathcal{S}}\\
& =\left\langle \mathcal{F}^{-1}\left[\widehat{\gamma^{\left(j\right)}}\cdot\widehat{f_{\mathcal{S}}}\right],\,\phi\right\rangle _{\mathcal{S}',\mathcal{S}}=\left\langle \gamma^{\left(j\right)}\ast f_{\mathcal{S}},\,\phi\right\rangle _{\mathcal{S}',\mathcal{S}}.
\end{align*}
Here, the last step uses the \emph{definition} $\gamma^{\left(j\right)}\ast f_{\mathcal{S}}:=\mathcal{F}^{-1}\left[\widehat{\gamma^{\left(j\right)}}\cdot\widehat{f_{\mathcal{S}}}\right]$
from above. This definition coincides with the usual one if $\gamma_{i}\in\mathcal{S}\left(\mathbb{R}^{d}\right)$
for all $i\in I$ (so that $\gamma^{\left(j\right)}\in\mathcal{S}\left(\mathbb{R}^{d}\right)$)
or (by Lemma \ref{lem:SpecialConvolutionConsistent} and since $\gamma^{\left(j\right)}\in L_{v_{0}}^{1}\left(\mathbb{R}^{d}\right)$
as seen in Remark \ref{rem:MainAssumptionsRemark}) if $f_{\mathcal{S}}\in\left(L_{v}^{1}+L_{v}^{\infty}\right)\left(\mathbb{R}^{d}\right)$,
which is satisfied in many cases.
Now, since $\mathcal{F}\left(\TestFunctionSpace{\mathbb{R}^{d}}\right)$
is dense in $\mathcal{S}\left(\mathbb{R}^{d}\right)$ (cf.\@ \cite[Proposition 9.9]{FollandRA})
and since we have $\mathcal{F}^{-1}f_{j}\in V_{j}\subset\mathcal{S}'\left(\mathbb{R}^{d}\right)$
and $\gamma^{\left(j\right)}\ast f_{\mathcal{S}}=\mathcal{F}^{-1}\left(\widehat{\gamma^{\left(j\right)}}\cdot\widehat{f_{\mathcal{S}}}\right)\in\mathcal{S}'\left(\mathbb{R}^{d}\right)$,
we conclude $\gamma^{\left(j\right)}\ast f_{\mathcal{S}}=\mathcal{F}^{-1}f_{j}$,
as claimed.
\end{rem*}
\begin{proof}[Proof of Theorem \ref{thm:ConvolvingDecompositionSpaceWithGammaJ}]
Let $f\in\DecompSp{\mathcal{Q}}p{\ell_{w}^{q}}v$ be arbitrary and let
$c_{i}:=\left\Vert \mathcal{F}^{-1}\left(\varphi_{i}^{\ast}\cdot\widehat{f}\right)\right\Vert _{L_{v}^{p}}$
for $i\in I$. Using the (quasi)-triangle inequality for $L^{p}\left(\mathbb{R}^{d}\right)$
and the uniform estimate $\left|i^{\ast}\right|\leq N_{\mathcal{Q}}$, we
obtain a constant $C_{1}=C_{1}\left(p,\mathcal{Q}\right)>0$ satisfying
\[
c_{i}=\left\Vert \mathcal{F}^{-1}\!\left(\varphi_{i}^{\ast}\cdot\widehat{f}\right)\right\Vert _{L_{v}^{p}}\leq C_{1}\cdot\sum_{\ell\in i^{\ast}}\left\Vert \mathcal{F}^{-1}\!\left(\varphi_{\ell}\cdot\widehat{f}\right)\right\Vert _{L_{v}^{p}}=C_{1}\cdot\left(\Gamma_{\mathcal{Q}}d\right)_{i}\quad\text{for}\quad d=\left(d_{i}\right)_{i\in I},\text{ with }d_{i}:=\left\Vert \mathcal{F}^{-1}\!\left(\varphi_{i}\cdot\widehat{f}\right)\right\Vert _{L_{v}^{p}}.
\]
In fact, as shown in \cite[Exercise 1.1.5(c)]{GrafakosClassicalFourierAnalysis},
we can choose
\[
C_{1}=\begin{cases}
1, & \text{if }p\in\left[1,\infty\right],\\
N_{\mathcal{Q}}^{\frac{1}{p}-1}, & \text{if }p\in\left(0,1\right).
\end{cases}
\]
Since $d\in\ell_{w}^{q}\left(I\right)$ with $\left\Vert d\right\Vert _{\ell_{w}^{q}}=\left\Vert f\right\Vert _{\DecompSp{\mathcal{Q}}p{\ell_{w}^{q}}v}$,
we get $c\in\ell_{w}^{q}\left(I\right)$ as well, and $\left\Vert c\right\Vert _{\ell_{w}^{q}}\leq C_{1}\cdot\vertiii{\Gamma_{\mathcal{Q}}}\cdot\left\Vert f\right\Vert _{\DecompSp{\mathcal{Q}}p{\ell_{w}^{q}}v}$.
Now, we distinguish the two cases $p\in\left[1,\infty\right]$ and
$p\in\left(0,1\right)$.
\medskip{}
In case of $p\in\left[1,\infty\right]$, we have $V_{j}=L_{v}^{p}\left(\mathbb{R}^{d}\right)$.
Here, the weighted Young inequality (equation (\ref{eq:WeightedYoungInequality}))
yields
\begin{align*}
\left\Vert \mathcal{F}^{-1}\left(\widehat{\gamma^{\left(j\right)}}\cdot\varphi_{i}\cdot\widehat{f}\right)\right\Vert _{L_{v}^{p}} & =\left\Vert \mathcal{F}^{-1}\left(\widehat{\gamma^{\left(j\right)}}\cdot\varphi_{i}\cdot\varphi_{i}^{\ast}\cdot\widehat{f}\right)\right\Vert _{L_{v}^{p}}\\
& =\left\Vert \mathcal{F}^{-1}\left(\widehat{\gamma^{\left(j\right)}}\cdot\varphi_{i}\right)\ast\mathcal{F}^{-1}\left(\varphi_{i}^{\ast}\cdot\widehat{f}\right)\right\Vert _{L_{v}^{p}}\\
& \leq\left\Vert \mathcal{F}^{-1}\left(\widehat{\gamma^{\left(j\right)}}\cdot\varphi_{i}\right)\right\Vert _{L_{v_{0}}^{1}}\cdot\left\Vert \mathcal{F}^{-1}\left(\varphi_{i}^{\ast}\cdot\widehat{f}\right)\right\Vert _{L_{v}^{p}}\\
& =A_{j,i}\cdot c_{i},
\end{align*}
with $A_{j,i}$ as in Assumption \ref{assu:MainAssumptions}. Hence,
we get
\begin{equation}
\sum_{i\in I}\left\Vert \mathcal{F}^{-1}\left(\widehat{\gamma^{\left(j\right)}}\cdot\varphi_{i}\cdot\widehat{f}\right)\right\Vert _{L_{v}^{p}}\leq\sum_{i\in I}\left[A_{j,i}\cdot c_{i}\right]=\left(\overrightarrow{A}\cdot c\right)_{j}<\infty,\label{eq:GammaAnalysisBanachCaseMainEstimate}
\end{equation}
since we have $c\in\ell_{w}^{q}\left(I\right)$ and since Assumption
\ref{assu:MainAssumptions} includes (for $p\in\left[1,\infty\right]$)
the assumption that $\overrightarrow{A}:\ell_{w}^{q}\left(I\right)\to\ell_{w}^{q}\left(I\right)$
is well-defined and bounded. This implies that the function
\[
F_{j}:=\sum_{i\in I}\mathcal{F}^{-1}\left(\widehat{\gamma^{\left(j\right)}}\cdot\varphi_{i}\cdot\widehat{f}\right)\in L_{v}^{p}\left(\smash{\mathbb{R}^{d}}\right)=V_{j}
\]
is well-defined, with normal convergence in $V_{j}$ and with absolute
convergence a.e.\@ of the defining series and such that $\left\Vert F_{j}\right\Vert _{L_{v}^{p}}\leq\left(\overrightarrow{A}\cdot c\right)_{j}$
for all $j\in I$.
\medskip{}
Next, in case of $p\in\left(0,1\right)$, define $e_{i}:=c_{i}^{p}$
for $i\in I$ and note $e=\left(e_{i}\right)_{i\in I}\in\ell_{w^{p}}^{q/p}\left(I\right)=\ell_{w^{\min\left\{ 1,p\right\} }}^{r}\left(I\right)$,
with
\[
\left\Vert e\right\Vert _{\ell_{w^{\min\left\{ 1,p\right\} }}^{r}\left(I\right)}=\left\Vert \left(w_{i}^{p}\cdot c_{i}^{p}\right)_{i\in I}\right\Vert _{\ell^{q/p}}=\left\Vert \left(w_{i}\cdot c_{i}\right)_{i\in I}\right\Vert _{\ell^{q}}^{p}=\left\Vert c\right\Vert _{\ell_{w}^{q}\left(I\right)}^{p}.
\]
Next, we note
\begin{align*}
\operatorname{supp}\left(\widehat{\gamma^{\left(j\right)}}\cdot\varphi_{i}\cdot\widehat{f}\right) & \subset\operatorname{supp}\varphi_{i}\subset\overline{Q_{i}}\subset T_{i}\overline{B_{R_{\mathcal{Q}}}\left(0\right)}+b_{i}\\
& \subset T_{j}\left[T_{j}^{-1}T_{i}\overline{B_{R_{\mathcal{Q}}}\left(0\right)}\right]+b_{i}\\
& \subset T_{j}\left[\left\Vert T_{j}^{-1}T_{i}\right\Vert \overline{B_{R_{\mathcal{Q}}}}\left(0\right)\right]+b_{i}\\
& \subset T_{j}\left[-\left\Vert T_{j}^{-1}T_{i}\right\Vert R_{\mathcal{Q}},\,\left\Vert T_{j}^{-1}T_{i}\right\Vert R_{\mathcal{Q}}\right]^{d}+b_{i},
\end{align*}
so that Theorem \ref{thm:BandlimitedWienerAmalgamSelfImproving} yields
for $C_{2}:=2^{4\left(1+\frac{d}{p}\right)}s_{d}^{\frac{1}{p}}\left(192\cdotd^{\frac{3}{2}}\cdot\left\lceil K\!+\!\frac{d+1}{p}\right\rceil \right)^{\left\lceil K+\frac{d+1}{p}\right\rceil +1}\cdot\Omega_{0}^{K}\Omega_{1}$
and $C_{3}:=C_{2}\cdot\left(1+R_{\mathcal{Q}}\right)^{d/p}$ that
\begin{align*}
\left\Vert \mathcal{F}^{-1}\left(\widehat{\gamma^{\left(j\right)}}\cdot\varphi_{i}\cdot\widehat{f}\right)\right\Vert _{V_{j}} & =\left\Vert \mathcal{F}^{-1}\left(\widehat{\gamma^{\left(j\right)}}\cdot\varphi_{i}\cdot\widehat{f}\right)\right\Vert _{W_{T_{j}^{-T}\left[-1,1\right]^{d}}\left(L_{v}^{p}\right)}\\
& \leq C_{2}\left(1+\left\Vert T_{j}^{-1}T_{i}\right\Vert R_{\mathcal{Q}}\right)^{d/p}\cdot\left\Vert \mathcal{F}^{-1}\left(\widehat{\gamma^{\left(j\right)}}\cdot\varphi_{i}\cdot\widehat{f}\right)\right\Vert _{L_{v}^{p}}\\
\left({\scriptstyle 1+ab\leq\left(1+a\right)\left(1+b\right)\text{ for }a,b\geq0}\right) & \leq C_{3}\cdot\left(1+\left\Vert T_{j}^{-1}T_{i}\right\Vert \right)^{d/p}\cdot\left\Vert \mathcal{F}^{-1}\left(\widehat{\gamma^{\left(j\right)}}\cdot\varphi_{i}\right)\ast\mathcal{F}^{-1}\left(\varphi_{i}^{\ast}\cdot\widehat{f}\right)\right\Vert _{L_{v}^{p}}\\
\left({\scriptstyle \text{Prop. }\ref{prop:BandlimitedConvolution}\text{ with }n=1}\right) & \leq C_{3}\!\cdot\!\left(12R_{\mathcal{Q}}C_{\mathcal{Q}}\right)^{d\left(\frac{1}{p}-1\right)}\!\cdot\!\left(1\!+\!\left\Vert T_{j}^{-1}T_{i}\right\Vert \right)^{d/p}\left|\det T_{i}\right|^{\frac{1}{p}-1}\left\Vert \mathcal{F}^{-1}\left(\varphi_{i}\,\widehat{\gamma^{\left(j\right)}}\right)\right\Vert _{L_{v_{0}}^{p}}\left\Vert \mathcal{F}^{-1}\left(\varphi_{i}^{\ast}\,\widehat{f}\right)\right\Vert _{L_{v}^{p}}\\
& \leq C_{3}\cdot\left(12R_{\mathcal{Q}}C_{\mathcal{Q}}\right)^{d\left(\frac{1}{p}-1\right)}\cdot A_{j,i}^{1/p}\cdot c_{i}\\
& =:C_{4}\cdot A_{j,i}^{1/p}\cdot c_{i}.
\end{align*}
Here, Proposition \ref{prop:BandlimitedConvolution} is applicable,
since $\varphi_{i}\in\TestFunctionSpace{\mathbb{R}^{d}}$ and $\widehat{\gamma^{\left(j\right)}}\in C^{\infty}\left(\mathbb{R}^{d}\right)$,
so that $\varphi_{i}\cdot\widehat{\gamma^{\left(j\right)}}\in\TestFunctionSpace{\mathbb{R}^{d}}$
and since clearly $\operatorname{supp}\left[\varphi_{i}\widehat{\gamma^{\left(j\right)}}\right]\subset\overline{Q_{i}^{\ast}}$
and $\operatorname{supp}\left[\varphi_{i}^{\ast}\widehat{f}\right]\subset\overline{Q_{i}^{\ast}}$.
Consequently, we arrive at
\begin{equation}
\sum_{i\in I}\left\Vert \mathcal{F}^{-1}\left(\widehat{\gamma^{\left(j\right)}}\cdot\varphi_{i}\cdot\widehat{f}\right)\right\Vert _{V_{j}}^{p}\leq C_{4}^{p}\cdot\sum_{i\in I}\left[A_{j,i}\cdot c_{i}^{p}\right]=C_{4}^{p}\cdot\left(\overrightarrow{A}\cdot e\right)_{j}<\infty,\label{eq:GammaAnalysisQuasiBanachCaseMainEstimate}
\end{equation}
since $\overrightarrow{A}:\ell_{w^{\min\left\{ 1,p\right\} }}^{r}\left(I\right)\to\ell_{w^{\min\left\{ 1,p\right\} }}^{r}\left(I\right)$
is well-defined and bounded and $e\in\ell_{w^{\min\left\{ 1,p\right\} }}^{r}\left(I\right)$.
Finally, we use the $p$-triangle inequality for $L^{p}\left(\mathbb{R}^{d}\right)$
(yielding the $p$-triangle inequality for $V_{j}=W_{T_{j}^{-T}\left[-1,1\right]^{d}}\left(L_{v}^{p}\right)$)
to conclude that $F_{j}:=\sum_{i\in I}\mathcal{F}^{-1}\left(\widehat{\gamma^{\left(j\right)}}\cdot\varphi_{i}\cdot\widehat{f}\right)\in V_{j}$
is well-defined, with normal convergence in $V_{j}$ and a.e.\@ absolute
convergence of the defining series and with $\left\Vert F_{j}\right\Vert _{V_{j}}\leq C_{4}\cdot\left(\smash{\overrightarrow{A}}\cdot e\right)_{j}^{1/p}$.
\medskip{}
Our next goal is to show that the previous results imply that $f_{j}\in\mathcal{S}'\left(\mathbb{R}^{d}\right)$
yields a well-defined tempered distribution. To this end, recall from
Lemma \ref{lem:SpecialConvolutionConsistent} that $V_{j}\hookrightarrow\mathcal{S}'\left(\mathbb{R}^{d}\right)$
for all $p\in\left(0,\infty\right]$. Consequently, we get $F_{j}\in\mathcal{S}'\left(\mathbb{R}^{d}\right)$
and $F_{j}=\sum_{i\in I}\mathcal{F}^{-1}\left(\widehat{\gamma^{\left(j\right)}}\cdot\varphi_{i}\cdot\widehat{f}\right)$
with unconditional convergence in $V_{j}\hookrightarrow\mathcal{S}'\left(\mathbb{R}^{d}\right)$,
which implies for $\phi\in\mathcal{S}\left(\mathbb{R}^{d}\right)$
that
\begin{align*}
\left\langle \mathcal{F} F_{j},\,\mathcal{F}^{-1}\phi\right\rangle _{\mathcal{S}',\mathcal{S}}=\left\langle F_{j},\,\phi\right\rangle _{\mathcal{S}',\mathcal{S}} & =\sum_{i\in I}\left\langle \mathcal{F}^{-1}\left(\widehat{\gamma^{\left(j\right)}}\cdot\varphi_{i}\cdot\widehat{f}\right),\,\phi\right\rangle _{\mathcal{S}',\mathcal{S}},\\
& =\sum_{i\in I}\left\langle \widehat{\gamma^{\left(j\right)}}\cdot\widehat{f},\,\varphi_{i}\cdot\mathcal{F}^{-1}\phi\right\rangle _{\DistributionSpace{\mathcal{O}},\TestFunctionSpace{\mathcal{O}}}\\
& =\left\langle f_{j},\,\mathcal{F}^{-1}\phi\right\rangle _{\mathcal{S}',\mathcal{S}},
\end{align*}
where the right-hand side is well-defined (with absolute convergence
of the series), since the left-hand side is. This shows that $f_{j}=\mathcal{F} F_{j}\in\mathcal{F} V_{j}\subset\mathcal{S}'\left(\mathbb{R}^{d}\right)$
is a well-defined tempered distribution, as claimed. Finally, we have
$\mathcal{F}^{-1}f_{j}=F_{j}=\sum_{i\in I}\mathcal{F}^{-1}\left(\widehat{\gamma^{\left(j\right)}}\cdot\varphi_{i}\cdot\widehat{f}\right)$,
where the series converges normally in $V_{j}$ and absolutely a.e.,
as claimed.
\medskip{}
It remains to verify boundedness of ${\rm Ana}_{\Gamma}$. But for
$p\in\left[1,\infty\right]$, we have by solidity of $\ell_{w}^{q}\left(I\right)$
and by the triangle inequality for $L^{p}\left(\mathbb{R}^{d}\right)$,
and since $C_{1}=1$ for $p\in\left[1,\infty\right]$, that
\begin{align*}
\left\Vert \left(\left\Vert \gamma^{\left(j\right)}\ast f\right\Vert _{V_{j}}\right)_{j\in I}\right\Vert _{\ell_{w}^{q}}=\left\Vert \left(\left\Vert F_{j}\right\Vert _{L_{v}^{p}}\right)_{j\in I}\right\Vert _{\ell_{w}^{q}} & \leq\left\Vert \left(\sum_{i\in I}\left\Vert \mathcal{F}^{-1}\left(\widehat{\gamma^{\left(j\right)}}\cdot\varphi_{i}\cdot\widehat{f}\right)\right\Vert _{L_{v}^{p}}\right)_{j\in I}\right\Vert _{\ell_{w}^{q}}\\
\left({\scriptstyle \text{eq. }\eqref{eq:GammaAnalysisBanachCaseMainEstimate}}\right) & \leq\left\Vert \left[\left(\overrightarrow{A}\cdot c\right)_{j}\right]_{j\in I}\right\Vert _{\ell_{w}^{q}}\\
& \leq\vertiii{\smash{\overrightarrow{A}}}\cdot\left\Vert c\right\Vert _{\ell_{w}^{q}}\\
& \leq\vertiii{\smash{\Gamma_{\mathcal{Q}}}}\cdot\vertiii{\smash{\overrightarrow{A}}}\cdot\left\Vert f\right\Vert _{\DecompSp{\mathcal{Q}}p{\ell_{w}^{q}}v}<\infty,
\end{align*}
as desired.
Finally, in case of $p\in\left(0,1\right)$, the $p$-triangle inequality
for $W_{T_{j}^{-T}\left[-1,1\right]^{d}}\left(L_{v}^{p}\right)$
yields
\begin{align*}
\left\Vert \gamma^{\left(j\right)}\ast f\right\Vert _{V_{j}}=\left\Vert F_{j}\right\Vert _{W_{T_{j}^{-T}\left[-1,1\right]^{d}}\left(L_{v}^{p}\right)} & \leq\left[\sum_{i\in I}\left\Vert \mathcal{F}^{-1}\left(\widehat{\gamma^{\left(j\right)}}\cdot\varphi_{i}\cdot\widehat{f}\right)\right\Vert _{W_{T_{j}^{-T}\left[-1,1\right]^{d}}\left(L_{v}^{p}\right)}^{p}\right]^{1/p}\\
\left({\scriptstyle \text{eq. }\eqref{eq:GammaAnalysisQuasiBanachCaseMainEstimate}}\right) & \leq C_{4}\cdot\left(\overrightarrow{A}\cdot e\right)_{j}^{1/p}.
\end{align*}
By solidity of $\ell_{w}^{q}\left(I\right)$, this implies
\begin{align*}
\left\Vert \left(\left\Vert \gamma^{\left(j\right)}\ast f\right\Vert _{V_{j}}\right)_{j\in I}\right\Vert _{\ell_{w}^{q}} & \leq C_{4}\cdot\left\Vert \left(\overrightarrow{A}\cdot e\right)^{1/p}\right\Vert _{\ell_{w}^{q}}\\
& =C_{4}\cdot\left\Vert \left(w^{p}\cdot\left[\overrightarrow{A}\cdot e\right]\right)^{1/p}\right\Vert _{\ell^{q}}\\
& =C_{4}\cdot\left\Vert w^{\min\left\{ 1,p\right\} }\cdot\left[\overrightarrow{A}\cdot e\right]\right\Vert _{\ell^{q/p}}^{1/p}\\
& =C_{4}\cdot\left\Vert \overrightarrow{A}\cdot e\right\Vert _{\ell_{w^{\min\left\{ 1,p\right\} }}^{r}}^{1/p}\\
& \leq C_{4}\cdot\vertiii{\smash{\overrightarrow{A}}}^{1/p}\cdot\left\Vert e\right\Vert _{\ell_{w^{\min\left\{ 1,p\right\} }}^{r}}^{1/p}\\
& =C_{4}\cdot\vertiii{\smash{\overrightarrow{A}}}^{1/p}\cdot\left\Vert c\right\Vert _{\ell_{w}^{q}}\\
& \leq C_{1}C_{4}\cdot\vertiii{\smash{\Gamma_{\mathcal{Q}}}}\cdot\vertiii{\smash{\overrightarrow{A}}}^{1/p}\cdot\left\Vert f\right\Vert _{\DecompSp{\mathcal{Q}}p{\ell_{w}^{q}}v}<\infty,
\end{align*}
which completes the proof.
\end{proof}
Next, we establish the estimate ``$\lesssim$'' in equation (\ref{eq:DesiredCharacterization}),
under suitable assumptions on $\left(\gamma_{i}\right)_{i\in I}$.
Notice that up to now we have not excluded the case $\gamma_{i}\equiv0$
for all $i\in I$. But if equation (\ref{eq:DesiredCharacterization})
was true, we would need at least that the family of frequency supports
$\operatorname{supp}\widehat{\gamma^{\left(i\right)}}$, with $i\in I$, covers
all of $\mathcal{O}$. To ensure this, we introduce the following additional
assumption:
\begin{assumption}
\label{assu:GammaCoversOrbit}We assume that for each $i\in I$ there
is some function $\theta_{i}^{\natural}\in\TestFunctionSpace{\mathbb{R}^{d}}$
such that the family $\theta=\left(\smash{\theta_{i}^{\natural}}\right)_{i\in I}$
satisfies the following properties:
\begin{enumerate}
\item We have $\theta_{i}^{\natural}\cdot\widehat{\gamma_{i}}\equiv1$ on
$Q_{i}'$ (and thus on $\overline{Q_{i}'}$) for all $i\in I$.
\item For each $p\in\left(0,\infty\right]$, the constant
\[
\Omega_{2}^{\left(p,K\right)}:=\Omega_{2}^{\left(p,K\right)}\left(\theta\right):=\begin{cases}
\sup_{i\in I}\left\Vert \mathcal{F}^{-1}\theta_{i}^{\natural}\right\Vert _{W_{\left[-1,1\right]^{d}}\left(L_{\left(1+\left|\bullet\right|\right)^{K}}^{p}\right)}, & \text{if }p\in\left(0,1\right),\\
\sup_{i\in I}\left\Vert \mathcal{F}^{-1}\theta_{i}^{\natural}\right\Vert _{L_{\left(1+\left|\bullet\right|\right)^{K}}^{1}}, & \text{if }p\in\left[1,\infty\right]
\end{cases}
\]
is finite.
\end{enumerate}
We fix such a family $\theta=\left(\smash{\theta_{i}^{\natural}}\right)_{i\in I}$
and the constant $\Omega_{2}^{\left(p,K\right)}$ for the remainder
of the paper. Finally, we recall $S_{i}\xi=T_{i}\xi+b_{i}$ and define
\[
\theta_{i}:=\theta_{i}^{\natural}\circ S_{i}^{-1}\in\TestFunctionSpace{\smash{\mathbb{R}^{d}}}\qquad\forall i\in I.\qedhere
\]
\end{assumption}
At least in the case where the set of prototypes $\left\{ \gamma_{i}\,\middle|\, i\in I\right\} $
is finite, the preceding assumption can be heavily simplified, as
we show now:
\begin{lem}
\label{lem:GammaCoversOrbitAssumptionSimplified}Assume that there
are $N$ functions $\gamma_{1}^{\left(0\right)},\dots,\gamma_{N}^{\left(0\right)}$
such that for each $i\in I$ we have $\gamma_{i}=\gamma_{n_{i}}^{\left(0\right)}$
for a suitable $n_{i}\in\underline{N}$. For $n\in\underline{N}$
let
\[
Q^{\left(n\right)}:=\bigcup\left\{ Q_{i}'\,\middle|\, i\in I\text{ and }n_{i}=n\right\} .
\]
If there is some $c>0$ satisfying $\left|\left(\mathcal{F}\smash{\gamma_{n}^{\left(0\right)}}\right)\left(\xi\right)\right|\geq c$
for all $\xi\in Q^{\left(n\right)}$, then the family $\left(\gamma_{i}\right)_{i\in I}$
satisfies Assumption \ref{assu:GammaCoversOrbit}.
In fact, for arbitrary $p_{0}\in\left(0,1\right]$ and $K^{\left(0\right)}\geq0$,
there is a constant $\Omega_{3}=\Omega_{3}\left(\mathcal{Q},\gamma_{1}^{\left(0\right)},\dots,\gamma_{N}^{\left(0\right)},p_{0},K^{\left(0\right)},d\right)>0$
satisfying
\[
\Omega_{2}^{\left(p,K\right)}\leq\Omega_{3}\qquad\forall p\geq p_{0}\text{ and }K\leq K^{\left(0\right)}.\qedhere
\]
\end{lem}
\begin{rem*}
If $\gamma_{i}=\gamma$ for all $i\in I$, then the above assumptions
reduce to $\left|\widehat{\gamma}\left(\xi\right)\right|\geq c>0$
for all $\xi\in Q:=\bigcup_{i\in I}Q_{i}'$.
\end{rem*}
\begin{proof}
Recall from Assumption \ref{assu:MainAssumptions} that we always
have $\widehat{\gamma_{i}}\in C^{\infty}\left(\mathbb{R}^{d}\right)$.
Now, by possibly dropping some elements of the family $\gamma_{1}^{\left(0\right)},\dots,\gamma_{N}^{\left(0\right)}$,
we can assume that for each $n\in\underline{N}$, there is some $i\in I$
satisfying $n_{i}=n$ and thus $\gamma_{n}^{\left(0\right)}=\gamma_{i}$.
In particular, this implies $\widehat{\gamma_{n}^{\left(0\right)}}\in C^{\infty}\left(\mathbb{R}^{d}\right)$
for all $n\in\underline{N}$.
By continuity of $\mathcal{F}\gamma_{n}^{\left(0\right)}$, we get $\left|\left(\mathcal{F}\smash{\gamma_{n}^{\left(0\right)}}\right)\left(\xi\right)\right|\geq c$
for all $\xi\in\overline{Q^{\left(n\right)}}$. Furthermore, recall
from Subsection \ref{subsec:DecompSpaceDefinitionStandingAssumptions}
that we have $Q_{i}'\subset\overline{B_{R_{\mathcal{Q}}}}\left(0\right)$
for all $i\in I$, so that each of the sets $Q^{\left(n\right)}$
is bounded. Hence, $\overline{Q^{\left(n\right)}}$ is compact. Again
by continuity of $\mathcal{F}\gamma_{n}^{\left(0\right)}$, each of the
sets
\[
U_{n}:=\left\{ \xi\in\mathbb{R}^{d}\,\middle|\,\left|\left(\mathcal{F}\smash{\gamma_{n}^{\left(0\right)}}\right)\left(\xi\right)\right|>\frac{c}{2}\right\}
\]
is open with $\overline{Q^{\left(n\right)}}\subset U_{n}$. Thus,
the $C^{\infty}$-Urysohn-Lemma (cf.\@ \cite[Lemma 8.18]{FollandRA})
yields some $\eta_{n}\in\TestFunctionSpace{U_{n}}$ with $\eta_{n}|_{Q^{\left(n\right)}}\equiv1$.
Now, note that $\eta_{n}/\widehat{\gamma_{n}^{\left(0\right)}}\in\TestFunctionSpace{U_{n}}$
is well-defined, since $\widehat{\gamma_{n}^{\left(0\right)}}\neq0$
on $U_{n}$. Thus, the function
\[
\theta^{\left(n\right)}:\mathbb{R}^{d}\to\mathbb{C},\xi\mapsto\begin{cases}
\frac{\eta_{n}\left(\xi\right)}{\widehat{\gamma_{n}^{\left(0\right)}}\left(\xi\right)}, & \text{if }\xi\in U_{n},\\
0, & \text{if }\xi\notin U_{n}
\end{cases}
\]
is a smooth function $\theta^{\left(n\right)}\in\TestFunctionSpace{\mathbb{R}^{d}}$
with $\operatorname{supp}\theta^{\left(n\right)}\subset U_{n}$ and with $\theta^{\left(n\right)}\cdot\widehat{\gamma_{n}^{\left(0\right)}}=\eta_{n}\equiv1$
on $Q^{\left(n\right)}$.
Now, define $\theta_{i}^{\natural}:=\theta^{\left(n_{i}\right)}\in\TestFunctionSpace{\mathbb{R}^{d}}$
for $i\in I$. Then, for each $i\in I$, we have $\theta_{i}^{\natural}\cdot\widehat{\gamma_{i}}=\theta^{\left(n_{i}\right)}\cdot\widehat{\gamma_{n_{i}}^{\left(0\right)}}\equiv1$
on $Q^{\left(n_{i}\right)}\supset Q_{i}'$, cf.\@ the definition
of $Q^{\left(n\right)}$.
Finally, Lemma \ref{lem:SchwartzFunctionsAreWiener} (with $N=K^{\left(0\right)}+\frac{d}{p_{0}}+1$)
yields for $p\geq p_{0}$ and $K\leq K^{\left(0\right)}$ the estimate
\begin{align*}
\left\Vert \mathcal{F}^{-1}\theta_{i}^{\natural}\right\Vert _{L_{\left(1+\left|\cdot\right|\right)^{K}}^{p}}\leq\left\Vert \mathcal{F}^{-1}\theta_{i}^{\natural}\right\Vert _{W_{\left[-1,1\right]^{d}}\left(L_{\left(1+\left|\cdot\right|\right)^{K}}^{p}\right)} & \leq\left(1+2\sqrt{d}\right)^{N}\cdot\left(\frac{1}{p}\frac{s_{d}}{N-K-\frac{d}{p}}\right)^{1/p}\cdot\left\Vert \mathcal{F}^{-1}\theta_{i}^{\natural}\right\Vert _{N}\\
& \leq\left(1+2\sqrt{d}\right)^{N}\cdot\left(1+\frac{s_{d}}{p_{0}}\right)^{1/p_{0}}\cdot\max_{n\in\underline{N}}\left\Vert \mathcal{F}^{-1}\theta^{\left(n\right)}\right\Vert _{N}=:\Omega_{3}.
\end{align*}
Since $N$ only depends on $K^{\left(0\right)},d,p_{0}$
and since $\theta^{\left(1\right)},\dots,\theta^{\left(N\right)}$
only depend on $\mathcal{Q}$ and on $\gamma_{1}^{\left(0\right)},\dots,\gamma_{N}^{\left(0\right)}$,
$\Omega_{3}$ is as claimed in the lemma. Note that each of the norms
$\left\Vert \mathcal{F}^{-1}\theta^{\left(n\right)}\right\Vert _{N}$
is finite, since $\theta^{\left(n\right)}\in\TestFunctionSpace{\mathbb{R}^{d}}$,
from which we get $\mathcal{F}^{-1}\theta^{\left(n\right)}\in\mathcal{S}\left(\mathbb{R}^{d}\right)$.
\end{proof}
Now, instead of just establishing equation (\ref{eq:DesiredCharacterization}),
we will actually show that the ``coefficient map'' ${\rm Ana}_{\Gamma}$
from Theorem \ref{thm:ConvolvingDecompositionSpaceWithGammaJ} yields
a semi-discrete Banach frame for $\DecompSp{\mathcal{Q}}p{\ell_{w}^{q}}v$.
By this we mean that there exists a bounded linear ``reconstruction''
map $R:V\to\DecompSp{\mathcal{Q}}p{\ell_{w}^{q}}v$ satisfying $R\circ{\rm Ana}_{\Gamma}=\operatorname{id}_{\DecompSp{\mathcal{Q}}p{\ell_{w}^{q}}v}$.
For the construction of $R$, the following result will turn out to
be helpful:
\begin{lem}
\label{lem:LocalInverseConvolution}Assume that $\Gamma=\left(\gamma_{i}\right)_{i\in I}$
satisfies Assumption \ref{assu:GammaCoversOrbit} and let $\left(\theta_{i}\right)_{i\in I}$
be defined as in that assumption.
Then $\widehat{\gamma^{\left(i\right)}}\cdot\theta_{i}\equiv1$ on
$\overline{Q_{i}}$ for each $i\in I$ and each of the maps
\[
I_{i}:V_{i}\to V_{i},f\mapsto\left(\mathcal{F}^{-1}\theta_{i}\right)\ast f
\]
is well-defined and bounded, with $\sup_{i\in I}\vertiii{I_{i}}\leq C<\infty$,
where
\[
C:=\begin{cases}
\Omega_{0}^{4K}\Omega_{1}^{4}\Omega_{2}^{\left(p,K\right)}\cdotd^{-\frac{d}{2p}}\cdot\left(972\cdotd^{5/2}\right)^{K+\frac{d}{p}}, & \text{if }p\in\left(0,1\right),\\
\Omega_{0}^{K}\Omega_{1}\Omega_{2}^{\left(p,K\right)}, & \text{if }p\in\left[1,\infty\right].
\end{cases}
\]
Hence, the map
\[
m_{\theta}:=\bigotimes_{i\in I}I_{i}:V\to V,\left(f_{i}\right)_{i\in I}\mapsto\left(\left(\mathcal{F}^{-1}\theta_{i}\right)\ast f_{i}\right)_{i\in I}
\]
is well-defined and bounded as well, with $\vertiii{m_{\theta}}\leq C$.
\end{lem}
\begin{proof}
First, observe
\[
\widehat{\gamma^{\left(i\right)}}\cdot\theta_{i}=\left(\widehat{\gamma_{i}}\circ S_{i}^{-1}\right)\cdot\left(\theta_{i}^{\natural}\circ S_{i}^{-1}\right)=\underbrace{\left(\widehat{\gamma_{i}}\cdot\theta_{i}^{\natural}\right)}_{\equiv1\text{ on }\overline{Q_{i}'}}\circ S_{i}^{-1}\equiv1\text{ on }S_{i}\overline{Q_{i}'}=\overline{Q_{i}},
\]
so that it remains to show that each of the maps $I_{i}$ is well-defined
and bounded, with the claimed estimate for the operator norm.
In case of $p\in\left[1,\infty\right]$, this is a consequence of
equation (\ref{eq:WeightedYoungInequality}), once we show that $\left\Vert \mathcal{F}^{-1}\theta_{i}\right\Vert _{L_{v_{0}}^{1}}$
is uniformly bounded. But we simply have
\begin{equation}
\theta_{i}=\theta_{i}^{\natural}\circ S_{i}^{-1}=L_{b_{i}}\left(\theta_{i}^{\natural}\circ T_{i}^{-1}\right)\qquad\text{ and hence }\qquad\mathcal{F}^{-1}\theta_{i}=\left|\det T_{i}\right|\cdot M_{b_{i}}\left[\left(\mathcal{F}^{-1}\theta_{i}^{\natural}\right)\circ T_{i}^{T}\right],\label{eq:LocalInverseClosedForm}
\end{equation}
which implies
\begin{align*}
\left\Vert \mathcal{F}^{-1}\theta_{i}\right\Vert _{L_{v_{0}}^{1}} & =\left|\det T_{i}\right|\cdot\left\Vert v_{0}\cdot\left[\left(\mathcal{F}^{-1}\theta_{i}^{\natural}\right)\circ T_{i}^{T}\right]\right\Vert _{L^{1}}\\
& =\left\Vert \left(v_{0}\circ T_{i}^{-T}\right)\cdot\left(\mathcal{F}^{-1}\theta_{i}^{\natural}\right)\right\Vert _{L^{1}}\\
\left({\scriptstyle \text{assumption on }v_{0}}\right) & \leq\Omega_{1}\cdot\left\Vert x\mapsto\left(1+\left|T_{i}^{-T}x\right|\right)^{K}\cdot\left(\mathcal{F}^{-1}\theta_{i}^{\natural}\right)\left(x\right)\right\Vert _{L^{1}}\\
\left({\scriptstyle \text{eq. }\eqref{eq:WeightLinearTransformationsConnection}}\right) & \leq\Omega_{0}^{K}\Omega_{1}\cdot\left\Vert \left(1+\left|\bullet\right|\right)^{K}\cdot\mathcal{F}^{-1}\theta_{i}^{\natural}\right\Vert _{L^{1}}\leq\Omega_{0}^{K}\Omega_{1}\cdot\Omega_{2}^{\left(p,K\right)}.
\end{align*}
Finally, for $p\in\left(0,1\right)$, we get from Corollary \ref{cor:WienerAmalgamConvolutionSimplified}
for $C_{1}:=\Omega_{0}^{3K}\Omega_{1}^{3}\cdotd^{-\frac{d}{2p}}\cdot\left(972\cdotd^{5/2}\right)^{K+\frac{d}{p}}$
that
\begin{align*}
\left\Vert \left(\mathcal{F}^{-1}\theta_{i}\right)\ast f\right\Vert _{W_{T_{i}^{-T}\left[-1,1\right]^{d}}\left(L_{v}^{p}\right)} & \leq C_{1}\cdot\left|\det T_{i}\right|^{\frac{1}{p}-1}\cdot\left\Vert \mathcal{F}^{-1}\theta_{i}\right\Vert _{W_{T_{i}^{-T}\left[-1,1\right]^{d}}\left(L_{v_{0}}^{p}\right)}\cdot\left\Vert f\right\Vert _{W_{T_{i}^{-T}\left[-1,1\right]^{d}}\left(L_{v}^{p}\right)}\\
\left({\scriptstyle \text{eq. }\eqref{eq:LocalInverseClosedForm}\text{ and }\left\Vert M_{b}f\right\Vert _{W_{Q}\left(L_{v_{0}}^{p}\right)}=\left\Vert f\right\Vert _{W_{Q}\left(L_{v_{0}}^{p}\right)}}\right) & =C_{1}\cdot\left|\det T_{i}\right|^{\frac{1}{p}-1}\cdot\left|\det T_{i}\right|\cdot\left\Vert \left(\mathcal{F}^{-1}\theta_{i}^{\natural}\right)\circ T_{i}^{T}\right\Vert _{W_{T_{i}^{-T}\left[-1,1\right]^{d}}\left(L_{v_{0}}^{p}\right)}\cdot\left\Vert f\right\Vert _{V_{i}}\\
\left({\scriptstyle \text{Lemma }\ref{lem:WienerTransformationFormula}}\right) & =C_{1}\cdot\left|\det T_{i}\right|^{\frac{1}{p}}\cdot\left\Vert \left(M_{\left[-1,1\right]^{d}}\left[\mathcal{F}^{-1}\theta_{i}^{\natural}\right]\right)\circ T_{i}^{T}\right\Vert _{L_{v_{0}}^{p}}\cdot\left\Vert f\right\Vert _{V_{i}}\\
& =C_{1}\cdot\left\Vert \left(v_{0}\circ T_{i}^{-T}\right)\cdot M_{\left[-1,1\right]^{d}}\left[\mathcal{F}^{-1}\theta_{i}^{\natural}\right]\right\Vert _{L^{p}}\cdot\left\Vert f\right\Vert _{V_{i}}\\
\left({\scriptstyle \text{asusmption on }v_{0}\text{ and eq. }\eqref{eq:WeightLinearTransformationsConnection}}\right) & \leq\Omega_{0}^{K}\Omega_{1}C_{1}\cdot\left\Vert \left(1+\left|\bullet\right|\right)^{K}\cdot M_{\left[-1,1\right]^{d}}\left[\mathcal{F}^{-1}\theta_{i}^{\natural}\right]\right\Vert _{L^{p}}\cdot\left\Vert f\right\Vert _{V_{i}}\\
& \leq\Omega_{0}^{K}\Omega_{1}\Omega_{2}^{\left(p,K\right)}\cdot C_{1}\cdot\left\Vert f\right\Vert _{V_{i}}.\qedhere
\end{align*}
\end{proof}
Our final ingredient for the construction of the ``reconstruction
map'' $R:V\to\DecompSp{\mathcal{Q}}p{\ell_{w}^{q}}v$ is the following
lemma.
\begin{lem}
\label{lem:DecompositionSynthesis}The map
\[
{\rm Synth}_{\mathcal{D}}:V\to\DecompSp{\mathcal{Q}}p{\ell_{w}^{q}}v,\left(f_{i}\right)_{i\in I}\mapsto\sum_{i\in I}\left[\left(\mathcal{F}^{-1}\varphi_{i}\right)\ast f_{i}\right]\overset{\text{Lemma }\ref{lem:SpecialConvolutionConsistent}}{=}\sum_{i\in I}\left[\mathcal{F}^{-1}\left(\varphi_{i}\cdot\widehat{f_{i}}\right)\right]
\]
is well-defined and bounded with unconditional convergence of the
series in $Z'\left(\mathcal{O}\right)$ and with $\vertiii{{\rm Synth}_{\mathcal{D}}}\leq\vertiii{\smash{\Gamma_{\mathcal{Q}}}}\cdot C$,
where
\[
C=\begin{cases}
\frac{\left(768/\sqrt{d}\right)^{d/p}}{59049\cdot12^{d}\cdotd^{5}}\cdot\!\left(\!186624\cdot\!d^{4}\!\cdot\!\left\lceil K\!+\!\frac{d+1}{p}\right\rceil \!\right)^{\!1+\left\lceil K+\frac{d+1}{p}\right\rceil }\!\cdot\!\left(1\!+\!R_{\mathcal{Q}}C_{\mathcal{Q}}\right)^{d\left(\frac{2}{p}-1\right)}\!\cdot\!\Omega_{0}^{4K}\Omega_{1}^{4}\cdot N_{\mathcal{Q}}^{\frac{1}{p}-1}C_{\mathcal{Q},\Phi,v_{0},p}^{2}, & \text{if }p\in\left(0,1\right),\\
C_{\mathcal{Q},\Phi,v_{0},p}^{2}, & \text{if }p\in\left[1,\infty\right]
\end{cases}
\]
and where $\Gamma_{\mathcal{Q}}:\ell_{w}^{q}\left(I\right)\to\ell_{w}^{q}\left(I\right),c\mapsto c^{\ast}$
denotes the $\mathcal{Q}$-clustering map, i.e., $c_{i}^{\ast}=\sum_{\ell\in i^{\ast}}c_{\ell}$.
\end{lem}
\begin{proof}
Recall that the Fourier transform $\mathcal{F}:Z'\left(\mathcal{O}\right)\to\DistributionSpace{\mathcal{O}}$
is an isomorphism that restricts to an isometric isomorphism $\mathcal{F}:\DecompSp{\mathcal{Q}}p{\ell_{w}^{q}}v\to\FourierDecompSp{\mathcal{Q}}p{\ell_{w}^{q}}v$.
Hence, it suffices to show that the map
\[
\Theta:=\mathcal{F}\circ{\rm Synth}_{\mathcal{D}}:V\to\FourierDecompSp{\mathcal{Q}}p{\ell_{w}^{q}}v,\left(f_{i}\right)_{i\in I}\mapsto\sum_{i\in I}\varphi_{i}\widehat{f_{i}}
\]
is well-defined and bounded, with unconditional convergence of the
series in $\mathcal{D}'\left(\mathcal{O}\right)$. Since the $\left(\varphi_{i}\right)_{i\in I}$
form a \emph{locally finite} partition of unity on $\mathcal{O}$, the series
\emph{does} converge unconditionally in $\mathcal{D}'\left(\mathcal{O}\right)$,
given that each term is a well-defined element of $\mathcal{D}'\left(\mathcal{O}\right)$.
But this is an easy consequence of the embedding $V_{i}\hookrightarrow\mathcal{S}'\left(\mathbb{R}^{d}\right)$,
which holds for each $i\in I$, according to Lemma \ref{lem:SpecialConvolutionConsistent}.
Now, define
\[
c_{i}:=\left\Vert f_{i}\right\Vert _{V_{i}}=\begin{cases}
\left\Vert f_{i}\right\Vert _{L_{v}^{p}}, & \text{if }p\in\left[1,\infty\right],\\
\left\Vert f_{i}\right\Vert _{W_{T_{i}^{-T}\left[-1,1\right]^{d}}\left(L_{v}^{p}\right)}, & \text{if }p\in\left(0,1\right).
\end{cases}
\]
By definition of $V$, we have $c:=\left(c_{i}\right)_{i\in I}\in\ell_{w}^{q}\left(I\right)$
and $\left\Vert \left(f_{i}\right)_{i\in I}\right\Vert _{V}=\left\Vert c\right\Vert _{\ell_{w}^{q}}$.
Furthermore, since the $\mathcal{Q}$-clustering map $\Gamma_{\mathcal{Q}}$ is
bounded, it suffices to show $\left\Vert \mathcal{F}^{-1}\left(\varphi_{j}\cdot\Theta\left(f_{i}\right)_{i\in I}\right)\right\Vert _{L_{v}^{p}}\leq C_{1}\cdot c_{j}^{\ast}$
for all $j\in I$ and a suitable constant $C_{1}>0$, since this implies
\[
\left\Vert \Theta\left(f_{i}\right)_{i\in I}\right\Vert _{\FourierDecompSp{\mathcal{Q}}p{\ell_{w}^{q}}v}=\left\Vert \left(\left\Vert \mathcal{F}^{-1}\!\left(\varphi_{j}\cdot\Theta\left(f_{i}\right)_{i\in I}\right)\right\Vert _{L_{v}^{p}}\right)_{j\in I}\right\Vert _{\ell_{w}^{q}}\!\leq C_{1}\cdot\left\Vert c^{\ast}\right\Vert _{\ell_{w}^{q}}\leq C_{1}\vertiii{\smash{\Gamma_{\mathcal{Q}}}}\cdot\left\Vert c\right\Vert _{\ell_{w}^{q}}=C_{1}\vertiii{\smash{\Gamma_{\mathcal{Q}}}}\cdot\left\Vert \left(f_{i}\right)_{i\in I}\right\Vert _{V}.
\]
To show $\left\Vert \mathcal{F}^{-1}\left(\varphi_{j}\cdot\Gamma\left(f_{i}\right)_{i\in I}\right)\right\Vert _{L_{v}^{p}}\leq C_{1}\cdot c_{j}^{\ast}$,
we distinguish two cases regarding $p$:
Let us start with the case $p\in\left[1,\infty\right]$. Since $\varphi_{j}\varphi_{\ell}\equiv0$
unless $\ell\in j^{\ast}$, we have
\begin{align*}
\left\Vert \mathcal{F}^{-1}\left[\varphi_{j}\cdot\Theta\left(f_{\ell}\right)_{\ell\in I}\right]\right\Vert _{L_{v}^{p}} & =\left\Vert \mathcal{F}^{-1}\left[\varphi_{j}\cdot\sum_{\ell\in I}\varphi_{\ell}\widehat{f_{\ell}}\right]\right\Vert _{L_{v}^{p}}\\
& =\left\Vert \sum_{\ell\in j^{\ast}}\mathcal{F}^{-1}\left[\varphi_{j}\varphi_{\ell}\widehat{f_{\ell}}\right]\right\Vert _{L_{v}^{p}}\\
& \leq\sum_{\ell\in j^{\ast}}\left\Vert \left(\mathcal{F}^{-1}\varphi_{j}\right)\ast\left(\mathcal{F}^{-1}\varphi_{\ell}\right)\ast f_{\ell}\right\Vert _{L_{v}^{p}}\\
\left({\scriptstyle \text{eq. }\eqref{eq:WeightedYoungInequality}}\right) & \leq\sum_{\ell\in j^{\ast}}\left\Vert \mathcal{F}^{-1}\varphi_{j}\right\Vert _{L_{v_{0}}^{1}}\left\Vert \mathcal{F}^{-1}\varphi_{\ell}\right\Vert _{L_{v_{0}}^{1}}\cdot\left\Vert f_{\ell}\right\Vert _{L_{v}^{p}}\\
& \leq C_{\mathcal{Q},\Phi,v_{0},p}^{2}\cdot\sum_{\ell\in j^{\ast}}\left\Vert f_{\ell}\right\Vert _{L_{v}^{p}}=C_{\mathcal{Q},\Phi,v_{0},p}^{2}\cdot c_{j}^{\ast},
\end{align*}
so that we can choose $C_{1}:=C_{\mathcal{Q},\Phi,v_{0},p}^{2}$.
Now, we consider the case $p\in\left(0,1\right)$. Here, we can first
argue as before:
\begin{align*}
\left\Vert \mathcal{F}^{-1}\left[\varphi_{j}\cdot\Theta\left(f_{\ell}\right)_{\ell\in I}\right]\right\Vert _{L_{v}^{p}} & =\left\Vert \sum_{\ell\in j^{\ast}}\mathcal{F}^{-1}\left[\varphi_{j}\varphi_{\ell}\widehat{f_{\ell}}\right]\right\Vert _{L_{v}^{p}}\\
\left({\scriptstyle \text{quasi-triangle inequality and }\left|j^{\ast}\right|\leq N_{\mathcal{Q}}}\right) & \leq N_{\mathcal{Q}}^{\frac{1}{p}-1}\cdot\sum_{\ell\in j^{\ast}}\left\Vert \mathcal{F}^{-1}\left(\varphi_{j}\varphi_{\ell}\right)\ast f_{\ell}\right\Vert _{L_{v}^{p}}\\
\left({\scriptstyle \text{Lemma }\ref{lem:MaximalFunctionDominatesF}}\right) & \leq N_{\mathcal{Q}}^{\frac{1}{p}-1}\cdot\sum_{\ell\in j^{\ast}}\left\Vert \mathcal{F}^{-1}\left(\varphi_{j}\varphi_{\ell}\right)\ast f_{\ell}\right\Vert _{W_{T_{\ell}^{-T}\left[-1,1\right]^{d}}\left(L_{v}^{p}\right)}.
\end{align*}
Then, we set $C_{2}:=\Omega_{0}^{3K}\Omega_{1}^{3}\cdotd^{-\frac{d}{2p}}\cdot\left(972\cdotd^{5/2}\right)^{K+\frac{d}{p}}$
and use Corollary \ref{cor:WienerAmalgamConvolutionSimplified} to
estimate each summand as follows:
\begin{align*}
& \left\Vert \mathcal{F}^{-1}\left(\varphi_{j}\varphi_{\ell}\right)\ast f_{\ell}\right\Vert _{W_{T_{\ell}^{-T}\left[-1,1\right]^{d}}\left(L_{v}^{p}\right)}\\
\left({\scriptstyle \text{Corollary }\ref{cor:WienerAmalgamConvolutionSimplified}}\right) & \leq C_{2}\cdot\left|\det T_{\ell}\right|^{\frac{1}{p}-1}\cdot\left\Vert \mathcal{F}^{-1}\left(\varphi_{j}\varphi_{\ell}\right)\right\Vert _{W_{T_{\ell}^{-T}\left[-1,1\right]^{d}}\left(L_{v_{0}}^{p}\right)}\,\left\Vert f_{\ell}\right\Vert _{W_{T_{\ell}^{-T}\left[-1,1\right]^{d}}\left(L_{v}^{p}\right)}\\
& =C_{2}\cdot\left|\det T_{\ell}\right|^{\frac{1}{p}-1}\cdot\left\Vert \mathcal{F}^{-1}\left(\varphi_{j}\varphi_{\ell}\right)\right\Vert _{W_{T_{\ell}^{-T}\left[-1,1\right]^{d}}\left(L_{v_{0}}^{p}\right)}\cdot c_{\ell}.
\end{align*}
Now, note
\[
\operatorname{supp}\left(\varphi_{j}\varphi_{\ell}\right)\subset\overline{Q_{\ell}}=T_{\ell}\overline{Q_{\ell}'}+b_{\ell}\subset T_{\ell}\left[-R_{\mathcal{Q}},R_{\mathcal{Q}}\right]^{d}+b_{\ell},
\]
so that Theorem \ref{thm:BandlimitedWienerAmalgamSelfImproving} (with
$v_{0}$ instead of $v$) shows for
\[
C_{3}:=2^{4\left(1+\frac{d}{p}\right)}s_{d}^{\frac{1}{p}}\left(192\cdotd^{3/2}\cdot\left\lceil K+\frac{d+1}{p}\right\rceil \right)^{\left\lceil K+\frac{d+1}{p}\right\rceil +1}\cdot\Omega_{0}^{K}\Omega_{1}\cdot\left(1+R_{\mathcal{Q}}\right)^{\frac{d}{p}}
\]
that
\begin{align*}
\left\Vert \mathcal{F}^{-1}\left(\varphi_{j}\varphi_{\ell}\right)\right\Vert _{W_{T_{\ell}^{-T}\left[-1,1\right]^{d}}\left(L_{v_{0}}^{p}\right)} & \leq C_{3}\cdot\left\Vert \mathcal{F}^{-1}\left(\varphi_{j}\varphi_{\ell}\right)\right\Vert _{L_{v_{0}}^{p}}\\
\left({\scriptstyle \text{Proposition }\ref{prop:BandlimitedConvolution}\text{ and }\operatorname{supp}\varphi_{j}\subset\overline{Q_{j}}\subset\overline{Q_{j}^{\ast}}\text{ and }\operatorname{supp}\varphi_{\ell}\subset\overline{Q_{\ell}}\subset\overline{Q_{j}^{\ast}}}\right) & \leq C_{3}\left(12R_{\mathcal{Q}}C_{\mathcal{Q}}\right)^{d\left(\frac{1}{p}-1\right)}\!\cdot\!\left|\det T_{j}\right|^{\frac{1}{p}-1}\!\cdot\!\left\Vert \mathcal{F}^{-1}\varphi_{j}\right\Vert _{L_{v_{0}}^{p}}\!\cdot\!\left\Vert \mathcal{F}^{-1}\varphi_{\ell}\right\Vert _{L_{v_{0}}^{p}}\\
& \leq C_{4}\cdot\left\Vert \mathcal{F}^{-1}\varphi_{\ell}\right\Vert _{L_{v_{0}}^{p}}
\end{align*}
for $C_{4}:=C_{3}\cdot\left(12R_{\mathcal{Q}}C_{\mathcal{Q}}\right)^{d\left(\frac{1}{p}-1\right)}\cdot C_{\mathcal{Q},\Phi,v_{0},p}$.
In total, we conclude
\begin{align*}
\left\Vert \mathcal{F}^{-1}\left(\varphi_{j}\varphi_{\ell}\right)\ast f_{\ell}\right\Vert _{W_{T_{\ell}^{-T}\left[-1,1\right]^{d}}\left(L_{v}^{p}\right)} & \leq C_{2}\cdot\left|\det T_{\ell}\right|^{\frac{1}{p}-1}\cdot\left\Vert \mathcal{F}^{-1}\left(\varphi_{j}\varphi_{\ell}\right)\right\Vert _{W_{T_{\ell}^{-T}\left[-1,1\right]^{d}}\left(L_{v_{0}}^{p}\right)}\cdot c_{\ell}\\
& \leq C_{2}C_{4}\cdot\left|\det T_{\ell}\right|^{\frac{1}{p}-1}\cdot\left\Vert \mathcal{F}^{-1}\varphi_{\ell}\right\Vert _{L_{v_{0}}^{p}}\cdot c_{\ell}\\
& \leq C_{2}C_{4}C_{\mathcal{Q},\Phi,v_{0},p}\cdot c_{\ell}
\end{align*}
and hence
\begin{align*}
\left\Vert \mathcal{F}^{-1}\left[\varphi_{j}\cdot\Theta\left(f_{\ell}\right)_{\ell\in I}\right]\right\Vert _{L_{v}^{p}} & \leq N_{\mathcal{Q}}^{\frac{1}{p}-1}\cdot\sum_{\ell\in j^{\ast}}\left\Vert \mathcal{F}^{-1}\left(\varphi_{j}\varphi_{\ell}\right)\ast f_{\ell}\right\Vert _{W_{T_{\ell}^{-T}\left[-1,1\right]^{d}}\left(L_{v}^{p}\right)}\\
& \leq N_{\mathcal{Q}}^{\frac{1}{p}-1}C_{2}C_{4}C_{\mathcal{Q},\Phi,v_{0},p}\cdot\sum_{\ell\in j^{\ast}}c_{\ell}\\
& =N_{\mathcal{Q}}^{\frac{1}{p}-1}C_{2}C_{4}C_{\mathcal{Q},\Phi,v_{0},p}\cdot c_{j}^{\ast},
\end{align*}
so that the desired estimate from the start of the proof holds with
$C_{1}:=N_{\mathcal{Q}}^{\frac{1}{p}-1}C_{2}C_{4}C_{\mathcal{Q},\Phi,v_{0},p}$.
Now, we finally set $N:=\left\lceil K+\frac{d+1}{p}\right\rceil $
and observe because of $N\geq K+\frac{d}{p}+1\geq\frac{d}{p}+1$
and $s_{d}\leq4^{d}$, as well as $C_{\mathcal{Q}}\geq\left\Vert T_{i}^{-1}T_{i}\right\Vert \geq1$
that
\begin{align*}
C_{1} & =d^{-\frac{d}{2p}}\!\cdot\!\left(972\cdotd^{\frac{5}{2}}\right)^{\!K+\frac{d}{p}}2^{4\left(1+\frac{d}{p}\right)}s_{d}^{\frac{1}{p}}\left(192\!\cdot\!d^{\frac{3}{2}}\cdot N\right)^{\!N+1}\!\cdot\left(1\!+\!R_{\mathcal{Q}}\right)^{\frac{d}{p}}\left(12R_{\mathcal{Q}}C_{\mathcal{Q}}\right)^{d\left(\frac{1}{p}-1\right)}\cdot\Omega_{0}^{4K}\Omega_{1}^{4}\cdot N_{\mathcal{Q}}^{\frac{1}{p}-1}C_{\mathcal{Q},\Phi,v_{0},p}^{2}\\
& \leq2^{4}\!\cdot\!\left(2^{6}/\sqrt{d}\right)^{d/p}\left(972\cdotd^{\frac{5}{2}}\right)^{K+\frac{d}{p}}\left(\!192\!\cdot\!d^{\frac{3}{2}}\cdot N\right)^{\!N+1}\!\cdot12^{d\left(\frac{1}{p}-1\right)}\left(1\!+\!R_{\mathcal{Q}}C_{\mathcal{Q}}\right)^{d\left(\frac{2}{p}-1\right)}\!\cdot\Omega_{0}^{4K}\Omega_{1}^{4}\cdot N_{\mathcal{Q}}^{\frac{1}{p}-1}C_{\mathcal{Q},\Phi,v_{0},p}^{2}\\
& \leq\frac{\left(768/\sqrt{d}\right)^{d/p}}{59049\cdot12^{d}\cdotd^{5}}\cdot\left(186624\cdotd^{4}\cdot N\right)^{N+1}\cdot\left(1+R_{\mathcal{Q}}C_{\mathcal{Q}}\right)^{d\left(\frac{2}{p}-1\right)}\cdot\Omega_{0}^{4K}\Omega_{1}^{4}\cdot N_{\mathcal{Q}}^{\frac{1}{p}-1}C_{\mathcal{Q},\Phi,v_{0},p}^{2}.\qedhere
\end{align*}
\end{proof}
Now, we can finally show existence of the reconstruction map $R$
and thus also derive the estimate ``$\lesssim$'' in equation (\ref{eq:DesiredCharacterization}).
\begin{thm}
\label{thm:SemiDiscreteBanachFrame}Assume that the family $\Gamma=\left(\gamma_{i}\right)_{i\in I}$
satisfies Assumptions \ref{assu:MainAssumptions} and \ref{assu:GammaCoversOrbit}.
Then, with $m_{\theta}$ as in Lemma \ref{lem:LocalInverseConvolution},
with ${\rm Synth}_{\mathcal{D}}$ as in Lemma \ref{lem:DecompositionSynthesis}
and with ${\rm Ana}_{\Gamma}$ as in Theorem \ref{thm:ConvolvingDecompositionSpaceWithGammaJ},
the map
\[
R:={\rm Synth}_{\mathcal{D}}\circ m_{\theta}:V\to\DecompSp{\mathcal{Q}}p{\ell_{w}^{q}}v
\]
is well-defined and bounded with $\vertiii R\leq\vertiii{\Gamma_{\mathcal{Q}}}C_{\mathcal{Q},\Phi,v_{0},p}^{2}\cdot C$
for
\[
C\!:=\!\!\begin{cases}
\frac{\left(768/d\right)^{d/p}}{2^{35}\cdot12^{d}\cdotd^{10}}\cdot\!\left(\!2^{28}\!\cdot\!d^{\frac{13}{2}}\!\cdot\!\left\lceil \!K\!+\!\frac{d+1}{p}\right\rceil \!\right)^{\!1+\left\lceil K+\frac{d+1}{p}\right\rceil }\!\!\cdot\!\left(1\!+\!R_{\mathcal{Q}}C_{\mathcal{Q}}\right)^{d\left(\frac{2}{p}-1\right)}\!\cdot\Omega_{0}^{8K}\Omega_{1}^{8}\Omega_{2}^{\left(p,K\right)}\cdot N_{\mathcal{Q}}^{\frac{1}{p}-1}, & \text{if }p<1,\\
\Omega_{0}^{K}\Omega_{1}\Omega_{2}^{\left(p,K\right)}, & \text{if }p\geq1,
\end{cases}
\]
where as usual $\Gamma_{\mathcal{Q}}:\ell_{w}^{q}\left(I\right)\to\ell_{w}^{q}\left(I\right)$
denotes the $\mathcal{Q}$-clustering map.
Furthermore, $R$ satisfies
\begin{equation}
R\circ{\rm Ana}_{\Gamma}=\operatorname{id}_{\DecompSp{\mathcal{Q}}p{\ell_{w}^{q}}v}.\label{eq:SemiDiscreteBanachFrame}
\end{equation}
In particular, equation (\ref{eq:DesiredCharacterization}) is satisfied,
i.e., we have
\[
\left\Vert f\right\Vert _{\DecompSp{\mathcal{Q}}p{\ell_{w}^{q}}v}\asymp\left\Vert \left(\smash{\gamma^{\left(i\right)}}\ast f\right)_{i\in I}\right\Vert _{V}\qquad\forall f\in\DecompSp{\mathcal{Q}}p{\ell_{w}^{q}}v.\qedhere
\]
\end{thm}
\begin{proof}
Boundedness of ${\rm Synth}_{\mathcal{D}}$ and $m_{\theta}$ and thus
of $R$ is a consequence of Lemmas \ref{lem:DecompositionSynthesis}
and \ref{lem:LocalInverseConvolution}, respectively, so that it suffices
to prove equation (\ref{eq:SemiDiscreteBanachFrame}).
To see this, we again use the isomorphism $\mathcal{F}:\DecompSp{\mathcal{Q}}p{\ell_{w}^{q}}v\to\FourierDecompSp{\mathcal{Q}}p{\ell_{w}^{q}}v$.
Recall from Lemma \ref{lem:DecompositionSynthesis} that we have
\[
\left(\mathcal{F}\circ{\rm Synth}_{\mathcal{D}}\right)\left(f_{i}\right)_{i\in I}=\sum_{i\in I}\left(\varphi_{i}\cdot\widehat{f_{i}}\right)\qquad\text{ for }\qquad\left(f_{i}\right)_{i\in I}\in V=\ell_{w}^{q}\left(\left[V_{i}\right]_{i\in I}\right),
\]
where it is used that $f_{i}\in V_{i}\hookrightarrow\mathcal{S}'\left(\mathbb{R}^{d}\right)$
for all $i\in I$.
Hence, for $f\in\DecompSp{\mathcal{Q}}p{\ell_{w}^{q}}v$, we have
\begin{align*}
\left(\mathcal{F}\circ R\circ{\rm Ana}_{\Gamma}\right)f & =\sum_{i\in I}\left[\varphi_{i}\cdot\mathcal{F}\left[\left(m_{\theta}\circ{\rm Ana}_{\Gamma}\right)f\right]_{i}\right]\\
& =\sum_{i\in I}\left[\varphi_{i}\cdot\mathcal{F}\left[\left(\mathcal{F}^{-1}\theta_{i}\right)\ast\left(\gamma^{\left(i\right)}\ast f\right)\right]\right]\\
\left({\scriptstyle \text{Lemma }\ref{lem:SpecialConvolutionConsistent}}\right) & =\sum_{i\in I}\left[\varphi_{i}\cdot\theta_{i}\cdot\widehat{\gamma^{\left(i\right)}\ast f}\right]\\
\left({\scriptstyle \text{Special Def. of }\gamma^{\left(i\right)}\ast f=\mathcal{F}^{-1}f_{i}\text{, cf. Theorem }\ref{thm:ConvolvingDecompositionSpaceWithGammaJ}}\right) & =\sum_{i\in I}\left[\varphi_{i}\cdot\theta_{i}\cdot f_{i}\right],
\end{align*}
where
\[
f_{i}:\mathcal{S}\left(\mathbb{R}^{d}\right)\to\mathbb{C},\phi\mapsto\sum_{\ell\in I}\left\langle \widehat{\gamma^{\left(i\right)}}\cdot\widehat{f},\,\varphi_{\ell}\phi\right\rangle _{\DistributionSpace{\mathcal{O}},\TestFunctionSpace{\mathcal{O}}}.
\]
Thus, we have for arbitrary $\phi\in\TestFunctionSpace{\mathcal{O}}$ that
\begin{align*}
\left\langle \left(\mathcal{F}\circ R\circ{\rm Ana}_{\Gamma}\right)f,\,\phi\right\rangle _{\DistributionSpace{\mathcal{O}},\TestFunctionSpace{\mathcal{O}}} & =\sum_{i\in I}\left\langle f_{i},\,\varphi_{i}\cdot\theta_{i}\cdot\phi\right\rangle _{\mathcal{S}',\mathcal{S}}\\
& =\sum_{i\in I}\sum_{\ell\in I}\left\langle \widehat{\gamma^{\left(i\right)}}\cdot\widehat{f},\,\varphi_{\ell}\varphi_{i}\cdot\theta_{i}\cdot\phi\right\rangle _{\DistributionSpace{\mathcal{O}},\TestFunctionSpace{\mathcal{O}}}\\
& =\sum_{i\in I}\sum_{\ell\in I}\left\langle \widehat{f},\,\varphi_{\ell}\varphi_{i}\cdot\widehat{\gamma^{\left(i\right)}}\cdot\theta_{i}\cdot\phi\right\rangle _{\DistributionSpace{\mathcal{O}},\TestFunctionSpace{\mathcal{O}}}\\
\left({\scriptstyle \widehat{\gamma^{\left(i\right)}}\cdot\theta_{i}\equiv1\text{ on }\overline{Q_{i}}\supset\operatorname{supp}\varphi_{i},\text{ cf. Lemma }\ref{lem:LocalInverseConvolution}}\right) & =\sum_{i\in I}\sum_{\ell\in I}\left\langle \widehat{f},\,\varphi_{\ell}\varphi_{i}\cdot\phi\right\rangle _{\DistributionSpace{\mathcal{O}},\TestFunctionSpace{\mathcal{O}}}\\
\left({\scriptstyle \phi\in\TestFunctionSpace{\mathcal{O}}\text{ and }\left(\varphi_{i}\right)_{i\in I}\text{ loc. finite part. of unity on }\mathcal{O}}\right) & =\sum_{i\in I}\left\langle \widehat{f},\,\varphi_{i}\cdot\phi\right\rangle _{\DistributionSpace{\mathcal{O}},\TestFunctionSpace{\mathcal{O}}}\\
\left({\scriptstyle \text{as above}}\right) & =\left\langle \widehat{f},\,\phi\right\rangle _{\DistributionSpace{\mathcal{O}},\TestFunctionSpace{\mathcal{O}}}.
\end{align*}
Hence, we have shown $\mathcal{F}\circ R\circ{\rm Ana}_{\Gamma}=\mathcal{F}$
on $\DecompSp{\mathcal{Q}}p{\ell_{w}^{q}}v$. Since $\mathcal{F}:\DecompSp{\mathcal{Q}}p{\ell_{w}^{q}}v\to\FourierDecompSp{\mathcal{Q}}p{\ell_{w}^{q}}v$
is an isomorphism, this implies $R\circ{\rm Ana}_{\Gamma}=\operatorname{id}_{\DecompSp{\mathcal{Q}}p{\ell_{w}^{q}}v}$,
as desired.
\end{proof}
\section{Fully Discrete Banach Frames}
\label{sec:FullyDiscreteBanachFrames}In the previous section, we
obtained \emph{semi-discrete} Banach frames for $\DecompSp{\mathcal{Q}}p{\ell_{w}^{q}}v$;
in particular, we showed
\[
\left\Vert f\right\Vert _{\DecompSp{\mathcal{Q}}p{\ell_{w}^{q}}v}\asymp\left\Vert \left(\left\Vert \smash{\gamma^{\left(i\right)}}\ast f\right\Vert _{V_{i}}\right)_{i\in I}\right\Vert _{\ell_{w}^{q}}.
\]
We call such a frame \textbf{semi-discrete}, because while the index
set $I$ is discrete, the convolutions $\gamma^{\left(i\right)}\ast f$
are treated as genuine functions, which are defined on the continuous
(non-discrete) index set $\mathbb{R}^{d}$.
In this section, our aim is a further discretization of these frames,
so that we will in the end obtain a (quasi)-norm equivalence of the
form
\begin{equation}
\left\Vert f\right\Vert _{\DecompSp{\mathcal{Q}}p{\ell_{w}^{q}}v}\asymp\left\Vert \left(\left\Vert \left[\left(\gamma^{\left[j\right]}\ast f\right)\!\!\left(\delta\cdot T_{j}^{-T}k\right)\right]_{k\in\mathbb{Z}^{d}}\right\Vert _{C_{j}^{\left(\delta\right)}}\right)_{\!\!j\in I}\right\Vert _{\ell_{u_{q}}^{q}}\qquad\text{ for a suitable weight }u_{q}\text{ on }I,\label{eq:DesiredBanachFrameNormEquivalence}
\end{equation}
where for each $\delta\in\left(0,1\right]$ and $j\in I$, the \textbf{coefficient
space} $C_{j}^{\left(\delta\right)}$ is given by
\begin{equation}
C_{j}^{\left(\delta\right)}:=\ell_{v^{\left(j,\delta\right)}}^{p}\left(\smash{\mathbb{Z}^{d}}\right)\quad\text{ with }\quad v_{k}^{\left(j,\delta\right)}=v\left(\delta\cdot T_{j}^{-T}k\right)\quad\text{ for }k\in\mathbb{Z}^{d}.\label{eq:CoefficientSpaceDefinition}
\end{equation}
For simplicity, the reader should keep in mind the important special
case $v\equiv1$, for which $C_{j}^{\left(\delta\right)}=\ell^{p}\left(\mathbb{Z}^{d}\right)$,
independently of $j,\delta$.
To ensure that equation (\ref{eq:DesiredBanachFrameNormEquivalence})
holds, we will introduce suitable assumptions on $\left(\gamma_{i}\right)_{i\in I}$
and $\delta>0$. In particular, for the formula above to make sense,
we also have to establish (at least) continuity of $\gamma^{\left(j\right)}\ast f$
(and thus of $\gamma^{\left[j\right]}\ast f$), so that the pointwise
evaluations $\left(\gamma^{\left[j\right]}\ast f\right)\left(\delta\cdot T_{j}^{-T}k\right)$
are meaningful; note that up to now, we only know (for $p\in\left[1,\infty\right]$)
that $\gamma^{\left[j\right]}\ast f\in L_{v}^{p}\left(\mathbb{R}^{d}\right)$.
To ensure this continuity, we introduce a new set of additional assumptions
and notations. In these assumptions, the $L_{v_{0}}^{p}$ (quasi)-norm
of certain vector valued functions $g:\mathbb{R}^{d}\to\mathbb{C}^{k}$
is considered. This has to be understood as $\left\Vert g\right\Vert _{L_{v_{0}}^{p}}:=\left\Vert \,\left|g\right|\,\right\Vert _{L_{v_{0}}^{p}}$,
where as usual $\left|g\right|\left(x\right):=\left|g\left(x\right)\right|=\left\Vert g\left(x\right)\right\Vert _{2}$
denotes the euclidean norm of $g\left(x\right)$. Furthermore, for
such a function $g=\left(g_{1},\dots,g_{k}\right)$, expressions as
the (inverse) Fourier transform $\mathcal{F}^{-1}g:=\left(\mathcal{F}^{-1}g_{1},\dots,\mathcal{F}^{-1}g_{k}\right)$
are always understood in a coordinatewise sense.
\begin{assumption}
\label{assu:DiscreteBanachFrameAssumptions}In addition to Assumption
\ref{assu:MainAssumptions}, we assume the following:
\begin{enumerate}
\item We have $\gamma_{i}\in C^{1}\left(\mathbb{R}^{d}\right)$ for all
$i\in I$ and the gradient $\phi_{i}:=\nabla\gamma_{i}$ satisfies
the following:
\begin{enumerate}
\item $\phi_{i}$ is bounded for each $i\in I$,
\item We have $\phi_{i}\in L_{v_{0}}^{1}\left(\mathbb{R}^{d};\mathbb{C}^{d}\right)\hookrightarrow L^{1}\left(\mathbb{R}^{d};\mathbb{C}^{d}\right)$
for all $i\in I$,
\item We have $\widehat{\phi_{i}}\in C^{\infty}\left(\mathbb{R}^{d};\mathbb{C}^{d}\right)$
for all $i\in I$.
\end{enumerate}
\item For $j\in I$, we define
\[
\phi^{\left(j\right)}:=\mathcal{F}^{-1}\left(\widehat{\phi_{j}}\circ S_{j}^{-1}\right)=\left|\det T_{j}\right|\cdot M_{b_{j}}\left[\phi_{j}\circ T_{j}^{T}\right],
\]
so that $\phi^{\left(j\right)}$ is to $\phi_{j}$ as $\gamma^{\left(j\right)}$
is to $\gamma_{j}$.
\item For $j,i\in I$, set
\[
B_{j,i}:=\begin{cases}
\left(1+\left\Vert T_{j}^{-1}T_{i}\right\Vert \right)^{K+d}\cdot\left\Vert \mathcal{F}^{-1}\left(\varphi_{i}\cdot\widehat{\phi^{\left(j\right)}}\right)\right\Vert _{L_{v_{0}}^{1}}, & \text{if }p\in\left[1,\infty\right],\\
\left(1+\left\Vert T_{j}^{-1}T_{i}\right\Vert \right)^{pK+d}\cdot\left|\det T_{i}\right|^{1-p}\cdot\left\Vert \mathcal{F}^{-1}\left(\varphi_{i}\cdot\widehat{\phi^{\left(j\right)}}\right)\right\Vert _{L_{v_{0}}^{p}}^{p}, & \text{if }p\in\left(0,1\right).
\end{cases}
\]
\item With
\[
r:=\max\left\{ q,\frac{q}{p}\right\} =\begin{cases}
q, & \text{if }p\in\left[1,\infty\right],\\
\frac{q}{p}, & \text{if }p\in\left(0,1\right)
\end{cases}
\]
as in Assumption \ref{assu:MainAssumptions}, we assume that the operator
$\overrightarrow{B}$ induced by $\left(B_{j,i}\right)_{j,i\in I}$,
i.e.\@
\[
\overrightarrow{B}\left(c_{i}\right)_{i\in I}:=\left(\sum_{i\in I}B_{j,i}\,c_{i}\right)_{j\in I}
\]
defines a well-defined, bounded operator $\overrightarrow{B}:\ell_{w^{\min\left\{ 1,p\right\} }}^{r}\left(I\right)\to\ell_{w^{\min\left\{ 1,p\right\} }}^{r}\left(I\right)$.
\item For $j\in I$ and $\delta\in\left(0,1\right]$, we let the $j$-th
\textbf{coefficient space} $C_{j}^{\left(\delta\right)}$ be defined
as in equation (\ref{eq:CoefficientSpaceDefinition}) and set
\[
W_{j}\!:=\!\left\{ f:\mathbb{R}^{d}\to\mathbb{C}\,\middle|\, f\text{ continuous and }\left\Vert f\right\Vert _{W_{j}}\!<\!\infty\right\} ,
\]
where
\[
\left\Vert f\right\Vert _{W_{j}}:=\left\Vert f\right\Vert _{V_{j}}+\sup_{0<\delta\leq1}\!\frac{1}{\delta}\left\Vert \osc{\delta\cdot T_{j}^{-T}\left[-1,1\right]^{d}}\left[M_{-b_{j}}f\right]\right\Vert _{V_{j}}.
\]
\item Finally, we define
\[
\ell_{w}^{q}\left(\smash{\left[W_{j}\right]_{j\in I}}\right):=\left\{ \left(f_{j}\right)_{j\in I}\,\middle|\,\left(\forall j\in I:\:f_{j}\in W_{j}\right)\text{ and }\left(\smash{\left\Vert f_{j}\right\Vert _{W_{j}}}\right)_{j\in I}\in\ell_{w}^{q}\left(I\right)\right\} ,
\]
equipped with the natural (quasi)-norm $\left\Vert \smash{\left(f_{j}\right)_{j\in I}}\right\Vert _{\ell_{w}^{q}\left(\left[W_{j}\right]_{j\in I}\right)}:=\left\Vert \left(\smash{\left\Vert f_{j}\right\Vert _{W_{j}}}\vphantom{\sum}\right)_{j\in I}\right\Vert _{\ell_{w}^{q}}$.\qedhere
\end{enumerate}
\end{assumption}
\begin{rem*}
Note that $\varphi_{i}\cdot\widehat{\phi^{\left(j\right)}}\in\TestFunctionSpace{\mathbb{R}^{d}}$
for all $i,j\in I$, since $\varphi_{i}\in\TestFunctionSpace{\mathbb{R}^{d}}$
and since $\widehat{\phi^{\left(j\right)}}=\widehat{\phi_{j}}\circ S_{j}^{-1}$
is smooth, because $\widehat{\phi_{j}}$ is. Hence, $\mathcal{F}^{-1}\left(\varphi_{i}\cdot\widehat{\phi^{\left(j\right)}}\right)\in\mathcal{S}\left(\mathbb{R}^{d}\right)$,
so that $B_{j,i}<\infty$ for all $i,j\in I$, cf.\@ Lemma \ref{lem:SchwartzFunctionsAreWiener}.
Although we again stated the assumption in the general case where
the prototype $\gamma_{i}$ may depend on $i\in I$, the reader should
keep in mind the most important special case where $\gamma_{i}=\gamma$
is independent of $i\in I$.
\end{rem*}
Given these assumptions, we now want to show in particular that $\gamma^{\left[j\right]}\ast f$
is continuous for each $f\in\DecompSp{\mathcal{Q}}p{\ell_{w}^{q}}v$. The
following lemma makes an important step in that direction.
\begin{lem}
\label{lem:MainOscillationEstimate}Assume that $\Gamma=\left(\gamma_{i}\right)_{i\in I}$
satisfies Assumption \ref{assu:DiscreteBanachFrameAssumptions}.
Then the following hold:
For $p\in\left[1,\infty\right]$ and
\[
C:=\frac{2^{6d}}{\sqrt{d}}\cdot\left(1152\cdotd^{5/2}\cdot\left\lceil K+d+1\right\rceil \right)^{\left\lceil K\right\rceil +d+2}\cdot\Omega_{0}^{2K}\Omega_{1}^{2}\cdot\left(1+R_{\mathcal{Q}}\right)^{d},
\]
we have
\[
\left\Vert {\rm osc}_{\delta\cdot T_{j}^{-T}\left[-1,1\right]^{d}}\left[M_{-b_{j}}\mathcal{F}^{-1}\left(\widehat{\gamma^{\left(j\right)}}\cdot\varphi_{i}\widehat{f}\right)\right]\right\Vert _{L_{v}^{p}}\leq C\cdot\delta\cdot\left(1+\left\Vert T_{j}^{-1}T_{i}\right\Vert \right)^{K+d}\cdot\left\Vert \mathcal{F}^{-1}\left(\varphi_{i}\cdot\smash{\widehat{\phi^{\left(j\right)}}}\right)\right\Vert _{L_{v_{0}}^{1}}\cdot\left\Vert \mathcal{F}^{-1}\left(\varphi_{i}^{\ast}\smash{\widehat{f}}\,\right)\right\Vert _{L_{v}^{p}}
\]
for all $0<\delta\leq1$, all $f\in Z'\left(\mathcal{O}\right)$ and all
$i,j\in I$.
Likewise, for $p\in\left(0,1\right)$ and
\[
C:=\frac{2^{16\frac{d}{p}}\cdot\left(1+C_{\mathcal{Q}}R_{\mathcal{Q}}\right)^{\frac{2d}{p}}}{370\cdotd^{11/2}\cdotd^{d/2p}}\cdot\left(4032\cdotd^{3}\cdot\left\lceil K+\frac{d+1}{p}\right\rceil \right)^{2\left\lceil K+\frac{d+1}{p}\right\rceil +2}\cdot\Omega_{0}^{5K}\Omega_{1}^{5},
\]
we have
\begin{align*}
& \left\Vert {\rm osc}_{\delta\cdot T_{j}^{-T}\left[-1,1\right]^{d}}\!\left[M_{-b_{j}}\mathcal{F}^{-1}\!\left(\varphi_{i}\widehat{\gamma^{\left(j\right)}}\smash{\widehat{f}}\right)\right]\right\Vert _{W_{T_{j}^{-T}\left[-1,1\right]^{d}}\left(L_{v}^{p}\right)}\\
& \leq\vphantom{\sum^{j}}C\cdot\delta\cdot\left|\det T_{i}\right|^{\frac{1}{p}-1}\cdot\left(1+\left\Vert T_{j}^{-1}T_{i}\right\Vert \right)^{K+\frac{d}{p}}\cdot\left\Vert \mathcal{F}^{-1}\!\left(\varphi_{i}\smash{\widehat{\phi^{\left(j\right)}}}\right)\right\Vert _{L_{v_{0}}^{p}}\cdot\left\Vert \mathcal{F}^{-1}\!\left(\varphi_{i}^{\ast}\smash{\widehat{f}}\,\right)\right\Vert _{L_{v}^{p}}
\end{align*}
for all $0<\delta\leq1$, all $f\in Z'\left(\mathcal{O}\right)$ and all
$i,j\in I$.
\end{lem}
\begin{proof}
First of all, note that $\widehat{f}\in\mathcal{D}'\left(\mathcal{O}\right)$
for $f\in Z'\left(\mathcal{O}\right)$. Because of $\varphi_{i}\in\TestFunctionSpace{\mathcal{O}}$,
this implies that $\varphi_{i}\cdot\widehat{f}\in\mathcal{S}'\left(\mathbb{R}^{d}\right)$
is a well-defined tempered distribution with compact support, so that
the same holds for $\varphi_{i}\cdot\widehat{\gamma^{\left(j\right)}}\cdot\widehat{f}$,
since $\widehat{\gamma^{\left(j\right)}}\in C^{\infty}\left(\mathbb{R}^{d}\right)$.
Hence, by the Paley-Wiener theorem, $\mathcal{F}^{-1}\left(\widehat{\gamma^{\left(j\right)}}\cdot\varphi_{i}\cdot\widehat{f}\right)$
is a smooth (even analytic) function with polynomially bounded derivatives
of all orders. In particular, expressions like $\left({\rm osc}_{\delta\cdot T_{j}^{-T}\left[-1,1\right]^{d}}\!\left[M_{-b_{j}}\mathcal{F}^{-1}\!\left(\varphi_{i}\widehat{\gamma^{\left(j\right)}}\smash{\widehat{f}}\right)\right]\right)\left(x\right)$
are well-defined for every $x\in\mathbb{R}^{d}$.
Let $f\in Z'\left(\mathcal{O}\right)$ be arbitrary. We can clearly assume
$\left\Vert \mathcal{F}^{-1}\left(\varphi_{i}^{\ast}\cdot\smash{\widehat{f}}\,\right)\right\Vert _{L_{v}^{p}}<\infty$,
since otherwise, the claim is trivial. Now, note that $\widehat{\gamma^{\left(j\right)}}\cdot\varphi_{i}\in\TestFunctionSpace{\mathbb{R}^{d}}\subset\mathcal{S}\left(\mathbb{R}^{d}\right)$
and $\varphi_{i}^{\ast}\widehat{f}\in\mathcal{S}'\left(\mathbb{R}^{d}\right)$,
as well as
\begin{align*}
M_{-b_{j}}\left[\mathcal{F}^{-1}\left(\widehat{\gamma^{\left(j\right)}}\cdot\varphi_{i}\cdot\widehat{f}\right)\right] & =\mathcal{F}^{-1}\left[L_{-b_{j}}\left(\widehat{\gamma^{\left(j\right)}}\varphi_{i}\cdot\varphi_{i}^{\ast}\widehat{f}\right)\right]\\
& =\mathcal{F}^{-1}\left[L_{-b_{j}}\left(\widehat{\gamma^{\left(j\right)}}\cdot\varphi_{i}\right)\right]\ast\mathcal{F}^{-1}\left[L_{-b_{j}}\left(\varphi_{i}^{\ast}\smash{\widehat{f}}\,\right)\right].
\end{align*}
In particular, the convolution is pointwise well-defined, so that
Lemma \ref{lem:OscillationConvolution} shows
\begin{equation}
{\rm osc}_{\delta\cdot T_{j}^{-T}\left[-1,1\right]^{d}}M_{-b_{j}}\left[\mathcal{F}^{-1}\left(\widehat{\gamma^{\left(j\right)}}\cdot\varphi_{i}\smash{\widehat{f}}\,\right)\right]\leq\left(\osc{\delta\cdot T_{j}^{-T}\left[-1,1\right]^{d}}\mathcal{F}^{-1}\left[L_{-b_{j}}\left(\smash{\widehat{\gamma^{\left(j\right)}}}\cdot\varphi_{i}\right)\right]\right)\ast\left|\mathcal{F}^{-1}\left[L_{-b_{j}}\left(\varphi_{i}^{\ast}\smash{\widehat{f}}\,\right)\right]\right|.\label{eq:MainOscillationEstimateOscillationToTheLeft}
\end{equation}
\medskip{}
Now, for $p\in\left(0,1\right)$, we want to apply Proposition \ref{prop:AlternativeWienerAmalgamConvolution}
with $Q_{1}=T_{i}^{-T}\left[-1,1\right]^{d}$, $Q_{2}=T_{j}^{-T}\left[-1,1\right]^{d}$
and
\[
g=\left|\mathcal{F}^{-1}\left[L_{-b_{j}}\left(\varphi_{i}^{\ast}\widehat{f}\right)\right]\right|,\qquad\text{ as well as }\qquad f=\osc{\delta\cdot T_{j}^{-T}\left[-1,1\right]^{d}}\mathcal{F}^{-1}\left[L_{-b_{j}}\left(\widehat{\gamma^{\left(j\right)}}\cdot\varphi_{i}\right)\right].
\]
To this end, first note just as in the proof of Corollary \ref{cor:WienerAmalgamConvolutionSimplified}
(cf.\@ equation (\ref{eq:LinearImageOfCubePartition})) that with
this choice of $Q_{1}$ and suitable choices of $\left(x_{i}\right)_{i\in\mathbb{Z}^{d}}$,
the constant $N$ from Theorem \ref{thm:WienerAmalgamConvolution}
satisfies $N\leq3^{d}$.
Next, we use the identities $Q_{1}-Q_{1}=T_{i}^{-T}\left[-2,2\right]^{d}$
and $\left|\mathcal{F}^{-1}\left[L_{b}h\right]\right|=\left|M_{b}\left[\mathcal{F}^{-1}h\right]\right|=\left|\mathcal{F}^{-1}h\right|$
and equation (\ref{eq:WienerLinearCubeEnlargement}), as well as
Theorem \ref{thm:BandlimitedWienerAmalgamSelfImproving} to get
\begin{align*}
\left\Vert \,\left|\mathcal{F}^{-1}\left[L_{-b_{j}}\left(\varphi_{i}^{\ast}\smash{\widehat{f}}\,\right)\right]\right|\,\right\Vert _{W_{Q_{1}-Q_{1}}\left(L_{v}^{p}\right)} & =\left\Vert \mathcal{F}^{-1}\left(\varphi_{i}^{\ast}\smash{\widehat{f}}\,\right)\right\Vert _{W_{T_{i}^{-T}\left[-2,2\right]^{d}}\left(L_{v}^{p}\right)}\\
\left({\scriptstyle \text{eq. }\eqref{eq:WienerLinearCubeEnlargement}}\right) & \leq\Omega_{0}^{K}\Omega_{1}\cdot\left(18d\right)^{K+\frac{d}{p}}\cdot\left\Vert \mathcal{F}^{-1}\left(\varphi_{i}^{\ast}\smash{\widehat{f}}\,\right)\right\Vert _{W_{T_{i}^{-T}\left[-1,1\right]^{d}}\left(L_{v}^{p}\right)}\\
\left({\scriptstyle \text{Theorem }\ref{thm:BandlimitedWienerAmalgamSelfImproving}}\right) & \leq\Omega_{0}^{K}\Omega_{1}\cdot\left(18d\right)^{K+\frac{d}{p}}C_{1}\cdot\left\Vert \mathcal{F}^{-1}\left(\varphi_{i}^{\ast}\smash{\widehat{f}}\,\right)\right\Vert _{L_{v}^{p}}
\end{align*}
for
\[
C_{1}:=2^{4\left(1+\frac{d}{p}\right)}s_{d}^{\frac{1}{p}}\left(192\cdotd^{3/2}\cdot\left\lceil K+\frac{d+1}{p}\right\rceil \right)^{\left\lceil K+\frac{d+1}{p}\right\rceil +1}\cdot\Omega_{0}^{K}\Omega_{1}\cdot\left(1+\left(1+2C_{\mathcal{Q}}\right)R_{\mathcal{Q}}\right)^{\frac{d}{p}},
\]
since \cite[Lemma 2.7]{DecompositionEmbedding} shows $\operatorname{supp}\left(\varphi_{i}^{\ast}\widehat{f}\right)\subset\overline{Q_{i}^{\ast}}\subset T_{i}\left[\overline{B_{R}}\left(0\right)\right]+b_{i}\subset T_{i}\left[-R,R\right]^{d}+b_{i}$
for $R=\left(1+2C_{\mathcal{Q}}\right)R_{\mathcal{Q}}$.
All in all, we now set $C_{2}:=\Omega_{0}^{K}\Omega_{1}\cdot\left(18d\right)^{K+\frac{d}{p}}C_{1}$
and use equation (\ref{eq:MainOscillationEstimateOscillationToTheLeft}),
Proposition \ref{prop:AlternativeWienerAmalgamConvolution}, and
the identity $\widehat{\gamma^{\left(j\right)}}=L_{b_{j}}\left(\widehat{\gamma_{j}}\circ T_{j}^{-1}\right)$
to conclude because of
\begin{align*}
\sup_{x\in Q_{1}}v_{0}\left(x\right) & \leq\Omega_{1}\cdot\sup_{x\in T_{i}^{-T}\left[-1,1\right]^{d}}\left(1+\left|x\right|\right)^{K}\\
\left({\scriptstyle \text{eq. }\eqref{eq:WeightLinearTransformationsConnection}}\right) & \leq\Omega_{0}^{K}\Omega_{1}\cdot\sup_{x\in T_{i}^{-T}\left[-1,1\right]^{d}}\left(1+\left|T_{i}^{T}x\right|\right)^{K}\\
& =\Omega_{0}^{K}\Omega_{1}\cdot\sup_{y\in\left[-1,1\right]^{d}}\left(1+\left|y\right|\right)^{K}\\
& \leq\Omega_{0}^{K}\Omega_{1}\left(1+\sqrt{d}\right)^{K}
\end{align*}
that
\begin{align*}
& \left\Vert {\rm osc}_{\delta\cdot T_{j}^{-T}\left[-1,1\right]^{d}}\left[M_{-b_{j}}\mathcal{F}^{-1}\left(\widehat{\gamma^{\left(j\right)}}\cdot\varphi_{i}\widehat{f}\right)\right]\right\Vert _{W_{T_{j}^{-T}\left[-1,1\right]^{d}}\left(L_{v}^{p}\right)}\\
& \leq3^{\frac{d}{p}}\Omega_{0}^{K}\Omega_{1}\left(1+\sqrt{d}\right)^{K}\cdot\left[\lambda_{d}\left(Q_{1}\right)\right]^{1-\frac{1}{p}}\cdot\left\Vert \left|\mathcal{F}^{-1}\left[L_{-b_{j}}\left(\varphi_{i}^{\ast}\smash{\widehat{f}}\,\right)\right]\right|\right\Vert _{W_{Q_{1}-Q_{1}}\left(L_{v}^{p}\right)}\\
& \phantom{\leq}\cdot\left\Vert {\rm osc}_{\delta\cdot T_{j}^{-T}\left[-1,1\right]^{d}}\left(\mathcal{F}^{-1}\left[L_{-b_{j}}\left(\widehat{\gamma^{\left(j\right)}}\cdot\varphi_{i}\right)\right]\right)\right\Vert _{W_{T_{j}^{-T}\left[-1,1\right]^{d}-T_{i}^{-T}\left[-1,1\right]^{d}}\left(L_{v_{0}}^{p}\right)}\\
& \leq3^{\frac{d}{p}}\Omega_{0}^{K}\Omega_{1}\left(1+\sqrt{d}\right)^{K}C_{2}\cdot2^{d\left(1-\frac{1}{p}\right)}\left|\det T_{i}\right|^{\frac{1}{p}-1}\cdot\left\Vert \mathcal{F}^{-1}\left(\varphi_{i}^{\ast}\smash{\widehat{f}}\,\right)\right\Vert _{L_{v}^{p}}\!\\
& \phantom{=}\cdot\left\Vert {\rm osc}_{\delta\cdot T_{j}^{-T}\left[-1,1\right]^{d}}\left(\mathcal{F}^{-1}\left[L_{-b_{j}}\left(L_{b_{j}}\left(\widehat{\gamma_{j}}\circ T_{j}^{-1}\right)\cdot\varphi_{i}\right)\right]\right)\right\Vert _{W_{T_{j}^{-T}\left[-1,1\right]^{d}-T_{i}^{-T}\left[-1,1\right]^{d}}\left(L_{v_{0}}^{p}\right)}\\
& \leq\left(2\sqrt{d}\right)^{K}3^{\frac{d}{p}}C_{2}\cdot\Omega_{0}^{K}\Omega_{1}\cdot\left|\det T_{i}\right|^{\frac{1}{p}-1}\cdot\left\Vert \mathcal{F}^{-1}\left(\varphi_{i}^{\ast}\smash{\widehat{f}}\,\right)\right\Vert _{L_{v}^{p}}\\
& \phantom{\leq}\cdot\left\Vert {\rm osc}_{\delta\cdot T_{j}^{-T}\left[-1,1\right]^{d}}\left(\mathcal{F}^{-1}\left[\left(\widehat{\gamma_{j}}\cdot\left[\left(L_{-b_{j}}\varphi_{i}\right)\circ T_{j}\right]\right)\circ T_{j}^{-1}\right]\right)\right\Vert _{W_{T_{j}^{-T}\left[-1,1\right]^{d}-T_{i}^{-T}\left[-1,1\right]^{d}}\left(L_{v_{0}}^{p}\right)}.
\end{align*}
Now, we recall $\phi_{j}=\nabla\gamma_{j}$ and estimate
\begin{align*}
& \left\Vert {\rm osc}_{\delta\cdot T_{j}^{-T}\left[-1,1\right]^{d}}\left(\mathcal{F}^{-1}\left[\left(\widehat{\gamma_{j}}\cdot\left[\left(L_{-b_{j}}\varphi_{i}\right)\circ T_{j}\right]\right)\circ T_{j}^{-1}\right]\right)\right\Vert _{W_{T_{j}^{-T}\left[-1,1\right]^{d}-T_{i}^{-T}\left[-1,1\right]^{d}}\left(L_{v_{0}}^{p}\right)}\\
& =\left|\det T_{j}\right|\cdot\left\Vert {\rm osc}_{\delta\cdot T_{j}^{-T}\left[-1,1\right]^{d}}\left[\left(\mathcal{F}^{-1}\left[\widehat{\gamma_{j}}\cdot\left(\left[L_{-b_{j}}\varphi_{i}\right]\circ T_{j}\right)\right]\right)\circ T_{j}^{T}\right]\right\Vert _{W_{T_{j}^{-T}\left[-1,1\right]^{d}-T_{i}^{-T}\left[-1,1\right]^{d}}\left(L_{v_{0}}^{p}\right)}\\
\left({\scriptstyle \text{Lem. }\ref{lem:OscillationLinearChange},\,\ref{lem:WienerTransformationFormula}}\right) & =\left|\det T_{j}\right|\cdot\left\Vert \left(M_{\left[-1,1\right]^{d}-T_{j}^{T}T_{i}^{-T}\left[-1,1\right]^{d}}\left[{\rm osc}_{\delta\left[-1,1\right]^{d}}\left(\mathcal{F}^{-1}\left[\widehat{\gamma_{j}}\cdot\left(\left[L_{-b_{j}}\varphi_{i}\right]\circ T_{j}\right)\right]\right)\right]\right)\circ T_{j}^{T}\right\Vert _{L_{v_{0}}^{p}}\\
& =\left|\det T_{j}\right|^{1-\frac{1}{p}}\left\Vert \left(v_{0}\circ T_{j}^{-T}\right)\cdot M_{\left[-1,1\right]^{d}-T_{j}^{T}T_{i}^{-T}\left[-1,1\right]^{d}}\left[{\rm osc}_{\delta\left[-1,1\right]^{d}}\left(\mathcal{F}^{-1}\left[\widehat{\gamma_{j}}\cdot\left(\left[L_{-b_{j}}\varphi_{i}\right]\circ T_{j}\right)\right]\right)\right]\right\Vert _{L^{p}}\\
\left({\scriptstyle \text{Lem. }\ref{lem:OscillationEstimatedByWienerDerivative}}\right) & \leq2\sqrt{d}\cdot\delta\cdot\left|\det T_{j}\right|^{1-\frac{1}{p}}\!\cdot\!\left\Vert \left(v_{0}\circ T_{j}^{-T}\right)\!\cdot M_{\left[-1,1\right]^{d}-T_{j}^{T}T_{i}^{-T}\left[-1,1\right]^{d}}\!\left[M_{\delta\left[-1,1\right]^{d}}\nabla\left(\mathcal{F}^{-1}\!\left[\widehat{\gamma_{j}}\cdot\!\left(\left[L_{-b_{j}}\varphi_{i}\right]\!\circ\!T_{j}\right)\right]\right)\right]\right\Vert _{L^{p}}\!\!\!.
\end{align*}
Since we have $\delta\leq1$, Lemma \ref{lem:IteratedMaximalFunction}
allows us to continue the estimate as follows:
\begin{align}
\dots & \leq2\sqrt{d}\cdot\delta\cdot\left|\det T_{j}\right|^{1-\frac{1}{p}}\!\cdot\!\left\Vert \left(v_{0}\circ T_{j}^{-T}\right)\!\cdot M_{\left[-2,2\right]^{d}-T_{j}^{T}T_{i}^{-T}\left[-1,1\right]^{d}}\left[\nabla\left(\gamma_{j}\ast\mathcal{F}^{-1}\left[\left(L_{-b_{j}}\varphi_{i}\right)\circ T_{j}\right]\right)\right]\right\Vert _{L^{p}}\nonumber \\
\left({\scriptstyle \nabla\left(f\ast g\right)=\left(\nabla f\right)\ast g}\right) & \overset{\left(\ast\right)}{=}2\sqrt{d}\cdot\delta\cdot\left|\det T_{j}\right|^{1-\frac{1}{p}}\!\cdot\!\left\Vert \left(v_{0}\circ T_{j}^{-T}\right)\!\cdot M_{\left[-2,2\right]^{d}-T_{j}^{T}T_{i}^{-T}\left[-1,1\right]^{d}}\left(\phi_{j}\ast\mathcal{F}^{-1}\left[\left(L_{-b_{j}}\varphi_{i}\right)\circ T_{j}\right]\right)\right\Vert _{L^{p}}\nonumber \\
& =2\sqrt{d}\cdot\delta\cdot\left|\det T_{j}\right|^{1-\frac{1}{p}}\!\cdot\!\left\Vert \left(v_{0}\circ T_{j}^{-T}\right)\!\cdot M_{\left[-2,2\right]^{d}-T_{j}^{T}T_{i}^{-T}\left[-1,1\right]^{d}}\!\left[\mathcal{F}^{-1}\!\left(\left[\left(\widehat{\phi_{j}}\circ T_{j}^{-1}\right)\!\cdot\!\left(L_{-b_{j}}\varphi_{i}\right)\right]\circ T_{j}\right)\right]\right\Vert _{L^{p}}\nonumber \\
& =2\sqrt{d}\cdot\delta\cdot\left|\det T_{j}\right|^{-\frac{1}{p}}\!\cdot\!\left\Vert \left(v_{0}\circ T_{j}^{-T}\right)\!\cdot M_{\left[-2,2\right]^{d}-T_{j}^{T}T_{i}^{-T}\left[-1,1\right]^{d}}\!\left[\left(\mathcal{F}^{-1}\!\left[\left(\widehat{\phi_{j}}\circ T_{j}^{-1}\right)\!\cdot\!\left(L_{-b_{j}}\varphi_{i}\right)\right]\right)\!\circ T_{j}^{-T}\right]\right\Vert _{L^{p}}\nonumber \\
\left({\scriptstyle \text{Lemma }\ref{lem:WienerTransformationFormula}}\right) & =2\sqrt{d}\cdot\delta\cdot\left|\det T_{j}\right|^{-\frac{1}{p}}\!\cdot\!\left\Vert \left[v_{0}\cdot M_{T_{j}^{-T}\left[-2,2\right]^{d}-T_{i}^{-T}\left[-1,1\right]^{d}}\left(\mathcal{F}^{-1}\left[\left(\widehat{\phi_{j}}\circ T_{j}^{-1}\right)\cdot\left(L_{-b_{j}}\varphi_{i}\right)\right]\right)\right]\circ T_{j}^{-T}\right\Vert _{L^{p}}\nonumber \\
& =2\sqrt{d}\cdot\delta\cdot\left\Vert \mathcal{F}^{-1}\left[\left(\widehat{\phi_{j}}\circ T_{j}^{-1}\right)\cdot\left(L_{-b_{j}}\varphi_{i}\right)\right]\right\Vert _{W_{T_{j}^{-T}\left[-2,2\right]^{d}-T_{i}^{-T}\left[-1,1\right]^{d}}^{d}\left(L_{v_{0}}^{p}\right)}\label{eq:MainOscillationEstimateQuasiBanachCalculationPart1}
\end{align}
Here, the step marked with $\left(\ast\right)$ is justified, since
$\mathcal{F}^{-1}\left[\left(L_{-b_{j}}\varphi_{i}\right)\circ T_{j}\right]\in\mathcal{S}\left(\mathbb{R}^{d}\right)$
and since $\gamma_{j}\in L^{1}\left(\mathbb{R}^{d}\right)\cap C^{1}\left(\mathbb{R}^{d}\right)$,
where $\phi_{j}=\nabla\gamma_{j}$ is bounded by Assumptions \ref{assu:MainAssumptions}
and \ref{assu:DiscreteBanachFrameAssumptions}.
Next, we observe
\begin{align*}
T_{j}^{-T}\left[-2,2\right]^{d}-T_{i}^{-T}\left[-1,1\right]^{d} & =T_{i}^{-T}\left(T_{i}^{T}T_{j}^{-T}\left[-2,2\right]^{d}-\left[-1,1\right]^{d}\right)\\
& \subset T_{i}^{-T}\left(2\left\Vert \left(T_{j}^{-1}T_{i}\right)^{T}\right\Vert _{\ell^{\infty}\to\ell^{\infty}}\left[-1,1\right]^{d}-\left[-1,1\right]^{d}\right)\\
& \subset T_{i}^{-T}\left[-\left(1+2\left\Vert T_{j}^{-1}T_{i}\right\Vert _{\ell^{1}\to\ell^{1}}\right),\,1+2\left\Vert T_{j}^{-1}T_{i}\right\Vert _{\ell^{1}\to\ell^{1}}\right]^{d}.
\end{align*}
Consequently, if we set $R:=1+2\left\Vert T_{j}^{-1}T_{i}\right\Vert _{\ell^{1}\to\ell^{1}}$
for brevity, then Corollary \ref{cor:WienerLinearCubeNormEstimate}
(with $v_{0}$ instead of $v$, with $i=j$ and with $L=1$) yields
for arbitrary measurable $h:\mathbb{R}^{d}\to\mathbb{C}^{k}$ the estimate
\begin{align*}
\left\Vert h\right\Vert _{W_{T_{j}^{-T}\left[-2,2\right]^{d}-T_{i}^{-T}\left[-1,1\right]^{d}}^{k}\left(L_{v_{0}}^{p}\right)} & \leq\left\Vert h\right\Vert _{W_{T_{i}^{-T}\left[-R,R\right]^{d}}^{k}\left(L_{v_{0}}^{p}\right)}\\
& \leq\Omega_{0}^{K}\Omega_{1}\cdot\left[3d\left(1+1+2\left\Vert T_{j}^{-1}T_{i}\right\Vert _{\ell^{1}\to\ell^{1}}\right)\right]^{K+\frac{d}{p}}\cdot\left(1+1\right)^{K+\frac{d}{p}}\cdot\left\Vert h\right\Vert _{W_{T_{i}^{-T}\left[-1,1\right]^{d}}^{k}\left(L_{v_{0}}^{p}\right)}\\
\left({\scriptstyle \text{since }\left\Vert A\right\Vert _{\ell^{1}\to\ell^{1}}\leq\sqrt{d}\left\Vert A\right\Vert }\right) & \leq\Omega_{0}^{K}\Omega_{1}\cdot\left[12\cdotd^{\frac{3}{2}}\left(1+\left\Vert T_{j}^{-1}T_{i}\right\Vert \right)\right]^{K+\frac{d}{p}}\cdot\left\Vert h\right\Vert _{W_{T_{i}^{-T}\left[-1,1\right]^{d}}^{k}\left(L_{v_{0}}^{p}\right)}.
\end{align*}
Now, we use this estimate and standard properties of the Fourier transform
to further estimate the right-hand side of equation (\ref{eq:MainOscillationEstimateQuasiBanachCalculationPart1})
as follows:
\begin{align}
\text{r.h.s.}\eqref{eq:MainOscillationEstimateQuasiBanachCalculationPart1} & =2\sqrt{d}\cdot\delta\cdot\left\Vert \mathcal{F}^{-1}\left(L_{-b_{j}}\left[\varphi_{i}\cdot L_{b_{j}}\left(\widehat{\phi_{j}}\circ T_{j}^{-1}\right)\right]\right)\right\Vert _{W_{T_{j}^{-T}\left[-2,2\right]^{d}-T_{i}^{-T}\left[-1,1\right]^{d}}^{d}\left(L_{v_{0}}^{p}\right)}\nonumber \\
\left({\scriptstyle \widehat{\phi^{\left(j\right)}}=L_{b_{j}}\left(\widehat{\phi_{j}}\circ T_{j}^{-1}\right)}\right) & =2\sqrt{d}\cdot\delta\cdot\left\Vert \mathcal{F}^{-1}\left(L_{-b_{j}}\left[\varphi_{i}\cdot\widehat{\phi^{\left(j\right)}}\right]\right)\right\Vert _{W_{T_{j}^{-T}\left[-2,2\right]^{d}-T_{i}^{-T}\left[-1,1\right]^{d}}^{d}\left(L_{v_{0}}^{p}\right)}\nonumber \\
\left({\scriptstyle \left|\mathcal{F}^{-1}\left[L_{b}h\right]\right|=\left|\mathcal{F}^{-1}h\right|}\right) & =2\sqrt{d}\cdot\delta\cdot\left\Vert \mathcal{F}^{-1}\left[\widehat{\phi^{\left(j\right)}}\cdot\varphi_{i}\right]\right\Vert _{W_{T_{j}^{-T}\left[-2,2\right]^{d}-T_{i}^{-T}\left[-1,1\right]^{d}}^{d}\left(L_{v_{0}}^{p}\right)}\nonumber \\
& \leq2\sqrt{d}\cdot\Omega_{0}^{K}\Omega_{1}\cdot\left[12\cdotd^{\frac{3}{2}}\left(1+\left\Vert T_{j}^{-1}T_{i}\right\Vert \right)\right]^{K+\frac{d}{p}}\cdot\delta\cdot\left\Vert \mathcal{F}^{-1}\left[\widehat{\phi^{\left(j\right)}}\cdot\varphi_{i}\right]\right\Vert _{W_{T_{i}^{-T}\left[-1,1\right]^{d}}^{d}\left(L_{v_{0}}^{p}\right)}.\label{eq:MainOscillationEstimateQuasiBanachCalculationPart2}
\end{align}
Recall that we are in the case $p\in\left(0,1\right)$. In particular,
we have $\left|y\right|\leq\left\Vert y\right\Vert _{\ell^{p}}$ for
each $y\in\mathbb{R}^{d}$. For a vector-valued function $f:\mathbb{R}^{d}\to\mathbb{R}^{k}$
and any (measurable) weight $u:\mathbb{R}^{d}\to\left(0,\infty\right)$,
this implies
\begin{align*}
\left\Vert f\right\Vert _{W_{Q}^{k}\left(L_{u}^{p}\right)}^{p} & =\int_{\mathbb{R}^{d}}\left[u\left(x\right)\cdot\left|\left(M_{Q}f\right)\left(x\right)\right|\right]^{p}\operatorname{d} x\\
& =\int_{\mathbb{R}^{d}}\left[u\left(x\right)\right]^{p}\cdot\esssup_{y\in x+Q}\left|f\left(y\right)\right|^{p}\operatorname{d} x\\
& \leq\int_{\mathbb{R}^{d}}\left[u\left(x\right)\right]^{p}\cdot\esssup_{y\in x+Q}\left\Vert f\left(y\right)\right\Vert _{\ell^{p}}^{p}\operatorname{d} x\\
& =\int_{\mathbb{R}^{d}}\left[u\left(x\right)\right]^{p}\cdot\esssup_{y\in x+Q}\sum_{\ell=1}^{k}\left|f_{\ell}\left(y\right)\right|^{p}\operatorname{d} x\\
& \leq\sum_{\ell=1}^{k}\int_{\mathbb{R}^{d}}\left[u\left(x\right)\right]^{p}\cdot\esssup_{y\in x+Q}\left|f_{\ell}\left(y\right)\right|^{p}\operatorname{d} x\\
& \leq k\cdot\max_{\ell\in\underline{k}}\left\Vert f_{\ell}\right\Vert _{W_{Q}\left(L_{u}^{p}\right)}^{p}.
\end{align*}
In other words, we have shown
\begin{equation}
\left\Vert f\right\Vert _{W_{Q}^{k}\left(L_{u}^{p}\right)}\leq k^{1/p}\cdot\max_{\ell\in\underline{k}}\left\Vert f_{\ell}\right\Vert _{W_{Q}\left(L_{u}^{p}\right)}.\label{eq:VectorValuedWienerEstimateQuasiBanach}
\end{equation}
Using this inequality (with $k=d$), we can further estimate
the right-hand side of equation (\ref{eq:MainOscillationEstimateQuasiBanachCalculationPart2})
as follows:
\begin{align*}
\text{r.h.s.}\eqref{eq:MainOscillationEstimateQuasiBanachCalculationPart2} & \leq2d^{\frac{1}{2}+\frac{1}{p}}\cdot\Omega_{0}^{K}\Omega_{1}\cdot\left[12\cdotd^{\frac{3}{2}}\left(1+\left\Vert T_{j}^{-1}T_{i}\right\Vert \right)\right]^{K+\frac{d}{p}}\cdot\delta\cdot\max_{\ell\in\underline{d}}\left\Vert \mathcal{F}^{-1}\left(\left[\vphantom{\phi^{\left(j\right)}}\smash{\widehat{\phi^{\left(j\right)}}}\right]_{\ell}\cdot\varphi_{i}\right)\right\Vert _{W_{T_{i}^{-T}\left[-1,1\right]^{d}}\left(L_{v_{0}}^{p}\right)}\\
\left({\scriptstyle \text{Theorem }\ref{thm:BandlimitedWienerAmalgamSelfImproving}}\right) & \leq2d^{\frac{1}{2}+\frac{1}{p}}\cdot\Omega_{0}^{K}\Omega_{1}\cdot\left[12\cdotd^{\frac{3}{2}}\left(1+\left\Vert T_{j}^{-1}T_{i}\right\Vert \right)\right]^{K+\frac{d}{p}}C_{3}\cdot\delta\cdot\max_{\ell\in\underline{d}}\left\Vert \mathcal{F}^{-1}\left(\left[\vphantom{\phi^{\left(j\right)}}\smash{\widehat{\phi^{\left(j\right)}}}\right]_{\ell}\cdot\varphi_{i}\right)\right\Vert _{L_{v_{0}}^{p}}\\
\left({\scriptstyle \text{since }d\leq2^{d}}\right) & \leq2d^{\frac{1}{2}}2^{\frac{d}{p}}\cdot\Omega_{0}^{K}\Omega_{1}\cdot\left[12\cdotd^{\frac{3}{2}}\left(1+\left\Vert T_{j}^{-1}T_{i}\right\Vert \right)\right]^{K+\frac{d}{p}}C_{3}\cdot\delta\cdot\left\Vert \mathcal{F}^{-1}\left[\widehat{\phi^{\left(j\right)}}\cdot\varphi_{i}\right]\right\Vert _{L_{v_{0}}^{p}}\\
& =:C_{4}\cdot\delta\cdot\left(1+\left\Vert T_{j}^{-1}T_{i}\right\Vert \right)^{K+\frac{d}{p}}\cdot\left\Vert \mathcal{F}^{-1}\left[\widehat{\phi^{\left(j\right)}}\cdot\varphi_{i}\right]\right\Vert _{L_{v_{0}}^{p}}.
\end{align*}
Here,
\[
C_{3}=2^{4\left(1+\frac{d}{p}\right)}s_{d}^{\frac{1}{p}}\left(192\cdotd^{3/2}\cdot\left\lceil K+\frac{d+1}{p}\right\rceil \right)^{\left\lceil K+\frac{d+1}{p}\right\rceil +1}\cdot\Omega_{0}^{K}\Omega_{1}\cdot\left(1+R_{\mathcal{Q}}\right)^{\frac{d}{p}},
\]
cf.\@ Theorem \ref{thm:BandlimitedWienerAmalgamSelfImproving}, since
we have for arbitrary $\ell\in\underline{d}$ that
\[
\operatorname{supp}\left(\left[\vphantom{\phi^{\left(j\right)}}\smash{\widehat{\phi^{\left(j\right)}}}\right]_{\ell}\cdot\varphi_{i}\right)\subset\overline{Q_{i}}\subset T_{i}\left[\overline{B_{R_{\mathcal{Q}}}}\left(0\right)\right]+b_{i}\subset T_{i}\left[-R_{\mathcal{Q}},R_{\mathcal{Q}}\right]^{d}+b_{i}.
\]
Putting everything together, we arrive at
\begin{align*}
& \left\Vert {\rm osc}_{\delta\cdot T_{j}^{-T}\left[-1,1\right]^{d}}\left[M_{-b_{j}}\mathcal{F}^{-1}\left(\widehat{\gamma^{\left(j\right)}}\cdot\varphi_{i}\smash{\widehat{f}}\,\right)\right]\right\Vert _{W_{T_{j}^{-T}\left[-1,1\right]^{d}}\left(L_{v}^{p}\right)}\\
& \leq\left(2\sqrt{d}\right)^{K}3^{\frac{d}{p}}C_{2}\Omega_{0}^{K}\Omega_{1}\cdot\left|\det T_{i}\right|^{\frac{1}{p}-1}\cdot\left\Vert \mathcal{F}^{-1}\!\left(\varphi_{i}^{\ast}\smash{\widehat{f}}\,\right)\right\Vert _{L_{v}^{p}}\\
& \phantom{\leq}\cdot\left\Vert {\rm osc}_{\delta\cdot T_{j}^{-T}\left[-1,1\right]^{d}}\left(\mathcal{F}^{-1}\!\left[\left(\widehat{\gamma_{j}}\cdot\left[\left(L_{-b_{j}}\varphi_{i}\right)\circ T_{j}\right]\right)\!\circ\!T_{j}^{-1}\right]\right)\right\Vert _{W_{T_{j}^{-T}\left[-1,1\right]^{d}-T_{i}^{-T}\left[-1,1\right]^{d}}\left(L_{v_{0}}^{p}\right)}\\
& \leq\left(2\sqrt{d}\right)^{K}3^{\frac{d}{p}}C_{2}C_{4}\cdot\Omega_{0}^{K}\Omega_{1}\cdot\left|\det T_{i}\right|^{\frac{1}{p}-1}\cdot\delta\cdot\left(1+\left\Vert T_{j}^{-1}T_{i}\right\Vert \right)^{K+\frac{d}{p}}\cdot\left\Vert \mathcal{F}^{-1}\left(\varphi_{i}^{\ast}\smash{\widehat{f}}\,\right)\right\Vert _{L_{v}^{p}}\cdot\left\Vert \mathcal{F}^{-1}\left[\widehat{\phi^{\left(j\right)}}\cdot\varphi_{i}\right]\right\Vert _{L_{v_{0}}^{p}}.
\end{align*}
This establishes the claim for $p\in\left(0,1\right)$, since we have
$C_{\mathcal{Q}}\geq\left\Vert T_{i}^{-1}T_{i}\right\Vert \geq1$ and $s_{d}\leq2^{2d}$
and hence
\begin{align*}
& \left(2\sqrt{d}\right)^{K}3^{\frac{d}{p}}C_{2}C_{4}\cdot\Omega_{0}^{K}\Omega_{1}\\
& =C_{1}\cdot2^{5}d^{1/2}\cdot\left(2\sqrt{d}\right)^{K}96^{\frac{d}{p}}\cdot\left(216\cdotd^{\frac{5}{2}}\right)^{K+\frac{d}{p}}\cdot s_{d}^{\frac{1}{p}}\left(192\cdotd^{3/2}\cdot\left\lceil K\!+\!\frac{d+1}{p}\right\rceil \right)^{\left\lceil K+\frac{d+1}{p}\right\rceil +1}\cdot\Omega_{0}^{4K}\Omega_{1}^{4}\cdot\left(1\!+\!R_{\mathcal{Q}}\right)^{\frac{d}{p}}\\
& \leq2^{9}d^{\frac{1}{2}}\cdot2^{15\frac{d}{p}}\left(2\sqrt{d}\right)^{-\frac{d}{p}}\!\cdot\left(432\cdotd^{3}\right)^{K+\frac{d}{p}}\!\cdot\!\left(192\cdotd^{3/2}\cdot\left\lceil K\!+\!\frac{d+1}{p}\right\rceil \right)^{2\left\lceil K+\frac{d+1}{p}\right\rceil +2}\cdot\Omega_{0}^{5K}\Omega_{1}^{5}\cdot\left(1\!+\!R_{\mathcal{Q}}\right)^{\frac{d}{p}}\left(1\!+\!3C_{\mathcal{Q}}R_{\mathcal{Q}}\right)^{\frac{d}{p}}\\
& \leq2^{9}d^{1/2}\cdot2^{17\frac{d}{p}}\left(2\sqrt{d}\right)^{-\frac{d}{p}}\cdot\left(21\cdotd^{3/2}\right)^{-4}\cdot\left(4032\cdotd^{3}\cdot\left\lceil K\!+\!\frac{d+1}{p}\right\rceil \right)^{2\left\lceil K+\frac{d+1}{p}\right\rceil +2}\cdot\Omega_{0}^{5K}\Omega_{1}^{5}\cdot\left(1\!+\!C_{\mathcal{Q}}R_{\mathcal{Q}}\right)^{\frac{2d}{p}}\\
& \leq\frac{2^{16\frac{d}{p}}\cdot\left(1+C_{\mathcal{Q}}R_{\mathcal{Q}}\right)^{\frac{2d}{p}}}{370\cdotd^{11/2}\cdotd^{d/2p}}\cdot\left(4032\cdotd^{3}\cdot\left\lceil K+\frac{d+1}{p}\right\rceil \right)^{2\left\lceil K+\frac{d+1}{p}\right\rceil +2}\cdot\Omega_{0}^{5K}\Omega_{1}^{5}.
\end{align*}
\medskip{}
For $p\in\left[1,\infty\right]$, the proof is simpler: We use the
weighted Young inequality (equation (\ref{eq:WeightedYoungInequality}))
and equation (\ref{eq:MainOscillationEstimateOscillationToTheLeft})
to derive
\begin{align*}
& \left\Vert {\rm osc}_{\delta\cdot T_{j}^{-T}\left[-1,1\right]^{d}}\left(M_{-b_{j}}\left[\mathcal{F}^{-1}\left(\widehat{\gamma^{\left(j\right)}}\cdot\varphi_{i}\widehat{f}\right)\right]\right)\right\Vert _{L_{v}^{p}}\\
\left({\scriptstyle \text{eqs. }\eqref{eq:MainOscillationEstimateOscillationToTheLeft},\eqref{eq:WeightedYoungInequality}}\right) & \leq\left\Vert \osc{\delta\cdot T_{j}^{-T}\left[-1,1\right]^{d}}\mathcal{F}^{-1}\left[L_{-b_{j}}\left(\widehat{\gamma^{\left(j\right)}}\cdot\varphi_{i}\right)\right]\right\Vert _{L_{v_{0}}^{1}}\cdot\left\Vert \mathcal{F}^{-1}\left[L_{-b_{j}}\left(\varphi_{i}^{\ast}\smash{\widehat{f}}\,\right)\right]\right\Vert _{L_{v}^{p}}\\
\left({\scriptstyle \widehat{\gamma^{\left(j\right)}}=L_{b_{j}}\left(\widehat{\gamma_{j}}\circ T_{j}^{-1}\right)}\right) & =\left\Vert \osc{\delta\cdot T_{j}^{-T}\left[-1,1\right]^{d}}\mathcal{F}^{-1}\left[\left(\widehat{\gamma_{j}}\circ T_{j}^{-1}\right)\cdot\left(L_{-b_{j}}\varphi_{i}\right)\right]\right\Vert _{L_{v_{0}}^{1}}\cdot\left\Vert \mathcal{F}^{-1}\left[L_{-b_{j}}\left(\varphi_{i}^{\ast}\smash{\widehat{f}}\,\right)\right]\right\Vert _{L_{v}^{p}}\\
\left({\scriptstyle \left|\mathcal{F}^{-1}\left[L_{b}h\right]\right|=\left|\mathcal{F}^{-1}h\right|}\right) & =\left\Vert \osc{\delta\cdot T_{j}^{-T}\left[-1,1\right]^{d}}\mathcal{F}^{-1}\left(\left[\widehat{\gamma_{j}}\cdot\left(\left[L_{-b_{j}}\varphi_{i}\right]\circ T_{j}\right)\right]\circ T_{j}^{-1}\right)\right\Vert _{L_{v_{0}}^{1}}\cdot\left\Vert \mathcal{F}^{-1}\left(\varphi_{i}^{\ast}\smash{\widehat{f}}\,\right)\right\Vert _{L_{v}^{p}}\\
& =\left|\det T_{j}\right|\cdot\left\Vert \osc{\delta\cdot T_{j}^{-T}\left[-1,1\right]^{d}}\left[\left(\mathcal{F}^{-1}\left[\widehat{\gamma_{j}}\cdot\left(\left[L_{-b_{j}}\varphi_{i}\right]\circ T_{j}\right)\right]\right)\circ T_{j}^{T}\right]\right\Vert _{L_{v_{0}}^{1}}\cdot\left\Vert \mathcal{F}^{-1}\left(\varphi_{i}^{\ast}\smash{\widehat{f}}\,\right)\right\Vert _{L_{v}^{p}}\\
\left({\scriptstyle \text{Lemma }\ref{lem:OscillationLinearChange}}\right) & =\left|\det T_{j}\right|\cdot\left\Vert \left(\osc{\delta\cdot\left[-1,1\right]^{d}}\mathcal{F}^{-1}\left[\widehat{\gamma_{j}}\cdot\left(\left[L_{-b_{j}}\varphi_{i}\right]\circ T_{j}\right)\right]\right)\circ T_{j}^{T}\right\Vert _{L_{v_{0}}^{1}}\cdot\left\Vert \mathcal{F}^{-1}\left(\varphi_{i}^{\ast}\smash{\widehat{f}}\,\right)\right\Vert _{L_{v}^{p}}\\
& =\left\Vert \left(v_{0}\circ T_{j}^{-T}\right)\cdot\osc{\delta\cdot\left[-1,1\right]^{d}}\left[\gamma_{j}\ast\mathcal{F}^{-1}\left(\left[L_{-b_{j}}\varphi_{i}\right]\circ T_{j}\right)\right]\right\Vert _{L^{1}}\cdot\left\Vert \mathcal{F}^{-1}\left(\varphi_{i}^{\ast}\smash{\widehat{f}}\,\right)\right\Vert _{L_{v}^{p}}\\
\left({\scriptstyle \text{Lemma }\ref{lem:OscillationEstimatedByWienerDerivative}}\right) & \leq2\delta\sqrt{d}\cdot\left\Vert \left(v_{0}\circ T_{j}^{-T}\right)\cdot M_{\delta\left[-1,1\right]^{d}}\left(\nabla\left[\gamma_{j}\ast\mathcal{F}^{-1}\left(\left[L_{-b_{j}}\varphi_{i}\right]\circ T_{j}\right)\right]\right)\right\Vert _{L^{1}}\cdot\left\Vert \mathcal{F}^{-1}\left(\varphi_{i}^{\ast}\smash{\widehat{f}}\,\right)\right\Vert _{L_{v}^{p}}\\
\left({\scriptstyle \text{since }\delta\leq1}\right) & \leq2\delta\sqrt{d}\cdot\left\Vert \left(v_{0}\circ T_{j}^{-T}\right)\cdot M_{\left[-1,1\right]^{d}}\left(\nabla\left[\gamma_{j}\ast\mathcal{F}^{-1}\left(\left[L_{-b_{j}}\varphi_{i}\right]\circ T_{j}\right)\right]\right)\right\Vert _{L^{1}}\cdot\left\Vert \mathcal{F}^{-1}\left(\varphi_{i}^{\ast}\smash{\widehat{f}}\,\right)\right\Vert _{L_{v}^{p}}\\
\left({\scriptstyle \nabla\left(\gamma\ast h\right)=\left(\nabla\gamma\right)\ast h}\right) & =2\delta\sqrt{d}\cdot\left\Vert \left(v_{0}\circ T_{j}^{-T}\right)\cdot M_{\left[-1,1\right]^{d}}\left[\left(\nabla\gamma_{j}\right)\ast\left(\mathcal{F}^{-1}\left[\left(L_{-b_{j}}\varphi_{i}\right)\circ T_{j}\right]\right)\right]\right\Vert _{L^{1}}\cdot\left\Vert \mathcal{F}^{-1}\left(\varphi_{i}^{\ast}\smash{\widehat{f}}\,\right)\right\Vert _{L_{v}^{p}}.
\end{align*}
Here, the last step is justified just as for $p\in\left(0,1\right)$.
Now, we recall $\phi_{j}=\nabla\gamma_{j}$ and continue our estimate:
\begin{align*}
\dots & =2\delta\sqrt{d}\cdot\left\Vert \left(v_{0}\circ T_{j}^{-T}\right)\cdot M_{\left[-1,1\right]^{d}}\left[\mathcal{F}^{-1}\left(\widehat{\phi_{j}}\cdot\left[\left(L_{-b_{j}}\varphi_{i}\right)\circ T_{j}\right]\right)\right]\right\Vert _{L^{1}}\cdot\left\Vert \mathcal{F}^{-1}\left(\varphi_{i}^{\ast}\smash{\widehat{f}}\,\right)\right\Vert _{L_{v}^{p}}\\
& =2\delta\sqrt{d}\cdot\left\Vert \left(v_{0}\circ T_{j}^{-T}\right)\cdot M_{\left[-1,1\right]^{d}}\left[\mathcal{F}^{-1}\left(\left[\left(\widehat{\phi_{j}}\circ T_{j}^{-1}\right)\cdot\left(L_{-b_{j}}\varphi_{i}\right)\right]\circ T_{j}\right)\right]\right\Vert _{L^{1}}\cdot\left\Vert \mathcal{F}^{-1}\left(\varphi_{i}^{\ast}\smash{\widehat{f}}\,\right)\right\Vert _{L_{v}^{p}}\\
& =2\delta\sqrt{d}\cdot\left|\det T_{j}\right|^{-1}\left\Vert \left(v_{0}\!\circ\!T_{j}^{-T}\right)\!\cdot\!M_{\left[-1,1\right]^{d}}\!\left[\!\left(\mathcal{F}^{-1}\!\left[L_{-b_{j}}\!\left[\varphi_{i}\cdot L_{b_{j}}\!\left(\widehat{\phi_{j}}\circ T_{j}^{-1}\right)\!\right]\right]\right)\!\circ\!T_{j}^{-T}\right]\!\right\Vert _{L^{1}}\left\Vert \mathcal{F}^{-1}\left(\varphi_{i}^{\ast}\smash{\widehat{f}}\,\right)\right\Vert _{L_{v}^{p}}\\
\left({\scriptstyle \text{Lem. }\ref{lem:WienerTransformationFormula}}\right) & =2\delta\sqrt{d}\cdot\left|\det T_{j}\right|^{-1}\left\Vert \left[v_{0}\!\cdot\!M_{T_{j}^{-T}\left[-1,1\right]^{d}}\left(\mathcal{F}^{-1}\!\left[L_{-b_{j}}\left[\varphi_{i}\cdot L_{b_{j}}\!\left(\widehat{\phi_{j}}\circ T_{j}^{-1}\right)\right]\right]\right)\right]\circ T_{j}^{-T}\right\Vert _{L^{1}}\left\Vert \mathcal{F}^{-1}\left(\varphi_{i}^{\ast}\smash{\widehat{f}}\,\right)\right\Vert _{L_{v}^{p}}\\
& =2\delta\sqrt{d}\cdot\left\Vert \mathcal{F}^{-1}\!\left(L_{-b_{j}}\left[\varphi_{i}\cdot\widehat{\phi^{\left(j\right)}}\right]\right)\right\Vert _{W_{T_{j}^{-T}\left[-1,1\right]^{d}}^{d}\left(L_{v_{0}}^{1}\right)}\left\Vert \mathcal{F}^{-1}\left(\varphi_{i}^{\ast}\smash{\widehat{f}}\,\right)\right\Vert _{L_{v}^{p}}\\
& =2\delta\sqrt{d}\cdot\left\Vert \mathcal{F}^{-1}\left[\varphi_{i}\cdot\widehat{\phi^{\left(j\right)}}\right]\right\Vert _{W_{T_{j}^{-T}\left[-1,1\right]^{d}}^{d}\left(L_{v_{0}}^{1}\right)}\cdot\left\Vert \mathcal{F}^{-1}\left(\varphi_{i}^{\ast}\smash{\widehat{f}}\,\right)\right\Vert _{L_{v}^{p}}.
\end{align*}
Here, the last step used that $\left|\mathcal{F}^{-1}\left[L_{b}h\right]\right|=\left|\mathcal{F}^{-1}h\right|$.
Now, we need an analog of equation (\ref{eq:VectorValuedWienerEstimateQuasiBanach})
for the case $p\in\left[1,\infty\right]$. But for an arbitrary (measurable)
weight $u:\mathbb{R}^{d}\to\left(0,\infty\right)$ and any $q\in\left[1,\infty\right]$,
the solidity of $W_{Q}\left(L_{u}^{q}\right)$ and the triangle inequality
for the associated norm yield for any measurable vector-valued function
$f=\left(f_{1},\dots,f_{k}\right):\mathbb{R}^{d}\to\mathbb{C}^{k}$ that
\begin{equation}
\left\Vert f\right\Vert _{W_{Q}^{k}\left(L_{u}^{q}\right)}=\left\Vert \left|f\right|\right\Vert _{W_{Q}\left(L_{u}^{q}\right)}\leq\left\Vert \sum_{\ell=1}^{k}\left|f_{\ell}\right|\right\Vert _{W_{Q}\left(L_{u}^{q}\right)}\leq\sum_{\ell=1}^{k}\left\Vert f_{\ell}\right\Vert _{W_{Q}\left(L_{u}^{q}\right)}\leq k\cdot\max_{\ell\in\underline{k}}\left\Vert f_{\ell}\right\Vert _{W_{Q}\left(L_{u}^{q}\right)}.\label{eq:VectorValuedWienerEstimateBanach}
\end{equation}
We now use this estimate (with $k=d$), as well as equation
(\ref{eq:WienerLinearCubeTransformationChange}) and Theorem \ref{thm:BandlimitedWienerAmalgamSelfImproving}
(both with $v_{0}$ instead of $v$) to conclude
\begin{align*}
\left\Vert \mathcal{F}^{-1}\left[\varphi_{i}\cdot\widehat{\phi^{\left(j\right)}}\right]\right\Vert _{W_{T_{j}^{-T}\cdot\left[-1,1\right]^{d}}^{d}\left(L_{v_{0}}^{1}\right)} & \leq\Omega_{0}^{K}\Omega_{1}\cdot\left(6d\right)^{K+d}\!\cdot\!\left(1\!+\!\left\Vert T_{j}^{-1}T_{i}\right\Vert \right)^{\!K+d}\cdot\left\Vert \mathcal{F}^{-1}\left[\varphi_{i}\cdot\widehat{\phi^{\left(j\right)}}\right]\right\Vert _{W_{T_{i}^{-T}\left[-1,1\right]^{d}}^{d}\left(L_{v_{0}}^{1}\right)}\\
& \leqd\cdot\Omega_{0}^{K}\Omega_{1}\cdot\left(6d\right)^{K+d}\!\cdot\!\left(1\!+\!\left\Vert T_{j}^{-1}T_{i}\right\Vert \right)^{\!K+d}\cdot\max_{\ell\in\underline{d}}\left\Vert \mathcal{F}^{-1}\left[\varphi_{i}\cdot\left(\widehat{\phi^{\left(j\right)}}\right)_{\ell}\right]\right\Vert _{W_{T_{i}^{-T}\left[-1,1\right]^{d}}\left(L_{v_{0}}^{1}\right)}\\
\left({\scriptstyle \text{Thm. }\ref{thm:BandlimitedWienerAmalgamSelfImproving}}\right) & \leq C_{5}\cdotd\cdot\Omega_{0}^{K}\Omega_{1}\cdot\left(6d\right)^{K+d}\!\cdot\!\left(1\!+\!\left\Vert T_{j}^{-1}T_{i}\right\Vert \right)^{\!K+d}\cdot\left\Vert \mathcal{F}^{-1}\left[\varphi_{i}\cdot\widehat{\phi^{\left(j\right)}}\right]\right\Vert _{L_{v_{0}}^{1}}.
\end{align*}
Here, Theorem \ref{thm:BandlimitedWienerAmalgamSelfImproving} is
applicable, since we have $\operatorname{supp}\left(\varphi_{i}\cdot\left(\widehat{\phi^{\left(j\right)}}\right)_{\ell}\right)\subset\overline{Q_{i}}\subset T_{i}\left[-R_{\mathcal{Q}},R_{\mathcal{Q}}\right]^{d}+b_{i}$.
Hence, that theorem justifies the last step in the estimate above,
with constant
\[
C_{5}:=2^{4\left(1+d\right)}s_{d}\left(192\cdotd^{3/2}\cdot\left\lceil K+d+1\right\rceil \right)^{\left\lceil K+d+1\right\rceil +1}\cdot\Omega_{0}^{K}\Omega_{1}\cdot\left(1+R_{\mathcal{Q}}\right)^{d}.
\]
It is not hard to see that this implies the claim for $p\in\left[1,\infty\right]$.
\end{proof}
Next, we show that the map ${\rm Ana}_{\Gamma}$ considered in Theorem
\ref{thm:ConvolvingDecompositionSpaceWithGammaJ} is not merely bounded
as a map into $\ell_{w}^{q}\left(\left[V_{j}\right]_{j\in I}\right)$,
but even as a map into the smaller space $\ell_{w}^{q}\left(\left[W_{j}\right]_{j\in I}\right)$.
In particular, this establishes continuity of $\gamma^{\left(j\right)}\ast f$
for every $j\in I$ and arbitrary $f\in\DecompSp{\mathcal{Q}}p{\ell_{w}^{q}}v$.
\begin{lem}
\label{lem:OscillationForFree}Let $p,q\in\left(0,\infty\right]$
and assume that $\Gamma=\left(\gamma_{i}\right)_{i\in I}$ fulfills
Assumption \ref{assu:DiscreteBanachFrameAssumptions}.
Then, the map
\[
{\rm Ana}_{{\rm osc}}:\DecompSp{\mathcal{Q}}p{\ell_{w}^{q}}v\to\ell_{w}^{q}\left(\left[W_{j}\right]_{j\in I}\right),f\mapsto\left(\gamma^{\left(j\right)}\ast f\right)_{j\in I}
\]
is well-defined and bounded, with
\[
\vertiii{{\rm Ana}_{{\rm osc}}}\leq C\cdot2^{\max\left\{ 0,\frac{1}{q}-1\right\} }\vertiii{\smash{\Gamma_{\mathcal{Q}}}}\cdot\left(\vertiii{\smash{\overrightarrow{A}}}^{\max\left\{ 1,\frac{1}{p}\right\} }+\vertiii{\smash{\overrightarrow{B}}}^{\max\left\{ 1,\frac{1}{p}\right\} }\right),
\]
where $\Gamma_{\mathcal{Q}}:\ell_{w}^{q}\left(I\right)\to\ell_{w}^{q}\left(I\right),c\mapsto c^{\ast}$
denotes the $\mathcal{Q}$-clustering map, i.e., $c_{i}^{\ast}=\sum_{\ell\in i^{\ast}}c_{\ell}$
and where
\[
C:=\begin{cases}
N_{\mathcal{Q}}^{\frac{1}{p}-1}\cdot\frac{2^{16\frac{d}{p}}\cdot\left(1+C_{\mathcal{Q}}R_{\mathcal{Q}}\right)^{\frac{2d}{p}}}{370\cdotd^{11/2}\cdotd^{d/2p}}\cdot\left(4032\cdotd^{3}\cdot\left\lceil K+\frac{d+1}{p}\right\rceil \right)^{2\left\lceil K+\frac{d+1}{p}\right\rceil +2}\cdot\Omega_{0}^{5K}\Omega_{1}^{5}, & \text{if }p\in\left(0,1\right),\\
\frac{2^{6d}}{\sqrt{d}}\cdot\left(1152\cdotd^{5/2}\cdot\left\lceil K+d+1\right\rceil \right)^{\left\lceil K\right\rceil +d+2}\cdot\Omega_{0}^{2K}\Omega_{1}^{2}\cdot\left(1+R_{\mathcal{Q}}\right)^{d}, & \text{if }p\in\left[1,\infty\right].
\end{cases}
\]
Furthermore, we have
\begin{equation}
\left(\gamma^{\left(j\right)}\ast f\right)\left(x\right)=\sum_{i\in I}\mathcal{F}^{-1}\left(\widehat{\gamma^{\left(j\right)}}\cdot\varphi_{i}\cdot\widehat{f}\right)\left(x\right)\qquad\forall x\in\mathbb{R}^{d}\qquad\forall f\in\DecompSp{\mathcal{Q}}p{\ell_{w}^{q}}v,\label{eq:SpecialConvolutionPointwiseDefinition}
\end{equation}
with \emph{locally} uniform convergence of the series.
\end{lem}
\begin{proof}
Recall from Theorem \ref{thm:ConvolvingDecompositionSpaceWithGammaJ}
and from the ensuing remark (which contains the definition of $\gamma^{\left(j\right)}\ast f$)
that
\begin{equation}
\left(\gamma^{\left(j\right)}\ast f\right)\left(x\right)=\sum_{i\in I}\mathcal{F}^{-1}\left(\widehat{\gamma^{\left(j\right)}}\cdot\varphi_{i}\cdot\widehat{f}\right)\left(x\right)\qquad\forall f\in\DecompSp{\mathcal{Q}}p{\ell_{w}^{q}}v,\label{eq:OscillationForFreeSeriesRepresentation}
\end{equation}
where we already know that the series converges absolutely almost
everywhere. Next, note that each of the summands of the series above
is a smooth function; this follows from the Paley-Wiener theorem,
since $\widehat{\gamma^{\left(j\right)}}\cdot\varphi_{i}\cdot\widehat{f}$
is a (tempered) distribution with compact support. Thus, to prove
continuity of $\gamma^{\left(j\right)}\ast f$, it suffices to show
that the series actually converges \emph{locally} uniformly; by continuity
of the summands, for this it suffices to have convergence in $L_{\left(1+\left|\bullet\right|\right)^{-K}}^{\infty}\left(\mathbb{R}^{d}\right)$.
We will prove this convergence in $L_{\left(1+\left|\bullet\right|\right)^{-K}}^{\infty}\left(\mathbb{R}^{d}\right)$
simultaneously with the boundedness of ${\rm Ana}_{{\rm osc}}$.
\medskip{}
Let us first consider the case $p\in\left[1,\infty\right]$. Here,
we let $C_{1}>0$ be the constant provided by Lemma \ref{lem:MainOscillationEstimate}
(for $p\in\left[1,\infty\right]$), so that we get for arbitrary
$0<\delta\leq1$ the estimate
\begin{align}
& \frac{1}{\delta}\sum_{i\in I}\left\Vert \osc{\delta\cdot T_{j}^{-T}\left[-1,1\right]^{d}}\left(M_{-b_{j}}\left[\mathcal{F}^{-1}\left(\widehat{\gamma^{\left(j\right)}}\cdot\varphi_{i}\cdot\widehat{f}\right)\right]\right)\right\Vert _{L_{v}^{p}}\nonumber \\
\left({\scriptstyle \text{Lemma }\ref{lem:MainOscillationEstimate}}\right) & \leq C_{1}\cdot\sum_{i\in I}\left[\left(1+\left\Vert T_{j}^{-1}T_{i}\right\Vert \right)^{K+d}\cdot\left\Vert \mathcal{F}^{-1}\left[\varphi_{i}\cdot\widehat{\phi^{\left(j\right)}}\right]\right\Vert _{L_{v_{0}}^{1}}\cdot\left\Vert \mathcal{F}^{-1}\left(\varphi_{i}^{\ast}\smash{\widehat{f}}\,\right)\right\Vert _{L_{v}^{p}}\right]\nonumber \\
& =C_{1}\cdot\sum_{i\in I}\left[B_{j,i}\cdot c_{i}\right]=C_{1}\cdot\left(\overrightarrow{B}c\right)_{j},\label{eq:OscillationSumEstimateBanachCase}
\end{align}
where we defined $c_{i}:=\left\Vert \mathcal{F}^{-1}\left(\varphi_{i}^{\ast}\smash{\widehat{f}}\,\right)\right\Vert _{L_{v}^{p}}$
for all $i\in I$.
Setting $d_{i}:=\left\Vert \mathcal{F}^{-1}\left(\varphi_{i}\smash{\widehat{f}}\,\right)\right\Vert _{L_{v}^{p}}$
for $i\in I$ and using the triangle inequality for $L_{v}^{p}$,
we get $c_{i}\leq\left(\Gamma_{\mathcal{Q}}\,d\right)_{i}$ for $i\in I$.
By solidity of $\ell_{w}^{q}\left(I\right)$, this allows us to conclude
\begin{align}
C_{1}\cdot\left\Vert \overrightarrow{B}c\right\Vert _{\ell_{w}^{q}} & \leq C_{1}\cdot\vertiii{\smash{\overrightarrow{B}}}\cdot\left\Vert c\right\Vert _{\ell_{w}^{q}}\nonumber \\
& \leq C_{1}\cdot\vertiii{\smash{\overrightarrow{B}}}\cdot\left\Vert \Gamma_{\mathcal{Q}}\,d\right\Vert _{\ell_{w}^{q}}\nonumber \\
& \leq C_{1}\cdot\vertiii{\smash{\Gamma_{\mathcal{Q}}}}\cdot\vertiii{\smash{\overrightarrow{B}}}\cdot\left\Vert d\right\Vert _{\ell_{w}^{q}}\nonumber \\
& =C_{1}\cdot\vertiii{\smash{\Gamma_{\mathcal{Q}}}}\cdot\vertiii{\smash{\overrightarrow{B}}}\cdot\left\Vert f\right\Vert _{\DecompSp{\mathcal{Q}}p{\ell_{w}^{q}}v}<\infty.\label{eq:OscillationForFreeSequenceNormEstimateBanachCase}
\end{align}
In particular, we get $\left(\smash{\overrightarrow{B}}c\right)_{j}<\infty$
for all $j\in I$, so that the right-hand side of equation (\ref{eq:OscillationSumEstimateBanachCase})
is finite. We now use this estimate for $\delta=1$: For arbitrary
$x\in\mathbb{R}^{d}$ and $a\in T_{j}^{-T}\left[-1,1\right]^{d}$,
we have $x,\,x+a\in x+T_{j}^{-T}\left[-1,1\right]^{d}$ and
hence
\begin{align*}
& \left|\left[\mathcal{F}^{-1}\left(\widehat{\gamma^{\left(j\right)}}\cdot\varphi_{i}\cdot\widehat{f}\right)\right]\left(x+a\right)\right|\\
& \leq\left|\left(M_{-b_{j}}\left[\mathcal{F}^{-1}\left(\widehat{\gamma^{\left(j\right)}}\cdot\varphi_{i}\cdot\widehat{f}\right)\right]\right)\left(x+a\right)-\left(M_{-b_{j}}\left[\mathcal{F}^{-1}\left(\widehat{\gamma^{\left(j\right)}}\cdot\varphi_{i}\cdot\widehat{f}\right)\right]\right)\left(x\right)\right|+\left|\left(M_{-b_{j}}\left[\mathcal{F}^{-1}\left(\widehat{\gamma^{\left(j\right)}}\cdot\varphi_{i}\cdot\widehat{f}\right)\right]\right)\left(x\right)\right|\\
& \leq\left[\osc{T_{j}^{-T}\left[-1,1\right]^{d}}\left(M_{-b_{j}}\left[\mathcal{F}^{-1}\left(\widehat{\gamma^{\left(j\right)}}\cdot\varphi_{i}\cdot\widehat{f}\right)\right]\right)\right]\left(x\right)+\left|\left[\mathcal{F}^{-1}\left(\widehat{\gamma^{\left(j\right)}}\cdot\varphi_{i}\cdot\widehat{f}\right)\right]\left(x\right)\right|,
\end{align*}
which yields
\[
M_{T_{j}^{-T}\left[-1,1\right]^{d}}\left[\mathcal{F}^{-1}\left(\widehat{\gamma^{\left(j\right)}}\cdot\varphi_{i}\cdot\widehat{f}\right)\right]\leq\osc{T_{j}^{-T}\left[-1,1\right]^{d}}\left(M_{-b_{j}}\left[\mathcal{F}^{-1}\left(\widehat{\gamma^{\left(j\right)}}\cdot\varphi_{i}\cdot\widehat{f}\right)\right]\right)+\left|\mathcal{F}^{-1}\left(\widehat{\gamma^{\left(j\right)}}\cdot\varphi_{i}\cdot\widehat{f}\right)\right|.
\]
Using the triangle inequality for $L_{v}^{p}\left(\mathbb{R}^{d}\right)$
and solidity of $L_{v}^{p}\left(\mathbb{R}^{d}\right)$, this yields
\begin{align*}
& \sum_{i\in I}\left\Vert \mathcal{F}^{-1}\left(\widehat{\gamma^{\left(j\right)}}\cdot\varphi_{i}\cdot\widehat{f}\right)\right\Vert _{W_{T_{j}^{-T}\left[-1,1\right]^{d}}\left(L_{v}^{p}\right)}\\
& =\sum_{i\in I}\left\Vert M_{T_{j}^{-T}\left[-1,1\right]^{d}}\left[\mathcal{F}^{-1}\left(\widehat{\gamma^{\left(j\right)}}\cdot\varphi_{i}\cdot\widehat{f}\right)\right]\right\Vert _{L_{v}^{p}}\\
& \leq\sum_{i\in I}\left(\left\Vert \mathcal{F}^{-1}\left(\widehat{\gamma^{\left(j\right)}}\cdot\varphi_{i}\cdot\widehat{f}\right)\right\Vert _{L_{v}^{p}}+\left\Vert \osc{T_{j}^{-T}\left[-1,1\right]^{d}}\left(M_{-b_{j}}\left[\mathcal{F}^{-1}\left(\widehat{\gamma^{\left(j\right)}}\cdot\varphi_{i}\cdot\widehat{f}\right)\right]\right)\right\Vert _{L_{v}^{p}}\right)<\infty.
\end{align*}
Here, finiteness of the right-hand side follows from equation (\ref{eq:OscillationSumEstimateBanachCase})
(with $\delta=1$) and from Theorem \ref{thm:ConvolvingDecompositionSpaceWithGammaJ},
where we saw that the series $\sum_{i\in I}\mathcal{F}^{-1}\left(\widehat{\gamma^{\left(j\right)}}\cdot\varphi_{i}\cdot\widehat{f}\right)$
converges normally in $L_{v}^{p}$.
But it follows from equation (\ref{eq:WeightedWienerAmalgamTemperedDistribution})
that $W_{T_{j}^{-T}\left[-1,1\right]^{d}}\left(L_{v}^{p}\right)\hookrightarrow L_{\left(1+\left|\bullet\right|\right)^{-K}}^{\infty}\left(\mathbb{R}^{d}\right)$,
where the norm of the embedding might heavily depend on $j$. Setting
$\left\Vert h\right\Vert _{\ast}:=\sup_{x\in\mathbb{R}^{d}}\left(1+\left|x\right|\right)^{-K}\left|h\left(x\right)\right|$,
this allows us to conclude by continuity that
\[
\sum_{i\in I}\left\Vert \mathcal{F}^{-1}\left(\widehat{\gamma^{\left(j\right)}}\cdot\varphi_{i}\cdot\widehat{f}\right)\right\Vert _{\ast}=\sum_{i\in I}\left\Vert \mathcal{F}^{-1}\left(\widehat{\gamma^{\left(j\right)}}\cdot\varphi_{i}\cdot\widehat{f}\right)\right\Vert _{L_{\left(1+\left|\bullet\right|\right)^{-K}}^{\infty}}\lesssim_{j}\,\,\sum_{i\in I}\left\Vert \mathcal{F}^{-1}\left(\widehat{\gamma^{\left(j\right)}}\cdot\varphi_{i}\cdot\widehat{f}\right)\right\Vert _{W_{T_{j}^{-T}\left[-1,1\right]^{d}}\left(L_{v}^{p}\right)}<\infty,
\]
so that the series in equation (\ref{eq:OscillationForFreeSeriesRepresentation})
indeed converges \emph{locally} uniformly. Hence, $\gamma^{\left(j\right)}\ast f$
is continuous for every $j\in I$ and arbitrary $f\in\DecompSp{\mathcal{Q}}p{\ell_{w}^{q}}v$.
\medskip{}
Now, it is not hard to see $\osc Q\left(\sum_{i\in I}f_{i}\right)\leq\sum_{i\in I}\left(\osc Qf_{i}\right)$
for each pointwise convergent series $\sum_{i\in I}f_{i}$. Hence,
equation (\ref{eq:OscillationForFreeSeriesRepresentation}) and the
triangle inequality for $L_{v}^{p}\left(\mathbb{R}^{d}\right)$ imply
\begin{align*}
\frac{1}{\delta}\left\Vert \osc{\delta\cdot T_{j}^{-T}\left[-1,1\right]^{d}}\left[M_{-b_{j}}\left(\gamma^{\left(j\right)}\ast f\right)\right]\right\Vert _{L_{v}^{p}} & \leq\frac{1}{\delta}\sum_{i\in I}\left\Vert \osc{\delta\cdot T_{j}^{-T}\left[-1,1\right]^{d}}\left(M_{-b_{j}}\left[\mathcal{F}^{-1}\left(\widehat{\gamma^{\left(j\right)}}\cdot\varphi_{i}\cdot\widehat{f}\right)\right]\right)\right\Vert _{L_{v}^{p}}\\
\left({\scriptstyle \text{eq. }\eqref{eq:OscillationSumEstimateBanachCase}}\right) & \leq C_{1}\cdot\left(\overrightarrow{B}c\right)_{j}<\infty
\end{align*}
for all $j\in I$ and $\delta\in\left(0,1\right]$. By equation (\ref{eq:OscillationForFreeSequenceNormEstimateBanachCase})
and by solidity of $\ell_{w}^{q}\left(I\right)$, this yields
\[
\left\Vert \left(\sup_{0<\delta\leq1}\frac{1}{\delta}\left\Vert \osc{\delta\cdot T_{j}^{-T}\left[-1,1\right]^{d}}\left[M_{-b_{j}}\left(\gamma^{\left(j\right)}\ast f\right)\right]\right\Vert _{L_{v}^{p}}\right)_{j\in I}\right\Vert _{\ell_{w}^{q}}\leq C_{1}\cdot\left\Vert \overrightarrow{B}c\right\Vert _{\ell_{w}^{q}}\leq C_{1}\cdot\vertiii{\smash{\Gamma_{\mathcal{Q}}}}\cdot\vertiii{\smash{\overrightarrow{B}}}\cdot\left\Vert f\right\Vert _{\DecompSp{\mathcal{Q}}p{\ell_{w}^{q}}v}<\infty.
\]
Finally, Theorem \ref{thm:ConvolvingDecompositionSpaceWithGammaJ}
shows
\[
\left\Vert \left(\left\Vert \gamma^{\left(j\right)}\ast f\right\Vert _{L_{v}^{p}}\right)_{j\in I}\right\Vert _{\ell_{w}^{q}}\leq\vertiii{\smash{\Gamma_{\mathcal{Q}}}}\cdot\vertiii{\smash{\overrightarrow{A}}}\cdot\left\Vert f\right\Vert _{\DecompSp{\mathcal{Q}}p{\ell_{w}^{q}}v}<\infty.
\]
It is not hard to see that this implies boundedness of ${\rm Ana}_{{\rm osc}}$,
with a bound for the operator norm as in the statement of the lemma,
since $2^{\max\left\{ 0,\frac{1}{q}-1\right\} }$ is a valid triangle
constant for $\ell_{w}^{q}\left(I\right)$ and since $C_{1}\geq1$.
\medskip{}
In case of $p\in\left(0,1\right)$, we first note that equation (\ref{eq:WeightedWienerAmalgamTemperedDistribution})
yields $V_{j}=W_{T_{j}^{-T}\left[-1,1\right]^{d}}\left(L_{v}^{p}\right)\hookrightarrow L_{\left(1+\left|\bullet\right|\right)^{-K}}^{\infty}\left(\mathbb{R}^{d}\right)$,
where again the norm of the embedding might depend heavily on the
choice of $j\in I$. But as seen in Theorem \ref{thm:ConvolvingDecompositionSpaceWithGammaJ},
the series in equation (\ref{eq:OscillationForFreeSeriesRepresentation})
converges in $V_{j}$ and hence in $L_{\left(1+\left|\bullet\right|\right)^{-K}}^{\infty}\left(\mathbb{R}^{d}\right)$,
which yields \emph{locally} uniform convergence, since each summand
of the series is continuous. In particular, we get continuity of $\gamma^{\left(j\right)}\ast f$
for each $j\in I$.
The remainder of the argument is similar as that for $p\in\left[1,\infty\right]$.
Nevertheless, it needs to be modified slightly, since for $p\in\left(0,1\right)$,
$L_{v}^{p}\left(\mathbb{R}^{d}\right)$ does not satisfy the triangle
inequality, but instead the so-called $p$-triangle inequality, i.e.,
$\left\Vert f+g\right\Vert _{L_{v}^{p}}^{p}\leq\left\Vert f\right\Vert _{L_{v}^{p}}^{p}+\left\Vert g\right\Vert _{L_{v}^{p}}^{p}$.
Precisely, using equation (\ref{eq:OscillationForFreeSeriesRepresentation})
and the estimates $\osc Q\left(\sum_{i\in I}f_{i}\right)\leq\sum_{i\in I}\left(\osc Qf_{i}\right)$
and $M_{Q}\left(\sum_{i\in I}f_{i}\right)\leq\sum_{i\in I}\left(M_{Q}f_{i}\right)$,
as well as the $p$-triangle inequality for $L_{v}^{p}\left(\mathbb{R}^{d}\right)$,
we get for arbitrary $0<\delta\leq1$ that
\begin{align*}
& \left(\frac{1}{\delta}\left\Vert \osc{\delta\cdot T_{j}^{-T}\left[-1,1\right]^{d}}\left[M_{-b_{j}}\left(\gamma^{\left(j\right)}\ast f\right)\right]\right\Vert _{W_{T_{j}^{-T}\left[-1,1\right]^{d}}\left(L_{v}^{p}\right)}\right)^{p}\\
& \leq\frac{1}{\delta^{p}}\sum_{i\in I}\left\Vert \osc{\delta\cdot T_{j}^{-T}\left[-1,1\right]^{d}}\left(M_{-b_{j}}\left[\mathcal{F}^{-1}\left(\widehat{\gamma^{\left(j\right)}}\cdot\varphi_{i}\cdot\smash{\widehat{f}}\,\right)\right]\right)\right\Vert _{W_{T_{j}^{-T}\left[-1,1\right]^{d}}\left(L_{v}^{p}\right)}^{p}\\
\left({\scriptstyle \text{Lemma }\ref{lem:MainOscillationEstimate}}\right) & \leq C_{2}^{p}\cdot\sum_{i\in I}\left[\left|\det T_{i}\right|^{1-p}\cdot\left(1+\left\Vert T_{j}^{-1}T_{i}\right\Vert \right)^{pK+d}\cdot\left\Vert \mathcal{F}^{-1}\left(\varphi_{i}^{\ast}\smash{\widehat{f}}\,\right)\right\Vert _{L_{v}^{p}}^{p}\cdot\left\Vert \mathcal{F}^{-1}\left[\widehat{\phi^{\left(j\right)}}\cdot\varphi_{i}\right]\right\Vert _{L_{v_{0}}^{p}}^{p}\right]\\
\left({\scriptstyle \text{with }\theta_{i}:=c_{i}^{p}=\left\Vert \mathcal{F}^{-1}\left(\varphi_{i}^{\ast}\smash{\widehat{f}}\,\right)\right\Vert _{L_{v}^{p}}^{p}}\right) & =C_{2}^{p}\cdot\sum_{i\in I}\left[B_{j,i}\theta_{i}\right]=C_{2}^{p}\cdot\left(\overrightarrow{B}\theta\right)_{j},
\end{align*}
where the constant $C_{2}>0$ is provided by Lemma \ref{lem:MainOscillationEstimate}.
We conclude using the solidity of $\ell_{w}^{q}\left(I\right)$ that
\begin{align*}
& \left\Vert \left(\sup_{0<\delta\leq1}\frac{1}{\delta}\left\Vert \osc{\delta\cdot T_{j}^{-T}\left[-1,1\right]^{d}}\left[M_{-b_{j}}\left(\gamma^{\left(j\right)}\ast f\right)\right]\right\Vert _{W_{T_{j}^{-T}\left[-1,1\right]^{d}}\left(L_{v}^{p}\right)}\right)_{j\in I}\right\Vert _{\ell_{w}^{q}}\\
& \leq C_{2}\cdot\left\Vert \left(\overrightarrow{B}\theta\right)^{1/p}\right\Vert _{\ell_{w}^{q}}=C_{2}\cdot\left\Vert \left(w^{p}\cdot\overrightarrow{B}\theta\right)^{1/p}\right\Vert _{\ell^{q}}\\
& =C_{2}\cdot\left\Vert w^{p}\cdot\overrightarrow{B}\theta\right\Vert _{\ell^{q/p}}^{1/p}=C_{2}\cdot\left\Vert \overrightarrow{B}\theta\right\Vert _{\ell_{w^{\min\left\{ 1,p\right\} }}^{r}}^{1/p}\\
& \leq C_{2}\cdot\left(\vertiii{\smash{\overrightarrow{B}}}\cdot\left\Vert \theta\right\Vert _{\ell_{w^{\min\left\{ 1,p\right\} }}^{r}}\right)^{1/p}\\
& =C_{2}\cdot\vertiii{\smash{\overrightarrow{B}}}^{1/p}\cdot\left\Vert w^{p}\cdot\theta\right\Vert _{\ell^{q/p}}^{1/p}\\
& =C_{2}\cdot\vertiii{\smash{\overrightarrow{B}}}^{1/p}\cdot\left\Vert w\cdot\theta^{1/p}\right\Vert _{\ell^{q}}\\
& =C_{2}\cdot\vertiii{\smash{\overrightarrow{B}}}^{1/p}\cdot\left\Vert c\right\Vert _{\ell_{w}^{q}}.
\end{align*}
Finally, using the quasi-triangle inequality $\left\Vert \sum_{i=1}^{N}f_{i}\right\Vert _{L^{p}}\leq N^{\frac{1}{p}-1}\cdot\sum_{i=1}^{N}\left\Vert f_{i}\right\Vert _{L^{p}}$
(cf.\@ \cite[Exercise 1.1.5(c)]{GrafakosClassicalFourierAnalysis})
and the estimate $\left|i^{\ast}\right|\leq N_{\mathcal{Q}}$ for all $i\in I$,
we also get $c_{i}\leq N_{\mathcal{Q}}^{\frac{1}{p}-1}\cdot\left(\Gamma_{\mathcal{Q}}\,d\right)_{i}$
for all $i\in I$ and $d_{i}:=\left\Vert \mathcal{F}^{-1}\left(\varphi_{i}\cdot\widehat{f}\right)\right\Vert _{L_{v}^{p}}$.
Hence,
\[
\left\Vert \left(\sup_{0<\delta\leq1}\frac{1}{\delta}\left\Vert \osc{\delta\cdot T_{j}^{-T}\left[-1,1\right]^{d}}\left[M_{-b_{j}}\left(\gamma^{\left(j\right)}\ast f\right)\right]\right\Vert _{W_{T_{j}^{-T}\left[-1,1\right]^{d}}\left(L_{v}^{p}\right)}\right)_{j\in I}\right\Vert _{\ell_{w}^{q}}\leq C_{2}\cdot N_{\mathcal{Q}}^{\frac{1}{p}-1}\cdot\vertiii{\smash{\Gamma_{\mathcal{Q}}}}\cdot\vertiii{\smash{\overrightarrow{B}}}^{1/p}\cdot\left\Vert d\right\Vert _{\ell_{w}^{q}}.
\]
Because of $\left\Vert d\right\Vert _{\ell_{w}^{q}}=\left\Vert f\right\Vert _{\DecompSp{\mathcal{Q}}p{\ell_{w}^{q}}v}$
and in combination with Theorem \ref{thm:ConvolvingDecompositionSpaceWithGammaJ},
we can now derive the claim using the same arguments as for $p\in\left[1,\infty\right]$.
Here, we use that $C_{\mathcal{Q}}\geq\left\Vert T_{i}^{-1}T_{i}\right\Vert =1$,
so that the constant $C_{3}>0$ provided by Theorem \ref{thm:ConvolvingDecompositionSpaceWithGammaJ}
(for $p\in\left(0,1\right)$) satisfies
\begin{align*}
C_{3} & =N_{\mathcal{Q}}^{\frac{1}{p}-1}\cdot\left(12288\cdotd^{3/2}\cdot\left\lceil K+\frac{d+1}{p}\right\rceil \right)^{\left\lceil K+\frac{d+1}{p}\right\rceil +1}\cdot\left(1+R_{\mathcal{Q}}\right)^{d/p}\left(12R_{\mathcal{Q}}C_{\mathcal{Q}}\right)^{d\left(\frac{1}{p}-1\right)}\cdot\Omega_{0}^{K}\Omega_{1}\\
\left({\scriptstyle \text{since }\Omega_{0},\Omega_{1}\geq1}\right) & \leq N_{\mathcal{Q}}^{\frac{1}{p}-1}\cdot12^{d\left(\frac{1}{p}-1\right)}\left(12288\cdotd^{3/2}\cdot\left\lceil K+\frac{d+1}{p}\right\rceil \right)^{\left\lceil K+\frac{d+1}{p}\right\rceil +1}\cdot\left(1+C_{\mathcal{Q}}R_{\mathcal{Q}}\right)^{d\left(\frac{2}{p}-1\right)}\cdot\Omega_{0}^{5K}\Omega_{1}^{5},\\
& \leq N_{\mathcal{Q}}^{\frac{1}{p}-1}\cdot12^{d\left(\frac{1}{p}-1\right)}\left(12288\cdotd^{3/2}\cdot\left\lceil K+\frac{d+1}{p}\right\rceil \right)^{\left\lceil K+\frac{d+1}{p}\right\rceil +1}\cdot\left(1+C_{\mathcal{Q}}R_{\mathcal{Q}}\right)^{\frac{2d}{p}}\cdot\Omega_{0}^{5K}\Omega_{1}^{5},
\end{align*}
so that
\begin{align*}
\frac{C_{3}}{N_{\mathcal{Q}}^{\frac{1}{p}-1}C_{2}} & \leq370\cdotd^{11/2}\cdotd^{d/2p}\cdot\frac{12^{d\left(\frac{1}{p}-1\right)}\left(12288\cdotd^{3/2}\cdot\left\lceil K+\frac{d+1}{p}\right\rceil \right)^{\left\lceil K+\frac{d+1}{p}\right\rceil +1}}{2^{16\frac{d}{p}}\cdot\left(4032\cdotd^{3}\cdot\left\lceil K+\frac{d+1}{p}\right\rceil \right)^{2\left\lceil K+\frac{d+1}{p}\right\rceil +2}}\\
& =\frac{370\cdotd^{11/2}\cdotd^{d/2p}\cdot12^{d\left(\frac{1}{p}-1\right)}}{2^{16\frac{d}{p}}\left(4032\cdotd^{3}\cdot\left\lceil K+\frac{d+1}{p}\right\rceil \right)^{\left\lceil K+\frac{d+1}{p}\right\rceil +1}}\cdot\left(\frac{12288\cdotd^{3/2}}{4032\cdotd^{3}}\right)^{\left\lceil K+\frac{d+1}{p}\right\rceil +1}\\
& \leq\frac{370\cdotd^{11/2}\cdotd^{d/2p}\cdot12^{\frac{d}{p}}}{2^{16\frac{d}{p}}}\cdot\left(\frac{4}{4032\cdotd^{9/2}}\right)^{\left\lceil K+\frac{d+1}{p}\right\rceil +1}\\
\left({\scriptstyle \text{since }\left\lceil K+\frac{d+1}{p}\right\rceil \geq\frac{d+1}{p}\geq\frac{d}{p}+1}\right) & \leq\frac{370\cdotd^{11/2}\cdotd^{d/2p}}{2^{12\frac{d}{p}}}\cdot\left(\frac{1}{1000\cdotd^{9/2}}\right)^{\frac{d}{p}+2}\\
& \leq\frac{370}{1000000}\frac{d^{11/2}\cdotd^{d/2p}}{2^{12\frac{d}{p}}}\cdotd^{-\frac{9}{2}\frac{d}{p}}d^{-9}=\frac{370}{1000000}\frac{1}{2^{12\frac{d}{p}}}\cdotd^{-4\frac{d}{p}}d^{-7/2}\leq1.\qedhere
\end{align*}
\end{proof}
In view of the preceding lemma, we know that (if $\Gamma=\left(\gamma_{i}\right)_{i\in I}$
fulfills Assumption \ref{assu:DiscreteBanachFrameAssumptions}) each
of the functions $\gamma^{\left(j\right)}\ast f$ and hence also $\gamma^{\left[j\right]}\ast f=\left|\det T_{j}\right|^{-1/2}\cdot\gamma^{\left(j\right)}\ast f$
is continuous, so that the coefficient mapping
\[
\DecompSp{\mathcal{Q}}p{\ell_{w}^{q}}v\ni f\mapsto\left[\left(\gamma^{\left[j\right]}\ast f\right)\!\left(\delta\cdot T_{j}^{-T}k\right)\right]_{j\in I,\,k\in\mathbb{Z}^{d}}\in\mathbb{C}^{I\times\mathbb{Z}^{d}}
\]
is well-defined. But eventually, we want to show that this map yields
a Banach frame for $\DecompSp{\mathcal{Q}}p{\ell_{w}^{q}}v$, so that we
also need to construct a suitable ``reconstruction mapping'', which
can recover $f$ from these coefficients. The following lemma is an
important ingredient for the construction of this reconstruction map.
\begin{lem}
\label{lem:DiscreteSynthesisOperatorIsAlmostIsometric}For $i\in I$
and $0<\delta\leq1$, let
\[
{\rm Synth}_{\delta,i}:\mathbb{C}^{\mathbb{Z}^{d}}\to\left\{ f:\mathbb{R}^{d}\to\mathbb{C}\,\middle|\, f\text{ measurable}\right\} ,\left(c_{k}\right)_{k\in\mathbb{Z}^{d}}\mapsto M_{b_{i}}\left[\sum_{k\in\mathbb{Z}^{d}}c_{k}\cdot e^{-2\pi i\left\langle b_{i},\delta\cdot T_{i}^{-T}k\right\rangle }{\mathds{1}}_{\delta\cdot T_{i}^{-T}\left(k+\left[0,1\right)^{d}\right)}\right].
\]
Then ${\rm Synth}_{\delta,i}$ is well-defined and yields a bounded
operator ${\rm Synth}_{\delta,i}:C_{i}^{\left(\delta\right)}\to V_{i}$,
where the $i$-th coefficient space $C_{i}^{\left(\delta\right)}$
is defined as in equation (\ref{eq:CoefficientSpaceDefinition}).
More precisely, we have
\[
\frac{\delta^{d/p}}{\Omega_{0}^{K}\Omega_{1}\cdot\left(1+\sqrt{d}\right)^{K}}\cdot\left|\det T_{i}\right|^{-1/p}\cdot\left\Vert c\right\Vert _{C_{i}^{\left(\delta\right)}}\leq\left\Vert {\rm Synth}_{\delta,i}\,c\right\Vert _{V_{i}}\leq C_{d,p,\delta,K}\cdot\left|\det T_{i}\right|^{-1/p}\cdot\left\Vert c\right\Vert _{C_{i}^{\left(\delta\right)}}\qquad\forall c\in\mathbb{C}^{\mathbb{Z}^{d}},
\]
with
\[
C_{d,p,\delta,K}=\begin{cases}
\left(1+\sqrt{d}\right)^{K}\cdot\Omega_{0}^{K}\Omega_{1}\cdot\delta^{d/p}, & \text{if }p\in\left[1,\infty\right],\\
4^{d/p}\cdot\left(1+2\sqrt{d}\right)^{K}\cdot\Omega_{0}^{K}\Omega_{1}, & \text{if }p\in\left(0,1\right).
\end{cases}\qedhere
\]
\end{lem}
\begin{proof}
First note that ${\rm Synth}_{\delta,i}$ is well-defined, since the
sets $\left(\delta\cdot T_{i}^{-T}\left(k+\left[0,1\right)^{d}\right)\right)_{k\in\mathbb{Z}^{d}}$
are pairwise disjoint. Also, we can ignore the modulation $M_{b_{i}}$
in the following, since $\left\Vert M_{b_{i}}f\right\Vert _{V_{i}}=\left\Vert f\right\Vert _{V_{i}}$,
because of $\left\Vert f\right\Vert _{V_{i}}=\left\Vert g\right\Vert _{V_{i}}$
for measurable $f,g$ satisfying $\left|f\right|=\left|g\right|$.
Furthermore, since we have for $x\in\delta\cdot T_{i}^{-T}\left(k+\left[0,1\right)^{d}\right)$,
i.e., for $x=\delta\cdot T_{i}^{-T}k+\delta T_{i}^{-T}q$ with $q\in\left[0,1\right)^{d}$
that
\begin{align*}
v_{k}^{\left(i,\delta\right)} & =v\left(\delta\cdot T_{i}^{-T}k\right)=v\left(x-\delta T_{i}^{-T}q\right)\leq v\left(x\right)\cdot v_{0}\left(-\delta T_{i}^{-T}q\right)\\
\left({\scriptstyle \text{assump. on }v_{0}}\right) & \leq\Omega_{1}\cdot\left(1+\left|\delta T_{i}^{-T}q\right|\right)^{K}\cdot v\left(x\right)\\
\left({\scriptstyle \text{eq. }\eqref{eq:WeightLinearTransformationsConnection}}\right) & \leq\Omega_{0}^{K}\Omega_{1}\cdot\left(1+\left|\delta q\right|\right)^{K}\cdot v\left(x\right)\\
\left({\scriptstyle \text{since }\delta\leq1}\right) & \leq\Omega_{0}^{K}\Omega_{1}\cdot\left(1+\sqrt{d}\right)^{K}\cdot v\left(x\right),
\end{align*}
Lemma \ref{lem:MaximalFunctionDominatesF} implies
\begin{equation}
\begin{split}\left\Vert {\rm Synth}_{\delta,i}\left(c_{k}\right)_{k\in\mathbb{Z}^{d}}\right\Vert _{V_{i}} & \geq\left\Vert {\rm Synth}_{\delta,i}\left(c_{k}\right)_{k\in\mathbb{Z}^{d}}\right\Vert _{L_{v}^{p}}\\
\left({\scriptstyle \text{pairwise disjointness}}\right) & =\left[\sum_{k\in\mathbb{Z}^{d}}\left|c_{k}\right|^{p}\int_{\delta\cdot T_{i}^{-T}\left(k+\left[0,1\right)^{d}\right)}\left[v\left(x\right)\right]^{p}\operatorname{d} x\right]^{1/p}\\
& \geq\frac{1}{\Omega_{0}^{K}\Omega_{1}\cdot\left(1+\sqrt{d}\right)^{K}}\cdot\left[\sum_{k\in\mathbb{Z}^{d}}\left|v_{k}^{\left(i,\delta\right)}\cdot c_{k}\right|^{p}\cdot\lambda_{d}\left(\delta\cdot T_{i}^{-T}\left[k+\left[0,1\right)^{d}\right]\right)\right]^{1/p}\\
& =\frac{\delta^{d/p}\cdot\left|\det T_{i}\right|^{-1/p}}{\Omega_{0}^{K}\Omega_{1}\cdot\left(1+\sqrt{d}\right)^{K}}\cdot\left\Vert \left(c_{k}\right)_{k\in\mathbb{Z}^{d}}\right\Vert _{C_{i}^{\left(\delta\right)}}.
\end{split}
\label{eq:DiscreteSynthesisOperatorLowerBound}
\end{equation}
This proves the lower bound.
\medskip{}
Now, we establish the reverse inequality for $p\in\left[1,\infty\right]$:
For $x=\delta\cdot T_{i}^{-T}k+\delta T_{i}^{-T}q\in\delta\cdot T_{i}^{-T}\left(k+\left[-L,L\right]^{d}\right)$
with $L\geq1$, we have as above that
\begin{align}
v\left(x\right) & =v\left(\delta\cdot T_{i}^{-T}k+\delta\cdot T_{i}^{-T}q\right)\nonumber \\
& \leq v\left(\delta\cdot T_{i}^{-T}k\right)\cdot v_{0}\left(\delta\cdot T_{i}^{-T}q\right)\nonumber \\
\left({\scriptstyle \text{assump. on }v_{0}\text{ and eq. }\eqref{eq:WeightLinearTransformationsConnection}}\right) & \leq\Omega_{0}^{K}\Omega_{1}\cdot v_{k}^{\left(i,\delta\right)}\cdot\left(1+\left|\delta\cdot q\right|\right)^{K}\nonumber \\
\left({\scriptstyle \text{since }\delta\leq1}\right) & \leq\left(1+L\sqrt{d}\right)^{K}\cdot\Omega_{0}^{K}\Omega_{1}\cdot v_{k}^{\left(i,\delta\right)}.\label{eq:DiscreteSynthesisOperatorUpperWeightEstimate}
\end{align}
Furthermore, since $p\in\left[1,\infty\right]$, the first inequality
in estimate (\ref{eq:DiscreteSynthesisOperatorLowerBound}) from above
is actually an equality, so that
\begin{align*}
\left\Vert {\rm Synth}_{\delta,i}\left(c_{k}\right)_{k\in\mathbb{Z}^{d}}\right\Vert _{V_{i}} & =\left[\sum_{k\in\mathbb{Z}^{d}}\left|c_{k}\right|^{p}\int_{\delta\cdot T_{i}^{-T}\left(k+\left[0,1\right)^{d}\right)}\left[v\left(x\right)\right]^{p}\operatorname{d} x\right]^{1/p}\\
& \leq\left(1+\sqrt{d}\right)^{K}\cdot\Omega_{0}^{K}\Omega_{1}\cdot\left[\sum_{k\in\mathbb{Z}^{d}}\left|v_{k}^{\left(i,\delta\right)}c_{k}\right|^{p}\cdot\lambda_{d}\left(\delta\cdot T_{i}^{-T}\left[k+\left[0,1\right)^{d}\right]\right)\right]^{1/p}\\
& =\left(1+\sqrt{d}\right)^{K}\cdot\Omega_{0}^{K}\Omega_{1}\cdot\delta^{d/p}\cdot\left|\det T_{i}\right|^{-1/p}\cdot\left\Vert \left(c_{k}\right)_{k\in\mathbb{Z}^{d}}\right\Vert _{C_{i}^{\left(\delta\right)}}.
\end{align*}
\medskip{}
Finally, for $p\in\left(0,1\right)$, we use the estimate $M_{Q}\left(\sum_{i\in I}f_{i}\right)\leq\sum_{i\in I}M_{Q}f_{i}$
and the $p$-triangle inequality for $L_{v}^{p}\left(\mathbb{R}^{d}\right)$
to deduce
\begin{align*}
\left\Vert {\rm Synth}_{\delta,i}\left(c_{k}\right)_{k\in\mathbb{Z}^{d}}\right\Vert _{V_{i}}^{p} & \leq\left\Vert \sum_{k\in\mathbb{Z}^{d}}M_{T_{i}^{-T}\left[-1,1\right]^{d}}\left(c_{k}\cdot e^{-2\pi i\left\langle b_{i},\delta\cdot T_{i}^{-T}k\right\rangle }{\mathds{1}}_{\delta\cdot T_{i}^{-T}\left(k+\left[0,1\right)^{d}\right)}\right)\right\Vert _{L_{v}^{p}}^{p}\\
& \leq\sum_{k\in\mathbb{Z}^{d}}\left[\left|c_{k}\right|^{p}\cdot\left\Vert M_{T_{i}^{-T}\left[-1,1\right]^{d}}{\mathds{1}}_{\delta\cdot T_{i}^{-T}\left(k+\left[0,1\right)^{d}\right)}\right\Vert _{L_{v}^{p}}^{p}\right].
\end{align*}
Next, observe
\[
\left(M_{T_{i}^{-T}\left[-1,1\right]^{d}}{\mathds{1}}_{\delta\cdot T_{i}^{-T}\left(k+\left[0,1\right)^{d}\right)}\right)\left(x\right)=\left\Vert {\mathds{1}}_{\delta\cdot T_{i}^{-T}\left(k+\left[0,1\right)^{d}\right)}\cdot{\mathds{1}}_{x+T_{i}^{-T}\left[-1,1\right]^{d}}\right\Vert _{L^{\infty}}\leq1.
\]
Furthermore, if the function inside the $\left\Vert \bullet\right\Vert _{L^{\infty}}$
norm does not vanish identically, we have
\[
x\in\delta T_{i}^{-T}k+T_{i}^{-T}\left(\delta\left[0,1\right)^{d}-\left[-1,1\right]^{d}\right)\subset\delta T_{i}^{-T}k+T_{i}^{-T}\left[-2,2\right]^{d}.
\]
Hence, equation (\ref{eq:DiscreteSynthesisOperatorUpperWeightEstimate})
yields $v\left(x\right)\leq\left(1+2\sqrt{d}\right)^{K}\Omega_{0}^{K}\Omega_{1}\cdot v_{k}^{\left(i,\delta\right)}$
and thus
\begin{align*}
\left\Vert M_{T_{i}^{-T}\left[-1,1\right]^{d}}{\mathds{1}}_{\delta\cdot T_{i}^{-T}\left(k+\left[0,1\right)^{d}\right)}\right\Vert _{L_{v}^{p}}^{p} & \leq\left(\left(1+2\sqrt{d}\right)^{K}\Omega_{0}^{K}\Omega_{1}\right)^{p}\cdot\left[v_{k}^{\left(i,\delta\right)}\right]^{p}\cdot\lambda_{d}\left(\delta T_{i}^{-T}k+T_{i}^{-T}\left[-2,2\right]^{d}\right)\\
& =4^{d}\cdot\left(\left(1+2\sqrt{d}\right)^{K}\Omega_{0}^{K}\Omega_{1}\right)^{p}\cdot\left|\det T_{i}\right|^{-1}\cdot\left[v_{k}^{\left(i,\delta\right)}\right]^{p},
\end{align*}
so that we get
\[
\left\Vert {\rm Synth}_{\delta,i}\left(c_{k}\right)_{k\in\mathbb{Z}^{d}}\right\Vert _{V_{i}}^{p}\leq4^{d}\cdot\left(\left(1+2\sqrt{d}\right)^{K}\Omega_{0}^{K}\Omega_{1}\right)^{p}\cdot\left|\det T_{i}\right|^{-1}\cdot\left\Vert \left(c_{k}\right)_{k\in\mathbb{Z}^{d}}\right\Vert _{C_{i}^{\left(\delta\right)}}^{p},
\]
as claimed.
\end{proof}
Below, we will employ a Neumann series argument to construct the reconstruction
operator $R$. To this end, we need to know that the space $\ell_{w}^{q}\left(\left[V_{i}\right]_{i\in I}\right)$
is a Quasi-Banach space.
\begin{lem}
\label{lem:IteratedSequenceSpaceComplete}Let $\left(X_{i}\right)_{i\in I}$
be a sequence of Quasi-Banach spaces, each with $\left\Vert x+y\right\Vert _{X_{i}}\leq C_{i}\cdot\left(\left\Vert x\right\Vert _{X_{i}}+\left\Vert y\right\Vert _{X_{i}}\right)$
for all $x,y\in X_{i}$ and suitable $C_{i}\geq1$. Assume that $C:=\sup_{i\in i}C_{i}$
is finite and that each quasi-norm $\left\Vert \bullet\right\Vert _{X_{i}}$
is continuous.
Define
\[
\ell_{w}^{q}\left(\left[X_{i}\right]_{i\in I}\right):=\left\{ x=\left(x_{i}\right)_{i\in I}\in\prod_{i\in I}X_{i}\,\middle|\,\left\Vert x\right\Vert :=\left\Vert \left(\left\Vert x_{i}\right\Vert _{X_{i}}\right)_{i\in I}\right\Vert _{\ell_{w}^{q}\left(I\right)}<\infty\right\} .
\]
Then $\left(\ell_{w}^{q}\left(\left[X_{i}\right]_{i\in I}\right),\left\Vert \bullet\right\Vert \right)$
is a Quasi-Banach space.
\end{lem}
\begin{rem*}
The lemma applies in particular with the choice $X_{i}=V_{i}$. Indeed,
$\left\Vert \bullet\right\Vert _{L_{v}^{p}}$ is an $s$-norm for
$s:=\min\left\{ 1,p\right\} $; since $M_{Q}\left(f+g\right)\leq M_{Q}f+M_{Q}g$,
we get $\left\Vert f+g\right\Vert _{W_{Q}\left(L_{v}^{p}\right)}^{s}\leq\left\Vert f\right\Vert _{W_{Q}\left(L_{v}^{p}\right)}^{s}+\left\Vert g\right\Vert _{W_{Q}\left(L_{v}^{p}\right)}^{s}$,
so that $\left\Vert \bullet\right\Vert _{W_{Q}\left(L_{v}^{p}\right)}$
is also an $s$-norm and hence continuous, since $\left|\left\Vert x_{n}\right\Vert ^{s}-\left\Vert x\right\Vert ^{s}\right|\leq\left\Vert x_{n}-x\right\Vert ^{s}\xrightarrow[n\to\infty]{}0$
for any $s$-norm $\left\Vert \bullet\right\Vert $ if $\left\Vert x_{n}-x\right\Vert \xrightarrow[n\to\infty]{}0$.
Furthermore, in case of $p\in\left[1,\infty\right]$, one can choose
$C_{i}=1$ for all $i\in I$. Finally, for $p\in\left(0,1\right)$,
Remark \ref{rem:MainAssumptionsRemark} shows that each $V_{i}$ is
a Quasi-Banach space and that we can choose $C_{i}=2^{\frac{1}{p}-1}$
for all $i\in I$. Hence, $V=\ell_{w}^{q}\left(\left[V_{i}\right]_{i\in I}\right)$
is a Quasi-Banach space.
\end{rem*}
\begin{proof}
For brevity, let $X:=\ell_{w}^{q}\left(\left[X_{i}\right]_{i\in I}\right)$.
It is clear that $X$ is closed under multiplication with scalars
and that $\left\Vert \alpha\cdot x\right\Vert =\left|\alpha\right|\cdot\left\Vert x\right\Vert $
for $\alpha\in\mathbb{K}$ (with $\mathbb{K}\in\left\{ \mathbb{R},\mathbb{C}\right\} $)
and $x\in X$. Furthermore, if $\left\Vert x\right\Vert =0$ for $x=\left(x_{i}\right)_{i\in I}$,
then $\left\Vert x_{i}\right\Vert _{X_{i}}=0$ for all $i\in I$,
so that $x=0$. Finally, for $x,y\in X$, we have by solidity of $\ell_{w}^{q}\left(I\right)$
that
\begin{align*}
\left\Vert x+y\right\Vert & =\left\Vert \left(\left\Vert x_{i}+y_{i}\right\Vert _{X_{i}}\right)_{i\in I}\right\Vert _{\ell_{w}^{q}}\leq\left\Vert \left(C\cdot\left[\left\Vert x_{i}\right\Vert _{X_{i}}+\left\Vert y_{i}\right\Vert _{X_{i}}\right]\right)_{i\in I}\right\Vert _{\ell_{w}^{q}}\\
& \leq C\cdot C_{q}\cdot\left[\left\Vert \left(\left\Vert x_{i}\right\Vert _{X_{i}}\right)_{i\in I}\right\Vert _{\ell_{w}^{q}}+\left\Vert \left(\left\Vert y_{i}\right\Vert _{X_{i}}\right)_{i\in I}\right\Vert _{\ell_{w}^{q}}\right]\\
& =C\cdot C_{q}\cdot\left[\left\Vert x\right\Vert +\left\Vert y\right\Vert \right]<\infty,
\end{align*}
where $C_{q}$ is a triangle constant for $\ell_{w}^{q}\left(I\right)$.
Hence, $X$ is closed under addition (and thus a vector space as a
subspace of $\prod_{i\in I}X_{i}$) and $\left\Vert \bullet\right\Vert $
is a quasi-norm on $X$.
\medskip{}
Now, let $\left(x^{\left(n\right)}\right)_{n\in\mathbb{N}}=\left[\left(\smash{x_{i}^{\left(n\right)}}\right)_{i\in I}\right]_{n\in\mathbb{N}}$
be a Cauchy sequence in $X$. It is not hard to see that each of the
projections $\pi_{i}:X\to X_{i},\left(x_{j}\right)_{j\in I}\mapsto x_{i}$
is a bounded linear map, so that each sequence $\left(\smash{x_{i}^{\left(n\right)}}\right)_{n\in\mathbb{N}}$
is Cauchy in $X_{i}$ and hence convergent to some $x_{i}\in X_{i}$.
Now, let $\varepsilon>0$ be arbitrary. There is some $N_{0}\in\mathbb{N}$
satisfying $\left\Vert x^{\left(n\right)}-x^{\left(m\right)}\right\Vert \leq\varepsilon$
for all $n,m\geq N_{0}$. By Fatou's lemma and by continuity of $\left\Vert \bullet\right\Vert _{X_{i}}$,
this implies for $m\geq N_{0}$ that
\begin{align*}
\left\Vert \left(\left\Vert x_{i}-\smash{x_{i}^{\left(m\right)}}\right\Vert _{X_{i}}\right)_{i\in I}\right\Vert _{\ell_{w}^{q}} & =\left\Vert \left(\liminf_{n\to\infty}\left\Vert \smash{x_{i}^{\left(n\right)}}-\smash{x_{i}^{\left(m\right)}}\right\Vert _{X_{i}}\right)_{i\in I}\right\Vert _{\ell_{w}^{q}}\leq\liminf_{n\to\infty}\left\Vert \left(\left\Vert \smash{x_{i}^{\left(n\right)}}-\smash{x_{i}^{\left(m\right)}}\right\Vert _{X_{i}}\right)_{i\in I}\right\Vert _{\ell_{w}^{q}}\\
& =\liminf_{n\to\infty}\left\Vert \smash{x^{\left(n\right)}}-\smash{x^{\left(m\right)}}\right\Vert \leq\varepsilon<\infty.
\end{align*}
Since $X$ is a vector space, this implies $x=\left(x_{i}\right)_{i\in I}=\left(x-\smash{x^{\left(m\right)}}\right)+x^{\left(m\right)}\in X$,
as well as $\left\Vert x-\smash{x^{\left(m\right)}}\right\Vert \xrightarrow[m\to\infty]{}0$.
\end{proof}
The next lemma is our final preparation for proving that the coefficient
map
\[
f\mapsto\left[\left(\gamma^{\left[j\right]}\ast f\right)\!\left(\delta\cdot T_{j}^{-T}k\right)\right]_{j\in I,\,k\in\mathbb{Z}^{d}}
\]
indeed yields a Banach frame for $\DecompSp{\mathcal{Q}}p{\ell_{w}^{q}}v$.
This lemma essentially yields a replacement for the usual reproducing
kernel property which is used in the theory of coorbit spaces (cf.\@
\cite{FeichtingerCoorbit0,FeichtingerCoorbit1,FeichtingerCoorbit2,RauhutCoorbitQuasiBanach}
and \cite[Section 2]{VoigtlaenderPhDThesis}).
\begin{lem}
\label{lem:SpecialProjection}Assume that $\Gamma=\left(\gamma_{i}\right)_{i\in I}$
satisfies Assumptions \ref{assu:DiscreteBanachFrameAssumptions} and
\ref{assu:GammaCoversOrbit}. We clearly have a norm-decreasing embedding
$W_{j}\hookrightarrow V_{j}$ and hence also $\iota:\ell_{w}^{q}\left(\left[W_{j}\right]_{j\in I}\right)\hookrightarrow\ell_{w}^{q}\left(\left[V_{j}\right]_{j\in I}\right)$.
Let
\[
{\rm Ana}_{{\rm osc}}:\DecompSp{\mathcal{Q}}p{\ell_{w}^{q}}v\to\ell_{w}^{q}\left(\left[W_{j}\right]_{j\in I}\right),f\mapsto\left(\gamma^{\left(j\right)}\ast f\right)_{j\in I}
\]
as in Lemma \ref{lem:OscillationForFree}, let
\[
{\rm Synth}_{\mathcal{D}}:\ell_{w}^{q}\left(\left[V_{j}\right]_{j\in I}\right)\to\DecompSp{\mathcal{Q}}p{\ell_{w}^{q}}v,\left(f_{i}\right)_{i\in I}\mapsto\sum_{i\in I}\left[\mathcal{F}^{-1}\left(\varphi_{i}\cdot\widehat{f_{i}}\right)\right]
\]
be defined as in Lemma \ref{lem:DecompositionSynthesis} and let
\[
m_{\theta}:\ell_{w}^{q}\left(\left[V_{j}\right]_{j\in I}\right)\to\ell_{w}^{q}\left(\left[V_{j}\right]_{j\in I}\right),\left(f_{j}\right)_{j\in I}\mapsto\left[\left(\mathcal{F}^{-1}\theta_{j}\right)\ast f_{j}\right]_{j\in I}
\]
be defined as in Lemma \ref{lem:LocalInverseConvolution}.
Then, the map
\[
F:\ell_{w}^{q}\left(\left[V_{j}\right]_{j\in I}\right)\to\ell_{w}^{q}\left(\left[V_{j}\right]_{j\in I}\right),\quad F:=\iota\circ{\rm Ana}_{{\rm osc}}\circ{\rm Synth}_{\mathcal{D}}\circ m_{\theta}
\]
is well-defined and bounded and satisfies the following additional
properties:
\begin{enumerate}
\item \label{enu:CompactAnalysisMapsIntoVernal}$F\left[\left(\gamma^{\left(j\right)}\ast f\right)_{j\in I}\right]=\left(\gamma^{\left(j\right)}\ast f\right)_{j\in I}$
for all $f\in\DecompSp{\mathcal{Q}}p{\ell_{w}^{q}}v$.
\item ${\rm Synth}_{\mathcal{D}}\circ m_{\theta}\circ\iota\circ{\rm Ana}_{{\rm osc}}=\operatorname{id}_{\DecompSp{\mathcal{Q}}p{\ell_{w}^{q}}v}$.
\item $F\circ F=F$.
\item The space $\vernal:=\left\{ \left(f_{i}\right)_{i\in I}\in\ell_{w}^{q}\left(\left[V_{i}\right]_{i\in I}\right)\,\middle|\, F\left(f_{i}\right)_{i\in I}=\left(f_{i}\right)_{i\in I}\right\} $
is a closed subspace of $\ell_{w}^{q}\left(\left[V_{i}\right]_{i\in I}\right)$.
\item For each $f=\left(f_{i}\right)_{i\in I}\in\vernal$, we have that
each $f_{i}:\mathbb{R}^{d}\to\mathbb{C}$ is continuous and furthermore
\begin{equation}
\left\Vert \left[\osc{\delta\cdot T_{i}^{-T}\left[-1,1\right]^{d}}\left(M_{-b_{i}}f_{i}\right)\right]_{i\in I}\right\Vert _{\ell_{w}^{q}\left(\left[V_{i}\right]_{i\in I}\right)}\leq\vertiii{F_{0}}\cdot\delta\cdot\left\Vert f\right\Vert _{\ell_{w}^{q}\left(\left[V_{i}\right]_{i\in I}\right)}\qquad\forall\delta\in\left(0,1\right]\label{eq:OscillationEstimateOnVernal}
\end{equation}
for $F_{0}:={\rm Ana}_{{\rm osc}}\circ{\rm Synth}_{\mathcal{D}}\circ m_{\theta}:\ell_{w}^{q}\left(\left[V_{i}\right]_{i\in I}\right)\to\ell_{w}^{q}\left(\left[W_{i}\right]_{i\in I}\right)$.
Here, we have
\[
\vertiii{F_{0}}\leq2^{\frac{1}{q}}C_{\mathcal{Q},\Phi,v_{0},p}^{2}\cdot\vertiii{\smash{\Gamma_{\mathcal{Q}}}}^{2}\cdot\left(\vertiii{\smash{\overrightarrow{A}}}^{\max\left\{ 1,\frac{1}{p}\right\} }+\vertiii{\smash{\overrightarrow{B}}}^{\max\left\{ 1,\frac{1}{p}\right\} }\right)\cdot C,
\]
for $N:=\left\lceil K+\frac{d+1}{\min\left\{ 1,p\right\} }\right\rceil $
and
\[
C\!:=\!\!\begin{cases}
\frac{\left(2^{16}\cdot768/d^{\frac{3}{2}}\right)^{\frac{d}{p}}}{2^{42}\cdot12^{d}\cdotd^{15}}\!\cdot\!\left(2^{52}\!\cdot\!d^{\frac{25}{2}}\!\cdot\!N^{3}\right)^{N+1}\!\!\!\cdot\!N_{\mathcal{Q}}^{2\left(\frac{1}{p}-1\right)}\!\left(1\!+\!R_{\mathcal{Q}}C_{\mathcal{Q}}\right)^{d\left(\frac{4}{p}-1\right)}\!\!\cdot\Omega_{0}^{13K}\Omega_{1}^{13}\Omega_{2}^{\left(p,K\right)}, & \text{if }p<1,\\
\frac{1}{\sqrt{d}\cdot2^{12+6\left\lceil K\right\rceil }}\cdot\left(2^{17}\cdotd^{5/2}\cdot N\right)^{\left\lceil K\right\rceil +d+2}\cdot\left(1+R_{\mathcal{Q}}\right)^{d}\cdot\Omega_{0}^{3K}\Omega_{1}^{3}\Omega_{2}^{\left(p,K\right)}, & \text{if }p\geq1.
\end{cases}\qedhere
\]
\end{enumerate}
\end{lem}
\begin{proof}
As a consequence of Lemmas \ref{lem:DecompositionSynthesis}, \ref{lem:LocalInverseConvolution}
and \ref{lem:OscillationForFree}, we see that $F_{0}:\ell_{w}^{q}\left(\left[V_{i}\right]_{i\in I}\right)\to\ell_{w}^{q}\left(\left[W_{i}\right]_{i\in I}\right)$
is bounded with $\vertiii{F_{0}}\leq\vertiii{{\rm Ana}_{{\rm osc}}}\cdot\vertiii{{\rm Synth}_{\mathcal{D}}}\cdot\vertiii{m_{\theta}}$.
By plugging in the estimates for the norms of these operators which
were obtained in the respective lemmas and using elementary estimates,
we easily get the stated estimate for $\vertiii{F_{0}}$. With $F_{0}$,
also $F=\iota\circ F_{0}$ is bounded. We now verify the different
claims individually.
\begin{enumerate}
\item The assumptions of the current lemma include those of Theorem \ref{thm:SemiDiscreteBanachFrame},
where it was shown (cf.\@ equation (\ref{eq:SemiDiscreteBanachFrame}))
that $\operatorname{id}_{\DecompSp{\mathcal{Q}}p{\ell_{w}^{q}}v}={\rm Synth}_{\mathcal{D}}\circ m_{\theta}\circ{\rm Ana}_{\Gamma}$,
where ${\rm Ana}_{\Gamma}=\iota\circ{\rm Ana}_{{\rm osc}}$. Hence,
\begin{equation}
{\rm Synth}_{\mathcal{D}}\circ m_{\theta}\circ\iota\circ{\rm Ana}_{{\rm osc}}=\operatorname{id}_{\DecompSp{\mathcal{Q}}p{\ell_{w}^{q}}v},\label{eq:SpecialIdentitiy}
\end{equation}
which proves the second part of the current lemma.
Furthermore, for $f\in\DecompSp{\mathcal{Q}}p{\ell_{w}^{q}}v$, we have
$\left(\gamma^{\left(j\right)}\ast f\right)_{j\in I}={\rm Ana}_{{\rm osc}}f\in\ell_{w}^{q}\left(\left[W_{i}\right]_{i\in I}\right)\subset\ell_{w}^{q}\left(\left[V_{i}\right]_{i\in I}\right)$,
so that $F\left(\left[\gamma^{\left(j\right)}\ast f\right]_{j\in I}\right)\in\ell_{w}^{q}\left(\left[V_{i}\right]_{i\in I}\right)$
is well-defined. Finally, we get
\begin{align*}
F\left[\left(\gamma^{\left(j\right)}\ast f\right)_{j\in I}\right] & =\left(\iota\circ{\rm Ana}_{{\rm osc}}\right)\left[\left({\rm Synth}_{\mathcal{D}}\circ m_{\theta}\right)\left[\gamma^{\left(j\right)}\ast f\right]_{j\in I}\right]\\
& =\left(\iota\circ{\rm Ana}_{{\rm osc}}\right)\left[\left({\rm Synth}_{\mathcal{D}}\circ m_{\theta}\circ\iota\circ{\rm Ana}_{{\rm osc}}\right)f\right]\\
\left({\scriptstyle \text{eq. }\eqref{eq:SpecialIdentitiy}}\right) & =\iota\left({\rm Ana}_{{\rm osc}}f\right)=\iota\left[\left(\gamma^{\left(j\right)}\ast f\right)_{j\in I}\right]=\left(\gamma^{\left(j\right)}\ast f\right)_{j\in I},
\end{align*}
as claimed in the first part.
\item This was proved just above.
\item As a consequence of equation (\ref{eq:SpecialIdentitiy}) (i.e., of
the second part of the lemma), we get
\[
F\circ F=\iota\circ{\rm Ana}_{{\rm osc}}\circ\underbrace{{\rm Synth}_{\mathcal{D}}\circ m_{\theta}\circ\iota\circ{\rm Ana}_{{\rm osc}}}_{=\operatorname{id}_{\DecompSp{\mathcal{Q}}p{\ell_{w}^{q}}v}}\circ{\rm Synth}_{\mathcal{D}}\circ m_{\theta}=\iota\circ{\rm Ana}_{{\rm osc}}\circ{\rm Synth}_{\mathcal{D}}\circ m_{\theta}=F.
\]
\item This trivially follows from continuity and linearity of $F$.
\item For $\left(f_{i}\right)_{i\in I}\in\vernal$, we have $\left(f_{i}\right)_{i\in I}=F\left(f_{i}\right)_{i\in I}=\iota\circ F_{0}\left(f_{i}\right)_{i\in I}$
and hence $\left(f_{i}\right)_{i\in I}=F_{0}\left(f_{i}\right)_{i\in I}$,
where—strictly speaking—on the left-hand side, $\left(f_{i}\right)_{i\in I}$
is interpreted as an element of $\ell_{w}^{q}\left(\left[W_{j}\right]_{j\in I}\right)$
and on the right-hand side as an element of $\ell_{w}^{q}\left(\left[V_{i}\right]_{i\in I}\right)$.
In particular, since $W_{j}\leq C\left(\mathbb{R}^{d}\right)$, we
see that each $f_{i}:\mathbb{R}^{d}\to\mathbb{C}$ is continuous. Finally,
using boundedness of $F_{0}$, we get
\begin{align*}
\sup_{0<\delta\leq1}\frac{1}{\delta}\left\Vert \left(\osc{\delta\cdot T_{i}^{-T}\left[-1,1\right]^{d}}\left[M_{-b_{i}}f_{i}\right]\right)_{i\in I}\right\Vert _{\ell_{w}^{q}\left(\left[V_{i}\right]_{i\in I}\right)} & \leq\left\Vert \left(f_{i}\right)_{i\in I}\right\Vert _{\ell_{w}^{q}\left(\left[W_{i}\right]_{i\in I}\right)}\\
& =\left\Vert F_{0}\left(f_{i}\right)_{i\in I}\right\Vert _{\ell_{w}^{q}\left(\left[W_{i}\right]_{i\in I}\right)}\\
& \leq\vertiii{F_{0}}\cdot\left\Vert \left(f_{i}\right)_{i\in I}\right\Vert _{\ell_{w}^{q}\left(\left[V_{i}\right]_{i\in I}\right)},
\end{align*}
which easily yields the claim.\qedhere
\end{enumerate}
\end{proof}
Given all of these preparations, we can finally show that we obtain
Banach frames for $\DecompSp{\mathcal{Q}}p{\ell_{w}^{q}}v$ in the expected
way:
\begin{thm}
\label{thm:DiscreteBanachFrameTheorem}Assume that $\Gamma=\left(\gamma_{i}\right)_{i\in I}$
satisfies Assumptions \ref{assu:DiscreteBanachFrameAssumptions} and
\ref{assu:GammaCoversOrbit}. Then there is some $\delta_{0}>0$ such
that for every $0<\delta\leq\delta_{0}$, the family $\left(L_{\delta\cdot T_{i}^{-T}k}\widetilde{\gamma^{\left[i\right]}}\right)_{i\in I,k\in\mathbb{Z}^{d}}$
forms a Banach frame for $\DecompSp{\mathcal{Q}}p{\ell_{w}^{q}}v$, with
$\widetilde{\gamma^{\left[i\right]}}\left(x\right)=\gamma^{\left[i\right]}\left(-x\right)$
and
\[
\gamma^{\left[i\right]}=\left|\det T_{i}\right|^{1/2}\cdot M_{b_{i}}\left[\gamma_{i}\circ T_{i}^{T}\right]\qquad\forall i\in I.
\]
In fact, one can choose $\delta_{0}=\frac{1}{1+2\vertiii{F_{0}}^{2}}$,
with $F_{0}$ as in Lemma \ref{lem:SpecialProjection}.
Precisely, the Banach frame property has to be understood as follows:
\begin{itemize}
\item The \textbf{analysis operator}
\begin{alignat*}{2}
A_{\delta}:\: & \DecompSp{\mathcal{Q}}p{\ell_{w}^{q}}v\to\ell_{\left(\left|\det T_{i}\right|^{\frac{1}{2}-\frac{1}{p}}\cdot w_{i}\right)_{i\in I}}^{q}\!\!\!\!\!\left(\left[\vphantom{\sum}\smash{C_{i}^{\left(\delta\right)}}\right]_{i\in I}\right), & f\mapsto\left(\left[\gamma^{\left[i\right]}\ast f\right]\left(\delta\cdot T_{i}^{-T}k\right)\right)_{k\in\mathbb{Z}^{d},i\in I}
\end{alignat*}
is well-defined and bounded for each $\delta\in\left(0,1\right]$.
\item As long as $0<\delta\leq\delta_{0}$, there is a bounded linear \textbf{reconstruction
operator}
\[
R_{\delta}:\ell_{\left(\left|\det T_{i}\right|^{\frac{1}{2}-\frac{1}{p}}\cdot w_{i}\right)_{i\in I}}^{q}\!\!\!\!\!\left(\left[\vphantom{\sum}\smash{C_{i}^{\left(\delta\right)}}\right]_{i\in I}\right)\to\DecompSp{\mathcal{Q}}p{\ell_{w}^{q}}v
\]
satisfying $R_{\delta}\circ A_{\delta}=\operatorname{id}_{\DecompSp{\mathcal{Q}}p{\ell_{w}^{q}}v}$.
\end{itemize}
Finally, we also have the following \textbf{consistency property}:
If $p_{1},p_{2},q_{1},q_{2}\in\left(0,\infty\right]$, if $w^{\left(1\right)}=\left(\smash{w_{i}^{\left(1\right)}}\right)_{i\in I}$
and $w^{\left(2\right)}=\left(\smash{w_{i}^{\left(2\right)}}\right)_{i\in I}$
are $\mathcal{Q}$-moderate weights and if $v^{\left(1\right)},v^{\left(2\right)}:\mathbb{R}^{d}\to\mathbb{C}$
are weights such that the assumptions of the current theorem are satisfied
for $\DecompSp{\mathcal{Q}}{p_{i}}{\ell_{w^{\left(i\right)}}^{q_{i}}}{v^{\left(i\right)}}$
for $i\in\left\{ 1,2\right\} $ and if $0<\delta\leq\min\left\{ \delta_{1},\delta_{2}\right\} $,
where the constant $\delta_{i}$ is equal to the constant $\delta_{0}$
for the choices $p=p_{i},q=q_{i}$, $w=w^{\left(i\right)}$ and $v=v^{\left(i\right)}$,
then we have
\[
\forall f\!\in\DecompSp{\mathcal{Q}}{p_{2}}{\ell_{w^{\left(2\right)}}^{q_{2}}}{v^{\left(2\right)}}:\,f\!\in\DecompSp{\mathcal{Q}}{p_{1}}{\ell_{w^{\left(1\right)}}^{q_{1}}}{v^{\left(1\right)}}\Longleftrightarrow\left[\!\left(\gamma^{\left[j\right]}\!\ast f\right)\!\!\left(\delta\cdot T_{j}^{-T}k\right)\!\right]_{k\in\mathbb{Z}^{d},j\in I}\!\in\!\ell_{\left(\left|\det T_{j}\right|^{\frac{1}{2}-\frac{1}{p_{1}}}w_{j}^{\left(1\right)}\right)_{j\in I}}^{q_{1}}\!\!\!\!\!\!\!\!\!\!\left(\!\left[C_{j}^{\left(1,\delta\right)}\right]_{j\in I}\!\right)\!,
\]
with $C_{j}^{\left(1,\delta\right)}=\ell_{\left(v^{\left(1\right)}\right)^{\left(j,\delta\right)}}^{p}\left(\mathbb{Z}^{d}\right)$
and $\left(v^{\left(1\right)}\right)_{k}^{\left(j,\delta\right)}=v^{\left(1\right)}\!\left(\delta\cdot T_{j}^{-T}k\right)$
for $j\in I$ and $k\in\mathbb{Z}^{d}$.
\end{thm}
\begin{rem*}
\begin{itemize}[leftmargin=0.4cm]
\item The statement of the theorem that the family $\left(L_{\delta\cdot T_{i}^{-T}k}\widetilde{\gamma^{\left[i\right]}}\right)_{i\in I,k\in\mathbb{Z}^{d}}$
forms a Banach frame for the decomposition space $\DecompSp{\mathcal{Q}}p{\ell_{w}^{q}}v$
has to be taken with a grain of salt (i.e., as saying that $A_{\delta},R_{\delta}$
as in the statement of the theorem are bounded and $R_{\delta}\circ A_{\delta}=\operatorname{id}_{\DecompSp{\mathcal{Q}}p{\ell_{w}^{q}}v}$).
But if we have $\mathcal{O}=\mathbb{R}^{d}$, $\gamma_{i}\in\mathcal{S}\left(\mathbb{R}^{d}\right)$
for all $i\in I$ and $\DecompSp{\mathcal{Q}}p{\ell_{w}^{q}}v\hookrightarrow\mathcal{S}'\left(\mathbb{R}^{d}\right)$,
then this statement can be taken literally: As seen in the remark
after Theorem \ref{thm:ConvolvingDecompositionSpaceWithGammaJ}, the
definition of $\gamma^{\left[i\right]}\ast f$ given there coincides
with the usual interpretation for $f\in\mathcal{S}'\left(\mathbb{R}^{d}\right)\supset\DecompSp{\mathcal{Q}}p{\ell_{w}^{q}}v$,
so that we indeed have
\[
A_{\delta}f=\left(\left(\gamma^{\left[i\right]}\ast f\right)\left(\delta\cdot T_{i}^{-T}k\right)\right)_{k\in\mathbb{Z}^{d},i\in I}=\left(\left\langle f,\,L_{\delta\cdot T_{i}^{-T}k}\widetilde{\gamma^{\left[i\right]}}\right\rangle _{\mathcal{S}',\mathcal{S}}\right)_{k\in\mathbb{Z}^{d},i\in I}.
\]
\item For the consistency statement, note that we only claim that an equivalence
of the form
\[
f\!\in\DecompSp{\mathcal{Q}}{p_{1}}{\ell_{w^{\left(1\right)}}^{q_{1}}}{v^{\left(1\right)}}\Longleftrightarrow\left[\left(\gamma^{\left[j\right]}\!\ast f\right)\!\!\left(\delta\cdot T_{j}^{-T}k\right)\right]_{k\in\mathbb{Z}^{d},j\in I}\!\in\!\ell_{\left(\left|\det T_{j}\right|^{\frac{1}{2}-\frac{1}{p_{1}}}w_{j}^{\left(1\right)}\right)_{j\in I}}^{q_{1}}\!\!\!\!\!\!\!\!\!\!\left(\!\left[C_{j}^{\left(1,\delta\right)}\right]_{j\in I}\!\right)
\]
holds under the \emph{assumption} that we \emph{already know} $f\in\DecompSp{\mathcal{Q}}{p_{2}}{\ell_{w^{\left(2\right)}}^{q_{2}}}{v^{\left(2\right)}}$
for suitable $p_{2},q_{2},v^{\left(2\right)},w^{\left(2\right)}$.
In other words, we require that we already know that $f$ has a certain
\emph{minimal amount of regularity}. This is quite natural, since
for an arbitrary $f\in Z'\left(\mathcal{O}\right)$, there is no reason
why $\gamma^{\left[j\right]}\ast f$ should be defined at all.
\item As the proof will show, the action of $R_{\delta}$ on a given sequence
$\left(\smash{c_{k}^{\left(i\right)}}\right)_{i\in I,k\in\mathbb{Z}^{d}}\in\ell_{\left(\left|\det T_{i}\right|^{\frac{1}{2}-\frac{1}{p}}\cdot w_{i}\right)_{i\in I}}^{q}\!\!\!\!\!\!\!\!\left(\left[\smash{C_{i}^{\left(\delta\right)}}\right]_{i\in I}\right)$
is actually \emph{independent} of $p,q,v,w$. The only thing which
depends on these quantities is $\delta_{0}$, so that $R_{\delta}\left(\smash{c_{k}^{\left(i\right)}}\right)_{i\in I,k\in\mathbb{Z}^{d}}$
is only defined for $0<\delta\leq\delta_{0}=\delta_{0}\left(p,q,v,w,\gamma\right)$.
But once this is satisfied, the definition is independent of $p,q,v,w$.\qedhere
\end{itemize}
\end{rem*}
\begin{proof}
First of all, we remark that the $L^{2}$-normalized functions $\gamma^{\left[i\right]}$
yield a nice statement of the theorem, while the proof can be formulated
easier in terms of the $L^{1}$-normalized functions $\gamma^{\left(i\right)}$.
Hence, we introduce the isometric isomorphism
\[
J:\ell_{\left(\left|\det T_{i}\right|^{\frac{1}{2}-\frac{1}{p}}\cdot w_{i}\right)_{i\in I}}^{q}\!\!\!\!\!\!\!\!\!\left(\!\left[C_{i}^{\left(\delta\right)}\right]_{i\in I}\!\right)\to\ell_{\left(\left|\det T_{i}\right|^{-1/p}\cdot w_{i}\right)_{i\in I}}^{q}\!\!\left(\!\left[C_{i}^{\left(\delta\right)}\right]_{i\in I}\!\right),\left(\smash{c_{k}^{\left(i\right)}}\right)_{k\in\mathbb{Z}^{d},i\in I}\mapsto\left(\left|\det T_{i}\right|^{1/2}\cdot c_{k}^{\left(i\right)}\right)_{k\in\mathbb{Z}^{d},i\in I}.
\]
Then, we define $A_{\delta}^{\left(0\right)}:=J\circ A_{\delta}$
and note
\[
A_{\delta}^{\left(0\right)}f=\left(\left(\gamma^{\left(i\right)}\ast f\right)\left(\delta\cdot T_{i}^{-T}k\right)\right)_{k\in\mathbb{Z}^{d},i\in I}\qquad\forall f\in\DecompSp{\mathcal{Q}}p{\ell_{w}^{q}}v,
\]
so it suffices to show that $A_{\delta}^{\left(0\right)}:\DecompSp{\mathcal{Q}}p{\ell_{w}^{q}}v\to\ell_{\left(\left|\det T_{i}\right|^{-1/p}\cdot w_{i}\right)_{i\in I}}^{q}\!\!\!\!\!\!\left(\left[\smash{C_{i}^{\left(\delta\right)}}\right]_{i\in I}\right)$
is well-defined and bounded. Further, if there is a bounded operator
$R_{\delta}^{\left(0\right)}:\ell_{\left(\left|\det T_{i}\right|^{-1/p}\cdot w_{i}\right)_{i\in I}}^{q}\!\!\!\!\!\!\left(\left[\smash{C_{i}^{\left(\delta\right)}}\right]_{i\in I}\right)\to\DecompSp{\mathcal{Q}}p{\ell_{w}^{q}}v$
satisfying $R_{\delta}^{\left(0\right)}\circ A_{\delta}^{\left(0\right)}=\operatorname{id}_{\DecompSp{\mathcal{Q}}p{\ell_{w}^{q}}v}$,
then a suitable definition of the reconstruction operator $R_{\delta}$
in the statement of the theorem is given by $R_{\delta}:=R_{\delta}^{\left(0\right)}\circ J$,
because of $R_{\delta}\circ A_{\delta}=R_{\delta}^{\left(0\right)}\circ J\circ J^{-1}\circ A_{\delta}^{\left(0\right)}=R_{\delta}^{\left(0\right)}\circ A_{\delta}^{\left(0\right)}$.
These considerations also apply to the consistency statement at the
end of the theorem. All in all, we can thus replace $\gamma^{\left[i\right]}$
by $\gamma^{\left(i\right)}$ in the proof, as long as we replace
all occurrences of $\smash{\ell_{\left(\left|\det T_{i}\right|^{\frac{1}{2}-\frac{1}{p}}\cdot w_{i}\right)_{i\in I}}^{q}}\!\!\!\!\!\!\left(\left[\smash{C_{i}^{\left(\delta\right)}}\right]_{i\in I}\right)$
by $\ell_{\left(\left|\det T_{i}\right|^{-1/p}\cdot w_{i}\right)_{i\in I}}^{q}\!\!\!\!\!\!\left(\left[\smash{C_{i}^{\left(\delta\right)}}\right]_{i\in I}\right)$.
\medskip{}
In the whole proof, we will use the nomenclature introduced in Lemma
\ref{lem:SpecialProjection}. As noted in that lemma, every function
$f_{i}$ is continuous if $\left(f_{i}\right)_{i\in I}\in\vernal$.
Hence, for each $i\in I$, the operator
\[
{\rm Samp}_{\delta,i}:\vernal\to\mathbb{C}^{\mathbb{Z}^{d}},\left(f_{j}\right)_{j\in I}\mapsto\left[f_{i}\left(\delta\cdot T_{i}^{-T}k\right)\right]_{k\in\mathbb{Z}^{d}}
\]
is well-defined. Now, note with ${\rm Synth}_{\delta,i}$ as in Lemma
\ref{lem:DiscreteSynthesisOperatorIsAlmostIsometric} that
\begin{align*}
& \left|f_{i}\left(x\right)-\left[\left({\rm Synth}_{\delta,i}\circ{\rm Samp}_{\delta,i}\right)\left(f_{j}\right)_{j\in I}\right]\left(x\right)\right|\\
& =\left|f_{i}\left(x\right)-\left(M_{b_{i}}\left[\sum_{k\in\mathbb{Z}^{d}}f_{i}\left(\delta\cdot T_{i}^{-T}k\right)\cdot e^{-2\pi i\left\langle b_{i},\delta\cdot T_{i}^{-T}k\right\rangle }{\mathds{1}}_{\delta\cdot T_{i}^{-T}\left(k+\left[0,1\right)^{d}\right)}\right]\right)\left(x\right)\right|\\
& =\left|\left(M_{-b_{i}}f_{i}\right)\left(x\right)-\sum_{k\in\mathbb{Z}^{d}}\left(M_{-b_{i}}f_{i}\right)\left(\delta\cdot T_{i}^{-T}k\right)\cdot{\mathds{1}}_{\delta\cdot T_{i}^{-T}\left(k+\left[0,1\right)^{d}\right)}\left(x\right)\right|\\
\left({\scriptstyle \mathbb{R}^{d}=\biguplus_{k\in\mathbb{Z}^{d}}\delta T_{i}^{-T}\left(k+\left[0,1\right)^{d}\right)}\right) & =\left|\sum_{k\in\mathbb{Z}^{d}}{\mathds{1}}_{\delta\cdot T_{i}^{-T}\left(k+\left[0,1\right)^{d}\right)}\left(x\right)\cdot\left[\left(M_{-b_{i}}f_{i}\right)\left(x\right)-\left(M_{-b_{i}}f_{i}\right)\left(\delta\cdot T_{i}^{-T}k\right)\right]\right|\\
& \leq\sum_{k\in\mathbb{Z}^{d}}{\mathds{1}}_{\delta\cdot T_{i}^{-T}\left(k+\left[0,1\right)^{d}\right)}\left(x\right)\cdot\left|\left(M_{-b_{i}}f_{i}\right)\left(x\right)-\left(M_{-b_{i}}f_{i}\right)\left(\delta\cdot T_{i}^{-T}k\right)\right|.
\end{align*}
Now, note that ${\mathds{1}}_{\delta\cdot T_{i}^{-T}\left(k+\left[0,1\right)^{d}\right)}\left(x\right)\neq0$
implies $\delta\cdot T_{i}^{-T}k\in x-\delta T_{i}^{-T}\left[0,1\right)^{d}\subset x+\delta T_{i}^{-T}\left[-1,1\right]^{d}$.
Since we trivially have $x\in x+\delta T_{i}^{-T}\left[-1,1\right]^{d}$,
we obtain
\[
\left|\left(M_{-b_{i}}f_{i}\right)\left(x\right)-\left(M_{-b_{i}}f_{i}\right)\left(\delta\cdot T_{i}^{-T}k\right)\right|\leq\left(\osc{\delta T_{i}^{-T}\left[-1,1\right]^{d}}\left[M_{-b_{i}}f_{i}\right]\right)\left(x\right).
\]
Using again that $\mathbb{R}^{d}=\biguplus_{k\in\mathbb{Z}^{d}}\delta T_{i}^{-T}\left(k+\left[0,1\right)^{d}\right)$,
we conclude
\[
\left|f_{i}\left(x\right)-\left[\left({\rm Synth}_{\delta,i}\circ{\rm Samp}_{\delta,i}\right)\left(f_{j}\right)_{j\in I}\right]\left(x\right)\right|\leq\left(\osc{\delta T_{i}^{-T}\left[-1,1\right]^{d}}\left[M_{-b_{i}}f_{i}\right]\right)\left(x\right)\qquad\forall i\in I\;\forall x\in\mathbb{R}^{d}\;\forall\left(f_{j}\right)_{j\in I}\in\vernal.
\]
Consequently, using the solidity of $V_{i}$, we get for
\begin{align*}
{\rm Samp}_{\delta} & :=\prod_{i\in I}{\rm Samp}_{\delta,i}:\vernal\to\left(\mathbb{C}^{\mathbb{Z}^{d}}\right)^{I},\left(f_{j}\right)_{j\in I}\mapsto\left({\rm Samp}_{\delta,i}\left(f_{j}\right)_{j\in I}\right)_{i\in I},\\
{\rm Synth}_{\delta} & :=\bigotimes_{i\in I}{\rm Synth}_{\delta,i}:\left(\smash{\mathbb{C}^{\mathbb{Z}^{d}}}\right)^{I}\!\to\left\{ \left(f_{i}\right)_{i\in I}\,\middle|\, f_{i}:\mathbb{R}^{d}\to\mathbb{C}\text{ measurable }\forall i\in I\right\} ,\left(\smash{c_{k}^{\left(i\right)}}\right)_{k\in\mathbb{Z}^{d},i\in I}\mapsto\left({\rm Synth}_{\delta,i}\left(\smash{c_{k}^{\left(i\right)}}\right)_{k\in\mathbb{Z}^{d}}\right)_{i\in I}
\end{align*}
that
\begin{align}
\left\Vert \left(f_{i}\right)_{i\in I}-\left({\rm Synth}_{\delta}\circ{\rm Samp}_{\delta}\right)\left(f_{i}\right)_{i\in I}\right\Vert _{\ell_{w}^{q}\left(\left[V_{i}\right]_{i\in I}\right)} & =\left\Vert \left(\left\Vert f_{i}-{\rm Synth}_{\delta,i}\circ{\rm Samp}_{\delta,i}\left(f_{j}\right)_{j\in I}\right\Vert _{V_{i}}\right)_{i\in I}\right\Vert _{\ell_{w}^{q}}\nonumber \\
& \leq\left\Vert \left(\left\Vert \osc{\delta T_{i}^{-T}\left[-1,1\right]^{d}}\left[M_{-b_{i}}f_{i}\right]\right\Vert _{V_{i}}\right)_{i\in I}\right\Vert _{\ell_{w}^{q}}\nonumber \\
\left({\scriptstyle \text{eq. }\eqref{eq:OscillationEstimateOnVernal}}\right) & \leq\vertiii{F_{0}}\cdot\delta\cdot\left\Vert \left(f_{i}\right)_{i\in I}\right\Vert _{\ell_{w}^{q}\left(\left[V_{i}\right]_{i\in I}\right)}\quad\forall\left(f_{i}\right)_{i\in I}\in\vernal\:\forall\delta\in\left(0,1\right].\label{eq:UnsmoothedDiscretizationIsClose}
\end{align}
Using the (quasi)-triangle inequality for $\ell_{w}^{q}\left(\left[V_{i}\right]_{i\in I}\right)$
(where $2^{\frac{1}{p}+\frac{1}{q}}$ is a valid triangle constant,
thanks to \cite[Exercise 1.1.5(c)]{GrafakosClassicalFourierAnalysis}
and to (the proof of) Lemma \ref{lem:IteratedSequenceSpaceComplete}),
we conclude that
\[
T_{0}^{\left(\delta\right)}:={\rm Synth}_{\delta}\circ{\rm Samp}_{\delta}:\vernal\to\ell_{w}^{q}\left(\left[V_{i}\right]_{i\in I}\right)
\]
is well-defined and bounded, with $\vertiii{T_{0}^{\left(\delta\right)}}\leq2^{\frac{1}{p}+\frac{1}{q}}\cdot\left(1+\vertiii{F_{0}}\delta\right)\leq2^{\frac{1}{p}+\frac{1}{q}}\left(1+\vertiii{F_{0}}\right)$
for all $\delta\in\left(0,1\right]$.
\medskip{}
Boundedness of $T_{0}^{\left(\delta\right)}$—together with estimate
(\ref{eq:UnsmoothedDiscretizationIsClose})—is almost sufficient for
our purposes, but not quite: In general, it need not be the case that
$T_{0}^{\left(\delta\right)}$ maps $\vernal$ into $\vernal$. But
since Lemma \ref{lem:SpecialProjection} shows $F\circ F=F$, it is
easy to see $F:\ell_{w}^{q}\left(\left[V_{i}\right]_{i\in I}\right)\to\vernal$,
so that $T^{\left(\delta\right)}:=F\circ T_{0}^{\left(\delta\right)}:\vernal\to\vernal$.
Furthermore, since $F|_{\vernal}=\operatorname{id}_{\vernal}$, we get
\begin{align}
\left\Vert \left(f_{i}\right)_{i\in I}-T^{\left(\delta\right)}\left(f_{i}\right)_{i\in I}\right\Vert _{\ell_{w}^{q}\left(\left[V_{i}\right]_{i\in I}\right)} & =\left\Vert F\left(f_{i}\right)_{i\in I}-FT_{0}^{\left(\delta\right)}\left(f_{i}\right)_{i\in I}\right\Vert _{\ell_{w}^{q}\left(\left[V_{i}\right]_{i\in I}\right)}\nonumber \\
& \leq\vertiii F\cdot\left\Vert \left(f_{i}\right)_{i\in I}-T_{0}^{\left(\delta\right)}\left(f_{i}\right)_{i\in I}\right\Vert _{\ell_{w}^{q}\left(\left[V_{i}\right]_{i\in I}\right)}\nonumber \\
\left({\scriptstyle \text{eq. }\eqref{eq:UnsmoothedDiscretizationIsClose}}\right) & \leq\vertiii{F_{0}}^{2}\cdot\delta\cdot\left\Vert \left(f_{i}\right)_{i\in I}\right\Vert _{\ell_{w}^{q}\left(\left[V_{i}\right]_{i\in I}\right)}\qquad\forall\left(f_{i}\right)_{i\in I}\in\vernal\:\forall\delta\in\left(0,1\right].\label{eq:SmoothedDiscretizationIsClose}
\end{align}
But for $0<\delta\leq\delta_{0}=\frac{1}{1+2\vertiii{F_{0}}^{2}}$,
we have $\vertiii{F_{0}}^{2}\cdot\delta\leq\frac{1}{2}$ and hence
$\vertiii{\operatorname{id}_{\vernal}-T^{\left(\delta\right)}}\leq\frac{1}{2}$.
Using a Neumann-series argument (which is also valid for Quasi-Banach
spaces, cf.\@ e.g.\@ \cite[Lemma 2.4.11]{VoigtlaenderPhDThesis}
and thus for the closed subspace $\vernal$ of the Quasi-Banach space
$\ell_{w}^{q}\left(\left[V_{i}\right]_{i\in I}\right)$ thanks to
Lemmas \ref{lem:IteratedSequenceSpaceComplete} and \ref{lem:SpecialProjection}),
we conclude that $T^{\left(\delta\right)}:\vernal\to\vernal$ is boundedly
invertible, as long as $0<\delta\leq\delta_{0}$.
\medskip{}
Now, for arbitrary $f\in\DecompSp{\mathcal{Q}}p{\ell_{w}^{q}}v$, Lemma
\ref{lem:SpecialProjection} shows that $\left({\rm Synth}_{\mathcal{D}}\circ m_{\theta}\circ\iota\circ{\rm Ana}_{{\rm osc}}\right)f=f$.
The same lemma also shows $F\left[\left(\gamma^{\left(j\right)}\ast f\right)_{j\in I}\right]=\left(\gamma^{\left(j\right)}\ast f\right)_{j\in I}$,
i.e., $\left(\iota\circ{\rm Ana}_{{\rm osc}}\right)f=\left(\gamma^{\left(j\right)}\ast f\right)_{j\in I}\in\vernal$.
Hence,
\begin{align*}
f & =\left({\rm Synth}_{\mathcal{D}}\circ m_{\theta}\circ\iota\circ{\rm Ana}_{{\rm osc}}\right)f\\
& =\left[\left({\rm Synth}_{\mathcal{D}}\circ m_{\theta}\right)\circ\left(T^{\left(\delta\right)}\right)^{-1}\circ T^{\left(\delta\right)}\circ\iota\circ{\rm Ana}_{{\rm osc}}\right]f\\
\left({\scriptstyle \text{def. of }T^{\left(\delta\right)}}\right) & =\left[\left(\left[{\rm Synth}_{\mathcal{D}}\circ m_{\theta}\right]\circ\left(T^{\left(\delta\right)}\right)^{-1}\circ F\circ{\rm Synth}_{\delta}\right)\circ{\rm Samp}_{\delta}\circ\iota\circ{\rm Ana}_{{\rm osc}}\right]f.
\end{align*}
Now, note
\[
\left[\left(\left[{\rm Samp}_{\delta}\circ\iota\circ{\rm Ana}_{{\rm osc}}\right]f\right)_{i}\right]_{k}=\left(\left[{\rm Samp}_{\delta}\left(\gamma^{\left(j\right)}\ast f\right)_{j\in I}\right]_{i}\right)_{k}=\left(\gamma^{\left(i\right)}\ast f\right)\left(\delta\cdot T_{i}^{-T}k\right)
\]
and hence ${\rm Samp}_{\delta}\circ\iota\circ{\rm Ana}_{{\rm osc}}=A_{\delta}^{\left(0\right)}$
on $\DecompSp{\mathcal{Q}}p{\ell_{w}^{q}}v$. Thus, if we define $R_{\delta}^{\left(0\right)}:=\left[{\rm Synth}_{\mathcal{D}}\circ m_{\theta}\right]\circ\left(T^{\left(\delta\right)}\right)^{-1}\circ F\circ{\rm Synth}_{\delta}$,
we have shown $\operatorname{id}_{\DecompSp{\mathcal{Q}}p{\ell_{w}^{q}}v}=R_{\delta}^{\left(0\right)}\circ A_{\delta}^{\left(0\right)}$,
as claimed. All that remains to show is that $R_{\delta}^{\left(0\right)},A_{\delta}^{\left(0\right)}$
are indeed well-defined and bounded with domains and codomains as
stated at the beginning of the proof.
\medskip{}
To this end, note that Lemma \ref{lem:DiscreteSynthesisOperatorIsAlmostIsometric}
easily implies that ${\rm Synth}_{\delta}:\ell_{\left(\left|\det T_{i}\right|^{-1/p}\cdot w_{i}\right)_{i\in I}}^{q}\!\!\!\!\!\left(\left[\smash{C_{i}^{\left(\delta\right)}}\right]_{i\in I}\right)\to\ell_{w}^{q}\left(\left[V_{i}\right]_{i\in I}\right)$
is well-defined and bounded. In fact, the lemma even shows that
\[
\left(\smash{c_{k}^{\left(i\right)}}\right)_{k\in\mathbb{Z}^{d},i\in I}\in\ell_{\left(\left|\det T_{i}\right|^{-1/p}\cdot w_{i}\right)_{i\in I}}^{q}\!\!\left(\left[\vphantom{F}\smash{C_{i}^{\left(\delta\right)}}\right]_{i\in I}\right)\Longleftrightarrow{\rm Synth}_{\delta}\left(\smash{c_{k}^{\left(i\right)}}\right)_{k\in\mathbb{Z}^{d},i\in I}\in\ell_{w}^{q}\left(\left[V_{i}\right]_{i\in I}\right)
\]
and
\[
\left\Vert {\rm Synth}_{\delta}\left(\smash{c_{k}^{\left(i\right)}}\right)_{k\in\mathbb{Z}^{d},i\in I}\right\Vert _{\ell_{w}^{q}\left(\left[V_{i}\right]_{i\in I}\right)}\asymp\left\Vert \left(\smash{c_{k}^{\left(i\right)}}\right)_{k\in\mathbb{Z}^{d},i\in I}\right\Vert _{\ell_{\left(\left|\det T_{i}\right|^{-1/p}\cdot w_{i}\right)_{i\in I}}^{q}\!\!\!\!\!\left(\left[\vphantom{F}\smash{C_{i}^{\left(\delta\right)}}\right]_{i\in I}\right)},
\]
where the implied constant may depend on $\delta$. Consequently,
$R_{\delta}^{\left(0\right)}:\ell_{\left(\left|\det T_{i}\right|^{-1/p}\cdot w_{i}\right)_{i\in I}}^{q}\!\!\!\left(\left[\vphantom{F}\smash{C_{i}^{\left(\delta\right)}}\right]_{i\in I}\right)\to\DecompSp{\mathcal{Q}}p{\ell_{w}^{q}}v$
is indeed well-defined and bounded for $0<\delta\leq\delta_{0}$.
Furthermore, we see (now for arbitrary $\delta\in\left(0,1\right]$)
that
\begin{align*}
\left\Vert \smash{A_{\delta}^{\left(0\right)}}f\right\Vert _{\ell_{\left(\left|\det T_{i}\right|^{-1/p}\cdot w_{i}\right)_{i\in I}}^{q}\!\!\!\left(\left[\vphantom{F}\smash{C_{i}^{\left(\delta\right)}}\right]_{i\in I}\right)} & =\left\Vert \left({\rm Samp}_{\delta}\circ\iota\circ{\rm Ana}_{{\rm osc}}\right)f\right\Vert _{\ell_{\left(\left|\det T_{i}\right|^{-1/p}\cdot w_{i}\right)_{i\in I}}^{q}\!\!\!\left(\left[\vphantom{F}\smash{C_{i}^{\left(\delta\right)}}\right]_{i\in I}\right)}\\
& \asymp_{\delta}\:\left\Vert \left({\rm Synth}_{\delta}\circ{\rm Samp}_{\delta}\circ\iota\circ{\rm Ana}_{{\rm osc}}\right)f\right\Vert _{\ell_{w}^{q}\left(\left[V_{i}\right]_{i\in I}\right)}\\
& =\left\Vert \left(\smash{T_{0}^{\left(\delta\right)}}\circ\iota\circ{\rm Ana}_{{\rm osc}}\right)f\right\Vert _{\ell_{w}^{q}\left(\left[V_{i}\right]_{i\in I}\right)}\\
\left({\scriptstyle \iota\circ{\rm Ana}_{{\rm osc}}:\DecompSp{\mathcal{Q}}p{\ell_{w}^{q}}v\to\vernal,\text{ as seen above}}\right) & \lesssim\vertiii{\smash{T_{0}^{\left(\delta\right)}}}_{\vernal\to\ell_{w}^{q}\left(\left[V_{i}\right]_{i\in I}\right)}\cdot\vertiii{\iota\circ{\rm Ana}_{{\rm osc}}}_{\DecompSp{\mathcal{Q}}p{\ell_{w}^{q}}v\to\ell_{w}^{q}\left(\left[V_{i}\right]_{i\in I}\right)}\cdot\left\Vert f\right\Vert _{\DecompSp{\mathcal{Q}}p{\ell_{w}^{q}}v}<\infty
\end{align*}
for all $f\in\DecompSp{\mathcal{Q}}p{\ell_{w}^{q}}v$. This finally shows
that $A_{\delta}^{\left(0\right)}:\DecompSp{\mathcal{Q}}p{\ell_{w}^{q}}v\to\ell_{\left(\left|\det T_{i}\right|^{-1/p}\cdot w_{i}\right)_{i\in I}}^{q}\!\!\!\left(\left[\vphantom{F}\smash{C_{i}^{\left(\delta\right)}}\right]_{i\in I}\right)$
is well-defined and bounded for each $\delta\in\left(0,1\right]$
and thus completes the proof of the Banach frame property.
\medskip{}
It remains to verify the consistency property stated above. To this
end, first define
\[
V_{j}^{\left(i\right)}:=\begin{cases}
L_{v^{\left(i\right)}}^{p_{i}}\left(\smash{\mathbb{R}^{d}}\right), & \text{if }p_{i}\in\left[1,\infty\right],\\
W_{T_{j}^{-T}\left[-1,1\right]^{d}}\left(L_{v^{\left(i\right)}}^{p_{i}}\right), & \text{if }p_{i}\in\left(0,1\right),
\end{cases}
\]
as well as $C_{j}^{\left(i,\delta\right)}=\ell_{\left(v^{\left(i\right)}\right)^{\left(j,\delta\right)}}^{p_{i}}\left(\mathbb{Z}^{d}\right)$
with $\left(v^{\left(i\right)}\right)_{k}^{\left(j,\delta\right)}=v^{\left(i\right)}\left(\delta\cdot T_{j}^{-T}k\right)$
for $k\in\mathbb{Z}^{d}$, $i\in\left\{ 1,2\right\} $ and $j\in I$.
Next, we observe that the domain and codomain of the reconstruction/analysis
operators
\[
R_{\delta}^{\left(0,i\right)}:\ell_{\left(\left|\det T_{j}\right|^{-1/p_{i}}\cdot w_{j}^{\left(i\right)}\right)_{j\in I}}^{q_{i}}\!\!\!\left(\left[\vphantom{F}\smash{C_{j}^{\left(i,\delta\right)}}\right]_{j\in I}\right)\to\DecompSp{\mathcal{Q}}{p_{i}}{\ell_{w^{\left(i\right)}}^{q_{i}}}{v^{\left(i\right)}}
\]
and
\[
A_{\delta}^{\left(0,i\right)}:\DecompSp{\mathcal{Q}}{p_{i}}{\ell_{w^{\left(i\right)}}^{q_{i}}}{v^{\left(i\right)}}\to\ell_{\left(\left|\det T_{j}\right|^{-1/p_{i}}\cdot w_{j}^{\left(i\right)}\right)_{j\in I}}^{q_{i}}\left(\left[\vphantom{F}\smash{C_{j}^{\left(i,\delta\right)}}\right]_{j\in I}\right)
\]
do depend on $i\in\left\{ 1,2\right\} $, but the actual \emph{action}
of these mappings do not: We always have
\[
A_{\delta}^{\left(0,i\right)}f=\left[\left(\gamma^{\left(j\right)}\ast f\right)\left(\delta\cdot T_{j}^{-T}k\right)\right]_{k\in\mathbb{Z}^{d},j\in I}\quad\overset{\text{eq. }\eqref{eq:SpecialConvolutionPointwiseDefinition}}{=}\quad\left[\sum_{\ell\in I}\mathcal{F}^{-1}\left(\widehat{\gamma^{\left(j\right)}}\cdot\varphi_{\ell}\cdot\widehat{f}\right)\left(\delta\cdot T_{j}^{-T}k\right)\right]_{k\in\mathbb{Z}^{d},j\in I}
\]
and
\begin{align*}
R_{\delta}^{\left(0,i\right)}\left(c_{k}^{\left(j\right)}\right)_{k\in\mathbb{Z}^{d},j\in I} & =\left(\left[{\rm Synth}_{\mathcal{D}}\circ m_{\theta}\right]\circ\left(T^{\left(\delta\right)}\right)^{-1}\circ F\circ{\rm Synth}_{\delta}\right)\left(c_{k}^{\left(j\right)}\right)_{k\in\mathbb{Z}^{d},j\in I}\\
& =\left(\left[{\rm Synth}_{\mathcal{D}}\circ m_{\theta}\right]\circ\left(T^{\left(\delta\right)}\right)^{-1}\circ\iota\circ{\rm Ana}_{{\rm osc}}\circ{\rm Synth}_{\mathcal{D}}\circ m_{\theta}\circ{\rm Synth}_{\delta}\right)\left(c_{k}^{\left(j\right)}\right)_{k\in\mathbb{Z}^{d},j\in I},
\end{align*}
where
\begin{align*}
{\rm Synth}_{\mathcal{D}}\left(f_{j}\right)_{j\in I} & =\sum_{j\in I}\left[\mathcal{F}^{-1}\left(\varphi_{j}\cdot\widehat{f_{j}}\right)\right]\quad\text{ with unconditional convergence in }Z'\left(\mathcal{O}\right),\\
m_{\theta}\left(f_{j}\right)_{j\in I} & =\left[\left(\mathcal{F}^{-1}\theta_{j}\right)\ast f_{j}\right]_{j\in I},\\
\iota\left(f_{j}\right)_{j\in I} & =\left(f_{j}\right)_{j\in I},\\
{\rm Ana}_{{\rm osc}}f & =\left(\gamma^{\left(j\right)}\ast f\right)_{j\in I}\quad\text{with }\gamma^{\left(j\right)}\ast f\text{ as in equation }\eqref{eq:SpecialConvolutionPointwiseDefinition},\\
{\rm Synth}_{\delta}\left(\smash{c_{k}^{\left(j\right)}}\right)_{k\in\mathbb{Z}^{d},j\in I} & =\left(M_{b_{j}}\left[\sum_{k\in\mathbb{Z}^{d}}c_{k}^{\left(j\right)}\cdot e^{-2\pi i\left\langle b_{j},\delta\cdot T_{j}^{-T}k\right\rangle }{\mathds{1}}_{\delta\cdot T_{j}^{-T}\left(k+\left[0,1\right)^{d}\right)}\right]\right)_{j\in I}
\end{align*}
for all $\left(f_{j}\right)_{j\in I}\in\ell_{w^{\left(i\right)}}^{q_{i}}\!\!\left(\left[\smash{V_{j}^{\left(i\right)}}\right]_{j\in I}\right)$,
all $\left(\smash{c_{k}^{\left(j\right)}}\right)_{k\in\mathbb{Z}^{d},j\in I}\in\ell_{\left(\left|\det T_{j}\right|^{-1/p_{i}}\cdot w_{j}^{\left(i\right)}\right)_{j\in I}}^{q_{i}}\!\!\!\left(\left[\vphantom{F}\smash{C_{j}^{\left(i,\delta\right)}}\right]_{j\in I}\right)$,
and all $f\in\DecompSp{\mathcal{Q}}{p_{i}}{\ell_{w^{\left(i\right)}}^{q_{i}}}{v^{\left(i\right)}}$.
Finally, we also have (since $\left(T^{\left(\delta\right)}\right)^{-1}$
can be computed by a Neumann series, as shown above)
\[
\left(T^{\left(\delta\right)}\right)^{-1}\left(f_{j}\right)_{j\in I}=\left(\operatorname{id}-\left[\operatorname{id}-T^{\left(\delta\right)}\right]\right)^{-1}\left(f_{j}\right)_{j\in I}=\sum_{n=0}^{\infty}\left(\operatorname{id}-T^{\left(\delta\right)}\right)^{n}\left(f_{j}\right)_{j\in I},
\]
where
\[
T^{\left(\delta\right)}\left(f_{j}\right)_{j\in I}=\left(F\circ T_{0}^{\left(\delta\right)}\right)\left(f_{j}\right)_{j\in I}=\left(\iota\circ{\rm Ana}_{{\rm osc}}\circ{\rm Synth}_{\mathcal{D}}\circ m_{\theta}\right)\circ\left({\rm Synth}_{\delta}\circ{\rm Samp}_{\delta}\right)\left(f_{j}\right)_{j\in I}
\]
for
\[
\left(f_{j}\right)_{j\in I}\in\vernal_{i}:=\left\{ \left(g_{j}\right)_{j\in I}\in\ell_{w^{\left(i\right)}}^{q_{i}}\left(\left[\vphantom{F}\smash{V_{j}^{\left(i\right)}}\right]_{j\in I}\right)\,\middle|\, F\left(g_{j}\right)_{j\in I}=\left(g_{j}\right)_{j\in I}\right\} .
\]
In summary, we have shown $R_{\delta}^{\left(0,1\right)}\left(\smash{c_{k}^{\left(j\right)}}\right)_{k\in\mathbb{Z}^{d},j\in I}=R_{\delta}^{\left(0,2\right)}\left(\smash{c_{k}^{\left(j\right)}}\right)_{k\in\mathbb{Z}^{d},j\in I}$
and $A_{\delta}^{\left(0,1\right)}f=A_{\delta}^{\left(0,2\right)}f$,
as long as both sides of the respective equations are defined. Now,
let $f\in\DecompSp{\mathcal{Q}}{p_{2}}{\ell_{w^{\left(2\right)}}^{q_{2}}}{v^{\left(2\right)}}$
be arbitrary. The implication ``$\Rightarrow$'' of the consistency
statement follows immediately from the main statement of the theorem,
so that we only need to show ``$\Leftarrow$''. Hence, assume
\[
c:=A_{\delta}^{\left(0,2\right)}f=\left[\left(\gamma^{\left(j\right)}\ast f\right)\left(\delta\cdot T_{j}^{-T}k\right)\right]_{k\in\mathbb{Z}^{d},j\in I}\in\ell_{\left(\left|\det T_{j}\right|^{-1/p_{1}}\cdot w_{j}^{\left(1\right)}\right)_{j\in I}}^{q_{1}}\!\!\!\left(\left[\vphantom{F}\smash{C_{j}^{\left(1,\delta\right)}}\right]_{j\in I}\right).
\]
We know from above that $f=R_{\delta}^{\left(0,2\right)}A_{\delta}^{\left(0,2\right)}f=R_{\delta}^{\left(0,2\right)}c$.
But we have $c\in\ell_{\left(\left|\det T_{j}\right|^{-1/p_{i}}\cdot w_{j}^{\left(i\right)}\right)_{j\in I}}^{q_{i}}\!\!\left(\left[\vphantom{F}\smash{C_{j}^{\left(i,\delta\right)}}\right]_{j\in I}\right)$
for both $i=1$ and $i=2$, so that we get $f=R_{\delta}^{\left(0,2\right)}c=R_{\delta}^{\left(0,1\right)}c$.
Since
\[
R_{\delta}^{\left(0,1\right)}:\ell_{\left(\left|\det T_{j}\right|^{-1/p_{1}}\cdot w_{j}^{\left(1\right)}\right)_{j\in I}}^{q_{1}}\!\!\!\left(\left[\vphantom{F}\smash{C_{j}^{\left(1,\delta\right)}}\right]_{j\in I}\right)\to\DecompSp{\mathcal{Q}}{p_{1}}{\ell_{w^{\left(1\right)}}^{q_{1}}}{v^{\left(1\right)}}
\]
is well-defined and bounded, we get $f\in\DecompSp{\mathcal{Q}}{p_{1}}{\ell_{w^{\left(1\right)}}^{q_{1}}}{v^{\left(1\right)}}$,
as claimed.
\end{proof}
The main limitation of Theorem \ref{thm:DiscreteBanachFrameTheorem}
is its somewhat opaque set of assumptions regarding $\Gamma=\left(\gamma_{i}\right)_{i\in I}$.
In Section \ref{sec:SimplifiedCriteria} (see in particular Corollary
\ref{cor:BanachFrameSimplifiedCriteria}), we will derive more transparent
criteria which ensure that Theorem \ref{thm:DiscreteBanachFrameTheorem}
is applicable.
But before that, we first consider the ``dual'' problem to the Banach
frame property, i.e., we show that the family $\left(L_{\delta\cdot T_{i}^{-T}k}\:\gamma^{\left[i\right]}\right)_{k\in\mathbb{Z}^{d},i\in I}$
forms an \emph{atomic decomposition} for the decomposition space $\DecompSp{\mathcal{Q}}p{\ell_{w}^{q}}v$,
under suitable assumptions on $\Gamma=\left(\gamma_{i}\right)_{i\in I}$.
Proving this is the main goal of the next section.
\section{Atomic decompositions}
\label{sec:AtomicDecompositions}In this section, we show the dual
statement to the preceding section, i.e., we show that the (discretely
translated) $\gamma^{\left[j\right]}$ not only form a Banach frame
for $\DecompSp{\mathcal{Q}}p{\ell_{w}^{q}}v$, but also an atomic decomposition.
For this, we introduce still another set of assumptions:
\begin{assumption}
\label{assu:AtomicDecompositionAssumption}We assume that for each
$i\in I$, we are given functions $\gamma_{i},\gamma_{i,1},\gamma_{i,2}$
with the following properties:
\begin{enumerate}
\item We have $\gamma_{i,1},\gamma_{i,2}\in L_{\left(1+\left|\bullet\right|\right)^{K}}^{1}\left(\mathbb{R}^{d}\right)\hookrightarrow L_{v_{0}}^{1}\left(\mathbb{R}^{d}\right)\hookrightarrow L^{1}\left(\mathbb{R}^{d}\right)$
for all $i\in I$.
\item We have $\gamma_{i}=\gamma_{i,1}\ast\gamma_{i,2}$ for all $i\in I$.
\item We have $\widehat{\gamma_{i,1}},\widehat{\gamma_{i,2}}\in C^{\infty}\left(\mathbb{R}^{d}\right)$
for all $i\in I$ and all partial derivatives of $\widehat{\gamma_{i,1}},\widehat{\gamma_{i,2}}$
have at most polynomial growth.
\item We have $\gamma_{i,2}\in C^{1}\left(\mathbb{R}^{d}\right)$ with
$\nabla\gamma_{i,2}\in L_{v_{0}}^{1}\left(\mathbb{R}^{d}\right)$
for all $i\in I$.
\item The constant
\begin{equation}
\Omega_{4}^{\left(p,K\right)}:=\sup_{i\in I}\left\Vert \gamma_{i,2}\right\Vert _{K_{0}}+\sup_{i\in I}\left\Vert \nabla\gamma_{i,2}\right\Vert _{K_{0}}\label{eq:AtomicDecompositionGamma2ConstantDefinition}
\end{equation}
is finite. Here, $K_{0}:=K+\frac{d}{\min\left\{ 1,p\right\} }+1$
and
\[
\left\Vert f\right\Vert _{K_{0}}:=\sup_{x\in\mathbb{R}^{d}}\left(1+\left|x\right|\right)^{K_{0}}\left|f\left(x\right)\right|\in\left[0,\infty\right].
\]
\item We have $\left\Vert \gamma_{i}\right\Vert _{K_{0}}<\infty$ for all
$i\in I$.
\item For $\ell\in\left\{ 1,2\right\} $ and $i\in I$, define
\begin{equation}
\gamma_{\ell}^{\left(i\right)}:=\mathcal{F}^{-1}\left(\widehat{\gamma_{i,\ell}}\circ S_{i}^{-1}\right)=\left|\det T_{i}\right|\cdot M_{b_{i}}\left[\gamma_{i,\ell}\circ T_{i}^{T}\right],\label{eq:AtomicDecompositionFamilyDefinition}
\end{equation}
so that $\gamma_{\ell}^{\left(i\right)}$ is to $\gamma_{i,\ell}$
as $\gamma^{\left(i\right)}$ is to $\gamma_{i}$.
\item For $i,j\in I$ set
\begin{equation}
C_{i,j}:=\begin{cases}
\left\Vert \mathcal{F}^{-1}\left(\varphi_{i}\cdot\widehat{\gamma_{1}^{\left(j\right)}}\right)\right\Vert _{L_{v_{0}}^{1}}, & \text{if }p\in\left[1,\infty\right],\\
\left(1+\left\Vert T_{j}^{-1}T_{i}\right\Vert \right)^{pK+d}\cdot\left|\det T_{j}\right|^{1-p}\cdot\left\Vert \mathcal{F}^{-1}\left(\varphi_{i}\cdot\widehat{\gamma_{1}^{\left(j\right)}}\right)\right\Vert _{L_{v_{0}}^{p}}^{p}, & \text{if }p\in\left(0,1\right).
\end{cases}\label{eq:GammaSynthesisMatrixEntries}
\end{equation}
\item With $r:=\max\left\{ q,\frac{q}{p}\right\} $, we assume that the
operator $\overrightarrow{C}$ induced by $\left(C_{i,j}\right)_{i,j\in I}$,
i.e.\@
\[
\overrightarrow{C}\left(c_{j}\right)_{j\in I}:=\left(\sum_{j\in I}C_{i,j}c_{j}\right)_{i\in I}
\]
defines a well-defined, bounded operator $\overrightarrow{C}:\ell_{w^{\min\left\{ 1,p\right\} }}^{r}\left(I\right)\to\ell_{w^{\min\left\{ 1,p\right\} }}^{r}\left(I\right)$.\qedhere
\end{enumerate}
\end{assumption}
\begin{rem*}
In the following, we will often use $\Gamma_{\ell}$ as a short notation
for the family $\Gamma_{\ell}=\left(\gamma_{i,\ell}\right)_{i\in I}$
($\ell\in\left\{ 1,2\right\} $), similar to the notation $\Gamma=\left(\gamma_{i}\right)_{i\in I}$.
The assumptions above are slightly redundant. In particular, since
$v_{0}\left(x\right)\leq\Omega_{1}\cdot\left(1+\left|x\right|\right)^{K}$,
it is an easy consequence of equation (\ref{eq:StandardDecayLpEstimate})
that $\left\Vert \gamma_{i,2}\right\Vert _{K_{0}}<\infty$ and $\left\Vert \nabla\gamma_{i,2}\right\Vert _{K_{0}}<\infty$
already imply $\gamma_{i,2}\in L_{\left(1+\left|\bullet\right|\right)^{K}}^{1}\left(\mathbb{R}^{d}\right)\hookrightarrow L_{v_{0}}^{1}\left(\mathbb{R}^{d}\right)$
and $\nabla\gamma_{i,2}\in L_{\left(1+\left|\bullet\right|\right)^{K}}^{1}\left(\mathbb{R}^{d}\right)\hookrightarrow L_{v_{0}}^{1}\left(\mathbb{R}^{d}\right)$,
respectively.
Exactly as in Remark \ref{rem:MainAssumptionsRemark}, we see that
$\gamma_{i,1},\gamma_{i,2}\in L_{\left(1+\left|\bullet\right|\right)^{K}}^{1}\left(\mathbb{R}^{d}\right)$
entails $\gamma_{1}^{\left(j\right)},\gamma_{2}^{\left(j\right)}\in L_{\left(1+\left|\bullet\right|\right)^{K}}^{1}\left(\mathbb{R}^{d}\right)\hookrightarrow L_{v_{0}}^{1}\left(\mathbb{R}^{d}\right)$
for all $j\in I$.
\end{rem*}
Part of the definition of an atomic decomposition $\left(\theta_{\ell}\right)_{\ell\in L}$
is that the synthesis map $\left(c_{\ell}\right)_{\ell\in L}\mapsto\sum_{\ell\in L}c_{\ell}\theta_{\ell}$
is bounded, when defined on a suitable sequence space. Our next lemma
establishes a variant of this property for a certain \emph{continuous}
(as opposed to discrete) synthesis operator. This lemma should be
compared to Lemma \ref{lem:DecompositionSynthesis}.
\begin{lem}
\label{lem:GammaSynthesisBounded}Assume that the family $\Gamma_{1}=\left(\gamma_{i,1}\right)_{i\in I}$
satisfies Assumption \ref{assu:AtomicDecompositionAssumption}. Then,
the operator
\[
{\rm Synth}_{\Gamma_{1}}\::\ell_{w}^{q}\left(\left[V_{j}\right]_{j\in I}\right)\to\DecompSp{\mathcal{Q}}p{\ell_{w}^{q}}v\leq Z'\left(\mathcal{O}\right),\left(g_{j}\right)_{j\in I}\mapsto\sum_{j\in I}\gamma_{1}^{\left(j\right)}\ast g_{j}\:\overset{\text{Lem. }\ref{lem:SpecialConvolutionConsistent}}{=}\:\sum_{j\in I}\mathcal{F}^{-1}\left(\widehat{\gamma_{1}^{\left(j\right)}}\cdot\widehat{g_{j}}\right)
\]
is well-defined and bounded with
\[
\vertiii{{\rm Synth}_{\Gamma_{1}}}\leq C\cdot\vertiii{\smash{\overrightarrow{C}}}^{\max\left\{ 1,\frac{1}{p}\right\} }\;\text{with}\;C=\begin{cases}
1, & \text{if }p\geq1\\
\frac{\left(2^{6}/\sqrt{d}\right)^{\frac{d}{p}}}{2^{21}\cdotd^{7}}\!\cdot\!\left(2^{21}\!\cdot\!d^{5}\!\cdot\!\left\lceil K\!+\!\frac{d+1}{p}\right\rceil \right)^{\!\left\lceil K+\frac{d+1}{p}\right\rceil +1}\!\!\!\cdot\!\left(1\!+\!R_{\mathcal{Q}}\right)^{\frac{d}{p}}\!\cdot\!\Omega_{0}^{5K}\Omega_{1}^{5}, & \text{if }p<1.
\end{cases}
\]
Here, ${\rm Synth}_{\Gamma_{1}}\left(g_{j}\right)_{j\in I}$ is the
linear functional
\begin{equation}
Z\left(\mathcal{O}\right)\to\mathbb{C},f\mapsto\sum_{j\in I}\left\langle \widehat{\gamma_{1}^{\left(j\right)}}\cdot\widehat{g_{j}},\,\mathcal{F}^{-1}f\right\rangle _{\mathcal{S}',\mathcal{S}}=\sum_{j\in I}\left\langle \mathcal{F}^{-1}\left(\widehat{\gamma_{1}^{\left(j\right)}}\cdot\widehat{g_{j}}\right),\,f\right\rangle _{\mathcal{S}',\mathcal{S}}\:\overset{\text{Lem. }\ref{lem:SpecialConvolutionConsistent}}{=}\:\sum_{j\in I}\left\langle \gamma_{1}^{\left(j\right)}\ast g_{j},\,f\right\rangle _{\mathcal{S}',\mathcal{S}},\label{eq:GammaSynthesisFunctionalExplicitDefinition}
\end{equation}
where each of the series converges absolutely for each $f\in Z\left(\mathcal{O}\right)$.
\end{lem}
\begin{proof}
First of all, recall from Lemma \ref{lem:SpecialConvolutionConsistent}
that $V_{j}\hookrightarrow\mathcal{S}'\left(\mathbb{R}^{d}\right)$
for all $j\in I$. Thus, for $\left(g_{j}\right)_{j\in I}\in\ell_{w}^{q}\left(\left[V_{j}\right]_{j\in I}\right)$,
we see that $\widehat{g_{j}}\in\mathcal{S}'\left(\mathbb{R}^{d}\right)$
is a well-defined tempered distribution for all $j\in I$. In view
of the inclusion $Z\left(\mathcal{O}\right)\hookrightarrow\mathcal{S}\left(\mathbb{R}^{d}\right)$,
we thus see that every \emph{individual} term of each of the series
in equation (\ref{eq:GammaSynthesisFunctionalExplicitDefinition})
is well-defined. Here, we use that $\widehat{\gamma_{j,1}}\in C^{\infty}\left(\mathbb{R}^{d}\right)$
with all derivatives of at most polynomial growth, so that the same
holds for $\widehat{\gamma_{1}^{\left(j\right)}}=\widehat{\gamma_{j,1}}\circ S_{j}^{-1}$.
We still have to show, however, that (each of) the series in equation
(\ref{eq:GammaSynthesisFunctionalExplicitDefinition}) converges (absolutely)
for every $f\in Z\left(\mathcal{O}\right)$ and defines a continuous linear
functional.
Since $Z\left(\mathcal{O}\right)=\mathcal{F}\left(\TestFunctionSpace{\mathcal{O}}\right)$,
this is equivalent to showing for arbitrary $\left(g_{j}\right)_{j\in I}\in\ell_{w}^{q}\left(\left[V_{j}\right]_{j\in I}\right)$
that the series defining the functional
\[
\phi:\TestFunctionSpace{\mathcal{O}}\to\mathbb{C},f\mapsto\sum_{j\in I}\left\langle \widehat{\gamma_{1}^{\left(j\right)}}\cdot\widehat{g_{j}},\,f\right\rangle _{\mathcal{S}',\mathcal{S}}
\]
converges absolutely for each $f\in\TestFunctionSpace{\mathcal{O}}$ and
that $\phi\in\DistributionSpace{\mathcal{O}}$. With the same reasoning
as above, we see at least that each term in the series is well-defined.
In the following, we will show that $\phi\in\DistributionSpace{\mathcal{O}}$
is indeed well-defined, with absolute convergence of the series.
But first, let us \emph{assume} that this is the case. Then we have,
for fixed $i\in I$ (by the usual formula for the (inverse) Fourier
transform of a compactly supported distribution, see e.g.\@ \cite[Theorem 7.23]{RudinFA})
\begin{align}
\left|\left[\mathcal{F}^{-1}\left(\varphi_{i}\phi\right)\right]\left(x\right)\right| & =\left|\left\langle \phi,\,\varphi_{i}\cdot e^{2\pi i\left\langle x,\cdot\right\rangle }\right\rangle _{\DistributionSpace{\mathcal{O}},\TestFunctionSpace{\mathcal{O}}}\right|\nonumber \\
& =\left|\sum_{j\in I}\left\langle \widehat{\gamma_{1}^{\left(j\right)}}\cdot\widehat{g_{j}},\,\varphi_{i}\cdot e^{2\pi i\left\langle x,\cdot\right\rangle }\right\rangle _{\mathcal{S}',\mathcal{S}}\right|\nonumber \\
& \leq\sum_{j\in I}\left|\left\langle \widehat{\gamma_{1}^{\left(j\right)}}\cdot\widehat{g_{j}},\,\varphi_{i}\cdot e^{2\pi i\left\langle x,\cdot\right\rangle }\right\rangle _{\mathcal{S}',\mathcal{S}}\right|\nonumber \\
& =\sum_{j\in I}\left|\left[\mathcal{F}^{-1}\left(\varphi_{i}\cdot\widehat{\gamma_{1}^{\left(j\right)}}\cdot\widehat{g_{j}}\right)\right]\left(x\right)\right|\nonumber \\
\left({\scriptstyle \text{Lemma }\ref{lem:SpecialConvolutionConsistent}}\right) & =\sum_{j\in I}\left|\left[\mathcal{F}^{-1}\left(\varphi_{i}\cdot\widehat{\gamma_{1}^{\left(j\right)}}\right)\ast g_{j}\right]\left(x\right)\right|\qquad\forall x\in\mathbb{R}^{d},\label{eq:GammaSynthesisLocalizedFourier}
\end{align}
where all but the first three terms always make sense (as elements
of $\left[0,\infty\right]$), even without assuming that $\phi$ is
a well-defined distribution.
Now, we invoke Theorem \ref{thm:BandlimitedWienerAmalgamSelfImproving}
to obtain for all $f\in\mathcal{S}'\left(\mathbb{R}^{d}\right)$ with
$\operatorname{supp}\widehat{f}\subset\overline{Q_{i}}\subset T_{i}\left[-R_{\mathcal{Q}},R_{\mathcal{Q}}\right]^{d}+b_{i}$
that $\left\Vert f\right\Vert _{W_{T_{i}^{-T}\left[-1,1\right]^{d}}\left(L_{v}^{p}\right)}\lesssim\,\left\Vert f\right\Vert _{L_{v}^{p}}$,
where the implied constant depends on $p,d$ and on $K,R_{\mathcal{Q}}$,
which are fixed throughout. In combination with the embedding $W_{T_{i}^{-T}\left[-1,1\right]^{d}}\left(L_{v}^{p}\right)\hookrightarrow L_{v}^{\infty}\left(\mathbb{R}^{d}\right)\hookrightarrow L_{\left(1+\left|\bullet\right|\right)^{-K}}^{\infty}\left(\mathbb{R}^{d}\right)$
from equation (\ref{eq:WeightedWienerAmalgamTemperedDistribution})
(where now the norm of the embedding depends on $i$ (and on $p,d,K,v$)),
we thus get for every $i\in I$ some constant $C^{\left(i\right)}=C^{\left(i\right)}\left(p,d,K,v,R_{\mathcal{Q}}\right)>0$
such that $\left\Vert f\right\Vert _{\ast}\leq C^{\left(i\right)}\cdot\left\Vert f\right\Vert _{L_{v}^{p}}$
for all $f\in\mathcal{S}'\left(\mathbb{R}^{d}\right)$ satisfying $\operatorname{supp}\widehat{f}\subset\overline{Q_{i}}$,
with $\left\Vert f\right\Vert _{\ast}:=\sup_{x\in\mathbb{R}^{d}}\left(1+\left|x\right|\right)^{-K}\left|f\left(x\right)\right|$.
Since $\operatorname{supp}\mathcal{F}\left[\mathcal{F}^{-1}\left(\varphi_{i}\cdot\widehat{\gamma_{1}^{\left(j\right)}}\right)\ast g_{j}\right]=\operatorname{supp}\left(\varphi_{i}\cdot\widehat{\gamma_{1}^{\left(j\right)}}\cdot\widehat{g_{j}}\right)\subset\overline{Q_{i}}$,
this yields
\[
\left\Vert \mathcal{F}^{-1}\left(\varphi_{i}\cdot\widehat{\gamma_{1}^{\left(j\right)}}\right)\ast g_{j}\right\Vert _{\ast}\leq C^{\left(i\right)}\cdot\left\Vert \mathcal{F}^{-1}\left(\varphi_{i}\cdot\widehat{\gamma_{1}^{\left(j\right)}}\right)\ast g_{j}\right\Vert _{L_{v}^{p}}\qquad\forall j\in I.
\]
Now, we distinguish two cases:
\textbf{Case 1}: We have $p\in\left[1,\infty\right]$. In this case,
we can simply use the weighted Young inequality (equation (\ref{eq:WeightedYoungInequality}))
to derive
\[
\left\Vert \mathcal{F}^{-1}\left(\varphi_{i}\cdot\widehat{\gamma_{1}^{\left(j\right)}}\right)\ast g_{j}\right\Vert _{L_{v}^{p}}\leq\left\Vert \mathcal{F}^{-1}\left(\varphi_{i}\cdot\widehat{\gamma_{1}^{\left(j\right)}}\right)\right\Vert _{L_{v_{0}}^{1}}\cdot\left\Vert g_{j}\right\Vert _{L_{v}^{p}}=C_{i,j}\cdot\left\Vert g_{j}\right\Vert _{V_{j}}.
\]
But since we have $c=\left(c_{j}\right)_{j\in I}\in\ell_{w}^{q}\left(I\right)=\ell_{w^{\min\left\{ 1,p\right\} }}^{r}\left(I\right)$
for $c_{j}:=\left\Vert g_{j}\right\Vert _{V_{j}}$, we get by boundedness
of $\overrightarrow{C}$ that $\overrightarrow{C}c\in\ell_{w}^{q}\left(I\right)$.
In particular,
\begin{equation}
\frac{1}{C^{\left(i\right)}}\cdot\sum_{j\in I}\left\Vert \mathcal{F}^{-1}\left(\varphi_{i}\cdot\widehat{\gamma_{1}^{\left(j\right)}}\right)\ast g_{j}\right\Vert _{\ast}\leq\sum_{j\in I}\left\Vert \mathcal{F}^{-1}\left(\varphi_{i}\cdot\widehat{\gamma_{1}^{\left(j\right)}}\right)\ast g_{j}\right\Vert _{L_{v}^{p}}\leq\sum_{j\in I}C_{i,j}\left\Vert g_{j}\right\Vert _{V_{j}}=\left(\smash{\overrightarrow{C}}c\right)_{i}<\infty,\label{eq:GammaSynthesisBanachCaseEstimate}
\end{equation}
from which it follows that the series $\sum_{j\in I}\left[\mathcal{F}^{-1}\left(\varphi_{i}\cdot\widehat{\gamma_{1}^{\left(j\right)}}\right)\ast g_{j}\right]\left(x\right)$
converges absolutely for all $x\in\mathbb{R}^{d}$ (even locally
uniformly in $x$).
Furthermore, for arbitrary $\theta\in\TestFunctionSpace{\mathcal{O}}$,
we have
\begin{align*}
\sum_{j\in I}\left|\left\langle \widehat{\gamma_{1}^{\left(j\right)}}\cdot\widehat{g_{j}},\,\varphi_{i}\theta\right\rangle _{\mathcal{S}',\mathcal{S}}\right| & =\sum_{j\in I}\left|\left\langle \mathcal{F}^{-1}\!\!\left(\varphi_{i}\cdot\widehat{\gamma_{1}^{\left(j\right)}}\cdot\widehat{g_{j}}\right),\,\widehat{\theta}\right\rangle _{\mathcal{S}',\mathcal{S}}\right|\\
& \leq\sum_{j\in I}\left\Vert \mathcal{F}^{-1}\!\!\left(\varphi_{i}\cdot\widehat{\gamma_{1}^{\left(j\right)}}\right)\!\ast\!g_{j}\right\Vert _{\ast}\left\Vert \smash{\widehat{\theta}}\right\Vert _{L_{\left(1+\left|\bullet\right|\right)^{K}}^{1}}\leq C^{\left(i\right)}\cdot\left(\smash{\overrightarrow{C}}c\right)_{i}\cdot\left\Vert \smash{\widehat{\theta}}\right\Vert _{L_{\left(1+\left|\bullet\right|\right)^{K}}^{1}}\!<\infty,
\end{align*}
so that the series $\sum_{j\in I}\left\langle \widehat{\gamma_{1}^{\left(j\right)}}\cdot\widehat{g_{j}},\,\varphi_{i}\theta\right\rangle $
defining $\phi\left(\varphi_{i}\theta\right)$ converges absolutely.
The same estimate also shows that $\theta\mapsto\phi\left(\varphi_{i}\theta\right)$
is a distribution on $\mathcal{O}$, since $\theta\mapsto\left\Vert \smash{\widehat{\theta}}\right\Vert _{L_{\left(1+\left|\bullet\right|\right)^{K}}^{1}}$
is a continuous seminorm on $\TestFunctionSpace{\mathcal{O}}\hookrightarrow\mathcal{S}'\left(\mathbb{R}^{d}\right)$.
But since $\left(\varphi_{i}\right)_{i\in I}$ is a locally finite
partition of unity on $\mathcal{O}$, we have $\theta=\sum_{i\in I_{\Upsilon}}\varphi_{i}\theta$
for every $\theta\in\TestFunctionSpace{\mathcal{O}}$ with $\operatorname{supp}\theta\subset\Upsilon$,
where $\Upsilon\subset\mathcal{O}$ is an arbitrary compact set and where
$I_{\Upsilon}\subset I$ is \emph{finite}. Hence, $\theta\mapsto\phi\left(\theta\right)=\sum_{i\in I_{\Upsilon}}\phi\left(\varphi_{i}\theta\right)$
is a continuous linear functional on $\left\{ \theta\in\TestFunctionSpace{\mathcal{O}}\,\middle|\,\operatorname{supp}\theta\subset\Upsilon\right\} $
for arbitrary compact $\Upsilon\subset\mathcal{O}$ and the defining series
converges absolutely (as a \emph{finite} sum of absolutely convergent
series). This shows that $\phi\in\DistributionSpace{\mathcal{O}}$ is well-defined
(with absolute convergence of the defining series), so that equation
(\ref{eq:GammaSynthesisLocalizedFourier}) is valid.
\medskip{}
As a consequence of equations (\ref{eq:GammaSynthesisLocalizedFourier})
and (\ref{eq:GammaSynthesisBanachCaseEstimate}) and of the triangle
inequality for $L_{v}^{p}\left(\mathbb{R}^{d}\right)$, we finally
get
\[
\left\Vert \mathcal{F}^{-1}\left(\varphi_{i}\phi\right)\right\Vert _{L_{v}^{p}}\leq\sum_{j\in I}\left\Vert \mathcal{F}^{-1}\left(\varphi_{i}\cdot\widehat{\gamma_{1}^{\left(j\right)}}\right)\ast g_{j}\right\Vert _{L_{v}^{p}}\leq\left(\smash{\overrightarrow{C}}c\right)_{i}\qquad\forall i\in I,
\]
so that solidity of $\ell_{w}^{q}\left(I\right)$ yields
\[
\left\Vert \phi\right\Vert _{\FourierDecompSp{\mathcal{Q}}p{\ell_{w}^{q}}v}=\left\Vert \left(\left\Vert \mathcal{F}^{-1}\left(\varphi_{i}\phi\right)\right\Vert _{L_{v}^{p}}\right)_{i\in I}\right\Vert _{\ell_{w}^{q}}\leq\left\Vert \smash{\overrightarrow{C}}c\right\Vert _{\ell_{w}^{q}}\leq\vertiii{\smash{\overrightarrow{C}}}\cdot\left\Vert c\right\Vert _{\ell_{w}^{q}}=\vertiii{\smash{\overrightarrow{C}}}\cdot\left\Vert \left(g_{j}\right)_{j\in I}\right\Vert _{\ell_{w}^{q}\left(\left[V_{i}\right]_{i\in I}\right)}<\infty.
\]
All in all, we see that $\phi\in\FourierDecompSp{\mathcal{Q}}p{\ell_{w}^{q}}v\leq\DistributionSpace{\mathcal{O}}$
is well-defined. But by definition of ${\rm Synth}_{\Gamma_{1}}$,
we have ${\rm Synth}_{\Gamma_{1}}\left(g_{j}\right)_{j\in I}=\mathcal{F}^{-1}\phi$
for the isometric isomorphism $\mathcal{F}^{-1}:\FourierDecompSp{\mathcal{Q}}p{\ell_{w}^{q}}v\to\DecompSp{\mathcal{Q}}p{\ell_{w}^{q}}v,\psi\mapsto\psi\circ\mathcal{F}^{-1}$.
As a consequence, ${\rm Synth}_{\Gamma_{1}}\left(g_{j}\right)_{j\in I}\in\DecompSp{\mathcal{Q}}p{\ell_{w}^{q}}v$
is well-defined and
\[
\left\Vert {\rm Synth}_{\Gamma_{1}}\left(g_{j}\right)_{j\in I}\right\Vert _{\DecompSp{\mathcal{Q}}p{\ell_{w}^{q}}v}=\left\Vert \phi\right\Vert _{\FourierDecompSp{\mathcal{Q}}p{\ell_{w}^{q}}v}\leq\vertiii{\smash{\overrightarrow{C}}}\cdot\left\Vert \left(g_{j}\right)_{j\in I}\right\Vert _{\ell_{w}^{q}\left(\left[V_{i}\right]_{i\in I}\right)},
\]
as desired.
\medskip{}
\textbf{Case 2}: We have $p\in\left(0,1\right)$. In this case, we
replace the application of the weighted Young inequality (equation
(\ref{eq:WeightedYoungInequality})) by an application of Corollary
\ref{cor:WienerAmalgamConvolutionSimplified} to get for $C_{1}:=d^{-\frac{d}{2p}}\cdot\left(972\cdotd^{5/2}\right)^{K+\frac{d}{p}}\cdot\Omega_{0}^{3K}\Omega_{1}^{3}$
that
\begin{align}
& \left\Vert \mathcal{F}^{-1}\left(\varphi_{i}\cdot\widehat{\gamma_{1}^{\left(j\right)}}\right)\ast g_{j}\right\Vert _{L_{v}^{p}}\nonumber \\
\left({\scriptstyle \text{Lemma }\ref{lem:MaximalFunctionDominatesF}}\right) & \leq\left\Vert \mathcal{F}^{-1}\left(\varphi_{i}\cdot\widehat{\gamma_{1}^{\left(j\right)}}\right)\ast g_{j}\right\Vert _{W_{T_{j}^{-T}\left[-1,1\right]^{d}}\left(L_{v}^{p}\right)}\nonumber \\
\left({\scriptstyle \text{Cor. }\ref{cor:WienerAmalgamConvolutionSimplified}}\right) & \leq C_{1}\cdot\left|\det T_{j}\right|^{\frac{1}{p}-1}\cdot\left\Vert \mathcal{F}^{-1}\left(\varphi_{i}\cdot\widehat{\gamma_{1}^{\left(j\right)}}\right)\right\Vert _{W_{T_{j}^{-T}\left[-1,1\right]^{d}}\left(L_{v_{0}}^{p}\right)}\cdot\left\Vert g_{j}\right\Vert _{W_{T_{j}^{-T}\left[-1,1\right]^{d}}\left(L_{v}^{p}\right)}\nonumber \\
\left({\scriptstyle \text{eq. }\eqref{eq:WienerLinearCubeTransformationChange}}\right) & \leq C_{1}\left(6d\right)^{K+\frac{d}{p}}\cdot\Omega_{0}^{K}\Omega_{1}\cdot\left(1+\left\Vert T_{j}^{-1}T_{i}\right\Vert \right)^{K+\frac{d}{p}}\cdot\left|\det T_{j}\right|^{\frac{1}{p}-1}\cdot\left\Vert \mathcal{F}^{-1}\left(\varphi_{i}\cdot\widehat{\gamma_{1}^{\left(j\right)}}\right)\right\Vert _{W_{T_{i}^{-T}\left[-1,1\right]^{d}}\left(L_{v_{0}}^{p}\right)}\cdot\left\Vert g_{j}\right\Vert _{V_{j}}\nonumber \\
\left({\scriptstyle \text{Thm. }\ref{thm:BandlimitedWienerAmalgamSelfImproving}}\right) & \overset{\left(\dagger\right)}{\leq}C_{1}C_{2}\left(6d\right)^{K+\frac{d}{p}}\cdot\Omega_{0}^{K}\Omega_{1}\cdot\left(1+\left\Vert T_{j}^{-1}T_{i}\right\Vert \right)^{K+\frac{d}{p}}\cdot\left|\det T_{j}\right|^{\frac{1}{p}-1}\cdot\left\Vert \mathcal{F}^{-1}\left(\varphi_{i}\cdot\widehat{\gamma_{1}^{\left(j\right)}}\right)\right\Vert _{L_{v_{0}}^{p}}\cdot\left\Vert g_{j}\right\Vert _{V_{j}}\nonumber \\
& =C_{1}C_{2}\left(6d\right)^{K+\frac{d}{p}}\cdot\Omega_{0}^{K}\Omega_{1}\cdot C_{i,j}^{1/p}\cdot\left\Vert g_{j}\right\Vert _{V_{j}}=:C_{3}\cdot C_{i,j}^{1/p}\cdot\left\Vert g_{j}\right\Vert _{V_{j}},\label{eq:GammaSynthesisQuasiBanachCaseMainEstimate}
\end{align}
where the step marked with $\left(\dagger\right)$ used that
\[
\operatorname{supp}\left(\varphi_{i}\cdot\widehat{\gamma_{1}^{\left(j\right)}}\right)\subset\overline{Q_{i}}\subset T_{i}\left[\overline{B_{R_{\mathcal{Q}}}}\left(0\right)\right]+b_{i}\subset T_{i}\left[-R_{\mathcal{Q}},R_{\mathcal{Q}}\right]^{d}+b_{i},
\]
so that Theorem \ref{thm:BandlimitedWienerAmalgamSelfImproving} (with
$v_{0}$ instead of $v$) yields $\left\Vert \mathcal{F}^{-1}\left(\varphi_{i}\cdot\widehat{\gamma_{1}^{\left(j\right)}}\right)\right\Vert _{W_{T_{i}^{-T}\left[-1,1\right]^{d}}\left(L_{v_{0}}^{p}\right)}\leq C_{2}\cdot\left\Vert \mathcal{F}^{-1}\left(\varphi_{i}\cdot\widehat{\gamma_{1}^{\left(j\right)}}\right)\right\Vert _{L_{v_{0}}^{p}}$
for
\[
C_{2}:=2^{4\left(1+\frac{d}{p}\right)}s_{d}^{\frac{1}{p}}\left(192\cdotd^{3/2}\cdot\left\lceil K+\frac{d+1}{p}\right\rceil \right)^{\left\lceil K+\frac{d+1}{p}\right\rceil +1}\cdot\Omega_{0}^{K}\Omega_{1}\cdot\left(1+R_{\mathcal{Q}}\right)^{\frac{d}{p}}.
\]
Next, set $c_{j}:=\left\Vert g_{j}\right\Vert _{V_{j}}^{p}$ for $j\in I$
and note that $\left(g_{j}\right)_{j\in I}\in\ell_{w}^{q}\left(\left[V_{j}\right]_{j\in I}\right)$
yields $c=\left(c_{j}\right)_{j\in I}\in\ell_{w^{p}}^{q/p}\left(I\right)=\ell_{w^{\min\left\{ 1,p\right\} }}^{r}\left(I\right)$.
Hence, we get because of $\ell^{p}\hookrightarrow\ell^{1}$ that
\begin{align*}
\frac{1}{C^{\left(i\right)}}\cdot\sum_{j\in I}\left\Vert \mathcal{F}^{-1}\left(\varphi_{i}\cdot\widehat{\gamma_{1}^{\left(j\right)}}\right)\ast g_{j}\right\Vert _{\ast} & \leq\left(\sum_{j\in I}\left\Vert \mathcal{F}^{-1}\left(\varphi_{i}\cdot\widehat{\gamma_{1}^{\left(j\right)}}\right)\ast g_{j}\right\Vert _{L_{v}^{p}}^{p}\right)^{1/p}\\
& \leq C_{3}\cdot\left(\sum_{j\in I}C_{i,j}\cdot c_{j}\right)^{1/p}\\
& =C_{3}\cdot\left(\smash{\overrightarrow{C}}\cdot c\right)_{i}^{1/p}<\infty,
\end{align*}
which is a slight variation of equation (\ref{eq:GammaSynthesisBanachCaseEstimate}).
Now, we see exactly as in case of $p\in\left[1,\infty\right]$ that
$\phi$ is a well-defined distribution $\phi\in\DistributionSpace{\mathcal{O}}$,
so that equation (\ref{eq:GammaSynthesisLocalizedFourier}) is valid.
Using this equation and the $p$-triangle inequality for $L_{v}^{p}\left(\mathbb{R}^{d}\right)$,
we derive
\begin{align*}
\left\Vert \mathcal{F}^{-1}\left(\varphi_{i}\phi\right)\right\Vert _{L_{v}^{p}} & \leq\left(\sum_{j\in I}\left\Vert \mathcal{F}^{-1}\left(\varphi_{i}\cdot\widehat{\gamma_{1}^{\left(j\right)}}\right)\ast g_{j}\right\Vert _{L_{v}^{p}}^{p}\right)^{1/p}\\
\left({\scriptstyle \text{eq. }\eqref{eq:GammaSynthesisQuasiBanachCaseMainEstimate}}\right) & \leq C_{3}\cdot\left(\sum_{j\in I}C_{i,j}\cdot\left\Vert g_{j}\right\Vert _{V_{j}}^{p}\right)^{1/p}\\
& =C_{3}\cdot\left(\smash{\overrightarrow{C}}\cdot c\right)_{i}^{1/p}<\infty
\end{align*}
and hence
\begin{align*}
\left\Vert \phi\right\Vert _{\FourierDecompSp{\mathcal{Q}}p{\ell_{w}^{q}}v} & =\left\Vert \left(\left\Vert \mathcal{F}^{-1}\left(\varphi_{i}\phi\right)\right\Vert _{L_{v}^{p}}\right)_{i\in I}\right\Vert _{\ell_{w}^{q}}\\
& \leq C_{3}\cdot\left\Vert \left(\smash{\overrightarrow{C}}\cdot c\right)^{1/p}\right\Vert _{\ell_{w}^{q}}\\
& =C_{3}\cdot\left\Vert w^{p}\cdot\left[\smash{\overrightarrow{C}}\cdot c\right]\right\Vert _{\ell^{q/p}}^{1/p}\\
& =C_{3}\cdot\left\Vert \smash{\overrightarrow{C}}\cdot c\right\Vert _{\ell_{w^{\min\left\{ 1,p\right\} }}^{r}}^{1/p}\\
& \leq C_{3}\cdot\vertiii{\smash{\overrightarrow{C}}}^{1/p}\cdot\left\Vert c\right\Vert _{\ell_{w^{\min\left\{ 1,p\right\} }}^{r}}^{1/p}\\
& =C_{3}\cdot\vertiii{\smash{\overrightarrow{C}}}^{1/p}\cdot\left\Vert \left(w_{j}\cdot\left\Vert g_{j}\right\Vert _{V_{j}}\right)_{j\in I}\right\Vert _{\ell^{q}}\\
& =C_{3}\cdot\vertiii{\smash{\overrightarrow{C}}}^{1/p}\cdot\left\Vert \left(g_{j}\right)_{j\in I}\right\Vert _{\ell_{w}^{q}\left(\left[V_{j}\right]_{j\in I}\right)}<\infty.
\end{align*}
Now, we see as for $p\in\left[1,\infty\right]$ that ${\rm Synth}_{\Gamma_{1}}$
is a bounded linear operator with $\vertiii{{\rm Synth}_{\Gamma_{1}}}\leq C_{3}\cdot\vertiii{\smash{\overrightarrow{C}}}^{1/p}$.
Finally, we observe, using $s_{d}\leq2^{2d}$, that
\begin{align*}
C_{3} & =C_{1}C_{2}\left(6d\right)^{K+\frac{d}{p}}\cdot\Omega_{0}^{K}\Omega_{1}\\
& =d^{-\frac{d}{2p}}\cdot\left(972\cdotd^{5/2}\right)^{K+\frac{d}{p}}\left(6d\right)^{K+\frac{d}{p}}\cdot2^{4\left(1+\frac{d}{p}\right)}s_{d}^{\frac{1}{p}}\left(192\cdotd^{3/2}\cdot\left\lceil K+\frac{d+1}{p}\right\rceil \right)^{\left\lceil K+\frac{d+1}{p}\right\rceil +1}\cdot\left(1+R_{\mathcal{Q}}\right)^{\frac{d}{p}}\cdot\Omega_{0}^{5K}\Omega_{1}^{5}\\
& \leqd^{-\frac{d}{2p}}\cdot\left(5832\cdotd^{7/2}\right)^{K+\frac{d}{p}}\cdot2^{4+6\frac{d}{p}}\left(192\cdotd^{3/2}\cdot\left\lceil K+\frac{d+1}{p}\right\rceil \right)^{\left\lceil K+\frac{d+1}{p}\right\rceil +1}\cdot\left(1+R_{\mathcal{Q}}\right)^{\frac{d}{p}}\cdot\Omega_{0}^{5K}\Omega_{1}^{5}\\
& \leqd^{-\frac{d}{2p}}\cdot\left(5832\cdotd^{7/2}\right)^{-2}\cdot2^{4+6\frac{d}{p}}\left(2^{21}\cdotd^{5}\cdot\left\lceil K+\frac{d+1}{p}\right\rceil \right)^{\left\lceil K+\frac{d+1}{p}\right\rceil +1}\cdot\left(1+R_{\mathcal{Q}}\right)^{\frac{d}{p}}\cdot\Omega_{0}^{5K}\Omega_{1}^{5}\\
& \leq\frac{\left(2^{6}/\sqrt{d}\right)^{d/p}}{2^{21}\cdotd^{7}}\cdot\left(2^{21}\cdotd^{5}\cdot\left\lceil K+\frac{d+1}{p}\right\rceil \right)^{\left\lceil K+\frac{d+1}{p}\right\rceil +1}\cdot\left(1+R_{\mathcal{Q}}\right)^{\frac{d}{p}}\cdot\Omega_{0}^{5K}\Omega_{1}^{5},
\end{align*}
which completes the proof.
\end{proof}
In order to switch from the continuous synthesis operator from the
preceding lemma to a discrete one, our next technical result is helpful.
\begin{lem}
\label{lem:SchwartzTranslationSynthesis}Let $p\in\left(0,\infty\right]$
and assume that $\varrho:\mathbb{R}^{d}\to\mathbb{C}$ is measurable
and satisfies $\left\Vert \varrho\right\Vert _{K_{0}}<\infty$ with
$K_{0}$ and $\left\Vert \bullet\right\Vert _{K_{0}}$ as in Assumption
\ref{assu:AtomicDecompositionAssumption}. Let $i\in I$ and $\delta\in\left(0,1\right]$
and let $V_{i}$ be defined as in Assumption \ref{assu:MainAssumptions}.
Furthermore, let the \textbf{coefficient space} $C_{i}^{\left(\delta\right)}$
be defined as in equation (\ref{eq:CoefficientSpaceDefinition}).
Then, the maps
\[
\Psi_{\left|\varrho\right|}^{\left(i,\delta\right)}:C_{i}^{\left(\delta\right)}\to V_{i},\left(c_{k}\right)_{k\in\mathbb{Z}^{d}}\mapsto\left(\sum_{k\in\mathbb{Z}^{d}}c_{k}\cdot L_{\delta\cdot k}\left|\varrho\right|\right)\circ T_{i}^{T}=\sum_{k\in\mathbb{Z}^{d}}c_{k}\cdot L_{\delta\cdot T_{i}^{-T}k}\left|\varrho\circ T_{i}^{T}\right|
\]
and
\[
\Psi_{\varrho}^{\left(i,\delta\right)}:C_{i}^{\left(\delta\right)}\to V_{i},\left(c_{k}\right)_{k\in\mathbb{Z}^{d}}\mapsto\left(\sum_{k\in\mathbb{Z}^{d}}c_{k}\cdot L_{\delta\cdot k}\varrho\right)\circ T_{i}^{T}=\sum_{k\in\mathbb{Z}^{d}}c_{k}\cdot L_{\delta\cdot T_{i}^{-T}k}\left[\varrho\circ T_{i}^{T}\right]
\]
are well-defined and bounded, with pointwise absolute convergence
of the series and with
\[
\vertiii{\Psi_{\varrho}^{\left(i,\delta\right)}}\leq\vertiii{\Psi_{\left|\varrho\right|}^{\left(i,\delta\right)}}\leq\begin{cases}
\Omega_{0}^{K}\Omega_{1}\cdot\left(1+2\sqrt{d}\right)^{K_{0}}\left(\frac{s_{d}}{p}\right)^{1/p}\cdot\left\Vert \varrho\right\Vert _{K_{0}}\cdot\left|\det T_{i}\right|^{-\frac{1}{p}}, & \text{if }p<1,\\
\Omega_{0}^{K}\Omega_{1}\cdot12^{d+1}\cdot\delta^{-\left(1-\frac{1}{p}\right)\left(d+1\right)}\cdot\left\Vert \varrho\right\Vert _{K_{0}}\cdot\left|\det T_{i}\right|^{-\frac{1}{p}}, & \text{if }p\geq1.
\end{cases}
\]
In particular, if $g\in L_{v_{0}}^{1}\left(\mathbb{R}^{d}\right)$,
then
\begin{equation}
g\ast\left[\Psi_{\varrho}^{\left(i,\delta\right)}\left(c_{k}\right)_{k\in\mathbb{Z}^{d}}\right]=\sum_{k\in\mathbb{Z}^{d}}\left(c_{k}\cdot\left[g\ast L_{\delta\cdot T_{i}^{-T}k}\left[\varrho\circ T_{i}^{T}\right]\right]\right).\qedhere\label{eq:SchwartzTranslationSynthesisConvolution}
\end{equation}
\end{lem}
\begin{proof}
Clearly, since $V_{i}$ and $C_{i}^{\left(\delta\right)}$ are solid,
boundedness of $\Psi_{\left|\varrho\right|}^{\left(i,\delta\right)}$
implies that of $\Psi_{\varrho}^{\left(i,\delta\right)}$, with $\vertiii{\Psi_{\varrho}^{\left(i,\delta\right)}}\leq\vertiii{\Psi_{\left|\varrho\right|}^{\left(i,\delta\right)}}$.
Furthermore, again by solidity and since $\left|\varrho\left(x\right)\right|\leq\left\Vert \varrho\right\Vert _{K_{0}}\cdot\left(1+\left|x\right|\right)^{-K_{0}}$
for all $x\in\mathbb{R}^{d}$, it suffices to prove the claim (except
for equation (\ref{eq:SchwartzTranslationSynthesisConvolution}))
for the special case $\varrho\left(x\right)=\left(1+\left|x\right|\right)^{-K_{0}}$,
so that $\left\Vert \varrho\right\Vert _{K_{0}}=1$.
Recall from equation (\ref{eq:CoefficientSpaceDefinition}) that $v_{k}^{\left(j,\delta\right)}=v\left(\delta\cdot T_{j}^{-T}k\right)$.
Now, we first observe
\begin{equation}
v\left(x\right)=v\left(\delta\cdot T_{i}^{-T}k+x-\delta\cdot T_{i}^{-T}k\right)\leq v\left(\delta\cdot T_{i}^{-T}k\right)\cdot v_{0}\left(x-\delta\cdot T_{i}^{-T}k\right)=v_{k}^{\left(i,\delta\right)}\cdot v_{0}\left(x-\delta\cdot T_{i}^{-T}k\right),\label{eq:SchwartzTranslationSynthesisWeightTrick}
\end{equation}
so that
\[
0\leq\frac{v\left(x\right)}{v_{k}^{\left(i,\delta\right)}}\cdot\left(L_{\delta\cdot T_{i}^{-T}k}\left[\varrho\circ T_{i}^{T}\right]\right)\left(x\right)\leq v_{0}\left(x-\delta\cdot T_{i}^{-T}k\right)\cdot\left(\varrho\circ T_{i}^{T}\right)\left(x-\delta\cdot T_{i}^{-T}k\right).
\]
By translation invariance of $\left\Vert \bullet\right\Vert _{L^{1}}$,
this implies for $p\in\left[1,\infty\right]$ (which entails $K_{0}=K+d+1$)
that
\begin{align*}
\left\Vert \frac{L_{\delta\cdot T_{i}^{-T}k}\left[\varrho\circ T_{i}^{T}\right]}{v_{k}^{\left(i,\delta\right)}}\right\Vert _{L_{v}^{1}} & \leq\left\Vert x\mapsto v_{0}\left(x-\delta\cdot T_{i}^{-T}k\right)\cdot\left(\varrho\circ T_{i}^{T}\right)\left(x-\delta\cdot T_{i}^{-T}k\right)\right\Vert _{L^{1}}\\
& =\left\Vert v_{0}\cdot\left(\varrho\circ T_{i}^{T}\right)\right\Vert _{L^{1}}\\
\left({\scriptstyle \text{standard change of variables}}\right) & =\left|\det T_{i}^{T}\right|^{-1}\cdot\left\Vert \left(v_{0}\circ T_{i}^{-T}\right)\cdot\varrho\right\Vert _{L^{1}}\\
\left({\scriptstyle \text{assumptions on }v_{0}}\right) & \leq\Omega_{1}\cdot\left|\det T_{i}\right|^{-1}\cdot\left\Vert x\mapsto\left(1+\left|T_{i}^{-T}x\right|\right)^{K}\cdot\varrho\left(x\right)\right\Vert _{L^{1}}\\
\left({\scriptstyle \text{eq. }\eqref{eq:WeightLinearTransformationsConnection}}\right) & \leq\Omega_{0}^{K}\Omega_{1}\cdot\left|\det T_{i}\right|^{-1}\cdot\left\Vert x\mapsto\left(1+\left|x\right|\right)^{K}\cdot\varrho\left(x\right)\right\Vert _{L^{1}}\\
\left({\scriptstyle K_{0}=K+d+1\text{ since }p\in\left[1,\infty\right]}\right) & =\Omega_{0}^{K}\Omega_{1}\cdot\left|\det T_{i}\right|^{-1}\cdot\left\Vert x\mapsto\left(1+\left|x\right|\right)^{-\left(d+1\right)}\right\Vert _{L^{1}}\\
\left({\scriptstyle \text{eq. }\eqref{eq:StandardDecayLpEstimate}}\right) & \leq\Omega_{0}^{K}\Omega_{1}s_{d}\cdot\left|\det T_{i}\right|^{-1}.
\end{align*}
Hence, we get in case of $p=1$ that
\begin{align*}
\left\Vert \Psi_{\left|\varrho\right|}^{\left(i,\delta\right)}\left(c_{k}\right)_{k\in\mathbb{Z}^{d}}\right\Vert _{L_{v}^{1}} & \leq\sum_{k\in\mathbb{Z}^{d}}v_{k}^{\left(i,\delta\right)}\left|c_{k}\right|\cdot\left\Vert \frac{L_{\delta\cdot T_{i}^{-T}k}\left[\varrho\circ T_{i}^{T}\right]}{v_{k}^{\left(i,\delta\right)}}\right\Vert _{L_{v}^{1}}\\
& \leq\Omega_{0}^{K}\Omega_{1}s_{d}\cdot\left|\det T_{i}\right|^{-1}\cdot\sum_{k\in\mathbb{Z}^{d}}v_{k}^{\left(i,\delta\right)}\left|c_{k}\right|\\
& =\Omega_{0}^{K}\Omega_{1}s_{d}\cdot\left|\det T_{i}\right|^{-1}\cdot\left\Vert \left(c_{k}\right)_{k\in\mathbb{Z}^{d}}\right\Vert _{C_{i}^{\left(\delta\right)}}<\infty.
\end{align*}
This establishes boundedness of $\Psi_{\left|\varrho\right|}^{\left(i,\delta\right)}$
for $p=1$.
\medskip{}
As our next step, we first note
\[
\left[M_{Q}\left(L_{x}f\right)\right]\left(y\right)=\left\Vert {\mathds{1}}_{y+Q}\cdot L_{x}f\right\Vert _{L^{\infty}}=\left\Vert \left(L_{-x}{\mathds{1}}_{y+Q}\right)\cdot f\right\Vert _{L^{\infty}}=\left\Vert {\mathds{1}}_{y-x+Q}\cdot f\right\Vert _{L^{\infty}}=\left(M_{Q}f\right)\left(y-x\right)=\left(L_{x}\left[M_{Q}f\right]\right)\left(y\right)
\]
for arbitrary measurable $f:\mathbb{R}^{d}\to\mathbb{C}$ and $Q\subset\mathbb{R}^{d}$.
Hence,
\begin{align*}
g_{i}^{\left(\delta\right)}\left(x\right) & :=v\left(x\right)\cdot\left[M_{T_{i}^{-T}\left[-1,1\right]^{d}}\left(\frac{L_{\delta\cdot T_{i}^{-T}k}\left[\varrho\circ T_{i}^{T}\right]}{v_{k}^{\left(i,\delta\right)}}\right)\right]\left(x\right)\\
& =\frac{v\left(x\right)}{v_{k}^{\left(i,\delta\right)}}\cdot\left[M_{T_{i}^{-T}\left[-1,1\right]^{d}}\left(\varrho\circ T_{i}^{T}\right)\right]\left(x-\delta\cdot T_{i}^{-T}k\right)\\
\left({\scriptstyle \text{eq. }\eqref{eq:SchwartzTranslationSynthesisWeightTrick}}\right) & \leq\left[v_{0}\cdot M_{T_{i}^{-T}\left[-1,1\right]^{d}}\left(\varrho\circ T_{i}^{T}\right)\right]\left(x-\delta\cdot T_{i}^{-T}k\right)\\
\left({\scriptstyle \text{Lemma }\ref{lem:WienerTransformationFormula}}\right) & =\left(v_{0}\cdot\left[\left(M_{\left[-1,1\right]^{d}}\varrho\right)\circ T_{i}^{T}\right]\right)\left(x-\delta\cdot T_{i}^{-T}k\right)\\
\left({\scriptstyle \text{assumptions on }v_{0}\text{ and eq. }\eqref{eq:WeightLinearTransformationsConnection}}\right) & \leq\Omega_{0}^{K}\Omega_{1}\cdot\left[\left(1+\left|\bullet\right|\right)^{K}\cdot\left(M_{\left[-1,1\right]^{d}}\varrho\right)\right]\left(T_{i}^{T}x-\delta\cdot k\right)\\
\left({\scriptstyle \text{Lemma }\ref{lem:SchwartzFunctionsAreWiener}}\right) & \leq\Omega_{0}^{K}\Omega_{1}\cdot\left(1+2\sqrt{d}\right)^{K_{0}}\cdot\left[\left(1+\left|\bullet\right|\right)^{K-K_{0}}\right]\left(T_{i}^{T}x-\delta k\right).
\end{align*}
But this implies for $p\in\left(0,1\right)$ that
\begin{align*}
\left\Vert \frac{L_{\delta\cdot T_{i}^{-T}k}\left[\varrho\circ T_{i}^{T}\right]}{v_{k}^{\left(i,\delta\right)}}\right\Vert _{V_{i}} & =\left\Vert g_{i}^{\left(\delta\right)}\right\Vert _{L^{p}}\\
& \leq\Omega_{0}^{K}\Omega_{1}\cdot\left(1+2\sqrt{d}\right)^{K_{0}}\cdot\left\Vert \left[L_{\delta k}\left(1+\left|\bullet\right|\right)^{K-K_{0}}\right]\circ T_{i}^{T}\right\Vert _{L^{p}}\\
& =\Omega_{0}^{K}\Omega_{1}\cdot\left(1+2\sqrt{d}\right)^{K_{0}}\cdot\left|\det T_{i}^{T}\right|^{-1/p}\cdot\left\Vert \left(1+\left|\bullet\right|\right)^{K-K_{0}}\right\Vert _{L^{p}}\\
\left({\scriptstyle \text{eq. }\eqref{eq:StandardDecayLpEstimate}\text{ and }K-K_{0}=-\left(\frac{d}{p}+1\right)\text{ since }p\in\left(0,1\right)}\right) & \leq\Omega_{0}^{K}\Omega_{1}\cdot\left(1+2\sqrt{d}\right)^{K_{0}}\left(\frac{s_{d}}{p}\right)^{1/p}\cdot\left|\det T_{i}\right|^{-1/p}.
\end{align*}
Now, we recall that for $p\in\left(0,1\right)$, we have the $p$-triangle
inequality $\left\Vert f+g\right\Vert _{L^{p}}^{p}\leq\left\Vert f\right\Vert _{L^{p}}^{p}+\left\Vert g\right\Vert _{L^{p}}^{p}$.
By solidity and because of $M_{Q}\left(f+g\right)\leq M_{Q}f+M_{Q}g$,
this also yields $\left\Vert f+g\right\Vert _{V_{i}}^{p}\leq\left\Vert f\right\Vert _{V_{i}}^{p}+\left\Vert g\right\Vert _{V_{i}}^{p}$,
so that we get
\begin{align*}
\left\Vert \Psi_{\left|\varrho\right|}^{\left(i,\delta\right)}\left(c_{k}\right)_{k\in\mathbb{Z}^{d}}\right\Vert _{V_{i}}^{p} & \leq\sum_{k\in\mathbb{Z}^{d}}\left[v_{k}^{\left(i,\delta\right)}\left|c_{k}\right|\right]^{p}\cdot\left\Vert \frac{L_{\delta\cdot T_{i}^{-T}k}\left[\varrho\circ T_{i}^{T}\right]}{v_{k}^{\left(i,\delta\right)}}\right\Vert _{V_{i}}^{p}\\
& \leq\left[\Omega_{0}^{K}\Omega_{1}\cdot\left(1+2\sqrt{d}\right)^{K_{0}}\left(\frac{s_{d}}{p}\right)^{1/p}\cdot\left|\det T_{i}\right|^{-1/p}\right]^{p}\cdot\left\Vert \left(c_{k}\right)_{k\in\mathbb{Z}^{d}}\right\Vert _{C_{i}^{\left(\delta\right)}}^{p},
\end{align*}
which yields the desired boundedness for $p\in\left(0,1\right)$.
\medskip{}
Next, we consider the case $p=\infty$. Here, we note because of
$\left|\varrho\left(x\right)\right|=\varrho\left(x\right)=\left(1+\left|x\right|\right)^{-K_{0}}$
that
\begin{align*}
v\left(x\right)\cdot\left|\left[\Psi_{\left|\varrho\right|}^{\left(i,\delta\right)}\left(c_{k}\right)_{k\in\mathbb{Z}^{d}}\right]\left(x\right)\right| & =\left|\sum_{k\in\mathbb{Z}^{d}}c_{k}v_{k}^{\left(i,\delta\right)}\cdot\frac{v\left(x\right)}{v_{k}^{\left(i,\delta\right)}}\cdot\left(\varrho\circ T_{i}^{T}\right)\left(x-\delta\cdot T_{i}^{-T}k\right)\right|\\
\left({\scriptstyle \text{for }c=\left(c_{k}\right)_{k\in\mathbb{Z}^{d}}\text{ and since }p=\infty}\right) & \leq\left\Vert c\right\Vert _{C_{i}^{\left(\delta\right)}}\cdot\sum_{k\in\mathbb{Z}^{d}}\left[\frac{v\left(x\right)}{v_{k}^{\left(i,\delta\right)}}\cdot\left(\varrho\circ T_{i}^{T}\right)\left(x-\delta\cdot T_{i}^{-T}k\right)\right]\\
\left({\scriptstyle \text{eq. }\eqref{eq:SchwartzTranslationSynthesisWeightTrick}\text{ and assumption on }v_{0}}\right) & \leq\Omega_{1}\cdot\left\Vert c\right\Vert _{C_{i}^{\left(\delta\right)}}\cdot\sum_{k\in\mathbb{Z}^{d}}\left[\left(1+\left|x-\delta\cdot T_{i}^{-T}k\right|\right)^{K}\cdot\left(\varrho\circ T_{i}^{T}\right)\left(x-\delta\cdot T_{i}^{-T}k\right)\right]\\
\left({\scriptstyle \text{eq. }\eqref{eq:WeightLinearTransformationsConnection}\text{ and }\varrho\left(x\right)=\left(1+\left|x\right|\right)^{-K_{0}}}\right) & \leq\Omega_{0}^{K}\Omega_{1}\cdot\left\Vert c\right\Vert _{C_{i}^{\left(\delta\right)}}\cdot\sum_{k\in\mathbb{Z}^{d}}\left[\left(1+\left|T_{i}^{T}\left(x-\delta\cdot T_{i}^{-T}k\right)\right|\right)^{K-K_{0}}\right]\\
& =\Omega_{0}^{K}\Omega_{1}\cdot\left\Vert c\right\Vert _{C_{i}^{\left(\delta\right)}}\cdot h\left(T_{i}^{T}x\right),
\end{align*}
where we introduced $h\left(y\right):=\sum_{k\in\mathbb{Z}^{d}}\left[\left(1+\left|y-\delta\cdot k\right|\right)^{K-K_{0}}\right]$
for $y\in\mathbb{R}^{d}$ in the last step. Now, we recall $0<\delta\leq1$
and $K-K_{0}\leq-\left(d+1\right)<0$, so that
\begin{align*}
h\left(y\right) & \leq\sum_{k\in\mathbb{Z}^{d}}\left(1+\left|\delta\cdot\left(\frac{y}{\delta}-k\right)\right|\right)^{-\left(d+1\right)}\\
& \leq\delta^{-\left(d+1\right)}\cdot\sum_{k\in\mathbb{Z}^{d}}\left(1+\left|\frac{y}{\delta}-k\right|\right)^{-\left(d+1\right)}\\
\left({\scriptstyle \text{since }\left|x\right|\geq\left\Vert x\right\Vert _{\infty}}\right) & \leq\delta^{-\left(d+1\right)}\cdot\sum_{k\in\mathbb{Z}^{d}}\left(1+\left\Vert \frac{y}{\delta}-k\right\Vert _{\infty}\right)^{-\left(d+1\right)}=:\delta^{-\left(d+1\right)}\cdot\widetilde{h}\left(\frac{y}{\delta}\right).
\end{align*}
Now, we note that $\widetilde{h}$ is $\mathbb{Z}^{d}$-periodic,
so that $\left\Vert \smash{\widetilde{h}}\right\Vert _{\sup}=\left\Vert \smash{\widetilde{h}}\right\Vert _{\sup,\left[0,1\right)}$.
But for arbitrary $x\in\left[0,1\right)^{d}$, we have
\[
1+\left\Vert k\right\Vert _{\infty}\leq1+\left\Vert k-x\right\Vert _{\infty}+\left\Vert x\right\Vert _{\infty}\leq2+\left\Vert x-k\right\Vert _{\infty}\leq2\left(1+\left\Vert x-k\right\Vert _{\infty}\right)
\]
and thus
\[
\widetilde{h}\left(x\right)=\sum_{k\in\mathbb{Z}^{d}}\left(1+\left\Vert x-k\right\Vert _{\infty}\right)^{-\left(d+1\right)}\leq2^{d+1}\cdot\sum_{k\in\mathbb{Z}^{d}}\left(1+\left\Vert k\right\Vert _{\infty}\right)^{-\left(d+1\right)}.
\]
Next, we use the layer-cake formula (cf.\@ \cite[Proposition (6.24)]{FollandRA})
with the counting measure $\mu$ on $\mathbb{Z}^{d}$ to estimate
for $\theta\left(k\right):=\left(1+\left\Vert k\right\Vert _{\infty}\right)^{-\left(d+1\right)}$
the series
\begin{equation}
\begin{split}\sum_{k\in\mathbb{Z}^{d}}\left(1+\left\Vert k\right\Vert _{\infty}\right)^{-\left(d+1\right)}=\int_{\mathbb{Z}^{d}}\theta\left(k\right)\operatorname{d}\mu\left(k\right) & =\int_{0}^{\infty}\mu\left(\left\{ k\in\mathbb{Z}^{d}\,\middle|\,\theta\left(k\right)>\lambda\right\} \right)\operatorname{d}\lambda\\
\left({\scriptstyle \text{since }\theta\left(k\right)>\lambda\Longleftrightarrow\left\Vert k\right\Vert _{\infty}<\lambda^{\frac{-1}{d+1}}-1\text{ can only hold for }\lambda<1}\right) & \leq\int_{0}^{1}\mu\left(\left[-\lambda^{-\frac{1}{d+1}},\lambda^{-\frac{1}{d+1}}\right]^{d}\cap\mathbb{Z}^{d}\right)\operatorname{d}\lambda\\
& \leq\int_{0}^{1}\mu\left(\left\{ -\left\lfloor \lambda^{-\frac{1}{d+1}}\right\rfloor ,\dots,\left\lfloor \lambda^{-\frac{1}{d+1}}\right\rfloor \right\} ^{d}\right)\operatorname{d}\lambda\\
& =\int_{0}^{1}\left(1+2\left\lfloor \lambda^{-\frac{1}{d+1}}\right\rfloor \right)^{d}\operatorname{d}\lambda\\
& \leq3^{d}\cdot\int_{0}^{1}\lambda^{-\frac{d}{d+1}}\operatorname{d}\lambda=\left(d+1\right)\cdot3^{d}\cdot\lambda^{\frac{1}{d+1}}\bigg|_{0}^{1}\\
\left({\scriptstyle \text{since }1+d\leq2^{d}}\right) & \leq6^{d}.
\end{split}
\label{eq:StandardDecayLatticeSeries}
\end{equation}
Hence, we get $\widetilde{h}\left(x\right)\leq2^{d+1}\cdot6^{d}\leq2\cdot12^{d}$
for all $x\in\left[0,1\right)^{d}$ and thus for all $x\in\mathbb{R}^{d}$,
by $\mathbb{Z}^{d}$-periodicity. In view of the estimates from above,
this entails for $c=\left(c_{k}\right)_{k\in\mathbb{Z}^{d}}$ that
\[
v\left(x\right)\cdot\left|\left[\Psi_{\left|\varrho\right|}^{\left(i,\delta\right)}\left(c_{k}\right)_{k\in\mathbb{Z}^{d}}\right]\left(x\right)\right|\leq\Omega_{0}^{K}\Omega_{1}\cdot\left\Vert c\right\Vert _{C_{i}^{\left(\delta\right)}}\cdot h\left(T_{i}^{T}x\right)\leq\delta^{-\left(d+1\right)}\cdot2\cdot12^{d}\cdot\Omega_{0}^{K}\Omega_{1}\cdot\left\Vert c\right\Vert _{C_{i}^{\left(\delta\right)}}<\infty
\]
for all $x\in\mathbb{R}^{d}$. In particular, the series defining
$\Psi_{\left|\varrho\right|}^{\left(i,\delta\right)}$ converges pointwise
absolutely in case of $p=\infty$. But since we have $\ell_{v^{\left(i,\delta\right)}}^{p}\left(\mathbb{Z}^{d}\right)\hookrightarrow\ell_{v^{\left(i,\delta\right)}}^{\infty}\left(\mathbb{Z}^{d}\right)$
for all $p\in\left(0,\infty\right]$, this implies absolute pointwise
convergence for arbitrary $p\in\left(0,\infty\right]$.
\medskip{}
Next, for $p\in\left[1,\infty\right]$, it is easy to see that the
operator
\[
\Lambda_{\left|\varrho\right|}^{\left(i,\delta\right)}:\ell^{p}\left(\smash{\mathbb{Z}^{d}}\right)\to L^{p}\left(\smash{\mathbb{R}^{d}}\right),\left(\zeta_{k}\right)_{k\in\mathbb{Z}^{d}}\mapsto v\cdot\sum_{k\in\mathbb{Z}^{d}}\frac{\zeta_{k}}{v_{k}^{\left(i,\delta\right)}}\cdot L_{\delta\cdot T_{i}^{-T}k}\left[\left|\varrho\right|\circ T_{i}^{T}\right]
\]
is well-defined and bounded if and only if $\Psi_{\left|\varrho\right|}^{\left(i,\delta\right)}$
is, with $\vertiii{\Lambda_{\left|\varrho\right|}^{\left(i,\delta\right)}}=\vertiii{\Psi_{\left|\varrho\right|}^{\left(i,\delta\right)}}$.
Hence, since we have already shown boundedness for $p\in\left\{ 1,\infty\right\} $,
we can use complex interpolation (the Riesz-Thorin theorem, \cite[Theorem (6.27)]{FollandRA})
to derive for $p\in\left[1,\infty\right]$ that $\Psi_{\left|\varrho\right|}^{\left(i,\delta\right)}$
is bounded, with
\begin{align*}
\vertiii{\Psi_{\left|\varrho\right|}^{\left(i,\delta\right)}} & \leq\left[\Omega_{0}^{K}\Omega_{1}s_{d}\cdot\left|\det T_{i}\right|^{-1}\right]^{\frac{1}{p}}\cdot\left[\delta^{-\left(d+1\right)}\cdot2\cdot12^{d}\cdot\Omega_{0}^{K}\Omega_{1}\right]^{1-\frac{1}{p}}\\
\left({\scriptstyle \text{since }s_{d}\leq4^{d}}\right) & \leq\Omega_{0}^{K}\Omega_{1}\cdot12^{d+1}\cdot\delta^{-\left(1-\frac{1}{p}\right)\left(d+1\right)}\cdot\left|\det T_{i}\right|^{-\frac{1}{p}}.
\end{align*}
\medskip{}
Finally, for $g\in L_{v_{0}}^{1}\left(\mathbb{R}^{d}\right)$ and
$c=\left(c_{k}\right)_{k\in\mathbb{Z}^{d}}\in C_{i}^{\left(\delta\right)}=\ell_{v^{\left(i,\delta\right)}}^{p}\left(\mathbb{Z}^{d}\right)\hookrightarrow\ell_{v^{\left(i,\delta\right)}}^{\infty}\left(\mathbb{Z}^{d}\right)$,
our previous considerations, in combination with $v\left(x\right)=v\left(x-y+y\right)\leq v\left(x-y\right)v_{0}\left(y\right)$,
show for arbitrary measurable $\varrho:\mathbb{R}^{d}\to\mathbb{C}$
with $\left\Vert \varrho\right\Vert _{K_{0}}<\infty$ that
\begin{align*}
\int_{\mathbb{R}^{d}}\left|g\left(y\right)\right|\!\cdot\!\sum_{k\in\mathbb{Z}^{d}}\left|c_{k}\right|\!\cdot\!\left|\left(\!L_{\delta\cdot T_{i}^{-T}k}\left[\varrho\circ T_{i}^{T}\right]\!\right)\!\!\left(x-y\right)\right|\operatorname{d} y & \leq\frac{1}{v\left(x\right)}\!\cdot\!\int_{\mathbb{R}^{d}}\left|v_{0}\left(y\right)\!\cdot\!g\left(y\right)\right|\cdot v\left(x\!-\!y\right)\cdot\left[\Psi_{\left|\varrho\right|}^{\left(i,\delta\right)}\left(\left|c_{k}\right|\right)_{k\in\mathbb{Z}^{d}}\right]\!\!\left(x-y\right)\operatorname{d} y\\
& \leq\frac{1}{v\left(x\right)}\cdot\left\Vert \Psi_{\left|\varrho\right|}^{\left(i,\delta\right)}\left(\left|c_{k}\right|\right)_{k\in\mathbb{Z}^{d}}\right\Vert _{L_{v}^{\infty}}\cdot\left\Vert g\right\Vert _{L_{v_{0}}^{1}}<\infty
\end{align*}
for all $x\in\mathbb{R}^{d}$, so that the interchange of summation
and integration in
\begin{align*}
\left[g\ast\Psi_{\varrho}^{\left(i,\delta\right)}\left(c_{k}\right)_{k\in\mathbb{Z}^{d}}\right]\left(x\right) & =\int_{\mathbb{R}^{d}}g\left(y\right)\cdot\left(\sum_{k\in\mathbb{Z}^{d}}c_{k}\cdot L_{\delta\cdot T_{i}^{-T}k}\left[\varrho\circ T_{i}^{T}\right]\right)\left(x-y\right)\operatorname{d} y\\
& =\sum_{k\in\mathbb{Z}^{d}}c_{k}\cdot\int_{\mathbb{R}^{d}}g\left(y\right)\cdot\left(L_{\delta\cdot T_{i}^{-T}k}\left[\varrho\circ T_{i}^{T}\right]\right)\left(x-y\right)\operatorname{d} y\\
& =\sum_{k\in\mathbb{Z}^{d}}c_{k}\cdot\left(g\ast L_{\delta\cdot T_{i}^{-T}k}\left[\varrho\circ T_{i}^{T}\right]\right)\left(x\right)
\end{align*}
is justified by the dominated convergence theorem.
\end{proof}
As a further ingredient, we will need the following ``sampling theorem''
for bandlimited functions. A very similar statement already appears
in \cite[Proposition in §1.3.3]{TriebelTheoryOfFunctionSpaces},
so no originality at all is claimed. Note, however, that in \cite{TriebelTheoryOfFunctionSpaces},
it is assumed that $\varphi\in\mathcal{S}\left(\mathbb{R}^{d}\right)$
instead of $\varphi\in\mathcal{S}'\left(\mathbb{R}^{d}\right)$ and
furthermore, the statement in \cite{TriebelTheoryOfFunctionSpaces}
is restricted to the \emph{unweighted} case.
\begin{lem}
\label{lem:BandlimitedSampling}For each $i\in I$, $R>0$ and $p\in\left(0,\infty\right]$,
as well as
\[
C:=2^{\max\left\{ 1,\frac{1}{p}\right\} }\cdot\Omega_{0}^{3K}\Omega_{1}^{3}\cdot\left(1+\sqrt{d}\right)^{K}\cdot\left(23040\cdotd^{3/2}\cdot\left(K+1+\frac{d+1}{\min\left\{ 1,p\right\} }\right)\right)^{K+2+\frac{d+1}{\min\left\{ 1,p\right\} }}\cdot\left(1+R\right)^{1+\frac{d}{\min\left\{ 1,p\right\} }}
\]
we have
\[
\left\Vert \left[\varphi\left(\delta\cdot T_{i}^{-T}k\right)\right]_{k\in\mathbb{Z}^{d}}\right\Vert _{C_{i}^{\left(\delta\right)}}\leq C\cdot\delta^{-d/p}\cdot\left|\det T_{i}\right|^{1/p}\cdot\left\Vert \varphi\right\Vert _{L_{v}^{p}}
\]
for all $\delta\in\left(0,1\right]$ and all $\varphi\in\mathcal{S}'\left(\mathbb{R}^{d}\right)$
with $\operatorname{supp}\widehat{\varphi}\subset T_{i}\left[-R,R\right]^{d}+\xi_{0}$,
for arbitrary $\xi_{0}\in\mathbb{R}^{d}$.
\end{lem}
\begin{proof}
Clearly, we can assume without loss of generality that $\left\Vert \varphi\right\Vert _{L_{v}^{p}}<\infty$.
Let us first consider the case $p=\infty$. Since we have $v\left(x\right)\cdot\left|\varphi\left(x\right)\right|\leq\left\Vert \varphi\right\Vert _{L_{v}^{\infty}}$
for almost all $x\in\mathbb{R}^{d}$, and thus for a dense subset
of $\mathbb{R}^{d}$, there is for arbitrary $k\in\mathbb{Z}^{d}$
a sequence $\left(x_{n}\right)_{n\in\mathbb{N}}$ satisfying $x_{n}\xrightarrow[n\to\infty]{}\delta\cdot T_{i}^{-T}k$
as well as $v\left(x_{n}\right)\cdot\left|\varphi\left(x_{n}\right)\right|\leq\left\Vert \varphi\right\Vert _{L_{v}^{\infty}}$.
But since $\varphi$ is given by (integration against) a continuous
function by the Paley-Wiener theorem, this implies
\begin{align*}
v\left(\delta\cdot T_{i}^{-T}k\right)\cdot\left|\varphi\left(\delta\cdot T_{i}^{-T}k\right)\right| & =\lim_{n\to\infty}v\left(\delta\cdot T_{i}^{-T}k\right)\cdot\left|\varphi\left(x_{n}\right)\right|\\
& \leq\liminf_{n\to\infty}v\left(x_{n}+\delta\cdot T_{i}^{-T}k-x_{n}\right)\cdot\left|\varphi\left(x_{n}\right)\right|\\
& \leq\liminf_{n\to\infty}v\left(x_{n}\right)\cdot\left|\varphi\left(x_{n}\right)\right|\cdot v_{0}\left(\delta\cdot T_{i}^{-T}k-x_{n}\right)\\
& \leq\Omega_{1}\cdot\left\Vert \varphi\right\Vert _{L_{v}^{\infty}}\cdot\liminf_{n\to\infty}\left(1+\left|\delta\cdot T_{i}^{-T}k-x_{n}\right|\right)^{K}\\
& =\Omega_{1}\cdot\left\Vert \varphi\right\Vert _{L_{v}^{\infty}}<\infty.
\end{align*}
Since $C\geq\Omega_{1}$, since $\delta^{-d/p}\cdot\left|\det T_{i}\right|^{1/p}=1$
for $p=\infty$ and since $k\in\mathbb{Z}^{d}$ was arbitrary, this
establishes the claim for $p=\infty$. Hence, we can assume $p\in\left(0,\infty\right)$
in what follows.
\medskip{}
Let $C>0$ as in the statement of the theorem and $C_{1}:=2^{-\max\left\{ 1,\frac{1}{p}\right\} }\cdot\left[\Omega_{0}^{K}\Omega_{1}\cdot\left(1+\sqrt{d}\right)^{K}\right]^{-1}\cdot C$.
Let $\varrho:=M_{-\xi_{0}}\varphi$ and note $\left|\varrho\right|=\left|\varphi\right|$.
Now, Theorem \ref{thm:BandlimitedOscillationSelfImproving} shows
\[
\left\Vert \osc{\delta\cdot T_{i}^{-T}\left[-1,1\right]^{d}}\,\varrho\right\Vert _{W_{T_{i}^{-T}\left[-1,1\right]^{d}}\left(L_{v}^{p}\right)}=\left\Vert \osc{\delta\cdot T_{i}^{-T}\left[-1,1\right]^{d}}\left[M_{-\xi_{0}}\varphi\right]\right\Vert _{W_{T_{i}^{-T}\left[-1,1\right]^{d}}\left(L_{v}^{p}\right)}\leq C_{1}\cdot\delta\cdot\left\Vert \varphi\right\Vert _{L_{v}^{p}}
\]
for all $\delta\in\left(0,1\right]$ and $\varphi$ as in the statement
of the lemma.
Now, notice for arbitrary $k\in\mathbb{Z}^{d}$ and $x\in\delta T_{i}^{-T}\left(k+\left[0,1\right)^{d}\right)$
that $\delta T_{i}^{-T}k\in x-\delta T_{i}^{-T}\left[0,1\right)^{d}\subset x+\delta T_{i}^{-T}\left[-1,1\right]^{d}$
and hence
\begin{align*}
\left|\varphi\left(\delta\cdot T_{i}^{-T}k\right)\right|=\left|\varrho\left(\delta\cdot T_{i}^{-T}k\right)\right| & \leq\left|\varrho\left(x\right)\right|+\left|\varrho\left(x\right)-\varrho\left(\delta\cdot T_{i}^{-T}k\right)\right|\\
& \leq\left|\varphi\left(x\right)\right|+\left(\osc{\delta\cdot T_{i}^{-T}\left[-1,1\right]^{d}}\varrho\right)\left(x\right).
\end{align*}
By multiplying this estimate with ${\mathds{1}}_{\delta T_{i}^{-T}\left(k+\left[0,1\right)^{d}\right)}\left(x\right)$,
summing over $k\in\mathbb{Z}^{d}$ and using $\mathbb{R}^{d}=\biguplus_{k\in\mathbb{Z}^{d}}\delta T_{i}^{-T}\left(k+\smash{\left[0,1\right)^{d}}\right)$,
we obtain
\[
\sum_{k\in\mathbb{Z}^{d}}\left({\mathds{1}}_{\delta T_{i}^{-T}\left(k+\left[0,1\right)^{d}\right)}\left(x\right)\cdot\left|\varphi\left(\delta T_{i}^{-T}k\right)\right|\right)\leq\left|\varphi\left(x\right)\right|+\left(\osc{\delta\cdot T_{i}^{-T}\left[-1,1\right]^{d}}\,\varrho\right)\left(x\right)\qquad\forall x\in\mathbb{R}^{d}.
\]
By solidity of $L_{v}^{p}\left(\mathbb{R}^{d}\right)$, we conclude
\begin{align*}
& \left\Vert \sum_{k\in\mathbb{Z}^{d}}\left({\mathds{1}}_{\delta T_{i}^{-T}\left(k+\left[0,1\right)^{d}\right)}\cdot\left|\varphi\left(\delta T_{i}^{-T}k\right)\right|\right)\right\Vert _{L_{v}^{p}}\\
& \leq\left\Vert \left|\varphi\right|+\osc{\delta\cdot T_{i}^{-T}\left[-1,1\right]^{d}}\,\varrho\right\Vert _{L_{v}^{p}}\\
\left({\scriptstyle C_{2}:=2^{\max\left\{ 0,\frac{1}{p}-1\right\} }\text{ is triangle const. for }L_{v}^{p}}\right) & \leq C_{2}\cdot\left(\left\Vert \varphi\right\Vert _{L_{v}^{p}}+\left\Vert \osc{\delta\cdot T_{i}^{-T}\left[-1,1\right]^{d}}\,\varrho\right\Vert _{L_{v}^{p}}\right)\\
& \leq C_{2}\cdot\left(\left\Vert \varphi\right\Vert _{L_{v}^{p}}+C_{1}\cdot\delta\cdot\left\Vert \varphi\right\Vert _{L_{v}^{p}}\right)\\
\left({\scriptstyle \text{since }\delta\leq1}\right) & \leq C_{2}\left(1+C_{1}\right)\cdot\left\Vert \varphi\right\Vert _{L_{v}^{p}}\\
\left({\scriptstyle \text{since }C_{1}\geq1}\right) & \leq2^{\max\left\{ 1,\frac{1}{p}\right\} }\cdot C_{1}\cdot\left\Vert \varphi\right\Vert _{L_{v}^{p}}.
\end{align*}
Finally, we note for $x\in\delta\cdot T_{i}^{-T}\left(k+\left[0,1\right)^{d}\right)$,
i.e., for $x=\delta T_{i}^{-T}k+\delta T_{i}^{-T}q$ with $q\in\left[0,1\right)^{d}$
that
\begin{align*}
v_{k}^{\left(i,\delta\right)} & =v\left(\delta\cdot T_{i}^{-T}k\right)=v\left(x-\delta T_{i}^{-T}q\right)\leq v\left(x\right)\cdot v_{0}\left(\delta\cdot T_{i}^{-T}q\right)\\
\left({\scriptstyle \text{assump. on }v_{0}\text{ and eq. }\eqref{eq:WeightLinearTransformationsConnection}}\right) & \leq\Omega_{1}\cdot v\left(x\right)\cdot\left(1+\left|\delta\cdot T_{i}^{-T}q\right|\right)^{K}\leq\Omega_{0}^{K}\Omega_{1}\cdot v\left(x\right)\cdot\left(1+\left|\delta\cdot q\right|\right)^{K}\\
\left({\scriptstyle \text{since }\delta\leq1}\right) & \leq\Omega_{0}^{K}\Omega_{1}\cdot\left(1+\sqrt{d}\right)^{K}\cdot v\left(x\right).
\end{align*}
Using this estimate and the pairwise disjointness of $\left(\delta\cdot T_{i}^{-T}\left(k+\left[0,1\right)^{d}\right)\right)_{k\in\mathbb{Z}^{d}}$,
we conclude
\begin{align*}
\left\Vert \sum_{k\in\mathbb{Z}^{d}}\!\!\left({\mathds{1}}_{\delta T_{i}^{-T}\!\left(k+\left[0,1\right)^{d}\right)}\cdot\left|\varphi\!\left(\delta T_{i}^{-T}k\right)\right|\right)\right\Vert _{L_{v}^{p}}\!\!\! & =\left(\sum_{k\in\mathbb{Z}^{d}}\left|\varphi\left(\delta\cdot T_{i}^{-T}k\right)\right|^{p}\!\int_{\delta T_{i}^{-T}\left(k+\left[0,1\right)\right)^{d}}\left[v\left(x\right)\right]^{p}\operatorname{d} x\right)^{1/p}\\
& \geq\left[\Omega_{0}^{K}\Omega_{1}\!\cdot\!\left(1\!+\!\sqrt{d}\right)^{\!K}\right]^{-1}\!\!\!\cdot\!\left(\sum_{k\in\mathbb{Z}^{d}}\!\left[v_{k}^{\left(i,\delta\right)}\!\cdot\left|\varphi\!\left(\delta T_{i}^{-T}k\right)\right|\right]^{p}\!\!\cdot\!\lambda_{d}\left(\!\delta T_{i}^{-T}\!\left(k\!+\!\left[0,1\right)^{d}\right)\!\right)\!\right)^{\!\!\frac{1}{p}}\\
& =\left[\Omega_{0}^{K}\Omega_{1}\cdot\left(1+\sqrt{d}\right)^{K}\right]^{-1}\cdot\delta^{d/p}\cdot\left|\det T_{i}\right|^{-1/p}\cdot\left\Vert \left[\varphi\left(\delta\cdot T_{i}^{-T}k\right)\right]_{k\in\mathbb{Z}^{d}}\right\Vert _{C_{i}^{\left(\delta\right)}}.
\end{align*}
Putting everything together, we conclude
\begin{align*}
\left\Vert \left[\varphi\left(\delta\cdot T_{i}^{-T}k\right)\right]_{k\in\mathbb{Z}^{d}}\right\Vert _{C_{i}^{\left(\delta\right)}} & \leq\delta^{-\frac{d}{p}}\cdot\left|\det T_{i}\right|^{\frac{1}{p}}\cdot\Omega_{0}^{K}\Omega_{1}\cdot\left(1+\sqrt{d}\right)^{K}\cdot\left\Vert \sum_{k\in\mathbb{Z}^{d}}\left({\mathds{1}}_{\delta T_{i}^{-T}\left(k+\left[0,1\right)^{d}\right)}\cdot\left|\varphi\left(\delta T_{i}^{-T}k\right)\right|\right)\right\Vert _{L_{v}^{p}}\\
& \leq\delta^{-\frac{d}{p}}\cdot\left|\det T_{i}\right|^{\frac{1}{p}}\cdot\Omega_{0}^{K}\Omega_{1}\cdot\left(1+\sqrt{d}\right)^{K}\cdot2^{\max\left\{ 1,\frac{1}{p}\right\} }\cdot C_{1}\cdot\left\Vert \varphi\right\Vert _{L_{v}^{p}}\\
& =\delta^{-\frac{d}{p}}\cdot\left|\det T_{i}\right|^{\frac{1}{p}}\cdot C\cdot\left\Vert \varphi\right\Vert _{L_{v}^{p}}<\infty,
\end{align*}
as desired.
\end{proof}
In the proof of Theorem \ref{thm:AtomicDecomposition}, we will employ
a Neumann series argument for an operator defined on $\DecompSp{\mathcal{Q}}p{\ell_{w}^{q}}v$.
For this to be justified, we need to know that this space is a Quasi-Banach
space, i.e., complete. For $v\equiv1$, this was already shown in
\cite[Theorem 3.21]{DecompositionEmbedding}, but for $v\not\equiv1$
and general $p,q\in\left(0,\infty\right]$, it seems that the following
lemma is a novel (though not too surprising) result:
\begin{lem}
\label{lem:WeightedDecompositionSpaceComplete}The decomposition space
$\DecompSp{\mathcal{Q}}p{\ell_{w}^{q}}v$ is a Quasi-Banach space.
\end{lem}
\begin{proof}
Verifying the quasi-norm properties of $\left\Vert \bullet\right\Vert _{\DecompSp{\mathcal{Q}}p{\ell_{w}^{q}}v}$
is relatively straightforward (and essentially identical to the verification
in \cite[Theorem 3.21]{DecompositionEmbedding}), so we only prove
completeness.
Instead of verifying completeness directly, we use a slightly more
abstract approach, employing other results from the paper. The main
new ingredient that we need to provide is boundedness of
\[
{\rm Ana}_{\ast}:\DecompSp{\mathcal{Q}}p{\ell_{w}^{q}}v\to V=\ell_{w}^{q}\left(\left[V_{i}\right]_{i\in I}\right),f\mapsto\left[\mathcal{F}^{-1}\left(\varphi_{i}^{\ast}\widehat{f}\right)\right]_{i\in I}.
\]
To this end, let $f\in\DecompSp{\mathcal{Q}}p{\ell_{w}^{q}}v$ be arbitrary
and define $c_{i}:=\left\Vert \mathcal{F}^{-1}\left(\varphi_{i}\widehat{f}\right)\right\Vert _{L_{v}^{p}}$
for $i\in I$. Note $c=\left(c_{i}\right)_{i\in I}\in\ell_{w}^{q}\left(I\right)$
and $\left\Vert c\right\Vert _{\ell_{w}^{q}}=\left\Vert f\right\Vert _{\DecompSp{\mathcal{Q}}p{\ell_{w}^{q}}v}$.
Recall that the clustering map $\Gamma_{\mathcal{Q}}:\ell_{w}^{q}\left(I\right)\to\ell_{w}^{q}\left(I\right)$
with $\Gamma_{\mathcal{Q}}\left(e_{i}\right)_{i\in I}=\left(e_{i}^{\ast}\right)_{i\in I}$
and $e_{i}^{\ast}=\sum_{\ell\in i^{\ast}}e_{\ell}$ is bounded.
Now, we distinguish the cases $p\in\left[1,\infty\right]$ and $p\in\left(0,1\right)$.
In case of $p\in\left[1,\infty\right]$, the triangle inequality for
$L_{v}^{p}\left(\mathbb{R}^{d}\right)$ yields because of $V_{i}=L_{v}^{p}\left(\mathbb{R}^{d}\right)$
for all $i\in I$ that
\begin{align*}
\left\Vert {\rm Ana}_{\ast}f\right\Vert _{V} & =\left\Vert \left(\left\Vert \mathcal{F}^{-1}\left(\varphi_{i}^{\ast}\cdot\smash{\widehat{f}}\:\right)\right\Vert _{V_{i}}\right)_{i\in I}\right\Vert _{\ell_{w}^{q}}\leq\left\Vert \left(\sum_{\ell\in i^{\ast}}\left\Vert \mathcal{F}^{-1}\left(\varphi_{\ell}\cdot\smash{\widehat{f}}\:\right)\right\Vert _{L_{v}^{p}}\right)_{i\in I}\right\Vert _{\ell_{w}^{q}}\\
& =\left\Vert \Gamma_{\mathcal{Q}}\,c\right\Vert _{\ell_{w}^{q}}\leq\vertiii{\smash{\Gamma_{\mathcal{Q}}}}\cdot\left\Vert c\right\Vert _{\ell_{w}^{q}}=\vertiii{\smash{\Gamma_{\mathcal{Q}}}}\cdot\left\Vert f\right\Vert _{\DecompSp{\mathcal{Q}}p{\ell_{w}^{q}}v}<\infty.
\end{align*}
Now, we consider the case $p\in\left(0,1\right)$. We observe that
\cite[Lemma 2.7]{DecompositionEmbedding} yields some $R=R\left(R_{\mathcal{Q}},C_{\mathcal{Q}}\right)>0$
satisfying $\overline{Q_{i}^{\ast}}\subset T_{i}\overline{B_{R}\left(0\right)}+b_{i}\subset T_{i}\left[-R,R\right]^{d}+b_{i}$
for all $i\in I$. Because of $\operatorname{supp}\left(\varphi_{i}^{\ast}\widehat{f}\right)\subset\overline{Q_{i}^{\ast}}$,
Theorem \ref{thm:BandlimitedWienerAmalgamSelfImproving} thus yields
a constant $C_{1}=C_{1}\left(d,p,R,\Omega_{0},\Omega_{1},K\right)>0$
such that
\[
\left\Vert \mathcal{F}^{-1}\left(\varphi_{i}^{\ast}\cdot\smash{\widehat{f}}\:\right)\right\Vert _{V_{i}}=\left\Vert \mathcal{F}^{-1}\left(\varphi_{i}^{\ast}\cdot\smash{\widehat{f}}\:\right)\right\Vert _{W_{T_{i}^{-T}\left[-1,1\right]^{d}}\left(L_{v}^{p}\right)}\leq C_{1}\cdot\left\Vert \mathcal{F}^{-1}\left(\varphi_{i}^{\ast}\cdot\smash{\widehat{f}}\:\right)\right\Vert _{L_{v}^{p}}\qquad\forall i\in I.
\]
Next, since $L_{v}^{p}\left(\mathbb{R}^{d}\right)$ is a Quasi-Banach
space and since we have the uniform estimate $\left|i^{\ast}\right|\leq N_{\mathcal{Q}}$
for all $i\in I$, there is a constant $C_{2}=C_{2}\left(N_{\mathcal{Q}},p\right)>0$
satisfying
\[
\left\Vert \mathcal{F}^{-1}\left(\varphi_{i}^{\ast}\cdot\smash{\widehat{f}}\:\right)\right\Vert _{L_{v}^{p}}\leq C_{2}\cdot\sum_{\ell\in i^{\ast}}\left\Vert \mathcal{F}^{-1}\left(\varphi_{\ell}\cdot\smash{\widehat{f}}\:\right)\right\Vert _{L_{v}^{p}}=C_{2}\cdot\left(\Gamma_{\mathcal{Q}}\,c\right)_{i}\qquad\forall i\in I.
\]
All in all, this entails by solidity of $\ell_{w}^{q}\left(I\right)$
that
\[
\left\Vert {\rm Ana}_{\ast}f\right\Vert _{V}=\left\Vert \left(\left\Vert \mathcal{F}^{-1}\left(\varphi_{i}^{\ast}\cdot\smash{\widehat{f}}\:\right)\right\Vert _{V_{i}}\right)_{i\in I}\right\Vert _{\ell_{w}^{q}}\leq C_{1}C_{2}\cdot\left\Vert \Gamma_{\mathcal{Q}}\,c\right\Vert _{\ell_{w}^{q}}\leq C_{1}C_{2}\vertiii{\smash{\Gamma_{\mathcal{Q}}}}\cdot\left\Vert f\right\Vert _{\DecompSp{\mathcal{Q}}p{\ell_{w}^{q}}v}<\infty,
\]
as above. In summary, ${\rm Ana}_{\ast}$ is well-defined and bounded
for all $p\in\left(0,\infty\right]$.
Now, using the map ${\rm Synth}_{\mathcal{D}}$ from Lemma \ref{lem:DecompositionSynthesis},
we have because of $\varphi_{i}^{\ast}\varphi_{i}=\varphi_{i}$ that
\[
\left({\rm Synth}_{\mathcal{D}}\circ{\rm Ana}_{\ast}\right)f=\sum_{i\in I}\mathcal{F}^{-1}\left(\varphi_{i}\cdot\mathcal{F}\left[{\rm Ana}_{\ast}f\right]_{i}\right)=\sum_{i\in I}\mathcal{F}^{-1}\left(\varphi_{i}\varphi_{i}^{\ast}\cdot\smash{\widehat{f}}\:\right)=\sum_{i\in I}\mathcal{F}^{-1}\left(\varphi_{i}\cdot\smash{\widehat{f}}\:\right)=f
\]
for all $f\in\DecompSp{\mathcal{Q}}p{\ell_{w}^{q}}v$. Finally, recall from
(the remark after) Lemma \ref{lem:IteratedSequenceSpaceComplete}
that $V=\ell_{w}^{q}\left(\left[V_{i}\right]_{i\in I}\right)$ is
complete.
Now, let $\left(f_{n}\right)_{n\in\mathbb{N}}$ be Cauchy in $\DecompSp{\mathcal{Q}}p{\ell_{w}^{q}}v$.
Since ${\rm Ana}_{\ast}$ is bounded, the sequence $\left(g_{n}\right)_{n\in\mathbb{N}}$
with $g_{n}:={\rm Ana}_{\ast}f_{n}$ is Cauchy in $V$. Hence, $g_{n}\to g$
for some $g\in V$. Define $f:={\rm Synth}_{\mathcal{D}}g\in\DecompSp{\mathcal{Q}}p{\ell_{w}^{q}}v$
and observe
\[
\left\Vert f_{n}-f\right\Vert _{\DecompSp{\mathcal{Q}}p{\ell_{w}^{q}}v}=\left\Vert {\rm Synth}_{\mathcal{D}}{\rm Ana}_{\ast}f_{n}-{\rm Synth}_{\mathcal{D}}g\right\Vert _{\DecompSp{\mathcal{Q}}p{\ell_{w}^{q}}v}\leq\vertiii{{\rm Synth}_{\mathcal{D}}}\cdot\left\Vert g_{n}-g\right\Vert _{V}\xrightarrow[n\to\infty]{}0.\qedhere
\]
\end{proof}
Using all of the technical lemmata collected in this section, we can
finally prove that the family $\left(L_{\delta\cdot T_{j}^{-T}k}\gamma^{\left[j\right]}\right)_{k\in\mathbb{Z}^{d},j\in I}$
yields an atomic decomposition of $\DecompSp{\mathcal{Q}}p{\ell_{w}^{q}}v$,
if $\delta>0$ is chosen small enough.
\begin{thm}
\label{thm:AtomicDecomposition}Assume that the families $\Gamma=\left(\gamma_{i}\right)_{i\in I}$
and $\Gamma_{\ell}=\left(\gamma_{i,\ell}\right)_{i\in I}$ with $\ell\in\left\{ 1,2\right\} $
satisfy Assumption \ref{assu:AtomicDecompositionAssumption} and that
$\Gamma=\left(\gamma_{i}\right)_{i\in I}$ satisfies Assumption \ref{assu:GammaCoversOrbit}.
Define $\delta_{0}>0$ by
\[
\delta_{0}^{-1}\!:=\!\begin{cases}
\!\frac{2s_{d}}{\sqrt{d}}\cdot\left(2^{17}\!\cdot\!d^{2}\!\cdot\!\left(K\!+\!2\!+\!d\right)\right)^{\!K+d+3}\!\!\!\cdot\left(1\!+\!R_{\mathcal{Q}}\right)^{d+1}\cdot\Omega_{0}^{4K}\Omega_{1}^{4}\Omega_{2}^{\left(p,K\right)}\Omega_{4}^{\left(p,K\right)}\cdot\vertiii{\smash{\overrightarrow{C}}}\,, & \text{if }p\geq1,\\
\frac{\left(2^{14}/d^{\frac{3}{2}}\right)^{\!\frac{d}{p}}}{2^{45}\cdotd^{17}}\!\cdot\!\left(\frac{s_{d}}{p}\right)^{\!\frac{1}{p}}\left(2^{68}\!\cdot\!d^{14}\!\cdot\!\left(K\!+\!1\!+\!\frac{d+1}{p}\right)^{3}\right)^{\!K+2+\frac{d+1}{p}}\!\!\!\cdot\!\left(1\!+\!R_{\mathcal{Q}}\right)^{1+\frac{3d}{p}}\!\cdot\!\Omega_{0}^{16K}\Omega_{1}^{16}\Omega_{2}^{\left(p,K\right)}\Omega_{4}^{\left(p,K\right)}\cdot\vertiii{\smash{\overrightarrow{C}}}^{\frac{1}{p}}, & \text{if }p<1.
\end{cases}
\]
Then, for each $0<\delta\leq\min\left\{ 1,\delta_{0}\right\} $, the
family $\left(L_{\delta\cdot T_{j}^{-T}k}\:\gamma^{\left[j\right]}\right)_{j\in I,k\in\mathbb{Z}^{d}}$
forms an \textbf{atomic decomposition} of $\DecompSp{\mathcal{Q}}p{\ell_{w}^{q}}v$.
Precisely, this means the following:
\begin{enumerate}
\item The \textbf{synthesis map}
\begin{align*}
\qquad\qquad S^{\left(\delta\right)}:\ell_{\left(\left|\det T_{j}\right|^{\frac{1}{2}-\frac{1}{p}}w_{j}\right)_{\!\!j\in I}}^{q}\!\!\!\!\!\!\!\left(\left[\vphantom{F}\smash{C_{j}^{\left(\delta\right)}}\right]_{j\in I}\right) & \to\DecompSp{\mathcal{Q}}p{\ell_{w}^{q}}v,\\
\left(\smash{c_{k}^{\left(j\right)}}\right)_{j\in I,k\in\mathbb{Z}^{d}} & \mapsto\sum_{j\in I}\sum_{k\in\mathbb{Z}^{d}}\left(\left|\det T_{j}\right|^{-\frac{1}{2}}c_{k}^{\left(j\right)}\cdot L_{\delta\cdot T_{j}^{-T}k}\gamma^{\left(j\right)}\right)=\sum_{j\in I}\sum_{k\in\mathbb{Z}^{d}}\left(c_{k}^{\left(j\right)}\cdot L_{\delta\cdot T_{j}^{-T}k}\gamma^{\left[j\right]}\right)
\end{align*}
is well-defined and bounded for each $\delta\in\left(0,1\right]$.
\item For $0<\delta\leq\min\left\{ 1,\delta_{0}\right\} $, there is a bounded
linear \textbf{coefficient map}
\[
C^{\left(\delta\right)}:\DecompSp{\mathcal{Q}}p{\ell_{w}^{q}}v\to\ell_{\left(\left|\det T_{j}\right|^{\frac{1}{2}-\frac{1}{p}}w_{j}\right)_{\!\!j\in I}}^{q}\!\!\!\!\!\!\left(\left[\vphantom{F}\smash{C_{j}^{\left(\delta\right)}}\right]_{j\in I}\right)
\]
satisfying $S^{\left(\delta\right)}\circ C^{\left(\delta\right)}=\operatorname{id}_{\DecompSp{\mathcal{Q}}p{\ell_{w}^{q}}v}$.\qedhere
\end{enumerate}
\end{thm}
\begin{rem*}
As the proof shows, convergence of the series $\sum_{j\in I}\sum_{k\in\mathbb{Z}^{d}}\left(\left|\det T_{j}\right|^{-\frac{1}{2}}c_{k}^{\left(j\right)}\cdot L_{\delta\cdot T_{j}^{-T}k}\gamma^{\left(j\right)}\right)$
has to be understood as follows: Each of the series
\[
\sum_{k\in\mathbb{Z}^{d}}\left(\left|\det T_{j}\right|^{-\frac{1}{2}}c_{k}^{\left(j\right)}\cdot L_{\delta\cdot T_{j}^{-T}k}\gamma^{\left(j\right)}\right)
\]
converges pointwise absolutely to a function $g_{j}\in V_{j}\hookrightarrow\mathcal{S}'\left(\mathbb{R}^{d}\right)$
and the series $\sum_{j\in I}g_{j}$ converges in the weak-$\ast$-sense
in $Z'\left(\mathcal{O}\right)$, i.e., for every $\phi\in Z\left(\mathcal{O}\right)$,
the series $\sum_{j\in I}\left\langle g_{j},\,\phi\right\rangle _{\mathcal{S}',\mathcal{S}}$
converges absolutely and the functional $\phi\mapsto\sum_{j\in I}\left\langle g_{j},\,\phi\right\rangle _{\mathcal{S}',\mathcal{S}}$
is continuous on $Z\left(\mathcal{O}\right)$.
Furthermore, the proof shows that the definition of $C^{\left(\delta\right)}$
is independent of the precise choice of $p,q,v,w$, as long as $\delta>0$
is chosen small enough that $C^{\left(\delta\right)}$ is defined
at all. In fact, the proof shows that $C^{\left(\delta\right)}=D^{\left(\delta\right)}\cdot\left(T^{\left(\delta\right)}\right)^{-1}$
where $T^{\left(\delta\right)}=S^{\left(\delta\right)}\circ D^{\left(\delta\right)}:\DecompSp{\mathcal{Q}}p{\ell_{w}^{q}}v\to\DecompSp{\mathcal{Q}}p{\ell_{w}^{q}}v$
is invertible (using a Neumann series) for $0<\delta\leq\min\left\{ 1,\delta_{0}\right\} $,
with
\begin{align*}
D^{\left(\delta\right)}:\DecompSp{\mathcal{Q}}p{\ell_{w}^{q}}v & \to\ell_{\left(\left|\det T_{j}\right|^{\frac{1}{2}-\frac{1}{p}}w_{j}\right)_{\!\!j\in I}}^{q}\!\!\!\!\!\!\!\!\left(\left[\vphantom{F}\smash{C_{j}^{\left(\delta\right)}}\right]_{j\in I}\right),\\
f & \mapsto\left[\left(\delta^{d}\cdot\left|\det T_{j}\right|^{-1/2}\cdot\left[\mathcal{F}^{-1}\!\left(\theta_{j}\varphi_{j}\cdot\widehat{f}\right)\right]\!\!\left(\delta\cdot T_{j}^{-T}k\right)\right)_{k\in\mathbb{Z}^{d}}\right]_{j\in I},
\end{align*}
where $\theta_{j}$ for $j\in I$ is defined as in Assumption \ref{assu:GammaCoversOrbit}.
\end{rem*}
\begin{proof}
We first study for arbitrary $j\in I$ and $\delta\in\left(0,1\right]$
boundedness (and well-definedness) of the map
\[
S_{\Gamma_{2}}^{\left(\delta,j\right)}:C_{j}^{\left(\delta\right)}\to V_{j},\left(c_{k}\right)_{k\in\mathbb{Z}^{d}}\mapsto\left|\det T_{j}\right|^{-1/2}\cdot\sum_{k\in\mathbb{Z}^{d}}c_{k}\cdot L_{\delta\cdot T_{j}^{-T}k}\:\gamma_{2}^{\left(j\right)}.
\]
Recall $\gamma_{2}^{\left(j\right)}=\left|\det T_{j}\right|\cdot M_{b_{j}}\left[\gamma_{j,2}\circ T_{j}^{T}\right]$,
so that
\begin{align}
\left[S_{\Gamma_{2}}^{\left(\delta,j\right)}\left(c_{k}\right)_{k\in\mathbb{Z}^{d}}\right]\left(x\right) & =\left|\det T_{j}\right|^{-1/2}\cdot\sum_{k\in\mathbb{Z}^{d}}c_{k}\cdot\gamma_{2}^{\left(j\right)}\left(x-\delta\cdot T_{j}^{-T}k\right)\nonumber \\
& =\left|\det T_{j}\right|^{1/2}\cdot\sum_{k\in\mathbb{Z}^{d}}e^{2\pi i\left\langle b_{j},x-\delta\cdot T_{j}^{-T}k\right\rangle }\cdot c_{k}\cdot\gamma_{j,2}\left(T_{j}^{T}x-\delta\cdot k\right)\nonumber \\
& =\left|\det T_{j}\right|^{1/2}\cdot e^{2\pi i\left\langle b_{j},x\right\rangle }\cdot\left(\sum_{k\in\mathbb{Z}^{d}}e^{-2\pi i\left\langle b_{j},\delta\cdot T_{j}^{-T}k\right\rangle }c_{k}\cdot L_{\delta\cdot k}\gamma_{j,2}\right)\!\!\left(T_{j}^{T}x\right)\nonumber \\
\left({\scriptstyle \text{with }\Psi_{\gamma_{j,2}}^{\left(j,\delta\right)}\text{ as in Lemma }\ref{lem:SchwartzTranslationSynthesis}}\right) & =\left|\det T_{j}\right|^{1/2}\cdot e^{2\pi i\left\langle b_{j},x\right\rangle }\cdot\Psi_{\gamma_{j,2}}^{\left(j,\delta\right)}\left[\left(e^{-2\pi i\left\langle b_{j},\delta\cdot T_{j}^{-T}k\right\rangle }c_{k}\right)_{k\in\mathbb{Z}^{d}}\right].\label{eq:AtomicDecompositionTranslationSynthesisNormalization}
\end{align}
Thus, in terms of the isometric isomorphism $m_{j}^{\left(\delta\right)}:C_{j}^{\left(\delta\right)}\to C_{j}^{\left(\delta\right)},\left(c_{k}\right)_{k\in\mathbb{Z}^{d}}\mapsto\left(e^{-2\pi i\left\langle b_{j},\delta\cdot T_{j}^{-T}k\right\rangle }\cdot c_{k}\right)_{k\in\mathbb{Z}^{d}}$
and of the map $\Psi_{\gamma_{j,2}}^{\left(j,\delta\right)}$ defined
in Lemma \ref{lem:SchwartzTranslationSynthesis}, the preceding calculations
show
\[
S_{\Gamma_{2}}^{\left(\delta,j\right)}c=\left|\det T_{j}\right|^{1/2}\cdot M_{b_{j}}\left[\Psi_{\gamma_{j,2}}^{\left(j,\delta\right)}\left(m_{j}^{\left(\delta\right)}c\right)\right]\qquad\forall c\in C_{j}^{\left(\delta\right)}.
\]
As a consequence of the solidity of $V_{j}$ and of Lemma \ref{lem:SchwartzTranslationSynthesis}
(which is applicable, since $\left\Vert \gamma_{j,2}\right\Vert _{K_{0}}\leq\Omega_{4}^{\left(p,K\right)}<\infty$
for all $j\in I$, cf.\@ Assumption \ref{assu:AtomicDecompositionAssumption}),
we thus get
\begin{equation}
\vertiii{S_{\Gamma_{2}}^{\left(\delta,j\right)}}\leq C_{K,\delta,d,p,\Gamma_{2}}\cdot\left|\det T_{j}\right|^{\frac{1}{2}-\frac{1}{p}}<\infty\qquad\forall j\in I\label{eq:DiscreteSynthesisUniformBound}
\end{equation}
for a suitable constant $C_{K,\delta,d,p,\Gamma_{2}}>0$
which is independent of $j\in I$. In particular, each map $S_{\Gamma_{2}}^{\left(\delta,j\right)}$
is well-defined with pointwise absolute convergence of the defining
series.
\medskip{}
Now, we can establish boundedness of the synthesis map $S^{\left(\delta\right)}$
as follows: In view of equation (\ref{eq:DiscreteSynthesisUniformBound}),
it follows that
\[
\bigotimes_{j\in I}S_{\Gamma_{2}}^{\left(\delta,j\right)}:\ell_{\left(\left|\det T_{j}\right|^{\frac{1}{2}-\frac{1}{p}}\cdot w_{j}\right)_{\!\!j\in I}}^{q}\!\!\!\!\!\!\!\!\left(\left[\vphantom{F}\smash{C_{j}^{\left(\delta\right)}}\right]_{j\in I}\right)\to\ell_{w}^{q}\left(\left[V_{j}\right]_{j\in I}\right),\left(\smash{c_{k}^{\left(j\right)}}\right)_{j\in I,k\in\mathbb{Z}^{d}}\mapsto\left[S_{\Gamma_{2}}^{\left(\delta,j\right)}\left(\smash{c_{k}^{\left(j\right)}}\right)_{k\in\mathbb{Z}^{d}}\right]_{j\in I}
\]
is well-defined and bounded, with $\vertiii{\bigotimes_{j\in I}S_{\Gamma_{2}}^{\left(\delta,j\right)}}\leq C_{K,\delta,d,p,\Gamma_{2}}$.
Furthermore, using $\gamma_{j}=\gamma_{j,1}\ast\gamma_{j,2}$, it
follows easily that $L_{x}\gamma^{\left(j\right)}=\gamma_{1}^{\left(j\right)}\ast L_{x}\gamma_{2}^{\left(j\right)}$
for arbitrary $x\in\mathbb{R}^{d}$ and $j\in I$, from which it
follows (with ${\rm Synth}_{\Gamma_{1}}$ as in Lemma \ref{lem:GammaSynthesisBounded})
that
\begin{align}
\left[{\rm Synth}_{\Gamma_{1}}\circ\bigotimes_{j\in I}S_{\Gamma_{2}}^{\left(\delta,j\right)}\right]\left(\smash{c_{k}^{\left(j\right)}}\right)_{j\in I,k\in\mathbb{Z}^{d}} & =\sum_{j\in I}\left[\gamma_{1}^{\left(j\right)}\ast S_{\Gamma_{2}}^{\left(\delta,j\right)}\left(\smash{c_{k}^{\left(j\right)}}\right)_{k\in\mathbb{Z}^{d}}\right]\nonumber \\
& =\sum_{j\in I}\left[\left|\det T_{j}\right|^{-1/2}\cdot\gamma_{1}^{\left(j\right)}\ast\sum_{k\in\mathbb{Z}^{d}}c_{k}^{\left(j\right)}\cdot L_{\delta\cdot T_{j}^{-T}k}\gamma_{2}^{\left(j\right)}\right]\nonumber \\
\left({\scriptstyle \gamma_{1}^{\left(j\right)}\in L_{v_{0}}^{1}\left(\mathbb{R}^{d}\right)\text{ and (proof of) Lemma }\ref{lem:SchwartzTranslationSynthesis}}\right) & =\sum_{j\in I}\left[\left|\det T_{j}\right|^{-1/2}\cdot\sum_{k\in\mathbb{Z}^{d}}c_{k}^{\left(j\right)}\cdot L_{\delta\cdot T_{j}^{-T}k}\left(\gamma_{1}^{\left(j\right)}\ast\gamma_{2}^{\left(j\right)}\right)\right]\nonumber \\
& =\sum_{j\in I}\left[\sum_{k\in\mathbb{Z}^{d}}\left|\det T_{j}\right|^{-1/2}\cdot c_{k}^{\left(j\right)}\cdot L_{\delta\cdot T_{j}^{-T}k}\gamma^{\left(j\right)}\right]\nonumber \\
& =S^{\left(\delta\right)}\left(\smash{c_{k}^{\left(j\right)}}\right)_{j\in I,k\in\mathbb{Z}^{d}}.\label{eq:AtomicDecompositionSynthesisOperatorAsComposition}
\end{align}
This shows for arbitrary $\delta\in\left(0,1\right]$ that $S^{\left(\delta\right)}={\rm Synth}_{\Gamma_{1}}\circ\bigotimes_{j\in I}S_{\Gamma_{2}}^{\left(\delta,j\right)}$
is bounded, as a composition of bounded maps.
\medskip{}
Finally, we prove that convergence of the series defining $S^{\left(\delta\right)}\left(\smash{c_{k}^{\left(j\right)}}\right)_{j\in I,k\in\mathbb{Z}^{d}}$
occurs in the sense described in the remark following the theorem:
First of all, we get exactly as in equation (\ref{eq:AtomicDecompositionTranslationSynthesisNormalization})
(but with $\gamma_{j}$ instead of $\gamma_{j,2}$) that
\[
g_{j}:=\sum_{k\in\mathbb{Z}^{d}}\left(\left|\det T_{j}\right|^{-\frac{1}{2}}c_{k}^{\left(j\right)}\cdot L_{\delta\cdot T_{j}^{-T}k}\gamma^{\left(j\right)}\right)=\left|\det T_{j}\right|^{1/2}\cdot M_{b_{j}}\left[\Psi_{\gamma_{j}}^{\left(j,\delta\right)}\left(m_{j}^{\left(\delta\right)}\left(\smash{c_{k}^{\left(j\right)}}\right)_{k\in\mathbb{Z}^{d}}\right)\right].
\]
Here, it is worth mentioning that the conditions in Assumption \ref{assu:AtomicDecompositionAssumption}
imply that Lemma \ref{lem:SchwartzTranslationSynthesis} is applicable
with $\varrho=\gamma_{j,2}$, as well as with $\varrho=\gamma_{j}$.
Hence, that lemma shows that the series defining $g_{j}$ converges
pointwise absolutely and that $g_{j}\in V_{j}\hookrightarrow\mathcal{S}'\left(\mathbb{R}^{d}\right)$,
where the last embedding is justified by Lemma \ref{lem:SpecialConvolutionConsistent}.
Finally, Lemma \ref{lem:GammaSynthesisBounded} shows that for $\left(h_{j}\right)_{j\in I}:=\left(\bigotimes_{j\in I}S_{\Gamma_{2}}^{\left(\delta,j\right)}\right)\left(\smash{c_{k}^{\left(j\right)}}\right)_{j\in I,k\in\mathbb{Z}^{d}}\in\ell_{w}^{q}\left(\left[V_{j}\right]_{j\in I}\right)$
and each $\phi\in Z\left(\mathcal{O}\right)$, the series
\begin{align*}
\left\langle {\rm Synth}_{\Gamma_{1}}\left(h_{j}\right)_{j\in I},\,\phi\right\rangle _{Z'\left(\mathcal{O}\right),Z\left(\mathcal{O}\right)} & =\sum_{j\in I}\left\langle \gamma_{1}^{\left(j\right)}\ast h_{j},\,\phi\right\rangle _{\mathcal{S}',\mathcal{S}}\\
& =\sum_{j\in I}\left\langle \gamma_{1}^{\left(j\right)}\ast\left[\left|\det T_{j}\right|^{-1/2}\cdot\sum_{k\in\mathbb{Z}^{d}}c_{k}^{\left(j\right)}L_{\delta\cdot T_{j}^{-T}k}\gamma_{2}^{\left(j\right)}\right],\,\phi\right\rangle _{\mathcal{S}',\mathcal{S}}\\
\left({\scriptstyle \text{cf. eq. }\eqref{eq:AtomicDecompositionSynthesisOperatorAsComposition}}\right) & =\sum_{j\in I}\left\langle \left|\det T_{j}\right|^{-1/2}\cdot\sum_{k\in\mathbb{Z}^{d}}c_{k}^{\left(j\right)}L_{\delta\cdot T_{j}^{-T}k}\gamma^{\left(j\right)},\,\phi\right\rangle _{\mathcal{S}',\mathcal{S}}\\
& =\sum_{j\in I}\left\langle g_{j},\,\phi\right\rangle _{\mathcal{S}',\mathcal{S}}
\end{align*}
converges absolutely and defines a continuous functional on $Z\left(\mathcal{O}\right)$.
\medskip{}
Next, we want to show existence of the coefficient map $C^{\left(\delta\right)}$,
for $0<\delta\leq\min\left\{ 1,\delta_{0}\right\} $. To this end,
first note that Theorem \ref{thm:BandlimitedWienerAmalgamSelfImproving}
shows for
\[
C_{1}:=\begin{cases}
1, & \text{if }p\geq1,\\
2^{4\left(1+\frac{d}{p}\right)}s_{d}^{\frac{1}{p}}\left(192\cdotd^{3/2}\cdot\left\lceil K+\frac{d+1}{p}\right\rceil \right)^{\left\lceil K+\frac{d+1}{p}\right\rceil +1}\cdot\Omega_{0}^{K}\Omega_{1}\cdot\left(1+R_{\mathcal{Q}}\right)^{\frac{d}{p}}, & \text{if }p<1
\end{cases}
\]
that
\[
\left\Vert \mathcal{F}^{-1}\left(\varphi_{j}\cdot\widehat{f}\right)\right\Vert _{V_{j}}\leq C_{1}\cdot\left\Vert \mathcal{F}^{-1}\left(\varphi_{j}\cdot\widehat{f}\right)\right\Vert _{L_{v}^{p}}\qquad\forall j\in I\qquad\forall f\in Z'\left(\mathcal{O}\right),
\]
since we have $\operatorname{supp}\left(\varphi_{j}\cdot\widehat{f}\right)\subset\overline{Q_{j}}\subset T_{j}\left[\overline{B_{R_{\mathcal{Q}}}}\left(0\right)\right]+b_{j}\subset T_{j}\left[-R_{\mathcal{Q}},R_{\mathcal{Q}}\right]^{d}+b_{j}$.
This easily shows that the map
\[
{\rm Ana}_{\varphi}:\DecompSp{\mathcal{Q}}p{\ell_{w}^{q}}v\to\ell_{w}^{q}\left(\left[V_{j}\right]_{j\in I}\right),f\mapsto\left(\mathcal{F}^{-1}\left[\varphi_{j}\cdot\widehat{f}\right]\right)_{j\in I}
\]
is well-defined and bounded, with $\vertiii{{\rm Ana}_{\varphi}}\leq C_{1}$.
Furthermore, Lemma \ref{lem:LocalInverseConvolution} shows that the
map
\[
m_{\theta}:\ell_{w}^{q}\left(\left[V_{j}\right]_{j\in I}\right)\to\ell_{w}^{q}\left(\left[V_{j}\right]_{j\in I}\right),\left(f_{j}\right)_{j\in I}\mapsto\left[\left(\mathcal{F}^{-1}\theta_{j}\right)\ast f_{j}\right]_{j\in I}\;\overset{\text{Lem. }\ref{lem:SpecialConvolutionConsistent}}{=}\;\left[\mathcal{F}^{-1}\left(\theta_{j}\cdot\widehat{f_{j}}\right)\right]_{j\in I}
\]
is well-defined and bounded, with $\vertiii{m_{\theta}}\leq C_{2}<\infty$
for
\[
C_{2}:=\begin{cases}
\Omega_{0}^{4K}\Omega_{1}^{4}\Omega_{2}^{\left(p,K\right)}\cdotd^{-\frac{d}{2p}}\cdot\left(972\cdotd^{5/2}\right)^{K+\frac{d}{p}}, & \text{if }p\in\left(0,1\right),\\
\Omega_{0}^{K}\Omega_{1}\Omega_{2}^{\left(p,K\right)}, & \text{if }p\in\left[1,\infty\right].
\end{cases}
\]
Next, it follows from Lemma \ref{lem:GammaSynthesisBounded} that
the map
\[
{\rm Synth}_{\Gamma_{1}}:\ell_{w}^{q}\left(\left[V_{j}\right]_{j\in I}\right)\to\DecompSp{\mathcal{Q}}p{\ell_{w}^{q}}v,\left(g_{j}\right)_{j\in I}\mapsto\sum_{j\in I}\gamma_{1}^{\left(j\right)}\ast g_{j}=\sum_{j\in I}\mathcal{F}^{-1}\left(\widehat{\gamma_{1}^{\left(j\right)}}\cdot\widehat{g_{j}}\right)
\]
is well-defined and bounded with $\vertiii{{\rm Synth}_{\Gamma_{1}}}\leq C_{3}\cdot\vertiii{\smash{\overrightarrow{C}}}^{\max\left\{ 1,1/p\right\} }$,
with
\[
C_{3}:=\begin{cases}
1, & \text{if }p\geq1\\
\frac{\left(2^{6}/\sqrt{d}\right)^{\frac{d}{p}}}{2^{21}\cdotd^{7}}\!\cdot\!\left(2^{21}\!\cdot\!d^{5}\!\cdot\!\left\lceil K\!+\!\frac{d+1}{p}\right\rceil \right)^{\!\left\lceil K+\frac{d+1}{p}\right\rceil +1}\!\!\!\cdot\!\left(1\!+\!R_{\mathcal{Q}}\right)^{\frac{d}{p}}\!\cdot\!\Omega_{0}^{5K}\Omega_{1}^{5}, & \text{if }p<1.
\end{cases}
\]
Finally, we will show below that the map
\[
m_{\Gamma_{2}}:\ell_{w}^{q}\left(\left[V_{j}\right]_{j\in I}\right)\to\ell_{w}^{q}\left(\left[V_{j}\right]_{j\in I}\right),\left(f_{j}\right)_{j\in I}\mapsto\left(\gamma_{2}^{\left(j\right)}\ast f_{j}\right)_{j\in I}
\]
is also well-defined and bounded. Once this is shown, note that we
have
\begin{align}
\left({\rm Synth}_{\Gamma_{1}}\circ m_{\Gamma_{2}}\circ m_{\theta}\circ{\rm Ana}_{\varphi}\right)f & =\sum_{j\in I}\left(\gamma_{1}^{\left(j\right)}\ast\gamma_{2}^{\left(j\right)}\ast\mathcal{F}^{-1}\theta_{j}\ast\mathcal{F}^{-1}\left(\varphi_{j}\cdot\widehat{f}\right)\right)\nonumber \\
\left({\scriptstyle \text{Lemma }\ref{lem:SpecialConvolutionConsistent}}\right) & =\sum_{j\in I}\mathcal{F}^{-1}\left(\widehat{\gamma_{1}^{\left(j\right)}}\cdot\widehat{\gamma_{2}^{\left(j\right)}}\cdot\theta_{j}\cdot\varphi_{j}\cdot\widehat{f}\right)\nonumber \\
\left({\scriptstyle \text{easy consequence of }\gamma_{j}=\gamma_{j,1}\ast\gamma_{j,2}}\right) & =\sum_{j\in I}\mathcal{F}^{-1}\left(\widehat{\gamma^{\left(j\right)}}\cdot\theta_{j}\cdot\varphi_{j}\cdot\widehat{f}\right)\nonumber \\
\left({\scriptstyle \text{since }\widehat{\gamma^{\left(j\right)}}\cdot\theta_{j}\equiv1\text{ on }\overline{Q_{j}}\supset\operatorname{supp}\varphi_{j}}\right) & =\sum_{j\in I}\mathcal{F}^{-1}\left(\varphi_{j}\cdot\widehat{f}\right)\nonumber \\
\left({\scriptstyle \text{since }\widehat{f}\in\DistributionSpace{\mathcal{O}}\text{ and }\left(\varphi_{j}\right)_{j\in I}\text{ is locally finite part. of unity on }\mathcal{O}}\right) & =f\label{eq:AtomicDecompositionReproducingFormula}
\end{align}
for all $f\in\DecompSp{\mathcal{Q}}p{\ell_{w}^{q}}v$. Thus, our goal in
the remainder of the proof—once we have shown boundedness of $m_{\Gamma_{2}}$—will
be to discretize this \textbf{reproducing formula}.
But first of all, let us verify boundedness of $m_{\Gamma_{2}}$.
To this end, it suffices to show that each map
\[
J_{j}:V_{j}\to V_{j},f\mapsto\gamma_{2}^{\left(j\right)}\ast f
\]
is bounded, with $\sup_{j\in I}\vertiii{J_{j}}<\infty$. But for $p\in\left[1,\infty\right]$,
this simply follows from the weighted Young inequality (equation (\ref{eq:WeightedYoungInequality})),
since in this case, we have $K_{0}=K+d+1$ and thus (cf.\@
equation (\ref{eq:AtomicDecompositionFamilyDefinition}))
\begin{align}
\left\Vert \gamma_{2}^{\left(j\right)}\right\Vert _{L_{v_{0}}^{1}} & =\left\Vert v_{0}\cdot\left|\det T_{j}\right|\cdot M_{b_{j}}\left[\gamma_{j,2}\circ T_{j}^{T}\right]\right\Vert _{L^{1}}\nonumber \\
& =\left\Vert \left(v_{0}\circ T_{j}^{-T}\right)\cdot\gamma_{j,2}\right\Vert _{L^{1}}\nonumber \\
\left({\scriptstyle \text{assump. on }v_{0}\text{ and eq. }\eqref{eq:WeightLinearTransformationsConnection}}\right) & \leq\Omega_{0}^{K}\Omega_{1}\cdot\left\Vert \left(1+\left|\bullet\right|\right)^{K}\cdot\gamma_{j,2}\right\Vert _{L^{1}}\nonumber \\
& \leq\Omega_{0}^{K}\Omega_{1}\cdot\left\Vert \gamma_{j,2}\right\Vert _{K_{0}}\cdot\left\Vert \left(1+\left|\bullet\right|\right)^{K-K_{0}}\right\Vert _{L^{1}}\nonumber \\
\left({\scriptstyle \text{eq. }\eqref{eq:StandardDecayLpEstimate}}\right) & \leq\Omega_{0}^{K}\Omega_{1}\Omega_{4}^{\left(p,K,1\right)}\cdot s_{d}<\infty.\label{eq:AtomicDecompositionGamma2WeightedL1Norm}
\end{align}
Here, we defined $\Omega_{4}^{\left(p,K,1\right)}:=\sup_{j\in I}\left\Vert \gamma_{j,2}\right\Vert _{K_{0}}$
in the last step. Note that $\Omega_{4}^{\left(p,K,1\right)}\leq\Omega_{4}^{\left(p,K\right)}$,
cf.\@ Assumption \ref{assu:AtomicDecompositionAssumption}, equation
(\ref{eq:AtomicDecompositionGamma2ConstantDefinition}).
Likewise, for $p\in\left(0,1\right)$, we can simply use Corollary
\ref{cor:WienerAmalgamConvolutionSimplified} to derive for $C_{4}:=\Omega_{0}^{3K}\Omega_{1}^{3}d^{-\frac{d}{2p}}\cdot\left(972\cdotd^{5/2}\right)^{K+\frac{d}{p}}$
that
\begin{align*}
\left\Vert \gamma_{2}^{\left(j\right)}\ast f\right\Vert _{V_{j}} & =\left\Vert \gamma_{2}^{\left(j\right)}\ast f\right\Vert _{W_{T_{j}^{-T}\left[-1,1\right]^{d}}\left(L_{v}^{p}\right)}\\
\left({\scriptstyle \text{Cor. }\ref{cor:WienerAmalgamConvolutionSimplified}}\right) & \leq C_{4}\cdot\left|\det T_{j}\right|^{\frac{1}{p}-1}\cdot\left\Vert \gamma_{2}^{\left(j\right)}\right\Vert _{W_{T_{j}^{-T}\left[-1,1\right]^{d}}\left(L_{v_{0}}^{p}\right)}\cdot\left\Vert f\right\Vert _{W_{T_{j}^{-T}\left[-1,1\right]^{d}}\left(L_{v}^{p}\right)}\\
\left({\scriptstyle \text{eq. }\eqref{eq:AtomicDecompositionFamilyDefinition}}\right) & =C_{4}\cdot\left|\det T_{j}\right|^{\frac{1}{p}}\cdot\left\Vert M_{b_{j}}\left[\gamma_{j,2}\circ T_{j}^{T}\right]\right\Vert _{W_{T_{j}^{-T}\left[-1,1\right]^{d}}\left(L_{v_{0}}^{p}\right)}\cdot\left\Vert f\right\Vert _{V_{j}}\\
& =C_{4}\cdot\left|\det T_{j}\right|^{\frac{1}{p}}\cdot\left\Vert v_{0}\cdot M_{T_{j}^{-T}\left[-1,1\right]^{d}}\left[\gamma_{j,2}\circ T_{j}^{T}\right]\right\Vert _{L^{p}}\cdot\left\Vert f\right\Vert _{V_{j}}\\
\left({\scriptstyle \text{Lemma }\ref{lem:WienerTransformationFormula}}\right) & =C_{4}\cdot\left|\det T_{j}\right|^{\frac{1}{p}}\cdot\left\Vert v_{0}\cdot\left(\left[M_{\left[-1,1\right]^{d}}\gamma_{j,2}\right]\circ T_{j}^{T}\right)\right\Vert _{L^{p}}\cdot\left\Vert f\right\Vert _{V_{j}}\\
& =C_{4}\cdot\left\Vert \left(v_{0}\circ T_{j}^{-T}\right)\cdot M_{\left[-1,1\right]^{d}}\gamma_{j,2}\right\Vert _{L^{p}}\cdot\left\Vert f\right\Vert _{V_{j}}\\
\left({\scriptstyle \text{assump. on }v_{0}\text{ and eq. }\eqref{eq:WeightLinearTransformationsConnection}}\right) & \leq\Omega_{0}^{K}\Omega_{1}\cdot C_{4}\cdot\left\Vert \left(1+\left|\bullet\right|\right)^{K}\cdot M_{\left[-1,1\right]^{d}}\gamma_{j,2}\right\Vert _{L^{p}}\cdot\left\Vert f\right\Vert _{V_{j}}\\
\left({\scriptstyle \text{Lemma }\ref{lem:SchwartzFunctionsAreWiener}}\right) & \leq\Omega_{0}^{K}\Omega_{1}\cdot C_{4}\cdot\left(1+2\sqrt{d}\right)^{K_{0}}\cdot\left\Vert \gamma_{j,2}\right\Vert _{K_{0}}\cdot\left\Vert \left(1+\left|\bullet\right|\right)^{K-K_{0}}\right\Vert _{L^{p}}\cdot\left\Vert f\right\Vert _{V_{j}}\\
\left({\scriptstyle \text{eq. }\eqref{eq:StandardDecayLpEstimate}\text{ and }K-K_{0}=-\left(\frac{d}{p}+1\right)}\right) & \leq\Omega_{0}^{K}\Omega_{1}\Omega_{4}^{\left(p,K,1\right)}\cdot C_{4}\cdot\left(1+2\sqrt{d}\right)^{K_{0}}\cdot\left(\frac{s_{d}}{p}\right)^{1/p}\cdot\left\Vert f\right\Vert _{V_{j}}.
\end{align*}
Here, we used the same definition of $\Omega_{4}^{\left(p,K,1\right)}$
as above. We have thus established boundedness of $m_{\Gamma_{2}}$
in all cases.
\medskip{}
In order to discretize the reproducing formula from equation (\ref{eq:AtomicDecompositionReproducingFormula}),
we define for $\delta\in\left(0,1\right]$ the map
\begin{align*}
D^{\left(\delta\right)}:\DecompSp{\mathcal{Q}}p{\ell_{w}^{q}}v & \to\ell_{\left(\left|\det T_{j}\right|^{\frac{1}{2}-\frac{1}{p}}w_{j}\right)_{\!\!j\in I}}^{q}\!\!\!\!\!\!\!\!\left(\left[\vphantom{F}\smash{C_{j}^{\left(\delta\right)}}\right]_{j\in I}\right),\\
f & \mapsto\left[\left(\delta^{d}\cdot\left|\det T_{j}\right|^{-1/2}\cdot\left[\mathcal{F}^{-1}\!\left(\theta_{j}\varphi_{j}\cdot\widehat{f}\right)\right]\!\!\left(\delta\cdot T_{j}^{-T}k\right)\right)_{k\in\mathbb{Z}^{d}}\right]_{j\in I}.
\end{align*}
This map is indeed well-defined and bounded, since Lemma \ref{lem:BandlimitedSampling}
yields for
\[
C_{5}:=2^{\max\left\{ 1,\frac{1}{p}\right\} }\cdot\Omega_{0}^{3K}\Omega_{1}^{3}\cdot\left(1+\sqrt{d}\right)^{K}\cdot\left(23040\cdotd^{3/2}\cdot\left(K+1+\frac{d+1}{\min\left\{ 1,p\right\} }\right)\right)^{K+2+\frac{d+1}{\min\left\{ 1,p\right\} }}\cdot\left(1+R_{\mathcal{Q}}\right)^{1+\frac{d}{\min\left\{ 1,p\right\} }}
\]
that
\begin{align*}
\left\Vert D^{\left(\delta\right)}f\right\Vert _{\ell_{\left(\left|\det T_{j}\right|^{\frac{1}{2}-\frac{1}{p}}w_{j}\right)_{j\in I}}^{q}\!\!\!\!\!\!\!\!\left(\vphantom{F}\smash{C_{j}^{\left(\delta\right)}}\right)} & =\delta^{d}\cdot\left\Vert \left(\left|\det T_{j}\right|^{-1/p}\cdot\left\Vert \left(\left[\mathcal{F}^{-1}\!\left(\theta_{j}\varphi_{j}\cdot\widehat{f}\right)\right]\!\left(\delta\cdot T_{j}^{-T}k\right)\right)_{k\in\mathbb{Z}^{d}}\right\Vert _{C_{j}^{\left(\delta\right)}}\right)_{j\in I}\right\Vert _{\ell_{w}^{q}\left(I\right)}\\
\left({\scriptstyle \text{since }\operatorname{supp}\left(\theta_{j}\varphi_{j}\cdot\widehat{f}\right)\subset\overline{Q_{j}}\subset T_{j}\left[-R_{\mathcal{Q}},R_{\mathcal{Q}}\right]^{d}+b_{j}}\right) & \leq C_{5}\cdot\delta^{d\left(1-\frac{1}{p}\right)}\cdot\left\Vert \left(\left\Vert \mathcal{F}^{-1}\left(\theta_{j}\varphi_{j}\cdot\widehat{f}\right)\right\Vert _{L_{v}^{p}}\right)_{j\in I}\right\Vert _{\ell_{w}^{q}\left(I\right)}\\
& \leq C_{5}\cdot\delta^{d\left(1-\frac{1}{p}\right)}\cdot\left\Vert \left(\left\Vert \mathcal{F}^{-1}\left(\theta_{j}\varphi_{j}\cdot\widehat{f}\right)\right\Vert _{V_{j}}\right)_{j\in I}\right\Vert _{\ell_{w}^{q}\left(I\right)}\\
& =C_{5}\cdot\delta^{d\left(1-\frac{1}{p}\right)}\cdot\left\Vert \left(m_{\theta}\circ{\rm Ana}_{\varphi}\right)f\right\Vert _{\ell_{w}^{q}\left(\left[V_{j}\right]_{j\in I}\right)}\\
& \leq C_{5}\cdot\vertiii{m_{\theta}}\cdot\vertiii{{\rm Ana}_{\varphi}}\cdot\delta^{d\left(1-\frac{1}{p}\right)}\cdot\left\Vert f\right\Vert _{\DecompSp{\mathcal{Q}}p{\ell_{w}^{q}}v}<\infty.
\end{align*}
Now, our goal is to show for
\[
E^{\left(\delta\right)}:=\left(m_{\Gamma_{2}}\circ m_{\theta}\circ{\rm Ana}_{\varphi}\right)-\left(\left[\smash{\bigotimes_{j\in I}}\vphantom{\sum_{i}}S_{\Gamma_{2}}^{\left(\delta,j\right)}\right]\circ D^{\left(\delta\right)}\right):\DecompSp{\mathcal{Q}}p{\ell_{w}^{q}}v\to\ell_{w}^{q}\left(\left[V_{j}\right]_{j\in I}\right)
\]
that we have $\vertiii{{\rm Synth}_{\Gamma_{1}}}\cdot\vertiii{E^{\left(\delta\right)}}\leq\frac{1}{2}$
for all $0<\delta\leq\min\left\{ 1,\delta_{0}\right\} $.
To this end, let $f\in\DecompSp{\mathcal{Q}}p{\ell_{w}^{q}}v$ be arbitrary
and for brevity, let
\[
f_{j}:=\mathcal{F}^{-1}\left(\theta_{j}\varphi_{j}\widehat{f}\right)=\left[\left(m_{\theta}\circ{\rm Ana}_{\varphi}\right)f\right]_{j}\in V_{j},
\]
as well as $f_{j}^{\left(2\right)}:=M_{-b_{j}}f_{j}$ and $\gamma_{2}^{\left(j,2\right)}:=M_{-b_{j}}\gamma_{2}^{\left(j\right)}=\left|\det T_{j}\right|\cdot\gamma_{j,2}\circ T_{j}^{T}$.
Note that since $f_{j}\in V_{j}$ is bandlimited with $\operatorname{supp}\widehat{f_{j}}\subset\overline{Q_{j}}\subset T_{j}\left[-R_{\mathcal{Q}},R_{\mathcal{Q}}\right]^{d}+b_{j}$,
Theorem \ref{thm:BandlimitedWienerAmalgamSelfImproving} and equation
(\ref{eq:WeightedWienerAmalgamTemperedDistribution}) yield $f_{j}\in W_{T_{j}^{-T}\left[-1,1\right]^{d}}\left(L_{v}^{p}\right)\hookrightarrow L_{v}^{\infty}\left(\mathbb{R}^{d}\right)$.
Since $\gamma_{2}^{\left(j\right)}\in L_{v_{0}}^{1}\left(\mathbb{R}^{d}\right)$,
this implies that the integral defining $\left(\gamma_{2}^{\left(j\right)}\ast f_{j}\right)\left(x\right)$
exists for every $x\in\mathbb{R}^{d}$, cf.\@ equation (\ref{eq:WeightedLInftyConvolution}).
Hence, using our newly introduced notation, we have
\begin{align*}
& \left|\left[E^{\left(\delta\right)}f\right]_{j}\left(x\right)\right|\\
& =\left|\left[\gamma_{2}^{\left(j\right)}\ast\mathcal{F}^{-1}\left(\theta_{j}\varphi_{j}\cdot\widehat{f}\right)\right]\left(x\right)-\left|\det T_{j}\right|^{-\frac{1}{2}}\sum_{k\in\mathbb{Z}^{d}}\delta^{d}\left|\det T_{j}\right|^{-\frac{1}{2}}\cdot\left[\mathcal{F}^{-1}\!\left(\theta_{j}\varphi_{j}\cdot\widehat{f}\right)\right]\!\left(\delta\cdot T_{j}^{-T}k\right)\cdot\left(L_{\delta\cdot T_{j}^{-T}k}\gamma_{2}^{\left(j\right)}\right)\left(x\right)\right|\\
& =\left|\sum_{k\in\mathbb{Z}^{d}}\left[\int_{\delta T_{j}^{-T}\!\left(k+\left[0,1\right)^{d}\right)}\gamma_{2}^{\left(j\right)}\left(x-y\right)\cdot f_{j}\left(y\right)\operatorname{d} y-\delta^{d}\left|\det T_{j}^{-T}\right|\cdot f_{j}\left(\delta\cdot T_{j}^{-T}k\right)\cdot\gamma_{2}^{\left(j\right)}\left(x-\delta\cdot T_{j}^{-T}k\right)\right]\right|\\
& \leq\sum_{k\in\mathbb{Z}^{d}}\int_{\delta T_{j}^{-T}\!\left(k+\left[0,1\right)^{d}\right)}\left|\gamma_{2}^{\left(j\right)}\left(x-y\right)\cdot f_{j}\left(y\right)-f_{j}\left(\delta\cdot T_{j}^{-T}k\right)\cdot\gamma_{2}^{\left(j\right)}\left(x-\delta\cdot T_{j}^{-T}k\right)\right|\operatorname{d} y\\
& \overset{\left(\ast\right)}{=}\sum_{k\in\mathbb{Z}^{d}}\int_{\delta T_{j}^{-T}\!\left(k+\left[0,1\right)^{d}\right)}\left|\gamma_{2}^{\left(j,2\right)}\left(x-y\right)\cdot f_{j}^{\left(2\right)}\left(y\right)-\gamma_{2}^{\left(j,2\right)}\left(x-\delta\cdot T_{j}^{-T}k\right)f_{j}^{\left(2\right)}\left(\delta\cdot T_{j}^{-T}k\right)\right|\operatorname{d} y\\
& \leq\!\sum_{k\in\mathbb{Z}^{d}}\int_{\delta T_{j}^{-T}\!\left(k+\left[0,1\right)^{d}\right)}\!\left|\gamma_{2}^{\left(j,2\right)}\!\left(x\!-\!y\right)\left[f_{j}^{\left(2\right)}\!\left(y\right)-f_{j}^{\left(2\right)}\!\left(\delta T_{j}^{-T}\!k\right)\right]\right|+\left|f_{j}^{\left(2\right)}\!\left(\delta T_{j}^{-T}\!k\right)\left[\gamma_{2}^{\left(j,2\right)}\!\left(x\!-\!y\right)-\gamma_{2}^{\left(j,2\right)}\!\left(x-\delta T_{j}^{-T}\!k\right)\right]\right|\operatorname{d} y.
\end{align*}
In this calculation, we used at $\left(\ast\right)$ the easily verifiable
identity $\left(M_{b}f\right)\left(x-y\right)\cdot\left(M_{b}g\right)\left(y\right)=e^{2\pi i\left\langle b,x\right\rangle }\cdot f\left(x-y\right)g\left(y\right)$.
Next, note for arbitrary $y\in\delta T_{j}^{-T}\left(k+\left[0,1\right)^{d}\right)$
that $y=\delta T_{j}^{-T}k+\delta T_{j}^{-T}u$ for some $u\in\left[-1,1\right]^{d}$.
This implies $\delta T_{j}^{-T}k=y-\delta T_{j}^{-T}u\in y+\delta T_{j}^{-T}\left[-1,1\right]^{d}$
and hence
\[
\left|f_{j}^{\left(2\right)}\left(y\right)-f_{j}^{\left(2\right)}\left(\delta\cdot T_{j}^{-T}k\right)\right|\leq\left(\osc{\delta T_{j}^{-T}\left[-1,1\right]^{d}}f_{j}^{\left(2\right)}\right)\left(y\right).
\]
Likewise, we have $x-\delta T_{j}^{-T}k=x-\left(y-\delta T_{j}^{-T}u\right)\in x-y+\delta T_{j}^{-T}\left[-1,1\right]^{d}$,
which yields
\[
\left|\gamma_{2}^{\left(j,2\right)}\left(x-y\right)-\gamma_{2}^{\left(j,2\right)}\left(x-\delta\cdot T_{j}^{-T}k\right)\right|\leq\left(\osc{\delta T_{j}^{-T}\left[-1,1\right]^{d}}\gamma_{2}^{\left(j,2\right)}\right)\left(x-y\right).
\]
Finally, we also have
\begin{align*}
\left|f_{j}^{\left(2\right)}\left(\delta\cdot T_{j}^{-T}k\right)\right| & \leq\left|f_{j}^{\left(2\right)}\left(\delta\cdot T_{j}^{-T}k\right)-f_{j}^{\left(2\right)}\left(y\right)\right|+\left|f_{j}^{\left(2\right)}\left(y\right)\right|\\
& \leq\left|f_{j}^{\left(2\right)}\left(y\right)\right|+\left(\osc{\delta T_{j}^{-T}\left[-1,1\right]^{d}}f_{j}^{\left(2\right)}\right)\left(y\right)\\
& =:e_{j}\left(y\right),
\end{align*}
so that we see
\begin{align}
& \left|\left[E^{\left(\delta\right)}f\right]_{j}\left(x\right)\right|\nonumber \\
& \leq\sum_{k\in\mathbb{Z}^{d}}\int_{\delta T_{j}^{-T}\left(k+\left[0,1\right)^{d}\right)}\left|\gamma_{2}^{\left(j,2\right)}\left(x-y\right)\right|\cdot\left[\osc{\delta T_{j}^{-T}\left[-1,1\right]^{d}}f_{j}^{\left(2\right)}\right]\left(y\right)+e_{j}\left(y\right)\cdot\left[\osc{\delta T_{j}^{-T}\left[-1,1\right]^{d}}\gamma_{2}^{\left(j,2\right)}\right]\left(x-y\right)\operatorname{d} y\nonumber \\
& =\left(\left|\gamma_{2}^{\left(j,2\right)}\right|\ast\left[\osc{\delta T_{j}^{-T}\left[-1,1\right]^{d}}f_{j}^{\left(2\right)}\right]\right)\left(x\right)+\left(e_{j}\ast\left[\osc{\delta T_{j}^{-T}\left[-1,1\right]^{d}}\gamma_{2}^{\left(j,2\right)}\right]\right)\left(x\right).\label{eq:AtomicDecompositionEDeltaPointwiseEstimate}
\end{align}
We now distinguish two cases: For $p\in\left[1,\infty\right]$, first
note $\left|\smash{\gamma_{2}^{\left(j,2\right)}}\right|=\left|\smash{\gamma_{2}^{\left(j\right)}}\right|$
and hence, thanks to equation (\ref{eq:AtomicDecompositionGamma2WeightedL1Norm}),
$\left\Vert \smash{\gamma_{2}^{\left(j,2\right)}}\right\Vert _{L_{v_{0}}^{1}}=\left\Vert \smash{\gamma_{2}^{\left(j\right)}}\right\Vert _{L_{v_{0}}^{1}}\leq\Omega_{0}^{K}\Omega_{1}\Omega_{4}^{\left(p,K,1\right)}\cdot s_{d}=:C_{6}$.
Hence, we get using the triangle inequality, the weighted Young inequality
(equation (\ref{eq:WeightedYoungInequality})), the definition of
$e_{j}$ and since $K_{0}=K+d+1$ that
\begin{align*}
& \left\Vert \left[E^{\left(\delta\right)}f\right]_{j}\right\Vert _{V_{j}}=\left\Vert \left[E^{\left(\delta\right)}f\right]_{j}\right\Vert _{L_{v}^{p}}\\
\left({\scriptstyle \text{def. of }e_{j}}\right) & \leq\left\Vert \gamma_{2}^{\left(j,2\right)}\right\Vert _{L_{v_{0}}^{1}}\!\cdot\left\Vert \osc{\delta T_{j}^{-T}\left[-1,1\right]^{d}}f_{j}^{\left(2\right)}\right\Vert _{L_{v}^{p}}+\left\Vert \osc{\delta T_{j}^{-T}\left[-1,1\right]^{d}}\gamma_{2}^{\left(j,2\right)}\right\Vert _{L_{v_{0}}^{1}}\!\cdot\left(\left\Vert f_{j}^{\left(2\right)}\right\Vert _{L_{v}^{p}}\!+\left\Vert \osc{\delta T_{j}^{-T}\left[-1,1\right]^{d}}f_{j}^{\left(2\right)}\right\Vert _{L_{v}^{p}}\right)\\
& \leq C_{6}\!\left\Vert \osc{\delta T_{j}^{-T}\left[-1,1\right]^{d}}\!\left[M_{-b_{j}}f_{j}\right]\right\Vert _{L_{v}^{p}}\!\!\!+\!\left\Vert \osc{\delta T_{j}^{-T}\left[-1,1\right]^{d}}\!\left[M_{-b_{j}}\gamma_{2}^{\left(j\right)}\right]\right\Vert _{L_{v_{0}}^{1}}\!\!\left(\!\left\Vert f_{j}\right\Vert _{L_{v}^{p}}\!+\!\left\Vert \osc{\delta T_{j}^{-T}\left[-1,1\right]^{d}}\!\left[M_{-b_{j}}f_{j}\right]\right\Vert _{L_{v}^{p}}\right)\\
\left({\scriptstyle \text{Thm. }\ref{thm:BandlimitedOscillationSelfImproving}}\right) & \leq C_{6}C_{7}\cdot\delta\cdot\left\Vert f_{j}\right\Vert _{L_{v}^{p}}+\left\Vert \osc{\delta T_{j}^{-T}\left[-1,1\right]^{d}}\left[M_{-b_{j}}\gamma_{2}^{\left(j\right)}\right]\right\Vert _{L_{v_{0}}^{1}}\cdot\left(\left\Vert f_{j}\right\Vert _{L_{v}^{p}}+C_{7}\cdot\delta\cdot\left\Vert f_{j}\right\Vert _{L_{v}^{p}}\right)\\
\left({\scriptstyle \text{eq. }\eqref{eq:AtomicDecompositionFamilyDefinition}}\right) & =C_{6}C_{7}\cdot\delta\cdot\left\Vert f_{j}\right\Vert _{L_{v}^{p}}+\left|\det T_{j}\right|\cdot\left\Vert \osc{\delta T_{j}^{-T}\left[-1,1\right]^{d}}\left[\gamma_{j,2}\circ T_{j}^{T}\right]\right\Vert _{L_{v_{0}}^{1}}\cdot\left(\left\Vert f_{j}\right\Vert _{L_{v}^{p}}+C_{7}\cdot\delta\cdot\left\Vert f_{j}\right\Vert _{L_{v}^{p}}\right)\\
\left({\scriptstyle \text{Lem. }\ref{lem:OscillationLinearChange}}\right) & =C_{6}C_{7}\cdot\delta\cdot\left\Vert f_{j}\right\Vert _{L_{v}^{p}}+\left|\det T_{j}\right|\cdot\left\Vert v_{0}\cdot\left(\left[\osc{\delta\left[-1,1\right]^{d}}\gamma_{j,2}\right]\circ T_{j}^{T}\right)\right\Vert _{L^{1}}\cdot\left(\left\Vert f_{j}\right\Vert _{L_{v}^{p}}+C_{7}\cdot\delta\cdot\left\Vert f_{j}\right\Vert _{L_{v}^{p}}\right)\\
& \overset{\left(\dagger\right)}{\leq}C_{6}C_{7}\cdot\delta\cdot\left\Vert f_{j}\right\Vert _{L_{v}^{p}}+\Omega_{0}^{K}\Omega_{1}\cdot\left\Vert \left(1+\left|\bullet\right|\right)^{K}\cdot\osc{\delta\left[-1,1\right]^{d}}\gamma_{j,2}\right\Vert _{L^{1}}\cdot\left(\left\Vert f_{j}\right\Vert _{L_{v}^{p}}+C_{7}\cdot\delta\cdot\left\Vert f_{j}\right\Vert _{L_{v}^{p}}\right)\\
\left({\scriptstyle \text{Lem. }\ref{lem:OscillationSchwartzFunction}}\right) & \leq C_{6}C_{7}\cdot\delta\cdot\left\Vert f_{j}\right\Vert _{L_{v}^{p}}+\Omega_{0}^{K}\Omega_{1}\cdot\left(3\sqrt{d}\right)^{K_{0}+1}\cdot\delta\cdot\left\Vert \nabla\gamma_{j,2}\right\Vert _{K_{0}}\cdot\left\Vert \left(1+\left|\bullet\right|\right)^{K-K_{0}}\right\Vert _{L^{1}}\cdot\left(\left\Vert f_{j}\right\Vert _{L_{v}^{p}}+C_{7}\cdot\delta\cdot\left\Vert f_{j}\right\Vert _{L_{v}^{p}}\right)\\
\left({\scriptstyle \text{eq. }\eqref{eq:StandardDecayLpEstimate}}\right) & \leq C_{6}C_{7}\cdot\delta\cdot\left\Vert f_{j}\right\Vert _{L_{v}^{p}}+\Omega_{0}^{K}\Omega_{1}\cdot s_{d}\left(3\sqrt{d}\right)^{K_{0}+1}\cdot\delta\cdot\left\Vert \nabla\gamma_{j,2}\right\Vert _{K_{0}}\cdot\left(\left\Vert f_{j}\right\Vert _{L_{v}^{p}}+C_{7}\cdot\delta\cdot\left\Vert f_{j}\right\Vert _{L_{v}^{p}}\right)\\
\left({\scriptstyle \text{since }\delta\leq1}\right) & \leq\delta\cdot\left\Vert f_{j}\right\Vert _{L_{v}^{p}}\cdot\left(C_{6}C_{7}+\Omega_{0}^{K}\Omega_{1}\cdot s_{d}\left(3\sqrt{d}\right)^{K_{0}+1}\cdot\Omega_{4}^{\left(p,K,2\right)}\cdot\left(1+C_{7}\right)\right)\\
& =:C_{8}\cdot\delta\cdot\left\Vert f_{j}\right\Vert _{L_{v}^{p}}=C_{8}\cdot\delta\cdot\left\Vert f_{j}\right\Vert _{V_{j}}.
\end{align*}
Here, we defined $\Omega_{4}^{\left(p,K,2\right)}:=\sup_{j\in I}\left\Vert \nabla\gamma_{j,2}\right\Vert _{K_{0}}$,
which is finite thanks to equation (\ref{eq:AtomicDecompositionGamma2ConstantDefinition}).
The step marked with $\left(\dagger\right)$ used a simple change
of variables and our assumption $v_{0}\left(x\right)\leq\Omega_{1}\cdot\left(1+\left|x\right|\right)^{K}$
in combination with estimate (\ref{eq:WeightLinearTransformationsConnection}).
Furthermore, our application of Theorem \ref{thm:BandlimitedOscillationSelfImproving}
is justified, since we have $f_{j}=\mathcal{F}^{-1}\left(\theta_{j}\varphi_{j}\widehat{f}\right)$,
which implies $\operatorname{supp}\widehat{f_{j}}\subset\operatorname{supp}\varphi_{j}\subset\overline{Q_{j}}\subset T_{j}\left[-R_{\mathcal{Q}},R_{\mathcal{Q}}\right]^{d}+b_{j}$,
so that Theorem \ref{thm:BandlimitedOscillationSelfImproving} yields
\begin{equation}
\left\Vert \osc{\delta T_{j}^{-T}\left[-1,1\right]^{d}}\left[M_{-b_{j}}f_{j}\right]\right\Vert _{V_{j}}\leq C_{7}\cdot\delta\cdot\left\Vert f_{j}\right\Vert _{L_{v}^{p}}\label{eq:AtomicDecompositionOscillationEstimate}
\end{equation}
for
\[
C_{7}:=\Omega_{0}^{2K}\Omega_{1}^{2}\cdot\left(\!23040\!\cdot\!d^{\frac{3}{2}}\!\cdot\!\left(\!K\!+\!1\!+\!\frac{d+1}{\min\left\{ 1,p\right\} }\right)\right)^{\!\!K+2+\frac{d+1}{\min\left\{ 1,p\right\} }}\!\!\!\cdot\left(1\!+\!R_{\mathcal{Q}}\right)^{1+\frac{d}{\min\left\{ 1,p\right\} }}\!=\!\left[\!2^{\max\left\{ 1,\frac{1}{p}\right\} }\Omega_{0}^{K}\Omega_{1}\left(1\!+\!\sqrt{d}\right)^{\!K}\right]^{-1}\!\cdot C_{5}.
\]
In case of $p\in\left(0,1\right)$, we let $C_{9}:=2^{\frac{1}{p}-1}$,
so that $C_{9}$ is a triangle constant for $L^{p}\left(\mathbb{R}^{d}\right)$.
Furthermore, we set $V_{j}^{\natural}:=W_{T_{j}^{-T}\left[-1,1\right]^{d}}\left(L_{v_{0}}^{p}\right)$
for brevity. Then, we use Corollary \ref{cor:WienerAmalgamConvolutionSimplified}
to get for $C_{10}:=C_{9}\cdot\Omega_{0}^{3K}\Omega_{1}^{3}\cdotd^{-\frac{d}{2p}}\cdot\left(972\cdotd^{5/2}\right)^{K+\frac{d}{p}}$
that
\begin{align*}
& \left\Vert \left[E^{\left(\delta\right)}f\right]_{j}\right\Vert _{V_{j}}\\
\left({\scriptstyle \text{eq. }\eqref{eq:AtomicDecompositionEDeltaPointwiseEstimate}}\right) & \leq C_{9}\cdot\left[\left\Vert \left|\gamma_{2}^{\left(j,2\right)}\right|\ast\osc{\delta T_{j}^{-T}\left[-1,1\right]^{d}}f_{j}^{\left(2\right)}\right\Vert _{V_{j}}+\left\Vert e_{j}\ast\left[\osc{\delta T_{j}^{-T}\left[-1,1\right]^{d}}\gamma_{2}^{\left(j,2\right)}\right]\right\Vert _{V_{j}}\right]\\
\left({\scriptstyle \text{Cor. }\ref{cor:WienerAmalgamConvolutionSimplified}}\right) & \leq C_{10}\!\cdot\!\left|\det T_{j}\right|^{\frac{1}{p}-1}\left(\left\Vert \gamma_{2}^{\left(j,2\right)}\right\Vert _{V_{j}^{\natural}}\cdot\left\Vert \osc{\delta T_{j}^{-T}\left[-1,1\right]^{d}}f_{j}^{\left(2\right)}\right\Vert _{V_{j}}+\left\Vert e_{j}\right\Vert _{V_{j}}\cdot\left\Vert \osc{\delta T_{j}^{-T}\left[-1,1\right]^{d}}\gamma_{2}^{\left(j,2\right)}\right\Vert _{V_{j}^{\natural}}\right)\\
\left({\scriptstyle \text{eq. }\eqref{eq:AtomicDecompositionFamilyDefinition}\text{, def. of }e_{j}}\right) & \leq C_{10}\!\cdot\!\left|\det T_{j}\right|^{\frac{1}{p}}\cdot\left(\left\Vert \gamma_{j,2}\circ T_{j}^{T}\right\Vert _{V_{j}^{\natural}}\cdot\left\Vert \osc{\delta T_{j}^{-T}\left[-1,1\right]^{d}}\left[M_{-b_{j}}f_{j}\right]\right\Vert _{V_{j}}\right.\\
& \phantom{\leq C_{10}\cdot\left|\det T_{j}\right|^{\frac{1}{p}}\cdot\bigg(}\left.+C_{9}\!\left[\left\Vert f_{j}\right\Vert _{V_{j}}\!+\!\left\Vert \osc{\delta T_{j}^{-T}\left[-1,1\right]^{d}}\left[M_{-b_{j}}f_{j}\right]\right\Vert _{V_{j}}\right]\!\cdot\left\Vert \osc{\delta T_{j}^{-T}\left[-1,1\right]^{d}}\left[\gamma_{j,2}\!\circ\!T_{j}^{T}\right]\right\Vert _{V_{j}^{\natural}}\right)\\
\left({\scriptstyle \text{Lem. }\ref{lem:OscillationLinearChange}\text{, eq. }\eqref{eq:AtomicDecompositionOscillationEstimate}}\right) & \leq C_{10}\!\cdot\!\left|\det T_{j}\right|^{\frac{1}{p}}\cdot\left(\left\Vert v_{0}\cdot M_{T_{j}^{-T}\left[-1,1\right]^{d}}\left[\gamma_{j,2}\circ T_{j}^{T}\right]\right\Vert _{L^{p}}\cdot C_{7}\cdot\delta\cdot\left\Vert f_{j}\right\Vert _{V_{j}}\right.\\
& \phantom{\leq C_{10}\!\cdot\!\left|\det T_{j}\right|^{\frac{1}{p}}\cdot\bigg(}\left.+C_{9}\!\left[\left\Vert f_{j}\right\Vert _{V_{j}}\!+\!C_{7}\cdot\delta\cdot\left\Vert f_{j}\right\Vert _{V_{j}}\right]\cdot\left\Vert v_{0}\!\cdot\!M_{T_{j}^{-T}\left[-1,1\right]^{d}}\left[\!\left(\osc{\delta\left[-1,1\right]^{d}}\gamma_{j,2}\right)\!\circ\!T_{j}^{T}\right]\right\Vert _{L^{p}}\right)\\
\left({\scriptstyle \text{Lem. }\ref{lem:WienerTransformationFormula}}\right) & =C_{10}\cdot\left(\left\Vert \left(v_{0}\circ T_{j}^{-T}\right)\cdot M_{\left[-1,1\right]^{d}}\gamma_{j,2}\right\Vert _{L^{p}}\cdot C_{7}\cdot\delta\cdot\left\Vert f_{j}\right\Vert _{V_{j}}\right.\\
& \phantom{=C_{10}\cdot\bigg(}\left.+C_{9}\left[\left\Vert f_{j}\right\Vert _{V_{j}}+C_{7}\cdot\delta\cdot\left\Vert f_{j}\right\Vert _{V_{j}}\right]\cdot\left\Vert \left(v_{0}\circ T_{j}^{-T}\right)\cdot M_{\left[-1,1\right]^{d}}\left[\osc{\delta\left[-1,1\right]^{d}}\gamma_{j,2}\right]\right\Vert _{L^{p}}\right)\\
\left({\scriptstyle \text{since }\delta\leq1}\right) & \leq C_{10}\Omega_{0}^{K}\Omega_{1}\cdot\left(\left\Vert \left(1+\left|\bullet\right|\right)^{K}\cdot M_{\left[-1,1\right]^{d}}\gamma_{j,2}\right\Vert _{L^{p}}\cdot C_{7}\cdot\delta\cdot\left\Vert f_{j}\right\Vert _{V_{j}}\right.\\
& \phantom{=C_{10}\Omega_{0}^{K}\Omega_{1}\cdot\bigg(}\left.+C_{9}\left\Vert f_{j}\right\Vert _{V_{j}}\left(1+C_{7}\right)\cdot\left\Vert \left(1+\left|\bullet\right|\right)^{K}\cdot M_{\left[-1,1\right]^{d}}\left[\osc{\delta\left[-1,1\right]^{d}}\gamma_{j,2}\right]\right\Vert _{L^{p}}\right).
\end{align*}
Here, the last step used as usual our assumption $v_{0}\left(x\right)\leq\Omega_{1}\cdot\left(1+\left|x\right|\right)^{K}$,
in combination with equation (\ref{eq:WeightLinearTransformationsConnection}).
We now combine Lemmas \ref{lem:OscillationSchwartzFunction} and
\ref{lem:SchwartzFunctionsAreWiener} to obtain
\begin{align*}
\left\Vert \left(1+\left|\bullet\right|\right)^{K}\cdot M_{\left[-1,1\right]^{d}}\left[\osc{\delta\left[-1,1\right]^{d}}\gamma_{j,2}\right]\right\Vert _{L^{p}} & \leq\left(3\sqrt{d}\right)^{K_{0}+1}\cdot\delta\cdot\left\Vert \nabla\gamma_{j,2}\right\Vert _{K_{0}}\cdot\left\Vert \left(1+\left|\bullet\right|\right)^{K}\cdot M_{\left[-1,1\right]^{d}}\left(1+\left|\bullet\right|\right)^{-K_{0}}\right\Vert _{L^{p}}\\
& \leq\left(1+2\sqrt{d}\right)^{K_{0}}\left(3\sqrt{d}\right)^{K_{0}+1}\cdot\delta\cdot\left\Vert \nabla\gamma_{j,2}\right\Vert _{K_{0}}\cdot\left\Vert \left(1+\left|\bullet\right|\right)^{K-K_{0}}\right\Vert _{L^{p}}\\
\left({\scriptstyle \text{eq. }\eqref{eq:StandardDecayLpEstimate}}\right) & \leq\left(3\sqrt{d}\right)^{2K_{0}+1}\Omega_{4}^{\left(p,K,2\right)}\cdot\delta\cdot\left(\frac{s_{d}}{p}\right)^{1/p}.
\end{align*}
Likewise, Lemma \ref{lem:SchwartzFunctionsAreWiener} also yields
\begin{align*}
\left\Vert \left(1+\left|\bullet\right|\right)^{K}\cdot M_{\left[-1,1\right]^{d}}\gamma_{j,2}\right\Vert _{L^{p}} & \leq\left\Vert \gamma_{j,2}\right\Vert _{K_{0}}\cdot\left\Vert \left(1+\left|\bullet\right|\right)^{K}\cdot M_{\left[-1,1\right]^{d}}\left(1+\left|\bullet\right|\right)^{-K_{0}}\right\Vert _{L^{p}}\\
& \leq\left\Vert \gamma_{j,2}\right\Vert _{K_{0}}\cdot\left(1+2\sqrt{d}\right)^{K_{0}}\cdot\left\Vert \left(1+\left|\bullet\right|\right)^{K-K_{0}}\right\Vert _{L^{p}}\\
& \leq\left(3\sqrt{d}\right)^{K_{0}}\Omega_{4}^{\left(p,K,1\right)}\cdot\left(\frac{s_{d}}{p}\right)^{1/p}.
\end{align*}
Combining these estimates with our estimate for $\left\Vert \left[E^{\left(\delta\right)}f\right]_{j}\right\Vert _{V_{j}}$,
we arrive at
\begin{align*}
\left\Vert \left[E^{\left(\delta\right)}f\right]_{j}\right\Vert _{V_{j}} & \leq C_{10}\left(\frac{s_{d}}{p}\right)^{1/p}\left(3\sqrt{d}\right)^{2K_{0}+1}\cdot\Omega_{0}^{K}\Omega_{1}\cdot\left(C_{7}\Omega_{4}^{\left(p,K,1\right)}+C_{9}\left(1+C_{7}\right)\Omega_{4}^{\left(p,K,2\right)}\right)\cdot\delta\cdot\left\Vert f_{j}\right\Vert _{V_{j}}\\
& =:C_{11}\cdot\delta\cdot\left\Vert f_{j}\right\Vert _{V_{j}},
\end{align*}
where $C_{11}$ is independent of $\delta$ and $j$.
\medskip{}
All in all, if we set $C_{12}:=C_{8}$ for $p\in\left[1,\infty\right]$
and $C_{12}:=C_{11}$ for $p\in\left(0,1\right)$, we have $\left\Vert \left[E^{\left(\delta\right)}f\right]_{j}\right\Vert _{V_{j}}\leq C_{12}\cdot\delta\cdot\left\Vert f_{j}\right\Vert _{V_{j}}$
for all $j\in I$ and $\delta\in\left(0,1\right]$. But this entails
\begin{align*}
\left\Vert E^{\left(\delta\right)}f\right\Vert _{\ell_{w}^{q}\left(\left[V_{j}\right]_{j\in I}\right)} & \leq C_{12}\cdot\delta\cdot\left\Vert \left(f_{j}\right)_{j\in I}\right\Vert _{\ell_{w}^{q}\left(\left[V_{j}\right]_{j\in I}\right)}\\
\left({\scriptstyle \text{since }f_{j}=\left[\left(m_{\theta}\circ{\rm Ana}_{\varphi}\right)f\right]_{j}}\right) & =C_{12}\cdot\delta\cdot\left\Vert \left(m_{\theta}\circ{\rm Ana}_{\varphi}\right)f\right\Vert _{\ell_{w}^{q}\left(\left[V_{j}\right]_{j\in I}\right)}\\
& \leq C_{12}\cdot\vertiii{m_{\theta}}\cdot\vertiii{{\rm Ana}_{\varphi}}\cdot\delta\cdot\left\Vert f\right\Vert _{\DecompSp{\mathcal{Q}}p{\ell_{w}^{q}}v}\\
& \leq C_{1}C_{2}C_{12}\cdot\delta\cdot\left\Vert f\right\Vert _{\DecompSp{\mathcal{Q}}p{\ell_{w}^{q}}v}.
\end{align*}
But since $\vertiii{{\rm Synth}_{\Gamma_{1}}}\leq C_{3}\cdot\vertiii{\smash{\overrightarrow{C}}}^{\max\left\{ 1,\frac{1}{p}\right\} }$,
this means in view of equation (\ref{eq:AtomicDecompositionReproducingFormula})
that
\begin{align*}
\vertiii{\operatorname{id}_{\DecompSp{\mathcal{Q}}p{\ell_{w}^{q}}v}-{\rm Synth}_{\Gamma_{1}}\circ\bigotimes_{j\in I}S_{\Gamma_{2}}^{\left(\delta,j\right)}\circ D^{\left(\delta\right)}} & =\vertiii{{\rm Synth}_{\Gamma_{1}}\circ m_{\Gamma_{2}}\circ m_{\theta}\circ{\rm Ana}_{\varphi}-{\rm Synth}_{\Gamma_{1}}\circ\bigotimes_{j\in I}S_{\Gamma_{2}}^{\left(\delta,j\right)}\circ D^{\left(\delta\right)}}\\
& \leq\vertiii{{\rm Synth}_{\Gamma_{1}}}\cdot\vertiii{\smash{E^{\left(\delta\right)}}}\\
& \leq C_{1}C_{2}C_{3}C_{12}\cdot\vertiii{\smash{\overrightarrow{C}}}^{\max\left\{ 1,\frac{1}{p}\right\} }\cdot\delta.
\end{align*}
\medskip{}
Now, we estimate the constant $C_{1}C_{2}C_{3}C_{12}$ to see that
$\delta\leq\delta_{0}$ implies $C_{1}C_{2}C_{3}C_{12}\cdot\vertiii{\smash{\overrightarrow{C}}}^{\max\left\{ 1,\frac{1}{p}\right\} }\cdot\delta\leq\frac{1}{2}$.
First, in case of $p\in\left[1,\infty\right]$, we have because of
$C_{7}\geq1$ and $\max\left\{ 1,\frac{1}{p}\right\} =1$, as well
as $K_{0}=K+d+1$ that
\begin{align*}
C_{1}C_{2}C_{3}C_{12} & =\Omega_{0}^{K}\Omega_{1}\Omega_{2}^{\left(p,K\right)}\cdot C_{8}\\
& =\left(C_{6}C_{7}+\Omega_{0}^{K}\Omega_{1}\cdot s_{d}\left(3\sqrt{d}\right)^{K_{0}+1}\cdot\Omega_{4}^{\left(p,K,2\right)}\cdot\left(1+C_{7}\right)\right)\cdot\Omega_{0}^{K}\Omega_{1}\Omega_{2}^{\left(p,K\right)}\\
\left({\scriptstyle \text{since }\Omega_{0},\Omega_{1}\geq1}\right) & \leq C_{7}s_{d}\left(\Omega_{4}^{\left(p,K,1\right)}+2\cdot\left(3\sqrt{d}\right)^{K_{0}+1}\Omega_{4}^{\left(p,K,2\right)}\right)\cdot\Omega_{0}^{2K}\Omega_{1}^{2}\Omega_{2}^{\left(p,K\right)}\\
& \leq\left(2^{17}\!\cdot\!d^{2}\!\cdot\!\left(K\!+\!2\!+\!d\right)\right)^{\!K+d+3}\!\!\!\cdot2s_{d}\cdot\left(3\!\cdot\!d^{\frac{1}{2}}\right)^{-1}\!\!\cdot\!\left(1\!+\!R_{\mathcal{Q}}\right)^{d+1}\!\cdot\!\left(\Omega_{4}^{\left(p,K,1\right)}\!+\!\Omega_{4}^{\left(p,K,2\right)}\right)\!\cdot\!\Omega_{0}^{4K}\Omega_{1}^{4}\Omega_{2}^{\left(p,K\right)}\\
& \leq\frac{s_{d}}{\sqrt{d}}\cdot\left(2^{17}\!\cdot\!d^{2}\!\cdot\!\left(K\!+\!2\!+\!d\right)\right)^{\!K+d+3}\!\!\!\cdot\left(1\!+\!R_{\mathcal{Q}}\right)^{d+1}\cdot\Omega_{0}^{4K}\Omega_{1}^{4}\Omega_{2}^{\left(p,K\right)}\Omega_{4}^{\left(p,K\right)}.
\end{align*}
Next, for $p\in\left(0,1\right)$, we get because of $\max\left\{ 1,\frac{1}{p}\right\} =\frac{1}{p}$
and $s_{d}\leq2^{2d}$ that
\begin{align*}
C_{1}C_{3} & =\frac{\left(2^{10}/d^{\frac{1}{2}}\right)^{\frac{d}{p}}}{2^{21}\cdotd^{7}}\cdot2^{4}s_{d}^{\frac{1}{p}}\left(192\!\cdotd^{\frac{3}{2}}\!\cdot\!\left\lceil K\!+\!\frac{d+1}{p}\right\rceil \right)^{\!\left\lceil K+\frac{d+1}{p}\right\rceil +1}\!\cdot\!\left(2^{21}\!\cdot\!d^{5}\!\cdot\!\left\lceil K\!+\!\frac{d+1}{p}\right\rceil \right)^{\!\left\lceil K+\frac{d+1}{p}\right\rceil +1}\!\!\!\cdot\!\left(1\!+\!R_{\mathcal{Q}}\right)^{\frac{2d}{p}}\!\cdot\!\Omega_{0}^{6K}\Omega_{1}^{6}\\
& \leq2^{4}\cdot\frac{\left(2^{12}/\sqrt{d}\right)^{\frac{d}{p}}}{2^{21}\cdotd^{7}}\cdot\left(2^{29}\cdotd^{\frac{13}{2}}\cdot\left\lceil K+\frac{d+1}{p}\right\rceil ^{2}\right)^{\left\lceil K+\frac{d+1}{p}\right\rceil +1}\!\!\!\cdot\!\left(1\!+\!R_{\mathcal{Q}}\right)^{\frac{2d}{p}}\!\cdot\!\Omega_{0}^{6K}\Omega_{1}^{6}
\end{align*}
and thus, since $\left\lceil K+\frac{d+1}{p}\right\rceil +1\geq K+\frac{d}{p}+2$,
\begin{align*}
C_{1}C_{2}C_{3} & \leqd^{-\frac{d}{2p}}\cdot\left(972\cdotd^{\frac{5}{2}}\right)^{-2}\!\cdot\!2^{4}\!\cdot\!\frac{\left(2^{12}/\sqrt{d}\right)^{\frac{d}{p}}}{2^{21}\cdotd^{7}}\!\cdot\!\left(2^{39}\!\cdot\!d^{9}\!\cdot\!\left\lceil K\!+\!\frac{d+1}{p}\right\rceil ^{2}\right)^{\!\left\lceil K+\frac{d+1}{p}\right\rceil +1}\!\!\!\cdot\!\left(1\!+\!R_{\mathcal{Q}}\right)^{\frac{2d}{p}}\!\cdot\!\Omega_{0}^{10K}\Omega_{1}^{10}\Omega_{2}^{\left(p,K\right)}\\
& \leq\frac{\left(2^{12}/d\right)^{\frac{d}{p}}}{2^{36}\cdotd^{12}}\cdot\left(2^{39}\cdotd^{9}\cdot\left\lceil K+\frac{d+1}{p}\right\rceil ^{2}\right)^{\left\lceil K+\frac{d+1}{p}\right\rceil +1}\!\!\!\cdot\!\left(1\!+\!R_{\mathcal{Q}}\right)^{\frac{2d}{p}}\!\cdot\!\Omega_{0}^{10K}\Omega_{1}^{10}\Omega_{2}^{\left(p,K\right)}.
\end{align*}
Now, recall that $C_{7}\geq1$ and $K_{0}=K+\frac{d}{p}+1$,
so that
\begin{align*}
C_{12} & =C_{11}=C_{10}\left(\frac{s_{d}}{p}\right)^{1/p}\left(3\sqrt{d}\right)^{2K_{0}+1}\cdot\Omega_{0}^{K}\Omega_{1}\cdot\left(C_{7}\Omega_{4}^{\left(p,K,1\right)}+C_{9}\Omega_{4}^{\left(p,K,2\right)}\left(1+C_{7}\right)\right)\\
& \leq C_{7}\cdot2^{\frac{1}{p}-1}\cdotd^{-\frac{d}{2p}}\cdot\left(972\cdotd^{5/2}\right)^{K+\frac{d}{p}}\cdot\left(\frac{s_{d}}{p}\right)^{1/p}\left(9d\right)^{K_{0}+1}\cdot\Omega_{0}^{4K}\Omega_{1}^{4}\cdot\left(\Omega_{4}^{\left(p,K,1\right)}+2^{\frac{1}{p}}\Omega_{4}^{\left(p,K,2\right)}\right)\\
& \leq\frac{1}{2}C_{7}\cdot4^{\frac{1}{p}}\cdotd^{-\frac{d}{2p}}\cdot\left(972\cdotd^{5/2}\right)^{-2}\cdot\left(8748\cdotd^{7/2}\right)^{K_{0}+1}\cdot\left(\frac{s_{d}}{p}\right)^{1/p}\cdot\Omega_{0}^{4K}\Omega_{1}^{4}\Omega_{4}^{\left(p,K\right)}
\end{align*}
and hence because of $K_{0}+1=K+\frac{d}{p}+2\leq\left\lceil K+\frac{d+1}{p}\right\rceil +1$,
\begin{align*}
& 2C_{1}C_{2}C_{3}C_{12}\\
& \leq C_{7}\!\cdot\!\frac{\left(2^{14}/d^{\frac{3}{2}}\right)^{\frac{d}{p}}}{2^{45}\cdotd^{17}}\!\cdot\!\left(\frac{s_{d}}{p}\right)^{\frac{1}{p}}\!\cdot\!\left(2^{53}\!\cdot\!d^{\frac{25}{2}}\!\cdot\!\left\lceil K\!+\!\frac{d+1}{p}\right\rceil ^{2}\right)^{\!\left\lceil K+\frac{d+1}{p}\right\rceil +1}\!\!\!\cdot\!\left(1\!+\!R_{\mathcal{Q}}\right)^{\frac{2d}{p}}\!\cdot\!\Omega_{0}^{14K}\Omega_{1}^{14}\Omega_{2}^{\left(p,K\right)}\Omega_{4}^{\left(p,K\right)}\\
& \leq\frac{\left(2^{14}/d^{\frac{3}{2}}\right)^{\frac{d}{p}}}{2^{45}\cdotd^{17}}\!\cdot\!\left(\frac{s_{d}}{p}\right)^{\frac{1}{p}}\!\cdot\!\left(2^{68}\!\cdot\!d^{14}\!\cdot\!\left[K\!+\!1\!+\!\frac{d+1}{p}\right]^{3}\right)^{\!K+\frac{d+1}{p}+2}\!\!\!\cdot\!\left(1\!+\!R_{\mathcal{Q}}\right)^{1+\frac{3d}{p}}\!\cdot\!\Omega_{0}^{16K}\Omega_{1}^{16}\Omega_{2}^{\left(p,K\right)}\Omega_{4}^{\left(p,K\right)}.
\end{align*}
These considerations easily show that $\delta\leq\delta_{0}$ indeed
implies $C_{1}C_{2}C_{3}C_{12}\cdot\vertiii{\smash{\overrightarrow{C}}}^{\max\left\{ 1,\frac{1}{p}\right\} }\cdot\delta\leq\frac{1}{2}$.
\medskip{}
All in all, our considerations show for
\[
T^{\left(\delta\right)}:={\rm Synth}_{\Gamma_{1}}\circ\bigotimes_{j\in I}S_{\Gamma_{2}}^{\left(\delta,j\right)}\circ D^{\left(\delta\right)}\:\overset{\text{eq. }\eqref{eq:AtomicDecompositionSynthesisOperatorAsComposition}}{=}\:S^{\left(\delta\right)}\circ D^{\left(\delta\right)}:\DecompSp{\mathcal{Q}}p{\ell_{w}^{q}}v\to\DecompSp{\mathcal{Q}}p{\ell_{w}^{q}}v
\]
that $\vertiii{\operatorname{id}_{\DecompSp{\mathcal{Q}}p{\ell_{w}^{q}}v}-T^{\left(\delta\right)}}\leq\frac{1}{2}$
for all $0<\delta\leq\min\left\{ 1,\delta_{0}\right\} $. Hence, since
$\DecompSp{\mathcal{Q}}p{\ell_{w}^{q}}v$ is a Quasi-Banach space by Lemma
\ref{lem:WeightedDecompositionSpaceComplete}, a Neumann series argument
(which is also valid for Quasi-Banach spaces, cf.\@ e.g.\@ \cite[Lemma 2.4.11]{VoigtlaenderPhDThesis}),
shows that $T^{\left(\delta\right)}$ is invertible for all $0<\delta\leq\min\left\{ 1,\delta_{0}\right\} $.
But then, $C^{\left(\delta\right)}:=D^{\left(\delta\right)}\circ\left(T^{\left(\delta\right)}\right)^{-1}:\DecompSp{\mathcal{Q}}p{\ell_{w}^{q}}v\to\smash{\ell_{\left(\left|\det T_{j}\right|^{\frac{1}{2}-\frac{1}{p}}w_{j}\right)_{\!\!j\in I}}^{q}\!\!\!}\!\!\!\left(\left[\vphantom{F}\smash{C_{j}^{\left(\delta\right)}}\right]_{j\in I}\right)$
is well-defined and bounded and we have for arbitrary $f\in\DecompSp{\mathcal{Q}}p{\ell_{w}^{q}}v$
that
\[
f=\left[T^{\left(\delta\right)}\circ\left(T^{\left(\delta\right)}\right)^{-1}\right]f=\left(\left[S^{\left(\delta\right)}\circ D^{\left(\delta\right)}\right]\circ\left(T^{\left(\delta\right)}\right)^{-1}\right)f=\left[S^{\left(\delta\right)}\circ C^{\left(\delta\right)}\right]f,
\]
as desired.
\end{proof}
\section{Simplified Criteria}
\label{sec:SimplifiedCriteria}In this section, we will derive simplified
conditions which ensure boundedness of the operators $\overrightarrow{A},\overrightarrow{B}$
and $\overrightarrow{C}$, mentioned in Assumptions \ref{assu:MainAssumptions},
\ref{assu:DiscreteBanachFrameAssumptions} and \ref{assu:AtomicDecompositionAssumption},
respectively.
One such general criterion is given by \textbf{Schur's test}, which
we state below. Afterwards, we will provide a convenient standard
estimate for the main term $\left\Vert \mathcal{F}^{-1}\left(\varphi_{i}\cdot\widehat{\gamma^{\left(j\right)}}\right)\right\Vert _{L_{v_{0}}^{p}}$
occurring in the entries of $\overrightarrow{A},\overrightarrow{B}$
and $\overrightarrow{C}$. Then we use these results to formulate
simplified criteria which allow to apply Theorems \ref{thm:DiscreteBanachFrameTheorem}
(leading to Banach frames) and \ref{thm:AtomicDecomposition} (leading
to atomic decompositions).
But first of all, we introduce certain additional assumptions regarding
the partition of unity $\Phi=\left(\varphi_{i}\right)_{i\in I}$.
Recall that in the preceding sections, we only assumed $\Phi$ to
be a $\mathcal{Q}$-$v_{0}$-BAPU, but in this section we will make the
following stronger assumption:
\begin{assumption}
\label{assu:RegularPartitionOfUnity}We assume that $\Phi=\left(\varphi_{i}\right)_{i\in I}$
is a \textbf{regular partition of unity} for $\mathcal{Q}$. This means
\begin{enumerate}
\item $\varphi_{i}\in\TestFunctionSpace{\mathcal{O}}$ with $\operatorname{supp}\varphi_{i}\subset Q_{i}$
for all $i\in I$,
\item $\sum_{i\in I}\varphi_{i}\equiv1$ on $\mathcal{O}$,
\item the \textbf{normalized family} $\Phi^{\natural}:=\left(\smash{\varphi_{i}^{\natural}}\right)_{i\in I}$—given
by $\varphi_{i}^{\natural}:=\varphi_{i}\circ S_{i}$ for $S_{i}\xi:=T_{i}\xi+b_{i}$—satisfies
\begin{equation}
C^{\left(\alpha\right)}:=\sup_{i\in I}\left\Vert \partial^{\alpha}\smash{\varphi_{i}^{\natural}}\right\Vert _{\sup}<\infty\qquad\text{ for all }\alpha\in\mathbb{N}_{0}^{d}.\qedhere\label{eq:RegularBAPUCondition}
\end{equation}
\end{enumerate}
\end{assumption}
\begin{rem*}
As seen in \cite[Lemma 2.5]{DecompositionEmbedding}, every regular
partition of unity is also a $\mathcal{Q}$-$v_{0}$-BAPU, as long as $v_{0}\lesssim1$.
As we will see in Corollary \ref{cor:RegularBAPUsAreWeightedBAPUs},
the same also remains true for general $v_{0}$.
Furthermore, it was shown in \cite[Theorem 2.8]{DecompositionIntoSobolev}
that every \emph{structured} admissible covering $\mathcal{Q}$ admits a
regular partition of unity. Here, the semi-structured covering $\mathcal{Q}=\left(T_{i}Q_{i}'+b_{i}\right)_{i\in I}$
is called \textbf{structured} if $Q_{i}'=Q$ for all $i\in I$ and
some fixed open set $Q\subset\mathbb{R}^{d}$ and if additionally,
there is an open set $P\subset\mathbb{R}^{d}$, compactly contained
in $Q$, such that the family $\left(T_{i}P+b_{i}\right)_{i\in I}$
covers all of $\mathcal{O}$.
\end{rem*}
Now that we have clarified our assumptions for this section, we state
a version of Schur's test which is suitable for our setting. We remark
that this lemma is in no way new; for example, it already appears
in \cite[Lemma 4.4]{ParabolicMolecules}.
\begin{lem}
\label{lem:SchursLemma}Let $I,J\neq\emptyset$ be two nonempty sets
and let $A=\left(A_{i,j}\right)_{\left(i,j\right)\in I\times J}\in\mathbb{C}^{I\times J}$.
Let $p\in\left(1,\infty\right)$ and assume that
\[
C_{1}:=\sup_{i\in I}\sum_{j\in J}\left|A_{i,j}\right|\qquad\text{ and }\qquad C_{2}:=\sup_{j\in J}\sum_{i\in I}\left|A_{i,j}\right|
\]
are finite. Then the operator
\[
\overrightarrow{A}:\ell^{p}\left(J\right)\to\ell^{p}\left(I\right),\left(c_{j}\right)_{j\in J}\mapsto\left(\sum_{j\in J}A_{i,j}c_{j}\right)_{i\in I}
\]
is well-defined and bounded with $\vertiii{\smash{\overrightarrow{A}}}\leq\max\left\{ C_{1},C_{2}\right\} $.
In case of $p\in\left(0,1\right]$, it suffices if
\[
C_{3}^{\left(p\right)}:=\sup_{j\in J}\sum_{i\in I}\left|A_{i,j}\right|^{p}
\]
is finite. In this case, $\vertiii{\smash{\overrightarrow{A}}}\leq\left(C_{3}^{\left(p\right)}\right)^{1/p}$.
Finally, in case of $p=\infty$, it suffices if
\[
C_{4}:=\sup_{i\in I}\sum_{j\in J}\left|A_{i,j}\right|
\]
is finite. In this case, $\vertiii{\smash{\overrightarrow{A}}}\leq C_{4}$.
\end{lem}
\begin{proof}
The statement for $p\in\left(1,\infty\right)$ follows from the more
general form of Schur's test as given e.g.\@ in \cite[Theorem 6.18]{FollandRA},
by considering $I$ and $J$ as measure spaces by equipping them with
the counting measure. Strictly speaking, that lemma assumes the underlying
measure spaces to be $\sigma$-finite (i.e., $I,J$ have to be countable),
but since Tonelli's theorem is applicable to uncountable sets equipped
with the counting measure, the proof given in \cite{FollandRA} still
works even for uncountable $I,J$.
Now, let us assume $p\in\left(0,1\right]$. In this case, we have
\begin{align*}
\left\Vert \overrightarrow{A}\left(c_{j}\right)_{j\in J}\right\Vert _{\ell^{p}}^{p}=\sum_{i\in I}\left|\left(\overrightarrow{A}\left(c_{j}\right)_{j\in J}\right)_{i}\right|^{p} & =\sum_{i\in I}\left|\sum_{j\in J}A_{i,j}\cdot c_{j}\right|^{p}\\
\left({\scriptstyle \text{since }\left(\sum a_{j}\right)^{p}\leq\sum a_{j}^{p}\text{ for }p\in\left(0,1\right]\text{ and }a_{j}\geq0}\right) & \leq\sum_{i\in I}\:\sum_{j\in J}\left|A_{i,j}\right|^{p}\left|c_{j}\right|^{p}\\
& =\sum_{j\in J}\left(\left|c_{j}\right|^{p}\,\sum_{i\in I}\left|A_{i,j}\right|^{p}\right)\\
& \leq C_{3}^{\left(p\right)}\cdot\sum_{j\in J}\left|c_{j}\right|^{p}\\
& =C_{3}^{\left(p\right)}\cdot\left\Vert \left(c_{j}\right)_{j\in J}\right\Vert _{\ell^{p}}^{p}<\infty,
\end{align*}
so that $\overrightarrow{A}:\ell^{p}\left(J\right)\to\ell^{p}\left(I\right)$
is bounded with $\vertiii{\smash{\overrightarrow{A}}}_{\ell^{p}\to\ell^{p}}\leq\left(C_{3}^{\left(p\right)}\right)^{1/p}$.
Finally, let $p=\infty$. For arbitrary $i\in I$, we have
\begin{align*}
\left|\left(\overrightarrow{A}\left(c_{j}\right)_{j\in J}\right)_{i}\right| & =\left|\sum_{j\in J}A_{i,j}\cdot c_{j}\right|\\
& \leq\sum_{j\in J}\left(\left|A_{i,j}\right|\cdot\left|c_{j}\right|\right)\\
& \leq\left\Vert \left(c_{j}\right)_{j\in J}\right\Vert _{\ell^{\infty}}\cdot\sum_{j\in J}\left|A_{i,j}\right|\\
& \leq C_{4}\cdot\left\Vert \left(c_{j}\right)_{j\in J}\right\Vert _{\ell^{p}}.
\end{align*}
As a further remark, we observe that the case $p\in\left(1,\infty\right)$
can be obtained by complex interpolation (i.e., by the Riesz-Thorin
Theorem \cite[Theorem 6.27]{FollandRA}) from the cases $p=1$ and
$p=\infty$, since $C_{1}=C_{4}$ and $C_{2}=C_{3}^{\left(1\right)}$.
\end{proof}
In Lemma \ref{lem:GramMatrixEstimates} below, we will derive a convenient
estimate for the main term of the ``infinite matrices'' $A,B,C$
from Assumptions \ref{assu:MainAssumptions}, \ref{assu:DiscreteBanachFrameAssumptions}
and \ref{assu:AtomicDecompositionAssumption}, namely for the term
$\left\Vert \mathcal{F}^{-1}\left(\varphi_{i}\widehat{\gamma^{\left(j\right)}}\right)\right\Vert _{L_{v_{0}}^{p}}$.
To derive this estimate, the following lemma will be useful. It makes
precise the notion that \emph{smoothness of $f$ yields decay of $\widehat{f}$}.
The statement itself is probably folklore, so no originality is claimed.
\begin{lem}
\label{lem:PointwiseFourierDecayEstimate}Let $N\in\mathbb{N}_{0}$ and $g\in W^{N,1}\left(\mathbb{R}^{d}\right)$.
Then
\begin{equation}
\left(1+\left|x\right|\right)^{N}\cdot\left|\mathcal{F}^{-1}g\left(x\right)\right|\leq\left(1+d\right)^{N}\cdot\left(\left|\mathcal{F}^{-1}g\left(x\right)\right|+\sum_{m=1}^{d}\left|\left[\mathcal{F}^{-1}\left(\partial_{m}^{N}g\right)\right]\left(x\right)\right|\right)\qquad\forall x\in\mathbb{R}^{d}.\qedhere\label{eq:PointwiseFourierDecayEstimate}
\end{equation}
\end{lem}
\begin{rem*}
Here, $W^{N,1}\left(\mathbb{R}^{d}\right)$ is the \textbf{Sobolev
space} of all functions $g\in L^{1}\left(\mathbb{R}^{d}\right)$
for which all weak derivatives $\partial^{\alpha}g$ with $\left|\alpha\right|\leq N$
satisfy $\partial^{\alpha}g\in L^{1}\left(\mathbb{R}^{d}\right)$.
It is a Banach space when equipped with the norm $\left\Vert g\right\Vert _{W^{N,1}}:=\sum_{\left|\alpha\right|\leq N}\left\Vert \partial^{\alpha}g\right\Vert _{L^{1}}$.
For $N=0$, we use the convention $W^{N,1}\left(\mathbb{R}^{d}\right)=L^{1}\left(\mathbb{R}^{d}\right)$.
\end{rem*}
\begin{proof}
It is well-known (see e.g. \cite[Corollary 3.23]{AdamsSobolevSpaces})
that $\TestFunctionSpace{\mathbb{R}^{d}}\subset W^{N,1}\left(\mathbb{R}^{d}\right)$
is dense. Furthermore, since $\mathcal{F}^{-1}:L^{1}\left(\mathbb{R}^{d}\right)\to C_{0}\left(\mathbb{R}^{d}\right)$
is well-defined and bounded, where the space $C_{0}\left(\mathbb{R}^{d}\right)$
of continuous functions vanishing at infinity is equipped with the
norm $\left\Vert h\right\Vert _{\sup}:=\sup_{x\in\mathbb{R}^{d}}\left|h\left(x\right)\right|$,
it is not hard to see that $\mathcal{F}^{-1}g\left(x\right)$ and $\left[\mathcal{F}^{-1}\left(\partial_{m}^{N}g\right)\right]\left(x\right)$
all depend continuously on $g\in W^{N,1}\left(\mathbb{R}^{d}\right)$,
for arbitrary $x\in\mathbb{R}^{d}$ and $m\in\underline{d}$.
Hence, we can without loss of generality assume $g\in\TestFunctionSpace{\mathbb{R}^{d}}$.
But under this assumption, we have (see e.g.\@ \cite[Theorem 8.22]{FollandRA})
the standard identity
\[
\left[\mathcal{F}^{-1}\left(\partial_{m}^{N}g\right)\right]\left(x\right)=\left(-2\pi ix_{m}\right)^{N}\cdot\left(\mathcal{F}^{-1}g\right)\left(x\right)\qquad\forall x\in\mathbb{R}^{d}.
\]
In particular, since
\[
\left(\sum_{i=1}^{k}a_{i}\right)^{N}\leq\left(k\cdot\max\left\{ a_{i}\,\middle|\, i\in\underline{k}\right\} \right)^{N}\leq k^{N}\cdot\sum_{i=1}^{k}a_{i}^{N}
\]
holds for arbitrary $a_{1},\dots,a_{k}\geq0$ and because of $\left|x\right|\leq\left\Vert x\right\Vert _{1}$,
we get
\begin{align*}
\left(1+\left|x\right|\right)^{N}\cdot\left|\left(\mathcal{F}^{-1}g\right)\left(x\right)\right| & \leq\left(1+\sum_{m=1}^{d}\left|x_{m}\right|\right)^{N}\cdot\left|\left(\mathcal{F}^{-1}g\right)\left(x\right)\right|\\
& \leq\left(d+1\right)^{N}\cdot\left|\left(\mathcal{F}^{-1}g\right)\left(x\right)\right|\cdot\left(1+\sum_{m=1}^{d}\left|x_{m}^{N}\right|\right)\\
& =\left(d+1\right)^{N}\cdot\left(\left|\left(\mathcal{F}^{-1}g\right)\left(x\right)\right|+\sum_{m=1}^{d}\left|\frac{\left[\mathcal{F}^{-1}\left(\partial_{m}^{N}g\right)\right]\left(x\right)}{\left(2\pi\right)^{N}}\right|\right)\\
& \leq\left(1+d\right)^{N}\cdot\left(\left|\left(\mathcal{F}^{-1}g\right)\left(x\right)\right|+\sum_{m=1}^{d}\left|\left[\mathcal{F}^{-1}\left(\partial_{m}^{N}g\right)\right]\left(x\right)\right|\right)
\end{align*}
for arbitrary $x\in\mathbb{R}^{d}$, as desired.
\end{proof}
Now, we are finally in a position to derive the promised estimate
for $\left\Vert \mathcal{F}^{-1}\left(\varphi_{i}\widehat{\gamma^{\left(j\right)}}\right)\right\Vert _{L_{v_{0}}^{p}}$.
\begin{lem}
\label{lem:GramMatrixEstimates}Suppose that $\left(\varphi_{i}\right)_{i\in I}$
satisfies Assumption \ref{assu:RegularPartitionOfUnity}. Let $\gamma\in L^{1}\left(\mathbb{R}^{d}\right)$
and assume that $\widehat{\gamma}\in C^{\infty}\left(\mathbb{R}^{d}\right)$.
For $j\in I$, define
\[
\gamma^{\left\llbracket j\right\rrbracket }:=\mathcal{F}^{-1}\left(\widehat{\gamma}\circ S_{j}^{-1}\right)=\left|\det T_{j}\right|\cdot M_{b_{j}}\left[\gamma\circ T_{j}^{T}\right].
\]
Then we have for arbitrary $\varepsilon>0$, $i,j\in I$ and $p\in\left(0,\infty\right)$
the estimate
\begin{align*}
\left\Vert \mathcal{F}^{-1}\left(\widehat{\gamma^{\left\llbracket j\right\rrbracket }}\cdot\varphi_{i}\right)\right\Vert _{L_{v_{0}}^{p}} & \leq C_{0}\cdot\left(1\!+\!\left\Vert T_{j}^{-1}T_{i}\right\Vert \right)^{\!\left\lceil K+\frac{d+\varepsilon}{p}\right\rceil }\!\cdot\left|\det T_{i}\right|^{-\frac{1}{p}}\!\cdot\!\int_{Q_{i}}\:\max_{\left|\alpha\right|\leq\left\lceil K+\frac{d+\varepsilon}{p}\right\rceil }\left|\left(\partial^{\alpha}\widehat{\gamma}\right)\!\left(S_{j}^{-1}\eta\right)\right|\operatorname{d}\eta\\
\left({\scriptstyle \xi=S_{i}^{-1}\eta}\right) & \leq C_{0}\cdot\left(1\!+\!\left\Vert T_{j}^{-1}T_{i}\right\Vert \right)^{\!\left\lceil K+\frac{d+\varepsilon}{p}\right\rceil }\!\cdot\left|\det T_{i}\right|^{1-\frac{1}{p}}\!\cdot\!\int_{Q_{i}'}\:\max_{\left|\alpha\right|\leq\left\lceil K+\frac{d+\varepsilon}{p}\right\rceil }\left|\left(\partial^{\alpha}\widehat{\gamma}\right)\!\left(S_{j}^{-1}S_{i}\xi\right)\right|\operatorname{d}\xi,
\end{align*}
with
\[
C_{0}:=\Omega_{0}^{K}\Omega_{1}\cdot\left(4\cdotd\right)^{1+2\left\lceil K+\frac{d+\varepsilon}{p}\right\rceil }\cdot\left(\frac{s_{d}}{\varepsilon}\right)^{1/p}\cdot\max_{\left|\alpha\right|\leq\left\lceil K+\frac{d+\varepsilon}{p}\right\rceil }C^{\left(\alpha\right)},
\]
where the constants $C^{\left(\alpha\right)}$ are defined in Assumption
\ref{assu:RegularPartitionOfUnity}, equation (\ref{eq:RegularBAPUCondition}).
\end{lem}
\begin{rem*}
With the notation $\gamma^{\left\llbracket j\right\rrbracket }$,
the usual notation $\gamma^{\left(j\right)}$ for a family $\Gamma=\left(\gamma_{i}\right)_{i\in I}$
takes the form $\gamma^{\left(j\right)}=\gamma_{j}^{\left\llbracket j\right\rrbracket }$.
\end{rem*}
\begin{proof}
Set $N:=\left\lceil K+\frac{d+\varepsilon}{p}\right\rceil $.
Now, recall from \cite[Lemma 2.6]{DecompositionIntoSobolev} the
identity
\[
\left(\partial^{\alpha}\left[f\circ A\right]\right)\left(x\right)=\sum_{\ell_{1},\dots,\ell_{k}\in\underline{d}}\left[A_{\ell_{1},i_{1}}\cdots A_{\ell_{k},i_{k}}\cdot\left(\partial_{\ell_{1}}\cdots\partial_{\ell_{k}}f\right)\left(Ax\right)\right]\qquad\forall x\in\mathbb{R}^{d}
\]
for arbitrary $A\in\mathrm{GL}\left(\mathbb{R}^{d}\right)$, $k\in\mathbb{N}$, $f\in C^{k}\left(\mathbb{R}^{d}\right)$
and $\alpha=\sum_{m=1}^{k}e_{i_{m}}\in\mathbb{N}_{0}^{d}$, where
$\left(e_{1},\dots,e_{d}\right)$ is the standard basis of
$\mathbb{R}^{d}$. In particular, this implies for arbitrary $k\in\mathbb{N}$
that
\[
\left|\left(\partial^{\alpha}\left[f\circ A\right]\right)\left(x\right)\right|\leqd^{k}\cdot\left\Vert A\right\Vert ^{k}\cdot\max_{\left|\beta\right|=k}\left|\left(\partial^{\beta}f\right)\left(Ax\right)\right|\qquad\forall\,x\in\mathbb{R}^{d}\,\forall\,\alpha\in\mathbb{N}_{0}^{d}\text{ with }\left|\alpha\right|=k\text{ and }f\in C^{k}\left(\smash{\mathbb{R}^{d}}\right)
\]
and this estimate obviously also holds for $k=0$.
Thus, using the notation $h^{\heartsuit}\left(\xi\right):=\max_{\left|\alpha\right|\leq N}\left|\left(\partial^{\alpha}h\right)\left(\xi\right)\right|$
for $\xi\in\mathbb{R}^{d}$ and $h\in C^{\infty}\left(\mathbb{R}^{d}\right)$,
we get for arbitrary $i\in I$, $m\in\underline{d}$ and
$\ell\in\left\{ 0\right\} \cup\underline{N}=\left\{ 0,\dots,\left\lceil K+\frac{d+\varepsilon}{p}\right\rceil \right\} $
as well as $T\in\mathrm{GL}\left(\mathbb{R}^{d}\right)$ that
\begin{align*}
\left|\left[\partial_{m}^{\ell}\left(h\circ S_{i}^{-1}\circ T\right)\right]\left(\xi\right)\right| & =\left|\left[\partial_{m}^{\ell}\left(\eta\mapsto h\left(T_{i}^{-1}T\eta-T_{i}^{-1}b_{i}\right)\right)\right]\left(\xi\right)\right|\\
& \leqd^{\ell}\cdot\left\Vert T_{i}^{-1}T\right\Vert ^{\ell}\cdot\max_{\left|\alpha\right|\leq\ell}\left|\left(\partial^{\alpha}h\right)\left(T_{i}^{-1}T\xi-T_{i}^{-1}b_{i}\right)\right|\\
\left({\scriptstyle \text{since }\ell\leq N}\right) & \leqd^{\ell}\cdot\left\Vert T_{i}^{-1}T\right\Vert ^{\ell}\cdot\left(h^{\heartsuit}\circ S_{i}^{-1}\circ T\right)\left(\xi\right).
\end{align*}
Now, set $g:=\varphi_{i}\cdot\widehat{\gamma^{\left\llbracket j\right\rrbracket }}\in\TestFunctionSpace{\mathbb{R}^{d}}$
and apply Leibniz's rule to the product
\[
g\circ T=\left(\widehat{\gamma^{\left\llbracket j\right\rrbracket }}\circ T\right)\cdot\left(\varphi_{i}\circ T\right)=\left(\widehat{\gamma}\circ S_{j}^{-1}\circ T\right)\cdot\left(\varphi_{i}^{\natural}\circ S_{i}^{-1}\circ T\right),
\]
to see using the binomial theorem that
\begin{align*}
\left|\partial_{m}^{N}\left(g\circ T\right)\right| & =\left|\sum_{\ell=0}^{N}\binom{N}{\ell}\cdot\partial_{m}^{\ell}\left(\widehat{\gamma}\circ S_{j}^{-1}\circ T\right)\cdot\partial_{m}^{N-\ell}\left(\varphi_{i}^{\natural}\circ S_{i}^{-1}\circ T\right)\right|\\
& \leq\left[\widehat{\gamma}^{\heartsuit}\circ S_{j}^{-1}\circ T\right]\cdot\left[\left(\smash{\varphi_{i}^{\natural}}\right)^{\heartsuit}\circ S_{i}^{-1}\circ T\right]\cdot\sum_{\ell=0}^{N}\left[\binom{N}{\ell}\cdotd^{N}\cdot\left\Vert T_{j}^{-1}T\right\Vert ^{\ell}\left\Vert T_{i}^{-1}T\right\Vert ^{N-\ell}\right]\\
& =d^{N}\cdot\left(\left\Vert T_{j}^{-1}T\right\Vert +\left\Vert T_{i}^{-1}T\right\Vert \right)^{N}\cdot\left[\widehat{\gamma}^{\heartsuit}\circ S_{j}^{-1}\circ T\right]\cdot\left[\left(\smash{\varphi_{i}^{\natural}}\right)^{\heartsuit}\circ S_{i}^{-1}\circ T\right].
\end{align*}
Now, set $C_{2}:=\max_{\left|\alpha\right|\leq N}C^{\left(\alpha\right)}$,
with $C^{\left(\alpha\right)}$ as in Assumption \ref{assu:RegularPartitionOfUnity},
equation (\ref{eq:RegularBAPUCondition}). Because of $\operatorname{supp}\varphi_{i}^{\natural}\subset Q_{i}'$,
this yields $\left(\smash{\varphi_{i}^{\natural}}\right)^{\heartsuit}\leq C_{2}\cdot{\mathds{1}}_{Q_{i}'}$
and thus
\[
\left(\smash{\varphi_{i}^{\natural}}\right)^{\heartsuit}\circ S_{i}^{-1}\circ T\leq C_{2}\cdot{\mathds{1}}_{T^{-1}\left(Q_{i}\right)}=C_{2}\cdot{\mathds{1}}_{Q_{i}}\circ T.
\]
Hence,
\begin{align*}
\left|\partial_{m}^{N}\left(g\circ T\right)\right| & \leqd^{N}C_{2}\cdot\left(\left\Vert T_{j}^{-1}T\right\Vert +\left\Vert T_{i}^{-1}T\right\Vert \right)^{N}\cdot\left[\widehat{\gamma}^{\heartsuit}\circ S_{j}^{-1}\circ T\right]\cdot\left({\mathds{1}}_{Q_{i}}\circ T\right)\\
& =d^{N}C_{2}\cdot\left(\left\Vert T_{j}^{-1}T\right\Vert +\left\Vert T_{i}^{-1}T\right\Vert \right)^{N}\cdot\left[\left(\widehat{\gamma}^{\heartsuit}\circ S_{j}^{-1}\right)\cdot{\mathds{1}}_{Q_{i}}\right]\circ T,
\end{align*}
as well as
\[
\left|g\circ T\right|=\left|\left(\widehat{\gamma}\circ S_{j}^{-1}\circ T\right)\cdot\left(\varphi_{i}^{\natural}\circ S_{i}^{-1}\circ T\right)\right|\leq C_{2}\cdot\left[\left(\widehat{\gamma}^{\heartsuit}\circ S_{j}^{-1}\right)\cdot{\mathds{1}}_{Q_{i}}\right]\circ T.
\]
By combining Lemma \ref{lem:PointwiseFourierDecayEstimate}, equation
(\ref{eq:PointwiseFourierDecayEstimate}) (with $g\circ T$ instead
of $g$) with the preceding estimates, we arrive at
\begin{align*}
\left(1+\left|x\right|\right)^{N}\cdot\left|\left[\mathcal{F}^{-1}\left(g\circ T\right)\right]\left(x\right)\right| & \leq\left(1+d\right)^{N}\cdot\left(\left|\left[\mathcal{F}^{-1}\left(g\circ T\right)\right]\left(x\right)\right|+\sum_{m=1}^{d}\left|\left[\mathcal{F}^{-1}\left(\partial_{m}^{N}\left(g\circ T\right)\right)\right]\left(x\right)\right|\right)\\
& \leq\left(1+d\right)^{N}\cdot\left(\left\Vert g\circ T\right\Vert _{L^{1}}+\sum_{m=1}^{d}\left\Vert \partial_{m}^{N}\left(g\circ T\right)\right\Vert _{L^{1}}\right)\\
& \leq\left(1\!+\!d\right)^{N}C_{2}\cdot\left\Vert \left[\left(\widehat{\gamma}^{\heartsuit}\circ S_{j}^{-1}\right)\cdot{\mathds{1}}_{Q_{i}}\right]\circ T\right\Vert _{L^{1}}\cdot\left(1\!+\!\sum_{m=1}^{d}d^{N}\cdot\left(\left\Vert T_{j}^{-1}T\right\Vert \!+\!\left\Vert T_{i}^{-1}T\right\Vert \right)^{N}\right)\\
& \overset{\left(\dagger\right)}{\leq}d^{N+1}\left(1+d\right)^{N}C_{2}\cdot\left(1+\left\Vert T_{j}^{-1}T\right\Vert +\left\Vert T_{i}^{-1}T\right\Vert \right)^{N}\cdot\left|\det T\right|^{-1}\cdot\left\Vert \left(\widehat{\gamma}^{\heartsuit}\circ S_{j}^{-1}\right)\cdot{\mathds{1}}_{Q_{i}}\right\Vert _{L^{1}}\\
& \leq\left(1+d\right)^{1+2N}\cdot C_{2}\cdot\left(1+\left\Vert T_{j}^{-1}T\right\Vert +\left\Vert T_{i}^{-1}T\right\Vert \right)^{N}\cdot\left|\det T\right|^{-1}\cdot\left\Vert \left(\widehat{\gamma}^{\heartsuit}\circ S_{j}^{-1}\right)\cdot{\mathds{1}}_{Q_{i}}\right\Vert _{L^{1}}\\
& =:\left|\det T\right|^{-1}\cdot C_{3}^{\left(i,j,T\right)},
\end{align*}
where the step marked with $\left(\dagger\right)$ used that $1+a^{N}\leq\left(1+a\right)^{N}$
for $a\geq0$, as can be seen by expanding the right-hand side using
the binomial theorem.
\medskip{}
Now, we choose $T=T_{i}$ and note $\left[\mathcal{F}^{-1}\left(g\circ T_{i}\right)\right]\left(x\right)=\left|\det T_{i}\right|^{-1}\cdot\left(\mathcal{F}^{-1}g\right)\left(T_{i}^{-T}x\right)$,
so that we have shown $\left|\left(\mathcal{F}^{-1}g\right)\left(T_{i}^{-T}x\right)\right|\leq C_{3}^{\left(i,j,T_{i}\right)}\cdot\left(1+\left|x\right|\right)^{-N}$
and thus $\left|\left(\mathcal{F}^{-1}g\right)\left(y\right)\right|\leq C_{3}^{\left(i,j,T_{i}\right)}\cdot\left(1+\left|T_{i}^{T}y\right|\right)^{-N}$
for all $y\in\mathbb{R}^{d}$. In conjunction with equation (\ref{eq:WeightLinearTransformationsConnection})
and because of $v_{0}\left(y\right)\leq\Omega_{1}\cdot\left(1+\left|y\right|\right)^{K}$,
we arrive at
\begin{align*}
v_{0}\left(y\right)\cdot\left|\left(\mathcal{F}^{-1}g\right)\left(y\right)\right| & \leq\Omega_{1}\cdot\left(1+\left|y\right|\right)^{K}\cdot\left|\left(\mathcal{F}^{-1}g\right)\left(y\right)\right|\\
\left({\scriptstyle \text{eq. }\eqref{eq:WeightLinearTransformationsConnection}}\right) & \leq\Omega_{0}^{K}\Omega_{1}\cdot C_{3}^{\left(i,j,T_{i}\right)}\cdot\left(1+\left|T_{i}^{T}y\right|\right)^{K-N}.
\end{align*}
By taking the $L^{p}$-quasi-norm of this estimate, we arrive at
\begin{align*}
\left\Vert \mathcal{F}^{-1}g\right\Vert _{L_{v_{0}}^{p}} & \leq\Omega_{0}^{K}\Omega_{1}\cdot C_{3}^{\left(i,j,T_{i}\right)}\cdot\left\Vert \left(1+\left|T_{i}^{T}\bullet\right|\right)^{K-N}\right\Vert _{L^{p}}\\
& =\Omega_{0}^{K}\Omega_{1}\cdot C_{3}^{\left(i,j,T_{i}\right)}\cdot\left|\det T_{i}\right|^{-1/p}\cdot\left\Vert \left(1+\left|\bullet\right|\right)^{-\left(N-K\right)}\right\Vert _{L^{p}}\\
\left({\scriptstyle \text{eq. }\eqref{eq:StandardDecayLpEstimate}}\right) & \leq\Omega_{0}^{K}\Omega_{1}\cdot C_{3}^{\left(i,j,T_{i}\right)}\cdot\left|\det T_{i}\right|^{-1/p}\cdot\left(\frac{s_{d}}{\varepsilon}\right)^{1/p},
\end{align*}
where the last step used our choice of $N=\left\lceil K+\frac{d+\varepsilon}{p}\right\rceil $.
This proves the claim, since
\begin{align*}
C_{3}^{\left(i,j,T_{i}\right)} & =\left(1+d\right)^{1+2N}\cdot C_{2}\cdot\left(2+\left\Vert T_{j}^{-1}T_{i}\right\Vert \right)^{N}\cdot\left\Vert \left(\widehat{\gamma}^{\heartsuit}\circ S_{j}^{-1}\right)\cdot{\mathds{1}}_{Q_{i}}\right\Vert _{L^{1}}\\
& \leq\left(4d\right)^{1+2N}\cdot C_{2}\cdot\left(1+\left\Vert T_{j}^{-1}T_{i}\right\Vert \right)^{N}\cdot\int_{Q_{i}}\:\max_{\left|\alpha\right|\leq\left\lceil K+\frac{d+\varepsilon}{p}\right\rceil }\left|\left(\partial^{\alpha}\widehat{\gamma}\right)\left(S_{j}^{-1}\xi\right)\right|\operatorname{d}\xi.\qedhere
\end{align*}
\end{proof}
As a consequence of the preceding estimate, we see in particular that
every regular $\mathcal{Q}$-BAPU is also a $\mathcal{Q}$-$v_{0}$-BAPU, even
for $v_{0}\not\equiv1$.
\begin{cor}
\label{cor:RegularBAPUsAreWeightedBAPUs}Every regular $\mathcal{Q}$-BAPU
$\Phi=\left(\varphi_{i}\right)_{i\in I}$ is a $\mathcal{Q}$-$v_{0}$-BAPU.
In fact, there is some $\varrho\in\TestFunctionSpace{\mathbb{R}^{d}}$,
depending only on $Q:=\overline{\bigcup_{i\in I}Q_{i}'}$ (and thus
only on $\mathcal{Q}$), such that
\[
C_{\mathcal{Q},\Phi,v_{0},p}\leq\Omega_{0}^{K}\Omega_{1}\cdot\left(4\cdotd\right)^{1+2\left\lceil K+\frac{d+\varepsilon}{p}\right\rceil }\cdot\left(\frac{s_{d}}{\varepsilon}\right)^{1/p}\cdot2^{\!\left\lceil K+\frac{d+\varepsilon}{p}\right\rceil }\cdot\lambda_{d}\left(Q\right)\cdot\max_{\left|\alpha\right|\leq\left\lceil K+\frac{d+\varepsilon}{p}\right\rceil }\left\Vert \partial^{\alpha}\varrho\right\Vert _{\sup}\cdot\max_{\left|\alpha\right|\leq\left\lceil K+\frac{d+\varepsilon}{p}\right\rceil }C^{\left(\alpha\right)},
\]
where $\varepsilon>0$ can be chosen arbitrarily.
\end{cor}
\begin{proof}
The set $Q\subset\mathbb{R}^{d}$ is compact, so that there is some
$\gamma\in\mathcal{S}\left(\mathbb{R}^{d}\right)$ satisfying $\widehat{\gamma}\in\TestFunctionSpace{\mathbb{R}^{d}}$
and $\gamma\equiv1$ on $Q$. In the notation of Lemma \ref{lem:GramMatrixEstimates},
this entails $\widehat{\gamma^{\left\llbracket j\right\rrbracket }}=\widehat{\gamma}\circ S_{j}^{-1}\equiv1$
on $S_{j}Q\supset S_{j}\overline{Q_{j}'}=\overline{Q_{j}}$. But because
of $\varphi_{j}\equiv0$ outside of $\overline{Q_{j}}$, this implies
$\widehat{\gamma^{\left\llbracket j\right\rrbracket }}\cdot\varphi_{j}=\varphi_{j}$,
so that Lemma \ref{lem:GramMatrixEstimates} yields (with $C_{0}$
as in that lemma) that
\begin{align*}
\left\Vert \mathcal{F}^{-1}\varphi_{j}\right\Vert _{L_{v_{0}}^{p}} & =\left\Vert \mathcal{F}^{-1}\left(\widehat{\gamma^{\left\llbracket j\right\rrbracket }}\cdot\varphi_{j}\right)\right\Vert _{L_{v_{0}}^{p}}\\
& \leq C_{0}\cdot\left(1\!+\!\left\Vert T_{j}^{-1}T_{j}\right\Vert \right)^{\!\left\lceil K+\frac{d+\varepsilon}{p}\right\rceil }\!\cdot\left|\det T_{j}\right|^{1-\frac{1}{p}}\!\cdot\!\int_{Q_{j}'}\:\max_{\left|\alpha\right|\leq\left\lceil K+\frac{d+\varepsilon}{p}\right\rceil }\left|\left(\partial^{\alpha}\widehat{\gamma}\right)\!\left(S_{j}^{-1}S_{j}\xi\right)\right|\operatorname{d}\xi\\
& \leq C_{0}\cdot2^{\!\left\lceil K+\frac{d+\varepsilon}{p}\right\rceil }\!\cdot\left|\det T_{j}\right|^{1-\frac{1}{p}}\!\cdot\!\lambda_{d}\left(Q_{j}'\right)\cdot\max_{\left|\alpha\right|\leq\left\lceil K+\frac{d+\varepsilon}{p}\right\rceil }\left\Vert \partial^{\alpha}\widehat{\gamma}\right\Vert _{\sup}\\
& \leq C_{0}\cdot2^{\!\left\lceil K+\frac{d+\varepsilon}{p}\right\rceil }\cdot\lambda_{d}\left(Q\right)\cdot\max_{\left|\alpha\right|\leq\left\lceil K+\frac{d+\varepsilon}{p}\right\rceil }\left\Vert \partial^{\alpha}\widehat{\gamma}\right\Vert _{\sup}\cdot\left|\det T_{j}\right|^{1-\frac{1}{p}}\\
& =:C\cdot\left|\det T_{j}\right|^{1-\frac{1}{p}},
\end{align*}
where $C>0$ is independent of $j\in I$. Recalling the definition
of a $\mathcal{Q}$-$v_{0}$-BAPU from Subsection \ref{subsec:DecompSpaceDefinitionStandingAssumptions},
this yields the claim, with $\varrho:=\widehat{\gamma}$.
\end{proof}
Using Schur's test as well as the estimates given in Lemma \ref{lem:GramMatrixEstimates},
we can now derive simplified sufficient criteria which ensure that
a given family $\Gamma=\left(\gamma_{i}\right)_{i\in I}$ of prototypes
indeed generates a Banach frame (as in Theorem \ref{thm:DiscreteBanachFrameTheorem})
or an atomic decomposition (as in Theorem \ref{thm:AtomicDecomposition}).
We start with a simplified criterion for Banach frames.
\begin{cor}
\label{cor:BanachFrameSimplifiedCriteria}Assume that $\left(\varphi_{i}\right)_{i\in I}$
satisfies Assumption \ref{assu:RegularPartitionOfUnity}. Then, for
each $p,q\in\left(0,\infty\right]$, there are
\[
N\in\mathbb{N},\qquad\sigma>0,\qquad\text{ and }\qquad\tau>0
\]
with the following property: If the family $\Gamma=\left(\gamma_{i}\right)_{i\in I}$
satisfies the following:
\begin{enumerate}
\item We have $\gamma_{i}\in L_{\left(1+\left|\bullet\right|\right)^{K}}^{1}\left(\mathbb{R}^{d}\right)$
and $\widehat{\gamma_{i}}\in C^{\infty}\left(\mathbb{R}^{d}\right)$
for all $i\in I$, where all partial derivatives of $\widehat{\gamma_{i}}$
are polynomially bounded.
\item We have $\gamma_{i}\in C^{1}\left(\mathbb{R}^{d}\right)$ and $\partial_{\ell}\gamma_{i}\in L_{v_{0}}^{1}\left(\mathbb{R}^{d}\right)\cap L^{\infty}\left(\mathbb{R}^{d}\right)$
for all $\ell\in\underline{d}$ and $i\in I$.
\item The family $\Gamma=\left(\gamma_{i}\right)_{i\in I}$ satisfies Assumption
\ref{assu:GammaCoversOrbit}.
\item We have
\[
C_{1}:=\sup_{i\in I}\,\sum_{j\in I}M_{j,i}<\infty\qquad\text{ and }\qquad C_{2}:=\sup_{j\in I}\sum_{i\in I}M_{j,i}<\infty
\]
with
\[
M_{j,i}:=\left(\frac{w_{j}}{w_{i}}\right)^{\tau}\cdot\left(1+\left\Vert T_{j}^{-1}T_{i}\right\Vert \right)^{\sigma}\cdot\max_{\left|\beta\right|\leq1}\left(\left|\det T_{i}\right|^{-1}\cdot\int_{Q_{i}}\max_{\left|\alpha\right|\leq N}\left|\left(\partial^{\alpha}\widehat{\partial^{\beta}\gamma_{j}}\right)\!\!\left(S_{j}^{-1}\xi\right)\right|\operatorname{d}\xi\right)^{\tau}.
\]
\end{enumerate}
Then $\Gamma$ fulfills Assumptions \ref{assu:DiscreteBanachFrameAssumptions}
and \ref{assu:GammaCoversOrbit} and thus all assumptions of Theorem
\ref{thm:DiscreteBanachFrameTheorem}.
In fact, the following choices are possible, for an arbitrary $\varepsilon>0$:
\begin{align*}
N & =\left\lceil K+\frac{d+\varepsilon}{\min\left\{ 1,p\right\} }\right\rceil \,,\\
\tau & =\min\left\{ 1,p,q\right\} =\begin{cases}
\min\left\{ 1,q\right\} , & \text{if }p\in\left[1,\infty\right],\\
\min\left\{ q,p\right\} , & \text{if }p\in\left(0,1\right),
\end{cases}\\
\sigma & =\tau\cdot\left(\frac{d}{\min\left\{ 1,p\right\} }+K+\left\lceil K+\frac{d+\varepsilon}{\min\left\{ 1,p\right\} }\right\rceil \right)=\begin{cases}
\min\left\{ 1,q\right\} \cdot\left(d+K+\left\lceil K+d+\varepsilon\right\rceil \right), & \text{if }p\in\left[1,\infty\right],\\
\min\left\{ p,q\right\} \cdot\left(\frac{d}{p}+K+\left\lceil K+\frac{d+\varepsilon}{p}\right\rceil \right), & \text{if }p\in\left(0,1\right).
\end{cases}
\end{align*}
With these choices, we even have $\vertiii{\smash{\overrightarrow{A}}}^{\max\left\{ 1,\frac{1}{p}\right\} }\leq C\cdot\left(C_{1}^{1/\tau}+C_{2}^{1/\tau}\right)$
and $\vertiii{\smash{\overrightarrow{B}}}^{\max\left\{ 1,\frac{1}{p}\right\} }\leq C\cdot\left(C_{1}^{1/\tau}+C_{2}^{1/\tau}\right)$
for
\[
C:=\Omega_{0}^{K}\Omega_{1}\cdotd^{1/\min\left\{ 1,p\right\} }\cdot\left(4\cdotd\right)^{1+2\left\lceil K+\frac{d+\varepsilon}{\min\left\{ 1,p\right\} }\right\rceil }\cdot\left(\frac{s_{d}}{\varepsilon}\right)^{1/\min\left\{ 1,p\right\} }\cdot\max_{\left|\alpha\right|\leq\left\lceil K+\frac{d+\varepsilon}{\min\left\{ 1,p\right\} }\right\rceil }C^{\left(\alpha\right)}.\qedhere
\]
\end{cor}
\begin{rem*}
As usual, the most important special case is when $\gamma_{i}=\gamma$
is independent of $i\in I$. In this case, validity of Assumption
\ref{assu:GammaCoversOrbit} can be verified easily using Lemma \ref{lem:GammaCoversOrbitAssumptionSimplified}.
The same lemma is also highly helpful if $\left\{ \gamma_{i}\,\middle|\, i\in I\right\} $
is finite, i.e., if only a finite number of different prototypes is
used.
\end{rem*}
\begin{proof}
Since our assumptions clearly include those of Assumption \ref{assu:GammaCoversOrbit},
we only need to verify Assumption \ref{assu:DiscreteBanachFrameAssumptions}.
This means the following:
\begin{itemize}
\item We have $\gamma_{i}\in C^{1}\left(\mathbb{R}^{d}\right)$ and the
gradient $\phi_{i}:=\nabla\gamma_{i}$ is bounded and satisfies $\phi_{i}\in L_{v_{0}}^{1}\left(\mathbb{R}^{d};\mathbb{C}^{d}\right)$,
as well as $\widehat{\phi_{i}}\in C^{\infty}\left(\mathbb{R}^{d};\mathbb{C}^{d}\right)$.
All of these properties except the last are included in our assumptions.
But standard properties of the Fourier transform show $\widehat{\partial_{\ell}\gamma_{i}}\left(\xi\right)=2\pi i\xi_{\ell}\cdot\widehat{\gamma_{i}}\left(\xi\right)$
for $\xi\in\mathbb{R}^{d}$, so that $\widehat{\partial_{\ell}\gamma_{i}}\in C^{\infty}\left(\mathbb{R}^{d}\right)$,
since $\widehat{\gamma_{i}}\in C^{\infty}\left(\mathbb{R}^{d}\right)$.
\item Assumption \ref{assu:MainAssumptions} is satisfied. For this, it
remains—in view of our present assumptions—to check that the operator
$\overrightarrow{A}:\ell_{w^{\min\left\{ 1,p\right\} }}^{r}\left(I\right)\to\ell_{w^{\min\left\{ 1,p\right\} }}^{r}\left(I\right)$
is bounded, where $r:=\max\left\{ q,\frac{q}{p}\right\} $ and $A=\left(A_{j,i}\right)_{j,i\in I}$
is given by
\[
A_{j,i}:=\begin{cases}
\left\Vert \mathcal{F}^{-1}\left(\varphi_{i}\cdot\widehat{\gamma^{\left(j\right)}}\right)\right\Vert _{L_{v_{0}}^{1}}, & \text{if }p\in\left[1,\infty\right],\\
\left(1+\left\Vert T_{j}^{-1}T_{i}\right\Vert \right)^{d}\cdot\left|\det T_{i}\right|^{1-p}\cdot\left\Vert \mathcal{F}^{-1}\left(\varphi_{i}\cdot\widehat{\gamma^{\left(j\right)}}\right)\right\Vert _{L_{v_{0}}^{p}}^{p}, & \text{if }p\in\left(0,1\right).
\end{cases}
\]
\item The infinite matrix $B=\left(B_{j,i}\right)_{j,i\in I}$ defines a
bounded linear operator $\overrightarrow{B}:\ell_{w^{\min\left\{ 1,p\right\} }}^{r}\left(I\right)\to\ell_{w^{\min\left\{ 1,p\right\} }}^{r}\left(I\right)$,
where $r=\max\left\{ q,\frac{q}{p}\right\} $ as above, $\phi_{i}=\nabla\gamma_{i}$
for $i\in I$ and
\[
B_{j,i}:=\begin{cases}
\left(1+\left\Vert T_{j}^{-1}T_{i}\right\Vert \right)^{K+d}\cdot\left\Vert \mathcal{F}^{-1}\left(\varphi_{i}\cdot\widehat{\phi^{\left(j\right)}}\right)\right\Vert _{L_{v_{0}}^{1}}, & \text{if }p\in\left[1,\infty\right],\\
\left(1+\left\Vert T_{j}^{-1}T_{i}\right\Vert \right)^{pK+d}\cdot\left|\det T_{i}\right|^{1-p}\cdot\left\Vert \mathcal{F}^{-1}\left(\varphi_{i}\cdot\widehat{\phi^{\left(j\right)}}\right)\right\Vert _{L_{v_{0}}^{p}}^{p}, & \text{if }p\in\left(0,1\right).
\end{cases}
\]
Here, $\phi^{\left(j\right)}$ is defined as in equation (\ref{eq:NonCompactFilterDefinition}),
with $\gamma_{j}$ replaced by $\phi_{j}$.
\end{itemize}
Hence, in the following, we verify boundedness of $\overrightarrow{A}$
and $\overrightarrow{B}$.
\medskip{}
We first make the auxiliary observation that a matrix operator $\overrightarrow{C}:\ell_{v}^{q}\left(I\right)\to\ell_{v}^{q}\left(I\right)$
is bounded if and only if the operator $\overrightarrow{C_{v}}:\ell^{q}\left(I\right)\to\ell^{q}\left(I\right)$
is bounded, where
\[
\left(C_{v}\right)_{j,i}=\frac{v_{j}}{v_{i}}\cdot C_{j,i}.
\]
This simply comes from the fact that $m_{v}:\ell_{v}^{q}\left(I\right)\to\ell^{q}\left(I\right),\left(x_{i}\right)_{i\in I}\mapsto\left(v_{i}x_{i}\right)_{i\in I}$
is an isometric isomorphism and that
\[
\left[\overrightarrow{C}:\ell_{v}^{q}\left(I\right)\to\ell_{v}^{q}\left(I\right)\right]=m_{v}^{-1}\circ\left(m_{v}\circ\overrightarrow{C}\circ m_{v}^{-1}\right)\circ m_{v},
\]
where a direct calculation shows $m_{v}\circ\overrightarrow{C}\circ m_{v}^{-1}=\overrightarrow{C_{v}}$.
Since $m_{v}$ is isometric, we also get $\vertiii{\smash{\overrightarrow{C_{v}}}}=\vertiii{\smash{\overrightarrow{C}}}$.
\medskip{}
Now, let us first consider the case $p\in\left[1,\infty\right]$.
Here, we want to have $\overrightarrow{A}:\ell_{w}^{q}\left(I\right)\to\ell_{w}^{q}\left(I\right)$
and likewise for $\overrightarrow{B}$. Recall $\gamma^{\left(j\right)}=\gamma_{j}^{\left\llbracket j\right\rrbracket }$
in the notation of Lemma \ref{lem:GramMatrixEstimates}. Hence, an
application of that lemma (with $p=1$) yields, with $C$ as in the
statement of the present corollary,
\begin{align*}
\left(A_{w}\right)_{j,i}=\frac{w_{j}}{w_{i}}\cdot A_{j,i} & \leq\frac{C}{d}\cdot\frac{w_{j}}{w_{i}}\cdot\left[1+\left\Vert T_{j}^{-1}T_{i}\right\Vert \right]^{\left\lceil K+d+\varepsilon\right\rceil }\cdot\left|\det T_{i}\right|^{-1}\cdot\int_{Q_{i}}\max_{\left|\alpha\right|\leq\left\lceil K+d+\varepsilon\right\rceil }\left|\left(\partial^{\alpha}\widehat{\gamma_{j}}\right)\left(S_{j}^{-1}\xi\right)\right|\operatorname{d}\xi\\
& \leq C\cdot\frac{w_{j}}{w_{i}}\cdot\left[1+\left\Vert T_{j}^{-1}T_{i}\right\Vert \right]^{K+d+\left\lceil K+d+\varepsilon\right\rceil }\cdot\left|\det T_{i}\right|^{-1}\cdot\max_{\left|\beta\right|\leq1}\int_{Q_{i}}\max_{\left|\alpha\right|\leq N}\left|\left(\partial^{\alpha}\widehat{\partial^{\beta}\gamma_{j}}\right)\left(S_{j}^{-1}\xi\right)\right|\operatorname{d}\xi\\
& \leq C\cdot M_{j,i}^{1/\min\left\{ 1,q\right\} }.
\end{align*}
Likewise, using $\left|\phi_{j}\right|=\left|\widehat{\nabla\gamma_{j}}\right|\leq\sum_{\ell=1}^{d}\left|\widehat{\partial_{\ell}\gamma_{j}}\right|$,
we get
\begin{align*}
\left(B_{w}\right)_{j,i}=\frac{w_{j}}{w_{i}}\cdot B_{j,i} & \leqd\cdot\frac{w_{j}}{w_{i}}\cdot\left(1+\left\Vert T_{j}^{-1}T_{i}\right\Vert \right)^{K+d}\cdot\max_{\left|\beta\right|=1}\left\Vert \mathcal{F}^{-1}\left(\varphi_{i}\cdot\widehat{\left(\partial^{\beta}\gamma_{j}\right)^{\left\llbracket j\right\rrbracket }}\right)\right\Vert _{L_{v_{0}}^{1}}\\
\left({\scriptstyle \text{Lemma }\ref{lem:GramMatrixEstimates}\text{ with }\partial^{\beta}\gamma_{j}\text{ instead of }\gamma}\right) & \leq C\!\cdot\!\frac{w_{j}}{w_{i}}\!\cdot\!\left(1\!+\!\left\Vert T_{j}^{-1}T_{i}\right\Vert \right)^{K+d+\left\lceil K+d+\varepsilon\right\rceil }\!\max_{\left|\beta\right|=1}\left[\left|\det T_{i}\right|^{-1}\!\!\int_{Q_{i}}\,\max_{\left|\alpha\right|\leq N}\left|\left(\partial^{\alpha}\widehat{\partial^{\beta}\gamma_{j}}\right)\!\!\left(S_{j}^{-1}\xi\right)\right|\operatorname{d}\xi\right]\\
& \leq C\cdot M_{j,i}^{1/\min\left\{ 1,q\right\} }.
\end{align*}
But Lemma \ref{lem:SchursLemma} shows that $\overrightarrow{A_{w}}:\ell^{q}\left(I\right)\to\ell^{q}\left(I\right)$
is bounded as soon as we have $K_{1}:=\sup_{i\in I}\sum_{j\in I}\left(A_{w}\right)_{j,i}^{\min\left\{ 1,q\right\} }<\infty$
and $K_{2}:=\sup_{j\in I}\sum_{i\in I}\left(A_{w}\right)_{j,i}^{\min\left\{ 1,q\right\} }<\infty$.
Further, that lemma also shows $\vertiii{\smash{\overrightarrow{A_{w}}}}\leq\max\left\{ K_{1}^{1/\min\left\{ 1,q\right\} },K_{2}^{1/\min\left\{ 1,q\right\} }\right\} $.
Since we have by assumption that $C_{1}=\sup_{i\in I}\sum_{j\in I}M_{j,i}<\infty$
and $C_{2}=\sup_{j\in I}\sum_{i\in I}M_{j,i}<\infty$, we get $K_{1}^{1/\min\left\{ 1,q\right\} }\leq C\cdot C_{1}^{1/\min\left\{ 1,q\right\} }=C\cdot C_{1}^{1/\tau}$
and likewise $K_{2}^{1/\min\left\{ 1,q\right\} }\leq C\cdot C_{2}^{1/\min\left\{ 1,q\right\} }=C\cdot C_{2}^{1/\tau}$,
so that $\vertiii{\smash{\overrightarrow{A_{w}}}}\leq C\cdot\max\left\{ C_{1}^{1/\tau},C_{2}^{1/\tau}\right\} $.
The same arguments show that $\overrightarrow{B_{w}}:\ell^{q}\left(I\right)\to\ell^{q}\left(I\right)$
is bounded and satisfies $\vertiii{\smash{\overrightarrow{B_{w}}}}\leq C\cdot\max\left\{ C_{1}^{1/\tau},C_{2}^{1/\tau}\right\} $.
In view of the auxiliary observation from above, this completes the
proof in case of $p\in\left[1,\infty\right]$.
\medskip{}
Now, let $p\in\left(0,1\right)$. In this case, we want to have $\overrightarrow{A}:\ell_{w^{p}}^{q/p}\left(I\right)\to\ell_{w^{p}}^{q/p}\left(I\right)$
and likewise for $\overrightarrow{B}$. But Lemma \ref{lem:GramMatrixEstimates}
yields, because of $\gamma^{\left(j\right)}=\gamma_{j}^{\left\llbracket j\right\rrbracket }$,
\begin{align*}
\left(A_{w^{p}}\right)_{j,i} & =\left(\frac{w_{j}}{w_{i}}\right)^{p}\cdot A_{j,i}\\
& =\left(\frac{w_{j}}{w_{i}}\right)^{p}\cdot\left(1+\left\Vert T_{j}^{-1}T_{i}\right\Vert \right)^{d}\cdot\left|\det T_{i}\right|^{1-p}\cdot\left\Vert \mathcal{F}^{-1}\left(\varphi_{i}\cdot\widehat{\gamma^{\left(j\right)}}\right)\right\Vert _{L_{v_{0}}^{p}}^{p}\\
& \leq\left(C/d\right)^{p}\cdot\left(\frac{w_{j}}{w_{i}}\right)^{p}\left(1+\left\Vert T_{j}^{-1}T_{i}\right\Vert \right)^{d+p\left\lceil K+\frac{d+\varepsilon}{p}\right\rceil }\left|\det T_{i}\right|^{1-p}\left|\det T_{i}\right|^{-1}\left(\int_{Q_{i}}\max_{\left|\alpha\right|\leq\left\lceil K+\frac{d+\varepsilon}{p}\right\rceil }\left|\left(\partial^{\alpha}\widehat{\gamma_{j}}\right)\!\left(S_{j}^{-1}\xi\right)\right|\operatorname{d}\xi\right)^{p}\\
& =\left(C/d\right)^{p}\cdot\left(\frac{w_{j}}{w_{i}}\right)^{p}\left(1+\left\Vert T_{j}^{-1}T_{i}\right\Vert \right)^{d+p\left\lceil K+\frac{d+\varepsilon}{p}\right\rceil }\left(\left|\det T_{i}\right|^{-1}\int_{Q_{i}}\max_{\left|\alpha\right|\leq N}\left|\left(\partial^{\alpha}\widehat{\gamma_{j}}\right)\left(S_{j}^{-1}\xi\right)\right|\operatorname{d}\xi\right)^{p}\\
& \leq\left(C/d\right)^{p}\cdot M_{j,i}^{1/\min\left\{ 1,\,\frac{q}{p}\right\} },
\end{align*}
where the last step used
\[
\min\left\{ 1,\frac{q}{p}\right\} \cdot\left(d+p\left\lceil K+\frac{d+\varepsilon}{p}\right\rceil \right)=\min\left\{ p,q\right\} \cdot\left(\frac{d}{p}+\left\lceil K+\frac{d+\varepsilon}{p}\right\rceil \right)\leq\sigma.
\]
Furthermore, since $p\in\left(0,1\right)$, we have $\left(\sum_{\ell=1}^{d}a_{\ell}\right)^{p}\leq\sum_{\ell=1}^{d}a_{\ell}^{p}$
for $a_{1},\dots,a_{d}\geq0$, so that the $L^{p}$-norm
of a vector valued (measurable) function $f:\mathbb{R}^{d}\to\mathbb{C}^{d}$
can be estimated as follows:
\begin{align*}
\left\Vert \left(f_{1},\dots,f_{d}\right)\right\Vert _{L^{p}}^{p} & =\int_{\mathbb{R}^{d}}\left|\left(f_{1},\dots,f_{d}\right)\left(x\right)\right|^{p}\operatorname{d} x\leq\int_{\mathbb{R}^{d}}\left(\sum_{\ell=1}^{d}\left|f_{\ell}\left(x\right)\right|\right)^{p}\operatorname{d} x\\
& \leq\int_{\mathbb{R}^{d}}\sum_{\ell=1}^{d}\left|f_{\ell}\left(x\right)\right|^{p}\operatorname{d} x=\sum_{\ell=1}^{d}\left\Vert f_{\ell}\right\Vert _{L^{p}}^{p}\leqd\cdot\max_{\ell\in\underline{d}}\left\Vert f_{\ell}\right\Vert _{L^{p}}^{p}.
\end{align*}
Consequently,
\begin{align*}
\left(B_{w^{p}}\right)_{j,i} & =\left(\frac{w_{j}}{w_{i}}\right)^{p}\cdot B_{j,i}\\
& =\left(\frac{w_{j}}{w_{i}}\right)^{p}\cdot\left(1+\left\Vert T_{j}^{-1}T_{i}\right\Vert \right)^{pK+d}\cdot\left|\det T_{i}\right|^{1-p}\cdot\left\Vert \mathcal{F}^{-1}\left(\varphi_{i}\cdot\widehat{\phi^{\left(j\right)}}\right)\right\Vert _{L_{v_{0}}^{p}}^{p}\\
\left({\scriptstyle \text{since }\phi_{j}=\nabla\gamma_{j}}\right) & \leqd\cdot\left(\frac{w_{j}}{w_{i}}\right)^{p}\cdot\left(1+\left\Vert T_{j}^{-1}T_{i}\right\Vert \right)^{pK+d}\cdot\left|\det T_{i}\right|^{1-p}\cdot\max_{\left|\beta\right|=1}\left\Vert \mathcal{F}^{-1}\left(\varphi_{i}\cdot\widehat{\left(\partial^{\beta}\gamma_{j}\right)^{\left\llbracket j\right\rrbracket }}\right)\right\Vert _{L_{v_{0}}^{p}}^{p}\\
\left({\scriptstyle \text{Lem. }\ref{lem:GramMatrixEstimates}\text{ /w }\partial^{\beta}\gamma_{j}\text{ inst. of }\gamma_{j}}\right) & \leqd\cdot\left(C/d^{\frac{1}{p}}\right)^{p}\cdot\left(\frac{w_{j}}{w_{i}}\right)^{p}\left(1+\left\Vert T_{j}^{-1}T_{i}\right\Vert \right)^{pK+d}\cdot\left|\det T_{i}\right|^{1-p}\\
& \phantom{\lesssim}\cdot\max_{\left|\beta\right|\leq1}\left[\left(1+\left\Vert T_{j}^{-1}T_{i}\right\Vert \right)^{\left\lceil K+\frac{d+\varepsilon}{p}\right\rceil }\left|\det T_{i}\right|^{-\frac{1}{p}}\int_{Q_{i}}\max_{\left|\alpha\right|\leq\left\lceil K+\frac{d+\varepsilon}{p}\right\rceil }\left|\left(\partial^{\alpha}\widehat{\partial^{\beta}\gamma_{j}}\right)\!\left(S_{j}^{-1}\xi\right)\right|\operatorname{d}\xi\right]^{p}\\
& =C^{p}\cdot\left(\frac{w_{j}}{w_{i}}\right)^{p}\!\left(1\!+\!\left\Vert T_{j}^{-1}T_{i}\right\Vert \right)^{p\left(\frac{d}{p}+K+\left\lceil K+\frac{d+\varepsilon}{p}\right\rceil \right)}\\
& \phantom{=}\cdot\max_{\left|\beta\right|\leq1}\left[\left|\det T_{i}\right|^{-1}\!\int_{Q_{i}}\max_{\left|\alpha\right|\leq N}\left|\left(\partial^{\alpha}\widehat{\partial^{\beta}\gamma_{j}}\right)\!\left(S_{j}^{-1}\xi\right)\right|\operatorname{d}\xi\right]^{p}\\
& =C^{p}\cdot M_{j,i}^{1/\min\left\{ 1,\,\frac{q}{p}\right\} }.
\end{align*}
Now, the remainder of the proof is similar to the case $p\in\left[1,\infty\right]$:
Lemma \ref{lem:SchursLemma} shows that $\overrightarrow{A_{w^{p}}}:\ell^{q/p}\left(I\right)\to\ell^{q/p}\left(I\right)$
is bounded as soon as we have $K_{3}:=\sup_{i\in I}\sum_{j\in I}\left(A_{w^{p}}\right)_{j,i}^{\min\left\{ 1,\frac{q}{p}\right\} }<\infty$
and $K_{4}:=\sup_{j\in I}\sum_{i\in I}\left(A_{w^{p}}\right)_{j,i}^{\min\left\{ 1,\frac{q}{p}\right\} }<\infty$.
Further, that lemma also shows $\vertiii{\smash{\overrightarrow{A_{w^{p}}}}}\leq\max\left\{ K_{3}^{1/\min\left\{ 1,q/p\right\} },K_{4}^{1/\min\left\{ 1,q/p\right\} }\right\} $.
Since we have by assumption that $C_{1}=\sup_{i\in I}\sum_{j\in I}M_{j,i}<\infty$
and $C_{2}=\sup_{j\in I}\sum_{i\in I}M_{j,i}<\infty$, we get
\begin{align*}
\vertiii{\smash{\overrightarrow{A_{w^{p}}}}}^{\max\left\{ 1,\frac{1}{p}\right\} } & =\vertiii{\smash{\overrightarrow{A_{w^{p}}}}}^{1/p}\\
& \leq\max\left\{ K_{3}^{\frac{1}{p}\cdot\frac{1}{\min\left\{ 1,q/p\right\} }},\,K_{4}^{\frac{1}{p}\cdot\frac{1}{\min\left\{ 1,q/p\right\} }}\right\} \\
& =\max\left\{ K_{3}^{1/\min\left\{ p,q\right\} },K_{4}^{1/\min\left\{ p,q\right\} }\right\} \\
& \leq\max\left\{ \left(\sup_{i\in I}\sum_{j\in I}\left[C^{p\cdot\min\left\{ 1,\frac{q}{p}\right\} }\cdot M_{j,i}\right]\right)^{1/\min\left\{ p,q\right\} },\,\left(\sup_{j\in I}\sum_{i\in I}\left[C^{p\cdot\min\left\{ 1,\frac{q}{p}\right\} }\cdot M_{j,i}\right]\right)^{1/\min\left\{ p,q\right\} }\right\} \\
& \leq\max\left\{ C\cdot C_{1}^{1/\tau},\,C\cdot C_{2}^{1/\tau}\right\} =C\cdot\max\left\{ C_{1}^{1/\tau},C_{2}^{1/\tau}\right\} .
\end{align*}
Exactly the same arguments also yield $\vertiii{\smash{\overrightarrow{B_{w^{p}}}}}^{\max\left\{ 1,\frac{1}{p}\right\} }\leq C\cdot\max\left\{ C_{1}^{1/\tau},C_{2}^{1/\tau}\right\} $.
\end{proof}
Our next result yields simplified criteria for the application of
Theorem \ref{thm:AtomicDecomposition}, which yields atomic decompositions
for $\DecompSp{\mathcal{Q}}p{\ell_{w}^{q}}v$.
\begin{cor}
\label{cor:AtomicDecompositionSimplifiedCriteria}Assume that $\left(\varphi_{i}\right)_{i\in I}$
satisfies Assumption \ref{assu:RegularPartitionOfUnity}. Then, for
each $p,q\in\left(0,\infty\right]$, there are
\[
N\in\mathbb{N},\qquad\sigma>0,\qquad\vartheta\geq0\qquad\text{ and }\qquad\tau>0
\]
with the following property: If the families $\Gamma=\left(\gamma_{i}\right)_{i\in I}$
and $\Gamma_{\ell}=\left(\gamma_{i,\ell}\right)_{i\in I}$ (with $\ell\in\left\{ 1,2\right\} $)
satisfy the following properties:
\begin{enumerate}
\item All $\gamma_{i},\gamma_{i,1},\gamma_{i,2}$ are measurable functions
$\mathbb{R}^{d}\to\mathbb{C}$,
\item We have $\gamma_{i,1}\in L_{\left(1+\left|\bullet\right|\right)^{K}}^{1}\left(\mathbb{R}^{d}\right)$
for all $i\in I$.
\item We have $\gamma_{i,2}\in C^{1}\left(\mathbb{R}^{d}\right)$ for all
$i\in I$.
\item With $K_{0}:=K+\frac{d}{\min\left\{ 1,p\right\} }+1$, we
have
\[
\Omega_{4}^{\left(p,K\right)}:=\sup_{i\in I}\left\Vert \gamma_{i,2}\right\Vert _{K_{0}}+\sup_{i\in I}\left\Vert \nabla\gamma_{i,2}\right\Vert _{K_{0}}<\infty,
\]
where $\left\Vert f\right\Vert _{K_{0}}:=\sup_{x\in\mathbb{R}^{d}}\left(1+\left|x\right|\right)^{K_{0}}\left|f\left(x\right)\right|$.
\item We have $\left\Vert \gamma_{i}\right\Vert _{K_{0}}<\infty$ for all
$i\in I$.
\item We have $\gamma_{i}=\gamma_{i,1}\ast\gamma_{i,2}$ for all $i\in I$.
\item We have $\widehat{\gamma_{i,1}},\widehat{\gamma_{i,2}}\in C^{\infty}\left(\mathbb{R}^{d}\right)$
and all partial derivatives of $\widehat{\gamma_{i,1}},\widehat{\gamma_{i,2}}$
are polynomially bounded for all $i\in I$.
\item $\Gamma=\left(\gamma_{i}\right)_{i\in I}$ satisfies Assumption \ref{assu:GammaCoversOrbit}.
\item \label{enu:AtomicDecompositionSimplifiedMainCondition}We have
\[
K_{1}:=\sup_{i\in I}\,\sum_{j\in I}N_{i,j}<\infty\qquad\text{ and }\qquad K_{2}:=\sup_{j\in I}\sum_{i\in I}N_{i,j}<\infty
\]
with
\[
\qquad\qquad N_{i,j}:=\left(\frac{w_{i}}{w_{j}}\cdot\left(\left|\det T_{j}\right|\big/\left|\det T_{i}\right|\right)^{\vartheta}\right)^{\tau}\cdot\left(1+\left\Vert T_{j}^{-1}T_{i}\right\Vert \right)^{\sigma}\cdot\left(\left|\det T_{i}\right|^{-1}\cdot\int_{Q_{i}}\:\max_{\left|\alpha\right|\leq N}\left|\left(\partial^{\alpha}\widehat{\gamma_{j,1}}\right)\left(S_{j}^{-1}\xi\right)\right|\operatorname{d}\xi\right)^{\tau}\!.
\]
\end{enumerate}
Then the families $\Gamma,\Gamma_{1},\Gamma_{2}$ fulfill Assumption
\ref{assu:AtomicDecompositionAssumption} and the family $\Gamma$
satisfies Assumption \ref{assu:GammaCoversOrbit}, so that Theorem
\ref{thm:AtomicDecomposition} is applicable to $\Gamma$.
In fact, the following choices are possible, for an arbitrary $\varepsilon>0$:
\begin{align*}
N & =\left\lceil K+\frac{d+\varepsilon}{\min\left\{ 1,p\right\} }\right\rceil ,\\
\tau & =\min\left\{ 1,p,q\right\} =\begin{cases}
\min\left\{ 1,q\right\} , & \text{if }p\in\left[1,\infty\right],\\
\min\left\{ p,q\right\} , & \text{if }p\in\left(0,1\right),
\end{cases}\\
\sigma & =\begin{cases}
\min\left\{ 1,q\right\} \cdot\left\lceil K+d+\varepsilon\right\rceil , & \text{if }p\in\left[1,\infty\right],\\
\min\left\{ p,q\right\} \cdot\left(\frac{d}{p}+K+\left\lceil K+\frac{d+\varepsilon}{p}\right\rceil \right), & \text{if }p\in\left(0,1\right),
\end{cases}\\
\vartheta & =\begin{cases}
0, & \text{if }p\in\left[1,\infty\right],\\
\frac{1}{p}-1, & \text{if }p\in\left(0,1\right).
\end{cases}
\end{align*}
With these choices, we even have $\vertiii{\smash{\overrightarrow{C}}}^{\max\left\{ 1,\frac{1}{p}\right\} }\leq\Omega\cdot\left(K_{1}^{1/\tau}+K_{2}^{1/\tau}\right)$,
where $\overrightarrow{C}:\ell_{w^{\min\left\{ 1,p\right\} }}^{\max\left\{ q,q/p\right\} }\left(I\right)\to\ell_{w^{\min\left\{ 1,p\right\} }}^{\max\left\{ q,q/p\right\} }\left(I\right)$
is defined as in Assumption \ref{assu:AtomicDecompositionAssumption}
and where
\[
\Omega:=\Omega_{0}^{K}\Omega_{1}\cdot\left(4\cdotd\right)^{1+2\left\lceil K+\frac{d+\varepsilon}{\min\left\{ 1,p\right\} }\right\rceil }\cdot\left(\frac{s_{d}}{\varepsilon}\right)^{1/\min\left\{ 1,p\right\} }\cdot\max_{\left|\alpha\right|\leq N}C^{\left(\alpha\right)}\qedhere
\]
\end{cor}
\begin{proof}
First, our assumptions clearly imply that Assumption \ref{assu:GammaCoversOrbit}
is satisfied. Hence, we only need to verify Assumption \ref{assu:AtomicDecompositionAssumption},
which means the following:
\begin{itemize}
\item We have $\gamma_{i,1},\gamma_{i,2}\in L_{\left(1+\left|\bullet\right|\right)^{K}}^{1}\left(\mathbb{R}^{d}\right)$
for all $i\in I$. For $\gamma_{i,1}$, this is part of our assumptions.
But for $\gamma_{i,2}$, we have $\left\Vert \gamma_{i,2}\right\Vert _{K_{0}}<\infty$,
which implies $\left\Vert \left(1+\left|\bullet\right|\right)^{K}\cdot\gamma_{i,2}\right\Vert _{K_{0}-K}<\infty$.
Because of $K_{0}-K=\frac{d}{\min\left\{ 1,p\right\} }+1\geqd+1$,
this easily implies $\gamma_{i,2}\in L_{\left(1+\left|\bullet\right|\right)^{K}}^{1}\left(\mathbb{R}^{d}\right)$;
see also the remark after Assumption \ref{assu:AtomicDecompositionAssumption}.
\item We have $\gamma_{i}=\gamma_{i,1}\ast\gamma_{i,2}$ for all $i\in I$,
which is part of our assumptions.
\item We have $\widehat{\gamma_{i,1}},\widehat{\gamma_{i,2}}\in C^{\infty}\left(\mathbb{R}^{d}\right)$,
with all partial derivatives of $\widehat{\gamma_{i,1}},\widehat{\gamma_{i,2}}$
being polynomially bounded. Again, this is part of our assumptions.
\item We have $\gamma_{i,2}\in C^{1}\left(\mathbb{R}^{d}\right)$ with
$\nabla\gamma_{i,2}\in L_{v_{0}}^{1}\left(\mathbb{R}^{d}\right)$.
The first of these properties is part of our assumptions and the second
property follows easily from $\left\Vert \nabla\gamma_{i,2}\right\Vert _{K_{0}}\leq\Omega_{4}^{\left(p,K\right)}<\infty$,
cf.\@ the remark after Assumption \ref{assu:AtomicDecompositionAssumption}.
\item We have $\Omega_{4}^{\left(p,K\right)}<\infty$, where the constant
$\Omega_{4}^{\left(p,K\right)}$ is defined as in the present corollary.
Hence, this prerequisite is part of our assumptions.
\item We have $\left\Vert \gamma_{i}\right\Vert _{K_{0}}<\infty$ for all
$i\in I$, which is again part of our assumptions.
\item The operator $\overrightarrow{C}:\ell_{w^{\min\left\{ 1,p\right\} }}^{r}\left(I\right)\to\ell_{w^{\min\left\{ 1,p\right\} }}^{r}\left(I\right)$
is bounded, where $r=\max\left\{ q,\frac{q}{p}\right\} $ and where
\[
C_{i,j}:=\begin{cases}
\left\Vert \mathcal{F}^{-1}\left(\varphi_{i}\cdot\widehat{\gamma_{1}^{\left(j\right)}}\right)\right\Vert _{L_{v_{0}}^{1}}, & \text{if }p\in\left[1,\infty\right],\\
\left(1+\left\Vert T_{j}^{-1}T_{i}\right\Vert \right)^{pK+d}\cdot\left|\det T_{j}\right|^{1-p}\cdot\left\Vert \mathcal{F}^{-1}\left(\varphi_{i}\cdot\widehat{\gamma_{1}^{\left(j\right)}}\right)\right\Vert _{L_{v_{0}}^{p}}^{p}, & \text{if }p\in\left(0,1\right).
\end{cases}
\]
\end{itemize}
Thus, in the remainder of the proof, we only need to verify boundedness
of $\overrightarrow{C}$. First of all, we recall from the proof
of Corollary \ref{cor:BanachFrameSimplifiedCriteria} that we have
$\vertiii{\smash{\overrightarrow{C}}}=\vertiii{\overrightarrow{C_{w^{\min\left\{ 1,p\right\} }}}}_{\ell^{r}\to\ell^{r}}$,
where
\[
\left(C_{w^{\min\left\{ 1,p\right\} }}\right)_{i,j}:=\frac{w_{i}}{w_{j}}\cdot C_{i,j}.
\]
To prove boundedness of $\overrightarrow{C_{w^{\min\left\{ 1,p\right\} }}}$,
we distinguish the cases $p\in\left[1,\infty\right]$ and $p\in\left(0,1\right)$.
\medskip{}
For $p\in\left[1,\infty\right]$, we want to have $\overrightarrow{C_{w}}:\ell^{q}\left(I\right)\to\ell^{q}\left(I\right)$.
But Lemma \ref{lem:GramMatrixEstimates} (with $p=1$ and with $\gamma_{j,1}$
instead of $\gamma$) yields because of $\gamma_{1}^{\left(j\right)}=\gamma_{j,1}^{\left\llbracket j\right\rrbracket }$
that
\begin{align*}
\left(C_{w}\right)_{i,j} & =\frac{w_{i}}{w_{j}}\cdot C_{i,j}=\frac{w_{i}}{w_{j}}\cdot\left\Vert \mathcal{F}^{-1}\left(\varphi_{i}\cdot\widehat{\gamma_{1}^{\left(j\right)}}\right)\right\Vert _{L_{v_{0}}^{1}}\\
& \leq\Omega\cdot\frac{w_{i}}{w_{j}}\cdot\left(1+\left\Vert T_{j}^{-1}T_{i}\right\Vert \right)^{\left\lceil K+d+\varepsilon\right\rceil }\cdot\left|\det T_{i}\right|^{-1}\cdot\int_{Q_{i}}\max_{\left|\alpha\right|\leq N}\left|\left(\partial^{\alpha}\widehat{\gamma_{j,1}}\right)\left(S_{j}^{-1}\eta\right)\right|\,\operatorname{d}\eta\\
& =\Omega\cdot N_{i,j}^{1/\min\left\{ 1,q\right\} }.
\end{align*}
Finally, Lemma \ref{lem:SchursLemma} shows that
\begin{align*}
\vertiii{\smash{\overrightarrow{C_{w}}}}_{\ell^{q}\to\ell^{q}} & \leq\max\left\{ \left(\sup_{i\in I}\sum_{j\in I}\left|\left(C_{w}\right)_{i,j}\right|^{\min\left\{ 1,q\right\} }\right)^{1/\min\left\{ 1,q\right\} },\,\left(\sup_{j\in I}\sum_{i\in I}\left|\left(C_{w}\right)_{i,j}\right|^{\min\left\{ 1,q\right\} }\right)^{1/\min\left\{ 1,q\right\} }\right\} \\
& \leq\Omega\cdot\max\left\{ \left(\sup_{i\in I}\sum_{j\in I}N_{i,j}\right)^{1/\min\left\{ 1,q\right\} },\,\left(\sup_{j\in I}\sum_{i\in I}N_{i,j}\right)^{1/\min\left\{ 1,q\right\} }\right\} \\
& =\Omega\cdot\max\left\{ K_{1}^{1/\tau},\,K_{2}^{1/\tau}\right\} ,
\end{align*}
as desired.
\medskip{}
In case of $p\in\left(0,1\right)$, we want to have $\overrightarrow{C_{w^{p}}}:\ell^{q/p}\left(I\right)\to\ell^{q/p}\left(I\right)$.
But Lemma \ref{lem:GramMatrixEstimates} (with $\gamma_{j,1}$ instead
of $\gamma$) yields
\begin{align*}
\left(C_{w^{p}}\right)_{i,j} & =\left(\frac{w_{i}}{w_{j}}\right)^{p}\cdot C_{i,j}=\left(\frac{w_{i}}{w_{j}}\right)^{p}\cdot\left(1+\left\Vert T_{j}^{-1}T_{i}\right\Vert \right)^{pK+d}\cdot\left|\det T_{j}\right|^{1-p}\cdot\left\Vert \mathcal{F}^{-1}\left(\varphi_{i}\cdot\widehat{\gamma_{1}^{\left(j\right)}}\right)\right\Vert _{L_{v_{0}}^{p}}^{p}\\
& \leq\Omega^{p}\cdot\left(\frac{w_{i}}{w_{j}}\right)^{p}\left(1\!+\!\left\Vert T_{j}^{-1}T_{i}\right\Vert \right)^{p\left(\frac{d}{p}+K+\left\lceil K+\frac{d+\varepsilon}{p}\right\rceil \right)}\left|\det T_{j}\right|^{1-p}\left|\det T_{i}\right|^{-1}\left(\int_{Q_{i}}\,\max_{\left|\alpha\right|\leq N}\left|\left(\partial^{\alpha}\widehat{\gamma_{j,1}}\right)\!\left(S_{j}^{-1}\eta\right)\right|\,\operatorname{d}\eta\right)^{p}\\
& =\Omega^{p}\cdot\left(\frac{w_{i}}{w_{j}}\cdot\left(\frac{\left|\det T_{j}\right|}{\left|\det T_{i}\right|}\right)^{\frac{1}{p}-1}\left(1+\left\Vert T_{j}^{-1}T_{i}\right\Vert \right)^{\frac{d}{p}+K+\left\lceil K+\frac{d+\varepsilon}{p}\right\rceil }\left|\det T_{i}\right|^{-1}\int_{Q_{i}}\,\max_{\left|\alpha\right|\leq N}\left|\left(\partial^{\alpha}\widehat{\gamma_{j,1}}\right)\left(S_{j}^{-1}\eta\right)\right|\,\operatorname{d}\eta\right)^{p}\\
& \leq\Omega^{p}\cdot N_{i,j}^{1/\min\left\{ 1,\frac{q}{p}\right\} }.
\end{align*}
Finally, Lemma \ref{lem:SchursLemma} shows that
\begin{align*}
\vertiii{\smash{\overrightarrow{C_{w^{p}}}}}_{\ell^{q/p}\to\ell^{q/p}}^{1/p} & \leq\left[\max\left\{ \left(\sup_{i\in I}\sum_{j\in I}\left(C_{w^{p}}\right)_{i,j}^{\min\left\{ 1,\frac{q}{p}\right\} }\right)^{1/\min\left\{ 1,\frac{q}{p}\right\} },\left(\sup_{j\in I}\sum_{i\in I}\left(C_{w^{p}}\right)_{i,j}^{\min\left\{ 1,\frac{q}{p}\right\} }\right)^{1/\min\left\{ 1,\frac{q}{p}\right\} }\right\} \right]^{1/p}\\
& \leq\Omega\cdot\max\left\{ \left(\sup_{i\in I}\sum_{j\in I}N_{i,j}\right)^{1/\min\left\{ p,q\right\} },\left(\sup_{j\in I}\sum_{i\in I}N_{i,j}\right)^{1/\min\left\{ p,q\right\} }\right\} \\
& =\Omega\cdot\max\left\{ K_{1}^{1/\tau},K_{2}^{1/\tau}\right\} ,
\end{align*}
as desired.
\end{proof}
One remaining limitation of Corollary \ref{cor:AtomicDecompositionSimplifiedCriteria}
is the assumption $\gamma_{i}=\gamma_{i,1}\ast\gamma_{i,2}$ with
certain assumptions on $\gamma_{i,1}$ and $\gamma_{i,2}$. For a
given function $\gamma$ (or $\gamma_{i}$), it can be cumbersome
to verify that it can be factorized as the convolution product of
two such functions.
Hence, we close this section by providing more readily verifiable
criteria which ensure that such a factorization is possible. For reasons
that will become clear later, we begin with the following technical
result:
\begin{lem}
\label{lem:ChineseBracketDerivative}For $\xi\in\mathbb{R}^{d}$,
set $\left\{ \xi\right\} :=1+\left|\xi\right|^{2}$. Then, for each
$\theta\in\mathbb{R}$ and each $\alpha\in\mathbb{N}_{0}^{d}$, there is
a polynomial $P_{\theta,\alpha}\in\mathbb{R}\left[\xi_{1},\dots,\xi_{d}\right]$
of degree $\deg P_{\theta,\alpha}\leq\left|\alpha\right|$ satisfying
\[
\partial^{\alpha}\left\{ \xi\right\} ^{\theta}=\left\{ \xi\right\} ^{\theta-\left|\alpha\right|}\cdot P_{\theta,\alpha}\left(\xi\right)\qquad\forall\xi\in\mathbb{R}^{d},
\]
as well as $\left|P_{\theta,\alpha}\left(\xi\right)\right|\leq C\cdot\left(1+\left|\xi\right|\right)^{\left|\alpha\right|}$
for all $\xi\in\mathbb{R}^{d}$, where
\[
C=\left|\alpha\right|!\cdot\left[2\cdot\left(1+d+\left|\theta\right|\right)\right]^{\left|\alpha\right|}.
\]
In particular, we have
\[
\left|\partial^{\alpha}\left\{ \xi\right\} ^{\theta}\right|\leq2^{\left|\theta\right|+\left|\alpha\right|}C\cdot\left(1+\left|\xi\right|\right)^{2\theta-\left|\alpha\right|}\leq2^{\left|\theta\right|+\left|\alpha\right|}C\cdot\left(1+\left|\xi\right|\right)^{2\theta}\qquad\forall\xi\in\mathbb{R}^{d}.\qedhere
\]
\end{lem}
\begin{proof}
We prove existence of the polynomial $P_{\theta,\alpha}$ by induction
on $N=\left|\alpha\right|\in\mathbb{N}_{0}$. But to do this, we need a slightly
different formulation of the claim: For $P=\sum_{\sigma\in\mathbb{N}_{0}^{d}}c_{\sigma}\xi^{\sigma}\in\mathbb{R}\left[\xi_{1},\dots,\xi_{d}\right]$,
we define $\left\Vert P\right\Vert _{\ast}:=\sum_{\sigma\in\mathbb{N}_{0}^{d}}\left|c_{\sigma}\right|$.
Below, we will prove by induction on $N=\left|\alpha\right|\in\mathbb{N}_{0}$
that the polynomial $P_{\theta,\alpha}$ satisfying $\partial^{\alpha}\left\{ \xi\right\} ^{\theta}$=$\left\{ \xi\right\} ^{\theta-\left|\alpha\right|}\cdot P_{\theta,\alpha}\left(\xi\right)$
can be chosen to satisfy $\left\Vert P_{\theta,\alpha}\right\Vert _{\ast}\leq C$
with $C$ as in the statement of the lemma. This will imply the claim,
since, for suitable coefficients $c_{\sigma}=c_{\sigma}\left(P_{\theta,\alpha}\right)$,
we have
\begin{align*}
\left|P_{\theta,\alpha}\left(\xi\right)\right| & \leq\sum_{\left|\sigma\right|\leq\left|\alpha\right|}\left|c_{\sigma}\right|\cdot\left|\xi^{\sigma}\right|\leq\sum_{\left|\sigma\right|\leq\left|\alpha\right|}\left|c_{\sigma}\right|\cdot\left(1+\left|\xi\right|\right)^{\left|\sigma\right|}\\
& \leq\left(1+\left|\xi\right|\right)^{\left|\alpha\right|}\cdot\left\Vert P_{\theta,\alpha}\right\Vert _{\ast}\leq C\cdot\left(1+\left|\xi\right|\right)^{\left|\alpha\right|}
\end{align*}
for all $\xi\in\mathbb{R}^{d}$.
It remains to prove the modified claim by induction on $N$. For $N=0$,
all properties are trivially satisfied for $P_{\theta,\alpha}\equiv1$.
For the induction step, we observe that each $\alpha\in\mathbb{N}_{0}^{d}$
with $\left|\alpha\right|=N+1$ can be written as $\alpha=\beta+e_{j}$
for some $j\in\underline{d}$, where $e_{j}$ is the $j$-th
standard basis vector and where $\beta\in\mathbb{N}_{0}^{d}$ with
$\left|\beta\right|=N$. Now, a direct calculation yields
\begin{align*}
\partial^{\alpha}\left\{ \xi\right\} ^{\theta} & =\partial_{j}\partial^{\beta}\left\{ \xi\right\} ^{\theta}=\partial_{j}\left[\left\{ \xi\right\} ^{\theta-\left|\beta\right|}\cdot P_{\theta,\beta}\left(\xi\right)\right]\\
& =\left\{ \xi\right\} ^{\theta-\left|\beta\right|}\cdot\partial_{j}P_{\theta,\beta}\left(\xi\right)+P_{\theta,\beta}\left(\xi\right)\cdot\left(\theta-\left|\beta\right|\right)\cdot\left\{ \xi\right\} ^{\theta-\left|\beta\right|-1}\cdot\partial_{j}\left\{ \xi\right\} \\
& =\left\{ \xi\right\} ^{\theta-\left|\alpha\right|}\left[\left\{ \xi\right\} \cdot\partial_{j}P_{\theta,\beta}\left(\xi\right)+2\left(\theta-\left|\beta\right|\right)\cdot\xi_{j}\cdot P_{\theta,\beta}\left(\xi\right)\right]\\
& =:\left\{ \xi\right\} ^{\theta-\left|\alpha\right|}\cdot P_{\theta,\alpha}\left(\xi\right).
\end{align*}
Since $\deg\left[\left\{ \xi\right\} \cdot\partial_{j}P_{\theta,\beta}\right]\leq2+\deg P_{\theta,\beta}-1\leq\left|\beta\right|+1=\left|\alpha\right|$,
it is not hard to see that $\deg P_{\theta,\alpha}\leq\left|\alpha\right|$.
Next, observe for $\sigma\in\mathbb{N}_{0}^{d}$ with $\sigma_{j}\geq1$
that
\[
\partial_{j}\xi^{\sigma}=\prod_{\ell\neq j}\xi_{\ell}^{\sigma_{\ell}}\cdot\partial_{j}\xi_{j}^{\sigma_{j}}=\sigma_{j}\cdot\xi^{\sigma-e_{j}}\qquad\forall\xi\in\mathbb{R}^{d}.
\]
Furthermore, $\partial_{j}\xi^{\sigma}\equiv0$ in case of $\sigma_{j}=0$.
For $P_{\theta,\beta}\left(\xi\right)=\sum_{\left|\sigma\right|\leq\left|\beta\right|}c_{\sigma}\xi^{\sigma}$,
this implies
\[
\left\Vert \partial_{j}P_{\theta,\beta}\right\Vert _{\ast}\leq\sum_{\left|\sigma\right|\leq\left|\beta\right|}\left|c_{\sigma}\right|\cdot\left\Vert \partial_{j}\xi^{\sigma}\right\Vert _{\ast}\leq\sum_{\substack{\left|\sigma\right|\leq\left|\beta\right|\\
\sigma_{j}\geq1
}
}\sigma_{j}\cdot\left|c_{\sigma}\right|\leq\left|\beta\right|\cdot\left\Vert P_{\theta,\beta}\right\Vert _{\ast}.
\]
Furthermore, since $\left\Vert \xi^{\sigma}\cdot P\right\Vert _{\ast}=\left\Vert P\right\Vert _{\ast}$
for each polynomial $P\in\mathbb{R}\left[\xi_{1},\dots,\xi_{d}\right]$
and each $\sigma\in\mathbb{N}_{0}^{d}$ and since we have $\left\{ \xi\right\} =1+\sum_{\ell=1}^{d}\xi^{2e_{\ell}}$,
we get
\begin{align*}
\left\Vert P_{\theta,\alpha}\right\Vert _{\ast} & \leq\left\Vert \partial_{j}P_{\theta,\beta}\right\Vert _{\ast}+\sum_{\ell=1}^{d}\left\Vert \xi^{2e_{\ell}}\cdot\partial_{j}P_{\theta,\beta}\right\Vert _{\ast}+2\left|\theta-\left|\beta\right|\right|\cdot\left\Vert \xi_{j}\cdot P_{\theta,\beta}\right\Vert _{\ast}\\
& \leq\left(1+d\right)\cdot\left\Vert \partial_{j}P_{\theta,\beta}\right\Vert _{\ast}+2\left(\left|\theta\right|+\left|\beta\right|\right)\cdot\left\Vert P_{\theta,\beta}\right\Vert _{\ast}\\
& \leq\left\Vert P_{\theta,\beta}\right\Vert _{\ast}\left[\left(1+d\right)\cdot\left|\beta\right|+2\left(\left|\theta\right|+\left|\beta\right|\right)\right]\\
& \leq\left\Vert P_{\theta,\beta}\right\Vert _{\ast}\left[\left(1+d\right)\cdot\left(1+\left|\beta\right|\right)+2\left(1+\left|\theta\right|\right)\left(1+\left|\beta\right|\right)\right]\\
& \leq\left|\alpha\right|\cdot\left[\left(1+d\right)+2\left(1+\left|\theta\right|\right)\right]\cdot\left\Vert P_{\theta,\beta}\right\Vert _{\ast}\\
\left({\scriptstyle \text{since }d\geq1}\right) & \leq\left|\alpha\right|\cdot2\left(1+d+\left|\theta\right|\right)\cdot\left\Vert P_{\theta,\beta}\right\Vert _{\ast}.
\end{align*}
Since $\left\Vert P_{\theta,\beta}\right\Vert _{\ast}\leq\left|\beta\right|!\cdot\left[2\left(1+d+\left|\theta\right|\right)\right]^{\left|\beta\right|}$
by induction and since $\left|\alpha\right|=\left|\beta\right|+1$,
the induction is complete.
It remains to verify the final statement of the lemma. To this end,
note
\[
\frac{1}{2}\left(1+\left|\xi\right|\right)^{2}\leq\left\{ \xi\right\} =1+\left|\xi\right|^{2}\leq\left(1+\left|\xi\right|\right)^{2}\leq2\cdot\left(1+\left|\xi\right|\right)^{2},
\]
so that $\left\{ \xi\right\} ^{\varrho}\leq2^{\left|\varrho\right|}\cdot\left(1+\left|\xi\right|\right)^{2\varrho}$
for all $\varrho\in\mathbb{R}$ and $\xi\in\mathbb{R}^{d}$. Hence,
\begin{align*}
\left|\partial^{\alpha}\left\{ \xi\right\} ^{\theta}\right|=\left|\left\{ \xi\right\} ^{\theta-\left|\alpha\right|}\cdot P_{\theta,\alpha}\left(\xi\right)\right| & \leq2^{\left|\theta-\left|\alpha\right|\right|}\cdot\left(1+\left|\xi\right|\right)^{2\theta-2\left|\alpha\right|}\cdot\left|P_{\theta,\alpha}\left(\xi\right)\right|\\
& \leq2^{\left|\theta\right|+\left|\alpha\right|}C\cdot\left(1+\left|\xi\right|\right)^{2\theta-\left|\alpha\right|},
\end{align*}
as claimed.
\end{proof}
\begin{lem}
\label{lem:ConvolutionFactorization}Let $\varrho\in L^{1}\left(\mathbb{R}^{d}\right)$
with $\varrho\geq0$. Let $N\geqd+1$ and assume that $\gamma\in L^{1}\left(\mathbb{R}^{d}\right)$
satisfies $\widehat{\gamma}\in C^{N}\left(\mathbb{R}^{d}\right)$
with
\[
\left|\partial^{\alpha}\widehat{\gamma}\left(\xi\right)\right|\leq\varrho\left(\xi\right)\cdot\left(1+\left|\xi\right|\right)^{-\left(d+1+\varepsilon\right)}\qquad\forall\xi\in\mathbb{R}^{d}\quad\forall\alpha\in\mathbb{N}_{0}^{d}\text{ with }\left|\alpha\right|\leq N
\]
for some $\varepsilon\in\left(0,1\right]$.
Then there are functions $\gamma_{1}\in C_{0}\left(\mathbb{R}^{d}\right)\cap L^{1}\left(\mathbb{R}^{d}\right)$
and $\gamma_{2}\in C^{1}\left(\mathbb{R}^{d}\right)\cap W^{1,1}\left(\mathbb{R}^{d}\right)$
with $\gamma=\gamma_{1}\ast\gamma_{2}$ and with the following additional
properties:
\begin{enumerate}
\item We have $\left\Vert \gamma_{2}\right\Vert _{K}\leq s_{d}\cdot2^{1+d+3K}\cdot K!\cdot\left(1+d\right)^{1+2K}$
and $\left\Vert \nabla\gamma_{2}\right\Vert _{K}\leq\frac{s_{d}}{\varepsilon}\cdot2^{4+d+3K}\cdot\left(1+d\right)^{2\left(1+K\right)}\cdot\left(K+1\right)!$
for all $K\in\mathbb{N}_{0}$, where as usual $\left\Vert g\right\Vert _{K}:=\sup_{x\in\mathbb{R}^{d}}\left(1+\left|x\right|\right)^{K}\left|g\left(x\right)\right|$.
\item We have $\widehat{\gamma_{2}}\in C^{\infty}\left(\mathbb{R}^{d}\right)$
with all partial derivatives of $\widehat{\gamma_{2}}$ being polynomially
bounded (even bounded).
\item If $\widehat{\gamma}\in C^{\infty}\left(\mathbb{R}^{d}\right)$ with
all partial derivatives being polynomially bounded, the same also
holds for $\widehat{\gamma_{1}}$.
\item We have $\left\Vert \gamma_{1}\right\Vert _{N}\leq\left(1+d\right)^{1+2N}\cdot2^{1+d+4N}\cdot N!\cdot\left\Vert \varrho\right\Vert _{L^{1}}$
and $\left\Vert \gamma\right\Vert _{N}\leq\left(1+d\right)^{N+1}\cdot\left\Vert \varrho\right\Vert _{L^{1}}$.
\item We have $\left|\partial^{\alpha}\widehat{\gamma_{1}}\left(\xi\right)\right|\leq2^{1+d+4N}\cdot N!\cdot\left(1+d\right)^{N}\cdot\varrho\left(\xi\right)$
for all $\xi\in\mathbb{R}^{d}$ and $\alpha\in\mathbb{N}_{0}^{d}$
with $\left|\alpha\right|\leq N$.\qedhere
\end{enumerate}
\end{lem}
\begin{rem*}
For concrete special cases of $\mathcal{Q}$, this lemma will be applied
as follows: In most cases, one can find a suitable function $\varrho$
as above such that property (\ref{enu:AtomicDecompositionSimplifiedMainCondition})
of Corollary \ref{cor:AtomicDecompositionSimplifiedCriteria} is satisfied
as soon as all $\gamma_{j,1}\in L^{1}\left(\mathbb{R}^{d}\right)$
satisfy $\left|\partial^{\alpha}\widehat{\gamma_{j,1}}\left(\xi\right)\right|\lesssim\varrho\left(\xi\right)$
uniformly in $\left|\alpha\right|\leq N$, $j\in I$ and $\xi\in\mathbb{R}^{d}$.
In this case, the preceding lemma shows that if we instead assume
$\left|\partial^{\alpha}\widehat{\gamma_{j}}\left(\xi\right)\right|\leq\varrho\left(\xi\right)\cdot\left(1+\left|\xi\right|\right)^{-\left(d+1+\varepsilon\right)}$
for all $\alpha,j,\xi$ as above, then we can write $\gamma_{j}=\gamma_{j,1}\ast\gamma_{j,2}$
such that $\left|\partial^{\alpha}\widehat{\gamma_{j,1}}\left(\xi\right)\right|\lesssim\varrho\left(\xi\right)$
uniformly in $\alpha,j,\xi$ as above, so that the family $\left(\gamma_{j,1}\right)_{j\in I}$
satisfies property (\ref{enu:AtomicDecompositionSimplifiedMainCondition}).
Furthermore, the family $\left(\gamma_{j,2}\right)_{j\in I}$ satisfies
all assumptions of Corollary \ref{cor:AtomicDecompositionSimplifiedCriteria}.
By possibly enlarging $N$ for the application of the lemma, it is
then not hard to ensure that all prerequisites of Corollary \ref{cor:AtomicDecompositionSimplifiedCriteria}
are satisfied. Hence, Lemma \ref{lem:ConvolutionFactorization} essentially
solves the \emph{factorization problem} mentioned before Lemma \ref{lem:ChineseBracketDerivative}.
\end{rem*}
\begin{proof}
We will use the notation $\left\{ \xi\right\} =1+\left|\xi\right|^{2}$
from Lemma \ref{lem:ChineseBracketDerivative}, as well as $\left\langle \xi\right\rangle :=\left\{ \xi\right\} ^{1/2}$.
With this notation, define $g\in C^{\infty}\left(\mathbb{R}^{d}\right)$
by $g:\mathbb{R}^{d}\to\left(0,\infty\right),\xi\mapsto\left\{ \xi\right\} ^{-\frac{d+1+\varepsilon}{2}}=\left\langle \xi\right\rangle ^{-\left(d+1+\varepsilon\right)}$.
In view of equation (\ref{eq:StandardDecayLpEstimate}) and since
$\left\langle \xi\right\rangle \geq\frac{1}{2}\left(1+\left|\xi\right|\right)$,
it is not hard to see $g\in L^{1}\left(\mathbb{R}^{d}\right)$, so
that $\gamma_{2}:=\mathcal{F}^{-1}g\in C_{0}\left(\mathbb{R}^{d}\right)$
is well-defined.
Next, let $K\in\mathbb{N}_{0}$ be arbitrary. For $\alpha\in\mathbb{N}_{0}^{d}$
with $\left|\alpha\right|\leq K$, Lemma \ref{lem:ChineseBracketDerivative}
(with $\theta=-\frac{d+1+\varepsilon}{2}$) shows
\begin{equation}
\left|\partial^{\alpha}g\left(\xi\right)\right|\leq2^{1+d+K}\cdot K!\cdot\left[4\cdot\left(1+d\right)\right]^{K}\cdot\left(1+\left|\xi\right|\right)^{-\left(d+1+\varepsilon\right)}=:C_{d,K}\cdot\left(1+\left|\xi\right|\right)^{-\left(d+1+\varepsilon\right)}\qquad\forall\xi\in\mathbb{R}^{d}.\label{eq:ChineseBracketDerivativeSelfBound}
\end{equation}
In particular, this implies $g\in W^{K,1}\left(\mathbb{R}^{d}\right)$.
In view of Lemma \ref{lem:PointwiseFourierDecayEstimate}, we thus
get
\begin{align*}
\left(1+\left|x\right|\right)^{K}\cdot\left|\gamma_{2}\left(x\right)\right|=\left(1+\left|x\right|\right)^{K}\cdot\left|\mathcal{F}^{-1}g\left(x\right)\right| & \leq\left(1+d\right)^{K}\cdot\left(\left|\left(\mathcal{F}^{-1}g\right)\left(x\right)\right|+\sum_{m=1}^{d}\left|\left[\mathcal{F}^{-1}\left(\partial_{m}^{K}g\right)\right]\left(x\right)\right|\right)\\
\left({\scriptstyle \text{since }\left|\mathcal{F}^{-1}h\right|\leq\left\Vert h\right\Vert _{L^{1}}}\right) & \leq\left(1+d\right)^{K}\cdot C_{d,K}\cdot\left\Vert \left(1+\left|\bullet\right|\right)^{-\left(d+1\right)}\right\Vert _{L^{1}}\cdot\left(1+d\right)\\
\left({\scriptstyle \text{eq. }\eqref{eq:StandardDecayLpEstimate}}\right) & \leq s_{d}\cdot2^{1+d+3K}\cdot K!\cdot\left(1+d\right)^{1+2K}
\end{align*}
for all $x\in\mathbb{R}^{d}$, and thus $\left\Vert \gamma_{2}\right\Vert _{K}\leq s_{d}\cdot2^{1+d+3K}\cdot K!\cdot\left(1+d\right)^{1+2K}<\infty$
for arbitrary $K\in\mathbb{N}_{0}$, as desired. In particular, $\gamma_{2}\in L^{1}\left(\mathbb{R}^{d}\right)$,
so that $\widehat{\gamma_{2}}=\mathcal{F}\Fourier^{-1}g=g$ by Fourier
inversion. Hence, equation (\ref{eq:ChineseBracketDerivativeSelfBound})
shows that all partial derivatives of $\widehat{\gamma_{2}}=g$ are
bounded.
Next, we want to estimate $\left\Vert \nabla\gamma_{2}\right\Vert _{K}$.
To this end, we observe for arbitrary $j\in\underline{d}$
that
\[
\left|\xi_{j}\cdot g\left(\xi\right)\right|\leq\left\{ \xi\right\} ^{1/2}\cdot\left|g\left(\xi\right)\right|=\left\{ \xi\right\} ^{-\frac{d+\varepsilon}{2}}=\left\langle \xi\right\rangle ^{-\left(d+\varepsilon\right)}\in L^{1}\left(\mathbb{R}^{d}\right),
\]
so that we can differentiate under the integral in the definition
of $\gamma_{2}\left(x\right)=\left(\mathcal{F}^{-1}g\right)\left(x\right)$
to conclude for $g_{j}\left(\xi\right):=\xi_{j}\cdot g\left(\xi\right)$
that $\gamma_{2}\in C^{1}\left(\mathbb{R}^{d}\right)$ with derivative
\[
\partial_{j}\gamma_{2}\left(x\right)=2\pi i\cdot\int_{\mathbb{R}^{d}}\xi_{j}\cdot g\left(\xi\right)\cdot e^{2\pi i\left\langle x,\xi\right\rangle }\operatorname{d}\xi=2\pi i\cdot\left(\mathcal{F}^{-1}g_{j}\right)\left(x\right).
\]
Now, we want to apply Lemma \ref{lem:PointwiseFourierDecayEstimate}
again to derive a bound for $\partial_{j}\gamma_{2}\left(x\right)$,
which requires us to bound the derivatives $\partial^{\alpha}g_{j}$.
To this end, we observe $\partial^{\alpha}\xi_{j}\equiv0$ unless
$\alpha=0$, in which case we have $\partial^{\alpha}\xi_{j}=\xi_{j}$,
or unless $\alpha=e_{j}$, in which case we have $\partial^{\alpha}\xi_{j}=1$.
In combination with Leibniz's rule, this yields for $\left|\alpha\right|\leq K$
the estimate
\begin{align*}
\left|\partial^{\alpha}g_{j}\left(\xi\right)\right| & \leq\sum_{\beta\leq\alpha}\binom{\alpha}{\beta}\left|\partial^{\beta}\xi_{j}\right|\left|\partial^{\alpha-\beta}g\left(\xi\right)\right|\\
& =\begin{cases}
\left|\xi_{j}\right|\cdot\left|\partial^{\alpha}g\left(\xi\right)\right|, & \text{if }\alpha_{j}=0,\\
\left|\xi_{j}\right|\cdot\left|\partial^{\alpha}g\left(\xi\right)\right|+\binom{\alpha}{e_{j}}\cdot\left|\partial^{\alpha-e_{j}}g\left(\xi\right)\right|, & \text{if }\alpha_{j}\geq1
\end{cases}\\
\left({\scriptstyle \text{eq. }\eqref{eq:ChineseBracketDerivativeSelfBound}}\right) & \leq\begin{cases}
C_{d,K}\cdot\left(1+\left|\xi\right|\right)^{-\left(d+1+\varepsilon\right)}\cdot\left|\xi_{j}\right|, & \text{if }\alpha_{j}=0,\\
C_{d,K}\cdot\left(\left|\xi_{j}\right|+\binom{\alpha}{e_{j}}\right)\cdot\left(1+\left|\xi\right|\right)^{-\left(d+1+\varepsilon\right)}, & \text{if }\alpha_{j}\geq1
\end{cases}\\
\left({\scriptstyle \text{since }\binom{\alpha}{e_{j}}=\binom{\alpha_{j}}{1}=\alpha_{j}\leq\left|\alpha\right|\leq K}\right) & \leq C_{d,K}\cdot\left(1+K\right)\cdot\left(1+\left|\xi\right|\right)^{-\left(d+\varepsilon\right)}.
\end{align*}
In particular, this implies $g_{j}\in W^{K,1}\left(\mathbb{R}^{d}\right)$.
Hence, another application of Lemma \ref{lem:PointwiseFourierDecayEstimate}
and equation (\ref{eq:StandardDecayLpEstimate}) yields
\begin{align*}
\left(1+\left|x\right|\right)^{K}\cdot\left|\partial_{j}\gamma_{2}\left(x\right)\right| & =2\pi\cdot\left(1+\left|x\right|\right)^{K}\cdot\left|\mathcal{F}^{-1}g_{j}\left(x\right)\right|\\
\left({\scriptstyle \text{eq. }\eqref{eq:PointwiseFourierDecayEstimate}}\right) & \leq8\cdot\left(1+d\right)^{K}\cdot\left(\left|\left(\mathcal{F}^{-1}g_{j}\right)\left(x\right)\right|+\sum_{m=1}^{d}\left|\left[\mathcal{F}^{-1}\left(\partial_{m}^{K}g_{j}\right)\right]\left(x\right)\right|\right)\\
& \leq8\cdot\left(1+d\right)^{K}\cdot C_{d,K}\cdot\left(1+K\right)\cdot\left(1+d\right)\cdot\left\Vert \left(1+\left|\bullet\right|\right)^{-\left(d+\varepsilon\right)}\right\Vert _{L^{1}}\\
\left({\scriptstyle \text{eq. }\eqref{eq:StandardDecayLpEstimate}}\right) & \leq\frac{s_{d}}{\varepsilon}\cdot2^{4+d+3K}\cdot\left(1+d\right)^{1+2K}\cdot\left(K+1\right)!
\end{align*}
and hence $\left\Vert \nabla\gamma_{2}\right\Vert _{K}\leq\frac{s_{d}}{\varepsilon}\cdot2^{4+d+3K}\cdot\left(1+d\right)^{2\left(1+K\right)}\cdot\left(K+1\right)!$
for arbitrary $K\in\mathbb{N}_{0}$, as claimed. In particular, $\nabla\gamma_{2}\in L^{1}\left(\mathbb{R}^{d}\right)$,
so that $\gamma_{2}\in C^{1}\left(\mathbb{R}^{d}\right)\cap W^{1,1}\left(\mathbb{R}^{d}\right)$,
as claimed.
It remains to construct $\gamma_{1}$ with the desired properties.
To this end, define $h:\mathbb{R}^{d}\to\mathbb{C},\xi\mapsto\widehat{\gamma}\left(\xi\right)\cdot\left\{ \xi\right\} ^{\frac{d+1+\varepsilon}{2}}$,
note $h\in C^{N}\left(\mathbb{R}^{d}\right)$ and observe that Lemma
\ref{lem:ChineseBracketDerivative} shows for arbitrary $\beta\in\mathbb{N}_{0}^{d}$
with $\left|\beta\right|\leq N$ that
\begin{equation}
\left|\partial^{\beta}\left\{ \xi\right\} ^{\frac{d+1+\varepsilon}{2}}\right|\leq C_{d,N}\cdot\left(1+\left|\xi\right|\right)^{d+1+\varepsilon}\label{eq:FactorizationLemmaChineseBracketPositiveExponentDerivative}
\end{equation}
with the same constant $C_{d,N}$ (with $N=K$) as in equation
(\ref{eq:ChineseBracketDerivativeSelfBound}). In combination with
Leibniz's rule and the $d$-dimensional binomial theorem
(cf.\@ \cite[Section 8.1, Exercise 2.b]{FollandRA}), this yields
for arbitrary $\alpha\in\mathbb{N}_{0}^{d}$ with $\left|\alpha\right|\leq N$
that
\begin{align}
\left|\partial^{\alpha}h\left(\xi\right)\right| & \leq\sum_{\beta\leq\alpha}\binom{\alpha}{\beta}\cdot\left|\partial^{\beta}\left\{ \xi\right\} ^{\frac{d+1+\varepsilon}{2}}\right|\cdot\left|\partial^{\alpha-\beta}\widehat{\gamma}\left(\xi\right)\right|\nonumber \\
\left({\scriptstyle \text{assump. on }\widehat{\gamma}\text{ and eq. \eqref{eq:FactorizationLemmaChineseBracketPositiveExponentDerivative}}}\right) & \leq\varrho\left(\xi\right)\cdot\left(1+\left|\xi\right|\right)^{-\left(d+1+\varepsilon\right)}C_{d,N}\left(1+\left|\xi\right|\right)^{d+1+\varepsilon}\cdot\sum_{\beta\leq\alpha}\binom{\alpha}{\beta}\nonumber \\
& \leq2^{N}C_{d,N}\cdot\varrho\left(\xi\right)\in L^{1}\left(\smash{\mathbb{R}^{d}}\right).\label{eq:FactorizationLemmaFirstFactorFourierEstimate}
\end{align}
In particular, $h\in L^{1}\left(\mathbb{R}^{d}\right)$, so that
$\gamma_{1}:=\mathcal{F}^{-1}h\in C_{0}\left(\mathbb{R}^{d}\right)$
is well-defined. Furthermore, we get $h\in W^{N,1}\left(\mathbb{R}^{d}\right)$.
Hence, we can invoke Lemma \ref{lem:PointwiseFourierDecayEstimate}
once again to derive
\begin{align*}
\left(1+\left|x\right|\right)^{N}\cdot\left|\gamma_{1}\left(x\right)\right| & =\left(1+\left|x\right|\right)^{N}\cdot\left|\left(\mathcal{F}^{-1}h\right)\left(x\right)\right|\\
\left({\scriptstyle \text{eq. }\eqref{eq:PointwiseFourierDecayEstimate}}\right) & \leq\left(1+d\right)^{N}\cdot\left(\left|\mathcal{F}^{-1}h\left(x\right)\right|+\sum_{m=1}^{d}\left|\left[\mathcal{F}^{-1}\left(\partial_{m}^{N}h\right)\right]\left(x\right)\right|\right)\\
& \leq\left(1+d\right)^{N+1}\cdot2^{N}C_{d,N}\cdot\left\Vert \varrho\right\Vert _{L^{1}}\\
& \leq\left(1+d\right)^{1+2N}\cdot2^{1+d+4N}\cdot N!\cdot\left\Vert \varrho\right\Vert _{L^{1}}<\infty
\end{align*}
for all $x\in\mathbb{R}^{d}$, so that $\left\Vert \gamma_{1}\right\Vert _{N}\leq\left(1+d\right)^{1+2N}\cdot2^{1+d+4N}\cdot N!\cdot\left\Vert \varrho\right\Vert _{L^{1}}$,
as claimed. Since $N\geqd+1$ by assumption, equation (\ref{eq:StandardDecayLpEstimate})
implies in particular that $\gamma_{1}\in L^{1}\left(\mathbb{R}^{d}\right)$.
Hence, Fourier inversion yields $\widehat{\gamma_{1}}=\mathcal{F}\Fourier^{-1}h=h$,
so that equation (\ref{eq:FactorizationLemmaFirstFactorFourierEstimate})
yields the claimed estimate for $\left|\partial^{\alpha}\widehat{\gamma_{1}}\right|$.
Next, the convolution theorem yields
\[
\mathcal{F}\left[\gamma_{1}\ast\gamma_{2}\right]\left(\xi\right)=\widehat{\gamma_{1}}\left(\xi\right)\cdot\widehat{\gamma_{2}}\left(\xi\right)=h\left(\xi\right)\cdot g\left(\xi\right)=\widehat{\gamma}\left(\xi\right)\qquad\forall\xi\in\mathbb{R}^{d}
\]
and thus $\gamma=\gamma_{1}\ast\gamma_{2}$, by injectivity of the
Fourier transform.
Next, note that if $\widehat{\gamma}\in C^{\infty}\left(\mathbb{R}^{d}\right)$
with all derivatives being polynomially bounded, we clearly get $\widehat{\gamma_{1}}=h\in C^{\infty}\left(\mathbb{R}^{d}\right)$,
again with all derivatives being polynomially bounded, thanks to Lemma
\ref{lem:ChineseBracketDerivative} and the Leibniz rule.
It remains to establish the estimate for $\left\Vert \gamma\right\Vert _{N}$.
But since $\left|\widehat{\gamma}\right|\leq\varrho\in L^{1}\left(\mathbb{R}^{d}\right)$,
we get $\gamma=\mathcal{F}^{-1}\widehat{\gamma}$ by Fourier inversion.
Furthermore, our assumptions easily yield $\partial^{\alpha}\widehat{\gamma}\in L^{1}\left(\mathbb{R}^{d}\right)$
for all $\alpha\in\mathbb{N}_{0}^{d}$ with $\left|\alpha\right|\leq N$.
Hence, a final application of Lemma \ref{lem:PointwiseFourierDecayEstimate},
together with our assumptions on $\widehat{\gamma}$, yields
\begin{align*}
\left(1+\left|x\right|\right)^{N}\cdot\left|\gamma\left(x\right)\right| & =\left(1+\left|x\right|\right)^{N}\cdot\left|\left(\mathcal{F}^{-1}\widehat{\gamma}\right)\left(x\right)\right|\\
& \leq\left(1+d\right)^{N}\cdot\left(\left|\left(\mathcal{F}^{-1}\widehat{\gamma}\right)\left(x\right)\right|+\sum_{m=1}^{d}\left|\left[\mathcal{F}^{-1}\left(\partial_{m}^{N}\widehat{\gamma}\right)\right]\left(x\right)\right|\right)\\
& \leq\left(1+d\right)^{N}\cdot\left(\left\Vert \widehat{\gamma}\right\Vert _{L^{1}}+\sum_{m=1}^{d}\left\Vert \partial_{m}^{N}\widehat{\gamma}\right\Vert _{L^{1}}\right)\\
& \leq\left(1+d\right)^{N+1}\cdot\left\Vert \varrho\right\Vert _{L^{1}}<\infty,
\end{align*}
which easily yields the claim.
\end{proof}
\section{Existence of compactly supported Banach frames and atomic decompositions
for \texorpdfstring{$\alpha$}{α}-modulation spaces}
\label{sec:CompactlySupportedAlphaModulationFrames}In this section,
we show that the general theory developed in this paper can be used
to prove existence of compactly supported, structured Banach frames
for $\alpha$-modulation spaces. A brief discussion of the history
and the applications of $\alpha$-modulation spaces, as well as a
comparison of our results with the established literature will be
given at the end of the section.
We begin our considerations by recalling the definition of $\alpha$-modulation
spaces, as given by Borup and Nielsen\cite{BorupNielsenAlphaModulationSpaces}.
First of all, we have to define the associated covering. Its admissibility
was established in \cite[Theorem 2.6]{BorupNielsenAlphaModulationSpaces};
precisely, the following was shown:
\begin{thm}
\label{thm:AlphaModulationCoveringDefinition}(cf. \cite[Theorem 2.6]{BorupNielsenAlphaModulationSpaces})
Let $d\in\mathbb{N}$ and $\alpha\in\left[0,1\right)$ be arbitrary.
Define $\alpha_{0}:=\frac{\alpha}{1-\alpha}$. Then there is a constant
$r_{1}=r_{1}\left(d,\alpha\right)$ such that the family
\begin{equation}
\mathcal{Q}^{\left(\alpha\right)}:=\mathcal{Q}_{r}^{\left(\alpha\right)}:=\left(Q_{r,k}^{\left(\alpha\right)}\right)_{k\in\mathbb{Z}^{d}\setminus\left\{ 0\right\} }:=\left(B_{r\cdot\left|k\right|^{\alpha_{0}}}\left(\left|k\right|^{\alpha_{0}}k\right)\right)_{k\in\mathbb{Z}^{d}\setminus\left\{ 0\right\} }\label{eq:AlphaModulationCovering}
\end{equation}
is an admissible covering of $\mathbb{R}^{d}$ for every $r>r_{1}$.
The covering $\mathcal{Q}_{r}^{\left(\alpha\right)}$ is called the \textbf{$\alpha$-modulation
covering} of $\mathbb{R}^{d}$. If the values of $r$ and $\alpha$
are clear from the context, we also write $Q_{k}:=Q_{r,k}^{\left(\alpha\right)}$.
\end{thm}
The associated weight is defined in our next lemma. There, and in
the remainder of this section, we use the notation $\left\langle \xi\right\rangle :=\left(1+\smash{\left|\xi\right|^{2}}\right)^{1/2}$
for $\xi\in\mathbb{R}^{d}$.
\begin{lem}
\label{lem:AlphaModulationFrequencyWeight}(cf. \cite[Lemma 9.2]{DecompositionEmbedding})
Let $d\in\mathbb{N}$ and $\alpha\in\left[0,1\right)$ and let $r>0$
be chosen such that the $\alpha$-modulation covering $\mathcal{Q}_{r}^{\left(\alpha\right)}=\left(Q_{r,k}^{\left(\alpha\right)}\right)_{k\in\mathbb{Z}^{d}\setminus\left\{ 0\right\} }$
is an admissible covering of $\mathbb{R}^{d}$. We then have
\[
\left\langle \xi\right\rangle \asymp\left\langle k\right\rangle ^{\frac{1}{1-\alpha}}\qquad\text{ for all }\qquad k\in\mathbb{Z}^{d}\setminus\left\{ 0\right\} \text{ and }\xi\in Q_{r,k}^{\left(\alpha\right)},
\]
where the implied constant only depends on $r,\alpha$.
Now, for $s\in\mathbb{R}$, we define the weight $w^{\left(s\right)}$ on
$\mathbb{Z}^{d}\setminus\left\{ 0\right\} $ by
\[
w_{k}^{\left(s\right)}:=\left\langle k\right\rangle ^{s}\qquad\text{ for }k\in\mathbb{Z}^{d}\setminus\left\{ 0\right\} .
\]
Then $w^{\left(s\right)}$ is $\mathcal{Q}_{r}^{\left(\alpha\right)}$-moderate
(cf.\@ equation (\ref{eq:IntroductionModerateWeightDefinition})).
\end{lem}
Note that Theorem \ref{thm:AlphaModulationCoveringDefinition} only
claims that $\mathcal{Q}_{r}^{\left(\alpha\right)}$ is an \emph{admissible}
covering of $\mathbb{R}^{d}$. The next result shows that it is actually
a \emph{structured} admissible covering of $\mathbb{R}^{d}$ (cf.\@
the remark after Assumption \ref{assu:RegularPartitionOfUnity}) and
thus in particular a semi-structured covering.
\begin{lem}
\label{lem:AlphaModulationStructuredAndBAPU}Let $d\in\mathbb{N}$
and $\alpha\in\left[0,1\right)$ and let $r_{1}=r_{1}\left(d,\alpha\right)$
be as in Theorem \ref{thm:AlphaModulationCoveringDefinition} and
let $r>r_{1}$. For $k\in\mathbb{Z}^{d}\setminus\left\{ 0\right\} $,
set $T_{k}:=\left|k\right|^{\alpha_{0}}\cdot\operatorname{id}$ and $b_{k}:=\left|k\right|^{\alpha_{0}}k$
and let $Q:=B_{r}\left(0\right)$. Then we have
\[
\mathcal{Q}_{r}^{\left(\alpha\right)}=\left(T_{k}Q+b_{k}\right)_{k\in\mathbb{Z}^{d}\setminus\left\{ 0\right\} }
\]
and with these choices, $\mathcal{Q}_{r}^{\left(\alpha\right)}$ is a semi-structured
admissible covering of $\mathbb{R}^{d}$.
Finally, $\mathcal{Q}_{r}^{\left(\alpha\right)}$ admits a regular partition
of unity $\left(\varphi_{k}\right)_{k\in\mathbb{Z}^{d}\setminus\left\{ 0\right\} }$
(which thus fulfills Assumption \ref{assu:RegularPartitionOfUnity})
and $\mathcal{Q}_{r}^{\left(\alpha\right)}$ fulfills the standing assumptions
from Section \ref{subsec:DecompSpaceDefinitionStandingAssumptions};
in particular, $\left\Vert T_{k}^{-1}\right\Vert \leq1=:\Omega_{0}$
for all $k\in\mathbb{Z}^{d}\setminus\left\{ 0\right\} $.
\end{lem}
\begin{proof}
The fact that $\mathcal{Q}_{r}^{\left(\alpha\right)}$ is a structured admissible
covering of $\mathbb{R}^{d}$ for $r>r_{1}$ was shown in \cite[Lemma 9.3]{DecompositionEmbedding}.
Since this is the case, \cite[Theorem 2.8]{DecompositionIntoSobolev}
shows that there is a regular partition of unity $\Phi=\left(\varphi_{k}\right)_{k\in\mathbb{Z}^{d}\setminus\left\{ 0\right\} }$
for $\mathcal{Q}_{r}^{\left(\alpha\right)}$. In view of Corollary \ref{cor:RegularBAPUsAreWeightedBAPUs},
$\Phi$ is thus also a $\mathcal{Q}_{r}^{\left(\alpha\right)}$-$v_{0}$-BAPU
for every weight $v_{0}$ satisfying the general assumptions of Section
\ref{subsec:DecompSpaceDefinitionStandingAssumptions}.
Finally, we clearly have $\left\Vert T_{k}^{-1}\right\Vert =\left\Vert \left|k\right|^{-\alpha_{0}}\operatorname{id}\right\Vert =\left|k\right|^{-\alpha_{0}}\leq1$
for all $k\in\mathbb{Z}^{d}\setminus\left\{ 0\right\} $, since $\left|k\right|\geq1$
and $\alpha_{0}=\frac{\alpha}{1-\alpha}\geq0$.
\end{proof}
Finally, we introduce the weights $v=v^{\left(\mu\right)}$ that we
will use for the weighted $L^{p}$ spaces $L_{v}^{p}\left(\mathbb{R}^{d}\right)$.
\begin{lem}
\label{lem:AlphaModulationSpaceWeight}For $\mu\in\mathbb{R}$ let
\begin{align*}
v^{\left(\mu\right)}: & \mathbb{R}^{d}\to\left(0,\infty\right),x\mapsto\left\langle x\right\rangle ^{\mu}=\left(1+\left|x\right|^{2}\right)^{\mu/2},\\
v_{0}: & \mathbb{R}^{d}\to\left(0,\infty\right),x\mapsto\left[2\cdot\left(1+\left|x\right|\right)\right]^{\left|\mu\right|}
\end{align*}
and set $K:=\left|\mu\right|$ and $\Omega_{1}:=2^{\left|\mu\right|}$.
With these choices, $v=v^{\left(\mu\right)}$ satisfies the standing
assumptions of Section \ref{subsec:DecompSpaceDefinitionStandingAssumptions}.
\end{lem}
\begin{proof}
First of all, assume $\mu=1$. In this case, we get
\begin{align*}
v^{\left(\mu\right)}\left(x+y\right) & =\left|\left(\begin{matrix}1\\
x+y
\end{matrix}\right)\right|\leq1+\left|x+y\right|\\
& \leq1+\left|x\right|+\left|y\right|\leq\left(1+\left|x\right|\right)\left(1+\left|y\right|\right)\\
& \leq2\cdot\left|\left(\begin{matrix}1\\
x
\end{matrix}\right)\right|\cdot\left(1+\left|y\right|\right)=v^{\left(\mu\right)}\left(x\right)\cdot v_{0}\left(y\right),
\end{align*}
where the last step used $\mu=1$. Now, for arbitrary $\mu\geq0$,
we likewise get $v^{\left(\mu\right)}\left(x+y\right)\leq v^{\left(\mu\right)}\left(x\right)\cdot v_{0}\left(y\right)$
by taking the $\mu$-th power of the preceding estimate.
Finally, if $\mu<0$, we have
\[
v^{\left(-\mu\right)}\left(x\right)=v^{\left(-\mu\right)}\left(x+y+\left(-y\right)\right)\leq v^{\left(-\mu\right)}\left(x+y\right)\cdot\left[2\cdot\left(1+\left|-y\right|\right)\right]^{\left|-\mu\right|}.
\]
Rearranging yields
\[
v^{\left(\mu\right)}\left(x+y\right)=\left[v^{\left(-\mu\right)}\left(x+y\right)\right]^{-1}\leq\left[v^{\left(-\mu\right)}\left(x\right)\right]^{-1}\cdot\left[2\cdot\left(1+\left|y\right|\right)\right]^{\left|\mu\right|}=v^{\left(\mu\right)}\left(x\right)\cdot v_{0}\left(y\right).
\]
Hence, we have shown for all $\mu\in\mathbb{R}$ that $v^{\left(\mu\right)}$
is $v_{0}$-moderate.
It is clear that $v_{0}\geq1$ and that $v_{0}$ is symmetric. Furthermore,
$v_{0}\left(x\right)=2^{\left|\mu\right|}\cdot\left(1+\left|x\right|\right)^{\left|\mu\right|}=\Omega_{1}\cdot\left(1+\left|x\right|\right)^{K}$
for all $x\in\mathbb{R}^{d}$. We do not necessarily have $K=0$,
but in Lemma \ref{lem:AlphaModulationStructuredAndBAPU}, we already
saw $\left\Vert T_{k}^{-1}\right\Vert \leq1=\Omega_{0}$ for all $k\in\mathbb{Z}^{d}\setminus\left\{ 0\right\} $.
The only thing which remains to be verified is that $v_{0}$ is submultiplicative.
But we have
\[
2\cdot\left(1+\left|x+y\right|\right)\leq2\cdot\left(1+\left|x\right|+\left|y\right|\right)\leq2\cdot\left(1+\left|x\right|\right)\left(1+\left|y\right|\right)\leq2\cdot\left(1+\left|x\right|\right)\cdot2\cdot\left(1+\left|y\right|\right).
\]
Taking the $\left|\mu\right|$-th power of this estimate yields $v_{0}\left(x+y\right)\leq v_{0}\left(x\right)\cdot v_{0}\left(y\right)$,
as desired.
\end{proof}
Having verified all these assumptions, we conclude from Proposition
\ref{prop:WeightedDecompositionSpaceWellDefined} and Lemma \ref{lem:WeightedDecompositionSpaceComplete}
that the $\alpha$-modulation spaces defined below are indeed well-defined
Quasi-Banach spaces.
\begin{defn}
\label{def:AlphaModulationSpaces}For $d\in\mathbb{N}$ and $\alpha\in\left[0,1\right)$,
choose some $r>r_{1}\left(d,\alpha\right)$ with $r_{1}\left(d,\alpha\right)$
as in Theorem \ref{thm:AlphaModulationCoveringDefinition}. Then,
for $p,q\in\left(0,\infty\right]$ and $s,\mu\in\mathbb{R}$, we define the
associated \textbf{(weighted) $\alpha$-modulation space} as
\[
M_{\left(s,\mu\right),\alpha}^{p,q}\left(\smash{\mathbb{R}^{d}}\right):=\DecompSp{\mathcal{Q}_{r}^{\left(\alpha\right)}}p{\ell_{w^{\left(s/\left(1-\alpha\right)\right)}}^{q}}{v^{\left(\mu\right)}}
\]
with $w^{\left(s/\left(1-\alpha\right)\right)}$ and $v^{\left(\mu\right)}$
as in Lemmas \ref{lem:AlphaModulationFrequencyWeight} and \ref{lem:AlphaModulationSpaceWeight},
respectively.
Furthermore, we define the \textbf{classical $\alpha$-modulation
space} as $M_{s,\alpha}^{p,q}\left(\mathbb{R}^{d}\right):=M_{\left(s,0\right),\alpha}^{p,q}\left(\mathbb{R}^{d}\right)$.
\end{defn}
\begin{rem*}
\begin{itemize}[leftmargin=0.4cm]
\item The classical $\alpha$-modulation spaces $M_{s,\alpha}^{p,q}\left(\mathbb{R}^{d}\right)$
defined above coincide with the $\alpha$-modulation spaces defined
in \cite[Definition 2.4]{BorupNielsenAlphaModulationSpaces}, up to
trivial identifications: The quasi-norms used in the two definitions
are precisely the same; the only difference between the two definitions
is that in \cite[Definition 2.4]{BorupNielsenAlphaModulationSpaces},
the $\alpha$-modulation spaces are defined as subspaces of $\mathcal{S}'\left(\mathbb{R}^{d}\right)$.
In contrast, with our definition as a decomposition space, $M_{s,\alpha}^{p,q}\left(\mathbb{R}^{d}\right)$
is a subspace of $Z'\left(\mathbb{R}^{d}\right)=\left[\mathcal{F}\left(\TestFunctionSpace{\mathbb{R}^{d}}\right)\right]'$,
cf.\@ Section \ref{subsec:DecompSpaceDefinitionStandingAssumptions}.
But \cite[Lemma 9.15]{DecompositionEmbedding} and \cite[Theorem 9.13]{DecompositionEmbedding}
show that each $f\in M_{s,\alpha}^{p,q}\left(\mathbb{R}^{d}\right)$
extends to a (uniquely determined) tempered distribution, which implies
that the two different definitions of $\alpha$-modulation spaces
indeed yield the same spaces, up to trivial identifications.
\item Observe that the parameter $r>r_{1}\left(d,\alpha\right)$
is suppressed on the left-hand side of the definition above. This
is justified, as we show now: Since any two coverings $\mathcal{Q}_{r}^{\left(\alpha\right)},\mathcal{Q}_{t}^{\left(\alpha\right)}$
(with $r,t>r_{1}\left(d,\alpha\right)$) use the same families
$\left(T_{k}\right)_{k\in\mathbb{Z}^{d}\setminus\left\{ 0\right\} }$
and $\left(b_{k}\right)_{k\in\mathbb{Z}^{d}\setminus\left\{ 0\right\} }$,
it follows that every regular partition of unity $\Phi=\left(\varphi_{k}\right)_{k\in\mathbb{Z}^{d}\setminus\left\{ 0\right\} }$
(cf.\@ Assumption \ref{assu:RegularPartitionOfUnity}) for $\mathcal{Q}_{r}^{\left(\alpha\right)}$
is also a regular partition of unity for $\mathcal{Q}_{t}^{\left(\alpha\right)}$,
at least for $t\geq r$, which we can always assume by symmetry. Thus,
by choosing the \emph{same} BAPU $\Phi$ for both coverings, we see
$\DecompSp{\mathcal{Q}_{r}^{\left(\alpha\right)}}p{\ell_{w^{\left(s^{\ast}\right)}}^{q}}{v^{\left(\mu\right)}}=\DecompSp{\mathcal{Q}_{t}^{\left(\alpha\right)}}p{\ell_{w^{\left(s^{\ast}\right)}}^{q}}{v^{\left(\mu\right)}}$,
with equivalent quasi-norms. Here, $s^{\ast}:=s/\left(1-\alpha\right)$.
We finally note that this argument implicitly uses that different
choices of the BAPU yield the same space (with equivalent quasi-norms),
cf.\@ Proposition \ref{prop:WeightedDecompositionSpaceWellDefined}.\qedhere
\end{itemize}
\end{rem*}
In the remainder of this section, we will determine conditions on
the prototype $\gamma$ which ensure that Corollary \ref{cor:BanachFrameSimplifiedCriteria}
(leading to Banach frames) or Corollary \ref{cor:AtomicDecompositionSimplifiedCriteria}
(leading to atomic decompositions) is applicable to $\gamma$. We
will see that this is the case for arbitrary Schwartz functions $\gamma$,
as long as $\widehat{\gamma}$ fulfills a certain non-vanishing condition.
To be precise, recall that in Corollaries \ref{cor:BanachFrameSimplifiedCriteria}
and \ref{cor:AtomicDecompositionSimplifiedCriteria}, we allowed the
prototype to depend on $i\in I$, i.e., we used a family $\left(\gamma_{i}\right)_{i\in I}$
of prototypes. But in this section, we will only consider the case
where $\gamma_{i}=\gamma$ is independent of $i\in I$.
To begin with, we recall that in Corollary \ref{cor:BanachFrameSimplifiedCriteria},
we are imposing certain summability conditions on
\[
M_{j,i}:=\left(\frac{w_{j}^{\left(s/\left(1-\alpha\right)\right)}}{w_{i}^{\left(s/\left(1-\alpha\right)\right)}}\right)^{\tau}\cdot\left(1+\left\Vert T_{j}^{-1}T_{i}\right\Vert \right)^{\sigma}\cdot\max_{\left|\beta\right|\leq1}\left(\left|\det T_{i}\right|^{-1}\cdot\int_{Q_{i}}\max_{\left|\alpha\right|\leq N}\left|\left(\partial^{\alpha}\widehat{\partial^{\beta}\gamma}\right)\!\!\left(S_{j}^{-1}\xi\right)\right|\operatorname{d}\xi\right)^{\tau}
\]
for suitable values of $\tau,\sigma>0$ and $N\in\mathbb{N}$. To slightly
simplify this expression, we will use the following notation for the
remainder of the section:
\begin{equation}
s^{\ast}:=s/\left(1-\alpha\right).\label{eq:AlphaModulationSmoothnessExponent}
\end{equation}
For our application of Corollary \ref{cor:BanachFrameSimplifiedCriteria},
we will assume $\gamma\in L^{1}\left(\mathbb{R}^{d}\right)\cap C^{1}\left(\mathbb{R}^{d}\right)$
with $\partial_{\ell}\gamma\in L^{1}\left(\mathbb{R}^{d}\right)$
for all $\ell\in\underline{d}$ and with $\widehat{\gamma}\in C^{\infty}\left(\mathbb{R}^{d}\right)$.
Under these assumptions, elementary properties of the Fourier transform
yield for $\beta=e_{\ell}$ (the $\ell$-th unit vector) that
\[
\widehat{\partial^{\beta}\gamma}\left(\xi\right)=2\pi i\xi_{\ell}\cdot\widehat{\gamma}\left(\xi\right)\qquad\forall\xi\in\mathbb{R}^{d}.
\]
Since we clearly have $\left|\frac{\partial^{\theta}}{\partial\eta^{\theta}}\eta_{\ell}\right|\leq1+\left|\eta\right|$
for all $\eta\in\mathbb{R}^{d}$ and $\theta\in\mathbb{N}_{0}^{d}$,
the Leibniz rule and the $d$-dimensional binomial theorem
(cf.\@ \cite[Section 8.1, Exercise 2.b]{FollandRA}) yield
\begin{align}
\left|\left(\partial^{\alpha}\widehat{\partial^{\beta}\gamma}\right)\left(\eta\right)\right| & =\left|2\pi i\cdot\sum_{\theta\leq\alpha}\binom{\alpha}{\theta}\cdot\partial^{\theta}\eta_{\ell}\cdot\left(\partial^{\alpha-\theta}\widehat{\gamma}\right)\left(\eta\right)\right|\nonumber \\
& \leq2\pi\cdot\left(1+\left|\eta\right|\right)\cdot\sum_{\theta\leq\alpha}\binom{\alpha}{\theta}\cdot\left|\left(\partial^{\alpha-\theta}\widehat{\gamma}\right)\left(\eta\right)\right|\nonumber \\
\left({\scriptstyle \text{eq. }\eqref{eq:AlphaModulationBanachFrameDecayAssumption}}\right) & \leq\left(1+\left|\eta\right|\right)^{1-N_{0}}\cdot2\pi C\cdot\sum_{\theta\leq\alpha}\binom{\alpha}{\theta}\nonumber \\
& \leq2^{N+1}\pi\cdot C\cdot\left(1+\left|\eta\right|\right)^{1-N_{0}},\label{eq:AlphaModulationIteratedDerivativeEstimate}
\end{align}
where we used $\left|\alpha-\theta\right|\leq\left|\alpha\right|\leq N$
and assumed that there is some $N_{0}\in\mathbb{R}$ satisfying
\begin{equation}
\max_{\left|\alpha\right|\leq N}\left|\left(\partial^{\alpha}\widehat{\gamma}\right)\left(\eta\right)\right|\leq C\cdot\left(1+\left|\eta\right|\right)^{-N_{0}}\qquad\forall\eta\in\mathbb{R}^{d}.\label{eq:AlphaModulationBanachFrameDecayAssumption}
\end{equation}
Recall that equation (\ref{eq:AlphaModulationIteratedDerivativeEstimate})
holds for $\beta=e_{\ell}$, with arbitrary $\ell\in\underline{d}$.
But for $\beta=0$, we simply have
\[
\left|\left(\partial^{\alpha}\widehat{\partial^{\beta}\gamma}\right)\left(\eta\right)\right|=\left|\left(\partial^{\alpha}\widehat{\gamma}\right)\left(\eta\right)\right|\leq C\cdot\left(1+\left|\eta\right|\right)^{-N_{0}}\leq2^{N+1}\pi\cdot C\cdot\left(1+\left|\eta\right|\right)^{1-N_{0}},
\]
so that we have verified equation (\ref{eq:AlphaModulationIteratedDerivativeEstimate})
for arbitrary $\alpha,\beta\in\mathbb{N}_{0}^{d}$ with $\left|\beta\right|\leq1$
and $\left|\alpha\right|\leq N$.
Hence, we have shown for $C':=2^{N+1}\pi\cdot C$ that
\begin{equation}
M_{j,i}\leq\left(C'\right)^{\tau}\cdot\left(\frac{w_{j}^{\left(s^{\ast}\right)}}{w_{i}^{\left(s^{\ast}\right)}}\right)^{\tau}\cdot\left(1+\left\Vert T_{j}^{-1}T_{i}\right\Vert \right)^{\sigma}\cdot\left(\left|\det T_{i}\right|^{-1}\cdot\int_{Q_{i}}\left(1+\left|S_{j}^{-1}\xi\right|\right)^{1-N_{0}}\operatorname{d}\xi\right)^{\tau}=:\left(C'\right)^{\tau}\cdot M_{j,i}^{\left(0\right)}\label{eq:AlphaModulationStandardEstimate}
\end{equation}
for all $i,j\in\mathbb{Z}^{d}\setminus\left\{ 0\right\} $. In view
of this estimate, the following lemma is crucial:
\begin{lem}
\label{lem:AlphaModulationMainLemma}Let $d\in\mathbb{N}$ and $\alpha\in\left[0,1\right)$
and set $\alpha_{0}:=\frac{\alpha}{1-\alpha}$. With $\mathcal{Q}=\mathcal{Q}_{r}^{\left(\alpha\right)}$
and with $M_{j,i}^{\left(0\right)}$ as in equation (\ref{eq:AlphaModulationStandardEstimate}),
assume
\[
N_{0}\geqd+2+\frac{d+1}{\tau}+\max\left\{ \left|s^{\ast}+d\alpha_{0}\right|,\left|s^{\ast}+\left(d-\frac{\sigma}{\tau}\right)\alpha_{0}\right|\right\} .
\]
Then we have
\[
\sup_{i\in\mathbb{Z}^{d}\setminus\left\{ 0\right\} }\:\sum_{j\in\mathbb{Z}^{d}\setminus\left\{ 0\right\} }M_{j,i}^{\left(0\right)}\leq\Omega\qquad\text{ and }\qquad\sup_{j\in\mathbb{Z}^{d}\setminus\left\{ 0\right\} }\:\sum_{i\in\mathbb{Z}^{d}\setminus\left\{ 0\right\} }M_{j,i}^{\left(0\right)}\leq\Omega
\]
for
\[
\Omega:=6^{d}2^{1+\sigma+\tau\left|s^{\ast}\right|}\!\cdot\max\left\{ 4^{\alpha_{0}\left(\sigma+d\tau\right)+\tau\left|s^{\ast}\right|}\cdot\left(12^{N_{0}}s_{d}\right)^{\tau},\:\left(2\!+\!4r\right)^{\tau\left|s^{\ast}\right|+\alpha_{0}\left[\sigma+\tau\left(d+N_{0}\right)\right]}\!\cdot\!2^{\taud}\!\cdot\!\left(1\!+\!\left(2\!+\!4r\right)^{\alpha_{0}}\!\cdot r\right)^{\tau\left(N_{0}+d\right)}\!\right\} .\qedhere
\]
\end{lem}
\begin{proof}
For brevity, set $M:=N_{0}-1$. Recall $T_{j}=\left|j\right|^{\alpha_{0}}\cdot\operatorname{id}$
and $b_{j}=\left|j\right|^{\alpha_{0}}j$, so that
\[
S_{j}^{-1}\xi=T_{j}^{-1}\left(\xi-b_{j}\right)=\left|j\right|^{-\alpha_{0}}\left(\xi-\left|j\right|^{\alpha_{0}}j\right)=\left|j\right|^{-\alpha_{0}}\xi-j.
\]
Hence,
\begin{align*}
\int_{Q_{i}}\left(1+\left|S_{j}^{-1}\xi\right|\right)^{1-N_{0}}\operatorname{d}\xi & =\int_{B_{\left|i\right|^{\alpha_{0}}r}\left(\left|i\right|^{\alpha_{0}}i\right)}\:\left(1+\left|\left|j\right|^{-\alpha_{0}}\xi-j\right|\right)^{-M}\operatorname{d}\xi\\
\left({\scriptstyle \text{with }\eta=\left|j\right|^{-\alpha_{0}}\xi}\right) & =\left|j\right|^{d\alpha_{0}}\cdot\int_{B_{\left(\left|i\right|/\left|j\right|\right)^{\alpha_{0}}\cdot r}\left(\left(\left|i\right|/\left|j\right|\right)^{\alpha_{0}}i\right)}\:\left(1+\left|\eta-j\right|\right)^{-M}\operatorname{d}\eta\\
\left({\scriptstyle \text{with }\xi=\eta-j}\right) & =\left|j\right|^{d\alpha_{0}}\cdot\int_{B_{\left(\left|i\right|/\left|j\right|\right)^{\alpha_{0}}\cdot r}\left(\left(\left|i\right|/\left|j\right|\right)^{\alpha_{0}}i-j\right)}\:\left(1+\left|\xi\right|\right)^{-M}\operatorname{d}\xi\\
& =\left|j\right|^{d\alpha_{0}}\cdot\int_{B_{R_{i,j}}\left(\xi_{i,j}\right)}\:\left(1+\left|\xi\right|\right)^{-M}\operatorname{d}\xi,
\end{align*}
where we defined
\[
\xi_{i,j}:=\left(\frac{\left|i\right|}{\left|j\right|}\right)^{\alpha_{0}}i-j\qquad\text{ and }\qquad R_{i,j}:=\left(\frac{\left|i\right|}{\left|j\right|}\right)^{\alpha_{0}}\cdot r
\]
for $i,j\in I=\mathbb{Z}^{d}\setminus\left\{ 0\right\} $. Here,
$r>r_{1}\left(d,\alpha\right)$ comes from the covering $\mathcal{Q}_{r}^{\left(\alpha\right)}$.
All in all, since $\left\Vert T_{j}^{-1}T_{i}\right\Vert =\left\Vert \left|i\right|^{\alpha_{0}}/\left|j\right|^{\alpha_{0}}\cdot\operatorname{id}\right\Vert =\left(\left|i\right|/\left|j\right|\right)^{\alpha_{0}}$
and $\left|\det T_{i}\right|=\left|\det\left|i\right|^{\alpha_{0}}\operatorname{id}\right|=\left|i\right|^{d\alpha_{0}}$,
and because of $\left|i\right|\leq\left\langle i\right\rangle \leq1+\left|i\right|\leq2\left|i\right|$
for $i\in\mathbb{Z}^{d}$, so that $\frac{1}{2}\frac{\left|j\right|}{\left|i\right|}\leq\frac{\left\langle j\right\rangle }{\left\langle i\right\rangle }\leq2\frac{\left|j\right|}{\left|i\right|}$,
we get the following estimate for $M_{j,i}^{\left(0\right)}$:
\begin{align}
M_{j,i}^{\left(0\right)} & =\left(\frac{w_{j}^{\left(s^{\ast}\right)}}{w_{i}^{\left(s^{\ast}\right)}}\right)^{\tau}\cdot\left(1+\left\Vert T_{j}^{-1}T_{i}\right\Vert \right)^{\sigma}\cdot\left(\left|\det T_{i}\right|^{-1}\cdot\int_{Q_{i}}\left(1+\left|S_{j}^{-1}\xi\right|\right)^{1-N_{0}}\operatorname{d}\xi\right)^{\tau}\nonumber \\
& =\left(\frac{\left\langle j\right\rangle }{\left\langle i\right\rangle }\right)^{s^{\ast}\cdot\tau}\cdot\left(1+\left(\frac{\left|i\right|}{\left|j\right|}\right)^{\alpha_{0}}\right)^{\sigma}\cdot\left[\left(\frac{\left|j\right|}{\left|i\right|}\right)^{d\alpha_{0}}\cdot\int_{B_{R_{i,j}}\left(\xi_{i,j}\right)}\left(1+\left|\xi\right|\right)^{-M}\operatorname{d}\xi\right]^{\tau}\nonumber \\
\left({\scriptstyle \text{since }\left(1+a\right)^{\sigma}\leq2^{\sigma}\cdot\left(1+a^{\sigma}\right)}\right) & \leq2^{\sigma+\tau\left|s^{\ast}\right|}\cdot\left(\frac{\left|j\right|}{\left|i\right|}\right)^{\tau\left(s^{\ast}+d\alpha_{0}\right)}\cdot\left(1+\left(\frac{\left|j\right|}{\left|i\right|}\right)^{-\sigma\alpha_{0}}\right)\cdot\left(\int_{B_{R_{i,j}}\left(\xi_{i,j}\right)}\left(1+\left|\xi\right|\right)^{-M}\operatorname{d}\xi\right)^{\tau}\nonumber \\
& =2^{\sigma+\tau\left|s^{\ast}\right|}\cdot\sum_{\lambda\in\left\{ 0,1\right\} }\left[\left(\frac{\left|j\right|}{\left|i\right|}\right)^{k_{\lambda}}\cdot\left(\int_{B_{R_{i,j}}\left(\xi_{i,j}\right)}\left(1+\left|\xi\right|\right)^{-M}\operatorname{d}\xi\right)^{\!\tau}\,\right],\label{eq:AlphaModulationMXiConnection}
\end{align}
where we defined $k_{\lambda}:=\tau\left(s^{\ast}+d\alpha_{0}\right)-\lambda\sigma\alpha_{0}$
for $\lambda\in\left\{ 0,1\right\} $.
Thus, our main goal is to estimate the term
\[
\Xi_{i,j}^{\left(k\right)}:=\left(\frac{\left|j\right|}{\left|i\right|}\right)^{k}\cdot\left(\int_{B_{R_{i,j}}\left(\xi_{i,j}\right)}\left(1+\left|\xi\right|\right)^{-M}\operatorname{d}\xi\right)^{\tau}
\]
for arbitrary $i,j\in\mathbb{Z}^{d}\setminus\left\{ 0\right\} $,
$k\in\mathbb{R}$ and $\tau>0$. To this end, we will distinguish three cases
concerning $i,j$ below. But before that, we introduce a useful notation
and some related estimates that will be used in several of the cases:
For $x\in\mathbb{R}^{d}$, we set $\left[x\right]:=1+\left|x\right|$.
We then have
\begin{equation}
\left[x\right]^{z}\leq\left[y\right]^{z}\cdot\left[x-y\right]^{\left|z\right|}\qquad\forall z\in\mathbb{R}\quad\forall x,y\in\mathbb{R}^{d}.\label{eq:AlphaModulationBracketModerate}
\end{equation}
Indeed, since $\left[x\right]=1+\left|x\right|\leq1+\left|y\right|+\left|x-y\right|\leq\left(1+\left|y\right|\right)\left(1+\left|x-y\right|\right)=\left[y\right]\cdot\left[x-y\right]$,
we get the claim for $z\geq0$. Finally, for $z<0$, we have
\begin{align*}
\left[x\right]^{z}=\left[x\right]^{-\left|z\right|} & =\left(\left[x\right]^{-\left|z\right|}\left[x-y\right]^{-\left|z\right|}\right)\cdot\left[x-y\right]^{\left|z\right|}\\
\left({\scriptstyle \text{eq. }\eqref{eq:AlphaModulationBracketModerate}\text{ rearranged, with }x,y\text{ interchanged and }\left|z\right|\text{ instead of }z}\right) & \leq\left[y\right]^{-\left|z\right|}\cdot\left[x-y\right]^{\left|z\right|}=\left[y\right]^{z}\cdot\left[x-y\right]^{\left|z\right|},
\end{align*}
as desired. Now, note for $i,j\in\mathbb{Z}^{d}\setminus\left\{ 0\right\} $
that $\left|i\right|\leq\left[i\right]\leq2\left|i\right|$ and likewise
for $j$, so that
\begin{equation}
\left|i\right|^{z}\leq2^{\left|z\right|}\cdot\left[i\right]^{z}\leq2^{\left|z\right|}\cdot\left[j\right]^{z}\cdot\left[i-j\right]^{\left|z\right|}\leq4^{\left|z\right|}\cdot\left|j\right|^{z}\cdot\left[i-j\right]^{\left|z\right|}.\label{eq:AlphaModulationAbsoluteValueBracketModerate}
\end{equation}
We will also need the following estimate, which I learned from \cite{EmbeddingsOfAlphaModulationIntoSobolev}:
\begin{equation}
\left|\beta\cdot x-y\right|\geq\left|x-y\right|\qquad\text{ if }\beta\in\mathbb{R}_{\geq1}\text{ and }x,y\in\mathbb{R}^{d}\text{ with }\left|x\right|\geq\left|y\right|.\label{eq:AlphaModulationKatoEstimate}
\end{equation}
For $\beta=1$, this estimate is trivial, so that we can assume $\beta>1$.
Next, note that both sides are nonnegative, so that the estimate is
equivalent to $\left|\beta\cdot x-y\right|^{2}\geq\left|x-y\right|^{2}$
and thus to
\begin{align*}
& \beta^{2}\left|x\right|^{2}-2\beta\cdot\left\langle x,y\right\rangle +\left|y\right|^{2}\overset{!}{\geq}\left|x\right|^{2}-2\left\langle x,y\right\rangle +\left|y\right|^{2}\\
\Longleftrightarrow & \left|x\right|^{2}\cdot\left(\beta^{2}-1\right)\overset{!}{\geq}2\cdot\left\langle x,y\right\rangle \cdot\left(\beta-1\right)\\
\left({\scriptstyle \text{since }\beta-1>0}\right)\Longleftrightarrow & \left|x\right|^{2}\cdot\left(\beta+1\right)\overset{!}{\geq}2\cdot\left\langle x,y\right\rangle .
\end{align*}
But the Cauchy-Schwarz inequality yields $2\cdot\left\langle x,y\right\rangle \leq2\cdot\left|\left\langle x,y\right\rangle \right|\leq2\cdot\left|x\right|\left|y\right|\leq2\cdot\left|x\right|^{2}\leq\left(1+\beta\right)\cdot\left|x\right|^{2}$,
since $\left|y\right|\leq\left|x\right|$ and since $\beta\geq1$.
Hence, we have established equation (\ref{eq:AlphaModulationKatoEstimate}).
We remark that for this estimate, it is crucial to use a norm which
is induced by a scalar product. For other norms, equation (\ref{eq:AlphaModulationKatoEstimate})
can fail.
\medskip{}
Now, we distinguish three cases depending on $i,j$:
\textbf{Case 1}: We have $\left|i\right|\geq2\left|j\right|+4r$.
In this case, we get
\begin{align*}
\left|\left(\frac{\left|i\right|}{\left|j\right|}\right)^{\alpha_{0}}i\right| & \leq\left|\left(\frac{\left|i\right|}{\left|j\right|}\right)^{\alpha_{0}}i-j\right|+\left|j\right|\\
& \leq\left|\xi_{i,j}\right|+\frac{\left|i\right|}{2}\\
\left({\scriptstyle \text{since }\frac{\left|i\right|}{\left|j\right|}\geq2\geq1\text{ and }\alpha_{0}\geq0}\right) & \leq\left|\xi_{i,j}\right|+\frac{1}{2}\left|\left(\frac{\left|i\right|}{\left|j\right|}\right)^{\alpha_{0}}i\right|
\end{align*}
and thus, since $\left|i\right|\geq4r$,
\begin{equation}
\left|\xi_{i,j}\right|\geq\frac{1}{2}\left|\left(\frac{\left|i\right|}{\left|j\right|}\right)^{\alpha_{0}}i\right|\geq2r\cdot\left(\frac{\left|i\right|}{\left|j\right|}\right)^{\alpha_{0}}=2\cdot R_{i,j}.\label{eq:AlphaModulationILargeCenterLarge}
\end{equation}
Hence, for arbitrary $\xi\in B_{R_{i,j}}\left(\xi_{i,j}\right)$,
we have $\left|\xi\right|\geq\left|\xi_{i,j}\right|-\left|\xi-\xi_{i,j}\right|\geq\left|\xi_{i,j}\right|-R_{i,j}\geq\frac{1}{2}\left|\xi_{i,j}\right|$
and thus
\begin{equation}
\begin{split}\int_{B_{R_{i,j}}\left(\xi_{i,j}\right)}\left(1+\left|\xi\right|\right)^{-M}\operatorname{d}\xi & \leq\left[\sup_{\xi\in B_{R_{i,j}}\left(\xi_{i,j}\right)}\left(1+\left|\xi\right|\right)^{d+1-M}\right]\cdot\int_{B_{R_{i,j}}\left(\xi_{i,j}\right)}\left(1+\left|\xi\right|\right)^{-\left(d+1\right)}\operatorname{d}\xi\\
\left({\scriptstyle \text{since }d+1-M\leq0}\right) & \leq\left(1+\frac{1}{2}\left|\xi_{i,j}\right|\right)^{d+1-M}\cdot\int_{\mathbb{R}^{d}}\left(1+\left|\xi\right|\right)^{-\left(d+1\right)}\operatorname{d}\xi\\
\left({\scriptstyle \text{eq. }\eqref{eq:StandardDecayLpEstimate}}\right) & \leq2^{M}\cdot s_{d}\cdot\left|\xi_{i,j}\right|^{d+1-M}\\
\left({\scriptstyle \text{eq. }\eqref{eq:AlphaModulationILargeCenterLarge}\text{ and }d+1-M\leq0}\right) & \leq4^{M}\cdot s_{d}\cdot\left|\left(\frac{\left|i\right|}{\left|j\right|}\right)^{\alpha_{0}}i\right|^{d+1-M}\\
\left({\scriptstyle \text{since }d+1-M\leq0\text{ and }\frac{\left|i\right|}{\left|j\right|}\geq1}\right) & \leq4^{M}\cdot s_{d}\cdot\left|i\right|^{d+1-M}.
\end{split}
\label{eq:AlphaModulationIntegralSupremumEstimate}
\end{equation}
Next, we observe $\left|i\right|\geq2\left|j\right|+4r\geq\left|j\right|$
and $i\in\mathbb{Z}^{d}\setminus\left\{ 0\right\} $, so that $\left|i\right|\geq1$.
This implies
\[
\left[i-j\right]=1+\left|i-j\right|\leq1+\left|i\right|+\left|j\right|\leq1+2\left|i\right|\leq3\left|i\right|,
\]
so that we finally arrive, again using $d+1-M\leq0$, at
\[
\int_{B_{R_{i,j}}\left(\xi_{i,j}\right)}\left(1+\left|\xi\right|\right)^{-M}\operatorname{d}\xi\leq12^{M}\cdot s_{d}\cdot\left[i-j\right]^{d+1-M}.
\]
Thus, using equation (\ref{eq:AlphaModulationAbsoluteValueBracketModerate}),
we conclude
\begin{equation}
\Xi_{i,j}^{\left(k\right)}=\left(\frac{\left|j\right|}{\left|i\right|}\right)^{k}\cdot\left(\int_{B_{R_{i,j}}\left(\xi_{i,j}\right)}\left(1+\left|\xi\right|\right)^{-M}\operatorname{d}\xi\right)^{\tau}\leq4^{\left|k\right|}\cdot\left(12^{M}\cdot s_{d}\right)^{\tau}\cdot\left[j-i\right]^{\left|k\right|+\tau\left(d+1-M\right)}.\label{eq:AlphaModulationXiEstimateFirstCase}
\end{equation}
\textbf{Case 2}: We have $\left|j\right|\geq2\left|i\right|+4r$.
Here, we first observe $\left|j-i\right|\geq\left|j\right|-\left|i\right|\geq\left|i\right|+4r\geq4r$.
Hence, we get
\begin{align*}
\left|\xi_{i,j}\right| & =\left(\frac{\left|i\right|}{\left|j\right|}\right)^{\alpha_{0}}\cdot\left|\left(\frac{\left|j\right|}{\left|i\right|}\right)^{\alpha_{0}}j-i\right|\\
\left({\scriptstyle \text{eq. }\eqref{eq:AlphaModulationKatoEstimate}\text{ and }\left(\left|j\right|/\left|i\right|\right)^{\alpha_{0}}\geq1,\text{ as well as }\left|j\right|\geq\left|i\right|}\right) & \geq\left(\frac{\left|i\right|}{\left|j\right|}\right)^{\alpha_{0}}\cdot\left|j-i\right|\\
& \geq4\cdot\left(\frac{\left|i\right|}{\left|j\right|}\right)^{\alpha_{0}}r=4\cdot R_{i,j}.
\end{align*}
Further, $\left|j\right|\geq2\left|i\right|+4r\geq2\left|i\right|$
implies $\left|i\right|\leq\frac{\left|j\right|}{2}$ and thus $\left|j\right|-\left|i\right|\geq\frac{1}{2}\left|j\right|$.
Consequently,
\[
\left|j-i\right|\leq\left|j\right|+\left|i\right|\leq2\left|j\right|\leq4\cdot\left(\left|j\right|-\left|i\right|\right),
\]
so that we get
\begin{align*}
\left|\xi_{i,j}\right| & =\left|\left(\frac{\left|i\right|}{\left|j\right|}\right)^{\alpha_{0}}i-j\right|\\
& \geq\left|j\right|-\left(\frac{\left|i\right|}{\left|j\right|}\right)^{\alpha_{0}}\left|i\right|\\
\left({\scriptstyle \text{since }\left|i\right|/\left|j\right|\leq1}\right) & \geq\left|j\right|-\left|i\right|\geq\frac{\left|j-i\right|}{4}.
\end{align*}
Now, for arbitrary $\xi\in B_{R_{i,j}}\left(\xi_{i,j}\right)$, the
two preceding displayed estimates yield $\left|\xi\right|\geq\left|\xi_{i,j}\right|-R_{i,j}\geq\frac{3}{4}\left|\xi_{i,j}\right|\geq\frac{3}{16}\cdot\left|j-i\right|$
and hence $1+\left|\xi\right|\geq\frac{3}{16}\cdot\left[j-i\right]$.
With an estimate entirely analogous to that in equation (\ref{eq:AlphaModulationIntegralSupremumEstimate}),
this implies
\begin{align*}
\int_{B_{R_{i,j}}\left(\xi_{i,j}\right)}\left(1+\left|\xi\right|\right)^{-M}\operatorname{d}\xi & \leq\left[\sup_{\xi\in B_{R_{i,j}}\left(\xi_{i,j}\right)}\left(1+\left|\xi\right|\right)^{d+1-M}\right]\cdot\int_{B_{R_{i,j}}\left(\xi_{i,j}\right)}\left(1+\left|\xi\right|\right)^{-\left(d+1\right)}\operatorname{d}\xi\\
\left({\scriptstyle \text{eq. }\eqref{eq:StandardDecayLpEstimate}\text{ and }d+1-M\leq0}\right) & \leq s_{d}\cdot\left(\frac{16}{3}\right)^{M}\cdot\left[j-i\right]^{d+1-M}.
\end{align*}
In view of equation (\ref{eq:AlphaModulationAbsoluteValueBracketModerate})
and since $\frac{16}{3}\leq12$, we conclude
\[
\Xi_{i,j}^{\left(k\right)}=\left(\frac{\left|j\right|}{\left|i\right|}\right)^{k}\cdot\left(\int_{B_{R_{i,j}}\left(\xi_{i,j}\right)}\left(1+\left|\xi\right|\right)^{-M}\operatorname{d}\xi\right)^{\tau}\leq4^{\left|k\right|}\cdot\left(12^{M}\cdot s_{d}\right)^{\tau}\cdot\left[j-i\right]^{\left|k\right|+\tau\left(d+1-M\right)},
\]
as in the previous case.
\textbf{Case 3}: The remaining case, i.e., $\left|i\right|<2\left|j\right|+4r$
\textbf{and} $\left|j\right|<2\left|i\right|+4r$. Since $i,j\in\mathbb{Z}^{d}\setminus\left\{ 0\right\} $,
we have $\left|i\right|,\left|j\right|\geq1$ and thus $\left|i\right|\leq\left|j\right|\cdot\left(2+4r\right)$
and $\left|j\right|\leq\left|i\right|\cdot\left(2+4r\right)$. In
particular, we have
\[
R_{i,j}=\left(\frac{\left|i\right|}{\left|j\right|}\right)^{\alpha_{0}}\cdot r\leq r\cdot\left(2+4r\right)^{\alpha_{0}}=:C_{r,\alpha_{0}}\:,
\]
so that every $\xi\in B_{R_{i,j}}\left(\xi_{i,j}\right)$ satisfies
\begin{align*}
1+\left|\xi_{i,j}\right| & \leq1+\left|\xi_{i,j}-\xi\right|+\left|\xi\right|\leq1+R_{i,j}+\left|\xi\right|\\
& \leq\left(1+R_{i,j}\right)\left(1+\left|\xi\right|\right)\leq\left(1+C_{r,\alpha_{0}}\right)\left(1+\left|\xi\right|\right).
\end{align*}
Consequently,
\begin{align*}
\int_{B_{R_{i,j}}\left(\xi_{i,j}\right)}\,\left(1+\left|\xi\right|\right)^{-M}\operatorname{d}\xi & \leq\left(1+C_{r,\alpha_{0}}\right)^{M}\cdot\lambda_{d}\left(B_{R_{i,j}}\left(\xi_{i,j}\right)\right)\cdot\left(1+\left|\xi_{i,j}\right|\right)^{-M}\\
\left({\scriptstyle \text{since }\lambda_{d}\left(B_{1}\left(0\right)\right)\leq\lambda_{d}\left(\left[-1,1\right]^{d}\right)=2^{d}}\right) & \leq2^{d}\cdot\left(1+C_{r,\alpha_{0}}\right)^{M}R_{i,j}^{d}\cdot\left(1+\left|\xi_{i,j}\right|\right)^{-M}\\
\left({\scriptstyle \text{since }R_{i,j}\leq C_{r,\alpha_{0}}}\right) & \leq2^{d}\cdot\left(1+C_{r,\alpha_{0}}\right)^{M+d}\cdot\left(1+\left|\xi_{i,j}\right|\right)^{-M}=:C_{d,M,r,\alpha_{0}}\cdot\left(1+\left|\xi_{i,j}\right|\right)^{-M}.
\end{align*}
Furthermore, since $\frac{1}{2+4r}\leq\frac{\left|j\right|}{\left|i\right|}\leq2+4r$,
we get $\left(\frac{\left|j\right|}{\left|i\right|}\right)^{k}\leq\left(2+4r\right)^{\left|k\right|}$
and thus
\[
\Xi_{i,j}^{\left(k\right)}\leq\left(2+4r\right)^{\left|k\right|}\cdot C_{d,M,r,\alpha_{0}}^{\tau}\cdot\left(1+\left|\xi_{i,j}\right|\right)^{-\tau M}.
\]
To further estimate the right-hand side, we distinguish two sub-cases:
\begin{enumerate}
\item We have $\left|i\right|\geq\left|j\right|$. In this case, equation
(\ref{eq:AlphaModulationKatoEstimate}) yields
\[
\left|\xi_{i,j}\right|=\left|\left(\frac{\left|i\right|}{\left|j\right|}\right)^{\alpha_{0}}i-j\right|\geq\left|i-j\right|
\]
and hence
\begin{align*}
\Xi_{i,j}^{\left(k\right)} & \leq\left(2+4r\right)^{\left|k\right|}\cdot C_{d,M,r,\alpha_{0}}^{\tau}\cdot\left[i-j\right]^{-\tau M}\\
& \leq\left(2+4r\right)^{\left|k\right|}\cdot C_{d,M,r,\alpha_{0}}^{\tau}\cdot\left[i-j\right]^{\left|k\right|+\tau\left(d+1-M\right)}.
\end{align*}
\item We have $\left|j\right|\geq\left|i\right|$. In this case, we can
again—after some rearranging—use equation (\ref{eq:AlphaModulationKatoEstimate})
to obtain
\begin{align*}
\left|\xi_{i,j}\right| & =\left|\left(\frac{\left|i\right|}{\left|j\right|}\right)^{\alpha_{0}}i-j\right|=\left(\frac{\left|i\right|}{\left|j\right|}\right)^{\alpha_{0}}\left|\left(\frac{\left|j\right|}{\left|i\right|}\right)^{\alpha_{0}}j-i\right|\\
\left({\scriptstyle \text{eq. }\eqref{eq:AlphaModulationKatoEstimate}}\right) & \geq\left(2+4r\right)^{-\alpha_{0}}\cdot\left|j-i\right|
\end{align*}
and hence
\begin{align*}
\Xi_{i,j}^{\left(k\right)} & \leq\left(2+4r\right)^{\left|k\right|}\cdot C_{d,M,r,\alpha_{0}}^{\tau}\cdot\left(1+\left(2+4r\right)^{-\alpha_{0}}\cdot\left|j-i\right|\right)^{-\tau M}\\
& \leq\left(2+4r\right)^{\left|k\right|+\alpha_{0}\tau M}\cdot C_{d,M,r,\alpha_{0}}^{\tau}\cdot\left[j-i\right]^{-\tau M}\\
& \leq\left(2+4r\right)^{\left|k\right|+\alpha_{0}\tau M}\cdot C_{d,M,r,\alpha_{0}}^{\tau}\cdot\left[j-i\right]^{\left|k\right|+\tau\left(d+1-M\right)}.
\end{align*}
\end{enumerate}
All in all, the preceding case distinction has established the bound
\begin{align}
\Xi_{i,j}^{\left(k\right)} & \leq\max\left\{ 4^{\left|k\right|}\cdot\left(12^{M}\cdot s_{d}\right)^{\tau},\:\left(2+4r\right)^{\left|k\right|+\alpha_{0}\tau M}\cdot C_{d,M,r,\alpha_{0}}^{\tau}\right\} \cdot\left[j-i\right]^{\left|k\right|+\tau\left(d+1-M\right)}\nonumber \\
& =:C_{d,M,r,\alpha_{0},k,\tau}\cdot\left[j-i\right]^{\left|k\right|+\tau\left(d+1-M\right)}\label{eq:AlphaModulationFinalXiEstimate}
\end{align}
for all $i,j\in\mathbb{Z}^{d}\setminus\left\{ 0\right\} $ and all
$k\in\mathbb{R}$, with $C_{d,M,r,\alpha_{0}}=2^{d}\cdot\left(1+C_{r,\alpha_{0}}\right)^{M+d}$
and $C_{r,\alpha_{0}}=r\cdot\left(2+4r\right)^{\alpha_{0}}$.
Now, we want to utilize this estimate for $k=k_{\lambda}=\tau\left(s^{\ast}+d\alpha_{0}\right)-\lambda\sigma\alpha_{0}$
for $\lambda\in\left\{ 0,1\right\} $, cf.\@ equation (\ref{eq:AlphaModulationMXiConnection}).
Note the equivalence
\begin{align*}
& \left|k_{\lambda}\right|+\tau\left(d+1-M\right)=\left|k_{\lambda}\right|+\tau\left(d+2-N_{0}\right)\overset{!}{\leq}-\left(d+1\right)\\
\Longleftrightarrow & N_{0}\overset{!}{\geq}d+2+\frac{\left|k_{\lambda}\right|+d+1}{\tau},
\end{align*}
where the last condition is satisfied for $\lambda\in\left\{ 0,1\right\} $
by definition of $k_{\lambda}$ and our assumptions regarding $N_{0}$.
Hence, we get—in view of equations (\ref{eq:StandardDecayLatticeSeries})
and (\ref{eq:AlphaModulationFinalXiEstimate}) and because of $\left|j-i\right|\geq\left\Vert j-i\right\Vert _{\infty}$—that
\begin{align*}
\sum_{j\in\mathbb{Z}^{d}\setminus\left\{ 0\right\} }\Xi_{i,j}^{\left(k_{\lambda}\right)} & \leq\left[\max_{\lambda\in\left\{ 0,1\right\} }C_{d,M,r,\alpha_{0},k_{\lambda},\tau}\right]\cdot\sum_{j\in\mathbb{Z}^{d}}\left[j-i\right]^{-\left(d+1\right)}\\
\left({\scriptstyle \text{with }\ell=j-i}\right) & \leq\left[\max_{\lambda\in\left\{ 0,1\right\} }C_{d,M,r,\alpha_{0},k_{\lambda},\tau}\right]\cdot\sum_{\ell\in\mathbb{Z}^{d}}\left(1+\left\Vert \ell\right\Vert _{\infty}\right)^{-\left(d+1\right)}\leq6^{d}\cdot\max_{\lambda\in\left\{ 0,1\right\} }C_{d,M,r,\alpha_{0},k_{\lambda},\tau}.
\end{align*}
The same estimate also holds when taking the sum over $i\in\mathbb{Z}^{d}\setminus\left\{ 0\right\} $
instead of over $j\in\mathbb{Z}^{d}\setminus\left\{ 0\right\} $.
In view of equation (\ref{eq:AlphaModulationMXiConnection}), we thus
get
\[
\sup_{i\in\mathbb{Z}^{d}\setminus\left\{ 0\right\} }\:\sum_{j\in\mathbb{Z}^{d}\setminus\left\{ 0\right\} }M_{j,i}^{\left(0\right)}\leq2^{1+\sigma+\tau\left|s^{\ast}\right|}\cdot6^{d}\cdot\max_{\lambda\in\left\{ 0,1\right\} }C_{d,M,r,\alpha_{0},k_{\lambda},\tau}
\]
and the same estimate also holds for $\sup_{j\in\mathbb{Z}^{d}\setminus\left\{ 0\right\} }\:\sum_{i\in\mathbb{Z}^{d}\setminus\left\{ 0\right\} }M_{j,i}^{\left(0\right)}$.
This easily yields the claim.
\end{proof}
Now, we can derive readily verifiable conditions which ensure that
the structured family generated by $\gamma$ yields a Banach frame
for a given $\alpha$-modulation space.
\begin{thm}
\label{thm:AlphaModulationBanachFrame}Let $d\in\mathbb{N}$, $\alpha\in\left[0,1\right)$
and choose $r>r_{1}\left(d,\alpha\right)$ with $r_{1}\left(d,\alpha\right)$
as in Theorem \ref{thm:AlphaModulationCoveringDefinition}.
Let $s_{0},\mu_{0}\geq0$ and $p_{0},q_{0}\in\left(0,1\right]$, as
well as $\varepsilon\in\left(0,1\right)$. Assume that $\gamma:\mathbb{R}^{d}\to\mathbb{C}$
satisfies the following:
\begin{enumerate}
\item We have $\gamma\in L_{\left(1+\left|\bullet\right|\right)^{\mu_{0}}}^{1}\left(\mathbb{R}^{d}\right)$
and $\widehat{\gamma}\in C^{\infty}\left(\mathbb{R}^{d}\right)$,
where all partial derivatives of $\widehat{\gamma}$ are polynomially
bounded.
\item We have $\gamma\in C^{1}\left(\mathbb{R}^{d}\right)$ and $\partial_{\ell}\gamma\in L_{\left(1+\left|\bullet\right|\right)^{\mu_{0}}}^{1}\left(\mathbb{R}^{d}\right)\cap L^{\infty}\left(\mathbb{R}^{d}\right)$
for all $\ell\in\underline{d}$.
\item \label{enu:AlphaModulationFrameFourierNonVanishing}We have $\left|\widehat{\gamma}\left(\xi\right)\right|\geq c>0$
for all $\xi\in\overline{B_{r}}\left(0\right)$.
\item \label{enu:AlphaModulationFrameFourierDecay}We have
\[
\left|\left(\partial^{\beta}\widehat{\gamma}\right)\left(\xi\right)\right|\lesssim\left(1+\left|\xi\right|\right)^{-N_{0}}
\]
for all $\xi\in\mathbb{R}^{d}$ and all $\beta\in\mathbb{N}_{0}^{d}$
with $\left|\beta\right|\leq\left\lceil \mu_{0}+\frac{d+\varepsilon}{p_{0}}\right\rceil $,
where
\[
\qquad\qquad N_{0}=d+2+\frac{d+1}{\min\left\{ p_{0},q_{0}\right\} }+\frac{1}{1-\alpha}\cdot\max\left\{ s_{0}+\alphad,\:s_{0}+\alpha\left(\frac{d}{p_{0}}-d+\mu_{0}+\left\lceil \mu_{0}+\frac{d+\varepsilon}{p_{0}}\right\rceil \right)\right\} .
\]
\end{enumerate}
Then there is some $\delta_{0}>0$ such that for all $0<\delta\leq\delta_{0}$,
the family
\[
\Gamma^{\left(\delta\right)}:=\left(L_{\delta\cdot k/\left|i\right|^{\alpha_{0}}}\:\widetilde{\gamma^{\left[i\right]}}\right)_{i\in\mathbb{Z}^{d}\setminus\left\{ 0\right\} ,k\in\mathbb{Z}^{d}},\quad\text{ with }\quad\gamma^{\left[i\right]}=\left|i\right|^{\frac{d\cdot\alpha_{0}}{2}}\cdot M_{\left|i\right|^{\alpha_{0}}\cdot i}\left[\gamma\circ\left|i\right|^{\alpha_{0}}\operatorname{id}\right]\quad\text{ and }\quad\tilde{g}\left(x\right):=g\left(-x\right)
\]
forms a Banach frame for $M_{\left(s,\mu\right),\alpha}^{p,q}\left(\mathbb{R}^{d}\right)$
for all $\left|s\right|\leq s_{0}$, $\left|\mu\right|\leq\mu_{0}$
and all $p,q\in\left(0,\infty\right]$ with $p\geq p_{0}$ and $q\geq q_{0}$.
Precisely, this means the following: Define the coefficient space
\[
\mathscr{C}_{p,q,s,\mu}^{\left(\alpha\right)}:=\ell_{\left[\left|i\right|^{\frac{1}{1-\alpha}\left(s+\alphad\left(\frac{1}{2}-\frac{1}{p}\right)\right)}\right]_{i\in\mathbb{Z}^{d}\setminus\left\{ 0\right\} }}^{q}\!\!\!\!\!\!\!\!\left(\left[\ell_{\left[\left(1+\left|k\right|/\left|i\right|^{\alpha_{0}}\right)^{\mu}\right]_{k\in\mathbb{Z}^{d}}}^{p}\left(\mathbb{Z}^{d}\right)\right]_{i\in\mathbb{Z}^{d}\setminus\left\{ 0\right\} }\right).
\]
Then the following hold:
\begin{enumerate}
\item The \textbf{analysis map}
\[
A^{\left(\delta\right)}:M_{\left(s,\mu\right),\alpha}^{p,q}\left(\smash{\mathbb{R}^{d}}\right)\to\mathscr{C}_{p,q,s,\mu}^{\left(\alpha\right)},f\mapsto\left[\left(\gamma^{\left[i\right]}\ast f\right)\left(\delta\cdot k/\left|i\right|^{\alpha_{0}}\right)\right]_{i\in\mathbb{Z}^{d}\setminus\left\{ 0\right\} ,k\in\mathbb{Z}^{d}}
\]
is well-defined and bounded for all $0<\delta\leq1$. Here, the convolution
$\left(\gamma^{\left[i\right]}\ast f\right)\left(x\right)$ has to
be understood similar to equation (\ref{eq:SpecialConvolutionPointwiseDefinition}).
\item For $0<\delta\leq\delta_{0}$, there is a bounded linear map \textbf{reconstruction
map} $R^{\left(\delta\right)}:\mathscr{C}_{p,q,s,\mu}^{\left(\alpha\right)}\to M_{\left(s,\mu\right),\alpha}^{p,q}\left(\mathbb{R}^{d}\right)$
satisfying $R^{\left(\delta\right)}\circ A^{\left(\delta\right)}=\operatorname{id}_{M_{\left(s,\mu\right),\alpha}^{p,q}\left(\mathbb{R}^{d}\right)}$.
Furthermore, the action of $R^{\left(\delta\right)}$ on a given sequence
is independent of the precise choice of $p,q,s,\mu$.
\item We have the following \textbf{consistency statement}: If $f\in M_{\left(s,\mu\right),\alpha}^{p,q}\left(\mathbb{R}^{d}\right)$
and if $p_{0}\leq\tilde{p}\leq\infty$ and $q_{0}\leq\tilde{q}\leq\infty$
and if furthermore $\left|\tilde{s}\right|\leq s_{0}$ and $\left|\tilde{\mu}\right|\leq\mu_{0}$,
then the following equivalence holds:
\[
f\in M_{\left(\tilde{s},\tilde{\mu}\right),\alpha}^{\tilde{p},\tilde{q}}\left(\smash{\mathbb{R}^{d}}\right)\qquad\Longleftrightarrow\qquad A^{\left(\delta\right)}f\in\mathscr{C}_{\tilde{p},\tilde{q},\tilde{s},\tilde{\mu}}^{\left(\alpha\right)}.\qedhere
\]
\end{enumerate}
\end{thm}
\begin{proof}
First of all, we remark that it is comparatively easy to show that
the family $\Gamma^{\left(\delta\right)}$ forms a Banach frame for
$M_{\left(s,\mu\right),\alpha}^{p,q}\left(\mathbb{R}^{d}\right)$
if $0<\delta\leq\delta_{0}$, where $\delta_{0}$ might depend on
$p,q,s,\mu$. About half of the proof will be spent on showing that
$\delta_{0}$ can actually be chosen \emph{independently} of $p,q,s,\mu$,
as long as these satisfy the restrictions mentioned in the statement
of the theorem.
Recall from Lemma \ref{lem:AlphaModulationStructuredAndBAPU} that
there is a family $\Phi=\left(\varphi_{i}\right)_{i\in\mathbb{Z}^{d}\setminus\left\{ 0\right\} }$
associated to $\mathcal{Q}=\mathcal{Q}_{r}^{\left(\alpha\right)}$ satisfying
Assumption \ref{assu:RegularPartitionOfUnity}. Furthermore, Corollary
\ref{cor:RegularBAPUsAreWeightedBAPUs} yields a function $\varrho\in\TestFunctionSpace{\mathbb{R}^{d}}$
such that, for $v_{0}\left(x\right)=\left[2\cdot\left(1+\left|x\right|\right)\right]^{\left|\mu\right|}$
as in Lemma \ref{lem:AlphaModulationSpaceWeight}, with $K=\left|\mu\right|\leq\mu_{0}$
and $Q=B_{r}\left(0\right)$, as well as $p\geq p_{0}$, we have
\begin{align}
C_{\mathcal{Q}_{r}^{\left(\alpha\right)},\Phi,v_{0},p} & \leq\Omega_{0}^{K}\Omega_{1}\cdot\left(4\cdotd\right)^{1+2\left\lceil K+\frac{d+\varepsilon}{p}\right\rceil }\cdot\left(\frac{s_{d}}{\varepsilon}\right)^{1/p}\cdot2^{\!\left\lceil K+\frac{d+\varepsilon}{p}\right\rceil }\cdot\lambda_{d}\left(Q\right)\cdot\max_{\left|\beta\right|\leq\left\lceil K+\frac{d+\varepsilon}{p}\right\rceil }\left\Vert \partial^{\beta}\varrho\right\Vert _{\sup}\cdot\max_{\left|\beta\right|\leq\left\lceil K+\frac{d+\varepsilon}{p}\right\rceil }C^{\left(\beta\right)}\nonumber \\
& \leq2^{\mu_{0}}\lambda_{d}\left(Q\right)\!\cdot\!\left(8\cdotd\right)^{1+2\left\lceil \mu_{0}+\frac{d+\varepsilon}{p_{0}}\right\rceil }\!\cdot\!\left(1\!+\!\frac{s_{d}}{\varepsilon}\right)^{\frac{1}{p_{0}}}\!\cdot\!\max_{\left|\beta\right|\leq\left\lceil \mu_{0}+\frac{d+\varepsilon}{p_{0}}\right\rceil }\left\Vert \partial^{\beta}\varrho\right\Vert _{\sup}\cdot\!\max_{\left|\beta\right|\leq\left\lceil \mu_{0}+\frac{d+\varepsilon}{p_{0}}\right\rceil }C^{\left(\beta\right)}=:L_{0}.\label{eq:AlphaModulationFrameBAPUConstantEstimate}
\end{align}
Now, assume that $\gamma$ satisfies all the stated properties and
let $\left|s\right|\leq s_{0}$, $\left|\mu\right|\leq\mu_{0}$ and
$p,q\in\left(0,\infty\right]$ with $p\geq p_{0}$ and $q\geq q_{0}$.
We want to verify the assumptions of Corollary \ref{cor:BanachFrameSimplifiedCriteria}
for the family $\left(\gamma_{i}\right)_{i\in\mathbb{Z}^{d}\setminus\left\{ 0\right\} }$,
with $\gamma_{i}:=\gamma$ for all $i\in\mathbb{Z}^{d}\setminus\left\{ 0\right\} $
and with $\mathcal{Q}=\mathcal{Q}_{r}^{\left(\alpha\right)}$. To this end, let
$\gamma_{1}^{\left(0\right)}:=\gamma$ and set $n_{i}:=1$, so that
$\gamma_{i}=\gamma=\gamma_{n_{i}}^{\left(0\right)}$ for all $i\in\mathbb{Z}^{d}\setminus\left\{ 0\right\} $.
In the notation of Lemma \ref{lem:GammaCoversOrbitAssumptionSimplified},
we then have $Q^{\left(1\right)}=\bigcup\left\{ Q_{i}'\,\middle|\, i\in\mathbb{Z}^{d}\setminus\left\{ 0\right\} \text{ and }n_{i}=1\right\} =B_{r}\left(0\right)$,
since $Q_{i}=T_{i}\left[B_{r}\left(0\right)\right]+b_{i}$ and thus
$Q_{i}'=B_{r}\left(0\right)$ for all $i\in\mathbb{Z}^{d}\setminus\left\{ 0\right\} $.
In view of part (\ref{enu:AlphaModulationFrameFourierNonVanishing})
of our assumptions, we thus see that Lemma \ref{lem:GammaCoversOrbitAssumptionSimplified}
is applicable, so that the family $\left(\gamma_{i}\right)_{i\in\mathbb{Z}^{d}\setminus\left\{ 0\right\} }$
satisfies Assumption \ref{assu:GammaCoversOrbit}, with $\Omega_{2}^{\left(p,K\right)}\leq L_{1}$
for some constant $L_{1}=L_{1}\left(\gamma,\mu_{0},p_{0},r,\alpha,d\right)>0$,
all $p\geq p_{0}$ and all $K\leq\mu_{0}$. Recall from Lemma \ref{lem:AlphaModulationSpaceWeight}
that we can choose $K=\left|\mu\right|\leq\mu_{0}$ in our present
setting.
Finally, recall from part (\ref{enu:AlphaModulationFrameFourierDecay})
of our assumptions that there is some $L_{2}>0$ (independent of $p,q,s,\mu$)
satisfying $\left|\partial^{\beta}\widehat{\gamma}\left(\xi\right)\right|\leq L_{2}\cdot\left(1+\left|\xi\right|\right)^{-N_{0}}$
for all $\xi\in\mathbb{R}^{d}$ and all $\beta\in\mathbb{N}_{0}^{d}$
with $\left|\beta\right|\leq\left\lceil \mu_{0}+\frac{d+\varepsilon}{p_{0}}\right\rceil $.
With these preparations, we can now verify the prerequisites of Corollary
\ref{cor:BanachFrameSimplifiedCriteria}:
\begin{enumerate}
\item As we have seen at the beginning of this section, the covering $\mathcal{Q}=\mathcal{Q}_{r}^{\left(\alpha\right)}$,
the weight $v=v^{\left(\mu\right)}$ (with $v_{0}\left(x\right)=\left[2\cdot\left(1+\left|x\right|\right)\right]^{\left|\mu\right|}$,
$\Omega_{0}=1$ and $\Omega_{1}=2^{\left|\mu\right|}\leq2^{\mu_{0}}$,
as well as $K=\left|\mu\right|\leq\mu_{0}$) satisfy all standing
assumptions of Section \ref{subsec:DecompSpaceDefinitionStandingAssumptions}.
Furthermore, the family $\Phi$ satisfies Assumption \ref{assu:RegularPartitionOfUnity}.
\item By our assumptions, we have $\gamma_{i}=\gamma\in L_{\left(1+\left|\bullet\right|\right)^{\mu_{0}}}^{1}\left(\mathbb{R}^{d}\right)\hookrightarrow L_{\left(1+\left|\bullet\right|\right)^{K}}^{1}\left(\mathbb{R}^{d}\right)$
and $\widehat{\gamma_{i}}=\widehat{\gamma}\in C^{\infty}\left(\mathbb{R}^{d}\right)$
for all $i\in\mathbb{Z}^{d}\setminus\left\{ 0\right\} $, where all
partial derivatives of $\widehat{\gamma_{i}}=\widehat{\gamma}$ are
polynomially bounded.
\item By our assumptions, we have $\gamma_{i}=\gamma\in C^{1}\left(\mathbb{R}^{d}\right)$
and $\partial_{\ell}\gamma\in L_{\left(1+\left|\bullet\right|\right)^{\mu_{0}}}^{1}\left(\mathbb{R}^{d}\right)\cap L^{\infty}\left(\mathbb{R}^{d}\right)$
and consequently also $\partial_{\ell}\gamma_{i}=\partial_{\ell}\gamma\in L_{\left(1+\left|\bullet\right|\right)^{K}}^{1}\left(\mathbb{R}^{d}\right)\cap L^{\infty}\left(\mathbb{R}^{d}\right)$
for all $\ell\in\underline{d}$ and $i\in\mathbb{Z}^{d}\setminus\left\{ 0\right\} $.
\item As seen above, the family $\left(\gamma_{i}\right)_{i\in\mathbb{Z}^{d}\setminus\left\{ 0\right\} }=\left(\gamma\right)_{i\in\mathbb{Z}^{d}\setminus\left\{ 0\right\} }$
satisfies Assumption \ref{assu:GammaCoversOrbit}.
\item Let
\[
C_{1}:=\sup_{i\in\mathbb{Z}^{d}\setminus\left\{ 0\right\} }\:\sum_{j\in\mathbb{Z}^{d}\setminus\left\{ 0\right\} }\:M_{j,i}\in\left[0,\infty\right]\qquad\text{ and }\qquad C_{2}:=\sup_{j\in\mathbb{Z}^{d}\setminus\left\{ 0\right\} }\:\sum_{i\in\mathbb{Z}^{d}\setminus\left\{ 0\right\} }\:M_{j,i}\in\left[0,\infty\right]
\]
as in Corollary \ref{cor:BanachFrameSimplifiedCriteria}, i.e., with
\[
M_{j,i}:=\left(\frac{w_{j}^{\left(s^{\ast}\right)}}{w_{i}^{\left(s^{\ast}\right)}}\right)^{\tau}\cdot\left(1+\left\Vert T_{j}^{-1}T_{i}\right\Vert \right)^{\sigma}\cdot\max_{\left|\beta\right|\leq1}\left(\left|\det T_{i}\right|^{-1}\cdot\int_{Q_{i}}\max_{\left|\theta\right|\leq N}\left|\left(\partial^{\theta}\widehat{\partial^{\beta}\gamma}\right)\!\!\left(S_{j}^{-1}\xi\right)\right|\operatorname{d}\xi\right)^{\tau},
\]
where, $s^{\ast}=\frac{s}{1-\alpha}$ and, since $K=\left|\mu\right|$,
\begin{align*}
N & =\left\lceil \left|\mu\right|+\frac{d+\varepsilon}{\min\left\{ 1,p\right\} }\right\rceil \,,\\
\tau & =\min\left\{ 1,p,q\right\} ,\\
\sigma & =\tau\cdot\left(\frac{d}{\min\left\{ 1,p\right\} }+\left|\mu\right|+\left\lceil \left|\mu\right|+\frac{d+\varepsilon}{\min\left\{ 1,p\right\} }\right\rceil \right).
\end{align*}
Note $\min\left\{ 1,p\right\} \geq\min\left\{ 1,p_{0}\right\} =p_{0}$
and $\left|\mu\right|\leq\mu_{0}$, so that $N\leq\left\lceil \mu_{0}+\frac{d+\varepsilon}{p_{0}}\right\rceil $.
Hence, we see that equation (\ref{eq:AlphaModulationBanachFrameDecayAssumption})
(with $C=L_{2}$) and hence also equation (\ref{eq:AlphaModulationStandardEstimate})
is satisfied, i.e., we have
\[
M_{j,i}\leq\left(2^{N+1}\pi\cdot L_{2}\right)^{\tau}\cdot M_{j,i}^{\left(0\right)}\qquad\forall i,j\in\mathbb{Z}^{d}\setminus\left\{ 0\right\} ,
\]
with $M_{j,i}^{\left(0\right)}$ as in equation (\ref{eq:AlphaModulationStandardEstimate}).
We now want to apply Lemma \ref{lem:AlphaModulationMainLemma}. To
this end, we have to verify that
\[
N_{0}\geqd+2+\frac{d+1}{\tau}+\max\left\{ \left|s^{\ast}+d\alpha_{0}\right|,\left|s^{\ast}+\left(d-\frac{\sigma}{\tau}\right)\alpha_{0}\right|\right\} .
\]
But we have $\tau=\min\left\{ 1,p,q\right\} \geq\min\left\{ 1,p_{0},q_{0}\right\} =\min\left\{ p_{0},q_{0}\right\} =:\tau_{0}$.
Furthermore, with $s_{0}^{\ast}:=s_{0}/\left(1-\alpha\right)$, we
have $\left|s^{\ast}+d\alpha_{0}\right|\leq s_{0}^{\ast}+d\alpha_{0}=\frac{1}{1-\alpha}\left(s_{0}+\alphad\right)$
and
\[
\frac{\sigma}{\tau}=\frac{d}{\min\left\{ 1,p\right\} }+\left|\mu\right|+\left\lceil \left|\mu\right|+\frac{d+\varepsilon}{\min\left\{ 1,p\right\} }\right\rceil \geq\frac{d}{\min\left\{ 1,p\right\} }\geqd,
\]
so that
\begin{align*}
\left|s^{\ast}+\left(d-\frac{\sigma}{\tau}\right)\alpha_{0}\right| & \leq s_{0}^{\ast}+\alpha_{0}\left|d-\frac{\sigma}{\tau}\right|=s_{0}^{\ast}+\alpha_{0}\left(\frac{\sigma}{\tau}-d\right)\\
& =\frac{1}{1-\alpha}\left[s_{0}+\alpha\left(\frac{d}{\min\left\{ 1,p\right\} }+\left|\mu\right|+\left\lceil \left|\mu\right|+\frac{d+\varepsilon}{\min\left\{ 1,p\right\} }\right\rceil -d\right)\right]\\
& \leq\frac{1}{1-\alpha}\left[s_{0}+\alpha\left(\frac{d}{p_{0}}-d+\mu_{0}+\left\lceil \mu_{0}+\frac{d+\varepsilon}{p_{0}}\right\rceil \right)\right].
\end{align*}
Hence, our assumptions easily yield
\begin{align*}
N_{0} & =d+2+\frac{d+1}{\tau_{0}}+\frac{1}{1-\alpha}\cdot\max\left\{ s_{0}+\alphad,\:s_{0}+\alpha\left(\frac{d}{p_{0}}-d+\mu_{0}+\left\lceil \mu_{0}+\frac{d+\varepsilon}{p_{0}}\right\rceil \right)\right\} \\
& \geqd+2+\frac{d+1}{\tau}+\max\left\{ \left|s^{\ast}+d\alpha_{0}\right|,\left|s^{\ast}+\left(d-\frac{\sigma}{\tau}\right)\alpha_{0}\right|\right\} ,
\end{align*}
as desired.
For brevity, set $L_{3}:=2^{d}\cdot\left(1+r\cdot\left(2+4r\right)^{\alpha_{0}}\right)^{N_{0}+d}$.
Since Lemma \ref{lem:AlphaModulationMainLemma} is applicable, and
since $\tau_{0}\leq\tau\leq1$, we get
\begin{align*}
\quad\quad C_{1}^{1/\tau} & \leq2^{N+1}\pi\cdot L_{2}\cdot\left[\sup_{i\in\mathbb{Z}^{d}\setminus\left\{ 0\right\} }\:\sum_{j\in\mathbb{Z}^{d}\setminus\left\{ 0\right\} }M_{j,i}^{\left(0\right)}\right]^{1/\tau}\\
\quad\quad\quad\quad & \leq2^{N+1}\pi L_{2}\cdot2^{\left|s^{\ast}\right|+\frac{1+\sigma}{\tau}}6^{\frac{d}{\tau}}\cdot\max\left\{ 4^{\alpha_{0}\left(\frac{\sigma}{\tau}+d\right)+\left|s^{\ast}\right|}\cdot12^{N_{0}}\cdot s_{d},\:\left(2+4r\right)^{\left|s^{\ast}\right|+\alpha_{0}\left[\frac{\sigma}{\tau}+d+N_{0}\right]}L_{3}\right\} \\
\quad\quad\quad\quad & \overset{\left(\ast\right)}{\leq}6^{\frac{d}{\tau_{0}}}2^{1+\left\lceil \mu_{0}+\frac{d+\varepsilon}{p_{0}}\right\rceil }\pi L_{2}\cdot2^{\frac{s_{0}}{1-\alpha}+\frac{1}{\tau_{0}}+\frac{\sigma_{0}}{\tau_{0}}}\cdot\max\left\{ \!4^{\frac{1}{1-\alpha}\left[s_{0}+\alpha\left(\frac{\sigma_{0}}{\tau_{0}}+d\right)\right]}\!\cdot12^{N_{0}}s_{d},\:\left(2\!+\!4r\right)^{\frac{1}{1-\alpha}\left[s_{0}+\alpha\left(\frac{\sigma_{0}}{\tau_{0}}+d+N_{0}\right)\right]}\!L_{3}\!\right\} \\
\quad\quad\quad\quad & =:L_{4}.
\end{align*}
Here, we recall $\tau_{0}=\min\left\{ p_{0},q_{0}\right\} $ and observe
that the step marked with $\left(\ast\right)$ used the estimate
\[
\frac{\sigma}{\tau}=\frac{d}{\min\left\{ 1,p\right\} }+\left|\mu\right|+\left\lceil \left|\mu\right|+\frac{d+\varepsilon}{\min\left\{ 1,p\right\} }\right\rceil \leq\frac{d}{p_{0}}+\mu_{0}+\left\lceil \mu_{0}+\frac{d+\varepsilon}{p_{0}}\right\rceil =:\frac{\sigma_{0}}{\tau_{0}}
\]
and $\sigma=\frac{\sigma}{\tau}\cdot\tau\leq\frac{\sigma}{\tau}\leq\frac{\sigma_{0}}{\tau_{0}}$.
Hence, we have shown $C_{1}^{1/\tau}\leq L_{4}$, where $L_{4}$ is
\emph{independent} of $p,q,s,\mu$. Exactly the same argument also
shows $C_{2}^{1/\tau}\leq L_{4}$, with the same constant $L_{4}$.
In particular, $C_{1}<\infty$ and $C_{2}<\infty$, which was the
last part of Corollary \ref{cor:BanachFrameSimplifiedCriteria} that
we needed to verify.
\end{enumerate}
All in all, Corollary \ref{cor:BanachFrameSimplifiedCriteria} shows
that the family $\left(\gamma_{i}\right)_{i\in\mathbb{Z}^{d}\setminus\left\{ 0\right\} }=\left(\gamma\right)_{i\in\mathbb{Z}^{d}\setminus\left\{ 0\right\} }$
satisfies all assumptions of Theorem \ref{thm:DiscreteBanachFrameTheorem}
and also that $\vertiii{\smash{\overrightarrow{A}}}^{\max\left\{ 1,\frac{1}{p}\right\} }\leq2L_{5}^{\left(0\right)}\cdot L_{4}$,
as well as $\vertiii{\smash{\overrightarrow{B}}}^{\max\left\{ 1,\frac{1}{p}\right\} }\leq2L_{5}^{\left(0\right)}\cdot L_{4}$,
where $\overrightarrow{A}$ and $\overrightarrow{B}$ are defined
as in Assumptions \ref{assu:MainAssumptions} and \ref{assu:DiscreteBanachFrameAssumptions}
and where
\begin{align*}
L_{5}^{\left(0\right)} & =\Omega_{0}^{K}\Omega_{1}\cdotd^{1/\min\left\{ 1,p\right\} }\cdot\left(4\cdotd\right)^{1+2\left\lceil K+\frac{d+\varepsilon}{\min\left\{ 1,p\right\} }\right\rceil }\cdot\left(\frac{s_{d}}{\varepsilon}\right)^{1/\min\left\{ 1,p\right\} }\cdot\max_{\left|\beta\right|\leq\left\lceil K+\frac{d+\varepsilon}{\min\left\{ 1,p\right\} }\right\rceil }C^{\left(\beta\right)}\\
& \leq2^{\mu_{0}}\cdotd^{1/p_{0}}\cdot\left(4\cdotd\right)^{1+2\left\lceil \mu_{0}+\frac{d+\varepsilon}{p_{0}}\right\rceil }\cdot\left[1+\frac{s_{d}}{\varepsilon}\right]^{\frac{1}{p_{0}}}\cdot\max_{\left|\beta\right|\leq\left\lceil \mu_{0}+\frac{d+\varepsilon}{p_{0}}\right\rceil }C^{\left(\beta\right)}=:L_{5},
\end{align*}
where the constants $C^{\left(\beta\right)}=C^{\left(\beta\right)}\left(\Phi\right)$
are defined as in Assumption \ref{assu:RegularPartitionOfUnity}.
Now, Theorem \ref{thm:DiscreteBanachFrameTheorem} shows that the
family $\Gamma^{\left(\delta\right)}$ is a Banach frame for $M_{\left(s,\mu\right),\alpha}^{p,q}\left(\mathbb{R}^{d}\right)=\DecompSp{\mathcal{Q}_{r}^{\left(\alpha\right)}}p{\ell_{w^{\left(s^{\ast}\right)}}^{q}}{v^{\left(\mu\right)}}$,
as soon as $0<\delta\leq\frac{1}{1+2\vertiii{F_{0}}^{2}}$ with $F_{0}$
as in Lemma \ref{lem:SpecialProjection}. But that lemma yields the
estimate
\begin{align*}
\vertiii{F_{0}} & \leq2^{\frac{1}{q}}C_{\mathcal{Q}_{r}^{\left(\alpha\right)},\Phi,v_{0},p}^{2}\cdot\vertiii{\smash{\Gamma_{\mathcal{Q}_{r}^{\left(\alpha\right)}}}}^{2}\cdot\left(\vertiii{\smash{\overrightarrow{A}}}^{\max\left\{ 1,\frac{1}{p}\right\} }+\vertiii{\smash{\overrightarrow{B}}}^{\max\left\{ 1,\frac{1}{p}\right\} }\right)\cdot L_{6}^{\left(0\right)}\\
\left({\scriptstyle \text{eqs. }\eqref{eq:WeightedSequenceSpaceClusteringMapNormEstimate}\text{ and }\eqref{eq:AlphaModulationFrameBAPUConstantEstimate}}\right) & \leq2^{\frac{1}{q_{0}}}\cdot C_{\mathcal{Q}_{r}^{\left(\alpha\right)},w^{\left(s^{\ast}\right)}}^{2}\cdot N_{\mathcal{Q}_{r}^{\left(\alpha\right)}}^{2\left(1+\frac{1}{q}\right)}\cdot4L_{0}^{2}L_{4}L_{5}\cdot L_{6}^{\left(0\right)}
\end{align*}
for
\[
L_{6}^{\left(0\right)}=\!\!\begin{cases}
\frac{\left(2^{16}\cdot768/d^{\frac{3}{2}}\right)^{\frac{d}{p}}}{2^{42}\cdot12^{d}\cdotd^{15}}\!\cdot\!\left(2^{52}\!\cdot\!d^{\frac{25}{2}}\!\cdot\!N^{3}\right)^{N+1}\!\!\!\cdot\!N_{\mathcal{Q}_{r}^{\left(\alpha\right)}}^{2\left(\frac{1}{p}-1\right)}\!\left(1\!+\!R_{\mathcal{Q}_{r}^{\left(\alpha\right)}}C_{\mathcal{Q}_{r}^{\left(\alpha\right)}}\right)^{d\left(\frac{4}{p}-1\right)}\!\!\cdot\Omega_{0}^{13K}\Omega_{1}^{13}\Omega_{2}^{\left(p,K\right)}, & \text{if }p<1,\\
\frac{1}{\sqrt{d}\cdot2^{12+6\left\lceil K\right\rceil }}\cdot\left(2^{17}\cdotd^{5/2}\cdot N\right)^{\left\lceil K\right\rceil +d+2}\cdot\left(1+R_{\mathcal{Q}_{r}^{\left(\alpha\right)}}\right)^{d}\cdot\Omega_{0}^{3K}\Omega_{1}^{3}\Omega_{2}^{\left(p,K\right)}, & \text{if }p\geq1.
\end{cases}
\]
But above we saw $\Omega_{2}^{\left(p,K\right)}\leq L_{1}$ since
$K=\left|\mu\right|\leq\mu_{0}$ and $p\geq p_{0}$. Using this estimate
and the bounds $N\leq\left\lceil \mu_{0}+\frac{d+\varepsilon}{p_{0}}\right\rceil $
and $0\leq K=\left|\mu\right|\leq\mu_{0}$, as well as $\Omega_{0}=1$
and $\Omega_{1}=2^{\left|\mu\right|}\leq2^{\mu_{0}}$, we see $L_{6}^{\left(0\right)}\leq L_{6}$,
where $L_{6}$ is independent of $p,q,s,\mu$.
Finally, it is not hard to see—because of $w^{\left(s^{\ast}\right)}=\left(w^{\left(1/\left(1-\alpha\right)\right)}\right)^{s}$—that
\[
N_{\mathcal{Q}_{r}^{\left(\alpha\right)},w^{\left(s^{\ast}\right)}}\leq N_{\mathcal{Q}_{r}^{\left(\alpha\right)},w^{\left(1/\left(1-\alpha\right)\right)}}^{\left|s\right|}\leq N_{\mathcal{Q}_{r}^{\left(\alpha\right)},w^{\left(1/\left(1-\alpha\right)\right)}}^{s_{0}}=:L_{7},
\]
where $L_{7}>0$ is independent of $p,q,s,\mu$. By putting everything
together, we get $\vertiii{F_{0}}\leq L_{8}$, with $L_{8}$ independent
of $p,q,s,\mu$. Hence, $\Gamma^{\left(\delta\right)}$ is a Banach
frame for $M_{\left(s,\mu\right),\alpha}^{p,q}\left(\mathbb{R}^{d}\right)$
in the sense of Theorem \ref{thm:DiscreteBanachFrameTheorem} as soon
as $0<\delta\leq\delta_{0}:=\frac{1}{1+2L_{8}^{2}}$, with $\delta_{0}$
independent of $p,q,s,\mu$.
\medskip{}
All that remains to verify is that the space $\ell_{\left(\left|\det T_{i}\right|^{\frac{1}{2}-\frac{1}{p}}\cdot w_{i}\right)_{i\in I}}^{q}\!\!\!\!\!\left(\left[\vphantom{\sum}\smash{C_{i}^{\left(\delta\right)}}\right]_{i\in I}\right)$
(with $w_{i}=w_{i}^{\left(s^{\ast}\right)}=\left\langle i\right\rangle ^{s^{\ast}}$)
appearing in Theorem \ref{thm:DiscreteBanachFrameTheorem} coincides
with the space $\mathscr{C}_{p,q,s,\mu}^{\left(\alpha\right)}$ from
the statement of the current theorem. To this end, first note that
$\left|i\right|\asymp\left\langle i\right\rangle $ for all $i\in\mathbb{Z}^{d}\setminus\left\{ 0\right\} $,
so that
\[
\left|\det T_{i}\right|^{\frac{1}{2}-\frac{1}{p}}\cdot w_{i}=\left|i\right|^{d\alpha_{0}\left(\frac{1}{2}-\frac{1}{p}\right)}\cdot\left\langle i\right\rangle ^{s^{\ast}}\asymp_{s}\;\left|i\right|^{\frac{1}{1-\alpha}\left[s+\alphad\left(\frac{1}{2}-\frac{1}{p}\right)\right]}\qquad\forall i\in\mathbb{Z}^{d}\setminus\left\{ 0\right\} .
\]
Finally, recall from equation (\ref{eq:CoefficientSpaceDefinition})
that
\[
C_{i}^{\left(\delta\right)}=\ell_{v^{\left(j,\delta\right)}}^{p}\left(\smash{\mathbb{Z}^{d}}\right)\quad\text{ with }\quad v_{k}^{\left(j,\delta\right)}=v\left(\delta\cdot T_{j}^{-T}k\right)
\]
where $v=v^{\left(\mu\right)}$ with $v^{\left(\mu\right)}\left(x\right)=\left\langle x\right\rangle ^{\mu}\asymp_{\mu}\;\left(1+\left|x\right|\right)^{\mu}$
for all $x\in\mathbb{R}^{d}$. Furthermore, since $0<\delta\leq1$,
we have
\[
\delta\cdot\left(1+\left|x\right|\right)\leq1+\left|\delta\cdot x\right|\leq1+\left|x\right|\qquad\forall x\in\mathbb{R}^{d},
\]
which yields
\[
v_{k}^{\left(j,\delta\right)}=v\left(\delta\cdot T_{j}^{-T}k\right)\:\asymp_{\mu}\:\left(1+\left|\delta\cdot T_{j}^{-T}k\right|\right)^{\mu}\:\asymp_{\mu,\delta}\:\left(1+\left|T_{j}^{-T}k\right|\right)^{\mu}=\left(1+\left|k\right|/\left|j\right|^{\alpha_{0}}\right)^{\mu}
\]
for all $k\in\mathbb{Z}^{d}$ and $j\in\mathbb{Z}^{d}\setminus\left\{ 0\right\} $.
Here, the implied constant might depend on $\mu,\delta$, but not
on $j,k$. Combining these facts, we conclude $\ell_{\left(\left|\det T_{i}\right|^{\frac{1}{2}-\frac{1}{p}}\cdot w_{i}\right)_{i\in I}}^{q}\!\!\!\!\!\left(\left[\vphantom{\sum}\smash{C_{i}^{\left(\delta\right)}}\right]_{i\in I}\right)=\mathscr{C}_{p,q,s,\mu}^{\left(\alpha\right)}$,
with equivalent quasi-norms, as desired. Now all claims follow from
Theorem \ref{thm:DiscreteBanachFrameTheorem}.
\end{proof}
Having established convenient criteria for the existence of Banach
frames, we finally consider nice criteria which ensure that a prototype
$\gamma$ generates an atomic decomposition for a given $\alpha$-modulation
space.
\begin{thm}
\label{thm:AlphaModulationAtomicDecomposition}Let $d\in\mathbb{N}$,
$\alpha\in\left[0,1\right)$ and choose $r>r_{1}\left(d,\alpha\right)$
with $r_{1}\left(d,\alpha\right)$ as in Theorem \ref{thm:AlphaModulationCoveringDefinition}.
Let $s_{0},\mu_{0}\geq0$ and $p_{0},q_{0}\in\left(0,1\right]$, as
well as $\varepsilon\in\left(0,1\right)$. Assume that $\gamma:\mathbb{R}^{d}\to\mathbb{C}$
is measurable and satisfies the following conditions:
\begin{enumerate}
\item We have $\left\Vert \gamma\right\Vert _{\mu_{0}+\frac{d}{p_{0}}+1}<\infty$,
where as usual $\left\Vert g\right\Vert _{M}=\sup_{x\in\mathbb{R}^{d}}\left(1+\left|x\right|\right)^{M}\left|g\left(x\right)\right|$.
In particular, $\gamma\in L^{1}\left(\mathbb{R}^{d}\right)$.
\item We have $\widehat{\gamma}\in C^{\infty}\left(\mathbb{R}^{d}\right)$
and all partial derivatives of $\widehat{\gamma}$ are polynomially
bounded.
\item \label{enu:AlphaModulationAtomicFourierNonVanishing}We have $\left|\widehat{\gamma}\left(\xi\right)\right|\geq c>0$
for all $\xi\in\overline{B_{r}}\left(0\right)$.
\item \label{enu:AlphaModulationAtomicFourierDecay}We have
\[
\left|\left(\partial^{\beta}\widehat{\gamma}\right)\left(\xi\right)\right|\lesssim\left(1+\left|\xi\right|\right)^{-M_{0}}
\]
for all $\xi\in\mathbb{R}^{d}$ and all $\beta\in\mathbb{N}_{0}^{d}$
with $\left|\beta\right|\leq\left\lceil \mu_{0}+\frac{d+\varepsilon}{p_{0}}\right\rceil $,
where
\[
M_{0}=\left(d+1\right)\cdot\left(2+\varepsilon+\frac{1}{\min\left\{ p_{0},q_{0}\right\} }\right)+\Lambda,
\]
with
\[
\Lambda=\begin{cases}
\frac{1}{1-\alpha}\max\left\{ s_{0}+d\alpha,\,s_{0}+\alpha\left(\left\lceil \mu_{0}+d+\varepsilon\right\rceil -d\right)\right\} , & \text{if }p_{0}=1,\\
\frac{1}{1-\alpha}\left[s_{0}+\alpha\left(\mu_{0}+\left\lceil \mu_{0}+p_{0}^{-1}\cdot\left(d+\varepsilon\right)\right\rceil \right)\right], & \text{if }p_{0}\in\left(0,1\right).
\end{cases}
\]
\end{enumerate}
Then there is some $\delta_{0}>0$ such that for all $0<\delta\leq\delta_{0}$,
the family
\[
\Gamma^{\left(\delta\right)}:=\left(L_{\delta\cdot k/\left|i\right|^{\alpha_{0}}}\:\gamma^{\left[i\right]}\right)_{i\in\mathbb{Z}^{d}\setminus\left\{ 0\right\} ,k\in\mathbb{Z}^{d}},\qquad\text{ with }\qquad\gamma^{\left[i\right]}=\left|i\right|^{\frac{d\cdot\alpha_{0}}{2}}\cdot M_{\left|i\right|^{\alpha_{0}}\cdot i}\left[\gamma\circ\left|i\right|^{\alpha_{0}}\operatorname{id}\right]
\]
forms an atomic decomposition for $M_{\left(s,\mu\right),\alpha}^{p,q}\left(\mathbb{R}^{d}\right)$
for all $\left|s\right|\leq s_{0}$, $\left|\mu\right|\leq\mu_{0}$
and all $p,q\in\left(0,\infty\right]$ with $p\geq p_{0}$ and $q\geq q_{0}$.
Precisely, this means the following: Define the coefficient space
\[
\mathscr{C}_{p,q,s,\mu}^{\left(\alpha\right)}:=\ell_{\left[\left|i\right|^{\frac{1}{1-\alpha}\left(s+\alphad\left(\frac{1}{2}-\frac{1}{p}\right)\right)}\right]_{i\in\mathbb{Z}^{d}\setminus\left\{ 0\right\} }}^{q}\!\!\!\!\!\!\!\!\left(\left[\ell_{\left[\left(1+\left|k\right|/\left|i\right|^{\alpha_{0}}\right)^{\mu}\right]_{k\in\mathbb{Z}^{d}}}^{p}\left(\mathbb{Z}^{d}\right)\right]_{i\in\mathbb{Z}^{d}\setminus\left\{ 0\right\} }\right).
\]
Then the following hold:
\begin{enumerate}
\item For arbitrary $\delta\in\left(0,1\right]$, the \textbf{synthesis
map}
\[
S^{\left(\delta\right)}:\mathscr{C}_{p,q,s,\mu}^{\left(\alpha\right)}\to M_{\left(s,\mu\right),\alpha}^{p,q}\left(\smash{\mathbb{R}^{d}}\right),\left(\smash{c_{k}^{\left(i\right)}}\right)_{i\in\mathbb{Z}^{d}\setminus\left\{ 0\right\} ,k\in\mathbb{Z}^{d}}\mapsto\sum_{i\in\mathbb{Z}^{d}\setminus\left\{ 0\right\} }\:\sum_{k\in\mathbb{Z}^{d}}\left[c_{k}^{\left(i\right)}\cdot L_{\delta\cdot k/\left|i\right|^{\alpha_{0}}}\:\gamma^{\left[i\right]}\right]
\]
is well-defined and bounded. Convergence of the series has to be understood
as described in the remark following Theorem \ref{thm:AtomicDecomposition}.
\item For $0<\delta\leq\delta_{0}$, there is a bounded linear \textbf{coefficient
map} $C^{\left(\delta\right)}:M_{\left(s,\mu\right),\alpha}^{p,q}\left(\smash{\mathbb{R}^{d}}\right)\to\mathscr{C}_{p,q,s,\mu}^{\left(\alpha\right)}$
satisfying $S^{\left(\delta\right)}\circ C^{\left(\delta\right)}=\operatorname{id}_{M_{\left(s,\mu\right),\alpha}^{p,q}\left(\smash{\mathbb{R}^{d}}\right)}$.
Furthermore, the action of $C^{\left(\delta\right)}$ on a given $f\in M_{\left(s,\mu\right),\alpha}^{p,q}\left(\mathbb{R}^{d}\right)$
is independent of the precise choice of $p,q,s,\mu$.\qedhere
\end{enumerate}
\end{thm}
\begin{rem*}
Choose $M_{0}$ as in the theorem above. If $\gamma\in C_{c}^{\left\lceil M_{0}\right\rceil }\left(\mathbb{R}^{d}\right)$,
then $\widehat{\gamma}\in C^{\infty}\left(\mathbb{R}^{d}\right)$
with all partial derivatives being polynomially bounded. Furthermore,
for arbitrary $\alpha\in\mathbb{N}_{0}^{d}$, we have $\gamma_{\alpha}\in C_{c}^{\left\lceil M_{0}\right\rceil }\left(\mathbb{R}^{d}\right)\hookrightarrow W^{\left\lceil M_{0}\right\rceil ,1}\left(\mathbb{R}^{d}\right)$
for
\[
\gamma_{\alpha}:\mathbb{R}^{d}\to\mathbb{C},x\mapsto\left(-2\pi ix\right)^{\alpha}\cdot\gamma\left(x\right).
\]
But by differentiation under the integral, it is not hard to see $\partial^{\alpha}\widehat{\gamma}\left(\xi\right)=\widehat{\gamma_{\alpha}}\left(\xi\right)$
for all $\xi\in\mathbb{R}^{d}$. Hence, Lemma \ref{lem:PointwiseFourierDecayEstimate}
yields $\left|\partial^{\alpha}\widehat{\gamma}\left(\xi\right)\right|=\left|\left(\mathcal{F}^{-1}\gamma_{\alpha}\right)\left(-\xi\right)\right|\lesssim\left(1+\left|\xi\right|\right)^{-\left\lceil M_{0}\right\rceil }\leq\left(1+\left|\xi\right|\right)^{-M_{0}}$.
Finally, we clearly have $\left\Vert \gamma\right\Vert _{\mu_{0}+\frac{d}{p_{0}}+1}<\infty$,
since $\gamma$ has compact support.
All in all, these considerations show that every prototype $\gamma\in C_{c}^{\left\lceil M_{0}\right\rceil }\left(\mathbb{R}^{d}\right)$
with $\widehat{\gamma}\left(\xi\right)\neq0$ for all $\xi\in\overline{B_{r}}\left(0\right)$
generates an atomic decomposition for $M_{\left(s,\mu\right),\alpha}^{p,q}\left(\mathbb{R}^{d}\right)$,
where $M_{0}=M_{0}\left(d,p,q,s,\mu,\alpha,\varepsilon\right)$
has to be chosen suitably. Very similar considerations apply for the
case of Banach frames: Here, it suffices to have $\gamma\in C_{c}^{\left\lceil N_{0}\right\rceil }\left(\mathbb{R}^{d}\right)$
with $\widehat{\gamma}\left(\xi\right)\neq0$ for all $\xi\in\overline{B_{r}}\left(0\right)$,
where $N_{0}=N_{0}\left(d,p,q,s,\mu,\alpha,\varepsilon\right)$
is chosen as in Theorem \ref{thm:AlphaModulationBanachFrame}.
\end{rem*}
\begin{proof}
First of all, we remark as in the proof of Theorem \ref{thm:AlphaModulationBanachFrame}
that it is comparatively easy to show that the family $\Gamma^{\left(\delta\right)}$
forms an atomic decomposition for $M_{\left(s,\mu\right),\alpha}^{p,q}\left(\mathbb{R}^{d}\right)$
if $0<\delta\leq\delta_{0}$, where $\delta_{0}$ might depend on
$p,q,s,\mu$. About half of the proof will be spent on showing that
$\delta_{0}$ can actually be chosen \emph{independently} of $p,q,s,\mu$,
as long as these satisfy the restrictions mentioned in the statement
of the theorem.
Our assumptions yield $L_{0}>0$ satisfying $\left|\partial^{\beta}\widehat{\gamma}\left(\xi\right)\right|\leq L_{0}\cdot\left(1+\left|\xi\right|\right)^{-M_{0}}$
for all $\xi\in\mathbb{R}^{d}$ and all multiindices $\beta\in\mathbb{N}_{0}^{d}$
with $\left|\beta\right|\leq\left\lceil \mu_{0}+\frac{d+\varepsilon}{p_{0}}\right\rceil =:N_{0}$.
As our first step, we invoke Lemma \ref{lem:ConvolutionFactorization}
with $N=N_{0}$ and
\[
\varrho:\mathbb{R}^{d}\to\left(0,\infty\right),\xi\mapsto L_{0}\cdot\left(1+\left|\xi\right|\right)^{-\left[M_{0}-\left(d+1\right)\left(1+\varepsilon\right)\right]}.
\]
To this end, we observe $N_{0}\geq\left\lceil \left(d+\varepsilon\right)/p_{0}\right\rceil \geq\left\lceil d+\varepsilon\right\rceil =d+1$
and furthermore
\[
M_{0}-\left(d+1\right)\left(1+\varepsilon\right)\geq\left(d+1\right)\cdot\left(2+\varepsilon\right)-\left(d+1\right)\left(1+\varepsilon\right)=d+1>d,
\]
so that $\varrho\in L^{1}\left(\mathbb{R}^{d}\right)$. Finally,
as we just saw, we indeed have
\[
\left|\partial^{\beta}\widehat{\gamma}\left(\xi\right)\right|\leq L_{0}\cdot\left(1+\left|\xi\right|\right)^{-M_{0}}=\varrho\left(\xi\right)\cdot\left(1+\left|\xi\right|\right)^{-\left(d+1\right)\left(1+\varepsilon\right)}\leq\varrho\left(\xi\right)\cdot\left(1+\left|\xi\right|\right)^{-\left(d+1+\varepsilon\right)}
\]
for all $\xi\in\mathbb{R}^{d}$ and all $\beta\in\mathbb{N}_{0}^{d}$
with $\left|\beta\right|\leq N_{0}$, so that all assumptions of Lemma
\ref{lem:ConvolutionFactorization} are satisfied. Hence, there are
functions $\gamma_{1},\gamma_{2}\in L^{1}\left(\mathbb{R}^{d}\right)$
with the following properties:
\begin{enumerate}
\item We have $\gamma=\gamma_{1}\ast\gamma_{2}$.
\item We have $\gamma_{2}\in C^{1}\left(\mathbb{R}^{d}\right)$ with $L_{1}^{\left(M\right)}:=\left\Vert \gamma_{2}\right\Vert _{M}+\left\Vert \nabla\gamma_{2}\right\Vert _{M}<\infty$
for arbitrary $M\in\mathbb{N}_{0}$.
\item We have $\widehat{\gamma_{1}},\widehat{\gamma_{2}}\in C^{\infty}\left(\mathbb{R}^{d}\right)$,
where all partial derivatives of these functions are polynomially
bounded.
\item We have $\left\Vert \gamma_{1}\right\Vert _{N_{0}}<\infty$ and $\left\Vert \gamma\right\Vert _{N_{0}}<\infty$.
\item We have $\left|\partial^{\beta}\widehat{\gamma_{1}}\left(\xi\right)\right|\leq L_{2}\cdot\varrho\left(\xi\right)=L_{0}L_{2}\cdot\left(1+\left|\xi\right|\right)^{-M_{00}}$
for all $\xi\in\mathbb{R}^{d}$ and all $\beta\in\mathbb{N}_{0}^{d}$
with $\left|\beta\right|\leq N_{0}$. Here, $L_{2}:=2^{1+d+4N_{0}}\cdot N_{0}!\cdot\left(1+d\right)^{N_{0}}$
and
\[
M_{00}:=M_{0}-\left(d+1\right)\left(1+\varepsilon\right)=\left(d+1\right)\cdot\left(1+\frac{1}{\min\left\{ p_{0},q_{0}\right\} }\right)+\Lambda.
\]
\end{enumerate}
Next, recall from Lemma \ref{lem:AlphaModulationStructuredAndBAPU}
that there is a family $\Phi=\left(\varphi_{i}\right)_{i\in\mathbb{Z}^{d}\setminus\left\{ 0\right\} }$
associated to $\mathcal{Q}=\mathcal{Q}_{r}^{\left(\alpha\right)}$ satisfying
Assumption \ref{assu:RegularPartitionOfUnity}. As in the proof of
Theorem \ref{thm:AlphaModulationBanachFrame} (cf.\@ equation (\ref{eq:AlphaModulationFrameBAPUConstantEstimate})),
we get as a consequence of Corollary \ref{cor:RegularBAPUsAreWeightedBAPUs}
a constant $L_{3}>0$ satisfying
\begin{equation}
C_{\mathcal{Q}_{r}^{\left(\alpha\right)},\Phi,v_{0},p}\leq L_{3},\quad\forall p\geq p_{0}\,\forall K=\left|\mu\right|\leq\mu_{0},\label{eq:AlphaModulationAtomicDecompositionBAPUConstantEstimate}
\end{equation}
where $v_{0}\left(x\right)=\left[2\cdot\left(1+\left|x\right|\right)\right]^{\left|\mu\right|}$
is as in Lemma \ref{lem:AlphaModulationSpaceWeight}.
Now, let $p,q,s,\mu$ as in the statement of the Theorem. We want
to show that the family $\left(\gamma_{i}\right)_{i\in\mathbb{Z}^{d}\setminus\left\{ 0\right\} }$
with $\gamma_{i}:=\gamma$ for all $i\in\mathbb{Z}^{d}\setminus\left\{ 0\right\} $
satisfies all assumptions of Corollary \ref{cor:AtomicDecompositionSimplifiedCriteria},
for $\mathcal{Q}=\mathcal{Q}_{r}^{\left(\alpha\right)}$. To this end, let $\gamma_{1}^{\left(0\right)}:=\gamma$
and $n_{i}:=1$, so that $\gamma_{i}=\gamma=\gamma_{n_{i}}^{\left(0\right)}$
for all $i\in\mathbb{Z}^{d}\setminus\left\{ 0\right\} $. As in the
proof of Theorem \ref{thm:AlphaModulationBanachFrame}, we then see
that all assumptions of Lemma \ref{lem:GammaCoversOrbitAssumptionSimplified}
are satisfied. Hence, the family $\left(\gamma_{i}\right)_{i\in\mathbb{Z}^{d}\setminus\left\{ 0\right\} }$
satisfies Assumption \ref{assu:GammaCoversOrbit} and we also get
$\Omega_{2}^{\left(p,K\right)}\leq L_{4}$ for some constant $L_{4}=L_{4}\left(\gamma,\mu_{0},p_{0},r,\alpha,d\right)>0$,
all $p\geq p_{0}$ and all $K\leq\mu_{0}$.
Set $\gamma_{i,1}:=\gamma_{1}$ and $\gamma_{i,2}:=\gamma_{2}$ for
$i\in\mathbb{Z}^{d}\setminus\left\{ 0\right\} $. Let us verify the
assumptions of Corollary \ref{cor:AtomicDecompositionSimplifiedCriteria}
for these choices:
\begin{enumerate}
\item As we have seen at the beginning of this section, the covering $\mathcal{Q}=\mathcal{Q}_{r}^{\left(\alpha\right)}$,
the weight $v=v^{\left(\mu\right)}$ (with $v_{0}\left(x\right)=\left[2\cdot\left(1+\left|x\right|\right)\right]^{\left|\mu\right|}$,
$\Omega_{0}=1$ and $\Omega_{1}=2^{\left|\mu\right|}\leq2^{\mu_{0}}$,
as well as $K=\left|\mu\right|\leq\mu_{0}$) satisfy all standing
assumptions of Section \ref{subsec:DecompSpaceDefinitionStandingAssumptions}.
Furthermore, the family $\Phi=\left(\varphi_{i}\right)_{i\in I}$
from above satisfies Assumption \ref{assu:RegularPartitionOfUnity}.
\item Our choice of $\gamma_{1},\gamma_{2}$ from above ensures that all
$\gamma_{i}=\gamma$, $\gamma_{i,1}=\gamma_{1}$ and $\gamma_{i,2}=\gamma_{2}$
are measurable functions.
\item As seen above, we have $\left\Vert \gamma_{1}\right\Vert _{N_{0}}<\infty$.
Since $N_{0}=\left\lceil \mu_{0}+\frac{d+\varepsilon}{p_{0}}\right\rceil \geq\mu_{0}+\frac{d+\varepsilon}{p_{0}}\geq\mu_{0}+d+\varepsilon$,
equation (\ref{eq:StandardDecayLpEstimate}) easily yields $\gamma_{i,1}=\gamma_{1}\in L_{\left(1+\left|\bullet\right|\right)^{\mu_{0}}}^{1}\left(\mathbb{R}^{d}\right)\hookrightarrow L_{\left(1+\left|\bullet\right|\right)^{K}}^{1}\left(\mathbb{R}^{d}\right)$
for all $i\in\mathbb{Z}^{d}\setminus\left\{ 0\right\} $.
\item As seen above, we have $\gamma_{i,2}=\gamma_{2}\in C^{1}\left(\mathbb{R}^{d}\right)$
for all $i\in\mathbb{Z}^{d}\setminus\left\{ 0\right\} $.
\item With $K_{0}:=K+\frac{d}{\min\left\{ 1,p\right\} }+1$, we
have
\begin{align*}
\Omega_{4}^{\left(p,K\right)} & =\sup_{i\in\mathbb{Z}^{d}\setminus\left\{ 0\right\} }\left\Vert \gamma_{i,2}\right\Vert _{K_{0}}+\sup_{i\in\mathbb{Z}^{d}\setminus\left\{ 0\right\} }\left\Vert \nabla\gamma_{i,2}\right\Vert _{K_{0}}\\
\left({\scriptstyle \text{since }\gamma_{i,2}=\gamma_{2}\text{ and }K_{0}\leq\mu_{0}+\frac{d}{p_{0}}+1}\right) & \leq2\cdot\sup_{i\in\mathbb{Z}^{d}\setminus\left\{ 0\right\} }\left(\left\Vert \gamma_{2}\right\Vert _{\left\lceil \mu_{0}+\frac{d}{p_{0}}+1\right\rceil }+\left\Vert \nabla\gamma_{2}\right\Vert _{\left\lceil \mu_{0}+\frac{d}{p_{0}}+1\right\rceil }\right)\\
& =2\cdot L_{1}^{\left(\left\lceil \mu_{0}+\frac{d}{p_{0}}+1\right\rceil \right)}=:L_{5}<\infty.
\end{align*}
\item By our assumptions, we have $\left\Vert \gamma_{i}\right\Vert _{K_{0}}=\left\Vert \gamma\right\Vert _{K_{0}}\leq\left\Vert \gamma\right\Vert _{\mu_{0}+\frac{d}{p_{0}}+1}<\infty$
for all $i\in\mathbb{Z}^{d}\setminus\left\{ 0\right\} $.
\item By choice of $\gamma_{1},\gamma_{2}$, we have $\gamma_{i}=\gamma=\gamma_{1}\ast\gamma_{2}=\gamma_{i,1}\ast\gamma_{i,2}$
for all $i\in\mathbb{Z}^{d}\setminus\left\{ 0\right\} $.
\item By the properties of $\gamma_{1},\gamma_{2}$ from above, we have
$\widehat{\gamma_{i,\ell}}=\widehat{\gamma_{\ell}}$ and all partial
derivatives of this function are polynomially bounded, for arbitrary
$i\in\mathbb{Z}^{d}\setminus\left\{ 0\right\} $ and $\ell\in\left\{ 1,2\right\} $.
\item As seen above, the family $\left(\gamma_{i}\right)_{i\in\mathbb{Z}^{d}\setminus\left\{ 0\right\} }$
satisfies Assumption \ref{assu:GammaCoversOrbit}.
\item Let
\[
K_{1}:=\sup_{i\in\mathbb{Z}^{d}\setminus\left\{ 0\right\} }\:\sum_{j\in\mathbb{Z}^{d}\setminus\left\{ 0\right\} }\:N_{i,j}\in\left[0,\infty\right]\qquad\text{ and }\qquad K_{2}:=\sup_{j\in\mathbb{Z}^{d}\setminus\left\{ 0\right\} }\:\sum_{i\in\mathbb{Z}^{d}\setminus\left\{ 0\right\} }\:N_{i,j}\in\left[0,\infty\right]
\]
as in Corollary \ref{cor:AtomicDecompositionSimplifiedCriteria},
i.e., with
\[
\qquad N_{i,j}=\left[\frac{w_{i}^{\left(s^{\ast}\right)}}{w_{j}^{\left(s^{\ast}\right)}}\cdot\left(\left|\det T_{j}\right|/\left|\det T_{i}\right|\right)^{\vartheta}\right]^{\tau}\cdot\left(1+\left\Vert T_{j}^{-1}T_{i}\right\Vert \right)^{\sigma}\cdot\left(\left|\det T_{i}\right|^{-1}\cdot\int_{Q_{i}}\max_{\left|\beta\right|\leq N}\left|\left(\partial^{\beta}\widehat{\gamma_{1}}\right)\left(S_{j}^{-1}\xi\right)\right|\operatorname{d}\xi\right)^{\tau},
\]
where $s^{\ast}=\frac{s}{1-\alpha}$ and, since $K=\left|\mu\right|$,
\begin{align*}
N & =\left\lceil \left|\mu\right|+\frac{d+\varepsilon}{\min\left\{ 1,p\right\} }\right\rceil \,,\\
\tau & =\min\left\{ 1,p,q\right\} ,\\
\vartheta & =\begin{cases}
0, & \text{if }p\in\left[1,\infty\right],\\
\frac{1}{p}-1, & \text{if }p\in\left(0,1\right),
\end{cases}\\
\sigma & =\begin{cases}
\tau\cdot\left\lceil \left|\mu\right|+d+\varepsilon\right\rceil , & \text{if }p\in\left[1,\infty\right],\\
\tau\cdot\left(\frac{d}{p}+\left|\mu\right|+\left\lceil \left|\mu\right|+\frac{d+\varepsilon}{p}\right\rceil \right), & \text{if }p\in\left(0,1\right).
\end{cases}
\end{align*}
Note that $\vartheta\geq0$ and $\tau>0$. Furthermore, since $\alpha_{0}\geq0$
and since $\left|j\right|\geq1$ for all $j\in\mathbb{Z}^{d}\setminus\left\{ 0\right\} $,
we have
\[
\left(2\left\langle j\right\rangle \right)^{d\alpha_{0}}\geq\left\langle j\right\rangle ^{d\alpha_{0}}\geq\left|\det T_{j}\right|=\left|j\right|^{d\alpha_{0}}\geq\left(\frac{1}{2}\left\langle j\right\rangle \right)^{d\alpha_{0}}
\]
and thus
\[
\qquad\quad\left[\frac{w_{i}^{\left(s^{\ast}\right)}}{w_{j}^{\left(s^{\ast}\right)}}\cdot\left(\frac{\left|\det T_{j}\right|}{\left|\det T_{i}\right|}\right)^{\!\vartheta}\,\right]^{\tau}\leq\left[\frac{w_{i}^{\left(s^{\ast}\right)}}{w_{j}^{\left(s^{\ast}\right)}}\cdot\left(\frac{2\left\langle j\right\rangle }{\frac{1}{2}\left\langle i\right\rangle }\right)^{\!\varthetad\alpha_{0}}\right]^{\tau}\leq4^{\tau\varthetad\alpha_{0}}\!\cdot\!\left[\frac{w_{i}^{\left(s^{\ast}-\varthetad\alpha_{0}\right)}}{w_{j}^{\left(s^{\ast}-\varthetad\alpha_{0}\right)}}\right]^{\tau}=4^{\tau\varthetad\alpha_{0}}\!\cdot\!\left[\frac{w_{j}^{\left(\varthetad\alpha_{0}-s^{\ast}\right)}}{w_{i}^{\left(\varthetad\alpha_{0}-s^{\ast}\right)}}\right]^{\tau}
\]
for all $i,j\in\mathbb{Z}^{d}\setminus\left\{ 0\right\} $. Moreover,
$N\leq\left\lceil \mu_{0}+\frac{d+\varepsilon}{p_{0}}\right\rceil =N_{0}$
and hence $\left|\left(\partial^{\beta}\widehat{\gamma_{1}}\right)\left(\xi\right)\right|\leq L_{0}L_{2}\cdot\left(1+\left|\xi\right|\right)^{-M_{00}}$
for all $\left|\beta\right|\leq N$. Combining these estimates and
noting $\vartheta\leq\frac{1}{p_{0}}$, we arrive at
\[
\qquad N_{i,j}\leq\left(L_{0}L_{2}\cdot4^{\frac{d\alpha_{0}}{p_{0}}}\right)^{\tau}\cdot\left[\frac{w_{j}^{\left(\varthetad\alpha_{0}-s^{\ast}\right)}}{w_{i}^{\left(\varthetad\alpha_{0}-s^{\ast}\right)}}\right]^{\tau}\cdot\left(1+\left\Vert T_{j}^{-1}T_{i}\right\Vert \right)^{\sigma}\cdot\left(\left|\det T_{i}\right|^{-1}\cdot\int_{Q_{i}}\left(1+\left|S_{j}^{-1}\xi\right|\right)^{-M_{00}}\operatorname{d}\xi\right)^{\tau}
\]
for all $i,j\in\mathbb{Z}^{d}\setminus\left\{ 0\right\} $. For brevity,
set $L_{6}:=\left(L_{0}L_{2}\cdot4^{\frac{d\alpha_{0}}{p_{0}}}\right)^{\tau}$.
Note that with this notation, the preceding estimate shows $N_{i,j}\leq N_{6}\cdot M_{j,i}^{\left(0\right)}$,
where $M_{j,i}^{\left(0\right)}$ is defined as in equation (\ref{eq:AlphaModulationStandardEstimate}),
but with $s^{\ast}$ replaced by $s^{\natural}:=\varthetad\alpha_{0}-s^{\ast}$
and with $N_{0}$ replaced by $M_{00}+1$.
Now, we want to apply Lemma \ref{lem:AlphaModulationMainLemma} to
estimate $M_{j,i}^{\left(0\right)}$. To this end, we have to verify
\begin{align*}
M_{00}+1 & \overset{!}{\geq}d+2+\frac{d+1}{\tau}+\max\left\{ \left|s^{\natural}+d\alpha_{0}\right|,\left|s^{\natural}+\left(d-\frac{\sigma}{\tau}\right)\alpha_{0}\right|\right\} \\
& =d+2+\frac{d+1}{\tau}+\max\left\{ \left|s^{\ast}-d\alpha_{0}\left(1+\vartheta\right)\right|,\left|s^{\ast}+\left(\frac{\sigma}{\tau}-d\left(1+\vartheta\right)\right)\alpha_{0}\right|\right\} \\
& =d+2+\frac{d+1}{\tau}+\max\left\{ \left|s^{\ast}-\frac{d\alpha_{0}}{\min\left\{ 1,p\right\} }\right|,\left|s^{\ast}+\alpha_{0}\left(\frac{\sigma}{\tau}-\frac{d}{\min\left\{ 1,p\right\} }\right)\right|\right\} ,
\end{align*}
where the last line used that
\[
1+\vartheta=\begin{cases}
1=\frac{1}{\min\left\{ 1,p\right\} }, & \text{if }p\in\left[1,\infty\right],\\
1+\frac{1}{p}-1=\frac{1}{p}=\frac{1}{\min\left\{ 1,p\right\} }, & \text{if }p\in\left(0,1\right).
\end{cases}
\]
But we have $\tau=\min\left\{ 1,p,q\right\} \geq\min\left\{ 1,p_{0},q_{0}\right\} =\min\left\{ p_{0},q_{0}\right\} =:\tau_{0}$
and furthermore
\begin{align*}
\frac{\sigma}{\tau} & =\begin{cases}
\left\lceil \left|\mu\right|+d+\varepsilon\right\rceil , & \text{if }p\in\left[1,\infty\right],\\
\frac{d}{p}+\left|\mu\right|+\left\lceil \left|\mu\right|+\frac{d+\varepsilon}{p}\right\rceil , & \text{if }p\in\left(0,1\right)
\end{cases}\\
& \leq\begin{cases}
\left\lceil \mu_{0}+d+\varepsilon\right\rceil , & \text{if }p_{0}=1,\\
\frac{d}{p_{0}}+\mu_{0}+\left\lceil \mu_{0}+\frac{d+\varepsilon}{p_{0}}\right\rceil , & \text{if }p_{0}\in\left(0,1\right)
\end{cases}\\
& =:\frac{\sigma_{0}}{\tau_{0}}.
\end{align*}
Hence, in case of $p\in\left[1,\infty\right]$, we have
\[
\left|s^{\ast}-d\alpha_{0}\left(1+\vartheta\right)\right|=\left|s^{\ast}-\frac{d\alpha_{0}}{\min\left\{ 1,p\right\} }\right|\leq\frac{1}{1-\alpha}\left(s_{0}+\alphad\right)\leq\Lambda,
\]
as well as
\begin{align*}
\left|s^{\ast}+\left(\frac{\sigma}{\tau}-d\left(1+\vartheta\right)\right)\alpha_{0}\right|=\left|s^{\ast}+\alpha_{0}\left(\frac{\sigma}{\tau}-\frac{d}{\min\left\{ 1,p\right\} }\right)\right| & \leq\frac{1}{1-\alpha}\left(s_{0}+\alpha\cdot\left|\left\lceil \left|\mu\right|+d+\varepsilon\right\rceil -d\right|\right)\\
\left({\scriptstyle \text{since }\left\lceil \left|\mu\right|+d+\varepsilon\right\rceil \geqd}\right) & =\frac{1}{1-\alpha}\left[s_{0}+\alpha\cdot\left(\left\lceil \left|\mu\right|+d+\varepsilon\right\rceil -d\right)\right]\leq\Lambda,
\end{align*}
as can be easily verified by distinguishing the cases $p_{0}=1$ and
$p_{0}\in\left(0,1\right)$.
Similarly, in case of $p\in\left(0,1\right)$, we necessarily have
$p_{0}\in\left(0,1\right)$ and thus
\[
\left|\frac{\sigma}{\tau}-\frac{d}{\min\left\{ 1,p\right\} }\right|=\left|\frac{d}{p}+\left|\mu\right|+\left\lceil \left|\mu\right|+\frac{d+\varepsilon}{p}\right\rceil -\frac{d}{p}\right|\leq\mu_{0}+\left\lceil \mu_{0}+\frac{d+\varepsilon}{p_{0}}\right\rceil ,
\]
which easily implies
\[
\left|s^{\ast}-d\alpha_{0}\left(1+\vartheta\right)\right|=\left|s^{\ast}-\frac{d\alpha_{0}}{\min\left\{ 1,p\right\} }\right|\leq\frac{1}{1-\alpha}\left(s_{0}+\frac{\alphad}{p_{0}}\right)\leq\Lambda,
\]
as well as
\begin{align*}
\left|s^{\ast}+\left(\frac{\sigma}{\tau}-d\left(1+\vartheta\right)\right)\alpha_{0}\right| & =\left|s^{\ast}+\alpha_{0}\left(\frac{\sigma}{\tau}-\frac{d}{\min\left\{ 1,p\right\} }\right)\right|\\
& \leq\frac{1}{1-\alpha}\left(s_{0}+\alpha\left|\frac{\sigma}{\tau}-\frac{d}{\min\left\{ 1,p\right\} }\right|\right)\\
& \leq\frac{1}{1-\alpha}\left[s_{0}+\alpha\left(\mu_{0}+\left\lceil \mu_{0}+\frac{d+\varepsilon}{p_{0}}\right\rceil \right)\right]\leq\Lambda.
\end{align*}
All in all, our assumptions on $M_{0}$ thus yield
\begin{align*}
M_{00}+1 & =d+2+\frac{d+1}{\min\left\{ p_{0},q_{0}\right\} }+\Lambda\\
& \geqd+2+\frac{d+1}{\tau}+\max\left\{ \left|s^{\ast}-d\alpha_{0}\left(1+\vartheta\right)\right|,\left|s^{\ast}+\left(\frac{\sigma}{\tau}-d\left(1+\vartheta\right)\right)\alpha_{0}\right|\right\} \\
& =d+2+\frac{d+1}{\tau}+\max\left\{ \left|s^{\natural}+d\alpha_{0}\right|,\left|s^{\natural}+\left(d-\frac{\sigma}{\tau}\right)\alpha_{0}\right|\right\} .
\end{align*}
Hence, Lemma \ref{lem:AlphaModulationMainLemma} is applicable. For
brevity, set $L_{7}:=2^{d}\cdot\left(1+\left(2+4r\right)^{\alpha_{0}}\cdot r\right)^{M_{00}+1+d}$.
Using Lemma \ref{lem:AlphaModulationMainLemma} and the estimate $\left|s^{\natural}\right|\leq\left|s^{\ast}\right|+d\alpha_{0}\left|\vartheta\right|\leq\frac{1}{1-\alpha}\left(s_{0}+\alpha\frac{d}{p_{0}}\right)$
and setting $L_{8}:=\frac{\sigma_{0}}{\tau_{0}}+d\left(1+\frac{1}{p_{0}}\right)$,
we now get
\begin{align*}
\quad\qquad K_{1}^{\frac{1}{\tau}} & \leq L_{0}L_{2}\cdot4^{\frac{d\alpha_{0}}{p_{0}}}\cdot\left(\sup_{i\in\mathbb{Z}^{d}\setminus\left\{ 0\right\} }\sum_{j\in\mathbb{Z}^{d}\setminus\left\{ 0\right\} }M_{j,i}^{\left(0\right)}\right)^{1/\tau}\\
& \leq L_{0}L_{2}\cdot4^{\frac{d\alpha_{0}}{p_{0}}}\cdot2^{\frac{1}{\tau}+\frac{\sigma}{\tau}+\left|s^{\natural}\right|}6^{d/\tau}\cdot\max\left\{ 4^{\alpha_{0}\left(\frac{\sigma}{\tau}+d\right)+\left|s^{\natural}\right|}\cdot12^{M_{00}+1}s_{d},\:\left(2+4r\right)^{\left|s^{\natural}\right|+\alpha_{0}\left[\frac{\sigma}{\tau}+d+1+M_{00}\right]}L_{7}\right\} \\
& \leq L_{0}L_{2}\cdot6^{\frac{d}{\tau_{0}}}2^{\frac{1}{\tau_{0}}+\frac{\sigma_{0}}{\tau_{0}}+\frac{1}{1-\alpha}\left(s_{0}+3\alpha\frac{d}{p_{0}}\right)}\!\!\cdot\max\left\{ 4^{\alpha_{0}L_{8}+\frac{s_{0}}{1-\alpha}}\cdot12^{M_{00}+1}s_{d},\:\left(2\!+\!4r\right)^{\frac{s_{0}}{1-\alpha}+\alpha_{0}\left[L_{8}+M_{00}+1\right]}L_{7}\right\} \!.
\end{align*}
Hence, $K_{1}^{1/\tau}\leq L_{9}$, for a constant $L_{9}$ which
is independent of $p,q,s,\mu$. Completely similar, we also get $K_{2}^{1/\tau}\leq L_{9}$,
with the same constant. In particular, $K_{1},K_{2}<\infty$.
\end{enumerate}
Having verified all prerequisites of Corollary \ref{cor:AtomicDecompositionSimplifiedCriteria},
we get $\vertiii{\smash{\overrightarrow{C}}}^{\max\left\{ 1,\frac{1}{p}\right\} }\leq\Omega\cdot\left(K_{1}^{1/\tau}+K_{2}^{1/\tau}\right)\leq2\Omega\cdot L_{9}$,
where $\overrightarrow{C}:\ell_{\left[w^{\left(s^{\ast}\right)}\right]^{\min\left\{ 1,p\right\} }}^{r}\left(\mathbb{Z}^{d}\setminus\left\{ 0\right\} \right)\to\ell_{\left[w^{\left(s^{\ast}\right)}\right]^{\min\left\{ 1,p\right\} }}^{r}\left(\mathbb{Z}^{d}\setminus\left\{ 0\right\} \right)$
is defined as in Assumption \ref{assu:AtomicDecompositionAssumption}
(with $r:=\max\left\{ q,\frac{q}{p}\right\} $) and where
\begin{align*}
\Omega & =\Omega_{0}^{K}\Omega_{1}\cdot\left(4\cdotd\right)^{1+2\left\lceil K+\frac{d+\varepsilon}{\min\left\{ 1,p\right\} }\right\rceil }\cdot\left(\frac{s_{d}}{\varepsilon}\right)^{1/\min\left\{ 1,p\right\} }\cdot\max_{\left|\beta\right|\leq\left\lceil K+\frac{d+\varepsilon}{\min\left\{ 1,p\right\} }\right\rceil }C^{\left(\beta\right)}\\
& \leq2^{\mu_{0}}\cdot\left(4\cdotd\right)^{1+2\left\lceil \mu_{0}+\frac{d+\varepsilon}{p_{0}}\right\rceil }\cdot\left(1+\frac{s_{d}}{\varepsilon}\right)^{1/p_{0}}\cdot\max_{\left|\beta\right|\leq\left\lceil \mu_{0}+\frac{d+\varepsilon}{p_{0}}\right\rceil }C^{\left(\beta\right)}=:L_{10},
\end{align*}
where the constants $C^{\left(\beta\right)}=C^{\left(\beta\right)}\left(\Phi\right)$
are defined as in Assumption \ref{assu:RegularPartitionOfUnity}.
Note again that $L_{10}$ is independent of $p,q,s,\mu$. Finally,
Corollary \ref{cor:AtomicDecompositionSimplifiedCriteria} shows that
the families $\left(\gamma_{i}\right)_{i\in\mathbb{Z}^{d}\setminus\left\{ 0\right\} },\left(\gamma_{i,1}\right)_{i\in\mathbb{Z}^{d}\setminus\left\{ 0\right\} }$
and $\left(\gamma_{i,2}\right)_{i\in\mathbb{Z}^{d}\setminus\left\{ 0\right\} }$
fulfill Assumption \ref{assu:AtomicDecompositionAssumption} and that
$\left(\gamma_{i}\right)_{i\in\mathbb{Z}^{d}\setminus\left\{ 0\right\} }$
satisfies Assumption \ref{assu:GammaCoversOrbit}, so that Theorem
\ref{thm:AtomicDecomposition} is applicable.
This theorem shows that $\Gamma^{\left(\delta\right)}$ indeed forms
an atomic decomposition for $M_{\left(s,\mu\right),\alpha}^{p,q}\left(\mathbb{R}^{d}\right)=\DecompSp{\mathcal{Q}_{r}^{\left(\alpha\right)}}p{\ell_{w^{\left(s^{\ast}\right)}}^{q}}{v^{\left(\mu\right)}}$,
as soon as $0<\delta\leq\min\left\{ 1,\delta_{00}\right\} $, with
\[
\delta_{00}^{-1}\!:=\!\begin{cases}
\!\frac{2s_{d}}{\sqrt{d}}\!\cdot\!\left(2^{17}\!\cdot\!d^{2}\!\cdot\!\left(K\!+\!2\!+\!d\right)\right)^{\!K+d+3}\cdot\left(1\!+\!R_{\mathcal{Q}_{r}^{\left(\alpha\right)}}\right)^{d+1}\!\cdot\!\Omega_{0}^{4K}\Omega_{1}^{4}\Omega_{2}^{\left(p,K\right)}\Omega_{4}^{\left(p,K\right)}\cdot\vertiii{\smash{\overrightarrow{C}}}\,, & \text{if }p\geq1,\\
\!\frac{\left(2^{14}/d^{\frac{3}{2}}\right)^{\!\frac{d}{p}}}{2^{45}\cdotd^{17}}\!\cdot\!\left(\frac{s_{d}}{p}\right)^{\!\frac{1}{p}}\left(2^{68}\!\cdot\!d^{14}\!\cdot\!\left(K\!+\!1\!+\!\frac{d+1}{p}\right)^{\!\!3}\,\right)^{\!K+2+\frac{d+1}{p}}\!\!\!\cdot\!\left(1\!+\!R_{\mathcal{Q}_{r}^{\left(\alpha\right)}}\right)^{1+\frac{3d}{p}}\!\cdot\!\Omega_{0}^{16K}\Omega_{1}^{16}\Omega_{2}^{\left(p,K\right)}\Omega_{4}^{\left(p,K\right)}\!\cdot\!\vertiii{\smash{\overrightarrow{C}}}^{\frac{1}{p}}, & \text{if }p<1.
\end{cases}
\]
Thus, to establish the claim of the current theorem, it suffices to
show that $\delta_{00}^{-1}$ can be bounded independently of $p,q,s,\mu$.
Strictly speaking, we then still have to verify that the coefficient
space used in Theorem \ref{thm:AtomicDecomposition} is just $\mathscr{C}_{p,q,s,\mu}^{\left(\alpha\right)}$,
but this can be done precisely as in the proof of Theorem \ref{thm:AlphaModulationBanachFrame}.
Now, for $p\geq1$, we have because of $K=\left|\mu\right|\leq\mu_{0}$
and $\Omega_{0}=1$, as well as $\Omega_{1}=2^{\mu_{0}}$ and $\Omega_{4}^{\left(p,K\right)}\leq L_{5}$,
as well as $\Omega_{2}^{\left(p,K\right)}\leq L_{4}$ that
\[
\delta_{00}^{-1}\leq\frac{2s_{d}}{\sqrt{d}}\cdot\left(2^{17}\!\cdot\!d^{2}\!\cdot\!\left(\mu_{0}\!+\!2\!+\!d\right)\right)^{\!\mu_{0}+d+3}\!\!\!\cdot\left(1\!+\!R_{\mathcal{Q}_{r}^{\left(\alpha\right)}}\right)^{d+1}\cdot L_{4}L_{5}\cdot16^{\mu_{0}}\cdot2L_{9}L_{10},
\]
which is independent of $p,q,s,\mu$.
Finally, in case of $p\in\left(0,1\right)$, we get similarly that
\[
\delta_{00}^{-1}\leq\frac{\left(2^{14}\right)^{\!\frac{d}{p_{0}}}}{2^{45}\cdotd^{17}}\!\cdot\!\left(1+\frac{s_{d}}{p_{0}}\right)^{\!\frac{1}{p_{0}}}\left(2^{68}\!\cdot\!d^{14}\!\cdot\!\left(\mu_{0}+1+\frac{d+1}{p_{0}}\right)^{3}\right)^{\!\mu_{0}+2+\frac{d+1}{p_{0}}}\!\!\!\cdot\!\left(1\!+\!R_{\mathcal{Q}_{r}^{\left(\alpha\right)}}\right)^{1+\frac{3d}{p_{0}}}\!\cdot\!2^{16\mu_{0}}\!\cdot\!L_{4}L_{5}\!\cdot\!2L_{9}L_{10},
\]
which is independent of $p,q,s,\mu$, as desired.
\end{proof}
We close this section with an overview over the history and applications
of $\alpha$-modulation spaces and with a comparison of our results
to the established literature.
\begin{rem}
\label{rem:AlphamModulationLiteratureDiscussion}$\alpha$-modulation
spaces were originally introduced in Gröbner's PhD.\@ thesis \cite{GroebnerAlphaModulationSpaces},
see also \cite{DecompositionSpaces1}. The definition of these spaces
was motivated by the realization that both modulation spaces and (inhomogeneous)
Besov spaces fit into the common framework of decomposition spaces,
which Feichtinger and Gröbner developed at the time\cite{DecompositionSpaces1,DecompositionSpaces2}.
The underlying frequency coverings are the uniform covering for the
modulation spaces and the dyadic covering for the Besov spaces. Given
these two ``end-point'' types of spaces, the $\alpha$-modulation
spaces provide a continuous family of smoothness spaces which ``lie
between'' modulation and Besov spaces in the sense that the associated
$\alpha$-modulation coverings $\mathcal{Q}_{r}^{\left(\alpha\right)}$
form a continuously indexed family of coverings which yields the uniform
covering for $\alpha=0$ and the (inhomogeneous) dyadic covering in
the limit $\alpha\uparrow1$. Note though that the $\beta$-modulation
space $M_{s,\beta}^{p,q}\left(\mathbb{R}^{d}\right)$ \emph{can not}
be obtained by complex interpolation between two $\alpha$-modulation
spaces $M_{s_{i},\alpha_{i}}^{p_{i},q_{i}}\left(\mathbb{R}^{d}\right)$
with $\alpha_{1}\neq\alpha_{2}$, except in a few trivial special
cases\cite{AlphaModulationNotInterpolation}. We also remark that
the frequency covering associated to the $\alpha$-modulation spaces
was independently introduced by Päivärinta and Somersalo\cite[Lemma 2.1]{SomersaloAlphaModulation},
in order to generalize the Calderón-Vaillancourt boundedness result
for pseudodifferential operators to the local Hardy spaces $h_{p}$.
\medskip{}
As for the classical modulation spaces, one application of $\alpha$-modulation
spaces is that they are suitable domains for the study of pseudodifferential
operators: For the one-dimensional case, it was shown in \cite{BorupPsiDOsOnAlphaModulation}
that if $T\in{\rm OPS}_{\alpha,\delta}^{s_{0}}$, then $T:M_{s,\alpha}^{p,q}\left(\mathbb{R}\right)\to M_{s+\alpha-s_{0}-1,\alpha}^{p,q}\left(\mathbb{R}\right)$
is continuous, if $p,q\in\left(1,\infty\right)$, $0<\alpha\leq1$,
$0\leq\delta\leq\alpha$ and $\delta<1$. The multivariate case was
considered in \cite{BorupNielsenPsiDOsOnMultivariateAlphaModulationSpaces},
where it was shown for symbols $\sigma\in S_{\varrho,0}^{m}$ and
for $0\leq\alpha\leq\varrho\leq1$ that $\sigma\left(x,D\right):M_{s,\alpha}^{p,q}\left(\mathbb{R}^{d}\right)\to M_{s-m,\alpha}^{p,q}\left(\mathbb{R}^{d}\right)$
is continuous for $p,q\in\left(1,\infty\right)$. For related results,
see also \cite{NazaretBrunoPsiDOsOnAlphaModulation}.
\medskip{}
The embeddings between $\alpha$-modulation spaces and other function
spaces have been considered by a number of authors: Already in Gröbner's
PhD.\@ thesis \cite{GroebnerAlphaModulationSpaces}, embeddings between
$M_{s_{1},\alpha_{1}}^{p,q}\left(\mathbb{R}^{d}\right)$ and $M_{s_{2},\alpha_{2}}^{p,q}\left(\mathbb{R}^{d}\right)$
for $\alpha_{1},\alpha_{2}\in\left[0,1\right]$ are considered, but
the resulting criteria are not sharp. These non-sharp conditions were
improved by Toft and Wahlberg\cite{ToftWahlbergAlphaModulationEmbeddings},
shortly before the question was completely solved by Han and Wang\cite{HanWangAlphaModulationEmbeddings}.
Note, however, that the preceding results only considered the embedding
$M_{s_{1},\alpha_{1}}^{p,q}\left(\mathbb{R}^{d}\right)\hookrightarrow M_{s_{2},\alpha_{2}}^{p,q}\left(\mathbb{R}^{d}\right)$,
where the exponents $p,q$ are \emph{the same} on both sides. The
existence of the completely general embedding $M_{s_{1},\alpha_{1}}^{p_{1},q_{1}}\left(\mathbb{R}^{d}\right)\hookrightarrow M_{s_{2},\alpha_{2}}^{p_{2},q_{2}}\left(\mathbb{R}^{d}\right)$
was characterized completely in my PhD.\@ thesis \cite[Theorems 6.1.7 and 6.2.8]{VoigtlaenderPhDThesis}.
The same results also appear in my recent paper \cite[Theorems 9.7, 9.13, 9.14 and Corollary 9.16]{DecompositionEmbedding}.
We finally remark that the complete characterization of the embedding
$M_{s_{1},\alpha_{1}}^{p_{1},q_{1}}\left(\mathbb{R}^{d}\right)\hookrightarrow M_{s_{2},\alpha_{2}}^{p_{2},q_{2}}\left(\mathbb{R}^{d}\right)$
was also obtained independently in \cite{AlphaModulationEmbeddingFullCharacterization}.
The existence of embeddings $M_{s_{1},\alpha}^{p,q}\left(\mathbb{R}^{d}\right)\hookrightarrow L_{s_{2}}^{p}\left(\mathbb{R}^{d}\right)$
and $L_{s_{1}}^{p}\left(\mathbb{R}^{d}\right)\hookrightarrow M_{s_{2},\alpha}^{p,q}\left(\mathbb{R}^{d}\right)$
between $\alpha$-modulation spaces and Sobolev spaces (or Bessel
potential spaces) has been fully characterized in \cite{EmbeddingsOfAlphaModulationIntoSobolev};
in the same paper, the author also considers these embeddings when
the Sobolev spaces $L_{s}^{p}\left(\mathbb{R}^{d}\right)$ are replaced
by the local hardy spaces $h_{p}\left(\mathbb{R}^{d}\right)$, for
$p\in\left(0,1\right)$. Embeddings between $\alpha$-modulation spaces
$M_{s,\alpha}^{p_{1},q}\left(\mathbb{R}^{d}\right)$ and the classical
Sobolev spaces $W^{k,p_{2}}\left(\mathbb{R}^{d}\right)$ are also
considered in \cite[Example 7.3]{DecompositionIntoSobolev}, as an
application of a more general theory: For $p_{2}\in\left[1,2\right]\cup\left\{ \infty\right\} $,
a complete characterization of the existence of this embedding is
obtained, but for $p_{2}\in\left(2,\infty\right)$, the given sufficient
criteria are strictly stronger than the given necessary criteria.
\medskip{}
Finally, we discuss the existing results concerning Banach frames
and atomic decompositions for $\alpha$-modulation spaces. A large
number of results in this direction were obtained by Borup and Nielsen:
In \cite{AlphaModulationNonlinearApproximation}, they showed that
certain \textbf{brushlet orthonormal bases} of $L^{2}\left(\mathbb{R}\right)$
form unconditional bases for the $\alpha$-modulation spaces $M_{s,\alpha}^{p,q}\left(\mathbb{R}\right)$,
for $p,q\in\left(1,\infty\right)$. Furthermore, a characterization
of the $\alpha$-modulation (quasi)-norm in terms of the brushlet
coefficients is obtained for arbitrary $p,q\in\left(0,\infty\right]$.
In fact, Borup and Nielsen showed that brushlet bases even yield \textbf{greedy
bases} (i.e., they are unconditional bases which satisfy the so-called
\textbf{democracy condition}), which was then used to characterize
the associated approximation spaces. As a further application of these
brushlet bases for $\alpha$-modulation spaces, Borup and Nielsen
derived boundedness results for certain pseudodifferential operators,
as briefly discussed above. In \cite{NielsenOrthonormalBasesForAlphaModulation},
Nielsen generalized the results concerning brushlet bases from the
one-dimensional case to the case $d=2$. Despite their great
utility, we remark that brushlet bases are \emph{not} generated by
a single prototype function; furthermore, brushlets are bandlimited
and can thus \emph{not} be compactly supported.
In addition to these brushlet bases for $\alpha$-modulation spaces
in dimensions $d=1$ and $d=2$, Borup and Nielsen
also proved existence of Banach frames for $\alpha$-modulation spaces
for the general case $d\in\mathbb{N}$. A first construction of a
\emph{non-tight} frame with explicitly given dual was obtained in
\cite{BorupNielsenAlphaModulationSpaces} and then generalized in
\cite[Section 6.1]{BorupNielsenDecomposition} to obtain \emph{tight}
Banach frames, even for the case of general decomposition spaces,
not just for $\alpha$-modulation spaces. But again, the frames constructed
in these two papers are \emph{not} generated by a single prototype
function and are bandlimited.
These limitations were partly overcome in later papers: In \cite[Theorem 1.1]{CompactlySupportedFramesForDecompositionSpaces},
Nielsen and Rasmussen obtained compactly supported Banach frames for
$\alpha$-modulation spaces. These frames $\left(\psi_{k,n}\right)_{k\in\mathbb{Z}^{d}\setminus\left\{ 0\right\} ,n\in\mathbb{Z}^{d}}$,
however, are \emph{not} of the same structured form as in Theorems
\ref{thm:AlphaModulationBanachFrame} and \ref{thm:AlphaModulationAtomicDecomposition}.
Instead, $\psi_{k,n}\left(x\right)=e^{i\left\langle x,d_{k}\right\rangle }\sum_{\ell=1}^{K}a_{k,\ell}\cdot g\left(c_{k}x+b_{k,n,\ell}\right)$
for suitable (but unknown) $K\in\mathbb{N}$, $a_{k,\ell}\in\mathbb{C}$, $d_{k},b_{k,n,\ell}\in\mathbb{R}^{d}$
and $c_{k}\in\mathbb{R}^{\ast}$, which makes it hard to say anything about
e.g.\@ the time-frequency concentration of the family $\left(\psi_{k,n}\right)_{k,n}$.
This limitation was finally removed by Nielsen in \cite{NielsenSinglyGeneratedFrames},
where it was shown—for a special class of coverings $\mathcal{Q}$, which
includes the $\alpha$ coverings $\mathcal{Q}_{r}^{\left(\alpha\right)}$
for $0\leq\alpha<1$, cf.\@ \cite[Lemma 2.9]{NielsenSinglyGeneratedFrames}—that
one can choose a single, compactly supported generator (or prototype)
$\gamma$ such that the resulting family $\Gamma^{\left(\delta\right)}$
defined as in Theorem \ref{thm:AlphaModulationBanachFrame} yields
a Banach frame for the $\alpha$-modulation space $M_{s,\alpha}^{p,q}\left(\mathbb{R}^{d}\right)$.
The main difference in comparison to Theorems \ref{thm:AlphaModulationBanachFrame}
and \ref{thm:AlphaModulationAtomicDecomposition} is that \cite{NielsenSinglyGeneratedFrames}
merely establishes \emph{existence} of a suitable generator $\gamma$;
it does \emph{not} provide readily verifiable conditions on $\gamma$
which allow to decide if $\gamma$ is suitable. This is due to the
employed proof technique: Nielsen first shows that a certain bandlimited
generator $\gamma_{0}$ generators a Banach frame and then uses a
perturbation argument to show that $\gamma$ generates a Banach frame,
provided that $\gamma$ is close enough to $\gamma_{0}$ in a certain
sense. The most concrete criterion in \cite{NielsenSinglyGeneratedFrames}
concerning $\gamma$ is that under certain \emph{readily verifiable}
conditions on a ``prototypical prototype'' $g$ (cf.\@ \cite[equations (3.13), (3.14)]{NielsenSinglyGeneratedFrames}),
one can always obtain a \emph{suitable} $\gamma$ by taking linear
combinations of translations of $g$. In stark contrast, Theorems
\ref{thm:AlphaModulationBanachFrame} and \ref{thm:AlphaModulationAtomicDecomposition}
(plus the associated remark) show that \emph{any} compactly supported
prototype $\gamma$ generates Banach frames and atomic decompositions
for the $\alpha$-modulation spaces, assuming that $\gamma$ is sufficiently
smooth and has nonvanishing Fourier transform on a certain neighborhood
of the origin.
\medskip{}
In addition to the constructions by Borup, Nielsen and Rasmussen,
Banach frames for $\alpha$-modulation spaces have also been considered
by Fornasier\cite{FornasierFramesForAlphaModulation}, by Dahlke et
al\@.\cite{QuotientCoorbitTheoryAndAlphaModulationSpaces} and finally
by Speckbacher et al.\cite{SpeckbacherAlphaModulation}, all for the
case $d=1$ and $p,q\in\left[1,\infty\right]$. The idea
in \cite{FornasierFramesForAlphaModulation} is to show that a family
$\Gamma_{\alpha}^{\left(\delta\right)}$ similar to $\Gamma^{\left(\delta\right)}$
from Theorem \ref{thm:AlphaModulationBanachFrame} is \textbf{intrinsicly
self-localized}, under suitable readily verifiable assumptions on
the generator $\gamma$, so that, for a sufficiently small sampling
density, the family $\Gamma_{\alpha}^{\left(\delta\right)}$ forms
a Banach frame and an atomic decomposition for the $\alpha$-modulation
space $M_{s,\alpha}^{p,q}\left(\mathbb{R}\right)$. In particular, $\gamma$
can be taken to have compact support, since any Schwartz function
with $\widehat{\gamma}\left(\xi\right)\neq0$ on $\left[-1,1\right]$
is suitable, cf.\@ \cite[Theorem 3.4]{FornasierFramesForAlphaModulation}.
Hence, Fornasier's results are very similar to Theorems \ref{thm:AlphaModulationBanachFrame}
and \ref{thm:AlphaModulationAtomicDecomposition}; the main difference
is that the results in this paper apply for the full range $p,q\in\left(0,\infty\right]$
and also for $d>1$. In this context, we remark that Fornasier
notes that ``We expect that the approach illustrated in this paper
{[}i.e., in \cite{FornasierFramesForAlphaModulation}{]} can be useful
also for a frame characterization of $M_{p,q}^{s,\alpha}\left(\mathbb{R}^{d}\right)$
for $d>1$, \emph{with major technical difficulties}.''
A further difference is that Fornasier only requires decay of $\widehat{\gamma}$,
not of $\partial^{\alpha}\widehat{\gamma}$, cf.\@ \cite[equation (29)]{FornasierFramesForAlphaModulation}.
The approach by Dahlke et al\@.\cite{QuotientCoorbitTheoryAndAlphaModulationSpaces}
is still different: They consider \textbf{coorbit spaces} of certain
quotients of the \textbf{affine Weyl-Heisenberg group}, based on the
general theory of coorbit spaces of homogeneous spaces developed in
\cite{CoorbitOnHomogenousSpaces,CoorbitOnHomogenousSpaces2}. It is
shown in \cite{QuotientCoorbitTheoryAndAlphaModulationSpaces} that
the general theory applies in this setting and (for suitable quotients)
that the associated coorbit spaces coincide with certain $\alpha$-modulation
spaces. Precisely, \cite[Theorem 6.1]{QuotientCoorbitTheoryAndAlphaModulationSpaces}
shows that $\mathcal{H}_{p,v_{s-\alpha\left(1/p-1/2\right),\alpha}}=M_{s,\alpha}^{p,p}\left(\mathbb{R}\right)$,
up to trivial identifications. Note that this only includes $\alpha$-modulation
spaces with $p=q$. The same theorem also shows—as a consequence of
the general discretization theory for coorbit spaces—that one obtains
Banach frames and atomic decompositions for the spaces $\mathcal{H}_{p,v_{s-\alpha\left(1/p-1/2\right),\alpha}}=M_{s,\alpha}^{p,p}\left(\mathbb{R}\right)$
which are of a similar form as the family $\Gamma^{\left(\delta\right)}$
considered in Theorems \ref{thm:AlphaModulationBanachFrame} and \ref{thm:AlphaModulationAtomicDecomposition}.
The main difference of these two theorems to the results in \cite{QuotientCoorbitTheoryAndAlphaModulationSpaces}
is that \cite{QuotientCoorbitTheoryAndAlphaModulationSpaces} is only
applicable to \emph{bandlimited generators}, only for $p=q\in\left[1,\infty\right]$
and only for $d=1$. Furthermore, while in Theorems \ref{thm:AlphaModulationBanachFrame}
and \ref{thm:AlphaModulationAtomicDecomposition} only the sampling
density \emph{in the space domain} has to be sufficiently dense, in
\cite{QuotientCoorbitTheoryAndAlphaModulationSpaces} one needs to
adjust the sampling density in space \emph{and in frequency}. Note
though that the assumption $\widehat{\gamma}\left(\xi\right)\neq0$
on $\overline{B_{r}}\left(0\right)$ is not present in \cite{QuotientCoorbitTheoryAndAlphaModulationSpaces}.
Finally, in \cite{SpeckbacherAlphaModulation}, Speckbacher et al.\@
extended the results of \cite{QuotientCoorbitTheoryAndAlphaModulationSpaces}
by showing that the theory developed in \cite{CoorbitOnHomogenousSpaces,CoorbitOnHomogenousSpaces2,QuotientCoorbitTheoryAndAlphaModulationSpaces}
is also applicable for suitable \emph{compactly supported} generators,
only subject to certain decay and smoothness conditions, cf.\@ \cite[Theorem 5.9]{SpeckbacherAlphaModulation}.
Note though that this does not remove the assumptions $p=q\in\left[1,\infty\right]$
and $d=1$. Precisely, the condition on the generator $\gamma\in L^{2}\left(\mathbb{R}\right)$
imposed in \cite{SpeckbacherAlphaModulation} to generate a Banach
frame and an atomic decomposition for $\mathcal{H}_{p,v_{s-\alpha\left(1/p-1/2\right),\alpha}}=M_{s,\alpha}^{p,p}\left(\mathbb{R}\right)$
is that $\widehat{\gamma}\in C^{3}\left(\mathbb{R}\right)$ with $\left|\partial^{j}\widehat{\gamma}\left(\xi\right)\right|\lesssim\left(1+\left|\xi\right|\right)^{-r}$
for $j\in\left\{ 0,1,2,3\right\} $ and
\[
r>1+\frac{2+2\left[s-\alpha\left(1/p-1/2\right)\right]+7\alpha-4\alpha^{2}}{2\left(1-\alpha\right)^{2}}=1+\frac{1+s+\alpha\left(4-\frac{1}{p}\right)-2\alpha^{2}}{\left(1-\alpha\right)^{2}}\geq\frac{2+s+\alpha\left(1-\alpha\right)}{\left(1-\alpha\right)^{2}}.
\]
In comparison, at least for $\gamma\in C_{c}^{1}\left(\mathbb{R}\right)$,
Theorem \ref{thm:AlphaModulationBanachFrame} (with $\mu_{0}=0$,
$p_{0}=q_{0}=1$, $s_{0}=s_{1}=s$ and $\varepsilon=\frac{1}{2}$)
only requires $\left|\partial^{j}\widehat{\gamma}\left(\xi\right)\right|\lesssim\left(1+\left|\xi\right|\right)^{-N_{0}}$
for $j\in\left\{ 0,1,2\right\} $ and
\[
N_{0}=5+\frac{1}{1-\alpha}\cdot\max\left\{ s+\alpha,\:s+2\alpha\right\} =5+\frac{s+2\alpha}{1-\alpha}=\frac{5+s-3\alpha}{1-\alpha}
\]
and Theorem \ref{thm:AlphaModulationAtomicDecomposition} requires
the same, but with
\[
N_{0}=2\cdot\left(3+\varepsilon\right)+\frac{s+\alpha}{1-\alpha}=6+2\varepsilon+\frac{s+\alpha}{1-\alpha},
\]
where $\varepsilon\in\left(0,1\right)$ can be chosen arbitrarily
small. Hence, in particular for $\alpha\approx1$ or for $\alpha>0$
and large $s$, the prerequisites of Theorems \ref{thm:AlphaModulationBanachFrame}
and \ref{thm:AlphaModulationAtomicDecomposition} are easier to fulfill
than those in \cite{SpeckbacherAlphaModulation}. We finally remark
that Fornasier\cite{FornasierFramesForAlphaModulation} requires $\left|\widehat{\gamma}\left(\xi\right)\right|\lesssim\left(1+\left|\xi\right|\right)^{-M_{0}}$
where $M_{0}$ satisfies $M_{0}\geq5+\frac{1}{2}+\frac{2s+2\alpha}{1-\alpha}$,
which is very close to the conditions in Theorems \ref{thm:AlphaModulationBanachFrame}
and \ref{thm:AlphaModulationAtomicDecomposition}.
Hence, our very general theory yields results comparable to specialized
treatments like \cite{FornasierFramesForAlphaModulation} and (in
many cases) better results than those obtained by using general coorbit
theory. Further, it is applicable for general $d\in\mathbb{N}$ and
also for general $p,q\in\left(0,\infty\right]$ instead of only for
$p=q\in\left[1,\infty\right]$.
\end{rem}
\section{Existence of compactly supported Banach frames and atomic decompositions
for inhomogeneous Besov spaces}
\label{sec:BesovFrames}In this section, we investigate assumptions
on the \textbf{scaling function} $\varphi:\mathbb{R}^{d}\to\mathbb{C}$
and the \textbf{mother wavelet} $\psi:\mathbb{R}^{d}\to\mathbb{C}$ which
ensure that the associated inhomogeneous \textbf{wavelet system} with
sampling density $c>0$,
\[
W\left(\varphi,\psi;c\right):=\left(\varphi\left(\bullet-c\cdot k\right)\right)_{k\in\mathbb{Z}^{d}}\cup\left(2^{j\frac{d}{2}}\cdot\psi\left(2^{j}\bullet-c\cdot k\right)\right)_{j\in\mathbb{N},k\in\mathbb{Z}^{d}},
\]
generates Banach frames or atomic decompositions for a subclass of
the class of inhomogeneous \textbf{Besov spaces}.
The inhomogeneous Besov spaces are decomposition spaces which are
defined using a certain dyadic covering of $\mathbb{R}^{d}$, which
we introduce now.
\begin{lem}
\label{lem:InhomogeneousBesovCovering}For $j\in\mathbb{N}$, let $T_{j}:=2^{j}\cdot\operatorname{id}$
and $b_{j}:=0$, as well as $Q_{j}':=B_{4}\left(0\right)\setminus\overline{B_{1/4}\left(0\right)}$.
Furthermore, set $T_{0}:=\operatorname{id}$ and $b_{0}:=0$, as well as $Q_{0}':=B_{2}\left(0\right)$.
The \textbf{(inhomogeneous) Besov covering} of $\mathbb{R}^{d}$
is given by
\[
\mathscr{B}:=\left(Q_{j}\right)_{j\in\mathbb{N}_{0}}:=\left(T_{j}Q_{j}'+b_{j}\right)_{j\in\mathbb{N}_{0}}.
\]
This covering is a semi-structured admissible covering of $\mathbb{R}^{d}$.
Furthermore, $\mathscr{B}$ admits a regular partition of unity $\Phi=\left(\varphi_{j}\right)_{j\in\mathbb{N}_{0}}$
(which thus fulfills Assumption \ref{assu:RegularPartitionOfUnity}),
which we fix for the remainder of the section.
Finally, $\mathscr{B}$ fulfills the standing assumptions from Section
\ref{subsec:DecompSpaceDefinitionStandingAssumptions}; in particular,
$\left\Vert T_{j}^{-1}\right\Vert \leq1=:\Omega_{0}$ for all $j\in\mathbb{N}_{0}$.
\end{lem}
\begin{proof}
It was shown in \cite[Example 7.2]{DecompositionIntoSobolev} that
$\mathscr{B}$ is a semi-structured covering of $\mathbb{R}^{d}$.
In the same example, it was also shown that $\mathscr{B}$ is in fact
a regular covering of $\mathbb{R}^{d}$, i.e., $\mathscr{B}$ admits
a regular partition of unity $\Phi$, as claimed. Thanks to Corollary
\ref{cor:RegularBAPUsAreWeightedBAPUs}, $\Phi$ is also a $\mathscr{B}$-$v_{0}$-BAPU
for each weight $v_{0}$ satisfying the assumptions from Section \ref{subsec:DecompSpaceDefinitionStandingAssumptions}.
To verify the standing assumptions from Section \ref{subsec:DecompSpaceDefinitionStandingAssumptions}
pertaining to the covering $\mathcal{Q}=\mathscr{B}$, we thus only have
to verify $\left\Vert T_{j}^{-1}\right\Vert \leq1$ for all $j\in\mathbb{N}_{0}$.
But since $T_{j}=2^{j}\cdot\operatorname{id}$ for all $j\in\mathbb{N}_{0}$, we simply
have $\left\Vert T_{j}^{-1}\right\Vert =2^{-j}\leq1$, as claimed.
\end{proof}
Now, we can define the inhomogeneous Besov spaces:
\begin{defn}
\label{def:InhomogeneousBesovSpaces}For $p,q\in\left(0,\infty\right]$
and $s,\mu\in\mathbb{R}$, we define the associated \textbf{(weighted) inhomogeneous
Besov space} as
\[
\mathcal{B}_{s,\mu}^{p,q}\left(\smash{\mathbb{R}^{d}}\right):=\DecompSp{\mathscr{B}}p{\ell_{\left(2^{js}\right)_{j\in\mathbb{N}_{0}}}^{q}}{v^{\left(\mu\right)}},
\]
with $v^{\left(\mu\right)}$ as in Lemma \ref{lem:AlphaModulationSpaceWeight}.
The \textbf{classical inhomogeneous Besov spaces} are given by $\mathcal{B}_{s}^{p,q}\left(\mathbb{R}^{d}\right):=\mathcal{B}_{s,0}^{p,q}\left(\mathbb{R}^{d}\right)$.
\end{defn}
\begin{rem*}
\begin{itemize}[leftmargin=0.4cm]
\item It is not hard to see that the weight $\left(2^{js}\right)_{j\in\mathbb{N}_{0}}$
is indeed $\mathscr{B}$-moderate (cf.\@ equation (\ref{eq:IntroductionModerateWeightDefinition})).
Furthermore, we saw in Lemma \ref{lem:AlphaModulationSpaceWeight}
that the weight $v^{\left(\mu\right)}$ satisfies all assumptions
from Section \ref{subsec:DecompSpaceDefinitionStandingAssumptions},
with $K:=\left|\mu\right|$ and $\Omega_{1}:=2^{\left|\mu\right|}$,
as well as $v_{0}:\mathbb{R}^{d}\to\left(0,\infty\right),x\mapsto\left[2\cdot\left(1+\left|x\right|\right)\right]^{\left|\mu\right|}$.
All in all, we thus see that all standing assumptions from Section
\ref{subsec:DecompSpaceDefinitionStandingAssumptions} are satisfied.
In particular, Lemma \ref{lem:WeightedDecompositionSpaceComplete}
and Proposition \ref{prop:WeightedDecompositionSpaceWellDefined}
show that the spaces $\mathcal{B}_{s,\mu}^{p,q}\left(\mathbb{R}^{d}\right)$
are well-defined Quasi-Banach spaces.
\item It is not hard to see that the usual inhomogeneous Besov spaces (e.g.\@
as defined in \cite[Definition 6.5.1]{GrafakosModernFourierAnalysis})
coincide with the spaces $\mathcal{B}_{s}^{p,q}\left(\mathbb{R}^{d}\right)$
defined above, up to trivial identifications. The main difference
is that the usual Besov spaces are defined as subspaces of the space
of tempered distributions, while $\mathcal{B}_{s}^{p,q}\left(\mathbb{R}^{d}\right)$
is a subspace of $Z'\left(\mathbb{R}^{d}\right)=\left[\mathcal{F}\left(\TestFunctionSpace{\mathbb{R}^{d}}\right)\right]'$,
cf.\@ Subsection \ref{subsec:DecompSpaceDefinitionStandingAssumptions}.
Claiming that the two spaces coincide amounts to claiming that each
$f\in\mathcal{B}_{s}^{p,q}\left(\mathbb{R}^{d}\right)$ extends to
a (uniquely determined) tempered distribution. As shown in \cite[Lemma 9.15]{DecompositionEmbedding},
this is indeed fulfilled.\qedhere
\end{itemize}
\end{rem*}
Note that if we define the family $\Gamma=\left(\gamma_{j}\right)_{j\in\mathbb{N}_{0}}$
by $\gamma_{0}:=\varphi$ and $\gamma_{j}:=\psi$ for $j\in\mathbb{N}$, where
$\varphi,\psi:\mathbb{R}^{d}\to\mathbb{C}$ are given, then the family
$\Gamma^{\left(\delta\right)}=\left(L_{\delta\cdot T_{j}^{-T}k}\,\gamma^{\left[j\right]}\right)_{j\in\mathbb{N}_{0},k\in\mathbb{Z}^{d}}$
considered in Theorem \ref{thm:AtomicDecomposition} (and in a slightly
modified form also in Theorem \ref{thm:DiscreteBanachFrameTheorem})
satisfies
\[
\Gamma^{\left(\delta\right)}=\left(\varphi\left(\bullet-\delta k\right)\right)_{k\in\mathbb{Z}^{d}}\cup\left(2^{j\frac{d}{2}}\cdot\psi\left(2^{j}\bullet-\delta k\right)\right)=W\left(\varphi,\psi;\delta\right),
\]
at least up to an obvious re-indexing. Consequently, we can use Corollaries
\ref{cor:BanachFrameSimplifiedCriteria} and \ref{cor:AtomicDecompositionSimplifiedCriteria}
to derive conditions on $\varphi,\psi$ which ensure that the wavelet
system $W\left(\varphi,\psi;\delta\right)$ yields a Banach frame,
or an atomic decomposition for the (weighted) inhomogeneous Besov
spaces $\mathcal{B}_{s,\mu}^{p,q}\left(\mathbb{R}^{d}\right)$. We
begin with the case of Banach frames.
\begin{prop}
\label{prop:BesovBanachFrames}Let $p_{0},q_{0}\in\left(0,1\right]$,
$\varepsilon>0$, $\mu_{0}\geq0$ and $-\infty<s_{0}\leq s_{1}<\infty$.
Assume that $\varphi,\psi:\mathbb{R}^{d}\to\mathbb{C}$ satisfy the following
conditions:
\begin{enumerate}
\item We have $\varphi,\psi\in L_{\left(1+\left|\bullet\right|\right)^{\mu_{0}}}^{1}\left(\mathbb{R}^{d}\right)$
and $\widehat{\varphi},\widehat{\psi}\in C^{\infty}\left(\mathbb{R}^{d}\right)$,
where all partial derivatives of $\widehat{\varphi}$ and $\widehat{\psi}$
are polynomially bounded.
\item We have $\varphi,\psi\in C^{1}\left(\mathbb{R}^{d}\right)$ and $\nabla\varphi,\nabla\psi\in L_{\left(1+\left|\bullet\right|\right)^{\mu_{0}}}^{1}\left(\mathbb{R}^{d}\right)\cap L^{\infty}\left(\mathbb{R}^{d}\right)$.
\item We have $\widehat{\varphi}\left(\xi\right)\neq0$ for all $\xi\in\overline{B_{2}}\left(0\right)$
and $\widehat{\psi}\left(\xi\right)\neq0$ for all $\xi\in\overline{B_{4}\left(0\right)}\setminus B_{1/4}\left(0\right)$.
\item We have
\begin{align*}
\left|\partial^{\alpha}\widehat{\varphi}\left(\xi\right)\right| & \leq G_{1}\cdot\left(1+\left|\xi\right|\right)^{-L},\\
\left|\partial^{\alpha}\widehat{\psi}\left(\xi\right)\right| & \leq G_{2}\cdot\left(1+\left|\xi\right|\right)^{-L_{1}}\cdot\min\left\{ 1,\left|\xi\right|^{L_{2}}\right\}
\end{align*}
for all $\xi\in\mathbb{R}^{d}$ and all $\alpha\in\mathbb{N}_{0}^{d}$
with $\left|\alpha\right|\leq\left\lceil \mu_{0}+\frac{d+\varepsilon}{p_{0}}\right\rceil $
for suitable $G_{1},G_{2}>0$ and certain $L_{2}\geq0$ and $L,L_{1}\geq1$
which satisfy
\[
L>1-s_{0}+\vartheta,\qquad L_{1}>1-s_{0}+\vartheta,\qquad\text{ and }\qquad L_{2}>s_{1},
\]
where $\vartheta:=\frac{d}{p_{0}}+\mu_{0}+\left\lceil \mu_{0}+\frac{d+\varepsilon}{p_{0}}\right\rceil $.
\end{enumerate}
Then there is some $\delta_{0}=\delta_{0}\left(p_{0},q_{0},s_{0},s_{1},\mu_{0},\varepsilon,d,\varphi,\psi\right)>0$
such that for each $0<\delta\leq\delta_{0}$, the family
\[
\Gamma^{\left(\delta\right)}=\left(2^{j\frac{d}{2}}\cdot\widetilde{\gamma_{j}}\left(2^{j}\bullet-\delta k\right)\right)_{j\in\mathbb{N}_{0},k\in\mathbb{Z}^{d}},\qquad\text{ with }\qquad\widetilde{\gamma_{0}}:=\varphi\left(-\bullet\right)\text{ and }\widetilde{\gamma_{j}}:=\psi\left(-\bullet\right)\text{ for }j\in\mathbb{N},
\]
forms a Banach frame for $\mathcal{B}_{s,\mu}^{p,q}\left(\mathbb{R}^{d}\right)$,
for arbitrary $p,q\in\left(0,\infty\right]$ and $s,\mu\in\mathbb{R}$ satisfying
$p\geq p_{0}$, $q\geq q_{0}$, $s_{0}\leq s\leq s_{1}$ and $\left|\mu\right|\leq\mu_{0}$.
Precisely, this means the following: Define the coefficient space
\[
\mathscr{C}_{p,q,s,\mu}:=\ell_{\left(2^{j\left(s+d\left(\frac{1}{2}-\frac{1}{p}\right)\right)}\right)_{j\in\mathbb{N}_{0}}}^{q}\!\!\!\!\!\!\!\!\!\!\!\!\left(\left[\ell_{\left[\left(1+\left|k\right|/2^{j}\right)^{\mu}\right]_{k\in\mathbb{Z}^{d}}}^{p}\left(\mathbb{Z}^{d}\right)\right]_{j\in\mathbb{N}_{0}}\right)\leq\mathbb{C}^{\mathbb{N}_{0}\times\mathbb{Z}^{d}}.
\]
Then the following hold:
\begin{itemize}
\item The \textbf{analysis operator}
\[
A^{\left(\delta\right)}:\mathcal{B}_{s,\mu}^{p,q}\left(\smash{\mathbb{R}^{d}}\right)\to\mathscr{C}_{p,q,s,\mu},f\mapsto\left[\left(\left[2^{j\frac{d}{2}}\cdot\gamma_{j}\left(2^{j}\bullet\right)\right]\ast f\right)\left(\delta\cdot\frac{k}{2^{j}}\right)\right]_{j\in\mathbb{N}_{0},k\in\mathbb{Z}^{d}}
\]
is well-defined and bounded for all $0<\delta\leq1$. Here, $\gamma_{0}:=\varphi$
and $\gamma_{j}:=\psi$ for $j\in\mathbb{N}$. The convolution considered
here is defined as in equation (\ref{eq:SpecialConvolutionPointwiseDefinition}).
\item For $0<\delta\leq1$, there is a bounded linear \textbf{reconstruction
operator} $R^{\left(\delta\right)}:\mathscr{C}_{p,q,s,\mu}\to\mathcal{B}_{s,\mu}^{p,q}\left(\mathbb{R}^{d}\right)$
satisfying $R^{\left(\delta\right)}\circ A^{\left(\delta\right)}=\operatorname{id}_{\mathcal{B}_{s,\mu}^{p,q}}$.
Furthermore, the action of $R^{\left(\delta\right)}$ on a given sequence
is independent of the precise choice of $p,q,s,\mu$.
\item We have the following \textbf{consistency statement}: If $f\in\mathcal{B}_{s,\mu}^{p,q}\left(\mathbb{R}^{d}\right)$
and if $p_{0}\leq\tilde{p}\leq\infty$ and $q_{0}\leq\tilde{q}\leq\infty$
and if furthermore $s_{0}\leq\tilde{s}\leq s_{1}$ and $\left|\tilde{\mu}\right|\leq\mu_{0}$,
then the following equivalence holds:
\[
f\in\mathcal{B}_{\tilde{s},\tilde{\mu}}^{\tilde{p},\tilde{q}}\left(\smash{\mathbb{R}^{d}}\right)\qquad\Longleftrightarrow\qquad A^{\left(\delta\right)}f\in\mathscr{C}_{\tilde{p},\tilde{q},\tilde{s},\tilde{\mu}}.\qedhere
\]
\end{itemize}
\end{prop}
\begin{proof}
Let $p,q,s,\mu$ as in the statement of the proposition. Our first
goal is to provide suitable estimates for the quantity $M_{j,i}$
which appears in Corollary \ref{cor:BanachFrameSimplifiedCriteria},
i.e.,
\[
M_{j,i}=\left(\frac{w_{j}}{w_{i}}\right)^{\tau}\cdot\left(1+\left\Vert T_{j}^{-1}T_{i}\right\Vert \right)^{\sigma}\cdot\max_{\left|\beta\right|\leq1}\left(\left|\det T_{i}\right|^{-1}\cdot\int_{Q_{i}}\max_{\left|\alpha\right|\leq N}\left|\left(\partial^{\alpha}\widehat{\partial^{\beta}\gamma_{j}}\right)\!\!\left(S_{j}^{-1}\xi\right)\right|\operatorname{d}\xi\right)^{\tau},
\]
where $K=\left|\mu\right|$ (cf.\@ the remark after Definition \ref{def:InhomogeneousBesovSpaces})
and
\begin{align*}
N & =\left\lceil K+\frac{d+\varepsilon}{\min\left\{ 1,p\right\} }\right\rceil \,,\\
\tau & =\min\left\{ 1,p,q\right\} ,\\
\sigma & =\tau\cdot\left(\frac{d}{\min\left\{ 1,p\right\} }+K+\left\lceil K+\frac{d+\varepsilon}{\min\left\{ 1,p\right\} }\right\rceil \right).
\end{align*}
We immediately observe $N\leq N_{0}:=\left\lceil \mu_{0}+\frac{d+\varepsilon}{p_{0}}\right\rceil $,
so that our estimates regarding $\left|\partial^{\alpha}\widehat{\varphi}\left(\xi\right)\right|$
and $\left|\partial^{\alpha}\widehat{\psi}\left(\xi\right)\right|$
can be applied.
Indeed, since $\varphi,\psi\in C^{1}\left(\mathbb{R}^{d}\right)$
with $\nabla\varphi,\nabla\psi\in L_{\left(1+\left|\bullet\right|\right)^{\mu_{0}}}^{1}\left(\mathbb{R}^{d}\right)\hookrightarrow L^{1}\left(\mathbb{R}^{d}\right)$
and since $\gamma_{j}=\varphi$ for $j=0$ and $\gamma_{j}=\psi$
otherwise, standard properties of the Fourier transform show
\[
\widehat{\partial^{\beta}\gamma_{j}}\left(\xi\right)=\left(2\pi i\xi\right)^{\beta}\cdot\widehat{\gamma_{j}}\left(\xi\right)\qquad\forall\xi\in\mathbb{R}^{d}\qquad\forall\beta\in\mathbb{N}_{0}^{d}\text{ with }\left|\beta\right|\leq1.
\]
Consequently, we get for $i\in\mathbb{N}$ that
\begin{equation}
M_{j,i}\leq2^{\tau s\left(j-i\right)}\cdot\left(1+2^{i-j}\right)^{\sigma}\cdot\left(2\pi\cdot\max_{\left|\beta\right|\leq1}2^{-i\cdotd}\int_{2^{i-2}<\left|\eta\right|<2^{i+2}}\:\max_{\left|\alpha\right|\leq N}\left|\left(\partial^{\alpha}\left[\xi\mapsto\xi^{\beta}\cdot\widehat{\gamma_{j}}\left(\xi\right)\right]\right)\left(\eta/2^{j}\right)\right|\operatorname{d}\eta\right)^{\tau}\!\!.\label{eq:BesovBanachFrameBasicEstimate}
\end{equation}
Next, we observe for $\beta\in\mathbb{N}_{0}^{d}$ with $\left|\beta\right|=1$,
i.e., $\beta=e_{j}$ for some $j\in\underline{d}$, that
\[
\left|\partial^{\nu}\xi^{\beta}\right|=\begin{cases}
\left|\xi^{\beta}\right|\leq\left|\xi\right|\leq1+\left|\xi\right|, & \text{if }\nu=0,\\
1\leq1+\left|\xi\right|, & \text{if }\nu=e_{j},\\
0\leq1+\left|\xi\right| & \text{otherwise}.
\end{cases}
\]
Likewise, in case of $\beta=0$, we have
\[
\left|\partial^{\nu}\xi^{\beta}\right|=\begin{cases}
1\leq1+\left|\xi\right|, & \text{if }\nu=0,\\
0\leq1+\left|\xi\right|, & \text{otherwise}.
\end{cases}
\]
In connection with Leibniz's rule and the $d$-dimensional
binomial theorem (cf.\@ \cite[Section 8.1, Exercise 2.b]{FollandRA}),
this yields for $j=0$ and $\alpha\in\mathbb{N}_{0}^{d}$ with $\left|\alpha\right|\leq N\leq N_{0}$
that
\begin{equation}
\begin{split}\left|\partial^{\alpha}\left[\xi\mapsto\xi^{\beta}\cdot\widehat{\gamma_{0}}\left(\xi\right)\right]\left(\eta\right)\right| & \leq\sum_{\nu\leq\alpha}\binom{\alpha}{\nu}\cdot\left|\partial^{\nu}\eta^{\beta}\right|\cdot\left|\partial^{\alpha-\nu}\widehat{\gamma_{0}}\left(\eta\right)\right|\\
\left({\scriptstyle \text{assumption for }\widehat{\gamma_{0}}=\widehat{\varphi}}\right) & \leq G_{1}\cdot\left(1+\left|\eta\right|\right)^{1-L}\cdot\sum_{\nu\leq\alpha}\binom{\alpha}{\nu}\\
& \leq2^{N_{0}}G_{1}\cdot\left(1+\left|\eta\right|\right)^{1-L}.
\end{split}
\label{eq:BesovBanachFrameLowPassEstimate}
\end{equation}
Likewise, for $j\in\mathbb{N}$ and $\alpha\in\mathbb{N}_{0}^{d}$ with $\left|\alpha\right|\leq N\leq N_{0}$,
we get
\begin{equation}
\begin{split}\left|\partial^{\alpha}\left[\xi\mapsto\xi^{\beta}\cdot\widehat{\gamma_{j}}\left(\xi\right)\right]\left(\eta\right)\right| & \leq\sum_{\nu\leq\alpha}\binom{\alpha}{\nu}\cdot\left|\partial^{\nu}\eta^{\beta}\right|\cdot\left|\partial^{\alpha-\nu}\widehat{\gamma_{j}}\left(\eta\right)\right|\\
\left(\text{\ensuremath{{\scriptstyle \text{assumption for }\widehat{\gamma_{j}}=\widehat{\psi}}}}\right) & \leq G_{2}\cdot\left(1+\left|\eta\right|\right)^{1-L_{1}}\cdot\min\left\{ 1,\left|\eta\right|^{L_{2}}\right\} \cdot\sum_{\nu\leq\alpha}\binom{\alpha}{\nu}\\
& \leq2^{N_{0}}G_{2}\cdot\left(1+\left|\eta\right|\right)^{1-L_{1}}\cdot\min\left\{ 1,\left|\eta\right|^{L_{2}}\right\} .
\end{split}
\label{eq:BesovBanachFrameMotherWaveletEstimate}
\end{equation}
Now, we first consider the case $i\in\mathbb{N}$ and note
\begin{equation}
\begin{split}\lambda_{d}\left(\left\{ \eta\in\mathbb{R}^{d}\with2^{i-2}<\left|\eta\right|<2^{i+2}\right\} \right) & =v_{d}\cdot\left(2^{d\left(i+2\right)}-2^{d\left(i-2\right)}\right)\\
& =2^{i\cdotd}\cdot v_{d}\left(4^{d}-4^{-d}\right)\\
& \leq2^{i\cdotd}\cdot4^{d}v_{d},
\end{split}
\label{eq:BesovAnnulusMeasureEstimate}
\end{equation}
so that
\begin{equation}
2^{-i\cdotd}\cdot\int_{2^{i-2}<\left|\eta\right|<2^{i+2}}h\left(\eta\right)\operatorname{d}\eta\leq4^{d}v_{d}\cdot\sup_{2^{i-2}<\left|\eta\right|<2^{i+2}}\;h\left(\eta\right)\label{eq:BesovBanachFrameMeanEstimate}
\end{equation}
for each nonnegative (measurable) function $h$. Now, we distinguish
two subcases for estimating $M_{j,i}$.
\textbf{Case 1}: We have $j\in\mathbb{N}$. In this case, we distinguish two
additional subcases:
\begin{enumerate}
\item We have $j\leq i$. For $2^{i-2}<\left|\eta\right|<2^{i+2}$, this
implies $\left|\eta/2^{j}\right|\geq2^{i-2-j}=\frac{2^{i-j}}{4}=\frac{2^{\left|i-j\right|}}{4}$.
Since we have $L_{1}\geq1$, a combination of this estimate with equations
(\ref{eq:BesovBanachFrameBasicEstimate}), (\ref{eq:BesovBanachFrameMotherWaveletEstimate})
and (\ref{eq:BesovBanachFrameMeanEstimate}) yields
\begin{align*}
M_{j,i} & \leq2^{\sigma}\cdot2^{-\tau s\left|j-i\right|}\cdot2^{\sigma\left|i-j\right|}\cdot\left(2^{N_{0}}G_{2}\cdot2\pi\cdot4^{d}v_{d}\cdot4^{L_{1}-1}\cdot2^{\left|i-j\right|\left(1-L_{1}\right)}\right)^{\tau}\\
& \leq2^{\sigma}\cdot\left(2^{N_{0}}G_{2}\cdot2\pi\cdot4^{d}v_{d}\cdot4^{L_{1}}\right)^{\tau}\cdot2^{\left|i-j\right|\left[\sigma-\tau\left(L_{1}-1+s\right)\right]}\\
& =:2^{\sigma}\cdot H_{1}^{\tau}\cdot2^{\left|i-j\right|\left[\sigma-\tau\left(L_{1}-1+s\right)\right]}.
\end{align*}
\item We have $i\leq j$. For $2^{i-2}<\left|\eta\right|<2^{i+2}$, this
implies because of $L_{2}\geq0$ that
\[
\min\left\{ 1,\left|\eta/2^{j}\right|^{L_{2}}\right\} \leq\left(2^{i+2-j}\right)^{L_{2}}\leq4^{L_{2}}\cdot2^{-L_{2}\left|i-j\right|}.
\]
Furthermore, $\left(1+\left|\eta/2^{j}\right|\right)^{1-L_{1}}\leq1$,
since $L_{1}\geq1$. Hence, as in the previous case, we can combine
equations (\ref{eq:BesovBanachFrameBasicEstimate}), (\ref{eq:BesovBanachFrameMotherWaveletEstimate})
and (\ref{eq:BesovBanachFrameMeanEstimate}) to derive
\begin{align*}
M_{j,i} & \leq2^{\tau s\left|j-i\right|}\cdot2^{\sigma}\cdot\left(2^{N_{0}}G_{2}\cdot2\pi\cdot4^{d}v_{d}\cdot4^{L_{2}}\cdot2^{-L_{2}\left|i-j\right|}\right)^{\tau}\\
& =:2^{\sigma}\cdot H_{2}^{\tau}\cdot2^{\tau\left|j-i\right|\left(s-L_{2}\right)}.
\end{align*}
\end{enumerate}
\textbf{Case 2}: We have $j=0$. In this case, we have for $2^{i-2}<\left|\eta\right|<2^{i+2}$
that $\left|\eta/2^{j}\right|=\left|\eta\right|\geq2^{i-2}$ and hence
$\left(1+\left|\eta/2^{j}\right|\right)^{1-L}\leq4^{L-1}2^{-i\left(L-1\right)}\leq4^{L}\cdot2^{-\left(L-1\right)\left|i-j\right|}$.
Here, we used $L\geq1$.
Now, a combination of equations (\ref{eq:BesovBanachFrameBasicEstimate}),
(\ref{eq:BesovBanachFrameLowPassEstimate}) and (\ref{eq:BesovBanachFrameMeanEstimate})
yields
\begin{align*}
M_{0,i} & \leq2^{-\tau s\left|i-j\right|}\cdot2^{\sigma}2^{\sigma\left|i-j\right|}\cdot\left(2^{N_{0}}G_{1}\cdot2\pi\cdot4^{d}v_{d}\cdot4^{L}\cdot2^{-\left(L-1\right)\left|i-j\right|}\right)^{\tau}\\
& =:2^{\sigma}\cdot H_{3}^{\tau}\cdot2^{\left|i-j\right|\left[\sigma-\tau\left(L-1+s\right)\right]}.
\end{align*}
These are the desired estimates in case of $i\in\mathbb{N}$. It remains to
consider the case $i=0$. Here, equation (\ref{eq:BesovBanachFrameBasicEstimate})
takes on the slightly modified form
\begin{equation}
\begin{split}M_{j,0} & \leq2^{\tau sj}\cdot\left(1+2^{-j}\right)^{\sigma}\cdot\left(2\pi\cdot\max_{\left|\beta\right|\leq1}\int_{B_{2}\left(0\right)}\max_{\left|\alpha\right|\leq N}\left|\left(\partial^{\alpha}\left[\xi\mapsto\xi^{\beta}\cdot\widehat{\gamma_{j}}\left(\xi\right)\right]\right)\left(\eta/2^{j}\right)\right|\operatorname{d}\eta\right)^{\tau}\\
& \leq2^{\sigma}\cdot2^{\tau sj}\cdot\left(2\pi\cdot\lambda_{d}\left(B_{2}\left(0\right)\right)\cdot\max_{\left|\beta\right|\leq1}\sup_{\left|\eta\right|<2}\left|\left(\partial^{\alpha}\left[\xi\mapsto\xi^{\beta}\cdot\widehat{\gamma_{j}}\left(\xi\right)\right]\right)\left(\eta/2^{j}\right)\right|\right)^{\tau}.
\end{split}
\label{eq:BesovBanachFrameBasicEstimateAtOrigin}
\end{equation}
Now, we again distinguish two cases:
\textbf{Case 1}: We have $j\in\mathbb{N}$. Here, we observe for $\left|\eta\right|<2$
that
\[
\min\left\{ 1,\left|\eta/2^{j}\right|^{L_{2}}\right\} \leq2^{\left(1-j\right)L_{2}}=2^{L_{2}}\cdot2^{-L_{2}\left|i-j\right|}.
\]
Since we also have $L_{1}\geq1$, a combination of equations (\ref{eq:BesovBanachFrameBasicEstimateAtOrigin})
and (\ref{eq:BesovBanachFrameMotherWaveletEstimate}) yields
\begin{align*}
M_{j,0} & \leq2^{\sigma}\cdot2^{\tau s\left|i-j\right|}\cdot\left(2^{N_{0}}G_{2}\cdot2\pi\cdot\lambda_{d}\left(B_{2}\left(0\right)\right)\cdot2^{L_{2}}\cdot2^{-L_{2}\left|i-j\right|}\right)^{\tau}\\
& =:2^{\sigma}\cdot H_{4}^{\tau}\cdot2^{\tau\left|i-j\right|\left(s-L_{2}\right)}.
\end{align*}
\textbf{Case 2}: We have $j=0$. Because of $L\geq1$, we have $\left(1+\left|\eta/2^{j}\right|\right)^{1-L}\leq1$
for arbitrary $\eta\in\mathbb{R}^{d}$, so that a combination of
equations (\ref{eq:BesovBanachFrameBasicEstimateAtOrigin}) and (\ref{eq:BesovBanachFrameLowPassEstimate})
yields
\begin{align*}
M_{0,0} & \leq2^{\sigma}\cdot\left(2^{N_{0}}G_{1}\cdot2\pi\cdot\lambda_{d}\left(B_{2}\left(0\right)\right)\right)^{\tau}\\
& =:2^{\sigma}\cdot H_{5}^{\tau}\\
& =2^{\sigma}\cdot H_{5}^{\tau}\cdot2^{-\left|i-j\right|\zeta},
\end{align*}
where $\zeta\in\mathbb{R}$ can be chosen arbitrarily, since $\left|i-j\right|=0$.
\medskip{}
All in all, if we set $H_{6}:=\max\left\{ H_{1},\dots,H_{5}\right\} $,
a combination of the preceding cases shows
\begin{equation}
M_{j,i}\leq2^{\sigma}\cdot H_{6}^{\tau}\cdot2^{-\tau\left|i-j\right|\min\left\{ 1,L_{2}-s,\,L-1+s-\frac{\sigma}{\tau},\,L_{1}-1+s-\frac{\sigma}{\tau}\right\} }.\label{eq:BesovBanachFrameSummary}
\end{equation}
Note that $H_{1},\dots,H_{5}$, and hence also $H_{6}$, are all independent
of $p,q,\mu,s$.
Furthermore, we have
\[
\frac{\sigma}{\tau}=\frac{d}{\min\left\{ 1,p\right\} }+K+\left\lceil K+\frac{d+\varepsilon}{\min\left\{ 1,p\right\} }\right\rceil \leq\frac{d}{p_{0}}+\mu_{0}+\left\lceil \mu_{0}+\frac{d+\varepsilon}{p_{0}}\right\rceil =\vartheta
\]
and $\sigma=\tau\cdot\frac{\sigma}{\tau}\leq\tau\vartheta$, as well
as $s_{0}\leq s\leq s_{1}$. Hence,
\[
M_{j,i}\leq\left(2^{\vartheta}\cdot H_{6}\right)^{\tau}\cdot2^{-\tau\left|i-j\right|\min\left\{ 1,\,L_{2}-s_{1},\,L-1+s_{0}-\vartheta,\,L_{1}-1+s_{0}-\vartheta\right\} }.
\]
But our assumptions on $L,L_{1},L_{2}$ imply that the exponent $\lambda:=\min\left\{ 1,\,L_{2}-s_{1},\,L-1+s_{0}-\vartheta,\,L_{1}-1+s_{0}-\vartheta\right\} $
is positive. Hence, we get, for the constants $C_{1},C_{2}$ defined
in Corollary \ref{cor:BanachFrameSimplifiedCriteria},
\begin{align*}
C_{1}^{1/\tau}=\left(\sup_{i\in\mathbb{N}_{0}}\sum_{j\in\mathbb{N}_{0}}M_{j,i}\right)^{1/\tau} & \leq2^{\vartheta}\cdot H_{6}\cdot\sup_{i\in\mathbb{N}_{0}}\left(\sum_{j\in\mathbb{N}_{0}}2^{-\tau\lambda\left|i-j\right|}\right)^{1/\tau}\\
\left({\scriptstyle \text{for }\ell=i-j}\right) & \leq2^{\vartheta}\cdot H_{6}\cdot\left(\sum_{\ell\in\mathbb{Z}}2^{-\tau\lambda\left|\ell\right|}\right)^{1/\tau}\\
& \leq2^{\vartheta}\cdot H_{6}\cdot\left(2\cdot\sum_{\ell=0}^{\infty}2^{-\tau\lambda\ell}\right)^{1/\tau}\\
\left({\scriptstyle \text{since }\ell^{\tau_{0}}\hookrightarrow\ell^{\tau}\text{ is norm-decreasing for }\tau_{0}:=\min\left\{ p_{0},q_{0}\right\} }\right) & \leq2^{\vartheta}\cdot H_{6}\cdot2^{1/\tau}\cdot\left(\sum_{\ell=0}^{\infty}2^{-\tau_{0}\lambda\ell}\right)^{1/\tau_{0}}\\
& \leq2^{\vartheta+\frac{1}{\tau_{0}}}\cdot H_{6}\cdot\left(\frac{1}{1-2^{-\tau_{0}\lambda}}\right)^{1/\tau_{0}}=:H_{7}<\infty.
\end{align*}
Observe again that $H_{7}$ is independent of $p,q,\mu,s$. Exactly
the same estimate also yields $C_{2}^{1/\tau}\leq H_{7}$.
\medskip{}
Next, we set $\gamma_{1}^{\left(0\right)}:=\varphi$ and $\gamma_{2}^{\left(0\right)}:=\psi$.
Furthermore, we set $n_{j}:=2$ for $j\in\mathbb{N}$ and $n_{0}:=1$, so
that $\gamma_{j}=\gamma_{n_{j}}^{\left(0\right)}$ for all $j\in\mathbb{N}_{0}$.
In the notation of Lemma \ref{lem:GammaCoversOrbitAssumptionSimplified},
these definitions entail
\begin{align*}
Q^{\left(1\right)} & =\bigcup\left\{ Q_{i}'\,\middle|\, i\in\mathbb{N}_{0}\text{ and }n_{i}=1\right\} =Q_{0}'=B_{2}\left(0\right),\\
Q^{\left(2\right)} & =\bigcup\left\{ Q_{i}'\,\middle|\, i\in\mathbb{N}_{0}\text{ and }n_{i}=2\right\} =\bigcup_{j\in\mathbb{N}}Q_{j}'=B_{4}\left(0\right)\setminus\overline{B_{1/4}}\left(0\right).
\end{align*}
But the prerequisites of the current proposition include the assumptions
$\smash{\widehat{\gamma_{1}^{\left(0\right)}}}\left(\xi\right)=\widehat{\varphi}\left(\xi\right)\neq0$
for all $\xi\in\overline{Q^{\left(1\right)}}$ and $\widehat{\gamma_{2}^{\left(0\right)}}\left(\xi\right)=\widehat{\psi}\left(\xi\right)\neq0$
for all $\xi\in\overline{Q^{\left(2\right)}}$. By continuity of $\widehat{\varphi},\widehat{\psi}$
and by compactness of $\overline{Q^{\left(1\right)}},\overline{Q^{\left(2\right)}}$,
we thus see that all assumptions of Lemma \ref{lem:GammaCoversOrbitAssumptionSimplified}
are satisfied. Consequently, the family $\Gamma=\left(\gamma_{i}\right)_{i\in I}$
satisfies Assumption \ref{assu:GammaCoversOrbit} and there is a constant
$\Omega_{3}=\Omega_{3}\left(\mathscr{B},\varphi,\psi,p_{0},\mu_{0},d\right)>0$
satisfying $\Omega_{2}^{\left(p,K\right)}\leq\Omega_{3}$ for all
$K\leq\mu_{0}$ and $p\geq p_{0}$, with $\Omega_{2}^{\left(p,K\right)}$
as in Assumption \ref{assu:GammaCoversOrbit}. Recall that in our
case, we indeed have $K=\left|\mu\right|\leq\mu_{0}$.
In view of the assumptions of the proposition, it is now not hard
to see that all prerequisites for Corollary \ref{cor:BanachFrameSimplifiedCriteria}
are satisfied. Hence, that corollary implies that $\Gamma^{\left(\delta\right)}$
forms a Banach frame (in the sense of Theorem \ref{thm:DiscreteBanachFrameTheorem})
for $\mathcal{B}_{s,\mu}^{p,q}\left(\mathbb{R}^{d}\right)=\DecompSp{\mathcal{Q}}p{\ell_{\left(2^{js}\right)_{j}}^{q}}{v^{\left(\mu\right)}}$,
as soon as $0<\delta\leq\delta_{00}$, where (cf.\@ Lemma \ref{lem:SpecialProjection}
for the definition of $F_{0}$ and the estimate for $\vertiii{F_{0}}$
used here)
\[
\delta_{00}=\frac{1}{1+2\vertiii{F_{0}}^{2}}
\]
and, with $w=\left(2^{js}\right)_{j\in\mathbb{N}_{0}}$,
\begin{align*}
\vertiii{F_{0}} & \leq2^{\frac{1}{q}}C_{\mathscr{B},\Phi,v_{0},p}^{2}\cdot\vertiii{\smash{\Gamma_{\mathscr{B}}}}_{\ell_{w}^{q}\to\ell_{w}^{q}}^{2}\cdot\left(\vertiii{\smash{\overrightarrow{A}}}^{\max\left\{ 1,\frac{1}{p}\right\} }+\vertiii{\smash{\overrightarrow{B}}}^{\max\left\{ 1,\frac{1}{p}\right\} }\right)\cdot C_{3}\\
\left({\scriptstyle \text{Corollary }\ref{cor:BanachFrameSimplifiedCriteria}}\right) & \leq2^{\frac{1}{q_{0}}}C_{\mathscr{B},\Phi,v_{0},p}^{2}\cdot\vertiii{\smash{\Gamma_{\mathscr{B}}}}_{\ell_{w}^{q}\to\ell_{w}^{q}}^{2}\cdot4H_{7}\cdot C_{3}C_{4}\\
\left({\scriptstyle \text{eq. }\eqref{eq:WeightedSequenceSpaceClusteringMapNormEstimate}}\right) & \leq4\cdot2^{\frac{1}{q_{0}}}C_{\mathscr{B},\Phi,v_{0},p}^{2}\cdot\left[C_{\mathscr{B},\left(2^{js}\right)_{j\in\mathbb{N}_{0}}}\cdot N_{\mathscr{B}}^{1+\frac{1}{q}}\right]^{2}\cdot H_{7}\cdot C_{3}C_{4}\\
& \leq4\cdot2^{\frac{1}{q_{0}}}C_{\mathscr{B},\Phi,v_{0},p}^{2}\cdot\left[C_{\mathscr{B},\left(2^{j}\right)_{j\in\mathbb{N}_{0}}}^{\left|s\right|}\cdot N_{\mathscr{B}}^{1+\frac{1}{q_{0}}}\right]^{2}\cdot H_{7}\cdot C_{3}C_{4}\\
& \leq4\cdot2^{\frac{1}{q_{0}}}C_{\mathscr{B},\Phi,v_{0},p}^{2}\cdot\left[C_{\mathscr{B},\left(2^{j}\right)_{j\in\mathbb{N}_{0}}}^{\max\left\{ s_{1},-s_{0}\right\} }\cdot N_{\mathscr{B}}^{1+\frac{1}{q_{0}}}\right]^{2}\cdot H_{7}\cdot C_{3}C_{4},
\end{align*}
where
\[
C_{3}=\begin{cases}
\frac{\left(2^{16}\cdot768/d^{\frac{3}{2}}\right)^{\frac{d}{p}}}{2^{42}\cdot12^{d}\cdotd^{15}}\!\cdot\!\left(2^{52}\!\cdot\!d^{\frac{25}{2}}\!\cdot\!\tilde{N}^{3}\right)^{\tilde{N}+1}\!\!\!\cdot\!N_{\mathscr{B}}^{2\left(\frac{1}{p}-1\right)}\!\left(1\!+\!R_{\mathscr{B}}C_{\mathscr{B}}\right)^{d\left(\frac{4}{p}-1\right)}\!\!\cdot\Omega_{0}^{13K}\Omega_{1}^{13}\Omega_{2}^{\left(p,K\right)}, & \text{if }p<1,\\
\frac{1}{\sqrt{d}\cdot2^{12+6\left\lceil K\right\rceil }}\cdot\left(2^{17}\cdotd^{5/2}\cdot\tilde{N}\right)^{\left\lceil K\right\rceil +d+2}\cdot\left(1+R_{\mathscr{B}}\right)^{d}\cdot\Omega_{0}^{3K}\Omega_{1}^{3}\Omega_{2}^{\left(p,K\right)}, & \text{if }p\geq1,
\end{cases}
\]
with $\tilde{N}=\left\lceil K+\frac{d+1}{\min\left\{ 1,p\right\} }\right\rceil \leq\left\lceil \mu_{0}+\frac{d+1}{p_{0}}\right\rceil $
and
\[
C_{4}=\Omega_{0}^{K}\Omega_{1}\cdotd^{1/\min\left\{ 1,p\right\} }\cdot\left(4\cdotd\right)^{1+2\left\lceil K+\frac{d+\varepsilon}{\min\left\{ 1,p\right\} }\right\rceil }\cdot\left(\frac{s_{d}}{\varepsilon}\right)^{1/\min\left\{ 1,p\right\} }\cdot\max_{\left|\alpha\right|\leq\left\lceil K+\frac{d+\varepsilon}{\min\left\{ 1,p\right\} }\right\rceil }C^{\left(\alpha\right)},
\]
where the constants $C^{\left(\alpha\right)}=C^{\left(\alpha\right)}\left(\Phi\right)$
are as in Assumption \ref{assu:RegularPartitionOfUnity}.
To establish that $\delta_{0}$ can be chosen independently of $p,q,s,\mu$,
it thus suffices to estimate $C_{3}C_{4}$ and $C_{\mathscr{B},\Phi,v_{0},p}$
independently of these quantities. But above, we estimated $\Omega_{2}^{\left(p,K\right)}\leq\Omega_{3}$
with $\Omega_{3}$ independent of $p,q,s,\mu$. Since we also have
$K=\left|\mu\right|\leq\mu_{0}$ and $0\leq\frac{1}{p}\leq\frac{1}{p_{0}}$,
as well as $\Omega_{0}=1$ and $\Omega_{1}=2^{\left|\mu\right|}\leq2^{\mu_{0}}$,
it is straightforward to see that $C_{3}$ can be estimated independently
of $p,q,s,\mu$. The same arguments also allow us to estimate $C_{4}$
independently of these quantities.
Finally, Corollary \ref{cor:RegularBAPUsAreWeightedBAPUs} shows that
there is a suitable $\varrho\in\TestFunctionSpace{\mathbb{R}^{d}}$
(depending only on $\mathscr{B}$) satisfying
\[
C_{\mathscr{B},\Phi,v_{0},p}\leq\Omega_{0}^{K}\Omega_{1}\cdot\left(4\cdotd\right)^{1+2\left\lceil K+\frac{d+\varepsilon}{p}\right\rceil }\cdot\left(\frac{s_{d}}{\varepsilon}\right)^{1/p}\cdot2^{\!\left\lceil K+\frac{d+\varepsilon}{p}\right\rceil }\cdot\lambda_{d}\left(Q\right)\cdot\max_{\left|\alpha\right|\leq\left\lceil K+\frac{d+\varepsilon}{p}\right\rceil }\left\Vert \partial^{\alpha}\varrho\right\Vert _{\sup}\cdot\max_{\left|\alpha\right|\leq\left\lceil K+\frac{d+\varepsilon}{p}\right\rceil }C^{\left(\alpha\right)},
\]
where $Q:=\overline{\bigcup_{i\in\mathbb{N}_{0}}Q_{i}'}\subset\overline{B_{4}}\left(0\right)$.
As above, since $0\leq\frac{1}{p}\leq\frac{1}{p_{0}}$ and $K=\left|\mu\right|\leq\mu_{0}$,
it is then not hard to see that $C_{\mathscr{B},\Phi,v_{0},p}$ can
be estimated independently of $p,q,s,\mu$.
\medskip{}
It remains to show that the sequence space $\mathscr{C}_{p,q,s,\mu}$
is identical to the coefficient space $\ell_{\left(\left|\det T_{i}\right|^{\frac{1}{2}-\frac{1}{p}}\cdot w_{i}\right)_{i\in I}}^{q}\!\!\!\!\!\!\!\!\left(\left[\vphantom{\sum}\smash{C_{i}^{\left(\delta\right)}}\right]_{i\in I}\right)$
mentioned in Theorem \ref{thm:DiscreteBanachFrameTheorem}. To this
end, recall from equation (\ref{eq:CoefficientSpaceDefinition}) that
$C_{j}^{\left(\delta\right)}=\ell_{v^{\left(j,\delta\right)}}^{p}\left(\smash{\mathbb{Z}^{d}}\right)$
with $v=v^{\left(\mu\right)}$ and
\begin{align*}
v_{k}^{\left(j,\delta\right)} & =v^{\left(\mu\right)}\left(\delta\cdot T_{j}^{-T}k\right)\\
& =\left(1+\left|\delta\cdot T_{j}^{-T}k\right|\right)^{\mu}\\
& =\left(1+\left|\delta\cdot k/2^{j}\right|\right)^{\mu}.
\end{align*}
But since $0<\delta\leq1$, we have $\delta\cdot\left(1+\left|k/2^{j}\right|\right)\leq1+\left|\delta\cdot\frac{k}{2^{j}}\right|\leq1+\left|k/2^{j}\right|$,
which implies
\[
\delta^{\mu_{0}}\cdot\left(1+\left|k/2^{j}\right|\right)^{\mu}\leq\delta^{\left|\mu\right|}\cdot\left(1+\left|k/2^{j}\right|\right)^{\mu}\leq v_{k}^{\left(j,\delta\right)}\leq\delta^{-\left|\mu\right|}\cdot\left(1+\left|k/2^{j}\right|\right)^{\mu}\leq\delta^{-\mu_{0}}\cdot\left(1+\left|k/2^{j}\right|\right)^{\mu}
\]
for all $k\in\mathbb{Z}^{d}$, $j\in\mathbb{N}_{0}$ and $0<\delta\leq1$.
Finally, since $w_{i}=2^{si}$ and $\left|\det T_{i}\right|=2^{i\cdotd}$
for $i\in\mathbb{N}_{0}$, we see
\[
\left|\det T_{i}\right|^{\frac{1}{2}-\frac{1}{p}}\cdot w_{i}=2^{i\left(s+d\left(\frac{1}{2}-\frac{1}{p}\right)\right)}.
\]
Taken together, these considerations easily show $\ell_{\left(\left|\det T_{i}\right|^{\frac{1}{2}-\frac{1}{p}}\cdot w_{i}\right)_{i\in I}}^{q}\!\!\!\!\!\left(\left[\vphantom{\sum}\smash{C_{i}^{\left(\delta\right)}}\right]_{i\in I}\right)=\mathscr{C}_{p,q,s,\mu}$
with equivalent quasi-norms. Here, the implicit constant is allowed
to depend on $\delta$.
\end{proof}
Next, we derive concrete conditions on $\varphi,\psi$ which ensure
that the generated wavelet system yields atomic decompositions for
the (weighted) Besov spaces $\mathcal{B}_{s,\mu}^{p,q}\left(\mathbb{R}^{d}\right)$.
\begin{prop}
\label{prop:BesovAtomicDecomposition}Let $p_{0},q_{0}\in\left(0,1\right]$,
$\varepsilon>0$, $\mu_{0}\geq0$ and $-\infty<s_{0}\leq s_{1}<\infty$.
Assume that $\varphi,\psi\in L^{1}\left(\mathbb{R}^{d}\right)$ satisfy
the following conditions:
\begin{enumerate}
\item We have $\left\Vert \varphi\right\Vert _{K_{00}}<\infty$ and $\left\Vert \psi\right\Vert _{K_{00}}<\infty$
for $K_{00}:=\mu_{0}+\frac{d}{p_{0}}+1$, where $\left\Vert g\right\Vert _{M}=\sup_{x\in\mathbb{R}^{d}}\left(1+\left|x\right|\right)^{M}\cdot\left|g\left(x\right)\right|$.
\item We have $\widehat{\varphi},\widehat{\psi}\in C^{\infty}\left(\mathbb{R}^{d}\right)$,
with all partial derivatives of $\widehat{\varphi},\widehat{\psi}$
being polynomially bounded.
\item We have $\widehat{\varphi}\left(\xi\right)\neq0$ for all $\xi\in\overline{B_{2}}\left(0\right)$
and $\widehat{\psi}\left(\xi\right)\neq0$ for all $\xi\in\overline{B_{4}}\left(0\right)\setminus B_{1/4}\left(0\right)$.
\item We have
\begin{align*}
\left|\partial^{\alpha}\widehat{\varphi}\left(\xi\right)\right| & \leq G_{1}\cdot\left(1+\left|\xi\right|\right)^{-L},\\
\left|\partial^{\alpha}\widehat{\psi}\left(\xi\right)\right| & \leq G_{2}\cdot\left(1+\left|\xi\right|\right)^{-L_{1}}\cdot\min\left\{ 1,\left|\xi\right|^{L_{2}}\right\}
\end{align*}
for all $\alpha\in\mathbb{N}_{0}^{d}$ with $\left|\alpha\right|\leq N_{0}:=\left\lceil \mu_{0}+\frac{d+\varepsilon}{p_{0}}\right\rceil $,
all $\xi\in\mathbb{R}^{d}$, suitable $G_{1},G_{2}>0$ and certain
$L,L_{1}\geq2d+1+2\varepsilon$ and $L_{2}\geq0$ which furthermore
satisfy
\[
L>s_{1}+\kappa+d+1+\varepsilon,\qquad L_{1}>s_{1}+\kappa+d+1+\varepsilon,\qquad\text{ and }\qquad L_{2}>\vartheta_{0}d-s_{0}
\]
for
\[
\vartheta_{0}:=\begin{cases}
0, & \text{if }p_{0}=1\\
\frac{1}{p_{0}}-1, & \text{if }p_{0}\in\left(0,1\right),
\end{cases}\qquad\text{ and }\qquad\kappa:=\begin{cases}
\left\lceil \mu_{0}+d+\varepsilon\right\rceil , & \text{if }p_{0}=1,\\
d+\mu_{0}+\left\lceil \mu_{0}+\frac{d+\varepsilon}{p_{0}}\right\rceil , & \text{if }p_{0}\in\left(0,1\right).
\end{cases}
\]
\end{enumerate}
Then there is some $\delta_{0}=\delta_{0}\left(d,p_{0},q_{0},\varepsilon,\mu_{0},s_{0},s_{1},\varphi,\psi\right)>0$
such that for each $0<\delta\leq\delta_{0}$, the family
\[
\Gamma^{\left(\delta\right)}=\left(2^{j\frac{d}{2}}\cdot\gamma_{j}\left(2^{j}\bullet-\delta k\right)\right)_{j\in\mathbb{N}_{0},k\in\mathbb{Z}^{d}},\qquad\text{ with }\qquad\gamma_{0}:=\varphi\quad\text{ and }\quad\gamma_{j}:=\psi\text{ for }j\in\mathbb{N},
\]
forms an atomic decomposition for $\mathcal{B}_{s,\mu}^{p,q}\left(\mathbb{R}^{d}\right)$,
for arbitrary $p,q\in\left(0,\infty\right]$ and $s,\mu\in\mathbb{R}$ satisfying
$p\geq p_{0}$, $q\geq q_{0}$ as well as $s_{0}\leq s\leq s_{1}$
and $\left|\mu\right|\leq\mu_{0}$.
Precisely, this means the following: With the space $\mathscr{C}_{p,q,s,\mu}\leq\mathbb{C}^{\mathbb{N}_{0}\times\mathbb{Z}^{d}}$
as in Proposition \ref{prop:BesovBanachFrames}, the following are
true:
\begin{enumerate}
\item The \textbf{synthesis map}
\[
S^{\left(\delta\right)}:\mathscr{C}_{p,q,s,\mu}\to\mathcal{B}_{s,\mu}^{p,q}\left(\smash{\mathbb{R}^{d}}\right),\left(\smash{c_{k}^{\left(i\right)}}\right)_{i\in\mathbb{N}_{0},k\in\mathbb{Z}^{d}}\mapsto\sum_{i\in\mathbb{N}_{0}}\sum_{k\in\mathbb{Z}^{d}}\left[c_{k}^{\left(i\right)}\cdot2^{i\frac{d}{2}}\cdot\gamma_{i}\left(2^{i}\bullet-\delta k\right)\right]
\]
is well-defined and bounded for each $0<\delta\leq1$.
Convergence of the series has to be understood as described in the
remark after Theorem \ref{thm:AtomicDecomposition}.
\item For $0<\delta\leq\delta_{0}$, there is a bounded linear \textbf{coefficient
map}
\[
C^{\left(\delta\right)}:\mathcal{B}_{s,\mu}^{p,q}\left(\smash{\mathbb{R}^{d}}\right)\to\mathscr{C}_{p,q,s,\mu}
\]
satisfying $S^{\left(\delta\right)}\circ C^{\left(\delta\right)}=\operatorname{id}_{\mathcal{B}_{s,\mu}^{p,q}}$.
Furthermore, the action of $C^{\left(\delta\right)}$ on a given $f\in\mathcal{B}_{s,\mu}^{p,q}\left(\mathbb{R}^{d}\right)$
is independent of the precise choice of $p,q,s,\mu$.\qedhere
\end{enumerate}
\end{prop}
\begin{proof}
Define $\widetilde{L}:=L-\left(d+1+\varepsilon\right)$ and
$\widetilde{L_{1}}:=L_{1}-\left(d+1+\varepsilon\right)$.
Now, an application of Lemma \ref{lem:ConvolutionFactorization} (with
$\gamma=\psi$, $N=N_{0}\geq\left\lceil \frac{d+\varepsilon}{p_{0}}\right\rceil \geq\left\lceil d+\varepsilon\right\rceil \geqd+1$
and $\varrho\left(\xi\right):=G_{2}\cdot\left(1+\left|\xi\right|\right)^{-\widetilde{L_{1}}}\cdot\min\left\{ 1,\left|\xi\right|^{L_{2}}\right\} $,
where $\varrho\in L^{1}\left(\mathbb{R}^{d}\right)$ since $\widetilde{L_{1}}\geqd+\varepsilon$)
yields functions $\psi_{1},\psi_{2}\in L^{1}\left(\mathbb{R}^{d}\right)$
with the following properties:
\begin{enumerate}
\item We have $\psi=\psi_{1}\ast\psi_{2}$.
\item We have $\psi_{2}\in C^{1}\left(\mathbb{R}^{d}\right)$ with $H_{1}^{\left(M\right)}:=\left\Vert \psi_{2}\right\Vert _{M}+\left\Vert \nabla\psi_{2}\right\Vert _{M}<\infty$
for all $M\in\mathbb{N}_{0}$.
\item We have $\widehat{\psi_{1}},\widehat{\psi_{2}}\in C^{\infty}\left(\mathbb{R}^{d}\right)$,
where all partial derivatives of these functions are polynomially
bounded.
\item We have $\left\Vert \psi_{1}\right\Vert _{N_{0}}<\infty$ and $\left\Vert \psi\right\Vert _{N_{0}}<\infty$.
In particular, since $N_{0}\geq\mu_{0}+\frac{d+\varepsilon}{p_{0}}\geq\mu_{0}+d+\varepsilon$,
we have $\psi_{1}\in L_{\left(1+\left|\bullet\right|\right)^{\mu_{0}+d+\varepsilon}}^{\infty}\left(\mathbb{R}^{d}\right)\hookrightarrow L_{\left(1+\left|\bullet\right|\right)^{\mu_{0}}}^{1}\left(\mathbb{R}^{d}\right)\hookrightarrow L_{\left(1+\left|\bullet\right|\right)^{K}}^{1}\left(\mathbb{R}^{d}\right)$
for all $K=\left|\mu\right|\leq\mu_{0}$.
\item We have
\begin{equation}
\begin{split}\left|\partial^{\alpha}\widehat{\psi_{1}}\left(\xi\right)\right| & \leq2^{1+d+4N_{0}}\cdot N_{0}!\cdot\left(1+d\right)^{N_{0}}\cdot\varrho\left(\xi\right)\\
& \leq H_{2}\cdot\left(1+\left|\xi\right|\right)^{-\widetilde{L_{1}}}\cdot\min\left\{ 1,\left|\xi\right|^{L_{2}}\right\}
\end{split}
\label{eq:BesovAtomicWaveletConvolutionFactorEstimate}
\end{equation}
with $H_{2}:=G_{2}\cdot2^{1+d+4N_{0}}\cdot N_{0}!\cdot\left(1+d\right)^{N_{0}}$
for all $\xi\in\mathbb{R}^{d}$ and all $\alpha\in\mathbb{N}_{0}^{d}$
with $\left|\alpha\right|\leq N_{0}$.
\end{enumerate}
Likewise, another application of Lemma \ref{lem:ConvolutionFactorization}
(this time with $\gamma=\varphi$, $N=N_{0}\geqd+1$ and
$\varrho\left(\xi\right):=G_{1}\cdot\left(1+\left|\xi\right|\right)^{-\widetilde{L}}$,
where $\varrho\in L^{1}\left(\mathbb{R}^{d}\right)$, since $\widetilde{L}\geqd+\varepsilon$)
yields certain functions $\varphi_{1},\varphi_{2}\in L^{1}\left(\mathbb{R}^{d}\right)$
with the following properties:
\begin{enumerate}
\item We have $\varphi=\varphi_{1}\ast\varphi_{2}$.
\item We have $\varphi_{2}\in C^{1}\left(\mathbb{R}^{d}\right)$ with $H_{3}^{\left(M\right)}:=\left\Vert \varphi_{2}\right\Vert _{M}+\left\Vert \nabla\varphi_{2}\right\Vert _{M}<\infty$
for all $M\in\mathbb{N}_{0}$.
\item We have $\widehat{\varphi_{1}},\widehat{\varphi_{2}}\in C^{\infty}\left(\mathbb{R}^{d}\right)$,
where all partial derivatives of these functions are polynomially
bounded.
\item We have $\left\Vert \varphi_{1}\right\Vert _{N_{0}}<\infty$ and $\left\Vert \varphi\right\Vert _{N_{0}}<\infty$.
As for $\psi_{1}$, this implies $\varphi_{1}\in L_{\left(1+\left|\bullet\right|\right)^{\mu_{0}}}^{1}\left(\mathbb{R}^{d}\right)\hookrightarrow L_{\left(1+\left|\bullet\right|\right)^{K}}^{1}\left(\mathbb{R}^{d}\right)$
for all $K=\left|\mu\right|\leq\mu_{0}$.
\item We have
\begin{equation}
\begin{split}\left|\partial^{\alpha}\widehat{\varphi_{1}}\left(\xi\right)\right| & \leq2^{1+d+4N_{0}}\cdot N_{0}!\cdot\left(1+d\right)^{N_{0}}\cdot\varrho\left(\xi\right)\\
& \leq H_{4}\cdot\left(1+\left|\xi\right|\right)^{-\widetilde{L}}
\end{split}
\label{eq:BesovAtomicScalingConvolutionFactorEstimate}
\end{equation}
with $H_{4}:=G_{1}\cdot2^{1+d+4N_{0}}\cdot N_{0}!\cdot\left(1+d\right)^{N_{0}}$
for all $\xi\in\mathbb{R}^{d}$ and all $\alpha\in\mathbb{N}_{0}^{d}$
with $\left|\alpha\right|\leq N_{0}$.
\end{enumerate}
Now, set $\gamma_{0}:=\varphi$ and $\gamma_{0,\ell}:=\varphi_{\ell}$,
as well as $\gamma_{j}:=\psi$ and $\gamma_{j,\ell}:=\psi_{\ell}$
for $\ell\in\left\{ 1,2\right\} $ and $j\in\mathbb{N}$.
As a further preparation, set $\gamma_{1}^{\left(0\right)}:=\varphi$
and $\gamma_{2}^{\left(0\right)}:=\psi$, as well as $n_{0}:=1$ and
$n_{j}:=2$ for $j\in\mathbb{N}$, so that $\gamma_{j}=\gamma_{n_{j}}^{\left(0\right)}$
for all $j\in\mathbb{N}_{0}$. Then, in the notation of Lemma \ref{lem:GammaCoversOrbitAssumptionSimplified},
we have $Q^{\left(1\right)}=B_{2}\left(0\right)$ and $Q^{\left(2\right)}=B_{4}\left(0\right)\setminus\overline{B_{1/4}}\left(0\right)$,
cf.\@ the proof of Proposition \ref{prop:BesovBanachFrames}. Exactly
as in that proof, we see that all prerequisites of Lemma \ref{lem:GammaCoversOrbitAssumptionSimplified}
are satisfied, so that the family $\Gamma=\left(\gamma_{i}\right)_{i\in\mathbb{N}_{0}}$
satisfies Assumption \ref{assu:GammaCoversOrbit}, where we furthermore
have $\Omega_{2}^{\left(p,K\right)}\leq\Omega_{3}$ for all $K\leq\mu_{0}$
and all $p\geq p_{0}$, for a suitable constant $\Omega_{3}=\Omega_{3}\left(\mathscr{B},\varphi,\psi,p_{0},\mu_{0},d\right)>0$.
Observe that indeed $K=\left|\mu\right|\leq\mu_{0}$ in the cases
which are of interest to us.
\medskip{}
Now, let $p,q,s,\mu$ be as in the statement of the proposition. We
want to verify the prerequisites of Corollary \ref{cor:AtomicDecompositionSimplifiedCriteria}
for the choices which we just made. For most of these assumptions,
this is not hard:
\begin{enumerate}
\item All $\gamma_{i},\gamma_{i,1},\gamma_{i,2}$ are measurable functions,
as required.
\item Since $K=\left|\mu\right|\leq\mu_{0}$, we have $\gamma_{i,1}=\psi_{1}\in L_{\left(1+\left|\bullet\right|\right)^{K}}^{1}\left(\mathbb{R}^{d}\right)$
for any $i\in\mathbb{N}$ and also $\gamma_{0,1}=\varphi_{1}\in L_{\left(1+\left|\bullet\right|\right)^{K}}^{1}\left(\mathbb{R}^{d}\right)$,
as seen above.
\item We have $\gamma_{i,2}=\psi_{2}\in C^{1}\left(\mathbb{R}^{d}\right)$
for any $i\in\mathbb{N}$ and also $\gamma_{0,2}=\varphi_{2}\in C^{1}\left(\mathbb{R}^{d}\right)$,
as noted above.
\item With $K_{0}:=K+\frac{d}{\min\left\{ 1,p\right\} }+1\leq\mu_{0}+\frac{d}{p_{0}}+1=K_{00}$,
we have
\begin{equation}
\begin{split}\Omega_{4}^{\left(p,K\right)} & =\sup_{i\in I}\left\Vert \gamma_{i,2}\right\Vert _{K_{0}}+\sup_{i\in I}\left\Vert \nabla\gamma_{i,2}\right\Vert _{K_{0}}\\
& \leq\max\left\{ \left\Vert \varphi_{2}\right\Vert _{K_{00}},\left\Vert \psi_{2}\right\Vert _{K_{00}}\right\} +\max\left\{ \left\Vert \nabla\varphi_{2}\right\Vert _{K_{00}},\left\Vert \nabla\psi_{2}\right\Vert _{K_{00}}\right\} \\
& \leq H_{1}^{\left(\left\lceil K_{00}\right\rceil \right)}+H_{3}^{\left(\left\lceil K_{00}\right\rceil \right)}=:H_{5}<\infty.
\end{split}
\label{eq:BesovAtomicDecompositionOmega4Estimate}
\end{equation}
\item We have $\left\Vert \gamma_{i}\right\Vert _{K_{0}}\leq\left\Vert \gamma_{i}\right\Vert _{K_{00}}<\infty$
for all $i\in\mathbb{N}_{0}$, since $\left\Vert \varphi\right\Vert _{K_{00}}<\infty$
and $\left\Vert \psi\right\Vert _{K_{00}}<\infty$ by assumption.
\item We have $\gamma_{i}=\psi=\psi_{1}\ast\psi_{2}=\gamma_{i,1}\ast\gamma_{i,2}$
for all $i\in\mathbb{N}$ and likewise $\gamma_{0}=\varphi=\varphi_{1}\ast\varphi_{2}=\gamma_{0,1}\ast\gamma_{0,2}$.
\item We have $\widehat{\gamma_{i,1}},\widehat{\gamma_{i,2}}\in C^{\infty}\left(\mathbb{R}^{d}\right)$
for all $i\in\mathbb{N}_{0}$, and all partial derivatives of these functions
are polynomially bounded.
\item As we showed above, the family $\Gamma=\left(\gamma_{i}\right)_{i\in\mathbb{N}_{0}}$
satisfies Assumption \ref{assu:GammaCoversOrbit}.
\end{enumerate}
Hence, the only prerequisite of Corollary \ref{cor:AtomicDecompositionSimplifiedCriteria}
which still needs to be verified is that
\[
K_{1}:=\sup_{i\in\mathbb{N}_{0}}\,\sum_{j\in\mathbb{N}_{0}}N_{i,j}<\infty\qquad\text{ and }\qquad K_{2}:=\sup_{j\in\mathbb{N}_{0}}\:\sum_{i\in\mathbb{N}_{0}}N_{i,j}<\infty,
\]
where
\[
N_{i,j}:=\left(\frac{w_{i}}{w_{j}}\cdot\left(\left|\det T_{j}\right|\big/\left|\det T_{i}\right|\right)^{\vartheta}\right)^{\tau}\cdot\left(1+\left\Vert T_{j}^{-1}T_{i}\right\Vert \right)^{\sigma}\cdot\left(\left|\det T_{i}\right|^{-1}\cdot\int_{Q_{i}}\max_{\left|\alpha\right|\leq N}\left|\left(\partial^{\alpha}\widehat{\gamma_{j,1}}\right)\left(S_{j}^{-1}\xi\right)\right|\operatorname{d}\xi\right)^{\tau},
\]
with
\begin{align*}
N & =\left\lceil K+\frac{d+\varepsilon}{\min\left\{ 1,p\right\} }\right\rceil ,\\
\tau & =\min\left\{ 1,p,q\right\} ,\\
\sigma & =\begin{cases}
\min\left\{ 1,q\right\} \cdot\left\lceil K+d+\varepsilon\right\rceil , & \text{if }p\in\left[1,\infty\right],\\
\min\left\{ p,q\right\} \cdot\left(\frac{d}{p}+K+\left\lceil K+\frac{d+\varepsilon}{p}\right\rceil \right), & \text{if }p\in\left(0,1\right),
\end{cases}\\
\vartheta & =\begin{cases}
0, & \text{if }p\in\left[1,\infty\right],\\
\frac{1}{p}-1, & \text{if }p\in\left(0,1\right).
\end{cases}
\end{align*}
To prove this, we begin with several auxiliary observations: First
of all, we observe $N\leq N_{0}$. Furthermore, we have $\vartheta\leq\vartheta_{0}$,
since $p_{0}=1$ implies $p\in\left[1,\infty\right]$. Finally, we
also have
\begin{equation}
\begin{split}\frac{\sigma}{\tau}-\varthetad & =\begin{cases}
\left\lceil K+d+\varepsilon\right\rceil , & \text{if }p\in\left[1,\infty\right],\\
\frac{d}{p}+K+\left\lceil K+\frac{d+\varepsilon}{p}\right\rceil -d\left(\frac{1}{p}-1\right), & \text{if }p\in\left(0,1\right)
\end{cases}\\
& \leq\begin{cases}
\left\lceil \mu_{0}+d+\varepsilon\right\rceil , & \text{if }p_{0}=1,\\
d+\mu_{0}+\left\lceil \mu_{0}+\frac{d+\varepsilon}{p_{0}}\right\rceil , & \text{if }p_{0}\in\left(0,1\right)
\end{cases}\\
& =\kappa,
\end{split}
\label{eq:BesovAtomicDecompositionKappaSignificance}
\end{equation}
where we used that $K=\left|\mu\right|\leq\mu_{0}$ and also that
$p_{0}=1$ entails $p\in\left[1,\infty\right]$.
We divide our estimates of $N_{i,j}$ into two main cases. The first
case is $i\in\mathbb{N}$. Here, we recall from the proof of Proposition \ref{prop:BesovBanachFrames}
(cf.\@ equation (\ref{eq:BesovAnnulusMeasureEstimate})) for $Q_{i}=\left\{ \xi\in\mathbb{R}^{d}\with2^{i-2}<\left|\xi\right|<2^{i+2}\right\} $
that $\lambda_{d}\left(Q_{i}\right)\leq2^{i\cdotd}\cdot4^{d}v_{d}$.
Since we also have $N\leq N_{0}$, we conclude
\begin{equation}
N_{i,j}\leq\left[2^{s\left(i-j\right)+\varthetad\left(j-i\right)}\right]^{\tau}\cdot\left(1+2^{i-j}\right)^{\sigma}\cdot\left(4^{d}v_{d}\cdot\sup_{2^{i-2}<\left|\xi\right|<2^{i+2}}\;\max_{\left|\alpha\right|\leq N_{0}}\left|\left(\partial^{\alpha}\widehat{\gamma_{j,1}}\right)\left(\xi/2^{j}\right)\right|\right)^{\tau}.\label{eq:BesovAtomicDecompositionMainTerm}
\end{equation}
To further estimate this term, we consider two subcases:
\textbf{Case 1}: We have $j\in\mathbb{N}$. Here, we again consider two subcases:
\begin{enumerate}
\item We have $i\leq j$. For $2^{i-2}<\left|\xi\right|<2^{i+2}$, equation
(\ref{eq:BesovAtomicWaveletConvolutionFactorEstimate}) implies because
of $\widetilde{L_{1}}\geq0$ and $L_{2}\geq0$ that
\begin{align*}
\max_{\left|\alpha\right|\leq N_{0}}\left|\left(\partial^{\alpha}\widehat{\gamma_{j,1}}\right)\left(\xi/2^{j}\right)\right|=\max_{\left|\alpha\right|\leq N_{0}}\left|\left(\partial^{\alpha}\widehat{\psi_{1}}\right)\left(\xi/2^{j}\right)\right| & \leq H_{2}\cdot\min\left\{ 1,\left|\xi/2^{j}\right|^{L_{2}}\right\} \\
& \leq H_{2}\cdot4^{L_{2}}\cdot2^{L_{2}\left(i-j\right)}\\
& =4^{L_{2}}H_{2}\cdot2^{-L_{2}\left|i-j\right|}.
\end{align*}
In combination with equation (\ref{eq:BesovAtomicDecompositionMainTerm}),
we get
\[
N_{i,j}\leq2^{\sigma}\cdot\left(4^{d+L_{2}}v_{d}\cdot H_{2}\right)^{\tau}\cdot2^{-\tau\left|i-j\right|\left(L_{2}+s-\varthetad\right)}.
\]
\item We have $j\leq i$. Here, equation (\ref{eq:BesovAtomicWaveletConvolutionFactorEstimate})
implies for $2^{i-2}<\left|\xi\right|<2^{i+2}$ that
\begin{align*}
\max_{\left|\alpha\right|\leq N_{0}}\left|\left(\partial^{\alpha}\widehat{\gamma_{j,1}}\right)\left(\xi/2^{j}\right)\right|=\max_{\left|\alpha\right|\leq N_{0}}\left|\left(\partial^{\alpha}\widehat{\psi_{1}}\right)\left(\xi/2^{j}\right)\right| & \leq H_{2}\cdot\left(1+\left|\xi/2^{j}\right|\right)^{-\widetilde{L_{1}}}\\
& \leq H_{2}\cdot\left(2^{i-j}/4\right)^{-\widetilde{L_{1}}}\\
& =4^{\widetilde{L_{1}}}H_{2}\cdot2^{-\widetilde{L_{1}}\left|i-j\right|}.
\end{align*}
In combination with equation (\ref{eq:BesovAtomicDecompositionMainTerm}),
we get
\[
N_{i,j}\leq2^{\sigma}\cdot\left(4^{d+\widetilde{L_{1}}}v_{d}\cdot H_{2}\right)^{\tau}\cdot2^{-\tau\left|i-j\right|\left(-\frac{\sigma}{\tau}-s+\varthetad+\widetilde{L_{1}}\right)}.
\]
\end{enumerate}
\textbf{Case 2}: We have $j=0$. Here, equation (\ref{eq:BesovAtomicScalingConvolutionFactorEstimate})
shows for $2^{i-2}<\left|\xi\right|<2^{i+2}$ that
\begin{align*}
\max_{\left|\alpha\right|\leq N_{0}}\left|\left(\partial^{\alpha}\widehat{\gamma_{j,1}}\right)\left(\xi/2^{j}\right)\right|=\max_{\left|\alpha\right|\leq N_{0}}\left|\partial^{\alpha}\widehat{\varphi_{1}}\left(\xi\right)\right| & \leq H_{4}\cdot\left(1+\left|\xi\right|\right)^{-\widetilde{L}}\\
& \leq H_{4}\cdot\left(2^{i}/4\right)^{-\widetilde{L}}\\
& =4^{\widetilde{L}}H_{4}\cdot2^{-i\widetilde{L}}.
\end{align*}
In combination with equation (\ref{eq:BesovAtomicDecompositionMainTerm}),
this implies
\[
N_{i,j}\leq2^{\sigma}\cdot\left(4^{d+\widetilde{L}}v_{d}\cdot H_{4}\right)^{\tau}\cdot2^{-\tau i\left(\widetilde{L}-s+\varthetad-\frac{\sigma}{\tau}\right)}=2^{\sigma}\cdot\left(4^{d+\widetilde{L}}v_{d}\cdot H_{4}\right)^{\tau}\cdot2^{-\tau\left|i-j\right|\left(\widetilde{L}-s+\varthetad-\frac{\sigma}{\tau}\right)}.
\]
These are the desired estimates for the case $i\in\mathbb{N}$.
If otherwise $i=0$, so that $Q_{i}=T_{0}Q_{0}'+b_{0}=Q_{0}'=B_{2}\left(0\right)$,
estimate (\ref{eq:BesovAtomicDecompositionMainTerm}) takes the modified
form
\begin{equation}
\begin{split}N_{i,j} & \leq2^{\tau j\left(\varthetad-s\right)}\cdot\left(1+2^{-j}\right)^{\sigma}\cdot\left(\int_{B_{2}\left(0\right)}\max_{\left|\alpha\right|\leq N}\left|\left(\partial^{\alpha}\widehat{\gamma_{j,1}}\right)\left(S_{j}^{-1}\xi\right)\right|\operatorname{d}\xi\right)^{\tau}\\
& \leq2^{\sigma}\cdot\left[\lambda_{d}\left(B_{2}\left(0\right)\right)\right]^{\tau}\cdot2^{\tau j\left(\varthetad-s\right)}\cdot\sup_{\left|\xi\right|<2}\,\left(\max_{\left|\alpha\right|\leq N_{0}}\left|\left(\partial^{\alpha}\widehat{\gamma_{j,1}}\right)\left(\xi/2^{j}\right)\right|\right)^{\tau}.
\end{split}
\label{eq:BesovAtomicDecompositionModifiedMainTerm}
\end{equation}
To further estimate this quantity, we again distinguish two cases:
\textbf{Case 1}: We have $j\in\mathbb{N}$. In this case, equation (\ref{eq:BesovAtomicWaveletConvolutionFactorEstimate})
shows for $\left|\xi\right|<2$ that
\begin{align*}
\max_{\left|\alpha\right|\leq N_{0}}\left|\left(\partial^{\alpha}\widehat{\gamma_{j,1}}\right)\left(\xi/2^{j}\right)\right|=\max_{\left|\alpha\right|\leq N_{0}}\left|\left(\partial^{\alpha}\widehat{\psi_{1}}\right)\left(\xi/2^{j}\right)\right| & \leq H_{2}\cdot\min\left\{ 1,\left|\xi/2^{j}\right|^{L_{2}}\right\} \\
& \leq2^{L_{2}}H_{2}\cdot2^{-jL_{2}}.
\end{align*}
In combination with equation (\ref{eq:BesovAtomicDecompositionModifiedMainTerm}),
we get
\begin{align*}
N_{i,j} & \leq2^{\sigma}\cdot\left[2^{L_{2}+d}v_{d}\cdot H_{2}\right]^{\tau}\cdot2^{\tau j\left(\varthetad-s-L_{2}\right)}\\
& =2^{\sigma}\cdot\left[2^{L_{2}+d}v_{d}\cdot H_{2}\right]^{\tau}\cdot2^{-\tau\left|i-j\right|\left(L_{2}+s-\varthetad\right)}.
\end{align*}
\textbf{Case 2}: We have $j=0$. In this case, equation (\ref{eq:BesovAtomicScalingConvolutionFactorEstimate})
shows for $\left|\xi\right|<2$ that
\[
\max_{\left|\alpha\right|\leq N_{0}}\left|\left(\partial^{\alpha}\widehat{\gamma_{j,1}}\right)\left(\xi/2^{j}\right)\right|=\max_{\left|\alpha\right|\leq N_{0}}\left|\left(\partial^{\alpha}\widehat{\varphi_{1}}\right)\left(\xi\right)\right|\leq H_{4},
\]
since $\widetilde{L}\geq0$. In combination with equation (\ref{eq:BesovAtomicDecompositionModifiedMainTerm}),
this yields
\[
N_{i,j}\leq2^{\sigma}\cdot\left[2^{d}v_{d}\cdot H_{4}\right]^{\tau}=2^{\sigma}\cdot\left[2^{d}v_{d}\cdot H_{4}\right]^{\tau}\cdot2^{-\tau\left|i-j\right|\zeta}
\]
for arbitrary $\zeta\in\mathbb{R}$, since $\left|i-j\right|=0$.
\medskip{}
All in all, our considerations show that there is some constant $H_{6}$
which is independent of $p,q,s,\mu$, such that
\[
N_{i,j}\leq2^{\sigma}\cdot H_{6}^{\tau}\cdot2^{-\tau\left|i-j\right|\lambda}\quad\text{ where }\quad\lambda:=\min\left\{ 1,\,L_{2}+s-\varthetad,\,\widetilde{L}-s+\varthetad-\frac{\sigma}{\tau},\,\widetilde{L_{1}}-s+\varthetad-\frac{\sigma}{\tau}\right\} .
\]
But in view of equation (\ref{eq:BesovAtomicDecompositionKappaSignificance})
and since $s_{0}\leq s\leq s_{1}$, as well as $\vartheta\leq\vartheta_{0}$,
we have
\[
\lambda\geq\min\left\{ 1,\,L_{2}+s_{0}-\vartheta_{0}d,\,\widetilde{L}-s_{1}-\kappa,\,\widetilde{L_{1}}-s_{1}-\kappa\right\} =:\lambda_{0}>0,
\]
by our assumptions regarding $L,L_{1},L_{2}$. Note that $\lambda_{0}$
is independent of $p,q,s,\mu$.
All in all, we thus get, for $K_{1},K_{2}$ as in Corollary \ref{cor:AtomicDecompositionSimplifiedCriteria},
\begin{align*}
K_{1}^{1/\tau}=\sup_{i\in\mathbb{N}_{0}}\,\left(\sum_{j\in\mathbb{N}_{0}}N_{i,j}\right)^{1/\tau} & \leq2^{\sigma/\tau}\cdot H_{6}\cdot\sup_{i\in\mathbb{N}_{0}}\,\left(\sum_{j\in\mathbb{N}_{0}}2^{-\tau\lambda\left|i-j\right|}\right)^{1/\tau}\\
\left({\scriptstyle \text{since }\ell^{\tau_{0}}\hookrightarrow\ell^{\tau}\text{ is norm-decreasing for }\tau_{0}:=\min\left\{ 1,p_{0},q_{0}\right\} \leq\tau}\right) & \leq2^{\sigma/\tau}\cdot H_{6}\cdot\sup_{i\in\mathbb{N}_{0}}\,\left(\sum_{j\in\mathbb{N}_{0}}2^{-\tau_{0}\lambda\left|i-j\right|}\right)^{1/\tau_{0}}\\
\left({\scriptstyle \text{since }\lambda\geq\lambda_{0}\text{ and with }\ell=i-j}\right) & \leq2^{\sigma/\tau}\cdot H_{6}\cdot\left(\sum_{\ell\in\mathbb{Z}}2^{-\tau_{0}\lambda_{0}\left|\ell\right|}\right)^{1/\tau_{0}}\\
& \leq2^{\frac{\sigma}{\tau}+\frac{1}{\tau_{0}}}\cdot H_{6}\cdot\left(\sum_{\ell=0}^{\infty}2^{-\tau_{0}\lambda_{0}\ell}\right)^{1/\tau_{0}}\\
\left({\scriptstyle \text{since }\frac{\sigma}{\tau}\leq\kappa+\varthetad\leq\kappa+\vartheta_{0}d\text{ by eq. }\eqref{eq:BesovAtomicDecompositionKappaSignificance}}\right) & \leq2^{\kappa+\vartheta_{0}d+\frac{1}{\tau_{0}}}\cdot H_{6}\cdot\left(\frac{1}{1-2^{-\tau_{0}\lambda_{0}}}\right)^{1/\tau_{0}}=:H_{7},
\end{align*}
where $H_{7}$ is independent of $p,q,s,\mu$. Precisely the same
estimate also yields $K_{2}^{1/\tau}\leq H_{7}$.
All in all, we see that Corollary \ref{cor:AtomicDecompositionSimplifiedCriteria}
is applicable, so that the operator $\overrightarrow{C}$ from Assumption
\ref{assu:AtomicDecompositionAssumption} satisfies $\vertiii{\smash{\overrightarrow{C}}}^{\max\left\{ 1,\frac{1}{p}\right\} }\leq\Omega\cdot\left(K_{1}^{1/\tau}+K_{2}^{1/\tau}\right)\leq2\Omega H_{7}$
for $\Omega=\Omega_{0}^{K}\Omega_{1}\cdot\left(4\cdotd\right)^{1+2\left\lceil K+\frac{d+\varepsilon}{\min\left\{ 1,p\right\} }\right\rceil }\cdot\left(\frac{s_{d}}{\varepsilon}\right)^{1/\min\left\{ 1,p\right\} }\cdot\max_{\left|\alpha\right|\leq N}C^{\left(\alpha\right)}$,
where the constants $C^{\left(\alpha\right)}$ are as in Assumption
\ref{assu:RegularPartitionOfUnity}. Since $N\leq N_{0}$, $K=\left|\mu\right|\leq\mu_{0}$
and $\frac{1}{\min\left\{ 1,p\right\} }\leq\frac{1}{p_{0}}$, as well
as $\Omega_{0}=1$ and $\Omega_{1}=2^{\left|\mu\right|}\leq2^{\mu_{0}}$,
it is not hard to see $\Omega\leq H_{8}$, where $H_{8}$ is independent
of $p,q,s,\mu$.
Corollary \ref{cor:AtomicDecompositionSimplifiedCriteria} ensures
that Theorem \ref{thm:AtomicDecomposition} is applicable, i.e., the
family $\Gamma^{\left(\delta\right)}$ from the statement of the current
proposition is indeed an atomic decomposition for $\mathcal{B}_{s,\mu}^{p,q}\left(\mathbb{R}^{d}\right)=\DecompSp{\mathscr{B}}p{\ell_{\left(2^{js}\right)_{j\in\mathbb{N}_{0}}}^{q}}{v^{\left(\mu\right)}}$
as soon as $0<\delta\leq\min\left\{ 1,\delta_{00}\right\} $, where
$\delta_{00}$ is defined by
\[
\delta_{00}^{-1}\!:=\!\begin{cases}
\!\frac{2s_{d}}{\sqrt{d}}\cdot\left(2^{17}\!\cdot\!d^{2}\!\cdot\!\left(K\!+\!2\!+\!d\right)\right)^{\!K+d+3}\!\!\!\cdot\left(1\!+\!R_{\mathscr{B}}\right)^{d+1}\cdot\Omega_{0}^{4K}\Omega_{1}^{4}\Omega_{2}^{\left(p,K\right)}\Omega_{4}^{\left(p,K\right)}\cdot\vertiii{\smash{\overrightarrow{C}}}\,, & \text{if }p\geq1,\\
\frac{\left(2^{14}/d^{\frac{3}{2}}\right)^{\!\frac{d}{p}}}{2^{45}\cdotd^{17}}\!\cdot\!\left(\frac{s_{d}}{p}\right)^{\!\frac{1}{p}}\left(2^{68}\!\cdot\!d^{14}\!\cdot\!\left(K\!+\!1\!+\!\frac{d+1}{p}\right)^{3}\right)^{\!K+2+\frac{d+1}{p}}\!\!\!\cdot\!\left(1\!+\!R_{\mathscr{B}}\right)^{1+\frac{3d}{p}}\!\cdot\!\Omega_{0}^{16K}\Omega_{1}^{16}\Omega_{2}^{\left(p,K\right)}\Omega_{4}^{\left(p,K\right)}\cdot\vertiii{\smash{\overrightarrow{C}}}^{\frac{1}{p}}, & \text{if }p<1.
\end{cases}
\]
The verification that the discrete sequence space from Theorem \ref{thm:AtomicDecomposition}
coincides with $\mathscr{C}_{p,q,s,\mu}$ is exactly as in the proof
of Proposition \ref{prop:BesovBanachFrames}. Hence, to complete the
proof, we only have to verify $\delta_{00}^{-1}\leq\delta_{0}^{-1}$,
where $\delta_{0}>0$ is independent of $p,q,s,\mu$.
But above, we showed $\Omega_{2}^{\left(p,K\right)}\leq\Omega_{3}$,
with $\Omega_{3}$ independent of $p,q,s,\mu$, since $K=\left|\mu\right|\leq\mu_{0}$
and $p\geq p_{0}$. Furthermore, equation (\ref{eq:BesovAtomicDecompositionOmega4Estimate})
shows $\Omega_{4}^{\left(p,K\right)}\leq H_{5}$ for $K=\left|\mu\right|\leq\mu_{0}$
and with $H_{5}$ independent of $p,q,s,\mu$. Using these estimates,
the estimate for $\vphantom{\overrightarrow{C}}\vertiii{\smash{\overrightarrow{C}}}$
from above, and the straightforward inequalities $0\leq\frac{1}{p}\leq\frac{1}{p_{0}}$
and $K=\left|\mu\right|\leq\mu_{0}$, as well as $\Omega_{0}=1$ and
$\Omega_{2}=2^{\left|\mu\right|}\leq2^{\mu_{0}}$, we see that indeed
$\delta_{00}^{-1}\leq\delta_{0}^{-1}$ for some $\delta_{0}>0$ independent
of $p,q,s,\mu$.
\end{proof}
\begin{rem}
\label{rem:BesovRemarks}We close this section by showing that our
results indeed imply the existence of compactly supported Banach frames
and atomic decompositions for Besov spaces. Finally, we compare our
results with the literature.
\begin{itemize}[leftmargin=0.4cm]
\item The conditions in Propositions \ref{prop:BesovBanachFrames} and
\ref{prop:BesovAtomicDecomposition} are still not completely straightforward
to verify. Thus, let $k,N\in\mathbb{N}$ and $L_{1},L_{2}\geq0$ be arbitrary.
We will show how one can construct a compactly supported function
$\psi\in C_{c}^{k}\left(\mathbb{R}^{d}\right)$ satisfying $\widehat{\psi}\left(\xi\right)\neq0$
for $\xi\in\overline{B_{4}}\left(0\right)\setminus B_{1/4}\left(0\right)$
as well as
\[
\left|\partial^{\alpha}\widehat{\psi}\left(\xi\right)\right|\lesssim\left(1+\left|\xi\right|\right)^{-L_{1}}\cdot\min\left\{ 1,\left|\xi\right|^{L_{2}}\right\} \qquad\forall\alpha\in\mathbb{N}_{0}^{d}\text{ with }\left|\alpha\right|\leq N.
\]
To this end, let $\ell:=\max\left\{ k,\left\lceil L_{1}\right\rceil \right\} $
and $L_{3}:=\max\left\{ 0,\left\lceil \left(L_{2}-N-1\right)/2\right\rceil \right\} $
and choose $\psi_{0}\in C_{c}^{\ell+2\left(L_{3}+N+1\right)}\left(\mathbb{R}^{d}\right)$
with $\psi_{0}\geq0$ and $\psi_{0}\not\equiv0$. This implies $\widehat{\psi_{0}}\left(0\right)=\left\Vert \psi_{0}\right\Vert _{L^{1}}>0$.
By continuity of $\widehat{\psi_{0}}$, there is thus some $c_{0}>0$
and some $\varepsilon>0$ satisfying $\left|\smash{\widehat{\psi_{0}}}\left(\xi\right)\right|\geq c_{0}$
for $\left|\xi\right|\leq\varepsilon$.
Define $\psi_{1}:=\psi_{0}\circ\frac{4}{\varepsilon}\operatorname{id}$ and
note $\vphantom{\widehat{\psi_{1}}}\left|\smash{\widehat{\psi_{1}}}\left(\xi\right)\right|=\left(\varepsilon/4\right)^{d}\cdot\left|\smash{\widehat{\psi_{0}}}\left(\frac{\varepsilon}{4}\xi\right)\right|\geq c_{0}\cdot\left(\varepsilon/4\right)^{d}=:c_{1}$
as long as $\left|\xi\right|\leq4$. Next, set
\[
\psi:=\Delta^{L_{3}+N+1}\psi_{1}\in C_{c}^{\ell}\left(\smash{\mathbb{R}^{d}}\right)\subset C_{c}^{k}\left(\smash{\mathbb{R}^{d}}\right),
\]
where $\Delta=\sum_{j=1}^{d}\frac{\partial^{2}}{\partial x_{j}^{2}}$
denotes the Laplace operator. An easy calculation using partial integration
shows $\mathcal{F}\left[\Delta g\right]\left(\xi\right)=-4\pi^{2}\cdot\left|\xi\right|^{2}\cdot\widehat{g}\left(\xi\right)$
for $g\in C_{c}^{2}\left(\mathbb{R}^{d}\right)$, so that we get
\begin{equation}
\widehat{\psi}\left(\xi\right)=\left(-4\pi^{2}\right)^{L_{3}+N+1}\cdot\left|\xi\right|^{2\left(L_{3}+N+1\right)}\cdot\widehat{\psi_{1}}\left(\xi\right)\qquad\forall\xi\in\mathbb{R}^{d}.\label{eq:BesovExplicitConstructionDerivative}
\end{equation}
In particular, $\widehat{\psi}\left(\xi\right)\in o\left(\smash{\left|\xi\right|^{2\left(L_{3}+N\right)+1}}\right)$
as $\xi\to0$. Furthermore, since $\psi\in C_{c}\left(\mathbb{R}^{d}\right)$,
we have $\widehat{\psi}\in C^{\infty}\left(\mathbb{R}^{d}\right)$,
so that Lemma \ref{lem:HighOrderVanishingYieldVanishingDerivatives}
shows $\partial^{\alpha}\widehat{\psi}\left(\xi\right)\in\vphantom{\left|\xi\right|^{2L_{3}}}o\left(\smash{\left|\xi\right|^{2\left(L_{3}+N\right)+1-\left|\alpha\right|}}\right)\subset o\left(\smash{\left|\xi\right|^{2L_{3}+N+1}}\right)\subset o\left(\smash{\left|\xi\right|^{L_{2}}}\right)$
as $\xi\to0$, for $\left|\alpha\right|\leq N$. By continuity of
$\partial^{\alpha}\widehat{\psi}$, this implies that there is a constant
$C>0$ satisfying
\[
\left|\partial^{\alpha}\widehat{\psi}\left(\xi\right)\right|\leq C\cdot\left|\xi\right|^{L_{2}}\leq2^{L_{1}}C\cdot\left(1+\left|\xi\right|\right)^{-L_{1}}\cdot\min\left\{ 1,\left|\xi\right|^{L_{2}}\right\}
\]
for all $\left|\xi\right|\leq1$ and all $\alpha\in\mathbb{N}_{0}^{d}$
with $\left|\alpha\right|\leq N$.
Furthermore, since $\psi\in C_{c}^{\ell}\left(\mathbb{R}^{d}\right)$
and thus also $\psi^{\left(\alpha\right)}:=\left[x\mapsto\left(-2\pi ix\right)^{\alpha}\cdot\psi\left(x\right)\right]\in C_{c}^{\ell}\left(\mathbb{R}^{d}\right)$,
Lemma \ref{lem:PointwiseFourierDecayEstimate} and elementary properties
of the Fourier transform imply
\[
\left|\partial^{\alpha}\widehat{\psi}\left(\xi\right)\right|=\left|\left(\mathcal{F}^{-1}\smash{\psi^{\left(\alpha\right)}}\right)\left(-\xi\right)\right|\lesssim\left(1+\left|\xi\right|\right)^{-\ell}\leq\left(1+\left|\xi\right|\right)^{-L_{1}}=\left(1+\left|\xi\right|\right)^{-L_{1}}\cdot\min\left\{ 1,\left|\xi\right|^{L_{2}}\right\} \qquad\text{ for }\left|\xi\right|\geq1
\]
for arbitrary $\alpha\in\mathbb{N}_{0}^{d}$. Hence, $\partial^{\alpha}\widehat{\psi}$
satisfies the desired decay properties.
Finally, note that eq.\@ (\ref{eq:BesovExplicitConstructionDerivative})
also yields $\left|\smash{\widehat{\psi}}\left(\xi\right)\right|=\left(4\pi^{2}\right)^{L_{3}+N+1}\cdot\left|\xi\right|^{2\left(L_{3}+N+1\right)}\cdot\left|\smash{\widehat{\psi_{1}}}\left(\xi\right)\right|\geq c_{1}\cdot\left(\pi^{2}/4\right)^{L_{3}+N+1}\geq c_{1}$
for all $\xi\in\overline{B_{4}}\left(0\right)\setminus B_{1/4}\left(0\right)$.
We have thus constructed $\psi$ as desired. Similarly, but easier,
one can construct $\varphi\in C_{c}^{k}\left(\mathbb{R}^{d}\right)$
satisfying $\widehat{\varphi}\left(\xi\right)\neq0$ for $\xi\in\overline{B_{2}}\left(0\right)$
and $\left|\partial^{\alpha}\widehat{\varphi}\left(\xi\right)\right|\lesssim\left(1+\left|\xi\right|\right)^{-L}$
for all $\alpha\in\mathbb{N}_{0}^{d}$ with $\left|\alpha\right|\leq N$.
It is then straightforward to check that $\varphi,\psi$ satisfy all
assumptions of Propositions \ref{prop:BesovBanachFrames} and \ref{prop:BesovAtomicDecomposition}
(for proper choices of $N,L,L_{1},L_{2}$). Hence, our general theory
indeed yields \emph{compactly supported} wavelet systems that form
atomic decompositions and Banach frames for inhomogeneous Besov spaces.
\item We observe that the assumptions of Propositions \ref{prop:BesovBanachFrames}
and \ref{prop:BesovAtomicDecomposition} are \emph{structurally} very
similar, but the precise values of $L,L_{1},L_{2}$ differ greatly.
Indeed, in order to get a \emph{Banach frame} for $\mathcal{B}_{s}^{p,q}\left(\mathbb{R}^{d}\right)$
using Proposition \ref{prop:BesovBanachFrames}, the mother wavelet
$\psi$ has to have at least $L_{2}>s$ vanishing moments, which increases
with the \emph{smoothness parameter} $s\in\mathbb{R}$. In contrast, in order
to obtain an \emph{atomic decomposition} of $\mathcal{B}_{s}^{p,q}\left(\mathbb{R}^{d}\right)$
using Proposition \ref{prop:BesovAtomicDecomposition}, the mother
wavelet $\psi$ only has to have $L_{2}>\vartheta_{0}d-s$
vanishing moments, where $\vartheta_{0}=\left(p^{-1}-1\right)_{+}$.
In particular, once the smoothness parameter $s$ satisfies $s>d\left(p^{-1}-1\right)_{+}$,
one can choose $L_{2}=0$, so that it is possible for $\psi$ to have
\emph{no vanishing moments at all}, i.e., $\widehat{\psi}\left(0\right)\neq0$
is allowed.
In this case, one can even choose $\varphi=\psi$ to show that the
system $\left(2^{j\frac{d}{2}}\cdot\varphi\left(2^{j}\bullet-\delta k\right)\right)_{j\in\mathbb{N}_{0},k\in\mathbb{Z}^{d}}$
yields an atomic decompositions of $\mathcal{B}_{s}^{p,q}\left(\mathbb{R}^{d}\right)$.
A peculiar property of this system is that it \emph{does not even
form a frame for $L^{2}\left(\mathbb{R}^{d}\right)$}, due to the
missing vanishing moments.
\item Wavelet characterizations of inhomogeneous Besov spaces have already
been considered by many other authors: In \cite[equations (10.1) and (10.2)]{MeyerWaveletsAndOperators},
as well as in \cite[Theorem 3.5(i)]{TriebelTheoryOfFunctionSpaces3},
it is shown that certain wavelet \emph{orthonormal bases} yield atomic
decompositions and Banach frames for the Besov spaces $\mathcal{B}_{s}^{p,q}\left(\mathbb{R}^{d}\right)$.
We remark that of the two mentioned books, only Triebel's book \cite{TriebelTheoryOfFunctionSpaces3}
covers the whole range $p,q\in\left(0,\infty\right]$, while Meyer\cite{MeyerWaveletsAndOperators}
only considers the case $p,q\in\left[1,\infty\right]$.
As explained in \cite[Theorem 1.61(ii)]{TriebelTheoryOfFunctionSpaces3},
the wavelet bases considered by Triebel in \cite[Theorem 3.5]{TriebelTheoryOfFunctionSpaces3}
are \emph{compactly supported} and are $C^{k}$, with $k$ vanishing
moments, where it is assumed that
\[
k>\max\left\{ s,\frac{2d}{p}+\frac{d}{2}-s\right\} .
\]
Hence, Triebel needs a large amount of vanishing moments if $s$ is
large, but also if $-s$ is large. As observed in the previous point,
this is not needed for the theory developed in this paper, at least
if one only wants to have \emph{either} Banach frames or atomic decompositions.
But since Triebel uses wavelet orthonormal bases, he obtains atomic
decompositions and Banach frames \emph{simultaneously}, which explains
the dependence of $k$ on $s$ observed above.
In addition to orthonormal bases, Triebel also considers wavelet \emph{frames},
cf.\@ \cite[Sections 1.8 and 3.2]{TriebelTheoryOfFunctionSpaces3}.
But for these, Triebel restricts to the case $p=q$. Then, for $s>\sigma_{p}=d\left(p^{-1}-1\right)_{+}$,
he derives atomic decomposition results using certain compactly supported
wavelet frames (cf.\@ \cite[Theorem 1.69]{TriebelTheoryOfFunctionSpaces3}).
As seen above, this is the range in which Proposition \ref{prop:BesovAtomicDecomposition}
does not need any vanishing moments. Additional atomic decomposition
results are obtained in \cite[Theorem 1.71]{TriebelTheoryOfFunctionSpaces3},
but these use \emph{bandlimited} wavelets and require $p>1$ as well
as $s<0$.
In a different approach, Rauhut and Ullrich showed \cite{GeneralizedCoorbit2}
(based upon previous work by Ullrich\cite{UllrichContinuousCharacterizationsOfBesovTriebelLizorkin})
that the inhomogeneous Besov spaces $\mathcal{B}_{s}^{p,q}\left(\mathbb{R}^{d}\right)$
can be obtained as certain \emph{generalized coorbit spaces}. Using
the theory of these spaces (cf.\@ \cite{GeneralizedCoorbit1,GeneralizedCoorbit2}),
they then again show that suitable wavelet \emph{orthonormal bases}
yield Banach frames and atomic decompositions for the spaces $\mathcal{B}_{s}^{p,q}\left(\mathbb{R}^{d}\right)$,
cf.\@ \cite[Theorem 5.8 and Remark 5.9]{GeneralizedCoorbit2}. Their
assumptions on the scaling function $\varphi$ and the mother wavelet
$\psi$ are very similar to the ones imposed in this paper: $\psi$
needs to have a suitable number of vanishing moments, and $\varphi,\psi$
are required to have a suitable decay in space, as well as in Fourier
domain. Furthermore, the decay in Fourier domain also needs to hold
for certain derivatives of $\widehat{\varphi},\widehat{\psi}$, cf.\@
\cite[Definition 1.1]{GeneralizedCoorbit2}. We remark, however, that
in \cite[Theorem 5.8]{GeneralizedCoorbit2}, only the range $p,q\in\left[1,\infty\right]$
is considered.
Finally, Frazier and Jawerth\cite{FrazierJawerthDecompositionOfBesovSpaces,FrazierJawerthDiscreteTransform,FrazierJawerthThePhiTransform}
also obtained atomic decompositions for Besov spaces, cf.\@ \cite[Theorem 7.1]{FrazierJawerthDecompositionOfBesovSpaces}.
In contrast to our approach, Frazier and Jawerth use a sampling density
which is fixed \emph{a priori}. This, however, requires the mother
wavelet $\psi$ to be \emph{bandlimited} to $\left\{ \xi\in\mathbb{R}^{d}\,\middle|\,\left|\xi\right|\leq\pi\right\} $
(cf.\@ \cite[between eq. (1.8) and eq. (1.9)]{FrazierJawerthDecompositionOfBesovSpaces});
in particular, $\psi$ can \emph{not} be compactly supported. We remark
that Frazier and Jawerth assume $\psi$ to have $N$ vanishing moments
with $N\geq\max\left\{ -1,\,d\left(p^{-1}-1\right)_{+}-s\right\} $.
This is very similar to the vanishing moment condition which we impose
in Proposition \ref{prop:BesovAtomicDecomposition}, cf.\@ the preceding
point. Finally, we mention the so-called \textbf{\emph{generalized}}\textbf{
$\varphi$-transform} of Frazier and Jawerth (cf.\@ \cite[Section 4]{FrazierJawerthDiscreteTransform})
which yields results that are very similar to Propositions \ref{prop:BesovBanachFrames}
and \ref{prop:BesovAtomicDecomposition}, but for the case of (homogeneous)
Triebel-Lizorkin spaces instead of inhomogeneous Besov spaces, cf.\@
\cite[Theorem 4.5]{FrazierJawerthThePhiTransform} and \cite[Corollaries 4.5 and 4.3]{FrazierJawerthDiscreteTransform}.
For the case of \emph{inhomogeneous} Triebel-Lizorkin spaces, see
\cite[Section 12, page 132]{FrazierJawerthDiscreteTransform}.
In summary, we have seen that the description of (inhomogeneous) Besov
spaces through wavelet systems—in particular through wavelet orthonormal
bases—was very well developed prior to this paper. Nevertheless, it
seems that in the case of compactly supported wavelet \emph{frames}
(as opposed to orthonormal bases), our results slightly improve the
state of the art: In \cite{TriebelTheoryOfFunctionSpaces3}, comparable
results are only derived for $p=q$ and $s>d\left(p^{-1}-1\right)_{+}$
and in \cite{FrazierJawerthDecompositionOfBesovSpaces}, only bandlimited
wavelet systems are considered. Finally, in \cite{FrazierJawerthDiscreteTransform},
the authors allow compactly supported wavelet frames, but consider
Triebel-Lizorkin spaces instead of Besov spaces.
We close our comparison with the literature by comparing the advantages
and disadvantages of wavelet orthonormal bases compared to more general
wavelet systems. As noted in \cite[Example 5.6(a)]{GroechenigDescribingFunctions},
``\emph{both types of description are useful {[}...{]}: The orthogonal
bases, when a concise characterization of a function without redundancy
is important, but the form of the basic wavelet $g$ is not essential;
the non-orthogonal expansions and frames, when the basic function
$g$ is given by the problem and flexibility is required.}'' Indeed,
if one is willing to sample sufficiently densely, Propositions \ref{prop:BesovBanachFrames}
and \ref{prop:BesovAtomicDecomposition} allow a \emph{very wide variety}
of scaling functions $\varphi$ and mother wavelets $\psi$ to be
used. In contrast, to obtain an orthonormal wavelet basis, $\varphi$
and $\psi$ need to be selected \emph{very carefully}. However, using
such an orthonormal basis has several advantages\cite{TriebelTheoryOfFunctionSpaces3}
that frames lack:
\begin{itemize}
\item the sampling density is known and fixed a priori,
\item the synthesis coefficients are uniquely determined and equal to the
analysis coefficients,
\item the analysis map yields an isomorphism of $\mathcal{B}_{s}^{p,q}\left(\mathbb{R}^{d}\right)$
\textbf{onto} the associated sequence space $b_{s}^{p,q}$.
\end{itemize}
\item Finally, we remark that we discussed inhomogeneous Besov spaces in
the general framework presented here mainly to indicate that—\emph{and
how}—the framework can be applied in concrete cases. More novel and
interesting applications of the general theory, in particular to shearlets,
will be discussed in the companion paper \cite{StructuredBanachFrames2}.\qedhere
\end{itemize}
\end{rem}
|
1612.08675
|
\section{Introduction}\label{sec:intro}
Under propitious circumstances, quantum vacuum fluctuations
produce macroscopically observable consequences. Such is the case when a
quantum field, and hence its fluctuations, satisfy non-trivial boundary
conditions.
One of the most celebrated physical realizations of this is the Casimir
force between two neutral bodies having non-trivial EM response
functions (which, in some cases, behave as approximate realizations of
idealized boundary conditions).
This effect has been predicted and experimentally measured for several
different geometries~\cite{book_bordag,book_milonni,book_milton,lamoreaux2004casimir,milton2004casimir,reynaud2001quantum}.
Qualitatively different effects, also due to the vacuum fluctuations, may
arise when the bodies are set into motion or, more generally, when some
external agent renders the boundary condition(s) time-dependent. The
resulting effect may involve dissipation and, when the boundary conditions
experiment non-vanishing accelerations, \textit{real} photons can be
excited out of the quantum vacuum. This embodies the most frequently
considered version of the so called dynamical Casimir effect (DCE)
\cite{review_dyncas}, also known as `motion induced radiation'.
A more startling situation appears when a purely quantum, dissipative,
frictional force arises between bodies moving with {\em constant\/}
relative speed. Here, the effect is due to the quantum degrees of freedom,
living on the moving media, which are excited out of the vacuum, while the EM
field is nevertheless required as a mediator for those fluctuations. The
resulting effect, termed {\em Casimir friction} has been extensively
studied and some of the issues involved in its calculation have spurred
some debate~\cite{review_friction,volokitin_persson,pendry97,pendry_debate}.
We recall that Casimir friction predictions have been obtained mostly
for dielectric materials. In this paper, we study the same effect, but for two graphene sheets. We
argue that graphene has unusual properties which render its theoretical
study more interesting. Indeed, because of graphene's low dimensionality
and particular crystalline structure, its low-energy excitations behave as
massless Dirac fermions (with the Fermi velocity $v_F$ playing the role of
light's speed). This yields an unusual semi-metallic
behavior~\cite{rev_graphene}, as well as peculiar transport and optical
properties~\cite{opacgraphene,opacexpgraphene, farias2013}.
In natural units (which we adopt here) the mass dimensions of the response
function of graphene in momentum space can only be given by
the momentum itself. Indeed, the only other ingredients: $v_F$ and the
effective electric charge of the fermions, are dimensionless. And, when a sheet is
moving at a constant speed $v$, another dimensionless object, $v$ itself,
enters into the game (see below). Thus the non-trivial dependence of the
macroscopic, Casimir friction observables, will exhibit the remarkable
property of being a function of $v$ and $v_F$, the overall (trivial)
dimensions of the respective magnitude being determined purely by geometry:
size and distance between
sheets, like in the static Casimir effect between perfect mirrors.
A somewhat related but different effect, also termed `quantum friction', has
been studied for graphene in Ref. \cite{volokitin2011quantum}. Note, however,
that in that work the system consists of a single static graphene sheet
over an SiO 2 substrate. The frictional force acts, in this case, on graphene’s
charge carriers, which are assumed to have a constant drift velocity $v$
with respect to the substrate.
In our study below, we start from a consideration of the microscopic model
for two graphene sheets coupled to the EM field. Those microscopic degrees
of freedom correspond to Dirac fields in $2+1$ dimensions which, in a
functional integral formulation, are integrated out. That integration, plus
the free gauge field action, produces an in-out effective action for the
latter. Integrating the gauge field, we finally get an effective action for
the full system, the imaginary part of which accounts for the dissipative
effects in the system, a procedure we have followed in our previous
works~\cite{farias_friction,farias2016}.
We perform our calculations within a functional integral
formalism~\cite{fosco2007_moving_mirrors,fosco2011_sidewise}, and after
evaluating the probability of vacuum decay, we relate the imaginary part
of the in-out effective action to the frictional force on the
plates, and plot the latter as a function of the velocity $v$.
The structure of this paper is as follows: in Section~\ref{sec:themodel}, we
introduce the microscopic model considered in this article. Then we derive
an `effective action' for the EM field, namely, an Euclidean action which,
in our description, is a functional of $A_\mu$, the gauge field
corresponding to the vacuum EM field. In order to achieve that, we need to
find the form of the vacuum polarization tensor for moving graphene (as seen
from rest) assuming relativistic effects can be
neglected.
In Section~\ref{sec:effective}, we calculate the full effective action resulting
from the integration of the EM field. That effective action, when rotated
to Minkowski space, is applied to the calculation of the probability of
vacuum decay, as a function of the velocity of the sliding graphene sheet.
In Section~\ref{sec:potencia}, we relate the imaginary part of the in-out
effective action to the dissipated power, and thereby to the frictional
force on the moving plate.
Section~\ref{sec:conclu} contains our conclusions.
\section{The model}\label{sec:themodel}
We first introduce the Euclidean action ${\mathcal S}$, for the EM field
plus the two graphene sheets, one of them static, the other moving at a
constant velocity (which is assumed to be parallel to the sheets). The
action depends on the gauge field and on the Dirac fields, the latter confined to
the mirrors. ${\mathcal S}$ naturally decomposes into three terms:
\begin{equation}\label{eq:defs}
{\mathcal S}[A; \bar\psi, \psi] \;=\; {\mathcal S}^{(0)}_{\rm g}[A] \,+\,
{\mathcal S}^{(0)}_{\rm d}[\bar\psi, \psi] \,+\, {\mathcal S}^{({\rm
int})}_{\rm dg}[\bar\psi, \psi,A] \;,
\end{equation}
where ${\mathcal S}^{(0)}_{\rm g}$ is the free (i.e., empty-space) action
for the EM field:
\begin{equation}\label{eq:defsg0}
{\mathcal S}^{(0)}_{\rm g}[A]\;=\; \frac{1}{4} \int d^4x \, F_{\mu\nu} F_{\mu\nu} \;,
\end{equation}
with $F_{\mu\nu} = \partial_\mu A_\nu - \partial_\nu A_\mu$,
whilst ${\mathcal S}^{(0)}_{\rm d}$ and ${\mathcal S}_{\rm d g}^{({\rm
int})}$ are the actions for the free Dirac matter fields and for their
interactions with the gauge field, respectively. Indices from the middle of
the Greek alphabet ($\mu, \nu, \ldots$) run from $0$ to $3$, with $x_0
\equiv c t$.
Both ${\mathcal S}^{(0)}_{\rm d}$ and ${\mathcal S}_{\rm d g}^{({\rm
int})}$ are localized on the regions occupied by the two sheets, which we
denote by $L$ and $R$ (each letter will be used to denote both a
mirror and the spatial region it occupies). Our choice of Cartesian
coordinates is such that $L$ is defined by $x_3=0$ and $R$ by $x_3=a$.
We adopt conventions such that $\hbar=c=1$.
We introduce $\Gamma$, the effective action for the full system defined
in (\ref{eq:defs}) by ${\mathcal S}$. It can be written in
terms of ${\mathcal Z}$, the zero-temperature partition function, which may
be represented as a functional integral:
\begin{equation}\label{eq:defgamma}
e^{-\Gamma} \;\equiv\; \mathcal{Z} \;\equiv\;\int [{\mathcal D} A]
{\mathcal D}\bar\psi {\mathcal D}\psi e^{-{\mathcal S}[A; \bar\psi,\psi]}
\;,
\end{equation}
where $[{\mathcal D} A]$ is the gauge field functional integration measure
including gauge fixing.
The effect of the Dirac fields on the gauge field is taken into
account by integrating out the former, we introduce ${\mathcal
S}_{\text{eff}}$, as follows:
\begin{equation}\label{eq:defseffa}
e^{- {\mathcal S}_{\text{eff}}[A]} \;\equiv\; \int {\mathcal D}\bar\psi
{\mathcal D}\psi \, e^{-{\mathcal S}[A; \bar\psi,\psi]} \;,
\end{equation}
so that:
\begin{equation}\label{eq:defgamma1}
e^{-\Gamma} \;\equiv\;\int [{\mathcal D} A] \, e^{-{\mathcal
S}_{\text{eff}}[A]}\;.
\end{equation}
Recalling our previous discussion and conventions, we write the effective action as
\begin{equation}
{\mathcal S}_{\text{eff}}[A] \;=\; {\mathcal S}^{(0)}_{\rm g}[A] \,+\,
{\mathcal S}^{(\text{int})}_{\rm g}[A] \;.
\end{equation}
In the next two subsections, we deal with the determination of ${\mathcal
S}^{(\text{int})}_{\rm g}$, which is the result of the integration of the
fermionic degrees of freedom.
\subsection{Effective action contribution due to the static sheet}
\label{sec:statmir}
As in~\cite{dynamical_matter}, the effective interaction term for the
gauge field in the presence of graphene sheets stems from two essentially
$2+1$ dimensional theories, coupled to the $3+1$ dimensional gauge field.
Therefore, ${\mathcal S}^{(\text{int})}_{\rm g} = {\mathcal
S}^{(\text{L})}_{\rm g} + {\mathcal S}^{(\text{R})}_{\rm g}$, where each
term is due to the respective plate. The fact that one of the
sheets is moving is irrelevant to the dimensionality of those theories,
since the surface it occupies is invariant under the sliding motion.
Let us first consider ${\mathcal S}^{(\text{L})}_{\rm g}[A]$, due to the
static sheet at $x_3 = 0$. Up to the quadratic order in the gauge field,
following~\cite{dynamical_matter}, we write such contribution as follows:
\begin{equation}\label{eq:defpil}
{\mathcal S}^{(\text{L})}_{\rm g}[A] \,=\,
\frac{1}{2} \,\int d^3x_\parallel \int d^3y_\parallel \,
A_\alpha(x_\parallel,0) \, \Pi_{\alpha\beta}(x_\parallel,y_\parallel) \,
A_\beta(y_\parallel,0) \;,
\end{equation}
where indices from the beginning of the Greek alphabet ($\alpha, \beta,
\ldots$) are assumed to take the values $0, 1, 2$ and are used here to
label spacetime coordinates on the $2+1$ dimensional world-volume of each
sheet. Those coordinates have been denoted collectively by $x_\parallel$.
Regarding the corresponding $2+1$ dimensional Fourier momentum, we use $k_{\parallel}
\equiv (k_0,k_1, k_2)$, and ${\mathbf k_\shortparallel} \equiv
(k_1,k_2)$ for its spatial part.
The tensor kernel $\Pi_{\alpha\beta}$ is the vacuum polarization tensor
(VPT) for the matter field on the $L$ plane. Under the assumptions of
time-independence, as well as invariance under spatial rotations and
translations, this tensor can be conveniently decomposed in Fourier space
into orthogonal projectors. Indeed, since it has to verify the
Ward identity:
\begin{equation}\label{eq:trans}
k_\alpha \widetilde{\Pi}_{\alpha \beta}(k)\;=\; 0 \;,
\end{equation}
(the tilde is used to denote Fourier transformation) the irreducible
tensors (projectors) along which
$\widetilde{\Pi}_{\alpha\beta}$ may be decomposed must satisfy the
condition above and may be constructed using as building blocks the
objects: $\delta_{\alpha\beta}$, $k_\alpha$, and \mbox{$n_\alpha =
(1,0,0)$}. By performing simple combinations among them, we also
introduce: \mbox{$\breve{k}_\alpha \equiv k_\alpha - k_0 n_\alpha$}, and
\mbox{$\breve{\delta}_{\alpha\beta} \equiv \delta_{\alpha\beta} - n_\alpha
n_\beta$}.
Since we cannot guarantee that the VPT will be proportional to
${\mathcal P}^\perp_{\alpha\beta} \,\equiv\, \delta_{\alpha\beta}- \frac{k_\alpha
k_\beta}{k^2}$, we construct two independent tensors satisfying the transversality
condition (\ref{eq:trans}), ${\mathcal P}^t$ and ${\mathcal P}^l$, defined
as follows:
\begin{equation}
{\mathcal P}^t_{\alpha\beta} \,\equiv\, \breve{\delta}_{\alpha\beta} -
\frac{\breve{k}_\alpha\breve{k}_\beta}{\breve{k}^2}
\end{equation}
and
\begin{equation}
{\mathcal P}^l_{\alpha\beta} \,\equiv\, {\mathcal
P}^\perp_{\alpha\beta}\,-\, {\mathcal P}^t_{\alpha\beta} \;.
\end{equation}
Defining also:
\begin{equation}
{\mathcal P}^\shortparallel_{\alpha\beta} \,\equiv\, \frac{k_\alpha
k_\beta}{k^2} \;,
\end{equation}
we verify the algebraic properties:
$$
{\mathcal P}^\perp + {\mathcal P}^\shortparallel = I \;,\;\;
{\mathcal P}^t + {\mathcal P}^l = {\mathcal P}^\perp
$$
$$
{\mathcal P}^t {\mathcal P}^l = {\mathcal P}^l {\mathcal P}^t = 0
\;,\;\; {\mathcal P}^\shortparallel {\mathcal P}^t = {\mathcal P}^t {\mathcal P}^\shortparallel = 0
\;,$$
$$\;\; {\mathcal P}^\shortparallel {\mathcal P}^l = {\mathcal P}^l {\mathcal P}^\shortparallel = 0 \;,
$$
\begin{equation}
\big({\mathcal P}^\perp\big)^2 = {\mathcal P}^\perp \;,
\big({\mathcal P}^\shortparallel\big)^2 = {\mathcal P}^\shortparallel \;,
\big({\mathcal P}^t\big)^2 = {\mathcal P}^t \;,
\big({\mathcal P}^l\big)^2 = {\mathcal P}^l \;.
\end{equation}
Note that $\delta_{\alpha\beta}$, ${\mathcal P}^\perp_{\alpha\beta}$,
and ${\mathcal P}^\shortparallel_{\alpha\beta}$ are second order Lorentz
tensors.
The other projectors, ${\mathcal P}^t$ and ${\mathcal P}^l$ are not: they
explicitly single out the time-like coordinate in their definition. On the
other hand, Lorentz tensors will tend to Galilean ones in the low speed
limit.
For a general medium, one has:
\begin{equation}
\widetilde{\Pi}_{\alpha\beta}(k_{\parallel}) \,=\,
g_t\big(k_0, {\mathbf k}_\parallel\big) \, {\mathcal P}^t_{\alpha\beta} \,+\,
g_l\big(k_0, {\mathbf k}_\parallel\big) \, {\mathcal P}^l_{\alpha\beta}
\;,
\label{piscalar}
\end{equation}
where $g_t$ and $g_l$ are model-dependent scalar functions.
If the matter-field action were relativistic, we would have $g_t=g_l\equiv
g$, a scalar function of $k_{\parallel}$, and the VPT would be proportional to a single projector:
\begin{equation}
{\widetilde \Pi}_{\alpha \beta}(k_\parallel) \;=\; g(k_\parallel)
\, {\mathcal P}^\perp_{\alpha\beta} \; .
\end{equation}
On the other hand, for the case of graphene, we may present the well-known
results for its VPT \cite{rev_graphene}, as follows:
\begin{equation}
\widetilde{\Pi}_{\alpha\beta}(k_{\parallel}) =\frac{e^2 N |m|}{4\pi}
F\big(\frac{k_0^2 + v_F^2 {\textbf{k}_\parallel}^2}{4 m^2}\big)
\Big[ {\mathcal P}^t_{\alpha\beta} +
\frac{k_0^2+{\textbf{k}_\parallel}^2}{k_0^2+ v_F^2 {\textbf{k}_\parallel}^2} {\mathcal P}^l_{\alpha\beta}
\Big]
\end{equation}
where:
\begin{equation}
F(x) \;=\; 1 - \frac{1 - x}{\sqrt{x}} \, \arcsin[ ( 1 +
x^{-1})^{-\frac{1}{2}}]\;,
\end{equation}
$m$ is the mass (gap), $N$ the number of $2$-component Dirac fermion
fields, and $v_F$ the Fermi velocity (in units where $c=1$).
Here we will consider gapless graphene ($m=0$) and define
$\alpha_N \equiv \frac{e^2 N}{16}$, so that
\begin{align}
\widetilde{\Pi}_{\alpha\beta} &= \alpha_N \,
\sqrt{k_0^2 + v_F^2 {\textbf{k}_\parallel}^2}\;
\Big[ {\mathcal P}^t_{\alpha\beta} \,+\,
\frac{k_0^2+{\textbf{k}_\parallel}^2}{k_0^2+ v_F^2 {\textbf{k}_\parallel}^2} {\mathcal P}^l_{\alpha\beta}
\Big] \nonumber\\
&= \alpha_N \, \sqrt{k_0^2+ {\textbf{k}_\parallel}^2} \,
\Big[ \, \sqrt{\frac{k_0^2 + v_F^2 {\textbf{k}_\parallel}^2}{k_0^2 +
{\textbf{k}_\parallel}^2}} \;
{\mathcal P}^t_{\alpha\beta} +
\sqrt{\frac{k_0^2 + {\textbf{k}_\parallel}^2}{k_0^2 + v_F^2
{\textbf{k}_\parallel}^2}} \; {\mathcal P}^l_{\alpha\beta}
\Big] \;.
\end{align}
We see explicitly that the mass
dimension of the VPT is given by the momentum, as mentioned in the Introduction.
We conclude the discussion on ${\mathcal S}_{\text{g}}^{(\text{L})}$ (we
work at the second order in the coupling constant) by writing it in a $3+1$
dimensional looking form,
\begin{equation}
\label{eq:seffl}
{\mathcal S}_{\text{g}}^{(\text{L})}[A] \;=\; \frac{1}{2}\, \int d^4x
\int d^4y \; A_\alpha(x) V^{({\text L})}_{\alpha\beta}(x,y) A_\beta(y) \, ,
\end{equation}
where $V^{(\text{L})}_{\alpha\beta}(x,y)$ is given by
\begin{equation}
\label{eq:vl}
V^{(\text{L})}_{\alpha\beta}(x,y) \;=\; \delta(x_3) \,
\Pi_{\alpha\beta}(x_\parallel,y_\parallel) \, \delta(y_3) \;.
\end{equation}
\subsection{Effective action due to the moving graphene sheet}
We already know the expression for the effective action due to the static
mirror at $x_3=0$; let us see now how to derive from it the corresponding object for
the moving sheet at $x_3=a$. We assume that its constant velocity is much smaller than
$c$, so that the form it adopts in two different inertial systems may be
derived using Galilean transformations. Besides, the material media
descriptions are usually restricted to the same regime, namely, small
speeds with respect to the laboratory system (the response functions are
usually defined in a comoving system).
Since we need to write the effective action in one and the same system, we
need to write the gauge field appearing in ${\mathcal S}_{\text{g}}^{(\text{R})}$
in the laboratory system, the one used in the previous subsection. We also need to
refer them to the same choice of coordinates. Thus,
\begin{equation}
\label{eq:seffr}
{\mathcal S}_{\text{g}}^{(\text{R})}[A] \;=\; \frac{1}{2}\, \int d^4x
\int d^4y \; A_\alpha(x) V^{(\text{R})}_{\alpha\beta}(x,y) A_\beta(y) \, ,
\end{equation}
where
\begin{equation}
\label{eq:rpot}
V^{(\text{R})}_{\alpha\beta}(x,y) \;=\; \delta(x_3 - a)
\Pi'_{\alpha\beta}(x_\parallel,y_\parallel) \delta(y_3 - a) \;,
\end{equation}
where the prime in an object denotes its form in the comoving system. To
write the expression above more explicitly, we need to introduce the
transformations $x_\parallel' = \Lambda(v) x_\parallel$, where
$x_\parallel$ is the column vector $x_\parallel = \left(\begin{array}{c}
x_0\\ x_1\\ x_2 \end{array}\right)$.
Those transformations can be obtained by
keeping the first non-trivial term in an expansion in powers of $v$. Since
we have adopted conventions such that $c=1$, and our metric is Euclidean,
we see that:
\begin{equation}
\Lambda(v)=\left(\begin{array}{c c c c }
1 & v & 0 \\
-v & 1 & 0 \\
0 & 0& 1
\end{array}
\right) \,
\end{equation}
(i.e., they are rotation matrices expanded for small angles). We have only
kept the three spacetime coordinates corresponding to the sheets, since the
role of the $x_3$ coordinate is irrelevant here.
Note that the matrix includes the transformation of the time coordinate, while
Galilean transformations do not include that transformation and are given by:
\begin{equation}
\Lambda_G(v)=\left(\begin{array}{c c c c }
1 & 0 & 0 \\
-v & 1 & 0 \\
0 & 0& 1
\end{array}
\right) \,.
\end{equation}
The EM field, on the other hand, transforms as
$A'_\alpha(x')=\Lambda_{\alpha\beta} A_\beta(x)$.
Regarding the VPT, we have
\begin{equation}
\Pi'_{\alpha\beta}(x_{\parallel} ', y_{\parallel} ') \;=\;\Lambda_{\alpha\gamma}
\Lambda_{\beta\delta} \, \Pi_{\gamma\delta}(x_{\parallel}, y_{\parallel}) \,.
\end{equation}
Thus,
\begin{equation}
\Pi'_{\alpha\beta}(x_{\parallel}, y_{\parallel}) \;=\;\Lambda_{\alpha\gamma}
\Lambda_{\beta\delta} \, \Pi_{\gamma\delta}(\Lambda^{-1}x_{\parallel},\Lambda^{-1}y_{\parallel}) \,.
\end{equation}
Then we see that:
\begin{equation}
\label{eq:pot}
V^{(\text{R})}_{\alpha\beta}(x,y) \;=\; \delta(x_3 - a)
\,\Lambda_{\alpha\gamma}
\Lambda_{\beta\delta} \, \Pi_{\gamma\delta}(\Lambda^{-1}x_{\parallel},\Lambda^{-1}y_{\parallel})
\, \delta(y_3 - a) \;,
\end{equation}
In momentum space, we can write
\begin{eqnarray}\label{eq:aux1}
\widetilde{\Pi}'_{\alpha\beta}(k_\parallel) &=&
\Lambda_{\alpha\gamma} \Lambda_{\beta\delta} \,
\widetilde{\Pi}_{\gamma\delta}(\Lambda^{-1} k_{\parallel}) \nonumber\\
&=&
\Lambda_{\alpha\gamma} \Lambda_{\beta\delta} \,
\widetilde{\Pi}_{\gamma\delta}(k_0 - v k_1, k_1 + v k_0, k_2) \;.
\end{eqnarray}
\subsection{The full effective action ${\mathcal S}_{\text
g}^{(\text{int})}$ for graphene}
Putting together the previous results, we have that:
\begin{equation}
\label{eq:seffint}
{\mathcal S}_{\text g}^{(\text{int})}\;=\;
\frac{1}{2}\, \int d^4x
\int d^4y \; A_\alpha(x) \big[ V^{(\text{L})}_{\alpha\beta}(x,y) +
V^{(\text{R})}_{\alpha\beta}(x,y) \big] A_\beta(y) \;,
\end{equation}
or
\begin{equation}
{\mathcal S}_{\text g}^{(\text{int})}[A] \;=\; \frac{1}{2} \, \int
dx_3 \int dy_3 \int \frac{d^3k_{\parallel}}{(2\pi)^3} \,
\tilde{A}^*_\alpha(k_{\parallel},x_3) \left[
\delta(x_3) \tilde{\Pi}_{\alpha\beta}(k_{\parallel}) \delta(y_3) + \delta(x_3 - a)
\tilde{\Pi}'_{\alpha\beta}(k_{\parallel}) \delta(y_3-a) \right]
\tilde{A}_\beta(k_{\parallel},y_3) \,,
\end{equation}
with $\tilde{\Pi}'_{\alpha\beta}(k_{\parallel})$ as defined in (\ref{eq:aux1}).
Let us now study in more detail the form of
$\widetilde{\Pi}'_{\alpha\beta}(k_\parallel)$.
We have:
\begin{equation}
\widetilde{\Pi}'_{\alpha\beta}(k_{\parallel}) \,=\, g_t\big(\Lambda^{-1}
k_\parallel\big) \,{\mathcal P}'^t_{\alpha\beta}
\,+\, g_l\big(\Lambda^{-1}
k_\parallel\big) \,{\mathcal P}'^l_{\alpha\beta} \;.
\end{equation}
Now, we will see that the two projectors remain invariant under Galilean
transformations.
Indeed, we first note that the Lorentz projectors, which enter into the
definition of the Galilean ones, are indeed invariant (we
use the approximate Lorentz form for the transformation matrix):
\begin{align}
{\mathcal P}'^\perp_{\alpha\beta}(k_\parallel)
&=\,\Lambda_{\alpha\gamma} \,
\Lambda_{\beta\delta} {\mathcal P}^\perp_{\gamma\delta}(\Lambda^{-1}
k_\parallel) = {\mathcal P}^\perp_{\alpha\beta}(k_\parallel) \\
{\mathcal P}'^\shortparallel_{\alpha\beta}(k_\parallel) &=\,
\Lambda_{\alpha\gamma} \, \Lambda_{\beta\delta} \,
{\mathcal P}^\shortparallel_{\gamma\delta}(\Lambda^{-1} k_\parallel)
={\mathcal P}^\shortparallel_{\alpha\beta} (k_\parallel)\;,
\end{align}
while for the Galilean tensor $P^t$ we verify explicitly that:
\begin{equation}
{\mathcal P}'^t_{\alpha\beta}(k_\parallel)
\,=\,(\Lambda_G)_{\alpha\gamma} \,
(\Lambda_G)_{\beta\delta} {\mathcal P}^t_{\gamma\delta}((\Lambda_G)^{-1}
k_\parallel) = {\mathcal P}^t_{\alpha\beta}(k_\parallel) \;.
\end{equation}
Since ${\mathcal P}^l$ is defined in terms of the previously considered
three projectors, we see that:
\begin{equation}
{\mathcal P}'^l_{\alpha\beta} = {\mathcal P}^l_{\alpha\beta} \;.
\end{equation}
Thus, we conclude that:
\begin{equation}
\widetilde{\Pi}'_{\alpha\beta}(k_\parallel) \,=\, g_t\big(\Lambda^{-1}
k_\parallel\big) \,{\mathcal P}^t_{\alpha\beta}
\,+\, g_l\big(\Lambda^{-1}
k_\parallel\big) \,{\mathcal P}^l_{\alpha\beta} \;.
\end{equation}
We are interested in small relative velocities between the plates, so we are able to use the simpler expression
\begin{equation}
\widetilde{\Pi}'_{\alpha\beta}(k_{\parallel}) \,=\, g_t\big(k_0-v k_1, k_1 + v k_0, k_2\big) \,{\mathcal P}^t_{\alpha\beta}
\,+\, g_l\big(k_0-v k_1, k_1 + v k_0, k_2 \big) \,{\mathcal P}^l_{\alpha\beta} \;.
\end{equation}
where
\begin{align}
\label{eq:gs}
g_t(k_\parallel)&=\alpha_N \sqrt{k_0^2 + v_F^2 \textbf{k}_\parallel ^2} \\
g_l(k_\parallel)&=\alpha_N \sqrt{k_0^2 + v_F^2 \textbf{k}_\parallel ^2} \, \frac{k_0^2 + \textbf{k}_\parallel ^2}{k_0^2+ v_F^2\textbf{k}_\parallel ^2}\nonumber
\end{align}
\section{Effective action}\label{sec:effective}
With all the previous considerations, we are now in a position to write the
total action for the gauge field, containing the effective influence of the
graphene plates. In Fourier space:
\begin{equation}
S_{\text{g}}[A] \,=\, \frac{1}{2} \int dx_3 \int dy_3 \int \frac{d^3k_{\parallel}}{(2\pi)^3} \tilde{A}^*_\alpha (k_{\parallel}, x_3) M_{\alpha\beta} (k_{\parallel},x_3,y_3) \tilde{A}_\beta(k_{\parallel},y_3)
\end{equation}
where the kernel $M_{\alpha\beta} (k_{\parallel},x_3,y_3)$ can be written as
\begin{equation}
M_{\alpha\beta} (k_{\parallel},x_3,y_3)=M^0_{\alpha \beta}(k_{\parallel},x_3,y_3)+M^{\text{int}}_{\alpha\beta} (k_{\parallel},x_3,y_3)
\end{equation}
where $M^0$ is the free kernel for the vacuum EM field
\begin{equation}
M^0_{\alpha \beta}(k_{\parallel},x_3,y_3)=-\partial_3^2 \delta(x_3-y_3) \mathcal{P}^\parallel_{\alpha \beta} + (-\partial_3^2+k_{\parallel}^2) \delta(x_3-y_3) \left[ \mathcal{P}^l_{\alpha\beta}+\mathcal{P}^t_{\alpha\beta} \right] \, ,
\end{equation}
and $M^{\text{int}}$ contains the effective interaction with the plates' internal degrees of freedom
\begin{equation}
M^{\text{int}}_{\alpha\beta} (k_{\parallel},x_3,y_3)= \tilde{V}^{(\text{L})}_{\alpha\beta}(k_{\parallel},x_3,y_3)+\tilde{V}^{(\text{R})}_{\alpha\beta}(k_{\parallel},x_3,y_3) \, .
\end{equation}
The generating functional for the system is defined by
\begin{equation}
\mathcal{Z}=\int \left[ \mathcal{D}A\right] e^{-S_{\text g}[A]}
\end{equation}
where $\left[ \mathcal{D}A\right]$ is gauge-fixed. Formally, it is equivalent to writting
\begin{equation}
\mathcal{Z}=\left[ \det \left( M_{\alpha \beta}(k_{\parallel},x_3,y_3)\right)\right]^{-\frac{1}{2}}
\end{equation}
Now, since we have chosen a complete set of projectors $\left\lbrace \mathcal{P}^\parallel, \mathcal{P}^t, \mathcal{P}^l \right\rbrace$, we can uniquely decompose the gauge field in their directions $\tilde{A}_\alpha \equiv \tilde{A}^\parallel_\alpha + \tilde{A}^t_\alpha + \tilde{A}^l_\alpha$, thus writing the functional integral over $A$ as three independent functional integrals
\begin{equation}
[ \mathcal{D}\tilde{A} ] = \mathcal{D }\tilde{A}^\parallel \, \mathcal{D} \tilde{A}^t \, \mathcal{D} \tilde{A}^l \, .
\end{equation}
This means that the integrating functional for the system can be written as the direct product of three independent integrating functionals:
\begin{equation}
\mathcal{Z}=\big[ \det \big( M^\parallel (k_{\parallel},x_3,y_3) \big)\big]^{-\frac{1}{2}} \left[ \det \left( M^t (k_{\parallel},x_3,y_3)\right)\right]^{-\frac{1}{2}} \left[ \det \left( M^l(k_{\parallel},x_3,y_3)\right)\right]^{-\frac{1}{2}} \equiv \mathcal{Z}^\parallel \mathcal{Z}^t \mathcal{Z}^l
\end{equation}
where we have defined the kernels:
\begin{equation}
\label{eq:mpar}
M^\parallel (k_{\parallel}, x_3, y_3) = - \partial_3^2 \delta(x_3-y_3) \mathcal{P}^\parallel \, ,
\end{equation}
\begin{align}
\label{eq:ml}
M^l(k_{\parallel},x_3,y_3)=&\big\lbrace (-\partial_3^2 + k_{\parallel}^2) \delta(x_3-y_3) + g_l (k_0,k_1,k_2) \delta(x_3) \delta(y_3) \nonumber \\
&+ g_l (k_0-v k_1,k_1+v k_0,k_2) \delta(x_3-a) \delta(y_3-a ) \big\rbrace \mathcal{P}^l \, ,
\end{align}
and
\begin{align}
\label{eq:mt}
M^t(k_{\parallel},x_3,y_3)=&\big\lbrace (-\partial_3^2 + k_{\parallel}^2) \delta(x_3-y_3) + g_t (k_0,k_1,k_2) \delta(x_3) \delta(y_3) \nonumber \\
&+ g_t (k_0-v k_1,k_1+v k_0,k_2) \delta(x_3-a) \delta(y_3-a ) \big\rbrace \mathcal{P}^t \, .
\end{align}
Given Eq. \eqref{eq:mpar}, it is easy to see that $\mathcal{Z}^\parallel$ is a free contribution that does not account for the presence of the plates. It is thus simply a normalization factor, and we shall not take it into account in the following. The remaining factors $\mathcal{Z}^t$ and $\mathcal{Z}^l$ are formally equivalent but different, except for relativistic materials
Regarding the effective action, it is easy to see that it shall have two independent contributions
\begin{equation}
\Gamma \equiv \Gamma^t + \Gamma^l = \frac{1}{2} \text{tr} \log M^t + \frac{1}{2} \text{tr} \log M^l \, .
\end{equation}
We shall now work out the formal expression for $\Gamma^t$; the
corresponding expression for $\Gamma^l$ is obtained by the substitutions
$g_t \to g_l$, $\mathcal{P}^t \to \mathcal{P}^l$. As in previous works
\citep{farias_friction,fosco2011_sidewise}, we will perform a perturbative
expansion in
the coupling constant, $e \ll 1$, and keep only the lowest-order
non-trivial term.
Explicitly taking
the trace over all discrete and continuous indices in this term we get a $T
\Sigma$ global factor, $T$ denoting the elapsed time and $\Sigma$ the sheets' area (this
is a reflection of the time and (parallel) space translation invariances of
the system). Since $\Gamma^t$ is extensive in those magnitudes, we
work instead with $\gamma^t \equiv \frac{\Gamma}{T \Sigma}$, which is
given by
\begin{equation}
\gamma^t = -\frac{ 1}{4} \int \frac{d^3k_{\parallel}}{(2\pi)^3} \int dx_3
\int dy_3 \int du_3 \int dv_3 G_{\alpha \gamma}(k_{\parallel},x_3,y_3)
V^t_{\gamma \delta}(k_{\parallel},y_3,u_3) G_{\delta \beta}(k_{\parallel},u_3,v_3)
V^t_{\beta \alpha} (k_{\parallel}, v_3, x_3) \;.
\end{equation}
Here, $G_{\alpha \gamma}(k_{\parallel},x_3,y_3)$ denotes the respective components
of the free Euclidean propagator for the gauge field, and we have
introduced
\begin{equation}
V^t \equiv \left[ g_t (k_0,k_1,k_2) \delta(x_3) \delta(y_3) + g_t
(k_0-v k_1,k_1+v k_0,k_2) \delta(x_3-a) \delta(y_3-a ) \right]
\mathcal{P}^t \;.
\end{equation}
We only consider in what follows the `crossed' terms, namely, those
involving both $g_t (k_0,k_1,k_2)$ and $g_t (k_0-v k_1,k_1+v k_0,k_2)$,
since they are the only ones that lead to friction (the others can be shown
to be $v$-independent).
Taking into account that, in the Feynman gauge:
\begin{equation}
G_{\alpha \beta}(k_{\parallel},x_3,y_3) \equiv \delta_{\alpha\beta} G(k_{\parallel},x_3,y_3) =
\delta_{\alpha\beta} \int \frac{dk_3}{2\pi} \frac{e^{i k_3 (x_3 -
y_3)}}{k_{\parallel}^2 + k_3^2} \,,
\end{equation}
and the properties of the projectors, we see that
\begin{equation}
\gamma^t \,=\, -\frac{1}{2} \int \frac{d^3k_{\parallel}}{(2\pi)^3} G(k_{\parallel},a,0) G(k_{\parallel},0,a)
g_t(k_0,k_1,k_2) g_t(k_0-v k_1, k_1+v k_0,k_2) \,.
\end{equation}
The procedure and outcome for the $\Gamma^l$ contribution are entirely
analogous, thus we may write ($s=t,l$):
\begin{equation}
\label{eq:effacts}
\gamma^s \;=\; -\frac{1}{2} \int \frac{d^3k_{\parallel}}{(2\pi)^3} \; G(k_{\parallel},a,0) G(k_{\parallel},0,a)
g_s(k_0,k_1,k_2) g_s(k_0-v k_1, k_1+v k_0,k_2) \, .
\end{equation}
Thus,
\begin{equation}
\label{eq:gammas}
\gamma^s \;=\; -\frac{1}{8\, a^3} \int \frac{d^3k_{\parallel}}{(2\pi)^3} \;
\frac{e^{-2\sqrt{k_0^2 + k_1^2 + k_2^2}}}{k_0^2 + k_1^2 + k_2^2}
g_s(k_0,k_1,k_2) g_s(k_0-v k_1, k_1+v k_0,k_2) \, ,
\end{equation}
where we have rescaled the momenta $ak_\alpha\to k_\alpha$ in order to factorize the dependence of the effective action
with the distance between sheets. Note that $\gamma_s$ is the effective action per unit time and area, and therefore has units
of $(length)^{-3}$.
Before evaluating the imaginary part of the real time (in-out) effective action, we would like to stress that the Euclidean effective action
$\gamma$, when evaluated at $v=0$, gives the usual Casimir interaction energy per unit area $E_C$ between the graphene sheets.
As described in the Appendix A, the result is
\begin{equation}
\label{eq:casfor}
E_\text{C} \approx - \frac{\alpha_N^2}{128 \pi} \frac{1}{a^3} \frac{1}{v_F} \, .
\end{equation}
As expected, due to the absence of dimensionful constants in the microscopic description of graphene, the Casimir energy has the
usual $1/a^3$ dependence of the static vacuum interaction energy for perfect conductors. {Eq. \eqref{eq:casfor} is quadratic in the coupling constant $\alpha_N$, while the Casimir force found in \cite{bordag_casimir_graphene} is linear. The reason is that
we calculate the force between two graphene plates,
while in \cite{bordag_casimir_graphene} the interaction
between a perfect conductor and a graphene sheet is considered.}
\subsection{Imaginary part of the effective action}
In order to compute the imaginary part of the in-out effective action, we have to rotate the Euclidean result to real time.
To that end, we will rewrite each contribution in a way that simplifies the forthcoming
discussion. Note that we can write the two functions $g_t$ and $g_l$ as follows:
\begin{align}
g_t(k_\parallel)&= \alpha_N \int_{-\infty}^{+\infty}\frac{dk_3}{\pi}
\frac{k_0^2 + v_F^2 \textbf{k}_\parallel^2}{k_0^2 + v_F^2 \textbf{k}_\parallel ^2 + k_3^2} \\
g_l(k_\parallel)&=\alpha_N \int_{-\infty}^{+\infty} \frac{dk_3}{\pi}
\frac{k_0^2 + \textbf{k}_\parallel ^2}{k_0^2 + v_F^2 \textbf{k}_\parallel ^2 + k_3^2} \;.
\end{align}
Then we see that,
\begin{equation}
\gamma^t = -\frac{ \alpha_N^2 }{8\, a^3}
\int \frac{dk_3}{\pi}\int\frac{dp_3}{\pi} \int
\frac{d^3k_\parallel}{(2\pi)^3} \,\frac{e^{-2\sqrt{k_0^2 + k_1^2 +
k_2^2}}}{k_0^2 + k_1^2 + k_2^2}
\frac{k_0^2 + v_F^2 \textbf{k}_\parallel ^2}{k_0^2 + v_F^2 \textbf{k}_\parallel ^2 + k_3^2}
\frac{(k_0-k_1 v)^2 + v_F^2 [(k_1+k_0 v)^2 + k_2^2]}{(k_0-k_1 v)^2 + v_F^2 [(k_1+k_0 v)^2 + k_2^2] + p_3^2}
\,,
\end{equation}
and
\begin{equation}
\gamma^l = -\frac{\alpha_N^2}{8\, a^3}
\int \frac{dk_3}{\pi}\int\frac{dp_3}{\pi} \int
\frac{d^3k_\parallel}{(2\pi)^3} \, \frac{e^{-2\sqrt{k_0^2 + k_1^2 +
k_2^2}}}{k_0^2 + k_1^2 + k_2^2}
\frac{k_0^2 +\textbf{k}_\parallel^2}{k_0^2 + v_F^2 \textbf{k}_\parallel ^2
+ k_3^2} \frac{k_0^2 +\textbf{k}_\parallel^2}{(k_0-k_1 v)^2 + v_F^2
[(k_1+k_0 v)^2 + k_2^2] + p_3^2} \,.
\end{equation}
In real-time, the longitudinal contribution to the effective action is
\begin{equation}
\label{eq:inoutl}
\gamma^l = \frac{i \alpha_N^2}{8\, a^3} \,
\int \frac{dk_3}{\pi} \int \frac{dp_3}{\pi} \int\frac{d^3k_{\parallel}}{(2\pi)^3}
\frac{e^{2 i \sqrt{k_0^2 - \textbf{k}_\parallel^2 + i \epsilon}}}{k_0^2 - \textbf{k}_\parallel^2 + i
\epsilon} \times
\frac{k_0^2 - \textbf{k}_\parallel^2}{k_0^2 - v_F^2 \textbf{k}_\parallel ^2 - k_3^2 + i \epsilon} \times
\frac{k_0^2 - \textbf{k}_\parallel^2}{(k_0-k_1 v)^2 - v_F^2 [(k_1-k_0 v)^2 + k_2^2] - p_3^2 + i \epsilon}
\,.
\end{equation}
We shall be concerned first with the integral along $k_0$, which may be
first conveniently written as follows:
\begin{equation}
\int_0^\infty dk_0 \frac{e^{2i \sqrt{k_0^2-\textbf{k}_\parallel^2+i\epsilon}}}{k_0^2-\textbf{k}_\parallel^2+i\epsilon} \left[ f_1(k_0) f_2(k_0)+f_1(-k_0) f_2(-k_0)\right]
\end{equation}
where
\begin{align}
f_1(k_0)&\equiv \frac{k_0^2 - \textbf{k}_\parallel^2}{k_0^2 - v_F^2 \textbf{k}_\parallel ^2 - k_3^2+ i \epsilon}\\
f_2(k_0)&\equiv \frac{k_0^2 -\textbf{k}_\parallel^2}{(k_0-k_1 v)^2 - v_F^2 [(k_1-k_0 v)^2 + k_2^2] - p_3^2 + i \epsilon} \nonumber \; .
\end{align}
In order to perform this integral, we proceed along a similar line to the
one followed in \cite{farias_friction}, namely, to study the analytical structure
of the functions $f_1$ and $f_2$ in order to perform a Wick-rotation by
means of a Cauchy integration on the quarter of a circle located in the first
quadrant. Note that the rest of the integrand is the same as the
one dealt with in~\cite{farias_friction}: it presents two branch-cuts and
two poles, none of them in the first quadrant, hence they do not contribute to
the Cauchy integral.
Let us then consider the poles of $f_1(k_0)=f_1(-k_0)$; they are located at:
\begin{equation}
k_0=\pm \sqrt{v_F^2 \textbf{k}_\parallel^2+k_3^2-i \epsilon}\approx \pm \sqrt{v_F^2 \textbf{k}_\parallel^2+k_3^2} \mp \frac{i \epsilon}{2 \sqrt{v_F^2 \textbf{k}_\parallel^2+k_3^2}} \, .
\end{equation}
Since none of them is located in the first quadrant, they will not
contribute to the Cauchy integral either. For the $f_2(k_0)$ function,
they are located at:
\begin{equation}
k_0^{(\pm)} = \frac{1}{(1-v_F^2 v^2)} \left\lbrace v k_1 (1 - v_F^2) \pm \sqrt{v^2 k_1^2(1-v_F^2)^2+(1-v_F^2v^2)\left[(v_F^2 - v^2) k_1^2 + v_F^2 k_2^2 + p_3^2 - i \epsilon \right]}\right\rbrace
\end{equation}
It can be seen that only $k_0^{(-)}$ may have a positive imaginary
part (and thus be located in the first quadrant). We shall denote its
position by $\Lambda_A\equiv k_0^{(-)}$.
The condition for it to belong to the first quadrant is
$\text{Re}\Lambda_A > 0$. We first note that, if $k_1 < 0$, then
$\text{Re}\Lambda_A < 0$ and there is no pole located on the first
quadrant. On the other hand, for positive values of $k_1$, one can show
that:
\begin{align*}
\text{Re}\Lambda_A >0 \Leftrightarrow -(v_F^2 - v^2) k_1^2 - (v_F^2 k_2^2 + p_3^2) >0 \, .
\end{align*}
Clearly, when $v<v_F$, the LHS of the last equation is negative-definite, and
the inequality can never be fulfilled. Hence, for velocities smaller than
the Fermi velocity of the material, this pole can never be located in the
first quadrant. Finally, when $v>v_F$, we will have a pole in the first
quadrant when
\begin{equation}
k_1 > \sqrt{\frac{v_F^2 k_2^2 + p_3^2}{v^2-v_F^2}} \, .
\end{equation}
Proceeding in a completely analogous way for the $f_2(-k_0)$ term, one can also check
that just one pole may belong to the
first quadrant when $v>v_F$. The position of that pole is given by:
\begin{equation}
\Lambda_B=\Lambda_A- 2 v k_1\frac{1-v_F^2}{1-v_F^2v^2}\, .
\end{equation}
The pole is located on the first quadrant for momenta such that
\begin{equation}
k_1< - \sqrt{\frac{v_F^2 k_2^2 + p_3^2}{v^2-v_F^2}}\, .
\end{equation}
Based on the previous analysis, we are now ready to perform the
Cauchy-integral along the quarter of a circle, in a rather similar fashion
as we did in~\cite{farias_friction}. The result is:
\begin{align}
\label{eq:antesdejuntarAB}
\gamma^l =& \frac{i \alpha_N^2}{8\, a^3 (2 \pi)^3} \int \frac{dk_3}{\pi}\int \frac{dp_3}{\pi} \int dk_2 \int dk_1 \left\lbrace -i\int_0^\infty dp_0 \frac{e^{-2k_{\parallel}}}{k_{\parallel}^2} f_1(i p_0) \left[ f_2(i p_0) + f_2(-ip_0) \right] \right. \nonumber\\
&+ 2 \left. \pi i \theta(v-v_F) \left[ \theta\left(k_1- \sqrt{\frac{v_F^2 k_2^2 + p_3^2}{v^2-v_F^2}}\right)\text{Res}(F_A(k_0),\Lambda_A) + \theta\left(-k_1- \sqrt{\frac{v_F^2 k_2^2 + p_3^2}{v^2-v_F^2}}\right)\text{Res}(F_B(k_0),\Lambda_B) \right] \right\rbrace \, ,
\end{align}
where
\begin{equation}
F_A(k_0)=F_B(-k_0)= \frac{e^{2i \sqrt{k_0^2-\textbf{k}_\parallel^2+i \epsilon}}}{k_0^2-\textbf{k}_\parallel^2+i \epsilon} f_1(k_0) f_2(k_0) \, .
\end{equation}
Since we are interested in computing the dissipative effects on the system, we shall take the imaginary part of the effective action. It is easy to see that $f_1(p_0) \in \mathbb{R}$ and that $f_2(i p_0) + f_2(-i p_0) \in \mathbb{R}$ also. Hence, the imaginary part of the longitudinal contribution to the effective action will be given by
\begin{align}
\text{Im} \gamma^l = -\frac{\alpha_N^2}{16 \pi^2\, a^3} \,\theta(v-v_F)
\int
\frac{dk_3}{\pi} \int \frac{dp_3}{\pi} \int dk_2
\int dk_1\text{Im} &\left\lbrace \, \text{Res}(F_A(k_0),\Lambda_A) \,
\theta\left(k_1- \sqrt{\frac{v_F^2 k_2^2 + p_3^2}{v^2-v_F^2}}\right)
\right\rbrace \, .
\end{align}
From this equation, we see that there is no longitudinal
contribution to the quantum friction for plates moving with a relative
velocity smaller than the Fermi velocity of the material.\\
Regarding the transversal contribution to the effective action, let us
first rotate it back to real time:
\begin{equation}
\label{eq:inoutt}
\gamma^t = \frac{i \alpha_N^2}{8\, a^3} \,
\int \frac{dk_3}{\pi}\int \frac{dp_3}{\pi}\int \frac{ d^3k_{\parallel}}{(2\pi)^3}\frac{e^{2 i \sqrt{k_0^2 - \textbf{k}_\parallel^2 + i \epsilon}}}{k_0^2 - \textbf{k}_\parallel^2 + i \epsilon} \times
\frac{k_0^2 - v_F^2 \textbf{k}_\parallel^2}{k_0^2 - v_F^2 \textbf{k}_\parallel ^2 - k_3^2 + i \epsilon} \times
\frac{(k_0-k_1 v)^2 - v_F^2 (k_1-k_0 v)^2 - v_F^2 k_2^2}{(k_0-k_1 v)^2 - v_F^2 [(k_1-k_0 v)^2 + k_2^2] - p_3^2 + i \epsilon}
\,.
\end{equation}
The calculation is entirely similar to the previous case.
The imaginary part of the transversal
contribution to the in-out effective action reads
\begin{align}\label{eq:resultt}
\text{Im} \gamma^t = -\frac{\alpha_N^2}{16 \pi^2\, a^3} \,\theta(v-v_F) \int \frac{dk_3}{\pi} \int \frac{dp_3}{\pi} \int dk_2 dk_1 \text{Im} &\left\lbrace \text{Res}(F_C(k_0),\Lambda_A) \, \theta\left(k_1- \sqrt{\frac{v_F^2 k_2^2 + p_3^2}{v^2-v_F^2}}\right)
\right\rbrace \, ,
\end{align}
with
\begin{equation}
F_C(k_0)= \frac{e^{2i\sqrt{k_0^2-\textbf{k}_\parallel^2+i \epsilon}}}{k_0^2-\textbf{k}_\parallel^2+i \epsilon} f_3(k_0) f_4(k_0) \, ,
\end{equation}
and
\begin{align}
f_3(k_0)&=\frac{k_0^2 - v_F^2 \textbf{k}_\parallel^2}{k_0^2 - v_F^2 \textbf{k}_\parallel ^2 - k_3^2 + i\epsilon}\\
f_4(k_0)&=\frac{(k_0-k_1 v)^2 - v_F^2 (k_1-k_0 v)^2 - v_F^2 k_2^2}{(k_0-k_1 v)^2 - v_F^2 [(k_1-k_0 v)^2 + k_2^2] - p_3^2 + i \epsilon} \nonumber \, .
\end{align}
Hence we arrive to the important conclusion that there will not
be quantum friction between two graphene plates unless they move at a
relative velocity larger than the Fermi velocity of the internal
excitations in graphene. Note that a velocity threshold effect has also been shown to appear in dielectric materials~\cite{cherenkov}, as a consequence of a different, Cerenkov-like effect. \\
The remaining integrals and the limit $\epsilon \rightarrow 0$, needed to obtain the imaginary part of the effective action,
can be performed with some analytical and numerical calculations that are detailed in the Appendix \ref{ap1}.
The results are shown in Fig.~\ref{fig:plots}, where it may be seen that the
transverse contribution is much smaller than the longitudinal one. Then, the first plot on Fig. \ref{fig:plots}
shows the behavior of the leading contribution to the imaginary part of the effective action as a
function of the relative velocity $v$, for a Fermi velocity of $v_F=0.003$.
\begin{figure}[h]
\includegraphics[scale=2]{plots.pdf}
\caption{\label{fig:plots} Imaginary part of the effective action per unit of time and area, as a function of the relative velocity of the plate, for a typical graphene Fermi velocity $v_F=0.003$, in units of $A= \frac{\alpha_N^2}{32 \pi^2} \frac{1}{a^3}$.}
\end{figure}
\section{Frictional force}\label{sec:potencia}
In order to quantify the dissipation, a rather convenient observable is the
dissipated power (and its related dissipative force). Let us see how
that power is related to the imaginary part of the in-out effective action.
Dissipation arises here when the Dirac vacuum becomes unstable against the
production of a
real (i.e., on shell) fermion pair.
The probability $\mathcal P$ of
such an event, during the whole history of the plates, is related to the
effective action by
\begin{equation}
\label{eq:gammaP}
2 \text{Im} \Gamma =\mathcal{P} = T \int d^3 k_{\parallel} p(k_{\parallel}) \, ,
\end{equation}
where $p(k_{\parallel})$ is the probability per unit time of creating a pair of fermions on the plates with total momentum $k_{\parallel}$. The result is
proportional to the whole time elapsed
$T$, since we are in a stationary regime (we assume this time to be a very long one after the mirror was set to motion). Note that $k_{\parallel}$ is the three-momentum injected on the system by the external conditions, i.e.
the motion of the R-mirror.
The explicit expression for $p(k_{\parallel})$ can be read from Eqs. (\ref{eq:inoutl}) and (\ref{eq:inoutt}). It can be written as
\begin{equation}
p(k_{\parallel})=\int dk_3 \int dp_3\, \delta(k_0-\Lambda_A) h(k_{\parallel},k_3,p_3) \, ,
\end{equation}
for some function $h$. The presence of the $\delta$-function highlights the fact that the integration in the $k_0$-complex plane
captures the contribution of a single pole at $k_0=\Lambda_{A}$.
On the other hand, the total energy $E$ accumulated in the plates due to the excitation of the internal degrees of freedom is given by
\begin{equation}
E = T \int d^3k_{\parallel} \vert k_0\vert p(k_{\parallel})\, .
\end{equation}
This energy is provided by the external source that keeps the plate moving at a constant velocity, against the frictional force (per unit area) $F_{\text{fr}}$. The energy balance is
\begin{equation}
\frac{E}{T\Sigma}=v F_\text{fr}\, .
\end{equation}
From the reasoning above, it is easy to see
that in order to obtain the dissipated power we can simply insert $\left| k_0 \right|$
in Eqs. \eqref{eq:inoutl} and \eqref{eq:inoutt}, repeat the procedure of
the last section, and multiply the result by $2/v$. Note that the insertion of $\left| k_0 \right|$does not spoil the discussion
about the position of the poles, that remains unchanged. The results for the longitudinal and transverse contributions to the force are shown in Fig.
\ref{fig:fza}. We plot the frictional force normalized by the static Casimir force between the graphene sheets $F_C$, given in Eq. \eqref{eq:casfor}.
\begin{figure}[t]
\includegraphics[scale=2]{fuerzanorm.pdf}
\caption{\label{fig:fza} Modulus of the transversal and longitudinal contributions to the force per unit of area $F_\text{fr}$ acting on the plate as a function of its relative velocity, for a typical graphene Fermi velocity $v_F=0.003$. The force is normalized by the static Casimir force between the plates.}
\end{figure}
\begin{figure}[h!]
\includegraphics[scale=2]{fuerzanorm_zoom.pdf}
\caption{\label{fig:plotszoom} Modulus of the force per unit of area acting on the plate as a function of its relative velocity, for velocities close to the Fermi velocity of graphene, $v_F=0.003$. The force is normalized by the static Casimir force between the plates.}
\end{figure}
In Fig. \ref{fig:plotszoom} we show the force for velocities close to the
Fermi velocity. There, it can be seen that the system undergoes
three different regimes regarding dissipation. For $v<v_F$, as already mentioned, there are no
dissipative effects on the system, and the total frictional force vanishes.
For velocities $v_F<v<2 v_F$, a frictional
force appears, but it grows comparatively slow with the velocity. For
$v>2 v_F$, however, the frictional force starts growing rapidly when the
velocity increases.
The existence of a threshold may be justified as follows. Let us consider the momentum and energy balance in a small time
interval $\delta t$, assuming that both the frictional force and the
dissipated energy are driven by pair creation.
The only relevant component of the total momentum ${\mathbf P}$ of the pair
for the (momentum) balance is the one along the direction of the velocity
$v$. Relating that component of ${\mathbf P}$ to the frictional
force, we see that:
\begin{equation}\label{eq:bmom}
F_\text{fr} \, \delta t \; = \; P_x \;.
\end{equation}
On the other hand, the energy balance reads
\begin{equation}\label{eq:bener}
F_\text{fr} \, v \, \delta t \; = \; {\mathcal E}
\end{equation}
where ${\mathcal E}$ is the energy of the pair. But, since the fermions are both
on-shell, we have
\begin{equation}\label{eq:disper}
{\mathcal E} \geq v_F |P_x|
\end{equation}
(the equal sign corresponds to a pair with momentum along the direction of
$v$). Dividing Eq.(\ref{eq:bener}) and Eq.(\ref{eq:bmom}), and taking into account
Eq.(\ref{eq:disper}), we see that a necessary condition for friction to
happen is:
\begin{equation}
v \; \geq \; v_F \;.
\end{equation}
\section{Conclusions}\label{sec:conclu}
In this paper we computed the vacuum friction between graphene sheets subjected to a sidewise motion with constant relative velocity. The interaction between the $2+1$ Dirac fields in the graphene sheets and the electromagnetic field has been taken into account using the known results for the comoving vacuum polarization tensor, properly transformed to the laboratory system in the case of the moving sheet. We have seen that this interaction generates an imaginary part in the effective action, that in the nonrelativistic limit can be interpreted as due to the excitation of the internal degrees of freedom produced by the relative motion between sheets. Therefore, dissipation effect arises due to the fact the Dirac vacuum becomes unstable against the production of a real (i.e., on shell) fermion pair. We also computed the frictional force between plates using a slight modification of the calculation of the imaginary part of the effective action.
The results for the imaginary part of the effective action and for the frictional force show an interesting phenomenon: there is a threshold for quantum friction effects, that is, there is no quantum friction when the relative velocity between sheets is smaller than the Fermi velocity. We have presented a simple argument that justifies the existence of this threshold.
The frictional force computed in this paper is much smaller than the usual Casimir force between graphene sheets, which in turn is smaller than the Casimir force between perfect conductors (at least when considering gapped graphene, see Ref. \cite{bordag_casimir_graphene}). However, one may envisage situations in which the frictional force could be more relevant. Indeed, it has been pointed out that, at high temperatures, the Casimir force between a graphene sheet and a perfect conductor becomes comparable with that between perfect conductors \cite{fialkovsky2011finite}. Moreover, doping can strongly enhance the Casimir force between graphene sheets \cite{bordag2016enhanced}. It would be of interest to generalize the results of the present paper to compute the frictional force in those situations, and discuss whether the enhancement of the Casimir force have a corresponding effect in the frictional force or not.
{
On the other hand, we have found that the frictional force vanishes identically for speeds smaller than $v_F$.
From the point of view of applications, graphene has been regarded as one of the most promising new materials, both for its electronic and mechanical properties. Our results imply, for example, that when graphene is used in a micro-mechanical device, Casimir friction, and its concomitant energy dissipation, will not be present below the threshold, which presumably will be the best scenario for most applications.
}
\section*{Acknowledgements}
This work was supported by ANPCyT, CONICET, UBA and UNCuyo. M.B.F would like to thank Tom\'as S. Bortol\'in for valuable insights and discussions.
\begin{appendix}
\section{Static Casimir force between two graphene sheets}\label{ap2}
In absence of dissipative effects (i.e., for $v=0$), the Euclidean vacuum persistence amplitude is
\begin{equation}
\mathcal{Z}=e^{-E_0 T}
\end{equation}
where $T$ is the elapsed time, and $E_0$ is the zero-point energy of the EM field. This means that the Casimir energy per unit of area $E_C=E_0/\Sigma$ can be obtained from the Euclidean effective action of the plates when their relative velocity vanishes, that is $E_C=\gamma_{\text{Eucl}}(v=0)$.
Taking $v=0$ in Eq. \eqref{eq:gammas}, and recalling the definitions of $g_t$ and $g_l$ of Eq. \eqref{eq:gs}, the transversal contribution to the zero-point energy results
\begin{align}
E_C^t=-\frac{1}{8\, a^3} \int \frac{d^3k_{\parallel}}{(2 \pi)^3} \frac{e^{-2 \sqrt{k_0^2 + k_{\parallel}^2}}}{k_0^2+k_{\parallel}^2} g_t^2(k_0,k_{\parallel})=-\frac{1}{48}\frac{\alpha_N^2}{(2\pi)^2} \frac{1}{a^3} (1+2 v_F^2) \,.
\end{align}
Analogously, the longitudinal contribution is given by
\begin{align}
E_C^l=-\frac{1}{8\, a^3} \int \frac{d^3k_{\parallel}}{(2 \pi)^3} \frac{e^{-2 \sqrt{k_0^2 + k_{\parallel}^2}}}{k_0^2+k_{\parallel}^2} g_l^2(k_0,k_{\parallel})=-\frac{1}{16}\frac{\alpha_N^2}{(2\pi)^2} \frac{1}{a^3} \frac{\arccos(v_F)}{v_F \sqrt{1-v_F^2}} \, .
\end{align}
Considering that typical Fermi velocities are much smaller than the velocity of light, the leading contribution to the static Casimir energy between two graphene sheets comes from the longitudinal effective action and reads
\begin{equation}
E_\text{C} \approx - \frac{\alpha_N^2}{128 \pi} \frac{1}{a^3} \frac{1}{v_F} \, .
\end{equation}%
Therefore the Casimir attractive force acting on the sheets results
\begin{equation}
\label{eq:fzacas}
F_\text{C}\approx \frac{3 \alpha_N^2}{128 \pi} \frac{1}{a^4} \frac{1}{v_F} \, .
\end{equation}
\section{Details of the calculation of the imaginary part of the effective action}\label{ap1}
In order to obtain a final expression for the imaginary part of the
effective action, it is necessary to compute the desired residues. We will repeat the proceadure we did in \cite{farias_friction}. Let us start
with the longitudinal part:
\begin{align}
\label{eq:res}
\text{Res}(F_A(k_0),\Lambda_A)\equiv & \lim_{k_0 \rightarrow \Lambda_A} (k_0-\Lambda_A) F_A(k_0) \nonumber \\
=&\frac{e^{2i \sqrt{\Lambda_A^2-\textbf{k}_\parallel^2+i \epsilon}}}{\Lambda_A^2-\textbf{k}_\parallel^2+i \epsilon}\times \frac{\Lambda_A^2 - \textbf{k}_\parallel^2}{\Lambda_A^2 - v_F^2 \textbf{k}_\parallel ^2 - k_3^2+i \epsilon} \times \frac{\Lambda_A^2 -\textbf{k}_\parallel^2}{-2 \sqrt{v_F^2 k_1^2 (1-v^2)^2 +(1-v_F^2 v^2)(v_F^2 k_2^2 + p_3^2)}} \, ,
\end{align}
where in the last factor we have explicitly used the fact that the denominator is positive-definite and thus the limit $\epsilon \rightarrow 0$ can be taken with no further harm.
It could be shown that $\Lambda_A^2-\textbf{k}_\parallel^2$ is definite-negative in all the
integration region (that is, for $k_1>\sqrt{v_F^2 k_2^2 +p_3^2 /(v^2
-v_F^2)}$). This can be easily seen when
explicitly taking both $v$ and $v_F\ll1$, but the relation still holds for arbitrary
values of $v,v_F<1$. This means that we can set $\epsilon
= 0$ everywhere except in the second factor of \eqref{eq:res}, that can be written as:
\begin{equation}
\frac{1}{\Lambda_A^2 - v_F^2 \textbf{k}_\parallel ^2 - k_3^2+i \epsilon}=\frac{1}{g(k_1)+i \epsilon} \, ,
\end{equation}
with
\begin{equation}
g(k_1,k_2,k_3,p_3)=\Lambda_A^2 - v_F^2 \textbf{k}_\parallel ^2 - k_3^2 \, .
\end{equation}
Now we can explicitly take the limit $\epsilon \rightarrow 0$,
\begin{equation}
\frac{1}{g(k_1,k_2,k_3,p_3)+i \epsilon} \rightarrow \text{p.v.} \left(\frac{1}{g} \right) - i \pi \delta \left( g(k_1,k_2,k_3,p_3) \right) \, .
\end{equation}
Therefore, the longitudinal contribution to the imaginary part of the effective action reads
\begin{align}
\text{Im} \gamma^{l} = & \, \frac{\alpha_N^2}{32 \pi\, a^3} \,\theta(v-v_F) \int \frac{dk_3}{\pi} \int \frac{dp_3}{\pi} \int dk_1\int dk_2 \delta \left( g(k_1,k_2,k_3,p_3 \right) \theta\left(k_1- \sqrt{\frac{v_F^2 k_2^2 + p_3^2}{v^2-v_F^2}}\right) \\
& \times e^{-2 \sqrt{\textbf{k}_\parallel^2-\Lambda_A^2}} \frac{\textbf{k}_\parallel^2-\Lambda_A^2}{\sqrt{v_F^2 k_1^2 (1-v^2)^2 +(1-v_F^2 v^2)(v_F^2 k_2^2 + p_3^2)}} \, .
\end{align}
Note that we have is a 4-dimensional integration of a function multiplied by the Dirac-delta function composed with $g$:
\begin{equation}
\int d^4 \kappa \mathcal{F}(\bm{\kappa}) \delta(g(\bm{\kappa})) = \intop_{\mathcal{S}/g(\bm{\kappa})=0} d\sigma \frac{ \mathcal{F}(\bm{\kappa})}{|\nabla g(\bm{\kappa})|} \, ,
\end{equation}
where in our case $\bm{\kappa}=(k_1,k_2,k_3,p_3)$. The second hand is an
integration over the 3-dimensional surface defined by
$g(k_1,k_2,k_3,p_3)=0$. We can think this surface as the one defined by the
equations $k_1=x_1(k_2,k_3,p_3)$ and $k_1=x_2(k_2,k_3,p_3)$, with
\begin{align}
x_1(k_2,k_3,p_3)&= \sqrt{\frac{u(k_2,k_3,p_3)-2 \sqrt{w(k_2,k_3,p_3)}}{v^2 \left(v_F^2-1\right)^2 \left(v v_F^2+v-2 v_F\right) \left(v v_F^2+v+2 v_F\right)}} \\
x_2(k_2,k_3,p_3)&= \sqrt{\frac{u(k_2,k_3,p_3)+2 \sqrt{w(k_2,k_3,p_3)}}{v^2 \left(v_F^2-1\right)^2 \left(v v_F^2+v-2 v_F\right) \left(v v_F^2+v+2 v_F\right)}}
\end{align}
where
\begin{align}
\label{eq:ceros}
u(k_2,k_3,p_3)=&v^2 \left(1-v_F^2\right) \left\lbrace p_3^2 \left(v_F^2+1\right) + k_2^2 v_F^2 \left[v_F^2 \left(v^2 \left(v_F^2+1\right)-2\right)+2\right]+k_3^2 \left[1+ v_F^2 \left(v^2 \left(v_F^2+1\right)-3\right)\right]\right\rbrace \nonumber \\
w(k_2,k_3,p_2)=&v^2 \left(1-v_F^2\right)^2 \left\lbrace k_3^4 v_F^2 \left(v^2-1\right)^2 + k_3^2 \left[k_2^2 v^2 v_F^2 \left(2 \left(v^2-2\right) +v_F^4+1\right)+p_3^2 \left(v^2 \left(v_F^4+1\right)-2 v_F^2\right)\right] \right. \nonumber \\
& + \left. k_2^4 v^2 v_F^4 \left[1+\left(v^2-2\right) v_F^2+v_F^4\right]+k_2^2 p_3^2 v^2 v_F^2 \left(v_F^4+1\right)+p_3^4 v_F^2 \right\rbrace \, .
\end{align}
Then we have
\begin{align}
\text{Im} \gamma^{l} = & \frac{\alpha_N^2}{32 \pi\, a^3} \,\theta(v-v_F) \int \frac{dk_3}{\pi} \int \frac{dp_3}{\pi} \int dk_2 \int dk_1 \sum_{i=1,2} \frac{\delta \left( k_1 - x_i \right)}{\left| \nabla g(k_1,k_2,k_3,p_3)\right|_{k_1=x_i}} \, \theta\left(k_1- \sqrt{\frac{v_F^2 k_2^2 + p_3^2}{v^2-v_F^2}}\right) \\
& \times e^{-2 \sqrt{\textbf{k}_\parallel^2-\Lambda_A^2}} \frac{\textbf{k}_\parallel^2-\Lambda_A^2}{\sqrt{v_F^2 k_1^2 (1-v^2)^2 +(1-v_F^2 v^2)(v_F^2 k_2^2 + p_3^2)}} \, .
\end{align}
The result of the integration over $k_1$ can be written as a Heaviside
step-function of a rather involved expression depending on the rest of the
integration variables. This, and the remaining integrals have been
performed numerically.
The calculation of $\gamma^t$ proceeds in a similar way.
\end{appendix}
\input{graphene_friction_PRD.bbl}
\end{document}
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2111.04330
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\section{Introduction}
\label{sec:intro}
There is a huge imbalance between labeled and unlabeled data, as labeled data is hard to obtain, yet unlabeled data is everywhere.
Self-supervised learning (SSL) can take advantage of such large volumes of unlabeled data to mine general-purpose knowledge.
It is a new trend in natural language processing \cite{devlin2018bert} and computer vision \cite{newell2020useful} to pre-train a shared SSL upstream model followed by minimal adaptation to downstream tasks, and the features extracted by such upstream models will benefit the performance of downstream tasks.
Recently, a leaderboard named Speech processing Universal PERformance Benchmark (SUPERB) \cite{yang2021superb}, which aims at benchmarking the performance of a shared speech self-supervised learning (SSL) model across various downstream speech tasks with minimal modification of architectures and small amount of data, has fueled the research for speech representation learning.
Futhermore, SUPERB demonstrates that speech SSL upstream models can also improve and boost the performance of various downstream speech tasks through minimal adaptation.
As the paradigm of the self-supervised learning upstream model followed by downstream tasks brings significant performance gains and arouses increasing attention in the speech community, it remains to be investigated whether the paradigm is robust enough to adversarial attacks.
The concept of adversarial attack was first proposed by \cite{szegedy2013intriguing}, and the authors showed the state-of-the-art image classification models are vulnerable to adversarial attacks.
Adversarial attacks are usually indistinguishable from their genuine counterparts based on human perception, yet it can manipulate the AI models and then cause them to have catastrophic failures.
In this sense, adversarial attacks are particularly dangerous.
Speech processing models, including automatic speech recognition (ASR) \cite{carlini2018audio,yakura2018robust,taori2019targeted,qin2019imperceptible}, automatic speaker verification (ASV) \cite{villalba2020x,kreuk2018fooling,marras2019adversarial,wu2021improving,li2020adversarial,wu2021voting,wu2021adversarialasv,wu2021spotting}, anti-spoofing for ASV \cite{kassis2021practical,liu2019adversarial,wu2020defense,wu2020defense_2,zhang2020black} and voice conversion \cite{huang2021defending}, are also susceptible to adversarial attacks.
Given a piece of audio, whether music, silence or speech, the authors \cite{carlini2018audio} can generate adversarial audio, which is indistinguishable from the genuine version based on human's ears, but will cause the ASR to transcribe any adversary-desired transcriptions.
\cite{villalba2020x} and \cite{li2020adversarial} respectively illustrate that adversarial attacks can also manipulate state-of-the-art ASV systems into falsely accepting the imposters or falsely rejecting the authorized persons.
\cite{liu2019adversarial} is the first to show that the anti-spoofing model which shields ASV, are also vulnerable to adversarial attacks.
As SSL features attain the merits of generalizability and re-usability, and the paradigm equipped with SSL achieves competitive performance in speech processing tasks, such a paradigm naturally arouses keen interests from both academia and industry.
Whether such a paradigm would be an exception which can counter adversarial attacks remains an open question, and characterizing the adversarial robustness of such paradigm is of high priority.
In this paper, we make the first attempt to investigate the adversarial vulnerability of such a paradigm under the attacks from both zero-knowledge adversaries and limited-knowledge adversaries (see section \ref{subec:threat model}).
The experiments mainly focus on attacking the upstream models (Sec \ref{subsec:upstream}), including wav2vec 2.0 \cite{baevski2020wav2vec} and HuBERT \cite{hsu2021hubert}, without access to the downstream models (Sec \ref{subsec:downstream}) and task-specific procedures (Sec \ref{subsec:taks-specific module}), in order to issue the attack across tasks.
The experimental results show the paradigm proposed by SUPERB is vulnerable to limited-knowledge adversaries, and the attacks generated by zero-knowledge adversaries are transferable.
The XAB test verifies the imperceptibility of the carefully concocted adversarial attacks.
\section{Upstream-downstream paradigm}
\label{sec:background}
SUPERB first introduced the upstream-downstream paradigm for speech processing in a systematic view.
The paradigm is shown in Fig.~\ref{fig:background}~(b).
The self-supervised learning models which learn general-purpose knowledge from a large amount of unlabeled data play the role of upstream models (Sec~\ref{subsec:upstream}).
The upstream models are pre-trained and then the parameters are frozen during downstream models training and inference.
A task-specific module (Sec~\ref{subsec:taks-specific module}), consisting of a layer-wise weighted sum procedure and a pre-processing procedure, is designed for each downstream task.
The downstream models (Sec~\ref{subsec:downstream}) get the features $z$ and $\Tilde{z}$, rather than traditional acoustic features, e.g. MFCC.
\subsection{Upstream}
\label{subsec:upstream}
\subsubsection{HuBERT}
HuBERT adopts BERT-style token classification for pre-training.
Off-line unsupervised clustering is first applied to acoustic features, such as MFCC, to get frame-level noise labels.
Then the extracted features from convolutional layers are masked, and based on the noise labels, BERT-like predictive loss is applied to the masked regions.
The authors expect the pre-trained models can create better features than MFCC,
so they re-implement the clustering to the HuBERT features in the early training iterations to get better noise labels, then repeat the BERT-like pre-training.
HuBERT gets the best performance in the SUPERB.
\subsubsection{Wav2vec 2.0}
wav2vec 2.0 learns general-purpose knowledge by contrastive loss.
It firstly masks the hidden speech representations extracted by a multi-layer convolutional network from an utterance, followed by transformer layers to build contextualized representations given the hidden representations.
After quantization of the hidden representations to derive the latent for each hidden representations, a contrastive task is introduced to distinguish the true latent and the distractors.
wav2vec 2.0 achieves comparable performance with that of HuBERT across the SUPERB downstream tasks.
\subsection{Task-specific module}
\label{subsec:taks-specific module}
The task-specific module is composed of a layer-wise weighted sum procedure and a pre-processing procedure.
The layer-wise weighted sum procedure consists of task-specific weights applied to all the hidden features from different layers of the upstream model, and such weights are usually jointly trained with downstream models.
The pre-processing procedure is designed for each downstream task, such as pre-emphasis and voice activity detection.
As we proceed to training and testing for the downstream task, the audio data first undergo task-specific pre-processing, and are then fed into the upstream model to extract hidden features, followed by the layer-wise weighted sum to get the final features - these are then adopted to train the downstream model.
However, during adversarial attack, the adversaries generally do not pre-process the audio, and only use averaged embeddings from each layer of the upstream model to make the attack less task-specific.
\begin{figure*}[ht]
\centering
\centerline{\includegraphics[width=0.9\linewidth]{figs/attack-paradigm.png}}
\caption{(a) Attacking framework for SSL. $x$ and $\Tilde{x}$ are the original and adversarial samples respectively, $z_a$ and $\Tilde{z}_a$ are the features of the upstream model given $x$ and $\Tilde{x}$ as inputs respectively, and $\delta$ is the carefully designed adversarial noise. (b) Upstream-downstream paradigm. $x$ and $x'$ are the original and pre-processed samples respectively, $h_1,h_2,...,h_n$ are hidden features extracted from upstream model, where the subscript of $h_{n}$ denotes the layer number of the upstream model, and $z$ is the final features obtained from weighted sum of $h_1,h_2,...,h_n$ by the layer-wise weighted sum procedure. $\Tilde{x}$, $\Tilde{x}'$, $\Tilde{h}_1,\Tilde{h}_2,...,\Tilde{h}_n$ and $\Tilde{z}$ are the adversarial counterparts of $x$, $x'$, $h_1,h_2,...,h_n$ and $z$ respectively.}
\label{fig:background}
\end{figure*}
\subsection{Downstream Tasks}
\label{subsec:downstream}
\textbf{Phoneme Recognition (PR)} recognizes phonemes from an utterance. SUPERB selects train-clean-100, dev-clean, and test-clean subsets of LibriSpeech \cite{librispeech} as training, validation, and testing set, respectively. Phone error rate (PER) is the evaluation metric. \\
\textbf{Automatic Speech Recognition (ASR)} recognizes words from an utterance. SUPERB adopts train-clean-100, dev-clean, and test-clean subsets of LibriSpeech \cite{librispeech} as training, validation, and testing set, respectively. Word error rate (WER) is the evaluation metric.\\
\textbf{Keyword Spotting (KS)} identifies predefined keywords from an utterance. SUPERB uses Speech Commands dataset v1.0 \cite{warden2018speech}, which introduces 12 classes, including 10 for keywords, one for silence, and one for unknown words. Accuracy (ACC) is the evaluation metric. \\
\textbf{Speaker Identification (SID)} is a close-set multi-class classification task, which aims at identifying the speaker of a given utterance from a set of predefined speakers. VoxCeleb1 \cite{nagrani2017voxceleb} is adopted by SUPERB. Accuracy (ACC) is the evaluation metric.\\
\textbf{Automatic Speaker Verification (ASV)} verifies whether a pair of utterances belong to the same speaker. VoxCeleb1 \cite{nagrani2017voxceleb} is adopted in this task. Accuracy (ACC), rather than equal error rate (EER) is the evaluation metric during attack (refer to \ref{subsec:exp setup}).\\
\textbf{Speaker Diarization (SD)} identifies who is speaking during each timestamp in an utterance. LibriMix \cite{cosentino2020librimix} is used in this task, which is generated from LibriSpeech \cite{librispeech}. The speaker labels of each chunk are generated by alignments from Kaldi \cite{kaldi} LibriSpeech ASR model. Diarization error rate (DER) is the evaluation metric.\\
\textbf{Intent Classification (IC)} identifies predefined classes of intent from an utterance. Fluent Speech Commands \cite{lugosch2019speech} is used, where every utterance is labeled with one of the three intent classes: action, object, and location. Accuracy (ACC) is the evaluation metric.\\
\textbf{Slot Filling (SF)} identifies all semantic slots of an utterance with predefined slot-type and slot-value. Audio SNIPS \cite{lai2020semisupervised} is adopted by SUPERB to generate multi-speaker utterances from SNIPS \cite{coucke2018snips}. According to the standard split in SNIPS, samples of US-accent speakers are selected as training set, and others as validation and testing sets. Since slot-type and slot-value are both essential in SF task, F1 score and CER \cite{Tomashenko_2019} are adopted as evaluation metrics for slot-type and slot-value, respectively.\\
\textbf{Emotion Recognition (ER)} recognizes emotion class from an utterance. IEOMCAP \cite{Busso2008IEMOCAPIE} is used. Following the past evaluation protocol, SUPERB discards unbalanced classes, resulting in four emotion classes remaining: neutral, happy, sad, and angry. The evaluation metric is accuracy (ACC).
The models of downstream tasks are simply structured in SUPERB.
A linear network is used for \textbf{PR}, and optimized by CTC loss. For \textbf{KS}, \textbf{SID}, \textbf{IC}, and \textbf{ER}, linear transformation models after mean-pooling trained by cross-entropy loss are adopted. As for \textbf{ASR}, a 2-layer 1024-unit BLSTM is used, and optimized by CTC loss. For \textbf{SF}, slot-type labels are represented into an ordered pair wrapping the corresponding slot-value to form a sequence of tokens, and then \textbf{SF} is transformed into an \textbf{ASR} task using the same model as ASR. For \textbf{ASV}, x-vector \cite{snyder2018x} is the backbone model optimized with AMSoftmax loss \cite{wang2018cosface}, and cosine-similarity backend is used for scoring. For \textbf{SD}, a single-layer 512-unit LSTM is used with permutation-invariant training (PIT) loss \cite{fujita2019endtoend}.
\section{Adversarial attack for speech SSL}
\label{sec:method}
\subsection{Attacking scenarios}
\label{subec:threat model}
In this work, we distinguish different attack scenarios from the perspective of the knowledge accessed by adversaries.
\textbf{Limited-knowledge adversaries}: Attackers can access the internals of the target upstream model, including the detailed parameters and gradients.
But they do not know which downstream task will be conducted, not to mention the internals of downstream models, the weights of the layer-wise weighted sum procedure and task-specific pre-processing procedures.
Knowing the internals of the target model, the attackers will directly calculate the gradients and generate adversarial samples, as shown in Fig.~\ref{fig:background}.(a).
\textbf{Zero-knowledge adversaries}: In this scenario, while the attackers aim at the \emph{target model}, it is unavailable to the attackers.
In such a case, the \emph{substitute model} is used for approximating gradients for adversarial sample generation.
Zero-knowledge adversaries can get even less knowledge than limited-knowledge adversaries.
Zero-knowledge adversaries do not even know the details of the target upstream model internals.
In order to conduct adversarial attacks, they have to train another substitute model, adopt the gradients of the substitute model to generate adversarial samples, and finally use such adversarial samples to fool the upstream-downstream paradigm using the target model.
\subsection{Attack procedure}
\label{subsec:attacking method}
Under the scenarios of zero-knowledge attacks and limited-knowledge attacks, attackers only have access to the upstream model without knowing the task-specific module (Section~\ref{subsec:taks-specific module}) and the downstream model (Section~\ref{subsec:downstream}) to craft the adversarial attacks.
For limited-knowledge adversaries, they have access to the target upstream model internals, while the zero-knowledge adversaries will train a substitute upstream model and use it to generate adversarial attacks.
During an attack, the weights in layer-wise weighted sum procedure are set as equal to average the embeddings in each layer of the upstream model to derive $z_a$ and $\Tilde{z}_a$.
Whatever the downstream task is, we only manipulate the upstream model to generate adversarial samples.
So in such a scenario, the attack method introduced below is less task-specific.
The attacking framework is shown in Fig.~\ref{fig:background}(a).
Having fixed parameters in the upstream model, the attackers aim at crafting the adversarial noise $\delta$ to maximize the difference between $z_a$ and $\Tilde{z}_a$, while also let $x$ and $\Tilde{x}$ indistinguishable from human's ears.
In order to fulfill the above two objectives, we introduce the basic iterative method (BIM) \cite{kurakin2016adversarial_2} for attack.
BIM crafts the adversarial sample in an iterative manner. Starting from the genuine input $x^{0}=x$, the input is perturbed iteratively as
\begin{equation}
\begin{aligned}
x^{n+1} = & \text{ }clip_{x, \epsilon}(x^{n}+ \delta), \\
& for \text{ } n = 0, ..., N-1 \label{eq:bim}
\end{aligned}
\end{equation}
where $\delta$ is the carefully designed adversarial noise, derived as
\begin{equation}
\delta = \alpha \times sign(\nabla_{x^{n}} \Vert z_a-\Tilde{z}_a \Vert_2)
\label{eq:bim-solu-2}
\end{equation}
where $\alpha$ is the step size, $sign(\cdot)$ is a function which gets the sign of the gradient, $N$ is the number of iterations and $clip_{x, \epsilon}(t)$ is the norm constraint which conducts element-wise clipping such that $\Vert t-x \Vert_{\infty} \leq \epsilon$ to assure the original sample $x$ and the derived adversarial sample $x^{N}$ are indistinguishable. $x^{N}$ will be used for attacking the upstream-downstream paradigm.
\begin{table*}[th]
\caption{Adversarial attack performance on SSL representations for various downstream tasks.}
\label{tab:adv-attack-performance}
\centering
\scalebox{0.95}{
\begin{threeparttable}
\begin{tabular}{cc|c|c|c|c|c|c|c|c|c|c|c}
\hline
\hline
& \multirow{2}{*}{} & ASR & PR & KS & IC & \multicolumn{2}{c|}{SF} & SID & ER & \multicolumn{2}{c|}{SD} & ASV \\
\cline{3-13}
& & WER $\downarrow$ & PER $\downarrow$ & Acc $\uparrow$ & Acc $\uparrow$ & F1 $\uparrow$ & CER $\downarrow$ & Acc $\uparrow$ & Acc $\uparrow$ & Acc $\uparrow$ & DER $\downarrow$ & Acc $\uparrow$ \\
\hline
\multirow{2}{*}{(a)} & \multirow{2}{*}{w2v2-w2v2} & 19.20\tnote{1} & 28.32 & 65.67 & 55.67 & 88.55 & 20.19 & 81.33 & 79.33 & 88.48 & 17.48 & 91.67 \\
& & (\textpm2.01) & (\textpm2.03) & (\textpm6.51) & (\textpm5.77) & (\textpm1.33) & (\textpm2.05) & (\textpm3.06) & (\textpm3.79) & (\textpm0.19) & (\textpm0.55) & (\textpm2.31) \\
\cline{3-13}
\multirow{2}{*}{(b)} & \multirow{2}{*}{HuBERT-w2v2} & 5.54 & 5.09 & 91.00 & 88.33 & 95.36 & 8.70 & 87.67 & 87.33 & 94.56 & 8.08 & 97.00 \\
& & (\textpm0.71) & (\textpm0.47) & (\textpm3.00) & (\textpm1.15) & (\textpm1.26) & (\textpm0.55) & (\textpm4.16) & (\textpm6.03) & (\textpm0.36) & (\textpm0.41) & (\textpm2.00) \\
\cline{3-13}
\multirow{2}{*}{(c)} & \multirow{2}{*}{gau-w2v2} & 0.48 & 1.11 & 98.67 & 93.67 & 99.71 & 0.71 & 97.67 & 95.67 & 98.24 & 2.51 & 99 \\
& & (\textpm0.06) & (\textpm0.05) & (\textpm0.58) & (\textpm1.15) & (\textpm0.27) & (\textpm0.50) & (\textpm2.08) & (\textpm3.06) & (\textpm0.09) & (\textpm0.11) & (\textpm0.00) \\
\cline{3-13}
(d) & Clean-w2v2 & 0 & 0 & 100 & 100 & 100 & 0 & 100 & 100 & 98.24 & 2.51 & 100 \\
\hline
\multirow{2}{*}{(e)} & \multirow{2}{*}{HuBERT-HuBERT} & 26.76 & 18.67 & 64.33 & 69.67 & 76.91 & 36.54 & 76.33 & 78.33 & 87.78 & 18.39 & 88.33 \\
& & (\textpm0.82) & (\textpm1.54) & (\textpm0.58) & (\textpm5.03) & (\textpm1.67) & (\textpm1.83) & (\textpm4.93) & (\textpm2.08) & (\textpm0.83) & (\textpm1.65) & (\textpm2.08) \\
\cline{3-13}
\multirow{2}{*}{(f)} & \multirow{2}{*}{w2v2-HuBERT} & 1.94 & 2.21 & 96.67 & 98.33 & 99.42 & 1.62 & 93.67 & 91.00 & 95.13 & 7.17 & 96.67 \\
& & (\textpm0.06) & (\textpm0.28) & (\textpm1.15) & (\textpm1.15) & (\textpm0.37) & (\textpm0.16) & (\textpm1.15) & (\textpm2.65) & (\textpm0.20) & (\textpm0.47) & (\textpm1.53) \\
\cline{3-13}
\multirow{2}{*}{(g)} & \multirow{2}{*}{gau-HuBERT} & 0.05 & 0.42 & 99.67 & 99.67 & 99.89 & 0.25 & 98.67 & 99.00 & 98.36 & 2.32 & 99.67 \\
& & (\textpm0.08) & (\textpm0.12) & (\textpm0.58) & (\textpm0.58) & (\textpm0.19) & (\textpm0.24) & (\textpm2.31) & (\textpm0.00) & (\textpm0.09) & (\textpm0.13) & (\textpm0.58) \\
\cline{3-13}
(h) & Clean-HuBERT & 0 & 0 & 100 & 100 & 100 & 0 & 100 & 100 & 98.37 & 2.31 & 100 \\
\hline
\hline
\end{tabular}
\begin{tablenotes}
\footnotesize
\item[1] We show the mean and standard deviation. Here 19.20 \textpm2.01 means that the mean and standard deviation of WER are 19.20\% and 2.01\%.
\end{tablenotes}
\end{threeparttable}
}
\end{table*}
\section{Experiment}
\label{sec:expt}
\subsection{Experimental setup}
\label{subsec:exp setup}
We train the upstream models in Fig.~\ref{fig:background}.(b), and fix its parameters as the downstream models are trained.
We omit the implementation details due to space limitation, and readers can refer to \cite{yang2021superb} for more information.
Note that the adversarial attacks are conducted during inference.
As the adversarial attack is time- and resource-consuming, we randomly selected 100 genuine samples for attacking, do the experiments three times, and then report the mean and variance of results.
Note that for ASV, we randomly select 50 non-target and target trials.
The ACC for ASV is derived by the number of trials with the right decision over the total trial number.
The performance for the genuine samples is shown in rows (d) and (h) of Table~\ref{tab:adv-attack-performance}. The standard deviations are less than 0.1\%, so we only show the means in rows (d) and (h).
Then we craft adversarial samples according to the attacking methods as in Section~\ref{subsec:attacking method}.
Gaussian noise of the same noise-to-signal ratio (NSR) with adversarial perturbations is introduced for comparison (in rows (c) and (g) ).
\begin{table}[th]
\caption{Adversarial attack performance ASR.}
\label{tab:adv-attack-performance-asr}
\centering
\begin{tabular}{c|c|c|c}
\hline
\hline
& NSR & EDR & WER \\
\hline
w2v2-w2v2 & 0.67(\textpm0.01) & 81.15(\textpm0.27) & 19.20(\textpm2.01) \\
HuBERT-w2v2 & 0.71(\textpm0.00) & 58.09(\textpm0.09) & 5.54(\textpm0.71) \\
gau-w2v2 & 0.69(\textpm0.00) & 30.40(\textpm0.19) & 0.48(\textpm0.06) \\
\hline
\hline
\end{tabular}
\end{table}
\subsection{Experimental Results}
Table~\ref{tab:adv-attack-performance} illustrates the attack performance on SSL for 9 downstream tasks.
The direction of the arrow in the second row denotes the direction towards the better performance of the task.
For example, $\downarrow$ for ASR means the lower WER implies better ASR performance, yet the less effective the attacks are.
The first column in Table~\ref{tab:adv-attack-performance} lists \emph{the method to generate the attack model and the target model}.
For example, w2v2-w2v2 denotes that the substitute model for generating adversarial samples is wav2vec 2.0, and the target model is also wav2vec 2.0.
The rows (a) and (e) are the limited-knowledge scenarios.
The rows (b) and (f) are the zero-knowledge scenarios.
gau-w2v2 denotes using the samples perturbed by Gaussian noise to attack wav2vec 2.0.
We adopt gau-w2v2 and gau-HuBERT as our baseline, and the results are in rows (c) and (g).
We have these observations:
(1) Limited-knowledge attackers achieve the most effective attack on wav2vec 2.0 and HuBERT for all downstream tasks. Take the IC task as an example, ``w2v2-w2v2" degrades the Acc of wav2vec 2.0 from 100\% to 55.67\%, and ``HuBERT-HuBERT" degrades that of HuBERT from 100\% to 69.67\%. These results verify the severe threats that adversarial attacks can pose on the SSL models. We also observe similar trends for all other 8 tasks.
(2) Zero-knowledge attackers achieve relatively weaker attacks on downstream tasks than limited-knowledge attackers, but the attack is still effective. Specifically, in some downstream tasks, zero-knowledge attackers also seriously degrade the system performance. For instance, in the ER downstream task, ``HuBERT-w2v2" degrades the Acc of wav2vec 2.0 from 100\% to 87.33\%, and ``w2v2-HuBERT" degrades the Acc of HuBERT from 100\% to 91.00\%.
(3) To verify the effectiveness of adversarial perturbations, we leverage the Gaussian noise for comparison. We observe that Gaussian noise degrades the wav2vec 2.0 and HuBERT much less for all downstream tasks compared with both limited-knowledge and zero-knowledge attacks. For instance, in the ASR task, Gaussian noise has little effect on the wav2vec 2.0 WER degradation (0.48\%) and practically has no influence on the HuBERT WER degradation (0.05\%).
This suggests that simply adding Gaussian noise cannot degrade a well-trained system for the malicious attack purpose and verifies the effectiveness and transferability of our attacks.
Moreover, we also compare the NSR and embedding distance rate (EDR) for the limited-knowledge attackers, zero-knowledge attackers and Gaussian noise in Table~\ref{tab:adv-attack-performance-asr}.
EDR is calculated by $1/N \times \sum_{n=1}^{N} \Vert z_{n}-\Tilde{z}_{n} \Vert_{2} / \Vert z_{n} \Vert_{2}$, where $z_{n}$ and $\Tilde{z}_{n}$ are the adversarial-original embedding pair, $N$ is the total adversarial-original pair number.
Here we only show the results on the ASR task with wav2vec 2.0 as target model due to space limitation, while other tasks have similar trends.
Table~\ref{tab:adv-attack-performance-asr} illustrates the results.
We set the NSR of all perturbations at a similar scale for fair comparison.
We observe that the EDR has similar trends with the WER.
Specifically, limited-knowledge attackers make the embeddings of adversarial inputs most far away from the original ones, resulting in the largest EDR when compared with the zero-knowledge attackers and Gaussian noise. For instance, ``w2v2-w2v2" achieves an EDR of 81.15\%.
For zero-knowledge attackers, ``HuBERT-w2v2" achieves an EDR of 58.09\%, which is less than those of the limited-knowledge attackers.
While for Gaussian noise, ``gau-w2v2" only achieves an EDR of 30.40\%.
This suggests the less effectiveness of Gaussian noise.
Finally, we observe a consistency between EDR and the system performance WER.
\subsection{XAB test}
To illustrate the imperceptibility of adversarial noise, we conduct the XAB listening test. XAB listening test is a standard test to evaluate the detectability between two choices of sensory stimuli.
We randomly select 2 adversarial-genuine pairs for each upstream-downstream paradigm.
36 randomly selected adversarial-genuine pairs (i.e., A and B) are shown to the listeners.
We randomly choose one from each pair as the reference audio (i.e., X) and let the listeners select the audio, which sounds more similar to X, from A and B.
Five listeners take part in the XAB listening test.
The XAB test has a classification accuracy of 58.89\%, which illustrates that the adversarial samples are hard to be distinguished from genuine samples.
Audio demos with the attacking settings are made open here \footnote[2]{\href{https://bzheng1024.github.io/adv-audio-demo/index.html}{Audio demo}}.
\section{conclusion}
\label{sec:conclusion}
Adversarial robustness is an AI standard for trustworthy machine learning systems.
Though the paradigm proposed by SUPERB is going to penetrate all the speech processing tasks and gains good performance, the adversarial robustness has not received sufficient consideration.
This work is the first to expose the vulnerability of such paradigm to adversarial attacks.
In future work, we will investigate attacks with higher transferability and less imperceptibility.
As more sophisticated attacks continue to be developed, we will need to come up with defense methods to alleviate such attacks.
The long-term goal is to design adaptive defense methods that offer protection against increasingly dangerous attacks.
\section{acknowledgements}
\label{sec:acknowledges}
This work was done when H. Wu was a visiting student at Centre for Perceptual and Interactive Intelligence, The Chinese University of Hong Kong.
H. Wu is supported by Google PHD Fellowship Scholarship.
|
2003.13162
|
\section{Introduction}
This paper is concerned with explicitly proving the convergence of a scalar ensemble Kalman filter in three cases: finite step and asymptotic ensemble, finite ensemble and asymptotic step, and finite step and finite ensemble with optimal inflation.
Bayesian data assimilation~\cite{asch2016data,law2015data,evensen2009data,reich2015probabilistic}, in our view, is concerned with transforming our \textit{a priori} uncertainty about the state of a (often chaotic) dynamical system~\cite{strogatz2014nonlinear}, and our uncertainty about observations of some truth. Representatives of our uncertainty in this context are taken to be the distributions of some random variables, and an application of Bayesian inference would involve applying Bayes' rule in an exact manner. In the vast majority of cases this problem is intractable.
By the principle of maximum entropy~\cite{jaynes2003probability}, if an there exists an ensemble of states with support from all of real space, and there is no other prior information, then if the mean and the covariance of the ensemble are known, then the best distribution that we can prescribe to the ensemble to describe our uncertainty is Gaussian. The ensemble Kalman filter~\cite{evensen1994sequential,le2009large,kwiatkowski2015convergence,mandel2011convergence,butala2008asymptotic} (EnKF) abuses this notion by assuming that the first two statistical moments of the ensemble are good descriptions of the exact moments, and that no other moments are known. In this way, the EnKF attempts to utilize the Kalman filter~\cite{kalman1960new} (KF) framework by substituting exact moments for their ensemble-derived statistical estimates. The advantage of the EnKF is in both the utilization of ensemble propagation~\cite{kalnay2003atmospheric}., and in better estimation of model forecast uncertainty. As the transformations defined by the EnKF do not exactly solve the problem of Bayesian inference, unlike those of the KF, the EnKF is wrought with heuristic attempts to correct it. One such heuristic is inflation~\cite{anderson1999monte}, which is thought to separate the anomalies in space in order for the ensemble covariance estimate to not degenerate prematurely.
All previous attempts at the convergence of the EnKF have looked at the asymptotic case of a large ensemble~\cite{le2009large,kwiatkowski2015convergence}. To date there has not been a full comprehensive analysis of the EnKF in the case of a finite ensemble, and more importantly, in the case of finite steps. We do not claim to provide such an analysis in the general case, but instead in a simplified scalar case. Additionally we also aim to explicitly derive inflation as not a heuristic, but as a natural consequence to the EnKF analysis non-linear action on the ensemble.
The paper is organized as follows. In section~\ref{sec:bg} we review the Kalman filter, and an over-sampled square-root ensemble Kalman filter.
We then introduce the scalar Kalman filter in section~\ref{sec:skf} and prove that it correctly models the uncertainly in a scalar linear dynamical system.
We follow this with an introduction of a toy scalar ensemble Kalman filter meant for pedagogical purposes called the Scalar Pedagogical Ensemble Kalman Filter (SPEnKF) in section~\ref{sec:spenkf}.
We derive explicit probability distributions~\cite{mood1974introduction}, explicit formulations of their moments, and explicitly describe the asymptotic behavior in both ensemble size and steps of such a formulation.
We then show that the distributions of the SPEnKF's mean and variance estimates degenerate to that of the scalar Kalman filter in the asymptotic ensemble case (subsection~\ref{subsec:ppfe}).
We derive sequential step-wise variance inflation, and mean correction factors, such that when these factors are applied, the expected values of the mean and variance estimates of the SPEnKF are exactly the mean and variance estimate of the scalar Kalman filter (subsection~\ref{subsec:opti}).
Moreover, we show that in the step limit that the finite-ensemble SPEnKF converges in probability, regardless of model behavior, to that of the Kalman filter (subsection~\ref{subsec:cfe}).
We then use our framework to look as to why EnKF with perturbed observations can potentially behave in a suboptimal manner (subsection~\ref{subsec:po}). Next, we provide a trivial multivariate extension ot the SPEnKF, and show that a form of localization can indeed reduce the need for an oversampled ensemble in section~\ref{sec:multivariate}. We end with some final thoughts in section~\ref{sec:conclusions}.
\section{Background}%
\label{sec:bg}
Consider the case of capturing our uncertainty about an unknown dynamical system,
\begin{equation}
\*\mx^{\|t}_{i+1} = \!M^{\|t}_{i}(\*\mx^{\|t}_{i}),
\end{equation}
that evolves a true state, $\*\mx^{\|t}_{i}$, from step $i$ to step $i+1$.
Now, consider us having access to an imperfect model of this dynamical system,
\begin{equation}
\label{eq:model}
X_{i+1} = \!M_{i}(X_{i}) + \xi_{i}, \quad i=0,2,\dots
\end{equation}
where the distribution of the random variables $X_{i}$ and $X_{i+1}$ represent our uncertainty about the true state at the respective steps, and the distribution of the random variable $\xi_{i}$ represents our uncertainty in the model propagation, commonly referred to as model error.
\begin{assumption}[Initial state]
\label{ass:state}
We assume that we have uncertainty about the initial state, and that this uncertainty is described by a normal distribution $X_0 \sim \mathcal{N}(\overline{\mx}_0,\*P_0)$.
\end{assumption}
\begin{assumption}[Model]
\label{ass:model}
We make the following simplifying assumptions:
\begin{enumerate}
\item The model \eqref{eq:model} is linear, $\!M_{i}\coloneqq\*M_{i}$, and
\item The model error is an unbiased normal random variable, $\*\xi_{i} \sim \!N(0,\*Q_{i})$.
\end{enumerate}
\end{assumption}
Similarly, the observations, $\*Y_{i}$, at step $i$ correspond to a transformation of the state of our system into a (usually lower dimensional) observation space, through an observation operator, $\!H_i$. Thus an observation at time $i$ can be obtained from
\begin{equation}
\label{eq:observations}
\*Y_{i} = \!H_i(X_{i}) + \eta_{i},
\end{equation}
where similarly, the distribution of the random variable, $\eta_{i}$, represents our uncertainty in the observations, is commonly referred to as observation error, and is typically used to account for inaccuracies in our measurements.
\begin{assumption}[Observations]
\label{ass:observations}
We make the following simplifying assumptions:
\begin{enumerate}
\item The observation operator \eqref{eq:observations} is linear, $\!H_i\coloneqq\*H_i$, and
\item The observation error is an unbiased normal random variable, $\eta_i \sim \!N(0,\*R_i)$.
\end{enumerate}
\end{assumption}
Under the stated Assumptions \ref{ass:state}, \ref{ass:model}, \ref{ass:observations}, and a perfect application of Bayes' rule, our uncertainty in the state of our system remains Gaussian at all times. The \textit{a priori} (forecast) probability distribution of the uncertainty in the state at the current step $i$ is $\!N(\mxmean^{\|f}_{i},\P^{\|f}_{i})$, and the \textit{a posteriori} (analysis) probability distribution of the uncertainty in the state at step $i$ is $\!N(\mxmean^{\|a}_{i},\P^{\|a}_{i})$.
The {\it forecast step} propagates the mean and covariance of our uncertainty in the state through the model \eqref{eq:model} from step $i$ to $i+1$:
\begin{equation}
\label{eq:kff}
\begin{split}
\mxmean^{\|f}_{i+1} &= \*M_{i} \mxmean^{\|a}_{i},\\
\P^{\|f}_{i+1} &= \*M_{i} \P^{\|a}_{i} \*M_{i}^\intercal + \*Q_{i},
\end{split}
\end{equation}
where $\mxmean^{\|a}_0 \coloneqq \overline{\mx}_0$, and $\P^{\|a}_0 \coloneqq \*P_0$.
The corresponding previous {\it analysis step} applies the canonical Kalman filter equations \cite{kalman1960new,sarkka2013bayesian}
\begin{equation} \begin{split}
\begin{split}
\mxmean^{\|a}_{i} &= \mxmean^{\|f}_{i} - \*K_{i}(\*H_{i}\mxmean^{\|f}_{i} - \*Y_{i}),\\
\P^{\|a}_{i} &= (\*I - \*K_{i}\*H_{i})\P^{\|f}_{i},\\
\*K_{i} &= \P^{\|f}_{i}\*H_{i}^\intercal{(\*H_{i}\P^{\|f}_{i}\*H_{i}^\intercal + \*R_{i})}^{-1},
\end{split}\label{eq:kfa}
\end{split} \end{equation}
to obtain the best linear unbiased estimate of our uncertainty in the state of a linear dynamical system under Gaussian error assumptions. Equation~\eqref{eq:kfa} calculates the \text{a posteriori} uncertainty from the prior information and the information described by the observations (and our uncertainty in them).
The ensemble Kalman filter takes a Monte Carlo approach to represent the prior and posterior probability densities.
The ensemble Kalman filter, instead of representing our uncertainties by the first two empirical moments of a normal distribution, attempts to represent our uncertainty by the first two statistical moments of an ensemble of samples.
One replaces the analytical Gaussian density defined by the mean and covariance with an empirical distribution defined by an ensemble of $N$ states, $\En{X} = [\mxs{1},\mxs{2}, \dots, \mxs{N}]$. The ensemble mean $\overline{\mx}$ will now represent the mean estimate of the Kalman filter, and the sample covariance estimate will similarly represent the covariance estimate of the Kalman filter. Recall that a sample covariance $(1/(N-1)) \*A \*A^{\intercal}$ is calculated using the matrix of sample anomalies, $\*A = \En{X} - \overline{\mx}\,\1_N^\intercal$, which is the matrix of the differences between the ensemble members and the ensemble mean.
In our formulation of the ensemble Kalman filter, we ignore model error ($\*Q_i = \*0$ in \eqref{eq:model}), we set the observation error covariance matrix and the observation operator to be constant in time ($\*R_i = \*R$, $\*H_i = \*H$), and look at an ideal oversampled square-root filter, in which the covariance matrix estimates come from a distribution with finite variances. In a square-root filter\cite{tippett2003ensemble} the covariance is transported through the analysis step using a transformation of the probability distribution. This transformation is typically done on the ensemble anomalies. In a perfect square-root filter, with a linear model, both the mean and the anomalies can be completely decoupled from each other, thus we will take the anomalies to not be derived from the ensemble mean at all, thus getting an additional degree of freedom, making our statistical covariance estimate $(1/N) \*A \*A^{\intercal}$ instead.
We write the mean propagation by the EnKF in a similar manner to that of the Kalman filter:
\begin{equation} \begin{split}
\mxmean^{\|f}_{i+1} &= \*M_{i} \mxmean^{\|a}_{i},\\
\mxmean^{\|a}_{i+1} &= \mxmean^{\|f}_{i+1} - \hat{\K}_{i+1}(\*H\mxmean^{\|f}_{i+1} - \*Y_{i+1}).
\end{split}\label{eq:meansqrttransport}
\end{equation}
We propagate the ensemble anomalies in a way that follows the linear structure of the base Kalman filter, through an approximate transport of distributions. Let $\hat{\P}{\vphantom{\P}}^{\|f}$ denote the ensemble estimate of the \textit{a priori} covariance matrix , and $\hat{\P}{\vphantom{\P}}^{\|a}$ denote the ensemble estimate of the \textit{a posteriori} covariance matrix. The EnKF propagated anomalies and covariances at the corresponding previous step $i$ take the form:
\begin{equation} \begin{split}
\A^{\|f}_{i} &= \*M_{i-1} \A^{\|a}_{i-1},\\
\A^{\|a}_{i} &= {\left(\hat{\P}{\vphantom{\P}}^{\|a}_{i}\right)}^{\frac{1}{2}}{\left(\hat{\P}{\vphantom{\P}}^{\|f}_{i}\right)}^{-\frac{1}{2}}\A^{\|f}_{i},\\
\hat{\K}_{i} &= \hat{\P}{\vphantom{\P}}^{\|f}_{i}\*H^\intercal{(\*H\hat{\P}{\vphantom{\P}}^{\|f}_{i}\*H^\intercal + \*R)}^{-1},\\
\hat{\P}{\vphantom{\P}}^{\|f}_{i} &= \frac{1}{N}(\A^{\|f}_{i} \A^{\|f,\mathsf{T}}_{i}),\\
\hat{\P}{\vphantom{\P}}^{\|a}_{i} &= (\*I - \hat{\K}_{i}\*H)\hat{\P}{\vphantom{\P}}^{\|f}_{i}.
\end{split}\label{eq:anomalysqrttransport}
\end{equation}
Note that, as the anomalies undergo a non-linear transformation through equation~\eqref{eq:anomalysqrttransport}, the distribution of the ensemble-estimated analysis covariance matrix is not the (scaled) Wishart distribution.
\section{The Scalar Kalman Filter (SKF)}%
\label{sec:skf}
We focus on the analysis of a scalar Kalman filter as this allows us to obtain analytical results that are almost intractable in the multivariate case.
\subsection{Definition of the SKF}
\begin{assumption}[Perfect scalar model]\label{ass:pm}
Our model state comes from $x \in \mathbb{R}$, the linear model is exact (no model error), can vary at each step, and is non-trivial, $\*M_i\coloneqq m_i\not=0$.
\end{assumption}
We can think of this assumption as requiring that that truth is also propagated through the scalar linear model,
\begin{align}
\x^{\|t}_{i+1} = m_i \x^{\|t}_i.
\end{align}
\begin{assumption}[Direct observation]
We observe our only component directly, $\*H \coloneqq h = 1$.
\end{assumption}
\begin{assumption}[Constant observation error]\label{ass:constobs}
The distribution of the observation error will be taken to be the same at each step, and each observation, $y_i$, is to be drawn from a normal distribution with mean $\x^{\|t}_i$ and variance $\*R\coloneqq r>0$.
\end{assumption}
The filtering process starts with the initial values $\x^{\|f}_0 \coloneqq x_0$, and $\p^{\|f}_0 \coloneqq p_0$.
Our model propagation step (equation~\eqref{eq:kff} in the multivariate case) is:
\begin{equation}
\label{eq:skfvp}
\begin{split}
x_{i+1}^\|f &= m_i \x^{\|a}_i,\\
\p^{\|f}_{i+1} &= m_i^2 \p^{\|a}_i.
\end{split}
\end{equation}
The corresponding analysis step (equation~\eqref{eq:kfa} in the multivariate case) has the form:
\begin{equation}
\begin{split}
\label{eq:skfa}
\x^{\|a}_i &= \x^{\|f}_i + k_i(y_i - \x^{\|f}_i),\\
\p^{\|a}_i &= (1 - k_i)\p^{\|f}_i,\\
k_i &= \frac{\p^{\|f}_i}{\p^{\|f}_i + r}.
\end{split}
\end{equation}
Next we will prove some fundamental things about linear propagation in the scalar case.
\subsection{Properties of the SKF}
We first wish to analyze the propagation of variance through the filter. We will now prove that the only non-linear operation that happens to the variance is in the computation of the Kalman gain. Note that propagation of variance through the model is trivially linear from~\eqref{eq:skfvp}. We now prove that the analysis variance is a linear scaling of the Kalman gain.
\begin{lemma}\label{lem:par}
The analysis variance at the $i$-th step is $\p^{\|a}_i = r k_i$.
\end{lemma}
\begin{proof}
We manipulate the variance analysis in equation~\eqref{eq:skfvp}:
%
\begin{equation*}
\begin{split}
\p^{\|a}_i &= (1-k_i)\p^{\|f}_i = \left(1-\frac{\p^{\|f}_i}{\p^{\|f}_i + r}\right)\p^{\|f}_i\\
&= \left(\frac{\p^{\|f}_i + r}{\p^{\|f}_i + r} - \frac{\p^{\|f}_i}{\p^{\|f}_i + r}\right)\p^{\|f}_i = \frac{r \p^{\|f}_i}{\p^{\|f}_i + r} = r k_i.
\end{split}
\end{equation*}
%
\end{proof}
Moreover we can show that the Kalman gain at each step is a linear fractional function of the initial input variance.
\begin{lemma}\label{lem:kalmangain}
The Kalman gain in the scalar Kalman filter at the $i$-th step is
%
\begin{equation*}
k_i = \frac{\left(\prod_{j=0}^{i-1}m_j^2\right)p_0}{\left(\sum_{l=0}^{i}\prod_{j=0}^{l-1}m_j^2\right)p_0 + r}.
\end{equation*}
%
\end{lemma}
\begin{proof}
The Kalman gain at step $0$ is clearly $k_0=\frac{p_0}{p_0+r}$, now we manipulate in typical inductive fashion.
Assume that,
%
\begin{equation*}
\begin{split}
k_{q-1} &= \frac{\left(\prod_{j=0}^{q-2}m_j^2\right)p_0}{\left(\sum_{l=0}^{q-1}\prod_{j=0}^{l-1}m_j^2\right)p_0 + r},\\
\p^{\|a}_{q-1} &= r\frac{\left(\prod_{j=0}^{q-2}m_j^2\right)p_0}{\left(\sum_{l=0}^{q-1}\prod_{j=0}^{l-1}m_j^2\right)p_0 + r},
\end{split}
\end{equation*}
%
then, by~\eqref{eq:skfvp} and Lemma~\ref{lem:kalmangain},
%
\begin{equation*}
\begin{split}
\p^{\|f}_{q} &= m_{q-1}^2 r\frac{\left(\prod_{j=0}^{q-2}m_j^2\right)p_0}{\left(\sum_{l=0}^{q-1}\prod_{j=0}^{l-1}m_j^2\right)p_0 + r} = r\frac{\left(\prod_{j=0}^{q-1}m_j^2\right)p_0}{\left(\sum_{l=0}^{q-1}\prod_{j=0}^{l-1}m_j^2\right)p_0 + r}\\
k_q &= {\left[r\frac{\left(\prod_{j=0}^{q-1}m_j^2\right)p_0}{\left(\sum_{l=0}^{q-1}\prod_{j=0}^{l-1}m_j^2\right)p_0 + r}\right]} {\left[r\frac{\left(\prod_{j=0}^{q-1}m_j^2\right)p_0}{\left(\sum_{l=0}^{q-1}\prod_{j=0}^{l-1}m_j^2\right)p_0 + r}+r\right]}^{-1}\\
&= {\left[\frac{\left(\prod_{j=0}^{q-1}m_j^2\right)p_0}{\left(\sum_{l=0}^{q-1}\prod_{j=0}^{l-1}m_j^2\right)p_0 + r}\right]} {\left[\frac{\left(\prod_{j=0}^{q-1}m_j^2\right)p_0+\left(\sum_{l=0}^{q-1}\prod_{j=0}^{l-1}m_j^2\right)p_0 + r}{\left(\sum_{l=0}^{q-1}\prod_{j=0}^{l-1}m_j^2\right)p_0 + r}\right]}^{-1}\\
&= \frac{\left(\prod_{j=0}^{q-1}m_j^2\right)p_0}{\left(\sum_{l=0}^{q}\prod_{j=0}^{l-1}m_j^2\right)p_0 + r}.
\end{split}
\end{equation*}
%
\end{proof}
We can extend this approach the analysis mean as well, meaning that the analysis mean at each step is a linear fractional function of the initial input variance and the initial input mean.
\begin{lemma}\label{lem:analysismean}
The analysis at the $i$-th step is
\begin{equation}
\x^{\|a}_i = \frac{\left(\prod_{j=0}^{i-1}m_j\right)\left[\left(\sum_{l=0}^i y_l \prod_{j=0}^{l-1}m_j\right)p_0+ r x_0 \right]}{\left(\sum_{l=0}^{i}\prod_{j=0}^{l-1}m_j^2\right)p_0 + r}.
\end{equation}
\end{lemma}
\begin{proof}
Clearly $\x^{\|a}_0 = x_0 - \frac{p_0}{p_0 + r}(x_0 - y_0)=\frac{y_0 p_0 + r x_0}{p_0+r}$. We thus proceed by induction:
%
\begin{equation*}
\begin{split}
\x^{\|a}_{q-1} &= \frac{\left(\prod_{j=0}^{q-2}m_j\right)\left[\left(\sum_{l=0}^{q-1} y_l \prod_{j=0}^{l-1}m_j\right)p_0+ r, x_0\right]}{\left(\sum_{l=0}^{q-1}\prod_{j=0}^{l-1}m_j^2\right)p_0 + r}\\
\x^{\|a}_q &= x^\|f_q - k_q(\x^{\|f}_q-y_q)\\
&= (1-k_q)m_{q-1}\x^{\|a}_{q-1}+k_q y_q\\
&= \left[1-\frac{\left(\prod_{j=0}^{q-1}m_j^2\right)p_0}{\left(\sum_{l=0}^q\prod_{j=0}^{l-1}m_j^2\right)p_0 + r}\right] \frac{\left(\prod_{j=0}^{q-1}m_j\right)\left[\left(\sum_{l=0}^{q-1} y_l \prod_{j=0}^{l-1}m_j\right)p_0+ r x_0\right]}{\left(\sum_{l=0}^{q-1}\prod_{j=0}^{l-1}m_j^2\right)p_0 + r} + \frac{y_q\left(\prod_{j=0}^{q-1}m_j^2\right)p_0 }{\left(\sum_{l=0}^q\prod_{j=0}^{l-1}m_j^2\right)p_0 + r}\\
&= \frac{\left[\left(\sum_{l=0}^q\prod_{j=0}^{l-1}m_j^2\right)p_0 + r\right]\left(\prod_{j=0}^{q-1}m_j\right)\left[\left(\sum_{l=0}^{q-1} y_l \prod_{j=0}^{l-1}m_j\right)p_0+ r x_0 + y_q\left(\prod_{j=0}^{q-1}m_j^2\right)p_0\right]}{\left[\left(\sum_{l=0}^q\prod_{j=0}^{l-1}m_j^2\right)p_0 + r\right]\left[\left(\sum_{l=0}^{q+1}\prod_{j=0}^{l-1}m_j^2\right)p_0 + r\right]}\\
&= \frac{\left(\prod_{j=0}^{q-1}m_j\right)\left[\left(\sum_{l=0}^{q} y_l \prod_{j=0}^{l-1}m_j\right)p_0+ r x_0\right]}{\left(\sum_{l=0}^{q}\prod_{j=0}^{l-1}m_j^2\right)p_0 + r}.
\end{split}
\end{equation*}
\end{proof}
We define the following three useful sequences:
\begin{equation}
\begin{split}
M_i &= \prod_{j=0}^{i-1} m_j, \\
S_i &= \sum_{l=0}^i \prod_{j=0}^{l-1} m_j^2 = \sum_{l=0}^i M_l^2, \\
B_i &= \sum_{l=0}^i y_l \prod_{j=0}^{l-1} m_j = \sum_{l=0}^i M_l y_l.
\end{split}
\end{equation}
Intuitively we can think of $M_i$ as the forward model propagator from the initial step $0$ to the current step $i$, $S_i$ as the cumulative model variance propagator to step $i$, and $B_i$ as the cumulative observation propagator to step $i$.
We thus write:
\begin{align}
\p^{\|a}_i &= \frac{M_i^2 p_0}{S_i p_0 + r}, \label{eq:skfpa}\\
\x^{\|a}_i &= \frac{M_i(B_i p_0 + r x_0)}{S_i p_0 + r}. \label{eq:skfxa}
\end{align}
Note again that we have assumed that we have a perfect non-trivial model (assumption~\ref{ass:pm}), therefore:
\begin{equation}
\begin{split}
M_0 &= 1,\\
S_0 &= 1,\\
S_0 < S_1 < &\cdots < S_n,\\
M_i^2 &\leq S_i.
\end{split}
\end{equation}
\begin{remark}
Consider a dynamical system described by real valued initial value problem
\begin{align*}
\*y' = \*f(\*y(t)),\quad t_0\leq t\leq t_f,\quad \*y(t_0)=\*y_0.
\end{align*}
Taking a forward Euler step in time,
\begin{align*}
\*y_{i+1} = \*y_i + h_i \*f(\*y_i),
\end{align*}
a linearization of the model could then be written as
\begin{align*}
\*M_i = \*I + h_i \*J(\*y(t_i)),
\end{align*}
where $\*J(\*y(t)) = \left.\frac{d \*f}{d \*y}\right\rvert_{\*y(t)}$ is the Jacobian of $\*f$.
Bounded chaotic systems generally have the property that $\lVert \prod_{j=0}^\infty \*M_j \rVert \to \infty$, therefore the corresponding scalar case is of particular interest.
\end{remark}
\subsection{SKF convergence}
In the Bayesian approach to uncertainty quantification we seek to correctly describe our information about the truth.
In the language of the scalar Kalman filter, our information is described by the mean and variance of a normal distribution. Thus, we wish to both optimally describe the truth via the mean, and optimally describe our confidence in it, through the variance.
Thus, the ideal desired behavior for the scalar Kalman filter is for it to be an unbiased estimator of the truth, meaning that the expected value of the analysis tends towards the truth in the step limit,
\begin{align}
\lim_{i\to\infty}\mathbb{E}[\x^{\|a}_i - x^\|t_i] = 0,
\end{align}
and an unbiased estimator of the variance in that estimate, meaning that the variance in the mean tends towards our description of it,
\begin{align}
\lim_{i\to\infty} \left[\Var\left(\x^{\|a}_i - x^\|t_i\right)- \p^{\|a}_i\right] = 0.
\end{align}
Using equation~\eqref{eq:skfpa} we look at the deviation of the analysis mean from the truth at some arbitrary step $i$:
\begin{equation}\label{eq:xamxt}
\begin{split}
\x^{\|a}_i - x_i^\|t &= \left[\frac{M_i(B_i p_0 + r x_0)}{S_i p_0 + r} - M_i \x^{\|t}_0 \right] = \frac{M_i \left[(B_i - \x^{\|t}_0 S_i) p_0 + r (x_0 - \x^{\|t}_0)\right]}{S_i p_0 + r}\\
&= \frac{M_i r(x_0 - \x^{\|t}_0)}{S_i p_0 + r} +p_0 \frac{M_i(B_i - \x^{\|t}_0 S_i)}{S_i p_0 + r}.
\end{split}
\end{equation}
We therefore have to look at the asymptotic behavior of two terms. The first term is the ratio of model propagator to that of the variance propagator:
\begin{align}
\frac{M_i}{S_i}.
\end{align}
The second term is the propagated cumulative normalized observation deviation,
\begin{align}
\frac{M_i(B_i - \x^{\|t}_0 S_i)}{S_i}.\label{eq:cumulativenormalizedobservationdeviation}
\end{align}
\begin{lemma}
\label{lem:cummodelas}
The cumulative model variance propagator grows faster than the model propagator:
%
\begin{align}
\lim_{i\to\infty} \frac{M_i}{S_i} = 0.
\end{align}
\end{lemma}
\begin{proof}
Without loss of generality it suffices to look at $\lvert M_i \rvert$. We will examine the following exhaustive list of cases:
\begin{enumerate}
\item $\lim_{i\to\infty}\lvert M_i \rvert = 0$,\label{en:pp1}
\item $\lim_{i\to\infty}\lvert M_i \rvert = C>0$,\label{en:pp2}
\item $\lim_{i\to\infty}\lvert M_i \rvert = \infty$,\label{en:pp3}
\item $\lim_{i\to\infty}\lvert M_i \rvert$ does not exist.\label{en:pp4}
\end{enumerate}
For Case~\ref{en:pp1} it suffices to see that $S_i \geq 1$, thus
%
\begin{align*}
\lim_{i\to\infty}\frac{\lvert M_i \rvert}{S_i} \leq \lim_{i\to\infty}\frac{\lvert M_i \rvert}{1} = 0.
\end{align*}
For Case~\ref{en:pp2}, there exists a step, $q$, and $\delta$, such that $C + \delta \geq \lvert M_i \rvert \geq C - \delta > 0, \forall i>q$, therefore,
\begin{align*}
\lim_{i\to\infty}\frac{\lvert M_i \rvert}{S_i} \leq \lim_{i\to\infty}\frac{C+\delta}{(i-q){(C-\delta)}^2} = 0.
\end{align*}
One will note that this case also proves the case where $\lvert M_i \rvert$ is bounded but does not converge.
For Case~\ref{en:pp3}, observe that
%
\begin{align*}
\lim_{i\to\infty}\frac{\lvert M_i \rvert}{S_i} &= \lim_{i\to\infty} \frac{\lvert M_i \rvert}{\sum_{l=0}^i M_l^2}\\
&\leq \lim_{i\to\infty}\frac{\lvert M_i \rvert}{M_i^2}\\
&=\lim_{i\to\infty}\frac{1}{\lvert M_i \rvert} = 0.
\end{align*}
For case~\ref{en:pp4}, observe that,
\begin{equation}
\inf_{1\leq j\leq i} M_j \leq M_i \leq \sup_{1\leq j\leq i} M_j,
\end{equation}
and as the two bounds fall into one of our other three categories, the estimates collapse, and we regress to the former.
\end{proof}
\begin{lemma}\label{lem:msqsi}
The variance propagator is at least as large as the square of the model propagator,
%
\begin{align}
0 \leq \frac{M_i^2}{S_i} \leq 1.
\end{align}
\end{lemma}
\begin{proof}
This trivially follows from the definitions.
\end{proof}
\begin{lemma}\label{lem:podurv}
The propagated cumulative normalized observation deviation is an unbiased random variable with variance converging to the observation variance times the ratio of the square of the model propagator to the variance propagator. In particular,
%
\begin{align}
\frac{M_i(B_i - \x^{\|t}_0 S_i)}{S_i} &= \sum_{l=0}^i \varepsilon_{l,i},\\
\mathbb{E}\left[\frac{M_i(B_i - \x^{\|t}_0 S_i)}{S_i}\right] &= 0,\\
\lim_{i\to\infty}\Var\left(\frac{M_i(B_i - \x^{\|t}_0 S_i)}{S_i}\right) &= r \lim_{i\to\infty} \frac{M_i^2}{S_i},
\end{align}
%
where $\varepsilon_{l,i} \sim \!N\left(0, M_i^2 M_l^2 S_i^{-2} r\right)$.
\end{lemma}
\begin{proof}
Every observation $y_l$ is a sample from the distribution $\!N(x_l^\|t,r)$. Define $y_{0,l} = M_l^{-1}y_l$, and observe that $\varepsilon_{l} = M_l(y_{0,l} - \x^{\|t}_0) = y_l - \x^{\|t}_l \sim \!N(0, r)$. Additionally define $\varepsilon_{l,i} = M_i M_l S_i^{-1}\varepsilon_{l} \sim \!N\left(0, M_i^2 M_l^2 S_i^{-2} r\right)$. Now we manipulate:
%
\begin{align*}
\frac{M_i(B_i - \x^{\|t}_0 S_i)}{S_i} &= \frac{M_i}{S_i} \sum_{l=0}^i (M_l y_l - M_l^2 x_0^t)
= \frac{M_i}{S_i} \sum_{l=0}^i M_l^2(y_{0,l} - x_0^t) \\
&= \frac{M_i}{S_i}\sum_{l=0}^i M_l\varepsilon_{l}
= \sum_{l=0}^i \varepsilon_{l,i}.
\end{align*}
%
As for the expected value and variance,
%
\begin{align*}
\mathbb{E}\left[\frac{M_i(B_i - \x^{\|t}_0 S_i)}{S_i}\right] = \mathbb{E}\left[\sum_{l=0}^i \varepsilon_{l,i}\right] = \sum_{l=0}^i \mathbb{E}\left[\varepsilon_{l,i}\right] = 0,\\
\lim_{i\to\infty} \Var\left(\sum_{l=0}^i \varepsilon_{l,i} \right) = r \lim_{i\to\infty} \frac{M_i^2}{S_i^2} \sum_{l=0}^i M_l^2 = r \lim_{i\to\infty} \frac{M_i^2}{S_i^2} S_i = r \lim_{i\to\infty} \frac{M_i^2}{S_i},
\end{align*}
%
as required.
\end{proof}
\begin{corollary}\label{cor:weakisanalysisvar}
In the step limit, the analysis uncertainty estimate, $\p^{\|a}_i$, approaches the variance of the propagated cumulative normalized observation deviation,
%
\begin{align}
\lim_{i\to\infty} \p^{\|a}_i = \lim_{i\to\infty} \Var\left(\frac{M_i(B_i - \x^{\|t}_0 S_i)}{S_i}\right).
\end{align}
\end{corollary}
\begin{proof}
\begin{align*}
\lim_{i\to\infty} \p^{\|a}_i = r \lim_{i\to\infty} \frac{M_i^2 p_0}{S_i p_0 + r} = r \lim_{i\to\infty} \frac{M_i^2}{S_i} = \lim_{i\to\infty} \Var\left(\sum_{l=0}^i \varepsilon_{l,i} \right) = \lim_{i\to\infty} \Var\left(\frac{M_i(B_i - \x^{\|t}_0 S_i)}{S_i}\right).
\end{align*}
\end{proof}
Note that the analysis variance is not zero in the step limit for models that grow sufficiently fast. Take $M_i^2 = e^i$, then,
\begin{align*}
\lim_{i\to\infty} \frac{M_i^2}{S_i} = \lim_{i\to\infty} \frac{e^i}{\sum_{l=0}^i e^l} = \lim_{i\to\infty}\frac{e^i - e^{i+1}}{1-e^{i+1}} = \frac{e - 1}{e} > 0.
\end{align*}
As a consequence of this, we can therefore have non-zero uncertainty in the analysis in the step limit, even for perfect models!
\begin{theorem}
In the step limit, the mean of the scalar Kalman filter approaches the truth, and our description of the variance tends towards the variance in the mean, meaning that,
\begin{align}
\lim_{i\to\infty} \mathbb{E}[\x^{\|a}_i - x_i^\|t] &= 0,\\
\lim_{i\to\infty} \left[\Var(\x^{\|a}_i - x_i^\|t) - \p^{\|a}_i\right] &= 0.
\end{align}
\end{theorem}
\begin{proof}
Following equation~\eqref{eq:xamxt}, we manipulate:
%
\begin{align*}
\lim_{i\to\infty}\x^{\|a}_i - x_i^\|t &= \lim_{i\to\infty} \left[\frac{M_i(B_i p_0 + r x_0)}{S_i p_0 + r} - M_i \x^{\|t}_0 \right] = \lim_{i\to\infty} \frac{M_i \left[(B_i - \x^{\|t}_0 S_i) p_0 + r (x_0 - \x^{\|t}_0)\right]}{S_i p_0 + r}\\
&= \left[\lim_{i\to\infty} \frac{M_i r(x_0 - \x^{\|t}_0)}{S_i p_0 + r}\right] + \left[p_0 \lim_{i\to\infty} \frac{M_i(B_i - \x^{\|t}_0 S_i)}{S_i p_0 + r}\right].
\end{align*}
%
The term $\lim_{i\to\infty} \frac{M_i r(x_0 - \x^{\|t}_0)}{S_i p_0 + r}$ always converges to zero in the limit by Lemma~\ref{lem:cummodelas}. As for $p_0 \lim_{i\to\infty} \frac{M_i(B_i - \x^{\|t}_0 S_i)}{S_i p_0 + r}$, its expected value is zero by Lemma~\ref{lem:podurv}, and its variance is $\lim_{i\to\infty} \p^{\|a}_i$ by Corollary~\ref{cor:weakisanalysisvar}.
\end{proof}
This shows that in the step limit, the scalar Kalman filter description of the moments converges to the moments describing the uncertainty.
\section{The Scalar Pedagogical Ensemble Kalman Filter (SPEnKF)}%
\label{sec:spenkf}
\subsection{Definition of the SPEnKF}
What is the fundamental characteristic that defines the ensemble Kalman filter? We argue that the key component is the non-linear expression used to build the sampled covariance estimation, and seek to create the simplest possible version of the EnKF which still carries with it uncertain information from sampling the (co-)variance.
\begin{assumption}[Identical initial sampling]
We assume that now our two inputs are $\hat{x}^\|a_0\coloneqq x_0$ (the same mean input as to that of the scalar Kalman filter), and $\*a^\|a_0 = \*a$ the vector of $N$ anomalies about the mean, such that ${[\*a]}_{1\leq i\leq N}\sim \!N(0,p_0)$ (anomalies are sampled exactly from a distribution with the variance used by the exact scalar Kalman filter).
\end{assumption}
\begin{lemma}
If ${[\*a]}_{1\leq i\leq N}\sim \!N(0,p_0)$, is a collection of $N$ samples from the distribution, then
\begin{equation}
\label{eq:gamma}
\frac{1}{N}\,(\*a\cdot \*a) = \frac{1}{N}\,\sum_{i=1}^N a_i^2 \sim \Gamma\left(\frac{N}{2},\frac{N}{2 p_0}\right).
\end{equation}
\end{lemma}
\begin{proof}
Consider first the case of $\tilde{a}_i\sim~\!N(0,1)$, by the definition of the chi-square distribution, $\tilde{\*a}\cdot\tilde{\*a}\sim\chi^2_N=\Gamma\left(\frac{N}{2},\frac{1}{2}\right)$. Note also that if $x\sim\Gamma(\alpha,\beta)$, then $c x\sim\Gamma(\alpha,c^{-1}\beta)$, and that that $\sqrt{p_0}\,\tilde{a}_i=a_i\sim \!N(0,p_0)$. Therefore \eqref{eq:gamma} holds.
\end{proof}
In what follows we denote by ``hat'' the ensemble-estimated variances. For example, the initial sample variance for an over-sampled ensemble ($N > 1$) is
\begin{equation}
\label{eq:hatp0}
\hat{p}_0=\frac{1}{N} (\*a \cdot \*a) \sim \Gamma\left(\alpha, \frac{\alpha}{p_0}\right), \quad \alpha := \frac{N}{2}.
\end{equation}
We will again assume a perfect model (assumption~\ref{ass:pm}) and a constant observation error variance (assumption~\ref{ass:constobs}).
We then construct the filter to as closely as possible approximate the behavior of the exact scalar filter. Propagating the mean one step:
\begin{equation}
\label{eq:spenkff}
\begin{split}
\hat{x}^\|f_{i+1} &= m_i \hat{x}^\|a_i,\\
\hat{x}^\|a_i &= \hat{x}^\|f_i + \hat{k}_i(y_i - \hat{x}^\|f_i),
\end{split}
\end{equation}
is exactly the same as in the scalar case, with the exception of the Kalman gain, which is dependent on the anomalies. Propagating the anomalies forward one step therefore works as follows:
\begin{subequations}
\label{eq:spenkfa}
\begin{align}
\*a^\|f_{i} &= m_{i-1} \*a^\|a_{i-1},\\
\*a^\|a_i &= {\left(\hat{p}^\|a_i\right)}^{\frac{1}{2}}{\left(\hat{p}^\|f_i\right)}^{-\frac{1}{2}}\*a^\|f_i,\\
\hat{k}_i &= \frac{\hat{p}^\|f_i}{\hat{p}^\|f_i+r},\\
\hat{p}^\|f_i &= \frac{1}{N} (\*a^\|f_i \cdot \*a^\|f_i),\label{eq:spenkfpf}\\
\hat{p}^\|a_i &= (1-\hat{k}_i)\hat{p}^\|f_i. \label{eq:spenkfpa}
\end{align}
\end{subequations}
Here the transformation ${\left(\hat{p}^\|a_i\right)}^{\frac{1}{2}}{\left(\hat{p}^\|f_i\right)}^{-\frac{1}{2}}$ would be the optimal transport in the case where it was assumed that ${\left[a^\|f_i\right]}_j \sim \!N(0, \hat{p}^\|f_i)$ and ${\left[a^\|a_i\right]}_j \sim \!N(0, \hat{p}^\|a_i)$. This is however not the case.
\subsection{Properties of SPEnKF}
\begin{lemma}
The analysis variance, $\hat{p}^\|a_i$, computed by \eqref{eq:spenkfpa}, is exactly the sampled variance matrix of the analysis anomalies:
\begin{align}
\hat{p}^\|a_i &= \frac{1}{N} (\*a^\|a_i \cdot \*a^\|a_i),
\end{align}
and the forecast variance, $\hat{p}^\|f$, computed by \eqref{eq:spenkfpf}, at step $i+1$ is exactly the previous analysis variance propagated by the model:
\begin{align}
\hat{p}^\|f_{i+1} = m_i^2\hat{p}^\|a_i.
\end{align}
\end{lemma}
\begin{proof}
By simple manipulation of \eqref{eq:spenkff} and \eqref{eq:spenkfa}:
\begin{align*}
\frac{1}{N} (\*a^\|a_i \cdot \*a^\|a_i) &= \left(\frac{(1-\hat{k}_i)\hat{p}^\|f_i}{\hat{p}^\|f_i}\right)\hat{p}^\|f_i = (1-\hat{k}_i)\hat{p}^\|f_i = \hat{p}^\|a_i,\\
\hat{p}^\|f_{i+1} &= \frac{1}{N} (\*a^\|f_{i+1} \cdot \*a^\|f_{i+1}) = m_i^2 \frac{1}{N} (\*a^\|a_i \cdot \*a^\|a_i) = m_i^2 \hat{p}^\|a_i.
\end{align*}
\end{proof}
This implies that the underlying anomalies are not important to the resulting distribution after several steps of the algorithm. All that matters to determining the resulting distribution, and thus the information of the variance at step $i$ is the distribution of the initial variance estimate at the onset.
The problem therefore reduces from attempting to grasp the distribution of the anomalies at a certain step---which almost certainly is not normal and whose members are not independent---to one of looking at a simple scalar.
\begin{lemma}
The Kalman gain $\hat{k}_i$ of the SPEnKF is a random variable of the form $\frac{a_i \hat{p}_0}{c_i \hat{p}_0 + d_i}$, where $\hat{p}_0$ is distributed according to \eqref{eq:hatp0}.
\end{lemma}
\begin{proof}
As the evolution of the variance in the SPEnKF is identical to that of the exact scalar Kalman filter, by Lemma~\ref{lem:kalmangain},
\begin{align*}
\hat{k}_i = \frac{a_i \hat{p}_0}{c_i \hat{p} + d_i},\\
a_i = M_i^2,\quad c_i = S_i,\quad d_i = r,
\end{align*}
as required.
\end{proof}
\begin{lemma}
The analysis mean $\hat{x}^\|a_i$ of the SPEnKF is a random variable of the form $\frac{a_i \hat{p}_0 + b_i}{c_i \hat{p}_0 + d_i}$,where $\hat{p}_0$ is distributed according to \eqref{eq:hatp0}.
\end{lemma}
\begin{proof}
As the analysis mean evolves with the same exact principles as in the canonical exact Kalman filter, Lemma~\ref{lem:analysismean} applies, and as such,
\begin{gather*}
\hat{x}^\|a_i = \frac{a_i \hat{p}_0 + b_i}{c_i\hat{p}_0 + d_i},\\
a_i = M_i B_i,\quad b_i = M_i x_0 r,\quad c_i = S_i,\quad d_i= r.
\end{gather*}
\end{proof}
This algorithm is obviously very similar, but not equivalent to the canonical scalar Kalman filter.
\subsection{Analysis of the perturbed problem}
As we have proven that the scalar Kalman filter (with a perfect non-trivial model) moments converge to the actual moments inherent in the estimates in the step limit, it suffices for us to prove that SPEnKF converges to the scalar Kalman filter in some certain asymptotic, and finite cases. We will accomplish this by showing degeneracy of the resulting distribution of the differences between the first two moment estimates of the SPEnKF and the SKF.
\begin{assumption}[Perturbed problem]
Let the SPEnKF take the inexact perturbed inputs $\tilde{p}_0$ (resulting from some perturbed anomalies), and $\tilde{x}_0$, whilst the corresponding exact scalar Kalman filter takes the unperturbed inputs $p_0$ and $x_0$.
\end{assumption}
We now look at the discrepancy between the SPEnKF and the exact scalar KF. The discrepancy in the analysis variance, and analysis mean, at the $i$th step are random variables such that:
\begin{equation}\begin{split}
\Delta p_i &= \hat{\tilde{p}}^a_i - p^a_i\\
&= \frac{M_i^2 r\hat{\tilde{p}}_0}{S_i \hat{\tilde{p}}_0 + r} - \frac{M_i^2 r p_0}{S_i p_0 + r}\\
&= \frac{M_i^2 r^2 \hat{\tilde{p}}_0 - M_i^2 r^2 p_0}{S_i(S_i p_0 + r) \hat{\tilde{p}}_0 + r(S_i p_0 + r)},\\
\Delta x_i &= \hat{\tilde{x}}^a_i - x^a_i\\
&= \frac{M_i B_i \hat{\tilde{p}}_0 + M_i r \tilde{x}_0}{S_i \hat{\tilde{p}}_0 + r} - \frac{M_i B_i p_0 + M_i r x_0}{S_i p_0 + r}\\
&= \frac{M_i r(B_i - S_i x_0)\hat{\tilde{p}}_0 + M_i r(S_i p_0 \tilde{x}_0 + r \tilde{x}_0 - B_i p_0 - r x_0)}{S_i(S_i p_0 + r) \hat{\tilde{p}}_0 + r(S_i p_0 + r)}.
\end{split}
\end{equation}
Denote the generalized exponential integral function by:
\[
E_n(z) := \int_1^\infty \frac{e^{- z t}}{t^n}\diff{t}.
\]
By Lemma~\ref{lem:expandvar} we have that:
\begin{align}
\mathbb{E}[\Delta p_i] &= \frac{\alpha M_i^2 r^2 e^{\frac{\alpha r}{S_i \tilde{p}_0}}}{S_i (S_i p_0 + r)}\left[-\frac{p_0}{\tilde{p}_0}E_{\alpha}\left(\frac{\alpha r}{S_i \tilde{p}_0}\right) + E_{\alpha+1}\left(\frac{\alpha r}{S_i \tilde{p}_0}\right)\right],\label{eq:pppm}\\
\mathbb{E}[\Delta x_i] &= \frac{\alpha M_i r e^{\frac{\alpha r}{S_i \tilde{p}_0}}}{S_i (S_i p_0 + r)}\left[\frac{r (\tilde{x}_0 - x_0) - (B_i - \tilde{x}_0 S_i ) p_0 }{\tilde{p}_0}E_{\alpha}\left(\frac{\alpha r}{S_i \tilde{p}_0}\right) + (B_i - S_i x_0) E_{\alpha+1}\left(\frac{\alpha r}{S_i \tilde{p}_0}\right)\right],\label{eq:ppxm}\\
\mathbb{E}[\Delta p_i^2] &= \frac{\alpha M_i^4 r^4 e^{\frac{\alpha r}{S_i \tilde{p}_0}}}{S_i^2 {(S_i p_0 + r)}^2} \left[\begin{aligned}\phantom{+}&\frac{\alpha p_0^2}{\tilde{p}_0^2} E_{\alpha-1}\left(\frac{\alpha r}{S_i \tilde{p}_0}\right)
+ \frac{\alpha p_0 (2 \tilde{p}_0 - p_0)}{\tilde{p}_0^2}E_{\alpha}\left(\frac{\alpha r}{S_i \tilde{p}_0}\right)\\
+& \frac{\tilde{p}_0 + \alpha (\tilde{p}_0 - 2 p_0)}{\tilde{p}_0}E_{\alpha+1}\left(\frac{\alpha r}{S_i \tilde{p}_0}\right)
-(\alpha + 1) E_{\alpha+2}\left(\frac{\alpha r}{S_i \tilde{p}_0}\right)\end{aligned}\right],\label{eq:pppv}\\
\begin{split}
\mathbb{E}[\Delta x_i^2] &= \frac{\alpha r^2 M_i^2 e^{\frac{\alpha r}{S_i \tilde{p}_0}}}{ S_i^2 \tilde{p}_0^2 {\left(S_i p_0 + r \right)}^2}\\
&\phantom{=}\left[
\begin{aligned}
\phantom{+}&\alpha {\left((B_i-\tilde{x}_0 S_i)p_0 + r(x_0 - \tilde{x}_0)\right)}^2 E_{\alpha -1}\left(\frac{\alpha r}{S_i \tilde{p}_0}\right)\\
+& \alpha \left((B_i-\tilde{x}_0 S_i)p_0 + r(x_0 - \tilde{x}_0)\right) \left[\begin{aligned}
-&(p_0 + 2\tilde{p}_0)\left(B_i - \frac{p_0\tilde{x}_0 + 2\tilde{p}_0 x_0}{p_0 + 2\tilde{p}_0} S_i\right)\\
+&r\left(\tilde{x}_0-x_0\right)
\end{aligned}
\right] E_{\alpha }\left(\frac{\alpha r}{S_i \tilde{p}_0}\right)\\
+&\tilde{p}_0 \left(B_i-x_0 S_i\right) \left[\begin{aligned}
\phantom{+}&(2\alpha p_0 + (\alpha + 1) \tilde{p}_0)\left(B_i - \frac{2\alpha\tilde{x}_0 p_0 + (\alpha + 1)x_0 \tilde{p}_0}{2\alpha p_0 + (\alpha + 1) \tilde{p}_0}S_i\right)\\
+&2\alpha r(x_0 - \tilde{x}_0)
\end{aligned}
\right] E_{\alpha +1}\left(\frac{\alpha r}{S_i \tilde{p}_0}\right)\\
-&(\alpha +1) \tilde{p}_0^2{\left(B_i-x_0 S_i\right)}^2 E_{\alpha +2}\left(\frac{\alpha r}{\tilde{p}_0 S_i}\right)
\end{aligned}
\right].\end{split}\label{eq:ppxv}
\end{align}
We first show that for certain special cases both the mean and the variance of the discrepancy approach zero, then we would show degeneracy of the perturbed problem.
\subsection{Convergence of the SPEnKF to the scalar KF as ensemble size grows to infinity}%
\label{subsec:ppfe}
The first case that we can look at is the one of the limiting case of the ensemble size growing to infinity. For the two algorithms to converge in ensemble size, their initial inputs have to be identical, as the algorithms operating on arbitrarily different inputs would necessitate arbitrarily different output.
\begin{theorem}\label{thm:enslimvar}
When the initial inputs to the scalar Kalman filter and the SPEnKF are identical, $\tilde{x}_0=x_0$, $\tilde{p}_0 = p_0$, then for all steps $i$, in the limit of ensemble size $\alpha\to\infty$, the expected value and variance of the discrepancy in the analysis variance are zero:
%
\begin{equation}\begin{split}
\lim_{\alpha\to\infty} \mathbb{E}[\Delta p_i] &= 0\\
\lim_{\alpha\to\infty} \left(\mathbb{E}[\Delta p_i^2] - \mathbb{E}{[\Delta p_i]}^2\right) &= 0.
\end{split}
\end{equation}
%
\end{theorem}
\begin{proof}
by Corollary~\ref{cor:expectedconverge}, and Corollary~\ref{cor:varianceconverge},
\begin{equation}\begin{split}
%
\lim_{\alpha\to\infty} \mathbb{E}[\Delta p_i] &= \frac{M_i^2 r^2 \tilde{p}_0 - M_i^2 r^2 p_0}{S_i(S_i p_0 + r) \tilde{p}_0 + r(S_i p_0 + r)}\\
&= \frac{M_i^2 r^2 p_0 - M_i^2 r^2 p_0}{{(S_i p_0 + r)}^2} = 0,\\
\lim_{\alpha\to\infty} \left(\mathbb{E}[\Delta p_i^2] - \mathbb{E}{[\Delta p_i]}^2\right) &= \lim_{\alpha\to\infty} \mathbb{E}[\Delta p_i^2] = {\left(\frac{M_i^2 r^2 \tilde{p}_0 - M_i^2 r^2 p_0}{S_i(S_i p_0 + r) \tilde{p}_0 + r(S_i p_0 + r)}\right)}^2\\
&= {\left(\frac{M_i^2 r^2 p_0 - M_i^2 r^2 p_0}{{(S_i p_0 + r)}^2}\right)}^2 = 0.
\end{split}
\end{equation}
%
as required.
\end{proof}
\begin{theorem}\label{thm:enslimmean}
When the initial inputs to the scalar Kalman filter and the SPEnKF are identical, $\tilde{x}_0=x_0$, $\tilde{p}_0 = p_0$,then for all steps $i$, in the limit of ensemble size $\alpha\to\infty$, the expected value and variance of the discrepancy in the analysis mean are zero:
%
\begin{equation}\begin{split}
\lim_{\alpha\to\infty} \mathbb{E}[\Delta x_i] &= 0\\
\lim_{\alpha\to\infty} \left(\mathbb{E}[\Delta x_i^2] - \mathbb{E}{[\Delta x_i]}^2\right) &= 0,
\end{split}\end{equation}
\end{theorem}
\begin{proof}
by Corollary~\ref{cor:expectedconverge}, and Corollary~\ref{cor:varianceconverge},
%
\begin{equation}\begin{split}
\lim_{\alpha\to\infty} \mathbb{E}[\Delta x_i] &= \frac{M_i r(B_i - S_i x_0)\tilde{p}_0 + M_i r(S_i p_0 \tilde{x}_0 + r \tilde{x}_0 - B_i p_0 - r x_0)}{S_i(S_i p_0 + r) \tilde{p}_0 + r(S_i p_0 + r)}\\
&= \frac{M_i r(B_i - S_i x_0)p_0 - M_i r(B_i - S_i x_0)p_0 + M_i r(r x_0 - r x_0)}{{(S_i p_0 + r)}^2} = 0,\\
\lim_{\alpha\to\infty} \left(\mathbb{E}[\Delta x_i^2] - \mathbb{E}{[\Delta x_i]}^2\right) &= \lim_{\alpha\to\infty} \mathbb{E}[\Delta x_i^2] = {\left(\frac{M_i r(B_i - S_i x_0)\tilde{p}_0 + M_i r(S_i p_0 \tilde{x}_0 + r \tilde{x}_0 - B_i p_0 - r x_0)}{S_i(S_i p_0 + r) \tilde{p}_0 + r(S_i p_0 + r)}\right)}^2\\
&= {\left(\frac{M_i r(B_i - S_i x_0)p_0 - M_i r(B_i - S_i x_0)p_0 + M_i r(r x_0 - r x_0)}{{(S_i p_0 + r)}^2}\right)}^2 = 0,
\end{split}\end{equation}
as required.
\end{proof}
Theorem~\ref{thm:enslimvar} and theorem~\ref{thm:enslimmean} combined show that in the asymptotic case of large ensemble sizes the trivial SPEnKF converges in means to the exact scalar Kalman filter and that the variances collapse to zero.
\subsection{Analysis of the perturbed problem in the case of a finite ensemble}
Arbitrarily large ensembles are theoretically nice, but impractical. Running the data assimilation scheme for an arbitrarily large number of steps however, is practical. Assume now that we have a finite over-sampled ensemble, $1 < \alpha < \infty$.
Observe also that~\eqref{eq:ppxm} and~\eqref{eq:pppm},
\begin{equation}\begin{split}
\lim_{i\to\infty} \mathbb{E}[\Delta p_i] &=
\alpha r^2
\left[\lim_{i\to\infty} \frac{M_i^2}{S_i}\right]
\left[\lim_{i\to\infty} \frac{1}{S_i p_0 + r}\right]
\left[\lim_{i\to\infty}e^{\frac{\alpha r}{S_i \tilde{p}_0}}\left(
E_{\alpha+1}\left(\frac{\alpha r}{S_i \tilde{p}_0}\right)
-\frac{p_0}{\tilde{p}_0}E_{\alpha}\left(\frac{\alpha r}{S_i \tilde{p}_0}\right) \right)\right]\\
\lim_{i\to\infty} \mathbb{E}[\Delta x_i] &= \begin{aligned}\phantom{+}&
\alpha r
\left[\lim_{i\to\infty} e^{\frac{\alpha r}{S_i \tilde{p}_0}}\left(\frac{M_i(B_i - x_0 S_i)}{S_i(S_i p_0 + r)} E_{\alpha+1}\left(\frac{\alpha r}{S_i \tilde{p}_0}\right) - \frac{p_0}{\tilde{p}_0} \frac{M_i(B_i - \tilde{x}_0 S_i)}{S_i(S_i p_0 + r)} E_{\alpha}\left(\frac{\alpha r}{S_i \tilde{p}_0}\right)\right)\right]\\
+&
\frac{\alpha r^2 (\tilde{x}_0 - x_0)}{\tilde{p}_0}
\left[\lim_{i\to\infty} \frac{M_i}{S_i p_0 + r}\right]
\left[\lim_{i\to\infty} e^{\frac{\alpha r}{S_i \tilde{p}}} E_{\alpha}\left(\frac{\alpha r}{S_i \tilde{p}_0}\right)\right]
\end{aligned}
\end{split}\end{equation}
Note that in the terms above, the cumulative normalized observation deviation~\eqref{eq:cumulativenormalizedobservationdeviation}, is normalized by an additional $S_i$, meaning that we need to look at the cumulative doubly normalized observation deviation.
\begin{lemma}[Weak convergence of cumulative doubly normalized observation deviation]
\label{lem:weakconvpert}
The cumulative doubly normalized observation deviation converges to zero in probability if the step limit, meaning that:
%
\begin{equation}\begin{split}
\lim_{i\to\infty}\Pr\left[\left\lvert\frac{M_i(B_i - \x^{\|t}_0 S_i)}{S_i^2}\right\rvert > \epsilon\right] = 0, \quad \forall \epsilon > 0,
\end{split}\end{equation}
unconditionally on model behavior.
\end{lemma}
\begin{proof}
As in Lemma~\ref{lem:podurv}, observe that instead of $\varepsilon_{l,i}$, we deal with $S_i^{-1}\varepsilon_{l,i}$,
\begin{equation*}
\begin{split}
\frac{M_i(B_i - \x^{\|t}_0 S_i)}{S_i^2} = \sum_{l=0}^i S_i^{-1}\varepsilon_{l,i},
\end{split}
\end{equation*}
we require the variance of the mean to be zero,
\begin{equation*}\begin{split}
\lim_{i\to\infty} \Var\left( \sum_{l=0}^i S_i^{-1}\varepsilon_{l,i}\right) = r\lim_{i\to\infty} \frac{M_i^2}{S_i^4} \sum_{l=0}^i M_l^2 = r \lim_{i\to\infty} \frac{M_i^2}{S_i^4} S_i = r\left[\lim_{i\to\infty} \frac{1}{S_i}\right]{\left[\lim_{i\to\infty} \frac{M_i}{S_i}\right]}^2 = 0,
\end{split}\end{equation*}
as required.
\end{proof}
\begin{lemma}[Strong convergence of cumulative doubly normalized observation deviation]\label{lem:strongconvpert}
The cumulative doubly normalized observation deviation propagated forward by the model converges to zero almost surely,
\begin{equation}\begin{split}
\Pr \left[\lim_{i\to\infty} \frac{M_i(B_i - \x^{\|t}_0 S_i)}{S_i^2}\right] = 0,
\end{split}\end{equation}
conditionally, whenever $\limsup_{i\to\infty}\frac{i+1}{S_i} < \infty$.
\end{lemma}
\begin{proof}
The criteria for strong convergence are that
\begin{equation*}
\begin{split}
r \lim_{i\to\infty} \sup_{0\leq l \leq i} \left\{ \frac{M_i^2 M_l^2 {(i+1)}^2}{S_i^4}\right\} &< \infty,\\
r \lim_{i\to\infty} \frac{M_i^2 {(i+1)}^2}{S_i^4} \sum_{l=0}^{i} \frac{M_l^2}{{(l+1)}^2} < \infty.
\end{split}
\end{equation*}
The first condition ensures that the variances of all the individual random variables are finite, by stating that in the limit, their supremum is. The second condition is for the sufficient decay in their variances.
Note that $\frac{M_i^2}{S_i} \leq 1$ by Lemma~\ref{lem:msqsi} (moreover $\frac{M_l^2}{S_i}\leq 1$ for all $l\leq i$) therefore,
\begin{equation*}
\begin{gathered}
r \lim_{i\to\infty} \sup_{0\leq l \leq i} \left\{ \frac{M_i^2 M_l^2 {(i+1)}^2}{S_i^4}\right\} \leq r \lim_{i\to\infty} \frac{{(i+1)}^2}{S_i^2} < \infty,\\
r \lim_{i\to\infty} \frac{M_i^2 {(i+1)}^2}{S_i^4} \sum_{l=0}^{i} \frac{M_l^2}{{(l+1)}^2} \leq r \lim_{i\to\infty} \frac{{(i+1)}^2}{S_i^2} \sum_{l=0}^{i} \frac{1}{{(l+1)}^2} < \infty.
\end{gathered}
\end{equation*}
as required.
\end{proof}
\begin{corollary}
In the case of imperfect truth, when $\x^{\|t}_0$ is replaced with some arbitrary constant $c$, and with slight abuse of notation,
\begin{equation}\begin{split}
\lim_{i\to\infty} \frac{M_i(B_i - c S_i)}{S_i^2} = 0.
\end{split}\end{equation}
in probability always or almost surely whenever$\limsup_{i\to\infty}\frac{i+1}{S_i} < \infty$.
\end{corollary}
\begin{proof}
\begin{equation*}\begin{split}
\lim_{i\to\infty} \frac{M_i(B_i - c S_i)}{S_i^2}
&= \lim_{i\to\infty} \frac{M_i(B_i - \x^{\|t}_0 S_i) + M_i(\x^{\|t}_0 S_i - c S_i)}{S_i^2}\\
& = \lim_{i\to\infty} \frac{M_i(B_i - \x^{\|t}_0 S_i)}{S_i^2} + \lim_{i\to\infty} \frac{M_i(\x^{\|t}_0 S_i - c S_i)}{S_i^2}\\
&= \lim_{i\to\infty} \frac{M_i(B_i - \x^{\|t}_0 S_i)}{S_i^2} + \left[\lim_{i\to\infty} \frac{M_i}{S_i}\right] (\x^{\|t}_0 - c)
= 0.
\end{split}\end{equation*}
as required.
\end{proof}
\subsection{Optimal inflation factors}%
\label{subsec:opti}
From the form of \eqref{eq:pppm}, it can be surmised there exists a value of $\tilde{p}_0$ such that $\mathbb{E}[\Delta p_i]$ is zero for some particular value of $i$.
A natural thought is to find a multiplicative factor, $\theta$ such that $\tilde{p}_0 = \theta\, p_0$. In this context, $\theta$ is a heuristic multiplicative scaling factor that is applied to a covariance matrix, and is called inflation in the context of ensemble Kalman filters. We will use the term here to describe both initial (applied once at the beginning of the algorithm) and step-wise (applied at each step) scaling factors of our variances.
\begin{theorem}\label{thm:inflationinfinite}
There exists an initial inflation factor $\theta_{*}$ such that for the input variance value $\tilde{p}_0 = \theta_{*} p_0$, the expected value of the variance of the variance deviation in the perturbed problem, in the step limit, is zero, meaning that,
\begin{equation}
\lim_{i\to\infty}\mathbb{E}[\Delta p_i] = 0,
\end{equation}
which, from~\eqref{eq:pppm}, is equivalent to requiring that,
\begin{equation}\label{eq:thminfl}
\lim_{i\to\infty} e^{\frac{\alpha r}{S_i \tilde{p}_0}}\left( E_{\alpha+1}\left(\frac{\alpha r}{S_i \tilde{p}_0}\right) - \frac{p_0}{\tilde{p}_0} E_{\alpha}\left(\frac{\alpha r}{S_i \tilde{p}_0}\right)\right) = 0.
\end{equation}
\end{theorem}
\begin{proof}
It is trivially evident that the solution to~\eqref{eq:thminfl} is the root of the function,
\begin{equation*}\begin{split}
\!J(\theta) = \theta - \lim_{i\to\infty}\frac{E_{\alpha}\left(\frac{\alpha r}{S_i \theta p_0}\right)}{E_{\alpha+1}\left(\frac{\alpha r}{S_i \theta p_0}\right)}
\end{split}\end{equation*}
There are only two cases, $S_i \to \infty$ and $S_i\to S_\infty < \infty$, as $S_i$ is a strictly monotonically increasing sequence.
When $S_i\to\infty$
\begin{equation*}\begin{split}
\theta - \lim_{i\to\infty}\frac{E_{\alpha}\left(\frac{\alpha r}{S_i \theta p_0}\right)}{E_{\alpha+1}\left(\frac{\alpha r}{S_i \theta p_0}\right)} = \theta - \frac{E_{\alpha}(0)}{E_{\alpha + 1 }(0)} = \theta - \frac{\alpha}{\alpha - 1},
\end{split}\end{equation*}
and the exact value for the inflation factor is
\[
\theta_{*} = \alpha\,{(\alpha - 1)}^{-1}.
\]
In the case when $S_i\to S_\infty < \infty$, we have to find the root of the function:
\begin{equation*}\begin{split}
\theta - \lim_{i\to\infty}\frac{E_{\alpha}\left(\frac{\alpha r}{S_i \theta p_0}\right)}{E_{\alpha+1}\left(\frac{\alpha r}{S_i \theta p_0}\right)} &= \theta - \frac{E_{\alpha}\left(\frac{\alpha r}{S_\infty \theta p_0}\right)}{E_{\alpha+1}\left(\frac{\alpha r}{S_\infty \theta p_0}\right)} = \theta - \frac{S_\infty \theta p_0}{\alpha r e^{\frac{\alpha r}{S_\infty \theta p_0}}E_{\alpha+1}\left(\frac{\alpha r}{S_\infty \theta p_0}\right)} + \frac{S_\infty \theta p_0}{r}\\
e^{\frac{\alpha r}{S_\infty \theta p_0}}E_{\alpha+1}\left(\frac{\alpha r}{S_\infty \theta p_0}\right) &= \frac{S_\infty p_0}{\alpha(S_\infty p_0 + r)}
\end{split}\end{equation*}
Let $\mathfrak{E}_{\alpha+1}(z) = e^z E_{\alpha+1}(z)$, and $\mathfrak{E}^{-1}_{\alpha+1}(z)$ be the corresponding inverse, which, as $\mathfrak{E}_{\alpha+1}(z)$ is a strictly monotonically decreasing function on $[0, \infty)$, is implicitly defined on $(0, \alpha^{-1}]$. As $0 < \frac{S_\infty p_0}{\alpha(S_\infty p_0 + r)} < \frac{1}{\alpha}$,
\begin{equation*}\begin{split}
\theta_{*} = {\left[\frac{S_\infty p_0}{\alpha r}\mathfrak{E}_{\alpha+1}^{-1}\left(\frac{S_\infty p_0}{\alpha(S_\infty p_0 + r)}\right)\right]}^{-1},
\end{split}\end{equation*}
%
is the unique inflation factor satisfying the criterion.
\end{proof}
Note that in the `interesting case', when $S_i\to\infty$, $\theta_{*}$ only depends on the size of the ensemble and not on the asymptotic model behavior!
Note also, that this implies that there exists an inflation factor, such that if it is applied at the beginning of the algorithm, the variance perturbation will be zero for a particular finite step $i$.
\begin{lemma}\label{lem:stepwiseinflation}
There exists a step-wise inflation factor, $\theta_i$ such that when $\tilde{p}_0 = \theta_i p_0$, the variance deviation at a particular step, $i$, is zero, equivalently,
\begin{equation}\begin{split}
e^{\frac{\alpha r}{S_i \tilde{p}_0}}\left( E_{\alpha+1}\left(\frac{\alpha r}{S_i \tilde{p}_0}\right)
- \frac{p_0}{\tilde{p}_0} E_{\alpha}\left(\frac{\alpha r}{S_i \tilde{p}_0}\right)\right) = 0.
\end{split}\end{equation}
\end{lemma}
\begin{proof}
By similar manipulation as in theorem~\ref{thm:inflationinfinite}, it is evident that
\begin{equation*}\begin{split}
\theta_i = {\left[\frac{S_i p_0}{\alpha r}\mathfrak{E}_{\alpha+1}^{-1}\left(\frac{S_i p_0}{\alpha(S_i p_0 + r)}\right)\right]}^{-1},
\end{split}\end{equation*}
as required.
\end{proof}
Applying the optimal inflation factor $\theta_i$ for a particular step at the start of the algorithm is impractical as the whole algorithm would have to be re-run to the current step. In practically implemented ensemble-based methods, inflation is applied at every step, therefore we must generalize our approach to such a methodology.
\begin{lemma}
The sequence of optimal step-wise initial inflation factors, defined by lemma~\ref{lem:stepwiseinflation}, is monotonically increasing and is bounded from below by one,
\begin{equation}\begin{split}
1\leq \theta_i \leq \theta_{i+1}.
\end{split}\end{equation}
\end{lemma}
\begin{proof}
Define the function $\!E(z) = {\left[z\mathfrak{E}^{-1}_{\alpha + 1}\left(\frac{1}{z^{-1} + \alpha}\right)\right]}^{-1}$, noting that $\!E\left(\frac{S_i p_0}{\alpha r}\right) = \theta_i$, and observe that
\begin{equation*}\begin{split}
\mathfrak{E}'_{\alpha + 1}(x) &= e^x E_{\alpha+1}(x)\left(1+\frac{\alpha}{x}\right) - \frac{1}{x},\\
\frac{\mathrm{d} \mathfrak{E}^{-1}}{\mathrm{d} z} &= \frac{1}{\mathfrak{E}'_{\alpha + 1}\left(\mathfrak{E}^{-1}_{\alpha+1}\left(\frac{1}{z^{-1} + \alpha}\right)\right)} = \frac{\mathfrak{E}_{\alpha+1}^{-1}\left(\frac{1}{z^{-1} + \alpha}\right)}{{(\alpha z + 1)}\left(\mathfrak{E}_{\alpha+1}^{-1}\left(\frac{1}{z^{-1} + \alpha}\right)(\alpha^2 z + z + \alpha) - \alpha z - 1\right)},\\
\frac{\mathrm{d} \!E}{\mathrm{d}z} &= \mathfrak{E}_{\alpha+1}^{-1}\left(\frac{1}{z^{-1} + \alpha}\right)\left[1+\frac{z}{{(\alpha z + 1)}^3\left(\mathfrak{E}_{\alpha+1}^{-1}\left(\frac{1}{z^{-1} + \alpha}\right)(\alpha^2 z + z + \alpha) - \alpha z - 1\right)}\right],
\end{split}\end{equation*}
and also observe that as $\mathfrak{E}^{-1}_{\alpha + 1}$ is monotonically decreasing, then by known inequalities,
\begin{equation*}\begin{split}
0 \leq \mathfrak{E}^{-1}_{\alpha + 1} \left(\frac{1}{z^{-1} + \alpha}\right) \leq z^{-1},
\end{split}\end{equation*}
therefore
\begin{equation*}\begin{split}
1 = \frac{\alpha r}{S_i p_0} \frac{S_i p_0}{\alpha r} \leq {\left[\frac{S_i p_0}{\alpha r} \mathfrak{E}^{-1}_{\alpha + 1}\left(\frac{1}{\frac{\alpha r}{S_i p_0} + \alpha}\right)\right]}^{-1} = {\left[\frac{S_i p_0}{\alpha r} \mathfrak{E}^{-1}_{\alpha + 1}\left(\frac{S_i p_0}{\alpha(S_i p_0 + r)}\right)\right]}^{-1} = \theta_i.
\end{split}\end{equation*}
A sufficient condition on $\frac{\mathrm{d} \!E}{\mathrm{d}z} > 0$ is that $1 - \frac{z}{{(\alpha z + 1)}^4} > 0$ which is evidently true $\forall z > 0$ when $\alpha \geq 1$. Thus as $\frac{S_i p_0}{\alpha r}$ is a monotonically increasing sequence by the definition of $S_i$,
\begin{equation*}\begin{split}
\!E\left(\frac{S_i p_0}{\alpha r}\right) = \theta_i \leq \theta_{i+1} = \!E\left(\frac{S_{i+1} p_0}{\alpha r}\right),
\end{split}\end{equation*}
as required.
\end{proof}
\begin{corollary}
The initial optimal inflation factor is the upper bound and the limit of the sequence of optimal inflation factors,
\begin{equation}\begin{split}
\theta_i &\leq \theta_{*},\\
\lim_{i\to\infty} \theta_i &= \theta_{*}.
\end{split}\end{equation}
\end{corollary}
We are now ready to describe sequential step-wise inflation factors, that can be applied continuously one after the other, while keeping both the expected value of the deviation of the variance, \eqref{eq:pppm}, and the expected value of the deviation of the mean, \eqref{eq:ppxm}, zero for every step $i$.
\begin{theorem}\label{thm:inf}
The application of the sequential step-wise inflation factors,
\begin{equation}\begin{split}
\phi_{i+1} = \frac{\theta_{i+1}(S_i\theta_i p_0+r)}{\theta_i(S_i\theta_{i+1} p_0+r)}\label{eq:optinfphi},
\end{split}\end{equation}
to the forecast variance at the $i+1$th step, $p_{i+1}^\|f \leftarrow \phi_{i+1}p_{i+1}^\|f$, for all $i$, with $\phi_0 = \theta_0$ being applied at the initial time, and the sequential addition of the true step-wise correction factor
\begin{equation}\begin{split}
\psi_{i+1} = \frac{M_{i+1} (B_i - S_i \tilde{x}_0)(\theta_{i+1} - \theta_i ) p_0 r }{(S_i\theta_{i+1} p_0+r)(S_i\theta_i p_0+r)}\label{eq:optcorpsi},
\end{split}\end{equation}
the forecast mean at the $i+1$th step, $x_{i+1}^\|f \leftarrow \psi_{i+1} + x_{i+1}^\|f$, is equivalent to applying $\theta_{i+1}$ at the initial onset of the algorithm.
\end{theorem}
\begin{proof}
Assume $\p^{\|a}_{i} = \frac{M_{i}\theta_{i}p_0}{S_{i} \theta_{i}p_0 + r}r$, and $\x^{\|a}_i=\frac{M_i(B_i\theta_i p_0 + r\tilde{x}_0)}{S_i p_0 + r}$ then
\begin{equation*}\begin{split}
\phi_{i+1}\p^{\|f}_{i+1} &= \frac{\theta_{i+1}(S_i\theta_i p_0+r)}{\theta_i(S_i\theta_{i+1} p_0+r)} \frac{M_{i+1}\theta_{i}p_0}{S_{i} \theta_{i}p_0 + r}r = \frac{M_{i+1}\theta_{i+1}p_0}{S_{i} \theta_{i+1}p_0 + r},\\
\psi_{i+1} + x_{i+1}^\|f &= \frac{M_{i+1} (B_i - S_i \tilde{x}_0)(\theta_{i+1} - \theta_i ) p_0 r }{(S_i\theta_{i+1} p_0+r)(S_i\theta_i p_0+r)} + \frac{M_{i+1}(B_i\theta_i p_0 + r\tilde{x}_0)}{S_i \theta_i p_0 + r} = \frac{M_{i+1}(B_i\theta_{i+1} p_0 + r\tilde{x}_0)}{S_i \theta_{i+1}p_0 + r},
\end{split}\end{equation*}
as required.
\end{proof}
In this way, we boot-strap step-wise correct inflation factors for sequentially applied inflation.
\begin{corollary}
The sequential step-wise inflation factors are bounded from below by 1, and from above by $\alpha{(\alpha - 1)}^{-1}$,
\begin{equation}\begin{split}
1\leq\phi_{i+1}\leq \frac{\alpha}{\alpha - 1}.
\end{split}\end{equation}
\end{corollary}
\begin{proof}
For the lower bound,
\begin{equation*}\begin{split}
\phi_{i+1} &= \frac{\theta_{i+1}(S_i\theta_i p_0+r)}{\theta_i(S_i\theta_{i+1} p_0+r)} = \frac{S_i\theta_i \theta_{i+1}p_0+r\theta_{i+1}}{S_i\theta_i\theta_{i+1} p_0+r\theta_i}\\
&\geq \frac{S_i\theta_i \theta_{i+1}p_0+r\theta_{i+1}}{S_i\theta_i\theta_{i+1} p_0+r\theta_{i+1}} = 1.
\end{split}\end{equation*}
For the upper bound,
\begin{equation*}\begin{split}
\phi_{i+1} &= \frac{\theta_{i+1}(S_i\theta_i p_0+r)}{\theta_i(S_i\theta_{i+1} p_0+r)} = \frac{S_i\theta_i \theta_{i+1}p_0+r\theta_{i+1}}{S_i\theta_i\theta_{i+1} p_0+r\theta_i}\\
&\leq \frac{\theta_{i+1}}{\theta_i} \leq \theta_{*} \leq \frac{\alpha}{\alpha - 1},
\end{split}\end{equation*}
as required.
\end{proof}
This means that there is concrete evidence for an inflation factor somewhere above one being applied sequentially, step-wise in various ensemble Kalman filters. Additionally, as applying the $\phi$ inflation, but ignoring the $\psi$ correction could potentially incur additional unbounded error, even if the sequence of corrections converges in probability in time, there is the potential for catastrophe, meaning that some time of sequential correction to the mean needs to be applied in Ensemble Kalman filtering. However, in a non-linear setting, the state is typically bounded, and therefore the absence of correction factors might dissipate in time (or be drowned out by the ensemble).
\subsection{Convergence of the perturbed problem in the case of a finite ensemble}%
\label{subsec:cfe}
We now have all the tools to prove the convergence of the SPEnKF in the case of a finite ensemble.
\begin{theorem}[Finite ensemble convergence of the analysis variance of the SPEnKF to that of the scalar KF]\label{thm:finvarconv}
In the case of a finite ensemble $(\alpha < \infty)$,
\begin{enumerate}
\item $\mathbb{E}[\Delta p_i]$ converges to zero in the step limit $i\to \infty$, and is always zero when optimal sequential step-wise inflation~\eqref{eq:optinfphi} is applied at each step,
and
\item $\mathbb{E}[\Delta p_i^2]$ converges to zero in the step limit.
\end{enumerate}
\end{theorem}
\begin{proof}
For $\mathbb{E}[\Delta p_i]$, recall from~\eqref{eq:pppm} that,
\begin{equation}\begin{split}
\mathbb{E}[\Delta p_i] &= \frac{\alpha M_i^2 r^2 e^{\frac{\alpha r}{S_i \tilde{p}_0}}}{S_i (S_i p_0 + r)}\left[-\frac{p_0}{\tilde{p}_0}E_{\alpha}\left(\frac{\alpha r}{S_i \tilde{p}_0}\right) + E_{\alpha+1}\left(\frac{\alpha r}{S_i \tilde{p}_0}\right)\right],
\end{split}\end{equation}
and observe that
\begin{equation*}\begin{split}
\lim_{i\to\infty} \frac{M_i^2}{S_i}
\end{split}\end{equation*}
is bounded by a constant above, but converges linearly to zero when $\lim_{i\to\infty} \lvert M_i\rvert < \infty$. Note that when $M_i\not\to0$, $S_i\to\infty$, and the term
\begin{equation*}\begin{split}
\lim_{i\to\infty} \frac{1}{S_i p_0 + r}
\end{split}\end{equation*}
trivially converges to zero, linearly. Note also that
\begin{equation*}\begin{split}
e^{\frac{\alpha r}{S_i \theta_i p_0}}\left(
E_{\alpha+1}\left(\frac{\alpha r}{S_i \theta_i p_0}\right)
-\frac{p_0}{\theta_i p_0}E_{\alpha}\left(\frac{\alpha r}{S_i \theta_i p_0}\right) \right) = 0.
\end{split}\end{equation*}
thus the last term converges to zero when optimal inflation is applied, at each step.
For $\mathbb{E}[\Delta p_i^2]$, \eqref{eq:pppv}, it is trivial to observe that either $\lim_{i\to\infty}\frac{M_i^4}{S_i^2} = 0$, or $\lim_{i\to\infty}\frac{1}{{(S_i p_0 + r)}^2} = 0$ (or both), with the remaining terms converging to constants.
\end{proof}
\begin{theorem}[Finite ensemble convergence of the analysis mean of the SPEnKF to that of the scalar KF]\label{thm:finmeanconv}
In the case of a finite ensemble $(\alpha < \infty)$, the term $\mathbb{E}[\Delta x_i]$:
%
\begin{enumerate}
\item converges to zero weakly always in the step limit,
\item converges strongly when $\limsup_{i\to\infty}\frac{i+1}{S_i} < \infty$ in the step limit, and
\item is always zero when $\tilde{x}_0 = x_0$, optimal inflation~\eqref{eq:optinfphi} and optimal correction~\eqref{eq:optcorpsi} are applied,
\end{enumerate}
and $\mathbb{E}[\Delta x_i^2]$ converges to zero weakly always in the step limit.
\end{theorem}
\begin{proof}
For $\mathbb{E}[\Delta x_i]$, from~\eqref{eq:ppxm},
\begin{equation}\begin{split}
\left[\lim_{i\to\infty} e^{\frac{\alpha r}{S_i \tilde{p}}}\left(\frac{M_i(B_i - x_0 S_i)}{S_i(S_i p_0 + r)} E_{\alpha+1}\left(\frac{\alpha r}{S_i \tilde{p}_0}\right) - \frac{p}{\tilde{p}_0} \frac{M_i(B_i - \tilde{x}_0 S_i)}{S_i(S_i p_0 + r)} E_{\alpha}\left(\frac{\alpha r}{S_i \tilde{p}_0}\right)\right)\right],
\end{split}\end{equation}
converges to zero in probability by Lemma~\ref{lem:weakconvpert}. If $\limsup_{i\to\infty}\frac{i+1}{S_i} < \infty$, by Lemma~\ref{lem:strongconvpert} this converges strongly to zero. The term
\begin{equation}\begin{split}
\frac{\alpha r^2 (\tilde{x}_0 - x_0)}{\tilde{p}_0}
\left[\lim_{i\to\infty} \frac{M_i}{S_i p_0 + r}\right]
\left[\lim_{i\to\infty} e^{\frac{\alpha r}{S_i \tilde{p}}} E_{\alpha}\left(\frac{\alpha r}{S_i \tilde{p}_0}\right)\right],
\end{split}\end{equation}
converges to zero in general as it has the term $\left[\lim_{i\to\infty} \frac{M_i}{S_i p_0 + r}\right]$, otherwise if $x_0=\tilde{x}_0$, with optimal sequential step-wise inflation and correction, the term is zero always by Theorem~\ref{thm:inf}.
For $\mathbb{E}[\Delta x_i^2]$, \eqref{eq:ppxv}, each term of the summation has two multiples of the term from Lemma~\ref{lem:weakconvpert}, thus converging to zero weakly always, and strongly if $\limsup_{i\to\infty}\frac{i+1}{S_i} < \infty$, by Lemma~\ref{lem:strongconvpert}.
\end{proof}
Theorems~\ref{thm:finvarconv} and~\ref{thm:finmeanconv} together show that there is strong evidence that the full ensemble Kalman filter can converge to the Kalman filter in expected value in the case of a finite ensemble, in finite time, provided that optimal corrections are made in the algorithm. Additionally we provide very strong evidence that sequential step-wise inflation, as performed in many flavours of the ensemble Kalman filter is not a heuristic, but in fact can be derived from the underlying distributions associated with it.
\subsection{SPEnKF with imaginary perturbations of observations}%
\label{subsec:po}
The idea of perturbed observations was first introduced in order to attempt to correct the ensemble Kalman filter~\cite{burgers1998analysis} from a statistical point of view under certain incorrect simlifications and assumptions. The wrongly assumed independence of the Kalman gain estimate from the anomalies and expected value of the Kalman gain estimate being the Kalman filter Kalman gain being just two. Augmenting the stochastic ensemble Kalman analysis update with a vector of `perturbed observations', $\*\Xi$, derived from the assumed distribution of the unbiased observation error, the update of the EnKF with perturbed observations, can be written as,
\begin{equation*}
\*x^\|a = \*x^\|f - \*K (\*H \*x^\|f + \*\Xi - \*y^o\*1^\intercal),
\end{equation*}
which we can decompose into the following two updates:
\begin{equation*}\begin{split}
\bar{\*x}^\|a &= \bar{\*x}^\|f - \*K(\*H\bar{\*x}^\|f - \*y^o),\\
\*A^\|a &= \*A^\|f - \*K(\*H\*A^\|f - \*\Xi),
\end{split}\end{equation*}
with the first just being the standard Kalman update, and the second being the unique stochastic EnKF anomaly update. In the scalar case we will again ignore $\*H$, as before and replace $\*A$ with $\*a$ and $\*\Xi$ with $\*\xi$.
In order to avoid difficulty with vector inner products, we will be looking at imaginary perturbed observations as a surrogate for true perturbed observations. Empirical results suggest that this is a better filter than that with real perturbations, thus we can say with some confidence that results about this filter will be a lower bound for the full SPEnKF with perturbed observations, though a full analysis is, as of yet, not in our reach. Additionally we will not be looking at the asymptotic case of steps. Instead, we will be computing a perturbed observation update and a normal SPEnKF update on the SPEnKF forecast, and looking at the discrepancy between the two.
We therefore assume that in the analysis update below, $\*a_i^\|f$ was obtained with an ideal square-root filter, run from step 0 to step $i$, and that the $\*a_i^\|a$ that is obtained via the imaginary perturbed observation approach will be discarded in favor of another square root update. We will thus look at the update,
\begin{equation*}
\*a_i^\|a = (1-\hat{k}_i) \*a_i^\|f + \mathrm{i}\hat{k}\*\xi_i,
\end{equation*}
where $\*\xi_i$ is an ensemble of $N$ samples from $\!N(0,r)$. We will also modify the analysis variance equation to account for complex conjugates, and observe:
\begin{equation*}\begin{split}
\hat{p}_i^\|a &= \frac{1}{N} \left(\*a_i^\|a \cdot \overline{\*a_i^\|a}\right)\\
&= {(1-\hat{k}_i)}^2 \hat{p}_i^\|f + \hat{k}_i^2 \hat{r}_i\\
&= \hat{k}_i r + \hat{k}_i^2 (\hat{r}_i - r)
\end{split}\end{equation*}
Representing the realizations in terms of random variables, we will arrive at the fact that the random variable representing the new analysis update can be written in the form: $P = r K + K^2 (R - r)$. It can be trivially shown that $R \sim \Gamma\left(\alpha, \frac{\alpha}{r}\right)$, and thus $\mathbb{E}[R] = r$.
Looking at the moments of $P$, we manipulate:
\begin{equation*}\begin{split}
\mathbb{E}[r K + K^2 (R - r)] &= \mathbb{E}[r K] + \mathbb{E}[K^2]\mathbb{E}[(R - r)],\\
&= r \mathbb{E}[K]\\
\text{Cov}(r K,K^2 (R - r)) &= \mathbb{E}[(r K - r\mathbb{E}[K])(K^2 (R - r) - \mathbb{E}[K^2 (R - r)])]\\
&= \mathbb{E}[r K^3 (R - r) - r K^2 (R - r)\mathbb{E}[K]]\\
&= \mathbb{E}[r K^3 - r K^2 \mathbb{E}[K]]\mathbb{E}[(R - r)]\\
&= 0,\\
\Var(r K + K^2 (R - r)) &= \Var(r K) + \Var(K^2 (R - r)) + 2\text{Cov}(r K, K^2 (R - r))\\
&= r^2 \Var(K) + \mathbb{E}[{(K^2 (R - r) - \mathbb{E}[K^2 (R - r)])}^2]\\
&= r^2 \Var(K) + \mathbb{E}[K^4]\mathbb{E}[{(R - r)}^2].
\end{split}\end{equation*}
Thus we see that the expected value of a perturbed observation filter is the same as of a perfect square root ensemble filter, however we do incur additional variance.
We can analyze this additional term, $\mathbb{E}[K^4]\mathbb{E}[{(R - r)}^2]$ in two different ways, in the asymptotic case of ensemble size, and in the step limit with a finite ensemble.
Note first that, without proof,
\begin{equation}\begin{split}
\mathbb{E}[{(R_i - r)}^2] &= \frac{r^2}{\alpha},\\
\mathbb{E}[K_i^4] &= \frac{M_i^8}{6 p^4 S_i^7}
\left[\begin{aligned}
\phantom{+}& p \left(p S_i \left(p S_i \left(6 p S_i+\alpha \
(\alpha (\alpha +7)+18) r\right)+(2 \alpha +9) \alpha ^2 \
r^2\right)+\alpha ^3r^3\right)\\
-& \frac{\alpha r e^{\frac{\alpha r}{p S_i}} \
\left((\alpha +3) p S_i \left((\alpha +2) p S_i \left((\alpha +1) p \
S_i+3 \alpha
r\right)+3 \alpha ^2 r^2\right)+\alpha ^3 r^3\right) E_{\alpha \
}\left(\frac{r \alpha }{p S_i}\right)}{S_i}
\end{aligned}\right].
\end{split}\end{equation}
The asymptotic case of ensemble size is by far the easiest:
\begin{equation}\begin{split}
\lim_{\alpha\to\infty} \mathbb{E}[{(R_i - r)}^2] &= 0\\
\lim_{\alpha\to\infty} \mathbb{E}[K_i^4] &= \frac{M_i^8 p_0 (2 r^2 - 3 S_i r p_0 + 3 S_i^2 p_0^2)}{3 S_i^3 {(S_i p_0 + r)}^3},
\end{split}\end{equation}
it is therefore the case that,
\begin{equation}\begin{split}
\lim_{\alpha\to\infty} \mathbb{E}[K_i^4]\mathbb{E}[{(R_i - r)}^2] = 0.
\end{split}\end{equation}
This shows that there is significant evidence that in the asymptotic case of ensemble size, perturbed observation filters are as good as square-root filters.
Let's now look at the case of a finite ensemble in the step limit, and the worst case where $S_i$ grows roughly as fast as $M_i^2$,
\begin{equation}\begin{split}
\lim_{i\to\infty}\mathbb{E}[K_i^4] = \text{const}.
\end{split}\end{equation}
This means that in the worst case, our variance has an additional constant term of $\frac{r^2}{\alpha}$, which can potentially be large. While we cannot claim that this will hold for non-imaginary perturbed observations, we postulate that this term is, in part responsible for some of the additional error that is seen in that type of filter compared to that of a square-root filter.
\section{Extending SPEnKF to Multivariate Case}
\label{sec:multivariate}
We will now attempt to extend the SPEnKF to a limited multivariate case. Assume now that we are looking at a multivariate state space, $\*x$ of size $n$, Assume additionally that we have a perfect model, whose step is represented by a matrix with independent action occurring in a constant basis throughout all time, that is,
\begin{align}
\*L_i = \*Z \*M_i \*Z^{-1},
\end{align}
with $\*M_i = \text{diag}(m_{i, 1}, \dots m_{i, n})$ being a diagonal matrix of real values, and $\*Z$ being any invertible constant matrix.
Let the initial input to our algorithm consist of a mean, $\bar{\*v}_0$, and a set of anomalies $\*B^\|f_0$ such that $\left[\*B^\|f_0\right]_{(:, {1\leq i\leq N})} \sim \!N(\*0, \*Z\*P_0\*Z^\intercal)$, where $\*P_0 = \text{diag}(p_{0, 1}, p_{0, 2}, \dots p_{0, n})$. Let all observations come from a normal distribution with a constant covariance matrix, $\*w_i \sim \!N(\*v_i^\|t, \*Z\*R\*Z^\intercal)$ with
$\*R = \text{diag}(r_1, \dots r_n)$. Converting out of the linear basis, we get the familiar notation,
\begin{align}
\begin{split}
\*x &= \*Z^{-1}\bar{\*v},\\
\*A &= \*Z^{-1}\*B,\\
\*y &= \*Z^{-1}\*w.
\end{split}
\end{align}
Note that this directly implies that the observations in the basis are distributed as $\*y_i \sim \!N(\*x_i^\|t, \*R)$, and the anomalies in the basis are distributed like $\left[\*A^\|f_0\right]_{(:, {1\leq i\leq N})} \sim \!N(\*0, \*P_0)$.
Note that long-term dynamics can be written in the form
\begin{align}
\prod_{j = 0}^i \*L_i = \*Z \left(\prod_{j=0}^i \*M_i\right) \*Z^{-1},
\end{align}
meaning that if we initialize the perfect square root EnKF in the model basis, we only have to look at independent model dynamics.
The SPEnKF formulas, \eqref{eq:spenkff} and \eqref{eq:spenkfa}, for the mean of the $j$th member of state space in the basis at the $i$th time, become,
\begin{equation}
\begin{split}
\bar{x}^\|f_{i+1,j} &= m_{i,j} \bar{x}^\|a_{i,j},\\
\bar{x}^\|a_{i,j} &= \bar{x}^\|f_{i,j} + \hat{k}_i(y_{i,j} - \bar{x}^\|f_{i,j}),
\end{split}
\end{equation}
and for the variances,
\begin{equation}
\begin{split}
\*a^\|f_{i+1,j} &= m_{i,j} \*a^\|a_{i,j},\\
\*a^\|a_{i,j} &= {\left(\hat{p}^\|a_{i,j}\right)}^{\frac{1}{2}}{\left(\hat{p}^\|f_{i,j}\right)}^{-\frac{1}{2}}\*a^\|f_{i,j},\\
\hat{k}_{i,j} &= \frac{\hat{p}^\|f_{i,j}}{\hat{p}^\|f_{i,j}+r_j},\\
\hat{p}^\|f_{i,j} &= \frac{1}{N} (\*a^\|f_{i,j} \cdot \*a^\|f_{i,j}),\\
\hat{p}^\|a_{i,j} &= (1-\hat{k}_{i,j})\hat{p}^\|f_{i,j}.
\end{split}
\end{equation}
Writing the mean formulas in matrix notation, we get,
\begin{equation}
\begin{split}
\bar{\*x}^\|f_{i+1,j} &= \*M_i \bar{\*x}^\|a_{i,j},\\
\bar{\*x}^\|a_{i,j} &= \bar{\*x}^\|f_{i,j} + \hat{\*K}_i(\*y_{i,j} - \bar{\*x}^\|f_{i,j}),
\end{split}
\end{equation}
and for the covariance,
\begin{equation}
\begin{split}
\*A^\|f_{i+1} &= \*M_i \*A^\|a_{i},\\
\*A^\|a_{i} &= {\left(\hat{\*P}^\|a_{i}\right)}^{\frac{1}{2}}{\left(\hat{\*P}^\|f_{i}\right)}^{-\frac{1}{2}}\*A^\|f_{i},\\
\hat{\*K}_{i} &= \hat{\*P}^\|f_{i}{(\hat{\*P}^\|f_{i}+\*R)}^{-1},\\
\hat{\*P}^\|f_{i} &= \*I \circ \frac{1}{N} (\*A^\|f_{i} \*A^{\|f,\intercal}_{i}),\\
\hat{\*P}^\|a_{i} &= (\*I-\hat{\*K}_i)\hat{\*P}^\|f_{i}.
\end{split}
\end{equation}
Note that these are almost identical to the ESRF formulas, \eqref{eq:meansqrttransport} and \eqref{eq:anomalysqrttransport}. The only difference comes in the covariance tapering, in this case commonly referred to as Schur-product localization in DA literature.
\section{Conclusions}
\label{sec:conclusions}
We introduce a toy idealized EnKF variant named the SPEnKF, for Scalar Pedagogical EnKF, about which we prove several results.
We show the trivial result that in the limit of ensemble size, the SPEnKF degenerates to that of the scalar Kalman filter.
We show that in the step limit, and with a finite ensemble, the SPEnKF converges to that of the scalar Kalman filter, weakly always, and strongly for ``useful'' problems.
We derive optimal sequential step-wise variance inflation and mean correction factors such that the expected values of the SPEnKF outputs converge exactly to that of the scalar Kalman filter in finite time and with a finite ensemble.
We thus provide an alternative explanation for the need for inflation in ensemble-based methods: it is the required in order for the EnKF estimates to be useful in the realistic finite step finite ensemble case.
We then apply this framework to a scalar imaginary perturbed observations Kalman filter and show that in the case of a finite ensemble, we introduce an additional variance proportional to the square of the observation error variance compared to that of the vanilla SPEnKF.
Future work would try to naturally generalize these results to the multivariate case. We believe that it is possible to show that methods such as Schur-product localization are also required for similar reasons. Moreover, there is evidence~\cite{bickel2008regularized} to suggest that this might be doable in the undersampled case as well.
\begin{acknowledgments}
This work was supported by awards AFOSR DDDAS FA9550--17--1--0015, AFOSR DDDAS 15RT1037, NSF CCF--1613905, NSF ACI--17097276, and by the Computational Science Laboratory at Virginia Tech.
\end{acknowledgments}
\section*{References}
|
0708.3019
|
\section{Introduction}
\label{sec1}
\vspace{-3mm}
The problem of fading and the ways to combat it through spatial diversity
techniques have been an active area of research. Multiple-input
multiple-output (MIMO) techniques have become popular in realizing spatial
diversity and high data rates through the use of multiple transmit antennas.
For such co-located multiple transmit antenna systems low maximum-likelihood
(ML) decoding complexity space-time block codes (STBCs) have been studied by
several researchers \cite{TJC}-\cite{KaR}
which include the well known complex orthogonal designs (CODs) and their
generalizations. Recent research has shown that the advantages of spatial
diversity could be realized in single-antenna user nodes through user
cooperation \cite{SEA},\cite{LaW} via relaying. A simple wireless relay
network of $N+2$ nodes consists of a single source-destination pair with
$N$ relays. For such relay channels, use of CODs
\cite{TJC},\cite{TiH} has been studied in \cite{JiJ}. CODs are attractive
for cooperative communications for the following reasons: $i)$ they offer
full diversity gain and coding gain, $ii)$ they are `scale free' in the
sense that deleting some rows does not affect the orthogonality, $iii)$
entries are linear combination of the information symbols and their
conjugates which means only linear processing is required at the relays,
and $iv)$ they admit very fast ML decoding (single-symbol decoding (SSD)).
However, it should be noted that the last property applies only to the
decode-and-forward (DF) policy at the relay node.
In a scenario where the relays amplify and forward (AF) the signal, it is
known that the orthogonality is lost, and hence the destination has to use
a complex multi-symbol ML decoding or sphere decoding \cite{JiJ},\cite{RaR1}.
It should be noted that the AF policy is attractive for two reasons: $i)$
the complexity at the relay is greatly reduced, and $ii)$ the restrictions
on the rate because the relay has to decode is avoided \cite{JiH}.
In order to avoid the complex ML decoding at the destination, in \cite{RaR2},
the authors propose an alternative code design strategy and propose a SSD
code for 2 and 4 relays. For arbitrary number of relays, recently in
\cite{YiK}, distributed orthogonal STBCs (DOSTBCs) have been studied
and it is shown that if the destination has the complete channel state
information (CSI) of all the source-to-relay channels and the
relay-to-destination channels, then the maximum possible rate is upper
bounded by $\frac{2}{N}$ complex symbols per channel use for $N$ relays.
Towards improving the rate of transmission and achieving simultaneously
both full-diversity as well as SSD at the destination, in this paper, we
study relay channels with the assumption that the relays have the phase
information of the source-to-relay channels and the destination has the
CSI of all the channels. Coding for partially-coherent relay channel
(PCRC, Section \ref{pcrc_sec}) has been studied in \cite{RaR3}, where a
sufficient condition for SSD has been presented.
The contributions of this paper can be summarized as follows:
\vspace{-4mm}
\begin{itemize}
\item First, a new set of necessary and sufficient conditions for a STBC
to be SSD for co-located multiple antenna communication is obtained. The
known set of necessary and sufficient conditions in \cite{KhR} is in terms
of the dispersion matrices (weight matrices) of the code, whereas our new
set of conditions is in terms of the column vector representation matrices
\cite{Lia} of the code and is a generalization of the conditions given in
\cite{Lia} in terms of column vector representation matrices for CODs.
\item A set of necessary and sufficient conditions for a distributed STBC
(DSTBC) to be SSD for a PCRC is obtained by identifying the additional
conditions. Using this, several SSD DSTBCs for PCRC are identified among
the known classes of STBCs for co-located multiple antenna system.
\item It is proved that even if a SSD STBC for a co-located MIMO channel
does not satisfy the additional conditions for the code to be SSD for a
PCRC, single-symbol decoding of it in a PCRC gives full-diversity and only
coding gain is lost.
\item
It is shown that when a DSTBC is SSD for a PCRC, then arbitrary
coordinate interleaving of the in-phase and quadrature-phase components
of the variables does not disturb its SSD property for PCRC.
\item It is shown that the possibility of {\em channel phase compensation}
operation at the relay nodes using partial CSI at the relays increases the
possible rate of SSD DSTBCs from $\frac{2}{N}$ when the relays do not have
CSI to $\frac{1}{2},$ which is independent of $N$.
\item Extensive simulation results are presented to illustrate the above
contributions.
\end{itemize}
\vspace{-5mm}
The remaining part of the paper is organized as follows: In Section
\ref{sec2}, the signal model for a PCRC is developed. Using this model, in
Section \ref{sec3}, a new set of necessary and sufficient conditions for a
STBC to be SSD in a co-located MIMO is presented. Several classes of SSD codes
are discussed and conditions for full-diversity of a subclass of SSD codes
is obtained. Then, in Section \ref{sec4}, SSD DSTBCs for PCRC are
characterized by identifying a set of necessary and sufficient conditions.
It is shown that the SSD property is invariant under coordinate interleaving
operations which leads to a class of SSD DSTBCs for PCRC. The class of rate
half CODs obtained from rate one real orthogonal designs (RODs) by stacking
construction \cite{TJC} is shown to be SSD for PCRC. Also, it is shown that
SSD codes for co-located MIMO, under suboptimal SSD decoder for PCRC offer
full diversity. Simulation results and discussion constitute Section
\ref{sec5}. Conclusions and scope for further work are presented in
Section \ref{sec6}.
\vspace{-4mm}
\section{System Model}
\label{sec2}
\vspace{-4mm}
Consider a wireless network with $N+2$ nodes consisting of a source, a
destination, and $N$ relays\footnote{In the system model considered here,
we assume that there is no direct link between source and destination.
However, whatever results we show here can be extended to a system model
with a direct link between source and destination.}, as shown in
Fig. \ref{fig1}. All nodes are
half-duplex nodes, i.e., a node can either transmit or receive at a time
on a specific frequency. We consider amplify-and-forward (AF) transmission
at the relays. Transmission from the source to the destination is carried
out in two phases. In the first phase, the source transmits information
symbols $x^{(i)}, 1 \leq i \leq T_1$ in $T_1$ time slots. All the $N$ relays
receive these $T_1$ symbols. This phase is called the {\em broadcast phase}.
In the second phase, all the $N$ relays\footnote{Here, we assume that all
the $N$ relays participate in the cooperative transmission. It is also
possible that some relays do not participate in the transmission based
on whether the channel is in outage or not. We do not consider such a
partial participation scenario here.} perform distributed space-time
block encoding on their received vectors and transmit the resulting
encoded vectors in $T_2$ time slots. That is, each relay will transmit
a column (with $T_2$ entries) of a distributed STBC matrix of size
$T_2\times N$. The destination receives a faded and noise added version
of this matrix. This phase is called the {\em relay phase}. We assume that
the source-to-relay channels remain static over $T_1$ time slots, and the
relay-to-destination channels remain static over $T_2$ time slots.
\subsection{No CSI at the relays}
\vspace{-4mm}
The received signal at the $j$th relay, $j=1,\cdots,N$, in the
$i$th time slot, $i=1,\cdots,T_1$, denoted by $v_{j}^{(i)}$,
can be written as\footnote{We use the following notation:
Vectors are denoted by boldface lowercase letters, and matrices are
denoted by boldface uppercase letters. Superscripts $T$ and $\mathcal{H}$
denote transpose and conjugate transpose operations, respectively and $*$
denotes matrix conjugation operation.}
\begin{eqnarray}
v_{j}^{(i)} & = & \sqrt{E_1} h_{sj} x^{(i)} + z_{j}^{(i)},
\label{rx_relay}
\end{eqnarray}
where $h_{sj}$ is the complex channel gain from the source $s$ to the $j$th
relay, $z_{j}^{(i)}$ is additive white Gaussian noise at relay $j$ with zero
mean and unit variance, $E_1$ is the transmit energy per symbol in the
broadcast phase, and $E\left[\left(x^{(i)}\right)^* x^{(i)}\right] = 1$.
But no channel knowledge is assumed at the relays. Under the assumption of
no CSI at the relays, the amplified $i$th received signal at the $j$th relay
can be written as \cite{JiJ}
\begin{eqnarray}
\widehat{v}_{j}^{(i)} & = & \underbrace{\sqrt{\frac{E_2}{E_1 + 1}}}_{\stackrel{\triangle}{=} \,\, G} \, {v}_{j}^{(i)},
\label{no_comp}
\end{eqnarray}
where $E_2$ is the transmit energy per transmission of a symbol in the
relay phase, and $G$ is the amplification factor at the relay that makes
the total transmission energy per symbol in the relay phase to be equal
to $E_2$. Let $E_t$ denote the total energy per symbol in both the phases
put together. Then, it is shown in \cite{JiH} that the optimum energy
allocation that maximizes the receive SNR at the destination is when half
the energy is spent in the broadcast phase and the remaining half in the
relay phase when the time allocations for the relay and broadcast phase are
same i.e., $T_1 = T_2$. We also assume that the energy is distributed equally
i.e., $E_1 = \frac{E_t}{2}$ and $E_2 = \frac{E_t}{2M}$, where $M$ is the
number of transmissions per symbol in the STBC. For the unequal-time
allocation ($T_1 \neq T_2$) this distribution might not be optimal.
At relay $j$, a $2T_1 \times 1$ real vector $\widehat{\bf v}_{j}$ given by
\begin{eqnarray}
\label{vhatx}
\widehat{\bf v}_{j} & = & \left[
\widehat{v}_{jI}^{(1)}, \widehat{v}_{jQ}^{(1)},
\widehat{v}_{jI}^{(2)}, \widehat{v}_{jQ}^{(2)},
\cdots,
\widehat{v}_{jI}^{(T_1)}, \widehat{v}_{jQ}^{(T_1)}\right]^T,
\end{eqnarray}
is formed,
where $\widehat{v}_{jI}^{(i)}$ and $\widehat{v}_{jQ}^{(i)}$,
respectively, are the in-phase (real part) and quadrature (imaginary part)
components of $\widehat{v}_{j}^{(i)}.$ Now, (\ref{vhatx}) can be written
in the form
\begin{eqnarray}
\label{vhat2x}
\widehat{\bf v}_{j} & = & G \, \sqrt{E_1} \, {\bf H}_{sj}\, {\bf x} \, + \, {\bf \widehat{z}}_{j},
\end{eqnarray}
where ${\bf x}$ is the $2T_1 \times 1$ data symbol real vector, given by
\begin{eqnarray}
\label{datavector}
{\bf x} = \left[
x_I^{(1)},x_Q^{(1)},
x_I^{(2)},x_Q^{(2)},
\cdots,
x_I^{(T_1)}, x_Q^{(T_1)}\right
]^T,
\end{eqnarray}
${\bf \widehat{z}}_{j}$ is the $2T_1 \times 1$ noise vector, given by
\begin{eqnarray*}
\widehat{\bf z}_{j} & = & \left[
\widehat{z}_{jI}^{(1)}, \widehat{z}_{jQ}^{(1)},
\widehat{z}_{jI}^{(2)}, \widehat{z}_{jQ}^{(2)},
\cdots,
\widehat{z}_{jI}^{(T_1)}, \widehat{z}_{jQ}^{(T_1)}
\right]^T,
\end{eqnarray*}
where $\widehat{z}_{j}^{(i)} = G \, z_{j}^{(i)}$, and
${\bf H}_{sj}$ is a $2T_1\times 2T_1$ block-diagonal matrix, given by
\begin{eqnarray}
\label{2bdiagonal}
{\bf H}_{sj} & = &
\left [ \begin{array}{ccc}
\left[ \begin{array}{cc}
h_{sjI}^{} & -h_{sjQ} \\ h_{sjQ} & h_{sjI}
\end{array} \right ]
& \cdots & {\bf 0} \\
\vdots & \ddots & \vdots \\
{\bf 0} & \cdots &
\left[ \begin{array}{cc}
h_{sjI} & -h_{sjQ} \\ h_{sjQ} & h_{sjI}
\end{array}\right ]
\end{array}\right ].
\end{eqnarray}
Let
\begin{eqnarray}
{\bf C} & = & \Big[{\bf c}_1, {\bf c}_2, \cdots, {\bf c}_N\Big]
\end{eqnarray}
denote the $T_2\times N$ distributed STBC matrix to be sent in the
relay phase jointly by all $N$ relays, where ${\bf c}_j$ denotes
the $j$th column of ${\bf C}$. The $j$th column ${\bf c}_j$ is
manufactured by the $j$th relay as
\begin{eqnarray}
\label{stackx}
{\bf c}_{j} & = & {\bf A}_j \widehat{\bf v}_{j} \nonumber \\
& = & \underbrace{G \, \sqrt{E_1} {\bf A}_j \, {\bf H}_{sj}}_{{\bf B}_j}\, {\bf x} \, + \,
{\bf A}_j \, {\bf \widehat{z}}_{j},
\end{eqnarray}
where ${\bf A}_j$ is the complex processing matrix of size $T_2\times 2T_1$
for the $j$th relay, called the {\em relay matrix} and ${\bf B}_j$ can be
viewed as the column vector representation matrix \cite{Lia} for the
distributed STBC with the difference that in our case the vector ${\bf x}$
is real whereas in \cite{Lia} it is complex. For example, for the 2-relay
case (i.e., $N = 2$), with $T_1=T_2=2,$ using Alamouti code, the relay
matrices are given by
\begin{eqnarray}
\label{AA}
{\bf A}_1 = \left [ \begin{array}{cccc} 1 & {\bf j} & 0 & 0 \\
0 & 0& -1 & {\bf j} \end{array} \right ]
& \mbox{ and } &
{\bf A}_2 = \left [ \begin{array}{cccc} 0 & 0 & 1 & {\bf j} \\
1 & -{\bf j} & 0 & 0 \end{array} \right ].
\end{eqnarray}
Let ${\bf y}$ denote the $T_2 \times 1$ received signal vector at the
destination in $T_2$ time slots. Then, ${\bf y}$ can be written as
\begin{eqnarray}
\label{rx1_no_csi}
{\bf y} & = & \sum_{j=1}^{N} h_{jd} {\bf c}_{j} + {\bf z}_d,
\end{eqnarray}
where $h_{jd}$ is the complex channel gain from the $j$th relay to the
destination, and ${\bf z}_d$ is the AWGN noise vector at the destination
with zero mean and $E[{\bf z}_d\, {\bf z}_d^*]={\bf I}$. Substituting
(\ref{stackx}) in (\ref{rx1_no_csi}), we can write
\begin{eqnarray}
\label{rx2_no_csi}
{\bf y} & = & G\, \sqrt{E_1} \left(\sum_{j=1}^{N} h_{jd} {\bf H}_{sj}
{\bf A}_j \right) {\bf x} \, + \,
\sum_{j=1}^{N} h_{jd} {\bf A}_j {\widehat{\bf z}}_{j} \, + \,{\bf z}_d.
\end{eqnarray}
\subsection{With phase only information at the relays}
\label{pcrc_sec}
\vspace{-3mm}
In this subsection, we obtain a signal model for the case of partial CSI
at the relays, where we assume that each relay has the knowledge of the
channel phase on the link between the source and itself in the broadcast
phase. That is, defining the channel gain from source to relay $j$ as
$h_{sj}=\alpha_{sj}e^{{\bf j}\theta_{sj}}$, we assume that relay $j$
has perfect knowledge of only $\theta_{sj}$ and does not have the
knowledge of $\alpha_{sj}.$
In the proposed scheme, we perform a phase compensation operation on the
amplified received signals at the relays, and space-time encoding is done
on these phase-compensated signals. That is, we multiply $\widehat{v}_j^{(i)}$
in (\ref{no_comp}) by $e^{-{\bf j}\theta_{sj}}$ before space-time encoding.
Note that multiplication by $e^{-{\bf j}\theta_{sj}}$ does not change the
statistics of ${z}_{j}^{(i)}$. Therefore, with this phase compensation, the
$\widehat{{\bf v}}_j$ vector in (\ref{vhat2x}) becomes
\begin{eqnarray}
\label{vhat2y}
\widehat{\bf v}_{j} & = & \left(G \, \sqrt{E_1} \, {\bf H}_{sj}\, {\bf x} \, + \, {\bf \widehat{z}}_{j}\right) \, e^{-{\bf j}\theta_{sj}} \nonumber \\
& = & G \, \sqrt{E_1} \, |h_{s_j}| \, {\bf x} \, + \, {\bf \widehat{z}}_{j}.
\end{eqnarray}
Consequently, the ${\bf c}_j$ vector generated by relay $j$ is given by
\begin{eqnarray}
\label{stacky}
{\bf c}_{j} & = & {\bf A}_j \widehat{\bf v}_{j} \nonumber \\
& = & \underbrace{G \, \sqrt{E_1} {\bf A}_j \, \left|h_{sj}\right|}_{\stackrel{\triangle}{=} \,\,\,{\bf B}_j^{'}}\, {\bf x} \, + \, {\bf A}_j \, {\bf \widehat{z}}_{j},
\end{eqnarray}
where ${\bf B}_j^{'}$ is the equivalent weight matrix with phase compensation.
Now, we can write the received vector $ {\bf y} $ as
\begin{eqnarray}
\label{rx2}
{\bf y} & = & G\, \sqrt{E_1} \left(\sum_{j=1}^{N} h_{jd} |h_{sj}| {\bf A}_j \right) {\bf x} \, + \, \underbrace{\sum_{j=1}^{N} h_{jd} {\bf A}_j {\widehat{\bf z}}_{j} \, + \,{\bf z}_d}_{{\tilde{{\bf z}}}_d \,\,\mbox{{\footnotesize : total noise}}}.
\end{eqnarray}
Figure \ref{fig2} shows the processing at the $j$th relay in the proposed
phase compensation scheme. Such systems will be referred as {\em
partially-coherent relay channels} (PCRC). A distributed STBC which is
SSD for a PCRC will be referred as SSD-DSTBC-PCRC.
\vspace{-4mm}
\section{Conditions for SSD and Full-Diversity for Co-located MIMO}
\label{sec3}
\vspace{-4mm}
The class of SSD codes, including the well known CODs, for co-located MIMO
has been studied in \cite{KhR}, where a set of necessary and sufficient
conditions for an arbitrary linear STBC to be SSD has been obtained in
terms of the dispersion matrices \cite{HaH}, also known as weight matrices.
In this section, a new set of necessary and sufficient conditions in terms
of the column vector representation matrices \cite{Lia} of the code is
obtained that are amenable for extension to PCRC. This is a generalization
of the conditions given in \cite{Lia} in terms of column vector
representation matrices for CODs.
Towards this end, the received vector ${\bf y}$ in a co-located MIMO setup
can be written as
\begin{eqnarray}
\label{rx_colocate}
{\bf y} & = & \sqrt{E_t} \left(\sum_{j=1}^{N} h_{jd} {\bf A}_j \right) {\bf x} \, + \,{\bf z}_d.
\end{eqnarray}
\begin{thm}
For co-located MIMO with $N$ transmit antennas, the linear STBC as given
in (\ref{rx_colocate}) is SSD {\em iff }
\begin{eqnarray}
\label{singlesymx}
{\bf A}_{jI}^T{\bf A}_{jI}+ {\bf A}_{jQ}^T{\bf A}_{jQ}&=&{\bf D}_{jj}^{(1)}; ~~~j=1,2,\cdots,N \nonumber \\
{\bf A}_{jI}^T{\bf A}_{iI} +{\bf A}_{jQ}^T{\bf A}_{iQ}+{\bf A}_{iI}^T{\bf A}_{jI}+{\bf A}_{iQ}^T{\bf A}_{jQ}&=&{\bf D}_{ij}^{(2)};~~~ 1 \leq i\neq j \leq N \nonumber \\
{\bf A}_{jI}^T{\bf A}_{iQ} +{\bf A}_{jQ}^T{\bf A}_{iI}-{\bf A}_{iI}^T{\bf A}_{jQ}-{\bf A}_{iQ}^T{\bf A}_{jI}&=&{\bf D}_{ij}^{(3)};~~~ 1 \leq i\neq j\leq N,
\end{eqnarray}
where
${\bf A}_j = {\bf A}_{jI} + {\bf j}{\bf A}_{jQ}, ~ j=1,2,\cdots,N$, where
${\bf A}_{jI}$ and ${\bf A}_{jQ}$ are real matrices, and
${\bf D}_{jj}^{(1)}, {\bf D}_{ij}^{(2)}$ and ${\bf D}_{ij}^{(3)}$
are block diagonal matrices of the form
\begin{eqnarray}
\label{blockdiagonal}
{\bf D}_{ij}^{(k)} &=&
\left [ \begin{array}{cccc}
\underbrace{
\left[ \begin{array}{cc}
a_{ij,1}^{(k)} & b_{ij,1}^{(k)} \\ b_{ij,1}^{(k)} & c_{ij,1}^{(k)}
\end{array} \right ]
}_{{\bf D}_{ij,1}^{(k)}}
& {\bf 0} & \cdots & {\bf 0} \\
{\bf 0} &
\underbrace{
\left[ \begin{array}{cc}
a_{ij,2}^{(k)} & b_{ij,2}^{(k)} \\ b_{ij,2}^{(k)} & c_{ij,2}^{(k)}
\end{array} \right ]
}_{{\bf D}_{ij,2}^{(k)}}
& \cdots & {\bf 0} \\
\vdots & \vdots & \ddots & \vdots \\
{\bf 0} & \cdots & \cdots &
\underbrace{
\left[ \begin{array}{cc}
a_{ij,T_1}^{(k)} & b_{ij,T_1}^{(k)} \\ b_{ij,T_1}^{(k)} & c_{ij,T_1}^{(k)}
\end{array}\right ]
}_{{\bf D}_{ij,T_1}^{(k)}}
\end{array}\right ],
\end{eqnarray}
where it is understood that whenever the superscript is (1) as in ${\bf D}_{ij}^{(1)},$ then $i=j.$
\end{thm}
{\em Proof: }
In (\ref{rx2_no_csi}), let
${\bf H}_{eq} = \sqrt{E_t} \sum_{j=1}^{N} h_{jd} {\bf A}_j $.
Then the ML optimal detection of ${\bf x}$ is given by
\begin{eqnarray*}
\widehat{\bf x} &=& \mbox{arg min} \,\,
|| {\bf y} - {\bf H}_{eq} {\bf x} ||^2.
\end{eqnarray*}
Since ${\bf x}$ is real,
\begin{eqnarray*}
|| {\bf y} - {\bf H}_{eq} {\bf x} ||^2 & = & || {\bf y} ||^2
- 2 {\bf x}^T \Re \left ( {\bf H}_{eq}^{\mathcal{H}} {\bf y} \right ) +
{\bf x}^T \Re \left ( {\bf H}_{eq}^{\mathcal{H}} {\bf H}_{eq} \right ) {\bf x},
\end{eqnarray*}
which can be written as the sum of several metrics each depending only on
one symbol {\em iff}
$\Re\left ( {\bf H}_{eq}^{\mathcal{H}} {\bf H}_{eq} \right )$ is a block
diagonal matrix of the form in (\ref{blockdiagonal}) for every possible
realization of $h_{jd}$. Now,
\begin{eqnarray*}
\Re\left ( {\bf H}_{eq}^{\mathcal{H}} {\bf H}_{eq} \right ) &=&
E_t\sum_{j=1}^{N} |h_{jd}|^2 \Re \left ({\bf A}_j^{\mathcal{H}} {\bf A}_j \right ) +
\nonumber \\
& &
E_t\sum_{j_1=1}^{N} \sum_{j_2=1,j_2 \neq j_1}^{N} \Re \left (
h^*_{j_1d} h_{j_2d} {\bf A}^{\mathcal{H}}_{j_1} {\bf A}_{j_2} +
h^*_{j_2d} h_{j_1d} {\bf A}^{\mathcal{H}}_{j_2} {\bf A}_{j_1}
\right ) \nonumber \\
&=&
E_t\sum_{j=1}^{N} |h_{jd}|^2 \Re \left ({\bf A}_j^{\mathcal{H}} {\bf A}_j \right ) +
\nonumber \\
& &
E_t\sum_{j_1=1}^{N} \sum_{j_2=1,j_2 \neq j_1}^{N}
\left (h_{j_1dI} h_{j_2dI} + h_{j_1dQ} h_{j_2dQ} \right )
\Re \left ({\bf A}^{\mathcal{H}}_{j_1} {\bf A}_{j_2} + {\bf A}^{\mathcal{H}}_{j_2} {\bf A}_{j_1}
\right ) +
\nonumber \\
& &
E_t\sum_{j_1=1}^{N} \sum_{j_2=1,j_2 \neq j_1}^{N}
\left (h_{j_1dI} h_{j_2dQ} - h_{j_1dQ} h_{j_2dI} \right )
\Im \left ({\bf A}^{\mathcal{H}}_{j_1} {\bf A}_{j_2} - {\bf A}^{\mathcal{H}}_{j_2} {\bf A}_{j_1}
\right ),
\end{eqnarray*}
which is block diagonal of the form in (\ref{blockdiagonal}) $\forall h_{jd}$
{\em iff} (\ref{singlesymx}) is satisfied\footnote{We note that, for the
co-located case, SSD conditions have been presented in \cite{KhR} in terms
of the linear dispersion matrices (also called weight matrices). Our SSD
conditions given in Theorem 1 is in terms of `column vector representation
matrices' \cite{Lia}. The significance of our version as in Theorem 1 is
that it is instrumental in proving Theorems 2 to 6.}. $\square$
Notice that ${\bf D}_{ij}^{(k)}= {\bf D}_{ji}^{(k)}$ for all $i,j,k.$ The
conditions for achieving maximum diversity depend on the ${\bf D}_{ij}^{(k)}$
matrices as well as the signal constellation used for the variables. Before
we discuss these conditions in Lemma \ref{lem1}, we illustrate the SSD
conditions (\ref{singlesymx}) for the following classes of SSD codes for
co-located MIMO.
\vspace{-3mm}
\subsection{SSD conditions for some known classes of codes}
\vspace{-3mm}
{\bf Complex Orthogonal designs (COD):} STBCs from CODs have been
extensively studied \cite{TJC},\cite{TiH},\cite{Lia}. A {\textit{Square
Complex Orthogonal Design}} (SCOD) $ {\bf G} (x_1,x_2,\cdots,x_K)$
(in short ${\bf G}$) of size $N$ is an $ N \times N $ matrix such that
$i)$ the entries of $ {\bf G} (x_1,x_2,\cdots,x_K)$ are complex linear
combinations of the variables $x_1,x_2,\cdots,x_K $ and their complex
conjugates $x_1^*,x_2^*,\cdots,x_K^*$, and
${\bf G}^{\mathcal{H}} {\bf G}=({\vert x_1\vert}^2 +\cdots+{\vert x_K\vert}^2) {\bf I}_N$,
where ${\bf I}_N$ is the $ N \times N $ identity matrix. The rate of
${\bf G}$ is $\frac{K}{N}$ complex symbols per channel use. SCODs
$COD_{2^a}$ for $2^a$ antennas, $a=2,3,\cdots$, can be recursively
constructed starting from
\begin{equation}
\label{itcod}
COD_2= \left[\begin{array}{rr}
x_1 &-x_2^* \\
x_2 & x_1^*
\end{array}\right],~~~
COD_{2^a}= \left[\begin{array}{rr}
{\bf G}_{a-1} & -x_{a+1}^*{\bf I}_{2^{a-1}} \\
x_{a+1}{\bf I}_{2^{a-1}} & {\bf G}_{a-1}^{\mathcal{H}}
\end{array}\right],
\end{equation}
\noindent
where $G_{2^a}$ is a $2^a\times 2^a$ complex matrix.
For example,
\begin{eqnarray}
\label{itcod4}
COD_4 &=& \left [\begin{array} {cccc}
x_1 & x_2 & -x_3^* & 0 \\
-x_2^* & x_1^* & 0 & -x_3^* \\
x_3 & 0 & x_1^* & -x_2 \\
0 & x_3 & x_2^* & x_1
\end{array} \right ],
\end{eqnarray}
\begin{eqnarray}
\label{itcod8}
COD_8 &=& \left [\begin{array} {cccccccc}
x_1 & x_2 & -x_3^* & 0 & -x_4^* & 0 & 0 & 0\\
-x_2^* & x_1^* & 0 &-x_3^* & 0 & -x_4^*& 0 & 0 \\
x_3 & 0 & x_1^* &-x_2 & 0 & 0 & -x_4^* & 0 \\
0 & x_3 & x_2^* & x_1 & 0 & 0 & 0 & -x_4^* \\
x_4 & 0 & 0 & 0 & x_1^* & -x_2 & x_3^* & 0 \\
0 & x_4 & 0 & 0 & x_2^* & x_1 & 0 &x_3^* \\
0 & 0 & x_4 & 0 & -x_3 & 0 & x_1 & x_2 \\
0 & 0 & 0 & x_4 & 0 & -x_3 & -x_2^* & x_1^*
\end{array} \right ].
\end{eqnarray}
Any COD, ${\bf G}$, can be written as
\begin{eqnarray}
{\bf G} = \left [ {\bf A}_1{\bf x},{\bf A}_2{\bf x},\cdots,{\bf A}_N {\bf x} \right ],
\end{eqnarray}
where $ {\bf A}_1, {\bf A}_2,\cdots, {\bf A}_N$ are the relay matrices.
By the definition of CODs, ${\bf G}^{\mathcal{H}} {\bf G} =
\left ( {\bf x}^T {\bf x} \right ) {\bf I}$,
which implies that
\begin{eqnarray}
\label{codd1}
{\bf x} ^T {\bf A}_j^{\mathcal{H}} {\bf A}_j {\bf x} &= &
{\bf x}^T {\bf x};~~~ \forall ~~~ j \\
\label{codd2}
{\bf x} ^T {\bf A}_j^{\mathcal{H}} {\bf A}_i {\bf x} &= & 0;
~~~ \forall ~~~ i\neq j.
\end{eqnarray}
Eqn. (\ref{codd1}) implies that $ \Re \left ( {\bf A}_j^{\mathcal{H}}
{\bf A}_j \right ) = {\bf I}\,\,\forall j$, i.e.,
$ {\bf D}^{(1)}_{jj} = {\bf I} \,\, \forall j$,
whereas Eqn. (\ref{codd2}) implies that
\begin{eqnarray}
\left ( {\bf A}_j^{\mathcal{H}} {\bf A}_i \right )^T &=&
- {\bf A}_j^{\mathcal{H}} {\bf A}_i;~~~ \forall ~~ i\neq j,
\end{eqnarray}
which implies that
\begin{eqnarray}
{\bf A}_{jI}^T{\bf A}_{iI} +{\bf A}_{jQ}^T{\bf A}_{iQ}+{\bf A}_{iI}^T{\bf A}_{jI}+{\bf A}_{iQ}^T{\bf A}_{jQ} & = & {\bf D}_{ij}^{(2)}
= {\bf 0};
~~~ \forall ~~~ i\neq j
\nonumber \\
{\bf A}_{jI}^T{\bf A}_{iQ} +{\bf A}_{jQ}^T{\bf A}_{iI}-{\bf A}_{iI}^T{\bf A}_{jQ}-{\bf A}_{iQ}^T{\bf A}_{jI} & = & {\bf D}_{ij}^{(3)}
= {\bf 0};
~~~ \forall ~~~ i\neq j.
\end{eqnarray}
Hence, for CODs $ {\bf D}^{(2)}_{ij} = {\bf D}^{(3)}_{ij} = {\bf 0} \,\,
\forall \,\, i,j $ and ${\bf D}_{jj}^{(1)}$ is the identity matrix $ \forall \,\, j$.
{\bf Coordinate Interleaved Orthogonal designs (CIOD) \cite{KhR}:}
A coordinate interleaved orthogonal design (CIOD)
in variables $x_{i}, i=0,\cdots, K-1$ (where $K$ is even) is a $2L \times 2N$
matrix ${\bf S}$, such that
\begin{equation}
\label{c2eq1}
{\bf S}({x}_{0},\cdots,{x}_{K-1})=\left[\begin{array}{cc} \Theta(\tilde{x}_0,\cdots,\tilde{x}_{K/2-1}) &0\\
0 & \Theta(\tilde{x}_{K/2},\cdots,\tilde{x}_{K-1})
\end{array}\right],
\end{equation}
where $\Theta(x_0,\cdots,x_{K/2-1})$ is generalized COD (GCOD) of size
$L \times N$ and rate $K/2L$ and
$\tilde{x}_i=\Re (x_i)+ {\bf j} \Im(x_{({i+K/2})_K})$ and
$(a)_K$ denotes $(a \,\, \mbox{mod} \,\, K)$.
Consider the four transmit antenna CIOD, denoted by
$CIOD_4$:
\begin{eqnarray}
\label{ciod4}
CIOD_4 &=& \left [\begin{array} {cccc}
\tilde{x}_0 & \tilde{x}_1 & 0 & 0 \\
-\tilde{x_1}^* & \tilde{x_0}^* & 0 & 0 \\
0 & 0 & \tilde{x}_2 & \tilde{x}_3 \\
0 & 0 & -\tilde{x}_3^* & \tilde{x}_2^*
\end{array} \right ],
\end{eqnarray}
where $ \tilde{x}_i = x_{iI} + {\bf j} x_{{((i + 2)~mod~4})Q}$, and the eight
transmit antenna CIOD, denoted by $CIOD_8$:
\begin{eqnarray}
\label{ciod8}
CIOD_8 &=& \left [\begin{array} {cccccccc}
\tilde{x}_1 & \tilde{x}_2 &\tilde{x}_3 & 0 & 0 & 0& 0 & 0\\
-\tilde{x}_2^* &\tilde{x}_1^* & 0 &\tilde{x}_3 & 0 & 0& 0 & 0 \\
-\tilde{x}_3^* & 0 & \tilde{x}_1 &\tilde{x}_2 & 0 & 0& 0 & 0 \\
0 & -\tilde{x}_3^* & -\tilde{x}_2^* & \tilde{x}_1^* & 0 & 0& 0 & 0 \\
0 & 0& 0 & 0 & \tilde{x}_4 & \tilde{x}_5 & \tilde{x}_6 & 0 \\
0 & 0& 0 & 0 & -\tilde{x}_5^* & \tilde{x}_4^* & 0 &\tilde{x}_6 \\
0 & 0& 0 & 0 & -\tilde{x}_6^* & 0 & \tilde{x}_4 & \tilde{x}_5 \\
0 & 0& 0 & 0 & 0 & -\tilde{x}_6^* & -\tilde{x}_5^* & \tilde{x}_4^*
\end{array} \right],
\end{eqnarray}
where $ \tilde{x}_i = x_{iI} + {\bf j} x_{{((i + 4)~mod~4})Q}.$
The data-symbol vector in (\ref{datavector}) after interleaving can be
written as
\begin{eqnarray}
\tilde{\bf x} & = & \tilde{\bf I} \, {\bf x},
\end{eqnarray}
where $\tilde{\bf I}$ is the interleaving matrix, which is a permutation
matrix obtained by permuting the rows (/columns) of the identity matrix
{\bf I} to reflect the coordinate interleaving operation. Hence, the
effective relay matrices of the design ${\bf S}, \, \bar{\bf A}_j $,
can be written as
$ \bar{\bf A}_j =
\left [ \begin{array} {cc} {\bf A}_{j} & {\bf 0}_{L \times K} \\
{\bf 0}_{L \times K} & {\bf 0}_{L \times K}
\end{array} \right ]
\tilde{\bf I},\,\, 1 \leq j \leq N $ and
$ \bar{\bf A}_j =
\left [ \begin{array} {cc} {\bf 0}_{L \times K} & {\bf 0}_{L \times K} \\
{\bf 0}_{L \times K} & {\bf A}_{j} \end{array} \right ]
\tilde{\bf I} ,\,\, N + 1 \leq j \leq 2N $, where ${\bf A}_{j}$'s
are relay matrices of the design $ \Theta$. It can be verified that
$ {\bf D}_{jj}^{(1)} = \tilde{\bf I}^T
\left [ \begin{array} {cc} {\bf I}_{K \times K} & {\bf 0}_{K \times K} \\
{\bf 0}_{K \times K} & {\bf 0}_{K \times K}
\end{array} \right ] \tilde{\bf I} $ for $ 1 \leq j \leq N $ and
$ {\bf D}_{jj}^{(1)} = \tilde{\bf I}^T
\left [ \begin{array} {cc} {\bf 0}_{K \times K} & {\bf 0}_{K \times K} \\
{\bf 0}_{K \times K} & {\bf I}_{K \times K}
\end{array} \right ] \tilde{\bf I} $ for $ N+1 \leq j \leq 2N$.
Also, ${\bf D}_{ij}^{(2)} = {\bf D}_{ij}^{(3)} = {\bf 0};\,\,
\forall \,i \neq j $.
Hence, $ {\bf D}^{(2)}_{ij} = {\bf D}^{(3)}_{ij} = {\bf 0}\, \forall \,\, i,j$
for CIODs also. But, ${\bf D}_{jj}^{(1)}$ is not the identity matrix
$\forall \,\, j.$
\vspace{-0mm}
{\bf Clifford UW-SSD codes \cite{KaR}:}
A $2^a-$Clifford Unitary Weight SSD (CUW-SSD) code, denoted by $CUW_{2^a},$
is a $2^a\times 2^a$ STBC, given by
{\footnotesize
\begin{equation}
\mbox{\hspace{-7mm}}
\begin{array}{l}
\sigma_{x_1} \bigotimes {\bf I}_2^{\otimes^{a-1}} +\rho_{x_{2a}} \bigotimes \sigma_3 ^{\otimes^{a-1}}
+ \sum_{i=1}^{a-1}
\left[
\sigma_{x_{2i}} \bigotimes {\bf I}_2^{\otimes^{a-i-1}} \bigotimes \sigma_1 \bigotimes \sigma_3^{\otimes^{i-1}}+\sigma_{x_{2i+1}} \bigotimes {\bf I}_2^{\otimes^{a-i-1}} \bigotimes \sigma_2 \bigotimes \sigma_3^{\otimes^{i-1}}
\right], \mbox{\hspace{-3mm}}
\end{array}
\label{cuwssdeq}
\end{equation}
}
\vspace{-8mm}
where
\[\begin{array}{c}
x_i = x_{iI}+jx_{iQ},~~
\sigma_{x_i} = \left[
\begin{array}{rr}
x_{iI} & jx_{iQ} \\
-jx_{iQ} & x_{iI}
\end{array}
\right], ~~
\rho_{x_i} = \left[
\begin{array}{rr}
-jx_{iQ} & jx_{iI} \\
-x_{iI} & -jx_{iQ}
\end{array}
\right],
\end{array}
\]
\begin{equation}
\label{paulimatrices}
\sigma_1 =\left[ \begin{array}{rr}
0 & 1 \\
-1 & 0
\end{array}
\right],~~
\sigma_2 =\left[ \begin{array}{rr}
0 & j \\
j & 0
\end{array}
\right],~~
\sigma_3 =\left[ \begin{array}{rr}
1 & 0 \\
0 & -1
\end{array}
\right],
\end{equation}
and
$\bigotimes$ stands for the tensor product of matrices.
Based on the above definition, the $2-$CUW-SSD code is given by
\begin{equation}
\begin{array}{lll}
CUW_2& = & \sigma_{x_1}+\rho_{x_2} = \left[
\begin{array}{rr}
x_{1I}-jx_{2Q} & x_{2I}+jx_{1Q} \\
-x_{2I}-jx_{1Q} & x_{1I}-jx_{2Q}
\end{array}
\right],
\end{array}
\end{equation}
and the $4-$CUW-SSD code is given by
\begin{eqnarray}
\begin{array}{lll}
CUW_4& = &\sigma_{x_1}\bigotimes {\bf I}_2 + \rho_{x_1} \bigotimes \sigma_3 +\sigma_{x_2} \bigotimes \sigma_1 +\sigma_{x_3}\bigotimes \sigma_2,
\end{array}
\end{eqnarray}
which is
\begin{eqnarray}
\label{cliff4}
CUW_4 &=& \left [ \begin{array} {cccc}
x_{1I} - {\bf j} x_{4Q} & x_{2I} + {\bf j} x_{3I} &
x_{4I} + {\bf j} x_{1Q} & -x_{3Q} + {\bf j} x_{2Q} \\
-x_{2I} - {\bf j} x_{3I} & x_{1I} + {\bf j} x_{4Q} &
-x_{3Q} - {\bf j} x_{2Q} & -x_{4I} + {\bf j} x_{1Q} \\
-x_{4I} - {\bf j} x_{1Q} & x_{3Q} - {\bf j} x_{2Q} &
x_{1I} - {\bf j} x_{4Q} & x_{2I} + {\bf j} x_{3I} \\
x_{3Q} + {\bf j} x_{2Q} & x_{4Q} - {\bf j} x_{1Q} &
-x_{2I} - {\bf j} x_{3I} & x_{1I} + {\bf j} x_{4Q}
\end{array} \right ].
\end{eqnarray}
It can be verified that for Clifford UW-SSD codes
${\bf D}^{(2)}_{ij} = {\bf 0} \, \forall \,i,j$, and the matrices
${\bf D}^{(3)}_{ij}\, \forall \,i,j$ and ${\bf D}_{jj}^{(1)} \, \forall \, j$
are of the form (\ref{blockdiagonal}). For example, for the $CUW_2$ code,
${\bf D}^{(1)}_{jj} = {\bf I} \,\, \forall j$,
${\bf D}^{(2)}_{ij} = {\bf 0} \,\, \forall i,j$,
\begin{footnotesize}
\begin{eqnarray}
{\bf D}^{(3)}_{1,2} = -{\bf D}^{(3)}_{2,1} =
\left [
\begin{array} {cccc}
0 & 2 & 0 & 0 \\
2 & 0 & 0 & 0 \\
0 & 0 & 0 & 2 \\
0 & 0 & 2 & 0 \\
\end{array}
\right ],
\end{eqnarray}
\end{footnotesize}
and ${\bf D}^{(3)}_{ij} = {\bf 0} $ for all other values of $i,j$.
For the $ CUW_4 $ code, $ {\bf D}^{(1)}_{jj} = {\bf I} \,\,
\forall j $, $ {\bf D}^{(2)}_{ij} = {\bf 0} \,\, \forall i,j$,
\begin{footnotesize}
\begin{eqnarray}
{\bf D}^{(3)}_{1,3} = {\bf D}^{(3)}_{2,4} = -{\bf D}^{(3)}_{3,1} =
-{\bf D}^{(3)}_{4,2} =
\left [
\begin{array} {cccccccc}
0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 \\
2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 \\
0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 \\
0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 \\
0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 \\
\end{array}
\right ],
\end{eqnarray}
\end{footnotesize}
and $ {\bf D}^{(3)}_{ij} = {\bf 0} $ for all other values of $i,j$.
\vspace{-3mm}
\subsection{Conditions for full-diversity}
\vspace{-3mm}
In the previous subsection, we saw several classes of SSD codes. The
problem of identifying all possible classes of SSD codes is a open
problem \cite{KaR}. Moreover, different classes of SSD codes may give
full-diversity for different sets of signal sets. The following lemma
obtains a set of necessary and sufficient conditions for the subclass
of SSD codes characterized by
${\bf D}_{ij}^{(2)} = {\bf D}_{ij}^{(3)} = {\bf 0}$ (CODs and CIODs,
for example) to offer full-diversity for all complex constellations.
\begin{lem}
\label{lem1}
For co-located MIMO, the linear STBC as given in (\ref{rx_colocate})
with the ${\bf D}_{ij}^{(k)}$ matrices in (\ref{singlesymx}) satisfying
${\bf D}_{ij}^{(2)} = {\bf D}_{ij}^{(3)} = {\bf 0}$ achieves maximum
diversity for all signal constellations {\em iff}
\begin{eqnarray}
\label{divcondn}
a^{(1)}_{jj,l} c^{(1)}_{jj,l} - {b^{(1)}_{jj,l}}^2 > 0, \,\,\,\, 1 \leq j \leq N;~~~1 \leq l \leq T_1,
\end{eqnarray}
i.e., ${\bf D}_{jj,l}^{(1)}$ is positive definite for all $j,l.$
\end{lem}
{\em Proof: } Consider the pairwise error probability that the
data vector ${\bf x}_1$ as in (\ref{datavector}) gets wrongly detected as
${\bf x}_2$. By Chernoff bound,
\begin{eqnarray}
\label{cb1}
P\left( {\bf x}_1 \rightarrow {\bf x}_2 \right ) \leq
E \left\{ e^{-d^2({\bf x}_1,{\bf x}_2) E_t/4 } \right\},
\end{eqnarray}
where, from (\ref{rx_colocate}),
\begin{eqnarray}
\label{cb2}
d^2({\bf x}_1,{\bf x}_2) =
({\bf x}_2 - {\bf x}_1)^T \Re \left ( {\bf H}_{eq}^{\mathcal{H}} {\bf H}_{eq} \right )({\bf x}_2 - {\bf x}_1).
\end{eqnarray}
Define $ \Delta {\bf x}^{(i)} = [ \Delta x_I^{(i)} \Delta x_Q^{(i)}]^T =
[ (x_{2I}^{(i)} - x_{1I}^{(j)}), (x_{2Q}^{(i)} - x_{1Q}^{(i)}) ]^T $.
Given that the conditions (\ref{singlesymx}) are satisfied, the distance
metric can be written as sum of $T_1$ terms as
\begin{eqnarray}
d^2({\bf x}_1,{\bf x}_2) &= &
\sum_{l=1}^{T_1} \Delta {{\bf x}^{(l)}}^T \left (
\sum_{j=1}^{N} |h_{jd}|^2 {\bf D}^{(1)}_{jj,l}
\right ) \Delta {\bf x}^{(l)} \nonumber \\
& = &
\sum_{j=1}^{N} |h_{jd}|^2 \left (
\sum_{l=1}^{T_1} \Delta {{\bf x}^{(l)}}^T
{\bf D}^{(1)}_{jj,l} \Delta {\bf x}^{(l)}
\right ).
\label{eq_here}
\end{eqnarray}
Substituting (\ref{eq_here}) in (\ref{cb1}) and evaluating the expectation,
we obtain
\begin{eqnarray}
\label{cb4}
P\left( {\bf x}_1 \rightarrow {\bf x}_2 \right ) \leq
\prod_{j=1}^N\left ( \frac{1}{1 +
\sum_{l=1}^{T_1} \Delta {{\bf x}^{(l)}}^T
{\bf D}^{(1)}_{jj,l} \Delta {\bf x}^{(l)} E_t/4
} \right ),
\end{eqnarray}
which, for high SNRs, can be written as
\begin{eqnarray}
\label{cb40}
P\left( {\bf x}_1 \rightarrow {\bf x}_2 \right ) \leq
\prod_{j=1}^N\left ( \frac{1}{
\sum_{l=1}^{T_1} \Delta {{\bf x}^{(l)}}^T
{\bf D}^{(1)}_{jj,l} \Delta {\bf x}^{(l)} E_t/4
} \right ).
\end{eqnarray}
Hence, for high SNRS, in order to achieve full diversity,
$\Delta {{\bf x}^{(l)}}^T {\bf D}^{(1)}_{jj,l} \Delta {\bf x}^{(l)}$
should be non-zero for all $j,l$, i.e., ${\bf D}^{(1)}_{jj,l}$ should be
a positive definite matrix $\forall j,l $, i.e.,
$ a^{(1)}_{j,l} c^{(1)}_{j,l} - {b^{(1)}_{j,l}}^2 > 0 \,\,\forall \,l,j.$
$\square$
For CODs, by definition, $b^{(1)}_{jj,l} = 0 $ and
$a^{(1)}_{jj,l} = c^{(1)}_{jj,i} = 1 \forall j,i $. Hence, the condition
in (\ref{divcondn}) is readily satisfied, and hence full diversity is
achieved for all signal constellations. However, for CIODs, the condition
(\ref{divcondn}) is not satisfied as shown below for the code $CIOD_4.$
For this code,
\begin{footnotesize}
\begin{eqnarray}
{\bf A}_{1I}^T{\bf A}_{1I}+ {\bf A}_{1Q}^T{\bf A}_{1Q} \,\,\, = \,\,\,
{\bf A}_{2I}^T{\bf A}_{2I}+ {\bf A}_{2Q}^T{\bf A}_{2Q} & = &
\left [
\begin{array} {cccccccc}
1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \\
\end{array}
\right ]; \\
{\bf A}_{3I}^T{\bf A}_{3I}+ {\bf A}_{3Q}^T{\bf A}_{3Q} \,\,\, = \,\,\,
{\bf A}_{4I}^T{\bf A}_{4I}+ {\bf A}_{4Q}^T{\bf A}_{4Q} & = &
\left [
\begin{array} {cccccccc}
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
\end{array}
\right ].
\end{eqnarray}
\end{footnotesize}
\vspace{-4mm}
Hence, none of the ${\bf D}^{(1)}_{jj} $ matrices are positive definite.
This does not mean that the code can not give full diversity; it only
means that it can not give full diversity for all complex constellations
as mentioned in Lemma \ref{lem1}. The constellations for which this code
offers full diversity can be obtained by choosing the signal constellation
such that for any two constellation points, $\Delta {{\bf x}_I^{(i)}}$ and
$\Delta {{\bf x}_Q^{(i)}}$ are both non-zero. Substituting these values
in the pair-wise error probability expression (\ref{cb4}), we get
\begin{eqnarray}
\label{cb5}
P\left( {\bf x}_1 \rightarrow {\bf x}_2 \right ) \leq
\prod_{i=1}^{2}
\left ( \frac{1}{1 + \Delta {{\bf x}_I^{(i)}}^2 E_t/4} \right )
\left ( \frac{1}{1 + \Delta {{\bf x}_Q^{(i)}}^2 E_t/4} \right ).
\end{eqnarray}
This has already been shown in \cite{KhR}.
\vspace{-4mm}
\section{SSD Codes for PCRC}
\label{sec4}
\vspace{-5mm}
In the previous section, we saw that SSD is achieved if the relay matrices
satisfy the condition (\ref{singlesymx}). However, to achieve SSD in the
case of distributed STBC with AF protocol, the equivalent weight matrices
${\bf B}_j$'s must satisfy the condition in (\ref{singlesymx}). It can be
seen that for any ${\bf A}_j $ that satisfies the condition in
(\ref{singlesymx}), the corresponding ${\bf B}_j$'s need not satisfy
(\ref{singlesymx}). For example, for the weight matrices in (\ref{AA}),
the corresponding equivalent weight matrices ${\bf B}_1$ and ${\bf B}_2$
do not satisfy the condition in (\ref{singlesymx}). That is, the Alamouti
code is not SSD as a distributed STBC with AF protocol. We note that, in
\cite{RaR2}, code designs which retain the SSD feature have been obtained
for no CSI at the relays, but only for $N=2$ and $4$. A key contribution
in this paper is that by using partial CSI at the relays (i.e., only the
channel phase information of the source-to-relay links), the SSD feature
at the destination can be restored for a large subclass of SSD codes for
co-located MIMO communication.
This key result is given in the following theorem, which
characterizes the class of SSD codes for PCRC.
\begin{thm}
\label{thm2}
A code as given by (\ref{stackx}) is SSD-DSTBC-PCRC {\em iff } the relay
matrices ${\bf A}_j,~ j=1,2,\cdots,N,$ satisfy (\ref{singlesymx}) (i.e.,
the code is SSD for a co-located MIMO set up), and, in addition,
for any three relays with indices $j_1$, $j_2$, $j_3$, where
$j_1,j_2,j_3 \in \{1,2,\cdots,N\}$,
{\small
\begin{eqnarray}
\label{ssddstbc1}
{{\bf A}_{j_1I}}^T\left(
{{\bf A}_{j_2I}} {{\bf A}_{j_2I}}^T +
{{\bf A}_{j_2Q}} {{\bf A}_{j_2Q}}^T
\right ){{\bf A}_{j_3I}} +
{{\bf A}_{j_3I}}^T\left(
{{\bf A}_{j_2I}} {{\bf A}_{j_2I}}^T +
{{\bf A}_{j_2Q}} {{\bf A}_{j_2Q}}^T
\right ){{\bf A}_{j_1I}} + & &\nonumber \\
{{\bf A}_{j_1Q}}^T\left(
{{\bf A}_{j_2I}} {{\bf A}_{j_2I}}^T +
{{\bf A}_{j_2Q}} {{\bf A}_{j_2Q}}^T
\right ){{\bf A}_{j_3Q}} +
{{\bf A}_{j_3Q}}^T\left(
{{\bf A}_{j_2I}} {{\bf A}_{j_2I}}^T +
{{\bf A}_{j_2Q}} {{\bf A}_{j_2Q}}^T
\right ){{\bf A}_{j_1Q}} &=& {\bf D}^{'}_{j_1,j_2,j_3}, \hspace{8mm}\\
\label{ssddstbc2}
{{\bf A}_{j_1I}}^T\left(
{{\bf A}_{j_2I}} {{\bf A}_{j_2Q}}^T +
{{\bf A}_{j_2Q}} {{\bf A}_{j_2I}}^T
\right ){{\bf A}_{j_3Q}} +
{{\bf A}_{j_3Q}}^T\left(
{{\bf A}_{j_2I}} {{\bf A}_{j_2Q}}^T +
{{\bf A}_{j_2Q}} {{\bf A}_{j_2I}}^T
\right ){{\bf A}_{j_1I}} + & &\nonumber \\
{{\bf A}_{j_1Q}}^T\left(
{{\bf A}_{j_2I}} {{\bf A}_{j_2Q}}^T +
{{\bf A}_{j_2Q}} {{\bf A}_{j_2I}}^T
\right ){{\bf A}_{j_3I}} +
{{\bf A}_{j_3I}}^T\left(
{{\bf A}_{j_2I}} {{\bf A}_{j_2Q}}^T +
{{\bf A}_{j_2Q}} {{\bf A}_{j_2I}}^T
\right ){{\bf A}_{j_1Q}} &=& {\bf D}^{''}_{j_1,j_2,j_3},
\end{eqnarray}
}
where $ {\bf D}^{'}_{j_1,j_2,j_3} $ and $ {\bf D}^{''}_{j_1,j_2,j_3}$
are block diagonal matrices of the form in (\ref{blockdiagonal}).
\end{thm}
{\em Proof:} First we show the sufficiency part. It can be
It can be
seen that the matrices
${\bf B}_j^{\prime} = G \sqrt{E_1} {\bf A}_j \, \left|h_{sj}\right|$,
$j=1,2,\cdots,N$
satisfy the condition (\ref{singlesymx}) in spite of the fact that
$\left|h_{sj} \right|$ are random variables
(since ${\bf B}_j^{\prime}$ matrices are scaled versions of the
${\bf A}_j$ matrices). Let
${\bf H}_{eq}^{(pc)} = G \sqrt{E_1} \sum_{j=1}^{N} |h_{sj}| h_{jd} {\bf A}_j$.
It can be seen that
$\Re\left({{\bf H}^{(pc)}_{eq}}^{\mathcal{H}} {\bf H}^{(pc)}_{eq} \right)$
is block diagonal of the form in (\ref{blockdiagonal}).
This implies that
each element of the $K\times 1$ vector
$\Re \left( {{\bf H}^{(pc)}_{eq}}^{\mathcal{H}} {\bf y} \right)$
is affected by only one information symbol (i.e., there will be no
information symbol entanglement in each element).
Hence, for SSD, it suffices
to show that noise in each of these terms are uncorrelated, i.e.,
the DSTBC is SSD {\em iff}
{\small
$E \left [ \Re \left ( {{\bf H}_{eq}^{(pc)}}^{\mathcal{H}} \tilde{\bf z}_d \right )
\Re \left ( {{\bf H}_{eq}^{(pc)}}^{\mathcal{H}} \tilde{\bf z}_d \right )^T \right ]$}
is a block diagonal matrix of the form (\ref{blockdiagonal}).
Expanding
{\small
$E \left [ \Re \left ( {{\bf H}_{eq}^{(pc)}}^{\mathcal{H}} \tilde{\bf z}_d \right ) \Re \left ( {{\bf H}_{eq}^{(pc)}}^{\mathcal{H}} \tilde{\bf z}_d \right )^T \right ] $}, we arrive, after some manipulation, at
{\footnotesize
\begin{eqnarray}
\label{longeqn}
E \left [ \Re \left ( {\bf H}_{eq}^{\mathcal{H}} \tilde{\bf z}_d \right )
\Re \left ( {\bf H}_{eq}^{\mathcal{H}} \tilde{\bf z}_d \right )^T \right ] &=&
\sum_{j_1 =1}^{N}\sum_{j_2 =1}^{N}\sum_{j_3 =1}^{N} |h_{sj_1}||h_{j_2d}|^2 |h_{sj_3}| \left ( h_{j_1dI} h_{j_3dI} + h_{j_1dQ} h_{j_3dQ} \right ) \nonumber \\
& &
\hspace{-20mm}
\Bigg [
{{\bf A}_{j_1I}}^T\left(
{{\bf A}_{j_2I}} {{\bf A}_{j_2I}}^T +
{{\bf A}_{j_2Q}} {{\bf A}_{j_2Q}}^T
\right ){{\bf A}_{j_3I}}
+ \,\, {{\bf A}_{j_3I}}^T\left(
{{\bf A}_{j_2I}} {{\bf A}_{j_2I}}^T +
{{\bf A}_{j_2Q}} {{\bf A}_{j_2Q}}^T
\right ){{\bf A}_{j_1I}} \nonumber \\
& &
\hspace{-20mm}
+ \,\, {{\bf A}_{j_1Q}}^T\left(
{{\bf A}_{j_2I}} {{\bf A}_{j_2I}}^T +
{{\bf A}_{j_2Q}} {{\bf A}_{j_2Q}}^T
\right ){{\bf A}_{j_3Q}} +
{{\bf A}_{j_3Q}}^T\left(
{{\bf A}_{j_2I}} {{\bf A}_{j_2I}}^T +
{{\bf A}_{j_2Q}} {{\bf A}_{j_2Q}}^T
\right ){{\bf A}_{j_1Q}}
\Bigg] \nonumber \\
& &
\hspace{-20mm}
+ \,\,
\sum_{j_1 =1}^{N}\sum_{j_2 =1}^{N}\sum_{j_3 =1}^{N} |h_{sj_1}||h_{j_2d}|^2 |h_{sj_3}| \left ( h_{j_1dI} h_{j_3dQ} + h_{j_1dQ} h_{j_3dI} \right ) \nonumber \\
& &
\hspace{-20mm}
\Bigg [
{{\bf A}_{j_1I}}^T\left(
{{\bf A}_{j_2I}} {{\bf A}_{j_2Q}}^T +
{{\bf A}_{j_2Q}} {{\bf A}_{j_2I}}^T
\right ){{\bf A}_{j_3Q}}
+ \,\,
{{\bf A}_{j_3Q}}^T\left(
{{\bf A}_{j_2I}} {{\bf A}_{j_2Q}}^T +
{{\bf A}_{j_2Q}} {{\bf A}_{j_2I}}^T
\right ){{\bf A}_{j_1I}} \nonumber \\
& &
\hspace{-20mm}
+ \,\,
{{\bf A}_{j_1Q}}^T\left(
{{\bf A}_{j_2I}} {{\bf A}_{j_2Q}}^T +
{{\bf A}_{j_2Q}} {{\bf A}_{j_2I}}^T
\right ){{\bf A}_{j_3I}} +
{{\bf A}_{j_3I}}^T\left(
{{\bf A}_{j_2I}} {{\bf A}_{j_2Q}}^T +
{{\bf A}_{j_2Q}} {{\bf A}_{j_2I}}^T
\right ){{\bf A}_{j_1Q}}
\Bigg ] \nonumber \\
& = &
\sum_{j_1 =1}^{N}\sum_{j_2 =1}^{N}\sum_{j_3 =1}^{N} |h_{sj_1}||h_{j_2d}|^2 |h_{sj_3}| \left ( h_{j_1dI} h_{j_3dI} + h_{j_1dQ} h_{j_3dQ} \right )
{\bf D}^{'}_{j_1,j_2,j_3} \nonumber \\
& &
+\,\,
\sum_{j_1 =1}^{N}\sum_{j_2 =1}^{N}\sum_{j_3 =1}^{N} |h_{sj_1}||h_{j_2d}|^2 |h_{sj_3}| \left ( h_{j_1dI} h_{j_3dQ} + h_{j_1dQ} h_{j_3dI} \right )
{\bf D}^{''}_{j_1,j_2,j_3},
\end{eqnarray}
}
where, in terms of notation, $h_{j_1dI}$ and $h_{j_1dQ}$ denote the
real and imaginary parts of the channel gains from the relay $j_1$ to
destination $d$ (i.e., the real and imaginary parts of $h_{j_1d}$),
respectively. Since (\ref{longeqn}) turns out to be a linear combination
of the ${\bf D}^{'}_{j_1,j_2,j_3}$ and ${\bf D}^{''}_{j_1,j_2,j_3}$
matrices in (\ref{ssddstbc1}) and (\ref{ssddstbc2}),
the covariance matrix
is of the form (\ref{blockdiagonal}). Hence, along with (\ref{singlesymx})
the conditions in (\ref{ssddstbc1}) and (\ref{ssddstbc2}) constitute a set
of sufficient conditions.
To show the ``necessary part,'' since the terms
$h_{sj_1}||h_{rj_2}|^2 |h_{sj_3}|
(h_{rj_1I} h_{rj_3I} + h_{rj_1Q} h_{rj_3Q} ) $ and
$h_{sj_1}||h_{rj_2}|^2 |h_{sj_3}|
(h_{rj_1I} h_{rj_3Q} + h_{rj_1Q} h_{rj_3I} ) $
are independent and if the co-variance matrix has to be block diagonal for
all the realizations of $h_{sj}$ and $ h_{rj} $, then the conditions in
(\ref{ssddstbc1}) and (\ref{ssddstbc2}) have to be necessarily satisfied.
Also, in the similar lines of the proof for {\em Theorem 1}, it can be
deduced that ${\bf B}_j^{\prime}$ satisfying condition (\ref{singlesymx})
is necessary
to achieve un-entanging of information symbols in the elements
of the vector
$ \Re \left ( {{\bf H}^{(pc)}_{eq}}^{\mathcal{H}} {\bf y} \right )$.
$\square$
In \cite{RaR3}, partially-coherent distributed set up has been studied
and a sufficient condition has been identified for a distributed STBC
to be SSD. In the following corollary, it is shown that Theorem \ref{thm2}
subsumes this sufficient condition as a special case.
\vspace{-2mm}
\begin{cor} The sufficient condition in \cite{RaR3}, i.e., the noise
co-variance to be a scaled identity matrix, is a subset of the conditions
(\ref{ssddstbc1}) and (\ref{ssddstbc2}).
\end{cor}
{\em Proof: } It can be observed that the ${\bf Z}_j $ matrix in \cite{RaR3},
when written in our notation, is $ {\bf Z}_j =
\left [ \begin{array} {c} {\bf A}_{jI} \\ {\bf A}_{jQ} \end{array} \right ]$.
Hence, if $ {\bf Z}_j {\bf Z}_j^T = \alpha {\bf I}\,\, \forall j $,
where $\alpha$ is a scalar, then,
${\bf A}_{jI} {{\bf A}_{jI}}^T = \alpha{\bf I} $,
${\bf A}_{jQ} {{\bf A}_{jQ}}^T = \alpha{\bf I} $,
${\bf A}_{jQ} {{\bf A}_{jI}}^T = {\bf 0},$ and
${\bf A}_{jI} {{\bf A}_{jQ}}^T = {\bf 0} $.
Substituting this in (\ref{ssddstbc1}) and (\ref{ssddstbc2}), we get the left
hand side of (\ref{ssddstbc1}) to be
\begin{eqnarray*}
\alpha \left( {{\bf A}_{j_1I}}^T {{\bf A}_{j_3I}} +
{{\bf A}_{j_3I}}^T {{\bf A}_{j_1I}} +
{{\bf A}_{j_1Q}}^T {{\bf A}_{j_3Q}} +
{{\bf A}_{j_3Q}}^T {{\bf A}_{j_3Q}} \right),
\end{eqnarray*}
which, by (\ref{singlesymx}), is always a block diagonal matrix of the form
(\ref{blockdiagonal}). Also, the left hand side of (\ref{ssddstbc2}) is
{\bf 0}.
Hence, ${\bf A}_{jI} {{\bf A}_{jI}}^T =
{\bf A}_{jQ} {{\bf A}_{jQ}}^T = \alpha{\bf I} $ and
$ {\bf A}_{jI} {{\bf A}_{jQ}}^T =
{\bf A}_{jQ} {{\bf A}_{jI}}^T = {\bf 0} \,\,
\forall j $ is a sufficient
condition for a DSTBC to be SSD. $\square$
In \cite{RaR3}, it is shown that the 8-antenna code given by (\ref{cliff8}),
which we
denote by $RR_8,$ does not satisfy the sufficient condition discussed in
that paper for SSD in PCRC, and hence not claimed to be SSD. However, it
can be verified that $RR_8$ satisfies (\ref{singlesymx}),
(\ref{ssddstbc1}) and (\ref{ssddstbc2}), and hence SSD-DSTBC-PCRC.
\vspace{-2mm}
\begin{scriptsize}
\begin{eqnarray}
\label{cliff8}
\mbox{\hspace{-10mm}}
RR_8 & \mbox{\hspace{-3mm}} = & \mbox{\hspace{-3mm}}
\left ( \begin{array} {cccccccc}
x_{1I} - {\bf j} x_{4Q} & x_{2I} + {\bf j} x_{3I} &
x_{4I} + {\bf j} x_{1Q} & -x_{3Q} + {\bf j} x_{2Q} & 0&0&0&0\\
-x_{2I} - {\bf j} x_{3I} & x_{1I} + {\bf j} x_{4Q} &
-x_{3Q} - {\bf j} x_{2Q} & -x_{4I} + {\bf j} x_{1Q} & 0&0&0&0 \\
-x_{4I} - {\bf j} x_{1Q} & x_{3Q} - {\bf j} x_{2Q} &
x_{1I} - {\bf j} x_{4Q} & x_{2I} + {\bf j} x_{3I} & 0&0&0&0 \\
x_{3Q} + {\bf j} x_{2Q} & x_{4Q} - {\bf j} x_{1Q} &
-x_{2I} - {\bf j} x_{3I} & x_{1I} + {\bf j} x_{4Q} & 0&0&0&0\\
0 & 0 & 0 & 0 &
x_{1I} - {\bf j} x_{4Q} & x_{2I} + {\bf j} x_{3I} &
x_{4I} + {\bf j} x_{1Q} & -x_{3Q} + {\bf j} x_{2Q} \\
0 & 0 & 0 & 0 &
-x_{2I} - {\bf j} x_{3I} & x_{1I} + {\bf j} x_{4Q} &
-x_{3Q} - {\bf j} x_{2Q} & -x_{4I} + {\bf j} x_{1Q} \\
0 & 0 & 0 & 0 &
-x_{4I} - {\bf j} x_{1Q} & x_{3Q} - {\bf j} x_{2Q} &
x_{1I} - {\bf j} x_{4Q} & x_{2I} + {\bf j} x_{3I} \\
0 & 0 & 0 & 0 &
x_{3Q} + {\bf j} x_{2Q} & x_{4Q} - {\bf j} x_{1Q} &
-x_{2I} - {\bf j} x_{3I} & x_{1I} + {\bf j} x_{4Q}
\end{array} \right). \mbox{\hspace{6mm}}
\end{eqnarray}
\end{scriptsize}
\vspace{-4mm}
\subsection{Invariance of SSD under coordinate interleaving}
\vspace{-4mm}
In this subsection, we show that the property of SSD of a DSTBC for PCRC is
invariant under coordinate interleaving of the data symbols. To illustrate
the usefulness of this result we first show the following lemma.
\vspace{-2mm}
\begin{lem}
\label{lem2}
If ${\bf G}(x_1,\cdots,x_{T_1}) $ is a SSD design
in $ T_1$ variables and $ N $ transmit nodes that satisfies
(\ref{singlesymx}), (\ref{ssddstbc1}) and (\ref{ssddstbc2}),
then the design in $ 2T_1$ variables and $ 2N $ transmit nodes
given by
\begin{eqnarray}
\label{gbar}
\bar{\bf G}(x_1,\cdots,x_{2T_1}) = \left [
\begin{array} {cc}
{\bf G}(x_1,\cdots,x_{T_1}) & {\bf 0} \\
{\bf 0} & {\bf G}(x_{T_1 + 1},\cdots,x_{2T_1})
\end{array}
\right ]
\end{eqnarray}
also satisfies (\ref{singlesymx}), (\ref{ssddstbc1}) and (\ref{ssddstbc2}).
\end{lem}
{\em Proof: } If $ {\bf A}_j, \,\, 1 \leq j \leq N $ are the relay matrices
of ${\bf G}$, then the corresponding $ \bar{\bf A}_j $
matrices for $ \bar{{\bf G}}$ are
$ \bar{\bf A}_j =
\left [ \begin{array} {cc} {\bf A}_j & {\bf 0} \\ {\bf 0} & {\bf 0}
\end{array} \right ]
,\,\, 1 \leq j \leq N $ and
$ \bar{\bf A}_j =
\left [ \begin{array} {cc} {\bf 0} & {\bf 0} \\ {\bf 0} & {\bf A}_j
\end{array} \right ]
,\,\, N + 1 \leq j \leq 2N $. It is easily verified that if $ {\bf A}_j $
satisfies (\ref{singlesymx}), (\ref{ssddstbc1}) and (\ref{ssddstbc2}),
then so do the matrices $ \bar{\bf A}_j. $ $\square$
As an example, if we choose ${\bf G}(x_1,x_2)$ to be the Alamouti code in
the lemma above then we get the code
\begin{eqnarray}
\label{doubleala}
\left [
\begin{array} {rrrr}
x_1 & x_2 & 0 & 0 \\
-x_2^* & x_1^* & 0 & 0 \\
0 & 0 & x_3 & x_4 \\
0 & 0 & -x_4^* & x_3^*
\end{array}
\right ].
\end{eqnarray}
This code is SSD for PCRC. Note that a 4-antenna COD has only rate only
$\frac{3}{4}$ whereas this code has rate 1. However, it is easily shown
that this code does not give full-diversity. But, coordinate interleaving
for this example results in $CIOD_4$ which gives full-diversity for any
signal set with coordinate product distance zero, and we have already
seen that $CIOD_4$ has the SSD property for PCRC.
The following theorem shows that it is the property of coordinate interleaving to leave the SSD property of any arbitrary STBC for PCRC intact.
\vspace{-2mm}
\begin{thm}
\label{thm4}
If an STBC with $K$ variables $x_1,x_2,\cdots,x_K,$ satisfy
(\ref{singlesymx}), (\ref{ssddstbc1}) and (\ref{ssddstbc2}), the SSD property
is unaffected by doing arbitrary coordinate interleaving among all real
and imaginary components of $x_i$.\footnote{It should be noted that neither
the source nor the relay does an
explicit interleaving, but the net effect of the relay matrices is such that
the output of relays is an interleaved version of the information symbols.}
\end{thm}
\vspace{-2mm}
{\em Proof:} The data-symbol vector in (\ref{datavector}) after interleaving
can be written as
\begin{eqnarray*}
\tilde{\bf x} & = & \tilde{\bf I} \, {\bf x}
\end{eqnarray*}
where $\tilde{\bf I}$ is the interleaving matrix which is a permutation
matrix obtained by permuting the rows (/columns) of the identity matrix
{\bf I} to reflect the coordinate interleaving operation. It can be easily
checked that $ \tilde{\bf I}^2={\bf I}.$ Also, if $ {\bf D} $ is a block
diagonal matrix of the form (\ref{2bdiagonal}), then so is the matrix
$ \tilde{\bf I} {\bf D} \tilde{\bf I}.$ Hence, for PCRC with co-ordinate
interleaving (\ref{stacky}) can be written as
\begin{eqnarray}
\label{stackxnew}
{\bf c}_{j} & = & {\bf A}_j \widehat{\bf v}_{j} \nonumber \\
& = & \underbrace{G \, \sqrt{E_1} {\bf A}_j |h_{sj}|\tilde{\bf I}}_{{\bf B}_j^{'}}\, {\bf x} \, + \,{\bf A}_j \, {\bf \widehat{z}}_{j},
\end{eqnarray}
which means that after interleaving, the equivalent linear processing
matrix is $ {\bf A}_j \tilde{\bf I} $. It is easily verified that
if $ {\bf A}_j $ satisfies (\ref{singlesymx}),
(\ref{ssddstbc1}) and (\ref{ssddstbc2}), then so does
${\bf A}_j \tilde{\bf I}$ also.
$\square$
As an example, consider the Alamouti code
$\left[\begin{array}{rr}
x_1 & x_2 \\
-x_2^* & x_1^*
\end{array}\right],$
whose relay matrices are given by (\ref{AA}). For this case,
$N=T_1=T_2=2.$ The permutation matrix $\tilde{\bf I}$ for the
coordinate interleaving operation is
$\left[\begin{array}{cccc}
1 & 0 & 0 & 0 \\
0 & 0 & 0 & 1 \\
0 & 0 & 1 & 0 \\
0 & 1 & 0 & 0
\end{array}\right].$
The relay matrices for the coordinate interleaved code are
\begin{eqnarray}
\label{AAnew}
{\bf A}_1 \tilde{\bf I}= \left [ \begin{array}{cccc} 1 & 0 & 0 & {\bf j} \\
0 & {\bf j} & -1 & 0 \end{array} \right ]
& \mbox{ and } &
{\bf A}_2 \tilde{\bf I}= \left [ \begin{array}{cccc} 0 & {\bf j} & 1 & 0\\
1 & 0& 0 & -{\bf j} \end{array} \right ],
\end{eqnarray}
and the resulting code is
$\left[\begin{array}{rr}
x_{1I}+jx_{2Q} & x_{2I}+jx_{1Q} \\
-x_{2I}+jx_{1Q} & x_{1I}-jx_{2Q}
\end{array}\right]=
\left[\begin{array}{rr}
\tilde{x}_1 & \tilde{x}_2 \\
-\tilde{x}_2^* & \tilde{x}_1^*
\end{array}\right].$
Also, for the code in (\ref{doubleala}) which is SSD for PCRC, if we choose
the permutation matrix $\tilde{\bf I}$ as
\begin{eqnarray}
\tilde{\bf I} = \left [ \begin{array} {cccccccc}
1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\
0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \\
0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\
0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\
\end{array} \right ],
\end{eqnarray}
the resulting code is given by
\begin{eqnarray}
\left[\begin{array}{rrrr}
x_{1I}+jx_{3Q} & x_{2I}+jx_{4Q} & 0 & 0\\
-x_{2I}+jx_{4Q} & x_{1I}-jx_{3Q} & 0 & 0 \\
0 & 0 & x_{3I}+jx_{1Q} & x_{4I}+jx_{2Q} \\
0 & 0 & -x_{4I}+jx_{2Q} & x_{3I}-jx_{1Q} \\
\end{array}\right]=
\left[\begin{array}{rrrr}
\tilde{x}_1 & \tilde{x}_2 & 0 & 0\\
-\tilde{x}_2^* & \tilde{x}_1^* & 0 & 0 \\
0 & 0 & \tilde{x}_3^* & \tilde{x}_4^* \\
0 & 0 & -\tilde{x}_4^* & \tilde{x}_3^* \\
\end{array}\right],
\end{eqnarray}
which is $CIOD_4$. Hence, $CIOD_4 $ is also SSD for PCRC. In general, if we
have a code with $K$ complex information symbols which is SSD for PCRC,
then we can generate $(2K)!$ codes which are SSD for PCRC by coordinate
interleaving.
\vspace{-4mm}
\subsection{A class of rate-$\frac{1}{2}$ SSD DSTBCs}
\vspace{-4mm}
All the classes of codes discussed so far are STBCs from square designs.
It is well known that the rate of square SSD codes for co-located MIMO
systems falls exponentially as
the number of antennas increases. In this subsection, it is shown that if
non-square designs are used then SSD codes for PCRCs can be achieved with
rate $\frac{1}{2}$ for any number of antennas.
It is well known \cite{TJC} that real orthogonal designs (RODs) with rate
one exist for any number of antennas and these are non-square designs for
more than 2 antennas and the delay increases exponentially with the number
of antennas. Using these RODs, in \cite{TJC}, a class of rate $\frac{1}{2}$
complex orthogonal designs for any number of antennas is obtained as follows:
If ${\bf G}$ is a $p \times N$ rate one ROD, where $p$ denotes the delay and
$N$ denotes the number of antennas with variables $x_1,x_2, \cdots, x_p,$
then, denoting by ${\bf G}^*$ the complex design obtained by replacing
$x_i$ with $x_i^*, ~i=1,2,\cdots,p,$ the design
$\left[
\begin{array} {c}
{\bf G} \\ {\bf G}^*
\end{array}
\right]$
is a $2p \times N$ rate-$\frac{1}{2}$ COD. We refer to this construction
as stacking construction. The following theorem asserts that the rate
$\frac{1}{2}$ CODs by stacking construction are SSD for PCRC.
\vspace{-2mm}
\begin{thm}
\label{ratehalfthm}
The rate-1/2 CODs, constructed from rate one RODs by stacking construction
\cite{TJC} are SSD-DSTBC-PCRC.
\end{thm}
\vspace{-2mm}
{\em Proof: } Let ${\bf G}_c$ be the rate-1/2 COD obtained from a
$p \times N$ ROD ${\bf G}$ by stacking construction, i.e.,
\begin{eqnarray}
{\bf G}_c = \left [ \begin{array} {c} {\bf G} \\ {\bf G}^*
\end{array} \right ].
\end{eqnarray}
Let the $ p \times p $ real matrices $\hat{{\bf A}}_j \, j=1,\cdots,N $
generate the columns of ${\bf G}$, i.e.,
\begin{eqnarray}
{\bf G}& = & \left [ \hat{{\bf A}}_1 {\bf x}, \hat{{\bf A}}_2 {\bf x}, \cdots,
\hat{{\bf A}}_N {\bf x} \right ],
\end{eqnarray}
where ${\bf x}$ is the $p \times 1$ real data vector and the matrices
$ \hat{{\bf A}}_j $ denote the column vector representation matrices used
in \cite{Lia}.
By the definition of RODs, ${\bf G}^T {\bf G}=\left ({\bf x}^T{\bf x} \right){\bf I}$.
This implies that
\begin{eqnarray}
\hat{\bf A}_j^T \hat{\bf A}_j &=& {\bf I}, \,\, j = 1,\cdots,N \nonumber \\
\hat{\bf A}_j^T \hat{\bf A}_i &=& - \hat{\bf A}_i^T \hat{\bf A}_j,
\,\, i,j = 1,\cdots,N, i \neq j.
\label{rod_condn}
\end{eqnarray}
It is noted that the Hurwitz-Radon family of matrices satisfy
(\ref{rod_condn}) and explicit construction for any $N$ is given in
\cite{TJC}. It is noted that the representation in \cite{TJC} is
different from the column vector representation used in this paper. An
important consequence is that the Hurwitz-Radon family of matrices
satisfy the conditions
\begin{eqnarray}
\hat{\bf A}_j^T \hat{\bf A}_j &=& {\bf I}, \,\, j = 1,\cdots,N \nonumber \\
\hat{\bf A}_j^T &=& -\hat{\bf A}_j, \,\, j = 1,\cdots,N \nonumber \\
\hat{\bf A}_j \hat{\bf A}_i &=& - \hat{\bf A}_i \hat{\bf A}_j,
\,\, i,j = 1,\cdots,N, i \neq j,
\end{eqnarray}
and hence $ \hat{\bf A}_j \hat{\bf A}_j^T = {\bf I} \,\, \forall j,$ which we
will use in our proof. Viewing ${\bf G}_c$ as a $T_2 \times N$ distributed
STBC with $T_1=p$ and $T_2=2p$, the $T_2 \times 2T_1$ relay matrices
$ {\bf A}_j $ of ${\bf G}_c $ have the structure
\begin{eqnarray}
{\bf A}_{jI} =
\left ( \begin{array} {c} {\bf U}_j \\ {\bf U}_j \end{array} \right ) \,\,\,
\mbox{and} \,\,\,
{\bf A}_{jQ} =
\left ( \begin{array} {r} {\bf V}_j \\ {\bf -V}_j \end{array} \right ).
\end{eqnarray}
Since ${\bf G}_c$ is constructed from a ROD, the coefficients of real and
imaginary components are same, i.e., the matrices ${\bf U}_j$ and ${\bf V}_j$
have the form
\begin{equation}
{\bf U}_j = \left[{\bf\gamma}_{1,j},{\bf 0},{\bf\gamma}_{2,j}, {\bf 0},
\cdots,{\bf\gamma}_{T_1,j}, {\bf 0 }\right ],\,\,\,\,\,\,\,\,\,\,\,
{\bf V}_j = \left[{{\bf 0}, \bf\gamma}_{1,j},{\bf 0},
{\bf\gamma}_{2,j},\cdots,{\bf 0},{\bf\gamma}_{T_1,j}\right ],
\end{equation}
with $ \gamma_{i,j}$ are column vectors of $\hat{\bf A}_j$. Since
$\hat{\bf A}_j \hat{\bf A}_j^T = {\bf I} \,\, \forall j$, it is easily
verified that $ {\bf U}_j {\bf U}_j^T = {\bf I}$ and
$ {\bf V}_j {\bf V}_j^T = {\bf I} \,\, \forall j $. It is also easily seen
that $ {\bf U}_j {\bf V}^T_j = {\bf 0} $ and $ {\bf V}_j {\bf U}^T_j =
{\bf 0} $. Hence, we have
\begin{eqnarray}
{\bf A}_{jI} {{\bf A}_{jI}}^T + {\bf A}_{jQ} {{\bf A}_{jQ}}^T
& = & 2 {\bf I} \nonumber \\
{\bf A}_{jI} {{\bf A}_{jQ}}^T + {\bf A}_{jQ} {{\bf A}_{jI}}^T
& = & {\bf 0}
\end{eqnarray}
Substituting this in (\ref{ssddstbc1}), we get the left hand side of
(\ref{ssddstbc1}) to be
\begin{eqnarray}
2 \left( {{\bf A}_{j_1I}}^T {{\bf A}_{j_3I}} +
{{\bf A}_{j_3}}^T {{\bf A}_{j_1I}} +
{{\bf A}_{j_1Q}}^T {{\bf A}_{j_3Q}} +
{{\bf A}_{j_3Q}}^T {{\bf A}_{j_3Q}} \right),
\end{eqnarray}
which, by (\ref{singlesymx}), is always a block diagonal matrix of the form
(\ref{blockdiagonal}). Also the left hand side of (\ref{ssddstbc2}) is
${\bf 0}$. Hence, ${\bf G}_c$ is SSD for PCRC. $\square$ \\
\vspace{-6mm}
In \cite{YiK}, it is shown that if the $N$ relays do not have any CSI
and the destination has all the CSI, then an upper bound on the rate
of distributed SSD codes is $\frac{2}{N},$ which decreases rapidly as
the number of relays increases. However, Theorem \ref{ratehalfthm} shows
that, if the relay knows only the phase information of the source-relay
channels then the lower bound on the rate of the distributed SSD codes is
$\frac{1}{2}$ which is independent of the number of relays. For example,
the ROD part of such rate-1/2 SSD DSTBCs for PCRC for 10 and 12
relays are given in (\ref{rod10}) and (\ref{rod12}), respectively, where
Hurwitz-Radon construction yields the $32\times 10$ matrix in (\ref{rod10})
for 10 relays and the $64\times 12$ matrix in (\ref{rod12}) for 12 relays.
\vspace{-4mm}
\subsection{Full-diversity, single-symbol non-ML detection }
\vspace{-4mm}
\begin{thm}
\label{nonMLSSD}The PCRC system given by
(\ref{rx2}) achieves full diversity irrespective of whether the total noise
($\tilde{{\bf z}}_d $) is correlated or not, if the STBC achieves full
diversity in the co-located case and condition (\ref{singlesymx}) is
satisfied.
\end{thm}
\vspace{-4mm}
{\em Proof: }
Since the noise $\tilde{{\bf z}}_d $ is not assumed to be uncorrelated,
the optimal detection of ${\bf x}$ in the maximum likelihood sense is given by
\begin{eqnarray}
\label{ml_opt}
\widehat{\bf x} &=& \mbox{arg min} \,\,
( {\bf y} - {\bf H}^{(pc)}_{eq}{\bf x} )^{\mathcal{H}} {\bf \Omega}^{-1}
( {\bf y} - {\bf H}^{(pc)}_{eq} {\bf x} ),
\end{eqnarray}
where ${\bf \Omega}$ is co-variance matrix of the noise, given by
${\bf \Omega} = E\{ \tilde{{\bf z}}_d \tilde{{\bf z}}_d ^{\mathcal{H}} \} $. We consider
the sub-optimal metric (ignoring ${\bf \Omega}^{-1}$)
\begin{eqnarray}
\label{ml_sub_opt}
\widehat{\bf x} &=& \mbox{arg min} \,\,
({\bf y} - {\bf H}^{(pc)}_{eq}{\bf x})^{\mathcal{H}}
({\bf y} - {\bf H}^{(pc)}_{eq}{\bf x}),
\end{eqnarray}
and show that this decision metric achieves full diversity. Proceeding on
the similar lines for the proof for the co-located case, the pair-wise
error probability is upper bounded by
\begin{eqnarray}
\label{d_cb1}
P\left( {\bf x}_1 \rightarrow {\bf x}_2 \right ) \leq
E \left\{ e^{-d^2({\bf x}_1,{\bf x}_2) E_t/4 } \right\},
\end{eqnarray}
where the Euclidean distance in (\ref{d_cb1}) can be
written as
\begin{eqnarray}
\label{d_cb2}
d^2({\bf x}_1,{\bf x}_2) =
({\bf x}_2 - {\bf x}_1)^T \Re \left (
{{\bf H}^{(pc)}_{eq}}^{\mathcal{H}} {\bf H}^{(pc)}_{eq} \right )({\bf x}_2 - {\bf x}_1).
\end{eqnarray}
Since (\ref{singlesymx}) is satisfied, this can be written as sum of
$T_1$ terms as
\begin{eqnarray}
d^2({\bf x}_1,{\bf x}_2) &= &
\sum_{i=1}^{T_1} \Delta {{\bf x}^{(i)}}^T \left (
\sum_{j=1}^{N} |h_{sj}|^2 |h_{jd}|^2 {\bf D}^{(1)}_{j,i}
\right ) \Delta {\bf x}^{(i)} \\
& = &
\sum_{j=1}^{N} |h_{sj}|^2 |h_{jd}|^2 \left (
\sum_{i=1}^{T_1} \Delta {{\bf x}^{(i)}}^T
{\bf D}^{(1)}_{j,i} \Delta {\bf x}^{(i)}
\right ).
\label{eq_here2}
\end{eqnarray}
Substituting (\ref{eq_here2}) in (\ref{d_cb1}) and evaluating the expectation
with respect to $ |h_{jd}|^2 $, we get
\begin{eqnarray}
\label{d_cb4}
P\left( {\bf x}_1 \rightarrow {\bf x}_2 \right | h_{sj} ) \leq
\prod_{j=1}^N\left ( \frac{1}{1 +
|h_{sj}|^2 \sum_{i=1}^{T_1} \Delta {{\bf x}^{(i)}}^T
{\bf D}^{(1)}_{j,i} \Delta {\bf x}^{(i)} E_t/4,
} \right ),
\end{eqnarray}
which, for high SNRs, could be approximated as
\begin{eqnarray}
P\left( {\bf x}_1 \rightarrow {\bf x}_2 | h_{sj} \right ) \leq
\prod_{j=1}^N\left ( \frac{1}{ \sum_{i=1}^{T_1} \Delta {{\bf x}^{(i)}}^T
{\bf D}^{(1)}_{j,i} \Delta {\bf x}^{(i)} E_t/4
} \right )
\prod_{j=1}^N\left ( \frac{1}{|h_{sj}|^2 } \right ).
\end{eqnarray}
Now, evaluating the expectation with respect to $ |h_{sj}|, $ we get
\begin{eqnarray}
\label{d_cb5}
P\left( {\bf x}_1 \rightarrow {\bf x}_2 \right ) \leq
\prod_{j=1}^N\left ( \frac{1}{ \sum_{i=1}^{T_1} \Delta {{\bf x}^{(i)}}^T
{\bf D}^{(1)}_{j,i} \Delta {\bf x}^{(i)} E_t/4
} \right )
\left ( {\bf Ei}(0) \right )^N,
\end{eqnarray}
where $ {\bf Ei}(x) $ is the exponential integral
$ \int_x^{\infty} \frac{e^{-t}}{t} dt $. From (\ref{d_cb5}), it is clear that
the condition for achieving maximum diversity is identical to that of
co-located MIMO (\ref{cb4}). $\square$
Theorem \ref{nonMLSSD} means that by using any STBC
which satisfies the conditions (\ref{singlesymx}) and achieves full
diversity in co-located MIMO system, it is possible to do decoding of
one symbol at a time and achieve full diversity, though not optimal in
the ML sense, in a distributed setup with phase compensation done at
the relay, even if {\it (\ref{ssddstbc1}) and (\ref{ssddstbc2}) are not
satisfied }. For example, the $CIOD_8$ is SSD and gives full-diversity
in a co-located 8-transmit antenna system for any signal set with
coordinate product distance (CPD) not equal to zero, and is not SSD
for PCRC since it does not satisfy the (\ref{ssddstbc1}) and
(\ref{ssddstbc2}). However, according to Theorem \ref{nonMLSSD} a SSD
decoder for $CIOD_8$ in a PCRC will result in full-diversity of order 8.
\vspace{-6mm}
\section{Discussion and Simulation Results}
\label{sec5}
\vspace{-6mm}
The results of our necessary and sufficient conditions (\ref{singlesymx}),
(\ref{ssddstbc1}) and (\ref{ssddstbc2}) as well as the sufficient condition
in \cite{RaR3}, evaluated for various classes of codes for PCRC are shown
in Table \ref{tab1}. As can be seen from the last column of Table \ref{tab1},
the sufficient condition in \cite{RaR3} identifies only $COD_2$ (Alamouti)
and $CUW_4$ as SSDs for PCRC. However, our conditions (\ref{singlesymx},
(\ref{ssddstbc1}) and (\ref{ssddstbc2}) identify $CIOD_4$, $RR_8$, and
$COD$s from $ROD$s, in addition to $COD_2$ and $CUW_4$, as SSDs for PCRC
(4th column of Table \ref{tab1}). It is noted that, $CIOD_4$ being a
construction by using ${\bf G}=COD_2$ in (\ref{gbar}) and coordinate
interleaving, it is SSD for PCRC from {\em Lemma \ref{lem2}} and {\em
Theorem \ref{thm4}}. Similarly, since $RR_8$ code is constructed by using
${\bf G}=CUW_4$ in (\ref{gbar}), it follows from {\em Lemma \ref{lem2}}
that $RR_8$ is also SSD for PCRC. Also, CODs from RODs are SSD for PCRC
from {\em Theorem \ref{ratehalfthm}}. Since $COD_4$, $COD_8$, and $CIOD_8$
do not satisfy our conditions, they are not SSD for PCRC.
Next, we present the bit error rate (BER) performance of various classes
of codes without and with phase compensation at the relays (i.e., PCRC).
For the purposes of the simulation results and discussions in this section,
we classify the decoding of codes for PCRC into two categories: $i)$ codes
for which single symbol decoding is ML-optimal; we refer to this decoding
as ML-SSD; we consider ML-SSD of $COD_2$ and $CIOD_4$, and $ii)$ codes
which when decoded using single symbol decoding are not ML-optimal, but
achieve full diversity; we refer to this decoding as non-ML-SSD;
we consider non-ML-SSD of $COD_4$, $COD_8$, and $CIOD_8$. When no phase
compensation is done at the relays, we consider ML decoding.
In Fig. \ref{fig3}, we plot the BER performance for $COD_2$, $COD_4$, and
$COD_8$ without and with phase compensation at the relays (i.e., PCRC) for
16-QAM. Note that $COD_2$ is SSD for PCRC whereas $COD_4$ and $COD_8$
are not SSD for PCRC. So decoding of $COD_2$ with PCRC is ML-SSD, whereas
decoding of $COD_4$ and $COD_8$ with PCRC is non-ML-SSD. When no phase
compensation is done at the relays, we do ML decoding for all $COD_2$,
$COD_4$, and $COD_8$. The following observations can be made from Fig.
\ref{fig3}: $i)$ $COD_2$ without and with phase compensation at the
relays (PCRC) achieve the full diversity order of 2, $ii)$ $COD_2$ with
PCRC and ML-SSD achieves better performance by about 3 dB at a BER of
$10^{-2}$ compared to ML decoding of $COD_2$ without phase compensation,
and $iii)$ even the non-ML-SSD of $COD_4$ and $COD_8$ with PCRC achieves
full diversity of 4 and 8, respectively (but not the ML performance
corresponding to PCRC), and even with this suboptimum decoding, PCRC
achieves about 1 dB and 0.5 dB better performance at a BER of $10^{-2}$,
respectively, compared to ML decoding of $COD_4$ and $COD_8$ without
phase compensation at the relays.
In Fig. \ref{fig4}, we present a similar BER performance comparison for
CIODs without and with phase compensation at the relays. QPSK modulation
with $30^\circ$ rotation of the constellation is used. Here again,
both $CIOD_4$ and $CIOD_8$ achieve their full diversities of 4 and 8,
respectively. We further observe that $CIOD_4$ (which is SSD for PCRC)
with PCRC and ML-SSD achieves better performance by about 3 dB at a BER
of $10^{-3}$ compared to ML decoding of $CIOD_4$ without phase compensation.
Likewise, $CIOD_8$ (which is not SSD for PCRC) with PCRC and non-ML-SSD
achieves better performance by about 1 dB at a BER of $10^{-3}$ compared
to ML decoding of $CIOD_8$ without phase compensation.
Finally, a performance comparison between CODs and CIODs with PCRC
for a given spectral efficiency is presented in Fig. \ref{fig5}.
A comparison at a spectral efficiency of 3 bps/Hz is made between
$i)$ $COD_4$ with rate-3/4 and 16-PSK (spectral efficiency =
$\frac{3}{4} \times \log_2 16 = 3$ bps/Hz), and $ii)$ $CIOD_4$ with rate-1
and 8-PSK with $10^\circ$ rotation (spectral efficiency =
$1\times \log_2 8 = 3$ bps/Hz). Likewise, a comparison is made at
a spectral efficiency of 1.5 bps/Hz between $COD_8$ and $CIOD_8$.
It can be observed that, as in the case of co-located MIMO \cite{KhR},
in distributed STBCs with PCRC also, CIODs perform better than COD, i.e.,
coordinate interleaving improves performance. All these simulation
results reinforce the claims made in the paper in Sec. \ref{sec1}.
\begin{eqnarray}
ROD_{10} = \left [
\begin{array} {cccccccccc}
x_{1} &x_{9} &x_{17} &x_{18} &x_{19} &x_{20} &x_{21} &x_{22} &x_{23} &x_{24}\\
x_{2} &x_{10} &x_{18} &-x_{17} &x_{20} &-x_{19} &x_{22} &-x_{21} &-x_{24} &x_{23}\\
x_{3} &x_{11} &x_{19} &-x_{20} &-x_{17} &x_{18} &-x_{23} &-x_{24} &x_{21} &x_{22}\\
x_{4} &x_{12} &x_{20} &x_{19} &-x_{18} &-x_{17} &-x_{24} &x_{23} &-x_{22} &x_{21}\\
x_{5} &x_{13} &x_{21} &-x_{22} &x_{23} &x_{24} &-x_{17} &x_{18} &-x_{19} &-x_{20}\\
x_{6} &x_{14} &x_{22} &x_{21} &x_{24} &-x_{23} &-x_{18} &-x_{17} &x_{20} &-x_{19}\\
x_{7} &x_{15} &x_{23} &x_{24} &-x_{21} &x_{22} &x_{19} &-x_{20} &-x_{17} &-x_{18}\\
x_{8} &x_{16} &x_{24} &-x_{23} &-x_{22} &-x_{21} &x_{20} &x_{19} &x_{18} &-x_{17}\\
x_{9} &-x_{1} &x_{25} &x_{26} &x_{27} &x_{28} &x_{29} &x_{30} &x_{31} &x_{32}\\
x_{10} &-x_{2} &x_{26} &-x_{25} &x_{28} &-x_{27} &x_{30} &-x_{29} &-x_{32} &x_{31}\\
x_{11} &-x_{3} &x_{27} &-x_{28} &-x_{25} &x_{26} &-x_{31} &-x_{32} &x_{29} &x_{30}\\
x_{12} &-x_{4} &x_{28} &x_{27} &-x_{26} &-x_{25} &-x_{32} &x_{31} &-x_{30} &x_{29}\\
x_{13} &-x_{5} &x_{29} &-x_{30} &x_{31} &x_{32} &-x_{25} &x_{26} &-x_{27} &-x_{28}\\
x_{14} &-x_{6} &x_{30} &x_{29} &x_{32} &-x_{31} &-x_{26} &-x_{25} &x_{28} &-x_{27}\\
x_{15} &-x_{7} &x_{31} &x_{32} &-x_{29} &x_{30} &x_{27} &-x_{28} &-x_{25} &-x_{26}\\
x_{16} &-x_{8} &x_{32} &-x_{31} &-x_{30} &-x_{29} &x_{28} &x_{27} &x_{26} &-x_{25}\\
-x_{17} &-x_{25} &x_{1} &x_{2} &x_{3} &x_{4} &x_{5} &x_{6} &x_{7} &x_{8}\\
-x_{18} &-x_{26} &x_{2} &-x_{1} &x_{4} &-x_{3} &x_{6} &-x_{5} &-x_{8} &x_{7}\\
-x_{19} &-x_{27} &x_{3} &-x_{4} &-x_{1} &x_{2} &-x_{7} &-x_{8} &x_{5} &x_{6}\\
-x_{20} &-x_{28} &x_{4} &x_{3} &-x_{2} &-x_{1} &-x_{8} &x_{7} &-x_{6} &x_{5}\\
-x_{21} &-x_{29} &x_{5} &-x_{6} &x_{7} &x_{8} &-x_{1} &x_{2} &-x_{3} &-x_{4}\\
-x_{22} &-x_{30} &x_{6} &x_{5} &x_{8} &-x_{7} &-x_{2} &-x_{1} &x_{4} &-x_{3}\\
-x_{23} &-x_{31} &x_{7} &x_{8} &-x_{5} &x_{6} &x_{3} &-x_{4} &-x_{1} &-x_{2}\\
-x_{24} &-x_{32} &x_{8} &-x_{7} &-x_{6} &-x_{5} &x_{4} &x_{3} &x_{2} &-x_{1}\\
-x_{25} &x_{17} &x_{9} &x_{10} &x_{11} &x_{12} &x_{13} &x_{14} &x_{15} &x_{16}\\
-x_{26} &x_{18} &x_{10} &-x_{9} &x_{12} &-x_{11} &x_{14} &-x_{13} &-x_{16} &x_{15}\\
-x_{27} &x_{19} &x_{11} &-x_{12} &-x_{9} &x_{10} &-x_{15} &-x_{16} &x_{13} &x_{14}\\
-x_{28} &x_{20} &x_{12} &x_{11} &-x_{10} &-x_{9} &-x_{16} &x_{15} &-x_{14} &x_{13}\\
-x_{29} &x_{21} &x_{13} &-x_{14} &x_{15} &x_{16} &-x_{9} &x_{10} &-x_{11} &-x_{12}\\
-x_{30} &x_{22} &x_{14} &x_{13} &x_{16} &-x_{15} &-x_{10} &-x_{9} &x_{12} &-x_{11}\\
-x_{31} &x_{23} &x_{15} &x_{16} &-x_{13} &x_{14} &x_{11} &-x_{12} &-x_{9} &-x_{10}\\
-x_{32} &x_{24} &x_{16} &-x_{15} &-x_{14} &-x_{13} &x_{12} &x_{11} &x_{10} &-x_{9}\\
\end{array} \right ]
\label{rod10}
\end{eqnarray}
\newpage
\begin{scriptsize}
\begin{eqnarray}
ROD_{12} = \left [
\begin{array} {cccccccccccc}
x_{1} &x_{9} &x_{17} &x_{25} &x_{33} &x_{34} &x_{35} &x_{36} &x_{37} &x_{38} &x_{39} &x_{40}\\
x_{2} &x_{10} &x_{18} &x_{26} &x_{34} &-x_{33} &x_{36} &-x_{35} &x_{38} &-x_{37} &-x_{40} &x_{39}\\
x_{3} &x_{11} &x_{19} &x_{27} &x_{35} &-x_{36} &-x_{33} &x_{34} &-x_{39} &-x_{40} &x_{37} &x_{38}\\
x_{4} &x_{12} &x_{20} &x_{28} &x_{36} &x_{35} &-x_{34} &-x_{33} &-x_{40} &x_{39} &-x_{38} &x_{37}\\
x_{5} &x_{13} &x_{21} &x_{29} &x_{37} &-x_{38} &x_{39} &x_{40} &-x_{33} &x_{34} &-x_{35} &-x_{36}\\
x_{6} &x_{14} &x_{22} &x_{30} &x_{38} &x_{37} &x_{40} &-x_{39} &-x_{34} &-x_{33} &x_{36} &-x_{35}\\
x_{7} &x_{15} &x_{23} &x_{31} &x_{39} &x_{40} &-x_{37} &x_{38} &x_{35} &-x_{36} &-x_{33} &-x_{34}\\
x_{8} &x_{16} &x_{24} &x_{32} &x_{40} &-x_{39} &-x_{38} &-x_{37} &x_{36} &x_{35} &x_{34} &-x_{33}\\
x_{9} &-x_{1} &x_{25} &-x_{17} &x_{41} &x_{42} &x_{43} &x_{44} &x_{45} &x_{46} &x_{47} &x_{48}\\
x_{10} &-x_{2} &x_{26} &-x_{18} &x_{42} &-x_{41} &x_{44} &-x_{43} &x_{46} &-x_{45} &-x_{48} &x_{47}\\
x_{11} &-x_{3} &x_{27} &-x_{19} &x_{43} &-x_{44} &-x_{41} &x_{42} &-x_{47} &-x_{48} &x_{45} &x_{46}\\
x_{12} &-x_{4} &x_{28} &-x_{20} &x_{44} &x_{43} &-x_{42} &-x_{41} &-x_{48} &x_{47} &-x_{46} &x_{45}\\
x_{13} &-x_{5} &x_{29} &-x_{21} &x_{45} &-x_{46} &x_{47} &x_{48} &-x_{41} &x_{42} &-x_{43} &-x_{44}\\
x_{14} &-x_{6} &x_{30} &-x_{22} &x_{46} &x_{45} &x_{48} &-x_{47} &-x_{42} &-x_{41} &x_{44} &-x_{43}\\
x_{15} &-x_{7} &x_{31} &-x_{23} &x_{47} &x_{48} &-x_{45} &x_{46} &x_{43} &-x_{44} &-x_{41} &-x_{42}\\
x_{16} &-x_{8} &x_{32} &-x_{24} &x_{48} &-x_{47} &-x_{46} &-x_{45} &x_{44} &x_{43} &x_{42} &-x_{41}\\
x_{17} &-x_{25} &-x_{1} &x_{9} &x_{49} &x_{50} &x_{51} &x_{52} &x_{53} &x_{54} &x_{55} &x_{56}\\
x_{18} &-x_{26} &-x_{2} &x_{10} &x_{50} &-x_{49} &x_{52} &-x_{51} &x_{54} &-x_{53} &-x_{56} &x_{55}\\
x_{19} &-x_{27} &-x_{3} &x_{11} &x_{51} &-x_{52} &-x_{49} &x_{50} &-x_{55} &-x_{56} &x_{53} &x_{54}\\
x_{20} &-x_{28} &-x_{4} &x_{12} &x_{52} &x_{51} &-x_{50} &-x_{49} &-x_{56} &x_{55} &-x_{54} &x_{53}\\
x_{21} &-x_{29} &-x_{5} &x_{13} &x_{53} &-x_{54} &x_{55} &x_{56} &-x_{49} &x_{50} &-x_{51} &-x_{52}\\
x_{22} &-x_{30} &-x_{6} &x_{14} &x_{54} &x_{53} &x_{56} &-x_{55} &-x_{50} &-x_{49} &x_{52} &-x_{51}\\
x_{23} &-x_{31} &-x_{7} &x_{15} &x_{55} &x_{56} &-x_{53} &x_{54} &x_{51} &-x_{52} &-x_{49} &-x_{50}\\
x_{24} &-x_{32} &-x_{8} &x_{16} &x_{56} &-x_{55} &-x_{54} &-x_{53} &x_{52} &x_{51} &x_{50} &-x_{49}\\
x_{25} &x_{17} &-x_{9} &-x_{1} &x_{57} &x_{58} &x_{59} &x_{60} &x_{61} &x_{62} &x_{63} &x_{64}\\
x_{26} &x_{18} &-x_{10} &-x_{2} &x_{58} &-x_{57} &x_{60} &-x_{59} &x_{62} &-x_{61} &-x_{64} &x_{63}\\
x_{27} &x_{19} &-x_{11} &-x_{3} &x_{59} &-x_{60} &-x_{57} &x_{58} &-x_{63} &-x_{64} &x_{61} &x_{62}\\
x_{28} &x_{20} &-x_{12} &-x_{4} &x_{60} &x_{59} &-x_{58} &-x_{57} &-x_{64} &x_{63} &-x_{62} &x_{61}\\
x_{29} &x_{21} &-x_{13} &-x_{5} &x_{61} &-x_{62} &x_{63} &x_{64} &-x_{57} &x_{58} &-x_{59} &-x_{60}\\
x_{30} &x_{22} &-x_{14} &-x_{6} &x_{62} &x_{61} &x_{64} &-x_{63} &-x_{58} &-x_{57} &x_{60} &-x_{59}\\
x_{31} &x_{23} &-x_{15} &-x_{7} &x_{63} &x_{64} &-x_{61} &x_{62} &x_{59} &-x_{60} &-x_{57} &-x_{58}\\
x_{32} &x_{24} &-x_{16} &-x_{8} &x_{64} &-x_{63} &-x_{62} &-x_{61} &x_{60} &x_{59} &x_{58} &-x_{57}\\
-x_{33} &-x_{41} &-x_{49} &-x_{57} &x_{1} &x_{2} &x_{3} &x_{4} &x_{5} &x_{6} &x_{7} &x_{8}\\
-x_{34} &-x_{42} &-x_{50} &-x_{58} &x_{2} &-x_{1} &x_{4} &-x_{3} &x_{6} &-x_{5} &-x_{8} &x_{7}\\
-x_{35} &-x_{43} &-x_{51} &-x_{59} &x_{3} &-x_{4} &-x_{1} &x_{2} &-x_{7} &-x_{8} &x_{5} &x_{6}\\
-x_{36} &-x_{44} &-x_{52} &-x_{60} &x_{4} &x_{3} &-x_{2} &-x_{1} &-x_{8} &x_{7} &-x_{6} &x_{5}\\
-x_{37} &-x_{45} &-x_{53} &-x_{61} &x_{5} &-x_{6} &x_{7} &x_{8} &-x_{1} &x_{2} &-x_{3} &-x_{4}\\
-x_{38} &-x_{46} &-x_{54} &-x_{62} &x_{6} &x_{5} &x_{8} &-x_{7} &-x_{2} &-x_{1} &x_{4} &-x_{3}\\
-x_{39} &-x_{47} &-x_{55} &-x_{63} &x_{7} &x_{8} &-x_{5} &x_{6} &x_{3} &-x_{4} &-x_{1} &-x_{2}\\
-x_{40} &-x_{48} &-x_{56} &-x_{64} &x_{8} &-x_{7} &-x_{6} &-x_{5} &x_{4} &x_{3} &x_{2} &-x_{1}\\
-x_{41} &x_{33} &-x_{57} &x_{49} &x_{9} &x_{10} &x_{11} &x_{12} &x_{13} &x_{14} &x_{15} &x_{16}\\
-x_{42} &x_{34} &-x_{58} &x_{50} &x_{10} &-x_{9} &x_{12} &-x_{11} &x_{14} &-x_{13} &-x_{16} &x_{15}\\
-x_{43} &x_{35} &-x_{59} &x_{51} &x_{11} &-x_{12} &-x_{9} &x_{10} &-x_{15} &-x_{16} &x_{13} &x_{14}\\
-x_{44} &x_{36} &-x_{60} &x_{52} &x_{12} &x_{11} &-x_{10} &-x_{9} &-x_{16} &x_{15} &-x_{14} &x_{13}\\
-x_{45} &x_{37} &-x_{61} &x_{53} &x_{13} &-x_{14} &x_{15} &x_{16} &-x_{9} &x_{10} &-x_{11} &-x_{12}\\
-x_{46} &x_{38} &-x_{62} &x_{54} &x_{14} &x_{13} &x_{16} &-x_{15} &-x_{10} &-x_{9} &x_{12} &-x_{11}\\
-x_{47} &x_{39} &-x_{63} &x_{55} &x_{15} &x_{16} &-x_{13} &x_{14} &x_{11} &-x_{12} &-x_{9} &-x_{10}\\
-x_{48} &x_{40} &-x_{64} &x_{56} &x_{16} &-x_{15} &-x_{14} &-x_{13} &x_{12} &x_{11} &x_{10} &-x_{9}\\
-x_{49} &x_{57} &x_{33} &-x_{41} &x_{17} &x_{18} &x_{19} &x_{20} &x_{21} &x_{22} &x_{23} &x_{24}\\
-x_{50} &x_{58} &x_{34} &-x_{42} &x_{18} &-x_{17} &x_{20} &-x_{19} &x_{22} &-x_{21} &-x_{24} &x_{23}\\
-x_{51} &x_{59} &x_{35} &-x_{43} &x_{19} &-x_{20} &-x_{17} &x_{18} &-x_{23} &-x_{24} &x_{21} &x_{22}\\
-x_{52} &x_{60} &x_{36} &-x_{44} &x_{20} &x_{19} &-x_{18} &-x_{17} &-x_{24} &x_{23} &-x_{22} &x_{21}\\
-x_{53} &x_{61} &x_{37} &-x_{45} &x_{21} &-x_{22} &x_{23} &x_{24} &-x_{17} &x_{18} &-x_{19} &-x_{20}\\
-x_{54} &x_{62} &x_{38} &-x_{46} &x_{22} &x_{21} &x_{24} &-x_{23} &-x_{18} &-x_{17} &x_{20} &-x_{19}\\
-x_{55} &x_{63} &x_{39} &-x_{47} &x_{23} &x_{24} &-x_{21} &x_{22} &x_{19} &-x_{20} &-x_{17} &-x_{18}\\
-x_{56} &x_{64} &x_{40} &-x_{48} &x_{24} &-x_{23} &-x_{22} &-x_{21} &x_{20} &x_{19} &x_{18} &-x_{17}\\
-x_{57} &-x_{49} &x_{41} &x_{33} &x_{25} &x_{26} &x_{27} &x_{28} &x_{29} &x_{30} &x_{31} &x_{32}\\
-x_{58} &-x_{50} &x_{42} &x_{34} &x_{26} &-x_{25} &x_{28} &-x_{27} &x_{30} &-x_{29} &-x_{32} &x_{31}\\
-x_{59} &-x_{51} &x_{43} &x_{35} &x_{27} &-x_{28} &-x_{25} &x_{26} &-x_{31} &-x_{32} &x_{29} &x_{30}\\
-x_{60} &-x_{52} &x_{44} &x_{36} &x_{28} &x_{27} &-x_{26} &-x_{25} &-x_{32} &x_{31} &-x_{30} &x_{29}\\
-x_{61} &-x_{53} &x_{45} &x_{37} &x_{29} &-x_{30} &x_{31} &x_{32} &-x_{25} &x_{26} &-x_{27} &-x_{28}\\
-x_{62} &-x_{54} &x_{46} &x_{38} &x_{30} &x_{29} &x_{32} &-x_{31} &-x_{26} &-x_{25} &x_{28} &-x_{27}\\
-x_{63} &-x_{55} &x_{47} &x_{39} &x_{31} &x_{32} &-x_{29} &x_{30} &x_{27} &-x_{28} &-x_{25} &-x_{26}\\
-x_{64} &-x_{56} &x_{48} &x_{40} &x_{32} &-x_{31} &-x_{30} &-x_{29} &x_{28} &x_{27} &x_{26} &-x_{25}\\
\end{array} \right ]
\label{rod12}
\end{eqnarray}
\end{scriptsize}
\section{Conclusions}
\label{sec6}
We summarize the conclusions in this paper and future work as follows.
Amplify-and-forward (AF) schemes in cooperative communications are
attractive because of their simplicity. Full diversity (FD),
linear-complexity single symbol decoding (SSD), and high rates of
DSTBCs are three important attributes to work towards AF cooperative
communications. Earlier work in \cite{YiK} has shown that, without
assuming phase knowledge at the relays, FD and SSD can be achieved
in AF distributed orthogonal STBC schemes; however, the rate achieved
decreases linearly with the number of relays $N$. Our work in this paper
established that if phase knowledge is exploited at the relays in the
way we have proposed, then FD, SSD, and high rate can be achieved
simultaneously; in particular, the rate achieved in our scheme can be
$\frac{1}{2}$, which is independent of the number of relays $N$. We
proved the SSD for our scheme in Theorem 2. FD was proved in Theorem 6.
Rate-1/2 construction for any $N$ was presented in Theorem 5. In addition
to these results, we also established other results regarding $i)$
invariance of SSD under coordinate interleaving (Theorem 4), and $ii)$
retention of FD even with single-symbol non-ML decoding. Simulation results
confirming the claims were presented. All these important results have not
been shown in the literature so far. These results offer useful insights
and knowledge for the designers of future cooperative communication based
systems (e.g., cooperative communication ideas are being considered in
future evolution of standards like IEEE 802.16).
In this work, we have assumed only phase knowledge at the relays. Of course,
one can assume that both amplitude as well as the phase of source-to-relay
are known at the relay. A natural question that can arise then is `what can
amplitude knowledge at the relay (in addition to phase knowledge) buy?'
Since we have shown that phase knowledge alone is adequate to achieve FD,
some extra coding gain may be possible with amplitude knowledge. This
aspect of the problem is beyond the scope of this paper; but it is a valid
topic for future work.
{\footnotesize
\bibliographystyle{IEEE}
|
0708.3784
|
\section{Introduction}
Localization of quantum states in phase space is a prerequisite
in some semiclassical treatments of quantum evolution. In the classically chaotic case,
the width of an initially localized Gau{\ss}ian increases exponentially
\cite{CR97} up to the so called Ehrenfest time $T_{E}$, i.e., the time up to
which quantum dynamics can be approximated by classical dynamics. In the case
of regular classical motion it can be shown that this width grows algebraically
in the semiclassical parameter $\hbar$ up to $T_{E}$.
In certain applications, it is necessary to propagate states semiclassically for long times.
This demands control of the width of the state. An example is the construction of quasi-modes
proposed by Paul and Uribe \cite{PaUr93}. One refers to a quasi-mode as a state $\psi$
which is a solution of a corresponding spectral problem of an operator $\widehat{H}$ up to some small discrepancy $\delta$, i.e.,
\begin{eqnarray*}
\left\Vert\widehat{H}\psi-E\psi\right\Vert & < & \delta
\end{eqnarray*} which insures, in the case of discrete spectrum, that there exist at least one eigenvalue of $\widehat{H}$
in the interval $\left[E-\delta,\,E+\delta\right]$.
The approximation proposed by Paul and Uribe uses the semiclassical propagation of coherent
states over closed classical trajectories leading to the well known Bohr-Sommerfeld quantization
rule in one dimension. The eigenvalues $E_{n}$ of the Hamiltonian $\widehat{H}$ are given
by the quantization condition
\begin{eqnarray}
\int_{\mathscr{C}_{E_{n}}}p\,\mathrm{d}q & = & 2\pi\hbar\left(n+\frac{1}{2}\right)+\mathcal{O}\left(\hbar^{2}\right),\quad n\in\mathds{N},
\end{eqnarray}
where the energy shells $\mathscr{C}_{E}=\left\lbrace \left(p,\,q\right):\,H\left(p,\, q\right)=E\right\rbrace $ are closed curves.
Certain examples suggest that there exist systems (other than the harmonic oscillator)
where the propagated width of an initial Gau{\ss}ian remains small for long times.
Such a behavior is exhibited by the propagation of Gau{\ss}ians generated by some
perturbed periodic Schr\"{o}dinger operators like the Wannier-Stark Hamiltonian
\begin{eqnarray*}
\widehat{H}_{\mathrm{WS}}\left( \varepsilon \right) & = & -\frac{\hbar^{2}}{2}\Delta_{x}+V_{\Gamma}\left( x\right)+\varepsilon x
\end{eqnarray*}
in the limit of small perturbations $\varepsilon$. Here $V_{\Gamma}\left(x \right)$ denotes a
periodic potential with respect to a lattice $\Gamma\cong \mathds{Z}^{d}$. It is known that
the band structure of the spectrum of the unperturbed operator is preserved for small enough
perturbations $\varepsilon$. Numerical studies \cite{Har04,Witt04} show that an initially
localized Gau{\ss}ian in momentum space defined on such an energy band apparently remains
Gau{\ss}ian for long times. The evolving states carry out oscillations. The center of the
Gau{\ss}ian can oscillate in position space describing so called Bloch oscillations \cite{Blo28}.
Alternatively, the width of the Gaussian can oscillate, whereupon the center remains fixed,
describing so called breathing modes. In both cases, the state returns to the initial
state after an oscillation period up to a small error.
We study the evolution generated by a class of operators on
$L^{2}\left(\mathds{R}^{d}\right)$
defined as Weyl quantizations of classical symbols $H\left(X\right)$
with the property\begin{eqnarray}
H^{\prime\prime}\left(X_{t}\right) & = &
H^{\prime\prime}\left(X_{t+T}\right)
,\quad\forall t\in\mathds{R},\label{eq:percond}
\end{eqnarray}
where $X_{t}$ denotes the solutions to Hamilton's equations of motion and
$H^{\prime\prime}\left(X\right)$ is the Hessian of $H$ with respect to $X$.
In one dimension, the condition (\ref{eq:percond}) is satisfied by bounded
classical motion or unbounded
motion in a periodic potential. We particularly study the evolution
of initial Gau{\ss}ian (or squeezed) states semi-classically, i.e.,
asymptotically as $\hbar\searrow0$, when $t\nearrow\infty$.
We focus our attention to the spreading of such wave packets.
In sections \ref{sec:Preliminaries:-semiclassical-propagation} and
\ref{sec:A semiclassical propagation theorem} we shortly
review the semiclassical propagation of Gau{\ss}ian coherent states. In
section \ref{EhrT} we give some known results on the validity of the
approximation. We then make, in section \ref{sec:Floquet-theory} and
section \ref{sec:A-Uniform-bound}, statements about the approximate
Gau{\ss}ian state given by this semiclassical propagation using Floquet
theory. These properties are then brought back to the true evolution
in section \ref{sec:Discussion}.
\subsection{\label{sec:Preliminaries:-semiclassical-propagation}Preliminaries:
semiclassical propagation of wave packets}
We will work in the context of self adjoint operators defined on $L^{2}\left(\mathds{R}^{d}\right)$
that are $\hbar-$Weyl quantizations of symbols. To a smooth
$\left(C^{\infty}\right)$ classical
symbol $b\left( X\right)$, i.e., a function on the phase space
$T^{*}\mathds{R}^{d}\cong\mathds{R}^{2d}$, there corresponds an operator
on $L^{2}\left(\mathds{R}^{d}\right)$, $\widehat{b}:=\mathrm{Op}_{\hbar}^{w}\left[b\right]$,
defined by\begin{eqnarray*}
\mathrm{Op}_{\hbar}^{w}\left[b\right]\psi\left(x\right) & := & \frac{1}{\left(2\pi\hbar\right)^{d}}\int_{\mathds{R}^{2d}}b\left(\frac{x+y}{2},\,\xi\right)\psi\left(y\right)e^{\frac{\mathrm{i}}{\hbar}\left(x-y\right)\xi}\,\mathrm{d}y\mathrm{d}\xi.\end{eqnarray*}
The following conditions will be assumed \label{conditions}:
\begin{enumerate}
\item \vspace{\enumskip}The classical Hamiltonian $H:\,\mathds{R}^{2d}\rightarrow\mathds{R}$ is a smooth function.
\item \vspace{\enumskip}$H\in S\left(m\right)$, i.e., for all multi-indices
$\alpha$, $\beta$, there exists $K_{\alpha ,\,\beta}>0$ such that \[
\left|\partial_{p}^{\alpha}\partial_{q}^{\beta}H\left(p,\, q\right)\right|\leq K_{\alpha ,\,\beta}\left(1+\left|p\right|^{2}+\left|q\right|^{2}\right)^{\frac{m}{2}}. \]
\item \vspace{\enumskip}The corresponding classical equation of motion is given
by \begin{eqnarray*}
\frac{\mathrm{d}X_{t}}{\mathrm{d}t} & = & \mathcal{J}H^{\prime}\left(X_{t}\right),
\end{eqnarray*}
where $\mathcal{J}$ is the symplectic
unity \begin{eqnarray*}
\mathcal{J} & := & \left(\begin{array}{cc}
0 & -\mathds{1}_{d\times d}\\
\mathds{1}_{d\times d} & 0\end{array}\right),
\end{eqnarray*}
and $H^{\prime}$ is the gradient of $H$ with respect to $X$. Furthermore,
we denote by $\Phi_{H}^{t}:\,\mathds{R}^{2d}\rightarrow\mathds{R}^{2d},
\,X_{0}\mapsto X_{t}=\Phi_{H}^{t}\left(X_{0}\right)$ the corresponding classical flow.
\item \vspace{\enumskip}The $\hbar-$Weyl quantization of $H\left(X\right)$,
$\widehat{H}:=\mathrm{Op}_{\hbar}^{w}\left[H\right]$, is an essentially self-adjoint
operator on $L^{2}\left(\mathds{R}^{d}\right)$
and generates a unitary time evolution $\forall t\in\mathds{R},$\begin{eqnarray*}
\widehat{U}\left(t \right) & : & L^{2}\left(\mathds{R}^{d}\right)\rightarrow L^{2}\left(\mathds{R}^{d}\right)\\
& & \psi\left(0\right)\mapsto\psi\left(t\right)\end{eqnarray*}
corresponding to the Schr\"{o}dinger equation\begin{eqnarray}
\mathrm{i}\hbar\frac{\partial\psi}{\partial t} & = & \widehat{H}\psi,\label{eq:Schroedinger}\end{eqnarray}
i.e., we will write $\widehat{U}\left(t\right)=e^{-\frac{\mathrm{i}}{\hbar}\widehat{H}t}.$\vspace{\enumskip}
\end{enumerate}
The Weyl calculus also allows a representation of quantum mechanical wave
functions on phase space. This is given by the Wigner function of
the state $u\in L^{2}\left(\mathds{R}^{d}\right),$
\begin{eqnarray}
W\left[u\right]\left(p,\, q\right) & := & \int_{\mathds{R}^{d}}\overline{u}\left(q+\frac{y}{2}\right)u\left(q-\frac{y}{2}\right)e^{\frac{\mathrm{i}}{\hbar} p y}\,\mathrm{d}y\label{WigRep}.\end{eqnarray}
The Wigner function of a Gau{\ss}ian defines a positive measure on
phase space.
\subsection{\label{sec:A semiclassical propagation theorem}A semiclassical propagation theorem}
With these assumptions we give a short summary of the method
of semiclassical propagation of coherent states introduced by Combescure
and Robert \cite{CR97}. Similar constructions have also been considered
in the past. See, e.g., Hagedorn \cite{Ha81,Ha85} and references
therein.
The idea is to expand the exact Hamiltonian $\widehat{H}$ along the
classical flow generated by the symbol $H$ up to second order, whereupon
the approximate time dependent Hamiltonian, $\widehat{H}_{2}\left(t\right)$,
is the $\hbar-$Weyl quantization of
\begin{eqnarray*}
H_{2}\left(t,\,Y\right) & := & H\left(X_{t}\right)+\left(Y-X_{t}\right)^{T}
H^{\prime}\left(X_{t}\right)+\frac{1}{2}\left(Y-X_{t}\right)^{T}H^{\prime\prime}
\left(X_{t}\right)\left(Y-X_{t}\right).
\end{eqnarray*}
The propagation of normalized wave functions (squeezed states)%
\footnote{We will, with some lack of rigor, call these states Gau{\ss}ian,
coherent or squeezed without distinction.%
} of the form
\begin{eqnarray}
\psi_{Z}^{\left(p,\, q\right)}\left(x\right) & := &
\frac{\det\left(\Im\left(Z\right)\right)^{\frac{1}{4}}}
{\left(\pi\hbar\right)^{\frac{d}{4}}}e^{\frac{\mathrm{i}}{\hbar}
\left(p^{T}\left(x-q\right)+\left(x-q\right)^{T}\frac{Z}{2}\left(x-q\right)\right)},
\,Z\in\Sigma_{d},\,\left(\begin{array}{c}p \\ q\end{array}\right)\in\mathds{R}^{2d}
\label{eq:squstate}\end{eqnarray}
by quadratic Hamiltonians is well known \cite{Fol89}. By $\Sigma_{d}$,
we mean the $d-$dimensional Siegel upper half space, i.e., the set of symmetric
$d\times d$ matrices with positive, non-degenerate imaginary
part \cite{Fol89}. The quadratic form $Z$
describes the shape of the wave packet and it should be underlined
that it is independent of $\hbar$.
In its Wigner representation (see eq.(\ref{WigRep}))
$\psi_{Z}^{\left(p,\, q\right)}$
is a Gau{\ss}ian centered around $X=\left(\begin{array}{c}
p\\
q\end{array}\right).$ This Wigner function is given by
\begin{eqnarray*}
W\left[\psi_{Z}^{X}\right]\left(Y\right) & = & \left(\frac{1}
{\pi\hbar}\right)^{d}\exp\left(-\frac{1}{\hbar}
\left(Y-X\right)^{T}G\left(Y-X\right)\right)
\end{eqnarray*}
where
\begin{eqnarray}
G & := & \left(\begin{array}{ccc}
\Im\left(Z\right)^{-1} & \quad & -\Im\left(Z\right)^{-1}\Re \left(Z\right)\\
\quad & \quad & \quad\\
-\Re\left(Z \right)\Im\left(Z\right)^{-1} & \quad &
\Im\left(Z \right)+\Re\left(Z \right)\Im
\left(Z \right)^{-1}\Re\left(Z \right) \end{array}\right)
\end{eqnarray}
is independent of $\hbar$.
The unitary evolution, $\widehat{U}_{2}\left(t\right)$, generated
by $\widehat{H}_{2}\left(t\right)$, acts on a squeezed state by translation and
metaplectic action, i.e.,
\begin{eqnarray}
\widehat{U}_{2}\left(t\right) & = & e^{\frac{\mathrm{i}}{\hbar}\Theta\left(t\right)}
\widehat{\mathcal{T}}\left( X_{t}\right)
\widehat{\mathcal{M}}\left(S_{t}\right)
\label{eq:metaplect}\end{eqnarray}
where $\widehat{\mathcal{T}}$ is the translation operator on $\mathds{R}^{2d}$, i.e., $\forall Y \in\mathds{R}^{2d},\,
=:\left(
\begin{array}{c}
\xi \\
q
\end{array}
\right)$ and $u\in L^{2}\left(\mathds{R}^{d}\right)$ we have
\begin{eqnarray*}
\widehat{\mathcal{T}}\left(Y\right)W\left[u\right]\left(Z\right)
& := & e^{\frac{\mathrm{i}}{\hbar}\left(\xi^{T} \widehat{x}-q^{T}\widehat{p}\right)}W\left[u\right]\left(Z\right)\\
& = & W\left[u\right]\left(Z-Y\right),
\end{eqnarray*}
where we denote by $\widehat{p}$ the momentum operator and by $\widehat{x}$ the position operator. The metaplectic
operator $\widehat{\mathcal{M}}\left( F\right)$ is the quantization of a linear symplectomorphism on $\mathds{R}^{2d}$ given by the symplectic matrix $F$. These operators form a double-valued unitary representation\footnote{A thorough description of the action of the evolution generated by quadratic operators on Gau{\ss}ians and the metaplectic representation can be found in \cite{Fol89}.} of the linear symplectomorphism of $\mathds{R}^{2d}$. $S_{t}$ denotes the flow differential. The classical flow $\Phi_{H}^{t}$ is a symplectomorphism which ensures that the flow differential is a symplectic matrix. Furthermore, $S_{t}$ satisfies
\begin{eqnarray}
\frac{\mathrm{d}S_{t}}{\mathrm{d}t} & = & \mathcal{J}H^{\prime\prime}
\left(X_{t}\right)S_{t},\label{eq:action}\\
\,S_{0} & = & \mathds{1}_{2d\times2d}.\nonumber
\end{eqnarray} In the prefactor of eq. (\ref{eq:metaplect}) we have used
\begin{eqnarray*}
\Theta\left(t\right)&:=&\mathcal{W}\left(t\right)+\hbar\mu,
\end{eqnarray*}
where
\begin{eqnarray*}
\mathcal{W}\left(t\right) & := & \int_{0}^{t} \left(p_{\tau}^{T}\dot{q}_{\tau}-H\left(p_{\tau},\,
q_{\tau}\right)\right)\, d\tau,
\end{eqnarray*}
is the action of the classical trajectory and $\mu$ is the Maslov
index of the classical trajectory. We have here expressed the solution
of Hamilton's equations in terms of the canonical variables
\begin{eqnarray*}
\left(\begin{array}{c}
p_{t}\\
q_{t}\end{array}\right) & := & X_{t}.
\end{eqnarray*}
Acting with (\ref{eq:metaplect}) on (\ref{eq:squstate}), one obtains at time $t$ a new Gau{\ss}ian state
$e^{\frac{\mathrm{i}}{\hbar}\Theta\left(t\right)}\psi_{Z_{t}}^{X_{t}}$ up to a
phase. The Gau{\ss}ian is centered around $X_{t}$ in phase space
and has a quadratic form $Z_{t}\in\Sigma_{d}$, is given explicitly by the group action
\cite{Fol89}
\begin{eqnarray*}
Z_{t} & = & S_{t}\left[Z_{0}\right]\\
& = & \left(A_{t}Z_{0}+B_{t}\right)\left(C_{t}Z_{0}+D_{t}\right)^{-1}\quad
,\,Z_{0}\in\Sigma_{d},
\end{eqnarray*}
i.e., linear fractional transformation on $\Sigma_{d}$. The matrix
$S_{t}$ is here written by means of the $d\times d$ blocks $A_{t}$, $B_{t}$, $C_{t}$ and $D_{t}$, i.e.,
\begin{eqnarray}
S_{t} & = & \left(\begin{array}{cc}
A_{t} & B_{t}\\
C_{t} & D_{t}\end{array}\right).
\end{eqnarray}
The quadratic form of the Wigner transform of $\psi_{Z_{t}}^{X_{t}}$, is explicitly given by
\begin{eqnarray}
G_{t} & = & \left(S_{t}^{-1}\right)^{T}G_{0}S_{t}^{-1}\label{eq:width}.
\end{eqnarray}
The evolution of the approximate wave packet can be perceived as being
generated by rotation and scaling of the Gau{\ss}ian profile in
$\mathds{R}^{2d}$.
The difficulty of this scheme resides in the control of errors made
by the approximation of $\widehat{U}\left(t\right)$ in terms of
$\widehat{U}_{2}\left(t\right)$. We do not dwell on
the details, but just state the result (for proof and a thorough description see
\cite{CR97}). The approximation can be checked to all orders in
$\hbar$. For this purpose, one defines the following approximating state
\begin{eqnarray*}
\Psi^{\left(N\right)}\left(t,\, x\right) & := & e^{\frac{\mathrm{i}}{\hbar}\Theta\left(t\right)}\sum_{0\leq j\leq N}\hbar^{\frac{j}{2}}\pi_{j}\left(X_{t},\,\frac{x}{\sqrt{\hbar}},\, t\right)\psi_{Z_{t}}^{X_{t}}\left(x\right),
\end{eqnarray*}
where $\pi_{j}\left(X_{t},\,\frac{x}{\sqrt{\hbar}},\, t\right)$
are polynomials in $x\left/\sqrt{\hbar}\right.$ and $X_{t}$
of degree smaller than or equal to $3j$ with time dependent coefficients.
\begin{theorem}
\label{thm:propagation_control_CR}(Combescure, Robert) Under the above
mentioned assumptions and for an initial Gau{\ss}ian state $\psi_{Z_{0}}^{X_{0}}$
centered in the phase space representation at $X_{0}\in\mathds{R}^{2d}$, for every
$N\in\mathds{N}$ there exists $C<\infty$ such that $\forall\hbar\in\left(0,\,\hbar_{0}\right],\,\hbar_{0}>0$,
\begin{eqnarray}
\left\Vert \widehat{U}\left(t\right)\psi_{Z_{0}}^{X_{0}}-\Psi^{\left(N\right)}\left(t\right)\right\Vert &
\leq & C_{N}\hbar^{\frac{N+1}{2}}te^{3\gamma t}\label{eq:normunitaries}
\end{eqnarray}
where $0\leq\gamma<\infty$ is the Lyapunov exponent of the classical motion.
\end{theorem}
We recall that a Lyapunov exponent is a measure of the exponential stability of the solutions of a
differential equation upon change of initial conditions. In the case of classical motion, this is given by the Lyapunov
exponent defined as
\begin{eqnarray}
\gamma & := & \max_{k}\left[ \lim_{t\rightarrow\infty}\sup\left( \frac{\ln\left(s_{k}\left( t\right)\right)}{t}\right) \right] \label{defclasslyapunov}
\end{eqnarray}
where $s_{k}\left( t\right)$ are the singular values of $S_{t}$. The Lyapunov exponent $\gamma$ hence satisfies
\begin{eqnarray}
\left\Vert S_{t}\right\Vert_{\mathrm{HS}} & < & c_{0}e^{\gamma \left\vert t\right\vert}\label{lyapunovinequality}
\end{eqnarray}
where $c_{0}<\infty$ is a positive constant. We denote by $\left\Vert M\right\Vert _{\mathrm{HS}}=
\sqrt{\mathrm{tr}\left(M^{\dagger}M\right)}$
the Hilbert-Schmidt norm of the matrix $M$. Hermitian conjugation
is denoted by $M^{\dagger}$, and transposition by $M^{T}$.
\subsection{Ehrenfest time and spreading of wave packets\label{EhrT}}
It is customary, in this context, to define what is known as the Ehrenfest
time. The latter is a time scale up to which the above approximation
is valid.
We define the Ehrenfest time, denoted $T_{E}\left( \hbar \right)$, as the maximal time up to which
\begin{itemize}
\item \vspace{\enumskip}the error $\left\Vert \widehat{U}\left(t\right)
\psi_{Z_{0}}^{X_{0}}-e^{\frac{\mathrm{i}}{\hbar}\Theta\left(t\right)}
\psi_{Z_{t}}^{X_{t}}\right\Vert$
remains small, \vspace{\enumskip}
\item the exact state remains localized.\vspace{\enumskip}
\end{itemize}
The latter ensures that the approximation retains a physical meaning, i.e.,
the above classical approximation makes no sense if the state does not remain
localized.
The total width of the semiclassically evolved Gau{\ss}ian is
\begin{eqnarray}
\sigma_{t}\left(X_{0};\,\hbar\right):=\Delta x^{2}\left(\hbar,\, t
\right)+\Delta p^{2}\left(\hbar,\, t\right) & = & \frac{\hbar}{2}
\mathrm{tr}\left(G_{t}\right),\label{eq:semclawidth}
\end{eqnarray}
where we have used the variance
\begin{eqnarray*}
\Delta x^{2} & = & \left\langle e^{\frac{\mathrm{i}}{\hbar}
\Theta\left(t\right)}\psi_{Z_{t}}^{X_{t}},
\, \widehat{x}^{2}e^{\frac{\mathrm{i}}{\hbar}\Theta\left(t\right)}
\psi_{Z_{t}}^{X_{t}}\right\rangle _{L^{2}\left(\mathds{R}^{d}\right)}-
\left\langle e^{\frac{\mathrm{i}}{\hbar}\Theta\left(t\right)}
\psi_{Z_{t}}^{X_{t}},\, \widehat{x}e^{\frac{\mathrm{i}}{\hbar}\Theta
\left(t\right)}\psi_{Z_{t}}^{X_{t}}\right\rangle_{L^{2}
\left(\mathds{R}^{d}\right)}^{2},
\end{eqnarray*}
and similarly for the momentum operator.
From eq. (\ref{eq:width}) one obtains that
\begin{eqnarray*}
\sigma_{t} & \leq & \sigma_{0}\left\Vert S_{t}\right\Vert^{2}_{\mathrm{HS}}.
\end{eqnarray*}
By Theorem \ref{thm:propagation_control_CR}, there will
exist a constant $c_{1}$ such that
\begin{eqnarray*}
\left\Vert\widehat{U}\left(t\right)\psi_{Z_{0}}^{X_{0}}-e^{\frac{\mathrm{i}}{\hbar}
\Theta\left(t\right)}\psi_{Z_{t}}^{X_{t}}\right\Vert & \leq &
c_{1} \sqrt{\hbar} t e^{3\gamma t}.
\end{eqnarray*}
The time scale for which the errors are small is thus algebraic in $\hbar$ if $\gamma=0$.
In the generic case, the errors remain small for logarithmic times in $\hbar$.
With this result, the errors
\begin{eqnarray}
\Delta\left(t\right) & := & \left\langle \widehat{U}\left(t\right)\varphi_{0},\,
\widehat{s}\left(\widehat{U}\left(t\right)
\varphi_{0}\right)\right\rangle_{L^{2}\left(\mathds{R}^{d}\right)} -\left\langle \widehat{U}_{2}\left(t\right)\varphi_{0},\,\widehat{s}\left(\widehat{U}_{2}
\left(t\right)\varphi_{0}\right)\right\rangle_{L^{2}\left(\mathds{R}^{d}\right)} \label{eq:obserror}
\end{eqnarray}
for propagating observables $\widehat{s}$ can be approximated explicitly. Of particular interest to us is the width
operator $\widehat{s}:=\mathrm{Op}_{\hbar}^{w}\left[\left|Y\right|^{2}\right]$. One can characterize
the times for which the error
$\Delta\left(t\right)$ is small, e.g., $\Delta\left(t\right)=\mathcal{O}\left(\hbar^{\alpha}\right)$
for some $\alpha>0$. Again, using the Lyapunov inequality (\ref{lyapunovinequality}) one finds that the width
of the approximate state is bounded by
\begin{eqnarray*}
\sigma_{t}\left(\hbar\right) & \leq & c_{3} e^{2\gamma\left|t\right|},
\quad c_{3}>0,
\end{eqnarray*}
and the error remains $\mathcal{O}\left(\hbar^{\alpha}\right)$ up to times
of order $\frac{\left|\ln\left(\hbar\right)\right|}{6\gamma}.$ We may thus
generically state that the Ehrenfest time is
\begin{eqnarray}
T_{E}\left( \hbar \right) & \propto & \frac{\left|\ln\left(\hbar\right)\right|}{6\gamma}.
\end{eqnarray}
In the integrable case, implying $\gamma=0$, the width grows like the square of the
Hilbert-Schmidt norm of the flow differential, i.e., at most polynomially in time. The
error remains small for times up to $\hbar^{-\frac{1}{2}}$. The Ehrenfest time thus
is algebraic in $\hbar$.
Our aim is to characterize the spreading of the approximate state
for a more specific class of classical motions.
\section{Results}
In addition to the conditions imposed on symbols $H$ above (see section
\ref{conditions}), we assume that the Hessian
$H^{\prime\prime}\left(X_{t}\right)$ is $T-$periodic, i.e.,
\begin{eqnarray}
H^{\prime\prime}\left(X_{t}\right) & = & H^{\prime\prime}\left(X_{t+T}\right),
\quad\forall t\in\mathds{R}.
\label{perhess}
\end{eqnarray}
We will also utilize the following definition.
\begin{definition}
A \textbf{classical revival} at a time $t>0$ is the event that
the approximate Gau{\ss}ian given above is the
initial one up to a phase factor, i.e.,
\begin{eqnarray*}
\psi_{Z_{t}}^{X_{t}} & = & e^{\mathrm{i}\alpha_{t}}\psi_{Z_{0}}^{X_{0}}.
\end{eqnarray*}
\end{definition}
\subsection{\label{sec:Floquet-theory}Floquet theory}
According to the Floquet theorem \cite{F1883}, any linear differential
equation with continuous $T-$periodic coefficients has a periodic
solution of the second type, i.e., a solutions which satisfy
\begin{eqnarray*}
f\left(t+T\right) & = & \upsilon f\left(t\right),\quad
\upsilon\in\mathds{C},\,\forall t\in\mathds{R}.
\end{eqnarray*}
In particular, the linear vector differential equation
\begin{eqnarray*}
\frac{\mathrm{d}}{\mathrm{d}t}f & = & A\left(t\right)f,
\end{eqnarray*}
where $A\left( t\right) $ is continuous and satisfies $A\left(t+T\right)=A\left(t\right),\,\forall t\in\mathds{R}$, has a
fundamental matrix of the form (see e.g. \cite{CoL55})
\begin{eqnarray*}
F_{t} & = & M^{-1}e^{L t}MU\left(t\right),\quad M\in\mathrm{GL}\left(n,\,
\mathds{C}\right),
\end{eqnarray*}
where $L$ is a diagonal matrix and $U\left(t\right)$ is a $T-$periodic matrix.
By definition, a fundamental matrix is a full rank matrix whose columns satisfy
the differential equation, i.e., the linear combinations of the columns of the
fundamental matrix span the full space of solutions of the differential equation.
We will call the elements of $L$ the \textit{Floquet exponents} of the fundamental system.
The result of this section will be summarized in the following way.
\begin{proposition}
\label{pro:periodicwidth} \textbf{If} $H^{\prime\prime}\left(X_{t}\right)$
is periodic \textbf{and} the Floquet exponents of $\,S_{t}$
are purely complex \textbf{then} the width of a Gau{\ss}ian propagated semiclassically
by $\widehat{U}_{2}\left(t\right)$ will remain unchanged at multiples
of the smallest classical period $T$. Furthermore, \textbf{if} the
classical flow is periodic \textbf{then} classical revivals will occur.
\end{proposition}
\begin{bew}{}
Under the condition (\ref{perhess}), Floquet theory states the
existence of a fundamental Floquet matrix
for the linear differential eq. (\ref{eq:action}),
\begin{eqnarray*}
S_{t} & = & M^{-1}e^{\Lambda t}M F_{t},\quad
M\in\mathrm{GL}\left(2d,\,\mathds{C}\right),
\end{eqnarray*}
where $e^{\Lambda t}$ is the diagonal matrix with entries
$e^{2\pi\lambda_{i}\frac{t}{T}},\,\lambda_{i}\in\mathds{C}$,
$F_{t}$ has minimal period $T$, and we have chosen
\begin{eqnarray*}
F_{0} & = & S_{0} = \mathds{1}_{2d\times2d}.
\end{eqnarray*}
One directly concludes that
\begin{eqnarray*}
S_{k T} & = & M^{-1}e^{k T\Lambda}M,\quad\forall k\in\mathds{Z}.
\end{eqnarray*}
for multiples $k$ of the classical period $T$.
Furthermore, the real fundamental matrix defined by
$S_{t}\in\mathrm{Sp}\left(2d,\,\mathds{R}\right)$
has a unique polar decomposition, i.e., there exists \cite{Fol89}
an orthogonal matrix $\mathcal{Q}_{t}\in\mathrm{O}\left(2d\right)\cap\mathrm{Sp}\left(2d,\,\mathds{R}\right)$
and a positive definite matrix $\mathcal{P}_{t}\in\mathrm{Sp} \left(2d,\,\mathds{R}\right)$
such that
\begin{eqnarray*}
S_{t} & = & \mathcal{Q}_{t}\mathcal{P}_{t}.
\end{eqnarray*}
The width of the approximate squeezed state at time $t$ is
(see eq. (\ref{eq:semclawidth}))
\begin{eqnarray*}
\sigma_{t}\left(\hbar\right) & = & \mathrm{tr}\left(
\left(S_{t}^{-1}\right)^{T}G_{0}
S_{t}^{-1}\right),
\end{eqnarray*}
and since any symplectic matrix $A$ satisfies \cite{Fol89}
\begin{eqnarray*}
A^{-1} & = & \mathcal{J}A^{T}\mathcal{J}^{-1},
\end{eqnarray*}
we can write by eq. (\ref{eq:width}),
\begin{eqnarray*}
\sigma_{t}\left(\hbar\right) & = & \frac{\hbar}{2}\mathrm{tr}
\left(\mathcal{Q}_{t}\mathcal{P}_{t}\mathcal{J}^{T}
G_{0}\mathcal{JP}_{t}^{T}\mathcal{Q}_{t}^{T}\right)=
\frac{\hbar}{2}\mathrm{tr}
\left(\mathcal{P}_{t}\mathcal{J}^{T}G_{0}\mathcal{J}
\mathcal{P}_{t}^{T}\right),
\end{eqnarray*}
since $\mathcal{Q}_{t}\in\mathrm{O}\left(2d\right)$ .
We are hence confronted with two cases.
\begin{itemize}
\item \vspace{\enumskip} Either $S_{k T}$ is orthogonal, i.e.,
$\mathcal{P}_{k T}=\mathds{1}_{2d\times2d}$; this corresponds to
\begin{eqnarray*}
\sigma_{k T}\left(\hbar\right) & = & \frac{\hbar}{2}\mathrm{tr}
\left(G_{0}\right) = \sigma_{0}\left(\hbar\right),\quad\forall k\in\mathds{Z}.
\end{eqnarray*}
The orthogonality of $S_{k T}$ implies that it has $2d$ singular
values $1$. It is furthermore similar to $e^{k T\Lambda}$
$\left( F_{k T}=\mathds{1}_{2d\times2d}\right)$, which
hence is unitary so the Floquet
exponents $\frac{2\pi\lambda_{i}}{T}$ are purely complex or zero.
By definition, the Lyapunov exponent of the classical trajectories
is
\begin{eqnarray*}
\gamma & := & \max_{k}\left[ \lim_{t\rightarrow\infty}\sup\left(\frac{\ln\left(
s_{k}\left(X_{t}\right)\right)}{t}\right)\right] ,
\end{eqnarray*}
where $s_{k}\left( t\right)$ are the singular values of the flow differential
$S_{t}$, i.e., the eigenvalues of
\begin{eqnarray*}
F_{t}^{\dagger}\left(M^{-1}e^{t\Lambda}M\right)^{\dagger}
M^{-1}e^{t\Lambda}M
F_{t} & = & F_{t}^{\dagger}F_{t}.
\end{eqnarray*}
Noting that $F_{t}$ is bounded (and periodic),
we see directly that if
\begin{eqnarray*}
\max_{i}\left(\Re\left( \lambda_{i}\right)\right) & = & 0,
\end{eqnarray*}
the classical motion is linearly stable.\vspace{\enumskip}
\end{itemize}
\begin{itemize}
\item \vspace{\enumskip}$\mathcal{P}_{k T}\neq\mathds{1}_{2d\times2d},$ which
corresponds to
\begin{eqnarray*}
\sigma_{k T}\left(\hbar\right) & = & \frac{\hbar}{2}\mathrm{tr}
\left(\mathcal{P}_{k T}\mathcal{J}^{T}G_{0}\mathcal{JP}_{k T}^{T}
\right)\\
& > & \sigma_{0}\left(\hbar\right).\end{eqnarray*}
since $\mathcal{P}_{t}\in\mathrm{Sp}\left(2d,\,\mathds{R}\right)$
is strictly positive definite. This corresponds to the case when $\Re\left(
\lambda_{i}\right)\neq0,$ hence $\max_{i}\left( s_{i}\right)>0$. The classical motion
is hence not linearly stable in this case.\vspace{\enumskip}
\end{itemize}
Furthermore, if the (purely complex) Floquet exponents are
rationally dependent, there exist some multiples
\begin{eqnarray*}
T_{R} & = & n_{R} T,\quad n_{R}\in\mathds{Z},
\end{eqnarray*}
of the classical period $T$ such that the orthogonal transformation
at times $T_{R}$ reduces to unity. Indeed, if $n_{R}$ is the smallest common multiple
of the denominators of the sequence $\left\{ \lambda_{i}\right\} $
defined by the Floquet exponents, i.e.,
\begin{eqnarray*}
\lambda_{i} n_{R} & \in & \mathds{N},\,\forall i,
\end{eqnarray*}
we find
\begin{eqnarray*}
M^{-1}e^{\Lambda T_{R}}M & = & M^{-1}\mathrm{diag}\left\{ e^{2\pi\mathrm{i}
n_{i}}\right\} M,\quad n_{i}\in\mathds{Z},\\
& = & \mathds{1}_{2d\times2d}.
\end{eqnarray*}
The semiclassical approximation then is the initial Gau{\ss}ian, if it
is localized at the initial point $X_{0}$. This is the case if the flow is
periodic. In this case we have a classical revival. \qed
\end{bew}
\begin{remark}
Note that these revivals are purely classical in the sense that
the conditions only reflect properties of the classical motion
and are hence independent of $\hbar$.
\end{remark}
\begin{remark}
If $S_{k T}$ is orthogonal, the approximate Gau{\ss}ian profile will
be the initial one at classical periods. It is just rotated and
translated in phase space.
\end{remark}
\subsection{\label{sec:A-Uniform-bound}A uniform bound}
We can further characterize the approximate state in the case when
the Floquet exponents are purely complex.
\begin{lemma}
\label{lem:Localization} \textbf{If} $H^{\prime\prime}\left(X_{t}\right)$
is $T-$periodic \textbf{and} the Floquet exponents of $S_{t}$
are purely complex \textbf{then} the width $\sigma_{t}\left(\hbar\right)$
of a Gau{\ss}ian state propagated by $\widehat{U}_{2}\left(t\right)$
satisfies the following uniform bound
\begin{eqnarray}
\sigma_{t}\left(\hbar\right) & \leq &
K\sigma_{0}\left(\hbar\right),\,\forall t\in\mathds{R}.
\label{eq:regularboundedwidth}
\end{eqnarray}
Furthermore, $K:=e^{\kappa}$, where $\kappa$ is defined by
\begin{eqnarray*}
\kappa & := & 2 T\sup_{t\in\left[0,\, T\right]}\left\Vert \mathcal{J}
H^{\prime\prime}\left(X_{t}\right)\right\Vert _{\mathrm{HS}}.\end{eqnarray*}
\end{lemma}
\begin{bew}{}
We consider eq. (\ref{eq:action}) \[
\left\{
\begin{array}{rll}
\frac{\mathrm{d}S_{t}}{\mathrm{d}t} & = &
\mathcal{J}H^{\prime\prime}\left(X_{t}\right)S_{t}\\
\\S_{0} & = & \mathds{1}.
\end{array}\right.\]
Starting at some initial time $k T,\, k\in\mathds{Z}$, $S_{t+k T}$
satisfies the Gr\"{o}nwall inequality
\begin{eqnarray*}
\left\Vert S_{t+k T}\right\Vert _{\mathrm{HS}} & \leq &
\left\Vert S_{k T}\right\Vert _{\mathrm{HS}} e^{\kappa},\,\forall t\in\left[0,\, T\right],
\end{eqnarray*}
during a classical period $T$ where
\begin{eqnarray*}
\kappa & = & 2 T\sup_{t\in\left[0,\, T\right]}\left\Vert
\mathcal{J}H^{\prime\prime}\left(X_{t}\right)\right\Vert _{\mathrm{HS}}.
\end{eqnarray*}
Furthermore,
\begin{eqnarray*}
\sigma_{t}\left(\hbar\right) & = & \frac{\hbar}{2}\mathrm{tr}
\left(S_{t}\mathcal{J}^{T}G_{0}\mathcal{J}
S_{t}^{T}\right)\\
& = & \frac{\hbar}{2}\mathrm{tr}\left(G_{0}
S_{t}^{T}S_{t}\right)\leq\frac{\hbar}{2}\left|
\mathrm{tr}\left(G_{0}\right)\right|\left\Vert
S_{t}\right\Vert _{\mathrm{HS}}^{2}.
\end{eqnarray*}
Hence
\begin{eqnarray}
\sigma_{t}\left(\hbar\right) & \leq & \frac{\hbar}{2}\left|
\mathrm{tr}\left(\mathbf{g}_{0}\right)\right| e^{\kappa},\label{eq:boundbetweenpewriods}
\end{eqnarray}
if $S_{t}$ has purely complex Floquet exponents since in
this case $\sigma_{0}\left(\hbar\right)=\sigma_{k T}\left(\hbar\right),$
according to proposition \ref{pro:periodicwidth}.
We conclude with this choice that
\begin{eqnarray*}
\sigma_{t}\left(\hbar\right) & \leq & K\sigma_{0}\left(\hbar\right)\end{eqnarray*}
uniformly in time. \qed
\end{bew}
If $S_{t}$ is not orthogonal at the classical periods $T$,
the latter bound takes the form
\begin{eqnarray*}
\sigma_{t}\left(\hbar\right) & \leq & K\sigma_{0}\left(\hbar\right)
e^{2\max_{i}\left(\Re\left(l_{i}\right)\right)\left|t\right|}
\end{eqnarray*}
where $l_{i}:=\frac{2\pi\lambda_{i}}{T}$ are the corresponding
Floquet exponents of $S_{t}$ which is nothing else than a
Lyapunov inequality where $\max_{i}\left(\Re\left(l_{i}\right)\right)$
plays the role of the Lyapunov exponent. Using the fact that
what we are approximating is the Hilbert-Schmidt norm of the flow
differential one can furthermore state the existence of a constant
$c_{4}>0$ such that
\begin{eqnarray*}
\frac{e^{2\max_{i}\left(\Re\left(l_{i}\right)\right)\left|t\right|}}{c_{4}} & \leq\sigma_{t}
\left(\hbar\right)\leq & c_{4} e^{2\max_{i}\left(\Re\left(l_{i}\right)\right)\left|t\right|}.
\end{eqnarray*}
which determines the asymptotic behavior of the width in that case.
\begin{remark}
We note that the equality in eq. (\ref{eq:regularboundedwidth}) is
reached if the classical period is null, i.e., if the Hamiltonian
has a constant Hessian $H^{\prime\prime}\left(X_{t}\right)$.
This is satisfied by the harmonic oscillator. The flow differential
$S_{t}$ is then an orthogonal matrix for all
times. It is well known that the propagation is dispersion-less in
this case.
\end{remark}
\section{\label{sec:Discussion}Discussion}
Theorem \ref{thm:propagation_control_CR} allows us to trace back
the properties of the approximation to the exact state
$\mathcal{U}\left(t\right)\psi_{Z_{0}}^{X_{0}}$. According to the discussion in
\cite{CR97} (see also section \ref{EhrT}), this can be done with an error
of order $\mathcal{O}\left(\hbar^{\alpha}\right),\,\alpha>0$ up to
times $\hbar^{-\frac{1}{2}}$ if the Floquet exponents have zero real
part (stable classical dynamics) and up to times $\frac{\left|\ln\left(
\hbar\right)\right|}{6 \nu},\,\nu=\max_{i}\Re\left(l_{i}\right)$
otherwise (unstable classical dynamics) as
$\hbar \searrow 0$ and $t \nearrow \infty$.
In the stable case, the approximate state remains localized ad infinitum
if this state remains localized between classical periods since
the width of such a state is the same at classical periods. We wish to
stress that we have defined the classical period as the minimal period
of the Hessian $H^{\prime\prime}\left(X_{t}\right)$ which is not necessarily
the period of the classical flow $\Phi_{H}^{t}$. An example is classical motion
in a one dimensional periodic potential $V_{\Gamma}$ with energies $E$ such that
\begin{eqnarray*}
E & > & \sup_{x\in\mathds{R}}
\left(V_{\Gamma}\left(x\right)\right).
\end{eqnarray*}
In such a case $H^{\prime\prime}\left( X_{t}\right)$ is periodic although the flow
$\Phi_{H}^{t}\left( X_{0}\right) $ is not. Furthermore, the approximation remains
localized between classical periods if
\begin{eqnarray*}
\kappa & = & 2T
\sup_{t\in\left[0,\,T\right]}\left\Vert \mathcal{J}H^{\prime\prime}\left(X_{t}\right)
\right\Vert_\mathrm{HS},
\end{eqnarray*}
the exponent in eq. (\ref{eq:boundbetweenpewriods}),
is small enough. We can hence state in that case, up to small errors
$\mathcal{O}\left(\hbar^{\alpha}\right)$, localization of the exact
propagation up to the Ehrenfest time $T_{E}\propto
\hbar^{-\frac{1}{2}}$. Note that the situation includes
unbounded motion.
If the flow differential $S_{t}$ is periodic, it is clear that
the shape will be the initial one at classical periods up to a small
error. The resonance condition allows to state that this will also
occur at some time if the flow differential is merely periodic of the
second type and if the Floquet exponents are rationally
dependent and purely imaginary. We may state this since the exact state remains
localized and that our asymptotic considerations are valid as $\hbar
\searrow 0$ and $t \nearrow \infty$. The property extends also to the
general stable case, i.e., in the case of rationally independent
Floquet exponents. Recall that the quadratic form $Z_{t}$ is given by
linear fractional transformation. In particular,
\begin{eqnarray*}
Z_{kT} & = & \left(A_{kT}Z_{0}+B_{kT}\right)\left(C_{kT}Z_{0}+D_{kT}\right)^{-1}
\end{eqnarray*}
where we have used the notation of section \ref{sec:A semiclassical propagation theorem}.
Since $S_{kT}\sim e^{\Lambda kT}$ which is unitary with
eigenvalues $e^{\mathrm{i}\ell_{i}\frac{2\pi}{T}}=e^{\lambda_{i}\frac{2\pi}{T}}$ where
$\ell_{i}\in\mathds{R}$, there exist $k-$independent matrices
$a_{i}\in\mathds{C}^{d\times d}$ such that $Z_{kT}$ can be viewed as the image of the vector
$\ell\in\mathds{T}^{d}$ with entries $\ell_{i}$ under the continuous map
\begin{eqnarray*}
\mathsf{e} & : & \mathds{T}^{d}\rightarrow\Sigma_{d}\\
& & \omega\mapsto\sum_{i=0}^{d}a_{i}e^{2\pi\mathrm{i}\omega_{i}}.
\end{eqnarray*}
Furthermore, the endomorphism
\begin{eqnarray*}
\tau & : & \mathds{T}^{d}\rightarrow\mathds{T}^{d}\\
& & L \mapsto L+\ell,
\end{eqnarray*}
where $\left( L_{i}+\ell_{i}\right)$ is defined modulo 1, is known to be ergodic with respect to Lebesgue measure on the $d-$torus since the frequencies $\ell_{i}$
are rationally independent. In this notation we have $Z_{0}=\mathsf{e}\left(0\right)=
\mathsf{e}\left(L\left( 0\right) \right)$.
From the ergodicity of $\tau$ on $\mathds{T}^{d}$ we may state that
for every $\varepsilon>0$ there exist $K\in\mathds{Z}$ such
that $\left\Vert \tau^{K}\left(L\left( 0\right) \right)-L\left( 0\right) \right\Vert <\varepsilon$.
It follows that for every $\epsilon>0$ there exist
some $n\in\mathds{Z}$ such that $\left\Vert \mathsf{e}\circ\tau^{n}\left(0\right)-\mathsf{e}\left(0\right)\right\Vert <\epsilon$ implying
\begin{eqnarray*}
\forall \epsilon > 0 , \, \exists n\in\mathds{Z},\quad\left\Vert Z_{n T}-Z_{0}\right\Vert < \epsilon,
\end{eqnarray*}
i.e., $Z_{kT}$ is quasi-periodic.
One concludes that the initial shape of the approximated state will up to some small
error reoccur at some multiple of the classical period if the Floquet exponents of the
flow differential are purely imaginary. We have again no {\`a} priori reasons to exclude unbounded motion.
In the case of a periodic flow, we have localization at the initial point and the
initial profile (up to some small error) at classical periods. For
bounded classical motion, those recurrences define revivals at periods
of the classical flow that have already been described \cite{APe89}
in the past.
We summarize our findings in the following theorem.
\begin{theorem}
Under the following assumptions
\begin{enumerate}
\item $H\left(X\right)$ satisfies the conditions of section \ref{conditions}
\vspace{\enumskip}
\item $H^{\prime\prime}\left(X_{t}\right)$ is $T-$periodic and
$\nu$ is the maximal real part of the Floquet exponents of the flow
differential, solution of eq. (\ref{eq:action}),\vspace{\enumskip}
\vspace{\enumskip}
\end{enumerate}
and with $\kappa:=2 T\sup_{t\in\left[0,\, T\right]}\left\Vert \mathcal{J}H^{\prime\prime}\left(X_{t}\right)\right\Vert _{\mathrm{HS}}$ where $K:=e^{\kappa},$
\vspace{\enumskip}
\noindent we can make the following statements up to $\mathcal{O}\left(\hbar^{\alpha}\right),\,\alpha>0$
as $\hbar\searrow0$.
\textbf{If} $K$ is small enough \textbf{then} the approximation described by theorem \ref{thm:propagation_control_CR}
will hold up to times $\frac{\left|\ln\left(\hbar\right)\right|}{6\nu}$
and the approximate width of the state will behave like $e^{2\nu\left|t\right|}$.
\textbf{In particular,} \textbf{if} the Floquet exponents are purely
complex or zero \textbf{then} the semiclassical propagation described
in theorem \ref{thm:propagation_control_CR} will hold to times $\hbar^{-\frac{1}{2}}$.
The width of the approximated state will be bounded and recurrences will, up to small errors,
take place at a multiple of $T$.
\end{theorem}
\begin{acknowledgement}
We would particularly like to thank Jens Bolte for valuable comments
and discussions.
\end{acknowledgement}
\bibliographystyle{unsrt}
|
0708.2747
|
\section{Introduction and Summary}
The proposed duality between Yang-Mills theories and string theories
has been a fascinating avenue for explorations since Maldacena's '97
conjecture\cite{malda}. The duality relating the weak-coupling
regime of a ten dimensional string theory to the strongly coupled
sector of a four dimensional super Yang-Mills (SYM) theory has been
notoriously hard to test. However, some fascinating new discoveries
have led to a remarkable amount of progress on the problem. The
basic insight that has allowed on to make some headway into the
problem is that both the ten dimensional string theory and the four
dimensional gauge theory appear to be two dimensional integrable
models in disguise! That there is an underlying two dimensional
dynamical system underlying the string theory is perhaps no so
surprising, as one is naturally led to a world sheet sigma model in
the analysis of the string dynamics. What is rather counterintuitive
is that for the purposes of computing it's spectrum, the four
dimensional gauge theory can be regarded as a two dimensional
integrable system as well. The closest thing that one has to a
spectrum of masses for this four dimensional conformal field theory
is the spectrum of anomalous dimensions i.e the spectrum of the
dilatation operator. In a remarkable paper, Minahan and Zarembo
\cite{minz} were able to show that the large $N$ dilatation operator
of the gauge theory, when restricted to the sector of operators
built out of scalars can be regarded as the Hamiltonian of an
integrable quantum spin chain. Subsequent analyses have shown that
this is a feature that is restricted neither to the one loop level
nor to operators built out of scalars\cite{bs1,bs2,bs3}.
Since, the original results of Minahan and Zarembo the understanding
of integrability in the gauge theory has only deepened. Due to
extremely impressive gauge theory computations\cite{bern1}, the spin
chain describing the dilatation operator of the Yang-Mills theory is
now known up to four loop orders. To the extent that we have a
formula for the spin chain, there is a great deal of evidence that
it is integrable\cite{bs4}.
Given a quantum spin chain, the surest test of its integrability can
probably be formulated in terms of algebraically independent
commuting higher charges and/or the existence of underlying quantum group
symmetries which can ultimately be understood in terms of an
appropriate transfer matrix. Such a detailed microscopic
understanding of the integrability of the quantum spin chains
emerging from the Yang-Mills theory exist at the first few orders in
perturbation theory. This has to do with the fact the we know the
Hamiltonian of the quantum spin chain relevant to $\mathcal{N}$ =4
SYM only to the first few orders in perturbation theory. However, a
lot of progress is possible even without the explicit knowledge of
the Hamiltonian. Beisert has recently shown that the fact that the
underlying gauge theory is maximally supersymmetric, i.e possess a
$psu(2,2|4)$ invariance, can be used to almost uniquely fix the two
particle scattering matrix of its dilatation operator, even without
the knowledge of the explicit form of the appropriate spin chain
Hamiltonian. It turned out that by considering the superconformal
group as a particular contraction of one of its non-central
extensions, the dispersion relation of the underlying spin chain
could be uniquely fixed. Furthermore, the two particle $S$ matrix is
determined up to a single undetermined function/phase, when one uses
this approach\cite{bss1,bss2}. He was also able to show that this
particular two body $S$ matrix also satisfies the Yang-Baxter
algebra: a necessary condition for integrability. This result brings
one very close to an exact solution of the large $N$ limit of the
gauge theory, as far as its spectrum is concerned. Indeed, Beisert's
result on the $S$ matrix and the undetermined phase, have been
largely constrained now and some extrapolations of the results to
the strong coupling regime have also been successfully carried
out\cite{bs4,kleb1}. These exciting chain of developments allow one
to make some very concrete statements about the validity of the
AdS/CFT correspondence.
However, various open questions still remain. The fact that the $S$
matrix for the spin chain underlying the gauge theory satisfies the
factorization conditions, implies that the underlying Hamiltonian
has a very good chance of being integrable. If that is indeed the
case, then it should be, at least in principle, possible to
understand its integrability in terms of symmetries. While that is
not possible yet in complete generality, important steps in that
direction have already been taken. In the first part of the review,
we shall briefly sketch out the nature of some of the non-trivial
conservation laws and symmetry principles that can help us
understand the perturbative integrability of the gauge theory.
Much of the success regarding the understanding of integrability of
$\mathcal{N}$=4 SYM, alluded to above, has had to do with studies of
gauge theory operators \begin{equation} \mathcal{O} =
\mbox{Tr}(\mathcal{W}_1\cdots \mathcal{W}_m).\end{equation} i.e. operators
constructed out of traces. $\mathcal{W}_i$s stand for generic
Yang-Mills fields. However, the gauge theory also possess many other
degrees of freedom, such as operators built out of multiple traces,
determinants and sub-determinants etc. Such degrees of freedom allow
us to probe various corners of the AdS/CFT correspondence, which are
not necessarily accessible by straightforward applications of the
techniques developed in the studies of single trace operators. Gauge
theory operators which are dual to D branes, for instance, cannot be
built from traces. The second part of the review will be devoted to
a brief overview of some of the developments in our understanding of
Baryonic/D-Brane degrees of freedom on the gauge theory side and the
relevant integrable systems.
This brief overview is by no means meant to be a complete synopsis
of the rapidly growing body of exiting work relating supersymmetric
Yang-Mills theories, two dimensional integrable systems and string
theory. For instance, most of the recent developments regarding the
understanding of the world-sheet $S$ matrices of the gauge and string
theories will not be covered in the article. However, given our
knowledge of world-sheet scattering in the gauge theory, the
spectrum of single trace operators to all orders in perturbation
theory, etc, it might be argued that the time is ripe for a more
detailed understanding of integrability in the gauge theory. The
review focusses on some particular aspects of integrability in
$\mathcal{N}$ =4 SYM about which our understanding remains
incomplete. In particular we will focus on the role of Yangian
symmetries in the gauge theory and the gauge theoretic description
of D-Brane degrees of freedom. The review is based on a seminar
given at the Institute for Advanced Studies, Princeton.
\section{Gauge Theory Set Up:}
$\mathcal{N}$ =4 supersymmetric Yang-Mills theory in four dimensions is a conformal field theory with a
vanishing beta function. The closest thing to the spectrum of masses for this truly massless theory is
its spectrum of anomalous dimensions. For the purposes of computing the
spectrum it is extremely convenient to regard the local composite
operators, formed out of considering fundamental constituent
Yang-Mills degrees of freedom at the same spacetime point inside a
color trace, as the states of an appropriate quantum spin chain.
For instance, if we were to restrict ourselves to the closed $su(2)$
sub-sector of the gauge theory made up of two complex scalars $Z,W$,
which are charged under different $U(1)$ subgroups of the $so(6)$ R
symmetry group, one gets $su(2)$ spin chains with spins in the
fundamental representation. \begin{equation} Tr(ZZWWZ) \Leftrightarrow
(\Uparrow\Uparrow\Downarrow\Downarrow\Uparrow).\end{equation}
As one turns on the 't Hooft coupling of the gauge theory, $\lambda
$, the effects of various interaction vertices, will lead to the
this operator mixing with various other operators that carry the
same quantum numbers. However, in the large $N$ limit, since,
spitting of single traces into multiple traces is suppressed, all
that the gauge theory vertex insertions can bring about is a
rearrangement of the fields inside the trace. One can thus think of
the effect of the vertex insertion i.e the action of the dilatation
operator $D$, as the action of an appropriate spin chain
Hamiltonian on the state of interest. In the $su(2)$ sector of the
gauge theory, the dilatation operator takes on the following form up
to the three loop level\cite{ser1,ser2}. \begin{eqnarray} D =
\sum_i((1-P_{i.i+1}) + \lambda
((1-P_{i.i+2})-4(1-P_{i.i+1}))\hspace{2cm}\nonumber\\
+ \lambda ^2(28(I-P_{i,i+1}) -8(I-P_{i,i+2})
-2(P_{i,i+3}P_{i+1,i+2}-P_{i,i+2}P_{i+1,i+3}))+
\cdots)\label{dilop}\end{eqnarray} $P_{i,j}$ is the permutation operator that
exchanges the spins at the sites $i$ and $j$. The one loop $su(2)$
dilatation operator (the leading therm in the above formula), is of
course nothing but the celebrated ferromagnetic Heisenberg spin
chain, a well known integrable many-body system, providing us with a
first contact between integrable structures and the gauge theory.
Interpreting the gauge theory dilatation operator as a quantum spin
chain greatly facilitates understanding many aspects of the AdS/CFT
correspondence. Most importantly, it is possible to apply Bethe
ansatz techniques to compute the gauge theory spectrum by purely
algebraic means. As mentioned earlier, much progress has been
achieved understanding the higher loop spectrum of the gauge theory
in various exciting recent papers, which we shall not review in the
present work. We shall instead now focus on the present
understanding of the symmetries and conservation laws responsible
for integrability on the gauge theory side.
\section{Symmetries and Conservation Laws:} To motivate the
non-trivial conservation laws leading to our present understanding
of the perturbative integrability of the gauge theory, it is
instructive to look at the one loop $su(2)$ dilatation operator, i.e
the Heisenberg Hamiltonian, which we re-write as, \begin{equation} D_1 = \sum_i
(1- S^a_b(i)S^b_a(i+1)).\end{equation} $S^a_b(i)$ is the Weyl operator that
acts as $|a><b|$ at the lattice site $i$. The Hamiltonian has an
obvious $su(2)$ symmetry, generated by \begin{equation} (\mathcal{Q}^0)^a_b =
\sum_i S^a_b(i).\end{equation} However, this does not, by itself, generate
enough conservation laws to render the system solvable. However
there exists another non-local charge that does commute with the
Hamiltonian. \begin{equation} (\mathcal{Q}^1)^a_b =
\frac{1}{2}\sum_{i<j}\theta(i,j)\left(S^a_k(i)S^k_b(j)-
(i\leftrightarrow j)\right).\end{equation} For the Heisenberg model, $\theta
(i,j) = 1 \forall i,j$, however for more complicated spin chains
that we shall consider, $\theta $ can have a more complicated
functional dependence. These two charges, known as the Yangian
charges, are ultimately responsible for the integrability of the
system. It turns out, that by forming repeated commutators, of the
charges with each other, one can generate an infinite tower of
algebraically independent charges, all of which commute with the
Heisenberg Hamiltonian. The symmetry underlying the dynamical system
is not a Lie algebraic one, as the charges do not close on each
other under commutation. As a matter of fact they form a Hopf
algebra with co-products given by
\begin{eqnarray}\Delta(\mathcal{Q}^0)^a_b = (\mathcal{Q}^0)^a_b
\otimes \mathcal{I} + \mathcal{I}\otimes (\mathcal{Q}^0)^a_b\\
\Delta(\mathcal{Q}^1)^a_b = (\mathcal{Q}^1)^a_b \otimes
\mathcal{I}+ \mathcal{I}\otimes (\mathcal{Q}^0)^a_b
\nonumber \\
+( (\mathcal{Q}^0)^a_d\otimes (\mathcal{Q}^0)^d_b
-(\mathcal{Q}^0)^d_b\otimes (\mathcal{Q}^0)^a_d)\nonumber
\end{eqnarray}
The co-product must be an algebra homomorphism. Physically, this may
be understood as the requirement that the algebra not change, if one
adds more spins to the spin chain state. The requirement restricts
the function $\theta $ which is required to satisfy the following
constraint, also known as the Serre relation \begin{equation}
\theta(j,k)\theta(j,n) + \theta(j,k)\theta(n,k) -
\theta(j,n)\theta(n,k) = \theta(j,k),\end{equation} which is trivially
satisfied if $\theta =1$
As far as the Heisenberg model is concerned, all the algebraically
independent charges obtained from the iterated commutators of the
two Yangian charges can be captured in a ($2\times 2$) transfer
matrix, which can be expressed as a path ordered exponential of the
Weyl matrix. \begin{equation} T^a_b(u) =
\left[e^{\wp\frac{1}{u}\sum_iS(i)}\right]^a_b = \sum_i
\frac{1}{u^n}(T_n)^a_b\end{equation} $u$ is the spectral parameter, and the
model expansion around $u\rightarrow \infty $ reads as \begin{equation}
T(u)^a_b = \mathcal{I} + \frac{1}{u}(\mathcal{Q}^0)^a_b +
\frac{1}{u^2}(\mathcal{Q}^1)^a_b + \cdots\end{equation} The commutation
relations between the various modes of the transfer matrix give us
the famous Yang-Baxter algebra \begin{equation} [T^{ab}_s, T^{cd}_{p+1}] -
[T^{ab}_{p+1}, T^{cd}_{s}] =\left( T^{cb}_{p} T^{ad}_{s} -
T^{cb}_{s} T^{ad}_{p}\right)\end{equation} while the co-product is reflected
as \begin{equation} \Delta T^{ab} = \sum_d T^{ad}T^{db}\end{equation} The Heisenberg
Hamiltonian is an element of the center of the Yang-Baxter algebra
and it commutes with $T$ i.e \begin{equation}[D_1, T^{ab}_n]=0.\end{equation}
\footnote{The above statement is strictly true if the spin chains
are infinitely long. For chains of finite length, one has to
consider an appropriate quotient of the algebra of the spin-chain observables by
a proper ideal to realize the non-local conservation laws. See for
example \cite{aa1}}Perhaps a more familiar way to write the
Yang-Baxter algebra is by using the $R$ matrix \begin{equation}
R(u-v)(T(u)\otimes \mathcal{I})(\mathcal{I}\otimes T(v))=
(\mathcal{I}\otimes T(v))(T(u)\otimes \mathcal{I})R(u-v) \end{equation} where,
\begin{equation} R = u\mathcal{I} - P.\end{equation} The key point here is that the
transfer matrix and the Yangian charges carry precisely the same
information. As a matter of fact, the Yang-Baxter relations for the
matrix elements of the transfer matrix are nothing but the Serre
relations in disguise. A detailed algebraic construction relating
all the matrix elements of transfer matrix of the Heisenberg model
to the Yangian charges, along with a great deal of pedagogical
exposition, may be found for example in\cite{ber,ge}. Given a
transfer matrix satisfying the Yang-Baxter relations, a purely
algebraic way of diagonalizing the entire transfer matrix also
exists due to the work of Faddeev and collaborators \cite{fadrev}.
Thus, to sum up the lessons from the Heisenberg model, if one is
given a quantum spin chain, a fairly conclusive way to establish its
integrability consists of showing the existence of Yangian charges
that commute with it.
That there might be an underlying Yangian symmetry in the gauge
thoery was first suggested in \cite{witten}. We can now focus on
how the above understanding of Yangian symmetries applies to the
gauge theory. If one were to focus on the $su(2)$ sector, then one
sees that, there are corrections to the Heisenberg Hamiltonian, with
the range of the spin chain increasing with the loop order. The
three loop Hamiltonian, for instance is given by ({\ref{dilop}).
Given the knowledge of the Dilatation generator up to a given loop
order, (say $\mathcal{O}(\lambda ^m)$), $D_m$, one would like to
find $\lambda $ dependent deformations of the Yangian charges, that
commute with the Hamiltonian up to terms of $\mathcal{O}(\lambda
^{m+1})$. In the $su(2)$ sector, we are aided by the realization
that the fist Yangian charge, i.e the $su(2)$ generator remains an
exact symmetry of the dilatation operator to any order, since the
$su(2)$ 'flavor' symmetry is manifestly realized in perturbation
theory. To find the second charge, we can use the fact that the
Serre relations have non-trivial solutions. For example, \begin{equation} \theta
(i,j) = \frac{t^{-i}}{t^{-i} - t^{-j}} \end{equation} solves the Serre
relations, for arbitrary, complex values of 't'. In particular
choosing \begin{equation} t = \sum_{n=1}^\infty c_n \lambda ^n, c_1=1, c_2=-3,
c_3=14 \end{equation} establishes the three loop Yangian invariance in the
$su(2)$ sector of the gauge theory. In explicit form, the second
Yangian charge looks like
\begin{eqnarray}
(\mathcal{Q}^1)^a_b &=& \sum_{i<j}S^a_d(i)S^d_b(j)
+\lambda \sum_i S^a_d(i)S^d_b(i+1)\nonumber \\
& &-\lambda ^2 \sum_i\left(2S^a_d(i)S^d_b(i+1) -
S^a_d(i)S^d_b(i+2)\right)+\lambda ^3\sum_i\left(9S^a_d(i)S^d_b(i+1)
+
S^a_d(i)S^d_b(i+3)\right)\nonumber \\
& &-((a,d)\leftrightarrow (d,b))\nonumber \end{eqnarray} This
construction establishes three loop integrability in the $su(2)$
sector of the gauge theory. These charges were computed order by
order in perturbation theory in \cite{aa1} For this particular
sector, one can also appeal to other constructions. For instance, it
was shown in\cite{ser1} that the three loop $su(2)$ dilatation
operator of the gauge theory can be embedded in the Inozemtsev spin
chain, which does have a Yangian symmetry. The Yangian charges,
reported above, can also be derived, by considering the appropriate
limits of the Yangian charges of the Inozemtsev model.
In proceeding beyond the $su(2)$ sector, one has to take into
account that even the flavor symmetry, (usually generated by the
first Yangian charge), is not always manifest. Nevertheless, in such
sectors of the gauge theory, constructions similar to the above
discussion are indeed possible, at least to the first two orders in
perturbation theory. For instance, for the $su(1|1)$ and $su(2|1)$
sectors, some recent results have been possible in \cite{aa2,bz}.
Finally, it is worth mentioning, that, as far the $S$ matrix of the
gauge spin chain is concerned, it is basically known to all loops in
perturbation theory, up to a single overall phase. This $S$ matrix
has recently been shown to possess Yangian invariance by Beisert
\cite{byan}. For related work, see also\cite{toryan}. One would of
course expect that the Yangian symmetry of the $S$ matrix would
intimately be related to that of the Hamiltonian discussed above. It
remains a fascinating subject of research to explain and tie
together all these glimpses of an underlying Yangian structure
obtained in the gauge theory so far.
\section{Protected Operators and Calogero Models:}
A different class of integrable dynamical systems emerge as the
effective Hamlitonians of the gauge theory when one focusses on
special operators that are protected against renormalization. One of
the best studied example is that of the dynamics of $\frac{1}{2}$
BPS, operators of the gauge theory, which are scalars $Z$ charged
under a $U(1)$ subgroup of the $R$ symmetry group. If one considered
the theory on $R\times S^3$, then the non-renormalization condition
reduces the dynamics of the this sector of the theory to that of the
zero models of these scalars. \begin{equation} Tr(Z)^J \Leftrightarrow
Tr(A^{\dagger})^J|0>, H_{SYM} \Leftrightarrow Tr(A^\dagger A)\end{equation} In
other words, the operators, distinguished by theory total $R$ charge
can be mapped to states of Hamiltonian matrix model, and the gauge
theory effective Hamiltonian takes on the form of a gauged matrix
Harmonic oscillator. The Hamiltonian, has a residual $U(N)$ gauge
invariance. One can remove that by going to the space of the
eigenvalues of the matrix model. As is well known from the study of
matrix models, this turns the system into a system of $N$
non-relativistic Fermions, which are otherwise free, and described
by the Hamiltonian \begin{equation} H = \sum_i\left(-\frac{\partial ^2}{\partial
x_i^2} + x_i^2\right)\end{equation} Recently, in a remarkable paper\cite{llm},
it was shown that the phase space of the free Fermion system, is
precisely the same as the space of all supergravity geometries that
are also half BPS, preserve an $O(4)\times O(4)$ isometry and are
asymptotically AdS.
Inspired by this result an attempt was made in\cite{aaa} to
understand the gauge fixed dynamics of a more general class of
operators within the gauge theory, which are protected, but not
necessarily because of supersymmetry reasons. The motivation being
that these will then be the prototypical operators that have a
chance of existing in theories without as much supersymmetry as
$\mathcal{N}$ =4 SYM.
One can focus on gauge theory operators built out of two Yang-Mills
fields, which we take to be a complex scalar $X$ and a Fermion $\Psi
^1$, which together make up the closed $su(1|1)$ subsector of the
gauge theory. Restricting to operators that are protected implies
that the effective Hamiltonian will be a sum of two Harmonic
oscillators \begin{equation} H = Tr(A^\dagger A + B^{\dagger} B), A,B
\leftrightarrow Z,\Psi^{1} \end{equation} To gauge fix it, we can go to a
basis such that, \begin{equation} A = \frac{1}{2}(X + iP) , X = U^\dagger x U, x
= diag(x_1 \cdots x_N)\end{equation} and we can denote the impurity of
Fermionic fields in this basis by \begin{equation} (b )^i_j = (UB U^\dagger
)^i_j \end{equation} The Hamiltonian in this basis takes on the form \begin{equation} H =
\sum_i\left(-\frac{\partial }{\partial x_i^2} + x_i^2\right) +
\sum_{i\neq j}\left( \frac{\mathcal{L}^i_j\mathcal{L}^j_i}{(x_i -
x_j)^2}\right) + tr b^{\dagger}b \end{equation} where, \begin{equation} \mathcal{L}^i_j =
U^i_m\frac{\partial }{\partial U^m_j}, [\mathcal{L}^i_j,
\mathcal{L}^k_l] = \delta ^k_j \mathcal{L}^i_l -\delta ^i_l
\mathcal{L}^k_j.\end{equation} $\mathcal{L}^i_j$ are generators of the $U(N)$
rotations. Because, we have a system of two matrices, gauge fixing,
does not completely remove the $U(N)$ degree of freedom,
nevertheless i writing the Hamiltonian in the above form, we have
been able to separate out the 'radial' $x_i$ and the angular
$\mathcal{L}^i_j$ degrees of freedom. The Hamiltonian describes a
system of particles on the line, that carry an internal spin $U(N)$
degree of freedom, and interact with each other by exchanging spins
on top of having an inverse square interaction. In other words it is
nothing but the $SU(N)$ generalization of the celebrated Calogero
model. The states are functions of the kind \begin{equation}
\Psi^{i_1 \cdots i_n}_{j_i \cdots j_n}(x)\Pi _{k= 1}^n (b^{\dagger
}) ^{j_k}_{i_k} |0> \end{equation} , where, $|0>$ is the vacuum with no
Fermions. It is important to note that because of the global $U(N)$
invariance of the problem, the $U(N)$ generators may be expressed in
terms of the $b$ fields as \begin{equation} \mathcal{L}^i_j =
\sum _\beta\left( (b^{\dagger})^i_l(b^{ })^l_j -
(b^{\dagger })^l_j(b^{})^i_l\right) .\end{equation} Not all states of the
Matrix model are protected gauge theory operators, however one could
focus on states such as
\begin{equation}\frac{1}{\sqrt{N^{n_1+1}}}tr\left((A^{\dagger })^{n_1}B^{\dagger
}\right) \frac{1}{\sqrt{N^{n_2}}}tr \left((A^\dagger )^{n_2}\right)
\cdots \frac{1}{\sqrt{N^{n_i+1}}}tr\left((A^{\dagger
})^{n_i}B^{\dagger }\right)|0>\end{equation} These states have the curious
property, that they are protected in the Large $N$ limit, while for
finite $N$ they correspond to like BMN Like Near Chiral Primariry
operators. Thus in the large $N$ limit, they are protected, without
being protected due to supersymmetry. In the gauge fixed language
they take on the form \begin{equation} \prod_m\Psi(x_1 \cdots x_N)^{i_1 \cdots
i_m}(b^{\dagger})_{i_1}\cdots (b^{\dagger })_{i_m}|0> +
O(\frac{1}{N}), (b^{\dagger \alpha })_i = (b^{\dagger \alpha
})^i_i.\end{equation} The spin degrees of freedom, Of which there were
approximately $N$, in the Calogero model, now reduce to just two.
These correspond to whether or not one does or does not have a
fermionic excitation within a trace. A detailed derivation of how
this comes about is given in\cite{aaa}, however, the key result is
that on these states of interest, the spin-spin interaction term can
be expressed as \begin{equation} \mathcal{L}^i_j\mathcal{L}^j_i= \frac{1}{2}(1
- \Pi_{i,j}),\end{equation} where, $\Pi$ is a (graded) permutation operator
for the spin degrees of freedom. The Hamiltonian, in turn, can be
written as \begin{equation} D = \sum_i\left(-\frac{1}{2}\frac{\partial
}{\partial x_i^2} + b^{\dagger i}b_i + \frac{1}{2}x_i^2\right) +
\frac{1}{2}\sum_{i\neq j}\left( \frac{1-\Pi_{i,j}}{(x_i -
x_j)^2}\right).\label{cal}\end{equation} i.e the Hamiltonian of the rational
Super-Calogero Model.
The Free fermion system corresponding to half BPS operators is a
special case of the Calogero system, it simply corresponds to states
with no impurity, $'b'$, type excitations. A detailed description of the
spectrum and degeneracies of the Calogero model in terms of Young
diagrams has also been carried out in\cite{aaa}.
Since, the Calogero model, which is an integrable system, gives us a
window into the strong coupling dynamics of the gauge theory, it is
instructive to look at the algebraic structure underlying its
integrability and compare it with the ones known to be present in
perturbative, weakly coupled, SYM analysis.
The Lax operator for the Calogero system given in({\ref{cal}) can be
expressed as\cite{ak}
\begin{equation} L_{j,k} = \delta _{j,k}\frac{\partial }{\partial x_j} + \hbar
(1-\delta _{j,k}) \theta(j,k)\Pi_{j,k}\end{equation} and \begin{equation} \theta(j,k) =
\frac{e^{-\frac{\hbar}{2}(x_i - x_j)}}{\sinh \frac{\hbar}{2}(x_i -
x_j)}\end{equation} This form of the Lax operator is more general than what is
needed, as a matter of fact, only the $\hbar \rightarrow 0$ limit
corresponds to the rational Calogero model given above. However, it
is instructive to keep this general form, with the fictitious
parameter $\hbar $ arbitrary. The non-local conserved charges for
the model can be constructed as \begin{equation} T_{n}^{ab} = \sum _{j,k}
S^{ab}(j)(L^n)_{j,k}\end{equation} and explicit computations show that
$T^{ii}_n$ are conserved. The Hamiltonian that commutes with the
non-local charges is expressible as \begin{equation} H_\hbar =
\frac{1}{2}\sum_{j,k}(-\partial ^2_j + x_j^2 + b^\dagger (j)b(j) +
\hbar \Pi _{j,k}\partial _j \theta(j,k) + \hbar
^2\theta_{j,k}\theta_{k,j}).\end{equation} The algebra of charges can be
computed to be \begin{eqnarray}[T^{ab}_s, T^{cd}_{p+1}]_\pm -
[T^{ab}_{p+1}, T^{cd}_{s}]_\pm = \hbar (-1)^{\epsilon (c) \epsilon
(a) + \epsilon (c) \epsilon (b) + \epsilon (b) \epsilon (a)}\left(
T^{cb}_{p} T^{ad}_{s} - T^{cb}_{s}
T^{ad}_{p}\right)\nonumber\end{eqnarray} $\epsilon (0) = 0,
\epsilon(1) = 1$. This is nothing but the supersymmetric Yangian
algeba encountered previously in the discussion of the spin chains.
It is important to note that the non-linearity of the Yang-Baxter
algebra is proportional to $\hbar$, and that in the $\hbar
\rightarrow 0$ limit, which is what is relevant for the Yang-Mills
theory, the Yangian algera degenerates into a loop algebra, \begin{equation}
[T^{ab}_s, T^{cd}_{p}]_\pm = \delta_{b,c} T^{ad}_{p+s} -
(-1)^{(\epsilon (a) + \epsilon (b))(\epsilon (c) + \epsilon
(d)}\delta_{a,d} T^{cb}_{p+s}.\end{equation} It has been known for sometime,
that loop algebras can be regarded as contractions or classical
limits of Yangian algebras\cite{ber}. Given the appearance of the
Yangian symmetry in perturbative gauge theory analyses, and that of
its classical limit in the small window into the strong coupling
sector provided by the Calogero model, it seems suggestive that a
contraction of the symmetry algebra takes place as one moves frmo
the weak to the strongly coupled regime of the gauge theory. It
would be extremely exciting if this possibility be explored further.
It may also be noted that for the rational super Calogero model, the
supercharges, and all the mutually commuting Hamiltonians can be
embedded in the loop algebra. \begin{equation} T^{21}_1 = Q, T^{12}_1 =
Q^\dagger, H = [ T^{21}_1, T^{12}_1]_+\end{equation}. The higher conserved
Hamiltonians are given as \begin{equation} H_{n+m} = [ T^{12}_n, T^{21}_m]_+\end{equation}
while \begin{equation} [T^{11}_m, T^{11}_n] = [T^{11}_m, T^{22}_n] = [T^{22}_m,
T^{22}_n] = 0 \forall m,n\end{equation} The discussion here is only a brief
summary of some of the connections between in the Yang-Mills theory
and Calogero systems presented in\cite{aaa}. For example it is
possible to extend the connection to the closed $su(2|3)$ sector of
the gauge theory which involves more than just two SYM fields. We
shall refer to\cite{aaa} for further reading.
\section{Open Spin Chains and Branes in SYM}
Apart from operators formed out of traces and products of traces of
Yang-Mills fields the gauge theory also has Baryonic operators whose
bare engineering dimensions are of $O(N)$. The simplest such
operator is \begin{equation} \mathcal{O} = \epsilon _{i_1 \cdots i_N}\epsilon
^{j_1 \cdots j_N}(Z^{i_1}_{j_1} \cdots Z^{i_N}_{J_N}).\end{equation} This
operator, being made up of $Z$ fields only is half BPS and
protected. From the point of view of the AdS/CFT correspondence,
such operators are important to study as they provide us with the
gauge theory analogs of D-Brane excitations. The above operator, for
example, is the SYM dual of a D3 brane. One can also construct the
dual of an open string ending ending on the giant graviton, simply
by removing one of the $Z$ fields from within the trace, and
replacing it by a matrix product of local Yang-Mills fields. \begin{equation}
\mathcal{O} \rightarrow \epsilon _{i_1 \cdots i_{N-1}
i_N}\epsilon^{j_1 \cdots j_{N-1}j_N}((Z^{i_1}_{j_1} \cdots
Z^{i_{N-1}}_{J_{N-1}})(WZZWZWW\cdots W)^{i_N}_{j_N}.\end{equation} One may ask
whether or not it makes sense to study the dynamics of these non-BPS
excitations as those of open quantum spin chains, with the string of
of Yang-Mills fields that replaces the $Z$ field playing the role of
an open quantum spin chain. These being operators of $O(N)$, and
keeping in mind that in the study of closed chains the spin chain
only emerged in the large $N$ limit, one has to carefully
investigate whether a sensible large $N$ limit leading to a spin
chain description of these operators is possible. That it is indeed
possible at the one loop level was shown by Berenstein and
collaborators in \cite{dber1}. In the $su(2)$ sector, the relevant
spin chain is given by \begin{equation}
D_1 = \sum_{l=1}^{L-1}(\lambda )(I - P_{l,l+1}) + q_1^Z + q_1^L.\end{equation}
In the above formula, we have identifies all the $L$ fields lying
between the two boundary $W$ fields to be the spin chain. The 'bulk'
Hamiltonian of the spin chain is the same as the one for closed spin
chains, but the interactions between the spins in the chain with the
ones in the determinant introduce some boundary terms, denoted by
the $q$'s.$q^Z_i$ is a projection operator, that checks is the spin
at the $i$Ith site is equal to $Z$. If it is not, then it
annihilates the site, otherwise, it just acts as the identity. Put
differently, the present of the brane imposes Dirichlet boundary
conditions which dictate that the first and the last fields in the
chain cannot be $Z$.
This spin chain is integrable, and it can be solved by Bethe ansatz
techniques. Its ground state is given by an the state \begin{equation} |0> =
\epsilon_{i_i \cdots i_{N-1}i_N}\epsilon^{j_i \cdots
j_{N-1}j_N}(Z^{i_1}_{j_1}\cdots Z^{i_{N-1}}_{j_{N-1}})(WWWW\cdots
WWW)^{i_N}_{j_{N}}.\end{equation} An eigenstate with two magnons can be
constructed as \begin{equation}|\Psi_2> = \sum_{x<y}\Psi(x,y)|x,y>\end{equation} with \begin{equation}
\Psi (x,y) = \sum _p \sigma (p)A(k_i,k_2)e^{i(k_1x_1 +k_2x_2)}.\end{equation}
In the above equation $x,y$ denote the positions of the flipped
spins i.e the $Z$ fields. The sum extends over all negations of the
momenta, and $\sigma $ in the sum indicates that we add a negative
sign each time the momenta are negated or permuted. i.e we make a
superposition of the incoming and outgoing plane waves. Unlike the
case of closed spin chains one has to account for the effect of
scattering from the boundary in addition to the mutual scattering of
the magnons. The Bethe equations, derived by following the magnons
around the spin chain and requiring that nothing change in doing so,
determine the momenta by the equations\begin{equation} \frac{\alpha (k_i)\beta
(k_i)}{\alpha (-k_i)\beta (-k_i)} = \prod_{j\neq
i}\frac{S(-k_i,k_j)}{S(k_i,k_j)},\end{equation} where $\alpha$ and $\beta $
encode the information about scattering from the boundary and $S$ is
the two magnon 'bulk' scattering matrix, given by:
\begin{eqnarray} \alpha (-k) = 1,
\beta (k) &=& e^{i(L+1)k},\end{eqnarray} \begin{equation} S(k_1,k_2) =
1-2e^{ik_2} + e^{i(k_1+k_2)}.\end{equation} The energy, in terms of the
momenta is obtained by the dispersion relation \begin{equation} E = 4\lambda
\sum_i \left(\sin ^2(\frac{k_i}{2})\right).\end{equation}
One might ask if the situation changes at higher loops. It turns out
that the spin chain description of the non-BPS excitations around
the giant graviton background continues to hold at higher loops as
well. Furthermore, the 'bulk' Hamiltonian of the two loop spin chain
is precisely what one had in the closed string sector. The crucial
question has to do with the nature of the boundary interactions. In
the one loop case, the boundary Hamiltonian $q_1, q_L$ came about
from the condition that the boundary fields cannot be $Z$ i.e. the
Dirichlet boundary conditions. At two loops the situation is more
subtle and non-trivial boundary terms stemming from the interaction
of the bulk and boundary degrees of freedom. The two loop spin chain
Hamiltonian and the appropriate boundary terms have recently been
computed in \cite{maldaopen} and the result is:
\begin{eqnarray}D_2
=\sum_{l=1}^{L-1}(1-4\lambda)(I-P_{l,l+1}) +
\sum_{l=1}^{L-2}\lambda(I-P_{l,l+2})\nonumber \\
+(1-2\lambda)q_1^z + \lambda q_2^Z + (1-2\lambda)q_L^Z + \lambda
q_{L-1}^Z.\label{open22}
\end{eqnarray}
As a historical aside, it is worth noting that in\cite{aaopen},
where a first attempt was made at computing the two loop open-chain
Hamiltonian, the result reported is the same as the one obtained if
one assumes that Dirichlet boundary conditions are the only
source of boundary interactions at two loops. The bulk Hamiltonian
in \cite{aaopen} is the same as (\ref{open22}), however the
boundary Hamiltonian is $(1-4\lambda)q_1^z + \lambda q_2^Z +
(1-4\lambda)q_L^Z + \lambda q_{L-1}^Z,$ which differs from the one
above by a factor of two in one of the terms. This boundary
Hamiltonian does not account for a subtle but extremely crucial
contribution of a particular class of Feynman diagrams, which,
though apparently sub-leading order in $\frac{1}{N}$, end up giving
$\mathcal{O}(1)$ contributions due to contributions from the overall
scaling dimensions of the operators involved\cite{maldaopen}. A
careful analysis of the Bethe ansatz applied to the Hamiltonian with
the boundary contribution given above \cite{aaopen} suggested that,
it is not solvable by Bethe-ansatz techniques. The difficulty had to
do with the existence of the non-trivial boundary conditions which
led to non-factorizable interactions between the magnons and the
boundary.
It turns out that once the added boundary contributions are
included, the Hamiltonian (\ref{open22}) is solvable by the same
Bethe ansatz techniques that were applied in \cite{aaopen}! The
Bethe equations determine the magnon momenta $k_i$ by the
equations\begin{equation} \frac{\alpha (k_i)\beta (k_i)}{\alpha (-k_i)\beta
(-k_i)} = \prod_{j\neq i}\frac{S(-k_i,k_j)}{S(k_i,k_j)},\end{equation} where
\begin{equation} S(p,p') = \frac{\phi(p) - \phi(p') +i}{\phi(p)- \phi(p') -i},
\phi(p) = \frac{1}{2}\cot \left(\frac{p}{2}\right) \sqrt{1+ 8\lambda
^2\sin ^2\left(\frac{p^2}{2}\right)}.\end{equation} $\alpha, \beta $ remain
the same as they were at the one loop level, while the formula for
the bulk scattering matrix above is to be understood as accurate to
$O\lambda ^2$. The two loop dispersion relation being given by \begin{equation}
E(p) = 4\sin ^2\left(\frac{k}{2}\right) - 16\lambda \sin
^4\left(\frac{k}{2}\right).\end{equation} It is thus extremely encouraging to
see that integrability probably does continue to be present even in
the case of giant-graviton boundary conditions. Evidence for
classical open string integrability in the giant graviton
background, from the world-sheet point of view has also recently
been presented in\cite{mann}. For a detailed analysis of various
other recently discovered aspects of open string dynamics and their
gauge theory duals we shall refer to\cite{maldaopen}. It would of
course be extremely interesting to comprehensively establish the
perturbative integrability of the open string sector of the gauge
theory from the point of symmetries and higher conserved charges
discussed before. The role of non-trivial boundaries/backgrounds in
our understanding of integrability of the gauge theory at higher
loops clearly are an exciting avenue
of exploration.\\
{\bf Acknowledgements:} We are extremely indebted to Diego Hofman
and Juan Maldacena for discussions and for sharing the manuscript of their recent
work\cite{maldaopen} before it was published. It is a great pleasure to thank
A.P.Polychronakos and S.G.Rajeev for their collaborations on
refs\cite{aaa} and\cite{aa1} respectively.
|
0708.4345
|
\section{ Introduction }\label{Introduction}
Detection of electron antineutrinos from SN~1987A (see \cite{H87,H88,Bi87})
confirmed the previous theoretical ideas about neutrinos playing crucial role
in the core collapse of type~II supernovae. According to well established
estimates, only about one percent of the gravitational binding energy (or $(1.4
\pm 0.4) \times 10^{51}$~erg for SN~1987A; see \cite{BP90}) is released in the
thermal and kinetic energies of the expanding ejecta; the remaining part is
carried away by different types of neutrinos \cite{Nad78}, which are known to
be trapped effectively within nuclear matter during the last stages of core
collapse (see \cite{ShaTyu} and references therein). Therefore, initially it
did not seem hard to explain the observed values of kinetic energy of the
expanding ejecta using neutrino-nucleon interaction as an effective channel of
energy transfer from the emitted neutrinos to nuclear matter \cite{CW66,A66}.
However, concrete realization of this mechanism of supernova explosions
encountered with great difficulties. Thus, it was realized that the prompt
shock wave, generated during the core collapse, fades away at the time scale of
several milliseconds, losing its energy on nuclear dissociation
\cite{M82,CBB84} and on the radiation of $\nu\bar{\nu}$-pairs \cite{B85}. As a
result, in simulations, the shock starts to move inward, pushed down by the
infalling matter, leading to the formation of a black hole rather than to
explosion.
Considerable efforts were made to remedy this situation. \citet{BW85} showed
that the shock wave could be revived about 100~msec after the core bounce if
neutrino luminosity were sufficiently enhanced. The way of such enhancement of
neutrino luminosity was opened after neutrino diffusion inside the center of
the proto-neutron star (PNS) was discovered \cite{BBC87} and found to be
convectively unstable (see \cite{Bu87,BJRK06} and references therein).
Nevertheless, numerical simulations of the neutrino-driven shock revival still
yield contradictory results. While explosions with kinetic energy up to $1.72
\times 10^{51}$\,erg are obtained in \cite{Sch06} ($0.5 \cdot 10^{51}$\,erg in
\cite{BJRK06}, $10^{50}$\,erg in \cite{KJH06}, $0.94 \cdot 10^{51}$ erg in
\cite{SJFK}), there is also a number of simulations where explosions are not
observed at all \cite{Lieb03,j04}.
This inconsistency between the observational data and the results
of core-collapse numerical simulations motivated some researches
to look for alternative mechanisms. The magnetorotational
mechanism, proposed by \citet{BK70}, can produce explosions with
energies up to $0.61 \times 10^{51}$\,erg at the time scale of
$\sim 0.5$~sec after the core bounce \cite{BKMA05,BKMA06}. The
acoustic mechanism, developed by \citet{BLD06}, also produces
explosion; although its energy is rather uncertain in their
numerical simulations, it develops for a time range of several
hundred milliseconds. Thus, even with new mechanisms and effects
taken into account, the resulting energy release is marginally
short of the observed values.
One can conclude that all mechanisms of core collapse require some additional
engine to provide the observable explosions. Such a new engine is the subject
of the present paper. By analogy with the work of \citet{BBC87}, we consider
an additional type of transport of nuclear matter in the core which is
different from diffusion and convection. It occurs in the form of {\em
boiling\/}, i.e., first order transition between different phases of nuclear
matter.
We start with the observation that supernova matter at the inner core can exist
in several phases and their mixtures \cite{RPW83,HSY84,WK85,LFBHS,WSYE,WMSYE}.
Among them are the following (in the order of increasing density):
\begin{itemize}
\item
the spherical \textit{nuclei\/} phase
\item
the phase consisting of elongated nuclei (often called \textit{pasta\/})
\item
the \textit{slab-like\/} nuclei
\item
the phase with \textit{cylindrical holes\/}
\item
the phase with charged \textit{microscopic\/} bubbles, or \textit{cheesed\/}
phase (we will use this last term below not to confuse it with
\textit{macroscopic\/} bubbles of which we will speak later)
\item
the \textit{homogeneous\/} supernova matter
\end{itemize}
It is natural to expect that several phase transitions can occur during the
evolution of nuclear matter during as well as after the core collapse.
Numerical simulations of the nuclear-matter phase transitions in supernovae
were usually aimed at determining the thermodynamic properties at the
pre-bounce stage of collapse, and were needed to understand the development of
the prompt shock wave. Such transitions taking place in a rapidly changing
environment during collapse can be called ``short-term'' phase transitions. In
our paper, we discuss what we call ``long-term'' phase transitions, which occur
in nuclear matter after the bounce under the condition of relative mechanical
and \textit{local thermodynamic equilibrium\/}. The main goal of this paper is
to examine the potential importance of such ``long-term'' phase transitions on
the dynamics of supernova explosion.
In the ``normal'' case of mechanical and local thermodynamic equilibrium,
different phases occupy the corresponding radial shells of the PNS, and this
spatial phase picture evolves continuously and relatively slowly in time.
However, if heating of the nuclear matter (which is mainly due to diffusion of
neutrinos) is sufficiently strong and inhomogeneous, it can lead to a condition
similar to that in an ordinary teakettle. Specifically, a bulk of particular
phase can overheat and become unstable with respect to phase transition, and
small bubbles of the lighter phase can spontaneously appear in its volume. The
bubbles will grow and, due to the Archimedean force, rise upwards. Henceforth,
such a process is called \textit{boiling\/} by analogy. In this paper, we argue
that this process can provide more efficient mechanism of heating the outer
parts of the PNS (compared with the neutrino-diffusion and convection
mechanisms) and generate additional pressure wave.
The paper is organized as follows. In Sec.~\ref{Transition}, we give the basic
thermodynamic description of the co-existence of various phases in supernova
matter after the bounce. In Sec.~\ref{Boiling}, we derive the criterion of
boiling to occur, and, in Sec.~\ref{Estimates}, we provide our numerical
estimates for the values involved in this criterion using the tabulated
equation of state from \cite{suraud85}. In Sec.~\ref{Model}, we consider a
simple model of the boiling mechanism and derive numerical estimates
characterizing the efficiency of this process. We summarize and discuss our
results in Sec.~\ref{Discussion}.
\section{Phase equilibrium}\label{Transition}
Just after the core bounce, the inward movement of the inner core significantly
decreases, and the prompt shock wave moves outwards losing its energy mostly on
nuclear dissociation and radiation of $\nu\bar{\nu}$-pairs. The material
within and around the PNS approaches a state of mechanical equilibrium (while
the velocities of the convective motion are much smaller than the appropriate
first cosmic velocities). In contrast with the rapid collapse during the
pre-bounce stage of contraction, local thermal equilibrium is a good
approximation for the post-bounce stage.
Because of the radial density gradient, the PNS at this stage has onion-like
structure. The density of its inner core is higher than the nuclear saturation
density, and the homogeneous matter phase is thermodynamically preferred. In
the outer layers of the PNS, the density is much less than the nuclear
saturation density, and matter exists in the form of ordinary nuclei. A number
of intermediate phases can exist between these two shells. Adjacent phases are
separated by the surfaces of coexistence. Here we derive a simple condition of
coexistence of phases applied to the situation under consideration.
We can consider a phase transition characterized by fixed pressure $p$,
temperature $T$, baryon number $B$, electric charge $C$ and lepton number $L$.
The conservation laws of the last three quantities imply the existence of the
corresponding chemical potentials: $\mu_{B}$, $\mu_{C}$ and $\mu_{L}$. We can
determine their values from the chemical potentials of the supernova matter
components (we suppose that the supernova matter consists only of neutrons,
protons, electrons and electron neutrinos): \eq{\mu_{n} = \mu_{B}\, , \quad
\mu_{p} = \mu_{B} + \mu_{C}\, , \quad \mu_{e} = \mu_{L} - \mu_{C}\, , \quad
\mu_{\nu} = \mu_{L}\, .} Since the number of particle species is greater than
the number of independent charges, we have one more relation on the chemical
potentials (the so-called beta-equilibrium condition): \eq{\mu_{p} + \mu_{e} =
\mu_{n} + \mu_{\nu}\, .\label{equilibrium}}
For fixed values of $p$ and $T$, our thermodynamic system tends to a state with
a minimum value of the Gibbs free energy (see, e.g., \cite{landau5}) \eq{\Phi =
\sum_{i}\mu_{i}N_{i} = \mu_{B}B + \mu_{C}C + \mu_{L}L \, .\label{fi}} Since we
are interested only in electrically neutral phases, we have \eq{C\equiv 0\, ,}
and the second term in (\ref{fi}) vanishes. The third term, $\mu_{L}L$, does
not change during the phase transition because the electron neutrinos interact
with nuclear matter very weakly.
Finally, the condition of equilibrium between phases 1 and 2 is \eq{\mu_{1n}
\left( p_{0},T_{0},\mu_{\nu 0} \right) = \mu_{2n} \left( p_{0},T_{0},\mu_{\nu
0} \right) \, ,\label{coex}} where the subscript ``{\small 0}'' marks the
values of the parameters right on the interface between the two phases.
\section{Criterion of boiling}\label{Boiling}
Local thermal equilibrium described in the previous section is continuously
disturbed by the process of diffusion of electron neutrinos away from the
central region of the PNS. This causes inhomogeneous heating of the bulk of
nuclear matter. If this heating is sufficiently strong, it can lead to an
overheat of a particular phase, which may result in its {\em boiling\/}
--- emergence of bubbles of the adjacent lighter phase in its bulk.
These bubbles will then raise and grow, effectively transferring heat and
lepton number. Boiling is a particular case of a non-equilibrium first order
phase transition. In this section, we derive a necessary condition of boiling
in the form of a relation on the parameters of the supernova matter.
According to Eq.~(\ref{coex}), in the process of external heating, a heavier
phase 1 becomes metastable with respect to its transition to a lighter phase 2
if the condition is reached such that \eq{\mu_{1n}(p_{0}+\delta p, T_{0}+\delta
T, \mu_{\nu 0} + \delta \mu_{\nu})> \mu_{2n}(p_{0}+\delta p, T_{0}+\delta T,
\mu_{\nu 0} + \delta \mu_{\nu}) \, ,} where the positive increments of the
thermodynamic variables correspond to their radial gradients. From this, we
obtain the condition
\begin{eqnarray} \label{main}
\left[\left(\frac{\partial \mu_{2n}}{\partial T}\right)_{p,\mu_{\nu}} - \left(
\frac{\partial \mu_{1n}}{\partial T}\right)_{p,\mu_{\nu}}\right]\frac{dT}{dr}
&+& \left[\left(\frac{\partial \mu_{2n}}{\partial \mu_{\nu}}\right)_{p,T} -
\left(\frac{\partial \mu_{1n}}{\partial \mu_{\nu}}\right)_{p,T}\right]
\frac{d\mu_{\nu}}{dr} \nonumber \\ &+& \left[\left(\frac{\partial
\mu_{2n}}{\partial p}\right)_{T,\mu_{\nu}} - \left(\frac{\partial
\mu_{1n}}{\partial p}\right)_{T,\mu_{\nu}}\right]\frac{dp}{dr}>0 \, .
\end{eqnarray}
In the important particular case of hydrostatic equilibrium, we have
\eq{\frac{dp}{dr} = -\rho_{m}g\, ,} where $\rho_{m}$ is the mean density, and
$g$ is the local acceleration of free fall. Substituting this into
(\ref{main}), we obtain
\begin{eqnarray} \label{main1}
\left[\left(\frac{\partial \mu_{2n}}{\partial T}\right)_{p,\mu_{\nu}} - \left(
\frac{\partial \mu_{1n}}{\partial T}\right)_{p,\mu_{\nu}}\right]\frac{dT}{dr}
&+& \left[\left(\frac{\partial \mu_{2n}}{\partial \mu_{\nu}}\right)_{p,T} -
\left(\frac{\partial \mu_{1n}}{\partial \mu_{\nu}}\right)_{p,T}\right]
\frac{d\mu_{\nu}}{dr} \nonumber \\ &>& \left[\left(\frac{\partial
\mu_{2n}}{\partial p}\right)_{T,\mu_{\nu}} - \left(\frac{\partial
\mu_{1n}}{\partial p}\right)_{T,\mu_{\nu}}\right]\rho_{m}g \, .
\end{eqnarray}
\section{Numerical estimates}\label{Estimates}
It remains to check whether the condition of boiling derived in the previous
section can be realized in the usual supernova core-collapse. For this
purpose, we estimate the values of the partial derivatives in (\ref{main1})
using the numerical simulations of the EoS of supernova matter given in
\cite{suraud85}.
The authors of \cite{suraud85} tabulate all essential parameters describing the
supernova matter in three phases: nuclei, cheesed phase and homogeneous matter.
In Table~\ref{input}, we show the parameters which are required in our
analysis. Because the required values of derivatives are not listed in
\cite{suraud85}, we try to obtain them from interpolation. Namely, we use the
finite differences: \eq{\mu_{n}(p_{2},T_{2},\mu_{\nu
2})-\mu_{n}(p_{1},T_{1},\mu_{\nu 1}) \approx \frac{\partial \mu_{n}}{\partial
p} (p_{2}-p_{1}) + \frac{\partial \mu_{n}}{\partial T} (T_{2}-T_{1}) +
\frac{\partial \mu_{n}}{\partial \mu_{\nu}} (\mu_{\nu 2}-\mu_{\nu 1})\, ,}
neglecting the higher derivatives. For each phase, we solve a system of three
equations for three unknown variables. The numerical values obtained in such a
manner should be considered as estimates. They are presented in
Table~\ref{results}.
\begin{table}
{
\begin{tabular}{|l|c|c|c|c|c|}
\hline
Phase & $n_{B}$, fm$^{-3}$ & $\mu_{n}$, MeV & $p$,
MeV\,$\cdot$\,fm$^{-3}$ & $T$, MeV & $\mu_{\nu}$, MeV\\
\hline \hline
Nuclei & 0.02 & $-2.230$ & 0.1807 & 4.00 & 81.613\\
\hline
& 0.04 & $-1.953$ & 0.4556 & 4.88 & 105.363\\
\hline
& 0.05 & $-1.837$ & 0.6111 & 5.17 & 114.092\\
\hline
& 0.06 & $-1.818$ & 0.7738 & 5.44 & 121.699\\
\hline \hline
Cheesed & 0.05 & $-2.660$ & 0.5615 & 5.07 & 114.149\\
\hline
& 0.07 & $-2.189$ & 0.9086 & 5.65 & 127.934\\
\hline
& 0.09 & $-1.967$ & 1.2726 & 6.08 & 139.143\\
\hline
& 0.10 & $-2.093$ & 1.4432 & 6.23 & 144.213\\
\hline \hline
Homogeneous & 0.10 & $-2.536$ & 1.4241 & 6.18 & 144.530\\
\hline
& 0.11 & $-1.202$ & 1.7203 & 6.64 & 147.952\\
\hline
& 0.12 & 0.154 & 2.2267 & 7.09 & 169.312\\
\hline
& 0.16 & 7.525 & 4.227 & 8.84 & 188.917\\
\hline
\end{tabular}
} \caption{The parameters from \cite{suraud85} used in the estimate of the
partial derivatives in (\ref{main1}).} \label{input}
\end{table}
\begin{table}
{
\begin{tabular}{|l|c|c|c|}
\hline Phase & $\left(\frac{\partial \mu_{n}}{\partial
p}\right)_{T,\mu_{\nu}}$, fm$^{3}$ & $\left(\frac{\partial \mu_{n}}{\partial
T}\right)_{p,\mu_{\nu}}$
& $\left(\frac{\partial \mu_{n}}{\partial \mu_{\nu}}\right)_{p,T}$\\
\hline \hline
Nuclei & $-1.353$ & $-2.626$ & 0.125\\
\hline
Cheesed & 0.666 & 5.182 & $-0.201$\\
\hline
Homogeneous & 2.87 & 1.289 & $-0.0317$\\
\hline
\end{tabular}
} \caption{The results for the partial derivatives.} \label{results}
\end{table}
To check whether the conditions of boiling are realized, we use the
conventional values of the other parameters. Numerical simulations (see, e.g.,
\cite{Lieb03}) give the following estimates: \eq{g \approx 1.0 \times 10^{14}\,
\mbox{cm\,$\cdot$\,sec$^{-2}$}\, , \quad \frac{d\mu_{\nu}}{dr}\approx
-(\mbox{10$-$20})\, \mbox{MeV\,$\cdot$\,km$^{-1}$} \, , \quad \frac{dT}{dr}<0\,
.\label{data}} Using this data together with Table~\ref{results}, we obtain the
following conditions for boiling:
\begin{itemize}
\item for the transition between the nuclear and cheesed phase
($\rho_{m} \approx 0.8 \times 10^{14}$\,g\,$\cdot$\,cm$^{-3}$),
\eq{ \frac{dT}{dr} < -(\mbox{0.4$-$0.8})\,
\mbox{MeV\,$\cdot$\,km$^{-1}$}\, ; \label{nch}}
\item for the transition between homogeneous matter and cheesed
phase ($\rho_{m} \approx 1.6 \times
10^{14}$~g\,$\cdot$\,cm$^{-3}$), \eq{ \frac{dT}{dr} >
-(\mbox{1.0$-$1.5})\, \mbox{MeV\,$\cdot$\,km$^{-1}$} \, .
\label{hch}}
\end{itemize}
The difference in the inequality signs in estimates (\ref{nch}) and (\ref{hch})
is connected with the difference in the signs of the coefficients of the
temperature gradients in (\ref{main}) or (\ref{main1}) for the two phase
transitions under consideration. Note that conditions (\ref{nch}) and
(\ref{hch}) are overlapping and complementary, so it is very likely that one of
them is satisfied. We, therefore, can expect that boiling of nuclear matter can
take place inside the supernova core. However, this only demonstrates the
possibility of principle, calling for additional thorough investigation of this
issue.
\section{A model}\label{Model}
In this section, we construct a simplified model of phase transition between a
heavier phase 1 and a lighter phase 2, which, for definiteness, we take to be
the cheesed phase and the phase of nuclei, respectively.
We assume spherical symmetry of the PNS, so that the cheesed phase is located
in a spherical shell with radial coordinate from $R - H$ to $R$. The free-fall
acceleration on the upper boundary of this region is equal to
\eq{g=\frac{GM}{R^{2}}=1.33\times 10^{14}
\left(\frac{M}{M_{\odot}}\right)\left(\frac{10 \, \mbox{km}}{R}\right)^{2}
\mbox{cm\,$\cdot$\,sec$^{-2}$} \, ,} where $M$ is the total mass inside the
sphere of radius $R$.
We can estimate $H$ using the condition of hydrostatic equilibrium:
\begin{equation}
\rho g H \approx \Delta p\, ,
\end{equation}
where $\Delta p$ is the dimension of the pressure interval in which the cheesed
phase exists. According to Table~\ref{input}, this pressure interval is
approximately equal to 0.83~MeV$\,\cdot$\,fm$^{-3}$. We thus have
\begin{equation}
H \approx \frac{\Delta p}{\rho g} \approx 1.0 \times \left( \frac{\Delta
p}{0.83\, \mbox{MeV\,$\cdot$\,fm$^{-3}$}}\right)\left(\frac{10^{14}\,
\mbox{g\,$\cdot$\,cm$^{-3}$}}{\rho}\right)\left(\frac{1.33 \times 10^{14}\,
\mbox{cm\,$\cdot$\,sec$^{-2}$}}{g}\right)\, \mbox{km} \, .
\end{equation}
If we assume that boiling takes place in the whole volume of phase 1, then the
total mass of the boiling matter can be estimated as\footnote{Recent
simulations \cite{Sonoda07} show that the total mass of different phases
between nuclei and homogeneous nuclear matter (collectively called ``pasta
phases'' in \cite{Sonoda07}) inside the supernova just {\em before\/} the
bounce can amount to $0.13$$-$$0.30\, M_{\odot}$. This estimate somewhat
differs from (\ref{Mb-estim}) because, before the bounce, one deals with {\em
small\/} pressure gradients in an (almost) free-falling matter, while, after
the bounce, we have matter close to hydrostatic equilibrium with its large
pressure gradients. It is possible, in principle, that the
\textit{pre-bounce\/} boiling, if it occurs, can affect the propagation of
prompt shock wave, but we do not consider this issue here.} \eq{M_{b} \approx
4\pi\rho R^{2}H \lesssim 6.1 \times 10^{-2} \left(\frac{\Delta p}{0.83\,
\mbox{MeV\,$\cdot$\,fm$^{-3}$}}\right) M\, . \label{Mb-estim}}
The densities of phases 1 and 2 are related by $\rho_{1}$ and $\rho_{2} =
\rho_{1}(1 - \epsilon)$, where $\epsilon \ll 1$. According to \cite{OS82},
$\epsilon$ is equal to $0.1$ for the phase transition between the cheesed phase
and homogeneous matter phase, and to $0.2$ for the transition between nuclei
and cheesed phase. Lower estimates for $\epsilon$ are present in
\cite{suraud85}: it equals to $0.07$$-$$0.08$ for both phase transitions if the
entropy per baryon (which is conserved in the simulations) $S/A = 1.0$; for
$S/A = 1.5$, the value $\epsilon = 0.05$ is obtained for the transition between
nuclei and cheesed phase, and $\epsilon = 0.01$ for the transition between
cheesed phase and homogeneous matter. To be conservative, we use the lowest
estimate in this paper, namely, $\epsilon = 0.01$.
The maxi\-mum acceleration which can be reached by a raising bubble (neglecting
the liquid resistance) is \eq{a_{\rm max} = \frac{\epsilon GM}{R^{2}}\, .} The
maximum velocity that can be reached by the bubble is then \eq{v_{\rm max} \sim
\left( \frac{2\epsilon \Delta p}{\rho} \right)^{1/2} = 5100 \sqrt{\left(
\frac{\epsilon}{10^{-2}} \right) \left( \frac{\Delta
p}{0.83\,\mbox{Mev\,$\cdot$\,fm$^{-3}$}} \right)\left(\frac{10^{14}\, \mbox{g
$\cdot$cm$^{-3}$}}{\rho}\right)} \, \mbox{km\,$\cdot$\,sec$^{-1}$} \, .}
Matter in the boiling volume will convectively move in both directions. We can
expect that roughly half of matter moves upwards (the bubbles and the
surrounding matter) and the other half moves downwards with the same average
velocity (from the momentum conservation). Therefore, the ``convective
boiling'' overturn will be established. We can try to estimate the maximum
efficiency of this overturn.
The first effect to be discussed is heat transfer. If the bubbles fill half of
the boiling volume, the heating rate at the surface of the boiling layer is
\begin{eqnarray} \label{heating}
\dot{Q}_{\rm max} &\sim& \frac{1}{2}\times 4\pi R^{2} v_{\rm max}\,q
\nonumber \\
&=& 7.7 \times 10^{52}\, \left(\frac{R}{10\, \mbox{km}}
\right)^{2}\left(\frac{v_{\rm max}}{5100\,
\mbox{km\,$\cdot$\,sec$^{-1}$}}\right)\left(\frac{q}{0.015\,
\mbox{MeV\,$\cdot$fm$^{-3}$}}\right)\,\mbox{erg\,$\cdot$\,sec$^{-1}$}\, .
\end{eqnarray}
Here, $q$ is the specific volume heat of evaporation, and its value is
estimated as $q=0.015$\,MeV\,$\cdot$\,fm$^{-3}$ from \cite{RPW83}. The value
(\ref{heating}) is comparable to the neutrino luminosity. This is, in
principle, the upper estimate of heating rate which corresponds to maximal
workload of the ``boiling machine.'' In reality, the boiling heat transfer
should work together with diffusion and/or convection below and above the
boiling shell, increasing the {\em net\/} heat transfer rate.\footnote{This can
be compared to an electric circuit with a series of resistances. As one of the
resistances in the series is shortened out, the total current increases.} This
should lead to an increase in pressure behind the shock, providing more
efficient conditions for shock revival.
The second effect is the momentum transfer. The maximum mechanical pressure
that the bubbles can exert is estimated as \eq{p_{\rm max} \sim \rho v_{\rm
max}^{2} = 2\epsilon \Delta p = 1.7 \times 10^{-2}\,
\mbox{MeV\,$\cdot$\,fm$^{-3}$}\, ,} where the numerical value corresponds to
the cheesed phase. This is much smaller than {\bf $\Delta p$} since $\epsilon
\ll 1$; therefore, including this contribution to pressure does not seriously
affect our previous calculation. But it can provide an additional barrier for
the infalling matter to reach the inner core, reducing the final mass of the
neutron star.
We should admit that most of the above estimates represent upper limits
corresponding to the maximal workload of the ``boiling machine.'' In a
subsequent paper, we will discuss these processes in greater detail.
\section{Summary and conclusions}\label{Discussion}
In this paper, we have demonstrated the possibility of boiling of nuclear
liquid in the supernova core after the bounce. If it occurs, it can lead to
effects increasing the efficiency of the neutrino-driven mechanism of supernova
explosions. Among these effects are the following:
\begin{itemize}
\item The increase of heat transfer rate from the inner core to the neutrinosphere. This
increases the mean neutrino energy, making the delayed-shock mechanism more
efficient.
\item The increase of pressure in the boiling region. It provides an
additional barrier between the infalling matter and the inner core, thereby
reducing the mass of the neutron-star remnant.
\end{itemize}
We expect that taking into account the effect of boiling in the conventional
delayed-shock/acoustic mechanism is important and will enable one to explain
the energetics of the supernova explosions in a more simple and self-consistent
way.
\section*{Acknowledgements}
D.~I.\@ is grateful to the Scientific and Educational
Center\footnote{http://sec.bitp.kiev.ua} of the Bogolyubov Institute for
Theoretical Physics in Kiev, Ukraine, and especially to Dr.~Vitaly Shadura for
creating wonderful atmosphere for young scientists, and to T.~Foglizzo,
M.~Liebend\"{o}rfer and D.~K.~Nadyozhin for their helpful comments. This work
was supported in part by grant No.~5-20 of the ``Cosmomicrophysics'' programme
of the Ukrainian Academy of Sciences and by the INTAS grant
No.~05-1000008-7865.
|
0708.4322
|
\section{Introduction}
\label{sec:intro}
We study the ratio of the unintegrated gluon distribution of a nucleus
$\varphi_A(k_{\perp},Y)$ over the unintegrated gluon distribution of a proton
$\varphi_p(k_{\perp},Y)$ scaled up by $A^{1/3}$
\begin{equation}
R_{pA} = \frac{\varphi_{A}\left( k_{\perp },Y\right) }{A^{\frac{1}{3}}\ \varphi_{p}\left(
k_{\perp },Y\right) } \ .
\label{R_pA}
\end{equation}
This ratio is a measure of the number of particles produced in a
proton-nucleus collision versus the number of particles in proton-proton
collisions times the number of collisions. The transverse
momentum of gluons is denoted by $k_{\perp}$ and the rapidity variable by $Y=\ln(1/x)$.
The ratio $R_{pA}$ has been widely studied~\cite{Iancu:2004bx+X1,
*Iancu:2004bx+X2, *Iancu:2004bx+X3, *Iancu:2004bx+X4, *Iancu:2004bx+X5,
*Iancu:2004bx+X6, *Iancu:2004bx+X7,*Iancu:2004bx+X8} in the framework of the
BK-equation~\cite{Balitsky:1995ub+X1, *Balitsky:1995ub+X2,
*Balitsky:1995ub+X3, *Balitsky:1995ub+X4} which describes the small-$x$
physics in the {\em mean field approximation}. Using the BK-equation one finds
in the geometric scaling regime (transition from high to low gluon density,
see Fig.\ref{had_wf}) in the fixed coupling case that the shape of the
unintegrated gluon distribution of the nucleus and proton as a function of
$k_{\perp}$ is preserved with increasing $Y$, see Fig.\ref{GS}(a), because of
the geometric scaling behaviour
$\varphi_{p,A}(k_{\perp},Y)=\varphi_{p,A}(k_{\perp}^2/Q^2_s(Y))$, and
therefore the leading contribution to the ratio $R_{pA}$ is basically
$k_{\perp}$ and $Y$ independent, scaling with the atomic number $A$
as~\cite{Mueller:2003bz,Iancu:2004bx}
\begin{equation}
R_{pA} \simeq \frac{1}{A^{\frac{1}{3}(1-\gamma_{_0})}} \ ,
\label{eq:RGS}
\end{equation}
where $\gamma_{_0}=0.6275$. This means that gluons inside the nucleus and
proton are somewhat shadowed since $\varphi_A/\varphi_p = A^{\gamma_{_0}/3}$
lies between total ($\varphi_A/\varphi_p=1$) and zero
($\varphi_A/\varphi_p=A^{1/3}$) gluon shadowing. The {\em partial gluon
shadowing} comes from the anomalous behaviour of the unintegrated gluon
distributions which stems from the BFKL evolution. The partial gluon shadowing
may explain why particle production in heavy ion collisions scales, roughly,
like $N_{part}$~\cite{Kharzeev:2002pc}.
Over the last few years, it has been understood how to deal with small-$x$
physics at high energy {\em beyond the mean field approximation}, i.e., beyond
the BK~\cite{Balitsky:1995ub+X1, *Balitsky:1995ub+X2, *Balitsky:1995ub+X3,
*Balitsky:1995ub+X4} and JIMWLK~\cite{Jalilian-Marian:1997jx+X1,
*Jalilian-Marian:1997jx+X2, *Jalilian-Marian:1997jx+X3,
*Jalilian-Marian:1997jx+X4, *Jalilian-Marian:1997jx+X5} equations. We have
learned how to account for the elements missed in the mean field evolution,
such as the descreteness and fluctuations of gluon
numbers~\cite{Mueller:2004sea,Iancu:2004es} or the Pomeron
loops~~\cite{Mueller:2005ut+X1, *Mueller:2005ut+X2, *Mueller:2005ut+X3,
*Mueller:2005ut+X4, *Mueller:2005ut+X5}. The main result as a consequence
of the above is the emerging of a new scaling behaviour for the
dipole-hadron/nucleus scattering amplitude at high
rapidities~\cite{Mueller:2004sea,Iancu:2004es}, the so-called diffusive
scaling. This is different from the geometric scaling behaviour which is the
hallmark of the "mean-field" evolution equations (JIMWLK and BK equations).
The effects of fluctuations on the scattering
amplitude~\cite{Brunet:2005bz+X1, *Brunet:2005bz+X2, *Brunet:2005bz+X3,
*Brunet:2005bz+X4, *Brunet:2005bz+X5, *Brunet:2005bz+X6, *Brunet:2005bz+X7,
*Brunet:2005bz+X8, *Brunet:2005bz+X9, *Brunet:2005bz+X10,
*Brunet:2005bz+X11, *Brunet:2005bz+X12, *Brunet:2005bz+X13,
*Brunet:2005bz+X14}, the diffractive scattering processes\cite{Hatta:2006hs,
Kovner:2006ge+X1, *Kovner:2006ge+X2, *Kovner:2006ge+X3} and forward gluon
production in hadronic scattering processes~\cite{Iancu:2006uc,Kovner:2006wr}
has been studied so far. In this work we show how the behaviour of $R_{pA}$ as
a function of $k_{\perp}$ and $Y$ in the fixed coupling case is completely
changed due the effects of gluon number fluctuations or Pomeron loops at high
rapidity~\cite{KSX_1,*KSX_2}.
\begin{figure}[htb]
\setlength{\unitlength}{1.cm}
\par
\begin{center}
\epsfig{file=nucl_wf1_ite.eps, width=11cm}
\end{center}
\caption{Phase diagram of a highly evolved nucleus/proton.}
\label{had_wf}
\end{figure}
\section{$R_{pA}$ ratio in the diffusive scaling regime}
\label{sec:Rpa}
According to the statistical physics/high energy QCD
correspondence~\cite{Iancu:2004es} the influence of fluctuations on the
unintegrated gluon distribution of a nucleus/proton is as follows: Starting
with an intial gluon distribution of the nucleus/proton at zero rapidity, the
stochastic evolution generates an ensamble of distributions at rapidity $Y$,
where the individual distributions seen by a probe typically have different
saturation momenta and correspond to different events in an experiment. To
include gluon number fluctuations one has to average over all individual
events,
\begin{equation}
\langle \varphi_{p,A}(\rho-\rho_s)\rangle = \int d\rho_s\ \varphi_{p,A}(\rho-\rho_s) \
P(\rho_s-\langle\rho_s\rangle) \ ,
\label{av_gd}
\end{equation}
where $\varphi_{p,A}(\rho-\rho_s)$ is the distribution for a single event with
$\rho = \ln(k_{\perp}^2/k_0^2)$ and
$ P(\rho_s-\langle\rho_s\rangle)$ the probability distribution of the
logarithm of the saturation momentum, $\rho_s(Y)=\ln(Q^2_s(Y)/k_0^2)$, which
is argued to have a Gaussian form~\cite{Marquet:2006xm},
\begin{equation}
P(\rho _{s})\simeq \frac{1}{\sqrt{2\pi \sigma ^{2}}}\exp \left[ -\frac{%
\left( \rho _{s}-\langle \rho _{s}\rangle \right) ^{2}}{2\sigma ^{2}}\right]
\quad \mbox{for} \quad \rho-\rho_s \ll \gamma_c^2 \sigma^2 \ ,
\label{proba_gauss}
\end{equation}
with the dispersion
\begin{equation}
\sigma^2 = \langle \rho_s^2 \rangle- \langle \rho_s \rangle^2 = D Y.
\label{sigma_v}
\end{equation}
The main consequence of fluctuations is the replacement of the geometric
scaling, $\varphi_{p,A}(k_{\perp},Y)=\varphi_{p,A}(k^2_{\perp}/Q_s^2(Y))$, by
a new scaling, the diffusive scaling~\cite{Mueller:2004sea,Iancu:2004es},
namely, $\langle \varphi_{p,A}(k_{\perp},Y) \rangle$ is a function of another
scaling variable ($\langle Q_s \rangle$ is the average saturatin momentum),
\begin{equation}
\langle \varphi_{p,A}(k_{\perp},Y)\rangle =
F_{p,A}\left(\frac{\ln(k^2_{\perp}/\langle Q_s(Y)\rangle^2)}{DY}\right) \ .
\end{equation}
The diffusive scaling, see Fig.~\ref{had_wf}, sets in when the dispersion of
the different events is large, $\sigma^2 = \langle \rho_s^2 \rangle- \langle
\rho_s \rangle^2 = D Y \gg 1$, i.e., $Y \gg Y_{DS} =1/D$ where $D$ is the
diffusion coefficient, and is valid in the region $\sigma \ll
\ln(k^2_{\perp}/\langle Q_s(Y)\rangle^2) \ll \gamma_{_0}\, \sigma^2$.
The diffusive scaling means that the shape of the unintegrated gluon
distribution of the nucleus/proton changes with increasing $Y$ because of the
additional $DY$ dependence as compared with the geometric scaling. The shape
becomes flatter and flatter with increasing rapidity $Y$, as shown in
Fig.\ref{GS}(b), in contrast to the preserved shape in the geometric scaling
regime shown in Fig.\ref{GS}(a). This flattening will lead to a new phenomenon
for $R_{pA}$ as discussed below.
Using Eq.(\ref{av_gd}) for the averaging over all events and the result from
the BK-equation for the the single event distribution one obtains~\cite{KSX_1}
for the ratio
\begin{equation}
R_{pA} \simeq \frac{1}{A^{\frac{1}{3}(1 - \frac{\Delta\rho_s}{2\sigma^2})}}\
\left[\frac{k_{\perp}^2}{\langle
Q^2_s(A,y)\rangle}\right]^{\frac{\Delta\rho_s}{\sigma^2}}
\label{eq:RPA1}
\end{equation}
with the difference between the average saturation
lines of the nucleus and the proton
\begin{equation}
\Delta \rho_s \; \equiv \; \langle \rho_s(A,Y)\rangle - \langle \rho_s(p,Y)\rangle
\; = \; \ln \frac{\langle Q_s(A,Y)\rangle^2}{\langle Q_s(p,Y)\rangle^2} \
\label{del_rho}
\end{equation}
where $\langle Q_s(A,Y)\rangle$ ($\langle Q_s(p,Y)\rangle$) is the average
saturation momentum of the nucleus (proton). The difference $\Delta \rho_s$
is fixed by the inital conditions for the average saturation momenta of the
nucleus and proton and is $Y$-indipendent in the fixed coupling case. For
example, using the known assumption $\langle Q_s(A)\rangle^2 = A^{1/3}\,\langle
Q_s(p)\rangle^2$ one obtains $\Delta \rho_s = \ln A^{1/3}$.
The ratio $R_{pA}$ in Eq.~(\ref{eq:RPA1}) shows the following very different
features as compared with the ratio in the geometric scaling regime given in
Eq.~(\ref{eq:RGS}):
\begin{itemize}
\item In the diffusive scaling regime where $k^2_{\perp}$ is close to $\langle
Q_s^2(A,Y)\rangle$, the gluon shadowing characterized by
$A^{\frac{1}{3}(\frac{\Delta\rho_s}{2\sigma^2}-1)}$ increases as the
rapidity grows (at fixed $A$ or $\Delta\rho_s$) because of $\sigma^2 = DY$.
At asymptotic rapidity one obtains {\em total gluon shadowing},
$R_{pA}=A^{1/3}$, which means that the unintegrated gluon distribution of
the nucleus and that of the proton become the same in the diffusive scaling
regime at $Y \to \infty$. The phenomenon of total gluon shadowing is
universal since it does not depend on the initial conditions ($\Delta\rho_s$).
\vskip 3mm
Total gluon shadowing is an effect of gluon number fluctuations (or Pomeron
loops) since fluctuations make the unintegrated gluon distributions of the
nucleus and of the proton flatter and flatter~\cite{Iancu:2004es} and their
ratio closer and closer to $1$ (at fixed $\Delta\rho_s$) with rising rapidity,
as shown in Fig.\ref{GS}(b). Total gluon shadowing is not possible in the
geometric scaling regime in the fixed coupling case since the shapes of the
gluon distributions of the nucleus and of the proton remain the same with
increasing $Y$ giving a constant ratio unequal one, as shown in
Fig.\ref{GS}(a). In the absence of fluctuations one can expect only partial
gluon shadowing, see Eq.~(\ref{eq:RGS}), in the fixed coupling case.
\item The ratio $R_{pA}$ increases with rising $k^2_{\perp}$ within the
diffusive scaling region. Since the exponent $\Delta\rho_s/\sigma^2$
decreases with rapidity, the slope of $R_{pA}$ as a function of
$k_{\perp}^2$ becomes smaller with growing $Y$. The result for $R_{pA}$ in
the diffusive scaling regime in Eq.(\ref{R_DS}) is very different from the
result obtained in the mean field approximation given in Eq.~(\ref{eq:RGS}),
where gluon number fluctuations are not included, which is basically
$k_{\perp}$ and $Y$-independent.
\end{itemize}
\vskip 3mm
The qualitative behaviour of $R_{pA}$ as a function of $k_{\perp}$ at four
different rapidities, $Y_1 \leq Y_2 \leq Y_3 \leq Y_4$, in the diffusive
scaling regime and for a fixed coupling is shown in Fig.~\ref{R_DS}. Note that
$R_{pA}$ is always smaller than one for values of $k_{\perp}$ in the diffusive
scaling regime.
\begin{figure}[htb]
\setlength{\unitlength}{1.cm}
\par
\begin{center}
\epsfig{file=T_Shad_Flu_GS_ite.eps, width=7.0cm} ~~
\epsfig{file=T_Shad_Flu_ite.eps, width=7.0cm} \\
\hspace*{0.3cm} (a) \hspace*{9cm} (b)
\end{center}
\caption{The qualitative behaviour of the unintegrated gluon distribution of a
nucleus (A) and a proton (p) at two different rapidities in the geometric
scaling regime (a) and diffusive scaling regime (b). }
\label{GS}
\end{figure}
\begin{figure}[htb]
\setlength{\unitlength}{1.cm}
\par
\begin{center}
\epsfig{file=R_pA_ite.eps, width=14cm}
\end{center}
\caption{The qualitative behaviour of the ratio $R_{pA}$ as a function of $k_{\perp}$ at four different
rapidities, $Y_1 \leq Y_2 \leq Y_3 \leq Y_4$, in the diffusive scaling regime.
$R_{pA}$ is always smaller than one for values of $k_{\perp}$ in the diffusive
scaling regime.}
\label{R_DS}
\end{figure}
The above effects of fluctuations on $R_{pA}$ are valid in the fixed coupling
case and at very large energy. It isn't clear yet whether the energy at LHC is
high enough for them to become important. Recently, while in
Ref.~\cite{Kozlov:2007wm} a possible evidence of gluon number fluctuations in
the HERA data has been found, in Ref.~\cite{Dumitru:2007ew}, using a toy
model, it has been argued that in case of a running coupling fluctuations can
be neglected in the range of HERA and LHC energies. See also Refs.~\cite{Mueller:2004sea,Beuf:2007qa} for
more studies on running coupling plus fluctuation effects.
Moreover, the running of the coupling~\cite{Albacete:2007yr+X1,
*Albacete:2007yr+X2, *Albacete:2007yr+X3, *Albacete:2007yr+X4,
*Albacete:2007yr+X5,*Albacete:2007yr+X6} may become more important than the effect of gluon
number fluctuations~\cite{Dumitru:2007ew}. In case of a running coupling, the
gluon shadowing increases with rising rapidity in the geometric scaling
regime~\cite{Iancu:2004bx+X1, *Iancu:2004bx+X2, *Iancu:2004bx+X3,
*Iancu:2004bx+X4, *Iancu:2004bx+X5, *Iancu:2004bx+X6, *Iancu:2004bx+X7,*Iancu:2004bx+X7}, as
opposed to the (roughly) fixed value (partial shadowing) in the fixed-coupling
case, and would lead to total gluon shadowing~\cite{Iancu:2004bx} at very high
rapidities even if fluctuations were absent. In case fluctuations are
important at LHC energy, in addition to the theoretically interesting
consequences of fluctuations on $R_{pA}$, the features of $R_{pA}$ worked out
here, as the increase of the gluon shadowing and the decrease as a function of
the gluon momentum with rising rapidity, may be viewed as signatures for
fluctuation effects in the LHC data. More work remains to be done in
order to clarify how important fluctuation or running coupling effects are at
given energy, e.g., at LHC energy. An extension of this work by the
running coupling may help to clarify some of the open questions.
\vskip 3mm
\leftline{\bf Acknowledgments}
\vskip 2mm
\noindent A. Sh. and M. K. acknowledge financial support by the Deutsche
Forschungsgemeinschaft under contract Sh 92/2-1.
\begin{footnotesize}
\bibliographystyle{blois07}
{\raggedright
|
0708.2894
|
\section{Statement of Results}\label{S:intro}
Let $\Omega\subset\mathbb{C}^2$ be a pseudoconvex domain (not necessarily bounded) having a smooth boundary.
Let $p\in\partial\Omega$ be a point of infinite type: by this we mean that for {\em each} $N\in\zahl_+$,
there exists a germ of a $1$-dimensional complex-analytic variety through $p$ whose order of
contact with $\partial\Omega$ at $p$ is at least $N$. If $\partial\Omega$ is not Levi-flat around $p$,
there exist holomorphic coordinates $(z,w;V_p)$ centered at $p$ such that
\begin{equation}\label{E:local}
\Omega\bigcap V_p \ = \ \{(z,w)\in V_p:\mi w > F(z)+R(z,\mathfrak{Re} w)\},
\end{equation}
where $F$ is a smooth, subharmonic, non-harmonic function defined in a neighbourhood of $z=0$,
that vanishes to infinite order at $z=0$; $R(\boldsymbol{\cdot} \ ,0)$ vanishes to
infinite order at $z=0$; and
$R$ is $O(|z||\mathfrak{Re} w|,|\mathfrak{Re} w|^2)$. Given the infinite order of vanishing of $F$ at $z=0$, how
does one find estimates for the Bergman kernel of $\Omega$ near $p$~? In many cases --- for instance:
when $\partial\Omega\cap V_p$ is pseudoconvex of strict type, in the sense of
\cite{kohn:bb/dbar/wpmd2:72}, away from $p\in\partial\Omega$ --- the function $F$ in \eqref{E:local}
can be extended to a global subharmonic function. In such situations, the model domain
\begin{equation}\label{E:mod}
\Omega_F \ := \ \{(z,w)\in\mathbb{C}^2:\mi w > F(z)\},
\end{equation}
approximates $\partial\Omega$ to infinite order along the complex-tangential directions at $p$.
One is thus motivated to investigate estimates for the Bergman kernel for domains of the
form \eqref{E:mod}. In this paper, we shall find estimates for the Bergman kernel of $\Omega_F$
on the diagonal as one approaches $(0,0)\in\partial\Omega_F$. More specifically:
\begin{itemize}
\item[\textbullet] We shall derive estimates
that hold {\em not just} in a non-tangential interior cone
with vertex at $(0,0)$, but for a family of much larger approach regions that comprises regions with
{\em arbitrarily high} orders of contact at $(0,0)$; and
\item[\textbullet] We shall find optimal estimates for the
growth of the kernel (evaluated on the diagonal
of $\Omega_F\times\Omega_F$) as $(z,w)\longrightarrow(0,0)$ through any of the aforementioned approach regions.
\end{itemize}
\smallskip
Pointwise estimates, and a lot more, have been obtained for finite-type domains in $\mathbb{C}^2$;
see for instance \cite{diedHerOhs:Bkuepd86} by Diederich {\em et al};
\cite{nagelRosaySteinWainger:eBkSkcwpd88} and \cite{nagelRosaySteinWainger:eBSkC289}
by Nagel {\em et al}; and \cite{mcneal:bbBkfC289} by McNeal. In \cite{kimLee:abBkaicitpd02},
Kim and Lee provide some estimates on the diagonal for the Bergman kernel, as one approaches
an infinite-type boundary point, for a class of convex, infinite-type domains in $\mathbb{C}^2$. However,
to the best of our knowledge, not even pointwise estimates are known for any reasonably general
class of pseudoconvex (not necessarily convex) domains of infinite type. Determining such
estimates even for model domains of the form \eqref{E:mod} is not so easy. For instance,
techniques analogous to the scaling methods used in the papers \cite{mcneal:bbBkfC289} and
\cite{nagelRosaySteinWainger:eBSkC289}, in \cite{boasStraubeYu:blBkm95} by Boas {\em et al},
and in \cite{krantzYu:BicBm96} by Krantz-Yu do not seem to yield optimal estimates. Another
problem is that we do not know {\em a priori} whether $\Omega_F$ --- recall that our models
{\em do not} arise as limits of scalings of bounded domains --- even has a non-trivial
Bergman space. Things become tractable if we impose a simplifying condition on $F$:
\[
(*) \ \begin{cases}
\ \text{$F$ is a radial function, i.e. $F(z)=F(|z|) \ \forall z\in\cplx$, and}\\
\ \text{$\exists\eta>0$ such that $F(z)\geq C|z|^\eta$ when $|z|\geq R$ for some $R>0$ and
$C>0$.}
\end{cases}
\]
Under this condition, $\Omega_F$ has a non-trivial Bergman space; see for instance
\cite{haslinger:BHsmd} by Haslinger.
However, given the condition $(*)$, we can say more: under this condition, $\Omega_F$ has
a bounded realization and thus admits a localization principle for the Bergman kernel.
To state this precisely, we recall that the Bergman projection for $\Omega$ is the orthogonal
projection $B_\Omega:\mathbb{L}^2(\Omega)\longrightarrow\mathcal{O}(\Omega)\cap\mathbb{L}^2(\Omega)$, and the Bergman kernel is the kernel
representing this projection. Let us denote the Bergman kernel of
$\Omega_F$ as $B_F(Z,Z^\prime), \ (Z,Z^\prime)\in\Omega_F\times\Omega_F$. We will denote the kernel
restricted to the diagonal by $K_F(z,w):=B_F((z,w),(z,w))$. We can now state
\smallskip
\begin{proposition}\label{P:locali}
Let $F$ be a $\mathcal{C}^\infty$-smooth subharmonic function that vanishes to infinite order at
$0\in\cplx$ and satisfies the condition $(*)$. Assume that the boundary of the domain
$\Omega_F:=\{(z,w)\in\mathbb{C}^2:\mi w > F(z)\}$ is not Levi-flat around the origin. Then:
\begin{enumerate}
\item[1)] There exists an injective holomorphic map $\Psi$ defined in a neighbourhood
of $\overline{\Omega}_F$ such that $\Psi(\Omega_F)$ is a bounded pseudoconvex domain.
\item[2)] For each polydisc $\triangle$ centered at the origin, there exists a
constant $\delta\equiv\delta(\triangle)>0$ such that
\begin{equation}\label{E:locali}
\delta K_{\Omega_F\cap\triangle}(z,w) \ \leq \ K_F(z,w) \;\; \forall
(z,w)\in\Omega_F\cap(\tfrac{1}{2}\triangle).
\end{equation}
\end{enumerate}
\end{proposition}
Yet, does the condition $(*)$ confer any degree of control on the decay of $F$ near
$z=0$ that is sufficient for optimal estimates; estimates on $K_F(z,w)$ {\em from below} in
particular~? Even if $F$ is radial, $K_F(z,w)\gtrsim\|(z,w)\|^{-2}$ is the best that one
expects (for non-tangential approach) without any {\em additional} information on $F$. To
illustrate: the condition that $(0,0)\in\partial\Omega_F$ is of finite type facilitates optimal
estimates because, with this extra information:
\begin{itemize}
\item[\textbullet] one can find constants $C,\delta>0$, and a $M\in\zahl_+$ such that
\begin{align}
(**) \;\; \ball{2}(0;\delta)\cap\{(z,w):\mi{w}>C|z|^{2M}\}
\subset \Omega_F&\cap \ball{2}(0;\delta) \notag \\
\subset \ball{2}(0;\delta)&\cap\{(z,w):\mi{w}>(1/C)|z|^{2M}\}; \notag
\end{align}
\item[\textbullet] one can now make precise estimates by exploiting the simplicity of the prototypal
defining function $z\longmapsto |z|^{2M}$.
\end{itemize}
Some condition that enables one to work --- in the spirit of $(**)$ --- with easier-to-handle
prototypes of $F$ is called for if one wants optimal estimates in the
infinite-type case. It turns out that we do have useful information if the infinite-type
$F$ satisfies the condition \eqref{E:flatness} spelt out in Theorem \ref{T:main} below.
While this condition might look rather arbitrary, it is in fact a mild restriction.
It is, in some sense, a signature of $F$ being of infinite type: a domain $\Omega_F$ satisfying
\eqref{E:flatness} is necessarily of infinite type at $(0,0)$. The condition
\eqref{E:flatness} encompasses a large class of domains, ranging from the
``mildly infinite type'' to the very flat at $(0,0)$ (refer to the observations
following Theorem~\ref{T:main}).
\smallskip
We need one further piece of notation. Let
$f:[0,\infty)\longrightarrow\mathbb{R}$ be a strictly increasing function, and let $f(0)=0$. We
define the function $\Lambda_f$ as
\[
\Lambda_f(x) \ := \ \begin{cases}
-1/\log(f(x)), &\text{if $0<x<f^{-1}(1)$}, \\
0, &\text{if $x=0$}.
\end{cases}
\]
We can now state our main theorem.
\begin{theorem}\label{T:main}
Let $F$ be a $\mathcal{C}^\infty$-smooth subharmonic function that vanishes
to infinite order at $0\in\cplx$ and satisfies the condition $(*)$. Suppose the
boundary of the domain $\Omega_F:=\{(z,w)\in\mathbb{C}^2:\mi w > F(z)\}$ is not Levi-flat around the
origin.
\begin{enumerate}
\item[1)] Define $f$ by the relation $f(|z|)=F(z)$. Then, $f$ is a strictly
increasing function on $[0,\infty)$.
\item[2)] Assume that $F$ satisfies the following condition:
\begin{align}
\text{$\exists$ constants} \ &B,\varepsilon_0>0,
\text{and a function $\chi\in\mathcal{C}([0,\varepsilon_0];\mathbb{R})$ s.t.} \notag \\
\; & \chi^p \ \text{is convex on $(0,\varepsilon_0)$ for some $p>0$, and}\qquad\quad \label{E:flatness} \\
(1/B)\chi(x) \ \leq \ &\Lambda_f(x) \ \leq \ B\chi(x) \;\; \forall x\in[0,\varepsilon_0]. \notag
\end{align}
Then, for each $\alpha>0$ and $N\in\zahl_+$, there exists a constant $H_{N,\alpha}>0$,
which depends only on $\alpha$ and $N$; and $C_0, C_1>0$, which
are independent of all parameters, such that:
\begin{align}\label{E:approach}
C_0(\mi w)^{-2}\left[f^{-1}(\mi w)\right]^{-2} \leq K_F&(z,w)
\leq \ C_1(\mi w)^{-2}\left[f^{-1}(\mi w)\right]^{-2} \\
&\forall (z,w)\in\mathscr{A}_{\alpha,N}, \ 0<\mi w<H_{N,\alpha}, \notag
\end{align}
where $\mathscr{A}_{\alpha,N}$ denotes the approach region
\[
\mathscr{A}_{\alpha,N} \ := \ \left\{(z,w)\in\Omega_F:\sqrt{|z|^2+|\mathfrak{Re} w|^2}<\alpha(\mi w)^{1/N}\right\}.
\]
\item[3)] Under the assumptions of (2), there exists a constant $H_0>0$ that is
independent of all parameters such that the left-hand inequality in \eqref{E:approach}
in fact holds for all $(z,w)\in \Omega_F\cap\{(z,w):\mi w<H_0\}$.
\end{enumerate}
\end{theorem}
\smallskip
The reader might like to see examples of domains that satisfy all the hypotheses of
Theorem~\ref{T:main}. We discuss two examples, beginning with a very familiar
example.
\smallskip
\begin{example}\label{Ex:exp} {\em Estimates for the pseudoconvex domain
\[
\Omega^\beta \ := \ \{(z,w)\in\mathbb{C}^2:\mi w > F_\beta(z)\}
\]
where:
\begin{itemize}
\item[\textbullet] $F_\beta$ is subharmonic;
\item[\textbullet] $F_\beta(z)=\exp(-1/|z|^\beta), \ \beta>0$, in a neighbourhood of $z=0$; and
\item[\textbullet] $F_\beta(z)$ grows like $|z|^2$ for $|z|\gg 1$.
\end{itemize}}
\noindent{We just have to check whether $F$ satisfies the condition \eqref{E:flatness}.
There exists an $\varepsilon_0>0$ such that $\Lambda_f(x)=x^\beta \ \forall x\in[0,\varepsilon_0]$.
We pick
\[
p \ = \ \begin{cases}
\text{any number $q$ such that $q\beta>1$}, &\text{if $0<\beta\leq 1$,} \\
1, &\text{if $\beta>1$.}
\end{cases}
\]
With such a choice for $p$, $(\Lambda_f)^p$ itself is convex on $(0,\varepsilon_0)$.
Hence, Theorem~\ref{T:main}
tells us that for each $\alpha>0$ and $N\in\zahl_+$, there exists a constant
$H_{N,\alpha}>0$; and $C_0,C_1>0$, which are independent of all parameters,
such that:
\begin{align}
C_0t^{-2}\left(\log(1/t)\right)^{2/\beta} \leq K_{\Omega^\beta}(z,s+it)
\leq &C_1 t^{-2}\left(\log(1/t)\right)^{2/\beta} \notag \\
&\forall (z,s+it)\in\mathscr{A}_{\alpha,N} \ \text{and} \ 0< t <H_{N,\alpha}. \qed\notag
\end{align}}
\end{example}
\smallskip
\begin{remark}\label{rem:Lam_f=inf}
We would like to emphasize here that {\em $\Lambda_f$ is allowed to vanish to infinite order at the origin,}
provided it satisfies condition~\eqref{E:flatness}. So, for example, Theorem~\ref{T:main} will provide
optimal growth estimates for $K_F$ for a domain $\Omega_F$ of the form \eqref{E:mod} where
\begin{itemize}
\item[\textbullet] $F(z)=\exp\left\{-e^{1/|z|}\right\}$ if $z:0\leq |z|\leq 1/4$; and
\item[\textbullet] $F(z)$ is so defined for $|z|\geq 1/4$ that $F$ satisfies condition $(*)$ and
$\Omega_F$ is pseudoconvex with non-Levi-flat boundary.
\end{itemize}
Domains like these are what we informally termed above as ``very flat at $(0,0)$''. The methods
used by Kim and Lee in \cite{kimLee:abBkaicitpd02} do not seem to work for domains like these
precisely because $\Lambda_f$ vanishes to infinite order.
\end{remark}
\smallskip
A few technical preliminaries are needed before a proof of Theorem~\ref{T:main} can
be given. It would be helpful to get a sense of the key ideas of our proof. A discussion of our
methodology, plus two lemmas, are presented in Section~\ref{S:BergPrelim}. The proof
itself is given in Section~\ref{S:proofMain}. In some sense, our key technical
preliminary --- without which sharp lower bounds would be tricky to derive --- is
the proof of Proposition~\ref{P:locali}. This proof will form our next section.
\medskip
\section{The proof of Proposition~\ref{P:locali}}
Let $\eta>0$ be as given in the condition $(*)$. We define
\[
{\kappa_\eta} \ := \ \begin{cases}
\text{the least positive integer $\kappa$ such that $\kappa>1/\eta$},
&\text{if $0<\eta\leq 1$,} \\
1, &\text{if $\eta > 1$}.
\end{cases}
\]
Define the objects
\begin{align}
\Psi=(\psi_1,\psi_2) \ &: \ (z,w)\ \longmapsto \ \left(\frac{(2i)^{\kappa_\eta} z}{(i+w)^{\kappa_\eta}},
\frac{i-w}{i+w}\right), \notag \\
\Pi \ &:= \ \cplx\times\{w\in\cplx:\mi w > -1\}. \notag
\end{align}
Note that $\Psi\in\mathcal{O}(\Pi;\mathbb{C}^2)$ and that $\Psi$ is injective on $\Pi$.
Define $f$ by the relation $f(|z|)=F(z)$. Then, under our hypotheses, $f$ is
strictly increasing, whence $F(z)\geq 0 \ \forall z\in\cplx$.
The reader is directed to Lemma~\ref{L:inc} for a proof of this
fact. Thus $\Omega_F\varsubsetneq\Pi$, whence $\Psi$ is injective on $\Omega_F$.
\smallskip
We now claim that $\Psi(\Omega_F)$ is bounded. To see this, note that any
$(z,w)\in\Omega_F$ can be written as $(z,\mathfrak{Re}{w}+i(F(z)+h)$, where $h>0$. Thus
\begin{equation}\label{E:psi2bd}
|\psi_2(z,w)|^2 \ = \ \frac{(F(z)+h-1)^2+(\mathfrak{Re}{w})^2}{(F(z)+h+1)^2+(\mathfrak{Re}{w})^2} \ \leq
\ 1 \;\; \forall(z,w)\in\Omega_F.
\end{equation}
We used the fact that $F(z)\geq 0 \ \forall z\in\cplx$ to deduce this estimate.
Now note that
\[
|\psi_1(z,w)|^2 \ = \ \frac{4^{\kappa_\eta}|z|^2}{((F(z)+h+1)^2+(\mathfrak{Re}{w})^2)^{\kappa_\eta}}.
\]
Let $R>0$ and $C>0$ be exactly as given in the condition $(*)$. Then
\[
|\psi_1(z,w)|^2 \ \leq \ \frac{4^{\kappa_\eta} R^2}{(F(z)+h+1)^{2{\kappa_\eta}}} \ \leq \ 4^{\kappa_\eta} R^2 \;\;
\forall(z,w)\in\Omega_F \ \text{and $|z|\leq R$.}
\]
On the other hand
\[
|\psi_1(z,w)|^2 \ \leq \ \frac{4^{\kappa_\eta}|z|^2}{(F(z))^{2{\kappa_\eta}}} \ \leq \
\frac{4^{\kappa_\eta}|z|^2}{C^{2{\kappa_\eta}}|z|^{2\eta{\kappa_\eta}}} \;\; \forall(z,w)\in\Omega_F \ \text{and $|z|\geq R$.}
\]
From the last two inequalities, we conclude that
\begin{equation}\label{E:psi1bd}
|\psi_1(z,w)|^2 \ \leq \ \max\left\{4^{\kappa_\eta} R^2, \
\left(\frac{4}{C^2}\right)^{\kappa_\eta} R^{-2(\eta{\kappa_\eta}-1)}\right\} \;\; \forall(z,w)\in\Omega_F.
\end{equation}
From \eqref{E:psi2bd} and \eqref{E:psi1bd}, our claim, and hence Part~(1), follows.
\smallskip
To demonstrate Part~(2), we will need a localization principle established by Ohsawa:
\begin{itemize}
\item[{}] {\bf {\em Localization Lemma} (Ohsawa, \cite{ohsawa:bbBkfpd84})}
{\em Let $D$ be a bounded pseudoconvex domain
in $\mathbb{C}^n$, $p$ be a boundary point, and $V\Subset U$ be two open neighbourhoods of $p$.
Then, there is a constant $\delta\equiv\delta(U,V)>0$ such that
\[
\delta K_{D\cap U}(Z) \ \leq \ K_D(Z) \;\; \forall Z\in D\cap V.
\]}
\end{itemize}
\noindent{Substituting
\begin{align}
D \ &= \ \Psi(\Omega_F), && \qquad p \;\; = \ (0,1), \notag \\
D\cap U \ &= \ \Psi(\Omega_F\cap\triangle), &&D\cap V \ = \
\Psi\left(\Omega_F\cap\left(\tfrac{1}{2}\triangle\right)\right) \notag
\end{align}
into the localization lemma, we conclude that there exists a $\delta\equiv\delta(\triangle)>0$
such that (here $G_F$ stands for $\Psi(\Omega_F)$)
\begin{equation}\label{E:locali1}
\delta K_{G_F\cap U}(\Psi(z,w)) \ \leq \ K_{G_F}(\Psi(z,w)) \;\; \forall
(z,w)\in\Omega_F\cap(\tfrac{1}{2}\triangle).
\end{equation}
Recall, however, the transformation rule for the Bergman kernel:
\[
K_{\Omega^j}(z,w) \ = \ |{\rm Jac}_\cplx(\Psi)(z,w)|^2 K_{\Psi(\Omega^j)}(\Psi(z,w)) \;\;
\forall (z,w)\in\Omega^j, \ j=1,2,
\]
where, in the present case, $\Omega^1=\Omega_F$ and $\Omega^2=\Omega_F\cap\triangle$. Applying this
to \eqref{E:locali1} gives us the inequality \eqref{E:locali}.\qed
\medskip
\section{Preliminary remarks and lemmas}\label{S:BergPrelim}
The idea behind the upper bound in \eqref{E:approach} is quite standard. Given a point $(z_0,w_0)\in\Omega_F$,
the quantity $K_F(z_0,w_0)$ is dominated by the reciprocal of the volume of the largest polydisc centered
at $(z_0,w_0)$ that is contained in $\Omega_F$. The volume of this polydisc will be influenced by the curvature
of $\partial\Omega_F$ at the point on $\partial\Omega_F$ that is closest to $(z_0,w_0)$. However, if $(z_0,w_0)$ is confined
to any of the approach regions $\mathscr{A}_{\alpha,N}$, then this volume is controlled by the boundary geometry
at $(0,0)\in\partial\Omega_F$. That one has this control for any approach region $\mathscr{A}_{\alpha,N}$ --- regardless
of $\alpha$ and $N$ --- is a consequence of the fact that $(0,0)$ is of infinite type.
\smallskip
The derivation of the lower bound in \eqref{E:approach} relies on the construction of a
suitable square-integrable holomorphic function. In this construction, we are aided by the
localization principle stated in Proposition~\ref{P:locali}.
The three main ingredients in the derivation of the lower bound are:
\begin{itemize}
\item[{\em i)}] We choose a suitable polydisc $\triangle$ centered at $(0,0)$ and estimate
$K_{\Omega_F\cap\triangle}(z,w)$ for $(z,w)\in\mathscr{A}_{\alpha,N}\cap\left(\tfrac{1}{2}\triangle\right)$. We rely
on the fact that $K_{\Omega_F\cap\triangle}(z,w)$ is given by
\[
K_{\Omega_F\cap\triangle}(z,w) \ = \ \sup\left\{\frac{|\phi(z,w)|^2}{\|\phi\|^2_{\mathbb{L}^2({\OM_F\cap\triangle})}}:
\phi\in A^2(\Omega_F\cap\triangle)\right\}.
\]
\item[{\em ii)}] To obtain a lower bound, we select a suitable function $\phi_t\in A^2(\Omega_F\cap\triangle)$
and estimate $\|\phi_t\|^2_{A^2}, \ t>0$. This reduces
finding a lower bound for $K_{\Omega_F\cap\triangle}(z,s+it)$,
$(z,s+it)\in\mathscr{A}_{\alpha,N}\cap\left(\tfrac{1}{2}\triangle\right)$, to estimating
an integral over a region in $\mathbb{R}^4$ whose boundaries are determined by the function $f$.
\item[{\em iii)}] The difficult issue is to find the {\em desired bound in terms of $t$} for
the latter integral. The condition~\eqref{E:flatness} is used to break up the aforementioned
region of integration into sub-domains on which the integral admits the desired estimate.
\end{itemize}
\smallskip
We now present two lemmas that will be necessary to complete the proof of
Theorem~\ref{T:main}. Lemma~\ref{L:inc} constitutes the proof of Part~(1) of
Theorem~\ref{T:main}.
\smallskip
\begin{lemma}\label{L:inc}
Let $F$ be a smooth subharmonic function on $\cplx$ such that $F(0)=0$ and $F$ is
radial. Define $f$ by the relation
$f(|z|)=F(z)$, and write $\Omega_F:=\{(z,w)\in\mathbb{C}^2:\mi w > F(z)\}$.
Assume that $\Omega_F$ is not Levi-flat in a neighbourhood of $(0,0)$. Then $f$ is a strictly
increasing function on $[0,\infty)$.
\end{lemma}
\begin{proof}
Suppose there exist $r_1<r_2$, with $r_1,r_2\in[0,\infty)$, such that $f(r_1)\geq f(r_2)$.
Then, by our hypothesis on $F$
\[
\sup_{z\in\partial D(0;r_2)}F(z) \ = \ f(r_2) \ \leq \ f(r_1).
\]
By the Maximum Principle, therefore, $F|_{D(0,r_2)}\equiv 0$. But then, this would imply
that the portion $\partial\Omega_F\cap\ball{2}(0;r_2)$ of $\partial\Omega_F$ is Levi-flat; i.e. a
contradiction. Hence $f$ is strictly increasing.
\end{proof}
\smallskip
\begin{lemma}\label{L:bigKey}
Let $F$ and $\Omega_F$ have all the properties listed in Lemma~\ref{L:inc}. Let $\Lambda_f$ satisfy
the condition~\eqref{E:flatness}, and let (since, in view of Lemma~\ref{L:inc}, $f$
is increasing) $G_f:=\Lambda_f^{-1}$. Then, there exist constants $T>0$ and $K>0$ such that
\begin{equation}\label{E:compare}
0 \ < \ G_f(2t)^2-G_f(t)^2 \ \leq \ KG_f(t)^2 \;\; \forall t\in(0,T).
\end{equation}
\end{lemma}
\begin{proof}
We just have to show that there exist $T>0$ and $M>0$ such that
\begin{equation}\label{E:prelim}
0 \ < \ G_f(2t)-G_f(t) \ \leq \ MG_f(t) \;\; \forall t\in(0,T).
\end{equation}
If we could show this, then it would follow that
\[
G_f(2t)^2-G_f(t)^2 \ \leq \ M(M+2)G_f(t)^2 \quad \forall t\in(0,T). \notag
\]
To proceed further, we need the following:
\smallskip
\noindent{{\bf Fact.} {\em Let $g$ be a continuous, strictly increasing function on $[0,R]$
satisfying $g(0)=0$, and assume $g^p$ is convex on $(0,R)$ for some $p>0$. Define
$G:=g^{-1}$, and let $B>1$. Then:}
\begin{equation}\label{E:ratio}
\frac{G(Bt)}{G(t)} \ \leq \ B^p \;\; \forall t\in (0,g(R)/B).
\end{equation}}
\noindent{To verify this fact, set $\Phi:=(g^p)^{-1}$. Then:
\begin{equation}\label{E:simplify}
\Phi(t) \ = \ G(t^{1/p}) \quad \forall t\in[0,g(R)^p],
\end{equation}
By hypothesis, $\Phi$ is concave on $(0,g(R)^p)$. But since $\Phi$ is also continuous,
\[
\frac{\Phi(B^{p}t^p)}{B^{p}t^p} \ \leq \ \frac{\Phi(t^p)}{t^p} \;\; \forall
t\in (0,g(R)/B).
\]
The above fact now follows simply by rearranging the terms in the above inequality, and
applying \eqref{E:simplify}.}
\smallskip
Now let $\varepsilon_0$, $B$, $\chi$, and $p$ be as in \eqref{E:flatness}. Let us also define
\begin{align}
\varkappa_0 &:= (\chi)^{-1} : [0,\chi(\varepsilon_0)]\longrightarrow \mathbb{R} \notag \\
\varkappa_1 &:= (B\chi)^{-1} : [0,B\chi(\varepsilon_0)]\longrightarrow \mathbb{R} \notag \\
\varkappa_2 &:= \ \left[(1/B)\chi\right]^{-1} : [0,(1/B)\chi(\varepsilon_0)]\longrightarrow \mathbb{R} \notag
\end{align}
Since $\Lambda_f$ and $\chi$ are strictly increasing (in view of Lemma~\ref{L:inc}),
condition~\eqref{E:flatness} implies
that:
\[
\varkappa_1(t) \ \leq \ G_f(t) \ \leq \ \varkappa_2(t) \;\; \forall t\in[0,T_1],
\]
where $T_1:=(1/B)\chi(\varepsilon_0)$. Therefore, we get
\begin{equation}\label{E:sandwich}
\frac{G_f(2t)}{G_f(t)} \ \leq \
\frac{\varkappa_2(2t)}{\varkappa_1(t)} \ = \
\frac{\varkappa_2(2t)}{\varkappa_2(t)} \ \frac{\varkappa_2(t)}{\varkappa_1(t)} \;\;
\forall t\in(0,T_1/2).
\end{equation}
We now note that
\[
\varkappa_1(t) \ = \ \varkappa_0(B^{-1}t), \quad
\varkappa_2(t) \ = \ \varkappa_0(Bt) \quad \forall t\in[0,T_1].
\]
Given this last piece of information, we can apply the inequality \eqref{E:ratio} to
the ratios on the right-hand side of \eqref{E:sandwich}. Set $T:=\min(T_1/2,(1/B)T_1)$.
Then,
\[
\frac{G_f(2t)}{G_f(t)} \ \leq \ (2B^2)^p
\;\; \forall t\in(0,T).
\]
From this, the estimate \eqref{E:prelim} clearly follows if we take
$M=(2B^2)^p-1$. Hence, by our earlier remarks, the result follows.
\end{proof}
\medskip
\section{The proof of Theorem~\ref{T:main}}\label{S:proofMain}
Part~(1) of Theorem~\ref{T:main} has already been established in Lemma~\ref{L:inc}.
Therefore, $f^{-1}$ is a well-defined function. Observe that
\begin{equation}\label{E:inverse}
f^{-1}(t) \ = \ G_f\left(\frac{1}{\log(1/t)}\right), \quad 0<t<1.
\end{equation}
Let $R>0$ be so small that
\[
f^{-1}(3t/4) \ = \ G_f\left(\frac{1}{\log(4/3)+\log(1/t)}\right) \ \geq \
G_f\left(\frac{1}{2\log(1/t)}\right) \;\; \forall t\in(0,R).
\]
Let $M$ and $T$ be as given by \eqref{E:prelim} above. Shrinking
$R>0$ if necessary so that $0<1/2\log(1/t)<T \ \forall t\in(0,R)$, we get
\begin{equation}\label{E:compare1}
\frac{f^{-1}(3t/4)}{f^{-1}(t)} \ \geq \ \frac{G_f(1/2\log(t^{-1}))}{G_f(1/\log(t^{-1}))} \
\geq \ (M+1)^{-1} \;\;\forall t\in[0,R).
\end{equation}
Write $\mu:=(M+1)^{-1}$. We are given $\alpha>0$ and $N\in\zahl_+$. Since $f(x)$ vanishes to infinite order
at $x=0$, there exists a $H_{N,\alpha}>0$ such that $R\geq H_{N,\alpha}$ and
\begin{equation}\label{E:compare2}
\alpha t^{1/N} \ \leq \ \frac{\mu}{4}f^{-1}(t) \quad\forall t\in[0,H_{N,\alpha}).
\end{equation}
From \eqref{E:compare1} and \eqref{E:compare2}, we see that
\[
|z|+\frac{\mu}{2}f^{-1}(t) \ < \ f^{-1}(3t/4)\;\;\forall z:0\leq|z|<\alpha t^{1/N}, \
0<t<H_{N,\alpha},
\]
whence the polydisc
\begin{multline}
\triangle(z,t) \ := \ \mathbb{D}\left(z;\frac{\mu}{2}f^{-1}(t)\right)\times\mathbb{D}(it;t/4) \ \subset \ \Omega_F \\
\forall z:0\leq|z|<\alpha t^{1/N}, \ 0<t<H_{N,\alpha}.
\end{multline}
Now note that that translations $T_s:(z,w)\longmapsto (z,s+w), \ s\in\mathbb{R}$, are all
automorphisms of $\Omega_F$. Thus, by the transformation rule for the Bergman
kernel, and by monotonicity, we get
\begin{align}
K_F(z,s+it) \ &= \ K_F(z,it) \notag \\
&\leq \ K_{\triangle(z,t)}(z,it) \notag \\
&= \ \frac{1}{{\rm vol}\left(\triangle(z,t)\right)} \;\;
\forall (z,s+it)\in\mathscr{A}_{\alpha,N}, \ 0<t<H_{N,\alpha}. \notag
\end{align}
The last equality follows from the fact that $\triangle(z,t)$ is a Reinhardt domain
centered at $(z,t)$. Hence, we have one half of the estimate~\eqref{E:approach}:
\begin{align}
K_F(z,w)
\leq \ C_1(\mi w)^{-2}\left[f^{-1}(\mi w)\right]^{-2}\;\;
\forall &(z,w)\in\mathscr{A}_{\alpha,N}, \label{E:1stHalf} \\
&0<\mi w<H_{N,\alpha}, \notag
\end{align}
where $C_1=64/\mu^2\pi^2$.
\smallskip
We will now derive a lower bound. We set $A:=\min(f^{-1}(1),1)$. For the remainder of
this proof, $\triangle$ will denote the polydisc $\mathbb{D}(0;A)\times\mathbb{D}(0;1)$. In view of
the inequality \eqref{E:locali} of Proposition~\ref{P:locali}, it suffices to find
a lower bound for $K_{\Omega_F\cap\triangle}(z,w)$ for $(z,w)\in\left(\tfrac{1}{2}\triangle\right)$.
It is well known that
\begin{equation}\label{E:maxim}
K_{\Omega_F\cap\triangle}(z,w) \ = \ \sup_{\phi\in A^2(\Omega_F\cap\triangle)}
\frac{|\phi(z,w)|^2}{\|\phi\|^2_{\mathbb{L}^2({\OM_F\cap\triangle})}}.
\end{equation}
Once again, we use the fact that the translations $T_u:(z,w)\longmapsto (z,u+w), \ u\in\mathbb{R}$,
are all automorphisms of $\Omega_F$, whence
\begin{equation}\label{E:translate}
K_F(z,s+it) \ = \ K_F(z,(u+s)+it) \quad\forall(z,s+it)\in\Omega_F \ \text{and $\forall u\in\mathbb{R}$}.
\end{equation}
Set $\phi_t(z,w):=-4t^2/(w+it)^2, \ t>0.$ Then, from the localization principle \eqref{E:locali}, and
from \eqref{E:maxim} and \eqref{E:translate}, we get
\begin{align}\label{E:lbdK1}
K_F(z,s+it) \ &\geq \ \delta K_{\Omega_F\cap\triangle}(z,it) \\
&\geq \ \frac{\delta}{\|\phi_t\|^2_{\mathbb{L}^2({\OM_F\cap\triangle})}}\;\;
\forall (z,t)\in\left(\tfrac{1}{2}\triangle\right).\notag
\end{align}
Let us write $w=u+iv$. We leave the reader to verify that we can apply Fubini's theorem
wherever necessary in the following computation:
\begin{align}
\|\phi_t\|^2_{\mathbb{L}^2({\OM_F\cap\triangle})} \ &= \
\lint{|z|<A}{{}} \ \lint{-1}{1}\lint{F(z)}{\sqrt{1-u^2}}\frac{16t^4}{|u+i(v+t)|^4}dv \ du \ dA(z)
\notag \\
&\leq \
16t^4\lint{|z|<A}{{}} \ \lint{F(z)}{\infty} \ \lint{-1}{1}(v+t)^{-4}
\left(1+\left(\frac{u}{v+t}\right)^2\right)^{-2}du \ dv \ dA(z) \notag \\
&\leq \
8t^4\left(\lint{\mathbb{R}}{{}}\frac{dX}{(1+X^2)^2}\right)
\lint{|z|<A}{{}}(t+F(z))^{-2}dA(z)\notag \\
&= \ Ct^4 \lint{0}{A}\frac{r}{(t+f(r))^2}dr, \notag
\end{align}
where $C>0$ is a universal constant. In what follows, we shall denote $f^{-1}(s)$ by
$R_{s}$. By equation \eqref{E:inverse} we have
\begin{equation}\label{E:inverse2}
R_{\sqrt{t}} \ = \ G_f\left(\frac{2}{\log(1/t)}\right), \quad 0<t<1.
\end{equation}
We break up the interval of integration
of the last integral into three sub-intervals to compute:
\begin{align}
\|\phi_t\|^2_{\mathbb{L}^2({\OM_F\cap\triangle})} \ &= \
Ct^4 \left(\int_0^{R_t} + \ \int_{R_t}^{R_{\sqrt{t}}} + \ \int_{R_{\sqrt{t}}}^A\frac{r}{(t+f(r))^2}dr\right)
\notag \\
&\leq \ Ct^4\int_0^{R_t}\frac{r}{t^2}dr +
Ct^4 \left(\int_{R_t}^{R_{\sqrt{t}}} + \ \int_{R_{\sqrt{t}}}^A
\frac{r}{4tf(r)}dr\right) \notag \\
&\leq \ \frac{C}{2}t^2(R_t)^2 + \frac{C}{4}t^2\int_{R_t}^{R_{\sqrt{t}}}r \ dr +
\frac{C}{4}t^{5/2}A(A-R_{\sqrt{t}}) \notag \\
&\leq \ \frac{C}{2}t^2(R_t)^2 + \frac{C}{4}t^{5/2}A(A-R_{\sqrt{t}}) \label{E:integ1} \\
&\qquad\quad+\frac{C}{8}t^{2}\left(G_f^2\left(\frac{2}{\log(1/t)}\right)-
G_f^2\left(\frac{1}{\log(1/t)}\right)\right),\;\; 0<t<1. \notag
\end{align}
We used the relation \eqref{E:inverse2} in the estimate for the middle integral above.
\smallskip
We now apply Lemma~\ref{L:bigKey} to the third term in \eqref{E:integ1}. Let
$T>0$ and $K>0$ be defined as given by Lemma~\ref{L:bigKey}. Let $H_0$ be so small that
$1/\log(t^{-1})<T \ \forall t\in(0,H_0)$, {\em and} so that the second inequality below
holds true:
\begin{align}\label{E:integ2}
\|\phi_t\|^2_{\mathbb{L}^2({\OM_F\cap\triangle})} \ &\leq \
\frac{C}{2}\left(1+\frac{K}{4}\right)t^2(R_t)^2 + \frac{C}{4}t^{5/2} \\
&\leq \ C(1+K/4)t^2(R_t)^2 \;\; \forall t\in(0,H_0).\notag
\end{align}
Since $f(x)$ vanishes to infinite order at $x=0$, we can lower $H_0$ --- and this is independent
of parameters like $\alpha>0$ and $N\in\zahl_+$ --- so that the first term of the first inequality
above dominates the second for $t\in(0,H_0)$, giving us \eqref{E:integ2}. Lowering the value of
$H_0$ further if necessary, we also ensure:
\[
\Omega_F\cap\{(z,w):\mi w<H_0\} \ \subset \ \Omega_F\cap\triangle.
\]
From \eqref{E:lbdK1} and \eqref{E:integ2}, we conclude that there exists a constant
$C_0$, which is independent of all parameters, such that
\[
C_0(\mi w)^{-2}\left[f^{-1}(\mi w)\right]^{-2} \leq K_F(z,w) \;\;
\forall (z,w)\in \Omega_F\cap\{(z,w):\mi w<H_0\}.
\]
This establishes Part~(3) of our theorem. As a special case, we get the lower bound
on $K_F(z,w)$ in the estimate \eqref{E:approach}. Along with \eqref{E:1stHalf}, this
establishes Part~(2) of our theorem. \qed
\bigskip
\noindent{{\bf Acknowledgements.} First, and foremost, I thank Alexander Nagel for his
interest in an earlier version of this article. My discussion with him led to the
simplification of several arguments in that version; now the arXiv preprint
\texttt{arXiv:0708.2894v1}. I also thank all my colleagues who noticed this preprint and
offered their comments during my stay at the Institut Mittag-Leffler during the Special
Semester on complex analysis in several variables. The hospitality of the Institut
Mittag-Leffler is greatly appreciated. Finally, I thank the anonymous referee of this
article for his/her valuable suggestions.}
|
2207.13801
|
\section{INTRODUCTION}
Sleep is known to play a significant role in the mental and physical health of an individual \cite{siegel2005clues} and therefore the development of tools to diagnose the quality of sleep and common sleep pathologies is fundamental.
Sleep is monitored through polysomnography (PSG), \emph{i.e.}, the analysis of electrical bio-signals, such as the electroencephalogram (EEG), the electromyograph (EMG), the electrooculograph (EOG), and the electrocardiograph (ECG). The recorded bio-signals are split into 30-second intervals (epochs), and annotated by clinicians into several categories, such as: wake (W), non-rapid eye movement (NREM: N1, N2, and N3), and rapid eye movement (REM). This task is difficult and time-consuming. Thus, the ability to carry it out consistently and at a large scale through automation has an important impact on medical research and clinical practice \cite{wulff2010sleep}.
Towards this purpose, machine learning methods have been introduced as a way to obtain automatic sleep scoring \cite{fiorillo2021deepsleepnet, phan2019seqsleepnet,kemp2000sleepedf}. However, these methods are still not widely adopted amongst sleep practitioners. One limitation is that the current methods for sleep stage classification typically experience a decay in performance on data obtained from new cohorts of patients.
The main reason behind this decay is the large variability of the bio-signals across subjects and sessions. This variability stems from experimental factors such as differences in the recording equipment and protocol (e.g., the number and placement of the electrodes) or physiological factors such as age, prognosis and medication.
While this problem is commonly known among practitioners through direct experience, we illustrate it quantitatively in detail in our experimental analysis.
An approach to overcome this limitation is transfer learning, where a classification model is adapted to the target cohort through further training. This approach also motivated the recent work MetaSleepLearner \cite{banluesombatkul2020metasleeplearner}, which builds on meta-learning in the case of few-shot adaptation. MetaSleepLearner requires only a small set of annotations of the target dataset and a limited amount of training whenever new data becomes available.
However, as we show in our experiments, the scheme used in MetaSleepLearner, \emph{i.e.}, MAML, still runs the risk of overfitting even if the adaptation is done on a large dataset. Moreover, we find the few-shot learning scenario or, more in general, the transfer learning case, not practical, because practitioners would need further training and/or to provide annotation to adapt a classifier to new target data.
Thus, in this paper we propose to build a single sleep staging model and then use it ``as is'' on new data. To avoid the overfitting issues of the MetaSleepLearner we combine the Model Agnostic Meta-Learning (MAML) framework with self-supervised learning (SSL) \cite{lemkhenter2020boosting}. SSL has the advantage of not requiring annotation and it can be designed to train models that overfit less to the training data. With a slight abuse of notation we refer to the proposed setting as \emph{zero-shot learning}, to emphasize that no new training or annotation is needed with new data.
To the best of our knowledge, the zero-shot learning scenario has not been explored so far in the literature for sleep scoring.
We test our proposed method on several datasets and find that the use of SSL with MAML yields state of the art performance in zero-shot learning.
\section{RELATED WORK}
The application of deep learning together with the steady increase in available public sleep data have resulted in a dramatic improvement of the performance of methods for automated sleep staging. These methods have now reached high levels of accuracy and robustness. Moreover, thanks to their computational efficiency they can be employed at a large scale and they can work directly on the raw data, instead of requiring hand-crafted guessing of useful pre-processing procedures.
\subsection{Automatic Sleep Scoring}
Recent methods, such as SeqSleepNet \cite{phan2019seqsleepnet}, have focused on exploiting the context of the data by staging sequences rather than single epochs, or aimed at reducing the model parameters and introducing the estimation of the uncertainty of the prediction \cite{fiorillo2021deepsleepnet}.
One important limitation that has emerged is the lack of generalization, \emph{i.e.}, the drop in performance when trained models are used on new data. As mentioned in the Introduction, this phenomenon is currently attributed to the large diversity of the data across subjects/patients and sessions.
To address this problem, U-Sleep \cite{perslev2021u} introduces a u-net architecture for high frequency sleep staging. However, the generalization across datasets still remains an open problem.
In our method, we do not seek for an optimal architecture, but rather for a training scheme that can be easily adapted to other methods. Thus, for simplicity, we adopt the DeepSleepNet-Lite \cite{fiorillo2021deepsleepnet} as the classification model.
\subsection{Meta Learning}
A more fundamental approach towards generalization is the meta-learning framework \cite{finn2017model}. The main objective of meta-learning is to \emph{learn to learn}. In other words, rather than just learning from a single dataset to generalize to new data from the same distribution (as in the training set), one aims to train a model that can generalize well across other datasets.
A recent approach in this domain is Model Agnostic Meta Learning (MAML) \cite{finn2017model}. This method has been the employed successfully in recent work on sleep staging
\cite{banluesombatkul2020metasleeplearner}, brain-computer interfacing \cite{li2021model} and for emotion prediction \cite{miyamoto2021meta}.
In our method, we combine it with self-supervised learning and show experimentally that this yields a significant boost in performance.
\subsection{Self-Supervised Learning}
Self-supervised learning (SSL) is a relatively recent technique in machine learning that has emerged as a very promising and powerful unsupervised learning approach. SSL allows one to train a model on data without annotation by specifying an artificial task, also called \emph{pseudo-task}. A model trained with such a pseudo-task can then be adapted to some target data through transfer learning. Recent work has defined pseudo-tasks based on detecting the phase-swap in EEG \cite{lemkhenter2020boosting} and the context prediction and contrastive predictive coding \cite{banville2021uncovering}.
In our method, we propose to combine the ability to generalize of an SSL method \cite{lemkhenter2020boosting}
with the meta-learning framework.
\section{METHODS}
\subsection{Datasets}
In this work, we use 5 different sleep scoring datasets.
\subsubsection{Sleep Cassette (SC)} It is a subset of the Expanded Sleep-EDF Database \cite{kemp2000sleepedf} . It contains PSG sleep recordings obtained between 1981 and 1991. It includes recordings from 78 healthy subjects between the age of 25 and 101 with two recordings per person for most of them.
\subsubsection{Sleep Telemetry (ST)} It is another subset of the Expanded Sleep-EDF Database \cite{kemp2000sleepedf}. It was collected as part of a 1994 study of the effect of temazepam on sleep. It contains PSG recordings from 22 subjects with one session per individual.
Old datasets like SC \& ST allows us to investigate generalization from/to recordings with different signal quality.
\subsubsection{ISRUC} It is a publicly available sleep dataset \cite{khalighi2016isruc}. It consists of PSG recordings obtained at the Sleep Medicine Centre of the Hospital of Coimbra University (CHUC) between 2009 and 2013. This database has three different subsets:
\begin{itemize}
\item Subgroup-I contains one recording per subject for 100 individuals with sleep disorders;
\item Subgroup-II contains two recordings per subject for 8 individuals with sleep disorders;
\item Subgroup-III contains one recording per subject for 10 healthy individuals.
\end{itemize}
\subsubsection{University College Dublin Sleep Apnea Database (UCD)} It is a 2011 database collected at St. Vincent's University Hospital \cite{heneghan2011ucd}. It contains one PSG recording per subject for 25 individuals with suspected sleep-disordered breathing.
\subsubsection{Cyclic Alternating Pattern (CAP) Sleep Database} It is a collection of one recording per subject for 108 individuals with varying conditions. It contains 10 healthy subjects, 40 diagnosed with NFLE, 22 affected by RBD, 10 with PLM, 9 insomniac, 5 narcoleptic, 4 affected by SDB and 2 by bruxism \cite{terzano2001cap}. It was published in 2001.
\subsection{Data Preprocessing}
Out of all the signals available in each recording, we keep the EEG , EMG and EOG channels. All signals are re-sampled at 102.4Hz. This allows us to represent a 30sec epoch with 3072 time points, which is more compact and closer to the original sampling frequency compared to the commonly adopted 128Hz. This is sufficient since most spectral features classically used for sleep scoring are at lower frequency bands.
Since the convolution architecture we adopted in our experiments requires a constant number of channels as input, we fix that number to 9. If the recording contains more than 9 channels, which is the case for ISRUC and CAP, we randomly select a subset of them. Otherwise, if the recording does not have enough channels, we add dummy ones that are all zeros. The channels are shuffled before being fed to the model. We normalize each channel to have zero mean and a unit standard deviation.
\subsection{Data Split}
\label{sec:seenunseen}
To evaluate our models, we choose two different train/evaluation splits. We first split each dataset by \textbf{subjects}, then we randomly split each recording into samples of $3 \times 30$sec. This allows us to have an evaluation set containing subjects that were \textbf{seen} during training and another evaluation set containing \textbf{unseen} ones. Both splits follow a 75\%-25\% ratio. An illustration of both splits is shown in Table~\ref{tab:split}.
\begin{table}[t]
\centering
\caption{Diagram illustrating our two evaluation sets in a setting with 4 subjects and 4 samples per subject.}
\label{tab:split}
\begin{tabular}{|l|l|l|l|l|}
\hline
& \textbf{Sample 1} & \textbf{Sample 2} & \textbf{Sample 3} & \textbf{Sample 4} \\ \hline
\textbf{Subject 1} & Train & Eval. Seen & Train & Train \\ \hline
\textbf{Subject 2} & Train & Train & Train & Eval. Seen \\ \hline
\textbf{Subject 3} & Train & Train & Train & Eval. Seen \\ \hline
\textbf{Subject 4} & - & - & - & Val. Unseen \\ \hline
\end{tabular}
\end{table}
\subsection{Notation}
We define the mapping $E: x \mapsto h$ as the encoding of the input signal $x$ into a feature vector $h$. The associated trainable parameters are denoted by $\Theta^E$. We denote the mapping from the feature vector $h$ to the predicted class label $\hat{y}$ for the supervised and self-supervised settings as $C^{SL}$ and $C^{SSL}$ respectively. Their associated trainable parameters are $\Theta_{SL}^C$ and $\Theta_{SSL}^C$ respectively. The predicted labels of the model are therefore \begin{equation}\label{eq:y_hat}
\hat{y}^{SL/SSL} = C^{SL/SSL}(E(x)).
\end{equation}
In all experiments, models are trained by minimizing the average cross-entropy loss given by
\begin{equation}\label{eq:ce}
\mathcal{L}(T, \Theta^E, \Theta^C) = \frac{1}{|T|}\sum_{t \in T} \frac{1}{|t|}\sum_{(x, y) \in t}- \sum_{c=1}^{N_c} y_c log(\hat{y}_c),
\end{equation}
where $y$, $\hat{y}$, $N_c$ and $T$ are respectively the true labels, the model predictions, the numbers of classes and a set of tasks $t$ consisting of signal-label pairs. Note that we represent the true labels as a one-hot encoding vector so $v_c$ refers to the $c$-th entry in the vector $v \in \mathbb{R}^{N_c}$. We frame the sleep scoring problem as a 5-way classification with the 5 classes being: Wake (W), N1, N2, N3 and REM.
\subsection{Self-Supervised MAML (S2MAML)}
Model Agnostic Meta Learning (MAML) is a meta-learning algorithm, where a given model is trained on a large variety of tasks with the goal of generalizing to novel tasks through fast-adaptation, \emph{i.e.}, few-shot learning or with no adaptation, \emph{i.e.}, zero-shot learning. In this work, we investigate the benefit of using meta-learning jointly with self-supervised learning to improve generalization to unseen subjects and datasets.
The problem that our proposed model solves can be described using the following bilevel formulation
\begin{align}
\Theta^{*E}, \Theta^{*C} &= \argmin_{\Theta^{E}, \Theta^{C}_{SL}} \mathcal{L}( T_{SL}, \hat{\Theta}^{E}, \Theta^{C}_{SL})\\
\text{s.t. } &\hat{\Theta}^{E}, \hat{\Theta}^{C}_{SSL}
= \argmin_{\Theta^{E}, \Theta^{C}_{SSL}} \mathcal{L}( T_{SSL}, \Theta^{E}, \Theta^{C}_{SSL}), \nonumber
\end{align}
where the model $E$ is optimized to learn useful self-supervised representations of the set of tasks $T_{SSL}$. The bilevel optimization favors representations that generalize well to the supervised outer problem on tasks $T_{SL}$.
Algorithm~\ref{alg:main} outlines our adaptation of MAML, which we call \textbf{S2MAML}. Given $K$ datasets $\{D_k\}_{k=1}^K$, we randomly sample $n_{tasks}$ tasks from each one of them. Each task $t = \{(x_j, y_j)\}_{j=1}^{N_s}$ is defined as a set signal and label pairs belonging to the same subject in a given dataset. The total set of tasks $T$ is then split into a meta-training set $T^{tr}$ and a meta-validation set $T^{val}$. Each MAML iteration consists of an inner and an outer optimization problem. In the inner problem, $\Theta_{in}^{E}$ is initialized with the values of $\Theta^E$. Both $\Theta_{in}^{E}$ and $\Theta^{C}_{SSL}$ are optimized for $n_{in}$ iterations with respect to the self-supervised loss $\mathcal{L}^{in}$ computed on the meta-training set. For that, we need to generate a set of self-supervised tasks $T^{SSL}$ based on the $T^{tr}$. The details of this step are outlined in section~\ref{sec:ps}.
The weights $\Theta^E$ are then updated in the outer problem by minimizing the supervised loss $\mathcal{L}^{out}$ (see Algorithm~\ref{alg:main}) computed on the meta-validation set. The gradient for the $\Theta^E$ update is computed at $\Theta_{in}^E$, and not at $\Theta^E$, because we use the first-order approximation version of MAML \cite{finn2017model}.
The goal of this design is to encourage the model to learn general purpose self-supervised features in the inner problem that would generalize well to the outer supervised problem computed on novel tasks, \emph{i.e.}, unseen subjects.
\begin{algorithm}[h]
\caption{S2MAML}\label{alg:main}
\begin{algorithmic}
\Require $\{D_k\}_{k=1}^K, \Theta^E, \Theta_{SSL}^C, \Theta_{Sup}^C, \lambda_{in}, \lambda_{out}$
\While{not converged}
\State $T \gets \{\}$
\For{ $k$ in $1..K$}
\For{$i$ in $1..n_{tasks}$}
\State $\mathrlap{t}\hphantom{T} \gets $ sample\_task($D_k$)
\State $T \gets T \cup \{t\}$
\EndFor
\EndFor
\State $T^{tr}, T^{val} \gets $ split($T$)
\State $\Theta^E_{in} \gets \Theta^E$
\For{$i$ in $1..n_{in}$}
\State $T_{SSL}^{tr} \gets$ generate\_ssl\_task($T^{tr}$)
\State $\mathrlap{\mathcal{L}^{in}}\hphantom{T_{SSL}^{tr}} \gets \mathcal{L}(T_{SSL}^{tr}, \Theta^E_{in}, \Theta_{SSL}^C)$
\State $\mathrlap{\Theta^E_{in}}\hphantom{T_{SSL}^{tr}} \gets \Theta^E_{in} - \lambda_{in}\nabla_{\Theta^E_{in}}\mathcal{L}^{in}$
\State $\mathrlap{\Theta_{SSL}^C}\hphantom{T_{SSL}^{tr}} \gets \Theta_{SSL}^C - \lambda_{in}\nabla_{\Theta_{SSL}^C}\mathcal{L}^{in} $
\EndFor
\State $\mathrlap{\mathcal{L}^{out}}\hphantom{T_{SSL}^{tr}} \gets \mathcal{L}(T^{val}, \Theta^E_{in}, \Theta_{Sup}^C)$
\State $\mathrlap{\Theta^E}\hphantom{T_{SSL}^{tr}} \gets \Theta^E - \lambda_{out}\nabla_{\Theta^E_{in}}\mathcal{L}^{out}$
\State $\mathrlap{\Theta_{Sup}^C}\hphantom{T_{SSL}^{tr}} \gets \Theta_{Sup}^C - \lambda_{out}\nabla_{\Theta_{Sup}^C}\mathcal{L}^{out} $
\EndWhile
\end{algorithmic}
\end{algorithm}
\subsection{PhaseSwap}\label{sec:ps}
For our self-supervised training, we choose \emph{PhaseSwap} (PS) introduced in \cite{lemkhenter2020boosting}. Our choice is motivated by two reasons. Firstly, PS has been shown to improve generalization to unseen subjects and this makes it a strong candidate for our approach. Secondly, PS can be defined on the same time scale as the supervised task. Indeed, other self-supervised methods such Relative Positioning (RP) or Contrastive Positional Coding (CPC) \cite{banville2021uncovering} require a longer temporal context, which would complicate the training loop. Since the main focus of this work is to highlight the potential of using self-supervised learning in a meta-learning setting, we opted for the simplest self-supervised loss.
More specifically, PS is defined as a binary classification problem, where a model is trained to distinguish between samples $x$ and $x_{PS}$ defined as
\begin{equation}\label{eq:ps}
x_{PS} = \mathcal{F}^{-1} \left[ \left|\mathcal{F}\left(x\right)\right| \odot \measuredangle \mathcal{F}\left(x'\right)\right],
\end{equation}
where $x$ and $x'$ are two different samples. For a complex scalar $z \in \mathbf{C}^*$, the absolute value $|.|$ and angle $\measuredangle$ operators are defined such that $z = |z| e^{i\measuredangle z}$.
In Algorithm~\ref{alg:main}, $T^{tr} = \{t_i\}_{i=1}^{K \times n_{tasks} / 2}$ is a set of supervised tasks. For each task $t \in T^{tr}$, the function \mbox{\bf generate\_ssl\_task} generates a new task $t_{SSL}$ to be included in $T^{tr}_{SSL}$. For each signal-label pair $(x, y) \in t$, $t_{SSL}$ includes $(x, y_{SSL}=0)$ and its phase-swapped counterpart $(x_{PS}, y_{SSL}=1)$.
\subsection{Architecture Choice}
For our experiment, we use DeepSleepNet-Lite \cite{fiorillo2021deepsleepnet} as our architecture of choice. It consists of two parallel convolutional neural networks using sets of small and large filters for the first layer respectively. The output of the two networks is concatenated into a single vector $h$ and fed into a softmax layer that maps it to the predicted class. The input $x$ to the network is a $90$sec segment, \emph{i.e.}, 3 consecutive epochs of $30$sec each. We chose this architecture for its simplicity, its shorter temporal context and the fact that it does not require the power spectrum as input.
\subsection{Baselines and Training Hyper-parameters}
In all experiments, we compare the performance of our S2MAML model to two other baselines: A supervised classification model without meta-learning and a MAML based training similar to ours, but where we replace the self-supervised problem in the inner loop with a supervised one. We refer to these models as \textbf{SL} and \textbf{MAML} respectively.
Unless stated otherwise, each task $t$ contains 8 samples from the same subjects. $n_{tasks}$, $n_{in}$, $\lambda_{out}$ and $\lambda_{in}$ are set to $32$, $1$, $10^{-4}$ and $5\cdot 10^{-5}$ respectively and each model is trained for 20 full iterations over all the databases considered for training. We use Adam \cite{kingma2015adam} as optimizer with its default hyperparameters. Our models are implemented using Pytorch\footnote{https://pytorch.org/} and ran on a single NVIDIA 1080Ti GPU. We observe no significant differences between the computation times off all models both in inference and training.
We adopt the same label smoothing regularization used by~\cite{fiorillo2021deepsleepnet} with their suggested tuning.
\subsection{Evaluation Metrics}
We use macro F1 (MF1) as an evaluation metric for our experiments. Macro F1 is defined as
\begin{equation}\label{eq:macro}
\text{MF1} = \frac{1}{N_c} \sum_{c=1}^{N_c} \text{F1}_c = \frac{1}{N_c} \sum_{c=1}^{N_c} \frac{2 \text{P}_c \times \text{R}_c}{\text{P}_c + \text{R}_c}
\end{equation}
where $N_c$, $P_c$ and $R_c$ are respectively the number of classes, the precision and recall for the class $c$. It is the average F1 score per class, where the F1 score is defined as the harmonic mean of precision and recall. We choose MF1, instead of the classic F1 score, as it is a better metric when the data has significant class imbalance, which is the case for sleep scoring. All reported MF1 Scores are averaged across a 4-way cross validation split.
\section{RESULTS}
\subsection{Generalization to Novel Databases: \textbf{3 vs 5}}\label{sec:3vs5}
\begin{table*}[h!]
\caption{Cross-validation MF1 Scores for the 3 vs 5 setting. Avg(S) and Avg(U) refer to the average MF1s across all seen and unseen evaluation sets marked with (S) and (U) respectively. Avg(U1) and Avg(U2) refer to the averages MF1s for unseen subjects across seen (CAP(U), ISRUC(U), ST(U)) and unseen (UCD(U), SC(U)) datasets respectively. Avg(U) is the average MF1 across all unseen sets.}
\label{tab:3vs5}
\centering
\setlength{\tabcolsep}{5pt}
\begin{tabular}{|lcccc|cccc|ccc|c|}
\hline
\textbf{Run} & \textbf{CAP(S)} & \textbf{ST(S)} & \textbf{ISRUC(S)} & \textbf{Avg(S)} & \textbf{CAP(U)} & \textbf{ST(U)} & \textbf{ISRUC(U)} & \textbf{Avg(U1)} & \textbf{SC(U)} & \textbf{UCD(U)} & \textbf{Avg(U2)} & \textbf{Avg(U)} \\ \hline
S2MAML & \textbf{68.8} & 74.8 & \textbf{74.7} & \textbf{72.8} & \textbf{56.5} & 65.2 & \textbf{70.3} & \textbf{64.0} & \textbf{41.1} & \textbf{43.7} & \textbf{42.4} & \textbf{55.4} \\
MAML & 66.4 & 71.3 & 73.3 & 70.3 & 55.0 & \textbf{65.3} & 68.9 & 63.1 & 34.4 & 42.1 & 38.2 & 53.1 \\
SL & 55.0 & \textbf{75.5} & 66.7 & 65.7 & 46.4 & 63.2 & 63.0 & 57.5 & 30.3 & 37.9 & 34.1 & 48.1 \\ \hline
\end{tabular}
\end{table*}
In this set of experiments, we compare the performance of S2MAML to the two baselines when training on 3 out of 5 of the considered databases. This allows us to evaluate the performance of our model on completely unseen cohorts of subjects belonging to different databases (see section~\ref{sec:seenunseen}). More specifically, we train using ST, CAP and ISRUC and evaluate on all five datasets. We report the performance of all models on both evaluation sets with \textbf{seen} and \textbf{unseen} subjects in Table~\ref{tab:3vs5}.
For seen subjects, we observe that our model outperforms the two baselines (supervised and MAML training) on average as well as on CAP and ISRUC. The performance gap on average is of $2.5\%$ compared to the MAML and $7.1\%$ compared to supervised training. This shows that meta-learning based methods are generally better suited for overcoming intra-subject variability, and that self-supervision is a powerful tool to further reduce that performance gap.
For unseen subjects, we observe that our model outperforms both baselines on most datasets as well as on average. We also find that the meta-learning based models outperform the supervised based baseline, a feature that we noticed also with seen subjects. More importantly, the performance gap between our S2MAML and MAML is wider on held out datasets. Although MAML generalizes better to unseen subjects from the databases used for training compared to the supervised baseline, it generalizes less to held out databases compared to our model. In other words, our S2MAML is not only better suited for dealing with inter-subject variability, but it is also better suited for dealing with inter-cohort variability.
We discuss the low performance on ST in in section~\ref{sec:onevsall}.
\subsection{Generalization in a Data Abundant Setting: \textbf{All vs All}}\label{sec:allvsall}
In this set of experiments we compare the performance of S2MAML to our two baselines when trained on all databases jointly. This allows us to highlight the benefit of our algorithm in a setting where a large quantity of labeled recordings are available. The MF1 Scores of all models on both seen and unseen subjects are reported in Table~\ref{tab:allvsall}.
\begin{table}[h]
\caption{Cross-validation MF1 Scores for the All vs All setting. We also report the average MF1 across all databases.}
\label{tab:allvsall}
\centering
\setlength{\tabcolsep}{5pt}
\begin{tabular}{|lccccccc|}
\hline
\textbf{Run} & \textbf{Subjects} & \textbf{CAP} & \textbf{ST} & \textbf{ISRUC} & \textbf{SC} & \textbf{UCD} & \textbf{Avg} \\ \hline
S2MAML & Seen & \textbf{82.1} & \textbf{85.0} & \textbf{88.8} & \textbf{86.3} & \textbf{90.4} & \textbf{86.5} \\
MAML & Seen & 80.2 & 81.5 & 86.1 & 84.1 & 89.4 & 84.3 \\
SL & Seen & 59.3 & 83.2 & 71.2 & 82.1 & 68.6 & 72.9 \\ \hline
S2MAML & Unseen & \textbf{67.9} & \textbf{73.7} & \textbf{82.7} & \textbf{83.8} & \textbf{70.7} & \textbf{75.8} \\
MAML & Unseen & 65.7 & 69.2 & 80.8 & 80.8 & 69.8 & 73.3 \\
SL & Unseen & 50.8 & 70.0 & 67.2 & 79.5 & 55.0 & 64.5 \\ \hline
\end{tabular}
\end{table}
We observe that our model outperforms both baselines on all datasets as well as on average for both evaluation settings. This shows that the generalization advantage of our model does not disappear when scaling up the amount of available data. In the deep learning literature, scaling up the amount of training data is a common practice used to improve the generalization of artificial neural networks. This relies on the implicit assumption that with enough data, one is able to obtain a training set that is similar in distribution to the evaluation set and contains most sources of variability that can be encountered. However, in the case of physiological signals this assumption may not hold as well. A new individual will always have subject-specific sleep patterns and the inter-dataset variability will always remain a challenge as long as hardware/software recording pipelines keep evolving.
In the previous section, we have split the data by subjects, which is not a common practice. We did so to illustrate a more extreme setting for generalization. In this section, we also obtain the performance on seen subjects, as done in the literature, so that it is easier to compare to prior work. Although the main focus of our work is to reduce the generalization gap between subjects and datasets, the MF1s reported on seen subjects are comparable or better than state of the art methods in the literature. Our model achieves an MF1 score of $86.3\%$ and $85.0\%$ compared to $79\%$ and $76\%$ for U-Sleep \cite{perslev2021u} on SC and ST respectively. The numbers are however not directly comparable due to the difference in the randomness of the splits. For this reason, and in order to keep our results focused on the generalization problem, we chose to omit numbers reported by other prior works from our tables.
\subsection{Disparity Between Datasets: \textbf{One vs All}}\label{sec:onevsall}
In this set of experiments, we compare the performance of the different models on unseen databases when trained only on a single one. This represents a worst case scenario, where one has access to a very limited number of subjects and therefore learning to generalize becomes much more challenging. Since the different databases considered in this study have different sizes, we choose to equalize experiments by training for a fixed number of gradient updates, $5000$, instead of looping through the training set $20$ times. The goal of these experiments is to gauge how similar or dissimilar the databases considered in this work are. In other words, our goal is to confirm that generalizing from one set to the others is indeed a challenging task and that each database has its particularities.
We report the obtained MF1 scores on Table~\ref{tab:onevsall}. For all datasets and all three models considered, we observe that the performance drops significantly on unseen datasets. One additional noteworthy observation is that out of all combinations, models trained on SC/ST and tested on others and vice versa seem to generalize the least. On the other hand, generalizing between ST and SC seems more feasible. This may be due to fact that ST/SC were collected a few decades ago or the fact that they include EEG electrodes that are not common in the other three databases. We believe that this observation may explain why both meta-learning models struggle compared to the SL baseline on ST as reported in Table~\ref{tab:3vs5}.
Overall in this setting the performance across different methods does not indicate a clear winner. Given the restricted number of subjects per dataset, all methods struggle to learn features that generalize well to new cohorts. However, on average across all possible combinations, S2MAML and MAML comes out slightly on top with $29.8 \%$ and $29.9 \%$ respectively compared to $29.4 \%$ for the SL baseline.
\begin{table}[t]
\caption{Cross-validation MF1 Scores for models trained on one dataset and evaluated on unseen subjects/datasets. Each row block corresponds to models trained on a single dataset.}
\label{tab:onevsall}
\centering
\begin{tabular}{|llccccc|}
\hline
\diagbox[width=6em]{\textbf{Train}}{\textbf{Test}} & \textbf{Model} & \textbf{ISRUC} & \textbf{SC} & \textbf{ST} & \textbf{CAP} & \textbf{UCD} \\ \hline
& S2MAML & \cellcolor[HTML]{34A853}\textbf{76.0} & \cellcolor[HTML]{EAD465}27.4 & \cellcolor[HTML]{F07244}9.7 & \cellcolor[HTML]{F0D665}\textbf{25.8} & \cellcolor[HTML]{F3864B}12.2 \\
& MAML & \cellcolor[HTML]{3AAA54}74.7 & \cellcolor[HTML]{F5D766}\textbf{24.5} & \cellcolor[HTML]{F17746}\textbf{10.3} & \cellcolor[HTML]{FAD866}23.3 & \cellcolor[HTML]{F69E53}15.0 \\
\multirow{-3}{*}{ISRUC} & SL & \cellcolor[HTML]{45AC55}71.7 & \cellcolor[HTML]{FAB65A}17.7 & \cellcolor[HTML]{F17746}10.3 & \cellcolor[HTML]{FED765}21.6 & \cellcolor[HTML]{F8A856}\textbf{16.1} \\ \hline
& S2MAML & \cellcolor[HTML]{ED5C3D}7.2 & \cellcolor[HTML]{3CAA54}74.1 & \cellcolor[HTML]{B6C860}41.5 & \cellcolor[HTML]{EC543A}6.3 & \cellcolor[HTML]{ED5E3D}7.4 \\
& MAML & \cellcolor[HTML]{EE643F}8.1 & \cellcolor[HTML]{38A954}75.2 & \cellcolor[HTML]{ABC55F}\textbf{44.3} & \cellcolor[HTML]{FCC45F}\textbf{19.4} & \cellcolor[HTML]{ED5F3E}7.5 \\
\multirow{-3}{*}{SC} & SL & \cellcolor[HTML]{F17545}\textbf{10.1} & \cellcolor[HTML]{37A954}\textbf{75.5} & \cellcolor[HTML]{ADC65F}43.7 & \cellcolor[HTML]{EC573B}6.6 & \cellcolor[HTML]{EE6540}\textbf{8.2} \\ \hline
& S2MAML & \cellcolor[HTML]{F69E52}14.9 & \cellcolor[HTML]{E6D364}28.7 & \cellcolor[HTML]{54B056}67.7 & \cellcolor[HTML]{F9B259}17.2 & \cellcolor[HTML]{FDCB61}\textbf{20.2} \\
& MAML & \cellcolor[HTML]{FCC35F}19.3 & \cellcolor[HTML]{E5D364}\textbf{28.9} & \cellcolor[HTML]{52B056}68.2 & \cellcolor[HTML]{FABC5C}\textbf{18.4} & \cellcolor[HTML]{FBC25E}19.2 \\
\multirow{-3}{*}{ST} & SL & \cellcolor[HTML]{FED865}\textbf{21.7} & \cellcolor[HTML]{E7D464}28.3 & \cellcolor[HTML]{51AF56}\textbf{68.3} & \cellcolor[HTML]{FABA5C}18.2 & \cellcolor[HTML]{FCC660}19.6 \\ \hline
& S2MAML & \cellcolor[HTML]{E5D364}28.8 & \cellcolor[HTML]{EAD465}\textbf{27.4} & \cellcolor[HTML]{F59550}13.9 & \cellcolor[HTML]{78B95A}\textbf{58.1} & \cellcolor[HTML]{9CC15D}\textbf{48.5} \\
& MAML & \cellcolor[HTML]{E6D364}28.5 & \cellcolor[HTML]{F5D765}24.6 & \cellcolor[HTML]{F48F4E}13.2 & \cellcolor[HTML]{86BC5B}54.3 & \cellcolor[HTML]{A3C35E}46.4 \\
\multirow{-3}{*}{CAP} & SL & \cellcolor[HTML]{E2D264}\textbf{29.8} & \cellcolor[HTML]{F2D665}25.4 & \cellcolor[HTML]{F69A51}\textbf{14.4} & \cellcolor[HTML]{9AC15D}49.0 & \cellcolor[HTML]{B5C760}41.7 \\ \hline
& S2MAML & \cellcolor[HTML]{FAD866}23.2 & \cellcolor[HTML]{FBBD5D}18.6 & \cellcolor[HTML]{EA4535}4.5 & \cellcolor[HTML]{F6D766}24.2 & \cellcolor[HTML]{68B558}\textbf{62.2} \\
& MAML & \cellcolor[HTML]{FCCA61}20.0 & \cellcolor[HTML]{FBBD5C}\textbf{18.5} & \cellcolor[HTML]{EA4836}4.9 & \cellcolor[HTML]{FDD163}20.9 & \cellcolor[HTML]{73B859}59.4 \\
\multirow{-3}{*}{UCD} & SL & \cellcolor[HTML]{E1D264}29.9 & \cellcolor[HTML]{FBBD5C}18.5 & \cellcolor[HTML]{EC533A}\textbf{6.1} & \cellcolor[HTML]{F1D665}\textbf{25.7} & \cellcolor[HTML]{79B95A}57.7 \\ \hline
\end{tabular}
\end{table}
\subsection{Effect of $\lambda_{in}$}
In this section we study the effect of $\lambda_{in}$ on our model and the MAML baseline. We train both our model and the MAML baseline in the \textbf{3 vs 5} setting described in Section~\ref{sec:3vs5} for $\lambda_{in} \in \{10^{-3}, 5\cdot 10^{-5}\}$. Tables~\ref{tab:lr-seen}
and \ref{tab:lr-unseen} report the obtained MF1 scores for seen and unseen subjects respectively. We observe that while the value of $\lambda_{in}$ has little effect on the performance of our model, setting it to $10^{-3}$ greatly reduces the performance of the MAML baseline. By setting $\lambda_{in}$ to a higher value, we put more emphasis on the convergence on the meta-train set, \emph{i.e.}, in the inner loop. This confirms that using \emph{PhaseSwap} as a self-supervised task in the inner loop, \emph{i.e.}, on the meta-train set, is less prone to learning subject-specific features and thus generalizes better compared to its supervised counterpart. Additionally, this shows that our methods is more robust to the choice of the hyper-parameter $\lambda_{in}$.
\begin{table}[t]
\centering
\caption{Cross-validation MF1 Scores on unseen subjects for models trained in the 3 vs 5 setting for different values of $\lambda_{in}$.}
\label{tab:lr-unseen}
\setlength{\tabcolsep}{4pt}
\begin{tabular}{|lccccc|ccc|}
\hline
\textbf{Run} & $\lambda_{in}$ & \textbf{CAP} & \textbf{ST} & \textbf{ISRUC} & \textbf{Avg} & \textbf{SC} & \textbf{UCD} & \textbf{Avg} \\ \hline
S2MAML & $10^{-3}$ & 60.0 & \textbf{70.2} & 67.4 & 65.9 & \textbf{34.3} & \textbf{49.0} & \textbf{41.7} \\
S2MAML & $5\cdot 10^{-5}$ & \textbf{61.9} & 70.1 & \textbf{68.4} & \textbf{66.8} & 32.6 & 46.9 & 39.8 \\ \hline
MAML & $10^{-3}$ & 23.1 & 25.4 & 33.1 & 27.2 & 15.9 & 10.4 & 13.2 \\
MAML & $5\cdot 10^{-5}$ & \textbf{59.2} & \textbf{67.9} & \textbf{65.4} & \textbf{64.2} & \textbf{25.9} & \textbf{49.4} & \textbf{37.7} \\ \hline
\end{tabular}
\end{table}
\begin{table}[t]
\centering
\caption{Cross-validation MF1 Scores on seen subjects for models trained in the 3 vs 5 setting for different values of $\lambda_{in}$.}
\label{tab:lr-seen}
\begin{tabular}{|llcccc|}
\hline
\textbf{Run} & $\lambda_{in}$ & \textbf{CAP} & \textbf{ST} & \textbf{ISRUC} & \textbf{Avg} \\ \hline
S2MAML & $10^{-3}$ & 68.3 & 70.8 & 73.4 & 71.0 \\
S2MAML & $5\cdot 10^{-5}$ & \textbf{68.8} & \textbf{74.8} & \textbf{74.7} & \textbf{72.8} \\ \hline
MAML & $10^{-3}$ & 16.6 & 24.5 & 20.5 & 20.5 \\
MAML & $5\cdot 10^{-5}$ & \textbf{66.4} & \textbf{71.3} & \textbf{73.2} & \textbf{70.8} \\ \hline
\end{tabular}
\end{table}
\section{DISCUSSIONS}
With the increasing popularity of deep learning methods, more and more artificial neural network architectures have been proposed in the literature for automatic sleep scoring. Reliable automatic sleep scoring models have the potential to speed up sleep research and make it more accessible by reducing the cost of manual annotations and enable more advanced closed-loop system. However, one important requirement for such models is that they should maintain their level of performance across sessions, subjects and hardware/software recording settings. Our work positions itself as a step forward toward achieving this goal. By leveraging both meta-learning and self-supervised learning, our S2MAML is able to reduce the performance drop associated with both intra-subject variability, \emph{i.e.}, unseen subjects from seen datasets, and intra-database variability, \emph{i.e.}, on unseen datasets.
\section{CONCLUSIONS}
In this work, we introduce a novel deep learning model for automatic sleep scoring. By leveraging meta-learning and robust self-supervised features, our model is able to better cope with intra-subject and intra-dataset variabilities in the zero-shot setting. We show through extensive experiments that our model outperforms all baselines in terms of generalization capabilities both on seen and unseen subjects. Our work presents itself as an important milestone toward the wide adoption of automatic sleep scoring in sleep research by bridging the performance gap present when deploying such models on new datasets and cohorts of subjects.
\section*{ACKNOWLEDGMENT}
This research was supported by the Interfaculty Research Cooperation ``Decoding Sleep: From Neurons to Health \& Mind'' of the University of Bern.
\bibliographystyle{ieeetr}
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